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abstract: 'Results are presented on searches for standard model and non-standard model production of a Higgs boson in [$p\bar{p}$]{} collisions at $\sqrt{s}= 1.96$ TeV with the CDF II detector at the Fermilab Tevatron. Using data corresponding to 2-3.6 1/fb of integrated luminosity, searches are performed in a number of different production and decay modes. No excess in data above that expected from backgrounds is observed; therefore, we set upper limits on the production cross section times branching fraction as a function of the Higgs boson mass.'
address: |
Fermilab, Batavia, IL, USA\
(On behalf of the CDF Collaboration)
author:
- Craig Group
bibliography:
- 'LLWI\_Higgs\_group.bib'
title: Higgs Boson Searches at CDF
---
Introduction
============
Although the Higgs mechanism [@Higgs:1964pj] was proposed in the 1960’s, the fundamental particle it predicts, the Higgs boson ($h$), has yet to be discovered. Direct limits from the LEP experiments exclude Higgs boson masses below $114.4$ [GeV/$c^2$]{} [@Barate:2003sz] at 95% Confidence Level (CL), while electroweak precision measurements place an indirect upper limit on the mass of a SM Higgs boson of $154$ [GeV/$c^2$]{} [@Collaboration:2008ub] at 95% CL.
Here we will summarize the status of the search for the Higgs boson using the the CDF II detector [@Acosta:2004yw] to analyze the [$p\bar{p}$]{} collision data from the Fermilab Tevatron.
Summary of Standard Model Search Efforts
========================================
For Higgs boson masses below $135$ [GeV/$c^2$]{}, [$b\bar{b}$]{} is the main decay mode [@Djouadi:1997yw]. For higher masses the Higgs boson decays primarily into a pair of $W$ bosons. Due to these different decay modes, it is natural to also split the discussion of the analysis effort at CDF into the low mass and high mass categories.
The search for a Higgs boson is quite challenging due to the large backgrounds and small signal expectation. In order to increase signal-background discrimination, analyses make maximal use of the information in each event by employing multivariate techniques to collect the discriminating power of multiple input variables into a single more powerful output variable. When using these techniques it is crucial to prove that all of the multivariate inputs and the discriminant outputs are described well by the background models. These checks are performed by comparing the background model with the data in various control regions carefully defined to test the modeling of major background components. Over the years CDF has built confidence in their detector modeling and has had success with the use of multivariate techniques in SM analyses such as the recent evidence and observation of single top quark production [@Aaltonen:2009jj; @Aaltonen:2008sy].
The Higgs boson search strategy is to perform dedicated analyses for each distinct final state with a significant production rate and then combine these results to make a statement about the sensitivity of the CDF experiment to the SM Higgs boson.
Low mass Higgs boson searches
-----------------------------
In the [$h \rightarrow b\bar{b}$]{} decay, each $b$ quark fragments into a jet of hadrons and the Higgs boson signal appears as a peak in the invariant mass distribution of these two jets. The two-jet signature alone is not useful to reject the dijet QCD background which is produced at the Tevatron with a rate about ten orders of magnitude higher than the Higgs boson. To handle this problem the low mass Higgs boson searches focus on production processes where the Higgs boson is produced in association with a $W$ or $Z$ boson ($Vh$).
The requirements of a high charged lepton candidate, missing transverse energy (), and at least one $b$-tagged jet reduce the background in the [$Wh \rightarrow \ell \nu b\bar{b}$]{} [@wh; @whPRD] search channel. In the past CDF has used an analysis based on an artificial neural network (NN) which includes the dijet invariant mass, the total system and, the event imbalance to improve the discrimination between the Higgs signal and background. Recently, CDF has combined this analysis with a new analysis which incorporates matrix element (ME) calculations into a boosted decision tree. This combination was done using another NN optimized based on genetic algorithms [@Whiteson:2006ws]. The combined result is the most sensitive low mass analysis obtaining observed (expected) 95% C.L. limit of 5.6 (4.8) times the SM prediction of the production cross section for a Higgs boson mass of 115 [GeV/$c^2$]{} using 2.7 [fb$^{-1}$]{} of integrated luminosity..
The [$Zh \rightarrow \ell \ell b \bar{b}$]{} [@Aaltonen:2008wj] has a smaller background but also a smaller signal expectation do to the reduced cross section of $Zh$ production. The reach of this analysis has been improved by loosening the lepton identification requirements to increase signal acceptance. In addition, this final state is fully constrained by the reconstructed objects and the can be used to correct the jet energies to improve the dijet mass resolution. CDF has a two-dimensional NN analysis and an analysis based on ME probabilities. For a Higgs boson mass of 115 [GeV/$c^2$]{} the NN analysis obtains an observed (expected) 95% C.L. limit of 7.7 (9.1) times the SM prediction of the production cross section using 2.7 [fb$^{-1}$]{} of integrated luminosity.
The third major low mass Higgs analysis is based on a requirement and identifying $b$ jets but does not allow any reconstructed leptons in the events [@Aaltonen:2008mi]. It is sensitive to [$Zh \rightarrow \nu \nu b \bar{b}$]{} but also [$Wh \rightarrow \ell \nu b\bar{b}$]{} when the lepton escapes detection. Without the charged lepton requirement the dominant background is QCD events where mismeasured jets fake the requirement. A NN selection tool is derived which uses information about the angular correlations between the jets and the as well as a “missing ” variable based on tracking information. By cutting on this NN the signal to background ratio is improved to the level of the lepton based analyses, making this channel competitive as one of the most sensitive low mass Higgs boson search channels. The +$bb$ analysis obtains observed (expected) 95% C.L. limit of 6.9 (5.6) times the SM prediction of the production cross section for a Higgs boson mass of 115 [GeV/$c^2$]{} using 2.1 [fb$^{-1}$]{} of integrated luminosity.
High mass Higgs boson searches
------------------------------
At high mass the [$h \rightarrow WW \rightarrow \ell \nu \ell \nu$]{} channel dominates the sensitivity to the Higgs boson. The leptonic decay mode of the $W$ bosons is chosen to reduce background and improve signal purity. The CDF analysis [@Aaltonen:2008ec] includes all significant production modes (gluon fusion, $Vh$, and vector boson fusion (VBF)), and splits the analysis up based on the number of jets observed in the final state. This is useful since the background and signal composition change considerably depending on the number of jets. The [$h \rightarrow WW$]{}analysis is the most sensitive single analysis at CDF.
Results
-------
The observed (expected) limits on the Higgs boson cross section in units of the SM prediction for all of the CDF analyses are shown in Fig. \[fig:CDF\_limits\](a). These results are combined into a single limit on the Higgs boson production rate for each Higgs boson mass hypothesis. In addition to the most sensitive low mass analyses, CDF also has analyses that focus on the all hadronic final state ($Vh$ where the vector boson decays into two jets), and an inclusive $h
\rightarrow \tau \tau$ analysis that are included in the combination. The result of the combination of CDF results is shown as the garnet line in Fig. \[fig:CDF\_limits\](a). The limits range from 1.4 to about 5 times the prediction of the SM rate.
The CDF results have also been combined with the results of the DØ experiment. This is shown in Fig. \[fig:CDF\_limits\](b). The combined result excludes Higgs bosons with masses between 160 and 170 [GeV/$c^2$]{} at the 95% C.L. This is the first new exclusion of a standard model Higgs boson based on a direct search using Tevatron data.
{width="48.00000%"} {width="48.00000%"}
Beyond the Standard Model Example: Fermiophobic Higgs Boson Search
==================================================================
The CDF experiment also searches for Higgs bosons produced in many extensions to the SM. One example which has been recently updated [@hgg] is a search in the diphoton final state. The SM prediction for the [$h \rightarrow \gamma \gamma$]{} branching fraction is extremely small (reaching a maximal value of only about 0.2% at a Higgs boson mass ([$m_{h}$]{})$~\sim 120$ [GeV/$c^2$]{}) [@Djouadi:1997yw]; however, in “fermiophobic” models, where the coupling of the Higgs boson to fermions is suppressed, the diphoton decay can be greatly enhanced. This phenomenon has been shown to arise in a variety of extensions to the SM [@Haber:1978jt; @Gunion:1989ci; @Basdevant:1992nb; @Barger:1992ty; @Akeroyd:1995hg], and the resulting collider phenomenology has been described [@Dobrescu:1999gv; @Matchev1; @Mrenna]. For this fermiophobic case, the decay into the diphoton final state dominates at low Higgs boson masses and is therefore the preferred search channel.
A benchmark fermiophobic model is considered in which the Higgs boson does not couple to fermions, yet retains its SM couplings to bosons. In this model, the fermiophobic Higgs boson production is dominated by two processes: $Vh$, and VBF. Two identified photon candidates are required in the analysis. In addition, a cut is applied on the transverse momentum of the diphoton pair ([$P_{T}^{~\gamma \gamma}$]{}) designed to optimize sensitivity (VBF and $Vh$ have more [$P_{T}^{~\gamma \gamma}$]{} on average than the SM diphoton production and QCD backgrounds).
The decay of a Higgs boson into a diphoton pair appears as a very narrow peak in the invariant mass distribution of these two photons ($\sigma _{m}/m < 3~\%$). The search can therfore be performed by looking for a narrow bump on an otherwise smooth background distribution. No narrow resonance is observed. In order to set limits on the Higgs boson production rate, a sideband fit excluding the hypothetical mass window is performed to estimate the background in the search window (see Fig. \[fig:Fermiophobic\](a)). The analysis results in 95 % C.L. limits on the production cross section ($\sigma\times$[$\cal{B}$r($h \rightarrow \gamma \gamma$)]{}) and on the branching fraction ([$\cal{B}$r($h \rightarrow \gamma \gamma$)]{}) as shown in Fig. \[fig:Fermiophobic\](b). The result excludes the benchmark model predictions for [$m_{h}$]{} of less than 106 [GeV/$c^2$]{}.
{width="40.00000%"} {width="40.00000%"}
Conclusions, and Outlook
========================
The CDF experiment is carefully searching for SM and BSM Higgs bosons. A combination of results with DØ excludes Higgs boson masses between 160 and 170 [GeV/$c^2$]{}, and this is the first exclusion of SM Higgs bosons based on Tevatron data. At low mass, sensitivity is better than three times the standard model prediction. With more data on tape to analyze, and improvements still being added to analysis techniques, the Higgs boson search results will be an exciting topic until the end of the Tevatron run. With the full dataset expected to be on the order of 10 [fb$^{-1}$]{}, there is a significant chance that the Tevatron will see some evidence of the elusive Higgs boson.
|
---
abstract: 'Conductance of an Au mono atomic contact was investigated under the electrochemical potential control. The Au contact showed three different behaviors depending on the potential: 1 $G_{0}$ ($G_{0}$ = $2e^{2}/h$), 0.5 $G_{0}$ and not-well defined values below 1 $G_{0}$ were shown when the potential of the contact was kept at -0.6 V (double layer potential), -1.0 V (hydrogen evolution potential), and 0.8 V (oxide formation potential) versus Ag/AgCl in 0.1 M Na$_{2}$SO$_{4}$ solution, respectively. These three reversible states and their respective conductances could be fully controlled by the electrochemical potential. These changes in the conductance values are discussed based on the proposed structure models of hydrogen adsorbed and oxygen incorporated on an Au mono atomic contact.'
address:
- '$^1$Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo, 060-0810, Japan'
- '$^2$Precursory Research for Embryonic Science and Technology (PRESTO), Japan Science and Technology Agency, Sapporo, Hokkaido 060-0810, Japan'
author:
- 'Manabu Kiguchi$^{1,2}$, Tatsuya Konishi$^1$, Kouta Hasegawa$^1$, Satoshi Shidara$^1$, and Kei Murakoshi$^1$'
title: Three reversible states controlled on a gold monoatomic contact by the electrochemical potential
---
INTRODUCTION {#sec1}
============
Study of charge transport in atomic scale metal or molecular nano wires is of fundamental interest, with potential applications for ultra-small electronic devices [@1; @2]. In order to use these nano wires for devices, their conductance should be tunable by some external field, such as light or magnetic field. There are some experimental challenges to controlling the conductance of nano wires by an external field [@3; @4; @5; @6; @7]. The switching of a photochromic molecular wire from the conducting state to the insulating state was observed by radiation of visible light [@3]. A change in the conductance behavior by molecular adsorption was observed for Au, Pt, Cu, Fe, Co, and Ni nano wires [@4; @5; @6; @7]. Here, it should be noted that the conductance can be switched from the off state to the on state or vice versa (among two states) in most studies. If the conductance of the nano wires could be controlled among three different conductance states, it would have attracted wide attention for fundamental science and technological applications.
In the present study, we paid attention to the electrochemical potential as an external field [@8; @9; @10-1; @10-2; @10]. The electrochemical potential can control the procession of two individual electrochemical reactions on a metal surface by changing the electrochemical potential of the metal electrode by only several volts. For example, hydrogen or oxygen evolution proceeds when the electrochemical potential of Pt electrodes is maintained at more negative than -0.2 V or more positive than +1.3 V versus Ag/AgCl in 0.1 M H$_{2}$SO$_{4}$ solution, respectively [@11]. The clean hydrogen or oxygen adsorbed states can be selectively prepared by controlling the electrochemical potential. Several states of the surface condition, at least three, can be maintained under the electrochemical potential control. If the conductance of the metal nano wire can be defined by the surface condition of the wire, the conductance of the metal nano wire may be controlled by the electrochemical potential. Au was investigated in the present study for the following reasons. First, the hydrogen and oxygen evolution can proceed without dissolving the Au ions from the Au electrode; this is a characteristic of the Au electrode in solution. In the case of Ni, Ni rapidly dissolves at the oxygen evolution potential [@11]. It is thus difficult to prepare stable Ni nano wires under oxygen evolution. Second, the Au mono atomic contact shows a well defined fixed conductance value. It normally shows 1 $G_{0}$ ($2e^{2}/h$), and it shows 0.5 $G_{0}$ under hydrogen evolution [@8; @10]. Showing well defined conductance values is useful for the understanding and utilization of the metal nano wire. Under ultra high vacuum (UHV) condition, the Au mono atomic contact does not show a well-defined fractional conductance value by the introduction of hydrogen molecule [@4]. In solution, the Au mono atomic contact shows well defined fractional conductance. Hydrogen incorporated wire, dimerized wire, and other atomic and electronic structures of the wire, were proposed to be the origin of the fractional conductance value [@8; @10]. However, the actual origin is not clear at present. In addition, the effect of oxygen on the Au mono atomic contact, such as the structure, stability, and conductance, in solution is not clear. In the present study, we have studied the Au mono atomic contact under the electrochemical potential control to reveal the effect of hydrogen and oxygen on the conductance behavior and to control the conductance of the Au mono atomic contact among three different states.
EXPERIMENTAL {#sec2}
============
The experiments were performed with the modified scanning tunneling microscope (STM: Pico-SPM, Molecular Imaging Co.) with a Nano ScopeIIIa controller (Digital Instruments Co.) in an electrochemical cell (see Ref [@10-1; @10-2; @10] for a detailed description of the experimental setup and conditions). The STM tip was made of an Au wire (diameter 0.25 mm, $>$ 99 $\%$) coated with wax to eliminate ionic conduction. The Au(111) substrate was prepared by a flame annealing and quenching method. Figure \[fig1\](a) shows a schematic view of the experimental setup. The electrochemical potential ($\phi$) of the Au tip and substrate was controlled using a potentiostat (Pico-Stat, Molecular Imaging Co.) with an Ag/AgCl reference electrode. A 0.50 mm diameter Pt wire was used as a counter electrode. The STM tip was repeatedly moved into and out of contact with the Au substrate at a rate of 50 nm/s in an electrolyte solution. The electrolyte solution was 0.1 M Na$_{2}$SO$_{4}$ or 50 mM H$_{2}$SO$_{4}$. Conductance was measured during the breaking process under an applied bias of 20 mV between the tip and substrate. All statistical data was obtained from a large number (over 1000) of individual conductance traces. Figure \[fig1\](b) shows the cyclic voltammogram (CV) of an Au electrode in the 0.1 M Na$_{2}$SO$_{4}$ solution. The double layer regime extended from $\phi$ = -0.7 V to 0.8 V. When the electrochemical potential of the Au electrode was kept at a potential more positive than $\phi$ = 1.3 V, the oxygen evolution proceeded via the formation of an oxide layer on the Au electrode. The hydrogen evolution proceeded at a potential more negative than $\phi$ = -0.7 V [@11; @12; @13]. Conductance of the Au nano contacts was measured at $\phi$ = -0.6V (double layer potential), $\phi$ = -1.0 V (hydrogen evolution potential) and $\phi$ = 0.8 V (oxide formation potential).
RESULTS AND DISCUSSION {#sec3}
======================
Figure \[fig2\] shows the typical conductance traces and histograms of the Au nano contacts in 0.1 M Na$_{2}$SO$_{4}$ solution during breaking the contact at the double layer, hydrogen evolution and oxide formation potentials. At the double layer potential (Fig. \[fig2\](b), (e)), the conductance decreased in a stepwise fashion with each step occurring at integer multiples of $G_{0}$. The corresponding conductance histograms showed a well-defined peak near 1 $G_{0}$, which corresponded to a clean Au atomic contact or wire [@2]. At the hydrogen evolution potential (Fig. \[fig2\](a), (d)), reversible transition of the conductance between 1 $G_{0}$ and 0.5 $G_{0}$ was observed. This conductance fluctuation resulted in the 0.5 $G_{0}$ peak in the conductance histogram. At the oxide formation potential (Fig. \[fig2\](c), (f)), the conductance continuously decreased stepwise at various conductance values below 1 $G_{0}$. The corresponding histogram showed a broad feature at 1 $G_{0}$ together with a large background. The conductance of the Au mono atomic contact could be controlled among three different conductance states (1 $G_{0}$, 0.5 $G_{0}$ and conductance below 1 $G_{0}$) by the electrochemical potential of the contact. These electrochemical potential dependent changes were fully reversible. The potential in which the 0.5 $G_{0}$ peak (or large background) appeared in the conductance histogram agreed with the onset potential of the hydrogen evolution (or oxide formation) reaction on the Au electrode. In the conductance histogram, the intensity of the 0.5 $G_{0}$ peak normalized by the 1 $G_{0}$ peak increased, and then saturated, as the potential was scanned from the double layer potential to the hydrogen evolution potential. The intensity of the background continuously increased, as the potential was scanned from the double layer potential to the oxide formation potential. The 1 $G_{0}$ peak was not apparent in the conductance histogram at more positive than $\phi$ = 1.0 V.
The conductance behavior of the Au mono atomic contact was investigated in other electrolytes, 0.1 M NaCl, 0.1 M NaOH, 50 mM H$_{2}$SO$_{4}$, and 0.1 M HClO$_{4}$. In the conductance histogram, a large background below 1 $G_{0}$ was observed at the oxide formation potential, while the 0.5 $G_{0}$ peak was observed at the hydrogen evolution potential. In acid solution (50 mM H$_{2}$SO$_{4}$, 0.1 M HClO$_{4}$), the 0.1 $G_{0}$ peak was observed in the conductance histogram, in addition to the 0.5 $G_{0}$ peak at the hydrogen evolution potential. Figure \[fig3\] shows the typical conductance traces and histogram of the Au nano contacts in 50 mM H$_{2}$SO$_{4}$ solution during breaking the contact at the hydrogen evolution potential. Reversible transitions of the conductance between 1 $G_{0}$ and 0.5 $G_{0}$, 0.5 $G_{0}$ and 0.1 $G_{0}$ were observed in the conductance traces. This conductance fluctuation resulted in the 0.5 $G_{0}$ and 0.1 $G_{0}$ peaks in the conductance histogram. The 0.1 $G_{0}$ peak in Fig. \[fig3\](b) seems very sharp compared to other 1 $G_{0}$ and 0.5 $G_{0}$ peaks. The half width at half maximum ($W$) of the histogram was 0.14, 0.15, 0.08 $G_{0}$ for the 1, 0.5, 0.1 $G_{0}$ peaks, respectively. On the other hand, the relative peak width ($W$ / conductance ratio) was 0.14, 0.28, 0.74 for the 1, 0.5, 0.1 $G_{0}$ peaks, showing that the 0.1 $G_{0}$ peak became broader than the 0.5 $G_{0}$ and 1 $G_{0}$ peaks.
The electrochemical potential dependence of the stability of the Au atomic contact was investigated in 0.1 M Na$_{2}$SO$_{4}$ solution. Figure \[fig4\](a) shows the distribution of lengths for the last conductance plateau ($D_{last}$). The length of the last plateau was defined as the distance between the points at which the conductance dropped below 1.3 $G_{0}$ and 0.05 $G_{0}$ , respectively. At the double layer potential, the contact broke within 0.2 nm. The contact could be stretched a longer distance at the hydrogen evolution and oxide formation potential. The average length of the Au mono atomic contact was 0.08 nm, 0.15 nm, and 0.18nm at the double layer, hydrogen evolution, and oxide formation potentials, respectively. At the hydrogen and oxygen potential, the mono atomic contact could be stretched 0.4 nm in length. Considering the Au-Au distance of 0.25 nm in the Au mono atomic contact obtained at low temperature in UHV [@14], the Au mono atomic wire could be formed in solution at the hydrogen evolution and oxide formation potential. Of course, the long stretched length did not directly indicate the formation of a mono atomic wire; the stem part of the Au contact might be deformed during the stretching. Although the formation of the Au mono atomic wire was not clear, the Au mono atomic contact was found to be stabilized at the hydrogen evolution and oxide formation potential.
In order to investigate the transformation process of the Au mono atomic contact in more detail, the distribution of lengths was analyzed for the 1 $G_{0}$ plateau, which corresponded to a clean Au mono atomic contact [@2] ($D_{1G}$: see Fig. \[fig4\](b)). The length of the 1 $G_{0}$ plateau was defined as the distance between the points at which the conductance dropped below 1.3 $G_{0}$ and 0.8 $G_{0}$, respectively. At the double layer and hydrogen evolution potential, the $D_{1G}$ was similar to the $D_{last}$. The 1 $G_{0}$ plateau could be stretched 0.4 nm in length at the hydrogen evolution potential. On the other hand, the 1 $G_{0}$ plateau was very short ($<$ 0.2 nm) at the oxide formation potential, although the last conductance plateau could be stretched 0.4 nm in length. The $D_{1G}$ at the oxide formation potential was close to the $D_{1G}$ and $D_{last}$ at the double layer potential. At the hydrogen evolution potential, the close agreement between the $D_{1G}$ and $D_{last}$ indicated that the formation of the structure showing 0.5 $G_{0}$ would not relate with the stabilization of the Au mono atomic contact at the hydrogen evolution potential. At the oxide formation potential, the conductance of the Au contact continuously decreased stepwise at various conductance values after showing 1 $G_{0}$. Combing with this continuous change in conductance value, the close agreement between the $D_{1G}$ at the oxide formation potential and the $D_{1G}$ at the double layer potential suggested the following transformation process of the Au atomic contact. Initially, the clean Au mono atomic contact was formed during the stretching the contact. Then, the structure showing conductance value below 1 $G_{0}$ would be formed at the oxide formation potential.
Here, the present experimental results are compared with our previously reported results measured in 0.1 M Na$_{2}$SO$_{4}$ and 50 mM H$_{2}$SO$_{4}$ solution [@10]. In our previous study, it was revealed that the electrochemical potential significantly affected the stability and conductance of the Au mono atomic contact. At the hydrogen evolution potential, the conductance histogram showed the 0.5 $G_{0}$ peak, whose intensity could be tuned by the electrochemical potential. The stability of the Au mono atomic contact could be also tuned by the electrochemical potential. As the potential of the Au electrodes was scanned from $\phi$ = 0.5 V to negative, the average length of the last conductance plateau decreased at $\phi$ = -0.2 V, and then reached a minimum value at $\phi$ = -0.4 V in 0.1 M Na$_{2}$SO$_{4}$ solution. Polarization more negative than $\phi$ = -0.6 V led to the recovery of the length. At the hydrogen evolution potential ($\phi$ $<$ -0.6 V), the 1 nm long Au mono atomic wire could be occasionally fabricated. The distribution of lengths for the last conductance plateau was investigated at the hydrogen evolution potential. The same behavior was observed again in the present system. In addition to previously reported characteristics, several interesting features appeared in the present system. First, the Au atomic contact showed the third conductance value (various conductance values below 1 $G_{0}$) at the oxide formation potential. Second, the Au mono atomic contact showed not only 0.5 $G_{0}$ but also 0.1 $G_{0}$ in acid solution at the hydrogen evolution potential. This 0.1 $G_{0}$ peak was not apparent in the conductance histogram in Fig. 6 of Reference [@10], because the intensity of the background around 0.1 $G_{0}$ was beyond the vertical axis range of the conductance histogram in Fig. 6 of Reference [@10]. Third, the Au atomic contact was stabilized at the oxide formation potential. Fourth, the distribution of lengths for the last conductance plateau was investigated at the double layer and oxide formation potential. The distribution of lengths was also shown for the 1 $G_{0}$ plateau, which provided information about the transformation process of the Au mono atomic contact during the stretching the contact. Fifth, the distribution of lengths for the conductance plateau was precisely determined by the statistical analysis with a large number of measurements for 10 different samples (20000 traces). In our previous report, the distribution of lengths for the last conductance plateau was obtained from limited number of conductance traces ($\sim$1000 traces) for one extraordinarily stable sample at the hydrogen evolution potential [@10]. Although very stable Au mono atomic wire occasionally could be fabricated, most of the Au atomic contact broke within 0.4 nm in length. So, the obtained distribution of lengths for the last conductance plateau shifted to the shorter distance compared to the previous result. The effect of hydrogen and oxygen on the Au mono atomic contacts is discussed based on these new findings and improvements.
The conductance behavior of the Au mono atomic contact is compared with that in UHV. The effect of hydrogen and oxygen on the Au mono atomic contact was investigated at low temperature in UHV [@4; @5]. The broad feature appeared below 1 $G_{0}$ in the conductance histogram of the Au contacts after the introduction of hydrogen gas. The feature below 1 $G_{0}$ was much smaller than the 1 $G_{0}$ peak [@4]. The conductance histogram did not change with the introduction of oxygen gas [@5]. On the other hand, in solution, clear 0.5 $G_{0}$ and 0.1 $G_{0}$ peaks appeared in the conductance histogram at the hydrogen evolution potential. A broad feature appeared below 1 $G_{0}$ at the oxide formation potential. These results suggest that a specific structure of the Au mono atomic contact with a well-defined fractional conductance value (0.5 $G_{0}$ or 0.1 $G_{0}$) was formed at the hydrogen evolution potential, and various structures in which oxygen strongly interacted with the atomic contact were formed at the oxide formation potential. The electrochemical potential determines the potential energy of electrons of the metal nano contact, resulting in the control of the bonding strength between the metal atoms, and of the interaction of the metals with molecules in the surrounding medium. These facts make it possible to use the environment to set the metal contacts, which cannot be set in other environments such as in UHV and in air, leading to successful fabrication of very stable metal nano structures showing the conductance quantization which cannot be observed in UHV.
The structure of the Au mono atomic contact at the hydrogen evolution and oxide formation potential is discussed based on the previously reported experimental result of a flat Au surface and theoretical calculation result. The hydrogen and oxygen evolution reaction on the flat Au electrodes in solution has been investigated by the analysis of current-potential curves [@11; @12; @13]. When the electrochemical potential of the Au electrode is kept positive ($\phi$ $>$ 1.6 V) in acidic solution, oxygen evolution has been proposed to proceed through the following process. First, hydroxyl ions adsorb onto the Au surface forming surface hydroxides (1), second, surface hydroxides convert to oxides (2), and third, O$_{2}$ (gas) desorbs from the surface (3).
Au+H$_{2}$O $\rightarrow$ Au-OH + H$^{+}$ + e$^{-}$ (1)
2Au-OH $\rightarrow$ Au-O + Au + H$_{2}$O (2)
2Au-O $\rightarrow$ 2Au+O$_{2}$ (3)
When the electrochemical potential of the Au electrode is kept negative ($\phi$ $<$ -0.3V) in acid solution, hydrogen evolution proceeds through the following process. First, protons adsorb onto the Au surface (4), and H$_{2}$ (gas) desorbs via surface diffusion and recombination of two adsorbed H atoms (5) or a combination of adsorbed H atoms and proton (5’).
Au+H$_{3}$O$^{+}$ + e$^{-}$ $\rightarrow$ Au-H +H$_{2}$O (4)
2Au-H $\rightarrow$ 2Au+H$_{2}$ (5)
Au-H+ H$_{3}$O$^{+}$ + e$^{-}$ $\rightarrow$ Au + H$_{2}$ + H$_{2}$O (5’)
In the CV of the Au electrode in 0.1 M Na$_{2}$SO$_{4}$ solution (Fig. \[fig1\](b)), the main anodic peak above $\phi$= 0.8 V and an increase in the oxidation current above $\phi$= 1.3 V correspond to the oxide formation process (2) and the O$_{2}$ (gas) desorption process (3). As for the surface oxide, higher oxide (Au$_{2}$O$_{3}$) or hydroxide (Au(OH)$_{3}$) was proposed [@13]. The increase in redox current below $\phi$= -0.7 V corresponds to the H$_{2}$ (gas) desorption process (5). The surface coverage of hydrogen was estimated to be very small ($<$ 0.3 $\%$)[@12]. Here, it should be noticed that the detail reaction mechanism of the oxygen and hydrogen evolution reaction on the Au electrode has not been fully understood up to now. In addition, the reaction on the Au mono atomic contact might not be the same as that on the flat Au electrode. However, the observed conductance behavior and the reaction on the flat metal surface strongly suggested that the oxygen or hydrogen is adsorbed on or incorporated into the Au mono atomic contact at the oxide formation and hydrogen evolution potential.
The structure of the Au mono atomic contact is discussed with the previously reported theoretical calculation result. The interaction between the Au mono atomic wire and oxygen or hydrogen has been investigated with theoretical calculations [@15; @16; @17]. Theoretical calculation results showed that the hydrogen 1s orbital or oxygen 2p orbital effectively hybridized with the Au 5d and Au 6s orbitals. Due to the strong interaction, oxygen and hydrogen molecules dissociated on the Au mono atomic wire, and then were stably incorporated into the Au mono atomic wire. In the hydrogen (oxygen) incorporated wire, we can expect electron transfer from hydrogen to Au (from Au to oxygen). The Au wire would be, thus, positively (negatively) charged for the hydrogen (oxygen) incorporated wire. Because of the formation of hydrogen or oxygen incorporation into the wire, the conductance of the Au mono atomic wire decreased to 0.6-0.01 $G_{0}$, possibly due to scattering or interference of conducting electrons in the wire. No preferential atomic configurations were found for the hydrogen or oxygen incorporated Au mono atomic wire. Jelinek et al. calculated the conductance of the Au mono atomic wire in which a hydrogen atom or an undissociated molecule adsorbed on it [@16]. The conductance value was calculated to be 0.7-0.5 $G_{0}$ and 1.05-0.95 $G_{0}$ for the Au atomic contact in which a hydrogen atom and molecule adsorbed on it, respectively. The adsorbed atomic hydrogen effectively affected the conductance of the Au mono atomic wire, as is the case for the incorporation of hydrogen into the Au wire. On the other hand, the conductance value of the Au atomic wire with an adsorbed undissociated hydrogen molecule did not change from that of the clean Au atomic wire (1 $G_{0}$). The adsorbed undissociated hydrogen molecule would not affect the conductance of the Au mono atomic wire, in contrast with the adsorbed hydrogen atom and incorporated hydrogen.
Now, we try to propose the structure model of the Au mono atomic contact at the hydrogen evolution and oxide formation potential based on the above discussion. At the oxide formation potential, various conductance values were observed for the Au mono atomic contact (see Fig. \[fig2\](c,f)). This conductance behavior agreed with the calculated conductance behavior of the oxygen incorporated Au wire [@15; @16; @17]. So, the conductance value below 1 $G_{0}$ would originate from the oxygen incorporated Au mono atomic contact. Combing with the result of the plateau length analysis showing the long last conductance plateau and short 1 $G_{0}$ plateau which was comparable to the 1 $G_{0}$ plateau at the double layer potential, the following transformation process of the Au atomic contact could be proposed. Initially, the clean Au mono atomic contact was formed during the stretching the contact. Then, the oxygen incorporated Au mono atomic contact would be formed. In contrast to the oxide formation potential, reversible transition between 1 $G_{0}$ and the well defined fractional conductance value (0.5 $G_{0}$) were observed in the conductance trace at the hydrogen evolution potential (see Fig. \[fig2\](a)). The reversible transition suggests the successive adsorption and desorption of hydrogen on the Au mono atomic contact. In the case of the atom or molecular adsorption on flat metal surfaces, atoms or molecules often adsorb with a fixed atomic configuration [@17]. Therefore, hydrogen could also adsorb on the Au mono atomic contact with a certain fixed atomic configuration showing a fixed conductance value. When the hydrogen incorporates into the Au contact, the Au mono atomic contact would not show a fixed conductance value, in contrast to the present experimental result. So, the hydrogen adsorbed Au mono atomic contact would be formed at the hydrogen evolution potential. As discussed in the previous section, the surface coverage of hydrogen is very low on the Au electrode [@12], and thus, the number of hydrogen atoms or molecules adsorbed on the Au mono atomic contact would be very small. The conductance value was calculated to be about 1.0 $G_{0}$ and 0.6 $G_{0}$ for the Au atomic contact in which one hydrogen molecule and atom adsorbed on it, respectively [@16]. Therefore, 0.5 $G_{0}$ would originate from the Au mono atomic contact, in which one hydrogen atom adsorbed on it.
The above discussion about the Au mono atomic contact with an adsorbed hydrogen atom could be supported by the conductance behavior of the Au nano contacts in acid solution. In acid solution, the rate determining process of the hydrogen evolution reaction on Au electrodes is the surface diffusion and desorption process of H$_{2}$ gas (5). The surface coverage of hydrogen atoms on the Au surface in acid solution is expected to be higher than that in neutral solution (Na$_{2}$SO$_{4}$) [@11]. Therefore, more than one hydrogen atom occasionally could adsorb on the Au mono atomic contact in acid solution under the hydrogen evolution reaction. The Au mono atomic contact with two adsorbed hydrogen atoms would show a smaller conductance value than that with one hydrogen atom. In 50mM H$_{2}$SO$_{4}$ and 0.1 M HClO$_{4}$, the 0.1 $G_{0}$ peak appeared in the conductance histogram in addition to the 0.5 $G_{0}$ peak at the hydrogen evolution potential (see Fig. \[fig3\](b)). The 0.1 $G_{0}$ peak would originate from the Au mono atomic contact in which two atomic hydrogen atoms adsorbed on it. This hypothesis about the origin of 0.1 $G_{0}$ peak can be supported by the conductance behavior of the Au atomic contact in acid solution at the hydrogen evolution potential. In the conductance traces, the reversible transitions of the conductance between 0.5 $G_{0}$ and 0.1 $G_{0}$, 1 $G_{0}$ and 0.5 $G_{0}$ were observed (see Fig. \[fig3\](a)). The reversible transitions of the conductance between 0.5 $G_{0}$ and 0.1 $G_{0}$ suggested the successive adsorption and desorption of hydrogen atom on the Au mono atomic contact, as is the case for the reversible transitions of the conductance between 1 $G_{0}$ and 0.5 $G_{0}$. The appearance of the 0.1 $G_{0}$ peak in the conductance histogram also suggested that the origin of the fractional conductance value was not the dimerized wire proposed in previous study [@10]. It is because the dimerized wire would show one fixed conductance value ($\sim$0.5 $G_{0}$ ). But, the Au mono atomic contact showed both 0.1 $G_{0}$ and 0.5 $G_{0}$ in acid solution under the hydrogen evolution.
Improved stability of the Au mono atomic contact at the hydrogen evolution and oxide formation potential is discussed. In UHV at 4 K, stabilization of Au or Ag mono atomic wire was observed by oxygen incorporation into the wire. As for the Ag, a 2nm long mono atomic wire could be formed by oxygen incorporation into the wire, while a clean Ag forms only short atomic contact [@5]. The theoretical calculation results showed that the incorporated oxygen strengthened Au-Au and Ag-Ag bonds in the wire, leading to stabilization of the Au and Ag mono atomic contact [@18]. The stabilization of the metal nano wires by molecular adsorption was also observed for Au, Fe, Co, and Ni nano wires [@19]. This stabilization was explained by the decrease in the surface energy caused by the molecular adsorption on the metal nano wires. Therefore, the stabilization of the Au mono atomic contact at the oxygen formation potential could be explained by the increase in the bond strength in the atomic contact caused by the oxygen incorporation into the contact. The stabilization of the contact at the hydrogen evolution potential could be explained by the decrease in the surface energy caused by the hydrogen adsorption on the Au mono atomic contact. The plateau length analysis revealed that the 1 $G_{0}$ plateau was much longer at the hydrogen evolution potential than the double layer potential. Since the conductance of the Au mono atomic contact with an adsorbed hydrogen molecule was calculate to be about 1 $G_{0}$ [@16], the long 1 $G_{0}$ plateau at the hydrogen evolution potential suggested that the Au mono atomic contact would be stabilized by the adsorption of a hydrogen molecule on the Au mono atomic contact.
In the present study, the structure models of the Au mono atomic contact at the hydrogen evolution and oxide formation potential were proposed based on the experimental and theoretical calculation result. Although our model of the hydrogen adsorbed and oxygen incorporated Au mono atomic contact could explain the experimental results, we did not obtain direct evidence for the formation of the hydrogen adsorbed and oxygen incorporated Au mono atomic contact proposed in this study. In addition, it is not completely clear whether a hydrogen (or oxygen) molecule or atom adsorbed on (or incorporated into) the Au mono atomic contact. The model proposed in this study is one of the possible models. Further investigation is needed to fix the structure formed at the hydrogen evolution and oxide formation potential. Although the structure model is not defined yet, our experimental results clearly showed the existence of hydrogen or oxygen at the Au mono atomic contact as a change of conductance. Up to now, there is little experimental results which directly show the existence of hydrogen on the Au surface under the hydrogen evolution reaction. Based on the qualitative analysis of current-potential curves, the surface coverage of hydrogen has been estimated. Thus, our present study might shed light on the understanding the mechanism of the hydrogen evolution reaction on the Au surface, as well as the conductance modulation of metal nanowire.
CONCLUSIONS {#sec4}
===========
The conductance behavior of the Au mono atomic contact was studied under the electrochemical potential control. The stability and conductance of the Au contact could be fully controlled by the electrochemical potential. The conductance could be defined among three different conductance states at respective potentials: 1 $G_{0}$ at the double layer potential, 0.5 $G_{0}$ at hydrogen evolution and conductance below 1 $G_{0}$ at the oxide formation potential. Based on the comparison between the conductance behavior and previously documented surface processes, we proposed the structural model. An oxygen would incorporate into the contact at the oxide formation potential. Atomic hydrogen would adsorb on the Au mono atomic contact at the hydrogen evolution potential. At the hydrogen evolution and oxide formation potential, the Au mono atomic contact was stabilized. The stabilization could be explained by the oxygen incorporation into the contact or hydrogen adsorption on the contact. Control of the three conductance states, stabilization of a fixed conductance state with hydrogen, and the appearance of the structure showing strong interaction between Au or oxygen were observed only for the metal contact under the electrochemical potential control, which could not be observed in other environments. The present results showed that electrochemical potential is one of the promising external fields for controlling the properties of nano scale materials.
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abstract: 'Recently some experimental evidences have been obtained in favour of the existence of the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state in heavy-fermion superconductor CeCoIn$_{5}$ and organic superconductor $\lambda$-(BETS)$_{2}$FeCl$_{4}$. However the unambiguous identification of FFLO state remains very difficult. We present the theoretical studies of the Gaussian fluctuations near the tricritical point (where the FFLO modulation appears) and demonstrate that the behavior of the fluctuational specific heat, paraconductivity and diamagnetism is qualitatively different from the usual superconducting transition. Special values of the critical exponent and the crossovers between different fluctuational regimes may provide a unique test for the FFLO state appearance.'
author:
- François Konschelle
- Jérôme Cayssol
- 'Alexandre I. Buzdin'
title: Anomalous fluctuation regimes at the FFLO transition
---
Introduction
============
The long-standing hunt for the inhomogenous superconducting state predicted by Fulde and Ferrell [@fulde_ferrell(1964)] and by Larkin and Ovchinnikov [@larkin_ovchinnikov(1964)] (the so-called FFLO state) has recently known a revival in the study of fermionic cold atoms [@zwierlein(science).2006; @partridge.2006], quantum chromodynamics color superconductivity [@casalbuoni.nardulli.2004] and heavy fermion superconductor CeCoIn$_{5}$ [@radovan_murphy.2003; @bianchi_movshovich.2002; @miclea.nicklas.2006]. The FFLO phase consists in a condensate of finite momentum Cooper pairs in contrast to the zero momenta pairs of the usual BCS state. Hence the FFLO superconducting order parameter acquires a spatial variation. Such a state may be induced when a chemical potential difference is applied between two species of fermions with mutual pairing attraction.
In superconductors the different chemical potentials are obtained by a Zeeman splitting $h=\mu_{B}H$ between spin up and down states, $\mu_{B}$ and $H$ being respectively the Bohr magneton and an external magnetic field (in magnetic superconductors $h$ is the internal exchange field). A characteristic feature of the field-temperature phase diagram is the existence of a tricritical point (TCP) which is the meeting point of three transition lines separating the normal metal, the uniform superconductor and the FFLO state. For clean $s$-wave superconductors the TCP is located at $T^{\ast}=0.56T_{c0}$ and $h^{\ast}=1.07k_{B}T_{c0}$, $T_{c0}$ being the zero field critical temperature. Unfortunately, the FFLO state is only energetically favorable in a small part of the phase diagram, located at low temperatures $T<T^{\ast}$ and high fields, see Fig.\[FIG\_phase\_diagram\]. Up to now, there is no compelling experimental evidence for this mysterious FFLO phase although some heavy fermion superconductors like CeCoIn$_{5}$ [@radovan_murphy.2003; @bianchi_movshovich.2002; @miclea.nicklas.2006], organic superconductors like $\lambda$-(BETS)$_{2}$FeCl$_{4}$ [@balicas.2001; @uji_graf_brooks.2006], and rare-earth magnetic superconductors like ErRh$_{4}$B$_{4}$ [@bulaevskii_buzdin.1985] are promising candidates.
In the neighborhood of the TCP, the FFLO transition may be described within a modified Ginzburg-Landau approach (MGL). The MGL free energy functional, which follows from BCS theory with Zeeman pair breaking, was first obtained for $s$-wave superconductors [@buzdin_kachkachi(1997)] and then generalized to $d$-wave pairing with/without impurities [@yang_agterberg.2001]. The salient features of the MGL functional are the presence of higher order derivatives of the order parameter $\Psi$ than the usual $g\left\vert
\mathbf{\nabla}\Psi\right\vert ^{2}$ term, and the fact that the coefficient $g$ of the latter term changes sign at the TCP being positive for $h/2\pi
k_{B}T<0.3$ and negative otherwise. When the stiffness $g$ becomes negative, the spatially varying configurations of the order parameter are favored leading to the FFLO transition. As a result of the anisotropy contained in higher order derivatives, the degeneracy over the FFLO modulation is removed even in the cubic lattice. Another source of anisotropy is related to momentum-dependent pairing. In the case of a $d$-wave Pauli limited superconductor both lattice and pairing anisotropies are of primary importance for the FFLO [@yang_sondhi.1998; @vorontsov_sauls_graf.2005; @shimahara.2002].
Beside these mean-field studies, the fluctuations have rarely been considered in the context of the FFLO transition, although they may be accounted for in a quite simple and accurate manner within the framework of the MGL. Using the standard Ginzburg-Landau functional, the effect of fluctuations on the properties of the normal state were extensively studied for the homogeneous BCS superconductors [@skocpol_tinkham.1975; @b.tinkham] and high-$T_{c}$ superconductors [@b.larkin_varlamov]. Though relatively small, thermodynamical and transport properties were actually measured in the Gaussian regime.
Up to now, a detailed investigation of the fluctuations near the FFLO transition was missing. A first attempt to fill this flaw was realized recently by Marenko and Samokhin who addressed the issue of the quantum critical fluctuations at the zero temperature FFLO transition [@samokhin.marenko.2006]. The aim of this paper is to perform the complementary investigation of the fluctuations in the neighborhood of the tricritical point.
In this Letter, we study the effect of the fluctuations close to the FFLO transition. We focus on the specific heat $C$ and on the excess conductivity $\sigma$, the so-called paraconductivity, induced by the superconducting fluctuations within the normal state. We have identified three regimes in the vicinity of the tricritical point $\left( T^{\ast},h^{\ast}\right) $ which differ by the nature of the soft fluctuation modes.
Within the first regime, depicted as region I in Fig.\[FIG\_phase\_diagram\], the soft modes have the usual quadratic dispersion and are located around the origin of momentum space. This familiar situation yields the well-established power law behaviors $\left( C,\sigma\right) \sim\left( T-T_{c}\right)
^{(d-4)/2}$ reviewed in [@skocpol_tinkham.1975; @b.tinkham; @b.larkin_varlamov].
{width="3in"}
In contrast, the second regime represented as region II in Fig.\[FIG\_phase\_diagram\], is characterized by anomalous power laws in isotropic or weakly anisotropic materials. Here anomalous simply means distinct from $\left( C,\sigma\right) \sim\left( T-T_{c}\right)
^{(d-4)/2}$. This discrepancy is related to the fact that the minima of the dispersion are attained on a finite dimensional surface, a sphere or a ring, see Fig.\[FIG\_crossovers\].$\left( c\right) $. Hence the available phase space volume of the low energy fluctuations is larger than in the case of the standard BCS transition towards a uniform ground state, as it was first pointed out by Brazovski [@brazovski]. On the other hand, among the host of promising candidates for the FFLO state, crystal or gap symmetry effects are important. In such case the sphere or ring containing the low energy modes is partially removed and the low energy fluctuations are located within small islands centered at a few number of isolated points in momentum space, as it is shown in Fig.\[FIG\_crossovers\].$\left( a\right) $. Thus for large anisotropy, the anomaly disappears and one recovers the usual power laws $\left( C,\sigma\right) \sim\left( T-T_{c}\right) ^{(d-4)/2}$. In summary the exponents in regime II depend drastically on the anisotropy of the system.
The most interesting regime, region III of Fig.\[FIG\_phase\_diagram\] is at the immediate vicinity of the TCP where the FFLO and the uniform superconducting phases compete. The power laws of the specific heat $C\sim\left( T-T_{c}\right) ^{(d-8)/4}$ and paraconductivity $\sigma
\sim\left( T-T_{c}\right) ^{(d-6)/4}$ are anomalous and quite universal since they pertain in both isotropic and anisotropic models. These predictions rely on the fact that the dispersion of the soft fluctuation modes becomes quartic within this regime.
Indeed, these three regimes and the related crossovers between them may serve as a powerfull tool to identify FFLO state.
General formalism
=================
We investigate the fluctuation properties at the transition between normal and FFLO states using a quite general approach based on the MGL functional [@buzdin_kachkachi(1997); @yang_agterberg.2001]. Note that we assume a second order phase transition to the FFLO state. In the framework of isotropic model this transition is of second order for one and two dimensional systems while it is of first order in three dimensional case [@buzdin_kachkachi(1997)]. However in the real superconductors the crystalline anisotropy (even a cubic one) is of primary importance and can modify this conclusion. Nevertheless, we may expect that in the case of weakly first order FFLO transition, the fluctuation regimes studied here would be still observable. Such description is adequate near the tricritical point where the most striking behavior is expected and where the wave-vector of the FFLO modulation is small. Looking for Gaussian fluctuations in the normal state it is sufficient to consider MGL functional only in the quadratic $\left\vert \Psi_{\mathbf{k}}\right\vert ^{2}$ approximation $$H\left[ \Psi\right] =\sum_{\mathbf{k}}\text{ }\underset{i,j=1}{{\displaystyle\sum^{d}}
}\left( \alpha+gk_{i}^{2}+\gamma_{ij}k_{i}^{2}k_{j}^{2}\right) \left\vert
\Psi_{\mathbf{k}}\right\vert ^{2} \label{EQ_our_model}$$ where $\Psi_{\mathbf{k}}$ are the Fourier components of the complex order parameter and $d$ the dimensionality of the sample. The coefficient $\alpha=a(T-\widetilde{T}_{c})$, the coefficients $g$ and $\gamma_{ij}$ depend on the magnetic field and temperature, $\widetilde{T}_{c}\left( h\right) $ being the critical temperature for the second order transition between normal and uniform superconducting states. Although qualitative conclusions are quite general, explicit calculations are performed within the cubic anisotropy model defined by $\gamma_{ij}=\gamma\delta_{ij}+\gamma\eta\left( 1-\delta
_{ij}\right) $, where $\gamma>0$. The case $\eta=1$ corresponds to the isotropic model since then the general fluctuation spectrum $\varepsilon
_{\mathbf{k}}=$ $\alpha+gk^{2}+\Sigma_{i,j}\gamma_{ij}k_{i}^{2}k_{j}^{2}$ depends only on $k^{2}=\Sigma_{i}k_{i}^{2}$ and on $k^{4}$. Hence $\left\vert
\eta-1\right\vert $ is a measure of the deviation from the ideally isotropic material with a spherical Fermi surface.
Integrating out the Gaussian modes $\Psi_{\mathbf{k}}$, one obtains the fluctuation free energy density $$F=k_{B}T\frac{1}{L^{d}}\sum_{\mathbf{k}}\ln\frac{\varepsilon_{\mathbf{k}}}{\pi
k_{B}T} \label{Free}$$ where $L$ is the size of the system. The specific heat may be readily deduced from the free energy (\[Free\]) via $$C=-T_{c}\frac{\partial^{2}F}{\partial T^{2}}=a^{2}k_{B}T_{c}^{2}\frac{1}{L^{d}}\sum_{\mathbf{k}}\text{ }\frac{1\text{ }}{\varepsilon_{\mathbf{k}}^{2}}\text{ \ .} \label{EQ_heat_capa}$$ In evaluating the paraconductivity within the Kubo formalism, one should use the time-dependent Ginzburg-Landau equation to obtain the current-current correlator at different times[@b.tinkham]. Moreover attention should be paid to the fact that, owing to the presence of high order derivatives $\gamma_{ij}k_{i}^{2}k_{j}^{2}$ in Eq.(\[Free\]), the expression for the current operator differs from the usual one. However the general expression[@b.larkin_varlamov] relating the paraconductivity to the fluctuation spectrum,$$\sigma_{xx}=\sigma_{yy}=\frac{\pi e^{2}ak_{B}T_{c}}{4\hslash}\frac{1}{L^{d}}\sum_{\mathbf{k}}\frac{v_{\mathbf{k}x}^{2}}{\varepsilon_{\mathbf{k}}^{3}},
\label{EQ_sigmagene}$$ still holds for the FFLO case although expressions for the velocities $v_{\mathbf{k}x}=\partial\varepsilon_{\mathbf{k}}/\partial k_{x}$ and for the current are very different from the usual quantum mechanical formula.
Owing to the denominators in Eqs.(\[EQ\_heat\_capa\],\[EQ\_sigmagene\]) the main contributions to the specific heat and conductivity originate from the modes with the lowest energies $\varepsilon_{\mathbf{k}}$. The above expressions lead to singular power laws which diverge at the transition with characteristic exponents. The exponents depend on the nature of the low energy modes: their location in momentum space and their dispersion. In the usual case of a uniform superconducting state, namely for $g>0$ and $\gamma_{ij}=0,$ the low energy modes are concentrated around the center of the Brillouin zone and propagate with a quadratic dispersion. The resulting power-law divergencies of the thermodynamical and transport properties are well-known [@skocpol_tinkham.1975; @b.tinkham; @b.larkin_varlamov]. In the vicinity of the tricritical point, the situation is much more interesting since three different kinds of spectra $\varepsilon_{\mathbf{k}}$, namely (\[EQ\_spectreiso\]), (\[EQ\_spectreaniso\]) or (\[EQ\_spectrum\_g\_0\]), lead to different power laws.
{width="3in"}
Isotropic model and large stiffness
===================================
We first consider the isotropic case $\eta=1$. At low fields and high temperatures $h/2\pi k_{B}T<0.3$, the positive stiffness $g$ leads to the standard spatially uniform superconducting state (referred as BCS state on Fig.\[FIG\_phase\_diagram\]). Conversely when $h/2\pi k_{B}T>0.3$, the stiffness becomes negative and the fluctuation spectrum $\varepsilon
_{\mathbf{k}}^{iso}=\alpha+gk^{2}+\gamma k^{4}$ may be rewritten as $$\varepsilon_{\mathbf{k}}^{iso}=\tau+\gamma\left( k^{2}-q_{0}^{2}\right)
^{2}\text{ \ } \label{EQ_spectrum}$$ where $q_{0}^{2}=-g/2\gamma$ and $\tau=\alpha-g^{2}/4\gamma$. Below $T_{c}=\widetilde{T}_{c}+g^{2}/4\gamma a\geq\widetilde{T}_{c}$ the normal state is instable towards the formation of the FFLO state with modulation wave vector $\boldsymbol{q}_{0}$. Owing to the isotropy of the model, all directions of $\boldsymbol{q}_{0}$ are degenerate. Above $T_{c}$ the minima of $\varepsilon_{\mathbf{k}}^{iso}$ are attained on a finite dimensional surface: a sphere for $d=3$ or a ring for $d=2,$ whose radius is exactly $q_{0}.$ The one dimensional case is special in the sense that the minima consist in two points with coordinates $\pm q_{0}$.
At the vicinity of the sphere/ring defined by $k=q_{0}$ the approximation$$\varepsilon_{\mathbf{k}}^{iso}=a\left( T-T_{c}\right) +4\gamma q_{0}^{2}\left( k-q_{0}\right) ^{2}\text{ \ } \label{EQ_spectreiso}$$ may be used to evaluate Eqs.(\[EQ\_heat\_capa\],\[EQ\_sigmagene\]) leading to the following singular behaviors for the specific heat $$~C_{iso}=A_{d}k_{B}T_{c}^{2}\left( \frac{a}{\gamma}\right) ^{1/2}\left(
\frac{\left\vert g\right\vert }{2\gamma}\right) ^{\left( d-2\right)
/2}\left( T-T_{c}\right) ^{-3/2}~~~~, \label{Heatiso}$$ and for the paraconductivity $$\sigma_{iso}=\frac{\pi A_{d}e^{2}k_{B}T_{c}}{\hslash d}\left( \frac{\gamma
}{a}\right) ^{1/2}\left( \frac{\left\vert g\right\vert }{2\gamma}\right)
^{d/2}\left( T-T_{c}\right) ^{-3/2}, \label{Condiso}$$ with $A_{d}=1/4,1/8,1/8\pi$ for $d=1,2,3$ respectively. Both $C_{iso}$ and $\sigma_{iso}$ diverge at the FFLO critical temperature $T_{c}$ with the same $d$-independent exponent in contrast to the usual uniform superconductivity case. In particular for the quasi-two and three dimensional cases, the divergencies of the fluctuation contributions near the FFLO transition are much stronger than the $\left( T-T_{c}\right) ^{(d-4)/2}$ power law characterizing the usual superconducting transition. In this sense, the anomalous behavior of the Gaussian fluctuations is a signature of the FFLO state in such ideal isotropic systems.
Anisotropic models and large stiffness
======================================
The previous results $\left( \ref{Heatiso},\ref{Condiso}\right) $ rely deeply on the perfect isotropy of the model. Nevertheless real superconductors are anisotropic owing to either crystalline effect or $d$-wave pairing. Hence the minima of the dispersion occur at a few number $N_{\alpha\text{ }}$of isolated points $\boldsymbol{q}_{0}^{(\alpha)}$ which are located at a finite distance from the origin. These points belong to high symmetry lines of the crystal. More specifically in the model $\gamma_{ij}=\gamma\delta_{ij}+\gamma\eta\left( 1-\delta_{ij}\right) $ the wave vectors $\boldsymbol{q}_{0}^{(\alpha)}$ are along the diagonals when $\eta<1$ and along the crystalline axis for $\eta>1$, whereas $\eta=1 $ corresponds to the isotropic case. The relevant low energy fluctuations form small islands around the points $\boldsymbol{q}_{0}^{(\alpha)}$, as shown in Fig.\[FIG\_crossovers\].$\left( a\right) $. In the case of a very weak anisotropy $\left\vert
\eta-1\right\vert \ll1$, these pockets may collapse within a finite dimensional surface and one recovers the isotropic model discussed above, see Fig.\[FIG\_crossovers\].$\left( b,c\right) $.
Here we consider the opposite case of significant anisotropies $\left\vert
\eta-1\right\vert \sim1$ which is expected in real compounds. Then the favored modes belong to $N_{\alpha\text{ }}$disconnected islands. Hence one may focus on each of the $\boldsymbol{q}_{0}^{(\alpha)}$ obtained by solving $\partial_{i}\varepsilon_{\mathbf{k}}(\boldsymbol{q}_{0}^{(\alpha)})=0$ and expand the fluctuation spectrum of $\left( \ref{EQ_our_model}\right) $ around $\boldsymbol{q}_{0}^{(\alpha)}$ as$$\varepsilon_{\mathbf{k}}^{ani}=a\left( T-T_{c}\right) +\underset{i,j=1}{{\displaystyle\sum^{d}}
}\lambda_{ij}^{(\alpha)}\left( k_{i}-q_{0i}^{(\alpha)}\right) \left(
k_{j}-q_{0j}^{(\alpha)}\right) \label{EQ_spectreaniso}$$ where $\lambda_{ij}^{(\alpha)}=\left( 1/2\right) (\partial_{i}\varepsilon_{\mathbf{k}}\partial_{j}\varepsilon_{\mathbf{k}})(\boldsymbol{q}_{0}^{(\alpha)})$ is the effective mass tensor for the fluctuating modes located around $\boldsymbol{q}_{0}^{(\alpha)}$. Owing to the symmetry of the crystal, the eigenvalues $\lambda_{i}$ of the tensors $\lambda_{ij}^{(\alpha
)}$ are the same for the $N_{\alpha}$ islands.
Evaluating $\left( \ref{EQ_heat_capa}\right) $ and $\left(
\ref{EQ_sigmagene}\right) $ with the spectrum $\left( \ref{EQ_spectreaniso}\right) $ and collecting the contributions from the $N_{\alpha\text{ }}$minima yield the specific heat $$C_{ani}\simeq N_{\alpha}k_{B}T_{c}^{2}\frac{a^{d/2}}{\sqrt{\det\lambda_{ij}}}\left( T-T_{c}\right) ^{\left( d-4\right) /2}\text{ } \label{HeatAniso}$$ and the diagonal paraconductivity $\sigma_{ani}(=\sigma_{xx}=\sigma_{yy})$ in the plane $xy$ $$\sigma_{ani}\simeq\frac{N_{\alpha}}{2}\frac{e^{2}k_{B}T_{c}}{\hslash}\left(
\frac{(\lambda_{1}+\lambda_{2})a^{d/2-1}}{\sqrt{\det\lambda_{ij}}}\right)
\left( T-T_{c}\right) ^{\left( d-4\right) /2} \label{CondAniso}$$ where $\lambda_{1}$ and $\lambda_{2}$ are the eigenvalues of $\lambda
_{ij}^{(\alpha)}$ whose eigenvectors span the $xy$ plane. For the conductivity $\sigma_{zz}$ one should replace $\lambda_{1}+\lambda_{2}$ by $2\lambda_{z}$. In sharp contrast to the isotropic model results (\[Heatiso\],\[Condiso\]), the temperature dependence of both $C_{ani}$ and $\sigma_{ani}$ becomes sensitive to the dimension with the same exponents as in the homogeneous case [@skocpol_tinkham.1975; @b.larkin_varlamov]. Thus in the anisotropic model, the exponents themselves no longer provide a valuable hallmark for the FFLO state. Naturally when $\eta\rightarrow1$ one should recover the isotropic results (\[Heatiso\],\[Condiso\]). The crossover from anisotropic to isotropic models is provided by the $\eta$ dependence of $\det\lambda_{ij}.$ The latter determinant vanishes in the limit $\eta\rightarrow1$, as an example like $\det\lambda_{ij}=g^{2}(1-\eta^{2})$ in the quasi-two dimensional case.
Three fluctuation regimes
=========================
Let us check now the validity of the approximations (\[EQ\_spectreiso\]) and (\[EQ\_spectreaniso\]) used so far in deriving the specific heat and paraconductivity. We aim to identify the region of the phase diagram where the previous results (\[Heatiso\],\[Condiso\]) and (\[HeatAniso\],\[CondAniso\]) apply. The modes which participate to the singular parts of the specific heat and conductivity are spread within a shell of width $\delta
k$ around the circle $k=q_{0}$ (isotropic case) or in islands of size $\delta
k$ centered at isolated points (anisotropic case). The width $\delta k$, which is defined by the condition $4q_{0}^{2}\gamma\delta k^{2}\approx\tau
=a(T-T_{c})$, must be smaller than $q_{0}$ in order to have well defined shell or islands (like in Fig.\[FIG\_crossovers\].$\left( a\right) $). Hence the common range of validity of Eqs.(\[Heatiso\],\[Condiso\]) and (\[HeatAniso\],\[CondAniso\]) is given by $\tau\ll\gamma q_{0}^{4}$ or equivalently $\left\vert T-T_{c}\right\vert \ll g^{2}/a\gamma.$ This corresponds to the area of the phase diagram which is sufficiently apart from the tricritical point (large enough $g$) and close to critical line (small enough $\left\vert T-T_{c}\right\vert $), schematically the region II on Fig.\[FIG\_phase\_diagram\].
Conversely when $\delta k\gtrsim q_{0}$ the relevant low energy modes merge towards the center of the Brillouin zone and the approximations (\[EQ\_spectreiso\]) and (\[EQ\_spectreaniso\]) fail. Hence the specific heat and paraconductivity must be reevaluated in the regime $\left\vert
T-T_{c}\right\vert \gg g^{2}/a\gamma$. Then the physics is dominated by extremely soft fluctuation modes having a quartic dispersion instead of the more usual quadratic one, yielding very special behaviors in the region III of Fig.\[FIG\_phase\_diagram\].
Finally we investigate the crossover regime between region I and III, see Fig.\[FIG\_phase\_diagram\]. When $g>0$ the specific heat is given by $$C_{iso}=\frac{a^{2}k_{B}T_{c}^{2}}{\gamma^{d/4}}\alpha^{(d-8)/4}{\displaystyle\int\nolimits_{0}^{\infty}}
\frac{dx\Omega_{d}x^{d-1}}{\left( 1+x^{4}+gx^{2}/\sqrt{\alpha\gamma}\right)
^{2}}\text{ \ .} \label{EQ_crossgposi}$$ in the isotropic case, $\Omega_{d}$ being the surface of the $d$-dimensional unit sphere. The parameter $g$ controls the relative importance of quartic and quadratic terms. For large $g$ the usual BCS results are recovered while the result Eq.(\[Heattri\]) is obtained in the opposite limit $g\rightarrow0.$ Both quartic and quadratic terms are equally important when $g^{2}\sim
\alpha\gamma$ yielding to the same criterion $\left\vert T-T_{c}\right\vert
\sim g^{2}/a\gamma$ for the borderline between regions I and III than between regions II and III obtained above.
Small stiffness for both isotropic/anisotropic cases
====================================================
We now consider the model (\[EQ\_our\_model\]) in the regime $\left\vert
T-T_{c}\right\vert \gg g^{2}/a\gamma$ where the most salient features are expected.
*Approaching the tricritical point from the line defined by* $h/2\pi
T=0.3$*, see Fig.\[FIG\_phase\_diagram\]*, one has $g=0$ and the spectrum of the fluctuations $$\varepsilon_{\mathbf{k}}^{\ast}=a\left( T-T^{\ast}\right) +\underset{i,j=1}{{\displaystyle\sum^{d}}
}\gamma_{ij}k_{i}^{2}k_{j}^{2} \label{EQ_spectrum_g_0}$$ becomes purely quartic inducing the following anomalous temperature dependences of specific heat and paraconductivity: $$C^{\ast}=k_{B}T^{\ast2}B_{d}\left( \frac{a}{\gamma}\right) ^{d/4}f_{1}(\eta)(T-T^{\ast})^{(d-8)/4} \label{Heattri}$$ with $B_{d}=3\sqrt{2}/16,1/16,1/16\sqrt{2}\pi$ for $d=1,2,3$ respectively, and $$\sigma^{\ast}=\frac{\pi}{4}\frac{e^{2}}{\hslash}k_{B}T^{\ast}C_{d}\left(
\frac{a}{\gamma}\right) ^{(d-2)/4}f_{2}(\eta)\left( T-T^{\ast}\right)
^{\left( d-6\right) /4} \label{Condtri}$$ with $C_{d}=3\sqrt{2}/32,1/8\pi,5/96\sqrt{2}\pi$ for $d=1,2,3$ respectively. The functions $f_{1}(\eta)$ and $f_{2}(\eta)$ are weakly dependent on the anisotropy factor and are equal to unity in the isotropic case $\eta=1.$ Interestingly, the $d$-dependent exponents differ from those which are typical of the uniform superconductivity (region I on Fig.\[FIG\_phase\_diagram\]) and of the FFLO transition at larger $g$ (region II).
Consequently, the regime $\left\vert T-T_{c}\right\vert \gg g^{2}/a\gamma$ (region III) and the related crossovers with regions I and II should reveal the existence of the tricritical point and FFLO phase. While they were derived along the line $g=0,$ the very unusual power laws $C\sim(T-T^{\ast})^{\left(
d-8\right) /4}$ and $\sigma\sim(T-T^{\ast})^{\left( d-6\right) /4}$ must emerge independently on the way the TCP is approached.
*Following the normal/FFLO transition line* $T_{c}(h)$ *to reach the TCP*, one has $g\sim\tau^{1/2\text{ }}$where $\tau=a(T^{\ast}-T)$. We shall demonstrate that the expressions (\[Heatiso\],\[Condiso\],\[HeatAniso\],\[CondAniso\]) applying in region II yield naturally the power laws (\[Heattri\],\[Condtri\]) accounting for the region III when $T\rightarrow T^{\ast}$. First from Eqs.(\[Heatiso\],\[HeatAniso\]), the specific heat is proportional to the product $g^{(d-2)/2}\tau^{-3/2}$ in the isotropic model and to $g^{-d/2}$ $\tau^{\left( d-4\right) /2\text{ }}$in presence of sensible anisotropy, since the eigenvalues $\lambda_{i}$ of the effective mass tensor scale as $g$ and $\det\lambda\sim g^{d}.$ Taking in mind that $g\sim\tau^{1/2}$ along the transition line, one obtains the more divergent $(T^{\ast}-T)^{(d-8)/4}$ behavior of the specific heat already predicted as Eq.(\[Heattri\]) in region III for both isotropic and anisotropic models. We have used the fact that the FFLO and the second order uniform transition lines intersect themselves at the tricritical point. Furthermore similar scaling on Eqs.(\[Condiso\],\[CondAniso\]) reveals that the conductivity in region II is proportional to $g^{d/2}\tau^{-3/2}$ for the isotropic model and to $g^{(2-d)/2}\tau^{\left( d-4\right) /2}$ for anisotropic models. In both situations, the substitution $g\sim\tau^{1/2}$ leads to the same power ($T^{\ast}-T)^{\left( d-6\right) /4}$, predicted above as Eq.(\[Condtri\]), for the paraconductivity. In summary and more generally, quartic terms in $\varepsilon_{\mathbf{k}}$ dominate the quadratic ones when $\left\vert T-T_{c}\right\vert \gg g^{2}/a\gamma,$ and $\left\vert
T-T_{c}\right\vert \ll T_{c}$, see region III in Fig.1.
Orbital effect
==============
Up to now the orbital effect was neglected to emphasize the influence of the Zeeman pair-breaking effect on the superconducting fluctuations. This is exact in ultracold fermionic atoms since there is no orbital effect at all for neutral objects. This is also a good approximation in superconductors with large Maki parameter which are the ones where the FFLO state is expected. Such a situation is generally encountered for i) layered superconductors under in plane magnetic field when the weakness of the interplane coupling quenches the orbital motion of the electrons (or for thin films of 3D superconductors), and ii) magnetic superconductors where the field originates from internal ordered magnetic moments.
Beside these favorable situations, one should evaluate the effect of orbital pair-breaking on the results obtained previously, especially for three dimensional superconductors under external magnetic field. Under strong magnetic field, the electronic motion is described by the lowest Landau level wherein the kinetic energy is totally quenched except along the direction of the field. This provides a realization of an effective one-dimensional system.
Coming from the BCS side with $g>0$ we evaluate how the fluctuation specific heat and paraconductivity scale as one approaches the tricritical point along the critical line $T_{c}(h)$, namely in the limit $g\rightarrow0^{+}.$ For strong enough fields or sufficiently close to the transition line, the main contribution to the specific heat is given by the lowest Landau level as $$C=a^{2}k_{B}T_{c}^{2}\left( \dfrac{2eB}{\hslash}\right) \int_{-\infty
}^{\infty}\frac{dk_{z}\text{ }}{\left( \tau+\frac{1}{2}\hslash\omega
_{c}+gk_{z}^{2}\right) ^{2}}\text{\ }$$ transforming the three-dimensional fluctuations into one dimensional fluctuations characterized by $C=g^{-1/2}\tau_{c}^{-3/2}$, $\hslash\omega
_{c}=4geB/\hslash$ being the Landau level spacing and $\tau_{c}=\tau
+\hslash\omega_{c}/2=0$ yielding the transition line under magnetic field. In a similar manner one may demonstrate that the magnetoconductivity along $z
$-axis is given by $\sigma_{zz}\sim g^{1/2}\tau_{c}^{-3/2}$. In sum one encounters a divergency of the specific heat while the magnetoconductivity $\sigma_{zz}$ is not singular in the limit $g\rightarrow0^{+}$ performed along the BCS transition line.
Approaching the critical point along the line $g=0$ provides an interesting field-induced crossover. In the isotropic model the specific heat is then given by $$\begin{aligned}
C & =a^{2}k_{B}T_{c}^{2}\dfrac{2eB}{\hslash}\int_{-\infty}^{\infty}\frac{dk_{z}\text{ }}{\left[ \tau+\gamma\left( \frac{2eB}{\hslash}+k_{z}^{2}\right) ^{2}\right] ^{2}}\text{\ ,}\\
& =\dfrac{2eB}{\hslash}\frac{a^{2}k_{B}T_{c}^{2}}{\gamma^{1/4}\tau^{\ast7/4}}\int_{-\infty}^{\infty}\frac{dx_{{}}\text{ }}{\left( 1+x^{4}+\zeta
x^{2}\right) ^{2}}$$ where $\tau^{\ast}=\tau+\gamma(2eB/\hslash)^{2}.$ The dimensionless parameter $\zeta=4(eB/\hslash)\sqrt{\gamma/\tau^{\ast}}$ controls the relative importance of quartic and quadratic terms yielding a crossover field $B^{\ast
}=$ $(\hslash/e)\sqrt{\tau^{\ast}/\gamma}$. A similar crossover between quartic and quadratic momentum dependence was already encountered at the borderlines of the region III, see Eq.(\[EQ\_crossgposi\]). Here the crossover is tuned by the magnetic field along the line $g=0$. Similarly the conductivity $$\sigma_{zz}\text{\ }\sim\frac{e^{2}}{\hslash}\gamma^{2}\frac{\gamma^{1/4}}{\tau^{\ast5/4}}\int_{-\infty}^{\infty}\frac{x^{6}dx\text{ }}{\left(
1+x^{4}+\zeta x^{2}\right) ^{3}}$$ behaves as $\tau^{\ast-5/4}$ for fields below $B^{\ast}=$ $\hslash(\tau^{\ast
}/\gamma)^{1/2}/e$ and $\tau^{\ast-3/2\text{ }}$ for larger fields.
Ginzburg-Levanyuk criterion
===========================
Now let us address the issue of the validity of the Gaussian approximation. It is well known that Gaussian approximation breaks down in the critical regime wherein the interactions between different fluctuation modes of the order parameter become sizeable. In the isotropic model, Brazovski demonstrated that due to the critical fluctuations the transition to the non-uniform state becomes of the first order [@brazovski]. Recently the same conclusion has been obtained in the anisotropic model with isolated minima of $\varepsilon
_{\mathbf{k}}$ [@dalidovich_yang.2004]. However performing a simple evaluation of the $\left\vert \Psi\right\vert ^{4}$ and $\left\vert
\Psi\right\vert ^{6}$ interaction terms contained in the nonlinear MGL functional [@buzdin_kachkachi(1997); @yang_agterberg.2001; @dalidovich_yang.2004], it can be proved that the relative width of the critical region (in temperature or magnetic field) is given by the usual Ginzburg-Levanyuk parameter $(T_{c}/E_{F})^{4}$, $E_{F}$ being the Fermi energy [@konschelle.press]. More precisely and using our notations, the width $\tau_{G}$ of the FFLO critical regime is given by $$(T-T_{c})/T_{c}=\tau_{G}\sim\left\vert \frac{g_{0}}{g}\right\vert \left(
\frac{T_{c}}{E_{F}}\right) ^{4}\text{ }$$ when the fluctuations propagate quadratically, namely when $\left\vert
g/g_{0}\right\vert >(T_{c}/E_{F})^{4/3}$ (regions I and II of Fig.\[FIG\_phase\_diagram\]). Here $g_{0}$ is the value of the coefficient $g$ far from the tricritical point. Conversely when $\left\vert g/g_{0}\right\vert
<(T_{c}/E_{F})^{4/3}$ (region III of Fig.\[FIG\_phase\_diagram\]) the critical region is somewhat broadened as $$(T-T_{c})/T_{c}=\tau_{G}\sim\left( \frac{T_{c}}{E_{F}}\right) ^{8/3}\text{ }$$ owing to the presence of quartic soft modes (region III of Fig.\[FIG\_phase\_diagram\]). However even the largest value of the critical region at $g=0$ is still very small, $(T_{c}/E_{F})^{8/3}\sim10^{-8}-10^{-10}$. Therefore the Gaussian approximation provides an excellent description of the fluctuation phenomena which may be observed at the superconducting FFLO transition. On the other hand the analysis [@brazovski; @dalidovich_yang.2004] may be relevant to the case of fermionic cold atoms where the ratio $T_{c}/E_{F}$ is not so small.
Conclusion
==========
We have demonstrated that the transition towards the FFLO state reveals very unusual and rich fluctuation regimes which may serve as a smoking gun of the FFLO state. Atomic Fermi gases provide a unique realization of an isotropic system while any real superconductor always presents some degree of anisotropy owing either to crystalline effects and/or $d$-wave pairing. The fluctuation contribution vary non-monotonously near the tricritical point when we go from the uniform to FFLO state. In the vicinity of the tricritical point (region III) the very special exponents are expected for specific heat and paraconductivity divergencies owing to the presence of soft modes with unusual quartic dispersion. The presence of the tricritical point is signaled by very pronounced crossovers regimes which might be observed in experiments, by varying temperature or magnetic field. Finally, the fluctuation behavior of the magnetic susceptibility is also very peculiar at the FFLO transition and warrants detailed analysis [@konschelle.press].
We are grateful to M. H<span style="font-variant:small-caps;">ouzet</span>, J.-N. F<span style="font-variant:small-caps;">uchs</span> and K. Y<span style="font-variant:small-caps;">ang</span> for useful comments. This work was supported by ANR Extreme Conditions Correlated Electrons (ANR-06-BLAN-0220).
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---
abstract: 'We review recent progress of understanding and resolving instabilities driven by mismatch between the Fermi surfaces of the pairing quarks in 2-flavor color superconductor. With the increase of mismatch, the 2SC phase exhibits chromomagnetic instability as well as color neutral baryon current instability. We describe the 2SC phase in the nonlinear realization framework, and show that each instability indicates the spontaneous generation of the corresponding pseudo Nambu-Golstone current. The Nambu-Goldstone currents generation state covers the gluon phase as well as the one-plane wave LOFF state. We further point out that, when charge neutrality condition is required, there exists a narrow unstable LOFF (Us-LOFF) window, where not only off-diagonal gluons but the diagonal 8-th gluon cannot avoid the magnetic instability. In this Us-LOFF window, the diagonal magnetic instability cannot be cured by off-diagonal gluon condensate in the color superconducting phase.'
address: |
Physics Department, University of Tokyo,\
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\
E-mail: [email protected]
author:
- Mei Huang
---
SPONTANEOUS NAMBU-GOLDSTONE CURRENTS GENERATION DRIVEN BY MISMATCH
==================================================================
Introduction
------------
Studying QCD at finite baryon density is the traditional subject of nuclear physics. The behaviour of QCD at finite baryon density and low temperature is central for astrophysics to understand the structure of compact stars, and conditions near the core of collapsing stars (supernovae, hypernovae). It is known that sufficiently cold and dense baryonic matter is in the color superconducting phase. This was proposed several decades ago by Frautschi [@Frautschi] and Barrois [@Barrois] by noticing that one-gluon exchange between two quarks is attractive in the color antitriplet channel. From BCS theory [@BCS], we know that if there is a weak attractive interaction in a cold Fermi sea, the system is unstable with respect to the formation of particle-particle Cooper-pair condensate in the momentum space. Studies on color superconducting phase in 1980’s can be found in Ref. [@Bailin-Love]. The topic of color superconductivity stirred a lot of interest in recent years [@cs; @cfl; @weak; @weak-cfl]. For reviews on recent progress of color superconductivity see, for example, Ref. [@reviews].
The color superconducting phase may exist in the central region of compact stars. To form bulk matter inside compact stars, the charge neutrality condition as well as $\beta$ equilibrium are required [@absence2sc; @neutral_steiner; @neutral_huang]. This induces mismatch between the Fermi surfaces of the pairing quarks. It is clear that the Cooper pairing will be eventually destroyed with the increase of mismatch.
Without the constraint from the charge neutrality condition, the system may exhibit a first order phase transition from the color superconducting phase to the normal phase when the mismatch increases [@asym-2sc]. It was also found that the system can experience a spatial non-uniform LOFF (Larkin-Ovchinnikov-Fudde-Ferrell) state [@loff-orig; @LOFF] in a certain window of moderate mismatch.
It is still not fully understood how the Cooper pairing will be eventually destroyed by increasing mismatch in a charge neutral system. The charge neutrality condition plays an essential role in determining the ground state of the neutral system. If the charge neutrality condition is satisfied globally, and also if the surface tension is small, the mixed phase will be favored [@mixed-phase]. It is difficult to precisely calculate the surface tension in the mixed phase, thus in the following, we would like to focus on the homogeneous phase when the charge neutrality condition is required locally.
It was found that homogeneous neutral cold-dense quark matter can be in the gapless 2SC (g2SC) phase [@SH] or gapless CFL (gCFL) phase [@gCFL], depending on the flavor structure of the system. The gapless state resembles the unstable Sarma state [@Sarma; @gABR]. However, under a natural charge neutrality condition, i.e., only neutral matter can exist, the gapless phase is indeed a thermal stable state as shown in [@SH; @gCFL]. The existence of thermal stable gapless color superconducting phases was confirmed in Refs. [@gapless-C] and generalized to finite temperatures in Refs. [@gapless-T]. Recent results based on more careful numerical calculations show that the g2SC and gCFL phases can exist at moderate baryon density in the color superconducting phase diagram [@phase-dia] .
One of the most important properties of an ordinary superconductor is the Meissner effect, i.e., the superconductor expels the magnetic field [@Meissner-cond]. In ideal color superconducting phases, e.g., in the 2SC and CFL phases, the gauge bosons connected with the broken generators obtain masses, which indicates the Meissner screening effect [@Meissner]. The Meissner effect can be understood using the standard Anderson-Higgs mechanism. Unexpectedly, it was found that in the g2SC phase, the Meissner screening masses for five gluons corresponding to broken generators of $SU(3)_c$ become imaginary, which indicates a type of chromomagnetic instability in the g2SC phase [@chromo-ins-g2SC-1; @chromo-ins-g2SC-2]. The calculations in the gCFL phase show the same type of chromomagnetic instability [@chromo-ins-gCFL]. Remembering the discovery of superfluidity density instability [@Wu-Yip] in the gapless interior-gap state [@Liu-Wilczek], it seems that the instability is a inherent property of gapless phases. (There are several exceptions: 1) It is shown that there is no chromomangetic instability near the critical temperature [@no-ins-T] ; 2) It is also found that the gapless phase in strong coupling region is free of any instabilities [@Strong-gapless].)
The chromomagnetic instability in the gapless phase still remains as a puzzle. By observing that, the 8-th gluon’s chromomagnetic instability is related to the instability with respect to a virtual net momentum of diquark pair, Giannakis and Ren suggested that a LOFF state might be the true ground state [@LOFF-Ren-1]. Their further calculations show that there is no chromomagnetic instability in a narrow LOFF window when the local stability condition is satisfied [@LOFF-Ren-2; @GHR-LOFF-Neutral]. Latter on, it was found in Ref. [@GHM-LOFF-Neutral] that a charge neutral LOFF state cannot cure the instability of off-diagonal 4-7th gluons, while a gluon condensate state [@GHM-gluon] can do the job. In Ref. [@NG-Huang] we further pointed out that, when charge neutrality condition is required, there exists another narrow unstable LOFF window, not only off-diagonal gluons but the diagonal 8-th gluon cannot avoid the magnetic instability.
In a minimal model of gapless color supercondutor, Hong showed in Ref. [@hong] that the mismatch can induce a spontaneous Nambu-Goldstone current generation. The Nambu-Goldstone current generation state in U(1) case resembles the one-plane wave LOFF state or diagonal gauge boson’s condensate. We extended the Nambu-Goldstone current generation picture to the 2SC case in the nonlinear realization framework in Ref. [@NG-Huang]. We show that five pseudo Nambu-Goldstone currents can be spontaneously generated by increasing the mismatch between the Fermi surfaces of the pairing quarks. The Nambu-Goldstone currents generation state covers the gluon phase as well as the one-plane wave LOFF state.
This article is organized as follows. In Sec. \[sec-gNJL\], we describe the framework of the gauged SU(2) Nambu–Jona-Lasinio (gNJL) model in $\beta$-equilibrium. We review chromomagnetic instabilities in Sec. \[sec-Mag-Ins\]. We discuss neutral baryon current instability and the LOFF state in Sec. \[sec-bc\]. Then Sec. \[sec-NG-current\] gives a general Nambu-Goldstone currents generation description in the non-linearization framework. At the end, we give the discussion and summary in Sec. \[sec-sum\].
The gauged SU(2) Nambu–Jona-Lasinio model {#sec-gNJL}
-----------------------------------------
We take the gauged form of the extended Nambu–Jona-Lasinio model [@Huang-NJL], the Lagrangian density has the form of $$\begin{aligned}
\label{lagr}
\mathcal{L} &=& {\bar q}(i\fsl{D}+\hat{\mu}\gamma^0)q +
G_S[({\bar q}q)^2 + ({\bar q}i\gamma_5{\bf {\bf \tau}}q)^2 ] \nonumber \\
& + & G_D[(i {\bar q}^C \varepsilon \epsilon^{b} \gamma_5 q )
(i {\bar q} \varepsilon \epsilon^{b} \gamma_5 q^C)] ,
\label{lg}\end{aligned}$$ with $D_\mu \equiv \partial_\mu - ig A_\mu^{a} T^{a}$. Here $A_\mu^{a}$ are gluon fields and $T^a$ with $a=1,\cdots, 8$ are generators of $SU(3)_{\rm c}$ gauge groups. Please note that we regard all the gauge fields as external fields, which are weakly interacting with the system. The property of the color superconducting phase characterized by the diquark gap parameter is determined by the unkown nonperturbative gluon fields, which has been simply replaced by the four-fermion interaction in the NJL model. However, the external gluon fields do not contribute to the properties of the system. Therefore, we do not have the contribution to the Lagrangian density from gauge field part ${\cal L}_g$ as introduced in Ref. [@GHM-gluon]. (In Sec. \[sec-NG-current\], by using the non-linear realization in the gNJL model, we will derive one Nambu-Goldstone currents state, which is equivalent to the so-called gluon-condensate state.)
In the Lagrangian density Eq. (\[lg\]), $q^C=C {\bar q}^T$, ${\bar q}^C=q^T C$ are charge-conjugate spinors, $C=i \gamma^2 \gamma^0$ is the charge conjugation matrix (the superscript $T$ denotes the transposition operation). The quark field $q \equiv q_{i\alpha}$ with $i=u,d$ and $\alpha=r,g,b$ is a flavor doublet and color triplet, as well as a four-component Dirac spinor, ${\bf \tau}=(\tau^1,\tau^2,\tau^3)$ are Pauli matrices in the flavor space, where $\tau^2$ is antisymmetric, and $(\varepsilon)^{ik} \equiv \varepsilon^{ik}$, $(\epsilon^b)^{\alpha \beta} \equiv \epsilon^{\alpha \beta b}$ are totally antisymmetric tensors in the flavor and color spaces. $\hat{\mu}$ is the matrix of chemical potentials in the color and flavor space. In $\beta$-equilibrium, the matrix of chemical potentials in the color-flavor space ${\hat \mu}$ is given in terms of the quark chemical potential $\mu$, the chemical potential for the electrical charge $\mu_e$ and the color chemical potential $\mu_8$, $$\begin{aligned}
\mu_{ij}^{\alpha\beta} = (\mu \delta_{ij} - \mu_e Q_{ij})\delta^{\alpha\beta}
+ \frac{2}{\sqrt{3}} \mu_8 \delta_{ij} (T_8)^{\alpha\beta}.\end{aligned}$$ $G_S$ and $G_D$ are the quark-antiquark coupling constant and the diquark coupling constant, respectively. In the following, we only focus on the color superconducting phase, where $< {\bar q} q >=0$ and $<{\bar q}\gamma^5 {\bf \tau} q>=0$.
After bosonization, one obtains the linearized version of the model for the 2-flavor superconducting phase, $$\begin{aligned}
\label{lagr2}
{\cal L}_{2SC} & = & {\bar q}(i\fsl{D}+\hat \mu \gamma^0)q
-\frac{\Delta^{*b}\Delta^{b}}{4G_D}
%- {\bar q}(\sigma+i \gamma^5{\bf \tau}{\bf \pi}) q
\nonumber \\
&-& \frac{1}{2}\Delta^{*b} (i{\bar q}^C \varepsilon \epsilon^{b}\gamma_5 q )
-\frac{1}{2}\Delta^b (i {\bar q} \varepsilon \epsilon^{b} \gamma_5 q^C)
% \nonumber \\ \frac{\sigma^2+{\bf \pi}^2}{4G_S}
\label{lagr-2sc}\end{aligned}$$ with the bosonic fields $$\begin{aligned}
\Delta^b \sim i {\bar q}^C \varepsilon \epsilon^{b}\gamma_5 q, \ \
\Delta^{*b} \sim i {\bar q} \varepsilon \epsilon^{b} \gamma_5 q^C.
% & & \sigma \sim {\bar q} q , \ \ {\bf \pi} \sim i {\bar q}\gamma^5 {\bf \tau} q.\end{aligned}$$ In the Nambu-Gor’kov space, $$\Psi = \left(\begin{array}{@{}c@{}} q \\ q^C \end{array}\right),$$ the inverse of the quark propagator is defined as $$\left[{\cal S}(P)\right]^{-1} = \left(\begin{array}{cc}
\left[G_0^{+}(P)\right]^{-1} & \Delta^- \\
\Delta^+ & \left[G_0^{-}(P)\right]^{-1}
\end{array}\right),
\label{prop}$$ with the off-diagonal elements $$\Delta^- \equiv -i\epsilon^b\varepsilon\gamma_5 \Delta, \qquad
\Delta^+ \equiv -i\epsilon^b\varepsilon\gamma_5 \Delta^*,$$ and the free quark propagators $G_0^{\pm}(P)$ taking the form of $$\left[G_0^{\pm}(P)\right]^{-1} =
\gamma^0 (p_0 \pm \hat{\mu}) - \vec{\gamma} \cdot \vec{p}.
\label{freep}$$ The 4-momenta are denoted by capital letters, e.g., $P=
(p_0,\vec{p})$. We have assumed the quarks are massless in dense quark matter, and the external gluon fields do not contribute to the quark self-energy.
The explicit form of the functions $G^{\pm}_I$ and $\Xi^{\pm}_{IJ}$ reads $$\begin{aligned}
\label{quark-propagator}
{\rm G}^{\pm}_1&=&
\frac{(k_0-E_{dg}^{\pm})\gamma^0{\tilde \Lambda}_{k}^+}{(k_0\mp\delta\mu)^2-{E_{\Delta}^{\pm}}^2}
+ \frac{(k_0+E_{dg}^{\mp}) \gamma^0{\tilde \Lambda}_{k}^-}{(k_0\mp\delta\mu)^2-{E_{\Delta}^{\mp}}^2},
\nonumber \\
{\rm G}^{\pm}_2&=&
\frac{(k_0-E_{ur}^{\pm})\gamma^0{\tilde \Lambda}_{k}^+}{(k_0\pm\delta\mu)^2-{E_{\Delta}^{\pm}}^2} + \frac{(k_0+E_{ur}^{\mp})\gamma^0{\tilde \Lambda}_{k}^- }{(k_0\pm\delta\mu)^2-{E_{\Delta}^{\mp}}^2}, \nonumber \\
{\rm G}^{\pm}_{3}&=&
\frac{1}{k_0+E_{bu}^{\pm}} \gamma^0{\tilde \Lambda}_{k}^+ +
\frac{1}{k_0-E_{bu}^{\mp}} \gamma^0{\tilde \Lambda}_{k}^- , \nonumber \\
{\rm G}^{\pm}_{4}&=&
\frac{1}{k_0+E_{bd}^{\pm}} \gamma^0{\tilde \Lambda}_{k}^+ +
\frac{1}{k_0-E_{bd}^{\mp}} \gamma^0{\tilde \Lambda}_{k}^- , \end{aligned}$$ \[G\_I\] with $E^{\pm}_{i\alpha}\equiv E_{k}\pm \mu_{i\alpha}$ and $$\begin{aligned}
\Xi^{\pm}_{12} &=& \left(\frac{-i \Delta \gamma^5{\tilde \Lambda}_{k}^+}
{(k_0\pm\delta\mu)^2-{E_{\Delta}^{\pm}}^2} +
\frac{-i \Delta\gamma^5 {\tilde \Lambda}_{k}^- }{(k_0\pm\delta\mu)^2-{E_{\Delta}^{\mp}}^2} \right) ,
\nonumber \\
\Xi^{\pm}_{21} &=&\left(\frac{ -i \Delta\gamma^5{\tilde \Lambda}_{k}^+ }
{(k_0\mp\delta\mu)^2-{E_{\Delta}^{\pm}}^2} +\frac{ -i \Delta\gamma^5 {\tilde \Lambda}_{k}^- }
{(k_0\mp\delta\mu)^2-{E_{\Delta}^{\mp}}^2} \right), \end{aligned}$$ \[Xi\_I\] where $$\begin{aligned}
\tilde{\Lambda}^{\pm}_{k} &=& \frac{1}{2}\left(1\pm
\gamma^0\frac{\boldsymbol{\gamma}\cdot\mathbf{k}-m}{E_{k}} \right)\label{t-Lambda-k}\end{aligned}$$ is an alternative set of energy projectors, and the following notation was used: $$\begin{aligned}
E_{k}^{\pm}&\equiv& E_{k} \pm \bar{\mu},\nonumber\\
E_{\Delta,k}^{\pm} &\equiv& \sqrt{(E_{k}^{\pm})^2 +\Delta^2}, \label{g-disp} \\
\bar{\mu} &\equiv&
\frac{\mu_{ur} +\mu_{dg}}{2}
=\frac{\mu_{ug}+\mu_{dr}}{2}
=\mu-\frac{\mu_{e}}{6}+\frac{\mu_{8}}{3}, \nonumber \label{mu-bar}\\
\delta\mu &\equiv&
\frac{\mu_{dg}-\mu_{ur}}{2}
=\frac{\mu_{dr}-\mu_{ug}}{2}
=\frac{\mu_{e}}{2}.
\label{delta-mu}\end{aligned}$$ From the dispersion relation of the quasiparticles Eq. (\[g-disp\]), we can read that, when $\delta\mu > \Delta$, there will be excitations of gapless modes in the system. The thermodynamic potential corresponding to the solution of the gapless state $\delta\mu > \Delta$ is a local maximum. However, under certain constraint, e.g., the charge neutrality condition, the gapless 2SC phase can be a thermal stable state [@SH].
Chromomagnetic instabilities driven by mismatch {#sec-Mag-Ins}
-----------------------------------------------
In this section, we review the chromomagnetic instabilities driven by mismatch in 2SC and g2SC phases as shown in Ref. [@chromo-ins-g2SC-1; @chromo-ins-g2SC-2]. The polarization tensor in momentum space has the following general structure: $$\Pi^{\mu \nu}_{AB} (P) =
-\frac{i}{2} \int\frac{d^{4}K}{(2\pi)^{4}}
\mbox{Tr}_{\rm D} \left[ \hat{\Gamma}^\mu_A
{\cal S} (K) \, \hat{\Gamma}^\nu_B {\cal S}(K-P) \right].
\label{PiPscfNG}$$ The trace here runs over the Dirac indices and the vertices $\hat{\Gamma}^\mu_A \equiv
{\rm diag}(g\,\gamma^\mu T_A,-g\,\gamma^\mu T_A^T)$ with $A=1,\ldots,8$.
The Debye masses $m_{D,A}^2$ and the Meissner masses $m_{M,A}^2$ of gauge bosons are defined as $$\begin{aligned}
m_{D,A}^2 &\equiv & - \lim_{p\to 0} {\Pi}_{AA}^{00}(0,p),
\label{def-Debye}\\
m_{M,A}^2 &\equiv & - \frac{1}{2}\lim_{p\to 0}
\left(g_{ij}+\frac{p_i p_j}{p^2}\right) {\Pi}_{AA}^{ji}(0,p) .
\label{def-Meissner}\end{aligned}$$
### Screening masses of the gluons $A=1,2,3$
Gluons $A=1,2,3$ of the unbroken $SU(2)_c$ subgroup couple only to the red and green quarks. The general expression for the polarization tensor $\Pi_{AB}^{\mu\nu}(0,p)$ with $A,B=1,2,3$ is diagonal. After performing the traces over the color, flavor and Nambu-Gorkov indices, the expression has the form of $$\begin{aligned}
\Pi_{11}^{\mu\nu}(P) &=& \frac{g^2T}{4}\sum_n\int \frac{d^3 {\mathbf k}}{(2\pi)^3}
\mbox{Tr}_{\rm D} \left[
\gamma^{\mu} G_{1}^+(K) \gamma^{\nu} G_{1}^+(K')
+ \gamma^{\mu} G_{1}^-(K) \gamma^{\nu}G_{1}^-(K')\right.\nonumber\\
&+& \left.
\gamma^{\mu} G_{2}^+(K) \gamma^{\nu}G_{2}^+(K')
+ \gamma^{\mu} G_{2}^-(K) \gamma^{\nu}G_{2}^-(K') \right.\nonumber\\
&+& \left.
\gamma^{\mu}\Xi_{12}^-(K) \gamma^{\nu}\Xi_{21}^+(K')
+ \gamma^{\mu}\Xi_{12}^+(K) \gamma^{\nu}\Xi_{21}^-(K')\right.\nonumber\\
&+& \left.
\gamma^{\mu}\Xi_{21}^-(K) \gamma^{\nu}\Xi_{12}^+ (K')
+ \gamma^{\mu}\Xi_{21}^+(K) \gamma^{\nu}\Xi_{12}^-(K')\right],
\label{Pi11}\end{aligned}$$
By making use of the definition in Eq. (\[def-Debye\]) and Eq. (\[def-Meissner\]). we arrive at the following result for the threefold degenerate Debye mass square: $$\begin{aligned}
m_{D,1}^2
&\simeq & \frac{4\alpha_s \bar\mu^2 \delta\mu}
{\pi\sqrt{(\delta\mu)^2-\Delta^2}}\theta(\delta\mu-\Delta),
\label{m_D_1}\end{aligned}$$ with $\alpha_s\equiv g^2/4\pi$. The Meissner mass square reads $$m_{M,1}^2 =0.
\label{m_M_1}$$
The Debye screening mass in Eq. (\[m\_D\_1\]) vanishes in the gapped phase (i.e., $\Delta/\delta\mu>1$). As in the case of the ideal 2SC phase, this reflects the fact that there are no gapless quasiparticles charged with respect to the unbroken SU(2)$_c$ gauge group. In the gapless 2SC phase, such quasiparticles exist and the value of the Debye screening mass is proportional to the density of states at the corresponding “effective” Fermi surfaces.
The Meissner screening mass of the gluons of the unbroken SU(2)$_c$ are vanishing in the gapped and gapless 2SC phases. This is in agreement with the general group-theoretical arguments.
### Screening masses of diagonal gluon $A=8$
The 8-th gluon can probe the Cooper-paired red and green quarks, as well as the unpaired blue quarks. After the traces over the color, the flavor and the Nambu-Gorkov indices are performed, the polarization tensor for the 8th gluon can be expressed as
$$\begin{aligned}
\Pi_{88}^{\mu\nu}(P) &=& \frac{1}{3}{\tilde \Pi}_{88}^{\mu\nu}(P)+ \frac{2}{3}
\Pi_{88,b}^{\mu\nu}(P), \\ \label{Pi88-2}
{\tilde \Pi}_{88}^{\mu\nu}(P)
&=& \frac{g^2T}{4}\sum_n\int \frac{d^3 {\mathbf k}}{(2\pi)^3}
\mbox{Tr}_{\rm D} \left[
\gamma^{\mu} G_{1}^+(K) \gamma^{\nu} G_{1}^+(K')
+ \gamma^{\mu} G_{1}^-(K) \gamma^{\nu}G_{1}^-(K')\right.\nonumber\\
&+& \left.
\gamma^{\mu} G_{2}^+(K) \gamma^{\nu}G_{2}^+(K')
+ \gamma^{\mu} G_{2}^-(K) \gamma^{\nu}G_{2}^-(K') \right.\nonumber\\
&-& \left.
\gamma^{\mu}\Xi_{12}^-(K) \gamma^{\nu}\Xi_{21}^+(K')
- \gamma^{\mu}\Xi_{12}^+(K) \gamma^{\nu}\Xi_{21}^-(K')\right.\nonumber\\
&-& \left.
\gamma^{\mu}\Xi_{21}^-(K) \gamma^{\nu}\Xi_{12}^+ (K')
- \gamma^{\mu}\Xi_{21}^+(K) \gamma^{\nu}\Xi_{12}^-(K')\right],
\label{Pi88} \\
\Pi_{88,b}^{\mu\nu}(P) &=& \frac{g^2T}{4}\sum_n\int \frac{d^3 {\mathbf k}}{(2\pi)^3}
\mbox{Tr}_{\rm D} \left[ \gamma^{\mu} G_{3}^+(K) \gamma^{\nu}
G_{3}^+(K') + \gamma^{\mu} G_{3}^-(K) \gamma^{\nu}G_{3}^-(K')
\right.\nonumber\\
&+& \left. \gamma^{\mu} G_{4}^+(K)
\gamma^{\nu}G_{4}^+(K') + \gamma^{\mu} G_{4}^-(K) \gamma^{\nu}G_{4}^-(K')
\right].
\label{Pi88b}\end{aligned}$$
The expressions for the Debye screening mass reads $$\begin{aligned}
m_{D,8}^2 &=& \frac{4\alpha_s\bar\mu^2}{\pi} ,
\label{m_D_8} \end{aligned}$$ and the Meissner screening mass takes the form of $$\begin{aligned}
m_{M,88}^2 &=& \frac{4\alpha_s\bar\mu^2}{9\pi}
\left(1-\frac{\delta\mu~\theta(\delta\mu-\Delta)}
{\sqrt{(\delta\mu)^2-\Delta^2}} \right).
\label{m_M_88} \end{aligned}$$
As is easy to see from Eq. (\[m\_M\_88\]), the Meissner screening mass squared of 8-th gluon is [*negative*]{} when $0<\Delta/\delta\mu<1$, this indicates a magnetic plasma instability in the gapless 2SC phase.
### Screening masses of the gluons with $A=4,5,6,7$
After performing the traces over the color, the flavor and the Nambu-Gorkov indices, the diagonal components of the polarization tensor $\Pi_{AB}^{\mu\nu}(P)$ with $A=B=4,5,6,7$ have the form $$\begin{aligned}
\Pi_{44}^{\mu\nu}(P) & = &
\frac{g^2 T}{8} \int \frac{d^3 {\mathbf k}}{(2\pi)^3}{\rm Tr}_{\rm D}
\big[ \gamma^{\mu} G_{3}^+(K) \gamma^{\nu}G_{1}^+(K')
+ \gamma^{\mu} G_{1}^+(K) \gamma^{\nu} G_{3}^+(K') \nonumber \\
& + & \gamma^{\mu} G_{4}^+(K) \gamma^{\nu}G_{2}^+(K')
+ \gamma^{\mu} G_{2}^+(K) \gamma^{\nu}G_{4}^+(K') \nonumber \\
& + & \gamma^{\mu} G_{3}^-(K) \gamma^{\nu}G_{1}^-(K')
+ \gamma^{\mu} G_{1}^-(K) \gamma^{\nu}G_{3}^-(K')\nonumber \\
& + & \gamma^{\mu} G_{4}^-(K) \gamma^{\nu}G_{2}^-(K')
+ \gamma^{\mu} G_{2}^-(K) \gamma^{\nu}G_{4}^-(K') \big].\end{aligned}$$ Note that $\Pi_{44}^{\mu\nu}(P)=\Pi_{55}^{\mu\nu}(P)
=\Pi_{66}^{\mu\nu}(P)=\Pi_{77}^{\mu\nu}(P)$. Apart from the diagonal elements, there are also nonzero off-diagonal elements, $$\begin{aligned}
\Pi_{45}^{\mu\nu}(P) = - \Pi_{54}^{\mu\nu} (P)
= \Pi_{67}^{\mu\nu}(P) = - \Pi_{76}^{\mu\nu}(P)
= i {\hat \Pi}^{\mu\nu}(P),
\label{odd-diag}\end{aligned}$$ with $$\begin{aligned}
{\hat \Pi}^{\mu\nu}(P) &=&
\frac{g^2 T}{8} \int \frac{d^3 {\mathbf k}}{(2\pi)^3}{\rm Tr}_{\rm D}
\big[ \gamma^{\mu} G_{3}^+(K) \gamma^{\nu}G_{1}^+(K')
- \gamma^{\mu} G_{1}^+(K) \gamma^{\nu} G_{3}^+(K') \nonumber \\
&+& \gamma^{\mu} G_{4}^+(K) \gamma^{\nu}G_{2}^+(K')
- \gamma^{\mu} G_{2}^+(K) \gamma^{\nu}G_{4}^+(K') \nonumber \\
& - & \gamma^{\mu} G_{3}^-(K) \gamma^{\nu}G_{1}^-(K')
+ \gamma^{\mu} G_{1}^-(K) \gamma^{\nu}G_{3}^-(K')\nonumber \\
& - & \gamma^{\mu} G_{4}^-(K) \gamma^{\nu}G_{2}^-(K')
+ \gamma^{\mu} G_{2}^-(K) \gamma^{\nu}G_{4}^-(K') \big].\end{aligned}$$ The physical gluon fields in the 2SC/g2SC phase are the following linear combinations: $\tilde{A}_{4,5}^{\mu}=(A_{4}^{\mu}\pm i A_{5}^{\mu})/\sqrt{2}$ and $\tilde{A}_{6,7}^{\mu}=(A_{6}^{\mu}\pm i A_{7}^{\mu})/\sqrt{2}$. These new fields, $\tilde{A}_{4,6}^{\mu}$ and $\tilde{A}_{7,5}^{\mu}$ describe two pairs of massive vector particles with well defined electomagnetic charges, $\tilde{Q}=\pm 1$. The components of the polarization tensor in the new basis read
$$\begin{aligned}
\tilde{\Pi}_{44}^{\mu\nu}(P) & = & \tilde{\Pi}_{66}^{\mu\nu}(P) = \Pi_{44}^{\mu\nu}(P) +
{\hat \Pi}^{\mu\nu}(P) \nonumber \\
& = &
\frac{g^2 T}{4} \int \frac{d^3 {\mathbf k}}{(2\pi)^3}{\rm Tr}_{\rm D}
\big[
\gamma^{\mu} G_{3}^+(K)
\gamma^{\nu}G_{1}^+(K') + \gamma^{\mu} G_{1}^-(K)
\gamma^{\nu}G_{3}^-(K') \nonumber \\
& & + \gamma^{\mu} G_{4}^+(K) \gamma^{\nu}G_{2}^+(K') +
\gamma^{\mu} G_{2}^-(K) \gamma^{\nu}G_{4}^-(K') \big], \end{aligned}$$
and $$\begin{aligned}
\tilde{\Pi}_{55}^{\mu\nu}(P) & = & \tilde{\Pi}_{77}^{\mu\nu}(P) = \Pi_{44}^{\mu\nu} (P)
- {\hat \Pi}^{\mu\nu}(P) \nonumber \\
& = &
\frac{g^2 T}{4} \int \frac{d^3 {\mathbf k}}{(2\pi)^3}{\rm Tr}_{\rm D}
\big[
\gamma^{\mu} G_{1}^+(K)\gamma^{\nu}G_{3}^+(K') + \gamma^{\mu}
G_{3}^-(K) \gamma^{\nu}G_{1}^-(K') \nonumber \\
& & + \gamma^{\mu} G_{2}^+(K) \gamma^{\nu}G_{4}^+(K') +
\gamma^{\mu} G_{4}^-(K) \gamma^{\nu}G_{2}^-(K') \big].\end{aligned}$$
In the static limit, all four eigenvalues of the polarization tensor are degenerate. By making use of the definition in Eq. (\[def-Debye\]), we derive the following result for the corresponding Debye masses: $$m_{D,4}^2 = \frac{4\alpha_s\bar\mu^2}{\pi} \left[
\frac{\Delta^2+2\delta\mu^2}{2\Delta^2}
-\frac{\delta\mu\sqrt{\delta\mu^2-\Delta^2}}{\Delta^2}
\theta(\delta\mu-\Delta)\right] \!\! .
\label{m_D_4}$$ Here we assumed that $\mu_8$ is vanishing which is a good approximation in neutral two-flavor quark matter.
The fourfold degenerate Meissner screening mass of the gluons with $A=4,5,6,7$ reads $$m_{M,4}^2 = \frac{4\alpha_s\bar\mu^2}{3\pi} \left[
\frac{\Delta^2-2\delta\mu^2}{2\Delta^2}
+\frac{\delta\mu\sqrt{\delta\mu^2-\Delta^2}}{\Delta^2}
\theta(\delta\mu-\Delta)\right] \!\! .
\label{m_M_4}$$ Both results in Eqs. (\[m\_D\_4\]) and (\[m\_M\_4\]) interpolate between the known results in the normal phase (i.e., $\Delta/\delta\mu=0$) and in the ideal 2SC phase (i.e., $\Delta/\delta\mu=\infty$). The instability of off-diagonal gluons appears in the whole region of gapless 2SC phases (with $0<\Delta/\delta\mu<1$) and even in some gapped 2SC phases (with $1<\Delta/\delta\mu<\sqrt{2}$).
It is noticed that the Meissner screening mass square for the off-diagonal gluons decreases monotonously to zero when the mismatch increases from zero to $ \delta\mu= \Delta/\sqrt{2}$, then goes to negative value with further increase of the mismatch in the gapped 2SC phase. However, the behavior of diagonal 8-th gluon’s Meissner mass square is quite different. It keeps as a constant in the gapped 2SC phase. In the gapless 2SC phase, all these five gluons’ Meissner mass square are negative.
Color neutral baryon current instability {#sec-bc}
----------------------------------------
It is not understood why gapless color superconducting phases exhibit chromomagnetic instability. It sounds quite strange especially in the g2SC phase, where it is the electrical neutrality not the color neutrality playing the essential role. It is a puzzle why the gluons can feel the instability by requiring the electrical neutrality on the system. In order to understand what is really going ‘wrong’ with the homogeneous g2SC phase, we want to know whether there exists other instabilities except the chromomagnetic instability. For that purpose, we probe the g2SC phase using different external sources, e.g., scalar and vector diquarks, mesons, vector current, and so on. Here we report the most interesting result regarding the response of the g2SC phase to an external vector current $V^{\mu}={\bar \psi} \gamma^{\mu} \psi$, the time-component and spatial-components of this current correspond to the baryon number density and baryon current, respectively.
From the linear response theory, the induced current and the external vector current is related by the response function $\Pi^{\mu\nu}_{V}(P)$, $$\label{PiVNG}
\Pi^{\mu \nu}_{V} (P) = \frac{1}{2} \, \frac{T}{V}
\sum_K {\rm Tr} \left[ \hat{\Gamma}^\mu_V
{\cal S} (K) \hat{\Gamma}^\nu_V {\cal S}(K-P) \right] .$$ The trace here runs over the Nambu-Gorkov, flavor, color and Dirac indices. The explicit form of vertices is $\hat{\Gamma}_V^\mu \equiv {\rm diag}( \gamma^\mu, -\gamma^\mu)$.
The explicit expression of the vector current response function takes the form of $$\begin{aligned}
\Pi_{V}^{\mu\nu}(P) &=& {\tilde \Pi}_{V}^{\mu\nu}(P)+ \Pi_{V,b}^{\mu\nu}(P),
\label{PiV-2}\end{aligned}$$ with $$\begin{aligned}
{\tilde \Pi}_{V}^{\mu\nu}(P)
&=& \frac{T}{2}\sum_n\int \frac{d^3 {\mathbf k}}{(2\pi)^3}
\mbox{Tr}_{\rm D} \left[ \right. \nonumber \\
& & \left. \gamma^{\mu} G_{1}^+(K) \gamma^{\nu} G_{1}^+(K')
+ \gamma^{\mu} G_{1}^-(K) \gamma^{\nu}G_{1}^-(K')\right.\nonumber\\
&+& \left.
\gamma^{\mu} G_{2}^+(K) \gamma^{\nu}G_{2}^+(K')
+ \gamma^{\mu} G_{2}^-(K) \gamma^{\nu}G_{2}^-(K') \right.\nonumber\\
&-& \left.
\gamma^{\mu}\Xi_{12}^-(K) \gamma^{\nu}\Xi_{21}^+(K')
- \gamma^{\mu}\Xi_{12}^+(K) \gamma^{\nu}\Xi_{21}^-(K')\right.\nonumber\\
&-& \left.
\gamma^{\mu}\Xi_{21}^-(K) \gamma^{\nu}\Xi_{12}^+ (K')
- \gamma^{\mu}\Xi_{21}^+(K) \gamma^{\nu}\Xi_{12}^-(K')\right], \\
%\label{PiV}
\Pi_{V,b}^{\mu\nu}(P) &=& \frac{T}{2}\sum_n\int \frac{d^3 {\mathbf k}}{(2\pi)^3}
\mbox{Tr}_{\rm D} \left[ \right. \nonumber\\
& & \left. \gamma^{\mu} G_{3}^+(K) \gamma^{\nu}
G_{3}^+(K') + \gamma^{\mu} G_{3}^-(K) \gamma^{\nu}G_{3}^-(K')
\right.\nonumber\\
&+& \left. \gamma^{\mu} G_{4}^+(K)
\gamma^{\nu}G_{4}^+(K') + \gamma^{\mu} G_{4}^-(K) \gamma^{\nu}G_{4}^-(K')
\right],
%\label{PiVb}\end{aligned}$$ here the trace is over the Dirac space.
Comparing the explicit expression of $\Pi_{V}^{\mu\nu}(P)$ with that of the 8-th gluon’s self-energy $\Pi_{88}^{\mu\nu}(P)$, i.e., Eq. (\[Pi88\]), it can be clearly seen that, $\Pi_{V}^{\mu\nu}(P)$ and $\Pi_{88}^{\mu\nu}(P)$ almost share the same expression, except the coefficients. This can be easily understood, because the color charge and color current carried by the 8-th gluon is proportional to the baryon number and baryon current, respectively. In the static long-wavelength ($p_0=0$ and $\vec{p} \to 0$ ) limit, the time-component and spatial component of $\Pi_{V}^{\mu\nu}(P)$ give the baryon number susceptibility $\xi_n$ and baryon current susceptibility $\xi_c$, respectively, $$\begin{aligned}
\xi_n &\equiv & - \lim_{\vec{p}\to 0} \tilde{\Pi}_{V}^{00}(0,\vec{p}) \propto m_{8,D}^2,
\label{def-xin}\\
\xi_c &\equiv & - \frac{1}{2}\lim_{\vec{p}\to 0} \left(g_{ij}+\frac{p_i p_j}{p^2}\right) {\Pi}_{V}^{ij}(0,\vec{p}) \propto m_{8,M}^2.
\label{def-xic}\end{aligned}$$
In the g2SC phase, $m_{8,M}^2$ as well as $\xi_c$ become negative. This means that, except the chromomagnetic instability corresponding to broken generators of $SU(3)_c$, and the instability of a net momentum for diquark pair, the g2SC phase is also unstable with respect to an external color neutral baryon current $\bar{\psi}\vec{\gamma}\psi$.
The 8-th gluon’s magnetic instability, the diquark momentum instability and the color neutral baryon current in the g2SC phase can be understood in one common physical picture. The g2SC phase exhibits a paramagnetic response to an external baryon current. Naturally, the color current carried by the 8-th gluon, which differs from the baryon current by a color charge, also experiences the instability in the g2SC phase. The paramagnetic instability of the baryon current indicates that the quark can spontaneously obtain a momentum, because diquark carries twice of the quark momentum, it is not hard to understand why the g2SC phase is also unstable with respect to the response of a net diquark momentum.
It is noticed that, the instability of $\bar{\psi}\vec{\gamma}\psi$ will be induced by mismatch in all the asymmetric Fermi pairing systems, including superfluid systems, where $\bar{\psi}\vec{\gamma}\psi$ can be interpreted as particle current.
### Spontaneous baryon current generation {#sec-UsLOFF}
The paramagnetic response to an external vector current naturally suggests that a vector current can be spontaneously generated in the system. The generated vector current behaves as a vector potential, which modifies the quark self-energy with a spatial vector condensate $\vec{\gamma} \cdot \vec{\Sigma}_V$, and breaks the rotational symmetry of the system. It can also be understood that the quasiparticles in the gapless phase spontaneously obtain a superfluid velocity, and the ground state is in an anisotropic state. The quark propagator $G_0^{\pm}(P)$ in Eq. (\[freep\]) is modified as $$\left[G_{0,V}^{\pm}(P)\right]^{-1} =
\gamma^0 (p_0 \pm \hat{\mu}) - \vec{\gamma} \cdot \vec{p} \mp \vec{\gamma} \cdot \vec{\Sigma}_V,
\label{freep-m}$$ with a subscript $V$ indicating the modified quark propagator. Correspondingly, the inverse of the quark propagator $[{\cal S}(P)]^{-1}$ in Eq. (\[prop\]) is modified as $$\left[{\cal S}_V(P)\right]^{-1} = \left(\begin{array}{cc}
\left[G_{0,V}^{+}(P)\right]^{-1} & \Delta^- \\
\Delta^+ & \left[G_{0,V}^{-}(P)\right]^{-1}
\end{array}\right).
\label{prop-m}$$ It is noticed that the expression of the modified inverse quark propagator $[S_V(P)]^{-1}$ takes the same form as the inverse quark propagator in the one-plane wave LOFF state shown in Ref. [@LOFF-Ren-2]. The net momentum $\vec{q}$ of the diquark pair in the LOFF state [@LOFF-Ren-2] is replaced here by a spatial vector condensate $\vec{\Sigma}_V$. The spatial vector condensate $\vec{\gamma} \cdot \vec{\Sigma}_V$ breaks rotational symmetry of the system. This means that the Fermi surfaces of the pairing quarks are not spherical any more.
It has to be pointed out, the baryon current offers one Doppler-shift superfluid velocity for the quarks. A spontaneously generated Nambu-Goldstone current in the minimal gapless model [@hong] or a condensate of 8-th gluon’s spatial component can do the same job. All these states mimic the one-plane wave LOFF state. In the following, we just call all these states as the single-plane wave LOFF state.
In order to determine the deformed structure of the Fermi surfaces, one should self-consistently minimize the free energy $\Gamma(\Sigma_V, \Delta, \mu,\mu_e,\mu_8)$. The explicit form of the free energy can be evaluated directly using the standard method, in the framework of Nambu–Jona-Lasinio model [@neutral_huang; @SH], it takes the form of $$\begin{aligned}
\Gamma= - \frac{T}{2} \sum_n\int\frac{d^3\vec{p}}{(2\pi)^3} {\rm Tr} \ln ( [{\cal S}_V(P)]^{-1}) + \frac{\Delta^2}{4 G_D},\end{aligned}$$ where $T$ is the temperature, and $G_D$ is the coupling constant in the diquark channel.
When there is no charge neutrality condition, the ground state is determined by the thermal stability condition, i.e., the local stability condition. The ground state is in the 2SC phase when $\delta\mu<0.706 \Delta_0$ with $\Delta \simeq \Delta_0$, in the LOFF phase when $0.706 \Delta_0 < \delta\mu <0.754 \Delta_0$ correspondingly $ 0<\Delta/\Delta_0<0.242$, and then in the normal phase with $\Delta=0$ when the mismatch is larger than $0.754 \Delta_0$. Here $\Delta, \Delta_0$ indicate the diquark gap in the case of $\delta\mu \neq 0$ and $\delta\mu=0$, respectively.
### Unstable neutral LOFF window
Now come to the charge neutral LOFF state, and investigate whether the LOFF state can resolve all the magnetic instabilities.
When charge neutrality condition is required, the ground state of charge neutral quark matter should be determined by solving the gap equations as well as the charge neutrality condition, i.e., $$\begin{aligned}
\frac{\partial \Gamma}{\partial \Sigma_V} =0, \ \, \frac{\partial \Gamma}{\partial \Delta} =0, \ \
\frac{\partial \Gamma}{\partial \mu_e} =0, \ \, \frac{\partial \Gamma}{\partial \mu_8} =0.\end{aligned}$$ By changing $\Delta_0$ or coupling strength $G_D$, the solution of the charge neutral LOFF state can stay everywhere in the full LOFF window, including the window not protected by the local stability condition, as shown explicitly in Ref. [@GHM-LOFF-Neutral].
From the lesson of charge neutral g2SC phase, we learn that even though the neutral state is a thermal stable state, i.e., the thermodynamic potential is a global minimum along the neutrality line, it cannot guarantee the dynamical stability of the system. The stability of the neutral system should be further determined by the dynamical stability condition, i.e., the positivity of the Meissner mass square.
The polarization tensor for the gluons with color $A=4,5,6,7,8$ should be evaluated using the modified quark propagator ${\cal S}_V$ in Eq. (\[prop-m\]), i.e., $$\Pi^{\mu \nu}_{AB} (P) = \frac{1}{2} \, \frac{T}{V} \sum_K
\mbox{Tr} \left[ \hat{\Gamma}^\mu_A
{\cal S}_V (K) \, \hat{\Gamma}^\nu_B {\cal S}_V (K-P) \right],
\label{PiAB}$$ with $A,B=4,5,6,7,8$ and the explicit form of the vertices $\hat{\Gamma}^\mu_A$ has the form $\hat{\Gamma}_A^\mu
\equiv {\rm diag}(g\,\gamma^\mu T_A,-g\,\gamma^\mu T_A^T) $. In the LOFF state, the Meissner tensor can be decomposed into transverse and longitudinal component. The transverse and longitudinal Meissner mass square for the off-diagonal 4-7 gluons and the diagonal 8-th gluon have been performed explicitly in the one-plane wave LOFF state in Ref. [@LOFF-Ren-2].
According to the dynamical stability condition, i.e., the positivity of the transverse as well as longitudinal Meissner mass square, we can devide the LOFF state into three LOFF windows [@Hai-cang]:
1\) The stable LOFF (S-LOFF) window in the region of $0<\Delta/\Delta_0<0.39$, which is free of any magnetic instability. Please note that this S-LOFF window is a little bit wider than the window $0<\Delta/\Delta_0<0.242$ protected by the local stability condition.
2\) The stable window for diagonal gluon characterized by Ds-LOFF window in the region of $0.39<\Delta/\Delta_0<0.83$, which is free of the diagonal 8-th gluon’s magnetic instability but not free of the off-diagonal gluons’ magnetic instability;
3\) The unstable LOFF (Us-LOFF) window in the region of $0.83<\Delta/\Delta_0< r_c$, with $r_c \, \equiv \, \Delta(\delta\mu=\Delta)/\Delta_0 \simeq 1$. In this Us-LOFF window, all the magnetic instabilities exist. Please note that, it is the longitudinal Meissner mass square for the 8-th gluon is negative in this Us-LOFF window, the transverse Meissner mass square of 8-th gluon is always zero in the full LOFF window, which is guaranteed by the momentum equation.
Us-LOFF is a very interesting window, it indicates that the LOFF state even cannot cure the 8-th gluon’s magnetic instability. In the charge neutral 2-flavor system, it seems that the diagonal gluon’s magnetic instability cannot be cured in the gluon phase, because there is no direct relation between the diagonal gluon’s instability and the off-diagonal gluons’ instability. (Of course, it has to be carefully checked, whether all the instabilities in this Us-LOFF window can be cured by off-diagonal gluons’ condensate in the charge neutral 2-flavor system.) It is also noticed that in this Us-LOFF window, the mismatch is close to the diquark gap, i.e., $\delta\mu \simeq \Delta$. Therefore it is interesting to check whether this Us-LOFF window can be stabilized by a spin-1 condensate [@spin-1] as proposed in Ref. [@hong].
In the charge neutral 2SC phase, though it is unlikely, we might have a lucky chance to cure the diagonal instability by the condensation of off-diagonal gluons. It is expected that this instability will show up in some constrained Abelian asymmetric superfluidity system, e.g., in the fixed number density case [@He-Jin-Zhuang]. It will be a new challenge for us to really solve this problem.
Spontaneous Nambu-Goldstone currents generation {#sec-NG-current}
-----------------------------------------------
We have seen that, except chromomagnetic instability corresponding to broken generators of $SU(3)_c$, the g2SC phase is also unstable with respect to the external neutral baryon current. It is noticed that all the instabilities are induced by increasing the mismatch between the Fermi surfaces of the Cooper pairing. In order to understand the instability driven by mismatch, in the following, we give some general analysis.
A superconductor will be eventually destroyed and goes to the normal Fermi liquid state, so one natural question is: how an ideal BCS superconductor will be destroyed by increasing mismatch? To answer how a superconductor will be destroyed, one has to firstly understand what is a superconductor. The superconducting phase is characterized by the order parameter $\Delta(x)$, which is a complex scalar field and has the form of e.g., for electrical superconductor, $\Delta (x) = |\Delta| e^{i\varphi (x)}$, with $|\Delta| $ the amplitude and $\varphi$ the phase of the gap order parameter or the pseudo Nambu-Goldstone boson.
1\) The superconducting phase is charaterized by the nonzero vacuum expectation value, i.e., $<\Delta>\neq 0$, which means the amplitude of the gap is finite, and the phase coherence is also established.
2\) If the amplitude is still finite, while the phase coherence is lost, this phase is in a phase decoherent pseudogap state characterized by $|\Delta|\neq 0$, but $<\Delta> =0$ because of $ <e^{i\varphi(x)}> = 0$.
3\) The normal state is characterized by $|\Delta|=0$.
There are two ways to destroy a superconductor. One way is by driving the amplitude of the order parameter to zero. This way is BCS-like, because it mimics the behavior of a conventional superconductor at finite temperature, the gap amplitude monotonously drops to zero with the increase of temperature; Another way is non-BCS like, but Berezinskii-Kosterlitz-Thouless (BKT)-like [@BKT], even if the amplitude of the order parameter is large and finite, superconductivity will be lost with the destruction of phase coherence, e.g. the phase transition from the $d-$wave superconductor to the pseudogap state in high temperature superconductors [@Emery-Kivelson].
Stimulating by the role of the phase fluctuation in the unconventional superconducting phase in condensed matter, we follow Ref. [@NonLinear] to formulate the 2SC phase in the nonlinear realization framework in order to naturally take into account the contribution from the phase fluctuation or pseudo Nambu-Goldstone current.
In the 2SC phase, the color symmetry $G=SU(3)_c$ breaks to $H=SU(2)_c$. The generators of the residual $SU(2)_c$ symmetry H are $\{S^a=T^a\}$ with $a=1,2,3$ and the broken generators $\{X^b=T^{b+3}\}$ with $b=1, \cdots, 5$. More precisely, the last broken generator is a combination of $T_8$ and the generator ${\bf 1}$ of the global $U(1)$ symmetry of baryon number conservation, $B \equiv ({\bf 1} + \sqrt{3} T_8)/3$ of generators of the global $U(1)_B$ and local $SU(3)_c$ symmetry.
The coset space $G/H$ is parameterized by the group elements $$\label{phase}
{\cal V}(x) \equiv \exp \left[ i \left( \sum_{a=4}^7 \varphi_a(x) T_a
+ \frac{1}{\sqrt{3}}\, \varphi_8(x) B \right) \right]\,\,,$$ here $\varphi_a (a=4,\cdots,7)$ and $\varphi_8$ are five Nambu-Goldstone diquarks, and we have neglected the singular phase, which should include the information of the topological defects [@FT; @Topo]. Operator ${\cal V}$ is unitary, ${\cal V}^{-1} = {\cal V}^\dagger$.
Introducing a new quark field $\chi$, which is connected with the original quark field $q$ in Eq. (\[lagr-2sc\]) in a nonlinear transformation form, $$\label{chi}
q = {\cal V}\, \chi
\,\,\,\, , \,\,\,\,\,
\bar{q} = \bar{\chi}\, {\cal V}^\dagger\,\, ,$$ and the charge-conjugate fields transform as $$q_{C} = {\cal V}^* \, \chi_{C}
\,\,\,\, , \,\,\,\,\,
\bar{q}_{C} = \bar{\chi}_{C} \, {\cal V}^T\,\, .$$ In high-$T_c$ superconductor, this technique is called charge-spin separation, see Ref. [@FT]. The advantage of transforming the quark fields is that this preserves the simple structure of the terms coupling the quark fields to the diquark sources, $$\bar{q}_{C}\, \Delta^+ \, q
\equiv \bar{\chi}_{C}\, \Phi^+ \, \chi
\,\,\,\, , \,\,\,\,\,
\bar{q}\, \Delta^- \, q_{C}
\equiv \bar{\chi} \, \Phi^- \, \chi_{C} \,\, .$$ In mean-field approximation, the diquark source terms are proportional to $$\label{mfa2}
\Phi^+
\sim \langle \, \chi_{C} \, \bar{\chi}\, \rangle
\,\,\,\, , \,\,\,\,\,
\Phi^-
\sim \langle \, \chi \, \bar{\chi}_{C} \, \rangle\,\, .$$
Introducing the new Nambu-Gor’kov spinors $$X \equiv \left( \begin{array}{c}
\chi \\
\chi_{C}
\end{array}
\right) \,\,\, , \,\,\,\,
\bar{X} \equiv ( \bar{\chi} \, , \, \bar{\chi}_{C} ),$$ the nonlinear realization of the original Lagrangian density Eq.(\[lagr-2sc\]) takes the form of $${\cal L}_{nl} \equiv
\bar{X} \, {\cal S}_{nl}^{-1} \, X
- \frac{\Phi^+\Phi^-}{4 G_D} \,\, ,
\label{lagr-nl}$$ where $${\cal S}_{nl}^{-1} \equiv
\left( \begin{array}{cc}
[G^+_{0,nl}]^{-1} & \Phi^- \\
\Phi^+ & [G^-_{0,nl}]^{-1}
\end{array} \right)\,\, .$$ Here the explicit form of the free propagator for the new quark field is $$\begin{aligned}
[G^+_{0,nl}]^{-1} & = & i\, \fsl{D} + {\hat \mu} \, \gamma_0 + \gamma_\mu \, V^\mu,
%[G^+_{0,nl}]^{-1} & = & i\, \gamma^\mu \partial_\mu + {\hat \mu} \, \gamma_0 + \gamma_\mu \, V^\mu, \end{aligned}$$ and $$\begin{aligned}
[G^-_{0,nl}]^{-1} & = & i\, \fsl{D}^T - {\hat \mu} \, \gamma_0 + \gamma_{\mu} \, V_C^\mu .\end{aligned}$$ Comparing with the free propagator in the original Lagrangian density, the free propagator in the non-linear realization framework naturally takes into account the contribution from the Nambu-Goldstone currents or phase fluctuations, i.e., $$\begin{aligned}
V^\mu & \equiv & {\cal V}^\dagger \, \left( i \, \partial^\mu \right) \, {\cal V}, \nonumber \\
V^\mu_C & \equiv & {\cal V}^T \, \left( i \, \partial^\mu \right) \, {\cal V}^*,\end{aligned}$$ which is the $N_c N_f \times N_c N_f$-dimensional Maurer-Cartan one-form introduced in Ref. [@NonLinear]. The linear order of the Nambu-Goldstone currents $V^\mu$ and $V_C^\mu $ has the explicit form of $$\begin{aligned}
V^\mu & \simeq & - \sum_{a=4}^7
\left( \partial^\mu \varphi_a \right)\, T_a -
\frac{1}{\sqrt{3}}\, \left(\partial^\mu \varphi_8\right)\, B\,\, , \\
V_C^\mu & \simeq & \sum_{a=4}^7
\left( \partial^\mu \varphi_a \right) \, T_a^T +
\frac{1}{\sqrt{3}}\, \left( \partial^\mu \varphi_8\right) \, B^T\,\, .\end{aligned}$$
The Lagrangian density Eq. (\[lagr-nl\]) for the new quark fields looks like an extension of the theory in Ref. [@FT] for high-$T_c$ superconductor to Non-Abelian system, except that here we neglected the singular phase contribution from the topologic defects. The advantage of the non-linear realization framework Eq. (\[lagr-nl\]) is that it can naturally take into account the contribution from the phase fluctuations or Nambu-Goldstone currents.
The task left is to correctly solve the ground state by considering the phase fluctuations. The free energy $\Gamma(V_{\mu},\Delta, \mu,\mu_8, \mu_e)$ can be evaluated directly and it takes the form of $$\begin{aligned}
\Gamma= - \frac{T}{2} \sum_n\int\frac{d^3\vec{p}}{(2\pi)^3} {\rm Tr} \ln ( [{\cal S}_{nl}(P)]^{-1}) + \frac{\Phi^2}{4 G_D}.
\label{free-energy-NG}\end{aligned}$$ To evaluate the ground state of $\Gamma(V_{\mu},\Delta, \mu,\mu_8, \mu_e)$ as a function of mismatch is tedious and still under progress. In the following we just give a brief discussion on the Nambu-Goldstone current generation state [@hong], one-plane wave LOFF state [@LOFF-Ren-1; @LOFF-Ren-2], as well as the gluon phase [@GHM-gluon].
If we expand the thermodynamic potential $\Gamma(V_{\mu},\Delta, \mu,\mu_8, \mu_e)$ of the non-linear realization form in terms of the Nambu-Goldstone currents, we will naturally have the Nambu-Goldstone currents generation in the system with the increase of mismatch, i.e., $<\sum_{a=4}^7 {\vec \triangledown} \varphi_a>\neq 0$ and/or $<{\vec \triangledown} \varphi_8 > \neq 0$ at large $\delta\mu$. This is an extended version of the Nambu-Goldstone current generation state proposed in a minimal gapless model in Ref. [@hong; @Huang-PKU]. From Eq. (\[lagr-nl\]), we can see that ${\vec \triangledown} \varphi_8$ contributes to the baryon current. $<{\vec \triangledown} \varphi_8 > \neq 0$ indicates a baryon current generation or 8-th gluon condensate in the system, it is just the one-plane wave LOFF state. This has been discussed in Sec. \[sec-UsLOFF\]. The other four Nambu-Goldstone currents generation $<\sum_{a=4}^7 {\vec \triangledown} \varphi_a>\neq 0$ indicates other color current generation in the system, and is equivalent to the gluon phase described in Ref. [@GHM-gluon].
We do not argue whether the system will exprience a gluon condensate phase or Nambu-Goldstone currents generation state. We simply think they are equivalent. In fact, the gauge fields and the Nambu-Goldstone currents share a gauge covariant form as shown in the free propogator. However, we prefer to using Nambu-Goldstone currents generation than the gluon condensate in the gNJL model. As mentioned in Sec. \[sec-gNJL\], in the gNJL model, all the information from unkown nonperturbative gluons are hidden in the diquark gap parameter $\Delta$. The gauge fields in the Lagrangian density are just external fields, they only play the role of probing the system, but do not contribute to the property of the color superconducting phase. Therefore, there is no gluon free-energy in the gNJL model, it is not clear how to derive the gluon condensate in this model. In order to investigate the problem in a fully self-consistent way, one has to use the ambitious framework by using the Dyson-Schwinger equations (DSE) [@DSE] including diquark degree of freedom [@DSE-SC] or in the framework of effective theory of high-density quark matter as in Ref. [@EFT-HDQ].
Conclusion and discussion {#sec-sum}
-------------------------
In this article, we reviewed the instabilities driven by mismatch and recent progress of resolving instabilities in the 2SC phase.
Except the chromomagnetic instability, the g2SC phase also exhibits a paramagnetic response to the perturbation of an external baryon current. This suggests a baryon current can be spontaneously generated in the g2SC phase, and the quasiparticles spontaneously obtain a superfluid velocity. The spontaneously generated baryon current breaks the rotational symmetry of the system, and it resembles the one-plane wave LOFF state.
We further describe the 2SC phase in the nonlinear realization framework, and show that each instability indicates the spontaneous generation of the corresponding pseudo Nambu-Goldstone current. We show this Nambu-Goldstone currents generation state can naturally cover the gluon phase as well as the one-plane wave LOFF state.
We also point out that, when charge neutrality condition is required, there exists a narrow unstable LOFF (Us-LOFF) window, where not only off-diagonal gluons but the diagonal 8-th gluon cannot avoid the magnetic instability. The diagonal gluon’s magnetic instability in this Us-LOFF window cannot be cured by off-diagonal gluon condensate in color superconducting phase. More interestingly, this Us-LOFF window will also show up in some constrained Abelian asymmetric superfluid system.
The Us-LOFF window brings us a new challenge. We need new thoughts on understanding how a BCS supercondutor will be eventually destroyed by increasing the mismatch, we also need to develop new methods to really resolve the instability problem. Some methods developed in unconventional superconductor field, e.g., High-$T_c$ superconductor, might be helpful.
Till now, the results on instabilities are based on mean-field (MF) approximation. The BCS theory at MF can describe strongly coherent or rigid superconducting state very well. However, as we pointed out in [@Huang-PKU], with the increase of mismatch, the low degrees of freedom in the system have been changed. For example, the gapless quasi-particle excitations in the gapless phase, and the small Meissner mass square of the off-diagonal gluons around $\delta\mu=\Delta/\sqrt{2}$. This indicates that these quasi-quarks and gluons become low degrees of freedom in the system, their fluctuations become more important. In order to correctly describe the system, the low degrees of freedom should be taken into account properly. The work toward this direction is still in progress.
In this article, we did not discuss the magnetic instabilty in the gCFL phase. For more discussion on solving the magnetic instabilty in the gCFL phase, please refer to Ref. [@gCFL-LOFF].
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author thanks M. Alford, F.A. Bais, K. Fukushima, E. Gubankova, M. Hashimoto, T. Hatsuda, L.Y. He, D.K. Hong, W.V. Liu, M. Mannarelli, T. Matsuura, Y. Nambu, K. Rajagopal, H.C. Ren, D. Rischke, T. Schafer, A. Schmitt, I. Shovkovy, D. T. Son, M. Tachibana, Z.Tesanovic, X. G. Wen, Z. Y. Weng, F. Wilczek and K. Yang for valuable discussions. The work is supported by the Japan Society for the Promotion of Science fellowship program.
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|
---
abstract: 'We present details of the discovery of [XLSSJ022303.0-043622]{}, a $z=1.2$ cluster of galaxies. This cluster was identified from its X-ray properties and selected as a $z>1$ candidate from its optical/near-IR characteristics in the XMM Large-Scale Structure Survey (XMM-LSS). It is the most distant system discovered in the survey to date. We present ground-based optical and near IR observations of the system carried out as part of the XMM-LSS survey. The cluster has a bolometric X-ray luminosity of $ 1.1\pm0.7 \times 10^{44}$ [erg s$^{-1}$]{}, fainter than most other known $z>1$ X-ray selected clusters. In the optical it has a remarkably compact core, with at least a dozen galaxies inside a 125 kpc radius circle centred on the X-ray position. Most of the galaxies within the core, and those spectroscopically confirmed to be cluster members, have stellar masses similar to those of massive cluster galaxies at low redshift. They have colours comparable to those of galaxies in other $z>1$ clusters, consistent with showing little sign of strong ongoing star formation. The bulk of the star formation within the galaxies appears to have ceased at least 1.5 Gyr before the observed epoch. Our results are consistent with massive cluster galaxies forming at $z>1$ and passively evolving thereafter. We also show that the system is straightforwardly identified in Spitzer/IRAC $3.6\micron$ and $4.5\micron$ data obtained by the SWIRE survey emphasising the power and utility of joint XMM and Spitzer searches for the most distant clusters.'
author:
- |
M.N. Bremer$^1$[^1], I. Valtchanov$^{2,3}$, J. Willis$^4$, B. Altieri$^3$, S. Andreon$^5$,P.A Duc$^6$, F. Fang$^{7,8}$, C. Jean$^9$, C. Lonsdale$^8$, F. Pacaud$^6$, M. Pierre$^6$,J.A. Surace$^{7,8}$, D.L. Scupe$^{7,8}$, I. Waddington$^{10}$\
$^1$ H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK.\
$^{2}$Blackett Laboratory, Astrophysics Group, Imperial College, London SW7 2BW, UK.\
$^{3}$ESA, Villafranca del Castillo, Spain.\
$^{4}$Department of Physics and Astronomy, University of Victoria, Elliot Building, 3800 Finnerty Road, Victoria, BC, V8P 1A1 Canada.\
$^{5}$INAF Osservatorio Astronomico di Brera, Milan, Italy.\
$^{6}$CEA/DSM/DAPNIA, Service d’Astrophysique, Saclay, F-91191 Gif sur Yvette, France.\
$^7$Spitzer Science Center, California Institute of Technology, MS 220-6, Pasadena, CA, 91125, USA.\
$^{8}$Infrared Processing and Analysis center, California Institute of Technology, MS 100-22, Pasadena, CA, 91125.\
Center, Caltech, USA.\
$^{9}$Institut d Astrophysique et de Géophysique, ULg, Allée du 6 AoÆut 17, B5C, 4000 Sart Tilman (Liège), Belgium.\
$^{10}$Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK.
date: 'Accepted ??. Received ; in original form '
title: 'XMM-LSS discovery of a $z=1.22$ galaxy cluster'
---
\[firstpage\]
galaxies – clusters; large-scale structure
Introduction
============
Determining the properties of clusters and their constituent galaxies at $z>1$ is central to understanding how such systems evolve with cosmic time. Distant clusters are identifiable using multiple techniques across a number of wavebands, from optical and near IR imaging surveys that identify emission from their galaxies to X-ray surveys that identify emission from the gaseous Intra Cluster Medium (ICM). As the X-ray identification of a cluster is straightforwardly related to important physical parameters such as the mass of a system, X-ray selected samples have been the most heavily studied to date. Although X-ray work had been carried out beforehand (notably with the EMSS, @gioia90), it was work using the ROSAT satellite which allowed the routine discovery and study of clusters out to $z\sim
0.8$, ([[*e.g. *]{}]{}@boh04 [@ros02; @vik98; @mul03]). Confirmed $z>1$ clusters remain rare beasts despite well over a decade of searches, with only a handful known from the ROSAT era ([[*e.g. *]{}]{}@vik98 [@ros99; @stanford02; @blak03]). The potential of XMM-Newton to detect clusters out to $z\sim 2$ was demonstrated as early as a year and a half before XMM launched [@val00]. Recently [@mul05] and [@stan06] have shown that clusters out to $z\sim 1.5$ are serendipitously detectable in moderately deep XMM exposures.
Work to date on X-ray selected clusters has shown that there is little evidence for evolution in their comoving space density (except for the most luminous systems) or $L_X-T_X$ relation out to $z\sim 1$ [@ros02]. Similarly, there is little evidence for anything other than passive evolution in the stellar populations of the massive galaxies in the cores of these clusters ([[*e.g. *]{}]{}@blak03) over the currently observed redshift range. Strong evolution of the ICM and galaxy populations in clusters seems to have occurred only at $z>1$. Increasing the number of known and well-studied $z>1$ clusters is therefore key to identifying this epoch and understanding the physical processes that drove the evolution of clusters of galaxies.
We are carrying out the XMM Large-Scale Structure Survey (XMM-LSS, @pie04) in order to study the large-scale distribution of matter in the Universe as traced through X-ray emission from clusters of galaxies and active galactic nuclei (AGN). One of the survey goals is to search for clusters of galaxies out to $z>1$ to luminosities $L_X([0.5-2.0]~\rm{keV}) > 5 \times 10^{43}$ [erg s$^{-1}$]{}([[*i.e. *]{}]{}a flux limit of $f_X([0.5-2.0]~\rm{keV}) \sim 10^{-14}$[erg cm$^{-2}$s$^{-1}$]{}). Here we present details of the discovery of a cluster with the highest spectroscopically-confimed redshift, $z=1.22$, so far found in the XMM-LSS. Some initial photometric data on this cluster was presented in [@andreon05].
This source was discovered in a survey of a relatively large contiguous area of sky ($\sim 5$ deg$^2$), specifically designed to detect clusters to $z>1$. This has the advantage that existing multi-wavelength datasets covering the same area can be used in combination to identify clusters from among the faint, extended X-ray sources in the XMM data, whereas for sources serendipitously detected in archived non-contiguous X-ray data, the multi-wavelength data have to be obtained after the identification of X-ray candidates. The disadvantage of the survey approach is that the X-ray exposure times are necessarily limited in order to cover sufficient area of sky (in the case of the XMM-LSS to typically 10-20 ksec), whereas the serendipitous approach can make use of far deeper exposures.
In the following we present the X-ray, optical and near-IR data for [XLSSJ022303.0-043622]{}, including Spitzer/IRAC 3.6 and 4.5 band imaging from the SWIRE survey. We discuss the cluster properties, briefly comparing them to those of other known $z>1$ clusters and demonstrate that a combined XMM and Spitzer survey is an efficient way of identifying further high redshift clusters. The cluster is also catalogued as [XLSSC 046]{}[^2]. We use $\Lambda$CDM cosmology ($\Omega_m = 0.3,\ \Omega_{\Lambda}=0.7,\ H_0 =
70$ km s$^{-1}$ Mpc$^{-1}$).
Observations
============
X-ray
-----
![Sofi $Ks-$band image of the central $\sim$ 1.5 by 1.5 arcmin$^2$ region of [XLSSJ022303.0-043622]{} smoothed with a gaussian of FWHM $0.9\arcsec$. Overlayed as contours is the wavelet filtered XMM \[0.5-2\] keV image. The lowest contour has a diameter of $\sim
1\arcmin$, contours increment logarithmically in surface brightness with and integrated total flux of 58 counts. Galaxies with measured redshifts are labelled as in table 1. North is up, East to the left.[]{data-label="fig:cls"}](mnbfig1_small.ps){width="8cm"}
The XMM X-ray data for the field of this cluster (XMM exposure ID 0109520601) were obtained as part of the first set of XMM-LSS pointings during GTO and AO-1 time. The resulting event list for the field was filtered for high background periods following @pra02 with a resulting exposure time of 51.6 ks across the three XMM detectors (MOS1: 22.5 ks; MOS2: 22.7 ks; pn: 16.4 ks). Less than 1% of the MOS and $\sim 4$% of the pn exposure was affected by periods of high background. The “cleaned” event list was then used to create images in different energy bands for the pn and MOS detectors. Using the procedure in @val01, potential candidate clusters were selected as extended X-ray sources in the \[0.5-2\] keV band, taking into account the effect of vignetting.
{width="15.5cm"}
The cluster was detected as an extended X-ray source at an off-axis angle of $8.8\arcmin$ using an early version of the XMM-LSS analysis pipeline. At this angle an incoming X-ray photon sees 60 per cent of the telescope area due to vignetting. Placing a $34\arcsec$ radius aperture over the source, and correcting for background counts gave 40 counts for the object in the \[0.3-10\] keV band of the MOS detectors and 45 in the PN detector. In the intervening time between the detection of the X-ray source and now, we have evolved our X-ray analysis based on the experience gained from the early XMM-LSS data. In section 3 we discuss the X-ray luminosity and temperature of the cluster as derived from our latest analysis methods (discussed in detail in @willis05 and @d1). Here we note that the source is detected and classified by the latest version of the XMM-LSS pipeline as a “C2” source (see @pacaud05 for details).
Optical and Near-IR data
------------------------
![The top and middle panels show spectra of the two brightest galaxies in the cluster (solid line). Wavelength regions associated with bright sky emission lines have been excised from the spectra – which have subsequently been resampled to match the spectrograph resolution. Total exposure time for each spectrum is 2 hours. The smooth line in each panel shows a low redshift elliptical galaxy template [@kin96] that has been redshifted and scaled to match the data on wavelength scales $> 500$Å. The vertical dotted line in the middle panel indicates the observed frame location of \[OII\] $\lambda$3727 emission. The lower panel displays the sky spectrum which is shown as a dotted line in wavelength regions excised from the spectra plotted above. The spectra are corrected for relative flux calibration, although this correction is uncertain longward of 9000 Å. []{data-label="fig:spec"}](mnbfig3.ps){width="8cm"}
On their own, the observed X-ray properties of the source are compatible with it being a nearby galaxy, a moderate redshift group or a high redshift cluster. Additional multi-wavelength data is required to determine its nature. The initial XMM-LSS sky area was imaged in several optical bands with the CFH12k camera on CFHT [@vvds; @mac03]. The lack of a clear identification for [XLSSC 046]{} in this relatively shallow optical data made the object a strong distant cluster candidate. In particular, there was no clear overdensity of galaxies (or an individual low redshift galaxy) at the X-ray position down to $I_{AB}=22$, indicating that the X-ray source was a potential $z>1$ cluster. Subsequently, the field of the cluster was observed during the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) using Megacam on CFHT [@boul03]. The $i'$ and $z'$ photometry used in this paper is derived from the CFHTLS data.
As a result of the lack of a clear optical ID in the CFH12K data, the source was targetted as part of an ESO NTT/SOFI observing run to follow up potential distant clusters (programme ID 70.A-0733A) on 2002 November 19 and 20. The cluster was observed for 40 minutes in each of $J$ and $Ks$ in photometric conditions at an airmass below 1.5, with observations divided into multiple, dithered exposures. These data were calibrated by observations of photometric standards sj9105 and sj9106 [@persson98]. Five 2MASS-catalogued objects were detected in the frames, the photometry for each had a typical uncertainty of 0.1 magnitude in the 2MASS catalogue; our calibration matches that of 2MASS within these uncertainties. All $J$ and $Ks$ magnitudes presented in this paper have been converted to AB magnitudes assuming $J_{AB}=J_{Vega}+0.9$ and $Ks_{AB}=Ks_{Vega}+1.9$ for consistency with the CFHTLS and IRAC magnitudes, which are measured in this system.
A high surface density of faint galaxies ($19.15<Ks<21.65$, or $17.25<Ks<19.75$ in Vega) – fifteen in a $30\arcsec$ diameter circle (a projected scale of 250 kpc at $z=1.22$) – was identified in both near-IR bands at the position of the X-ray detection (see figures \[fig:cls\] and \[tci\]). A circle of twice that diameter includes only a further eight objects in the same magnitude range, indicating the compactness of the galaxy distribution. All of this is compatible with a compact cluster of galaxies at $z>1$.
Spectroscopic observations of the cluster were obtained using the ESO/VLT FORS2 spectrograph on 2004 December 14 and 15. Multi-slit spectroscopic data were obtained using the 600z$+$23 grism and OG590$+$32 order-sorting filter. The slit width on each MXU mask was typically $1.2\arcsec$ and resulted in a spectral resolution of 8Åover the wavelength interval 7400 to 10000Å. The cluster was observed with two slit masks oriented at 90 degrees to each other in order to maximise the sampling of candidate cluster members. Each mask was observed for $4 \times 30$ minute exposures at an airmass below 1.5 and typical atmospheric seeing of $0.8\arcsec$. Any given object was observed through one slit mask giving total exposure times of two hours per galaxy. Spectral data were processed employing standard reduction techniques within [IRAF]{}. Zero level, flat–field and cosmic ray corrections were applied to all data prior to the identification, sky subtraction and extraction of individual spectra employing the [apextract]{} package. The dispersion solution for each extracted spectrum was determined employing HeNeAr lamp exposures with a typical residual scatter of $0.5-0.8$Å and the final spectra were resampled to a linear wavelength scale. A spectrophotometric standard star from the atlas of [@hamuy92] was observed during each night and was employed to correct for the relative instrumental efficiency as a function of wavelength.
Reliable redshift values consistent with a cluster redshift of $z=1.22
\pm 0.01$ were estimated for seven galaxy spectra (see table 1). Initial redshift estimates obtained from a visual inspection of prominent absorption and emission features in individual spectra were refined via cross–correlation with a representative early–type galaxy template [@kin96] employing the [IRAF]{} routine [xcsao]{} [@tonry79]. Examples of extracted spectra are displayed in figure \[fig:spec\]. The location of all galaxies confirmed at the cluster redshift are displayed in figure \[fig:cls\].
SWIRE Spitzer observations
--------------------------
The field of the cluster was imaged by the IRAC camera on the Spitzer satellite as part of the SWIRE [@lon03] survey of the XMM-LSS sky area. Images of the field at 3.6 and $4.5\micron$ – reduced using the standard SWIRE processing [@lon04] – clearly show the cluster (the 3.6 image is used as the red channel in figure \[tci\]). The high surface density of the galaxies at the very centre of the cluster, coupled with the relatively poor spatial resolution of IRAC at 3.6 and $4.5\micron$ leads to high crowding at the cluster centre and so to difficult photometry in this region. The rest of the field is uncrowded making for straightforward photometry. The field was also imaged at 24 by the MIPS camera. No cluster members were detected to a limit of 0.15mJy (fainter than 18.6 in AB). Unfortunately this limit hardly constrains the star formation activity in the galaxies. At the cluster redshift it translates to a luminosity of less than $L_{IR}<10^{12}$L$_\odot$, so no cluster galaxies host obscured ULIRG-like starbursts.
Photometry
----------
In order to carry out photometry on our ground-based optical and near-IR data sets, we used Sextractor [@bert96] to identify individual objects and determine their magnitudes. Results for the six spectroscopically-confirmed cluster galaxies brighter than $K_s<21.65$ are given in table 1. We selected the catalogue in the $Ks-$band, using the $Ks$ SOFI image as the finding image for other bands.
ID Redshift RA(2000) Dec(2000) $Ks$ $i'-Ks$ $i'-z'$ $J-Ks$ $Ks-3.6$ $3.6-4.5$
---- ---------- ---------- ----------- ----------------- ----------------- ----------------- ----------------- ----------------- -----------------
G1 1.219 35.7739 -4.6030 $ 20.31\pm0.05$ $ 3.07\pm0.10$ $ 0.90\pm0.06$ $ 0.94\pm0.09$ $ 0.66\pm0.12$ $ -0.25\pm0.11$
G2 1.215 35.7725 -4.6102 $ 20.80\pm0.08$ $ 2.75\pm0.12$ $ 0.79\pm0.07$ $ 0.98\pm0.14$ $ 0.74\pm0.14$ $ -0.26\pm0.11$
G3 1.215 35.7696 -4.6039 $ 21.10\pm0.10$ $ 3.04\pm0.14$ $ 0.64\pm0.13$ $ 0.68\pm0.15$ $ 0.69\pm0.14$ $ -0.26\pm0.12$
G4 1.210 35.7655 -4.6051 $ 19.78\pm0.04$ $ 3.01\pm0.08$ $ 0.72\pm0.06$ $ 1.13\pm0.07$ $ 0.69\pm0.17$ $ -0.17\pm0.10$
G5 1.221 35.7635 -4.6043 $ 19.62\pm0.04$ $ 3.21\pm0.07$ $ 0.87\pm0.06$ $ 0.97\pm0.05$ $ 0.58\pm0.19$ $ -0.19\pm0.10$
G7 1.210 35.7539 -4.6140 $ 20.31\pm0.05$ $ 2.56\pm0.10$ $ 0.88\pm0.06$ $ 0.80\pm0.08$ $ 0.55\pm0.20$ $ -0.28\pm0.11$
SEXTRACTOR was used in dual mode using the $Ks$ image to identify objects and to define the apertures that were then applied to the other optical and near-IR images. The $Ks-$band detected number counts peaked between $Ks=21.4$ and 21.65 and dropped to 50 per cent of the peak between $Ks=22.15$ and 22.4. We carried out the photometry in two ways: firstly by using 2 arcsec diameter fixed apertures with appropriate aperture corrections in each band and secondly by using the corrected isophotal “AUTO” magnitude (after Kron 1980) having previously smoothed all the data to 0.8 arcsec FWHM, matching the worst seeing image (the $I-$band). Both methods gave comparable results, in the following we report the 2 arcsec aperture magnitudes as the smoothing process complicates the determination of uncertainties for the corrected isophotal magnitudes.
Using the $Ks$ frame as the finding image was straightforward for the $J-$band photometry because the data shared the same plate scale and astrometric solution. For the Megacam data, we resampled the $Ks$ image to the Megacam plate scale to act as the finding image. The same objects and apertures were re-identified in the resampled image as in the original $Ks$ frame and applied to the $i'$ and $z'$ data. The resampling meant that the SEXTRACTOR coordinates (and therefore the aperture centre) for each object changed slightly, but by no more than 0.15 arcsec (and usually $<0.1$ arcsec, less than half a pixel) in the case of the cluster members listed in table 1. As a check on the effect of resampling on the determination of magnitudes, we compared the results of photometry on the original and resampled $Ks$ image. For the galaxies of comparable magnitudes to those in listed in table 1, this introduced a scatter of no more than 0.05 magnitudes. Additionally we shifted the resampled image by 0.5 pixels in both RA and DEC to see what effect that had on the determination of the $i'$ and $z'-$band magnitudes. For galaxies of similar magnitudes to those in table 1, a typical scatter of 0.025 magnitudes was induced by this process. To account for these the uncertainties quoted in table 1 for $i'-Ks$ and $z'-Ks$ include the statistical uncertainties from SEXTRACTOR and a 0.05 magnitude systematic uncertainty added in quadrature.
For the IRAC photometry, we used the catalogue made by the SWIRE collaboration from the SWIRE data release 2 (described in @surace05) as this provides aperture photometry with appropriate aperture corrections. Following the recommendations in Surace etal we used apertures of diameter 3.8 arcsec (twice the instrumental FWHM) which provide the most accurate photometry and have had aperture corrections for unresolved sources (like the cluster galaxies) already applied. The IRAC aperture centroids matched those in the K-band to within 0.2 arcsec for all objects in table 1 except for G3 which matched within 0.4 arcsec. For galaxies G4,5 and 7, the 3.8 arcsec apertures included emission from fainter close companion galaxies. To correct for this, we determined the difference in $Ks$-band flux between 2 and 3.8 arcsec apertures centred on these objects, assumed that the companions had similar $K-3.6\micron$ colours to the brighter galaxies and adjusted this colour for each of the objects accordingly (a correction of between 0.3 and 0.37 magnitudes in each case). The assumption of similar colours for the companion objects is reasonable as they are likely cluster members, nevertheless this is a source of uncertainty. Assuming a potential colour differential of $-0.3<\Delta(K-3.6)<0.3$ between the main galaxy and a companion contributing 30 per cent of the flux in an aperture, this would give an uncertainty of just les than 0.1 magnitudes in the colur determined in this way. This is reflected in the uncertainties quoted for the $K-3.6\micron$ colours of these objects in table 1. The absolute accuracy quoted by the SPITZER Science Centre for IRAC is 10% in flux density. The $K-3.6\micron$ and $3.6-4.5\micron$ uncertainties listed in table 1 include this and the statistical uncertainties combined in quadrature.
Discussion
==========
X-ray luminosity of the cluster
-------------------------------
With the redshift of the cluster known, the X-ray properties of the cluster can be estimated, although with of order 85 detected photons, uncertainties are large. The temperature of the X-ray emitting gas was estimated using XSPEC. Photons from all detectors were extracted from a 34 arcsec radius circle centred on the X-ray centroid. Following [@willis05], a spectral fit was performed with XSPEC, using an APEC model [@smith01], abundances from [@grevesse99] and a Galactic neutral hydrogen column of $2.64 \times 10^{20}$ cm$^{-2}$ [@dic90]. This resulted in a best fit temperature of 3.8keV, with a 1$\sigma$ lower limit of 1.9keV and an effectively unconstrained upper limit.
In order to determine a count-rate and hence a luminosity for the cluster, an aperture over which the rate is to be measured must be defined. The low number of counts from the object limits the useful size of the aperture to be around 50-60, comparable to those used in previous X-ray studies of distant clusters. To support this choice we estimated r500, the radius at which the cluster density is 500 times the critical density at its redshift. Using the best-fit temperature and following the method in [@willis05], we estimated this to be 55(albeit with considerable uncertainty due to the uncertainty on the temperature). We used this radius for the subsequent analysis. The background-subtracted count rate in the \[0.5-2\]keV band within this radius was $6.8 \times 10^{-3}$ cts/s, corresponding to a flux of $6.2\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$. Using this to normalise the spectral fit gives an unabsorbed bolometric luminosity of $1.1\times 10^{44}$ [erg s$^{-1}$]{}with a statistical error of approximately 20 per cent. Ideally we would apply an aperture correction for flux beyond 55 , but given the limited signal-to-noise this would be highly uncertain for this object, so we only quote the luminosity within this aperture. From simulations and experience with other low count-rate groups and clusters in the XMM-LSS, we found that statistical errors are usually under-estimates of the true errors on the luminosities in such cases ([[*e.g. *]{}]{}see @d1), with more realistic errors twice that of the raw statistical values. Consequently, we quote an unabsorbed bolometric luminosity of $1.1\pm 0.7 \times 10^{44}$ [erg s$^{-1}$]{} and, for comparison to other high redshift clusters, a rest-frame \[0.1-2.4\]keV luminosity of $7.7\pm 0.3
\times 10^{43}$ [erg s$^{-1}$]{} for [XLSSC 046]{}.
Given that only seven cluster galaxies have confirmed redshifts, the velocity dispersion of the galaxies add little to this, all we can say here is that the dispersion in these redshifts is consistent with the derived X-ray luminosity. The above luminosity is low in comparison to those of other known $z>1$ clusters. RX J0848+4453 at $z=1.27$ has the lowest luminosity of the previously known systems. Using the values in [@stan01] converted to the cosmology used here gives a \[0.1-2.4\]keV luminosity of $5.9 \pm 2 \times 10^{43}$ [erg s$^{-1}$]{}measured in a 35 radius aperture. Using an aperture of the same size for [XLSSC 046]{} results in a luminosity of $3 \pm 1.5 \times
10^{43}$ [erg s$^{-1}$]{}. Thus [XLSSC 046]{} is about as a luminous as RX J0848+4453 (and possibly slightly less luminous), given the quoted uncertainties.
![$J-Ks$ colour magnitude diagram. Objects from across the $5\arcmin$ by $5\arcmin$ SOFI field detected in both bands plotted as small symbols with no error bars. Objects in table 1 and those detected in the central 250 kpc of the cluster are plotted with error bars. Overplotted is the best-fit colour-magnitude relation from Lidman etal., (2004) for RDCS J1252.9-2927 at a similar redshift, indicating the similarity in the colours of the galaxies in the two systems.[]{data-label="cmd1"}](mnbfig4a.ps){width="8cm"}
Galaxy properties
-----------------
Figure \[fig:cls\] shows the $Ks-$band image of the cluster, overlayed with the X-ray contours from the XMM data, with the spectroscopically-confirmed cluster members labelled G1 to G7. Figure \[tci\] shows a wider-field pseudo true-colour image of the cluster made from $z-$band VLT/FORS2 data, $Ks-$band NTT/SOFI data and the 3.6 Spitzer/IRAC image. The cluster is easily identified in this image as an obvious overdensity of relatively red sources at the centre. As noted above, there are fifteen $19.15<Ks<21.65$ objects identified in a 250 kpc ($30 \arcsec$) diameter circle centred close to the X-ray peak. Simple visual inspection of figure \[tci\] demonstrates that this is an order of magnitude or more overdensity of such sources in comparison to the surrounding field.
Figure \[tci\] also indicates that many of the galaxies in the central 250 kpc region have similar colours, redder than the average faint source between $z$ and $3.6\micron$. Figure \[cmd1\] shows the $J-Ks$ colour-magnitude diagram of the objects in the SOFI data. The symbols with error bars denote the data for objects in the central 250 kpc and those listed in table 1. Overlayed is the colour-magnitude relation found for RDCS1252.9-2927, another z=1.22 cluster, by [@lidman04]. The colours of most of these galaxies are consistent with this relation, and following the arguments in [@lidman04] indicate that the galaxies have colours consistent with passive evolution, having formed the bulk of their stellar populations at higher redshift.
Figure \[cmd2\] shows a similar colour-magnitude diagram for $I-Ks$. This shows a tight relation between colour and magnitude for the confirmed cluster members and those in the central 250 kpc. Overplotted is the $J-Ks$ colour-magnitude relation from [@lidman04], shifted by 2.2 magnitudes (a reasonable shift for a z=1.22 elliptical). Most of the cluster members fall on this relation within their uncertainties. The tightness of the relation and the very red colours again indicates that the majority of these galaxies appear to have predominantly passively evolving stellar populations dominated by old stars with little significant ongoing star formation. For example, in figure \[sed\] we plot the rest-frame spectral energy distribution of G5, the brightest galaxy in the central $30\arcsec$ which has colours typical of other galaxies in the central region of the cluster. We overplot the best-fit instantaneous single burst, solar metallicity simple stellar population (SSP) synthesis model from [@bc03]. This has an age of 3 Gyr, implying a redshift of formation of $z=3.2$. However, metallicity and reddening can effect the inferred age of the best-fit to broad-band SEDs ( [[*e.g. *]{}]{}@lidman04 and @brem02). We attempted to fit younger models reddened by a $\lambda^{-1.3}$ power law normalised to $E(B-V)$=0.1. Models as young as 1.5 Gyr gave reasonable fits given the uncertainties in the photometry. As found by other authors, [[*e.g. *]{}]{}@lidman04, models with more extended, but exponentially-declining star formation episodes give comparable ages for the bulk of stars in the galaxies. As noted in section 2.2 and in figure \[fig:spec\], the absorption line spectra of the cluster members is well–matched by a [@kin96] elliptical spectrum, again consistent with a substantial old population of stars. Clearly there is some star formation (or AGN) activity in at least one of the galaxies, given the presence of the \[OII\] $\lambda$3727 line in the spectrum of G4. This may be due to emission from the galaxy itself, or contamination from another cluster member in the spectroscopic slit, possibly the object that is included in the IRAC aperture.
Where the colours of the galaxies indicate that they have relatively old stellar populations, their IRAC and $Ks-$band magnitudes indicate that these galaxies are already massive at $z=1.22$. If they evolved passively to $z=0$ they would be approximately 1-1.3 magnitudes fainter in the rest-frame $K-$band (depending on the age of the galaxies at $z=1.2$). At $z=1.22$, the 4.5 band maps almost directly to the rest-frame $K$. If the system was placed at the distance and look-back time of Coma, taking into account passive evolution, galaxy G5 would have $Ks \sim 10.6$ (8.7 in Vega), within a few tenths of a magnitude of the value for NGC 4889 in Coma. Similarly, the tenth brightest galaxy in figure \[cmd1\] is less than two magnitudes fainter than this, again comparable to the tenth brightest member in Coma [@depropris98]. In common with other recent studies ([[*e.g. *]{}]{}@depropris99 [@toft04; @andreon04]), these results are consistent with a scenario where the more massive cluster galaxies are largely in place within a cluster at $z>1$, subsequently evolving passively with no significant new star formation or substantial growth by mergers.
How do the properties of these cluster galaxies compare to those in other known clusters at similar redshifts? [@lidman04] and [@blak03] have carried out ground-based near IR and HST-based optical studies respectively of another z=1.2 cluster, RDCS J1252.9-2927. We have already shown that figure \[cmd1\] is consistent with figure 2 in [@lidman04] – a similar colour-magnitude diagram based on SOFI data for that cluster. If we assume that the three brightest galaxies in the $J-K$ colour-magnitude sequence of RDCS J1252.9-2927 are the same as the three brightest in the $i-z$ sequence [@blak03], we infer similar colours for galaxies in that cluster as for the galaxies in [XLSSC 046]{}. [@stanford97] and [@stanford02] found similar colours for galaxies in the $z=1.27$ cluster RX J0848+4453 and the $z=1.16$ cluster around 3C210. All of these authors conclude that such colours are consistent with a high formation redshift for the stellar populations of the bulk of the identified bright/massive cluster galaxies and passive evolution thereafter.
![$I-Ks$ colour magnitude diagram for the same objects as figure \[cmd1\]. The J-K colour-magnitude relation from Lidman etal., 2004 has been shifted by 2.2 magnitudes and overplotted on the data and proves a good fit to the photometry for the galaxies at the centre of the cluster. []{data-label="cmd2"}](mnbfig4b.ps){width="8cm"}
Taking all of the above into account, our current data is consistent with most of the identified cluster galaxies at the centre of the system having dominant stellar populations at least 1.5 Gyr old, and possibly as old as 3 Gyr, having evolved passively after an initial burst or a short period of star formation. Their stellar masses are comparable to those of bright cluster ellipticals at low redshift, appearing to rule out significant growth by merger for these galaxies at lower redshifts. That is not to say that that other less massive galaxies within this cluster are not undergoing significant merger and star formation activity and can continue to do so, our $Ks-$band selection naturally selects the most massive galaxies at the observed epoch.
The spatial distribution of these galaxies appears more compact than in clusters of a similar redshift such as RX J0848+4453 (@stanford97 [@ros99]) and RDCS J1252.9-2927 [@lidman04]. The surface density of sources brighter than $Ks<21.65$ in the central 30 of [XLSSC 046]{} is $\sim 75$ per arcmin$^2$, this compares to $\sim 25$ and $ \sim 45$ per arcmin$^2$ in the central 70 and 40 arcseconds of these two clusters, estimated from the colour magnitude diagrams in [@ros99] and [@lidman04]. The published images of these clusters do not show as sharp a drop in surface density of sources beyond 15 radius. However, it is unclear whether the cluster and its mass distribution is truely compact, or whether our current data only trace the distribution of the most massive galaxies within it. Deeper optical, near-IR and X-ray imaging is clearly required to determine the spatial distribution of matter within the cluster, as is further optical spectroscopy in order to probe its mass.
Detectability of high redshift clusters with combined XMM-LSS and SWIRE data
----------------------------------------------------------------------------
The availability of deep CFHTLS optical imaging data over much of the initial $\sim 5$ deg$^2$ of the XMM-LSS area means that it is straightforward to identify groups and clusters of galaxies out to $z\sim 0.8-1$ as extended X-ray sources associated with overdensities of faint galaxies with similar optical colours ([[*e.g. *]{}]{}@valt04 [@willis05]). The same data can be used to identify clusters as galaxy overdensities at higher redshifts, but this becomes increasingly difficult at $z>1$ and potentially impossible at $z>1.4$. So long as XMM sources can be reliably identified as extended, one can have confidence that the absense of an identification in deep optical data indicates that the source is likely to be a high redshift cluster. However, given typical $z>1$ clusters with luminosities around $10^{44}$ [erg s$^{-1}$]{}will produce only a few tens of counts in typical XMM-LSS exposures, confirming that these sources are extended is challenging. Given the red colours of galaxies in cluster cores, a sufficiently deep ground-based near-IR or spaced-based Spitzer/IRAC survey which covers the same area of sky as XMM-LSS can potentially play the same role at the highest redshifts as the CFHTLS data does at $z<1$. The clear detection of [XLSSJ022303.0-043622]{} in the SWIRE data is therefore significant, as this survey covers much of the initial XMM-LSS area. A similar cluster at even higher redshift would have been detectable as an excess of galaxies in data as deep as that as SWIRE ([[*e.g. *]{}]{}the $z=1.4$ cluster detected by [@stan05] would be detectable in IRAC data of comparable depth to that of SWIRE). Clusters out to $z\sim
1.5$ discovered by X-ray emission in the XMM-LSS can therefore be photometrically confirmed by a combination of existing SWIRE/IRAC and CFHTLS data. A search for such clusters is now underway (Bremer et al., in prep.)
![Rest-frame spectral energy distribution for the cluster galaxy G5. The asterisks denote the $i'$ and $z'$ phrotometry from CFHTLS, the SOFI $J$ and $Ks$ values and the IRAC $3.6$ and $4.5$ photometry. Overplotted is a Bruzual & Charlot (2003) instantaneous burst, simple stellar population model with solar metallicity seen 3 Gyr after the burst. Similar models as young as 1.5 Gyrs can give an acceptable fit to the data assuming some intrinsic reddening (see text for details).The statistical uncertainties quoted in table 1 are smaller than the size of the asterisks.[]{data-label="sed"}](mnbfig5.ps){width="9cm"}
Conclusions
===========
We have reported details of the discovery of the $z=1.22$ cluster [XLSSJ022303.0-043622]{}, presenting multiband imaging and initial spectroscopy of the system. We spectroscopically confirm seven galaxies with redshifts of $z=1.22\pm 0.01$ within an arcminute of the X-ray position. The cluster appears to have a centrally-condensed galaxy distribution, with fifteen galaxies with $17.25<Ks<19.75$ within $15 \arcsec$ of the centre and only a further eight in an annulus between $15\arcsec$ and $30\arcsec$ from the centre. The spectroscopically-confirmed cluster members have the colours of passively evolving ellipticals indicating the bulk of their star formation occurred at least 1.5 Gyr before $z=1.22$ ([[*i.e. *]{}]{}at $z>2$). Based on their $Ks$ and IRAC magnitudes, they have stellar masses comparable with those of massive galaxies in clusters at low redshift, indicating that massive cluster galaxies may be in place at $z>1$ and passively evolve at lower redshift with little significant star formation or growth through mergers. The straightforward detectability of this cluster in Spitzer/IRAC data demonstrates that the combination of SWIRE and XMM-LSS datasets allow for efficient searches for the most distant clusters.
Acknowledgments {#acknowledgments .unnumbered}
===============
XMM is an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This work was based on observations made with ESO Telescopes at the La Silla and Paranal Observatories under programme IDs 70.A-0733 and 074.A-0360 and with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. Support for SWIRE, part of the Spitzer Space Telescope Legacy Science Program, was provided by NASA through an award issued by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407. This work was also supported by the European Community RTN Network POE (grant nr. HPRN-CT-2000-00138). MNB acknowledges Leverhulme Trust funding for the early part of this work. SA acknowledges financial contribution from contract ASI-INAF I/023/05/0. We thank Roberto de Propris for useful discussions.
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: See http://vizier.u-strasbg.fr/cgi-bin/Dic-Simbad?XLSSC
|
---
abstract: |
We review recent progress in the explorations of topological quantum states of matter in iron-based superconductors. In particular, we focus on the nontrivial topology existing in the band structures and superconducting states of iron’s 3d orbitals. The basic concepts, models, materials and experimental results are reviewed. The natural integration between topology and high-temperature superconductivity in iron-based superconductors provides great opportunities to study topological superconductivity and Majorana modes at high temperature.
**Keywords:** iron-based superconductor, topological insulator, topological superconductor, spin-orbit coupling
author:
- Ning Hao
- Jiangping Hu
title: 'Topological quantum states of matter in iron-based superconductors: From concepts to material realization'
---
Introduction
============
In the past decade, topology becomes an essential ingredient to classify various types of materials, including insulators/semiconductors, semimetals and superconductors[@Hasan_rmp_2010; @Qi_rmp_2011; @Armitage_rmp_2018]. The physical consequence in a topological material is the existence of topologically protected surface states, which can be measured directly in transport, angle resolved photoemission spectrum(ARPES), scanning tunneling microscopy(STM) and other experiments[@Hasan_rmp_2010; @Qi_rmp_2011; @Armitage_rmp_2018]. In particular, in a topological superconductor, there are surface bound states, Majorana modes, which can be used to realize topological quantum computing because of their topological protection and non-Abelian braiding statistics[@Nayak_rmp_2008].
While naturally-born topological superconductors are very rare, the realization of Majorana modes can be achieved in many artificial hybrid systems. Recently, a wealth of proposals for such experimental designs has been proposed, including the superconducting surface states of a topological insulator in proximity to conventional superconductors[@Fu_prl_2008], quantum wires with strong spin-orbit coupling in proximity to conventional superconductors[@Lutchyn_prl_2010], semiconductor-superconductor heterostructures[@Sau_prl_2010], and spin-chains embedded in conventional superconductors[@Stevan_science_2014] etc. However, these hybrid systems, in general, have two shortcomings. First, it is always difficult to manage the interface between two different structures. Second, in all these proposals, as the proximity effect requires a long superconducting coherent length, high temperature superconductors, such as cuprates and iron-based superconductors, have never been candidates in those integration processes because of their extreme short coherent lengths and structural incompatibility. Thus, all devices require to be operated at very low temperature.
The above shortcomings can be overcome if we can find a high temperature superconductor which hosts nontrivial topological band structures. Specifically, to differentiate them from topological superconductors as well as the above hybrid superconducting systems, we refer this type of superconductors specifically as *connate* topological superconductors[@Shi_sb_2017]. The connate topological superconductor can be viewed as an internal hybrid system which has conventional superconductivity in bulk but topological superconductivity on surface caused by the nontrivial topology on some part of band structures[@Shi_sb_2017; @Xu_prl_2016]. Because of this intrinsic hybridization, the superconductor, in general, must be a multiple band electronic system. As iron-based high temperature superconductors are known to be multi-orbital electronic systems, they become promising candidates.
During the past several years, starting from theoretical understanding, the research of iron-based superconductors as connate topological superconductors has gradually been materialized. The first theoretical study of nontrivial band topology was carried out by us for the single layer FeSe/STO, in which a band inversion can take place at M points[@Hao_prx_2014] to create nontrivial topology. Very quickly, it was found that the band inversion can easily take place at $\Gamma$ point if the anion height from Fe layers are high enough. For FeSe, the height can be increased by substituting Se with Te[@Wu_prb_2016; @Wang_prb_2015]. For iron-pnicitides, the As height is predicted to be high enough in the 111 series, LiFeAs to host nontrivial topology[@Zhang_arxiv_2018-1]. Besides these intrinsic topological properties from the Fe d-orbitals, nontrivial topology can also stem from bands outside Fe layers. For example, the As p-orbitals in the As layers of the 122 CaFeAs$_{2}$ are shown to be described by a model similar to the Kane-Mele model in graphene[@Wu_prb_2015-1]. Most recently, because of the improvement of sample quality and experimental resolutions, there have been increasing experimental evidence for topological properties in iron-based superconductors[@Zhang_science_2018; @Wang_science_2018; @Liu_arxiv_2018]. The theoretically predicted band inversions, together with the topologically protected surface states, have been directly observed. The Majorana-like modes are observed in several iron-chalcogenide materials[@Wang_science_2018; @Liu_arxiv_2018]. All these progresses have made iron-based superconductors to be a new research frontier for topological superconductivity.
In this paper, we give a brief review of both theoretical and experimental results regarding of the topological properties of iron-based superconductors. In section II, we discuss theoretical concepts and models for the topological band structure in iron-based superconductors and recent experimental evidence. In section III, we review topological superconductivity that can be emerged from the topological bands of iron-based superconductors and experimental evidence of Majorana-like modes in these materials. Finally, we will address open issues in this field.
Topology in iron d-orbital bands
================================
Concepts and models
-------------------
Since the discovery of iron-based superconductors in 2008, there has been remarkable progress in material growth and synthesis about the iron-based compounds. According to the element composition, the iron-based superconductors are classified into different categories denoted with 1111, 122, 111, 11, etc[@Paglione_np_2010]. All categories possess the kernel substructure of X-Fe-X trilayer with X denoting As, P, S, Se, Te, as shown in Fig.\[fig1\] (a). The X-Fe-X trilayer is the basic unit cell to give arise to magnetism and superconductivity, and play a similar role as Cu-O plane in cuprates. Following the principle from complexity to simplicity, the X-Fe-X trilayer skips the specificity among all the compounds in iron-based superconductors and brings the intrinsic physics to the surface. However, along the opposite logic, the diversity may include important subtle surprising differences. For iron-based superconductors, such kinds of accidental surprises can be intuitively demonstrated through evaluating the sensitivity of the electronic structures upon the tiny change of the structure of the X-Fe-X trilayer[@Guterding_prb_2017]. Fig.\[fig1\] (c) gives such intuitive demonstration. The band structures sensitively depend on the fine tune of the distances between Fe-Fe and Fe-X. In particular, the bands switch orders near $\Gamma$ point, a band gap opens near $M$ point and the bands become strongly dispersive along $\Gamma-Z$ direction when the third dimension is considered. Indeed, the layered structures of the iron-based superconductors provide the possibilities to tune the distances between Fe-Fe and Fe-X. For example, the La-O layer in LaOFeAs and the Ba-As layer in BaFe$_{2}$As$_{2}$ naturally cause different lattice constants for Fe-X layers[@Singh_prl_2008; @Singh_prb_2008]. A variety of materials in the family of iron-based superconductors provide different fine-tuned X-Fe-X trilayers.
{width="1.0\linewidth"}
The band fine tuning would become nontrivial if there exist a topological phase transition. The discovery of topological insulators has established a standard paradigm about the topological quantum states of matter, which includes band inversion, bulk-boundary correspondence and relationship between symmetry and topological invariant etc[@Kane_prl_2005-1; @Bernevig_science_2006; @Konig_science_2007; @Moore_prb_2007; @Fu_prl_2007; @Roy_prb_2009; @Fu_prb_2007; @Chen_science_2009; @Hasan_rmp_2010; @Qi_rmp_2011]. For example, the first experimentally confirmed two-dimensional topological insulator, HgTe/CdTe quantum well, has a band inversion induced by the large spin-orbit coupling from Hg, depending on the thickness of the well, to gives arise to a topological insulating state[@Bernevig_science_2006; @Konig_science_2007]. The well-known three-dimensional topological insulators, Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$, has a band inversion caused by a strong spin-orbit coupling that switch two p$_{z}$-type bands with opposite spacial-inversion-symmetry parities at the $\Gamma$ point[@Zhang_np_2009; @Chen_science_2009; @Xia_np_2009]. The picture of band inversion can be further simplified into the energy level shift in atomic limit through the adiabatic deformations[@Zhang_np_2009]. Fig. \[fig1\] (d) gives the typical picture of energy level shift under the influence of several kinds of interactions in Bi$_{2}$Se$_{3}$[@Liu_prb_2010].
Interestingly, a similar picture also exists in some specific iron-based superconductors with fine-tuned X-Fe-X layers. The typical picture of energy level shift of iron $d$-orbitals are shown in Fig. \[fig1\] (e) and (f) for $\Gamma$ point in (e) and $M$ point in (f), respectively. Note that the space group of Fe-X-Fe trilayer is P4/nmm, in which the glide-plane mirror symmetry operation {$m_{z}|\frac{1}{2}\frac{1}{2}0$} and inversion symmetry operation {$i|\frac{1}{2}\frac{1}{2}0$} are essential[@Cvetkovic_prb_2013; @Hao_prb_2014; @Hao_prx_2014; @Hao_prb_2015]. First, the Bloch states can be classified according to the parities of {$m_{z}|\frac{1}{2}\frac{1}{2}0$}, i.e., $|d_{o/e,\alpha}\rangle$ or $|d_{o/e},m_{l},m_{j}\rangle$ with $o$, $e$, $\alpha$, $m_{l}$, $m_{j}$ denoting the odd or even parity of {$m_{z}|\frac{1}{2}\frac{1}{2}0$}, the $\alpha$th $d$ orbital, and two magnetic quantum numbers, respectively. Second, under the inversion symmetry operation {$i|\frac{1}{2}\frac{1}{2}0$}, the inversion parities of $|d_{o/e,\alpha}\rangle$ and $|d_{o/e},m_{l},m_{j}\rangle$ for $\alpha=xz/yz$, $m_{l}=\pm1$ are opposite to the inversion parities of $|d_{o/e,\alpha}\rangle$ for $\alpha=xy$. Focus on the green rectangles in Fig. \[fig1\] (e) and (f), the spin-orbit coupling can switch the order of the energy levels with opposite inversion parities and induce a topological phase transition[@Hao_prb_2014; @Hao_prx_2014; @Hao_prb_2015].
In early 2014, the authors of this paper noted that a tiny band gap around $M$ point in the band structure of monolayer FeSe/SrTiO$_{3}$(FeSe/STO)[@Wang_cpl_2012] from measurement of the ARPES[@He_nm_2013; @Tan_nm_2013; @Liu_nc_2012; @Lee_nature_2014; @Miyata_nm_2015; @Peng_nc_2014] and predicted the topological phase transition in this two-dimensional system [@Hao_prx_2014]. It is the first proposal to discuss the topological quantum state of matter in iron-based superconductors. Corresponding to Fig. \[fig1\] (f), an effective **k**$\cdot$**p** model can be constructed in the basis set of $[\{|\psi_{o}\rangle\},\{|\psi_{e}\rangle\}]$ with $\{|\psi_{o}\rangle\}=\{|d_{o,xy,\uparrow}\rangle,|d_{o},1,\frac{3}{2}\rangle,|d_{o,xy,\downarrow}\rangle,|d_{o},-1,-\frac{3}{2}\rangle\}$ and $\{|\psi_{o}\rangle\}=\{|d_{e,xy,\uparrow}\rangle,|d_{e},-1,\frac{1}{2}\rangle,|d_{e,xy,\downarrow}\rangle,|d_{e},1,-\frac{1}{2}\rangle\}$ $$H_{M}(k)=\left[
\begin{array}
[c]{cc}H_{M}^{o}(k) & H_{c}\\
H_{c} & H_{M}^{e}(k)
\end{array}
\right] . \label{HM1}$$ Here, $$H_{M}^{o}(k)=\left[
\begin{array}
[c]{cc}H(k) & 0\\
0 & H^{\ast}(-k)
\end{array}
\right] , \label{Hm2}$$ $H_{M}^{e}(k)=$ $H_{M}^{o,\ast}(-k)$, $H(k)=\varepsilon(k)+d_{i}(k)\sigma_{i}$ with $\varepsilon(k)=C-D(k_{x}^{2}+k_{y}^{2})$, $d_{1}(k)+id_{2}(k)=A(k_{x}+ik_{y})$, and $d_{3}(k)=M-B(k_{x}^{2}+k_{y}^{2})$ with $MB>0$. In the absence of $H_{c}$ term, the Hamiltonian in Eq.(\[HM1\]) reduces into two copies of Bernevig-Hughes-Zhang (BHZ) model[@Bernevig_science_2006], which is the standard model for quantum spin Hall effect. In each subspace with odd or even parity, a topological invariant $Z_{2}=1$ can be defined. Actually, $H_{c}$ term is from the spin-flipped term $\lambda_{so}(L_{x}s_{x}+L_{y}s_{y})$, which mixes the orbitals with odd and even parities of {$m_{z}|\frac{1}{2}\frac{1}{2}0$}. As a consequence, the parity of {$m_{z}|\frac{1}{2}\frac{1}{2}0$} is no longer a good quantum number. The two subspaces couple with each other. The topological states is more like weak type. However, if the two iron sublattices have different on-site potential, i.e., the staggered sublattice potential, which is introduced by the substrate, the weak topological state can be tuned into the strong topological states, because the potential can renormalize the mass term $M$ in $d_{3}(k)$, and change its sign in only one copy. Now, the band inversion condition with $MB>0$ is satisfied only in another copy. The topological state becomes strong type and is robust against the $H_{c}$ coupling without breaking time reversal symmetry[@Hao_prx_2014].
In late 2014, the topological phase transition around $\Gamma$ point was proposed in Fe(Te$_{1-x}$Se$_{x}$) thin film[@Wu_prb_2016], as well as in the bulk materials[@Wang_prb_2015]. The first-principle calculations indicated that the proper ratio between Te and Se could induced the band inversion around $\Gamma$ point. Refer to Fig. \[fig1\] (e), an effective **k**$\cdot$**p** model can be constructed in the basis set $\{|d_{o},1,\frac{3}{2}\rangle,|d_{o,xy,\uparrow}\rangle,|d_{o},-1,-\frac
{3}{2}\rangle,|d_{o,xy,\downarrow}\rangle\}$, $$H_{\Gamma}(k)=\varepsilon_{0}+\left[
\begin{array}
[c]{cccc}-M(k) & Ak_{+} & & \\
Ak_{-} & M(k) & & \\
& & -M(k) & -Ak_{-}\\
& & -Ak_{+} & M(k)
\end{array}
\right] . \label{HG}$$ Here, $\varepsilon_{0}=C-D(k_{x}^{2}+k_{y}^{2})$, $M(k)=M-B(k_{x}^{2}+k_{y}^{2})$. In the band inversion regime, $MB>0$. Likewise, effective **k**$\cdot$**p** model around the $\Gamma$ point in Eq. (\[HG\]) restores the famous BHZ model which describes the quantum spin Hall effect in HgTe/CdTe quantum well. In the original paper[@Wu_prb_2016], the author considered the hybridization between $p$ orbitals of Te/Se with $d$ orbitals of Fe. The basis functions for the **k**$\cdot$**p** model would be complex. Here, we use the only $d$ orbitals of Fe to construct the basis functions through downfolding the $p$ orbital parts without changing the symmetries. Therefore, the effective **k**$\cdot$**p** models in the basis sets involving $d$ and $p$ orbitals or only $d$ orbitals have the identical forms.
The topological phase transition around $\Gamma$ point in the Fe(Te$_{1-x}$Se$_{x}$) thin film can be generalized into the bulk Fe(Te$_{1-x}$Se$_{x}$) single crystal. Correspondingly, the two-dimensional topological state is generalized into three-dimensional topological states, which is similar to topological insulator in Bi$_{2}$Se$_{3}$. The topological nature of the band structures of bulk Fe(Te$_{1-x}$Se$_{x}$) single crystal was proposal through the first-principles calculations[@Wang_prb_2015]. The band inversion and $Z_{2}$ topological invariant was revealed. Following the picture of topological phase transition at $\Gamma$ point shown in Fig. \[fig1\] (e), the topological phase transition in bulk Fe(Te$_{1-x}$Se$_{x}$) single crystal is a little different from that in FeTe$_{1-x}$Se$_{x}$ thin film. The spin-orbit coupling in the latter case does not play a primary role to the topological phase transition[@Wu_prb_2016]. The spin-orbit coupling, however, is indispensable in the former case. Because the small band gap between $\Gamma_{6}^{+}$ and $\Lambda_{6}$ between $\Gamma-Z$ points is from the transmission effect, which transmits the coupling between $\Gamma_{4}^{+}$ and $\Gamma_{5}^{+}$ to the coupling between $\Gamma_{6}^{+}$ and $\Lambda_{6}$ through the medium of spin-orbit coupling (See Ref.[@Wang_prb_2015] for the relevant band labeling). The transmission effect can be revealed by a tight-binding model only involving the five $d$ orbitals of irons (the weight of $|p_{z},\mathbf{k}+\mathbf{Q}\rangle$ state in $\Gamma_{2}^{-}$ band can be renormalized to the $|d_{xy},\mathbf{k}\rangle$ state). The interlayer couplings include the parity-conserved terms and the parity-mixing terms[@Hao_prx_2014]. Note that the $\Gamma_{2}^{-}$ state in the first-principles calculations is captured by the band 4 in Fig. \[fig2\](i). Without interlayer parity-mixing term, even the spin-orbit coupling cannot open a gap between band 4 and bands 1, 2. Only when both interlayer parity-mixing term and spin-orbit coupling are tuned on, a small band gap opens as shown in Fig. \[fig2\](i). The key interlayer parity-mixing term is the hopping between the $d_{xz}$ and $d_{yz}$, i.e., $-4it_{xz,yz}^{c}(\cos
k_{x}+\cos k_{y})\sin k_{z}$. The effect of interlayer parity-mixing term can be renormalized to obtain an effective spin-orbit coupling under the second-order perturbation approximation, $$\tilde{H}_{soc}=\left[
\begin{array}
[c]{cc}0 & \tilde{h}_{soc}\\
\tilde{h}_{soc}^{\dag} & 0
\end{array}
\right] , \label{Hsoc_eff}$$$$\tilde{h}_{soc}\propto\lambda_{soc}[H_{c}^{\dag}L^{-}+L^{-}H_{c}].
\label{Hsoc_eff1}$$ Here, $\lambda_{soc}$ is the strength of spin-orbit coupling. $L^{-}$ is the matrix of $d$ orbitals. $H_{c}$ is the interlayer parity-mixing term. Along the $\Gamma-Z$ line, $(k_{x},k_{y})=(0,0)$, we have $$\tilde{h}_{soc}\propto it_{xz,yz}^{c}\lambda_{soc}\sin k_{z}\left[
\begin{array}
[c]{ccccc}0 & 0 & -i & 1 & 0\\
0 & 0 & -1 & -i & 0\\
-i & -1 & 0 & 0 & -\sqrt{3}\\
1 & -i & 0 & 0 & \sqrt{3}i\\
0 & 0 & -\sqrt{3} & \sqrt{3}i & 0
\end{array}
\right] . \label{Hsoc_eff2}$$ Based on the information from the tight-binding Hamiltonian, the effective **k**$\cdot$**p** Hamiltonian around the $\Gamma-Z$ line can be constructed under the basis spanned by the states $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$ in Fig. \[fig2\](i)[@Xu_prl_2016]. The detailed form of the effective **k**$\cdot$**p** Hamiltonian can be constructed in the basis set, $[\{|\psi_{\uparrow}\rangle\},\{|\psi
_{\downarrow}\rangle\}]$ with $\{|\psi_{\uparrow}\rangle\}=\{|d_{e,xy,\uparrow
}\rangle,|d_{o},1,\frac{3}{2}\rangle,|d_{o},-1,-\frac{1}{2}\rangle
,|d_{o,xy,\uparrow}\rangle,\}$ and $\{|\psi_{\downarrow}\rangle
\}=\{|d_{e,xy,\downarrow}\rangle,|d_{o},1,\frac{1}{2}\rangle,|d_{o},-1,-\frac{3}{2}\rangle,|d_{o,xy,\downarrow}\rangle\},$ $$\begin{aligned}
H_{\Gamma Z}(k) & =\left[
\begin{array}
[c]{cccc}M_{1}(k) & \gamma\sin k_{z}k_{-} & \gamma\sin k_{z}k_{+} & 0\\
& M_{2}(k) & \alpha k_{+}^{2}+\beta k_{-}^{2} & i\delta k_{-}\\
& & M_{2}(k) & i\delta k_{+}\\
& & & M_{4}(k)
\end{array}
\right] \otimes I_{2\times2}\nonumber\\
& +H_{\Gamma Z}^{soc}(k) \label{Hgz}$$ Here, the mass terms $M_{n}(k)=E_{n}+\frac{k_{\parallel}^{2}}{2m_{nx}}+t_{nz}(1-\cos k_{z})$ with $n=1$, 2, 4. $H_{\Gamma Z}^{soc}$ are some components of $\tilde{h}_{soc}$ in Eq. (\[Hsoc\_eff2\]) and have the following form, $H_{\Gamma Z}^{soc}(k)=[h_{11},h_{12};h_{12}^{\ast},-h11]$ with $h_{11}=\frac{\lambda_{soc}}{2}[(\sigma_{z}-1)\oplus(\sigma_{z}+1)]$, $h_{12}=\frac{\sqrt{2}\lambda_{soc}}{2}[i\sigma_{x}-\sigma_{y},(1-\sigma
_{z})k_{z};1+\sigma_{z},(i\sigma_{x}+\sigma_{y})k_{z}]$.
Note that the band 4 and band 2 cross along $\Gamma-Z$ line without gap opening in Fig. \[fig2\] (i). Actually, this can be called topological Dirac semimetal states when the chemical potential is moved to the cross point. This state can also be described by the effective model in Eq. (\[Hgz\]). The target materials include Fe(Te,Se) and Li(Fe,Co)As[@Zhang_arxiv_2018; @Zhang_arxiv_2018-1].
Materials and experiments
-------------------------
{width="1.0\linewidth"}
The three typical materials to realize the aforementioned topological quantum states of matter described by the three effective **k**$\cdot$**p** Hamiltonian are monolayer FeSe/STO, monolayer FeTe$_{1-x}$Se$_{x}$/STO and FeTeSe single crystal. To experimentally identify these topological states, scanning tunneling microscopy/spectroscopy (STM/S) and ARPES are very powerful tools. STM/S is a real space surface measurement technique that measures the density of states as a function of position, and can be used to distinguish the edge states from bulk states[@Yang_prl_2012; @Drozdov_np_2014]. ARPES is a momentum space measurement technique that can directly read out the band structure, and can be used to evaluate the band evolution. The experimental results from STM/S and ARPES for these three materials are summarized in Fig. \[fig2\].
For monolayer FeSe/STO, the idea is based on comparing the gap (band gap and superconducting gap) from dI/dV of STS with the energy distribution curve (EDC) of ARPES in Fig. \[fig2\] (a) to determine the bulk gap. Then, the topological states possess the edge states, which cross the bulk band gap and are different from the trivial normal chemical edge states[@Wang_nm_2016-1]. The contribution to the density of states from the topological edge states can be extracted by comparing the STS spectra between the bulk regime and the edge regime. Fig. \[fig2\] (c) and (d) are theoretical and experimental results, respectively. The key experimental observations are shown in Fig. \[fig2\] (c), from which, one can find that there exists some additional states from the edges after subtracting the contributions from the bulk background. However, only this feature is not enough to prove the nontrivial characteristics of the edge states. The trivial edge states can also have similar dI/dV behaviors. In Ref.[@Wang_nm_2016-1], the checkerboard antiferromagnetic order is assumed to exist to open a trivial gap around $M$ point in the monolayer FeSe/SrTO. In Ref.[@Hao_prx_2014], the trivial band gap at $M$ point is natural by taking into account the tension from SrTiO$_{3}$ substrate. Furthermore, the coexistence of antiferromagnetic order and superconducting order is doubtable in monolayer FeSe/STO, because the gap from the antiferromagnetic order is about 50meV, which should be easily to detect. For example, the gap should disappear above the antiferromagnetic transition temperature T$_{N}$. Thus, the nontrivial characteristics of the edge states should be further tested by other experimental method such as the spin-resolved STM or nonlocal transport[@Roth_science_2009].
In Fe(Te,Se) thin films, the topological phase transition appears when increasing the Te substitution of Se. Pictorial band evolution as change as Te substitution is a promising evidence to testify the topological phase transition in Fe(Te,Se) thin film. Therefore, ARPES experiment is the primary choice. Fig. \[fig2\] (h) summarizes the band dispersions at $\Gamma$ point for the samples with different $x$. The experiment results show that a down-shifting electron-like band move towards the hole-like and the band gap between them decreases rapidly when the Se content remains shrinks. Eventually the bands touch each other at a Se concentration of approximately 33%, which is further revealed in the plots of the constant energy contours and momentum distribution curves, as shown in Fig. \[fig2\] (e)-(g). The touch point corresponds to the critical point of band inversion. The ARPES experimental results give the indirect evidence for the topological band structure in monolayer FeTe$_{1-x}$Se$_{x}$/STO[@Shi_sb_2017].
For the bulk FeTe$_{1-x}$Se$_{x}$ single crystal, the emergence of electron band No. 4 in Fig. \[fig2\] (i) is the key ingredient to produce topological states when increasing the Te substitution of Se. The early ARPES experiment proved its existence through introducing the electron doping with $in$ $situ$ K evaporation[@Wang_prb_2015]. The newly high energy and momentum resolution ARPES (HR-ARPES) (Energy resolution$\sim$70 $\mu$eV) and the spin-resolved ARPES (SARPES) (Energy resolution$\sim$1.7 meV) provide powerful tools to directly observe the topological surface states and their spin polarization. Fig. \[fig2\] (j) and (k) clearly demonstrate the topological surface states with Dirac cone structure. Fig. \[fig2\] (n)-(q) identify the helical spin structure of the topological surface states. The combination of HR-ARPES and SARPES results directly proved the topological band structure in the bulk FeTe$_{1-x}$Se$_{x}$ single crystal[@Zhang_science_2018]. Recently, the similar topological band structure has also been identified in Li(Fe,Co)As[@Zhang_arxiv_2018-1], which not only confirms theoretical predictions but also proves the generic existence of tunable topological states in iron-based superconductors.
Connate topological superconductivity
=====================================
Material proposals
------------------
![(Color Online) Schematic illustrations of three kinds of strategies to realize topological superconducting states. (a) The hetero-structure involving conventional s-wave superconductor and topological insulator film. (b) The iron-based superconductors with topological surface states. (c) The unconventional superconductors with odd-parity pairing, i.e., the spin-polarized $p+ip$ pairing here. []{data-label="fig3"}](fig3.eps){width="1.0\linewidth"}
As we have mentioned in the introduction, a standard topological superconductor requires an odd-parity pairing, as shown in Fig. \[fig3\] (c). The famous representative materials including Sr$_{2}$RuO$_{4}$[@Mackenzie_rmp_2003] and doped topological insulators Cu$_{x}$Bi$_{2}$Se$_{3}$ and Sr$_{x}$Bi$_{2}$Se$_{3}$[@Fu_prl_2010; @Hor_prl_2010; @Kriener_prl_2011; @Sasaki_prl_2011; @Levy_prl_2013; @Mizushima_prb_2014; @Lahoud_prb_2013; @Liu_jacs_2015] are proposed to be potential topological superconductors. However, the experimental situation is far from definitive, because the odd-parity pairing imposes restrictions to the pairing in spin-triplet channel, which is very rare in solid-state materials. Therefore, the recent research mainly focuses on some artificial structures which use the proximity effect from conventional superconductors on the surface/edge states of the three/two-dimensional topological insulator, on semiconductor film/nanowire with strong Rashba spin-orbit coupling, and on iron atom chain[@Fu_prl_2008; @Sau_prl_2010; @Lutchyn_prl_2010; @Alicea_prb_2010; @Mourik_science_2012; @Nadj_science_2014; @Albrecht_nature_2016; @Xu_np_2014; @Xu_prl_2015; @Sun_prl_2016], as shown in Fig. \[fig3\] (a). Effectively, the model described the structure in Fig. \[fig3\] (a) eventually reduce into the simpler model in Fig. \[fig3\] (c). The ultra-low superconducting transition temperature and the uncontrollability and uncertainty induced by the mismatch between different materials in the artificial structures take many undetermined problems and make these structures far beyond practicability[@Xu_prl_2015; @Sun_prl_2016].
The superconductivity in iron-based superconductors is very robust against the fine tuning the band structures. Furthermore, the aforementioned topological phase transitions around $\Gamma$, $M$ and $\Gamma$-$Z$ line have no overall band gap because the iron-based superconductors are multi-orbital type and there exist other trivial bands across the Fermi energy besides the topological bands. When the temperature decreases below the superconducting transition temperature, the trivial bands across the Fermi energy open a superconducting gap due the formation of the cooper pairs. At the boundaries of the materials, the topological bands support the surface/edge states, which also cross the Fermi energy. In comparison with trivial or extrinsic proximity effect involving two different kinds of materials in Fig. \[fig3\] (a), the inducing superconductivity from trivial bulk bands to topological boundary bands happens in a single material, and can also be called intrinsic or self-proximity effect, as shown in Fig. \[fig3\] (b). When the Fermi energy is close to the surface Dirac point to guarantee the good approximation of the linear dispersion of the surface Dirac band, the superconducting single Dirac band can be reduced into a spinless $p_{x}+ip_{y}$ superconductor[@Fu_prl_2008; @Xu_prl_2016-1], which is a topological superconductor, as shown in Fig. \[fig3\] (c). When the $\pi$-flux vortex is formed in the magnetic field, the effective topological superconductor can support the zero-energy vortex-line end states, which are called Majorana modes.
Keeping the aforementioned picture in mind, one can find that all iron-based superconductors with topological band structures can support topological superconductors. For the monolayer FeSe/STO, the heavy hole-doped case can support the topological edge states while the electron-doped case can support extremely high-temperature superconductivity. Then, the boundary between the hole-doped and electron-doped regimes in a single monolayer sample can produce one-dimensional topological superconductor. For monolayer FeTe$_{1-x}$Se$_{x}$/STO, the superconductivity is robust in the whole doping regime[@Shi_sb_2017]. The topological edge states emerge when $x<0.33$, and the cooper pairs from the electron bands near $M$ point can be scattered into the topological edge states from topological bands near $\Gamma$ point. Then, the system spontaneously transform into the topological superconductor. For (Ca,Pr)FeAs$_{2}$ and Ca$_{1-x}$La$_{x}$FeAs$_{2}$, the distorted As chains in CaAs layers support topological edge states through the topological bands near $B$ points, while the FeAs layers support superconductivity through the trivial bulk band near both $\Gamma$ and $M$ points. The self-proximity effect can induce the one-dimensional topological superconductivity in both (Ca,Pr)FeAs$_{2}$ and Ca$_{1-x}$La$_{x}$FeAs$_{2}$[@Wu_prb_2014-1; @Wu_prb_2015-1]. For bulk FeTe$_{1-x}$Se$_{x}$ single crystal, the topological Dirac-cone type surface states emerge at $\bar
{\Gamma}$ point in the (001) surface Brillouin zone in the topological doped regime. Then, the cooper pairs from the trivial bulk bands near $\Gamma$-$Z$ line and the $M$-$A$ line can be scattered into the topological Dirac-cone type surface states. These primary and secondary self-proximity effect can drive the bulk FeTe$_{1-x}$Se$_{x}$ single crystal into the two-dimensional topological superconductor.
Experiments and open questions
------------------------------
For the monolayer FeSe/STO and FeTe$_{1-x}$Se$_{x}$/STO, the monolayer FeSe and FeTe$_{1-x}$Se$_{x}$ grow on the substrate STO through the assistant of molecular beam epitaxy (MBE). Until now, both systems have the highest superconducting transition temperature among all iron-based superconductors, whereas they are unstable in the air. The shortcoming takes challenges to the devices fabrication and relevant transport measurement. On a contrary, the bulk FeTe$_{1-x}$Se$_{x}$ single crystal is quite stable and has nice (001) cleavage surface. More importantly, the topological superconducting states are two-dimensional. The spontaneously generated vortex under external magnetic field could bound Majorana zero-energy mode if the superconducting state is topological. Then, some experimental methods like ARPES and STM/S can be used to verify the topological superconducting state and detect the Majorana zero-energy modes. Based on these upsides, most experimental progresses are mainly made in the bulk FeTe$_{1-x}$Se$_{x}$ single crystal and (Li$_{0.84}$Fe$_{0.16}$)OHFeSe single crystal[@Yin_np_2015; @Zhang_science_2018; @Wang_science_2018; @Chen_nc_2018; @Liu_arxiv_2018]. Along time line, we review these experiments in the following.
{width="1.0\linewidth"}
The first unexpected experiment is about the impurity bound states in FeTe$_{0.57}$Se$_{0.43}$ single crystals[@Yin_np_2015]. FeTe$_{0.57}$Se$_{0.43}$ single crystals contain a large amount of excess iron that as single iron atoms randomly situate at the interstitial sites between two (Te, Se) atomic planes[@Taen_prb_2009]. The STM/S spectrum observed a strong zero-energy bound state at the centre of the single interstitial Fe impurity. The experimental results are summarized in Fig. \[fig4\] (a)-(f). The zero-energy bound state has the following features. (1) The spatial pattern of the zero-energy bound state is almost circular, which is different from the cross-shape pattern of the Zn impurity in Bi$_{2}$Sr$_{2}$Ca(Cu,Zn)$_{2}$O$_{8+\delta}$[@Pan_nature_2000]. (2) The intensity of the zero-energy bound state exponentially decays with a characteristic length of $\xi
=3.5$Å, which is almost one order of magnitude smaller than the typical coherent length of $25$Å in the iron-based superconductor[@Yin_prl_2009; @Shan_np_2011]. (3) The bound state is strictly at zero even the external magnetic field increases to 8T. (4) The zero-energy bound state peak remains at zero energy even when two interstitial Fe impurity atoms are located near each other ($\sim15$Å). It is a serious challenge to consistently explain these features of the zero-energy bound state induced by interstitial Fe impurity. The d-wave pairing symmetry scenario can result in a zero-energy bound state at unitary limit[@Balatsky_rmp_2006], but violates feature (1). The Kondo impurity resonance scenario can give an accidental zero-energy bound state[@Balatsky_rmp_2006], but violates feature (4). A fascinating scenario is that the mode is Majorana zero-energy mode[@Read_prb_2000; @Kitaev_pu_2001], which captures features (1)-(3). Recently, a theoretical work claimed that an interstitial Fe impurity could bound an quantum anomalous vortex without magnetic field, and the quantum anomalous vortex can bound a Majorana zero-energy mode when topological surface states of FeTe$_{0.57}$Se$_{0.43}$ become superconducting[@Jiang_arxiv_2018]. However, it is still hard to explain feature (4) by the Majorana zero-energy mode scenario. Until now, the origin of the zero-energy bound state trapped by interstitial Fe impurity is still underdetermined. Topological or other reasons need further experimental and theoretical explorations.
The second experimental breakthrough is about the vortex bound states on the surface of FeTe$_{0.55}$Se$_{0.45}$ single crystals[@Wang_science_2018; @Chen_nc_2018]. FeTe$_{0.55}$Se$_{0.45}$ belongs to type II superconductor. Once a small external magnetic field is applies along c-axis, magnetic vortex structures are formed due to the small lower critical field $H_{c1}$. The high-resolution STM/S can measure the bound states trapped by the vortex. Two experimental group claimed completely different results for the same material FeTe$_{0.55}$Se$_{0.45}$ single crystals. The former group claimed that they observed a sharp zero-bias peak inside a vortex core that does not split when moving away from the vortex center, which could be attributed to the nearly pure Majorana bound state[@Wang_science_2018]. The experimental results are summarized in Fig. \[fig4\] (g)-(j). The vortex bound states exhibit the following features. (1) Statistically, there are about 20% success rate in observing the isolated pure Majorana bound state during more than 150 measurements. (2) Across a large range of magnetic fields the observed zero-bias peak does not split when moving away from a vortex center. (3) Most of the observed zero-bias peak vanish around 3K. (4) Robust zero-bias peaks can be observed over two orders of magnitude in tunneling barrier conductance, with the width barely changing. Feature (1) is argued to attribute to the disorder effect and/or inhomogeneous distribution of Te/Se. Feature (2) is attributed to the large $\Delta
_{sc}/E_{F}$ ratio in this system. Feature (3) is attributed to that the Caroli-de-Gennes-Matricon (CdGM) state[@Caroli_pl_1964] is protected by a mini-energy gap with with a temperature about $\Delta_{sc}^{2}/E_{F}\sim3$K, and the thermal excitation around and beyond 3K can kill the CdGM state. Feature (4) indicated the line width of zero-bias peaks is almost completely limited by the combined broadening of energy resolution and STM thermal effect, suggesting that the intrinsic width of the Majorana bound state is much smaller in the weak tunnelling regime[@Setiawan_prb_2017; @Colbert_prb_2014]. The detailed experimental measurements eliminate some scenarios to cause a zero-bias peak in tunneling experiments, such as antilocalization, reflectionless tunneling, Kondo effect, Josephson supercurrent and packed CdGM states near zero energy[@Pikulin_njp_2012; @Bagrets_prl_2012; @Wees_prl_1992; @Eduardo_prl_2012; @Churchill_prb_2013; @Levy_prl_2013; @Hess_prl_1990; @Gygi_prb_1990]. Features (2)-(4) can be well understood with the Majorana bound state scenario, it is probable that the observed zero-bias peaks correspond to Majorana bound state. However, the feature (1) is a serious problem, which is different from other proposals to realize Majorana bound states. In the present experiments, it seems no comprehensive evidences of the disorder effect and/or influence of inhomogeneous distribution of Te/Se are provided. Furthermore, if the observed zero-bias peaks are from Majorana bound states, the non-Abelian statistics can be demonstrated by move a vortex with a STM tip. This kind of experiment is the smoking gun for Majorana modes. Another experimental group claimed that they only observed the trivial CdGM bound state trapped by vortex in the same FeTe$_{0.55}$Se$_{0.45}$ single crystals. For statistics, the energies of bound state peaks close to the zero bias are collected from all measured nine vortices presented[@Chen_nc_2018]. The experimental results are summarized in Fig. \[fig4\] (k)-(l). In principle, there should be a special vortex to bound the zero-bias peak according to the 20% success rate claimed in the former experiment. Unfortunately, two experiments for the same material from two groups give the inconsistent results[@Wang_science_2018; @Chen_nc_2018]. The argument about the difference being attributed to the different annealed process is not very convincing. It seems that the appearance of zero-bias peaks is selective. The behaviors challenge the topological origin, which is usually universal and robust.
The third subsequent experiment is about the vortex bound states on the FeSe cleavage plane of (Li$_{0.84}$Fe$_{0.16}$)OHFeSe single crystal[@Liu_arxiv_2018]. In compared with FeTe$_{0.55}$Se$_{0.45}$, the superconducting FeSe layers in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe are stoichiometric. Therefore, there exist defect-free areas, which support the un-pinned or free vortex. The STM/S measurements show that (1) The free vortex cores bound zero-bias modes, which do not shift with varying underlying superconducting gap as the other peaks do. (2) the zero-bias modes survive to high magnetic field due to the short coherence length. (3) The zero-bias mode coexists with other low-lying CdGM states but separates from each other. These features are similar to those of the zero-bias modes observed in FeTe$_{0.55}$Se$_{0.45}$. Therefore, the zero-bias modes can be also attributed to Majorana zero-energy modes, and can be argued to have topological origin in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe. However, the topological origin in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe is underdetermined, unlike FeTe$_{0.55}$Se$_{0.45}$ with solid experimental evidences for the topological band structure. Recall the discussions about the band inversion along $\Gamma$-$Z$ line in FeTe$_{0.55}$Se$_{0.45}$ in Section II A, the strong dispersion of band 4 in Fig. \[fig2\] (j) benefits from the quite small layer distance and large size of Te atoms. The band 4 in pure FeSe is flat[@Wang_prb_2015]. It is very strange that the band 4 in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe has strong dispersion. Furthermore, the band gap open is due to the strong spin-orbit coupling from Te atom not Se atom. Another critical condition to obtain the topological surface states is that the chemical potential must properly lie in the quite small band gap. However, the chemical potential in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe is far from the band gap. In this situation, the top and bottom surfaces start to communicate with each other and break the zero-bias mode. At last, it lacks the smoking gun ARPES experiment to prove the helical structure of the claimed observed topological surface states in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe. In summary, the experimental observations of the zero-bias modes in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe is more clear, but the topological origin needs to be understood.
Summary and Perspectives
========================
The discovery of topological insulators has established a standard paradigm to guide the communities to pursue the topological states of matter in quantum materials. Such pursuits cause intersections between the topology and iron-based superconductors. As emphasized in this review, important principles for the theoretical understandings of the energy-band topology in new materials include applying general concepts with the help of symmetry analysis and constructing the effective models. For the iron-based superconductors, the multi-orbital band structures and the diversity of materials provide opportunities to realize the effective theoretical models. These topological materials include monolayer FeSe/STO, monolayer Fee$_{1-x}$Se$_{x}$/STO, FeTe$_{1-x}$Se$_{x}$, and LiFe$_{1-x}$Co$_{x}$As etc.
In the superconducting states, it is naturally expected to obtain the topological superconducting states with the help of self-proximity effect. However, different from the energy-band topology, the expected topological superconducting states exhibit many unexpected experimental phenomena, including the surprising robust zero-energy mode trapped by Fe impurity in FeTe$_{0.57}$Se$_{0.43}$, the selective appearance of zero-bias mode trapped by vortex in FeTe$_{0.57}$Se$_{0.43}$, and the coexistence of zero-bias mode and CdGM states trapped by free vortex in (Li$_{0.84}$Fe$_{0.16}$)OHFeSe. Even if all these phenomena are attributed to Majorana zero-energy modes, there are deep inconsistencies within different experiments as well as between experiments and theories. In this respect, clarifying the creation mechanism of these so-called Majorana zero-energy modes are worth pursuing. For such efforts, the availability of the high-quality single crystal, whose chemical potential can be artificially fine tuned, would be crucial. Once the physics of the so-called Majorana zero-energy modes are clarified, finding ways to manipulate the non-Abelian statistics of the Majorana zero-energy modes is a significant challenge for future applications in quantum computing.
Iron-based superconductor owns a rich phase diagram. Beside the normal and superconducting phases, there include nematic phase, orbital ordering phase and various antiferromagnetic phases. Searching the topology embedded in these ordered phases would be interesting. For the theoretical aspect, there have been some studies[@Wu_arxiv_2016; @Hao_prb_2017], but the experimental exploration is blank. In future, a stronger collaboration between theory and experiment is required to explore topological quantum states in new materials of iron-based superconductors.
Finally, it can not be entirely ruled out that the superconducting states of iron-based superconductors themselves could be highly unconventional. In the ten years of the research of iron-based superconductors, there still are many unsolved puzzles[@Hirschfeld_rpp_2011; @Chubukov_arxiv_2014] observed by a variety of different experimental methods, such as transport, Raman spectrum, neutron scattering, nuclear magnetic resonance, electron spin resonance, STM/S, and ARPES etc. For example, the interplay between spin, orbital, lattice and charge degrees of freedom is not fully understood, not only is there no smoking gun proof for the s$_{\pm}$ pairing yet but also it is clear that the s$_{\pm}$ pairing symmetry cannot be valid for many iron-chalcogenide systems, whether there is a sign change in the superconducting states of iron-chalcogenide systems without hole pockets or not are highly debated, and the origin of the enhancement of the transition temperature found in single layer FeSe remains to be understood. The topological exploration in iron-based superconductors may help us to discover surprising characters and mechanism hidden behind superconducting pairing, and leads to answers to these unsolved puzzles.
N. Hao thanks the KITS, UCAS for the hospitality during his visit. This work was supported by the National Key R&D Program of China (No.2015CB921300, No.2017YFA0303100, 2017YFA0303201),National Natural Science Foundation of China (No. 11674331, No. 11334012), the ‘100 Talents Project’ of Chinese Academy of Sciences, and the Strategic Priority Research Program of CAS (No.XDB07000000).
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---
abstract: 'We present observations of the center of the Galactic globular cluster NGC 6273, obtained with the HST/WFPC2 camera as part of the snapshot program GO-7470. A $B$, $V$ color–magnitude diagram (CMD) for $\sim$28,000 stars is presented and discussed. The most prominent feature of the CMD, identified for the first time in this paper, is the extended horizontal branch blue tail (EBT) with a clear double-peaked distribution and a significant gap. The EBT of NGC 6273 is compared with the EBTs of seven other globular clusters for which we have a CMD in the same photometric system. From this comparison we conclude that all the globular clusters in our sample with an EBT show at least one gap along the HB, which could have similar origins. A comparison with theoretical models suggests that at least some of these gaps may be occuring at a particular value of the stellar mass, common to a number of different clusters. From the CMD of NGC 6273 we obtain a distance modulus $(m-M)_V=16.27\pm0.20$. We also estimate an average reddening E($B-V$)=0.47$\pm$0.03, though the CMD is strongly affected by differential reddening, with the relative reddening spanning a $\Delta E(B-V)\sim 0.2$ magnitude in the WFPC2 field. A luminosity function for the evolved stars in NGC 6273 is also presented and compared with the most recent evolutionary models.'
author:
- 'G. Piotto, M. Zoccali, I. R. King, S. G. Djorgovski, C. Sosin, R. M. Rich, and G. Meylan'
title: 'HST Observations of Galactic Globular Cluster Cores. II. NGC 6273 and the Problem of Horizontal-Branch Gaps.'
---
\#1[ ]{} å[A&A]{}
6[$V_{606}$]{} 5[$V_{555}$]{} 8[$I_{814}$]{}
Introduction {#intro}
============
A growing number of Galactic globular clusters (GGC) is found to show extended-horizontal-branch blue tails (EBT), which sometimes reach the He-burning main sequence, indicating that some of the stars must have lost (almost) all of their envelope during the red giant branch (RGB) phase. The EBTs represent a puzzle in the stellar evolution models. In many GGCs there is another peculiarity: a discontinuity in the stellar distribution along the EBT, which sometimes results in a sort of gap, i.e., a region clearly underpopulated in stars (Sosin et al.1997a, Catelan et al. 1998). No clear explanation for the origin of the gaps is available at present. These abnormalities in the HB probably represent one of the most extreme of the mixed bag of anomalies that are sometimes lumped under the term “second-parameter problem” (cf. Ferraro et al. 1998).
In this paper, we present a new cluster with both an EBT and a significant gap on it: NGC 6273. This object is one of the 46 clusters we are hoping to observe within our ongoing snapshot programs. NGC 6273 (M19) ($\alpha_{1950}=16^{{\rm h}}59^{{\rm m}},
\delta_{1950}=-26^\circ11'$) is a medium-concentration ($c=1.5$, Djorgovski 1993), intermediate-metallicity (\[Fe/H\] $=-1.68$, Zinn & West 1984) cluster, located toward the direction of the Galactic bulge ($l=357^\circ$, $b=+9^\circ$). NGC 6273 is the second most luminous cluster, after $\omega$ Cen, in the Djorgovski (1993) compilation. In a photographic image NGC 6273 looks very similar to $\omega$ Cen. The other noteworthy property of NGC 6273 is its high ellipticity, $\epsilon=0.28$, which makes M19 the most flattened of the GGC in the White & Shawl (1987) catalogue, and probably in the entire Galaxy. The only available color–magnitude diagram (CMD) is by Harris, Racine, & De Roux (1976), and it barely reaches the HB level. The CMD looks quite dispersed due to field contamination and differential reddening, estimated by the same authors to be around 0.2 magnitude in $B-V$. Here we present a new CMD, extending to $\sim$2 magnitudes below the turnoff (TO), and corrected for differential reddening..
Observations and Analysis {#obs}
=========================
The center of the GGC NGC 6273 was observed with the WFPC2 camera through the filters F439W and F555W ($=$ $B,V).$ A series of short and long exposures was secured in order to cover a large magnitude range (see Table 1 for a detailed log of the observations). The longer F439W exposures were repeated in order to allow removal of cosmic rays. In both cases, the cluster center was placed in the PC field.
The images were pre-processed by the standard pipeline. Following the procedures described by Silbermann et al. (1996), we masked out the vignetted pixels and bad pixels and columns using a vignetting mask created by P. B. Stetson, together with the appropriate data-quality file for each frame. We have also multiplied each frame by a pixel area map (also provided by P. B. Stetson) in order to correct for the geometric distortion (Silbermann et al.1996). The saturated pixels of the brightest stars in the longest exposures were corrected as in Cool, Piotto, & King (1996), using the shorter exposures.
The photometric reduction was carried out following the same procedure that is described in detail in Piotto et al. (1999a) for a similar set of images acquired within the same HST program. The raw instrumental magnitudes were transformed into the standard Johnson $BV$ system using Eq. (8) in Holtzman et al. (1995) and the coefficents in their Table 7.
As this paper makes extensive use of star counts, particular attention was devoted to estimating the completeness in all the principal branches of the CMD. The completeness corrections were determined by artificial-star experiments, again as described in Piotto et al. (1999a).
Color–Magnitude Diagrams {#cmds}
========================
The CMD for $\sim$28,000 stars in NGC 6273 is shown in Fig.\[cmdor\]. The diagram extends from almost the tip of the giant branch (stars brighter than $V \approx 15$ are saturated even in the shortest-exposure frames) to $\sim$2 magnitudes below the TO. All the relevant branches of the CMD are well defined. As in all the CMDs of the GGC cores, a well defined sequence of blue stragglers is clearly seen, extending from the TO up to the HB. The most prominent features in Fig. \[cmdor\] are the extended blue HB and the striking gap centered at $V\sim19.5$. Neither of these features was reached by the previously published CMD.
Reddening {#red}
---------
All the sequences in Fig. \[cmdor\] are broadened. The broadening is much larger than the typical photometric errors, at all magnitudes. According to the artificial-star experiments, the dispersion in color due to photometric errors along the RGB (for $17<V<19$) is $\Delta
(B-V)=0.03$ (standard deviation around the mean value). In view of the position of NGC 6273 ($l=357^\circ$, $b=+9^\circ$), we can expect some differential reddening. In order to test this hypothesis, we divided our field into 86 $15\times15$ arcsec$^2$ regions (Fig.\[rgrid\]). The dimensions of the regions are a compromise between the need to have the highest spatial resolution and the need to have a sufficient number of stars to be able to identify the giant-branch position. For each sub-image we obtained a CMD. We selected one of these CMDs as the reference CMD, and extracted its fiducial points by drawing a line by hand. In Fig. \[rgrid\] the CMDs for each region are compared with the fiducial line obtained by fitting a spline to the fiducial points. The effects of the differential reddening are clearly visible. For each region the reddening relative to the fiducial sequence is indicated. The relative reddening has been calculated as follows. In each spatial region, for each star with $15<V<20$ and $(B-V)>0.6$, we calculated its distance from the fiducial CMD along the reddening line \[defined by the relation $V=3.2(B-V)$\]. Each one of these distances is the resultant of two components: $E(B-V)$ on the abscissa and $A_V$ on the ordinate. The relative reddening is the median value of the abscissas of all the distances. Fig. \[rgrid\] shows a radial trend of the relative reddening, with the central region less reddened. The relative reddening spans a $\Delta E(B-V)\sim 0.2$ magnitude in the WFPC2 field, confirming the results by Harris et al. (1976). The standard deviation around the average reddening \[the mode of the reddening distribution for the WFPC2 field is $E(B-V)=0.41$\] is 0.05 magnitude. We also determined the average reddening of NGC 6273 by comparing the location of its EBT with the EBTs of two other clusters: NGC 1904 (Sosin et al. 1997b, Piotto et al. 1999b) and NGC 6205 (M13, Zoccali & Piotto 1999). The turning down of the blue part of the HB is due to the saturation of the $B-V$ color as the bolometric correction increases sharply at $T_{\rm eff}\sim 10,000$ K. It is thus not very sensitive to metallicity or age. We selected these two clusters for the comparison because they show the same peculiar HB as NGC 6273. Besides, in Zinn & West (1984) the three clusters have the same metallicity (formally within 0.03 dex). The data of NGC 1904 and NGC 6205 are in the same HST photometric system as NGC 6273. By comparing the HBs, we find that NGC 6273 is $0.45\pm0.02$ magnitude redder than NGC 1904 and $0.47\pm0.02$ magnitude redder than NGC 6205. Assuming $E(B-V)=0.01$ for NGC 1904 (Stetson & Harris 1977) and $E(B-V)=0.02$ for NGC 6205 (Djorgovski 1993), we have a mean reddening for the reference fiducial points of Fig. \[rgrid\] $E(B-V)=0.47\pm0.03$ for our field in NGC 6273.
The reddening estimates in Fig. \[rgrid\] can be used to (partially) correct the CMD of Fig. \[cmdor\] for the effects of differential reddening. To the CMDs in each of the regions shown in Fig. \[rgrid\] we applied the relative reddening values indicated, and corrected the color and the magnitude of each star for reddening and the corresponding absorption $A_V=3.2E(B-V)$. The resulting CMD for the entire WFPC2 field is shown in Fig. \[cmdco\]. All the sequences in the CMD of Fig. \[cmdco\] are sensibly narrower than in Fig. \[cmdor\], but still broader than expected from the artificial-star experiments. There are two possible explanations: [*i)*]{} there might be some differential reddening on spatial scales smaller than $15\times15$ arcsec$^2$; [*ii)*]{} there might be some dispersion in the metal content of the stars in NGC 6273. Though the first hypothesis seems the more likely, nevertheless in view of the similarity of NGC 6273 to $\omega$ Cen discussed in Section \[intro\], a study of the metallicity of its stars would be worth doing. The spread in the combined equivalent width of the lines in the Ca II triplet ($\Sigma Ca$) in Fig. 7 of Rutledge et al. (1997) might be an indication of a possible metallicity dispersion among the NGC 6273 stars, but the number of measured stars is too small to allow any conclusion.
Distance {#dist}
--------
NGC 6273 has a blue HB without a horizontal portion, making the distance determination quite uncertain. So far, only three RR Lyrae have been identified in the direction of NGC 6273 (Clement and Sawyer Hogg 1978). Two of them fall within the RR Lyrae instability strip of NGC 6273, indicating possible cluster membership. The third one is 1.5 mag. brighter than the HB, and Clement and Sawyer Hogg (1978) consider it to be a non-member.
A more accurate distance estimate comes again from a comparison with NGC 1904 and NGC 6205. In particular, the large statistical sample of stars in the CMD of NGC 6273 and of the reference clusters allows us to derive an accurate luminosity function (LF), where the red-giant bump (Iben 1968) can be easily identified at $V_{\rm
bump}=16.80\pm0.10$. In view of the similarity in metallicity (and the similarity of the CMDs) we expect the bump to be located at the same absolute magnitude in the three clusters. From the LFs of NGC 1904 and NGC 6205 (Zoccali & Piotto 1999) we measured $V_{\rm
bump}=16.00\pm0.10$ for NGC 1904 and $V_{\rm bump}=14.85\pm0.10$ for NGC 6205. Adopting an apparent distance modulus ($m-M$)$_V=15.45$ (Ferraro et al. 1992) for NGC 1904 and ($m-M$)$_V=14.35$ (Ferraro et al. 1997) for NGC 6205, we obtain for NGC 6273 ($m-M$)$_V=16.25$ and ($m-M$)$_V=16.30$, respectively. From here on, we will adopt for NGC 6273 ($m-M$)$_V=16.27\pm0.20$. As there is no error associated with the distance determination of NGC 1904 and NGC 6205, we have assumed a typical error of 0.2 magnitude, though the method would allow relative distance determination with sensibly smaller uncertainties.
Assuming $E(B-V)=0.47$, and adopting $A_V=3.2E(B-V)$, the absolute distance modulus of NGC 6273 is $(m-M)_0=14.77\pm0.20$, i.e., NGC 6273 is at $d_\odot=9.0$ kpc from the Sun, at $Z=1.4$ kpc from the Galactic plane.
The distribution along the HB {#HB}
=============================
Certainly, the most prominent features of the CMDs of both Fig.\[cmdor\] and Fig. \[cmdco\] are the extended blue tail of the HB and the gap along it. As discussed in Section \[intro\], there are other clusters which have HBs apparently similar to that of NGC 6273. One question that is still open is whether these features originate from the same (though still unknown) physical process. Certainly, the lack of clear observational inputs makes more difficult the physical interpretation of both the EBT and the gaps. There are two problems which complicate the analysis of the presently available data:\
[*i)*]{} the heterogeneity of the data gathered so far, in particular the different photometric bands used;\
[*ii)*]{} the shape of the HB, particularly complicated in the $V$, $B-V$ plane because of the saturation of the $B-V$ color as a temperature index when $B-V<0$.
In order to try to shed some light on this problem, we have collected from the HST archive all the public WFPC2 images of GGC cores observed with the same filters (F439W and F555W) that we used in our previous HST GO-6095 program, and that we are still using in the presently running GO-7470 and will use in GO-8118. At the present time, we have CMDs for 29 GGCs in this homogeneous data set. Among these, at least eight GGCs show blue tails with more or less pronounced gaps.
A detailed comparison of the positions of peaks and gaps as a function of stellar mass or temperature for different theoretical models is beyond the scope of the present paper, and will be addressed elsewhere. On a more empirical basis, as originally suggested by Rood & Crocker (1985, 1989), in order to analyze the distribution of the HB stars it is useful to define a new coordinate, $l_{\rm HB}$, which is linear along the HB ridge line. This coordinate removes the saturation of the $(B-V)$ color as a function of the temperature. Unfortunately, different authors (Ferraro et al. 1992, Dixon et al.1996, Catelan et al. 1998) have given different definitions of $l_{\rm HB}$, besides the fact that sometimes the definition itself has not been clear nor the measurement of $l_{\rm HB}$ easily reproducible.
In order to linearize [*all*]{} the HBs in our data base, which is the largest one presently available, we were led to define an $l_{\rm HB}$ which is itself slightly different from the previous ones. However, in order to avoid confusion, and in the attempt to introduce a “standard” $l_{\rm HB}$, we describe in detail the recipe we used.
- First of all, the observed CMDs were de-reddened, as described in Section \[red\] for NGC 6273, by comparison with NGC 6205, adopting an $E(B-V)=0.02$ for the latter.
- The HB colors and magnitudes were then transformed according to the following relations: $$c = 204.8 (B-V)_0 + 102.4\qquad\hbox{and}\qquad b = -42.67 M_V + 281.6.$$
These new “color” and “brightness” coordinates allow mapping the CMD onto a plane where 1 unit in the abscissa has the same length (in centimeters) as 1 unit in the ordinate, mantaining the same scale as in Ferraro et al. (1992) and Fusi Pecci et al. (1993). This last condition (which is at the basis of the choice of the numerical factors in the above equation) is not strictly necessary for the comparison among our clusters, but we preferred to keep our $l_{\rm HB}$ as close as possible to the previous definitions.
- The HB ridge lines (HBRL) were obtained by a spline interpolation of hand-drawn fiducial points.
- As shown in Fig. \[pro\], the projection of each HB star on the HBRL was determined as the point ($c_R$, $b_R$) that minimizes the distance $$\sqrt{(c-c_R)^2 + (b-b_R)^2}.$$
- For each star, the length $l$ of the HBRL from a fixed zero point of the coordinate, at $c=300$, to the point ($c_R$,$b_R$) was then computed by dividing the HBRL into $N$ pieces of $\Delta c=0.01$, and approximating each of these pieces by a straight line.
- Finally, the $l_{\rm HB}$ coordinate was obtained applying the scaling relation: $$l_{\rm HB}=0.1086 l$$ as in Dixon et al. (1996) for NGC 1904.
This procedure gives an $l_{\rm HB}$ on the same scale as in Ferraro et al. (1992) and Dixon et al. (1996), with a different zero point for $l_{\rm HB}$, now set at $c=300$ (corresponding to $B-V=0.965$). The zero point corresponds to a point redder than the reddest HB star in NGC 6441 (the most metal-rich of our clusters), in order to avoid negative values for $l_{\rm HB}$.
The HB of NGC 6273 {#HB19}
------------------
The $l_{\rm HB}$ distribution for NGC 6273 is shown in Fig.\[histo\]. The HB shows two remarkable peculiarities. First of all, there is a blue tail which in the CMD extends well below the cluster turnoff. Second, the EBT is clearly bimodal, with two peaks centered on $l_{\rm
HB}=26.5$ and on $l_{\rm HB}=41.3$ and a large gap, centered on $l_{\rm
HB}=36$. Apart from NGC 2808, none of the HBs with EBTs in our sample (cf. Section \[otherhb\]) shows such a clear bimodality in the $l_{\rm HB}$ distribution. Fig. \[histo\] shows that the two peaks are quite sharp. It is questionable whether the distribution along the EBT of NGC 6273 is peculiar because of the presence of these two peaks or rather because of the gap. The two peaks can be fitted with gaussian functions centered on $l_{\rm HB}=26.5$ and on $l_{\rm
HB}=41.3$ with a sigma of 2.1 and 2 in $l_{\rm HB}$ units, respectively. According to the new horizontal-branch models by Bono, Cassisi, & Castellani (1999), this dispersion in $l_{\rm HB}$ corresponds to a mass dispersion $\Delta m = 0.017\> m_\odot$ on the HB. Figure \[modelhb\] shows an enlargment of the HB of NGC 6273 with overplotted the HB model by Bono et al. (1999). The model has been fitted adopting the average reddening obtained in Section \[red\] and allowing a vertical shift such that the model is the lower envelope of the red HB stars. According to these models, the gap in NGC 6273 corresponds to a $T_{\rm eff}\sim19,200$ K, and to a mass of $\sim0.54\>
m_\odot$.
Multimodal HBs in other clusters {#otherhb}
--------------------------------
Though the gap in Fig. \[histo\] is remarkable, it is rather difficult to assess its statistical significance. None of the methods suggested by Catelan et al. (1998) can be applied to NGC 6273, as it is not possible to know what is the underlying true $l_{\rm HB}$ distribution. Surely it cannot be uniform, as supposed by Ferraro et al. (1998) for NGC 6093 and NGC 6205. On the other hand, there are other clusters which have blue tails and show gaps in the HB which might be related to the gap in NGC 6273. It is of some interest to check whether these gaps are located in similar positions on the HBs of different clusters. Ferraro et al. (1998) have shown that all the clusters with EBTs have a gap on the lower part of the blue tail. They have also suggested that at least one gap seems to be present at the same location in all clusters, at $T_{\rm eff}\sim18,000$ K. The models in Fig. \[modelhb\] seem to suggests a slightly hotter temperature for the gap in NGC 6273. However, the absolute value of the temperature depends on the adopted models (Ferraro et al. used the models by Dorman et al. 1997), and the adopted transformation from the theoretical to the observational plane. Moreover, the $V$ vs. $B-V$ plane is not the best one for estimating the $T_{\rm eff}$ of such hot stars, which can be better estimated with ultraviolet observations, as in Ferraro et al. (1997).
In the following, we will try to take advantage of the fact that we have eight clusters with blue HB tails observed exactly in the same photometric system and with homogeneous photometry. We already noted that our photometric bands are far from ideal for a study of the hot population in the EBT of the GGCs, and that, due to the saturation of the ($B-V$) color for stars hotter than $\sim 10,000$K the HB becomes almost vertical, making more difficult the identification of all the possible gaps (real or statistical fluctuations as they might be). A typical example is NGC 1904, for which Hill et al. (1996) suggested a possible gap at $T_{\rm eff}\sim
9,990$K, which is not visible in our HST CMD (Sosin et al. 1997b). Still, in Figs. \[cmd\_Tcost\] and \[histo\_Tcost\], there is clear evidence of gaps (the most significant ones are indicated by arrows), so we can still try to address the question of whether there is anything systematic about their positions.
One possible way to compare the location of the gaps in different clusters is to compare their $l_{\rm HB}$ directly. This method has the advantage of being totally empirical. However, in turn it makes very hard the interpretation of the different locations of the gaps in terms of physical parameters like $T_{\rm eff}$ or mass. This is mainly due to the fact that, by definition, $l_{\rm HB}$ does not run over the ZAHB, but over the mean HB line; therefore the comparison between observed data and models (which refer to the ZAHB) is not completely consistent. On the other hand, forcing $l_{\rm HB}$ to run over the ZAHB (, the model) instead of on the mean ridge line, may introduce strong biases in the projections of the stars in those regions where the shape of the model does not reproduce exactly the observed HB. Finally, the models often do not reach magnitudes as faint as the data do, in the EBT. For these reasons, and since all the gaps visible in our CMDs are actually located in the vertical part of the HB, we decided that a more direct analysis could be done by simply comparing the distribution in absolute magnitude of the vertical part of the HBs of the different clusters.
To this end, we proceeded as follows. We registered the blue bend of all the HBs to that of NGC 6205, in order to constrain the relative reddening. We then assumed an $E(B-V)=0.02$ for NGC 6205 and de-reddened accordingly the HBs of all the other clusters. Allowing only a vertical shift we then matched the lower envelope of the stars in the observed HBs with the models by Bono et al. (1999), as shown by Fig. \[cmd\_Tcost\]. In this figure, the most prominent HB gaps are indicated with the horizontal arrows. Following the suggestion by Ferraro et al. (1998) we marked with a dashed horizontal strip the region corresponding to $T_{\rm eff}=19,200\pm1000$K (i.e., the temperature at the center of the gap of NGC 6273). At least one gap of the intermediate-metallicity and metal-poor clusters is always inside this strip, confirming their hypothesis that in all the EBTs there is a gap corresponding to this temperature. However, for the more metal-rich clusters, namely NGC 6441, NGC 6388, and NGC 2808, there is no gap corresponding to the dashed areas in Fig \[cmd\_Tcost\]. While in the case of NGC 6388 the region at $T_{\rm eff}=19,200\pm1000$K is in any case in a sparsely populated part of the CMD, in NGC 6441 and NGC 2808 it corresponds to local peaks in the distribution along the HB. Note that in the case of NGC 2808, all the gaps visible in the ultraviolet regions ($B$ vs. F214W$-B$ diagram) are also visible in the $V$ vs. $B-V$ CMD (Sosin et al. 1997). We do not have suitable UV data for NGC 6388 and NGC 6441. Shall we conclude that gaps are at random positions in the HB? Of course, the uncertainties in the models and in the transformation from the theoretical to the observational plane, and the errors in fitting the models to the observed HB, play an important role in the interpretation of Fig. \[cmd\_Tcost\]; we cannot exclude that the hypothesis in Ferraro et al. (1998) is still valid. On the other hand, it is possible that some of the gaps marked in Figs. \[cmd\_Tcost\] and \[histo\_Tcost\] are a random fluctuation in the distribution of the stars along the HB (Catelan et al. 1998). We have also tested an alternative hypothesis. Since the main parameter governing the position of a star along the HB is the ratio of the envelope mass to the core mass (a ratio that follows from the total mass, assuming that the core mass is the same for all these stars), we have investigated the possibility that the gaps appear at costant mass on the HB. Fig.\[cmd\_Mcost\] shows the same HBs of Fig. \[cmd\_Tcost\] with a shaded region indicating the magnitude corresponding to $M=0.535\pm0.005M_\odot$. At least one gap of each cluster falls inside this area, suggesting that the possibility of the gaps being physically associated with a “forbidden” value of the mass is compatible with the observations. The analysis of the histograms, in absolute magnitude, of the stars in the vertical portion of the HBs gives the same result. Figures \[histo\_Tcost\] and \[histo\_Mcost\] show the distribution of the same stars plotted in Figs. \[cmd\_Tcost\] and \[cmd\_Mcost\] as a function of magnitude. Again, the shaded region indicates the constant-temperature loci (Fig.\[histo\_Tcost\]) and the constant-mass loci (Fig.\[histo\_Mcost\]). And again, the hypothesis of a set of gaps at constant mass for all the clusters with EBT is in agreement with all the data at our disposal.
In conclusion, the distribution of the stars along the blue HB tails is clearly not uniform and differs from cluster to cluster. There are both peaks and gaps. All the clusters with an EBT have at least one gap, as suggested by Ferraro et al. (1998). Of course a sample of only seven clusters is not sufficient to draw any conclusion at this point. Moreover, as mentioned above, the $V,B-V$ plane is not the ideal one for these studies. Still, Figures \[cmd\_Tcost\]–\[histo\_Mcost\] suggest that the possibility that gaps are at constant mass is at least as plausible as the hypothesis of gaps at constant temperature as suggested by Ferraro et al. (1998). There are also other gaps, which apparently are not directly related to either of the two proposed hypotheses.
Luminosity Functions {#lf}
====================
>From the CMD of Fig. \[cmdco\] it is possible to extract a LF for the evolved stars, which is an important direct test of the evolutionary clock (Renzini & Fusi Pecci 1988). It has been recently pointed out that, at least for the metal-poor clusters, the agreement of the models with the observed LFs for the GGC stars is far from satisfactory (Faulkner & Swenson 1993, Bolte 1994, and references therein). Sandquist et al. (1998), by comparing the LF of M30 with the theoretical LFs from Bergbusch & VandenBerg (1992) and VandenBerg et al. (1999), confirm this result, at least for the most metal-poor clusters, and suggest that it might be the consequence of some deep mixing events. On the other hand, Silvestri et al. (1998) claim that their set of models strongly reduces the discrepancy with the observed LFs, though they cannot provide the reasons for the differences among the different models.
The rich fields in the centers of the GGCs whose CMDs we are collecting are suitable for a detailed study of the evolution time scales along the red giant branch. In Piotto et al.(1999a) we already presented a comparison between the Bergbusch & VandenBerg (1992) LFs and the observed LFs for NGC 6362 and NGC 6934, with conflicting results. While the LF of NGC 6934 (\[Fe/H\] $=-1.03$) could be reasonably well fitted with the theoretical LFs by Bergbusch & VandenBerg (1992), NGC 6362 (\[Fe/H\] $=-1.48$) showed the same excess of bright giants as suggested by Bolte (1994) for the most metal-poor clusters.
We first tried the same fit as in Piotto et al. (1999a), using the appropriate set of isochrones from Bergbusch & VandenBerg (1992), on the observed LF of NGC 6273, after the crowding corrections. Again, we found an overabundance of bright RGB stars with respect to the MS, as for NGC 6362 or M30 (Sandquist et al. 1998).
A better fit could be obtained using both the most recent models by Silvestri et al. (1998), and the models by Straniero, Chieffi & Limongi (1997). In order to mimic an $\alpha$-element enhancement of 0.4 (Sneden et al. 1991), we adopted a higher metallicity than discussed in Section \[intro\], following the method described by Salaris et al. (1993). We also adopted a flat mass function ($x=-0.5$, where the Salpeter slope is 1.35), which is more appropriate for the GC center. In both cases there is a good agreement between the observed and theoretical LFs (Fig. \[lfdan\] and Fig. \[lfchi\]), which mostly removes the claimed discrepancies with the calculated evolutionary times along the RGB. We note that the models by Straniero et al. (1997) seem to reproduce the observed stellar distribution better.
Piotto et al. (1999a) also suggested that the large number of evolved stars in the LF from the cluster cores could be used to infer the cluster ages independently. Silvestri et al. (1998) pointed out that the method can work, provided we have an independent distance estimate. This is also evident from Figs. \[lfdan\] and \[lfchi\]. Each panel shows the fit of the NGC 6273 LF with the theoretical LFs for 4 different ages. The inset in each panel shows the $\chi^2$ trend as a function of the distance. The minimum value of the $\chi^2$ is similar in the three panels corresponding to an age $\geq12$ Gyr, while it is higher for 10 Gyr. Note that if we allow the distance to vary, there is no preferred age between 12 and 16 Gyr. However, this would imply an uncertainty in the apparent distance modulus at a level of more than 0.4 magnitude, which is not realistic. On the other hand, if we have some estimate of the distance, the uncertainty in the age is drastically reduced. For example, if we take the distance estimate from Section \[dist\] we are forced to conclude that the age of NGC 6273 is $15\pm2$ Gyr (from the Straniero et al. 1997 models, slightly older than from the Silvestri et al. models). Note that the distance obtained in Section \[dist\] is still based on the old, shorter distance scale. If we adopt the longer distance scale from the HIPPARCOS data, as in Gratton et al. (1997), we should reduce the ages by 2 Gyr.
It remains still to verify the suggestion by Bergbusch & VandenBerg (1992) that the detailed structure of the LF around the TO can constrain both the age and the distance. The smearing of the LF due to the differential reddening does not allow applying the method to NGC 6273, however.
Acknowledgments {#ack}
===============
We thank P. Stetson for his generosity with software. It is a pleasure to thank M. Catelan for useful discussions. We are indebted to S. Cassisi for providing us a set of HB models specifically calculated for the metallicities of our clusters. Support for this work was provided to I.R.K., S.G.D., and R.M.R. by NASA through grant GO-7470 from the Space Telescope Science Institute. G.P. and M.Z. acknowledge partial support by the Ministero dell’Università e della Ricerca Scientifica.
Bolte, M. 1994, , 431, 223 Bono, G., Cassisi, S., & Castellani, V. 1999, in prepation Bergbusch, P. A., & VandenBerg, D. A. 1992, , 81, 163 Catelan, M., Borissova, J., Sweigart, A. V., Spassova, N., 1998, , 494, 265 Clement, C., & Sawyer Hogg, H., 1978, , 83, 167 Cool, A. M., Piotto, G., & King, I. R. 1996, , 468, 655 Dixon, W. V. D., Davidsen, A. F., Dorman, B., & Ferguson, H. C., 1996, , 111, 1936 Djorgovski, S. 1993, in Structure and Dynamics of Globular Clusters, eds. S. G. Djorgovski & G. Meylan (San Francisco: ASP),p. 373 Ferraro, F. R., Clementini, G., Fusi Pecci, F., Sortino, R., & Buonanno, R. 1992, , 256, 391 Ferraro, F. R., et al. 1997, å, 324, 915 Ferraro, F. R., Paltrinieri, B., Fusi Pecci, F., Rood, R. T., & Dorman, B., 1998, , 311, 319 Faulkner, J., & Swenson, F. J. 1993, , 411, 200 Fusi Pecci, F., Ferraro, F. R., Bellazzini, M., Djorgovski, S. G., Piotto, G., & Buonanno, R., 1993, , 105, 1145 Gratton R. E., Fusi Pecci, F., Carretta, E., Clementini, G., Corsi, C., & Lattanzi, M. 1997, , 491, 749 Harris, W. E., Racine, R., & de Roux, J., 1976, , 31, 13 Hill, R. S., 1996, , 112, 601 Holtzman, J. A., Burrows, C. J., Casertano, S., Hester, J. J., Watson, A. M., & Worthey, G. S. 1995, PASP, 107, 1065 Iben, I. 1968, Nature, 220, 143 Norris, J., 1983, , 272, 245 Piotto, G., Zoccali, M., King, I. R., Djorgovski, S. G., Sosin, C., Dorman, B., & Rich, R. M. 1999a, , 117, 264 Piotto, G., et al. 1999b, in preparation Renzini, A., & Fusi Pecci, F. 1988, ARA&A, 26, 199 Rood, R. T., & Crocker, D. A., 1985, in Horizontal Branch and UV-Bright Stars, ed. A. G. D. Philip (Schenectady: L. Davis Press), 99 Rood, R. T., & Crocker, D. A. 1989, in The Use of Pulsating Stars in Fundamental Problems of Astronomy (IAU Colloq. 111), ed. E. G. Schmidt (Cambridge: Cambridge Univ. Press),p. 103 Rutledge, G. A., Hesser, J. E., Stetson, P. B., Mateo, M., Simard, L., Bolte, M., Friel, E. D., & Copin, Y., 1997, PASP, 109, 883 Salaris, M., Chieffi, A.,& Straniero, O., 1993, , 414, 580 Sandquist, E. L., Bolte, M., & Hernquist, L. 1997, , 477, 335 Silbermann, N. A., et al. 1996, , 470, 1 Silvestri, F., Ventura, P., D’Antona, F., & Mazzitelli, I., 1998, , submitted Sneden, C., Kraft, R. P., Prosser, C. F., & Langer, G. E., 1991, , 102, 2001 Sosin, C., Dorman, B., Djorgovski, S. G., Piotto, G., Rich, R. M., King, I. R., Liebert, J., Phinney, E. S., & Renzini, A. 1997a, 480, L35 Sosin, C., Piotto, G., King, I. R., Djorgovski, S. G., Rich, R. M., King, I. R., Dorman, B., Liebert, J., & Renzini, A. 1997b in Advances in Stellar Evolution, eds. R. T. Rood and A. Renzini (Cambridge: Cambridge Univ. Press), p. 92 Stetson, P. B., & Harris, W. E., 1977, , 82, 954 Straniero, O., Chieffi, A., & Limongi, M., 1997, , 490, 425 VandenBerg D. A., Swenson F. J., Rogers F. J., Iglesias C. A., & Alexander D. R., 1998, , submitted White, R. E., & Shawl, S. J., 1987, , 317, 246 Zinn, R., & West, M. 1984, , 55, 45 Zoccali, M., & Piotto, G., 1999, in preparation
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IPMU11-0147
1.2cm Yuji Tachikawa 1.2cm
IPMU, University of Tokyo,\
5-1-5 Kashiwa-no-ha, Kashiwa, Chiba, 277-8583 Japan
1.5cm
**abstract**
A relationship between 4d gauge theory and 2d CFT will be reviewed from the very basics. We will first cover the introductory material on the 2d CFT and on the instantons of 4d gauge theory. Next we will explicitly calculate and check the agreement of the norm of a coherent state on the 2d side and the instanton partition function on the 4d side. We will then see how this agreement can be understood from the perspective of string and M theory.
*to appear on the proceedings of the\
“Summer School on Mathematical Physics 2011”, Komaba*
IPMU11-0147
1.2cm 立川 裕二 1.2cm
〒277-8583 千葉県柏市柏の葉 5-1-5\
東京大学数物連携宇宙研究機構(IPMU)
1.5cm
**概要**
近年発見された、四次元ゲージ理論と二次元共形場理論の関係を非常に基礎から解説します。まず、二次元の共形場理論および四次元のインスタントンの入門をした後、 二次元側でコヒーレント状態のノルムの計算、四次元側でインスタントンの分配関数を具体的に計算を行って確認します。その後、一致が超弦理論/M理論の枠内でどのように自然に解釈されるかを見ます。
はじめに
========
超弦理論は、現時点では厳密に数学的に構築されてもいませんし、実験的に物理的に確認されたわけでもありません。しかし、いろいろな状況証拠から判断するに、超弦理論が自己矛盾の無い数学的対象として存在するのは間違いないと思われます[^1]。
超弦理論は、数学的には非常に大雑把には、10次元の多様体 $M$ に対して分配関数と呼ばれる複素数 $Z(M)$ を対応させる手続きです。この手続きは完全には未だ定義されていませんので、弦理論屋は、これをまず何とか数学的にきちんと定義され、計算できる量まで変形します。しばしば、$Z(M)$ を計算できる量にする方法は何通りもあります。それが二つあるとし、$Z_A$、 $Z_B$ と呼びましょう。単に同じものを二通りの仕方で計算しただけですから、 $Z_A$ と $Z_B$ は等しいはずです(図\[sketch\])。$Z_A$ も $Z_B$ も数学的にきちんと定義できる量ですから、$Z_A=Z_B$ は現在の数学で扱える主張です。しかし、超弦理論 $Z(M)$ は未だ満足いくように定義されていませんから、しばしば $Z_A=Z_B$ は数学者には何故成立するのか俄には判らない主張として現れることになります。
$$\includegraphics[width=.5\textwidth]{sketch}$$
このような現象の例の最も著名で、非常に深いものとして90年代初頭に発見されたミラー対称性があげられます。また、Donaldson 不変量と Seiberg-Witten 不変量の一致が物理で云う Seiberg-Witten 理論から従うのもこの現象の一例と思うことができます。さらに、ゲージ・重力対応からも箙の構造と佐々木アインシュタイン多様体の幾何が対応するなど数学的現象を取りだすことが出来ます。
今回の講義では、以上挙げた例に比べるとかなり浅いですが、手を動かして計算しやすい $$\begin{aligned}
Z_A &= \text{二次元共形場理論のコヒーレント状態のノルム}\hbox{、} \\
Z_B &= \text{四次元インスタントンの分配関数}\end{aligned}$$ の二つが一致するという事実を説明します。$Z(M)$ は、六次元の $\cN=(2,0)$ 理論と呼ばれるものになります。具体的には、$Z_A$ は $c$ を中心電荷、$\Delta$ を $L_0$ の固有値、$\lambda$ をコヒーレント状態のパラメタとして $$Z_A=\vev{\Delta,\lambda|\Delta,\lambda}=1+\frac{\lambda^2}{2\Delta}+\frac{\lambda^4(c+8\Delta)}{4\Delta((1+\Delta)c-10\Delta+16\Delta^2)}+\cdots \hbox{、}$$ $Z_B$ は $q$, $\epsilon_{1,2}$, $a$ をそれぞれインスタントン数、角運動量、ゲージ荷に対する化学ポテンシャルとして、$$\begin{gathered}
Z_B=Z^\text{instanton}_{\epsilon_1,\epsilon_2,a}=1+ \frac{q}{\epsilon_1\epsilon_2} \frac{2}{(\epsilon_1+\epsilon_2)^2 -4a^2} \\
+ \frac{q^2}{\epsilon_1^2\epsilon_2^2} \frac{( 8(\epsilon_1+\epsilon_2)^2 +\epsilon_1\epsilon_2 - 8a^2)}
{ ((\epsilon_1+\epsilon_2)^2-4a^2)
((2\epsilon_1+\epsilon_2)^2-4a^2) ((\epsilon_1+2\epsilon_2)^2-4a^2) } + \cdots\end{gathered}$$となります。これらが $$\lambda^2=\frac{q}{(\epsilon_1\epsilon_2)^2}\hbox{、} \qquad
\Delta=\frac1{\epsilon_1\epsilon_2}(\frac{(\epsilon_1+\epsilon_2)^2}{4} -a^2) ,\qquad
c=1+6\frac{(\epsilon_1+\epsilon_2)^2}{\epsilon_1\epsilon_2}$$ と対応付けると完全に一致する、というものです。
講義の順序は、まずは超弦理論のことはすっかり忘れて、$Z_A$ および $Z_B$ をかなり基礎的なところから説明し、上でつかった用語をまず説明します。次に、$Z_A$ と $Z_B$ を具体的に計算して一致する事を確認したのち、それがどのように自然に超弦理論から理解できるかを説明する、というようにしたいと思います。 このノートは昨年書きました [@Gakkaishi] の増補版になっておりますので、もっと短いものをお好みの方はそちらをご参照ください。
二次元共形場理論とそのコヒーレント状態
======================================
二次元理論の例
--------------
まずは二次元の数理物理からはじめましょう。(この節の詳細は、例えば教科書 [@ID; @KY; @Yamada; @ItoText] を参照のこと。)
二次元の物理系の簡単な例として、イジング模型を考えます。この模型は、二次元面の各格子点 $(i,j)$ に自由度 $\sigma_{i,j}=\pm 1$ があり、ある配位の実現される確率は $T$ を正の数として $$\frac1Z \exp \left[\frac1{T} \sum_{i,j} (\sigma_{i,j}\sigma_{i+1,j} + \sigma_{i,j}\sigma_{i,j+1})\right]$$ である、というものです。$Z$ は全体の確率が1になるための規格化の定数です。明らかに、$$Z=\sum_{\sigma_{i,j}} \exp \left[\frac1{T} \sum_{i,j} (\sigma_{i,j}\sigma_{i+1,j} + \sigma_{i,j}\sigma_{i,j+1})\right] \label{isingZ}$$ です。$Z$ を分配関数と呼びます。 $\sigma$ をスピンと呼び、$\sigma=+1$ を上向き、$\sigma=-1$ を下向きといいましょう。すると、隣り合った格子点 $p$ と $q$ に対して、スピン $\sigma_p$ と $\sigma_q$ が揃っているほうが揃っていない場合より $\exp(2/T)$ だけ可能性が高いわけです。 $T$ を温度と呼びましょう。この用語は次のように考える事ができます。$T=0$ では、隣り合った格子点のスピンは揃う確率が揃わない確率に対して無限に可能性が高い。ですから、全ての格子点のスピン $\sigma$ が一致します。これは、スピンを各格子点にある微小な磁石の向きだと思うと、温度が非常に低いときは系全体が自発的に一方向に磁化している、と思えます。一方、$T=\infty$ では、各格子点は独立ですから、系全体としては完全に乱雑になっています。これは高温では磁石の磁化が消えた、と思う事ができます。$T$ を $0$ から $\infty$ まで徐々に変化させた場合、一体どこで磁化が消えるのだろうか? 磁化が消える瞬間はどのようになるのか、ということを考えましょう。
$$\begin{array}{ccc}
\includegraphics[width=.2\textwidth]{hi} &
\includegraphics[width=.2\textwidth]{critical} &
\includegraphics[width=.2\textwidth]{low} \\
\text{高温相} &
\text{臨界} &
\text{低温相}
\end{array}$$
$$\vcenter{\hbox{\includegraphics[width=.2\textwidth]{spin1024framed}}}
\longrightarrow
\vcenter{\hbox{\includegraphics[width=.2\textwidth]{spin512framed}}}
\longrightarrow
\vcenter{\hbox{\includegraphics[width=.2\textwidth]{spin256}}}$$
この模型は厳密に解かれているので、それを解説することも出来ますが、最近はパソコンが非常に強力なので、スピン配位をこの確率に従って生成して、何が起こっているかをみるのも悪くありません。しばらく遊ぶと、$T=2.2$ あたりで磁化が消滅することがわかります (図\[ising\])。厳密解では、$$T_c=\frac{2}{\log(1+\sqrt{2})}$$となることが知られています。さて、丁度温度をこの臨界点 $T_c$ にすると、スピンの揃っている塊の大きさにいろいろなものがあることが判ります。ひとつ配位を取って、二倍、二倍と拡大してみると、だいたい見た目が一定であることが判ります (図\[scaling\])。もっと定量的には、$\vev{\sigma_{0,0} \sigma_{x,0}}$ を測定すると、両対数グラフで直線上にのることが判ります (図\[spinspin\])。厳密解では $$\vev{\sigma(x)\sigma(0)}\propto 1/x^{1/4}\hbox{。}$$ ですから、スケールの変更 $\vec x\to \alpha \vec x$ に伴って $$\sigma \to \alpha^{1/8} \sigma \label{spintr}$$ としてやれば系は不変です。 これを、系のスケール変換 $\vec x\to \alpha\vec x$ のもとでの不変性と言います。
$$\includegraphics[width=.4\textwidth]{spinspin}$$
$$\begin{array}{cc}
\vcenter{\hbox{\includegraphics[width=.2\textwidth]{rotation}}}&
\vcenter{\hbox{\includegraphics[width=.2\textwidth]{quadratic}}} \\
z\mapsto e^{i\theta}\cdot z &
z\mapsto z+ a z^2
\end{array}$$
ある緩い条件を満たす二次元系では、もし系がスケール変換で不変ならば、自動的にもっと一般の局所的な角度を保つ変換のもとで理論が不変になることが知られています。二次元の座標を $z=x+iy$ と書くと、一般に正則関数 $f(z)$ を用いて $$z \mapsto z'=f(z)$$ という操作が局所的な角度を保ちます (図\[conformal\])。微小変換は $$z \mapsto z'=z+ \sum_n \epsilon_n z^{n+1}$$ ですから、無限個のベクトル場 $\xi_n = z^{n+1} \partial_z $ で系が不変であることになります。 複素共役の $\bar\xi_n = \bar z^{n+1}\bar\partial_{\bar z}$ もあります。これらの交換関係は、計算すると $$[\xi_m,\xi_n] = (m-n) \xi_{m+n}\hbox{、} \quad
[\bar\xi_m,\bar\xi_n] = (m-n) \bar\xi_{m+n}\hbox{、} \quad
[\xi_m,\bar\xi_n]=0
\label{Witt}$$ となります。
$$\includegraphics[width=.3\textwidth]{ising-on-cyl}$$
さて、すこし見方を変えて、この二次元系のうち $x$ 方向は空間で、$y$ 方向は時間だと思いましょう: $t=y$。 無限に広いとややこしいので、$x$ 方向は $N_x$ 個、$t$ 方向は $N_t$ 個格子点があるとします(図\[ising-on-cyl\])。 ある時刻 $t$ を固定すると、スピン $\sigma_i$ ($-\infty<i<\infty$) がありますが、一つ一つの配位に対してベクトル空間の基底 $\ket{\sigma_x}$ を考えましょう。この空間 $\cH$ には合計 $2^{N_x}$ 個基底があります。$\cH$ に作用する行列を二つ考えます: $$\begin{aligned}
A\ket{\sigma_x} &= \exp\left[\frac1T\sum_{x}\sigma_x\sigma_{x+1} \right] \ket{\sigma_x}\hbox{、} \\
B\ket{\sigma_x} &= \sum_{(\sigma'_x)}\exp\left[\frac1T\sum_{x}\sigma_x\sigma'_{x} \right] \ket{\sigma'_x}\end{aligned}$$ すると、分配関数 は $$Z=\tr_\cH (AB)^{N_y} = \tr_\cH e^{-N_y H} \quad \text{但し}\quad H=\log (AB)$$ となり、時間 $N_t$ だけ量子力学的ハミルトニアン $H$ で系が発展したと思うことが出来ます。 いいかえれば、時間方向に動かすベクトル場 $\partial_t$ が、$\cH$ に作用する行列 $H$ に変わったわけです。
$N_x$ を非常に大きく取っておいて、この書き換えを臨界点 $T=T_c$ のときに行うと、時間並進だけでなく、共形変換 $\xi_n$、 $\bar\xi_n$ 全体が $\cH$ に作用する行列になります。それを $L_n$、 $\bar L_n$ と書きましょう。一般に、古典的なベクトル場を量子力学的な行列に焼き直すと、補正が入り得ます。無限個ある $\xi_n$ の交換関係 を矛盾無く変更するには、次のようにするしかないと知られています: $$[L_m,L_n] = (m-n)L_{m+n} + \frac c{12}(m^3-m) \delta_{n,-m}\label{virasoro}$$但し $-\infty<n,m<\infty$ は整数で $c$ は正の定数。これがビラソロ代数です。 $\bar\xi_n$ も全て演算子 $\bar L_n$ になり、同じ交換関係を満たします。 $L_n$ のエルミート共役は $L_{-n}=L_n^\dagger$ とします。以下簡単のため通常 $\bar L_{n}$ は忘れることにします。
ビラソロ代数の表現
------------------
ビラソロ代数の表現を調べる前に、調和振動子の量子化を復習しましょう。 運動量演算子 $p$ と位置演算子 $q$ が $[q,p]=i$ という交換関係を満たす際に、ハミルトニアン $H=(p^2+q^2)/2$ を調べたい。勿論 $p=\partial/\partial q $ として、微分演算子 $\partial^2/\partial q^2 + q^2$ の固有関数を特殊関数として調べても良いですが、代数的に考えましょう。そのために $a=p+iq$、$a^\dagger = p-iq$ を定義すると、$$H= a^\dagger a + \frac12$$ と書き直せます。$[a,a^\dagger]=-1$ ですから、$[H,a]=-a$ です。さて、$H$ に固有値 $E$ の固有状態 $\ket{E}$ があったとしましょう: $H\ket E=E \ket E$。すると、$$H a \ket E = (a H - a) \ket E = (E-1) a\ket E\hbox{、}$$ すなわち、$a\ket E$ は 固有値 $E-1$ の固有ベクトルです。 この操作は何度でも繰り返せます。 一方で、状態空間が正値、すなわち勝手な状態 $\ket\psi$ に対して $\| \ket \psi \|^2 = \vev{\psi|\psi}\ge 0$ とすると、 $$E \| \ket E \|^2 = \vev{E | H | E} = \| p \ket E \|^2 + \|q \ket E \|^2 \ge 0$$ですから、固有値 $E$ は非負です。よって、$a^n\ket E$ はいずれ消滅しないといけない、すなわち何か状態 $\ket{\text{vac}}$ があって $$a \ket{\text{vac}} = 0\hbox{。}$$ すると $$H \ket{\text{vac}} = \frac12 \ket{\text{vac}}\hbox{。}$$ また、同様に固有値 $E$ の状態に $a^\dagger$ を掛けると固有値 $E+1$ の状態になるのも示せます。よって、$$\ket{n+\frac12} = (a^\dagger)^n \ket{\text{vac}}$$ が固有値 $\frac12+n$ の固有状態になります。
ですから、ハミルトニアン $H$ に対して、生成演算子 $a^\dagger$ はエネルギーを $1$ あげ、消滅演算子 $a$ はエネルギーを $1$ 下げ、消滅演算子 $a$ で消される状態が最低エネルギー状態 $\ket{\text{vac}}$ です。
ビラソロ代数の場合は、$L_0$ をハミルトニアンと思うのが都合が良いです: $H=L_0$。すると、$$[L_0,L_n] = -n L_n$$ ですから、$L_{-n}$ がエネルギーを $n$ あげる生成演算子で、$L_{n}=L_{-n}^\dagger$ がエネルギーを $n$ 下げる消滅演算子です。全ての消滅演算子で消される状態が最低エネルギーですが、そこでの $H=L_0$ の固有値を $\Delta$ としましょう: $$L_{0} \ket{\Delta}=\Delta \ket{\Delta}\hbox{、} \qquad
L_{n} \ket{\Delta}=0 \qquad (n>0)\hbox{。}$$ 調和振動子の場合と異なり、$\Delta$ はこれだけでは定まりません。
$$\includegraphics[width=.8\textwidth]{virasoroverma}$$
一般に、$L_0$ の固有値が $\Delta+N$ である状態は $L_{-n_1}L_{-n_2}\cdots L_{-n_k}\ket\Delta$ で $\sum n_i=N$ となるようなものです(図\[verma\])。 これを簡略に $$\ket{\Delta;N;n_1,\ldots,n_k} =L_{-n_1}L_{-n_2}\cdots L_{-n_k}\ket\Delta$$ と書きましょう、但し $n_1\ge n_2 \ge \cdots \ge n_k$ としておきます。 これらが全て線形独立な場合、この表現をビラソロ代数の Verma 表現と言います。
$N$ は通常次数 (grade) と呼ばれます。次数が 1 の状態は $L_{-1}\ket{\Delta}$ のみです。これのノルムは、交換関係をつかうと $$\vev{\Delta|L_{1}L_{-1}|\Delta}=2\Delta \vev{\Delta|\Delta}$$ となりますから、$\Delta$ は正です。
次に次数が 2 の状態は一般に $$\ket{\psi}=c_{11} \ket{\Delta;2;1,1} + c_2 \ket{\Delta;2;2}$$と書けますが、これのノルムは交換関係 をつかって計算すると $$\vev{\psi|\psi} = (\bar c_{11},\bar c_{2}) M_{N=2} \begin{pmatrix}
c_{11} \\ c_2
\end{pmatrix}\quad\text{但し}\quad
M_2= \begin{pmatrix}
4\Delta(2\Delta+1) & 6\Delta \\
6\Delta & 4\Delta+c/2
\end{pmatrix}$$ となります。これが負にならないためには、$$4\Delta(2\Delta+1)(4\Delta+c/2) \ge (6\Delta)^2$$ でなければいけません。書き換えると、$$\Delta(\Delta-\Delta_{1,2})(\Delta-\Delta_{2,1})\ge 0$$ です、ただし $$\Delta_{r,s}=\frac{c-1}{24}+\frac14(r\alpha_++s\alpha_-)^2\hbox{、}\qquad \alpha_\pm=\frac{\sqrt{1-c}\pm\sqrt{25-c}}{\sqrt{24}}\hbox{。}$$ $c<1$ ですと $\Delta_{1,2}$、 $\Delta_{2,1}$ は実ですから、$\Delta_{1,2}< \Delta < \Delta_{2,1}$ だと駄目なわけです。
一般に次数 $N$ のところにある状態の数は、$N$ を正の整数の和として書く場合の数だけあります。それを $p_N$ と書く事にしますと、 $M_{N}$ は $p_N\times p_N$ 行列になります。状態空間が正定値であるためには、全ての $N$ に対して $\det M_N\ge 0$ でないといけません。この行列式は Kac によって計算されており、$$\det M_N \propto \prod_{r,s\ge 1;\ rs\le N} (\Delta-\Delta_{r,s})^{p_{N-rs}}$$ となります。$c\ge 1$ のときはほぼ自動的にこれは正になりますが、$c<1$ のときは $\Delta$ が $\Delta_{r,s}$ のどれかでない限り、いずれ何らかの $\Delta_{r,s}$、 $\Delta_{s,r}$ に挟まれて駄目になってしまいます。 これらの条件を丁寧に調べると、状態空間が正定値であるためには、$$c\ge 1 \quad \text{もしくは $m\ge2$ なる正の整数を取って} c=1-\frac{6}{m(m+1)}$$となることが知られています。さらに、後者の場合は $\Delta$ の値は $r,s$ を $1\le s \le r <m$ なる正の整数として$\Delta = \Delta_{r,s}$ に限られます。$$\alpha_+=\frac{m+1}{\sqrt{m(m+1)}}\hbox{、}\quad
\alpha_-=\frac{-m}{\sqrt{m(m+1)}}$$ですから、 $$\Delta_{r,s}=\frac{((m+1)r-ms)^2-1}{4m(m+1)}$$ となります。 $m=2$ のときは $c=0$、 許される $\Delta$ は $\Delta=0$ のみで面白くありません。 次の $m=3$ の場合は、$c=1/2$、 許される $\Delta$ は $$\Delta_{1,1}=0\hbox{、} \quad \Delta_{2,2}=\frac1{16}\hbox{、} \quad \Delta_{1,2}=\frac12 \label{isingdims}$$の三種類です。
イジング模型は臨界点では丁度この $c=1/2$ のビラソロ代数が $\xi_n$ から来る $L_n$ と $\bar\xi_n$ から来る $\bar L_n$ とがあります。 スケール変換 $(x,y)\to \alpha(x,y)$ は微小変換 $\alpha=1+\epsilon$ に対しては $z=x+iy$ で書いて $$z\frac{\partial}{\partial z}
+\bar z\frac{\partial}{\partial \bar z} = \xi_0+\bar \xi_0$$ で与えられます。すなわち $L_0+\bar L_0$ です。 スピン演算子の変換性を に書きましたが、微小変換を考えると $L_0+\bar L_0$ の固有値が $1/8$ であることになります。これは で $L_0$ 及び $\bar L_0$ の固有値が両方とも $1/16$ になっていることに対応します。
ビラソロ代数のコヒーレント状態
------------------------------
さて、話をまた調和振動子に戻して、コヒーレント状態を考えましょう。 $p$ と $q$ は交換しませんから、同時固有状態をとることはできません。 ある状態 $\ket\psi$ に対して、$\vev{O}=\vev{\psi|O|\psi}$ と略記することにすると、$p$ と $q$ の広がりは $$(\delta p)^2=\vev{(p-\vev{p})^2}\hbox{、} \qquad
(\delta q)^2=\vev{(q-\vev{q})^2}$$ と思えますから、$$\delta p^2 \delta q^2 = \| (p-\vev{p}) \ket\psi \|^2 \| (q-\vev{q}) \ket\psi \|^2 \ge \left[\mathrm{Im} \vev{\psi| (p-\vev{p})(q-\vev{q}) |\psi} \right]^2= \frac14$$ となるのでした。不確定性を最小にする状態は可能な限り古典的な状態と言っても良いでしょう。上の式変形を等式にする一つの方法は $$i(p-\vev{p})\ket\psi
=
(q-\vev{q})\ket\psi$$ とすればよいです。$\lambda=\vev{p+iq}$ とすると、$\psi$ が消滅演算子 $a=p+iq$ の固有状態であることがわかります: $$a\ket{\psi}=\lambda\ket{\psi}\hbox{。}$$ これをコヒーレント状態と呼ぶのでした。以下固有値 $\lambda$ のコヒーレント状態を $\ket{\lambda}$ と書く事にしましょう。
調和振動子の固有状態はコヒーレント状態の一例です: $\ket{\text{vac}}=\ket{0}$。 $[a,a^\dagger]=1$ であることを利用して、$$a e^{\lambda a^\dagger} \ket{0} = \lambda e^{\lambda a^\dagger} \ket{0}$$ すなわち $\ket{\lambda}=e^{\lambda a^\dagger}\ket 0$ です。状態を規格化するには、$$\vev{\lambda|\lambda}=\vev{0| e^{\bar\lambda a}e^{\lambda a^\dagger} |0} = e^{\bar\lambda\lambda} \vev{0|0}$$ とすればよいです。
調和振動子のコヒーレント状態はいろいろな応用があります。ビラソロ代数も重要です。ですから、ビラソロ代数のコヒーレント状態を考える事も意味が無くはないでしょう。$\ket{\Delta}$ で生成される Verma 表現の中のベクトル $\ket\psi$ で、消滅演算子 $L_n$ ($n>0$) が固有値を持つものを考えます: $$L_n \ket\psi = \lambda_n \ket\psi\hbox{。}$$ 交換関係から、すぐに $n\ge 3$ なら $\lambda_n=0$ とわかります。簡単のために $\lambda_2$ もゼロとしてしまって、$\lambda\equiv \lambda_1$ で指定される状態 $\ket{\Delta,\lambda}$ を考えましょう: $$L_1\ket{\Delta,\lambda}= \lambda \ket{\Delta,\lambda}, \quad L_2\ket{\Delta,\lambda}=0\hbox{。}$$ すると $n>2$ について $L_n\ket{\Delta,\lambda}=0$ は自動的に従います。 調和振動子にならって、$e^{\lambda L_{-1}}\ket{\Delta}$ を考えたいところですが、ビラソロ代数の交換関係はそれほど簡単でないため、それでは安直すぎます。
まず愚直に計算してみましょう。欲しい状態は、なんにせよ展開できる筈ですから、$$\ket{\Delta,\lambda}=\ket{\Delta} + c_1 L_{-1} \ket{\Delta} + c_{11} L_{-1}^2 \ket{\Delta} + c_2 L_{-2} \ket{\Delta} + \cdots$$と書きます。すると、 $$L_1 c_1 L_{-1} \ket{\Delta}= \lambda \ket{\Delta}$$ から $c_1= \lambda/(2\Delta)$ と定まり、$$\begin{aligned}
L_1 (c_{11} L_{-1} ^2 \ket{\Delta} + c_2 L_{-2} \ket{\Delta})&= \lambda c_1 L_{-1} \ket{\Delta}, &
L_2 (c_{11} L_{-1} ^2 \ket{\Delta} + c_2 L_{-2} \ket{\Delta})&=0\end{aligned}$$ から $c_{11}$、 $c_{2}$ が定まります。長さの二乗は、これより $$\vev{\Delta,\lambda|\Delta,\lambda}=1+\frac{\lambda^2}{2\Delta}+\frac{\lambda^4(c+8\Delta)}{4\Delta((1+\Delta)c-10\Delta+16\Delta^2)}+\cdots$$ となります。
さて、この調子で下から係数はすべて決められるのでしょうか? 次数 2 では、未知数が $c_{11}$ と $c_{2}$ が丁度二つ、方程式は次数 1 に状態が 1 つ、次数 0 にも状態が 1 つあったので、こちらも 2 つで無事解くことができました。これは $p_2=p_1+p_0$ だったという偶然に基づくもので、一般には $N$ が大きくなると $p_N \ll p_{N-1}+p_{N-2}$ となり、方程式の数が過剰になって解くのが一見困難になります。実際に計算をしてみると、それでも無事に解く事ができることがわかります。
その理由はこうです[@Marshakov:2009gn]。欲しいコヒーレント状態があったとして、それを次数毎にまとめて $$\ket{\Delta,\lambda}=\ket{\psi_0}+\lambda\ket{\psi_1}+\lambda^2\ket{\psi_2}+\cdots$$ とします。すると、 $$L_1\ket{\psi_N} = \ket{\psi_{N-1}}\hbox{、} \quad
L_2\ket{\psi_N}=0$$ となっているはずです。すると、簡単にわかるように、$$\vev{\Delta| (L_1)^N |\psi_N}=1\hbox{、}$$ また、それ以外の組み合わせでは $$\vev{\Delta| (L_1)^{N-2} L_2 |\psi_N} =0$$ 等となります。すなわち、$\ket{\psi_N}$ は、次数 $N$ の状態の中で、 $L_{-1}^N \ket{\Delta}$ とのみ内積を持って、他の基底とは直交しているわけです。そこで、$$\ket{\psi_N}=\sum_{i_1+\cdots+i_k=N} (M^{-1}_N)^{11\cdots1,i_1i_2\cdots i_k} L_{-i_1}L_{-i_2} \cdots L_{-i_k} \ket{\Delta}$$ と取ればよいことがわかりました。 これから、$\ket{\Delta,\lambda}$ の内積もすぐ計算でき、$$\vev{\Delta,\lambda | \Delta,\lambda}= 1 + \lambda^2 (M^{-1}_1)^{1,1}+ \lambda^4 (M^{-1}_2)^{11,11}+ \lambda^6 (M^{-1}_3)^{111,111} +\cdots\label{russian}$$ となります。 すぐわかるように $e^{tL_0} \ket{\Delta,\lambda}= e^{t\Delta} \ket{\Delta,e^t \lambda}$ ですから、$$\vev{\Delta,\lambda | \Delta,\lambda}= \vev{\Delta,1 | \lambda^{2(L_0-\Delta)} | \Delta,1}$$ と書き直せます。 $L_0$ が円柱に二次元共形場理論を置いたときの時間発展の演算子だったことを思い出しますと、この節で計算した量は、図 \[HH\] のように図示することができます。
$$\includegraphics[width=.3\textwidth]{cyl}$$
四次元ゲージ理論とインスタントンの統計力学
==========================================
非可換ゲージ理論
----------------
さて、すっかり話を変えて、この節では四次元のゲージ理論の話をします。(この節の内容の詳細は、例えば教科書 [@Coleman] や、講義録[@tHooft:1999au]を参照のこと。) ゲージ理論の一番簡単な例は Maxwell の電磁気学です。電場 $\vec E$ と磁場 $\vec B$ は相対論的形式では $F_{\mu\nu}=-F_{\nu\mu}$ にまとまるのでした: $$F_{0i}=E_i\hbox{、}\quad
F_{12}=B_3\hbox{、} \ F_{23}=B_1\hbox{、}\ F_{31}=B_2\hbox{。}$$すると、方程式は $$\partial_\mu F_{\mu\nu}=0\hbox{、} \qquad \partial_{\mu} F_{\nu\rho} + \partial_{\nu} F_{\rho\mu} + \partial_{\rho} F_{\mu\nu}=0 \label{MaxwellEOM}$$と書けます。但し前者では拡張された Einstein の規約、すなわち同じ添字が二度現れると適切に計量 $\eta^{\mu\nu}$ を入れて足し上げるという規約を使いました: $$\partial_\mu F_{\mu\nu}= \sum_{\rho,\mu}\partial_\rho \eta^{\rho\mu} F_{\mu\nu}\hbox{。}$$ の後者は四元ベクトルポテンシャル $A_\mu$ を用いて $$F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$$とすると自動的に解くことが出来ます。但し、$F_{\mu\nu}$ から $A_\mu$ は一意的には定まらず、$\chi$ を勝手なスカラー関数として $$A_\mu \mapsto A_\mu+\partial_\mu \chi \label{MaxwellGauge}$$ としても $F_{\mu\nu}$ は変わりません。これをゲージ変換と言います。 もうひとつ重要な点は、 $A_\mu$ を基本的な力学変数だと思うと作用を書く事ができます、すなわち $$S=\int d^4x \frac14 F_{\mu\nu} F_{\mu\nu}$$ を変分することによって が出て来ます。 Maxwell 方程式の簡単でうれしいところは、その線形性です。すなわち、$A_\mu$ と $A'_\mu$ を 2 つの解とすると、$A_\mu+A_\mu'$ も自動的に解になります。
以上は古典論でしたが、これを量子論にするには、 Feynman の経路積分をする必要があります、すなわち $$Z=\int [DA_\mu] e^{iS}$$ として、可能なすべてのベクトルポテンシャル $A_\mu$ の配位に対して、位相 $e^{iS}$ をつけて積分せよ、という操作です。 以上は計量が $\eta=\diag(-1,+1,+1,+1)$ のミンコフスキ空間での議論でしたが、以下簡単のため計量が $\delta=\diag(+1,+1,+1,+1)$ のユークリッド空間に話を変えることにします。すると経路積分は $$Z=\int [DA_\mu] e^{-S}$$となって、未だ無限次元の積分ですが少しは扱いやすくなります。
20世紀後半の物理の大きな発見のひとつは、電磁気以外にあるこの世の他の2つの力、「強い力」と「弱い力」がどちらもこの Maxwell 理論の拡張である非可換ゲージ場の理論で書かれるということでした。 まず、非可換群 $\SU(N)$ をおさらいしましょう。$g\in \SU(N)$ は複素 $N\times N$ 行列で、ユニタリ $g^\dagger g=1$、 さらに $\det g=1$ としたものです。$\SU(2)$ は特に $$g = \begin{pmatrix}
z & -\bar w \\
w & \bar z
\end{pmatrix}$$ で且つ $|z|^2+|w|^2=1$ というものですから、$\SU(2)\simeq S^3$ であることがわかります。
$g$ が単位元に近いとして、 $g=1+\epsilon+\cdots $ と書きますと、$g^\dagger g=1$ から $\epsilon+\epsilon^\dagger=0$、 更に $\det g=1$ から $\tr \epsilon=0$ となります。 というわけで、反エルミートでトレースがゼロの行列全体を $\SU(N)$ のリー代数と言います。
これをつかって、ゲージポテンシャル $A_\mu$ が $\mu=1,2,3,4$ に対して $\SU(N)$ のリー代数に入っているとしましょう。 ゲージ場の強さを $$F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu +[A_\mu,A_\nu]$$ と定めると、運動方程式は $$\partial_\mu F_{\mu\nu} + [A_\mu,F_{\mu\nu}]=0 \label{YMEOM}$$ となります。これが Yang-Mills 場の方程式です。これを与える作用は $$S=-\frac{1}{2 g^2} \int \tr F_{\mu\nu} F_{\mu\nu}$$です。実際に変分してみてください。$N\times N$ 行列の場合に、これを $\SU(N)$ ゲージ理論と呼びます。$g$ は結合定数と呼ばれます。
「弱い力」はこれで $N=2$ としたもの、「強い力」は $N=3$ として、さらに量子論にしたもので書かれることがわかっています。 実際に計算するには、やはり経路積分をします: $$Z=\int [DA_\mu] e^{-S} \label{YMpathintegral}$$ 勿論、これは無限次元積分になって、厳密な定義が出来ると Clay 賞の半分を取ったようなものですが、数学的厳密さにそれほどこだわらなければ、例えば時空 $\bR^4$ を非常に細かな格子 $\bZ^4$ で近似して、超巨大スーパーコンピューターで積分を計算してやることができます。そうすると、たとえばいろいろなハドロンの質量比が計算誤差の範囲できちんと再現できることがわかっています。
さて、Maxwell 場の作用がゲージ変換 で不変だったように、非可換ゲージ場の作用もゲージ変換で不変です。まず、$A_\mu$ に対するゲージ変換は時空から $\SU(N)$ への写像 $g(x)$ を使って $$A_\mu \mapsto g A_\mu g^{-1} + g \partial_\mu g^{-1} \label{YMgauge}$$とするものとします。すると、$F_{\mu\nu}$ に対しては $$F_{\mu\nu} \mapsto g F_{\mu\nu} g^{-1}$$ となり、作用の被積分関数は $$\tr F_{\mu\nu} F_{\mu\nu} \mapsto \tr g F_{\mu\nu} g^{-1} g F_{\mu\nu} g^{-1} =\tr F_{\mu\nu} F_{\mu\nu}$$ となることが示せます。 ですから、通常 $$\lim_{|x|\to \infty} g(x) = 1$$ となる $g(x)$ でのゲージ変換で結びつく二つの Yang-Mills 方程式の解は同一視します。これを、局所ゲージ変換での同一視と呼びましょう。 一方で、 $g(x)\equiv g$ と場所に依らない場合は での微分項が落ちます。 これを大域ゲージ変換と呼び、大域ゲージ変換で結びつく二つの解は同一視しないことにします。
Yang-Mills 場の運動方程式 には $A$ について三次の項がありますから、非線形です。解 $A_\mu$ と $A'_\mu$ とが二つ得られても、$A_\mu + A'_\mu$ は解にはなりません。ですから、フーリエ変換すれば解が求まるというわけでもありません。しかし、インスタントンと呼ばれる一連の具体的な解が知られており、それらは非常に詳細に調べる事ができます。そこで、それについて次に説明しましょう。
インスタントン
--------------
Yang-Mills 場の経路積分 をすることを考えましょう。$S$ が小さいほうが寄与が大きいですから、$S$ を最小化する配位が重要そうです。 それを調べる為に、まず双対場 $$\tilde F_{\mu\nu} = \epsilon_{\mu\nu\rho\sigma} F_{\mu\nu}/2$$ を導入しましょう。これは、非相対論的に成分を電場 $\vec E$ と磁場 $\vec B$ に分ければ、ちょうど $\vec E$ と $\vec B$ を入れ替えたものです。
勝手な配位 $A_\mu$ が与えられた際、作用が発散しないように無限遠で充分速く $F_{\mu\nu}\to 0$ となっているとします。このとき、$k$ を整数として $$-\int d^4 x \tr F_{\mu\nu} \tilde F_{\mu\nu} = -4 \int d^4 x \tr \vec E\cdot \vec B=16\pi^2 k$$ となることが知られています。 ですから、$A_\mu$ の配位の空間は無限次元ですが、それは整数 $k$ でラベルされる部分空間にわかれているわけです (図\[components\])。 これは下ですぐに示しますので、それまで認めて頂きましょう。この $k$ は物理ではインスタントン数、数学では第二チャーン数と呼ばれます。
$$\includegraphics[width=.8\textwidth]{components}$$
さて、 $$\begin{aligned}
-\tr F_{\mu\nu}F_{\mu\nu} &= -2\tr (\vec E^2+\vec B^2) \\
&=-2\tr( \vec E\pm \vec B)^2 \pm 4 \tr \vec E\cdot\vec B \\
&\ge \pm 4 \tr \vec E\cdot\vec B = \pm\tr F_{\mu\nu}\tilde F_{\mu\nu}\end{aligned}$$ ですから、 $$\int d^4 x \tr F_{\mu\nu} F_{\mu\nu} \ge \left|\int d^4 x \tr F_{\mu\nu} \tilde F_{\mu\nu}\right| = 16\pi^2 |k|\hbox{、}$$ この等号を満たすには $k$ が正ならば $$\vec B+\vec E=0, \quad\hbox{もしくは}\quad F_{\mu\nu} + \tilde F_{\mu\nu}=0 \label{SD}$$ であればよいことがわかります(再度図\[components\]を参照下さい)。これを反自己双対方程式といいます。 $k$ が負の場合は $F_{\mu\nu}=\tilde F_{\mu\nu}$ という自己双対方程式を考えれば良いですが、本質的に同じですので、今後は $k$ は非負としましょう。
さて、反自己双対方程式を調べる前に、$k$ が整数になることを確かめましょう。まず、$\tr F_{\mu\nu} \tilde F_{\mu\nu}$ が全微分であることに注意します: $$\tr F_{\mu\nu} \tilde F_{\mu\nu} = \partial_{\mu} \epsilon_{\mu\nu\rho\sigma} \tr( A_\nu F_{\rho\sigma} -\frac13 A_\nu A_\rho A_\sigma)\hbox{。}$$ よって、$$-\int d^4x \tr F_{\mu\nu} \tilde F_{\mu\nu}
= \int_{S^3} dn_\mu \epsilon_{\mu\nu\rho\sigma} \tr ( -A_\nu F_{\rho\sigma} + \frac13 A_\nu A_\rho A_\sigma)$$ です。無限遠で $F_{\mu\nu}=0$ としましたから、ゲージ変換 を $F_{\mu\nu}=0$、 $A_{\mu}=0$ から逆に使って、$$= \frac13\int_{S^3} dn_\mu \epsilon_{\mu\nu\rho\sigma} (g^{-1}\partial_\nu g)(g^{-1}\partial_\rho g)(g^{-1}\partial_\sigma g)\hbox{、}$$ ただし、$g$ は無限遠の $S^3$ から群 $\SU(N)$ への写像です。もっとも簡単な $N=2$ のときを考えますと、$\SU(2)\sim S^3$ ですから、$g$ は $$g: S^3\to S^3$$ と思うことができ、上記積分はこの写像が何回巻き付いているかを測ったものになります。比例係数を計算すると、まきつき数を $k$ として$$= 16\pi^2 k\hbox{。}$$
反自己双対方程式の良いところは、まず、これを満たせば自動的に Yang-Mills 方程式を満たすことがあります: $$\partial_\mu F_{\mu\nu}+ [A_\mu, F_{\mu\nu}] = -\partial_\mu \tilde F_{\mu\nu} - [A_\mu,\tilde F_{\mu\nu}]$$ですが、右辺に $F_{\mu\nu}$ の定義を代入すると自動的にゼロになることがわかります。 また、Yang-Mills 方程式は二階の微分方程式ですが、反自己双対方程式は一階です。さらに、非線形項も三次でなくて二次でおさまります。
### 1-インスタントン解
では、インスタントン数が $k$ の反自己双対解はどんな形をしているのでしょうか? まず一番簡単な $k=1$ の場合を見ましょう。これは一般に $$A_\mu(x) = \frac{H_{\mu\nu} (x-x_0)_\nu}{|x-x_0|^2 +\rho^2}$$ という形をしていると知られています。ここで $x_0$ はインスタントンの中心、$\rho$ はインスタントンの大きさを決めます。 $H_{\mu\nu}$ はどう取ればいいでしょうか? 答えを反自己双対にしたいので、$H_{\mu\nu}=-H_{\nu\mu}$、$H_{\mu\nu}=-\tilde H_{\mu\nu}$ という $N\times N$ 行列をとることにします。この形を反自己双対方程式に代入すると、$H_{01}$、 $H_{02}$、 $H_{03}$ が $$H_{01}=[H_{02},H_{03}]\hbox{、} \quad
H_{02}=[H_{03},H_{01}]\hbox{、}\quad
H_{03}=[H_{01},H_{02}]$$ と、$\SO(3)$ の交換関係をみたすべきことがわかります。さらに、インスタントン数を計算すると $$k=2\tr H_{03}{}^2$$ となることもわかります。ですから、一番簡単な解は、$$H_{0i}=\frac12 i\sigma_i \oplus O_{N-2}$$とすることです。但し $\sigma_{1,2,3}$ は通常のパウリ行列 $$\sigma_1=\begin{pmatrix}
0&1 \\
1&0
\end{pmatrix}, \quad
\sigma_2=\begin{pmatrix}
0&i \\
i&0
\end{pmatrix}, \quad
\sigma_3=\begin{pmatrix}
1&0 \\
0&-1
\end{pmatrix}$$で、これは $2\times 2$ 行列ですから、$O_{N-2}$ は$(N-2)\times (N-2)$ 行列で全てゼロなものとして、それを付け足して $N\times N$ 行列にすることにします。$\oplus$ は行列をブロック対角にならべる操作です。
ひとつ解ができると、勝手な $\SU(N)$ 行列 $g$ を取って $$H_{0i}=g^{-1}(\frac12 i \sigma_i \oplus O_{N-2}) g\label{g}$$としても当然解になります。$g$ は $ (h \, \mathrm{Id}_{2\times 2}) \oplus g_{(N-2)\times (N-2)} $ という形の行列だと空回りしますので、これで $$N^2-1-(N-2)^2= 4N-5$$ 自由度だけ解が得られたことになります。$x_0$ にある四つの自由度および $\rho$ にある一つの自由度を足すと、合計 $4N$ 自由度あることがわかりました。 これらの自由度のことをインスタントンのモジュライと呼びます。
特に $N=2$ の場合は、$4\cdot 2-5=3$ 自由度は $\SU(2)$ 行列 $g$ でまわす自由度そのものです。この自由度は $\SU(2)\simeq S^3$ だけあります。ただし、$$g=\begin{pmatrix}
-1 & 0\\
0& -1
\end{pmatrix}$$ の場合は交換してしまって $H_{\mu\nu}$ に影響がないので、実際に意味があるのは $\SU(2) / \{\pm 1\} \simeq S^3/\bZ_2$ だけです。 結論として、$\SU(2)$ の1-インスタントン解のモジュライの空間 $\cM_{2,1}$ は $$\cM_{2,1}\simeq \bR^4 \times \bR_+ \times S^3/\bZ_2 \simeq \bR^4 \times \bR^4/\bZ_2$$ であたえられることがわかります。但し、右辺では大きさのパラメタ $\rho$ とゲージの向きのパラメタ $S^3/\bZ_2$ をくみあわせて $\bR^4/\bZ_2$ としました。
### 多重インスタントン解
さて、前節では 1-インスタントン解には中心の位置に $4$ パラメタ、サイズに $1$ パラメタ、ゲージの向きに $4N-5$ パラメタあることを学びました。具体的に作用密度 $\tr F_{\mu\nu}^2$ を図示すると図 \[instanton\] のようになります。これからもわかるように、中心から離れるとゲージ場の強さ $F_{\mu\nu}$ は十分に小さくなります。
$$\includegraphics[width=.3\textwidth]{1instanton}$$
$$\begin{array}{cc}
\raisebox{0cm}{\includegraphics[width=.3\textwidth]{2instanton-far-parallel}}&
\raisebox{0cm}{\includegraphics[width=.3\textwidth]{2instanton-far-antiparallel}}\\
\downarrow & \downarrow \\
\raisebox{0cm}{\includegraphics[width=.3\textwidth]{2instanton-near-parallel}}&
\raisebox{0cm}{\includegraphics[width=.3\textwidth]{2instanton-near-antiparallel}}
\end{array}$$
ですから、1-インスタントン解をふたつ、片方 $A'_\mu$ は中心が $x_0'$ サイズが $\rho'$、 もうひとつ $A_\mu''$は中心が $x_0''$ サイズが $\rho''$ であるように取ると、 $|x_0'-x_0''| \gg \rho'+\rho''$ であれば、両者が同時に大きな値にはなりませんから、$A_\mu' + A_\mu''$ もほとんど反自己双対方程式の解になります。 勿論 $A_\mu' A_\mu''$ からくる補正があるので、それを修正してやらないといけません: $$A_\mu = A_\mu' + A_\mu'' + \text{小さな補正}\hbox{。}$$ この領域では、2-インスタントン解には $8N$ パラメタがあることがわかります。インスタントンが二つ近づいた場合は、このような安直な解析ではいけませんが、その場合でも方程式をきちんと解けることが知られています(図\[2instanton\])。
一般にインスタントン数が $k$ の場合は、同様にして全てのインスタントンの中心が離れていると、重ね合わせることによって $4Nk$個モジュライをもった解がつくれます。 いいかえると、このモジュライ空間を $\cM_{N,k}$ と書くと、モジュライ空間の外の方ではおおよそ $$\cM_{N,k} \sim (\cM_{N,1})^k / S_k$$ となっている、すなわち 1-インスタントンのモジュライのコピーが $k$ 個あってそれを置換群 $S_k$ で同一視したものになっていますが、中心部はもっと複雑になっています。
$\cM_{N,k}$ を具体的に書き下す方法も Atiyah-Drinfeld-Hitchin-Manin [@Atiyah:1978ri] によって示されています。それをきちんと説明する時間は到底ありませんが、雰囲気だけは説明したいと思います(この節の詳細はレビュー [@Dorey:2002ik] や講義録 [@Ito] 等を参照のこと。)。 まず、$k=1$ の場合に戻りますと、解の微妙な部分は $H_{0i}$ で与えられていました。$H_{0i}$ は、$X=H_{01}+iH_{02}$ で から復元できますが、条件 $X^2=0$ と $\tr |X|^2=1$ が必要です。ここで、 $\rho$ の自由度を $X$ に含めてしまえば、$\tr |X|^2 $に関する条件は落とすことができます。そこで、$X^2=0$ にだけ注目しましょう。すると、$X$ の階数は $1$ なので、$$X^i_j=B^iA_j$$ と書く事ができます。但し $i,j=1,\ldots,N$。 $X^2=0$ を満足させるために $$A_i B^i=0 \label{foo}$$ を要求して、さらに勝手な $c\in \bC$ に対して $(A_i,B_i) \to (cA_i,c^{-1} B_i)$ としても $X$ が変わらないので、$$A_i \overline{A}^i -B^j \overline{B}_j =0\label{bar}$$としてせめて$c\in \bR$ の自由度は消すことにしましょう。おしまいに、中心の自由度 $x_0\in \bR^4$ を $(z,w)\in \bC^2$ と書くとすると、結局 1-インスタントン解は $(z, w, A_i,B^i)$ で 、 を満たし、さらに $$(A_i,B^i)\to (e^{i\theta} A_i,e^{-i\theta}B^i) \label{baz}$$ という作用に対して同一視をしたもの、と書く事ができます。 Atiyah-Drinfeld-Hitchin-Manin は、これの自然な拡張が $k$-インスタントン解のモジュライを記述する事を見いだしました。 答えだけ書きますと、$1,\ldots,k$ を走る添字 $a,b$ を用意し、上記の $z,w,A,B$ に添字を追加して $z^a_b$, $w^a_b$, $A_i^a$, $B^j_a$ とします。そうして、 の拡張として $$A_i^a B^i_b +[z,w]^a_b =0\hbox{、} \label{FOO}$$ の拡張として $$A_i^a \overline{A}^i_b - B^i_b \overline{B}^a_i + [z,z^\dagger]^a_b + [w,w^\dagger]^a_b=0\label{BAR}$$ を課し、 の拡張として、$$(A_i^a,B^i_b,z,w)\to (g^a_b A_i^b,B^i_a (g^{-1})^a_b, gzg^{-1}, gwg^{-1}) \label{BAZ}$$ という $k\times k$ ユニタリ行列 $g^a_b$ の作用で同一視することにせよ、というのが彼らの見つけた表示です。
モジュライの数を勘定しましょう。$A$ $B$ には合計 $4Nk$ 自由度があり、$z$ $w$ には $4k^2$ 個自由度があります。 で $2k^2$ 個条件を課し、 で $k^2$ 個、さらに で $k^2$ 個自由度を取り除くので、結局 $4Nk$ 個自由度があることになります。 1-インスタントン解を $k$ 個とってきた場合も、対応する $(A_i,B^i,z,w)$ を $k$ 組とってくれば、 をおおよそ解く行列を作るのはブロック対角に並べればいいですが、$[z,w]$ の交換子のあたりからインスタントン間の相互作用が出てくるわけです。
インスタントンの統計力学
------------------------
さて、インスタントン解について多少学んだところで、経路積分の評価にもどりましょう。作用はインスタントン数が $k$ の配位の中では反自己双対な配位で最小になり、経路積分にもっとも寄与するのですから、一般の配位 $A_\mu$ を $$A_\mu = A_\mu^\text{ASD} + \delta A_\mu$$のように反自己双対部分 $A_\mu^\text{ASD}$ とそこからのずれ $\delta A_\mu$ に分解することを考えます(図\[decomp\])。
$$\includegraphics[width=.3\textwidth]{decomp}$$
$\delta A_\mu$ が小さければ、$$S = \frac{8\pi^2 k}{ g^2} + \int d^4 x \text{($\delta A_\mu$の二次式)} + \text{(高次項)}$$となり、経路積分の積分変数も分解すると $$Z=\int [DA_\mu] e^{-S} =\sum_k \int [DA_\mu^\text{ASD}] \int [\delta A_\mu] e^{-\frac{8\pi^2 k}{g^2} + \cdots }$$となります。この展開を利用して量子 Yang-Mills 理論を調べようとはじめにがんばったのが ’t Hooft の論文[@'tHooft:1976fv]ですが、揺らぎ $\delta A_\mu$ の処理は非常に大変なので、今回はそれをすっかり省略して、以下のトイモデルを考えましょう: $$Z_\text{toy}^\text{instanton}=\sum_k q^k \int [DA_\mu^\text{SD}] = \sum_k q^k \int_{\cM_{N,k}} d\vol\hbox{、}$$ 但し $d\vol$ $\cM_{N,k}$ の上の自然な体積形式です。 $q=\exp(-{8\pi/ g^2})$ はインスタントン数の化学ポテンシャル、すなわちをインスタントンを一つ系に導入する際のコストだと思う事ができます。 あとは自然な体積形式以外は何も積分していませんから、どのパラメタの反自己双対配位も同じ確率で起こりうるという状況を考えています。
勿論このままでは $Z$ は時空の積分のために発散します。すなわち、$d^{4Nk}s$ の中には、インスタントンの中心の位置 $\bR^4$ に関する積分があるので、その分の発散があります。統計力学を学びますと、通常この問題は時空を一辺 $L$ の大きな超立方体の箱に入れて計算して、$\log Z \sim L^4$ となるその比例定数を取りだすことで処理しますが、インスタントンは箱にいれると更に解析が難しくなるという性質がありますので、ちょっと別のことをしてみます。
そのために、もっと簡単な点粒子の模型を考えましょう、単に粒子が $(x,y)\in \bR^2$ のどの箇所にも同じ確率で存在しうるとします。すると、勿論分配関数は $$Z=\int_{-\infty}^\infty\int_{-\infty}^\infty dxdy =\infty$$となって発散します。そのかわりに、ガウス型の因子を手でいれて収束させましょう: $$Z_{\epsilon}= \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-\pi \epsilon (x^2+y^2)}dxdy = \frac1{\epsilon}\hbox{。}$$ $\epsilon\to 0$ とすると収束因子を取り払うことになって、それにともなって $Z$ も発散します。ですから $1/\epsilon$ は収束因子を入れた際の実効的な $\bR^2$ の面積と思うことが出来ます。
これはどうみても手でむりやり収束させたような感じですが、もうすこし意味付けをすることができます。$(x,y)$ は空間の場所としましたが、ポアソン括弧を入れて力学系の相空間の座標と思うことにします: $$\{x,y\}_\text{P.B}=1\hbox{。}$$ そうすると、$J=(x^2+y^2)/2$ と書くと、$$\{H,x\}_\text{P.B.}=y\hbox{、}\quad
\{H,y\}_\text{P.B.}=-x$$ですから、$J$ は $(x,y)$ 平面の回転のハミルトニアンですね(図\[harmonic\])。ですから、上記の $Z_\epsilon$ は、 $$Z_\epsilon = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-2\pi \epsilon J}dxdy$$ と、回転のハミルトニアン $J$ が大きくなると $\epsilon$ の重みで損をするように積分しなさい、ということだと思えます。今回はこの方法で系を箱に入れましょう。
$$\includegraphics[width=.3\textwidth]{harmonic}$$
インスタントンに話を戻しますと、今は時空は四次元で $x_1,x_2,x_3,x_4$ とありますから、ポアソン括弧を $$\{x_1,x_2\}_\text{P.B.}=1\hbox{、}\qquad
\{x_3,x_4\}_\text{P.B.}=1$$ その他はゼロ、として位相空間と思うことにしましょう。このように $\bR^4$ に位相空間の構造をいれると、自然にインスタントンのモジュライ空間 $\cM_{N,k}$ にも相空間の構造が入ることが知られています。 $\cM_{N,k}$ の一点をとりましょう。すると、ひとつ $A_\mu(x)$ というインスタントン配位が定まりますから、 $\bR^4$ の回転をすると、$A_\mu'(x)$ という別個のインスタントン配位が定まり、$\cM_{N,k}$ の別の一点になります。ですから、$\bR^4$ の回転は自然に $\cM_{N,k}$ に働きます。 そこで、$\bR^4$ の $(x_1,x_2)$ 平面を回す回転が $\cM_{N,k}$ に引き起こす作用を考え、これのハミルトニアンを $J_1$, $\bR^4$ の $(x_3,x_4)$ を回す回転に対応するハミルトニアンを $J_2$ と呼ぶことにしましょう。それぞれに対する重みを $\epsilon_{1,2}$ と呼ぶと、$$e^{-2\pi (\epsilon_1 J_1 + \epsilon_2 J_2)}$$ という重みをつけることにします。
また、大域ゲージ変換も $\cM_{N,k}$ の変換になります。簡単のため $\SU(2)$ を考えると、$\sigma_3$ で生成されるゲージ変換に対応するハミルトニアン $K$ を考え、重み $a$ をつけましょう。すると、指数関数の肩に $aK$ という因子をいれることになります。
一般の $\SU(N)$ では、対角行列のゲージ変換 $\diag(a_1,\ldots,a_N)$ に対して、ハミルトニアン $K_1,\ldots, K_N$ を考え、指数関数の肩に $\sum_ia_i K_i$ という因子をいれることにします。
というわけで、我々が考えたいインスタントンの統計力学模型は次のようなものです: $$Z^\text{instanton}_{\epsilon_1,\epsilon_2;a_i} = \sum_k q^k Z_{N,k}\hbox{、} \quad \text{但し}\quad Z_{N,k}=\int_{\cM_{N,k}} e^{-2\pi(\epsilon_1 J_1 + \epsilon_2 J_2 + \sum_i a_i K_i)} d\vol\hbox{。}$$
随分ややこしいことを言いましたが、結局は、インスタントンを沢山いれると $q$ だけ損をする、 $(x_1,x_2)$ 平面内の角運動量が大きいと $\epsilon_1 J_1$ だけ損をする、 $(x_3,x_4)$ 平面内の角運動量が大きいと $\epsilon_2 J_2$ だけ損をする、 ゲージ変換性が大きいと パラメタ $a_i$ に応じて $\sum_i a_i K_i$ だけ損をする、という、インスタントンが沢山あったばあいに統計力学的に扱おうとする際に考えることの出来るもっとも簡単な模型になっています。$J_1$、 $J_2$、 $K_i$ も具体形は結局は単に調和ポテンシャルになるだけです。
それを実際に $\SU(2)$ の 1-インスタントンの場合に確かめてみましょう。モジュライ空間は先に $\bR^4 \times \bR^4/\bZ_2$ だといいました。ひとつめの $\bR^4$ はインスタントンの中心 $(x_1,x_2,x_3,x_4)$ をパラメタしていて、ふたつめの $\bR^4$ はゲージの向きをあらわす行列 $H_{0i}$ と インスタントンのサイズ $\rho$ をあわせたものでした。 ふたつめの $\bR^4/\bZ_2$ を複素数ふたつ $(z,w)$ を $(z,w)\to -(z,w)$で同一視したものと思うことにしましょう。
ゲージ回転は単に $(z,w)$ が$\SU(2)$ の二次元表現ですから $$(z,w)\mapsto (e^{ia}z,e^{-ia} w)$$ と働きます。一方、 $x_1,x_2$ 平面の $\epsilon_1$ 回転および $x_3,x_4$ 平面の $\epsilon_2$ 回転は $$(x_1+ ix_2,x_3+ix_4,z,w) \mapsto (e^{i\epsilon_1}(x_1+ix_2), e^{i\epsilon_2}(x_3+ix_4), e^{i(\epsilon_1+\epsilon_2)/2}z,e^{i(\epsilon_1+\epsilon_2)/2}w)$$ と作用します。ここで、$(z,w)$ への作用は、時空の回転が $H_{\mu\nu}$ を混ぜることから、式 にあらわれる $g$ を変換する必要があり、$g$ が $(z,w)\in\bC^2$ の角度部分であったことから生じます。
以上から、対応するハミルトニアンは $$\begin{gathered}
\epsilon_1 J_1 + \epsilon_2 J_2 + a K = \frac{\epsilon_1}2 (x_1^2+x_2^2)+\frac{\epsilon_2}2 (x_3^2+x_4^2)\\
+ \frac{(\epsilon_1+\epsilon_2)/2+a}2 |z|^2 + \frac{(\epsilon_1+\epsilon_2)/2-a}2 |w|^2 \end{gathered}$$ ですね。 すると $$Z_{2,1}
= \frac12 \frac{1}{\epsilon_1}\frac{1}{\epsilon_2} \frac{1}{(\epsilon_1+\epsilon_2)/2 +a} \frac{1}{(\epsilon_1+\epsilon_2)/2 -a}\label{M21}$$ となります。先頭の $1/2$ は $\bR^4/\bZ_2$ の $/\bZ_2$ から来ます。
すこしだけこの計算結果の解釈をしておきましょう。$1/(\epsilon_1\epsilon_2)$ は時空の箱の大きさですから、箱を大きくするために $\epsilon_{1,2}\to 0$ とします。するとガウス積分 の収束性が悪くなりますから、$a$ は純虚として計算が問題ないようにします。 すると、時空の箱が大きい極限では、インスタントンが一個あることによる効果は単位時空体積あたり $$\epsilon_1\epsilon_2\log Z_{2,1}\sim \frac12 \frac{1}{(\epsilon_1+\epsilon_2)/2 +a} \frac{1}{(\epsilon_1+\epsilon_2)/2 -a}
\sim -\frac12\frac1{a^2}$$ となるわけです、ただし $|a|\gg |\epsilon_{1,2}|$ としました。
積分の局所化
------------
さて、$\SU(2)$ の場合の $Z^\text{instanton}$ を計算するには、この調子で $\cM_{2,2}$, $\cM_{2,3}$, …上の積分を計算できればいいのですが、それを直接行うのはなかなか大変です。そこで、Duistermaat-Heckman の局所化公式というものをつかいます。(この節の内容の詳細は、教科書[@GS]を参照のこと。)
この公式は、丁度我々が計算したいような、ハミルトニアンを指数関数の肩にのせたものの積分をたちどころに出来てしまうものです。一番簡単な例として、二次元球面を考えます。緯度を $-\frac\pi2<\theta<\frac\pi 2$、 経度を $0<\psi<2\pi$ として、面要素を $\cos\theta d\theta d\psi$ とします。経度方向回転のハミルトニアンは $H=\sin\theta $です。そこで $$Z=\iint e^{-2\pi\epsilon H} \cos\theta d\theta d\psi$$を考えましょう。これは直接計算できて、 $$=\frac{e^{-H(\theta=\pi/2)}}{\epsilon}-\frac{e^{-H(\theta=-\pi/2)}}{\epsilon}$$ となりますが、これを次のように書きます[^2] : $$=\sum_{p=\text{北極,南極}} \frac{e^{-H(p)}}{\text{($p$ での回転角)}}\hbox{。}$$
Duistermaat-Heckman の公式は、これが一般になりたつ、というもので、$M$ が $2n$次元の滑らかな相空間で、$H$ がハミルトニアンとしてその上に流れを引き起こし、流れの固定点 $p$ が孤立しているとすると、$$Z=\int e^{-2\pi\epsilon H} d\mathrm{vol} = \sum_p \frac{e^{-H(p)}}{\prod_{i=1}^{n} \theta_{i,p}}\label{DH}$$ が成り立つ、というものです。ただし、各固定点 $p$ のまわりでは、流れは点 $p$ まわりの回転になりますから、$\bR^{2n}$ を $n$ 個の $\bR^2$ にわけてそれぞれの面が角度 $\theta_{i,p}$ ($i=1,\ldots,n$)で回転している、というようにしました。
もうひとつ実例を見ましょう。 $M=\bR^4$ として、$(x_1,x_2)$ を角速度 $\epsilon_1$ で、$(x_3,x_4)$ を角速度 $\epsilon_2$ で回すことを考えると、ハミルトニアンは $$J=\frac{\epsilon_1}2(x_1^2+x_2^2)+\frac{\epsilon_2}2(x_3^2+x_4^2)$$です。公式を適用するには、固定点を探さねばなりませんが、勿論それは原点だけで、そこでの回転角は $\epsilon_1$ と $\epsilon_2$ ですね。よって$$\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty
e^{-2\pi J }d^4 x = \frac{1}{\epsilon_1\epsilon_2}$$となるはずですが、これは単にガウス積分です。これは先ほどやった積分の $\bR^4$ 部分です。
この計算を $\bR^4/\bZ_2$ に適用するにはどうすればよいでしょうか? ひとつ問題は、定理は滑らかな多様体に対してのみ成り立つのですが、$\bR^4/\bZ_2$ は先端がとがっています。 とがりを解消する為、ブローアップという操作をします。まず、$(z,w)\in\bC^2\simeq \bR^4$ を座標として、$(z,w)\mapsto -(z,w)$ という $\bZ_2$ 作用で割っていますから、$u=z^2$, $v=w^2$, $t=zw$ が $\bZ_2$ 作用で不変です。これらは $uv=t^2$ を満たします。$u=v=t=0$ のあたりが非常に尖っています。
$$\begin{array}{c@{\qquad}c}
\includegraphics[width=.3\textwidth]{blowup}&
\includegraphics[width=.3\textwidth]{blowup-lines}\\
\downarrow\\
\includegraphics[width=.3\textwidth]{blowdown}&
\includegraphics[width=.3\textwidth]{blowdown-lines}
\end{array}$$
そこで、尖ったところに $S^2$ を差し込んで滑らかにします(図\[blowup\])。 $S^2$ の北極では局所的には $(u,v')$ 但し $uv'=t$、 $S^2$ の南極では局所的には $(u',v)$ 但し $u'v=t$ が良い座標になっています。 $z$ の回転角が $(\epsilon_1+\epsilon_2)/2+a$、 $w$ の回転角が $(\epsilon_1+\epsilon_2)/2-a$ でしたから、 北極での回転角は $$(\epsilon_1+\epsilon_2)+2a, -2a$$ 南極での回転角は $$2a, (\epsilon_1+\epsilon_2)-2a$$となります。そこで、Duistermaat-Heckman の公式をつかうと、$$\int e^{-2\pi H} d\mathrm{vol} = \frac{e^{-H_\text{北極}}}{((\epsilon_1+\epsilon_2)+2a)(-2a)}+
\frac{e^{-H_\text{南極}}}{(2a)((\epsilon_1+\epsilon_2)-2a)}$$ です。ブローアップするまえの $\bR^4/\bZ_2$ の積分を知るには、$S^2$ が小さくなる極限をとりますが、そうすると分子の $H$ は何にせよ原点に行ってゼロになりますので、$$\begin{aligned}
&\to \frac{1}{((\epsilon_1+\epsilon_2)+2a)(-2a)}+
\frac{1}{(2a)((\epsilon_1+\epsilon_2)-2a)}\\
&=\frac2{((\epsilon_1+\epsilon_2)+2a)((\epsilon_1+\epsilon_2)-2a)}\\
&=\frac12\frac1{((\epsilon_1+\epsilon_2)/2+a)((\epsilon_1+\epsilon_2)/2-a)}\end{aligned}$$となり、先ほどの計算を再現しました。
$\cM_{2,2}$ 全体の積分にするには、$\bR^4$ の積分を掛ければよいです。 すると、 では一項だったものを、$$Z_{2,1}= \frac{1}{\epsilon_1\epsilon_2}\frac{1}{((\epsilon_1+\epsilon_2)+2a)(-2a)}
+\frac{1}{\epsilon_1\epsilon_2}\frac{1}{(2a)((\epsilon_1+\epsilon_2)-2a)}\label{fubar}$$ と二項に分割したことになります。
多重インスタントン計算
----------------------
以上前節では$\bR^4\times \bR^4/\bZ_2$上のガウス積分を非常にまわりくどく行ったわけですが、この方法の良いところは、前節で紹介した ADHM 構成と組み合わせると、一般の $N$, $k$ に対して計算が出来るところです[@Moore:1997dj; @Nekrasov:2002qd; @Flume:2002az; @Nakajima:2003uh]。 導出は難しいのできちんとはできませんが、雰囲気だけ説明しましょう。$\SU(2)$ 1-インスタントンの場合、ブローアップは $uv=t^2$ の尖っているところに $S^2$ を埋め込みました。これは、ADHM 構成でいうと、尖っている状況は $$A_1 B_1+A_2B_2=0\hbox{、}\quad \sum_{i=1,2} |A_i|^2 - |B^i|^2=0$$ で $(A_i,B_i)\mapsto (e^{i\theta} A_i,e^{-i\theta} B_i) $ という同一視をしていたものを、$$A_1 B_1+A_2B_2=0\hbox{、} \quad \sum_{i=1,2} |A_i|^2 - |B^i|^2= r^2$$ と変えることに相当します。$B_i=0$ とすると、$$|A_1|^2+|A_2|^2=r^2$$という $S^3$ がありますが、これを $A_i\mapsto e^{i\theta}A_i$ で割ったのが、はめ込んだ $S^2$ になっていたわけです。$S^2$の北極南極は、$$(A_1,A_2)=(r,0)\hbox{、}\qquad
(A_1,A_2)=(0,r)$$ にそれぞれ対応します。前者をとりますと、$\SU(2)$ 変換をすると勿論回ってしまいます $$(A_1,A_2)=(r,0) \mapsto ( e^{ia}r,0)$$が、これはどうせ $(A_1,A_2)\simeq e^{i\theta}(A_1,A_2)$ と同一視するのでしたから、$$\simeq (r,0)$$ となって、$e^{i\theta}$ の範囲で固定されていることになります。$(A_1,A_2)=(0,r)$ の場合も同様ですね。しかし、$A_1,A_2$ の両方が $0$ でないと、$\SU(2)$ 回転すると $$(A_1,A_2)\mapsto (e^{ia} A_1,e^{-ia}A_2)$$と変換しますから、$e^{i\theta}$ で元に戻すことが出来ません。 ですから、肝は、$\SU(2)$ ゲージ回転および、$\bR^4$ の $(x_1,x_2)$ 平面と $(x_3,x_4)$平面の回転がどのように $e^{i\theta}$ 回転で相殺できるかということです。
一般には、を保ったまま、 の右辺を $r^2$ にすることが $\cM_{N,k}$ を滑らかにすることになります。固定点は $k=1$ のときと同様、$B^i=0$ のところにあります。 固定点も同様に$\SU(N)$ ゲージ回転および、$\bR^4$ の $(x_1,x_2)$ 平面と $(x_3,x_4)$平面の回転がどのように $\U(k)$ 回転で相殺できるかで決まっています。 $\SU(N)$ ゲージ回転を $\diag(a_1,\ldots,a_N)$ で、$\bR^4$ の回転を $\epsilon_{1,2}$ ですることにすると、固定点はヤング図の $N$ 個組 $Y_1,\ldots,Y_N$ で、箱の数が合計 $k$ であるようなものでラベル付けがなされます。その際、相殺に用いる$\U(k)$ 回転は、対角成分が $$a_i + j\epsilon_1 + k \epsilon_2$$ 但し $0\le j< (Y_i \text{の列の数})$, $0\le k<(Y_i \text{の $j$ 列目の高さ})$ となります。
$$\includegraphics[width=.2\textwidth]{young}$$
ですから、$\cM_{N,k}$ をブローアップしたもの上での積分は、公式 による固定点 $p=(Y_1,\ldots,Y_N)$ 上の足し上げになります。元の$\cM_{N,k}$ 上での積分は、分子の $e^{-H}$ の項が全て $1$ になって、 $$Z_{N,k}=\int_{\cM_{N,k}} e^{-2\pi(\epsilon_1 J_1+\epsilon_2 J_2+a_i K_i)} d\vol=
\sum_{p=(Y_1,\ldots,Y_N)} \prod_{i} \frac{1}{\theta_{i,p}}$$ となります。各固定点での角度の計算 $\theta_{i,p}$ を実行すると、 $$\begin{gathered}
Z_{N,k}= \sum_{Y_1,\ldots,Y_N} \prod_{i,j=1}^N
\prod_{s\in Y_i} (-L_{Y_j}(s)\epsilon_1+(A_{Y_i}(s)+1)\epsilon_2 + a_j-a_i )^{-1}\\
\times\prod_{t\in Y_j}((L_{Y_i}(t)+1)\epsilon_1-A_{Y_j}(s)\epsilon_2 + a_j-a_i )^{-1} \label{instantoncounting}\end{gathered}$$ という具体的な式で与えられます。 但し、$\sum a_i=0$とし、 $Y_1,\ldots,Y_N$ はヤング図で、箱の数が合計 $k$ 個であるようにします。$A_Y(s)$、$L_Y(s)$ はヤング図 $Y$ の箱 $s$ の腕長、足長と言われ、図 \[length\] のように決めます。$\cM_{N,k}$ は $4Nk$ 次元ですから、分母には $2Nk$ 個回転角が並んでいるはずですので、それを確認してみてください。 $N=2$ の場合は $(a_1,a_2)=(a,-a)$ とすることにします。 $k=1$ とすると、$(Y_1,Y_2)=(\Young{1},0)$ か $=(0,\Young{1})$です。公式の組み合わせ論的な式を展開すると、 の二項をそれぞれ再現するのがわかると思います。
$k=2$ とすると、$(Y_1,Y_2)$ は $(\Young{2},0)$, $(\Young{11},0)$, $(\Young{1},\Young{1})$, $(0,\Young{2})$, $(0,\Young{11})$ の五通りあります。例えば $(\Young{2},0)$ からの寄与は $$\left[-4\epsilon_1^2\epsilon_2 (\epsilon_1-\epsilon_2) a (2a+\epsilon_1)(2a+\epsilon_1+\epsilon_2) (2a+2\epsilon_1+\epsilon_2) \right]^{-1}$$ となります。他の項も頑張って計算しますと、$$Z_{2,2}=\frac{( 8(\epsilon_1+\epsilon_2)^2 +\epsilon_1\epsilon_2 - 8a^2)}
{\epsilon_1^2\epsilon_2^2 ((\epsilon_1+\epsilon_2)^2-4a^2)
((2\epsilon_1+\epsilon_2)^2-4a^2) ((\epsilon_1+2\epsilon_2)^2-4a^2) }\label{Z2}$$ となります。
なんだかややこしい結果になりましたが、もうすこしだけ物理的内容をとりだしてみましょう。インスタントン一つの場合は でした。インスタントンふたつが遠く離れていれば、1-インスタントン解を二つ重ね合わせれば2-インスタントン解ができます。ですから、 モジュライ空間の外部では、おおよそ $\cM_{2,2} \sim (\cM_{2,1})^2/\bZ_2 $ でした。 ですから、$\cM_{2,2}$上の積分は、おおよそ$\cM_{2,1}$上の積分のほぼ二乗で与えられて、差はインスタントン二つが互いに近づいた際の相互作用からくるはずです。それを実際に確かめましょう。
$|a|\gg \epsilon_{1,2}$ としたとします。すると、 にある $Z_{2,2}$ も $^2$ で与えられる $Z_{2,1}{}^2$もそれぞれ $(\epsilon_1\epsilon_2)^{-2}$ の因子があります、すなわち時空の箱のサイズの二乗の因子があります。これは、時空をふたつ物体が動いている積分から来ます。一方、相互作用の効果は $$Z_{2,2}-\frac12 Z_{2,1}{}^2=\frac{20a^2+7\epsilon_1^2+16\epsilon_1\epsilon_2 + 7\epsilon_2^2}{\epsilon_1\epsilon_2((\epsilon_1+\epsilon_2)^2-4a^2)((2\epsilon_1+\epsilon_2)^2-4a^2)((\epsilon_1+2\epsilon_2)^2-4a^2)} \label{hoge}$$となって、 $(\epsilon_1\epsilon_2)^{-1}$ の項が相殺し、 $(\epsilon_1\epsilon_2)^{-1}$ の寄与しかありません。これは、時空の各点を相互作用点として、そこに二つのインスタントンが近づいてきたときにおこる寄与がある、ということになっています。別の言い方をしますと、$$Z^\text{instanton}_{\epsilon_1,\epsilon_2;a} = 1 + q Z_{2,1} + q^2 Z_{2,2} +\cdots$$としますと、 $q^k$ の項はインスタントンが $k$ 個ありますから $(\epsilon_1\epsilon_2)^{-k}$ の因子がありますが、対数をとってやると $$\begin{aligned}
\log Z^\text{instanton}_{\epsilon_1,\epsilon_2;a}&= \frac{q}{\epsilon_1\epsilon_2}
\frac12\frac1{((\epsilon_1+\epsilon_2)/2+a)((\epsilon_1+\epsilon_2)/2-a)} \\
&+\frac{q^2}{\epsilon_1\epsilon_2}\frac{20a^2+7\epsilon_1^2+16\epsilon_1\epsilon_2 + 7\epsilon_2^2}{((\epsilon_1+\epsilon_2)^2-4a^2)((2\epsilon_1+\epsilon_2)^2-4a^2)((\epsilon_1+2\epsilon_2)^2-4a^2)}+\cdots \\
&\to \frac{q}{\epsilon_1\epsilon_2} \frac{-1}{a^2} + \frac{q^2}{\epsilon_1\epsilon_2} \frac{-5}{16a^4} + \cdots \label{bosh}\end{aligned}$$ となって、$q$ のべきにかかわらず時空体積の因子 $(\epsilon_1\epsilon_2)^{-1}$ が毎回でることになります。これは、$k$ インスタントンが相互作用するのは各時空点においてである、という様子を捉えているわけです。 では $|a|\gg \epsilon_{1,2}$ という極限をとりました。
二次元と四次元の関係
====================
対応関係
--------
さて、もうそろそろはじめに何をやったのかお忘れではないかと思いますが、前々節では、二次元共形場理論で、状態 $\ket{\Delta}$ から生成される Verma 表現を考えて、その中でコヒーレント状態 $$L_1\ket{\Delta,\lambda}=\lambda \ket{\Delta,\lambda}\hbox{、}\qquad
L_2\ket{\Delta,\lambda}=0$$を取り、$$\vev{\Delta,\lambda|\Delta,\lambda}=\vev{\Delta,1| \lambda^{2(L_0-\Delta)} |\Delta,1}=1+\frac{\lambda^2}{2\Delta}+\frac{\lambda^4(c+8\Delta)}{4\Delta((1+\Delta)c-10\Delta+16\Delta^2)}+\cdots \label{A}$$を計算しました。
また前節では、インスタントンの統計力学を考えました。時空を超立方体の箱にいれる替わりに、時空の回転およびゲージの回転に対して化学ポテンシャルを入れ、$$Z^\text{instanton}_{\epsilon_1,\epsilon_2,a}=\sum_k q^k \int_{\cM_{2,k}} e^{-2\pi(\epsilon_1 J_1 + \epsilon_2 J_2 + a K)} d\vol \hbox{。}$$を考えました。但し $N=2$ の $\SU(2)$ の場合を考えます。これを頑張って計算すると、$$\begin{gathered}
=1+ \frac{q}{\epsilon_1\epsilon_2} \frac{2}{(\epsilon_1+\epsilon_2)^2 -4a^2} \\
+ \frac{q^2}{\epsilon_1^2\epsilon_2^2} \frac{( 8(\epsilon_1+\epsilon_2)^2 +\epsilon_1\epsilon_2 - 8a^2)}
{ ((\epsilon_1+\epsilon_2)^2-4a^2)
((2\epsilon_1+\epsilon_2)^2-4a^2) ((\epsilon_1+2\epsilon_2)^2-4a^2) } + \cdots\label{B}\end{gathered}$$ となりました。
ふたつの結果, を見比べますと、第二項までは $$\lambda^2=\frac{q}{(\epsilon_1\epsilon_2)^2}\hbox{、} \qquad
\Delta=\frac1{\epsilon_1\epsilon_2}(\frac{(\epsilon_1+\epsilon_2)^2}{4} -a^2) \label{co}$$ とすれば合致します。第三項は、さらに $$c=1+6\frac{(\epsilon_1+\epsilon_2)^2}{\epsilon_1\epsilon_2}$$とすると合います。ここまでははじめの三項を使って3つの変数を関係付けただけのように見えるかもしれませんが、第四項、第五項とどんどん計算すると、上記の変数の対応で と がいつまでも合致するということがわかります[@Alday:2009aq; @Gaiotto:2009ma]。前節で説明した と を計算するアルゴリズムは前節と前々節に書きましたので、是非次の項を計算してみてください。 手計算でやると 3-インスタントンのあたりから突然面倒になりますので、プログラムを書いたほうがいいでしょう。Mathematica で実装したものが プレプリントページの ancillary files のところに置いてありますので、それをご覧ください。
量 $\vev{\Delta,\lambda|\Delta,\lambda}$ は二次元の物理から出て来たものです。量 $Z^\text{instanton}_{\epsilon_1,\epsilon_2,a}$ は四次元の物理から出て来たものです。これらが一致するという現象を、どのように理解すれば良いでしょうか。 双方とも計算方法は判っています。コヒーレント状態の長さの二乗は公式 で具体的に与えられていますし、インスタントンの分配関数は公式 で具体的に与えられています。 ですから、背景は忘れて、これらの公式を睨んで証明しよう、とすることが出来、実際そのような証明が既に知られています[@Poghossian:2009mk; @Fateev:2009aw; @Hadasz:2010xp]。 また、モジュライ空間に作用する無限次元代数を構成するというのは、数学では幾何学的表現論という一分野をなすほどです。たとえば、$\bC^2/\Gamma$ 但し $\Gamma\subset \SU(2)$ という空間上でのインスタントンのモジュライ空間を考えると、$\Gamma$ 型の Kac-Moody 代数の表現があらわれるというのが [@Nakajima] で示されています。 ですから、上記等式は$\Gamma$ で割る替わりに $\epsilon_{1,2}$ を入れた自然な拡張になっていて、幾何学的表現論からの証明がそろそろ発表されるという噂です[@MO]。 これら二つの手法は、図\[relation\] で言えば、数学の中に留まって関係を理解しよう、というものです。 しかし、僕の専門は数学ではなく弦理論ですので、厳密ではない弦理論の世界を通過して、どのようにこの関係が理解できるか、というのを説明したいと思います。
$$\includegraphics[width=.6\textwidth]{relation}$$
まず五次元へ
------------
我々の理解したい式は $$\vev{\Delta,1|\lambda^{2(L_0-\Delta_0)}|\Delta,1} = \sum_k q^k \int_{\cM_{2,k}} e^{-2\pi(\epsilon_1 J_1 +\epsilon_2 J_2 + aK)} d\vol$$というものでした。二次元と四次元の量を比較しているという以前に、 左辺は波動関数の二乗の形をしており、右辺は単なる積分です。これをどう比較すればよいでしょう? そのために、右辺を以下のように考え直してみましょう。$\psi_k(t)$ を、$\cM_{2,k}$ 上の関数で $$\psi_k(t)=e^{-\pi(\epsilon_1 J_1+\epsilon_2 J_2 + aK)}$$ で定めます。具体的には、$k=1$ では $$\psi_1(x_1,x_2,x_3,x_4,z,w)=e^{-\pi(\epsilon_1(x_1^2+x_2^2)+\epsilon_2(x_3^2+x_4^2) +
(\frac{\epsilon_1+\epsilon_2}2+a)|z|^2+(\frac{\epsilon_1+\epsilon_2}2-a)|w|^2 )}$$で、ガウス型の関数です。すると、右辺は $$=\sum_k q^k \int_{\cM_{2,k}} |\psi_k(t)|^2 d\vol$$です。 $\phi_k(t)$を、$\cM_{2,k}$ 上を動いている量子力学的粒子の波動関数だと思って $\ket{\phi_k}$ と書きましょう: $$\ket{\phi_k} \in \cH(\cM_{2,k})$$ ただし $\cH(\cM_{2,k})$ は $\cM_{2,k}$ 上の波動関数のなすヒルベルト空間です。すると、右辺はさらに $$=\sum_k q^k \vev{\phi_k|\phi_k}$$と書けます。そこで、さらに $$\ket{\phi}=\ket{\phi_0}\oplus \ket{\phi_1} \oplus \cdots
\in \cH(\cM_{2,0}) \oplus \cH(\cM_{2,1}) \oplus \cdots \label{phi}$$ というベクトルを考え、ハミルトニアン $H$ を $\cH(\cM_{2,k})$ で固有値 $k$ をもつような演算子だとすると、関係式は$$\vev{\Delta,1|\lambda^{2L_0-\Delta}|\Delta,1}=\vev{\phi | q^H | \phi}$$となります。$\ket{\Delta,1}$ も $$\ket{\Delta,1}=\ket{\psi_0} \oplus \ket{\psi_1} \oplus \ket{\psi_2} \oplus \cdots
\in V_0 \oplus V_1 \oplus V_2 \oplus \cdots$$ という展開がありました、但し $V_k$ は次数が $k$ の成分で、そこでは $L_0-\Delta=k$ なのでした。
$V_k$ は有限次元でした: $\dim V_k = p_k$ ただし $p_k$ は $k$ を正整数の和として書く方法の数。一方で $\cH(\cM_{2,k})$ は当然無限次元です。しかし、何らかの意味で $$V_k \subset \cH(\cM_{2,k})$$と自然に埋め込まれており、その埋め込みのもとで $$\ket{\psi_k} = \ket{\phi_k}$$ となっているならば、我々の関係式は自然に従います。
しかし、インスタントンのモジュライ空間を動く粒子というのはどういうことでしょうか? これは、四次元のゲージ理論でなく、五次元のゲージ理論を考えると自然に現れます。 ヤンミルズ場を $\bR^4$ でなく $\bR^5 =\bR^4\times \bR_t $ 上で考えましょう。付け加えた一方向を時間だと思うことにします: $(x_1,x_2,x_3,x_4,t) \in \bR^5$ 作用は $$\int \frac1{2g_\text{5d}^2} \tr F_{\mu\nu} F_{\mu\nu} dt d^4x \label{5daction}$$です。 五次元の配位をひとつ $\cA_\mu(x_1,x_2,x_3,x_4,t)$ をとると、各時刻 $t$ 毎に $\bR^4$ 上の配位 $$A_\mu(x_i;t) = \cA_\mu(x_1,x_2,x_3,x_4,t)$$ が定まっていると思えます。時刻 $t=t_1$ での $A_\mu(x_i;t_1)$インスタントン数を $k$ とすると、インスタントン数は整数ですから、他のいつの時刻 $t=t_2$ でもインスタントン数は $k$ になります。 量子力学にするために経路積分をすることを考えますと、各時刻でエネルギーを極小にするために、各時刻 $t$ で $A_\mu(x_i;t)$ が反自己双対であるような配位がもっとも寄与が大きくなります(図\[5d\])。 各時刻でのエネルギーは $$\int \frac1{2g_\text{5d}^2} \tr F_{\mu\nu} F_{\mu\nu} d^4x \sim \frac{8\pi^2 k}{g_\text{5d}^2}$$となります。エネルギーと質量は等価ですから、 インスタントン一つが質量 $8\pi^2/g_\text{5d}^2$ の粒子に見えるわけです。
$$\includegraphics[width=.6\textwidth]{5d}$$
さて、勝手なインスタントン数 $k$ の反自己双対解は $4Nk$ 個のパラメタ $s_i$ で決まっているはずですから、時刻 $t$ に依存して $s_i(t)$ が決まりました: $$s_i(t): \bR_t \to \cM_{N,k}\hbox{。}$$ ですから、五次元のゲージ理論で、経路積分に最も寄与の大きい部分は、インスタントンのモジュライ空間を動く量子力学的粒子の運動で捉えられることがわかりました。
しかし、我々は四次元のゲージ理論を考えていました。五次元のゲージ理論から四次元のゲージ理論をつくる簡単な方法は、一つの方向を「コンパクト化」することです。例えば、$t$ 方向を $[0,L]$ の線分にしてしまいましょう。すると、理論を $L$ より非常におおきなスケールでみる限りは、$t$ 方向を区別することはできず、実質四次元の理論になります。非常に安直には、五次元の作用 において、$F_{\mu\nu}$ が $t$ 方向に変化しなければ、$dt$ 積分をしてしまって $$\int \frac1{2g_\text{4d}^2} \tr F_{\mu\nu} F_{\mu\nu} d^4x$$ とできるというわけです。但し、$$\frac1{g_\text{4d}^2} = \frac{L}{g_\text{5d}^2}\hbox{。} \label{45}$$
一般に、このようなコンパクト化をすると Kaluza-Klein 粒子というものが現れます。五次元の時空に質量のない粒子があったとしましょう。すると、エネルギーと運動量は $$E^2 = \vec{p}\,^2 + p_5^2$$ を満たします、ただし $\vec p$ は $\bR^3$ 方向の運動量で、$p_5$ は第五方向の運動量とします。第五方向を $L$ にコンパクト化すると、量子力学的には運動量は波動関数の位相 $e^{ i \vec p\cdot \vec x}$ ですから、$2\pi p_5 L$ は整数でないといけません。それを $k$ とすると、$p_5 = 2\pi k /L$ となります。すると、四次元の立場からは、$$E^2 - \vec{p}\,^2= \left(\frac{2\pi k}{L}\right)^2$$となって、質量が $2\pi k/L$ の粒子が現れます。これは $L$ が小さければどんどん重くなるので、$L$ が小さいほど測りにくくなるわけです。
$$\includegraphics[width=.8\textwidth]{5d2d}$$
ゲージ理論に話を戻しますと、我々は四次元の結合定数が $g_{4d}$ のゲージ理論を考えたいので、五次元のゲージ理論を で定まる長さ $L$ の線分にコンパクト化します 図\[interval\]。 すると、この系の分配関数は、$$Z=\vev{\Phi |e^{-LH} |\Phi}$$ で与えられることになります。但し $\Phi$ は $t=0$ および $t=L$ での境界条件から定まる状態で、$H$ は $t$ 方向への時間発展の演算子です。
さらに六次元へ
--------------
ここまでは非常に一般的な考察でしたが、五次元のゲージ理論として、単にゲージ場だけの理論でなく、最大超対称 $\SU(N)$ ゲージ理論とよばれるものを取り、境界条件で半分の超対称を保つものを使うと、四次元の理論として純 $\cN=2$ $\SU(N)$ゲージ理論というものになります。この理論に更に Nekrasov の $\epsilon_{1,2}$ 変形というものを加えると、分配関数が我々の扱ってきた統計力学模型と一致することが知られています。いま我々は $N=2$ を考えています。すると、上記 $\ket{\Phi}$ は $\ket{\phi}$ と一致します。 さて、これは更に共形場理論のコヒーレント状態 $\ket{\Delta,1}$ と一致するのですが、状況を比べると五次元を線分において四次元理論をつくるのと、 コヒーレント状態のノルムを計算する状況はほとんど同じですね(図\[interval\])。 五次元方向の長さ $L$ は$$L\sim \frac{1}{g_\text{4d}^2} \sim \log q$$でしたが、円柱の横幅は $\sim\log \lambda$ で、 で見た対応関係から期待される通りです。
ですから、五次元最大超対称 $\SU(2)$ ゲージ理論の $t$ 方向の発展の演算子 $H$ は、二次元の理論の演算子 $L_0$ と同一視すべきです。同じことですが、図 \[interval\] の第五方向、すなわち $L$ 方向と、図 \[HH\] の円柱の $\log \lambda$ 方向は同一視すべきです。 五次元の理論の五方向はすべて $\SO(5)$ 回転で等価です: $$\text{$\bR^4$の一方向} \xrightarrow{\text{$\SO(5)$ 回転}} \text{第五方向}$$ 一方で、共形変換は $L_0$ だけでなく、$L_{n}$ を含み、特に $L_{\pm 1}$ は円柱の $\bR$ 方向を円柱の$S^1$方向に回す変換を含んでいます。物理的には、円柱の中の二方向は等価なわけです。$$\text{円柱の $\bR$ 方向} \xrightarrow{\text{$L_1$ 回転}} \text{円柱の$S^1$方向}$$ しかし、$$\text{五次元理論の第五方向} = \text{円柱の $\bR$ 方向}$$です。これらを組み合わせると、五次元理論の $\bR^4$ 方向は、円柱の空間方向へ回転させることが出来るという主張に至ります。そのためには、六次元の理論が必要です。 純粋に五次元最大超対称 $\SU(2)$ ゲージ理論を考えているつもりだったが、それは六次元のある理論を一周 $R$ の円周にコンパクト化したものであると思うべきである、ということです。
六次元の理論を円周にコンパクト化したのであれば、すぐ前におさらいしたように、整数 $k$ に対して重さ $2\pi k/R$ の Kaluza-Klein 粒子がでるはずですが、確かに、五次元理論にはインスタントン粒子があり、その重さは $8\pi^2 k/g_\text{5d}$ でした。ですから、五次元理論のインスタントン粒子は、実は六次元理論の Kaluza-Klein 粒子であり、 $$R= \frac{g_\text{5d}^2}{4\pi} \label{56}$$ である、ということがわかります。
ですから、我々は六次元の理論を、$S^1\times [0,L] \times \bR^4$ 上で考えていたのです。この系を解析する際に、
- ここで、$S^1$ が非常に小さいとすると、$[0,L]\times \bR^4$ 上で五次元ゲージ理論を、さらに $L$ も小さいとして、結局 $\bR^4$ で四次元ゲージ理論を考えるという方法があります。
- 一方、 $\bR^4$ 方向にはポテンシャル $\epsilon_1(x_1^2+x_2^2) + \epsilon_2(x_3^2+x_4^2)$ を入れているので、$\epsilon_{1,2}$ が大きいと、$\bR^4$ 方向の箱を非常に小さくすることが出来ます。すると、$S^1\times [0,L]$ 上で二次元理論を考える、ということになります。
この二通りの評価法を比較することにより、$$Z^\text{instanton}_{\epsilon_1,\epsilon_2,a} = \vev{\Delta,\lambda|\Delta,\lambda}$$ が得られるわけです。
では、この六次元理論はなんでしょうか? 先ほど、五次元ゲージ理論を長さ $L$ の線分にコンパクト化すれば、おおよそ四次元ゲージ理論になることを説明しましたから、同様に、六次元ゲージ理論を一周 $R$ の円周にコンパクト化しているのではないか、と安直には思えます。しかし、そうすると と同様にして、 $$\frac{1}{g_\text{5d}^2 } \propto R$$となってしまい、 とはまったく逆になってしまいます。
きちんと を出すような六次元理論は、我々の業界では「六次元 $\cN=(2,0)$ 理論」と呼ばれています。これの性質ははっきりとはわかっていませんが、ゲージ理論の $F_{\mu\nu}=\partial_\mu A_{\nu} -\partial_\nu A_\mu +[A_\mu,A_\nu] $ の替わりに、$F_{\mu\nu\rho}=\partial_\mu B_{\nu\rho} + \cdots $ という場があり、$$F_{\mu\nu\rho}=\tilde F_{\mu\nu\rho} \quad \text{但し} \quad \tilde F_{\mu\nu\rho}=
\frac16 \epsilon_{\mu\nu\rho\alpha\beta\gamma} F_{\alpha\beta\gamma}$$ が成り立っている「ようなもの」だと思われています。この六次元理論はいろいろな次元での超対称ゲージ理論の親玉だと思われており、最近活発に研究されています (数学者向けのまとめは [@Witten:2009at; @Witten:2009mh] 等を参照のこと)。
五次元の最大超対称ゲージ理論は、結合定数 $g_\text{5d}$ を大きくすると、インスタントン粒子が軽くなり、一周 $R\sim g_\text{5d}^2$ の円周を生成して六次元の理論になってしまうわけですが、この事実は [@Dijkgraaf:1997vv] に示唆されて 1997 年に [@Berkooz:1997cq] がはじめに指摘しました。 これは 1995 年から理解され始めた「超弦理論は M 理論である」という事実の一環です。 実際、Type IIA 超弦理論は10次元の理論なのですが、弦理論の結合定数を大きくすると、D0-ブレーンという粒子が軽くなり、円周を生成して11次元の理論になってしまいます。 Type IIA 超弦には D4-ブレーンという五次元にひろがった物体があり、この操作にともなって、M理論の M5-ブレーンという6次元にひろがった物体が生成された円周に巻き付いている状態になります(このあたりの詳細は、教科書 [@Polchinski] 等を参照のこと)。 D4-ブレーンが $N$ 枚重なっていると、その上には五次元最大超対称 $\SU(N)$ ゲージ理論が住みます。これを強結合にすると、M5-ブレーンが $N$ 枚重なっていることになります。この上に住んでいるのが、我々の知りたい6次元$\SU(N)$型 $\cN=(2,0)$ 理論です。
この六次元理論ははっきりとはわかっていませんが、これを一周 $R$ の円周上にコンパクト化し、さらにこれを長さ $L$ の線分にコンパクト化した系の分配関数を二通りの方法で計算しようとしたところ、僕と共著者はひとつの方法では $\vev{\Delta,\lambda|\Delta,\lambda}$ を、もうひとつの方法では $Z^\text{instanton}_{\epsilon_1,\epsilon_2,a}$ になることが分かったので、これらは等しいはずだ、という事実に辿り着いたのでした [@Alday:2009aq; @Gaiotto:2009ma][^3]。
拡張
----
さて、ここまでは $$\vev{\Delta,\lambda| \Delta,\lambda} = Z^\text{instanton}_{\epsilon_1,\epsilon_2,a}$$ というただ一つの関係式に絞ってなるべく具体的に説明をしてきたつもりですが、 前節での説明から、すこし弦理論の設定をかえると、幾らでも関連する関係式が得られることがわかるでしょう。それをいくつか述べておしまいにしたいと思います。
まず、この等式では左辺に二次元のビラソロ代数のコヒーレント状態があり、 右辺では $\SU(2)$ のインスタントンの分配関数があります。 $\SU(2)$ のかわりに $\SU(N)$ にするとどうなるでしょうか? 右辺を計算することは簡単で、公式は既に に書きました。 左辺はどうすればよいでしょう? 1980年代遅くから1990年代初頭にかけて、ビラソロ代数は $W_N$ 代数とよばれる無限次元代数のクラスの中で $N=2$ の一番簡単なもの、すなわち $W_2$ 代数であるということが認識されました。(W代数については、レビュー[@Bouwknegt:1992wg] や、「数理物理2004」の脇本先生の講義録 [@Wakimoto] 等を参照。) ですから、上式を一般化して $$\text{$W_N$ 代数のコヒーレント状態のノルム} = \text{$\SU(N)$インスタントンの分配関数}$$ということを考えるのは自然です[@Wyllard:2009hg; @Mironov:2009by]。例えば、$N=3$ の場合は、$L_n$ に加えて $W_n$ という演算子があり、交換関係は $$\begin{aligned}
&[L_n, L_m]=(n-m)L_{n+m}+\frac{c}{12}(n^3-n)\delta_{n,-m}\hbox{、}\\
&[L_n, W_m]=(2n-m)W_{n+m}\hbox{、}\\
&[W_n, W_m]=\frac{1}{48}\Big[ \frac{c(22+5c)}{3\cdot 5!}n(n^2-1)(n^2-4)\delta_{n,-m}
+{16}(n-m)\Lambda_{n+m}\nonumber\\
&\qquad\qquad\qquad +(22+5c)(n-m) \left( \frac{(n+m+2)(n+m+3)}{15}-\frac{(n+2)(m+2)}{6} \right)L_{n+m}\Big]\hbox{。}\end{aligned}$$で与えられます、但し $\Lambda_n$ は $$\begin{aligned}
\Lambda_n=\sum_{m\le -2}L_m L_{n-m}+\sum_{m\ge -1} L_{n-m}L_m-
\frac3{10}(n+2)(n+3)L_n\end{aligned}$$ で定義されます。ここで、$W_m$ の規格化は通常のものの $\sqrt{(22+5c)/216}$倍にしました。自由場表示からはこちらのほうが自然です。$W_3$代数の基本的な表現は $$L_0\ket{\Delta,w}=\Delta\ket{\Delta,w}, \
W_0\ket{\Delta,w}=w\ket{\Delta,w}$$ で $n>0$ なら$L_n$, $W_n$ を掛けると消えるような状態から生成されます。それらの線形結合から、コヒーレント状態を $$W_1\ket{\Delta,w,\lambda}=\lambda \ket{\Delta,w,\lambda}$$ で定めると、そのノルムが式で $N=3$ とした $Z^\text{instanton}$ と一致することが知られています [@Mironov:2009by; @Taki:2009zd]。ここで、パラメタの対応は $$c=2+24(b+\frac1b)^2,\quad
\Delta=(b+\frac1b)^2-a_1^2-a_1a_2-a_2^2,\quad
w=ia_1a_2(a_1+a_2)$$ とします。但し、簡単のため $\epsilon_1=b$, $\epsilon_2=1/b$ としさらに$(a_1,a_2,a_3)=(a_1,a_2,-a_1-a_2)$ と取りました。
$$\includegraphics[width=.7\textwidth]{matter}$$
また、$\SU(2)$ のままで、ゲージ理論に物質場を足すことも出来ます。例えば、現実の「弱い力」のもとではクォークは $\SU(2)$ の二重項ですから、それに倣って二重項のクォークの寄与をインスタントン分配関数に加えることができます。すると、 の分子にもいろいろ項が加わることが知られています。我々の設定でクォークをひとつ足す簡単な方法は、片方の境界にだけクォークを足すことです。すると、対応して二次元の共形場理論では、片側の境界条件が変更されます(図\[matter\])。すると、$$\vev{\Delta,\lambda|\Delta,\lambda,m} = Z^\text{instanton}_\text{with quark}$$という等式が成り立ちます[@Gaiotto:2009ma]。ただし、$\ket{\Delta,\lambda,m}$ は $L_1$ も $L_2$ もノンゼロの固有値をもつようなコヒーレント状態です: $$L_1\ket{\Delta,\lambda,m}=\lambda\ket{\Delta,\lambda,m}\hbox{、}\qquad
L_2\ket{\Delta,\lambda,m}=\sqrt{\lambda} m\ket{\Delta,\lambda,m}\hbox{。}$$ はじめの論文 [@Alday:2009aq] で扱われたのは、さらに複雑にクォーク二重項を四つ加えた場合でした。
別の拡張として、$W_N$ 代数はアファイン $\SU(N)$ 代数から量子 Drinfeld-Sokolov 還元という手法で作ることが出来ますが、還元にはデータとして $N$ の分割 $\rho=(N_1,\ldots,N_k)$ を指定してやることができます。$W_N$ は特に $\rho=(N)$ という場合ですが、一般の $\rho$ に対して $W(\SU(N),\rho)$ 代数というものがあります。 これを出すにはゲージ理論側にどのような変更を加えれば良いかというのも知られており、$\bR^4$ 上のインスタントンを考える際に、$x_1=x_2=0$ の平面に沿ってゲージ場に特異性 $$A_\mu dx^\mu \sim \mathrm{diag}(
\underbrace{\alpha_{(1)},\ldots,\alpha_{(1)}}_{\text{$n_1$ times}},
\underbrace{\alpha_{(2)},\ldots,\alpha_{(2)}}_{\text{$n_2$ times}},
\ldots,
\underbrace{\alpha_{(k)},\ldots,\alpha_{(k)}}_{\text{$n_k$ times}} ) i d\theta,$$ を入れれば良いです[@Braverman:2010ef; @Wyllard:2010vi; @Kanno:2011fw]。
また、$\bR^4$ 上でばかりインスタントンを考えてきましたが、そのかわりに $\bR^4/\Gamma$ 上でインスタントンを考えればどうなるでしょうか。ただし、$\Gamma\subset \SU(2)$ とします。$\epsilon_{1,2}$ を入れない状況は [@Nakajima] によって調べられ、$\Gamma$ 型のKac-Moody 代数がでることが知られていました。$\Gamma=\bZ_m$ の場合はさらに $\epsilon_1$, $\epsilon_2$ による変形を入れることが出来、$m$-次パラ $W_N$ 代数という恐ろしい代数が出てくると思われています[@Nishioka:2011jk]。特に $m=N=2$ の場合は、$2$-次 パラ$W_2$ 代数というのは通常の超対称ビラソロ代数になるので具体的な確認をいくつもすることが出来、最近いろいろと論文が出ています[@Belavin:2011pp; @Bonelli:2011jx; @Bonelli:2011kv]。
謝辞 {#謝辞 .unnumbered}
====
まずは夏の学校「数理物理2011」の世話人の皆様に、このような機会を与えてくださった事を感謝したいと思います。 著者の仕事は部分的にアメリカNSFのグラント番号 PHY-0969448 及び高等研究所の Marvin L. Goldberger membership からの援助を受けました。また、数物連携宇宙機構を通じて日本国文部科学省世界トップレベル研究拠点プログラムからの補助も受けています。
[^1]: 我々が生まれてくるのが二百年遅ければ、普通に教科書に定義が載っていて不思議ではないだろう、ということです。解析学だって Newton, Leibnitz のころはいろいろ問題がありました。
[^2]: 一変数関数の定積分の公式が固定点定理の一種であるというのは講演 [@NakajimaSougou] から学びました。
[^3]: 正直なことを言いますと僕が事実に辿り着いたというのは言い過ぎです。僕はインスタントンの分配関数の計算を修士論文にしていたのですが、それを知っていた D. Gaiotto がある日、 F. Alday との共同研究の過程で、$Z^\text{instanton}$ が二次元共形場理論のこういう量で掛ける筈だとなったんだが確かめてくれないか、と僕に言いました。そこで、数年前に書いた Mathematica プログラムをパソコンから掘り出してわけもわからず計算してみると確かに一致していたので驚愕した、というのが [@Alday:2009aq] の真相です。その後、さらに例を簡単化したのが [@Gaiotto:2009ma] で、この講義では簡単化したものだけを説明しました。
|
---
abstract: 'We present the results of a monitoring campaign using the KAT-7 and HartRAO 26m telescopes, of hydroxyl, methanol and water vapour masers associated with the high-mass star forming region G9.62+0.20E. Periodic flaring of the main line hydroxyl masers were found, similar to that seen in the 6.7 and 12.2 GHz methanol masers. The 1667 MHz flares are characterized by a rapid decrease in flux density which is coincident with the start of the 12.2 GHz methanol maser flare. The decrease in the OH maser flux density is followed by a slow increase till a maximum is reached after which the maser decays to its pre-flare level. A possible interpretation of the rapid decrease in the maser flux density is presented. Considering the projected separation between the periodic methanol and OH masers, we conclude that the periodic 12.2 methanol masing region is located about 1600 AU deeper into the molecular envelope compared to the location of the periodic OH masers. A single water maser flare was also detected which seems not to be associated with the same event that gives rise to the periodic methanol and OH maser flares.'
author:
- |
S. Goedhart$^{1,2}$[^1], R. van Rooyen$^{1,2}$, D.J. van der Walt$^{2}$, J.P. Maswanganye$^{2}$, A. Sanna$^{3}$, G.C. MacLeod$^{4,5}$ and S.P. van den Heever$^{4}$\
$^{1}$South African Radio Astronomy Observatory, 2 Fir Street, Black River Park, Observatory, 7925, South Africa\
$^{2}$Center for Space Research, North-West University, Potchefstroom campus, Private Bag X6001, Potchefstroom, 2520, South Africa\
$^3$ Max-Planck-Institut f[ü]{}r Radioastonomie, Auf dem H[ü]{}gel 69, D-53121 Bonn, Germany\
$^4$ Hartebeesthoek Radio Astronomy Observatory, PO Box 443, Krugersdorp, 1740, South Africa\
$^{5}$The University of Western Ontario, 1151 Richmond Street. London, ON N6A 3K7, Canada\
bibliography:
- 'goedhart\_oh\_mon\_rev1\_clean.bib'
date: 'Accepted 2019 March 7. Received 2019 February 27; in original form 2018 December 20'
title: 'Periodic variability of the mainline hydroxyl masers in G9.62+0.20E'
---
\[firstpage\]
masers, star:formation, ISM:clouds, regions, radio lines:ISM
Introduction
============
To date there are at least 20 known periodic methanol masers [@Goedhart2009; @Araya2010; @Szymczak2011a; @Goedhart2014; @Fujisawa2014; @Szymczak2015a; @Maswanganye2015; @Szymczak2016; @Maswanganye2016; @sugiyama2017] including one source which shows quasi-periodic variations in both methanol and formaldehyde [@Araya2010]. The formaldehyde and methanol variations were simultaneous and showed very close correspondence to each other, which the authors speculated was due to modulation of the infrared radiation field by periodic accretion in a binary system. @Green2012 found a possible indication of periodic variability in the hydroxyl masers in G12.89+0.49, where the hydroxyl masers may have undergone a periodic drop in intensity. However, their results were not conclusive since the time series was undersampled. @Szymczak2016 found anti-correlated variations in the water and methanol masers in G107.298+5.639 even though there are good spatial and velocity overlap between the two masers. Methanol and hydroxyl masers are thought to share a similar (radiative) pump mechanism [@Cragg2002] and are often found in close spatial proximity, while water masers are collisionally pumped and require high density environments. Thus, correlated variability may be expected between hydroxyl and methanol masers. However, since the two molecules have different energy level structures, they may not necessarily respond in exactly the same way to changes in the pumping radiation field. The fact that several of the periodic methanol maser sources also have associated hydroxyl masers allows us to investigate to what extent the hydroxyl masers also show periodic flaring and therefore to possibly determine whether the flaring is due to changes in the maser amplification or in the background seed photon flux.
A number of known periodic methanol masers were monitored on a weekly basis at 1665 and 1667 MHz as part of the seven-element Karoo Array Telescope (KAT-7) science verification programme. This paper focuses on G9.62+0.20E, which showed clear evidence of variability in the hydroxyl masers during the first year of monitoring. It was subsequently monitored on a daily basis during the expected June 2014 and February 2015 methanol maser flares while simultanously monitoring the 6.7 and 12.2 GHz methanol and 22.2 GHz water masers using the Hartebeesthoek Radio Astronomy Observatory (HartRAO) 26m telescope.
G9.62+0.20E is a high-mass star forming region with a hypercompact region harbouring at least one massive star in an early evolutionary phase [@Garay1993]. It has a number of masers projected against the region - Class II methanol masers, water masers and hydroxyl masers [@Sanna2015] and is located at a distance of 5.2 kpc from the Sun [@Sanna2009]. It was the first methanol maser source discovered to show periodic variations [@Goedhart2003], with a best-fit period of 243.3 days [@Goedhart2014]. Simultaneous flares in methanol have been observed at 6.7-, 12.2- and 107 GHz [@VanderWalt2009]. One explanation of the methanol maser flare profiles is by the variation of the free-free continuum flux in the background region due to variable ionizing radiation associated with shocked winds from a colliding wind binary system with a non-zero orbital eccentricity [@VanderWalt2011]. However, it has been argued by @Parfenov2014a that the same effect can be produced by variations of the dust temperature in an accretion disk around a forming binary system, while @Inayoshi2013 suggest that pulsational instabilities of massive protostars could arise during rapid accretion. @Singh2012 investigated the possibility of bipolar outflows in young binary systems as a possible explanation for some of the periodic methanol masers. See @vanderwalt2016 for a further discussion.
The OH monitoring of G9.62+0.20E revealed complex behaviour across multiple OH maser features, with some features showing a pronounced drop in power at the same time that the methanol masers started to flare. Not all of the hydroxyl or methanol maser features flare, and since the relative positions of all the maser species and the background region is known [@Sanna2015], the regions undergoing periodic flares can be isolated. It was found that the flaring methanol and OH masers are, in projection, about 1600 AU apart. The decay of the flaring OH masers can described in terms of the variation of the free-free emission from a recombining hydrogen plasma, similar to what is found for the methanol masers in G9.62+0.20E.
Observations and data reduction
===============================
KAT-7
------
The seven-dish Karoo Array Telescope [@Foley2016] was built as an engineering prototype for the 64-dish MeerKAT array in the Karoo region of the Northen Cape, South Africa. It consists of 7 12-m diameter dishes with prime focus linearly-polarised receivers covering a frequency range of 1.2 to 1.95 GHz. It is a compact array, with a maximum baseline of 186m and shortest baseline of 26m. The system temperature of the antennas is approximately 30 K, and apperture efficiency is on average 65%.
The OH maser monitoring observations started during the early stages of spectral line commissioning on KAT-7 in February 2013. We used the c16n2M4k correlator mode, which gives a velocity resolution of 68 m s$^{-1}$ at the OH rest frequencies. Since May 2013, observations at 1665 and 1667 MHz were interleaved in a single schedule block using an LST-based scheduling mechanism, which ensures consistent uv-coverage from one observation to the next and ensures that observations at both frequencies are executed quasi-simultaneously. Prior to this observations at each frequency were scheduled separately, preferably within the same day. The total integration time on source at each frequency was initially 20 minutes and subsequently increased to up to 50 minutes for the weekly monitoring programme and up to 80 minutes for the daily observations whenever possible. Daily observations during predicted flares were dynamically scheduled subject to telescope availability. Not all antennas were always available, and integration time was increased when possible to compensate. The typical rms noise achieved ranges from 0.15 to 0.2 Jy. The typical beam size is $\sim$ 3 arcmin thus the masers are unresolved and the relative positions of the masers cannot be measured due to the poor resolution.
PKS 1934-638 – the flux and bandpass calibrator – was observed for five minutes every hour. PKS1730-130 was used as the gain calibrator and was observed once every ten minutes for one minute.
The data were calibrated following standard interferometric calibration procedures using CASA [@McMullin2007]. The bandpass response turned out to be stable in time so all scans were combined to produce a single bandpass calibration table. In order to avoid introducing noise from the bandpass calibration into the spectrum (the continuum emission is similar in strength to the bandpass calibrator) the bandpass solution was smoothed using a 3rd-order polynomial in both phase and amplitude. We were unable to obtain a reliable polarisation calibration in the narrow band mode due to insufficient signal to noise on the polarisation calibrator 3C286 so only the Stokes I product is considered here.
KAT-7 has a 1 square degree field of view and excellent sensitivity to low surface brightness emission. This results in a complex continuum image. Spectral line data cubes were imaged after continuum subtraction in the u-v plane. The sheer volume of the data necessitated automated data reduction and imaging. The first ’quick-look’ image cubes produced had imaging artefacts (spurious source detections in isolated channels) and higher rms noise than expected in some channels. To address this problem, a deep image was made of the field by combining all available observations at the time - some 60 hours of data on-source - in order to optimise the imaging parameters. There are five spectral line sources in the field of view, including a deep absorption feature, which were fully characterised to determine the appropriate velocity ranges for continuum subtraction, and to create a velocity-dependent CLEAN mask for non-interactive image deconvolution. The clean threshold was determined by measuring the rms noise in an emission-free channel. This approach led to much better and more consistent imaging quality even when uv-coverage was reduced.
Data quality was severely compromised by solar interference during December and early January when first the gain calibrator and then the target were less than 12 degrees from the Sun. We have discarded all data taken during this period.
Single dish observations
------------------------
Concurrent monitoring of the methanol and water masers was done using the Hartebeesthoek Radio Astronomy HartRAO 26-m telescope, subject to scheduling constraints.
Flux calibration at 6.7 and 12.2 GHz was done by daily drift scans on Virgo A and Hydra A using the flux scale of @Ott1994. Amplitude corrections due to pointing errors were calculated by offset observations to the east-west and north-south half-power points. Bandpass calibration at 6.7 and 12.2 GHz was done by frequency switching, while position-switching was done at 22 GHz to accommodate a potentially wider velocity spread. The pointing offsets at 22 GHz were greater than half a beamwidth at times, making it impossible to derive reliable amplitude correction since we detect the source in only one or two of the offset pointing positions. These observations have been discarded from the timeseries. The 22 GHz flux densities were also corrected for atmospheric absorption using water vapour radiometer and environmental data. Jupiter was used as a flux calibrator.
Results
=======
Figure \[fig:spectra\] shows the spectra and range of variation of the OH masers at 1665 and 1667 MHz, the methanol masers at 6.7 and 12.2 GHz and the water masers at 22 GHz. Variability is seen in the same velocity ranges for the methanol and hydroxyl transitions, between 1 to 3 [km s$^{-1}$]{}. The water masers are highly variable but have a different and larger velocity range.
![Spectra of the maser transitions monitored, showing the range of variation in each velocity channel.[]{data-label="fig:spectra"}](spectra-eps-converted-to.pdf){width="\columnwidth"}
Hydroxyl maser time-series
--------------------------
Figure \[fig:cont\] shows the continuum image of the field with locations and identification of spectral line sources marked in red. Note that primary beam correction has not been applied to this image. At L-band we are detecting optically thin, evolved regions. Continuum emission is visible at the G9.62+0.20 site and is most likely dominated by components A and B [@Garay1993], which are more evolved and extended regions in the complex.
![Overview of sources detected in the field. The colour-scale shows the continuum image, Locations of spectral line sources are indicated by the red circles. The size of the circles indicates the synthesised beam size.[]{data-label="fig:cont"}](cont-eps-converted-to.pdf){width="\columnwidth"}
Time-series were generated by fitting a 2-dimensional Gaussian to channels of interest in the image plane. The uncertainty in the measured amplitude was calculated by measuring the rms noise in an outer quadrant of the image for that channel. We generated time-series for all velocity channels which had amplitudes greater than three times the rms noise. Figures \[fig:1665\_ts\] and \[fig:1667\_ts\] show the time-series for the 1665 and 1667 MHz data, respectively. The KAT-7 data suffers from heavy spectral blending, which could not be resolved by fitting individual Gaussian components to the spectra. Instead we plot the measured flux density in each channel. We have grouped channels by visual inspection of both the spectral structure and the characteristics of the variability, which appears markedly different as we move between spots in the general star forming region. The most notable behaviour is seen in the 1667 MHz transition in the peak velocity channels at $\sim$ 1.7 [km s$^{-1}$]{}. These features show a drop in intensity at the time that the methanol masers start to flare, followed by a sharp rise, and a gradual decay. In contrast, the features at $\sim$ 2 [km s$^{-1}$]{} show a flare profile that is more similar to that of the methanol masers.
Comparison with VLBI spectra and radio continuum positions
----------------------------------------------------------
Since the individual hydroxyl maser spots cannot be resolved by KAT-7 and there are multiple spots with overlapping velocities, we inspected the VLBI spectra more closely to see if they could be correlated with the temporal and velocity behaviour seen in the KAT-7 data. The radio continuum source E appears to have two components - E1 is the stronger peak over which the methanol masers are centred, while E2 is one-fifth of the strength of E1 and is located $\sim$ 1000 AU in projection to the north-east [see @Sanna2015].
Figure \[fig:spotmap\] shows the maser spotmap from @Sanna2015, with zoomed insets on the various OH maser clumps. The OH masers are not co-located with the methanol masers; instead they are distributed in several clumps around the region. The clump to the north-east is on the far side of component E2. Other clumps are distributed offset from the peak of E2 to the south-east, south and west. The water masers are located between E2 and E1. It is also known that not all of the methanol masers flare [@Goedhart2005], so comparison of the maser positions and their time-dependent behaviour may help to narrow down the origin of the periodic behaviour.
![Spectra of individual 1665 MHz maser features from @Sanna2015. The top panel shows the KAT-7 spectrum from the nearest epoch (2013-07-16) 14 days later.[]{data-label="fig:vlbi_spec1665"}](vlbi_spec_1665-eps-converted-to.pdf){width="\columnwidth"}
Figures \[fig:vlbi\_spec1665\] and \[fig:vlbi\_spec1667\] show the VLBA spectra, with the KAT-7 spectra taken at the closest epoch (16 July 2013) for comparison. These spectra are used to inform the allocation of spectral channels to mapped features. The following discussion will work through the enumerated panels in Figures \[fig:1665\_ts\] and \[fig:1667\_ts\]. Panel ‘a’ covers velocities in the range 0.67 to 1.08 [km s$^{-1}$]{}(the shoulder on the 1665 MHz spectrum) and shows relatively constant flux densities throughout the monitoring period. These channels may be part of features 18/19 in the western group but the VLBI spectra show a single channel of 2.5 Jy in this velocity range while the the KAT-7 spectra show far more flux. It is not completely clear where panel ‘b’ channels originate. It may be a blend of spatially separate features. The velocity range covered by panels ‘b’ and ‘c’ – 1.15 to 1.36 [km s$^{-1}$]{}– is the same as that of features 17, 18 and 19 in the western group, however the VLBI spectra recover significantly lower flux. It may be that the weak indication of flaring behaviour in panel ‘c’ is spectral blending with features 12 and 13 shown in panel ‘d’, and that the western group does not flare. The features in panel are unambiguously in the eastern group. Panel ‘e’ probably shows the blending of features 12/13 with features 18 and 19 in the west. In panel ‘f’, note the slight rise in flux density at 1.9 to 2.04 [km s$^{-1}$]{}during May-June 2014 and a possible similar event February 2015, prior to the expected periodic flare. These features are in the same velocity range as 16, 18, 19 to the west, or 20 and 21 in the east. The feature in panel ‘g’, 3.62 to 3.9 [km s$^{-1}$]{}, appears to arise from a region $\approx$ 3 arcsec to the south of the peak of the region [@Fish2005] and are not shown in @Sanna2015. These features do not show any sort of correlated variability with the other masers.
![Spectra of individual 1667 MHz maser features from @Sanna2015. The top panel shows the KAT-7 spectrum from the nearest epoch (2013-07-16) 14 days later.[]{data-label="fig:vlbi_spec1667"}](vlbi_spec_1667-eps-converted-to.pdf){width="\columnwidth"}
Now to consider the 1667 MHz transition. Here the spectrum is simpler and the variability much more pronounced. Panel ‘h’ does not show any significant variation. It is not entirely clear where this emission arises since the VLBI spectra at this velocity range, 1.16 to 1.38 [km s$^{-1}$]{}, do not recover much flux. It could belong to features 4, 5 or 10, all of which are in the south-eastern group. Panel ‘i’ in the range 1.44 to 1.78 [km s$^{-1}$]{}, shows very well demarcated drops in intensity, which occur at the onset of the methanol flare, as we will show in the next section. After the ‘dip’, the masers increase in strength to a level higher than the pre-flare level then slowly recover over a period of several months. Features in this velocity range include 1, 2, 4, 5, 7, 9 and 10. However, considering peak flux densities the contribution must be predominantly from features 1, 2, 7 and 9 which are in the east. Panel ‘j’ does not show much variation. This velocity range, 1.85 to 2.05 [km s$^{-1}$]{}, may be covered by features 7 and 9 but its not clear why they would not flare if all the other masers in this region are flaring. Panel ‘k’ shows very pronounced flaring behaviour in the velocity range 2.12 to 2.4 [km s$^{-1}$]{} but it is unclear where these features are in the VLBI map - they could be part of 7 & 9 or 4 & 5 in the south-east. The features in panel ‘l’, as with panel ‘g’, arise from a region 3 arcsec to the south [@Fish2005].
To summarise the most variable features appear to be in the eastern group, closer to region component E2 and it is likely that none of the other OH maser groups are flaring.
![More detailed comparison of the profiles of the OH and methanol maser flares. The solid blue lines are the fits of eq. A7 of @VanderWalt2009 to the decay of the flares. See @VanderWalt2009 for the interpretation of $u_0$ and $n_{e,\star}$.[]{data-label="fig:meth_comp2"}](fitohdecay-eps-converted-to.pdf){width="\columnwidth"}
Comparison of OH and methanol maser flare profiles
--------------------------------------------------
As already noted above, the OH maser flares have a “characteristic” dip in the flux density before the flare. Such behaviour has not been reported for the methanol maser flares. In Figure \[fig:meth\_comp2\] a more detailed comparison between the OH and methanol maser flares is shown. The dip is seen to be significantly more pronounced for the 1667 MHz maser than for the 1665 MHz maser. It is rather interesting to note that for the 1667 MHz maser the onset of the dip is very near to the start of the 12.2 GHz maser flare. After the dip, the OH masers increase to a maximum after which they decay to the same level as before the onset of the 12.2 GHz maser flare. The time intervals between the peak of the 12.2 GHz maser and when the 1665 and 1667 MHz masers reach their maxima are about 23 and 13 days respectively.
Given that after reaching a maximum, the OH masers recover to the same pre-flare level, a behaviour also seen for the methanol masers, we also tried to fit the decay part of the OH masers with equation A7 of @VanderWalt2009 as well as to the decay of the 12.2 GHz methanol maser flare shown in Figure \[fig:meth\_comp2\]. The fits are shown as the blue solid lines in Figure \[fig:meth\_comp2\]. It is seen that for all three masers the decay of the masers are described very well by eq. A7 of @VanderWalt2009. Assuming that the decay of the masers are indeed related to the recombination of a partially ionized hydrogen plasma, the fit allows us to obtain estimates of the quiescent state electron density ($n_{e,\star}$) as well as the ratio ($u_0$) of the electron density from where the recombination started and $n_{e,\star}$. The values of these two quantities obtained from the fits are shown in the respective panels of Figure \[fig:meth\_comp2\]. The derived quiescent state densities range from $5.7 \times 10^5~ \mathrm{cm^{-3}}$ for the 12.2 GHz maser at 1.25 kms$^{-1}$ to $1.54 \times 10^5~\mathrm{cm^{-3}}$ for the 1665 MHz OH maser. The order of magnitude of these densities are what is expected for very young regions. It is also seen that for the OH masers the value of $u_0$ as found from the fit is only 1.07, in agreement with the small amplitude of the flares. For comparison, $u_0 = 1.85$ for the strong 12.2 GHz flare.
Water maser monitoring
----------------------
Figure \[fig:h2o\] shows a dynamic spectrum of the water maser observations. While the masers are variable, there does not appear to be any particular correlation with the methanol or hydroxyl, but the time-coverage during the September 2013 flare is extremely sparse due to bad telescope pointing. There are some transient features at $\sim$ 5 [km s$^{-1}$]{} and $\sim$ 8 [km s$^{-1}$]{} in the second flare cycle, which also appear to have slight changes in peak velocities very typical of water masers in an outflow.
In Figure \[fig:methwater\] we show a more detailed comparison between the time series of the water- and 12.2 GHz methanol masers between MJD 56700 and 57400. It is seen that the duration of the water maser flare covers a time interval approximately equal to the period of the 12.2 GHz flares. The water and methanol maser flares do not seem to be related in the same way as seems to be the case for the OH and methanol maser flares.
![Comparison of water and 12.2 GHz methanol maser flares[]{data-label="fig:methwater"}](watermasers-eps-converted-to.pdf){width="\columnwidth"}
Discussion
==========
There are quite a number of interesting aspects to note in G9.62+0.20. We first note, using Figure 1 of @Sanna2015, that, in projection, the periodic methanol and OH masers are located approximately 330 and 1930 AU from the continuum peak of the VLA A-array 7mm emission. Although there are OH masers located closer to the continuum peak, these do not appear to be periodic at all. The question then is why the periodic OH masers are located so much further than the periodic methanol masers from the peak of continuum. As shown above, equation A7 of @VanderWalt2009 provides a very good fit to the decay of the 12.2 GHz methanol maser flare. The implication is that the variation in the electron density occurs at such a position that the region is optically thin for outward propagating 12.2 GHz photons in the direction of the masing gas. Since the optical depth is proportional to $\lambda^{2.1}$, it follows that at the projected position of the periodic methanol masers, the optical depth at 1665 MHz is about 65 times greater than at 12.2 GHz. If, for example, the optical depth at 12.2 GHz is 0.1, it means that it is 6.5 at 1665 MHz and therefore that all variability will be damped due to absorption in the region. Qualitatively, it is therefore expected that if there are periodic OH masers, that these will be located further from the core of the region compared to the periodic methanol masers. This might explain the relative projected positions of the periodic methanol and OH masers in G9.62+0.20E.
Perhaps the most intruiging aspect of the periodic OH masers in G9.62+0.20E is the pronounced sharp decrease (dip) in the maser flux density before the “flare”. That the dip is a real feature is clear from the fact that it is seen in two flares and is also present in the 1665 MHz masers. As noted above, the periodic methanol masers do not show this behaviour. If, as proposed by @VanderWalt2009, @VanderWalt2011 and, @vanderwalt2016, the periodicity of the methanol masers in G9.62+0.20E are driven by a colliding-wind binary system and that the flares are due to changes in the background free-free emission from the region, it is required that the dip should also be explained within the same framework. We consider two possibilities.
\(a) *The periodicity is due to a CWB associated with component E1:* Although no quantitative explanation can be given, qualitatively we note the following. First, from the maps of @Sanna2015 it is seen that the periodic OH masers are projected significantly ($\sim$1600 AU) further from the core of the region E1 than the periodic methanol masers. Should both masers amplify the free-free emission from the background region, it follows that they probe two completely different parts of the region. The ionization structure of an UC region as calculated with the photo-ionization code [Cloudy]{} [see eg. @VanderWalt2011] shows that partially ionized gas extends beyond the ionization front. Within the framework of the CWB scenario, the lower energy ionizing photons from the hot shocked gas are absorbed at the ionization front and gives rise to the flaring of the methanol masers. Higher energy ($>$ 100 eV) photons, on the other hand, can propagate beyond the ionization front due to the significantly lower photo-ionization cross section for HI. Ionization of hydrogen by these photons will result in higher average electron temperatures. In the optically thin case the free-free emission is proportional to the volume emissivity which, in turn, is proportional to $T_e^{-0.5}$. Raising the electron temperature of the plasma thus lowers the volume emissivity.
A *possible* explanation then for the dip in the 1667 MHz maser flux density is that, because of the higher energies of the ionizing photons, the electron temperature is raised significantly during the ionization event in the gas against which the OH maser is projected, resulting in a decrease in the free-free emission. After the pulse of ionizing photons has passed, the electrons cool to the equilibrium temperature giving rise to an increase in the free-free emission and therefore also in the maser emission. Simultaneous to the increase in the electron temperature, the electron density also increases due to the pulse of ionizing photons. The exact profile of the flare (which includes the dip) depends on the magnitude of the change in electron temperature, the cooling rate of the electrons and the ionization rate associated with the pulse of ionizing photons. The observed peak of the OH maser flares therefore does not necessarily indicate the time of the peak of the pulse of ionizing photons as in the case of the methanol maser flares. The difference in time between the peaks of the methanol and OH maser flares, might therefore not be due to a geometric delay only (see below) but also to other physical effects. The required change in electron temperature to explain the magnitude of the dip can be estimated by considering the ratio of flux densities of the 1667 MHz maser before the dip and at the minimum of the dip (Figure \[fig:meth\_comp2\]). It then follows that the electron temperature has to be raised by a factor of 1.8 to explain the observed decrease in the maser flux density. Thus, for example, if before the dip the electron temperature was 5000 K, it has to be raised to 9000 K in order to explain the decrease seen in the maser flux density.
\(b) *The periodicity is associated with component E2.* Given that the decay of all three masers are described very well in terms of the recombination of a hydrogen plasma, it is reasonable to assume that component E2 must then also be a periodic source of ionizing photons which influence the flux of seed photons for both the methanol and OH masers. Since the methanol masers are projected against E1, it is therefore required that the ionizing photons propagate from E2 to the position against which the periodic methanol masers are projected. However, it is clear from @Sanna2015 that there is very little ionized gas associated with E2, which means that a significant fraction of the flux of the periodically produced ionizing photons will be absorbed close to E2. It is therefore difficult to see how a source of periodically varying ionizing photons located at E2 will influence that part of E1 against which the periodic methanol masers are projected.
Having excluded E2 as the driving source for the periodic methanol and OH masers, we finally note the following. From Figure \[fig:meth\_comp2\] it is seen that although the peak of the 1667 MHz flare lags that of the 12.2 GHz flare by about 13 days, the dip of the 1667 MHz maser occurs almost simultaneously with the onset of the 12.2 GHz methanol maser flare. In view of the above possible explanation for the dip, it seems reasonable to regard the onset of the 12.2 GHz flare and the almost simultaneous sharp decrease of the 1667 MHz maser flux density to be caused by the same ionization event. However, due to the difference in projected distances from the core of the region, we then expect the dip of the OH maser flares to be delayed by about nine days with respect to the onset of the 12.2 GHz flare. Considering the very simple geometry in Figure \[fig:geometry\], the time difference between the methanol and hydroxyl flares is given by $\Delta t =
(r_{\mathrm{B}} - r_{\mathrm{A}} + z_{\mathrm{OH}} - z_{\mathrm{CH_3OH}})/c$. For the present case $\Delta t \sim 0$, from which follows that $z_{\mathrm{CH_3OH}}-
z_{\mathrm{OH}} \approx r_{\mathrm{B}}- r_{\mathrm{A}}$. Our observations of the methanol and OH maser flares therefore suggest that the masing region of the periodic 12.2 GHz methanol masers are located about 1600 AU further from the “surface” of the region into the molecular envelope compared to the location of the periodic OH masers.
![Simplified geometry for the locations of the periodic methanol and hydroxyl masers. The ionizing star is indicated by the star symbol. The “surface” of the region is assumed to be flat. The methanol and hydroxyl masers respectively respond to changes in the free-free emission from points A and B[]{data-label="fig:geometry"}](hiifigure-eps-converted-to.pdf){width="\columnwidth"}
As noted above, a water maser flare occured that overlapped in time with a 12.2 GHz methanol maser flare. It seems as if a comparison between the water and methanol maser flares similar to that between OH and methanol masers cannot be made. Although the water maser flare overlaps in time with the 12.2 GHz methanol maser, the water maser flare clearly starts before the methanol maser flare. Also, the decay time of the water maser flare is significantly different from that of the OH and methanol maser flares and does not suggest any causal relation with the methanol and OH flares.
To our knowledge, the results presented above are the first conclusive observational evidence of periodic variability of OH masers associated with a high-mass star forming region. @Green2012 found an indication of periodicity in the OH masers toward G12.889+0.489 (which has a 29.5 day period in methanol), but the time-series were too undersampled to obtain detailed cycle profiles and no contemporaneous methanol monitoring was done. It is interesting that the hydroxyl masers appear to undergo a drop in emission coincident with the expected minima of 6.7 GHz methanol masers but a key difference is that the hydroxyl masers do not show any flares. G12.889+0.489 also differs from G9.62+0.20E in that it has somewhat irregular flaring behaviour in the methanol, but shows a well-defined minimum, which seems to be periodic [@Goedhart2009], while the peaks of the methanol flares can occur any time in an 11-day window. It is quite likely, given the short period and the difference in the methanol maser light curves, that a different mechanism is modulating the maser intensity. It would undoubtedly be of great benefit to our understanding of these phenomena to conduct intensive monitoring of both maser species through one 29.5 day cycle in this source.
Summary and Conclusions
=======================
We presented the first conclusive observational evidence for the periodic variability of OH mainline masers associated with a high-mass star forming region. The 1667 MHz masers show a pronounced dip in flux density which closely coincide with the onset of the 12.2 GHz methanol maser flare. The decay of the OH maser flares, similar to that of the methanol maser flares, can be described very well by the decrease in the free-free emission from the background region as expected from a recombining hydrogen plasma. A possible explanation, within the framework of the colliding-wind binary scenario, for the dip in flux density of the 1667 MHz masers is that it is due to an increase in electron temperature following the ionization of the outer regions of the region by photons with energy greater than about 100 eV. We also argued, within this framework that the masing region for the periodic 12.2 GHz methanol masers are located about 1600 AU further from the “surface” of the region into the molecular envelope compared to the location of the periodic OH masers.
\[lastpage\]
[^1]: E-mail: [email protected]
|
---
abstract: 'Least-squares reverse time migration is well-known for its capability to generate artifact-free true-amplitude subsurface images through fitting observed data in the least-squares sense. However, when applied to realistic imaging problems, this approach is faced with issues related to overfitting and excessive computational costs induced by many wave-equation solves. The fact that the source function is unknown complicates this situation even further. Motivated by recent results in stochastic optimization and transform-domain sparsity-promotion, we demonstrate that the computational costs of inversion can be reduced significantly while avoiding imaging artifacts and restoring amplitudes. While powerfull, these new approaches do require accurate information on the source-time function, which is often lacking. Without this information, the imaging quality deteriorates rapidly. We address this issue by presenting an approach where the source-time function is estimated on the fly through a technique known as variable projection. Aside from introducing negligible computational overhead, the proposed method is shown to perform well on imaging problems with noisy data and problems that involve complex settings such as salt. In either case, the presented method produces high resolution high-amplitude fidelity images including an estimates for the source-time function. In addition, due to its use of stochastic optimization, we arrive at these images at roughly one to two times the cost of conventional reverse time migration involving all data.'
author:
- |
Mengmeng Yang^1\*\#^, Zhilong Fang^2\*^, Philipp Witte^3^ and Felix J. Herrmann^1,3^\
^1^ School of Earth and Atmospheric Sciences, Georgia Institute of Technology\
^2^ Department of Mathematics, Massachusetts Institute of Technology\
^3^ School of Computational Science and Engineering, Georgia Institute of Technology\
\* Equally contributed\
\# To whom correspondence should be addressed: [email protected]
bibliography:
- 'SE.bib'
title: 'Time-domain sparsity promoting least-squares reverse time migration with source estimation'
---
[ namedef[[email protected]]{}[9999/12/31]{} namedef[[email protected]]{}]{}
Introduction
============
Reverse-time migration (RTM) is a popular wave equation-based seismic imaging methodology where the inverse of the linearized Born scattering operator is approximated by applying its adjoint directly to the observed reflection data [@baysal1983reverse; @whitmore1983iterative]. Because the adjoint does not equal the pseudo inverse, conventional RTM produces images with incorrect amplitudes. Among the factors that contribute to low-fidelity amplitudes, the imprint of the temporal bandwidth limitation of the typically unknown source wavelet features prominently and so does the fact that the Born scattering operator is not inverted. To overcome these issues, we formulate our imaging problem as a linear least-squares inversion problem where the difference between observed and predicted data is minimized in the $\ell_2$-norm [@schuster1993least; @nemeth1999least; @dong2012least; @zeng2014least]. While least-squares migration is a powerful technique, its succesful application to industry-scale problems is hampered by three key issues. First, iterative demigrations (i.e. Born modeling) and migrations become computationally prohibitively expensive when carried out over all shots. Second, we run the risk of overfitting the data when minimizing the $\ell_2$-norm of the data residual. This overfitting may introduce noise-related artifacts in inverted images. Third, while the source location is generally well known, the temporal source function is often not known accurately. Because imaging relies on knowing the source function, this may have a detrimental effect on the image and makes it necessary to come up with source estimation methodology. Since we carry out our imaging iteratively, we propose to estimate the wavelet on the fly as we build up the image.
We address the issue of computational feasibility by combining techniques from stochastic optimization [@van2011seismic; @haber2012effective; @powell2014clearing], curvelet-domain sparsity-promotion [@herrmann2012efficient], and online convex optimization [@lorenz2014linearized] with linearized Bregman. Stochastic optimization allows us to work with small random subsets of shots, which limits the number of wave equation solves—i.e., passes through the data. Convergence is guaranteed [@herrmann2015fast; @yang2016time ; @witte2019compressive] by replacing the $\ell_1$-norm, by an elastic net consisting of a strongly convex combination of $\ell_1$- and $\ell_2$-norm objectives. Inclusion of the $\ell_2$-norm results in a greatly simplified algorithm involving linearized Bregman iterations, which corresponds to gradient descent on the dual variable supplemented by a simple soft thresholding operation [@cai2009convergence; @yin2010analysis] with a threshold that is fixed. We refer to this method as sparsity-promoting least-squares reverse-time migration (SPLS-RTM).
In addition to the high computational cost, the lack of accurate knowledge on the unknown temporal source signature may also adversely affect the performance of the inversion. Errors in the source signature lead to erroneous residuals, which in turn result in inaccurately imaged reflectors, which now may be positioned wrongly or may have the wrong amplitude or phase. To mitigate these errors, we need an embedded procedure where the source signature is updated along with the image during the inversion [@pratt1999seismic; @aravkin2012source; @aravkin2013sparse; @fang2018source] using a technique known as variable projection [@rickett2013variable; @van2014reply]. For time-harmonic imaging, variable projection involves the estimation of the source function by solving a least-squares problem for each frequency separately. Since the unknown for each frequency in that case is a single complex-valued variable, this process is simple and has resulted in an accurate estimation and compensation for the source-time function [@tu2013fast ; @fang2018source]. Unfortunately, the situation is more complicated during imaging in the time domain, where we have to estimate the complete source signature during each iteration. For this purpose, we integrate early work by @yang2016time and inverse-scattering method [@witte2019compressive] and achieve an approach that is suitable for realistic imaging scenarios that may include salt.
Our work is outlined as follows. First, we introduce the basic equations for time-domain reverse time migration and least-squares reverse time migration. To overcome the computational cost associated with the latter, we introduce a stochastic optimization method with sparsity promotion. This method is designed to provide an image at a fraction of the cost. Next, we extend this approach so it includes on-the-fly source estimation. This allows us to remove the requirement of the source function. We conclude by presenting a number of synthetic case studies designed to demonstrate robustness with respect to noisy data and to complex imaging scenarios that include salt.
Methodology
===========
Since our approach hinges on cost-effective least-squares imaging, we first introduce our formulation of sparsity-promoting least-squares migration with stochastic optimization, followed by our approach to on-the-fly source estimation during the iterations.
From RTM to LS-RTM
------------------
Reverse time migration derives from a linearization (see e.g. @mulder2004comparison) with respect to the background squared slowness. For the $i^\mathrm{th}$ source this linearization reads $${\delta{\mathbf{d}}}_{i} = \mathbf{F}_{i}({\mathbf{m}}_0 + {\delta{\mathbf{m}}}, {\mathbf{q}}) - \mathbf{F}_{i}({\mathbf{m}}_0, {\mathbf{q}}) \approx {\mathbf{J}}_{i}({\mathbf{m}}_0, {\mathbf{q}}) {\delta{\mathbf{m}}},
\label{BornModeling}$$ where the vectors ${\delta{\mathbf{m}}}$, ${\mathbf{q}}$, and ${\delta{\mathbf{d}}}$ denote the model perturbation, the source-time function, and the corresponding data perturbation, respectively. We model the data for $n_t$ time samples over a time interval of $T\,\mathrm{s}$. The number of receivers is $n_r$ so a single shot record is of size $n_t\times n_r$. The nonlinear forward modeling operator $\mathbf{F}_{i}({\mathbf{m}}, {\mathbf{q}})$ for the $i^\mathrm{th}$ source location involves the solution of the discretized acoustic wave equation $$\begin{aligned}
\left ({\mathbf{m}}\odot\frac{\partial^2}{\partial t^{2}} - \Delta\right){\mathbf{u}}_{i}&={\mathbf{P}}_{\text{s},i}^{\top}{\mathbf{q}}, \\
{\mathbf{P}}_{\text{r},i} {\mathbf{u}}_{i} &= {\mathbf{d}}_{i},
\end{aligned}
\label{AcousticWave}$$ parameterized by the squared slowness collected in the vector ${\mathbf{m}}$ (for simplicity, we kept the density constant and we used the symbol $\odot$ to denote elementwise multiplication.) The symbol $\Delta$ represents the discretized Laplacian and the linear operator ${\mathbf{P}}_{\text{r},i}$ restricts the wavefield for the $i^\mathrm{th}$ source to the corresponding receiver locations, while the linear operator ${\mathbf{P}}_{\text{s},i}^{\top}$ injects the source time function at the location of the $i^\mathrm{th}$ source in the computational grid. The Jacobian ${\mathbf{J}}_{i}({\mathbf{m}}_0, {\mathbf{q}})$ is known as the Born modeling operator and is given by the derivative of $\mathbf{F}_i({\mathbf{m}}, {\mathbf{q}})$ at the point of ${\mathbf{m}}_0$. Applying the Jacobian ${\mathbf{J}}_{i}({\mathbf{m}}_0, {\mathbf{q}})$ to the model perturbation ${\delta{\mathbf{m}}}$ requires the solution of the following linearized equation: $$\begin{aligned}
\left({\mathbf{m}}_0\odot\frac{\partial^2}{\partial t^{2}} - \Delta\right)\delta{\mathbf{u}}_{i}& = -\frac{\partial^2}{\partial t^2} \big( \delta{\mathbf{m}}\odot{\mathbf{u}}_{i} \big), \\
{\mathbf{P}}_{\text{r},i} \delta{\mathbf{u}}_{i} &= {\delta{\mathbf{d}}}_{i},
\end{aligned}
\label{BornWave}$$ where the vector $\delta{\mathbf{u}}_{i}$ corresponds to the wavefield perturbation for the $i^\mathrm{th}$ source.
The goal of seismic imaging is to estimate the model perturbations from observed data. We can expect this reconstruction process to be successful in situations where the above linear approximation is accurate—i.e., the background velocity model needs to be sufficiently accurate, which we assume it is. We also need accurate knowledge on the source function, an important aspect we will address below.
While the above linearization allows us to create an image via $${\delta{\mathbf{m}}}_{\text{RTM}}=\sum_{i=1}^{n_s} {\mathbf{J}}^\top_{i}{\delta{\mathbf{d}}}_i
\label{migration}$$ with $n_s$ the number of shots, the adjoint of the Jacobian (denoted by the symbol $^\top$) does not correspond to its inverse and ${\delta{\mathbf{m}}}_{\text{RTM}}$ will suffer from wavelet side lobes and inaccurate and unbalanced amplitudes [@mulder2004comparison; @bednar2006two; @hou2016accelerating]. Unlike RTM (Equation \[migration\]), LS-RTM [@aoki2009fast; @herrmann2012efficient; @tu2015fast] reconstructs the model perturbation by computing the pseudo-inverse of the Born modeling operator, which can significantly mitigate these defects. LS-RTM iteratively solves the following least squares data-fitting problem: $${\mathop{\mathrm{minimize}}}_{{\delta{\mathbf{m}}}}\frac{1}{2}\sum_{i=1}^{{n_{\text{s}}}}\|{\mathbf{J}}_{i}({\mathbf{m}}_0,{\mathbf{q}}){\delta{\mathbf{m}}}-{\delta{\mathbf{d}}}_{i}\|^2.
\label{Lineardata}$$ Compared to Equation \[migration\], the above minimization requires multiple evaluations of the Jacobian and its adjoint, which becomes rapidly computationally prohibitive for large 2D and 3D imaging problems as the number of sources $n_s$ grows. This in part explains the relatively slow adaptation of least-squares reverse time migration (cf. Equation \[Lineardata\]) by industry. As we show below, we overcome this problem by combining ideas from stochastic optimization and sparsity promotion [@herrmann2015fast; @yang2016time; @witte2019compressive], which allow us to obtain artifact-free images at the cost of two to three passes through the data.
Stochastic optimization with sparsity promotion
-----------------------------------------------
As we mentioned above, the minimization of Equation \[Lineardata\] over all $n_s$ shots is computationally prohibitively expensive. In addition, the minimization is unconstrained and misses regularization to battle the adverse effects of noise and the null space (missing frequencies and finite aperture) associated with solving the least-squares imaging problems of the type listed in Equation \[Lineardata\]. To address these two problems, we combine ideas from stochastic optimization, during which we only work on randomized subsets of shots during each iteration, and ideas from sparsity-promoting optimization designed to remove the imprint of the null space and source subsampling related artifacts. As we have learned from the field of compressive sensing [@candes2006compressive; @donoho2006compressed; @candes2008introduction], transform-domain sparsity promotion is a viable technique to remove subsample related noise in imaging via $$\begin{aligned}
{\mathop{\mathrm{minimize}}}_{{\mathbf{x}}} \quad &\|{\mathbf{x}}\|_{1}, \\
\text{subject to } \, &\sum_{i=1}^{n_s} \|{\mathbf{J}}_{i}( {\mathbf{m}}_0, \mathbf{q}) {\mathbf{C}}^{\top} {\mathbf{x}}- \delta {\mathbf{d}}_{i}\|_{2} \leq \sigma.
\end{aligned}
\label{BPDN}$$ In this formulation, known as the Basis Pursuit Denoise (BPDN, @chen2001atomic) problem, we include the sparsity-promoting $\ell_1$-norm as the objective on the curvelet coefficients ${\mathbf{x}}$ of the image. These coefficients are related to the linearized data via the adjoint of the curvelet transform $({\mathbf{C}}^{\top})$ and the above program seeks to find the sparsest curvelet coefficient vector that matches the data within the noise level $\sigma$. While the above problem is known to produce high-fidelity results, its solution relies on iterations that involve a loop over all $n_s$ shots.
Stochastic gradient descent [e.g. @haber2012effective in the context of seismic inversion] is a widely used tool to make unconstrained optimization problems of the type included in Equation \[Lineardata\] computationally feasible by computing the gradient of Equation \[Lineardata\] for randomized subsets of shots, using a given a batch size that corresponds to the number of shots used per iteration. This approach allows to minimize Equation \[Lineardata\] in very few epochs, using only few passes through data consisting of $n_s$ shot records, as long as the step lengths adhere to certain conditions to guarantee convergence. Unfortunately, this complicates the solution of BPDN. To avoid this complication, we reformulate, following @cai2009convergence, Equation \[BPDN\] by replacing its convex $\ell_1$-norm objective by the strongly convex objective involving $$\begin{aligned}
{\mathop{\mathrm{minimize}}}_{\mathbf{x}} \ & \lambda_{1} \Vert\mathbf{x} \Vert_1 + \frac{1}{2} \Vert\mathbf{x} \Vert^{2}_{2} \\
\text{subject to} \ &\sum_{i=1}^{n_s} \Vert {\mathbf{J}}_{i}( \mathbf{m}_0, \mathbf{q}) \mathbf{C}^{\top} \mathbf{x} - \delta \mathbf{d}_{i}\Vert_{2} \leq \sigma
\end{aligned}
\label{PlainLB}$$ with the estimate for the image given by $\delta\mathbf{\hat{m}}=\mathbf{C}^{\top} \mathbf{\hat{x}}$ where $\mathbf{\hat{x}}$ is the minimizer of the above optimization problem. The mixed objective in this problem is known as an elastic net in machine learning, which offers convergence guarantees (see @lorenz2014linearized) in situations where we work during each iteration with different randomized subsets of shots indexed by $\mathcal{I}_k\subset [1\cdots n_s]$ with cardinality $|\mathcal{I}|=n_s^\prime\ll n_s$. We choose these subsets without replacement.
For $\lambda\rightarrow\infty$, which in practice means $\lambda$ large enough, iterative solutions of Equation \[PlainLB\] as summarized in Algorithm \[alg:LSM1\] converge to the solution of Equation \[BPDN\], even in situations where we work with randomized subsets of shots. Compared to iterative solutions of Equation \[BPDN\], the iterations (lines 7–8 in Algorithm \[alg:LSM1\]) correspond to iterative thresholding with a fixed threshold $\lambda$ on the dual variable ($\mathbf{z_k}$) with a dynamic step length given by $t_k = \Vert \mathbf{A}_k \mathbf{x}_k - \mathbf{b}_k\Vert^{2}_{2} / \Vert \mathbf{A}_k^{\top} (\mathbf{A}_k \mathbf{x}_k - \mathbf{b}_k)\Vert^{2}_{2}$ [@lorenz2014linearized]. During each iteration, known as linerarized Bregman iterations, the residual is projected onto an $\ell_2$-norm ball with the radius $\sigma$ through a projection operator $\mathcal{P}_\sigma$. To avoid too many iterations, we set the threshold $\lambda$, related to the the tradeoff between the $\ell_1$ and $\ell_2$-norm objectives in Equation \[PlainLB\], to a value that is not too large—i.e., typically proportional to the maximum of $|\mathbf{z}_k|$ at the first iteration ($k=1$). As reported by @yang2016time and @witte2019compressive, high quality images can be obtained running Algorithm \[alg:LSM1\] for a few epochs as long as the source time function $\mathbf{q}$ and background velocity model are sufficiently accurate. As we will show below, the background velocity model also needs to be smooth so the tomography-related imaging is avoided.
1. Initialize $\mathbf{x}_0 = \mathbf{0}$, $\mathbf{z}_0 = \mathbf{0}$, $\mathbf{q}$, $\lambda_{1}$, batchsize $n^\prime_{s} \ll n_s$\
2. **for** $k=0,1, \cdots$\
3. Randomly choose shot subsets $\mathcal{I}_k \subset [1 \cdots n_s],\, \vert \mathcal{I} \vert = n^\prime_{s}$\
4. $\mathbf{A}_k = \{{\mathbf{J}}_i ( \mathbf{m}_0,\mathbf{q} ) \mathbf{C}^{\top}\}_{i\in\mathcal{I}_k}$\
5. $\mathbf{b}_k = \{\mathbf{\delta d}_i\}_{i\in\mathcal{I}_k}$\
6. $t_k = \Vert \mathbf{A}_k \mathbf{x}_k - \mathbf{b}_k\Vert^{2}_{2} / \Vert \mathbf{A}_k^{\top} (\mathbf{A}_k \mathbf{x}_k - \mathbf{b}_k)\Vert^{2}_{2}$\
7. $\mathbf{z}_{k+1} = \mathbf{z}_k - t_{k} \mathbf{A}^{\top}_k \mathcal{P}_\sigma (\mathbf{A}_k\mathbf{x}_k - \mathbf{b}_k)$\
8. $\mathbf{x}_{k+1}=S_{\lambda_{1}}(\mathbf{z}_{k+1})$\
9. **end**\
10. **Output:** $\hat{{\delta{\mathbf{m}}}}=\mathbf{C}^\top\mathbf{x}_{k+1}$\
note: $S_{\lambda_{1}}(\mathbf{z}_{k+1})=\mathrm{sign}(\mathbf{z}_{k+1})\max\{ 0, \vert \mathbf{z}_{k+1} \vert - \lambda_{1} \}$\
$\mathcal{P}_{\sigma}(\mathbf{A}_k \mathbf{x}_k - \mathbf{b}_k) = \max\{ 0,1-\frac{\sigma}{\Vert \mathbf{A}_k \mathbf{x}_k -\mathbf{b}_k \Vert}\} \cdot (\mathbf{A}_k \mathbf{x}_k -\mathbf{b}_k)$
On-the-fly source estimation
----------------------------
In practice, we unfortunately do not have access to the source time function $\mathbf{q}$ required by Algorithm \[alg:LSM1\]. Following our earlier work on source estimation in time-harmonic imaging and full-waveform inversion [@van2011seismic; @tu2014fis], we propose an approach during which we estimate the source-time signature after each model update by solving a least-squares problem that matches predicted and observed data via a time-domain filter.
To keep our time-domain wave equation solvers with finite differences numerically stable (In our implementation, we used Devito (<https://www.devitoproject.org>) for our time-domain finite difference simulations and gradient computations [@devito-compiler; @devito-api] and JUDI (<https://github.com/slimgroup/JUDI.jl>) as an abstract linear algebra interface to our Algorithms [@witte2018alf]), we introduce an initial guess for the source time function $\mathbf{q}_0$ with a bandwidth limited spectrum that is flat over the frequency range of interest. Under some assumptions on the source time function, we can write the true source time function as the convolution between the initial guess and the unknown filter $\mathbf{w}$—i.e., we have $\mathbf{q}=\mathbf{w}\ast\mathbf{q}_0$ where the symbol $\ast$ denotes the temporal convolution. Because we assume one and the same source time function for all shots, we can write $${\mathbf{J}}_{i}(\mathbf{m}_0, \mathbf{w} \ast \mathbf{q}_{0}) = \mathbf{w} \ast {\mathbf{J}}_{i}(\mathbf{m}_0, \mathbf{q}_{0})
\label{SrcApp}$$ for all sources $i=1\cdots n_s$. In this expression, we make use of linearity of the wave equation with respect to its source. To simplify the notation, we also overload the temporal convolution (denoted by the symbol $\ast$) to apply to all data—i.e. all traces in the shot records.
Based on the above relationship, we propose to solve for $\mathbf{w}$ after each linearized Bregman iteration (line 10 of Algorithm \[alg:LSM2\]) via $$\begin{aligned}
\min_{\mathbf{w}} \ & \sum_{i\in\mathcal{I}_k} \Vert \mathbf{w} \ast {\mathbf{J}}_{i}(\mathbf{m}_0, \mathbf{q}_{0}) \mathbf{C}^T \mathbf{x} - \delta \mathbf{d}_i\Vert ^2_{2} + \Vert \mathbf{r}\odot (\mathbf{w} \ast \mathbf{q}_{0}) \Vert^2_{2}
\end{aligned}
\label{sub22}$$ with $\mathcal{I}_k \subset [1 \cdots n_s], \vert \mathcal{I} \vert = n_{s}^\prime$ a randomly chosen shot subset of shot records.
To prevent overfitting while fitting the generated data $\mathbf{\tilde{b}}_k$ at the $k\text{th}$ iteration to the observed data $\mathbf{b}_k$, we include a penalty term $\mathbf{r}$ consisting of an exponential weighting vector as given by: $$\mathbf{r}(t)= \nu + \log (1+e^{\alpha (t-t_0)}).
\label{Weight}$$ In this expression, the scalar $\alpha$ determines the rate of growth after $t=t_0$. We choose $t_0$ such that oscillations related to overfitting are suppressed after this time. This prevents overfitting and ensures the filters $\mathbf{w}_k$ to be short such that the estimated source time function $\mathbf{q}=\mathbf{w}_k\ast\mathbf{q}_0$ remains short as well. The weight parameter $\nu$ penalizes the energy of the estimated source $\mathbf{q}$, which also helps to alleviate the ill-conditioning of this sub-problem.
We summarize the different steps of our approach in Algorithm \[alg:LSM2\] below. As earlier, we solve the sparsity-promoting optimization problem via linearized Bregman iterations, which now include a correlation (correlation denoted by the symbol $\star$ is the adjoint of convolution) with the current estimate for source time function correction ($\mathbf{w}_k$) in line 8. We initialize the source time function correction with a discrete Delta distribution ($\mathbf{w}_0=\mathbf{\delta}$). We refer to this method with on-the-fly source estimation as sparsity-promoting LS-RTM with source estimation (SPLS-RTM-SE).
1. Initialize $\mathbf{x}_0 = \mathbf{0}$, $\mathbf{w}_0=\mathbf{\delta}$, $\mathbf{z}_0 = \mathbf{0}$, $\mathbf{q}_0$, $\mathbf{r}(t)$, batchsize $n_{s}^\prime\ll n_s$, $\mathbf{r}$\
2. **for** $k=0,1, \cdots$\
3. Randomly choose shot subsets $\mathcal{I}_k \subset [1 \cdots n_s], \vert \mathcal{I} \vert = n_{s}^\prime$\
4. $\mathbf{A}_k = \{{\mathbf{J}}_i ( \mathbf{m}_0,\mathbf{q}_{0} ) \mathbf{C}^{\top}\}_{i\in\mathcal{I}_k}$\
5. $\mathbf{b}_k = \{\mathbf{\delta d}_i\}_{i\in\mathcal{I}_k}$ \
6. $\mathbf{\tilde{b}}_k = \mathbf{A}_k \mathbf{x}_k$\
7. $t_k = \Vert \mathbf{\tilde{b}}_k - \mathbf{b}_k\Vert^{2}_{2} / \Vert \mathbf{A}_k^{\top} (\mathbf{\tilde{b}}_k - \mathbf{b}_k)\Vert^{2}_{2}$\
8. $\mathbf{z}_{k+1} = \mathbf{z}_k - t_{k} \mathbf{A}^\top_{k} \Big( \mathbf{w}_k {\star} \mathcal{P}_\sigma (\mathbf{w}_k \ast \mathbf{\tilde{b}}_k - \mathbf{b}_k) \Big)$\
9. $\mathbf{x}_{k+1}=S_{\lambda}(\mathbf{z}_{k+1})$\
10. $\mathbf{w}_{k+1} = \argmin_{\mathbf{w}} \Vert \mathbf{w} \ast \tilde{\mathbf{b}}_{k} - \mathbf{b}_{k}\Vert ^2_{2} + \Vert \mathbf{r}\odot (\mathbf{w} \ast \mathbf{q}_0) \Vert^2_{2}$\
11. **end**\
12. **Output:** $\hat{\mathbf{q}}=\mathbf{w}_{k+1}\ast \mathbf{q}_0$ and $\hat{{\delta{\mathbf{m}}}}=\mathbf{C}^\top\mathbf{x}_{k+1}$
Numerical experiments
=====================
In this experiment section, we demonstrate the viability of our approach by means of carefully designed synthetic examples. We start by showing that linearized Bregman iterations with on-the-fly source estimation are indeed able to jointly estimate the source and the sparse vector of image curvelet coefficients. Next, we consider an imaging experiment on the Marmousi model emphasizing the importance of including the source function and the influence of noise. We conclude by introducing a practical workflow that is capable of handling salt-related imaging problems.
Stylized example
----------------
While imaging with linearized Bregman (LB) iterations has resulted in high-fidelity true-amplitude images in complex models [@witte2019compressive], the viability of this alternative sparsity-promoting approach has not yet been verified in combination with on-the-fly source estimation. For this purpose, we examine the performance of LB with source estimation on a simplified stylized example. As we can see, Equation \[SrcApp\] implies a bilinear dependence of the reflected data on both the unknown filter $\mathbf{w}$ and the curvelet coefficient vector $\mathbf{x}$. It is well known that this sort of bilinear dependence can give rise to ambiguities even though the vector $\mathbf{x}$ is sparse.
We exemplify this seismic bilinear relationship be defining $\mathbf{W}\mathbf{A}\mathbf{x}=\mathbf{b}$, where the matrix $\mathbf{A} \in \mathcal{R}^{20000 \times 10000}$ is ill-conditioned, with $\text{rank}(\mathbf{A})=500$. The sparse vector $\mathbf{x}\in \mathcal{R}^{10000\times 1}$ has only $20$ random non-zero elements. A block of the tall matrix, $\mathbf{A}_i \in \mathcal{R}^{500\times 10000}, i\in [1\dots 40]$ serves as a proxy for the LB modeling operator $\mathbf{J}_i$ for the $i^\mathrm{th}$ shot with only one single trace. We implement the trace-by-trace convolution via a Toeplitz matrix defined in terms of the filter $\mathbf{w}\in \mathcal{R}^{500\times 1}$ acting on each $\mathbf{A}_i\mathbf{x}$. The multiplication of the convolution matrix $\mathbf{W}\in \mathcal{R}^{20000 \times 20000}$ with $\mathbf{Ax}$ compactly represents the repeated convolutions of the filter with all traces.
This example, designed to jointly invert for $\mathbf{x}$ and $\mathbf{w}$, aims to exhibit the capability of our Algorithm \[alg:LSM2\] to carry out seismic imaging and on-the-fly source estimation. To demonstrate the effect of the penalty term in line 10 of Algorithm \[alg:LSM2\], we compare sparsity-promoting solutions for the fixed true wavelet to solutions with on-the-fly source estimation with and without the additional penalty. During each iteration, we randomly choose $10\%$ of the blocks of the tall matrix $\mathbf{A}$ and we run five passes through the data in total. After some parameter testing, we choose the following values for the penalty parameters: $\lambda=1,\nu=1$ and $\alpha=8$. We find that different choices for these penalty parameters have little effect on our inversion results. Finally, the time parameter $t_0$ is set according to the approximate duration of the filter $\mathbf{w}$, which in this case corresponds to a Ricker wavelet since we choose $\mathbf{q}_0$ to be a delta Dirac. We also initialize the filter $\mathbf{w}$ with a normalized Dirac. Because of the well-known amplitude ambiguity inherent to blind deconvolution problems, we normalize the $\ell_2$-norms of the estimated source and reflectivity.
Pairs of estimated sparse “reflectivities” ($\hat{{\delta{\mathbf{m}}}}$) and source functions ($\hat{\mathbf{q}}=\mathbf{w}_{final}\ast\mathbf{q}_0$) after normalization are included in Figure \[fig\_toy2\] . We can draw the following conclusions from these results. First, for the noise-free data, the LB iterations are able to recover the sparse “reflectivity” and source function well up to a constant single amplitude factor, which we correct by normalizing its $\ell_2$-norm. Second, the estimated source function and reflectivity become noisy (cf. the dotted line in Figure \[fig\_toy2\]a and the dot line in Figure \[fig\_toy2\]b ) when we do not include a penalty enforcing the estimated filter to be short in time. Finally, the method is robust with respect to noise as we can see from Figures \[fig\_toy2\]c and \[fig\_toy2\]d where $10\%$ Gaussian noise is added. This result stresses the importance of including the penalty.
\
Experiments on the modified Marmousi model
------------------------------------------
To illustrate the performance and robustness with respect to noise of the proposed SPLS-RTM-SE method, we consider a model with complex layered stratigraphy. We derive this imaging example from the well-known synthetic Marmousi model [@brougois1990marmousi], which is $3.2$ km deep and $8.0$ km wide, with a grid size of $5\times 5$ m. To avoid imaging artifacts, we use a background velocity that is kinematically correct. We simulate $320$ equally spaced sources positioned at a depth of $25$m. We use a minimum phase source time function with its significant spectrum ranging from $10$ to $40$ Hz as shown in Figure \[fig\_source\_Marmousi\]. We use this type of source to generate linear data by applying the demigration operator (${\mathbf{J}}_i ( \mathbf{m}_0,\mathbf{q} ),\, i=1\cdots n_s$) to a bandwidth limited medium perturbation ${\delta{\mathbf{m}}}$ given by the difference between two smoothings of the true medium [@huang2016flexibly]. We record data at $320$ equally spaced co-located receivers. To assess the sensitivity to noise, we create two additional data sets by adding zero-centered Gaussian noise whose energies are $50\%$ and $200\%$ of the simulated linear data respectively.
Contrary to source estimation in the frequency domain, we need an initial source function $\mathbf{q}_0$ for the source time function (see Figure \[fig\_source\_Marmousi\_a\] and \[fig\_source\_Marmousi\_b\] where the initial source time function and its amplitude spectrum are depicted by dashed black lines). We need this initial source function to make sure that the finite-difference propagators remain stable. To make sure we do not exceed the valid frequency range of our simulations, we choose the frequency band of the initial source time function broader than the true one. To circumvent bias, we initialize the time function with a flat amplitude spectrum between $20-50$ Hz. To allow for a realistic scenario, we apply a phase shift to this initial guess making it mixed phase and non symmetric .
To carry out the alternating inversion for the reflectivity and unknown filter $\mathbf{w}$, we run Algorithm \[alg:LSM2\] for $40$ iterations with a batch size of $8$—i.e., we use $8$ randomly selected sources per iteration without replacement. The total number of wave equation solves is equivalent to touching each shot only once—i.e., we make one pass through the data. To improve the convergence of the inversion, we employ preconditioners in both the data and model domains [see @herrmann2009curvelet for detail]. To remove the imprint of the sources/receivers on the image, we also include a top mute to our operators. Also, we apply a mute to the data to suppress the dominating water bottom reflection and long offsets. Finally, we choose the thresholding parameter $\lambda$ to be $10\%$ of the maximum value of the first gradient to avoid unnecessary extra iterations resulting from a threshold value that is too large or small.
The estimated source functions $\hat{\mathbf{q}}=\mathbf{w}_{final}\ast\mathbf{q}_0$ and their amplitude spectra after applying an $\ell_2$-norm normalization are included in Figure \[fig\_source\_Marmousi\]. Overall we can see that the source functions are well recovered despite the presence of noise. For a small amount of noise, the estimated spectrum is the same as the one obtained from the noise-free data while the source function obtained from data with a high noise level is less smooth, but closer to the true source function. Other than the fact that we are dealing with nonlinear blind deconvolution, we do not have an explanation for this behavior. While the noise dependence of the estimated source functions behaves somewhat aberrant, the recovered reflectivity behaves as expected (cf. Figures \[fig\_LSRTM\_Marmousi\_a\] and \[fig\_LSRTM\_Marmousi\_b\] for images obtained with the true source and with the initial guess and images \[fig\_LSRTM\_Marmousi\_estQ\_a\] – \[fig\_LSRTM\_Marmousi\_estQ\_c\] obtained with on-the-fly source estimation for noise-free and noisy data.) We can make the following observations from these experiments: first, it is important to image with the correct source even when the data is noise free. While our sparsity-promoting scheme is able to recover a high-resolution image (see Figure \[fig\_LSRTM\_Marmousi\_a\]) when the source function corresponds to the true source, the image quality deteriorates rapidly if the amplitude and phase spectra of the wavelet are wrong (see Figure \[fig\_LSRTM\_Marmousi\_b\]). Energy is no longer focussed and the shape and locations of the imaged reflectors are off. However, the results included in Figure \[fig\_LSRTM\_Marmousi\_estQ\] demonstrate that good results can be obtained when estimating the source function on the fly. The estimated reflectivity depicted in Figure \[fig\_LSRTM\_Marmousi\_estQ\_a\] is close to the reflectivity obtained when we image with the true source function (cf. Figures \[fig\_LSRTM\_Marmousi\_a\] and \[fig\_LSRTM\_Marmousi\_estQ\]). Moreover, the estimated images are, as expected, relatively insensitive to noise in the data albeit the imaged reflectivity for the high noise case somewhat deteriorated (cf. Figures \[fig\_LSRTM\_Marmousi\_estQ\_a\] – \[fig\_LSRTM\_Marmousi\_estQ\_c\]). Contrary to the imaging result for the wrong initial source function, the reflectors are positioned correctly and have the correct phase, shape, and amplitude, even in situations of substantial noise although at the expense of some remaining low- and high-frequency artifacts. The latter are related to the use of the curvelet transform and are to be expected. Overall, these results confirm the robustness of our imaging algorithm in the situation where there is significant noise in the data.
\
To arrive at the estimated images in Figure \[fig\_LSRTM\_Marmousi\_estQ\], we set the penalty parameters $\nu=1$ and $\alpha=8$ in Algorithm \[alg:LSM2\]. After the first source estimation in the second iteration, we reset the coefficients $\mathbf{z}$ and $\mathbf{x}$ to zero to avoid spending too many iterations on correcting wrongly located reflectors from the first iteration in which the initial guess of the source wavelet is used. In addition to the visual quality of the estimated images, convergence plots for the relative error for the data residual (the relative $\ell_2$-norm error between the observed data and the demigrated data for estimated reflectivity $\hat{{\delta{\mathbf{m}}}}$ convolved with the estimated filter,$\frac{\|\mathbf{w}_k\ast\tilde{\mathbf{b}}_k-\mathbf{b}_k \|_2}{\|\mathbf{b}_k\|_2}$) and the relative model error (the $\ell_2$-norm error between the true reflectivity and the recovered reflectivity,$\frac{\|\hat{\delta\mathbf{m}}_k -\delta\mathbf{m}\|_2}{\|\delta\mathbf{m}\|_2}$) confirm our observation that Algorithm \[alg:LSM2\] is capable of providing high quality images in the absence of precise knowledge on the source function and in the presence of substantial noise. Our approach arrives at these least-squares images at the cost of a single data pass. Understandably, the algorithm starts off with a large relative residual and model error due to the wrong initial guess for the source function. As Algorithm \[alg:LSM2\] progresses, these relative errors continue to decay and are comparable to the convergence plots for the true source function. Because on-the-fly source estimation improves our ability to adapt to the data, the relative data residual for the noise-free case (dashed line) is even better then the relative error in case the source function is known (solid line). While encouraging, these results are obtained for a relatively simple imaging experiment and for data that is obtained with linearized modeling via demigration. In other words, we commit an inversion crime. In the next section, we will show that the proposed method also performs well in more complicated settings with nonlinear data.
Experiments on the Sigsbee model
--------------------------------
Sparsity-promoting imaging algorithms such as SPLS-RTM (Algoritm \[alg:LSM2\]) are designed to handle complex imaging scenarios with strong velocity contrasts and strong lateral velocity variations. Examples of such scenarios are salt plays where reflections underneath the salt are of interest. To demonstrate the viability of our imaging approach with on-the-fly source estimation in this scenario, we consider the challenging Sigsbee2A model of size $24.4\times 9.2$ km. This model contains a large salt body and a number of faults and point diffractors. To demonstrate the capability of our approach to handel this challenging situation, we simulate nonlinear data for a marine acquisition without a free surface. We model $960$ sources in total, with each shot record being recorded by an array of $320$ receivers with $25$ m receiver spacing, a maximum offset of $8$ km and a towing depth of $15$ m. We use a source wavelet with a peak frequency at $15\text{Hz}$ (see Figure \[fig\_source\]) and we record for $10$ s.
As is customary during imaging under salt, we use a background velocity model that features salt with relatively strong and therefore reflecting boundaries. We approximate linear data by using this background velocity model to generate data, which we subtract from the simulated data in the true Sigsbee2A model (i.e. from the observed data). Due to the presence of salt in the background model, the incident wavefield contains reflections that give rise to unwanted tomographic low-frequency artifacts in the image. This problem is widely reported in the literature (e.g. @Yoon2006; @Guitton2007). To remove these imaging artifacts, we replace the conventional imaging condition for RTM by the inverse-scattering imaging condition [@stolk2012linearized; @whitmore2012applications; @witte2017EAGEspl]. While this condition has been proven capable of removing tomographic artifacts during RTM [@whitmore2012applications; @witte2017EAGEspl] and sparsity-promoting least-squares RTM [@witte2017EAGEspl], it changes the linearized forward operator (the Jacobian $\mathbf{J}_i$), resulting in an inconsistent system. Contrary to RTM with the conventional imaging condition, imaging with the inverse scattering imaging condition corresponds to estimating perturbations in the impedance, rather than in the velocity.
Unfortunately, the difference in which quantity is being imaged, is problematic for our proposed on-the-fly source estimation, which tries to correct for inconsistencies between observed and predicted data. Contrary to the situations where we use the conventional imaging condition, the data residual now contains contributions from the wrong wavelet and the linearized imaging condition, which leads to wrong estimates for the unknown source function. We overcome this problem via a hybrid iterative algorithm where we switch imaging conditions during the iterations as outlined in Algorithm \[alg:LSM2\]. To estimate the source function, we first iterate with the conventional imaging condition. Since the convergence to the source function is fast, we switch after five iterations to the scattering imaging, but keep the estimated source function fixed. Basically, we jump from Algorithm \[alg:LSM2\] to Algorithm \[alg:LSM1\].
Results of this hybrid approach are summarized in Figures \[fig\_source\] – \[fig\_dm\_traces\]. As before, we compare our results with on-the-fly source estimation to SPLS-RTM for the true source function. The initial guess and estimated wavelets in Figure \[fig\_source\] again confirm the validity of our approach, yielding a reasonably accurate estimate for the source after only five iterations and subsequent normalization of the $\ell_2$-norm. Imaging results obtained after twenty iterations with $10\%$ of the shots, which amounts to two data passes in total, are included in Figures \[fig\_Sigsbee\] and \[fig\_dm\_traces\]. Unlike a typical RTM image, images obtained by SPLS-RTM are well resolved and contain true amplitude. This is because we invert the linearized modeling operator, which compensates for the source, finite aperture, and propagation effects. As before, we include preconditioners and mutes to improve the conditioning number of the linear system. Comparisons of Figure \[fig\_LSRTM\_Sigsbee\_a\], obtained with Algoritm \[alg:LSM1\] with the true source function, and Figure \[fig\_LSRTM\_Sigsbee\_b\], which we compute with our hybrid method switching from Algoritm \[alg:LSM2\] to Algoritm \[alg:LSM1\] after five iterations, show near identical results, thus confirming the validity of the proposed approach. These observations are confirmed by trace-by-trace comparisons in Figure \[fig\_dm\_traces\]. Similar to the Marmousi experiments, we set the penalty parameters $\alpha=8$, and $\nu=1$, and the thresholding parameter $\lambda$ is set according to $10\%$ of the maximum absolute amplitude level of $\mathbf{z}_1$.
\
Discussion
==========
Due to its computational costs, sparsity-promoting LS-RTM has been an expensive proposition and is for this reason not yet widely adapted while this method is capable of achieving images with high-amplitude fidelity and fewer artifacts. With the proposed work, we are able to come up with an alternative low-cost approach combining techniques from stochastic optimization and sparsity-promotion with on-the-fly source estimation using the technique of variable projection. Compared to earlier work, addressing the memory demands of (LS-)RTM via on-fly-Fourier transforms [@witte2019compressive], we tackle the problem of on-the-fly source estimation in the time domain. Because we use industry-strength time-domain finite-difference propagators provided by Devito [@devito-compiler; @devito-api] and exposed in the Julia programming language by JUDI [@witte2018alf], our approach scales in principle to large 3D industrial problems. While we address the importance of estimating source function, we believe that the sensitivity of LS-RTM to errors in the background velocity model needs to be studied as well albeit early work on time-harmonic LS-RTM showing some robustness with respect to these errors [@tu2014fis]. Combining our approach with the method of on-the-fly Fourier transforms is also a topic that needs further study.
While carrying out full scale 3D (LS-)RTM experiments is generally out of reach for academia where access to large high-performance compute is often limited, recent work on a serverless implementation of RTM [@witte2019RHPCssi; @witte2019TPDedas] has shown that industry-scale workloads can be run in the cloud leveraging the power of Devito. In the not too distant future, we plan to demonstrate the presented method on an industry-scale imaging problem using tilted transversely-isotropic propagators. This would truly exemplify the power of modern code bases and linear algebra abstractions as utilized by JUDI [@witte2018alf]. This framework gives us flexibility for instance to switch to more involved 3D propagators or to estimating source-time functions that are allowed to vary along the survey.
Finally, since sparsity-promoting LS-RTM carries out inversions, we expect to be able to obtain images from sparsely sampled data, e.g. data collected with sparse ocean bottom nodes and (multi-)source vessel simultaneous recordings. We plan to report on these aspects in the not too distant future.
Conclusion
==========
We proposed a scalable time-domain approach to sparsity-promoting least-squared reverse time migration with on-the-fly source estimation in principle suitable for industrial 3D imaging problems. The presented approach leverages recently developed techniques from convex optimization and variable projection that greatly reduce costs and the necessity to provide an estimate for the source function. As a result, our approach is capable of generating high-fidelity true-amplitude images including source estimates at the cost of roughly one to two migrations involving all data.
By means of carefully designed experiments in 2D, we were able to demonstrate that our method is capable of handling noisy data and complex imaging settings such as salt. We were able to image under salt, which is often plagued by low-frequency tomographic artifacts, by switching between applying the conventional imaging condition initially, followed by iterations that apply the inverse-scattering condition. In this way, we estimated the source function first while creating an artifact-free image with later iterations during which the imaging condition was switched while keeping the source function fixed.
Because the presented method relies on time-domain propagators, we anticipate it will be able to scale to large 3D industrial imaging problems. Because 3D imaging with full-azimuthal sparse data typically provided good illumination of the reservoir, we expect the proposed methodology to produce high fidelity results at a cost of roughly one to two reverse time migrations involving all shots.
Acknowledgements
================
This research was funded by the Georgia Research Alliance and the Georgia Institute of Technology. And this work is a collaborative effort of all the co-authors. We confirm that there is no conflict of interest to declare.
Data availability statement {#data_availability_statement}
===========================
The data that support the findings of this study are available from the corresponding author upon reasonable request.
|
---
abstract: 'We investigate two special classes of two-mode Gaussian states of light that are important from both the experimental and theoretical points of view: the mode-mixed thermal states and the squeezed thermal ones. Aiming to a parallel study, we write the Uhlmann fidelity between pairs of states belonging to each class in terms of their defining parameters. The quantum Fisher information matrices on the corresponding four-dimensional manifolds are diagonal and allow insightful parameter estimation. The scalar curvatures of the Bures metric on both Riemannian manifolds of special two-mode Gaussian states are evaluated and discussed. They are functions of two variables, namely, the mean numbers of photons in the incident thermal modes. Our comparative analysis opens the door to further investigation of the interplay between geometry and statistics for Gaussian states produced in simple optical devices.'
author:
- 'Paulina Marian$^{1,2}$'
- 'Tudor A. Marian$^{1}$'
title: 'Quantum Fisher information on two manifolds of two-mode Gaussian states'
---
Introduction
============
Answering the question “How close are two states of a quantum system?” is basic to the theory of quantum information processing. In principle, the similarity of states is decided via quantum-mechanical measurements whose outcomes are random, but have definite state-depending probability distributions. There are several distance-type measures which are widely used to quantify the probabilistic distinguishability of two states [@NielChua; @BZ]. However, in this paper we focus on the most convenient of them, namely, the Bures distance that is defined in connection with quantum fidelity. Besides, the Bures metric for neighbouring states is proportional to the quantum Fisher information (QFI) metric known to be an essential ingredient in quantum metrology.
On the other hand, our paper deals with two-mode Gaussian states (GSs) of the quantum radiation field. There are two reasons for this choice. First, GSs are very useful tools in many quantum optics experiments. Some of them are interwoven with quantum information tasks. As an example, we mention that several quantum teleportation experiments were performed employing only GSs, starting with the first one carried out by Furusawa [*et al.*]{} [@Furusawa]. On the theoretical side, the GSs of continuous-variable systems have been intensely investigated in the last three decades. Excellent recent reviews cover research on their role in quantum optics [@Olivares] and in quantum information as well [@WPGCRSL; @Adesso].
Second, exact explicit formulae have been found for the quantum fidelity of arbitrary one-mode [@HS1998] and two-mode GSs [@PT2012]. These available compact-form expressions are adequate for various calculations. In particular, they allow one a straightforward derivation of the QFI metric for the one- and two-mode GSs. In what follows, we address this issue by exploiting the fidelity between two-mode GSs. More specifically, we make a parallel analysis for two classes of such states that are interesting in both experimental and theoretical research.
Below is presented the outline of the paper in some detail. In Sec. II, we start from the analogy between a classical probability distribution and that of the outcomes of a quantum-mechanical measurement. We recall the definition of the quantum fidelity between two states of a quantum system, its connection with their Bures distance, as well as the Uhlmann concise formula. Emphasis is laid on the proportionality of the infinitesimal Bures distance between neighbouring states to their QFI one. In Sec. III, we cast the expression of the fidelity between two arbitrary two-mode GSs [@PT2012] into an alternative form which seems to be more suitable for subsequent calculations. Section IV examines two important families of two-mode GSs, namely, the mode-mixed thermal states (MTSs) and the squeezed thermal states (STSs). We briefly review their optical generation, as well as their quantum-mechanical description. Despite some formal similarities in their natural parametrizations, their physical properties are quite different. However, this analysis has lead us to term them special two-mode GSs and has suggested that a parallel investigation of their closeness features would be appropriate. By applying our alternative formula for the fidelity of two-mode GSs, we find in Sec. V the fidelity between MTSs and that between STSs, expressed in terms of their specific parameters. Section VI is devoted to the derivation of the QFI metric tensors on the four-dimensional Riemannian manifolds of the MTSs and STSs. Taking advantage that both natural parametrizations are orthogonal ones, we evaluate in Sec. VII the scalar curvature of the Bures metric on each Riemannian manifold of special two-mode GSs. The scalar curvatures of all the MTSs and STSs depend only on their average numbers of photons in the incident thermal modes. We explain this specific property along with an alternative way of deriving both scalar curvatures. Their parallel presentation offers the possibility of some interesting comparisons. Section VIII summarizes the results and then gives a reliable statistical interpretation of the Riemannian scalar curvature based on the Bures metric. Three appendices detail two basic inequalities involving fidelity in the particular case of GSs. Appendix A deals with the inequality between the fidelity and the overlap of two $n$-mode GSs. In Appendix B, the property of fidelity of being less than one or at most equal to one is checked for thermal states (TSs). Explicit use of this conclusion is made in Appendix C in order to verify the same inequality for two-mode MTSs and STSs.
Quantum fidelity
================
To start on, let $P:=\{ p_b \}, \; (b=1,...,N),$ denote a probability distribution assigned to the sample space of an experiment with $N$ outcomes. We consider two arbitrary probability distributions, $P^{\prime}:=\{p_b^{\prime}\}$ and $P^{\prime \prime}:=\{p_b^{\prime \prime}\}, $ ascribed to the given sample space. Their classical fidelity is the square of the scalar product of two unit vectors in $\mathbb{R}^{N}$ with the components $\{ \sqrt{p_b^{\prime}} \}$ and $\{ \sqrt{p_b^{\prime \prime}} \}$:
$${\cal F}_{cl}(P^{\prime}, P^{\prime \prime}):
=\left( \sum_{b=1}^N \sqrt{ p_b^{\prime}\, p_b^{\prime \prime}} \right)^2.
\label{F_c}$$
These vectors define two points on a hyperoctant of the unit sphere $S^{N-1}$ embedded in the Euclidean space ${\mathbb R}^{N}.$ Their angle at the center of the sphere [@Bhatta; @Woot] is called the Bhattacharyya-Wootters statistical distance [@FC]:
$$D_{\rm BW}(P^{\prime}, P^{\prime \prime}):=\arccos{\left( \sqrt{ {\cal F}_{cl} }
(P^{\prime}, P^{\prime \prime} )\right) }.
\label{BW}$$
The square root of the classical fidelity is referred to as affinity of the probability distributions [@Luo]. Note that the Hellinger distance between the points $ P^{\prime}$ and $ P^{\prime \prime}$ [@Luo], $$D_{\rm H}(P^{\prime}, P^{\prime \prime}):=\left[ \sum_{b=1}^N \left( \sqrt{ p_b^{\prime}}
-\sqrt{ p_b^{\prime \prime}} \right)^2 \right]^{\frac{1}{2}},
\label{Hell}$$ coincides with their chordal distance and is in turn determined by the classical fidelity (\[F\_c\]): $$D_{\rm H}(P^{\prime}, P^{\prime \prime})=\left[ 2-2 \sqrt{ {\cal F}_{cl} }(P^{\prime},
P^{\prime \prime}) \right]^{\frac{1}{2}}.
\label{HD}$$
In order to estimate the closeness of two arbitrary states, $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$, of a given quantum system, Bures introduced a distance between them [@Bu] which is similar to the classical Hellinger distance (\[HD\]): $$D_{\rm B}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})
:=\left[ 2-2 \sqrt{ {\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})} \right]^{\frac{1}{2}}.
\label{BD}$$ Let ${\cal H}$ denote the Hilbert space associated to the quantum system. The quantity ${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})$ occurring in Eq. (\[BD\]) is defined as the maximal transition probability between any pair of purifications of the states $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$. However, it is sufficient to restrict our choice of purifications to pairs of state vectors, $|{\Psi}^{\prime}\rangle$ and $|{\Psi}^{\prime \prime}\rangle$, both belonging to the tensor product space ${\cal H} \otimes {\cal H}$:
$${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime}):
=\max_{\{|{\Psi}^{\prime}\rangle, |{\Psi}^{\prime \prime}\rangle\}}
|\langle{\Psi}^{\prime}|{\Psi}^{\prime \prime}\rangle|^2.
\label{maxprob}$$
Subsequently [@Uhl], Uhlmann succeeded in deriving the compact expression $${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
\left[ {\rm Tr}\left( \sqrt{\sqrt{\hat{\rho}^{\prime \prime}} \hat{\rho}^{\prime}
\sqrt{\hat{\rho}^{\prime \prime}}} \right) \right]^2
\label{F}$$ and interpreted it as a [*generalization*]{} of the quantum-mechanical transition probability between the above states. In a later paper [@Jo], Jozsa coined the name [*fidelity*]{} for the non-negative quantity (\[F\]) and gave elementary proofs of its main properties. Note that, when at least one of the quantum states is pure, the fidelity (\[F\]) reduces to the Hilbert-Schmidt scalar product of the states: ${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})
={\rm Tr}(\hat{\rho}^{\prime} \hat{\rho}^{\prime \prime})$. Let us assume, for instance, that $\hat{\rho}^{\prime \prime}$ is a pure state, i. e., $\hat{\rho}^{\prime \prime} =|{\psi}^{\prime \prime}\rangle\langle{\psi}^{\prime \prime}|$, while the state $\hat{\rho}^{\prime}$ is either pure or mixed. Then their fidelity, $${\cal F}(\hat{\rho}^{\prime}, |{\psi}^{\prime \prime}\rangle\langle{\psi}^{\prime \prime}|)
=\langle{\psi}^{\prime \prime}|\hat{\rho}^{\prime}|{\psi}^{\prime \prime}\rangle,
\label{pure}$$ is the probability of transforming the state $\hat{\rho}^{\prime}$ into the state $\hat{\rho}^{\prime \prime} =|{\psi}^{\prime \prime}\rangle\langle{\psi}^{\prime \prime}|$ via a selective measurement. By contrast, the fidelity (\[F\]) has no such operational meaning when both states are mixed ones. In this general case, its probabilistic significance relies merely on the Bures-Uhlmann definition (\[maxprob\]).
We mention three properties of the fidelity that are displayed by Eq. (\[maxprob\]).
1\) Fidelity is less than one or at most equal to one. This inequality saturates if and only if the states coincide: $${\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}) \leqq 1: \quad
{\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}) =1\; \iff \;
{\hat{\rho}}^{\prime \prime}= {\hat{\rho}}^{\prime}.
\label{F<1}$$
2\) Symmetry: $${\cal F}({\hat{\rho}}^{\prime \prime}, {\hat{\rho}}^{\prime}) =
{\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}).
\label{sym}$$
3\) Monotonicity. Fidelity doesn’t decrease under any trace-preserving quantum operation ${\mathcal E}$ performed on both states [@NielChua]: $${\cal F}({\mathcal E}({\hat{\rho}}^{\prime}), {\mathcal E}({\hat{\rho}}^{\prime \prime})) \geqq
{\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}).
\label{mon}$$
Besides, fidelity proved to be an appropriate indicator of the closeness of two quantum states via measurements. Recall that for any general measurement there is a Positive Operator-Valued Measure (POVM) [@NielChua], i. e., a set of positive operators ${\{\hat E_b\}}$ on the Hilbert space ${\cal H}$, which provides a resolution of the identity: ${\sum_{b} \hat E_b}=\hat I$. The subscript $b$ indexes the possible outcomes of the measurement whose probabilities in the quantum states $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$ are, respectively, $p_{\hat{\rho}^{\prime}}(b)={\rm Tr}(\hat{\rho}^{\prime} \hat E_b)$ and $p_{\hat{\rho}^{\prime \prime}}(b)={\rm Tr}(\hat{\rho}^{\prime \prime} \hat E_b).$ One can extend the formula (\[F\_c\]) to define the classical fidelity of these probability distributions in a given quantum measurement: $${\cal F}_{cl}\left( p_{\hat{\rho}^{\prime}},\, p_{\hat{\rho}^{\prime \prime}};\, \{\hat E_b\}\right):=\left[ \sum_{b}
\sqrt{p_{\hat{\rho}^{\prime}}(b)\, p_{\hat{\rho}^{\prime \prime}}(b)} \right]^2.
\label{POVM}$$ In Refs. [@FC; @Fuchs; @BCFJS] it was proven an important theorem stating that the quantum fidelity is the minimal classical one (\[POVM\]) over the collection of all POVMs: $${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
\min_{\{\hat E_b\}}\left[ {\cal F}_{cl}\left( p_{\hat{\rho}^{\prime}},\, p_{\hat{\rho}^{\prime \prime}};\,
\{\hat E_b\}\right) \right].
\label{min}$$
When the states $\hat{\rho}^{\prime}$ and $ \hat{\rho}^{\prime \prime}$ commute, then and only then their fidelity (\[F\]) is equal to the square of their quantum affinity [@PT2015]: $$[\hat{\rho}^{\prime},\, \hat{\rho}^{\prime \prime}]=\hat 0 \iff
{\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
\left[ {\rm Tr}\left( \sqrt{\hat{\rho}^{\prime}}\sqrt{\hat{\rho}^{\prime \prime}} \right) \right]^2.
\label{Fcom}$$ The spectral resolutions of the commuting density operators $\hat{\rho}^{\prime}$ and $ \hat{\rho}^{\prime \prime}$ are written in terms of the same complete set of orthogonal projections, albeit with specific spectra: $\{ {\lambda}_n^{\prime} \}$ and $\{ {\lambda}_n^{\prime \prime} \}$, respectively. Therefore, Eq. (\[Fcom\]) takes the form (\[POVM\]): $$[\hat{\rho}^{\prime},\, \hat{\rho}^{\prime \prime}]=\hat 0\; \iff \;
{\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
\left( \sum_{n} \sqrt{ {\lambda}_n^{\prime} {\lambda}_n^{\prime \prime}}\right)^2.
\label{Fspec}$$ Accordingly, in the properly termed [*classical*]{} situation of commuting states, the minimum (\[min\]) is reached for a clearly specified projective measurement.
We emphasize that, in the spirit of the correspondence principle, it is possible to [*guess*]{} the positive operator $\hat{\mathcal B}:
=\sqrt{\hat{\rho}^{\prime \prime}}\hat{\rho}^{\prime}\sqrt{\hat{\rho}^{\prime \prime}}$ [@PT2012] occurring in the Uhlmann formula (\[F\]). Indeed, the above structure of $\hat{\mathcal B}$ is the simplest one which leads to the classical limit (\[Fspec\]).
Another fidelity-based distance is the Bures angle [@NielChua], $$D_{\rm A}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime}):=\arccos{\left( \sqrt{ {\cal F}(\hat{\rho}^{\prime},
\hat{\rho}^{\prime \prime})}\right) }.
\label{D_A}$$ On account of Eqs. (\[F<1\]) and (\[sym\]), both the Bures distance and the Bures angle are genuine metrics since they fulfill in addition the triangle inequality, as shown in Refs. [@Bu; @MNMFL] and, respectively, [@NielChua]. Moreover, by virtue of the monotonicity property (\[mon\]) of the fidelity, they are contractive distances (monotone metrics).
Recall that the natural distance between two pure states, $|{\psi}^{\prime}\rangle \langle{\psi}^{\prime}|$ and $|{\psi}^{\prime \prime}\rangle \langle{\psi}^{\prime \prime}|$, on the manifold of the projective Hilbert space is the Fubini-Study metric [@BZ]: $$D_{\rm FS}{\left( |{\psi}^{\prime}\rangle \langle{\psi}^{\prime}|, |{\psi}^{\prime \prime}\rangle
\langle{\psi}^{\prime \prime}| \right) }
:=\arccos{\left(|\langle{\psi}^{\prime} |{\psi}^{\prime \prime}\rangle|\right) }.
\label{D_FS}$$ The Bures angle (\[D\_A\]) is just the generalization of the Fubini-Study metric to the case of mixed states.
We further concentrate on the squared Bures distance between two neighbouring quantum states denoted $\hat{\rho}$ and $\hat{\rho}+d\hat{\rho}$: $$(ds_{\rm B})^2:=[D_{\rm B}(\hat{\rho}, \hat{\rho}+d\hat{\rho})]^2.
\label{ds_B^2}$$ The above squared infinitesimal line element is built with the Bures metric tensor on a certain differentiable manifold of quantum states. An important result of Uhlmann’s school obtained long ago [@Hueb] is that the Bures metric (\[ds\_B\^2\]) is Riemannian.
In a seminal paper [@BC1994], Braunstein and Caves employ the theory of parameter estimation to find the optimal quantum measurement that resolves two neighbouring mixed states. Thus they generalize the Wootters statistical distance between pure states [@Woot]. These authors succeeded in deriving the QFI metric $(ds_{\rm F})^2$ as a reliable measure of statistical distinguishability between neighbouring quantum states. Furthermore, a comparison of the QFI metric with Hübner’s general expression of the infinitesimal Bures metric [@Hueb] allowed them to establish the basic proportionality formula $$(ds_{\rm B})^2=\frac{1}{4}(ds_{\rm F})^2.
\label{1:4}$$
A pertinent analysis of the infinitesimal metric on any finite-dimensional state space is carried out in Ref. [@PeSu] and then carefully reviewed in Ref. [@SZ]. It reveals a couple of distinctive features of the Bures metric (\[ds\_B\^2\]). Specifically, this is the minimal one among the monotone, Riemannian, and Fisher-adjusted metrics. In addition, it is the only metric from the above class whose extension to pure states yields precisely the Fubini-Study metric.
Uhlmann fidelity between two-mode Gaussian states
=================================================
In order to tackle the two-mode GSs, we arrange the canonical quadrature operators of the modes in a row vector: $$(\hat u)^T:=(\hat q_1,\; \hat p_1,\; \hat q_2,\; \hat p_2).
\label{u^T}$$ Their eigenvalues are the components of an arbitrary dimensionless vector $u \in {\mathbb R}^4$: $$u^T:=(q_1,\; p_1,\; q_2,\; p_2).
\label{uT}$$ Recall that any GS $\hat{\rho}$ is defined by its characteristic function (CF) $\chi (u)$. In turn, this is fully determined by the first- and second-order moments of the quadrature operators (\[u\^T\]) in the given state $\hat{\rho}$. As a matter of fact, the CF $\chi (u)$ is an exponential whose argument is a specific quadratic function of the current vector $u$, Eq. (\[uT\]): $$\chi (u)=\exp\left[-\frac{1}{2}\,(J u)^{T}{\mathcal V}\,(J u)+iv^{T}(J u)\right].\label{CF}$$ In Eq. (\[CF\]), the components of the vector $v \in \mathbb{R}^4$ are the expectation values of the quadrature operators (\[u\^T\]) in the chosen GS ${\hat \rho}$: $v:={\langle{\hat u}\rangle}_{\hat{\rho}}.$ The second-order moments of the deviations from the means of the canonical quadrature operators are collected as entries of the real and symmetric $4 \times 4$ covariance matrix (CM) ${\mathcal V}$ of the GS $\hat{\rho}$. In the sequel, we find it often useful to write the CM partitioned into the following $2\times 2$ submatrices: $$\begin{aligned}
{\mathcal V}=\left(
\begin{matrix}
\mathcal V_{1}\; & \; {\mathcal C} \\ \\
{\mathcal C}^ T \; & \; \mathcal V_{2}
\end{matrix}
\right).
\label{part}\end{aligned}$$ The submatrices $\mathcal V_{j}, \; (j=1,\, 2)$, are the CMs of the single-mode reduced GSs, while $ {\mathcal C}$ displays the cross-correlations between the modes. Further, $J$ denotes the standard $4\times 4$ matrix of the symplectic form on ${\mathbb R}^4$, which is block-diagonal and skew-symmetric: $$\begin{aligned}
J:=J_1 \oplus J_2, \quad J_1=J_2:=\left(
\begin{matrix}
0 & 1\\ -1 & 0
\end{matrix}
\right).
\label{J}\end{aligned}$$
Any [*bona fide*]{} CM ${\mathcal V}$ fulfills the concise Robertson-Schrödinger uncertainty relation: $${\zeta}^{\dag}\left({\mathcal V}+\frac{i}{2}J \right){\zeta} \geqq 0, \qquad
\left( \zeta \in \mathbb{C}^4 \right).
\label{R&S}$$ Briefly stated, the matrix ${\mathcal V}+\frac{i}{2}J$ has to be positive semidefinite: ${\mathcal V}+\frac{i}{2}J \geq 0 $. This requirement is a necessary and sufficient condition for the very existence of the Gaussian quantum state ${\hat \rho}$ [@Simon1; @HS1989; @Simon2]. It implies the inequality $\det{\left( {\mathcal V}+\frac{i}{2}J \right) \geqq 0}$ and, in addition, that the CM ${\mathcal V}$ is positive definite: ${\mathcal V}>0$. The limit property $\det{\left( {\mathcal V}+\frac{i}{2}J \right) =0}$ is therefore quite special. However, it is shared by all the pure GSs, as well by some interesting mixed ones. All these states are said to be at the physicality edge.
In the paper [@PT2012] we derived an explicit expression of the fidelity between a pair of two-mode GSs, $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$, with the mean quadratures $v^{\prime}:={\langle {\hat u} \rangle}_{\hat{\rho}^{\prime}}$ and $v^{\prime \prime}:={\langle {\hat u} \rangle}_{\hat{\rho}^{\prime \prime}}$, and the CMs ${\mathcal V}^{\prime}$ and ${\mathcal V}^{\prime \prime}$, respectively. Let us denote their relative average displacement $\delta v:=v^{\prime}-v^{\prime \prime}$. We have found it convenient to employ three determinants satisfying the following inequalities: $$\begin{aligned}
& \Delta:=\det \left({\mathcal V}^{\prime}+{\mathcal V}^{\prime \prime}\right) \geqq 1 \, ;
\label{Delta} \\
& \Gamma:=2^4{}\det \left[ (J{\mathcal V}^{\prime})
\,(J{\mathcal V}^{\prime \prime})-\frac{1}{4}I\right] \geqq \Delta \, ;
\label{Gamma} \\
& \Lambda:=2^4\det \left({\mathcal V}^{\prime }+\frac{i}{2}\,J \right)
\det \left({\mathcal V}^{\prime \prime}+\frac{i}{2}\,J\right) \geqq 0.
\label{Lambda} \end{aligned}$$ In Eq. (\[Gamma\]), $I$ denotes the $4 \times 4$ identity matrix. The above determinants are manifestly symmetric with respect to the states $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$ and so are the exact expressions of their overlap, $${\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} )
=\frac{1}{ \sqrt{\Delta}}
\exp{\left[-\frac{1}{2}\left(\delta v \right)^T
\left({\mathcal V}^{\prime}+{\mathcal V}^{\prime\prime}\right)^{-1}
\delta v \right]}>0,
\label{overlap}$$ and their fidelity [@PT2012], $$\begin{aligned}
& {\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime})
= \left[ \left( \sqrt{\Gamma}+\sqrt{\Lambda} \right)
-\sqrt{\left( \sqrt{\Gamma}+\sqrt{\Lambda} \right)^2-\Delta} \right]^{-1} \notag \\
& \times \exp{\left[-\frac{1}{2}\left(\delta v \right)^T
\left({\mathcal V}^{\prime}+{\mathcal V}^{\prime\prime}\right)^{-1}
\delta v \right]}.
\label{2F}\end{aligned}$$
It is useful to introduce a pair of non-negative quantities: $$\begin{aligned}
& K_{\pm}:=\sqrt{\Gamma}+\sqrt{\Lambda}\pm \sqrt{\Delta}: \notag\\
& K_{-} \geqq 0, \quad K_{+}-K_{-} \geqq 2.
\label{Kpm}\end{aligned}$$ The proportionality relation (\[AB\]) has the explicit form $$\begin{aligned}
{\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})
=\left[ 1+\sqrt{\frac{K_{-}}{\Delta}} \left( \sqrt{K_{+}}+ \sqrt{K_{-}}\right) \right]
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} )>0,
\label{fo2}\end{aligned}$$ which exhibits the general inequality (\[F>O\]). The corresponding saturation condition,
$$\begin{aligned}
& {\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} ) \; \iff \;
K_{-} =0, \notag\\
& \text{i. e.,} \;\; \Gamma=\Delta \; \; \text{and} \;\; \Lambda =0,
\label{sat2}\end{aligned}$$
is partly redundant in comparison with the general conclusion (\[G=D\]). Indeed, as shown in Appendix A, the latter equality, $\Lambda =0$, is just a consequence of the former, $\Gamma=\Delta$.
We finally write down an alternative form of the fidelity (\[2F\]) that we find appropriate to what follows: $$\begin{aligned}
& {\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime})
=2 \left( \sqrt{K_{+}}-\sqrt{K_{-}} \right)^{-2} \notag\\
& \times \exp{\left[-\frac{1}{2}\left( \delta v \right)^T
\left( {\mathcal V}^{\prime}+{\mathcal V}^{\prime\prime} \right)^{-1} \delta v \right]}.
\label{2F(K)}\end{aligned}$$
Special two-mode Gaussian states
================================
We focus on two important families of two-mode GSs that are obtained by employing simple optical instruments such as beam splitters and non-degenerate parametric down-converters. Two input light modes interact with the device and their coupling results in two output modes [@Ulf]. When the incoming beams are chosen to be in TSs, the outgoing ones are in a two-mode undisplaced GS. In a lossless beam splitter, a linear interaction mixes the incident waves to generate a MTS. By contrast, in a non-degenerate parametric amplifier, pumping of photons produces a non-linear interaction whose outcome is a STS.
Let us summarize the features of preparation and then recall a concise characterization of the above output states.
1. Each incident light wave is in a single-mode TS, so that the global input is their product, i.e., a two-mode TS on the Hilbert space ${\mathcal H}_1 \otimes {\mathcal H}_2$: $$\begin{aligned}
& \hat{\rho}_{\rm T}(\bar{n}_1, \bar{n}_2):= \left( \hat{\rho}_{\rm T} \right)_1 (\bar{n}_1)
\otimes \left( \hat{\rho}_{\rm T} \right)_2 (\bar{n}_2): \notag\\
& \left( \hat{\rho}_{\rm T} \right)_j (\bar{n}_j):=\frac{1}{\bar{n}_j+1}
\exp{\left( -{\eta}_j {\hat{a}}_j^{\dag} {\hat{a}}_j \right)}, \notag\\
& (j=1, 2),
\label{TS} \end{aligned}$$ with $ {\hat{a}}_j:=\frac{1}{\sqrt{2}}(\hat{q}_j+i\hat{p}_j)$ denoting the photon annihilation operator of the mode $j$. In Eq. (\[TS\]), $\bar{n}_j$ is the Bose-Einstein mean photon number in the mode $j$, $$\bar{n}_j=\left[ \exp{({\eta}_j)}-1 \right]^{-1},
\label{BE}$$ and ${\eta}_j$ is the positive dimensionless ratio $${\eta}_j:=\frac{\hbar\, {\omega}_j}{k_BT_j}=\ln{\left( \frac{\bar{n}_j+1}{\bar{n}_j} \right) }.
\label{eta}$$
2. The final effect of the mode coupling in both optical devices is modeled by a specific unitary operator that induces a linear transformation of the amplitude operators of the modes.
3. It is well known [@Duan] that any two-mode GS $\hat{\rho}$ is similar, via a local unitary, to another one whose CM has a unique standard form which consists of a partitioning (\[part\]) into special diagonal $2\times 2$ submatrices: $$\begin{aligned}
& {\mathcal V}_j=b_j{\sigma}_0, \qquad \left( b_j \geqq \frac{1}{2} \right), \qquad (j=1, 2), \notag\\
& {\mathcal C}=\left(
\begin{matrix}
c\; & 0\; \\ 0\; & d\;
\end{matrix}
\right), \qquad (c \geqq |d| \geqq 0).
\label{standard}\end{aligned}$$ In Eq. (\[standard\]) and further on, ${\sigma}_0$ designates the $2\times 2$ identity matrix. The four numbers $b_1, b_2, c,\, \text{and} \; d$ are called the standard-form parameters of the given two-mode GS $\hat{\rho}$. However, there are special classes of two-mode GSs with a smaller number of such independent parameters. In particular, for TSs, $c=d=0$, while for MTSs, $c=d>0$, and for STSs, $c=-d>0$.
Mode-mixed thermal states
-------------------------
The optical interference of two modes in a reversible, lossless beam splitter is described by a mode-mixing operator [@Bonny]: $$\hat M_{12}(\theta,\phi):=\exp{\left[ \frac{\theta}{2}
\left( {\rm e}^{i\phi} \hat a_1 \hat a^{\dag}_2-{\rm e}^{-i\phi}
\hat a^{\dag}_1 \hat a_2 \right) \right] }.
\label{M_12}$$ Its parameters are the spherical polar angles $\theta$ and $\phi: \theta\in [0,\pi),\; \phi\in (-\pi,\pi]$. The co-latitude $\theta$ determines the intensity transmission and reflection coefficients of the device, which are $T=\left[ \cos \left( \frac{\theta}{2}\right) \right]^2$ and $R=\left[ \sin \left( \frac{\theta}{2}\right) \right]^2$, respectively. The longitude $\phi$ accounts for a phase shifting. As a matter of fact, in view of the Jordan-Schwinger two-mode bosonic realization of angular momentum [@Jordan; @Schwinger], $$\hat{J}_{+}=\hat a^{\dag}_1 \hat a_2, \;\;\; \hat{J}_{-}= \hat a_1 \hat a^{\dag}_2, \;\;\,
\hat{J}_3=\frac{1}{2}\left( \hat a^{\dag}_1 \hat a_1-\hat a^{\dag}_2 \hat a_2 \right),
\label{JJJ}$$ the unitary operator (\[M\_12\]) is a $SU(2)$ displacement operator [@Radcliffe; @Arecchi] , $$\hat D(\eta):=\exp{\left( \eta\hat{J}_{-}-{\eta}^{\ast} \hat{J}_{+} \right) }, \quad
\left( \eta:=\frac{\theta}{2}\, e^{i\phi} \right),
\label{DSU(2)}$$ acting on the two-mode Fock space ${\mathcal H}_1 \otimes {\mathcal H}_2$. At the same time, we employ the Euler angle parametrization to write it as a $SU(2)$ unitary representation operator whose carrier Hilbert space is ${\mathcal H}_1 \otimes {\mathcal H}_2$: $$\begin{aligned}
& \hat M_{12}(\theta,\phi)=\hat{\mathcal D} \left[ \, U(\phi, \, \theta, \, -\phi) \, \right] \notag\\
& =\exp{\left( -i\phi \hat{J}_3 \right) }\, \exp{\left( -i\theta \hat{J}_2 \right) }\,
\exp{\left( i\phi \hat{J}_3 \right) }.
\label{SU(2)rep}\end{aligned}$$
When choosing an asymmetrical two-mode TS as input to the beam splitter, then we get an emerging MTS as its output: $$\hat \rho_{\rm MT}=\hat M_{12}(\theta,\phi)\hat \rho_{\rm T}(\bar{n}_1, \bar{n}_2)
\hat M^{\dag}_{12}(\theta,\phi), \;\; (\bar{n}_1 > \bar{n}_2).
\label{MTS}$$ To the unitary state evolution (\[MTS\]) in the Schrödinger picture is associated the $SU(2)$ matrix $U(\phi, \, \theta, \, -\phi)$ that transforms the annihilation operators in the Heisenberg picture, $$\begin{aligned}
& \left(
\begin{matrix}
\hat{a}_1^{\prime} \\ \\ \hat{a}_2^{\prime}
\end{matrix}
\right)
=U(\phi, \, \theta, \, -\phi)
\left(
\begin{matrix}
\hat{a}_1 \\ \\ \hat{a}_2
\end{matrix}
\right): \notag\\ \notag\\
& U(\phi, \, \theta, \, -\phi)=
\left(
\begin{matrix}
\cos{\left( \frac{\theta}{2} \right) } & -\sin{\left( \frac{\theta}{2} \right) }\, e^{-i\phi} \\
\\ \sin{\left( \frac{\theta}{2} \right) }\, e^{i\phi} & \cos{\left( \frac{\theta}{2} \right) }
\end{matrix}
\right).
\label{U}\end{aligned}$$ In turn, the $SU(2)$ transformation (\[U\]) gives rise to a symplectic orthogonal one of the quadratures (\[u\^T\]). Its matrix $S(\theta, \phi) \in Sp(4, \mathbb{R}) \cap O(4)$ has the following partition into $2 \times 2$ submatrices:
$$\begin{aligned}
S(\theta, \phi)=
\left(
\begin{matrix}
\cos{\left( \frac{\theta}{2} \right) }\,{\sigma}_0 \ & -\sin{\left( \frac{\theta}{2} \right) } R(-\phi) \\
\\ \sin{\left( \frac{\theta}{2} \right) }R(\phi) & \cos{\left( \frac{\theta}{2} \right) }\, {\sigma}_0
\end{matrix}
\right).
\label{S(U)}\end{aligned}$$
We have employed the two-dimensional rotation matrix $$\begin{aligned}
& R(\phi):=
\left(
\begin{matrix}
\cos{\left(\phi \right) }\; & -\sin{\left( \phi \right) }\, \\ \\
\sin{\left( \phi \right) }\; & \cos{\left( \phi \right) }\,
\end{matrix}
\right), \quad (-\pi < \phi \leqq \pi): \notag\\ \notag\\
& R(\phi)=\cos{\left( \phi \right) }\, {\sigma}_0 -i\sin{\left( \phi \right) }\, {\sigma}_2,
\label{R(phi)}\end{aligned}$$ where ${\sigma}_2$ is a Pauli matrix. In view of Eq. (\[VT1\]), the CM of a two-mode TS (\[TS\]) is diagonal: $${\mathcal V}_{\rm T}(\bar{n}_1, \bar{n}_2)= \left( \bar{n}_1+\frac{1}{2} \right) {\sigma}_0
\oplus \left( \bar{n}_2+\frac{1}{2} \right) {\sigma}_0.
\label{V_in}$$ The unitary similarity (\[MTS\]) of the input and output two-mode GSs is equivalent to the symplectic congruence of their CMs:
$$\begin{aligned}
& {\mathcal V}_{\rm MT}(\bar{n}_1, \bar{n}_2, \theta, \phi)
=S(\theta, \phi)\, {\mathcal V}_{\rm T}(\bar{n}_1, \bar{n}_2)\, S^T(\theta, \phi), \notag \\ \notag \\
& (\bar{n}_1 > \bar{n}_2).
\label{MTSC}\end{aligned}$$
Accordingly, the CM of the output MTS (\[MTS\]) has the $2 \times 2$ block structure
$$\begin{aligned}
{\mathcal V}_{\rm MT}=
\left(
\begin{matrix}
b_1 {\sigma}_0 \; & cR(-\phi) \, \\ \\
cR(\phi) \; & b_2 {\sigma}_0 \,
\end{matrix}
\right),
\label{VMT}\end{aligned}$$
with the standard-form entries: $$\begin{aligned}
& b_{1}= \left( \bar{n}_1+\frac{1}{2} \right) \left[ \cos{ \left( \frac{\theta}{2} \right) } \right]^2
+ \left( \bar{n}_2+\frac{1}{2} \right) \left[ \sin{ \left( \frac{\theta}{2} \right) } \right]^2, \notag \\
& b_{2}= \left( \bar{n}_1+\frac{1}{2} \right) \left[ \sin{ \left( \frac{\theta}{2} \right) } \right]^2
+ \left( \bar{n}_2+\frac{1}{2} \right) \left[ \cos{ \left( \frac{\theta}{2} \right) } \right]^2, \notag \\ \notag\\
& c=d=\left( \bar{n}_{1}- \bar{n}_{2} \right) \cos{\left( \frac{\theta}{2} \right) }
\sin{\left( \frac{\theta}{2} \right) }>0.
\label{sfVMT}\end{aligned}$$ Needless to say, one gets the standard form (\[standard\]) of the CM ${\mathcal V}_{\rm MT}(\bar{n}_1, \bar{n}_2, \theta, \phi)$ of a MTS by setting $\phi=0$ in Eq. (\[VMT\]). Note also that, in the limit case $ \bar{n}_1= \bar{n}_2=: \bar{n}$, a two-mode MTS reduces to the input symmetric two-mode TS with the standard-form parameters $$b_{1}=b_{2}=:b= \bar{n}+\frac{1}{2}, \quad c=0.
\label{sfTS}$$ This happens because the beam splitter has no influence upon two incident light beams whose one-mode states are identical.
Squeezed thermal states
-----------------------
The coupling of the modes in a non-degenerate parametric amplifier is modelled by the action of a two-mode squeeze operator [@SC1985], $$\begin{aligned}
& \hat S_{12}(r, \phi):=\exp{\left[ \,r \left( {\rm e}^{i\phi}
\hat a^{\dag}_1 \hat a^{\dag}_2-{\rm e}^{-i\phi} \hat a_1
\hat a_2 \right) \right] }, \notag\\
& \left( r>0,\;\; \phi\in (-\pi,\pi] \right).
\label{S_12}\end{aligned}$$ The positive dimensionless quantity $r$ is called squeeze parameter [@GK2005]. Long ago, in a remarkable paper [@YCK1986], Yurke, McCall, and Klauder introduced a two-mode bosonic realization of the $su(1,1)$ algebra: $$\begin{aligned}
& \hat{K}_{+}=\hat a^{\dag}_1 \hat a^{\dag}_2, \quad \hat{K}_{-}= \hat a_1 \hat a_2, \notag \\
& \hat{K}_3=\frac{1}{2}\left( \hat a^{\dag}_1 \hat a_1+\hat a_2 \hat a^{\dag}_2 \right) .
\label{KKK}\end{aligned}$$ Starting from these formulae, a $SU(1,1)$ unitary representation on the Hilbert space ${\mathcal H}_1 \otimes {\mathcal H}_2$ can be decomposed into irreducible unitary representations of $SU(1,1)$ belonging to the positive discrete series [@BL2]. Note that the Casimir operator $$\hat{C}:= -\hat{K}_{+}\hat{K}_{-}- \hat{K}_3+ {\hat{K}_3}^2
\label{C}$$ and the generator $\hat{K}_3$ are diagonal in the standard Fock basis of the Hilbert space ${\mathcal H}_1 \otimes {\mathcal H}_2$. Indeed, their eigenvalue equations have the solutions: $$\begin{aligned}
& \hat{C}\mid k,m \rangle =k(k-1)\mid k,m \rangle, \quad
\left( k=\frac{1}{2}, \, 1, \, \frac{3}{2}, \, 2, \, \frac{5}{2}, \dots \right) , \notag\\
& \hat{K}_3\mid k,m \rangle =m\mid k,m \rangle, \quad
(m=k+l, \;\; l=0, 1, 2, 3, \dots ): \notag\\
& {\mid k,m \rangle}_{\pm}:=\mid \bar{n}_1, \bar{n}_2 \rangle, \quad
k:=\frac{1}{2}\left( \mid \bar{n}_1- \bar{n}_2 \mid +1 \right), \notag\\
& m:=\frac{1}{2}\left( \bar{n}_1+ \bar{n}_2 +1 \right), \:
\mid \bar{n}_1- \bar{n}_2 \mid ={\pm}\left( \bar{n}_1- \bar{n}_2 \right).
\label{km}\end{aligned}$$ The two-mode Fock space ${\mathcal H}_1 \otimes {\mathcal H}_2$ is therefore an orthogonal sum of infinite-dimensional invariant subspaces which are labelled with the Bargmann index $k$:
$${\mathcal H}_1 \otimes {\mathcal H}_2={\mathcal H}^{+\left (\frac{1}{2}\right )} \oplus
\bigoplus_{k> \frac{1}{2}} \left[ {\mathcal H}_{+}^{+(k)} \oplus {\mathcal H}_{-}^{+(k)} \right] .
\label{HH}$$
This property enables us to write the above-mentioned decomposition of a unitary representation of $SU(1,1)$:
$$\begin{aligned}
& \hat{\mathcal D}(V) =\hat {\mathcal D}^{+\left (\frac{1}{2}\right )}(V) \oplus
\bigoplus_{k> \frac{1}{2}} \left[ \hat{\mathcal D}^{+(k)}(V) \oplus \hat{\mathcal D}^{+(k)}(V)\right] ,
\notag\\
& \left [ V \in SU(1, 1) \right].
\label{D(V)}\end{aligned}$$
Analogously to $SU(2)$, the corresponding $SU(1,1)$ displacement operator acting on the Hilbert space ${\mathcal H}_1 \otimes {\mathcal H}_2$ [@Brif; @Novaes], $$\begin{aligned}
& \hat D^{+}(\zeta):=\exp{\left( \zeta \hat{K}_{+}-{\zeta}^{\ast} \hat{K}_{-} \right ) }, \notag\\
& \left( \zeta:=\frac{\tau}{2}\, e^{i\phi}, \quad \tau \geqq 0, \quad -\pi< \phi \leqq \pi \right),
\label{DSU(11)}\end{aligned}$$ is a $SU(1,1)$ unitary representation operator as well: $$\begin{aligned}
& \hat D^{+}(\zeta):= \hat{\mathcal D}{\left[ \, V(\chi, \, \tau, \, -\chi) \, \right] } \notag\\
& =\exp{\left( -i\chi \hat{K}_3 \right) }\, \exp{\left( -i\tau \hat{K}_2 \right) }\,
\exp{\left( i\chi \hat{K}_3 \right) }, \notag\\
& \left( \chi:=-\phi {\pm}\pi: \quad -\pi< \chi \leqq \pi \right).
\label{DD}\end{aligned}$$
Owing to the formulae (\[KKK\]), any two-mode squeeze operator (\[S\_12\]) is at the same time a $SU(1,1)$ displacement operator (\[DSU(11)\]) with the positive parameter $\tau =2r$: $$\hat S_{12}(r, \phi)=\hat D^{+}(r\, e^{i\phi}).
\label{S=D}$$
When the input to a non-degenerate parametric amplifier is a two-mode thermal radiation at optical frequencies, then its output is light in a STS: $$\hat \rho_{\rm ST}=\hat S_{12}(r, \phi)\hat \rho_{\rm T}(\bar{n}_1, \bar{n}_2)
\hat S^{\dag}_{12}(r, \phi).
\label{STS}$$ The unitary transformation (\[STS\]) of the state in the Schrödinger picture determines the $SU(1,1)$ matrix $V(\chi, \, 2r, \, -\chi)$ corresponding to a Bogoliubov transformation of the amplitude operators in the Heisenberg picture: $$\begin{aligned}
& \left(
\begin{matrix}
\hat{a}_1^{\prime} \\ \\ (\hat{a}_2^{\prime})^{\dag}
\end{matrix}
\right)
=V(\chi, \, 2r, \, -\chi)
\left(
\begin{matrix}
\hat{a}_1 \\ \\ {\hat{a}_2}^{\dag}
\end{matrix}
\right): \notag\\ \notag\\
& V(\chi, \, 2r, \, -\chi)=
\left(
\begin{matrix}
\cosh(r) & \sinh(r) \, e^{i\phi} \\
\\ \sinh(r) \, e^{-i\phi} & \cosh(r)
\end{matrix}
\right).
\label{V}\end{aligned}$$ Further, the Bogoliubov transformation (\[V\]) is equivalent to a symplectic one of the quadratures (\[u\^T\]). Its matrix $S(r,\, \phi) \in Sp(4, \mathbb{R})$ has the following $2 \times 2$ blocks expressed in terms of the identity and Pauli matrices: $$\begin{aligned}
& S(r,\, \phi)=
\left(
\begin{matrix}
S_a \, & S_b \, \\
S_b \, & S_a \,
\end{matrix}
\right): \quad
S_a:=\cosh(r) {\sigma}_0, \notag\\ \notag\\
& S_b:= \sinh(r) {\left[ \cos(\phi) {\sigma}_3+ \sin(\phi){\sigma}_1 \right] }.
\label{S(V)}\end{aligned}$$ This symmetric matrix accomplishes a symplectic congruence of the type (\[MTSC\]), $$\begin{aligned}
{\mathcal V}_{\rm ST}(\bar{n}_1, \bar{n}_2, r, \phi)
=S(r, \phi)\: {\mathcal V}_{\rm T}(\bar{n}_1, \bar{n}_2)\: S^T(r, \phi).
\label{STSC}\end{aligned}$$ We apply Eq. (\[STSC\]) to write the CM ${\mathcal V}_{\rm ST}$ of the output STS (\[STS\]). This has the usual partition (\[part\]) with the symmetric $2 \times 2$ submatrices: $$\begin{aligned}
& {\mathcal V}_j=b_j{\sigma}_0, \qquad \left( b_j \geqq \frac{1}{2} \right), \qquad (j=1, 2), \notag\\ \notag\\
& {\mathcal C}= c\left[ \cos(\phi) {\sigma}_3+ \sin(\phi){\sigma}_1 \right] , \qquad (c>0).
\label{VST}\end{aligned}$$ The standard form of the CM ${\mathcal V}_{\rm ST}(\bar{n}_1, \bar{n}_2, r, \phi)$ of a STS is obtained by setting $\phi=0$ in Eq. (\[VST\]) and has the following parameters: $$\begin{aligned}
& b_{1}= \left( \bar{n}_1+\frac{1}{2} \right) \left[ \cosh(r) \right]^2
+ \left( \bar{n}_2+\frac{1}{2} \right) \left[ \sinh(r) \right]^2, \notag\\
& b_{2}= \left( \bar{n}_1+\frac{1}{2} \right) \left[ \sinh(r) \right]^2
+ \left( \bar{n}_2+\frac{1}{2} \right) \left[ \cosh(r) \right]^2, \notag\\ \notag\\
& c=-d=\left( \bar{n}_{1}+\bar{n}_{2}+1 \right) \cosh(r) \sinh(r)>0 .
\label{sfVST}\end{aligned}$$
The only pure states belonging to the class of the STSs are the two-mode squeezed vacuum states (SVSs). Such a state is the output of a non-degenerate parametric amplifier when there is no photon at its input ports, that is, when both incoming field modes are in the vacuum state: $$\begin{aligned}
\hat{\rho}_{\rm SV}=|{\Psi}_{\rm SV}\rangle \langle {\Psi}_{\rm SV}| : \quad
|{\Psi}_{\rm SV}\rangle=\hat S_{12}(r, \phi) | 0,0 \rangle.
\label{SVS}\end{aligned}$$ Note that the two-mode SVSs make up a two-parameter family of pure symmetric STSs, $\left( \bar{n}_{1}=0, \, \bar{n}_{2}=0 \right)$, with the standard-form parameters: $$b_1=b_2=:b=\frac{1}{2}\cosh(2r), \quad c=\frac{1}{2}\sinh(2r).
\label{sfVSV}$$
We finally mention that a comprehensive study of the transformation of the two-mode GSs in a a non-degenerate parametric amplifier, including a detailed analysis of the conditions of separability and classicality of the output state, was carried out in an earlier paper [@PTH2001]. Quite recently, we employed the sets of two-mode MTSs and STSs in a comparative investigation of the Hellinger distance as a Gaussian measure of all the correlations between the modes [@PT2015].
Fidelity between special two-mode Gaussian states
=================================================
We consider a pair of special two-mode GSs of the same kind, $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$. Being undisplaced, the states are determined, respectively, by their CMs, ${\mathcal V}^{\prime}$ and ${\mathcal V}^{\prime \prime}$, whose standard-form parameters are denoted $\{ b_1^{\prime}, \, b_2^{\prime}, \, c^{\prime}\}$ and $\{ b_1^{\prime \prime}, \, b_2^{\prime \prime}, \, c^{\prime \prime}\}$. At the same time, their fidelity (\[2F(K)\]) has a simpler form: $${\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime})
=2 \left( \sqrt{K_{+}}-\sqrt{K_{-}} \right)^{-2}.
\label{F2(K)}$$
Mode-mixed thermal states
-------------------------
Let $\{ \bar{n}_1^{\prime}, \, \bar{n}_2^{\prime}, \, {\theta}^{\prime}, \, {\phi}^{\prime}\} \,$ and $\{ \bar{n}_1^{\prime \prime}, \, \bar{n}_2^{\prime \prime}, \, {\theta}^{\prime \prime}, \,
{\phi}^{\prime \prime}\} \,$ stand for the parameters of the MTSs $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$, respectively. Making use of the CM (\[VMT\]), we have evaluated the determinants $\Delta$, Eq. (\[Delta\]) and $\Gamma$, Eq. (\[Gamma\]), via the partitions of the corresponding $4\times 4$ matrices into $2\times 2$ blocks. We have applied the Schur determinant factorization (as the product of the determinant of a $2\times 2$ principal submatrix by that of its Schur complement) to obtain the formulae:
$$\begin{aligned}
& \Delta=\left\{ \left( b^{\prime}_{1}+b^{\prime \prime}_{1} \right)
\left( b^{\prime}_{2}+b^{\prime \prime}_{2} \right) \right. \notag \\
& \left. -\left[ \left( c^{\prime} \right)^2+\left( c^{\prime \prime} \right)^2
+2c^{\prime}c^{\prime \prime}\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right]
\right\}^2 ;
\label{DeltaM}\end{aligned}$$
$$\begin{aligned}
& \Gamma =16\left\{ \left[ b^{\prime}_{1}b^{\prime}_{2}-\left( c^{\prime} \right)^2 \right]
\left[ b^{\prime \prime}_{1}b^{\prime \prime}_{2}-\left( c^{\prime \prime} \right)^2 \right] \right. \notag \\
& \left. +\frac{1}{4}\left[ b^{\prime}_{1}b^{\prime \prime}_{1}
+b^{\prime}_{2}b^{\prime \prime}_{2}+2c^{\prime}c^{\prime \prime}
\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right] +\frac{1}{16} \right\}^2.
\label{GammaM}\end{aligned}$$
The determinant $\Lambda$ is the product (\[Lambda\]) of two similar symplectic invariants: $$\begin{aligned}
& \Lambda=16\left\{ \left[ b^{\prime}_{1}b^{\prime}_{2}-\left( c^{\prime}\right)^2 \right]^2 \right. \notag \\
& \left. -\frac{1}{4}\left[ \left( b_{1}^{\prime}\right)^2+\left( b_{2}^{\prime}\right)^2
+2\left( c^{\prime}\right)^2 \right]+\frac{1}{16} \right\}
\left\{ \left[ b^{\prime \prime}_{1}b^{\prime \prime}_{2}-\left( c^{\prime \prime}\right)^2 \right]^2 \right. \notag \\
& \left. -\frac{1}{4}\left[ \left( b_{1}^{\prime \prime}\right)^2+\left( b_{2}^{\prime \prime}\right)^2
+2\left( c^{\prime \prime}\right)^2 \right]+\frac{1}{16} \right\}.
\label{LambdaM} \end{aligned}$$ In the resulting functions $K_{\pm}$, Eq. (\[Kpm\]), we substitute the specific expressions (\[sfVMT\]) of the standard-form parameters and get the following couple of formulae:
$$\begin{aligned}
& K_{+}=2\left\{ \left( \bar{n}_1^{\prime} \, \bar{n}_2^{\prime} \, \bar{n}_1^{\prime \prime} \,
\bar{n}_2^{\prime \prime} \, \right)^{\frac{1}{2}} \right. \notag \\
& \left. +\left[ \left( \bar{n}_1^{\prime}+1 \right) \left( \bar{n}_2^{\prime}+1 \right)
\left( \bar{n}_1^{\prime \prime}+1 \right) \left( \bar{n}_2^{\prime \prime}+1 \right) \right]^{\frac{1}{2}}
\right\}^2.
\label{K(+)M}\end{aligned}$$
$$\begin{aligned}
& K_{-}=2\left\{ \left[ \bar{n}_1^{\prime} \left( \bar{n}_2^{\prime}+1 \right) \bar{n}_1^{\prime \prime}
\left( \bar{n}_2^{\prime \prime}+1 \right) \right]^{\frac{1}{2}} \right. \notag \\
& \left. +\left[ \left( \bar{n}_1^{\prime}+1 \right) \bar{n}_2^{\prime}
\left( \bar{n}_1^{\prime \prime}+1 \right) \bar{n}_2^{\prime \prime} \right]^{\frac{1}{2}} \right\}^2 \notag \\
& -\left( \bar{n}_1^{\prime}-\bar{n}_2^{\prime} \right)\, \left( \bar{n}_1^{\prime \prime}
-\bar{n}_2^{\prime \prime} \right) \left\{ 1-\cos{\left( {\theta}^{\prime}-{\theta}^{\prime \prime} \right) }\right. \notag \\
& \left. +\sin{\left( {\theta}^{\prime} \right) } \sin{\left( {\theta}^{\prime \prime} \right) }
\left[ 1-\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right] \right\} .
\label{K(-)M}\end{aligned}$$
Insertion of Eqs. (\[K(+)M\]) and (\[K(-)M\]) into Eq. (\[F2(K)\]) gives the fidelity of two MTSs. When all the other parameters of both states are kept fixed, this fidelity is an even function of the phase difference ${\phi}^{\prime}-{\phi}^{\prime \prime}$, which is strictly decreasing in the interval $[0, \pi]$.
Squeezed thermal states
-----------------------
We focus on a pair of STSs, $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$, and designate their sets of parameters as $\{ \bar{n}_1^{\prime}, \, \bar{n}_2^{\prime}, \, r^{\prime},
\, {\phi}^{\prime} \} \,$ and, respectively, $\{ \bar{n}_1^{\prime \prime}, \, \bar{n}_2^{\prime \prime}, \,
r^{\prime \prime}, \, {\phi}^{\prime \prime} \} \,$. Starting from the CM ${\mathcal V}_{\rm ST}(\bar{n}_1, \bar{n}_2, r, \phi)$, specified by Eqs. (\[part\]) and (\[VST\]), and employing the same technique as for MTSs, we have evaluated the determinants (\[Delta\])- (\[Lambda\]):
$$\begin{aligned}
& \Delta=\left\{ \left( b^{\prime}_{1}+b^{\prime \prime}_{1} \right)
\left( b^{\prime}_{2}+b^{\prime \prime}_{2} \right) \right. \notag \\
& \left. -\left[ \left( c^{\prime} \right)^2+\left( c^{\prime \prime} \right)^2
+2c^{\prime}c^{\prime \prime}\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right]
\right\}^2 ;
\label{DeltaS}\end{aligned}$$
$$\begin{aligned}
& \Gamma =16\left\{ \left[ b^{\prime}_{1}b^{\prime}_{2}-\left( c^{\prime} \right)^2 \right]
\left[ b^{\prime \prime}_{1}b^{\prime \prime}_{2}-\left( c^{\prime \prime} \right)^2 \right] \right. \notag \\
& \left. +\frac{1}{4}\left[ b^{\prime}_{1}b^{\prime \prime}_{1}
+b^{\prime}_{2}b^{\prime \prime}_{2}-2c^{\prime}c^{\prime \prime}
\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right] +\frac{1}{16} \right\}^2;
\label{GammaS}\end{aligned}$$
$$\begin{aligned}
& \Lambda=16\left\{ \left[ b^{\prime}_{1}b^{\prime}_{2}-\left( c^{\prime}\right)^2 \right]^2
-\frac{1}{4}\left[ \left( b_{1}^{\prime}\right)^2+\left( b_{2}^{\prime}\right)^2
-2\left( c^{\prime}\right)^2 \right]+\frac{1}{16} \right\} \notag \\
& \times \left\{ \left[ b^{\prime \prime}_{1}b^{\prime \prime}_{2}-\left( c^{\prime \prime}\right)^2 \right]^2
-\frac{1}{4}\left[ \left( b_{1}^{\prime \prime}\right)^2+\left( b_{2}^{\prime \prime}\right)^2
-2\left( c^{\prime \prime}\right)^2 \right]+\frac{1}{16} \right\}.
\label{LambdaS} \end{aligned}$$
Substitution of the above formulae into Eq. (\[Kpm\]) and subsequent insertion of the specific standard-form parameters (\[sfVST\]) yield the following functions:
$$\begin{aligned}
& K_{+}=2\left\{ \left[ \bar{n}_1^{\prime} \, \bar{n}_2^{\prime}\left( \bar{n}_1^{\prime \prime} +1 \right)
\left( \bar{n}_2^{\prime \prime}+1 \right) \right]^{\frac{1}{2}} \right. \notag \\
& \left. +\left[ \left( \bar{n}_1^{\prime}+1 \right) \left( \bar{n}_2^{\prime}+1 \right)
\bar{n}_1^{\prime \prime} \, \bar{n}_2^{\prime \prime} \right]^{\frac{1}{2}} \right\}^2 \notag \\
& +\left( \bar{n}_1^{\prime}+\bar{n}_2^{\prime}+1 \right) \left( \bar{n}_1^{\prime \prime}
+\bar{n}_2^{\prime \prime}+1 \right) \left\{ 1+\cosh{\left[ 2\left( r^{\prime}-r^{\prime \prime} \right) \right] }\right.
\notag \\
& \left. +\sinh{\left( 2r^{\prime} \right) } \sinh{\left( 2r^{\prime \prime} \right) }
\left[ 1-\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right] \right\} .
\label{K(+)S}\end{aligned}$$
$$\begin{aligned}
& K_{-}=2\left\{ \left[ \bar{n}_1^{\prime} \left( \bar{n}_2^{\prime} +1 \right) \bar{n}_1^{\prime \prime}
\left( \bar{n}_2^{\prime \prime} +1 \right) \right]^{\frac{1}{2}} \right. \notag \\
& \left. +\left[ \left( \bar{n}_1^{\prime}+1 \right) \bar{n}_2^{\prime}
\left( \bar{n}_1^{\prime \prime}+1 \right) \bar{n}_2^{\prime \prime} \right]^{\frac{1}{2}}
\right\}^2.
\label{K(-)S}\end{aligned}$$
By introducing the functions (\[K(+)S\]) and (\[K(-)S\]) into Eq. (\[F2(K)\]), we recover the expression of the fidelity between two STSs that has previously been used to quantify the Gaussian entanglement of such a two-mode state [@PTH2003]. Concerning its dependence on the phase difference ${\phi}^{\prime}-{\phi}^{\prime \prime}$, the fidelity between two STSs is an even function of this variable and strictly decreases with it in the interval $[0, \pi]$.
Quantum Fisher information tensor on the manifolds of special two-mode Gaussian states
======================================================================================
Let us look at a quantum system which has a manifold of states that are characterized by a finite set of continuous real variables $\{ \xi \}$. We concentrate on a pair of neighbouring states, $\hat{\rho}(\xi)$ and $\hat{\rho}(\xi+t d\xi)$, where $t$ is a real non-negative variable. We apply Eq. (\[BD\]): $$\begin{aligned}
& \frac{1}{2}\left\{ D_{\rm B}{ \left[ \hat{\rho}(\xi), \,\hat{\rho}(\xi+t d\xi) \right] } \right\}^2
=1-\sqrt{ {\cal F}(t) }: \notag \\
& {\cal F}(t):={\cal F}{ \left[ \hat{\rho}(\xi), \, \hat{\rho}(\xi+t d\xi) \right] }, \quad
\left( t \geqq 0 \right),
\label{BDn}\end{aligned}$$ Note the general properties: $${\cal F}(0)=1, \qquad -\left[ \frac{d}{dt}\sqrt{ {\cal F}(t) } \right]_{t=0}=0.
\label{F(0)}$$ The former identity in Eq. (\[F(0)\]) represents the sufficiency part of the saturation case in Eq. (\[F<1\]), while the latter was proven by Hübner in Ref. [@Hueb]. Therefore, the first non-vanishing term in the Maclaurin series of the squared Bures distance (\[BDn\]) is the $t^2$ term: $$\left\{ D_{\rm B}{ \left[ \hat{\rho}(\xi), \,\hat{\rho}(\xi+t d\xi) \right] } \right\}^2
=t^2 \left( ds_{\rm B} \right)^2+{\rm O}(t^3).
\label{t^2}$$ Its coefficient is the squared infinitesimal Bures line element on the above-specified manifold, $$\left( ds_{\rm B} \right)^2= \sum_{\alpha} \sum_{\beta} \, g_{\alpha \beta}(\xi)
d{\xi}_{\alpha} d{\xi}_{\beta},
\label{u^2}$$ where $g_{\alpha \beta}(\xi)$ are the components of the affiliated Riemannian metric tensor. We will evaluate it for two-mode MTSs and STSs as the second-order derivative $$\left( ds_{\rm B} \right)^2=-\left[ \frac{d^2}{dt^2}\sqrt{ {\cal F}(t) } \right]_{t=0} \geqq 0.
\label{dsB}$$ In the realm of the GSs, this method was first applied by Twamley to evaluate the Bures geodesic metric for one-mode STSs [@Twamley]. Then it was used to evaluating and studying the QFI metric for single-mode displaced TSs [@PS1998] and quite recently for arbitrary one-mode GSs [@Pinel]. Here we take advantage of the natural parametrizations for both families of two-mode states, MTSs and STSs, as well as of the convenient formulae for their fidelity established in Sec. V.
Mode-mixed thermal states
-------------------------
The fidelity (\[BDn\]) between neighbouring MTSs, $$\begin{aligned}
& {\cal F}(t):={\cal F} \left[ \hat{\rho} \left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right), \right. \notag \\
& \left. \hat{\rho} \left( \bar{n}_1+t d \bar{n}_1, \, \bar{n}_2+t d \bar{n}_2, \,
\theta+t d\theta, \, \phi+t d\phi \right) \right], \notag \\
& \left( t \geqq 0 \right),
\label{FnMTS}\end{aligned}$$ has the expression (\[F2(K)\]), $${\cal F}(t)=2 \left[ \sqrt{K_{+}(t)}-\sqrt{K_{-}(t)} \right]^{-2},
\label{F2(Kt)}$$ with the functions $K_{\pm}(t)$ obviously introduced in the manner of Eq. (\[FnMTS\]). Hence the Bures metric (\[dsB\]) for MTSs reads:
$$\left( ds_{\rm B} \right)^2=-\sqrt{2}\left\{ \frac{d^2}{dt^2}\left[ \sqrt{K_{+}(t)}
-\sqrt{K_{-}(t)} \right]^{-1} \right\}_{t=0}.
\label{dsSG}$$
One can readily check the identities (\[F(0)\]), the first one being precisely the sufficient condition in Eq. (\[FMT=1\]). A straightforward calculation then yields the formula: $$\begin{aligned}
& \left[ \left( ds_{\rm B} \right)_{\rm MT} \right]^2=\frac{1}{4}\left( \frac{1}{\bar{n}_1(\bar{n}_1+1)}
\left( d\bar{n}_1 \right)^2
+\frac{1}{\bar{n}_2(\bar{n}_2+1)}\left( d\bar{n}_2 \right)^2 \right. \notag \\
& \left. +\frac{\left( \bar{n}_1-\bar{n}_2 \right)^2 }{2\bar{n}_1 \bar{n}_2+ \bar{n}_1+\bar{n}_2 }
\left\{ (d\theta)^2 +\left[ \sin(\theta) \right]^2 (d\phi)^2 \right\} \right).
\label{dsMT}\end{aligned}$$ We have thus checked that the differentiable manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$ of the two-mode MTSs equipped with the Bures metric (\[dsMT\]) is a Riemannian one. Besides, its metric tensor has a diagonal matrix $g_{\rm MT}{\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$. Because the first two terms in Eq. (\[dsMT\]) are not influenced by the action of the beam splitter, they depend only on the input two-mode TS $\hat \rho_{\rm T}(\bar{n}_1, \bar{n}_2)$. Their sum defines therefore the Bures metric on the two-dimensional manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2 \right) }$ of the two-mode TSs (\[TS\]): $$\begin{aligned}
& \left[ \left( ds_{\rm B} \right)_{\rm T} \right ]^2=\frac{1}{4}\left[ \frac{1}{\bar{n}_1(\bar{n}_1+1)}
\left( d\bar{n}_1 \right)^2 \right. \notag \\
& \left. +\frac{1}{\bar{n}_2(\bar{n}_2+1)}\left( d\bar{n}_2 \right)^2 \right],
\label{dsT}\end{aligned}$$ With the reparametrization $\sqrt{\bar{n}_j}:=\sinh(x_j), \, (j=1,2), $ the metric (\[dsT\]) becomes an Euclidean flat one: $$\left[ \left( ds_{\rm B} \right)_{\rm T} \right ]^2=(dx_1)^2+(dx_2)^2.
\label{dsTR^2}$$ This means that the Riemannian manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2 \right) }$ is locally isometric with the first quadrant $\mathbb{R}_{+}^2$ of the Euclidean plane. The last two terms in Eq. (\[dsMT\]) originate in the interaction of the incoming thermal modes with the beam splitter resulting in the $SU(2)$ unitary state evolution (\[MTS\]). Their sum is proportional to the Euclidean round metric on the two-dimensional unit sphere $S^2$: $$\left( ds_{\theta, \phi} \right)^2= (d\theta)^2 \, +\left[ \sin(\theta) \right]^2 (d\phi)^2.
\label{S2}$$ In this line, $S^2$ can be viewed as a compact homogeneous space: $S^2=SU(2)/U(1)$.
Owing to the general relation (\[1:4\]), Eq. (\[dsMT\]) provides additionally the statistical distance $$\begin{aligned}
& \left[ \left( ds_{\rm F} \right)_{\rm MT} \right]^2=H_{\bar{n}_1} \left( d\bar{n}_1 \right)^2
+H_{\bar{n}_2} \left( d\bar{n}_2 \right)^2 \notag \\
& +H_{\theta} (d\theta)^2 +H_{\phi} (d\phi)^2.
\label{dsFMT}\end{aligned}$$ The components of the diagonal QFI metric tensor are independent of the phase $\phi$: $$\begin{aligned}
& H_{\bar{n}_1}=\frac{1}{\bar{n}_1(\bar{n}_1+1)}, \qquad
H_{\bar{n}_2}=\frac{1}{\bar{n}_2 (\bar{n}_2+1)}, \notag \\
& H_{\theta}=\frac{\left( \bar{n}_1-\bar{n}_2 \right)^2 }
{2\bar{n}_1 \bar{n}_2+ \bar{n}_1+\bar{n}_2 }, \notag \\
& H_{\phi}=\frac{\left( \bar{n}_1-\bar{n}_2 \right)^2 \, \left[ \sin(\theta) \right]^2 }
{2\bar{n}_1 \bar{n}_2+ \bar{n}_1+\bar{n}_2 }.
\label{HMT}\end{aligned}$$ Since the above QFI matrix is diagonal, the natural parameters $\left\{ \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right \} $ of the MTSs are said to be orthogonal. According to Eqs. (\[dsFMT\]) and (\[HMT\]), the quantum Cramér-Rao lower bound for the variance of such a state estimator ${\xi}_{\alpha}$ reads [@Paris]: $$\left( \Delta {\xi}_{\alpha} \right)^2 \geqq \frac{1}{{\cal N} H_{{\xi}_{\alpha}}}, \qquad
({\xi}_{\alpha} =\bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi),
\label{CRBM}$$ where ${\cal N}$ is the number of measurements.
To sum up, the Bures metric (\[dsMT\]) has the structure: $$\begin{aligned}
& \left[ \left( ds_{\rm B} \right)_{\rm MT} \right]^2=\left[ \left( ds_{\rm B} \right)_{\rm T} \right ]^2
+\left[ f_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right) } \right]^2 \left( ds_{\theta, \phi} \right)^2: \notag \\
& f_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right) }:=\frac{1}{2}\sqrt{H_{\theta}}.
\label{dsMT+}\end{aligned}$$ In addition, let us write down the Bures-metric volume element on the Riemannian manifold of the two-mode MTSs: $$d{\mathcal V}_{\rm B}:=\sqrt{\det{\left[ g_{\rm MT}
{\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)} \right] }} \,
d\bar{n}_1 \, d\bar{n}_2 \, d\theta \, d\phi.
\label{dVMT}$$ This volume element is an invariant quantity under any change of parametrization. Moreover, by virtue of the formula $$\sqrt{\det{\left[ g_{\rm MT}{\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)} \right] }}
=\frac{1}{16}\sqrt{H_{\bar{n}_1} H_{\bar{n}_2}H_{\theta}H_ {\phi}},
\label{JPMT}$$ it is proportional to the square root of the determinant of the QFI matrix: $$\begin{aligned}
& {\mathcal J}_{\rm MT}{\left( \bar{n}_1, \bar{n}_2, \theta \right) }:
=\sqrt{H_{\bar{n}_1} H_{\bar{n}_2}H_{\theta}H_ {\phi}}: \notag\\
& {\mathcal J}_{\rm MT}{\left( \bar{n}_1, \bar{n}_2, \theta \right) }
=\frac{1}{\sqrt{\bar{n}_1(\bar{n}_1+1)\bar{n}_2(\bar{n}_2+1)} } \notag\\
& \times \frac{\left( \bar{n}_1-\bar{n}_2 \right)^2 \,\sin(\theta) }
{2\bar{n}_1 \bar{n}_2+ \bar{n}_1+\bar{n}_2 }.
\label{QJPMT}\end{aligned}$$ The function (\[QJPMT\]) is called quantum Jeffreys’ prior [@Slater] due to its role in Bayesian statistical inference [@CFS2002]. Indeed, by extension of Jeffreys’ geometric rule [@Kass], when properly normalized, it is a reliable [*a priori*]{} probability density on any compact part of the state manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$.
Squeezed thermal states
-----------------------
The fidelity (\[BDn\]) between neighbouring STSs, $$\begin{aligned}
& {\cal F}(t):={\cal F} \left[ \hat{\rho} \left( \bar{n}_1, \, \bar{n}_2, \, r, \, \phi \right), \right. \notag \\
& \left. \hat{\rho} \left( \bar{n}_1+t d \bar{n}_1, \, \bar{n}_2+t d \bar{n}_2, \,
r+t dr, \, \phi+t d\phi \right) \right], \notag \\
& \left( t \geqq 0 \right),
\label{FnSTS}\end{aligned}$$ is given by the formula (\[F2(Kt)\]), where the functions $K_{\pm}(t)$ are consistent with Eq. (\[FnSTS\]). Accordingly, the Bures metric (\[dsB\]) for STSs has the expression (\[dsSG\]). It is easy to recover the identities (\[F(0)\]), the first one being included in Eq. (\[FST=1\]). We are subsequently lead to the formula: $$\begin{aligned}
& \left[ \left( ds_{\rm B} \right)_{\rm ST} \right]^2=\frac{1}{4}\left( \frac{1}{\bar{n}_1(\bar{n}_1+1)}
\left( d\bar{n}_1 \right)^2
+\frac{1}{\bar{n}_2(\bar{n}_2+1)}\left( d\bar{n}_2 \right)^2 \right. \notag \\
& \left. +\frac{\left( \bar{n}_1+\bar{n}_2+1 \right)^2 }
{2\bar{n}_1 \bar{n}_2+\bar{n}_1+\bar{n}_2+1}
\left\{ [d(2r)]^2 +\left[ \sinh(2r) \right]^2 (d\phi)^2 \right\} \right).
\label{dsST}\end{aligned}$$ Equation (\[dsST\]) actually defines the Bures metric on the Riemannian manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$ of the two-mode STSs. Note that the associated metric tensor has a diagonal matrix $g_{\rm ST}{\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$. The sum of the first two terms in the r. h. s. of Eq. (\[dsST\]) is the squared line element (\[dsT\]) on the manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2 \right) }$ of the two-mode TSs (\[TS\]). The interaction of the incident thermal modes with the non-degenerate parametric amplifier produces the $SU(1,1)$ unitary state evolution (\[STS\]). This is represented by the last two terms in the r. h. s. of Eq. (\[dsST\]). Remarkably, their sum is proportional to the Minkowski metric on the hyperboloid of two sheets $x^2+y^2-z^2=-1$: $$\left( ds_{\tau, \phi} \right)^2= (d\tau)^2 \, +\left[ \sinh(\tau) \right]^2 (d\phi)^2: \quad \tau=2r.
\label{H2}$$ The upper sheet $z>0$ of the hyperboloid is a two-dimensional Riemannian manifold denoted $H_{-1}^2$, which is a non-compact homogeneous space: $H_{-1}^2=SU(1,1)/U(1)$. At the same time, $H_{-1}^2$ is an analytic model of the hyperbolic plane $H^2$ [@Coxeter; @Cannon].
By reason of the general relation (\[1:4\]), Eq. (\[dsST\]) supplies the infinitesimal statistical distance $$\begin{aligned}
& \left[ \left( ds_{\rm F} \right)_{\rm ST} \right]^2=H_{\bar{n}_1} \left( d\bar{n}_1 \right)^2
+H_{\bar{n}_2} \left( d\bar{n}_2 \right)^2 \notag \\
& +H_{2r} \left[ d(2r) \right]^2 +H_{\phi} (d\phi)^2,
\label{dsFST}\end{aligned}$$ whose QFI matrix is diagonal, with entries independent of the phase $\phi$: $$\begin{aligned}
& H_{\bar{n}_1}=\frac{1}{\bar{n}_1(\bar{n}_1+1)} \, , \qquad
H_{\bar{n}_2}=\frac{1}{\bar{n}_2 (\bar{n}_2+1)} \, , \notag \\
& H_{2r}=\frac{\left( \bar{n}_1+\bar{n}_2 +1\right)^2 }
{2\bar{n}_1 \bar{n}_2+\bar{n}_1+\bar{n}_2+1}\, , \notag \\
& H_{\phi}=\frac{\left( \bar{n}_1+\bar{n}_2 +1\right)^2 \, \left[ \sinh(2r) \right]^2 }
{2\bar{n}_1 \bar{n}_2+\bar{n}_1+\bar{n}_2+1}.
\label{HST}\end{aligned}$$ This diagonal form of the QFI tensor shows that the natural parameters $\left\{ \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right \} $ of the STSs are orthogonal. It allows one to write directly the quantum Cramér-Rao bound for the variance of any such a state estimator ${\xi}_{\alpha}$ [@Paris]: $$\left( \Delta {\xi}_{\alpha} \right)^2 \geqq \frac{1}{{\cal N} H_{{\xi}_{\alpha}}}, \qquad
({\xi}_{\alpha}=\bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi).
\label{CRBS}$$ In Eq. (\[CRBS\]), ${\cal N}$ denotes the number of the performed measurements.
To recapitulate, we point out that the Bures metric (\[dsST\]) has a decomposition similar to that shown by Eq. (\[dsMT+\]): $$\begin{aligned}
& \left[ \left( ds_{\rm B} \right)_{\rm ST} \right]^2=\left[ \left( ds_{\rm B} \right)_{\rm T} \right ]^2
+\left[ f_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right) } \right]^2 \left( ds_{2r, \phi} \right)^2: \notag \\
& f_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right) }:=\frac{1}{2}\sqrt{H_{2r}}.
\label{dsST+}\end{aligned}$$ Besides, we indicate the Bures-metric volume element on the Riemannian manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$ of the two-mode STSs: $$d{\mathcal V}_{\rm B}:=\sqrt{\det{\left[ g_{\rm ST}
{\left( \bar{n}_1, \bar{n}_2, 2r, \phi \right)} \right] }} \, d\bar{n}_1 \, d\bar{n}_2 \, d(2r) d\phi.
\label{dVST}$$ The formula $$\sqrt{\det{\left[ g_{\rm ST}{\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)} \right] }}
=\frac{1}{16}\sqrt{ H_{\bar{n}_1} H_{\bar{n}_2}H_{2r}H_ {\phi} }
\label{JPST}$$ demonstrates that the parametrization-invariant volume element (\[dVST\]) is proportional to the quantum Jeffreys’ prior: $$\begin{aligned}
& {\mathcal J}_{\rm ST}{\left( \bar{n}_1, \bar{n}_2, 2r \right) }:
=\sqrt{ H_{\bar{n}_1} H_{\bar{n}_2} H_{2 r} H_ {\phi} }.
\label{QJPST}\end{aligned}$$ We recall the separability threshold $r_s$ for a two-mode STS, introduced in Ref. [@PTH2001]: $$\sinh(r_s):=\sqrt{ \frac{\bar{n}_1 \, \bar{n}_2}{ \bar{n}_1+\bar{n}_2+1} }.
\label{r_s}$$ Noticeably, the quantum Jeffreys’ prior (\[QJPST\]) depends only on two variables, $r_s$ and $r$: $${\mathcal J}_{\rm ST}{\left( 2r_s, 2r \right) }=\frac{4\sinh(2r)}{\sinh(4r_s)}.
\label{JST}$$ At a fixed value of the squeeze parameter $r$, the function (\[JST\]) strictly decreases with the variable $r_s$ from the limit ${\mathcal J}_{\rm ST}(0, 2r) =+\infty$ to zero, for $r_s \to +\infty$. The starting limit is reached by any two-mode STS at the physicality edge $(r_s=0)$ and, in particular, by any two-mode SVS. The value at the separability threshold ${\mathcal J}_{\rm ST}{\left( 2r_s, 2r_s \right) }=2\,{\rm sech}(2r_s)$ is itself a decreasing function of the variable $r_s$.
Discussion
----------
We stress that the explicit formula (\[2F\]) for the fidelity of two-mode GSs allows one the evaluation of QFI for estimating various parameters via Eq. (\[dsB\]). This method has efficiently been exploited in some recent applications [@Adesso1; @Fu1]. For instance, the concept of interferometric power, introduced and evaluated in Ref. [@Adesso1], reduces in the particular case of an STS to the QFI matrix element $H_{\phi}$, Eq. (\[HST\]). A productive research [@Fu2; @Fu3] in relativistic quantum metrology is based on QFI obtained by using Eq. (\[2F\]).
However, there are few cases when one could use an explicit expression of the Uhlmann fidelity to derive the QFI metric via Eq. (\[dsB\]). The most widespread approach to evaluating the QFI is based on a central quantity in parameter estimation theory, namely, the symmetric logarithmic derivative (SLD) [@BC1994; @Paris]. In particular, some important results have been obtained for GSs by employing the SLD-method. An interesting example is the optimal estimation of entanglement for two-mode symmetric STSs in Ref. [@GGP]. The QFI for one-parameter estimation in the case of multi-mode Gaussian channels and states was recently derived [@MI; @Mo; @Jiang]. The general result for an $n$-mode GS obtained in Refs. [@Mo; @Jiang] is a compact expression in terms of the CM, the displacement vector, and their first-order derivatives with respect to the estimated parameter.
In order to check on Eqs. (\[HMT\]) and (\[HST\]), we apply the QFI formula from Ref. [@Jiang] together with all the necessary ingredients involved. Making use of the CMs (\[VMT\]) and (\[VST\]), as well as of the corresponding symplectic matrices (\[S(U)\]) and (\[S(V)\]) that diagonalize them by congruence, a routine calculation allows us to retrieve the QFI matrices for both manifolds of two-mode MTSs and STSs. However, the key point is the knowledge of the diagonalizing symplectic transformations.
Scalar curvatures of the Bures metric on the manifolds of special two-mode Gaussian states
==========================================================================================
The scalar curvature is the simplest invariant derived from the metric of a Riemannian or pseudo-Riemannian manifold. This is a real function $R$ defined on such a manifold $\mathcal M$ which is determined [*solely*]{} by its intrinsic geometry. The value $R(p)$ at each point $p \in {\mathcal M}$ depends on the local features of the metric.
We evaluate the scalar curvatures on the four-dimensional Riemannian manifolds of the two-mode MTSs and STSs starting from their Bures metric tensors. Such a calculation exploits standard formulae from Riemannian Geometry [@Carmo] and consists of the following compulsory steps:
1. Evaluation of the Christoffel symbols of the Levi-Civita connection;
2. Calculation of the required components of the Riemann curvature tensor by employing the Christoffel symbols and their first-order derivatives;
3. Calculation of the diagonal components of the Ricci tensor, which is defined as a contraction of the Riemann tensor;
4. Evaluation of the Riemannian scalar curvature as the trace of the Ricci tensor with respect to the metric.
We mention that calculations of the scalar curvature of the Bures metric tensor along the same lines were carried out previously for Riemannian manifolds of two kinds of single-mode GSs, namely, the STSs [@Twamley] and the displaced TSs [@PS1998].
Mode-mixed thermal states
-------------------------
By carrying out the above-sketched program, we have found the scalar curvature on the Riemannian manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$ of the two-mode MTSs:
$$\begin{aligned}
& R_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right)}=\frac{2}{\left( 2 \bar{n}_1 \bar{n}_2
+ \bar{n}_1 +\bar{n}_2 \right)^2} \notag \\
& \times \left[ \left( \bar{n}_1-\bar{n}_2 \right)^2 -24\, \bar{n}_1\left( \bar{n}_1 +1 \right)
\bar{n}_2 \left( \bar{n}_2 +1 \right) \right. \notag \\
& \left. +9\left( 2 \bar{n}_1 \bar{n}_2 + \bar{n}_1 +\bar{n}_2 \right) \right].
\label{RMT}\end{aligned}$$
The scalar curvature (\[RMT\]) does not depend on the parameters $\{ \theta, \, \phi \}$ of the beam splitter. Its expression is valid for any values of the mean thermal photon occupancies $ \bar{n}_1, \bar{n}_2$, and displays the symmetry property $$R_{\rm MT}{\left( \bar{n}_2, \bar{n}_1 \right)}= R_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right)}.
\label{symRMT}$$ Therefore, Eq. (\[RMT\]) describes a two-dimensional surface in ${\mathbb R}^3$ which is represented in Fig. 1. It looks like a descending symmetric valley whose talweg is precisely the intersection with its symmetry plane $\bar{n}_1= \bar{n}_2$, i.e., the vertical plane that bisects the first octant.
![(Color online) The scalar curvature $R_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right)}$, Eq. (\[RMT\]), of the two-mode MTSs. This is a convex surface whose general aspect is that of a symmetric valley that descends and widens continuously. Its talweg belongs to the symmetry plane $\bar{n}_1= \bar{n}_2$ and is drawn in Fig. 2a.](Fig1-rev.eps){width="7cm"}
In the limit case $\bar{n}_1= \bar{n}_2=:\bar{n}$ of an emerging two-mode TS, Eq. (\[RMT\]) simplifies to: $$R_{\rm MT}{\left( \bar{n}, \bar{n} \right)}= \frac{9}{\bar{n}\left( \bar{n}+1 \right) }-12.
\label{RT}$$ The graph of the above function is the intersection of the two-dimensional surface (\[RMT\]) and its symmetry plane $\bar{n}_1= \bar{n}_2$. The function (\[RT\]) strictly decreases with the variable $\bar{n}$ from $+\infty$ at $\bar{n}=0$ to the negative asymptotic value $\lim_{\bar{n} \to \infty} R_{\rm MT}{\left( \bar{n}, \bar{n} \right) }=-12$. Besides, this is a convex function which has a unique zero, $\bar{n}=\frac{1}{2}$.
![(Color online) a) The vertical intersection (\[RT\]) of the surface (\[RMT\]) and its symmetry plane $\bar{n}_1= \bar{n}_2$. This is the talweg of the surface (\[RMT\]) and is made up of the symmetric two-mode TSs.\
b) The vertical intersection (\[RMTperp\]) of the surface (\[RMT\]) and the plane $\bar{n}_1+\bar{n}_2=1$. The vertical sections a) and b) which are orthogonal and meet at their unique zero, $\bar{n}_1=\frac{1}{2}$.](Fig2a-rev.eps "fig:"){width="4.4cm"} ![(Color online) a) The vertical intersection (\[RT\]) of the surface (\[RMT\]) and its symmetry plane $\bar{n}_1= \bar{n}_2$. This is the talweg of the surface (\[RMT\]) and is made up of the symmetric two-mode TSs.\
b) The vertical intersection (\[RMTperp\]) of the surface (\[RMT\]) and the plane $\bar{n}_1+\bar{n}_2=1$. The vertical sections a) and b) which are orthogonal and meet at their unique zero, $\bar{n}_1=\frac{1}{2}$.](Fig2b-rev.eps "fig:"){width="4cm"}
It is instructive to examine further the intersection of the surface (\[RMT\]) and a vertical plane perpendicular to its symmetry plane. As an example, we choose the plane $\bar{n}_1+\bar{n}_2=1$ which meets the symmetry plane in the vertical line $\bar{n}_1=\bar{n}_2=\frac{1}{2}$. This straight line contains the above-mentioned zero of the function (\[RT\]). The intersection of the surface (\[RMT\]) and the vertical plane $\bar{n}_1+\bar{n}_2=1$ is the graph of the following function of the variable $\bar{n}_1 \in [0, \, 1]:$ $$\begin{aligned}
& R_{\rm MT}{\left( \bar{n}_1, 1- \bar{n}_1 \right)}
=-4\, \frac{12 {\alpha}^2+17\alpha-5}{\left( 2\alpha+1 \right)^2} \geqq 0 , \notag \\
& \alpha:=\bar{n}_1 \left( 1- \bar{n}_1 \right) \in \left[ 0, \, \frac{1}{4} \right].
\label{RMTperp}\end{aligned}$$ The function (\[RMTperp\]) strictly decreases from the limit value $R_{\rm MT}(0, \, 1)=20$ to its minimum $R_{\rm MT}{\left( \frac{1}{2}, \, \frac{1}{2} \right) }=0$ and then has a mirror increase on the interval $\bar{n}_1 \in \left[ \frac{1}{2}, \, 1 \right].$ Its graph exhibits the profile of a symmetric valley. Such a vertical section is typical for the the two-dimensional surface (\[RMT\]). The vertical sections (\[RT\]) and (\[RMTperp\]) are plotted in Figs. 2a and 2b.
A noteworhy limit situation arises when one incoming mode is in the vacuum state and the other is not $\left( \bar{n}_1>0, \; \bar{n}_2=0 \right)$. Then the output two-mode MTS is at the physicality edge and has the scalar curvature $$R_{\rm MT}{\left( \bar{n}_1, 0 \right)}=2+ \frac{18}{\; \bar{n}_1}.
\label{RMTedge}$$ The function (\[RMTedge\]) is positive, strictly decreasing and convex. Figure 5 presents its graph, as well as that of the function (\[RSTedge\]).
Squeezed thermal states
-----------------------
In a similar way we have evaluated the scalar curvature on the Riemannian manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$ of the two-mode STSs:
$$\begin{aligned}
& R_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right)}=\frac{2}{\left( 2 \bar{n}_1 \bar{n}_2
+ \bar{n}_1 +\bar{n}_2 +1 \right)^2} \notag \\
& \times \left[ \left( \bar{n}_1+\bar{n}_2 +1\right)^2 -24\, \bar{n}_1\left( \bar{n}_1 +1 \right)
\bar{n}_2 \left( \bar{n}_2 +1 \right) \right. \notag \\
& \left. -9\left( 2 \bar{n}_1 \bar{n}_2 + \bar{n}_1 +\bar{n}_2+1 \right) \right].
\label{RST}\end{aligned}$$
The scalar curvature (\[RST\]) does not depend on the parameters $\{ 2r, \, \phi \}$ of the non-degenerate parametric amplifier. Its expression is valid for any values of the mean thermal photon occupancies $ \bar{n}_1, \bar{n}_2$, and displays the symmetry property $$R_{\rm ST}{\left( \bar{n}_2, \bar{n}_1 \right)}= R_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right)}.
\label{symRST}$$ Accordingly, the two-dimensional surface (\[RST\]) in ${\mathbb R}^3$ has the vertical symmetry plane $\bar{n}_1= \bar{n}_2$ that bisects the first octant. Figure 3 displays the general aspect of this surface. It looks like a pair of opposite symmetric valleys. The ascending valley is narrow and steep, while the descending one is broader and slower. The talweg is the intersection (\[RsST\]) with the symmetry plane $\bar{n}_1= \bar{n}_2$, while the watershed is the intersection (\[RSTperp\]) with the vertical plane $\bar{n}_1+\bar{n}_2=2\bar{n}_s$, which is perpendicular to the first one. The talweg and the watershed are tangent at the unique saddle point $S$ on the surface, which is located at the point $(\bar{n}_s, \bar{n}_s)$ with $\bar{n}_s:=-0.5+\sqrt{1.15} \approxeq 0.5724$.
![(Color online) The scalar curvature $R_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right)}$, Eq. (\[RST\]), of the two-mode STSs. This surface consists of two opposite symmetric valleys separated by a watershed: a narrow, steeply ascending valley and a wider one which is slowly descending. It possesses a unique saddle point $S$ at the intersection of the talweg and the watershed. The talweg is the normal section (\[RsST\]) in the symmetry plane $\bar{n}_1= \bar{n}_2$, while the watershed is the normal section (\[RSTperp\]) in the plane $\bar{n}_1+\bar{n}_2=2\bar{n}_s$. These normal sections, which are vertical and orthogonal, are represented in Figs. 4a and 4b.](Fig3-rev.eps){width="8cm"}
For symmetric two-mode STSs, $\left( \bar{n}_1= \bar{n}_2=:\bar{n} \right)$, the scalar curvature (\[RST\]) is negative: $$\begin{aligned}
& R_{\rm ST}{\left( \bar{n}, \bar{n} \right) }
=-4\, \frac{12 {\beta}^2+7\beta+4}{\left( 2\beta+1 \right)^2} < 0 , \notag \\
& \beta:=\bar{n} \left( \bar{n}+1 \right) \geqq 0.
\label{RsST}\end{aligned}$$ The graph of the above function is reproduced in Fig. 4a and is the talweg of the surface (\[RST\]). On its ascending side, the function (\[RsST\]) has a steep rise from the limit value $ R_{\rm ST}(0, \, 0)=-16$, reached for any two-mode SVS, to a maximum $R_{\rm ST}{\left( \bar{n}_s, \bar{n}_s \right) }=-\frac{143}{14} \approxeq -10.2143,$ reached at the point $\bar{n}_s=-0.5+\sqrt{1.15} \approxeq 0.5724$. Then, on the descending side, it has a moderate fall toward the asymptotic value $\lim_{\bar{n} \to \infty} R_{\rm ST}{\left( \bar{n}, \bar{n} \right) }=-12$. As expected on intuitive grounds, this asymptotic limit coincides with the similar one for MTSs, displayed in Fig. 2a: $\lim_{\bar{n} \to \infty} R_{\rm MT}{\left( \bar{n}, \bar{n} \right) }=-12$.
Let us contemplate next the intersection of the surface (\[RST\]) and the vertical plane $\bar{n}_1+\bar{n}_2=2\bar{n}_s$, where $\bar{n}_s$ is the maximum point of the function (\[RsST\]). This plane is perpendicular to the symmetry plane $\bar{n}_1= \bar{n}_2$ and meets it in the vertical straight line $\bar{n}_1=\bar{n}_2=\bar{n}_s$. The aforementioned intersection is the graph of the following function of the variable $\bar{n}_1 \in \left[ 0, \, 2\bar{n}_s \right]:$ $$\begin{aligned}
& R_{\rm ST}{\left( \bar{n}_1, 2\bar{n}_s- \bar{n}_1 \right)}
=-\frac{4}{\left[ 2\left( \gamma +\bar{n}_s \right) +1 \right]^2 } \notag \\
& \times \left[ 12\left( \gamma +\bar{n}_s \right)^2
+21\left( \gamma +\bar{n}_s \right) -8.6 \right] < 0\, , \notag \\
& \gamma:=\bar{n}_1\left( 2\bar{n}_s- \bar{n}_1 \right)
\in \left[ 0, \, {\left( \bar{n}_s \right) }^2 \right], \quad
{\left( \bar{n}_s \right) }^2=0.3276 , \notag \\
& 4-14\bar{n}_s\left( \bar{n}_s+1 \right) =8.6.
\label{RSTperp}\end{aligned}$$ The function (\[RSTperp\]) decreases from the limit value $R_{\rm ST}(0, 2\bar{n}_s) \approxeq -6.3925$ to its minimum, $R_{\rm ST}{\left( \bar{n}_s, \bar{n}_s \right) }=-\frac{143}{14} \approxeq -10.2143,$ and then has a mirror increase on the interval $\bar{n}_1 \in \left[ \bar{n}_s, \, 2\bar{n}_s \right].$ Its graph is therefore the profile of a symmetric valley and is drawn in Fig. 4b.
![(Color online) a) The vertical intersection (\[RsST\]) of the surface (\[RST\]) and its symmetry plane $\bar{n}_1= \bar{n}_2$. This is the talweg of the surface (\[RST\]) and is made up of the symmetric two-mode STSs.\
The function (\[RsST\]) has a steep rise from its lowest value $R_{\rm ST}(0, \, 0)=-16$, reached for any two-mode SVS, to a maximum $R_{\rm ST}{\left( \bar{n}_s, \bar{n}_s \right) }=-\frac{143}{14} \approxeq -10.2143,$ reached at the point $\bar{n}_s=-0.5+\sqrt{1.15} \approxeq 0.5724$. Then it has a moderate fall toward the asymptotic value $\lim_{\bar{n} \to \infty} R_{\rm ST}{\left( \bar{n}, \bar{n} \right) }=-12$. The function (\[RsST\]) changes concavity at the unique inflection point $\bar{n}_i \approxeq 0.9565$, where it has the value $R_{\rm ST}{\left( \bar{n}_i, \bar{n}_i \right) } \approxeq -10.5140$.\
b) The vertical intersection (\[RSTperp\]) of the surface (\[RST\]) and the plane $\bar{n}_1+\bar{n}_2=2\bar{n}_s$. This is the watershed on the surface (\[RST\]), which is orthogonal to the talweg and touches it at their common extremum point, $\bar{n}_1=\bar{n}_s$.](Fig4a-rev.eps "fig:"){width="4.4cm"} ![(Color online) a) The vertical intersection (\[RsST\]) of the surface (\[RST\]) and its symmetry plane $\bar{n}_1= \bar{n}_2$. This is the talweg of the surface (\[RST\]) and is made up of the symmetric two-mode STSs.\
The function (\[RsST\]) has a steep rise from its lowest value $R_{\rm ST}(0, \, 0)=-16$, reached for any two-mode SVS, to a maximum $R_{\rm ST}{\left( \bar{n}_s, \bar{n}_s \right) }=-\frac{143}{14} \approxeq -10.2143,$ reached at the point $\bar{n}_s=-0.5+\sqrt{1.15} \approxeq 0.5724$. Then it has a moderate fall toward the asymptotic value $\lim_{\bar{n} \to \infty} R_{\rm ST}{\left( \bar{n}, \bar{n} \right) }=-12$. The function (\[RsST\]) changes concavity at the unique inflection point $\bar{n}_i \approxeq 0.9565$, where it has the value $R_{\rm ST}{\left( \bar{n}_i, \bar{n}_i \right) } \approxeq -10.5140$.\
b) The vertical intersection (\[RSTperp\]) of the surface (\[RST\]) and the plane $\bar{n}_1+\bar{n}_2=2\bar{n}_s$. This is the watershed on the surface (\[RST\]), which is orthogonal to the talweg and touches it at their common extremum point, $\bar{n}_1=\bar{n}_s$.](Fig4b-rev.eps "fig:"){width="4cm"}
Although such a vertical section is typical for the two-dimensional surface (\[RST\]), the section (\[RSTperp\]) is a rather special one. Indeed, the common extremum point $S: \left\{ \bar{n}_s, \, \bar{n}_s, \, R_{\rm ST}{\left( \bar{n}_s, \bar{n}_s \right) } \right\} $ of the curves (\[RsST\]) and (\[RSTperp\]) is a saddle point on the two-dimensional surface (\[RST\]) in ${\mathbb R}^3$, having an upward vertical normal. Moreover, this turns out to be its unique stationary point. The curves (\[RsST\]) and (\[RSTperp\]) are precisely the normal sections that include the principal directions tangent to the surface (\[RST\]) at the saddle point $S$ [@K].
When one of the incoming modes is in the vacuum state and the other is not, $\left( \bar{n}_1>0, \; \bar{n}_2=0 \right)$, then the output two-mode STS is at the physicality edge. It has the scalar curvature $$R_{\rm ST}{\left( \bar{n}_1, 0 \right)}=2- \frac{18}{ \bar{n}_1+1}.
\label{RSTedge}$$ The function (\[RSTedge\]) strictly increases with the variable $\bar{n}_1$ from the SVS value $R_{\rm ST}(0, \, 0)=-16$ to the positive asymptotic value $\lim_{\bar{n}_1 \to \infty} R_{\rm ST}{\left( \bar{n}_1, 0 \right) }=2$. Besides, it is a concave function which has a unique zero, $\bar{n}_1=8$. The curves (\[RMTedge\]) and (\[RSTedge\]) have a common asymptote and are represented together in Fig. 5.
![(Color online) The scalar curvatures (\[RMTedge\]) and (\[RSTedge\]) of the two-mode MTSs and STSs at the physicality edge. They are figured as the marginal vertical curves on the surfaces (\[RMT\]) and (\[RST\]), respectively, which meet the plane $ \bar{n}_2=0$ along them. The former is decreasing and convex, while the latter is increasing and concave. Their starting values are, respectively, $R_{\rm MT}(0,0) = +\infty$ for the vacuum state and $R_{\rm ST}(0,0) = -16$ for any two-mode SVS. Nevertheless, they have a common asymptotic limit: $R_{\rm MT}(+\infty,0)=R_{\rm ST}(+\infty,0)=2.$](Fig5-rev.eps){width="8cm"}
Alternative derivation
----------------------
The most intriguing feature of both scalar curvatures $R_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right) }$, Eq. (\[RMT\]), and $R_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right) }$, Eq. (\[RST\]), is that they do not depend on the parameters $\{ \theta, \, \phi \}$ and, respectively, $\{ 2r, \, \phi \}$. These parameters specify the unitary transformations modeling the interactions of the incident thermal-light beams with the optical devices under discussion. Nevertheless, the scalar curvatures originate precisely in the mentioned unitary transformations. Since the scalar curvature of a given metric on a Riemannian manifold is determined solely by the metric itself, any explanation should start from the decompositions (\[dsMT+\]) and (\[dsST+\]) of the squared Bures line elements on the Riemannian manifolds of special two-mode GSs. They can be written in terms of the metric tensors as follows: $$\begin{aligned}
& g_{\rm MT}{\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right) } \notag \\
& = g_{\rm T}{\left( \bar{n}_1, \, \bar{n}_2 \right) } \oplus
\left[ f_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right) } \right]^2 \,
g_{S^2} {\left( \theta, \, \phi \right)};
\label{gMT+}\end{aligned}$$ $$\begin{aligned}
& g_{\rm ST}{\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right) } \notag \\
& = g_{\rm T}{\left( \bar{n}_1, \, \bar{n}_2 \right) } \oplus
\left[ f_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right) } \right]^2 \,
g_{H_{-1}^2} {\left( 2r, \, \phi \right) }.
\label{gST+}\end{aligned}$$ In the above equations, $g_{S^2} $ and $g_{H_{-1}^2}$ designate the metric tensors on the two-dimensional Riemannian manifolds $S^2$ and $H_{-1}^2$, as given by Eqs. (\[S2\]) and (\[H2\]), respectively. In contrast to the Euclidean plane, these surfaces have a constant scalar curvature which is not zero. Moreover, up to a scale factor, they are the only connected ones to share this intrinsic geometric property: $$R{\left( {\mathbb R}^2 \right) }=0, \quad R{\left( S^2 \right) }=2, \quad
R{\left( H_{-1}^2 \right) }=-2.
\label{RSH}$$
We further need the notion of warped product of two Riemannian manifolds, which was introduced in Ref. [@BO'N]. Let $({\mathcal B}, \, g_{\rm B})$ and $({\mathcal F},\, g_{\rm F})$ be Riemannian manifolds of dimensions $m$ and $n$, respectively, and $f:{\mathcal B} \to {\mathbb R}_{+}\setminus \{0\}$ a smooth function: $f \in C^{\infty}({\mathcal B})$. The warped product ${\mathcal M}:={\mathcal B} {\times}_f {\mathcal F}$ is the differentiable product manifold ${\mathcal B} \times {\mathcal F}$, of dimension $m+n$, endowed with the Riemannian metric $g_{\rm M}:=g_{\rm B} \oplus f^2 g_{\rm F}$. By means of this rule, the warping function $f>0$ determines the Riemannian structure of the warped product $({\mathcal M},\, g_{\rm M})$.
The relationship between the scalar curvatures $R({\mathcal B}), \, R({\mathcal F})$, and $R({\mathcal M})$ was established in Ref. [@DD]: $$\begin{aligned}
& -\frac{4n}{n+1}{\Delta}_{g_{\rm B}}u+R({\mathcal B})u+R({\mathcal F})u^{\frac{n-3}{n+1}}
=R({\mathcal M})u, \notag\\
& u:=f^{\frac{n+1}{2}}.
\label{R(M)}\end{aligned}$$ In Eq. (\[R(M)\]), ${\Delta}_{g_{\rm B}}$ is the Laplace-Beltrami operator on the Riemannian manifold $({\mathcal B}, \, g_{\rm B})$. This second-order differential operator is the divergence of the gradient: $$\begin{aligned}
& {\Delta}_{g_{\rm B}}v =\frac{1}{\sqrt{\det{\left( g_{\rm B}\right) } } }
{\partial}_j \left[ \sqrt{\det{\left( g_{\rm B}\right) } } \left( g_{\rm B}\right)^{jk}
{\partial}_k v \right], \notag\\
& v \in C^2({\mathcal B}).
\label{LB}\end{aligned}$$
We mention two important consequences of Eq. (\[R(M)\]).
First, when $f=1$, it reduces to the familiar addition law $$R{\left( {\mathcal B}\times {\mathcal F} \right) }=R{\left( {\mathcal B} \right) }
+R{\left( {\mathcal F} \right) }
\label{f=1}$$ for the scalar curvature of the Riemannian product manifold $\left( {\mathcal B}\times {\mathcal F}, \, g_{\rm B} \oplus g_{\rm F} \right)$.
Second, the scalar curvature $R({\mathcal M})$ of the warped product ${\mathcal M}:={\mathcal B} {\times}_f {\mathcal F}$ does not depend on the parameters of the manifold ${\mathcal F}$ if and only if ${\mathcal F}$ has a constant scalar curvature.
Coming back to our current problem, the structure of the metric tensors (\[gMT+\]) and (\[gST+\]) shows that the four-dimensional Riemannian manifolds ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$ and ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$ are warped products: $${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}
={\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2 \right) }
{\times}_{ f_{\rm MT} } \, S^2( \theta, \, \phi);
\label{wMTS}$$ $${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}
={\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2 \right) }
{\times}_{ f_{\rm ST} } \, H_{-1}^2(2r, \, \phi).
\label{wSTS}$$ In view of the isometry (\[dsTR\^2\]), the two-dimensional Riemannian manifold ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2 \right) } \equiv {\mathbb R}_{+}^2$ of the two-mode TSs (\[TS\]) has a vanishing scalar curvature. However, the essential issue is that both $S^2$ and $H_{-1}^2$ are surfaces of constant scalar curvature. Then the second consequence stated above explains why the scalar curvatures $R_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right) }$ and $R_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right) }$ depend only on the mean thermal photon numbers in the incoming modes. Needless to say, this dependence is specific for each of the warped products (\[wMTS\]) and (\[wSTS\]).
Let us check on our formulae (\[RMT\]) and (\[RST\]) by specializing Eqs. (\[R(M)\]) and (\[LB\]) for the warped products (\[wMTS\]) and (\[wSTS\]), respectively. Taking account of Eqs. (\[HMT\]), (\[dsMT+\]), (\[HST\]), (\[dsST+\]), and (\[RSH\]), we find the following pair of equations for the scalar curvatures: $$\begin{aligned}
& R_{\rm MT}{\left( \bar{n}_1, \bar{n}_2 \right) }=\frac{8}{H_{\theta}}
-2\sum_{j=1}^2 \bar{n}_j\left( \bar{n}_j +1 \right) \left\{ 4\frac{ {\partial}^2}{\partial \bar{n}_j^2}
\ln{\left( H_{\theta} \right)} \right. \notag\\
& \left. +3\left[ \frac{\partial}{\partial \bar{n}_j}\ln{\left( H_{\theta} \right)} \right]^2 \right\}
-4\sum_{j=1}^2 \left( 2\bar{n}_j +1 \right) \frac{\partial}{\partial \bar{n}_j}
\ln{\left( H_{\theta} \right)}, \notag\\
& \left( \bar{n}_1 \ne \bar{n}_2 \right);
\label{eqRMT}\end{aligned}$$ $$\begin{aligned}
& R_{\rm ST}{\left( \bar{n}_1, \bar{n}_2 \right) }=-\frac{8}{H_{2r} }
-2\sum_{j=1}^2 \bar{n}_j\left( \bar{n}_j +1 \right) \left\{ 4\frac{ {\partial}^2}{\partial \bar{n}_j^2}
\ln{\left( H_{2r} \right)} \right. \notag\\
& \left. +3\left[ \frac{\partial}{\partial \bar{n}_j}\ln{\left( H_{2r} \right)} \right]^2 \right\}
-4\sum_{j=1}^2 \left( 2\bar{n}_j +1 \right) \frac{\partial}{\partial \bar{n}_j}
\ln{\left( H_{2r} \right)}.
\label{eqRST}\end{aligned}$$ Substitution of the functions $H_{\theta}$ and $H_{2r}$ into the above equations yields the expected formulae (\[RMT\]) and (\[RST\]).
Summary and conclusions
=======================
We start this overview by stressing the main results we have obtained in the present work. First, we have established an alternative expression of the fidelity between two-mode GSs, Eq. (\[2F(K)\]). On the one hand, this is efficient in evaluating the fidelity between special states, as it happens with the two-mode MTSs, Eqs. (\[K(+)M\])- (\[K(-)M\]), and the two-mode STSs, Eqs. (\[K(+)S\])- (\[K(-)S\]). On the other hand, it is flexible enough to have checked with ease, in Appendix C, the inequality ${\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}) \leqq 1$ for both families of special two-mode GSs.
Second, taking advantage of the above-cited formulae, we have derived the Bures infinitesimal geodesic distances on the Riemannian manifolds ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$ of the two-mode MTSs, Eq. (\[dsMT\]), and ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$ of the two-mode STSs, Eq. (\[dsST\]). They are statistically relevant due to the proportionality between the Bures and QFI metric tensors [@BC1994]. This endows the Bures metric with the general feature of statistical distinguishability between neigbouring states on a Riemannian manifold when performing suitable quantum measurements. In addition, the diagonal form of the QFI metric tensors (\[HMT\]) of MTSs and (\[HST\]) of STSs with respect to their natural parameters simplifies the corresponding quantum Cramér-Rao inequalities.
Third, we have employed a standard procedure to evaluate the scalar curvature associated to the Bures metric on each of the Riemannian manifolds ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$ of the two-mode MTSs and ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$ of the two-mode STSs. The formulae (\[RMT\]) and (\[RST\]) are the corresponding exact analytic results. Both scalar curvatures are merely functions of the mean photon numbers in the incident thermal modes, $\bar{n}_1$ and $\bar{n}_2$. In spite of being determined by the interaction of thermal radiation with the optical instruments described previously, neither of them depends on the specific parameters of the optical device involved. This particular property stems from the symmetry nature of the unitary operators (\[M\_12\]) and (\[S\_12\]) describing the optical processes in question: $SU(2)$ and, respectively, $SU(1,1)$. In addition, we have exploited these symmetries to recover the scalar curvatures (\[RMT\]) and (\[RST\]) by an alternative method. Figures 1 and 3 allow one to visualize each of them as a function of the mean photon occupancies of the incoming thermal modes.
In order to reveal the significance of the Bures scalar curvature, we follow closely Petz’s exposition in Ref. [@Petz]. Let us consider an $n$-dimensional Riemannian manifold $({\mathcal M}, g_{\rm B})$ of quantum states, which is equipped with the Bures metric $g_{\rm B}$. Then the geodesic distance $D_{\rm B}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})$ between two points on the manifold ${\mathcal M}$ is interpreted as the statistical distinguishability of the two states by means of the optimal quantum measurement.
We focus on a given state $\hat{\rho}_0 \in ({\mathcal M}, \, g_{\rm B})$. The geodesic ball $$B_n{ \left( \hat{\rho}_0; \, \varepsilon \right) }:=\{\hat{\rho} \in ({\mathcal M}, \, g_{\rm B}):
\;\; D_{\rm B}(\hat{\rho}_0, \hat{\rho})<\varepsilon\}
\label{B_n}$$ contains all the states that can be distinguished from $\hat{\rho}_0$ by an information effort smaller than that corresponding to the radius $\varepsilon >0$. According to Jeffreys’ rule [@Kass], the size of the statistical inference region (\[B\_n\]), which measures the uncertainty in the information acquired about the state $\hat{\rho}_0$, is precisely the Bures volume $ {\mathcal V}_{\rm B}{\left[ B_n{ \left( \hat{\rho}_0; \, \varepsilon \right) }\right] }$. Therefore, this volume can be interpreted as the [*average statistical uncertainty*]{} of the state $\hat{\rho}_0 \in ({\mathcal M}, \,g_{\rm B})$. In order to improve the accuracy in identifying the state $\hat{\rho}_0$, one has to contract the geodesic ball (\[B\_n\]). Ideal asymptotic inference means reducing its radius as much as possible, that is, making $ \varepsilon \to 0$. This asymptotic behaviour is described by the following geometric formula [@GHL]: $$\begin{aligned}
& {\mathcal V}_{\rm B}{ \left[ B_n{ \left( \hat{\rho}_0; \, \varepsilon \right) }\right] }
= V_{n}(1)\, {\varepsilon}^n
-\frac{ V_{n}(1) }{n+2} \, R{ \left( \hat{\rho}_0 \right) }\, {\varepsilon}^{n+2}
+{\rm o}{ \left( {\varepsilon}^{n+2} \right) }: \notag \\
& V_{n}(1)=\frac{ {\pi}^{ \frac{n}{2} } }{\Gamma{ \left( \frac{n}{2}+1 \right) } },
\qquad \left( \varepsilon \ll 1 \right).
\label{VB_n}\end{aligned}$$ In Eq. (\[VB\_n\]), $V_{n}(1)$ is the volume of the unit ball $B_n(0;1)$ in the Euclidean space ${\mathbb R}^n$, while $R{ \left( \hat{\rho}_0 \right) }$ denotes the Bures scalar curvature at the state $\hat{\rho}_0$. What Eq. (\[VB\_n\]) shows us is that, under the condition $\varepsilon \ll 1$, the average statistical uncertainty is fully determined by the scalar curvature $R{ \left( \hat{\rho}_0 \right) }$ and, namely, is a decreasing function of it.
To conclude, we come back to our four-dimensional Riemannian manifolds of special two-mode GSs, ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, \theta, \, \phi \right)}$ and ${\mathcal M} {\left( \bar{n}_1, \, \bar{n}_2, \, 2r, \, \phi \right)}$. Figures 1 and 3, where their scalar curvatures are plotted, offer a global view on the average statistical uncertainty of these noteworthy states. Remark that almost all of them are noisy, except for the pure states, $\left( \bar{n}_1=0, \, \bar{n}_2=0 \right) $, i. e., the vacuum and, respectively, all the two-mode SVSs. The figured values, albeit not completely intuitive, nevertheless display several regularities and provide some interesting comparisons. It is our opinion that these results urge a deeper understanding from a quantum information perspective.
Fidelity between $n$-mode Gaussian states: The inequality ${\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}) \geqq
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} )$
================================================================================================================================
We make a digression intended for a pair of arbitrary $n$-mode Gaussian states, $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$. It is straightforward to extend Eqs. (\[Delta\])- (\[overlap\]) to the multi-mode case [@PT2012]. Since the overlap ${\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} )$ of two GSs never vanishes, we have been lead to introduce a key GS [@PT2012],
$$\begin{aligned}
\hat{\rho}_B:=[{\rm Tr}(\hat{\mathcal B})]^{-1}\hat{\mathcal B}, \qquad
\hat{\mathcal B}:=\sqrt{\hat{\rho}^{\prime \prime}}\hat{\rho}^{\prime}
\sqrt{\hat{\rho}^{\prime \prime}}.
\label{rhoB}\end{aligned}$$
Its CM, denoted ${\mathcal V}_B$, has the symplectic invariants [@PT2012]:
$$\det \left({\mathcal V}_B \right)=2^{-2n}\frac{\Gamma}{\Delta}, \quad
\det \left({\mathcal V}_B +\frac{i}{2}J \right)=2^{-2n}\frac{\Lambda}{\Delta}.
\label{invar}$$
Obviously, the fidelity (\[F\]) of two GSs is proportional to their overlap given by an $n$-mode analogue of Eq. (\[overlap\]) [@PT2012]: $$\begin{aligned}
{\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})
=\left[ {\rm Tr}\left( \sqrt{\hat{\rho}_B} \right) \right]^2
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} )>0.
\label{AB}\end{aligned}$$ Equation (\[AB\]) displays the general inequality
$$\begin{aligned}
{\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime}) \geqq
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} ),
\label{F>O}\end{aligned}$$
as well as its saturation, which is achieved if and only if the state $\hat{\rho}_B$ is pure:
$$\begin{aligned}
{\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} ) \; \iff \;
{\rm Tr}\left[ \left( {\hat{\rho}_B} \right)^2 \right]=1.
\label{F=O}\end{aligned}$$
Owing to the formulae (\[invar\]), the purity condition (\[F=O\]) reads
$${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} ) \; \iff \; \Gamma=\Delta,
\label{G=D}$$
and implies the equation
$$\det \left({\mathcal V}_B +\frac{i}{2}J \right)=0 \; \iff \; {\Lambda}=0.
\label{edge}$$
The necessary condition (\[edge\]) signifies that at least one of the GSs, for instance $\hat{\rho}^{\prime}$, is at the physicality edge: $\det \left({\mathcal V}^{\prime}+\frac{i}{2}J \right)=0$. Specifically, in the single-mode case $(n=1)$, the corresponding state $\hat{\rho}^{\prime}$ is pure.
Conversely, if one of the above $n$-mode GSs, say $\hat{\rho}^{\prime}$, is pure, then so is the GS $\hat{\rho}_B$, Eq. (\[rhoB\]). Indeed, the required equality $\Gamma=\Delta$ is a consequence of the assumed purity conditions $\det \left({\mathcal V}^{\prime} \right)=2^{-2n}$ and ${\mathcal V}^{\prime}=-\frac{1}{4}J \left({\mathcal V}^{\prime} \right)^{-1}J$ [@PT2012]. As shown in Sec. II, this sufficient condition for the saturation property ${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} )$ is a general one. We stress that, for single-mode GSs, it is both necessary and sufficient.
However, the one-mode case can readily be handled by making direct use of the explicit fidelity formula which is available for a long time [@HS1998]:
$$\begin{aligned}
{\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime})
& =\left( \sqrt{\Delta+\Lambda}-\sqrt{\Lambda} \right)^{-1} \notag \\
& \times \exp{\left[-\frac{1}{2}\left(\delta v \right)^T
\left({\mathcal V}^{\prime}+{\mathcal V}^{\prime\prime}\right)^{-1}
\delta v \right]}.
\label{F1}\end{aligned}$$
Equation (\[AB\]) has the specific form
$$\begin{aligned}
{\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})
=\left( \sqrt{1+\frac{\Lambda}{\Delta}} +\sqrt{\frac{\Lambda}{\Delta}}\right)
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} )>0.
\label{fo1}\end{aligned}$$
Accordingly, the general inequality (\[F>O\]) is manifest and so is the saturation condition
$${\cal F}(\hat{\rho}^{\prime}, \hat{\rho}^{\prime \prime})=
{\rm Tr}({\hat{\rho}}^{\prime} {\hat{\rho}}^{\prime \prime} ) \; \iff \; \Lambda =0,
\label{pure1}$$
meaning that at least one of the single-mode GSs $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$ is pure.
Fidelity between thermal states: The inequality ${\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}) \leqq 1$
========================================================================================================================
Let us introduce the positive function $$\begin{aligned}
& {\cal Q}(x,y):=\sqrt{(x+1)(y+1)} -\sqrt{xy}\, , \notag \\
& \left( x \geqq 0, \;\; y \geqq 0 \right).
\label{Q}\end{aligned}$$ Remark that the equivalent inequalities $${\left( \sqrt{x}-\sqrt{y} \right)}^2 \geqq 0 \; \iff \; {\cal Q}(x,y) \geqq 1
\label{Q>1}$$ become saturate if and only if $x=y$: $${\cal Q}(x,y)=1 \; \iff \; x=y.
\label{Q=1}$$
Single-mode thermal states
--------------------------
A single-mode TS, $\hat{\rho}_{\rm T}(\bar{n})$, is an unshifted GS whose explicit expression is written in Eq. (\[TS\]). Recall that its $2 \times 2$ CM is a multiple of the identity: $${\mathcal V}_{\rm T}(\bar{n})= \left( \bar{n}+\frac{1}{2} \right) {\sigma}_0.
\label{VT1}$$ The fidelity (\[F1\]) of a pair of one-mode TSs, $\hat{\rho}_{\rm T}(\bar{n}^{\prime}) $ and $\hat{\rho}_{\rm T}(\bar{n}^{\prime \prime})$, is therefore
$${\cal F}{\left[ \hat{\rho}_{\rm T}(\bar{n}^{\prime}), \, \hat{\rho}_{\rm T}(\bar{n}^{\prime \prime}) \right] }
=\left( \sqrt{\Delta+\Lambda}-\sqrt{\Lambda} \right)^{-1}
\label{FTS1}$$
with the determinants: $$\Delta =\left( \bar{n}^{\prime}+\bar{n}^{\prime \prime}+1 \right)^2, \quad
\Lambda=4 \bar{n}^{\prime}(\bar{n}^{\prime}+1)\, \bar{n}^{\prime \prime}(\bar{n}^{\prime \prime}+1).
\label{DL}$$ From Eqs. (\[DL\]) and (\[Q\]) we get the identity $$\sqrt{\Delta+\Lambda}-\sqrt{\Lambda}={\left[ {\cal Q}{ \left( \bar{n}^{\prime},
\bar{n}^{\prime \prime} \right) } \right] }^2
\label{DLQ}$$ and the fidelity (\[FTS1\]) reads thereby explicitly: $${\cal F}{\left[ \hat{\rho}_{\rm T}(\bar{n}^{\prime}), \, \hat{\rho}_{\rm T}(\bar{n}^{\prime \prime}) \right] }
=\left[ {\cal Q}{ \left( \bar{n}^{\prime}, \bar{n}^{\prime \prime} \right) } \right]^{-2}.
\label{FT1}$$ Accordingly, Eq. (\[Q>1\]) displays the inequality $${\cal F}{\left[ \hat{\rho}_{\rm T}(\bar{n}^{\prime}), \, \hat{\rho}_{\rm T}(\bar{n}^{\prime \prime}) \right] }
\leqq 1,
\label{FT1<1}$$ while Eq. (\[Q=1\]) ascertains its saturation: $${\cal F}{\left[ \hat{\rho}_{\rm T}(\bar{n}^{\prime}), \, \hat{\rho}_{\rm T}(\bar{n}^{\prime \prime}) \right] }=1
\; \iff \; \bar{n}^{\prime}=\bar{n}^{\prime \prime}.
\label{FT1=1}$$
Two-mode thermal states
-----------------------
Let $K_{\pm}(\{\bar{n}\})$ designate the functions (\[Kpm\]) for a pair of two-mode TSs. They are limit cases of the similar functions for both corresponding pairs of MTSs and STSs. In order to write them, it is sufficient to set either ${\theta}^{\prime}={\theta}^{\prime \prime}, \,
{\phi}^{\prime}={\phi}^{\prime \prime}$ in Eq. (\[K(-)M\]) or $r^{\prime}=r^{\prime \prime}, \, {\phi}^{\prime}={\phi}^{\prime \prime}$ in Eq. (\[K(+)S\]): $$\begin{aligned}
& K_{+}(\{\bar{n}\})=2\left\{ \left( \bar{n}_1^{\prime} \, \bar{n}_2^{\prime} \,
\bar{n}_1^{\prime \prime} \, \bar{n}_2^{\prime \prime} \, \right)^{\frac{1}{2}} \right. \notag \\
& \left. +\left[ \left( \bar{n}_1^{\prime}+1 \right) \left( \bar{n}_2^{\prime}+1 \right)
\left( \bar{n}_1^{\prime \prime}+1 \right) \left( \bar{n}_2^{\prime \prime}+1 \right) \right]^{\frac{1}{2}}
\right\}^2.
\label{K(+)T}\end{aligned}$$ $$\begin{aligned}
& K_{-}(\{\bar{n}\})=2\left\{ \left[ \bar{n}_1^{\prime} \left( \bar{n}_2^{\prime} +1 \right)
\bar{n}_1^{\prime \prime} \left( \bar{n}_2^{\prime \prime} +1 \right) \right]^{\frac{1}{2}} \right. \notag \\
& \left. +\left[ \left( \bar{n}_1^{\prime}+1 \right) \bar{n}_2^{\prime}
\left( \bar{n}_1^{\prime \prime}+1 \right) \bar{n}_2^{\prime \prime} \right]^{\frac{1}{2}}
\right\}^2.
\label{K(-)T}\end{aligned}$$ The difference between the square roots of the above functions factors as follows: $$\begin{aligned}
& \sqrt{ K_{+}(\{\bar{n}\}) }-\sqrt{ K_{-}(\{\bar{n}\}) } \notag \\
& =\sqrt{2}\, {\cal Q}{ \left( \bar{n}_1^{\prime}, \bar{n}_1^{\prime \prime} \right) }
\, {\cal Q}{ \left( \bar{n}_2^{\prime}, \bar{n}_2^{\prime \prime} \right) }.
\label{dif}\end{aligned}$$ Substitution of Eq. (\[dif\]) into Eq. (\[F2(K)\]) gives the fidelity of a pair of two-mode TSs: $$\begin{aligned}
& {\cal F}{\left[ \hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \right) }, \,
\hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \right) }\right] } \notag \\
& =\left[ {\cal Q}{ \left( \bar{n}_1^{\prime}, \bar{n}_1^{\prime \prime} \right) } \,
{\cal Q}{ \left( \bar{n}_2^{\prime}, \bar{n}_2^{\prime \prime} \right) } \right]^{-2}.
\label{FT2}\end{aligned}$$ Equation (\[Q>1\]) confirms therefore the inequality $${\cal F}{\left[ \hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \right) }, \,
\hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \right) }\right] } \leqq 1,
\label{FT2<1}$$ which saturates as stated by Eq. (\[Q=1\]): $$\begin{aligned}
& {\cal F}{\left[ \hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \right), } \,
\hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \right) } \right] }=1
\; \iff \; \bar{n}_j^{\prime}=\bar{n}_j^{\prime \prime}, \notag \\
& (j=1,2).
\label{FT2=1}\end{aligned}$$
Taking account of the formula (\[FT1\]) for one-mode TSs, the structure (\[FT2\]) of the fidelity between two-mode TSs (\[TS\]) checks the multiplicativity of fidelity in this particular case. The reason for which the formulae (\[FT2\])- (\[FT2=1\]) can be extended to $n$-mode TSs, regardless of the number of modes, is precisely the above-mentioned multiplication rule.
Fidelity of special two-mode Gaussian states: The inequality ${\cal F}({\hat{\rho}}^{\prime}, {\hat{\rho}}^{\prime \prime}) \leqq 1$
=====================================================================================================================================
In addition to a pair of special two-mode GSs of the same kind, $\hat{\rho}^{\prime}$ and $\hat{\rho}^{\prime \prime}$, we envisage the pair of two-mode TSs, $\hat{\rho}_{\rm T}^{\prime}$ and $\hat{\rho}_{\rm T}^{\prime \prime}$, with the same mean thermal photon occupancies: $\{ \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \}$ and $\{ \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \}$, respectively.
Mode-mixed thermal states
-------------------------
We define the MTSs $\hat{\rho}_{\rm \, MT}^{\prime}$ and $\hat{\rho}_{\rm \, MT}^{\prime \prime}$ by the sets of their usual parameters, $\{ \bar{n}_1^{\prime}, \, \bar{n}_2^{\prime}, \, {\theta}^{\prime},
\, {\phi}^{\prime}\} \,$ and $\{ \bar{n}_1^{\prime \prime}, \, \bar{n}_2^{\prime \prime}, \, {\theta}^{\prime \prime}, \,
{\phi}^{\prime \prime}\} \,$, respectively. Inspection of Eqs. (\[K(+)M\])- (\[K(-)M\]) and (\[K(+)T\])- (\[K(-)T\]) provides the identities: $$\begin{aligned}
& K_{+}=K_{+}(\{\bar{n}\}), \qquad K_{-}=K_{-}(\{\bar{n}\}) \notag \\
& -\left( \bar{n}_1^{\prime}-\bar{n}_2^{\prime} \right)\, \left( \bar{n}_1^{\prime \prime}
-\bar{n}_2^{\prime \prime} \right) \left\{ 1-\cos{\left( {\theta}^{\prime}-{\theta}^{\prime \prime} \right) }\right. \notag \\
& \left. +\sin{\left( {\theta}^{\prime} \right) } \sin{\left( {\theta}^{\prime \prime} \right) }
\left[ 1-\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right] \right\} .
\label{KMT}\end{aligned}$$ The emerging inequality $$\sqrt{K_{+}}-\sqrt{K_{-}} \geqq \sqrt{ K_{+}(\{\bar{n}\})}-\sqrt{K_{-}(\{\bar{n}\}) }
\label{KMT1}$$ generates via Eq. (\[F2(K)\]) an inequality for fidelities, $${\cal F}{\left( \hat{\rho}_{\rm \, MT}^{\prime}, \, \hat{\rho}_{\rm \, MT}^{\prime \prime} \right) } \leqq
{\cal F}{\left[ \hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \right) }, \,
\hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \right) }\right] },
\label{FMT<FT}$$ with the saturation condition $$\begin{aligned}
& {\cal F}{\left( \hat{\rho}_{\rm \, MT}^{\prime}, \, \hat{\rho}_{\rm \, MT}^{\prime \prime} \right) }
={\cal F}{\left[ \hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \right) },
\hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \right) }\right] } \notag \\
& \; \iff \; {\theta}^{\prime}={\theta}^{\prime \prime}, \,
{\phi}^{\prime}={\phi}^{\prime \prime}.
\label{FMT=FT}\end{aligned}$$ By use of Eqs. (\[FMT<FT\])- (\[FMT=FT\]) and (\[FT2<1\])- (\[FT2=1\]), we get the expected property (\[F<1\]) for MTSs, that is, the inequality $${\cal F}{\left( \hat{\rho}_{\rm \, MT}^{\prime}, \, \hat{\rho}_{\rm \, MT}^{\prime \prime} \right) } \leqq 1
\label{FMT<1}$$ and its saturation condition as well: $${\cal F}{\left( \hat{\rho}_{\rm \, MT}^{\prime}, \, \hat{\rho}_{\rm \, MT}^{\prime \prime} \right) }=1
\; \iff \; \hat{\rho}_{\rm \, MT}^{\prime}=\hat{\rho}_{\rm \, MT}^{\prime \prime}.
\label{FMT=1}$$
Squeezed thermal states
-----------------------
Let $\{ \bar{n}_1^{\prime}, \, \bar{n}_2^{\prime}, \, r^{\prime}, \, {\phi}^{\prime} \} \,$ and $\{ \bar{n}_1^{\prime \prime}, \, \bar{n}_2^{\prime \prime}, \, r^{\prime \prime}, \, {\phi}^{\prime \prime} \} \,$ be the parameters of the STSs $\hat{\rho}_{\rm \, ST}^{\prime}$ and $\hat{\rho}_{\rm \, ST}^{\prime \prime}$, respectively. By looking at Eqs. (\[K(+)S\])- (\[K(-)S\]) and (\[K(+)T\])- (\[K(-)T\]), we get the formulae: $$\begin{aligned}
& K_{+}=K_{+}(\{\bar{n}\}) \notag \\
& +\left( \bar{n}_1^{\prime}+\bar{n}_2^{\prime}+1 \right)
\left( \bar{n}_1^{\prime \prime}+\bar{n}_2^{\prime \prime}+1 \right)
\left\{ \cosh{\left[ 2\left( r^{\prime}-r^{\prime \prime} \right) \right] -1}\right. \notag \\
& \left. +\sinh{\left( 2r^{\prime} \right) } \sinh{\left( 2r^{\prime \prime} \right) }
\left[ 1-\cos{\left( {\phi}^{\prime}-{\phi}^{\prime \prime} \right) }\right] \right\} , \notag \\
& K_{-}=K_{-}(\{\bar{n}\})
\label{KST}\end{aligned}$$ The obvious inequality $$\sqrt{K_{+}}-\sqrt{K_{-}} \geqq \sqrt{ K_{+}(\{\bar{n}\})}-\sqrt{K_{-}(\{\bar{n}\}) }
\label{KST1}$$ gives rise, via Eq. (\[F2(K)\]), to an inequality for fidelities, $${\cal F}{\left( \hat{\rho}_{\rm \, ST}^{\prime}, \, \hat{\rho}_{\rm \, ST}^{\prime \prime} \right) } \leqq
{\cal F}{\left[ \hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \right) }, \,
\hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \right) }\right] },
\label{FST<FT}$$ with the saturation case $$\begin{aligned}
& {\cal F}{\left( \hat{\rho}_{\rm \, ST}^{\prime}, \, \hat{\rho}_{\rm \, ST}^{\prime \prime} \right) }
={\cal F}{\left[ \hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime}, \bar{n}_2^{\prime} \right) },
\hat{\rho}_{\rm T}{ \left( \bar{n}_1^{\prime \prime}, \bar{n}_2^{\prime \prime} \right) }\right] } \notag \\
& \; \iff \; r^{\prime}=r^{\prime \prime}, \, {\phi}^{\prime}={\phi}^{\prime \prime}.
\label{FST=FT}\end{aligned}$$ By making combined use of Eqs. (\[FST<FT\])- (\[FST=FT\]) together with Eqs. (\[FT2<1\])- (\[FT2=1\]), we get the expected property (\[F<1\]) for STSs, consisting of the inequality $${\cal F}{\left( \hat{\rho}_{\rm \, ST}^{\prime}, \, \hat{\rho}_{\rm \, ST}^{\prime \prime} \right) } \leqq 1
\label{FST<1}$$ and its saturation case: $${\cal F}{\left( \hat{\rho}_{\rm \, ST}^{\prime}, \, \hat{\rho}_{\rm \, ST}^{\prime \prime} \right) }=1
\; \iff \; \hat{\rho}_{\rm \, ST}^{\prime}=\hat{\rho}_{\rm \, ST}^{\prime \prime}.
\label{FST=1}$$
[**Acknowledgments**]{} This work was supported by the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, through Project PN-II-ID-PCE-2011-3-1012 for the University of Bucharest.
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|
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abstract: |
Interband absorption and luminescence of quasi-two-dimensional, circularly symmetric, $N_{e}$-electron quantum dots are studied at high magnetic fields, $8\leq B\leq 60$ T, and low temperatures, $T\ll 2$ K. In the $%
N_{e}=0 $ and 1 dots, the initial and final states of such processes are fixed, and thus the dependence on $B$ of peak intensities is monotonic. For larger systems, ground state rearrangements with varying magnetic field lead to substantial modifications of the absorption and luminescence spectra. Collective effects are seen in the $N_{e}=2$ and 3 dots at “filling fractions” 1/2, 1/3 and 1/5.
address:
- |
Centro de Matemáticas y Física Teórica, Calle E No. 309, Ciudad Habana, Cuba and\
Departamento de Física, Universidad de Antioquia, AA 1226, Medellin, Colombia\
and
- 'Instituto de Materiales y Reactivos, Universidad de La Habana San Lázaro y L, Vedado 10400, Ciudad Habana, Cuba'
author:
- 'Augusto Gonzalez[@augusto]'
- 'Eduardo Menéndez-Proupin[@eduardo]'
title: Interband absorption and luminescence in small quantum dots under strong magnetic fields
---
Introduction
============
The quasi-two-dimensional electron gas in a high magnetic field is a strongly correlated system exhibiting very complicated dynamics. At special values of the filling factor, the essential features of the ground state are captured by the Laughlin wave function [@Laughlin], or its composite fermion generalization[@CF]. The low-lying excitations can be described in the single-mode approximation of Girvin [*et al*]{}, [@GMP; @Chakraborty] or in the composite fermion picture [@CF; @CF2].
Many experiments have been designed and carried out in order to test the excitation spectrum of this highly correlated system. Inelastic (Raman) light scattering experiments have tested basically the excitation gap at wavevector ${\bf k}=0$. [@Raman] Spin-flipped states and the magnetoroton minimum at $k\approx 1/l_{B}$ ($l_{B}$ is the magnetic length) have also been observed , although they should be activated by impurities or other mechanism to produce a trace in the Raman spectra. Evidence of the magnetoroton minimum comes also from the absorption of ballistic acoustic phonons[@fonones].
On the other hand, experiments on photoluminescence (PL) related to interband electronic transitions around filling factor $\nu =1$ have tested the excited states with an additional electron-hole (e-h) pair[@nu1]. Recently, the observations have been extended to lower filling factors by increasing the magnetic field up to 60 T. [@Xminus1; @Xminus] The related theory is not in complete agreement with the experiment. In the infinite magnetic field limit, it was predicted that only the exciton ($X^{0}$) and the negatively charged triplet exciton ($X_{t}^{-}$) are bound [@PYM96]. The latter is expected to be dark in luminescence [@PYM96] as a result of a hidden symmetry related to magnetic translations [@LL81]. In the experiments, however, very distinct singlet and triplet peaks ($X_{s}^{-}$ and $X_{t}^{-}$) are observed. A realistic calculation of ground state energies was presented in Ref. , where Landau level and quantum well (qwell) sub-band mixing were taken into account. The $%
X_{t}^{-}$ peak position was reproduced, but in theory this state is dark. The problem was recently revisited by Wojs [*et al*]{}[@Wojs], who showed that in a narrow (10 nm wide) well a second bright $X_{t}^{-}$ state becomes bound, thus interpreting the observed luminescence as coming from the bright state. We shall notice that both Refs. and deal with isolated three-particle systems, and thus are not able to describe the filling factor dependence of observed magnitudes for $\nu \geq 1/5$.
In the present paper, we study small quantum dots (qdots) under conditions similar to the experiments reported in Refs. , [*i.e.*]{} quasi-two-dimensional motion, magnetic fields in the interval $8$ T$\leq B\leq 60$ T, and temperatures well below 2 K. The laser excitation power is assumed to be low (a few mW/cm$^{2}$), thus the dot works under a linear regime. The lateral confinement is modelled by a harmonic potential. Energy levels, charge densities and dipole matrix elements for absorption and luminescence are computed by exact diagonalization in the first Landau level (1LL) approximation.
Absorption or PL experiments on electron-hole qdots under very high magnetic fields are lacking. To the best of our knowledge, there is only one experiment [@CRL99], in which the luminescence at higher (4 K) temperature and $B\leq 45$ T is measured in order to estimate the e-h correlation energy.
Breaking of the magnetic translation symmetry by a lateral confinement in a qdot makes the lowest $X^-$ triplet state bright. Highly nontrivial PL and absorption spectra arise even in the 1LL approximation. These spectra contain information about the energy levels and particle correlations in the system. Let us stress that a calculation of $X^0$ and $X^-$ energy levels of in a qdot, which includes LL mixing, is available [@WH95]. The absorption coefficient is also reported in that paper. The differences with our work are the following. First, we consider both absorption and PL. Second, we trace the changes in the ground-state (g.s.) wave function and charge rearrangements as the magnetic field is varied. Finally, we consider larger qdots with $X^{2-}$ and $X^{3-}$ complexes (unbound in a qwell). It will be seen below that indications of collective effects are evident even in these relatively small systems.
The plan of the paper is as follows. The model and certain general statements are explained in section II. The next section presents results for particular systems. We start with the exciton and end up with the $X^{3-} $ complex. Finally, a few concluding remarks are given.
The model
=========
We consider the two-dimensional motion of $N_e$ electrons and $N_h$ holes in an external parabolic potential and a perpendicular magnetic field (along the $z$ axis). In particular, we will study the $N_h=1$ and 2 systems, which are the ones participating in interband absorption and recombination processes. The unit of length is $\sqrt{2}$ times the magnetic length. In the 1LL approximation, the Hamiltonian is written as
$$\begin{aligned}
H(N_e,N_h)&=&\left (\frac{\hbar\omega_c^e}{2}+E_z^e\right )N_e+ \left (\frac{%
\hbar\omega_c^h}{2}+E_z^h+E_{gap}\right )N_h \nonumber \\
&+& E_{Zeeman}^e+E_{Zeeman}^h+V_{conf}+V_{coul}. \label{hamiltoniano}\end{aligned}$$
The Hamiltonian (\[hamiltoniano\]) is intended to model a GaAs qdot with a thickness of 20 nm in the $z$-direction. The meaning of the different terms entering $H$ is evident. The specific qdot characteristics are reflected in the confinement energies along the $z$-direction, $E_{z}$, the in-plane confinement potential, $V_{conf}=\sum_{i}v_{conf}(r_{i})$, and the $z$-averaged Coulomb interactions, $V_{coul}=\sum_{i,j}v_{coul}(r_{ij})$. We will use the expression
$$v_{coul}(r)=\pm 3.316~\beta~\sqrt{B}~\frac{1}{r}~[{\rm meV}],$$
for the pair Coulomb interactions ($B$ in Teslas), and
$$v_{conf}(r)=\frac{3.316}{B}~\beta~K~r^2~[{\rm meV}],$$
for the one-particle confinement potential. Even these simple expressions lead to very interesting physical results. Notice that $1/\sqrt{2}$ times the characteristic Coulomb energy, $e^2 /(\kappa l_B)$, equals 3.316 $\sqrt{B}$ in our units. The constant $\beta =0.6$ is used to simulate the $z$-averaging of the Coulomb interactions in the 20 nm - wide qdot [@Chakraborty; @MA84]. We fixed it by requiring the binding energy of the unconfined ($v_{conf}$ set to zero) $X_{t}^{-}$ relative to the $X^{0}$ to be 0.6 meV at $B=17$ T. This is a representative value[@Xminus]. On the other hand, the dimensionless constant $K$ will be fixed to 7.0 in order to obtain a “filling factor” around 1/3 for $B\approx 30$ T, also a common situation met in the experiments [@Xminus].
The only nontrivial terms entering (\[hamiltoniano\]) are $V_{conf}$ and $V_{coul}$. They should be diagonalized in a basis of Slater 1LL functions. The energies coming from the diagonalization processes will be denoted $\epsilon$, and the wave functions will be used to compute physical observables. Note that, in the 1LL, the electron (hole) angular momentum is a non-positive (non-negative) number. Thus, the total angular momentum is written $M=M_e+M_h=-|M_e|+M_h$.
In a GaAs electron system, the validity of the 1LL approximation can be stated by comparing the excitation energy to the 2LL, $\hbar \omega
_{c}^{e}=1.728~B$ meV, with the Coulomb energy, $3.316~\beta ~\sqrt{B}$ meV. Thus, for $B\gg 1$ T the 1LL approximation works. Spin excitations are lower in energy, $\Delta E_{Zeeman}\approx 0.025~B$ meV. However, at temperatures below 2 K and for $B>8$ T, they can not be thermally excited. It means that in both absorption and luminescence the transition starts from the lowest optically active state. When holes are created, the 1LL approximation becomes valid at higher fields. If we take for the heavy hole mass in the $xy$ plane the value $\mu_{h}=0.11~m_{0}$[@mh], then $\hbar
\omega _{c}^{h}\approx 1B$ meV. The 1LL approximation works for $B\gg 4$ T. Below, we present results obtained in the 1LL approximation for $8$ T$\leq
B\leq 60$ T.
On the other hand, the expression (\[hamiltoniano\]) assumes that the particles are sitting on the first qwell sub-band. As it was stressed in Ref. , this may be a rough approximation. For a 20 nm qwell, the second electronic sub-band is around 30 meV higher, but the second hole sub-band is only 6 meV higher (a heavy hole mass $\mu_{h}^{z}\approx 0.38~m_{0}$ is assumed). Our first sub-band approximation is qualitatively valid in the present situation, and will improve for narrower wells.
Interband absorption and luminescence (general grounds)
-------------------------------------------------------
Interband absorption and luminescence will be studied at temperatures $T\ll 2$ K, [*i.e.*]{} typically lower than spin excitation gaps. Thus, the processes proceed from a unique initial state, which is the g.s. of the polarized $(N_{e},0)$ system in absorption, and the lowest optically active state of the $(N_{e}+1,1)$ system in emission. In general, these processes take place in different angular momentum channels. For absorption, the incident light is supposed to be circularly polarized and propagating along the $z$-direction. Also circularly polarized light is supposed to be measured from the qdot luminescence.
A simple two-band model, with bands split by the Zeeman energy, will be used. The conduction-band ($m_{s}=\pm 1/2$) mass is $\mu_{e}=0.067~m_{0}$, and the heavy hole band, $m_{j}=\pm 3/2$, shows anisotropic effective masses, $\mu_{h}=\mu_{h}^{xy}=0.11~m_{0}$, $\mu_{h}^{z}=0.38~m_{0}$. LL mixing in the $m_{j}=3/2$ branch [@RLZ59] will be neglected. $m_{j}=-3/2$ will be called the spin-up hole branch, and $m_{j}=3/2$ – the spin-down branch. For propagation along the $z$ axis, the allowed transitions are $m_{j}=-3/2\rightarrow m_{s}=-1/2$ for right-handed circular polarization (RHCP), and $m_{j}=3/2\rightarrow m_{s}=1/2$ for left-handed circular polarization (LHCP) [@RLZ59; @SPRS95].
The dipole approximation is used for the interaction Hamiltonian, [*i.e.*]{} $-{\cal E} \cdot {\bf D}$. In the 1LL, the interband dipole operator takes the form
$${\bf D}=\frac{e{\bf p}_{cv}}{m_{0}\omega }\sum\limits_{l\geq 0}\left(
e_{-l,\downarrow }^{\dagger }h_{l,\uparrow }^{\dagger }+e_{-l,\uparrow
}^{\dagger }h_{l,\downarrow }^{\dagger }\right) +H.C., \label{hint}$$
where ${\bf p}_{cv}$ is the GaAs interband constant. The reason for not including the light hole in (\[hint\]) is twofold. First, $E_{z}$ is around 6 meV higher ($\mu_{lh}^{z}\approx 0.09~m_{0}$), thus its absorption or luminescence lines are shifted. Second, the constant ${\bf p}_{cv}^{2}$ is three times smaller for light holes. Notice that the interaction Hamiltonian preserves total angular momentum.
In our $(N_e,N_h)$ systems with $N_h=0,~1$, the states may be classified according to the symmetry of the electronic subsystem. For example, the $N_e=2$ system may be in a spatially antisymmetric (triplet) state, or in a spatially symmetric (singlet) state. We will present calculations only for spatially antisymmetric states. They are the only ones appearing in LHCP, and the ones associated to the most intense lines in RHCP [@WH95]. The wave functions may be written as $\psi=\phi_{coord}^{antisymm}
\chi_{spin}^{symm}$, or in a second quantization formalism,
$$\left| \psi (N_{e},0)\right\rangle =\sum C_{l_{1}l_{2}\dots
l_{N_{e}}}e_{-l_{1},\uparrow }^{\dagger }e_{-l_{2},\uparrow }^{\dagger
}\dots e_{-l_{N_{e}},\uparrow }^{\dagger }|0\rangle ,$$
$$\begin{aligned}
&&\left| \psi _{LHCP}(N_{e}+1,1)\right\rangle = \nonumber \\
&&\sum C_{l_{1}l_{2}\dots l_{N_{e}+1},l_{h}}e_{-l_{1},\uparrow }^{\dagger
}e_{-l_{2},\uparrow }^{\dagger }\dots e_{-l_{N_{e}+1},\uparrow }^{\dagger
}h_{l_{h},\downarrow }^{\dagger }|0\rangle ,\end{aligned}$$
$$\begin{aligned}
&&\left| \psi _{RHCP}(N_{e}+1,1)\right\rangle =\frac{1}{\sqrt{N_{e}}}\sum
C_{l_{1}l_{2}\dots l_{N_{e}+1},l_{h}} \nonumber \\
&&\times \left( e_{-l_{1},\downarrow }^{\dagger }e_{-l_{2},\uparrow
}^{\dagger }\dots e_{-l_{N_{e}+1},\uparrow }^{\dagger }\right.\nonumber \\
&&+e_{-l_{1},\uparrow }^{\dagger }e_{-l_{2},\downarrow }^{\dagger }\dots
e_{-l_{N_{e}+1},\uparrow }^{\dagger } \nonumber \\
&&+\dots +\left. e_{-l_{1},\uparrow }^{\dagger }e_{-l_{2},\uparrow
}^{\dagger }\dots e_{-l_{N_{e}+1},\downarrow }^{\dagger }\right)
h_{l_{h},\uparrow }^{\dagger }|0\rangle .\end{aligned}$$
$\psi _{LHCP}$ corresponds to a spin-polarized electronic subsystem, and $\psi _{RHCP}$ to a not completely polarized state. In the pure electron system, the sum runs over angular momentum states obeying $0\leq l_{1}<l_{2}<\dots <\l _{N_{e}}$ and fixed $M=-l_{1}-l_{2}-\dots
-l_{N_{e}}$. In the one-hole system, the total angular momentum $M=-l_{1}-l_{2}-\dots -l_{N_{e}+1}+l_{h}$ is fixed.
Diagonalization of $V_{conf}+V_{coul}$ in (\[hamiltoniano\]) leads to the determination of eigenenergies and wave functions. Transition energies, transition probabilities and charge densities of the relevant states are computed from these results. The transition energies are given by
$$\begin{aligned}
\hbar\omega&=&E_{gap}+E_z^e+E_z^h+\frac{\hbar\omega_c^e}{2}+
\frac{\hbar\omega_c^h}{2} \nonumber \\
&+& E_{Zeeman}^e+E_{Zeeman}^h+ \epsilon (N_e+1,1)-\epsilon (N_e,0),
\label{homega}\end{aligned}$$
where $\epsilon $ are the energies coming from $V_{conf}+V_{coul}$. We took the values $E_{gap}=1510$, $E_{z}^{e}=11$, $E_{z}^{h}=2$, $\hbar \omega _{c}^{e}/2=0.864~B$, $\hbar \omega _{c}^{h}/2=0.526~B$, $E_{Zeeman}^{e}=-0.025~m_{s}^{e}~B$, $E_{Zeeman}^{h}=-0.016~m_{s}^{h}~B$, for the quantities entering (\[homega\]), where energies are given in meV and $B$ in Teslas. Our treatment of Zeeman energies of both electrons and holes is very simple. We used the value $g_{e}=-0.44$ for the electron Land[é]{} factor and extracted the hole energy from the observed splitting of $X^{0}$ luminescence lines in RHCP and LHCP [@Xminus]. The hole spin projection is conventionally written as $m_{s}^{h}=\pm 1/2$. Actually, the Zeeman energy shows a nonlinear dependence on $B$. [@SPW97] Notice, however, that $E_{gap}$, $\hbar \omega _{c}$ and $E_{Zeeman}$ are important in determining the absolute position of a given absorption or PL line, but not its relative position with respect to $X^{0}$ in the same polarization.
The absorption coefficient of a dot is given by
$$\alpha (\omega )=\frac{4\pi ^{2}\omega }{\hbar cV}\sum_{f}|\langle f|
{\bf e}\cdot {\bf D}|i\rangle |^{2}\delta (\omega -\omega _{fi}),$$
where $|i\rangle $ is the g.s. of the $(N_{e},0)$ system, $f$ are the states of the $(N_{e}+1,1)$ system in the same angular momentum tower and $\hbar \omega _{fi}$ is their energy difference computed from (\[homega\]). ${\bf e}$ is the light polarization vector, $c$ – the light velocity, and $V$ is the volume of absorption. We have used a phenomenological width, $\Gamma =0.8$ meV, to replace the delta function by a Lorentzian
$$\delta (x) \to \frac{\Gamma/\pi}{\Gamma^2+x^2}.$$
In luminescence, we compute the matrix elements $|\langle f|{\bf e}\cdot
{\bf D}|i\rangle |^{2}$, assuming that $|i\rangle $ is the lowest state of the $N_{h}=1$ system.
Results
=======
We present results in the following interval of magnetic field values, 8 T $\leq B\leq 60$ T. Computations are carried out for spin polarized electronic systems, with total spin $M_{s}^{e}=N_{e}/2$, which contribute to the LHCP spectra. The energies of the incompletely polarized states with $M_{s}^{e}=N_{e}/2-1$, entering the RHCP spectra, are obtained by adding the corresponding Zeeman shifts.
Binding energies of excitonic complexes
---------------------------------------
We draw in Fig. \[fig1\] the g.s. energies, $\epsilon$, coming from the diagonalization of $V_{conf}+V_{coul}$ in (\[hamiltoniano\]) as a function of the applied magnetic field. The polarized systems $(N_{e}+1,N_{h})$= (1,1), (2,1), (3,1) and (4,1) are shown. The common notation for the excitonic systems (1,1) and (2,1) are $X^{0}$ and $X^{-}$, so that the charged complex (4,1) may be denoted $X^{3-}$. Note that the slopes of the (2,1), (3,1) and (4,1) curves are very similar. It means that the relative binding energies vary smoothly with $B$, and that the magnetic moments of these states take almost the same values. For example, $X^{3-}$ is 14.77 meV above $X^{-}$ at $B=30$ T, and 14.29 meV above $X^{-}$ at $B=50$ T.
The total angular momenta in the g.s. is a constant, independent of $B$, in the smallest systems. It is $M_{gs}=0$ in the exciton, and $M_{gs}=-1$ in the triplet $X^-$ at any $B$. The larger systems, however, undergo abrupt rearrangements at particular $B$ values. The interplay between g.s. rearrangements in the $(N_e,0)$ and $(N_e+1,1)$ systems as $B$ is varied has direct consequences on absorption and luminescence, as will be seen below.
Note that, unlike pure electron systems, when holes are present the Hilbert space in a given $M=-|M_{e}|+M_{h}$ sector is not finite. We enlarged the included subspace until convergence is reached. For example, in the (4,1) system at $B=40$ T, 2374 many-particle states ([*i.e.*]{} all of the states in $15\leq |M_{e}|\leq 35$) are enough to reach convergence for the lowest energy eigenvalue in the $M=-15$ tower.
The low-lying energy levels of $X^{3-}$ at $B=35$ T are shown in Fig. \[fig2\] as an example. Energy distances between the lowest adjacent levels are around 0.5 meV, the same as in the three-electron system at this value of the magnetic field.
Interband absorption
--------------------
As previously stated, temperatures are low enough for absorption to proceed from the g.s. of the $N_{e}$-electron system. It means that spin flips should not be thermally induced, [*i.e.*]{} $T\ll 2$ K for $B>8$ T.
We show in Fig. \[fig3\] the absorption coefficient for the $N_{e}=0$ qdot at $B=40$ T. The process under consideration, $(0,0)\rightarrow (1,1)$, goes through the $M=0$ channel. The main properties of the curve drawn in Fig. \[fig3\], [*i.e.*]{} dominance of the exciton g.s. and monotony, are visible also at any other value of the magnetic field. The main effect of $B$ is to reinforce the dominance of the first line. The threshold for absorption is determined by the exciton g.s. energy, and the maximum dipole squared behaves like $B^{0.78}$.
The absorption coefficient of the negatively charged dot, $N_{e}=1$, is shown in Fig. \[fig4\]. The $(1,0)\rightarrow (2,1)$ process takes place in the $M=0$ sector. At $B=8$ T, a structure of isospaced bands is seen in the spectrum at higher energies. Most of these lines are suppressed already at $B=40$ T. The threshold for absorption and maximum strength transition are determined by the lowest $X^{-}$ state in the $M=0$ tower. As a function of $B$, we get $D^{2}\sim B^{0.79}$ at the maximum.
The absorption thresholds for the smallest systems, $N_{e}=0$ and 1, are smooth functions of $B$, signalling that the states entering the transition $%
(N_{e},0)\rightarrow (N_{e}+1,1)$ do not change qualitatively as $B$ is raised. For larger systems, however, there is an abrupt decrease in the threshold for fields around 10 T (“filling factor” near one), and small steps at higher fields . The steps are originated by the different rates of change of $M_{gs}$ in the $(N_{e},0)$ and $(N_{e}+1,1)$ systems (see Table \[tab1\]). Let us consider, for example, the $(3,0)\rightarrow (4,1)$ process. For $B\leq 10$ T, the process goes from the g.s. of (3,0) to the excited states of $X^{3-}$ with $M=-3$. For $B>10$ T, the g.s. of (3,0) moves to $M=-6$, a sector which contains the g.s. of (4,1). Thus, the threshold is lowered. Every time one of the systems rearranges, there is a step like change in the absorption threshold. The actual (experimental) profile is expected to be smoothed because of temperature effects.
Of course, not only the threshold changes, but the whole spectrum is restructured. We show in Fig. \[fig5\] the absorption in the $N_{e}=2$ dot ($X^{2-}$ formation) at $B=8$ T and 50 T. At $B=8$ T, the spectrum is similar to the $X^{-}$ spectrum. The added electron is placed in an outer orbit because the inner orbitals are filled. For higher fields, there is place for the new electron in the core region, but the minimization of energy causes a global restructuration of the charge density in the dot, as will be seen below. The added pair losses its identity. Notice that for $B>10$ T there are two very distinct lines in the spectrum. One is the threshold (the transition to the lowest state of (3,1)), and the second is the maximum, which is 7-4 meV above the threshold.
The dipole squared at maxima as a function of $B$ are drawn in Fig. \[fig6\]. Besides lowest state rearrangements, there are manifestations of collective effects even in these small systems. A decrease of absorption in the $N_{e}=2$ and 3 systems at “filling factors” $\nu \approx 1/2$, 1/3 and 1/5 is evident from Fig. \[fig6\].
Magnetoluminescence
-------------------
The second part of Fig. \[fig5\] shows the square of the dipole matrix elements corresponding to the luminescence of the $N_{e}=2$ dot at $B=40$ T. Only transitions starting from the g.s. of (3,1) are considered. Notice that the lowest state of (2,0) gives the strongest line, approximately 50 times higher than the next one. This is the common situation in our luminescence calculations for any of the systems under study. The strongest line corresponds to the transition from the g.s. of $(N_{e}+1,1)$ to the lowest state of $(N_{e},0)$ in the same angular momentum tower. The higher states of $(N_{e},0)$ give negligible contributions.
Luminescence in the $N_{e}=0$, and 1 dots is monotonic with $B$ because the initial and final states participating in it are fixed. Exciton luminescence proceeds in the $M=0$ channel, and $X^{-}$ luminescence in the $M=-1$ sector. In the latter case, the absorption and luminescence channels are different. With increasing $B$, the $X^{0}$ peak intensity increases, as in absorption, but the $X^{-}$ intensity decreases. We obtained $D^{2}\sim $ Exp$(-0.018~B)$ at the maximum.
For larger systems, the luminescence shows non monotonic behaviour because of lowest state rearrangements and collective effects, as in absorption. As a rule, the channels for absorption and PL are different in these systems. The luminescence maxima as a function of $B$ are drawn in Fig. \[fig7\].
Charge densities
----------------
Electron and hole charge densities inside the dot for the relevant states participating in absorption and luminescence are presented in this section. For electrons, we found more convenient to draw the difference $\rho_e^{\prime}=\rho_e(N_e+1,1)-\rho_e(N_e,0)$, which gives the density “added” to the dot.
Figure \[fig8\] shows the final-state densities in the absorption situations discussed in Fig. \[fig5\]. For the $N_e=2$ dot at $B=8$ T, the added electron and hole densities are almost identical. The exciton keeps its identity inside the dot. At $B=50$ T, however, the added pair causes a redistribution of the charge density of the initial two-electron state.
On the other hand, as shown above, the relevant states participating in luminescence transitions are the g.s. of $(N_{e}+1,1)$ and the lowest state of the $N_{e}$-electron system in the same angular momentum sector. We show in Fig. \[fig9\] the densities of these states in the $N_{e}=2$ dot at $B=40$ T. These curves are typical. The exciton is annihilated from a distribution very similar to the isolated exciton g.s. (also shown in the figure for comparison).
Concluding remarks
==================
We have studied few-electron systems and excitonic complexes (with one hole) in qdots under intense magnetic fields and low temperatures. In 1- and 2-electron qdots the g.s. angular momentum is independent of the magnetic field intensity. However, larger systems undergo abrupt rearrangements at particular $B$ values, a fact that is reflected in the optical absorption and PL.
We computed the interband optical properties of these systems. In absorption, the initial state is the polarized ground state of $N_{e}$ electrons (for temperatures $\ll 2$ K), and the final states are the states of $N_{e}+1$ electrons and one hole. The main result of these computations is the non monotonic behaviour of the absorption maxima in the larger ($N_{e}=2$ and 3) systems as the field is varied (Fig. \[fig6\]). This result can be understood as a consequence of ground state rearrangements and collective effects. We have presented typical charge densities in support of this picture. We found a reduction of absorption at “filling factors” 1/2, 1/3 and 1/5.
For luminescence events, we have considered the recombination from the g.s. of $N_{e}+1$ electrons and a hole. At a given magnetic field intensity, the angular momentum of this state may be different from the $N_{e}$-electron g.s. angular momentum. Thus, intrinsic absorption and luminescence may proceed through different channels. Of particular interest is that, opposite to the qwell case, the ground state of the negatively charged exciton $X_t^{-}$ is bright in luminescence. This is a consequence of the qdot lateral confinement. Furthermore, for very high $B$ the $X_t^{-}$ state recovers its dark character as compared with the other complexes here studied. On the other hand, the maximum of the recombination oscillator strength is a monotonic function of $B$ for qdots with 1 or 2 electrons and a hole, but it is nonmonotonic for qdots with more electrons, showing collective effects even in these small dots.
Although our calculations for finite systems with a smooth lateral confinement can not be easily extrapolated to the infinite limit, our results suggest that many-body effects should be taken into account in the computation of the $X^{-}$ luminescence in a qwell. Whittaker and Shields [@WS97], and Wojs [*et al*]{} [@Wojs] have used a three-particle model for the $X^{-}$. This model is indeed useful at very high magnetic fields. At intermediate values of $B$, the magnetoexciton size, which is $\sim 2~l_{B}\sim 50/\sqrt{B}$ nm, becomes comparable to the inter-electronic distance, around 20 nm for a typical carrier density of 1-2$ \times 10^{11}$ cm$^{-2}$. Many-body effects should take care of the observed dependence of the PL maximum with the filling factor.
We have not attempted a more sophisticated calculation in these systems because of the absence of experimental results for qdots in very intense magnetic fields. Nevertheless, our simple approach (1LL, one qwell sub-band, parabolic lateral confinement, unrealistic Zeeman energies and $z$-averaged Coulomb interactions) captures the essential physics and indicates the importance of collective effects even in small qdots.
A. G. acknowledges support by the Caribbean Network for Theoretical Physics. E. M-P acknowledges the Abdus Salam ICTP, where part of this work was done. The authors are grateful to C. Trallero-Giner for many useful discussions.
Electronic mail: [email protected] Electronic mail: [email protected] R. B. Laughlin, Phys. Rev. Lett. [**50**]{}, 1395 (1983). J. K. Jain and R. K. Kamilla, in [*Composite Fermions*]{}, edited by O. Heinonen, World Scientific, New York, 1998. S. M. Girvin, A. H. MacDonald and P. M. Platzman, Phys. Rev. B [** 33**]{}, 2481 (1986). T. Chakraborty and P. Pietilainen, [*The Quantum Hall Effects*]{}, Springer, New-York (1996). V. W. Scarola, K. Park and J. K. Jain, cond-mat/9910491; K. Park and J. Jain, cond-mat/9910460. A. Pinczuk [*et al*]{}, Phys. Rev. Lett. [**70**]{}, 3983 (1993); H. D. M. Davies, J. C. Harris, J. F. Ryan and A. J. Turberfield, Phys. Rev. Lett. [**78**]{}, 4095 (1997); A. Pinczuk [*et al*]{}, in [ *Proceedings of 12th Int. Conf. on High Magnetic Fields in Phys. of Semicond.*]{}, World Scientific, Singapore (1997), page 83; M. Kang, A. Pinczuk, B. S. Dennis, M. A. Eriksson, L. N. Pfeiffer, and K. W. West, cond-mat/9911350. C. J. Mellor [*et al*]{}, Phys. Rev. Lett. [**74**]{}, 2339 (1995); U. Zeitler [*et al*]{}, Phys. Rev. Lett. [**82**]{}, 5333 (1999). A. J. Shields, M. Pepper, M. Y. Simmons and D. A. Ritchie, Phys. Rev. B[** 52**]{}, 7841 (1995); G. Finkelstein, H. Shtrikman and I. Bar-Joseph, Phys. Rev. B[** 53**]{}, R1709 (1996); A. J. Shields, J. L. Osborne, D. M. Whittaker, M. Y. Simmons, M. Pepper and D. A. Ritchie, Phys. Rev. B[** 55**]{}, 1318 (1997); Y. Kim, C. H. Perry, K. S. Lee and D. G. Rickel, Phys. Rev. B[** 59**]{}, 1641 (1999). M. Hayne, C. L. Jones, R. Bogaerts [*et al*]{}, Phys. Rev. B[** 59**]{}, 2927 (1999). Y. Kim, F. M. Muntcanu, C. H. Perry, D. G. Rickel, J. A. Simmons and J. L. Reno, Phys. Rev. B[** 61**]{}, 4492 (2000); [*ibid*]{} [**61**]{}, 4731 (2000). J. J. Palacios, D. Yoshioka and A. H. MacDonald, Phys. Rev. B[** 54**]{}, R2296 (1996). I. V. Lerner and Yu. E. Lozovik, Zh. Eksp. Teor. Fiz. [**80**]{}, 1488 (1981) \[Sov. Phys. JETP [**53**]{}, 763 (1981)\]. D. M. Whittaker and A. J. Shields, Phys. Rev. B[** 56**]{}, 15185 (1997). A. Wojs, J. J. Quinn and P. Hawrylak, cond-mat/0001327; cond-mat/0001328. R. Cingolani, R. Rinaldi, H. Lipsalen [*et al*]{}, Phys. Rev. Lett. [**83**]{}, 4832 (1999). A. Wojs and P. Hawrylak, Phys. Rev. B[** 51**]{}, 10880 (1995). A. H. MacDonald and G. C. Aers, Phys. Rev. B[** 29**]{}, 5976 (1984). , ed. by O. Madelung, H. Schulz, and H. Weiss, Vol. [**III/17a**]{} and [**22a**]{}, Springer, Berlin, 1982. L. M. Roth, B. Lax and S. Zwerdling, Phys. Rev. [**114**]{}, 90 (1959). A. J. Shields, M. Pepper, D. A. Ritchie and M. Y. Simmons, Adv. Phys. [**44**]{}, 47 (1995). M. Seck, M. Potemski and P. Wyder, Phys. Rev. B[** 56**]{}, 7422 (1997).
$B [T]$ $M_{gs}(2,0)$ $M_{gs}(3,1)$ $M_{gs}(3,0)$ $M_{gs}(4,1)$
--------- --------------- --------------- --------------- ---------------
8 1 3 3 6
16 - - 6 6
20 3 3 6 9
25 - - 9 9
30 3 3 9 12
35 5 3 12 12
40 5 5 12 15
45 - - 15 15
50 7 5 15 18
58 7 7 - -
: Ground-state orbital angular momentum in the $N_e$=2 and 3 dots.[]{data-label="tab1"}
|
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author:
- 'Bar Light[^1] '
bibliography:
- 'Concent.bib'
title: 'Improving Bennett’s and Hoeffding’s inequalities using higher moments information'
---
<span style="font-variant:small-caps;">Abstract</span>:
We derive upper bounds on the moment-generating function of a random variable that depends on the random variable’s first $p$ moments. We use these bounds to generalize and improve the classical Hoeffding’s and Bennett’s inequalities for the case where there is some information on the random variables’ first $p$ moments for every positive integer $p$. Our generalized Hoeffding’s inequality is tighter than Hoeffding’s inequality and is given in a simple closed-form expression for every positive integer $p$. Hence, the generalized Hoeffding’s inequality is easy to use in applications. Our generalized Bennett’s inequality is given in terms of the generalized Lambert $W$-function and is tighter than Bennett’s inequality.
[Keywords: Concentration inequalities, Hoeffding’s inequality, Bennett’s inequality. ]{}\
Introduction
============
Concentration inequalities provide bounds on the probability that a random variable differs from some value, typically the random variable’s expected value (see [@boucheron2013concentration] for a textbook treatment of concentration inequalities). Besides their importance in probability theory, concentration inequalities are an important mathematical tool in statistics (see [@massart2000some]), machine learning theory (see [@mohri2018foundations]) and many other fields. Two of the most important and useful concentration inequalities are Hoeffding’s inequality [@hoeffding1994probability] and Bennett’s inequality [@bennett1962probability].[^2] These are inequalities that bound the probability that the sum of independent random variables differs from its expected value. The bound derived in Hoeffding’s inequality holds for bounded random variables and uses information on the random variables’ first moment. The bound derived in Bennett’s inequality holds for random variables that are bounded from above and uses information on the random variables’ first and second moments.
In this paper we generalize and significantly improve Bennett’s and Hoeffding’s inequalities. We provide bounds that use information on the random variables’ higher moments. More precisely, we provide bounds on the probability that the sum of independent random variables differs from its expected value where the bounds depend on the random variables’ first $p$ moments for every integer $p \geq 1$. We provide two families of concentration inequalities, one that generalizes Hoeffding’s inequality and one that generalizes Bennett’s inequality. Importantly, the bounds that we derive are tighter than Bennett’s and Hoeffding’s inequalities and are given as closed-form expressions in most cases. In our generalized Hoeffding’s inequality, our bounds hold for bounded random variables and are given as simple closed-form expressions (see Theorem \[Thm: concent hof\]) for every integer $p \geq 1$. For any integer $p \geq 1$ the bound uses information on the random variables’ first $p$ moments and is always tighter than Hoeffding’s inequality. We also show, using a numerical example, that our bound can significantly improve Hoeffding’s inequality. In our generalized Bennett’s inequality, our bounds hold for random variables that are bounded from above. For $p=3$, our bound is given in a closed-form expression in terms of the Lambert $W$-function. This bound uses information on the random variables’ first three moments and is tighter than Bennett’s inequality. For $p > 3$ our bounds are given in terms of the generalized Lambert $W$-function (see Theorem \[Thm: concent bennett\]).
For every positive integer $p$, independent random variables $X_{1},\ldots,X_{n}$ such that $\mathbb{P}(X_{i} \in [a_{i},b_{i}] = 1)$, and all $t>0$, our generalized Hoeffding’s inequality is given by $$\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t)
\leq \exp \left (-\frac{2t^{2}}{\sum _{i=1}^{n} (b_{i} - a_{i})^{2} C_{p}(t,X_{i}) } \right )$$ where $S_{n}= \sum _{i=1}^{n} X_{i}$ and $C_{p}(t,X_{i})$ is a function that depends on $t$, on the first $p$ moments of $X_{i}$, and on $X_{i}$’s support: $[a_{i},b_{i}]$. We show that for every positive integer $p$ we have $C_{p} \leq 1$. Thus, our generalized Hoeffding’s inequality is tighter than Hoeffding’s inequality which corresponds to $p=1$ and $C_{1}=1$. We provide a simple closed-form expression for the function $C_{p}$ for any integer $p \geq 1$. For example, suppose that the support of a random variable $X$ is $[0,b]$ for some $X=X_{i}$, $i=1,\ldots,n$. Then $C_{p}(t,X)$ is given by $$C_{p}(t,X) = \left ( \frac{\mathbb{E}X^{p} \exp(y) + \sum _{j=0} ^{p-3} \frac{y^{j}} {j!} \left ( b^{p-j-2}\mathbb{E}X^{j+2} - \mathbb{E}X^{p} \right )}{\mathbb{E}X^{p}\exp(y) + \sum _{j=0} ^{p-2} \frac{y^{j}} {j!} \left ( b^{p-j-1}\mathbb{E}X^{j+1} - \mathbb{E}X^{p} \right )} \right )^{2}$$ where $y = 4tb/\sum_{i=1}^{n} d (X_{i})$ and $d(X_{i}) = \left (\mathbb{E}X_{i}^{2} / \mathbb{E}X_{i} \right )^{2} $ (see Theorem \[Thm: concent hof\]). We note that our generalized Hoeffding’s bounds are exponential bounds, and hence, these bounds are not optimal in the sense that there is a missing factor in those bounds (see [@talagrand1995missing]). However, in many applications it is convenient to use exponential bounds that are given in closed-form expressions such as the bound given in Hoeffding’s inequality and in our generalized Hoeffding’s inequality. Therefore, we believe that our bounds will be useful for future research and applications.
Main results
============
In this section we state our main results. In Section \[Sec: bounds on the MGF\] we derive upper bounds on the moment-generating function of a random variable that is bounded from above. In Section \[Sec: concert Hoeffiding\] we derive our generalized Hoeffding’s inequalities. In Section \[Sec: concert Bennet\] we derive our generalized Bennett’s inequalities.
We first introduce some notations.
Throughout the paper we consider a fixed probability space $\left (\Omega ,\mathcal{F} ,\mathbb{P}\right )$. A random variable $X$ is a measurable real-valued function from $\Omega $ to $\mathbb{R}$. We denote the expectation of a random variable on the probability space $\left (\Omega ,\mathcal{F} ,\mathbb{P}\right )$ by $\mathbb{E}$. For $1 \leq p \leq \infty $ let $L^{p} : =L^{p}\left (\Omega ,\mathcal{F} ,\mathbb{P}\right )$ be the space of all random variables $X :\Omega \rightarrow \mathbb{R}$ such that $\left \Vert X\right \Vert _{p}$ is finite, where $\left \Vert X\right \Vert _{p} =\left (\int _{\Omega }\left \vert X(\omega )\right \vert ^{p} \mathbb{P}(d \omega) \right )^{1/p}$ for $1 \leq p <\infty $ and $\left \Vert X\right \Vert _{p} =\sup _{\omega \in \Omega }\left \vert X(\omega )\right \vert $ for $p =\infty $. We say that $X$ is a random variable on $[a,b]$ for some $a<b$ if $\mathbb{P}(X \in [a,b]) = 1$.
For $k\geq 1$, we denote by $f^{(k)}$ the $k$th derivative of a $k$ times differentiable function $f :[a ,b] \rightarrow \mathbb{R}$ and for $k=0$ we define $f^{(0)}:=f$. As usual, the derivatives at the extreme points $f^{(k)}(a)$ and $f^{(k)}(b)$ are defined by taking the left-side and right-side limits, respectively.
For the rest of the paper we define $$T_{p}(x):= \exp (x) - \sum _{j=0} ^{p-2} \frac { x^{j} } {j!}$$ to be the Taylor remainder of the exponential function of order $p-2$ at the point $0$. The function $T_{p}$ plays an important role in our analysis.
Upper bounds on the moment-generating function {#Sec: bounds on the MGF}
----------------------------------------------
In this section we provide upper bounds on the moment-generating function of a random variable that is bounded from above.
We show that $$\label{Ineq: Wp and Wb}
\frac{T_{p+1}(x)}{T_{p+1}(b)} \leq \frac{\max (x^{p},0)}{b^{p}}$$ for all $x \leq b$, $b > 0$ and every positive integer $p$. This bound on the ratio of the Taylor remainders is the key ingredient in deriving the upper bounds on the moment-generating function. The proof of Bennett’s inequality uses inequality (\[Ineq: Wp and Wb\]) with $p=2$ to bound the moment-generating function (see [@boucheron2013concentration]). We use inequality (\[Ineq: Wp and Wb\]) to provide upper bounds on the moment-generating function using information on the random variable’s first $p$ moments for every positive integer $p$. The Appendix contains the proofs not presented in the main text.
\[Coro: exp upper bound\] Let $X \in L^{p-1}$ be a random variable on $(-\infty ,b]$ for some $b>0$ where $p$ is a positive integer. For all $s \geq 0$ we have $$\begin{aligned}
\label{Ineq: MGF bound}
\begin{split}
\mathbb{E} \exp(sX) & \leq \frac{\mathbb{E} \max(X^{p},0)}{b^{p}}\left (\exp(sb) - \sum _{j=0} ^{p-1} \frac{s^{j}b^{j}}{j!} \right )+ \mathbb{E} \left ( \sum _{j=0} ^{p-1} \frac{s^{j}X^{j}} {j!} \right ) \\
& = \frac{\mathbb{E} \max (X^{p},0) }{b^{p}} T_{p+1}(sb) + \mathbb{E} \left ( \sum _{j=0} ^{p-1} \frac{s^{j}X^{j}} {j!} \right ).
\end{split}\end{aligned}$$
Theorem \[Coro: exp upper bound\] provides a unified approach for seemingly independent bounds on the moment-generating function that were derived in previous literature and used to prove concentration inequalities.
For $p=2$, and for a random variable $X$ on $(-\infty,b]$, Theorem \[Coro: exp upper bound\] yields the inequality $$\label{Ineq: Bound on MGF p=2}
\mathbb{E}\exp(sX) \leq \frac{\mathbb{E}X^{2}}{b^{2}}\left (\exp(sb) - 1 - sb \right ) + 1 + s\mathbb{E}(X)$$ which is fundamental in proving Bennett’s inequality (see [@bennett1962probability]). For $p=3$, denoting $\mu^{3}=\mathbb{E} \max(X^{3},0)$, we have $$\begin{aligned}
\begin{split}
\frac{\mu^{3}}{b^{3}}T_{4}(sb) + \mathbb{E} \left ( \sum _{j=0} ^{2} \frac{s^{j}X^{j}} {j!} \right ) & = \frac{\mu^{3}}{b^{3}}T_{3}(sb) + 1 + s\mathbb{E}(X) + \frac{s^{2}}{2}\left (\mathbb{E}X^{2} - \frac{\mu^{3}}{b} \right ) \\
& \leq \exp \left (\frac{\mu^{3}}{b^{3}}T_{3}(sb) + s\mathbb{E}(X) + \frac{s^{2}}{2}\left (\mathbb{E}X^{2} - \frac{\mu^{3}}{b} \right ) \right ).
\end{split}\end{aligned}$$ The last inequality follows from the elementary inequality $1 + x \leq \exp(x) $ for all $x \in \mathbb{R}$. Thus, Theorem \[Coro: exp upper bound\] implies $$\mathbb{E} \exp(sX) \leq \exp \left (\frac{\mathbb{E} \max(X^{3},0)}{b^{3}}T_{3}(sb) + s\mathbb{E}(X) + \frac{s^{2}}{2}\left (\mathbb{E}X^{2} - \frac{\mathbb{E} \max(X^{3},0)}{b^{3}} \right ) \right )$$ which is proved in Theorem 2 in [@pinelis1990exact].
Let $$m_{X}(p) := \frac{\mathbb{E} \max (X^{p},0) }{b^{p}} T_{p+1}(sb) + \mathbb{E} \left ( \sum _{j=0} ^{p-1} \frac{s^{j}X^{j}} {j!} \right )$$ be the right-hand side of inequality (\[Ineq: MGF bound\]). The next proposition shows that for every even number $p$ we have $m_{X}(p) \geq m_{X}(p+1)$. If, in addition, the random variable $X$ is non-negative, then we also have $m_{X}(p+1) \geq m_{X}(p+2)$, and hence, $m_{X}(p)$ is decreasing. Thus, for non-negative random variables, inequality (\[Ineq: MGF bound\]) is tighter when $p$ increases. In particular, we have $m_{X}(2) \geq m_{X}(p)$ for every integer $p \geq 3$, i.e., the bound on the moment-generating function given in inequality (\[Ineq: MGF bound\]) is tighter than Bennett’s bound (\[Ineq: Bound on MGF p=2\]) for every integer $p \geq 3$ when $X$ is non-negative.
\[prop: m(p)\] Let $X \in L^{p}$ be a random variable on $(-\infty,b]$. Let $p \geq 2$ be an even number. The following statements hold:
\(i) $m_{X}(p) \geq m_{X}(p+1)$.
\(ii) If $X \geq 0$ then $m_{X}(p+1) \geq m_{X}(p+2)$.
The upper bound on the moment-generating function (\[Ineq: MGF bound\]) is optimal in the sense that there exists a random variable that achieves equality. Even for $p=1$ there exists a random variable that achieves equality in (\[Ineq: MGF bound\]). For example, a Bernoulli random variable that yields $1$ with probability $q$ and $0$ with probability $1-q$ achieves equality in (\[Ineq: MGF bound\]) for $p=1$. Note that for the Bernoulli random variable all the moments are equal to $q$ which is the highest value that the higher moments can have given that the first moment equals $q$ and the support is $[0,1]$. Thus, higher moments do not provide any useful information and for every integer $p>1$ inequality (\[Ineq: MGF bound\]) reduces to the case of $p=1$.
For any random variable $X$ on $[0,b]$ and $s \geq 0$ Theorem \[Coro: exp upper bound\] implies that $$\mathbb{E}T_{p+1}(sX) \leq \frac{\mathbb{E} X^{p} }{b^{p}}T_{p+1}(sb)$$ where $p$ is a positive integer. The last inequality can be easily applied to any bounded random variable. For example, suppose that $Y$ is a random variable on $[a,b]$ for some $a<b$. Defining the random variable $X=Y-a$ and using inequality (\[Ineq: MGF bound\]) yields the following upper bound on the moment-generating function: $$\mathbb{E} \exp(s(Y-a)) \leq \frac{\mathbb{E} (Y-a) ^{p}}{(b-a)^{p}}\left (\exp(s(b-a)) - \sum _{j=0} ^{p-1} \frac{s^{j}(b-a)^{j}}{j!} \right )+ \mathbb{E} \left ( \sum _{j=0} ^{p-1} \frac{s^{j}(Y-a)^{j}} {j!} \right ).$$
Concentration inequalities: Hoeffding type inequalities {#Sec: concert Hoeffiding}
-------------------------------------------------------
In this section we derive Hoeffding type concentration inequalities that provide exponential bounds on the probability that the sum of independent bounded random variables differs from its expected value. We improve Hoeffding’s inequality by using information on the random variables’ first $p$ moments and by using a refined upper bound on the moment-generating function of a bounded random variable (see Theorem \[Coro: exp upper bound\]). We derive a tighter bound than the standard Hoeffding’s bound for every integer $p \geq 2$ (see Theorem \[Thm: concent hof\] part (ii)). Importantly, for every $p$ the bound is given as a simple closed-form expression that depends on the random variables’ first $p$ moments. Thus, the bound can be easily used in applications. We also provide a bound for the case that $p$ tends to infinity. This bound depends on all of the random variables’ moments (see Theorem \[Thm: concent hof\] part (iii)).
\[Thm: concent hof\] Let $X_{1},\ldots,X_{n}$ be independent random variables where $X_{i}$ is a random variable on $[0,b_{i}]$, $b_{i} >0$. Let $S_{n} = \sum _{i=1} ^{n} X_{i}$. Let $p \geq 1$ be an integer. Denote $\mathbb{E}(X^{k}_{i}) = \mu ^{k}_{i}>0$ for all $k=1,\ldots,p$ and all $i = 1,\ldots, n$. Let $D_{n} = \sum _{i=1} ^{n} d(X_{i}) $ where $d(X_{i}) = \left (\mu^{2}_{i} / \mu ^{1}_{i}\right )^{2}$.
\(i) For all $t>0$ we have $$\label{Ineq: Hoef bound}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left (- \frac{2t^{2}}{\sum_{i=1}^{n} b^{2} _{i} C_{p} \left (\frac{4tb_{i}}{D_{n} },b_{i}, \mu ^{1}_{i}, \ldots , \mu ^{p}_{i} \right ) } \right ).$$ where $$C_{p} \left (y ,b_{i} , \mu^{1} _{i} , \ldots , \mu^{p}_{i} \right ) = \left ( \frac{\exp(y) + \sum _{j=0} ^{p-3} \frac{y^{j}} {j!} \left ( \frac{b^{p-j-2}_{i}\mu^{j+2}_{i}}{\mu^{p}_{i}} - 1 \right )}{\exp(y) + \sum _{j=0} ^{p-2} \frac{y^{j}} {j!} \left ( \frac{b^{p-j-1}_{i}\mu^{j+1}_{i}}{\mu^{p}_{i}} - 1 \right )} \right )^{2}$$ for all $i=1,\ldots,n$ and all $y > 0$.
\(ii) For every integer $p \geq 1$ we have $0 < C_{p} \leq 1$. Thus, inequality (\[Ineq: Hoef bound\]) is tighter than Hoeffding’s inequality: $$\label{Ineq: Hoef bound p=1}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left (-\frac{2t^{2}}{\sum _{i=1}^{n} b^{2} _{i} } \right )$$ which corresponds to $p=1$ and $C_{1} = 1$.
\(iii) When $p$ tends to infinity we have $$\begin{aligned}
\lim _{p \rightarrow \infty} C_{p}(x,b_{i},\mu^{1}_{i},\ldots,\mu^{p}_{i})= \frac{1}{b^{2}_{i}}\left (\frac{\mathbb{E}X^{2}_{i}\exp(xX_{i}/b_{i})}{\mathbb{E}X_{i}\exp(xX_{i}/b_{i})} \right ) ^{2}
\end{aligned}$$ for all $i=1,\ldots,n$ and all $x \geq 0$. Using part (i) implies that for all $t>0$ we have $$\label{Ineq: Hoef p=infty}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left (-\frac{2t^{2}}{\sum _{i=1}^{n}\left (\frac{\mathbb{E}X^{2}_{i}\exp(4tX_{i}/D_{n})}{\mathbb{E}X_{i}\exp(4tX_{i}/D_{n})} \right ) ^{2} } \right ).$$ Note that inequality (\[Ineq: Hoef p=infty\]) does not depend on $b_{i}$.
\(i) Theorem \[Thm: concent hof\] can be easily applied to bounded random variables that are not necessarily positive. If $Y_{i}$ is a random variable on $[a_{i},b_{i}]$ and $Y_{1},\ldots,Y_{n}$ are independent, we can define the random variables $X_{i}=Y_{i}-a_{i}$ on $[0,b_{i}-a_{i}]$ and use Theorem \[Thm: concent hof\] to conclude that $$\begin{aligned}
\label{Ineq: Hoef bounded random variables}
\begin{split}
\mathbb{P} \left ( \sum _{i=1}^{n} Y_{i}-\mathbb{E} \left (\sum _{i=1}^{n} Y_{i} \right ) \geq t \right) & = \mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) \\
& \leq \exp \left (-\frac{2t^{2}}{\sum _{i=1}^{n} (b_{i} - a_{i})^{2} C_{p} \left (4t(b_{i}-a_{i})/D_{n} ,b_{i}-a_{i} , \mu^{1} _{i} , \ldots , \mu^{p}_{i} \right )} \right ).
\end{split}
\end{aligned}$$ Note that $\mu^{k}_{i} = \mathbb{E}(Y_{i}-a_{i})^{k}$ for all $i=1,\ldots,n$ and all $k=1,\ldots,p$.
\(ii) If $X_{1},\ldots,X_{n}$ are identically distributed then inequality (\[Ineq: Hoef bound\]) yields $$\label{Ineq: Hoef IID}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq nt) \leq \exp \left (-\frac{2nt^{2}}{ b^{2} _{i} C_{p} \left (\frac{4tb_{i}}{ d(X_{i}) } ,b_{i} , \mu^{1} _{i} , \ldots , \mu^{p}_{i} \right )} \right ).$$
We now discuss the sketch of the proof of Theorem \[Thm: concent hof\] part (i). The full proof is in the Appendix.
Fix a positive integer $p$. We start with a random variable $X$ on $[0,b]$. Assume for simplicity that $b=1$. From Theorem \[Coro: exp upper bound\] we have $\mathbb{E}\exp(sX) \leq v(y)$ where $v(y)$ is the right-hand side of inequality (\[Ineq: MGF bound\]). Let $g(y)=\ln(v(y))$. Then using Taylor’s theorem we can show (using a standard argument) that $\mathbb{E}\exp(sX-s\mathbb{E}(X)) \leq \exp(0.5s^{2}\max _{0 \leq y \leq s} g^{(2)}(y) )$. Let $C(s)=\max _{0 \leq y \leq s} (v^{(2)}(y)/v^{(1)}(y))^{2}$. A key step in the proof of Theorem \[Thm: concent hof\] is to show that $v^{(1)}$ is a log-convex function, and hence, $v^{(2)}/v^{(1)}$ is increasing which implies that $C(s)=(v^{(2)}(s)/v^{(1)}(s))^{2}$ is given in a closed-form expression. We have $$\begin{aligned}
\max _{0 \leq z \leq s} g^{(2)}(y) = \max _{0 \leq y \leq s} \frac{v^{(2)}(y)}{v(y)}\left (1 - \frac{v^{(2)}(y)(v^{(1)}(y))^{2}}{v(y)(v^{(2)}(y))^{2}} \right ) \leq \max _{0 \leq y \leq s} \frac{v^{(2)}(y)}{v(y)}\left (1 - \frac{v^{(2)}(y)}{v(y)C(s)} \right ) \leq 0.25C(s)\end{aligned}$$ where the second inequality follows from the elementary inequality $x(1-x/z) \leq 0.25z$ for all $z>0$ and $x>0$. With this bound we can conclude that $\mathbb{E}\exp(sX-s\mathbb{E}(X)) \leq \exp(s^{2}C(s)/8)$. Applying the Chernoff bound and choosing a specific value for $s$ proves Theorem \[Coro: exp upper bound\] part (i).
The calculation of $C_{p}$ in inequality (\[Ineq: Hoef bound\]) is immediate. For example, for $p=2$ we have $$C_{2}(x,b_{i},\mu^{1}_{i},\mu^{2}_{i}) = \left ( \frac{\mu^{2} _{i}\exp(x)}{\mu^{2} _{i}\exp(x) + b_{i}\mu^{1} _{i} - \mu^{2} _{i}} \right ) ^{2}$$ and for $p=3$ we have $$\begin{aligned}
C_{3}(x,b_{i},\mu^{1}_{i},\mu^{2}_{i},\mu^{3}_{i}) = \left ( \frac{\mu^{3} _{i}\exp(x) + b_{i} \mu^{2} _{i}- \mu^{3} _{i} }{\mu^{3} _{i}\exp(x) + b^{2} _{i} \mu^{1} _{i} - \mu^{3} _{i} + (b_{i} \mu^{2} _{i} - \mu^{3} _{i} )x} \right ) ^{2}
\end{aligned}$$ for all $i=1,\ldots,n$ and all $x \geq 0$.
We now provide a numerical example where the results in Theorem \[Thm: concent hof\] significantly improve Hoeffding’s inequality.
\[Exam: uniform\] (i) Suppose that $X_{1},\ldots,X_{n}$ are independent continuous uniform random variables on $[0,1]$, i.e., $\mathbb{P}(X_{i} \leq t) = t$ for $0 \leq t \leq 1$. In this case, a straightforward calculation shows that $$\mathbb{E}X_{i}\exp(sX_{i})= \frac{\exp(s)(s-1)+1}{s^{2}} \text{ and } \mathbb{E}X_{i}^{2}\exp(sX_{i})= \frac{\exp(s)(s^{2}-2s+2)-2}{s^{3}}.$$ Using Theorem \[Thm: concent hof\] part (iii) we have $$\lim _{p \rightarrow \infty} C_{p} \left (x ,1 , \mu^{1} _{i} , \ldots , \mu^{p}_{i} \right ) = \left ( \frac{-2 + \exp(x)(2-2x+x^2)}{x(1+\exp(x)(x-1))} \right )^{2} : = C_{\infty}(x).$$ Using the fact that $d (X_{i}) = \left (\mathbb{E}X_{i}^{2} / \mathbb{E}X_{i} \right )^{2} = 4/9$ inequality (\[Ineq: Hoef p=infty\]) yields $$\label{Ineq: uniform}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq nt) \leq \exp \left ( -\frac{2nt^{2}}{C_{\infty}(9t)} \right ).$$
In Figure \[Fig: uniform\] we plot the bound given in Hoeffding’s inequality (see Theorem \[Thm: concent hof\] inequality (\[Ineq: Hoef bound p=1\])) for $\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq nt)$ divided by the bound given in (\[Ineq: uniform\]) as a function of $t$ on the interval $[0.1,0.4]$ for $n=40$. We see that the bound given in (\[Ineq: uniform\]) significantly improves Hoeffding’s bound.
\[ht\]
Concentration inequalities: Bennett type inequalities {#Sec: concert Bennet}
-----------------------------------------------------
In this section we derive Bennett type concentration inequalities that provide bounds on the probability that the sum of independent and bounded from above random variables differs from its expected value. The bounds depend on the random variables’ first $p$ moments and are given in terms of the generalized Lambert $W$-function [@scott2006general]. For real numbers $\alpha_{i}$, $i=0,\ldots,p$, $\alpha _{0} > 1$, consider the one dimensional transcendental equation: $$\label{Eq: POLY-EXPO}
\alpha_{0} - \sum _{j=1} ^{p} \alpha _{j} x^{i} = \exp(x).$$ The solutions to equation (\[Eq: POLY-EXPO\]) are a special case of the generalized Lambert $W$-function [@scott2006general]. Because $\alpha_{0} > 1$ it is easy to see that equation (\[Eq: POLY-EXPO\]) has a positive solution. We denote the non-empty set of positive solutions of equation (\[Eq: POLY-EXPO\]) by $G_{p}(\alpha_{0},\ldots,\alpha_{p})$. The bounds given in Theorem \[Thm: concent bennett\] depend on the elements of the set $G_{p}(\alpha_{0},\ldots,\alpha_{p})$ where $\alpha_{i}$ depends on the random variables’ moments. When $p=0$ the set $G_{0}(\alpha_{0})$ consists of one element $\ln (\alpha_{0})$. When $p=1$ and assuming that $\alpha _{1} >0$, the set $G_{1}(\alpha_{0},\alpha_{1})$ consists of one element that is given in terms of the Lambert W-function. Recall that for $x \geq 0$, $y \exp (y) = x$ holds if and only if $y=W(x)$ where $W$ is the principal branch of the Lambert W-function (see [@corless1996lambertw]). Because $\alpha _{0} >1$ and assuming $\alpha _{1} >0$, the unique positive solution to the equation $\exp (x) = \alpha _{0} - \alpha_{1}x$ is given by $$\frac{\alpha _{0}}{\alpha_{1}} - W\left ( \frac{\exp (\alpha_{0} / \alpha_{1}) }{\alpha_{1}} \right )$$ (see [@corless1996lambertw]). We leverage this observation to derive a bound on the probability that the sum of independent random variables differs from its expected value when we use information on the random variables’ first three moments. The bound is given as a closed-form expression in terms of the $W$-Lambert function (see Theorem \[Thm: concent bennett\] part (iv)). We show that this bound is tighter than the bound given in Bennett’s inequality (see Proposition \[prop: Tighter bennett\]). We provide an example of the magnitude of improvement (see Example \[Exam: expon\]).
Finding the positive solutions of the transcendental equation (\[Eq: POLY-EXPO\]) for $p \geq 2$ can be done using a computer program. It involves solving an exponential polynomial equation of order $p$ that has at least one positive solution. When the random variables have non-negative moments we show that the transcendental equation (\[Eq: POLY-EXPO\]) has a unique positive solution (see Theorem \[Thm: concent bennett\] part (ii)).
\[Thm: concent bennett\] Let $X_{1},\ldots,X_{n}$ be independent random variables on $(-\infty,b]$ for some $b>0$ and let $S_{n} = \sum _{i=1} ^{n} X_{i}$. Let $p \geq 2$ be an integer and assume that $X_{i} \in L^{p}$ for all $i=1,\ldots,n$. Denote $\mathbb{E}(X_{i}) = \mu ^{1}_{i}$, and assume that $\mathbb{E}(X ^{k}_{i}) \leq \mu ^{k}_{i}$ and $0<\mathbb{E}(\max (X ^{p}_{i},0)) \leq \mu ^{p}_{i}$ for all $k=1,\ldots,p-1$ and all $ i=1,\ldots,n$.
\(i) For all $t>0$ we have $$\label{Ineq: Bennt concent}
\mathbb{P} (S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left ( - \max _{y \in G_{p-2}(\alpha_{0},\ldots,\alpha_{p-2})} \left ( \frac{t}{b} - \left ( \frac{t}{b} + \frac{\mu^{2} }{b^{2} }\right) y + \sum_{j=2}^{p-1} \left (\frac{\mu^{j}}{b^{j}j!} - \frac{\mu ^{j+1}}{b^{j+1}j!} \right ) y ^{j} \right ) \right )$$ where $$\alpha_{0} = 1 + \frac{tb^{p-1}}{\mu^{p}} > 1 \text{ and } \alpha_{j} = \frac{b^{p-j-1} \mu ^{j+1}}{\mu^{p} j!} - \frac{1}{j!}$$ for all $j=1,\ldots,p-2$ and $\mu ^{k} = \sum _{i=1} ^{n} \mu _{i} ^{k} $ for all $k=1,\ldots,p$.
\(ii) If $\mu^{j} \geq 0$ for every odd number $j \geq 3$ then $G_{p-2}$ consists of one element and inequality (\[Ineq: Bennt concent\]) reduces to $$\label{Ineq: Ben useful}
\mathbb{P} (S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left ( - \left ( \frac{t}{b} - \left ( \frac{t}{b} + \frac{\mu^{2} }{b^{2} }\right) y + \sum_{j=2}^{p-1} \left (\frac{\mu^{j}}{b^{j}j!} - \frac{\mu ^{j+1}}{b^{j+1}j!} \right ) y ^{j} \right ) \right )$$ where $y$ is the unique element of $G_{p-2}$, i.e., $y$ is the unique positive solution of the equation $\alpha_{0} - \sum _{j=1}^{p-2} \alpha_{j}x^{j} = \exp(x)$.
\(iii) Suppose that $p=2$. Then $G_{0}(\alpha_{0}) = \{\ln(\alpha_{0} ) \}$ consists of one element and inequality (\[Ineq: Bennt concent\]) reduces to Bennett’s inequality: $$\begin{aligned}
\label{Ineq: Bennett LOG p=2 CONCENT}
\begin{split}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) & \leq \exp \left ( - \left ( \frac{t}{b} - \left ( \frac{t}{b} + \frac{\mu^{2} }{b^{2} }\right) \ln \left (\frac{tb}{\mu^{2}} +1 \right ) \right ) \right ) \\
& =\exp \left (- \frac {\mu ^{2} }{b^{2}} \left ( \left (\frac{bt}{\mu^{2}} + 1 \right ) \ln \left (\frac{bt}{\mu^{2}} + 1 \right ) - \frac{bt}{\mu^{2}} \right ) \right ).
\end{split}\end{aligned}$$
\(iv) Suppose that $p=3$, $\alpha_{1} \neq 0$, and $\mathbb{E}(\max (X ^{3}_{i},0)) = \mu ^{3}_{i}$ for all $ i = 1,\ldots,n$. Then $G_{1}(\alpha_{0},\alpha_{1}) = \left \{ \frac{\alpha _{0}}{\alpha_{1}} - W\left ( \frac{\exp (\alpha_{0} / \alpha_{1}) }{\alpha_{1}} \right ) \right \}$ consists of one element and inequality (\[Ineq: Bennt concent\]) reduces to $$\begin{aligned}
\label{Ineq: Bennett p=3 W-lamb}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t)
\leq \exp \left ( - \left ( \frac{t}{b} - \left ( \frac{t}{b} + \frac{\mu^{2} }{b^{2} }\right) y + \left (\frac{\mu^{2}}{2b^{2}} - \frac{\mu ^{3}}{2b^{3}} \right ) y ^{2} \right ) \right )\end{aligned}$$ where $y=\frac{\alpha _{0}}{\alpha_{1}} - W\left ( \frac{\exp (\alpha_{0} / \alpha_{1}) }{\alpha_{1}} \right ) $ and $W$ is the Lambert $W$-function.
The proof of Theorem \[Thm: concent bennett\] consists of three steps. In the first step we bound the moment-generating function of a random variable $X$ that is bounded from above using the first $p$ moments of $X$. We use Theorem \[Coro: exp upper bound\] to prove the first step. In the second step we derive an exponential bound on the moment-generating function using the elementary inequality $1+x \leq \exp(x)$ for all $x \in \mathbb{R}$. We note that in some cases this inequality is loose and and so the second step may potentially be improved (for example see [@jebara2018refinement] and [@zheng2018improved]). In the third step we apply the Chernoff bound to derive the concentration inequality.
In applications, it is more convenient to use inequality (\[Ineq: Ben useful\]) than inequality (\[Ineq: Bennt concent\]). Using a Taylor series approximation, one can easily calculate the unique and positive solution to the equation $\alpha_{0} - \sum _{j=1}^{p-2} \alpha_{j}x^{j} = \exp(x)$. To use inequality (\[Ineq: Ben useful\]) when there is information on the random variables’ first $p$ moments, we can choose a non-negative $\mu^{j}$ for every odd number $j \geq 3$. This is the essence of Corollary \[Coro: Ben\]. For $p \geq 4$, Corollary \[Coro: Ben\] can be used instead of Theorem \[Thm: concent bennett\] part (i). The proof of Corollary \[Coro: Ben\] follows immediately from part (ii) of Theorem \[Thm: concent bennett\].
\[Coro: Ben\] Assume that the notations and conditions of Theorem \[Thm: concent bennett\] hold. Suppose that $p \geq 2$. For every odd number $j \geq 3$ such that $j \neq p$ let $\mu ^{j} = \max (\sum^{n}_{i=1} \mu _{i}^{j} , 0)$. Then $$\mathbb{P} (S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left ( - \left ( \frac{t}{b} - \left ( \frac{t}{b} + \frac{\mu^{2} }{b^{2} }\right) y + \sum_{j=2}^{p-1} \left (\frac{\mu^{j}}{b^{j}j!} - \frac{\mu ^{j+1}}{b^{j+1}j!} \right ) y ^{j} \right ) \right )$$ where $y$ is the unique positive solution of the equation $\alpha_{0} - \sum _{j=1}^{p-2} \alpha_{j}x^{j} = \exp(x)$.
The next proposition shows that the concentration inequality (\[Ineq: Bennett p=3 W-lamb\]) that we derive in Theorem \[Thm: concent bennett\] part (iv) is always tighter than Bennett’s inequality.
\[prop: Tighter bennett\] Assume that the conditions of Theorem \[Thm: concent bennett\] hold. Assume that $\mathbb{E}(X ^{2}_{i}) = \mu ^{2}_{i}$ and $\mathbb{E}(\max (X ^{3}_{i},0)) = \mu ^{3}_{i}$ for all $ i = 1,\ldots,n$. Then the right-hand side of inequality (\[Ineq: Bennett p=3 W-lamb\]) is smaller than the right-hand side of inequality (\[Ineq: Bennett LOG p=2 CONCENT\]). That is, inequality (\[Ineq: Bennett p=3 W-lamb\]) is tighter than Bennett’s inequality.
We provide a numerical example that compares inequality (\[Ineq: Bennett p=3 W-lamb\]) which uses information on the random variables’ third moment to Bennett’s inequality (\[Ineq: Bennett LOG p=2 CONCENT\]). For simplicity, we consider a one-sided concentration inequality for exponential random variables.
\[Exam: expon\] Let $G_{1},\ldots,G_{n}$ be independent exponential random variables with rate $1$, i.e., $G_{i}$ is a random variable on $[0,\infty)$ and $\mathbb{P}(G_{i} \leq x) = 1 - \exp(-x)$ for all $i=1,\ldots,n$ and $x \geq 0$. Define the random variables $X_{i} = \mathbb{E}(G_{i}) - G_{i} = 1 - G_{i}$ on $(-\infty,1]$. We have $\mu ^{1} _{i} = 0$, $\mu ^{2} _{i} = 1$, and $$\mu ^{3} _{i} = \int _{0} ^{\infty} \max ((1-x)^{3},0)\exp(-x)dx = \frac{6}{\exp(1)} - 2$$ for all $i=1,\ldots,n$. In Figure \[Fig: Exp\] we plot the bound for $\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq nt)$ given in Bennett’s inequality (see Theorem \[Thm: concent bennett\] part (iii)) divided by the bound derived in Theorem \[Thm: concent bennett\] part (iv) as a function of $t$ for $n=30$. We use the program Mathematica [@Mathematica] to plot Figure \[Fig: Exp\] where the Lambert $W$-function is implemented.
\[ht\]
Conclusions
===========
We provide upper bounds on the moment-generating function of a random variable that is bounded from above using information on the random variable’s higher moments (see Theorem \[Coro: exp upper bound\]). Using these bounds we generalize and improve Hoeffding’s inequality (see Theorem \[Thm: concent hof\]) and Bennett’s inequality (see Theorem \[Thm: concent bennett\]) for the case that some information on the random variables’ higher moments is available. Our bounds are simple to use and are given as closed-form expressions in most cases. Our results can be extended in a standard way to martingales and to their maximal functions. Other inequalities and results that use Hoeffding’s or Bennett’s inequalities can also be improved using our results.
Appendix
========
Proofs of the results in Section \[Sec: bounds on the MGF\]
-----------------------------------------------------------
\[Proof of Theorem \[Coro: exp upper bound\]\] Clearly Theorem \[Coro: exp upper bound\] holds for $s=0$. Fix $s > 0$, $b>0$ and a positive integer $p$. Consider the function $g(x):=T_{p+1}(x)/x^{p}$ on $(-\infty,\infty)$ where we define $$g(0) := \frac{1}{p!} = \lim _{x \rightarrow 0 } g(x).$$
The proof proceeds with the following steps:
**Step 1.** We have $g(x) \leq g(0)$ for all $x<0$.
**Proof of Step 1.** First note that for $x \leq 0$ we have $T_{p} \leq 0$ if $p$ is an even number and $T_{p} \geq 0$ if $p$ is an odd number (to see this note that $T_{1}(x) = \exp(x) \geq 0$, $T_{p}(0)=0$ and $T^{(1)}_{p} = T_{p-1}$ for all $p \geq 2$). We now show that $g(x) \leq g(b)$ for all $x \leq b$.
Suppose first that $x < 0$. If $p$ is an even number then $T_{p+1} (x)/ x ^{p} \leq 1/p!$ if and only if $T_{p+1} (x) - x ^{p}/p! \leq 0$. The last inequality is equivalent to $T_{p+2}(x) \leq 0$ which holds because $p$ is an even number. Similarly, if $p$ is an odd number then $T_{p+1} (x)/ x ^{p} \leq 1/p!$ if and only if $T_{p+2}(x) \geq 0$ which holds because $p$ is an odd number. Thus, $g(x) \leq g(0)$ for all $x < 0$.
**Step 2.** Let $f,k:[a,b) \rightarrow \mathbb{R}$ be continuously differentiable functions such that $k^{(1)}(x) \neq 0$ for all $x \in (a,b)$. If $f^{(1)}/k^{(1)}$ is increasing on $(a,b)$ then $(f(x)-f(a))/(k(x)-k(a))$ is increasing in $x$ on $(a,b)$.
**Proof of Step 2.** Step 2 is known as the L’Hospital rule for monotonicity. For a proof see Lemma 2.2 in [@anderson1993inequalities].
**Step 3.** The function $g$ is increasing on $(0,y)$ for all $y>0$.
**Proof of Step 3.** Let $y>0$ and note that the function $T_{1}(x)/p!=\exp(x)/p!$ is increasing on $(0,y)$. Using Step 2 with $f(x)=\exp(x)$ and $k(x)=p!x$ implies that the function $$\frac{\exp(x)-1}{p!x}= \frac{T_{2}(x)}{p!x}$$ is increasing on $(0,y)$. Applying again Step 2 and using the facts that $T_{k+1}^{(1)} = T_{k}$ and $T_{k}(0)=0$ for all $k=2,\ldots$ implies that the function $T_{k}(x)/(x^{k-1}p!/(k-1)!)$ is increasing in $x$ on $(0,y)$ for all $k=2,\ldots$. Choosing $k=p+1$ shows that $g$ is increasing on $(0,y)$.
**Step 4.** We have $$T_{p+1}(sx) \leq \frac{\max (x^{p},0)}{b^{p}}T_{p+1}(sb)$$ for all $ x \leq b$.
**Proof of Step 4.** Step 3 shows that $g$ is an increasing function on $(0,b]$. Hence, $g(x) \leq g(b)$ for all $x \in (0,b]$. Because $g$ is a continuous function we have $g(0) \leq g(b)$. Using Step 1 implies that $g(x) \leq g(b)$ for all $x \leq b$.
Let $x \leq b$ and assume $x \neq 0$. Multiplying each side of the inequality $g(sx) \leq g(sb)$ by the positive number $\max (x^{p},0)$ yields $$\frac{\max (x^{p},0)}{x^{p}}T_{p+1}(sx) \leq \frac{\max (x^{p},0)}{b^{p}}T_{p+1}(sb).$$ Note that $$T_{p+1}(sx) \leq \frac{\max (x^{p},0)}{x^{p}}T_{p+1}(sx).$$ The last inequality holds as equality if $x > 0$ or if $p$ is an even number. If $x<0$ and $p$ is an odd number, then $T_{p+1}(sx) \leq 0$ (see Step 1), so the last inequality holds. We conclude that $$T_{p+1}(sx) \leq \frac{\max (x^{p},0)}{b^{p}}T_{p+1}(sb)$$ for all $ x \leq b$.
To prove Theorem \[Coro: exp upper bound\] apply Step 4 to conclude that $$\exp(sx) \leq \frac{\max (x^{p},0)}{b^{p}}T_{p+1}(sb) + \sum_{j=0}^{p-1} \frac{s^{j}x^{j}}{j!}$$ for all $x \leq b$. Taking expectations in both sides of the last inequality proves Theorem \[Coro: exp upper bound\].
Let $p \geq 2$ be an even number.
\(i) We have $$\begin{aligned}
& \frac{\mathbb{E}X^{p}}{b^{p}} T_{p+1}(sb) + \mathbb{E} \left ( \sum _{j=0} ^{p-1} \frac{s^{j}X^{j}} {j!} \right ) \geq \frac{\mathbb{E} \max (X,0)^{p+1}}{b^{p+1}} T_{p+2}(sb) + \mathbb{E} \left ( \sum _{j=0} ^{p} \frac{s^{j}X^{j}} {j!} \right ) \\
& \Longleftrightarrow \frac{\mathbb{E} X^{p}}{b^{p}} \left (T_{p+1}(sb) - \frac{s^{p}b^{p}}{p!} \right ) \geq \frac{\mathbb{E} \max (X,0)^{p+1}}{b^{p+1}} T_{p+2}(sb) \\
& \Longleftrightarrow b{\mathbb{E} X^{p}} \geq {\mathbb{E} \max (X,0)^{p+1}}
\end{aligned}$$ which holds for a random variable $X$ on $(-\infty,b]$ and an even number $p$ because $bx^{p} \geq \max(x,0)^{p+1}$ for all $x \leq b$.
\(ii) Similarly to part (i) we have $m_{X}(p+1) \geq m_{X}(p+2)$ if and only if $b{\mathbb{E} X^{p+1}} \geq {\mathbb{E} X^{p+2}}$ which holds for a non-negative random variable because $b x^{p+1} \geq x^{p+2}$ for all $0 \leq x \leq b$.
Proofs of the results in Section \[Sec: concert Hoeffiding\]
------------------------------------------------------------
\[Proof of Theorem \[Thm: concent hof\]\] Let $p \geq 2$ be an integer. We will use the following notations in proof. Let $X$ be a random variable on $[0,b]$. Denote $\mathbb{E}(X^{k}) = \mu ^{k}$ for all $k=1,\ldots,p$.
For every integer $p \geq 1$ we define the function $$v(y,b,\mu^{1},\ldots,\mu^{p}) := \frac{\mu^{p}}{b^{p}} T_{p+1}(y) +
\sum _{j=0} ^{p-1} \frac{y^{j}\mu^{j}} {b^{j}j!}.$$ For all $x \geq 0$ we define the function $$C_{p}(x,b,\mu ^{1},\ldots, \mu ^{p})=\max _{0 \leq y \leq x} \frac{(v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}{(v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}.$$
We denote by $v^{(k)}(y,b,\mu^{1},\ldots,\mu^{p})$ the $k$th derivative of $v$ with respect to its first argument. A straightforward calculation shows that $$v^{(k)}(y,b,\mu^{1},\ldots,\mu^{p}) = \frac{\mu^{p} } {b^{p}} T_{p+1-k}(y) + \sum _{j=0} ^{p-1-k} \frac{\mu^{j+k}y^{j}} {b^{j+k}j!}$$ Thus, $$v^{(1)}(0,b,\mu^{1},\ldots,\mu^{p})= \frac{\mathbb{E}(X)}{b} > 0 \text{ and } v^{(2)}(0,b,\mu^{1},\ldots,\mu^{p})=\frac{\mathbb{E}(X^{2})}{b^{2}} > 0.$$ Because $v^{(2)}$ and $v^{(1)}$ are increasing in the first argument as the sum of increasing functions, we conclude that $v^{(2)}(y,b,\mu^{1},\ldots,\mu^{p})$ and $v^{(1)}(y,b,\mu^{1},\ldots,\mu^{p})$ are positive for every $y \in [0,x]$ and all $x>0$.
The proof proceeds with the following steps:
**Step 1.** We have $\mu^{d+2}\mu^{d} \geq (\mu^{d+1})^{2}$ for every positive integer $d$.
**Proof of Step 1.** Let $d$ be a positive integer. From the Cauchy-Schwarz inequality for the (positive) random variables $X^{(d+2)/2}$ and $X^{d/2}$ we have $$\mathbb{E}X^{d/2} X^{(d+2)/2} \leq \sqrt{\mathbb{E}X^{d} \mathbb{E} X^{d+2}}.$$ That is, we have $\mu^{d+2}\mu^{d} \geq (\mu^{d+1})^{2}$ which proves Step 1.
**Step 2.** For every positive integer $p$ and all $x>0$ the function $v^{(1)}(y,b,\mu^{1},\ldots,\mu^{p})$ is log-convex in $y$ on $(0,x)$ (i.e., $\log(v^{(1)}(y,b,\mu^{1},\ldots,\mu^{p}))$ is a convex function on $(0,x)$).
**Proof of Step 2.** Fix a positive integer $p$ and $x>0$. Let $$w(y):= T_{p}(y) + \frac{b^{p}}{\mu^{p}}\sum _{j=0} ^{p-2} \frac{\mu^{j+1}y^{j}} {b^{j+1}j!} = \frac{b^{p} v^{(1)}(y,b,\mu^{1},\ldots,\mu^{p})}{\mu^{p}}.$$ To prove Step 2 it is enough to prove that $w$ is log-convex on $(0,x)$. Note that $$w(y) = \exp(y) + \sum _{j=0} ^{p-2} \frac{y^{j}} {j!}\beta_{j+1}$$ where $$\beta_{j} = \frac{b^{p-j}\mu^{j}}{\mu^{p}} - 1$$ for $j=1,\ldots,p$. We have $\beta_{j} \geq 0$ for all $j=1,\ldots,p-1$. To see this note that $x^{j}b^{p-j} \geq x^{p}$ for all $x \in [0,b]$ so taking expectations implies that $\beta_{j} \geq 0$.
$w$ is log-convex on $(0,x)$ if and only if $w^{(1)}/w$ is increasing on $(0,x)$. For every integer $k=0,\ldots,p-2$ define the function $$w_{k}(y) = \exp(y) + \sum _{j=0} ^{k} \frac{y^{j}} {j!}\beta_{p-1+j-k}$$ and note that $w_{p-2} = w$. By construction we have $w_{k}^{(1)} = w_{k-1}$ We now show that $w_{k}$ is log-convex on $(0,x)$ for all $k=0,\ldots,p-2$. The proof is by induction.
For $k=0$ the function $$\frac{w^{(1)}_{0}(y)}{w_{0}(y)} = \frac{\exp(y)}{\exp(y) + \beta_{p-1}}$$ is increasing because $ \beta_{p-1} \geq 0$ and the function $x/(x+d)$ is increasing in $x$ on $[0,\infty)$ when $d \geq 0$. We conclude that the function $w_{0} = \exp(y) + \beta_{p-1}$ is log-convex on $(0,x)$.
Assume that $w_{k}$ is log-convex on $(0,x)$ for some integer $0 \leq k \leq p-3$. We show that $w_{k+1}$ is log-convex on $(0,x)$. Log-convexity of $w_{k}$ implies that the function $$\label{Log-convex1}
\frac{w^{(1)}_{k}(y)}{w_{k}(y)} = \frac{\exp(y) + \sum _{j=0} ^{k-1} \frac{y^{j}} {j!}\beta_{p+j-k}}{\exp(y) + \sum _{j=0} ^{k} \frac{y^{j}} {j!}\beta_{p-1+j-k}}$$ is increasing on $(0,x)$. Using the fact that $w^{(1)}_{k+1}=w_{k}$ and applying Step 2 in the proof of Theorem \[Coro: exp upper bound\] we conclude that the function $$m(y):= \frac{w_{k}(y) - w_{k}(0)}{w_{k+1}(y) - w_{k+1}(0)} = \frac{\exp(y) + \sum _{j=0} ^{k} \frac{y^{j}} {j!}\beta_{p-1+j-k} - (1+\beta_{p-1-k})}{\exp(y) + \sum _{j=0} ^{k+1} \frac{y^{j}} {j!}\beta_{p-2+j-k} - (1+\beta_{p-2-k})}$$ is increasing on $(0,x)$. Thus, $m^{(1)}(y) \geq 0$ for all $y \in (0,x)$. That is, $$\label{Log-convex2}
w_{k+1}^{(2)}(y) w_{k+1}(y) -w_{k+1}^{(2)}(y)(1+\beta_{p-2-k}) \geq (w_{k+1}^{(1)}(y))^{2} - w_{k+1}^{(1)}(y) (1+\beta_{p-1-k})$$ for all $y \in (0,x)$. We now show that $w_{k+1}^{(2)}(y) w_{k+1}(y) \geq (w_{k+1}^{(1)}(y))^{2}$. Because $w_{k+1}^{(2)}/w_{k+1}^{(1)}$ is increasing and positive (see (\[Log-convex1\])) we have $$\label{Log-convex3} \frac{w_{k+1}^{(2)}(y)}{w_{k+1}^{(1)}(y)}(1+\beta_{p-2-k}) \geq (1+\beta_{p-1-k})$$ for all $y \in (0,x)$ if the last inequality holds for $y=0$, i.e., if $$\begin{aligned}
(1+\beta_{p-k}) (1+ \beta_{p-2-k}) \geq (1+ \beta_{p-1-k})^{2} & \Longleftrightarrow \left (\frac{b^{k}\mu^{p-k}}{\mu^{p}} \right ) \left (\frac{b^{k+2}\mu^{p-2-k}}{\mu^{p}} \right ) \geq \left (\frac{b^{k+1}\mu^{p-k-1}}{\mu^{p}} \right )^{2} \\
& \Longleftrightarrow \mu^{p-k} \mu^{p-k-2} \geq (\mu^{p-k-1})^{2}\end{aligned}$$ which holds from Step 1. We conclude that inequality (\[Log-convex3\]) holds. Using inequality (\[Log-convex2\]) we have $$w_{k+1}^{(2)}(y) w_{k+1}(y) - (w_{k+1}^{(1)}(y))^{2} \geq w_{k+1}^{(2)}(y)(1+\beta_{p-2-k}) - w_{k+1}^{(1)}(y)(1+\beta_{p-1-k}) \geq 0.$$ That is, $w_{k+1}^{(2)}(y) w_{k+1}(y) \geq (w_{k+1}^{(1)}(y))^{2}$ for all $y \in (0,x)$. We conclude that $w^{(1)}_{k+1}/w_{k+1}$ is increasing on $(0,x)$, i.e., $w_{k+1}$ is log-convex. This shows that $w_{k}$ is log-convex for all $k=0,\ldots,p-2$. In particular, $w_{p-2}:=w$ is log-convex which proves Step 2.
**Step 3.** We have $$C_{p}(x,b,\mu^{1},\ldots,\mu^{p}) = \left ( \frac{\exp(x) + \sum _{j=0} ^{p-3} \frac{x^{j}} {j!} \left ( \frac{b^{p-j-2}\mu^{j+2}}{\mu^{p}} - 1 \right )}{\exp(x) + \sum _{j=0} ^{p-2} \frac{x^{j}} {j!} \left ( \frac{b^{p-j-1}\mu^{j+1}}{\mu^{p}} - 1 \right )} \right )^{2}.$$ for all $x \geq 0$.
**Proof of Step 3.** Let $x \geq 0$. From Step 2 the function $v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p})b^{p}/\mu^{p}:=w(y)$ is log-convex on $(0,x)$ where $w$ is defined in the proof of Step 2. This implies that $$C_{p}(x,b,\mu ^{1},\ldots, \mu ^{p})=\max _{0 \leq y \leq x} \frac{(v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}{(v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}} = \max _{0 \leq y \leq x} \left (\frac{w^{(1)}(y)} {w(y)} \right )^{2} = \left (\frac{w^{(1)}(x)} {w(x)} \right )^{2}$$ which proves Step 3.
**Step 4.** For all $x \geq 0$ we have $$\max _{ 0 \leq y \leq x} \left ( \frac{v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})}{v(y,b,\mu ^{1},\ldots, \mu ^{p})} - \frac{(v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}{(v(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}} \right ) \leq \frac{1}{4} C_{p}(x,b,\mu ^{1},\ldots, \mu ^{p}).$$ **Proof of Step 4.** For all $x \geq 0$ we have $$\begin{aligned}
& \max _{ 0 \leq y \leq x} \left ( \frac{v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})}{v(y,b,\mu ^{1},\ldots, \mu ^{p})} - \frac{(v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}{(v(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}} \right )\\
& = \max _{ 0 \leq y \leq x} \left ( \frac{v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})}{v(y,b,\mu ^{1},\ldots, \mu ^{p})} \left (1 - \frac{v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})(v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}{v(y,b,\mu ^{1},\ldots, \mu ^{p})(v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}} \right ) \right ) \\
& \leq \max _{0 \leq y \leq x} \left ( \frac{v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})}{v(y,b,\mu ^{1},\ldots, \mu ^{p})} \left (1 - \frac{v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})}{v(y,b,\mu ^{1},\ldots, \mu ^{p}) C_{p}(x,b,\mu ^{1},\ldots, \mu ^{p})} \right ) \right ) \\
& \leq \frac{1}{4} C_{p}(x,b,\mu ^{1},\ldots, \mu ^{p}).\end{aligned}$$ The first inequality follows from the definition of $C_{p}$ and because $v^{(2)} > 0$ and $v^{(1)} > 0$. The second inequality follows from the elementary inequality $x(1-x/z) \leq 0.25z$ for all $z>0$ and $x>0$.
**Step 5.** We have $$\begin{aligned}
\mathbb{E} \exp (s(X-\mathbb{E}(X)) \leq \exp \left ( \frac{s^2 b^{2}C_{p}(sb,b,\mu ^{1}, \ldots , \mu ^{p} ) }{8} \right )\end{aligned}$$ **Proof of Step 5.** From Theorem \[Coro: exp upper bound\] for all $s \geq 0$ we have $$\begin{aligned}
\mathbb{E}\exp (sX) & \leq \frac{\mathbb{E} X ^{p} } {b^{p}} T_{p+1}(sb) + \mathbb{E} \left ( \sum _{j=0} ^{p-1} \frac{s^{j}X^{j}} { j!} \right ) \\
& = v(y,b,\mu^{1},\ldots,\mu^{p})\end{aligned}$$ where $y=sb \geq 0$. Define the function $$\begin{aligned}
g(y) & = \ln \left ( v(y,b,\mu^{1},\ldots,\mu^{p}).
\right )\end{aligned}$$ Clearly $v$ is a positive function so the function $g :\mathbb{R}_{ +} \rightarrow \mathbb{R}$ is well defined. Note that $\mathbb{E}\exp (sX) \leq \exp (g(y))$. Recall that $$v^{(1)}(0,b,\mu^{1},\ldots,\mu^{p})= \frac{\mathbb{E}(X)}{b} > 0 \text{ and } v^{(2)}(0,b,\mu^{1},\ldots,\mu^{p})=\frac{\mathbb{E}(X^{2})}{b^{2}} > 0.$$ Because $v(0,b,\mu^{1} , \ldots , \mu^{p}) = 1$ we have $g(0) =\ln (1) =0$. We have $$\begin{aligned}
g^{(1)}(y) = \frac{v^{(1)}(y,b, \mu^{1},\ldots,\mu^{p})}{v(y,b,\mu^{1},\ldots,\mu^{p})}.\end{aligned}$$ Thus, $g^{(1)}(0) = \mathbb{E}(X)/b$. Differentiating again yields $$\begin{aligned}
g^{ (2) }(y) & =
\frac{v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})}{v(y,b,\mu ^{1},\ldots, \mu ^{p})} - \frac{(v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}{(v(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}. \end{aligned}$$
From Taylor’s theorem for all $y \geq 0$ there exists a $z \in [0,y]$ such that $g(y) = g(0) + yg^{(1)}(0) + 0.5 y^2 g^{(2)}(z) $. Thus, using the fact that $y=sb$ we have $$\begin{aligned}
g(y) = g(0) + yg^{(1)}(0) + 0.5 y^2 g^{(2)}(z) = s\mathbb{E}(X)+ 0.5 s^2 b^{2} g^{(2)}(z) \leq s\mathbb{E}(X) + 0.5 s^2 b^{2} V(sb,b,\mu ^{1}, \ldots , \mu ^{p} ) \end{aligned}$$ where $$V(y,b,\mu ^{1}, \ldots , \mu ^{p} ) = \sup _{0 \leq z \leq y} g^{(2)} (z).$$ Using $\mathbb{E}\exp(sX) \leq \exp (g(y))$ and Step 4 imply $$\begin{aligned}
\mathbb{E} \exp (s(X-\mathbb{E}(X)) & \leq \exp \left ( \frac{ s^2 b^{2}V(sb,b,\mu ^{1}, \ldots , \mu ^{p} ) } {2} \right ) \\
& \leq \exp \left ( \frac{s^2 b^{2}C_{p}(sb,b,\mu ^{1}, \ldots , \mu ^{p} ) }{8} \right )\end{aligned}$$
**Step 6.** For all $t>0$ we have $$\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left (- \frac{t^{2}}{2 \sum_{i=1}^{n} b^{2} _{i} C_{p} \left (\frac{4tb_{i}}{\sum _{i=1} ^{n} d(X_{i}) },b_{i}, \mu ^{1}_{i}, \ldots , \mu ^{p}_{i} \right ) } \right ).$$
**Proof of Step 6.** Using independence, Step 5, and Markov’s inequality, a standard argument shows that: $$\begin{aligned}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) & \leq \exp(-st) \mathbb{E} \exp (s(S_{n}-\mathbb{E}(S_{n})) \\
& = \exp(-st) \prod_{i=1}^{n} \mathbb{E} \exp (s(X_{i}-\mathbb{E}(X_{i})) \\
& \leq \exp(-st) \prod_{i=1}^{n} \exp \left ( \frac{s^2 b^{2}_{i} C_{p}(sb_{i},b_{i},\mu ^{1}_{i}, \ldots , \mu ^{p}_{i} ) }{8} \right )\\
& = \exp \left (-st + \frac{s^{2} } {8} \sum_{i=1}^{n} b^{2}_{i} C_{p} (sb_{i},b_{i}, \mu ^{1}_{i}, \ldots , \mu ^{p}_{i} ) \right ) \end{aligned}$$ Let $$s = \frac{4t}{\sum _{i=1} ^{n} b^{2}_{i} C_{p} \left (\frac {4tb_{i} } { \sum _{i=1}^{n} d(X_{i})},b_{i}, \mu ^{1}_{i}, \ldots , \mu ^{p}_{i} \right )}$$ Note that $$C_{p}(0,b_{i},\mu ^{1}_{i}, \ldots , \mu ^{p}_{i} ) = \left ( \frac{\mu^{2}_{i}}{\mu^{1}_{i} b_{i}} \right )^{2} = \frac {d(X_{i})} {b_{i}^{2}}$$ Because $C_{p}$ is increasing in the first argument we have $$\label{Eq: Vp proof H}
C_{p}(y,b_{i},\mu ^{1}_{i}, \ldots , \mu ^{p}_{i} ) \geq \frac {d(X_{i})} {b_{i}^{2}} > 0$$ for all $y \geq 0$. Thus, $$sb_{i} = \frac{4tb_{i}}{\sum _{i=1} ^{n} b^{2} _{i} C_{p} \left (\frac {4tb_{i} } { \sum _{i=1}^{n} d(X_{i})},b_{i}, \mu ^{1}_{i}, \ldots , \mu ^{p}_{i} \right )} \leq \frac{4tb_{i}}{\sum _{i=1}^{n} d(X_{i})}.$$ Using again the fact that $C_{p}$ is increasing in the first argument implies $$\begin{aligned}
-st + \frac{ s^{2} } {8} \sum_{i=1}^{n} b^{2} _{i} C_{p}(sb_{i},b_{i} , \mu ^{1}_{i}, \ldots , \mu ^{p}_{i} ) & \leq -st + \frac{s^{2} }{8} \sum_{i=1}^{n} b^{2} _{i} C_{p} \left (\frac{4tb_{i}}{\sum _{i=1} ^{n} d(X_{i}) },b_{i},\mu ^{1}_{i}, \ldots , \mu ^{p}_{i} \right ) \\
& = - \frac{2t^{2}}{\sum_{i=1}^{n} b^{2} _{i} C_{p} \left (\frac{4tb_{i}}{\sum _{i=1} ^{n} d(X_{i}) },b_{i},\mu ^{1}_{i}, \ldots , \mu ^{p}_{i} \right ) }.\end{aligned}$$ We conclude that $$\label{Hof theorem Conc1}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) \leq \exp \left (- \frac{2t^{2}}{\sum_{i=1}^{n} b^{2} _{i} C_{p} \left (\frac{4tb_{i}}{\sum _{i=1} ^{n} d(X_{i}) },b_{i}, \mu ^{1}_{i}, \ldots , \mu ^{p}_{i} \right ) } \right ).$$ which proves Step 6.
Combining Steps 3 and 6 proves part (i).
\(ii) Let $X$ be a random variable on $[0,b]$. Denote $\mathbb{E}(X^{k}) = \mu ^{k}$ for all $k=1,\ldots,p$. Clearly $0 < C_{p}$ because $v^{(2)}$ and $v^{(1)}$ are positive functions (see part (i)).
We show that $v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}) \leq v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p})$ for all $y \geq 0$.
We have $v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}) \leq v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p})$ if and only if $$\frac{\mu ^{p}}{b^{p}}T_{p-1}(y) + \sum _{j=0} ^{p-3} \frac{y^{j}\mu^{j+2}} {b^{j+2}j!} \leq \frac{\mu ^{p}}{b^{p}}T_{p}(y) + \sum _{j=0} ^{p-2} \frac{y^{j}\mu^{j+1}} {b^{j+1}j!}.$$ The last inequality holds if and only if $$\begin{aligned}
& \frac{ \mu ^{p} y^{p-2}}{b^{p}(p-2)!} + \sum _{j=0} ^{p-3} \frac{y^{j}\mu ^{j+2}} {b^{j+2}j!} - \sum _{j=0} ^{p-2} \frac{y^{j}\mu^{j+1}} {b^{j+1}j!} \leq 0 \\
& \iff \sum _{j=0} ^{p-2} \frac{y^{j}\mu^{j+2}} {b^{j+2}j!} - \sum _{j=0} ^{p-2} \frac{y^{j}\mu ^{j+1}} {b^{j+1}j!} \leq 0. \end{aligned}$$ To see that the last inequality holds let $0 \leq x \leq b$. We have $b^{j+1}x^{j+2} \leq x^{j+1}b^{j+2}$. Taking expectations and multiplying by $y^{j} / j!$ show that $$\frac{y^{j}\mu^{j+2}} {b^{j+2}j!} \leq \frac{y^{j}\mu^{j+1}} {b^{j+1}j!}$$ for all $1 \leq j \leq p-2$ and all $y \geq 0$. We conclude that $v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}) \leq v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p})$ for all $y \geq 0$. Thus, $$C_{p}(x,b,\mu ^{1},\ldots, \mu ^{p})=\max _{0 \leq y \leq x} \frac{(v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}}{(v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}))^{2}} \leq 1$$ which immediately implies that inequality (\[Ineq: Hoef bound\]) is tighter then Hoeffding’s inequality which corresponds to $C_{p}=1$ (when $p=1$ the argument above shows that $v^{2} = v^{1}$ so $C_{1} = 1$ and we derive Hoeffding’s inequality (\[Ineq: Hoef bound p=1\])).
\(iii) Let $Z$ be a random variable on $[0,b]$. Denote $\mathbb{E}(Z^{k}) = \mu ^{k}$ for all $k=1,\ldots$ and let $y \in [0,x]$ for some $x \geq 0$.
First note that $0 \leq \lim _{p \rightarrow \infty} b^{-p}\mu^{p}T_{p}(y) \leq \lim _{p \rightarrow \infty} T_{p}(y) = 0$. In addition, for every $z \in [0,b]$ we have $$\lim _{p \rightarrow \infty} b^{-1} z \exp(yz/b) - b^{-1} z \sum _{j=0} ^{p-2} \frac{y^{j} z^{j} } {b^{j}j!} =0.$$ Because $Z$ is a random variable on $[0,b]$ we can use the bounded convergence theorem to conclude that $$\begin{aligned}
\lim _{p \rightarrow \infty} v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p}) = \lim _{p \rightarrow \infty} \frac{\mu ^{p}}{b^{p}}T_{p}(y) + \sum _{j=0} ^{p-2} \frac{y^{j}\mu^{j+1}} {b^{j+1}j!} = b^{-1}\mathbb{E}Z \exp(yZ/b). \end{aligned}$$
Similarly, $$\begin{aligned}
\lim _{p \rightarrow \infty} v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p}) = \lim _{p \rightarrow \infty} \frac{\mu ^{p}}{b^{p}}T_{p-1}(y) + \sum _{j=0} ^{p-3} \frac{y^{j}\mu^{j+2}} {b^{j+2}j!} = b^{-2}\mathbb{E}Z^{2} \exp(yZ/b). \end{aligned}$$ We conclude that $$\lim _{p \rightarrow \infty} \left ( \frac{ v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})} {v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p})} \right ) ^{2} = \left ( \frac{b^{-2}\mathbb{E}Z^{2} \exp(yZ/b)}{b^{-1}\mathbb{E}Z \exp(yZ/b)} \right )^{2} = b^{-2} \left ( \frac{\mathbb{E}Z^{2} \exp(yZ/b)}{\mathbb{E}Z \exp(yZ/b)} \right )^{2}$$
Using Step 3 in the proof of part (i) yields $$\begin{aligned}
\lim _{p \rightarrow \infty} C_{p} (x,b_{i},\mu^{1},\ldots,\mu^{p}) & = \lim _{p \rightarrow \infty} \max _{0 \leq y \leq x} \left ( \frac{ v^{(2)}(y,b,\mu ^{1},\ldots, \mu ^{p})} {v^{(1)}(y,b,\mu ^{1},\ldots, \mu ^{p})} \right ) ^{2} \\
& = \lim _{p \rightarrow \infty} \left ( \frac{ v^{(2)}(x,b,\mu ^{1},\ldots, \mu ^{p})} {v^{(1)}(x,b,\mu ^{1},\ldots, \mu ^{p})} \right )^{2} \\
& = b^{-2} \left ( \frac{\mathbb{E}Z^{2} \exp(xZ/b)}{\mathbb{E}Z \exp(xZ/b)} \right )^{2}\end{aligned}$$ which proves part (iii).
Proofs of the results in Section \[Sec: concert Bennet\]
--------------------------------------------------------
\(i) Let $ s \geq 0$ and let $p \geq 2$ be an integer. We first assume that $b=1$ so that $X_{i}$ is a random variable on $(-\infty,1]$ for all $i=1,\ldots,n$.
For any random variable $X_{i}$ on $(-\infty,1]$ we have $$\begin{aligned}
\mathbb{E}\exp (sX_{i}) & \leq \mu ^{p}_{i} \left (\exp(s) - \sum _{j=0} ^{p-1} \frac{s^{j}}{j!} \right ) + 1 + \sum_{j=1}^{p-1} \frac{s^{j}\mu ^{j}_{i}}{j!} \\
& \leq \exp \left (\mu ^{p}_{i}\left (\exp(s) - \sum _{j=0} ^{p-1} \frac{s^{j}}{j!} \right ) + \sum_{j=1}^{p-1} \frac{s^{j}\mu ^{j}_{i}}{j!} \right ) \\
& = \exp \left (\mu ^{p}_{i} T_{p+1}(s) + \sum_{j=1}^{p-1} \frac{s^{j}\mu ^{j}_{i}}{j!} \right )\end{aligned}$$ The first inequality follows from Theorem \[Coro: exp upper bound\] and the fact that $T_{p+1}(s) \geq 0$ for $s \geq 0$. The second inequality follows from the elementary inequality $1 + x \leq e^{x}$ for all $x \in \mathbb{R}$. Thus, $$\mathbb{E}\exp (s(X_{i}-\mu ^{1}_{i})) \leq \exp \left (\mu ^{p}_{i} T_{p+1}(s) + \sum_{j=2}^{p-1} \frac{s^{j}\mu ^{j}_{i}}{j!} \right )$$ and $$\begin{aligned}
\prod_{i=1}^{n} \mathbb{E} \exp (s(X_{i}-\mathbb{E}(X_{i})) & \leq \prod_{i=1}^{n} \exp \left (\mu ^{p}_{i} T_{p+1}(s) + \sum_{j=2}^{p-1} \frac{s^{j}\mu ^{j}_{i}}{j!} \right ) \\
& = \exp \left (\mu ^{p}T_{p+1}(s) + \sum_{j=2}^{p-1} \frac{s^{j}\mu ^{j}}{j!} \right )\end{aligned}$$
From the Chernoff bound and the fact that $X_{1},\ldots,X_{n}$ are independent random variables, for all $t > 0$, we have $$\begin{aligned}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) & \leq \inf _{s \geq 0} \exp(-st) \mathbb{E} \exp (s(S_{n}-\mathbb{E}(S_{n})) \\
& = \inf _{s \geq 0} \exp(-st) \prod_{i=1}^{n} \mathbb{E} \exp (s(X_{i}-\mathbb{E}(X_{i})) \\
& \leq \inf _{s \geq 0} \exp \left (-st + \sum_{j=2}^{p-1} \frac{s^{j}\mu ^{j}}{j!} + \mu ^{p} T_{p+1}(s) \right ) \\
& = \exp \left ( - \sup _{s \geq 0} \left ( st - \sum_{j=2}^{p-1} \frac{s^{j}\mu ^{j}}{j!} - \mu ^{p} T_{p+1}(s) \right ) \right ) \\
& = \exp \left (- \mu^{p} \sup _{x \geq 0} h_{p} (x,t, \mu ^{2},\ldots,\mu^{p} ) \right )\end{aligned}$$ where $$\begin{aligned}
h_{p}(x,t,\mu ^{2},\ldots,\mu^{p} ) & = \frac{t}{\mu^{p}}x - \frac{1}{\mu ^{p}} \sum_{j=2}^{p-1} \frac{x^{j}\mu ^{j}}{ j!} - T_{p+1}(x) \\
& = 1 + \left ( \frac{t}{\mu^{p}} + 1\right ) x - \sum_{j=2}^{p-1} \left (\frac{\mu ^{j}}{\mu^{p} j!} - \frac{1}{j!} \right ) x^{j} - \exp(x)
\end{aligned}$$ Because $h_{p}$ is continuous, $h_{p}(0,t,\mu ^{2},\ldots,\mu^{p}) = 0$, and $\lim _{x \rightarrow \infty} h_{p}(x,t, b,\mu ^{2},\ldots,\mu^{p}) = -\infty$, the function $h_{p}$ has a maximizer. Let $h^{(j)}_{p}$ the $j$th derivative of $h_{p}$ with respect to $x$.
Note that $$\begin{aligned}
h^{(1)}_{p}(x,t,\mu ^{2},\ldots,\mu^{p} ) & = \frac{t}{\mu^{p}} + 1 - \sum_{j=1}^{p-2} \left (\frac{\mu ^{j+1}}{\mu ^{p} j!} - \frac{1}{j!} \right ) x^{j} - \exp(x) \\
& = \alpha_{0} - \sum _{j=1}^{p-2} \alpha_{j} x^{j} - \exp(x)
\end{aligned}$$ Thus, $h^{(1)}_{p} (0,t,\mu ^{2},\ldots,\mu^{p} ) = \alpha_{0} - \exp(0) > 0 $ and $h^{(1)}_{p} (x,t,\mu ^{2},\ldots,\mu^{p} ) < 0$ for all $x \geq \overline{x}$ for some large $\overline{x}$. Because $h^{(1)}_{p}$ is continuous we conclude that the maximizer $y$ of $h_{p}$ on $[0,\infty)$ satisfies $h^{(1)}_{p} (y,t,\mu ^{2},\ldots,\mu^{p} ) = 0$, that is, $y \in G_{p-2}(\alpha_{0},\ldots,\alpha_{p-2})$. Plugging $y$ into $h_{p}$ yields $$\begin{aligned}
& \left ( \frac{t}{\mu^{p}} + 1\right ) y - \sum_{j=2}^{p-1} \left (\frac{\mu ^{j}}{\mu^{p} j!} - \frac{1}{j!} \right ) y^{j} + 1 - \exp(y) \\
& = \left ( \frac{t}{\mu^{p}} + 1\right ) y - \sum_{j=2}^{p-1} \left (\frac{\mu ^{j}}{\mu^{p} j!} - \frac{1}{j!} \right ) y^{j} - \frac{t}{\mu^{p}} + \sum_{j=1}^{p-2} \left (\frac{\mu ^{j+1}}{ \mu^{p} j!} - \frac{1}{j!} \right ) y^{j} \\
& = - \frac{t}{\mu^{p}} + \left ( \frac{t}{\mu^{p}} + \frac{\mu^{2}}{\mu^{p}}\right ) y - \frac{1}{\mu^{p}} \sum_{j=2}^{p-1} \left (\frac{\mu ^{j}}{ j!} - \frac{\mu^{j+1}}{j!} \right ) y^{j}.\end{aligned}$$ In the first equality we used the fact that $h^{(1)}_{p} (y,t,\mu ^{2},\ldots,\mu^{p} ) = 0$. Thus, $$\begin{aligned}
\label{Bennett Proof1}
\begin{split}
\mathbb{P}(S_{n}-\mathbb{E}(S_{n}) \geq t) & \leq \exp \left ( - \mu^{p} \left ( - \frac{t}{\mu^{p}} + \left ( \frac{t}{\mu^{p}} + \frac{\mu^{2}}{\mu^{p}}\right ) y - \frac{1}{\mu^{p}} \sum_{j=2}^{p-1} \left (\frac{\mu ^{j}}{ j!} - \frac{\mu^{j+1}}{j!} \right ) y^{j}\right ) \right ) \\
& = \exp \left ( - \max _{y \in G_{p-2}(\alpha_{0},\ldots,\alpha_{p-2})} \left ( - t + \left ( t + \mu^{2}\right ) y - \sum_{j=2}^{p-1} \left (\frac{\mu ^{j}}{ j!} - \frac{\mu^{j+1}}{j!} \right ) y^{j} \right ) \right )
\end{split}\end{aligned}$$ which proves part (i) for the case that $b=1$. Now suppose that $b \neq 1$ and $X_{i} \leq b$ for some $b>0$. Define the random variable $Y_{i} = X_{i}/b$ and note that $Y_{i} \leq 1$ and $ \mathbb{E}Y_{i}^{k} \leq \mu ^{k}_{i}/b^{k}$. Thus, we can apply inequality (\[Bennett Proof1\]) for the random variables $Y_{1},\ldots,Y_{n}$ to conclude that for all $t>0$ we have $$\begin{aligned}
\begin{split}
& \mathbb{P} (S_{n}-\mathbb{E}(S_{n}) \geq t) = \mathbb{P} \left (\sum _{i=1} ^{n} Y_{i} -\mathbb{E}\left (\sum _{i=1} ^{n} Y_{i} \right ) \geq \frac{t}{b} \right ) \\
& \leq \exp \left ( - \max _{y \in G_{p-2}(\alpha_{0},\ldots,\alpha_{p-2})} \left ( \frac{t}{b} - \left ( \frac{t}{b} + \frac{\mu^{2} }{b^{2} }\right) y + \sum_{j=2}^{p-1} \left (\frac{\mu^{j}}{b^{j}j!} - \frac{\mu ^{j+1}}{b^{j+1}j!} \right ) y ^{j} \right ) \right )
\end{split}\end{aligned}$$ where $$\alpha_{0} = 1 + \frac{tb^{p-1}}{\mu^{p}} > 1 \text{ and } \alpha_{j} = \frac{b^{p-j-1} \mu ^{j+1}}{\mu^{p} j!} - \frac{1}{j!}$$ for all $j=1,\ldots,p-2$. This proves part (i).
\(ii) Suppose for simplicity that $b=1$ (as in part (i) part (ii) holds for any $b>0$ when it holds for $b=1$). Note that $$h^{(1)}_{p}(x,t,\mu ^{2},\ldots,\mu^{p} ) = \frac{t}{\mu^{p}} - \frac{1}{\mu ^{p}} \sum_{j=1}^{p-2} \frac{x^{j}\mu ^{j+1}}{ j!} - T_{p}(x)$$ so if $\mu^{j} \geq 0$ for every odd number $j \geq 3$, $j \neq p$, then $h^{(1)}_{p}$ is strictly decreasing on $(0,\infty)$. Hence, there is a unique positive solution for the equation $h^{(1)}_{p}(x,t,\mu ^{2},\ldots,\mu^{p} ) = 0$ which implies that the set $G_{p-2}(\alpha_{0},\ldots,\alpha_{p-2})$ consists only one element (see the proof of part (i)).
\(iii) Assume that $p=2$. Then the unique solution to the equation $\alpha_{0} = \exp(x)$ is $\ln(\alpha_{0})$. Thus, $G_{2}(\alpha_{0}) = \{ y \} $ where $$y=\ln \left (1 + \frac{tb}{\mu^{2}} \right ).$$ Plugging $y$ into equation (\[Thm: concent bennett\]) proves part (iii).
\(iv) Assume that $p=3$. From part (ii) $G_{3}$ consists of one element. Note that $bx^{2} \geq \max (x^{3},0)$ for all $x \leq b$. Thus, $b\mu^{2}_{i} \geq \mu^{3}_{i}$ for all $i=1,\ldots,n$. Hence, $\alpha _{1}$ is non-negative. Because $\alpha _{0} >1$ and $\alpha _{1} >0$ (if $\alpha_{1}=0$ we get Bennett’s inequality as in part (iii)), $G_{3}(\alpha_{0},\alpha_{1}) = \{ y \} $ where $y$ is the unique and positive solution to the equation $\exp (x) = \alpha _{0} - \alpha_{1}x$ that is given by $$y=\frac{\alpha _{0}}{\alpha_{1}} - W\left ( \frac{\exp (\alpha_{0} / \alpha_{1}) }{\alpha_{1}} \right )$$ where $W$ is the Lambert $W$-function (see [@corless1996lambertw]). Plugging $y$ into equation (\[Thm: concent bennett\]) proves part (iv).
As in Theorem \[Thm: concent bennett\] we denote $\sum _{i=1} ^{n} \mathbb{E}(\max (X ^{3}_{i},0)) = \mu ^{3}$ and $\sum _{i=1} ^{n} \mathbb{E} X ^{2}_{i} = \mu ^{2}$. We can assume without loss of generality that $b=1$ so that $X_{i}$ is a random variable on $(-\infty,1]$ for all $i=1,\ldots,n$ (see the proof of Theorem \[Thm: concent bennett\] part (i)).
Let $$\mu ^{p} h_{p}(x,t,\mu ^{2},\ldots,\mu^{p} ) := tx - \sum_{j=2}^{p-1} \frac{x^{j}\mu ^{j}}{ j!} - \mu ^{p} T_{p+1}(x)$$ for $p=2$ and $p=3$ (see the proof of Theorem \[Thm: concent bennett\] for the definition of $h_{p}$).
For all $x \geq 0$, we have $$\begin{aligned}
&\mu^{2} h_{2}(x,t,\mu ^{2} ) \leq \mu^{3} h_{3}(x,t,\mu ^{2}, \mu^{3} ) \\
& \Longleftrightarrow tx - \mu ^{2} T_{3}(x) \leq tx - \frac{x^{2}\mu ^{2}}{ 2} - \mu ^{3} T_{4}(x) \\
& \Longleftrightarrow \mu ^{3} T_{4}(x) \leq \mu^{2} T_{4}(x) \\
& \Longleftrightarrow \mu ^{3} \leq \mu ^{2}\end{aligned}$$ which holds because $\max (x^3,0) \leq x^2$ for all $x \leq 1$. We conclude that $\mu^{2} h_{2} \leq \mu ^{3} h_{3}$. Thus, $$\exp (- \mu^{2} \sup _{x \geq 0} h_{2}(x,t,\mu ^{2} ) ) \geq \exp ( - \mu^{3} \sup _{x \geq 0} h_{3}(x,t,\mu ^{2}, \mu^{3} ) ).$$ From the proof of Theorem \[Thm: concent bennett\] the left-hand side of the last inequality equals the right-hand side of inequality (\[Ineq: Bennett LOG p=2 CONCENT\]) and the right-hand side of the last inequality equals the right-hand side of inequality (\[Ineq: Bennett p=3 W-lamb\]).
[^1]: Graduate School of Business, Stanford University, Stanford, CA 94305, USA. e-mail:
[^2]: There are many extensions and generalizations of Hoeffding’s and Bennett’s inequalities. For example see [@freedman1975tail], [@pinelis1994optimum], [@talagrand1995missing], [@roussas1996exponential], [@cohen1999optimal], [@victor1999general], [@bousquet2002bennett], [@bentkus2004hoeffding], [@klein2005concentration], [@kontorovich2008concentration], [@fan2012hoeffding], [@junge2013noncommutative], [@pinelis2014bennett], and [@pelekis2015bernstein].
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abstract: 'We study a modification of electromagnetism which violates Lorentz invariance at large distances. In this theory, electromagnetic waves are massive, but the static force between charged particles is Coulomb not Yukawa. At very short distances the theory looks just like QED. But for distances larger than $1/m$ the massive dispersion relation of the waves can be appreciated, and the Coulomb force can be used to communicate faster than the speed of light. In fact, electrical signals are transmitted instantly, but take a time $\sim 1/m$ to build up to full strength. After that, undamped oscillations of the electric field are set in and continue until they are dispersed by the arrival of the Lorentz-obeying part of the transmission. We study experimental constraints on such a theory and find that the Compton wavelength of the photon may be as small as 6000 km. This bound is weaker than for a Lorentz-invariant mass, essentially because in our case the Coulomb constraint is removed.'
author:
- Gia Dvali
- Michele Papucci
- 'Matthew D. Schwartz$^2$'
title: Infrared Lorentz Violation and Slowly Instantaneous Electricity
---
Quantum Electrodynamics and Lorentz invariance are firmly established to astounding accuracy. But most of the evidence for both of these ideas comes from short distance tests. These tests are justified, because we expect new ideas to become relevant at short distances, but there are plenty of signs that something strange may happen at large distances too (for example, dark energy), where the constraints are much weaker. In fact, even drastic modifications of the standard model may not be ruled out. To explore this possibility, we consider a scenario in which the photon has a Lorentz-violating mass $m$. We find a number of counterintuitive results. The photon mass does not slow the field down, but speeds it up. In fact, the electric field can now transmit signals instantly, although it takes a fixed time $\sim 1/m$ for the signal to mature. We also find that the bound on the photon mass is actually weaker than for the Lorentz invariant case.
The simplest infrared Lorentz-violating modification of an Abelian gauge theory is the addition of rotationally invariant mass terms. (During the preparation of this work, we learned about an independent work [@Gabadadze:2004iv], which introduces a similar system.) The most general possibility is $$\mathcal{L_{\mathrm{gen}}} = - \frac{1}{4} F_{\mu \nu}^2 + \frac{1}{2} m_0^2
A_0^2 - \frac{1}{2} m^2 A_i^2 + A_{\mu} J^{\mu} .$$ For $m_0 = m$ this is the Lorentz-invariant Proca theory [@Proca]. The equations of motion imply the generalized Proca constraint $m_0^2 \partial_t A_0
= m^2 \partial_i A_i$ and for the static case let us immediately solve for the scalar potential $$A_0 = \frac{1}{\triangle - m_0^2} J_0 . \label{scalpot}$$ Thus, for $m_0 \neq 0$ the static electrical force is Yukawa. That is, except for the special case $m=0$, when the theory degenerates to the massless one [@footnote1].
The main subject of this paper is a different special case, $m_0 = 0$, for which the Lagrangian reads $$\mathcal{L} = - \frac{1}{4} F_{\mu \nu}^2 - \frac{1}{2} m^2 A_i^2 + A_{\mu} J^{\mu} .
\label{ourlag}$$ Using current conservation, $\partial_{\mu} J^{\mu} = 0$, the equations of motion can be reduced to $$A_i =- \frac{ \delta_{ij}-\frac{\partial_i \partial_j}{\triangle}}
{\Box + m^2} J_i
\hspace{2em} A_0 = \frac{J_0}{\triangle} . \label{aeqs}$$ These seem to imply that electromagnetic waves are transverse but massive, while the force between charges is Coulomb and instantaneous. However, for $m
= 0$ these equations are precisely those of massless electrodynamics in the Coulomb gauge, where we know the Coulomb force is not instantaneous, but causality is obscure.
To clarify the situation, it is helpful to work directly with the field strengths, which are the only degrees of freedom that couple to matter. From follow the wave equations for the electric and magnetic fields: $$\begin{aligned}
( \square + m^2 ) \vec{B} &=& \vec{\nabla} \times {\vec{J}} \\
( \square + m^2 ) E_i &=& \partial_i J_0 - \partial_t J_i - \frac{m^2}{\triangle}
\partial_i J_0 \label{efield}\end{aligned}$$ where $\vec{B} =- \vec{\nabla} \times \vec{A} $ and $E_i = \partial_t A_i - \partial_i A_0$. To see the Coulomb force, take a static situation where $\partial_t = 0$. Then $\square = -\triangle $ and the equation of motion for the electric field is the same as in the static massless case: $\triangle E_i = - \partial_i J_0$. This implies that there is a long distance Coulomb force in our theory, but tells us nothing about its dynamics.
To understand the dynamics, it is helpful to separate the electric field into two parts: $$\begin{aligned}
E_i &=& E_i^{\mathrm{proca}} + \partial_i \Phi \label{esplit}\\
( \square + m^2 ) E_i^{\mathrm{proca}} &=& \partial_i J_0 - \partial_t J_i \\
( \square + m^2 ) \Phi &=& - \frac{m^2}{\triangle} J_0 \label{instant}\end{aligned}$$ Now we see that $E^{\mathrm{proca}}_i$ and $\vec{B}$ are identical to the electric and magnetic fields in the Proca theory: transverse electromagnetic waves are massive, and the potential they induce between charges has Yukawa’s form $\frac{1}{r} e^{- m r}$. All the new effects are therefore contained in the new scalar potential $\Phi$, which propagates like a massive field and has an effective source $\rho_{\mathrm{eff}} = - \frac{m^2}{\triangle} J_0$. For a single point charge at the origin, $J_0 =4 \pi q \delta^3 ( r )$ and so $\rho_{\mathrm{eff}} = q \frac{m^2}{r}$. Then the solution to , in the static case, is $$( -\triangle + m^2 ) \Phi = q \frac{m^2}{r} \quad \Rightarrow \quad \Phi =
\frac{q}{r} - \frac{q}{r} e^{- m r} \label{plaspot} .$$ The Yukawa part of this potential precisely cancels the Yukawa force mediated by $E^{\mathrm{proca}}_i$, leaving the Coulomb force which confirms the calculation directly from . So the cartoon we have is of regular massive electric and magnetic fields enhanced by another electric field induced by a spurious plasma which has a charge distribution $\rho_{\mathrm{eff}} = q \frac{m^2}{r}$ around each source.
Now suppose we wiggle the source, how long does it take for the Coulomb field at a distance $r$ to be affected? We shall now show that the system exhibits the following behavior. First, for a localized source the response is “slowly-instantaneous”. That is, the electric field at arbitrarily large distances $r \gg m^{-1}$ starts responding immediately to the time variation of a source at $r=0$, but the amplitude of the response takes time $t \sim m^{-1}$ to build up to full strength. Second, the time duration of the response can be arbitrarily longer than the time-dependence of the source. To demonstrate these effects, it is enough to consider the contribution to the electric field from $\Phi$, which mediates all the Lorentz violation. We suppose the source $J_0$ is localized and moves for only a finite time interval: $\partial_t J_0\, \neq 0$ only for $0 \leqslant t \leqslant \Delta t$. In such a case, a good order parameter for analyzing the retardation properties of the system is the time derivative $\dot{E}^{\Phi}_j\, =\, \partial_j\dot \Phi$.
From (\[instant\]), by performing a Fourier transformation in the spatial directions but keeping time-dependence explicit, we arrive at $$\label{psi}
\partial_t^2 \tilde{\dot{E}}^\Phi_j (p,t)\, + \, (p^2 \, + \, m^2)\tilde{\dot{E}}^\Phi_j(p,t) \,
= -i {p_j \over p^2} \, m^2 \dot{\tilde{J_0}}(p,t)$$ where $p = |\vec{p}|$. For each momentum mode, this is the equation of a harmonic oscillator subjected to an external force of the duration $\Delta t$. The oscillator will start responding immediately to the force, but it takes a time $\sim m^{-1}$ for the response to develop. After the external force is switched off, the oscillator generically will be left excited and will continue to oscillate.
It is instructive to demonstrate the above effects by solving an explicit example of a physical situation. Suppose we have two opposite charges at the origin, and separate them into a dipole of strength $\mu$ in the $z$ direction at time $t=0$. (As far as $\dot E$ is concerned, this is the same moving a charge $q$ a distance $\Delta L = \mu /q$.) The current is $$\begin{aligned}
J_0 &=& - 4 \pi \mu \partial_z \delta^3(r) \Theta(t)\\
J_z &=& - 4 \pi \mu \delta^3(r)\delta(t),\quad J_x = J_y=0\end{aligned}$$ Then $\Phi$ satisfies $$( \partial_t^2 - \triangle + m^2 ) \Phi = - \mu \partial_z \frac{m^2}{r} \Theta ( t ) \label{source}$$ Instead of solving for $\Phi$ exactly, we will pull of the important part of the solution $$\begin{aligned}
\Phi &=& \Phi_a + \Phi_b\\
\Phi_a &=& \mu \partial_z \frac{1}{r} \left[ 1 - \cos ( m t ) \right] \Theta ( t )\end{aligned}$$ which leaves $$( \partial_t^2 - \nabla^2 + m^2 ) \Phi_b = - 4 \pi
\mu \partial_z \delta^3 ( r ) [ 1 - \cos ( m t )] \Theta ( t )$$ The point of this is that now the source for $\Phi_b$ is localized, at $r =
0$, and so $\Phi_b$ will vanish outside the light cone: $\Phi_b=0$ for $t > r$. Thus the Lorentz-violating effects are contained entirely $\Phi_a$, for which we know the complete solution. So we get for the order parameter $\dot{E}_j$ $$\dot{E}_j = \mu m \partial_j \partial_z \frac{1}{r} \sin(m t) \Theta(t), \quad r>t \label{exsol} .$$ This is an exact result. The electric field outside the light-cone is that of a dipole in massless electrodynamics, but is modulated by undamped oscillations of frequency $\omega = m$.
After time $t=r$, when the Lorentz-invariant signal arrives, we do not have a simple closed form expression for the electric field. But it is not hard to show that the oscillations will become damped, matching smoothly onto the static solution for $t \gg r$. An exact expression for $\dot \Phi$ can be derived by integrating the retarded Green’s function for a massive scalar against the non-local source in . We reserve for the reader the pleasure of demonstrating that in this case $$\begin{aligned}
\dot{E}^{\Phi}_j &=&\mu m^2 \partial_j \partial_z\frac{ t}{r}
\int_0^{r/t} dx {\cal J}_0 (m t \sqrt{1-x^2} ),
\quad t>r \nonumber\\
&\approx& \mu m^3 \partial_z \partial_j \frac{ r^2}{t} {\cal J}_1(m t) + \cdots, \quad t \gg r\end{aligned}$$ The Bessel function ${\cal J}_1(m t)$ dies as $ (m t)^{-1/2}$ for large $t$. Similar large time behavior holds for the contribution to $\dot E$ from the Proca field in and we conclude that all the oscillations eventually die off.
Returning to , we can observe all the essential properties of the slowly-instantaneous signal propagation in our system: at arbitrarily large distances, $r\gg m^{-1}$, the full response sets in only after the time $m^{-1}$; and the oscillations continue for arbitrarily long times, until the point $r$ enters the light cone of the source.
The appearance of the oscillating electric field is not so unexpected. Because the space-like photons are massive bosons the homogeneous equation for the electric field has free massive wave solutions with arbitrary momenta, and in particular an oscillating solution $$\label{osce}
E_j \, = \, a_j \cos (mt)\, + \, b_j \sin (mt).$$ These oscillations are not any different from the coherent oscillations of any other massive boson field, and describe a state with non-zero occupation number of zero momentum photons.
With this understanding we can proceed to study constraints on the model, to see what the experimental bound is on $m$. First, observe that the mass term $m^2 A_j^2$ contributes to the Hamiltonian, and will generically lead to extremely strong constraints, from limits on the known energy density of the universe. For example, the contribution from the vector potential of the galactic magnetic field would force $m<10^{-27}$ eV. However, it has been shown [@Adelberger:2003qx] that in the Lorentz-invariant Proca case, this constraint is removed completely if the photon mass is spontaneously generated, due to the creation of vortices, like in a type II superconductor. We will now review this argument and show that it holds for a Lorentz-violating mass as well.
The mass in can be generated spontaneously from a Lorentz-violating but gauge invariant Lagrangian $${\cal L} = -\frac{1}{4} F_{\mu \nu}^2 + |D_i H|^2 + V(H) + A_\mu J^\mu \label{higgs}$$ where $D_i H = \partial_i H + q_H i A_i H$ and $V$ is some potential for Higgs field $H$. If the vacuum has $\langle H \rangle = v$, we are returned to at low energy, with $m \sim q_H v$. Note that the charge of the Higgs, its mass, and the mass of the photon are all free parameters.
Equivalently, we can rewrite the mass term as $m^2A_j^2 \rightarrow |H|^2 (q_H A_j \, + \, \partial_j\theta)^2$, where $H$ is now the modulus field, and $\theta$ its Goldstone phase. By separating out $\theta$ as an independent field, we see that it is free to assume a configuration which tries to cancel the contribution to the Proca energy from $A_j$. Suppose there is a uniform magnetic field $B_z$, then we can take $A_\phi = B_z r$, where $(r,\phi,z)$ are cylindrical coordinates. It follows that a minimum energy configuration would have the Higgs winding around the $z$ axis with $1/r \partial_\phi \theta = q_H B_z r$. Of course, $\theta$ and $\phi$ are periodic, so we can only have vortex configurations, such as $\theta = c \phi$ where $c \approx q_H B_z r^2$ is an integer. These vortices will form and distribute, resulting in a total vortex flux $n/q_H$, which should be close to $\sim B_z r^2$ for a region of size $r$. In particular, if there are very many vortices in the region with $r<1/m$, that is if $n = q_H B/m^2 \gg 1$, then effect of the mass on the magnetic field will be completely screened.
The vortex configurations are only possible if $H$ has isolated zeros at the vortex cores, which is why this argument fails without the modulus scalar. The cores will have a size $\sim 1/m_H$, giving a gradient energy $\sim m_H^2 v^2$. This should be smaller than the Proca energy $\sim q_H^2 v^2 B^2_z r^2$ if the vortices are to be created at all. But if the system satisfies $q_H B_z r/m_H \gg 1$ then the vortices can be created classically at almost no cost. On the other hand, as $m_H \to \infty$, the vortices disappear, which is, of course, consistent with decoupling the modulus.
The only difference between this case and one discussed in [@Adelberger:2003qx] is the absence of a time derivative for $H$ in the Lagrangian. But since the diffusion of the Proca energy and the screening of the magnetic field is from static vortices, the argument goes through unchanged. So we conclude that also in this case the constraint from the Proca energy of the universe can be ignored. Moreover, all constraints derived from the decay of planetary or galactic magnetic fields, become irrelevant if the vortices can proliferate. For example, on the earth $B \sim 10^{-2} {\mathrm eV}^2$, and $r \sim 10^{13} {\mathrm eV}^{-1}$. For a photon mass of $10^{-13} eV$, we find $n=q_H B/m^2 = 10^{24} q_H$ and $q_H B_z r/m_H \sim 10^{11}{\mathrm eV} q_H/m_H$. We can easily arrange for both of these numbers to be huge because $q_H$ and $m_H$ are free parameters. Note that including the Higgs does not to take us out of the Proca phase, except in tiny regions near the vortex cores; waves are still massive and the slowly instantaneous behavior of electricity persists.
The next strongest constraints on the Proca theory come from precision tests of Coulomb’s law. One tries to fit the electric force to a form $r^{-2+\delta}$ and by bounding $\delta$, one bounds $m$. In the Proca case, this gives $m < 5\times 10^{-15} {\mathrm eV}$ [@Tu:2005ge]. However, in our case, Coulomb’s law is unchanged, and so these bounds simply do not apply. In fact, the only experimental constraints which will apply are dynamical ones, involving electromagnetic waves. For example, there is a constraint on the Proca theory coming from the time delay dispersal in the arrival of light from distant optical pulses [@Tu:2005ge]. This puts a limit $m \lesssim 2 \times 10^{-13} \mathrm{eV} \sim ( 6000 \mathrm{km} )^{- 1} \sim (20 \mathrm{ms})^{-1}$.
It is worth briefly considering how the Lorentz violation in or might come about. It is possible to formally restore Lorentz invariance to the Lagrangian, by introducing a new vector field $B_\mu$ and replacing the $(D_iH)^2$ term with $${\cal L}_B = D_\mu H(B^\mu B^\nu - \eta^{\mu \nu} B_\alpha B^\alpha)D_\nu H^\star$$ We can think of a $B_\mu$ as a spurion with the fixed value $B_\mu = (1,0,0,0)$. Or $B_\mu$ can be related to some cosmological vector which happens to have a time-like vacuum expectation value. For example, if we write $B_\mu = \partial_\mu \phi$ then the term becomes $${\cal L}_\phi = D_{\mu} H [ (
\partial^{\mu} \phi ) ( \partial^{\nu} \phi ) - \eta^{\mu \nu} (
\partial_{\alpha} \phi ) ( \partial^{\alpha} \phi ) ] D_{\nu}
H^{\star} \label{phicond}$$ Then if $\phi$ is some background scalar field with expectation value $\langle \phi \rangle \sim t$ we are brought back to . Such a situation can be achieved, for example, from ghost condensation [@Arkani-Hamed:2003uy] in which derivatives of $\phi$ are given a potential with non-zero minimum. But we may also think of $\phi$ as being related to other background fields, such as the scale factor of the universe $a(t)$. We should also mention that although the form of looks unnatural, it is in fact protected by the symmetry $H \to exp(i f(\phi)) H$, for an arbitrary function $f$. In the vacuum, this symmetry and gauge invariance are almost completely broken, leaving only $A_\mu \to A_\mu + \partial_\mu f(t)$, which is the residual symmetry keeping $m_0=0$ in .
Next, we can consider making this type of Lorentz-violating modification to a gauge theory with self-interactions. If we drop the Higgs, then for a non-Abelian theory, the biggest change is the appearance of a scale where perturbation theory breaks down, $\Lambda \sim 4 \pi m / g$. This is the same scale as for a Lorentz-invariant theory, and can be traced to the scattering of longitudinal modes. In our case there are no propagating longitudinal modes, but there are additional degrees of freedom $\Phi^a$ which are the physical cause of the superluminal effects, and we must take their interactions seriously. For QCD, there is no legitimate constraint from low energy, where it [*is*]{} non-perturbative, but asymptotic freedom up to $\sim 100$ GeV forces the gluon to be very heavy, which is ruled out, for example, by the observation of jets. The strong coupling could be moderated by a Higgs, but this new Higgs would have to be colored and it is unlikely that such a colored particle could have avoided characterization until now. For the weak interactions, the gauge bosons are already massive. If we changed the mass term to the Lorentz-violating form, tree level results would be unchanged, because the longitudinal mode does not couple to matter, but 1-loop corrections would contradict electroweak precision data. Moreover, the weak force would be Coulomb, not Yukawa – it would not be weak! So we conclude that the non-Abelian gauge theories in the standard model cannot accommodate Lorentz-violating masses. But there is no inconsistency in the non-Abelian generalization, and Lorentz-violating extensions of the standard model may be worthwhile to explore.
For gravity, the most general Lagrangian with Lorentz-violating mass terms for the graviton is [@Rubakov:2004eb] $$\begin{aligned}
&& \mathcal{L} = gR + \\
&& m_0^2 h_{00} + 2 m_1^2 h_{0j}^2 - m_2^2 h_{i j}^2 + m_3^2 h_{i i}^2 - 2
m_4^2 h_{00} h_{i i} \nonumber
% \qquad\qquad m_4^2 = m_0^2 ( 3 m_3^2 - m_2^2 )\end{aligned}$$ The numerical coefficients are chosen so that if $m_0^2 =m_1^2=m_2^2=m_3^2=m_4^2$ the model reduces to the Lorentz-invariant Fierz-Pauli theory [@Pauli:1939xp] (see also [@Arkani-Hamed:2002sp]). Some other mass tunings have been discussed in [@Arkani-Hamed:2003uy; @Rubakov:2004eb; @Dubovsky:2004sg]. In particular [@Dubovsky:2004ud] (see also [@Gabadadze:2004iv]) describes a model with many similarities to our electrodynamical system – there are massive transverse gravitational waves, and a seemingly instantaneous Newton’s law. We will not go into the details of the construction here, but merely comment that in this model it seems the physical analysis we have done for the photon should apply: the gravitational potential will be slowly-instantaneous, producing oscillations outside the light-cone with a characteristic frequency $m$. However, in massive gravity, one cannot avoid strong coupling, because a Higgs mechanism is not known. Moreover, without an explicit construction of gravitational vortices, it may not be possible to avoid an analog the Proca energy constraint for a background gravitational field.
As a final word, it is perhaps worth briefly mentioning the sensitivity of our classical model to quantum corrections. Naturally, standard model loops will contribute to additional relevant Lorentz-violating operators. But coefficients of these terms are at most around $m^2/m_e^2 \sim 10^{-34}$, which is well below even the strong bounds $\sim 10^{-17}$ [@Lipa:2003mh] which already exist.
We thank S. Dubovsky, T. Gregoire, A. Gruzinov, J. D. Jackson, R. Rattazzi, and T. Watari for helpful discussions. G.D. is supported in part by David and Lucile Packard Foundation Fellowship for Science and Engineering, and by NSF grant PHY-0245068.
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A. Proca, Compt. Rend. Phys. [**190**]{}, 1377 (1930).
For $m=0$, the Proca constraint reads $\partial_t A_0=0$; this implies that $A_0$ cannot mediate the electric force around a charge, because it cannot move. Thus the Yukawa solution is not the physical one. L. C. Tu, J. Luo and G. T. Gillies, Rept. Prog. Phys. [**68**]{}, 77 (2005).
E. Adelberger, G. Dvali and A. Gruzinov, arXiv:hep-ph/0306245. W. Pauli and M. Fierz, Helv. Phys. Acta [**12**]{}, 297 (1939). N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals Phys. [**305**]{}, 96 (2003) \[arXiv:hep-th/0210184\].
N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, JHEP [**0405**]{}, 074 (2004) \[arXiv:hep-th/0312099\].
V. Rubakov, arXiv:hep-th/0407104. S. L. Dubovsky, JHEP [**0410**]{}, 076 (2004) \[arXiv:hep-th/0409124\]. S. L. Dubovsky, P. G. Tinyakov and I. I. Tkachev, arXiv:hep-th/0411158. G. Gabadadze and L. Grisa, arXiv:hep-th/0412332. We thank the authors for communication. J. A. Lipa, J. A. Nissen, S. Wang, D. A. Stricker and D. Avaloff, Phys. Rev. Lett. [**90**]{}, 060403 (2003) \[arXiv:physics/0302093\].
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abstract: 'We use a Feshbach resonance to tune the scattering length $a$ of a Bose-Einstein condensate of $^7$Li in the $|F = 1, m_F = 1\rangle$ state. Using the spatial extent of the trapped condensate we extract $a$ over a range spanning 7 decades from small attractive interactions to extremely strong repulsive interactions. The shallow zero-crossing in the wing of the Feshbach resonance enables the determination of $a$ as small as $0.01$ Bohr radii. Evidence of the weak anisotropic magnetic dipole interaction is obtained by comparison with different trap geometries for small $a$.'
author:
- 'S. E. Pollack'
- 'D. Dries'
- 'M. Junker[^1]'
- 'Y. P. Chen[^2]'
- 'T. A. Corcovilos'
- 'R. G. Hulet'
bibliography:
- 'zerocrossing.bib'
title: 'Extreme tunability of interactions in a $^7$Li Bose-Einstein condensate'
---
The ability to control the parameters of ultracold atomic gases and to impose external potentials upon them provides unique opportunities to create model systems for exploring complex phenomena in condensed matter and nuclear physics. Control of atomic interactions using Feshbach resonances has proven to be particularly productive in studies involving Bose-Einstein condensates (BECs) or paired Fermi gases [@frucg1]. While strong interactions are usually the focus of these studies, interesting phenomena also occur in the weakly-interacting regime. An example of such a phenomenon is Anderson localization in disordered media [@PhysRev.109.1492], which was recently observed in weakly repulsive BECs [@AspectAL; @ModugnoAL]. Another example is the formation of bright solitons in BECs with weakly attractive interactions, which have been created in condensates of $^7$Li [@Khaykovich05172002; @StreckerSolitons] and $^{85}$Rb [@cornish:170401]. Atom interferometers may also benefit by the increased coherence times afforded by weakly interacting gases [@gustavsson:080404; @fattori:080405], or even by a dispersionless atomic soliton laser [@StreckerSolitons; @PhysRevA.70.033607].
Several atomic species exhibit Feshbach resonances where the $s$-wave scattering length $a$ changes sign at a certain field in the wings of the resonance. These zero-crossings are useful in applications requiring weak interactions. In addition to $^7$Li [@Khaykovich05172002; @StreckerSolitons], such zero-crossings have been studied in $^{85}$Rb [@PhysRevLett.85.1795], $^{52}$Cr [@Lahaye], $^{39}$K [@roati:010403], and $^{133}$Cs [@gustavsson:080404], In this Letter, we report the measurement of $a$ for $^7$Li in the $|F = 1, m_F = 1\rangle$ state for fields near the Feshbach resonance at 737G [@Khaykovich05172002; @StreckerSolitons; @PhysRevA.51.4852; @junker]. By measuring the *in situ* size of the confined condensate, $a$ is measured over a range of 7 decades. We find that the slope of the zero-crossing is only $\sim$0.1$a_0$/G, where $a_0$ is the Bohr radius. This is the shallowest known zero-crossing, requiring only modest field stability to achieve an essentially non-interacting gas. We explore the effects of the magnetic dipole interaction (MDI) in this regime. Unlike Cr, which has a large magnetic moment of $6\,\mu_\mathrm{B}$ resulting in a relatively large MDI [@Cr; @koch:218; @lahaye:080401], the MDI in alkali atoms is weak due to their small magnetic moments of $\sim$$1\,\mu_\mathrm{B}$. Nonetheless, the MDI has been recently detected in alkali atoms [@fattori:190405; @vengalattore:170403]. We explore the role of the MDI by modifying the confining geometry of the BEC.
Our experimental apparatus for generating a BEC of $^7$Li has been described previously [@StreckerSolitons; @Disorder]. Atoms in the $|F=1, m_F=1\rangle$ state are confined in an optical trap formed from a single focused laser beam with wavelength of 1.03$\mu$m. A bias magnetic field, directed along the trap axis, is used to tune $a$ via the Feshbach resonance. We create condensates at a field where $a$ is large to facilitate rapid rethermalization of the atoms during evaporation from the optical dipole trap. After a condensate is formed we slowly ($\sim$4s) ramp the field to the desired value and determine the scattering length, as described below. There is no discernable thermal part to the density distributions and we estimate that $T/T_c < 0.5$, where $T_c$ is the condensation temperature. The final trapping potential is a combination of the optical field and a residual axial magnetic curvature from the bias field. The trap is cylindrically symmetric with measured radial and axial trapping frequencies of $\omega_r/2\pi = 193\,$Hz and $\omega_z/2\pi = 3\,$Hz, respectively. We use *in situ* polarization phase-contrast imaging [@PhysRevLett.78.985] to acquire the column density distribution of the condensate at the desired magnetic field. When the $s$-wave interactions are large and repulsive they inflate the size of the condensate well above the harmonic oscillator size. As the interactions decrease the size of the condensate becomes smaller, approaching the harmonic oscillator ground state near zero interactions. Figure \[fig:images\] shows representative images of condensates with various repulsive or attractive interaction strengths. Solitons form when $a < 0$, either a single one for a slow magnetic field ramp or multiple solitons for ramps fast compared to the axial trap period.
![ (color online) Representative *in situ* polarization phase-contrast images of condensates with various interaction strengths. (a) $B=719.1\,$G, $a = 396\,a_0$, $N=1.7 \times 10^5$; (b) $B=597.4\,$G, $a = 8\,a_0$, $N=2.9 \times 10^5$; (c) $B=544.7\,$G, $a = 0.1\,a_0$, $N=2.0 \times 10^5$; (d) $B=542.4\,$G, $a = -0.1\,a_0$, $N=1.2 \times 10^5$; (e) same as (d) but with a faster field ramp from $710\,$G to $542.4\,$G, resulting in multiple solitons with $N\approx10^4$ per soliton. The probe laser detuning from resonance is adjusted to keep a nearly constant signal level, and varies between $20\,\gamma$ for large $a$ to $150\,\gamma$ for small $a$, where $\gamma/2\pi \approx 5.9\,$MHz is the excited state linewidth. The color map is adjusted to maximize contrast for each image. \[fig:images\]](bec_interactions_zoom.eps){width="1.0\columnwidth"}
We integrate the image of the condensate in the remaining radial dimension to produce an axial density profile. The $1/e$ radius of this profile is used as a measure of the condensate size, as shown in Fig. \[fig:axialSize\] for a range of magnetic field values. In the Thomas-Fermi regime, the axial size of the condensate is dependent on the product of $a$ and the number of atoms in the condensate $N$. The average number per condensate is $N_0 = 3 \times 10^5$ atoms, with a shot-to-shot variation of 20%. The inset of Fig. \[fig:axialSize\] shows the axial size scaled by $(N/N_0)^{1/5}$ to account for these fluctuations. Several condensates are found to have axial sizes smaller than the axial harmonic oscillator size due to net attractive interactions, as discussed below.
To determine $a$ for each image requires a mapping from the measured axial size and $N$ to $a$. We model the system using the three-dimensional (3D) Gross-Pitaevskii equation $$\begin{aligned}
\label{eqn:gpe}
\mu \psi =&& -\frac{\hbar^2}{2 m}\nabla^2 \psi + V \psi + \frac{4 \pi \hbar^2 a}{m} |\psi|^2 \psi \nonumber\\
&& + \frac{\mu_0 \mu_m^2}{4 \pi} \int \frac{1 - 3 \cos^2 \theta}{|\mathbf{r}-\mathbf{r}'|^3} |\psi (\mathbf{r}')|^2 d\mathbf{r}'\psi,\end{aligned}$$ where we account for the MDI in addition to the $s$-wave contact interaction and the trapping potential. At 540G, $\mu_m \approx 0.94\,\mu_\mathrm{B}$ for $^7$Li in the $|1,1\rangle$ state. Mapping is accomplished by performing a variational calculation using a 3D cylindrically symmetric Gaussian wavefunction as a trial solution to Eq. (\[eqn:gpe\]). Minimizing the corresponding energy functional results in equations for the radial and axial sizes of the condensate [@PhysRevA.63.053607], which are solved to give the desired mapping function. Figure \[fig:mapping\] shows this mapping with and without inclusion of the MDI, as well as the corresponding mapping from the Thomas-Fermi approximation. In our geometry, the magnetic moments are aligned with the long axis of the trap. This causes the MDI to be effectively attractive, making the condensate smaller axially for a given value of $a$. We have verified the accuracy of the variational calculation by exact numerical solution of Eq. (\[eqn:gpe\]) for various values of $a$ and find good agreement between the two methods. Since the variational calculation is much faster computationally, we use this method to analyze the data.
Figure \[fig:ascatt\] shows the axial size data of Fig. \[fig:axialSize\] mapped onto $a$. The general shape follows that of a typical Feshbach resonance with $a = a_{BG} [1 + \Delta / (B - B_{\infty})]$, where $a_{BG} = -24.5^{+3.0}_{-0.2}\,a_0$, $\Delta = 192.3(3)\,\mathrm{G}$, and $B_{\infty} = 736.8(2)\,\mathrm{G}$. The uncertainties in these derived values are a result of the systematic uncertainty in field calibration of 0.1G and a systematic uncertainty in $a$ of $\sim$20%, primarily due to uncertainty in measuring the axial size and determination of $\omega_z$. A linear fit to the data for $B < 550\,$G gives a slope of $0.08(1)\,a_0/\mathrm{G}$ and a zero-crossing at $B_0 = 543.6(1)$G [@error]. The smallest mean positive scattering length of a collection of shots was $0.01(2)\,a_0$ at 543.6(1)G with $\sim$$3\times 10^5$ atoms. Under these conditions the peak density is $3 \times 10^{14}\,\mathrm{cm}^{-3}$ and the corresponding condensate healing length is comparable to the length of the condensate itself. Although Eq. (\[eqn:gpe\]) assumes the mean field approximation, beyond mean field corrections are expected to be important when $n a^3 \gtrsim 1$ [@HY; @LY; @LHY; @PhysRevA.63.063601; @papp:135301]. The leading order correction to the interaction term in Eq. (\[eqn:gpe\]), the Lee-Huang-Yang parameter, is $\alpha = 32/(3\sqrt{\pi}) \sqrt{n a^3} \gg 1$ for the most strongly interacting condensates observed. We have accounted for this correction in extracting $a$ for data where $\alpha < 1$. For the four data points with $\alpha > 1$, this low-density expansion is not valid. We are unaware of a theoretical treatment that addresses the density distribution in the strongly interacting regime. While we extract a value a $a$ for these four data points by fitting to a Thomas-Fermi profile ignoring beyond mean-field effects, and plot them in Fig. 4, we exclude them in the Feshbach resonance fit. Using this method the largest mean positive scattering length was $\sim$$2 \times 10^5\,a_0$ at 736.9(1)G with $\sim$$2 \times 10^4$ atoms, which has a peak density $n \approx 5 \times 10^{10}\,\mathrm{cm}^{-3}$. The comparatively smaller number of atoms close to resonance is likely due to large inelastic collisional losses in this regime [@PhysRevLett.77.2921].
Figure \[fig:ascatt\] also shows a comparison between a coupled-channels calculation and the experimentally extracted values of $a$. The coupled-channels calculation requires the ground-state singlet and triplet potentials of $^7$Li$_2$ as input, as described previously [@PhysRevLett.74.1315; @PhysRevA.55.R3299]. We have updated the potentials to be consistent with the new measurements of $B_\infty$ and $B_0$ reported here, as well as the previously measured binding energy of the least bound triplet vibrational level [@PhysRevLett.74.1315; @footnote]. The updates involve adjusting the singlet and triplet dissociation energies to $D_e(X^1\Sigma^+_g) = 8516.68(10)\,\mathrm{cm}^{-1}$ and $D_e(a^3\Sigma^+_u) = 333.714(40)\,\mathrm{cm}^{-1}$, where the stated uncertainties account for uncertainties in the remaining portions of the model potentials. These values are consistent with previous determinations [@PhysRevA.55.R3299; @linton:6036; @colavecchia:5484]. The agreement between the calculated and measured values of $a$, while not perfect over the entire range of fields, is reasonably accurate over a range spanning 7 decades.
![ (color online) Extracted values of $a$ near the zero-crossing for trapping potentials with $\omega_z/2\pi = 3$Hz (filled squares) or $\omega_z/2\pi = 16$Hz (unfilled squares), when (a) neglecting or (b) including the MDI in the mapping function. The MDI has a negligible effect on the extracted values of $a$ for the 16Hz trap, but neglecting the MDI in analysis of the 3Hz trap systematically lowers the mapped values of $a$, especially for $a \lesssim 0.15\,a_0$. \[fig:dipole\]](zerocrossing_trapsV.eps){width="1.03\columnwidth"}
The effects of the MDI are strongly dependent on geometry. To better distinguish their role, we increased the axial trapping frequency from 3Hz to 16Hz by applying magnetic curvature. Figure \[fig:dipole\] compares the extracted values of $a$ for both trap geometries when the MDI is included or neglected in the mapping function. As expected, neglecting the MDI in the analysis systematically lowers the extracted values of $a$. This effect is most noticeable near the zero-crossing where a systematic geometry-dependent discrepancy appears in the derived values of $a$. Inclusion of the MDI in the analysis produces a consistent value of $a$ for a given magnetic field regardless of the trapping potential. The data show that the magnetic dipole interaction, although weak, is discernible in $^7$Li despite having a magnetic moment of only $\sim$$1\,\mu_\mathrm{B}$.
We have mapped the Feshbach resonance from the regime of small attractive interactions far from the resonance to extremely strong repulsive interactions very close to resonance. The zero-crossing and resonance positions have been precisely located, enabling experimental access to a broad range of accurately known interactions. Of particular interest will be explorations of atom and soliton transport through a disordered potential in the weakly interacting regime.
We thank James Hitchcock and Chris Welford for their contributions to this project. Support for this work was provided by the NSF, ONR, the Keck Foundation, and the Welch Foundation (C-1133).
[^1]: Current Address: School of Physics, University of Melbourne, Victoria 3010, Australia
[^2]: Current Address: Dept of Physics, Purdue University, 525 Northwestern Ave., West Lafayette, IN 47907
|
---
abstract: 'Born and Oppenheimer reported an approximate separation of molecular eigenfunctions into electronic, vibrational, and rotational parts, but at the end of their paper showed that the two angles describing rotation of the nuclei in a diatomic molecule are exactly separable. A year later in a two-part work devoted strictly to diatomic molecules, Wigner and Witmer published (1) an exact diatomic eigenfunction and (2) the rules correlating the electronic state of a diatomic molecule to the orbital and spin momenta of the separated atoms. The second part of the Wigner-Witmer paper became famous for its correlation rules, but, oddly, the exact eigenfunction from which their rules were obtained received hardly any attention. Using three fundamental symmetries, we give a derivation of the Wigner-Witmer diatomic eigenfunction. Applications of our derivations are fundamental to predicting accurate diatomic molecular spectra that we compare with recorded spectra for diagnostic purposes, such as measurements of molecular spectra following generation of laser-induced plasma.'
address: |
The Center for Laser Applications,\
The University of Tennessee Space Institute,\
Tullahoma, TN 37388 - 9700,\
U.S.A.
author:
- 'J. O. Hornkohl, A. C. Woods and C. G. Parigger[^1]'
title: 'The Wigner-Witmer diatomic eigenfunction'
---
Introduction
============
In the introduction to their paper, Born & Oppenheimer allude to an exact separation of two rotational coordinates in the diatomic molecule. In their next section, which is applicable to polyatomic molecules, they introduce a coordinate system attached to the nuclei whose orientation is set by the Euler angles, and note that there are terms in the molecular Hamiltonian in which both electronic and nuclear coordinates appear thereby preventing the exact separation of the total eigenfunction into a product of electronic and nuclear eigenfunctions. In their final section, Born and Oppenheimer return to the diatomic molecule and give the details of the exact separation of two of the Euler angles, angles $\theta$ and $\phi$ that describe rotation of the two nuclei. The spherical harmonic $Y_{\ell \, m}(\theta, \phi)$ is the angular momentum part of Born-Oppenheimer diatomic eigenfunction. A year after, Wigner & Witmer published a two-part article on diatomic theory in which they replace the spherical harmonic $Y_{\ell \, m}(\theta, \phi)$ with the Wigner $D$-function $D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \chi)$. The Wiger-Witmer paper was the introduction of the rotation matrix elements into diatomic theory. After the Wigner-Witmer paper, the Wigner $D$-function mostly disappeared from diatomic literature for about four decades. Hirschfelder & Wigner used the $D$-function in their separation of 6 coordinates (3 for the total linear momentum, 3 for the total angular momentum) for $N$ particle systems, but do not explicitly mention the diatomic molecule. Again not specifically mentioning the diatomic molecule, Curtis & Hirschfelder & Adler repeat the separation of 6 coordinates for $N$ particle systems, and consider the three-body system in detail. Davydov [@Davydov] in his quantum mechanics textbook used the $D$-function in his discussion of diatomic theory. At about the same time, Rubin [@Rubin] employed it for his calculations of Hönl-London factors. Pack & Hirschfelder used the $D$-function to separate two angular rotational coordinates in the diatomic eigenfunction but failed to notice that their Eq. (2.35) holds for all values of the third Euler angle, not just $\gamma=0$. Zare [@Zare1973] explicitly used the $D$-function in their case (a) basis function. Judd [@Judd] and Mizushima [@Mizushima], in their treatments of diatomic theory, introduce the $D$-function and discuss its mathematical properties, but do not explicitly display it in their Hund’s cases (a) and (b) basis functions. The Wigner $D$-function has since become a vital mathematical component of diatomic theory, as comprehensively collated by Varshalovich & Moskalev & Khersonskii [@Varsha]. However, the exact separation of $D_{M \Omega}^{J^{\scriptstyle *}} (\phi \, \theta \, \chi)$ in the diatomic eigenfunction where the nuclear coordinates $\phi$ and $\theta$ appear with the electronic coordinate $\chi$ has remained essentially forgotten for eight decades. We give a derivation of the Wigner-Witmer diatomic eigenfunction based upon three fundamental symmetries and the geometrical symmetry of a molecule possessing precisely two nuclei.
Derivation of the Wigner-Witmer diatomic eigenfunction. {#WWequation}
=======================================================
Here we obtain the Wigner-Witmer eigenfunction by applying three symmetry principles to the eigenfunction of a free conservative system composed of $N$ electrons and precisely two nuclei. Energy is the generator of translations in time, the time translation (evolution) operator $U(t,t_0)$ is a continuous unitary operator, the total energy is a constant of the motion, and the dependence of the eigenfunction on the physical variable time $t$ is exactly separable if the time origin $t_0$ can be associated with some physical event. Linear momentum is the generator of translations in space, the spatial translation operator $\mathcal{T}(\mathbf{R},\mathbf{R}_0)$ is a continuous unitary operator, the total linear momentum is a constant of the motion, and the total linear momentum is exactly separable if the coordinates $\mathbf{R}_{\rm CM}$ of the center of mass can be introduced as physical variables of the system. Angular momentum is the generator of rotations, the rotation operator $\mathcal{R}(\alpha,\beta,\gamma)$ is a continuous unitary operator, the total angular momentum is a constant of the motion, but the total angular momentum $\mathbf{J}(\phi,\theta,\chi)$ is *not*, in general, exactly separable because except for very simple systems one cannot find physical rotations $\phi$, $\theta$, and $\chi$ which duplicate the angles $\alpha$, $\beta$, and $\gamma$ of coordinate rotation. The diatomic molecule deserves a special place in the quantum theory of angular momentum because it is one of the most complicated systems for which the Euler angles $\alpha$, $\beta$, and $\gamma$ of coordinate rotation are also the angles of physical rotation describing the total angular momentum $\mathbf{J}$
Quantum mechanical descriptions of the diatomic molecule typically begin with the Hamiltonian, but minutia in the Hamiltonian tend to obscure the few fundamentals at play. For example, Brown & Carrington write a diatomic Hamiltonian, their Eq. (2.297), containing 32 types of Hamiltonian terms. We begin our discussion of diatomic theory with the eigenfunction $$\Psi_{nvJM}(\mathbf{R}_1, \mathbf{R}_2, \dots, \mathbf{R}_N, \mathbf{R}_{\rm a}, \mathbf{R}_b,t) \equiv \langle \mathbf{R}_1, \mathbf{R}_2, \dots, \mathbf{R}_N, \mathbf{R}_{\rm a}, \mathbf{R}_b,t \, |nvJM\rangle
\label{dm1}$$
in which $\mathbf{R}_1, \mathbf{R}_2, \dots, \mathbf{R}_N$ are the spatial coordinates of the $N$ electrons and $\mathbf{R}_{\rm a}$ and $\mathbf{R}_b$ are those of the nuclei. The total angular momentum quantum numbers $J$ and $M$ refer to the true total. That is, in spectroscopic nomenclature they would be replaced by $F$ and $M_F$. The symbol $n$ represents all other required quantum numbers and continuous indices except the vibrational quantum number $v$.
The symmetries of translation in time and translation in space produce a separation of the time coordinate $t$ and the spatial coordinates $\mathbf{R}_{\rm CM}$ of the center of mass. A two-body reduction of the motion of the nuclei requires placement of the coordinate origin a the center of mass of the nuclei, and then replaces $\mathbf{R}_{\rm a}$ and $\mathbf{R}_{\rm b}$ with the internuclear vector $\mathbf{r}$. Of the $3N+7$ dynamical variables in the total eigenfunction (\[dm1\]), $3N+3$ remain in the internal eigenfunction $\langle \mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N,\mathbf{r} \, |nvJM\rangle$. The axes of the translated coordinates $\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N,\mathbf{r}$ are parallel to those of the original laboratory coordinates $\mathbf{R}_1, \mathbf{R}_2, \dots, \mathbf{R}_N, \mathbf{R}_{\rm a}, \mathbf{R}_{\rm b}$. We now address how rotational symmetry influences the internal eigenfunction.
Operation of the rotation operator $\mathcal{R}(\alpha,\beta,\gamma)$ on the internal eigenfunction yields $$\langle \mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N,\mathbf{r} \, | \mathcal{R}(\alpha,\beta,\gamma) |nvJM\rangle = \langle \mathbf{r}'_1, \mathbf{r}'_2, \dots, \mathbf{r}'_N,\mathbf{r}' \, | nvJM\rangle$$
where primes denote rotated coordinates given by $$\mathcal{D}(\alpha, \beta, \gamma)
= \left[ \begin{matrix} \cos\alpha \cos\beta \cos\gamma - \sin\alpha \sin\gamma & \sin\alpha \cos\beta \cos\gamma + \cos\alpha \sin\gamma & -\sin\beta \cos\gamma \\
-\cos\alpha \cos\beta \sin\gamma - \sin\alpha \cos\gamma & -\sin\alpha \cos\beta \sin\gamma + \cos\alpha \cos\gamma & \sin\beta \sin\gamma \\
\cos\alpha \sin\beta & \sin\alpha \sin\beta & \cos\beta
\end{matrix} \right], \label{Drot}$$ $$\left[ \begin{matrix}
x' \\
y' \\
z' \end{matrix} \right]
= \mathcal{D}(\alpha, \beta, \gamma)
\left[ \begin{matrix}
x \\
y \\
z \end{matrix} \right].
\label{r'}$$
The effect of $\mathcal{R}(\alpha,\beta,\gamma)$ on the eigenfunction can be rewritten as $$\begin{aligned}
\langle \mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N,\mathbf{r} \, | nvJM\rangle &= \langle \mathbf{r}'_1, \mathbf{r}'_2, \dots, \mathbf{r}'_N,\mathbf{r}' \, | \mathcal{R}^{\dagger}(\alpha,\beta,\gamma) |nvJM\rangle \\
&= \sum_{\Omega=-J}^J \langle \mathbf{r}'_1, \mathbf{r}'_2, \dots, \mathbf{r}'_N,\mathbf{r}' \, |nvJ\Omega\rangle \, \langle J\Omega \, | \mathcal{R}^{\dagger}(\alpha,\beta,\gamma) \, | JM\rangle \\
&= \sum_{\Omega=-J}^J \langle \mathbf{r}'_1, \mathbf{r}'_2, \dots, \mathbf{r}'_N,\mathbf{r}' \, |nvJ\Omega\rangle \, D_{M \Omega}^{J^{\scriptstyle *}} (\alpha, \beta, \gamma).\end{aligned}$$
When the spherical coordinates of the internuclear vector $\mathbf{r} = \mathbf{r}(r,\theta,\phi)$ are introduced, the equation becomes $$\langle \mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N,r,\theta,\phi \, | nvJM\rangle \\
= \sum_{\Omega=-J}^J \langle \mathbf{r}'_1, \mathbf{r}'_2, \dots, \mathbf{r}'_N,r,\theta',\phi' \, |nvJ\Omega\rangle \, D_{M \Omega}^{J^{\scriptstyle *}} (\alpha, \beta, \gamma).$$
The internuclear distance $r$ is unprimed on the right because it is a scalar. Because physical rotation $\phi$ and coordinate rotation $\alpha$ are both counterclockwise rotations about the $z$ axis, the physical angles $\phi'$ is given by $$\phi' = \phi - \alpha.$$
Similarly, physical rotation $\theta$ and coordinate rotation $\beta$ are counterclockwise rotations about the first intermediate $y$ axis of the total coordinate rotation. $$\theta' = \theta - \beta.$$
Rotational symmetry gives us the option to view the molecule at any orientation we choose, and we choose $\alpha=\phi$ and $\beta=\theta$. $$\langle \mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N,r,\theta,\phi \, | nvJM\rangle = \sum_{\Omega=-J}^J \langle \mathbf{r}'_1, \mathbf{r}'_2, \dots, \mathbf{r}'_N,r \, |nvJ\Omega\rangle \, D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \gamma).$$
The rotated coordinates of the one of the electrons, we arbitrarily select the electron labeled 1, are expressed in cylindrical coordinates $\rho'_1$, $\zeta'_1$, and $\chi'_1$. $$\langle \mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N,r,\theta,\phi \, | nvJM\rangle = \sum_{\Omega=-J}^J \langle \rho'_1, \zeta'_1, \chi'_1, \mathbf{r}'_2, \dots, \mathbf{r}'_N,r \, |nvJ\Omega\rangle \, D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \gamma).$$
The chosen electron is distance $\rho'_1$ from the internuclear vector and signed distance $\zeta'_1$ above or below the plane perpendicular to the internuclear vector and passing through the coordinate origin. The angle $\chi'$ describes rotation of this electron about the internuclear distance. Like the internuclear distance $r$, primes on $\rho'_1$ and $\zeta'_1$ are unnecessary because they are scalars whose values are unchanged by coordinate rotation. Because $\chi'_1$ and $\gamma$ are rotations about the same axis, coordinate rotation changes, of course, the value of $\chi'_1$, but this also means that this angle has a value $\chi_1$ in laboratory coordinates. The coordinate rotation angle $\gamma$ is chosen to make $\chi'_1$ zero, $$\chi'_1 = \chi_1 - \gamma .$$
The equation for the eigenfunction now reads $$\langle \rho, \zeta, \chi, \mathbf{r}_2, \dots, \mathbf{r}_N, r, \theta, \phi \, | nvJM\rangle = \sum_{\Omega=-J}^J \langle \rho, \zeta, \mathbf{r}'_2, \dots, \mathbf{r}'_N,r \, |nv\rangle \, D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \chi)
\label{WWeq}$$
after the subscripts on $\rho_1$, $\zeta_1$, and $\chi_1$ have been dropped. This result is the Wigner-Witmer diatomic eigenfunction. The total diatomic eigenfunction is given as the sum of $2J+1$ products of electronic-vibrational basis functions $\langle \rho, \zeta, \mathbf{r}'_2, \dots, \mathbf{r}'_N,r \, |nv\rangle$ and total angular momentum basis functions $D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \chi)$ or Wigner D-functions. The Born-Oppenheimer approximation separates the electronic-vibrational basis into the product of electronic and vibrational basis functions, but the separation of $D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \chi)$ from the electronic-vibrational basis is exact. Many individual orbital and spin momenta are contained in the electronic-vibrational basis, but $D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \chi)$ is the total angular momentum basis.
It is noteworthy that eigenfunctions for a rotational state of the diatomic molecule usually contain parity, in other words, parity symmetrization is customary. Inclusion of the discrete parity symmetry is accomplished after the construction of the Wigner-Witmer diatomic eigenfunction. The parity operator can be constructed from proper and improper rotations that have a determinant of the transformation matrix of +1 and -1, respectively. Subsequently, Equation 2.14 for the eigenfunction can be split for specific values of $\Omega$ followed by parity symmetrization. The approach of including parity for specific values of $\Omega$ has been utilized in the literature in order to reduce the size of the Hamiltonian matrix prior to finding eigenvalues by diagonalization. In our work [@JALSpaper], the parity operation is considered after establishment of the eigenfunction in terms of $J$ and $M$ as sum over $\Omega$ in Eq. 2.14.
Clearly, as electronic states for the diatomic molecule are considered, parity is paramount for building these states utilizing the Wigner-Witmer correlation rules [@Bellary]. Yet in this work we focus on the use of the Wigner-Witmer eigenfunction for computation of spectra, rather than molecular structure predictions in non-Born-Oppenheimer calculations for molecules [@Adamowicz1; @Adamowicz2].
Discussion and conclusion
=========================
The mixing of the electronic coordinate $\chi$ with the two nuclear coordinates $\phi$ and $\theta$ in $D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \chi)$ is an obvious departure from current expositions of diatomic theory. However, the exact separation of the total angular momentum basis from the electronic-vibrational basis proves useful. For example, writing the parity operator $\mathcal{P}$ as the product of a proper rotation $\mathcal{P}_{\alpha \beta \gamma}$ and an improper rotation $P_{\Sigma}$, $$\mathcal{P} = \mathcal{P}_{\alpha \beta \gamma} \, P_{\Sigma},$$
one obtains a simple equation for the parity of diatomic states. The eigenvalue of $\mathcal{P}$ is, of course, $\pm 1$, and the product of eigenvalues $p_{\Sigma} \, p_{\alpha \beta \gamma}$, $$p = p_{\Sigma} \, (-)^{J+2M},
\label{diatomicParity1}$$
is always $\pm 1$ as required. Sign changes due to parity can show different effects on the Wigner D-function. Depending on the specific rotation group, the effect can be expressed in terms of $J$ and $M$ or in terms of $J$ and $\Omega$. Table \[C2\] summarizes the sign changes due to parity. These results are consistent with the ones presented by Varshalovich & Moskalev & Khersonskii [@Varsha].
Transformation group Euler angles Coordinates Effect on $D_{M \Omega}^{J^{\scriptstyle *}} (\alpha, \beta, \gamma)$
---------------------- ----------------------------------- ---------------------- --------------------------------------------------------------------------------
$C_2(x')$ $\alpha \rightarrow \pi + \alpha$ $x' \rightarrow x'$
$\beta \rightarrow \pi - \beta$ $y' \rightarrow -y'$ $(-)^{J+2M} \, D_{M,-\Omega}^{J^{\scriptstyle *}} (\alpha, \beta, \gamma)$
$\gamma \rightarrow -\gamma$ $z' \rightarrow -z'$
$C_2(y')$ $\alpha \rightarrow \pi + \alpha$ $x' \rightarrow -x'$
$\beta \rightarrow \pi - \beta$ $y' \rightarrow y'$ $(-)^{J-\Omega} \, D_{M,-\Omega}^{J^{\scriptstyle *}} (\alpha, \beta, \gamma)$
$\gamma \rightarrow \pi - \gamma$ $z' \rightarrow -z'$
$C_2(z')$ $\alpha \rightarrow \alpha$ $x' \rightarrow -x'$
$\beta \rightarrow \beta$ $y' \rightarrow -y'$ $(-)^{-\Omega} \, D_{M,-\Omega}^{J^{\scriptstyle *}} (\alpha, \beta, \gamma) $
$\gamma \rightarrow \pi + \gamma$ $z' \rightarrow z'$
: The sign changes on the components $x'$, $y'$ and $z'$ of a coordinate vector ${\bf r}'(x',y',z')$ produced by three different discrete Euler angle transformations, and the effect of these Euler angle transformations on $D_{M \Omega}^{J^{\scriptstyle *}} (\alpha, \beta, \gamma)$.
\[C2\]
Note that for half-integer $J$ in Equation (3.2), the individual eigenvalues $p_{\Sigma}$ and $p_{\alpha \beta \gamma}$ are purely imaginary. A widely accepted convention allows one to treat the parity eigenvalues $p_{\Sigma}$ and $p_{\alpha \beta \gamma}$ as real for both integer and half-integer $J$ [@Brown_ef1975]. If one agrees to always subtract $1/2$ from half-integer values of $J$, then the diatomic parity can be written as
$$\begin{aligned}
p &= + p_{\Sigma} \, (-)^J \ \ \ \ \ \ \ \ \qquad J\text{ integer} \\
&= - p_{\Sigma} \, (-)^{J-1/2} \qquad J\text{ half-integer}\end{aligned}$$
\[diatomicParity2\]
in which $p_{\Sigma} = \pm1$ is always real.
The Wigner-Witmer diatomic eigenfunction simplifies the process in which one infers molecular parameters such as the rotational parameter $B_v$ and the spin-orbit parameter $A_v$ from experimentally measured line positions. Application of our detailed Wigner Witmer eigenfunctions include analyses of low- and high-temperature spectra of diatomic carbon spectra [@JObook], or as another example, development of line strengths for specific transitions of the aluminium monoxide (AlO) diatomic molecule [@SAA]. The parameters we use are electronic-vibrational matrix elements. In turn, the Born-Oppenheimer approximation separates these into the products of electronic matrix elements times vibrational matrix elements, and introduces a large number of differential equations which couple the many Born-Oppenheimer vibrational states thereby producing a large Hamiltonian matrix. Van Vleck transformations reduce the dimension of the Hamiltonian to yield an effective Hamiltonian. If in the fitting process, one deals with electronic-vibrational matrix elements such as $B_v$ and $A_v$ instead of breaking them into the products of electronic matrix elements and vibrational matrix elements, Van Vleck transformations are no longer required.
Conversely, the Wigner-Witmer eigenfunction does not reveal how the total angular momentum is built from its components. One must use an angular momentum coupling model which has a complete basis. For example, the Hund’s case [*a*]{} basis appropriate to the Wigner-Witmer eigenfunction is $$|a\rangle = |nJM\Omega \Lambda S \Sigma \rangle
=\sqrt{\frac{2J+1}{8 \pi^2}} \, \langle \rho, \zeta, \mathbf{r}'_2, \dots, \mathbf{r}'_N, r \, | nv \rangle \, D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, \chi) \, |S \Sigma \rangle ,$$
where $\Omega = \Lambda + \Sigma$. Written in our notation, current practice replaces the above with
$$|a\rangle = % |nvJM\Omega \Lambda S \Sigma \rangle = \sqrt{\frac{2J+1}{8 \pi^2}}
\sum_{v_{BO}=0} \langle \rho, \zeta, \mathbf{r}'_2, \dots, \mathbf{r}'_N; r \, | n v_{BO} JM \Omega \Lambda \rangle \, \langle v_{BO} \, | v \rangle
D_{M \Omega}^{J^{\scriptstyle *}} (\phi, \theta, 0) \, |S \Sigma \rangle.
\label{BO}$$
The total angular momentum is not exactly separated in this equation and there is the sum over a large number of Born-Oppenheimer vibrational states $|v_{BO}\rangle$. It is often said that the Born-Oppenheimer approximation separates the diatomic eigenfunction into electronic, vibrational, and rotational states, but this does not hold with spectroscopic accuracy. In the modeling of upper and lower Hamiltonians whose term differences accurately agree with measured line positions, one must deal with the large set of coupled differential equations that result when Eq. (\[BO\]) is inserted in the Schrödinger equation. Analytical techniques that employ Van Vleck transformations and parity symmetrization have been developed to yield much smaller effective Hamiltonians.
Except for the simplest of diatomic molecules, *ab initio* computations are usually not as accurate as results obtained in experimental spectroscopy. For an accurate prediction of a diatomic spectrum, one must have values for the molecular parameters such as $B_v$, $A_v$, $\lambda_v$, $\gamma_v$, $\dots,$ and their centrifugally stretched forms. Computer programs have been developed which find the molecular parameters by fitting upper and lower term differences from model Hamiltonians to accurately measured line positions. Such a program begins with trial values for the molecular parameters, computes theoretical line positions as eigenvalue differences between upper and lower Hamiltonians, computes corrections to the trial values of the molecular parameters from the differences between the computed and measured line positions, and iterates until corrections to the parameters become negligibly small. There are now many examples of this algorithm for which the errors in the computed line positions, *i.e.,* computed vacuum wavenumbers, are not significantly larger then the estimated experimental accuracy. With two exceptions, replacement of the Born-Oppenheimer approximation with the Wigner-Witmer eigenfunction does not significantly alter the flow charts for these programs. First, the matrix elements of the effective Hamiltonian are replaced with Hund’s case [*a*]{} matrix elements unmodified by Van Vleck transformations and parity symmetrization. Second, the manual enforcement of selection rules is replaced by computation of the Hönl-London factors [@HLfactors]. The exact separation of the total angular momentum in the Wigner-Witmer eigenfunction provides simple, accurate computation of the Hönl-London factors, and there is but a single diatomic selection rule: Transitions for which the Hönl-London factor is non-vanishing are allowed. Thus, use of the Wigner-Witmer diatomic eigenfunction represents a significant departure from current practices in diatomic theory.
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[^1]: Corresponding author: Christian G Parigger, [email protected]
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author:
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[Kevin Hsieh[[$^\dag$]{}]{}[[$^\mathsection$]{}]{} Ganesh Ananthanarayanan[[$^\mathsection$]{}]{} Peter Bodik[[$^\mathsection$]{}]{} Paramvir Bahl[[$^\mathsection$]{}]{} Matthai Philipose[[$^\mathsection$]{}]{} ]{}\
[ Phillip B. Gibbons[[$^\dag$]{}]{}Onur Mutlu[[$^*$]{}]{}[[$^\dag$]{}]{}]{}\
[**[[$^\dag$]{}]{}Carnegie Mellon University [[$^\mathsection$]{}]{}Microsoft [[$^*$]{}]{}ETH Z[ü]{}rich**]{}
title: 'Focus: Querying Large Video Datasets with Low Latency and Low Cost'
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abstract: 'This work presents a novel breast cancer imaging approach that uses compressive sensing in a hybrid Digital Breast Tomosynthesis (DBT) / Nearfield Radar Imaging (NRI) system configuration. The non-homogeneous tissue distribution of the breast, described in terms of dielectric constant and conductivity, is extracted from the DBT image, and it is used by a full-wave Finite Difference in the Frequency Domain (FDFD) method to build a linearized model of the non-linear NRI imaging problem. The inversion of the linear problem is solved using compressive sensing imaging techniques, which lead to a reduction on the required number of sensing antennas and operational bandwidth without loss of performance.'
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Introduction
============
It has been shown [@Kopans2011] that X-ray based Digital Breast Tomosynthesis (DBT) enhances the imaging capabilities of Conventional Mammography (CM) by producing volumetric imaging. Unfortunately, DBT still suffers from the limited radiological contrast existing between cancerous and healthy fibroglandular tissue, which is of the order of 1%. At microwave frequencies, the contrast between cancerous and healthy fibroglandular tissue is of the order of 10% [@Lazebnik2007]. For this reason, Nearfield Radar Imaging (NRI), working at these frequencies, is an appealing technology to detect cancerous tissue within the breast. Unfortunately, NRI by itself fails to detect cancerous tissue embedded in a random, heterogeneous matrix of fibroglandular and fatty breast tissue. A recent paper [@MartinezLorenzo2013] demonstrated that it is possible to fuse DBT with NRI to achieve improved breast cancer detection. The successful detection of breast cancer using this configuration required a high performance NRI sensor operating with the following requirements: 1) $17$ transmitting/receiving antennas in a multi-static configuration, and 2) a wideband system, having a $1$GHz bandwidth at a $1.5$GHz center frequency. These high performance requirements make the NRI sensor too expensive for widespread use in a clinical setting.
This paper presents preliminary 2D results of a new 3D imaging approach for the NRI sensor [@molaei2015], based on techniques from compressive sensing theory. This approach utilizes a multiple monostatic configuration with fewer antennas and a smaller operating bandwidth than the NRI system presented in [@MartinezLorenzo2013]. The new approach reduces the cost of the NRI system without loss of performance, thereby making the technology more suitable for clinical use.
Methodology {#sec:method}
===========
System Configuration
--------------------
The concept of a fused DBT / NRI imaging system was first introduced in [@MartinezLorenzo2013; @Obermeier2014], and the configuration of the 3D imaging mechatronics device was presented in [@molaei2015]. In this approach, the breast is placed under clinical compression, and the DBT measurements are recorded using low-dosage X-rays. The DBT measurements are used to create a 3D reconstruction of the fat distribution in the breast. Simultaneously, a set of NRI measurements are collected using a series of microwave antennas, which are placed along the breast in a bolus material in order to maximize its coupling with the breast surface.
The Sensing Problem {#sec:sensing_form}
-------------------
The NRI system operates using a multiple monostatic configuration, in which each antenna utilizes stepped-frequency waveforms. Measurements of $n_f$ frequencies are taken by $n_a$ antennas for a total of $M =n_a\cdot n_f$ measurements. The relationship between the electric fields $\mathbf{E}(\mathbf{r},\omega)$ and the unknown complex permittivity $\epsilon(\mathbf{r}, \omega)$ of the breast tissues can be expressed as: $$\begin{aligned}
\mathbf{E}_s(\mathbf{r},\omega) &= \int \mathbf{G}_b(\mathbf{r}, \mathbf{r}', \omega) k_b^2(\mathbf{r}', \omega)\mathbf{E}(\mathbf{r}',\omega)\chi(\mathbf{r}',\omega) d\mathbf{r}' \label{eq:cont_src} \\
\mathbf{E}(\mathbf{r},\omega) &= \mathbf{E}_b(\mathbf{r},\omega) + \mathbf{E}_s(\mathbf{r}, \omega)\end{aligned}$$ where $\mathbf{G}_b(\mathbf{r}, \mathbf{r}', \omega)$ are the Green’s functions of the background medium, $\mathbf{E}_b(\mathbf{r},\omega)$ are the incident electric fields, $\mathbf{E}_s(\mathbf{r},\omega)$ are the scattered electric fields, and $\chi(\mathbf{r},\omega)=\frac{\epsilon(\mathbf{r},\omega)-\epsilon_b(\mathbf{r},\omega)}{\epsilon_b(\mathbf{r},\omega)}$ are the contrast variables [@VanDenBerg2001; @Abubakar2011; @Zakaria2013].
Eq. \[eq:cont\_src\] is a nonlinear function of the contrast variable $\chi(\mathbf{r}, \omega)$ and total electric field $\mathbf{E}(\mathbf{r}, \omega)$, and so nonlinear programming techniques such as the Contrast Source Inversion (CSI) algorithm [@VanDenBerg2001; @Abubakar2011; @Zakaria2013] must be applied in order to recover $\chi(\mathbf{r}, \omega)$. These types of nonlinear algorithms typically require several calls to a forward model solver in each iteration, which makes them computationally expensive. As a result, it is desirable to make some simplifying assumptions in order to reduce the computation time. This work makes two such assumptions. First, the Born approximation is applied, $\mathbf{E}(\mathbf{r},\omega) \approx \mathbf{E}_b(\mathbf{r},\omega)$, in order to linearize Eq. \[eq:cont\_src\]. Second, the complex permittivities are assumed to be approximately constant over the frequency range of the NRI system, i.e. $\epsilon(\mathbf{r}, \omega) \approx \epsilon(\mathbf{r})$ and $\epsilon_b(\mathbf{r}, \omega) \approx \epsilon_b(\mathbf{r})$, so that the contrast variable is also approximately constant over frequency. With these two modifications, Eq. \[eq:cont\_src\] can be rewritten in the following form: $$\begin{aligned}
\mathbf{E}_s(\mathbf{r},\omega) &= \int\mathbf{G}_b(\mathbf{r}, \mathbf{r}', \omega) k_b^2(\mathbf{r}', \omega)\mathbf{E}_b(\mathbf{r}',\omega)\chi(\mathbf{r}') d\mathbf{r}' \label{eq:born_approx} \\
&+ \mathbf{\hat{e}}_s(\mathbf{r}, \omega)\nonumber\end{aligned}$$ where $\mathbf{\hat{e}}_s(\mathbf{r}, \omega)$ is the error introduced by the approximating assumptions.
Eq. \[eq:born\_approx\] can be discretized as $\mathbf{y} = \mathbf{A}\mathbf{x} + \mathbf{e} + \boldsymbol{\eta}$, where $\mathbf{x} \in \mathbb{C}^N$ are the contrast variables, $\mathbf{y} \in \mathbb{C}^M$ are the measured fields, $\mathbf{A} \in \mathbb{C}^{M\times N}$ is the sensing matrix constructed from the incident fields and Green’s functions of the background medium, $\mathbf{e} \in \mathbb{C}^M$ is the error vector, and $\boldsymbol{\eta} \in \mathbb{C}^M$ is the random noise introduced by the measurement system. In practice, $M < N$, and so this system has an infinite number of solutions satisfying $\mathbf{y} = \mathbf{A}\mathbf{x} + \mathbf{e} + \boldsymbol{\eta}$. When $\|\mathbf{e}\|_{\ell_2} \ll \|\boldsymbol{\eta}\|_{\ell_2}$, the performance of linear inverse techniques only depends upon the vector $\boldsymbol{\eta}$. When $\|\mathbf{e}\|_{\ell_2} \sim \|\boldsymbol{\eta}\|_{\ell_2}$, then the performance of linear inverse techniques depends upon both $\mathbf{e}$ and $\boldsymbol{\eta}$. In practice, the statistics of the measurement noise $\boldsymbol{\eta}$ can be estimated in order to tune the proposed inverse algorithm accordingly. However, it is difficult to estimate $\mathbf{e}$, since it requires a-priori knowledge of the unknown contrast variable $\mathbf{x}$.
Stand-alone NRI systems tend to perform poorly in breast cancer imaging applications because they lack a suitable background model $\epsilon_b(\mathbf{r},\omega)$. Without any prior knowledge, NRI systems select a homogeneous background medium that has a dielectric constant derived from averaging that of low-water-content (LWC) fatty tissue and high-water-content (HWC) fibroglandular tissue. Choosing a homogeneous dielectric constant leads to a contrast variable $\mathbf{x}$ that has a significant number of large, non-zero elements. This violates the assumptions made by the Born approximation and produces an error vector $\mathbf{e}$ with a large norm, which ultimately challenges the accurate inversion of the linear system of equations.
The hybrid DBT / NRI system utilizes the prior knowledge obtained from the DBT image in order to construct the heterogeneous background model, thereby overcoming the aforementioned limitations of the homogeneous model used by stand-alone NRI systems. In this hybrid system, the DBT image is segmented into three types of tissue, skin, muscle (pectoralis major), and breast tissue, and it is assumed that the latter only contains healthy tissue. Additionally, each pixel of breast tissue consists of $p\%$ of fatty tissue and $(100-p)\%$ of fibroglandular tissue, and its constitutive properties $\epsilon_r(\mathbf{r}, \omega)$ and $\sigma(\mathbf{r}, \omega)$ are determined from a composite model, which considers the dispersive properties of the breast tissues [@MartinezLorenzo2013a]. Fig. \[fig:segment\] displays the fat content segmented from a single slice of a 3D DBT image. The segmented geometry is input to an electromagnetic numerical simulation based on Finite Differences in the Frequency Domain (FDFD) [@Rappaport2001] in order to model the NRI sensing process. The FDFD is a full-wave model that accounts for all mutual interactions within the breast, and is used to generate the incident fields $\mathbf{E}_b(\mathbf{r}, \omega)$ and the non-uniform, cancer free Green’s functions $\mathbf{G}_b(\mathbf{r}, \mathbf{r}', \omega)$.
![Fat content of breast segmented from a 3D DBT image. High intensity indicates a high percentage of fat.[]{data-label="fig:segment"}](./dbt_fat.png){height="\figsizea"}
Imaging with Compressive Sensing
================================
Ideally, the hybrid DBT / NRI system would segment the healthy breast tissue perfectly and would classify any cancerous tissue as HWC fibroglandular tissue, producing a contrast variable $\mathbf{x}$ that is non-zero only at the locations of cancerous lesions. In this case, the reconstruction process becomes a sparse recovery problem, which can be solved using techniques from compressive sensing (CS) theory. CS theory is a relatively novel signal processing technique, which was first introduced by Candes et al. in [@Candes2006a], and it has since been refined in many other works [@Candes2006; @Donoho2006; @Becker2011; @massa2015]. CS establishes that sparse signals can be recovered using far fewer measurements than required by the Nyquist sampling criterion. In order to apply such principles, certain mathematical conditions must be satisfied by the sensing matrix $\mathbf{A}$ and the reconstructed image $\mathbf{x}$. The sensing matrix must satisfy the Restricted Isometry Property condition, which is related to the independence of its columns, and the number of non-zero entries, $N_{nz}$, of the reconstructed image must be much smaller than the total number of elements, $N$. If the two aforementioned conditions are satisfied, then the reconstruction of the unknown vector can be performed with a small number of measurements by solving a norm-$1$ optimization problem. In this work, the contrast variables $\mathbf{x}$ are recovered by solving the modified basis pursuit denoising problem: $$\begin{aligned}
\underset{\mathbf{x}}{\text{minimize}} ~~&\frac{1}{2}\|\mathbf{A}\mathbf{x}-\mathbf{y}\|_{\ell_2}^2 + \lambda \|\mathbf{x}\|_{\ell_1} \label{eq:l1opt} \\
\text{subject to} ~~&\operatorname{\mathbf{Re}}(\operatorname{diag}({\boldsymbol\epsilon}_b)\mathbf{x} + {\boldsymbol\epsilon}_b) \ge \mathbf{1} \nonumber \\
~~&\operatorname{\mathbf{Im}}(\operatorname{diag}({\boldsymbol\epsilon}_b)\mathbf{x} + {\boldsymbol\epsilon}_b) \ge \mathbf{0} \nonumber\end{aligned}$$ where the constraints on the variable $\mathbf{x}$ ensure that the solution is physically realizable, i.e. $\epsilon_r \ge 1$ and $\sigma \ge 0$. In practice, the weighting factor $\lambda$ must be selected based upon the expected error in the measurement vector $\mathbf{y}$ and sparsity of the vector $\mathbf{x}$. This problem can be efficiently solved using Nesterov’s accelerated gradient method for non-smooth convex functions [@Nesterov2005], which was first applied to the basis pursuit denoising problem in [@Becker2011]. In this method, the norm-$1$ term is replaced by the smooth approximation: $$g_\mu(\mathbf{x}) = g_\mu(x_1, \ldots, x_N) = \sum_{n=1}^N f_\mu(x_n)$$ where the Huber function $f_\mu(\cdot)$ is defined as: $$f_\mu(x) = \begin{cases}
|x| & |x| \ge \mu \\
\frac{1}{2\mu}|x|^2 & |x| < \mu
\end{cases}$$ The Huber function is Lipschitz continuously differentiable, so first-order techniques such as Nesterov’s method can be used to minimize the smoothed objective function $\frac{1}{2}\|\mathbf{A}\mathbf{x}-\mathbf{y}\|_{\ell_2}^2 + \lambda g_\mu(\mathbf{x})$. At each step in Nesterov’s method, the desired variable is updated using a step direction derived from the gradients up to and including the current iteration. This updated point is then projected onto the feasible set to ensure that all of the constraints are satisfied. For the imaging problem of Eq. \[eq:l1opt\], the projection operator can be expressed as the solution to the following convex optimization problem: $$\begin{aligned}
\underset{\mathbf{x}}{\text{minimize}} ~~&\|\mathbf{x}-\mathbf{z}\|_{\ell_2}^2 \label{eq:prox} \\
\text{subject to} ~~&\operatorname{\mathbf{Re}}(\operatorname{diag}({\boldsymbol\epsilon}_b)\mathbf{x} + {\boldsymbol\epsilon}_b) \ge \mathbf{1} \nonumber \\
~~&\operatorname{\mathbf{Im}}(\operatorname{diag}({\boldsymbol\epsilon}_b)\mathbf{x} + {\boldsymbol\epsilon}_b) \ge \mathbf{0} \nonumber\end{aligned}$$ This problem is separable in the elements $x_1,\ldots,x_N$ of the vector $\mathbf{x}$. For a scalar contrast $x$, it can be shown that this projection problem has the following closed form solution: $$\begin{aligned}
P_{Q_p}(x) &= \frac{P_\epsilon(\epsilon_b x + \epsilon_b) - \epsilon_b}{\epsilon_b}\label{eq:proj1} \\
P_{\epsilon}(\epsilon) &= \operatorname{max}(\operatorname{Re}(\epsilon),1) + \jmath \operatorname{max}(\operatorname{Im}(\epsilon), 0) \label{eq:proj2}\end{aligned}$$ The reader is referred to the literature [@Nesterov2005; @Becker2011] for further details on Nesterov’s method
Numerical Results {#sec:results}
=================
Following the process of Section \[sec:sensing\_form\], a 2D model of a healthy breast was generated by segmenting a 2D slice from a 3D DBT image. In order to simulate data from a cancerous case, a lesion with frequency-dependent electrical properties modeled after [@Lazebnik2007] was added to the healthy breast. A 2D version of the FDFD code was used to generate the synthetic NRI measurements of the healthy breast, the synthetic NRI measurements of the cancerous breast, and the sensing matrix of the healthy breast $\mathbf{A}$ according to Eq. \[eq:born\_approx\]. Note that the FDFD model accounted for the dispersive properties of both the healthy breast tissue and the cancerous tissue; only the inversion process utilized the simplifying assumptions of Section \[sec:sensing\_form\]. In the simulation, the NRI system used six transmitting and receiving antennas operating in a multiiple monostatic configuration. Each antenna was excited with three different frequencies, $500$MHz, $600$MHz, and $700$MHz, for a total of $18$ measurements among the antennas.
![Real and imaginary parts of true contrast variable $\chi_{{\epsilon}}$ obtained when the DBT image is segmented perfectly.[]{data-label="fig:true_contrast_000"}](./true_contrast_dbt_000.png){height="\figsizeb"}
Figure \[fig:true\_contrast\_000\] displays the true contrast variable obtained when the fat percentage is perfectly segmented from the DBT image. In this plot, the white dots represent the antenna positions and the green curves represent the breast and lesion borders. Since the fat percentage was segmented perfectly, the contrast variable is non-zero only at the location of the cancerous lesion. Figure \[fig:cs\_contrast\_000\_000\] displays the estimated contrast variable obtained using the perfect fat percentage segmentation and noiseless measurements. The artifacts within the image are due to the error vector $\mathbf{\hat{e}}_s(\mathbf{r}, \omega)$ that is introduced to the measurement vector when the simplifying assumptions of Section \[sec:sensing\_form\] are applied. Despite these artifacts, the algorithm is able to locate the cancerous lesion. This represents a significant improvement over the phase-based results presented in [@MartinezLorenzo2013], which utilized $17$ antennas in a multi-static configuration and $11$ frequencies over a $1$GHz bandwidth in order image the cancerous lesion.
![Real and imaginary parts of reconstructed contrast variable $\hat{\chi}_{{\epsilon}}$ obtained when the DBT image is segmented perfectly and there is no measurement noise.[]{data-label="fig:cs_contrast_000_000"}](./res_dbt_000_nse_000.png){height="\figsizeb"}
Figure \[fig:true\_contrast\_100\] displays the true contrast variable obtained when the fat percentage is segmented from the DBT image with $10\%$ random error. More specifically, the fat percentage values were corrupted by i.i.d. random noise following a uniform distribution, taking values between $\pm 10\%$ with equal probability. Since the fat percentage is not segmented correctly, the true contrast variable is non-zero within the healthy tissue. Nevertheless, the true contrast variable is approximately compressible, and so Eq. \[eq:l1opt\] can still be used to image the breast. This result can be seen in Figure \[fig:cs\_contrast\_100\_000\], which displays the estimated contrast variable obtained using the noisy fat percentage segmentation and noiseless measurements.
![Real and imaginary parts of true contrast variable $\chi_{{\epsilon}}$ obtained when the fat percentage is segmented from the DBT image with $10\%$ error.[]{data-label="fig:true_contrast_100"}](./true_contrast_dbt_100.png){height="\figsizeb"}
![Real and imaginary parts of reconstructed contrast variable $\hat{\chi}_{{\epsilon}}$ obtained when the fat percentage is segmented from the DBT image with $10\%$ error and there is no measurement noise.[]{data-label="fig:cs_contrast_100_000"}](./res_dbt_100_nse_000.png){height="\figsizeb"}
Figure \[fig:cs\_contrast\_100\_300\] displays the estimated contrast variable obtained using the noisy fat percentage segmentation and measurements whose SNR $=49\text{dB}$. It is important to note that the signals in this SNR calculation are the electric fields scattered by the entire breast, and not just the fields scattered by the cancerous lesion. In this example, the fields scattered by the lesion are approximately $40\text{dB}$ lower in magnitude than the fields scattered by the rest of the breast, so that the “lesion signal to noise ratio” is on the order of $10\text{dB}$. Therefore, the NRI system must have a significant SNR to ensure that the fields produced by cancerous lesions are not overwhelmed by the noise, or it must have antennas with higher directivity in order to improve the SNR - the latter case may require the CS algorithm to use additional measurements. With this high SNR, the CS algorithm is able to image the cancerous lesion with some additional artifacts compared to the noiseless case. However, when the SNR is decreased to $43\text{dB}$, the algorithm is no longer able to image the lesion, as can be seen in Figure \[fig:cs\_contrast\_100\_500\].
![Real and imaginary parts of reconstructed contrast variable $\hat{\chi}_{{\epsilon}}$ obtained when the fat percentage is segmented from the DBT image with $10\%$ error and and the measurement SNR $=49\text{dB}$.[]{data-label="fig:cs_contrast_100_300"}](./res_dbt_100_nse_300.png){height="\figsizeb"}
![Real and imaginary parts of reconstructed contrast variable $\hat{\chi}_{{\epsilon}}$ obtained when the fat percentage is segmented from the DBT image with $10\%$ error and and the measurement SNR $=43\text{dB}$.[]{data-label="fig:cs_contrast_100_500"}](./res_dbt_100_nse_500.png){height="\figsizeb"}
Conclusions {#sec:conclusions}
===========
This work presents a novel approach for imaging breast cancer using compressive sensing techniques in a hybrid DBT / NRI imaging system. Using the prior knowledge obtained from the DBT reconstruction, the hybrid system creates a background tissue distribution of the assumed healthy breast, thereby overcoming a common pitfall in stand-alone NRI imaging systems. Since cancerous lesions tend to be localized to a relatively small region of the breast, image reconstruction can be expressed as a sparse recovery problem, which is solved using techniques from compressive sensing. Numerical results show that the CS imaging algorithm can localize cancerous lesions within in the breast, even when corrupted by DBT segmentation and measurement errors.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work has been funded by the U.S. National Science Foundation award number 1347454.
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---
abstract: 'Symmetry arguments are used to show that a boundary of a magnetoelectric antiferromagnet has an equilibrium magnetization. This magnetization is coupled to the bulk antiferromagnetic order parameter and can be switched along with it by a combination of $\mathbf{E}$ and $\mathbf{B}$ fields. As a result, the antiferromagnetic domain state of a magnetoelectric can be used as a non-volatile switchable state variable in nanoelectronic device applications. Mechanisms affecting the boundary magnetization and its temperature dependence are classified. The boundary magnetization can be especially large if the boundary breaks the equivalence of the antiferromagnetic sublattices.'
author:
- 'K. D. Belashchenko'
title: Equilibrium magnetization at the boundary of a magnetoelectric antiferromagnet
---
Magnetoelectric antiferromagnets (AFM) develop a magnetization $\mathbf{M}$ (or electric polarization $\mathbf{P}$) in the bulk when an electric (or magnetic) field is applied [@LL8; @Fiebig; @Schmid]. This property is due to the presence of a magnetoelectric term in the free energy, $F_{\mathrm{ME}}=-\alpha_{ik}E_iH_k$, where $\alpha_{ik}$ is the magnetoelectric tensor; the latter is odd under time reversal. An AFM is magnetoelectric if the presence of an invariant polar vector $\mathbf{E}$ can reduce its magnetic point group to a ferromagnetic one [@Dzyal-Cr2O3; @Schmid].
Magnetoelectric and multiferroic materials can provide the necessary response to allow electrical switching of the magnetic state [@Fiebig; @Eerenstein; @RS; @Cheong] and potentially enable fast, high-density, low-power, and non-volatile memory devices (magnetoelectric memory) [@BD; @Bibes; @Ramesh; @Chu]. To enable easy readout of the magnetic state, the magnetoelectric or multiferroic layer needs to be coupled to a proximate ferromagnetic layer. This coupling requires an exchange bias [@Meiklejohn; @Nogues1; @Nogues2] at the interface, which is the time-reversal-breaking shift the hysteresis loop of the ferromagnet along the magnetic field axis. Much attention in this context has been focused on the room-temperature multiferroic BiFeO$_3$, but the destabilizing effects of ferroelastic strains and depolarizing fields need to be circumvented for non-volatile operation [@Baek]. Ferroelastic strain could be avoided by using a multiferroic material with linear coupling of $\mathbf{P}$ and $\mathbf{M}$, but suitable materials for room-temperature operation are not available [@Fennie].
An alternative approach to electric magnetization control uses the AFM order parameter of a magnetoelectric material as the switchable state variable. Magnetoelectric switching of Cr$_2$O$_3$ was shown [@Binek] to induce a reversible switching of the exchange bias polarity in the proximate ferromagnetic Pd/Co multilayer on the macroscopic scale. It was argued [@Binek] that this effect is a manifestation of the equilibrium boundary magnetization of a magnetoelectric, which is required by symmetry and couples to the bulk AFM order parameter. Essentially, the boundary reduces the symmetry in a similar way to the electric field. Another manifestation of this effect is the spin polarization of the photoelectron current emitted from the free Cr$_2$O$_3$ (0001) surface [@Binek].
Macroscopic signatures of boundary magnetization of Cr$_2$O$_3$ [@Binek] show that the lack of macroscopic time-reversal symmetry in a magnetoelectric can translate into strong spin asymmetry at its boundary. However, the microscopic mechanisms of this effect are not understood. In this Letter the salient features of boundary magnetism of magnetoelectrics are analyzed from the general point of view. A rigorous microscopic proof of the existence of equilibrium boundary magnetization is given, and its microscopic mechanisms are classified. In particular, it is shown that the effects can be very large if the boundary breaks the equivalence of the AFM sublattices.
Consider a *macroscopically* flat boundary (surface or interface) of an AFM with an external normal $\mathbf{n}$, which is allowed to have roughness and all possible terminations distributed with equilibrium Gibbs probabilities. The magnetic structure of the boundary is also assumed to be equilibrium, subject to the constraint that the bulk of the crystal is in the single AFM domain state [@note-staggered].
We are generally interested in the response of the boundary on the macroscopic scale to an external probe which couples to the magnetic moment density $\mathbf{m}(\mathbf{r})$. This probe can represent spin-resolved photoemission, magneto-optic Kerr effect, exchange bias with a ferromagnet, or magnetometry. For typical probes the measured quantity is an *odd* functional $G\{m_i(\mathbf{r})\}$, such that $G\{m_i(\mathbf{r})\}=G\{m_i(\mathbf{r}+\mathbf{t})\}$ for any shift $\mathbf{t}$ (or at least for any translation vector of the bulk lattice treated as non-magnetic, such that $\mathbf{t}\cdot\mathbf{n}$ is not large compared to the equilibrium roughness). Hereafter a probe is assumed to satisfy this condition [@probe-note]. The component $m_i$ is selected by the polarization of the probe.
The following arguments do not depend on the nature of the probe. For definiteness, let us select the *boundary moment* $\vec\mathcal{M}$ as the probe, defined in a way that satisfies the above requirement of translational invariance. Specifically, if the magnetic unit cell is larger than the structural unit cell, the magnetic moment $\vec\mathcal{M}$ of the boundary region must be averaged over the different ways of separating this region from the bulk, related to each other by purely structural translations (see Fig. 1). Surface-sensitive probes like exchange bias or spin-polarized spectroscopies are free from this complication.
The macroscopic boundary magnetization is given by the equilibrium Gibbs average $\mathbf{M}_{b}=\langle \vec\mathcal{M} \rangle/A$, where $A$ is the boundary area, and the thermodynamic limit of large $A$ is assumed. Its $i$-th component vanishes only if any boundary configuration (termination and magnetic structure) has an energetically degenerate one with a reversed $\mathcal{M}_i$; otherwise it is finite. Such degeneracy occurs only if the bulk magnetic *space* group of the crystal contains an operation under which the vector $\mathbf{n}$ is invariant, and $\mathcal{M}_i$ is odd. All degenerate boundary configurations can be generated by bulk space group operations; this is the boundary generalization of the Aizu procedure [@Aizu]. In particular, energetically degenerate atomic steps are automatically accounted for, as shown in Fig. 1 and 2b. However, since both $\mathbf{n}$ and $\mathcal{M}_i$ are invariant with respect to any translation, the latter can be disregarded, and we are led to consider only the magnetic [*point*]{} group. The presence of an invariant polar vector $\mathbf{n}$ selects the same subgroup of the bulk magnetic point group as a homogeneous $\mathbf{E}$ field. It follows that the boundary acquires the same magnetization components as the bulk with an applied $\mathbf{E}$ field in the direction of $\mathbf{n}$. Therefore, the boundary develops finite equilibrium magnetization only if the bulk is magnetoelectric. This conclusion is equally valid for a metallic AFM whose magnetic point group would make an insulator magnetoelectric.
{width="40.00000%"}
This conclusion reflects the fact that the free energy of the system with a boundary depends on the polar vector $\mathbf{n}$ as a macroscopic parameter. Just as in the bulk, the existence of the magnetization at the boundary can be deduced from the reduction of the bulk magnetic point group by the presence of the boundary, because translations do not affect $\mathbf{n}$ or $\mathbf{M}_b$. From a different angle, a boundary can generate a magnetization only if its zero value is not protected by macroscopic time-reversal symmetry in the bulk; this singles out the magnetoelectrics.
Thus, equilibrium boundary magnetization of a magnetoelectric is finite unless $\alpha_{zk}=0$ for all $k$ in the reference frame where $\mathbf{n}$ is parallel to the $z$ axis. If this magnetization is finite for the given $\mathbf{n}$, it is also necessarily finite for any particular termination with this orientation, because the magnetic symmetry group of the latter is a subgroup of the former.
Probes that are not surface-sensitive, such as magnetometry with $G\{m_i(\mathbf{r})\}=\int m_i(\mathbf{r})d^3r$, measure the sum of contributions of two film boundaries. The total magnetization is non-zero unless there is a bulk symmetry operation interchanging the boundaries. It is always non-zero if these boundaries are with different materials.
The exchange bias induced in a proximate ferromagnetic film by a magnetoelectric is fundamentally different from the conventional exchange bias, which is due to a small excess magnetic moment “frozen-in” in the AFM during field-cooling. In particular, this non-equilibrium character typically leads to an irreversible decline of the exchange bias as the magnetization of the ferromagnetic layer is repeatedly reversed — the so-called training effect [@Nogues1; @Nogues2]. By contrast, the switchable exchange bias observed in Ref. is an equilibrium property and does not exhibit the training effect.
Since the effect of the boundary is comparable to that of $\mathbf{E}$ of the crystal-field scale, even simple extrapolation suggests that the induced magnetic moments at the boundary can be a few orders of magnitude larger than those achievable due to the bulk magnetoelectric effect. In fact, some mechanisms do not contain any small parameters and are capable of producing magnetizations of the order of a few Bohr magnetons per boundary site. I now classify these mechanisms.
All mechanisms producing linear magnetoelectric response in the bulk [@Fiebig] can generate boundary magnetization as well; these include changes in (A) the $g$-tensor (here we can also include hybridization-induced changes of the spin moment), (B) the single-ion anisotropy tensor, (C) the *intrasublattice* symmetric coupling (including Heisenberg exchange and dipolar interactions), and (D) the Dzyaloshinsky-Moriya [@Dzyal; @Moriya] exchange coupling induced by $\mathbf{E}$ or by the boundary. All of these except C involve relativistic terms in the Hamiltonian.
Consider the usual case of a collinear AFM. Mechanism C is active only if the perturbation breaks the equivalence of the AFM sublattices. This symmetry breaking can be identified by analyzing the so-called black-and-white (Heesch-Shubnikov) point group based on the decoration of the magnetic sites with Ising spin variables instead of axial vectors. If the perturbation (polar vector $\mathbf{E}$ or $\mathbf{n}$) removes all symmetry operations mapping black and white sites onto each other, the AFM sublattices become inequivalent; otherwise they do not. If the black-and-white point group allows Ising ferromagnetism, the true magnetic point group is also ferromagnetic, but the reverse is not necessarily true. In the first case mechanism C is allowed, but in the second case the magnetoelectric response occurs only through spin canting due to a relativistic perturbation. Thus, certain components of the magnetoelectric tensor, and likewise the boundary magnetization for certain directions of $\mathbf{n}$, may have no contribution from mechanism C. For example, consider Cr$_2$O$_3$ (magnetic point group $\underline{\overline{3}m}$) with $\mathbf{E}$ or $\mathbf{n}$ oriented parallel to one of the three $U_2$ axes or to one of the three $\sigma_d$ planes bisecting them. The corresponding symmetry operation is not removed by $\mathbf{E}$ or $\mathbf{n}$; since both $U_2$ and $\sigma_d$ interchange the AFM sublattices, the latter remain equivalent. However, the appearance of $\mathbf{M}$ parallel to $\mathbf{E}$ or $\mathbf{M}_b$ parallel to $\mathbf{n}$ through spin canting is allowed.
{width="85.00000%"}
If the equivalence of the AFM sublattices *is* broken by the boundary, the consequences are far more drastic than in the bulk mechanism C. For any particular boundary termination, even without reconstruction, the sites corresponding to different AFM sublattices are structurally different. For example, all the sites closest to the boundary can have spins “down,” while there is no equivalent termination with boundary spins “up.” The situation is illustrated in Fig. 2b using Fe$_2$TeO$_6$ (magnetic point group $4/\underline{mmm}$) as an example. In this figure, terminations A (with boundary spins up) and B (with boundary spins down) are structurally distinct, and therefore they occur with different probabilities in equilibrium. Even if they did appear with equal weights, they are inequivalent magnetically, and don’t generally add up to zero. Indeed, there are several mechanisms leading to the deviation of the average magnetic moments on the boundary sites from the bulk ones (see Fig. 2c-2e): (S1) Different local environment of the magnetic sites near the boundary leads to a different local magnetic moment, and perhaps even a different atomic multiplet. (S2) Since the translational symmetry is broken by the boundary, any symmetric exchange interaction (even purely *intersublattice* one) leads to different thermal averages at the boundary. (S3) The exchange coupling near the boundary can be so different from the bulk that the AFM ordering pattern can change to ferrimagnetic there. Mechanism S1 can be viewed as a boundary analog of bulk mechanism A, and S2 is the boundary counterpart of mechanism C. S1 and S2 are always present if the black-white symmetry is broken. Apart from these mechanisms affecting the magnitudes of the local moments, the *coupling* of these moments to the external probe can be different. For example, in the exchange bias setup, the sites closest to the boundary are expected to have the strongest exchange coupling to the proximate ferromagnet.
The bulk linear magnetoelectric effect appears in the free energy expansion as a second-rank tensor $\alpha_{ik}$. This is not the case for the equilibrium boundary magnetization. Since the free energy is a non-analytic function of $\mathbf{n}$ [@LL5], the boundary magnetization, given by its field derivative, is also non-analytic. Thus, even if the AFM sublattices are equivalent for some symmetric directions of $\mathbf{n}$, forbidding mechanisms S1-S3 for these orientations, these mechanisms can still generate large boundary magnetization for less symmetric orientations.
Boundary magnetization $\mathbf{M}_b$ vanishes at the bulk Néel temperature, but different mechanisms can be partially distinguished experimentally based on its temperature dependence. The thermal mechanism S2 can lead to a non-monotonic contribution with a maximum, similar to the bulk behavior of mechanism C (cf. $\alpha_{zz}$ in Cr$_2$O$_3$). Other mechanisms should lead to $\mathbf{M}_b$ monotonically decreasing with $T$. All these mechanisms do not contain any small parameters and can produce $\mathbf{M}_b$ of the order of a few Bohr magnetons per boundary site. The non-monotonic temperature dependence of the exchange bias field observed in the heterostructure of Ref. [@Binek] suggests that mechanism S2 plays an important role at the Cr$_2$O$_3$(0001)/Pd interface.
Roughness-insensitive mechanisms based on Dzyaloshinskii-Moriya interaction at the compensated interface were proposed [@Dagotto] to explain the exchange bias induced by single-domain multiferroic BiFeO$_3$ in a proximate FM [@Ramesh]. They are, however, fundamentally different from all the mechanisms discussed here, because BiFeO$_3$ is weakly ferromagnetic in the bulk, and also its $\mathbf{M}$ and $\mathbf{P}$ are not linearly coupled [@Spaldin]. Therefore, the boundary does not induce additional magnetization components that are forbidden in the bulk.
Much attention is devoted to the search of a room-temperature multiferroic material with linear coupling between the electric polarization $\mathbf{P}$ and magnetization $\mathbf{M}$, because its $\mathbf{M}$ could be switched along with $\mathbf{P}$ by electric field only [@Eerenstein; @RS; @Cheong; @Bibes; @Fennie]. The paraelectric phase of such a material is a magnetoelectric AFM [@Scott; @Fennie]. The desirable geometry involves voltage applied across the multiferroic film, with $\mathbf{E}$ normal to its surface [@Bibes]. Equilibrium magnetization $\mathbf{M}_b$ necessarily exists at the boundary of such a material. This $\mathbf{M}_b$ has the same components as the bulk $\mathbf{M}$ coupled to $\mathbf{P}$, but $\mathbf{M}_b$ is coupled to the bulk AFM order parameter. Ferroelectric switching directly switches only the part of $\mathbf{M}$ linearly coupled to $\mathbf{P}$, but not $\mathbf{M}_b$. In addition, the states with parallel or antiparallel $\mathbf{M}_b$ and $\mathbf{M}$ are non-degenerate, meaning that one of them is metastable or even unstable. Thus, the presence of intrinsic boundary magnetization may hamper purely electric magnetization switching using a multiferroic with linear coupling of $\mathbf{P}$ and $\mathbf{M}$.
Equilibrium boundary magnetization of magnetoelectric AFMs has far-reaching consequences for the design of magnetic nanostructures. First, very large exchange bias fields should be achievable in magnetoelectric/ferromagnet bilayers, comparable to the estimates for a fully uncompensated AFM interface [@Meiklejohn; @Kiwi]. Second, magnetoelectric AFMs are precisely the materials that can be switched between the time-reversed AFM domain variants by a simultaneous application of $\mathbf{E}$ and $\mathbf{B}$ fields [@Schmid; @Martin], thereby switching the boundary magnetization and the exchange bias field [@Binek]. The $\mathbf{B}$ field may be permanent, while $\mathbf{E}$ may be provided by a voltage pulse across the magnetoelectric film. Since no depolarizing fields or elastic strains are involved, the AFM domains are stable, and this switching is fully non-volatile. Some device architectures based on the magnetoelectric active layer, where the operation is based on the linear bulk magnetoelectric response, were described by Binek and Doudin [@BD]. These architectures are greatly facilitated by the existence of a switchable equilibrium boundary magnetization, which moreover allows the AFM domain state to be used as a switchable state variable.
In summary, symmetry requires that magnetoelectric antiferromagnets possess a finite boundary magnetization in thermodynamic equilibrium. This magnetization is particularly large if the boundary breaks the equivalence of the antiferromagnetic sublattices; specific microscopic mechanisms have been classified. This understanding will hopefully stimulate further studies of boundary magnetization of magnetoelectrics and its exploitation in nanoelectronic devices.
This work was supported by NSF MRSEC, the Nanoelectronics Research Initiative, and Nebraska Research Initiative. The author is a Cottrell Scholar of Research Corporation.
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---
abstract: |
In recent years machine learning (ML) took bio- and cheminformatics fields by storm, providing new solutions for a vast repertoire of problems related to protein sequence, structure, and interactions analysis. ML techniques, deep neural networks especially, were proven more effective than classical models for tasks like predicting binding affinity for molecular complex.
In this work we investigated the earlier stage of drug discovery process – finding druggable pockets on protein surface, that can be later used to design active molecules. For this purpose we developed a 3D fully convolutional neural network capable of binding site segmentation. Our solution has high prediction accuracy and provides intuitive representations of the results, which makes it easy to incorporate into drug discovery projects. The model’s source code, together with scripts for most common use-cases is freely available at <http://gitlab.com/cheminfIBB/kalasanty>.
author:
- 'Marta M. Stepniewska-Dziubinska^1^'
- 'Piotr Zielenkiewicz^1,2^'
- 'Pawel Siedlecki^1,2,\*^'
bibliography:
- 'references.bib'
date: |
^1^Institute of Biochemistry and Biophysics, Polish Academy of Sciences, Pawinskiego 5a, 02-106 Warsaw, Poland\
^2^Department of Systems Biology, Institute of Experimental Plant Biology and Biotechnology, University of Warsaw, Miecznikowa 1, 02-096 Warsaw, Poland\
^\*^[email protected]
title: 'Improving detection of protein-ligand binding sites with 3D segmentation.'
---
Introduction
============
The aim of rational drug design is to discover new drugs faster and cheaper. Much of the effort is put into improving docking and scoring methodologies. However, most techniques assume that the exact location of binding sites – also referred to as *pockets* or *binding cavities* – is known. Such pockets can be located both on a surface of a single protein (and be used to modulate its activity) or at protein-protein interaction (PPI) interfaces (and be used to disrupt the interaction). This task is very challenging and we lack a method that would predict binding sites with high accuracy – most methods are able to detect only 30%-40% of pockets [@benchmark; @deepsite].
Traditional approaches for binding cavity detection are typically geometry-based [@pass; @pocketpicker; @fpocket; @depth], but there are also examples of tools using binding energy to different chemical probes [@ftsite; @sitehound], sequence conservation (template or evolutionary methods) [@concavity; @glosa; @probis], or a combination of these [@ligsitecsc; @metapocket]. For example, ProBiS [@probis] – similarity-based tool, uses local surface alignment with sub-residue precision, allowing to find sites with similar physicochemical properties to the templates stored in the database. Such methods simultaneously detect binding sites and provide some insight into their expected properties – they are most probably similar to the templates they were matched to. Other approaches rely on a two-step algorithm, in which potential pockets are first identified and then scored to select the most probable binding sites. For example, Fpocket [@fpocket] is a geometry-based method, which first finds cavities in a protein’s structure and then scores them. The reverse approach is used in P2RANK [@p2rank], which uses a random forest (RF) model to predict “ligandibility” score for each point on a protein’s surface, to then cluster points with high scores.
The latter tool is an example of applying machine learning (ML) to detect pockets – supervised ML to score surface points and unsupervised ML to post-process these predictions. Unsupervised ML models are trained on unlabeled observations and aim to find patterns in the data in order to simplify their representation, remove the noise, and get better understanding of their nature. Supervised ML models, on the other hand, require the observations to be paired with their corresponding labels (expected output, class, etc.). The main purpose of this class of models is finding the relationship between the data and the labels that are actually desired. The data is relatively readily available (in case of P2RANK – the structure of a protein) but the desired information is typically much harder to acquire (e.g. location of binding sites).
Another axis of classification of ML models is based on their complexity, or *depth*. Deep learning (DL) is a branch of ML grouping more complex models of different types, both supervised and unsupervised. There is no clear border between “classical” ML and DL, but in general deep models are complex, multilayer neural networks capable of finding more sophisticated and convoluted relations between input and labels than their shallow counterparts. DL models require less manual work and feature engineering, and use model’s internal layers to extract features from the unprocessed data.
In the context of bio- and cheminformatics DL allows to predict *in silico* properties that require much effort to establish experimentally, like detecting functional motives in sequences [@dl_seq] or assessing binding affinity for protein-ligand complexes [@kdeep; @pafnucy].
A recent example of a deep model used for binding site detection is DeepSite [@deepsite]. Similarly to P2RANK, DeepSite classifies each point in a 3D space based on its local environment as belonging (or not) to a binding pocket. Probabilities for all points form a 3D density, that can then be post-processed to get the most probable locations and shapes of binding sites present in the structure. Unlike P2RANK however, DeepSite uses a deep 3D convolutional neural network, with an architecture typical for image classification problems – set of convolutional layers paired with max pooling layers, followed by fully connected layers and a final neuron with the predicted class. Such an architecture allows for the extraction of features – 3D structural patterns – that are immediately used by the model to make predictions.
DeepSite was proven superior to two other state-of-the-art approaches at the time: Fpocket [@fpocket] and Concavity [@concavity]. The first one is a geometry-based, whereas the second – a sequence-conservation-based method. But although DeepSite achieved better results, only approximately 50% of the predicted pockets is at most 4 from the actual position of the binding site. This is not an acceptable standard and calls for an improvement.
In this work we present a different DL-based approach for finding binding pockets, inspired by semantic image segmentation instead of classification. Image segmentation aims at locating an object, or multiple objects, in an image. Output of such a model is a set of scores assigned to each pixel, where the score denotes the probability that a given pixel belongs to the desired object.
In our case, the input is a 3D structure of a protein represented with a grid that can be analyzed in the same manner as 3D images, whereas the desired object is the binding pocket. Our model called Kalasanty is based on U-Net [@unet] – a state of the art model for image segmentation. We adapted this model to the problem of binding cavity detection, and added functionalities that allow to easily generate predictions for protein structures. The model takes protein structure as input, automatically converts it to a 3D grid with features, and outputs probability density – each point in the 3D space has assigned probability of being a part of a pocket. Predictions can then be saved as `.cmap` or `.cube` files, that can be later analyzed in molecular modeling software. Kalasanty can also output parts of the protein that form pockets and save them as `.mol2` files.
Methods
=======
Our approach
------------
In order to solve any problem with DL, it first needs to be specified in terms of how input and output are represented. In this work we formulate the pocket detection task as a 3D image segmentation problem. This allowed us to use well established DL methods, originally developed for 2D images.
With this approach both input and output are represented as 3D grids with the exact same dimensions – each grid point in the input has a corresponding point in the output. The input is a discretized protein structure with multiple feature channels describing atomic properties. The returned output has a single channel with probability of belonging to the pocket (details in section “Data”). Known pockets, which are used for training and evaluation, are represented with binary grids, where 1s denote grid points that belong to the pocket and 0s otherwise.
{width="\textwidth"}
The model is based on an architecture called U-Net (see Figure \[fig:model\]), which is a fully-convolutional, encoder-decoder model that pioneered skip connections (see text below). This kind of architecture prevents from loosing fine-grained information about the input which greatly increases the precision of the resulting segmentations.
Kalasanty was built using the same ideas and architecture design as the original U-Net. Similarly to the original network, it is composed of blocks consisting of two convolutional layers and a single max-pooling or up-sampling layer (depending on the side of the “U”). However, Kalasanty works on 3D data instead of 2D images, therefore 3D versions of convolutional, pooling and up-sampling layers were used. Also, the number of layers, number of convolutional filters, and patch sizes were adjusted to match the size of the input and difficulty of the task.
Kalasanty has 9 convolutional blocks – 4 in the encoder, one in the bottleneck, and 4 in the decoder part of the network. Each block consists of two convolutional layers with the same number of filters (32, 64, 128, 256, or 512), kernel size of 3x3x3 pixels and ReLU activation function, combined either with a max-pooling layer (encoder path, left side of the Figure \[fig:model\]) or with an up-sampling layer (decoder path, right side of the Figure \[fig:model\]). The two first max-pooling layers and the two last up-sampling layers have 2x2x2 patch sizes, while layers in the middle have 3x3x3 patch sizes. This way, for input of 36x36x36 pixels that was used in this work, feature maps in the middle of the network (bottleneck, bottom of the Figure \[fig:model\]) have spacial sizes of 1x1x1 and can be used as feature vectors.
What is unique about U-Net-like models when compared with other encoder-decoder networks it that the information between the two paths is not only passed through a bottleneck, but also after each block using so-called skip connections. The final feature map from each block in the encoder is copied and concatenated with the first feature map in the corresponding decoder block (orange dashed lines). This allows to better localize features and therefore return more accurate segmentations.
The model was defined with the Keras library [@keras]. Apart from methods needed for ML-related tasks, we implemented custom methods for working with the molecular data: making predictions for molecules; locating amino acids forming the pockets; saving the predicted probabilities as `.cmap` or `.cube` files; and saving parts of proteins forming pockets as `.mol2` files. Source code and network’s parameters are freely available at <http://gitlab.com/cheminfIBB/kalasanty>.
Models were trained with L2 regularization on each layer’s weights ($\lambda=10^{-5}$). Also, random translations and rotations were used during training to augment the dataset. As the objective function we used the negative Dice coefficient for continuous variables: $$C(y, t) = \frac{ -2 \sum_{i,j,k} (y_{i,j,k} \cdot t_{i,j,k}) + \epsilon}{\sum_{i,j,k} (y_{i,j,k} + t_{i,j,k}) + \epsilon}$$ where $y$ and $t$ are the predicted and target segmentations, $(i,j,k)$ are indices of a grid cell, and $\epsilon$ is a smoothing factor ($\epsilon=0.01$ was used). Minimization was performed for $1.5\cdot10^{6}$ steps with 10 samples in each batch, and the Adam optimizer with a learning rate of $10^{-6}$ and with default values for the remaining parameters [@adam].
Data {#sec:met_data}
----
For training and validation of the model the sc-PDB [@scpdb] dataset was used. The database consists of known binding sites, accompanied with prepared protein structures. Binding sites are represented both with 3D shapes of cavities generated with VolSite [@volsite] (which were used in this study) and amino-acids that form them. VolSite describes a binding cavity with a set of pharmacophoric properties arranged on a 3D grid, based on the properties of the neighboring protein atoms. Data is stored as mol2 file, with atoms encoding each property. In this project, we converted data in the grid to a binary information – whether a point in space is part of a pocket or not. Grid cell (see section “Our approach”) with such point inside was considered a part of the pocket.
The database (v 2017) consists of 17594 binding sites, corresponding to 16612 PDB structures and 5540 UniProt IDs. The test set was constructed from a different dataset, one used for benchmarking by Chen *et al.* [@benchmark] (see below). When assessing the quality of a ML model it is important to evaluate it using a separate dataset. Using a different dataset minimizes the risk that there were some database-related artifacts, that might have been exploited by the model. However, it is easy to make a mistake and have the same protein (with slightly different structures) in the training and the test set as well. This is a common error called *data leakage* which leads to overly optimistic assessment of a model. In order to avoid data leakage, all structures of proteins from the benchmark were removed from the sc-PDB database (481 structures). Also, 304 binding sites were discarded because of errors when loading their corresponding protein structures with Open Babel. Finally, 15860 structures, corresponding to 5473 UniProt entries, were used for training.
Number of structures per protein varied from 1 to 280, with median equal to 1 and mean equal to 3.26. The dataset contained protein structures originating from 952 different organisms, from which the most abundant were human (34.4%), *E. coli* (5.6%), Human immunodeficiency virus (4.2%), rat (2.9%), and mouse (2.4%). Also, diverse protein architectures were well represented, with 5171 structures of mainly alpha proteins, 2500 of mainly beta proteins, 11758 of alpha-beta proteins and 53 structures of proteins with few secondary structures [@cath]. The dataset was also diverse from a sequence perspective and contained proteins from 1983 different Pfam families and 982 superfamilies, protein kinases being the most frequent.
The data from sc-PDB were split into 10 folds (subgroups containing 1586 structures each) based on UniProt ID, i.e. all structures of a single protein must be in the same fold. This setup was necessary to avoid data leakage during validation. Putting different structures of the same protein in different folds would result in having almost identical training and validation examples, which might result in unnoticed overfitting, and as a consequence overoptimistic evaluation and possibly selection of incorrect hyperparameters.
We also analysed binding site similarity across obtained folds (using Shaper [@volsite] results provided by sc-PDB) to assure that there is no data leakage during validation. This setup was used to train the models with 10-fold cross-validation (CV). CV results were used to select model and optimization parameters and assess models’ stability. The final model was trained on all 10 folds combined to achieve the best possible performance.
As mentioned, for the test set we used structures from the Chen benchmark [@benchmark]. This benchmark set contains apo and holo structures for 104 proteins (208 in total). Structures were converted to the format used in sc-PDB to evaluate the models using the following steps. First, for each structure ligand(s) and protein were split into separate files using UCSF Chimera [@chimera]. Solvent and ions were assigned to the protein.
Then, we used VolSite [@volsite] (available in IChem toolkit) to describe a cavity for each ligand. In case of apo structures, they were aligned to their holo counterparts, and then the ligands were used to select pockets.
For 59 structures (31 apo and 28 holo) out of the 208 present in the benchmark, VolSite failed to find a cavity because of its insufficient buriedness. Still, the remaining part of the dataset (149 structures with 269 binding sites) offers a valuable test set because it contains diverse proteins not used for training.
We used the final 149 structures with 269 binding sites to assess the performance of our model and the performance of the DeepSite model. Note, that we have no control over the dataset that was used by [@deepsite] and 12 proteins from the test set have been used to train DeepSite. This might result in a slightly more optimistic evaluation for DeepSite, but we have decided to keep those structures so that the test set was larger.
It is important to note, that the results presented in this work cannot be directly compared to the ones presented by Chen *et al.* [@benchmark] because only a subset of the original dataset was used.
Finally, all the resulting protein structures and segmentations were represented with 3D grids with 2 resolution. The grids were centered on a protein center and had 70 in each direction. Proteins were described with 18 atomic features used in our previous project [@pafnucy]. Pockets were transformed to 3D binary masks with same size, center, and resolutions as grids representing their corresponding protein structures.
For 79 (0.5%) entries in the sc-PDB database this procedure lead to empty pocket grids. After manually inspecting several of such cases it turned out that it affects large protein complexes and less carefully prepared protein structures (e.g. 1zis, which contains two unbound protein chains, located far away from each other). Although such structures may arise in high-throughput experiments and analyses (like the one presented in this study), they are highly unlikely to occur in real-life studies aiming to discover binding cavities in a protein of interest. Also, this issue affects a neglectable fraction of the training data and none of the test examples. We therefore decided not to modify the structures, nor the procedure for data preparation. The data with empty pocket grids were used for training as negative examples, and skipped in the validation.
Results evaluation
------------------
Results obtained with Kalasanty and DeepSite were evaluated using two popular metrics: $DCC$ and $DVO$ (discretized volume overlap). $DCC$ is the distance between the predicted and the actual center of the pocket. It is typically used to describe a success rate for the method, i.e. the fraction of sites below the given $DCC$ threshold. Similarly to other authors [@benchmark; @deepsite; @pocketpicker], we analyzed success rates for thresholds up to 20 and considered pockets with $DCC$ below 4 as correctly located. $DVO$, on the other hand, is a more strict metric comparing shapes of the predicted and actual pockets. It is the volume of the intersection of the predicted and the actual segmentations, divided by volume of their union. The two metrics complement one another, highlighting different aspects of prediction quality – correct location ($DCC$) and shape ($DVO$) of predicted pockets. When used together, they provide concise yet rich description of the results, and allow to analyze them faster and more objectively than with visual investigation.
In order to calculate both metrics, predicted densities were converted to binary segmentations. For Kalasanty, probability threshold of 0.5 was used, which was selected based on models’ performance on the validation set. However, we note that using thresholds between 0.3 and 0.8 leads to very similar results.
DeepSite predictions were obtained using the playmolecule.org web-service. Predicted binding cavities were download as `.cube` files and converted to binary masks using the probability threshold of 0.4, which was recommended by DeepSite’s authors.
Following the work of Chen *et al.* [@benchmark], for each structure only $n$ predicted pockets with the highest scores were considered, where $n$ is the number of pockets present in the structure. Then, each predicted pocket was matched with the closest real pocket and $DCC$ and $DVO$ values were calculated. If no pocket was predicted, we used the worst possible values for the metrics, which were $DVO=0$ (no overlap) and $DCC=70\sqrt{3}$$=121.24$ (the biggest possible distance for a 70 cube).
Additionally we used F1 score – metric typically used to evaluate ML models for detection tasks. F1 score combines precision (positive predictive value) and recall (also called sensitivity) – its their harmonic mean. Conversely to other popular metrics used in ML, like accuracy or ROC AUC, F1 do not require notion of true negative, which is undefined for detection problems. We used $DCC$ of 4 as a threshold for true positives when calculating these metrics.
Results
=======
In this study we present two sets of results. In the first part, we describe cross-validation (CV) experiments, in which we tested Kalasanty’s stability and its general properties, using nearly 16k structures from the sc-PDB database. In the second part, we show results for the model trained on the whole training set and evaluated on the test set. We also compare it to another DL-based approach – DeepSite.
For CV experiments, the training set was divided into 10 parts. Then, 10 models were trained with one fold left out for validation. This way we were able to evaluate our approach on the whole dataset, without making predictions for structures that were used to train a particular model.
Also, the validation sets were used to monitor the training and to select optimal parameters defining the network architecture and optimization procedure. For each model we observed a plateau on the learning curve for the validation set. Results were stable across folds and we observed similar distributions of $DCC$ values for each fold (see Figure \[fig:cv\_scores\]). The variation for $DCC$ is so small, that 95% confidence interval for mean estimation almost melts into the mean curve (Figure \[fig:cv\_scores\]A). The model did not return any pockets for 4.7% of the structures (737 pockets, ranging from 31 in fold 6 to 113 in fold 2). F1 score was equal to 0.69, with recall of 0.66 and precision of 0.73.
![ Models’ performances on the validation sets. A) Success rate plot for different $DCC$ thresholds, averaged over 10 folds. Blue area around the curve depicts 95% confidence interval. 4 threshold is marked with gray dotted line. B) $DVO$ distribution for correctly located pockets ($DCC < 4$) in all folds combined. []{data-label="fig:cv_scores"}](figures/cv_scores.pdf){width="50.00000%"}
We have also looked for global trends in prediction accuracy and factors, that should be irrelevant for the model in order for it to generalize well. We had an unbalanced dataset and the model might have been biased in favor of the most prevalent types of proteins. Although the performance differs between different groups of proteins (see the “Discussion”), we did not observe any systemic differences between results for the most frequent types of proteins and average results. This also suggests that the model did not overfit to the training set, and that it generalizes well to new, unseen structures.
Next, we used the best set of parameters and trained the final model on the whole sc-PDB dataset (15860 structures). Finally, we evaluated the performance of this model and DeepSite’s on the test set. This was a more challenging dataset, containing proteins from a different source.
As expected, the performance was slightly worse than the one observed in CV experiments yet still promising (see Figure \[fig:test\_scores\], panels A and C). For only 3 structures (1.3%) no pockets were detected and for 120 of the binding sites (44.6%) center of the predicted pocket was at most 4 from the center of the real binding site. However, 5% of correctly located pockets (compared to 1.5% in the validation set) had incorrectly predicted shape, resulting in $DVO$ below 0.25. The F1 score was equal to 0.45 (precision=0.64, recall=0.35).
{width="\textwidth"}
DeepSite performance on the test set was also worse than CV results reported in [@deepsite] (see Figure \[fig:test\_scores\], panels B and D). When threshold of 0.4 (recommended by the authors) was used, pockets were detected in all structures, but only 64 (23.8%) of them had $DCC$ below 4. What is more, only 2 pockets in the entire test set had $DVO$ above 0.5. It is worth noting that similarly to DeepSite’s authors we did not observe significant changes in the results when different thresholds between 0.3 and 0.6 were used. The overall F1 score was equal to 0.26 with precision of 0.36 and recall of 0.20.
Discussion
==========
In order to better understand the difference between Kalasanty and DeepSite, we analyzed if the two models make similar mistakes (see Figure \[fig:scores\_comp\]). Although the general trends are similar and same proteins were problematic for the two models, Kalasanty correctly detected almost twice as many pockets as DeepSite (44.6% vs 23.8%, respectively). Also, for 84.6% (115 out of 136) of pockets detected by at least one of the models, Kalasanty had lower $DCC$ than DeepSite.
{width="\textwidth"}
This difference in performance is probably caused by the fact, that DeepSite tends to return more voluminous predictions (Figure \[fig:viz\]). Although there is no single definition of a binding site (should it be a group of amino-acids, or a void between them?), the two models were trained on the same dataset so this comparison is justified. After converting densities into binary predictions, pockets returned by DeepSite are on average twice as big as those predicted by our model. It also explains the discrepancy between $DCC$ and $DVO$ results for DeepSite – even correctly located pockets are usually too big and their shape is not modeled accurately.
{width="\textwidth"}
Interestingly, for two proteins – L-histidinol dehydrogenase (UniProt ID: P06988, PDB IDs: 1k75 and 1kae) and 7,8-dihydroneopterin aldolase (UniProt ID: P56740, PDB IDs: 1dhn and 2nm2) – DeepSite correctly located all pockets in both holo and apo structures, while Kalasanty completely missed them (see Figure \[fig:scores\_comp\]A).
We investigated what is the root of the observed differences in those two specific cases. We were interested whether poor results for the two proteins are a part of a bigger trend and can be explained by our model’s inability to predict pockets for a particular group of proteins. Unfortunately such analysis is hampered by the test set size which is too small to perform statistical analysis. We therefore analyzed relationships between protein properties (superfamily, fold, source organism, length and size of the binding site) and prediction quality using CV results. From these experiments we indeed can observe higher $DCC$ values for both superfamilies when compared to the rest of the dataset (Supplementary figure S1, panels A and B). We also observed significant differences for both source organisms (Supplementary figure S1, panels C and D). However, we cannot determine if those are causal relationships, or is there some other underlying factor, correlated with these two properties.
To summarize, we hypothesize that poor results for the two proteins might be related to some systematic errors in predictions made for such types of proteins.
Conclusions
===========
In this work we presented Kalasanty – a neural network model for detecting binding cavities on protein surfaces. Kalasanty was trained and validated with the sc-PDB database and additionally evaluated on an independent test set. We compared Kalasanty with DeepSite which was proven better than Fpocket and Concavity – one of the best conventional methods for binding site prediction. Results show that our model achieves high accuracy and is able to locate pockets more precisely than DeepSite (44.6% and 23.8% correctly located pockets from the test set, respectively). What is also important, Kalasanty is stable and cross-validation results are comparable to those obtained for new data (not used for training nor validation).
However, it should be noted that the sc-PDB dataset contains only deep cavities, which tend to have better properties (druggability). Model trained on such a dataset is not able to detect binding sites that are located on flat surfaces. To obtain such a model, different datasets should be acquired which is out of the scope of this study. \] Kalasanty is able to find multiple binding sites for a single protein. However, if they are closely located the current post-processing procedure might merge them into one pocket. A possible extension of this approach would be to look for local probability maxima around which candidate binding sites would be constructed.
Kalasanty was implemented in Python and the architecture was defined using the Keras library. Source code, together with trained model and helper scripts are freely available at <http://gitlab.com/cheminfIBB/kalasanty>. The repository can be also used to launch online demo, allowing to test Kalasanty through the web browser on a molecule of interest, without the necessity of installation nor registration. The network was supplemented with additional methods that allow for making predictions directly for molecules, and handle all necessary preprocessing under the hood. Predictions can then be saved as `.cmap` or `.cube` file and visualized in molecular modelling software.
Although Kalasanty is a deep neural network, using it does not require GPU. GPU is crucial for training, but not for inference. It takes 5 seconds to load the model and a second to make a prediction on a Intel Core i7 CPU. This makes Kalasanty accessible for all researchers.
Kalasanty is based on U-Net – a state-of-the-art neural network architecture for semantic segmentation, originally developed for 2D medical images. We adapted the U-Net to process 3D protein structures and provided the model with input relevant for the task of identifying binding cavities.
Deep learning methods gained popularity in the recent years because of their flexibility and potential for capturing complex relationships hidden in the data. The field of deep learning is ripe with noteworthy ideas that have already been tested in disciplines such as computer vision and sequence modelling. Therefore, this work can also be seen as an example of adapting deep learning methods developed in other fields to structural bioinformatics.
Supplementary Figures {#supplementary-figures .unnumbered}
=====================
![Relationship between prediction accuracy and protein properties: superfamily (A), fold (B), and source organism (C and D). “n” denotes number of binding sites in each group. p-values were calculated with Mann-Whitney U-test.[]{data-label="fig:metadata"}](figures/meta){width="50.00000%"}
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date: 'Received: date / Accepted: date'
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[**[Configuration and Self-averaging in disordered systems]{}**]{}\
[ $^2$Department of Physics, Presidency University, 86/1 College Street, Kolkata 700073, W.B., India.]{}\
[ $^3$Department of Condensed Matter and Materials Science, S.N. Bose National Centre for Basic Sciences, JD-III, Salt Lake, Kolkata 7000098, W.B., India.]{}\
[ $^4$Department of Physics, Lady Brabourne College, 1/2 Suhrawardy Street, Kolkata 700017, W.B., India.]{}
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0.1cm $^\ast$Corresponding Author, Email : [email protected]\
[**Abstract**]{}
The main aim of this work is to present two different methodologies for configuration averaging in disordered systems. The Recursion method is suitable for the calculation of spatial or self-averaging, while the Augmented space formalism averages over different possible configurations of the system. We have applied these techniques to a simple example and compared their results. Based on these, we have reexamined the concept of spatial ergodicity in disordered systems. The specific aspect, we have focused on, is the question “Why does an experimentalist often obtain the averaged result on a single sample ?" We have found that in our example of disordered graphene, the two lead to the same result within the error limits of the two methods.
[**Keywords:**]{} Recursion Method; Augmented space formalism, Spatial Ergodicity\
[**PACS Nos.:**]{}71.15.-m; 71.23.An; 73.22.Pr; 74.20.Pq
**Introduction : Averaging in disordered systems**
==================================================
The study of averaging over all possible different ‘configurations’ of a quenched disordered system has been a focused problem in the theory of measurements. Configuration averaging is ubiquitous both in quantum mechanics and statistical physics. For annealed disorder, where the thermal disorder driven fluctuations control the behavior of the system, the idea of spatial ergodicity is important. At finite temperatures different possible states of a canonical ensemble, for example, are occupied with Boltzmann probabilities, and observable physical properties are averaged over the ensemble. Similarly, when we wish to measure a given physical observable in a quantum system, the result of the measurement is spread over different possible states with probabilities given by squared amplitudes of their wave function projection onto those states. Our discussions will be essentially at 0K and we shall focus on frozen or quenched disorder as in a glass or disordered alloy.
During the last four decades considerable effort has gone into devising methods for carrying out averages of physical observables over different configurations realized by disordered systems.
Why do we wish to carry out such averages and is such a procedure meaningful [@kn:erg1]?
Let us examine a specific example. An experimentalist is carrying out energy resolved photo-emission studies on a disordered binary alloy A$_x$B$_y$. Varying the frequency of the incident photon and keeping the energy window of the excited outgoing electrons reasonably narrow, one can map out the density of states of the valence electrons for the alloy. Ideally, if the experiment is carried out on ten different samples of the same alloy, slightly different results should be obtained. Every different sample of the disordered alloy has different distributions of the A and B constituents and hence should give slightly different random results. Yet, in practice, the variation, the experimenter sees in the different samples, is well within experimental error bars. What is obtained is an average result, averaged over different realizable configurations of atomic arrangements in the alloy. The interesting fact is that averaged result is obtained in a single large sample. The same is true for other measured bulk properties like the specific heat, conductivity and different response functions.
Note that all these measured properties are global. Should there be a difference if we measure local properties with local probes ? Let us take another example of a magnetically disordered alloy AuFe (with $<$ 10$\%$ of Fe). If we measure the magnetization of a sample, it remains zero upto liquid He temperatures. Yet, if we carry out a Mössbauer study on the same alloy, there is a clear indication of a frozen local exchange field at low temperatures, indicating the existence of non-zero local magnetization. Configuration averaging is meaningless if we wish to look at local properties. Even here, a degree of averaging over the far environment is relevant. Although the radioactive Fe atom giving rise to the Mössbauer spectrum sits in different environments in different samples, yet experiments yield an average exchange field distribution.
We shall focus on the question, “Why does an experimentalist working on a disordered material often obtain the averaged result from a single sample ?"
Understanding self-averaging
============================
Why do we observe configuration averaged results in a particular macroscopic sample ? The answer lies in the idea of spatial ergodicity or self-averaging. We shall use these terms as synonyms. Let us look at the example of a random binary alloy as illustrated in Figs.\[fig1\] (a) - \[fig1\] (d). They show a number of different samples each with N atoms and its own A-B distribution. There are in all 2$^N$ distinct configurations. Let us label each configuration by ${\cal C}_n$, formally the configuration average of a physical quantity ${\cal A}$ is
$$\ll {\cal A} \gg_{conf} = \frac{1}{2^N} \sum_n {\cal A}({\cal C}_n) \gamma({\cal C}_n)$$
where $\gamma({\cal C}_n)$ is the number of times a given configuration ${\cal C}_n$ occurs in the collection (space) of configurations shown in Fig.\[fig1\](a)-(d). Now, if $N\rightarrow\infty$ then $\gamma({\cal C}_n)/2^N$ is the probability associated with the configuration and $$\ll {\cal A}\gg_{conf} = \sum_n {\cal A}({\cal C}_n) {\cal P}r({\cal C}_n)$$
Let us now take a large single sample with 2$^N$ atoms and partition it into subsystems each with $N$ atoms. The spatial average taken over this one single sample is :
$$\ll {\cal A}\gg_{spat} = (1/2^N)\sum^{2^N}_{n=1} {\cal A}(\vec{r}_n)$$
where $\vec{r}$ denotes the positions of the atoms. We shall now partition the sample as shown in Fig.\[fig1\] (e) and group these sites into microsystems of size $N$ atoms, so there are $2^N/N$ such partitions which we shall call ${\cal C}'_n$, then
$$\ll {\cal A}\gg_{spat} = (N/2^N) \sum_{m=1}^{2^N/N} \frac{1}{N} \sum_{m\in {\cal C}'_m} {\cal A}(\vec{r}_m)$$
In the collection ${\cal C}'_m$ all distributions are not distinct. Assume that there are $\gamma({\cal C}'_m)$ identically distributed microsystems, then the above equation can be written as :
$$\ll {\cal A}\gg_{spat} = (N/2^N) \sum_{m=1} {\cal A}({\cal C}'_m) \gamma({\cal C}'_m)$$
If we now we let $N \rightarrow\infty$ then we get : $$\ll {\cal A}\gg_{spat} = \sum_{m=1} {\cal A}({\cal C}'_m) {\cal P}r({\cal C}'_m)$$
Is it then true that it follows from Eqs. (1) and (5) that the configuration average taken over many different samples is the same as the spatial average over a single sample ? The answer is in general in the negative. We should note that :
- The statement is untrue for any finite system.
- The statement remains true if, as $N\rightarrow\infty$ in such a way that each partition of the single sample also becomes infinitely large, but for every configuration (shown on Fig.\[fig1\] (a)-\[fig1\] (d)) there is a one-to-one correspondence with a partition shown in Fig.\[fig1\] (e) and vice versa. This is a very strong statement and is known as the “Spatial Ergodic Principle” for quenched disordered systems.
- If these averages diverge as $N\rightarrow\infty$, but the variance diverges faster, then there is no point in talking about averages, since the fluctuations about the average dominate. Example of such a system is the intensity of starlight after it passes through a disordered dielectric medium. The fluctuations in intensity dominate over the average, causing star to twinkle even outside our atmosphere. [@kn:cs]. The same holds for conductance fluctuations in disordered media.[@mj].
Although numerous very detailed and rigorous works exist on temporal ergodicity [@kn:erg7; @kn:erg5], a similar detailed exposition on spatial ergodicity is scarce. The concept of spatial ergodicity is a conjecture; its mathematical proof involves many stringent pre-conditions. Many systems do not satisfy them and therefore are not spatially ergodic. In order to develop an algorithm for configuration averaging, we have to be careful to ensure that the assumption of spatial ergodicity remains valid. The aim of this paper is to introduce two different numerical techniques : one of which explicitly calculates the spatial average and the other the configuration average and then compare the two results for a simple example of disordered graphene. We have specifically chosen a two-dimensional model, as prior experience tells us that non-ergodicity appears more often in lower dimensional systems [@mj].
**Spatial averaging and the Recursion Method**
===============================================
Our systems of interest are disordered systems, and consequently the Bloch Theorem fails to hold. Strictly speaking reciprocal space approaches are invalid. There has been supercell approaches, where the local disorder has been incorporated in a large supercell and periodic boundary conditions were imposed on the surfaces of the supercell. Unfortunately, the fact that whenever there are imposed periodicities, the spectral function always leads to sharp bands and the disorder induced lifetimes which gives a spread to the bands cannot be obtained. In all such approaches one has to introduce a small imaginary part to the energy to smoothen these sharp features and this is introduced arbitrarily ‘by hand’. We have briefly discussed the effects of the superlattice approach and its attendant artificial periodicity imposed in Appendix 1.
We turn instead to real space techniques. We first propose a methodology to deal with spatial averaging in disordered systems. It has been argued that many of the properties of solids are crucially dependent on the local chemistry of the atoms constituting the solid [@ter2]. For such properties a Black Body Theorem essentially states that the very far environment of an atom in a solid has very little influence on its local chemistry. This local environment approach to the electronic structure of solids requires an alternative to band theory for solving the Schrödinger equation. Band theory is invalid in disordered systems. Physics is better understood by means of a solution that explicitly accounts for the role of local environment. The recursion method introduced by Haydock [*et al*]{}[@md27] is a lucid approach in this direction. It expresses the Hamiltonian in a form that couples an atom to its first nearest neighbor, then through them to its distant neighbors and so on. Mathematically, a new orthonormal basis set $\vert n \rangle$ is constructed by a three term recurrence formula to make the Hamiltonian tridiagonal.The starting state $\vert 0 \rangle$ of recursion is : 0= \_i \_iR\_i
where, $\eta_i$ take the values $\pm 1$ randomly, so that :
$$\begin{aligned}
\langle 0\vert G(z)\vert 0\rangle &
= \frac{\displaystyle 1}{\displaystyle N} \left\{ \sum_i \eta_i^2 \langle R_i\vert G(z)\vert R_i\rangle +
\sum_i\sum_j \eta_i\eta_j\langle R_i\vert G(z)\vert R_j\rangle \right\}\end{aligned}$$
Since $\eta_i^2=1$ and the second term $O(\sqrt{N})$, the spatial average of the total density of states (TDOS) is : - m Tr G(E+i0) = n(E)\_[sp]{}
The whole set of orthonormal states are generated by the following three term recurrence relation: $$\beta_{n+1}\vert n+1 \rangle = H\vert n \rangle - \alpha_{n} \vert n \rangle - \beta_{n} \vert n-1 \rangle
\label{rec1}$$
Since we cannot apply Bloch theorem for systems where periodic symmetry is lost, we take recourse to an alternative approach of obtaining physical properties from the averaged resolvent. Haydock [@md27] have showed that using Eq. (\[rec1\]) we can expand the resolvent as a continued fraction :
G\_[RR]{}(z)\_[sp]{} = \[cf\]
In practice, the continued fraction is evaluated to a finite number of steps.
**The far environment : terminators**
-------------------------------------
Right at the start we chose the real space algorithm over mean-field and supercell approaches because we do not wish to introduce artificial periodicity and miss out on the effects of long-ranged disorder. The problem with any numerical calculation is that we can deal with only a finite number of operations. In the recursion algorithm, we can go up to a finite number of steps and if we stop the recursion, or impose periodicity this would lead to exactly what we wish to avoid. The analysis of the asymptotic part of the continued fraction is therefore of prime interest to us. This is the “termination” procedure discussed by Haydock and Nex[@hn], Luchini and Nex[@ter3], Beer and Pettifor[@bp] and in considerable detail by Viswanath and Muller [@ter5]. This terminator $T(z)$ which accurately describes the far environment, must maintain the herglotz analytical properties. We have to incorporate not only the singularities at the band edges, but also those lying on the compact spectrum of $H$. Viswanath and Muller [@ter5] have proposed a terminator :
T(z) = z-E\_0\^p{(z-E\_1)(E\_2-z)}\^q
The spectral bounds are at $E_1$ and $E_2$ with square-root singularities, $E_m^2 = E_1E_2$ and there is a cusp singularity at $E_0$ if $p=1,q=1$ or infra-red divergence if $p=-1/2,q=0$. $E_0$ sits on the compact spectrum of $H$. Magnus [@mag] has cited a closed form of the convergent continued fraction coefficients of the terminator :
$$\begin{aligned}
\beta^2_{2n} & = & E_m^2\ \frac{4n(n+q)}{(4n+2q+p-1)(4n+2q+p+1)} \nonumber\\
\beta^2_{2n+1} & = & E_m^2\ \frac{(2n+2p+1)(2n+2q+p+1)}{(4n+2q+p+1)(4n+2q+p+3)}\nonumber\\\end{aligned}$$
The parameters of the terminator are estimated from the asymptotic part of the continued fraction coefficients calculated from our recursion. Viswanath-Müller termination was used and seamlessly enmeshed with the calculated coefficients as shown in Fig. \[fig2\] (a) and \[fig2\] (b). In this way both the near and the far environments are accurately taken into account.
**Ensemble averaging and the Augmented Space technique**
========================================================
The augmented space or configuration space method had been proposed as early as 1973 by Mookerjee [@mook]. Some of the most successful beyond single-site, mean-field averaging techniques in random systems are based on this method : e.g. the travelling Cluster Approximation (CA) [@tca] and the itinerant Coherent Potential Approximation (CPA) [@icpa]. In this approach we work not only with individual configurations of the sample but also with collection of all possible configurations. Let us examine the basic concepts in this methodology.
Suppose we measure a property of the system which is not disordered. In that case we may carry out repeated measurements of the property N and get a set of results which are all the same within error bars : $n \pm \delta$ and we get this with probability one. When the system is disordered the same set of measurements yield a set of values $n_1,n_2 \ldots n_m \pm \delta $ with probabilities $p_1,p_2\ldots p_m$. We cannot associate a scalar quantity with this property. Instead, we associate an operator $\tilde{N}$ whose eigenvalues are the measured quantities and whose spectral density is the probability density of $\tilde{N}$.
Take for example, a random variable $n_R$, which takes the value 1 if the site $R$ is occupied by a A type atom with probability x and 0 if the site $R$ is occupied by a B type atom with probability y. The next problem is the inverse problem of recursion. There the tri-diagonal representation of the Hamiltonian is given and the resolvent is obtained as a continued fraction. Here, the resolvent is given and by inspection, related to the probability density :
$$\begin{aligned}
P(n_R) & = & x \delta(n_R-1) + y \delta(n_R) = - (1/\pi) Im \left\{\frac{x}{n_R-1}+\frac{y}{n_R}\right\}\\
& = & - (1/\pi) Im \dfrac{1}{n_R - x - \dfrac{xy}{n_R - y}} \end{aligned}$$
So $$\tilde{N} = \left( \begin{matrix}
x & \sqrt{xy} \\
\sqrt{xy}& y
\end{matrix} \right)$$
The eigenvalues are 0 and 1 as expected. The above representation is in a basis :\
${\framebox{\color{white}{\rule{0.06cm}{0.06cm}}}}_R\ =\ \sqrt{x}|0\rangle + \sqrt{y} |1\rangle$ ; ${\color{black}{\rule{0.3cm}{0.3cm}}_R}\ =\ \sqrt{y}|0\rangle - \sqrt{x} |1\rangle$\
Each member of the basis, is a ‘configuration’ of $\tilde{N}$ at $R$. For later convenience we choose the first to be the ‘reference configuration’ and the second to be a disorder induced ‘configuration fluctuation’ whose creation from the reference configuration is described by $b^\dagger_{R}$. Since the disorder is binary and each site can have only one fluctuation, these fluctuations behave like fermions :
$$b^\dagger_{R}\ {\color{black}{\rule{0.3cm}{0.3cm}}}_R = 0\quad ;\quad b_{R}\ \framebox{{\color{white}{\rule{0.1cm}{0.1cm}}}}_R = 0$$
Each site $R$ has an operator and a configuration space associated with it. The system of N sites then has a configuration space $\Phi = \prod^\otimes \phi_{R}$ of rank 2$^N$. The number of fluctuations in a configuration is called its ‘cardinality’ and the sequence of sites where they occur is called its ‘cardinality sequence’. Some typical configurations are shown below with their cardinalities and cardinality sequences :
$$\begin{aligned}
\quad\left\{ \framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_1}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_2}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_3}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_4}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_5}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_6}\quad \ldots\right\} & \mbox{card} = 0,& \mbox{card. sequence} = \{\emptyset\}\\
\left\{ {\color{black}{\rule{0.5cm}{0.5cm}}}_{R_1}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_2}
\quad{\color{black}{\rule{0.5cm}{0.5cm}}}_{R_3}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_4}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_5}
\quad{\color{black}{\rule{0.5cm}{0.5cm}}}_{R_6} \ldots\right\}
& \mbox{card} = 3,& \mbox{card. sequence} = \{R_1,R_3,R_6\} \\
\left\{ \framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_1}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_2}
\quad{\color{black}{\rule{0.5cm}{0.5cm}}}_{R_3}
\quad{\color{black}{\rule{0.5cm}{0.5cm}}}_{R_4}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_5}
\quad\framebox{{\color{white}{\rule{0.15cm}{0.15cm}}}}_{R_6}\ldots\right\} &\mbox{card} = 2,& \mbox{card. sequence} = \{R_3,R_4\}\end{aligned}$$
These operators are in the ‘configuration’ space $\Phi$ of rank 2$^N$. They may be written as :
$$\tilde{N}_{R} = I + (y-x) b^\dagger_{R} b_{R} + \sqrt{xy} ( b^\dagger_{R} + b_{R})$$
The next thrust in this approach came with the Augmented Space Theorem stated by Mookerjee [@mook]. We shall try to understand this by an example :
P(n\_R) (-1/) m ( (n\_R +0)I N\_[R]{})\^[-1]{} (-1/) m g(z)
where $g(z) = \langle\emptyset\vert (z\tilde{I}-\tilde{N}_R)^{-1} \vert\emptyset\rangle$ is the resolvent of $\tilde{N}_R$\
and
$$|\emptyset\rangle \eq \left\vert \framebox{\color{white}{\rule{0.15cm}{0.15cm}}}_{R_1}\quad\framebox{\color{white}{\rule{0.15cm}{0.15cm}}}_{R_2}\quad\framebox{\color{white}{\rule{0.15cm}{0.15cm}}}_{R_3}\quad\framebox{\color{white}{\rule{0.15cm}{0.15cm}}}_{R_4}\quad\framebox{\color{white}{\rule{0.15cm}{0.15cm}}}_{R_5}\quad\framebox{\color{white}{\rule{0.15cm}{0.15cm}}}_{R_6}\quad \ldots\right.\rangle$$
Let $f(n_1,n_2 \ldots n_k \ldots)$ be a well behaved function of the set of independent random variables $\{n_1,n_2,\ldots n_k \ldots\}$, then
$$\begin{aligned}
\ll f(n_1,n_2 \ldots ) \gg & = & \int dn_1 \int dn_2 \ldots
f(n_1,n_2 \ldots ) P(n_1)P(n_2)\ldots \\
& = & \oint dz_1 \oint dz_2 \ldots f(z_1,z_2 \ldots )\ g(z_1) g(z_2) \ldots\end{aligned}$$
Using the completeness of the eigenbasis of $\{\tilde{N}_R\}$ $$\begin{aligned}
\langle\emptyset| \oint dz_1 \oint dz_2\ldots f(z_1,z_2,\ldots)\int d\rho(\mu_1) \int d\rho(\mu_2) \ldots |\mu_1,\mu_2 \ldots\rangle\langle \mu_1,\mu_2 \ldots | \ldots\\
\prod (zI-N_R)^{-1}\int d\rho(\mu'_1)\int d\rho(\mu'_2)\ldots |\mu'_1,\mu'_2 \ldots\rangle\langle \mu'_1,\mu'_2 \ldots |\emptyset\rangle \\\end{aligned}$$
Now using the orthogonality of the eigenstates of $\{\tilde{N}_R\}$ :
$$= \langle\emptyset| \int d\rho(\mu_1) \int d\rho(\mu_2) \ldots
|\mu_1,\mu_2 \ldots\rangle\left[ \oint dz_1 \oint dz_2 f(z_1,z_2,\ldots) \prod (z-\mu_k)^{-1} \right] \langle \mu_1,\mu_2 \ldots |\emptyset\rangle$$ $$= \langle\emptyset| \left[ \int d\rho(\mu_1) \int d\rho(\mu_2) \ldots
|\mu_1,\mu_2 \ldots\rangle f(\mu_1,\mu_2,\ldots) \langle\mu_1,\mu_2 \ldots \right] |\emptyset\rangle$$ Finally, $$\ll f(n_1,n_2 \ldots ) \gg = \langle\emptyset| \widetilde{f}(\tilde{N}_1,\tilde{N}_2 \ldots) |\emptyset\rangle$$
The operator on the right-hand side is the same operator function of the operators $N_1,N_2,\ldots$ as the function f was of $n_1,n_2 ,\ldots n_3$. This is the central result of the augmented space technique : configuration averages are specific matrix elements in the full augmented space, which carries not only the information about the underlying lattice, but also how that lattice is randomly decorated with the two constituents subject to their concentrations.
Let us return to the disordered binary alloy described by the Hamiltonian :
$$\begin{aligned}
H&\ =\ &\sum_{R}\ \left[ \epsilon_B + (\epsilon_B-\epsilon_A) n_{R}\right]\ a^\dagger_{R} a_{R} \ +\ldots\nonumber\\
&\ldots + & \sum_{R}\sum_{\chi} \ \left[t^{BB}(\chi) + t^{(2)}(\chi)\ n_{R}\ n_{R+\chi} + t^{(1)}(\chi)\ (n_{R}+n_{R+\chi}) \right]\ a^\dagger_{R+\chi} a_{R} \end{aligned}$$
Here, $t^{(1)}(\chi) = t^{AB}(\chi)-t^{BB}(\chi)$ and $t^{(2)}(\chi) = t^{AA}(\chi)+t^{BB}(\chi)-2 t^{AB}(\chi)$, where $\chi = R-R'$.
This Hamiltonian is an operator in a Hilbert space ${\cal H}$ spanned by the denumerable basis $\{|R\rangle\}$. We now turn to the augmented space formalism and associate with each random variable $n_{R}$ an operator $\tilde{N}_{R}$ such that its eigenvalues are the values attained by $n_{R}$ and the density of states of $N_{R}$ is the probability density of $n_{R}$ for attaining its values. For binary randomness (e.g. $n_{R}$ takes the values 0 and 1 with probabilities $x$ and $y$) the space $\phi_{R}$ in which $N_{R}$ acts is of rank 2.
The ‘augmented Hamiltonian’ can then be written as :
$$\begin{aligned}
\widetilde H&\ =\ &\sum_{R}\ \left[ \ll\epsilon\gg + \Delta\left\{(y-x) b^\dagger_{R} b_{R} + \sqrt{xy}(b^\dagger_{R}+b_{R})\right\} \right]\ a^\dagger_{R} a_{R} \ +\ldots\nonumber\\
& & \ldots + \sum_{R}\sum_\chi \ \left[\ll t(\chi)\gg + \sqrt{xy} t^{(2)}(\chi) (b^\dagger_{R} + b_{{R}+\chi})\ + t^{(1)}(\chi)(b^\dagger_{R}+b_{{R}+\chi}) \right]\ a^\dagger_{{R}+\chi} a_{R} \end{aligned}$$
Here, $a_R, a^\dagger_R$ are electron destruction and creation operators, $\Delta = (\epsilon_B-\epsilon_A)$.
The Augmented Space Theorem[@ast] then leads to :
G([R]{},[R]{}’,z)= {}(z -)\^[-1]{}’ {}
The result is significant, since we have reduced the calculation of averages to one of obtaining a particular matrix element of an operator in the configuration space of the variable. Physically, of course, the augmented Hamiltonian is the collection of all Hamiltonians. This augmented Hamiltonian is an operator in the augmented space $\Psi\eq {\cal H}\otimes \Phi$ where ${\cal H}$ is the space spanned by the tight binding basis and $\Phi$ the full space of all configurations. The result is exact. Approximations can now be introduced in the actual calculation of this matrix element in a controlled manner.
The same recursion method used for spatial averaging ( Haydock [@md27]) is ideally suited for obtaining matrix elements in augmented space. Since configuration averaging is an intrinsically difficult problem, we must pay the price for the above simplification. This comes in the shape of the enormous rank of the augmented space. For some time it was thought that recursion on the full augmented space was not a feasible proposition. However, if randomness is homogeneous in the sense that $p(n_R)$ is independent of the label $R$, then the full augmented space has a large number of local point group and lattice translational symmetries. These have been utilized to reduce vastly the rank of the effective space on which the recursion can be carried out. Recursion on augmented space can be done now with ease, even on desktop computers.
**A simple application to disordered graphene**
===============================================
It would be interesting to apply our two different techniques on a simple, yet interesting system. We have chosen disordered graphene. The choice of this example was both because of interest in graphene and because non-ergodicity seems to be more probable in lower dimensional systems [@mj]. Disorder in graphene can significantly modify the electronic properties because the interplay between electron-electron interactions and the disorder controls the low energy regime of electrons in graphene [@PRB06; @Alam2012; @Chowdhury2014; @Chowdhury2015]. The presence of disorder in graphene can emerge through various synthesis processes when there occurs an interaction with the substrates due to its exposed surface. Various kinds of nature of defects such as random impurities and vacancies present in graphene can change the electronic structure. Chemical substitution and irradiation are some of the effective methods to produce local disorder in graphene. In fact, recently, Pereira et al. [@PRB08] have shown the dramatic changes (such as localized zero modes, gap and pseudo-gap behavior and strong resonances) in the low energy spectrum of graphene within the tight binding approximation due to this local disorder.
With this motivation, let us introduce a simple, non-trivial model of a disordered graphene sheet. This is described by a tight-binding Hamiltonian with a single band $p_z$ per site. Anderson [@pwa] studied this model in his now celebrated 1958 paper and it goes under his name. $$H = \sum_{R} \varepsilon_{R} \proj_{R} + \sum_{R}\sum_{R^{\prime}} t_{RR^{\prime}} \trans_{RR^{\prime}}$$
Here $\proj_{R}$ = $\vert R\rangle\langle R\vert$ and $\trans_{\ij}$ = $\vert R\rangle\langle R^{\prime}\vert$ are projection and transfer operators on the space spanned by the tight-binding basis $\{\vert R\rangle\}$. The simplest model of a void is just the removal of an atom from a site $R$, putting $t_{R'R} = 0$ and $\varepsilon_R \rightarrow\ -\infty$ . We have used the tight-binding linear muffin-tin orbitals (TB-LMTO) technique followed by a N-th order muffin-tin orbitals (NMTO) downfolding to obtain the parameters of the single $p_z$ orbital per site model. The pristine graphene itself is a open structure and required inclusion of three empty spheres for proper space filling. When we randomly removed carbon atoms to produce voids, the space had to be replaced by empty spheres.
(eV) $\varepsilon$ t$(\chi_1)$ t$(\chi_2)$ t$(\chi_3)$
------ --------------- ------------- ------------- -------------
-0.291 -2.544 0.1668 -0.1586
: Tight-binding parameters generated by NMTO. $\chi_n$ refer to the n-th nearest neighbor vector on the lattice.
\[tab1\]
The tight-binding parameters are shown in Table \[tab1\]. The overlap $t$ decays with distance. In this work we have truncated at the nearest neighbor distances. The results of TB-LMTO band structure for graphene are shown in Figs.\[fig3\](a) and \[fig3\](b). For justification of $p_{z}$ only downfolding, we have compared the various orbital projected partial density of states (PDOS) in Fig.\[fig3\](c).
**Results and discussion**
==========================
We have applied the two above techniques : the real space recursion and augmented space to obtain the density of states (DOS) for graphene with random voids. The former yields the spatial averages, while the latter yields configuration averages. Figs.\[fig4\](a) - \[fig4\](f) compare the spatial and configuration averages of three void compositions. Other than the energy resolution difference in the two methods (discussed shortly in the Appendix) our conclusion is that disordered graphene is self-averaging.
However there is a subtle point which we should comment on :
If we examine the band edges, we note that whenever there is periodicity at any scale at all, the band edges are sharp and quadratic. Long range disorder leads to band tailing. Signs of this appear in the configuration averaged results. This disorder induced band tailing has been known for decades and it would be interesting to examine if the tail states are localized or not.
Again periodicity at whatever scale leads to structure in the DOS. This is smoothened out by the disorder scattering induced complex ‘self-energy’. Occasionally practitioners of super-cell techniques introduce an artificial imaginary part to the energy. This smoothes these structures too, but the procedure is entirely ad hoc. Augmented space leads to an energy dependent self-energy systematically.
It is important to note that the inverse of the imaginary part of the self-energy, called the “life time” is accessible to neutron scattering experiments. The spatial averages cannot access this “life-time" effect and it is essential to carry out configuration dependent averaging to calculate it.
Further insights into different averaging procedures is desirable. Besides, the TDOS results obtained here can be verified through scanning tunneling microscopy and the effect on the TDOS particularly at the Fermi energy / Dirac point should reflect on the electronic transport process.
Conclusions
===========
We have employed two techniques, the real space recursion and augmented space to obtain the density of states for graphene with random voids. The numerical results obtained here clearly indicate that spatial ergodicity holds good for graphene with random voids.
Appendix {#appendix .unnumbered}
========
It will be interesting to understand why different methods have different energy resolutions. Periodicity and the Bloch Theorem leads to the labeling of quantum states by a real vector $\vec{k}$ and a supplementary band label ’n’.
$$\langle \vec{k},n\vert G(z) \vert\vec{k},n\rangle =
\frac{1}{z-E_n(\vec{k})}$$
So that : $$-(1/\pi) \sum_n Im G(E+i0,\vec{k},n) = \sum_n \delta(E-E_n(\vec{k}))$$
For a fixed $\vec{k}$ this is a set of delta functions. If we abandon periodicity and adopt the augmented space formalism :
$$\begin{aligned}
-(1/\pi) \sum_n Im \ll G(E+i0,\vec{k},n)\gg & = & -(1/\pi) \sum_n Im
\frac{1}{E-E_n(\vec{k})- \Sigma_r(E,\vec{k})-i\Sigma_{im}(E,\vec{k})} \\
& = & \sum_n \frac{[1/\pi\tau(E,\vec{k},n)]}{[(E-E'_n(\vec{k})]^2 + [(1/\tau(E,\vec{k},n)]^2}\end{aligned}$$
This spectral density comes out to be much smoother and the smoothening lifetime has its origin in scattering by disorder fluctuations. Moreover the lifetime emerges out of the calculations and no external broadening or smoothening factors are required.
**Acknowledgments** {#acknowledgments .unnumbered}
===================
SC would like to thank DST, India for financial support through the Inspire Fellowship. This work was done under the HYDRA collaboration between our institutes.
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****
Figure 1 : (a)-(d) Ensemble of macrosystems in the ’configuration space’ of a binary alloy. (e) A single macrosystem constructed out of the microsystems shown in (a)-(d).
Figure 2 : (a) Asymptotic parts of the calculated graphene Green function continued fraction coefficients obtained by recursion for $5 \%$ voids and (b) the same for $1 \%$ voids.
Figure 3 : (a) The all orbital TB-LMTO band structure for graphene.(b) The downfolded $p_z$ bands based on the single band effective Hamiltonian. (c) The orbital projected partial density of states.
Figure 4 : (a),(b),(c):TDOS for Graphene using configuration averaging for 1, 2 and 10 % voids respectively. (d),(e),(f):TDOS for Graphene using spatial averaging for 1, 2 and 10 % voids respectively.
![(a)-(d) Ensemble of macrosystems in the ’configuration space’ of a binary alloy. (e) A single macrosystem constructed out of the microsystems shown in (a)-(d). \[fig1\]](fig1.eps "fig:"){width="10cm" height="10cm"}1.0cm
![(a) Asymptotic parts of the calculated graphene Green function continued fraction coefficients obtained by recursion for $5 \%$ voids and (b) the same for $1 \%$ voids. \[fig2\]](fig2.eps "fig:"){height="12cm" width="12cm"}1.2cm
![(a) The all orbital TB-LMTO band structure for graphene.(b) The downfolded p$_z$ bands based on the single band effective Hamiltonian. (c) The orbital projected partial density of states.\[fig3\]](fig3.eps "fig:"){width="12cm" height="18cm"}1.5cm
![(a),(b),(c):TDOS for Graphene using configuration averaging for 1, 2 and 10 % voids respectively. (d),(e),(f):TDOS for Graphene using spatial averaging for 1, 2 and 10 % voids respectively.\[fig4\]](fig4.eps "fig:"){width="18cm" height="17cm"}0.1cm
|
---
abstract: 'In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree.'
address: 'Department of Mathematics, UACEG, Sofia, Bulgaria'
author:
- Vladimir Samodivkin
title: Upper bounds for domination related parameters in graphs on surfaces
---
[**MSC 2012**]{}: 05C69
Introduction {#Intro}
============
All graphs considered in this paper are finite, undirected, loopless, and without multiple edges. We denote the vertex set and the edge set of a graph $G$ by $V(G)$ and $ E(G),$ respectively. For a vertex $x$ of $G$, $N(x)$ denotes the set of all neighbors of $x$ in $G$ and the degree of $x$ is $\deg(x) = |N(x)|$. The minimum and maximum degree among the vertices of $G$ is denoted by $\delta (G)$ and $\Delta (G)$, respectively. For $e=xy \in E(G)$, let $\xi (e) = deg (x) + deg (y) - 2$ and $\xi(G)= min\{\xi(e)$ : $e \in E(G)\}$. The parameter $\xi(G)$ is called the minimum edge-degree of $G$. We let $\left\langle U \right\rangle$ denote the subgraph of $G$ induced by a subset $U \subseteq V(G)$. The girth of a graph $G$, denoted as $g(G)$, is the length of a shortest cycle in $G$; if $G$ is a forest then $g(G) = \infty$.
An orientable compact 2-manifold $\mathbb{S}_h$ or orientable surface $\mathbb{S}_h$ (see [@Ringel]) of genus $h$ is obtained from the sphere by adding $h$ handles. Correspondingly, a non-orientable compact 2-manifold $\mathbb{N}_k$ or non-orientable surface $\mathbb{N}_k$ of genus $k$ is obtained from the sphere by adding $k$ crosscaps. Compact 2-manifolds are called simply surfaces throughout the paper. The Euler characteristic is defined by $\chi(\mathbb{S}_h) = 2 - 2h$, $h\geq 0$, and $\chi(\mathbb{N}_q) = 2 - q$, $q\geq 1$. A connected graph $G$ is embeddable on a surface $\mathbb{M}$ if it admits a drawing on the surface with no crossing edges. Such a drawing of $G$ on the surface $\mathbb{M}$ is called an embedding of $G$ on $\mathbb{M}$. Notice that there can be many different embeddings of the same graph $G$ on a particular surface $\mathbb{M}$. An embedding of a graph $G$ on surface $\mathbb{M}$ is said to be $2$-cell if every face of the embedding is homeomorphic to a disc. The set of faces of a particular embedding of $G$ on $\mathbb{M}$ is denoted by $F(G)$. The orientable genus of a graph $G$ is the smallest integer $g = g(G)$ such that $G$ admits an embedding on an orientable topological surface $\mathbb{M}$ of genus $g$. The non-orientable genus of $G$ is the smallest integer $\overline{g} = \overline{g}(G)$ such that $G$ can be embedded on a non-orientable topological surface $\mathbb{M}$ of genus $\overline{g}$. Note that (a) every embedding of a graph $G$ on $\mathbb{S}_{g(G)}$ is $2$-cell ([@yo]), and (b) if a graph $G$ has non-orientable genus $h$ then $G$ has $2$-cell embedding on $\mathbb{N}_h$([@pppv]). Let a graph $G$ be $2$-cell embedded on a surface $\mathbb{M}$. Set $|G| = |V (G)|$, $\|G\| = |E(G)|$, and $f(G) = |F(G)|$. The Euler’s formula states $$|G| - \|G\| + f(G) = \chi(\mathbb{M}).$$
Let $G$ be a non-trivial connected graph and $S \subseteq E(G)$. If $G - S$ is disconnected and contains no isolated vertices, then $S$ is called a restricted edge-cut of $G$. The restricted edge-connectivity of $G$, denoted by $\lambda^{\prime}(G)$, is defined as the minimum cardinality over all restricted edge-cuts of $G$. Besides the classical edge-connectivity $\lambda(G)$, the parameter $\lambda^{\prime}(G)$ provides a more accurate measure of fault-tolerance of networks than the classical edge-connectivity (see [@E]).
A subset $D$ of $V (G)$ is dominating in $G$ if every vertex of $V (G) - D$ has at least one neighbor in $D$. The domination number of $G$, denoted by $\gamma(G)$, is the size of its smallest dominating set. When $G$ is connected, we say $D$ is a connected dominating set if $\left\langle D \right\rangle$ is connected. The connected domination number of $G$ is the size of its smallest connected dominating set, and is denoted by $\gamma_c(G)$. For a connected graph $G$ and any non-empty $S \subseteq V (G)$, $S$ is called a weakly connected dominating set of $G$ if the subgraph obtained from $G$ by removing all edges each joining any two vertices in $V (G)-S$ is connected. The weakly connected domination number $\gamma_w(G)$ of $G$ is the minimum cardinality among all weakly connected dominating sets in $G$.
A set $R\subseteq V(G)$ is a restrained dominating set if every vertex not in $R$ is adjacent to a vertex in $R$ and to a vertex in $V(G) - R$. Every graph $G$ has a restrained dominating set, since $R = V(G)$ is such a set. The restrained domination number of $G$, denoted by $\gamma_r (G)$, is the minimum cardinality of a restrained dominating set of G. One measure of the stability of the restrained domination number of $G$ under edge removal is the restrained bondage number $b_r(G)$, defined in [@HP] by Hattingh and Plummer as the smallest number of edges whose removal from $G$ results in a graph with larger restrained domination number.
A set $S\subseteq V (G)$ is a total restrained dominating set, denoted $TRDS$, if every vertex is adjacent to a vertex in $S$ and every vertex in $V (G)-S$ is also adjacent to a vertex in $V (G)-S$. The total restrained domination number of $G$, denoted by $\gamma_{tr}(G)$, is the minimum cardinality of a total restrained dominating set of $G$. Note that any isolate-free graph $G$ has a TRDS, since $V(G)$ is a TRDS. The total restrained bondage number $b_{tr}(G)$ of a graph $G$ with no isolated vertex, is the cardinality of a smallest set of edges $E_1\subseteq E(G)$ for which (1) $G-E_1$ has no isolated vertex, and (2) $\gamma_{tr}(G-E_1) > \gamma_{tr}(G)$. In the case that there is no such subset $E_1$, we define $b_{tr}(G) = \infty$.
A labelling $f : V(G) \rightarrow \{0, 1, 2\}$ is a Roman dominating function (or simply an RDF), if every vertex $u$ with $f (u) = 0$ has at least one neighbour $v$ with $f (v) = 2$. Define the weight of an RDF $f$ to be $w( f ) = \Sigma_{v\in V(G)} f (v)$. The Roman domination number of G is $\gamma_R(G) = \min\{w( f ) : f \mbox{ is an RDF}\}$. The Roman bondage number $b_R(G)$ of a graph $G$ is defined to be the minimum cardinality of all sets $E \subseteq E(G)$ for which $\gamma_R(G - E) > \gamma_R(G)$.
The rest of the paper is organized as follows. Section 2 contains known results. In section 3 we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. In Section 4, we present upper bounds on the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree.
Known results {#prelimi}
=============
We make use of the following results in this paper.
\[rcon\] If $G$ is a connected graph with at least four vertices and it is not a star graph, then $\lambda^{\prime}(G) \leq \xi (G)$.
\[help1\] If $\delta (G) \geq 2$, then $b_r (G) \leq \xi (G)$.
\[help2\] Let $G$ be a connected graph of order $n$, $n\geq 5$. Assume that $G$ has a path $x,y,z$ such that $deg(x) > 1$, $deg(z) > 1$ and $G-\{x,y,z\}$ has no isolated vertex. Then $b_{tr}(G) \leq deg(x) + deg(y) + deg(z) - 4$.
\[help22a\] $b_{tr}(K_n) = n-1$ for $n \geq 4$.
\[help2a\] If $G$ is a graph, and $xyz$ a path of length $2$ in $G$, then $b_R(G) \leq deg(x) + deg(y) + deg(z) - 3$.
\[help66\] Let $G$ be a connected graph of order $n$ and size $m$.
- [(Sanchis [@Sanchist])]{} If $\gamma_t(G) = \gamma_t \geq 5$, then $m \leq \binom{n-\gamma_t +1}{2} + \left\lfloor \frac{\gamma_t}{2}\right\rfloor$.
- [(Sanchis [@Sanchis])]{} If $\gamma_w (G) = \gamma_w \geq 3$, then $m \leq \binom{n-\gamma_w +1}{2}$.
- [(Sanchis [@Sanchisc])]{} If $\gamma_c (G) = \gamma_c \geq 3$, then $m \leq \binom{n-\gamma_c +1}{2} + \gamma_c -1$.
\[dghhmThm\] [(Dunbar et al. [@dghhm])]{} If $G$ is a connected graph with $n \geq 2$ vertices then $\gamma_w (G) \leq n/2$.
The average degree $ad(G)$ of a graph $G$ is defined as $ad(G) = 2\|G\|/|G|$.
\[hra\] [(Hartnell and Rall [@hr])]{} For any connected nontrivial graph $G$, there exists a pair of vertices, say $u$ and $v$, that are either adjacent or at distance $2$ from each other, with the property that $deg(u) + deg(v) \leq 2ad(G)$.
\[SGZ\] Let $G$ be a connected graph embeddable on a surface $\mathbb{M}$ whose Euler characteristic $\chi$ is as large as possible and let $g(G) = g < \infty$. Then: $$ad(G) \leq \frac{2g}{g-2}(1 - \frac{\chi}{|G|}).$$
Let $$h_1(x) =
\begin{cases}
2x + 13 &\text{for $0 \leq x \leq 3$}\\
4x + 7 &\text {for $x \geq 3$}
\end{cases}
, \ \
h_2(x) =
\begin{cases}
8 &\text{for $x=0$}\\
4x + 5 &\text {for $x \geq 1$}
\end{cases}
,$$ $$k_1(x) =
\begin{cases}
2x + 11 &\text{for $1 \leq x \leq 2$}\\
2x + 9 &\text {for $3 \leq x \leq 5$}\\
2x + 7 &\text {for $x \geq 6$}.
\end{cases}
\ \ and \ \
k_2(x) =
\begin{cases}
8 &\text{for $x=1$}\\
2x+ 5 &\text {for $x \geq 2$}.
\end{cases}$$
\[help33\] If $G$ is a connected graph of orientable genus $g$ and minimum degree at least $3$, then $G$ contains an edge $e=xy$ such that $deg(x)+deg(y) \leq h_1(g)$. Furthermore, if $G$ does not contain $3$-cycles, then $G$ contains an edge $e=xy$ such that $deg(x)+deg(y) \leq h_2(g)$.
\[help3\] If $G$ is a connected graph of minimum degree at least $3$ on a nonorientable surface of genus $\overline{g} \geq 1$, then $G$ contains an edge $e=xy$ such that $deg(x)+deg(y) \leq k_1(\overline{g})$. Furthermore, if $G$ does not contain $3$-cycles, then $deg(x)+deg(y) \leq k_2(\overline{g})$.
A path $uvw$ is a path of type $(i, j, k)$ if $deg(u) \leq i$, $deg(v) \leq j$, and $deg(w) \leq k$.
\[help444\] Let $G$ be a planar graph with $\delta(G) \geq 3$. If no $2$ adjacent vertices have degree $3$ then $G$ has a $3$-path of one of the following types: $$\begin{array}{llllllll}
(3,4,11) & (3,7,5) & (3,10,4) & (3,15,3) & (4,4,9) & (6,4,8) & (7,4,7) & (6,5,6).
\end{array}$$
\[zzz\] Let $G$ be a connected graph with $n $ vertices and $q$ edges.
- [(Ringel [@Ringel], [Stahl]{} [@St])]{} If $G$ is not a tree then $G$ can be $2$-cell embedded on $\mathbb{N}_{q-n+1}$.
- [(Jungerman [@jun])]{} If $G$ is a $4$-edge connected, then $G$ can be $2$-cell embedded on $\mathbb{S}_{\left\lfloor \frac{q-n+1}{2}\right\rfloor}$.
Connected, weakly connected and total domination {#twcd}
================================================
\[total\] Let $G$ be a connected graph of order $n$ and total domination number $\gamma_t \geq 5$, which is $2$-cell embedded on a surface $\mathbb{M}$. Then:
- $n \geq \gamma_t + (1 + \sqrt{9+8(\left\lceil \gamma_t/2 \right\rceil - \chi(\mathbb{M}))})/2$;
- $\gamma_t \leq n - \sqrt{n + 2 - 2\chi(\mathbb{M})}$ when $\gamma_t$ is even and $\gamma_t \leq n - \sqrt{n + 3 - 2\chi(\mathbb{M})}$ when $\gamma_t$ is odd.
Note that $n > \gamma_t \geq 5$ and $\chi(\mathbb{M}) \leq 2$. Since $f(G) \geq 1$, Euler’s formula implies $n - \|G\| + 1 \leq \chi(\mathbb{M})$. By Theorem \[help66\](i) we have $\|G\| \leq (n-\gamma_t +1)(n-\gamma_t)/2 + \left\lfloor \gamma_t/2 \right\rfloor$. Hence $$2\chi(\mathbb{M}) \geq 2n+2 - (n-\gamma_t +1)(n-\gamma_t) -2\left\lfloor \gamma_t/2 \right\rfloor,$$ or equivalently $$n^2 - (2\gamma_t+1)n + \gamma_t^2 - \gamma_t +2\left\lfloor \gamma_t/2 \right\rfloor - 2 + 2\chi(\mathbb{M}) \geq 0, \ \mbox{and}$$ $$\gamma_t^2 - 2n\gamma_t + n^2 - n - \alpha + 2\chi(\mathbb{M}) \geq 0,$$ where $\alpha = 2$ when $\gamma_t$ is even and $\alpha = 3$ when $\gamma_t$ is odd. Solving these inequalities we respectively obtain the bounds stated in (i) and (ii).
Next we show that the bounds in Theorem \[total\] are tight. Let $n$, $d$ and $t$ be integers such that $n = d + 4t + 1$, $t \geq d \geq 6$ and $d \equiv 2 \textrm{ (mod 4)}$. Let us consider any graph $G$ which has the following form:
- $G$ is obtained from $K_{n-d}\cup \frac{d}{2}K_2$ by adding edges between the clique and the graph $\frac{d}{2}K_2$ in such a way that each vertex in the clique is adjacent to exactly one vertex in $\frac{d}{2}K_2$ and each component of $\frac{d}{2}K_2$ has at least one vertex adjacent to a vertex in the clique.
Clearly, $|G| = n$, $\gamma_t(G) = d$ and $\|G\| = \binom{n-d +1}{2} + \frac{d}{2}$. Hence $p = (\|G\| - |G| +1)/2 = 4t^2+t + (2-d)/4$ is an integer and $G$ can be $2$-cell embedded in $\mathbb{N}_{2p}$(by Theorem \[zzz\](i)). Now, let in addition, $\delta (G) \geq 5$. Then since $G$ is clearly $4$-edge connected, Theorem \[zzz\](ii) implies that $G$ can be embedded in $\mathbb{S}_p$. It is easy to see that, in both cases, the inequalities in Theorem \[total\] become equalities.
\[upper\] Let $G$ be a connected graph of order $n$ which is $2$-cell embedded on a surface $\mathbb{M}$. If $\gamma_w (G) = \gamma_w \geq 4$ then $$\label{nga}
n \geq \gamma_w + (1 + \sqrt{9+8\gamma_w-8\chi(\mathbb{M})})/2, \ \mbox{and}$$ $$\label{gan}
\gamma_w \leq n + (1 - \sqrt{8n + 9 - 8\chi(\mathbb{M})})/2.$$
Since $f(G) \geq 1$, Euler’s formula implies $n - \|G\| + 1 \leq \chi(\mathbb{M})$. Since $\|G\| \leq (n-\gamma_w +1)(n-\gamma_w)/2$ (by Theorem \[help66\](ii)), we have $2\chi(\mathbb{M}) \geq 2n - (n-\gamma_w +1)(n-\gamma_w) +2$, or equivalently $$n^2 - (2\gamma_w+1)n + \gamma_w^2 - \gamma_w - 2 + 2\chi(\mathbb{M}) \geq 0 \ \mbox{and}$$ $$\gamma_w^2 - (2n+1)\gamma_w + n^2 - n - 2 + 2\chi(\mathbb{M}) \geq 0.$$ Solving these inequalities we respectively obtain and , because $n \geq 2\gamma_w$ (by Theorem \[dghhmThm\]).
The bounds in Theorem \[upper\] are attainable. Let $n$, $d$ and $t$ be integers such that $t \geq d \geq 4$ and $n = d + 4t + i$, where $i = 1$ when $d$ is odd, and $i = 2$ when $d$ is even. We consider an arbitrary graph $G$ which has the following form:
- $G$ is the union of a clique of $n-d$ vertices, and an independent set of size $d$, such that each of the vertices in the $(n - d)$-clique is adjacent to exactly one of the vertices in the independent set, and such that each of these $d$ vertices has at least one vertex adjacent to it.
Obviously, $|G| = n$, $\gamma_w(G) = d$ and $\|G\| = \binom{n-d +1}{2}$. If $p = (\|G\| - |G| +1)/2$ then $p = 4t^2+t + (1-d)/2$ when $d$ is odd, and $p = 4t^2 +3t +1 - d/2$ when $d$ is even. Hence $p$ is an integer and $G$ can be $2$-cell embedded in $\mathbb{N}_{2p}$, which follows by Theorem \[zzz\](i). Now, let in addition, $\delta (G) \geq 4$. Then since $G$ is clearly $4$-edge connected, $G$ can be embedded in $\mathbb{S}_p$(by Theorem \[zzz\](ii)). It is easy to see that, in both cases, we have equalities in and .
\[connec\] Let $G$ be a connected graph of order $n$ which is $2$-cell embedded on a surface $\mathbb{M}$. If $ \gamma_c (G) = \gamma_c \geq 3$ then $$\label{ngaa}
\gamma_c \leq n - (1 + \sqrt{17 - 8\chi(\mathbb{M})})/2.$$
Note that $\gamma_c \geq 3$ implies $\gamma_c < n$. By Theorem \[help66\](iii) we have $2\|G\| \leq (n-\gamma_c +1)(n-\gamma_c) + 2\gamma_c -2$. Hence by Euler’s formula $$2\chi(\mathbb{M}) \geq 2n - 2\|G\| +2
\geq 2n +2 - (n-\gamma_c +1)(n-\gamma_c) -2\gamma_c +2,$$ or equivalently $$\gamma_c^2 - (2n-1)\gamma_c + n^2 - n - 4 - \chi(\mathbb{M}) \geq 0$$ Since $\gamma_c < n$, it immediately follows .
The bound in Theorem \[connec\] is sharp. Let $n$, $d$ and $t$ be integers such that $t \geq d \geq 4$ and $n = d + t$. We consider any graph $G$ which has the following form:
- $G$ is the union of a clique of $n-d$ vertices, and a path of $d$ vertices, where each vertex in the clique is adjacent to exactly one of the endpoints of the path, and each endpoint has at least one clique vertex adjacent to it.
Clearly $|G| = n$, $\gamma_c(G) = d$, $\|G\| = \binom{n-d +1}{2} + d -1$ and $k = \|G\| - |G| +1 = \binom{t}{2}$. Now Theorem \[zzz\](i) implies that $G$ can be $2$-cell embedded in $\mathbb{N}_{k}$. Finally, it is easy to check that $\gamma_c(G) = n - (1 + \sqrt{17 - 8\chi(\mathbb{N}_{k})})/2$.
In ending this section we note that in [@Samarxiv], the present author proved analogous results for the ordinary domination number.
Bondage numbers and restricted edge connectivity {#rbrc}
================================================
\[main3\] Let $G$ be a nontrivial connected graph of orientable genus $g$, non-orientable genus $\overline{g}$ and minimum degree at least $3$. Then
- $\max \{\lambda^{\prime}(G) ,b_r(G)\} \leq \min \{h_1 (g), k_1(\overline{g})\}-2$, and
- $\max \{\lambda^{\prime}(G) ,b_r(G)\} \leq \min \{h_2 (g), k_2(\overline{g})\}-2$, provided $G$ does not contain $3$-cycles.
\(i) By combining Theorem \[rcon\] and Theorem \[help1\] with Theorem \[help33\] we obtain the result.
\(ii) Theorem \[rcon\], Theorem \[help1\] and Theorem \[help3\] together immediately imply the required inequality.
\[restrtot\] Let $G$ be a connected graph with $\delta(G) \geq 4$.
- Then $b_{tr}(G) \leq \xi(G) + \Delta (G) - 2$.
- If $G$ is of orientable genus $g$, then $b_{tr}(G) \leq h_1(g)+ \Delta (G) - 4$. Furthermore, if $G$ does not contain $3$-cycles, then $b_{tr}(G) \leq h_2(g)+ \Delta (G) - 4$.
- If $G$ is of nonorientable genus $\overline{g} $, then $b_{tr}(G) \leq k_1(\overline{g})+ \Delta (G) - 4$. Furthermore, if $G$ does not contain $3$-cycles, then $b_{tr}(G) \leq k_2(\overline{g})+ \Delta (G) - 4$.
- Then $b_{tr}(G) \leq 2ad(G) + \Delta(G) -4$.
- Let $G$ be embeddable on a surface $\mathbb{M}$ whose Euler characteristic $\chi$ is as large as possible and let $g(G) = g$. Then: $$b_{tr}(G) \leq \frac{4g}{g-2}(1 - \frac{\chi}{|G|}) + \Delta(G) -4 \leq -\frac{12\chi}{|G|} + \Delta(G) +8.$$
\(i) Since $\delta (G) \geq 4$, there is a path $x,y,z$ in $G$ such that $G-\{x,y,z\}$ has no isolated vertices and $\xi(xy) = \xi (G)$. Now, by Theorem \[help2\] we have $b_{tr}(G) \leq deg(x) + deg(y) + deg(z) - 4 \leq \xi(G) + deg(z) - 2
\leq \xi(G) + \Delta (G) -2$.
\(ii) Combining (i) and Theorem \[help33\] we obtain the required.
\(iii) The result follows by combining Theorem \[help3\] and (i).
\(iv) By Theorem \[help22a\] we know that $b_{tr}(K_n) = n-1$ whenever $n \geq 4$. Hence we may assume $G$ has nonadjacent vertices. Theorem \[hra\] implies that there are $2$ vertices, say $x$ and $y$, that are either adjacent or at distance 2 from each other, with the property that $deg (x) + deg (y) \leq 2ad (G)$. Since $G$ is connected and $\delta (G) \geq 4$, there is a vertex $z$ such that $xyz$ or $xzy$ is a path. In either case by Theorem \[help2\] we have $b_{tr}(G) \leq deg(x) + deg(y) + deg(z) - 4 \leq 2ad(G) + \Delta(G) -4$.
\(v) Theorem \[SGZ\] and (iv) together imply the result.
It is well known that the minimum degree of any planar graph is at most $5$.
\[last\] Let $G$ be a planar graph with minimum degree $\delta(G) = 4+i$, $i \in \{0,1\}$. Then $b_{tr}(G) \leq 14-i$.
There is a path $x,y,z$ such that $deg(x) + deg(y) + deg(z) \leq 18-i$, because of Theorem \[help444\]. Since $\delta (G) \geq 4$, $G-\{x,y,z\}$ has no isolated vertices. The result now follows by Theorem \[help33\].
Find a sharp constant upper bound for the total restrained bondage number of a planar graph.
Akbari, Khatirinejad and Qajar [@akq] recently showed that the roman bondage number of any planar graph is not more than $15$. In case when the planar graph has minimum degree $5$, we improve this bound by $1$.
\[lastl\] Let $G$ be a planar graph with $\delta(G) = 5$. Then $b_{R}(G) \leq 14$.
By Theorem \[help444\] there is a path $x,y,z$ such that $deg(x) + deg(y) + deg(z) \leq 17$. Since $b_R(G) \leq deg(x) + deg(y) + deg(z) - 3$ (Theorem \[help2a\]), the result follows.
Results on the roman bondage number of graphs on surfaces may be found in [@samCz].
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---
abstract: 'A model Hamiltonian for the reaction CH$_4^+ \rightarrow$ CH$_3^+$ + H, parametrized to exhibit either early or late inner transition states, is employed to investigate the dynamical characteristics of the roaming mechanism. Tight/loose transition states and conventional/roaming reaction pathways are identified in terms of time-invariant objects in phase space. These are dividing surfaces associated with normally hyperbolic invariant manifolds (NHIMs). For systems with two degrees of freedom NHIMS are unstable periodic orbits which, in conjunction with their stable and unstable manifolds, unambiguously define the (locally) non-recrossing dividing surfaces assumed in statistical theories of reaction rates. By constructing periodic orbit continuation/bifurcation diagrams for two values of the potential function parameter corresponding to late and early transition states, respectively, and using the total energy as another parameter, we dynamically assign different regions of phase space to reactants and products as well as to conventional and roaming reaction pathways. The classical dynamics of the system are investigated by uniformly sampling trajectory initial conditions on the dividing surfaces. Trajectories are classified into four different categories: direct reactive and non reactive trajectories, which lead to the formation of molecular and radical products respectively, and roaming reactive and non reactive orbiting trajectories, which represent alternative pathways to form molecular and radical products. By analysing gap time distributions at several energies we demonstrate that the phase space structure of the roaming region, which is strongly influenced by non-linear resonances between the two degrees of freedom, results in nonexponential (nonstatistical) decay.'
author:
- 'Frédéric A. L. Mauguière'
- Peter Collins
- 'Gregory S. Ezra'
- 'Stavros C. Farantos'
- Stephen Wiggins
title: 'Roaming dynamics in ion-molecule reactions: phase space reaction pathways and geometrical interpretation'
---
Introduction {#sec:intro}
============
New experimental techniques for studying chemical reaction dynamics, such as imaging methods [@Ashfold06] and multidimensional infra-red spectroscopy [@Mukamel2000], have revealed unprecedented details of the mechanisms of chemical reactions. The temporal and spatial resolution achieved allows the measurement of reactant and product state distributions, thus providing data that challenge existing theory. Given that accurate quantum dynamical studies can be carried out only for small polyatomic molecules, most theoretical analyses of chemical reaction rates and mechanisms are formulated in terms of classical mechanics (trajectory studies) or statistical approaches, such as RRKM (Rice, Ramsperger, Kassel and Marcus) theory [@Forst03; @Baer96] or transition state theory (TST) [@Levine09].
A significant challenge to conventional approaches to reaction mechanism is provided by the recently discovered “[*roaming reactions*]{}”. This type of reaction was revealed in 2004 by Townsend *et al.* in a study of photodissociation of formaldehyde [@townsend2004roaming]. When excited by photons, the formaldehyde molecule can dissociate via two channels: H$_2$CO $\rightarrow$ H + HCO (radical channel) or H$_2$CO $\rightarrow$ H$_2$ + CO (molecular channel). Zee *et al.* [@zee:1664] found that, above the threshold for the H + HCO dissociation channel, the CO rotational state distribution exhibited an intriguing ‘shoulder’ at lower rotational levels correlated with a hot vibrational distribution of H$_2$ co-product. The observed product state distribution did not fit well with the traditional picture of the dissociation of formaldehyde via the well characterized saddle point transition state for the molecular channel. Instead, a new pathway is followed that is dynamical in nature, and such dynamical reaction paths or roaming mechanisms are the central topic of this paper.
The roaming mechanism, which explains the observations of Zee and co-workers, was demonstrated both experimentally and in classical trajectory simulations by Townsend *et al.* [@townsend2004roaming]. Following this work, roaming has been identified in the unimolecular dissociation of molecules such as CH$_3$CHO, CH$_3$OOH or CH$_3$CCH, and in ion-molecule reactions [@Yu11], and is now recognized as a general phenomenon in unimolecular decomposition (see Ref. \[\] and references therein).
Reactions exhibiting roaming pose a considerable challenge to basic understanding concerning the dynamics of molecular reactions. The standard picture in reaction dynamics is firmly based on the concept of the reaction coordinate [@Heidrich95], for example, the [*intrinsic reaction coordinate*]{} (IRC). The IRC is a minimum energy path (MEP) in configuration space that smoothly connects reactants to products and, according to conventional wisdom, it is the path a system follows (possibly modified by small fluctuations about this path) as reaction occurs. Roaming reactions, instead, avoid the IRC and involve alternative reaction pathways. (It is important to note that reactions involving dynamics that avoids the IRC, so-called non-MEP reactions, were extensively studied before the term “roaming” was coined [@Sun02; @Lopez07; @Mikosch08; @Zhang10].)
For the case of formaldehyde photodissociation, for example, the roaming effect manifests itself by a hydrogen atom nearly dissociating and starting to orbit the HCO fragment at long distances and later returning to abstract the other hydrogen and form the products H$_2$ and CO. Long-range interactions between dissociating fragments allow the possibility of reorientational dynamics that can result in a different set of products and/or energy distributions than the one expected from MEP intuition, while a dynamical bottleneck prevents facile escape of the orbiting H atom.
The roaming effect has now been identified in a variety of different types of reactions; for example, those involving excited electronic states [@NorthScience2012] or isomerization [@Ulusoy13; @Ulusoy13b]. These studies have identified some general characteristics of the roaming mechanism and point out the need for extending the theories of chemical reactions.
TST is a fundamental approach to calculating chemical reaction rate constants, and can take various forms, such as RRKM theory [@Forst03] or variational transition state theory (VTST)[@Truhlar1984]. The central ingredient of TST is the concept of a [*dividing surface*]{} (DS), which is a surface the system must cross in order to pass from reactants to products (or the reverse). By its very definition, the DS belongs neither to reactants nor to products but is located at the interface between these two species; this is the essence of the notion of [*transition state*]{}. Association of transition states with saddle points on the potential energy surface (PES) (and their vicinity) has a long history of successful applications in chemistry, and has provided great insight into reaction dynamics [@Levine09; @Carpenter84; @Wales03]. Accordingly, much effort has been devoted to connecting roaming reaction pathways with the existence (or not) of particular saddle points on the PES, as is evidenced by continued discussion of the role of the so-called “roaming saddle” [@shepler2011roaming; @Harding_et_al_2012].
It is in fact reasonable to expect that in cases where reactions proceed without a clear correlation to saddles of the PES, they are mediated by transition states that are dynamical in nature, i.e. [*phase space structures*]{}. Phase space formulations of TST have been known since the beginning of the theory [@Wigner38]. Only in recent years, however, has the phase space (as opposed to a configuration space) formulation of TST reached conceptual and computational maturity [@Wiggins08] for systems with more that two degrees of freedom. Fundamental to this development is the recognition of the role of phase space objects, namely [*normally hyperbolic invariant manifolds*]{} (NHIMs) [@Wiggins_book1994], in the construction of relevant DSs for chemical reactions. While the NHIM approach to TST has enabled a deeper understanding of reaction dynamics for systems with many ($\geq3$) degrees of freedom (DoF) [@Wiggins08; @ezra2009microcanonical], its practical implementation has relied strongly on mathematical techniques to compute NHIMs, such as the normal form theory [@Wiggins_book03]. Normal form theory, as applied to reaction rate theory, requires the existence of a saddle of index $\geq 1$ [@Wiggins08] on the PES to construct NHIMs and their attached DSs. For dynamical systems with two DoF the NHIMs are just unstable periodic orbits (PO), which have long been known in this context as Periodic Orbit Dividing Surfaces (PODS). (We recall that a PO is an invariant manifold. In phase space, an unstable PO forms the boundary of the dividing surface for 2 DoF. For natural Hamiltonian systems, kinetic plus potential energy, with 2 DoF, the PODS defines a dividing line in configuration space between reactants and products [@Pechukas81].) As we shall see, these particular hyperbolic invariant phase space structures (unstable POs) are appropriate for describing reaction dynamics in situations where there is no critical point of the potential energy surface in the relevant region of configuration space.
A common characteristic of systems exhibiting roaming reactions studied so far is the presence of long range interactions between the fragments of the dissociating molecule. This characteristic is typical of ion-molecule reactions and roaming is clearly expected to be at play in these reactions. The theory of ion-molecule reactions has a long history going back to Langevin [@Langevin1905], who investigated the interaction between an ion and a neutral molecule in the gas phase and derived an expression for ion-molecule collisional capture rates. As researchers have sought to develop models to account for data on ion-molecule reactions, there has been much debate in the literature concerning the interpretation of experimental results. Some results support a model for reactions taking place via the so-called loose or [*orbiting transition states*]{} (OTS), while others rather suggest that the reaction operates through a [*tight transition state*]{} (TTS) (for a review, see Ref. ). In order to explain this puzzling situation the concept of [*transition state switching*]{} was developed [@Chesnavich82], where both kinds of TS (TTS and OTS) are present and determine the reaction rate. (See also the unified statistical theory of Miller [@Miller76a].) Chesnavich presented a simple model to illustrate these ideas [@Chesnavich1986]. This relatively simple model has all the ingredients required to manifest the roaming effect [@MauguiereCPL2014], and the present work extends our investigations of the dynamics of Chesnavich’s model.
In a recent study [@MauguiereCPL2014] we have revisited the Chesnavich model [@Chesnavich1986] in light of recent developments in TST. We have shown that, for barrierless systems such as ion-molecule reactions, the concepts of OTS and TTS can be clearly formulated in terms of well defined phase space geometrical objects (for recent work on the phase space description of OTS, see Ref. ). We demonstrated how OTS and TTS can be identified with well defined [*phase space dividing surfaces*]{} attached to NHIMs. Moreover, this study showed that new reaction pathways, sharing all the characteristics of roaming reactions, may emerge and that they are associated with free rotor periodic orbits, which emanate from [*center-saddle bifurcations*]{} (CS).
The Chesnavich model for ion-molecule reactions, parametrised in such a way to represent different classes of molecules with early and late transition states, offers a useful theoretical laboratory for investigation of the evolution of phase space structures relevant to roaming dynamics, both in energy and as a function of additional potential function parameters; such a study is the aim of the present article. The birth of new reaction pathways in phase space associated with non-linear mechanical resonances raises questions concerning the applicability of statistical models. In the present work we investigate these questions with a detailed [*gap time analysis*]{} [@Thiele62; @Thiele62a; @ezra2009microcanonical] of direct and roaming dynamics.
The paper is organised as follows. In subsection \[subsec:syst\_Hamiltonian\] we introduce the Hamiltonian of the system to be studied. We then proceed to summarise work on the utility of NHIMs in the context of TST and discuss the DS associated with them in the subsection \[subsec:TST\]. Subsection \[subsec:NHIMs\] concludes section \[sec:Hamiltonian\] with a discussion of the application of the NHIM approach to the definition of the TTS and OTS in the Chesnavich model. Section \[sec:roamdyn\] presents a dynamical study of the roaming mechanism. In subsection \[subsec:roaminginterp\], we provide a discussion of the roaming phenomenon based on an analysis of the dynamics of the Chesnavich model for two different values of the parameter that controls the transition state switching in this model. The role of the so-called “roaming saddle” in the dynamical interpretation of the roaming phenomenon is also discussed. Section \[gaptime\] provides a gap time analysis for the Chesnavich model, while section \[sec:conclusion\] concludes.
Hamiltonian, NHIMs and transition states {#sec:Hamiltonian}
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System Hamiltonian {#subsec:syst_Hamiltonian}
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More than thirty years ago, the transition state switching model was proposed to account for the competition between multiple transition states in ion-molecule reactions (for a review, see Ref. ). Multiple transition states were studied by Chesnavich in the reaction CH$_4^+$ $\rightarrow$ CH$_3^+$ + H using a simple model Hamiltonian [@Chesnavich1986]. The model system consists of two parts: a rigid, symmetric top representing the CH$_3^+$ cation, and a mobile H atom. In the following, we employ a simplified version of Chesnavich’s model restricted to two DoF to study roaming.
The Hamiltonian for planar motion with zero overall angular momentum is: $$\label{eq:ham}
H = \frac{p_r^2}{2 \mu} +
\frac{p_{\theta}^2}{2} \left (\frac{1}{I_{CH_3}} + \frac{1}{ \mu r^2} \right ) + V(r,\theta),$$ where $r$ is the distance between the centre of mass of the CH$_3^+$ fragment and the hydrogen atom. The coordinate $\theta$ describes the relative orientation of the two fragments, CH$_3^+$ and H, in a plane. The momenta conjugate to these coordinates are $p_r$ and $p_{\theta}$, respectively, while $\mu$ is the reduced mass of the system and $I_{CH_3}$ is the moment of inertia of the CH$_3^+$ fragment. The potential $V(r, \theta)$ describes the so-called transitional mode. It is generally assumed that in ion-molecule reactions the different modes of the system separate into intramolecular (or conserved) and intermolecular (or transitional) modes [@VandeLinde90a; @Peslherbe95; @Klippenstein2011]. The potential $V(r,\theta)$ is made up of two terms: $$\label{eq:pot}
V(r,\theta) = V_{CH}(r)+V_{coup}(r,\theta),$$ with:
\[eq:subpot\] $$\begin{aligned}
V_{CH}(r) &= \frac{D_e}{c_1-6} \left\{ 2(3-c_2)\exp
\left[c_1(1-x)\right]
%\right. \nonumber \\ %% remove for single column
%& \left. %% remove for single column
- (4c_2-c_1c_2+c_1) x^{-6} - (c_1-6)c_2 x^{-4} \right\}, \\
V_{coup} (r,\theta) & = \frac{V_0(r)}{2} \left [ 1-\cos(2\theta) \right ], \\
V_0(r) & = V_e \exp \left [-\alpha(r-r_e)^2 \right ].\end{aligned}$$
Here $x=r/r_e$, and parameters fitted to reproduce data from CH$_4^+$ species are: dissociation energy $D_e = 47$ kcal/mol and equilibrium distance $r_e = 1.1$ Å. Parameters $c_1 = 7.37$, $c_2 = 1.61$, fit the polarizability of the H atom and yield a stretch harmonic frequency of 3000 cm$^{-1}$. Finally, $V_e = 55$ kcal/mol is the equilibrium barrier height for internal rotation, chosen so that at $r = r_e$ the hindered rotor has, in the low energy harmonic oscillator limit, a bending frequency of 1300 cm$^{-1}$. The masses are taken to be $m_H = 1.007825$ u, $m_C = 12.0$ u, and the moment of inertia $I_{CH_3} = 2.373409 $ uÅ$^2$. The parameter $\alpha$ controls the rate of conversion of the transitional mode from the angular to the radial mode. By adjusting this parameter one can control whether the conversion occurs ‘early’ or ‘late’ along the reaction coordinate $r$. For our study we will study the two cases $\alpha = 1$ Å$^{-2}$, which corresponds to a late conversion, and $\alpha = 4$ Å$^{-2}$, which corresponds to an early conversion.
Figs \[fig1\]a and \[fig2\]a show contour plots of the potential function as well as representative periodic orbits (see section \[sec:roamdyn\]) for $\alpha=1$ and $\alpha=4$, respectively. In Table \[table:equil\], the stationary points of the potential function for the two values of the parameter $\alpha=1$ and $\alpha = 4$ are tabulated and are labelled according to their stability properties. The minimum for CH$_4^+$ (EP1) is of center-center (CC) stability type, which means that it is stable in both coordinates, $r$ and $\theta$. The saddle, which separates two symmetric minima at $\theta= 0$ and $\pi$ (EP2), is of center-saddle (CS) type, i.e. stable in $r$ coordinate and unstable in $\theta$. The maximum in the PES (EP4) is a saddle-saddle (SS) equilibrium point. The outer saddle (EP3) is a CS equilibrium point.
--------------------- --------- ---------------- ----------- ------- --------------------- --------- ---------------- ----------- -------
E (kcal mol$^{-1}$) $r$ (Å) $\theta$ (rad) Stability Label E (kcal mol$^{-1}$) $r$ (Å) $\theta$ (rad) Stability Label
-47.0 1.1 0 CC EP1 -47.0 1.1 0 CC EP1
8.0 1.1 $\pi/2$ CS EP2 8.0 1.1 $\pi/2$ CS EP2
-0.63 3.45 $\pi/2$ CS EP3 -6.44 1.96 $\pi/2$ CS EP3
22.27 1.63 $\pi/2$ SS EP4 8.82 1.25 $\pi/2$ SS EP4
--------------------- --------- ---------------- ----------- ------- --------------------- --------- ---------------- ----------- -------
: \[table:equil\] Equilibrium points for the potential $V(r, \theta)$ ($\alpha = 1$ and $4$). (CC) means a center-center equilibrium point (EP), (CS) a center-saddle EP and (SS) a saddle-saddle EP.
The MEP connecting the minimum EP1 with the saddle EP2 at $r=1.1$ Å (see Figs \[fig1\]a and \[fig2\]a) describes a reaction involving ‘isomerisation’ between two symmetric minima. These two isomers cannot of course be distinguished physically for a symmetric Hamiltonian. The MEP for dissociation to radical products (CH$_{3}^{+}$ cation and H atom) follows the line $\theta=0$ with $r \rightarrow \infty$ and has no potential barrier (or, one might locate the barrier at infinity). Broad similarities between the Chesnavich model and the photodissociation of formaldehyde and other molecules for which the roaming reaction has been observed can readily be identified. In the Chesnavich model we recognize two reaction ‘channels’, one leading to a molecular product by passage through an inner TS, and one to radical products via dissociation. Moreover, a saddle (EP3) exists just below the dissociation threshold, just as has been found in molecules showing the roaming effect.
In the remainder of this article we show that, by adopting a phase space perspective and employing the appropriate transition states defined in phase space, not only is the dynamical meaning of the roaming mechanism revealed but, most importantly, this dynamics is shown to be intimately associated with the generic behavior of non-linear dynamical systems in parameter space, where bifurcations and resonances may occur and qualitatively different dynamics (reaction pathways) are born.
Transition states, dividing surfaces and statistical assumptions {#subsec:TST}
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TST is based on certain fundamental assumptions [@Wigner38; @Baer96; @Levine09]. Once these are accepted (or tested for the problem considered), TST provides a powerful and very simple tool for computing the rate constant of a given reaction. One of the assumptions is the existence of a DS having the property that classical trajectories originating in reactants (resp. products) cross this surface only once in proceeding to products (resp. reactants). Such a DS therefore separates the phase space into two distinct regions, reactants and products, and therefore constitutes the boundary between them. The definition of the DS given above is fundamentally dynamical in nature (the local non-recrossing condition). The DS is in general a surface in the phase space of the system under consideration. Computing, or locating, a phase space surface (hypersurface/volume) that realises the first assumption of TST is in general not an easy task as one has to find a codimension one hypersurface in a $2n$-dimensional space for an $n$ DoF system. As discussed below, the NHIM approach to TST provides a solution to this problem.
A comment on nomenclature: The term ‘transition state’ is sometimes used to designate a saddle point of the system potential energy surface. Identification of the *transition state* with a point in configuration space is of course misleading; a transition state is more precisely defined as the manifold of phase space points where the transition between reactants and products occurs. The phase space DS defined above is just such a collection of phase space transition points. Confusion between saddle points and TS (DS) arises in situations where the system has to overcome a barrier in the PES in order to react. In such a case, there is a saddle point (index one) at the top of the barrier and the DS (TS) originates (in phase space) in the vicinity of this saddle point. However, there are situations for which the reaction does not proceed via a potential barrier and in these cases one has to find other phase space structures that define DS (TS). We have seen that the Chesnavich model provides an example where the outer TS is not associated with any potential saddle.
In searching for the appropriate DS for which the (local) non-recrossing property applies, and thus the minimal reactive flux criterion, it is reasonable to start with stationary points on the PES. However, minimization of the flux by varying the DS in configuration space as in variational transition state theory (VTST)[@Truhlar1984] may give a better DS. In this approach, the DS is still defined in configuration space but its location along some reaction path is determined by a variational principle. One can also investigate the flux through surfaces of specified geometry to determine optimal dividing surfaces in a given family of such surfaces (for an example of such a surface applied to the roaming phenomenon see Ref. ).
The minimal flux through the DS requirement can be cast into a minimum of the sum of states at the DS. As we move along some reaction coordinate from reactants to products, there is two competing effects which affect the sum of states in the DS [@Miller76a]. First, as we move to the dissociation products the potential energy is constantly rising and the available kinetic energy is decreasing which has the effect of lowering the sum of states. The second effect is a lowering of the vibrational frequencies at the DS that tends to increase the sum of states. These two competing effects result in a minimum in the sum of states located at some value of the reaction coordinate. This minimum has been called an “entropic barrier” for the reaction or a tight transition state. On the other hand, in the orbiting model of a complex forming the DS is located at the centrifugal barrier induced by the effective potential (the orbiting TS) [@Miller76a; @Chesnavich82]. In general the TTS and OTS are not located at the same position along the reaction coordinate and so one can ask which of these two DS should be used to compute the rate of the reaction. This problem gives rise to the theory of multiple transition states where one has to decide which DS (TTS or OTS) to use in the computation of the rate of the reaction. The Chesnavich model provides an excellent example [@Chesnavich1981; @Chesnavich82; @Chesnavich1986]. In this model both TS (DS) exist simultaneously and the actual TS (DS) for the computation of the reaction rate in a naive TST calculation is the one giving the minimal flux or, equivalently, the minimal sum of states. Millers’s approach provides a unified theory approriate when the fluxes associated with each DS are of comparable magnitude. We will see in the next paragraph how the Chesnavich model is treated within the NHIM approach to TST and in the next section how this approach relates to the roaming phenomenon.
The other fundamental assumption of TST is that of statistical dynamics. If one considers the reaction at a specific energy, the statistical assumption requires that throughout the dissociation of the molecule all phase space points are equally probable on the timescale of reaction [@Baer96]. This assumption is equivalent to saying that the redistribution of the energy amongst the different DoF of the system on the reactant side of the DS is fast compared to the rate of the reaction, and guarantees a single exponential decay for the reaction (random lifetime assumption for the reactant part of the phase space [@Thiele62]). In section \[gaptime\] we will investigate this statistical assumption for the roaming phenomenon by studying the gap time distributions in the “roaming region” defined in section \[sec:roamdyn\].
Normally hyperbolic invariant manifolds and their related dividing surfaces {#subsec:NHIMs}
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In this section we summarise recent work which has shown the usefulness of NHIMs in the context of TST [@Wiggins08]. In the previous subsection we recalled that TST is build on the assumption of the existence of a DS separating the phase space into two parts, reactant and products. The construction of this surface has been the subject of many studies. As we emphasized, the DS is in general a surface in phase space, and the construction of such surfaces for systems with three and higher DoF has until recently been a major obstacle in the development of the theory.
For systems with two DoF described by a natural Hamiltonian, kinetic plus potential energy, the construction of the DS is relatively straightforward. This problem was solved during the 1970s by McLafferty, Pechukas and Pollak [@Pechukas73; @Pechukas77; @Pollak78; @Pechukas79]. They showed that the DS at a specific energy is intimately related to an invariant phase space object, an unstable PO. The PO defines the bottleneck in phase space through which the reaction occurs and the DS which intersects trajectories evolving from reactants to products can be shown to have the topology of a hemisphere whose boundary is the PO [@waalkens2004direct; @wiggins2001impenetrable]. The same construction can be carried out for a DS intersecting trajectories travelling from products to reactants and these two hemispheres form a sphere for which the PO is the equator.
Generalisation of the above construction to higher dimensional systems has been a major question in TST and has only received a satisfactory answer relatively recently [@waalkens2004direct; @wiggins2001impenetrable]. The key difficulty concerns the higher dimensional analogue of the unstable PO used in the two DoF problem for the construction of the DS. Results from dynamical systems theory show that transport in phase space is controlled by various high dimensional manifolds, Normally Hyperbolic Invariant Manifolds (NHIMs), which are the natural generalisation of the unstable PO of the two DoF case. Normal hyperbolicity of these invariant manifolds means that they are, in a precise sense, structurally stable, and possess stable and unstable invariant manifolds that govern the transport in phase space [@Fenichel1971; @Fenichel1977; @Fenichel1974; @Wiggins_book1994].
Existence theorems for NHIMs are well established [@Fenichel1971; @Fenichel1977; @Fenichel1974; @Wiggins_book1994], but for concrete examples one needs methods to compute them. One approach involves a procedure based on Poincaré-Birkhoff normalisation: the idea is to find a set of canonical coordinates by means of canonical transformations that put the Hamiltonian of the system in a “simple” form in a neighbourhood of an equilibrium point of saddle-centre$\cdots$-centre type (an equilibrium point at which the linearized vector field has one pair of real eigenvalues and $n-1$ imaginary eigenvalues for a system of $n$ DoF). The “simplicity” comes from the fact that, under non-resonance conditions among the imaginary frequencies at the saddle point, one can construct an integrable system valid in the neighbourhood of the equilibrium point and thereby describe the dynamics in this neighbourhood very simply. With this new Hamiltonian, the geometrical structures that govern reaction dynamics are revealed. For two DoF systems, the NHIM is simply a PO. For an $n > 2$ DoF system at a fixed energy, the NHIM has the topology of a $(2n-3)$ sphere. This $(2n-3)$-dimensional sphere is the equator of a $(2n-2)$-dimensional sphere which constitutes the DS. The DS divides the $(2n-1)$-dimensional energy surface into two parts, reactants and products, and one can show that it is a surface of [*minimal flux*]{} [@waalkens2004direct].
The NHIM approach to TST consists of constructing DSs for the reaction studied built from NHIMs, and constitutes a rigorous realisation of the local non-recrossing property. Once these geometrical objects (NHIM and DS) are computed the reactive flux from reactant to products through the DS can easily be expressed as the integral of a flux form over the DS. Furthermore, it is possible to sample the DS and use this knowledge to propagate classical trajectories initiated at the TS (DS). As noted above, for the two DoF case, the unstable PO used in the construction of the DS is just an example of such a NHIM. In this paper, we are concerned with an $n=2$ DoF problem and therefore the NHIMs we will be interested in are POs. Extension to $n >2$ DoF systems is in principle conceptually straightforward.
Roaming dynamics {#sec:roamdyn}
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Dynamical interpretation of the roaming mechanism {#subsec:roaminginterp}
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In the previous section, we discussed the notions of TTS and OTS in the context of a reaction occurring without a potential barrier. We also discussed how the NHIM approach to TST provides a rigorous way of constructing a DS that satisfies the local no-recrossing requirement of TST. To define DSs that are relevant for the description of reactions in the model Hamiltonian defined in Eq. , we need to locate unstable POs. Periodic orbits for conservative Hamiltonian systems exist in *families*, where the POs in a family depend on system parameters. In molecular systems, for example, it is very common to consider PO families obtained by variation of the energy of the system (see for example Refs. and ). At critical parameter values (the energy, for example) bifurcations take place and new families are born. Continuation/bifurcation (CB) diagrams are obtained by plotting a PO property as a function of the parameter. One important kind of elementary bifurcation is the center-saddle, which turns out to be ubiquitous in non-linear dynamical systems [@Wiggins_book03]. Although periodic orbits, being one dimensional objects, cannot reveal the full structure of phase space, they do provide a “skeleton’ around which more complex structures such as invariant tori develop. Numerous explorations of non-linear dynamical systems by construction of PO CB diagrams have been made. In particular, for molecules with multidimensional, highly anharmonic and coupled potential functions, software has been developed to locate POs based on multiple shooting algorithms [@Farantos98], and has successfully been applied to realistic models of small polyatomic molecules [@Farantos09]. In Figs. \[fig1\]b and \[fig2\]b such CB diagrams are shown for the Chesnavich model for the values of the parameter $\alpha=1$ and $\alpha=4$ respectively. Not all principal families of POs generated from all equilibria are shown, but only those which are relevant for our discussion of the roaming phenomenon.
We identify the DS constructed from the PO denoted TTS-PO in Figs. \[fig1\]b and \[fig2\]b with the TTS. These periodic orbits show hindered rotor behavior. The OTS is related to the centrifugal barrier appearing due to the presence of the centrifugal ($\approx r^{-2}$) term in the kinetic energy, Eq. . There is in fact a PO associated with the centrifugal barrier, referred to as a [*relative equilibrium*]{}. In Figs. \[fig1\]b and \[fig2\]b we refer to this PO as OTS-PO and we identify the DS constructed from this PO with the OTS. These relative equilibria POs and higher dimensional analogues have been studied by Wiesenfeld et *al.* [@Wiesenfeld05] in the context of capture theories of reaction rates.
We have therefore clearly identified the notions of TTS and OTS found in the literature with DSs constructed from NHIMs. These TSs (DSs) are surfaces which satisfy rigorously the requirement of local no-recrossing TST theory. These two TSs (DSs) exist simultaneously for our model Hamiltonian and in the following we discuss the dynamical consequences of this fact and how one can interpret roaming phenomenon in this setting.
### Roaming and non-linear mechanical resonances {#subsec:roaming}
The TTS and OTS are associated with different reactive bottlenecks in the system, and hence, in a certain sense, with different ‘reaction pathways’. In order to completely dissociate to CH$_3^+$+H, the system has to cross the OTS. This surface satisfies a global non-recrossing condition (as opposed to a local non-recrossing condition) in the sense that once the system crosses this surface in the outward sense the orbital momentum is an approximate constant of motion (for sufficiently large $r$), and the system enters an uncoupled free rotor regime. The TTS, on the other hand, is the DS associated with formation of the cation CH$_4^+$. These two DS delimit an intermediate region defining the association complex CH$_3^+\cdots$H. The picture here is similar to that discussed by Miller in his Unified Statistical Theory [@Miller76a], where a modified statistical theory is developed to describe association/dissociation dynamics in the presence of a complex. In Miller’s theory, the complex was associated with a well in the PES, whereas in our case, there is actually no potential well in the intermediate region between the TTS and OTS with which the complex can be unambiguously associated. Instead, there are non-linear mechanical resonances, which create ‘sticky’ regions in phase space (for rigorous results on the notion of stickiness in Hamiltonian systems see [@pw94; @mg95b]). These resonances are marked by the families of periodic orbits FR1 and its period doublings (for example FR12), so that the phase space region delimited by the TTS and the OTS can be thought of as a “dynamical complex”.
The two TTS and OTS form two phase space bottlenecks between which trajectories can be trapped for arbitrary long times. This trapping is responsible for the existence of trajectories for which the hydrogen atom winds around the CH$_3^+$ fragment and “roams” before exiting the dynamical complex, either to reform CH$_4^+$ or to dissociate to CH$_3^+$+H. Hence, the dynamical complex defines the *roaming region*. To study this trapping phenomenon we initiate classical trajectories on the OTS and follow them either until the CH$_4^+$ cation is formed or dissociation back to CH$_3^+$+H occurs. Trajectory propagation is then stopped shortly after crossing of the TTS or OTS occurs. We make such calculations for two different values of the parameter $\alpha$ which controls the location of the TS in the Chesnavich transition state switching model. In the next paragraph we describe these classical trajectory simulations.
### Classical trajectory simulations {#subsec:cltrajsimul}
To perform our classical trajectory simulation we uniformly sample trajectory initial conditons on the OTS at constant energy (microcanonical sampling). As explained in section \[sec:Hamiltonian\], the OTS-DS is composed of two parts: one hemisphere for which the trajectories cross from reactants to products (forward hemisphere) and the other for which the trajectories cross from products to reactants (backward hemisphere). For the OTS, if we define as reactants the complex CH$_3^+\cdots$H and as products CH$_3^+$+H, in our simulation we are interested only in trajectories lying on the backward hemisphere of the DS. We sample this hemisphere uniformly and numerically integrated the equations of motion until the trajectories cross either the OTS (forward hemisphere this time) or the TTS (backward hemisphere if CH$_4^+$ is defined as reactants and the complex CH$_3^+\cdots$H as the products for the TTS).
We wish to classify trajectories according to qualitatively different types of behavior, i.e., trajectories associated with different reactive events. Two obvious qualitatively different types of trajectories can be identified. First, there are trajectories which cross the TTS and form CH$_4^+$. These trajectories are ‘reactive’ trajectories. Second, there are trajectories which recross the OTS to form CH$_3^+$+H. These trajectories are ‘non reactive’.
Our classification scheme requires a precise definition of ‘roaming’ trajectories. In a previous publication [@MauguiereCPL2014] we proposed a classification of trajectories according to the number of turning points in the $r$ coordinate. Here, in light of subsequent investigations involving gap times (see the next section), we refine this definition of roaming. In the present system, roaming is intuitively associated with motions in which the hydrogen atom orbits the CH$_3^+$ fragment while undergoing oscillations in the $r$ coordinate. For such motions to occur, energy must be transferred from the radial to the angular mode and (see below) the mechanism for such an energy transfer involves non-linear resonances, which are manifest by the appearance of the FR1 POs. Just as we construct DS associated with the TTS-POs and the OTS-POs, it is possible to define a DS associated with the FR1 PO, which we denote the FR1-DS. To exhibit roaming character according to our revised definition, a trajectory must cross the FR1-DS several times. Such a trajectory will therefore involve exchange of energy between the radial and angular DoF before finding its way to a final state (either CH$_4^+$ or CH$_3^+$+H).
We now define the four categories of trajectories used in our analysis of the Chesnavich model:
- Direct reactive trajectories: these trajectories cross the FR1-DS only once before crossing the TTS to form CH$_4^+$.
- Roaming reactive trajectories: these trajectories cross the FR1-DS at least three times before crossing the TTS to form CH$_4^+$. Note that a reactive trajectory has to cross the FR1-DS an odd number of times.
- Direct non reactive trajectories: these trajectories cross the FR1-DS only twice before crossing the OTS to form CH$_3^+$+H.
- Roaming non reactive trajectories: these trajectories cross the FR1-DS at least four times before crossing the OTS to form CH$_3^+$+H. Note non reactive trajectories have to cross the FR1-DS an even number of times.
Note that, in principle there may be non reactive trajectories which never cross the FR1-DS but which return immediately to recross the OTS. The existence of these trajectories is perfectly conceivable as the stable and unstable manifolds of the period doubling bifurcated orbits of the FR1 family, could ‘reflect back’ the incoming trajectories. However, we find no such trajectories in our simulations.
Trajectories were propagated and classified into the four different classes according to the definitions given above for two different values of the parameter $\alpha$ of the Hamiltonian. In the Chesnavich model, this parameter controls the “switching” of the transition state from late to early. The switching model was developed in the context of variational TST where it is necessary to determine the optimal transition state to use in a statistical theory in order to compute the reaction rate. A variational criterion is used to select the relevant TS, tight or loose. In the system under study here, two phase space DS (TTS and OTS) exist simultaneously, and in order to analyze the roaming phenomenon in dynamical terms, both must be taken into account. In the next section, we investigate the question of the assumption of statistical dynamics in the roaming region.
In Fig. \[fig3\] we show the result of our classical trajectory simulations at energy $E=0.5$ kcal/mol for the case $\alpha=1$. The case $\alpha=1$ corresponds to the switching occurring late, which means that if one were to use variational TST an OTS would be used to compute the rate. In Fig. \[fig3\] the TTS-PO, the FR1 and the OTS-PO are represented as thick black curves. Each panel of the figure shows trajectories belonging to different classes of trajectories that we defined earlier. Fig. \[fig3\]a shows the direct reactive trajectories, Fig. \[fig3\]b the roaming reactive trajectories, Fig. \[fig3\]c the direct non reactive trajectories and Fig. \[fig3\]d the roaming non reactive trajectories. Similarly, Fig. \[fig4\] shows the results for the case $\alpha=4$ at the same energy.
Figure \[fig5\] shows the evolution of the TTS-PO with $\alpha$ at constant energy of 0.5 kcal.mol$^{-1}$. The location of the OTS remains practically unchanged with $\alpha$. In order to quantify the roaming effect we plot in Fig. \[fig6\] the fractions of the different classes of trajectories versus energy. Fig. \[fig6\]a is for the case $\alpha=1$ and Fig. \[fig6\]b for $\alpha=4$. It is also instructive to look at the rotor angular momentum ($p_{\theta}$) distributions for the direct and roaming non reactive trajectories at the beginning and the end of the trajectory propagation. The angular momentum distributions for direct trajectories and those exhibiting roaming are found to be qualitatively different in experiments [@townsend2004roaming] and Figs. \[fig7\] and \[fig8\] show that our classification scheme captures this aspect of the roaming phenomenon for our model system. Fig. \[fig7\] shows initial and final angular momentum distributions for the case $\alpha=1$ at several energies and similarly Fig. \[fig8\] for the case $\alpha=4$. Both initial and final distributions are identical within the statistical errors, as is expected since the OTS PO defines the only entrance (exit) portal for the association (dissociation) of radical reactants (products) to occur.
There has been an interesting discussion in the literature concerning the possible existence of a saddle in the PES responsible for the roaming reaction, often referred to as the “the roaming saddle” [@Harding07; @shepler2011roaming; @Klippenstein2011; @Harding10]. Indeed, for the Chesnavich model, there exists such a saddle point on the PES, labelled EP3 in Table \[table:equil\], which could be considered to be a roaming saddle. However, as has already been pointed out, transition states are in general not associated with particular potential saddles. We have shown that the TTS and OTS are the dividing surfaces associated with unstable periodic orbits, those of TTS-PO and OTS-PO families, which originate from center-saddle bifurcations (see Figs. \[fig1\]b and \[fig2\]b). On the other hand, equilibria of the PES, both stable and unstable, are essential in tracing the birth of time invariant objects in phase space, such as principal families of periodic orbits, tori, NHIMs and other invariant manifolds.
The existence of ubiquitous center–saddle bifurcations of periodic orbits is supported by the Newhouse theorem [@Newhouse79; @Wiggins_book03], which was initially proved for dissipative dynamical systems, and later extended to Hamiltonian systems [@Duarte99; @Gonchenko00]. The theorem states that tangencies of the stable and unstable manifolds associated with unstable equilibria and periodic orbits generate an infinite number of period doubling and center–saddle bifurcations. Hence, the unstable periodic orbits of the principal family of EP3 (Lyapunov POs) are expected to generate such CS bifurcations as their manifolds extend along the bend degree of freedom. Intersections of these manifolds, either self-intersections or with manifolds from different equilibria, generate homoclinic and heteroclinic orbits, respectively [@Wiggins_book03]. Such orbits can connect remote regions of phase space. This phenomenon has been repeatedly underlined in explorations of the phase space of a variety of small polyatomic molecules [@Farantos09; @Mauguiere10; @Farantos11]. The numerical location of such bi-aymptotic orbits is not easy, and this fact makes periodic orbit families even more precious in studying the complexity of the molecular phase space at high excitation energies.
The FR1 POs are associated with a 2:1 resonance region in phase space between stretch ($r$) and bend ($\theta$) modes. The CS bifurcation generates “out of nowhere” stable and unstable PO branches. In this way we can understand the trapping of (non) reactive trajectories in the roaming mechanism, and can also assign a DS attached to the NHIM FR1-PO.
Gap time analysis of the roaming region {#gaptime}
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In the preceding section, we described the dynamics of roaming reactions. We showed that the TTS and the OTS delimit a roaming region inside which some trajectories may be trapped for long times, and that roaming region can be seen as a dynamical complex even if there is no well in the PES with which this complex can be associated. The situation is very similar to that described in Miller’s unified treatment of statistical theory in the presence of a complex [@Miller76a]. Our analysis showed that the roaming region can be viewed as a dynamical complex, and it is therefore relevant to investigate the validity of the assumption of statistical dynamics within the roaming region.
To investigate the nature of dynamics in the roaming region, we perform a gap time analysis. We will first briefly review the gap time approach to reaction rates due to Thiele [@Thiele62; @Thiele62a]. Our exposition follows closely Ref. to which we refer the reader for more information.
Many theoretical investigations have been made of the validity of the statistical assumption underlying TST, focusing on the lifetime and gap time distributions of species involved. A non exhaustive list of important work includes the reserarch of Slater [@Slater56; @Slater59], Bunker [@Bunker62; @Bunker64], Bunker and Hase [@Bunker73], Thiele [@Thiele62; @Thiele62a], Dumont and Brumer [@Dumont86] and DeLeon and co-workers [@DeLeon81; @Berne82]. Broadly speaking, nonstatistical or non-RRKM behavior for a specific unimolecular dissociation reaction can arise in two essentially different ways. First, for a specific reaction, non-RRKM behavior can be observed because reactants are prepared in a specific state which violates the assumption of uniform phase space density in the reactant region. The second possible origin of non-RRKM behavior is due to inherent non-statistical intramolecular dynamics, so-called intrinsic non-RRKM behavior [@Bunker73].
Gap time approach to unimolecular reaction rates {#gaptime-approach}
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### Phase space volumes, gap times and microcanonical RRKM rates {#vol-gaptimes}
To introduce the essential concepts needed, we will first treat the case for which the reactant is described by a single well to which access is mediated by a single channel (DS). (For this case, cf. Fig. 1 of Ref. .) As noted previously, for a given energy, a DS constructed from NHIMs divides the energy surface into two distinct species, reactants and products. Furthermore, the DS is composed of two hemispheres, one of which intersects trajectories travelling from reactant to product, and controls exit from the well, the other of which intersects trajectories travelling from products to reactants, and controls the access to the well. The hemisphere which controls the access to the well is designated $DS_{in}(E)$ and that which controls the exit from the well $DS_{out}(E)$, where the (microcanonical) DS is defined at constant energy $E$. The distinct phase space regions corresponding to reactants and products are denoted $\mathcal{M}_{r}$ and $\mathcal{M}_{p}$, respectively. The microcanonical density of states for reactant species is: $$\label{eq:density}
\rho_r(E) = \int_{\mathcal{M}_{r}} d\boldsymbol{x} \; \delta(E-H(\boldsymbol{x})),$$ where $\boldsymbol{x} \in \mathbb{R}^{2n}$ designate a phase space point for an $n$ DoF system. A similar expression can be written for the product part of the phase space $\mathcal{M}_{p}$. The points on the reactant part of the phase space can be uniquely specified by coordinates $(\bar{q},\bar{p},\psi)$, where $(\bar{q},\bar{p}) \in DS_{in}(E)$ is a point on $DS_{in}(E)$ specified by $2(n-1)$ coordinates $(\bar{q} , \bar{p})$, and $\psi$ is a time variable. The point $\boldsymbol{x}(\bar{q} , \bar{p} , \psi)$ is reached by propagating the initial condition $(\bar{q} , \bar{p}) \in DS_{in}(E)$ forward for time $\psi$. As all initial conditions on $DS_{in}(E)$ will leave the reactant region in finite time by crossing $DS_{out}(E)$, for each $(\bar{q} , \bar{p}) \in DS_{in}(E)$ we can define the *gap time* $s = s(\bar{q} , \bar{p})$, which is the time it takes for the incoming trajectory to traverse the reactant region. That is, $\boldsymbol{x}(\bar{q} , \bar{p} , \psi = s(\bar{q} , \bar{p})) \in DS_{out}(E)$. For the phase point $\boldsymbol{x}(\bar{q},\bar{p},\psi)$, we therefore have $0 \leq \psi \leq s(\bar{q},\bar{p})$.
The coordinate transformation $\boldsymbol{x} \rightarrow (E, \psi, \bar{q}, \bar{p})$ is canonical [@Thiele62; @Binney85; @Meyer86] so that the phase space volume element is $$\label{eq:volelemt}
d^{2n}\boldsymbol{x}= dE \; d\psi \; d\sigma,$$ with $d\sigma \equiv d^{n-1} \bar{q} \; d^{n-1} \bar{p}$ an element of $2n-2$ dimensional area on the DS. We denote the flux across $DS_{in}(E)$ and $DS_{out}(E)$ by $\phi_{in}(E)$ and $\phi_{out}(E)$, respectively, and note that $\phi_{in}(E) + \phi_{out}(E) = 0$. For our purposes we only need the magnitude of the flux, and so set $|\phi_{in}(E)| = |\phi_{out}(E)| \equiv \phi(E)$ the magnitude $\phi(E)$ of the flux through dividing surface $DS_{in}(E)$ at energy E is given by $$\label{eq:flux}
\phi(E)= \left| \int_{DS_{in}(E)} d\sigma \right|,$$ where the element of area $d\sigma$ is precisely the restriction to $DS_{in}(E)$ of the appropriate flux $(2n-2)$-form, $\omega^{n-1}/(n-1)!$, corresponding to the Hamiltonian vector field associated with $H(\boldsymbol{x})$. The reactant phase space volume occupied by points initiated on the dividing surface $DS_{in}(E)$ with energies between $E$ and $E+dE$ is therefore $$\label{eq:volume}
dE \int_{DS_{in}(E)} d\sigma \int_0^{s} d\psi = dE \int_{DS_{in}(E)} d\sigma \; s = dE \; \phi(E) \; \bar{s},$$ where the *mean gap time* $\bar{s}$ is defined as $$\label{eq:meangap}
\bar{s} = \frac{1}{\phi(E)} \int_{DS_{in}(E)} d\sigma \; s.$$ From this we conclude that the reactant density of state associated with trajectories that enter and exit the well region is $$\label{eq:reac-density-state}
\rho_r^c(E) = \phi(E) \; \bar{s},$$ where the superscript $c$ denotes that this density refers to *crossing* trajectories (some trajectories may be trapped in the well region and never escape from it). Equation \[eq:reac-density-state\] is the content of the so-called spectral theorem [@Brumer80; @Pollak81; @Waalkens05; @Waalkens05a; @Binney85]. If all phase space points in the reactant region $\mathcal{M}_{r}$ were to react, we would have $\rho_r^c(E)=\rho_r(E)$, where $\rho_r(E)$ now denotes the density of states for the full reactant region $\mathcal{M}_{r}$. However, because of the existence of trapped trajectories, in general we have $\rho_r^c(E) \leq \rho_r(E)$. If $\rho_r^c(E) < \rho_r(E)$ it is then necessary to introduce corrections to the statistical estimate of the reaction rate [@Hase83; @Grebenshchikov03; @Berblinger94; @Berne82; @Gray87; @Stember07].
The statistical (RRKM) microcanonical rate for the forward reaction from reactant to products at energy $E$ is given by $$\label{eq:RRKMrate}
k_{RRKM}(E) = \frac{\phi(E)}{\rho_r(E)},$$ and if $\rho_r^c(E)=\rho_r(E)$, we then have $$\label{eq:RRKMrategaptime}
k_{RRKM}(E) = \frac{1}{\bar{s}}.$$ In general the inverse of the mean gap time is given by $$\label{eq:invmeangap}
\frac{1}{\bar{s}} = \frac{\phi(E)}{\rho_r^c(E)}=k_{RRKM}(E) \left[\frac{\rho_r(E)}{\rho_r^c(E)}\right] \equiv k_{RRKM}^c(E) \geq k_{RRKM}(E),$$ where the superscript in $k_{RRKM}^c(E)$ is for corrected $RRKM$ microcanonical rate.
Generalisation to situations for which the access/exit to the reactant region is controlled by $d$ DSs gives for the corrected RRKM rate (see Ref. \[\]) $$\label{eq:multirate}
k_{RRKM}^c(E) = \frac{\sum_{i=1}^d \phi_i(E)}{\sum_{i=1}^d \bar{s}_{DS_{i,in}(E)} \phi_i (E)}$$
### Gap time and lifetime distributions {#distributions}
An important notion in the gap time formulation of TST is the gap time distribution, $P(s;E)$: the probability that a phase space point on $DS_{in}(E)$ at energy $E$ has a gap time between $s$ and $s+ds$ is equal to $P(s;E)\;ds$. The statistical assumption of TST is equivalent to the requirement that the gap time distribution is the random, exponential distribution $$\label{eq:randgapdist}
P(s;E) = k(E) \exp(-k(E)s).$$ This distribution is characterised by a single exponential decay constant $k(E)$ function of the energy to which corresponds the mean gap time $\bar{s}(E)=k(E)^{-1}$.
The lifetime of a phase space point $\boldsymbol{x}(\bar{q},\bar{p},\psi)$ is the time needed for this point to exit the reactant region $\mathcal{M}_{r}$ by crossing $DS_{out}(E)$ and is then defined as $t=s(\bar{q},\bar{p})-\psi$. It can be shown (see Ref. \[\]) that the lifetime distribution function $\mathbb{P}(t;E)$ is related to the gap time distribution by $$\label{eq:randgapdist2}
\mathbb{P}(t;E) = \frac{1}{\bar{s}} \int_t^{+\infty} ds \; P(s;E).$$ An exponential gap time distribution (satisfying the statistical assumption) implies that the lifetime distribution is also exponential.
Finally, in addition to the gap time distribution itself, we also consider the integrated gap time distribution $F(t;E)$, which is defined as the fraction of trajectories on the DS with gap times $s \geq t$, and is simply the product of the normalized reactant lifetime distribution function $\mathbb{P}(t ; E)$ and the mean gap time $\bar{s}$ $$\label{eq:intgapdist}
F(t;E) = \bar{s} \mathbb{P}(t ; E) = \int_t^{+\infty} ds \; P(s;E).$$ For the random gap time distribution the integrated gap time distribution is exponential, $F(t;E)=\exp(-kt)$.
Trajectory simulations of gap time distributions {#gaptime-traj}
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In order to test the statistical assumption for the roaming region we analyze the gap time distributions for this phase space region. To do so, we sample the OTS microcanonically on the incoming hemisphere and integrate trajectories initiated at these sample points until they recross either the OTS or the TTS.
Gap time distributions obtained from these simulations are shown in Figs. \[fig9\] and \[fig10\] for $\alpha=1$ and $\alpha=4$, respectively. Each of these figures has 4 panels, corresponding to different energies. In each panel we show the normalised gap time distributions for each of the four classes of trajectories we defined earlier, as well as the gap time distribution for all the trajectories taken together. Details are given in the captions of the figures.
The integrated gap time distributions for the same samples used in Figs \[fig9\] and \[fig10\] are shown in Figs \[fig11\] and \[fig12\], respectively. As seen in these figures, the gap time distributions, as well as the integrated gap time distributions, exhibit significant deviation from random (exponential) distributions, indicating that the statistical dynamical assumption of TST is not satisfied for motion in the roaming region.
In Fig \[fig13\] we plot the sampled trajectory initial conditions on the OTS in the $(\theta, p_\theta)$ plane; different colors are used to represent trajectory intial conditions belonging to different classes. This plot reveals a succession of *bands* of different types on the DS (see, for example, ref. and references therein). The arrangement of bands can be very complicated (fractal) [@Grice87]. In Fig \[fig14\] we plot gap time versus $p_\theta$ for initial conditions on the OTS at fixed $\theta=0$ for a range of $p_\theta$ values. The plot shows the fractal nature of bands associated with different trajectory types, and indicates that gap times diverge at the boundary between bands associated with two different trajectory types [@Pechukas77; @Mauguiere13]. An infinitely fine sampling of the $p_\theta$ axis would presumably reveal a set of measure zero of initial conditions for which the gap times are infinite. Infinite gap times correspond to trajectories trapped forever in the roaming regions, and such trajectories are on the stable invariant manifolds of stationary objects in the roaming region, such as the FR1 PO and its period doubling bifurcations.
Summary and Conclusion {#sec:conclusion}
======================
The model Hamiltonian for the reaction CH$_4^+ \rightarrow$ CH$_3^+$ + H proposed by Chesnavich [@Chesnavich1986] to study transition state switching in ion-molecule reactions has been employed to investigate roaming dynamics. The Chesnavich model supports multiple transition states and, despite its simplicity, is endowed with all the essential characteristics of systems previously found to exhibit the roaming mechanism.
Using concepts and methods from non-linear mechanics, early/late or tight/loose transition states are identified with time invariant objects in phase space, which are dividing surfaces in phase space associated with NHIMs – normally hyperbolic invariant manifolds. For two degree of freedom systems NHIMs are unstable periodic orbits which define the boundaries of locally non-recrossing dividing surfaces assumed in statistical reaction rate theories such as TST. The roaming region of phase space is itself unambiguously defined by these dividing surfaces.
By constructing continuation/bifurcation diagrams of periodic orbits for two values of the parameter in the Chesnavich Hamiltonian model controlling the early versus late nature of the transition state, and using the total energy as a second parameter, we identify phase space regions associated with roaming reaction pathways (i.e., trapping in the roaming region). The classical dynamics of the system are investigated by microcanonically sampling the outer OTS DS and assigning trajectories to four different classes: direct reactive and direct non-reactive, which describe the formation of molecular and radical products respectively, and roaming reactive and roaming non reactive, which folow alternative pathways to formation of molecular and radical products.
We identify the TTS and OTS with dividing surfaces associated with unstable periodic orbits of the TTS-PO and OTS-PO families. Additional PO families such as the FR1 POs reveal alternative reaction pathways, the roaming pathway, and define region in phase space associated with a 2:1 resonance between the stretch ($r$) and the bend ($\theta$) modes. The CS bifurcations generate “out of nowhere” a branch with stable and a branch with unstable periodic orbits. In this way we can understand the dynamical origin of the trapping of (non) reactive trajectories in the roaming region.
To investigate the validity of the assumption of statistical dynamics for the microcanonical ensembles we consider, we have analysed gap time distributions at several energies. Lifetime distributions exhibit multiple exponential dissociation rates at long times, violating the assumption of random gap times underlying statistical theory.
By plotting the outcome for trajectory initial conditions initiated on the $(\theta, p_\theta)$ at the OTS DS, we observe a regular succession of ‘bands’ of different types of trajectories. Our numerical results indicate the existence of a fractal band structure, where the gap time diverges at the boundary between distinct trajectory types. Such divergent gap times are associated with initial conditions on the stable manifold of invariant objects in the roaming region such as the FR1 PO and its period doubling bifurcations.
It is worth emphasizing that the concepts, theory and algorithms described here for two degrees of freedom systems can in principle be straightforwardly extended to higher dimensional systems. Nevertheless, substantial technical difficulties need to be overcome for accurate computation of NHIM-DS for higher dimensional systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the National Science Foundation under Grant No. CHE-1223754 (to GSE). FM, PC, and SW acknowledge the support of the Office of Naval Research (Grant No. N00014-01-1-0769), the Leverhulme Trust, and the Engineering and Physical Sciences Research Council (Grant No. EP/K000489/1).
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![(a) Contour plot of the PES for $\alpha=1$ with representative POs. (b) Continuation/bifurcation diagram of families of periodic orbits for $\alpha=1$.[]{data-label="fig1"}](fig1.pdf)
![(a) Contour plot of the PES for $\alpha=4$ with representative POs. (b) Continuation/bifurcation diagram of families of periodic orbits for $\alpha=4$.[]{data-label="fig2"}](fig2.pdf)
![The four different types of trajectories for the case $\alpha=1$. The thick black curves correspond to the TTS-PO, FR1 PO and OTS PO, respectively. (a) direct reactive trajectories (red). (b) Roaming reactive trajectories (green). (c) Direct non reactive trajectories (blue). (d) Roaming non reactive trajectories (magenta).[]{data-label="fig3"}](fig3.pdf)
![The four different types of trajectories for the case $\alpha=4$. The thick black curves correspond to the TTS-PO, FR1 PO and OTS PO, respectively. (a) Direct reactive trajectories (red). (b) Roaming reactive trajectories (green). (c) Direct non reactive trajectories (blue). (d) Roaming non reactive trajectories (magenta).[]{data-label="fig4"}](fig4.pdf)
![Evolution of the TTS-PO with parameter $\alpha$. Constant energy of 0.5 kcal.mol$^{-1}$[]{data-label="fig5"}](fig5.pdf)
![(color online) Fractions of different types of trajectories versus energy. Red line is the fraction of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. (a) $\alpha=1$, (b) $\alpha=4$.[]{data-label="fig6"}](fig6a.pdf "fig:") ![(color online) Fractions of different types of trajectories versus energy. Red line is the fraction of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. (a) $\alpha=1$, (b) $\alpha=4$.[]{data-label="fig6"}](fig6b.pdf "fig:")
![Initial and final normalised $p_{\theta}$ distributions for $\alpha=1$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig7"}](fig7a.pdf "fig:") ![Initial and final normalised $p_{\theta}$ distributions for $\alpha=1$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig7"}](fig7b.pdf "fig:") ![Initial and final normalised $p_{\theta}$ distributions for $\alpha=1$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig7"}](fig7c.pdf "fig:") ![Initial and final normalised $p_{\theta}$ distributions for $\alpha=1$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig7"}](fig7d.pdf "fig:")
![Initial and final normalised $p_{\theta}$ distributions for $\alpha=4$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig8"}](fig8a.pdf "fig:") ![Initial and final normalised $p_{\theta}$ distributions for $\alpha=4$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig8"}](fig8b.pdf "fig:") ![Initial and final normalised $p_{\theta}$ distributions for $\alpha=4$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig8"}](fig8c.pdf "fig:") ![Initial and final normalised $p_{\theta}$ distributions for $\alpha=4$. Blue and magenta curves represent initial distributions for direct non reactive and roaming non reactive trajectories, respectively, and cyan and pink curves represent the final distributions for direct non reactive and roaming non reactive trajectories, respectively. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig8"}](fig8d.pdf "fig:")
![Gap time distributions for $\alpha=1$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig9"}](fig9a.pdf "fig:") ![Gap time distributions for $\alpha=1$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig9"}](fig9b.pdf "fig:")\
![Gap time distributions for $\alpha=1$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig9"}](fig9c.pdf "fig:") ![Gap time distributions for $\alpha=1$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig9"}](fig9d.pdf "fig:")
![Gap time distributions for $\alpha=4$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green that for roaming reactive trajectories, blue for direct non reactive trajectories and magenta for roaming non reactive trajectories. The thick black curve denote the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig10"}](fig10a.pdf "fig:") ![Gap time distributions for $\alpha=4$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green that for roaming reactive trajectories, blue for direct non reactive trajectories and magenta for roaming non reactive trajectories. The thick black curve denote the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig10"}](fig10b.pdf "fig:") ![Gap time distributions for $\alpha=4$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green that for roaming reactive trajectories, blue for direct non reactive trajectories and magenta for roaming non reactive trajectories. The thick black curve denote the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig10"}](fig10c.pdf "fig:") ![Gap time distributions for $\alpha=4$. In each panel, red line denotes the normalised gap time distribution of direct reactive trajectories, green that for roaming reactive trajectories, blue for direct non reactive trajectories and magenta for roaming non reactive trajectories. The thick black curve denote the normalised gap time distribution for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig10"}](fig10d.pdf "fig:")
![The logarithm of the lifetime distributions for $\alpha=1$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig11"}](fig11a.pdf "fig:") ![The logarithm of the lifetime distributions for $\alpha=1$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig11"}](fig11b.pdf "fig:") ![The logarithm of the lifetime distributions for $\alpha=1$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig11"}](fig11c.pdf "fig:") ![The logarithm of the lifetime distributions for $\alpha=1$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig11"}](fig11d.pdf "fig:")
![The logarithm of the lifetime distributions for $\alpha=4$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig12"}](fig12a.pdf "fig:") ![The logarithm of the lifetime distributions for $\alpha=4$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig12"}](fig12b.pdf "fig:") ![The logarithm of the lifetime distributions for $\alpha=4$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig12"}](fig12c.pdf "fig:") ![The logarithm of the lifetime distributions for $\alpha=4$. In each panel, red line denotes the normalised logarithm of the lifetime distribution of direct reactive trajectories, green for roaming reactive trajectories, blue for direct non reactive and magenta for roaming non reactive trajectories. The thick black curve denotes the normalised logarithm of the lifetime for all trajectories. (a) Energy E=0.5 kcal.mol$^{-1}$. (b) E=1.0 kcal.mol$^{-1}$. (c) E=1.5 kcal.mol$^{-1}$. (d) E=2.0 kcal.mol$^{-1}$.[]{data-label="fig12"}](fig12d.pdf "fig:")
![Distribution of the different types of trajectories on the OTS. (a) $\alpha=1$, energy E=0.5 kcal.mol$^{-1}$. (b) $\alpha=4$, energy E=0.5 kcal.mol$^{-1}$[]{data-label="fig13"}](fig13a.pdf "fig:") ![Distribution of the different types of trajectories on the OTS. (a) $\alpha=1$, energy E=0.5 kcal.mol$^{-1}$. (b) $\alpha=4$, energy E=0.5 kcal.mol$^{-1}$[]{data-label="fig13"}](fig13b.pdf "fig:")
![The fractal nature of the boundaries between different types of trajectories on the DS. Initial points are selected along the line $\theta=0$ on the OTS, with $\alpha=4$ and energy E=0.5 kcal.mol$^{-1}$. The vertical axis shows the gap time and sampling points are assigned a colour to mark their type. Red: direct reactive trajectories, green: roaming reactive trajectories, blue: direct non reactive and magenta: roaming non reactive trajectories.[]{data-label="fig14"}](fig14.pdf)
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abstract: 'A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized Shrödinger operator we describe all spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The space $X_{n,\lambda}$ carries a natural effective action of a compact $(n-1)$-torus. We describe the topology of its orbit space and, in particular, show that whenever the isospectral space is a manifold, its orbit space is homeomorphic to $S^4\times T^{n-3}$. There is a classical dynamical system: the flow of the periodic Toda lattice, acting on $X_{n,\lambda}$. Except for the degenerate locus $X_{n,\lambda}^0$, the Toda lattice exhibits Liouville–Arnold behavior, so that the space $X_{n,\lambda}\setminus X_{n,\lambda}^0$ is fibered into tori. The degenerate locus of the Toda system is described in terms of combinatorial geometry: its structure is encoded in the special cell subdivision of a torus, which is obtained from the regular tiling of the euclidean space by permutohedra. We apply methods of commutative algebra and toric topology to describe the cohomology and equivariant cohomology modules of $X_{n,\lambda}$.'
address: 'Faculty of computer science, Higher School of Economics'
author:
- Anton Ayzenberg
title: Space of isospectral periodic tridiagonal matrices
---
[^1]
Introduction
============
Let $\Gamma=(V,E)$ be a simple graph on a set $V=[n]=\{1,\ldots,n\}$. Let $M_\Gamma$ be the vector space of Hermitian $n\times n$-matrices $A=(a_{ij})$, such that $a_{ij}=0$ for $(i,j)\notin E$. We consider the space $M_{\Gamma,\lambda}\subset M_\Gamma$ of all such matrices, which have a given simple spectrum $\lambda=(\lambda_1<\lambda_2<\cdots<\lambda_n)$. Note that each space $M_{\Gamma,\lambda}$ carries the conjugation action of a compact torus $T^n$. The action is noneffective: scalar matrices commute with every matrix, hence the diagonal subgroup of $T^n$ acts trivially.
Several examples are well studied. The complete graph $\Gamma=K_n$ corresponds to the space of all isospectral matrices, which is diffeomorphic to the variety $\operatorname{Fl}_n$ of complete flags in ${\mathbb{C}}^n$. The path graph $\Gamma={\mathbb{I}}_n$ with $n+1$ vertices produces the space $M_{{\mathbb{I}}_n,\lambda}$ of isospectral tridiagonal matrices, which is known to be a smooth $2n$-manifold; its smooth type is independent of $\lambda$. The real version of $M_{{\mathbb{I}}_n,\lambda}$ is called the Tomei manifold: it was introduced and studied in [@Tomei]. The $T^n$-action on $M_{{\mathbb{I}}_n,\lambda}$ is locally standard and its orbit space is diffeomorphic to a simple polytope, the permutohedron [@Tomei; @DJ]. Note that $M_{{\mathbb{I}}_n,\lambda}$ is not a toric variety, although it is closely related to the permutohedral variety [@BFR].
More generally, the spaces $M_{\Gamma_h,\lambda}$ corresponding to indifferent graphs $\Gamma_h$ are the spaces of staircase matrices. It is more convenient to encode this type of spaces by Hessenberg functions. The Hessenberg function is a function $h\colon[n]\to[n]$ such that $h(i)\geqslant i$ and $h(i+1)\geqslant h(i)$. The space $M_{\Gamma_h}$ is the space of Hermitian matrices $A$ such that $a_{ij}=0$ for $j>h(i)$. Every space $M_{\Gamma_h,\lambda}$ is a smooth manifold independent of a simple spectrum $\lambda$. Its odd degree cohomology modules vanish, therefore $M_{\Gamma_h,\lambda}$ is equivariantly formal in the sense of Goresky–MacPherson (see Definition \[definEqForm\]). The equivariant cohomology ring of $M_{\Gamma_h,\lambda}$ can be described using GKM-theory [@GKM; @Kur]. See [@ABhess] for details on the the spaces $M_{\Gamma_h,\lambda}$ and their relation to regular semi-simple Hessenberg varieties.
For the star graph $\Gamma=\operatorname{St}_n$ (see Fig.\[figGraphs\]), the space $M_{\operatorname{St}_n,\lambda}$ is also a smooth manifold, and its diffeomorphism type does not depend on $\lambda$. The effective action of $T=T^{n+1}/\Delta(T^1)$ on $M_{\operatorname{St}_n,\lambda}$ is locally standard, therefore the orbit space $Q_{\operatorname{St}_n,\lambda}=M_{\operatorname{St}_n,\lambda}/T$ is a manifold with corners. Unlike the case of tridiagonal matrices, the orbit space $Q_{\operatorname{St}_n,\lambda}$ for $n\geqslant 3$ is not a simple polytope. The topology of $Q_{\operatorname{St}_n,\lambda}$ itself is quite complicated, and it is difficult to state any general result about the manifold $M_{\operatorname{St}_n,\lambda}$ itself. However, the topology can be described in details for $n=4$, which was done in [@ABarrow].
![Particular graphs, encoding important isospectral matrix spaces: the path graph ${\mathbb{I}}_n$, indifferent graphs $\Gamma_h$, the star graph $\operatorname{St}_n$, and the cycle graph $\operatorname{Cy}_n$[]{data-label="figGraphs"}](graphs.pdf)
In this paper we consider the case $\Gamma=\operatorname{Cy}_n$, the cyclic graph on $n$ vertices. The Hermitian matrices corresponding to $\operatorname{Cy}_n$ have the form $$\label{eqCyclicMatrix}
L=L(\underline{a},\underline{b})=\begin{pmatrix}
a_1 & b_1& 0 & \cdots & \overline{b}_n\\
\overline{b}_1& a_2 & b_2 & 0 & \vdots\\
0 & \overline{b}_2 & a_3 & \ddots & 0\\
\vdots & 0 & \ddots &\ddots& b_{n-1}\\
b_n& \cdots & 0 & \overline{b}_{n-1} &a_n
\end{pmatrix},$$ where $a_i\in {\mathbb{R}}$, $b_i\in {\mathbb{C}}$. Such matrices are called *periodic tridiagonal matrices* or *periodic Jacobi matrices*. We will simply call them *periodic*. It is assumed throughout the paper that $n\geqslant 3$.
The space ${X_{n,\lambda}}=M_{\operatorname{Cy}_n,\lambda}$ of all periodic matrices with a simple spectrum $\lambda$ has dimension $2n$, and carries an effective action of $T=T^{n-1}$. Hence the torus action has complexity one. The difference between half the real dimension of a manifold and the dimension of a torus is called *the complexity of the action*: this terminology naturally comes from both algebraic geometry and symplectic geometry.
We prove that under certain conditions on a simple spectrum, the space ${X_{n,\lambda}}$ is not a smooth manifold, not even a homology manifold, see Theorem \[thmNonSmooth\]. This gives a negative answer to our question, posed in [@ABarrow]. This also settles certain inaccuracy appearing in the work of van Moerbeke [@VanM], who studied the real analogue of ${X_{n,\lambda}}$.
For any simple spectrum $\lambda$, we describe the topology of the orbit space ${X_{n,\lambda}}/T$, see Corollary \[corOrbitSpaceGeneral\]. If ${X_{n,\lambda}}$ is a topological manifold, we prove that the orbit space ${X_{n,\lambda}}/T$ is homeomorphic to the product $S^4\times T^{n-3}$. When $n=3$, the space $X_{3,\lambda}$ is the space of all Hermitian matrices with the given spectrum $\lambda$. This space is diffeomorphic to the full complex flag variety $\operatorname{Fl}_3$. Hence, for $n=3$, we recover the result of Buchstaber–Terzic [@BTober; @BT; @BT2], which states that $\operatorname{Fl}_3/T^2\cong S^4$. Note that the action is not free, however the orbit space is still a topological manifold. This fact is consistent with the general theory developed in [@AyLoc].
The main ingredient of our arguments is the product of off-diagonal elements $$B=\prod\nolimits_{i=1}^nb_i\in {\mathbb{C}}$$ of the periodic matrix $L(\underline{a},\underline{b})$. We show that with the matrix spectrum fixed, the number $B$ takes values inside a compact convex subset ${\mathbb{B}}\subset{\mathbb{C}}$, lying between two confocal parabolas, see Theorem \[thmImage\]. This statement may be considered a folklore: its real version was proved in [@VanM; @Krich], and the complex version is not more complicated. In Section \[secSpectralCurve\] we briefly review the necessary facts about discrete [Schrödinger ]{}operator, needed for this result.
The value $B$ is preserved by the torus action, hence there is a map ${\tilde{p}}\colon{X_{n,\lambda}}/T\to{\mathbb{C}}$ from the orbit space, evaluating the number $B$. The set ${\tilde{p}}^{-1}({\mathbb{C}}\setminus\{0\})$ consists of free orbits. However the torus action has nontrivial $T$-equivariant skeleton, which is a proper subset of ${\tilde{p}}^{-1}(0)$. To describe the structure of the equivariant skeleton, we use combinatorial geometry.
It is well known that euclidean space can be tiled by parallel copies of a regular permutohedron. Taking quotients by lattices in a euclidean space, we may produce many interesting permutohedral cell subdivisions of a torus. We show that a certain lattice produces a regular cell subdivision ${\mathcal{PT}}^{n-1}$ of an $(n-1)$-dimensional torus, which we called *the wonderful subdivision*. It has several interesting properties. First, it models the equivariant skeleton of the torus action on ${X_{n,\lambda}}$. Second, this wonderful subdivision minimizes the number of facets among all possible regular cell subdivisions of a torus. Such subdivisions and their dual simplicial cell subdivisions for general PL-manifolds are known in combinatorial topology under the name of crystallizations [@FGG]. We briefly recall the required combinatorial geometry in Section \[secTilings\].
Next we describe the topology of the whole space ${X_{n,\lambda}}$. Let ${X_{n,\lambda}}^0={\tilde{p}}^{-1}(0)$ denote the subset of matrices with $B=0$. The space ${X_{n,\lambda}}$ is smooth in vicinity of ${X_{n,\lambda}}^0$: this actually follows from the properties of non-periodic Toda lattice, see Proposition \[propSmoothOverZero\]. Using the result of [@AyLoc] concerning the topological classification of complexity one torus actions, we describe the topology of a small neighborhood ${X_{n,\lambda}^{\leqslant\varepsilon}}$ of ${X_{n,\lambda}}^0$. It happens that, up to homeomorphism, the $T^{n-1}$-action on ${X_{n,\lambda}^{\leqslant\varepsilon}}$ can be extended to a locally standard $T^n$-action on this space. The necessary notions related to complexity one torus actions are given in Section \[secNeighborhoodGeneral\].
In a series of works [@Ay1; @Ay2; @Ay3; @AMPZ] we developed a toolbox to compute cohomology and equivariant cohomology of manifolds with locally standard torus whose orbit spaces have acyclic proper faces. This toolbox is applied to the subspace ${X_{n,\lambda}^{\leqslant\varepsilon}}$. The $T^n$-orbit space of ${X_{n,\lambda}^{\leqslant\varepsilon}}$ is a manifold with corners, whose face structure is the wonderful cell subdivision of a torus, hence all its proper faces are acyclic, so we are in position to apply the general technique. The algebro-topological invariants of ${X_{n,\lambda}^{\leqslant\varepsilon}}$ are computed in terms of combinatorial invariants of the wonderful cell subdivision ${\mathcal{PT}}^{n-1}$. We recall the theory of $h$-, $h'$-, and $h''$-numbers of simplicial posets and compute these invariants for the dual simplicial poset of the wonderful subdivision in Section \[secNeighborhoodCombinatorics\].
In Section \[secEquivCohom\] we describe the additive structure of $T^{n-1}$-equivariant cohomology modules of the neighborhood ${X_{n,\lambda}^{\leqslant\varepsilon}}$. The ordinary Betti numbers of ${X_{n,\lambda}}$ are calculated in Section \[secBetti\]. There we also prove that ${X_{n,\lambda}}$ is not equivariantly formal for $n\geqslant 4$ by comparing equivariant and ordinary Betti numbers of ${X_{n,\lambda}}$.
Torus action and Toda flow {#secActionAndFlow}
==========================
The element $t=(t_1,\ldots,t_n)\in T^n$ acts on a cyclic matrix by the formula $$\label{eqTorusActionCoords}
tL(\underline{a};b_1,\ldots,b_{n-1},b_n)=L(\underline{a};t_1t_2^{-1}\cdot
b_1,\ldots,t_{n-1}t_n^{-1}\cdot b_{n-1},t_nt_1^{-1}\cdot b_n).$$ It is easy to see that the torus action preserves the quantity $B=\prod_1^nb_i$. The action is non-effective: the scalar matrices act trivially. Hence we consider the effective action of the quotient torus $T^{n-1}=T^n/\Delta(S^1)$ on ${X_{n,\lambda}}$.
Apart from the torus action, there is a classical dynamical system acting on the space of periodic matrices: the *periodic Toda lattice*. We now briefly recall the definition and properties of this dynamical system.
For a matrix $L=L(\underline{a},\underline{b})$ consider the skew-Hermitian matrix $$P=P(L)=\begin{pmatrix}
0 & b_1& & & -\overline{b}_n\\
-\overline{b}_1& 0 & b_2 & & \\
& -\overline{b}_2 & 0 & \ddots & \\
& & \ddots &\ddots& b_{n-1}\\
b_n& & & -\overline{b}_{n-1} &0
\end{pmatrix}$$ The Toda flow (the flow of the periodic Toda lattice) is the flow $$\label{eqTodaFlow}
\dot{L}=[L,P]=LP-PL.$$ The solution $L(t)$ to remains similar to the initial matrix $L(0)$ at all times $t\in{\mathbb{R}}$, so the Toda flow preserves the spectrum. Therefore the flow acts on the isospectral space ${X_{n,\lambda}}$.
The Toda flow commutes with the torus action. Indeed, the action of $T$ on $L$ is given by $DLD^{-1}$, for diagonal Hermitian matrix $D$. We have $P(DLD^{-1})=DP(L)D^{-1}$ and therefore $[DLD^{-1},P(DLD^{-1})]=D[L,P(L)]D^{-1}$.
The periodic Toda system is well studied for real symmetric matrices. We need a more general Hermitian version of periodic Toda system in order to incorporate torus actions. However, the complex case is not more complicated than the real one. The equations of the flow have the coordinate form: $$\label{eqTodaCoords}
\begin{cases}
\dot{a}_i=2(|b_{i-1}|^2-2|b_i|^2),\quad i=1,\ldots,n;\\
\dot{b}_i=b_i(a_i-a_{i+1}),\quad i=1,\ldots,n,
\end{cases}$$ where $a_i,b_i$ are assumed cyclically ordered. Since $b_i\in{\mathbb{C}}$, each expression in the second line represents two real equations. We see that the arguments of $b_i\in{\mathbb{C}}$ remain constant along the flow. The equations on $a_i,|b_i|$ have the form $$\label{eqTodaCoordsAbsValues}
\begin{cases}
\dot{a}_i=2(|b_{i-1}|^2-2|b_i|^2),\quad i=1,\ldots,n;\\
\frac{d}{dt}|b_i|=|b_i|(a_i-a_{i+1}),\quad i=1,\ldots,n.
\end{cases}$$ which coincide with the real form of the periodic Toda flow.
\[conDegenerationPoints\] It is a simple exercise that the quantity $B=\prod_1^n b_i$ is preserved along the flow. In what follows we consider the exceptional subspace $${X_{n,\lambda}}^{0}=\{L\in {X_{n,\lambda}}\mid B=0 \}$$ This subspace can be represented as the union ${X_{n,\lambda}}^{0}=\bigcup_1^nY_i$, where $Y_i\subset {X_{n,\lambda}}$ is the subset of matrices having $b_i=0$ for a particular index $i\in[n]$. The set $Y_n$ is just the set of isospectral tridiagonal Hermitian matrices, which is known to be a smooth manifold whose smooth type is independent of a simple spectrum $\lambda$ [@Tomei]. Moreover, it is known that $Y_n$ is a quasitoric manifold over a permutohedron [@BFR; @DJ] (the reader is advised to consult [@BPnew] concerning the terminology of quasitoric manifolds). Each of $Y_i$ for $i\neq n$ is diffeomorphic to $Y_n$. This follows from the fact that the matrix with $b_i=0$ can be transformed to tridiagonal Hermitian matrix by a cyclic permutation of rows and columns.
Therefore ${X_{n,\lambda}}^{0}$ is the union of $n$ submanifolds of dimension $2n-2$, however, these submanifolds intersect nontrivially. In the intersection of $Y_i$ and $Y_j$ there lies the submanifold of matrices with $b_i=b_j=0$, which is a torus invariant codimension 2 submanifold of both $Y_i$ and $Y_j$. The combinatorial structure of these intersections will be described in detail in Section \[secTilings\].
The Toda flow degenerates to a Toda flow of non-periodic Toda lattice on the exceptional set ${X_{n,\lambda}}^{0}$. Each submanifold $Y_i$ is preserved by the flow. The Toda flow on $Y_i$ is a gradient flow (see e.g. [@TomeiON] or [@ChShS]), which means that asymptotically each trajectory on $Y_i$ tends to an equilibrium point. The equilibrium points are the diagonal matrices $$L_\sigma=\operatorname{diag}(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n)}),
\quad \sigma\in S_n$$ A direct check shows that the subspace ${X_{n,\lambda}}$ is a smooth manifold in a neighborhood of each equilibrium point $L_\sigma$ [@Tomei]. The asymptotical properties of the flow on the exceptional set imply that ${X_{n,\lambda}}$ is a smooth manifold in a neighborhood of ${X_{n,\lambda}}^{0}$. It will be shown in Section \[secOrbitSpace\] that ${X_{n,\lambda}}$ is not always smooth in points with large values of $B$.
\[remSard\] For generic spectrum $\lambda$ the whole space ${X_{n,\lambda}}$ is a smooth manifold. This easily follows from Sard’s theorem applied to the map sending the periodic tridiagonal matrix $L$ to the tuple $(\operatorname{tr}L,\operatorname{tr}L^2,\ldots,\operatorname{tr}L^n)$
The orbit space of the torus action {#secOrbitSpace}
===================================
The action of $T=T^{n-1}$ on ${X_{n,\lambda}}$ has $n!$ fixed points $L_\sigma$, $\sigma\in S_n$ which coincide with the equilibria of the Toda flow.
\[propSmoothOverZero\] The orbit space ${Q_{n,\lambda}}={X_{n,\lambda}}/T$ is a topological manifold in a neighborhood of ${X_{n,\lambda}}^{0}/T$. The space ${Q_{n,\lambda}}$ is a topological manifold for generic $\lambda$.
Note that $\dim {X_{n,\lambda}}=2n$ and $\dim T=n-1$. Consider any fixed point $L_\sigma$. The tangent representation of the action at a point $L_\sigma$ has the weight decomposition $$T_{L_\sigma}{X_{n,\lambda}}=V(\alpha_{1,\sigma})\oplus\cdots\oplus
V(\alpha_{n,\sigma}),\qquad \alpha_{i,\sigma}\in \operatorname{Hom}(T^n,S^1)$$ where $V(\alpha)$ is the 1-dimensional complex representation $$tz=\alpha(t)\cdot z.$$ In terms of the noneffective action of $n$-dimensional torus $T^n$ we have $$\alpha_{i,\sigma}=\epsilon_i-\epsilon_{i+1},\quad\mbox{ for any
}\sigma\in S_n$$ where $\{\epsilon_1=\epsilon_{n+1},\epsilon_2,\ldots,\epsilon_n\}$ is the standard basis of $\operatorname{Hom}(T^n,S^1)\cong{\mathbb{Z}}^n$, as follows from the explicit expression for the action.
The following fact was proved in [@AyLoc]. Suppose a torus $T$ of dimension $n-1$ acts effectively on a smooth manifold $X$ of dimension $2n$, and assume that each connected component of each equivariant skeleton $X_j$ contains a fixed point. Assume, moreover, that the action has finitely many fixed points, and, at each fixed point, any $n-1$ of $n$ weights $\alpha_1,\ldots,\alpha_n\in{\mathbb{Z}}^{n-1}$ of the tangent representation are linearly independent. Then $X/T$ is a closed topological $(n+1)$-manifold. Applying this result to ${X_{n,\lambda}}$ in a neighborhood of ${X_{n,\lambda}}^0$, we get the first part of the proposition.
The second part follows easily from Remark \[remSard\], since the action of $T$ outside ${X_{n,\lambda}}^0$ is free. Therefore, whenever ${X_{n,\lambda}}$ is a smooth manifold, the orbit space ${X_{n,\lambda}}/T$ is smooth outside ${X_{n,\lambda}}^0/T$, thus it is a topological manifold.
To describe the topology of ${Q_{n,\lambda}}$ and ${X_{n,\lambda}}$, we formulate an result of an independent interest. Let $p\colon{X_{n,\lambda}}\to{\mathbb{C}}$ be the map which associates the number $B=\prod_{i=1}^n b_i$ to a periodic tridiagonal matrix $L(\underline{a},\underline{b})$. Since the $T$-action preserves $B$, there is an induced continuous map ${\tilde{p}}\colon{Q_{n,\lambda}}\to{\mathbb{C}}$.
The aim of the following constructions is to describe the image of ${\tilde{p}}$ and all its preimages. The description is given in Theorem \[thmImage\] below.
Let a simple spectrum $(\lambda_1<\ldots<\lambda_n)$ be given. Consider the characteristic polynomial $F(x)=\prod_{i=1}^n(x-\lambda_i)$. Since the polynomial has $n$ real roots, we have the sequence of real numbers $$\tilde{x}_1<\tilde{x}_2<\cdots<\tilde{x}_{n-2}<\tilde{x}_{n-1},$$ where $x_{n-1},x_{n-3},x_{n-5},\ldots$ are the local minima, and $x_{n-2},x_{n-4},\ldots$ are the local maxima of $F$. Let $$\label{eqMMdefin}
M=\min_{i\mbox{ is even}}F(x_{n-i}),\qquad m=\min_{i\mbox{ is
odd}} -F(x_{n-i}).$$ We obviously have $m,M>0$.
![The values $M$ and $-m$ on the plot of a characteristic polynomial[]{data-label="figCharPolyPlot"}](polyInterval.pdf)
\[remMandMmeaning\] The interval $[-m,M]$ represents the set of all $s\in{\mathbb{R}}$ such that the polynomial $F(x)-s$ has $n$ real roots, see Fig.\[figCharPolyPlot\].
Let $n_+$ be the number of local maxima at which $M$ is achieved and, similarly, $n_-$ be the number of local minima at which $-m$ is achieved. For generic $\lambda$ there holds $n_+=n_-=1$. Fig.\[figCharPolyPlot\] shows the case $n_+=1$, $n_-=2$.
\[thmImage\] The image of $p\colon{X_{n,\lambda}}\to{\mathbb{C}}$ is the set $$\label{eqImage}
{\mathbb{B}}=\left\{z\in{\mathbb{C}}\left| |z|\leqslant
\frac12\min\left(\frac{m}{1+\cos\operatorname{Arg}z},\frac{M}{1-\cos\operatorname{Arg}z}\right)\right.\right\}$$ The preimages of the map ${\tilde{p}}\colon {Q_{n,\lambda}}\to{\mathbb{B}}$ are as follows. If $z\in {\mathbb{B}}^{\circ}$, then ${\tilde{p}}^{-1}(z)$ is homeomorphic to a compact torus ${\mathcal{T}}^{n-1}$. If $z\in{\partial}{\mathbb{B}}$ and minimum in is achieved at $\frac{M}{1-\cos\operatorname{Arg}z}$, then ${\tilde{p}}^{-1}(z)$ is a torus of dimension $n-1-n_+$. If $z\in{\partial}{\mathbb{B}}$ and minimum in is achieved at $\frac{m}{1+\cos\operatorname{Arg}z}$, then ${\tilde{p}}^{-1}(z)$ is a torus of dimension $n-1-n_-$. If $z\in{\partial}{\mathbb{B}}$ and $\frac{m}{1+\cos\operatorname{Arg}z}=\frac{M}{1-\cos\operatorname{Arg}z}$, then ${\tilde{p}}^{-1}(z)$ is a torus of dimension $n-1-n_+-n_-$.
The convex set ${\mathbb{B}}$ is shown on Fig.\[figBset\]: it is bounded by arcs of two confocal parabolas. The set ${\mathbb{B}}$ is a 2-dimensional manifold with corners: we denote by $F_+$ and $F_-$ its left and right sides respectively, and $F_+\cap F_-=\{z_{top},z_{bot}\}$. Note that the minimum is achieved at $\frac{M}{1-\cos\operatorname{Arg}z}$ whenever $z$ lies on the left side of the figure, which explains the notation.
It will be convenient to distinguish between the torus, which acts on spaces and the geometrical tori arising in Theorem \[thmImage\]. Hence toric groups are denoted by the symbol $T$, and tori appearing in geometrical considerations are denoted by the symbol ${\mathcal{T}}$.
plot\[domain=-2\*sqrt(3):2\*sqrt(3),smooth,variable=\] ([-(9-)/6]{},)–plot\[domain=2\*sqrt(3):-2\*sqrt(3),smooth,variable=\] ([-(-16)/8]{},);
(-3,0) – (3,0); (0,-5) – (0,5); plot ([-3\*cos(r)/(1+cos(r))]{},[3\*sin(r)/(1+cos(r))]{});
plot ([-4\*cos(r)/(1-cos(r))]{},[4\*sin(r)/(1-cos(r))]{});
at (-2,1) [$F_+$]{}; at (2.5,1) [$F_-$]{};
(0.5,[2\*sqrt(3)]{}) circle(3pt) (0.5,[-2\*sqrt(3)]{}) circle(3pt);
at (0.5,0.5) [${\mathbb{B}}$]{}; at (0.7,4.3) [$z_{top}$]{}; at (0.7,-4.3) [$z_{bot}$]{};
\[corOrbitSpaceGeneral\] With parameters $n_+$ and $n_-$ as above, the orbit space ${Q_{n,\lambda}}$ is homeomorphic to $\Sigma({\mathcal{T}}^{n_-}\ast {\mathcal{T}}^{n_+})\times
{\mathcal{T}}^{n-1-n_--n_+}$.
The space ${Q_{n,\lambda}}$ is foliated over the contractible space ${\mathbb{B}}$ by tori. Hence $${Q_{n,\lambda}}\cong {\mathbb{B}}\times {\mathcal{T}}^{n-1}/\sim,$$ where certain $n_+$-dimensional subtorus ${\mathcal{T}}_+$ is collapsed over $F_+$ and another $n_-$-dimensional subtorus ${\mathcal{T}}_-$ is collapsed over $F_-$ (the nature of these tori and their independence is clarified in Section \[secSpectralCurve\]). We have ${\mathcal{T}}={\mathcal{T}}_+\times {\mathcal{T}}_-\times {\mathcal{T}}^{n-1-n_--n_+}$. The subgroup ${\mathcal{T}}^{n-1-n_--n_+}$ separates as a direct factor of ${Q_{n,\lambda}}$. The remaining factor is the suspension space, with the suspension points being the preimages of the points $z_{top}$ and $z_{bot}$. This suspension is taken over the space ${\tilde{p}}^{-1}({\mathbb{B}}\cap{\mathbb{R}})/K$ which is homeomorphic to the join of ${\mathcal{T}}_+$ and ${\mathcal{T}}_-$.
\[corOrbitSpace\] For generic spectrum $\lambda$ there is a homeomorphism ${Q_{n,\lambda}}\cong S^4\times
{\mathcal{T}}^{n-3}$.
In generic case we have $n_+=n_-=1$. Therefore $\Sigma({\mathcal{T}}^1\ast{\mathcal{T}}^1)\cong\Sigma S^3~\cong~S^4$.
\[corOrbitFlags\] Consider the effective action of $T=T^3/\Delta(T^1)$ on the manifold $\operatorname{Fl}_3$ of complete complex flags in ${\mathbb{C}}^3$. The orbit space $\operatorname{Fl}_3/T$ is homeomorphic to $S^4$.
Note that $X_{3,\lambda}$ is just the set of all Hermitian matrices with the given spectrum. This manifold is diffeomorphic to the flag manifold $\operatorname{Fl}_3$. The statement is the particular case of Corollary \[corOrbitSpace\] with $n=3$.
\[thmNonSmooth\] If $\lambda$ is a simple spectrum such that either $n_+>1$ or $n_->1$, then ${X_{n,\lambda}}$ is not a homology manifold. In particular, this space is not a smooth manifold.
Assume $n_+>1$. The space ${Q_{n,\lambda}}\cong \Sigma(T^{n_-}\ast
T^{n_+})\times T^{n-1-n_--n_+}$ is not a homology manifold unless $n_+=n_-=1$. Its singular points lie over the face $F_+\subset{\partial}{\mathbb{B}}$. Let $q\in{Q_{n,\lambda}}$ be a singular point such that ${\tilde{p}}(q)\in F_+$ and let $U_q\subset{Q_{n,\lambda}}$ be a neighborhood of $q$. We have $H_i(U_q,U_q\setminus\{q\};{\mathbb{Z}})\neq 0$ for some $i<n+1$. The torus action is free over ${\partial}{\mathbb{B}}$. Hence, for any point $x\in{X_{n,\lambda}}$ lying in the orbit $q$, its neighborhood $U_x\ni
x$ is homeomorphic to $U_q\times{\mathbb{R}}^{n-1}$. Therefore $H_i(U_x,U_x\setminus\{x\};{\mathbb{Z}})\neq 0$ for some $i<2n$, so far ${X_{n,\lambda}}$ is not a homology manifold.
Van Moerbeke [@VanM] proves the real analogue of Theorem \[thmImage\]. In the real case, there is a family of tori, parametrized by real numbers from the interval $[-M/4,0)\subset{\mathbb{B}}\cap{\mathbb{R}}$. The dimension of all tori is $n-1$, except for the torus over the endpoint $-M/4$: its dimension decreases by $n_+$. Van Moerbeke calls the union of such family “an open $n$-dimensional torus”. This naming seems misleading, since this union is not even a manifold for $n_+>1$, which is proved similarly to Theorem \[thmNonSmooth\].
\[remDegenerateCheb\] The most degenerate case appears in the situation when the values at all local minima of the characteristic polynomial $F(x)=\prod_{i=1}^{n}(x-\lambda_i)$ coincide, and the values at all local maxima coincide. For example, this holds for the Chebyshev polynomials $T_n(x)$ (which are defined on the interval $[-1,1]$ by $T_n(x)=\cos(n\arccos x)$). It can be seen, that a polynomial gives the maximal possible degeneration if and only if it coincides with Chebyshev polynomial up to affine transformation of the image and the domain: $$F(x)=\gamma T_n(\alpha x+\beta)+\delta,$$ with the natural requirement that $F(x)$ has $n$ distinct real roots.
[Schrödinger ]{}equation and the spectral curve {#secSpectralCurve}
===============================================
In this section we prove the first part of Theorem \[thmImage\]. It will be assumed that $B=\prod_1^nb_i\neq 0$, i.e. $L\notin
{X_{n,\lambda}}^0$. The action of $T$ is free on such matrices. We may identify ${Q_{n,\lambda}}={X_{n,\lambda}}/T$ with the set of isospectral Hermitian matrices of the form $$\label{eqRotatedMatrix}
L(w)=\begin{pmatrix}
a_1 & b_1& 0&\cdots& w^{-1}b_n\\
b_1& a_2 & b_2 & & \vdots\\
0 & b_2 & a_3 & \ddots &\\
\vdots&&\ddots& \ddots& b_{n-1}\\
wb_n& \cdots&& b_{n-1} &a_n
\end{pmatrix}$$ where $b_1,\ldots,b_n$ are positive real numbers, and $w\in{\mathbb{C}}$, $|w|=1$. Indeed, the arguments of any $n-1$ off-diagonal terms of a periodic tridiagonal matrix $L(\underline{a},\underline{b})$ can be rotated to zero by the torus action . We continue denoting $\prod_{i=1}^n b_i$ by the letter $B$, although in the new notation $B$ is a positive real number.
\[propImageDescription\] For a matrix $L(w)$ with a simple spectrum $\lambda_1<\cdots<\lambda_n$ there holds $$B \leqslant \frac12\min\left(\frac{M}{1-\cos\operatorname{Arg}w},\frac{m}{1+\cos\operatorname{Arg}w}\right)$$ where $M$ and $m$ are defined by .
Matrices of the form $L(w)$ can be studied using algebro-geometric method in mathematical physics (we refer to [@Krich] for a brief exposition of this subject in relation to periodic Toda flow). Let $\l$ be the space of infinite to both sides sequences $\{\psi_k\}$: $$\psi_k\in{\mathbb{C}},\quad k\in{\mathbb{Z}}$$ Consider the *periodic discrete [Schrödinger ]{}* operator given by $$H\colon \l\to \l,\qquad
H(\psi)_k=b_{k-1}\psi_{k-1}+a_k\psi_k+b_k\psi_{k+1}$$ where we assume $a_{k+n}=a_k$ and $b_{k+n}=b_k$. The eigenfunction $\psi\in\l$ of the [Schrödinger ]{}operator with eigenvalue $x$ satisfies the equation $$\label{eqSchEq}
H(\psi)=x\psi$$ Since $b_i\neq 0$, every eigenfunction is determined by its initial values $(\psi_0,\psi_1)\in{\mathbb{C}}^2$. We can define the *monodromy operator* along the period: $$\label{eqMonodromy}
M(x)\colon {\mathbb{C}}^2\to{\mathbb{C}}^2, \qquad M(x)\colon (\psi_0,\psi_1)\mapsto
(\psi_n,\psi_{n+1}).$$ Note that the matrix $L(w)$ has eigenvalue $x$ if and only if there exists a solution $\psi$ to such that $$\psi_{k+n}=w\psi_k$$ Such functions are called *Bloch solutions*. We see that whenever there exists a nonzero Bloch solution with parameter $w$, the number $w$ is the eigenvalue of the monodromy operator $M(x)$, so we get a relation $$\label{eqSpecCurve}
\det(w-M(x))=0.$$ This equation defines a so called *spectral curve* of the periodic [Schrödinger ]{}equation in the space of parameters $(w,x)\in{\mathbb{C}}^2$. One can show that $\det M(x)=1$ (hint: the operator $M_i\colon
(\psi_{i-1},\psi_i)\mapsto (\psi_i,\psi_{i+1})$ has determinant $\frac{b_{i-1}}{b_i}$, therefore $M=M_nM_{n-1}\cdots
M_1=\frac{b_{n-1}}{b_n}\frac{b_{n-2}}{b_{n-1}}\cdots\frac{b_0}{b_1}=1$). Hence, the equation of the spectral curve can be rewritten in the form $$\label{eqSpecCurve2}
w^2-\operatorname{tr}M(x)w+1=0.$$ It can be shown that $\operatorname{tr}M(x)=\frac{1}{B}P(x)$, where $B=\prod_1^n b_i$ as before, and $P(x)$ is a monic polynomial in $x$ (Hint: decompose $M(x)$ as the product of operators $M_i$ along the period, $i=1,\ldots,n$, and count the terms of highest degree of $x$). Dividing by $w$ and denoting $t=\operatorname{Re}w=\frac12(w+w^{-1})$, we get $2t=\frac{1}{B}P(x)$.
The polynomial $P(x)-2Bt$ is monic and has the given sequence $\lambda_1,\ldots,\lambda_n$ as its roots, therefore $$P(x)-2Bt=\prod\nolimits_{i=1}^n(x-\lambda_i)=F(x),$$ $$\label{eqRelOnPolynomials}
P(x)=F(x)+2Bt$$ Consider the set $${\mathcal{A}}=\{s\in{\mathbb{R}}\mid P(x)=2Bs\mbox{ has }n \mbox{ real roots}\}.$$ Recalling the definition of $m$ and $M$ and remark \[remMandMmeaning\] as well as relation , we see that ${\mathcal{A}}$ is the closed interval $[-\frac{m}{2B}+t,\frac{M}{2B}+t]$.
Note that the polynomial $P(x)-2Bs$ is the characteristic polynomial of the matrix $L(w_s)$, where $|w_s|=1$, $\operatorname{Re}w_s=s$. Therefore, the equation $P(x)=2Bs$ necessarily has $n$ real roots for any $s\in[-1,1]$. Therefore, $$[-1;1]\subseteq \left[-\frac{m}{2B}+t,\frac{M}{2B}+t\right],$$ from which we deduce $B\leqslant \frac12\min(\frac{M}{1-t},
\frac{m}{1+t})$. Remembering $t=\operatorname{Re}w=\cos\operatorname{Arg}w$, we get the required inequality.
\[propPreimageOfNonzero\] For $z\in{\mathbb{B}}$, $z\neq 0$, the preimage ${\tilde{p}}^{-1}(z)$ is homeomorphic to a torus. The dimension of a torus is $n-1$ if $z$ lies in the interior of ${\mathbb{B}}$, $n-1-n_+$ if $z$ lies in the relative interior of $F_+$, $n-1-n_-$ if $z$ lies in the relative interior of $F_-$, and $n-1-n_+-n_-$ if $z$ is either $z_{top}$ or $z_{bot}$.
In short, this follows from the fact that the periodic Toda lattice is an integrable dynamical system and its energy levels are the compact submanifolds. Liouville–Arnold theorem then implies that these preimages ${\tilde{p}}^{-1}(z)$ are tori. To specify the dimensions we give more details on the theory, related to periodic tridiagonal matrices.
As before, consider $P(x)=B\operatorname{tr}M(x)$, the monic polynomial in $x$ with coefficients depending on $a_i,b_i\in{\mathbb{R}}$. As follows from the considerations above, the eigenvalues of matrices $L(1)$ and $L(-1)$ are the roots of the polynomials $P(x)-2B$ and $P(x)+2B$ respectively. Let $$x_1<x_2\leqslant x_3<\cdots<x_{2n-2}\leqslant x_{2n-1}<x_{2n}$$ be the union of all these roots, so that $x_{2n},x_{2n-3},x_{2n-4},x_{2n-7},x_{2n-8},\ldots$ are the roots of $P(x)-2B$ and $x_{2n-1},x_{2n-2},x_{2n-5},x_{2n-6},\ldots$ are the roots of $P(x)+2B$. The intervals $$I_1=[x_{2},x_{3}],\quad I_2=[x_{4},x_{5}],\quad\ldots\quad,
I_{n-1}=[x_{2n-2},x_{2n-1}]$$ are called the *forbidden zones*. We will call $I_{n-1},I_{n-3},\ldots$ lower forbidden zones, and $I_{n-2},I_{n-4},\ldots$ upper forbidden zones as motivated by Fig.\[figForbiddenZones\].
![Upper and lower forbidden zones[]{data-label="figForbiddenZones"}](polyUpperLower.pdf)
Consider the Riemannian surface $\Theta_g$ of the multivalued function $$g(x)=\sqrt{\prod\nolimits_{i=1}^{2n}(x-x_i)}$$ Over each forbidden interval $I_k$, $k=1,\ldots,n-1$, there lies a circle $S_k$ on $\Theta_g$. If an interval $I_k$ degenerates to a point (i.e. $x_{2k}=x_{2k+1}$), the circle $S_k$ also collapses to a point. Van Moerbeke [@VanM] proved
\[propMoerbeke\] Real periodic tridiagonal symmetric matrices $L(\underline{a},\underline{b})$ with the given spectrum $\lambda$, given $B=\prod_1^nb_i$, and $b_i>0$, are in one-to-one correspondence with $(n-1)$-tuples $(\mu_1,\ldots,\mu_{n-1})$, where $\mu_k\in S_k$.
Therefore, for real $z$, the preimage ${\tilde{p}}^{-1}(z)$ is diffeomorphic to a torus ${\mathcal{T}}=\prod_1^{n-1} S_i$. The dimension of this torus equals $n-1$ in general, however, when some forbidden intervals are collapsed, the dimension reduces by the number of collapsed intervals. The upper forbidden intervals collapse if and only if the value $2B$ reaches $M$. The number of collapses among upper intervals equals $n_+$. Similarly, the lower intervals collapse if $2B$ reaches $m$, and the number of collapses among lower intervals is $n_-$.
Now let $L(w)$ be an arbitrary matrix with $w\in{\mathbb{C}}$, $|w|=1$ and the given spectrum $\lambda$. Let $\widetilde{\lambda}$ be the set of roots of the polynomial $\prod(x-\lambda_i)+2B\operatorname{Re}w$. It was mentioned in the proof of Proposition \[propImageDescription\] that the matrix $L(w)$ has spectrum $\lambda$ if and only if $L(1)$ has spectrum $\widetilde{\lambda}$. Thus Proposition \[propMoerbeke\] implies the required statement for all matrices.
Permutohedral tilings {#secTilings}
=====================
In this section we study the degenerate locus of the periodic Toda lattice. Recall that ${X_{n,\lambda}}^{0}=p^{-1}(0)\subset {X_{n,\lambda}}$ is the set of all isospectral matrices with $B=\prod_1^nb_i=0$, and ${Q_{n,\lambda}}^{0}={X_{n,\lambda}}^{0}/T={\tilde{p}}^{-1}(0)$.
We recall some standard facts from combinatorial geometry. Let $\epsilon_1,\ldots,\epsilon_n$ be the standard basis of ${\mathbb{Z}}^n\cong\operatorname{Hom}(T^n,S^1)$. We assume that ${\mathbb{Z}}^n\subset {\mathbb{R}}^n$ and there is a fixed inner product on ${\mathbb{R}}^n$ such that $\epsilon_1,\ldots,\epsilon_n$ are orthonormal.
Consider the sublattice $N\subset {\mathbb{Z}}^n$ of rank $n-1$ given by $$N=\operatorname{Hom}(T^n/\Delta(T^1),S^1)=\left\{\left.\sum\nolimits_{i=1}^na_i\epsilon_i\right|
a_i\in{\mathbb{Z}}, \sum a_i=0,\mbox{ and } a_i-a_j\equiv 0\mod n\right\}$$ and let $N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}{\mathbb{R}}$ be its real span. Consider the vectors $\alpha_1,\ldots,\alpha_n$: $$\alpha_i=(n-1)\epsilon_i-\sum_{j\neq i}\epsilon_j,\qquad i=1,\ldots,n-1.$$ We see that $$\label{eqAlphaNotation}
\sum_{i=1}^n\alpha_i=0,$$ and any $n-1$ of $\alpha_1,\ldots,\alpha_n$ generate the lattice $N$. One can think about $\alpha_i$’s as the outward unit normal vectors to the facets of a regular simplex in ${\mathbb{R}}^{n-1}$.
For any subset $S\subset[n]=\{1,\ldots,n\}$ such that $S\neq\varnothing, [n]$, consider the vector $$\label{eqAlphaSconvention}
\alpha_S=\sum\nolimits_{i\in S}\alpha_i.$$ Let ${\mathcal{P}}_{n-1}$ be the Voronoi cell decomposition of $N_{\mathbb{R}}\cong {\mathbb{R}}^{n-1}$ generated by the lattice $N$. In other words, for any $\alpha\in
N$ we consider the Voronoi cell $$P_\alpha=\{x\in N_{\mathbb{R}}\mid \operatorname{dist}(x,\alpha)\leqslant \operatorname{dist}(x,\beta)
\mbox{ for any }\beta\in N,\beta\neq\alpha\},$$ where $\operatorname{dist}$ is the distance determined by the inner product on $N_{\mathbb{R}}\subset {\mathbb{R}}^n$. Each $P_\alpha$ is a convex $(n-1)$-dimensional polytope and all these polytopes are the parallel copies of each other, $P_\alpha=P_0+\alpha$.
It can be shown that $P_0$ is the $(n-1)$-dimensional permutohedron $\operatorname{Pe}^{n-1}$ determined by the inequalities $$\operatorname{Pe}^{n-1}=P_0=\left\{x\in N_{\mathbb{R}}\mid \langle \alpha_S,x\rangle\leqslant
\frac12\langle\alpha_S,\alpha_S\rangle, S\in 2^{[n]},
S\neq\varnothing,[n]\right\}$$ We recall the basic facts about the combinatorics of a permutohedron. The polytope $\operatorname{Pe}^{n-1}$ is simple, which means that every codimension $k$ face is contained in exactly $k$ facets. For a proper subset $S\subset[n]$ let $F_S$ denote the facet of $\operatorname{Pe}^{n-1}$ determined by the support hyperplane $\langle\alpha_S,x\rangle=
\frac12\langle\alpha_S,\alpha_S\rangle$. Note that $\operatorname{Pe}^{n-1}$ is centrally symmetric: the facets $F_S$ and $F_{\bar{S}}$ are opposite to each other whenever $\bar{S}=[n]\setminus S$.
Facets $F_{S_1},\ldots,F_{S_k}$ have nonempty intersection in $\operatorname{Pe}^{n-1}$ if and only if the subsets $\{S_1,\ldots,S_k\}$ form a chain in the Boolean lattice $2^{[n]}$. If $\sigma=(S_1\subset\cdots\subset S_k)$ is such a chain, we denote by $F_\sigma$ the face $F_{S_1}\cap\cdots\cap F_{S_k}$ of the permutohedron. Each face of $\operatorname{Pe}^{n-1}$ is known to be a product of permutohedra of smaller dimensions.
We denote by $F_S(P_\alpha)$ (resp. $F_\sigma(P_\alpha)$) the corresponding facets (resp. faces) of the the Voronoi cell $P_\alpha$ to distinguish different copies of a permutohedron in the Voronoi diagram. It can be seen that $$\label{eqNeighbors}
F_S(P_\alpha)=F_{\bar{S}}(P_{\alpha+\alpha_S}).$$ A facet of each cell is adjoint to an opposite facet of a neighboring cell.
We formulate a general construction to precede a particular case needed in the proof of Theorem \[thmImage\].
\[conGeneralVoronoiQuotient\] Let ${{\widehat{N}}}\subseteq N$ be a sublattice of finite index, i.e. $q=|N/{{\widehat{N}}}|<\infty$. Consider the quotient $N_{\mathbb{R}}/{{\widehat{N}}}$. Since ${{\widehat{N}}}$ is a cocompact lattice, this quotient is a torus ${\mathcal{T}}^{n-1}$. The action of ${{\widehat{N}}}$ by parallel shifts preserves the Voronoi diagram, therefore we have a cell subdivision of the torus ${\mathcal{T}}^{n-1}\cong N_{\mathbb{R}}/{{\widehat{N}}}$. There are $q$ maximal cells in this subdivision, each is a parallel copy of a permutohedron.
A natural example is ${{\widehat{N}}}=N$. In this case the torus is given by identifying the opposite facets of a single permutohedron. The cell structure on a torus given by this identification is known: the corresponding partially ordered set was introduced and studied by Panina [@Panina] under the name of *cyclopermutohedron*. This poset has a natural combinatorial description.
For the considerations of this paper we need another sublattice.
\[conWonderfulSublattice\] Let $N'\subset N$ be the sublattice generated by the vectors $$\beta_k=\alpha_k-\alpha_{k+1},\qquad k=1,\ldots,n-1.$$ Note that $\beta_{n-1}=\alpha_{n-1}-\alpha_n=2\alpha_{n-1}+\alpha_1+\cdots+\alpha_{n-2}$ according to . It can be shown that $N/N'$ is the cyclic group of order $n$. Indeed, in the quotient group $N/N'$ we have the identities $$\label{eqModLattice}
[\alpha_1]=\cdots=[\alpha_n],\qquad
n[\alpha_1]=[\alpha_1]+\cdots+[\alpha_{n-2}]+2[\alpha_{n-1}]=0.$$
\[definWonderfulDecomposition\] Let ${\mathcal{PT}}^{n-1}$ be the cell decomposition of a torus ${\mathcal{T}}^{n-1}$ obtained as a quotient of Voronoi diagram of the space $N_{\mathbb{R}}$ by the sublattice $N'$. We call ${\mathcal{PT}}^{n-1}$ the *wonderful cell decomposition* of a torus.
The wonderful decomposition ${\mathcal{PT}}^{n-1}$ has $n$ maximal cells. The cells $P_\alpha$ and $P_{\alpha+\beta}$ are identified in ${\mathcal{PT}}^{n-1}$ whenever $\beta\in N'$. We denote the resulting cell of ${\mathcal{PT}}^{n-1}$ by $P_{[\alpha]}$. Relations imply $$[\alpha_S]=|S|[\alpha_1],\qquad [n\alpha_1]=[0].$$
\[lemGluingRulesForZ\] Let $1\leqslant k<m\leqslant n$. In the cell complex ${\mathcal{PT}}^{n-1}$ we have $$F_S(P_{k[\alpha_1]})=F_{\bar{S}}(P_{m[\alpha_1]}),$$ where $S$ is any subset of $[n]$ of cardinality $m-k$.
Choose any subset $S'$ such that $|S'|=k$ and $S'$ is disjoint from $S$. According to we have $$F_S(P_{k[\alpha_1]})=F_S(P_{[\alpha_{S'}]})=F_{\bar{S}}(P_{[\alpha_{S'}+\alpha_S]})=
F_{\bar{S}}(P_{[\alpha_{S'\sqcup
S}]})=F_{\bar{S}}(P_{m[\alpha_1]}).$$ which proves the statement.
In the following, we denote the maximal cells $P_{k[\alpha_1]}$ by ${\mathcal{PT}}_k$. Now we return to the space of tridiagonal matrices. Recall that $Y_k$ denotes the space of all isospectral matrices $L(\underline{a},\underline{b})$ with $b_k=0$, for $k=1,\ldots,n$. Let $Q_k$ denote the orbit space $Y_k/T$. We have ${Q_{n,\lambda}}^{0}=\bigcup_1^nQ_k$. For convenience introduce the cyclic notation: $Q_k=Q_{k+n}$, for any $k\in {\mathbb{Z}}$.
\[thmTorusForZero\] The space ${Q_{n,\lambda}}^{0}$ can be identified with ${\mathcal{PT}}^{n-1}$ so that the subspaces $Q_k$ are identified with ${\mathcal{PT}}_k$.
The orbit space $Q_0=Q_n$ is identified with the space of all tridiagonal symmetric real matrices $$L=\begin{pmatrix}
a_1 & b_1& 0&\cdots& 0\\
b_1& a_2 & b_2 & & 0\\
0 & b_2 & a_3 & \ddots & \vdots \\
\vdots&& \ddots& \ddots &b_{n-1}\\
0& 0&\cdots& b_{n-1} &a_n
\end{pmatrix}$$ with $b_i\geqslant 0$ and the given simple spectrum $\lambda$. It is known (see [@Tomei]) that $Q_0$ is diffeomorphic to a permutohedron $\operatorname{Pe}^{n-1}$ as a manifold with corners. The facet $F_S(\operatorname{Pe}^{n-1})$ corresponds to the subset of $Q_0$, which consists of matrices $L$ such that $b_{|S|}=0$ and the eigenvalues $\{\lambda_i\mid i\in S\}$ are distributed in the first $(|S|\times
|S|)$-block.
Similar considerations are valid for other spaces $Q_k$: this can be shown by cyclic permutation of rows and columns of $L$. Indeed, the set $Q_k$ can be identified with $\operatorname{Pe}^{n-1}$ in such way that the facet $F_S(\operatorname{Pe}^{n-1})$ consists of all matrices with the property $$b_k=0,\qquad b_{k+|S|}=0,$$ and the block between $k$-th and $(k+|S|)$ rows and columns has eigenvalues $\{\lambda_i\mid i\in S\}$.
It can be seen that the faces $F_S(Q_k)$ and $F_{\bar{S}}(Q_{m})$ represent the same set of matrices for $1\leqslant k<m\leqslant n$ and $|S|=m-k$. Therefore, $F_S(Q_k)=F_{\bar{S}}(Q_{m})$ in ${Q_{n,\lambda}}^0$. These gluing rules for the cells in ${Q_{n,\lambda}}^0$ coincide with the gluing rules for ${\mathcal{PT}}_k$ in ${\mathcal{PT}}^{n-1}$ according to Lemma \[lemGluingRulesForZ\].
Right part of Fig.\[figHexes\] shows the space $Q_{3,\lambda}^{0}={\mathcal{PT}}^2$. This example was described in details by van Moerbeke [@VanM]. The 1-skeleton of ${\mathcal{PT}}^2$ is shown on the left. As an abstract graph, it is isomorphic to the complete bipartite graph $K_{3,3}$. This graph is a GKM-graph of the complete flag variety $\operatorname{Fl}_3$, see details in [@AyLoc].
![The wonderful cell decomposition $Q_{3,\lambda}^{0}={\mathcal{PT}}^2$.[]{data-label="figHexes"}](FlagGraph.pdf)
Let us briefly sketch the phase portrait of the Toda flow on the degenerate set of orbits ${Q_{n,\lambda}}^0$. Let $v\in N_{\mathbb{R}}^*$ be a generic linear function on $N_{\mathbb{R}}$. Take any face of any permutohedron of the Voronoi diagram in $N_{\mathbb{R}}$. On each such polytope consider a flow, which moves all points in the interior of $P$ to the vertex maximizing the linear function $v$. The flow looks the same on all Voronoi cells, thus we have an induced flow on the torus ${\mathcal{PT}}^{n-1}=N_{\mathbb{R}}/N'$.
This picture describes the Toda flow on ${Q_{n,\lambda}}^0\cong {\mathcal{PT}}^{n-1}$. Indeed, Toda flow degenerates to the flow of a non-periodic Toda lattice on each permutohedron $Q_i$, and its Morse-like behavior is well-known (see [@DNT]). For any block tridiagonal matrix, the Toda flow “sorts” the diagonal elements within each block [@Tomei].
The phase portrait for $n=3$ is shown on the left part of Fig.\[figWonderfulFlow\]. The oddity of the phase portrait near equilibria points is explained by the fact that the orbit space ${Q_{n,\lambda}}$ is not smooth at these points.
Note that for $B\neq 0$, the Toda flow exhibits Liouville–Arnold behavior. The equilibria points disappear, however the flow still follows some direction $v$ on a torus, see Fig.\[figWonderfulFlow\], right part.
![The Toda flow on the level set $B=\operatorname{const}$ of the orbit space $Q_{3,\lambda}^0$. Left part shows the case $B=0$. Right part shows the case $B=\operatorname{const}\neq 0$[]{data-label="figWonderfulFlow"}](wonderfulFlow2.pdf)
Each $k$-dimensional cell of the cell subdivision ${\mathcal{PT}}^{n-1}$ lies in exactly $n-k$ different maximal cells. This means there exists a dual simplicial cell subdivision ${K_{\mathcal{PT}}}^{n-1}$. In Section \[secNeighborhoodCombinatorics\] we recall the definition of a simplicial poset which is a useful combinatorial notion to study simplicial cell subdivisions. Note that the simplicial poset ${K_{\mathcal{PT}}}^{n-1}$ minimizes the number of vertices among all simplicial cell subdivisions of the torus ${\mathcal{T}}^{n-1}$. Indeed, any $(n-1)$-dimensional simplex of such subdivision has $n$ distinct vertices, therefore a simplicial cell subdivision of ${\mathcal{T}}^{n-1}$ should have at least $n$ vertices. This number is achieved at ${K_{\mathcal{PT}}}^{n-1}$.
Note that any closed connected $(n-1)$-manifold admits a simplicial cell subdivision with exactly $n$ vertices. This result was proved in [@FGG], where such subdivisions (or, their equivalent combinatorial representations) were called *the crystallizations*. Previous remark shows that ${K_{\mathcal{PT}}}^{n-1}$ provides an explicit crystallization for the torus ${\mathcal{T}}^{n-1}$.
Propositions \[propImageDescription\], \[propPreimageOfNonzero\], and Theorem \[thmTorusForZero\] conclude the proof of Theorem \[thmImage\].
Topology near degeneration locus {#secNeighborhoodGeneral}
================================
In this section we study the topology of a small neighborhood of ${X_{n,\lambda}}^0$. The space ${X_{n,\lambda}}$ is a smooth manifold in vicinity of ${X_{n,\lambda}}^0$, see Construction \[conDegenerationPoints\].
\[remFreeOutside\] Note that the $T$-action is free outside ${X_{n,\lambda}}^0$ and admits a section given by the formula . However, the free part of the action is larger than ${X_{n,\lambda}}^0$: the action is also free over the interiors of facets of ${Q_{n,\lambda}}^0\cong {\mathcal{PT}}^{n-1}$. The whole free action ${X_{n,\lambda}}^{\operatorname{free}}\to{X_{n,\lambda}}^{\operatorname{free}}/T$ does not admit a section, as explained below.
Recall that $p\colon {X_{n,\lambda}}\to {\mathbb{C}}$ maps a matrix $L(\underline{a},\underline{b})$ to the product $B=\prod_1^nb_i$. For a small $\varepsilon$ consider the preimage of points close to zero: $${X_{n,\lambda}^{\leqslant\varepsilon}}=p^{-1}(\{z\in{\mathbb{C}}\mid |z|\leqslant \varepsilon\}).$$ According to Proposition \[propPreimageOfNonzero\], ${X_{n,\lambda}^{\leqslant\varepsilon}}$ is a manifold with boundary, the boundary ${\partial}{X_{n,\lambda}^{\leqslant\varepsilon}}$ being the subset $$\label{eqEpsTorus}
{X_{n,\lambda}}^{={\varepsilon}}=p^{-1}(\{|z|=\varepsilon\})\cong S_{{\varepsilon}}^1\times
{\mathcal{T}}^{n-1}\times T^{n-1}\cong T^{2n-1},$$ where $S_{{\varepsilon}}^1=\{z\mid |z|={\varepsilon}\}$, ${\mathcal{T}}^{n-1}$ is the Liouville–Arnold torus, and $T^{n-1}$ it the acting torus.
It will be useful to incorporate the circle $S_{{\varepsilon}}^1$ into the action to obtain a $T^n$-action on ${X_{n,\lambda}^{\leqslant\varepsilon}}$.
\[conModelOfNeighb\] Consider a topological manifold with boundary $W={\mathcal{T}}^{n-1}\times[0,1]$. Its boundary consists of two connected components $${\partial}W={\partial}_0W\sqcup{\partial}_1W,\qquad
{\partial}_0W={\mathcal{T}}^{n-1}\times\{0\},\quad{\partial}_1 W=T^{n-1}\times\{1\}.$$ On the left component ${\partial}_0W$, we introduce the wonderful cell structure ${\mathcal{PT}}^{n-1}$, constructed in Section \[secTilings\]. This procedure subdivides ${\partial}_0W$ into $n$ permutohedra ${\mathcal{PT}}_1,\ldots,{\mathcal{PT}}_n$ of dimension $n-1$ so that every cell of dimension $k$ lies in $n-k$ top-dimensional cells. This makes $W$ a manifold with corners (understood in a broad topological sense). We leave the right boundary component ${\partial}_1 W$ unchanged: no face structure is imposed on ${\partial}_1 W$.
Let $T^n=\{t=(t_1,\ldots,t_n)\mid |t_i|=1\}$ be a compact $n$-torus and $T_I$, $I\subseteq[n]$ be its coordinate subtorus, $$T_I=\{t\in T^n\mid t_j=1, j\notin I\}.$$ Consider the space $$Y=W\times T^n/\sim$$ where $(r,t)$ and $(r',t')$ are identified whenever $r=r'$ lies in the intersection of facets $\{{\mathcal{PT}}_i\mid i\in I\}$ and $t^{-1}t'\in T_I$ for some subset $I\subseteq[n]$. This construction can be considered as particular case of either moment-angle manifold construction for simplicial posets (see [@PL; @BPnew]) or the construction of locally standard actions (see [@Yo]). The space $Y$ is a particular case of the collar models introduced in [@Ay3].
The space $Y$ is a manifold with boundary ${\partial}Y={\partial}_1 W\times
T^n\cong {\mathcal{T}}^{n-1}\times T^n$. It carries the action of $T^n$ which is free on the boundary and its orbit space is $W$.
Consider the induced action of the subtorus $$T^{n-1}=\{t_1t_2\cdots t_n=1\}\subset T^n$$ on the space $Y$. It can be checked (see details in [@AyLoc]) that the orbit space $Y/T^{n-1}$ is homeomorphic to ${\mathcal{T}}^{n-1}\times D^2$.
\[thmTopologyNeighb\] The space ${X_{n,\lambda}^{\leqslant\varepsilon}}$ is $T^{n-1}$-equivariantly homeomorphic to the collar model $Y$.
In [@AyLoc] we developed a topological theory of complexity one torus actions. The main concepts are recalled here. By definition, an effective action of $T\cong T^{n-1}$ on $X=X^{2n}$ is called a *strictly appropriate action in general position*, if the following conditions hold.
1. The action has finitely many fixed points.
2. Stabilizers of all points are connected.
3. Each connected component of each equivariant skeleton $X_j$ contains a fixed point.
4. For every fixed point $x$, the weights $\alpha_1,\ldots,\alpha_n\in \operatorname{Hom}(T^{n-1},S^1)\cong {\mathbb{Z}}^{n-1}$ of the tangent representation are in general position, which means that every $n-1$ of them are linearly independent.
For such actions we proved that the orbit space $Q=X/T^{n-1}$ is a topological manifold of dimension $n+1$. The orbits of dimensions less than $n-1$ form a subset $Z\subset Q$ which is called a *sponge*. A sponge is an $(n-2)$-dimensional subset of $Q$ locally modeled by a $(n-2)$-skeleton of ${\mathbb{R}_{\geqslant 0}}^n$. The free part of action gives the principal $T^{n-1}$-bundle $$X^{\operatorname{free}}\to Q\setminus Z.$$ This bundle is classified by the cohomology class $e\in
H^2(Q\setminus Z;H_1(T^{n-1}))$, which is called the *Euler class* of the action. Proposition 3.7 of [@AyLoc] asserts that equivariant topological type of $X$ is uniquely determined by the triple $(Q,Z,e)$ (which essentially means that the information on stabilizers of the action can be recovered from the class $e$).
The inclusion $i_x\colon U_x\to Q$ induces a homomorphism $$i_x^*\colon H^2(Q,Q\setminus Z;H_1(T^{n-1}))\to
H^2(U_x,U_x\setminus Z;H_1(T^{n-1})).$$ The class $e_x=i_x^*(e)\in H^2(U_x,U_x\setminus Z;H_1(T^{n-1}))$ is called the local Euler class at $x$. It was noted in [@AyLoc] that local Euler classes are always nonzero. In particular, the global Euler class is always non-zero for suitable actions of complexity one.
These constructions work similarly if $X$ is a manifold with boundary, and the torus action is free on the boundary. In this case, $Q=X/T^{n-1}$ is a manifold with boundary ${\partial}X/T^{n-1}$. The sponge of the action lies in the interior of $Q$. Under certain conditions the local Euler classes at fixed points determine the space $X$ uniquely.
Assume $Q$ has the form $Q_M=M\times D^2$, where $M$ is a closed $(n-1)$-manifold with a fixed simple cell decomposition. Assume that the sponge $Z_M$ is the $(n-2)$-skeleton of this cell structure, and we have $$Z_M=M^{(n-2)}=M^{(n-2)}\times\{0\}\subset M\times D^2=Q_M.$$
\[propClassificationEarlier\] Let $X$ be a manifold with boundary, which carries a strictly appropriate torus action in general position such that the orbit space and the sponge of the action are given by $(Q_M,Z_M)$. Assume that the free action of $T$ on the boundary is a trivial principal bundle. Then the local Euler classes at fixed points uniquely determine the $T^{n-1}$-equivariant homeomorphism type of $X$.
Apply this proposition to spaces ${X_{n,\lambda}^{\leqslant\varepsilon}}$ and $Y$. The orbit space is ${\mathcal{T}}^{n-1}\times D^2$ in both cases. The sponge of the action is the $(n-2)$-skeleton of the wonderful cell subdivision ${\mathcal{PT}}^{n-1}$, defined earlier. The free action on the boundary is a trivial principal bundle. This is true for ${X_{n,\lambda}^{\leqslant\varepsilon}}$ since there is a section of the action, see remark \[remFreeOutside\]. This is true for $Y$ since $Y=P\times T^n/\sim$, and the $T^n$-action over ${\partial}_1P$ is a trivial principal bundle.
Finally, consider any fixed point $x=L_\sigma$ of ${X_{n,\lambda}^{\leqslant\varepsilon}}$. The tangent representation at $x$ is isomorphic to the standard action of $$T^{n-1}=\{t_1\cdots t_n=1\}\subset T^n=\{(t_1,\ldots,t_n)\}$$ on ${\mathbb{C}}^n$ (the infinitesimal action just rotates off-diagonal entries, so that the angles of rotation sum to zero). However the action of $T^{n-1}$ in the neighborhood of the corresponding fixed point of $Y$ is exactly the same by the definition of $Y$. Therefore the local Euler classes of ${X_{n,\lambda}^{\leqslant\varepsilon}}$ and $Y$ coincide at each fixed point.
Proposition \[propClassificationEarlier\] then implies the existence of $T^{n-1}$-homeomorphism ${X_{n,\lambda}^{\leqslant\varepsilon}}\cong Y$.
Enumerative combinatorics of the wonderful subdivision {#secNeighborhoodCombinatorics}
======================================================
In this section we study the enumerative invariants of the permutoheral cell complex ${\mathcal{PT}}^{n-1}$ or, equivalently, its dual simplicial poset ${K_{\mathcal{PT}}}^{n-1}$. These invariants will be used further to describe the homological structure of ${X_{n,\lambda}}$. At first, we recall several standard definitions from commutative algebra and combinatorics.
\[definSimpPoset\] A finite partially ordered set $S$ is called *simplicial* if it has the minimal element ${\hat{0}}\in S$ and, for any $I\in
S$, the order interval $\{J\in S\mid J\leqslant I\}$ is isomorphic to the poset of faces of a $k$-dimensional simplex, for some number $k\geqslant 0$.
The elements of $S$ are called *simplices*. The number $k$ from the definition is called the dimension of a simplex $I$. A simplex of dimension $0$ is called a *vertex*. The geometrical realization of $S$ is the simplicial cell complex, obtained by gluing geometrical simplices according to the order relation in $S$, see [@BPposets] for details. In the following we only consider *pure* simplicial posets, which means that all maximal elements of $S$ have the same dimension. A simplicial poset is called *a homology sphere* (resp. *a homology manifold*) if its geometrical realization is a homology sphere (resp. a homology manifold).
Let $f_j$ denote the number of $j$-dimensional simplices of $S$ for $j=-1,0,\ldots,n-1$, in particular, $f_{-1}=1$ (the empty simplex ${\hat{0}}$ has dimension $-1$). $h$-numbers of $S$ are defined from the relation: $$\label{eqHvecDefin}
\sum_{j=0}^nh_jt^{n-j}=\sum_{j=0}^nf_{j-1}(t-1)^{n-j},$$ where $t$ is a formal variable. Let ${\widetilde{\beta}}_j(S)=\dim {\widetilde{H}}_j(S)$ be the reduced Betti number of the geometric realization of $S$. *$h'$- and $h''$-numbers* of $S$ are defined as follows $$\label{eqDefHprime}
h_j'=h_j+{n\choose
j}\left(\sum_{s=1}^{j-1}(-1)^{j-s-1}{\widetilde{\beta}}_{s-1}(S)\right)\mbox{ for
} 0\leqslant j\leqslant n;$$ $$\label{eqDefHtwoprimes}
h_j'' = h_j'-{n\choose j}{\widetilde{\beta}}_{j-1}(S) = h_j+{n\choose
j}\left(\sum_{s=1}^{j}(-1)^{j-s-1}{\widetilde{\beta}}_{s-1}(S)\right)$$ for $0\leqslant j\leqslant n-1$, and $h''_n=h'_n$. Sums over empty sets are assumed zero.
Let $[m]=\{1,\ldots,m\}$ be the vertex set of $S$, $m=f_0$. Let $R$ be a field or the ring ${\mathbb{Z}}$, and let $R[m]=R[v_1,\ldots,v_m]$, $\deg v_i=2$, denote the graded polynomial algebra with $m$ generators, corresponding to the vertices of $S$. Slightly abusing the terminology, we call the elements of degree 2 linear, when working with such polynomial rings. For a graded $R$-module $V^*=\bigoplus_{j=0}^\infty V_j$ we denote by $\operatorname{Hilb}(V^*;t)$ its Hilbert–Poincare function $\sum_{j=0}^\infty t^j\operatorname{rk}_RV_j\in {\mathbb{Z}}[[t]]$.
\[definFaceRing\] The face ring of a simplicial poset $S$ is the commutative associative graded algebra $R[S]$ over a ring $R$ generated by formal variables $v_I$, one for each simplex $I\in S$, with relations $$v_{I_1}\cdot v_{I_2}=v_{I_1\cap I_2}\cdot\sum_{J\in I_1\vee
I_2}v_J,\qquad v_{{\hat{0}}}=1.$$ Here $I_1\vee I_2$ denotes the set of least upper bounds of $I_1,I_2\in S$, and $I_1\cap I_2\in S$ is the intersection of simplices (it is well-defined and unique when $I_1\vee
I_2\neq{\hat{0}}$). We take the doubled grading on the ring, in which $v_I$ has degree $2(\dim I+1)$. The natural graded ring homomorphism $R[m]=R[v_1,\ldots,v_m]\to R[S]$ defines the structure of the $R[m]$-module on $R[S]$.
If $R$ is an infinite field, and $\dim S=n-1$, then a generic set of linear elements $\theta_1,\ldots,\theta_n\in R[S]_2$ is a linear system of parameters (we remark that linear systems of parameters can be constructed using characteristic functions on $S$, see e.g. [@BPnew Lm.3.5.8]). Let $\Theta$ denote the parametric ideal of $R[S]$ generated by $\theta_1,\ldots,\theta_n$
\[propStanley\] For a pure simplicial poset $S$ of dimension $n-1$ there holds $$\operatorname{Hilb}(R[S];t)=\dfrac{h_0+h_1t^2+\cdots+h_nt^n}{(1-t^2)^n}.$$ For a homology sphere $S$ there holds $\operatorname{Hilb}(R[S]/\Theta;t)=\sum_ih_it^{2i}$.
\(1) For a homology manifold $S$ there holds $$\operatorname{Hilb}(R[S]/\Theta;t)=\sum_ih'_it^{2i}.$$
\(2) Let $S$ be a connected $R$-orientable homology manifold of dimension $n-1$. The $2j$-th graded component of the module $R[S]/\Theta$ contains a vector subspace $(I_{NS})_{2j}\cong{n\choose j}{\widetilde{H}}^{j-1}(S;R)$, which is a trivial $R[m]$-submodule (i.e. $R[m]_+(I_{NS})_{2j}=0$). Let $I_{NS}=\bigoplus_{j=0}^{n-1}(I_{NS})_{2j}$ be the sum of all these submodules except the top-degree component. Then the quotient module $R[S]/\Theta/I_{NS}$ is a Poincare duality algebra, and there holds $$\operatorname{Hilb}(R[S]/\Theta/I_{NS};t)=\sum_ih''_it^{2i}.$$
We now compute the combinatorial characteristics of the simplicial poset ${K_{\mathcal{PT}}}^{n-1}$ dual to ${\mathcal{PT}}^{n-1}$. Combinatorially, the simplicial cell complex ${K_{\mathcal{PT}}}^{n-1}$ can be defined as a poset, whose elements are the faces of the wonderful cell decomposition ${\mathcal{PT}}^{n-1}$ and the order is given by the reversed inclusion. It can be seen that ${K_{\mathcal{PT}}}^{n-1}$ is a simplicial poset. Recall that ${{n\atopwithdelims\{\}k}}$ denotes the Stirling number of the second kind, that is the number of unordered partitions of the set $[n]$ into $k$ nonempty subsets.
For the simplicial poset ${K_{\mathcal{PT}}}^{n-1}$ there holds $$f_{k-1}=n(k-1)!{{n\atopwithdelims\{\}k}} \mbox{ for }k=1,2,\ldots,n;\quad
f_{-1}=1;$$ $$\label{eqHnumbers}
h_l=(-1)^l{n\choose n-l}+\sum_{k=1}^l(-1)^{l-k}{n-k\choose
n-l}n(k-1)!{{n\atopwithdelims\{\}k}}\quad\mbox{for }l=0,1,\ldots,n$$ $$\begin{gathered}
\label{eqHprimenumbers}
h_l'=(-1)^l{n\choose n-l}+\sum_{k=1}^l(-1)^{l-k}{n-k\choose
n-l}n(k-1)!{{n\atopwithdelims\{\}k}}+\\+{n\choose
l}\sum_{k=2}^{l-1}(-1)^{l-k-1}{n-1\choose k-1} \quad\mbox{ for
}l=0,1,\ldots,n\end{gathered}$$ $$\begin{gathered}
\label{eqHtwoprimenumbers}
h_l''=(-1)^l{n\choose n-l}+\sum_{k=1}^l(-1)^{l-k}{n-k\choose
n-l}n(k-1)!{{n\atopwithdelims\{\}k}}+\\+{n\choose
l}\sum_{k=2}^{l}(-1)^{l-k-1}{n-1\choose k-1}\quad \mbox{for
}l=0,1,\ldots,n-1,\quad\mbox{ and }h_{n}''=1.\end{gathered}$$
From the combinatorial description of a permutohedron it follows that the number $f_{n-k}(\operatorname{Pe}^{n-1})$ is equal to $k!{{n\atopwithdelims\{\}k}}$. The wonderful subdivision ${\mathcal{PT}}^{n-1}$ consists of $n$ permutohedra and each $(n-k)$-dimensional face of ${\mathcal{PT}}^{n-1}$ lies in exactly $k$ permutohedral cells, since the subdivision is simple. Therefore, $$f_{k-1}({K_{\mathcal{PT}}}^{n-1})=f_{n-k}({\mathcal{PT}}^{n-1})=\frac{n}{k}f_{n-k}(\operatorname{Pe}^{n-1})=n(k-1)!{{n\atopwithdelims\{\}k}}$$ for $k\geqslant 1$. The identity $f_{-1}=1$ holds authomatically.
Since ${K_{\mathcal{PT}}}^{n-1}$ is a simplicial cell subdivision of the torus ${\mathcal{T}}^{n-1}$, we have ${\widetilde{\beta}}_j({K_{\mathcal{PT}}}^{n-1})={n-1\choose j}$ for $j\geqslant 1$. Expressions ,, and follow from the general definitions of $h$-, $h'$-, and $h''$-numbers.
Equivariant cohomology {#secEquivCohom}
======================
Let $X$ be a $2n$-manifold with a locally standard action of $T^n$. The orbit space $P=X/T^n$ is a manifold with faces. It means that every codimension $k$ face of $P$ lies in exactly $k$ different facets of $P$. Let $S_P$ denote the simplicial poset dual to the poset of faces of $P$. In [@AMPZ] we proved
\[propAMPZ\] Assume that all proper faces of $P$ are acyclic and the projection map $X\to P$ admits a section. Then $H^*_{T^n}(X;{\mathbb{Z}})\cong
{\mathbb{Z}}[S_P]\oplus H^*(P;{\mathbb{Z}})$ as the rings, and as the modules over ${\mathbb{Z}}[n]\cong H^*(BT^n)$. The components of degree 0 are identified in the direct sum. The ring $H^*(P;{\mathbb{Z}})$ is considered a trivial ${\mathbb{Z}}[n]$-module.
Now we apply this statement to the space ${X_{n,\lambda}^{\leqslant\varepsilon}}$ which is $T^{n-1}$-equivariantly homeomorphic to $Y$ (see construction \[conModelOfNeighb\]).
\[thmEqBettiNeighb\] The Hilbert–Poincare series of the $T^{n-1}$-equivariant cohomology ring of ${X_{n,\lambda}^{\leqslant\varepsilon}}$ is given by $$\operatorname{Hilb}(H^*_{T^{n-1}}({X_{n,\lambda}^{\leqslant\varepsilon}});t)=\dfrac{\sum_{i=0}^nh_it^{2i}}{(1-t^2)^{n-1}}+(1+t)^{n}-1-t.$$ Here $h_i$, the $h$-numbers of the simplicial poset ${K_{\mathcal{PT}}}^{n-1}$, are given by .
Recall that $Y$ is the collar model, that is the locally standard $T^n$-space over ${\mathcal{T}}^{n-1}\times [0,1]$. Proposition \[propAMPZ\] implies the following isomorphism for the $T^n$-equivariant cohomology $$H^*_{T^n}(Y)\cong{\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]\oplus H^*({\mathcal{T}}^{n-1}).$$ There is an induced action of the $(n-1)$-dimensional subtorus $$\label{eqSubtorusDiag}
T^{n-1}=\{t_1\cdots t_n=1\}$$ on $Y$, and Theorem \[thmTopologyNeighb\] states that $Y$ and ${X_{n,\lambda}^{\leqslant\varepsilon}}$ are $T^{n-1}$-equivariantly homeomorphic. To compute the $T^{n-1}$-equivariant cohomology of $Y$, we first note that there is a Serre fibration $$Y_{T^{n-1}}\stackrel{S^1}{\longrightarrow} Y_{T^{n}}, \qquad
S^1=T^n/T^{n-1}$$ where $Y_{T^{n-1}}$ and $Y_{T^{n}}$ are the Borel constructions of $T^{n-1}$- and $T^n$-actions on $Y$ respectively. Consider the corresponding Serre spectral sequence: $$E_2^{p,q}=H_{T^n}^p(Y)\otimes H^q(S^1) \Rightarrow
H^{p+q}_{T^{n-1}}(Y).$$ The sequence has only two nonzero rows, hence it collapses at the $E_3$-term. Let $\omega$ denote a generator of $H^1(S^1)$. The second differential $d_2\colon H^1(S^1)\to H^2_{T^n}(Y)$ of the spectral sequence coincides with the composition $$H^1(T^{n}/T^{n-1})\cong H^2(B(T^{n}/T^{n-1}))\to H^2(BT^n) \to
H^2_{T^n}(Y),$$ where the middle map is induced by the projection $T^n\to
T^n/T^{n-1}$ and the right map is the defining map for the $H^*(BT^n)$-module structure on $H^*_{T^n}(Y)$. It follows that $$d_2(\omega)=\eta\in {\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]_2\subset
H^2_{T^n}(Y),\quad\mbox{ where }\eta=\sum\nolimits_{i=1}^nv_i,$$ according to the definition of the subtorus $T^{n-1}$.
$\eta$ is not a zero divisor in the face ring ${\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]$, or, equivalently, $\eta$ is a regular element.
We use the standard argument in the theory of face rings. For any non-empty simplex $I\cong \Delta^{k}$ in ${K_{\mathcal{PT}}}^{n-1}$ consider the epimorphism $$\varphi_\sigma\colon {\Bbbk}[{K_{\mathcal{PT}}}^{n-1}]\to {\Bbbk}[I]$$ defined by sending $v_J$ to $0$ for all $J\nleqslant I$. Notice that ${\Bbbk}[I]$ is just the polynomial algebra in $\dim I+1$ generators. The map $\varphi_I$ is a homomorphism of ${\Bbbk}[n]$-algebras, with the ${\Bbbk}[n]$-structure on ${\Bbbk}[I]$ is defined by an epimorphism $\psi_I$ sending the excess variables to zeroes.
Assume that there exists $\beta\in {\Bbbk}[{K_{\mathcal{PT}}}^{n-1}]$ such that $\eta\cdot\beta=0$. Then $\psi_I(\eta)\cdot\varphi_\sigma(\beta)=0$ in ${\Bbbk}[I]$. Since there are no zero divisors in ${\Bbbk}[I]$, and $\psi_I(\eta)=\sum_{i\leqslant I}v_i\neq 0$, we have $\varphi_I(\beta)=0$ for any simplex $I$ of ${K_{\mathcal{PT}}}^{n-1}$. The homomorphism $$\bigoplus\nolimits_\sigma\varphi_\sigma\colon {\Bbbk}[{K_{\mathcal{PT}}}^{n-1}]\to \bigoplus\nolimits_\sigma{\Bbbk}[I_\sigma]$$ is known to be injective [@BPnew Thm 3.5.6]. Therefore $\beta=0$.
According to the lemma, for any nonzero element $\beta\in {\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]\subset H^*_{T^n}(Y)$, there holds $$d_2(\omega \beta)=(d_2\omega)\beta\pm \omega d_2(\beta)=\eta
\beta\neq 0$$ In other words, $d_2$ is injective on the submodule ${\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]\otimes H^1(S^1)\subset E_2^{*,1}$.
On the other hand, for any element $\alpha\in H^i({\mathcal{T}}^{n-1})\subset
H^i_{T^n}(Y)$, $i>0$, we have $$d_2(\omega \alpha)=(d_2\omega)\alpha\pm \omega
d_2(\alpha)=\eta\cdot\alpha=0,$$ since the products of elements from the components $H^{+}({\mathcal{T}}^{n-1})$ and ${\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]_{+}$ of the ring $H^*_{T^n}(Y)$ vanish. Therefore the differential $d_2$ vanishes on $H^{+}({\mathcal{T}}^{n-1})$. Finally, we have $$E_3^{p,q}\cong \begin{cases} {\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]/(\eta)_p\oplus
H^p({\mathcal{T}}^{n-1}),\mbox{ for }q=0;\\
H^p({\mathcal{T}}^{n-1}), \mbox{ for }q=1.
\end{cases}$$ The Hilbert–Poincare series of ${\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]$ is given by $\left(\sum_{i=0}^nh_it^{2i}\right)/(1-t^2)^{n}$ according to Proposition \[propStanley\]. We have $$\operatorname{Hilb}({\mathbb{Z}}[{K_{\mathcal{PT}}}^{n-1}]/(\eta);t)=\dfrac{\sum_{i=0}^nh_it^{2i}}{(1-t^2)^{n-1}}.$$ The statement now follows from the degeneration of the spectral sequence at $E_3$-term.
\[corHilbertEqCohom\] For the whole isospectral space ${X_{n,\lambda}}$ there holds $$\operatorname{Hilb}(H^*_{T^{n-1}}({X_{n,\lambda}});t)=\dfrac{\sum_{i=0}^nh_it^{2i}}{(1-t^2)^{n-1}}+R(t),$$ where $R(t)$ is a polynomial, and $h_i$ are given by .
The space ${X_{n,\lambda}}$ is patched from ${X_{n,\lambda}^{\leqslant\varepsilon}}$ and ${X_{n,\lambda}^{\geqslant\varepsilon}}=\{p^{-1}(|z|\geqslant \varepsilon)\}$. Both subsets are preserved by the torus action, hence the equivariant cohomology groups can be computed via Mayer–Vietoris exact sequence. However, the torus action on ${X_{n,\lambda}^{\geqslant\varepsilon}}$ and ${X_{n,\lambda}^{\geqslant\varepsilon}}\cap {X_{n,\lambda}^{\leqslant\varepsilon}}$ is free, so the equivariant cohomology groups of these subsets coincide with the ordinary cohomology groups of their orbit spaces. Hence they are concentrated in a finite range of degrees. The statement now follows from Theorem \[thmEqBettiNeighb\].
We check the calculations for the case $n=3$. The isospectral space $X_{3,\lambda}$ coincides with the manifold $\operatorname{Fl}_3$ of complete complex flags. Its equivariant cohomology are known to satisfy $$\operatorname{Hilb}(H^*_{T^2}(\operatorname{Fl}_3);t)=\dfrac{\operatorname{Hilb}(H^*(\operatorname{Fl}_3);t)}{(1-t^2)^2}=\dfrac{1+2t^2+2t^4+t^6}{(1-t^2)^2}$$ Using formulas compute the h-numbers of ${K_{\mathcal{PT}}}^2$: $(h_0,h_1,h_2,h_3)=(1,0,6,-1)$. There holds $$\dfrac{1+2t^2+2t^4+t^6}{(1-t^2)^2}=\dfrac{1+6t^4-t^6}{(1-t^2)^2}+2t^2,$$ which confirms Corollary \[corHilbertEqCohom\].
Betti numbers {#secBetti}
=============
In this section we describe the additive structure of the cohomology ring of ${X_{n,\lambda}}$. As a first step, we compute the homological structure of the subset ${X_{n,\lambda}^{\leqslant\varepsilon}}$, containing the essential information on the torus action. It is assumed in this section, that all coefficients are taken in a fixed field. The next proposition follows from the general technique developed in [@Ay1].
\[propBettiNeighborhood\] The homology modules of $Y\cong {X_{n,\lambda}^{\leqslant\varepsilon}}$ admit the double grading: $H_j(Y)\cong \bigoplus_{p+q=j} H_{p,q}(Y)$. There holds
1. $H_{p,q}(Y)\cong H_{p}({\mathcal{T}}^{n-1})\otimes
H_q(T^n)$ for $q<p<n$.
2. $H_{p,q}(Y)=0$ for $q>p$.
3. The dimension of $H_{p,p}(Y)$ equals $$h_p+{n\choose p}\sum_{k=2}^{p+1}(-1)^{p+k-1}{n-1\choose k-1}$$ for $p=0,1,\ldots,n-1$. In particular, for $p\geqslant 2$ there holds $\dim H_{p,p}(Y)=h_p''+{n\choose p}{n-1\choose p}$.
The inclusion map $i\colon{\mathcal{T}}^{n-1}\times T^n\cong {\partial}Y\to Y$ induces the homomorphism in homology, which respects the double grading: $$i_*\colon H_p({\mathcal{T}}^{n-1})\otimes H_q(T^n)\to H_{p,q}(Y).$$ This homomorphism is an isomorphism for $q<p$, injective for $q=p$, and zero for $q>p$.
Note that the subspace ${X_{n,\lambda}^{\leqslant\varepsilon}}$ does not depend on the parameters $n_+,n_-$ discussed in the previous sections. Now we are in position to compute the Betti numbers of ${X_{n,\lambda}}$. Homology of ${X_{n,\lambda}}$ will certainly depend on parameters $n_+,n_-$, which encode “the degree of degeneration” of this space.
\[thmBettiTotal\] The Hilbert–Poincare series for homology of ${X_{n,\lambda}}$ is given by the formula $$\label{eqManySummands}
\sum_{i=0}^{2n}\beta({X_{n,\lambda}})t^i=H^{\geqslant{\varepsilon}}(t)+H^{\leqslant{\varepsilon}}(t)-H^{{\varepsilon}}(t)+(1+t)\cdot H^{\operatorname{Ker}}(t),$$ where $$\begin{aligned}
H^{={\varepsilon}}(t) &= (1+t)^{2n-1},\label{eqHomolEps} \\
H^{\geqslant{\varepsilon}}(t) &= (1+t)^{2n-n_+-n_--2}(1-t+t(1+t)^{n_+}+t(1+t)^{n_-}), \label{eqHomolGEps}\\
H^{\leqslant{\varepsilon}}(t) &= \sum_{p=0}^{n-1}\left(h_p+{n\choose p}\sum_{k=2}^{p+1}(-1)^{p+k-1}{n-1\choose k-1}\right)t^{2p} +\sum_{q<p<n}{n-1\choose p}{n\choose q}t^{p+q}, \label{eqHomolLEps}\\
H^{\operatorname{Ker}}(t) &= \sum_{(p,e,q,s,r)\in \Upsilon}{n-1-n_+-n_-\choose p}{1\choose e}{n_+\choose q}{n_-\choose s}{n-1\choose r}t^{p+e+q+s+r},\label{eqHomolKer}\end{aligned}$$ The indexing subset $\Upsilon$ in the last expression is defined by the conditions $$\label{eqConditions}
\begin{array}{c}
0\leqslant p\leqslant n-1-n_+-n_-;\quad 0\leqslant e\leqslant 1;\\
0\leqslant q\leqslant n_+;\quad 0\leqslant s \leqslant n_-;\quad 0\leqslant r \leqslant n-1; \\
r+e>p+q+s;\\
\mbox{either }(e=0 \mbox{ and } q+s>0) \mbox{ or }(e=1 \mbox{ and } q>0 \mbox{ and }s>0).
\end{array}$$ The $h$-numbers are given by .
Althouth the result looks awkward, the idea behind this calculation is straightforward: we analyze the Mayer–Vietoris sequence for the union ${X_{n,\lambda}}={X_{n,\lambda}^{\leqslant\varepsilon}}\cup{X_{n,\lambda}^{\geqslant\varepsilon}}$. Let ${X_{n,\lambda}}^{={\varepsilon}}(t)$ denote the intersection ${X_{n,\lambda}^{\leqslant\varepsilon}}\cap{X_{n,\lambda}^{\geqslant\varepsilon}}$. Then there is a long exact sequence $$\label{eqMVhomology}
\to H_i({X_{n,\lambda}}^{={\varepsilon}})\stackrel{\iota_i}{\to} H_i({X_{n,\lambda}^{\leqslant\varepsilon}})\oplus H_i({X_{n,\lambda}^{\geqslant\varepsilon}})\to
H_i({X_{n,\lambda}})\to H_{i-1}({X_{n,\lambda}}^{={\varepsilon}})\stackrel{\iota_{i-1}}{\to} H_{i-1}({X_{n,\lambda}}^{={\varepsilon}})\to$$ Note, that given an exact sequence $$A_i\stackrel{\alpha_i}{\to} B_i\to C_i\to A_{i-1}\stackrel{\alpha_{i-1}}{\to} B_{i-1},$$ the dimension of the vector space in the middle is given by $$\label{eqExactGen}
\dim C_i=\dim B_i-\dim A_i+\dim\operatorname{Ker}\alpha_i+\dim\operatorname{Ker}\alpha_{i-1}$$ After introducing a natural notation $$\begin{aligned}
H^{={\varepsilon}}(t) &= \sum\nolimits_{i}\dim H_i({X_{n,\lambda}}^{={\varepsilon}})t^i, \\
H^{\geqslant{\varepsilon}}(t) &= \sum\nolimits_{i}\dim H_i({X_{n,\lambda}^{\geqslant\varepsilon}})t^i, \\
H^{\leqslant{\varepsilon}}(t) &= \sum\nolimits_{i}\dim H_i({X_{n,\lambda}^{\leqslant\varepsilon}})t^i, \\
H^{\operatorname{Ker}}(t) &= \sum\nolimits_{i}\dim(\operatorname{Ker}\iota_i\colon H_i({X_{n,\lambda}}^{={\varepsilon}})\to H_i({X_{n,\lambda}^{\leqslant\varepsilon}})\oplus H_i({X_{n,\lambda}^{\geqslant\varepsilon}}))t^i.\end{aligned}$$ formula implies . We need to check formulas –.
\(1) According to , ${X_{n,\lambda}}^{={\varepsilon}}\cong T^{2n-1}$, which implies .
\(2) The space ${X_{n,\lambda}^{\geqslant\varepsilon}}$ supports a free action of $T^{n-1}$, which admits a section. Therefore, $$\label{eqProduct}
{X_{n,\lambda}^{\geqslant\varepsilon}}={Q_{n,\lambda}^{\geqslant\varepsilon}}\times T^{n-1},$$ where ${Q_{n,\lambda}^{\geqslant\varepsilon}}={X_{n,\lambda}^{\geqslant\varepsilon}}/T^{n-1}$. The structure of the orbit space was described in detail in Section \[secOrbitSpace\]: ${Q_{n,\lambda}^{\geqslant\varepsilon}}={X_{n,\lambda}^{\geqslant\varepsilon}}/T^{n-1}$ is fibered by Liouville–Arnold tori over ${\mathbb{B}}\setminus \{|z|<\varepsilon\}$, — the biangle with a hole. The homotopy type of the latter space is $$\label{eqHomotHoledFamily}
{Q_{n,\lambda}^{\geqslant\varepsilon}}\simeq {\mathcal{T}}^{n-n_+-n_--1}\times (S^1\vee\Sigma{\mathcal{T}}^{n_+}\vee \Sigma{\mathcal{T}}^{n_-}).$$ Formula follows from and .
\(3) Homology of ${X_{n,\lambda}^{\leqslant\varepsilon}}$ are described by Theorem \[propBettiNeighborhood\]. Formula is its simple consequence.
\(4) To derive , we need to count the dimension of the vector space of all homology cycles of ${X_{n,\lambda}}^{={\varepsilon}}$, annihilated by both maps $$\begin{aligned}
\iota_*^{\leqslant}\colon& H_*({X_{n,\lambda}}^{={\varepsilon}})\to H_*({X_{n,\lambda}^{\leqslant\varepsilon}}), \\
\iota_*^{\geqslant}\colon& H_*({X_{n,\lambda}}^{={\varepsilon}})\to H_*({X_{n,\lambda}^{\geqslant\varepsilon}}).\end{aligned}$$ The torus ${X_{n,\lambda}}^{={\varepsilon}}$ decomposes into the product $${X_{n,\lambda}}^{={\varepsilon}}\cong S_{\varepsilon}^1\times {\mathcal{T}}^{n-1}\times T^{n-1}\cong {\mathcal{T}}^{n-1-n_+-n_-}\times S_{\varepsilon}^1\times {\mathcal{T}}^{n_+}\times {\mathcal{T}}^{n_-}\times T^{n-1},$$ where
1. the component $T^{n-1}$ corresponds to the acting torus;
2. ${\mathcal{T}}^{n_+}$, ${\mathcal{T}}^{n_-}$ are the components of Liouville–Arnold tori, collapsing over the sides of the biangle ${\mathbb{B}}$;
3. ${\mathcal{T}}^{n-1-n_+-n_-}$ is the surviving component of the Liouville–Arnold torus.
4. $S^1$ is the circle $\{|z|=\varepsilon\}$, lifted to the total space.
These five components of the torus explain the appearance the five-element indexing subset in the statement. Let $\theta=\omega_p\otimes\alpha_e\otimes \omega_q^+\otimes \omega_s^-\otimes \nu_r$ be a homology cycle from $$H_*({X_{n,\lambda}}^{={\varepsilon}})\cong H_*({\mathcal{T}}^{n-1-n_+-n_-})\otimes H_*(S^1)\otimes H_*({\mathcal{T}}^{n_+})\otimes H_*({\mathcal{T}}^{n_-})\otimes H_*(T^{n-1}),$$ where the degrees of the factors are indicated in their subscripts. A straightforward calculation shows that both maps $\iota_*^{\leqslant}$ and $\iota_*^{\geqslant}$ annihilate $\theta$ if and only if the 5-tuple $(p,e,q,s,r)$ satisfies the conditions . This proves .
The Betti numbers computed for small values of $n$ are shown in the tables \[tableManifoldBetti\] and \[tableDegenBetti\].
n $\beta_0$ $\beta_1$ $\beta_2$ $\beta_3$ $\beta_4$ $\beta_5$ $\beta_6$ $\beta_7$ $\beta_8$ $\beta_9$ $\beta_{10}$ $\beta_{11}$ $\beta_{12}$
--- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -------------- -------------- --------------
3 1 0 2 0 2 0 1
4 1 1 6 2 16 2 6 1 1
5 1 2 13 9 65 16 65 9 13 2 1
6 1 3 23 25 203 67 456 67 203 25 23 3 1
n $\beta_0$ $\beta_1$ $\beta_2$ $\beta_3$ $\beta_4$ $\beta_5$ $\beta_6$ $\beta_7$ $\beta_8$ $\beta_9$ $\beta_{10}$ $\beta_{11}$ $\beta_{12}$
--- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -------------- -------------- --------------
4 1 0 3 1 16 3 9 2 1
5 1 0 4 2 57 16 77 22 24 4 1
6 1 0 5 4 167 55 471 115 276 61 39 5 1
\[corOddDeg\] The space ${X_{n,\lambda}}$ has nonzero Betti numbers in odd degrees for $n\geqslant 4$.
Theorem \[thmBettiTotal\] implies that $\beta_1({X_{n,\lambda}})=n-1-n_+-n_-$. This number is nonzero unless $n_+$, $n_-$ are the maximal possible, representing the most degenerate case. For the most degenerate case, and $n\geqslant 4$, $\beta_{2n-1}({X_{n,\lambda}})$ is nonzero. To prove this, it is sufficient to estimate the coefficient at $t^{2n-1}$ in by $2$ from below. Indeed, the only negative term in comes from $H^{\operatorname{Ker}}(t)$, and its coefficient at $t^{2n-1}$ is one. The estimation is straightforward.
Recall that with any action of a torus $T$ on a space $X$ one can associate a fibration $r\colon X\times_TET\stackrel{X}{\to}BT$, where $ET$ is a contractible space, carrying the free action of $T$ and $BT$ is the classifying space of a torus. It can be assumed that $ET=(S^\infty)^k$ and $BT=({\mathbb{C}P}^\infty)^k$, where $k=\dim T$.
\[definEqForm\] The space $X$ is called *equivariantly formal* in the sense of Goresky–Macpherson [@GKM] if the Serre spectral sequence $$\label{eqSerreSpecSeq}
E_2^{*,*}\cong H^*(BT)\otimes H^*(X)\Rightarrow H^*(X\times_TET)=H^*_T(X),$$ i.e. the spectral sequence of the fibration $r$, degenerates at its second page.
The space ${X_{n,\lambda}}$ is not equivariantly formal for $n\geqslant 4$.
Assume that ${X_{n,\lambda}}$ is equivariantly formal. The degeneration of the Serre spectral sequence at a second page then implies $$\operatorname{Hilb}(H^*_T({X_{n,\lambda}});t)=\operatorname{Hilb}(H^*({X_{n,\lambda}});t)\cdot \operatorname{Hilb}(H^*(BT^{n-1});t)=\operatorname{Hilb}(H^*({X_{n,\lambda}});t)\cdot (1+t^2)^{n-1}.$$ This identity and Corollary \[corOddDeg\] imply that $H^*_T({X_{n,\lambda}})$ has nontrivial components of arbitrarily high odd degree. This contradicts to Corollary \[corHilbertEqCohom\].
The fundamental group of ${X_{n,\lambda}}$ can be explicitly described as well. As in the computations above, we consider the decomposition ${X_{n,\lambda}}={X_{n,\lambda}^{\leqslant\varepsilon}}\cup{X_{n,\lambda}^{\geqslant\varepsilon}}$ and apply van Kampen theorem. For the intersection there holds $$\pi_1({X_{n,\lambda}}^{={\varepsilon}})=\pi_1(T^{n_+}\times T^{n_-}\times T^{n-1-n_+-n_-}\times S_{\varepsilon}^1\times T^{n-1})={\mathbb{Z}}^{n_+}\oplus {\mathbb{Z}}^{n_-}\oplus {\mathbb{Z}}^{n-1-n_+-n_-}\oplus {\mathbb{Z}}^{n}$$ The summands ${\mathbb{Z}}^{n_+}$ and ${\mathbb{Z}}^{n_-}$ vanish in $\pi_1({X_{n,\lambda}^{\geqslant\varepsilon}})$ since the corresponding components of Lioville–Arnold tori are collapsed over the facets of biangle ${\mathbb{B}}$. The summand ${\mathbb{Z}}^{n}=\pi_1(S_{\varepsilon}^1\times T^{n-1})$ vanishes in $\pi_1({X_{n,\lambda}^{\leqslant\varepsilon}})$ according to [@Zeng]. The result of this paper asserts that for a locally standard action of $T$, $\dim T=n$, on $M$, $\dim M=n$, having a fixed point, there holds $\pi_1(M)\cong \pi_1(M/T)$. In other words, any loop on the acting torus can be contracted via a fixed point. This result is applied to ${X_{n,\lambda}^{\leqslant\varepsilon}}\cong Y$, carrying the extended action of $T^n$.
No other loops appear in ${X_{n,\lambda}^{\geqslant\varepsilon}}$ and ${X_{n,\lambda}^{\leqslant\varepsilon}}$, therefore $$\pi_1({X_{n,\lambda}})\cong {\mathbb{Z}}^{n-1-n_+-n_-}.$$ Hence ${X_{n,\lambda}}$ is simply connected if and only if the spectrum $\lambda$ satisfies $n_-+n_+=n-1$. This corresponds to the most degenerate situation, considered in Remark \[remDegenerateCheb\].
Acknowledgements {#secThanks .unnumbered}
================
The author would like to thank Victor Buchstaber and Igor Krichever for their valuable comments on this work. I am also grateful to Dmitry Gugnin and Alexander Gaifullin who brought the subject of crystallizations and the paper [@FGG] to my attention.
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[^1]: This work is supported by the Russian Science Foundation under grant 18-71-00009.
|
---
abstract: 'In this paper, using different approaches we demonstrate the equality of the two mixed charge-spin response functions of a spin-polarized electron gas when orbital effects are negligible. Within a generalized STLS approximation we show that the two mixed responses are equal. We also present arguments for the equality of the two dynamic responses by considering a symmetry of the effective screened interaction between two opposite spin electrons. Furthermore, using the reflection symmetry of the system and the fact that the hamiltonian is real we prove rigorously that the two responses coincide identically.'
address: 'Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta'
author:
- Sudhakar Yarlagadda
title: 'Mixed charge-spin response functions of an arbitrarily polarized electron gas'
---
[2]{}
INTRODUCTION
============
An electron gas (EG) can be macroscopically characterized in terms of its response functions. For an unpolarized EG in a uniform positive background, using many-body local fields to account for the vertex corrections associated with the charge and spin density fluctuations, it has been shown in Refs. that only the charge response and the longitudinal spin response functions are sufficient to obtain the self-energy and the effective interaction between electrons. Again using the many-body local field formalism, for a system that is polarized it was however found that one has to also have knowledge of the two mixed charge-spin response functions (i.e., the charge response to an external magnetic field and the spin response to an external electric potential) and the transverse spin response [@YG2]. Earlier on Kim et al. [@kim] treated the two charge-spin mixed response functions within a random phase approximation and found them to be identical. More recently, Yi and Quinn [@YQ] studied these two responses in an uniform EG which is polarized by a DC magnetic field but which has negligible orbital effects. These authors conjectured that the two mixed response functions, in general, can be different.
In this paper we demonstrate the equality of the two mixed response functions in the polarized system considered in Ref. . In Sec. II, we use arguments of increasing sophistication/rigor to establish the coincidence of the two responses. We first present arguments for the equality of the responses within a generalized STLS approximation [@NgSingwi]. Next we show that the mixed charge-spin responses are equal by exploiting a symmetry of the effective screened interaction between two opposite spin electrons. Lastly by invoking the reflection symmetry of the system and the fact that the hamiltonian is not only hermitian but also real, we prove that the two responses are exactly identical. Finally in Sec. III we discuss the behavior of the mixed responses in the limit of large momentum or large frequency and also present our conclusions.
SYMMETRY BASED EQUALITY OF RESPONSES
====================================
In the analysis that follows we will ignore the effect of the induced magnetization. We will first derive expressions for the charge, the spin, and the two mixed response functions. Using arguments similar to those used in Ref. we note that the potential felt by an electron of spin $\sigma$ when an external potential is applied is given by $$\begin{aligned}
\Phi ^{\sigma}(\vec{q}, \omega ) =
\Delta n ^{\sigma} / \chi^{\sigma}_{0}=
\phi ^{\sigma} +
&& v(q) \left [
\Delta n^{-\sigma}(1 - 2 G^{\sigma , -\sigma} )
\right .
\nonumber \\
&&
\left . +
\Delta n^{\sigma}(1 - 2 G^{\sigma , \sigma} )
\right ] ,
\label{phiCS}\end{aligned}$$ where $v(q)$ is the Coulombic interaction between two electrons. Furthermore $ \chi^{\sigma}_{0}$ is the non-interacting polarizability, $\Delta n^{\sigma}$ is the density fluctuation, and $G^{\sigma , \sigma ^{\prime}}$ are the many-body local fields and are all functions of $\vec{q}$ and $\omega$. In Eq. (\[phiCS\]) for an applied spin symmetric \[spin antisymmetric\] electric potential $V_{ext} (\vec{q}, \omega )$ \[magnetic field $H_{ext} (\vec{q}, \omega )$ \], $ \phi ^{\sigma}
(\vec{q}, \omega )
= eV _{ext}$ $ [g \sigma \mu _{B} H _{ext}]$. On defining $G_{\pm} ^{\sigma} \equiv
G^{\sigma , \sigma}
\pm G^{\sigma ,-\sigma}$ we obtain, in agreement with Ref. the following expressions for the charge, the longitudinal spin, and the two mixed charge-spin response functions: $$\begin{aligned}
\chi_{C}(\vec{q} ,\omega ) && =
\frac{\Delta n^{\uparrow}
+\Delta n^{\downarrow}}{eV_{ext}}
\nonumber \\
&& =\frac{(\chi ^{\uparrow}_{0} +\chi^{\downarrow}_{0} ) +
2 v(q) \chi ^{\uparrow}_{0} \chi^{\downarrow}_{0} (
G^{\downarrow}_{-}
+ G^{\uparrow}_{-} )}{\cal{D}} ,
\label{chiC}\end{aligned}$$ $$\begin{aligned}
\frac{ \chi_{S}(\vec{q} ,\omega )}{-\mu_{B}^2} && =
\frac{\Delta n^{\uparrow}
-\Delta n^{\downarrow}}{\mu_{B} H_{ext}}
\nonumber \\
&& =\frac{(\chi ^{\uparrow}_{0} +\chi^{\downarrow}_{0} ) +
2 v(q) \chi ^{\uparrow}_{0} \chi^{\downarrow}_{0} (
G^{\downarrow}_{+}
+ G^{\uparrow}_{+} -2 )}{\cal{D}} ,
\label{chiS}\end{aligned}$$ $$\begin{aligned}
\chi_{CS}^{V}(\vec{q} ,\omega ) && =
\frac{\Delta n^{\uparrow}
-\Delta n^{\downarrow}}{eV_{ext}}
\nonumber \\
&& =\frac{(\chi ^{\uparrow}_{0} -\chi^{\downarrow}_{0} ) +
2 v(q) \chi ^{\uparrow}_{0} \chi^{\downarrow}_{0} (
G^{\downarrow}_{+}
- G^{\uparrow}_{+} )}{\cal{D}} ,
\label{chiCSV}\end{aligned}$$ and $$\begin{aligned}
\chi_{CS}^{H}(\vec{q} ,\omega ) && =
\frac{\Delta n^{\uparrow}
+\Delta n^{\downarrow}}{\mu_{B} H_{ext}}
\nonumber \\
&& =\frac{(\chi ^{\uparrow}_{0} -\chi^{\downarrow}_{0} ) +
2 v(q) \chi ^{\uparrow}_{0} \chi^{\downarrow}_{0} (
G^{\downarrow}_{-}
- G^{\uparrow}_{-} )}{\cal{D}} ,
\label{chiCSH}\end{aligned}$$ where $$\begin{aligned}
{\cal{D}} = &&
1-
v(q) \chi ^{\uparrow}_{0} [ 1 -
G^{\uparrow}_{+} -
G^{\uparrow}_{-} ]
- v(q) \chi ^{\downarrow}_{0} [ 1 -
G^{\downarrow}_{+} -
G^{\downarrow}_{-} ]
\nonumber \\
&&
- 2 v(q) ^2 \chi ^{\uparrow}_{0} \chi^{\downarrow}_{0} [
G^{\uparrow}_{-} (1 -
G^{\downarrow}_{+})
+ G^{\downarrow}_{-} (1 -
G^{\uparrow}_{+})].
\label{calD}\end{aligned}$$ Thus from Eqs. (\[chiCSV\]) and (\[chiCSH\]) we see that the mixed response functions are equal to each other if and only if $G^{\uparrow , \downarrow} (q, \omega) =
G^{\downarrow , \uparrow} (q, \omega )$.
We will now argue, within the generalized STLS approximation formulated by Ng and Singwi [@NgSingwi], that the two mixed response functions are equal. Within the generalized STLS scheme the density fluctuations $\Delta n ^{\sigma} (\vec{q} , \omega )$ induced in the EG by an external spin dependent potential $\phi ^{\sigma} (\vec{q}, \omega )$ are given by (see Ref. ) $$\begin{aligned}
\Delta n ^{\sigma} / \chi^{\sigma}_{0} (\vec{q} , \omega ) =
\phi ^{\sigma} +
v(q) && \left [
\Delta n^{-\sigma} \left \{ 1 - 2 {\cal G}^{\sigma , -\sigma} (q)
\right \}
\right .
\nonumber \\ && + \left . \left \{
\Delta n^{\sigma}(1 - 2 {\cal G}^{\sigma , \sigma} (q) \right \}
\right ] ,
\label{STLS}\end{aligned}$$ where ${\cal G}^{\sigma, \sigma^{\prime}}(q)$ are the [*static*]{} many-body local fields. Here we would like to reiterate the known fact that the static nature of the local fields ${\cal G}^{\sigma, \sigma^{\prime}}(q)$ in the STLS scheme is a consequence of the approximation made in obtaining the time-dependent two-particle distribution function and is not a matter of definition as might appear from Eq. (\[STLS\]). From the above considerations \[using Eqs. (\[phiCS\]) and (\[STLS\])\] we see that the mixed responses within the generalized STLS approach are obtained by replacing $ G^{\sigma, \sigma^{\prime}}(q , \omega )$ with ${\cal G}^{\sigma, \sigma^{\prime}}(q)$ in the expressions for the mixed responses given by Eqs. (\[chiCSV\]) and (\[chiCSH\]). It has been shown, by Ng and Singwi that $ V^{\sigma \sigma^{\prime}}_{\rm{eff}}(q)
\equiv v(q)[1- 2{\cal G}^{\sigma, \sigma^{\prime}}(q)]$ is related to the pair correlation function in real space as follows $$V^{\sigma \sigma^{\prime}}_{\rm{eff}}(r) =
- \int_{r} ^{\infty} \frac{d v(R)}{d R}
g_{\sigma \sigma^{\prime}}(R) d R.
\label{Veffgreltn}$$
We will now demonstrate that $g_{\sigma -\sigma } ( \vec{r} ) =
g_{- \sigma \sigma } ( \vec{r} )$. Firstly we observe, for an arbitrarily spin polarized EG in any dimension, that due to reflection symmetry the static pair correlation function $g_{\sigma \sigma ^{\prime}} ( \vec{r} )$ satisfies the relationship $g_{\sigma \sigma ^{\prime}} ( \vec{r} )
= g_{\sigma \sigma ^{\prime}} ( - \vec{r} )$. Next we note that the pair correlation function $g_{\sigma - \sigma } ( \vec{r} )$ is related to the instantaneous density-density correlation function through the relation $$g_{\sigma - \sigma } ( \vec{r} )
=
\frac {
<0|\rho ^{\sigma} (\vec{r} + \vec{r} ^{\prime})
\rho ^{{- \sigma}} ( \vec{r} ^{\prime}) |0> }
{N^{\sigma} N ^{- \sigma } } ,
\label{paircor}$$ where $N^{\sigma}$ is the number of spin $\sigma$ particles in a system of unit volume and $\rho ^{\sigma} (\vec{r})$ is the density operator in real space. Then, since the system has reflection symmetry and translational invariance, it follows that $g_{\sigma -\sigma } ( \vec{r} ) =
g_{- \sigma \sigma } ( \vec{r} )$ [@alternate]. Then from Eq. (\[Veffgreltn\]) it is clear that $ {\cal G}^{\sigma, -\sigma}(q)
= {\cal G}^{- \sigma, \sigma}(q)$ and thus we see that the two mixed responses are equal within the generalized STLS scheme. Here it should be pointed out that the many-body local fields ${\cal G}^{\sigma, \sigma^{\prime}}(q)$ in the generalized STLS approximation are not functions of frequency and thus are not the same as the true static many-body local fields $G^{\sigma, \sigma^{\prime}}(q, 0)$.
We will now proceed to give arguments for the equality of the two dynamic many-body local fields $G^{\uparrow , \downarrow} (q, \omega)$ and $ G^{\downarrow , \uparrow} (q, \omega )$. To this end we will derive the effective screened potential $W_{-\sigma \sigma}$ felt by a spin $-\sigma$ electron in the EG when a spin $\sigma$ electron sets up perturbations and then show that based on a symmetry of $W_{-\sigma \sigma}$ one obtains $G^{\uparrow , \downarrow} (q, \omega) =
G^{\downarrow , \uparrow} (q, \omega )$. Let $\rho ^{\sigma} _{e}$ be the number density of the perturbing electron setting up density fluctuations $\Delta n^{\sigma}$ and $\Delta n^{-\sigma}$. Then the potential felt by an electron of spin $\sigma ^{\prime}$ is given by $$\begin{aligned}
\Phi ^{\star} _{\sigma ^{\prime} \sigma } =
\Delta n ^{\sigma ^{\prime}} / \chi^{\sigma ^{\prime}}_{0}=
\phi _{\sigma ^{\prime} \sigma }
+
v(q) &&
[
\Delta n^{-\sigma}(1 - 2 G^{\sigma ^{\prime} , -\sigma} )
\nonumber \\
&& +
\Delta n^{\sigma}(1 - 2 G^{\sigma ^{\prime} , \sigma} )
] ,\end{aligned}$$ where $$\phi _{\sigma ^{\prime} \sigma } =
v(q) \rho^{\sigma} _{e}(1 - 2 G^{\sigma ^{\prime} , \sigma} ) .$$ On simplifying the above one gets $$\Phi ^{\star} _{- \sigma \sigma } =
\frac { \phi _{- \sigma \sigma } }
{\cal{D}} ,$$ where $\cal{D}$ is given by Eq. (\[calD\]). Now, as argued in Refs. and for an unpolarized system, the actual screened potential between the two electrons $ W_{\sigma ^{\prime} \sigma }$ is given by $$W _{\sigma ^{\prime} \sigma }
\rho^{\sigma} _{e} =
\Phi ^{\star} _{\sigma ^{\prime} \sigma } +
2 v(q) \rho^{\sigma} _{e}
{ G^{\sigma ^{\prime} , \sigma} } .
\label{Phisigmasigma}$$ The effective screened potential $ W _{\sigma ^{\prime} \sigma } $ obtained in such a fashion also agrees, for an infinitesimally polarized system, with the results of Ref. . Furthermore, in our treatment if $ \rho^{-\sigma}_{e}$ is a spectator electron which experiences an effective screened potential $W_{-\sigma \sigma} $, the interaction energy between the two electrons $ \rho ^{\sigma} _{e}$ and $ \rho^{-\sigma} _{e}$ is $W_{-\sigma \sigma}
\rho^{\sigma} _{e}
\rho^{-\sigma} _{e}$. From the fact that the interaction energy between the two electrons $ \rho ^{\sigma} _{e}$ and $ \rho^{-\sigma} _{e}$ is the same irrespective of whether $ \rho ^{\sigma} _{e}$ is the perturbing electron with $ \rho^{-\sigma} _{e}$ being the spectator electron or vice versa, we have $W_{\uparrow \downarrow}
= W_{\downarrow \uparrow} $. It then follows from Eq. (\[Phisigmasigma\]) that $G^{\uparrow , \downarrow} (q, \omega) =
G^{\downarrow , \uparrow} (q, \omega )$ and hence that $\chi_{CS}^{V}(\vec{q} , \omega ) =
\chi_{CS}^{H}(\vec{q} ,\omega )$.
We also note that the effective screened interaction between two electrons can be expressed as $$\begin{aligned}
W_{\sigma \sigma}
= && v(q)^2
\left \{
( 1 - G^{\sigma}_{+})^{2} \chi_{C}
+(G^{\sigma}_{-})^{2} \chi_{S}/(-\mu_{B}^2 )
\right .
\nonumber \\
&&
\left .
- \sigma ( 1 - G^{\sigma}_{+}) G^{\sigma}_{-}
[ \chi_{CS}^{V}+ \chi_{CS}^{H}] \right \}
\nonumber \\
+ v(q) && +2 \sigma v(q)^2 \chi ^{-\sigma}_{0}
( 1 - G^{\sigma}_{+} + G^{\sigma}_{-} )
\frac {G^{\uparrow \downarrow} -
G^{\downarrow \uparrow}}
{\cal{D}} ,\end{aligned}$$ and $$\begin{aligned}
W_{\downarrow \uparrow}
= v(q)^2 &&
\left \{
( 1 - G^{\uparrow}_{+})
( 1 - G^{\downarrow}_{+})
\chi_{C}
-G^{\uparrow}_{-}
G^{\downarrow}_{-}
\chi_{S}/(-\mu_{B}^2 ) \right .
\nonumber \\
&&
\left .
+( 1 - G^{\uparrow}_{+}) G^{\downarrow}_{-}
\chi_{CS}^{H}
-( 1 - G^{\downarrow}_{+}) G^{\uparrow}_{-}
\chi_{CS}^{V}
\right \}
\nonumber \\
+ v(q) && +2 v(q)^2 \chi ^{\downarrow}_{0}
( 1 - G^{\downarrow}_{+} - G^{\downarrow}_{-} )
\frac {G^{\uparrow \downarrow} -
G^{\downarrow \uparrow}}
{\cal{D}} .\end{aligned}$$ When $G^{\uparrow \downarrow} =
G^{\downarrow \uparrow}$ we see from the above two equations for $W_{\sigma \sigma}$ and $W_{\downarrow \uparrow}$ that the last term vanishes and the resulting expressions have a pleasing symmetry in terms of the response functions.
We will now prove that the two responses are identical to each other by invoking the reflection symmetry and the real valued property of the hamiltonian. For this purpose, we begin with the definitions of the mixed charge-spin response functions. $$\chi _{CS}^{V}(\vec{q}, t) \equiv -i \theta (t)
<0|[S^z_{-\vec{q}} (t),\rho_{\vec{q}}]|0> ,$$ and $$\chi _{CS}^{H}(\vec{q}, t) \equiv
- i \theta (t)
<0| [ \rho_{-\vec{q}} (t), S^z_{\vec{q}} ]|0> ,$$ where $\chi _{CS}^V$ $(\chi _{CS}^H)$ corresponds to the spin (charge) response when an external electric potential (magnetic field) is applied. Furthermore $\rho_{\vec{q}}$ and $ S^z_{-\vec{q}} $ represent respectively the charge and spin density fluctuation operators in momentum space, while $\theta (t)$ stands for the step function. From the above definitions one obtains (see Ref. for alternate derivations) $$\begin{aligned}
\chi _{CS}^{H} ( \vec{q} , \nu )
= \sum _{n} && \left \{
\frac {
\langle 0 | \rho _{-\vec{q}}| n \rangle \langle n | S
_{\vec{q}}^{z}|
0 \rangle }
{\nu -\omega _{n0} + i \eta}
\right .
\nonumber \\
&& ~~~~ \left .
- \frac {
\langle 0 | S _{\vec{q}} ^{z} | n \rangle \langle n |
\rho _{-\vec{q}}|
0 \rangle }
{\nu +\omega _{n0} + i \eta}
\right \} ,
\label{CSH}
\end{aligned}$$ and $$\begin{aligned}
\chi _{CS}^{V} ( \vec{q} , \nu )
= \sum _{n} && \left \{
\frac {
\langle 0 | S _{-\vec{q}} ^{z}| n \rangle \langle n |
\rho _{\vec{q}}|
0 \rangle }
{\nu -\omega _{n0} + i \eta}
\right .
\nonumber \\
&& ~~~~ \left .
- \frac {
\langle 0 | \rho _{\vec{q}}| n \rangle
\langle n | S _{-\vec{q}} ^{z} | 0 \rangle
}
{\nu +\omega _{n0} + i \eta}
\right \} .
\label{CSV}
\end{aligned}$$
Next, in the hamiltonian we note that the kinetic term ($p^2 _{i} /(2m)$) and the interaction term ($\sum_{i \neq j} v(\vec{r}_{i} -\vec{r}_{j})$) have inversion symmetry. Furthermore the Zeeman term ($g \mu _B \vec{\sigma}_{i} \cdot \vec{H} _{ext}$) also has reflection symmetry because both spin and magnetic field are pseudo vectors. Then, since reflection symmetry holds for all the terms of the hamiltonian, we have $\chi_{CS}^{V(H)}(\vec{q} ,\omega )
=\chi_{CS}^{V(H)}(-\vec{q} ,\omega )$ and hence we obtain the following equality: $$\begin{aligned}
\chi_{CS}^{V}(\vec{q} , \omega =0 )
= && \chi_{CS}^{H}(\vec{q} ,\omega =0 )
\nonumber \\
= &&
- \sum _{n}
\frac {1}{\omega_{n0}}
\left \{
<0|S^z_{\vec{q}} |n><n|\rho_{-\vec{q}}|0>
\right .
\nonumber \\ && ~~~~~~~~~~~~ +
\left .
<0|\rho_{-\vec{q}}|n> <n|S^z_{\vec{q}} |0> \right \} .
%\label{CSmag}\end{aligned}$$ Thus we see that reflection symmetry alone is sufficient to produce equality of the two mixed responses in the static case.
We will now prove that the two mixed responses coincide in the more general dynamic case as well. We first note that the hamiltonian is real when the external uniform magnetic field is taken to be in the z-direction leading to the Zeeman term being real. Then the eigenvectors of the hamiltonian can be taken to be real. Consequently we observe that $<n|\rho_{\vec{q}}|0> = <0|\rho_{\vec{q}}|n>$ and $<0|S^z_{\vec{q}} |n>= <n|S^z_{\vec{q}} |0>$. Based on the above equalities and reflection symmetry it follows from Eqs. (\[CSH\]) and (\[CSV\]) that $\chi_{CS}^{H}(\vec{q} ,\omega )
=\chi_{CS}^{V}(-\vec{q} ,\omega )
=\chi_{CS}^{V}(\vec{q} ,\omega )$. Here we would like to point out that reflection symmetry holds even if orbital effects are not negligible. However, when the orbital effects are included, the eigenvectors need not be real and hence the mixed responses need not be equal.
The arguments involving reflection symmetry and the hamiltonian being real can be generalized to give the equality of the response functions $ -i \theta (t) <0|[A_{-\vec{q}} (t),B_{\vec{q}}]|0> $ and $ -i \theta (t) <0|[B_{-\vec{q}} (t),A_{\vec{q}}]|0> $ where $A$ and $B$ are operators. Here, if in one response function we interchange the time independent operator that couples to the external field with the time-dependent operator that corresponds to the response of the system we get the other response function.
DISCUSSION AND CONCLUSIONS
==========================
In Ref. , Marinescu and Quinn obtained exact expressions for the many-body local fields of a polarized system in the large momentum limit or large frequency limit by using approaches similar to those used for unpolarized systems in Refs. and . In the large momentum limit, $G^{\uparrow , \downarrow}$ and $ G^{\downarrow , \uparrow}$ have the same functional dependence on the pair correlation functions $g^{\uparrow , \downarrow} (0) $ and $ g^{\downarrow , \uparrow} (0) $ respectively and hence it follows from reflection symmetry that $G^{\uparrow , \downarrow} (q \rightarrow \infty , \omega )
= G^{\downarrow , \uparrow} (q \rightarrow \infty , \omega ) $. Furthermore for finite $q$ and $\omega \rightarrow \infty $, we note that $\chi ^{\sigma} _0 (q , \omega )
\sim N^{\sigma} q^2/(m \omega ^2)$ and from Ref. that $G^{\sigma , -\sigma} (q , \omega \rightarrow \infty ) $ is even at least up to order $1/\omega ^2$. Then, from Eqs. (\[chiCSV\])–(\[calD\]) it follows that the difference between the two responses is of the form $$\begin{aligned}
\chi_{CS}^{V}(\vec{q} , \omega \rightarrow \infty )
- \chi_{CS}^{H}(\vec{q} ,\omega \rightarrow \infty )
\sim A/\omega ^4 + B/\omega ^6
+... ,\end{aligned}$$ and is thus an even function of $1/\omega $ to at least order $1/\omega ^6$. Next, from Eqs. (\[CSH\]) and (\[CSV\]), on using only reflection symmetry, we note that $$\begin{aligned}
\chi_{CS}^{V} (\vec{q} , \omega \rightarrow \infty )
- && \chi_{CS}^{H}(\vec{q} ,\omega \rightarrow \infty )
\nonumber \\
&&
=
2 \sum _{n}
\left \{
<0|S^z_{\vec{q}} |n><n|\rho_{-\vec{q}}|0> \right .
\nonumber \\
&& ~~~~~~~~
\left . - <0|\rho_{-\vec{q}}|n> <n|S^z_{\vec{q}} |0> \right \}
\nonumber \\ && ~~~~~~~~
\times
\left \{
\frac {1}{\omega} +
\frac {\omega_{n0}^2} {\omega ^3} +
\frac {\omega_{n0}^4} {\omega ^5} + ...\right \} .
%\label{CSmag}\end{aligned}$$ Hence, for $\omega \rightarrow \infty $ we see from the above two equations that the two mixed responses, when expanded as a series in powers of $1/\omega $, coincide at least to order $1/\omega ^6$.
In conclusion, within a generalized STLS framework we showed that the two charge-spin response functions coincide. Furthermore, we also presented arguments for their equality in the dynamic case by considering a symmetry of the effective screened interaction between two opposite spin electrons. Lastly, from the facts that the system has reflection symmetry and that the hamiltonian is real we established the equality of the two mixed response functions.
Extending the study of mixed responses to quantum Hall effect systems and including orbital effects could lead to some interesting insights.
[^1]
C. A. Kukkonen and A. W. Overhauser, Phys. Rev. B [**20**]{}, 550 (1979).
G. Vignale and K. S. Singwi, Phys. Rev. B [**32**]{}, 2156 (1985); K. S. Singwi, Physica Scripta [**32**]{}, 397 (1985).
S. Yarlagadda and G. F. Giuliani, Solid State Commun. [**69**]{}, 677 (1989).
S. Yarlagadda and G. F. Giuliani, Phys. Rev. B [**49**]{}, 7887 (1994).
D. J. Kim, B. B. Schwartz, and H. C. Praddaude, Phys. Rev. B [**7**]{}, 205 (1973).
K. S. Yi and J. J. Quinn, Phys. Rev. B [**54**]{}, 13398 (1996).
T. K. Ng and K. S. Singwi, Phys. Rev. B [**35**]{}, 6683 (1987).
Alternately, by defining the pair distribution function in terms of the many-body wavefunction, we can also show the equality $g_{\sigma \sigma ^{\prime}} ( \vec{r} ) =
g_{\sigma^{\prime} \sigma } ( \vec{r} )$.
S. Yarlagadda and G. F. Giuliani (unpublished).
D. C. Marinescu and J. J. Quinn, Phys. Rev. B [**56**]{}, 1114 (1997).
G. Niklasson, Phys. Rev. B [**10**]{}, 3052 (1974).
X. Zhu and A. W. Overhauser, Phys. Rev. B [**30**]{}, 3158 (1984).
[^1]: The author would like to thank Subrata Ray and A. Harindranath for discussions.
|
---
abstract: |
We discuss the role that dwarf galaxies may have played in the formation of the Galactic halo (Halo) using RR Lyrae stars (RRL) as tracers of their ancient stellar component. The comparison is performed using two observables (periods, luminosity amplitudes) that are reddening and distance independent. Fundamental mode RRL in six dwarf spheroidals and eleven ultra faint dwarf galaxies ($\sim$1,300) show a Gaussian period distribution well peaked around a mean period of $<$Pab$>=$0.610$\pm$0.001 days ($\sigma$=0.03). The Halo RRL ($\sim$15,000) are characterized by a broader period distribution.
The fundamental mode RRL in all the dwarf spheroidals apart from Sagittarius are completely lacking in High Amplitude Short Period (HASP) variables, defined as those having P$\ltsim$0.48 days and A$_V\ge$0.75mag. Such variables are not uncommon in the Halo and among the globular clusters and massive dwarf irregulars. To further interpret this evidence, we considered eighteen globulars covering a broad range in metallicity (-2.3$\lsim$\[Fe/H\]$\lsim$-1.1) and hosting more than 35 RRL each.
The metallicity turns out to be the main parameter, since only globulars more metal–rich than \[Fe/H\]$\sim$-1.5 host RRL in the HASP region. This finding suggests that dSphs similar to the surviving ones do not appear to be the major building–blocks of the Halo. Leading physical arguments suggest an [*extreme*]{} upper limit of $\sim50\%$ to their contribution. On the other hand, massive dwarfs hosting an old population with a broad metallicity distribution (Large Magellanic Cloud, Sagittarius) may have played a primary role in the formation of the Halo.
author:
- 'Giuliana Fiorentino$^{1}$, Giuseppe Bono$^{2,3}$, Matteo Monelli$^{4,5}$, Peter B. Stetson$^{6}$, Eline Tolstoy$^{7}$, Carme Gallart$^{4,5}$, Maurizio Salaris$^{8}$, Clara Mart[í]{}nez-V[á]{}zquez$^{4,5}$ & Edouard J. Bernard$^{9}$'
title: 'Weak Galactic halo–dwarf spheroidal connection from RR Lyrae stars'
---
Introduction {#intro}
============
{width="15.5cm"}
The early suggestion by [@searle78] that the Milky Way (MW) outer Halo formed from the aggregation of protogalactic fragments was supported i) theoretically, by $\Lambda$CDM simulations of galaxy formation in which small galaxies form first and then cluster to form larger galaxies, and ii) observationally, by the discovery of stellar streams and merging satellites in the MW [@ibata94] and in other galaxies. However, the characteristics of the halo building–blocks is still a matter of debate. In particular, the question of whether the current dwarf spheroidal (dSph) satellites of the MW are surviving representatives of the Halo’s building–blocks has been explored in several works [@tolstoy09]. The conclusions of these works differ in some details, but they suggest that there are difficulties in forming the Halo exclusively with dwarfs similar to current MW satellites. Among these, the studies using element ratios [@venn04], of stellar populations in dSphs and in the Halo are based on tracers (RGs) covering a wide range in age. Thus, they suffer from the drawback that some of the dSph present a complex evolution spanning several Gyr, while the evolution of the halo building–blocks was likely interrupted at early times, when they were accreted into the halo. This is the reason why old stellar tracers are crucial in the comparison between the MW halo and dSphs.
A real possibility to isolate the ancient (age $\ge$10Gyr) populations in these different stellar systems is offered by a special class of low mass, radial variables: RR Lyrae stars (RRLs). They pulsate in the fundamental (RRab) and in the first overtone mode (RRc). Due to their variability and relatively distinct light curves RRLs can be easily distinguished from other stars. Extensive variability surveys of our Galaxy have been performed and are nowadays releasing their final catalogues. We have compiled a huge catalogue ($\sim$14,700 stars) from QUEST [@vivas04; @zinn14], ASAS [@szczygiel09] and CATALINA [@drake13] surveys that have classified RRLs and provided Johnson $V$–band magnitudes and amplitudes. The final catalog is mainly based on CATALINA RRLs (85%) and covers a large range in galactocentric distances (5Kpc$\lsim$d$_G\lsim$60Kpc). Moreover, the sample radial distribution does not show evidence of gaps. This makes possible a direct comparison with dSphs where RRLs are always observed. We have gathered the results from accurate and quite complete photometric studies of classical dSphs (Draco, Carina, Tucana, Sculptor, Cetus and Leo I and some ultra faint dwarfs, hereinafter UFDs) that different research groups have carried out during the last ten years [see @stetson14b].
In @stetson14b we performed a first detailed analysis of the RRLs properties using these sizable samples. Very interestingly, comparing their period–amplitude (or Bailey) diagrams, we highlighted that, in the sample of six dSphs plus eleven UFDs that we considered, there are [*no*]{} RRab stars with A$_V\ge$0.75mag and P $\ltsim$0.48d. This was first found by @bersier02 in Fornax and not explained by the temporal sampling of the observations, since their probability to detect a period in such a range was always higher than 66%. The authors attributed it instead to the transition period between RRab and RRc–type variables. The same applies to the Draco dSph as discussed by @catelan09. We have observed this evidence in another five dSphs and eleven UFDs and named it the missing High Amplitude Short Period (HASP) RRab problem in dSphs. This evidence can not be related to photometric incompleteness of a single photometric dataset, particularly since high amplitude RRLs are the easiest to recognize among the variable candidates.
In this letter, using the properties of RRab in GGCs and taking advantage of predictions from theoretical models, we propose an explanation for the missing HASP problem in dwarfs. We also give a rough estimate of the upper limit to the contribution of dSph-like galaxies to the Halo stellar population. We close the letter extending the discussion to the possible contribution to the Halo of systems similar to the Large Magellanic Cloud (LMC) and the Sagittarius (Sgr) dwarfs.
[ccccccc]{} \[tabper\] $<$Pab$>$ & 0.584$\pm$0.001 & 0.576$\pm$0.001 & 0.610$\pm$0.001 & 0.575$\pm$0.002 & 0.576$\pm$0.001 & 0.580$\pm$0.002\
$\sigma_{Pab}$ & 0.08 & 0.07 & [**0.05**]{} & 0.07 & 0.07 & 0.07
The missing HASP RRab in dSphs {#hasp}
==============================
Fig. \[fig1\] shows the Bailey diagrams (left) and the period distributions (right) for the RRab observed in dSphs ([*a*]{} panel), in GGCs ([*b*]{} panel), in the inner (d$_G
% R_3
\lsim$16[^1] Kpc, [*c*]{} panel) and in the outer (d$_G \gsim$16 Kpc, [*d*]{} panel) halo. The boundary between inner and outer halo has been arbitrarily chosen to have comparable numbers of RRLs in the two samples and taking into consideration the value found in @carollo07 [d$_G \sim$15–20 Kpc]. We take advantage also of the exceptionally complete LMC sample from OGLE III [[*f*]{} panel, @soszynski10b] and of the recent release of OGLE IV that includes Sgr [[*e*]{} panel, @soszynski14]. The shaded grey area shows the location in Bailey diagram of the HASP region. Our analysis focuses on fundamental RRLs ($\log P\ge$-0.35). The shorter period first overtones will not be included, because they have smaller luminosity amplitudes and are, at fixed limiting magnitude, more affected than RRab by completeness problems.
The period distributions of RRab stars plotted in the right panels of Fig. \[fig1\] are quite different even if the mean periods achieve similar values to within 1$\sigma$ (Table \[tabper\]). In particular, the shape of the histogram corresponding to dSph and UFD is strikingly different from the rest. Not only the HASP are absent (except for two in Cetus, that appear to be peculiar for other reasons), but there is also a dearth (or a lower fraction) of short period variables (P$\lesssim$-0.25), compared to the other five samples. The symmetry of the dSphs period distribution can be fitted with a Gaussian function ($\sigma=$0.03, see Fig. \[fig1\]) and suggests also that metal–poor RRab in UFDs are still a minor fraction of the entire sample (NRR$_{UFD}$/NRR$_{UFD+dSphs}=$3%). Indeed, the RRab in UFDs tend to contribute significantly to the long period tail of dSphs [see Fig. 9 in @stetson14b]. The mean and the $\sigma$ of the period distribution observed in dSphs (in bold in Table \[tabper\]) does not increase when the RRab sample is almost doubled thanks to the inclusion of newly detected RRab stars (PBS, priv. comm.) in Fornax ($\gsim$1000 RRab) and Sculptor ($\sim$300).
{width="15.5cm"}
Why HASP RRLs are missing in dSph?
----------------------------------
Globular Clusters (GCs) are fundamental laboratories to constrain old stellar populations, since individually they host stars with similar age and chemical composition. To investigate the fine structure of the Instability Strips (IS) we selected 16 Galactic GCs hosting at least 35 RRLs according to the @clement01 catalog (2013 edition, see Fig. \[fig2\]). To extend the metallicity range covered by the selected GGCs we included two LMC globulars, namely NGC 1466 and NGC 2257 [@walker92; @walker93]. The entire sample covers a range in metallicity of \[Fe/H\] from $\sim$–2.3 to –1.1 [@harris96 2010 edition]. To constrain the metallicity dependence the entire sample of GCs was split into four arbitrary metallicity bins. Every bin in metallicity includes at least four GCs.
A glance at the data plotted in Fig \[fig2\] clearly shows that the HASP region starts to be filled only when RRLs have a metallicity $\gtrsim$–1.5dex. It becomes more populated when the metal content increases to –1dex. The above evidence suggests the hypothesis that metallicity is the key parameter causing the lack of HASPs in dSphs. It would imply that the maximum metallicity reached by the stellar population to which the RRL belong in dSphs is \[Fe/H\]$\lesssim$–1.5dex, and that the other stellar systems have reached a higher metallicity at the early time when they were still able to produce stars of a mass suitable for becoming today’s RRL. Additionally, a firm dependence of the [*mean*]{} periods on the metallicity can be observed in Figure \[fig2\]. The $\langle$Pab$\rangle$, when moving from the metal–poor to the metal–rich regime, decreases from 0.644$\pm$0.007 to 0.599$\pm$0.006d, while the $\langle$Pc$\rangle$ decreases from 0.364$\pm$0.003 to 0.300$\pm$0.004d. Moreover and even more importantly, the fraction of HASPs over the total number of RRab stars is vanishing in the two most metal–poor bins and becomes of the order of 3% and 14% in the two most metal–rich bins, in order of increasing metallicity.
[cccccccccccc]{} \[tabHASP\] N$_{HASP}$/NRRL&8% & 6% & 0% & 6% & 1% & 3% & 9–12%$^a$ & 7%$^a$ & 2%$^a$ & 6% & 17%\
<\[Fe/H\]>$\pm$$\sigma$ &n.c.& n.c.& $\lsim$-1.4$\pm$0.1$^{11}$ & -0.5$\pm$0.4$^{11}$ & -1.0$\pm$0.2$^{11}$ & n.c. & n.c.& -0.25$\pm$0.1$^{11}$ &-0.5$\pm$0.1$^{12}$ &-0.4$\pm$0.2$^{11}$ &-0.3$\pm$0.5$^{13}$\
To further validate the above trend we investigated the occurrence of HASPs in other nearby stellar systems (see Table \[tabHASP\]). For some of these systems, space observations (in F606W filter) are available. In order to select the HASP RRab we converted F606W amplitudes into the Johnson–Cousin photometric system, assuming A$_{F606W}/$A$_V$=0.92 [@brown04]. We selected RRab with periods shorter than 0.48d and luminosity amplitudes larger than A$_{F606W}=$0.69. We found that the ratio of HASP RRL to total number of RRab follows a trend similar to GCs, and indeed, they range from a few percent in systems where the mean metallicity is poor (SMC) to more than $\sim$10% in more metal–rich systems (Bulge).
In this context, the two peculiar RRab in Cetus [@bernard09], located in the HASPs region, might trace the tail of a metal–rich stellar component. Their luminosity is $\sim$0.1mag fainter than the remaining $\sim$500 RRLs, thus suggesting an important metallicity increase in the early star formation event experienced by this quite massive galaxy.
Insights from pulsation and evolutionary theory {#theory}
-----------------------------------------------
Non-linear, convective hydrodynamical models of radial variables indicate that RRab have their largest amplitudes close to the fundamental blue edge [FBE, @bono94].
The FBE boundary is almost constant over a broad range of metal abundances (–2.3$\lsim$\[Fe/H\]$\lsim$–1.3, @bono95a). This means that the pulsation properties of an RRL across the IS are dictated mostly by its evolution. A change in chemical composition causes a change in stellar mass and in luminosity, and in turn a change in the morphology of the evolutionary paths crossing the IS. Pulsation and evolutionary prescriptions indicate that the minimum period reached by RRab, i.e., the period at the FBE, decreases as the metal content increases. In particular, @bono97d showed that logP$_{ab}^{min}$ decreases from –0.26 to –0.37 when Z increases from 0.0001(\[Fe/H\]$\sim$–2.3, using $\alpha$–enahnced values) to 0.001(\[Fe/H\]$\sim$–1.3). These predictions agree quite well with the minimum period observed in M3 and in M15—globulars characterized by sizable samples of RRLs.
This scenario suggests that old stars in dSphs, in spite of their complex star formation and chemical enrichment history, are characterized by a narrow metallicity distribution when compared with relatively “simple” stellar systems in the MW such as GCs. A complementary conclusion was reached by @salaris13 studing in detail the Horizonthal Branch morphology of the Sculptor dSph. They found that this can be explained, at odds with GCs, without invoking He–enhanced models [@rood73]. This evidence further supports the above findings, since an increase in the helium content would imply, at fixed intrinsic parameters, a steady increase in the pulsation period [@marconi11], thus further exacerbating the HASPs problem.
![Same as in Fig. \[fig1\], but for the RRLs for which an estimate of metallicity exists from SDSS [@drake13]. \[fig3\]](fig3.ps){width="8.5cm"}
Evidence from SDSS optical spectra {#spectra}
----------------------------------
The recent evidence of multiple stellar populations in GCs [@monelli13 and reference therein] and the fact that the horizontal branch in dSphs can be reproduced without assuming any helium enrichment or complex mass–loss law [@salaris13], induce us to question whether we are using too complex stellar systems to understand the HASP dearth in dSph. In order to corroborate the hypothesis that low metallicity is the cause of the missing HASP in dSphs, we take advantage of the medium resolution SDSS spectra available for a sample of $\sim$1,400 fundamental mode RRLs [@drake13]. Given that the RRLs span a magnitude range from 14 and 20mag, the errorbar of each individual metallicity estimate has been estimated of $\lsim$0.2dex as discussed in [@yanny09]. In Fig. \[fig3\] we show the Bailey distribution of these stars grouped in four metallicity bins. Note that RRLs in the above metallicity bins cover similar ranges in Galactocentric distances (5$\lsim$d$_G\lsim60$ Kpc). We can clearly see that metal–rich groups of RRLs (\[Fe/H\]>–1.7, top panel) tend to populate the HASP region whereas the metal–poor ones (\[Fe/H\]$\lsim$–1.7) leave this region almost empty, starting from period lpgP$\lsim$–0.3. Although the absolute value of the metallicity at which the transition occurs would need a careful calibration of these data, we find a trend in agreement with that observed in GCs.
Implications for the early formation of the Halo
================================================
We have shown that the period distribution of RRL in nearby dSphs follows a Gaussian distribution with a smaller dispersion ($\sigma=$0.03) than the Halo one, which is more skewed to short periods. This peculiarity of the dSphs is not observed in GGCs, the LMC or the Sgr dwarf. Furthermore, we found evidence that, in order to populate the HASP region, an old component more metal–rich than \[Fe/H\]$\sim$–1.5 is required (see Fig. \[fig2\]). In the following, we will analyze the evidence provided by the RRL populations on the building of the Halo from the combination of different types of progenitors. In this exercise, we will assume that the Halo sample is statistically significant, i.e., that an increase in its size would not affect the shape of the period distribution[^2].
We will first try to obtain an upper limit on the Halo fraction originating in dSph-like systems. For that, the period distribution of RRab in dSph (panel [*a*]{} of Fig. \[fig1\]) has been rescaled to fill the maximum possible area under the curves representing the period distribution for both the inner and outer Halo (red histograms in Fig. \[fig1\], panels [*c*]{} and [*d*]{}). Assuming that the RRab falling inside the area covered by the red distribution have been entirely accreted from dSphs, we find that the maximum contribution of dSph-like systems into the inner and the outer Halo can not be more than $\sim$50%. This fraction has to be cautiously treated. In fact, it is [*an extreme upper limit*]{} since the [*difference*]{} between the black and red histograms consists almost entirely of short-period variables, completely unlike the observed LMC and Sgr distributions. Any admixture at all of these latter two population types to fill in the short periods would result in far too many halo variables with log P $\sim-0.22\,$d.
Even though the above results rely on rough preliminary estimates, they pose a serious question: [*Where does the rest (in fact most) of the Halo mass come from?*]{} There are two main proposed scenario: [*i)*]{} from few large and metal–rich stellar systems LMC or Sgr–like [as suggested by @zinn14; @tissera14]; [*ii)*]{} [*in situ*]{} stellar formation [@brusadin13; @vincenzo14].
From the exceptionally complete OGLE III and the new OGLE IV data for the LMC and the Sgr dwarf respectively, presented in Section \[hasp\], we have noticed that their RRab populations share similar properties to that of the Halo in terms of mean period and sigma, and HASP fraction (Table \[tabHASP\]). We remember here that the application of a one–dimensional Kolmogorov–Smirnov (KS) test on the LMC, dSphs, GGCs, inner and outer halo period distributions [@stetson14b] strongly support the evidence that their cumulative distributions are not drawn from the same parent population (probability $\lsim$0%). Interestingly enough, the one–sample KS–test applied to the new Sgr RRab population does support, with a not negligible probability (10%[^3] ), that the outer Halo and Sgr are drawn from the same distribution. We highlight here that the exceptional completeness of the huge OGLE LMC sample may partially hide similarities between the RRab Halo and LMC distributions.
We performed the same exercise described above, in order to estimate the fraction of the Halo that may have formed from systems similar to the LMC or Sgr. We apply a scaling factor to the LMC and Sgr RRab period distribution in order to match the largest possible fraction of the halo distribution. We find that $\sim$80–90% of the halo may have been formed from this kind of stellar systems, thus supporting the hypothesis [*i)*]{}. The HASP fraction of the Halo further supports previous conclusions [@venn04; @helmi06] that typical dSphs played a minor role, if any, in its early formation. In this paper we have provided evidence that HASP RRL are missing in dSph because these galaxies did not reach a high enough metallicity during the time they were able to produce RRLs. In other words, in their internal chemical evolution the dSphs achieved a metallicity \[Fe/H\]$\sim$–1.5 [*too recently in the past*]{} for stars of a mass suitable for making RR Lyraes to be currently evolving from the main sequence. The LMC, Sgr, and the Halo, in contrast, achieved these higher metallicities more quickly. This gives an indication that the early chemical enrichment histories of dSphs and more massive stellar systems are dissimilar, in the sense that chemical enrichment was faster in larger galaxies. This is in agreement with the well defined scaling relation between mass/luminosity and metallicity [@chilingarian11; @schroyen13; @kirby08] obeyed by dwarf galaxies, with more massive galaxies being more metal-rich, and having a broader metallicity distribution. It follows that the Halo was made primarily from progenitor galaxies larger than those that survived to become today’s dSphs. Future surveys like GAIA will provide a census of a significant fraction of the Halo, discovering more than 70,000 new RRLs [@eyer00]. This is the required new information to constrain whether the major contributors of the Halo should be sought in massive dwarf galaxies—LMC and/or Sgr–like—or in some different formation scenario.
Financial support for this work was provided by FIRB 2013 (RBFR13J716, PI G. Fiorentino), IAC (grant 310394), ESMS (grant AYA2010-16717) and PRIN–MIUR (2010LY5N2T, PI F. Matteucci). We thank an anonymous referee for her/his pertinent suggestions.
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[^1]: This distance is slightly smaller than the distance adopted by @stetson14b (14Kpc). The difference does not affect the conclusions of the paper.
[^2]: The validity of this assumption depends on the Galactocentric distance. The new variability surveys [CATALINA, @drake13 their Fig. 13] appear to be quite complete ($\sim$50%) out to 40 kpc (V$\sim$18mag). If this assumption is wrong—that is, if the Halo is affected by a significant amount of uncompleteness—this could affect the results of the Kolmogorov–Smirnov test presented in this section. However, we can be confident that it would not affect the HASP fractions: in all the stellar systems, including the Halo, the HASP fraction is not sensitive to the cut in amplitude used. This evidence suggests that the Halo sample incompleteness is far from severe.
[^3]: This result is not affected by possible presence of Sgr RRLs in the outer Halo. We neglected RRLs with d$_G\gsim30 Kpc$ [@zinn14; @drake13] and both the mean period (<Pab> = 0.576$\pm$0.001 \[0.07\]) and the correlation (10%) attain similar values.
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abstract: 'Point defect species and concentrations in metastable Fe-C alloys are determined using density functional theory and a constrained free-energy functional. Carbon interstitials dominate unless iron vacancies are in significant excess, whereas excess carbon causes greatly enhances vacancy concentration. Our predictions are amenable to experimental verification; they provide a baseline for rationalizing complex microstructures known in hardened and tempered steels, and by extension other technological materials created by or subjected to extreme environments.'
author:
- 'Clemens J. Först,$^{1,2}$ Jan Slycke,$^{3}$ Krystyn J. Van Vliet,$^{2,*}$ and Sidney Yip$^{1,2}$'
title: 'Point defect concentrations in metastable Fe-C alloys'
---
Many industrially significant alloys are intentionally processed as metastable microstructures comprising a supersaturation of crystal defects in various forms of aggregation [@Vehanen82; @Ogawa02]. How the defects are distributed and in what local concentrations they exist are fundamental questions which are not yet resolved experimentally, yet it is this microstructural complexity that governs the performance of the material. Hardened steels are an important example where deformation behavior is intrinsically coupled to the lattice defects: in this case, a body-centered cubic (bcc) Fe matrix with carbon content in excess of equilibrium values for ferrite (solid solution of carbon in bcc Fe) and cementite (Fe$_3$C), as well as high dislocation density and supersaturation of vacancies [@Vehanen82]. While it is generally known that carbon binds strongly to crystal defects such as vacancies, what is lacking is a robust computational strategy capable of linking the fundamental physics of crystal defects and their energetics with quantitative, microscopic details of the defect microstructure as it evolves during processing and service.
In this work we determine the concentration of point defects and defect clusters in Fe-C alloys using an approach based on first-principles calculation of the formation energies of specific defects, and a free-energy formulation allowing either the carbon or the iron vacancy concentration to be out of equilibrium. We find that the vacancy concentration in the form of carbon-vacancy clusters increases dramatically in the presence of carbon due to strongly exothermic clustering reactions. Nevertheless, the concentrations of carbon-vacancy clusters remain orders of magnitude below the concentration of interstitial carbon. Only under the assumption of a significant supersaturation of vacancies will carbon-vacancy clusters begin to dominate the defect spectrum, in accord with experimental observation of dominant carbon-vacancy clusters in irradiated bcc-Fe alloys [@Lavrentev98]. Ab-initio results on the structure and energetics of vacancies, self-interstitials and carbon interstitials in bcc-Fe have been reported previously [@Jiang03; @Domain01], as well as the interaction of one and two C interstitials with the Fe vacancies [@Domain04]. Our results at 0K are in close agreement with cluster formation energies reported previously [@Domain04]. Additionally, we have considered clusters with higher carbon contents and included di-vacancies in our considerations.
The supersaturation of carbon in such alloys has been established through several experimental approaches, although the degree of supersaturation in the bcc-Fe matrix ranges from 0.3at.% C upon 400$^\circ$C annealing [@Ohsaki05] to up to 5at.% C in volumes including ferritic grain boundaries [@Wilde00; @SKF]. While experiments can quantify local carbon concentrations (e.g., three-dimensional atom probe [@Wilde00]) or determine the carbon-vacancy binding energy [@McKee72] (a value of 0.85eV was reported by positron annihilation measurements [@Vehanen82]), it is not yet feasible to measure the relative concentrations and specific structures of C-vacancy clusters in bcc-Fe during thermal processing. The most likely C-related defects are: (1) simple interstitial supersaturation [@Ohsaki05]; (2) co-association with vacancies as inferred from reduced vacancy diffusion [@Vehanen82] and carbon segregation to vacancy rich regions [@Lavrentev98]; (3) direct observation of elevated carbon concentration up to 8at.% near dislocation cores [@Wilde00]; (4) up to 6.6at.% at grain boundaries [@Wilde00; @Ohsaki05].
We have performed a set of total energy calculations on the interaction of carbon with single and double vacancies and applied the results in a statistical mechanics model to obtain parameter-free estimates of the concentration of point defects in various structural configurations in the bcc-Fe matrix. We employed density functional theory (DFT) [@Kohn; @KohnSham; @PBE] using Blöchl’s projector augmented wave method [@Blo94; @Blo03] as implemented in the <span style="font-variant:small-caps;">VASP</span> code [@vasp2; @vasppaw] with a plane-wave cutoff of 400eV. The calculations were performed in 128-atom supercells with the theoretical lattice constant of 2.83Å, using a k-mesh of $2\times 2\times 2$ and a Methfessel Paxton Fermi-surface smearing parameter of 0.05eV [@Methfessel89]. No symmetry constraints were imposed. Screening calculations and vibrational frequency calculations were carried out in 54-atom supercells. The geometry optimization was terminated with a force cutoff of 5meV/Å. All calculations included spin polarization, starting with a ferromagnetic charge density. Because a wide variety of different bulk and defect structures are considered, we anticipate an error of 0.1–0.2eV for the defect formation energies. At the target temperature of 160$^\circ$C, where the body centered tetragonal phase transforms to bcc, this implies error in species concentration of one to two orders of magnitude. However, the overall conclusions regarding the dominant defect concentrations are not found to be affected by this level of uncertainty.
Table \[tab:eform\] summarizes the formation energies of the defects and defect clusters under consideration,
$$\begin{aligned}
\nonumber E^\mathrm{form}(T,\mu_{Fe}, \mu_C) &=&
E^\mathrm{D}(T) - E^\mathrm{0}(T) \\ && - \mu_{Fe}\cdot \Delta n_{Fe} -
\mu_C\cdot \Delta n_C, \label{eqn:eform} \end{aligned}$$
where $E^\mathrm{D}(T)$ and $E^\mathrm{0}(T)$ denote the DFT energies of the supercells with and without defects present, and $\mu_X$ and $\Delta
n_X$ are the chemical potential of species $X$ and the difference in atom number between the two supercells. Temperature dependence is taken into account through configurational and vibrational free energy contributions, the latter being calculated using the dynamical matrix of all atoms affected by the defect formation. To approximate the vibrational free energy we consider each atom as a three-dimensional harmonic oscillator with force-constants derived from ab-initio total energy calculations. The vibrational frequencies of carbon are found to be strongly dependent on the defect geometry, varying between 7 and 34THz. Variation of the frequencies of Fe atoms in the vicinity of a (double-) vacancy with respect to the bulk value is less than 3THz and is therefore neglected. The vibrational contributions to the formation energies can amount to 0.2eV at 160$^\circ$C.
defect species $E^\mathrm{form}$ \[eV\] defect species $E^\mathrm{form}$ \[eV\]
------------------------ -------------------------- ------------------------ --------------------------
$C_I$ octahedral 0.58 $(2Fe_V + 2C)$ \[100\] 3.31
$Fe_V$ 2.02 $(2Fe_V + 3C)$ \[100\] 3.21
$Fe_I$ 3.91 $(2Fe_V + 4C)$ \[100\] 2.77
$(Fe_V + 1C)$ 1.96 $(2Fe_V + 5C)$ \[100\] 3.23
$(Fe_V + 2C)$ 1.53 $(2Fe_V + 6C)$ \[100\] 3.74
$(Fe_V + 3C)$ 1.98 $(2Fe_V + 7C)$ \[100\] 5.78
$(Fe_V + 4C)$ 3.03 $(2Fe_V + 1C)$ \[111\] 3.66
$(Fe_V + 5C)$ 6.58 $(2Fe_V + 2C)$ \[111\] 3.06
$(Fe_V + 6C)$ 13.48 $(2Fe_V + 3C)$ \[111\] 2.91
$(2Fe_V)$ \[100\] 3.83 $(2Fe_V + 4C)$ \[111\] 2.43
$(2Fe_V)$ \[111\] 3.85 $(2Fe_V + 5C)$ \[111\] 3.50
$(2Fe_V + 1C)$ \[100\] 3.49 $(2Fe_V + 6C)$ \[111\] 4.76
: Formation energies for the different defect stoichiometries at 160$^\circ$C. The chemical potentials $\mu_{Fe/C}$ (compare Eqn. \[eqn:eform\]) are chosen to represent thermal equilibrium of bulk Fe and Fe$_3$C. $X_I$ and $X_V$ refer to an interstitial and a vacancy of species $X$, respectively. Parentheses (…) denote a defect cluster. The crystallographic directions \[…\] refer to the orientation of the iron double-vacancies.[]{data-label="tab:eform"}
Table \[tab:eform\] shows the most stable configuration for a given stoichiometry. While there are different ways to arrange a given number of C atoms around an Fe vacancy, we find that these typically have very different formation energies. A more extensive list of energetics of one and two carbon atoms near a single vacancy can be found in Ref. [@Domain04]. Here we regard structures with formation energies within 0.1eV (our approximate error) to be degenerate. Structures with energies greater than than 0.3eV relative to the lowest-energy configuration are not considered.
Figure \[fig:defectstrc1\] shows the stable geometries for an Fe vacancy surrounded by up to six carbon atoms. In the case of di-vacancies, we need to distinguish between two possible orientations \[100\] and \[111\], which are energetically practically degenerate in the carbon-free state (compare Fig. \[fig:defectstrc2\] and Table \[tab:eform\]). Around a \[111\] vacancy there are two types of adsorption sites: those with one of the six coordinating Fe atoms missing (shaded sides in Fig. \[fig:defectstrc2\]) and those with two missing coordinating atoms. We find that configurations with two missing coordinating Fe atoms are energetically unfavorable. Otherwise, the stable carbon geometries are analogous to that of the isolated vacancy. In the case of the \[100\] oriented di-vacancy, the configuration of a single carbon atom situated between the two vacancies is most favorable; however, this preference does not hold for pairs and triplets of C atoms. From screening a wide variety of C geometries, including chains and tetrahedra connecting two vacancies, no new, low-energy carbon configurations are found.
![Atomic structure of C interstitials and C-vacancy clusters in bcc-Fe. Fe atoms in open circles, carbon atoms filled circles. Left panel: octahedral and tetrahedral carbon interstitials; right panel: position of carbon atoms around a vacancy. A carbon-vacancy cluster with $n$ carbon atoms contains the C atoms with labels from 1 to $n$.[]{data-label="fig:defectstrc1"}](fig1 "fig:"){width="4cm"} ![Atomic structure of C interstitials and C-vacancy clusters in bcc-Fe. Fe atoms in open circles, carbon atoms filled circles. Left panel: octahedral and tetrahedral carbon interstitials; right panel: position of carbon atoms around a vacancy. A carbon-vacancy cluster with $n$ carbon atoms contains the C atoms with labels from 1 to $n$.[]{data-label="fig:defectstrc1"}](fig2 "fig:"){width="3.5cm"}\
![Atomic structure of C-double vacancy clusters oriented in \[100\] (left panel) and \[111\] (right panel) directions. Refer to Fig. \[fig:defectstrc1\] for labeling. In case of the \[100\] vacancy, the position ’(1)’ is only occupied if there is a single C atom.[]{data-label="fig:defectstrc2"}](fig3 "fig:"){height="5.2cm"} ![Atomic structure of C-double vacancy clusters oriented in \[100\] (left panel) and \[111\] (right panel) directions. Refer to Fig. \[fig:defectstrc1\] for labeling. In case of the \[100\] vacancy, the position ’(1)’ is only occupied if there is a single C atom.[]{data-label="fig:defectstrc2"}](fig4 "fig:"){height="4.5cm"}
The formation of a C-vacancy cluster by combining a vacancy with $n$ carbon interstitials,
$$Fe_V + n\cdot C_I \longrightarrow (Fe_V + n\cdot C),$$
is seen to give a negative reaction enthalpy $\Delta H$ for (Table \[tab:eform\]). The same is also predicted for two single vacancies/C-vacancy clusters forming a di-vacancy/C-di-vacancy cluster (e.g., $2(Fe_V + 2C) \rightarrow (2Fe_V
+ 4C)\ [111]$ with $\Delta H = -0.6$eV). However, one should note that the mass-action law for a clustering reaction such as $X+X
\rightarrow X_2$ relates the concentration of the cluster $c_{X_2}$ to the square of the concentration of the individual defects, $(c_X)^2$. Thus the defect complex is the dominating species only if $\exp(-\frac{\Delta H}{k_BT})$ is larger than $1/c_X$. For typical defects in solids, $c_X$ is often very small ($\ll$ 1ppm) and $\Delta H$ is on the order of $-0.1$eV, so this condition is not necessarily fulfilled. Nonetheless, enthalpy provides a measure of “thermal stability" of a complex once it is formed, as in the case of vacancy agglomerates [@Vehanen82].
The equilibrium concentrations of defects $c_i$ in a macroscopic crystal are given by minimizing the free energy $F(T,\mu_{Fe},\mu_C)$ at given temperature $T$ and chemical potentials $\mu_{Fe}$ and $\mu_C$:
$$\begin{aligned}
\nonumber
F(T,\mu_{Fe},\mu_C) &=&
\sum_{i=1}^{M}
c_i E^\mathrm{form}_i(T,\mu_{Fe},\mu_C)
\\
+ k_BT &&\hspace*{-6mm} \sum_{i=1}^M \left[\vphantom{\sum} c_i\ln c_i +
(1-c_i)\ln (1-c_i) \right]
,
\label{eqn:efree}\end{aligned}$$
where $E^\mathrm{form}$ is defined in Eqn. \[eqn:eform\], $M$ denotes the number of different defect species and $c_i$ is the concentration of defect species $i$ [@footnote1]. One obtains from Eqn. \[eqn:efree\]
$$c_i(T,\mu_{Fe},\mu_C) = \frac{1}{\exp\left(
\frac{E^\mathrm{form}_i(T,\mu_{Fe},\mu_C)}{k_B T}\right) +1 }.
\label{eqn:conc}$$
The dependence on the two chemical potentials, which enters through the formation energies, can be turned into an explicit dependence on the total carbon and/or vacancy concentrations if we impose the respective constraints:
$$\sum_{i=1}^M c_i\cdot \Delta n^i_{Va/C} =
c^\mathrm{tot}_{Va/C}.
\label{eqn:constraint}$$
This is useful for performing calculations under physically motivated hypotheses.
As discussed above, carbon and vacancy concentrations in realistic microstructures of Fe-C alloys need to be treated as nonequilibrium. To make the calculations tractable we consider two alternative hypotheses under the condition of partial thermodynamic equilibrium. That is, we have thermal equilibrium for one species (e.g., Fe) and non-equilibrium for the other (e.g., C), thus reducing the problem to a study of the effects of the concentration of the non-equilibrium species. In hypothesis (1) the carbon concentration is not in equilibrium with the Fe$_3$C/Fe system, whereas Fe is assumed to be in equilibrated with bulk Fe. This scenario is motivated by the experimental observation that carbide precipitation is kinetically hindered and requires tempering at elevated temperatures [@Ohsaki05]. It is also implied that the Fe matrix remains substantially undistorted. In hypothesis (2) the vacancy concentration is assumed to be non-equilibrium, whereas carbon is assumed to equilibrate with the bulk Fe/Fe$_3$C system. This reflects the anticipated processing conditions for hardening of Fe-C alloys; in the course of martensitic transformation, large plastic strain leads to high intrinsic defect concentrations, with vacancies binding strongly to carbon interstitials. Due to the increased diffusion barriers associated with carbon-vacancy clusters [@Vehanen82] and the absence of a comparable number of Fe interstitials (high formation energies), it is likely that a significant vacancy concentration is effectively frozen in. These two hypotheses represent extreme scenarios; any intermediate situation can also be studied using the present data.
Figure \[fig:ofcc\] shows the consequences of hypothesis (1), the variation of dominant defect species concentrations with the total carbon concentration. These results are obtained by taking the chemical potential $\mu_{Fe}$ to be the bulk Fe value, while $\mu_C$ is given by the total carbon concentration (Eqn. \[eqn:constraint\]). We observe that the presence of carbon in the matrix can cause the vacancy concentration (including carbon-vacancy clusters) to increase by 15 orders of magnitude relative to the intrinsic vacancy concentration ($Fe_V$) at a total carbon concentration of 1at.% Comparable increases in vacancy concentration have been reported in the presence of a different impurity atom, hydrogen [@Gavriljuk96; @McLellan97]. We also note that the concentration of carbon-free di-vacancies is negligible.
![Defect concentrations in a bcc-Fe matrix as a function of the total carbon concentration at 160$^\circ$C. Labeling as in Table \[tab:eform\]. The vertical dashed line marks thermal equilibrium with bulk cementite. Note that that small precipitates are expected to have less favorable energetics in terms of atomic and magnetic structure, which may shift the equilibrium carbon concentration to higher values.[]{data-label="fig:ofcc"}](fig5){width="7.5cm"}
Figure \[fig:ofcv\] shows the defect concentration variation with total iron vacancy concentration under hypothesis (2). In this case $\mu_C$ is fixed at the reference value, the coexistence of Fe$_3$C and bulk Fe. We observe that the number of carbon atoms associated with vacancies exceeds the number of carbon interstitials only when the vacancy concentrations are larger than $10^{-4}$at.%, which corresponds to a typical equilibrium vacancy concentration near the melting point [@RHA94]. This correlates with the experimental finding that C-vacancy clusters have been observed to be the dominant carbon-species in irradiated samples [@Lavrentev98], which cannot be explained under the assumption of an equilibrated vacancy concentration (hypothesis (1)).
![Defect concentrations in a bcc-Fe matrix as a function of the total vacancy concentration at 160$^\circ$C. []{data-label="fig:ofcv"}](fig6){width="7.5cm"}
In conclusion, our work provides the first quantitative assessment of the interplay of different defect species in Fe-C alloys. We predict defect phase diagrams as a function of total carbon and vacancy concentrations, results which are amenable to experimental verification. As a computational framework for addressing microstructure complexity, our approach should be applicable to other advanced technological materials subject to extreme environmental conditions.
The authors thank Alejandro Sanz and Fred Lucas for stimulating discussions, and acknowledge SKF for financial support. We have benefited from computational resources funded by NSF (IMR-0414849).
[99]{} contact author: Krystyn J. Van Vliet ([email protected]) A. Vehanen, P. Hautojärvi, J. Hohannson, J. Yli-Kauppila and P. Moser, Phys. Rev. B **25**, 762 (1982). E. T. Ogawa, J. W. McPherson, J. A. Rosal, K. J. Dickerson, T.-C. Chiu, L. Y. Tsung, M. K. Jain, T. D. Bonifield, J. C. Ondrusek, and W. R. McKee, Proc. 40th Int. Reliability Physics Symp., p312 (2002). V.I. Lavrent’ev, A.D. Pogrebnyak, A.D. Mikhalev, N.A. Pogrebnyak, R. Shandrik, Z. Zecca and Y.V. Tsvintarnaya, Tech.Phys.Lett. **24**, 334 (1998) D.E. Jiang and E.A. Carter, Phys.Rev.B. **67**, 214103 (2003). C. Domain and C.S. Becquart, Phys.Rev.B. **65**, 24103 (2001). C. Domain, C.S. Becquart and J. Foct, Phys.Rev.B. **69**, 144112 (2004). S. Ohsaki, K. Hono, H. Hidaka and S. Takaki, Scripta.Met. **52**, 271 (2005). J. Wilde, A. Cerezo and G.D.W. Smith, Scripta Mater. **43**, 39 (2000). J. Slycke, SKF Engineering Research Centre, The Netherlands, unpublished work. B. T. A. McKee, W. Triftshäuser, and A. T. Stewart, Phys. Rev. Lett. **28**, 358 (1972) P. Hohenberg and W. Kohn, Phys. Rev. **136**, B864 (1964). W. Kohn and L.J. Sham, Phys. Rev. **140**, A1133 (1965). J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. **77**, 3865 (1996). P.E. Blöchl, Phys. Rev. B **50**, 17953 (1994). Peter E. Blöchl, Clemens J. Först and Johannes Schimpl, Bull. Mater. Sci. **26**, 33 (2003) G. Kresse and J. Furthmüller, Phys. Rev. B. **54**, 11169 (1996). G. Kresse and D. Joubert, Phys. Rev. B **59**, 1758 (1999). M. Methfessel and A.T. Paxton, Phys.Rev.B **40**, 3616 (1989) Note that we have assumed here that different defect species do not compete for sites in the bcc lattice, i.e. the number of iron sites available is large compared to the number of defects in the system. Considering a competition for defect sites the numerator of Eqn. \[eqn:conc\] would be $(1 - \sum_{j\ne i} c_j)$. R.E. Reed-Hill and R. Abbaschian. Physical Metallurgy Principles. Boston: PWS Publishing Company (1994). V.G. Gavriljuk, V.N. Bugaev, Y.N. Petrov, A.V. Tarasenko and B.Z. Yanchitski, Scripta Mater. **34**, 903 (1996) R.B. McLellan and Z.R. Xu, Scripta Mater. **36**, 1201 (1997)
|
---
abstract: 'In this paper we report on LCG Monte-Carlo Data Base (MCDB) and software which has been developed to operate MCDB. The main purpose of the LCG MCDB project is to provide a storage and documentation system for sophisticated event samples simulated for the LHC collaborations by experts. In many cases, the modern Monte-Carlo simulation of physical processes requires expert knowledge in Monte-Carlo generators or significant amount of CPU time to produce the events. MCDB is a knowledgebase mainly dedicated to accumulate simulated events of this type. The main motivation behind LCG MCDB is to make the sophisticated MC event samples available for various physical groups. All the data from MCDB is accessible in several convenient ways. LCG MCDB is being developed within the CERN LCG Application Area Simulation project.'
address:
- 'Joint Institute for Nuclear Research, Dubna, Moscow region, Russia, 141980'
- 'Scobeltsyn Institute of Nuclear Physics of Lomonosov Moscow State University, Moscow, Russia, 119992'
- 'Institute For High Energy Physics, Protvino, Russia, 142281'
- 'CERN/SFT, CH-1211 Geneva 23, Switzerland'
- 'Cavendish Laboratory, University of Cambridge, CB3 0HE, UK'
author:
- 'S. Belov'
- 'L. Dudko'
- 'E. Galkin'
- 'A. Gusev'
- 'W. Pokorski'
- 'A. Sherstnev'
title: 'LCG MCDB – a Knowledgebase of Monte-Carlo Simulated Events.'
---
Program Summary {#progsum}
===============
[*Manuscript Title: LCG MCDB – a Knowledgebase of Monte-Carlo Simulated Events.*]{}\
[*Authors: S. Belov, L. Dudko, E. Galkin, A. Gusev, W. Pokorski, A. Sherstnev*]{}\
[*Program Title:LCG Monte-Carlo Data Base*]{}\
[*Journal Reference:* ]{}\
[*Catalogue identifier:*]{}\
[*Licensing provisions: GPL*]{}\
[*Programming language: Perl*]{}\
[*Computer: CPU: Intel Pentium 4, RAM: 1 Gb, HDD: 100 Gb*]{}\
[*Operating system: Scientific Linux CERN 3/4*]{}\
[*RAM: 1073741824 bytes (1 Gb)*]{}\
[*Number of processors used: 1*]{}\
[*Supplementary material:*]{}\
[*Keywords: MCDB, Monte-Carlo Simulation, Monte-Carlo Generators, LHC, Knowledgebase.*]{}\
[*PACS: 01.50.hv, 07.05.-t, 07.05.Tp, 07.05.Wr*]{}\
[*Classification: 9 Databases, Data Compilation and Information Retrieval*]{}\
[*External routines/libraries:\
perl $>=$ 5.8.5; Perl modules (DBD-mysql $>=$ 2.9004, File::Basename, GD::SecurityImage,\
GD::SecurityImage::AC, Linux::Statistics, XML::LibXML $>$ 1.6, XML::SAX,\
XML::NamespaceSupport); Apache HTTP Server $>=$ 2.0.59; mod\_auth\_external $>=$ 2.2.9; edg-utils-system RPM package; gd $>=$ 2.0.28; rpm package CASTOR-client $>=$ 2.1.2-4; arc-server (optional).* ]{}\
[*Subprograms used:*]{}\
[*Catalogue identifier of previous version:*]{}\
[*Journal reference of previous version:*]{}\
[*Does the new version supersede the previous version?:*]{}\
[*Nature of problem:\
Often, different groups of experimentalists prepare similar samples of particle collision events or turn to the same group of authors of Monte-Carlo (MC) generators to prepare the events. For example, the same MC samples of Standard Model (SM) processes can be employed for the investigations either in the SM analyses (as a signal) or in searches for new phenomena in Beyond Standard Model analyses (as a background). If the samples are made available publicly and equipped with corresponding and comprehensive documentation, it can speed up cross checks of the samples themselves and physical models applied. Some event samples require a lot of computing resources for preparation. So, a central storage of the samples prevents possible waste of researcher time and computing resources, which can be used to prepare the same events many times.* ]{}\
[*Solution method:\
Creation of a special knowledgebase (MCDB) designed to keep event samples for the LHC experimental and phenomenological community. The knowledgebase is realized as a separate web-server (mcdb.cern.ch). All event samples are kept on types at CERN. Documentation described the events is the main contents of MCDB. Users can browse the knowledgebase, read and comment articles (documentation), and download event samples. Authors can upload new event samples, create new articles, and edit own articles.* ]{}\
[*Reasons for the new version:*]{}\
[*Summary of revisions:*]{}\
[*Restrictions:\
The software is adopted to solve the problems, described in the article and there are no any additional restrictions.* ]{}\
[*Unusual features:\
The software provides a framework to store and document large files with flexibal authentication and authorization system. Different external storages with large capacity can be used to keep the files. The WEB Content Management System provides all of the necessary interfaces for the authors of the files, end-users and administrators.* ]{}\
[*Additional comments:*]{}\
[*Running time:\
Real time operations.* ]{}\
[*References:*]{}
The main LCG MCDB server `http://mcdb.cern.ch/`
P. Bartalini, L. Dudko, A. Kryukov, I. V. Selyuzhenkov, A. Sherstnev and A. Vologdin, “LCG Monte-Carlo data base,” \[arXiv:hep-ph/0404241\].
J. P. Baud, B. Couturier, C. Curran, J. D. Durand, E. Knezo, S. Occhetti and O. Barring, “CASTOR: status and evolution,” \[arXiv:cs.oh/0305047\].
Introduction {#intro}
============
The LCG MCDB project [@Dobbs:2004bu; @Bartalini:2004nd] has been created to facilitate communication between experts/authors of Monte-Carlo (MC) generators and users of the programs in the LHC collaborations.
The current version of LCG MCDB provides flexible infrastructure to share samples of events of particle collisions in accelerators prepared by the MC method (MC event samples) and keep the files in a reliable and convenient way. It has several interfaces, mainly Web-based, which help to carry out routine operations with stored samples by end-users and authors of the samples.
LCG MCDB is particularly useful in tasks where the preparation of event samples requires specific knowledge of the Monte-Carlo codes/techniques applied, significant computing power, and/or constant interaction between end-users and the authors of the events. In many standard tasks events can be produced “on the fly” keeping just initial “data-cards”, i.e. MC code parameter values which unambiguously define a concrete simulation run. But if the simulation time or exploitable resources become a significant factor, it would be worth considering the event sample as a candidate to keep in LCG MCDB. For instance, this situation can arise if we use such MC programs as ALPGEN [@Mangano:2002ea], CompHEP [@Boos:2004kh], GRACE [@Yuasa:1999rg], or MadGraph [@Maltoni:2002qb]. Even MC generators as PYTHIA or HERWIG sometimes require the keeping of event files themselves. Examples of this sort happen in simulations of rare processes and/or with strong pre-selection cuts.
The second motivation behind the project is to create a central database of MC events, where stored event samples are publicly available for all groups to use and/or validate. Often, different groups of experimentalists prepare similar event samples or turn to the same group of authors of MC generators in the simulation. For example, the same MC samples of Standard Model (SM) processes can be employed for the investigations either in the SM analyses (as a signal) or in searches for new phenomena in Beyond Standard Model analyses (as a background). Publicity of the event samples equipped with corresponding and comprehensive documentation can speed up cross checks of the samples themselves and physical models applied. It also prevents possible waste of researcher time and computing resources.
The previous version [@cms_mcdb_url] of MCDB was launched by the CMS collaboration in 2002. Several years of extensive use of the database have shown some limitations of a design applied in CMS MCDB. Storage of event files on AFS [^1] allows one to keep only small sized MC samples (basically parton level events prepared by Matrix Element tools), phonetic-based search turned out to be insufficient, and the database does not have simple tools to reuse information entered in MCDB earlier, such as physical parameters (masses, couplings, etc.), process information (name, PDF, particle content), generator information, etc. The new design of MCDB presented in this paper overcomes these problems and gives opportunities for further development of the idea. LCG MCDB is based on much more powerful, standardized and exportable software tools that are available to the LHC collaborations. Current migration of CMS physical groups from old CMS MCDB to the LCG framework gives us the motivation for the further development of the tool.
In the next sections we describe the LCG MCDB design and ideas in more detail and briefly portray subsystems and modules of LCG MCDB. Section \[howto\] reports how end-users can use the software. A more detailed manual and installation instructions are available on the LCG MCDB server ( [@lcg_mcdb_url], help section).
At present, LCG MCDB is a stable software package and ready to use for the LHC community. A dedicated web server is deployed in [@lcg_mcdb_url].
LCG MCDB as a knowledgebase {#concept}
===========================
Knowledgebase is a special kind of database for knowledge management. It provides the means for the computerized collection, organization, and retrieval of knowledge [@wikipedia_kb]. According to the definition one of the specific features of knowledgebase is that it keeps metadata or meta-information, i.e. information on data. Usually it is not possible to strictly distinguish between data and metadata, since the separation depends on situations where the data are exploited. In our concrete case we discriminate between events, as sets of particle 4-momenta (data), and information describing the events as one entity, an event sample (metadata). The latter is also not very strictly defined. For example, if the number of particles in an event is the same for the whole sample, we can add the parameter to metadata. But if it varies from event to event it is certainly a part of data. In our definition of metadata we try to single out the most common characteristic of event samples, which could be applied in most cases. Metadata form the main contents of MCDB. In this sense, MCDB can hold a path to an event sample only and the sample itself can be located somewhere else.
The benefit of the separation is the following. MCDB interfaces provide the means to manipulate with metadata only[^2]. This simplifies the structure and software of MCDB drastically. Thus by means of MCDB interfaces an end-user can search for a necessary event sample (according to given criteria), comprehend what the sample holds, and how the events were prepared. In other words, MCDB should let the end-user know how to reproduce the events. According to the idea the metadata must describe the corresponding event sample in a comprehensive manner. This information should be entered by the event sample author. In some cases, metadata are encoded inside the event file itself and can be inserted to MCDB (semi-)automatically.
Comprehensive description of an event sample requires a lot of information, which should be entered to the database. However, in practice, in this specific data domain a large part of the information is common for lots of samples. For example, the Standard Model processes are described by a large set of SM couplings and particle parameters (masses, widths, etc.)[^3], but usually, only few parameters are modified from one sample to another. In the MCDB conception we introduce “Model” – a set of parameters, which can be attached to an event sample. An author of events can choose an appropriate model and change a few parameters in the model and store the modified set of parameters as a new model with a new name. The same solution is used in the description of MC generators. We introduce a standard record to describe the programs. The author simply chooses one of the standard records and attaches it to a new article. In order to include the features in the author interface we developed our own Content Management System with a flexible structure, which can be extended in a simple manner.
The second idea behind the current design of MCDB is that MCDB is an area for interaction between two different communities, producers of events and consumers of the events. We call the groups “authors” and end-users respectively. Since the goals and tasks of the two groups are different, the corresponding MCDB interfaces intended for each group of users should also be different. Any researcher who feels his/her sample is worthy to come within MCDB can make a request to open a new author account on the MCDB server. It means MCDB does not assume to have a special team (of event producers) to prepare events according to end-user requests.
There are several blocks in LCG MCDB, which should be realized:
- Content Management System with a powerful and flexible Web interface for authors of event samples. It should have several types of templates to simplify the task of event sample description.
- A block of tree graph of physical categories with articles published by authors. This is the main part MCDB visible via Web browsers with no authentication in MCDB.
- A powerful search engine based on SQL/XML to search for contents of MCDB.
- A programming interface to CASTOR [@Baud:2003ys], which is used as a native storage of event samples.
- A block of direct uploading of files from Web/AFS/CASTOR/GRID to MCDB.
- Block of direct downloading of files from MCDB via Web/CASTOR/GRID (URI).
- A flexible and reliable authentication system based on CERN AFS/Kerberos logins or LCG GRID certificates
- Backup system for all stored samples and corresponding SQL information. g
- API to the LHC collaboration software environments.
- The standard record of an event sample. The record should be encoded to a set of SQL tables.
- A unified and flexible format of event files based on the LHEF agreement and the HepML language. A programming package which supports the format.
Metadata in LCG MCDB {#parameters}
--------------------
In general, metadata can hold very specific information and can be presented in an arbitrary form. In fact, it is one of the main problems of knowledgebases, since the arbitrariness results in problems in introducing effective and relevant search methods in knowledgebases. Our situation is simpler than the general case and we can limit ourselves by some general set of parameters which cover most parts of the necessary information on event samples. Owing to specific purposes and application area of MCDB, we can define the standard record for MCDB articles. Now the record corresponds to a set of parameters stored as a record in our relational DB and a comment written by the author in free form. All information which is not kept within the standard record can be and should be put in the comment. MCDB search requests use the standard records to retrieve information. If MCDB users, authors or end-users, request to add new parameters to the standard record it can be extended.
The standard information to describe event samples can be divided into several blocks. Each of the blocks corresponds to a definite set of parameters which are necessary to interpret a concrete event sample. The list below gives a short description of the main blocks:
- General information about a simulated event sample or a group of samples
- Title of physical process
- Physical Category (e.g. Higgs, Top physics or W+jets processes)
- Abstract (short description)
- List of authors
- Name of an experiment and/or a group (for which the sample was prepared or intended)
- Author comment on the sample (some additional unstructured information on the sample)
- Physical process
- Initial state (names of beam particle, energy, etc.)
- Final state (name of the final particles, etc.)
- QCD scale(s)
- Process PDF (parton distribution functions) applied
- Information on separate subprocesses, if they are distinguished
- Event file
- File name
- The number of events
- Cross section and cross section error(s)
- Author comment
- Used MC generator
- Name and version
- Short description
- Home page Web-address
- Theoretical model used to simulate the events
- Name
- Short Description
- A set of physical parameters and their values with the author’s descriptions
- Applied cuts
LCG MCDB Software Description {#soft}
=============================
This section describes shortly all subsystems and software technologies adopted in LCG MCDB. The current version of LCG MCDB is based on the following technologies: WWW, CGI, Perl, SQL, XML, CASTOR, and GRID. MCDB is a Web server written as a set of Perl CGI scripts. The scripts interact with relational DB by means of SQL requests and can generate either Web pages or XML documents. The main storage of event files is based on tape robots at CERN available via CASTOR. The MCDB software is organized as a set of Perl modules with the possibility of installing and customizing the software on other sites. All of the MCDB software has been developed from scratch and is available publicly in LCG CVS [@lcg_mcdb_cvs_savanna].
For the whole contents of LCG MCDB we provide a daily backup of the SQL DB and double mirroring of the samples in CASTOR. The main unit of MCDB is an [**article**]{}, a document describing one or several event samples. MCDB articles are distributed into [**categories**]{}, i.e. a set of articles concerning a particular type of physical process (e.g. top physics, Higgs physics) or theoretical model (e.g. supersymmetry, extra dimensions). Each category has its own branch in the main MCDB Web tree graph. The access system in MCDB reminds of a classical system used in the usual Internet forums or newsgroups. There are four different types of permissions to access MCDB. The [**end-user**]{} access is reserved for users who are interested in requesting a new event sample or in downloading or making comments to already published event samples. The [**author**]{} access is reserved for authorized users (MC experts) and requires registration on the main Web site. Only authors can upload and describe new event samples. The [**moderator**]{} access is reserved for users who manage author profiles and are responsible for general monitoring of information uploaded. The [**administrator**]{} access is reserved for software developers and maintainers who take care of the database itself. The scheme of LCG MCDB is shown in Fig. \[lcg\_mcdb\_scheme\].
=10.0cm
Web interface
-------------
The Web-interface consists of two parts:
- End-user area, where any user can search for a necessary event sample in the whole set of available samples. Requests can be done either via a search form or by browsing the main tree graph of categories where all articles are available. Users can read the description of the events, download the samples, ask questions about the samples and read the previous discussions on the particular event sample.
- Author area, where authors can upload new event files to the database and describe the events using the MCDB template system. As we mentioned, the users do not need to enter all the necessary information from scratch, since the templates have a lot of pre-entered information. Authors can interact with the end-users of their samples on public forums attached to each article. With the same interface, authors can edit his/her own previous articles or make the articles temporarily inaccessible in the end-user area.
SQL DB
------
LCG MCDB adopts MySQL. The SQL technology provides the possibility to keep information in a very structured way. Authors provide documentation on event samples through forms and MCDB scripts translate the information to records in the MCDB relational database.
Search engine
-------------
Since we use a relational DB, it is possible to provide a variety of complex search queries, which can use specific relations between DB records. The deployed Web search interface is realized as a dynamic query construction wizard which is based on the JavaScript XML-query constructor. The development of application programming interfaces to specific external software (for example a simulation framework of a LHC experiment) may benefit from similar tools in order to simplify the query construction.
Storage
-------
As a native storage interface for event samples we have selected CASTOR, because of the absence of serious space limitations on tapes and taking into account popularity of the interface in the LHC collaborations. We provide direct CASTOR paths for all LCG MCDB samples and also several options to obtain the samples through other interfaces (HTTP, GridFtp etc.). A local disk cache system is used to speed up the storage operations.
Authentication
--------------
We pay special attention to the security of transactions during all LCG MCDB operations. All of the transactions are encrypted by SSL technology. There are two ways to log in to MCDB. The first one relies on CERN AFS/Kerberos login/password. The second mechanism uses LCG GRID certificates. Authors can choose either or both of these ways. These authentication methods are the standard at CERN and any CERN user can use at least one of these two methods.
Documentation
-------------
Most of the LCG MCDB documentation is available from the dedicated web server. The information consists of different parts. Technical documentation describes the current implementation of LCG MCDB itself. The user documentation is organized as a set of HOW-TOs for end-users and authors. A separate documentation (available from the CVS repository) is intended for developers of the LCG MCDB software. A brief start-up manual for non-experienced LCG MCDB users is also available in the next section of this document. In addition, there are two freely accessible mailing lists dedicated to users and developers. Their addresses are available in the documentation section on the main web page of the server.
How to use LCG MCDB {#howto}
===================
An end-user who is going to look for and download events for a particular process can browse the MCDB categories and subcategories (the menu at the left side of the main LCG MCDB web page [@lcg_mcdb_url]) and verify, whether an appropriate sample has already been generated. If this is the case, the end-user may read the sample article describing how the events have been prepared (check parameters of the theoretical model, generator name and generation parameters, kinematic cuts, etc.). At the bottom of the page there is a link to the uploaded file(s), as well as a direct CASTOR path to the sample. The web page also contains a link to the “Users Comments” interface, where end-users can ask questions about the sample and browse the previous discussions on the article. Users do not need any special authorization to carry out all the steps described above. The search engine provides different possibilities of search queries based on the set of main parameters of the article and samples.
If someone wants to upload a new event sample or publish a new article in LCG MCDB (it means the user will become an author), (s)he should follow the following procedure:
- Register as a new author. There is a link to the registration interface on the right side menu of the main MCDB Web page. Wait for a confirmation e-mail.
- Login to the LCG MCDB authors area.
- Choose the option “Create New Article” in the authors menu. It appears at the right side after authentication.
- Fill all necessary forms in the documentation template (title, generator, theoretical model, cuts, etc.)
- Upload event files in the “Event Files” sub-window.
- Tick the box “Publish” and click “Preview/Save”.
As we mentioned above, for authentication the author needs a valid CERN AFS login or a LCG GRID digital certificate.
Authors can save unfinished articles in MCDB and resume to edit them later. Authors can edit their previous articles that are already published on the Web or make the articles publicly inaccessible for a while.
The LCG MCDB team appreciates any bug reports, feedback, comments or suggestions for possible new implementations and improvements of the service (LCG MCDB).
API to collaboration software {#api}
=============================
Apart from the MCDB server, LCG MCDB team provides application programming interfaces (APIs) specific to the simulation environments of the LHC collaborations. The main idea of these subsystems is to develop a set of routines for the collaboration software which would give a direct access to the LCG MCDB files during the MC production on computer farms.
The most simple way to access event samples is to use direct WEB, CASTOR or GRID path to the event samples. This way does not require any special software developments on the side of collaboration software. This way, however, does not provide any possibility for automatic access to event sample description. This is the reason we developed a more complicated interface which could be used for automatic processing of event samples and the corresponding documentation. According to our idea, MCDB team provides API based on XML representation of event sample metadata. The current version of the API is a C++ library, which can be added to collaboration software. The XML output from LCG MCDB is based on the HepML [@hepml],[@cedar] specifications (for more details, see the next section).
The current library contains C++ classes and provides routines to fill the class objects with information from a MCDB article, including CASTOR/GRID/HTTP paths to event files attached to the article. Such an interface has already been implemented in the CMS collaboration software environment. The software and documentation are available in LCG CVS [@lcg_mcdb_cvs_savanna].
=12.5cm
Fig. \[lcg\_api\_scheme\] reports a general scheme of an interaction between LCG MCDB to external user software via the API.
In the future, some emphasis will be put on the development of extensions of API specific to the automatic uploading of HepML information and event samples to MCDB. This development will be carried out in the context of the HepML project.
A unified XML format HepML {#hepmlsec}
==========================
At present, each MC generator supports its own output format of event files. Authors of Matrix Element tools (the term originates from [@Boos:2001cv]) provide interface programs to pass the events of a particular MC generator to the subsequent level of simulation (i.e. showering, hadronization, decays, simulation of detector response). The first step to standardize such interfaces has been described in the agreement “Les Houches Accord Number One” (LHA-I) [@Boos:2001cv], where a definite and strict structure of FORTRAN COMMON BLOCKS to transfer the necessary information from one code to another was fixed. The second step in this direction has been done in the agreement “Les Houches Event File” (LHEF) [@Alwall:2006yp], where the information fixed in LHA-I is translated to the event file structure. All other information can be kept in a specific place inside the header of the event file. The standard does not apply any limitations on the extra information and the structure of the block. The next natural step is to provide a unified format to keep the necessary information within the LHEF structure. In this context the other information means the metadata described in the section \[concept\] and some other information specific to the sample (parameters of matching in different schemes, information on specific NLO approximations, jet parameters, etc.).
Owing to the highly dissimilar nature of the information, the most appropriate technology for the unified representation could be XML-based format. In this case it can provide the possibility to describe the stored information in a very flexible and structured way.
The main idea behind the XML-based format is the flexibility to build and include a set of necessary parameters in an event file. For example, different MC generators may use the same tags for description of the physical parameters or they may need to keep specific information (through introduction of new dedicated tags). The new tags do not spoil the event file format and we do not need to re-write our routines which process these event files automatically. HepML is now being developed within the special LCG HepML project in collaboration with the CEDAR project [@Butterworth:2004mu]. More information is available on our wiki [@hepml-wiki].
As the first part of HepML we have prepared several XML Schemas. The main goal of the Schemas is to provide a general and formal description of event data structures which are kept in XML files. Adapting the idea authors of MC codes can use powerful XML tools in developing of I/O routines. If the routines are consistent with the Schemas, event files generated by the routines can be read by other programs without changes in input routines of the programs. Also the Schemas can be used for validation of event files if the files are written according to HepML specifications. Now we have three main Schemas. The first XML Schema [*lha1.xsd*]{} corresponds to the whole set of parameters composing the LHA-I agreement. The other two Schemas, [*sample-description.xsd*]{} and [*mcdb-article.xsd*]{}, describe parameters, which are necessary to generate an XML data for an event sample and to form an LCG MCDB article for the sample. It means it includes all parameters mentioned in the list in Sect. \[parameters\] except for some parameters from LHA-I. The CEDAR team develops other XML Schemas for other tasks arising in the problem of automatization of data processing in HEP. Now all the Schemas are unified in one general formal XML Schema [*hepml.xsd*]{}, which includes all the other Schemas as sub-Schemas. This solution leaves freedom to develop Schemas and software in independent groups, but to use Schemas of both groups in one software project. All the developed Schemas are available in [@hemplschemas].
Possible internal adaption of LHEF and HepML formats into the most popular MC generator projects would result in a significant improvement of the MC event sample documentation and book-keeping. Such adaption of the unified standards provides the possibility to develop new standard interfaces and utilities. The LCG MCDB project already implements a part of HepML specifications in MCDB API. A dedicated document discussing the details of the requirements and describing the HepML proposal will appear in the near future [@hepmlpaper].
Conclusion
==========
MCDB is a special knowledgebase designed to keep event samples for the LHC experimental and phenomenological community. Now, a new version of the software has been finished and the server is ready for use by the community. Some new important features are implemented in the software. The features simplify and improve the process of documentation of event samples.
In addition to the server, MCDB team has prepared an API for the LHC collaboration software environments. Implementation of the API to the software environments could give a possibility to use MCDB as a native storage in large-scale productions in collaborations.
Subsequent development of the software will rely on further standardization of event file formats and elaboration of the HepML specifications and software.
Acknowledgements
================
This work was partially supported by the RFBR (the RFBR grant 07-07-00365-a). We thank Seyi Latunda-Dada for discussions of the text. We also acknowledge the LCG collaboration for support and hospitality at CERN. Participation of A. S. in the project was partly supported by the UK Particle Physics and Astronomy Council.
[00]{}
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[^1]: The Andrew File System (AFS) is a distributed networked file system developed by Carnegie Mellon University as part of the Andrew Project.
[^2]: Except for interfaces which are responsible for downloading and uploading of event files. See more details in the next sections.
[^3]: A worse situation arises in Beyond SM models, where we can have hundreds of physical parameters in some models
|
---
abstract: 'Two-dimensional density-matrix renormalization group method is employed to examine the ground state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction $U$ at $U_{\rm c1}\sim 7.8t$ and $U_{\rm c2}\sim 9.9t$ ($t$ being the hopping integral), indicating that there are three phases separated by first order transitions. The absence of any singularity in physical quantities for $0 \le U < U_{\rm c1}$ implies a metallic phase in this regime. For $U > U_{\rm c2}$, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the $120^{\circ}$ Néel ordered state realized in the limit of $U \to \infty$. For $U_{\rm c1} < U < U_{\rm c2}$, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the ${\rm K}$ and ${\rm K}^{\prime}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing $U$ and thus the superconductivity is unlikely in the intermediate phase.'
author:
- 'Tomonori Shirakawa$^{1}$'
- 'Takami Tohyama$^{2}$'
- 'Jure Kokalj$^{3,4}$'
- 'Sigetoshi Sota$^{5}$'
- 'Seiji Yunoki$^{1,5,6}$'
title: 'Ground-state phase diagram of the triangular lattice Hubbard model by the density-matrix renormalization group method'
---
Introduction
============
There have been accumulating experimental evidences that several organic materials, $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ [@shimizu], EtMe$_3$Sb\[Pd(dmit)$_2$\]$_2$ [@itou; @yamashita1; @yamashita2], and $\kappa$-H$_3$(Cat-EDT-TTF)$_2$ [@isono], forming a quasi two-dimensional (2D) triangular structure and exhibit quantum spin liquid (QSL) [@patricklee; @balentz], where any spatial symmetry breaking does not occur due to the quantum fluctuation, even when it is cooled down to zero temperature. The realization of QSL against a symmetry-broken ordered state in higher spatial dimensions more than one dimension is one of the long standing issues in condensed matter physics [@patricklee] since the first proposal of resonating valence bonds (RVB) states by Anderson [@anderson]. It has been considered that one of the key ingredients for the emergence of stable QSL is geometrical frustration [@anderson], which increases quantum fluctuations and thus prevents symmetry breaking. In this context, the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice had been considered [@fazekas]. However, recent numerical studies, including two-dimensional density-matrix renormalization group (2D-DMRG) analysis, have suggested that the ground state of the spatially isotropic model is $120^{\circ}$ Néel ordered [@bernu; @capriotti; @zheng; @white3].
In addition to the geometrical frustration, other factors for stabilizing QSL have also been considered, such as i) the spatially anisotropic exchange interactions [@yunoki], ii) the higher order corrections of exchange interactions [@motrunich], and iii) the charge degree of freedom [@note; @hotta; @naka; @watanabe2017]. The later two are captured by the triangular lattice Hubbard model at half electron filling described by the Hamiltonian $$\begin{aligned}
\mathcal{H} = - t \sum_{\left< i,j \right>} \sum_{\sigma = \uparrow,\downarrow}
( c_{i,\sigma}^{\dagger} c_{j,\sigma} + {\rm h.c.} ) + U \sum_i n_{i,\uparrow} n_{i,\downarrow},
\label{eq:ham}\end{aligned}$$ where $c_{i,\sigma}$ ($c_{i,\sigma}^{\dagger}$) represents the annihilation (creation) operator of an electron with spin $\sigma\, (= \uparrow,\downarrow)$ at site $i$ on the triangular lattice, $n_{i,\sigma}= c_{i,\sigma}^{\dagger} c_{i,\sigma}$, and the sum $\left< i, j \right>$ runs over all pairs of nearest-neighbor sites $i$ and $j$. Indeed, the QSL phase in the organic materials appears next to the metallic phase, indicating that the QSL occurs close to the Mott metal-insulator transition [@kurosaki; @furukawa; @kandpal] where the above two factors ii) and iii) are important. In fact, it has been extensively argued that the triangular lattice Hubbard model is the simplest effective model to describe and understand the metal-insulator transition and the QSL phase in the organic materials [@powell].
Elucidating the ground state phase diagram of the triangular lattice Hubbard model at half-filling is a challenge for numerical techniques in strongly correlated electron systems. Various numerical methods [@koretsune; @clay; @kokalj; @sahebsara; @yamada; @morita; @yoshioka; @yang; @kyung; @liebsch; @galanakis; @ohashi; @sato; @lee; @dang; @li; @watanabe1; @watanabe2; @tocchio1; @tocchio2] have been applied so far, but the results are, nevertheless, controversial. The exact diagonalization techniques [@koretsune; @clay; @kokalj], the variational cluster approximation (VCA) [@sahebsara; @yamada], the path-integral renormalization group (PIRG) method [@morita; @yoshioka], as well as the recently proposed ladder dual-Fermion approach [@li] have suggested that there exist three phases in the ground state phase diagram with increasing $U/t$, i.e., a metallic phase, a nonmagnetic insulating phase, and the $120^{\circ}$ Néel ordered insulating phase. This is also supported by the numerical analysis of an effective strong coupling spin model [@yang]. On the other hand, the variational Monte Carlo methods [@watanabe1; @watanabe2; @tocchio1; @tocchio2] have suggested the absence of the nonmagnetic insulating phase. Besides the presence or absence of the intermediate insulating phase, the critical $U_c/t$ for the metal-insulator transition significantly varies among different methods [@kokalj].
Recently, the 2D-DMRG method has been applied to various 2D strongly correlated quantum systems [@stoudenmire; @jiang1; @gong1; @yan; @jiang2; @dependbrock; @nishimoto; @gong2; @gong3; @ganesh1; @ganesh2; @shinjo; @tohyama; @zhu; @hu; @sellmann; @shinjo2; @okubo2017], although the DMRG method is best performed for one-dimensional gapful systems [@white1; @white2; @schollwock; @hallberg1]. This is because the 2D-DMRG calculations with keeping large enough number of adapted density-matrix eigenstates to guarantee the desired numerical accuracy have become possible within reasonable computational resources, specially, for 2D spin-1/2 Heisenberg models [@stoudenmire].
Here, we employ the 2D-DMRG method to examine the ground state phase diagram of the repulsive Hubbard model on the triangular lattice at half electron filling. Our calculation reveals two discontinuities in the double occupancy of electrons with increasing $U/t$ at $U_{\rm c1}= 7.55t\sim8.05t$ and $U_{\rm c2}= 9.65t\sim10.15t$ for three different clusters up to 48 sites, strongly indicating that there are three phases separated by first order transitions at $U_{\rm c1}$ and $U_{\rm c2}$. The spin oscillation pattern for $U > U_{\rm c2}$ under a pinning magnetic field exhibits a three sublattice feature, compatible with the $120^{\circ}$ Néel ordered state. Moreover, the spatial distribution of the nearest-neighbor spin correlation is found to be quite different among the three phases. The suppression of oscillatory behavior in the intermediate phase at $U_{\rm c1} < U < U_{\rm c2}$ suggests this phase in neither bond order nor valence bond solid. In addition, the spin correlation length in the intermediate phase is found to be larger than that for $U < U_{\rm c1}$ but smaller than that for $U > U_{\rm c2}$. Furthermore, the response to a pinning magnetic field in the intermediate phase is rather comparable to that in the paramagnetic metallic state. These features in the intermediate phase resembles gapless spin liquid [@powell; @normand]. However, the spin structure factor in the intermediate phase shows a single maximum at the ${\rm K}$ and ${\rm K}^{\prime}$ points in the momentum space, which is not compatible with the expectation for the spinon Fermi sea state [@motrunich]. Superconductivity is also excluded in the intermediate phase.
The rest of this paper is organized as follows. First, the shape of 2D clusters studied here is introduced and the convergence of the DMRG calculations is discussed in Sec. \[sec:method\]. Section \[sec:cal\] is devoted to our results for the triangular lattice Hubbard model at half-filling. We first show the ground state energy and the double occupancy to reveal the existence of three phases in Sec. \[sec:energy\]. Next, we explore the properties of the ground state in each phase by calculating different quantities, including the response to a pinning magnetic field in Sec. \[sec:pinning\], the spin correlation function in Sec. \[sec:spincorrelation\], the spin structure factor in Sec. \[sec:sq\], the spatial distribution of the nearest-neighbor spin and bond correlations in Sec. \[sec:nn\_spin\] and Sec. \[sec:bond\], respectively, the chiral correlation function in Sec. \[sec:chiral\], and the pairing correlation function in Sec. \[sec:scpair\]. We then discuss possible relevance to the experimental observation and provide several remarks in Sec. \[sec:discussion\], before summarizing the paper in Sec. \[sec:summary\]. In Appendix \[app:egap\], we examine the entanglement gap of the ground state as a function of $U/t$.
\[sec:method\]Method
====================
We consider 32-, 36- and 48-site clusters depicted in Fig. \[cylinder\]. Since the results for these different clusters are qualitatively the same, we shall mainly show the results for the 36-site cluster. Following the notation in Refs. , clusters forming the triangular lattice can be classified as XC$n$ (YC$n$) where the bond direction of a cluster is parallel to the $x$ direction ($y$ direction), as shown in Fig. \[cylinder\], and $n$ in XC$n$ (YC$n$) represents the number of bonds in zigzag (vertical) $y$ direction. Accordingly, the 36- and 48-site clusters belong to XC6, and the 32-site cluster belongs to YC4.
![The (a) 36-, (b) 32-, and (c) 48-site clusters. Open (periodic) boundary conditions are imposed in the $x$ direction ($y$ direction). The indexing of bonds ($l=1,2,3,\cdots,21$) as well as elementary triangles (${\triangle_i}=0,1,2,...,9$) with their chiral directions (arrows) are indicated in (a). []{data-label="cylinder"}](fig01.png){width="\hsize"}
Figure \[extrapenergy\] shows the convergence of the ground state energy for the 36-site cluster, as a function of the discarded weight $\delta_m$ defined as $$\begin{aligned}
\delta_m = 1 - \sum_{n=1}^m \lambda_{n}, \end{aligned}$$ where $\lambda_{n}$ is the $n$th largest eigenvalue of the reduced density-matrix of the ground state. As shown in Fig. \[extrapenergy\], we find that the ground state energies for $m \geq 10\ 000$ scale linearly with $\delta_m$, implying that the convergence of our calculations is well controlled.
. The number $m$ of eigenstates of the reduced density-matrix kept in the DMRG calculations is indicated beside each data point. A red straight line shows a linear fit to the three data points with $m=10\ 000$, $12\ 000$, and $14\ 000$. []{data-label="extrapenergy"}](fig02.pdf){width="\hsize"}
Throughout the study, we set the $z$ component of total spin to be zero. We keep up to $m=10\ 000$ density-matrix eigenstates for the 32-site cluster, $m=14\ 000$ for the 36-site cluster, and $m=20\ 000$ for the 48-site cluster. As shown in Fig. \[extrapenergy\], when we use $m=14\ 000$ density-matrix eigenstates for the 36-site cluster, the typical orders of the discarded weight are $ 2.7 \times 10^{-5}$ for $U=6t$, $ 2.7 \times 10^{-6}$ for $U=8.5t$, and $ 7.6 \times 10^{-7}$ for $U=11t$. On the other hand, when we use $m=20\ 000$ density-matrix eigenstates for the 48-site cluster, the typical orders of the discarded weight are $2.6 \times 10^{-5}$ for $U=6t$, $2.3 \times 10^{-6}$ for $U=8.5t$, and $2.8 \times 10^{-7}$ for $U=11t$, thus obtaining the convergence similar to that for the 36-site cluster.
\[sec:cal\]Results
==================
\[sec:energy\]Energy and double occupancy
-----------------------------------------
We first study the $U/t$ dependence of the ground-state energy and double occupancy. Figures \[phase\](a) and \[phase\](b) show the ground-state energy per site, $$\begin{aligned}
\varepsilon_0=\langle\psi_0|\mathcal{H}|\psi_0\rangle/N\end{aligned}$$ and the site average of the double occupancy, $$\begin{aligned}
n_d = \frac{1}{N}\sum_i \left< \psi_0 \right| n_{i,\uparrow} n_{i,\downarrow} \left| \psi_0 \right>,\end{aligned}$$ where $\left| \psi_0 \right>$ is the ground-state obtained by the 2D-DMRG calculation and $N$ is the number of sites. As shown in Fig. \[phase\](b), there exist two discontinuities in the double occupancy. It should be noted that $\varepsilon_0$ and $n_d$ are related via $n_d=\partial\varepsilon_0/\partial U$. We have numerically verified this relation, supporting the satisfactory convergence of our results.
![(color online) (a) Ground state energy per site $\epsilon_0$, (b) double occupancy $n_d$, and (c) $n_d U^2$ for three different clusters. Insets in (b) and (c) show the results for small values of $U$ calculated in the 36-site cluster. The ground state phase diagram is shown schematically in the top panel, where QSL denotes quantum spin liquid. The phase boundaries are indicated by gray shades. []{data-label="phase"}](fig03.pdf){width="0.95\hsize"}
As shown in Fig. \[phase\](c), the discontinuities in the double occupancy are most apparent when $n_d U^2$ is plotted. The first discontinuity occurs at $U_{\rm c1}/t=7.55$-$8.05$ and the second one is located at $U_{\rm c2}/t=9.65$-$10.15$. We find that these discontinuities in $n_d U^2$ become sharper with increasing $m$, indicating the nature of the first-order transition. We also find in the insets of Figs. \[phase\](b) and \[phase\](c) that there is no additional discontinuity for $0\le U < U_{\rm c1}$. Therefore, we conclude that there exist two first order transitions separating three phases. In the following, we call the three regions phases I, II, and III, as indicated in Fig. \[phase\].
Let us now compare our results with the previous studies. Since it includes the noninteracting limit with $U=0$, phase I is regarded as the metallic phase. The exact diagonalization analysis of a 16-site cluster using a finite-temperature Lanczos method has found that the metal-insulator transition occurs at $U_{\rm c}/t = 7.5 \pm 0.5 $ [@kokalj]. The metal-insulator transition is also found at $U_{\rm c}/t \sim 7.4 \pm 0.1$ for clusters up to 36 sites by the PIRG method [@yoshioka]. These $U_{\rm c}$ values are rather similar to $U_{\rm c1}$ in our calculations. On the other hand, the metal-insulator transition found by the VCA is at $U_{\rm c}/t \sim 6.3$-$6.7$ [@sahebsara; @yamada], which is slightly smaller than $U_{\rm c1}$. This is probably due to smaller clusters used in these VCA calculations, which tend to enhance an insulating phase. We calculate in Appendix \[app:egap\] the entanglement spectrum of the ground state as a function of $U/t$ and find an abrupt increase of the entanglement gap in the charge sector, supporting that the transition between phases I and II can be regarded as the metal-insulator transition.
The analysis based on the strong coupling expansion of the triangular lattice Hubbard model for clusters up to 36 sites [@yang] finds that the phase transition from the $120^{\circ}$ Néel-ordered phase to an insulating QSL phase occurs at $U_{\rm c} \sim 10 t$, which is close to $U_{\rm c2}$ obtained in our calculations. The intermediate insulating phase with $U_{\rm c2}=9.2t\pm0.3t$ is also reported in the PIRG calculations [@yoshioka]. Based on the comparison with these previous studies, phases II and III found in our calculations should correspond to a QSL phase and the $120^{\circ}$ Néel-ordered phase, respectively. In the following, we shall examine the nature of these phases.
\[sec:pinning\]Response to a pinning magnetic field
---------------------------------------------------
Let us first explore a possible magnetic order by applying a pinning magnetic field along the $z$-direction at a single site located at the edge of the cluster (see Fig. \[pinning\]). The pinning magnetic field applied at site $i_{\rm imp}$ is described by the following Hamiltonian: $$\begin{aligned}
\mathcal{H}^{\prime} = -\sum_i h_i\mathcal{S}_i^z, \end{aligned}$$ where $h_i=h\delta_{i,i_{\rm imp}}$, $\delta_{i,j}$ is the Kronecker delta (i.e., $\delta_{i,j}=1$ only when $i=j$), and $\mathcal{S}_i^z = \frac{1}{2}(n_{i,\uparrow} - n_{i,\downarrow})$. The results of the local spin density $$S_i^z = \langle \psi_0 | \mathcal{S}_i^z | \psi_0 \rangle$$ are summarized in Fig. \[pinning\]. Note that the local spin density $S_i^z$ is zero in the absence of the pinning magnetic field.
![(color online) Local spin density $S^z_i$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$ when a pinning magnetic field $h = 0.005 t$ is applied along the $z$ direction at a single site located at the edge of the 36-site cluster (indicated by a green open circle). For comparison, the results for the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice with the nearest-neighbor exchange interaction $J$ is also shown in (d). Here, the same 36-site cluster is used as in (a)–(c) and the applied pinning magnetic field is $h=0.005J$. The dashed line in (c) and (d) indicates a domain wall separating the cluster into two pieces, each of which exhibits a three sublattice pattern, expected for the $120^{\circ}$ Néel order. Note that the $z$ component of total spin is kept zero, i.e., $\sum_{i=1}^NS_i^z=0$, in the calculations. []{data-label="pinning"}](fig04.png){width="0.85\hsize"}
As shown in Fig. \[pinning\](c), the local spin density for $U=11t$ in phase III exhibits a three sublattice pattern, compatible with the $120^{\circ}$ Néel order, except that there exists a domain wall at the center of the cluster running along the $y$ direction. Indeed, the spatial distribution of the local spin density found here, including the domain wall structure, is essentially identical to that for the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice with the nearest-neighbor exchange interaction \[see Fig. \[pinning\] (d)\], where the ground state is 120$^{\circ}$ Néel ordered [@bernu; @capriotti; @zheng; @white3]. Therefore, this is strong evidence that the ground state in phase III is also 120$^{\circ}$ Néel ordered. In contrast, as shown in Figs. \[pinning\](a) and \[pinning\](b), the local spin densities for $U=6t$ in phase I and $U=8.5t$ in phase II are less affected by the pinning magnetic field, indicating the absence of long-range magnetic order.
Applying the perturbation theory, the leading correction of the spin density $\Delta S_i^z $ at site $i$ is $$\begin{aligned}
\Delta S_i^z \sim -h \sum_{n\,(\ne0)}
\frac{\left< \psi_0 \right| \mathcal{S}_{i_{\rm imp}}^z \left| \psi_n \right> \left< \psi_n \right| \mathcal{S}_i^z \left| \psi_0 \right> }{E_0 - E_n},
\label{eq:pinning}\end{aligned}$$ where $\left| \psi_n \right>$ is the $n$th eigenstate of $\mathcal{H}$ (without the pinning magnetic field) with its eigenvalue $E_n$. Since $\mathcal{H}$ commutes with the total spin operator, the total spin $S_{\rm tot}$ is a good quantum number. We now assume that the ground state $\left| \psi_0 \right>$ is spin singlet with $S_{\rm tot} = 0$. Then, the Wigner-Eckert theorem states that the matrix elements in the numerator of Eq. (\[eq:pinning\]) satisfy $$\begin{aligned}
\langle\psi_n|\mathcal{S}_i^z|\psi_0\rangle = \left\{
\begin{array}{cc}
0 & {\rm if}\ |\psi_n\rangle \not\in S_{\rm tot}=1 \\
{\rm finite} & {\rm if}\ |\psi_n\rangle \in S_{\rm tot}=1 \\
\end{array}
\right. , \end{aligned}$$ where $|\psi_n\rangle \in S_{\rm tot}=1$ ($|\psi_n\rangle \not\in S_{\rm tot}=1$) indicates that $|\psi_n\rangle$ belongs (does not belong) to the $S_{\rm tot}=1$ subspace. Therefore, the different behavior of the local spin density under the applied pinning magnetic field should be attributed to the amount of the low-lying triplet excitations.
–\[pinning\](c) but in the logarithmic scale. []{data-label="fig:pinning_logplot"}](fig05.png){width="\hsize"}
Figure \[fig:pinning\_logplot\] shows the same result as in Fig. \[pinning\] but in the logarithmic scale. It clearly shows that the local spin density $S_i^z$ for $U=8.5t$ in phase II exhibits comparable amplitude with that for $U=6t$ in phase I where the ground state is the paramagnetic metal and thus no triplet excitation gap is expected. On the other hand, the local spin density $S_i^z$ for $U=8.5t$ is significantly smaller than that for $U=11t$ in phase III where the ground state is $120^{\circ}$ Néel ordered. The similarity to the $U=6t$ case thus indicates that there exist an extensive amount of gapless spin excitations in phase II expected in the thermodynamic limit.
\[sec:spincorrelation\]Spin correlation function
------------------------------------------------
Next, we calculate the spin correlation $$S_{i,j} = \langle \psi_0 | \mathcal{S}_i^z \mathcal{S}_j^z | \psi_0 \rangle$$ between a reference site $j$ located at the center of the cluster and other sites $i$. The representative results for the three different phases are shown in Fig. \[sscor\]. Figure \[sscor\](c) clearly shows that $S_{i,j}$ for $U=11t$ in phase III exhibits a three sublattice pattern, compatible with the $120^{\circ}$ Néel order. On the other hand, $S_{i,j}$ in phases I and II does not show such a three sublattice pattern \[see Figs. \[sscor\](a) and \[sscor\](b)\], strongly suggesting that these phases are not $120^{\circ}$ Néel ordered.
. []{data-label="sscor"}](fig06.png){width="\hsize"}
We notice, however, in Fig. \[sscor\] that the intensity of $S_{i,j}$ between distant sites for $U=8.5t$ is comparable to that for $U=11t$, indicating the relatively long spin correlation in phase II. For a quantitative comparison, we estimate the correlation length $\xi$ via $$\xi=2\sqrt{\frac{\sum_{i=1}^N|S_{i,j}{\bf r}_{i,j}|^2}{\sum_{i=1}^N|S_{i,j}|^2}},$$ where ${\bf r}_{i,j} = {\bf r}_i - {\bf r}_j$ and ${\bf r}_i$ is the position vector of site $i$. As shown in Fig. \[sscor\], we find that $\xi$ for $U=8.5t$ is shorter than that for $U=11t$ where $\xi$ diverges in the thermodynamic limit, but is longer than that for $U=6t$ where the spin correlation decays algebraically. The rather long range spin correlation in phase II should be contrasted with the exponential decay of spin correlation expected in gapped QSL. Recalling also the result of the response to the pinning magnetic field, the ground state in phase II resembles gapless QSL.
\[sec:sq\]Spin structure factor
-------------------------------
The spin structure factor $S({\bf q})$ is the Fourier transform of the real-space spin correlation function $S_{i,j}$ and defined as $$\begin{aligned}
S({\bf q}) = \sum_{i=1}^N S_{i,j} e^{{\rm i}{\bf q}\cdot ({\bf r}_i - {\bf r}_j)}, \end{aligned}$$ where site $j$ is a representative site chosen at the central site of the cluster as in Fig. \[sscor\]. Although the wave number ${\bf q}$ is not a good quantum number due to open boundary conditions in the $x$ direction, here we calculate $S({\bf q})$ for arbitrary ${\bf q}$.
The representative results for the three different phases are shown in Figs. \[fig:sq\](a)–\[fig:sq\](c). We find in Fig. \[fig:sq\](c) that $S({\bf q})$ for $U=11t$ in phase III displays sharp peaks at ${\bf q} = (2\pi/3,2\pi/\sqrt{3})$ (the ${\rm K}$ point) and other equivalent ${\bf q}$’s including the ${\rm K}^{\prime}$ point at ${\bm q} = (-2\pi/3,2\pi\sqrt{3})$, which is compatible with the $120^{\circ}$ Néel-ordered state. The $S({\bf q})$ for $U=8.5t$ in phase II shown in Fig. \[fig:sq\](b) also exhibits broad maxima at the ${\rm K}$ point and other equivalent ${\bf q}$’s, but the peak structure is softened as compared with $S({\bf q})$ for $U=11t$. In contrast, we find in Fig. \[fig:sq\](a) that $S({\bf q})$ for $U=6t$ in phase I shows enhanced intensities forming a ring-like structure around the K point (and other equivalent ${\bf q}$’s), which implies the presence of $2{\bf k}_{\rm F}$ scattering, where ${\bf k}_{\rm F}$ is the Fermi momentum in the noninteracting limit.
![(color online) Intensity plot of spin structure factor $S({\bf q})$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$ calculated in the 36-site cluster and for (d) $U=3t$ obtained by the random phase approximation. Notice that the intensity is doubled in (d) for clarity. The Brillouin zone boundaries are indicated by yellow-green lines. []{data-label="fig:sq"}](fig07.png){width="\hsize"}
In order to better understand $S({\bf q})$ in phases I and II, we calculate $S({\bf q})$ within the random phase approximation (RPA). In the RPA, the spin susceptibility $\chi ({\bf q},z)$ at zero temperature is given as $$\begin{aligned}
\chi ({\bf q},z) = \frac{\chi_0 ({\bf q},z)}{1 - U \chi_0 ({\bf q},z)},
\label{eq:rpa}\end{aligned}$$ where $z$ is the complex frequency. The susceptibility $\chi_0 ({\bf q},z)$ in the noninteracting limit is $$\begin{aligned}
\chi_0 ({\bf q},z) =
\frac{1}{N} \sum_{{\bf k}\in {\rm 1st\,BZ}} \frac{\Theta (-\varepsilon_{\bf k}) -
\Theta (-\varepsilon_{{\bf k}+{\bf q}})}{z - \varepsilon_{\bf k} + \varepsilon_{{\bf k}+{\bf q}}}, \end{aligned}$$ where the sum is taken over the first Brillouin zone (BZ) of the triangular lattice, $\Theta (x)$ is the Heaviside step function, and $\varepsilon_{\bf k}$ is the noninteracting band dispersion $$\begin{aligned}
\varepsilon_{\bf k} &=& - 2 t \cos k_x - 2 t \cos \left( \frac{k_x}{2} + \frac{\sqrt{3}k_y}{2} \right) \nonumber \\
&{}& - 2 t \cos \left( - \frac{k_x}{2} + \frac{\sqrt{3}k_y}{2} \right) - \mu. \end{aligned}$$ The chemical potential $\mu$ is tuned such that the electron density is $0.5$ per spin. We set the chemical potential $\mu \sim 0.8347t$ for the calculation in the thermodynamic limit.
From $\chi({\bf q},z)$ obtained above within the RPA, the spin structure factor $S({\bf q})$ is evaluated as $$\begin{aligned}
S({\bf q}) = \frac{1}{\pi} \int_0^{\infty} {\rm d}x {\rm Re} \chi ({\bf q},{\rm i}x). \end{aligned}$$ In deriving the above equation, we have assumed that the $z$ component of total spin is zero and the system is invariant under the global spin flip. Figure \[fig:sq\](d) shows $S({\bf q})$ for $U=3t$ within the RPA. Here we choose relatively small $U$ because $\chi({\bf q},0)$ diverges at $U=3.7$–$3.8t$. As shown in Fig. \[fig:sq\](d), $S({\bf q})$ exhibits a triangular shell-like structure around the K point (and other equivalent ${\bf q}$’s). The ridges of the shells lie exactly along the $2{\bf k}_{\rm F}$ lines and the local minimum in the center of the shell is located at the K point (and other equivalent ${\bf q}$’s). These features are indeed similar to those found in Fig. \[fig:sq\](a) for $U=6t$.
The spinon Fermi sea state is a kind of gappless spin liquid state and has been considered as a candidate for the ground state of triangular lattice systems [@motrunich]. Due to the presence of spinon Fermi surface, $S({\bf q})$ for the spinon Fermi sea state exhibits singularities along the $2{\bf k}_{\rm F}$ lines [@block] and is expected to be similar to those shown in Figs. \[fig:sq\](a) and \[fig:sq\](d). However, as shown in Fig. \[fig:sq\](b), we find that $S({\bf q})$ for $U=8.5t$ in phase II is quite different from those in Figs. \[fig:sq\](a) and \[fig:sq\](d). Therefore, the spinon Fermi sea state is unlikely to be the ground state in phase II.
The similarity of $S({\bf q})$ for $U=8.5t$ and $U=11t$ in Fig. \[fig:sq\] tempts us to conclude that also the ground state in phase II shows tendency towards the $120^{\circ}$ Néel order. However, we emphasize that the peak structure in $S({\bf q})$ for $U=8.5t$ in phase II are much smaller and broader than that for $U=11t$ in phase III. Indeed, as shown in Fig. \[pinning\] and Fig. \[sscor\], the response to the pinning magnetic field and the real-space spin correlation function are clearly different in phases II and III.
\[sec:nn\_spin\]Nearest-neighbor spin correlation
-------------------------------------------------
We calculate the nearest-neighbor spin correlation $S_{\left< i,j\right>}\,(=S_{i,j})$ for all nearest-neighbor sites $i$ and $j$, and the representative results for the three different phases are shown in Fig. \[nnspin\]. We first notice in Figs. \[nnspin\](a)–\[nnspin\](c) that the results are invariant under the translation along the $y$ direction, the reflection about mirror planes perpendicular to the $y$ direction, and the $180^{\circ}$ rotation around the center of the cluster, thus implying that the convergence of our results is satisfactory. For better quantitative comparison, we show in Fig. \[nnspin\](d) $S(l)=S_{\left< i,j \right>}$ along the $x$ direction, where the bond index $l$ connecting sites $i$ and $j$ is denoted in Fig. \[cylinder\](a).
. The 36-site cluster is used for all figures. []{data-label="nnspin"}](fig08.png){width="\hsize"}
It is clearly observed in Fig. \[nnspin\](d) that $S(l)$ for $U=11t$ in phase III is rather enhanced at $l=4n-3$ and suppressed at $l=4n-1$ for $n=1, 2, 3, \cdots$. More interestingly, the oscillations in $S(l)$ for $U=8.5t$ in phase II are smallest, especially around the center of the cluster. Note that spatial variation of $S(l)$ is an indication of spatial symmetry breaking, e.g., valence bond solid, or a tendency for it, and that in our case the spatial variation is induced by open boundary conditions in the $x$ direction. The strong suppression of oscillations in phase II is therefore a strong indication of the absence of valence bond solid and also other possible spatial symmetry breakings.
Let us now compare the nature of the ground state in phase II with the $Z_2$ spin liquid ground state of the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice with the next nearest-neighbor exchange interaction, recently reported in Refs. . The cluster used here (Fig. \[cylinder\]) is an odd cylinder [@zhu; @hu] with odd number of sites in the one dimensional unit cell. Therefore, according to the Lieb-Schultz-Mattis theorem [@lieb], the ground state for this cluster is degenerate if the excitation gap is finite as in the case for the $Z_2$ spin liquid. Reflecting this degeneracy, the spatial distribution of the nearest-neighbor spin correlation $S_{\left< i,j \right>}$ exhibits the strong alternating oscillation, induced by open boundary conditions along the $x$ direction [@zhu; @hu]. Although the similar oscillation pattern is found in phase III and around the edge of the cluster in phase II, the central region of the cluster in phase II does not show such feature (see Fig. \[nnspin\]).
\[sec:bond\]Nearest-neighbor bond correlation
---------------------------------------------
We also calculate the hopping amplitude $B_{\left< i,j\right>}$ between nearest-neighbor sites $i$ and $j$ defined as $$\begin{aligned}
B_{\left< i,j\right>} = \frac{1}{2}\left< \psi_0 \right| (c_{i,\sigma}^{\dagger} c_{j,\sigma}+c_{j,\sigma}^{\dagger}c_{i,\sigma}) \left| \psi_0 \right>. \end{aligned}$$ The representative results for the three different phases are shown in Fig. \[nnbond\]. Similarly to $S_{\left< i,j \right>}$, we find that $B_{\left< i,j \right>}$ is invariant under the translation, reflection, and rotation operations \[see Figs. \[nnbond\](a)–\[nnbond\](c)\], suggesting that our results are well converged. For better quantitative comparison, Fig. \[nnbond\](d) shows $B(l)=B_{\left< i,j \right>}$ along the $x$ direction for the $l$th bond connecting sites $i$ and $j$ \[for the indexing of bonds, see Fig. \[cylinder\](a)\].
![(color online) Nearest-neighbor hopping amplitude $B_{\left< i,j\right>}$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$. Here, $\max[B_{\left< i,j\right> }]$ is the maximum value of $B_{\left< i,j \right>}$ for each $U$. (d) $B(l)=B_{\left< i,j \right>}$ along the $x$ direction, where the bond index $l$ connecting neighboring sites $i$ and $j$ is denoted in Fig. \[cylinder\](a). Notice that the increase of kinetic energy, proportional to $-t\sum_l B(l)$, is nicely observed with increasing $U/t$. The 36-site cluster is used for all figures. []{data-label="nnbond"}](fig09.png){width="0.95\hsize"}
We find that $B_{\left< i,j \right>}$ exhibits the similar oscillation patterns to those observed in $S_{\left< i,j \right>}$ (see Fig. \[nnspin\]). The similarity between $B_{\left< i,j \right>}$ and $S_{\left< i,j \right>}$ is expected for large $U/t$ since the kinetic energy, proportional to $B_{\left< i,j \right>}$, can be in a strong coupling regime transferred to the Heisenberg exchange interaction, which is related to $S_{\left< i,j \right>}$. However, it is surprising that this similarity is present also in the coupling regimes shown in Fig. \[nnbond\], where there is in general no direct connection between $B_{\left< i,j \right>}$ and $S_{\left< i, j \right>}$. As shown in Fig. \[nnbond\](d), we find that the oscillations of $B(l)$ around the center of the cluster are most strongly reduced for $U=8.5t$ in phase II as compared with those in phases I and III. This implies that the ground state in phase II is not compatible with the nearest-neighbor valence bond solid.
\[sec:chiral\]Chiral correlation function
-----------------------------------------
Next, we calculate the chiral correlation function $C({\triangle_i},{\triangle_j})$ defined as $$\begin{aligned}
C ({\triangle_i},{\triangle_j}) = \left<\psi_0| C_{\triangle_i} C_{\triangle_j} |\psi_0\right> \end{aligned}$$ with $$\begin{aligned}
C_{\triangle_i} = \vec{\mathcal{S}}_{i_1} \cdot ( \vec{\mathcal{S}}_{i_2} \times \vec{\mathcal{S}}_{i_3} ), \end{aligned}$$ where $\triangle_i$ indicates the $i$th elementary triangle formed by three neighboring sites $i_1$, $i_2$, and $i_3$ in clockwise or counter clockwise order, and the indexing of elementary triangles as well as their chiral directions is indicated in Fig. \[cylinder\](a). $\vec{S}_i$ is the spin operator at site $i$ defined as $$\begin{aligned}
\vec{\mathcal{S}}_i = \frac{1}{2} \sum_{\sigma_1=\uparrow,\downarrow} \sum_{\sigma_2 = \uparrow,\downarrow} c_{i,\sigma_1}^{\dagger} \vec{\sigma}_{\sigma_1,\sigma_2} c_{i,\sigma_2}, \end{aligned}$$ where $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z) $ are Pauli matrices. Figure \[chiral\] shows the chiral correlation function $C ({\triangle_i},{\triangle_i}+l)=C(l)$ with ${\triangle_i}=0$ along the $x$ direction \[see Fig. \[cylinder\](a)\]. We find that the sign of $C (l)$ exhibits nontrivial oscillation \[Fig \[chiral\](a)\] and the amplitude of $C (l)$ decays exponentially \[Fig \[chiral\](b)\]. Therefore, we conclude that the chiral spin liquid is most unlikely to be the ground state in phase II [@gong2].
. []{data-label="chiral"}](fig10.pdf){width="\hsize"}
\[sec:scpair\]Pairing correlation function
------------------------------------------
Finally, let us discuss the possibility of superconductivity by calculating the pairing correlation function $P_{\nu}(i,j,k,l)$ defined as $$P_{\nu}(i,j,k,l) = \langle \psi_0 \vert \Delta_{\nu}(i,j) \Delta^{\dagger}_{\nu}(k,l) \vert \psi_0 \rangle
\label{eq:scpair}$$ with the nearest-neighbor singlet channel ($\nu = {\rm s}$) $$\Delta_{\rm s}(i,j) = \frac{1}{\sqrt{2}} \left( c_{i, \uparrow} c_{j, \downarrow} - c_{i, \downarrow} c_{j, \uparrow} \right)$$ and the nearest-neighbor triplet channel ($\nu = {\rm t}$) $$\Delta_{\rm t}(i,j) = \frac{1}{\sqrt{2}} \left( c_{i, \uparrow} c_{j, \downarrow} + c_{i, \downarrow} c_{j, \uparrow} \right).$$ Figure \[fig:scpair\] shows the representative results of the pairing correlation function $P_{\nu}(r) = P_{\nu}(i,j,k,l)$ for both singlet and triplet channels, where $r$ is the distance between the centers of two pairs of nearest-neighbor sites $(i,j)$ and $(k,l)$. We find that the pairing correlations in both channels are significantly suppressed for $U=8.5t$ in phase II and $U=11t$ in phase III as compared with those for $U=6t$ in phase I. Therefore, we conclude that the ground-state in phase II is unlikely to be superconducting. It should also be noted that the short range superconducting correlations in the spin triplet channel is stronger than those in the spin singlet channel for the three representative cases, although the superconducting correlations in the spin singlet channel dominates in the long distance.
It is also interesting to notice that the pairing correlations at long distances seem to be saturated for $U=6t$ in phase I. However, since the longest distances are calculated from sites close to the cluster edges, the upturn of the pairing correlations might be a finite size effect. Therefore, our calculations alone cannot support the presence of the superconducting phase. Larger systems with $U$ closer to $U_{\rm c1}$ might show stronger pairing correlations. This issue is left for the future study.
\], representing nearest-neighbor sites $(i,j)$ and $(k,l)$, respectively, for which the pairing operators $\Delta_\nu (i,j)$ and $\Delta^\dag_\nu (k,l)$ are chosen in Eq. (\[eq:scpair\]). $r$ is the distance between the centers of bonds $l_1$ and $l_2$. []{data-label="fig:scpair"}](fig11.pdf){width="\hsize"}
\[sec:discussion\]Discussion
============================
Let us briefly discuss implications of our results for the experiments. Our results show a rather small discontinuity $\Delta n_d$ ($\sim0.007$) in the double occupancy found at the metal-insulator transition $U_{\rm c1}$, which is in sharp contrast to the previous studies using other approaches [@yoshioka], where $\Delta n_d$ is typically much larger (from 0.02 to 0.06), and is even qualitatively different from the continuous transition discussed in Refs. and . Smallness of the jump might be the origin of the controversy about the order of the transition. We note, however, that the discontinuity $\Delta n_d$ corresponds to the interaction (or equivalently kinetic) energy jump of $U_{\rm c1}\Delta n_d \sim 0.05t$, giving $\sim 25$ K for the organic materials with $t \sim 50$ meV [@kokalj], which nicely compares with the temperature where the first-order transition line ends in the phase diagram of the organic materials [@furukawa].
Next, let us discuss the ground-state in phase II in terms of RVB states described by Gutzwiller projected fermionic wave functions. For this purpose, it is important to recall that the structure factor $S({\bm q})$ in phase II exhibits the maximum at the ${\rm K}$ and ${\rm K}^{\prime}$ points. This feature is not consistent with a projected Fermi sea with a large Fermi surface because the $2{\bf k}_{\rm F}$ structure is not found in $S({\bm q})$. Instead, this feature is rather comparable to a projected Fermi sea with gapless nodal points such as a projected Dirac fermion [@sindzingre; @yunoki2004].
Another interesting feature is that the superconducting fluctuations for the spin-triplet (singlet) channel dominates in the short (long) distance, although the superconducting correlation functions for both channels decay exponentially in the insulating phases. This suggests that the long wavelength behavior of the ground-state in phase II might be captured by a projected BCS wave function with a singlet pairing, but the strong modification of the wave function would be required to describe the short-range properties such as the ground-state energy.
\[sec:summary\]Summary
======================
In summary, we have performed large scale 2D-DRMG calculations, using up to $m=20\ 000$ density-matrix eigenstates, to examine the ground state phase diagram of the Hubbard model on the triangular lattice at half-filling. We have shown that the convergence of our results is well controlled and, therefore, our results can be regarded as the most accurate and unbiased results available at a moment, apart from the small cluster shape and size dependence.
We have found two first-order transitions separating the three different phases, which include the metallic phase in weak coupling region, the 120$^{\circ}$ Néel-ordered phase in strong coupling region, and the QSL like phase in the intermediate couplin region. The weak and intermediate coupling phases are less affected by the pinning magnetic fields, suggesting the absence of magnetic order in these two phases. The spin correlations in the intermediate phase is weaker than those in the 120$^{\circ}$ Néel-ordered phase and stronger than those in the metallic phase. The spin structure factor in the intermediate phase shows a maximum at the ${\rm K}$ and ${\rm K}^{\prime}$ points, which is not compatible with the spinon Fermi sea state [@motrunich]. The spatial distribution of the nearest-neighbor spin correlation in the intermediate phase is not comparable with the $Z_2$ spin liquid found in the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice with the next-nearest-neighbor exchange interaction [@zhu; @hu]. We have also calculated the chiral correlation function and found that the chiral spin liquid [@gong2] is unlikely in the intermediate phase. The pairing correlation function decreases monotonically with increasing $U/t$, suggesting that the superconductivity is also unlikely in the intermediate phase.
The clusters used here are much smaller than those employed for the 2D-DMRG studies of spin-1/2 antiferromagnetic Heisenberg models on the triangular lattice reported in Refs. . This is simply because the local degrees of freedom in the Hubbard model is two times larger than those in the spin-1/2 Heisenberg models. Therefore, the more detail analysis using larger clusters is highly desirable in order to determine the nature of the ground state in the intermediate phase and, in particular, to address the size of the spin gap in the thermodynamic limit and the experimental observation in EtMe$_3$Sb\[Pd(dmit)$_2$\]$_2$ where gapless QSL is suggested [@yamashita1]. Further properties of the intermediate phase, including the nature of excitations, remain to be firmly examined since that would greatly improve our understanding of spin liquid in general as well as of the organic materials in particular.
The authors are grateful to S. Nishimoto and T. Li for valuable discussion. This work has been supported by Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS) (Grant No. 24740269, No. 26287079, and No. 17K14148) and Slovenian Research Agency (Z1-5442), and in part by RIKEN Molecular Systems and RIKEN iTHES Project. T. T. and J. K. acknowledge the visiting program for young researchers at Yukawa Institute for Theoretical Physics, Kyoto University. The computation has been performed using the RIKEN Cluster of Clusters (RICC), the RIKEN supercomputer system (HOKUSAI GreatWave), and the K computer at RIKEN Advanced Institute for Computational Science (AICS) under MEXT HPCI Strategic Programs for Innovative Research (SPIRE) (Project Nos. hp120137, hp140128, hp150112, hp160122, and hp170324).
\[app:egap\]Entanglement Gap
============================
In this appendix, we examine the charge and spin gaps in the low-lying entanglement spectrum of the ground state [@lihaldane] to support our results in the main text. In the DMRG method, the system is divided into two regions, blocks $A$ and $B$, and thus the ground state $\vert \psi \rangle$ is represented as $$\vert \psi \rangle = \sum_{i,j} \psi_{ij} \vert i \rangle_A \vert j \rangle_B,$$ where $\vert i \rangle_A$ ($\vert j \rangle_B$) denotes a basis in block $A$ ($B$). The reduced density matrix $\rho_A$ for block $A$ is obtained by tracing out the degrees of freedom in block $B$, $$\rho_A = {\rm Tr}_B \vert \psi \rangle \langle \psi \vert,$$ where ${\rm Tr}_B$ indicates the trace over all bases in block $B$. The entanglement spectrum $\xi_n$ (where $n=1,2,3,\dots$) is defined as $$\xi_n = -\ln \lambda_n,
\label{eq:def:espec}$$ where $\lambda_n$ is the $n$th largest eigenvalue of the reduced density matrix $\rho_A$. Since $0<\lambda_n<1$ in general, $\xi_1 \leq \xi_2 \leq \xi_3 \leq\cdots$.
Equation (\[eq:def:espec\]) implies that $\xi_n$ can be considered as the eigenvalues of the entanglement Hamiltonian $H_{\rm E}$ defined as $$H_{\rm E} = - \ln \rho_A.$$ This in turn suggests that $H_{\rm E}$ can be regarded as an effective Hamiltonian to represent the density matrix $\rho_A$ with the Boltzmann distribution ${\rm e}^{-H_{\rm E}}$. Since the density matrix can be block diagonalized with respect to the number of electrons $N_e$ and the $z$-component $S^z$ of the total spin in block $A$, the entanglement spectrum $\xi_n$ is also labelled with these quantum numbers, i.e., $\xi_n = \xi(k, N_e, S^z)$, where $k\,(=0,1,2,\dots)$ is an index to distinguish the entanglement spectrum in the same quantum number sector: $\xi(0,N_e,S^z) \leq \xi(1,N_e,S^z) \leq \xi(2,N_e,S^z) \leq \cdots$. We can now define the entanglement gaps for the charge sector as $$\begin{aligned}
\Delta \xi_{\rm C} = {\rm min}[
& \xi(0,N/2+1,1/2) - \xi(0,N/2,0), \nonumber \\
& \xi(0,N/2-1,1/2) - \xi(0,N/2,0)]\end{aligned}$$ and for the spin sector as $$\begin{aligned}
\Delta \xi_{\rm S} = {\rm min}[
& \xi(0,N/2,1) - \xi(0,N/2,0), \nonumber \\
& \xi(0,N/2,-1)-\xi(0,N/2,0)],\end{aligned}$$ where the size of block $A$ is half of the cluster size $N$.
Figure \[fig:egaps\] shows the entanglement gaps $\Delta \xi_{\rm C}$ and $\Delta \xi_{\rm S}$ as a function of $U/t$. We indeed find that $\Delta \xi_{\rm C}$ increases abruptly at the phase boundaries. Since the value of $\Delta \xi_{\rm C}$ is related inversely to the global charge fluctuations between blocks A and B, the abrupt increase of $\Delta \xi_{\rm C}$ in the phase boundary between phases I and II suggests that this transition involves the opining of charge gap. Therefore, one would regard the transition between phases I and II as the Mott transition.
![(color online) Entanglement gaps $\Delta \xi_{\rm C}$ and $\Delta \xi_{\rm S}$ for the charge and spin sectors, respectively. A shaded region indicates phase II determined from the discontinuities of the double occupancy in the $48$-site cluster. []{data-label="fig:egaps"}](fig12.pdf){width="\hsize"}
We also find that the entanglement gap $\Delta\xi_{\rm S}$ for the spin sector increases abruptly at the phase boundary between phases II and III. Moreover, we find that the $\Delta \xi_{\rm S} \sim 0$ in the phase II. This tempts us to conclude that the ground state is spin gapless in phase II. However, this is not appropriate because a topologically non-trivial gapped state can induce characteristic low-lying edge states with the gapless entanglement spectrum for the spin sector. Therefore, we can only argue from Fig. \[fig:egaps\] that the spin structure of phase II is distinguishable from phase III.
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|
---
abstract: 'We consider functions defined by deep neural networks as definable objects in an o-miminal expansion of the real field, and derive an almost linear (in the number of weights) bound on sample complexity of such networks.'
address: |
Alexander Usvyatsov\
Universidade de Lisboa\
Centro de Matemática e Aplicaões Fundamentais\
Av. Prof. Gama Pinto,2\
1649-003 Lisboa\
Portugal
author:
- Alexander Usvyatsov
bibliography:
- 'common.bib'
title: On sample complexity of neural networks
---
Introduction and Preliminaries
==============================
Introduction
------------
Recall that a function $f\colon \setR^k \to \setR$ is called *restricted analytic* if there exists a function $\bar f\colon \setR^k\to\setR$ and a closed interval $[a.b]$ such that $\bar f$ is analytic in some neighborhood of $[a,b]^k$, and $f = \bar f$ on $[a,b]^k$ (and equals $0$ on the complement of $[a,b]^k$). Note that all activation functions of neural networks discussed in literature can be considered in this context.
Let $\cC$ be a binary neural network with arbitrary restricted analytic activation functions. Note that we do not require that the activations functions at different nodes are all the same. Then $\cC$ defines a binary function $\cF = \cF(x_1,\ldots,x_n,w_1,\ldots,w_m: \setR^n\times \setR^m \to \set{0,1}$, where $x_1,\ldots,x_n$ are the inputs and $w_1,\ldots.w_m$ are the weights. Given a fixed collection of weights $\bar w = w_1,\ldots, w_m \in \setR$ we therefore obtain a binary function $\cF_{\w} \colon \setR^n \to \set{0,1}$.
Consider the hypothesis class $\cH = \set{\cF_{\w}\colon \w \in \setR^m}$. It is well known (e.g. [@Koiran96neuralnetworks; @Bartlett_vapnik-chervonenkisdimension; @Sontag98vcdimension; @Karpinski95polynomialbounds]) that, depending on the activation functions of $\cC$, the VC-dimension of $\cH$ could be quadratic in $m$,even in quite simple and natural cases (e.g., linear activations, or a fixed sigmoid $\sigma$). This leads one to conclude that the best possible theoretical upper bound on sample complexity $k(\eps,\delta)$ of such $\cH$ is $O\left(\frac{m^2+\ln\frac 1\delta}{\eps}\right)$ for $(\eps,\delta)$-PAC learnability, or $O\left(\frac{m^2+\ln\frac 1\delta}{\eps^2}\right)$ for agnostic $(\eps,\delta)$-PAC learnability (see Theorem 6.8 in [@Shalev-shwartz_fromtheory]). In other words, for most activation functions used in practice, sample complexity of a neural network appears to be quadratic in the number of weights, and therefore $O(k^4)$ where $k$ is the size of $\cC$, i.e., the number of nodes in it. Moreover, for some non-algebraic, but still “tame”, activations, such as $\sigma = \tanh$, VC-dimension is known to be $m^4$ (Karpinski and Macyntire [@Karpinski95polynomialbounds]); that is, the sample complexity appears to be $O(k^8)$.
However, it is intuitively clear that the number of “degrees of freedom” of $\cH$ is the number of weights, and not the square of the number of weights. One would therefore expect the sample complexity to be linear in $m$, hence $O(k^2)$ where $k$ is the size of $\cC$. VC-dimension does not, therefore, seem to explain this phenomenon. Even if we restrict ourselves to a very limited class of threshold activation functions, the VC-dimension of $\cH$ is still going to be $m\log(m)$. One way of settling the issue is simply noting, as in [@Livni:2014:CET:2968826.2968922], that, since all real numbers involved in the computation of $\cF$ are in practice represented by a finite number of bits, one can without loss of generality restrict their attention to a subfamily of $\cH$ with a linear VC-dimension. This solution, however, still seems somewhat unsatisfying.
In this note we observe that one can obtain a much better bound on sample complexity in terms of the number weights, once the notion of VC-dimension is replaced with that of (combinatorial) VC-density. We shall recall that VC-density of any hypothesis set $\cH$ that arises from a neural network as above is $m$, and compute an upper bounds on sample complexity using combinatorial density. This will yield an $O(m\log(m)$ bound for any neural network $\fC$, provided that all the activation functions are restricted analytic. We will, however, have to pay a small price in dependence on either the confidence level $\delta$, or on the acceptable error $\eps$. This makes sense, since the bound $O\left(\frac{m^2+\ln\frac 1\delta}{\eps^2}\right)$ is known to be tight; however, the additional factor of $\log(1/\eps)$ seems insignificant in comparison with the gain ($m\log(m)$ as opposed to $m^2$ or even $m^4$), especially for very large networks used in practice today. We also hope that this factor can be improved further using a more careful analysis. In addition, we believe that using our approach and more sophisticated techniques, one can obtain a *linear* dependence on $m$, which would fully settle the issue raised above. We will return to this in a future work.
Let us also note that more general activation function can be allowed in our analysis. As observed in [@Karpinski95polynomialbounds], there are neural networks with a smooth activation function and infinite VC-dimension; in this case, by the general theory, VC-density will be infinite as well. However, one can allow *certain* unrestricted functions: e.g., the exponential function $e^x$ (and, more generally, any function “definable” from $e^x$), or the function $x \mapsto x^{-1}$ which is defined to be $\frac{1}{x}$ for $x\neq 0$ and $0$ for $x=0$. In general, the only requirement that we have on the collection of all the activation functions of $\cC$ is that they are all *simultaneously definable in a single o-minimal expansion of $\setR$*. In this context, this assumption is quite reasonable: all restricted analytic functions and $e^x$ are definable in $\setR_{exp,an}$, and $x \mapsto x^{-1}$ is definable in $(\setR_{an},{}^{-1})$; both of these structures are known to be o-minimal. In particular, the case of $\sigma=\tanh$ is also covered by our analysis. There are many references for o-minimality of various expansions of $\setR$, e.g., [@Wi; @vDDMil; @vDDSp; @vDDSp2; @vDDSp3].
O-minimality has already had many fruitful applications in mathematics and computer science (for example, in verification and control theory, e.g. Brihaye [@Wallonie-bruxelles_verificationand]). Techniques from o-minimality have already been used in the study of neural networks, particularly, in computations of VC-dimension by Karpinsky and Macyntire [@Karpinski95polynomialbounds]. We believe that incorporating the progress of the last 20 years may lead to more illuminating results and yield new ideas and techniques. This note is just a small step in that direction.
The setting {#sec:the_setting}
===========
First we recall some basic notions from statistical learning theory.
VC-dimension {#sub:vc_dimension}
------------
Let $\cX$ be a set. We denote by $2^{\cX}$ the power set of $\cX$. In our case, $\cX = \setR^n$.
Recall that the VC-dimension of the collection of subsets $\cA\subseteq 2^{\cX}$ of $\cX$ is defined to be the maximal size (if exists) of a finite subset of $\cX$ which is shattered by $\cA$, i.e.
$$\text{VC}(\cA) = \sup{|B|<\infty \colon B\subseteq \cX, |B\cap\cA|=2^{|B|}}$$
So if the maximum does not exist, we say that $\text{VC}(\cA)=\infty$.
In the definition above, $B\cap\cA = \set{B\cap A \colon A \in \cA}$. So $|B\cap\cA|=2^{|B|}$ if and only if for every subset $B' \subseteq B$ there exists $A' \in \cA$ so that $A'\cap B = B'$ (this is the origin of the term “shattered”). Hence infinite VC-dimension means that $\cA$ shatters arbitrarily large sets (but not necessarily all sets). See e.g. Sontag [@Sontag98vcdimension] for more details and examples.
Given $\cA$ as above, $B\subseteq \cX$ finite and $B' \subseteq B$, we will say that $\cA$ *recognizes* $B'$ in $B$ if for some $A' \in \cA$ we have $A'\cap B = B'$. So $\cA$ shatters $B$ if it recognizes all of its subsets.
The relevance of VC-dimension to learning theory lies in the following simple but brilliant observation. It turns out that there is a sharp dichotomy in the number of subsets of an arbitrary set finite set $B$ that any collection $\cA$ can recognize. Specifically, either $\cA$ shatters arbitrary large sets (so $\text{VC}(\cA) = \infty$) or for any set large enough finite $B$, $\cA$ only recognizes a polynomial number of subsets of $B$. Moreover, if $\text{VC}(A) = d < \infty$, then for any finite $B \subseteq \cX$, the number of subsets of $B$ that $\cA$ can recognize is $O(n^d)$. This fact is known as the Sauer-Shelah Lemma, and it was proven independently by Sauer, Shelah, Perles, and Vapnik and Chervonenkis in slightly different contexts for different purposes around the same time. In other words,
\[lem:sauer\](Sauer-Shelah Lemma) Let $\cA$ be a collection of subsets of a set $\cX$. Then either $\text{VC}(A) = \infty$, or, if $\text{VC}(A) = d < \infty$, then for every finite $B \subseteq \cX$ we have $$|B\cap \cA| = O(|B|^d)$$
A more precise formula can be given, but it is of no interest to us here.
VC-density {#sub:vc_density}
----------
Motivated by the Sauer-Shelah Lemma, one can make the following definition:
\[dfn:growth function\] Let $\cX$, $\cA$ be as above. We define the *growth function* of $\cA$, $\tau_{\cA}:\setN\to\setN$ as follows: $$\tau_{\cA}(n) = \max{|B\cap \cA|\colon B\subseteq \cX, |B|=n}$$
In other words, $\tau_{\cA}(n)$ measures the maximal number of subsets of a set of size $n$ that $\cA$ can recognize. By the Sauer-Shelah Dichotomy Lemma, we have either $\tau_{\cA}(n) = 2^n$ or all $n$ (this case corresponds to infinite VC-dimension), or $\tau_{\cA}(n)$ is sub-polynomial, and in fact, $\tau_{\cA}(n) = O(n^d)$ where $d = \text{VC}(\cA)$.
It is natural to ask whether the exponent $d$ above is optimal. And indeed, it turns out that in most cases it is really not. The “true” measure of the exponent in the growth function is called the *combinatorial density* or the *VC-density* of $\cA$, and it is denoted by $\text{vc}(A)$. More precisely:
\[dfn:VCdensity\] Let $\cX$, $\cA$ as above. Then the VC-density of $\cA$ is defined as follows: $$\text{vc}(A) = \inf\set{q\in \setQ\colon \tau_{\cA}(n) = O(n^q)}$$
Note:
\[obs:basicproperties\]
1. $\text{vc}(\cA)\le\text{VC}(\cA)$ for all $\cA$ \[this somewhat explains the notation\]
2. $\text{vc}(\cA) = \infty$ if and only if $\text{VC}(\cA) = \infty$ for all $\cA$
In general, VC-density is not particularly well-behaved. For instance, in Aschenbrenner et al [@Aschenbrenner11vapnik-chervonenkisdensity] examples of hypothesis classes of non-integer and even irrational VC-density are given. VC-density is also not known to be sub-additive (in [@KOU] Kaplan, Onshuus, and the author prove sub-additivity for a certain integer analogue of VC-density). However, in the particular examples that we are interested in, VC-density has been computed, and it turns out to be the minimal possible, as will be discussed in the next subsection.
O-minimality {#sub:the_model_theoretic_framework}
------------
Let $(\cR,0,1,<,+,\cdot,f_\al \colon \al \in \cI)$ be an o-minimal expansion of $(\setR, 0,1,<,+,\cdot)$ with a collection of functions $\set{f_\al\colon \al \in \cI}$. For the purpose of this paper, $\cR$ can be simply $\setR_{an}$ or $\setR_{an,exp}$. In a nutshell, O-minimality means that any set, definable in $\cR$, is a finite collection of intervals. We refer to \cite{} for a survey on o-minimality an o-minimal expansions of $\setR$.
Let $\cC$ be a neural network with activation functions all definable in $\cR$. As described in the introduction, it defines a family of binary functions $\cH = \set{\cF_{\w}\colon \w \in \setR^m}$, which is precisely the hypothesis class that we are interested in, where each $\cF_{\w}$ is a boolean function on $\setR^n$. ALternatively, we can, of course, think of $\cF_{\w}$ as a subset $X_{\w}$of $\setR^n$ (say, the set of all $\x\in\setR^n$ on which $\cF_{\w}$ takes the value $1$). Moreover, the family $\set{X_{\w}\colon \w \in \setR^m}$ is *uniformly definable* in $\cR$: there exists a formula, in fact, a *quantifier free* formula, $\ph(\x,\w) = \ph(x,1,\ldots,x_n,w_1,\ldots,w_n)$ so that $X_{\w} = \set{\x \in \setR^n \colon \cR\models \ph(\x,\w)}$.
In other words, our hypothesis class is exactly the class $\set{\ph(\x,\w)^{\cR^{n}} \colon \w \in \setR^m}$ where $\ph(\x,\w)^{\cR^{n}}$ is the set of “solutions” in $\cR^n$ of the formula $\ph(\x,\w)$ (where $\x$ are the variables, and $\w \in \setR^m$ is fixed). That is, $\ph(\x,\w)^{\cR^{n}} = \set{\x \in \setR^n \colon \cR\models \ph(\x,\w)}$. See Karpinski and Macintyre [@Karpinski95polynomialbounds] for more details.
As in the previous subsection, we now denote this collection by $\cA$. So $\cA=\set{X_{\w}\colon \w \in \setR^m} = \set{\ph(\x,\w)^{\cR^{n}} \colon \w \in \setR^m}$
As mentioned in the introduction, depending on the activations of $\cC$ that we started with, it is possible that $\text{VC}(\cA) = m\log(m)$ or $m^2$ or $m^4$, or even $\infty$. Again, we refer to Sontag [@Sontag98vcdimension] for details. The finite possibilities can all be realized in the o-minimal setting that we are working in (examples in [@Sontag98vcdimension] can all be defined in $\setR_{an,exp}$), however, any uniformly definable family of sets in an o-minimal structure has a finite VC-dimension, therefore the last possibility is impossible in our case. In fact, this is true in a much wider class of dependent (NIP) structures (which we will not discuss here).
However, a much stronger statement can be made concerning VC-density in an o-minimal structure. Specifically:
\[thm:vcdensity in ominimal\] Let $\cA$ be a uniformly definable family of sets in an o-minimal structure $\cR$. That is, assume that $\cA = \set{\ph(\x,\w)^{\cR^{n}} \colon \w \in \cR^m}$ for some formula $\ph(\x,\y) = \ph(x_1,\ldots,x_n,y_1,\ldots,y_m)$. Then $\text{vc}(\cA)\le m$.
Note that one can not expect better: the simple formula $x=y_1 \lor x=y_2 \lor \ldots \lor x=y_m$ with one variable $x$ defines the family $\cA = \set{A \subseteq \cX \colon |A| \le m}$, so $|\cA\cap B|$ is all the subsets of $B$ of size at most $m$, hence roughly of the size $|B|^m$, at least for $B$ large enough.
This theorem is due to Johnson and Laskowski [@Johnson09c.:compression]. It was obtained earlier for o-minimal expansions of the reals (which is the context we are considering) by Karpinski and Macyntyre [@Karpinski:1997:AVI:895414]. A more recent and general approach that applies in a much wider context can be found in [@Aschenbrenner11vapnik-chervonenkisdensity].
Sample complexity I
===================
We now turn to computing the desired bound on sample complexity. In this section, we show an elementary computation, which provides a loose bound.
We refer to Ben David and Shalev-Shwartz [@Shalev-shwartz_fromtheory] for basic concepts of statistical learning.
Let $\cH$ be a binary hypothesis class on a sample space $\cX$, and let $\cD$ be a probability distribution on $\cX$. Denote by $L_{\cD} \colon \cH\to[0,1]$ the 0-1 loss function function with respect to $\cD$. Essentially, $L_{\cD}(h)$ measures the propability (with respect to $\cD$) of a sample to be misclassified by $h$. See [@Shalev-shwartz_fromtheory] for details. Recall that given a sample $S \in \cX^k$ of size $k$ and $h \in \cH$, we denote by $L_S(h)$ the 0-1 loss of $h$ with respect $S$. This is simply the percentage of elements of $S$ that $h$ misclassifies - the most natural estimate for $L_{\cD}(h)$, if all one sees is $S$). Recall also that $\cD^k$ denotes the product measure on $\cX^k$ that arises from $\cD$.
Recall that if we succeed in showing that, given $\eps,\delta>0$, there exists $k = k(\eps,\delta)$ such that with probability $1-\delta$ over the choice of a sample $S$ of size $k$, we have $\left|L_{\cD}(h)-L_S(h)\right| \le \eps$, then in particular this $k$ provides an upper bound on sample complexity for agnostic PAC learnability. In fact, such $k$ witnesses a stronger property called “uniform convergence” for $\cH$.
The following basic fact is Theorem 6.11 in [@Shalev-shwartz_fromtheory]:
\[fct:theorem growth function\] For every $h\in\cH$ and $\delta \in (0,1)$, with probability of at least $1-\delta$ over the choice of $S \sim \cD^m$ we have $$\left|L_{\cD}(h)-L_S(h)\right| \le \frac{4+\sqrt{\log(\tau_{\cH}(2k)}}{\delta\sqrt{2k}}$$
Where $\tau_{\cH}$ denotes the growth function of $\cH$, as defined in subsection \[sub:vc\_density\].
Now let us combine Fact \[fct:theorem growth function\] with Theorem \[thm:vcdensity in ominimal\], i.e. recall that $\tau_{\cH}(2k) \le (2k)^m$:
$$\left|L_{\cD}(h)-L_S(h)\right| \le \frac{4+\sqrt{\log(2^mk^m)}}{\delta\sqrt{2k}} = \frac{4+\sqrt{m+m\log(k)}}{\delta\sqrt{2k}}$$
Let $\eps>0$. We want the expression on the left side of the equation to be at most $\eps>0$. For this (assuming $k$ is large enough) it is enough to find $k$ such that
$$\frac{2\sqrt{\log(2^mk^m)}}{\delta\sqrt{2k}} = \frac{2\sqrt{m+m\log(k)}}{\delta\sqrt{2k}} \le \eps$$
That is,
$$\frac{4{m\log(2k)}}{\delta^2{2k}}\le \eps^2$$
$$\frac{\log(2k)}{{2k}}\le \frac{\eps^2\delta^2}{4m}$$
Or
$$\frac{{2k}}{\log(2k)}\ge \frac{4m}{\eps^2\delta^2}$$
So we want $k$ large enough so that the following inequality holds:
$${2k}\ge \frac{4m}{\eps^2\delta^2}\log(2k)$$
For this (see Lemma A.1 in [@Shalev-shwartz_fromtheory]) it is enough to have
$$2k \ge 2\frac{4m}{\eps^2\delta^2}\log(\frac{4m}{\eps^2\delta^2})$$
Or
$$k \ge \frac{4m}{\eps^2\delta^2}\log(\frac{4m}{\eps^2\delta^2})$$
We have therefore shown
\[thm:sample complexity\] A neural network with $m$ weights and activations simultaneously definable in some o-minimal structure admits the property of uniform convergence, and is therefore agnostic PAC-learnable, with sample complexity
$$k = k(\eps,\delta) \le \frac{4m}{\eps^2\delta^2}\log(\frac{4m}{\eps^2\delta^2})$$
In the case of high VC-dimension, we get a much better dependence on $m$ at the expense of a worse dependence on $\delta$ than in the classical bounds. It is clear, however, that more careful analysis will yield better bounds. We confirm this in the next section.
Sample complexity II
====================
In this section we provide a tighter bound, significantly improving the dependence on $\delta$, at the expense of a worse multiplicative constant, and an additional factor of $\log(1/\eps)$.
As in the previous section, we refer the [@Shalev-shwartz_fromtheory] for background, specifically, for the discussion of Rademacher complexity and its properties.
The setting is the same as in the previous section: $\cH$ is a binary hypothesis class of VC-density $m$ on a sample space $\cX$, $\cD$ is a probability distribution on $\cX$; $L_{\cD} \colon \cH$ denotes the 0-1 loss function function with respect to $\cD$, $L_S(h)$ denotes the 0-1 loss of a hypothesis $h$ with respect to a sample $S \in \cX^k$.
Our main result is the following:
\[thm:sample complexity bound\] Let $\cC$ be a binary neural network with $m$ weights and activation functions definable in an o-minimal structure. Let $\cH$ be the hypothesis set given by $\cC$. Then there exists a multiplicative constant $\hat C$ such that for every $\eps,\delta>0$ we have $\left|L_{\cD}(h)-L_S(h)\right| \le \eps$ with probability at least $1-\delta$ over the choice of $S$ for all $h \in \cH$, provided that $$|S|\ge \hat C \left[\frac{m}{\eps^2}\log\left(\frac{2m}{\eps^2}\right) + \frac{\log\left(\frac{4}{\delta}\right)}{\eps^2}\right]$$
Given a classification training set $\set{(x_i,y_i)\colon i\le k}$, let $\setA$ denote the set of all binary vectors in $\setR^k$ of the form $([h(x_1) = y_1],[h(x_2) = y_2],\ldots,[h(x_k) = y_k])$ for $h \in \cH$; where we naturally interpret the value of a boolean expression as 1 if it is true, and 0 if it is false. By Fact \[fct:linear\], the size of $\setA$ is bounded by $Ck^m$ for some multiplicative constant $C$. Hence by Massart’s Lemma (see Lemma 26.8 in [@Shalev-shwartz_fromtheory]), the Rademacher complexity of $\setA$ is bounded above by $$\sqrt{\frac{2\log(Ck^m)}{k}} = \sqrt{\frac{2m\log(C'k)}{k}}$$
For some constant $C'$ (and note that the norm of a binary vector in $\setR^k$ is at most $\sqrt{k}$, hence we get $\sqrt{k}$ in the denominator).
Now let $\delta>0$. By Theorem 26.5 in [@Shalev-shwartz_fromtheory], with probability at most $1-\delta$, we get
$$\left|L_{\cD}(h)-L_S(h)\right| \le \sqrt{\frac{8m\log(C'k)}{k}} + \sqrt{\frac{2\log(\frac{4}{\delta})}{m}} \le 2\sqrt{\frac{8m\log(C'k)+2\log(\frac{4}{\delta})}{m}}$$
And so, given $\eps>0,\delta>0$, we need the sample size $k = k(\eps,\delta)$ to satisfy:
$$4{\frac{8m\log(C'k)+2\log(\frac{4}{\delta})}{k}} \le \eps^2$$
Or
$${\frac{k}{16m\log(C'k)+8\log(\frac{4}{\delta})}} \ge \frac{1}{\eps^2}$$
Rewriting the desired inequality in the following form
$${\frac{C'k}{8C'\left[2m\log(C'k)+\log(\frac{4}{\delta})\right]}} \ge \frac{1}{\eps^2}$$
We need
$${C'k} \ge \frac{8C'\left[2m\log(C'k)+\log(\frac{4}{\delta})\right]}{\eps^2} = 16C'\frac{m}{\eps^2}\log(C'k) + 8C'\frac{\log(\frac{4}{\delta})}{\eps^2}$$
Applying Lemma A.2 from [@Shalev-shwartz_fromtheory], we can now find a constant $\hat C$ (which can be explicitly computed from $C'$, hence from $C$) so that the above inequality holds provided that
$$k \ge \hat C \left[\frac{m}{\eps^2}\log\left(\frac{2m}{\eps^2}\right) + \frac{\log\left(\frac{4}{\delta}\right)}{\eps^2}\right]$$
This gives us a much better dependence on $\delta$ at the expense of the much less significant factor $\log(1/\eps)$ (and a worse multiplicative constant). It is possible that this factor can be improved using more sophisticated techniques; we intend to examine this question in future works.
|
---
abstract: 'The assembly of the Milky Way bulge is an old topic in astronomy, one now in a period of renewed and rapid development. That is due to tremendous advances in observations of bulge stars; motivating observations of both local and high-redshift galaxies; and increasingly sophisticated simulations. The dominant scenario for bulge formation is that of the Milky Way as a nearly pure disk galaxy, with the inner disk having formed a bar and buckled. This can potentially explain virtually all bulge stars with \[Fe/H\] $\gtrsim -1.0$, which comprise 95% of the stellar population. The evidence is the incredible success in N-body models of this type in making non-trivial, non-generic predictions, such as the rotation curve and velocity dispersion measured from radial velocities, and the spatial morphologies of the peanut/X-shape and the long bar. The classical bulge scenario, whereby the bulge formed from early dissipative collapse and mergers, remains viable for stars with \[Fe/H\] $\lesssim -1.0$ and potentially a minority of the other stars. A classical bulge is expected from $\Lambda$-CDM cosmological simulations, can accentuate the properties of an existing bar in a hybrid system, and is most consistent with the bulge abundance trends such as \[Mg/Fe\], which are elevated relative to both the thin and thick disks. Finally, the clumpy-galaxy scenario is considered, as it is the correct description of most Milky Way precursors given observations of high-redshift galaxies. Simulations predict that these star-forming clumps will sometimes migrate to the centres of galaxies where they may form a bulge, and galaxies often include a bulge clump as well. They will possibly form a bar with properties consistent with those of the Milky Way, such as the exponential profile and metallicity gradient. Given the relative successes of these scenarios, the Milky Way bulge is plausibly of composite origin, with a classical bulge and/or inner halo numerically dominant for stars with \[Fe/H\] $\lesssim -1.0$, a buckling thick disk for stars with $-1.0 \lesssim \rm{[Fe/H]]} \lesssim -0.50$ perhaps descended from the clumpy galaxy phase, and a buckling thin disk for stars with \[Fe/H\]$\gtrsim -0.50$. Overlaps from these scenarios are uncertain throughout.'
author:
- 'David M. Nataf$^1$[^1]\'
title: 'Was the Milky Way Bulge Formed From The Buckling Disk Instability, Hierarchical Collapse, Accretion of Clumps, or All of the Above?'
---
Galaxy: Bulge – Galaxy: kinematics and dynamics
INTRODUCTION {#sec:intro}
============
Formation scenarions for the Galactic bulge (and bulges in general) have been around for some time.
@1962ApJ...136..748E suggested the scenario of early, rapid, monolithic formation by dissipative collapse, this is now referred to as the “classical bulge" scenario. The hierarchical merging of smaller objects, typical of the earliest phases of galaxy formation as predicted by $\lambda$-CDM cosmological simulations [@2010ApJ...708.1398T; @2011ApJ...729...16K], is now often included as part of the definition of “classical bulges" [@2005MNRAS.358.1477A]. It is also known that bars can form from the dynamical evolution of a disk galaxy via the “buckling instability" scenario, where the bulge would be predominantly composed of disk stars now on bar orbits [@1978ApJ...223..811M; @1979ApJ...227..785M; @1979AJ.....84..585H], for example the x1 orbital family [@2002MNRAS.333..847S]. Finally, the migration of star-forming clumps toward the centres of disk galaxies due to dynamical friction emerges naturally from simulations, this is the “clump-origin" scenario [@1998Natur.392..253N; @1999ApJ...514...77N].
Given the wide availability of plausible models, it is no surprise that it has long been acknowledged that the formation of the bulge may be a composite process. For example, in a previous review (of bulges in general), @2000bgfp.conf..413C wrote:
> *A fraction of bulges could have formed early (at first collapse); then secular dynamical evolution enrich them; in parallel, according to environment, accretion and minor mergers contribute to raise their mass.*
The first and third process mentioned are the two components of the classical bulge scenario, the second is the buckling instability scenario. Relative contributions of the different scenarios of bulge formation were not assigned precise bounds, given that this is an intrinsically difficult problem.
Astronomers need an accurate, precise and thorough census of the current Milky Way bulge before we can seriously consider disentangling its history.
That census is now slowly, but surely, becoming available. These include ground-based photometric surveys such as OGLE-IV in the optical [@2015AcA....65....1U], $VVV$ in the near-infrared and *Wide-field Infrared Survey Explorer (WISE)* in the mid-infrared [@2010AJ....140.1868W]; Deep, multi-wavelength *Hubble Space Telescope (HST)* photometry which measures the main-sequence and thus constrains ages [@2010ApJ...725L..19B]. At the spectroscopic end, surveys such as BRAVA [@2012AJ....143...57K], ARGOS [@2013MNRAS.428.3660F], Gaia-ESO , EMBLA [@2014MNRAS.445.4241H], APOGEE [@2016ApJ...819....2N], BRAVA-RR [@2016ApJ...821L..25K], and GIBS , collectively provide kinematics and sometimes detailed chemistries for tens of thousands of bulge stars.
Concurrent with this, there have been vast improvements in the theory and simulations as well. For example, the N-body simulation of @2017MNRAS.467L..46A contains 17.5 million particles, whereas that of @1979ApJ...227..785M had 96,017 particles. The level of detail, and sophistication of current models allows more robust analysis. Further, more questions are being asked of models, such as how a classical bulge will evolve when embedded within a massive disk [@2012MNRAS.421..333S], where younger stars might be distributed within a bar [@2016arXiv161109023D], or how the separation between the two arms of an X-shape will appear as a function of direction [@2015MNRAS.447.1535N]. These questions, largely motivated by the observations, should yield more predictive and discriminatory power in evaluating models, with the caveat that it is scientifically misguided to expect a perfect quantitative match between a simulation and an actual galaxy.
In this review, three scenarios for bulge formation are discussed as well as the current evidence in their favour.
THE MILKY WAY AS A NEARLY PURE DISK GALAXY
==========================================
This is the scenario where the Milky Way bulge is largely or nearly entirely a bar. This bar would have first formed from a disk, and then extended vertically by one or more buckling instability episodes. This scenario arguably has the most evidence in its favour.
The evidence presented is that:
- Pure disk galaxies are observed to exist in the local universe, and the bar buckling process occurs naturally in N-body simulations.
- The radial velocity measurements from large spectroscopic surveys are consistent with the theoretical predictions from buckling disk models.
- The peanut/X-shaped morphology for the distribution of bulge stars is a prediction of these models, and is constrained to represent a large fraction of the bulge mass.
- The morphology of the long bar is another specific, precise prediction of these models.
- An old argument against this theory, that of the metallicity gradient, has been shown to be invalid as it can also be produced by the buckling instability model.
Pure disk galaxies exist, and simulations show that they can naturally evolve to be barred galaxies
---------------------------------------------------------------------------------------------------
Pure disk galaxies exist. @2010ApJ...723...54K obtained *HST* photometry of six nearby galaxies and measured their surface brightness profiles. They found an upperbound on the total (classical bulge + pseudobulge) stellar mass fraction of $\sim$3% of the total disk mass. They also took an inventory of galaxies within 8 Mpc with $v_{circ} > 150\,\, \rm{km\, s}^{-1}$, and found that 11 of 19 showed no evidence for a classical bulge, and four of the remaining eight may contain classical bulges contributing 5% - 12% of the stellar mass. It is the case, however surprisingly, that pure disk galaxies are common in the low redshift universe, and thus it is plausible for the Milky Way to be one as well.
Simulations of disk galaxies consistently show that bars and subsequently buckling bars can occur spontaneously in disk galaxies (e.g. ). The rate of evolution of the bar is sensitive to various variables, such as the ratio of the disk to halo mass . The bar may have a north-south asymmetry at first, but this rapidly dissipates [@1991Natur.352..411R]. Overtime, the bars tend to grow in vertical extent, for example via multiple, recurrent buckling episodes (@2006ApJ...637..214M, see Figure 3).
The combination of these two facts, that pure disk galaxies are common in the local universe, and that simulations predict that pure disk galaxies can spontaneously evolve to have bars, render it a plausible model for the Milky Way’s bulge as well. The necessary initial conditions – a disk galaxy – are ubiquitous, and the necessary evolution is natural.
Radial velocity measurements from large spectroscopic surveys
-------------------------------------------------------------
The Milky Way is a barred galaxy [@1995ApJ...445..716D], however it has not been clear until recently how quantitatively dominant the bar is relative to the hypothetical classical bulge contribution. The advent of large spectroscopic surveys has allowed us to disentangle the bulge into plausible subcomponents.
@2010ApJ...720L..72S fit a suite of N-body models to bulge radial velocity data from the BRAVA survey, specifically the mean radial velocity (and thus rotation curve) and radial velocity dispersion as a function of longitude and latitude. They recover a best-fit viewing angle, betweeen the bar’s major axis and the line of sight between the Sun and the Galactic centre, of $\alpha_{\rm{Bar}}=20^{\circ}$, consistent with more recent measurements. More significantly, they show that models with significant classical bulges, with parameters defined to lie on the fundamental plane of ellipticals and bulges [@2009ApJS..182..216K], are not well-fit by the data. They constrain the classical bulge mass of the Milky Way to be no more than 8% of the total mass of the disk.
@2013MNRAS.432.2092N analyzed the data from the larger ARGOS survey, specifically 17,400 red giants with best-fit parameters including $R_{GC} \leq 3.5$ Kpc, which were spread over 30 degrees of longitude and 20 degrees of latitude. They find several features in the data consistent with N-body expectations for disk origin to the bulge. Bulge stars located between 5 and 10 degrees from the plane ($0.7\, \rm{Kpc} \lesssim |z| \lesssim 1.5\, \rm{Kpc}$) are cylindrically rotating, with the exception of the 5% of stars with \[Fe/H\] $\leq -1.0$. They suggest that those stars are the inner extensions of the halo and the metal-weak thick disk. It is also the case that the thin disk includes very, very few stars with \[Fe/H\] $\leq -0.50$ [@2017arXiv170504349D], so it was always an implausible origin source for the metal-poor component of the bulge.
compared the ARGOS data to three different N-body models and found that the bulge could not be explained as having resulted from the evolution of a pure thin disk galaxy, and thus a joint origin with the thick disk is likely required as well. This may appear as requiring too many free parameters from models, but this is a parameter that exists in nature, the Milky Way is not a pure thin disk galaxy, it also has a thick disk. show that if the bulge was purely due to a thin disk buckling event, the curve of mean radial velocity versus longitude would decrease in amplitude, and the velocity dispersions would decrease, with increasing metallicity. The decrease in the amplitude of the rotation curve is seen for stars satisfying \[Fe/H\] $\gtrsim -0.50$, but not for lower metallicities. The velocity dispersions do decrease, but not as quickly as predicted from a pure thin disk model. suggest that the solution to this issue is that bulge stars with \[Fe/H\] $\lesssim -0.50$ formed from the thick disk, rather than from the more metal-poor component of the thin disk.
@2016ApJ...832..132Z investigated the skewness (third moment) and kurtosis (fourth moments) of the radial velocity distribution functions in APOGEE data toward the bulge. The correlation between skewness and mean velocity (first moment), a known diagnostic of bars, is only observed for the more metal-rich fraction of the stars, defined in that paper as stars with \[Fe/H\] $\gtrsim -0.40$. The data have a flat kurtotsis, Kurt(V) $\approx 0$, consistent with that expected from models. The data have slightly higher velocity dispersion and slightly lower skewness than expected from N-body models of a simple single disk galaxy undergoing buckling.
The recent review of @2016PASA...33...27D goes over these lines of evidence in greater detail than possible here, and concludes that bulge stars with \[Fe/H\] $\gtrsim -1.0$ formed from buckling thick and thin disks.
The peanut/X-shaped spatial distribution of bulge stars
-------------------------------------------------------
![Figure 1 from @2016AJ....152...14N, the X-shaped bulge is unambiguous in the integrated mid-infrared photometry of the Milky Way.[]{data-label="NessLang"}](xbulge-00.pdf)
![Figure 3 from @2006ApJ...637..214M. An X-shaped bulge is a natural outcome of N-body models of disk galaxies undergoing the buckling instability.[]{data-label="MV2006"}](fig3MV2006.pdf)
The sightline-dependent bifurcation of the apparent magnitude distribution of red clump stars (comprising $\sim$99% of bulge horizontal branch stars, see @2013ApJ...769...88N) was first reported by @2007MNRAS.378.1064R. They speculated that it might be due to a distinct population lying at a different distance. @2010ApJ...721L..28N and @2010ApJ...724.1491M showed that the double peak was spread across a large swath of the bulge, at large separations ($|b| \gtrsim 5^{\circ}$) from the plane. @2010ApJ...724.1491M argued convincingly, that the feature had to be due to an X-shaped bulge (a very strong peanut shape), a feature of strong bars in N-body models. Subsequent analyses confirmed the claim. @2012ApJ...756...22N found that the bright and faint red clumps had the same difference in mean radial velocity as expected from N-body models. @2012ApJ...757L...7L showed that the N-body model of @2010ApJ...720L..72S, already demonstrated to be a good match to radial velocity observations, also includes an X-shape. The relative brightness, and the dependence of the relative number counts on direction, was qualitatively similar to that seen in the data. showed that the proper motion distributions were consistent. @2015MNRAS.447.1535N showed that precision measurements of the brightness difference, relative number counts, and total number density of stars in the data from OGLE-III [@2008AcA....58...69U] was qualitatively matched by N-body models from @2003MNRAS.341.1179A and @2010ApJ...720L..72S. An unambiguous image of the X-shaped bulge in integrated mid-infrared photometry of the Milky Way is shown in Figure \[NessLang\], and the development of one from an N-body model is shown in Figure \[MV2006\].
Precision modelling of the photometric data by @2013MNRAS.435.1874W has been followed by detailed dynamical modelling by @2015MNRAS.450L..66P [@2015MNRAS.448..713P; @2017MNRAS.465.1621P]. A suite of N-body models coarsely consistent with Milky Way constraints were adjusted to be more consistent using the made-to-measure method [@1996MNRAS.282..223S], whereby weights of the particles in the model are shifted up or down to match the observational data. The end models always contain a significant X-shape, comprising 40-50% of the stellar mass of the bulge, but only dominant for stars with \[Fe/H\] $\gtrsim -0.50$. That is not a repetition of the results mentioned in the previous subsection, as @2015MNRAS.450L..66P [@2015MNRAS.448..713P; @2017MNRAS.465.1621P] also constrained their N-body models to match the spectroscopic data from the APOGEE survey, which goes much closer to the plane [@2016ApJ...819....2N], and the global photometric parameters of the bulge which trace the number density and distance distribution function of stars [@2013MNRAS.435.1874W].
Reiterating, this is an impressive series of observational tests for models to have passed. First, the alternative hypotheses (RGB bump, metallicity differences, etc) do not work. Second, N-body model predictions qualitatively match the radial velocity offsets, the proper motion measurements, the mean brightness difference, and the dependence of the relative and total number counts as a function of direction. This picture survives, and is in fact dominant for stars with \[Fe/H\] $\gtrsim -0.50$, when models are required to match observations from several different surveys.
The morphology of the long bar is another specific, precise prediction of these models.
----------------------------------------------------------------------------------------
An issue of Galactic modelling in the last two decades is that of the long bar of the Milky Way . Evidence has accumulated from multiple investigations of a long, in-plane bar, with a half-length of $\sim$4 Kpc, and an orientation angle of $\alpha_{\rm{Bar}}=45^{\circ}$. This represented a challenge to Galactic structure studies, as the orientation angle is far larger than the value found for the triaxial bulge toward the inner few Kpc, and the morphology is not consistent with the traxial ellipsoid models used to fit for the bulge [@1995ApJ...445..716D; @1997ApJ...477..163S]. It appears as a double-barred system where the two bars are of comparable length and are not aligned, which is an unstable configuration [@2012EPJWC..1906004A].
A suggestion to this resolution was independently proposed by @2011ApJ...734L..20M and @2012EPJWC..1906004A. Their idea was the analytical triaxial ellipsoid models then widely used to model the bulge were only (somewhat) suitable for the peanut/X-shaped component, whereas bars in simulations often have long, and thin extensions. This hypothesis made a specific prediction, that the difference in measured angle ${\Delta}\alpha_{\rm{Bar}}$, then around 20 degrees, was due to observational errors. A smaller difference might remain if the ends of the bar develop interactions with spiral arms @2011ApJ...734L..20M.
This prediction was confirmed by @2015MNRAS.450.4050W, who combined photometry from Spitzer-GLIMPSE [@2005ApJ...630L.149B], 2MASS [@2006AJ....131.1163S], VVV [@2012MNRAS.422.1902I], and UKIDSS [@2008MNRAS.391..136L] to make a global map of the bulge. The long bar had an angle of $\alpha_{\rm{Bar}}=28-33^{\circ}$, consistent with the values of $\alpha_{\rm{Bar}}=27 \pm 2^{\circ}$ and $\alpha_{\rm{Bar}} \approx 29^{\circ}$ then recently measured by @2013MNRAS.435.1874W and @2013MNRAS.434..595C respectively for the inner peanut/X-shaped component.
Thus, the issue of the long bar, which was previously a challenge to Galactic structure models, ended up being a triumph. The long bar is not only not surprising, but expected from simulations of buckling disk galaxies. Models predicted that the difference in orientation angle should shrink with better analysis, and it did.
The metallicity gradient can also be reproduced by buckling disk
-----------------------------------------------------------------
The Galactic bulge has a vertical metallicity gradient, with stars further from the plane having a lower mean metallicity. @1995AJ....110.2788T used near-IR photometry and estimated a gradient in ${\nabla}$\[Fe/H\] $= -0.06\pm0.03$ dex/deg or $-0.43\pm0.21$ dex/kpc between $b=-3^{\circ}$ and $b=-12^{\circ}$. The metallicity gradient was independently and concurrently confirmed by @1995MNRAS.277.1293M, and since confirmed numerous times with spectroscopic data (e.g. )
This was argued by @1995MNRAS.277.1293M to be evidence for a dissipative collapse, as models of spheroid formation predicted a radial metallicity gradient [@1984ApJ...286..403C]. However, @1995MNRAS.277.1293M acknowledged that “*the alternative interpretation that the gradient itself is caused by the mixing of different components in the inner Galaxy cannot be ruled out.*”
That second point, that the bulge has different components with both different metallicity distribution functions and scale heights, is very much the case, as should be clear from the literature evidence summarized in this review. Further, it also turns out that a metallicity gradient for the bulge can emerge from a pure disk galaxy. @2013ApJ...766L...3M evolved an N-body model of a pure disk galaxy which included an initial radial metallicity gradient. They chose $\rm{[M/H]}(R)=+0.60-0.40(R)$/Kpc, where “R” is the galactocentriuc radius of particles at the start of the simulation. Given that particles from initial radii are scattered to different orbits, a metallicity gradient emerges which is qualitatively consistent with that observed for the whole bulge – there is impressive agreement with the global photometric metallicity maps from , which are derived from the morphology of the red giant branch. That said, it is worth noting that the metallicity gradient required by @2013ApJ...766L...3M is extremely large.
This result is not altogether surprising as it has been known for a while that dynamical mixing processes do not erase gradients. For example, @1980MNRAS.191P...1W demonstrated that simulations of mergers were expected to reduce, but not erase, gradients in metallicity. finds that the metallicity gradients in interacting galaxies, measured in terms of oxygen abundance of HII regions versus effective radius, is reduced by $\sim 1/2$ relative to non-interacting galaxies.
The metallicity gradient of the bulge, first largely argued to be evidence for dissipative collapse and thus evidence against a disk-galaxy origin, turns out not to be discriminating. A metallicity gradient can be matched by models of dissipative collapse, pure disk galaxies undergoing the buckling instability, a combination of the two, or as we will see, the clump-origin bulge scenario [@2012MNRAS.422.1902I]. The existence of a metallicity gradient, by itself, yields no significant constraints bulge formation.
Caveat against the bar scenario: Gas fractions and the challenge of initial conditions
--------------------------------------------------------------------------------------
The properties and evolution of bars in N-body simulations of pure exponential disks are a well-researched subject. However, real galaxies (such as the Milky Way) contain gas, and generally contained more gas in the past.
@2013MNRAS.429.1949A investigated the relative predicted properties of bars in N-body simulations with and without gas. They found that the gas-rich galaxies remain axisymmetric for longer. When they do develop bars, they do so at a slower rate, and end up much weaker. Their Figure 7 show that bar strengths in gas-rich galaxies are predicted to end up lower even after the gas has been depleted. In contrast, as discussed in this section, the Milky Way has a very strong bar.
Indeed, the bar fraction is considerably lower at high redshift. @2014MNRAS.438.2882M finds that it drops to 11% at $z=1$, corresponding to a lookback time of 7.8 Gyr. @2014MNRAS.445.3466S also estimates a bar-fraction of 11%, in the redshift range $0.50 < z < 2.0$. By that time most Milky Way bulge stars were already formed [@2011ApJ...735...37C; @2017arXiv170202971B] and thus they already have a kinematic distribution.
The combination of these issues, should give pause to the notion that a bar+buckling instability from a pure exponential disk is independently sufficient to explain the inner Milky Way’s dynamics. Separately from the issue of gas weakening bars, subsequent sections of this review will discuss how a minor classical bulge can actually strengthen the bar, and why the Milky Way is expected to be a former “clumpy galaxy" which has implications for bulge formation.
THE CLASSICAL BULGE SCENARIO
============================
The classical bulge scenario is one where the bulge is formed a combination of early, dissipative collapse and accretion of objects via minor or major mergers.
The evidence we present is that:
- It is expected from theory.
- The metal-poor bulge stars are kinematically consistent with a classical bulge behaviour.
- The theoretical interaction between classical bulges and buckling disks is consistent with observations.
- The bulge trends in the $\alpha$-elements are not consistent with those of the disk.
A classical bulge is expected from theory
-----------------------------------------
The most obvious advantages of this scenario are that it is predicted by straightforward theory of gravitational, dissipative collapse [@1962ApJ...136..748E], and hierarchical clustering in a cold dark matter universe [@1978MNRAS.183..341W; @1993MNRAS.264..201K].
@1978MNRAS.183..341W proposed that most of the matter in the universe condensed early into small “dark” objects. They suggested a model with $\Omega_{m}=0.20$ and dark matter making up 80% of the matter, impressively similar to the modern values . Within this picture, the “pure disk galaxy" that has particles and no gas is not a viable initial condition, as the universe and star formation begin with larger number of small haloes that coalesce and accrete additional small haloes.
As far as current, more up-to-date models are concerned, a classical bulge can be considered a requirement. @2011ApJ...729...16K simulate 150 galaxies from various, cosmologically-motivated initial conditions. They formed disk structures in 48 of their galaxies, including 5 galaxies that had masses comparable to the Milky Way. Though some of their simulated galaxies had small classical bulges, none completely lacked a classical bulge. It is a concern that many local galaxies are found not to have classical bulges locally [@2010ApJ...723...54K], but one can argue against that by saying that information is lost when looking at those galaxies in integrated light. That cannot be argued for the Milky Way.
The kinematics of metal-poor bulge stars
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![Left panel of Figure 2 from @2016ApJ...821L..25K. RR Lyrae stars show null or negligible Galactic rotation, as well as very high velocity dispersion, in contrast to the majority of bulge stars.[]{data-label="Kunder2016"}](fig2bKunder_crop.pdf)
There has been much written in this review of the spectacular consistency between N-body models of buckling disks and the dynamics of bulge stars with \[Fe/H\] $\gtrsim -1.0$ or $\gtrsim -0.50$, depending on the study. These consistencies are not found for the metal-poor bulge stars. For example, the Galactic bar is either null [@2013ApJ...776L..19D] or weak [@2012ApJ...750..169P; @2015ApJ...811..113P] in the RR Lyrae stars, variable stars of standardizable distance that can be used to directly probe the metal-poor bulge. @2013ApJ...776L..19D investigated 7,663 fundamental-mode RR Lyrae in $I$ and $K_{s}$ bands, and found that they do not trace a strong bar, but rather a more spheroidal and centrally concentrated distribution. No correlation is observed between dereddened distance and longitude, a correlation detected at high significance in the more metal-rich red clump stars which trace a bar [@1997ApJ...477..163S].
@2012ApJ...750..169P and @2015ApJ...811..113P obtained different results in their analysis of 16,836 and 27,258 RR Lyrae in the OGLE-III and OGLE-IV surveys respectively. They do find a bar, but it is kinematically hotter than the bar measured in red clump stars, and does not show a peanut/X-shape at large separations from the plane.
Analysis of RR Lyrae kinematics further these findings. @2016ApJ...821L..25K analyze spectroscopic data for 947 RR Lyrae as part of the ongoing BRAVA-RR survey. These RR Lyrae, measured toward $|l| \lesssim 4^{\circ}$ and $-6^{\circ} \lesssim b \lesssim -3^{\circ}$ show higher velocity dispersions and weaker rotation than the metal-rich M-giants studied as part of the BRAVA survey. The velocity dispersion is $\sim$15% higher, and the rotation is null or negligible. When the RR Lyrae are split into two metallicity bins with \[Fe/H\]$=-$0.75 marking the bifurcation point, no difference in rotation is measured though the metal-poor RR Lyrae do have a higher velocity dispersion. The RR Lyrae radial velocity measurements are shown in Figure \[Kunder2016\].
A further issue with the RR Lyrae is the very fact that they are RR Lyrae. The bulge RR Lyrae are measured to have a mean metallicity of \[Fe/H\]$\approx -1.0$, whereas metallicities of \[Fe/H\]$\approx -1.60$ are more typical of RR Lyrae in the globular clusters and in the halo [@2010ApJ...708..698D]. More metal-rich stars have a higher turnoff mass at fixed age, and thus for these horizontal branch stars to be on the instability strip at a higher metallicity, they need to have a lower turnoff mass by other means, and thus likely a greater age. @1992AJ....104.1780L estimated that the bulge RR Lyrae had to be ${\Delta}t \sim 1.3 \pm 0.30$ Gyr older than halo stars, for stars formed with metallicities $-1.5 \lesssim \rm{[Fe/H]} \lesssim -1.0$. A bulge which is older than the halo is more consistent with a classical bulge then say, a buckling disk origin.
This argument also applies to bulge globular clusters. For example, NGC 6522, has a blue horizontal branch with a mean metallicity of \[Fe/H\]$=-1.0$ , necessitating an extremely old age. This is a consistent pattern of bulge globular clusters . This information, combined with that of the RR Lyrae, strongly suggests that bulge stars of metallicity \[Fe/H\]$=-1.0$ are the oldest or among the oldest stellar populations in the Galaxy, and thus were around prior to there being a massive disk that could form a bar.
The theoretical interaction between classical bulges and buckling disks is consistent with observations.
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There is relatively sparse research on the theoretically predicted interaction of a central concentration of stars on a buckling disk, but what research is available now turns out to be consistent with the data. This is true both of the predictions of the classical bulge behaviour and the bar behaviour.
@2002MNRAS.330...35A studied three different N-body models of disk galaxies with variable initial central concentrations. The disk mass and disk-to-halo mass ratios are fixed. Two models have a non-centrally concentrated halo, one of those two also has a bulge. A third model has a centrally concentrated halo, but no bulge. Within this suite of models, the galaxy with a non-centrally concentrated halo and no bulge ends up forming the weakest bar, it does not develop cylindrical rotation, and its boxy shape does not evolve to an X-shape. Interestingly, the model with a centrally concentrated halo but no bulge ended up with the strongest bar. This is suggestive that greater central concentration was needed given the exceptionally strong bar of the Milky Way, though in and of itself it is not sufficient as more models would be needed for a more convincing picture.
Conversely, have investigated how a joint classical bulge / buckling disk origin impacts the development of the bulge by means of N-body models. @2012MNRAS.421..333S finds that a small (7% of the total dis mass) classical bulge can pick up angular momentum from the larger rotating bar, and thus even develop into a triaxial object with cylindrical rotation. @2013MNRAS.430.2039S show that the composite bulge always ends up rotating cylindrically, but may have deviations from cylindrical rotation at specific moments in its evolution. Curiously, they find that the final size of the composite bulges are reduced if the initial classical bulge has its own angular momentum. extend their prior results to show that even massive classical bulges can pick up as much specific angular momentum as low-mass classical bulges, but that the resulting rotation is non-cylindrical. All composite systems eventually form a boxy/peanut bulge.
@2017MNRAS.464L..80P model the evolution of a disk galaxy with a halo via an N-body model and find that the properties of the RR Lyrae are consistent with that of an inner halo, specifically their number density, their slow rotation, the lack of a peanut/X-shape in their spatial distribution, and their higher velocity dispersion. Within their model, only 12% of RR Lyrae end trapped on bar-like orbits, which can be considered a prediction that the fraction will be very small once proper motions are available and once astronomers can compute orbits for these stars.
It been established in previous sections of this review that observations and analysis thereof have ruled out a predominantly classical bulge for the Milky Way bulge. However, the N-body models are also consistent with either a centrally concentrated inner halo, or an additional classical bulge in addition to that. Such a feature could help explain the Milky Way’s very strong bar as well as the behaviour of low-metallicity bulge stars. Further analysis, and more data of metal-poor bulge stars are needed.
The bulge trends in the $\alpha$-elements are not consistent with those of the thin and thick disks.
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The abundances of the $\alpha$-elements relative to iron, \[$\alpha$/Fe\], are a historic argument for a distinct formation to the bulge and the disk. That is because the \[$\alpha$/Fe\] abundance ratios trace the efficiency of star formation and possibly the initial mass function of a stellar population . A higher level of \[$\alpha$/Fe\] can be due to a lower contribution to the chemical enrichment of interstellar gas from type Ia SNe (which take longer to form), and thus star formation would need to be faster.
The landmark study of @1994ApJS...91..749M found that the trends for magnesium and titanium were enhanced by $\approx$0.30 dex relative to the solar neighbourhood trends over the full range of \[Fe/H\], whereas the trends for calcium and silicon were consistent with those of the disk. These different ratios suggested a dfferent origin, and @1994ApJS...91..749M said it may reflect a common enrichment process between bulges and ellipticals. @2007ApJ...661.1152F compiled a more sophisticated analysis of a higher resolution, higher signal-to-noise sample. They found that the bulge has an magnesium trend elevated by $\sim$0.30 dex relative to the disk, whereas the trends of oxygen, silicon, calcium, and titanium are slightly elevated relative to the disk. The bulge also has higher aluminum abundances at fixed iron abundance. @2007ApJ...661.1152F conclude that the relative abundance offsets between the bulge and the disk are inconsistent with models where the bulge forms from the buckling of the disk. Separately, @2007ApJ...661.1152F found that the metal-poor bulge stars also show higher mean abundances of silicon, calcium, and titanium than the halo. They concluded that the metal-poor bulge stars could not have formed from gas with the present-day halo composition.
The discussion shifted with the work of . That investigation used high-resolution of optical spectra of 25 bulge giants and 55 comparison giants and analyzed abundances in a homogeneous manner to minimize systematic offsets. Their results were that metal-poor bulge stars (\[Fe/H\] $\lesssim -0.50$) have the same abundances as the thick disk, and more metal-rich stars have the same abundances as the thin disk.
![From Figure 21 of @2017arXiv170202971B. The \[Mg/Fe\] vs \[Fe/H\] abundance trend for bulge stars (coloured points) are shown superimposed on the disk trends (grey points). The magnesium abundances are elevated with respect to the disk abundances at all \[Fe/H\] values. []{data-label="Bensby2017"}](haltplot_mgfe_crop.pdf)
The relative abundance offsets between the bulge and the disk have not converged to zero as more data have come in. @2017arXiv170202971B compiled what is among the best datasets of bulge abundances, as they have high-resolution, high signal-to-noise abundances for 90 bulge stars located on the main-sequence turnoff and subgiant branch, analyzed using the same methods as their comparison disk sample, which is composed of stars in the solar neighbourhood. Some elements trace the same abundance trends as the disk, but magnesium, titanium, and aluminum do not. They typically trace the upper end of the larger distributions spanned by the thick and thin disks. The abundance trend of \[Mg/Fe\] vs \[Fe/H\] is shown in Figure \[Bensby2017\].
Relative abundance offsets between the bulge and the disk remain, which is a challenge to models of the Galaxy where the bulge is simply due to buckling thin and thick disks. As the abundance ratios are higher in magnesium in particular, suggesting a more rapid star formation, the suggestion is that the bulge formed faster than the disk, characteristic of an early, dissipative collapse.
THE CLUMP-ORIGIN BULGE SCENARIO
===============================
Before proceeding, a brief description of star-forming clumps will be given, though the description is itself a matter of active research. Whereas “smooth exponential disks" are decent approximations to local disk galaxies and exact descriptions of many N-body models, star-forming galaxies at high-redshift are generally clumpy and gas-rich [@1995AJ....110.1576C; @1996AJ....112..359V; @2006Natur.442..786G], with off-centre clumps accounting for 7% of the stellar mass and 20% of the star formation in massive, star-forming galaxies [@2012ApJ...753..114W].
The clumps are regions of excess star-formation seen in the disks and proto-disks of high-redshift galaxies. @2012MNRAS.422.3339W studied the properties of eight clumps in three redshift $z \sim 1.3$ observed as part of the WiggleZ Dark Energy Survey. They had an average size of 1.5 Kpc and average Jeans mass of $4.2 \times 10^9 M_{\odot}$, and accounted for roughly half the stellar mass of the disks.
Within this scenario, motivated by predictions from simulations [@1998Natur.392..253N; @1999ApJ...514...77N], some of these clumps will migrate to the centre of their galaxies due to dynamical friction, and thus form a bulge.
The evidence presented here is that:
- High-redshift galaxies largely appear as clumpy galaxies, and thus this is the plausible set of “initial" conditions for the Milky Way.
- The simulations succeed at predicting many of the observations.
This is less evidence than for the other two scenarios, but that is plausibly simply due to this being less researched topic. It is hoped that there will be further research testing whether or not the Milky Way bulge may be a clump-origin bulge. also discussed the issue. They pointed out that the mean age of bulge stars corresponds to an epoch of gas-rich disks, with gas fractions sometimes exceeding 50% [@2010Natur.463..781T; @2010ApJ...713..686D], which are more consistent with simulations of clumpy-galaxies than the usually gas-free N-body simulations of bar formation in disks.
Readers interested in a more thorough review of bulge growth in high-redshift galaxies are referred to the excellent review by @2016ASSL..418..355B. In particular, Section 3, “Mechanisms of bulge growth through high-redshift disk instabilities".
This is what high-redshift galaxies actually look like
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@2015ApJ...800...39G analyzed 3,239 high-redshift, star-forming galaxies studied as part of the *Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey (CANDELS)*. They use a conservative definition for clumps – a clump has to contribute at least 8% of the UV light of a galaxy, which excludes smaller clumps. They also require clumps to be off-centre, which excludes central clumps and is thus limiting in our context as a central clump could obviously contribute to bulge formation. One of their mosaics of clumpy galaxies is shown in Figure \[Guo2015\].
They find that galaxies with $\log(M_{\star}./M_{\odot}) > 9.8$ have a 55% probability of being clumpy at redshift $z \sim3$, down to 15% at $z \sim0.5$, which is largely due to the fact star formation declines with decreasing redshift. At all redshifts, the clump contribution to rest-frame UV light peaks at $\log(M_{\star}./M_{\odot}) > 10.5$ – the current stellar mass of the Milky Way. Integrating over both clumpy and non-clumpy galaxies, they find that 4%-10% of the star formation takes place within these massive clumps.
In other words, the clumpy galaxy is a very plausible assumption for the initial conditions of the Milky Way. Much has been said in this review that the pure disk galaxy can work, as @2010ApJ...723...54K has pointed out. However, the smooth and massive exponential disks are widely seen in observations of the local universe. That clumpy galaxies are the norm for high-redshift observations suggests that most massive galaxies have passed through a clumpy phase.
Promising insights from simulations
-----------------------------------
The prevalence of clumps in high-redshift galaxies renders them a legitimate point of discussion for the origin of the Milky Way.
A way to test this is with comparison to the Milky Way, which @2012MNRAS.422.1902I did. They used an N-body/SPH model to study the evolution of an isolated disk galaxy where the clumps migrate to the centre via dynamical friction and form a clump-origin bulge. The final bulge resembles what they call a pseudo bulge, and is referred to as a bar throughout this paper and most of the Milky Way literature. The surface density profile is nearly exponential, the final shape of the bar is boxy, and the rotation is significant. The resulting stars are old and metal-rich, with a flat star-formation history in an interval of $\Delta t \sim 2$ Gyr followed by a rapid decline in star formation. They obtain a metallicity gradient that stretches across the full vertical extent of the bulge. All of these properties are qualitatively consistent with what is observed for the Milky Way bulge. However, these properties can be matched by other models, and further this is a comparison of a single clump-origin bulge model to those of the bulge. It is worthy of consideration, but it is far too premature to declare victory.
One concern is that of whether or not the clumps actually do migrate to the centre, as they do the simulations previously discussed in this review. This is largely a question for theory, as the migration duration is too long for the baseline of observations. In the simulations of @2012MNRAS.427..968H, the inclusion of their prescription for stellar feedback disrupted the clumps, and prevented them from migrating to the bulge where they can coalesce. This prediction is not reproduced by the simulations of @2014ApJ...780...57B, who find that the ejection of stars in clumps due to stellar feedback is compensated by their accretion of gas from the gas-rich disks in which they are contained.
@2014MNRAS.443.3675M studied 770 snapshots of 20 simulated galaxies using adaptive-mesh refinement cosmological models of Galaxy evolution. The global number of clumps were consistent with those in observations, an important check of the violent-disk instability hypothesis of clump formation. They did not study the properties of the final resulting bulge, but did say that if the clumps can survive accretion onto the centre of the galaxy, they are expected to accrete gas from the surrounding interstellar medium (similarly to @2014ApJ...780...57B), and will thus show gradients in their mean properties with respect to separation from the centre of their galaxy, such as those measured by @2011ApJ...739...45F that clumps closer to the centres of their disks are redder, older, and more massive. Further, they found in their simulations was that a full 91% of galaxies develop a bulge clump. These are massive, typically equivalent to 40% of the disk mass, with 20% of the star formation, and gas fractions of less than 1%.
@2017MNRAS.464..635M study 34 galaxies with more sophisticated prescriptions. Among their findings, they find that the inclusion of radiation pressure disrupts the smaller clumps, reducing their lifetimes to a few free-fall times, but that the more massive and dense clumps still nevertheless survive and migrate to the centre. The inclusion of radiation pressure reduces the number of long-lived clumps by 81%. Radiation pressure has little to no effect on the bulge clumps, with $\sim$83% of simulated galaxies hosting a bulge clump.
A CLUE AS TO THE ORIGINS OF THE BULGE FROM APOGEE
=================================================
A significant clue as to the origin of the metal-poor stars in the bulge has been identified by the APOGEE collaboration. Previously, the detailed chemical abundance trends for the bulge have only been interpreted in their mean, due to the large observational error. The mean \[X/Fe\] vs \[Fe/H\] can and has be compared to the thin disk, thick disk, and halo, but the scatter has not yet been of particular insight.
@2017MNRAS.465..501S found a population of nitrogen-rich stars in the bulge, predominantly at \[Fe/H\] $\leq -1.0$. These stars have enhanced nitrogen, aluminum, and depleted carbon, characteristic of the “second-generation" stars in globular clusters . Given that even surviving globular clusters must have been far more massive at birth [@2012ApJ...758...21C] to produce their second generation, this suggests that between 50% and 100% of bulge stars with \[Fe/H\] $\leq -1.0$ formed in disassociated globular clusters.
This is a clue to the origin of the bulge, but it is not clear which line of evidence it can be used to support. That is why it is left as a separate section.
DISCUSSION AND CONCLUSION {#sec:Conclusion}
=========================
The level of inputs new to the last decade, both observational and theoretical, that can inform and constrain bulge formation scenarios is truly spectacular. Global photometric maps are now available from the optical through to the mid-infrared, with substantial coverage in the variability domaine. Spectroscopic data sets are now available toward a large fraction of bulge stars, and toward the full metallicity range. Knowledge of what high-redshift galaxies look like, including plausible Milky Way precursors, is greater than it’s ever been. The breadth and depth of models if constantly increasing.
It would be tempting to say that the situation remains one of uncertainty between different scenarios, but that would be so limiting as to be inaccurate. In a competition between the buckling disk and the classical bulge, the buckling disk is winning. The peanut/X-shape, the long bar, the correlation between the mean and skewness of the velocity distribution functions, and so on are non-trivial predictions that are required of any theory of bulge formation and at this time require a buckling disk. The classical bulge may dominate for the 5% of stars with \[Fe/H\] $\lesssim -1.0$ and a minority of more metal-rich stars, but that is an upper limit on its contribution.
The clump-origin bulge scenario may prove to be a viable alternative. It cannot be ignored given the ubiquity of star-forming, gas-rich clumps in high-redshift galaxies. It is more likely than not that the Milky Way was, at one time, a clumpy galaxy. More simulations and comparisons of said simulations to observations are needed to ascertain whether or not this describes the bulge assembly history. One plausible hybrid scenario, is if the clumpy phase of the Milky Way led to the thick disk [@2014MNRAS.441..243I], with the thick disk predominantly responsible for bulge stars with $-1.0 \lesssim \rm{[Fe/H]} \lesssim -0.50$ .
One thing is certain, the Milky Way bulge is a sensitive probe of Galactic assembly history, and research of its properties will continue yielding insights thereof.
Acknowledgments {#acknowledgments .unnumbered}
===============
DMN was supported by the Allan C. and Dorothy H. Davis Fellowship. Chiaki Kobayashi, Rosemary Wyse, and Alex de la Vega are thanked for helpful discussions.
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[^1]: [email protected], [email protected]
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abstract: |
We consider the problem of finding (possibly non connected) discrete surfaces spanning a finite set of discrete boundary curves in the three-dimensional space and minimizing (globally) a discrete energy involving mean curvature. Although we consider a fairly general class of energies, our main focus is on the Willmore energy, i.e. the total squared mean curvature.
Most works in the literature have been devoted to the approximation of a surface evolving by the Willmore flow and, in particular, to the approximation of the so-called Willmore surfaces, i.e., the critical points of the Willmore energy. Our purpose is to address the delicate task of approximating [*global*]{} minimizers of the energy under boundary constraints.
The main contribution of this work is to translate the nonlinear boundary value problem into an integer linear program, using a natural formulation involving pairs of elementary triangles chosen in a pre-specified dictionary and allowing self-intersection.
The reason for such strategy is the well-known existence of algorithms that can compute [*global minimizers*]{} of a large class of linear optimization problems, however at a significant computational and memory cost. The case of integer linear programming is particularly delicate and usual strategies consist in relaxing the integral constraint $x\in\{0,1\}$ into $x\in[0,1]$ which is easier to handle. Our work focuses essentially on the connection between the integer linear program and its relaxation. We prove that:
- One cannot guarantee the total unimodularity of the constraint matrix, which is a sufficient condition for the global solution of the relaxed linear program to be always integral, and therefore to be a solution of the integer program as well;
- Furthermore, there are actually experimental evidences that, in some cases, solving the relaxed problem yields a fractional solution.
These facts prove that the problem cannot be tackled with classical linear programming solvers, but only with pure integer linear solvers. Nevertheless, due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design ad-hoc integer solvers that yield high-definition approximations to solutions of several boundary value problems involving mean curvature, in particular the Willmore boundary value problem.
author:
- Thomas Schoenemann
- Simon Masnou
- Daniel Cremers
title: On a linear programming approach to the discrete Willmore boundary value problem and generalizations
---
Introduction
============
The Willmore energy of an immersed compact oriented surface $f:\Sigma\to{\mathbb{R}}^N$ with boundary $\partial\Sigma$ is defined as $${\cal W}(f)=\int_\Sigma|H|^2dA+\int_{\partial\Sigma}\kappa\,ds$$ where $H$ is the mean curvature vector on $\Sigma$, $\kappa$ the geodesic curvature on $\partial\Sigma$, and $dA$, $ds$ the induced area and length metrics on $\Sigma$, $\partial\Sigma$. The Willmore energy of surfaces with or without boundary plays an important role in geometry, elastic membranes theory, strings theory, and image processing. Among the many concrete optimization problems where the Willmore functional appears, let us mention for instance the modeling of biological membranes, the design of glasses, and the smoothing of meshed surfaces in computer graphics. The Willmore energy is the subject of a long-standing research not only due to its relevance to some physical situations but also due to its fundamental property of being conformal invariant, which makes it an interesting substitute to the area functional in conformal geometry. Critical points of ${\cal
W}$ with respect to interior variations are called Willmore surfaces. They are solutions of the Euler-Lagrange equation $\delta{\cal W}=0$ whose expression is particularly simple when $N=3$: $\Delta H+2H(H^2-K)=0$, being $K$ the Gauss curvature. It is known since Blaschke and Thomsen [@pinkall-sterling] that stereographic projections of compact minimal surfaces in ${\mathbb{S}}^3\subset{\mathbb{R}}^4$ are always Willmore surfaces in ${\mathbb{R}}^3$. However, Pinkall exhibited in [@Pinkall] an infinite series of compact embedded Willmore surfaces that are not stereographic projections of compact embedded minimal surfaces in ${\mathbb{S}}^3$. Yet Kusner conjectured [@Kusner] that stereographic projections of Lawson’s $g$-holed tori in ${\mathbb{S}}^3$ should be global minimizers of ${\cal W}$ among all genus $g$ surfaces. This conjecture is still open, except of course for the case $g=0$ where the round sphere is known to be the unique global minimizer.
The existence of smooth surfaces that minimize the Willmore energy spanning a given boundary and a conormal field has been proved by Sch[ä]{}tzle in [@Sch]. Following the notations in [@Sch], we consider a smooth embedded closed oriented curve $\Gamma\subset{\mathbb{R}}^N$ together with a smooth unit normal field $n_\Gamma\in N_\Gamma$ and we denote as $\pm\Gamma$ and $\pm n_\Gamma$ their possible orientations. We assume that there exist oriented extensions of $\pm \Gamma$, $\pm n_\Gamma$, that is, there are compact oriented surfaces $\Sigma_-,\,\Sigma_+\subset{\mathbb{R}}^N$ with boundary $\partial\Sigma_\pm=\pm\Gamma$ and conormal vector field $\operatorname{co}_{\Sigma_\pm}=\pm n_\Gamma$ on $\partial\Sigma_\pm$. We also assume that there exists a bounded open set $B\supset\Gamma$ such that the set $$\begin{gathered}
\{\Sigma_\pm\mbox{ oriented extensions of } (\Gamma, n_\Gamma),\; \Sigma_+\mbox{ connected },\\
\Sigma_+\cup\Sigma_-\subset B,\;{\cal W}(\Sigma_+\cup\Sigma_-)< 8\pi\mbox\}\end{gathered}$$ is not empty. The condition on energy ensures that $\Sigma_+\cup\Sigma_-$ is an embedding.
It follows from [@Sch], Corollary 1.2, that the Willmore boundary problem associated with $(\Gamma,n_\Gamma)$ in $B$ has a solution, i.e., there exists a compact, oriented, connected, smooth surface $\Sigma\subset B$ with $\partial\Sigma=\Gamma$, $\operatorname{co}_\Sigma=n_\Gamma$ on $\partial\Sigma$, and $$W(\Sigma)=\min\{W(\tilde\Sigma),\,\tilde\Sigma\mbox{ smooth},\;\tilde\Sigma\subset B,\;\partial\tilde\Sigma=\Gamma,\;\operatorname{co}_{\tilde\Sigma}=n_\Gamma\mbox{ on }\partial\tilde\Sigma\}$$
There have been many contributions to the numerical simulation of Willmore surfaces in space dimension $N=3$. Among them, Hsu, Kusner and Sullivan have tested experimentally in [@HsuKusnerSullivan92] the validity of Kusner’s conjecture: starting from a triangulated polyhedron in ${\mathbb{R}}^3$ that is close to a Lawson’s surface of genus $g$, they let it evolve by a discrete Willmore flow using Brakke’s Surface Evolver [@Brakke-92] and check that the solution obtained after convergence is ${\cal W}$-stable. Recent updates that Brakke brought to its program give now the possibility to test the flow with various discrete definitions of the mean curvature. Mayer and Simonett [@MayerSim] introduce a finite difference scheme to approximate axisymmetric solutions of the Willmore flow. Rusu [@Rusu] and Clarenz et al. [@Clarenz-et-al-04] use a finite elements approximation of the flow to compute the evolution of surfaces with or without boundary. In both works, position and mean curvature vector are taken as independent variables, which is also the case of the contribution by Verdera et al. [@Verdera03], where a triangulated surface with a hole in it is restored using the following approach: by the coarea formula, the Willmore energy (actually a generalization to other curvature exponents) is replaced with the energy of an implicit and smooth representation of the surface, and the mean curvature term is replaced by the divergence of an unknown field that aims to represent the normal field. Droske and Rumpf [@DroskeRumpf04] propose a finite element approach to the Willmore flow but replace the standard flow equation by its level set formulation. The contribution of Dziuk [@dziuk] is twofold: it provides a finite element approximation to the Willmore flow with or without boundary conditions that can handle as well embedded or immersed surfaces (turning the surface problem into a quasi-planar problem), and a consistency result showing the convergence of both the discrete surface and the discrete Willmore energy to the continuous surface and its energy when the approximated surface has enough regularity. Bobenko and Schr[ö]{}der [@BobenkoSchroeder05] use a difference strategy: they introduce a discrete notion of mean curvature for triangulated surfaces computed from the circles circumscribed to each triangle that shares with the continuous definition a few properties, in particular the invariance with respect to the full M[ö]{}bius group in ${\mathbb{R}}^3$. This discrete definition is vertex-based and a discrete flow can be derived. Based also on several axiomatic constraints but using a finite elements framework, Wardetzky et al. [@Wardetzky-et-al-07] introduce an edge-based discrete Willmore energy for triangulated surfaces. Olischläger and Rumpf [@Olischlager:2009] introduce a two step time discretization of the Willmore flow that extends to the Willmore case, at least formally, the discrete time approximation of the mean curvature motion due to Almgren, Taylor, and Wang [@Almgren-Taylor-Wang-1993], and Luckhaus and Sturzenhecker [@Luckhaus-Styrzenhecker-1995]. The strategy consists in using the mean curvature flow to compute an approximation of the mean curvature and plug it in a time discrete approximation of the Willmore flow. Grzibovskis and Heintz [@Heintz03], and Esedoglu et al. [@EsedogluRuuthTsai-06] discuss how 4th order flows can be approximated by iterative convolution with suitable kernels and thresholding.
While all the previous approaches yield approximations of critical points of the Willmore energy, our motivation in this paper is to approximate global minimizers of the energy. This is an obviously nontrivial task due to the high nonlinearity and nonconvexity of the energy. Yet, for the simpler area functional, Sullivan [@Sullivan-94] has shown with a calibration argument that the task of finding minimal surfaces can be turned into a linear problem. Even more, when a discrete solution is seeked among surfaces that are union of faces in a cubic grid partition of ${\mathbb{R}}^3$, he proved that the minimization of the linear program is equivalent to solving a minimum-cost circulation network flow problem, for which efficient codes have been developed by Boykov and Kolmogorov [@Boykov-Kolmogorov-01] after Ford and Fulkerson [@Ford-Fulkerson-62]. Sullivan [@Sullivan-94] did not provide experiments in his paper but this was done recently by Grady [@Grady-09], with applications to the segmentation of medical images.
The linear formulation that we propose here is based on two key ideas: the concept of surface continuation constraints that has been pioneered by Sullivan [@Sullivan-94] and Grady [@Grady-09], and the representation of a triangular surface using pairs of triangles. With this representation and a suitable definition of discrete mean curvature, we are able to turn into a linear formulation the task of minimizing discrete representations of any functional of the form $$W_\varphi(\Sigma)=\int_\Sigma \varphi(x,n,H)dA$$ among discrete immersed surfaces with boundary constraints: $$\partial\Sigma=\Gamma,\quad\operatorname{co}_{\tilde\Sigma}=n_\Gamma\mbox{ on }\partial\Sigma.$$ In the expression of $W_\varphi(\Sigma)$, $x$ denotes the space variable, $n$ the normal vector field on $\Sigma$ and $H$ the mean curvature vector. The linear problem we obtain involves integer-valued unknowns and does not seem to admit any simple graph-based equivalent. We will therefore discuss whether classical strategies for linear optimization can be used.
The paper is organized as follows: in section \[sec:1\] we discuss both the chosen representation of surfaces and the definition of discrete mean curvature. In section \[sec:2\] we present a first possible approach yielding a quadratic energy. We present in section \[sec:3\] our linear formulation and discuss whether it can be tackled by classical linear optimization techniques.
Discrete framework {#sec:1}
==================
Triangular meshes from a set of pre-defined triangles {#sec:mesh}
-----------------------------------------------------
The equivalence shown by Sullivan between finding minimal surfaces and solving a flow problem holds true for discrete surfaces defined as a connected set of cell faces in a cellular complex discrete representation of the space. We will consider here polyhedral surfaces defined as union of triangles with vertices in (a finite subset of) the cubic lattice $\epsilon{\mathbb{Z}}^3$ where $\epsilon=\frac 1 n$ is the resolution scale. Not all possible triangles are allowed but only those respecting a specified limit on the maximal edge length. We assume that each triangle, as well as each triangle edge, is represented twice, once for each orientation. We let ${\cal I}$ denote the collection of oriented triangles, $N=|{\cal I}|$ its cardinality, and $M$ the number of oriented triangle edges. The constrained boundary is given as a contiguous oriented set of triangle edges. The orientation of the boundary constrains the spanning surfaces since we will allow only spanning triangles whose orientation is compatible.
In this framework, one can represent a triangular mesh as a binary indicator vector $x = \{0,1\}^N$ where $1$ means that the respective triangle is present in the mesh, $0$ that it is not. Obviously, not all binary indicator vectors can be associated with a triangular surface since the corresponding triangles may not be contiguous. However, as discussed by Grady [@Grady-09] and, in a slightly different setting, by Sullivan [@Sullivan-thesis; @Sullivan-94], it is possible to write in a linear form the constraint that only binary vectors that correspond to surfaces spanning the given boundary are considered. We will see that using the same approach here turns the initial boundary value problem into a quadratic program. Another formulation will be necessary to get a linear problem.
Admissible indicator vectors: a first attempt
---------------------------------------------
To define the set of admissible indicator vectors, we first consider a relationship between oriented triangles and oriented edges which is called *incidence*: a triangle is positive incident to an edge if the edge is one of its borders and the two agree in orientation. It is negative incident if the edge is one of its borders, but in the opposite orientation. Otherwise it is not incident to the edge. For example, the triangle in Figure \[fig:incidence\] is positive incident to the edge $e_1$, negative incident to $e_2$ and $e_3$ and not incident to $e_4$.
Being defined as above the set of $N$ oriented triangles and their $M$ oriented edges, we introduce the matrix $B=(b_{ij})_{\begin{subarray}{l}
i\in\{1,\cdots,N\}\\
j\in\{1,\cdots,M\}\end{subarray}}$ whose element $b_{ij}$ gives account of the incidence between triangle $i$ and edge $j$. More precisely
$$b_{ij} =
\begin{cases}
1 & \mbox{if edge $i$ is an edge of triangle $j$ with same orientation}\\
-1 & \mbox{if edge $i$ is an edge of triangle $j$ with opposite orientation}\\
0 & \mbox{otherwise}
\end{cases}$$ The knowledge of which edges are present in the set of prescribed boundary segments is expressed as a vector $r \in \{-1,0,1\}^{M}$ with $$r_j =
\begin{cases}
1 & \mbox{if the oriented boundary contains the edge $j$}
\\&\mbox{\qquad with agreeing orientation}\\
-1 & \mbox{if the oriented boundary contains the edge $-j$}\\
&\mbox{\qquad with opposing orientation}\\
0 & \mbox{otherwise}
\end{cases}$$
With these notations set up we can now describe the equation system defining that a vector $x \in \{0,1\}^N$ encodes an oriented triangular mesh with the pre-specified oriented boundary. This system has one equation for each edge. If the edge is not contained in the given boundary, this equation expresses that, among all triangles indicated by $x$ that contain the edge, there are as many triangles with same orientation as the edge as triangles with opposite orientation. If the edge is contained in the boundary with coherent orientation, there must be one more positive incident triangle than negative incident. If it is contained with opposite orientation, there is one less positive than negative incident. Altogether the constraint for edge $j$ can be expressed as the linear equation $$\sum\limits_{i} b_{ij}\, x_i = r_j$$ and the entire system as $$B\, x = r.$$ So far, we did not incorporate the conormal constraint. Actually not all conormal constraints are possible, exactly like not all discrete curves can be spanned in our framework but only union of edges of dictionary triangles, i.e. the collection of triangles defined in the previous section that determine the possible surfaces. For the conormal constraint, only the conormal vectors that are tangent to dictionary triangles sharing an edge with the boundary curve are allowed. Then the conormal constraint can be easily plugged into our formulation by simply imposing the corresponding triangles to be part of the surface, see Figure \[fig:tribound\], and by defining accordingly a new boundary indicator vector $\tilde r$.
Denoting as ${\cal J}$ the collection of those additional triangles, the complete constraint reads $$\label{eq:const}
\left\{\begin{array}{l}
B\, x = \tilde r\\
x_j=1,\;j\in {\cal J}
\end{array}\right.$$ We discuss in the next section how discrete mean curvature can be evaluated in this framework.
Discrete mean curvature on triangular meshes
--------------------------------------------
The various definitions of discrete mean curvature that have been proposed in the literature obviously depend on the chosen discrete representations of surfaces. Presenting and discussing all possible definitions is out of the scope of the present paper. The important thing to know is that there is no fully consistent definition: the pointwise convergence of mean curvature cannot be guaranteed in general but only in specific situations [@Hildebrandt05; @Morvan-2008]. Among the many possible definitions, we will use the edge-based one proposed by Polthier [@Polthier-Hab] for it suits with our framework. Recalling that, in the smooth case but also for generalized surfaces like varifolds [@Simon-83], the first variation of the area can be written in terms of the mean curvature, the definition due to Polthier of the mean curvature vector at an interior edge $e$ of a simplicial surface reads $$H(e)=|e|\cos\frac{\theta_e}2N_e\label{eq:meancurv}$$ where $|e|$ is the edge-length, $\theta_e$ is the dihedral angle between the two triangles adjacent to $e$, and $N_e$ is the angle bisecting unit normal vector, i.e., the unit vector collinear to the half sum of the two unit vectors normal to the adjacent triangles (see figure \[fig:local-triangle\]). Remark that this formula is a discrete counterpart of the continuous $H=\kappa_1+\kappa_2$ depending on the principal curvatures, which is used in many papers for simplicity as definition of mean curvature. When the correct continuous definition $H=\frac 1 2(\kappa_1+\kappa_2)$ is used, the formulas above and hereafter should be adapted.
The justification of formula by Polthier [@Polthier-Hab; @Polthier-Clay] is as follows: it is exactly the gradient at any point $m\in e$ of the area of the two triangles $T_1$ and $T_2$ adjacent to $e$, and this gradient does not depend on the exact position of $m$. Indeed, one can subdivide $T_1$, $T_2$ in four triangles $T'_i$, $i\in\{1,\cdots,4\}$ having $m\in e$ as a vertex and such that $T_1=T'_1\cup T'_2$ and $T_2=T'_3\cup T'_4$. The area of each triangle is half the product of the opposite edge’s length and the height. Therefore, if $e_i$ is the positively oriented edge opposite to $m$ in the triangle $T'_i$ and $J_1$, $J_2$ the rotations in the planes of $T_1$, $T_2$ by $\frac\pi 2$, the area gradients of $T'_i$, $i\in\{1,\cdots, 4\}$ at $m$ are $\frac 1 2 J_1e_1$, $\frac 1 2 J_1e_2$, $\frac 1 2 J_2e_3$, $\frac 1 2 J_2e_4$. The sum is the total area gradient of $T_1\cup T_2$ at $m$ and equals $\frac 1 2 (J_1 e+J_2 e)$, which coincides with the formula above.
As discussed by Wardetsky et al. using the Galerkin theory of approximation, this discrete mean curvature is an integrated quantity: it scales as $\lambda$ when each space dimension is rescaled by $\lambda$. A pointwise discrete mean curvature rescaling as $\frac 1{\lambda}$ is given by (see [@Wardetzky-et-al-07]) $$H^{{\rm pw}}(e)=\frac{3|e|}{A_e}\cos\frac{\theta_e}2N_e,$$ where $A_e$ denotes the total area of the two triangles adjacent to $e$. The factor $3$ comes from the fact that, when the mean curvatures are summed up over all edges, the area of each triangle is counted three times, once for each edge. Then a discrete counterpart of the energy $\ds\int_\Sigma \varphi(H)\, dA$ is given by $$\sum_{{\rm edges}\; e}\frac{A_{e}}{3} \varphi(\frac{3|e|}{A_{e}}\cos\frac{\theta_{e}}2N_{e}).\label{eq:globenerg}$$
In particular, the edge-based total squared mean curvature is $$\label{eq:willm}
\sum_{{\rm edges}\; e}\frac{3|e|^2}{A_{e}}(\cos\frac{\theta_e}2)^2.$$
A quadratic program for the minimization of the discrete Willmore energy {#sec:2}
========================================================================
Ultimately we are aiming at casting the optimization problem in a form that can be handled by standard linear optimization software. Having in mind the framework described above where a discrete surface spanning the prescribed discrete boundary is given as a collection of oriented triangles satisfying equation and chosen among a pre-specified collection of triangles, a somewhat natural direction at first glance seems to be solving a *quadratic program*. Like in section \[sec:mesh\], let us indeed denote as $(x_i)$ the collection of binary variables associated to the “dictionary” of triangles $(T_i)$ and define
- $e_{ij}$ the common edge to two adjacent triangles $T_i$ and $T_j$;
- $\theta_{ij}$ the corresponding dihedral angle;
- $N_{ij}$ the angle bisecting unit normal;
- $A_{ij}$ the total area of both triangles.
Then a continuous energy of the form $\ds\int_\Sigma \varphi(x,n,H)dA$ can be discretized as $$\label{eq:formgen}
\sum_{i,j} q_{ij}\, x_i\,
x_j$$ with $\ds\quad q_{ij}=\left\{\begin{array}{ll}
\ds\frac 1 2\frac{A_{ij}}{3} \varphi(e_{ij},N_{ij},\frac{3|e_{ij}|}{A_{ij}}\cos\frac{\theta_{ij}}2N_{ij})&\mbox{if $i\not=j$ are adjacent}\\[1mm]
\tilde\varphi(T_i,N_i)&\mbox{if $i=j$}\\[1mm]
0&\mbox{otherwise}\end{array}\right.$
where $\tilde\varphi$ allows to incorporate dependences on each triangle $T_i$’s position and unit normal $N_i$. In particular, the discrete Willmore energy is $$\sum_{i,j}q^w_{ij}x_i\,
x_j\label{eq:formwill}$$ with $$q^w_{ij}=\left\{\begin{array}{ll}
\ds\frac{3|e_{ij}|^2}{2A_{ij}}(\cos\frac{\theta_{ij}}2)^2&\mbox{if $i\not=j$ are adjacent}\\[1mm]
0&\mbox{otherwise}\end{array}\right.$$
Assuming that the maps $\varphi$ and $\tilde\varphi$ are positive-valued, both energy matrices $Q=(q_{ij})$ and $Q^w=(q^w_{ij})$ are symmetric matrices in ${{{\mathbb{R}}}^+}^{N \times
N}$, and the minimization of either or with boundary constraints turns to be the following quadratic program with linear and integrality constraints: $$\begin{aligned}
& \min\limits_x & \langle Q\, x,x\rangle \\
& \mbox{such that} & B\, x = r\\
&& x_i = 1\ \, \forall i \in {\mathcal{J}}\\
& & x \in \{0,1\}^N \quad \quad .\end{aligned}$$ We know of no solution to solve this problem efficiently due to the integrality constraint. What is worse, even the relaxed problem where one optimizes over $x \in [0,1]^N$ is very hard to solve: terms of the form $x_i x_j$ with $i\neq j$ are indefinite, so (unless $Q$ has a dominant diagonal) the objective function is a non-convex one.
Moreover, a solution to the relaxed problem would not be of practical use: already for the 2D-problem of optimizing curvature energies over curves in the plane, the respective quadratic program favors fractional solutions. The relaxation would therefore not be useful for solving the integer program. However, in this case Amini et al. [@Amini-et-al-90] showed that one can solve a linear program instead. This inspired us for the major contribution of this work: to cast the problem as an integer linear program.
An integer linear programming approach {#sec:3}
======================================
Augmented indicator vectors
---------------------------
The key idea of the proposed integer linear program is to consider additional indicator vectors. Aside from the indicator variables $x_i$ for basic triangles, one now also considers entries $x_{ij}$ corresponding to *pairs* of adjacent triangles. Such a pair is called *quadrangle* in the following. We will denote $\Hat x$ the augmented vector $(x_1,\cdots,x_N,\cdots, x_{ij},\cdots)$ where $i\not=j$ run over all indices of adjacent triangles. The cost function can be easily written in a linear form with this augmented vector, i.e. it reads $$\sum w_k\Hat x_k$$ with (see the notations of the previous section) $$w_k=\left\{\begin{array}{ll}
q_{ii}&\mbox{if }\Hat x_k=x_i\\
q_{ij}&\mbox{if }\Hat x_k=x_{ij}\end{array}\right.$$
The major problem to overcome is how to set up a system of constraints that guarantees consistency of the augmented vector: the indicator variable $x_{ij}$ for the pair of triangles $i$ and $j$ should be $1$ if and only if both the variables $x_i$ and $x_j$ are $1$. Otherwise it should be $0$. In addition, one again wants to optimize only over indicator vectors that correspond to a triangular mesh.
To encode this in a linear constraint system, a couple of changes are necessary. First of all, we will now have a constraint for each pair of triangle and adjacent edge. Secondly, edges are no longer oriented. Still, the set of pre-specified indices ${\mathcal{J}}$ implies that the orientation of the border is fixed - we still require that for each edge of the boundary an adjacent (oriented) triangle is fixed to constrain the conormal information.
To encode the constraint system we introduce a modified notion of incidence. We are no longer interested in incidence of triangles and edges. Instead we now consider the incidence of both triangles and quadrangles to pairs of triangles and (adjacent) edges.
For convenience, we define that triangles are positive incident to a pair of edge and triangle, whereas all quadrangles are negative incident.
We propose an incidence matrix where lines correspond to pairs (triangle, edge) and columns to either triangles or quadrangles. The entries of this incidence matrix are either the incidence of a pair (triangle, edge) with a triangle, defined as $$d((\mbox{triangle }k,\mbox{edge } e),\mbox{triangle } i) = \begin{cases}
1 & \mbox{if } i=k,\ e \mbox{ is an edge of triangle } i\\
0 & \mbox{otherwise}
\end{cases},$$ or the incidence of a pair (triangle, edge) with a quadrangle, defined as $$d((\mbox{triangle }k,\mbox{edge } e),\mbox{quadrangle }ij) =
\begin{cases}
-1 & \mbox{if } i\!=\!k\mbox{ or } j\!=\!k \mbox{ and }
i, j \mbox{ share } e
\\
\ 0 & \mbox{otherwise}
\end{cases}.$$ The columns of this incidence matrix are of two types: either with only 0’s and exactly three $1$ (a column corresponding to a triangle $T$, whose three edges are found at lines $(T,e_1)$, $(T,e_2)$, $(T,e_3)$), or with only 0’s and exactly two $(-1)$’s (a column corresponding to a quadrangle $(T_1,T_2)$ that matches with lines $(T_1,e_{12})$ and $(T_2,e_{12})$).
Again, both the conormal constraints and the boundary edges can be imposed by imposing additional triangles indexed by a collection ${\mathcal{J}}$ of indices. The general constraint has the form $$\sum_i d((x_k,e),x_i)\, +\, \sum_{i,j} d((x_k,e),x_{ij}) = r'_{(k,e)},$$ where the right-hand side depends whether the edge $e$ is shared by two triangles of the surface (and even several quadrangles in case of self-intersection), or belongs to the new boundary indicated by the additional triangles. If $e$ is an inner edge, then the sum must be zero due to our definition of $d$, otherwise there is an adjacent triangle, but no adjacent quadrangle, so the right-hand side should be $1$: $$r'_{(k,e)} = \begin{cases} 1 & \mbox{if } k \in {\mathcal{J}}, e
\mbox{ is part of the modified boundary}\\ 0 & \mbox{otherwise}
\end{cases}$$ To sum up, we get the following integer linear program: $$\begin{aligned}
& \min\limits_{\Hat x} & \langle w,\Hat x\rangle \label{eq:ilp}\\
&\mbox{such that} & D\,\Hat x = r' \nonumber \\
&& \Hat x_j = 1\quad \forall j \in {\mathcal{J}}\nonumber \\
&& \Hat x_i \in \{0,1\}\quad \forall i \in \{1,\ldots, \Hat N\} \nonumber \end{aligned}$$ where $\Hat N$ is the total number of entries in $\Hat x$, namely all triangles plus all pairs of adjacent triangles. It is worth noticing that such formulation allows triangle surfaces with self-intersection.
On the linear programming relaxation
------------------------------------
Solving integer linear programs is an NP-complete problem, see e.g. [@Schrijver-book chapter 18.1]. This implies that, to the noticeable exception of a few particular problems [@Schrijver-book], no efficient solutions are known. As a consequence one often resorts to solving the corresponding linear programming (LP) relaxation, i.e. one drops the integrality constraints. In our case this means to solve the problem: $$\begin{aligned}
& \min\limits_{\Hat x} & \langle w, \Hat x\rangle \label{eq:lpf}\\
&\mbox{such that} & D\, \Hat x = r' \nonumber \\
&& \Hat x_j = 1\quad \forall j \in {\mathcal{J}}\nonumber \\
&& 0 \le \Hat x_i \le 1\quad \forall i \in \{1,\ldots, \Hat N\} \nonumber \end{aligned}$$ or, equivalently, by suitably augmenting $D$ and $r'$ in order to incorporate the second constraint $\Hat x_j = 1$, $\forall j \in {\mathcal{J}}$: $$\begin{aligned}
& \min\limits_{\Hat x} & \langle w, \Hat x\rangle \label{eq:lp}\\
&\mbox{such that} & \Hat D \Hat x = \Hat r \nonumber \\
&& 0 \le \Hat x_i \le 1\quad \forall i \in \{1,\ldots, \Hat N\} \nonumber \end{aligned}$$ There are various algorithms for solving this problem, the most classical being the simplex algorithm and several interior point algorithms. Let us now discuss the conditions under which these relaxed solutions are also solutions of the original integer linear program. Recalling the basics of LP-relaxation [@Schrijver-book], the set of admissible solutions $$P=\{\Hat x\in {\mathbb{R}}^{\Hat N},\;\Hat D \Hat x= \Hat r,\; 0\leq x\leq 1\}$$ is a polyhedron, i.e. a finite intersection of half-spaces in ${\mathbb{R}}^{\Hat N}$. A classical result states that minimizing solutions for the linear objective functions can be seeked among the extremal points of $P$ only, i.e. its vertices. Denoting $P_e$ the integral envelope of $P$, that is the convex envelope of $P\cap{\mathbb{Z}}^{\Hat N}$, another classical result states that $P$ has integral vertices only (i.e. vertices with integral coordinates) if and only if $P=P_e$
Since $P=\{\Hat x\in {\mathbb{R}}^{\Hat N},\,\Hat D \Hat x= \Hat r, 0\leq \Hat x\leq 1\}$, according to Theorem 19.3 in [@Schrijver-book], a [*sufficient*]{} condition for having $P=P_e$ is the property of $B$ being totally unimodular, i.e. any square submatrix has determinant either $0$, $-1$ or $1$. Under this condition, any extremal point of $P$ that is a solution of $$\min_{\Hat D \Hat x= \Hat r,\,\Hat x_i\in[0,1]}\langle w,\Hat x\rangle$$ has integral coordinates therefore is a solution of the original integer linear program $$\min_{\Hat D \Hat x= \Hat r,\,\Hat x_i\in\{0,1\}}\langle w,\Hat x\rangle.$$ Theorem 19.3 in [@Schrijver-book] mentions an interesting characterization of total unimodularity due to Paul Camion [@camion-65]: a matrix is totally unimodular if, and only if, the sum of the entries of every Eulerian square submatrix (i.e. with even rows and columns) is divisible by four.
Unfortunately, we can prove that, as soon as the triangle space is rich enough, the incidence matrix $\Hat D$ does not satisfy Camion’s criterion, therefore is not totally unimodular, and neither are the matrices for richer triangles spaces. As a consequence, there are choices of the triangle space for which the polyhedron $P=\{\Hat x\in {\mathbb{R}}^{\Hat N},\,\Hat D \Hat x= \Hat r, 0\leq \Hat x\leq 1\}$ may have not only integral vertices, or more precisely one cannot guarantee this property thanks to total unimodularity. This is summarized in the following theorem.
The incidence matrix associated with any triangle space where each triangle has a large enough number of adjacent neighbors is not totally unimodular.
We show in Figure \[fig:counterex\] a configuration and, in Table \[tab\], an associated square submatrix of the incidence matrix. The sum of entries over each line and the sum over each column are even, though the total sum of the matrix entries is not divisible by four. By a result of Camion [@camion-65], the incidence matrix is not totally unimodular which yields the conclusion according to [@Schrijver-book]\[Thm 19.3\]. Clearly, any triangle space for which this configuration can occur is also associated to an incidence matrix that is not totally unimodular.
It is worth noticing that the previous theorem does not imply that the extremal points of the polyhedron $P$ are necessarily not all integral. It only states that this cannot be guaranteed as usual by the criterion of total unimodularity. We will discuss in the next section what additional informations about integrality can be obtained from a few experiments that we have done using classical solvers for addressing the relaxed linear problem.
$T_1$ $T_5$ $T_9$ $T_{1,2}$ $T_{2,3}$ $T_{3,4}$ $T_{4,5}$ $T_{2,5}$ $T_{5,6}$ $T_{1,6}$ $T_{1,7}$ $T_{7,8}$ $T_{2,8}$ $T_{1,9}$ $T_{9,10}$ $T_{10,11}$ $T_{1,11}$ $T_{1,12}$ $T_{12,13}$ $T_{9,13}$ $T_{9,5}$ $T_{5,14}$ $T_{14,15}$ $T_{9,15}$ $T_{9,16}$ $T_{16,17}$ $T_{5,17}$ $\sum$
---------------- ------- ------- ------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ------------ ------------- ------------ ------------ ------------- ------------ ----------- ------------ ------------- ------------ ------------ ------------- ------------ --------
$(T_1,e_1)$ 1 -1 -1 -1 -2
$(T_2,e_1)$ -1 -1 -1 -1 -4
$(T_3,e_1)$ -1 -1 -2
$(T_4,e_1)$ -1 -1 -2
$(T_5,e_1)$ 1 -1 -1 -1 -2
$(T_6,e_1)$ -1 -1 -2
$(T_7,e_1)$ -1 -1 -2
$(T_8,e_1)$ -1 -1 -2
$(T_1,e_2)$ 1 -1 -1 -1 -2
$(T_9,e_2)$ 1 -1 -1 -1 -2
$(T_{10},e_2)$ -1 -1 -2
$(T_{11},e_2)$ -1 -1 -2
$(T_{12},e_2)$ -1 -1 -2
$(T_{13},e_2)$ -1 -1 -2
$(T_9,e_3)$ 1 -1 -1 -1 -2
$(T_5,e_3)$ 1 -1 -1 -1 -2
$(T_{14},e_3)$ -1 -1 -2
$(T_{15},e_3)$ -1 -1 -2
$(T_{16},e_3)$ -1 -1 -2
$(T_{17},e_3)$ -1 -1 -2
$\sum$ 2 2 2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -42
: A square incidence matrix associated with the configuration in Figure \[fig:counterex\]. It is Eulerian, i.e. the sum along each line and the sum along each column are even, but the total sum is not divisible by four. According to Camion [@camion-65], the matrix is not totally unimodular.[]{data-label="tab"}
Testing the relaxed linear problem
----------------------------------
We have tested the relaxed formulation on a few examples at low-resolution using the dual simplex method implemented in the CLP solver. The main reason for using low-resolution is that the number of triangles becomes significantly important as the resolution increases, and both the computational cost and the memory requirements tend to become large. Another reason for working at low-resolution is that there is no need to go high before finding a case of non-integrality. Indeed, consider the examples in figure \[fig:example1\]: integral solutions are obtained when the resolution is very low (i.e. when there is no risk to have configurations like in figure \[fig:counterex\]). In the last configuration, however, the optimal solution of the relaxed problem has fractional entries. This confirms that our initial problem cannot be addressed though the classical techniques of relaxation, and with usual LP solvers.
-- --
-- --
On integer linear programming
-----------------------------
Our results above indicate that, necessarily, integer linear solvers [@Schrijver-book; @achterberg-07] should be used. These commonly start with solving the linear programming relaxations, then derive further valid inequalities (called *cuts*) and/or apply a branch-and-bound scheme. Due to the small number of fractional values that we have observed in our experiments, it is quite likely that the derivation of a few cuts only would give integral solutions. However, we did not test this so far because of the running times of this approach: in cases where we get fractional solutions the dual simplex method often needs as long as two weeks and up to $12$ GB memory! From experience with other linear programming problems we consider it likely that the interior point methods implemented in commercial solvers will be much faster here (we expect less than a day). At the same time, we expect the memory consumption to be considerably much higher, so the method would most probably be unusable in practice.
We strongly believe that a specific integer linear solver should be developed rather than using general implementations. It is well known that, for a few problems like the knapsack problem [@Schrijver-book]\[chapter 24.6\], their specific structure gives rise to ad-hoc efficient approaches. Recalling that our incidence matrix is very sparse and well structured (the nonzero entries of each column are either exactly two $(-1)$, or exactly three $1$) we strongly believe that an efficient integer solver can be developed and our approach can be amenable to higher-resolution results in the near future.
Conclusion
==========
We have shown that the minimization under boundary constraints of mean curvature based energies over surfaces, and in particular the Willmore energy, can be cast as an integer linear program. Unfortunately, this integer program is not equivalent to its relaxation so the classical LP algorithms offer no warranty that the integer optimal solution will be found. This implies that pure integer linear algorithms must be used, which are in general much more involved. We believe however that the particular structure of the problem paves the way to a dedicated algorithm that would provide high-resolution [*global*]{} minimizers of the Willmore boundary problem and generalizations. This is the purpose of future research.
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---
abstract: 'This paper compares the auto-repressed gene to a simple one (a gene without auto-regulation) in terms of response time and output noise under the assumption of fixed metabolic cost. The analysis shows that, in the case of non-vanishing leak expression rate, the negative feedback reduces both the switching on and switching off times of a gene. The noise of the auto-repressed gene will be lower than the one of the simple gene only for low leak expression rates. Summing up, for low, but non-vanishing leak expression rates, the auto-repressed gene is both faster and less noisier compared to the simple one.'
address: |
$^1$School of Computing, University of Kent, CT2 7NF, Canterbury, UK\
$^{2}$ Present address: Cambridge Systems Biology Centre and Department of Genetics, University of Cambridge, Tennis Court Road, Cambridge CB2 1QR, UK\
$^*$ Corresponding author: [email protected]
author:
- 'Nicolae Radu Zabet$^{1,2,*}$'
bibliography:
- 'nar\_paper\_jtb\_rev2.bib'
title: Negative Feedback and Physical Limits of Genes
---
gene regulatory networks ,auto-repression ,output noise ,response time ,metabolic cost ,trade-off
Introduction
============
Genes which maintain a functional relationship between the concentration of the regulatory input protein(s) and the concentration of the output protein can be thought of as being capable of performing computations [@weiss_2003; @buchler_2003; @yokobayashi_2002; @mayo_2006; @fernando_2008]. For instance, often cells need to respond to the presence or the absence of various chemical factors. In this case, one can say that the cell performs some sort of “computations” on the input chemical factors and, based on the result of these computations, the cell will produce a specific chemical or physical response (output). A classic example is the lac operon in *E.coli* where the proteins associated with the lactose metabolism are produced only when the glucose is absent and the lactose is present [@setty_2003]. This is often approximated by an AND NOT gate, which suggests that the operon performs logical computations.
These “computations” performed by genes are characterised by several properties (such as accuracy, speed and energy cost) which are significantly influenced by the specificity of the environment (the cell). For instance, in the case of low number of molecules, inherent fluctuations in reaction rates, caused by thermal noise, induce stochastic fluctuations (noise) in the copy number of molecules [@lei_2008]. Usually, in living cells, there are few copies of mRNA molecules, and one or two copies per gene [@arkin_1998]. Consequently, the gene expression process is affected by *noise* [@kaern_2005]. In the context of genes as computational units, stochastic fluctuations can hide useful signals in noise and, thus, the accuracy of the response is reduced. In addition to accuracy, computations are also characterised by the speed at which they are performed [@bennett_1982]. The response of a gene to a change of input is not instantaneous, but rather affected by a time delay. This time delay is often called the *response time* of the gene [@alon_book_2007] and is connected to the speed at which the genes “compute”, in the sense that higher response times translate in slower computations, while lower response times in faster computations. In the case of genes, the speed at which a gene responds to changes in the transcription factors abundance is of high importance. In particular, a cell that is able to respond faster to changes in the environment can have certain advantages over slower cells. For example, a cell that is able to uptake food faster can consume more nutrients compared to a slower cell and, consequently, have an energy advantage over it.
Ideally, one would want to increase both speed and accuracy as much as possible, but this is often limited by the available energy supply [@bennett_1973; @bennett_1982; @lloyd_2000]. Each cellular process (protein production, protein decay and maintenance processes) has a *metabolic cost* attached to it, which is, usually, measured in number of ATP molecules [@akashi_2002]. The notion of cost used in this paper is not the exact quantitative measure of the actual metabolic cost, but rather a number which describes how the actual metabolic cost scales when the parameters of the genes are changed.
With few exceptions, these three properties (speed, accuracy and cost) were investigated previously in a stand-alone fashion. These exceptions include studies which analysed only the speed and accuracy and disregarded the cost in several molecular systems, such as: DNA based logic gates [@stojanovic_2003b], protein-protein interaction networks [@wang_2010] and gene regulatory networks [@rosenfeld_2005; @isaacs_2005; @hooshangi_2005; @shahrezaei_2008]. A few other studies examined all three properties (speed, accuracy and cost) in various gene regulatory networks, like: auto-repressed genes [@stekel_2008], toggle switches [@mehta_2008] or gene networks that used frequency encoded signals [@tan_2007b]. Nevertheless, these studies which integrate all three properties (speed, accuracy and cost) addressed different scenarios and used a different measure of cost compared to the one used in this contribution (for a discussion on the measure of cost see below). For a comprehensive review on computational properties of molecular systems see [@zabet_2010_thesis].
In addition to the aforementioned studies, [@zabet_2009] showed that, in the case of a simple gene (a gene without auto-regulation), the speed, accuracy and cost properties are interconnected. In particular, they found that under fixed metabolic cost there is a speed-accuracy trade-off, which is controlled by the decay rate, i.e., high decay rates lead to faster and less accurate responses while lower decay rates, to slower and more accurate ones.
One of the central results of this previous work by [@zabet_2009] stated that the speed-accuracy trade-off is optimal for systems with zero leak rates, i.e., there are no solutions that have better speed-accuracy characteristics. The vanishing leak expression rate represents a theoretical performance limit of a gene under fixed cost, but it is difficult to achieve in real systems and would require high metabolic cost [@zabet_2009]. The current paper will investigate whether the performance of a gene can be improved beyond this optimal configuration (of zero leak rate), i.e., if a gene can display faster response times and less noise at the output without increasing the metabolic cost.
One candidate mechanism to enhance the performance of a gene is the negative feedback, which is a network motif in bacterial cells [@savageau_1974; @thieffry_1998; @shen_orr_2002; @alon_book_2007], in the sense that it is a sub-network which is encountered with high occurrence, e.g. in *E.coli* $40\%$ of the genes are auto-repressed [@austin_2006]. The fact that it is encountered with high occurrence suggests that auto-repressed genes have certain advantages compared to simple ones. This paper aims to investigate whether negative feedback can enhance both response time and output noise while keeping the metabolic cost fixed (equal to the one of the gene without auto-repression).
The auto-repressed gene is a well studied system, which received great attention from the community over the past decade [@alon_2007]. [@rosenfeld_2002] showed that negative feedback reduces only the response time of switching on (when the gene goes from a low expression to a high one) . In the current setting (genes as computational units), the switching direction is not important and, thus, the mechanism should be capable of reducing the response times of both switching on and switching off (when the gene goes from a high expression to a low one). One of the assumptions of the aforementioned study of [@rosenfeld_2002] is that genes do not display leaky expression, which is obviously not true for all genes. Actually, most of the genes will display non-zero leak expression rates, because the metabolic cost associated with removing the leak rate would be very high [@zabet_2009]. Thus, this paper aims to investigate whether, under the assumption of leaky expression, the negative feedback can reduce the response time of both switching on and off.
Furthermore, experimental evidence suggested that a negatively auto-regulated gene displays lower noise compared with the gene without any type of auto-regulation [@becskei_2000]. Analytical results confirmed that the noise of the auto-repressed gene is lower than the one of the simple gene [@thattai_2001; @paulsson_2004]. However, these two studies [@thattai_2001; @paulsson_2004] did not consider fixed metabolic cost. Other studies derived the equation of noise analytically under the assumption that the simple gene and the auto-repressed one display equal average number of molecules at steady state [@paulsson_2000b; @stekel_2008; @zhang_2009b]. Their results confirmed that the auto-repressed gene reduces the output noise. Nevertheless, they assumed that if the two systems have an equal average number of molecules at steady state, they also have an equal metabolic cost.
A better measure for the metabolic cost is the production rate of a gene [@zabet_2009; @chu_2011]. This is justified by the fact that a measure of metabolic cost should describe the energy consumption per time unit. For example, consider the case of two proteins ($X1$ and $X2$) that have the same average number of molecules at steady state, but the first one ($X1$) is produced and decayed faster compared with the second one ($X2$). Then, more molecules of the first protein ($X1$) will be produced and decayed compared with the second one ($X2$) and, consequently, the metabolic cost associated with the first protein ($X1$) will be higher compared with the one of the second protein ($X2$). Hence, the production rate will describe better the scaling properties of the metabolic cost compared to the average number of molecules at steady state.
This paper aims to compare systems that have equal metabolic cost. In the case of fixed decay rate, imposing the production rates (the measure of metabolic cost) of two systems to be equal leads to the output steady state abundances of the systems to be equal as well. This means that, when production rate is kept fixed, it is ensured that a previously used measure of cost (steady state abundance) is also kept fixed. Nevertheless, the current analysis is not limited to cases where the decay rate is kept fixed (see for example Figure \[fig:speedAccuracyInstant\]) and, in those cases, attempting to keep production rates fixed will lead to variable steady state abundances. Due to the reasons mentioned above, the production rate will be used as the only indicator of metabolic cost when the decay rate is not fixed. Note however that, for fixed decay rate, the two measures are equivalent and, consequently, the results obtained under the assumption of fixed production rates (metabolic cost) are also valid for equal steady state abundances.
The results from this contribution show that negative feedback reduces the response time in the case of leaky expression for both switching on and switching off. In addition, for low leak rates, negative feedback reduces the noise, while for high leak rates it increases the noise. Both these results were obtained under the assumption of fixed metabolic cost. Furthermore, the analysis identified a subset in the parameter space (low but non-vanishing leak rates) where the negatively auto-regulated gene outperforms the simple one in both speed and accuracy, thus, setting a new theoretical performance limit for genes.
Model
=====
Simple Gene
-----------
The model of the simple gene consists of a single gene $G_y$, which has an output $y$ and is regulated by a single input species $x$; see Figure \[fig:model-simple\]. This system is described by the following set of chemical reactions $$\emptyset \xrightarrow{\beta f(x)}y, \quad y \xrightarrow{\mu}\emptyset\label{eq:modelSimpleReact}$$ Here, $\beta$ is the maximal expression rate of the gene, $f(x)$ the regulation function, $x$ the concentration of the regulatory input, and $\mu$ is the degradation rate of the product of the gene; see Table \[tab:nomenclature\].
Gene regulation functions are often approximated by Hill functions [@ackers_1982; @bintu_2005_model; @chu_2009], which are sigmoid functions characterized by two parameters, namely the threshold ($K$) and the Hill coefficient ($l$). The latter determines the steepness of the function, whereas $K$ represents the required input concentration to achieve half activation of the gene. This paper considers two families of regulation functions, namely the Hill activation function ($f\equiv\phi$) and the Hill repression function ($f\equiv\bar{\phi}$). $$\label{eq:modelHill}
{\phi}(x) = \frac{x^l}{K^l + x^l}\qquad\textrm{and}\qquad \bar{\phi}(x) = \frac{K^l}{K^l + x^l}$$
[|>m[1.7cm]{}| m[11.8cm]{} |]{} $x$ & abundance of the regulatory input protein $y$ & output abundance of the simple gene $z$ & output abundance of the auto-repressed gene $\beta$ & maximum expression rate for both genes $\mu$ & decay rate of the output proteins $f(x)$ & gene activation function $K$ & threshold of the gene activation function $l$ & Hill coefficient of the gene activation function $g(z)$ & gene auto-repression function $K_n$ & threshold of the gene auto-repression function $x_L$ & abundance level of the input that leads to low gene expression $x_H$ & abundance level of the input that leads to high gene expression $H$ & high abundance level of the output for both the simple and auto-repressed genes $L$ & low abundance level of the output for the simple gene $L_n$ & low abundance level of the output for the auto-repressed gene $m$ & relative leak rate, $m=L/H$ or $m=L_n/H$ $x_0,\ y_0,\ z_0$ & initial steady state abundance of the specific protein $x^*,\ y^*,\ z^* $ & new steady state abundance of the specific protein $y_\theta,\ z_\theta$ & fraction $\theta$ of the new steady state abundance $\tilde{t},\ \tilde{\beta},\ \tilde{T}$ & specific parameters when the time is measured in $1/\mu$ (change of variable) $T_y^{LH},\ T_z^{LH}$ & switching on time $\ T_y^{HL},\ T_z^{HL}$ & switching off time $T_d^{LH},\ T_d^{HL}$ & difference between the switching time of the simple gene and the one of the auto-repressed gene $\sigma^2_{y},\ \sigma^2_{z}$ & variance of the steady state abundance $\eta_{y},\ \eta_{z}$ & normalized variance (noise) of the steady state abundance $\eta^{\textrm{in}}_{y},\ \eta^{\textrm{in}}_{z}$ & intrinsic component of noise $\eta^{\textrm{up}}_{y},\ \eta^{\textrm{up}}_{z}$ & upstream (from input) component of noise $\eta_{c}^{\textrm{in}},\ \eta_{c}^{\textrm{up}}$ & ratio between the noise of the auto-repressed gene and the one of the simple gene
Usually, the equilibrium state (steady-state) and the kinetic behaviour (time evolution) of a system are defined using the differential equation associated to the system (the ODE) [@b_murray_2002]. In the case of the simple gene, the ODE has the following form $$\label{eq:modelSimpleDiff}
\frac{d y(t)}{dt} = \beta f(x) - \mu y(t)$$ This system is completely determined by the input $x$, which is assumed to change instantaneously between two abundance levels, $x=x_L$ and $x=x_H$ [@zabet_2009; @chu_2011]. These two levels of abundances of $x$ will lead to two concentration levels of the output $y$, namely a high concentration state ($y=H$ when $x=x_H$) and a low one ($y=L$ when $x=x_L$); see Figure \[fig:modelSimple-ss\]. For convenience, the low state (also called the leak rate) will be denoted as the fraction $m$ from the high state $$L = m H, \quad m\in[0,1] \label{eq:modelSimpleSSL}$$ where by restricting $m$ to the interval $[0,1]$ it is ensured that $L\leq H$.
Note that [@rosenfeld_2002] assumed that the concentration of the output evolved from an initial value of zero ($y_0=L=0$) to a new non-zero one ($y^{*}=H>0$), which led to the following limitations: [($i$) ]{}they only consider the switching on (or raise) time and [($ii$) ]{}they assume that there is no leaky gene expression. The current paper assumes a more general scenario, in which both the initial ($y_0$) and the new steady state ($y^{*}$) have positive values ($y_0 \geq 0$ and $y^{*} \geq 0$) and the system can evolve in both directions, i.e., in the case of switching on, the gene evolves from a low output state ($y_0=L$) to high one ($y^*=H$), while, in the case of switching off, the gene evolves from a high output state ($y_0=H$) to a low one ($y^*=L$); see Figure \[fig:modelSimple-dynamic\].
Auto-Repressed Gene
-------------------
A negatively auto-regulated gene is a gene which synthesises a protein that represses its own synthesis; see Figure \[fig:model-nar\]. The auto-repressed gene is denoted by $G_z$ and this gene synthesises protein $z$. The differential equation that describes this system becomes $$\frac{dz}{dt}=\beta f(x)g(z) -\mu z \label{eq:modelNarDiff}$$ where $g(z)$ is a Hill repression function, $$g(z) = \gamma\frac{K_n^{h}}{K_n^{h}+z^{h}}.$$ The auto-repression function, $g(z)$, adds three new parameters, namely: $\gamma$ the maximum contribution of the auto-repression function ($\beta\gamma$ is the maximum production rate), $K_n$ the auto-repression threshold and $h$ the auto-repression Hill coefficient. In order to keep the mathematics trackable, this function needs to be brought to a simpler form. First, similarly as in the paper of [@rosenfeld_2002], it is assumed that the auto-repression Hill coefficient is $1$. This yields $$g(z) = \gamma\frac{K_n}{K_n+z}$$ Note that bacterial cells can implement auto-repression when the output protein binds to the promoter area of the gene and stops the RNAp molecules to transcribe the gene. In this case and assuming that only monomers auto-regulate the gene, the Hill coefficient of the auto-repression function is $1$ [@chu_2009].
Furthermore, to make the comparison “fair”, the two systems (the simple gene and the negatively auto-regulated one) will display equal metabolic costs. As previously [@zabet_2009; @chu_2011], it is assumed that the *metabolic cost* of a gene is approximated by the maximum production rate, which represents the pessimistic scenario. This maximal production rate is achieved when $x=x_H$, which results in the abundance of the output proteins reaching the maximum value of $y=z=H$. Note that the exact metabolic cost will depend on various additional factors, such as the length of the encoded protein, the average time the gene is active and so on. Nevertheless, these details are irrelevant in the current scenario, where only systems with single genes are considered.
Since the metabolic cost ($\zeta$) is measured as the maximum production rate (when $x=x_H$), imposing that the two systems have equal metabolic costs yields $$\zeta = \beta \cdot f(x_H)\cdot g(H) = \beta \cdot f(x_H)$$ which leads to $$g(z) = \frac{K_n + H}{K_n + z} \label{eq:modelNarG}$$ where $\gamma$ was set to $\gamma = (K_n+H)/K_n$ in order to ensure that the two systems have equal metabolic cost. Hence, the new parameter $\gamma$ was removed from the model.
The steady state abundance levels of the output $y$ which corresponds to the input concentration levels of $x=x_L$ and $x=x_H$ are computed as follows. First, for $x=x_H$, it results from equation that $g(z)=1$ and, thus, the differential equation becomes $$\frac{dz}{dt}=\beta f(x) -\mu z$$ This yields $$H_n = \frac{\beta}{\mu}f(x_H)=H\label{eq:modelNarSS-H}$$ , which led to the same steady state abundance as in the case of the simple gene.
Furthermore, feeding equation into the differential equation and solving at steady state for $x=x_L$ leads to only one positive solution: $$L_n = \frac{1}{2}\left(- K_n + \sqrt{ K_n^2 + 4\frac{\beta}{\mu}f(x_L)\left(K_n + H\right)}\right) \label{eq:modelNarSS-L}$$ The low output state of the negatively auto-regulated gene is denoted by $L_n$, as opposed to $L$, the low output state of the simple gene. Since both systems display the same abundance in the high output state, this high output state will be denoted by $H$.
From the form of equations and one can see that the high output state remains constant while changing the strength of the auto-repression, $K_n$, but the low state is increased if the auto-repression is strengthened ($K_n \searrow\ \Rightarrow\ L_n \nearrow$). It can be shown that the low state of the output varies between the following limits $$\begin{aligned}
\lim_{K_n\rightarrow \infty} L_n & = & \frac{\beta}{\mu}f(x_L) = L\nonumber\\
\lim_{K_n\rightarrow 0} L_n & = & \frac{\beta}{\mu}\sqrt{f(x_L)f(x_H)} = \sqrt{L \cdot H}=\sqrt{m} \cdot H \end{aligned}$$
Depending on the relationship between $K_n$ and $H$, the auto-regulation function can be rewritten in a simpler form. In particular, there are two extreme cases: [($i$) ]{}$K_n\gg H$ and [($ii$) ]{}$K_n\ll H$. In the limit of weak auto-repression ($K_n\gg H$), the system will be similar to the simple one. This case does not pose any interest, since this paper aims to compare the simple gene to a system that displays different behaviour.
Furthermore, for $K_n \ll H$, the auto-repression becomes strong and the auto-regulation function is approximated by $$g(z) = \frac{K_n + H}{K_n + z} \approx \frac{H}{z} \label{eq:modelNarGStrong}$$ Note that the current definition of strong auto-repression is slightly different from the one of [@stekel_2008], which considers strong auto-repression in absolute values, i.e., smaller than $< 10^{-4}\ \mu M$. The current definition is rather concerned with the relative repression strength ($K_n$) compared with the high abundance level of the output species ($H$) and aims to determine a parameter space where the auto-regulation function can be written in a simpler form .
In this case, $K_n \ll H$, the low output state will be approximated $$L_n \approx \sqrt{m} H \label{eq:modelNarSSL}$$
The auto-repression function specified in equation does not contain any of the three additional parameters ($\gamma$, $K_n$ and $h$) and, thus, it represents a simpler mathematical formulation of the system. This form of the auto-repression function will be used, in the next section, to compare the two systems (simple gene and auto-repressed one) in both speed and accuracy.
Response Time
=============
Generally, one would want to process information as fast as possible, but genes are very slow, in the sense that the time required to turn on/off a gene (the switching time) is of the order of tens of minutes, even for an instant input change. Thus, it is essential to investigate what constrains the speed at which genes function and whether there are any methods to increase this speed.
A common measure of the processing speed of genes is the *response time*; that is the time required for the output of a gene to reach a new steady state once the input was changed. Note that the regulatory input, $x$, evolves from the initial abundance state $x_0$ to the new one $x^*$ (switching between $x_L$ and $x_H$). The input $x$ can either change instantaneously or non-instantaneously and, below, both cases will be considered.
Instantaneous Change of Input
-----------------------------
The time of the simple gene to reach a fraction $\theta$ of the steady state when the input is changed instantaneously was computed previously by [@zabet_2009] as $$T_y=\frac{1}{\mu}\ln\frac{y^*-y_0}{y^*-y_\theta}$$ where $y_\theta = y_0+(y^*-y_0)\theta$.
In the case of the auto-repressed gene, knowing that the auto-repression function is the one from equation , the solution to the differential equation yields $$z(t) = \sqrt{\frac{\beta}{\mu}f(x)H + \left[(z_0)^2-\frac{\beta}{\mu}f(x)H\right]e^{-2\mu t}}$$ The time to reach a fraction $\theta$ of the steady state, $z_\theta = z_0+(z^*-z_0)\theta$, becomes $$T_z=\frac{1}{2\mu}\ln\frac{\frac{\beta}{\mu}f(x)H-\mu^2 (z_0)^2}{\frac{\beta}{\mu}f(x)H-\mu^2 (z_\theta)^2} = \frac{1}{2\mu}\ln\frac{(z^*)^2-\mu^2 (z_0)^2}{(z^*)^2-\mu^2 (z_\theta)^2}$$ where the steady state solution for the output protein $z$ was used, $z^* = \sqrt{\beta f(x) H/\mu}$.
To make the comparison easier, a change of variable will be performed to the differential equations attached to the two systems, and . The time $t$ will be measured in units of $1/\mu$ and this new time is denoted by $\tilde{t} = t/\mu$. The two differential equations become $$\begin{aligned}
\frac{dy}{d \tilde{t}} &=& \tilde{\beta} f(x) -y\\
\frac{dz}{d \tilde{t}} &=& \tilde{\beta} f(x)\frac{H}{z} -z\end{aligned}$$ Recomputing the time to reach a fraction $\theta$ of the steady state yields $$\begin{aligned}
\tilde{T}_y &=& \ln\frac{y_0 - y^*}{y_\theta-y^*} \label{eq:timeReduced-simple}\\
\tilde{T}_z &=& \frac{1}{2}\ln\frac{(z^*)^2- (z_0)^2}{(z^*)^2-(z_\theta)^2}\label{eq:timeReduced-nar}\end{aligned}$$
In the case of the simple gene, the switching on and switching off are equal. However, for the auto-repressed gene, these two response times are usually different. The response time of the gene is measured as the maximum between switching on time and switching off time [@zabet_2010b].
### Switching On
First, the case when the two systems are turned on will be considered, i.e., ($y_0 = L$, $y^* = H$) and ($z_0 = L_n$, $z^* = H$). The fraction $\theta$ of the steady state can be written as: $$\begin{aligned}
y_\theta&=&L+(H-L)\theta = H\left[m+(1-m)\theta\right]\nonumber\\
z_\theta&=&L_n+(H-L_n)\theta = H\left[\sqrt{m}+(1-\sqrt{m})\theta\right]\end{aligned}$$ where the following equations where used $L=m\cdot H$ and $L_n=\sqrt{m}\cdot H$
Using equations and the switching on times yield $$\begin{aligned}
\tilde{T}_y^{LH} &=& \ln{\frac{1}{1-\theta}}\nonumber\\
\tilde{T}_z^{LH} &=& \frac{1}{2}\ln{\frac{1-m}{1-\left[\sqrt{m}+(1-\sqrt{m})\theta\right]^2}}\end{aligned}$$
The two times to reach a fraction of the steady state are compared by computing the difference between the two $$\begin{aligned}
\tilde{T}_d^{LH} &=& \tilde{T}_y^{LH} - \tilde{T}_z^{LH} \nonumber\\
&=& \frac{1}{2}\ln{\left[\frac{1}{(1-\theta)^2} \frac{1-\left[\sqrt{m}+(1-\sqrt{m})\theta\right]^2}{1-m}\right]}\nonumber\\
&=& \frac{1}{2}\ln{\frac{1-m-2\sqrt{m}\theta+2m\theta-\theta^2+2\sqrt{m}\theta^2-m\theta^2}{1-m-2\theta+2m\theta+\theta^2-m\theta^2}}\end{aligned}$$ This time difference is positive (and consequently negative auto-regulation speeds up the switching on) when the fraction in the logarithm is higher than or equal to $1$, which takes place when $$\theta(1-\theta)(1-\sqrt{m})\geq 0$$ This is true for any $\theta, m \in[0,1]$. Hence, negative auto-regulation always speeds up the switching on time compared with the simple gene. Figure \[fig:timeNarSimpleDifference-on\] confirms this result and shows that higher fractions $\theta$ of the steady states display better increase in speed that lower ones.
In the special case of no leak rate (the optimum configuration for noise), $L=0$ and $L_n=0$, the time gain reduces to $$\tilde{T}_d^{LH} = \frac{1}{2}\ln{\frac{1+\theta}{1-\theta}}$$ which is positive as long as $\theta >0$.
### Switching Off
When the gene is turned off, ($y_0=H$, $y^*=L$) and ($z_0=H$, $z^*=L_n$), the output states ($H$, $L$ and $L_n$) remain the same as the ones for switching on, but the fractions $\theta$ of the steady state become $$\begin{aligned}
y_\theta&=&H+(L-H)\theta = H\left[1-(1-m)\theta\right]\nonumber\\
z_\theta&=&H+(L_n-H)\theta = H\left[1-(1-\sqrt{m})\theta\right]\end{aligned}$$ From equations and one can compute the switching off time as $$\begin{aligned}
\tilde{T}_y^{HL} &=& \ln{\frac{1}{1-\theta}}\nonumber\\
\tilde{T}_z^{HL} &=& \frac{1}{2}\ln{\frac{1-m}{\left[1-(1-\sqrt{m})\theta\right]^2-m}}\end{aligned}$$ The difference in time between $\tilde{T}^{HL}$ and $\tilde{T}^{HL}$ yields $$\begin{aligned}
\tilde{T}_d^{HL} &=& \tilde{T}_y^{HL} - \tilde{T}_z^{HL} \nonumber\\
&=& \frac{1}{2}\ln{\left[\frac{1}{(1-\theta)^2} \frac{\left[1-(1-\sqrt{m})\theta\right]^2-m}{1-m}\right]}\nonumber\\
&=& \frac{1}{2}\ln{\frac{1-2\theta+2\sqrt{m}\theta+\theta^2-2\sqrt{m}\theta^2+m\theta^2-m}{1-m-2\theta+2m\theta+\theta^2-m\theta^2}}\end{aligned}$$ Analogously, as in the case of switching on, one can determine whether the time difference is positive by verifying if the fraction in the logarithm is higher than or equal to $1$, which reduces to $$\theta\sqrt{m}(1-\theta)(1-\sqrt{m})\geq 0$$ This means that $\tilde{T}_d^{HL}$ is always positive and the time to switch off an auto-repressed gene is at most equal to the time to switch off the simple gene. Figure \[fig:timeNarSimpleDifference-off\] confirms these results.
In the case of vanishing leak rates $m = L = L_n=0$, the time difference between the two systems becomes zero $$\tilde{T}_d^{HL} = \frac{1}{2}\ln{\frac{(1-\theta)^2}{(1-\theta)^2}} = 0$$ Thus, for vanishing leak rates, the auto-repressed gene turns on faster compared with the simple gene, but has an equal speed when turning off. Vanishing leak rates are optimal in terms of noise and require $f(x_L)$ to be zero. This is usually difficult to achieve. For repressor genes, the gene can be turn off completely if either the Hill coefficient or $x_L$ have high values, which comes at a high metabolic cost [@zabet_2009]. Even for activator genes, having a gene completely turned off can be very difficult to achieve, i.e., the regulator molecule would need to be totally absent and the affinity of RNAp for the non activated promoter should be zero. Putting all together, in the case of non-vanishing leak rates (sub-optimal in terms of noise), the negative auto-regulation speeds the switching in both directions (on and off).
Non-Instantaneous Change of Input
---------------------------------
The assumption that $x$ changes instantaneously between $x_L$ and $x_H$ will be relaxed now. In the case of non-instantaneous input change, the solution of the differential equations and can only be computed numerically. For very fast, but non-instantaneous input change, one expects the behaviour to be similar to the one predicted by the instantaneous input change. Figure \[fig:timeNarSimpleDifferenceNonInstant-fast\] confirms that the difference between the switching off time of the simple gene and the one of the auto-repressed gene is always positive.
Usually, it is assumed that the input and the output are affected by the same decay rate (dilution) and, thus, they change at a similar speed. The numerical analysis reveals that for most of the parameter space the relationship between switching times (the signs of $T_d^{LH}$ and $T_d^{HL}$) is conserved (see Figure \[fig:timeNarSimpleDifferenceNonInstant-slow\]). However, for no or very small leak rate ($m\leq 0.0002$ in Figure \[fig:timeNarSimpleDifferenceNonInstant-slow\]) the switching off time seems to be increased by negative auto-regulation. This suggests that negative auto-regulation is beneficial for speed but only for non-vanishing leak rates of the output gene.
Noise
=====
Gene expression is affected by noise [@spudich_1976; @arkin_1998; @elowitz_2002]. This noise is a consequence of the fact that genes have low copy numbers and that they are slowly expressed [@kaern_2005]. In the context of genes as computational units, this output noise is undesirable because it makes difficult the assessment of the output of the gene as either low or high.
At steady state one can compute the variance of the output species $y$ of the simple gene as [@paulsson_2004; @shibata_2005; @kampen_2007]: $$\sigma_{y}^2 = y+ \left[\beta f'(x)\tau\right]^2\frac{1}{1 + \tau/\tau_x}x$$ Usually, gene expression is modelled as a three steps process (regulation, transcription and translation), but here it is modelled as a one step process. Consequently, the noise produced by regulation and translation steps is ignored and the output noise stems mainly from transcription. Nevertheless, this assumption is often valid as shown both theoretically and experimentally [@even_2006; @newman_2006].
As proposed previously [@zabet_2009; @zabet_2010b; @chu_2011], the noise will be measured by computing the variance in the high state, $y=H$, normalized by the square of the signal strength (the difference between the high and the low state), $\eta_y = \sigma_y^2/(H-L)^2 = \sigma_y^2/[H^2(1-m)^2]$. $$\begin{aligned}
\eta_{y} &=& \overbrace{\frac{1}{H(1-m)^2}}^{\eta_y^{\textrm{in}}} + \overbrace{\left[\frac{f'(x_H)}{f(x_H)(1-m)}\right]^2\frac{1}{1 + \tau/\tau_x}x_H}^{\eta_y^{\textrm{up}}} \nonumber\\
& = & \frac{1}{(1-m)^2}\left[ \frac{1}{H}+ \left(\frac{f'(x_H)}{f(x_H)}\right)^2\frac{1}{1 + \tau/\tau_x}x_H \right] \label{eq:noiseSimple}\end{aligned}$$ The noise consists of two components: the intrinsic noise, $\eta^{\textrm{in}}$, (generated by the randomness of the birth/death process) and the upstream (or extrinsic) noise, $\eta^{\textrm{up}}$, (propagated from the upstream component) [@elowitz_2002; @swain_2002; @pedraza_2005].
In the case of the auto-repressed gene, the steady state variance of the species $z$ yields [@paulsson_2004; @shibata_2005; @kampen_2007] $$\sigma_{z}^2 = \frac{z}{1+\tau \beta f(x)H/z^2} + \left[\frac{\tau\beta f'(x)H/z}{\tau\beta f(x)H/z^2+1}\right]^2\frac{1}{1 + \frac{\tau}{\tau_x}\frac{1}{1+\tau \beta f(x)H/z^2}}x$$ Then, the noise of the auto-repressed gene (the variance in the high state normalised by the square of the signal strength) becomes $$\begin{aligned}
\eta_{z} &=& \overbrace{\frac{1}{H2(1-\sqrt{m})^2}}^{\eta_z^{\textrm{in}}} + \overbrace{\left[\frac{f'(x_H)}{f(x_H)\sqrt{2}(1-\sqrt{m})^2}\right]^2 \frac{1}{2+\tau/\tau_x}x_H}^{\eta_z^{\textrm{up}}} \nonumber\\
& = & \frac{1}{2(1-\sqrt{m})^2}\left[ \frac{1}{H} + \left(\frac{f'(x_H)}{f(x_H)}\right)^2 \frac{1}{2+\tau/\tau_x}x_H \right]\label{eq:noiseNar}\end{aligned}$$ where the following two steady state equations were used: $\beta f(x_H)\tau/H = 1$ and $\tau \beta/H=f(x_H)$.
The reliability of the analytical method for noise was validated previously by a series of stochastic simulations. The results confirmed that LNA works well for both the simple gene [@zabet_2009] and the auto-repressed one [@thattai_2001; @hayot_2004; @stekel_2008]. However, for extremely strong auto-repression values ($K_n< 10^{-4}\ \mu M$), the analytical method underestimates the simulation results [@stekel_2008]. In this context, it is worthwhile noting that [@zhang_2009b] showed that, although the LNA underestimates the actual value of the noise in the auto-repressed gene, it still captures the dependence of noise on the parameters of the system.
The noise of the two systems (the simple and the auto-repressed gene) is compared by analysing the ratio for each component of the noise (intrinsic and upstream). $$\eta_c^{\textrm{in}} = \frac{\eta_{z}^{\textrm{in}}}{\eta_y^{\textrm{in}}} = \frac{(1+\sqrt{m})^2}{2}\quad \textrm{and}\quad \eta_c^{\textrm{up}} = \frac{\eta_{z}^{\textrm{up}}}{\eta_y^{\textrm{up}}} = \frac{(1+\sqrt{m})^2}{2}\frac{1+\tau/\tau_x}{2+\tau/\tau_x}$$ Examining the first equation, one can notice that, for low leak rates ($m\leq 0.17$), the intrinsic component of the noise is not amplified by negative feedback. $$\eta_c^{\textrm{in}} = \frac{(1+\sqrt{m})^2}{2} \leq 1\ \Rightarrow\ m\leq (\sqrt{2}-1)^2 \approx 0.17$$
Furthermore, the fraction $(1+\tau_y/\tau_x)/(2+\tau_y/\tau_x)$ can never be higher than one. This yields $$\eta_c^{\textrm{up}} \leq \frac{(1+\sqrt{m})^2}{2} \Rightarrow m\lesssim 0.17$$ Overall, for low leak rates, the intrinsic and the upstream components of the noise are reduced.
![*The leak rate influences the noise levels of negative auto-regulation*. Increasing the leak rate reduces the accuracy. The following set of parameters was used: $\theta=0.9$, $\mu=1 min^{-1}$, $x_H=0.9\ \mu M$, $K_n=0.01\ \mu M$ and $V=8\cdot 10^{-16}\ l$. $\beta$ was selected so that the cost remains fixed to $\zeta=1.2 \ \mu M min^{-1}$ and $K$ so that the threshold resides at the midpoint between the low and the high state of the input, $K = 0.5\cdot(x_H+x_L)$. The noise generated by $x$ is assumed to be Poissonian. The gene is activated by a regulatory input $x$. []{data-label="fig:noiseComparison"}](graph_2genes_noise_0m1_activator.eps)
Figure \[fig:noiseComparison\] confirms that, for low levels of leak rate, negative feedback enhances accuracy. This in conjunction with the fact that negative feedback increases speed for high leak rates suggest that there is a trade-off between speed and accuracy, in the sense that for lower leak rates the auto-repressed gene is more accurate than the simple gene while for higher leak rates the auto-repressed gene becomes faster. Nevertheless, these results indicate that there is a range of values for the leak rate for which the auto-repressed gene is both faster and more accurate compared to the simple one.
A gene with negative feedback and low but non zero leak rate will display less noise ($m<0.1$ in Figure \[fig:noiseComparison\]) and shorter response time ($m>0$ in Figure \[fig:timeNarSimpleDifference\]) compared to the optimal configuration of the simple gene with an equivalent metabolic cost. This shows that negative feedback is able to improve the performance of a gene beyond the one of the zero leak rate configuration and without increasing the metabolic cost.
Speed-Accuracy Trade-offs
=========================
Recently, [@zabet_2009] showed that the processing speed and accuracy of genes are connected, in the sense that there is a trade-off between speed and accuracy, which is controlled by the decay rate. Figure \[fig:speedAccuracyInstant\] confirms the existence of this trade-off also in the case of the auto-repressed gene. Furthermore, the graph confirms that for low leak rates the auto-repressed gene displays a better trade-off curve compared to the simple one; see Figure \[fig:speedAccuracyInstant-05\]. Note that not only the metabolic cost is fixed along the curve, but also the two curves display equal costs. In the case of high leak rate, the auto-repressed gene is still faster than the simple one, but this time the noise in the former is higher than the one in the latter.
Figure \[fig:speedAccuracyInstant\] shows that for low (but non-zero) leak rate it is always better for the gene to display negative feedback, i.e., the auto-repressed gene is both faster and more accurate than the simple one. However, for high leak rates, the cell has to choose between being fast or more accurate, by selecting whether it has negative feedback or not. This result reveals a new speed-accuracy trade-off, which is not controlled by a parameter of the system (as previously identified), but rather by the architecture of the system.
For zero leak rate, both auto-repressed and simple genes display the lowest noise configurations and this is denoted by $m=m_\eta$. It is worthwhile noting that, by increasing the leak rate, the speed gain is increased while the accuracy gain is reduced; see Figure \[fig:speedAccuracyLeakRate\]. However, the increase in speed is achieved up to a certain maximum speed ($m_T$), above which the speed cannot be increased further by negative feedback. This indicates that there is no advantage in having leak rates higher than this optimal point in speed, while for any leak rate in the interval $m \in[m_T,m_{\eta}]$ there is a speed-accuracy trade-off.
Discussion
==========
Negative auto-regulation was suggested as an alternative approach to reduce the response time of a single gene. [@rosenfeld_2002] showed theoretically that negative auto-regulation can speed up *only* the turn on response time, i.e., the turn on time of a negatively auto-regulated gene is five times smaller than the one of a simple gene. In the context of genes as computational units, the direction of switching is not important in the sense that the system needs to turn both on and off as fast as possible. One of the assumptions of [@rosenfeld_2002] is that the auto-repressed gene does not have a leaky expression, which is not always biologically plausible. In this contribution, it is shown that, in the case of leaky gene expression, auto-repressed genes are always faster compared to the simple ones in both turning on and off; see Figures \[fig:timeNarSimpleDifference\] and \[fig:timeNarSimpleDifferenceNonInstant\].
Furthermore, several theoretical studies showed that negative auto-regulation reduces the output noise [@thattai_2001; @paulsson_2004; @shahrezaei_2008; @hornung_2008; @bruggeman_2009]. These results were obtained without comparing two systems with equal cost and, in fact, the negatively auto-regulated system displays a lower metabolic cost. Nevertheless, several studies [@paulsson_2000b; @stekel_2008; @zhang_2009b] compared the noise in the simple and auto-repressed gene under the assumption of fixed average number of molecules at steady state, which, in their settings, could be thought as the metabolic cost. A better measure of metabolic cost is the energy consumption per time unit and the scaling properties of this are well given by the protein production rate.
Recently, [@chu_2011] found that negative feedback can reduce output noise in a gene cascade under the assumption of fixed production rate. However, the study of [@chu_2011] did not identify what parameter controls whether the auto-repressed gene has lower or higher noise compared to the simple one. This current paper considers the case of a single gene (not a cascade) and compares the auto-repressed gene to a simple one by keeping the metabolic cost (maximum production rate) fixed. The results revealed that the leak expression rate plays a crucial role for the output noise, in the sense that, for low leak rates, the auto-repressed gene is less noisier than the simple one, while, for high leak rates, the auto-repressed gene becomes noisier; see Figure \[fig:noiseComparison\]. This result can be explained by the fact that increasing the leak expression rate reduces the signal strength and, consequently, it reduces the normalisation term of the noise, $(H-L)^2$. The reduction in the signal strength is more pronounced in the case of the auto-repressed gene than in the case of the simple gene, because the low state increases faster in the former compared to the latter. Thus, negative feedback increases the sensitivity of noise to the leak expression rate. Note, however, that these analytical results of noise cannot be extended automatically to very strongly auto-repressed genes ($K_n<10^{-4}\ \mu M$), where the Linear Noise Approximation seems to fail to represent accurately the noise of the system [@stekel_2008].
Previously, [@zabet_2009] showed that the speed-accuracy properties of a gene can be enhanced by reducing the leak rate, which comes at an undesirable increase in the metabolic cost. Here, it is proved that negative feedback can further enhance the speed-accuracy properties of genes without increasing the metabolic cost, by actually exploiting the non-zero leak rate of genes; see Figures \[fig:timeNarSimpleDifference\] and \[fig:noiseComparison\]. Hence, negative feedback is able to enhance the performance of a gene and this can explain why a high proportion of the bacterial genes display this mechanism. For example, in *E.Coli*, $40\%$ of the genes are auto-repressed [@austin_2006]. Nevertheless, one question that still needs to be addressed is why only $40\%$ of the genes are negatively auto-regulated and not a higher proportion. One possible answer to this question is that for high leak rates the negative feedback only enhances speed, while at the same time reduces accuracy; see Figure \[fig:speedAccuracyInstant-30\]. Thus, a system that displays higher leak rates has to choose between being faster or being more accurate, in the sense that the simple gene is the most accurate system, while the auto-repressed one is the fastest. It might be the case that, sometimes, it is more important for the system to be more accurate that faster, while other times to be faster than more accurate.
Finally, the results revealed that the leak rate controls a speed-accuracy trade-off in a negatively auto-regulated gene. Particularly, there are two specific leak rate values: one which optimises the system in terms of accuracy ($m_\eta=0$) and another which optimises the system in speed ($m_T > 0$); see Figure \[fig:speedAccuracyLeakRate\]. Increasing the leak rate above $m_T$ results in a system that is not faster than the optimal system in speed, but which has higher noise levels. This means that the leak rate determines two speed-accuracy trade-off curves, namely: [($i$) ]{}$m\in[0,m_T]$ where the system can be optimised in speed or accuracy and [($ii$) ]{}$m\in[m_T,1]$ where the system is always suboptimal in terms of noise and speed compared to the first case. Hence, the optimal parameter set for an auto-repressed gene consists of leak rates smaller than or equal to $m_T$, while for higher leak rates the auto-repressed gene becomes suboptimal.
To sum up, negative feedback further introduces additional speed-accuracy trade-offs for genes as information processors. These trade-offs manifest through either sets of parameters (such as leak rate) or the actual architecture of the system (as either with or without feedback). Various configuration of parameters and architectures that are encountered in today’s living organisms are results of long time evolution towards an optimal design.
Acknowledgements
================
The author would like to thank Dr Dominique F.Chu, Professor Andrew N.W. Hone, Dr Boris Adryan and the anonymus reviewers for useful comments which lead to improvements of the manuscript and Felicia Dana Zabet for proofreading the manuscript.
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---
abstract: |
We study the time evolution of the survival probability $P(t)$ in open one-dimensional quasiperiodic tight-binding samples of size $L$, at critical conditions. We show that it decays algebraically as $P(t)\sim t^{-\alpha}$ up to times $t^*\sim L^{\gamma}$, where $\alpha = 1-D_0^E$, $\gamma=1/D_0^E$ and $D_0^E$ is the fractal dimension of the spectrum of the closed system. We verified these results for the Harper model at the metal-insulator transition and for Fibonacci lattices. Our predictions should be observable in propagation experiments with electrons or classical waves in quasiperiodic superlattices or dielectric multilayers.\
PACS numbers:05.60Gg,03.65Nk,72.15Rn
author:
- |
A. Ossipov, M. Weiss, Tsampikos Kottos, and T. Geisel\
Max-Planck-Institut für Strömungsforschung und Fakultät Physik der Universität Göttingen,\
Bunsenstraße 10, D-37073 Göttingen, Germany
title: Quantum mechanical relaxation of open quasiperiodic systems
---
[2]{}
The decay properties of open quantum mechanical systems, have been attracting considerable attention for several decades. Their study was motivated by various areas of physics, ranging from nuclear [@nuclear], atomic [@atomic] and molecular [@molecular] physics, to mesoscopics [@mesoscopics] and classical wave scattering [@KS00]. In recent years, the interest in quantum mechanical decay was stirred by mesoscopic cavities and microwave billiards where immediate experimental realizations have become feasible [@GSST99]. At the same time various analytical techniques have been developed to study the problem in more detail. One possible formulation of the problem is to consider the survival probability $P(t)$ of a wave packet localized initially inside the open system. In particular it was found that this quantity exhibits slower than exponential decay (i.e. long-time tails) for disordered wires in the localized (in one-dimension) [@AKL87] and metallic (in higher dimensions) regimes [@MK95; @M95; @FE95; @SA97]. Moreover, for ballistic systems, Random Matrix Theory (RMT) predicts an algebraic decay $P(t) \sim 1/ t^{\beta M/2}$, where $M$ is the number of channels and $\beta=1(2)$ for preserved (broken) time reversal symmetry [@SS97; @D00].
The investigation of the survival probability has recently been extended to quantum systems with a mixed classical phase space [@CMS99; @HKW00] where it was found that $P(t)\sim1/t$. The same algebraic decay was found for systems with exponential localization. In both cases, this law is related to localization and tunneling effects and applies for intermediate asymptotic times [@CMS99].
The subject of the present paper is the survival probability in a new setting, namely a class of systems whose closed analogues have fractal spectra. The latter exhibit level clustering [@GKP95] in strong contrast to the level repulsion predicted by RMT for systems in the ballistic regime and to the Poisson statistics in the localized regime [@P65]. Representatives of this class are quasi-periodic systems with a metal-insulator transition like the Harper model [@GKP95; @AA80], Fibonacci chains [@GKP95; @SBGC84], or quantum systems with a chaotic classical limit like the kicked Harper model [@GKP91].
Here for the first time we present consequences of the fractal nature of the spectrum for the quantum time evolution of open systems. In particular, we ask how they are encoded in the survival probability $P(t)$, which is the simplest quantity measured in laboratory experiments. We show that $P(t)$ decays as $$\begin{aligned}
\label{powlaw}
P(t) \sim 1/t^{\alpha}\,;\,\,\,\;\;\alpha=1-D_0^E,\end{aligned}$$ where $D_0^E$ is the fractal (box-counting) dimension of the spectrum of the closed system. Moreover, we determine the time scale $t^*$ up to which this power-law decay can be observed. It scales as $$\label{maxt}
t^* \sim L^{\gamma}\,;\,\,\,\;\; \gamma=1/D_0^E,$$ where $L$ is the sample size. Beyond this time scale $P(t)$ decays exponentially. Our results (\[powlaw\]),(\[maxt\]) are confirmed numerically for two types of quasi-periodic tight-binding models and are supported by analytical arguments.
[\
[**FIG. 1.**]{} The survival probability $P(t)$ of the Harper model $(\lambda = 2)$, for three different sample lengths $L=250, 500, 4000$ exhibits an inverse power-law $P(t)\sim t^{-\alpha}$. A least squares fit yields $\alpha =0.55\pm
0.05$ in accordance with $D_0^E\simeq 0.5$ and Eq. (\[powlaw\]). ]{}
The mathematical model we consider is the time-dependent Schrödinger equation $$\label{eqmo}
i\,{\frac{d\psi_n(t)}{{dt}}}=\,V_n \psi_n(t)+\psi_{n+1}(t)+\psi_{n-1}(t) ,$$ where $\psi_n(t)$ is the probability amplitude for an electron to be at site $n$ of a one-dimensional (1D) sample of length $L$. The on-site potential $V_n$ is determined by quasi-periodic sequences. We assume absorbing boundary conditions at the ends of the sample [@note] and initial excitations in the form of a $\delta-$like packet launched at one of the boundaries, i.e. $\psi_n(t=0)=\delta_{n,1}$. Equation (\[eqmo\]) has been integrated numerically using a Cayley scheme [@WKG00] with integration time step $dt=0.1$. We attached 15 additional sites at the ends of the sample and erased all components of the wave packet on these sites after each time step $dt$. The decay of the norm of the wave packet obtained in this way was not affected by a further decrease of $dt$.
[\
[**FIG. 2.**]{} (a) Survival probabilities $P(t)$ of the Fibonacci model, for three different potential strengths $V_1=0.5,~V_2=0.75$ and $V_3=1.5$ showing inverse power-laws $P(t)\sim t^{-\alpha}$ (dashed lines). The sample size is $L=2000$ in all cases. (b) Power-law exponents $\alpha$ of the survival probability obtained numerically as a function of the potential strength $V$ for the Fibonacci model. The solid line is the theoretical prediction $\alpha = 1-D_0^E$. ]{}
We motivate and numerically verify our results using first the well known Harper model, a paradigm of quasi-periodic 1D systems with a metal-insulator transition [@GKP95; @AA80]. It is described by a tight-binding Hamiltonian with an on-site potential given by $$\label{harper}
V_n=\lambda \cos (2\pi\sigma n ).$$ This system effectively describes a particle in a two-dimensional periodic potential in a uniform magnetic field with $\sigma=a^2eB/hc$ being the number of flux quanta in a unit cell of area $a^2$. When $\sigma$ is an irrational number, the period of the effective potential $V_n$ is incommensurate with the lattice period. The states of the corresponding closed system are extended when $\lambda<2$, and the spectrum consists of bands (ballistic regime). For $\lambda>2$ the spectrum is point-like and all states are exponentially localized (localized regime). The most interesting case corresponds to $\lambda=2$ of the metal-insulator transition. At this point, the spectrum is a Cantor set with fractal dimension $D_0^E\leq 0.5$ [@frank] while self-similar fluctuations of the eigenstates exist on all scales [@GKP95; @AA80].
[\
[**FIG. 3.**]{} Resonance widths $\Gamma_k$ as a function of the overlapping elements $|c_k|^2$. The data are obtained by direct diagonalization of the effective Hamiltonian ${\cal H}_{eff}$ (see Eq. (\[effec\])). The dashed lines of slope $1$ are shown for comparison demonstrating a linear relation for small $\Gamma$. In all cases the sample size is $L=1597$, corresponding to an approximant of the golden mean $\sigma = \frac {987}{1597}$. (a) Harper model for $\lambda=2$; (b) Fibonacci model for $V=0.1$; (c) Fibonacci model for $V=0.5$; and (d) Fibonacci model for $V=1.5$. ]{}
We investigate the survival probability $P(t)$ for the Harper model at the critical point $\lambda=2$. In our calculations we assume $\sigma$ equal to the golden mean $\sigma_G= ({\sqrt 5}+1)/2$. For this case it is known that $D_0^E\approx 0.5$ [@frank]. The results for various sample lengths $L$ are shown in Fig. 1. In all cases the survival probability clearly displays an inverse power law $$\label{int1a}
P(t)\equiv \sum_{n=1}^L |\psi_n(t)|^2 \sim t^{-\alpha}.$$ The best fit to the numerical data yields $\alpha = 0.55\pm 0.05$ in accordance with Eq. (\[powlaw\]).
For a further test of the validity of Eq. (\[powlaw\]) we now turn to the Fibonacci model, where $D_0^E$ can be varied. Here the potential $V_n$ only takes on two values $\pm V$ ($V\neq 0$) that are arranged in a Fibonacci sequence [@SBGC84]. It was shown that the spectrum is a Cantor set with fractal dimension $D_0^E$ depending on $V$ [@SBGC84]. In Fig. 2(a) we report some of our numerical results for $P(t)$. Again we find a power-law decay $P(t)\sim
t^{-\alpha}$, where the exponent depends on the potential strength $V$. The exponents $\alpha$ extracted for various $V$ are compared with the corresponding fractal dimension $D_0^E$ in Fig. 2(b) and confirm the validity of Eq. (\[powlaw\]).
[\
[**FIG. 4.**]{} The scattering autocorrelation function Eq. (\[sscor\]) (plotted as $1-|C(\chi)|$) for some representative $V$ values of the Fibonacci model compared to the theoretical expectation (dashed line). In these calculations the sample has length $L=10946$ and is attached to one lead. The phase of the corresponding scattering matrix $S(E)=
e^{i\Phi(E)}$ was calculated with the help of an iteration relation developed in Ref. [@OKG00]. ]{}
We now want to give a general argument for the validity of Eq. (\[powlaw\]). The effect of an open edge for the system described by Eq. (\[eqmo\]) can be simulated by adding the imaginary shift $i$ to the first diagonal element of the Hamiltonian matrix [@FS97]. Denoting our quasi-periodic tight-binding Hamiltonian by $H_L$, this approach yields an effective Hamiltonian $$\label{effec}
{\cal H}_{eff} = H_L -i\vec{e}\bigotimes\vec{e},$$ where $\vec{e}=(1,0,\ldots ,0)^{~T}$ is an $L-$dimensional vector that describes at which site we impose the absorbing boundary condition. The eigenenergies of the effective Hamiltonian are complex ${\cal E}_k = E_k - i\Gamma_k/2$ leading to the decay of the survival probability $P(t)$. When the on-site potential fulfills $|V_n|>1$ the imaginary shift can be considered as a small perturbation of the Hamiltonian of the closed system. In this case according to the perturbation theory, $\Gamma_k\sim |\psi_1^k|^2$ holds, where $\psi_n^k$ is an eigenstate of the closed system with energy $E_k$. The survival probability is then given by $P(t)\simeq\sum_k |c_k|^2 e^{-\Gamma_k t}$, where $c_k$ are overlapping elements of the initial state with the eigenstates $\psi_n^k$. Choosing the initial state to be concentrated at site $n=1$, we have $|c_k|^2=|\psi_1^k|^2\sim\Gamma_k$. Using this and converting the sum into an integral we obtain:
$$\label{gamint}
P(t) \sim \int \Gamma\; {\cal P}(\Gamma)\; e^{-\Gamma t}d\Gamma,$$
where ${\cal P}(\Gamma)$ is the resonance width distribution. In Ref. [@SOKG00] it was shown that ${\cal P}(\Gamma) \sim \Gamma^{-(1+D_0^E)}$ for small $\Gamma$. Inserting this expression into Eq. (\[gamint\]) one finds the asymptotic power-law decay stated in Eq. (\[powlaw\]). In order to check the validity of the perturbative arguments we numerically calculate the resonance widths $\Gamma_k$ and the overlapping elements $c_k$ for the Harper and Fibonacci models. As can be seen from Fig. 3 the prediction $|c_k|^2\sim\Gamma_k$ of the perturbation theory holds even for small $V$.
[\
[**FIG. 5.**]{} Dependence of the break-time $t^*$, on the system size $L$ and the box-counting dimension $D_0^E$ . ($\circ$) refer to the Harper model at $\lambda=2$ and various $L$’s while ($\Diamond$) refer to the Fibonacci model with various $V$’s and $L=2000$. The dashed line has slope $1$ and corresponds to the theoretical expectation Eq. (\[maxt\]). ]{}
An immediate consequence of Eq. (\[powlaw\]) is that the scattering matrix autocorrelation function $C(\chi)\equiv <S(E)^{\dagger}S(E+\chi)>_E$, decays in the form of a power law. In particular, using the relation between the survival probability $P(t)$ and $C(\chi)$ [@D00], we obtain $$\label{sscor}
1-C(\chi)\sim \chi \int dt P(t) exp(-i t\chi)\sim \chi^{\alpha},\quad \chi\ll 1.$$ Equation (\[sscor\]) is in contrast to the Lorentzian form of $C(\chi)$ predicted by RMT for chaotic/ballistic systems [@BS88]. Comparison of $C(\chi)$ for various $V$-values of the Fibonacci model with the theoretical prediction Eq. (\[sscor\]) in Fig. 4 shows a nice agreement and provides an additional check for the validity of Eq. (\[powlaw\]).
For finite samples the power-law decay of $P(t)$ (Eq. (\[powlaw\])) holds up to a break time $t^*$, beyond which it turns into an exponential decay. The rate of the latter is determined by the smallest resonance width $\Gamma_{min}$ and thus $t^*\sim 1/\Gamma_{min}$. An estimation for $t^*$ can be derived as follows. Imposing the normalization condition for the resonance width distribution and assuming that the power law ${\cal P}(\Gamma) \sim \Gamma^{-(1+D_0^E)}$ is valid for $\Gamma \ge
\Gamma_{min}$, i.e $\int_{\Gamma_{min}}^{\infty}d\Gamma {\cal P}(\Gamma) =
L$, we obtain Eq. (\[maxt\]). This prediction is verified numerically in Fig. 5 where we defined $t^*$ as the time where $P(t)$ deviates by 5% from the power-law decay. We want to point out that the increase of $t^*$ for decreasing $D_0^E$ is consistent with the enlargement of the interval where $|c_k|^2\sim\Gamma_k$ holds and its shift towards smaller values (notice the change of the axes scales in Fig. 3(b)-(d)).
In summary we find that systems with a fractal spectrum show a power-law decay of the survival probability $P(t)\sim t^{-(1-D_0^E)}$. For the finite systems of size $L$, this decay can be observed up to a time scale $t^*\sim
L^{1/D_0^E}$ beyond which an exponential decay sets in. Our predictions should be observable in propagation experiments with electrons or classical waves in quasi-periodic superlattices or dielectric multilayers [@exper]. Moreover the connection between $\alpha$, $\gamma$ and $D_0^E$ provides a new possibility for experimental measurements of the fractal dimension $D_0^E$ by studying current-relaxation processes.
We thank L. Hufnagel, for useful discussions. (T.K) thanks U. Smilansky for initiating his interest in quantum scattering.
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abstract: 'We investigate the existence of envelope soliton solutions in collisionless quantum plasmas, using the quantum-corrected Zakharov equations in the kinetic case, which describes the interaction between high frequency Langmuir waves and low frequency plasma density variations. We show the role played by quantum effects in the nonlinearity/dispersion balance leading to the formation of soliton solutions of the quantum-corrected nonlinear Schrodinger (QNLS) equation.'
address: |
$^{1}$School of Physics, University of Sydney, New South Wales 2006, Australia\
$^{2}$Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan\
$^{3}$Metamaterials Laboratory, National Research University of Information Technology, Mechanics, and Optics, St Petersburg 199034, Russia
author:
- 'F. Sayed$^{1}$, S. V. Vladimirov$^{1,2,3}$, Yu. Tyshetskiy$^{1}$, and O. Ishihara$^{2}$'
title: Soliton solution of the Zakharov equations in quantum plasmas
---
Introduction
============
A soliton is a special type of solitary wave (a hump- or dip - shaped nonlinear wave) which preserves its shape and speed, even after collisions with other solitary wave. It arises because of the balance between the effects of nonlinearity and dispersion (when the effect of dissipation is negligible in comparison with those of the nonlinearity and dispersion). One-dimensional solitons into plasma physics were first considered in 1963 by Gurevich and Pitaevskii [@r1]. Modulational instabilities are responsible for the formation of Langmuir envelope solitons (cavitons) [@r2; @r3; @r4; @r5; @r6]. Electromagnetic envelope solitons were investigated for the first time by Hasegawa [@r7] and Karpman [@r8]. Currently, many types of solitons in plasmas are known, e. g. one-dimensional solitons and two or three-dimensional solitons [@r9]. They are invoked in various physical theories, especially for the construction of different versions of strong turbulence theory. In order for solitons to be applicable in these theories it is necessary that they be stable. Therefore, the problem of soliton stability is of particular importance [@r10].
Quantum plasmas are ubiquitous and appear in different physical settings from laser-matter interaction experiments (e.g., the compressed hydrogen in the fast ignition scenario of inertial fusion is in a quantum plasma state), to astrophysical and cosmological objects [@r11; @r12; @r13] (e.g., interstellar or molecular clouds, planetary rings, comets, interiors of white dwarf stars, etc.), nanostructures [@r14], and microelectronic devices [@r15]. At room temperature and standard metallic densities, the electron gas in an ordinary metal is a good example of a quantum plasma system. In such plasmas, quantum mechanical effects (e.g., tunneling) are important since the de Broglie wavelength of the charge carriers (e.g., electrons and holes or positrons) is comparable to the dimension of the system. Recently, the topic of quantum plasmas has attracted considerable attention [@r16; @r17; @r18; @r19; @r20; @r21; @r22; @r23] and it is desirable that a good understanding of the basic phenomena of quantum plasmas and be achieved.
In classical plasmas, nonlinear phenomena are often formulated in terms of completely integrable evolution equations of the Korteweg-de Vries (KdV) or nonlinear Schrodinger equation (NLS) type [@r24; @r25; @r26]. These completely integrable equations, as is well known, admit N-soliton solutions, derivable from the Inverse Scattering Transform (IST) method.
In plasma, large amplitude Langmuir waves can initiate a number of nonlinear effects including decay and modulational instabilities, which are described by the Zakharov equations [@r26; @r27]. The Zakharov equations in a quantum plasma (which we hereby call the quantum-corrected Zakharov equations) were recently obtained in the electrostatic approximation using the Wigner kinetic approach for the electron quantum distribution function [@r28]. It was shown in [@r28] that in quantum plasma, the classical Zakharov equations are modified by quantum correction terms proportional to $\hbar^2$ (note that in the formal classical limit $\hbar=0$, the quantum-corrected Zakharov equations obtained in [@r28] reduce to the classical Zakharov equations [@r6]).
Nonlinear waves in quantum plasmas are not yet sufficiently well understood. For example, the classical Zakharov equations in the adiabatic limit reduce to the nonlinear Schrodinger (NLS) equation which is completely integrable via the IST (inverse scattering transform) method, but no such technique seems to be easily constructable in the quantum case.
In terms of the variational formulation of quantum-modified NLS equation, Haas et al. [@r29] investigated the coefficients of soliton solution in quantum-corrected form (when $v_0=0$, where $v_0$ is a constant speed at which the soliton solution propagates). In the present work, we will focus on the soliton solution of the quantum-corrected Zakharov equations which propagates with constant speed $v_0$. The purpose of this study is to investigate the role played by quantum effects in the quantum-corrected Zakharov system; these effects modify the dispersion-nonlinearity equilibrium, which is the ultimate factor responsible for the existence of solitons.
In this paper, we investigate the previously obtained quantum-corrected Zakharov equations relevant to quantum plasmas, and obtain their quantum-corrected soliton solutions, comparing them with those in the case of the classical Zakharov model.
Soliton solution of the quantum Zakharov equations
==================================================
In the previous work [@r28], we have generalized the formalism of modulational interactions to nonrelativistic quantum plasmas, based on the Wigner kinetic description of collisionless quantum plasmas. In particular, we derived kinetically the effective cubic response of a quantum plasma (which in general is a complex-valued function), which can be used for various modulational processes. We derived the quantum-corrected Zakharov equations for collisionless quantum plasmas by neglecting the imaginary part of the effective cubic response. The resulting quantum-corrected Zakharov equations describe the coupled nonlinear evolution of high-frequency fields and low-frequency density perturbations in collisionless quantum plasmas that can be written in the one dimensional case as $$\begin{aligned}
&&\hspace{-1.8cm}\left(i \frac{\partial}{\partial
t}+\frac{3}{2}\frac{v^2_e}{\omega_{pe}}\frac{\partial^2}{\partial x^2}
-\frac{\hbar^2}{8m^2_e\omega_{pe}}\frac{\partial^4}{\partial
x^4}\right)E(x,t) =\frac{\omega_{pe}}{2}\frac{\delta n
(x,t)}{n_0}E(x,t),\label{q1}\end{aligned}$$ $$\begin{aligned}
&&\hspace{-1.8cm}\left(\frac{\partial^2}{\partial
t^2}+\frac{\hbar^2}{12m^2_ev^2_e}\frac{\partial^4}{\partial t^2
\partial x^2}
-v^2_s\frac{\partial^2}{\partial x^2}\right)\frac{\delta n
(x,t) }{n_0} =\left(\frac{\partial^2}{\partial
x^2}+\frac{\hbar^2}{12m^2_ev^2_e}\frac{\partial^4}{\partial x^4}
\right)\frac{|E(x,t)|^2}{4\pi n_0 m_i}, \label{q2}\end{aligned}$$ where $E(x,t)$ is the envelope of the Langmuir wave packet modulated by the nonlinear interaction with plasma density variations, $\delta n(x,t)$ is the plasma density variations from its equilibrium value $n_0$, $\omega_{pe}$ is the electron plasma frequency (i.e. $\omega_{pe}=\sqrt{{4\pi e^2n_0/m_e}}$), $m_e$ and $m_i$ are the electron and ion masses, respectively, $v_e$ is the electrons thermal velocity and $v_s$ is the ion sound velocity, and $\hbar$ is the reduced Planck constant.
Upon introducing the dimensionless variables $$\begin{aligned}
&&\hspace{-1.8cm}{
x}=\frac{2}{3}\frac{m_e}{m_i}\frac{\omega_{pe}}{v_s}{X},~~~~~t= \frac{2}{3}\frac{m_e}{m_i}\omega_{pe}\tau,\nonumber\\
&&\hspace{-3.4cm} ~~~~~~{\it E}=\sqrt{16\pi
n_0T_em_e/3m_i}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
},~~~\frac{\delta n}{n_0}=\frac{4}{3}\frac{m_e}{m_i}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
},\nonumber\\
&&\hspace{-2.7cm}~~~~~~H=\frac{\hbar\omega_{pe}}{m_ev^2_s
},\label{q3}\end{aligned}$$ the corresponding dimensionless Zakharov equations in quantum plasma can be presented in the form $$\begin{aligned}
&&\hspace{-3.8cm}\left(i \frac{\partial}{\partial
\tau}+\frac{\partial^2}{\partial X^2}-H^2\frac{\partial^4}{\partial
X^4}\right){ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}={ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
} { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
},\label{q4}\end{aligned}$$ $$\begin{aligned}
&&\hspace{-2.8cm}\left(\frac{\partial^2}{\partial
\tau^2}+H^2\frac{\partial^4}{\partial \tau^2
\partial X^2}
-\frac{\partial^2}{\partial X^2}\right){ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}
=\frac{\partial^2}{\partial
X^2}\left(1+H^2\frac{\partial^2}{\partial X^2}\right){|
{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2},\label{q5}\end{aligned}$$ where the quantum parameter $H$ is given in Eq. (\[q3\]). In dense plasmas [@r11; @r13], the particle density is about $10^{25}$ – $10^{32}$ $m^{-3}$ and temperature is about $10^5$ – $10^7$ K. For a completely ionized hydrogen plasma in these density and temperature ranges, $H$ typically ranges from about $10^{-5}$ up to values of order unity. For large values of $H$ (i. e. $H\sim1$) particularly in astrophysical plasmas, quantum effects in the coupling between Langmuir and ion-acoustic modes become important [@r30].
If we ignore the quantum correction by setting $H=0$, we simply obtain the classical Zakharov equations [@r6]. At the classical level, a set of coupled nonlinear wave equations describing the interaction between high-frequency Langmuir waves with low frequency plasma density variations was first derived by Zakharov [@r26; @r27]. For the classical model, one can find many kinds of solitons by various methods [@r31; @r32; @r33]. In the quantum case, the analysis of existence of localized or soliton solutions is much more difficult. The basic difficulty follows from the fact that the new, quantum-corrected equations constitute a more complicated system of coupled, fourth-order nonlinear equations (\[q4\])-(\[q5\]).
In the adiabatic limit, by setting $[(\partial^2 /\partial \tau^2)({ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
})]=0$ in Eq. (\[q5\]), solutions of the classical ($H=0$) Zakharov system are found. In this situation, the envelope of the electric field satisfies a nonlinear Schrodinger equation, which is completely integrable yielding N-soliton solutions [@r6]. Solitons usually arise as a consequence of the detailed balance between dispersive and nonlinear contributions. The quantum effects may perturb or perhaps even destroy these localized solitonic solutions. Since quantum effects enhance dispersion, one should expect that quantum solitons will not be so easily found for the quantum Zakharov equations as for the classical case [@r26]. We investigate this assumption by taking $[(\partial^2 /\partial \tau^2)({ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
})]=0$ in Eq. (\[q5\]) and obtain $$\begin{aligned}
&&\hspace{-4.8cm}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}=-\frac{\mid { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\mid^2}{(1-v_0^2)}-\frac{H^2}{(1-v_0^2)^2}\frac{\partial^2}{\partial
X^2}|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2+O(H^4). \label{q6}\end{aligned}$$ The detailed calculation of Eq. (\[q6\]) is given in Appendix A.
Substituting Eq. (\[q6\]) into Eq. (\[q4\]), and keeping only terms of order up to $H^2$ (note that our approximate analysis is only valid for ‘weakly quantum’ plasmas with small $H$), yields the decoupled equation for the envelope ${ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}$ of the modulated Langmuir wave packet: $$\begin{aligned}
&&\hspace{-3.8cm}\left(i \frac{\partial}{\partial
\tau}+\frac{\partial^2}{\partial X^2}+\frac{|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2}{(1-v_0^2)}\right){ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\approx H^2\frac{\partial^4}{\partial
X^4}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}-\frac{H^2}{(1-v_0^2)^2}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\frac{\partial^2}{\partial X^2}|
{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2.\label{q7}\end{aligned}$$ Eq. (\[q7\]) is the quantum-corrected nonlinear Schrodinger equation which is derivable from a variational principle, $$\begin{aligned}
&&\hspace{-4.8cm}\delta S=\delta\int L dX d\tau=0, \label{q8}\end{aligned}$$ with the Lagrangian [@r29] $$\begin{aligned}
&&\hspace{-1.8cm}L=\frac{i}{2}\left({ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
\tau}-{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial \tau}\right)+\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial
X}\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial X}-\frac{|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^4}{2(1-v_0^2)}+H^2\left[\frac{\partial^2 { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial
X^2}\frac{\partial^2 { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
X^2}-\frac{|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2}{2(1-v_0^2)^2}\frac{\partial^2|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2}{\partial X^2}\right].\label{q9}\end{aligned}$$ The variational derivatives $\delta S/\delta { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast = \delta S/\delta { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
} = 0$ produce Eq. (\[q7\]) and its complex conjugate equation, respectively. The detailed calculation showing this is given in Appendix B.
In this section, we investigate the existence of quantum-modified Langmuir envelope solitons described by the quantum-corrected Zakharov equations. We start by proposing time-dependent singular solution [@r6] of the form: $$\begin{aligned}
&&\hspace{-1.8cm}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}=\alpha(\tau)\exp\bigg\{i\bigg[\frac{v_0}{2}X+\theta(\tau)\bigg]\bigg\}
\textrm{sech}[\beta(\tau)(X-v_0\tau)],\label{q10}\end{aligned}$$ where $\alpha$, $\beta$, and $\theta$ are considered as real-valued functions of time only. Below we use the variational principle Eq. (\[q8\]), requiring that the solution (\[q10\]) with arbitrary $\alpha$, $\beta$ and $\theta$ minimize the action, to derive the quantum-corrected form of the coefficients (i. e. $\alpha$, $\beta$, and $\theta$) of soliton solution (\[q10\]).
Substituting Eq. (\[q10\]) into Eq. (\[q9\]), we get a mechanical system governed by the action $$\begin{aligned}
&&\hspace{-4.8cm} S=\int L'(\alpha(\tau),\beta(\tau),\theta(\tau)) d\tau, \label{q11}\end{aligned}$$ where $$\begin{aligned}
&&\hspace{-4.8cm} L'(\alpha(\tau),\beta(\tau),\theta(\tau))=\int L dX. \label{qq11}\end{aligned}$$ Now $$\begin{aligned}
&&\hspace{-1.2cm}~~~~L'(\alpha, \beta, \theta)\nonumber\\
&&\hspace{-1.2cm}~~~=\frac{2\alpha^2\dot\theta}{\beta}+\frac{\alpha^2v_0^2}{2\beta}+\frac{2\alpha^2\beta}
{3}-\frac{2\alpha^4}
{3\beta(1-v_0^2)}+\frac{H^2\alpha^2 v_0^2}{8\beta}+H^2\alpha^2\beta v_0^4+\frac{8H^2\alpha^4\beta}{15(1-v_0^2)^2},\label{q12}\end{aligned}$$ where $\dot{\theta}=d\theta/d\tau$.
The variational principle Eq. (\[q8\]) requires that the variational derivatives $\delta S/\delta\theta=\delta S/\delta\alpha = \delta S/\delta\beta= 0$. Calculating the variational derivatives, we obtain: $$\begin{aligned}
&&\hspace{-10.8cm}\frac{\partial L'}{\partial
\theta}=0\Rightarrow\frac{d}{d\tau}\left(\frac{\alpha^2}{\beta}\right)=0.\label{q13}\end{aligned}$$
${\partial L'}/{\partial \alpha}=0$ $$\begin{aligned}
&&\hspace{-1.1cm}\Rightarrow
\dot{\theta}+\frac{v_0^2}{4}+\frac{\beta^2}{3}-\frac{2\alpha^2}
{3(1-v_0^2)}+\frac{7}{15}H^2\beta^4+\frac{1}{2}H^2\beta^2+\frac{1}{16}H^2v_0^4+\frac{8\alpha^2\beta^2H^2}{15(1-v_0^2)^2}=0.\label{q15}\end{aligned}$$
${\partial L'}/{\partial \beta}=0$ $$\begin{aligned}
&&\hspace{-1.1cm}\Rightarrow
\dot{\theta}+\frac{v_0^2}{4}-\frac{\beta^2}{3}-\frac{\alpha^2}
{3(1-v_0^2)}-\frac{7}{5}H^2\beta^4-\frac{1}{2}H^2\beta^2+\frac{1}{16}H^2v_0^4-\frac{4\alpha^2\beta^2H^2}{15(1-v_0^2)^2}=0.\label{q17}\end{aligned}$$
Solving Eqs. (\[q13\]) - (\[q17\]) for $\alpha$, $\beta$ and $\theta$ enables us to construct the quantum-modified soliton solution of the form Eq. (\[q10\]) that is a true solution of Eq. (\[q7\]), as it minimizes the action of the system. From Eq. (\[q13\]) we have $$\begin{aligned}
&&\hspace{-10.8cm}\dot\alpha=\frac{\alpha}{2}\frac{\dot\beta}{\beta}. \label{qq13}\end{aligned}$$ Differentiating Eq. (\[q15\]) and Eq. (\[q17\]) with respect to time and then subtracting, we have $$\begin{aligned}
&&\hspace{-1.0cm}\frac{4\beta\dot\beta}{3}-\frac{2\alpha\dot\alpha}
{3(1-v_0^2)}+\frac{112}{15}H^2\beta^3\dot\beta+2H^2\beta\dot\beta v_0^2+\frac{12}{15(1-v_0^2)^2}(2\alpha\dot\alpha\beta^2H^2+2\beta\dot\beta\alpha^2H^2)=0.
\label{b18}\end{aligned}$$ Now substituting the value of $\dot\alpha$ Eq. (\[qq13\]) in Eq. (\[b18\]) we obtain $$\begin{aligned}
&&\hspace{-1.0cm}\dot\beta\bigg[\frac{4\beta}{3}-\frac{\alpha^2}
{3\beta(1-v_0^2)}+\frac{112H^2\beta^3}{15}+2H^2\beta v_0^2+\frac{12}{15(1-v_0^2)^2}(2\alpha^2\beta H^2+2\alpha^2\beta H^2)\bigg]=0.
\label{b19}\end{aligned}$$ From Eq. (\[b19\]) we have the following two cases
1\) $\dot\beta=0$, $\beta$ is constant.
2\) $\big[{4\beta}/{3}-{\alpha^2}/
{3\beta(1-v_0^2)}+{112H^2\beta^3}/{15}+2H^2\beta v_0^2+{12}/{15(1-v_0^2)^2}(2\alpha^2\beta H^2+2\alpha^2\beta H^2)\big]=0$, $\dot\beta$ is constant.
We consider case 1 which implies that $\beta$ is a constant. Then from Eq. (\[q13\]), it follows that $\alpha$ is a constant as well. If we consider case 2 it leads to a contradiction with Eq. (\[q13\]), since Eq. (\[q13\]) implies that $\alpha^2\propto\beta$ whereas Eq. (\[b19\]) implies $\alpha\propto\beta$ (i. e. in classical case), which can only be resolved if $\beta$ is a constant. Therefore, case 1 is true.
In this section we derive expressions for these constants $\alpha$ (amplitude of the soliton), $\beta$ (inverse width), and the corresponding $\theta$ (phase shift) from Eqs. (\[q15\]) - (\[q17\]).
Subtracting Eq. (\[q17\]) from Eq. (\[q15\]) we have $$\begin{aligned}
&&\hspace{-2.0cm}\frac{2\beta^2}{3}-\frac{\alpha^2}
{3(1-v_0^2)}+\frac{28}{15}H^2\beta^4+H^2\beta^2v_0^2+\frac{12}{15}\frac{\alpha^2\beta^2H^2}{(1-v_0^2)^2}=0.\label{q18}\end{aligned}$$ From Eq. (\[q18\]), we obtain the quantum-corrected amplitude of the soliton solution (\[q10\]) in terms of $\beta$ and $v_0$ in the following form $$\begin{aligned}
&&\hspace{-1.8cm}\alpha=\sqrt{2(1-v_0^2)}\beta\left[1+\frac{H^2}{20(1-v)^2}(15v_0^2-15v_0^4+\beta^2[52-28v_0^2])\right]+O(H^4).\label{q23}\end{aligned}$$ Substituting the value of $\alpha$ (\[q23\]) into Eq. (\[q17\]) and integrating over time, we obtain the quantum-corrected phase shift of the soliton solution (\[q10\]) in terms of $\beta$ and $v_0$ in the following form $$\begin{aligned}
&&\hspace{-1.8cm}~~~~{\theta}(\tau)=\bigg[\bigg(\beta^2-\frac{v_0^2}{4}\bigg)-\bigg(\frac{v_0^4}{16}-\frac{\beta^2\{18v_0^2(1-v_0^2)+\beta^2(81-49v_0^2)\}}{15(1-v_0^2)}\bigg)H^2\bigg]\tau+O(H^4).\label{q24}\end{aligned}$$ Thus, we have $\alpha$ (\[q23\]) and $\theta(\tau)$ (\[q24\]) which are defined in terms of $\beta$ and $v_0$. On the other hand, the soliton solution depends on two important parameters, namely, amplitude $E_0$ and the speed $v_0$. Now we express $\alpha$, $\beta$, and $\theta(\tau)$ in terms of $E_0$ and $v_0$ instead of $\beta$ and $v_0$. From Eq. (\[q23\]) we obtain the equation for $\beta$ in terms of $E_0$ and $v_0$ as
{height="90mm" width="100mm"}
$$\begin{aligned}
&&\hspace{-1.1cm}\beta+\frac{H^2\beta}{20(1-v_0^2)}(15v_0^2-15v_0^4+\beta^2[52-28v_0^2])=\frac{E_0}{\sqrt{2(1-v_0^2)}}.\label{q27}\end{aligned}$$
In the formal classical limit $H=0$ in Eq. (\[q27\]), we obtain $\beta=E_0/\sqrt{2(1-v_0^2)}$, which reproduces the classical one-soliton solution Eq. (\[q10\]) with $\alpha=E_0$, $\beta=E_0/\sqrt{2(1-v_0^2)}$, and $\theta=E^2_0/2(1-v_0^2)-v_0^2/4$. In the quantum case $H>0$ (note that typically $H<<1$ in quantum plasmas at high densities) Eq. (\[q27\]) has the perturbative solution $$\begin{aligned}
&&\hspace{-1.8cm}\beta=\frac{E_0}{\sqrt{2(1-v_0^2)}}-\frac{H^2E_0(15[1-v_0^2]-2E_0[13-7v_0^2])}{20\sqrt{2}(1-v_0^2)^{5/2}}+O(H^4).\label{q28}\end{aligned}$$ Eq. (\[q28\]) represents the quantum-corrected inverse width of the soliton solution (\[q10\]) in terms of $E_0$ and $v_0$.
Substituting Eq. (\[q28\]) into Eq. (\[q24\]), we obtain the quantum-corrected phase shift of the soliton solution (\[q10\]) in terms of $E_0$ and $v_0$ in the following form $$\begin{aligned}
&&\hspace{-1.8cm}\theta(\tau)=\bigg[\frac{E_0^2}{2(1-v_0^2)}\bigg(1+\frac{H^2E_0^2}
{10(1-v_0^2)}\bigg)-\frac{v_0^2}{4}-\frac{H^2v_0^4}{16}\bigg]\tau
+O(H^4),\label{q29}\end{aligned}$$ or $$\begin{aligned}
&&\hspace{-12.1cm}~~~~~~~~~~\theta(\tau)=\Omega\tau,\label{q30}\end{aligned}$$ where $$\begin{aligned}
&&\hspace{-1.8cm}\Omega=\frac{E_0^2}{2(1-v_0^2)}\bigg(1+\frac{H^2E_0^2}{10(1-v_0^2)}\bigg)-\frac{v_0^2}{4}-\frac{H^2v_0^4}{16}
+O(H^4).\label{q31}\end{aligned}$$ We used the variational principle Eq. (\[q8\]), requiring that the solution (\[q10\]) with arbitrary $\alpha$, $\beta$ and $\theta$ minimize the action, to derive the quantum-corrected form of the coefficients (i. e. $\alpha$, $\beta$, and $\theta$) of the soliton solution (\[q10\]). Finally, we obtained the expressions for the constants $\alpha$ (amplitude), $\beta$ (inverse width) and the corresponding $\theta$ (phase shift) of the soliton solution (\[q10\]) that are expressed in terms of the two arbitrary parameters $v_0$ and $E_0$ in quantum-corrected form (i.e., $\beta$ and $\theta$ in Eqs. (\[q28\]) and (\[q29\])); note that by definition, $\alpha=E_0$. When $H=0$ in Eqs. (\[q28\]) and (\[q29\]) then the soliton solution (\[q10\]) reduces to the exact soliton solution of the classical NLS. Thus, $\alpha$, $\beta$ and $\theta$ in Eqs. (\[q28\]) and (\[q29\]), respectively, enable us to construct the quantum-modified soliton solution of the form Eq. (\[q10\]) that is a true solution of Eq. (\[q7\]), minimizing the action of the system.
We use the quantum-modified soliton solution Eq. (\[q10\]) with $\alpha$ , $\beta$ and $\theta$ defined by Eqs. (\[q28\]) and (\[q29\]), respectively, to examine cases with different values for $H$ while keeping $E_0$ and $v_0$ fixed. We plot these different cases in Fig.1. It is clear from Fig.1 that for the same amplitude (i.e., $\alpha=E_0$, which is same for both quantum and classical case) the inverse width of the soliton decreases with quantum correction terms proportional to $H^2$; i.e., the width increases with quantum-correction term $H$. In other words, the quantum effects lead to widening of the soliton in quantum plasma, compared to the classical soliton of the same amplitude.
Conclusion
==========
We have investigated the quantum-corrected Zakharov equations and the existence of quantum-corrected solitons in a fully nonlinear quantum plasma. We constructed a solution using a variational principle (requiring that the solution minimizes the action of the system) and also derived the quantum-modified coefficients of the soliton solution of the quantum-modified NLS equation. We found that for the same amplitude (i.e., $\alpha=E_0$, for both quantum and classical cases) the quantum effects change the inverse width $(\beta)$ and the phase shift $(\theta)$ given by Eqs. (\[q28\]) and (\[q29\]) for the soliton solution Eq. (\[q10\]). In Fig.1 we have shown that for the same amplitude (i.e., $\alpha=E_0$, is same for both quantum and classical case) the inverse width of the soliton decreases with quantum correction terms proportional to $H^2$; i.e., the width increases with the quantum-correction term $H$. We will study in a subsequent paper how the quantum correction terms proportional to $H^2$ in Eqs. (\[q28\]), (\[q29\]) affect the stability of such solutions. In particular, we will numerically study the ultimate fate of the solitons in the quantum plasmas and will also analyse the collision of two solitons for both cases where $H=0$ and $H>0$.
The dimensionless Zakharov equations in quantum plasma can be presented in the form $$\begin{aligned}
&&\hspace{-3.8cm}\left(i \frac{\partial}{\partial
\tau}+\frac{\partial^2}{\partial X^2}-H^2\frac{\partial^4}{\partial
X^4}\right){ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}={ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
} { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
},\label{a1}\end{aligned}$$ $$\begin{aligned}
&&\hspace{-2.8cm}\left(\frac{\partial^2}{\partial
\tau^2}+H^2\frac{\partial^4}{\partial \tau^2
\partial X^2}
-\frac{\partial^2}{\partial X^2}\right){ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}
=\frac{\partial^2}{\partial
X^2}\left(1+H^2\frac{\partial^2}{\partial X^2}\right){|
{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2}.\label{a2}\end{aligned}$$ The solution of these nonlinear equations is obtained by transforming the independent variables, using $$\begin{aligned}
&&\hspace{-2.8cm}\eta=X-v_0\tau,\label{a3}\end{aligned}$$ where $v_0$ is a constant speed at which the soliton solution propagates.
Differentiating Eq. (\[a3\]) with respect to $\tau$ and $X$ respectively, we obtain $$\begin{aligned}
&&\hspace{-2.8cm}\frac{\partial}{\partial\tau}=-v_0\frac{\partial}{\partial\eta},\label{a4}\end{aligned}$$ $$\begin{aligned}
&&\hspace{-2.8cm}\frac{\partial}{\partial X}=\frac{\partial}{\partial\eta}.\label{a5}\end{aligned}$$
Substituting the values of Eq. (\[a4\]) and Eq. (\[a5\]) in Eq. (\[a2\]), we obtain the following equation
$$\begin{aligned}
&&\hspace{-2.8cm}(v_0^2-1){ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}+H^2v_0^2\frac{\partial^2}{\partial \eta^2
}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
} n
=|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2+H^2\frac{\partial^2}{\partial \eta^2
}|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2.\label{a6}\end{aligned}$$
The iterative solution of Eq. (\[a6\]) is as follows:
When $H=0$ then Eq. (\[a6\]) becomes
$$\begin{aligned}
&&\hspace{-2.8cm}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(1)}
=-\frac{|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2}{(1-v_0^2)}.\label{a7}\end{aligned}$$
When $H\neq0$ then Eq. (\[a6\]) can be written as
$$\begin{aligned}
&&\hspace{-2.8cm}H^2v_0^2\frac{\partial^2}{\partial \eta^2
}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(1)}+{(v_0^2-1)}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
} n^{(2)}
=|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2+H^2\frac{\partial^2}{\partial \eta^2
}|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2,\label{a8}\end{aligned}$$
so that $$\begin{aligned}
&&\hspace{-2.8cm}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta}}
} n^{(2)}
={ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(1)}-\frac{H^2}{(1-v_0^2)}\frac{\partial^2}{\partial \eta^2
}|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2-\frac{H^2v_0^2}{(1-v_0^2)}\frac{\partial^2}{\partial \eta^2
}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(1)}.\label{a9}\end{aligned}$$ Substituting Eq. (\[a7\]) into Eq. (\[a9\]) we obtain $$\begin{aligned}
&&\hspace{-2.8cm}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(2)}=
-\frac{|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2}{(1-v_0^2)}-\frac{H^2}{(1-v_0^2)^2}\frac{\partial^2}{\partial \eta^2
}|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2.\label{a10}\end{aligned}$$ again $$\begin{aligned}
&&\hspace{-2.8cm}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(3)}
={ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(1)}-\frac{H^2}{(1-v_0^2)}\frac{\partial^2}{\partial \eta^2
}|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2-\frac{H^2v_0^2}{(1-v_0^2)}\frac{\partial^2}{\partial \eta^2
}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}^{(1)}+O(H^4).\label{a11}\end{aligned}$$ Finally, we can write $$\begin{aligned}
&&\hspace{-2.8cm}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{\delta n}}
}
=-\frac{|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2}{(1-v_0^2)}-\frac{H^2}{(1-v_0^2)^2}\frac{\partial^2}{\partial X^2
}|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2+O(H^4).\label{a12}\end{aligned}$$
The quantum-corrected nonlinear Schrodinger equation can be written as $$\begin{aligned}
&&\hspace{-3.8cm}\left(i \frac{\partial}{\partial
\tau}+\frac{\partial^2}{\partial X^2}+\frac{|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2}{(1-v_0^2)}\right){ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\approx H^2\frac{\partial^4}{\partial
X^4}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}-\frac{H^2}{(1-v_0^2)^2}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\frac{\partial^2}{\partial X^2}|
{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2,\label{b1}\end{aligned}$$ which is derivable from a variational principle, $$\begin{aligned}
&&\hspace{-4.8cm}\delta S=\delta\int L dX d\tau=0, \label{b2}\end{aligned}$$ based on the Lagrangian [@r29] $$\begin{aligned}
&&\hspace{-1.8cm}L=\frac{i}{2}\left({ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
\tau}-{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial \tau}\right)+\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial
X}\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial X}-\frac{|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^4}{2(1-v_0^2)}+H^2\left[\frac{\partial^2 { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial
X^2}\frac{\partial^2 { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
X^2}-\frac{|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2}{2(1-v_0^2)^2}\frac{\partial^2|{{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}|^2}{\partial X^2}\right],\label{b3}\end{aligned}$$ where $L=L({ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}, { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast, \partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}/\partial X, \partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast/\partial X,t)$. The variational derivatives $\delta S/\delta { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast = \delta S/\delta { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
} = 0$ produce Eq. (\[b1\]) and its complex conjugate equation, respectively. The corresponding Lagrange equation of the quantum-corrected nonlinear Schrodinger equation can be written as $$\begin{aligned}
&&\hspace{-1.8cm}\frac{\partial}{\partial
\tau}\frac{\partial L}{\partial \left(\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
\tau}\right)}+\frac{\partial}{\partial
\ X}\frac{\partial L}{\partial \left(\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
X}\right)}-\frac{\partial L}{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}=0.\label{b4}\end{aligned}$$ From Eq. (\[b3\]), we obtain $$\begin{aligned}
&&\hspace{-1.8cm}\frac{\partial L}{\partial \left(\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
X}\right)}=\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial X}-H^2\frac{\partial^3 { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial
X^3},\label{b6}\end{aligned}$$ $$\begin{aligned}
&&\hspace{-1.8cm}\frac{\partial L}{\partial \left(\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}{\partial
\tau}\right)}=\frac{i}{2}{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
},\label{b7}\end{aligned}$$ and $$\begin{aligned}
&&\hspace{-1.8cm}\frac{\partial L}{\partial
{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}^\ast}=-\frac{i}{2}\frac{\partial { \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}}{\partial \tau}-{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2-H^2{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}\frac{\partial^2 |{ \mathchoice
{\accentset{\displaystyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\textstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
{\accentset{\scriptscriptstyle{ \text{\smash{\raisebox{-1.3ex}{ $\widehatsym$}}}}}{E}}
}|^2}{\partial X^2}.\label{b8}\end{aligned}$$
Substituting the values of Eqs. (\[b6\]) - (\[b8\]) into Eq. (\[b4\]) one can obtain the quantum-corrected nonlinear Schrodinger equation Eq. (\[b1\]).\
\
[**Acknowledgments.**]{} This study was partially supported by the Australian Research Council (ARC) and by Asian Office of Aerospace R and D under grant number FA2386-12-1-4077 and JSPS Grant-in-Aid for Scientific Research (A) under Grant 23244110.
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|
---
abstract: 'We use group theory to construct infinite families of maps on surfaces which are invariant under Wilson’s map operations of order $3$ but not under the operations of order $2$, such as duality and Petrie duality.'
author:
- |
Gareth A. Jones\
School of Mathematics\
University of Southampton\
Southampton SO17 1BJ, U.K.\
[[email protected]]{}
- |
Andrew Poulton\
School of Mathematics\
University of Southampton\
Southampton SO17 1BJ, U.K.\
[[email protected]]{}
title: Maps admitting trialities but not dualities
---
[**MSC classification:**]{} Primary 05C25, secondary 05C10, 20B25. [**Keywords:**]{} Map, operation, duality, triality.
[**Running head:**]{} Maps admitting trialities.\
Introduction
============
In [@Wil], Wilson introduced a group $\Sigma$ of six operations on regular maps, isomorphic to the symmetric group $S_3$. In addition to the identity operation, $\Sigma$ contains the duality operation $D$, transposing vertices and faces, and the Petrie duality $P$, transposing faces and Petrie polygons (closed zig-zag paths); these two operations have order $2$, and they generate the group $\Sigma$. The remaining elements of $\Sigma$ are the ‘opposite’ operation $DPD=PDP$, which is a duality transposing vertices and Petrie polygons, and two mutually inverse operations $DP$ and $PD$ of order $3$, called [*trialities*]{}, each permuting vertices, faces and Petrie polygons in a cycle of length $3$.
The images of a map under these six operations are called its [*direct derivates*]{}, and Wilson divided regular maps into classes I – IV as they have $6$, $3$, $2$ or $1$ of these, up to isomorphism (the only possibilities). Maps $\cal M$ in class III are invariant under the triality operations $DP$ and $PD$, but not under the duality operations $D$, $P$ and $DPD$; thus $${\cal M}\cong DP({\cal M})\cong PD({\cal M})\quad {\rm and}\quad
D({\cal M})\cong P({\cal M})\cong DPD({\cal M}),$$ but ${\cal M}\not\cong D({\cal M})$. Wilson pointed out in [@Wil] that such maps seem to be very rare, and indeed at one stage he had suspected that they did not exist. The single example given in [@Wil] is a dual pair $\cal W$ and $D({\cal W})$ of non-orientable maps of type $\{9,9\}_9$ and characteristic $-70$, with the simple group $L_2(8)$ of order $504$ as their common automorphism group. These appear to be the only examples currently in the literature, although after Wilson raised the question of their existence at the SIGMAC 06 conference in Aveiro, Conder [@Con06] found several further examples by using a computer program.
In [@JT], Jones and Thornton showed that regular maps correspond to normal subgroups of a certain group $\Gamma$, the free product $V_4*C_2$ of a Klein four-group $V_4$ and a cyclic group $C_2$ of order $2$, and that the operations in $\Sigma$ are induced by the outer automorphism group ${\rm Out}\,\Gamma\cong {\rm Aut}\,V_4\cong S_3$ acting on such subgroups. Our aim here is to use this algebraic approach to reinterpret Wilson’s example, and to give several constructions which yield infinite series of regular maps in class III.
We thank Marston Conder and Steve Wilson for some valuable comments, and the Nuffield Foundation for supporting the second author with a Nuffield Science Bursary.
Maps
====
Here we outline the algebraic theory of maps. For simplicity of exposition we will restrict attention to maps without boundary and without free edges, since it is easy to show that there are no regular maps in class III possessing these features. For a more comprehensive account, see [@BS].
If $\cal M$ is a map then let $\Omega$ be the set of [*blades*]{} associated with $\cal M$: these are local flags $\alpha=(v,e,f)$ where $v$ is a vertex, $e$ is an edge, and $f$ is a face, all mutually incident. We define three permutations $r_0, r_1$ and $r_2$ of $\Omega$ as follows: $r_i$ sends each flag $\alpha$ to an adjacent flag $\alpha r_i$ by preserving its $j$-dimensional components for each $j\neq i$, while changing (in the only possible way) its $i$-dimensional component. These permutations are illustrated in Figure 1, where the small triangles represent blades.
(100,160)(0,10)
(120,60) (280,60) (110,50)[$v$]{} (195,120)[$f$]{} (195,65)[$e$]{}
(120,59.7)[(1,0)[160]{}]{} (120,59.8)[(1,0)[160]{}]{} (120,59.9)[(1,0)[160]{}]{} (120,60)[(1,0)[160]{}]{} (120,60.1)[(1,0)[160]{}]{} (120,60.2)[(1,0)[160]{}]{} (120,60.3)[(1,0)[160]{}]{}
(120,59.7)[(-1,2)[40]{}]{} (120,59.8)[(-1,2)[40]{}]{} (120,59.9)[(-1,2)[40]{}]{} (120,60)[(-1,2)[40]{}]{} (120,60.1)[(-1,2)[40]{}]{} (120,60.2)[(-1,2)[40]{}]{} (120,60.3)[(-1,2)[40]{}]{}
(120,60)[(2,1)[30]{}]{} (120,60)[(2,-1)[30]{}]{} (280,60)[(-2,1)[30]{}]{} (280,60)[(-2,-1)[30]{}]{} (150,45)[(0,1)[30]{}]{} (250,45)[(0,1)[30]{}]{}
(120,60)[(0,1)[35]{}]{} (105,88)[(2,1)[15]{}]{}
(145,80)[$\alpha$]{} (242,80)[$\alpha r_0$]{} (142,35)[$\alpha r_2$]{} (217,35)[$\alpha r_0r_2=\alpha r_2r_0$]{} (115,100)[$\alpha r_1$]{}
Figure 1: the permutations $r_0$, $r_1$ and $r_2$.
By connectivity these permutations generate a transitive subgroup $G$ of the symmetric group ${\rm Sym}\,\Omega$ on $\Omega$, known as the [*monodromy group*]{} ${\rm Mon}\,{\cal M}$ of $\cal M$. They satisfy $$r_0^2=r_1^2=r_2^2=(r_0r_2)^2=1,$$ so that if we define an abstract group $\Gamma$ by means of the presentation $$\Gamma=\langle R_0, R_1, R_2\mid R_0^2=R_1^2=R_2^2=(R_0R_2)^2=1\rangle$$ then there is an epimorphism $\theta:\Gamma\to G$ given by $R_i\mapsto r_i$. (The generators $R_i$ of $\Gamma$ were denoted by $l, r$ and $t$ in [@JT], but here we follow the notation of Coxeter and Moser [@CM].) The subgroups $M=\theta^{-1}(G_{\alpha})$ of $\Gamma$ fixing a blade $\alpha\in\Omega$, known as the [*map subgroups*]{}, form a conjugacy class of subgroups of index $|\Omega|$ in $\Gamma$. One can reverse this process: any transitive permutation representation of $\Gamma$ gives rise to a map in which the vertices, edges and faces correspond to the orbits of the dihedral subgroups $\langle R_1, R_2\rangle$, $\langle R_0, R_2\rangle$ and $\langle R_0, R_1\rangle$, with incidence given by non-empty intersection. A map $\cal M$ is compact if and only if $\Omega$ is finite, or equivalently $M$ has finite index in $\Gamma$; it is without boundary and free edges if and only if $M$ is torsion-free (i.e. contains no conjugate of any $R_i$ or of $R_0R_2$), in which case $\cal M$ is orientable if and only if $M$ is contained in the even subgroup $\Gamma^+$ of $\Gamma$, the subgroup of index $2$ consisting of all words of even length in the generators $R_i$.
The automorphism group $A={\rm Aut}\,{\cal M}$ of $\cal M$ is the set of permutations of $\Omega$ commuting with each $r_i$; equivalently, $A$ is the centraliser of $G$ in ${\rm Sym}\,\Omega$. It acts freely on $\Omega$, since an automorphism fixing a blade must fix all of them, and it is isomorphic to $N_{\Gamma}(M)/M$. We say that $\cal M$ is [*regular*]{} if $A$ acts transitively (and hence regularly) on $\Omega$, or equivalently $M$ is normal in $\Gamma$, in which case $$A\cong G\cong\Gamma/M,$$ and we can regard $A$ and $G$ as the left and right regular representations of the same group. Coverings ${\cal M}_1\to{\cal M}_2$ of maps correspond to inclusions $M_1\leq M_2$ of the corresponding map subgroups, with the index $|M_2:M_1|$ equal to the number of sheets. Normal inclusions correspond to regular coverings, induced by $M_2/M_1$ acting as a group of automorphism of ${\cal M}_2$.
The Petrie polygons of a map $\cal M$ are its closed zig-zag paths, turning alternately first right and first left at successive vertices; these can be identified with the orbits of the dihedral subgroup $\langle R_0R_2, R_1\rangle$ on $\Omega$. Following Coxeter and Moser [@CM Ch. 8], we say that a map $\cal M$ has [*type*]{} $\{p,q\}_r$, or simply $\{p,q\}$, if $p$, $q$ and $r$ are the orders of the permutations $r_0r_1$, $r_1r_2$ and $r_0r_1r_2$, or equivalently the least common multiples of the valencies of the faces, the vertices and the Petrie polygons of $\cal M$. In a regular map these all have the same valencies $p$, $q$ and $r$ respectively.
In [@Wil], Wilson described a group $\Sigma\cong S_3$ of operations on regular maps. It is generated by the classical duality $D$, which transposes the vertices and faces of a map, while preserving the underlying surface, and the Petrie duality $P$, which preserves the embedded graph while transposing the faces and the Petrie polygons, so that the surface may change both its orientability and its characteristic. These operations have order $2$. Apart from the identity operation, the remaining elements of $\Sigma$ are the ‘opposite’ operation $DPD=PDP$, also of order $2$, which transposes vertices and Petrie polygons by rejoining adjacent faces across common edges with reversed identifications, and two operations $DP$ and $PD$ of order $3$ called trialities.
In [@JT] it is shown that $\Sigma$ can be identified with the outer automorphism group $${\rm Out}\,\Gamma={\rm Aut}\,\Gamma/{\rm Inn}\,\Gamma\cong S_3$$ of $\Gamma$, acting on the normal subgroups $M$ of $\Gamma$ corresponding to regular maps $\cal M$. This is because $\Gamma$ is the free product $V*C$ of a Klein four-group $V=\langle R_0, R_2\rangle\cong V_4$ and a cyclic group $C=\langle R_1\rangle\cong C_2$ of order $2$, and ${\rm Aut}\,\Gamma$ is a semidirect product of the inner automorphism group ${\rm Inn}\,\Gamma\cong\Gamma$ by ${\rm Aut}\,V\cong S_3$, the latter permuting the three involutions $R_0, R_2$ and $R_0R_2$ of $V$. Since the faces, vertices and Petrie polygons of a map $\cal M$ can be identified with the orbits of the dihedral subgroups $\langle R_0, R_1\rangle$, $\langle R_2, R_1\rangle$ and $\langle R_0R_2, R_1\rangle$ of $\Gamma$ on $\Omega$, one can identify the action of $\Sigma$ on maps with that of ${\rm Aut}\,V$, and hence of ${\rm Out}\,\Gamma$, on normal subgroups $M$ of $\Gamma$.
The direct derivates of a regular map $\cal M$ are its images under $\Sigma$. The only transitive actions of $\Sigma$ are its regular action of degree $6$, its natural action of degree $3$, its action of degree $2$ as the quotient $S_3/A_3\cong S_2$, and its trivial action of degree $1$, so Wilson divided regular maps into classes I – IV respectively, as they have $6$, $3$, $2$ or $1$ derivates, admitting one of these four actions. The aim of this paper is to give constructions for the apparently rare maps in class III: these are the regular maps with just two direct derivates, so that they are invariant under the subgroup $\Sigma^+\cong C_3$ of $\Sigma$ generated by the triality $DP$ (or by its inverse $PD$), but not under $\Sigma$.
If $M$ is a normal subgroup of $\Gamma$, corresponding to a regular map $\cal M$, then let $$M_{\Sigma}=\bigcap_{\sigma\in\Sigma}\,\sigma(M) \quad{\rm and}\quad
M_{\Sigma^+}=\bigcap_{\sigma\in\Sigma^+}\,\sigma(M),$$ the largest $\Sigma$-invariant (i.e. characteristic) and $\Sigma^+$-invariant normal subgroups of $\Gamma$ contained in $M$. Let ${\cal M}_{\Sigma}$ and ${\cal M}_{\Sigma^+}$ be the corresponding maps, the smallest $\Sigma$- and $\Sigma^+$-invariant regular maps covering $\cal M$. Then $\cal M$ is in class IV if and only if ${\cal M}={\cal M}_{\Sigma}$. or equivalently $M=M_{\Sigma}$, and $\cal M$ is in class III if and only if ${\cal M}={\cal M}_{\Sigma^+}\neq{\cal M}_{\Sigma}$, or equivalently $M=M_{\Sigma^+}>M_{\Sigma}$.
In order to consider the orientability of class III maps, we need to understand the action of $\Sigma$ on the subgroups $\Delta$ of index $2$ in $\Gamma$. There are seven of these, each containing the commutator subgroup $\Gamma'$, which has quotient $\Gamma/\Gamma'\cong V_8$. Each such subgroup $\Delta$ is determined by which proper subset of the generators $R_0, R_1$ and $R_2$ it contains: we write $\Delta=\Gamma_{\emptyset}$, $\Gamma_i$ or $\Gamma_{ij}$ if it contains none of them, just $R_i$, or just $R_i$ and $R_j$ with $i<j$. Thus $\Gamma_{\emptyset}$ is the even subgroup $\Gamma^+$, consisting of those elements of $\Gamma$ which are words of even length in the generators $R_i$. Both $\Sigma$ and $\Sigma^+$ leave $\Gamma_{02}$ invariant, and have orbits $\{ \Gamma_{\emptyset},\Gamma_0, \Gamma_2\}$ and $\{\Gamma_1, \Gamma_{01},\Gamma_{12}\}$ on the remaining six subgroups. The groups $\Delta$ in the first of these two orbits satisfy $\Delta_{\Sigma}=\Delta_{\Sigma^+}=\Gamma'$, and those in the second satisfy $\Delta_{\Sigma}=\Delta_{\Sigma^+}=\Gamma^*:=\langle\Gamma', R_1\rangle$, subgroups of index $8$ and $4$ in $\Gamma$ respectively.
Each $\Sigma^+$-invariant regular map $\cal M$ has type $\{n,n\}_n$ for some $n$. If $\cal M$ is also orientable and without boundary, then $n$ is even
[*Proof.*]{} If $\cal M$ is $\Sigma^+$-invariant, then since $\Sigma^+$ transforms the vertices of $\cal M$ into its faces and Petrie polygons, these all have the same valency $n$, and thus $\cal M$ has type $\{n,n\}_n$. If $\cal M$ is also orientable and without boundary, then the corresponding map subgroup $M$ is contained in $\Gamma^+$. Being $\Sigma^+$-invariant, $M$ is contained $\Gamma^+_{\Sigma^+}=\Gamma'$. Now $R_0R_1$, $R_1R_2$ and $R_0R_1R_2$ have images of order $2$ in $\Gamma/\Gamma'$, and hence the order $n$ of their images in $\Gamma/M\cong {\rm Aut}\,{\cal M}$ is even. $\square$
If a regular map $\cal M$ has type $\{n,n\}_n$ and automorphism group $G$, then it has $|G|/2n$ vertices, $|G|/4$ edges and $|G|/2n$ faces, so it has Euler characteristic $$\chi=|G|\Bigl(\frac{1}{2n}-\frac{1}{4}+\frac{1}{2n}\Bigr)=|G|\frac{4-n}{4n}.$$ In particular, this applies if $\cal M$ is regular and $\Sigma^+$-invariant, in which case all direct derivates of $\cal M$ are on homeomorphic surfaces. These have genus $1-\frac{1}{2}\chi$ or $2-\chi$ as they are orientable or non-orientable.
Generalising Wilson’s example
=============================
Wilson’s example in [@Wil] is a dual pair $\cal W$ and $D({\cal W})$ of maps of type $\{9,9\}_9$ in class III with automorphism group $L_2(8)$. Here we will show that there are similar examples based on the group $L_2(2^e)$ whenever $e$ is divisible by $3$. Our constructions use properties of finite fields; see [@LN] for background.
Let $K$ be the field $F_q$ of order $q=2^e$, and let $F$ be the subfield $F_r$ where $r=2^f$ and $e=3f$. Then ${\rm Gal}\,K/F$ is a cyclic group of order $3$, generated by the automorphism $\alpha: x\mapsto x^r$. Let us call an element $x\in K$ a [*useful generator*]{} if it generates the field $K$ and satisfies $x+x^r+x^{r^2}=0$
There exists a useful generator $x\in K$.
[*Proof.*]{} The trace map ${\rm Tr}_{K/F}: x\mapsto x+x^r+x^{r^2}$ is a homomorphism $K\to F$ of additive groups, so its kernel has order at least $|K|/|F|=2^{2e/3}$. (In fact we have equality here, since the polynomial ${\rm Tr}_{K/F}(x)$ has at most $r^2=2^{2e/3}$ roots in $K$, but we do not need this.) The elements which do not generate $K$ as a field are those contained in maximal subfields of $K$, and there is one of these, of order $2^{e/p}$, for each prime $p$ dividing $e$. If there are $k$ such primes (including $3$) then $e\geq 3.2^{k-1}$ and so $k\leq 1+\log_2(e/3)$. Each subfield has order at most $2^{e/2}$, so the number of such non-generators is at most $2^{e/2}(1+\log_2(e/3))$. The result is therefore proved if $2^{2e/3}>2^{e/2}(1+\log_2(e/3))$, that is, $2^{e/6}>1+\log_2(e/3)$, and this is true for all $e>6$. The cases $e=3$ and $6$ are easily dealt with by more careful counting arguments (alternatively, see Example 3A). $\square$
The above method of proof has the disadvantages of not being constructive, and of not dealing with small values of $e$. The following constructive proof deals with all cases where $e$ is divisible by $3$ but not by $9$.
[**Example 3A.**]{} Suppose that $e=3f$ where $f$ is not divisible by $3$. There exists $t\in E=F_8\leq K$ such that $t^3+t+1=0$. Now $t^7=1$, so $t^{2^i}=t, t^2$ or $t^4$ as $i\equiv 0, 1$ or $2$ mod $(3)$. Since $r=2^f$ with $f\not\equiv 0$ mod $(3)$ we have ${\rm Tr}_{K/F}(t)=t+t^r+t^{r^2}=t+t^2+t^4=0$. Now let $x=tu$ where $u$ generates the field $F$. Then ${\rm Tr}_{K/F}(x)=x+x^r+x^{r^2}=tu+(tu)^r+(tu)^{r^2}=(t+t^r+t^{r^2})u=0$ since $u^{r^2}=u^r=u$. The multiplicative order of $u$ divides $2^f-1$, and is therefore coprime to $7$, whereas $t$ has order $7$, so $t$ and $u$ are powers of $x$. Since $t$ and $u$ generate the fields $E$ and $F$, and these subfields generate $K$, it follows that $x$ generates $K$.
For each positive integer $e$ divisible by $3$ there is a regular map in class III with automorphism group $L_2(2^e)$.
[*Proof.*]{} Let $q=2^e$ and let $x$ be a useful generator of $K=F_q$. We map the generators $R_i$ of $\Gamma$ into $G=L_2(q)=SL_2(K)$ by
$$R_0\mapsto r_0=\Big(\,\begin{matrix}1&x\cr 0&1\end{matrix}\,\Big),
\quad
R_1\mapsto r_1=\Big(\,\begin{matrix}1&0\cr 1&1\end{matrix}\,\Big)
\quad{\rm and}\quad
R_2\mapsto r_2=\Big(\,\begin{matrix}1&x^r\cr 0&1\end{matrix}\,\Big).$$ Since $r_0$, $r_1$, $r_2$ and $$r_0r_2=\Big(\,\begin{matrix}1&x+x^r\cr 0&1\end{matrix}\,\Big)
=\Big(\,\begin{matrix}1&x^{r^2}\cr 0&1\end{matrix}\,\Big)$$ all have order $2$, this mapping extends to a homomorphism $\theta:\Gamma\to G$. In order to show that $\theta$ is an epimorphism we need to show that no maximal subgroup of $G$ contains $r_0, r_1$ and $r_2$. In [@Dic Ch. XII], Dickson classified the subgroups of the groups $L_2(q)$ for all prime powers $q$, and in our case we see that the maximal subgroups of $G$ are the following:
1. normalisers of Sylow $2$-subgroups, isomorphic to the general affine group $AGL_1(q)$;
2. dihedral groups of order $2(q\pm 1)$;
3. subgroups conjugate to $SL_2(q')$ where $q'=2^{e/p}$ for primes $p$ dividing $e$.
Maximal subgroups of the first type are eliminated because their elements of order $2$ all commute, whereas $r_0$ and $r_1$ do not. A dihedral group of order $2n$, for odd $n$, does not contain a Klein four-group such as $\langle r_0, r_2\rangle$, so maximal subgroups of the second type are also eliminated. Since $$r_0r_1=\Big(\,\begin{matrix}x+1&x\cr 1&1\end{matrix}\,\Big),$$ has trace $x$ generating $K$, groups of the third type are eliminated. Thus $r_0, r_1$ and $r_2$ generate $G$, so $\theta$ is an epimorphism. The subgroup $M={\rm ker}\,\theta$ of $\Gamma$ is therefore the map subgroup corresponding to a regular map ${\cal M}={\cal M}(x)$ with ${\rm Aut}\,{\cal M}\cong G$.
The automorphism $\alpha$ of order $3$ of $K$ induces an automorphism of $G$ fixing $r_1$ and permuting $r_0$, $r_2$ and $r_0r_2$ in a $3$-cycle $(r_0, r_2, r_0r_2)$, so $\cal M$ is invariant under $\Sigma^+$. If $\cal M$ were invariant under $\Sigma$ then there would be an automorphism of $G$ fixing $r_1$ and transposing $r_0$ and $r_2$. Now ${\rm Aut}\,G\cong \Sigma L_2(K)$, an extension of $G$ by ${\rm Gal}\,K\cong C_e$, and the traces $x$, $x^r$ and $x^{r^2}$ of $r_0r_1$, $r_2r_1$ and $r_0r_2r_1$ each generate the field $K$, so no automorphism of $G$ can transpose $r_0r_1$ and $r_2r_1$ while fixing $r_0r_2r_1$. It follows that $\cal M$ is not invariant under $\Sigma$, so it is in class III. $\square$
The map ${\cal M}(x)$ constructed here has type $\{n,n\}_n$, where $n$ is the order of the element $r_0r_1$ in $G=L_2(K)$. This can be determined from the trace $x$ of this element. If $K$ is any field of characteristic $2$, then each non-identity element of $L_2(K)$ is conjugate to a matrix $$A=\Big(\,\begin{matrix}0&1\cr 1&x\end{matrix}\,\Big),$$ where $x$ is its trace. If we write $$A^n=\Big(\,\begin{matrix}a_n&b_n\cr c_n&d_n\end{matrix}\,\Big),$$ then the equation $A^n=A^{n-1}A$ gives $$a_n=b_{n-1},\quad b_n=a_{n-1}+xb_{n-1},\quad c_n=d_{n-1},\quad d_n=c_{n-1}+xd_{n-1}.$$ These, and the initial condition $A^0=I$, imply that the sequence $(d_n)$ is the solution of the recurrence relation $$f_n=f_{n-2}+xf_{n-1}$$ satisfying the initial conditions $f_0=1$ and $f_1=x$, and that $$b_n=c_n=d_{n-1}\quad{\rm and}\quad a_n=d_{n-2}.$$ The sequence defined by $t_n:={\rm Tr}\,A^n=a_n+d_n$ is therefore the solution of the recurrence relation (2) with initial conditions $f_0=0$ and $f_1=x$. This enables the successive powers of $A$, and in particular their traces, to be computed efficiently, so that one can determine the order of $r_0r_1$ and hence the type of $\cal M$.
[**Example 3B.**]{} Let $e=3$. Then $F=F_2$ and we can define $K=F_8$ as $F_2[t]/(t^3+t+1)$, so ${\rm Tr}_{K/F}\,(t)=t+t^2+t^4=0$ and the kernel of ${\rm Tr}_{K/F}$ is $\{0, t, t^2, t^4\}$. Taking $x=t$ as a useful generator for $K$, the proof of Theorem 3.2, gives $$r_0r_1=\Big(\,\begin{matrix}t+1&t\cr 1&1\end{matrix}\,\Big).$$ Applying the recurrence relation (1) we find that $(r_0r_1)^3$ has trace $t_3=1$, so it has order $3$ and hence $r_0r_1$ has order $9$. Thus the corresponding map ${\cal M}={\cal M}(x)$ and its dual $D({\cal M})$ have type $\{9,9\}_9$, so by equation (1) they have characteristic $\chi=-70$.
As we will now show, this map $\cal M$ is isomorphic to the class III map $\cal W$ described by Wilson in [@Wil]. His construction is completely different from ours, and it does not seem to generalise so easily to give other class III maps. He starts with the non-orientable regular map $\cal N$ denoted in [@CM] by $\{3,7\}_9$, since it is the largest regular map of this type. The opposite map $DPD({\cal N})$ is the non-orientable regular map $\{3,9\}_7$ of this corresponding type. Having defined an operation $H_2$, applicable to maps of odd vertex valency, which preserves the embedded graph but replaces the rotation $r_1r_2$ of edges around each vertex with its square, he notes that the regular map ${\cal W}=H_2(DPD({\cal N}))$ has type $\{9,9\}_9$. He then shows that ${\rm Aut}\,{\cal W}$ has generators $R$, $S$ and $T$ (corresponding to our $r_0r_1$, $r_2r_1$ and $r_0r_2r_1$), with defining relations $$R^9=S^9=T^9=(RS^3)^3=(ST^3)^3=(TR^3)^3=(SR^3)^7=(TS^3)^7=(RT^3)^7=1.$$ The $3$-cycle $(R,S,T)$ clearly induces an automorphism of ${\rm Aut}\,{\cal W}$ whereas the transposition $(R,S)$ does not, so $\cal W$ is in class III. Wilson does not specifically identify ${\rm Aut}\,{\cal W}$, but since $D$, $P$ and $H_2$ preserve automorphism groups we have ${\rm Aut}\,{\cal W}={\rm Aut}\,{\cal N}$, and the latter is identified by Coxeter and Moser in [@CM Table 8] as $LF(2,2^3)$, Dickson’s notation for $L_2(8)$. It is straightforward to check that the generators $r_i$ in Example 3B satisfy the relations corresponding to $(3)$, so ${\cal M}\cong{\cal W}$.
Alternatively, one can identify the maps in Example 3B with Wilson’s pair by using the character table and list of maximal subgroups of $L_2(8)$ in [@CCNPW], together with formula (7.3) of [@Ser], to show that the triangle group $$\Delta=\langle X, Y, Z\mid X^9=Y^2=Z^9=XYZ=1\rangle$$ has only two normal subgroups with quotient group $L_2(8)$. These lift back, through the epimorphism $R_1R_2\mapsto X$, $R_2R_0\mapsto Y$, $R_0R_1\mapsto Z$, to two normal subgroups of $\Gamma^+$ (which are, in fact, normal in $\Gamma$) corresponding to the canonical double covers of two regular maps of type $\{9,9\}$ with automorphism group $L_2(8)$. Any such map must arise in this way, so there are only two of them, namely $\cal W$ and its dual.
One can also verify this identification by using Conder’s lists of regular maps [@Con]. He used a computer to find all the orientable and non-orientable regular maps of characteristic $\chi=-1, \ldots, -200$ (those with $\chi\geq 0$ are well-known, see [@CM Ch. 8] for instance, and there are none in class III). In his notation, an entry R$g.i$ or N$g.i$ denotes the $i$-th orientable or non-orientable regular map (or dual pair of maps) of genus $g$, lexicographically ordered by their type $\{p,q\}$ with $p\leq q$. There is only one entry corresponding to regular maps of characteristic $-70$ and type $\{9,9\}$, namely the dual pair N72.9 of type $\{9,9\}_9$; their canonical double covers are the pair R71.15 of type $\{9,9\}_{18}$. The map ${\cal N}=\{3,7\}_9$ used by Wilson is Conder’s N8.1, while $DPD({\cal N})$ is N16.1.
[(a)]{} The only regular maps in class III with automorphism group $L_2(q)$ for $q=2^e\geq 4$ are the maps ${\cal M}(x)$ constructed in Theorem 3.2, together with their dual maps $D({\cal M}(x))$, where $e$ is divisible by $3$ and $x$ is a useful generator for the field $K=F_q$. [(b)]{} Two such maps ${\cal M}(x_1)$ and ${\cal M}(x_2)$ are isomorphic if and only if $x_1$ and $x_2$ are conjugate under ${\rm Gal}\,K$. [(c)]{} There are no isomorphisms between maps ${\cal M}(x_1)$ and dual maps $D({\cal M}(x_2))$.
[*Proof.*]{} (a) Let $\cal M$ be a regular map in class III with ${\rm Aut}\,{\cal M}\cong G=L_2(q)$, corresponding to an epimorphism $\Gamma\to G$, $R_i\mapsto r_i$. Since $G$ is neither dihedral nor cyclic, each $r_i$ is an involution. All involutions in $G$ are conjugate, so by composing $\theta$ with an inner automorphism of $G$ we may assume that $$r_1=\Big(\,\begin{matrix}1&0\cr 1&1\end{matrix}\,\Big).$$ Now $r_0$ and $r_2$ are commuting involutions. The centraliser of any involution in $G$ is the unique Sylow $2$-subgroup containing it, so $r_0$ and $r_2$ lie in a Sylow $2$-subgroup $S$, which must be distinct from the Sylow $2$-subgroup $T=C_G(r_1)$ containing $r_1$. Each Sylow $2$-subgroup acts regularly by conjugation on the $q$ others in $G$, so by further conjugating with a unique element of $T$ we may also assume that $$r_0=\Big(\,\begin{matrix}1&x\cr 0&1\end{matrix}\,\Big)
\quad{\rm and}\quad
r_2=\Big(\,\begin{matrix}1&y\cr 0&1\end{matrix}\,\Big),
\quad{\rm giving}\quad
r_0r_2=\Big(\,\begin{matrix}1&x+y\cr 0&1\end{matrix}\,\Big),$$ for some $x, y\in K$. Indeed, this argument shows that there is a [*unique*]{} inner automorphism of $G$ taking the original generators $r_0, r_1$ and $r_2$ to this form.
Since $\cal M$ is in class III there is an automorphism $\alpha$ of $G$ fixing $r_1$ and permuting $r_0, r_2$ and $r_0r_2$ in a $3$-cycle. Now ${\rm Aut}\,G=\Sigma L_2(q)$, a semidirect product of $G$ by ${\rm Gal}\,K$. The automorphisms fixing $r_1$ form the semidirect product of $T$ by ${\rm Gal}\,K$, and the above uniqueness result shows that such an automorphism preserves $S$ if and only if it is in ${\rm Gal}\,K$. Since $\alpha$ has order $3$ it follows that $e=3f$ for some $f$, and $\alpha$ acts on $K$ by $t\mapsto t^r$ or $t\mapsto t^{r^2}$, where $r=2^f$. Suppose first that $\alpha: t\mapsto t^r$. Then $y=x^r$ and $x+y=x^{r^2}$, so $x+x^r+x^{r^2}=0$. If $K_0$ is the subfield of $K$ generated by $x$ then $y\in\alpha(K_0)=K_0$, so $r_0, r_1, r_2\in L_2(K_0)$; however, these elements generate $G$, so $K_0=K$. Thus $x$ is a useful generator of $K$, and ${\cal M}\cong{\cal M}(x)$. We obtain a similar conclusion if $\alpha:t\mapsto t^{r^2}$, except that now $y=x^{r^2}$ and $x+y=x^r$; this differs from the preceding case by inverting $\alpha$, or equivalently by transposing $r_0$ and $r_2$, so in this case ${\cal M}\cong D({\cal M}(x))$.
\(b) This condition is sufficient, since an automorphism of $K$ taking $x_1$ to $x_2$ will give an automorphism of $G$ inducing an isomorphism from ${\cal M}(x_1)$ to ${\cal M}(x_2)$. The uniqueness result in the proof of (a) also shows that this condition is necessary.
\(c) The set of maps ${\cal M}(x)$ is distinguished from the set of their dual maps by the fact that the field automorphism inducing the triality sending $r_0$ to $r_2$ is respectively given by $t\mapsto t^r$ or its inverse, so there can be no isomorphisms between these two sets of maps. $\square$
Since ${\rm Gal}\,K$ has order $e$, and acts wthout fixed points of the set of useful generators of $K$, we immediately have the following corollary:
If $q=2^e$ with $e$ divisible by $3$, and $N_e$ is the number of useful generators of $K=F_q$, then there are, up to isomorphism, $2N_e/e$ regular maps in class III with automorphism group $L_2(q)$, forming $N_e/e$ dual pairs. $\square$
[**Example 3C.**]{} Let $e=6$, so that $G=L_2(64)$. The field $K=F_{64}$ has two maximal subfields $E=F_8=F_2[t]/(t^3+t+1)$ and $F=F_4=F_2[u]/(u^2+u+1)$. There are $2^{2e/3}=16$ elements $x$ in the kernel $Z$ of the homomorphism ${\rm Tr}_{K/F}:x\mapsto x+x^4+x^{16}$. If $x\in F$ then $x=x^4=x^{16}$, so only $x=0$ is in $Z$. If $x\in E$ then $x^8=x$ so $x^{16}=x^2$ and hence ${\rm Tr}_{K/F}(x)=x+x^2+x^4={\rm Tr}_{E/F_2}(x)$, giving $2^2=4$ elements of $Z$ (namely $0, t, t^2$ and $t+t^2$). Thus there are $N_6=16-4=12$ useful generators $x$ for $K$, giving $N_6/6=2$ dual pairs of class III maps with automorphism group $G$.
In order to construct such a map we can take $x=tu$ as a useful generator for $K$, as in Example 3A. Applying the recurrence relation (2), and using $t^3=t+1$ and $u^2=u+1$, we eventually find that $(r_0r_1)^{13}$ has trace $t_{13}=u+1$, so it is conjugate to an element of order $5$ in $L_2(F)$; since $t_5=t^2+u+t^2u\neq 0$ we have $(r_0r_1)^5\neq I$, so it follows that $r_0r_1$ has order $65$. Thus the resulting dual pair of class III maps have type $\{65,65\}_{65}$, and since they have automorphism group $G$ of order $63.64.65$, they have characteristic $\chi=-61488$ by equation (1). Alternatively, since ${\rm Tr}_{K/F}(t^2)={\rm Tr}_{K/F}(tu)=0$ we can take $x=t^2+tu$ as a useful generator. In this case $t_7=t^2+t$, which is the trace of an element of order $9$ in $L_2(E)$, so $r_0r_1$ has order $63$. This gives the second dual pair of class III maps, of type $\{63,63\}_{63}$ and characteristic $-61360$.
We can generalise the counting argument in Example 3C, as follows. Let $Z$ be the kernel of ${\rm Tr}_{K/F}$, the set of zeros of $x+x^r+x^{r^2}$, so $|Z|=2^{2e/3}$. An element $x\in K$ is a useful generator for $K$ if it is in $Z$ but not in any proper subfield $L$ of $K$. The automorphism $\alpha:x\mapsto x^r$ of $K$ has fixed field $F$, so ${\rm Tr}_{K/F}(x)=x$ for all $x\in F$; thus if $L$ is a subfield of $F$ then $L\cap Z=\{0\}$. If $L$ is not a subfield of $F$, then $\alpha$ induces an automorphism of order $3$ of $L$ with fixed field $M=L\cap F$, so ${\rm Tr}_{K/F}(x) = {\rm Tr}_{L/M}(x)$ for all $x\in L$, giving $|L\cap Z|=|L|/|M|=|L|^{2/3}$. Now $K$ has one subfield $L$, of order $2^d$, for each $d$ dividing $e$, and $L$ is a subfield of $F$ if and only if $d$ divides $f$. Möbius inversion over the lattice of subfields of $K$, partitioned into those contained or not contained in $F$, gives $$N_e=\sum_{d\mid e}\mu\Bigl(\frac{e}{d}\Bigr)|L\cap Z|=
\sum_{d\mid e,\, d\negthinspace\not\;\mid f}\mu\Bigl(\frac{e}{d}\Bigr)2^{2d/3}
+\sum_{d\mid f}\mu\Bigl(\frac{e}{d}\Bigr)$$ where $\mu$ is the Möbius function [@HW §§16.3–4]. Putting $c=e/d$ we can write this as $$N_e=\sum_{c\mid e'}\mu(c)2^{2e/3c}+\sum_{3\mid c\mid e}\mu(c)$$ where $e=3^ie'$ with $(e',3)=1$. Putting $c=3b$ in the second sum gives $$\sum_{3\mid c\mid e}\mu(c)=\sum_{b\mid f}\mu(3b)
=-\sum_{3\negthinspace\not\;\mid b\mid f}\mu(b)
=-\sum_{b\mid e'}\mu(b),$$ and this is $-1$ or $0$ as $e'=1$ or $e'>1$, that is, as $e$ is or is not a power of $3$. We have therefore proved:
Let $e=3^ie'$ with $e'$ coprime to $3$. Then $$N_e=\sum_{c\mid e'}\mu(c)2^{2e/3c}-\nu_e$$ where $\mu$ is the Möbius function and $\nu_e=1$ or $0$ as $e'=1$ or $e'>1$. $\square$
[**Example 3D.**]{} Together with Corollary 3.4, this shows that if $e=3, 6, 9, 12$ or $15$ then the number of dual pairs of class III maps with automorphism group $L_2(2^e)$ is respectively $1, 2, 7, 20$ or $68$. If $e=3^i>3$ we have $N_e=2^{2e/3}-1$, giving $(2^{2e/3}-1)/e$ such pairs.
The argument used to prove Lemma 3.1 shows that the formula in Proposition 3.5 is dominated by the term with $c=1$. Thus $N_e\sim 2^{2e/3}=q^{2/3}$ as $e\to\infty$, so the number $N_e/e$ of dual pairs in Corollary 3.4 is asymptotic to $q^{2/3}/e$, which grows quite rapidly. This shows that one can find arbitrarily many class III pairs with the same automorphism group $G=L_2(2^e)$ by taking $e$ sufficiently large. These maps have type $\{p,p\}_p$ where $p$ is the order of $r_0r_1$; now the orders of the elements of $G$ are $2$ and the divisors of $q\pm 1$, so the number of distinct orders is $d(q-1)+d(q+1)$ where $d(n)$ is the number of divisors of an integer $n$. As $n\to\infty$ this function grows more slowly than any positive power of $n$ (see [@HW Theorem 315], for instance), so by taking $e$ sufficiently large one can find arbitrarily many class III pairs with automorphism group $G$, and with mutually distinct types.
Maps in class III as coverings
==============================
Another method of finding maps in class III is to construct them as coverings of known maps in class III: for instance, several of the examples found by Conder in [@Con06] are coverings of Wilson’s maps $\cal W$ and $D({\cal W})$. We start with a normal subgroup $N$ of $\Gamma$, corresponding to a regular map $\cal N$ in class III, and we try to find subgroups $M$ of $N$, corresponding to maps $\cal M$ which cover $\cal N$, such that $\cal M$ is also in class III.
Let $N$ be the map subgroup of $\Gamma$ corresponding to a map in class III, and let $M$ be a characteristic subgroup of $N$ such that $N/M$ is a characteristic subgroup of $\Gamma/M$. Then the map $\cal M$ corresponding to $M$ is in class III.
[*Proof.*]{} Since $M$ is a characteristic subgroup of $N$, and $N$ is a $\Sigma^+$-invariant normal subgroup of $\Gamma$, it follows that $M$ is a $\Sigma^+$-invariant normal subgroup of $\Gamma$. The corresponding map $\cal M$ is therefore regular and invariant under $\Sigma^+$. If $\Sigma$ leaves $M$ invariant, it induces a group of automorphisms of $\Gamma/M$ which must leave the characteristic subgroup $N/M$ invariant. Thus $\Sigma$ must leave $N$ invariant, which is false, so $M$ is not $\Sigma$-invariant and the corresponding map $\cal M$ is in class III. $\square$
Now $N$ is the fundamental group of a punctured surface, namely the underlying surface of $\cal N$, with punctures at the vertices and face centres, so it is a free group of finite rank $r$ (equal to $V+F+2g-1$ if $\cal N$ is orientable, of genus $g$, and $V+F+g-1$ if it is non-orientable of genus $g$). For any integer $n\geq 2$, the subgroup $M=N'N^n$ of $N$ generated by its commutators and $n$th powers is a characteristic subgroup of index $n^r$ in $N$. In most cases, and certainly if $n$ is coprime to $|\Gamma:N|$, $N/M$ is a characteristic subgroup of $\Gamma/M$, so Lemma 4.1 gives a class III map $\cal M$ corresponding to $M$. In cases where ${\rm Aut}\,{\cal N}$ is a non-abelian simple group (as when $\cal N$ is Wilson’s map $\cal W$, for instance), this is true for all $n$, since $N/M$ is the only normal subgroup of $\Gamma/M$ with a non-abelian simple quotient group.
If we replace $\Gamma$ with the appropriate extended triangle group by adding the relations $(R_0R_1)^p=(R_1R_2)^q=1$, where $\cal N$ has type $\{p,q\}$, then $N$ becomes a surface group, of rank $r=2g$ or $g$ as $\cal N$ is orientable or not, and a similar construction can be used.
Choosing different values of $n$ gives infinitely many class III maps $\cal M$ as coverings of $\cal N$. Alternatively one could iterate this process, finding a suitable characteristic subgroup of $M$, and so on. Taking $M$ to be a subgroup of infinite index, such as $N'$, gives infinite maps in class III by the same argument.
Another way of constructing class III maps as coverings is to start with the map subgroup $N$ corresponding to a class III map $\cal N$, and to take its intersection $M=N\cap K$ with a $\Sigma^+$-invariant normal subgroup $K$ of $\Gamma$ such that $N/M$ is a characteristic subgroup of $\Gamma/M$. Since $N$ and $K$ are normal and $\Sigma^+$-invariant, so is $M$. If $M$ were $\Sigma$-invariant then $\Sigma$ would induce a group of automorphisms of $\Gamma/M$ preserving $N/M$, so $N$ would be $\Sigma$-invariant, which is false; thus $M$ is not $\Sigma$-invariant, so it corresponds to a map $\cal M$ in class III. For example, if $K=\Gamma_{02}$, $\Gamma^*$ or $\Gamma'$, of index $2$, $4$ or $8$, so that $N/M\cong NK/K$ is an elementary abelian $2$-group, and if ${\rm Aut}\,{\cal N}$ has no non-trivial normal $2$-subgroups, then $N/M$ is a characteristic subgroup of $\Gamma/M$, as required.
[**Example 4A.**]{} By taking ${\cal N}={\cal W}$ or $D({\cal W})$ and $K=\Gamma_{02}$ we obtain a pair of maps $\cal M$ in class III which are non-orientable double covers of Wilson’s maps; these are the dual pair N198.6 of type $\{18,18\}_{18}$ and characteristic $-196$ in [@Con], with automorphism group $L_2(8)\times C_2$. If, instead, we take $K=\Gamma^*$ we obtain a dual pair of non-orientable maps of type $\{18,18\}_{18}$ and characteristic $-392$ with automorphism group $L_2(8)\times V_4$, whereas taking $K=\Gamma'$ gives a dual pair of orientable maps of type $\{18,18\}_{18}$ and characteristic $-784$ with automorphism group $L_2(8)\times V_8$.
On the other hand, $\Gamma^+$ is not $\Sigma^+$-invariant, so the canonical double cover of a non-orientable class III map, corresponding to $M=N\cap\Gamma^+$, need not be in class III: Again, Wilson’s maps $\cal W$ and $D({\cal W})$ illustrate this, having canonical double covers in class I, namely the dual pair R71.15 in [@Con].
Maps in class III as parallel products
======================================
Let $\cal N$ be a regular map in class I, corresponding to a normal subgroup $N$ of $\Gamma$. Since $\cal N$ has six direct derivates, $N$ is one of an orbit $\{N_1,\ldots, N_6\}$ of six normal subgroups of $\Gamma$ under the action of $\Sigma$, all with quotient group $\Gamma/N_i\cong G={\rm Aut}\,{\cal N}$. We can choose the numbering so that $N_1=N$, and $N_2$ and $N_3$ are the remaining images of $N$ under $\Sigma^+$. The subgroup $M=N_{\Sigma^+}=N_1\cap N_2\cap N_3$ is then normal in $\Gamma$, and has finite index in $\Gamma$ provided $N$ has finite index, so it corresponds to a regular map ${\cal M}={\cal N}_{\Sigma^+}$ with ${\rm Aut}\,{\cal M}\cong\Gamma/M$. In Wilson’s terminology [@Wil94], $\cal M$ is the parallel product of the maps $\cal N$, $DP({\cal N})$ and $PD({\cal N})$; if $\cal N$ has type $\{p,q\}_r$ then $\cal M$ has type $\{n,n\}_n$ where $n={\rm lcm}\{p,q,r\}$. Clearly $M$ is $\Sigma^+$-invariant, and it is $\Sigma$-invariant if and only if it coincides with $N_{\Sigma}=N_1\cap\cdots\cap N_6$, so $\cal M$ is in class III or IV as $M>N_{\Sigma}$ or $M=N_{\Sigma}$ respectively. There are many cases where the former happens, and the following simple (and presumably well-known) lemma provides a sufficient condition for this.
Let $H$ be a group with distinct normal subgroups $K_1,\ldots, K_k$ such that each $H/K_i$ is isomorphic to a non-abelian simple group $G_i$. Then the normal subgroup $K=K_1\cap\cdots\cap K_k$ of $H$ has quotient $H/K\cong G_1\times\cdots\times G_k$.
[*Proof.*]{} We use induction on $k$. There is nothing to prove if $k=1$, so assume that $k>1$ and that the result has been proved for $k-1$ normal subgroups. Let $L=K_1\cap\cdots\cap K_{k-1}$. We first need to show that $L\not\leq K_k$, so suppose that $L\leq K_k$. Then $K_k/L$ is a maximal normal subgroup of $H/L$; however, $H/L\cong G_1\times\cdots\times G_{k-1}$ by the induction hypothesis, and since the groups $G_i$ are non-abelian and simple the maximal normal subgroups of $G_1\times\cdots\times G_{k-1}$ are just the $k-1$ products of all but one of the direct factors [@Hup I.9.12(b)], corresponding to the subgroups $K_i/L$ of $H/L$ for $i=1,\ldots, k-1$; thus $K_k/L=K_i/L$ and hence $K_k=K_i$ for some $i<k$, against the hypotheses. Thus $L\not\leq K_k$, so $H=K_kL$ since $K_k$ is a maximal normal subgroup of $H$; since $K=L\cap K_k$ we therefore have $$H/K = K_k/K\times L/K \cong H/L\times H/K_k \cong
(G_1\times\cdots\times G_{k-1})\times G_k,$$ as required. $\square$
This result cannot be extended to arbitrary quotient groups $G_i$: for instance, if $p$ is prime and $e>1$ then an elementary abelian group $H=C_p\times\cdots\times C_p$ of order $p^e$ has $k=(p^e-1)/(p-1) > e$ normal subgroups $K_i$ with quotient $H/K_i\cong C_p$ and intersection $K=1$. In general, all one can conclude is that the projections $H/K\to H/K_i$ induce an embedding of $H/K$ in $G_1\times\cdots\times G_k$.
A group $A$ is [*almost simple*]{} if it has a non-abelian simple normal subgroup $S$ such that $C_A(S)=1$, so that $A$ is embedded in ${\rm Aut}\,S$. Then $S$ is the unique minimal normal subgroup of $A$, since any other would centralise $S$. If $A$ is finite then $A/S$ is solvable by the Schreier Conjecture, which asserts that if $S$ is any non-abelian finite simple group then ${\rm Out}\,S$ is solvable; the general proof depends on the classification of finite simple groups (see [@Gor Theorem 1.46], for instance), though in any specific case it can be verified directly.
If $\cal N$ is a regular map in class I such that ${\rm Aut}\,{\cal N}$ is almost simple then the map ${\cal M}={\cal N}_{\Sigma^+}$ is in class III.
[*Proof.*]{} Let $\cal N$ correspond to an epimorphism $\theta:\Gamma\to A={\rm Aut}\,{\cal N}$ with kernel $N$, and let $M=N_{\Sigma^+}$. It is sufficient to show that $M\neq N_{\Sigma}$, so suppose for a contradiction that $M=N_{\Sigma}$. Let $T=\theta^{-1}(S)$, where $S$ is the unique non-abelian simple normal subgroup of $A$, and let $N_i$ and $T_i$, for $i=1,\ldots, 6$, be the images of $N$ and $T$ under $\Sigma$, numbered as above. Thus $T_i$ is a normal subgroup of $\Gamma$ containing $N_i$, with $\Gamma/T_i\cong A/S$ solvable and $T_i/N_i\cong S$ non-abelian and simple. Now $T_{\Sigma^+}$ and $T_{\Sigma}$ are the smallest normal subgroups of $\Gamma$, respectively containing $N_{\Sigma^+}$ and $N_{\Sigma}$, with solvable quotient groups, so they are equal since $N_{\Sigma^+}=N_{\Sigma}$. Let $H=T_{\Sigma^+}=T_{\Sigma}$. The six normal subgroups $K_i=H\cap N_i$ of $H$ are all distinct, since if $K_i=K_j$ with $i\neq j$ then $N_iN_j/N_i$ is a non-trivial solvable normal subgroup of the almost simple group $\Gamma/N_i$, which is impossible. Now $M$ is the intersection of both three and six of these groups $K_i$, and they have non-abelian simple quotient groups $H/K_i\cong HN_i/N_i=T_i/N_i\cong S$, so Lemma 5.1 gives $S^3\cong H/M\cong S^6$, which is impossible. Thus $M\neq N_{\Sigma}$, so the corresponding map ${\cal M}={\cal N}_{\Sigma^+}$ is in class III. $\square$
If $\cal N$ is orientable then $\cal M$, which covers $\cal N$, is also orientable. If $A=S$, that is, ${\rm Aut}\,{\cal N}$ is simple, then $H=\Gamma$ and the above proof shows that ${\rm Aut}\,{\cal M} = S^3$. However, if $A>S$ then all one can conclude is that $S^3\unlhd {\rm Aut}\,{\cal M}\leq A^3$, and in general more work is required to identify ${\rm Aut}\,{\cal M}$ precisely (see Examples 5A and 5C).
There are many instances of maps $\cal N$ satisfying the hypotheses of Theorem 5.2, and here we give a few straightforward examples.
[**Example 5A.**]{} The smallest orientable map $\cal N$ satisfying the hypotheses of Theorem 5.2 is the map R3.1 in [@Con], of type $\{3,7\}_8$ and characteristic $-4$; it has automorphism group $A=PGL_2(7)$ with $S=L_2(7)$, the simple group of order $168$. (The underlying surface of $\cal N$ can be identified with Klein’s quartic curve of genus $3$, the Riemann surface of least genus attaining Hurwitz’s upper bound of $84(g-1)$ automorphisms for surfaces of genus $g\geq 2$.) Theorem 5.2 then gives an orientable class III map $\cal M$ of type $\{168,168\}_{168}$. Each $r_i$ is in $A\setminus S$, so $T=\theta^{-1}(S)$ is the group $\Gamma^+=\Gamma_{\emptyset}$ and hence $H=T_{\Sigma^+}=\Gamma'$ (see §2). The proof of Theorem 5.2 then shows that ${\rm Aut}\,{\cal M}=A^3=PGL_2(7)^3$. Since $|PGL_2(7)|=336$, equation (1) shows that $\cal M$ has characteristic $-9257472$. For the smallest non-orientable map $\cal N$ satisfying the hypotheses of Theorem 5.2, see the case $n=5$ of Example 5C.
[**Example 5B.**]{} Let $A$ be the simple group $L_2(p)=SL_2(p)/\{\pm 1\}$ for a prime $p\equiv 1$ mod $(4)$. Since $-1=i^2$ for some $i\in F_p$ we can define a homomorphism $\theta:\Gamma\to A$ by $$R_0\mapsto r_0=\pm\Big(\,\begin{matrix}0&i\cr i&0\end{matrix}\,\Big),
\quad
R_1\mapsto r_1=\pm\Big(\,\begin{matrix}i&i\cr 0&-i\end{matrix}\,\Big)
\quad{\rm and}\quad
R_2\mapsto r_2=\pm\Big(\,\begin{matrix}i&0\cr 0&-i\end{matrix}\,\Big).$$ Thus $\langle r_0, r_2\rangle$ is a Klein four-group, and $$r_1r_2=\pm\Big(\,\begin{matrix}1&-1\cr 0&1\end{matrix}\,\Big),$$ has order $p$. The maximal subgroups of $A$ are isomorphic to the unique subgroup of order $p(p-1)/2$ in $AGL_1(p)$, to dihedral groups of order $p\pm 1$, and (depending on $p$) to $A_4$, $S_4$ or $A_5$. If we take $p>5$ then only the first of these contain elements of order $p$, and these do not contain any Klein four-groups, so $r_0, r_1$ and $r_2$ generate $A$. Thus $\theta$ is an epimorphism, so its kernel $N$ corresponds to a regular map $\cal N$ with ${\rm Aut}\,{\cal N}\cong A$. Now $$r_0r_1=\pm\Big(\,\begin{matrix}0&1\cr -1&-1\end{matrix}\,\Big)
\quad{\rm and}\quad
r_0r_2r_1=\pm\Big(\,\begin{matrix}0&i\cr i&i\end{matrix}\,\Big),$$ with $i\neq\pm 1$ or $\pm 2$ for $p>5$, so these and $r_1r_2$ all have distinct traces. Since ${\rm Aut}\,A=PGL_2(p)$ preserves traces, it follows that $\cal N$ is in class I. We therefore obtain a non-orientable regular map $\cal M$ in class III with ${\rm Aut}\,{\cal M}\cong L_2(p)^3$ for each prime $p>5$ satisfying $p\equiv 1$ mod $(4)$. Since $r_0r_1$ and $r_1r_2$ have orders $3$ and $p$, $\cal N$ has type $\{3,p\}_r$ where $r$ is the order of $r_0r_1r_2$. It follows that $\cal M$ has type $\{n,n\}_n$ where $n$ is $pr$ or $3pr$ as $3$ does or does not divide $r$. In the smallest example, arising for $p=13$, we find that $r=7$, so $\cal M$ has type $\{273,273\}_{273}$ and characteristic $-320772816$; the map $\cal N$ is N51.1 in [@Con].
(The canonical double covers of the maps $\cal N$ used here are the regular triangular maps associated with the principal congruence subgroups of level $p$ in the modular group $PSL_2({\bf Z})$; these orientable maps have automorphism group $L_2(p)\times C_2$ for $p\equiv 1$ mod $(4)$.)
[**Example 5C.**]{} Let us define a homomorphism $\theta:\Gamma\to S_n$ for $n\geq 5$ by $$R_0\mapsto r_0=(1,n)(2,n-1)(3,n-2)\ldots,\quad R_1\mapsto r_1=(2,n)(3,n-1)(4,n-2)\ldots$$ and $$R_2\mapsto r_2=(1,n).$$ Here $r_0$ and $r_1$ are induced by two reflections of a regular $n$-gon with vertices $1, 2, \ldots, n$, while $r_2$ is chosen to commute with $r_0$. Then $$r_0r_1= (1,2,3,\ldots, n),\quad r_1r_2 = (1,n,2)(3,n-1)(4,n-2)\ldots$$ and $$r_0r_1r_2 = (1, 2,3,\ldots,n-1),$$ these permutations having orders $n$, $6$ and $n-1$. The $n$-cycle $r_0r_1$ and the transposition $r_2$ generate $S_n$, so $\theta$ is an epimorphism. The subgroup $N=\ker\theta$ of $\Gamma$ therefore corresponds to a regular map $\cal N$ of type $\{n,6\}_{n-1}$ with ${\rm Aut}\,{\cal N}\cong S_n$. If $n\neq 6$ then all automorphisms of $S_n$ are inner; now $r_0$, $r_2$ and $r_0r_2$ have different cycle-structures if $n\geq 7$, so they are mutually inequivalent under automorphisms of $S_n$, and hence $\cal N$ is in class I. The same applies for $n=5$ since then $\cal N$ has type $\{5,6\}_4$. If $n=6$, however, $\cal N$ is the map N62.3 of type $\{6,6\}_5$ in [@Con], and the outer automorphism of $S_6$, interchanging transpositions with products of three transpositions, induces a self-duality. Since $S_n$ is almost simple for $n\geq 5$ we therefore obtain a regular map ${\cal M}={\cal N}_{\Sigma^+}$ in class III for $n=5$ and for each $n\geq 7$.
Note that $r_0$ is even if and only if $n\equiv 0$ or $1$ mod $(4)$, that $r_1$ is even if and only if $n\equiv 1$ or $2$ mod $(4)$, and that $r_2$ is always odd. Thus the subgroup $T=\theta^{-1}(A_n)$ is $\Gamma_0$, $\Gamma_{01}$, $\Gamma_1$ or $\Gamma_{\emptyset}$ as $n\equiv 0, 1, 2$ or $3$ mod $(4)$. It follows from §2 that the subgroup $H=T_{\Sigma^+}$ is $\Gamma'$ if $n\equiv 0$ or $3$ mod $(4)$, and it is $\Gamma^*=\langle\Gamma', R_1\rangle$ if $n\equiv 1$ or $2$ mod $(4)$. In the first case $\cal M$ is orientable, with ${\rm Aut}\,{\cal M}=S_n^3$, whereas in the second case it is non-orientable, with ${\rm Aut}\,{\cal M}$ a subgroup of index $2$ in $S_n^3$. The smallest example, for $n=5$, is a non-orientable map of type $\{60,60\}_{60}$ and characteristic $-201600$; in this case $\cal N$ is a direct derivate of the map N5.1 of type $\{4,5\}_6$, an antipodal quotient of the map R4.2 of this type on Bring’s curve of genus $4$. (In fact $\cal N$ and its direct derivates are the smallest non-orientable maps satisfying the hypotheses of Theorem 5.2.)
[**Example 5D.**]{} In Example 5C let $n\equiv 1$ mod $(4)$, so that $r_0$ and $r_1$ are even. There are several choices of even permutations $r_2$ giving rise to class I maps $\cal N$ with ${\rm Aut}\,{\cal N}=A_n$, and hence to class III maps $\cal M$ with ${\rm Aut}\,{\cal M}=A_n^3$. For example, if $n>5$ and $$r_2=(1,n)(3,n-2)$$ then $r_1r_2=(1,n,2)(3,n-1,n-2,4)(5,n-3)\ldots$, so the elements $(r_1r_2)^4=(1,n,2)$ and $r_0r_1=(1,2,\ldots, n)$ generate $A_n$. Since $r_1r_2$ and $r_0r_1r_2=(1,2,n-2,n-1)(3,4,\ldots, n-3)$ have orders $12$ and $n-5$, $\cal N$ has type $\{n,12\}_{n-5}$. Thus $\cal N$ is in class I for $n\neq 17$, and if $n=17$ this follows from the fact that $r_0, r_2$ and $r_0r_2$ have different cycle structures. Since $A_n$ is simple for $n\geq 5$, Theorem 5.2 gives a non-orientable class III map $\cal M$ with ${\rm Aut}\,{\cal M}=A_n^3$. The smallest example, with $n=9$, has type $\{36,36\}_{36}$ and characteristic $-1327353495552000$.
As an alternative, if we take $r_2=(1,2)(n-1,n)$ with $n>5$ then $\cal N$ has type $\{n,10\}_{n-2}$ and is therefore in class I, with ${\rm Aut}\,{\cal N}=A_n$. If $n=9$, for instance, the resulting class III map $\cal M$ has type $\{630,630\}_{630}$ and characteristic $-1483791586099200$.
Maps of small genus in class III
================================
Each of the class III maps constructed in the preceding section has rather large genus, whereas it interesting to know the lowest genera which can arise. Conder’s lists of regular maps [@Con] give the type $\{p,q\}_r$ of each map, for characteristic $\chi=-1,\ldots, -200$. A map in class III must have $p=q=r$, so one can search through these lists for maps of such types and try to determine whether or not they are in class III. In the case of non-orientable maps, this leads to the following result:
The only non-orientable class III regular maps of characteristic $\chi\geq -200$ are Wilson’s dual pair $\cal W$ and $D({\cal W})$ with $\chi=-70$, and their double covers with $\chi=-196$ constructed in Example 4A.
[*Proof.*]{} In Conder’s list of non-orientable regular maps there are $21$ entries of type $\{p,p\}_p$, each representing a self-dual map or a dual pair. The information provided includes the multiplicities $mV$ and $mF$ of edges connecting adjacent pairs of vertices and adjacent pairs of faces. In a class III map these must be equal, since the map is invariant under an operation sending vertices to faces. One can therefore exclude any of these $21$ entries with $mV\neq mF$, namely N22.3, N86.15, N170.15, N182.10 and N200.22. Conder’s list also tells us whether or not an entry with $p=q$ represents a self-dual map: such a map cannot be in class III, and this further criterion excludes the entries N12.3, N35.3, N44.6, N50.8, N119.6, N119.7, N146.8, N162.9, N200.23 and N200.26. Each entry gives defining relations for the automorphism group $G={\rm Aut}\,{\cal M}$, in terms of generators $R=r_0r_1$, $S=r_1r_2$ and $T=r_1$. Adding the relations $[R,S]=[R,T]=[S,T]=1$ gives a presentation for the abelianisation $G^{\rm ab}=G/G'$, and hence a set of normal generators for $\theta^{-1}(G')$ where $\theta:\Gamma\to G$ is the epimorphism $R_i\mapsto r_i$ corresponding to $\cal M$. Since $G'$ is a characteristic subgroup of $G$, if $\cal M$ is in class III then $\theta^{-1}(G')$ must be a $\Sigma^+$-invariant subgroup of $\Gamma$, containing $\Gamma'$. As shown in §2, the only such subgroups are $\Gamma$, $\Gamma_{02}$, $\Gamma^*$ and $\Gamma'$, so if $\theta^{-1}(G')$ is not one of these then this entry can be excluded. This eliminates N62.4, N166.14 and N170.16. The remaining three entries are N72.9 of type $\{9,9\}_9$ corresponding to Wilson’s class III maps $\cal W$ and $D({\cal W})$, N198.6 of type $\{18,18\}_{18}$ corresponding to their class III double covers described in Example 4A, and N119.5 of type $\{7,7\}_7$ and characteristic $-117$, with automorphism group $L_2(13)$. This last dual pair are in fact in class II, having the self-dual map N119.6 as their third direct derivate; there is also another map N119.7 of type $\{7,7\}_7$ with automorphism group $L_2(13)$, but this is in class IV.
To verify the assertions in this last sentence one needs to construct the epimorphisms $\Gamma\to G=L_2(13)$ corresponding to regular maps of type $\{7,7\}_7$. By applying a suitable automorphism of $G$ one may assume that $$r_0=\pm\left(\begin{array}{cc}5&0\\ 0&-5\end{array}\right)
\quad{\rm and}\quad
r_2=\pm\left(\begin{array}{cc}0&1\\ -1&0\end{array}\right),
\quad{\rm while}\quad
r_1=\pm\left(\begin{array}{cc}a&b\\ c&-a\end{array}\right)$$ for some $a, b, c\in F_{13}$ with $a^2+bc=-1$. There are three conjugacy classes of elements of order $7$ in $G$, all self-inverse, consisting of the elements with traces $\pm 3$, $\pm 5$ and $\pm 6$. By applying this condition to the traces $\pm 3a$, $\pm(b-c)$ and $\pm 5(b+c)$ of the elements $r_0r_1$, $r_1r_2$ and $r_0r_1r_2$, and solving the resulting equations, one finds just the following solutions (unique up to multiplication by $-1$ and conjugation by an element of the Klein four-group $\langle r_0, r_2\rangle$):
1. $a=6$, $b=4$, $c=-6$, with $r_0r_1$, $r_1r_2$ and $r_0r_1r_2$ having traces $\pm 5$, $\pm 3$ and $\pm 3$;
2. $a=1$, $b=5$, $c=-3$, with $r_0r_1$, $r_1r_2$ and $r_0r_1r_2$ having traces $\pm 3$, $\pm 5$ and $\pm 3$;
3. $a=1$, $b=1$, $c=-2$, with $r_0r_1$, $r_1r_2$ and $r_0r_1r_2$ having traces $\pm 3$, $\pm 3$ and $\pm 5$;
4. $a=2$, $b=1$, $c=-5$, with $r_0r_1$, $r_1r_2$ and $r_0r_1r_2$ all having trace $\pm 6$.
The dihedral group $\langle r_0, r_1\rangle$ of order $14$ is maximal in $G$, and it does not contain a Klein four-group, so in each case the elements $r_i$ generate $G$. It follows that there are exactly four regular maps ${\cal M}_1,\ldots, {\cal M}_4$ of type $\{7,7\}_7$ with automorphism group $G$, corresponding to these four solutions. They are non-orientable since $G$ has no subgroups of index $2$. Since ${\rm Aut}\,G=PGL_2(13)$ preserves traces, ${\cal M}_1$ and ${\cal M}_2$ form a dual pair, which must correspond to Conder’s entry N119.5, while ${\cal M}_3$ and ${\cal M}_4$ are self-dual; similarly, ${\cal M}_3=P({\cal M}_1)$, so ${\cal M}_1$, ${\cal M}_2$ and ${\cal M}_3$ form a class II orbit of $\Sigma$, while ${\cal M}_4$ is in class IV. This completes the proof, but one can go on to show that ${\cal M}_3$ is the map N119.6, so that ${\cal M}_4$ is N119.7, by verifying that in case 3 the generators satisfy the relation $(r_0r_1(r_2r_1)^2r_0r_1)^2=(RS^{-2}R)^2=1$ given in [@Con] for N119.6, whereas in case 4 they do not. $\square$
By contrast, the list of orientable regular maps in [@Con] has $112$ entries with $p=q=r$, rather than the $21$ found above. Lemma 2.1 and its proof show that an orientable class III map must have type $\{p,p\}_p$ with $p$ even, and that it must satisfy $G^{\rm ab}\cong V_8$; these and the preceding criteria eliminate all but R17.26, R33.45, R49.34, R61.13, R65.81, R73.85, R82.21, R97.108, R97.121 and R97.126 as possibilities for class III maps. It would be possible, but very tedious, to deal with these cases by hand, as in the proof of Theorem 6.1. However, a computer search by Conder [@Con09] has eliminated them all, by showing that the least genus of any orientable class III map is $193$, attained by a dual pair of type $\{16,16\}_{16}$ with an automorphism group of order $2048$. His search (which also confirms Theorem 6.1) shows that this is the smallest possible automorphism group of such a map, and that the second smallest is the group $L_2(8)\times V_8$ of order $4032$ corresponding to the dual pair of maps of type $\{18,18\}_{18}$ and genus $392$ constructed in Example 4A.
It is interesting that in Conder’s example the automorphism group is solvable (in fact nilpotent, having order $2^{11}$), whereas the examples constructed in this paper all have non-solvable automorphism groups. The methods of §4 yield further class III maps with solvable groups, as coverings of Conder’s, but it would be interesting to find infinite families of such maps which do not arise as coverings of other class III maps. In particular one would like to see further examples with nilpotent groups (which are necessarily $2$-groups).
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abstract: 'We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the *second systole* of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated.'
address:
- 'CMAT, Facultad de Ciencias, Universidad de la República, Uruguay'
- 'IMERL, Facultad de Ingeniería, Universidad de la República, Uruguay'
- 'IMERL, Facultad de Ingeniería, Universidad de la República, Uruguay'
- 'CMAT, Facultad de Ciencias, Universidad de la República, Uruguay'
- 'IMJ, Université Sorbonne Paris Cité, France'
author:
- 'S[é]{}bastien Alvarez'
- Joaquín Brum
- Matilde Martínez
- Rafael Potrie
- an Appendix written with Maxime Wolff
- 'S[é]{}bastien Alvarez, Joaquín Brum, Matilde Martínez, Rafael Potrie'
- Maxime Wolff
title: Topology of leaves for minimal laminations by hyperbolic surfaces
---
Introduction
============
A *lamination* or [*foliated space*]{} of dimension $d$ is a compact metrizable space which is locally homeomorphic to a disk $D$ in $\mathbb{R}^d$ times a compact set $T$ called *transversal*. The space is required to have a compatibility condition between these local trivialisations, which guarantees that sets of the form $D\times\{t\}$ glue together to form $d$-dimensional manifolds called [*leaves*]{}. The space is therefore a disjoint union of the leaves, which can be embedded in the compact space in very complicated ways. When the space is a manifold, this structure is usually called a [*foliation*]{}. When the transversal $T$ is a Cantor set, it is called a [*solenoid*]{}. When all the leaves are dense, we say that the lamination is [*minimal*]{}. We refer the reader to [@candel-conlon] for more details and to [@Ghys_Laminations] for an excellent survey about the two-dimensional case.
For laminations by surfaces, i.e. when $d=2$, the topology and the geometry of leaves have been widely studied. Cantwell and Conlon proved that any surface appears as a leaf of a foliation by surfaces of a closed 3-manifold, see [@CC_realization]. Note that their construction does not produce a minimal foliation. In a very nice work [@MG], Gusm[ã]{}o and Meni[ñ]{}o have recently shown how to construct *minimal* foliations by surfaces on the product of any closed surface by the circle containing any prescribed countable familly of noncompact surfaces as leaves. On the other hand, Ghys proved in [@Ghys_generic] that for a lamination by surfaces, the generic leaf –in the sense of Garnett’s harmonic measures [@Garnett]– is either compact or in a list containing only six noncompact surfaces. An analogous statement holds for generic leaves in a topological sense under an assumption which is satisfied by minimal laminations [@Cantwell_Conlon]. Much work on the quasi-isometry class of leaves has led to many results concerning the topology of generic leaves. This is the subject of [@alvarez-candel], which contains most relevant results and references. Many remarkable examples have given us insights into which leaves can appear, or coexist, in a lamination by surfaces. It is worth mentioning the Ghys-Kenyon example, constructed in [@Ghys_Laminations] which is a *minimal* lamination containing leaves with different conformal types. Also one has the famous Hirsch’s foliation where all leaves have infinite topological types (see [@Hirsch1975] for the original construction and [@ADMV; @AlvarezLessa; @Cantwell_Conlon; @Ghys_generic] for the minimal construction). Other aspects of the study of this subject have been pursued in different works, a non exhaustive list is [@Clark_Hurder; @Hurder_matchbox; @Sibony_etc; @McCord; @Schori; @Sullivan; @Verjovsky].
In this paper, we are interested in the study of the topology of the leaves for minimal laminations by [*hyperbolic*]{} surfaces. Such laminations are quite ubiquituous after a very beautiful uniformisation result of Candel [@Candel] which shows that unless some natural obstruction appears, every lamination by surfaces admits a leafwise metric which makes every leaf of constant negative curvature.
The motivation for this work was to understand the possible topologies that can coexist in a minimal lamination by hyperbolic surfaces. In this setting, a strong dichotomy holds: either the generic leaf is simply connected or all leaves have a ‘big’ fundamental group –one which is not finitely generated– see [@ADMV Theorem 2]. We have found that there are no obstructions to having all surfaces simultaneously in the same lamination.
\[t.uno\] There exists a minimal lamination by hyperbolic surfaces so that for every non-compact open surface $S$ there is at least one leaf homeomorphic to $S$.
After proving this theorem, we were informed that a similar result had been announced in the late 90’s by Blanc [@Blanc_these], as part of his unpublished doctoral thesis. Nevertheless, the lamination described by Blanc, constructed with a refinement of Ghys-Kenyon’s method, does not seem to admit a leafwise metric of constant curvature $-1$.
We present a very flexible combinatorial method to construct minimal laminations by hyperbolic surfaces with prescribed surfaces as leaves. This yields, in a straightforward way, the example announced in Theorem \[t.uno\] and many others. For minimal laminations, there is always a residual set of leaves having 1, 2 or a Cantor set of ends, see [@Cantwell_Conlon]. In this paper we restrict ourselves to considering laminations for which there is a residual set of leaves which are planes. Blanc studied the two-end case in [@Blanc_2bouts].
A companion paper [@AB] by the first two authors will address the case where the generic leaf has a Cantor set of ends. It contains a refinment of [@ADMV Theorem 2]: *for every leaf of such a lamination, all isolated ends are accumulated by genus —this is called condition $(\ast)$ in* [@AB]. Using the formalism developped in the present paper, as well as new techniques, it is proven that this is the only obstruction. Better still: *there exists a minimal lamination ${\mathcal{L}}$ by hyperbolic surfaces whose generic leaf is a Cantor tree and such that for every non-compact surface $S$ satisfying condition $(\ast)$, there is at least one leaf homeomorphic to $S$.* The formalism used there and the spirit of the proof are very similar, but other difficulties appear, and new techniques are needed.
We are unable, in general, to prescribe exactly which noncompact surfaces of infinte type will appear as leaves in our examples. This difficulty arises from the fact that there are uncountably many topological types of surfaces, and sometimes a certain family of leaves will force the appearance of a new leaf, which is non-homeomorphic to the ones in the family. We do not know if this is a weakness of our method or if it reflects a general obstruction. On the other hand, we have a precise control on the topology of non planar leaves thanks to Theorem \[teo-mainabstract\] using the formalism of *forests of surfaces* which we introduce in this paper. When restricting ourselves to finite or countable families of surfaces, this allows us to get an optimal result.
\[t.tres\] Let ${\mathcal{A}}= \{S_n\}_{n}$ be a finite or countable sequence of non-compact open surfaces different from the plane. Then, there is a minimal lamination by hyperbolic surfaces for which the generic leaf is a plane, and the leaves which are not simply connected form a sequence $\{L_n\}$ such that $L_n$ is homeomorphic to $S_n$ for every $n$.
Notice that the sequence ${\mathcal{A}}$ can take any value more than once –even infinitely many times. So this theorem says that, given a finite or countable set of noncompact surfaces, there is a lamination having each element as a leaf exactly some prescribed number of times.
These techniques can produce a wider variety of examples. We refer the reader to Theorem \[teo-mainabstract\] and Proposition \[p.forestinclusion\] for the most general statements which allow in particular to show Theorems \[t.uno\] and \[t.tres\] (see section \[s.including\_forest\]).
\[rem.obstsusp\] Foliations of codimension one do not have this flexibility, at least in enough regularity. In fact, for a foliation by surfaces of a closed 3-manifold by surfaces of finite type, either all leaves are simply connected or there are infinitely many leaves which are not. This will be explained in Proposition \[rem.obstminimalfol\] (see also Remark \[rem.obssuspS1\]).
All of our examples are solenoids, obtained as the inverse limit $\mathcal{L}$ of an infinite tower $$\cdots \xrightarrow{p_n} \Sigma_n \xrightarrow{p_{n-1}} \Sigma_{n-1} \cdots \xrightarrow{p_0}\Sigma_0$$ of finite covers of a compact hyperbolic surface $\Sigma_0$. These solenoids are also the object of [@Sibony_etc Section 2]. There, Sibony, Fornaess and Wold prove, among other things, that there is a unique transverse holonomy-invariant measure, and also that there are no harmonic measures other than the one which is totally invariant. Also, they prove that the laminations we construct embed in $\mathbb{C}\mathbb{P}^3$ (see also [@Deroin_plongements] for other results of immersion of laminations inside projective complex spaces). On the other hand, it is easy to see that laminations obtained by inverse limits hardly ever embed in a 3-manifold. This follows from the more general fact that a minimal lamination by hyperbolic surfaces which admits a holonomy-invariant measure and embeds in a 3-manifold has the following property: either all its leaves are simply connected or none of them are, see Remark \[remark-embed\].
All covering maps $p_n:\Sigma_{n-1}\to \Sigma_n$ are local isometries, and an appropriate control of the geometry of the $\Sigma_n$ will allow us to prescribe the topology of the leaves of $\mathcal{L}$. Fornaess, Sibony and Wold construct a lamination where all leaves but one are simply connected. In the present work, to be able to construct every possible surface, we need a tighter grip on the properties of the tower, and therefore a better understanding of finite coverings of surfaces. The main technical tool is the following statement, of independent interest, concerning covering maps between compact hyperbolic surfaces (which appears in the Appendix, joint with M. Wolff).
\[teo.apendice\] Let $\Sigma$ be a closed hyperbolic surface, and let $\alpha\subset\Sigma$ be a simple closed geodesic. Then, for all $K>0$, there exists a finite covering $\pi:\hat \Sigma\to\Sigma$ such that
- $\hat \Sigma$ contains a non-separating simple closed geodesic such that $\pi(\hat \alpha)=\alpha$ and $\pi$ restricts to a homeomorphism on $\hat \alpha$;
- every simple closed geodesic which is not $\hat \alpha$ has length larger than $K$.
This result allows us to get a relative version of residual finiteness for surface groups which may be interesting by itself, see Theorem \[t.finitecovering\]. It is similar in spirit to the LERF property proved for surface groups by Scott in [@Scott]. To compare the two, suppose that $\Sigma$ and $\alpha$ are as above, and that $\beta_1,\ldots,\beta_n$ are the closed geodesics in $\Sigma$ of length smaller than $K$. The LERF property implies that there is a finite covering $\hat \pi:\hat \Sigma \to \Sigma$ where $\alpha$ is lifted to $\hat\alpha$ as above and where all the $\beta_i$ “open up”, but it does not guarantee that no new “short” geodesics appear. This could happen if, for example, a conjugate of $\beta_1$ belonged to the subgroup of $\pi_1(\Sigma)$ generated by $\alpha$.
Similar geometric and quantitative properties of surface groups have been proved with different motivations, for a recent such result see, e.g. [@lazarovich-levit-minsky].
#### [**Organization of the paper –**]{}
The paper is structured as follows: Section \[s.Preliminaries\] covers preliminary material related to compact hyperbolic surfaces, towers of coverings of such surfaces and properties of their inverse limits. In Section \[s.illustration\] we present some illustrative examples, motivating the techniques and pointing out some differences with the foliation setting, it closes with some explainations on how the general results will be obtained. Section \[s.toolbox\] develops the necessary tools to control the geometry of the finite covers (in particular, the general version: Theorem \[t.finitecovering\] of the relative version of residual finitness is obtained). In Section \[s.admissible\] we define an abstract object, an *admissible tower of coverings*, which enables us to control the topology of leaves of a lamination. The notion of forest of surfaces is also introduced. Section \[s.including\_forest\] is the technical heart of the paper: we prove there Proposition \[p.forestinclusion\], which allows us to construct towers of finite coverings admissible with respect to any forest of surfaces. Finally, in Section \[s.final\_proof\] we construct the necessary forests of surfaces in order to prove Theorems \[t.uno\] and \[t.tres\]; the constructions there are flexible and allow to make other examples that the reader can pursue if desired.
#### [**Acknowledgements –**]{}
It is a pleasure to thank Henry Wilton whose answer to our question in MathOverflow, which contained a first sketch of proof of Theorem \[teo.apendice\] (see [@HW]), has been very important for the completion of our work. Gilbert Hector kindly communicated to us Blanc’s thesis, we are thankful to him. Finally we thank Fernando Alcalde, Pablo Lessa, Jesús Álvarez Lopez, Paulo Gusmão and Carlos Meniño for useful discussions.
Preliminaries {#s.Preliminaries}
=============
Towers of coverings and minimal laminations
-------------------------------------------
#### [**Definitions –**]{}
Let ${\mathbb{T}}= \{p_n : \Sigma_{n+1} \to \Sigma_n \}$ where $\Sigma_n$ are closed hyperbolic surfaces and $p_n$ are finite (isometric) coverings. We define $\mathcal{L}$ to be the inverse limit of ${\mathbb{T}}$, that consists of sequences ${\mathbf{x}}=(x_n)_{n\in{\mathbb{N}}}\in\prod_n \Sigma_n$ such that for every $n\in{\mathbb{N}}$, $p_n(x_{n+1})=x_n$, and we endow it with the the topology induced by the product topology.
Let us emphasize that in this paper all covering maps are local isometries.
The set $\mathcal{L}$ is a compact space and possesses a lamination structure so that the leaf of a sequence ${\mathbf{x}}=(x_n)_{n\in{\mathbb{N}}}$, denoted by $L_{\mathbf{x}}$, is formed by those sequences ${\mathbf{y}}=(y_n)_{n\in{\mathbb{N}}}$ such that ${\mathrm{dist}}(x_n,y_n)$ is bounded.
\[r.distanciamonotona\] Let ${\mathbf{x}}=(x_n)_{n\in{\mathbb{N}}}$ and ${\mathbf{y}}=(y_n)_{n\in{\mathbb{N}}}$ be two different points in $\mathcal{L}$. Then the sequence of distances $d(x_n,y_n)$ is increasing with $n$. To see this notice that a path $\alpha$ between $x_n$ and $y_n$ that realizes the distance between them, projects down onto a path of the same length between $x_m$ and $y_m$ whenever $m\leq n$.
#### [**Leafwise metric –**]{}
Let ${\mathbf{x}}\in{\mathcal{L}}$ and $L_{\mathbf{x}}$ be the leaf of ${\mathbf{x}}$. Let us consider the following covering maps
- $\Pi_n:L_{\mathbf{x}}\to \Sigma_n$ associating to ${\mathbf{y}}$ the $n$-th coordinate $y_n$.
- $P_n=p_0\circ\ldots \circ p_{n-1}: \Sigma_n\to \Sigma_0$.
- $P_{n,m}=p_m\circ\ldots\circ p_{n-1} :\Sigma_n\to\Sigma_m$ for $m<n$.
Note that $P_n \circ \Pi_n =\Pi_0$ for every $n$ and that $p_{n-1} \circ \Pi_n=\Pi_{n-1}$. We can lift the metric of $\Sigma_0$ on each $\Sigma_n$ using maps $P_n$ (so all coverings $p_n$ are local isometries) and on $L_{\mathbf{x}}$ (so that all $\Pi_n$ are local isometries). We denote by $g_n$ the metric on $\Sigma_n$ and by $g_{L_{\mathbf{x}}}$ the metric on $L_{\mathbf{x}}$. This gives a *leafwise metric*, i.e. an assignment $L\mapsto g_L$ which is transversally continuous in local charts.
#### [**Minimality –**]{}
Recall that a lamination is said to be *minimal* if all of its leaves are dense.
\[p.minimality\] The lamination ${\mathcal{L}}$ defined by a tower ${\mathbb{T}}$ of finite coverings of closed hyperbolic surfaces is minimal.
Recall that the topology on ${\mathcal{L}}$ is induced by the product topology. So given an open set $\hat U$ of ${\mathcal{L}}$ there exists an integer $n_0>0$, a positive number $\delta_0$ and a finite sequence $(x_n)_{n=0,\ldots n_0}$ satisfying $p_n(x_{n+1})=x_n$ for every $n<n_0$, such that $\hat U$ contains every point ${\mathbf{y}}$ satisfying ${\mathrm{dist}}(x_n,y_n)<\delta_0$ for every $n\leq n_0$.
Let ${\mathbf{y}}\in{\mathcal{L}}$ and $\alpha_{n_0}$ be any path in $\Sigma_{n_0}$ starting at $y_{n_0}$ and ending at $x_{n_0}$. For every $n\geq n_0$ there exists a path $\alpha_n$ in $\Sigma_n$ starting at $y_n$ such that $P_{n,n_0}\circ\alpha_n=\alpha_{n_0}$. Note that for all $n\geq n_0$ we have $l_{\alpha_n}
=l_{\alpha_{n_0}}$. Let ${\mathbf{y}}'\in{\mathcal{L}}$ be defined as follows. For $n\leq n_0$, $y_n'=x_n$ and for $n>n_0$, $y_n'$ is the other extremity of $\alpha_n$. The first condition implies that ${\mathbf{y}}'\in\hat U
$. The second one implies that ${\mathrm{dist}}(y_n,y_n')\leq l_{\alpha_{n_0}}$ for every $n$ so that ${\mathbf{y}}'\in L_{\mathbf{y}}$. This proves that $\hat U\cap L_{\mathbf{y}}\neq{\emptyset}$. This proves the minimality of ${\mathcal{L}}$.
In fact, we know from [@Matsumoto] that laminations constructed in this way must be *uniquely ergodic*, since it can be shown that they are equicontinuous (see also [@Sibony_etc]).
\[remark-embed\] Laminations constructed this way that embed in 3-manifolds need to be quite special. Indeed, since they admit a transverse invariant measure, the codimension one property implies some local order preservation: If $\Lambda$ is compact lamination with a transverse invariant measure, $i: \Lambda \to M$ is an embedding in a 3-manifold and if $L$ is a non-simply connected leaf, then, one can consider a small transversal to a non trivial loop $\gamma$ and the holonomy of the lamination can be pushed to nearby leaves because the measure is preserved as well as the order. The loops in the nearby leaves cannot be homotopically trivial since that would imply that closed curves of a given length bound arbitrarily large disks contradicting the fact that leaves are hyperbolic. If the lamination is minimal this implies that every leaf has a non-trivial fundamental group. This implies that if a minimal lamination with hyperbolic leaves and a transverse invariant measure embeds in a 3-manifold then either every leaf is simply connected, or no leaf is.
Geometry and topology of the leaves {#s.geom_topo_leaves}
-----------------------------------
#### [**Cheeger-Gromov convergence –**]{}
A sequence $(\Sigma_n,g_n,x_n)_{n\in{\mathbb{N}}}$ of pointed Riemannian manifolds is said to converge towards the pointed Riemannian manifold $(L,g,x)$ in the *Cheeger-Gromov sense* whenever there exists a sequence of smooth mappings $\Pi_n:L\to\Sigma_n$ such that
1. for every $n\in{\mathbb{N}}$, $\Pi_n(x)=x_n$; and for every compact set $K\dans M$ there exists a integer $n_0=n_0(K)>0$ such that
2. for every $n\geq n_0$, $\Pi_n$ restricts to a diffeomorphism of $K$ onto its image;
3. the sequence of pull-back metrics $(\Pi_n^{\ast}g_n)_{n\geq n_0}$ converges to $g$ in the $C^{\infty}$-topology over $K$.
The sequence $(\Pi_n)_{n\in{\mathbb{N}}}$ is called a sequence of *convergence mappings* of $(L_n,g_n,x_n)_{n\in{\mathbb{N}}}$ with respect to $(L,g,x)$. This mode of convergence is sometimes called *smooth convergence*: [@Pablito; @Pet]. It appeared first in [@Gr], where Gromov proved that Cheeger’s finiteness theorem (see [@Ch]) was in fact a compactness result. We will refer to [@Pet] for more details about it.
#### [**Topology of the leaves –**]{}
Cheeger-Gromov convergence proves to be especially useful to identify the topology of leaves of a lamination coming from a tower of finite coverings.
Below, ${\mathcal{L}}$ denotes the inverse limit of a tower ${\mathbb{T}}=\{p_{n+1}:\Sigma_{n+1}\to\Sigma_n\}$ of finite coverings of closed hyperbolic surfaces.
\[p.Cheeger\_Gromov\] Let ${\mathbf{x}}\in M$. Then the pointed leaf $(L_{\mathbf{x}},g_{L_{\mathbf{x}}},{\mathbf{x}})$ is the Cheeger-Gromov limit of pointed Riemannian manifolds $(\Sigma_n,g_n,x_n)$.
Candidates for convergence mappings are given by the maps $\Pi_n:L_{\mathbf{x}}\to\Sigma_n$. These are indeed local isometries. By the inverse function theorem it is enough to prove that for every $R>0$ there exists $n_0$ such that $\Pi_n$ is injective on the ball $B_{L_{\mathbf{x}}}({\mathbf{x}},R)$ for every $n\geq n_0$.
Consider the groups $G_n=(P_n)_\ast(\pi_1(\Sigma_n,x_n))$ and $G_{\mathbf{x}}=(\Pi_0)_\ast(\pi_1(L_{\mathbf{x}},{\mathbf{x}}))$. By definition they form a decreasing sequence of subgroups of $\pi_1(\Sigma_0,x_0)$ and for every $R>0$ there exists $n_1(R)\geq 0$ such that for every $n\geq n_1(R)$ we have $$\label{eq.group_stat}
G_{\mathbf{x}}\cap D_R=G_n\cap D_R$$ where $D_R$ denotes the disc centered at the identity of radius $R$ inside $\pi_1(\Sigma_0,x_0)$ for the geometric norm, i.e. the one that associates to $\gamma$ the length of the associated geodesic loop based at $x_0$.
First note that if $\Pi_n({\mathbf{y}})=\Pi_n({\mathbf{z}})$ for some $n\in{\mathbb{N}}$ and ${\mathbf{y}},{\mathbf{z}}\in L_{\mathbf{x}}$ then we have $\Pi_m({\mathbf{y}})=\Pi_m({\mathbf{z}})$ for every $m\leq n$. Assume that for infinitely many integers $n$ there exists an open geodesic segment $\alpha_n\dans B({\mathbf{x}},R)$ so that $\Pi_n\circ\alpha_n$ is a geodesic loop (where $B({\mathbf{x}},R)$ denotes the ball of radius $R$ about ${\mathbf{x}}$). Using Ascoli’s theorem and the remark above, we see that there exists an open geodesic ray $\alpha\dans B({\mathbf{x}},R)$ such that $\Pi_n\circ\alpha$ is a closed geodesic loop (hence nontrivial in homotopy) for every $n\in{\mathbb{N}}$, contradicting .
Notice in particular that the leaves of ${\mathcal{L}}$ are hyperbolic surfaces, but all this discussion works equally well if one considers towers of coverings of Riemannian manifolds of any dimension.
Some hyperbolic geometry {#s.hyp_geom}
------------------------
#### [**Systoles, collars and injectivity radius –**]{}
Below we set definitions and notations of hyperbolic geometry that will be used throughout the paper.
\[d.collars1\]Let $X$ be a compact hyperbolic surface with geodesic boundary. The *systole* ${\mathrm{sys}}(X)$ of $X$ is the length of the shortest geodesic in $X$. The *internal systole* of $X$ is the shortest length of an essential and primitive closed curve in $X$ which is not isotopic to a boundary component. This is also the smallest length of a closed geodesic included in the interior ${\mathrm{Int}}(X)$.
Notice that if $X$ has no boundary, these two concepts coincide, but when $X$ has boundary, the systole could be achieved by a boundary component.
\[d.collars\] The *(maximal) half-collar width* $K_0$ at a boundary component $\alpha$ of $X$ is the minimal half-distance of two lifts of $\alpha$ to the Poincaré disk ${\mathbb{D}}$. It satisfies that for every $K<K_0$ the $K$-neighbourhood of $\alpha$ is an embedded half-collar.
We say that the boundary of $X$ has a *half-collar of width* $K_0$ if there exists a neighbourhood of $\partial X$ consisting of a disjoint union of embedded half-collars of width $K_0$.
We now give a series of lemmas that we will use later in the text.
\[l.halfcollars\] Let $X$ be a hyperbolic surface with geodesic boundary which is not a pair of pants and $\alpha\dans\partial X$ be a boundary component. Then $$K_0>\frac{\sigma-l_{\alpha}}{2},$$ where $\sigma$ and $K_0$ denote respectively the internal systole and the half-collar width at $\alpha$ of $X$.
By definition $2K_0$ is the minimal distance between two lifts of $\alpha$ to the upper half plane and it is the length of a geodesic segment $\gamma$ cutting $\alpha$ orthogonally at two points $x$ and $y$, included inside the pair of pants $P$ attached to $\alpha$.
The pair of pants $P$ has a boundary component $\beta$ disjoint from the boundary $\partial X$. This simple closed curve is isotopic to the concatenation of $\gamma$ with a geodesic segment $[x,y]$ included in $\alpha$. We find $$2K_0+l_\alpha>l_\beta\geq\sigma.$$
The lemma follows.
The injectivity radius at a point $x$ of a Riemannian manifold will be denoted by ${r_{\mathrm{inj}}}(x)$.
\[l.inj\_radius\] Let $X$ be a hyperbolic surface with geodesic boundary and $\sigma$ be its internal systole. Assume that boundary components of $X$ have disjoint collars of width $K_0>0$. Assume furthermore that we have $$K_0\leq {\mathrm{sys}}(X)\cosh\left(\frac{K_0}{2}\right).$$
Let $x\in X$ such that ${\mathrm{dist}}(x,\partial X)\geq K_0$. Then $${r_{\mathrm{inj}}}(x)\geq\min\left(\sigma,\frac{K_0}{2}\right).$$
Assume that the hypotheses of the lemma hold. Let $x$ be a point such that ${\mathrm{dist}}(x,\partial X)\geq K_0$. We must prove that a primitive geodesic loop $\gamma$ based at $x$ satisfies $l_\gamma\geq\min(\sigma,K_0/2)$. If $\gamma$ is not isotopic to a boundary component of $X$ then $l_\gamma\geq\sigma$. So assume that $\gamma$ is isotopic to a boundary component $\beta$ of $X$ and that $\gamma$ satisfies $l_\gamma<K_0/2$. Then $\gamma$ is entirely contained outside the $(K_0/2)$-neighbourhood of $\beta$.
Let $\tilde{\beta}$ be a lift of $\beta$ to the Poincaré disk ${\mathbb{D}}$, it is invariant by a hyperbolic isometry denoted by $h$ whose translation length is $l_\beta\geq{\mathrm{sys}}(X)$. There exists a geodesic segment $\tilde \gamma$ between $\tilde x$ (a lift of $x$) and $h(\tilde x)$ which projects down isometrically onto $\gamma$ and is located outside a $(K_0/2)$-neighbourhood of $\tilde \beta$. Since the orthogonal projection outside a $(K_0/2)$-neighbourhod of $\tilde \beta$ is a contraction of factor $1/\cosh(K_0/2)$ we must have $${\mathrm{sys}}(X)\leq l_{\tilde{\beta}}\leq \frac{l_{\tilde{\gamma}}}{\cosh(K_0/2)}\leq\frac{K_0}{2\cosh(K_0/2)},$$ which contradicts the hypothesis. We deduce that if $\gamma$ is isotopic to a boundary component it must satisfy $l_\gamma\geq K_0/2$.
\[l.crosscollars\] Consider $\Sigma$ and $X$ hyperbolic surfaces with geodesic boundary, and a map $$\varphi:X\to\Sigma$$ which is an isometric embedding in restriction to ${\mathrm{Int}}(X)$. Take $\alpha\dans\partial X$, a boundary component and denote by $\sigma$ and $K_0$ respectively the internal systole of $X$ and the half-collar width of $\alpha$. Then if $\gamma$ is a closed geodesic in $\Sigma$ that crosses $\varphi(\alpha)$, we have $l_{\gamma}>K_0$
Let us denote by $C$ the image by $\varphi$ of a half-collar at $\alpha$ with width $K_0$, as stated in the lemma. Let $\gamma_0$ be a connected component of $\gamma_0\cap C$ which meets $\varphi(\alpha)$. Since two geodesic arcs cannot bound a bigon, $\gamma_0\cap\varphi(\alpha)$ must be a singleton, so $\gamma_0$ must connect the two boundary components of $C$, therefore it must have length greater than $K_0$.
We will also need the following:
\[l.systole-gluing\] Let $X$ be a hyperbolic surface with geodesic boundary written as a union $$X=\bigcup_{i=0}^{n}X_i$$ where the $X_i$ are subsurfaces with geodesic boundary meeting each other at boundary components, and $K$ a positive number. Assume moreover that for $i=0,\ldots,n$ we have:
- the internal systole of $X_i$ is greater than $K$;
- the half collar width of every boundary component of $X_i$ included in the interior of $X$ is greater than $K$;
- the boundary components of $X_i$ included in the interior of $X$ have length greater than $K$.
Then, the internal systole of $X$ is greater than $K$.
Take a closed geodesic $\gamma\subseteq \textrm{Int}(X)$, we must check that its length is greater than $K$. For this, we distinguish three cases.
*Case 1. $\gamma\subseteq {\mathrm{Int}}(X_i)$ $i$.* In this case, the length of $\gamma$ must be greater or equal than the internal systole of $X_i$, and therefore is greater than $K$ by hypothesis.
*Case 2. $\gamma$ crosses a boundary component $b$ of $X_i$ for some $i$.* By hypothesis, the half-collar width of every boundary component of $X_i$ included in the interior of $X$ is greater than $K$, then Lemma \[l.crosscollars\] implies that $l_{\gamma}>K$.
*Case 3. $\gamma$ is a boundary component of some $X_i$ included in ${\mathrm{Int}}(X)$.* In this case the length of $\gamma$ is greater than $K$ by hypothesis. This finishes the proof of the lemma.
#### [**Retraction on subsurfaces –**]{}
We will need the following proposition to identify the topology of some complete hyperbolic surface knowing that of a subsurface.
\[p.recognize\_surface\] Let $L$ be a complete hyperbolic surface without cusps and $S\dans L$ a closed subsurface with geodesic boundary such that every connected component $C_i$ of $L{\setminus}S$ satisfies the following properties.
1. $C_i$ does not contain a closed geodesic.
2. The boundary $\partial \overline{C}_i$ is connected
Then $L$ is diffeomorphic to ${\mathrm{Int}}(S)$.
Let $C_i$ be a connected component of $L{\setminus}\overline{S}$ and $\widetilde{C}_i$ be a component of its preimage to the Poincaré disk ${\mathbb{D}}$ by uniformization. Its closure is geodesically convex and has geodesic boundary (argue like in the proof of [@Casson_Bleiler Lemma 4.1.]).
Moreover, we can prove that its boundary is connected so this is a half plane. In order to see this we use that the boundary $\partial\overline{C}_i$ is connected so if the closure of $\widetilde{C}_i$ had various boundary components, there would exist a geodesic ray between two of them projecting down to a geodesic loop inside $C_i$. Such a loop cannot be isotopic to the boundary of $C_i$, thus contradicting the first hypothesis.
Since $\overline{C}_i$ has no cusp, no interior closed geodesic and only one geodesic boundary component, its fundamental group (which equals the fundamental group of $C_i$) must be trivial or cyclic generated by the translation about the geodesic boundary.
We deduce from this that $\overline{C}_i$ must be a hyperbolic half-plane or a funnel with geodesic boundary. Using the transport on geodesics orthogonal to $\partial S$ and basic Morse theory we see that all manifolds defined as $\{x;{\mathrm{dist}}(x,S)\leq r\},\,r>0$ are diffeomorphic to $S$ so their interiors are all diffeomorphic to ${\mathrm{Int}}(S)$. We deduce that $L$ is diffeomorphic to ${\mathrm{Int}}(S)$.
Noncompact surfaces {#ss.classifsurfaces}
-------------------
#### [**Ends of a space –**]{}
Let us recall the definition of an end of a connected topological space $X$. Let $(K_n)_{n\in{\mathbb{N}}}$ be an exhausting and increasing sequence of compact subsets of $X$. An *end* of $X$ is a decreasing sequence $${\mathcal{C}}_1\supset{\mathcal{C}}_2\supset...\supset{\mathcal{C}}_n\supset...$$ where ${\mathcal{C}}_n$ is a connected component of $X{\setminus}K_n$. We denote by ${\mathcal{E}}(X)$ the *space of ends* of $X$. It is independent of the choice of $K_n$.
The space of ends of $X$ possesses a natural topology which makes it a compact subspace of a Cantor space. An open neighbourhood of an end $e=({\mathcal{C}}_n)_{n\in{\mathbb{N}}}$ is an open set $V\dans X$ such that ${\mathcal{C}}_n\dans V$ for all but finitely many $n\in{\mathbb{N}}$.
#### [**Classifying triples –**]{}
In what follows, a *classifying triple* is the data $\tau=(g,{\mathcal{E}}_0,{\mathcal{E}})$ of
- a number $g\in{\mathbb{N}}\cup\{\infty\}$;
- a pair of nested spaces ${\mathcal{E}}_0\dans{\mathcal{E}}$ where ${\mathcal{E}}$ is a nonempty, totally disconnected and compact topological space; which satisfy
- $g=\infty$ if and only if ${\mathcal{E}}_0\neq{\emptyset}$.
Say that two classifying triples $\tau=(g,{\mathcal{E}}_0,{\mathcal{E}})$ and $\tau'=(g',{\mathcal{E}}_0',{\mathcal{E}}')$ are *equivalent* if $g=g'$ and if there exists a homeomorphism $h:{\mathcal{E}}\to{\mathcal{E}}'$ such that $h({\mathcal{E}}_0)={\mathcal{E}}_0'$.
#### [**Noncompact surfaces –**]{} {#noncompact-surfaces}
We now recall the modern classification of surfaces as it appears in [@Ric]. The leaves of a hyperbolic surface laminations are orientable so we are only interested in the classification of *orientable* surfaces.
Recall that an end $e=({\mathcal{C}}_n)_{n\in{\mathbb{N}}}$ of $\Sigma$ is *accumulated by genus* if for every $n\in{\mathbb{N}}$, the surface ${\mathcal{C}}_n$ has genus. The ends accumulated by genus form a compact subset that we denote by ${\mathcal{E}}_0(\Sigma)\dans{\mathcal{E}}(\Sigma)$. In our terminology the triple $\tau(\Sigma)=(g(\Sigma),{\mathcal{E}}_0(\Sigma),{\mathcal{E}}(\Sigma))$ is a classifying triple.
\[classification\] Two orientable noncompact surfaces $\Sigma$ and $\Sigma'$ are homeomorphic if and only if their classifying triples $\tau(\Sigma)$ and $\tau(\Sigma')$ are equivalent.
Moreover for every classifying triple $\tau$ there exists an orientable noncompact surface $\Sigma$ such that $\tau(\Sigma)$ is equivalent to $\tau$.
As a direct consequence, there are uncountably many different topological types of open surfaces as there exists uncountably many closed subsets of the Cantor set.
Direct limits of surfaces
-------------------------
#### [**Inclusions of surfaces –**]{}
Let $S$ and $S'$ be two surfaces with boundary. We say that a map $f:S\to S'$ is an *inclusion* if the two conditions below are satisfied.
- $f$ is continuous and injective.
- $f$ maps every boundary component of $S$ to a boundary component of $S'$ or inside the interior of $S'$.
When $\overline{S'\setminus f(S)}$ admits a hyperbolic structure with geodesic boundary we call $f$ a *good inclusion*.
When we specify two points $x$ and $x'$ on $S$ and $S'$ respectively, a (good) inclusion $f:(S,x)\to(S',x')$ is supposed to map $x$ to $x'$.
\[rem\_isom\_embedding\_proper\_inclusion\] If $S$ and $S'$ are hyperbolic surfaces with geodesic boundary, then any isometric embedding $f:S\to S'$ is a good inclusion.
#### [**Direct limits –**]{}
Let $(S_n)_{n\in{\mathbb{N}}}$ be a sequence of surfaces with boundary and a chain of inclusions $\{j_n:S_n\to S_{n+1}\}$. The *direct limit* of this chain is the quotient $$S_\infty=\underrightarrow{\lim}\{j_n:S_n\to S_{n+1}\}= \left.\bigsqcup S_n\right/\sim$$ where $\sim$ is the equivalence relation generated by $\forall x\in S_n,\,x\sim j_n(x)$. The space $S_\infty$ is naturally a topological surface, possibly with boundary. By definition of inclusions, a point of $\partial S_\infty$ corresponds to a point $x$ of the boundary of some $S_{n_0}$ such that for every $n\geq n_0$ the map $j_{n-1}\circ\ldots\circ j_{n_0}(x)$ belongs to the boundary of $S_n$. Moreover there exists an inclusion $J_n:S_n\to S_\infty$.
Direct limits enjoy the following universal property.
\[t.universal\_property\] Let $L$ be a surface. Assume that there exists a sequence of inclusions $\phi_n:S_n\to L$ which satisfy the compatibility condition $$\phi_n=\phi_{n+1}\circ j_n.$$ Then there exists an inclusion $\phi:S_\infty\to L$ such that for every $n\in{\mathbb{N}}$ $$\phi_n=\phi\circ J_n.$$
#### [**Open direct limits –**]{}
Let $(S_n)_{n\in{\mathbb{N}}}$ be a sequence of surfaces with boundary and a chain of inclusions $\{j_n:S_n\to S_{n+1}\}$. The *open direct limit* of this chain is by definition the interior of the direct limit. This is by definition an open surface.
#### [**Geometric direct limits –**]{}
We now assume that we are given a sequence $(S_n)_{n\in{\mathbb{N}}}$ of *compact hyperbolic surfaces with geodesic boundary* and a chain of *isometric embeddings* $\{j_n:S_n\to S_{n+1}\}$. This is in particular a chain of good inclusions (see Remark \[rem\_isom\_embedding\_proper\_inclusion\]). The direct limit $S_\infty$ of this chain might not be complete: imagine the case of a sequence of surfaces obtained by gluing hyperbolic pairs of pants whose boundary components have length growing very fast.
\[d.geom\_direct\_limit\] The *geometric direct limit* of the chain $\{j_n:S_n\to S_{n+1}\}$ of isometric embeddings of compact hyperbolic surfaces with geodesic boundary is the surface denoted by $\overline{S}_\infty$ and defined as the metric completion of the direct limit $S_\infty$.
\[p:geom\_limit\] The geometric direct limit of a chain $\{j_n:S_n\to S_{n+1}\}$ of isometric embeddings of compact hyperbolic surfaces with geodesic boundary is a hyperbolic surface whose boundary components are disjoint geodesics (that can be closed or not) and satisfies that ${\mathrm{Int}}(\overline{S}_\infty)={\mathrm{Int}}(S_\infty)$, the open direct limit.
An inductive argument using the uniformization theorem shows the following. There exist an increasing sequence of Fuchsian groups $\Gamma_1<\Gamma_2<\ldots\Gamma_n<\ldots$, an increasing sequence of connected domains with geodesic boundary of the Poincaré disk ${\mathbb{D}}$, denoted by $D_1\dans D_2\dans\ldots\dans D_n\dans\ldots$ (defined as the *Nielsen cores* of the $\Gamma_n$, i.e. the convex hulls of their limits sets) and a sequence of isometries $$\phi_n:S_n\to D_n/\Gamma_n$$ satisfying the following compatibility condition $$\iota_n\circ\phi_n=\phi_{n+1}\circ j_n,$$ where $\iota_n:D_n/\Gamma_n\to D_{n+1}/\Gamma_{n+1}$ is the natural inclusion.
Let $D_\infty$ be the increasing union of the domains $D_n$ and $\Gamma_\infty$ be the increasing union of the groups $\Gamma_n$. The set $D_\infty$ is a convex set invariant by the Fuchsian group $\Gamma_\infty$ and $D_\infty/\Gamma$ is the direct limit of the inclusions $\iota_n:D_n/\Gamma_n\to D_{n+1}/\Gamma_{n+1}$. Using the universal property of direct limits (Theorem \[t.universal\_property\]) we see that $S_\infty$ is isometric to $D_\infty/\Gamma_\infty$.
The surface $D_\infty/\Gamma_\infty$ embeds isometrically inside the complete surface ${\mathbb{D}}/\Gamma_\infty$ so its metric completion, which is isometric to $\overline{S}_\infty$, is realized as the closure $\overline{D}_\infty/\Gamma_\infty$. The boundary of $D_\infty$ is geodesic (by convexity), its interior is precisely the interior of $D_\infty$, and these sets are $\Gamma_\infty$-invariant. This proves that the surface $\overline{D}_\infty/\Gamma_\infty$ satisfies the conclusion of the proposition, and hence $\overline{S}_\infty$ as well.
Illustrative examples {#s.illustration}
=====================
A lamination where every leaf is a disk
---------------------------------------
#### [**Residual finiteness and geometry –**]{}
Surface groups enjoy a property known as *residual finiteness* (see [@delaharpe III.18] ). More precisely, given a closed hyperbolic surface $\Sigma$, and a non-trivial element $\gamma \in \pi_1(\Sigma)$ there exists a finite group $G$ and a morphism $\phi: \pi_1(\Sigma) \to G$ so that $\phi(\gamma)\neq 1$. In other words, every non-trivial element is disjoint from some finite index normal subgroup in $\pi_1(\Sigma)$. It is clear that this also implies that for any finite subset of $F \subset \pi_1(\Sigma) \setminus \{e\}$ there is a finite index normal subgroup $N \lhd \pi_1(\Sigma)$ such that $N \cap F = \emptyset$. From the geometric point of view this easily implies:
\[l.res\_large\_rinj\] For every closed hyperbolic surface $\Sigma$ and every $K>0$ there exists a normal covering map $\hat p: \hat \Sigma \to \Sigma$ such that $\mathrm{sys}(\hat \Sigma) >K$.
Just take as finite set the elements of $\pi_1(\Sigma)$ corresponding to simple closed geodesics of length $\leq K$ and the covering associated to the normal finite index subgroup given by residual finiteness which will then *open* all short curves to give the desired statement.
Note that the covering constructed in Lemma \[l.res\_large\_rinj\] has injectivity radius $> K$ at every point.
#### [**Tower of coverings –**]{}
Thus we can consider a tower of regular covering maps ${\mathbb{T}}=\{p_n:\Sigma_{n+1}\to\Sigma_n \}$ such that the injectivity radius at every point of $\Sigma_n$ tends to infinity with $n$. See Figure \[fig:Residouille\].
![Residual finiteness.[]{data-label="fig:Residouille"}](Residually_finite-eps-converted-to.pdf)
The inverse limit of such a tower gives rise to a minimal lamination (see Proposition \[p.minimality\]). Moreover Proposition \[p.Cheeger\_Gromov\] states that for every $K>0$ the $K$-neighbourhood of an element ${\mathbf{x}}\in \mathcal{L}$ inside its leaf is a copy of the $K$-neighbourhood of $x_n$ inside $\Sigma_n$ for $n$ large enough, which is an embedded disc of radius $R$. This means in particular that the leaf of every ${\mathbf{x}}$ is an increasing union of disks: this must be a disk. This completes the construction of a minimal lamination all of whose leaves are hyperbolic disks.
Notice that from the topological point of view, such examples are quite well known. For instance, one can take a totally irrational linear foliation in $\mathbb{T}^3$ to get a minimal foliation by planes. Also, the universal inverse limit construction, obtained as the inverse limit of *all* finite coverings of a given surface gives rise to another minimal foliations by disks which is indeed the same as the one constructed above (as they are cofinal). We refer to [@Sullivan; @Verjovsky]. See also [@Kapovich] for a realization of some laminations by hyperbolic spaces as inverse limits of towers of coverings. From the point of view of our construction, nevertheless, this construction is quite illustrative, as it shows in the simplest possible context the strategy we want to follow to prove our main theorems, the key idea is to be able to construct a tower of regular coverings so that the injectivity radius grows in most places while we control that some other places get lifted carefully in order to get some leaves with topology. This will become clearer in our next example.
Realizing a cylinder as a leaf
------------------------------
Let us show next how to construct a minimal lamination by Riemann surfaces for which one leaf is a cylinder, and every other leaf is a disk. This is not as simple as it seems and it contains one of the key difficulties in our whole construction.
Using the same line of reasoning as above we would like to construct a tower of coverings such that for one sequence ${\mathbf{x}}=(x_n)_{n\in{\mathbb{N}}}$ of the inverse limits the $K$-neighbourhood of $x_n$ inside $\Sigma_n$ is an embedded annulus when $n$ is large enough. The corresponding leaf would be an annulus. And furthermore, when ${\mathrm{dist}}(x_n,y_n)\to\infty$ (so ${\mathbf{x}}$ and ${\mathbf{y}}$ lie on different leaves), we want the $K$-neighbourhood of $y_n$ to be a disk, when $n$ is large enough. The corresponding leaf would be a disk. See Figure \[fig:HW\].
![Creating a cylinder.[]{data-label="fig:HW"}](Relative_Residual_finiteness-eps-converted-to.pdf)
This is obtained by using our relative version of residual finiteness, Theorem \[teo.apendice\]. For every simple closed geodesic $\alpha$ there exists a finite cover $\pi:\hat \Sigma\to\Sigma$ such that $\alpha$ has a unique $(1:1)$-lift to $\hat \Sigma$, called $\hat \alpha$, and such that every other simple closed geodesic of $\hat \Sigma$ has length $\geq K$ where $K$ can be arbitrarily large. One could say that the *second systole* of $\hat \Sigma$ is arbitrarily large.
Some basic facts of hyperbolic geometry (see Lemmas \[l.halfcollars\] and \[l.inj\_radius\]) show that when $K$ is large enough, there is a collar about $\hat \alpha$ of width $\geq K/2$ and that points away from $\hat \alpha$ have injectivity radii larger than $K$. This allows to implement the desired tower of finite coverings.
\[rem.obssuspS1\] It is possible to see that representation $\rho: \pi_1(\Sigma) \to \mathrm{Homeo}(S^1)$ of a surface group cannot produce, via the suspension construction, a foliation such that there is a unique annulus and the rest of the leaves are planes. This is because this would imply the existence of a non-abelian free subgroup[^1] acting freely on $S^1$ which is impossible.
More generally, one can show:
\[rem.obstminimalfol\] Let $\mathcal{F}$ be a minimal[^2] foliation by surfaces in a closed 3 manifold $M$ with all leaves of finite type and so that not every leaf is a disk, then it must have infinitely many leaves which are not disks.
To see this, notice first that in this context there cannot be a transverse invariant measure: if a minimal foliation has a transverse invariant measure and one leaf is not a disk, then infinitely many leaves must have non-trivial fundamental group, notice that one can lift a non-trivial loop to nearby leaves, and these cannot become homotopically trivial in their leaves because of Novikov’s theorem (recall that a minimal foliation cannot have a Reeb-component).
Therefore Candel’s theorem applies and there is a smooth Riemannian metric on $M$ such that leaves have negative curvature everywhere (see for example [@Alvarez_Yang Theorem B]). Thus, one can apply [@Alvarez_Yang Theorem A] to get a hyperbolic measure for the *foliated geodesic flow* which produces an infinite number of periodic orbits and each corresponds to a non-trivial closed geodesic in some leaves. This is produced by a measure which has the SRB property (in particular its support is saturated by strong unstable manifolds) and therefore cannot be supported in finitely many leaves (see [@Alvarez_Yang Proposition 3.1 (3)]).
If all leaves have finite topological type and are hyperbolic, since the injectivity radius must be bounded from below it follows that all closed geodesic in the leaves must lie in a compact core inside each leaf. In particular, if one looks at an accumulation point along the transverse direction in the support of the measure not all periodic orbits of the foliated geodesic flow can belong to the same leaf.
This implies that there are infinitely many leaves with non-trivial topology.
It seems reasonable to expect the previous result to hold without any regularity assumption nor that assuming that all leaves have finite topological type, but we decided not to pursue this as it is not central for the results of this paper.
Realizing the Loch-Ness monster
-------------------------------
Assume now that we wish to construct a minimal lamination for which one leaf is a Loch-Ness monster (i.e. has one end and infinite genus) and every other leaf is a disk. In such an example, surfaces of finite and infinite topological type will coincide inside the same minimal hyperbolic surface lamination.
The strategy will be similar. We need to construct a tower of regular coverings $\mathbb{T} = \{ p_{n} : \Sigma_{n+1} \to \Sigma_n\}$ with the following properties (see Figure \[fig:lochnouille\]):
![Realizing a Loch-Ness monster.[]{data-label="fig:lochnouille"}](Loch_Ness_Monster-eps-converted-to.pdf)
- Each $\Sigma_n = S_n \cup X_n$ where $S_n$ and $X_n$ are sub-surfaces with geodesic boundary (compare with admissible decompositions defined in §\[ss.relative\]).
- The surface $S_n$ is connected, with genus $n$ and one boundary component and admits a $(1:1)$ lift into a subsurface of $S_{n+1}$. *This is the part we want to stabilize*.
- The internal systole $\sigma_n$ of $X_n$ (see Definition \[d.collars\]) grows to $\infty$ with $n$.
Notice that these conditions imply that the boundary component of $S_n$ has length going to infinity. This may produce some issue on the understanding of the topology of the limit leaf as the *direct limit* of the surfaces $S_n$ may not be complete, this is dealt with in Proposition \[p.recognize\_surface\]. The fact that such a tower can be constructed relies on a strengthening of the relative residual finiteness mentioned in the previous section that is obtained in Theorem \[t.finitecovering\]. A similar argument as above, provides the desired construction. Notice that the leaf corresponding to the Loch-Ness monster will be anyone obtained as a sequence ${\mathbf{x}}$ where ${\mathrm{dist}}(x_n, S_n)$ remains bounded. Any other sequence gets far from the $S_n$ and therefore the injectivity radius goes to infinity, so the corresponding leaf is a disk.
Combining both examples and some comments on cylinders {#ss.cylinders}
------------------------------------------------------
It can be noticed from the proof of Theorem \[t.finitecovering\] below that it is possible to adapt the relative residual finiteness in order to consider a tower of regular coverings $\mathbb{T}= \{ p_n : \Sigma_{n+1} \to \Sigma_n \}$ so that:
- Each $\Sigma_n = S_n \cup X_n$ where $S_n$ and $X_n$ are sub-surfaces with geodesic boundary.
- The surface $S_n$ is connected, with genus $n$ and one boundary component and admits a $(1:1$) covering into a subsurface of $S_{n+1}$.
- The surface $X_n$ has the property that it contains a unique simple closed geodesic $\alpha_n$ which has length smaller than $1$ but every other primitive closed geodesic of length smaller than $n$ has to be homotopic to either the boundary of $X_n$ or to $\alpha_n$ (we could say that the *second internal systole* grows to infinity).
- The distance between $\partial S_n$ and $\alpha_n$ goes to infinity with $n$.
Notice in particular that $p_n$ must map $\alpha_{n+1}$ as a (1:1) covering of $\alpha_n$ for sufficiently large $n$. This will imply that the leaf corresponding to the lifts of $\alpha_n$ will be an annulus, the leaf corresponding to points which remain at bounded distance from the surfaces $S_n$ will be a Loch-Ness monster and the rest of the leaves have to be disks.
In what follows we will extend this constructions in order to be able to produce several different possible laminations. As this example shows, the production of cylinders in the lamination is a bit different from the construction of the Loch-Ness monster as one requires the construction of surfaces in the coverings, and cylinders are detected by closed geodesics with large collar neighborhoods. It turns out that a procedure similar to the one used to construct the Loch-Ness monster works for *every* other surface (except the cylinder). It is possible to find a more cumbersome formalism that includes cylinders, but in order to simplify the presentation, we will ignore cylinder leaves and leave the construction of laminations which also have cylinder leaves to the reader.
Toolbox for constructing finite coverings {#s.toolbox}
=========================================
We now give the principal tool that we will use in order to implement the idea given in §\[s.illustration\]. This is a variation of the residual finiteness of surface groups.
A relative version of residual finiteness {#ss.relative}
-----------------------------------------
#### [**The key tool –**]{}
The following result will provide us with an essential tool for the proof of Theorem \[t.finitecovering\] and will be used in several points. Its proof is deferred to Appendix \[s.appendix\].
\[teo-HW\]
Let $\Sigma$ be a closed hyperbolic surface, and let $\alpha\subset\Sigma$ be a simple closed geodesic. Then, for all $K>0$, there exists a finite covering $\pi:\hat \Sigma\to\Sigma$ such that
- $\hat \Sigma$ contains a non-separating simple closed geodesic such that $\pi(\hat \alpha)=\alpha$ and $\pi$ restricts to a homeomorphism on $\hat \alpha$;
- every simple closed geodesic which is not $\hat \alpha$ has length larger than $K$.
We say that the *second systole* of $\hat \Sigma$ is large because the only short closed curves (i.e. shorter than $K$) in $\hat \Sigma$ need to be homotopic to a power of $\hat \alpha$.
\[rem.large\_collars\_tubes\] By Lemma \[l.halfcollars\] the surface $\hat \Sigma$ has a collar around $\hat \alpha$ of width $\geq K-l_\alpha$. If $K$ is large enough we can assume that the width of this collar is $\geq K/2$.
\[rem\_tubes\_genus\] The internal systole of the connected surface with boundary $T$ obtained by cutting $\hat \Sigma$ along $\hat \alpha$ is greater than $K$ so there must exist a point of $T$ with injectivity radius $\geq K$. We deduce that the area of $T$ (which equals that of $\hat \Sigma$) is $\geq 2\pi(\cosh K-1).$ By Gauss-Bonnet’s theorem, the genus $g$ of $T$ (note that it equals $genus(\hat \Sigma)-1$) satisfies the following inequality $$g\geq\frac{\cosh K -1}{2}.$$
#### [**Admissible decompositions –**]{}
Let $\Sigma$ be a closed hyperbolic surface. An *admissible decomposition* of $\Sigma$ is a pair $(X,S)$ of (possibly disconnected) compact hyperbolic subsurfaces of $\Sigma$ with geodesic boundary such that
- $\Sigma=X\cup S$;
- $X$ and $S$ meet at their common boundary.
The surface $X$ will be sometimes called the *admissible complement of $S$*.
#### [**Relative residual finiteness –**]{}
We now state a relative version of residual finiteness of the fundamental group of a given closed hyperbolic surface $\Sigma$ which is adapted to a given admissible decomposition $(X,S)$. More precisely, we want to find coverings of $\Sigma$ where we keep a copy of $S$ while increasing the internal systole and collar width of its admissible complement $X$. The following theorem in the case where $S$ is empty, can be deduced from the residual finiteness of surface groups.
\[t.finitecovering\] Let $\Sigma$ be a closed hyperbolic surface and $(X,S)$ be an admissible decomposition of $\Sigma$. Then, given $K>0$, there exists a closed hyperbolic surface $\hat \Sigma$ with an admissible decomposition $(\hat X,\hat S)$ as well as a finite covering $p: \hat \Sigma \to \Sigma$ such that
1. the restriction $p|_{\hat S}:\hat S\to S$ is a $(1:1)$ isometry;
2. the internal systole of $\hat X$ is larger than $K$;
3. the boundary components of $\hat X$ have disjoint half collars of width larger than $K$.
\[r.room\] Assume the hypothesis of Theorem \[t.finitecovering\], and consider $S_0$ a compact surface with boundary (not necessarily connected and possibly with degenerate components[^3]) and $g\in{\mathbb{N}}$. Then, we can perform the construction so that, in addition to the conclusion of Theorem \[t.finitecovering\], $\hat X$ contains a subsurface $X$ with geodesic boundary, written as a disjoint union $X=X_1\sqcup X_2$ such that:
- $X_1$ is homeomorphic to $S_0$
- for every boundary component $\alpha_j$ of $\hat X$ there exists a subsurface $A_j\dans X_2$ such that
1. $A_j$ has genus $g$ and two boundary components;
2. one of the boundary components of $A_j$ is $\alpha_j$;
3. for $i\neq j$ we have $A_i\cap A_j={\emptyset}$.
This means that boundary components of $\hat X$ are separated by topology.
The rest of the section is devoted to the proof of Theorem \[t.finitecovering\] and Remark \[r.room\]. For that purpose we will show how to perform surgeries of finite coverings.
Surgeries of finite coverings
-----------------------------
We now introduce a technique to construct new finite coverings of a surface from others. We call this technique *surgery* since it consists in ‘cutting and pasting’ different finite coverings (see Figure \[fig:surg\]).
![The surgery of two double covers.[]{data-label="fig:surg"}](Surgery-eps-converted-to.pdf)
Given a closed hyperbolic surface $\Sigma$ and a simple closed geodesic $\alpha \subset \Sigma$ we denote by $\Sigma_\alpha$ the (not necessarily connected) hyperbolic surface obtained by cutting along $\alpha$. Two new boundary components appear in $\Sigma_\alpha$ associated to $\alpha$ which we denote by $\alpha^+$ and $\alpha^-$ according to the orientation. We will fix a point $x\in\Sigma$ and its two copies $x^+\in\alpha^+$ and $x^-\in\alpha^-$.
Let $p_i: \hat \Sigma_i \to \Sigma$ be coverings of $\Sigma$ ($i=1,2$) such that for some simple closed geodesic $\alpha$ of $\Sigma$ there exists, for every $i$, a closed geodesic $\alpha_i \subset \hat \Sigma_i$ such that $p_i$ is a $(1:1)$ isometry from $\alpha_i$ to $\alpha$. Consider the two surfaces $\hat \Sigma_{\alpha_i}$, the four copies $\alpha_i^\pm$ of $\alpha$ as well as the four points $x^\pm_i$, which project down to $x$.
We define $\hat \Sigma_{\alpha_1,\alpha_2}$ to be the surface obtained from $\hat \Sigma_{\alpha_1}$ and $\hat \Sigma_{\alpha_2}$ by gluing $\alpha_1^+$ with $\alpha_2^-$ and $\alpha_1^-$ with $\alpha_2^+$. To completely describe the gluing one must require that it sends respectively $x_1^+$ and $x_1^-$ on $x_2^-$ and $x_2^+$ and that it is an isometry: notice that the four curves are isometric lifts of $\alpha$.
Together with $\hat \Sigma_{\alpha_1,\alpha_2}$ one can define a map $p: \hat \Sigma_{\alpha_1,\alpha_2} \to \Sigma$ which is obtained by applying $p_1$ or $p_2$. We let $\hat \alpha^+$ and $\hat \alpha^-$ be these two distinguished curves, which are both isometric to $\alpha$
The map $p: \hat \Sigma_{\alpha_1, \alpha_2} \to \Sigma$ is called the *surgery* of $p_1$ and $p_2$ *along the pair* $(\alpha_1,\alpha_2)$.
\[propo\_connected\_covering\] The map $p$ is a finite cover of $\Sigma$ and the surface $\hat \Sigma_{\alpha_1, \alpha_2}$ is connected if the $\hat \Sigma_i$ are connected and at least one of the $\alpha_i$ is non-separating.
Suppose $\hat \Sigma_i$ are connected and $\alpha_1$ is non separating. Then $\hat \Sigma_{\alpha_1,\alpha_2}$ is a union of $\hat \Sigma_{\alpha_1}$, a connected surface, with two (if $\alpha_2$ separates) or one (if it does not) connected surfaces intersecting $\hat \Sigma_{\alpha_1}$. It must be connected.
Moreover the surface $\hat \Sigma_{\alpha_1,\alpha_2}$ is compact so it suffices to prove that $p$ is a local isometry. Since $p_1$ and $p_2$ are local isometries, it is enough to verify this in a neighbourhood of $\hat \alpha^+$ and $\hat \alpha^-$.
Recall that a point $y$ inside a sufficiently thin collar in $\Sigma$ about $\alpha$ is described by its *Fermi coordinates* (based at $x$) $(\rho(y),\theta(y))$ where $\rho(y)$ is the signed distance from $y$ to $\alpha$ (positive on the right side, negative on the left side) and where the orthogonal projection of $y$ onto $\alpha$ is $\alpha(\theta(y))$ (here we choose $\alpha(0)=x$). See [@Bu].
Since they are isometries from a collar of $\alpha_i$ onto a collar onto $\alpha$, the maps $p_i$ preserve Fermi coordinates (based at $x_i$ respectively). So by construction the map $p$ preserves Fermi coordinates in a collar of $\hat \alpha^\pm$. This means that it is an isometry from these open sets onto a collar about $\alpha$, concluding the proof.
Attaching tubes
----------------
To construct the desired coverings we will use Theorem C several times and perform surgeries from this. To simplify the structure we will give a name to the building blocks of the surgeries provided by Theorem C.
Given $\alpha \subset \Sigma$ a simple closed geodesic and $K>0$ we say that $T_\alpha$ is a $(\alpha,K)$-tube if it is a surface given by Theorem C for the curve $\alpha$ and the constant $K$.
Given an admissible decomposition $(X,S)$ of $\Sigma$ we will “attach tubes” to boundary components of $S$ in order to isolate them one from the other.
#### [**Attaching tubes at a closed geodesic –**]{}
Given a surface $\Sigma$ with a simple closed geodesic $\alpha$. Consider $T_\alpha$, a $(\alpha,K)$-tube and $\pi:T_\alpha\to\Sigma$ the covering map defined by Theorem C and $\hat \alpha$, the unique $(1:1)$-lift of $\alpha$.
![Attaching a tube.[]{data-label="fig:tubouille"}](Attaching_tubes-eps-converted-to.pdf)
We will say that a covering $p: \hat \Sigma \to \Sigma$ is obtained from $\Sigma$ by *attaching a $(\alpha,K)$-tube* at $\alpha$ if it is the surgery of $\pi$ and the identity ${\mathrm{Id}}: \Sigma \to \Sigma$ along the pair $(\hat \alpha,\alpha)$. Since the curve $\hat \alpha$ is non-separating, the surface $\hat \Sigma$ is connected (see Proposition \[propo\_connected\_covering\]).
In $\hat \Sigma$ there are two distinguished curves $\hat \alpha^+$ and $\hat \alpha^-$ which are the unique $(1:1)$-lifts of $\alpha$ and have at least one half collar of width $\geq K/4$. See Figure \[fig:tubouille\].
Proof of Theorem \[t.finitecovering\].
--------------------------------------
The proof of Theorem \[t.finitecovering\] consists in starting with $\Sigma$ and attaching several $(\alpha,K)$-tubes. The construction has several stages. Consider a closed hyperbolic surface $\Sigma$ with admissible decomposition $(X,S)$. Let $\alpha_1,\ldots,\alpha_k$ denote the boundary components of $S\dans\Sigma$. Let $K>0$ and $L=4K$.
\[r.collars\_admissible\] Let $1\leq i\leq k$. If $\alpha_i$ is a boundary component of $S$ then it has one half-collar included inside $S$ and another one included in $X$.
#### [**Isolating components of $S$ –**]{}
Denote $\Sigma_0= \Sigma$ and consider a finite cover $p_1: \Sigma_1 \to \Sigma_0$ obtained by attaching an $(\alpha_1,L)$-tube along $\alpha_1$.
There is exactly one $(1:1)$-lift of $\alpha_j$ for $j\geq 2$, and exactly two $(1:1)$-lifts of $\alpha_1$. Moreover only one of these two lifts has a half-collar that projects down into $S$ (see Remark \[r.collars\_admissible\]). This lift bounds a $(1:1)$-copy of the corresponding connected component of $S$ and has one half collar of width $\geq K$. Let $\hat \alpha_1$ denote the lift of $\alpha_1$ that we distinguished. As a consequence, the surface $\Sigma_1$ possesses an admissible decomposition $(X_1,S_1)$ $p_1|_{S_1}:S_1\to S$ is an isometry and where $\hat \alpha_1$ has one half-collar of width $\geq K$ included in $X_1$.
We can continue this process and construct $p_j : \Sigma_j \to \Sigma_{j-1}$ for $j = 2, \ldots, k$ by attaching $(\alpha_j,L)$-tubes along $\alpha_j \subset \Sigma_{j-1}$. This produces a finite cover $\hat p_k: \Sigma_k \to \Sigma$ where $\Sigma_k$ has an admissible decomposition $\Sigma_k=X_k\cup S_k$ such that
- each component of $S$ has a $(1:1)$-lift to $S_k$ by $\hat p_k$;
- boundary components of $X_k$ have disjoint half-collars of width $\geq K$.
\[rem\_no-pants\] Tubes have large genus by Remark \[rem\_tubes\_genus\] so we can assume that $X_k$ is a finite union of compact connected surfaces with boundary, none of which is a pair of pants.
#### [**Enlarging the internal systole of $X$ –**]{}
To complete the proof we need to take a finite cover that lifts $S_k$ while enlarging the internal systole of $X_k$, the admissible complement of $S_k$. This will be done by attaching tubes to large simple closed geodesics intersecting those curves of $X_k$ that have length $<K$.
Let $Y$ be a connected compact hyperbolic surface with geodesic boundary. Say that two simple closed geodesics $\gamma_1$ and $\gamma_2$ *fill* $Y$, a connected compact hyperbolic surface, possibly with geodesic boundary, if $Y{\setminus}(\gamma_1\cup\gamma_2)$ is a union of discs and of annuli isotopic to the boundary of $Y$.
\[p.fill\_large\_curves\] Let $Y$ be a connected compact hyperbolic surface with geodesic boundary which is not a pair of pants. For every $K>0$ there exists two simple closed geodesics $\gamma_1$ and $\gamma_2$ that fill $Y$ and such that $l_{\gamma_i}\geq K$ for $i=1,2$.
The set $Y$ is not a pair of pants so we can find two simple closed geodesics $\alpha_1$ and $\alpha_2$ that fill $Y$: see [@Farb_Marg Proposition 3.5.]. In particular they satisfy $i(\alpha_1,\alpha_2)>0.$ Let $T_{\alpha_1}:Y\to Y$ be the Dehn twist about $\alpha_1$. Then for every $m\geq 1$ the curves $\alpha_1$ and $T_{\alpha_1}^m(\alpha_2)$ fill $Y$. Moreover for $m$ large enough the curve $\gamma_2=T_{\alpha_1}^m(\alpha_2)$ has length $\geq K$. The same argument allows one to prove that there exists $n\geq 0$ such that $\gamma_1=T_{\gamma_2}^n(\alpha_1)$ has length $\geq K$, and moreover, the geodesics $\gamma_1$ and $\gamma_2$ fill $Y$.
Since $X_k$ is a compact hyperbolic manifold its length spectrum (c.f. Appendix \[s.appendix\]) is discrete and there are finitely many closed geodesics (not necessarily simple) $\beta_1,\ldots,\beta_l\dans X_k$ with length $< K$ and which are not isotopic to the boundary of $X_k$.
*Assume for the moment that $X_k$ is connected*. It is not a pair of pants by Remark \[rem\_no-pants\]. Since the curves $\beta_i$ which are included inside $X_k$ are not isotopic to the boundary, they must intersect the union $\gamma_1\cup\gamma_2$.
Consider now $\hat \Sigma_0=\Sigma_k$, $\hat S_0=S_k$ and $\hat X_0= X_k$. Consider the simple closed geodesic $\gamma_1$ and the covering map $\hat p_1: \hat \Sigma_1 \to \hat \Sigma_0$ defined by attaching a $(\gamma_1, K)$-tube along $\gamma_1$. Each connected component of $\hat S_0$ has a unique $(1:1)$-lift to $\hat \Sigma_1$, this defines an admissible decomposition $(\hat X_1,\hat S_1)$ of $\hat \Sigma_1$. The surface $\hat X_1$ has half-collars of width $\geq K$, its boundary components are separated by surfaces of high genus and the unique closed geodesics inside $\hat X_1$ with length $< K$ are those unique isometric lifts of the curves $\beta_1,...,\beta_l$ that don’t intersect $\gamma_1$. These must intersect a lift of $\gamma_2$ (still denoted by $\gamma_2$) which in particular has length $\geq K$ and is disjoint from $\hat S_1$.
Attaching a $(\gamma_2,K)$-tube to $\gamma_2$, we increase the length of the $\beta_i$ cutting $\gamma_2$. So we obtain the desired cover $\hat p:\hat \Sigma\to\Sigma$ where the admissible decomposition $\hat \Sigma=\hat X\cup\hat \Sigma$ satisfies the four Items of Theorem \[t.finitecovering\].
If $X_k$ is not connected perform the operation decribed above with all of its connected components to get the desired covering map $\hat p:\hat \Sigma\to\Sigma$.
Note that since tubes have large genus (see once again Remark \[rem\_tubes\_genus\]) we can, up to attaching tubes to each boundary component of $\hat S\dans\hat \Sigma$, ensure that the conclusion of Remark \[r.room\] holds: there is enough topological room inside $\hat X$.
Forests of surfaces and towers of finite coverings {#s.admissible}
==================================================
Organizing surfaces in forests {#ss.surfaces-forests}
------------------------------
We now use the combinatorial description of surfaces and the concept of open direct limit to organize a family of open surfaces. This seemingly complicated way to organize the surfaces gives us more flexibility to control the topology of the leaves of a lamination constructed as a tower of coverings (see Remark \[r.forestjustif\]).
#### [**Forests –**]{}
A forest will be defined as a countable union of disjoint rooted trees. Let us be more precise and state some notations.
Let $G=(V,E)$ be an oriented graph where $V$ is the set of vertices of $G$ and $E\subseteq V^2$ is the set of edges. We define the *origin* and *terminal* functions $o:E\to V$ and $t:E\to V$ so that $e=(o(e),t(e))$ for every $e\in E$.
\[d.forest\] A *forest* is an oriented graph $\mathcal{T}=(V(\mathcal{T}),E(\mathcal{T}))$ where the set $V({\mathcal{T}})$ of vertices and the set $E({\mathcal{T}})\dans V({\mathcal{T}})^2$ of oriented edges satisfy
- The set of vertices $V(\mathcal{T})$ has a countable partition $V(\mathcal{T})=\bigsqcup_{n\in{\mathbb{N}}}V_n(\mathcal{T})$ were the $V_n(\mathcal{T})$ are finite sets. We call $V_n(\mathcal{T})$ the $n$-th floor of $\mathcal{T}$.
- $E(\mathcal{T})$ is contained in $\bigcup_{n\in{\mathbb{N}}} (V_n(\mathcal{T})\times V_{n+1}(\mathcal{T}))$. In other words, given any edge, its terminal vertex is one floor above its origin vertex.
- Every vertex is the terminal vertex of at most one edge. This implies that $\mathcal{T}$ has no cycles.
- Every vertex is the origin vertex of at least one edge.
We will write $E(\mathcal{T})=\bigsqcup_{n\in{\mathbb{N}}}E_n(\mathcal{T})$ where $E_n(\mathcal{T})=\{e\in E(\mathcal{T}):o(e)\in V_n(\mathcal{T})\}$.
A *root* of $\mathcal{T}$ is a vertex $v\in V(\mathcal{T})$ that is not the terminal vertex of any edge: a root can be located at an arbitrary level. We note $R(\mathcal{T})$ the set of roots of $\mathcal{T}$ and $R_k(\mathcal{T})$ the set of roots of $\mathcal{T}$ that belong to $V_k(\mathcal{T})$. Notice that $\mathcal{T}=\bigsqcup_{v\in R(\mathcal{T})} \mathcal{T}_{v}$ where $\mathcal{T}_{v}$ is the maximal connected subtree of $\mathcal{T}$ containing the root $v$.
On the other hand, we define the *ends* of $\mathcal{T}$ as the union of the ends of its sub-trees, that is $${\mathcal{E}}(\mathcal{T})=\bigsqcup_{v\in R(\mathcal{T})} {\mathcal{E}}(\mathcal{T}_{v})$$ A *ray* of $\mathcal{T}$ is a concatenation of edges $r=(e_n)$ starting at a root. We will index those edges according to the floor to which they belong, that is: if a ray $r$ starts at a root $v\in R_k(\mathcal{T})$, we will denote its edges as $e_k e_{k+1}\ldots$ Also, we will consider $r_{\alpha}$ as a graph morphism $r_{\alpha}:[k,+\infty)\to\mathcal{T}$ where $[k,+\infty)$ is the half-line with one vertex for each integer greater or equal than $k$ and $r_{\alpha}([n,n+1])\in E_n(\mathcal{T})$.
Notice that the set of rays is in correspondence with ${\mathcal{E}}(\mathcal{T})$, we will note $r_{\alpha}$ the ray converging to $\alpha$.
#### [**Forests of surfaces –**]{}
A *forest of surfaces* is a triple $$\mathcal{S}=(\mathcal{T},\{S_v\}_{v\in V(\mathcal{T})},\{i_e\}_{e\in E(\mathcal{T})})$$ where $\mathcal{T}$ is a forest , $\{S_v\}_{v\in V(\mathcal{T})}$ is a family of pointed compact surfaces with boundary and $\{i_e\}_{e\in E(\mathcal{T})}$ is a family of good inclusions $i_e:S_{o(e)}\to S_{t(e)}$. When necessary, we will note the pointed surface $(S_v,q_v)$, however we will omit the pointing whenever it is possible.
We associate to $\mathcal{S}$ a family of pointed surfaces $\{{\mathbf{S}}^{\alpha}\}_{\alpha\in{\mathcal{E}}(\mathcal{T})}$ that is called the set of *limit surfaces* of $\mathcal{S}$ and is defined as follows. For an end $\alpha\in{\mathcal{E}}(\mathcal{T})$ with the corresponding ray $r_{\alpha}=(e_n)_{n\geq k}$ ($k$ being the floor of the corresponding root) and the chain of inclusions $\{i_{e_{n}}:S_{o(e_n)}\to S_{t(e_n)}\}_{n\geq k}$ associated to it. We define ${\mathbf{S}}^\alpha$ as the open direct limit of this chain.
As mentioned in §\[s.illustration\], we need to be very careful in our construction of towers of coverings if we want to control the topology of leaves in the inverse limit. The next definition gives the correct way to organize the towers.
Admissible towers and forests
-----------------------------
#### [**Forests of surfaces included in towers –**]{}
A forest of surfaces $$\mathcal{S}=(\mathcal{T},\{Z_v\}_{v\in V(\mathcal{T})},\{j_e\}_{e\in E(\mathcal{T})})$$ is said to be included in a tower $\mathbb{T}=\left\{p_n:\Sigma_{n+1}\to\Sigma_n\right\}$ (as in Figure \[fig:tree\_tower\]) if there exists
- subsurfaces with geodesic boundary $S_n=\bigsqcup_{v\in V_n({\mathcal{T}})} S_v$ included in $\Sigma_n$;
- a family of homeomorphisms $\{h_v:Z_v\to S_v:v\in V(\mathcal{T})\}$ and
- a family of embeddings $\{i_e:S_{o(e)}\to S_{t(e)}:e\in E(\mathcal{T})\}$
such that
- $p_n\circ i_e={\mathrm{Id}}$ for every $e\in E_n(\mathcal{T})$
- $i_e\circ h_{o(e)}=h_{t(e)}\circ j_e$ for every $e\in E(\mathcal{T})$
For every $n\in{\mathbb{N}}$ we define the subsurface $S_n^\ast\dans S_n$ by $$S_n^{\ast}=\bigcup_{e\in E_{n-1}(\mathcal{T})}i_e(S_{o(e)}).$$ The (not necessarily connected) surface $S_n$ consists precisely of those surfaces that we want to stabilize (as was illustrated in §\[s.illustration\]) whereas the surface $S_n^\ast$ is the isometric lift to the level $n$ of the surfaces we constructed at the level $n-1$.
We will let $X_n$ denote the admissible complement of $S_n$ and $X_n^{\ast}$ the admissible complement of $S_n^{\ast}$.
![Including a forest inside a tower of coverings.[]{data-label="fig:tree_tower"}](Forest_tower-eps-converted-to.pdf)
#### [**Admissible towers –**]{}
The definition of admissible tower provides the geometric formalization of the intuition explained in §\[s.illustration\].
\[def:admissible\] Following the previous notation, we say that *the tower of finite coverings $\mathbb{T}$ is admissible with respect to the forest of surfaces $\mathcal{S}$* if ${\mathcal{S}}$ is included in ${\mathbb{T}}$ and if furthermore
1. the internal systole $\sigma_n$ of $X_n^{\ast}$ tends to infinity with $n$;
2. the boundary of $X_n$ has a half-collar of width $K_n\to\infty$.
When a tower $\mathbb{T}$ is admissible with respect to some surface forest, we call it *an admissible tower*.
The following result encapsules the main abstract criteria to control the topology of the leaves of a lamination made by a tower of finite coverings. Let $\mathcal{L}/_{\sim}$ denote the set of leaves of $\mathcal{L}$.
\[teo-mainabstract\] Consider a forest of surfaces $$\mathcal{S}=(\mathcal{T},\{S_v\}_{v\in V(\mathcal{T})},\{i_e\}_{e\in E(\mathcal{T})})$$ and $\mathbb{T}=\{p_n:\Sigma_{n+1}\to\Sigma_n\}$ an admissible tower with respect to $\mathcal{S}$ with inverse limit $\mathcal{L}$. Then, the generic leaf of $\mathcal{L}$ is a disk and there exists an injective map ${\mathcal{E}}(\mathcal{T})\hookrightarrow \mathcal{L}/_{\sim}$ such that
- the leaf corresponding to an end $\alpha\in{\mathcal{E}}(\mathcal{T})$ is homeomorphic to ${\mathbf{S}}^\alpha$, the open direct limit defined in \[ss.surfaces-forests\];
- every leaf which is not included in the image of this map is a disk.
The rest of the section is devoted to the proof of Theorem \[teo-mainabstract\]. We first give some geometric properties of admissible towers and deduce that the generic leaf of the solenoid defined by an admissible tower is a disk.
\[r.cyl\] Once again we point out that it is possible to adapt this formalism to include cylinder leaves which are not taken into account in the way we have presented forests of surfaces. To do this one should allow degenerate surfaces which involves no new difficulty but makes the presentation more dense. In Section \[s.illustration\] we already illustrated how cylinders can be embedded and we leave to the reader the adaptations needed to include them in the formalism presented here.
\[r.forestjustif\] The main difference between trees and forests is that the set of ends of a tree is *compact* and the set of ends of a forest may be *non compact*. For example assume that we want to find a lamination such that all leaves are disks except countably many leaves with prescribed topology (cf Theorem \[t.tres\]) by including a *tree* of surfaces inside a tower. After Theorem \[teo-mainabstract\] countably many ends $\alpha_n$ of the tree would provide leaves homeomorphic to the desired surfaces. But this sequence must accumulate to other ends of the tree: undesired surfaces can appear as leaves of the lamination. In order to prove Theorem \[t.tres\] with our method we will need to use a countable union of trees.
Injectivity radius, decompositions and systoles {#ss.decomposition}
-----------------------------------------------
Consider a tower of finite coverings $\mathbb{T}=\{p_n:\Sigma_{n+1}\to\Sigma_n\}$ admissible with respect to the forest of surfaces $$\mathcal{S}=(\mathcal{T},\{S_v\}_{v\in V(\mathcal{T})},\{i_e\}_{e\in E(\mathcal{T})}).$$
#### [**Large injectivity radius –**]{}
We first use the definition of admissibility to prove that there exist points of $X_n$ with arbitrarily large injectivity radii when $n\to\infty$.
\[l.large\_inj\_radius\] There exists $n_0>0$ such that for every $n\geq n_0$ and $x_n\in X_n$ such that ${\mathrm{dist}}(x_n,\partial X_n)\geq K_n$ we have $${r_{\mathrm{inj}}}(x_n)\geq\min\left(\sigma_n,\frac{K_n}{2}\right).$$ In particular ${r_{\mathrm{inj}}}(x_n)\to\infty$ for such a sequence of points $(x_n)_{n\in{\mathbb{N}}}$.
Note first that for every $n\in{\mathbb{N}}$, ${\mathrm{sys}}(X_n)\geq{\mathrm{sys}}(\Sigma_0)>0$ and that $K_n\to\infty$ as $n\to\infty$. Hence there exists $n_0>0$ such that for every $n\geq n_0$, $$K_n\leq{\mathrm{sys}}(X_n)\cosh\left(\frac{K_n}{2}\right).$$ Finally $\sigma_n$ is smaller that the internal systole of $X_n$ (which is contained inside $X_n^\ast$). Therefore we can use Lemma \[l.inj\_radius\] and order to prove the result.
#### [**Level $i$ subsurfaces –**]{}
A difficulty that we have to deal with in order to prove our main theorems is that there could exist subtrees of the forest ${\mathcal{T}}$ with a root appearing at an arbitrarily large floor.
So let $V_{n}^{k}(\mathcal{T})$ denote the set of vertices in $V_n(\mathcal{T})$ that belong to a subtree $\mathcal{T}_v$ with $v\in R_k(\mathcal{T})$. We define the family of *level $i$ surfaces* $\{S_{v,i}:v\in\mathcal{T},i\in{\mathbb{N}}\}$ as follows.
- If $v\in V_{n}^{k}(\mathcal{T})$ and $i<k$ or $i>n$ define $S_{v,i}=\emptyset$.
- If $v\in V_n^{k}(\mathcal{T})$ and $i=n$ define $S_{v,i}$ as
- $S_v$ if $k=n$ (i.e. if $v$ is a root appearing at floor $k$)
- $\overline{S_v\setminus j_e(S_{o(e)})}$ with $v=t(e)$ if $n>k$ (this is one of the building blocks defined in §\[ss.classifsurfaces\]).
- The family satisfies the recurrence relation $S_{t(e),i}=j_e(S_{o(e),i})$ for $e\in E_m(\mathcal{T})$ and every $m\in{\mathbb{N}}$ and $i\leq m$.
Notice that $$S_v=\bigcup_{i\in{\mathbb{N}}} S_{v,i}$$ is a decomposition by subsurfaces with geodesic boundary meeting each other along boundary components. We define $S_{n,k}$ as the union of all the subsurfaces $S_{v,i}$ with $v\in V_n(\mathcal{T})$ and $i\leq k$
Then we define $X_{n,k}$ as the admissible complement of $S_{n,k}$. Note that by definition $X_{n,n-1}=X_n^{\ast}$. Note that these surface have the following decomposition $$\label{eq.deco_Xnk}
X_{n,k}=X_n\cup\bigcup_{v\in V_n({\mathcal{T}})}\left(\bigcup_{i=k}^n S_{v,i}\right).$$
\[rem\_int\_boundary\_level\] Recall that when $i<n$ we denoted $P_{n,i}=p_i\circ\ldots\circ p_{n-1}:\Sigma_n\to\Sigma_i$. Then for every $v\in V_n({\mathcal{T}})$
1. the interior of $S_{v,i}$ is mapped inside of ${\mathrm{Int}}(X_i^\ast)$ by $P_{n,i}$;
2. a boundary component of $S_{v,i}$ is either mapped isometrically onto a boundary component of $X_{i}$ by $P_{n,i}$ or onto a boundary component of $X_{i-1}$ by $P_{n,i-1}$.
#### [**Increasing internal systoles –**]{}
We will need the following result.
\[p:systole\_internal\] Let $k_n$ be a sequence of integers satisfying $k_n\leq n$ for every $n$ and $\lim_n k_n=\infty$. Then the internal systole of $X_{n,k_n}$ tends to infinity with $n$.
Define $m_n=\textrm{min}_{i\geq n}\{\sigma_i, K_i\}$ where $\sigma_i$ and $K_i$ are as in the definition of admissible tower. Then $m_n\to\infty$ as $n$ goes to $\infty$.
We consider a sequence $(k_n)_{n\in{\mathbb{N}}}$ as in the statement of the lemma. We will prove that every closed geodesic of ${\mathrm{Int}}(X_{n,k_n})$ has a length $\geq m_{k_n}$, which is enough to prove the lemma. For every $n\in{\mathbb{N}}$, $X_{n,k_n}$ has a decomposition as in . Recall that this is a decomposition by subsurfaces with geodesic boundary and disjoint interiors. We deduce that there are four possibilities for a closed geodesic $\gamma\dans{\mathrm{Int}}(X_{n,k_n})$.
1. $\gamma$ is included inside ${\mathrm{Int}}(X_n)$.
2. $\gamma$ is included inside ${\mathrm{Int}}(S_{v,i})$ for some $v\in V_n({\mathcal{T}})$ and $k_n\leq i\leq n$.
3. $\gamma$ is a boundary component of $S_{v,i}$ for some $v\in V_n({\mathcal{T}})$ and $k_n<i\leq n$.
4. $\gamma$ crosses $\partial S_{v,i}$ for some $v\in V_n({\mathcal{T}})$ and $k_n<i\leq n$.
In Case 1, we automatically have that $l_\gamma\geq\sigma_n\geq m_{k_n}$.
In Case 2, we use Item 1 of Remark \[rem\_int\_boundary\_level\] to prove that $P_{n,i}(\gamma)$ is included in ${\mathrm{Int}}(X_i^\ast)$ so $l_\gamma\geq\sigma_i\geq m_{k_n}$.
In Case 3, we use Item 2 of Remark \[rem\_int\_boundary\_level\] to get that $P_{n,j}(\gamma)$ is a boundary component of $X_j$ and hence belong to ${\mathrm{Int}}(X_j^\ast)$ for $j=i$ or $i-1$. This implies that $l_\gamma\geq\sigma_j\geq m_{k_n}$.
And finally in Case 4, we also use Item 2 of Remark \[rem\_int\_boundary\_level\]. The projection $P_{n,j}(\gamma)$ crosses a boundary component of $X_j$ inside $\Sigma_j$ for $j=i$ or $i-1$. Using Item 2 of Lemma \[l.crosscollars\] we see that $l_\gamma\geq K_j\geq m_{k_n}$.
Proof of Theorem \[teo-mainabstract\]
--------------------------------------
#### [**Topology of the generic leaf –**]{}
The first and easiest step in the proof of Theorem \[teo-mainabstract\] is to prove that the generic leaf of ${\mathcal{L}}$ defined by an admissible tower is a disk. Then we will need a further analysis using the forest structure to identify the topology of all leaves.
The generic leaf of $\mathcal{L}$ is simply connected.
For $k\in{\mathbb{N}}$ define $$U_k=\{{\mathbf{x}}\in\mathcal{L}:r_{\textrm{inj}}(x_{m_0})> k \textrm{ for some }m_0\in{\mathbb{N}}\}.$$ First notice that $r_{\textrm{inj}}(x_n)$ is increasing with $n$. Therefore, Proposition \[p.Cheeger\_Gromov\] implies that if ${\mathbf{x}}\in U_k$ then the injectivity radius of $L_{{\mathbf{x}}}$ at ${\mathbf{x}}$ is greater than $k$.
We will show that $U_k$ is open and dense for every $k\in{\mathbb{N}}$, getting that $\bigcap_{k\in{\mathbb{N}}}U_k$ is a generic and saturated set all whose leaves are disks.
[*Step 1. $U_k$ is open for every $k\in{\mathbb{N}}$.* ]{} Take ${\mathbf{x}}\in U_k$ and $m_0\in{\mathbb{N}}$ so that $r_{\textrm{inj}}(x_{m_0})>k$. Since the injectivity radius function is lower semi-continuous, we can take a neighbourhood $W$ of $x_{m_0}$ in $\Sigma_{m_0}$ such that every point in $W$ has injectivity radius greater than $k$. Then, the set of ${\mathbf{y}}\in{\mathcal{L}}$ satisfying $y_{m_0}\in W$ is an open neighbourhood of ${\mathbf{x}}$ contained in $U_k$.
[*Step 2. $U_k$ is dense for every $k\in{\mathbb{N}}$.* ]{} Fix two integers $k,m_0\in{\mathbb{N}}$, as well as a sequence $(x_n)_{n=0,\ldots m_0}$ satisfying $p_n(x_{n+1})=x_n$ when $n<m_0$. We will construct a sequence ${\mathbf{y}}\in{\mathcal{L}}$ such that $y_n=x_n$ when $n\leq m_0$ and ${r_{\mathrm{inj}}}(y_n)>k$ for $n$ large enough (so ${\mathbf{y}}\in U_k$).
Fix $D\geq\textrm{diam}(\Sigma_n)$ for all $n\leq m_0$. Using Lemma \[l.large\_inj\_radius\] we see that there exists $m$ and a point $y_m'\in\Sigma_m$ such that ${r_{\mathrm{inj}}}(y_m')> k+D$. Arguing as in the proof of the minimality of ${\mathcal{L}}$ (see Lemma \[p.minimality\]) we find a sequence ${\mathbf{y}}\in{\mathcal{L}}$ such that $y_n=x_n$ for $n\leq m_0$ and ${\mathrm{dist}}(y_m,y_m')\leq D$, which implies that ${r_{\mathrm{inj}}}(y_m)>k$.
Now we will need to go further and associate a marking to some leaves such that the following dichotomy holds. Unmarked leaves are disks, and the topology of marked leaves is prescribed by the forest.
#### [**Associated markings –** ]{}
We note $(S_v,q_v)$ the pointed surface associated to the vertex $v$. Recall that inclusions appearing in the forest of surfaces respect the base points, i.e. $$i_e\left(q_{o(e)}\right)=q_{t(e)}.$$
We can naturally associate a point ${\mathbf{x}}^\alpha$ in the inverse limit of $\mathbb{T}$ to every end $\alpha$ in ${\mathcal{E}}(\mathcal{T})$. For this consider the associated ray $r_\alpha$ (recall Definition \[d.forest\]) and let $k_0$ denote the floor of $r_\alpha(0)$. The sequence ${\mathbf{x}}^\alpha=(x_n^\alpha)_{n\in{\mathbb{N}}}\in{\mathcal{L}}$ is defined by
$$x_n^\alpha=\left\{
\begin{array}{lr}
q_{r_\alpha(n-k_0)} & n\geq k_0\\
P_{k_0,k_0-n}\left(q_{r_\alpha(0)}\right) & n<k_0
\end{array}\right. .$$
\[d.markouille\] We define *the set of markings associated to the admissible tower* as the subset $({\mathbf{x}}^\alpha)_{\alpha\in{\mathcal{E}}(\mathcal{T})}$ included in $\mathcal{L}$. The leaves of points ${\mathbf{x}}^\alpha$ will be called *marked*.
\[l:dif\_comp\] Consider $\alpha$ and $\beta$ different ends of ${\mathcal{E}}(\mathcal{T})$. Then $${\mathrm{dist}}(x_n^{\alpha},x_n^{\beta})\To_{n\to\infty}\infty$$ and $L_{{\mathbf{x}}^\alpha}\neq L_{{\mathbf{x}}^\beta}$.
If $\alpha\neq\beta$, there exists $n_0$ such that when $n\geq n_0$ the points $x_n^{\alpha}$ and $x_n^{\beta}$ belong to distinct connected components of $S_n$. So any geodesic path between these two points must cross two disjoint half collars of boundary components of $X_n$. Thus, the length of this geodesic path must be greater than $2K_n$. This quantity goes to infinity with $n$ by definition and the lemma is proven.
#### [**Topology of non-marked leaves –**]{}
A non-marked leaf is by definition the leaf of a sequence ${\mathbf{x}}\in{\mathcal{L}}$ satisfying ${\mathrm{dist}}(x_n,x_n^\alpha)\to\infty$ for every end of the forest $\alpha\in{\mathcal{E}}({\mathcal{T}})$. We want to prove that such a leaf exists and that it is a disk. We will have to face a difficulty: new roots of the forest can appear at an arbitrary floor and we want to prove that a sequence defining a non-marked leaf goes away from all those roots.
Recall that $V_n^k(\mathcal{T})$ consists of those vertices of $V_n(\mathcal{T})$ that belong to a subtree whose root is at floor $k$. We set $$Q^k_n=\{q_v:v\in V_n^k(\mathcal{T})\}$$ Note that $Q^k_n\dans S_n\dans \Sigma_n$. Recall that for two integers $n\geq m$, $P_{n,m}$ denotes the projection $p_m\circ\ldots\circ p_{n-1}$.
\[l.dicotomia\] Let ${\mathbf{x}}\in\mathcal{L}$. We have the following dichotomy.
- Either there exists $\alpha\in{\mathcal{E}}(\mathcal{T})$ such that ${\mathrm{dist}}(x_n,x_n^{\alpha})$ is uniformly bounded.
- Or, for every $k\in{\mathbb{N}}$ we have $${\mathrm{dist}}(x_n,Q^k_n)\To_{n\to\infty}\infty.$$
Suppose there exist $k\in{\mathbb{N}}$ and $C>0$ so that ${\mathrm{dist}}(x_n,Q^k_n)\leq C$ for every $n\in{\mathbb{N}}$. Then, there exists a sequence of points $q_n\in Q^k_n$ such that for every $n\in{\mathbb{N}}$, ${\mathrm{dist}}(x_n,q_n)\leq C$.
Fix $m\geq k$ and define for $n\geq m$ the sequence $q_n^m=P_{n,m}(q_n)\in\Sigma_m$. For such a pair $(m,n)$ we have by definition $q_n^m\in Q_m^k$. The set $Q_m^k$ is finite so for a given $m\geq k$ infinitely many of the points $q_n^m$ coincide. Hence a diagonal argument provides an infinite subsequence of integers $(n_i)_{i\in{\mathbb{N}}}$ such that for every $m\geq k$ the sequence $(q^m_{n_i})_{i\in{\mathbb{N}}}$ of points of $Q^k_m$ is eventually constant (we denote $y_m$ the common value), and satisfies $p_m(q^{m+1}_{n_i})=q^{m}_{n_i}$ and ${\mathrm{dist}}(x_m,q^m_{n_i})\leq C$ for $i$ large enough.
Hence the sequence $(y_m)_{m\geq k}$ is the tail of a point of ${\mathcal{L}}$ which must be marked by some end $\alpha\in{\mathcal{E}}({\mathcal{T}})$ (this is because for every $m\geq k$, $y_m$ is the marked point $q_v$ of some surface $S_v$). This implies that $y_m=x_m^{\alpha}$ for every $m$ and finally that ${\mathrm{dist}}(x_m,x_m^\alpha)\leq C$ and the lemma follows.
\[l:discs\] Let ${\mathbf{x}}=(x_n)_{n\in{\mathbb{N}}}\in{\mathcal{L}}$ such that for every $\alpha\in{\mathcal{E}}(\mathcal{T})$ we have $${\mathrm{dist}}(x_n,x_n^{\alpha})\To_{n\to\infty}\infty.$$ Then we have $${r_{\mathrm{inj}}}(x_n)\to\infty.$$ In particular $L_{{\mathbf{x}}}$ is a disk by Proposition \[p.Cheeger\_Gromov\].
We must prove that the length of every geodesic loop based at $x_n$ tends to infinity with $n$. Arguing by contradiction, suppose there exists a sequence of geodesic loops $\gamma_{n}$ based at $x_{n}$ with uniformly bounded lengths $l_{\gamma_{n}}$.
Applying the dichotomy of Lemma \[l.dicotomia\] and the fact that $l_{\gamma_n}$ is uniformly bounded, we get ${\mathrm{dist}}(\gamma_n,Q^k_n)\to\infty$ for every $k\in{\mathbb{N}}$. This implies in particular that for every $k\in{\mathbb{N}}$ we have $\gamma_n\dans X_n^k$ for every $n$ large enough.
Then, there exists $k_n\to\infty$ such that $\gamma_n$ is included in $X_{n,k_n}$ for every $n\in{\mathbb{N}}$. There are two possibilities.
*Case 1. $\gamma_n$ is not isotopic to a boundary component of $X_{n,k_n}$.* Applying Proposition \[p:systole\_internal\] we get that the internal systole of $X_{n,k_n}$ goes to $+\infty$. On the other hand, since $\gamma_n$ is not isotopic to a boundary component of $X_{n_k}$, $l_{\gamma_n}$ is greater than its internal systole. Therefore, Case 1 happens for finitely many $n$.
*Case 2. $\gamma_n$ is isotopic to a boundary component of $X_{n,k_n}$.* Denote by $\beta_n$ the boundary component of $X_{n,k_n}$ isotopic to $\gamma_n$. We have $l_{\gamma_n}\geq l_{\beta_n}$ so $\beta_n$ has bounded length. In particular $\beta_n$ bounds a surface of uniformly bounded level (hence with bounded diameter) so the distance $D_n$ of $\gamma_n$ to $\beta_n$ tends to infinity. Arguing as in the proof of Lemma \[l.inj\_radius\] we obtain a lower bound
$$l_{\gamma_n}\geq \cosh(D_n) l_{\beta_n}\geq \cosh(D_n){\mathrm{sys}}(\Sigma_0).$$ This contradicts that $l_{\gamma_n}$ is uniformly bounded.
We will now end the proof of Theorem \[teo-mainabstract\] and characterize the topology of marked leaves.
#### [**Embedding direct limits –**]{}
Given $\alpha$, an end of $\mathcal{T}$ represented by a sequence $(e_n)_n$ of edges, we denote respectively by ${\mathbf{S}}^\alpha$ and $\overline{{\mathbf{S}}}^\alpha$ the open and geometric direct limits of the sequence of isometric embeddings $\{i_{e_n}:S_{o(e_n)}\to S_{t(e_n)}\}$ (recall definition \[d.geom\_direct\_limit\]). Recall that ${\mathbf{S}}^\alpha$ is diffeomorphic to the interior of $\overline{{\mathbf{S}}}^\alpha$.
\[l.encaje\_Lalpha\] For every end $\alpha\in{\mathcal{E}}(\mathcal{T})$ there exists an isometric embedding $\phi:\overline{{\mathbf{S}}}^\alpha\to L_{{\mathbf{x}}^\alpha}$.
Set $\alpha=(e_n)_n$ and $v_n=r_{\alpha}(n)$ for every $n$ greater or equal than the floor where $r_{\alpha}$ starts. By Proposition \[p.Cheeger\_Gromov\] the pointed leaf $(L_{{\mathbf{x}}^\alpha},{\mathbf{x}}^\alpha)$ is the Cheeger-Gromov limit of the sequence of pointed surfaces $(\Sigma_n,x^{\alpha}_n)_{n\in{\mathbb{N}}}$. More precisely, the proof of that Proposition shows that for every compact domain $D\dans L_{{\mathbf{x}}^\alpha}$ there exists $n_0$ such that for every $n\geq n_0$, the projection on the $n$-th coordinate induces an isometric embedding $$\Pi_n:(D,{\mathbf{x}}^\alpha)\to(\Sigma_n,x^{\alpha}_n).$$
For $n<m$, set $i_{m,n}=i_{m-1}\circ\ldots\circ i_n:S_{v_n}\to S_{v_m}$. This is an isometric embedding whose image is included inside the $R_n$-neighbourhood of $x_m^{\alpha}$ for some $R_n$ independent of $m>n$. Using the property stated above, for $m$ large enough the inverse of $\Pi_m$ induces a isometric embedding $\phi_n:S_{v_n}\to L_{{\mathbf{x}}^\alpha}$. This yields a sequence of isometric embeddings which satisfy $\phi_{n+1}\circ i_n=\phi_n$ (note that we have $i_{m,n}\circ\Pi_n=\Pi_m$ when the left-hand term is defined).
Using the universal property of direct limits and the fact that the leaf $L_{{\mathbf{x}}^\alpha}$ is a complete Riemannian surface, we see there exists an isometric embedding of the geometric direct limit $\phi:\overline{{\mathbf{S}}}^\alpha\to L_{{\mathbf{x}}^\alpha}$.
#### [**Topology of the marked leaf –**]{}
The embedding obtained in Lemma \[l.encaje\_Lalpha\] might not be surjective. In fact, its image can be complicated from the geometric point of view since we don’t control the lengths of the boundary components of the surfaces $S_{v_n}$. Nevertheless, using Proposition \[p.recognize\_surface\] and the geometric properties of admissible towers, we will prove that this embedding contains all the topological information of the complete hyperbolic surface $L_{{\mathbf{x}}^\alpha}$.
Recall the definition of open direct limit in §\[ss.surfaces-forests\]. Given an end $\alpha=(e_n)$ we define ${\mathbf{S}}^{\alpha}$ as the open direct limit of the sequence of embeddings $(j_{e_n})$.
\[p:topology\_leaves\] The leaf of a sequence ${\mathbf{x}}^{\alpha}$ is diffeomorphic to the open direct limit ${\mathbf{S}}^\alpha$.
This proposition finishes the topological characterization of all leaves of ${\mathcal{L}}$ and the proof of Theorem \[teo-mainabstract\].
We will fix $\alpha\in{\mathcal{E}}(\mathcal{T})$ and note $v_n=r_{\alpha}(n)$. Consider the closed surface defined in Lemma \[l.encaje\_Lalpha\] $$S=\phi(\overline{{\mathbf{S}}}^\alpha).$$ This is a closed surface with (possibly) geodesic boundary (see Proposition \[p:geom\_limit\]).
We shall prove Proposition \[p:topology\_leaves\] in two steps, by checking that the pair $(L_{{\mathbf{x}}^\alpha},S)$ satisfies the hypotheses of Proposition \[p.recognize\_surface\]. For this, we need two lemmas.
\[l.no\_closed\_geod\_outside\] Let $C$ be a connected component of $L_{{\mathbf{x}}^\alpha}{\setminus}S$. Then $C$ contains no closed geodesic.
Suppose there exists a closed geodesic $\gamma$ included in $C$. The projections $\gamma_n=\Pi_n(\gamma)$ define a sequence of closed geodesics in $\Sigma_n$ disjoint from $S_{v_n}$ with the same length and located at a uniform distance to $x_n^{\alpha}$ (by definition of the leaf $L_{{\mathbf{x}}^\alpha}$). In particular it must be disjoint from all other subsurfaces $S_{v}$ with $v\in V_n(\mathcal{T})$, when $n$ is large enough (by Lemma \[l:dif\_comp\]). Hence it must be completely included in ${\mathrm{Int}}(X_n)$ for $n$ large enough, contradicting that the internal systole of $X_n$ goes to infinity as $n$ grows.
\[l.no\_connexion\] Let $C$ be a connected component of $L_{{\mathbf{x}}^\alpha}{\setminus}S$. Then the boundary $\partial C$ is connected.
Assume that $\partial C$ has more than one boundary component (that can be a closed geodesic or a complete geodesic). Then there exists a simple geodesic arc $\gamma$ contained in $\overline{C}$ and connecting two points $y$ and $z$ of these two connected components. Consider $D_1,D_2$, two disks inside $L_{{\mathbf{x}}^\alpha}$ centered at $y$ and $z$ respectively.
Now note that $S$ is an increasing union of compact surfaces with boundary $S_n$ such that $\Pi_n:S_n\to S_{v_n}$ is an isometry for every $n$. For $n$ large enough, there exist two points $y_n\in D_1\cap\partial S_n$ and $z_n\in D_2\cap \partial S_n$ and a simple geodesic arc $\gamma_n$ between them, that is outside $\textrm{Int}(S_n)$ and whose length is bounded independently of $n$. As a consequence there exists a compact domain $D$ containing all geodesics $\gamma_n$. For $n$ large enough the projection $\Pi_n|_D:D\to\Sigma_n$ is an isometric embedding and the projection of the $\gamma_n$ (still denoted by $\gamma_n$) to $\Sigma_n$ is a path in $\Sigma_n{\setminus}S_{v_n}$ connecting two points, denoted abusively by $y_n,z_n$, of $\partial S_{v_n}$. Fix such an $n$ and assume that the quantity $K_n$ (recall that it denotes a lower bounds of the width of half-collars of boundary components of $X_n$) is $>l_{\gamma_n}$. There are two possibilities.
If $y_n$ and $z_n$ belong to two distincts connected components of $\partial S_{v_n}$ then, arguing as in Lemma \[l:dif\_comp\] (two large and disjoint half collars are attached to these components) we see that $l_{\gamma_n}\geq 2K_n$ which is a contradiction.
If $y_n$ and $z_n$ belong to the same boundary component, called $\alpha_n$, then $\gamma_n$, which has length $<K_n$, must be completely included inside a collar about $\alpha_n$. This implies that the geodesics $\alpha_n$ and the simple geodesic arc $\gamma_n$ form a bigon, which is absurd.
Including forests of surfaces in towers {#s.including_forest}
=======================================
The rest of the paper is devoted to the proof of Theorem \[t.uno\] and Theorem \[t.tres\]. Both theorems will be deduced from the more general result
\[p.forestinclusion\]Given a forest of surfaces $\mathcal{S}$ there exists a tower of finite coverings $\mathbb{T}$ which is admissible with respect to $\mathcal{S}$.
Consider a forest of surfaces $$\mathcal{S}=(\mathcal{T},\{Z_v\}_{v\in V(\mathcal{T})},\{j_e\}_{e\in E(\mathcal{T})}).$$ Proceeding inductively, we will construct a tower of finite coverings $\mathbb{T}$ which is admissible with respect to $\mathcal{S}$.
#### [**The base case –**]{}
Consider a hyperbolic surface $\Sigma_0$ containing a set of pairwise disjoint subsurfaces with geodesic boundary $\{S_v:v\in V_0(\mathcal{T})\}$, so that each $S_v$ is homeomorphic to $Z_v$. Define $S_0:=\bigcup_{v\in V_0(\mathcal{T})} S_v$ and $X_0$ as it admissible complement.
#### [**The induction hypothesis –**]{}
Suppose we have already included $\mathcal{S}$ up to the floor $k$. Namely, we have:
- Finite coverings $p_i:\Sigma_{i+1}\to\Sigma_i$ for $i=0,\ldots,k-1$
- Subsurfaces $S_i\dans \Sigma_i$ with $S_i=\bigsqcup_{v\in V_i(\mathcal{T})} S_v$ where each $S_v$ is homeomorphic to $Z_v$ and $i=1,\ldots,k$
- A family of lifts $\{j_e:S_{o(e)}\to S_{t(e)}\}$ where $e$ ranges over $E_i(\mathcal{T})$ with $i\leq k-1$
- A family of homeomorphisms $\{h_v:Z_v\to S_v;\,v\in V_i(\mathcal{T}),i\leq k\}$
such that for every $e\in E_{i}(\mathcal{T})$ we have $h_{t(e)}\circ i_e=j_e\circ h_{o(e)}$
- The internal systole of $X_i^{\ast}$ and half-collar width of the boundary components of $X_i$ are greater than $i$ for $i=1,\ldots,k$; where $X_i^{\ast}$ is the admissible complement of $$S_{i}^{\ast}=\bigcup_{e\in E_{i-1}}j_e(S_{o(e)})$$ and $X_i$ is the admissible complement of $S_i$.
#### [**The induction step –**]{}
We will now use the tools described in Section \[s.toolbox\] in order to construct the desired covering map $p_k:\Sigma_{k+1}\to\Sigma_k$.\
[*Step 1. Creating new roots and space for extending level $k$ surfaces.*]{} Take $g\in{\mathbb{N}}$ such that every surface in $\{Z_{t(e)};\,e\in E_{k}(\mathcal{T})\}$ can be realized a subsurface with geodesic boundary of a hyperbolic surface with geodesic boundary with genus $g$ and two boundary components. Define $$S_0=\bigsqcup_{v\in R_{k+1}(\mathcal{T})}Z_v$$ the union of surfaces corresponding to the roots of ${\mathcal{T}}$ appearing at floor $k+1$.
Applying Theorem \[t.finitecovering\] and Remark \[r.room\] we can construct a finite covering $q_1:\Sigma^{(1)}\to \Sigma_{k}$ such that $\Sigma^{(1)}$ admits an admissible decomposition $(X^\ast,S^\ast)$ satisfying
- $S^\ast$ decomposes as $$S^{\ast}=\bigsqcup_{e\in E_k(\mathcal{T})} S^{\ast}_{t(e)}$$ such that $q_1$ restricted to $S^{\ast}_{t(e)}$ is an isometry onto $S_{o(e)}$ for every $e\in E_{k}(\mathcal{T})$. Denote by $j^{(1)}_e:S_{o(e)}\to S^{\ast}_{t(e)}$ the inverses of these restrictions;
- the internal systole of $X^\ast$ is $\geq k+1$;
- $X^{\ast}$ contains a subsurface with geodesic boundary $X=X_1\sqcup X_2$ such that:
- $X_1$ is homeomorphic to $S_0$;
- For every boundary component $\alpha_j$ of $X^{\ast}$ there exists a genus $g$ surface $A_j$ included in $X_2$ with two boundary components one of which is $\alpha_j$. Moreover, subsurfaces corresponding to different boundary components are disjoint.
Note that by this last condition, $X_1\dans{\mathrm{Int}}(X^\ast)$.\
[*Step 2. Recognizing level $k+1$ subsurfaces –*]{} We shall now proceed to construct both families $S^{(1)}=\{S^{(1)}_v:v\in V_{k+1}(\mathcal{T})\}$ and $h^{(1)}=\{h_v^{(1)}:v\in V_{k+1}(\mathcal{T})\}$ simultaneously.
- [*Case $v\in R_{k+1}(\mathcal{T})$.* ]{}
By construction, we have a decomposition $X_1=\bigsqcup_{v\in R_{k+1}(\mathcal{T})}S^{(1)}_{v}$ where each $S^{(1)}_{v}$ is homeomorphic to $Z_v$. For every $v\in R_{k+1}(\mathcal{T})$ define $h_{v}$ as any homeomorphism between $S^{(1)}_{v}$ and $Z_v$.
- [*Case $v=t(e)$ for some $e\in E_k(\mathcal{T})$.*]{}
Set $A={\mathrm{Int}}(X_2)\dans\Sigma^{(1)}$. This is a key point of the whole construction. Recall that by definition of good inclusion we have that $\overline{Z_{t(e)}\setminus i_e(Z_{o(e)})}$ admits a hyperbolic structure with geodesic boundary. Therefore, the choice of $g$ and the construction of $A$, imply that there is enough room to construct a pairwise disjoint family of subsurfaces with geodesic boundary $\{S^{(1)}_{t(e)};\,e\in E_{k}(\mathcal{T})\}$ and a family of homeomorphisms $\{h^{(1)}_{t(e)}:Z_{t(e)}\to S^{(1)}_{t(e)};e\in E_{k}(\mathcal{T})\}$ so that
1. $S^{(1)}_{t(e)}\subseteq S^{\ast}_{t(e)}\cup A$
2. $h^{(1)}_{t(e)}\circ i_e=j^{(1)}_e\circ h_{o(e)}$
Define $$S^{(1)}=\bigcup_{v\in V_{k+1}(\mathcal{T})}S^{(1)}_v.$$ Using notations coherent with §\[ss.decomposition\], we define $S^{(1)}_{v,k+1}$ as follows:
- if $v\in R_{k+1}(\mathcal{T})$, define $$S^{(1)}_{v,k+1}=S^{(1)}_{v}$$
- Otherwise $v=t(e)$ for some $e\in E_k(\mathcal{T})$ and we define $$S^{(1)}_{v,k+1}=\overline{S^{(1)}_{v}\setminus S^{\ast}_{t(e)}}$$
Notice that, since the internal systole of $X^{\ast}$ is greater than $k+1$, so are those of surfaces $S^{(1)}_{v,k+1}$. Also for the same reason, the boundary components of these surfaces which are not in the interior of the $S_v^{(1)}$ have length greater than $k+1$.\
[*Step 3. Increasing collars of the admissible complement –*]{} Applying Theorem \[t.finitecovering\] again, we can construct a covering $q_2:\Sigma_{k+1}\to \Sigma^{(1)}$ with a subsurface $S_{k+1}$ such that
- $S_{k+1}$ decomposes as $$S_{k+1}=\bigsqcup_{v\in V_{k+1}(\mathcal{T})}S_v$$ where $q$ restricted to $S_v$ is an isometry onto $S^{(1)}_v$ for every $v\in V_{k+1}(\mathcal{T})$;
- The admissible complement of $S_{k+1}$, denoted by $X_{k+1}$, satisfies:
1. The internal systole of $X_{k+1}$ is greater than $k+1$
2. The boundary of $X_{k+1}$ has a collar of width $k+1$
Let $j_v^{(2)}$ denote the inverse of $q_2|_{S_v}$ for every $v\in V_{k+1}(\mathcal{T})$. We define
- $p_k$ as the composition $q_2\circ q_1$;
- $j_e$ as the composition $j_{t(e)}^{(2)}\circ j_e^{(1)}$ for every $e\in E_{k}(\mathcal{T})$;
- $h_v$ as the composition $j_v^{(2)}\circ h^{(1)}_v$ for every $v\in V_{k+1}(\mathcal{T})$;
- $S_{v,k+1}$ as the images by the maps $j_v^{(2)}$ of the surfaces $S^{(1)}_{v,k+1}$;
We define $S_{k+1}^{\ast}=\bigcup_{e\in E_{i-1}}j_e(S_{o(e)})$ and $X_{k+1}^{\ast}$ as its admissible complement.\
[*Step 4. The internal systole of $X_{k+1}^{\ast}$ –*]{} It remains to check that the internal systole of $X^{\ast}_{k+1}$ is greater than $k+1$. For this, we consider the decomposition stated in $$X_{k+1}^\ast=X_{k+1}\cup\bigcup_{v\in V_{k+1}}S_{v,k+1}.$$
The following properties are satisfied.
- The boundary components of subsurfaces in the decomposition lying on the interior of $X_{k+1}^{\ast}$ satisfy
1. They are contained in the boundary of $X_{k+1}$ and therefore have a half-collar width greater than $k+1$ by construction.
2. They belong to a boundary component of $S_{v,k+1}$ not contained in the interior of the $S_v$, for some $v\in V_{k+1}(\mathcal{T})$. Therefore they have length greater than $k+1$ (see the end of Step 2)
- The internal systole of $X_{k+1}$ is greater than $k+1$
- The internal systole of each surface $S_{v,k+1}$ is greater than $k+1$
Now we are in condition to apply Lemma \[l.systole-gluing\] to the decomposition of $X_{k+1}^{\ast}$ and get that the internal systole of $X_{k+1}^{\ast}$ is greater than $k+1$ as desired. This finishes the proof of the Proposition.
Proof of main theorems {#s.final_proof}
======================
Combinatorial representation of surfaces
----------------------------------------
In order to prove Theorems \[t.uno\] and \[t.tres\] we will apply Proposition \[p.forestinclusion\] and Theorem \[teo-mainabstract\] to particular choices of surface forests. We proceed to define coding trees which will give us a combinatorial framework to construct surface forests. We recall here that we will prove this theorems ignoring the cylinder leafs, but that the arguments may be easily addapted to include cylinders as leafs (c.f. Remark \[r.cyl\]).
#### [**Coding trees –**]{}
In [@BWal], Bavard and Walker define a combinatorial object (they call it *core tree*), which is a rooted and colored tree that gives a combinatorial framework for describing open surfaces. We give a variation of their setting that is more suited to our purposes.
A *coding tree* is a connected and rooted tree $\Lambda$ having two types of vertices
- *boundary vertices* that must have valency $1$ or $2$
- *simple vertices* that must have valency $1$, $2$ or $3$
and satisfying the following properties:
- The *root* must be a simple vertex.
- All simple vertices must have valency strictly greater than one, with the only possible exception of the root.
- Edges must join a boundary vertex with a simple vertex (they cannot joint two vertices of the same type).
Any coding tree $\Lambda$ can be written as the union of the balls of radius one around its simple vertices. Notice that this balls meet at boundary vertices, and consist of simple vertices with one, two or three adjacent vertices, that will be calles valency one, two or three basic pieces respectively. Call these sub-graphs the *basic sub-graphs* of $\Lambda$.
#### [**From coding trees to pointed surfaces –**]{}
We proceed to describe how to associate a pointed surface to coding tree $\Lambda$. First, associate to each basic subgraph of $\Lambda$ a surface with boundary as follows
- Valency one simple vertex $\leftrightarrow$ surface of genus one with one boundary component
- Valency two simple vertex $\leftrightarrow$ surface of genus one with two boundary components
- Valency three simple vertex $\leftrightarrow$ surface of genus zero with three boundary components
![From coding trees to surfaces.[]{data-label="fig:tree_surf"}](Core_tree_surface-eps-converted-to.pdf)
Consider also a correspondence between the boundary vertices of each basic piece and the boundary components of its associated surface. Then, glue the boundary components of two different surfaces if their associated boundary vertices are equal. To obtain a pointed surface, take any point in the interior of the sub-surface corresponding to the root of $\Lambda$. See Figure \[fig:tree\_surf\].
Notice that the topological type of the resulting pointed surface does not depend on the choices that we have made. Abusing notation we will denote $\Sigma_{\Lambda}$ to either the topological type obtained by the previous construction or to a particular representative of this equivalence class of topological surfaces.
#### [**Every surface can be obtained from a coding tree –**]{}
The proof of the following Lemma is a straightforward adaptation of [@BWal Lemma 2.3.1] which relies on the classification of open surfaces given by classifying triples.
\[t:bavad\_walker\] Let $S$ be an open orientable surface other than the disc and the annulus. Then there exists a coding tree $\Lambda$ such that $S$ is homeomorphic $\Sigma_{\Lambda}$.
#### [**Good inclusions between coding trees –**]{}
Consider coding trees $\Lambda_1$ and $\Lambda_2$. We say that an inclusion $j:\Lambda_1\to\Lambda_2$ is a *good inclusion of coding trees* if $j$ is an injective map that preserves the graph structure, the roots, the vertices types, and whose image boundary consists of a union of boundary vertices.
Notice that a good inclusion of coding trees $j:\Lambda_1\to\Lambda_2$ naturally induces a good inclusion of surfaces $j_{\Sigma}:\Sigma_{\Lambda_1}\to\Sigma_{\Lambda_2}$ that respects sub-surfaces coming from basic sub-graphs.
When there is a bijective good inclusion between two coding trees we will say they are isomorphic. Given a coding tree $\Lambda$ we will note $[\Lambda]$ its class of isomorphism.
#### [**Forests of coding trees –**]{}
In the rest of the section we want to show how to use this combinatorial description for surfaces in order to exhibit a combinatorial way to organize non-compact surfaces.
Analogously to the definition of forest of surfaces, we define a *forest of coding trees* as a triple $$\mathcal{T}^{\ast}=(\mathcal{T},\{\Lambda_v\}_{v\in V(\mathcal{T})},\{j_e\}_{e\in E(\mathcal{T})})$$ where $\{\Lambda_v\}_{v\in V(\mathcal{T})}$ is a family of finite coding trees and $\{j_e\}_{e\in E(\mathcal{T})}$ is a family of good inclusions $j_e:\Lambda_{o(e)}\to\Lambda_{t(e)}$.
There is also an analogous definition of set of *limit coding trees* that associates a coding tree $\Lambda^{\alpha}$ to each end $\alpha\in{\mathcal{E}}(\mathcal{T})$.
#### [**From forests of coding trees to forests of surfaces –**]{}
Consider $\mathcal{T}^{\ast}$ a forest of coding trees over $\mathcal{T}$. We define its associated forest of surfaces as follows (see Figure \[fig:forestouille\]):
![A forest of coding trees and its associated forest of surfaces.[]{data-label="fig:forestouille"}](Forest_core_surfaces-eps-converted-to.pdf)
For every vertex $v\in V(\mathcal{T})$ take a surface $S_v$ with the topological type of $\Sigma_{\Lambda_v}$. Then, for every edge $e\in E(\mathcal{T})$ consider a good inclusion of surfaces $({j_e})_{\Sigma}:S_{o(e)}\to S_{t(e)}$ preserving the sub-surfaces corresponding to basic subgraphs. We will note $\Sigma_{\mathcal{T}^{\ast}}$ the resulting forest of surfaces.
Although several choices were made in the previous construction, different choices give rise to *isomorphic trees of surfaces*. Namely, if
$$\Sigma^{(1)}_{\mathcal{T}^{\ast}}=(\mathcal{T},\{S_v\}_{v\in V(\mathcal{T})},\{{(i_e)}_{\Sigma}\}_{e\in E(\mathcal{T})})$$ and $$\Sigma^{(2)}_{\mathcal{T}^{\ast}}=(\mathcal{T},\{Z_v\}_{v\in V(\mathcal{T})},\{{(j_e)}_{\Sigma}\}_{e\in E(\mathcal{T})})$$ are trees of surfaces obtained taking different choices, we can find a family of homeomorphisms $\{h_v:S_v\to Z_v\}_{v\in V(\mathcal{T})}$ preserving roots and satisfying $j_e\circ h_{o(e)}=h_{t(e)}\circ i_e$ for every $e\in E(\mathcal{T})$. In particular, if $\alpha$ is an end of ${\mathcal{E}}(\mathcal{T})$, the corresponding limit surfaces $S^{\alpha}$ and $Z^{\alpha}$ are homeomorphic.
\[r.limitcorrespondence\] It follows directly from the definitions that $S^{\alpha}$ is homeomorphic to the interior of $\Sigma_{\Lambda^{\alpha}}$
Constructing surface forests {#s.constructingcodingtrees}
----------------------------
By Theorem \[teo-mainabstract\] and Proposition \[p.forestinclusion\], in order to prove Theorem \[t.uno\], it is enough to show that the family of all surfaces can be realized *inside* the set of ends a set of a forest of surfaces. In order to prove Theorem \[t.tres\], it is enough to show that every *finite or countable* family of open surfaces can be realied *as* the set of ends of of a forest of surfaces (see Remark \[r.forestjustif\]).
On the other hand, by Remark \[r.limitcorrespondence\] and Lemma \[t:bavad\_walker\] this reduces to realizing certain sets of coding trees as the set of limit coding trees of a forest.
#### [**The universal forest of coding trees–**]{}
In the particular case of Theorem \[t.uno\], it is enough to construct a forest of coding trees $\mathcal{T}^{\ast}$ so that every coding tree appears as a limit coding tree.
Consider $(\Lambda,v_0)$ a coding tree. Notice that even though not every ball included in $\Lambda$ is a coding tree, if $v_0$ si the root, then $B_{\Lambda}(v_0,2n+1)$ is also a coding tree for every $n\in{\mathbb{N}}$.
#### [**The construction of $\mathcal{T}^{\ast}$ –**]{}
We start defining the underlying forest $\mathcal{T}$, for this we define $$V_n(\mathcal{T})=\Bigg\{\bigg[B_{\Lambda}(v_0,2n+1)\bigg]:(\Lambda,v_0) \text{ coding tree}\Bigg\}$$
Since vertices in coding trees have bounded valency, $V_n(\mathcal{T})$ is a finite set. We define that $$([\Lambda_1],[\Lambda_2])\in E_n(\mathcal{T})\subseteq V_n(\mathcal{T})\times V_{n+1}(\mathcal{T})$$ if there exists a good inclusion $i:\Lambda_1\to\Lambda_2$ (recall that good inclusions preserve the root). Given $[\Omega]\in V(\mathcal{T})$ define $\Lambda_{[\Omega]}$ as any representative of $[\Omega]$, and given
$$e=([\Omega_1],[\Omega_2])\in E(\mathcal{T})$$
define $i_e$ as any good inclusion from $\Lambda_{[\Omega_1]}$ to $\Lambda_{[\Omega_2]}$. Summarizing, our forest of coding trees is $$\mathcal{T}^{\ast}=(\mathcal{T},\{\Lambda_{[\Omega]}\}_{[\Omega]\in V(\mathcal{T})},{\{ j_e\}}_{e\in E(\mathcal{T})})$$
Finally, notice that if $(\Omega,v_0)$ is a coding tree, the ray $r(n)=[B_{\Omega}(v_0,2n+1)]\subseteq\mathcal{T}$ represents an end $\alpha\in{\mathcal{E}}(\mathcal{T})$ satisfying $\Lambda^{\alpha}\cong (\Omega,v_0)$.
#### [**The countable forest of coding trees –**]{}
In order to prove Theorem \[t.tres\], we have to construct a forest of coding trees $\mathcal{T}^{\ast}$ that realizes any given countable family of coding trees $\{\Lambda_1,\ldots,\Lambda_i,\ldots\}$ as its set of limit coding trees.
#### [**The construction of $\mathcal{T}^{\ast}$:**]{}
We start defining the underlying forest. For this, set $$V_n(\mathcal{T})=\{B_{\Lambda_i}(v_i,r_{n,i}):i=1,\ldots,n\}$$ with $r_{n,i}=2(n-i)+1$. Notice that in this case the vertices are not equivalence classes of coding trees, but actual coding trees. Then, we define $\Lambda_v=v$ for every $v\in V(\mathcal{T})=\bigsqcup V_n(\mathcal{T})$ and set $$(\Omega_1,\Omega_2)\in E(\mathcal{T})$$ if and only if there exists $i,k\in{\mathbb{N}}$ so that $\Omega_1=B_{\Lambda_i}(v_i,r_{k,i})$ and $\Omega_2=B_{\Lambda_i}(v_i,r_{k+1,i})$.
Appendix: Coverings and the second systole {#s.appendix}
==========================================
In this appendix we prove the Theorem below. A proof was first suggested by Henry Wilton on MathOverflow, as an answer to a question posed by the authors in that site (see [@HW]), while the proof presented here grew out of conversations between the authors.
\[t.appendix\]
Let $\Sigma$ be a closed hyperbolic surface, and let $\alpha\subset\Sigma$ be a simple closed geodesic. Then, for all $K>0$, there exists a finite covering $\pi:\hat \Sigma\to\Sigma$ such that
- $\hat \Sigma$ contains a non-separating simple closed geodesic such that $\pi(\hat \alpha)=\alpha$ and $\pi$ restricts to a homeomorphism on $\hat \alpha$;
- every simple closed geodesic which is not $\hat \alpha$ has length larger than $K$.
#### [**Adapted metrics –**]{}
Recall that, on a compact manifold, any two Riemannian metrics are bi-Lipschitz equivalent. It should be observed that the statement of Theorem C does not depend on the metric chosen on $\Sigma$, up to rescaling $K$. Therefore, we may as well suppose that the metric on $\Sigma$ is adapted to the problem, and we will choose it in the following way. We will say that the metric on $\Sigma$ is [*adapted*]{} to $\alpha$ if the two following conditions are satisfied
1. $\alpha$ realizes the unique systole of $\Sigma$;
2. the width of the collar about $\alpha$ is larger than the length of $\alpha$.
#### [**Length spectrum –**]{}
If $\Sigma$ is a compact hyperbolic surface, we denote by $$LS(\Sigma) = \left( \ell_1(\Sigma),\ell_2(\Sigma),\ldots,\right)$$ the (unmarked) length spectrum of $\Sigma$ with multiplicity, but with the restriction that we do not take any higher power of any curve realizing the systole of $\Sigma$. In other words, we enumerate, up to making choices, the unoriented closed geodesic curves $\gamma_n\subset\Sigma$, with $n\geq 1$, that either realize the systole of $\Sigma$, or meet such a curve only transversally if at all, sort them by increasing length and set $\ell_k(\Sigma)$ to be the length of $\gamma_k$. Recall that the length spectrum of every compact hyperbolic surface is discrete, and $\ell_k(\Sigma)\to+\infty$ as $k$ goes to $+\infty$.
#### [**Increasing the second systole –**]{}
We now reduce Theorem C to the following statement.
\[teo:bis\] Let $\Sigma$ be a closed hyperbolic surface, with metric adapted to some nonseparating simple closed geodesic $\alpha$. Then there exists a finite covering $\pi\colon\Sigma'\to\Sigma$ such that $\alpha$ has a unique $(1:1)$-lift to $\Sigma'$ and $\ell_2(\Sigma')>\ell_2(\Sigma)$.
First, up to starting with a cover of degree two, we may assume without loss of generality that $\alpha$ is nonseparating, and then we may choose a hyperbolic metric adapted to $\alpha$. Now let $$d_1<d_2<\cdots$$ be the (unmarked) length spectrum of $\Sigma$, without multiplicity, and without restrictions (*i.e.*, this time we consider all geodesic curves). Let $\pi^{(1)}\colon\Sigma^{(1)}\to\Sigma$ be a covering of $\Sigma$ as in Theorem \[teo:bis\]. It follows from the statement of this theorem that $\alpha$ admits a unique lift $\alpha^{(1)}$ such that $\pi^{(1)}$ is $(1:1)$ in restriction to $\alpha^{(1)}$, and we have $\ell_2(\Sigma^{(1)})\geq d_2$. In particular $\alpha^{(1)}$ is the unique systole of $\Sigma^{(1)}$, and it follows that the metric on $\Sigma^{(1)}$ is adapted to $\alpha^{(1)}$. Hence, we may apply Theorem \[teo:bis\] to $(\Sigma^{(1)},\alpha^{(1)})$, getting a covering $\pi_{(2)}\colon\Sigma^{(2)}\to\Sigma^{(1)}$, and so on.
This yields a sequence of finite coverings $\Sigma^{(k)}\to\Sigma$. By construction, for all $k$, $\Sigma^{(k)}$ has a closed geodesic $\alpha^{(k)}$ mapping homeomorphically to $\alpha$, and the sequence of second systoles of $\Sigma^{(k)}$ is strictly increasing.
Now the geodesics of $\Sigma^{(k)}$ realizing this second systole project to geodesics of $\Sigma$; it follows that $\ell_2(\Sigma^{(k)})\geq d_k$ for all $k$. Since the length spectrum of $\Sigma$ is discrete, this sequence of coverings provides, for $k$ large enough, a covering satisfying the conclusion of Theorem C.
#### [**Product of coverings –**]{}
Before carrying out the proof, let us recall a construction of the smallest common covering associated to a finite family of coverings.
\[d:product\_cover\] Consider a finite family of finite covering maps $\pi^{(i)}:\Sigma^{(i)}\to\Sigma$, for $i=1,...,n$. The *product* of these coverings is the map $\widehat{\pi}:\widehat{\Sigma}\to\Sigma$ where $$\widehat{\Sigma}=\left\lbrace (x,y_1,\ldots,y_n)\in
\Sigma\times\Sigma^{(1)}\times\cdots\times\Sigma^{(n)}:\, \forall j,
\pi^{(j)}(y_j)=x \right\rbrace$$and $\widehat{\pi}(x,y_1,\cdots,y_n)=x$.
Notice that by the homotopy lifting property this cover is connected (though this will not be used in the proof).
This notion allows us to reduce the proof of Theorem \[teo:bis\] to that of the following technical result which can also be seen as a straightening of residual finiteness.
\[lem:2curvas\] Let $\Sigma$ be a closed hyperbolic surface with metric adapted to some nonseparating simple closed curve $\alpha$. Let $\beta$ be a geodesic of $\Sigma$ realizing the length $\ell_2(\Sigma)$, and intersecting $\alpha$ only transversally. Let $N>0$ be an integer.
Then there exists a finite covering $\pi_\beta\colon\Sigma'\to\Sigma$, such that $\beta$ admits no $(1:1)$ lifts, and such that $\alpha$ admits a unique $(1:1)$ lift $\hat \alpha$, and such that all other lifts of $\alpha$ are at least $(N:1)$.
Let $\gamma_1,\ldots,\gamma_n$ be all the geodesics of $\Sigma$ involved in the definition of the length $\ell_2(\Sigma)$, and let $N\geq 1$ be such that $N\ell_1(\Sigma)>\ell_2(\Sigma)$. For each $j\in\{1,\dots,n\}$ let $\pi_{\gamma_j}\colon\Sigma^{(j)}\to\Sigma$ be a finite covering as provided by Lemma \[lem:2curvas\]. We will prove that the product covering of all these coverings satisfies the conclusion of Theorem \[teo:bis\]. Thus, let us consider $\hat \pi:\widehat{\Sigma}\to\Sigma$, the product of all these coverings. Recall that a point of $\widehat{\Sigma}$ is denoted by $(x,y_1,\cdots y_n)$, $y_j\in\Sigma^{(j)}$ and that $\hat \pi$ is the projection on the first coordinate.
By construction, $\alpha$ has a unique lift to $\widehat{\Sigma}$. It consists of points $(x,y_1,\ldots,y_n)$ such that $x$ lies in $\alpha$ and such that $y_j$ lies in the unique $(1:1)$ lift of $\alpha$ to $\Sigma^{(j)}$, for all $j$. It follows that $\hat \alpha$ is the unique systole of $\widehat{\Sigma}$, and $\ell_2(\widehat{\Sigma})>\ell_2(\Sigma)$. Thus, let us consider a geodesic $\gamma$ of $\widehat{\Sigma}$ that may intersect $\hat \alpha$ only transversally (or equivalently, which is not a power of $\hat \alpha$), we have to prove that its length is $>\ell_2(\Sigma)$. If $\gamma$ projects to $\alpha$ in $\Sigma$, then, as $\gamma$ intersects $\hat \alpha$ only transversally, there exists $j\in\{1,\ldots,n\}$ such that the image of $\gamma$ in $\Sigma^{(j)}$ is not the $(1:1)$ lift of $\alpha$. Hence, the length of $\gamma$ is at least $N\ell_1(\Sigma)$, which is (strictly) larger than $\ell_2(\Sigma)$.
Otherwise, $\gamma$ projects to a curve $\hat \pi(\gamma)$ which may intersect $\alpha$ only transversally. By definition of $\ell_2(\Sigma)$, it follows that the length of $\hat \pi(\gamma)$ is at least $\ell_2(\Sigma)$, and equal to $\ell_2(\Sigma)$ only if $\hat \pi(\gamma)$ is one of the $\gamma_j$, $j\in\{1,\ldots,n\}$. But $\gamma_j$ does not have any $(1:1)$ lifts to $\Sigma^{(j)}$. It follows that $\gamma$ is not a $(1:1)$ lift of $\gamma_j$, hence the length of $\gamma$ is strictly larger than $\ell_2(\Sigma)$ in either case, and Theorem \[teo:bis\] is proven.
#### [**Curves realizing the second systole –**]{}
We now use the assumption made on the metric in order to study curves that realize the second systole.
\[lem:BetaCool\] Let $\Sigma$ be a closed hyperbolic surface with metric adapted to a closed geodesic $\alpha$. Let $\beta$ be a curve intersecting $\alpha$ only transversally, and realizing the length $\ell_2(\Sigma)$. Then $\beta$ is simple, and its geometric intersection number with $\alpha$ is at most one.
First, let us prove that $\beta$ is simple, by contradiction. Recall (see [@Bu Theorem 4.2.4]) that if a curve realizes the minimum of the length among all primitive non-simple curves, then it is a figure eight. Obviously, the curve $\beta$ is primitive by assumption, hence this theorem applies: $\beta$ has a unique self-intersection point. Let us choose this intersection point as a base point and write $\beta=\beta_1 \beta_2$ as the concatenation of two closed loops, both geodesic except at the base point. Also, $\beta_3=\beta_1 \beta_2^{-1}$ is a non-trivial loop, geodesic except at the base point.
By minimality of $\ell_2(\Sigma)$ and uniqueness of the curve realizing the systole on $\Sigma$, it follows that $\beta_1$, $\beta_2$ and $\beta_3$ are all freely homotopic to $\alpha$, hence $\beta_2$ and $\beta_3$ are conjugate to $\beta_1^{\pm 1}$. In the abelianization of $\pi_1(\Sigma)$, this gives a contradiction, as $\alpha$ was supposed to be non-separating, hence nontrivial in homology.
Now, again by contradiction, suppose that $\alpha$ and $\beta$ intersect at least twice. Then we may decompose $\beta$ as the concatenation of two geodesic segments, $\beta_1$ and $\beta_2$, with common endpoints on $\alpha$, and let $\alpha_1$ be a geodesic subpath of $\alpha$ of minimal length and joining these two endpoints of $\beta_1$ and $\beta_2$. As the collar around $\alpha$ is greater than its length, $\alpha_1$ is shorter than $\beta_2$. Hence the curve formed by $\alpha_1$ and $\beta_1$ is shorter than $\beta$, and it is essential and non-freely homotopic to $\alpha$ (for otherwise $\alpha$ and $\beta$ would form a bigon). This contradicts the minimality assumption on $\beta$.
#### [**Conclusion –**]{}
Now we finish the proof of the theorem.
Thanks to Lemma \[lem:BetaCool\], we are left with four possibilities:
1. $i(\alpha,\beta)=1$;
2. $i(\alpha,\beta)=0$ and $(\alpha,\beta)$ is free in the homology of $\Sigma$;
3. $i(\alpha,\beta)=0$ and $[\beta]=0$ in homology;
4. $i(\alpha,\beta)=0$ and $[\alpha]+[\beta]=0$ in homology mod 2;
these four cases are illustrated in Figure \[fig:4casos\].
![The curve $\alpha$ and the four possible cases for $\beta$.[]{data-label="fig:4casos"}](Apendice-1.pdf)
Cases (1) and (2) are the easiest to deal with: in both these cases, we can find two disjoint, simple geodesics $\delta_1$, $\delta_2$ with geometric intersection numbers $i(\alpha,\delta_2)=i(\beta,\delta_1)=0$ and $i(\alpha,\delta_1)=i(\beta,\delta_2)=1$. We can then cut $\Sigma$ along $\delta_1$ and $\delta_2$; this gives a surface $\Sigma_1$ with four boundary components. Finally we can glue $N+1$ pieces of $\Sigma_1$, along a graph as suggested in Figure \[fig:2PrimCasos\], thus obtaining a surface $\widehat{\Sigma}$ covering $\Sigma$ in a way that satisfies the lemma. Let us be a little more precise here. We may choose a co-orientation for the curves $\delta_1$ and $\delta_2$. This gives an orientation for the edges of the (figure eight) graph $\Gamma$ dual to the cutting system $(\delta_1,\delta_2)$.
![Left: cutting $\Sigma$. Right: gluing $\Sigma'$ with pieces of $\Sigma_1$.[]{data-label="fig:2PrimCasos"}](Apendice-2.pdf "fig:")![Left: cutting $\Sigma$. Right: gluing $\Sigma'$ with pieces of $\Sigma_1$.[]{data-label="fig:2PrimCasos"}](Apendice-3.pdf "fig:")
Denote by $\langle a,b\rangle$ the fundamental group of $\Gamma$, where $a$ is the oriented edge dual to $\delta_1$ and $b$ dual to $\delta_2$. Any covering of $\Gamma$ gives rise to a covering of $\Sigma$, either by pulling back a pinching map $\Sigma\to\Gamma$, or equivalently, by thinking of the covering of $\Gamma$ as a set of instructions for gluing as many copies of $\Sigma_1$ as the vertices of the covering graph. The covering of $\Gamma$ suggested in Figure \[fig:2PrimCasos\] is associated to a morphism $\sigma\colon\langle a,b\rangle\to\mathfrak{S}_{N+1}$ (the symmetric group on $N+1$ elements) where $\sigma(a)$ has one fixed point and one cycle of length $N$, and $\sigma(b)$ has a cycle of length $N+1$. Now, up to choosing an orientation on them, the closed curves $\alpha,\beta$ yield two loops in this figure eight oriented graph: $\alpha$ yields the path $a$, hence $\alpha$ has one $(1:1)$ lift and one $(N:1)$ lift, while $\beta$ yields the path $b$, hence it has one $(N+1:1)$ lift, and this covering satisfies the conclusion of Lemma \[lem:2curvas\].
Cases (3) and (4) are similar, except that the curve $\beta$ cannot be mapped to a single generator $b$ in $\Gamma$, but to a slightly longer word. In case (3), we can find two disjoint simple curves $\delta_1,\delta_2$ such that $i(\alpha,\delta_1)=1$, $i(\alpha,\delta_2)=0$, and $i(\beta,\delta_1)=i(\beta,\delta_2)=2$ as in Figure \[fig:2UltCasos\].
![The curve $\alpha$ and the four possible cases for $\beta$.[]{data-label="fig:2UltCasos"}](Apendice-4.pdf)
For some co-orientations of $\delta_1$ and $\delta_2$, and some orientations on $\alpha$ and $\beta$, the loop $\alpha$ gives the loop $a$ in the graph $\Gamma$ as above, while the loop $\alpha$ yields the word $aba^{-1}b^{-1}$. Thus as before, finding a cover satisfying the conclusion of Lemma \[lem:2curvas\] amounts to choosing two permutations $\sigma(a)$ and $\sigma(b)$, such that $\sigma(a)$ has one fixed point and one cycle of length $N$, say, $(2~3\cdots~N+1)$, and such that the commutator $\sigma(aba^{-1}b^{-1})$ has no fixed point: it suffices to choose $\sigma(b)$ in such a way that $\sigma(ba^{-1}b^{-1})=(1~2\cdots N)$.
Finally, in case (4), which may happen only if the genus of $\Sigma$ is at least three, it is best to cut $\Sigma$ along three curves $\delta_1$, $\delta_2$ and $\delta_3$, as pictured in Figure \[fig:2UltCasos\]. This time $\alpha$ yields the word $a$, while $\beta$ yields the word $abcb^{-1}c^{-1}$ in the fundamental group of the graph $\Gamma$, which is this time a bouquet of three circles. We pick again $\sigma(a)$ to be the cycle $(2~3\cdots N+1)$, as before. As long as $N\geq 4$, the cycle $(1~3~5)=(1~3)(3~5)$ is a commutator in $\mathfrak{S}_{N+1}$, hence we may pick $\sigma(b)$ and $\sigma(c)$ so that $\sigma(bcb^{-1}c^{-1})=(1~3~5)$, and then $\sigma(abcb^{-1}c^{-1})$ has no fixed point: this yields a covering of $\Sigma$ satisfying the conclusion of Lemma \[lem:2curvas\], in either case.
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[^1]: To see this it is enough to find two noncommuting elements in $\pi_1(\Sigma)$ which do not intersect the normal subgroup generated by an element which has a unique fixed point in $S^1$ (notice that only elements in the normal subgroup generated by this element can have fixed points since there is a unique non-planar leaf). To find such elements, one can look at the projection of $\pi_1(\Sigma)$ into its first homology group.
[^2]: Sufficiently smooth, $C^{1+}$ is enough to use [@Alvarez_Yang].
[^3]: This would be isolated simple closed geodesics with large embedded collars. This is needed if one wishes to construct cylinder leaves, but as we mentioned before, we will ignore this to avoid cumbersome notation.
|
---
abstract: 'The phenomenon of jet supression observed in highly energetic heavy ion collisions is discussed. The focus is devoted to the stunning applications of the AdS/CFT correspondence [@AdSCFT] to describe these real time processes, hard to be illuminated by other means. In particular, the introduction of as many flavors as colors into the quark-gluon plasma is scrutinized.'
author:
- 'José D. Edelstein'
- 'Carlos A. Salgado'
title: |
Jet Quenching in Heavy Ion Collisions\
\[.13em\] from AdS/CFT
---
[ address=[Department of Particle Physics and IGFAE, University of Santiago de Compostela\
E-15782, Santiago de Compostela, Spain]{} ,altaddress=[Centro de Estudios Científicos (CECS), Casilla 1469, Valdivia, Chile]{}]{}
[ address=[Department of Particle Physics and IGFAE, University of Santiago de Compostela\
E-15782, Santiago de Compostela, Spain]{}]{}
The Phenomenon of Jet Quenching
===============================
One of the most recent surprises in particle physics came from the results of the Relativistic Heavy Ion Collider (RHIC), indicating that hadronic matter at temperatures slightly above the crossover critical temperature, $T_c$, may be a strongly coupled quark-gluon plasma (sQGP). Naive expectations pointed towards a free gas of quarks and gluons or quasi-particles, a picture that can be understood from perturbative calculations of thermodynamic quantities such as the equation of state. These perturbative methods fail, however, at low temperatures, close to $T_c$, where lattice QCD simulations are employed instead. Although the underlying dynamical picture is difficult to infer from these numerical results, the findings at RHIC are in qualitative agreement with the $\sim 20\%$ departure from the ideal gas, Steffan-Boltzmann law, predicted from lattice (see e.g. [@Karsch:2001cy] for a review). This departure is indeed very close to the exact $25\%$ value found from AdS/CFT, which is rigorously valid at infinite coupling (see Igor Klebanov’s article in this volume [@IK10years]).
Several observable consequences of the creation of this new state of matter are measured at RHIC as probes to characterize its properties. Among them, we focus in the present article in the supression of the inclusive cross-section at large transverse momentum, $p_t$, so-called [*jet quenching*]{}. In the absence of any nuclear effect, the cross section to produce a particle with high transverse momentum in heavy ion collisions should scale with the number of elementary interactions. However, a strong supression is found experimentally, the resulting particle production being about 20$\%$ of the expected value. This dramatic departure from the naive expectation can be understood as being due to the energy loss of highly energetic partons traversing the medium created by the collision (see e.g. [@CasSal] for a recent review).
Another interesting observation in the high-$p_t$ region of the spectrum is a supression in the observed back-to-back high-$p_t$ jets in Au+Au vs. p+p collisions. High-$p_t$ quarks or gluons are produced predominantly in pairs in elementary, hard collisions, flying in opposite directions in the transverse plane. Correlation of azimuthal angles among high-$p_t$ particles produced in the same event is measured. In the absence of any final state effect, one is led to expect a peak at $\phi = 0$ for partners in the same jet as the trigger parton, and a recoil peak at $\phi = \pi$. These back-to-back jets are indeed observed in p+p and d+Au collisions. In central Au+Au collisions, though, the recoil jet is greatly distorted (absent) indicating large final state effects on the produced hard quarks and gluons due to the resulting medium.
The observed deficit of high energy jets seems to be the result of a slowing down, damping or quenching of the most energetic partons as they propagate through the quark-gluon plasma (QGP). The rate of energy loss should be spectacular: several GeV per fm, more than one order of magnitude larger than in cold nuclear matter. The energy loss of a hard parton in quantum chromodynamics (QCD) can be parameterized through the transport coefficient known as $\hat q$. If defined at weak coupling, it would measure the average square transverse momentum transferred to the hard parton per mean free path length. Fitting the current data it seems that, with a great degree of confidence, its measured value is generously within the range $$\hat q_{\rm exp} = (15 \pm 10)~ \frac{{\rm GeV}^2}{\rm fm} ~.
\label{qhatexp}$$ All estimates for this quantity in weak coupling computations lead to much lower values[^1] and the obvious question is whether a calculation at strong coupling could result in better agreement. Lattice techniques are not well suited to study this kind of phenomenon as it involves real-time dynamics which is fairly difficult to address in a Euclidean time formulation. Under these conditions, the use of AdS/CFT provides a powerful tool to study this type of physics, providing both valuable insights into the physical aspects of the underlying mechanism of energy loss, as well as on the applicability of these techniques to actual experimental situations.
The calculation of the jet quenching parameter is not the only example of this novel connection: attempts to compute quantities in AdS/CFT which could be of interest for the physics of heavy ion collisions are currently abundant. One of the greatest successes of this approach is the computation of the ratio of the shear viscosity to the entropy density [@KSS], $\eta/s = \hbar/4\pi$, which agrees with current fits of hydrodynamical models to RHIC data on the elliptic flow (see [@IK10years]). Other examples are the computation of thermodynamical quantities such as the free energy, the energy density, the heat capacity, the speed of sound, etc. Further calculations within the AdS/CFT framework aiming at providing a bridge towards experimental data include the drag force coefficient, the relaxation time, diffusion constants, thermal spectral functions, stability of heavy-quark bound states, and the hydrodynamical behavior of the collision.
A Parton Through the Quark-Gluon Plasma
=======================================
In order to study the jet quenching phenomenon, we must first provide an appropriate phenomenological description of the relevant physics. The original formulation of the induced emission by an extended medium goes back to the early fifties, when Landau and Pomeranchuk gave a framework in which to consider a charged particle moving through a classical electrodynamical environment [@LandauPom], and generalized shortly after to the quantum case by Migdal [@Migdal]. These results were extended to the case of QCD by Gyulassy, Plumer and Wang [@GPW], Zakharov [@Zakharov] and by Baier, Dokshitzer, Mueller, Peigné and Schiff [@BDMPS].
Let us briefly present the basics of medium induced gluon radiation for a highly energetic parton. A quark with energy $E$ emits a gluon with a fraction of momentum $x$ and transverse momentum with respect to the quark direction $k_\perp$. The interaction of this system is depicted by multiple scatterings with the medium which, in some models, can be considered as a collection of static scattering centers. In the eikonal approximation ($E\gg xE\gg k_\perp$), the particle trajectories can be written as Wilson lines in the light-cone coordinate. Using the approximation above, the quark is seen as traveling in a straight line while the gluon is allowed to move in transverse space by interaction with the medium. In the multiple soft scattering approximation, the transverse position of the gluon follows a Brownian motion, and the average transverse momentum after traveling a distance $L$ is characterized by the transport coefficient $\langle k^2_\perp\rangle\simeq \hat q\,L$. The origin of this parameter and the relation with the Wilson lines can be understood as follows: in order to compute the cross-section for gluon emission, a Wilson line ending at transverse position ${\bf x}$ appears in the amplitude corresponding to the gluon propagation, together with another Wilson line for the gluon in the conjugate amplitude, at transverse position $\bf y$. These Wilson lines then need to be averaged over all possible medium configurations, appearing only in combinations like [@Zakharov; @Wiedemann] $$\frac{1}{N_c^2-1}\, {\rm Tr}\langle W^A({\bf x})\, W^A({\bf y})
\rangle \simeq \exp\left[-\frac{({\bf x-y})^2}{4\sqrt{2}} \int
dx_-\, \hat q(x_-)\right] ~.
\label{eq:average}$$ This approximation is valid, up to logarithmic corrections, in the small distance limit $({\bf x-y})^2\ll1/\Lambda_{\rm QCD}^2$. The average $\langle \cdots \rangle$ and the corresponding medium properties are all encoded in a single jet quenching parameter, $\hat q$. When the medium does not vary along the light-cone trajectory of the gluon[^2], and assuming the transverse component to be much smaller than the longitudinal one [@LiuRajWie1], $$\langle W^A(\mathcal{C})\rangle \simeq \exp\left[-\frac{1}{4\sqrt{2}}\,
\hat{q}\, L^-\, L^2 \right] ~,
\label{defqhat}$$ for a rectangular Wilson loop with a large light-like side $L^-$ and a much smaller space-like separation $L$, $L\ll L^-$. Eq.(\[defqhat\]) can be naturally extrapolated to the strong coupling regime and considered as a non-perturbative definition of the transport coefficient $\hat q$.
The Jet Quenching Parameter in AdS/CFT
======================================
The Wilson loop computation leading to the jet quenching parameter can be readily performed, at large $N_c$, within the framework of the AdS/CFT correspondence[^3]. It amounts to the evaluation of the (regularized) Nambu–Goto action, $S_{\rm NG}(\mathcal{C})$, for a string hanging from the curve $\mathcal{C}$ at the boundary towards the bulk of AdS, $$\langle W^A(\mathcal{C})\rangle \simeq \exp\,[-2\, S_{\rm
NG}(\mathcal{C})] + {\cal O} \left( 1/N_c \right) ~.
\label{wilson}$$ Let us succintly cover the basics of this computation. The family of black brane metrics of interest for us have the following form [@Buchel]: $$ds^2 = - c_T^2\, dt^2 + c_X^2\, dx^i\, dx_i + c_R^2\, dr^2 +
G_{M n}\, dX^M\, dX^n ~,
\label{classmet}$$ where $X^M = (t, x^i, r, X^n), ~i=1,\dots,p, ~n=1,\dots,8-p$. We shall use light-cone coordinates $x^\pm$, and consider a rectangular Wilson loop $\mathcal{C}$ parameterized by the embedding $x^- = \tau
\in (0, L^-)$, $x^2 = \sigma \in (- L/2,L/2)$, and $r = r(\sigma)$. For a symmetric configuration around $\sigma = 0$, $r'(0) = 0$, the Nambu-Goto action takes the following form $$S_{\rm NG}(\mathcal{C}) = \frac{L^-}{\sqrt 2\pi \alpha'}
\int_{0}^{L/2} d\sigma ~\left( c_X^2
- c_T^2 \right)^{1/2} \left( c_X^2 + c_R^2\, r'(\sigma)^2 \right)^{1/2} ~,$$ $\alpha'$ being the inverse of the string tension. The energy is a first integral of motion, from which the profile $r(\sigma)$ can be extracted by inverting $$\sigma(r) = \int_{r_H}^r \frac{c_R}{c_X}\, \frac{dr}{\left( k\,
c_X^2\, (c_X^2 - c_T^2) - 1 \right)^{1/2}} ~.
\label{sigmar}$$ $k$ is an integration constant fixed by the relation $\sigma(\infty) = L/2$. It is more convenient to deal with a dimensionless radial coordinate $u = r/r_H$, where $r_H
= r(0)$ is the location of the black brane horizon, and perform the rescalings, $$\frac{\hat c_T^{\,2}}{c_T^2} =
\frac{\hat c_X^{\,2}}{c_X^2} =
\frac{c_R^2}{\hat c_R^{\,2}} =
\Bigg( \frac{(\alpha')^{5-p}\, \lambda}{r_H^{7-p}} \Bigg)^{1/2} ~,$$ $\lambda$ being the ’t Hooft coupling in the dual ($p+1$)-dimensional gauge theory. $L$ is inversely proportional to $k$. Thus, we have to explore the limit $k\to \infty$, and keep the leading term in $L^-\, L^2$ [@LiuRajWie1]. The action has to be regularized by substracting the Nambu-Goto action for a pair of Wilson lines that stretch straight from the boundary to the horizon. Therefore, the jet quenching parameter finally reads [@ArmEdelsMas]: $$\hat q = \frac{1}{\pi \lambda }
\left( \frac{r_H}{\alpha'} \right)^{6-p} \,
\left( \int_{1}^\infty \frac{\hat c_R\, du}{\hat c_X^2\, (\hat c_X^2 -
\hat c_T^2)^{1/2}} \right)^{-1} ~.
\label{gjetq}$$ This remarkably compact formula is valid for a vast family of gauge/gravity duals, some of which will be presently explored. As it stands, it calls for a translation of gravity parameters in terms of the field theoretical quantities. This is provided by the (nowadays standard) AdS/CFT dictionary.
Jet Quenching Parameter Bestiary
--------------------------------
In this subsection we would like to review some of the quantum field theories for whose plasmas the jet quenching parameter has been computed. In order to provide some numbers, we consider as a representative choice of average values, $\lambda = 6\pi$ (that is, $\alpha_{\rm s} = 1/2$) and $T = 300$ MeV. This is, of course, a crude simplification for several reasons: the real $\hat q$ varies with time and, moreover, it is hard to provide both a reliable dependence of the coupling on the temperature, $\lambda(T)$, as well as to choose a representative average value for these quantities. Thus, numbers should be taken as indicative.
### $\mathcal{N} = 4$ Super Yang–Mills Theory
The original computation in the AdS/CFT framework, performed by Liu, Rajagopal and Wiedemann [@LiuRajWie1] for $\mathcal{N} = 4$ super Yang–Mills theory, gives $$\hat q_{{\rm SYM}} = \frac{\pi^{3/2}\,
\Gamma(\frac{3}{4})}{\Gamma(\frac{5}{4})}\,
T^3\, \sqrt{\lambda} ~.
\label{n4sym}$$ Notice that, contrary to what happens at weak coupling [@qhatpert], it does not depend explicitly on the number of degrees of freedom. For the representative values, $\hat q_{{\rm SYM}}
= 4.48$ GeV$^2$/fm.
### Witten’s QCD
The jet quenching parameter in Witten’s construction [@WittenQCD] of a holographic string dual of 4d SU($N_c$) Yang–Mills theory can be easily computed by applying eq.(\[gjetq\]) to a D4–(black) brane background wrapping a Kaluza–Klein circle of radius $\ell$ with antiperiodic boundary conditions [@ArmEdelsMas], $$\hat q_{{\rm WQCD}} = \frac{16\, \pi^{3/2}\,
\Gamma(\frac{2}{3})}{81\, \Gamma(\frac{7}{6})}\, T^4\, \ell\, \lambda ~,
\label{dcuatro}$$ where $\lambda$ is the 4d (dimensionless) coupling. This background describes[^4] the finite temperature physics for $T > T_c$. Introducing representative values, $\ell\, T \sim 1.7$, we get $\hat q_{{\rm WQCD}} = 7.11$ GeV$^2$/fm, nicely within the allowed experimental range.
### Finite ’t Hooft Coupling
The AdS/CFT conjecture is a statement which goes beyond the classical limit of string theory, in which it maps classical solutions of supergravity to quantum field theory vacua in the strong coupling limit $\lambda \to \infty$. Corrections in $\lambda^{-1}$ are in direct correspondence with those in powers of $\alpha'$ in the string theory side[^5]. Considering the $\alpha'$ corrected near extremal D3–brane [@BlackD3alpha], it is not difficult to evaluate $$\frac{\hat q(\lambda)}{\hat q_{\rm SYM}} = 1 - \frac{\zeta(3)}{8}\,
\left[ 45 - \frac{30725 \pi}{512 \sqrt{2}\,
\Gamma(\frac54)\,\Gamma(\frac{15}4)} \right] \, \lambda^{-3/2}
+ \cdots$$ Finite coupling corrections tend to diminish the value of the jet quenching parameter. For the same representative values chosen above, $\hat q(6\pi) = 4.38$ GeV$^2$/fm. The decrease in the jet quenching parameter is suggestive of a smooth interpolation between the strong coupling regime and the perturbative results.
### Finite Chemical Potential
$\mathcal{N} = 4$ SYM theory has a global $SO(6)$ R–symmetry. Chemical potentials, $\kappa_i$, for the $U(1)^3 \subset SO(6)_R$, which amount to considering a rotating black D3–brane with maximal number of angular momenta, can be turned on. In spite of the fact that the relevant supergravity solution [@Russo:1998by] heavily depends on various angles, the jet quenching parameter reads [@ArmEdelsMas]: $$\frac{\hat q(\kappa_i)}{\hat q_{\rm SYM}} = \frac{8\, \pi^{1/2}\,
\Gamma(\frac54)\, \Delta(\kappa_i)}{\Gamma(\frac34)}\,
\left( \int_{1}^\infty \frac{u\, du}{\sqrt{u^2 - 1}
\sqrt{u^4 + (1 + \kappa_+) u^2
- \kappa_{123}}}\, \right)^{-1} ~,$$ where $\kappa_+ = \kappa_1 + \kappa_2 + \kappa_3$, $\kappa_{123} =
\kappa_1 \kappa_2 \kappa_3$, and $$\Delta(\kappa_i) = \frac{(1 + \kappa_1)^2 (1 + \kappa_2)^2
(1 + \kappa_3)^2}{(2 + \kappa_1 + \kappa_2 + \kappa_3 - \kappa_1
\kappa_2 \kappa_3)^3} ~, \label{qkappa}$$ and all the information about the internal coordinates has dissapeared. Instead of performing a detailed analysis of this result, we shall stress its most significant qualitative behaviour: the jet quenching parameter raises its value for nonzero values of the chemical potentials, $\hat q(\kappa_i) > \hat q_{\rm SYM}$ [@ArmEdelsMas] (see also [@chempot]). The increase is not monotonic across the whole parameter space. It is easy to check that the above ratio tends to one when the chemical potentials are turned off.
### $\mathcal{N} = 1$ Superconformal Quiver Theories
There is a generalization of AdS/CFT in which the S$^5$ is replaced by a Sasaki–Einstein manifold X$^5$. The resulting gauge theory ends up having reduced supersymmetry and the field content of an $\mathcal{N} = 1$ superconformal quiver theory (SQT) [@KW]. The jet quenching computation in this case proceeds as before [@Buchel], the only difference being at the last step where the relation between the radius of the manifold and the ’t Hooft coupling depends on the volume of X$^5$ which, in turn, is inversely proportional to the central charge of the gauge theory [@LiuRajWie3], $$\frac{\hat q_{\rm SQT}}{\hat q_{\rm SYM}} =
\sqrt{\frac{\vol\, {\rm S}^5}{\vol\, {\rm X}^5}} =
\sqrt{\frac{a_{\rm SQT}}{a_{\rm SYM}}} ~.$$ For the prototypical case, X$^5$ $=$ T$^{1,1}$, [*i.e.*]{} the Klebanov–Witten (KW) model, this equation implies $\hat q_{\rm KW} = \sqrt{27/32}~
\hat q_{\rm SYM} = 4.12$ GeV$^2$/fm. A mild version of this result can be extended to further superconformal field theories and, in particular, implies that if two such theories are connected by a renormalization group flow, then $\hat q$ for the ultraviolet (UV) theory is always larger than that for the infrared (IR) theory [@LiuRajWie3].
### Breaking of Conformal Invariance
A possible mechanism for conformal symmetry breaking is given by the introduction of fractional branes in a complex deformation of the Calabi–Yau cone over X$^5$ (see e.g. [@EdelsPort] for a review). This leads to cascading quiver gauge theories whose archetype is the Klebanov–Strassler (KS) model [@KS]. The jet quenching parameter can be seen to increase its value with respect to the conformal KW case [@Buchel].
Within the framework of bottom-up approaches like, so-called, AdS/QCD [@AdSQCD], a nonconformal gauge/gravity dual pair was studied in [@NakTerWen]. The nonconformal deformation is given by a single parameter $c$ that appears in a warp factor in front of the AdS metric. Finite $c$ raises the jet quenching parameter for fixed $\lambda$ and $T$. A detailed study of this behavior was recently performed in [@LiuRajShi], where it was shown that the enhancement could be as high as 30$\%$ of the $\hat q_{\rm SYM}$ value. These two examples suggest that breaking of conformal invariance might be associated to an increase of the jet quenching parameter.
A Call for Massless Dynamical Quarks
====================================
Quarks are prime ingredients of QCD. Up to this point, however, we have misleadingly used the acronym QGP for theories without quarks. This is quite generic in the literature since gravity duals including quantum field theoretical degrees of freedom in the fundamental representation of the gauge group are scarce. A notable exception is given by the case of quenched flavor, $N_f \ll N_c$, in which quarks can be represented by probe D–branes in the background sourced by a large number of color branes [@KaKa]. Besides the formal interest of this case, it is evident that, real quark–gluon plasmas demand massless quarks beyond the quenched approximation, [*i.e.*]{}, $N_f \sim N_c$. Moreover, we must cope with finite temperature gauge/gravity duals, with $T > T_c$, which means that we need to scrutinize (non-supersymmetric) black brane solutions which are rather elusive.
There is only one known analytic solution with all these ingredients in critical string theory. A one parameter family of black hole solutions in the background sourced by $N_c$ [*color*]{} wrapped D5–branes and $N_f$ (smeared) [*flavor*]{} D5–branes recently constructed by Casero, Núñez and Paredes (CNP) [@cnp]. This is conjectured to be the thermal deformation of the gauge/gravity dual of an ${\cal N}=1$ SQCD–like theory with quartic superpotential, at the conformal point, $N_f = 2 N_c$, coupled to Kaluza–Klein adjoint matter. The temperature of these black holes is independent of the horizon radius and, indeed, coincides with the (Hagedorn) temperature, $T_H$, of Little String Theory (LST). This is possibly related to the fact that the UV completion of this solution involves NS5–branes. We shall comment on this issue while reviewing the computation of $\hat q$ performed in [@BBCE].
In the context of non-critical string theory, gauge/gravity duals of 4d theories with large $N_c$ and $N_f$, both at zero and high temperature, have been considered in the last few years. We shall focus on two interesting cases: an AdS$_5$ black brane proposed as the non-critical dual of the thermal version of conformal QCD, and an AdS$_5$ $\times$ S$^1$ black brane solution conjectured to be dual to thermal ${\cal N}=1$ SQCD in the Seiberg conformal window [@CasParSon]. The dilaton is constant in both models. The zero temperature theories were constructed, respectively, in [@BCCKP] and [@km] . The color degrees of freedom are introduced via $N_c$ D3–brane sources and the back-reaction of $N_f$ flavor branes on the background is taken into account. These flavor branes are, roughly, spacetime filling brane–antibrane pairs (matching the classical $U(N_f) \times U(N_f)$ flavor symmetry). Properties of their quark-gluon plasmas were studied in [@BBCE] by means of the gauge/gravity correspondence[^6]. The relevant gravity solutions are generically strongly curved and $\alpha'$ corrections are not subleading. The optimistic prejudice, driven by the unexpected success of bottom-up approaches like the alluded to AdS/QCD, is that the non-critical solutions might capture at least qualitative information of the dual field theories, insensitive to these corrections.
Little String Theory Plasmas
----------------------------
A one parameter family of black brane solutions, conjectured to be the finite temperature gauge/gravity dual of an ${\cal N}=1$ SQCD–like theory at the conformal point, $N_f = 2 N_c$, was obtained in [@cnp]. Their Hawking temperature is given by the Hagedorn temperature. This suggests that there could be thermodynamical instabilities (negative specific heat) in this solution, in the very same way as happens in the standard LST case [@ThermoLST]. In order to test that the above statement is correct and thermodynamical instabilities cannot be cured by introducing IR cut-offs, we consider a generalization of CNP black branes, parameterized by $\xi \in (0,4)$, that should be gauge/gravity duals of the above LST plasma compactified on S$^{3}$. Their string frame metric is $$\begin{aligned}
ds^2&=&e^{\Phi_0 + r} \Big[ - {\cal F}dt^2 + R^2 d \Omega_3^2 +
\frac{R^2 N_c \alpha'}{R^2 + N_c\alpha'} {\cal F}^{-1} dr^2 +
N_c \alpha' \Big( \frac{1}{\xi} d \Omega_2^2 \nonumber \\
&&+\frac{1}{4-\xi}\, d \tilde\Omega_2^2 + \frac14 (d\psi + \cos
\theta d\varphi + \cos \tilde \theta d\tilde\varphi)^2
\Big) \Big] ~,
\label{simplebh}\end{aligned}$$ where ${\cal F}= 1- e^{2r_H - 2r}$, so the horizon is placed at $r=r_H$. This introduces a new scale into the system, the (quantized [@BBCE]) radius $R$ of the S$^3$, that indeed produces a departure in the black hole temperature from Hagedorn’s $$T(R) = T_H \, \sqrt{1 + \frac{N_c \alpha'}{R^2} } \, > \, T_H ~,
\label{TR}$$ albeit still independent of the horizon radius, seemingly a common feature of black holes obtained from NS5 and D5–brane configurations. These black holes seem to present analogous instabilities as their uncompactified counterparts[^7] [@CPT]. It is not clear, thus, if these solutions provide reliable descriptions of 4d finite temperature gauge/gravity duals.
While the energy loss of a probe quark due to drag force in the plasma [@DragForce] is found to be non-zero (and formally analogous to the ${\cal N}=4$ SYM theory case)[^8], the jet quenching parameter exactly vanishes. These quantities are related to the energy loss of a parton in two opposite regimes of the transverse momentum (large momentum for the quenching parameter and small momentum for the drag coefficient). Still, obtaining such completely different results is puzzling. The jet quenching parameter is definitely dependent on the UV behavior of the dual backgrounds, and so on the 6d LST asymptotics, while the same dependence for the drag force is not apparent. This fact suggests that the result $\hat q = 0$ has to be associated more to non-local LST modes than to their claimed local counterparts. These give total screening and, as a consequence, zero jet quenching parameter. Thus, these backgrounds do not seem to be useful in order to study UV properties of realistic plasmas. It is expected that the same should happen for possible $N_f \neq 2 N_c$ plasmas in the framework of [@cnp], even though the correponding gravity backgrounds are not currently known[^9].
Interestingly, the hydrodynamic properties of the QGP are not affected by the troublesome thermodynamic behavior mentioned above. This seems to be related to the fact that the ratio $\eta/s$ depends on universal properties of black hole horizons, and is not substantially altered by the presence of flavors.
QCD in the Conformal Window
---------------------------
A 5–dimensional model dual to QCD in the conformal window has been constructed in [@BCCKP] (for $T = 0$) and [@CasParSon] (for $T
\neq 0$). In $\alpha'=1$ units, the metric reads ds\^[2]{} = ( )\^[2]{} + ( )\^[2]{} (1 - )\^[-1]{} dr\^[2]{} , \[generalback\] the radius being an increasing function of $\rho \sim N_f/(2 \pi N_c)$, R\^[2]{} = . \[5drad\] The dilaton is related to the gauge theory coupling by g\_[QCD]{}\^2 N\_c = e\^[\_[0]{}]{} N\_c = ( - 7) . \[5ddil\] It decreases with $\rho$, which is consistent with the known fact that the zero temperature theory should be weakly coupled in the upper part of the conformal window, that is when $\rho$ is the largest. The behavior of the coupling is given by ${\cal F}(\rho)/N_c$ for a function ${\cal F}$ whose behavior for large $\rho$ is ${\cal F}(\rho)
\sim 1/\rho$, as expected in the Veneziano limit [@veneziano].
The computation of $\hat q$ proceeds as in the previous section, and results in a monotonically increasing function of $\rho$. Its asymptotics[^10] can be readily computed, q \~ T\^[3]{} {
[ll]{} 1 + + (\^2) & [for]{} 0 ,\
\[2ex\] + ( 1/\^2 ) & [for]{} .
\[asympqhat5d\] The transport coefficient displays a dependence on the “effective number of massless quarks” $\rho$. A representative value may be obtained by assuming the physically sensible value $\rho \approx 1$, and $T = 300$ MeV, leading to $\hat q_{N_f \sim N_c} = 5.29$ GeV$^2$/fm. The variation of $\hat q$ is very small in the whole range of $\rho$. It signals the fact that the flavor contribution is not drastically changing the properties of the plasma. This is compatible with (and in a sense gives a reason for) the evidence that the values of plasma properties computed with gravity duals including only adjoint fields are very similar to the experimental ones.
SQCD in Seiberg’s Conformal Window
----------------------------------
A 6–dimensional model dual to SQCD in Seiberg’s conformal window has been constructed by Klebanov and Maldacena [@km] (for $T = 0$) and [@CasParSon] (for $T \neq 0$). In $\alpha'=1$ units, the 6d metric reads ds\^[2]{} = + + d\^2 , \[generalback6d\] where the AdS radius is now independent[^11] of both $N_c$ and $N_f$. The coupling, instead, depends on the quotient $\rho$, = e\^[\_[0]{}]{} N\_c = , and satisfies Veneziano’s asymptotics. The jet quenching parameter also turns out to be independent of $\rho$, q = T\^[3]{} , \[JQP6d\] something that looks odd. Strikingly, its value for the representative temperature of the process is $\hat q_{N_f \sim N_c} = 6.19$ GeV$^2$/fm, slightly higher than the ${\cal N}=4$ SYM value and so more comfortably within the RHIC range.
Concluding Remarks
==================
The AdS/CFT correspondence embodies a powerful device to scrutinize the strong coupling regime of non-Abelian gauge theories. Its application to finite temperature quantum field theories has produced stunning results, mostly connected to the physics above the crossover in the phase diagram of Quantum Chromodynamics. This is the regime of QCD presently being explored at RHIC where increasing evidence points towards the formation of a short lived strongly coupled quark–gluon plasma that behaves like a nearly perfect fluid. Within the framework of the AdS/CFT correspondence, it has been proven that this is a universal behavior of plasmas of fairly generic gauge theories. Several other properties of this plasma have been studied.
We do not have a string dual of QCD at our disposal. And it is not clear if we will have something like this in the future. However, we can try to understand which results are universal, how different quantities depend on supersymmetry, dimensionality, field content, etc. Some attempts to extrapolate results from ${\cal N}=4$ SYM towards QCD have been explored recently in theories without fundamental degrees of freedom. Gubser argues that the extrapolation should be done by using the energy densities of both theories as an unambiguous quantity to be fixed for comparison [@Gubser1]. This leads to a map between temperatures of the sort $T_{\rm SYM} =
3^{-1/4}\, T_{\rm QCD}$ and, consequently, to the somehow disturbing conclusion that $\hat q_{\rm QCD} < \hat q_{\rm SYM}$ [@Gubser2]. Liu, Rajagopal and Wiedemann [@LiuRajWie3], instead, conjecture that, since QCD’s QGP is approximately conformal at $T \approx 2 T_c$, $$\frac{\hat q_{\rm QCD}}{\hat q_{\rm SYM}} \simeq
\sqrt{\frac{s_{\rm QCD}}{s_{\rm SYM}}} \approx 0.63 ~,$$ the same tendency as before. The results in [@BBCE] seem to indicate that the relation between ratios of jet quenching parameter and entropy densities ceases to be valid when quarks are introduced, even if the theory remains conformal. Those in [@Buchel; @LiuRajShi; @BBCE] tend to suggest, on the contrary, that $\hat q_{\rm QCD}$ may be higher than $\hat q_{\rm SYM}$.
A full understanding of the consequences of adding quarks to YM theory at strong coupling is still missing. In this presentation we attempted to give an insight into the behavior of $\hat q$ with respect to parameters that are relevant to QCD. However, it seems clear that elaborations on a critical string theory example not involving NS5–branes would be desirable. The dependence of all the observables on $\rho$ is mild, so that the numerical results for $\hat q$ (as well as other interesting quantities that are not detailed in this article) are always very similar to $\hat q_{\rm SYM}$. This provides an *a posteriori* explanation of why the latter is so similar to what is observed at RHIC.
The experimental program of the LHC will begin this year, with energies extending the present reach by more than one order of magnitude. The first heavy-ion collisions will start one year after. The use of the same machine to accelerate protons and nuclei implies that, for the first time, the energy frontier is the same for Standard Model (SM) or beyond the SM searches and for hot and dense QCD physics. This increase in energy translates into an increase in the temperature reached in the corresponding Pb+Pb collisions of a little less than a factor of two. A longer–lived plasma is also expected, reducing the hadronic matter effects which could obscure the interpretation of some of the measurements. Hopes exist, also, that this increase in the temperature could presumably be large enough to see a transition from the sQGP to the expected weakly coupled QGP.
String theory seems to have something to say, through the AdS/CFT correspondence, in real sQGP physics. Even if as of today the connection is still preliminary and not on sufficiently firm ground, no doubt this is a very promising avenue for future research in high energy theoretical physics.
We are very pleased to acknowledge our collaborators in the subject Néstor Armesto, Gaetano Bertoldi, Francesco Bigazzi, Jorge Casalderrey–Solana, Aldo Cotrone, Javier Mas and Urs Wiedemann. We are thankful to Jonathan Shock for several helpful comments on the manuscript. This work is supported in part by MEC and FEDER (grants FPA2005-00188 and FPA2005-01963), by the Spanish Consolider–Ingenio 2010 Programme CPAN (CSD2007-00042), by Xunta de Galicia (Consellería de Educación and grant PGIDIT06PXIB206185PR), and by the European Commission (grants MRTN-CT-2004-005104 and PERG02-GA-2007-224770). The authors are [*Ramón y Cajal*]{} Research Fellows. The Centro de Estudios Científicos (CECS) is funded by the Chilean Government through the Millennium Science Initiative and the Centers of Excellence Base Financing Program of Conicyt. CECS is also supported by a group of private companies which at present includes Antofagasta Minerals, Arauco, Empresas CMPC, Indura, Naviera Ultragas and Telefónica del Sur.
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[^1]: Parametrically, at large $N_c$, $\hat q\sim
\pi\, N_c^2\, \alpha_s^2\, T^3$, with a prefactor of order one which depends on the assumptions in the calculation, this leading to $\hat q \sim 1$ GeV$^2$/fm (see e.g. [@qhatpert]).
[^2]: This hypothesis does not seem appropriate in the experimental set up at RHIC. However, it has been shown in [@Salgado:2002cd] that it is always possible to write eq.(\[eq:average\]), and interpret $\hat q$ as a properly weighted average measure of the time dependent transport coefficient.
[^3]: This was first established by Rey and Yee [@WloopRY] and, independently, by Maldacena [@WloopM]. The case of partially light-like Wilson loops was first presented by Liu, Rajagopal and Wiedemann [@LiuRajWie1], while the schematic computation of the present section was discussed in full detail in [@ArmEdelsMas].
[^4]: The radius $\ell$ gives a further scale that triggers the confinement/deconfinement transition [@Kruczenski:2003uq]. Thus, it is natural to identify it with the inverse critical temperature, $\ell = T_c^{-1}$.
[^5]: There is another source of corrections given by world-sheet fluctuations of the string. These go like $\lambda^{-1/2}$. Thus, they are dominant at large $\lambda$ though much harder to compute (see e.g. [@DGT] for a similar problem in the case of the Wilson loop that corresponds to the quark antiquark potential).
[^6]: In order to study the dynamics of hard probes in the quark–gluon plasmas of these theories, it is assumed that the mass of the probes is related to the radial distance of a flavor brane from the center of the space, as it happens in the critical case, at least in some effective way. Therefore, the general results on the drag force and jet quenching parameter extend in a straightforward way to the non-critical setup.
[^7]: Indeed, the same problems are already present in a family of black brane solutions corresponding to finite temperature $\mathcal{N} = 1$ supersymmetric YM theory [@gtv].
[^8]: In this respect, it seems that the friction that a slow, heavy parton experiences in a strongly coupled plasma is in practice always the same, irrespective of the features of the dual field theory.
[^9]: The results of the forthcoming subsections, as well as in phenomenological 5d models [@NakTerWen], where both the drag force and the jet quenching parameter are found to be different from zero, indicate that dynamical quarks seem to introduce no specific problem to the evaluation of $\hat q$.
[^10]: These expressions must be taken with a grain of salt: since the background is expected to be corrected by order one terms, the numerical coefficients are not trustworthy. Moreover, since the dual field theory should be QCD in the conformal window for definite, finite values of $\rho$, the limit $\rho
\rightarrow 0$ ($\rho\rightarrow \infty$) is meaningful only as an indication of the behavior for small (large) $\rho$. The strict $\rho= 0$ case is dual to the finite temperature version of a YM theory without flavors first studied by Polyakov [@wall].
[^11]: The fact that the radius is independent of $N_c$ and $N_f$ is probably signaling that this model is incomplete.
|
---
abstract: 'We present one-dimensional non-local thermodynamic equilibrium time-dependent radiative transfer simulations (using [[cmfgen]{}]{}) of two sub-Chandrasekhar (sub-), one and one super- Type Ia SN ejecta models. Three originate from delayed detonation models, and the fourth is a sub- detonation model. Ejecta masses are 1.02, 1.04, 1.40, and 1.70 , and all models have 0.62 of [$^{56}{\rm Ni}$]{}. Sub- model light curves evolve faster, reaching bolometric maximum 2–3 days earlier and having 3–4 days shorter bolometric half light widths. The models vary by $\sim$12 per cent at maximum bolometric luminosity and by 0.17 mag in $B_{\rm max}$. While $\Delta M_{15}(B)$ increases with ejecta mass it only varies by $\sim$5 per cent around 1 mag. Sub- models are 0.25 mag bluer in $B-R$ at $B_{\rm max}$. Optical spectra share many similarities, but lower mass models exhibit less UV line blanketing during the photospheric phase. At nebular times, significant NIR spectroscopic differences are seen. In particular, emission lines of the NIR triplet; \[\] [$\lambda$]{}[$\lambda$]{}9068,9530; \[\] [$\lambda$]{}[$\lambda$]{}7291,7324; \[\] [$\lambda$]{}[$\lambda$]{}7135,7751; and \[\] 1.939 $\mu$m are stronger in higher mass models. The \[\] 1.939 $\mu$m line is absent in the sub- detonation model, and provides a valuable potential tool to distinguish sub- explosions from explosions. In general, the nebular phase models are too highly ionized. We attribute this to the neglect of clumping and/or the distribution of intermediate mass and iron group elements. The two sub- models, while exploded by different mechanisms, can be distinguished in the $J$ and $H$ bands at late times (e.g., $+200$ days).'
author:
- |
Kevin D. Wilk,$^{1}$[^1] D. John Hillier,$^1$[^2] Luc Dessart$^2$\
$^1$ Department of Physics and Astronomy & Pittsburgh Particle physics, Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh,\
Pittsburgh, PA 15260, USA\
$^2$ Unidad Mixta Internacional Franco-Chilena de Astronomía (CNRS UMI 3386), Departamento de Astronomía, Universidad de Chile,\
Camino El Observatorio 1515, Las Condes, Santiago, Chile
bibliography:
- 'SNeIa\_diag\_TEST.bib'
date: 'Accepted . Received '
title: Ejecta Mass Diagnostics of Type Ia Supernovae
---
\[firstpage\]
radiative transfer – supernovae: general – white dwarfs
Introduction
============
Type Ia supernovae (SNe Ia) are thought to be thermonuclear runaway explosions of carbon-oxygen (C/O) white dwarfs (WDs) [@Hoyle1960], but the formation channel of the progenitor remains uncertain. Determining the progenitor channel(s) of SNe Ia, and their diversity, is of crucial importance since it will improve their use as probes of cosmological parameters [@Riess1998; @Perlmutter1999]. Further, understanding the progenitor channel allows us to probe stellar evolution of binary systems prior to the SN occurring, and hence place constraints on both binary synthesis models and binary evolution.
There are two main progenitor channels invoked to explain the origin of SNe Ia – the single degenerate (SD) channel and double degenerate channel. In the “classic" SD channel, the WD accretes matter (usually hydrogen and/or helium) from a non-degenerate star due to binary interaction via Roche lobe overflow. However, mass transfer in the SD channel is not limited to Roche lobe overflow – it can also be mediated by wind mass loss as, for example, in symbiotic systems [see @Maoz2014 for a review]. In the SD channel matter accumulates on the white dwarf, where stable burning can occur, until the star approaches the Chandrasekhar mass ($\approx$1.4 $M_\odot$) and explodes leading to a SN Ia [@WhelanIben1973; @Nomoto1982a].
How the WD explodes has long been a matter of study within the astrophysics community. The different explosion mechanisms include pure detonation [@Arnett1969; @Sim_2010], pure deflagration [@Nomoto_1976], delayed detonation and pulsating delayed detonation [@Khokhlov1991a; @Khokhlov1991b; @Gamezo2005; @Livne2005; @Ropke2007; @Jackson2010; @Seitenzahl2013], and gravitationally-confined detonation (GCD) [@Plewa2004]. More recent research on the GCD, including 3D full star simulations, was presented by [@Seitenzahl2016]. For sub- WD masses, a double detonation scenario has been explored [@Woosley1994; @Livne1995; @Fink2007; @Fink2010].
One of the most successful theories at reproducing the properties of standard SNe Ia is the delayed-detonation transition (DDT) model where the WD first undergoes a subsonic deflagration phase. The resulting expansion of the WD creates lower density fuel, which is necessary for the production of intermediate mass elements (IMEs). At a particular density, the burning switches from subsonic to supersonic [@Khokhlov1991a; @Khokhlov1991b]. This scenario is required to recreate the observed chemical stratification. Pure detonation models produce too many iron group elements (IGEs) such as iron and nickel but not enough IMEs at high velocities to reproduce SN Ia spectra. On the other hand, deflagration models produce abundant amounts of IMEs but not enough [$^{56}{\rm Ni}$]{} to power the SN luminosity. This mechanism is a possible channel for under-luminous SNe Ia [@Travaglio2004; @Phillips2007; @Jordan2012; @Kromer2013]. One dimensional (1D) DDT models have been very successful in reproducing the observed properties of SNe Ia. However, these models describe the turbulent flame propagation only in a parametrized way. Three dimensional (3D) models better capture the fluid instabilities and mixing. This lead to different ejecta structures that do not reproduce observables as well as 1D models [see e.g. @Seitenzahl2013; @Sim2013MNRAS].
The double degenerate scenario corresponds to the merger of two WDs through loss of angular momentum by gravitational wave emission. It is not understood how the merger triggers the explosion, although it is thought that during the merger the smaller mass WD donates matter to the more massive WD. Recently, modeling by [@Sato2015] has shown that, depending on the masses of the two WDs, explosions can occur either during the merger phase, provided both WDs are within a mass range between 0.9–1.1 M$_\odot$, or within the merger remnant phase when the more massive object reaches . From their models, the authors estimate that 9 per cent of galactic SNe Ia can be attributed to mergers. One problem in this scenario is that rapid mass accretion of carbon and oxygen leads to an off-centre carbon ignition and subsequently an O/Ne/Mg WD [@Saio_Nomoto1985]. Accretion induced collapse creates additional problems for high accretion rates, leading to the formation of a neutron star instead of a SN Ia. For an extensive review about progenitors of SNe Ia, see [@Maoz2014].
When almost the entire C/O WD has burnt, releasing $\sim$$10^{51}$ ergs (more energy than the gravitational binding energy of a WD), the energy liberated unbinds the WD, producing an ejecta with no remnant. Ejecta velocities of the order of $\sim$10000 are produced, and within minutes, the ejecta reaches a homologous coasting phase (radiation pressure from [$^{56}{\rm Ni}$]{} decay produces second order effects). At early times the ejecta is radiation dominated and heated by the decay of [$^{56}{\rm Ni}$]{} ([$^{56}{\rm Co}$]{} for post-maximum evolution). It is because of this main decay chain of [$^{56}{\rm Ni}$]{}$\rightarrow$[$^{56}{\rm Co}$]{}$\rightarrow$[$^{56}{\rm Fe}$]{}, with roughly 1.7 MeV of energy release per decay for [$^{56}{\rm Ni}$]{} and 3.7 MeV per decay for [$^{56}{\rm Co}$]{}, that these objects are so luminous. However, [@Dessart2014a] show the importance of heating the outer ($\ge$20000 ) ejecta from other decay channels.
Early work by [@Stritzinger2006a] suggested sub- WDs as progenitors for some SNe Ia based on comparisons of $UVOIR$ light curves of 16 SNe with analytical models of nuclear decay luminosities and energy deposition. [@Scalzo2014a; @Scalzo2014b], [@Sim_2010; @Sim2013], and [@Blondin2017] have shown that there is both observational and theoretical evidence for sub- explosions. From photometric model fitting, [@Scalzo2014a; @Scalzo2014b] shows that the mass distribution for SNe Ia includes both sub- and super- events. [@Scalzo2014b] argue that 25–50 per cent of SN Ia events deviate from events, with most of these occurring as sub- explosions. Therefore, for given measured [$^{56}{\rm Ni}$]{} masses, one can compare SNe Ia to better understand how ejecta mass affects light curves and spectral evolution.
[@Sim_2010; @Sim2013] found good agreement with photometric observations at maximum and reproduced IME features in synthetic spectra at maximum with 1D pure detonations of sub- explosions. However, these models decline too rapidly post maximum. This idealized approach ignores the influence of any accreted helium layer. Previous works [@Woosley1994; @Livne1995; @HoeflichKhokhlov1996; @Hoeflich1996; @Nugent1997; @Kromer2010] found difficulties reproducing the observed light curves, colours and spectral evolution after treating the burnt helium layer, which synthesizes a significant amount of [$^{56}{\rm Ni}$]{}.
[@Woosley2007] explored a grid of SN Ia models coming from 1.38 WDs with varying amounts of mixing and [$^{56}{\rm Ni}$]{}, finding models of similar [$^{56}{\rm Ni}$]{} to have large variations of decline rates ([$\Delta M_{15}(B)$]{} – which is the change in $B$-band magnitude 15 days after $B$-band maximum [@Phillips1993]) and anti width-luminosity relationships. [@Woosley2011] computed hydrodynamic and radiative transfer simulations of sub- mass models for helium-accreting WDs, but the authors only found reasonable agreement with spectra and light curves of common SNe Ia for the most massive white dwarfs with the smallest helium layers they considered. [@Blondin2017] looks at broadening our understanding of the width-luminosity relation (WLR) using pure detonations of sub- WDs. Their work shows promising agreement with observations of faint SNe Ia, confirming the need for two WD populations to explain the full behavior of the WLR seen at high and low SN Ia brightnesses.
Observations have shown high-velocity features (HVFs) in early-time spectra of SNe Ia. These features have been used to constrain the explosion scenario [@Mazzali2005a; @Mazzali2005b; @Tanaka2006; @Blondin2013; @Childress2013; @Childress2014; @Silverman2015; @Zhao2015; @Pan2015]. Studying the formation of these features as well as their correlation with galaxy environment can improve our understanding of Type Ia progenitors. We can gain insights, for example, into the density and temperature structure of the outer layers. The latter will reveal itself through changes in ionization and hence the strength of spectral features.
In this paper we study two sub- models ($\sim$1 ), a model ($\sim$1.4 ), and a super- model ($\sim$1.7 ), all with the same [$^{56}{\rm Ni}$]{} mass by design, to determine the effects of ejecta mass. One model is a standard DDT model, two models are standard DDT models whose density have been scaled to give the desired ejecta mass, while the final model arises from a pure detonation in a sub- model. The original DDT models are also somewhat artificial. For example, the onset of detonation in a DDT model is a free parameter. Further, it is impossible in current models to resolve and adequately model the thermonuclear flame (see [@Ciaraldi-Schoolmann2013] and references therein). We consider evolution over more than two hundred days in time – from $\sim$1 day after the explosion until approximately 220 days after the explosion, and look for diagnostics of ejecta mass ($M_{\rm ej}$) for our [$^{56}{\rm Ni}$]{} mass that can be used to distinguish between the different models. Since the precise explosion mechanism is uncertain, the two sub- models were exploded using different assumptions. Spectra and light curves were computed using non-LTE and time-dependent radiative transfer. Looking for SN Ia diagnostics, we wanted to remove sensitivity of [$^{56}{\rm Ni}$]{} while focusing on ejecta mass to uncover insights in SN Ia evolution, both spectroscopically and photometrically.
The paper is organized as follows. In Section \[section\_technique\], we discuss the techniques used and initial ejecta properties. We compare the bolometric light curves as well as synthetic photometric light curves in Section \[section\_lc\]. We discuss the spectral evolution, highlighting the strong spectral differences, in Section \[section\_spec\_evo\]. In Section \[section\_sub1\_vs\_sub2\], we further highlight the distinctions between our two sub- models. In Section \[section\_HVF\] we comment on the lack of high velocity features. Section \[section\_data\_compare\] shows spectral comparison to a few SNe Ia that are close in measured $M_{\rm ej}$ and [$^{56}{\rm Ni}$]{} mass. In Section \[section\_problems\] we discuss shortcomings of our models in reproducing spectral characteristics of SNe Ia beyond 20 days post-maximum. Section \[Conclusion\] summarizes our results and conclusions.
Technique {#section_technique}
=========
To determine spectral and light curve diagnostics we utilize hydrodynamical models that have been evolved from explosion until 0.75 days. Homologous expansion of the ejecta is well established at 100 seconds, and it is assumed to strictly hold at all times thereafter. We allow for non-local thermodynamic equilibrium (non-LTE) and solve the spherically symmetric, time-dependent, relativistic radiative transfer equation in order to produce emergent synthetic spectra from which synthetic light curves (LCs) can be produced.
Ejecta and Radiative Transfer Modeling
--------------------------------------
Our models correspond to scaled ejecta of model DDC0 (density scaled by 0.73, model SUB2) and DDC15 (density scaled by 1.22, model SUP), complemented with models DDC10 (no scaling applied, model CHAN) – see [@Blondin2013] – and SCH5p5 (density scaled by 0.98, model SUB1) – see [@Blondin2017]. This density scaling is applied to produce ejecta with the same [$^{56}{\rm Ni}$]{} mass initially (0.62 ), but differing in ejecta mass so that they lie below, at, and above the . This scaling, applied exclusively to the density at 10 seconds after explosion, is obviously artificial. An advantage of this scaling method is that, while the mass varies, the chemical stratification does not. This ensures the models retain their fundamental characteristic of all SN Ia ejecta. We do not compute the combustion nor make any claim that a flame would behave in the way adopted for the corresponding WD mass (i.e., deflagration followed by detonation in the DDC models; pure detonation in the SCH model). Even in the original (unscaled) model, the treatment of combustion is largely imposed rather than computed from first principles. The purpose of the study is to investigate the signatures sensitive to variations in ejecta mass, keeping the [$^{56}{\rm Ni}$]{} mass the same between all models in order to retain only one variable quantity. Model masses, kinetic energies, and important species masses are summarized in Table \[model\_info\_abund\].
The use of scaled DDT models for SUB2 and SUP is problematical since the explosion properties will depend on the mass of the WD. For example, differences in the initial central densities before the explosion will likely lead to different abundance profiles within the ejecta. In particular, sub- WDs, unlike their counterpart with stable IGEs in their inner ejecta, do not exhibit a ‘$^{56}$Ni hole’ which is a low [$^{56}{\rm Ni}$]{} abundance inside an expansion velocity of 2500 . However, a comparison of SUB2 with SUB1 will allow us to test the sensitivity of the results to the adopted explosion model. The explosion mechanism for ejecta with super-Chandrasekhar masses is extremely uncertain, and any adopted model will have limitations.
The radiative transfer models have been computed using [[cmfgen]{}]{} [@Hillier1998; @Hillier2012; @Dessart2014a; @Dessart2014b], which solves the spherically symmetric, non-local thermodynamic equilibrium (non-LTE), time-dependent, relativistic radiative transfer equation in the co-moving frame. To advance in time, we used a 10 per cent time step for each model starting from 0.75 days until $\sim$100 days, after which we used a time step of 10 days. At early times ($\lesssim$7 days) during the time sequence, we assumed that $\gamma$-ray photons created from radioactive decays are locally deposited. Otherwise, we approximate the $\gamma$-ray deposition as grey, adopting the procedure from [@Swartz1995] and a $\kappa_\gamma=0.06Y_e$ cm$^2$ g$^{-1}$. The kinetic energy of decay positrons is locally deposited at all epochs. [[cmfgen]{}]{} currently treats both one- and two-step decay chains for calculating non-thermal heating.
![Illustration of the outward cumulative mass as a function of velocity. The cumulative mass begins to flatten off around 25000 for all models, the velocity at which the density begins to decrease rapidly. Less than 1 per cent of the total mass is exterior to this velocity.[]{data-label="mass_vs_v"}](plots/mass_vs_v_1.pdf)
---------------- ------------- ------------------ ---------------------- ------------- ------------- ------------------------------------------- --------------------- ------------- ------------- -------------
Model Mass $E_{\rm Kin}$ $\upsilon_{\rm rms}$ C O Na Mg Si S Ar
\[0.5ex\] () ($10^{51}$ ergs) () () () () () () () ()
\[0.5ex\] SUB1 1.04 1.22 10821 3.295$(-3)$ 5.196$(-2)$ 1.306$(-5)$ 2.441$(-3)$ 1.553$(-1)$ 1.022$(-1)$ 2.248$(-2)$
SUB2 1.02 1.17 10695 8.803$(-4)$ 3.906$(-2)$ 4.466(-6) 2.615$(-3)$ 1.185$(-1)$ 7.488$(-2)$ 1.654$(-2)$
CHAN 1.40 1.51 10415 2.419$(-3)$ 9.595$(-2)$ 1.244$(-5)$ 9.557$(-3)$ 2.551$(-1)$ 1.661$(-1)$ 3.693$(-2)$
SUP 1.70 1.81 10351 3.333$(-3)$ 1.293$(-1)$ 2.044$(-5)$ 1.387$(-2)$ 3.805$(-1)$ 2.443$(-1)$ 5.223$(-2)$
Model Ca Ti Cr Fe Co [$^{58}{\rm Ni}$]{} + [$^{60}{\rm Ni}$]{} [$^{56}{\rm Ni}$]{}
\[0.5ex\] () () () () () () ()
\[0.5ex\] SUB1 2.361$(-2)$ 2.665$(-5)$ 1.030$(-2)$ 2.226$(-2)$ 5.526$(-2)$ 1.1$(-2)$ 5.684$(-1)$
SUB2 1.822$(-2)$ 1.083$(-4)$ 1.516$(-2)$ 6.654$(-2)$ 5.680$(-2)$ 2.6$(-2)$ 5.710$(-1)$
CHAN 4.120$(-2)$ 1.486$(-4)$ 2.689$(-2)$ 1.020$(-1)$ 5.713$(-2)$ 2.5$(-2)$ 5.708$(-1)$
SUP 5.562$(-2)$ 1.828$(-4)$ 2.977$(-2)$ 1.224$(-1)$ 5.777$(-2)$ 3.0$(-2)$ 5.707$(-1)$
---------------- ------------- ------------------ ---------------------- ------------- ------------- ------------------------------------------- --------------------- ------------- ------------- -------------
Ejecta Conditions for Radiative Transfer
----------------------------------------
Table \[model\_info\_abund\] shows the yields for the most abundant species in our ejecta models at 0.75 days. Since the [$^{56}{\rm Ni}$]{} mass is fixed, there are substantial variations in the mass of the IMEs. In particular, the mass of individual IMEs in model SUP is a factor of 3 to 5 larger than in model SUB2; model SUB1 has $\sim$1.3 to 1.4 times the mass of IMEs and $\sim$1/3 times the iron mass of SUB2. The initial iron abundance in SUB1 is almost a factor of 6 lower than in SUP. The mass of stable nickel ([$^{58}{\rm Ni}$]{} and [$^{60}{\rm Ni}$]{}) is 0.011, 0.026, 0.025, and 0.030 for SUB1, SUB2, CHAN, and SUP. The presence of stable nickel features in nebular spectra, and the ability to measure the nickel abundance, is discussed in Sections \[section\_spec\_evo\] and \[section\_problems\].
In \[mass\_vs\_v\] we show the outward cumulative mass as a function of velocity. Less than 1 per cent of the mass lies beyond 25000 for all models, and hence we restrict future model ejecta comparisons to velocities less than 30000. Higher velocity material makes very minor contributions to synthetic spectra.
\[initial\_rho\_all\] compares the initial mass fraction at 0.75 days for all models. All but model SUB1 exhibit an ‘[$^{56}{\rm Ni}$]{} hole’ at velocities less than $\sim$2500. As noted earlier, the hole in SUB2 is artificial, and arises since the model was scaled from a model based on the explosion of a WD which has a higher central density.
Light Curves {#section_lc}
============
The light curves and colour evolution of Type Ia models depends on the progenitor system, and potentially offer a means to distinguish between progenitor systems. Due to differences in ejecta mass, the diffusion time varies between models, giving rise to morphological separations in both the width of each bolometric light curve and the peak luminosity. However, from work by [@Pinto_Eastman2000a; @Pinto_Eastman2000b], we expect this effect to be small.
\[LogLbol\] shows the bolometric light curves of all models relative to the time of explosion. Decreasing ejecta mass (shorter diffusion time) corresponds to a faster evolving supernova. In days since explosion, the bolometric luminosity maximum occurs at 14.4 (3.80$\times10^9$ ), 14.4 (3.96$\times10^9$ ), 15.84 (3.63$\times10^9$ ), and 17.42 (3.47$\times10^9$ ) for models SUB1, SUB2, CHAN, and SUP respectively. To characterize the bolometric light curves, we list $t_{\rm max}$ (time of bolometric maximum), $t_{-1/2}$ and $t_{+1/2}$ (the times to rise from half bolometric maximum luminosity to maximum and to decline from bolometric maximum back to half of maximum – see [@Contardo2000]) in Table \[lc\_data\]. Between $\sim$1–1.7 , we have roughly a 1 day difference in half light rise times ($t_{-1/2}$), with SUB ejecta models rising faster. After bolometric maximum, the SUP ejecta model takes roughly $\sim3.5$ days longer than the SUB ejecta models to decline to half light ($t_{+1/2}$). With precise measurement determinations of the rise time in $L_{\rm bol}$, we can put stronger constraints on the ejecta mass for a given [$^{56}{\rm Ni}$]{} mass.
As the [$^{56}{\rm Ni}$]{} mass is the same for all models, differences in the bolometric light curve evolution are primarily due to ejecta mass. However, despite differences in ejecta mass of $\sim$70 per cent, the differences are relatively small, though measurable. This effect of ejecta mass should be clearly visible from a statistical sample of deep high cadence observations of SNe Ia.
\[lc\_wrtb\] shows synthetic light curves plotted in days since $B$-band maximum. The light curves for the two sub- ejecta models tend to be more luminous at maximum (with the exception of the NIR bands) but have fainter nebular luminosities. The former arises because at bolometric maximum $L_{\rm bol}\approx L_{\rm decay}$ and since the peak is earlier, the decay rate is greater. The latter arises because of the less efficient trapping of . Table \[lc\_data\] also lists the peak LC absolute magnitudes for different Johnson bands ($M_U\rightarrow M_K$), rise times (in days since explosion), and decline parameter $\Delta M_{15}(X)$ for a given band $X$. All bands show a faster rise time for lower-mass ejecta models, just as they do for $L_{\rm bol}$.
Observational evidence suggests that $H$-band photometry for SNe Ia can provide higher accuracy than the $B$-band calibration galactic distances [@Krisciunas2004; @Krisciunas2007; @Wood-Vasey2008]. However, our models have a spread of almost one magnitude in the $H$-band at the time of $B$-band maximum light. Following the temperature separation between models (higher mass $\rightarrow$ cooler ejecta), the higher mass ejecta also show larger flux in the $H$-band at all epochs. Our light curves still show two peaks in the $H$-band as normal SNe Ia do, a consequence of the ionization shift in iron (and other IGEs) going from ionization states 2+$\rightarrow$1+ [@Kasen2006]. The second peak is of the same brightness for our SUB models compared to higher ejecta mass models. Past the second $H$-band peak, we see roughly a constant decline in all models until close to 150 days past maximum light.
\[LCcolor\] shows the $U-B$, $B-V$, and $B-R$ colour evolution relative to $B$-band maximum. It shows that the lower mass models are bluer at most epochs. There is over a magnitude difference in $B-R$ from sub- to super- around 25 days after $B_{\rm max}$. SUB models are bluest in color post $B$-band maximum compared to higher mass models. Post maximum, model SUP remains the most red of all the models, while SUB1 remains bluer than other models after $+40$ days, a result of higher temperatures and ionization due to larger $M(^{56}{\rm Ni})/M_{\rm ej}$. This is explored in \[TvsV\_all\] and \[IF\_frac\] and then in Section \[section\_spec\_evo\] where we discuss the spectral evolution.
\[TvsV\_all\] shows the temperature evolution of our models, with SUB1 and SUB2 maintaining higher temperatures at all epochs. \[IF\_frac\] shows the average ionization for a few IGEs such as iron, cobalt, and nickel. SUB models show a higher ionization compared to higher mass models, producing ejecta with bluer colors. The monotonic temperature distribution of SUB1 below 5000 is due to the lack of a ‘[$^{56}{\rm Ni}$]{} hole’. \[edep\_ratio\] shows the fraction of the energy deposition from positrons to the total energy deposition at about 216 days post explosion. At late times SUB1 maintains a higher ionization in the innermost ejecta due to more assumed local positron energy deposition. Generally speaking, SUB1 and SUB2 have higher temperatures at comparable epochs than those of CHAN and SUP. We further discuss the implications of higher temperatures in Section \[section\_spec\_evo\].
---------------- ------------ ------------------ -------------------- --------------- ------------------ -------------------- ---------- ------------------ -------------------- ---------- ------------------ --------------------
Model $M_U$ $t(U_{\rm max})$ $\Delta M_{15}(U)$ $M_B$ $t(B_{\rm max})$ $\Delta M_{15}(B)$ $M_V$ $t(V_{\rm max})$ $\Delta M_{15}(V)$ $M_R$ $t(R_{\rm max})$ $\Delta M_{15}(R)$
(mag) (days) (mag) (mag) (days) (mag) (mag) (days) (mag) (mag) (days) (mag)
\[0.5ex\] SUB1 $-$19.89 14.13 1.07 $-$19.39 16.97 0.95 $-$19.36 17.99 0.88 $-$19.17 17.40 0.89
SUB2 $-$19.96 13.35 1.08 $-$19.41 15.64 0.94 $-$19.36 16.48 0.96 $-$19.14 17.01 0.91
CHAN $-$19.78 14.90 1.06 $-$19.31 18.54 0.94 $-$19.31 19.91 0.68 $-$19.24 18.92 0.76
SUP $-$19.67 15.92 1.03 $-$19.25 19.86 0.99 $-$19.31 21.89 0.60 $-$19.29 20.48 0.59
$M_I$ $t(I_{\rm max})$ $\Delta M_{15}(I)$ $M_J$ $t(J_{\rm max})$ $\Delta M_{15}(J)$ $M_H$ $t(H_{\rm max})$ $\Delta M_{15}(H)$ $M_K$ $t(K_{\rm max})$ $\Delta M_{15}(K)$
(mag) (days) (mag) (mag) (days) (mag) (mag) (days) (mag) (mag) (days) (mag)
SUB1 $-$18.68 12.86 0.70 $-$18.24 9.60 1.54 $-$17.87 8.76 0.29 $-$17.90 9.62 0.08
SUB2 $-$18.60 13.76 0.64 $-$18.06 8.39 1.03 $-$17.70 7.81 0.09 $-$17.77 9.27 $-$0.08
CHAN $-$18.87 16.41 0.44 $-$18.48 12.73 1.26 $-$18.08 10.73 0.10 $-$18.10 12.40 $-$0.02
SUP $-$18.99 19.01 0.27 $-$18.67 15.29 1.17 $-$18.30 12.91 0.02 $-$18.28 15.25 $-$0.07
$t_{-1/2}$ $t_{+1/2}$ $t_{\rm max}$ $L_{\rm bol}$
(mag) (days) (mag) $(10^9$)
SUB1 8.18 13.0 14.4 3.80
SUB2 8.02 12.44 14.4 3.96
CHAN 8.83 14.75 15.84 3.63
SUP 9.27 16.57 17.42 3.47
---------------- ------------ ------------------ -------------------- --------------- ------------------ -------------------- ---------- ------------------ -------------------- ---------- ------------------ --------------------
![Ratio of the locally deposited energy from positrons (mainly from [$^{56}{\rm Co}$]{}) to the energy deposited by nuclear decays in the ejecta at about 216 days post explosion. SUB1 shows a much higher ratio shortward of 5000 due to the lack of a ‘[$^{56}{\rm Ni}$]{} hole’ and higher [$^{56}{\rm Ni}$]{} production in the core. This leads to higher ionization and a higher temperature in the inner region (in combination with lower densities) compared to other models.[]{data-label="edep_ratio"}](plots/edep_ratio_200d_1.pdf)
Spectral Diagnostics {#section_spec_evo}
====================
Spectra provide important constraints for distinguishing progenitor and explosion models. However, despite a 70 per cent difference in mass, model optical spectra ( \[premax\_max\_spec\], \[postmax\_1\], \[postmax\_2\], \[nebular\_1\], and \[nebular\_2\]) at most phases are similar, consistent with work by [@Blondin2013] who concluded that SNe Ia are mainly distinguished by their [$^{56}{\rm Ni}$]{} mass. This similarity is observed for classical SNe Ia where differences in optical spectra are generally rather subtle [@Filippenko1997], which presumably occurs because of similarities in composition. Interestingly, optical spectra of models SUB1 and SUB2 are remarkably similar for most phases of evolution, despite the different methods to produce these ejecta models. However, there are differences between models, particularly in the infrared, and these do give rise to useful diagnostics. We investigate these diagnostics by comparing model spectra at time steps of approximately $-10$, $-5$, $0$, $+5$, $+10$, $+20$, $+50$, $+100$, and $+200$ days from bolometric maximum ( \[premax\_max\_spec\], \[postmax\_1\], \[postmax\_2\], \[nebular\_1\], and \[nebular\_2\]). Thus, when comparing observational spectra of SNe Ia from comparable [$^{56}{\rm Ni}$]{} mass, these diagnostics will separate events by ejecta mass. Note that model spectra are plotted for vacuum wavelengths; however, wavelengths $\geq$2000 Å listed are quoted in air.
\[premax\_max\_spec\] shows the early spectral evolution ($-10$, $-5$, and 0 days relative to bolometric maximum) plotted in $\lambda F_{\lambda}$ (arbitrary units). We label the contributions of important species at bolometric maximum. \[postmax\_1\] and \[postmax\_2\] show the post maximum photospheric phase ($+5$, $+10$, $+20$, $+50$) plotted in $\lambda F_{\lambda}$ (arbitrary units). \[nebular\_1\] and \[nebular\_2\] show nebular spectra at $+100$ and $+200$ days post bolometric maximum plotted in $F_{\lambda}$ (arbitrary units). Contributions from important species are labelled. Notice the transition towards predominantly forbidden lines in nebular spectra. These figures are used to highlight important diagnostics.
\[\] 1.939 $\mu$m {#nickel_line_diag}
-----------------
In SNe Ia, the nickel abundance is sensitive to the progenitor mass and/or explosion scenario. In 1D explosion modeling, higher central densities have higher neutronisation that leads to more stable [$^{58}{\rm Ni}$]{} being produced during nuclear burning [@Nomoto1984; @Khokhlov1991a; @Khokhlov1991b]. This 1D modeling implies that sub- SNe Ia will show a lower abundance of [$^{58}{\rm Ni}$]{} compared to SNe Ia (for the same [$^{56}{\rm Ni}$]{} mass). However, 3D DDT modeling suggests that the [$^{56}{\rm Ni}$]{} hole predicted in 1D WD DDT models may be absent, and both [$^{56}{\rm Ni}$]{} and [$^{58}{\rm Ni}$]{} extend from the lowest velocities to about 10000 [@Kasen2009; @Seitenzahl2013]. [$^{22}{\rm Ne}$]{} settling in sub- has also been proposed as a way to enhance the neutronisation; however, the time-scale for gravitational settling can be $\sim10^9-10^{10}$ yrs [@Bildsten2001]. Therefore, nickel diagnostics, particularly at nebular times, may constrain the progenitor scenario, nucleosynthesis, and explosion mechanism [@Woosley1997; @Iwamoto1999; @Stehle2005; @Mazzali2006; @Gerardy2007; @Maeda2010; @Mazzali2011; @Mazzali2012; @Mazzali2015]. At nebular times most of the [$^{56}{\rm Ni}$]{} will have decayed, and any nickel emission features are due to stable nickel, and in particular [$^{58}{\rm Ni}$]{} and [$^{60}{\rm Ni}$]{}, which are expected to be underabundant in (1D) sub- DDT models compared with models. The width of any observed nebular nickel feature will constrain the hydrodynamic width of the emitting region, thus testing model predictions about the presence of a [$^{56}{\rm Ni}$]{} hole. Therefore, nickel features may offer the best diagnostic for ejecta masses below if ejecta do or do not have [$^{56}{\rm Ni}$]{} holes as predicted by 1D modeling.
In the optical nebular spectra at $+100$ and $+200$ days ( \[nebular\_1\] and \[nebular\_2\]), the \[\] [$\lambda$]{}[$\lambda$]{}7378,7412 lines are blended, and hence not very useful for abundance determinations. However, in the NIR there is a forbidden \[\] transition (3d8($^3$F)4s${}^2$F$_{7/2}\,-\,$3d8($^3$F)4s$^4$F$_{9/2}$) at 1.939 $\mu$m which in our synthetic spectra is relatively blend-free. It overlaps with telluric lines in low-redshift SNe, but higher redshift ($z>0.08$) SNe avoid telluric absorption. While observations of this line appear to be rare, [@Friesen2014] find evidence for this line in spectra of SN2011fe, SN2014J, and SN2003du.
In our models, SUB1 shows no evidence of \[\] 1.939 $\mu$m. On the other hand, SUB2, CHAN, and SUP show the line, with a strength that correlates with ejecta mass. The absence of \[\] 1.939 $\mu$m in SUB1 arises from two effects — SUB1 has a smaller amount of stable nickel (see Table \[model\_info\_abund\]) and a higher ionization than the other models. The presence of [$^{56}{\rm Co}$]{} under $\sim$3000 in SUB1 means that there is a great amount of heating from positrons, which deposit their energy locally ( \[edep\_ratio\]). This, combined with the lower densities, leads to both a higher temperature, and a higher ionization ( \[TvsV\_all\] and \[IF\_frac\]). Surprisingly, and despite their similar ionization potentials, cannot be used as an ionization tracer for . In SUB1 / is significantly larger than /. This arises because the photoionization of is dominated by large resonances in its photoionization cross-section. Therefore, the absence of \[\] 1.939 $\mu$m in SN Ia spectra at 100–200 days indicates that the mass of the progenitor is below . However, we re-emphasize that SUB2 is a scaled model, so it is not a true sub- model. SUB2 has a stable nickel core and shows \[\] 1.939 $\mu$m.
Ionization
----------
Once the [$^{56}{\rm Ni}$]{} mass is determined via “Arnett’s rule", which states that the luminosity at bolometric maximum is equal to the [$^{56}{\rm Ni}$]{} decay chain luminosity, or using LC fitting like that of [@Scalzo2014a; @Scalzo2014b], one can separate different SNe Ia based on ejecta mass using differences in ionization/temperature (see \[TvsV\_all\]). This result follows from the heating per gram available to the gas. The greater the ejecta mass is, the lower the heating rate per gram is. Consequently, ejecta with a larger $M(^{56}{\rm Ni})/M_{\rm ej}$ are hotter – see [@Blondin2017] for $\dot{e}_{\rm decay}\equiv L_{\rm decay}/M_{\rm tot}$. Indeed our models indicate that low mass WD models, for a given [$^{56}{\rm Ni}$]{} mass, maintain higher ionizations throughout their spectral evolution.
### UV-Blanketing
Previous studies focused on the UV variability and used the UV spectral region for understanding SNe Ia. These studies looked at the role of metallicity on UV blanketing [@Lentz2000; @Walker2012; @Wang2012; @Foley2013] finding that lower metallicities shift the blanketing blueward. [@Foley2016] looked at a sample of SNe Ia and found the UV diversity linearly correlates with the optical LC shape. In particular, the strength of UV line flux measurements ($\sim$2030 & 2535 Å) increases with increasing $\Delta m_{15}(B)$. Other studies of UV variation hope to use it as a cosmological utility [@Ellis2008; @Sullivan2009] to improve standardizability. Therefore, understanding how $M_{\rm ej}$ for a given [$^{56}{\rm Ni}$]{} mass influences the UV spectrum is important to the astronomical community.
Until the ejecta begins entering its nebular phase ($\sim$100 days), we see larger UV blanketing shortward of 4000 Å for larger mass ejecta. This effect is attributed to a temperature difference between models. We see in \[TvsV\_all\] and \[IF\_frac\] that below 25000 models with higher temperatures have higher ionizations, seen as a shift in the line blanketing to higher frequencies. Pre-maximum spectra show the feature (H & K lines near [$\lambda$]{}3500) is affected by UV blanketing, making it difficult to distinguish in SUP and CHAN ( \[premax\_max\_spec\]). contributes to much of the blanketing more than 5 days before maximum, while , , and shape the UV spectra just prior to maximum. Around maximum, contributes much of the UV blanketing (below 3500 Å) with the strongest blanketing occurring in model SUP. Looking inwards of 25000 , SUB1 and SUB2 show a higher ionization of cobalt than that of models SUP and CHAN. For there is about a half dex difference in ionization between SUB1 and SUP. These differences show up as absorption affecting the slope of the feature at $\sim3500$ Å. Post-maximum ( \[postmax\_1\] and \[postmax\_2\]), there is less variation in UV blanketing between the models.
If we compare the peak fluxes at bolometric maximum ( \[premax\_max\_spec\]) of three UV features (namely the features near $\sim$2850Å, $\sim$3150Å, and $\sim$3550Å), we can characterize the level of blanketing by comparing the flux at peak in each feature. For all ejecta models, the flux ratio $F(3150)/F(2850)$ is close to unity (0.93, 1.07, 1.03, and 1.11 for SUB1, SUB2, CHAN, and SUP). However, comparing these lines to the feature just short of the H&K and $\sim$[$\lambda$]{}3660 Å absorption profile, we see that the flux ratio $F(3550)/F(2850)$ is strongly dependent on ejecta mass. This flux ratio $F(3550)/F(2850)$ is 0.99, 1.01, 1.20, and 1.75 for SUB1, SUB2, CHAN, and SUP. These UV features reflect the temperature and ionization of the ejecta and offer a diagnostic of ejecta mass for a given [$^{56}{\rm Ni}$]{} mass.
### Optical and IR {#optical_NIR}
Besides variations in UV blanketing, other ionization diagnostics are seen in optical and infrared spectra. For instance, leading up to maximum ( \[premax\_max\_spec\]), each model shows a different strength of the triplet ([$\lambda$]{}[$\lambda$]{}4553,4568,4575) absorption, which is strongest in the models SUB1 and SUB2. SNe Ia typically classified as normal, such as SN2011fe, show the feature around 4400 Å [@Pereira2013] as our model CHAN does. SNe Ia such as SN2003hv, thought to be a sub- event [@Mazzali2011], show this absorption feature much more strongly [@Leloudas2009], as in our SUB models.
Post maximum ( \[postmax\_2\]), the near-infrared part of the spectrum begins to show prominent permitted (9997.58, 10501.50, 10862.64, 16787.18, and 16873.20 Å) and features (11829.72, 15758.43, 16064.48, 16360.46, 16687.30, 21344.70, 21503.28, 22202.92, 22475.63, and 23612.53 Å), as well as forbidden \[\] (22178.21, 22420.43, and 23478.80 Å) and \[\] (12724.19, 15483.56, 17408.66, 19575.24, 20022.57, and 20973.15 Å) lines. Many of the and features are absent in SUB1, a result of the higher ionization.
Optical nebular spectra typically exhibit emission lines of and ( \[nebular\_1\] and \[nebular\_2\]). In the NIR S$^{2+}$ and Ar$^{2+}$ show up in our model spectra as \[\] [$\lambda$]{}[$\lambda$]{}9068,9530, and \[\] [$\lambda$]{}[$\lambda$]{}7135,7751, with the strength of these features relative to \[\] [$\lambda$]{}4658 correlating with higher ejecta mass. As will be discussed in Section \[section\_problems\], our model spectra tend to exhibit too high an ionization, especially after 40 days. In particular, they lack strong \[\] (e.g., \[\] $\sim$4350 Å). However \[\] and \[\] features are readily identified in the IR, except for model SUB1. Since SUB1 comes from the explosion of a sub- WD, its inner density is lower throughout its evolution compared to SUB2, and this hinders recombination. Further, SUB1 lacks the ‘[$^{56}{\rm Ni}$]{} hole’ seen in the later models, and hence the temperature in the inner region is higher than in the other models (see \[TvsV\_all\], \[IF\_frac\], and \[edep\_ratio\]).
C/O and IMEs
------------
As the mass of C/O and IMEs is strongly correlated (by design from the density scalings) with the ejecta mass for a given [$^{56}{\rm Ni}$]{} mass (see Table \[model\_info\_abund\]), one should expect that lines from C/O and IMEs will provide a useful diagnostic tool for ejecta mass. As to be expected, our models show stronger absorption features for oxygen and IMEs for increasing ejecta mass. For example, the strength of absorption due to the [$\lambda$]{}[$\lambda$]{}7772,7774,7775 triplet absorption correlates with ejecta mass in pre-maximum spectrum ( \[premax\_max\_spec\]). The feature fades by a few weeks post bolometric maximum. [$\lambda$]{}[$\lambda$]{}9218,9244 is another feature whose strength correlates with high ejecta mass–see Table \[Sk2\_widths\] which lists the pseudo-equivalent widths (pEWs) measured by a straight line across the maxima of the absorption profile; it also fades within a few weeks post bolometric maximum.
[$\lambda$]{}[$\lambda$]{}5958,5979, observed roughly around $\sim$[$\lambda$]{}5750, is a spectroscopic classification diagnostic for SNe Ia ($\mathcal{R}$(Si) $\equiv$ pEW( [$\lambda$]{}5750)/pEW( [$\lambda$]{}6100)) and, like other IME features, its strength correlates with ejecta mass. Table \[Sk2\_widths\] highlights the correlation of pEWs of various features with ejecta mass of our models. Post maximum ( \[postmax\_1\] and \[postmax\_2\]), we see the strength of the emission increase, giving a large morphological separation between models. Calcium (as ) also shows the same behaviour as . The absorption and emission strength of the NIR triplet distinguishes models throughout the spectral evolution. We find that the strength of this feature correlates with ejecta mass. We further discuss the NIR triplet and the [$\lambda$]{}[$\lambda$]{}6347,6371 doublet in Section \[section\_HVF\].
Explosion Scenario: SUB1 versus SUB2 {#section_sub1_vs_sub2}
====================================
Since the explosion process and progenitor system are unknown, we highlight and summarize useful diagnostics for distinguishing our models of the same $M_{\rm ej}$. As mentioned earlier, SUB1 comes from a detonation model of a sub- WD, while SUB2 comes from the DDT of a WD, which was scaled in density to have the same mass as SUB1 and the same [$^{56}{\rm Ni}$]{} mass. Since SUB1 was detonated as a sub- WD, it had lower densities when exploded compared to SUB2 and lacks the ‘[$^{56}{\rm Ni}$]{} hole’. Without the ‘[$^{56}{\rm Ni}$]{} hole’, SUB1 has a larger (assumed) local deposition fraction from decay positrons compared to the total decay energy deposition ( \[edep\_ratio\]) for velocities less than 5000 . This keeps the inner region of SUB1 hotter than SUB2, which shows stronger features of higher ionization states of IGEs as the ejecta evolves past the photospheric phase and exposes the inner iron-rich material. At nebular times, the strength of \[\] 1.939 $\mu$m gives a clear distinction between SUB1 and SUB2, as lower density ejecta model SUB1 does not show this feature.
Up to maximum light, SUB1 and SUB2 possess very similar spectra ( \[premax\_max\_spec\]), especially in the optical. However, as the photosphere begins to recede inwards differences are seen in the NIR – and features are absent in SUB1 but present in SUB2 ( \[postmax\_1\] and \[postmax\_2\]). Below 5000 , the densities in SUB1 are roughly a factor of 3 lower than in SUB2. Further, SUB1 has a larger fraction of local radioactive heating from positrons. These factors inhibit recombination and a higher ionization persists in SUB1 compared to SUB2. The NIR region is potentially the best diagnostic for the ionization state of the ejecta in SNe Ia (shown in \[nebular\_1\] and \[nebular\_2\]).
This higher ionization, seen in post-maximum spectra ( \[postmax\_1\] and \[postmax\_2\]), yields lower fluxes in the NIR. We see roughly half a magnitude difference in the post maximum $I$, $J$, and $H$ bands. However, the magnitude difference between $J$ and $H$ grows to $\sim$2 mag difference by 200 days post maximum.
Additional Investigations
=========================
In this section we present additional investigations of our ejecta models focusing on high velocity features and comparisons to observational data. We also explore shortcomings with our ejecta models.
High Velocity Features – & {#section_HVF}
----------------------------
High velocity features (HVFs) are absorption features, seen in the strongest lines, that show a distinct difference in velocity (often early and prior to maximum), by more than a few thousand from the lower velocity, photospheric component [@Gerardy2004; @Mazzali2005a; @Mazzali2005b]. Note the two strongest components of the NIR triplet ([$\lambda$]{}8542 & [$\lambda$]{}8662) are separated by $\sim$4000. Thus, any single NIR profile may show an absorption feature with two components separated by a few thousand which is different from a HVF. In many SNe Ia, HVFs have even been observed at maximum for the NIR triplet but not for [$\lambda$]{}[$\lambda$]{}6347,6371 [@Childress2014]. There is no clear indication when HVFs start to disappear in all observed cases. [@Silverman2015] state that the HVF triplet begins to disappear around $-1$ days prior to maximum for $\Delta M_{15}(B)=1.4$ to 1.6 mag, however discoveries of HVFs are potentially biased towards those that persist closer to maximum light.
Shown in \[HVF\_Sk2\] and \[HVF\_Ca2\] is the evolution of the doublet and the NIR triplet, with a vertical line at $-15\,000$ as a reference. In all models, HVFs are seen before bolometric maximum ($\lesssim-11$ days). However, no HVF for [$\lambda$]{}[$\lambda$]{}6347,6371 is seen. Notice the striking difference in the pre-maximum triplet profile ($\lesssim -11$ days) and the profile at later dates. The lack of a doublet HVF could just be a byproduct of atomic physics. Although both the NIR triplet and the doublet are not resonance transitions, the lower level of the triplet is metastable. The lower level of the [$\lambda$]{}6355 doublet is the 4s state which is coupled to the ground state by a permitted transition. Therefore, when compared to the [$\lambda$]{}6355 doublet, the NIR triplet persists longer because the metastable lower level population persists longer.
By defining a straight line between the maxima on either side of the absorption profiles of these features, we are able to compute our models’ pEWs (listed in Table \[Sk2\_widths\]). Comparing our work to fig. 8 of [@Blondin2012], we find our spectra are clustered around those labelled broad-lined Ia as seen in \[BranchTypes\]. [@Branch2006] looked at the pEWs of features near $\lambda 6100$ and $\lambda 5750$ in these spectra at maximum in order to group these spectra in different classifications: “core-normal", “broad-line", “shallow-silicon", and “cool". For the most massive model (SUP), it might fall under the “cool" classification from [@Branch2006], but it lacks the strong absorption.
![Plot of the pEWs of the $\lambda$6355 and $\lambda$5972 features along with the data from [@Blondin2012]. CN, BL, SS, and CL correspond to “core normal", “broad line", “shallow silicon", and “cool" classifications defined by [@Branch2006]. Our models lie clustered near the BL classification.[]{data-label="BranchTypes"}](plots/branch_plot_5.pdf)
---------------- -------------------- -------------------- -------------------- --------------------
Model [$\lambda$]{}5750 [$\lambda$]{}6100 [$\lambda$]{}7400 [$\lambda$]{}8700
\[0.5ex\] SUB1 10 139 17 $\lesssim$ 1
SUB2 8 160 9 $\sim$ 1
CHAN 20 156 32 32
SUP 30 158 50 53
---------------- -------------------- -------------------- -------------------- --------------------
: Approximate pEW (Å) based on a straight line across the profile of the absorption feature.[]{data-label="Sk2_widths"}
Comparison to Data {#section_data_compare}
------------------
Here we present both light curve and spectral comparisons to data for a span of spectral epochs. We focus on SNe Ia that have claimed [$^{56}{\rm Ni}$]{} masses similar to that of our models (0.6 ) or similar [$\Delta m_{15}(B)$]{} and those tagged as sub- (SN2005el), (SN1995D), and similar Branch types (SN2001ay). We used the supernova identification program SNID [@Blondin2007] on models at bolometric maximum to find additional SNe Ia to compare (SN1994ae). The spectra are taken from the CfA Supernova Archive [@Blondin2012]. Archived light curve photometry is taken from Open Supernova Catalog [@Guillochon2017]. When comparing models to observations, the spectra are normalized between $\lambda_{\rm min}$=4000 Å and $\lambda_{\rm max}$=7000 Å, such that $$\label{normalize}
\frac{1}{\lambda_{\rm max}-\lambda_{\rm min}}\int^{\lambda_{\rm max}}_{\lambda_{\rm min}}F_\lambda d\lambda=1 \text{ erg cm$^{-2}$ s$^{-1}$ \AA$^{-1}$}$$ Normalizing spectra allows us to better compare spectral features, removes uncertainties in distance, and compensates for small differences in mass. To compare LCs, we correct for extinction using the CCM reddening law [@CCM1989] and literature $E(B-V)$ and $R_V$ values. We normalize the LCs by adding a constant offset (model and object dependent), such that $B_{\rm max}$ = 0 mag at $t(B_{\rm max})$. We also shift the LCs so that time of $B$-band maxima agree. Thus, uncertainties in distance and explosion time are reduced. A constant value of 0.05 mag is included with the photometric error bars for uncertainty in reddening. $K$-corrections, expected to be small, have not been applied. Photometric band magnitudes and bolometric luminosities of the models at maximum are provided in Table \[lc\_data\].
### SN1994ae {#SN1994ae}
SN1994ae exploded in NGC 3370 [$z=0.0043$ – @Riess1999CFA1; @Jha2007] and was first discovered on 14 November 1994 by [@vanDyk1994]. It reached $B$-band maximum ($m_B=13.21$ mag) on MJD 49685.5 with [$\Delta m_{15}(B)$]{}=0.96 mag [@Riess1999CFA1; @Jha2007]. For comparison, we reddened our models using $E(B-V)=0.0226$ mag and $R_V=3.1$ [@Jha2007]. \[SN1994ae\_spec\] shows the spectral comparison of SN1994ae at $+0.0$, $+10.0$, and $+152.7$ days after $B$-band maximum and normalized LCs relative to band maximum are shown in \[SN1994ae\_LC\].
At $+0$ days, our model spectra do not reproduce the velocity of the [$\lambda$]{}6355 doublet and UV triplet. Results by [@Dessart2014a] suggest SNe Ia resulting from pulsational-delayed detonations (PDD) retain more unburnt carbon and have little mass at high velocity ($\gtrsim$15000 ) due to pulsations. Therefore, spectral features of SN1994ae might be best explained by PDD modeling, and would resemble similar radiative properties of DDT models. The spectra also show evidence of the triplet ([$\lambda$]{}[$\lambda$]{}4553,4568,4575) absorption as in our SUB models (an indication of high ionization). Later spectra show cooler ejecta and model SUP is closest to reproducing the features. However, at nebular times (+152.7 d) our model optical \[\] lines appear too strong and \[\] [$\lambda$]{}[$\lambda$]{}9068,9530 are absent in the observational data.
For the light curve comparison, we shifted the LCs to give the same time of $B_{\rm max}$ and reddened the models with $A_B$ = 0.091, $A_V$ = 0.070, $A_R$ = 0.057, and $A_I$ = 0.041 mag, obtained using $E(B-V)=0.0226$ mag and $R_V=3.1$ from [@Jha2007]. We normalized the light curves to 0 mag at $B_{\rm max}$ and shifted the observational data by 12.98 mag. We see in \[SN1994ae\_LC\] that our $B$-band LC is consistent until 20 days post maximum, where our LCs begin showing roughly half a magnitude more flux. Model CHAN matches well the $V/R$-band observations. However, our models fail to reproduce the second peak in the $I$-band, and the disagreement is greater in lower mass models.
### SN1995D {#SN1995D}
SN1995D exploded in NGC 2962 and was discovered on 10 February 1995 [@Nakano1995]. Its redshift is $z=0.0067$, and it reached $B$-band maximum ($m_B$=13.44 mag) on MJD 49768.7 [@Riess1999CFA1; @Jha2007]. SN1995D has been argued as having a [$^{56}{\rm Ni}$]{} mass of about 0.58 and an ejecta mass around 1.45 [@Childress2015]. For comparison, we reddened our models using $E(B-V)=0.026$ mag and $R_V=3.1$ [@Jha2007].
\[SN1995D\_spec\] shows the spectral comparison of SN1995D at $+3.6$, $+42.5$, and $+93.5$ days after $B$-band maximum. The early epochs ($+3.6$ days) show good qualitative agreement with SUB1 except our model shows a larger blueshifted doublet. This may be best explained by a PDD model [@Dessart2014a]. SUB1 also matches the UV spectrum shortward of 4000 Å. At later epochs like $+42.5$ and $+93.5$ days, we see a better agreement to model SUP and to CHAN, due to lower temperatures and ionization. Roughly all features at $+93.5$ days are matched by SUP. Despite the calculated [$^{56}{\rm Ni}$]{} and ejecta mass being closest to CHAN, SN1995D shows only moderate qualitative agreement at later epochs. SN1995D transitions from looking like our SUB1 into that of SUP from early to late epochs.
To compare light curves, we shifted the LCs to give the same time of $B_{\rm max}$ and reddened the models with $A_B$ = 0.106, $A_V$ = 0.081, $A_R$ = 0.066, and $A_I$ = 0.048 mag, obtained using $E(B-V)=0.026$ mag and $R_V=3.1$ [@Jha2007]. We normalized the light curves to 0 mag at $B_{\rm max}$ and shift the observational data by 13.35 mag.
In \[SN1995D\_LC\], we see that our $B$-band LC is consistent with all models until $\sim$12 days post maximum, where our LCs then begin showing roughly half a magnitude more flux. Model CHAN matches well the $V/R$-band observations. Our models fail to reproduce the second peak in the $I$-band (it occurs 10 to 20 days too early) although the SUP model matches the data at late times. Model SUP also seems consistent with the peak flux ratios in SN1995D.
### SN2001ay {#SN2001ay}
SN2001ay exploded outside IC 4423 and was discovered on 18 April 2001 by [@SwiftLi2001]. [@Krisciunas2011] and references therein cite its redshift as $z=0.0302$ and indicate that it reached $B$-band maximum ($M_B=-19.19$ mag) on 23 April 2001. For spectral comparison, we reddened our model spectra using $E(B-V)_{MW}$ = 0.026 mag, $E(B-V)_{\rm host}$ = 0.072 mag, and $R_V$ = 3.1 [@Krisciunas2011]. [@Krisciunas2011] states a $M(^{56}\text{Ni})$ of (0.58$\pm$0.15)/$\alpha$ , for an $\alpha=L_{\rm max}/E_{\rm Ni}$, typically between 1–1.2. Given the close proximity between the estimated [$^{56}{\rm Ni}$]{} mass for SN2001ay and that of our model set, we explore the spectral similarities.
\[SN2001ay\_spec\] shows the spectral comparison for epochs $-1.5$, $+9.3$, and $+56.3$ days relative to $B$-band maximum. All models provide a good qualitative fit to the optical spectrum at $-1.5$ days, with SUP exhibiting the worst fit. While all models fit the [$\lambda$]{}[$\lambda$]{}6347,6371 doublet in absorption strength and velocity, our models show stronger absorption in the [$\lambda$]{}[$\lambda$]{}5041,5056,5056.3 triplet around 4800 Å. Blended with this feature is absorption arising from [$\lambda$]{}5018, and this is also somewhat too strong in the models. The biggest discrepancy between model and observation for the blend occurs for model SUP. SUB1 lacks absorption at $\sim$4000 Å, which is clearly present in the observations, and all of the other models. No model reproduces the shape of the UV absorption near 3700 Å, which could be due to a discrepancy with the H&K lines.
Later, model SUP qualitatively agrees the SN2001ay spectra at $+9.3$ and $+56.3$ days best. At $+9.3$ days, SUP shows agreement despite its stronger absorption lines around 4800 Å. At $+56.3$ days, the spectra is dominated by features, which SUP matches well given its cooler temperatures and lower ionization. Models SUB1, SUB2, and CHAN are too highly ionized, and exhibit too much emission from higher ionization states such as . Despite matching much of the optical spectrum, SUP does not match well the absorption features associated with the NIR triplet and the H&K lines. Given the discrepancy with calcium at $-1.5$ days, this may indicate that the calcium abundance is too high, or that the distribution in velocity space is incorrect.
For the light curve comparison, we reddened the models with $A_B$ = 0.397, $A_V$ = 0.307, $A_R$ = 0.148, and $A_I$ = 0.178 mag, by combining host and MW values as $E(B-V)$=0.098 mag and $R_V$=3.1 [@Krisciunas2011]. We normalize the light curves to 0 mag at $B_{\rm max}$ and shift the observed data by 16.35 mag. \[SN2001ay\_LC\] shows our models fail to reproduce the post maximum decline except for $U/B$-bands. Our models show too little $V/R/I$ flux in the decline post maximum, but SUP agrees in peak flux ratios between bands with SN2001ay.
### SN2005el {#SN2005el}
SN2005el exploded in NGC 1819 and was discovered on 19 September 2005 [@Madison2005] at a redshift of $z=0.0148$ [@Hicken2009]. It reached $B$-band maximum ($m_B=14.84$ mag) on MJD 53646.4 [@Hicken2009]. [@Scalzo2014a] classified SN2005el as having 0.9 of ejecta as well as 0.54 of [$^{56}{\rm Ni}$]{}, which, considering the errors in the determinations, are close to our models SUB1 and SUB2. For comparison, we applied reddening to our models using $E(B-V)=0.136$ mag and $R_V=3.1$ (an $E(B-V)$ value that is higher than that stated in [@Scalzo2014a]).
\[SN2005el\_spec\] shows our spectral comparison to SN2005el. The early epochs of $-5.9$ and $+2.1$ days show some qualitative agreement, mostly with SUB1 and SUB2. At this epoch, our models do not reproduce the [$\lambda$]{}[$\lambda$]{}6347,6371 doublet. Our models indicate a [$\lambda$]{}[$\lambda$]{}6347,6371 doublet formed at higher velocities. Therefore, spectral features of SN2005el may be best explained by a PDD model. Unlike the doublet, models CHAN and SUP do reproduce the ‘w’ feature. Since our models show a higher blue-shifted doublet, it is not surprising that our UV does not match, given other and H&K features in this region. If the H&K lines and [$\lambda$]{}[$\lambda$]{}3854,3856,3862 triplet are separated by thousands of , then it is likely to result in the spike seen at the bottom of the 3700Å absorption feature, whereas our models show one broad absorption feature around 3700 Å – seen in SN2008ec, for example). Given the strong absorption profile around 4400 Å, we suggest this is the [$\lambda$]{}[$\lambda$]{}4553,4568,4575 triplet, indicating a high ionization at this epoch.
At $+24.9$ days, we see that model SUP agrees qualitatively in almost all features. Other models are too blue compared to the cooler SUP model. This is surprising given the claim that SN2005el is a sub- SN with an ejecta mass of only 0.9 . The discrepancy around 5300 Å could be the result of differences in the or absorption.
One should expect SUB1 or SUB2 to resemble the spectral evolution of SN2005el; however, we only see that SUB1 matches prior to maximum and does not match SN2005el at late epochs, where SUP shows best agreement. There are several possible explanations for the inconsistencies. First, the [$^{56}{\rm Ni}$]{} mass may be lower than 0.54. Second, the poor agreement in the extent of the [$\lambda$]{}[$\lambda$]{}6347,6371 doublet could indicate a different explosion scenario (such as PDD mentioned earlier – little mass at high velocity).
To compare light curves, we shifted the LCs relative to the time of $B_{\rm max}$ and reddened the models with $A_B$ = 0.543, $A_V$ = 0.414, $A_R$ = 0.339, $A_I$ = 0.245, $A_J$ = 0.122, and $A_H$ = 0.077 mag, obtained using $E(B-V)=0.136$ mag and $R_V=3.1$ slightly higher than [@Scalzo2014a]. The light curves were normalized to 0 mag at $B_{\rm max}$ and we adjust the observational data by 14.76 mag. In \[SN2005el\_LC\], we see the optical bands are reproduced well with our SUB models (except $B$ beyond 20 days). Although the late time behaviour in the $H$-band is reproduced, the NIR LCs do not generally agree with the SUB models. The double peak structure in the $J$-band observations is well produced by the models, although in the $H$-band it less evident.
Model Setbacks and Theoretical Problems {#section_problems}
---------------------------------------
When compared to observation data, our models do show a higher ionization, especially in the nebular phase. The strength of the \[\] [$\lambda$]{}4658 feature is too strong compared to other optical/NIR features. Further, optical spectra lack emission such as \[\] $\sim$[$\lambda$]{}4350 emission, seen in nebular spectra of SNe Ia of [@Taubenberger2013] and [@Black2016], for example. Other researchers have also had difficulty modeling the feature near 4350 Å [Spyromillo 2016, private communication; Sim 2016, private communication; @Mazzali2015; @Friesen2017]. It is not surprising that these models struggle to get the ionization correct – there are no free parameters and the density structure and element distribution is set by the adopted initial model.
At late times the super- model was generally in better agreement with observation – a result of the model being cooler with lower ionization. Since we know that most of the observed SNe we discussed are not super-, there is a fundamental problem with the models. This problem might arise from the adopted explosion models, be related to assumptions about mixing and clumping, and/or be a problem in the ionization calculations. Since nebular spectra show strong \[\] and \[\], the Fe$^+$/Fe$^{2+}$ ratio must be of order unity, and consequently it is sensitive to the Fe atomic models (and the density structure).
The disconnect between early and late time modeling is not unexpected. Early time spectra are dependent on the outer ejecta whereas late time spectra are primarily dependent on the inner ejecta. Further, the processes determining the observed spectra in the photospheric and nebular phases are distinct, and subject to different uncertainties in the atomic data.
Another problem is the strong nebular \[\] [$\lambda$]{}[$\lambda$]{}9068,9530 and \[\] [$\lambda$]{}[$\lambda$]{}7135,7751 lines. The \[\] [$\lambda$]{}[$\lambda$]{}9068,9530 does not seem to appear in nebular spectra. However, it is not clear if \[\] [$\lambda$]{}7135 is present. There are three additional transitions contributing to that overall feature between 7000-7500 Å. There are two \[\] [$\lambda$]{}[$\lambda$]{}7155,7172 lines that overlap \[\] [$\lambda$]{}7135 and, depending on the ionization structure of the ejecta, it becomes difficult to determine the source of the feature in observations. However, atomic physics of the \[\] [$\lambda$]{}[$\lambda$]{}7135,7751 lines requires that the line ratio, $I(7135)/I(7751)$, should be a factor of 4.2, so if spectral detections of \[\] [$\lambda$]{}7751 are possible, then one can determine the strength of the blended \[\] [$\lambda$]{}7135 line. However, observed SN Ia nebular spectra appear absent of IME lines. This could be due to an absence of [$^{56}{\rm Ni}$]{} in the IME zone. In our models, the presence of some [$^{56}{\rm Ni}$]{} in the IME zone means that positrons are available as a heating source after the ejecta has become optically thin to photons. One would expect some level of mixing to occur through Rayleigh-Taylor instabilities between these layers – see [@Hicks2015] and references therein. To address the problem of too high an ionization, clumping, arising from radiation hydrodynamic instabilities, should be considered in future studies. Our preliminary work shows that, as expected, clumping lowers the ionization, and we will address this issue in a future paper.
Another possible explanation concerns the validity of the explosion models. We have considered only four models, and only two of the explosion models were obtained from “first principles", and even these were derived from 1D explosions. Alternative explosion mechanisms might give rise to different density and abundance profiles, and in particular, the spatial distribution of [$^{56}{\rm Ni}$]{}. The later will influence the amount of UV line blanketing, potentially introducing degeneracies with the ejecta mass. However, other diagnostics (e.g. the NIR nickel line) provide additional information, and can break the degeneracies. Further, despite the deficiencies, the models have highlighted important diagnostics and questions that can help facilitate future progress towards understanding Type Ia SNe.
Conclusion {#Conclusion}
==========
We have presented four 1D SN Ia models – three delayed detonation models with masses of 1.02, 1.40, and 1.70 and one detonation sub- model with a mass of 1.04 . By design, the models have the same [$^{56}{\rm Ni}$]{} mass of $\sim$0.62 which allows us to investigate the dependence of light curves and spectra on ejecta mass. Despite the smallness of the model grid they serve to highlight important diagnostics that can help facilitate future progress towards understanding Type Ia SNe.
Our results show that despite large differences in ejecta mass, the optical flux throughout the photospheric phase shows less than 0.3 mag difference in peak brightness in the LCs, as well as nearly identical spectral features. We have seen that the peak bolometric luminosity of each model is similar to within about 15 per cent, and the difference in rise time is less than $\sim$20 per cent. Due to differences in diffusion time, however, the two sub- mass models do evolve faster (pre-maximum) by a day as seen from the bolometric luminosity and synthetic $B$-band LCs. There is only a slight difference ($\sim$5 per cent) in the decline parameter, $\Delta M_{15}(B)$, between sub- and super- models. Our sub- models have much bluer colours at all epochs compared with SUP ($B-R$ difference of $\approx 0.3$ mag at maximum and a difference in $B-R\gtrsim$1 mag roughly 20 days post maximum). Our models show larger differences in NIR light curves, particularly with the $H$-band’s $\sim$1 mag difference at maximum light between sub- and super-.
Spectroscopically, at most photospheric phases, the optical spectra show the same gross features. However, the strength of UV blanketing between 2000-4000 Å is found to correlate with ejecta mass. Lower mass models have higher temperature and ionization (as more heating per gram), and hence lower UV blanketing between 2000-4000 Å. $M(^{56}{\rm Ni})/M_{\rm ej}$ is the leading parameter controlling this study. Higher mass models produce stronger IME features, such as the NIR triplet and the [$\lambda$]{}6347,6371 doublet prior to the nebular phase, and stronger \[\] [$\lambda$]{}[$\lambda$]{}9530,9068, \[\] [$\lambda$]{}[$\lambda$]{}7291,7324, and \[\] [$\lambda$]{}7135 in the nebular phase. Lower mass models have higher ionization, as indicated by the presence of the [$\lambda$]{}[$\lambda$]{}4553,4568,4575 triplet near maximum and the lack of strong and lines in the optical post-photospheric/nebular phase. Model SUB1, unlike SUB2, is dominated by strong \[\] and \[\] lines, such as \[\] [$\lambda$]{}4658, \[\] [$\lambda$]{}5270, \[\] [$\lambda$]{}5888, and \[\] 1.5484 $\mu$m.
In the nebular phase, the \[\] 1.939 $\mu$m line is absent in our sub- detonation model, but readily visible in the three other models. Potentially, the \[\] 1.939 $\mu$m line provides us with a diagnostic of the amount of stable nickel ([$^{58}{\rm Ni}$]{} & [$^{60}{\rm Ni}$]{}), unlike the blended optical \[\] [$\lambda$]{}[$\lambda$]{}7378,7412 lines. Its absence in NIR spectra would provide strong evidence for a lack of a ‘[$^{56}{\rm Ni}$]{} hole’ and potentially sub-Chandrasekhar mass ejecta (given 1D modeling). However, complex ionization issues can influence the strength of all \[\] lines, making absolute determinations of the abundance model-dependent. Overall the NIR provides the best diagnostics for distinguishing between our different SN Ia progenitor models.
In comparing our spectra to observation at times greater than 20 days post maximum, we consistently find better qualitative fits with our cooler, high mass super- model. Given that there is a $\sim$20-70 per cent difference in claimed ejecta mass between our compared observational objects and our super- model, we suggest clumping as a way to lower the high ionization and high temperatures observed in our models. While it is difficult to reproduce all observational features due to the diversity of SNe Ia, we are able to match some features shown in our comparison to observations. Prior to maximum, the best choice of model varies. Agreement depends on the velocity structure of the ejecta. For instance, the photospheric features [$\lambda$]{}[$\lambda$]{}6347,6371 and the NIR triplet expose the difficulty of reproducing the velocity structure of SN Ia ( \[SN2001ay\_spec\] and \[SN2005el\_spec\], for example). Future efforts to reproduce the diversity of these features requires a better understanding of the outer ejecta and explosion mechanism.
Parallel work has been undertaken by Blondin et al. (submitted to MNRAS) who studied SN 1999by using a low mass model (0.9) and a model with a Chandrasekhar mass (both with 0.12of ). They find that the lower mass model provides a better match to the light curve, and exhibits a faster rise and a brighter maximum. As in our study, the lower mass model does not show the \[\] 1.939 $\mu$m line, which is seen in their Chandrasekhar model
To determine more accurate diagnostic signatures of SN Ia progenitors we need to understand clumping and inhomogeneities in Ia ejecta. Some insights can be obtained from multi-dimensional explosion modeling, while additional insight might be obtained from studies of young SN remnants that are not interacting with the surrounding ISM. During the photospheric phase, more UV spectral data will help to constrain the ionization and temperature of the gas. More NIR spectral data will help to test our diagnostics, such as the NIR triplet (or nebular \[\] [$\lambda$]{}[$\lambda$]{}7291,7324), the nebular features between 9000 Å-1 $\mu$m (such as \[\] [$\lambda$]{}[$\lambda$]{}9068,9530), and the \[\] 1.939 $\mu$m line (requiring SNe Ia at a high enough redshift to avoid the telluric absorption). These nebular features can provide leverage on the progenitor channel by constraining initial densities (\[\]), the overlap between IMEs and IGEs (\[\]), and the ionization structure.
As many more SN spectra become available it will be possible to do systematic statistical comparisons between SNe which have a similar initial [$^{56}{\rm Ni}$]{} mass. As discussed above, our studies show that Type Ia SN will exhibit systematic differences in spectra and multi-band LCs as a function of ejecta mass, thus providing fundamental constraints on the nature of the progenitors.
acknowledgements {#acknowledgements .unnumbered}
================
We thank Stéphane Blondin for providing feedback for this work as well as everyone at the “Supernovae Through the Ages: Understanding the Past to Prepare for the Future" conference who presented work and talked with us about our research. We also thank the referee for providing very detailed and useful comments and questions. This research has made use of the CfA Supernova Archive, which is funded in part by the National Science Foundation through grant AST 0907903. DJH acknowledges partial support from STScI theory grant HST-AR-12640.01, and DJH and KDW thank NASA for partial support through theory grant NNX14AB41G.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
|
---
abstract: 'Grey-box fuzzers such as American Fuzzy Lop (AFL) are popular tools for finding bugs and potential vulnerabilities in programs. While these fuzzers have been able to find vulnerabilities in many widely used programs, they are not efficient; of the millions of inputs executed by AFL in a typical fuzzing run, only a handful discover unseen behavior or trigger a crash. The remaining inputs are redundant, exhibiting behavior that has already been observed. Here, we present an approach to increase the efficiency of fuzzers like AFL by applying machine learning to directly model how programs behave. We learn a forward prediction model that maps program inputs to execution traces, training on the thousands of inputs collected during standard fuzzing. This learned model guides exploration by focusing on fuzzing inputs on which our model is the most uncertain (measured via the entropy of the predicted execution trace distribution). By focusing on executing inputs our learned model is unsure about, and ignoring any input whose behavior our model is certain about, we show that we can significantly limit wasteful execution. Through testing our approach on a set of binaries released as part of the DARPA Cyber Grand Challenge, we show that our approach is able to find a set of inputs that result in more code coverage and discovered crashes than baseline fuzzers with significantly fewer executions.'
author:
- Siddharth Karamcheti
- Gideon Mann
- David Rosenberg
bibliography:
- 'references.bib'
title: 'Improving Grey-Box Fuzzing by Modeling Program Behavior'
---
<ccs2012> <concept> <concept\_id>10011007.10011074.10011099.10011102.10011103</concept\_id> <concept\_desc>Software and its engineering Software testing and debugging</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002978.10003022.10003023</concept\_id> <concept\_desc>Security and privacy Software security engineering</concept\_desc> <concept\_significance>100</concept\_significance> </concept> </ccs2012>
Introduction
============
The goal of fuzz-testing, or fuzzing, is to discover a set of test inputs that maximize code coverage in a given program, with the hope that doing so allows one to find bugs, crashes, or other potential vulnerabilities. While there are many tools for fuzzing, grey-box mutational fuzzers such as American Fuzzy Lop (AFL) are among the most successful. These fuzzers work by maintaining a queue of interesting program inputs, or “parents”, that cover different parts of the program, and mutating them iteratively, with a set of stochastic mutation functions (e.g. flip bits, delete bits, insert random bits, etc.) to generate new “children” inputs. These children are then fed to a version of the program that has been lightly instrumented to trace the execution for a given input. If the input takes a path through the program that has not been observed before, it is added to the queue. Otherwise, it is discarded. Unfortunately, discarding inputs comes at a cost; each execution takes time, ranging from a couple of nanoseconds, to longer than a second, depending on the program. Within a typical fuzzing run, on the order of billions of inputs are generated, with only a handful actually covering unseen code paths, leading to hundreds of minutes of unnecessary execution time. In this work, we propose a method to cut down on these redundant executions by using machine learning to model program behavior.
Specifically, we posit that to fuzz successfully, it is important to be able to correlate program inputs with the resulting execution paths. By using machine learning to predict the execution path from a given input, we introduce an approach that is complementary to grey-box fuzzing, allowing us to filter useless inputs prior to execution. The intuition behind our filtering approach is simple: we focus on executing the generated inputs on which our learned reasoning model expresses low confidence (if we cannot reason about what the input will do, chances are it is likely to do something different than what we have seen before). By focusing on **modeling program behavior** we show that we can significantly improve program coverage with a smaller number of program executions.
We build our approach on top of AFL, one of the preeminent grey-box fuzzers. We show through a series of experiments on the DARPA Cyber Grand Challenge binaries that our approach offers significant improvements to fuzzing efficiency, obtaining a higher rate of code coverage than many strong baselines, including the best performing version of AFL.
Related Work
============
While there are many classes of approaches for finding bugs in programs, we focus on two key types: white-box approaches [@Ganesh2009TaintbasedDW], including symbolic execution [@Li2013SteeringSE; @Ma2011DirectedSE; @Baldoni2018ASO; @Pasareanu2011SymbolicEW; @Wang2017AngrT; @David2016BINSECSEAD; @Cadar2008KLEEUA; @Godefroid2012SAGEWF], and grey-box approaches [@Rawat2017VUzzerAE; @Stephens2016DrillerAF; @AFL; @Lemieux2017FairFuzzTR; @Bhme2016CoveragebasedGF; @Li2017SteelixPB]. These names come from the amount of transparency into the underlying program required; white-box approaches require a great deal of transparency (often access to the program source, or the ability to lift a binary up to an intermediate representation), while grey-box approaches require very little (often it’s just enough to instrument the program at runtime, to collect small pieces of information, such as whenever the code enters a new basic block). There also exist black-box approaches [@Woo2013SchedulingBM; @Gascon2015PulsarSB; @Radamsa; @ZZUF] that randomly generate inputs very quickly, and execute programs to discover crashes. While useful for finding shallow bugs, they often fail to penetrate deeply into programs.
The core of white-box approaches is the ability to leverage full transparency into the program under test, in order to explicitly reason about the nature of control flow. Symbolic execution tools like KLEE [@Cadar2008KLEEUA] and SAGE [@Godefroid2012SAGEWF] lift programs up to an intermediate representation, where the entirety of the control flow graph is exposed. These approaches then treat inputs as variables, and branch conditions as constraints on these variables. To find an input that will pass a certain branch (e.g. enter the if condition of an if/else statement), a SAT, or constraint solver is used. By explicitly solving for inputs that satisfy constraints, one can theoretically find an input that will reach any statement in a program. That being said, the downside of such approaches is in their speed, and need for resources. Average programs can have hundreds to thousands of non-trivial branch conditions, and as such, it can take extended periods of time to solve the corresponding equations. This is exacerbated by several factors, including the size of the inputs, reliance on external libraries in the code, and the ease of which one can instrument a given program.
On the flipside, grey-box approaches assume minimal transparency into a program. Rather than reason about the entire control flow graph, for many grey-box approaches, it is enough to instrument the program at run time, solely for the purpose of tracking when an input hits a new (previously unseen) basic block, or a new edge between basic blocks. Approaches like American Fuzzy Lop [@AFL], and it’s many variants [@Stephens2016DrillerAF; @Lemieux2017FairFuzzTR; @Bhme2016CoveragebasedGF; @Bhme2017DirectedGF] track this information to guide a simple genetic algorithm that generates a series of inputs and executes them through the program.
The key here is speed; generating new inputs to test is near instantaneous, and the only real limiting factor is the speed of execution through the program. In many cases this speed of execution is not negligible - and furthermore, as many of the generated inputs are either redundant, or trivially malformed, AFL and related approaches are wasteful, spending thousands of cycles on inputs that add no new meaningful information. The goal of our work is to introduce a new approach that obtains similar speed and efficiency to AFL, while using machine learning techniques to obtain some of the precision and reasoning ability as white-box approaches. With such an approach, we limit wasted cycles, while retaining the ability to find bugs efficiently and at scale.
Approach
========
*//* **Algorithm for AFL + Program Modeling** *// afl: Instance of AFL for generating/executing inputs* *// iterations: Fixed number of generation iterations* *// num\_generate: New inputs to generate each iteration* *// $\alpha$: Fraction of generated inputs to execute each iteration* *// queue: Queue of inputs that exercise new code paths* *// model: Predicts distribution over execution paths for an input* *// ranker: Given predictions, ranks by entropy values (high - low)*
$generated \gets \emph{afl}\tt{.generate(\emph{queue}, \emph{num\_generate})}$ $execute := []$ $execute \gets \emph{model}\tt{.predict(g)}$ $execute \gets \emph{ranker}\tt{.rank(\emph{execute})}$ $queue, path \gets \emph{afl}\tt{.execute(\emph{queue, execute[j]})}$ $model \gets model\tt{.retrain(execute[j], path)}$
Modeling program behavior is key to improving fuzzing efficiency. While there are many ways to approach this modeling problem, in this work, we focus on learning forward prediction models: given an input, predict the corresponding execution path through the program. If we had a perfect execution model, we could simply skip inputs that lead to execution paths we have already seen, saving significant time. Our approach, described below, is based on the heuristic that the less confident our model is in the execution path it predicts for a given input, the more likely that input is to lead to an execution path that we have never seen before.
There is an additional benefit to this method. As we execute each input, we get additional training data for the prediction model. Selecting new inputs on the basis of uncertainty is a well-known active learning technique and so this input selection method also serves to hasten prediction model improvement and thus the ability of the system to find good candidates.
At a high level, our approach is to repeatedly perform the following steps:
1\) Use AFL to generate some number of possible children inputs, 2) Feed these inputs through our model to predict distributions over execution paths, 3) Rank these generated inputs by the confidence in the predictions, 4) Execute some fraction of those ranked inputs that we are the least confident about, and 5) Use the executed inputs to retrain our path prediction model. This process is graphically depicted in Figure \[fig:pipeline\], and logically depicted in Algorithm \[alg:entropy\]. In the remainder of this section, we provide additional details about each of the steps in the above inner loop: generating candidate inputs, learning a path prediction model, and using this model to rank and execute the candidates.
Generating Candidate Inputs {#sec:generation}
---------------------------
To select a set of promising inputs to execute through the program, we first need candidates. We obtain this set of candidates by applying AFL’s mutation logic. Specifically, given our input queue, we first sample a parent input, then we apply a set of mutation operators to obtain our new candidate. We repeat this process $K$ times to obtain a full batch of promising candidates. Note that this process is extremely fast, as we do not execute any of the generated inputs through the program.
Modeling Programs via Path Prediction {#sec:pathpred}
-------------------------------------
A crucial component of our approach is to understand program behavior by prediction execution paths from program inputs. When fuzzing starts, we have no examples on which to train our model, so we predict a uniform distribution over a single (null) path for each example, which effectively results in a random ranking over the batch when ranking based on confidence (as will be discussed in Section \[sec:execution\]). However, after executing the first batch of inputs, from the first iteration of Algorithm \[alg:entropy\], we have an initial set of labeled examples $\{x, p\}$, where $x$ corresponds to a featurized representation of the input (e.g. a bag of words representation of an input string), and $p$ represents the corresponding execution path. Note that for our purposes, the execution path $p$ is represented as a unique label (i.e. each observed execution path gets its own label), rather than as a sequence of basic blocks, or edges in the underlying control flow graph. This is because programs can have hundreds of basic blocks, while we note that in practice, only a handful of unique execution paths (sets of traversed basic blocks) are observed.
With these examples $\{x, p\}$, and a total number of unique observed execution paths $P$, we can then train a probabilistic classifier to predict a distribution over the $P$ paths for a given input $x$. In our experiments, we build the probabilistic classifier using multinomial logistic regression (via the one-vs-all reduction to $P$ separate binary logistic regression models, for efficiency). We featurize our inputs using bigram counts over the bytes of each input string — that is, we compute a histogram of how many times each unique bigram sequence appears in the input, and use that histogram as our representation. We choose not to utilize an L1 or L2 penalty, and train all models to convergence.
Estimating Uncertainty via Entropy {#sec:execution}
----------------------------------
The final piece of our approach is using our model to make decisions about which candidates to actually execute through the program. To do this, we apply our hypothesis that inputs with uncertain predictions are more likely to exhibit execution paths that have not been observed, while inputs with confident predictions are more likely to be redundant.
As a measure of uncertainty, we use the entropy of the predicted distribution over execution paths, with high entropy referring to high uncertainty, and low entropy referring to low uncertainty. Given an input $x$, let $\Pr(p_i \mid x)$ be the probability that $x$ exhibits execution path $p_i$. Given this distribution, we compute the entropy as: $$\begin{aligned}
H(x) = \sum_{i = 1}^P \Pr(p_i \mid x) \log(\Pr(p_i \mid x))\end{aligned}$$
With this formula, we then score each generated input in the batch, for the given iteration. Then we rank them by their entropy (highest - lowest). Finally, we select a fraction $\alpha$ of the highest entropy inputs to execute. A full breakdown of the process can be found in Algorithm \[alg:entropy\].
Dataset
=======
We run preliminary experiments on a subset of the DARPA Cyber Grand Challenge Binaries. This dataset consists of 200 separate programs released as part of a 2016 challenge to create tools for finding, verifying, and patching bugs. Many new tools building off of AFL and symbolic execution based approaches came out of this contest [@Stephens2016DrillerAF; @Cha2015ProgramAdaptiveMF], and this dataset has been used to benchmark similar tools ever since. Each program provides unique functionality, and was written by humans (affiliated with various DARPA programs). More importantly, each program was written with one or more human-written bugs, meant to mimic errors that developers might make when writing actual programs. Furthermore, the programs in this dataset range in complexity.
In our work, we utilize a subset of 24 randomly chosen programs from this dataset (due to time constraints, we could not run on the full 200). We utilized a version of DARPA CGC binaries compiled for x86 Linux (as opposed to the original DARPA-specific VM), released via this link: <https://github.com/trailofbits/cb-multios>.
Experimental Setup
==================
![Graph depicting relative coverage over number of executions for the program Flash File System (in the CGC binaries). Here, we see that the Program Modeling approach (ML) outperforms all the baselines by a significant margin. Furthermore, as we continue execution, the gap between the ML strategy and the others grows.[]{data-label="fig:flash"}](img/FlashGraph){width="\linewidth"}
We implement our program modeling approach using logistic regression as our prediction model, and featurize our input strings by collecting a bag of byte bigrams (a histogram of the number of times each unique pair of bytes appear in the input). We compare our program modeling approach with three strong baselines. The first baseline is that of AFL itself, as it ships out of the box. However, rather than use the standard AFL parameters, we run AFL with the “-d” flag, or “FidgetyAFL” [@FidgetyAFL; @Lemieux2017FairFuzzTR], as it performs better than standard AFL given a short fuzzing time period.
The second baseline we utilize is a batched version of AFL, which we refer to as Batched FidgetyAFL. The differences between Batched FidgetyAFL and FidgetyAFL are as follows: FidgetyAFL updates its state (its queue, and therefore which parent inputs are sampled to generate the next child) immediately after each input is generated and executed. The batched versions instead remove this consistent state update and replace it with a batched update, where multiple inputs are first generated all together without any state updates, and then executed (with state updates) all at once. We choose this baseline as it offers a better comparison to our program modeling approach. Recall that in our approach, we first generate a batch of examples (without execution), then rank the inputs among the batch to pick the fraction $\alpha$ to execute. Like Batched FidgetyAFL, we do not update the state of AFL’s queue until after we execute all the ranked inputs.
In addition to FidgetyAFL and Batched FidgetyAFL, we have a third baseline, Random Batched FidgetyAFL, which explores the effect the ranking over the generated inputs has relative to fuzzing performance. Unlike our program modeling approach, which generates a large number of inputs, then executes the top $\alpha$-fraction after ranking the predictions by their entropy, Random Batched FidgetyAFL randomly picks a fraction $\alpha$ of the batch to execute. In this way, this baseline lets us examine if our entropy-ranking approach is actually working. Note that Random Batched FidgetyAFL is markedly different from the Batched FidgetyAFL baseline — this is because AFL does not uniformly sample inputs from its queue — instead, it uses a heuristic *schedule* to sample queue inputs, first sampling elements from the queue that are more recent, then later (after generating many new inputs) starts sampling other elements from the queue. In this way, Random Batched FidgetyAFL exhibits slightly more random behavior than Batched FidgetyAFL, as it reflects a wide variety of different sampled parents.
Executions FidgetyAFL Batched FidgetyAFL Random Batched FidgetyAFL Logistic Regression with Bigram Features
------------ ----------------- -------------------- --------------------------- ------------------------------------------
10000 .623 $\pm$ .011 .624 $\pm$ .011 .632 $\pm$ .011 **.638 $\pm$ .011**
20000 .647 $\pm$ .011 .644 $\pm$ .011 .667 $\pm$ .011 **.688 $\pm$ .011**
30000 .671 $\pm$ .011 .660 $\pm$ .011 .699 $\pm$ .011 **.755 $\pm$ .009**
40000 .692 $\pm$ .010 .671 $\pm$ .010 .716 $\pm$ .011 **.791 $\pm$ .009**
50000 .706 $\pm$ .010 .680 $\pm$ .010 .755 $\pm$ .009 **.816 $\pm$ .009**
We run our experiments on 24 of the DARPA CGC binaries, for a total of 50,000 executions per binary. To jumpstart learning, and to eliminate most of the variance across fuzzing runs, we start all runs by letting FidgetyAFL run for a 3 minute period. We pre-train our logistic regression model on the inputs executed during this window. We then use the resulting AFL state, and the queue of inputs created as our initial queue, and start each of the 4 different strategies on top (FidgetyAFL, Batched FidgetyAFL, Random, and Logistic Regression). For all experiments, we utilize AFL version 2.52b.
![Summary Graph depicting relative coverage over number of executions for the 24 CGC binaries, at a 95% confidence interval. Again, we see the Program Modeling approach (ML) outperform the other approaches by a larger and larger margin as execution continues.[]{data-label="fig:summary"}](img/summary){width="\linewidth"}
To measure the relative performance across all strategies, we use a metric we refer to as relative coverage. Let $s$ correspond to a given strategy, $t$ the given execution iteration, $T$ the max number of execution iterations, and *$\text{code-paths}_t(s)$* the number of unique code-paths that strategy $s$ has discovered by execution iteration $t$. Then Relative Coverage *$\text{rel-cov}$* for a single program is defined as: $$\begin{aligned}
\text{rel-cov}_t(s) &= \dfrac{\text{code-paths}_t(s)}{\max_{s'} [\text{code-paths}_T(s')]}\end{aligned}$$ or the ratio between the number of code paths strategy $s$ has found, and the maximum number of code paths across all strategies by the final execution iteration $T$. We report the mean and standard error of across all 24 programs. Furthermore, to get a better sense of how the different strategies behave over time, we report relative coverage statistics at every 10,000 executions.
Results and Discussion
======================
Table \[tbl:results\] reports relative coverage statistics for each of the four strategies at each interval of 10,000 executions. Furthermore, Figure \[fig:flash\] provides a graph reporting number of code paths discovered vs. executions for program Flash File System (an example program from the CGC binaries). Finally, Figure \[fig:summary\] contains a summary graph aggregating relative coverage over all 24 binaries in our test set, at a 95% confidence interval across binaries.
From these results, there are two key conclusions to pick out. The first is to realize that at all time steps, the program modeling approach is a clear winner, obtaining higher coverage than any of the baseline strategies. This seems to indicate that the gains from Logistic Regression ranking are significantly higher than the losses suffered from the batched update procedure. As such, a possible avenue for future work would be to augment the Entropy Ranking based Logistic Regression with a thresholding operation, to allow the model to make choices about whether to execute an input or not, in an online fashion. Doing so would remove any need for batching, and allow the approach to incur the same benefits as traditional AFL, with its continuous state updates.
The second key observation is that the performance gap between the program modeling approach and the other baseline approaches grows as the number of executions rises. This is best exhibited by the graphs in Figures \[fig:flash\] and \[fig:summary\]. We see that at the beginning of fuzzing, there is just a small, almost negligible gap in performance, while as execution continues, the gap grows larger and larger. There are two possible conclusions to be drawn from this: the first is a rather simple one, that as the program modeling approach identifies more code paths, AFL’s queue is updated, and we begin sampling more of the recently discovered inputs — reasons for why AFL itself is successful. However, another possible explanation is that of how the program model behaves as more data becomes available. With more executions, the logistic regression is given more labeled examples. As such, it gets better at identifying patterns in the inputs, and the confidence scores assigned at inference time become more meaningful. Another avenue of future work is to examine the nature of the learned models, and how they change as execution continues. It may also be worthwhile to throw stronger learning algorithms and more program-specific features into the mix, seeing if there is a way to strengthen the reported confidence scores.
Conclusion
==========
In this work, we presented a system for improving the efficiency and precision of AFL, the premier grey-box fuzzer, utilizing techniques from machine learning to directly model program behavior. Specifically, we note that a major weakness of AFL and similar approaches is the number of program executions that are wasted on redundant or non-informative inputs. To remedy this problem, we proposed a two-phase approach that 1) learns a forward prediction model that maps inputs to execution paths, and 2) uses that model to identify inputs that are potentially interesting. The intuition we use is that if we can confidently model how a given input will behave when executed, then it is not worth executing. Instead, we should focus on the inputs for which our model exhibits low confidence — these are inputs that when executed, will potentially trigger new areas of code that have not yet been observed. Our results show that our ranking-based approach built with a simple logistic regression classifier obtains extremely strong performance, beating 3 strong baselines, including the standard, out-of-the-box implementation of AFL itself. Furthermore, our results show that as we continue fuzzing, our approach gets better and better, with the performance gap between our approach and baselines widening over time. These results indicate that there are strong benefits to be found in applying techniques from machine learning and pattern recognition to fuzzing, and that this is a very fruitful avenue of research.
|
---
abstract: 'By using of the Euler-Lagrange equations, we find a static spherically symmetric solution in the Einstein-aether theory with the coupling constants restricted. The solution is similar to the Reissner-Nordstrom solution in that it has an inner Cauchy horizon and an outer black hole event horizon. But a remarkable difference from the Reissner-Nordstrom solution is that it is not asymptotically flat but approaches a two dimensional sphere. The resulting electric potential is regular in the whole spacetime except for the curvature singularity. On the other hand, the magnetic potential is divergent on both Cauchy horizon and the outer event horizon.'
author:
- Changjun Gao
- 'You-Gen Shen'
title: 'A Static Spherically Symmetric Solution of the Einstein-aether Theory'
---
Introduction
============
The Einstein-aether theory [@ted:00; @ted:07] belongs to the vector-tensor theories in nature. Besides the ordinary matters and the metric tensor $g_{\mu\nu}$, the fundamental field in the theory is a timelike vector field $A_{\mu}$. Different from the usual vector-tensor theories, $A_{\mu}$ is constrained to have a constant norm. So the vector field $A_{\mu}$ cannot vanish anywhere. Therefore, a preferred frame is defined and the Lorentz symmetry is violated. The vector field is referred to as the “aether”. The Einstein-aether theory has become an interesting theoretical laboratory to explore both the Lorentz violation effects and the preferred frame effects. Up to now, the Einstein-aether theory has been widely studied in literature in various ways: the analysis of classical and quantum perturbations [@lim:05; @na:10; @car:00; @car:01; @car:02; @car:03], the cosmologies [@car:04; @bon:08], the gravitational collapse [@gar:08], the Einstein-aether waves [@wave:04], the radiation damping [@fos:06] and so on.
The purpose of the present paper is to seek for a static spherically symmetric solution of the Einstein-aether theory. The black hole solutions in the Einstein-aether theory have been investigated in Refs. [@bh1; @bh2; @bh3; @bh4; @bh5]. These investigations mainly focus on the numerical analysis of the solutions due to the complication of the Einstein equations. To our knowledge, one have not yet find the exact, static and spherically symmetric solution in the Einstein-aether theory. In this paper, instead of solving the Einstein equations, we are going to solve the Euler-Lagrange equations in order to derive the static spherically symmetric solution. We find it is relatively simple in the calculations. We shall use the system of units in which $16\pi G=c=\hbar=4\pi\varepsilon_0=1$ and the metric signature $(-,\ +,\ +,\ +)$ throughout the paper.
static spherically symmetric solution
=====================================
In the context of spherical symmetry and after the redefinitions of metric $g_{\mu\nu}$ and aether field $A_{\mu}$, the Lagrangian density of the Einstein-aether theory can be written as $$\begin{aligned}
\label{eq:lagM}
\mathscr{L}&=&-R-\frac{c_1}{2}F_{\mu\nu}F^{\mu\nu}-c_2\left(\nabla_{\mu}A^{\mu}\right)^2
+\lambda\left(A_{\mu}A^{\mu}+m^2\right)\;,\end{aligned}$$ with the field strength tensor $$\begin{aligned}
F_{\mu\nu}&=&\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}\;.\end{aligned}$$ Here $R$ is the Ricci scalar and the $c_i$ are dimensionless constants. We note that there is a sign difference from [@gar:08] in the definition of Ricci tensor. $\lambda$ is the Lagrange multiplier field which has the dimension of the square of inverse length, $l^{-2}$. $m$ is a positive dimensionless constant which has the physical meaning of the squared norm for the aether field. The requirement of $m^2>0$ ensures the aether to be timelike.
The static and spherically symmetric metric can always be written as $$\begin{aligned}
\label{eq:line}
ds^2=-U\left(r\right)dt^2+\frac{1}{U\left(r\right)}dr^2+f\left(r\right)^2d\Omega^2\;.\end{aligned}$$ Instead of solving the Einstein equations, we prefer to deal with the Euler-Lagrange equations from the Lagrangian Eq. (\[eq:lagM\]) for simplicity in calculations. Because of the static and spherically symmetric property of the spacetime, the vector field $A_{\mu}$ takes the form $$\begin{aligned}
\label{eq:A}
A_{\mu}=\left[\phi\left(r\right),\ \frac{1}{\psi\left(r\right)},\
0,\ 0\right]\;,\end{aligned}$$ where $\phi$ and $\psi^{-1}$ correspond to the electric and magnetic part of the electromagnetic potential. Then we have $$\begin{aligned}
F_{\mu\nu}F^{\mu\nu}=-2\phi^{'2}\;,\ \ \
\nabla_{\mu}A^{\mu}=\left(\frac{U}{\psi}\right)^{'}+2\frac{f^{'}U}{f\psi}\;.\end{aligned}$$ The prime here and in what follows denotes the derivative with respect to $r$. Taking into account the Ricci scalar, $R$, we have the total Lagrangian as follows
$$\begin{aligned}
\label{eq:lagMM}
\mathscr{L}&=&-U^{''}-4U^{'}\frac{f^{'}}{f}-4U\frac{f^{''}}{f}+\frac{2}{f^2}-2U\frac{f^{'2}}{f^2}
\nonumber\\&&+{c_1}\phi^{'2}-c_2\left[\left(\frac{U}{\psi}\right)^{'}+2\frac{f^{'}U}{f\psi}\right]^2
\nonumber\\&&+\lambda\left(-\frac{1}{U}\phi^2+\frac{U}{\psi^2}+m^2\right)\;.\end{aligned}$$
Let $$\begin{aligned}
\psi=\frac{Uf^2}{K}\;,\end{aligned}$$ we can rewrite the Lagrangian, Eq. (\[eq:lagMM\]), as follows
$$\begin{aligned}
\label{eq:lagMMM}
\mathscr{L}&=&-U^{''}-4U^{'}\frac{f^{'}}{f}-4U\frac{f^{''}}{f}+\frac{2}{f^2}-2U\frac{f^{'2}}{f^2}
\nonumber\\&&+{c_1}\phi^{'2}-c_2\frac{K^{'2}}{f^4}
\nonumber\\&&+\lambda\left(-\frac{1}{U}\phi^2+\frac{K^2}{Uf^4}+m^2\right)\;.\end{aligned}$$
Now there are $U,\ f,\ \phi,\ K,\ \lambda$ five variables in the Lagrangian which correspond to five equations of motion. Then using the Euler-Lagrange equation, we obtain the equation of motion for $\lambda$,
$$\begin{aligned}
\label{eq:lambda}
-\frac{1}{U}\phi^2+\frac{K^2}{Uf^4}+m^2=0\;,\end{aligned}$$
for $\phi$,
$$\begin{aligned}
\label{eq:phi} c_1Uf\phi^{''}+2c_1U\phi^{'}f^{'}+\lambda\phi
f=0\;,\end{aligned}$$
for $K$,
$$\begin{aligned}
\label{eq:K}
c_2UfK^{''}-2c_2U K^{'}f^{'}+\lambda K f=0\;,\end{aligned}$$
for $U$,
$$\begin{aligned}
\label{eq:U}
-2U^2f^3f^{''}-\lambda K^{2}+\lambda \phi^2 f^4=0\;,\end{aligned}$$
and for$f$, $$\begin{aligned}
\label{eq:f}
&&-c_1\phi^{'2}Uf^4-c_2UK^{'2}+\lambda\phi^2f^4+Uf^4U^{''}-\lambda
m^2 Uf^4\nonumber\\&&+2Uf^3U^{'}f^{'}+2U^2f^3f^{''}+\lambda
K^2=0\;,\end{aligned}$$ respectively. We have five independent differential equations and five variables, $U,\ f,\ \phi,\ K,\ \lambda$. So the system of equations is closed.
From Eq. (\[eq:lambda\]) and Eq. (\[eq:phi\]), we obtain $$\begin{aligned}
\label{eq:sU}
U=\frac{\phi^2f^4-K^2}{m^2f^4}\;,\end{aligned}$$ and $$\begin{aligned}
\label{eq:sphi}
\lambda=-\frac{c_1U\left(f\phi^{''}+2\phi^{'}f^{'}\right)}{f\phi}\;,\end{aligned}$$ respectively. Substituted Eq. (\[eq:sU\]) and Eq. (\[eq:sphi\]) into Eq. (\[eq:U\]), then Eq. (\[eq:U\]) becomes
$$\begin{aligned}
\label{eq:ssU}
2\phi f^{''}+c_1m^2f\phi^{''}+2c_1m^2\phi^{'}f^{'}=0\;.\end{aligned}$$
The norm of the aether field is usually constrained by the Lagrange multiplier to be unity, $m=1$. But in this paper, we would constrain the norm to meet
$$\begin{aligned}
\label{eq:norm}
c_1m^2=2\;.\end{aligned}$$
We note that this choice is consistent with the perturbation analysis of Lim [@lim:05]. He showed that in order to have a positive definite Hamiltonian, $c_1$ should satisfy $$\begin{aligned}
\label{eq:lim}
c_1>0\;.\end{aligned}$$ We stress that the choice of $c_1m^2=2$ corresponds to the special case that has been called $c_{14}=2$ [@eling04; @car:04] which leads the Newton’s gravitational constant to infinity[@eling04; @car:04]:
$$\begin{aligned}
\label{eq:G}
G_{N}=\frac{G}{1-c_{14}/2}\;.\end{aligned}$$
But Eq. (\[eq:G\]) should be taken with a grain of salt because it is derived with vanishing spatial components in $A^{\mu}$. We also stress that the choice of $c_1m^2=2$ is not “ for convenience ” but actually a restriction on the theory [^1].
Then Eq. (\[eq:ssU\]) gives the solution as follows $$\begin{aligned}
\label{eq:ssphi}
\phi=\frac{\phi_0+\phi_1 r}{f}\;,\end{aligned}$$ where $\phi_0,\ \phi_1$ are two integration constants. $\phi_1$ is dimensionless while $\phi_0$ has the dimension of length.
Keeping Eqs. (\[eq:sU\]), (\[eq:sphi\]) and (\[eq:ssphi\]) in mind, we find Eq.(\[eq:K\]) and Eq. (\[eq:f\]) are reduced to the following form
$$\begin{aligned}
\label{eq:33}
2c_2m^2 K^{'}f^{'}-c_2m^2f K^{''}-2K f^{''}=0\;,\end{aligned}$$
and $$\begin{aligned}
\label{eq:55}
&&2f^2K^{'2}+12K^2 f^{'2}-12fK f^{'}K^{'}+c_2m^2f^2
K^{'2}\nonumber\\&&-6fK^2 f^{''}+2Kf^2 K^{''}=0\;,\end{aligned}$$ respectively. Putting
$$\begin{aligned}
\label{eq:alpha}
&&c_2=\frac{1}{\alpha m^2}\;,\end{aligned}$$
with $\alpha$ a new dimensionless parameter, we obtain from Eq. (\[eq:33\]) and Eq. (\[eq:55\])
$$\begin{aligned}
\label{eq:eqa}
&&6\alpha f K^2 f^{''}-12\alpha K^2 f^{'2}+8\alpha f K
f^{'}K^{'}\nonumber\\&&+4\alpha^2f K^2 f^{''}-2\alpha f^2
K^{'2}-f^2 K^{'2}=0\;.\end{aligned}$$
In order that the spin-$0$ field does not propagate superluminally, Lim constrained $c_2$ to meet [@lim:05]
$$\begin{aligned}
c_2>0\;, \ \ \ \ \textrm{and}\ \ \ \frac{c_2}{c_1}\leq 1\;.\end{aligned}$$
Taking account of Eq. (\[eq:lim\]), we conclude that $\alpha$ should satisfy
$$\begin{aligned}
\alpha\geq\frac{1}{2}\;.\end{aligned}$$
Solving the differential equation, we obtain
$$\begin{aligned}
\label{eq:KKK}
&&K=K_0\exp\left\{{\int\frac{1}{\left(1+2\alpha\right)f}\left[4\alpha
f^{'}+\sqrt{2\alpha\left(2\alpha+3\right)\left(2\alpha f f^{''}+f
f^{''}-2f^{'2}\right)}\right]dr}\right\}\;,\end{aligned}$$
where $K_0$ is an integration constant which has the dimension of the square of the length, $l^{2}$. We may assume $K_0>0$. Substituting Eq. (\[eq:KKK\]) into Eq. (\[eq:55\]), we obtain
$$\begin{aligned}
\label{eq:chang}
&&\left(16\alpha f f^{''}+8 f f^{''}-16
f^{'2}\right)\sqrt{\alpha\left(2\alpha+3\right)\left(2\alpha f
f^{''}+ff^{''}-2f^{'2}\right)}\nonumber\\&&+\sqrt{2}\left(4\alpha^2f^2f^{'''}+12\alpha^2ff^{'}f^{''}-8\alpha
f^{'3}+4\alpha f^2 f^{'''}-12\alpha
ff^{'}f^{''}-9ff^{'}f^{''}+f^2f^{'''}+12f^{'3}\right)=0\;.\end{aligned}$$
At first glance, Eq. (\[eq:chang\]) is rather complicated. But using the calling sequence of “dsolve” in Maple Program, it is easy to find the solutions with $\alpha=1/2,\ \ \alpha=3/2, \ \
\alpha=5/2,\ \cdot\cdot\cdot$. For general $\alpha$, $f$ is found to be
$$\begin{aligned}
\label{eq:fs}
&&f=f_0\left(1-k^2r^2\right)^{\frac{2\alpha+1}{4\alpha-2}}e^{\frac{\sqrt{6\alpha+4\alpha^2}}{2\alpha-1}\tanh^{-1}k
r}\;,\end{aligned}$$
where $f_0$ and $k$ are integration constants. Both $f_0$ and $k^{-1}$ have the dimension of length. Without the loss of generality, the third integration constant with the dimension of length has been absorbed by $r$.
Eq. (\[eq:fs\]) forces $k r$ to satisfy $$\begin{aligned}
-1\leq k r\leq 1\;.\end{aligned}$$ Up to this point, we could present all the variables:
$$\begin{aligned}
\label{eq:all}
f&=&f_0\left(1-k^2r^2\right)^{\frac{2\alpha+1}{4\alpha-2}}e^{\frac{\sqrt{6\alpha+4\alpha^2}}{2\alpha-1}\tanh^{-1}k
r}\;,\\
K&=&K_0\left(1-k
r\right)^{\frac{4\alpha-\sqrt{2\alpha\left(2\alpha+3\right)}}{4\alpha-2}}\cdot\left(1+k
r\right)^{\frac{4\alpha+\sqrt{2\alpha\left(2\alpha+3\right)}}{4\alpha-2}}\;,\\
\phi&=&\frac{1}{f_0}\left(\phi_0+\phi_1
r\right)\left(1-k^2r^2\right)^{-\frac{2\alpha+1}{4\alpha-2}}e^{-\frac{\sqrt{6\alpha+4\alpha^2}}{2\alpha-1}\tanh^{-1}k
r}\;,\\
U&=&\frac{1}{m^2f_0^2}\left(\phi_0+\phi_1
r\right)^2\left(1-k^2r^2\right)^{-\frac{2\alpha+1}{2\alpha-1}}e^{-\frac{2\sqrt{6\alpha+4\alpha^2}\tanh^{-1}k
r}{2\alpha-1}}\nonumber\\&&-\frac{K_0^2}{m^2f_0^4}\left(1-k
r\right)^{\frac{4\alpha-\sqrt{2\alpha\left(2\alpha+3\right)}}{2\alpha-1}}\cdot\left(1+k
r\right)^{\frac{4\alpha+\sqrt{2\alpha\left(2\alpha+3\right)}}{2\alpha-1}}\left(1-k^2r^2\right)^{-\frac{4\alpha+2}{2\alpha-1}}
e^{-\frac{4\sqrt{6\alpha+4\alpha^2}\tanh^{-1}k
r}{2\alpha-1}}\;,\\
\psi&=&\frac{\left(\phi_0+\phi_1r\right)^2\left(1-k^2r^2\right)^{-\frac{4\alpha+\sqrt{6\alpha
+4\alpha^2}}{4\alpha-2}}\left(1-kr\right)^{\frac{\sqrt{6\alpha+4\alpha^2}}{2\alpha-1}}}{K_0m^2}\nonumber\\&&-\frac{K_0\left(1-k
r\right)^{\frac{4\alpha}{2\alpha-1}}\left(1+k
r\right)^{\frac{4\alpha+\sqrt{6\alpha+4\alpha^2}}{2\alpha-1}}}
{m^2f_0^2\left(1-k^2r^2\right)^{\frac{2+8\alpha+\sqrt{6\alpha+4\alpha^2}}{4\alpha-2}}e^{\frac{2\sqrt{6\alpha+4\alpha^2}\tanh^{-1}k
r}{2\alpha-1}}}\;,\\
\lambda&=&-\frac{2k^2\left(\phi_0+\phi_1 r\right)^2}
{f_0^2m^4\left(2\alpha-1\right)^2\left(1-k^2r^2\right)^{\frac{6\alpha-1}{2\alpha-1}}}\left[4k
r\sqrt{4\alpha^2+6\alpha}-6\alpha-2k^2r^2-1-4\alpha
k^2r^2\right]e^{-\frac{2\sqrt{6\alpha+4\alpha^2}\tanh^{-1}k
r}{2\alpha-1}}\nonumber\\&&+\frac{2k^2K_0^2}
{f_0^4m^4\left(2\alpha-1\right)^2\left(1-k^2r^2\right)^{\frac{6\alpha-1}{2\alpha-1}}}\left[4k
r\sqrt{4\alpha^2+6\alpha}-6\alpha-2k^2r^2-1-4\alpha
k^2r^2\right]\nonumber\\&&\cdot\left(1-k
r\right)^{\frac{4\alpha-\sqrt{2\alpha\left(2\alpha+3\right)}}{2\alpha-1}}\cdot\left(1+k
r\right)^{\frac{4\alpha+\sqrt{2\alpha\left(2\alpha+3\right)}}{2\alpha-1}}e^{-\frac{4\sqrt{6\alpha+4\alpha^2}\tanh^{-1}k
r}{2\alpha-1}}\;.\end{aligned}$$
If we define $$\begin{aligned}
\phi_0\equiv\alpha_0f_0\;,\ \ \ \phi_1\equiv\alpha_1\;,\ \ \
k\equiv \alpha_2\frac{1}{f_0}\;,\ \ \ K_0\equiv\alpha_3 f_0^2\;,\end{aligned}$$ then $\alpha_i$ are dimensionless constants. Together with $m$ and $\alpha$, we have totally six dimensionless constants and one dimensional parameter, $f_0$. We note that the seven parameters are not independent and there are six parameters in the solution in nature. In fact, Eling and Jacobson [@bh3] have argued that there is a $3$-parameter (corresponding to the mass, electric charge and magnetic charge, respectively) family of spherical, static solutions before asymptotic flatness and regularity are imposed. If one take into account the two coupling constants, that would be $5$ parameters in all. But our solution Eqs. (31-36) is not asymptotic flat. So there is an extra parameter of “cosmological-constant-like”. Then the total number of parameters is six. [^2] This could be understood from the expression of $f$ and $U$ with the replacements
$$\begin{aligned}
\frac{\phi_0}{mf_0}\rightarrow\bar{\phi_0}\;,\ \ \
\frac{\phi_1}{mf_0}\rightarrow\bar{\phi_1}\;,\frac{K_0}{mf_0^2}\rightarrow\bar{K_0}\;.\end{aligned}$$
Then the metric of spacetime is determined by six parameters.
structure of the spacetime
===========================
In this section, let’s numerically study the structure of spacetime described by the solution. Since $f$ is the physical length, we should rewrite the metric as follows
$$\begin{aligned}
ds^2=-U\left(r\right)dt^2+\frac{1}{V\left(r\right)}df^2+f\left(r\right)^2d\Omega^2\;,\end{aligned}$$
with $$\begin{aligned}
V\left(r\right)={U\left(r\right)f^{'2}}\;.\end{aligned}$$ Now $f$ plays the role of physical radius (proper length) of the static spherically symmetric space.
As an example, we put the dimensional constant $f_0=1$(for example, $f_0$ equals to one Schwarzschild radius). Five dimensionless constants are put $m=1,\ \alpha_0=\alpha_2=1,\ \
\alpha=3/2$. As for $\alpha_1$, we let $\alpha_1=0.15,\ 0.1,\ 0,\
-0.2,\ -0.4$, respectively.
There are usually two kinds of horizons in a static spherically symmetric spacetime, namely, the timelike limit surface (TLS) and the event horizon (EH). The timelike limit surface separates the timelike region of the Killing vector field from the spacelike part which is determined by [@Haw73] $$\begin{aligned}
g_{00}=U=0\;.\end{aligned}$$
In Fig. \[fig:fu\], we plot the evolution of $U$ with respect to the physical radius $\ln f$ for different $\alpha_1$. The figure shows that there are two TLS in the spacetime in general. One of them is the inner Cauchy horizon and the other is the black hole event horizon. This is very similar to the spacetime of Reissner-Nordstrom solution. [^3] On the other hand, the Reissner-Nordstrom spacetime is asymptotically flat in space. But this solution is asymptotically a two dimensional sphere. [^4] With the increasing of $\alpha_1$, the event horizon is shrinking. When $\alpha_1=0$, the inner Cauchy horizon and the black hole event horizon coincide and the solution corresponds to the extreme solution.
Compared to Fig. \[fig:fu\], the structure of Reissner-Nordstrom spacetime is shown in Fig. \[fig:RND\]. The metric of Reissner-Nordstrom spacetime takes the form of
$$\begin{aligned}
ds^2&=&-Udt^2+\frac{1}{U}df^2+f^2d\Omega^2\;,\ \ \ \ \nonumber\\
U&=&1-\frac{2M}{f}+\frac{Q^2}{f^2}\;.\end{aligned}$$
Without the loss of generality, we take the mass $M=1.0$ and the electric charge $Q=1.3,\ 1.0,\ 0.8,\ 0.7$, respectively. There are two horizons in the spacetime, the inner Cauchy horizon ($\textrm{CH}$) and the black hole event horizon ($\textrm{EH}$). (As an example, the $\textrm{CH}$ and $\textrm{EH}$ are given for $Q=0.7$). The space is asymptotically flat. With the increasing of electric charge $Q$, the EH is shrinking and the CH expanding. When $Q=M=1.0$, the inner Cauchy horizon and the black hole event horizon coincide and the solution corresponds to the extreme solution.
On the other hand, the EH is determined by [@Haw73]
$$\begin{aligned}
g^{11}=V=0\;.\end{aligned}$$
In Fig. \[fig:fv\], we plot the evolution of $V$ with respect to the physical radius $\ln f$ for different $\alpha_1$. The figure shows that there are two horizons in the spacetime in general, namely, the inner Cauchy horizon and the black hole event horizon. With the increasing of $\alpha_1$, the event horizon is shrinking. When $\alpha_1=0$, the inner Cauchy horizon and the black hole event horizon coincide and the solution corresponds to the extreme solution.
In Fig. \[fig:fphi\], we plot the evolution of the electric potential $\phi$ with respect to the physical radius $f$ for different $\alpha_1$. It shows that $\phi$ is regular in the spacetime except for $f=0$ (curvature singularity). The potential $\phi$ is divergent at $f=0$ and asymptotically approaches zero in the infinity of space. This behavior is the same as the electric potential in Reissner-Nordstrom solution.
In order to show $f=0$ is the curvature singularity, as an example, we plot the evolution of the Ricci scalar $R$ with respect to the physical radius $f$ in Fig. \[fig:fR\] with $m=1,\ \alpha_0=\alpha_2=1,\ \ \alpha=3/2,\ \ \ \alpha_1=-0.4$. It is apparent $R$ is divergent at $f=0$. This reveals $f=0$ is indeed the curvature singularity.
In Fig. \[fig:fpsi\], we plot the evolution of the inverse of magnetic potential $\psi$ with respect to the physical radius $
\ln f$ for different $\alpha_1$. It shows that the magnetic potential $\psi^{-1}$ is divergent on both horizons while asymptotically approaches zero in both the infinity of space and the curvature singularity.
![The evolution of $U$ with respect to the physical radius $\ln{f}$ for different $\alpha_1=-0.4,\ -0.2,\ 0, \, 0.1,\ 0.15$. There are two horizons in the spacetime in general, the inner Cauchy horizon ($\textrm{CH}$) and the black hole event horizon ($\textrm{EH}$) (As an example, the $\textrm{CH}$ and $\textrm{EH}$ are given for $\alpha_1=-0.4$). When $f\rightarrow
\infty$, we have $U=0$. So the solution is not asymptotically flat in space. With the increasing of $\alpha_1$, the event horizon is shrinking. When $\alpha_1=0$, the inner Cauchy horizon and the black hole event horizon coincide and the solution corresponds to the extreme solution.[]{data-label="fig:fu"}](fu.eps "fig:"){width="8.4cm"}\
![The evolution of $U$ with respect to the physical radius ${f}$ in the Reissner-Nordstrom solution for different electric charge $Q=1.3,\ 1.0,\ 0.8,\ 0.7$. There are two horizons in the spacetime, the inner Cauchy horizon ($\textrm{CH}$) and the black hole event horizon ($\textrm{EH}$). (As an example, the $\textrm{CH}$ and $\textrm{EH}$ are given for $Q=0.7$). The space is asymptotically flat. With the increasing of electric charge $Q$, the EH is shrinking and the CH expanding. When $Q=M=1.0$, the inner Cauchy horizon and the black hole event horizon coincide and the solution corresponds to the extreme solution.[]{data-label="fig:RND"}](RND.eps "fig:"){width="8.4cm"}\
![The evolution of $V$ with respect to the physical radius $\ln{f}$ for different $\alpha_1=-0.4,\ -0.2,\ 0, \, 0.1,\ 0.15$. There are two horizons in the spacetime in general, the inner Cauchy horizon and the black hole event horizon (As an example, the $\textrm{CH}$ and $\textrm{EH}$ are given for $\alpha_1=-0.4$). With the increasing of $\alpha_1$, the event horizon is shrinking. When $\alpha_1=0$, the inner Cauchy horizon and the black hole event horizon coincide and the solution corresponds to the extreme solution.[]{data-label="fig:fv"}](fv.eps "fig:"){width="8.4cm"}\
In Fig. \[fig:fu0\], we plot the evolution of $U$ with respect to the physical radius $ \ln f$ with values $m=1,\
\alpha_1=\alpha_2=1,\ \ \alpha=3/2$. As for $\alpha_0$, we let $\alpha_0=0.15,\ 0.1,\ 0,\ -0.2,\ -0.4$, respectively. Comparing with Fig. \[fig:fu\], we find the black hole event horizon is pushed to infinity in this case. We are left with only the inner Cauchy horizon. Keep the constants ($\alpha_0,\ \alpha_1,\ \
\alpha_2,\ \ m,\ \ f_0$) to be fixed and verify $\alpha$, we find the figures are similar to Fig. \[fig:fu\] or Fig. \[fig:fu0\].
![The evolution of the electric potential $\phi$ with respect to the physical radius ${f}$ for different $\alpha_1=-0.4,\ -0.2,\ 0, \, 0.1,\ 0.15$. It shows that $\phi$ is regular in the spacetime except for $f=0$ (curvature singularity). The potential $\phi$ is divergent at the curvature singularity and asymptotically approaches zero in the infinity of space.[]{data-label="fig:fphi"}](fphi.eps "fig:"){width="8.4cm"}\
![The evolution of the Ricci scalar $\ln R$ with respect to the physical radius ${f}$ with $m=1,\ \alpha_0=\alpha_2=1,\ \
\alpha=3/2,\ \ \ \alpha_1=-0.4$. It is apparent $R$ is divergent at $f=0$. This reveals that $f=0$ is indeed the curvature singularity of spacetime.[]{data-label="fig:fR"}](R.eps "fig:"){width="8.4cm"}\
![The evolution of the inverse of magnetic potential $\psi$ with respect to the physical radius $\ln{f}$ for different $\alpha_1=-0.4,\ -0.2,\ 0, \, 0.1,\ 0.15$. It shows that the magnetic potential $\psi^{-1}$ is divergent on the inner Cauchy horizon and the outer black hole event horizon. On the curvature singularity and the spatial infinity, it asymptotically approaches zero.[]{data-label="fig:fpsi"}](fpsi.eps "fig:"){width="8.4cm"}\
![The evolution of $U$ with respect to the physical radius $\ln{f}$ with values $m=1,\ \alpha_1=\alpha_2=1,\ \ \alpha=3/2$. As for $\alpha_0$, we let $\alpha_0=0.15,\ 0.1,\ 0,\ -0.2,\ -0.4$, respectively. Comparing with Fig. \[fig:fu\], we find the black hole event horizon is pushed to infinity in this case. We are left with uniquely the inner Cauchy horizon. As an example, the $\textrm{CH}$ and $\textrm{EH}$ are given for $\alpha_0=-0.4$. []{data-label="fig:fu0"}](fu0.eps "fig:"){width="8.4cm"}\
Finally, in order to understand the structure of horizons very well, it would be very helpful to investigate the trajectories of geodesic (free fall) paths in the spacetime. For simplicity, we shall restrict ourselves to timelike and radial geodesics. The equations of motion could be derived from the Lagrangian $$\begin{aligned}
\mathscr{L}=\frac{1}{2}\left[U\dot{t}^2-\frac{1}{U}\dot{r}^2-f^2\dot{r}^2-f^2\sin^2\theta\dot{\varphi}^2\right]\;,\end{aligned}$$ where the dot denotes the differentiation with respect to the proper time $\tau$. They could also be derived from the geodesic equation
$$\begin{aligned}
\frac{d^2X^{\mu}}{d\tau^2}+\Gamma^{\mu}_{\alpha\beta}\cdot\frac{dX^{\alpha}}{d\tau}\cdot\frac{dX^{\beta}}{d\tau}=0\;.\end{aligned}$$
The equations of motion are found to be
$$\begin{aligned}
\frac{dr}{d\tau}=-\sqrt{E^2-U}\;,\end{aligned}$$
for proper time and $$\begin{aligned}
\frac{dr}{dt}=-\frac{U}{E}\cdot\sqrt{E^2-U}\;,\end{aligned}$$ for coordinate time $t$, respectively. Here $E$ is a constant.
We shall consider the trajectories of particles which start from rest at some finite distance $r_0$ and fall towards the center. The constant $E$ is related to the starting distance $r_0$ by
$$\begin{aligned}
E=\sqrt{U\mid_{r=r_0}}\;, \ \ (r=r_0\ \ \ \textrm{when}\ \
\dot{r}=0)\end{aligned}$$
![The evolution of coordinate time $t$ and proper time $\tau$ along a timelike radial geodesics of a test particle, starting at rest at $r_0=0.9\;$ or $\ln f_0=1.46$ and falling towards the singularity.[]{data-label="fig:ff"}](ff.eps "fig:"){width="8.4cm"}\
In Fig. \[fig:ff\], we plot the evolution of coordinate time $t$ and proper time $\tau$ along the timelike radial geodesics. The test particle starts at rest at $r_0=0.9\;$ or $\ln f_0=1.46$ and falls towards the singularity. The same as the Figures (1-6), the parameters are assumed with $f_0=1$, $m=1,\ \alpha_0=\alpha_2=1,\
\ \alpha=3/2$, $\alpha_1=-0.4$. The circled lines denote the black hole event horizon (EH) and the Cauchy horizon (CH), respectively(also shown in Fig. 1 and Fig. 3). Line A denotes the coordinate time $t$. It shows that with respect to an observer stationed at infinity, a particle describing a timelike trajectory will take an infinite time to reach the black hole event horizon. The behavior is in sharp contrast with that of proper time. Line B denotes the evolution of proper time $\tau$. It shows that the particle crosses the black hole event horizon and the Cauchy horizon with finite proper time. And after crossing the Cauchy horizon, the particle will arrive at some finite distance with finite proper time.
check of the solution with the Einstein equations
==================================================
In section II, we construct the static spherically symmetric solution by imposing the symmetries of interest-rotational symmetry and statistic-on the action principle rather than on the field equations. Compared to the method of solving Einstein equations, it is relatively simple, but also seems questionable. The question is as follows. In imposing the symmetry before carrying the variation of the action principle, one generally loses field equations. So one may worry about that the solution maybe do not satisfy the lost equations contained in the Einstein equations. In this section, we shall check our solution with the Einstein equations. To this end, we should start from the total action of the theory which is given by
$$\begin{aligned}
S=\int
d^4x\sqrt{-g}\left[-{R}-\frac{c_1}{2}F_{\mu\nu}F^{\mu\nu}-c_2\left(\nabla_{\mu}A^{\mu}\right)^2+\lambda\left(A_{\mu}A^{\mu}+m^2\right)\right]\;.\end{aligned}$$
In the first place, variation of the action with respect to $\lambda$, we obtain the equation of motion for $\lambda$
$$\begin{aligned}
\label{eq:check1}
A_{\mu}A^{\mu}+m^2=0\;.\end{aligned}$$
Actually, it is the fixed-norm constraint on the aether field.
Secondly, variation of the action with respect to $A^{\mu}$ leads to the equation of motion for aether field $$\begin{aligned}
\label{eq:check2}
c_1\nabla_{\nu}F^{\nu}_{\ \
\mu}+c_2\nabla_{\mu}\left(\nabla_{\nu}A^{\nu}\right)+\lambda
A_{\mu}=0\;.\end{aligned}$$ This equation determines the dynamics of $A^{\mu}$. Finally, variation of the action with respect to the metric gives the Einstein equations
$$\begin{aligned}
\label{eq:check3}
G_{\mu\nu}=T_{\mu\nu}\;.\end{aligned}$$
We emphasize that the equation of fixed-norm constraint Eq. (\[eq:check1\]) could be followed from the equation of motion of aether field Eq. (\[eq:check2\]) and the Einstein equations Eq. (\[eq:check3\]) in view of the fact that: $$\begin{aligned}
\label{eq:cons}
\nabla_{\nu}T^{\mu\nu}=0\;.\end{aligned}$$ The energy-momentum tensor of the Einstein-aether field takes the form [@picon:09]
$$\begin{aligned}
T_{\mu\nu}&=&c_1F_{\mu\alpha}F_{\nu}^{\
\alpha}+c_2g_{\mu\nu}\left[A^{\alpha}\nabla_{\alpha}\left(\nabla_{\beta}A^{\beta}\right)+\left(\nabla_{\alpha}A^{\alpha}\right)^2\right]-2c_2A_{(\mu}\nabla_{\nu)}
\left(\nabla_{\alpha}A^{\alpha}\right)\nonumber\\&&-\lambda
A_{\mu}A_{\nu}+\frac{1}{2}g_{\mu\nu}\left[-\frac{c_1}{2}F_{\alpha\beta}F^{\alpha\beta}-c_2\left(\nabla_{\alpha}A^{\alpha}\right)^2
-\lambda\left(A_{\alpha}A^{\alpha}+m^2\right)\right]\;.\end{aligned}$$
Given the metric Eq. (\[eq:line\]) and the aether field Eq. (\[eq:A\]), we find the equation of motion for $\lambda$:
$$\begin{aligned}
\label{eq:lambda0}
-\frac{1}{U}\phi^2+\frac{K^2}{Uf^4}+m^2=0\;,\end{aligned}$$
the equation of motion for $A_{\mu}$: $$\begin{aligned}
\label{eq:phi0}
\label{eq:phi} c_1Uf\phi^{''}+2c_1U\phi^{'}f^{'}+\lambda\phi
f=0\;,\end{aligned}$$ $$\begin{aligned}
\label{eq:K0}
c_2UfK^{''}-2c_2U K^{'}f^{'}+\lambda K f=0\;.\end{aligned}$$ and the Einstein equations: $$\begin{aligned}
\label{eq:00E}
\left(1-2Uff^{''}-U^{'}ff^{'}-Uf^{'2}\right)\cdot\frac{1}{f^2}&=&\frac{1}{2}c_1\phi^{'2}-\frac{1}{2}c_2K^{'2}\frac{1}{f^4}+\lambda\left(\frac{K^2}{2Uf^4}-\frac{1}{2}m^2
-\frac{1}{2}\frac{\phi^{2}}{U}\right)\;,\\
\left(1-Uf^{'2}-U^{'}ff^{'}\right)\cdot\frac{1}{f^2}&=&\frac{1}{2}c_1\phi^{'2}-\frac{1}{2}c_2K^{'2}\frac{1}{f^4}+\lambda\left(-\frac{K^2}{2Uf^4}-\frac{1}{2}m^2
+\frac{1}{2}\frac{\phi^{2}}{U}\right)\;,\\
-\frac{1}{2}U^{''}-Uf^{''}\frac{1}{f}-U^{'}f^{'}\frac{1}{f}&=&-\frac{1}{2}c_1\phi^{'2}-\frac{1}{2}c_2K^{'2}\frac{1}{f^4}-\lambda\left(-\frac{K^2}{2Uf^4}+\frac{1}{2}m^2
-\frac{1}{2}\frac{\phi^{2}}{U}\right)\;,\end{aligned}$$ which correspond to $G_0^0=T_0^0$, $G_1^1=T_1^1$ and $G_2^2=T_2^2$ , respectively. Now we have six equations of motion but five variables, namely, $U,\ f,\ \phi,\ K,\ \lambda$. Therefore, among the six equations, only five of them are independent. It is indeed the case when we take into account the fact that the equation of motion for $\lambda$ Eq. (\[eq:check1\]) follows from the equation of motion of aether field Eq. (\[eq:check2\]) and the Einstein equations Eq. (\[eq:check3\]). In practice, one could show that Eq. (\[eq:lambda0\]) follows from Eqs.(56-60) by using of Eq. (\[eq:cons\]).
One may ask whether the five equations of motion Eqs. (9-13) derived with the Euler-Lagrange method could be derived from above six equations of motion Eqs. (55-66). The answer is yes. In fact, we have $$\begin{aligned}
&&\textrm{Eq.}\left(9\right)\Longleftrightarrow\textrm{Eq.}\left(55\right)\;,\nonumber\\
&&\textrm{Eq.}\left(10\right)\Longleftrightarrow\textrm{Eq.}\left(56\right)\;,\nonumber\\
&&\textrm{Eq.}\left(11\right)\Longleftrightarrow\textrm{Eq.}\left(57\right)\;,\nonumber\\
&&\textrm{Eq.}\left(12\right)\Longleftrightarrow\textrm{Eq.}\left(58\right)-\textrm{Eq.}\left(59\right)\;,\nonumber\\
&&\textrm{Eq.}\left(13\right)\Longleftrightarrow\textrm{Eq.}\left(60\right)\;.\nonumber\end{aligned}$$ Now we could understand that our solution satisfies all the equations: the fixed-norm constraint equation, the equation of motion of $A_{\mu}$ and the Einstein equations.
conclusion and discussion
=========================
In conclusion, a static spherically symmetric solution in the Einstein-aether is obtained. Due to the complication of the Einstein equations, we prefer to deal with the Euler-Lagrange equations. This method is relatively simple and the same as the Einstein equations in nature. By this way, an exact solution is constructed. The solution is similar to the Reissner-Nordstrom solution in that it has an inner Cauchy horizon and an outer black hole event horizon. But a remarkable difference from the Reissner-Nordstrom solution is that it is not asymptotically flat in space. We find the solution asymptotically approaches a two dimensional sphere. The resulting electric potential is regular in the whole spacetime except for the curvature singularity. On the other hand, the magnetic potential is divergent on both Cauchy horizon and the outer event horizon.
We sincerely thank the anonymous referee for the expert and insightful comments which have significantly improved the paper. We also thank Prof. Ted Jacobson for the very helpful discussions. This work is supported by the National Science Foundation of China under the Grant No. 10973014 and the 973 Project (No. 2010CB833004).
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[^1]: We thank Ted Jacobson for bringing these points to our notice.
[^2]: We thank Ted Jacobson for pointing out this point.
[^3]: A spacetime is globally hyperbolic if there exist Cauchy surfaces (not Cauchy horizon) in the spacetime. In the Reissner-Nordstrom (RN) spacetime, the timelike property of the curvature singularity reveals it is not globally hyperbolic. So there is no Cauchy surface in the RN spacetime. But there is an inner Cauchy horizon in the RN spacetime [@wald84]. Similarly, the singularity in our solution is also timelike because of $g_{00}<0$ and $g_{11}>0$ when $0<f<f_{CH}$ ($f_{CH}$ represents the radius of Cauchy horizon) and our solution is not globally hyperbolic. When $f_{CH}<f<f_{EH}$ ($f_{EH}$ represents the radius of black hole event horizon), we have $g_{00}>0$ and $g_{11}<0$. It is a spacelike region. Furthermore, there exists a curvature singularity within the event horizon. So the spacetime is for a black hole. When $f>f_{EH}$, we have $g_{00}<0$ and $g_{11}>0$ which is again a timelike region.
[^4]: When $f\rightarrow \infty$, we find $g_{00}=0$ and $g_{11}=0$. The metric becomes $ds^2=f^2d\Omega^2$ which is for a two dimensional sphere.
|
---
abstract: 'We report ac susceptibility, specific heat and neutron scattering measurements on a dipolar-coupled antiferromagnet . For the thermal transition, the order-parameter critical exponent is found to be 0.20(1) and the specific-heat critical exponent $-0.25(1)$. The exponents agree with the 2D XY$/h_4$ universality class despite the lack of apparent two-dimensionality in the structure. The order-parameter exponent for the quantum phase transitions is found to be 0.35(1) corresponding to $(2+1)$D. These results are in line with those found for which has the same crystal structure, but largely different , crystal field environment and hyperfine interactions. Our results therefore experimentally establish that the dimensional reduction is universal to quantum dipolar antiferromagnets on a distorted diamond lattice.'
author:
- 'P. Babkevich'
- 'M. Jeong'
- 'Y. Matsumoto'
- 'I. Kovacevic'
- 'A. Finco'
- 'R. Toft-Petersen'
- 'C. Ritter'
- 'M. M[å]{}nsson'
- 'S. Nakatsuji'
- 'H. M. Rønnow'
bibliography:
- 'shorttitles.bib'
- 'biblio\_v0p5.bib'
title: Dimensional Reduction in Quantum Dipolar Antiferromagnets
---
Critical phenomena near continuous phase transitions do not depend on the microscopic details of systems but only on the symmetry of the order parameter and interactions and the spatial dimensionality [@stanley1987introduction]. Such universality for classical thermal transitions has been thoroughly demonstrated with various physical systems over decades while nowadays a similar line of effort is actively pursued for zero-temperature quantum transitions [@sondhi1997continuous; @vojta2003quantum; @sachdev2007quantum]. Comparing experimental observations with theoretical models has been particularly successful for magnetic insulators that could be simply modeled by short-ranged, exchange-coupled spins on a lattice. Although dipolar interactions appear to be more classical than their exchange-coupled counterparts, it has been shown that on a square or diamond lattice, quantum fluctuations can map long-ranged dipolar interactions to a two-dimensional Ising model [@shender-jetp-1982; @henley-prl-1989; @orendacova-cz-2002]. The Li$R$F$_4$ family is special as the rare-earth ions are arranged in a slightly distorted diamond-like structure making them intriguing to study in relation to order by disorder phenomena [@schustereit-crystals-2011].
For the case of a dipolar-coupled Ising ferromagnet, the theoretical upper critical dimension D$^\ast=3$ and the mean-field calculations actually apply quite well as shown, for instance, in LiHoF$_4$ [@bitko-prl-1996]. This is despite the significant role of hyperfine interactions around the quantum phase transition [@ronnow-science-2005; @kovacevic-submitted]. Recently, quantum and classical critical properties of a long-range, dipolar-coupled antiferromagnet could be investigated for the first time with [@kraemer-science-2012]. It was discovered that the specific-heat and order-parameter critical exponents, $\alpha=-0.28(4)$ and $\beta_T=0.15(2)$, for the thermal transition are totally different from the mean-field predictions of $\alpha=0$ and $\beta_T=0.5$. Instead, these exponent values suggest a 2D XY$/h_4$ universality class, despite the absence of any apparent two-dimensionality in the structure of the system. This intriguing dimensional reduction was further corroborated by the $\beta_H=0.31(2)$ for the quantum transition induced by applying a longitudinal magnetic field, which corresponds to $(2+1)$D, as expected from quantum-classical mapping [@sachdev2007quantum]. Whether the dimensional reduction is universal to all dipolar quantum antiferromagnets or is special to , due to rather close (3meV) higher-lying crystal-field levels or weak hyperfine interactions, is to date unknown.
Among the Li$R$F$_4$ family where $R$ is a rare-earth ion, has been suggested to be an alternate candidate for a dipolar antiferromagnet [@babkevich-prb-2015]. However, there are marked differences between and . First, the electronic level scheme is quite different with crystalline electric field split first excited state an order of magnitude higher in . Second, in Yb$^{3+}$, there are two stable isotopes of Yb with strong hyperfine coupling – $11.0\,\mu$eV for $^{171}$Yb (14.3%) and $-3.0\,\mu$eV for $^{173}$Yb (16.1%). contains $^{167}$Er (22.8%) whose hyperfine coupling strength is weak, 0.5$\mu$eV. Therefore, could serve as an excellent candidate to test for the robustness of dimensional reduction in dipolar antiferromagnets arranged on a distorted diamond lattice.
![(a) Real part of ac susceptibility $\chi'$ as a function of temperature in zero field and (b) $\chi'$ as a function of field at different temperatures. (c) Magnetic phase diagram mapped out using the susceptibility. Inset shows the bilayer magnetic structure of .[]{data-label="fig1"}](fig1.eps){width="1\columnwidth"}
In this Letter, we present ac susceptibility, specific heat, and neutron scattering measurements on and demonstrate the thermal and quantum critical properties. The field-temperature ($H$-$T$) phase diagram is first mapped out and a bilayered XY antiferromagnetic order for the ground state is identified. Then we show that the critical exponents $\alpha$, $\beta_T$, and $\beta_H$ support the dimensional reduction as a universal feature of quantum dipolar antiferromagnets.
Large, high-quality single crystals were obtained from a commercial source. In order to reduce neutron absorption, the samples were enriched with the $^7$Li isotope. The ac susceptibility $\chi(T,H)$ was measured on a single crystal using mutual inductance method where the excitation field was 40mOe and the excitation frequency 545Hz. The specific heat $C_p(T)$ was measured by the relaxation method in a dilution refrigerator with a temperature stability of 0.1mK. Powder neutron diffraction was performed using the high-intensity D1B and high-resolution D2B diffractometers at ILL, France using incident neutron wavelength 2.52 and 1.59Å, respectively. The evolution of the magnetic Bragg peak intensities with temperature and field was followed by performing high-resolution single-crystal neutron scattering using the triple-axis spectrometer FLEXX at HZB, Germany [@le-flexx]. The instrument was set up with 40’ collimation before and after the sample and incident neutron wavelength of $\lambda = 4.05$Å. The corresponding wavevector and energy resolution (FWHM) was on the order of 0.014Å and 0.15meV, respectively.
Figure \[fig1\] shows bulk ac susceptibility data from a single-crystal . The temperature-field phase boundary was mapped for a transverse magnetic field applied along the $c$ axis. Figure \[fig1\](a) shows the real part of the ac susceptibility, $\chi'$, as a function of temperature in zero field. The peak in zero field reflects the antiferromagnetic transition at $\TN=130$mK. Figure \[fig1\](b) shows $\chi'(H)$ at 30-200mK. Below , a pronounced cusp is observed which corresponds to a quantum transition from the ordered to a quantum paramagnetic phase. At base temperature, a maximum in $\chi'(H)$ is found at $\Hc=0.48$T. The peak shifts to lower fields as temperature is increased. Based on these measurements, we can accurately map out the phase diagram shown in Fig. \[fig1\](c).
![(a) Specific heat in zero and finite fields as a function of temperature. Calculation of specific heat capacity in the single-ion limit for different fields are plotted by continuous lines. The data were displaced vertically by multiplying with scaling factors given in the figure. (b) Determination of the specific-heat critical exponent $\alpha$ for the thermal transition based on measurements above and below (dashed line). Scaling away from the critical region was fitted by the dotted line.[]{data-label="fig2"}](fig2.eps){width="0.9\columnwidth"}
The specific heat as a function of temperature is shown in Fig. \[fig2\](a). In zero field, a sharp peak in the specific heat capacity marks the second-order thermal transition [^1]. On applying a transverse field, we find the peak at decreases in amplitude and shifts to lower temperature at $H=0.45$T. Above , only a broad hump is found in the specific heat capacity. At such low temperatures, phonon and crystal-field-level contributions are frozen out. We model the specific heat capacity away from the QPT using a parameter-free model where the Hamiltonian $\mathcal{H}$ contains crystal field, hyperfine and Zeeman terms. From the diagonalized Hamiltonian $\langle n | \mathcal{H} | n \rangle = \epsilon_n$, we calculate for each isotope $i$ the Schottky specific heat, $C^{\rm Sch}_i = k_{\rm B}\beta^2\left[\langle\epsilon^2\rangle
- \langle\epsilon\rangle^2\right]$, where $k_{\rm B}$ is the Boltzmann factor and $\beta = 1/(k_{\rm B} T)$. The thermal ensemble average is denoted by $\langle \ldots \rangle$. The total specific heat capacity is found from the weighted sum of contributions from each Yb isotope. The comparison between the experiment and our simple model is remarkably good considering that this is a parameter-free calculation with all parameters fixed from other experiments. It is possible to improve the comparison by including quadrupolar operators, and by fine-tuning hyperfine coupling strengths and the crystal field parameters, etc. However, this would give too many adjustable parameters, and the calculation anyway ignores collective effects beyond the mean-field level. In zero applied field, close to , the heat capacity can be described by a universal power-law, $$C^{\rm crit}_p = A|t|^{\alpha} + B,$$ where the reduced temperature $t = 1-T/\TN$, $A$ and $B$ are free parameters which can have different values above and below . The results of our analysis are shown in Fig. \[fig2\](b). The contribution from the background term, $B$, is found to be small and is set to zero above and below . A good fit is found for $\alpha=-0.25(1)$, similar to the value of $-0.28(4)$ found in [@kraemer-science-2012]. The negative exponents imply that $C_p$ is finite at . Away from the phase transition we observe a change in the scaling. Above around 250mK and below 100mK the data can be fit to an exponent of around $-1.3(1)$. It is somewhat surprising that the critical scaling can be traced out all the way to 2 and is dramatically different to where a cross over was found above 1.03 [@kraemer-science-2012].
![(a) Magnetic powder diffraction pattern from the subtraction of paramagnetic background from 50mK measurements. (b) Magnetic Bragg peak from powder diffraction at different temperatures in zero field and (c) single-crystal measurements at 70 mK in different fields. Lines are fits to a Gaussian with additional contribution from critical scattering.[]{data-label="fig3"}](fig3.eps){width="\columnwidth"}
To elucidate the magnetic structure below , we performed neutron diffraction on a powder of . At 10K, in the paramagnetic phase, the crystal lattice was refined using the $I4_1/a$ space group where $a=5.13433(8)$Å and $c=10.5917(2)$Å. Below 140mK we find additional peaks which emerge from antiferromagnetic ordering corresponding to a $\mathbf{k}=(1,0,0)$ magnetic propagation wavevector. Figure \[fig3\](a) shows powder diffraction pattern obtained by subtracting measurements above from 50mK data. The magnetic peaks are well described by a bilayer antiferromagnetic structure with moments along the \[110\] direction, where moments related by $I$-centering are aligned antiparallel. An ordered moment of 1.9(1)$\mu_{\rm B}$ is found to reside on each Yb$^{3+}$ ion. A schematic of a possible magnetic structure is shown in Fig. \[fig1\](c). This differs from where the moments are parallel to the \[100\] direction. Although our data do not allow us to uniquely identify the magnetic structure, it is clear that and order differently (see Supplemental Material). The origin of this is not entirely obvious but could be attributed to the in-plane anisotropy set by the crystal field. This would depend primarily on the $B_4^4(c)\mathbf{O}_4^4(c)$ crystal field term and result in the configuration energy $E\sim B_4^4(c) \cos(4\phi)$ having minima rotated by 45$^\circ$ when changing the sign of $B_4^4(c)$ parameter. Indeed, our previously reported results show that $B_4^4(c)$ is significantly larger and of opposite sign in compared to [@babkevich-prb-2015].
The powder sample of was measured as a function of temperature in fine steps across the thermal phase transition. Figure \[fig3\](b) shows how the magnetic intensity of the $(001)$ reflection decreases with temperature. As expected from ac susceptibility and heat capacity measurements, magnetic order disappears above 136mK. Single-crystal measurements as a function of transverse field are shown in Fig. \[fig3\](c). At $T_{\rm base} = 70$mK, a field of around 0.43T suppresses the $(100)$ magnetic peak. A small contribution from critical scattering is observed as tails of the main peak. The neutron scattering measurements of reaffirm the phase diagram found from ac susceptibility in Fig. \[fig1\].
{width="90.00000%"}
The evolution of the magnetic Bragg peak intensities with temperature and field are shown in Fig. \[fig4\](a). Continuous onset and smooth evolution of the order parameter is observed with both temperature and field.
For both the powder and single-crystal data we have considered a model consisting of (i) a Lorentzian lineshape to describe the critical fluctuations close to the phase transition and (ii) a delta-function to account for long-range order. Both of these were then convoluted by a Gaussian, representing the instrumental resolution. The strength of scattering from critical fluctuations is rather weak and within the measured resolution and statistics cannot be refined to extract further exponents in either powder or single-crystal data. The amplitude of the convoluted delta-function $\sigma$ corresponds to the square of the order parameter, i.e., staggered magnetization. Therefore, sufficiently close to the phase boundary, $\sigma \propto t^{2\beta_T}$ for a zero-field measurement and $\sigma \propto h^{2\beta_H}$, where $h = 1-H/\Hc$ on sweeping magnetic field at constant temperature. From such treatment we obtain the results shown in Fig. \[fig4\](b), where squares and circles are for thermal and quantum critical exponents, respectively. Fitting the data to a power-law, we obtain $\beta_T = 0.20(1)$ and $\beta_H = 0.35(1)$.
The base temperature of 70mK at which the field was swept to cross the quantum phase transition may appear rather high as $T_\mathrm{base}\simeq 0.5\, \TN$. For , on the other hand, the $\beta_H$ was extracted at $T_\mathrm{base} \simeq 0.2\, \TN$. To ensure that the extracted $\beta_H = 0.35(1)$ is correct and not affected by thermal fluctuations, we followed the field evolution of the $(100)$ Bragg peak at a few higher temperatures. We find, as shown in Fig. \[fig4\](c), no appreciable change in $\beta_H$ in the temperature range studied. This assertion is further corroborated by the heat capacity measurements, shown in Fig. \[fig2\], where the thermal critical region is found above around 0.8. Comparing the critical exponents to tabulated results [@lovesey-book; @kagawa-nature-2005; @taroni-jpcm-2008], it is clear that the quantum transition falls in the $\beta=0.32$-0.36 range predicted for 3D models. While the 2D XY/$h_4$ model predicts $\beta=0.125$-0.23, bound by 2D Ising and XY transitions, which best describes the thermal phase transition [@taroni-jpcm-2008].
Such dimensional reduction has been hinted at from studies of other dipolar systems. A good example is $R$Ba$_2$Cu$_3$O$_{7-\delta}$ whose dipolar interactions were the focus of some theoretical work [@debell-jpcm-1991; @macisaac-prb-1992]. It was argued two-dimensional behavior is strongly related to the spacing of basal planes with a cross-over from three-dimensional behavior around $c/a > 2.5$. However, relatively strong exchange coupling as well as superconductivity makes this system more complicated to separate the influence of the dipolar interaction. We hypothesize that systems such as $R$PO$_4$(MoO$_3$)$_{12}\cdot$30H$_2$O where rare-earth ions form a diamond lattice would also be a good candidate to examine quantum criticality due to strong dipolar and weak exchange interactions [@noordaa-jlow-2000]. Quantum spin fluctuations of dipolar-coupled antiferromagnetism have already been suggested to play a major role in these systems [@white-prl-1993].
To conclude, dipolar-coupled undergoes a thermal transition into the bilayer, XY antiferromagnetically ordered phase, where the critical exponents follow the 2D XY$/h_4$ universality class despite the lack of apparent two-dimensionality in the structure. Applying a transverse magnetic field suppresses the order, inducing a quantum phase transition into a paramagnetic state, which scales according to $(2+1)$D universality. These observations are in accordance with those for with largely different crystal field environment, , and hyperfine interactions. Our results, therefore, experimentally establish that the dimensional reduction is a universal feature of dipolar-coupled quantum antiferromagnets on the distorted diamond-like lattice and are likely to be applicable to a vast range of seemingly different systems. While it may be premature to conclude that dimensional reduction is universal to other lattices, the challenge is now to find a dipolar-coupled antiferromagnet without it.
We are grateful to B. Klemke for his technical support. We would like to thank J. O. Piatek for his help in setting up the ac susceptibility measurements and I. Živković for helpful discussions. We are indebted to J. S. White, M. Zolliker and M. Bartkowiak for their assistance during preliminary measurements on the TASP spectrometer at SINQ, PSI. This work was funded by the Swiss National Science Foundation and its Sinergia network MPBH, Marie Curie Action COFUND (EPFL Fellows), and European Research Council Grant CONQUEST. This work is partially supported by Grants-in-Aid for Scientific Research (No. 25707030 and No. 15K13515) and Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers (No. R2604) from the Japanese Society for the Promotion of Science. We thank HZB for the allocation of neutron radiation beam time. This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under the NMI3-II Grant No. 283883. M. M. was partly supported by Marie Sklodowska Curie Action, International Career Grant through the European Commission and Swedish Research Council (VR), Grant No. INCA-2014-6426.
Supplemental Material
=====================
Crystallographic structure
--------------------------
![(Color online) High-resolution neutron powder diffraction measurements using D2B diffractometer. Data collected at 10K and refined to the structural model described in the text. Incident neutron wavelength was 1.594Å.[]{data-label="figS1"}](figS1.eps){width="0.95\columnwidth"}
It is well known that systems of the Li$R$F$_4$ family crystallize in a scheelite CaWO$_4$ type structure. To verify our sample, we have performed careful measurements using D2B diffractometer in the paramagnetic phase at 10K – well above magnetic ordering temperature. Our results are presented in Fig. \[figS1\]. A good fit to the diffraction pattern was found using Rietveld method in the Fullprof package [@carvajal-physicab-1993] which allows us to extract the atomic positions and $B_{\rm iso}$ isotropic Debye-Waller factors. In the case of $^7$Li, it was not possible to accurately refine the $B_{\rm iso}$ parameter and therefore it was fixed in the fitting. The detailed refinement, described in Table \[tab:nuc\_struct\], is in excellent agreement with that reported previously on the system in Ref. [@thoma-inchem-1970].
Atom site $x$ $y$ $z$ $B_{\rm iso}$ (Å$^2$)
-------- ------ ----------- ----------- ----------- -----------------------
$^7$Li 4a 0.0000 0.2500 0.1250 0.80
Yb 4b 0.0000 0.2500 0.6250 0.09(3)
F 16f 0.2186(4) 0.4169(4) 0.4571(2) 0.43(4)
: Nuclear structure refinement of shown in Fig. \[figS1\]. The Bragg peaks were indexed by $I4_1/a$ space group with lattice parameters of $a=5.13435(8)$Å and $c=10.5918(2)$Å. The fractional atomic positions using the second origin choice setting are listed in the table together with uncertainties given in brackets. \[tab:nuc\_struct\]
Magnetic structure
------------------
{width="70.00000%"}
Having confirmed the crystallographic structure of and the absence of impurities, we next consider the arrangement of the magnetic moments below . Previous study of found that magnetic moments are arranged into a bilayer structure where the moments connected by $I$-centering are antiparallel [@kraemer-science-2012]. Indeed, solving the Hamiltonian in the mean-field approximation quickly converges to this structure. Our previous mean-field simulations of and indicate that the groundstate magnetic structures should be the same, with the only difference that the moment on Yb$^{3+}$ ion is expected to be smaller than that on Er$^{3+}$ [@babkevich-prb-2015].
Neutron diffraction data from studies of is plotted in Figs. \[figS2\](a) and (b). Measurements were collected using DMC diffractometer with $\lambda = 2.457$Å. Antiferromagnetic ordering in sets in below 375mK [@kraemer-science-2012]. In order to obtain purely the magnetic contribution to the signal, we have subtracted measurements collected above 900mK. Surprisingly, some of the stronger peaks are found to sit on broad humps which could indicate some short-range correlations in the system but could also be some artifacts related to the background. The origin of these cannot be elucidated further.
In comparison, data collected using D1B at $\lambda = 2.52$Å examining show a slowly varying background with no signs of any additional features. We notice from the and diffraction patterns shown in Figs. \[figS2\](b) and (d) that the relative intensities of $(100)$ and $(102)$, close to 5.1 and 3.7Å, respectively, are clearly different for the two systems. The ratio of $\sigma(100)$ to $\sigma(102)$ intensity in is 3.36(7) and in is 1.241(4).
Since the incident neutron wavelengths are very similar and the instrumental resolution is not very different for the two diffractometers we would have expected from mean-field simulations that the magnetic powder patterns are nearly the same. Intriguingly this does not appear to be the case. Performing Rietveld refinement of the magnetic structure for gives a better fit when the moments are allowed to rotate to be along the $[110]$ direction. The simulations for the two different moment directions is shown in Figs. \[figS2\](e) and (f). In the model where the moments are along $[100]$, the $\sigma(100)/\sigma(102) = 4.14$ – close to what we find for . Repeating this analysis for moments along $[110]$, we find instead $\sigma(100)/\sigma(102) = 1.35$, viz .
Magnetic representation analysis
--------------------------------
$\nu$ $g_1$ $g_2$ $g_3$ $g_4$ $g_5$ $g_6$ $g_7$ $g_8$
------- ------------------------------------------------- --------------------------------------------------- -------------------------------------------------------------- -------------------------------------------------------------- ------------------------------------------------- --------------------------------------------------- -------------------------------------------------------------- --------------------------------------------------------------
1 $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ $\begin{pmatrix} 0 &-1 \\ 1 & 0 \end{pmatrix}$
2 $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ $\begin{pmatrix} {\rm i} & 0 \\ 0 & -{\rm i} \end{pmatrix}$ $\begin{pmatrix} -{\rm i} & 0 \\ 0 & {\rm i} \end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ $\begin{pmatrix} 0 & -{\rm i} \\ {\rm i} & 0 \end{pmatrix}$ $\begin{pmatrix} 0 & {\rm i} \\ -{\rm i} & 0 \end{pmatrix}$
$\nu$ $n$ $(\psi_x^1,\psi_y^1,\psi_z^1)$ $(\psi_x^2,\psi_y^2,\psi_z^2)$
------- ----- -------------------------------- --------------------------------
1 1 $(1,0,0)$ $(0,1,0)$
1 2 $(0,1,0)$ $(-1,0,0)$
1 3 $(0,-1,0)$ $(-1,0,0)$
1 4 $(1,0,0)$ $(0,-1,0)$
2 1 $(1,0,0)$ $(0,0,-{\rm i})$
2 2 $(0,0,{\rm i})$ $(0,0,-1)$
: Basis functions $\psi$ of irreducible representation $\Gamma_\nu$ for ions situated at 1. $(x,y,z)$ and 2. $(-y+3/4,x+1/4,z+1/4)$. \[tab:basis\_vectors\]
The magnetic structures of and can be described by the magnetic propagation wavevector $\kk = (1,0,0)$. From the paramagnetic space group $I4_1/a$, the little group $G_\kk$ contains 8 symmetry elements ($g_1$ – $g_8$) listed in Table \[tab:sym\_elements\]. The magnetic representation $\Gamma_{\rm mag}$ of $G_\kk$ reduces to $\Gamma_{\rm mag} = 2\Gamma_1 + \Gamma_2$. Both $\Gamma_1$ and $\Gamma_2$ are two dimensional and their characters are given in Table \[tab:sym\_elements\]. Using Basireps [@basireps], we obtain basis functions $\psi$, shown in Table \[tab:basis\_vectors\] for two symmetry-related sites. The two sites create an extinction condition which makes is possible to distinguish between magnetic moment directions even in the tetragonal cell with powder averaging. In general, the $n$th moment $\mathbf{m}_n$ can be expressed as a Fourier series, $$\mathbf{m}_n = \sum_\kk \mathbf{S}_n^\kk e^{-{\rm i}\kk\cdot\mathbf{t}},$$ where $\mathbf{t}$ is the real space translation vector. The vectors $\mathbf{S}_n^\kk$ are a linear sum of the basis vectors such that, $$\mathbf{S}_n^\kk = \sum_{m,p} c_{mp} \psi^{\kk}_{\nu mp},$$ where coefficients $c_{mp}$ can be complex. We label $\nu$ as the active irreducible representation $\Gamma_\nu$, $m=1\ldots n_\nu$, where $n_\nu$ is the number of times $\Gamma_\nu$ is contained in $\Gamma_{\rm mag}$. The index $p$ labels the component corresponding to the dimension of $\Gamma_\nu$.
{width="90.00000%"}
In the case of , the moments lie in the $ab$ plane, therefore $\Gamma_1$ is active (see Table \[tab:basis\_vectors\]). However, the neutron data does not allow us to uniquely identify the magnetic ordering as any of the four basis vectors can refine the measured data. All four arrangements result in moments which rotate by 90$^\circ$ along $c$, as for example shown in Fig. \[figS3\](a). It is also possible to use a combination of two basis vectors, such as 1 and 3 or 2 and 4 to describe a collinear magnetic structure as shown in Figs. \[figS3\](b–d). However, it is not possible to refine the measured data for using the same combination of basis vectors which appear to describe . Indeed a combination of all four basis vectors, as depicted in Fig. \[figS3\](e), is needed to describe the best possible solution for reported in Ref. [@kraemer-science-2012] which sees the moments along the $a$ axis
Crystal field interaction
-------------------------
ion $10^3B^0_2$ $10^3B^0_4$ $10^6B^0_6$ $10^3B^4_4$(c) $10^3B^4_6$(c) $10^6|B^4_6$(s)$|$
----- ------------- ------------- ------------- ---------------- ---------------- --------------------
Er 58.1 -0.536 -0.00625 **-5.53** -0.106 23.8
(3.4) (0.032) (0.00041) (0.31) (0.0061) (1.5)
Yb 457 7.75 0 **196** -9.78 0
(5.2) (0.12) (0) (0.65) (0.0094) (0)
: Crystal field parameters of and compounds determine by inelastic neutron scattering. Typically, a coordinate system with $B^4_4(s) = 0$ is chosen, while two possible equivalent coordinations of $R$ ion by F ions give different sign of $B_6^4(s)$. After [@babkevich-prb-2015]. \[tab:cf\_params\]
The in-plane anisotropy in and is largely determined by the single-ion crystal field and dipolar interactions. We would expect that as the magnetic moment size is very similar in and , the dipolar interactions in the two systems do not differ significantly. One possible arrangement in is shown in Fig. \[figS3\](c) where the moments are all rotated by 45$^\circ$ in the basal plane with respect to the magnetic structure. This structure (amongst others) fits well the measured data. From the crystal field whose Hamiltonian for $\bar{4}$ point group symmetry at the $R$ site is given by, $$\mathcal{H}_{\rm CEF} = \sum_{l =2,4,6} B_l^0{\mathbf O}_l^0 +
\sum_{l = 4,6} B_l^4(c){\mathbf O}_l^4(c) + B_l^4(s){\mathbf O}_l^4(s).
\label{eq:crystal_field}$$ The later $B_l^4$ terms play a role in the planar anisotropy where $B_4^4(c)$ term is found from experiments to be largest, see Table \[tab:cf\_params\]. Classically, one obtains the energy of rotating a moment of size $J_0$ in the plane by angle $\phi$ to be $E = J_0 B_4^4(c) \cos(4\phi)$. Hence, the minimum in energy for different signs of $B_4^4(c)$ is found to be 45$^\circ$ apart. While this appears to be a simple explanation for preferred moment direction, a strong crystal field interaction would result in an Ising-like system, which is not what we observe experimentally. Furthermore, dipolar interactions are not expected to favor such ordering. Therefore, further theoretical work is necessary to examine the mechanism by which the dipolar-coupled antiferromagnets order.
Conclusion
----------
While it is entirely possible that the diffraction patterns can be also described by other models including ones where moments are non-collinear, qualitatively our experimental data appears to suggest that the groundstate magnetic structure of is not the same as . Thus, this highlights the universality of antiferromagnetism on a distorted diamond lattice described in our Letter.
[^1]: We note that the different techniques gave slightly different values of , within 10mK. We attribute this to differing thermometer calibrations and possibly small thermal gradients between sample and thermometer. This does not affect the extracted exponents nor the main conclusions of this Letter.
|
---
abstract: 'We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in [@KP08]. These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, resp. exactly, one $1$-entry per row. They are important objects for symmetry reduction in certain integer programs. Using the extended formulations, we also derive a rather simple proof of the fact [@KP08] that basically shifted-column inequalities suffice in order to describe those orbitopes linearly.'
address:
- 'Dipartimento di Ingegneria dell’Impresa, Università di Roma “Tor Vergata”, Rome, Italy'
- 'Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik, Universitätsplatz 2, 39106 Magdeburg, Germany'
author:
- Yuri Faenza
- Volker Kaibel
title: Extended Formulations for Packing and Partitioning Orbitopes
---
Introduction {#sec:intro}
============
Exploitation of symmetries is crucial for many very difficult integer programming models. Over the last few years, significant progress has been achieved with respect to general techniques for dealing with symmetries within branch-and-cut algorithms. Very nice and effective procedures have been devised, like isomorphism pruning [@Mar02; @Mar03; @Mar03b; @Mar07] and orbital branching [@LiOsRoSm06; @LiOsRoSm08]. There has also been progress in understanding linear inequalities to be added to certain integer programs in order to remove symmetry. Towards this end, orbitopes have been introduced in [@KP08].
The *packing orbitope* ${\operatorname{O}^{\le}_{p,q}}$ and the *partitioning orbitope* ${\operatorname{O}^{=}_{p,q}}$ are the convex hulls of all 0/1-matrices of size $p\times q$ whose columns are in lexicographically decreasing order having at most or exactly, respectively, one $1$-entry per row. In [@KP08], complete descriptions with linear inequalities have been derived for these polytopes (see Thm. 16 and 17 in [@KP08]). Knowledge on orbitopes turns out to be quite useful in practical symmetry reduction for certain integer programming models. For instance, in a well-known formulation of the graph partitioning problem (for graphs having $p$ nodes to be partitioned into $q$ parts) the symmetry on the 0/1-variables $x_{ij}$ indicating whether node $i$ is put into part $j$ of the partitioning the symmetry arising from permuting the parts can be removed by requiring $x\in{\operatorname{O}^{=}_{p,q}}$. We refer to [@KaPePf07] and [@KP08] for a more detailed discussion of the practical use of orbitopes.
The topic of this paper are extended formulations for these orbitopes, i.e., (simple) linear descriptions of higher dimensional polytopes which can be projected to ${\operatorname{O}^{\le}_{p,q}}$ and ${\operatorname{O}^{=}_{p,q}}$. In fact, such extended formulations play important roles in polyhedral combinatorics and integer programming in general, because rather than solving a linear optimization problem over a polyhedron in the original space, one may solve it over a (hopefully simpler described) polyhedron of which the first one is a linear projection. For instance, the lift-and-project approach [@BCC93] and other general reformulation schemes (e.g., the ones due to Lovász and Schrijver [@LS91] as well as Adams and Sherali [@SA94]) are based on extended formulations. Recent examples include work on mixed integer programming for duals of network matrices [@CdSW08; @CGZ07; @CdSEW06]. More classical is the general theory on extended formulations obtained from (certain) dynamic programming algorithms [@MRC90]. The results of this paper are much in the spirit of the latter work.
In order to give an overview on the contributions of this paper let us first recall a few facts on orbitopes. As no 0/1-matrix with at most one $1$-entry per row and lexicographically (decreasing) sorted columns has a one above its main diagonal, we may assume without loss of generality that ${\operatorname{O}^{=}_{p,q}}\subseteq{\operatorname{O}^{\le}_{p,q}}\subseteq{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$ with ${\mathcal{I}_{{p},{q}}}={\{{(i,j)\in{[{p}]}\times{[{q}]}}\,:\,{i\ge
j}\}}$ (where ${[{n}]}=\{1,2,\dots,n\}$). In fact, ${\operatorname{O}^{=}_{p,q}}$ is the face of ${\operatorname{O}^{\le}_{p,q}}$ defined by requiring that all *row-sum inequalities* $x({\text{row}_{i}})\le 1$ for $i\in{[{p}]}$ are satisfied with equality, where ${\text{row}_{i}}={\{{(i,j)\in{\mathcal{I}_{{p},{q}}}}\,:\,{j\in{[{q}]}}\}}$.
The main result of [@KP08] is a complete description of ${\operatorname{O}^{\le}_{p,q}}$ and ${\operatorname{O}^{=}_{p,q}}$ by means of linear equations and inequalities. This system of constraints (the *SCI-system*) consists, next to nonnegativity constraints and row-sum inequalities (or row-sum equations for ${\operatorname{O}^{=}_{p,q}}$), of the exponentially large class of *shifted column inequalities* (*SCI*) that will be defined at the end of Sect. \[sec:setup\]. In [@KP08] it is also proved that, up to a few exceptions, these exponentially many SCIs define facets of the orbitopes. The proof given in [@KP08] of the fact that the SCI-system completely describes these orbitopes is rather lengthy and somewhat technical. Extending over pages $18$ to $27$, it hardly leaves (not only) the reader with a good idea of the reasons for the SCI-system being sufficient to describe the orbitopes.
In contrast to this, the contributions of the present work are the following: We provide a quite simple extended formulation for ${\operatorname{O}^{\le}_{p,q}}$ (along with a rather short proof establishing this) and, moreover, we show by some simple and natural (not technical) arguments that the SCI-system describes the projection of the feasible region of that extended formulation to the original space, thus providing a new proof showing that the SCI-system describes ${\operatorname{O}^{\le}_{p,q}}$.
This latter proof is much shorter than the original one, and it seems to provide much better insight into the reasons for the SCI-system to describe the orbitopes. Clearly, as ${\operatorname{O}^{=}_{p,q}}$ is a face of ${\operatorname{O}^{\le}_{p,q}}$, the results for the latter polytope immediately yield corresponding results for the first one. However, besides leading to that simpler proof, we believe that our extended formulation for ${\operatorname{O}^{\le}_{p,q}}$ is interesting itself. It provides a description of a quite natural polytope (the orbitope ${\operatorname{O}^{\le}_{p,q}}$) by a system of constraints in a space whose dimension is roughly twice the original dimension $|{\mathcal{I}_{{p},{q}}}|$ with only linearly (in $|{\mathcal{I}_{{p},{q}}}|$) many nonzero coefficients, while every linear description of the orbitope in the original space requires exponentially many inequalities. This may also turn out to be computationally attractive.
The basic idea of our extended formulation is to assign to each vertex of ${\operatorname{O}^{\le}_{p,q}}$ a directed path in a certain acyclic digraph. The additional variables in our extended formulations are used to suitably express these paths. The digraph we work with is set up in Sect. \[sec:setup\], where we also fix some notations and define SCIs. In Sect. \[sec:ext\] we then describe the extended formulations for ${\operatorname{O}^{\le}_{p,q}}$ and ${\operatorname{O}^{=}_{p,q}}$ (Thm. \[thm:extform\] and Cor. \[cor:extpartpath\]). The main work is done in Sect. \[subsec:ext:pack\], where the extended formulation for ${\operatorname{O}^{\le}_{p,q}}$ with additional variables encoding the paths mentioned above is introduced and proved to define an integral polyhedron (Thm. \[thm:integral\]). From this it is easy to conclude that the formulation indeed defines a polytope that projects down to ${\operatorname{O}^{\le}_{p,q}}$ (Thm. \[thm:extform\]). Both the extension of such results to the partitioning case ${\operatorname{O}^{=}_{p,q}}$ (Cor. \[cor:extpartpath\] in Sect. \[subsec:ext:part\]), and the transformations of the systems in order to reduce the numbers of variables and nonzero coefficients (Thm. \[thm:verycompact\] in Sect. \[subsec:ext:reduce\]) are obtained without much work. On the way, we also derive linear (in $|{\mathcal{I}_{{p},{q}}}|$) time algorithms for optimizing linear objective functions over ${\operatorname{O}^{\le}_{p,q}}$ and ${\operatorname{O}^{=}_{p,q}}$ (Cor. \[cor:alg\] and \[cor:alg:part\]). In Sect. \[sec:project\] we finally prove that the projection of the feasible region defined by the extended formulation contains the polytope defined by the SCI-system (Thm. \[thm:liftscipoly\]), thus providing the new proof of the fact (Thm. \[thm:orbipackSCI\]) that the latter polytope equals ${\operatorname{O}^{\le}_{p,q}}$. We conclude with a few remarks and acknowledgements in Sect. \[sec:remarks\].
The Setup {#sec:setup}
=========
Let us assume $p\ge q\ge 1$ throughout the paper. We define a directed acyclic graph ${\operatorname{D}_{{p},{q}}}=({\operatorname{V}_{{p},{q}}},{\operatorname{A}_{{p},{q}}})$ with node set $${\operatorname{V}_{{p},{q}}}={\mathcal{I}_{{p},{q}}}\uplus({[{p}]_0}\times\{0\})\uplus\{s\}\uplus\{t\}$$ (where ${[{n}]_0}={[{n}]}\cup\{0\}$ and $\uplus$ means disjoint union). Using the notation $q(i)=\min\{i,q\}$, the set of arcs of ${\operatorname{D}_{{p},{q}}}$ is $${\operatorname{A}_{{p},{q}}}={\operatorname{A}^{\shortdownarrow}_{{p},{q}}} \cup{\operatorname{A}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}_{{p},{q}}}\cup\{(s,(0,0))\}\cup{\{{((p,j),t)}\,:\,{j\in{[{q}]_0}}\}}\,,$$ where $${\operatorname{A}^{\shortdownarrow}_{{p},{q}}}={\{{((i,j),(i+1,j))}\,:\,{i\in{[{p-1}]_0},j\in{[{q(i)}]_0}}\}}$$ is the set of *vertical arcs* that are denoted by ${({i},{j})^{\shortdownarrow}}=((i,j),(i+1,j))$, and $${\operatorname{A}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}_{{p},{q}}}={\{{((i,j),(i+1,j+1))}\,:\,{i\in{[{p-1}]_0},j\in{[{q(i+1)-1}]_0}}\}}\,$$ is the set of *diagonal arcs* that are denoted by ${({i},{j})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}=((i,j),(i+1,j+1))$. The crucial property of ${\operatorname{D}_{{p},{q}}}$ is that every vertex of ${\operatorname{O}^{\le}_{p,q}}$ induces an $s$-$t$-path in ${\operatorname{D}_{{p},{q}}}$ as indicated in Fig. \[fig:digraph\]. Note that different vertices may induce the same path.
![The digraph ${\operatorname{D}_{{8},{6}}}$ and a vertex of ${\operatorname{O}^{\le}_{8,6}}$ along with its $s$-$t$-path.[]{data-label="fig:digraph"}](digraph "fig:"){width=".35\textwidth"} ![The digraph ${\operatorname{D}_{{8},{6}}}$ and a vertex of ${\operatorname{O}^{\le}_{8,6}}$ along with its $s$-$t$-path.[]{data-label="fig:digraph"}](vertexpath "fig:"){width=".35\textwidth"}
For a subset $W\subseteq{\operatorname{V}_{{p},{q}}}$ we use the following notation: $$\begin{aligned}
{\operatorname{out}({W})} & = & {\{{(w,u)\in{\operatorname{A}_{{p},{q}}}}\,:\,{w\in W,u\not\in W}\}}\\
{\operatorname{out}^{\shortdownarrow}({W})} & = & {\{{(w,u)\in{\operatorname{A}^{\shortdownarrow}_{{p},{q}}}}\,:\,{w\in W,u\not\in W}\}}\\
{\operatorname{in}({W})} & = & {\{{(u,w)\in{\operatorname{A}_{{p},{q}}}}\,:\,{w\in W,u\not\in W}\}}\\
{\operatorname{in}^{\shortdownarrow}({W})} & = & {\{{(u,w)\in{\operatorname{A}^{\shortdownarrow}_{{p},{q}}}}\,:\,{w\in W,u\not\in W}\}}\\
{\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({W})} & = & {\{{(u,w)\in{\operatorname{A}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}_{{p},{q}}}}\,:\,{w\in W,u\not\in W}\}}\end{aligned}$$ For a directed path $\Gamma$ in ${\operatorname{D}_{{p},{q}}}$, we denote by
--------------------------------------------------------------------- -----------------------------------------------------------------------------
${\operatorname{V}({\Gamma})}\subseteq{\operatorname{V}_{{p},{q}}}$ the set of nodes on the path $\Gamma$,
${\operatorname{S}({\Gamma})}\subseteq{\operatorname{V}({\Gamma})}$ the set of nodes on $\Gamma$ not entered by $\Gamma$ via diagonal arcs, and
${\operatorname{T}({\Gamma})}\subseteq{\operatorname{V}({\Gamma})}$ the set of nodes on $\Gamma$ left by $\Gamma$ via diagonal arcs
--------------------------------------------------------------------- -----------------------------------------------------------------------------
(see Fig. \[fig:stnodes\]).
![A path $\Gamma$ along with the sets ${\operatorname{S}({\Gamma})}$ (left) and ${\operatorname{T}({\Gamma})}$ (right).[]{data-label="fig:stnodes"}](snodes "fig:"){width=".35\textwidth"} ![A path $\Gamma$ along with the sets ${\operatorname{S}({\Gamma})}$ (left) and ${\operatorname{T}({\Gamma})}$ (right).[]{data-label="fig:stnodes"}](tnodes "fig:"){width=".35\textwidth"}
Note that ${\operatorname{S}({\Gamma})}$ always contains the start node of $\Gamma$, and ${\operatorname{T}({\Gamma})}$ always excludes the end node of $\Gamma$.
\[rem:inoutgamma\] For every directed path $\Gamma$ in ${\operatorname{D}_{{p},{q}}}$ with end node $(i,j)\in{\mathcal{I}_{{i},{j}}}$, we have $${\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{V}({\Gamma})}})}={\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{S}({\Gamma})}})}
\quad\text{and}\quad
{\operatorname{out}^{\shortdownarrow}({{\operatorname{V}({\Gamma})}\setminus\{(i,j)\}})}={\operatorname{out}^{\shortdownarrow}({{\operatorname{T}({\Gamma})}})}\,.$$
A subset $S\subseteq{\mathcal{I}_{{p},{q}}}$ is a *shifted column* if and only if $S={\operatorname{S}({\Gamma})}$ for some $(\ell,\ell)$-$(i-1,j-1)$-path $\Gamma$ in ${\operatorname{D}_{{p},{q}}}$ with $i\in{[{p}]}\setminus\{1\}$, $j\in{[{q}]}\setminus\{1\}$, and $\ell\in{[{q}]}$. The associated *shifted-column inequality* is $x({\operatorname{bar}_{{i},{j}}})\le x(S)$, where ${\operatorname{bar}_{{i},{j}}}={\{{(i,\ell)\in{\mathcal{I}_{{p},{q}}}}\,:\,{\ell \ge j}\}}$ and, as usual, we write $z(N)=\sum_{e\in N}z_e$ for some vector $z\in {\mathbbm{R}}^{M}$ and a subset $N\subseteq M$ (see Fig. \[fig:sci\]).
![Coefficient vectors of two SCIs with the same bar ${\operatorname{bar}_{{8},{5}}}$.[]{data-label="fig:sci"}](sci "fig:"){width=".35\textwidth"} ![Coefficient vectors of two SCIs with the same bar ${\operatorname{bar}_{{8},{5}}}$.[]{data-label="fig:sci"}](ci "fig:"){width=".35\textwidth"}
Extended formulations {#sec:ext}
=====================
The packing case {#subsec:ext:pack}
----------------
Denote by ${\operatorname{F}_{{p},{q}}}\subseteq{\mathbbm{R}}^{{\operatorname{A}_{{p},{q}}}}$ the set of all $s$-$t$-flows (without any capacity restrictions) in ${\operatorname{D}_{{p},{q}}}$ with flow value one. Clearly, ${\operatorname{F}_{{p},{q}}}$ is an integral polytope. Since ${\operatorname{D}_{{p},{q}}}$ is acyclic, the vertices of ${\operatorname{F}_{{p},{q}}}$ are the incidence vectors of the directed $s$-$t$-paths (viewed as subsets of arcs) in ${\operatorname{D}_{{p},{q}}}$.
For a flow $y\in{\operatorname{F}_{{p},{q}}}$ and a node $(i,j)\in{\mathcal{I}_{{p},{q}}}$, we denote by $$y(i,j)=y({\operatorname{in}({i,j})})=y({\operatorname{out}({i,j})})$$ the amount of flow passing node $(i,j)$. For a subset $W\subseteq{\text{row}_{i}}$ of nodes in the same row, $y(W)=\sum_{w\in W}y(w)$ is the total amount of flow entering $W$ (or, equivalently, leaving $W$).
\[lem:cuts\] For a directed $(k,\ell)$-$(i,j)$-path $\Gamma$ in ${\operatorname{D}_{{p},{q}}}$ with $(i,j)\in{\mathcal{I}_{{p},{q}}}$ the following statements hold for all $y\in{\operatorname{F}_{{p},{q}}}$ (see Fig. \[fig:lemma2\]):
1. If $k=\ell\ge 1$ then $$y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{S}({\Gamma})}})})-y({\operatorname{out}^{\shortdownarrow}({{\operatorname{T}({\Gamma})}})})=y({\operatorname{bar}_{{i},{j}}})\,.$$
2. If $\ell=0$ then $$1-y({\operatorname{out}^{\shortdownarrow}({{\operatorname{T}({\Gamma})}})})=y({\operatorname{bar}_{{i},{j}}})\,.$$
![Illustration of part (1) (left) and part (2) (right) of Lemma \[lem:cuts\] with $(i,j)=(7,4)$.[]{data-label="fig:lemma2"}](lemma2part1 "fig:"){width=".35\textwidth"} ![Illustration of part (1) (left) and part (2) (right) of Lemma \[lem:cuts\] with $(i,j)=(7,4)$.[]{data-label="fig:lemma2"}](lemma2part2 "fig:"){width=".35\textwidth"}
For both cases, let $$W={\{{(a',b)\in{\operatorname{V}_{{p},{q}}}\setminus\{s,t\}}\,:\,{a'\le a \text{ for some }(a,b)\in{\operatorname{V}({\Gamma})}\cup{\operatorname{bar}_{{i},{j}}}}\}}$$ be the set of all nodes (different from $s$ and $t$) in or above ${\operatorname{V}({\Gamma})}\cup{\operatorname{bar}_{{i},{j}}}$.
For case (1), we start by observing $$\label{eq:lem:cuts:1}
y({\operatorname{in}({W})})=y({\operatorname{out}({W})})$$ (as $y\in{\operatorname{F}_{{p},{q}}}$ is an $s$-$t$-flow). Because of $(0,0)\not\in W$ (thus $s$ being not adjacent to $W$) we have ${\operatorname{in}({W})}={\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{V}({\Gamma})}})}$, which according to Remark \[rem:inoutgamma\] equals ${\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{S}({\Gamma})}})}$, yielding $$\label{eq:lem:cuts:2}
y({\operatorname{in}({W})})=y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{S}({\Gamma})}})})\,.$$ Similarly, we have ${\operatorname{out}({W})}={\operatorname{out}^{\shortdownarrow}({{\operatorname{V}({\Gamma})}\setminus\{(i,j)\}})}\uplus{\operatorname{out}({{\operatorname{bar}_{{i},{j}}}})}$, where Remark \[rem:inoutgamma\] gives ${\operatorname{out}^{\shortdownarrow}({{\operatorname{V}({\Gamma})}\setminus\{(i,j)\}})}={\operatorname{out}^{\shortdownarrow}({{\operatorname{T}({\Gamma})}})}$. Thus, we obtain $$\label{eq:lem:cuts:3}
y({\operatorname{out}({W})}) =y({\operatorname{out}^{\shortdownarrow}({{\operatorname{T}({\Gamma})}})})+y({\operatorname{out}({{\operatorname{bar}_{{i},{j}}}})})\,.$$ Equations , , and imply the statement on case (1).
For case (2), we exploit the fact that the $s$-$t$-flow $y\in{\operatorname{F}_{{p},{q}}}$ of value one satisfies $$\label{eq:cuts:case2}
1 = y({\operatorname{out}({W\cup\{s\}})})-y({\operatorname{in}({W\cup\{s\}})})\,.$$ We have ${\operatorname{in}({W\cup\{s\}})}=\varnothing$ and ${\operatorname{out}({W\cup\{s\}})}={\operatorname{out}({W})}$ (due to $(0,0)\in W$ in this case). From $${\operatorname{out}({W})}={\operatorname{out}^{\shortdownarrow}({{\operatorname{V}({\Gamma})}\setminus\{(i,j)\}})}
\uplus
{\operatorname{out}({{\operatorname{bar}_{{i},{j}}}})}$$ and ${\operatorname{out}^{\shortdownarrow}({{\operatorname{V}({\Gamma})}\setminus\{(i,j)\}})}={\operatorname{out}^{\shortdownarrow}({{\operatorname{T}({\Gamma})}})}$ (see Remark \[rem:inoutgamma\]), we thus derive the statement on case (2) from .
For $(i,j)\in{\mathcal{I}_{{p},{q}}}$ we denote by $${{\operatorname{col}_{{i},{j}}}}={\{{(k,j)}\,:\,{j\le k\le i}\}}$$ the upper part of the $j$-th column from $(j,j)$ down to $(i,j)$, including both nodes. For the directed path $\Gamma$ with ${\operatorname{V}({\Gamma})}={{\operatorname{col}_{{i},{j}}}}$, we have ${\operatorname{S}({\Gamma})}={{\operatorname{col}_{{i},{j}}}}$ and ${\operatorname{T}({\Gamma})}=\varnothing$. Thus, part (1) of Lemma \[lem:cuts\] implies the following.
\[rem:cuts\] For all $y\in{\operatorname{F}_{{p},{q}}}$ and $(i,j)\in{\mathcal{I}_{{p},{q}}}$, we have $$y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{i},{j}}}}})})=y({\operatorname{bar}_{{i},{j}}})\,.$$
The central object of study of this paper is the polytope $${\operatorname{P}_{{p},{q}}}=
{\{{(x,y)\in{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\operatorname{A}_{{p},{q}}}}}\,:\,{y\in{\operatorname{F}_{{p},{q}}} \text{ and }(x,y) \text{ satisfies~\eqref{eq:bindxy} and~\eqref{eq:bindxybar} below}}\}}$$ with $$\label{eq:bindxy}
y_{{({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}}\le x_{ij}\quad\text{ for all }(i,j)\in{\mathcal{I}_{{p},{q}}}$$ and $$\label{eq:bindxybar}
x({\operatorname{bar}_{{i},{j}}})\le y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{i},{j}}}}})})\quad\text{ for all }(i,j)\in{\mathcal{I}_{{p},{q}}}$$ (see Fig. \[fig:eq56\]).
![Coefficient vectors of inequalities (left) and (right).[]{data-label="fig:eq56"}](eq5 "fig:"){width=".35\textwidth"} ![Coefficient vectors of inequalities (left) and (right).[]{data-label="fig:eq56"}](eq6 "fig:"){width=".35\textwidth"}
Since ${\operatorname{F}_{{p},{q}}}$ is the set of all $s$-$t$-flows of value one, all $(x,y)\in{\operatorname{P}_{{p},{q}}}$ satisfy ${\mathbf{0}}\le y\le{\mathbf{1}}$, and thus ${\mathbf{0}}\le x\le{\mathbf{1}}$ due to and . Furthermore, inequalities imply the row-sum inequalities $x({\text{row}_{i}})\le 1$ for all $(x,y)\in{\operatorname{P}_{{p},{q}}}$.
\[thm:integral\] The polytope ${\operatorname{P}_{{p},{q}}}$ is integral.
For the proof of this theorem, we need the following result.
\[lem:integral\] Let $x_1,\dots,x_n\ge 0$ and $y_1,\dots,y_n\in{\mathbbm{R}}$ with $$\sum_{\ell=j}^nx_{\ell}\le\sum_{\ell=j}^ny_{\ell}
\quad\text{for all }1\le j\le n\,.$$ For all numbers $\alpha_1,\dots,\alpha_n\in{\mathbbm{R}}$ and $0\le\beta_1\le\beta_2\le\cdots\le\beta_n$ with $$\alpha_j\le\beta_{j}
\quad\text{for all }1\le j\le n$$ the inequality $$\sum_{j=1}^n\alpha_jx_j\le\sum_{j=1}^n\beta_jy_j$$ holds.
We prove the claim by induction on $n$. The case $n=1$ is trivial, thus let $n\ge 1$. Ignoring index $1$ and decreasing the remaining $\alpha_j$ and $\beta_j$ by $\beta_1$, from the induction hypothesis we obtain $$\sum_{j=2}^n(\alpha_j-\beta_1)x_j\le\sum_{j=2}^n(\beta_j-\beta_1)y_j\,.$$ Due to $\alpha_1\le\beta_1$ and $x_1\ge 0$ we have $(\alpha_1-\beta_1)x_1\le 0$, thus we deduce $$\sum_{j=1}^n(\alpha_j-\beta_1)x_j\le\sum_{j=1}^n(\beta_j-\beta_1)y_j\,.$$ Hence we have $$\sum_{j=1}^n\alpha_jx_j-\sum_{j=1}^n\beta_jy_j\le
\beta_1\cdot\Big(\sum_{j=1}^nx_j-\sum_{j=1}^ny_j\Big)\,,$$ with a nonpositive right-hand side due to $\beta_1\ge 0$ and $\sum_{j=1}^nx_j\le\sum_{j=1}^ny_j$.
In order to show that ${\operatorname{P}_{{p},{q}}}$ is integral, we show that for an arbitrary objective function vector $c\in{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\operatorname{A}_{{p},{q}}}}$ the optimization problem $$\label{eq:intproof:optprob}
\max{\{{{\langle{c},{(x,y)}\rangle}}\,:\,{(x,y)\in{\operatorname{P}_{{p},{q}}}}\}}$$ has an optimal solution with 0/1-components.
We define two vectors $c^{(1)},c^{(2)}\in{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\operatorname{A}_{{p},{q}}}}$ which are zero in all components with the following exceptions: $$\begin{array}{lcll}
c^{(1)}_{{({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}} & = & c_{(i,j)} & \text{for all }(i,j)\in{\mathcal{I}_{{p},{q}}}\\
c^{(2)}_{{({i-1},{j})^{\shortdownarrow}}} & = &
\max\{0,c_{(i,1)},\dots,c_{(i,j)}\} &
\text{for all }(i,j)\in{\mathcal{I}_{{p},{q}}}\\
\end{array}$$ We are going to establish the following two claims:
1. For each $(x,y)\in{\operatorname{P}_{{p},{q}}}$ we have $${\langle{c},{(x,y)}\rangle}\le{\langle{c+c^{(1)}+c^{(2)}},{({\mathbf{0}},y)}\rangle}\,.$$
2. For each $s$-$t$-flow $y\in{\operatorname{F}_{{p},{q}}}\cap\{0,1\}^{{\operatorname{A}_{{p},{q}}}}$ there is some $x\in\{0,1\}^{{\mathcal{I}_{{p},{q}}}}$ with $$(x,y)\in{\operatorname{P}_{{p},{q}}}
\quad\text{and}\quad
{\langle{c+c^{(1)}+c^{(2)}},{({\mathbf{0}},y)}\rangle}={\langle{c},{(x,y)}\rangle}\,.$$
With these two claims, the existence of an integral optimal solution to can be established as follows: Let $\tilde{c}\in{\mathbbm{R}}^{{\operatorname{A}_{{p},{q}}}}$ be the $y$-part of $c+c^{(1)}+c^{(2)}$. As ${\operatorname{F}_{{p},{q}}}$ is a 0/1-polytope, there is a 0/1-flow $y^{\star}\in{\operatorname{F}_{{p},{q}}}\cap\{0,1\}^{{\operatorname{A}_{{p},{q}}}}$ with $${\langle{\tilde{c}},{y^{\star}}\rangle}=\max{\{{{\langle{\tilde{c}},{y}\rangle}}\,:\,{y\in{\operatorname{F}_{{p},{q}}}}\}}\,.$$ Due to ${\langle{c+c^{(1)}+c^{(2)}},{({\mathbf{0}},y)}\rangle}={\langle{\tilde{c}},{y}\rangle}$ for all $y\in{\operatorname{F}_{{p},{q}}}$, claim (1) implies that the optimal value of is at most ${\langle{\tilde{c}},{y^{\star}}\rangle}$. On the other hand, claim (2) ensures that there is some $x^{\star}\in\{0,1\}^{{\mathcal{I}_{{p},{q}}}}$ with $(x^{\star},y^{\star})\in{\operatorname{P}_{{p},{q}}}$ and $${\langle{c},{(x^{\star},y^{\star})}\rangle}
={\langle{c+c^{(1)}+c^{(2)}},{({\mathbf{0}},y^{\star})}\rangle}
={\langle{\tilde{c}},{y^{\star}}\rangle}\,.$$ Thus, $(x^{\star},y^{\star})$ is an integral optimal solution to .
In order to prove claim (2), let $y\in{\operatorname{F}_{{p},{q}}}\cap\{0,1\}^{{\operatorname{A}_{{p},{q}}}}$, i.e., $y$ is the incidence vector of an $s$-$t$-path in ${\operatorname{D}_{{p},{q}}}$. For the construction of some $x\in\{0,1\}^{{\mathcal{I}_{{p},{q}}}}$ as required we start by initializing $x={\mathbf{0}}$. For each $(i,j)\in{\operatorname{V}_{{p},{q}}}$ with $y_{{({i},{j})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}}=1$ set $x_{i+1,j+1}=1$. For every $(i,j)\in{\operatorname{V}_{{p},{q}}}$ with $j\ge 1$ and $y_{{({i},{j})^{\shortdownarrow}}}=1$ choose $\ell\in{[{j}]}$ with $$c_{i+1,\ell}=\max\{c_{(i+1),1},\dots,c_{(i+1),j}\}\,,$$ and set $x_{i+1,\ell}=1$ if $c_{i+1,\ell}\ge 0$.
For claim (1) let $(x,y)\in{\operatorname{P}_{{p},{q}}}$. Define $x'\in{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$ via $$x'_{ij}=x_{ij}-y_{{({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}}$$ for all $(i,j)\in{\mathcal{I}_{{p},{q}}}$. As $(x,y)$ satisfies , $x'\ge{\mathbf{0}}$ holds. Furthermore, we have $$\label{eq:inproof:1}
{\langle{c^{(1)}},{({\mathbf{0}},y)}\rangle}={\langle{c},{(x-x',{\mathbf{0}})}\rangle}\,.$$ Therefore, it suffices to show $$\label{eq:inproof:2}
{\langle{c^{(2)}},{({\mathbf{0}},y)}\rangle}\ge{\langle{c},{(x',{\mathbf{0}})}\rangle}\,,$$ because and yield $$\begin{aligned}
{\langle{c+c^{(1)}+c^{(2)}},{({\mathbf{0}},y)}\rangle}
& = & {\langle{c},{({\mathbf{0}},y)}\rangle}
+{\langle{c^{(1)}},{({\mathbf{0}},y)}\rangle}
+{\langle{c^{(2)}},{({\mathbf{0}},y)}\rangle}\\
& \ge & {\langle{c},{({\mathbf{0}},y)}\rangle}+{\langle{c},{(x-x',{\mathbf{0}})}\rangle}+{\langle{c},{(x',{\mathbf{0}})}\rangle}\\
& = & {\langle{c},{(x,y)}\rangle}\,.\end{aligned}$$
In order to establish , we prove for every $i\in{[{p}]}$ $$\label{eq:inproof:3}
\sum_{j=1}^{q(i)}c_{ij}x'_{ij}\le\sum_{j=1}^{q(i-1)}c^{(2)}_{{({i-1},{j})^{\shortdownarrow}}}y_{{({i-1},{j})^{\shortdownarrow}}}\,,$$ which by summation over $i\in{[{p}]}$ yields . To see for some $i\in{[{p}]}$, observe that, for every $j\in{[{q(i)}]}$, we have $$x'({\operatorname{bar}_{{i},{j}}})=x({\operatorname{bar}_{{i},{j}}})-y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{bar}_{{i},{j}}}})})
=x({\operatorname{bar}_{{i},{j}}})-y({\operatorname{bar}_{{i},{j}}})+y({\operatorname{in}^{\shortdownarrow}({{\operatorname{bar}_{{i},{j}}}})})\,.$$ Due to Remark \[rem:cuts\], and as $(x,y)$ satisfies , this implies $$x'({\operatorname{bar}_{{i},{j}}})\le y({\operatorname{in}^{\shortdownarrow}({{\operatorname{bar}_{{i},{j}}}})})\,.$$ Defining $y_{{({i},{q(i)})^{\shortdownarrow}}}=0$ in case of $q(i-1)<q(i)$, we thus have $$\sum_{\ell=j}^{q(i)}x'_{i\ell}\le\sum_{\ell=j}^{q(i)}y_{{({i-1},{\ell})^{\shortdownarrow}}}$$ for every $j\in{[{q(i)}]}$. Setting $c^{(2)}_{{({i-1},{q(i)})^{\shortdownarrow}}}$ to the biggest component of $c$ in case of $q(i-1)<q(i)$, we furthermore have $$0\le c^{(2)}_{{({i-1},{1})^{\shortdownarrow}}}\le\cdots\le c^{(2)}_{{({i-1},{q(i)})^{\shortdownarrow}}}$$ and $c_{ij}\le c^{(2)}_{{({(i-1)},{j})^{\shortdownarrow}}}$ for all $j\in{[{q(i)}]}$. Thus we can use Lemma \[lem:integral\] (with $n=q(i)$, $x_j=x'_{ij}\ge 0$, $y_j=y_{{({i-1},{j})^{\shortdownarrow}}}$, $\alpha_j=c_{ij}$, and $\beta_j=c^{(2)}_{{({i-1},{j})^{\shortdownarrow}}}$) to deduce $$\sum_{j=1}^{q(i)}c_{ij}x'_{ij}\le\sum_{j=1}^{q(i)}c^{(2)}_{{({i-1},{j})^{\shortdownarrow}}}y_{{({i-1},{j})^{\shortdownarrow}}}\,,$$ which yields .
From Theorem \[thm:integral\] one obtains that ${\operatorname{P}_{{p},{q}}}$ is an extended formulation for ${\operatorname{O}^{\le}_{p,q}}$.
\[thm:extform\] The orbitope ${\operatorname{O}^{\le}_{p,q}}\subseteq{\mathbbm{R}}^{\mathcal{I}_{{p},{q}}}$ is the orthogonal projection of the polytope ${\operatorname{P}_{{p},{q}}}\subseteq{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\operatorname{A}_{{p},{q}}}}$ to the space ${\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$.
Let $x\in{\operatorname{O}^{\le}_{p,q}}\cap\{0,1\}^{{\mathcal{I}_{{p},{q}}}}$ be an arbitrary vertex of ${\operatorname{O}^{\le}_{p,q}}$. The incidence vector $y\in\{0,1\}^{{\operatorname{A}_{{p},{q}}}}$ of the unique $s$-$t$-path using all arcs ${({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}$ with $(i,j)\in{\mathcal{I}_{{p},{q}}}$ and $x_{ij}=1$ satisfies $(x,y)\in{\operatorname{P}_{{p},{q}}}$. Thus, ${\operatorname{O}^{\le}_{p,q}}$ is contained in the projection of ${\operatorname{P}_{{p},{q}}}$.
To see that vice versa the projection of ${\operatorname{P}_{{p},{q}}}$ is contained in ${\operatorname{O}^{\le}_{p,q}}$, by Theorem \[thm:integral\] it suffices to observe that every 0/1-point $(x,y)\in{\operatorname{P}_{{p},{q}}}$ is contained in ${\operatorname{O}^{\le}_{p,q}}$. Clearly, for such a point $x$ has at most one one-entry per row (since the row-sum inequalities are implied by the fact $(x,y)\in{\operatorname{P}_{{p},{q}}}$). Furthermore, if the $j$-th column of $x$ was lexicographically larger than the $(j-1)$-st column of $x$ with $i$ being minimal such that $x_{ij}=1$ holds, then one would find that $y({{\operatorname{col}_{{i},{j-1}}}})=0$ holds (because of ), contradicting for $(i,j-1)$.
From the proof of Theorem \[thm:integral\], we derive a combinatorial algorithm for the linear optimization problem $$\label{optpack}
\max{\{{{\langle{d},{x}\rangle}}\,:\,{x\in{\operatorname{O}^{\le}_{p,q}}}\}}$$ with $d\in{\mathcal{I}_{{p},{q}}}$. Indeed, with $c=(d,{\mathbf{0}})\in{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\operatorname{A}_{{p},{q}}}}$ every optimal solution $(x^{\star},y^{\star})$ to $$\label{optpackext}
\max{\{{{\langle{c},{(x,y)}\rangle}}\,:\,{(x,y)\in{\operatorname{P}_{{p},{q}}}}\}}$$ yields an optimal solution $x^{\star}$ to . From the proof of Theorem \[thm:integral\] we know that we can compute an optimal solution $(x^{\star},y^{\star})$ to by first computing the incidence vector $y^{\star}\in\{0,1\}^{{\operatorname{A}_{{p},{q}}}}$ of a longest $s$-$t$-path in the digraph ${\operatorname{D}_{{p},{q}}}$ with respect to arc length given by $c^{(1)}+c^{(2)}$ (which can be done in linear time since ${\operatorname{D}_{{p},{q}}}$ is acyclic) and then setting $x^{\star}\in\{0,1\}^{{\mathcal{I}_{{p},{q}}}}$ as described in the proof of claim (2) (in the proof of Theorem \[thm:integral\]).
\[cor:alg\] Linear optimization over ${\operatorname{O}^{\le}_{p,q}}$ can be solved in time ${\operatorname{O}({pq})}$.
The partitioning case {#subsec:ext:part}
---------------------
The previous results can be easily extended to the partitioning case. Since the row-sum inequalities $x({\text{row}_{i}})\le 1$ are valid for ${\operatorname{P}_{{p},{q}}}$, $${\operatorname{P}^{=}_{{p},{q}}}=
{\{{(x,y) \in {\operatorname{P}_{{p},{q}}}}\,:\,{x({\text{row}_{i}})=1\text{ for all } i \in [p]}\}}$$ is a face of ${\operatorname{P}_{{p},{q}}}$. Clearly, due to Theorem \[thm:extform\] this face maps to the face (see Sect. \[sec:intro\]) $${\{{x\in{\operatorname{O}^{\le}_{p,q}}}\,:\,{x({\text{row}_{i}})=1\text{ for all } i \in [p]}\}}={\operatorname{O}^{=}_{p,q}}$$ of ${\operatorname{O}^{\le}_{p,q}}$ via the orthogonal projection onto the $x$-space ${\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$.
\[cor:extpartpath\] ${\operatorname{P}^{=}_{{p},{q}}}$ is an extended formulation for ${\operatorname{O}^{=}_{p,q}}$.
Suppose we want to solve $$\label{optpart}
\max{\{{{\langle{d},{x}\rangle}}\,:\,{x\in{\operatorname{O}^{=}_{p,q}}}\}}$$ for some $d\in{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$. As all points $x\in{\operatorname{O}^{=}_{p,q}}$ satisfy the row-sum equations $x({\text{row}_{i}})=1$ for all $i\in{[{p}]}$, we may add, for each $i$, an arbitrary constant to the objective function coefficients of the variables belonging to ${\text{row}_{i}}$ without changing the optimal solutions to (though, of course, changing the objective function values of the solutions). Therefore, we may assume that $d$ has only positive components. But then $$\max{\{{{\langle{d},{x}\rangle}}\,:\,{x\in{\operatorname{O}^{=}_{p,q}}}\}}=\max{\{{{\langle{d},{x}\rangle}}\,:\,{x\in{\operatorname{O}^{\le}_{p,q}}}\}}\,,$$ and all optimal solutions to the optimization problem over ${\operatorname{O}^{\le}_{p,q}}$ are points in ${\operatorname{O}^{=}_{p,q}}$. Thus we derive the following from Corollary \[cor:alg\].
\[cor:alg:part\] Linear optimization over ${\operatorname{O}^{=}_{p,q}}$ can be solved in time ${\operatorname{O}({pq})}$.
Reducing the number of variables and nonzero elements {#subsec:ext:reduce}
-----------------------------------------------------
Let us manipulate the defining system of ${\operatorname{P}_{{p},{q}}}$ in order to decrease the number of variables and nonzero coefficients. This may be advantageous for practical purposes. It furthermore emphasizes the simplicity of the extended formulation. For the sake of readability, we define ${\operatorname{bar}_{{i},{j}}}=\varnothing$ and ${{\operatorname{col}_{{i},{j}}}}=\varnothing$ whenever $j>q(i)$.
Since every $y\in{\operatorname{F}_{{p},{q}}}$ satisfies $y_{{({i},{j})^{\shortdownarrow}}} = y({\operatorname{bar}_{{i},{j}}}) -
y({\operatorname{bar}_{{i+1},{j+1}}})$, we deduce from Remark \[rem:cuts\] that for all $(x,y)\in{\operatorname{P}_{{p},{q}}}$ $$y_{{({i},{j})^{\shortdownarrow}}} =y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{i},{j}}}}})}) -
y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{i+1},{j+1}}}}})})$$ holds for all vertical arcs ${({i},{j})^{\shortdownarrow}}$ with $(i,j)\in{\mathcal{I}_{{p},{q}}}$ (and $i<p$) as well as $$y_{((p,j),t)}=y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{p},{j}}}}})})-
y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{p},{j+1}}}}})})$$ for all arcs $((p,j),t)$ with $j\in{[{q}]_0}$, where we defined $y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{p},{q+1}}}}})})=0$. Similarly to the derivation of Remark \[rem:cuts\], one furthermore deduces that every $(x,y)\in{\operatorname{P}_{{p},{q}}}$ satisfies $$y_{{({i},{0})^{\shortdownarrow}}}=1-y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{i+1},{1}}}}})})$$ for all $i\in{[{p-1}]_0}$. Finally, every $(x,y)\in{\operatorname{P}_{{p},{q}}}$ clearly satisfies $y_{(s,(0,0))}=1$. Therefore, we can eliminate from the system describing ${\operatorname{P}_{{p},{q}}}$ all arc variables except for the ones corresponding to diagonal arcs.
We finally apply the linear transformation defined by $$z_{ij}=x({\operatorname{bar}_{{i},{j}}}) \quad\text{and}\quad w_{ij}=y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{{\operatorname{col}_{{i},{j}}}}})})\quad\hbox{ for all } (i,j)\in{\mathcal{I}_{{p},{q}}}
\,,$$ to ${\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\operatorname{A}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}_{{p},{q}}}}$, whose inverse is given by $$x_{ij}=z_{i,j}-z_{i,j+1} \quad\text{for all }(i,j)\in{\mathcal{I}_{{p},{q}}}\,.$$ (defining $z_{i,q(i)+1}=0$, for all $i \in [p]$) and $$y_{{({i},{j})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}}=w_{i+1,j+1}-w_{i,j+1}\quad\text{for all } i \in [p-1]_0, j \in
[q(i+1)-1]_0\,.$$
Few calculations are needed to check that the previous transformation (bijectively) maps ${\operatorname{P}_{{p},{q}}}$ onto the polytope ${\operatorname{P}^{\text{comp}}_{{p},{q}}}\subseteq{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$ defined by the following ”very compact” set of constraints: $$\begin{aligned}
\label{NEW}
w_{i+1,j+1} - w_{i,j+1} & \geq 0 & \hbox{for } i \in [p-1]_0, j \in[q(i+1)-1]_0 \label{NEF:F}\\
w_{i,j} - w_{i+1,j+1} & \geq 0 & \hbox{for } (i,j)\in{\mathcal{I}_{{p},{q}}}, i<p \label{NEF:G}\\
w_{p,1} & \leq 1 \label{NEF:H}\\
w_{i,j} -w_{i-1,j} - z_{ij}+z_{i,j+1} &\le 0 & \hbox{for } (i,j)\in{\mathcal{I}_{{p},{q}}} \label{NEF:I}\\
z_{i,j} - w_{i,j}& \leq 0 & \hbox{for } (i,j)\in{\mathcal{I}_{{p},{q}}} \label{NEF:J}\\
w_{i,q(i)} & \geq 0& \hbox{for } i \in [p] \label{NEF:L}\end{aligned}$$ Here, represent the nonnegativity constraints on the diagonal arcs. Nonnegativity on the vertical arcs ${({i},{j})^{\shortdownarrow}}$ with $i\in{[{p-1}]_0}$ is reflected by for $j\in{[{q(i)}]}$ and by (together with the nonnegativity of $w$, which is implied by and ) for $j=0$. Finally, equations and translate to and , respectively.
Ignoring the nonnegativity constraints, system – has less than $2pq$ variables and $4pq$ constraints, for a total number of nonzero coefficients that is smaller than $10pq$.
Note that $w_{1,1}\le 1$ is a valid inequality for ${\operatorname{P}^{\text{comp}}_{{p},{q}}}$. The face of ${\operatorname{P}^{\text{comp}}_{{p},{q}}}$ defined by $w_{1,1}= 1$ is the image of the face ${\operatorname{P}^{=}_{{p},{q}}}$ of ${\operatorname{P}_{{p},{q}}}$. Thus, adding $w_{1,1}=
1$ to the system – one arrives at another extended formulation of ${\operatorname{O}^{=}_{p,q}}$.
We summarize the results of this subsection.
\[thm:verycompact\] The polytope ${\operatorname{P}^{\text{comp}}_{{p},{q}}}\subseteq{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}\times{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$ defined by – is an extended formulation of ${\operatorname{O}^{\le}_{p,q}}$. The face of ${\operatorname{P}^{\text{comp}}_{{p},{q}}}$ defined by $w_{1,1}= 1$ is an extended formulation of ${\operatorname{O}^{=}_{p,q}}$.
The projection {#sec:project}
==============
Let ${\operatorname{Q}_{{p},{q}}}\subseteq{\mathbbm{R}}^{{\mathcal{I}_{{p},{q}}}}$ be the polytope defined by the nonnegativity constraints $x\ge{\mathbf{0}}$, the row-sum inequalities $x({\text{row}_{i}})\le 1$ for all $i\in{[{p}]}$ and all shifted-column inequalities. By checking the vertices (0/1-vectors) of ${\operatorname{O}^{\le}_{p,q}}$ it is easy to see that ${\operatorname{O}^{\le}_{p,q}}\subseteq{\operatorname{Q}_{{p},{q}}}$ holds. Thus, in order to prove
\[thm:orbipackSCI\] ${\operatorname{O}^{\le}_{p,q}}={\operatorname{Q}_{{p},{q}}}$
(which is Prop. 13 in [@KP08]) it suffices (due to Theorem \[thm:extform\]) to show the following:
\[thm:liftscipoly\] For each $x\in{\operatorname{Q}_{{p},{q}}}$ there is some $y\in{\operatorname{F}_{{p},{q}}}$ with $(x,y)\in{\operatorname{P}_{{p},{q}}}$.
For $x\in{\operatorname{Q}_{{p},{q}}}$ consider the network ${\operatorname{D}_{{p},{q}}}$ with $$\text{capacity }x_{ij}\text{ on the diagonal arc }{({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}$$ for each $(i,j)\in{\mathcal{I}_{{p},{q}}}$ and infinite capacities on all other arcs. In this network, we construct a feasible flow $y\in{\operatorname{F}_{{p},{q}}}$ of value one with the property $$\label{eq:flowcond}
y_{{({i-1},{j-1})^{\shortdownarrow}}}>0\quad\Rightarrow\quad y_{{({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}}=x_{ij}$$ for all $(i,j)\in{\mathcal{I}_{{p},{q}}}$. Phrased verbally, the flow $y$ uses a vertical arc only if the diagonal arc emanating from its tail is saturated. Such a flow can easily be constructed in the following way: start by sending one unit of flow from $s$ to $t$ along the vertical path in column zero. At each step, if the flow $y$ constructed so far violates for some $(i,j)\in{\mathcal{I}_{{p},{q}}}$, choose such a pair $(i,j)$ with minimal $j$, breaking ties by choosing $i$ minimally as well. With $$\vartheta=\min\{y_{{({i-1},{j-1})^{\shortdownarrow}}},x_{ij}-y_{{({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}}\}$$ (i.e., the minimum of the flow on the vertical arc and the residual capacity on the diagonal arc starting at $(i-1,j-1)$) reroute $\vartheta$ units of the flow currently travelling on the vertical arc ${({i-1},{j-1})^{\shortdownarrow}}$ along the path starting with the diagonal arc ${({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}$ and then using the vertical arcs in column $j$. Note that this affects only arcs leaving nodes $(k,\ell)$ with $k\ge i$ and $\ell\ge j$. After this rerouting, holds for $(i,j)$. The minimality requirements in the choice of $(i,j)$ ensure that the flow on the two arcs leaving $(i,j)$ is not changed again afterwards. Thus, will always be satisfied for $(i,j)$ in the future. Therefore, the procedure eventually ends with a flow as required.
As $(x,y)$ satisfies by construction, it suffices to show in order to prove $(x,y)\in{\operatorname{P}_{{p},{q}}}$. To this end, let $(i,j)\in{\mathcal{I}_{{p},{q}}}$. Due to Remark \[rem:cuts\], we only need to prove $$\label{eq:barbar}
y({\operatorname{bar}_{{i},{j}}})\ge x({\operatorname{bar}_{{i},{j}}})\,.$$
We construct a directed $(k,\ell)$-$(i,j)$-path $\Gamma$ in the residual network with respect to the flow $y$ (containing only those arcs of ${\operatorname{D}_{{p},{q}}}$ that are not saturated by $y$) with $k=\ell$ or $\ell=0$ in the following way: Starting from the trivial (length zero) $w$-$(i,j)$-path with $w=(i,j)$, in each step we extend the path at its current start node $w$ by the diagonal arc entering $w$ if this arc is part of the residual network, and by the vertical arc entering $w$ otherwise. As the residual network contains all vertical arcs, we clearly can proceed this way until the start node of the current path is some node $(k,\ell)$ with $k=\ell$ or with $\ell=0$.
Since $\Gamma$ is a path in the residual network and due to , we have $$\label{eq:vertouttgamma}
y({\operatorname{out}^{\shortdownarrow}({{\operatorname{T}({\Gamma})}})})=0\,.$$
If $\ell=0$, part (2) of Lemma \[lem:cuts\] together with yields $y({\operatorname{bar}_{{i},{j}}})=1$, from which follows since $x$ satisfies the row-sum inequality $x({\text{row}_{i}})\le 1$ and the nonnegativity constraints.
If $\ell\ne 0$, then $k=\ell\ge 1$. Thus, according to part (1) of Lemma \[lem:cuts\] and due to , we have $$\label{eq:diaginsbarrr}
y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{S}({\Gamma})}})})=y({\operatorname{bar}_{{i},{j}}})\,.$$ Since we preferred diagonal arcs from the residual network in our backwards construction of $\Gamma$, we find that all arcs from ${\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{S}({\Gamma})}})}$ are saturated by the flow $y$. Therefore, for the shifted column $S={\operatorname{S}({\Gamma})}$, we have (using ) $$\label{eq:scdiaginsnodes}
x(S)=y({\operatorname{in}^{\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}\,({{\operatorname{S}({\Gamma})}})}=y({\operatorname{bar}_{{i},{j}}})\,.$$ Let $a\in{\operatorname{A}_{{p},{q}}}$ be the arc in $\Gamma$ entering $(i,j)$, and denote by $\Gamma'$ the path arising from $\Gamma$ by removing $a$.
If $a$ is diagonal, then the $(\ell,\ell)$-$(i-1,j-1)$-path $\Gamma'$ satisfies ${\operatorname{S}({\Gamma'})}=S$, and the shifted-column inequality $x({\operatorname{bar}_{{i},{j}}})\le x(S)$ (satisfied by $x$) establishes via .
If $a$ is vertical, by construction of $\Gamma$, we have $y_{{({i-1},{j-1})^{\,\begin{rotate}{45}\tiny{$\shortdownarrow$}\end{rotate}}}}=x_{ij}$. Furthermore, the $(\ell,\ell)$-$(i-1,j)$-path $\Gamma'$ satisfies ${\operatorname{S}({\Gamma'})}=S\setminus\{(i,j\})$. Thus using the shifted-column inequality $x({\operatorname{bar}_{{i},{j+1}}})\le x(S\setminus\{(i,j)\})$ one obtains from the inequality $$y({\operatorname{bar}_{{i},{j}}})=x(S)=x(S\setminus\{(i,j)\})+x_{ij}\ge x({\operatorname{bar}_{{i},{j+1}}})+x_{ij}=x({\operatorname{bar}_{{i},{j}}})\,.$$ Thus, is established also in this case, which finally proves Theorem \[thm:liftscipoly\].
Remarks {#sec:remarks}
=======
In our view, the extended formulations for the orbitopes ${\operatorname{O}^{\le}_{p,q}}$ and ${\operatorname{O}^{=}_{p,q}}$ presented in this paper once more demonstrate the power that lies in the concept of extended formulations. Not only do the extended formulations provide a very compact way of describing the orbitopes, but also do they allow to derive rather simple proofs of the fact that nonnegativity constraints, row-sum inequalities/equations, and SCIs suffice in order to linearly describe ${\operatorname{O}^{\le}_{p,q}}$ and ${\operatorname{O}^{=}_{p,q}}$.
To us it seems that these proofs better reveal the reason why SCIs are necessary and (basically) sufficient in these descriptions. The construction of the flow in the proof of Thm. \[thm:liftscipoly\] is quite natural. The rest of the proof (i.e., the backwards construction of the path $\Gamma$) one may also have done without knowing the SCIs in advance. Thus, knowing the extended formulation, one possibly could also have *detected* SCIs on the way trying to do this proof.
An interesting practical question is whether the very sparse and compact extended formulations for orbitopes lead to performance gains in branch-and-cut algorithms compared to versions that dynamically add SCIs via the linear time separation algorithm (described in [@KP08]).
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank Laura Sanità for useful discussions and Marc Pfetsch for valuable comments on an earlier version of this paper.
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|
---
abstract: |
We present the outcomes of a [*Chandra X-ray Observatory*]{} snapshot study of five nearby Herbig Ae/Be (HAeBe) stars which are kinematically linked with the Oph-Sco-Cen Association (OSCA). Optical photometric and spectroscopic followup was conducted for the HD 104237 field. The principal result is the discovery of a compact group of pre-main sequence (PMS) stars associated with HD 104237 and its codistant, comoving B9 neighbor $\epsilon$ Chamaeleontis AB. We name the group after the most massive member. The group has five confirmed stellar systems ranging from spectral type B9–M5, including a remarkably high degree of multiplicity for HD 104237 itself. The HD 104237 system is at least a quintet with four low mass PMS companions in nonhierarchical orbits within a projected separation of 1500 AU of the HAeBe primary. Two of the low-mass members of the group are actively accreting classical T Tauri stars. The $Chandra$ observations also increase the census of companions for two of the other four HAeBe stars, HD 141569 and HD 150193, and identify several additional new members of the OSCA.
We discuss this work in light of several theoretical issues: the origin of X-rays from HAeBe stars; the uneventful dynamical history of the high-multiplicity HD 104237 system; and the origin of the $\epsilon$ Cha group and other OSCA outlying groups in the context of turbulent giant molecular clouds. Together with the similar $\eta$ Cha cluster, we paint a portrait of sparse stellar clusters dominated by intermediate-mass stars $5-10$ Myr after their formation.
author:
- 'Eric D. Feigelson, Warrick A. Lawson, Gordon P. Garmire'
title: 'The $\epsilon$ Chamaeleontis young stellar group and the characterization of sparse stellar clusters'
---
Introduction
============
While much star formation research has concentrated on the origins of rich stellar clusters and of isolated individual stars, it is likely that a significant fraction of stars also form in groups of $N \sim
10-100$ stars which represent an intermediate scale in the hierarchical structure of star formation regions [@Clarke00; @Elmegreen00; @Adams01; @Kroupa02]. If their stellar population is drawn from a standard initial mass function (IMF), these stellar groups will typically be dominated by a few $3-\-30$ M$_\odot$ stars. These groups may emerge from relatively small molecular clouds, such as found in the Taurus-Auriga cloud complex, or in parts of giant molecular clouds (GMCs) which also produce the rich OB associations. In the latter case, some plausibly will appear dispersed around OB associations, propelled by motions inherited from the natal turbulent molecular material [@Feigelson96].
Such young stellar groups are often difficult to find as they are often dynamically unbound, dispersing into the Galactic field within a few million years [@Bonnell99]. Study of small stellar groups is thus largely confined to the pre-main sequence (PMS) phase. Searches have been pursued in two ways. First, excess stellar densities in the neighborhoods of nearby Herbig Ae/Be (HAeBe) stars – intermediate-mass PMS stars readily identified by their infrared-luminous disks and active accretion – are sought using optical and near-infrared imagery [@Aspin94; @Hillenbrand95; @Testi97; @Testi99; @Weinberger00]. They are often surrounded by small groups of $5-40$ lower mass stars, as expected from the IMF.
Second, sparse stellar groups kinematically convergent with the nearest OB association, the Oph-Sco-Cen Association [OSCA; this nomenclature is adopted from @Blaauw91], have been identified [@Mamajek00; @Ortega02]. These include one compact group – the $\eta$ Cha cluster dominated by a B8 star and three A stars [@Mamajek99; @Lawson01a] – and two dispersed groups – the $\beta$ Pic moving group dominated by five A stars [@Zuckerman01] and the TW Hya Association with a single A star (HR 4796; Song, Bessell & Zuckerman 2002 and references therein). The locations of these groups in relation to the OSCA are illustrated in Figure \[lb.fig\]. Membership of OSCA outlier groups are principally based on a combination of kinematical criteria and elevated X-ray emission, which is a ubiquitous characteristic throughout PMS evolution from Class I protostars through post-T Tauri stars [@Feigelson99].
In addition to clustering on $0.1-10$ pc scales, intermediate-mass stars may be accompanied by bound companions on $\leq 1000$ AU scales. The stellar multiplicity of older low-mass stars is established to be 57:38:4:1 singles:binaries:triples:quartets for every 100 stars [@Duquennoy91]. A few higher multiplicity field systems are known such as Castor, a sextet of three close binaries dominated by two early A stars. The situation is more confusing for PMS systems. The crowded and rich Orion Nebula Cluster simultaneously shows very high multiplicities around its massive OB stars, main sequence levels of close lower-mass binaries, and a deficiency of wide binaries [@Preibisch99; @Simon99]. The distribution of binary separations may depend on the local density of neighbors even within a single cluster [@Brandner98]. In less crowded environments like the Taurus-Auriga clouds, binarity is nearly a factor of 2 higher among lower-mass PMS systems than in main sequence stars such that virtually all stars appear to be born in multiple systems [@Mathieu94]. Binary fractions among HAeBe stars are also elevated above main sequence levels and a few triple systems are known (§\[hd104\_bound.sec\]).
The study of multiplicity in young stars is of critical importance [@Zinnecker01]. It first addresses primordial conditions of star formation, such as the fragmentation of molecular material and redistribution of angular momentum during gravitational collapse [@Klessen01; @Boss02; @Larson02 and the review by Bodenheimer et al. 2000]. But it also reveals subsequent dynamical evolutionary effects, such as the dissipation of molecular material, close stellar encounters, stellar evaporation, and the survival of protoplanetary disks (reviews by Bonnell 2000 and Kroupa 2000).
We report here an effort that combines the two observatioal approaches outlined above. We seek young low-mass PMS stars with elevated X-ray emission around intermediate-mass PMS HAeBe stars that are kinematically associated with the OSCA. This is a small study based on only five X-ray snapshots designed to test the efficacy of the method. Despite the exploratory nature of our effort, one new young OSCA outlying stellar group is found associated with the closely spaced B/A stars $\epsilon$ Cha and HD 104237, including a remarkably high multiplicity of HD 104237 itself. Section 5 details our X-ray, optical/infrared photometric and spectroscopic characterization of the $\epsilon$ Cha group[^1]. We also find new companions within $\sim 200$ AU of two of the other observed HAeBe stars, HD 141569 and HD 150193 (§6). A variety of theoretical issues are discussed in §7.
Our strategy for locating young stellar systems \[targets.sec\]
===============================================================
Figure \[lb.fig\] shows $\simeq 1/4$ of the celestial sphere featuring stars associated with the OSCA. The dashed lines outline the boundaries of the rich Upper Sco (US), Upper Centaurus Lupus (UCL) and Lower Centaurus Crux (LCC) subgroups of the OSCA defined by @deZeeuw99 in their detailed study of [*Hipparcos*]{} motions. The brighter members, complete to $V \sim 7$ and consisting mainly of intermediate-mass BAF stars, identified by them are plotted as small dots. The three thick arrows show subgroup average proper motions during the next 1 Myr without correction for solar reflex motion. [*Hipparcos*]{} parallaxes of these bright members establish the subgroup distances to be 145 pc (US), 140 pc (UCL) and 118 pc (LCC).
Several samples of fainter OSCA members within the traditional boundaries have been constructed based on various combinations of kinematical, spectroscopic and X-ray selection criteria [@Preibisch02; @Mamajek02], but these constitute only a few percent of the full population of the subgroups, estimated to be $\simeq 6000$ stars with $M > 0.1$ M$_\odot$ assuming a standard initial mass function and some stellar evaporation [@deGeus92]. The late-type members of these samples are particularly valuable in establishing ages from their positions on the PMS tracks in the HR diagram. The ages for Oph, US, UCL and LCC OSCA subgroups are found to be $0-5$, 5, 16 and 17 Myr respectively. This progression of ages led @Blaauw64 [@Blaauw91] to suggest that a sequence of distinct star forming events occurred in the ancestral giant molecular cloud where later events are triggered by the shocks and ionization fronts produced by the OB stars of previous events [@Elmegreen77]. The ages of the outlying OSCA groups are similarly established to be $\sim 9$, $\sim 12$, and $\sim
10$ Myr for the $\eta$ Cha, $\beta$ Pic and TW Hya groups respectively [@Lawson01b; @Zuckerman01; @Webb99].
In this preliminary study, we chose five HAeBe stars with present-day space positions in or near the OSCA, and with space motions consistent with an origin in the SCA giant molecular cloud (Table \[targets.tab\]). The first five columns of the table give the star identifiers, [*Hipparcos*]{} distances, and estimated masses and ages. See table notes for details.
The ninth column of Table \[targets.tab\] indicates that two of the stars are closely associated with small molecular clouds. HD 100546 is likely associated with the bright-rimmed dark cloud DC296.2-7.9 located 0.4$^\circ$ away [@Vieira99; @Mizuno01]. HD 104237, and its comoving non-emission line B9Vn companion $\epsilon$ Cha 2 away, lie among several small molecular clumps with $M_{gas}<1$ M$_\odot$ within $\sim$10 [@Knee96]. Knee & Prusti plausibly argue that these are dissipating remnants of the molecular cloud from which these two intermediate-mass stars formed. Indeed, the entire Chamaeleon-Musca region south of the OSCA has many widely dispersed small molecular clouds and filaments, some closely associated with relatively isolated PMS stars [@Mizuno98; @Mizuno01]. These authors (see also Mamajek et al. 2000) discuss the possibility that this molecular material remains after the passage of the interstellar supershells attributed to the OB winds and supernova remnants of the OSCA [@deGeus92].
Figure \[lb.fig\] qualitatively shows that the five targets have $Hipparcos$ proper motions similar to those of the principal OSCA subgroups. Quantitative assessment of an origin in the same molecular cloud complex, which is now largely dispersed, requires accurate radial velocities which are generally not available. We adopt here the approximate method described by @Mamajek00 (see also Mamajek & Feigelson 2001) where we evaluate the closest approach in three dimensions between a star and OSCA subgroups, $D_{OSCA}$, assuming linear motion and arbitrary radial velocities. This calculation of closest approach includes correction for solar reflex motion, and is similar to the measurement of proximity of ‘spaghetti’ in 6-dimensional phase space described by @Hoogerwerf99 and used by @deZeeuw99 to establish OSCA memberships. The resulting $D_{OSCA}$ distances for the target stars to OSCA subgroup centers over the past 10 Myr are given in the last column of Table \[targets.tab\]. In the two cases where radial velocity measurements are available (HD 141569 and 152404, Barbier-Brossat & Figon 2000), our method correctly gives the closest approach to OSCA subgroups for the measured value compared to other hypothetical values.
From this measure of past proximity, available kinematical data for three of the target stars (HD 100546, 104237 and 150193) are fully consistent with OSCA subgroup membership ($D_{OSCA} < 10$ pc), while for two stars (HD 141569 and 152404) SCA membership is less certain ($D_{OSCA} \simeq 30$ pc). For the three stars lying within the SCA boundaries (Figure \[lb.fig\]), these results agree with those obtained by @deZeeuw99 who assign HD 100546 and 150193 as high probability OSCA members and HD 152404 as a lower probability member.
Finally, we note that for three targets $-$ HD 100546, HD 141569 and HD 150193 $-$ searches for close low-mass companions to the bright HAeBe primary have been made at optical or near-infrared bands (§\[others.sec\]). This provides us opportunity to compare the effectiveness of finding companions through X-ray activity [*versus*]{} photospheric emission.
$Chandra$ observations and analysis \[Chandra.sec\]
===================================================
Table \[chandra.tab\] gives the log of X-ray observations. We used the $16\arcmin \times 16\arcmin$ Advanced CCD Imaging Spectrometer Imaging (ACIS-I) array on board the [*Chandra X-ray Observatory*]{}. The satellite and instrument are described by @Weisskopf02. The first stages of data reduction are described in the Appendix of @Townsley03. Briefly, we start with the Level 1 events from the satellite telemetry, correct event energies for charge transfer inefficiency, and apply a variety of data selection operations such as ASCA event grades and energies in the range $0.5-8$ keV. A small (typically $\sim 1$) correction to the image boresight is made so the X-ray bright Herbig Ae/Be stellar position agrees with the $Hipparcos$ position.
Candidate sources were located using a wavelet-based detection algorithm [@Freeman02]. We applied a low threshold ($P = 1 \times
10^{-5}$) so that some spurious sources are found which we exclude later. The image was visually examined for additional sources such as close companions to the bright HAeBe star. Events for each candidate source were extracted using the [*acis\_extract*]{}[^2] procedures which take into account the position-dependent point spread function. Background is negligible for sources of interest in these short exposures and was not subtracted. Candidate sources with $<$3 extracted counts are now rejected.
The valid sources are cross-correlated with the USNO-B1.0 star catalog derived from all-sky Schmidt survey photographic plates [@Monet03]. Sources with counterparts brighter than 16.0 magnitude in the $B$, $V$ or $R$ band are considered to be prime candidate young stars. This criterion eliminates virtually all X-ray sources which are extragalactic. The X-ray sources with stellar counterparts are listed in Tables \[EpsCha.tab\] and \[others.tab\].
Table \[xray.tab\] provide results from subsequent analysis of the X-ray properties of the sources most likely associated with PMS stars. The following software packages were used: CIAO 2.3 and $acis\_extract$ for photon extraction, XRONOS 5.19 for variability, and XSPEC 11.2 for spectral modeling. $C_{extr}$ events were extracted in the $0.5-8$ keV band from a circular region of radius $R_{extr}$ (in arcsec). $f_{PSF}$ gives the fraction of a point spread function lying within that radius at the source’s location in the ACIS field[^3].
The distribution of photon energies were modeled as emission from a thermal plasma with energy $kT$ based on MEKAL emissivities [@Kaastra00]. For two of the stronger sources, a two-temperature plasma model was needed. For the weaker sources with $C_{extr} \leq
30$ counts, the derived $kT$ values are unreliable and are provided only to indicate how broadband luminosities were derived. With one exception (HD 104237 E), successful fits were found without intervening absorption by interstellar or circumstellar material. While the derived plasma energies are often imprecise, broad-band fluxes integrated over the best-fit model are insensitive to spectral fitting uncertainties and have roughly $1/\sqrt{C_{extr}}$ errors. X-ray luminosities, $L_s$ in the soft $0.5-2$ keV band and $L_t$ in the total $0.5-8$ keV band, are obtained from these fluxes by multiplying by $4
\pi d^2$ using the distances in Table \[targets.tab\] and dividing by $f_{PSF}$.
Variability information is limited by our short exposures. No source exhibited significant variations within an observation, as measured with a Kolmogorov-Smirnov one-sample test. Virtually all stellar sources observed in the two widely separate exposures of the $\epsilon$ Cha/HD 104237 field showed long-term variability. Table \[xray.tab\] gives these luminosities separately assuming no variations in spectral shape.
Optical observations \[opt.sec\]
================================
Color-magnitude photometric study of the $\epsilon$ Cha/HD 104237 field \[optphot.sec\]
---------------------------------------------------------------------------------------
Optical color-magnitude diagrams are a powerful tool aiding the discovery and characterization of PMS stellar populations [@Walter00; @Lawson01a]. For nearby compact, coeval and codistant groups, PMS stars form an isochrone that is elevated in magnitude above the vast majority of field stars owing to a combination of youth, proximity and (for groups dispersed from their parent molecular cloud) the absence of significant reddening. For X-ray discovered groups of PMS stars such as the $\eta$ Cha cluster [@Mamajek99] and the $\epsilon$ Cha group announced here, optical photometric study also permits an independent evaluation of completeness within the X-ray field (except for low-mass stars located very close to the bright A and B stars). X-ray-faint stars with similar photometric properties to X-ray selected cluster members can be identified and subsequently observed using spectroscopy for confirmation of membership; e.g. for the $\eta$ Cha cluster, we identified two X-ray faint late-type members residing within the $ROSAT$ High-Resolution Imager discovery field including the strongest disk source in the cluster [@Lawson02; @Lyo03a].
We made a map covering most of the [*Chandra*]{} $\epsilon$ Cha/HD 104237 field in the Cousins [*VI*]{} photometric bands using the 1.0-m telescope and SITe CCD detector at the Sutherland field station of the South African Astronomical Observatory (SAAO) during 2002 January. Some later [*VRI*]{} observations were made in 2002 April and 2003 April to complete coverage of the field. A total field of $\approx 300$ square arcmin centered on HD 104237 was surveyed under photometric conditions in $\approx$ 1.5 seeing, with the observations transformed to the standard system using observations of southern photometric standard stars. Exposure times in the [*VRI*]{} bands of 90 s, 60 s and 30 s, respectively, permits detection – for $\sim 10$ Myr-old PMS stars at $\sim 100$ pc distance – down to M6 spectral type cluster members with $V \approx 18$ and ($V-I$) $\approx
4.5$. For the central regions of the [*Chandra*]{} field, we also obtained exposures of shorter duration ($1-30$ s) to minimize saturation effects from the bright $\epsilon$ Cha and HD 104237 on photometry of several spatially proximate late-type stars. Finally, we obtained deeper exposures ([*VRI*]{} exposures of 900s, 600s and 300s) to characterize the optical counterparts of the faintest [*Chandra*]{} sources.
Figure \[CMdiag.fig\] shows the placement of X-ray selected stars from the [*Chandra*]{} field (Table \[EpsCha.tab\]) in the ($V-I$) $vs.$ $V$ color-mag diagram (open and filled circles) along with several hundred X-ray-faint field stars representative of background sources in the shallow [*VI*]{} survey. The symbols used for these and other stars reflect the outcomes of photometric and spectroscopic studies that we detail in the following sections. We compare the location of these stars to isochrones from the PMS evolutionary models of @Siess00 scaled to a distance of 114 pc; the mean of [*Hipparcos*]{} distances to $\epsilon$ Cha ($112 \pm 7$ pc) and HD 140237 ($116 \pm 8$ pc). The location of the [*Chandra*]{} stellar counterparts in the color-mag diagram is our first evidence for stellar youth in the X-ray bright stars located nearby $\epsilon$ Cha and HD 140237, with many of these sources (open and filled circles in Figure \[CMdiag.fig\]) located between the 2- and 20-Myr isochrones. This is consistent with an independently estimated age for HD 104237 itself, of around $2-3$ Myr (Table \[targets.tab\]).
From Figure \[CMdiag.fig\], we identify two groups of stars for follow-up spectroscopic characterization for signs of stellar youth such as Li 6707 absorption and enhanced optical activity: (i) the X-ray-selected stars; and (ii) field stars with photometry broadly consistent with that of the [*Chandra*]{} counterparts (defined as stars with $V$ mags within $\approx 1$ magnitude, for a given color, of the sequence of [*Chandra*]{} sources proximate to $\epsilon$ Cha and HD 104237). These latter stars are indicated by the open and filled squares in Figure \[CMdiag.fig\].
Spectroscopic confirmation \[DBSspec.sec\]
------------------------------------------
Optical spectroscopy of most of the [*Chandra*]{} stellar counterparts and optical photometric candidates were obtained during 2002 March and 2003 April using the 2.3-m telescope and dual-beam spectrograph (DBS) at Mount Stromlo & Siding Spring Observatories (MSSSO). Using the 1200 linemm$^{-1}$ grating in the red beam resulted in spectra with coverage from $6325-7240$ Å at a 2-pixel resolution of 1.1 Å, with the slit projecting 2 on the sky. For the late-type stars, exposure times of $600-3000$ s yielded continuum signal-to-noise ratios of $20-100$ near H$\alpha$. The spectra were reduced using dome flats, bias and Fe-Ar arc frames, making use of standard [IRAF]{} routines such as [ccdproc]{}.
Analysis of the spectra showed four of the [*Chandra*]{}-selected stars were active, lithium-rich late-type objects (Figure \[DBSspec.fig\]). An additional T Tauri star not detected by [*Chandra*]{} was identified from the list of photometric candidates. The two optically-bright [*Chandra*]{} sources in the field with proper motions discordant with $\epsilon$ Cha and HD 104237, CPD $-77^{\rm
o}$773 and CPD $-77^{\rm o}$775, lack detectable Li 6707 (equivalent widths $EW < 0.05$ Å), as do the several field stars observed with the DBS that were either detected by [*Chandra*]{} or have photometric properties similar to the $\epsilon$ Cha/HD104237 group stars. We did not obtain spectroscopy of three very faint [*Chandra*]{} counterparts following the analysis of optical colors showing them not to be late-type stars.
The resulting H$\alpha$ and Li-region normalized spectra for $\epsilon$ Cha, HD 104237 and the five T Tauri stars located within the [*Chandra*]{} field are shown in Figure \[DBSspec.fig\], with H$\alpha$ and Li $EW$’s listed in Table \[EpsCha.tab\]. Surprisingly, for HD 104237 itself we detect Li I 6707 and nearby lines such as Ca I 6718 that are indicative of a late-type star; we discuss these spectra further in §\[members.sec\].
Rotation and variability studies \[rot.sec\]
--------------------------------------------
For several the candidate stars identified by [*Chandra*]{} and confirmed to be PMS in our early ground-based studies, we obtained multi-epoch differential photometry using the 1.0-m telescope at SAAO during 2002 April and May. Observations were made in the Cousins [*VRI*]{} bands for the central regions of the HD 104237 field, and two distant fields containing late-type sources. For each field, $\sim 15$ epochs were obtained over 10 nights in $1.5-3$ conditions. The differential magnitudes for each of the T Tauri stars were determined with respect to $4-6$ local ‘standard stars’ within each CCD field which were found to remain constant to $< 0.01$ mag. A few of the nightly datasets were also transformed to the standard system via the observation of southern photometric standard stars. Results from this study are presented in \[members.sec\].
The $\epsilon$ Cha young stellar group \[epscha\_gp.sec\]
=========================================================
We describe here individual stars in the $\epsilon$ Cha field found by these X-ray and/or optical methods. The running star number, positions, names and optical properties are given in Table \[EpsCha.tab\]. The ‘ID Type’ in column 2 summarizes whether a given star is selected as a PMS group member by virtue of its X-ray emission, its location on the PMS photometric isochrone (Figure \[CMdiag.fig\]), the presence of H$\alpha$ or Li 6707 spectroscopic indicators (Figure \[DBSspec.fig\]), and/or [*Hipparcos*]{} astrometric association with $\epsilon$ Cha/HD 104237 and the OSCA. The ‘Class’ in the final column of Table \[EpsCha.tab\] gives our assessment distinguishing accreting classical T Tauri (CTT) stars from non-accreting stars weak-lined T Tauri (WTT) stars based on the strength and width of the H$\alpha$ emission line.
Figure \[hd104\_dss.fig\] shows the large-scale optical field, and Figure \[hd104\_Ximg.fig\] shows the X-ray sources in the immediate vicinity of $\epsilon$ Cha and HD 104237. X-ray spectra, variability and luminosities are given in Table \[xray.tab\]. We give rough mass estimates based on the correlation between X-ray luminosity and mass seen in the large sample of Orion Nebula Cluster stars (see Figure 4 of Feigelson et al. 2003). Stars probably associated with $\epsilon$ Cha and HD 104237 (confirmed members) and non-members are listed in separate sections of Table \[EpsCha.tab\] and are described below.
Confirmed members \[members.sec\]
---------------------------------
[**\#1: CXOU 115908.2-781232**]{} Located $\approx$ 2 WNW of HD 104237 A, red beam DBS spectroscopy shows the star is of M5 spectral type (with an uncertainty of $0.5-1$ subtype) with $EW = -6.2$ Å H$\alpha$ emission and very prominent Li 6707 absorption. Its photometric colors ($V-R$) = 1.42 and ($V-I$) = 3.16 are consistent with this spectral type without reddening, and its low X-ray luminosity around $\log L_t \simeq 28.5$ erg s$^{-1}$ is typical for M-type PMS stars. The soft X-ray spectrum with most of the counts below 1 keV is reminiscent of the ultrasoft spectrum of the TW Hya brown dwarf TWA 5B [@Tsuboi03]. Multi-epoch differential $VRI$ photometry obtained at SAAO during 2002 April was inconclusive; the $V$-band data was found to be variable at the 2$\sigma$ level compared to stars of similar brightness within the CCD field (1$\sigma = 0.015$ mag at $V \approx
17$).
[**\#2: $\epsilon$ Cha AB**]{} The star shows strong H$\alpha$ absorption ($EW = +13$ Å) and has photometry consistent with the established spectral type of B9V given in the SIMBAD database. The star is a cataloged binary consisting of components with visual magnitudes of 5.4 (A) and 6.1 (B) with separation variously reported between 0.45 and 1.9 [@ESA97; @Worley97; @Dommanget02]. Assuming $\epsilon$ Cha A is spectral type B9 without significant reddening, then $\epsilon$ Cha B is an early-A star (and we adopt here A1). The projected separation of $50-200$ AU implies a many century-long period. However, the star is a radial velocity variable on far shorter timescales, inferring a higher order of multiplicity. @Buscombe62 lists four velocities obtained over 120 days that vary between $-9$ and $+16$ kms$^{-1}$, and an older measurement of $+22$ kms$^{-1}$. As further evidence of multiplicity, HR diagram placement of $\epsilon$ Cha A (Figure \[HRdiag.fig\]) suggests the star is over-luminous compared to other group members (see §\[HRdiag.sec\]). The absence of any X-ray emission gives a very low limit of $\log L_t < 27.7$ erg s$^{-1}$ ($<
3$ ACIS counts). This strongly suggests the $\epsilon$ Cha system contains no late-type PMS companions above mid-M. This conclusion is supported by inspection of the DBS red spectrum; no late-type stellar features are evident in the H$\alpha$ region. With its over-luminosity and absence of late-type companion, we suggest that $\epsilon$ Cha A is an unresolved binary between two BA-type stars, and thus the $\epsilon$ Cha system in its entirety may have three BA-type stars.
[**\#3: HD 104237 C**]{} This and the following $Chandra$ source lie within 5 of HD 104237 A and could not be seen with the acquisition camera of the MSSSO 2.3-m in grey conditions in $\approx
2$ seeing. They are also not listed in either the USNO-B1.0 or 2MASS catalogs. We therefore provide here no optical information for these sources. Component C, lying 5 to the NW, was variable in X-rays: all 6 of its events appeared during the second epoch exposure. Its low X-ray luminosity is consistent with a PMS M-type star or brown dwarf.
[**\#4: HD 104237 B**]{} Component B lies in the wings on the W side of HD 104237 A’s point spread function. Its X-ray luminosity of $\log
L_t = 29.1$ erg s$^{-1}$ is typical of a late-K or early-M star. This star may be the origin of the K-type spectral features seen in HD 104237 A (see below). An additional X-ray component may also be present; a weak source to the North of components A and B residing within A’s point spread function. In this particular case, we can not be confident of its existence or properties and do not assign it a component letter. A longer $Chandra$ exposure giving sufficient signal for subarcsecond deconvolution (see, for example, the procedures in Tsuboi et al. 2003) might clearly resolve this close component.
[**\#5: HD 104237 A**]{} Spectroscopy of the HD 104237 primary shows H$\alpha$ in emission as expected for an HAeBe star, with $EW = -20$ Å. @vandenAncker97 finds $A_{V} = 0.71$ mag for the star. Given the lack of absorption in most of the other $\epsilon$ Cha members described here, we suggest this absorption arises in the immediate environment (e.g., inflow, outflow or disk) of this star. Surprisingly, we also detect cool star features indicative of a K-type T Tauri star: Li I 6707 and Ca I 6718 are clearly detected in the expanded spectrum of Figure \[DBSspec.fig\]. Assuming a typical Ca I equivalent width for K-type stars of $0.2-0.3$ Å, we find a similar $EW$ for the Li I 6707 line. The DBS spectrum was obtained along Position Angle (PA) = 182$^\circ$ with a slit projection on the sky of 2. It seems likely that we have detected component B (or possibly a closer component). We tentatively associate a ‘K:’ spectral type to component B for this reason in Table \[EpsCha.tab\].
[**\#6: HD 104237 D**]{} This star, 10 ESE of the HAeBe star, is also invisible in on-line Digital Sky Surveys, but was clearly distinguished with the acquisition cameras of the 1.0-m telescope at SAAO and the 2.3-m telescope at MSSSO. DBS spectroscopy indicates a spectral type of M3 with weak H$\alpha$ emission and strong Li I 6707 absorption. SAAO colors are ($V-R$) = 1.19 and ($V-I$) = 2.66 with uncertainties of $\pm 0.05$ mag because of the strongly variable background from HD 104237 A. The 2MASS $JHK$ photometry for this star carries confusion flags due to the proximity of HD 104237 A, and is therefore not listed in Table \[EpsCha.tab\]. Within the uncertainty in the spectral type and the photometry, reddening is negligible for HD 104237 D. Multi-epoch $VRI$ photometry from SAAO during 2002 April provides little information on the variability of the star; the nearby bright HAeBe star spoiled most of the photometric observations which typically suffered $\geq 2$ seeing. However, several epochs obtained in $< 2$ conditions suggested low photometric variability over the 10 day time interval of the observations. The X-ray emission also did not change between our two widely-spaced observations. Its luminosity of $\log L_t \simeq 29.3$ erg s$^{-1}$ is consistent with that expected from an early-M PMS star, and its spectrum (as with member \#1) appears unusually soft.
[**\#7: HD 104237 E**]{} The star is clearly resolved from HD 104237 A in POSS-2 scans of the field. Spectroscopy obtained with the MSSSO 2.3-m telescope and both the red ($R \approx 6000$ at H$\alpha$; see Figure \[DBSspec.fig\]) and blue beams ($R \approx 2000$ at 4500 Å) of the DBS spectrograph show the star is an early K-type star. We adopt here a spectral type of K3. The red beam spectrum shows strong Li I 6707 absorption with $EW = 0.5$ Å and weak but broad H$\alpha$ with strong self-absorption. The H$\alpha$ $EW = -4.5$ Å, and is $\approx -6$ Å if we ‘correct’ for the self-absorption signature. While the unabsorbed H$\alpha$ $EW$ is below the historical 10 Å separator WTT and CTT stars, it lies above the $EW > 3$ Å boundary suggested by @White03 for accreting K-type stars. The velocity width is $\approx 500$ kms$^{-1}$ at the level 10 percent above the surrounding continuum, strongly suggestive of on-going accretion from a circumstellar disk.
Photometry obtained at SAAO shows the star is highly variable on a 2.45 day period with [*VRI*]{} values at maximum light of $V = 12.08$, ($V-R$) = 0.83 and ($V-I$) = 1.80. Adopting intrinsic colors (for main sequence stars) from Kenyon & Hartmann (1995) and the extinction corrections of Bessell & Brett (1988), we find $A_{V} = 1.8 \pm 0.3$ mag. (We plot the measured colours of the star in Figure \[CMdiag.fig\], and the de-reddened luminosity in Figure \[HRdiag.fig\].) In this respect, HD 104237 E is distinct from all the other T Tauri stars associated with $\epsilon$ Cha and HD 104237 A which show little or no optical reddening, including the very nearby HD 104237 D. Similarly, the X-ray spectrum of HD 104237 E uniquely shows significant soft X-ray absorption (Table \[xray.tab\]). The best fit gives $\log N_H = 22.4$ cm$^{-2}$, equivalent to $A_V = 16$. The star was also highly variable in X-rays with an 8-fold difference between the first to second epochs. At its peak and corrected for absorption, its X-ray luminosity exceeded that of the HAeBe primary HD 104237 A.
The optical reddening and X-ray absorption for this star suggests that we are viewing the surface of HD 104237 E through local intervening material. While it could be associated with patchy interstellar cloud material (dust emitting weakly at 100$\mu$ is seen with $IRAS$ within 10 of HD 104237; Knee & Prusti 1996), we suspect it arises from material within the stellar system. The 2MASS photometry for this star has either upper limits or confusion flags, and is not listed in Table \[EpsCha.tab\]. A future study will detail the variability and nature of this star.
[**\#8: USNO-B 120144.7-781926**]{} The star was outside the original photometric field surveyed at SAAO during 2002 January, but was found during 2002 April in a field centered on nearby CXOU 120152.8-781840.9. The star resides near the edge of the [*Chandra*]{} field, but was undetected in X-rays, indicating log $L_t <
28.0$ ergs$^{-1}$. It was observed spectroscopically because it has photometric properties similar to several of the [*Chandra*]{}-selected late-type members. DBS spectroscopy obtained in 2003 April confirms the star is an active M5 spectral type PMS star with optical classical T Tauri star characteristics such as strong H$\alpha$ ($EW = -23$ Å) and He I emission at 6678 Å. Lithium is present as levels typical for T Tauri stars ($EW = 0.6$ Å). Analysis of the optical and 2MASS photometry suggests no significant $K$-band excess. Multi-epoch photometry obtained in 2002 April indicates the star is highly variable with amplitude of $\sim 0.2$ mag but with no detected periodicity probably due to undersampling of the light curve.
[**\#9: CXOU 120152.8-781840**]{} DBS spectroscopy shows the star is of M5 spectral type with moderately strong H$\alpha$ emission and Li I 6707 absorption. [*VRI*]{} photometry indicates ($V-R$) = 1.49 and ($V-I$) = 3.26, consistent with the spectral type estimate. Our multi-epoch photometric observations indicate no significant level of variability compared to field stars of similar magnitude. The X-ray emission was low, typical for PMS M stars, but probably variable with 14 of 21 events arriving in the first epoch.
Non-members \[nonmembers.sec\]
------------------------------
[**CPD -77$^{\rm o}$773**]{} The H$\alpha$ line is seen weakly in absorption and Li I 6707 is undetected, indicating this star is not PMS. Our DBS spectroscopy and SAAO photometric color ($V-I$ = 1.34) is consistent with the star being a early K-type giant. The SIMBAD lists a spectral type of K0.
[**CPD -77$^{\rm o}$775**]{} The H$\alpha$ line is strongly in absorption and Li I 6707 is undetected, again indicating the star is older than PMS. Our spectroscopy and color ($V-I$ = 0.49) indicates the star is a (possibly mildly reddened) early F-type dwarf, supporting the SIMBAD spectral type of F0.
[**CXOU 120118.2-780252**]{} Located near the North edge of the [*Chandra*]{} field, the star is a weak but highly variable X-ray source; all 17 of its photons arrived during the second [*Chandra*]{} snapshot. DBS spectroscopy shows the star is a late K-type star, with weak H$\alpha$ emission ($EW = -1.5$ Å) and no detection of lithium. For its spectral type, the star is $\sim 4$ mag too faint to be associated with the $\epsilon$ Cha PMS stellar group. If the star is main-sequence, it lies at a distance of $\sim 250$ pc.
[**Three faint [*Chandra*]{} counterparts**]{} Optical CCD study of the faint counterparts associated with CXOU 115942.2-781836, CXOU 120101.4-780618 and CXOU 120135.3-780427 shows none have photometric properties consistent with late-M stars or brown dwarfs associated with $\epsilon$ Cha and HD 104237. [*VRI*]{} photometry obtained for these objects at SAAO during 2003 April indicate colours consistent with early K-type dwarfs. None of these objects were observed with the DBS spectrograph. If main-sequence K stars, they must lie at distances of $2-4$ kpc inferring unrealistically high log $L_t > 30$ ergs$^{-1}$. If instead these objects are active galaxies, they are unresolved at the $\approx$ 1.5 resolution of the SAAO CCD images.
[**Three additional photometric candidates**]{} The final three stars in Table \[EpsCha.tab\] have $V$ magnitudes that, for their ($V-I$) colour, fell within $\approx 1$ mag (fainter) of the sequence of [*Chandra*]{}-detected late-type stars in the $\epsilon$ Cha/HD 104237 field. We observed them with the DBS spectrograph and found two early K and one early M star. None were active, lithium-rich objects; all are likely field giants.
An HR diagram for the $\epsilon$ Cha group \[HRdiag.sec\]
---------------------------------------------------------
Based on these photometric and spectroscopic characterizations of the PMS stars associated with $\epsilon$ Cha and HD 104237, we produce in Figure \[HRdiag.fig\] an HR diagram of group members. The spectral type-$T_{\rm eff}$ and bolometric correction sequences for main-sequence stars given by @Kenyon95 are used. Locations of the $\epsilon$ Cha group stars are compared to the PMS evolutionary grids of @Siess00 from which several isochrone and isomass lines are shown.
We find that the HAeBe star HD 104237 A and $\epsilon$ Cha B lie near the 3 Myr isochrone, and the two well-characterised companions to HD 104237 A, HD 104237 D and E, lie near the 5 Myr isochrone. Since [*Chandra*]{} imaging and DBS optical spectroscopy for HD 104237 A shows evidence for one (or maybe two) very nearby late-type companions, its position in the HR diagram is likely slightly elevated. It is also dependent on the quality of the reddening estimate ($A_V = 0.71$ mag) given by @vandenAncker97 which we have applied to HD 104237 A[^4].
In Figure \[HRdiag.fig\], we show $\epsilon$ Cha AB as an B9+A1 system with a 0.7 mag brightness difference. As noted in §5.1, $\epsilon$ Cha A appears to be elevated in luminosity (and therefore appears younger) compared to the isochronal locus of $\epsilon$ Cha B, HD 104237 A, HD 104237 D and HD104237 E. The discrepant position of $\epsilon$ Cha A may be grid dependent or, as discussed in §\[members.sec\], that it is a close binary itself with detected radial velocity variations. We thus suspect that $\epsilon$ Cha A consists of two components which fall near the $3-5$ Myr isochrones.
The three active, lithium-rich M5 stars (CXOU 115908.2-781232, USNO-B 120144.7-781926 and CXOU 120152.8-781840) appear systematically older than the stars of earlier spectral type. Their HR diagram placement implies an age of $\sim 10$ Myr. In considering the ages of these stars compared to other group members, we discuss two possibilities:
1. The star-grid comparison may be flawed for late-M stars. Here comparison with the $\sim 9$ Myr-old $\eta$ Cha cluster stars is valuable. @Lawson01b found the grids of @Siess00 best achieved coevality across the then-known $\eta$ Cha cluster population ranging from spectral types B8–M3. Since then, on-going study of the cluster stellar population has discovered several M4–M5 stars residing $< 20$ of the cluster center that appear older than earlier-type cluster members when compared to @Siess00 tracks [@Lyo03b]. In the $\eta$ Cha cluster, the age discrepancy appears rapidly for stars with ($V-I$) $> 3$; i.e. for stars later than $\approx $ M3 (or $T_{\rm eff} < 3400$ K). The same phenomenon appears in the late-type $\epsilon$ Cha stars. We suggest a variety of possible causes: a deficient temperature calibration, either in the models or in the application of the Kenyon & Hartmann (1995) main-sequence temperature calibration to PMS stars; an incorrect treatment of the stellar luminosities perhaps due to the large bolometric corrections required to transform the observations; or theoretical errors in the treatment of opacities in the M star models.
2. It is likely that the rich $\simeq 15$ Myr old OSCA subgroups have evaporated members into the Chamaeleon vicinity [@Blaauw91; @deGeus92]. If we make the hypothetical and optimistic assumption that half of subgroup members have dispersed into a halo across the region of the sky shown in Figure \[lb.fig\], then a typical [*Chandra*]{} ACIS observation will contain on average only $\sim 0.04$ OSCA PMS stars. This explanation is clearly inadequate to explain three M stars in the $\epsilon$ Cha/HD 104237 ACIS field. We conclude that these three stars are probably $\epsilon$ Cha group members and thus coeval with the higher mass members.
In summary, the $\epsilon$ Cha PMS stellar group currently consists of: $\epsilon$ Cha with two confirmed, and quite possibly a third, late-B/early-A stars within $\simeq 200$ AU; HD 104237 with an A stars and four confirmed (and possibly a fifth) late-type companions within $\simeq 1500$ AU; and three mid-M stars distributed over $\simeq 0.5$ pc. Our best estimate for the group age is $3-5$ Myr.
PMS stars around the other Herbig Ae/Be stars \[others.sec\]
============================================================
Figure \[chandra\_imgs.fig\] and Table \[xray.tab\] summarize the X-ray results within 1000 AU of the four other HAeBe targets, and Table \[others.tab\] gives X-ray sources likely associated with stars in the full ACIS fields. We have not made any optical study of these fields.
Table \[others.tab\] shows that three of the four HAeBe stars each have several likely stellar X-ray sources dispersed in the ACIS fields; i.e. with projected distances $>1000$ AU and $<0.3$ pc of the targeted HAeBe star. HD 141569 is the exception with no additional stellar X-ray sources. This is easily interpretable by reference to their global positions with respect to the OSCA shown in Figure \[lb.fig\]: HD 100546, HD 150193 and HD 152404 lie within the traditional boundaries of the rich OSCA subgroups while HD 141569 does not. These new X-ray stars are thus likely members of the OSCA, and we suspect they are not dynamically linked to the HAeBe stars.
Details on these proposed OSCA X-ray stars are given Table \[others.tab\] and its notes. They include: several unstudied late-type stars with $K \simeq 11$ around HD 100546 in LCC subgroup; the CTT star CXOU 163945.5-240202 = IRAS 16367-2356 in the Ophiuchi cloud complex; the late-type star CXOU 164031.3-234915 and its (probably) intermediate-mass companion also in the Ophiuchi cloud complex; CXOU 165430.7-364924 = HD 152368 (B9V) and a probable late-type companion in the UCL subgroup.
Stars in the immediate vicinity of the HAeBe targets are as follows. Recall that we have made $\simeq 1$ alignments of the $Chandra$ images assuming the brightest source is coincident with the primary, which may not always be correct.
[**HD 100546**]{} The $Chandra$ image shows a single source with modest emission around $2 \times 10^{29}$ ergs$^{-1}$ typical of PMS K stars [@Feigelson03]. The photon distribution appears slightly extended from the usual point spread function, but any multiplicity must lie within 1 of the primary[^5]. Several non-X-ray-emitting stars lies within 10 of the primary, but these are most likely background stars unrelated to the HAeBe star [@Grady01].
[**HD 141569**]{} This system is clearly resolved into two components separated by 1.5 along PA = 300$^\circ$ where the secondary to the NW is only slightly fainter than the primary[^6]. The projected separation is 150 AU. We call this component ‘D’ because two other companions ‘B’ and ‘C’, established to share the primary’s proper motion, have been found in optical images [@Weinberger00]. From their HR diagram locations, the estimated masses of components B and C are 0.45 M$_\odot$ and 0.22 M$_\odot$ respectively with age of 3 Myr. The high X-ray luminosity of HD 141569 D, $\log L_t \simeq 29.9$ erg s$^{-1}$, suggests a mass around $\sim 1$ $M_\odot$. The absence of components B and C from the [*Chandra*]{} image is not surprising, as a large fraction of the Orion Nebula M-type stars fall below the $\log L_x \simeq 28.0$ erg s$^{-1}$ sensitivity limit of the brief exposure available here [@Feigelson03].
[**HD 150193**]{} This source is also double with a component ‘C’ lying 1.5 from the primary along PA = 55$^\circ$. The projected separation is 220 AU. Although its X-ray emission is $5-10$ times fainter than that of the primary, the luminosity is still consistent with a $\simeq 1$ M$_\odot$ PMS star. The [*Chandra*]{} image does not show component ‘B’ (unknown spectral type) 1.1 to the SW of the primary reported from K-band imagery [@Pirzkal97].
[**HD 152404**]{} This source is weak and unresolved in the $Chandra$ image. It is a double-lined spectroscopic binary with period 13.6 days and eccentricity 0.47 [@Andersen89]. The spectral type is F5 IVe and the components have equal masses around 1.5 M$_\odot$.
Discussion and Concluding Remarks \[discussion.sec\]
====================================================
The multiplicity and X-ray emission of Herbig Ae/Be stars \[mult\_haebe.sec\]
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[*Chandra*]{} imagery is clearly a useful complement to high-resolution optical and near-infrared imagery and spectroscopy in the study of the multiplicity (i.e. companions within $\simeq 1000$ AU) of intermediate-mass PMS stars. In the X-ray band, the primary is not orders of magnitude brighter than the companions so that coronographic methods are not necessary. Combining the results of §$5-6$ with optical-infrared studies, we find a quintet (or possibly sextet) in HD 104237, a quartet in HD 141569, a triple in HD 150193, a double in HD 100546, and a single in HD 152404. Half of these companions were discovered in the [*Chandra*]{} images. If the primary’s X-ray emission arises from an unresolved lower mass companion (see below), then the multiplicity of each star is increased by at least one. $Chandra$ imagery is limited in two respects: it detects only a fraction of PMS M-type and brown dwarfs (though it should be nearly complete for higher mass stars if sensitivities reach $\log L_t \simeq
28.0$ erg s$^{-1}$; Feigelson et al. 2003); and it can not resolve companions closer than $\simeq 1$ from the primary.
The X-ray emission from intermediate-mass HAeBe stars, and AB stars in general, has been a long-standing puzzle as stars without outer convection zones should not have a magnetic dynamo of the type known in lower mass stars. Recent $Chandra$ images of nearby main sequence B stars have confirmed that, in at least 4 of 5 cases, that the emission arises from late-type companions [@Stelzer03]. For HAeBe stars, it has been debated whether the X-rays are from companions or are produced by the primary through star-disk magnetic interaction [@Zinnecker94; @Skinner96].
Our results do not clearly solve this puzzle. The five primary HAeBe stars observed here have X-ray luminosities in the range $29.1 < \log
L_x < 30.7$ erg s$^{-1}$ in the $0.5-8$ keV band with plasma energies in the range $0.4 < kT < 5$ keV. These properties are consistent with intermediate-mass Orion Nebula Cluster A- and B-type stars, most of which are probably not actively accreting, as well as solar-mass PMS stars [@Feigelson02]. Our $Chandra$ images show that some of the X-rays attributed to HAeBe stars from low resolution $ROSAT$ and $ASCA$ studies are produced by resolved stellar companions, but most of the emission still arises from within 1 of the primary. This could be either a close unresolved companion or the accreting primary itself. In the former case, the companion must have roughly $\geq 1$ M$_\odot$ because substantially lower mass stars are fainter with X-ray luminosites in the $\log L_x<28$ to $29$ erg s$^{-1}$ range [@Feigelson03]. In the latter case, the mechanism of HAeBe X-ray production must give X-ray properties essentially indistinguishable from those of solar-mass PMS stars.
HD 104237 as a bound high multiplicity stellar system \[hd104\_bound.sec\]
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At least a quintet, HD 104237 is the highest multiplicity HAeBe star known. The majority of HAeBe stars lie in binaries [@Leinert97; @Pirzkal97; @Corporon99] and a few are in triple systems (TY CrA, Casey et al. 1995; NX Pup, Brandner et al. 1995). Several quartet of lower-mass PMS stars have been found including GG Tau, UZ Tau, UX Tau, V773 Tau, HD 98800 and BD +26$^\circ$718B. It is very unlikely that any of the companions HD 104237 B-E seen in the $Chandra$ image appears projected so close to the primary by chance: $P \simeq 1$% for a randomly located PMS star or (for components B and C without optical spectroscopic confirmation) $P \simeq 0.1$% for a randomly located extragalactic X-ray source.
The system must be bound. If the stars were formed independently with the $\simeq 0.5$ km s$^{-1}$ velocity dispersion characteristic of small molecular clouds [@Efremov98], it would disperse within a few thousand years. Unlike many other multiple systems (like HD 98800, GG Tau and Castor), the HD 104237 components do not exhibit a hierarchical orbital structure of two or three close binary pairs. Due to the fragility of its orbits, we can infer that the HD 104237 system as a whole has not been ejected from some larger stellar aggregate but rather was born in a dynamically quiescent environment [@Kroupa98].
Perhaps of greatest interest, the HD 104237 quintet has apparently not suffered from serious internal dynamical instabilities during the $10^2-10^3$ orbits of its $3-5$ Myr lifetime. Instabilities leading to ejection of some members are thought to be common in multiple PMS systems (e.g. Sterzik & Durisen 1995, 1998; Reipurth 2000). In their dynamical calculations of stellar systems with realistic mass distributions, @Sterzik98 find that 98% of quintiple systems with an intermediate-mass primary will eject two or more members within 300 orbits. Also, the disks of at least two of its constituent stars – HD 104237 A and E – have not been destroyed as expected from close dynamical encounters [@Armitage97]. HD 104237 thus appears to be an unusually stable high multiplicity system, probably born under quiescent conditions in the low density environment of a small molecular cloud.
Large-scale environment of the $\epsilon$ Cha group \[large\_scale.sec\]
------------------------------------------------------------------------
On a $\sim 10^\circ$ scale, the interstellar environment is relatively free of molecular material between the Cha I and Cha II clouds, which have 1000 M$_\odot$ and 1900 M$_\odot$ of molecular gas and lie at distances of 160 pc and 180 pc respectively [@Mizuno01]. The $\epsilon$ Cha group lies in front of a dusty screen that covers the entire Chamaeleon/Musca region at a distance of 150 pc [@Franco91; @Knude98]. On a smaller (10) scale, three small clumps of CO and far-infrared emission are found between and west of $\epsilon$ Cha and HD 104237 [@Knee96]. These cloudlets are probably translucent with masses below 0.5 M$_\odot$.
Recent studies report young stars on large-scales that may be associated with the compact $\epsilon$ Cha group discussed here. @Sartori03 place $\epsilon$ Cha and HD 104237 on the near edge of a proposed new Chamaeleon OB association with 21 identified B- and A-type stars spread over $\sim 10-20^\circ$ ($50-100$ pc). This new grouping appears as a nearly continuous extension of the well-known US-UCL-LCC OSCA subgroups. @Mamajek03 criticizes this finding on the grounds that there is no overdensity of B stars in this region and the derived velocity dispersion is consistent with random field stars. @Blaauw91 had earlier suggested an extension of the LCC subgroup B stars into the Carina-Volans region next to Chamaeleon. From an extensive spectroscopic survey of later-type southern stars, @Quast03 define a stellar association called “$\epsilon$ Cha A” with at least 15 K-type members spanning $10-15^\circ$ in Chamaeleon. These stars are Li-rich with ages around 10 Myr. Here again, it is difficult to distinguish between members of the proposed new grouping and an extended or evaporating LCC subgroup. @Frink98 previously reported 7 comoving T Tauri stars over a subregion of this association around $\epsilon$ Cha, but with different kinematic properties.
We do not derive a clear view of the large-scale young stellar environment of $\epsilon$ Cha from these confusing reports. Many young stars are present, but it is difficult or impossible to distinguish distinct clusters from the profusion of outlying and evaporated stars likely to surround the rich Oph-US-UCL-LCC OSCA concentrations. For the brighter B- and A-type stars, it is also difficult to distinguish $5-20$ Myr stars physically associated with the OSCA from somewhat older field stars unless late-type companions can be found and characterized.
Origin of the $\epsilon$ Cha group \[group\_origin.sec\]
--------------------------------------------------------
The link between the brightest members $\epsilon$ Cha and HD 104237 as comoving, likely coeval PMS stars has been repeatedly discussed in the past [@Hu91; @Knee96; @Shen99; @Mamajek00]. But there has been debate regarding their origin. Writing before [*Hipparcos*]{} parallactic measurement of 114 pc was available for the group, @Knee96 suggested the system lies around 140 pc away and the nearby interstellar cloudlets were part of the Cha II star forming region. Writing after the release of $Hipparcos$ measurements, @Eggen98 showed that the system is more likely a member of the Local Association (which includes the OSCA), though he does not list it as a OSCA member. We establish (Mamajek et al. 2000 and Table \[targets.tab\]) that the extrapolated motions of HD 104237 and $\epsilon$ Cha lie within 10 pc of (within measurement errors, consistent with exact coincidence with) the centroid of OSCA subgroups in the past $\sim 10$ Myr.
We believe that this kinematic link between the $\epsilon$ Cha group and the OSCA is reasonably convincing evidence that they originated in the same giant molecular cloud. The question then arises why HD 104237 is judged from its HR diagram location to have an age far younger than the nearest OSCA subgroup: based on isochrones in the HR diagram, we find an age of $3-5$ Myr for the $\epsilon$ Cha group (§\[HRdiag.sec\]) while the UCL subgroup has age of 17 Myr [@Mamajek02]. The same question can be raised about the $\eta$ Cha cluster which, with age no older than 9 Myr [@Lawson01a; @Lawson01b], is younger than the nearest OSCA subgroup, the LCC with age of 16 Myr.
Such age discrepancies can be explained within the dispersal scenario outlined by @Feigelson96. The scenario is based on the dispersion of different portions of a giant molecular cloud along velocity vectors established by turbulence processes. Some portions of the cloud complex form rich stellar clusters relatively early and dissipate their molecular material soon afterwards [@Kroupa00]. Other portions of the complex remain as gaseous clouds as they disperse, forming stars at different times and far from the OB-rich environments of the larger clusters. This corresponds to the supervirial cloudlet regime, where $Q = \| {\rm
Kinetic~energy/Gravitational~energy} \|$ $> 1$ and cloudlets fly apart without collisions, in the recent hydrodynamical study by @Gittins03. Thus relatively young sparse groups, like those around $\eta$ Cha and $\epsilon$ Cha, may be found in the vicinity of older clusters like the OSCA subgroups. Some of these dispersed groups may be compact (like the $\eta$ Cha and $\epsilon$ Cha groups) due to smaller local values of $Q$, while other groups with higher $Q$ may themselves appear widely dispersed (like the TW Hya Association and $\beta$ Pic moving group; see §\[targets.sec\] and Figure \[lb.fig\]). However, all of these systems would share space motions converging onto the same ancestral giant molecular cloud.
Comparison of the $\eta$ Cha and $\epsilon$ Cha groups \[comparison.sec\]
-------------------------------------------------------------------------
We have now made considerable progress in characterizing the populations of two nearby sparse PMS stellar clusters dominated by intermediate-mass stars and lying on the outskirts of a large and rich OB association:
1. The $\eta$ Cha cluster has three intermediate-mass systems: the B8 star $\eta$ Cha probably (due to its [*ROSAT*]{} X-ray detection) with a low mass companion; RS Cha, a A7+A8 hard binary with a likely lower mass companion; and the single A1 star HD 75505 [@Mamajek00]. These three systems have projected separations $\leq 0.25$ pc. They are accompanied by 14 late-type primaries of which several are binaries [@Lyo03b]. Disks are prevasive in the cluster: two of the intermediate-mass systems ($\eta$ Cha and HD 75505) have weak $L$-band excesses; $\simeq 4$ of the late-type members have both IR excesses and optical signatures of active accretion; and $\simeq 3$ additional late-type members have weak $L$-band excesses [@Lawson02; @Lyo03a].
2. The $\epsilon$ Cha group has two intermediate-mass systems: $\epsilon$ Cha with two or three $\sim$A0 stars, and HD 104237 with at least 4 lower-mass companions. Due to the absence of X-ray emission, we infer that $\epsilon$ Cha does not have any lower-mass companions above $M \simeq 0.3$ M$_\odot$. These high-multiplicity systems have a projected separation of 0.07 pc. The group also has three M5 stars distributed over 0.5 pc. Three members – HD 104237 A, HD 104237 E and USNO-B 120144.4-782936 – have CTT-type optical emission lines indicating active accretion. This group has not been surveyed in the $L$ band like $\eta$ Cha, so the apparent lower disk fraction may not be real.
These stellar systems are quite similar, and together paint a portrait of $N \sim 10-100$ member groups several million years after their formation. In each group, the total stellar mass is about $15-20$ M$_{\odot}$ and the size is $r \approx 0.5$ pc, giving an escape velocity of $\sim 0.5$ kms$^{-1}$. As this is about the expected velocity dispersion inherited from the molecular material (§\[hd104\_bound.sec\]), the outlying members of the groups could be unbound. It is likely that the original census was considerably higher and many members of the original clusters have already escaped into the stellar field. The survival of so many high-multiplicity systems and disks in these sparse groups implies that little or no close dynamical interactions have occurred among the stars.
[*Acknowledgements:*]{} We thank A-R. Lyo (UNSW@ADFA) for her very capable assistance with the optical spectroscopy, and L. Crause (Cape Town) for her expert reduction of the optical photometry. J. Skuljan (Canterbury) and E. Mamajek (Arizona) provided helpful assistance with kinematical calculations, and P. Broos and L. Townsley (Penn State) developed critical [*Chandra*]{} data analysis tools. E. Mamajek (Arizona), C. Torres (LNA Brazil) and an anonymous referee suggested helpful improvements to the manuscript. We thank the SAAO and MSSSO Telescope Allocation Committees for observing time, and EDF appreciates the University of New South Wales and Australian Defence Force Academy for hospitality during much of this work. We greatly benefitted from the SIMBAD, 2MASS and USNO databases. This study was supported by NASA contract NAS-8-38252 (GPG, PI) and UNSW@ADFA URSP, FRG and SRG research grants (WAL, PI).
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[ccrrcrcccr]{}
100546 & KR Mus & 11 33 25.44 & -70 11 41.2 & B9Vne & 103 & 2.5 &$\geq$10 & Yes & 2 \
104237 & DX Cha & 12 00 05.08 & -78 11 34.5 & A0Vpc & 116 & 2.5 & 2 & Yes & 9 \
141569 & ... & 15 49 57.75 & -03 55 16.4 & B9.5e & 99 && $5\pm3$ & No & 35 \
150193 & MWC 863& 16 40 17.92 & -23 53 45.2 & A1Ve & 150 & 2.3 & $>$2 & No & 5 \
152404 & AK Sco & 16 54 44.85 & -36 53 18.6 &F5 IVe & 145 &1.5+1.5&$\sim 6$ & No & 29 \
[rrc]{} 100546 & 4 Feb 2002 & 5.2\
104237 & 5 Jun 2001 & 3.0\
& 4 Feb 2002 & 2.8\
141569 & 23 Jun 2001 & 2.9\
150193 & 19 Aug 2001 & 2.9\
152404 & 19 Aug 2001 & 3.1\
[ccccclccrrrrrrcrrc]{}
\
1 &psx & 11 59 08.0 & -78 12 32.2 & 1 & CXOU 115908.2-781232 & +0.0 & +0.0 & 16.99 & 15.57 & 13.83 & 12.01 & 11.45 & 11.17 & M5 & $-6.2$ & +0.9 & WTT\
2 & a & 11 59 37.6 & -78 13 18.6 & 2 & $\epsilon$ Cha AB & & & 4.90 & & & 5.02 & 5.04 & 4.98 & B9Vn & +13 & & AB\
3 & x & 12 00 03.6 & -78 11 31.0 & 3 & HD 104237 C & & & & & & & & & & & &\
4 & x & 12 00 04.0 & -78 11 37.0 & 3 & HD 104237 B & & & & & & & & & K: & & &\
5 & asx & 12 00 05.1 & -78 11 34.6 & 2 & HD 104237 A & 0.0 & 0.0 & 6.59 & & & 5.81 & 5.25 & 4.59 & A0Vpe & $-20$ & & HAeBe\
6 & psx & 12 00 08.3 & -78 11 39.5 & 1 & HD 104237 D & +0.0 & +0.0 & 14.28 & 13.09 & 11.62 & & & & M3 & $-3.9$ & +0.6 & WTT\
7 & psx & 12 00 09.3 & -78 11 42.4 & 1 & HD 104237 E & +0.0 & +0.1 & 12.08 & 11.25 & 10.28 & & & & K2 & $-4.5$ & +0.5 & CTT\
8 & ps & 12 01 44.4 & -78 19 26.7 & 1 & USNO-B 120144.7-781926 & & & 17.18 & 15.61 & 13.72 & 11.68 & 11.12 & 10.78 & M5 & $-23$ & +0.6 & CTT\
9 & psx & 12 01 52.5 & -78 18 41.3 & 1 & CXOU 120152.8-781840 & -0.9 & +0.4 & 16.78 & 15.29 & 13.52 & 11.63 & 11.04 & 10.77 & M5 & $-7.8$ & +0.6 & WTT\
&&&&&&&&&&&&&&&&&\
\
& x & 11 59 48.1 & -78 11 45.0 & 2 & CPD $-77^{\circ}773$ & -0.1 & +0.0 & 8.87 & & 7.53 & 6.59 & 5.98 & 5.85 & K0 & +1.2 & $<$0.05 &\
& x & 12 00 49.5 & -78 09 57.2 & 2 & CPD $-77^{\circ}775$ & -0.2 & +0.2 & 9.62 & & 9.13 & 8.77 & 8.61 & 8.54 & F0 & +10 & $<$0.05 &\
& x & 12 01 18.1 & -78 02 52.2 & 1 & CXOU 120118.2-780252 & +0.6 & +1.4 & 15.58 & 14.62 & 13.80 & 12.53 & 11.84 & 11.61 & K7 & $-1.5$ & $<$0.05 &\
& x & 11 59 42.0 & -78 18 36.6 & 1 & CXOU 115942.2-781836 & +0.6 & +1.1 & 18.25 & 17.74 & 17.22 & 16.46 & 15.85 &$<$16.08 & & & &\
& x & 12 01 01.4 & -78 06 18.0 & 4 & CXOU 120101.4-780618 & +2.4 & $-1.9$ & 18.82 & 18.35 & 17.88 & & & & & & &\
& x & 12 01 35.3 & -78 04 27.6 & 4 & CXOU 120135.3-780427 & -0.3 & +1.1 & 19.61 & 19.07 & 18.42 & & & & & & &\
& p & 11 58 16.0 & -78 08 24.5 & 1 & USNO-B 115816.0-780824 & & & 12.00 & & 10.57 & 9.51 & 8.84 & 8.72 & K0 & +1.0 & $<$0.05 &\
& p & 11 58 40.3 & -78 12 29.2 & 1 & USNO-B 115840.3-781229 & & & 13.04 & & 11.37 & 10.19 & 9.47 & 9.24 & K2 & +1.1 & $<$0.05 &\
& p & 12 02 23.1 & -78 05 44.6 & 1 & USNO-B 120223.1-780544 & & & 15.38 & & 12.84 & 11.26 & 10.32 & 10.04 & M3 & +1.2 & $<$0.05 &\
[rrrrrrrrrrrrrc]{}
HD 100546 & 11 33 39.6 & -70 08 05.5 & +0.5 & -0.2 & 99 && 13.9 & 13.7 & 12.2 & 11.33 & 10.68 & 10.54 &\
& 11 33 42.6 & -70 21 09.4 & +0.6 & -0.4 & 82 && 13.6 & 12.9 & 11.8 & 11.27 & 10.86 & 10.78 & a\
& 11 33 43.7 & -70 16 25.9 & -0.1 & -1.4 & 17 && 15.7 & 14.7 & 14.0 & 12.68 & 12.09 & 11.93 &\
& 11 34 11.8 & -70 19 37.7 & +0.7 & -1.0 & 13 && 15.3 & 14.1 & 12.9 & 12.04 & 11.47 & 11.36 &\
& 11 34 27.4 & -70 13 21.5 & -0.3 & +0.2 & 21 && 13.6 & 13.3 & 12.9 & 11.74 & 11.38 & 11.26 &\
HD 141569 & & &&&&& &&&&&&\
HD 150193 & 16 39 45.5 & -24 02 02.7 & -1.0 & -1.3 & 111 && 18.0 & 14.0 & 14.3 & 10.17 & 8.66 & 7.63 & b\
& 16 40 31.3 & -23 49 15.1 & +2.3 & -1.4 & 27 && 17.7: & 15.6 & 13.7 &&&& c\
HD 152404 & 16 54 16.9 & -36 56 22.0 & -1.4 & +2.3 & 4 && 16.6 & 14.8 & 13.5 & 12.80 & 12.06 & 11.87 & d\
& 16 54 30.8 & -36 49 24.4 & +0.3 & +0.6 & 54 && 10.4 & 10.0 & 9.8 & 9.41 & 9.33 & 9.28 & e\
& 16 55 10.2 & -36 51 20.2 & +4.5 & -4.9 & 8 && 16.7 & 14.2 & 13.8 & 12.42 & 11.56 & 11.29 & f\
[rlcrcccrr]{}
\
1 & CXOU 115908.0 & 1 & 10 & 4.0 & 0.90 & 0.3 & 28.6 &\
& -781232 & 2 & 4 & & & & 28.2 &\
3 & HD 104237 C & 1 & 0 & 1.0 & 0.60 & 0.9 &$<$28.2 &\
& & 2 & 6 & & & & 28.5 &\
4 & HD 104237 B & 1 & 18 & 0.50 & 0.15 & 1 & 29.1 &\
& & 2 & 26 & & & & 29.2 &\
5 & HD 104237 A & 1 & 299 & 0.75 & 0.40 & 0.7/5.2 & 30.2 & 30.4\
& & 2 & 340 & & & & 30.2 & 30.5\
6 & HD 104237 D & 1 & 37 & 1.0 & 0.60 & 0.6 & 29.4 &\
& & 2 & 33 & & & & 29.3 &\
7 & HD 104237 E & 1 & 25 & 1.0 & 0.60 & 2.0 & 28.6 & 29.9\
& & 2 & 198 & & & & 29.5 & 30.8\
9 & CXOU 120152.5 & 1 & 14 & 7.5 & 0.75 & 1 & 28.5 &\
& -781841 & 2 & 7 & & & & 28.2 &\
&&&&&&&\
\
& HD 100546 & 1 & 59 & 2.5 & 0.75 & 2.5 & 29.2 & 29.4\
& HD 141569A & 1 & 161 & 1.0 & 0.60 & 0.7/4.5 & 29.9 & 30.1\
& HD 141569D & 1 & 49 & 0.5 & 0.15 & 1.0 & 29.9 &\
& HD 150193A & 1 & 152 & 1.0 & 0.60 & 5.0 & 30.2 & 30.7\
& HD 150193C & 1 & 13 & 0.5 & 0.15 & 0.8 & 29.6 &\
& HD 152404 & 1 & 17 & 2.5 & 0.75 & 0.4 & 29.1 &\
![The Oph-Sco-Cen Association (OSCA) and its comoving stellar groups: US, UCL and LCC rich subgroups (dots in dashed outline), $\rho$ Oph embedded cluster (filled triangle), TW Hya association (open squares), $\beta$ Pic moving group (crosses), $\eta$ Cha cluster (filled square), $\epsilon$ Cha group (with HD 104237) and the other four Herbig Ae/Be systems discussed here (5-pointed stars). Selected proper motion vectors show displacements over the next 1 Myr. \[lb.fig\]](f1.eps){width="100.00000%"}
![($V-I$) $vs.$ $V$ color-magnitude diagram for the $\epsilon$ Cha group. Large filled symbols are confirmed PMS stars resulting from optical spectroscopic study of [*Chandra*]{} detections (filled circles) and those resulting from ground-based photometric study (filled square). The large open symbols are [*Chandra*]{} and photometric candidates not found to be PMS. The small dots are representative background stars. The model isochrones (units of Myr) are from Siess, Dufour, & Forestini (2000). \[CMdiag.fig\]](f2.eps){width="100.00000%"}
![Normalised MSSSO 2.3-m/DBS spectra of $\epsilon$ Cha group members near H$\alpha$ and Li 6707. The spectrum for HD 104237 A is shown expanded by a factor of 5 to highlight the presence of cool star features; see §\[DBSspec.sec\] for details. \[DBSspec.fig\]](f3.eps){width="100.00000%"}
![A 26$\times$28 red POSS-2 all-sky survey image centered at $(\alpha, \delta) = (12^{\rm h}00^{\rm m}30^{\rm s},
-78^\circ12\arcmin00\arcsec)$ overlaid with the the two [*Chandra*]{} fields (Table \[chandra.tab\]). Members of the $\epsilon$ Cha group are identified; see Table \[EpsCha.tab\] for the locations and properties of these objects. The rectangle surrounding the HD 104237 A-E system delineates the region of the merged [*Chandra*]{} image shown at high spatial resolution in Figure \[hd104\_Ximg.fig\]. \[hd104\_dss.fig\]](f4.eps){width="100.00000%"}
![$Chandra$ ACIS image in the vicinity of HD 104237. The image is displayed with $0.25\arcsec \times 0.25\arcsec$ pixels, and the greyscale is logarithmic with the faintest level showing individual X-ray events. The circles show the extraction regions used in spectral analysis. \[hd104\_Ximg.fig\]](f5.eps){width="70.00000%"}
![An HR diagram for the $\epsilon$ Cha group members, identified by abbreviated names (see Table \[EpsCha.tab\] for details on the individual stars). The model isochrone (units of Myr) and isomass (units of M$_{\odot}$) lines are from Siess, Dufour, & Forestini (2000). \[HRdiag.fig\]](f6.eps){width="100.00000%"}
![$Chandra$ ACIS image in the vicinity of four Herbig Ae/Be objects. The image is displayed with $0.25\arcsec \times 0.25\arcsec$ pixels, and the greyscale is logarithmic with the faintest level showing individual X-ray events. The boxes are companions found at optical and infrared wavelengths. \[chandra\_imgs.fig\]](f7a.eps "fig:"){width="45.00000%"} ![$Chandra$ ACIS image in the vicinity of four Herbig Ae/Be objects. The image is displayed with $0.25\arcsec \times 0.25\arcsec$ pixels, and the greyscale is logarithmic with the faintest level showing individual X-ray events. The boxes are companions found at optical and infrared wavelengths. \[chandra\_imgs.fig\]](f7b.eps "fig:"){width="45.00000%"}
\
\[0.2in\]
![$Chandra$ ACIS image in the vicinity of four Herbig Ae/Be objects. The image is displayed with $0.25\arcsec \times 0.25\arcsec$ pixels, and the greyscale is logarithmic with the faintest level showing individual X-ray events. The boxes are companions found at optical and infrared wavelengths. \[chandra\_imgs.fig\]](f7c.eps "fig:"){width="45.00000%"} ![$Chandra$ ACIS image in the vicinity of four Herbig Ae/Be objects. The image is displayed with $0.25\arcsec \times 0.25\arcsec$ pixels, and the greyscale is logarithmic with the faintest level showing individual X-ray events. The boxes are companions found at optical and infrared wavelengths. \[chandra\_imgs.fig\]](f7d.eps "fig:"){width="45.00000%"}
[^1]: Following the practice of @Mamajek99 in naming the $\eta$ Chamaeleontis star cluster, we name the new $\epsilon$ Chamaeleontis group of PMS stars after the highest mass member. Components HD 104237 B-E are names in order of proximity to the primary HD 104237A.
[^2]: Description and code for [*acis\_extract*]{} are available at\
<http://www.astro.psu.edu/xray/docs/TARA/ae_users_guide.html>.
[^3]: These values are derived from the memo ‘An analysis of the ACIS-HRMA point response function’ by A. Ware and B. R. McNamara (1999) available at <http://cxc.harvard.edu/cal/Acis/Cal_prods/psf/Memo/abstract.html> and its associated data products. The subarcsecond on-axis values were derived from calibration run H-IAI-CR-1.001 and are not very certain because the point spread function under in-flight conditions may differ slightly from that seen during ground calibration.
[^4]: Our $3-5$ Myr estimate for $\epsilon$ Cha B and the HD 104237 A-E system using @Siess00 tracks can be compared to the 2 Myr estimate given by @vandenAncker97 for HD 104237A using @Palla93 tracks. Along with the observational considerations discussed above, the age difference is also likely model dependent. Comparing several sets of PMS evolutionary grids to the HR diagram location of members of the $\sim 9$ Myr-old $\eta$ Cha cluster, @Lawson01b found the more-recent @Palla99 models gave factor of $\sim 2$ younger inferred ages for early-type stars compared to the @Siess00 models.
[^5]: The disk of HD 100546 is seen in scattered light out to 4 and in millimeter emission out to $\sim 30$, extended in the SE-NW direction [@Clampin03; @Henning98].
[^6]: There is a hint of a third component 0.7 from the primary along P.A. 90$^\circ$ with around 10 photons, but it can not be clearly discriminated from the wings of the primary point spread function.
|
---
author:
- Bin Yang
- 'Damien Hutsem[é]{}kers'
- Yoshiharu Shinnaka
- Cyrielle Opitom
- Jean Manfroid
- 'Emmanu[ë]{}l Jehin'
- 'Karen J. Meech'
- 'Olivier R. Hainaut'
- 'Jacqueline V. Keane'
- 'Micha[ë]{}l Gillon'
title: 'Isotopic ratios in outbursting comet C/2015 ER61'
---
Introduction
============
Understanding how planetary systems form from protoplanetary disks remains one of the great challenges in astronomy. In our own solar system, a wealth of processes happened involving chemistry and dynamics at all scales before reaching its present state. Comets are among the most unaltered materials in the solar system, having preserved their primitive element abundances. Comets witnessed the formation processes of the solar system and retain chemical signatures of the solar nebula in which they formed. Among all the observable properties, isotopic abundances are key tracers for reconstructing the origin and evolution of cometary material. Isotopic fractionation is sensitive to nebular environmental conditions, such as temperature, density, radiation, and composition [@Bockelee-Morvan:2015]. However, because of the faint signatures of isotopologs, isotopic measurements are only obtained in a handful of bright comets.
Discovered by the Pan-STARRS1 telescope on Haleakala on 2015 March 14, comet C/2015 ER61 (PANSTARRS, hereafter ER61) is on a highly eccentric orbit ($e=0.9973$, $a=385$ au), suggesting its origin is in the inner Oort cloud [@Meech:2017]. At the time of discovery, ER61 appeared asteroidal with no visible coma detected around the nucleus and it was thought to be a potential so-called Manx comet, which is an observationally inactive or nearly inactive object that is coming in from the Oort cloud [@Meech:2016]. The significance of Manx objects is that they may represent volatile-poor material that formed in the early inner solar system and was subsequently ejected to the Oort cloud during the solar system formation [@Meech:2016]. The comet became visually active a few months after its initial discovery with the appearance of a faint coma, which was detected by the Gemini telescope in 2015 June when the comet was 7.7 au from the sun. [@Meech:2017].
![Pre- and post-outburst evolution of the brightness of comet C/2015 ER61. The stars are the $Af\rho$ values we measured using I-band images obtained with the TRAPPIST-South telescope. The $Af\rho$ value is a proxy for the dust production rate [@ahearn:1984]. The two red arrows indicate the dates (UT 2017 April 13 and 17) of the UVES observations, which were just one week after the outburst. The vertical dashed line indicates the time when the comet reached its perihelion.[]{data-label="lcv"}](ER61_dust_activity_n.pdf){width="9cm"}
On 2017 April 4, ER61 underwent a significant outburst, with a visual magnitude of 6.2, up from a pre-outburst brightness of 8.4 mag . Photometric observations with the 60 cm TRAPPIST-South telescope [@Jehin:2011] at La Silla on 2017 April 5 show that the gas production rates increased by a factor of 7 compared to the previous observations made on 2017 March 31 [@Jehin:2017], and the dust mass-loss rate increased at least by a factor of 4; see Figure 1.
The outburst of ER61 provides a rare opportunity to study interior materials, excavated and released to the coma, which have been well protected from surface alterations (i.e., space weathering and cometary activity). Most importantly, the enhanced brightness of ER61 has enabled measurements of isotopic ratios of several species in the coma of this comet via high-resolution spectroscopy. High resolution is needed for isotopic studies because the emission lines of isotopes are weak and they have to be distinguished from emission lines of other isotopes (e.g., NH$_2$ lines are blended with the C$_2$ Swan lines) and from the underlying dust-scattered solar continuum [@Manfroid:2009].
More than a month after the outburst, ER61 reached its perihelion on 2017 May 9.94 at 1.042 au from the Sun. Later in 2017 June, a fragment co-moving with the nucleus was detected by the 16-inch Tenagra robotic telescope (MPEC 2017-M09). This newly detected secondary is now officially named C/2015 ER61-B (PANSTARRS). The outburst that occurred in early 2017 April may be associated with this splitting event, but the cause of the fragmentation has not been investigated and is beyond the scope of this paper.
Observation and data analysis
=============================
Observations of comet ER61 were carried out in service mode with the Ultraviolet-Visual Echelle Spectrograph (UVES) mounted on the 8.2 m UT2 telescope of the European Southern Observatory. Using director’s discretionary time, the total 10,800s of science exposure was divided into two exposures of 6400s each on 2017 April 13 and 17, respectively. As shown in Figure 1, the outburst was short-lived. At the time of the UVES observations, the brightness of the comet was decreasing steadily, but was still much higher than the pre-outburst level. We used the atmospheric dispersion corrector and the UVES standard setting 346 + 580 with dichroic $\#$1 that covers roughly from 3000 to 3880 Å on the blue CCD and from 4760 to 6840 Å on the two red CCDs. We used a 0.5 $\times$ $10\farcs 0$ slit, providing a resolving power R $\approx$ 80,000.
The raw spectral data were reduced using the UVES Common Pipeline Library (CPL) data reduction pipeline [@Ballester:2000], modified to accurately merge individual orders into a two-dimensional spectrum. Subsequently, the echelle package of the IRAF software was used to calibrate the spectra and to extract one-dimensional spectra. In turn, the cosmic rays were removed and the comet spectra were rebinned and corrected for the velocity of the comet. Lastly, the continuum component, including the sunlight reflected by cometary dust grains and the telluric absorption features, was removed. The final comet spectrum contains the gas component only, meaning the emissions from photodissociated radicals. More details regarding data reduction procedures are described in [@Manfroid:2009] and [@Shinnaka:2016].
![UVES spectra (black line) compared to synthetic spectra of isotopic species: $^{12}$C$^{14}$N (green line), $^{12}$C$^{15}$N (red line), and $^{13}$C$^{14}$N (blue line). The synthetic spectra are computed with the adopted isotopic abundances. The lines of $^{12}$C$^{15}$N and $^{13}$C$^{14}$N are identified by the red ticks and the blue ticks, respectively, and a few R lines are indicated by their quantum number. []{data-label="cn"}](cn_er61_3.pdf){width="8.5cm"}
Results
=======
In Figure \[cn\], we present the combined UVES spectrum of comet ER61, taken at $r=$1.13 and $r=$1.11 au, shortly after the outbursting event. Only the R lines of the B-X (0, 0) band (i.e., shortward of 3875 Å) are used since the P lines of the three isotopologs of CN are strongly blended. The $^{12}$C/$^{13}$C and $^{14}$N/$^{15}$N ratios are obtained simultaneously by fitting the CN B-X (0, 0) band with theoretical models. More details of our models are described in [@Manfroid:2009]. Based on measurements of the CN band, we derived $^{12}$C/$^{13}$C=100 $\pm$ 15 and $^{14}$N/$^{15}$N=130 $\pm$ 15. The uncertainties of our measurements are not dominated by random errors, but come from the subtraction of the underlying continuum (i.e., the dust continuum plus the faint wings of the strong lines) and inaccuracies in the models we used. We estimated that they roughly correspond to 2$\sigma$ errors, beyond which reasonable fits of the isotopic species are not acceptable. We also attempted to measure $^{16}$O/$^{18}$O using the OH ultraviolet bands at 3063 Å (0, 0) and 3121 Å (1, 1). Some $^{18}$OH lines were marginally detected, but the signal-to-noise ratio of these lines is too low to derive a reliable $^{16}$O/$^{18}$O estimate.
In addition, we also derived the nitrogen isotopic ratio using NH$_2$ lines around 5700 Å. The advantage of using NH$_2$ over CN is that NH$_2$ is the dominant photodissociation product of NH$_3$, which is directly incorporated into the nucleus, whereas there are multiple possible parent molecules of CN besides HCN. The methodology and assumptions for measuring the $^{14}$N/$^{15}$N ratio of NH$_2$ are explained in [@Shinnaka:2016]. We derive the $^{14}$N/$^{15}$N ratio of NH$_2$ of 140 $\pm$ 28, which is consistent with the nitrogen isotopic ratio of CN within the uncertainties. The uncertainty of the NH$_2$ measurements consists of both random and systematic errors, where the random errors come from flux measurements of the isotopolog lines and the continuum subtraction and systematic errors come from theoretical assumptions. The two key assumptions are, first, the same transitions in $^{14}$NH$_2$ and $^{15}$NH$_2$ have consistent transition probabilities and, second, NH$_3$ and $^{15}$NH$_3$ have similar photodissociation rates to produce the two isotopologs.
Discussion
==========
ER61 developed a dust coma when it was near 8 au from the Sun, where the surface temperature is too cold for water ice to sublimate. This suggests that the coma was driven by more volatile species such as CO or CO$_2$. However, if these super volatiles were near the surface, the comet would have been active beyond 10 au, which is not the case for ER61. There are two possibilities to account for the distant activity: either these volatiles were buried beneath a refractory layer or they were trapped in amorphous water ice, which is thermodynamically unstable and converts exothermically to the crystalline form [@Prialnik:2004]. However, no near-infrared spectroscopy of ER61 is available and the physical state of water ice in ER61 is unknown. Although we cannot rule out the possibility that amorphous-to-crystalline transition of water ice is responsible for the distant activity of ER61, the analysis of the activity pattern of ER61 and thermal models indicates that CO or CO$_2$ on this comet were depleted down to depths of 6.9m and 0.4m, respectively [@Meech:2017]. Assuming water release is controlled only by insolation, @Meech:2017 have predicted that the comet would show relatively moderate activity when approaching the sun. This prediction is consistent with the constant gas production rates measured from 2.3 au to 1.3 au from the Sun by TRAPPIST-South telescope just before the outburst in 2017 March [@Jehin:2017]. The orbital characteristics of ER61 suggest that it is not a dynamically new comet. Given that it had entered the inner solar system before, the activity pattern of ER61 indicates that this comet has developed an aging surface that consists of an insulating porous dust layer. It has been previously shown that such a dust layer is efficient in protecting the subsurface materials from heat, solar wind, or minor impacts [@Schorghofer:2008; @Guilbert:2015]. The outburst in 2017 April, thus, is of particular importance because it enabled us to sample the well-preserved subsurface materials released into the coma via the outburst, which are otherwise hard to access.
![Comparison between the isotopic ratios of ER61 with those measured for other comets shows that ER61 is typical of other comets. Data are from [@Manfroid:2009], [@Bockelee-Morvan:2015], and references therein. The measurements of OCCs are shown as squares and JFCs as triangles. Isotopic ratios for C/1995 O1 (Hale-Bopp) are measured three times at different heliocentric distances. The terrestrial isotopic ratios are from [@Anders:1989]. []{data-label="C_2"}](ER61_comparison_n.pdf){width="9.0cm"}
Shown as the red star in Figure 4, our measurements of $^{14}$N/$^{15}$N and $^{12}$C/$^{13}$C in ER61 do not exhibit significant difference from those of other comets, even though the measured materials are expected to be pristine. Similarly, [@Bockelee-Morvan:2008] observed comet 17P/Holmes soon after its remarkable outburst in 2007 and measured $^{14}$N/$^{15}$N = 139 $\pm$ 26 in HCN, while [@Manfroid:2009] measured $^{14}$N/$^{15}$N = 165 $\pm$ 40 in CN, which are also unremarkable compared to other comets. Also, there is no evidence that the subsurface material released from comet 9P/Tempel 1, as a result of the Deep Impact event, was isotopically different from the surface material [@Jehin:2006]. The only outliers that have possibly different isotopic ratios are the fragments of the split nucleus of 73P/Schwassmann-Wachmann 3 (hereafter P/SW3). [@Shinnaka:2011] noted that the measured ortho-to-para abundance ratios (OPR) of 1.0 in NH$_3$ correspond to a high nuclear spin temperature. Other observations show that P/SW3 is strongly depleted in C$_2$ and NH$_2$ relative to CN and OH [@Schleicher:2011]. The high $^{14}$N/$^{15}$N ratio in CN and the extreme depletions consistently suggest that P/SW3 formed under relatively warm conditions. This is an interesting finding because P/SW3 is a Jupiter family comet (JFC) and should have formed at 30 - 50 au according to the classical picture [@Levison:1997].
Before the significance of the migration of the giant planets was recognized, JFCs and Oort cloud comets (OCCs) were thought to have formed in distinct regions, where JFCs formed beyond Neptune and OCCs formed between giant planets [@Dones:2004]. Except for P/SW3, we see little if any systematic difference between JFCs and OCCs, even when freshly exposed interior materials were measured. In terms of chemical composition, surveys of cometary volatiles have also shown no systematic difference between JFCs and OCCs, but that significant variations are present within each group [@mumma:2011; @AHearn:2012].
As growing observational evidence reveals that JFCs and OCCs formed within the same broad region, dynamical simulations, in the framework of planetary migration, also suggest that the Oort cloud and scattered disk are derived from a common parent population, i.e., the primordial trans-Neptunian disk [@Brasser:2013]. Recent measurements of HDO/H$_2$O in 67P [@Altwegg:2015] by the ROSINA mass spectrometer on board the Rosetta spacecraft, also suggest that JFCs and OCCs formed in largely overlapping regions where the giant planets are today [@AHearn:2017]. However, the proportions of comets formed at different heliocentric distances are not yet clear [@Dones:2015].
[@Furi:2015] studied the distribution of N isotopes in the solar system and found that it follows a rough trend with increased $^{15}$N abundances at larger radial distances from the Sun. Nitriles detected in cometary comae so far are consistently enriched in $^{15}$N by a factor of $\sim$3 relative to the protosolar nebula and a factor of 1.8 relative to the terrestrial planets [@Furi:2015]. The significant enrichment in $^{15}$N is unlikely to be due to primordial nucleosynthetic heterogeneities, but rather a result of isotope fractionation processes during the formation of the solar system [@Furi:2015]. Possible processes include ion-molecule reactions in the interstellar medium under cold temperature with sufficient density [@charnley:2002] or photodissociation of N$_2$ by UV light from the proto-Sun or nearby stars [@muskatel:2011].
Unlike nitrogen, the available carbon isotope ratios of comets, including the newly derived $^{12}$C/$^{13}$C ratio of ER61, are consistent with the solar system abundance ratio of 90 $\pm$ 10 [@Wyckoff:2000]. The in situ measurements of the CO$_2$ isotopologs in the coma of 67P/Churyumov-Gerasimenko, taken by the Double Focusing Mass Spectrometer (DFMS) of the ROSINA, find that $^{12}$C/$^{13}$C = 84 $\pm$ 4 [@Hassig:2017]. The DFMS measurements show a slight enrichment in $^{13}$C in terms of the $^{12}$C/$^{13}$C ratio compared to an average $^{12}$C/$^{13}$C ratio of 91.0 $\pm$ 3.6 [@Manfroid:2009; @Bockelee-Morvan:2015]. However, given the large uncertainties of other ground-based observations, this difference seen in the DFMS measurements of 67P is not statistically significant. Our observations and previous studies consistently suggest that comets have formed in the Sun’s protoplanetary disk and inherited the $^{12}$C/$^{13}$C ratio from the original protosolar nebula [@Hassig:2017] and the chemical fractionation did not significantly change the ratios after the molecules formed in the protosolar cloud [@Wyckoff:2000].
Conclusions
===========
We performed high-resolution spectroscopy on the outbursting comet C/2015 ER61 on 2017 April 13 and 17, approximately one week after the outburst event. Our main results are summarized as follows:
First, we derived $^{14}$N/$^{15}$N and $^{12}$C/$^{13}$C in CN to be 130 $\pm$ 15 and 100 $\pm$ 15, respectively. In addition, we derived $^{14}$N/$^{15}$N = 140 $\pm$ 28 in NH$_2$. Second, although it is likely that the fresh subsurface materials of ER61 were analyzed, both the nitrogen and carbon isotopic ratios of this comet do not deviate significantly from those of other comets. Finally, this work and previous studies consistently suggest that JFCs and OCCs originated in largely overlapping regions beyond proto-Neptune.
Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 299.C-5006(A). TRAPPIST-South is funded by the Belgian Fund for Scientific Research (Fond National de la Recherche Scientifique, FNRS) under the grant FRFC 2.5.594.09.F. D.H. and E.J. are Belgian FNRS Senior research Associates, and M.G. is FNRS research Associate. YS was supported by Grant-in-Aid for JSPS Fellows Grant No. 15J10864. KJM and JVK acknowledge support from NSF AST-1617015. The authors would like to thank the referee, Anita Cochran, for her thoughtful and constructive comments.
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abstract: 'The Loop Quantum Gravity (LQG) program is briefly reviewed and one of its main applications, namely the counting of black hole entropy within the framework is considered. In particular, recent results for Planck size black holes are reviewed. These results are consistent with an asymptotic linear relation (that fixes uniquely a free parameter of the theory) and a logarithmic correction with a coefficient equal to $-1/2$. The account is tailored as an introduction to the subject for non-experts.'
author:
- Alejandro Corichi
- 'Jacobo Díaz-Polo'
- 'Enrique Fernández-Borja'
title: 'Loop quantum gravity and Planck-size black hole entropy'
---
Introduction
============
[*Loop Quantum Gravity*]{} (LQG) has become in the past years a mayor player as a candidate for a quantum theory of gravity [@lqg; @AL:review]. On the one hand it has matured into a serious contender together with other approaches such as [*String/M Theory*]{}, but on the other is it not as well understood, neither properly credited as a real physical theory of quantum gravity. The purpose of this contribution is twofold. First, we would like to provide a starting point for those interested in learning the basics of the theory and to provide at the same time an introduction to one of its application, namely the treatment of black holes and its entropy counting.
Loop quantum gravity is based and two basic principles, namely the general principles of quantum theory and the lessons from general relativity: that physics is diffeomorphism invariant. This means that the field describing the gravitational interaction, and the geometry of spacetime is fully dynamical and interacting with the rest of the fields present; when one considers the description of the full gravity-matter system, this better be a background independent one. The fact that LQG is based in general principles of quantum mechanics means only that one is looking for a description based on the language of Quantum Mechanics: states are elements on a Hilbert space (well defined, one expects), observables will be Hermitian operators thereon, etc. This does not mean that one should use [*all*]{} that is already known about quantizing fields. Quite on the contrary, the tools needed to construct a background independent quantization (certainly not like the quantization we know), are rather new.
Another important feature of LQG is that it is the most serious attempt to perform a full [*non-perturbative*]{} quantization of the gravitational field by itself. It is an attempt to answer the following question: can we quantize the gravitational degrees of freedom without considering matter on the first place? Since LQG aims at being a physical theory, which means it better be falsifiable, one expects to answer that question unambiguously, whenever one has the theory fully developed. This is one of the main present challenges of the theory, namely to produce predictions that can be tested experimentally. On the other hand, one can consider situations in which the full knowledge of the quantum theory is not needed in order to describe a particular physical situation. This is precisely the case of black hole horizons and their entropy.
Black hole entropy is one of the most intriguing constructs of modern theoretical physics. On the one hand, it has a correspondence with the black hole horizon area through the laws of (classical) black hole mechanics [@BCH]. On the other hand it is assumed to have a quantum statistical origin given that the proper identification between entropy and area $S=A/4 \,\ell^2_p$ came only after an analysis of [*quantum*]{} fields on a fixed background [@BH]. One of the most crucial test that a candidate quantum theory of gravity must pass then is to provide a mechanism to account for the microscopic degrees of freedom of black holes. It is not unfair to say that at the moment we have only two candidates for quantum gravity that have offered such an explanation: string/brane theory [@vafa] and loop quantum gravity [@ABCK]. The LQG formalism uses as starting point isolated horizon (IH) boundary conditions at the classical level, where the interior of the BH is excluded from the region under consideration. In this sense, the description is somewhat effective, since part of the information about the interior is encoded in the boundary conditions at the horizon that in the quantum theory get promoted to a condition that horizon states must satisfy. There is also an important issue regarding this formalism. Loop quantum gravity possesses a one parameter family of inequivalent representations of the classical theory labelled by a real number $\gamma$, the so called Barbero-Immirzi (BI) parameter (it is the analogue of the $\theta$ ambiguity in QCD [@BI]). It turns out that the BH entropy calculation provides a linear relation between entropy and area for very large black holes (in Planck units) as, $$S=\lambda\,A(\gamma),$$ where the parameter $\lambda$ is independent of $\gamma$ and depends only in the counting. We have put the $\gamma$ dependence in the Area, since the parameter appears explicitly in the area spectrum. The strategy that has been followed within the LQG community is to regard the Bekenstein-Hawking entropy $S=A/4$ as a requirement that large black holes should satisfy. This fixes uniquely the value of $\gamma=\gamma_0$ once and for all, by looking at the asymptotic behavior, provided that one has the ‘correct counting’ that provides the right value for $\gamma_0$. The analytic counting has provided also an expression for the first correction term that turns out to be logarithmic [@meiss; @GM].
The purpose of the present article is to provide an introduction to the main ideas behind the loop quantum gravity program and to one of its main applications, namely the computation of black hole entropy. In this regard, we shall describe some recent results that have been performed for *Planck size* black holes and that complement in a precise way the analytical computations. In particular, as we shall show, even when the black holes considered are outside the original domain of applicability of the formalism, one can still learn from these considerations.
The structure of the paper is as follows. In Sec. \[sec:2\] we present some preliminaries, such as the standard geometrodynamical variables for canonical gravity, the passage to connections and triads and the choice of classical observables to be quantized. In Sec. \[sec:3\] we describe the loop quantum geometry formalism, including some relevant geometric operators. Sec. \[sec:4\] is devoted to the formalism of quantum isolated horizons. We recall the classical formalism and the basic steps to the quantization of the horizon theory. State counting and black hole entropy is the subject of Sec. \[sec:5\]. We and with a discussion in Sec. \[sec:6\].
Preliminaries {#sec:2}
=============
Geometrodynamics
----------------
The first step is to introduce the basic classical variables of the theory. Since the theory is described by a Hamiltonian formalism, this means that the 4-dim spacetime $M$ of the form $M=\Sigma\times\mathbb{R}$, where $\Sigma$ is a 3-dimensional manifold. The first thing to do is to start with the geometrodynamical phase space $\G_{\rm g}$ of Riemannian metrics $q_{ab}$ on $\Sigma$ and their canonical momenta $\tilde{\pi}^{ab}$ (related to the extrinsic curvature $K_{ab}$ of $\Sigma$ into $M$ by $\tilde{\pi}^{ab}=\sqrt{q}\,(K^{ab}-\frac{1}{2}q^{ab}\,K)$, with $q={\rm det}(q_{ab})$ and $K=q^{ab}K_{ab}$). Recall that they satisfy, {\^[ab]{}(x),q\_[cd]{}(y)}=2\^[a]{}\_[(c]{}\^b\_[d)]{} \^3(x,y);{q\_[ab]{}(x),q\_[cd]{}(y)}={\^[ab]{}(x),\^[cd]{}(y)}=0 General Relativity in these geometrodynamical variables is a theory with constraints, which means that the canonical variables $(q_{ab},\tilde{\pi}^{ab})$ do not take arbitrary values but must satisfy four constraints: \^b=D\_a(\^[ab]{})0 ,=0 The first set of constraints are known as the vector constraint and what they generate (its gauge orbit) are spatial diffeomorphisms on $\Sigma$. The other constraint, the scalar constraint (or super-Hamiltonian) generates “time reparametrizations". We start with 12 degrees of freedom, minus 4 constraints means that the constraint surface has 8 dimensions (per point) minus the four gauge orbits generated by the constraints giving the four phase space degrees of freedom, which corresponds to the two polarizations of the gravitational field.
Connection Dynamics
-------------------
In order to arrive at the connection formulation, we need first to enlarge the phase space $\G_{\rm g}$ by considering not metrics $q_{ab}$ but the co-triads $e_a^i$ that define the metric by, q\_[ab]{}=e\_a\^ie\_b\^j\_[ij]{} where $i,j=1,2,3$ are internal labels for the frames. These represent 9 variables instead of the 6 defining the metric $q_{ab}$, so we have introduced more variables, but at the same time a new symmetry in the theory, namely the $SO(3)$ rotations in the triads. Recall that a triad $e_a^i$ and a rotated triad $e^{\prime
i}_a(x)={U^i}_j(x)\,e^j_a(x)$ define the same metric $q_{ab}(x)$, with ${U^i}_j(x) \in SO(3)$ a local rotation. In order to account for the extra symmetry, there will be extra constraints (first class) that will get rid of the extra degrees of freedom introduced. Let us now introduce the densitized triad as follows: \^a\_i=\_[ijk]{}\^[abc]{}e\^j\_be\^k\_c where ${\eta}^{abc}$ is the naturally defined levi-civita density one antisymmetric object. Note that ${\tilde{E}}^a_i{\tilde{E}}^b_j\delta^{ij}=q\,q^{ab}$.
Let us now consider the canonical variables. It turns out that the canonical momenta to the densitized triad ${\tilde{E}}^a_i$ is closely related to the extrinsic curvature of the metric, K\^i\_a=\^[ij]{}\^b\_jK\_[ab]{} For details see [@Perez:2004hj]. Once one has enlarged the phase space from the pairs $(q_{ab},\tilde{\pi}^{ab})$ to $({\tilde{E}}^a_i,K_b^j)$, the next step is to perform the canonical transformation to go to the $Ashtekar-Barbero$ variables. First we need to introduce the so called [*spin connection*]{} $\G^i_a$, the one defined by the derivative operator that annihilates the triad $e_a^i$ (in complete analogy to the Christoffel symbol that defined the covariant derivative $D_a$ killing the metric). It can be inverted from the form, \_[\[a]{}e\^i\_[b\]]{} +[\^i]{}\_[jk]{}\^j\_a e\^k\_b=0 This can be seen as an extension of the covariant derivative to objects with mixed indices. The key to the definition of the new variables is to combine these two objects, namely the spin connection $\G$ with the object $K^i_a$ (a tensorial object), to produce a new connection \^A\_a\^i:=\^i\_a+K\^i\_a This is the [*Ashtekar-Barbero Connection*]{}. Similarly, the other conjugate variable will be the rescaled triad, \^\^a\_i=\^a\_i/Now, the pair $({}^{\gamma}\!A_a^i, {}^{\gamma}\!{\tilde{E}}^a_i)$ will coordinatize the new phase space $\G_\gamma$. We have emphasized the parameter $\g$ since this labels a one parameter family of different classically equivalent theories, one for each value of $\gamma$. The real and positive parameter $\g$ is known as the Barbero-Immirzi parameter [@barbero; @Immirzi]. In terms of these new variables, the canonical Poisson brackets are given by, { \^A\_a\^i(x), \^\^b\_j(y)}=\^b\_a\^i\_j\^3(x,y) . and, { \^A\_a\^i(x), \^A\_b\^j(y)}={\^\^a\_i(x), \^\^b\_j(y)}=0 The new constraint that arises because of the introduction of new degrees of freedom takes a very simple form, G\_i=[D]{}\_a\^a\_i0 that is, it has the structure of Gauss’ law in Yang-Mills theory. We have denoted by ${\cal D}$ the covariant defined by the connection ${}^{\gamma}\!A_a^i$, such that ${\cal
D}_a{\tilde{E}}^a_i=\partial_a{\tilde{E}}^a_i+{\epsilon_{ij}}^{k}\;{}^{\gamma}
\!A_a^j{\tilde{E}}^a_k$. The vector and scalar constraints now take the form, V\_a=F\^i\_[ab]{}\^b\_i-(1+\^2)K\^i\_aG\_i 0 where $F^i_{ab}=\partial_{a}{}^{\gamma}\!A_b^i-\partial_{b}{}^{\gamma}\!A_a^i+
{\epsilon^i}_{jk}\;{}^{\gamma}\!A_a^j\,{}^{\gamma}\!A_b^k$ is the curvature of the connection ${}^{\gamma}\!A_b^j$. The other constraint is, §=0 The next step is to consider the right choice of variables, now seen as functions of the phase space $\G_\g$ that are preferred for the non-perturbative quantization we are seeking. As we shall see, the guiding principle will be that the functions (defined by an appropriate choice of smearing functions) will be those that can be defined without the need of a background structure, i.e. a metric on $\Sigma$.
Holonomies and Fluxes
---------------------
Since the theory possesses these constraints, the strategy to be followed is to quantize first and then to impose the set of constraints as operators on a Hilbert space. This is known as the Kinematical Hilbert Space ${\cal H}_{\rm kin}$. One of the main achievements of LQG is that this space has been rigourously defined.
Let us start by considering the connection $A_a^i$. The most natural object one can construct from a connection is a holonomy $h_\a(A)$ along a loop $\a$. This is an element of the gauge group $G=SU(2)$ and is denoted by, h\_(A)=[P]{}( \_A\_as\^a ) The path-order exponential of the connection. Note that for notational simplicity we have omitted the ‘lie-algebra indices’. From the holonomy, it is immediate to construct a gauge invariant function by taking the trace arriving then at the Wilson loop $T[\a] := \frac{1}{2}\,
{\rm Tr}\,\, {\cal P}\,\exp\,(\oint_\a A_a\, \d s^a)$.
In recent years the emphasis has shifted from loops to consider instead closed graphs $\Upsilon$, that consist of $N$ edges $e_I$ ($I=1,2,\ldots, N$), and $M$ vertices $v_\mu$, with the restriction that there are no edges with ‘loose ends’. Given a graph $\Upsilon$, one can consider the parallel transport along the edges $e_I$, the end result is an element of the gauge group $g_I=h(e_I)\in G$ for each such edge. One can then think of the connection $A^i_a$ as a map from graphs to $N$-copies of the gauge group: $A^i_a: \Upsilon\rightarrow G^N$. Furthermore, one can think of ${\cal A}_{\Upsilon}$ as the configuration space for the graph $\Upsilon$, that is homeomorphic to $G^N$.
What we are doing at the moment is to construct relevant configuration functions. In particular, what we need is to consider generalizations of the Wilson loops $T[\a]$ defined previously, making use of the graphs and the space ${\cal A}_{\Upsilon}$. Every graph $\Upsilon$ can be decomposed into independent loops $\a_i$ and the corresponding Wilson loops $T[\a_i]$ are a particular example of functions defined over ${\cal A}_{\Upsilon}$. What we shall consider as a generalization of the Wilson loop are [*all*]{} possible functions defined over ${\cal A}_{\Upsilon}$. Thus, a function $c:G^N\rightarrow \mathbb{C}$ defines a [*cylindrical*]{} function $C_\Upsilon$ of the connection $A$ as, C\_:= c(h(e\_1),h(e\_2),…,h(e\_N)) By considering all possible functions $c$ and all possible embedded graphs $\Upsilon$, we generate the algebra of functions known as Cyl (it is closely related to the holonomy algebra, and it can be converted into a $C^*$-algebra $\overline{\rm Cyl}$, by suitable completion).
Let us now discuss why this choice of configuration functions is compatible with the basic guiding principles for the quantization we are building up, namely diffeomorphism invariance and background independence. Background independence is clear since there is no need for a background metric to define the holonomies. Diffeomorphism invariance is a bit more subtle. Clearly, when one applies a diffeomorphism $\phi:\Sigma\rightarrow \Sigma$, the holonomies transform in a covariant way \_\* h(e\_I)=h(\^[-1]{}e\_I) , that is, the diffeomorphism acts by moving the edge (or loop). How can we then end up with a diffeo-invariant quantum theory? The strategy in LQG is to look for a [*diffeomorphism invariant*]{} representation of the diffeo-covariant configuration functions. As we shall see later, this has indeed been possible and in a sense represents the present ‘success’ of the approach.
Let us now consider the functions depending of the momenta that will be fundamental in the (loop) quantization. The basic idea is again to look for functions that are defined in a background independent way, that are natural from the view point of the geometric character of the object (1-form, 2-form, etc), and that transform covariantly with respect to the gauge invariances of the theory. Just as the the connection $A_a^i$ can be identified with a one form that could be integrated along a one-dimensional object, one would like analyze the geometric character of the densitised triad ${\tilde{E}}$ in order to naturally define a smeared object. Recall that the momentum is a density-one vector field on $\Sigma$, ${\tilde{E}}^a_i$ with values in the dual of the lie-algebra $su(2)$. In terms of its tensorial character, it is naturally dual to a (lie-algebra valued) two form, E\_[abi]{}:= \_[abc]{}\^c\_i where ${\rlap{\lower2ex\hbox{$\,\tilde{}$}}\eta{}}_{abc}$ is the naturally defined Levi-Civita symbol. It is now obvious that the momenta is crying to be integrated over a two-surface $S$. It is now easy to define the objects E\[S,f\]:=\_S E\_[abi]{}f\^iS\^[ab]{} , where $f^i$ is a lie-algebra valued smearing function on $S$. This ‘Electric flux’ variable does not need a background metric to be defined, and it transforms again covariantly as was the case of the holonomies. The algebra generated by holonomies and flux variables is known as the [*Holonomy-Flux*]{} algebra ${\cal HF}$.
Perhaps the main reason why this Holonomy-Flux algebra ${\cal HF}$ is interesting, is the way in which the basic generators interact, when considering the classical (Poisson) lie-bracket. First, given that the configuration functions depend only on the connection and the connections Poisson-commute, one expects that $\{T[\a],T[\beta]\}=0$ for any loops $\a$ and $\beta$. The most interesting poisson bracket one is interested in is the one between a configuration and a momenta variable, {T\[\],E\[S,f\]}= \_f\^i(v)(,S|\_[v]{})[Tr]{} (\_i h()) \[poissonB\] where the sum is over the vertices $v$ and $\iota(\a,S|_{v})=\pm 1$ is something like the intersection number between the loops $\a$ and the surface $S$ at point $v$. The sum is over all intersection of the loop $\a$ and the surface $S$. The most important property of the Poisson Bracket is that it is completely topological. This has to be so if we want to have a fully background independent classical algebra for the quantization.
A remark is in order. The value of the constant $\iota|_{v}$ depends not only on the relative orientation of the tangent vector of the loop $\a$ with respect to the orientation of $\Sigma$ and $S$, but also on a further decomposition of the loop into edges, and whether they are ‘incoming’ or ‘outgoing’ to the vertex $v$. The end result is that is we have, for simple intersections, that the number $\iota|_{v}$ becomes insensitive to the ‘orientation’. This is different to the $U(1)$ case where the final result [*is*]{} the intersection number. For details see [@ACZ].
Let us now consider the slightly more involved case of a cylindrical function $C_\Upsilon$ that is defined over a graph $\Upsilon$ with edges $e_I$, intersecting the surface $S$ at points $p$. We have then, { C\_, E\[S,f\] } = \_[p]{} \_[I\_p]{} ([I\_p]{}) f\^i(p) X\^i\_[I\_p]{}c \[poissonbb\] where the sum is over the vertices $p$ of the graph that lie on the surface $S$, $I_p$ are the edges starting or finishing in $p$ and where $X^i_{I_P}\cdot c$ is the result of the action of the $i$-th left (resp. right) invariant vector field on the $I_p$-th copy of the group if the $I_p$-th edge is pointing away from (resp. towards) the surface $S$. Note the structure of the right hand side. The result is non-zero only if the graph $\Upsilon$ used in the definition of the configuration variable $C_\Upsilon$ intersects the surface $S$ used to smear the triad. If the two intersect, the contributions arise from the action of right/left invariant vector fields on the arguments of $c$ associated with the edges at the intersection.
Finally, the next bracket we should consider is between two momentum functions, namely $\{E[S,f],E[S^\prime,g]\}$. Just as in the case of holonomies, these functions depend only on one of the canonical variables, namely the triad ${\tilde{E}}$. One should then expect that their Poisson bracket vanishes. Surprisingly, this is [*not*]{} the case and one has to appropriately define the correct algebraic structure[^1].
We have arrived then to the basic variables that will be used in the quantization in order to arrive at LQG. They are given by, h(e\_I) and E\[S,f\], subject to the basic Poisson bracket relations given by Eqs. (\[poissonB\]) and (\[poissonbb\]). In the next section we shall take the Holonomy-Flux algebra ${\cal HF}$ as the starting point for the quantization.
Loop Quantum Geometry {#sec:3}
=====================
Let us now consider the particular representation that defines LQG. As we have discussed before, the basic observables are represented as operators acting on wave functions $\Psi_\Upsilon(\overline{A})\in{\cal H}_{\rm kin}$ as follows: (e\_I)()=(h(e\_I))(|[A]{}) and \[11\] \[S,f\]\_()=i { \_, \^2E\[S,f\] } = i8 \_[p]{} \_[I\_p]{} ([I\_p]{}) f\^i(p) X\^i\_[I\_p]{}where $\ell^2_{\rm P}=G\,\hbar$, the Planck area is giving us the scale of the theory (recall that the Immirzi parameter $\g$ does not appear in the basic Poisson bracket, and should therefore not play any role in the quantum representation). Here we have assumed that a cylindrical function $\Psi_\Upsilon$ on a graph $\Upsilon$ is an element of the Kinematical Hilbert space (which we haven’t defined yet!). This implies one of the most important assumption in the loop quantization prescription, namely, that objects such as holonomies and Wilson loops that are smeared in one dimension are well defined operators on the quantum theory[^2].
The basic idea for the construction of both the Hilbert space ${\cal H}_{\rm kin}$ (with its measure) and the quantum configuration space $\bar{\cal A}$, is to consider [*all*]{} possible graphs on $\Sigma$. For any given graph $\Upsilon$, we have a configuration space ${\cal A}_\Upsilon=(SU(2))^N$, which is $n$-copies of the (compact) gauge group $SU(2)$. Now, it turns out that there is a preferred (normalized) measure on any compact semi-simple Lie group that is left and right invariant. It is known as the Haar measure $\mu_{\rm H}$ on the group. We can thus endow ${\cal A}_\Upsilon$ with a measure $\mu_\Upsilon$ that is defined by using the Haar measure on all copies of the group. Given this measure on ${\cal A}_\Upsilon$, we can consider square integrable functions thereon and with them the graph-$\Upsilon$ Hilbert space ${\cal H}_\Upsilon$, which is of the form: \_=L\^2([A]{}\_,\_) If we were working with a unique, fixed graph $\Upsilon_0$ (which we are not), we would be in the case of a lattice gauge theory on an irregular lattice. The main difference between that situation and LQG is that, in the latter case, one is considering all graphs on $\Sigma$, and one has a family of configurations spaces $\{
{\cal A}_\Upsilon \,/ \Upsilon \,{\rm a\,graph\,in\,} \Sigma\}$, and a family of Hilbert spaces $\{ {\cal H}_\Upsilon \,/ \Upsilon
\,{\rm a\,graph\,in\,} \Sigma\}$.
The quantum configuration space $\overline{\cal A}$ is the configuration space for the “largest graph"; and similarly, the kinematical Hilbert space ${\cal H}_{\rm kin}$ is the largest space containing all Hilbert spaces in $\{ {\cal H}_\Upsilon \,/
\Upsilon \,{\rm a\,graph\,in\,} \Sigma\}$. The Ashtekar-Lewandowski measure $\mu_{\rm AL}$ on ${\cal H}_{\rm
kin}$ is then the measure whose projection to any ${\cal
A}_\Upsilon$ yields the corresponding Haar measure $\mu_\Upsilon$. The resulting Hilbert space can thus be written as $${\cal H}_{\rm kin}=L^2(\overline{\cal A},\d\mu_{\rm AL})$$ The cylindrical functions $\Psi_\Upsilon\in$ Cyl belong to the Hilbert space of the theory.
Let us then recall what is the structure of simple states in the theory. The vacuum or ‘ground state’ $|0\rangle$ is given by the unit function. One can then create excitations by acting via multiplication with holonomies or Wilson loops. The resulting state $|\a\rangle=\hat{T}[\a]\cdot|0\rangle$ is an excitation of the geometry but only along the one dimensional loop $\a$. Since the excitations are one dimensional, the geometry is sometimes said to be [*polymer like*]{}. In order to obtain a geometry that resembles a three dimensional continuum one needs a huge number of edges ($10^{68}$) and vertices.
A choice of basis: Spin Networks
--------------------------------
The purpose of this part is to provide a useful decomposition of the Hilbert space ${\cal H}_\Upsilon$, for all graphs. From our previous discussion we know that the Hilbert space ${\cal
H}_\Upsilon$ is the Cauchy completion of the space of cylinder functions on $\Upsilon$, ${\rm Cyl}_\Upsilon$ with respect to the norm induced by the Haar measure on the graph configuration space ${\cal A}_\Upsilon = (\SU(2))^N$. Thus, what we are looking for is a convenient basis for functions $F_\Upsilon(A)$ of the form F\_(A):= f(h(e\_1),h(e\_2),…,h(e\_N)) Let us for a moment consider just one edge, say $e_i$. What we need to do is to be able to decompose any function $F$ on $G$ (in this case we only have one copy of the group), in a suitable basis.
In the case of the group $G=SU(2)$, there is a decomposition of a function $f(g)$ of the group ($g\in G$). It reads, f(g)=\_jf\^[m m’]{}\_j\_[m m’]{}(g) where, f\^[m m’]{}\_j=\_G\_H \_[m m’]{}(g\^[-1]{})f(g) and is the equivalent of the Fourier component. The functions $\stackrel{j}{\Pi}_{m m'}\,(g)$ play the role of the Fourier basis. In this case these are unitary representation of the group, and the label $j$ labels the irreducible representations. In the $SU(2)$ case with the interpretation of spin, these represent the spin-$j$ representations of the group. In our case, we will continue to use that terminology (spin) even when the interpretation is somewhat different.
Given a cylindrical function $\Psi_\Upsilon[A]=\psi(h(e_1),h(e_2),\ldots,h(e_N))$, we can then write an expansion for it as, \_&=& (h(e\_1),h(e\_2),…,h(e\_N))\
& = & \_[j\_1j\_N]{}f\^[m\_1m\_N,n\_1n\_N]{}\_[j\_1j\_N]{} \^[j\_1]{}\_[m\_1n\_1]{}(h(e\_1))\^[j\_N]{}\_[m\_Nn\_N]{}(h(e\_N)), where $\phi^{j}_{mn}(g)=\sqrt{j(j+1)}\,\stackrel{j}{\Pi}_{m
n}(g)$ is the normalized function satisfying $$\int_G \d\mu_{\rm H}\,\overline{\phi^{j}_{mn}(g)}\;\phi^{j'}_{m'
n'}(g)= \delta_{j,j'} \delta_{m,m'}\delta_{n,n'}\, .$$ The expansion coefficients can be obtained by projecting the state $|\Psi_\Upsilon\rangle$, f\^[m\_1m\_N,n\_1n\_N]{}\_[j\_1j\_N]{} =\^[j\_1]{}\_[m\_1n\_1]{}\^[j\_N]{}\_[m\_Nn\_N]{}|\_This implies that the products of components of irreducible representations $\prod^{N}_{i=1}\phi^{j_i}_{m_i n_i}[h(e_i)]$ associated with the $N$ edges $e_I\in\Upsilon$, for all values of spins $j$ and for $-j\leq m,n\leq j$ and for any graph $\Upsilon$, is a complete orthonormal basis for ${\cal H}_{\rm kin}$. We can the write, \_=\_j\_[,j]{} where the Hilbert space ${\cal H}_{\a,j}$ for a single loop $\a$, and a label $j$ is the familiar $(2j+1)$ dimensional Hilbert space of a particle of ‘spin $j$’. For a complete treatment see [@Perez:2004hj].
In the case of geometry with group $SU(2)$, the graphs with labelling $j_I={\bf j}$ are known as spin networks. As the reader might have noticed, in the geometry case there are more labels than the spins for the edges. Normally these are associated to vertices and are known as intertwiners. This means that the Hilbert spaces ${\cal H}_{\Upsilon,{\bf j}}$ is finite dimensional. Its dimension being a measure of the extra freedom contained in the intertwiners. One could then introduce further labelling $\bf l$ for the graph, so we can decompose the Hilbert space as \_=\_j\_[,[**j**]{}]{}=\_[[**j**]{},[**l**]{}]{}\_[,[**j**]{},[**l**]{}]{} where now the spaces ${\cal H}_{\Upsilon,{\bf j},{\bf l}}$ are one-dimensional. For more details see [@ac:playa], [@Perez:2004hj] and [@AL:review]. With this convenient basis it is simple to consider geometrical operator. The most important one in the study of black holes is given by the flux and the area operators that we consider next.
Flux and Area operators
-----------------------
The operators $\hat{E}[S,f]$ corresponding to the electric flux observables, are in a sense the basic building blocks for constructing the quantum geometry. We have seen in Sec.\[sec:3\] the action of this operators on cylindrical functions, \[S,f\]\_()=-i { \_, \^2E\[S,f\] } = -i8 \_[p]{} \_[I\_p]{} ([I\_p]{}) f\^i(p) X\^i\_[I\_p]{}\[flux1\] Here the first sum is over the intersections of the surface $S$ with the graph $\Upsilon$, and the second sum is over all possible edges $I_p$ that have the vertex $v_p$ (in the intersection of $S$ and the graph) as initial of final point. In the simplest case of a loop $\a$, there are only simple intersections (meaning that there are two edges for each vertex), and in the simplest case of only one intersection between $S$ and $\a$ we have one term in the first sum and two terms in the second (due to the fact that the loop $\a$ is seen as having a vertex at the intersection point). In this simplified case we have \[S,f\]\_()= -i8\^2\_[P]{} f\^i(p) X\^i\_[I\_p]{} Note that the action of the operator is to ‘project’ the angular momentum in the direction given by $f^i$ (in the internal space associated with the Lie algebra). As we shall see, this operator is in a sense fundamental the fundamental entity for constructing (gauge invariant) geometrical operators. For this, let us rewrite the action of the flux operator (\[flux1\]), dividing the edges that are above the surface $S$, as ‘up’ edges, and those that lie under the surface as ‘down’ edges. \[S,f\]=8\_p f\^i(p)(\^[p]{}\_[i(u)]{}- \^[p]{}\_[i(d)]{})where the sum is over the vertices at the intersection of the graph and the surface, and where the ‘up’ operator $\hat{J}^{p}_{i(u)}=\hat{J}^{p,e_1}_{i}+
\hat{J}^{p,e_2}_{i}+\cdots+\hat{J}^{p,e_u}_{i}$ is the sum over all the up edges and the down operator $\hat{J}^{p}_{i(d)}$ is the corresponding one for the down edges.
The second simplest operator that can be constructed representing geometrical quantities of interest is the [*area operator*]{}, associated to surfaces $S$. The reason behind this is again the fact that the densitized triad is dual to a two form that is naturally integrated along a surface. The difference between the classical expression for the area and the flux variable is the fact that the area is a gauge invariant quantity. Let us first recall what the classical expression for the area function is, and then we will outline the regularization procedure to arrive at a well defined operator on the Hilbert space. The area $A[S]$ of a surface $S$ is given by $A[S]=\int_S\d^2 x\,\sqrt{h}$, where $h$ is the determinant of the induced metric $h_{ab}$ on $S$. When the surface $S$ can be parametrized by setting, say, $x^3=0$, then the expression for the area in terms of the densitized triad takes a simple form: A\[S\]=\_S\^2 x where $k^{ij}=\delta^{ij}$ is the Killing-Cartan metric on the Lie algebra, and $\gamma$ is the Barbero-Immirzi parameter (recall that the canonical conjugate to $A$ is ${}^\gamma\!\tilde{E}^a_i=\tilde{E}^a_i/\gamma$). Note that the functions is again smeared in two dimensions and that the quantity inside the square root is very much a square of the (local) flux. One expects from the experience with the flux operator, that the resulting operator will be a sum over the intersecting points $p$, so one should focus the attention to the vertex operator $$\Delta_{S,\Upsilon,p}=-\left[(\hat{J}^{p}_{i(u)}-
\hat{J}^{p}_{i(d)})(\hat{J}^{p}_{j(u)}-
\hat{J}^{p}_{j(d)})\right]k^{ij}$$ with this, the area operator takes the form, \[S\]=8\^2\_[P]{}\_p We can now combine both the form of the vertex operator with Gauss’ law $(\hat{J}^{p}_{i(u)}+ \hat{J}^{p}_{i(d)})\cdot \Psi=0$ to arrive at, |(\^[p]{}\_[i(u)]{} - \^[p]{}\_[i(d)]{})|\^2 = |2(\^[p]{}\_[i(u)]{})|\^2 where we are assuming that there are no tangential edges. The operator $\hat{J}^{p}_{i(u)}$ is an angular momentum operator, and therefore its square has eigenvalues equal to $j^u(j^u+1)$ where $j^u$ is the label for the total ‘up’ angular momentum. We can then write the form of the operator \[S\](,)=8\^2\_[P]{} \_[vV]{} (,) With these conventions, in the case of simple intersections between the graph $\Upsilon$ and the surface $S$, the area operator takes the well known form: $$\hat{A}[S]\cdot{\cal
N}(\Upsilon,\vec{j})=8\pi\,\gamma\,\ell^2_{\rm P} \sum_{v\in \,V}
\;\sqrt{j_v(j_v+1)}\cdot
{\cal N}(\Upsilon,\vec{j})$$ when acting on a [*spin network*]{} ${\cal N}(\gamma,\vec{j})$ defined over $\Upsilon$ and with labels $\vec{j}$ on the edges (we have not used a label for the intertwiners). As we shall se when we consider the quantum theory of isolated horizons, the two operators considered here will play an important role not only in the geometry of the horizon but in the entropy counting.
Quantum Isolated Horizons {#sec:4}
=========================
Let us focus on the sector of the theory consisting of space-times which admit a type I isolated horizon $\Delta$ with a fixed area $a_o$ as the internal boundary. Then $\Sigma$ is asymptotically flat and has an internal boundary $S$, topologically a 2-sphere, the intersection of $\Sigma$ with $\Delta$. The type I isolated horizon boundary conditions require that i) $\Delta$ be null, ii) Non-expanding, iii) The field equations be satisfied there and iv) the intrinsic geometry on $\Delta$ be left invariant by the null vecto $\ell^a$ generating $\Delta$. For details see [@Askris04].
Introduce on $S$ an internal, unit, radial vector field $r^i$ (i.e. any isomorphism from the unit 2-sphere in the Lie algebra of $\SU(2)$ to $S$). Then it turns out that *the intrinsic geometry of $S$ is completely determined by the pull-back $\ub{A}^ir_i =: W$ to $S$ of the (internal-radial component of the) connection $A^i$ on $\Sigma$* [@Askris04]. Furthermore, $W$ is in fact a spin-connection intrinsic to the 2-sphere $S$. Finally, the fact that $S$ is (the intersection of $\Sigma$ with) a type I isolated horizon is captured in a relation between the two canonically conjugate fields: \[bc1\] F:= W = - \^i r\_i .where $\underline{\Sigma}^i$ is the pull-back to $S$ of the 2-forms $\Sigma_{ab}^i=\eta_{abc}E^{i\,c}$ on $\Sigma$. (Throughout, $\=$ will stand for equality restricted to $\Delta$.) Thus, because of the isolated horizon boundary conditions, fields which would otherwise be independent are now related. In particular, the pull-backs to $S$ of the canonically conjugate fields $A^i,\, \Sigma^i$ are completely determined by the $U(1)$ connection $W$.
In absence of an internal boundary, the symplectic structure is given just by a volume integral [@Askris04]. In presence of the internal boundary under consideration, it now acquires a surface term [@ABCK]: \[sym1\] [****]{}(\_1, \_2) = , where $\delta \equiv (\delta A, \delta \Sigma)$ denotes tangent vectors to the phase space ${\bf \Gamma}$. Since $W$ is essentially the only ‘free data’ on the horizon, it is not surprising that the surface term of the symplectic structure is expressible entirely in terms of $W$. However, it is interesting that the new surface term is precisely the symplectic structure of a $\U(1)$-Chern Simons theory. *The symplectic structures of the Maxwell, Yang-Mills, scalar and dilatonic fields do not acquire surface terms and, because of minimal coupling, do not feature in the gravitational symplectic structure either.* Conceptually, this is an important point: this, in essence, is the reason why the black hole entropy depends just on the horizon area and not, in addition, on the matter charges [@ABCK].
One can systematically ‘quantize’ this sector of the phase space [@ABCK]. We can focus only on the gravitational field since the matter fields do not play a significant role. One begins with a Kinematic Hilbert space $\H = \H_{V}\otimes \H_{S}$ where $\H_V$ is the Hilbert space of states in the bulk as described before and $\H_S$ is the Hilbert space of surface states. Expression (\[sym1\]) of the symplectic structure implies that $\H_S$ should be the Hilbert space of states of a Chern-Simons theory on the punctured $S$, where the ‘level’, or the coupling constant, is given by: \[level1\] k = A pre-quantization consistency requirement is that $k$ be an integer [@ABCK].
Our next task is to encode in the quantum theory the fact that $\Delta$ is a type I horizon with area $a_o$. This is done by imposing the horizon boundary condition as an *operator equation*: \[qbc1\] (1) = - ( (r)1) , on admissible states $\Psi$ in $\H$. Now, a general solution to (\[qbc1\]) can be expanded out in a basis: $\Psi = \sum_n\,
\Psi_V^{(n)} \otimes \Psi_S^{(n)}$, where $\Psi_V^{(n)}$ is an eigenvector of the ‘triad operator’ $- ({2\pi\gamma}/{a_o})\,
(\hat{\underline{\Sigma}}\cdot r)(x)$ on $\H_V$ and $\Psi_S^{(n)}$ is an eigenvector of the ‘curvature operator’ $\hat{F}(x)$ on $\H_S$ *with same eigenvalues*. Thus, the theory is non-trivial only if a sufficiently large number of eigenvalues of the two operators coincide. Since the two operators act on entirely different Hilbert spaces and are introduced quite independently of one another, this is a *very* non-trivial requirement.
Now, in the bulk Hilbert space $\H_V$, the eigenvalues of the ‘triad operator’ are given by [@ACZ]: \[triadev1\] - () (8\^2 \_I m\_I \^3(x, p\_I) \_[ab]{}) , where $m_I$ are half integers and $\eta_{ab}$ is the natural, metric independent Levi-Civita density on $S$ and $p_I$ are points on $S$ at which the polymer excitations of the bulk geometry in the state $\Psi_V$ puncture $S$. A completely independent calculation [@ABCK], involving just the surface Hilbert space $\H_S$, yields the following eigenvalues of $\hat{F}(x)$: \[Fev1\] \_I n\_I \^3(x, p\_I) 2 \_I n\_I \^3(x, p\_I) where $n_I$ are integers modulo $k$. Thus, with the identification $-2m_I = n_I\, {\rm mod}\, k$, the two sets of eigenvalues match exactly. There is a further requirement or constraint that the numbers $m_I$ should satisfy, namely, \_I m\_I=0 \[46\] This constraint is sometimes referred to as the *projection constraint*, given that the ‘total projection of the angular momentum’ is zero. Note that in the Chern-Simons theory the eigenvalues of $F(x)$ are dictated by the ‘level’ $k$ and the isolated horizon boundary conditions tie it to the area parameter $a_o$ just in the way required to obtain a coherent description of the geometry of the quantum horizon.
In the classical theory, the parameter $a_{o}$ in the expression of the surface term of the symplectic structure (\[sym1\]) and in the boundary condition (\[bc1\]) is the horizon area. However in the *quantum theory*, $a_{o}$ has simply been a parameter so far; we have not tied it to the *physical area of the horizon*. Therefore, in the entropy calculation, to capture the intended physical situation, one constructs a suitable ‘micro-canonical’ ensemble. This leads to the last essential technical step.
Let us begin by recalling that, in quantum geometry, the area eigenvalues are given by, $$8\pi \gamma\lp^2\, \sum_I \sqrt{j_I(j_I +1)}\, .$$ We can therefore construct a micro-canonical ensemble by considering only that sub-space of the volume theory which, at the horizon, satisfies: \[micro1\] a\_[o]{} -8\^2 \_I a\_[o]{} + where $I$ ranges over the number of punctures, $j_I$ is the spin label associated with the puncture $p_I$ [@ABCK].[^3] Quantum Einstein’s equations can be imposed as follows. The implementation of the Gauss and the diffeomorphism constraints is the same as in [@ABCK]. The first says that the ‘total’ state in $\H$ be invariant under the $\SU(2)$ gauge rotations of triads and, as in [@ABCK], this condition is automatically met when the state satisfies the quantum boundary condition (\[qbc1\]). The second constraint says that two states are physically the same if they are related by a diffeomorphism. The detailed implementation of this condition is rather subtle because an extra structure is needed in the construction of the surface Hilbert space and the effect of diffeomorphisms on this structure has to be handled carefully [@ABCK]. However, the final result is rather simple: For surface states, what matters is only the number of punctures; their location is irrelevant. The last quantum constraint is the Hamiltonian one. In the classical theory, the constraint is differentiable on the phase space only if the lapse goes to zero on the boundary. Therefore, this constraint restricts only the volume states. However, there is an indirect restriction on surface states which arises as follows. Consider a set $(p_I,
j_I)$ with $I= 1,2,\ldots N$ consisting of $N$ punctures $p_I$ and half-integers $j_I$, real, satisfying (\[micro1\]). We will refer to this set as ‘surface data’. Suppose there exists a bulk state satisfying the Hamiltonian constraint which is compatible with this ‘surface data’. Then, we can find compatible surface states such that the resulting states in the total Hilbert space $\H$ lie in our ensemble. The space $S_{(p_I,j_I)}$ of these surface states is determined entirely by the surface data. In our state counting, we include the number ${\cal N}_{(p_I,j_I)}$ of these surface states, subject however, to the projection constraint that is purely intrinsic to the horizon.[^4]
Black Hole Entropy {#sec:5}
==================
In this section we shall deal with the issue of entropy counting. We have started with a type I isolated horizon of area $a_o$ (in vacuum this is the only multipole defining the horizon), and we have quantized the theory and arrived to a Hilbert space as described before. The question now is: How many microstates correspond to the given macrostate, defined uniquely by $a_o$?
Let us now pose the condition that the states in $S_{(p_I, j_I)}$ should satisfy:
- They belong to the *physical* Hilbert space on the surface $\H_S$.
- The condition (\[micro1\]) is satisfied.
- The quantum boundary condition (\[Fev1\]) is satisfied.
- The projection constraint (\[46\]) is satisfied.
In terms of a concrete counting the problem is posed a follows: We shall consider the lists $(p_I, j_I, m_I)$ corresponding to the allowed punctures, spins of the piercing edges, and ‘projected angular momentum’ labels, respectively.
The task is then to count these states and find ${\cal
N}_{(p_I,j_I, m_I)}$. The entropy will be then, S\_[BH]{}:= ([N]{}\_[(p\_I,j\_I, m\_I)]{}) This problem was systematically addressed in [@ABCK] in the approximation of *large* horizon area $a_o$. Unfortunately, the number of such states was underestimated in [@ABCK].[^5] In [@GM] the counting was completed and it was shown in detail that, for large black holes (in Planck units), the entropy behaves as: $$S_{\rm BH}=\frac{A}{4}-\frac{1}{2}\ln{A},$$ provided the Barbero-Immirzi parameter $\gamma$ is chosen to coincide with the value $\gamma_0$, that has to satisfy [@GM]: 1=\_I (2j\_I+1) . The solution to this transcendental equation is approximately $\gamma_0=0.27398\ldots$ (see [@GM; @CDF-3] for details).
Here we shall perform the counting in a different regime, namely for small black holes in the Planck regime [@CDF]. Thus we shall perform no approximations as in the previous results. Thus our results are complementary to the analytic treatments. On the other hand our counting will be exact, since the computer algorithm is designed to count all states allowed. Counting configurations for large values of the area (or mass) is extremely difficult for the simple reason that the number of states scales exponentially. Thus, for the computing power at our disposal, we have been able to compute states up to a value of area of about $a_o=550\; l^2_P$ (recall that the minimum area gap for a spin $1/2$ edge is about $a_{\rm min} \approx 6\,l^2_P$, so the number of punctures on the horizon is below 100). At this point the number of states exceeds $2.8\times 10^{58}$. In terms of Planck masses, the largest value we have calculated is $M=3.3 \,M_P$. When the projection constraint is introduced, the upper mass we can calculate is much smaller, given the computational complexity of implementing the condition. In this case, the maximum mass is about $1.7\,M_P$.
It is important to describe briefly what the program for counting does. What we are using is what it is known, within combinatorial problems, as a brute force algorithm. This is, we are simply asking the computer to perform all possible combinations of the labels we need to consider, attending to the distinguishability -indistinguishability criteria that are relevant [@ABCK; @CDF-3], and to select (count) only those that satisfy the conditions needed to be considered as permissible combinations, i.e., the area condition and the spin projection constraint. An algorithm of this kind has an important disadvantage: it is obviously not the most optimized way of counting and the running time increases rapidly as we go to little higher areas. This is currently the main limitation of our algorithm. But, on the other hand, this algorithm presents a very important advantage, and this is the reason why we are using it: its explicit counting guarantees us that, if the labels considered are correct, the result must be the right one, as no additional assumption or approximation is being made. It is also important to have a clear understanding that the algorithm does not rely on any particular analytical counting available. That is, the program counts states as specified in the original formalism [@ABCK]. The computer program has three inputs: i) the classical mass $M$ (or area $a_o=16\pi\,M^2$), ii) The value of $\gamma$ and iii) The size of the interval $\delta$.
![\[fig:1\] The entropy as a function of area is shown, where the projection constraint has not been imposed. The BI is taken as $\gamma=0.274$.](figure1)
Once these values are given, the algorithm computes the level of the horizon Chern Simons theory $k=\left[a_o/4\pi\gamma\right]$ and the maximum number of punctures possible $n_{\rm
max}=\left[a_o/4\pi\gamma\sqrt{3}\right]$ (where $[\cdot]$, indicates the principal integer value). At first sight we see that the two conditions we have to impose to permissible combinations act on different labels. The area condition acts over $j$’s and the spin projection constraint over $m$’s. This allows us to first perform combinations of $j$’s and select those satisfying the area condition. After that, we can perform combinations of $m$’s only for those combinations of $j$’s with the correct area, avoiding some unnecessary work. We could also be allowed to perform the counting without imposing the spin projection constraint, by simply counting combinations of $j$’s and including a multiplicity factor of $\prod_I (2 j_I + 1)$ for each one, accounting for all the possible combinations of $m$’s compatible with each combination of $j$’s. This would reduce considerably the running time of the program, as no counting over $m$’s has to be done, and will allow us to separate the effects of the spin projection constraint (that, as we will see, is the responsible of a logarithmic correction). It is very important to notice at this point that this separation of the counting is completely compatible with the distinguishability criteria.
The next step of the algorithm is to consider, in increasing order, all the possible number of punctures (from 1 to $n_{\rm
max}$) and in each case it considers all possible values of $j_I$. Given a configuration $(j_1,j_2,\ldots,j_n)$ ($n\leq n_{\rm
max}$), we ask whether the quantum area eigenvalue $A=\sum_I
8\pi\gamma\sqrt{{j_I}\left({j_I}+1\right)}$ lies within $[a_o-\delta ,a_o+\delta ]$. If it is not, then we go to the next configuration. If the answer is positive, then the labels $m$’s are considered as described before. That is, for each of them it is checked whether $\sum m_I=0$ is satisfied.
![\[fig:2\] Entropy [*vs*]{} Area with and without the projection constraint, with $\delta =0.5$.](Fig-2)
In Figure \[fig:1\], we have plotted the entropy, as $S=\ln(\#\,{\rm states})$ [*vs*]{} the area $a_o$, where we have counted all possible states without imposing the $\sum m_I$ constraint, and have chosen a $\delta=0.5$. As it can be seen, the relation is amazingly linear, even for such small values of the area. When we fix the BI parameter to be $\gamma=\gamma_0=0.274$, and approximate the curve by a linear function, we find that the best fit is for a slope equal to $0.2502$.
![\[fig:3\] Entropy [*vs*]{} Area with and without the projection constraint, with $\delta =2$.](Fig-3)
When we include the projection constraint, the computation becomes more involved and we are forced to consider a smaller range of values for the area of the black holes. In Figure \[fig:2\], we plot both the entropy without the projection and with the projection, keeping the same $\delta $. The first thing to note is that for the computation with the constraint implemented, there are some large oscillations in the number of states. Fitting a straight line gives a slope of $0.237$. In order to reduce the oscillations, we increased the size of $\delta $ to $\delta =2$. The result is plotted in Figure \[fig:3\]. As can be seen the oscillations are much smaller, and the result of implementing the constraint is to shift the curve down (the slope is now 0.241). In order to compare it with the expected behavior of $S=A/4-(1/2)\ln{A}$, we subtracted both curves of Figure \[fig:3\], in the range $a_o=[50,160]$, and compared the difference with a logarithmic function. The coefficient that gave the best fit is equal to $-0.4831$ (See Figure \[fig:5\]). What can we conclude from this? While it is true that the rapidly oscillating function is far from the analytic curve, it is quite interesting that the oscillatory function follows a logarithmic curve with the “right" coefficient. It is still a challenge to understand the meaning of the oscillatory phase. Even when not conclusive by any means, we can say that the counting of states is consistent with a (n asymptotic) logarithmic correction with a coefficient equal to (-1/2).
![\[fig:5\] The curves of Fig. \[fig:3\] are subtracted and the difference, an oscillatory function, shown in the upper figure. The curve is approximated by a logarithm curve in the lower figure.](figure4c "fig:") ![\[fig:5\] The curves of Fig. \[fig:3\] are subtracted and the difference, an oscillatory function, shown in the upper figure. The curve is approximated by a logarithm curve in the lower figure.](figure4b "fig:")
Discussion {#sec:6}
==========
In this contribution we have considered the approach to the quantum theory of gravity known as loop quantum gravity. We have presented a brief introduction to the main ideas behind this approach and have considered one of its main applications, black hole entropy. We have discussed the main features in the approach to black hole entropy, in particular in the implementation of the isolated horizon boundary conditions to the quantum theory and how this conditions tell us what states can be regarded as ‘black hole states’ that contribute to the entropy of the horizon. As we have seen, the fact that there is an intrinsic discreteness in the quantum horizon theory and that we are ignoring (tracing out) the states in the bulk, is the reason why the entropy becomes finite. It is sometimes believed that the fact that we do get an entropy proportional to area is natural are not surprising, given that on the horizon, the theory under consideration is a Chern-Simons theory with punctures, and the entropy of a two dimensional theory should be proportional to the total volume (area in this case). It is important to stress that the result is not as trivial as it sounds. To begin with, we do [*not*]{} have a given Chern-Simons theory on the horizon, for any macro-state of a given area $a_o$, there are many possible microstates that can be associated with it. They do [*not*]{} all live on the same ‘Chern-Simons Hilbert space’. The surface Hilbert space $\H_S$ is made out of the tensor product of all possible Chern-Simons states compatible with the constraints detailed in Sec. ref[sec:5]{}, which belong to different Chern-Simons states (characterized by, say, the total number of punctures). That the total number of states compatible with the (externally imposed) constraints is proportional to area is thus a rather non-trivial result.
One might also wonder about the nature of the entropy one is associating to the horizon. there has been some controversy about the origin and location of the degrees of freedom responsible for black hole entropy (see for instance [@MRT] for a recent discussion). It has been argued that the degrees of freedom lie behind the horizon, on the horizon and even on an asymptotic region at infinity. What is then our viewpoint on this issue? The viewpoint is that the IH boundary conditions implement in a consistent manner an effective description, as horizon data, of the degrees of freedom that might have formed the horizon. These degrees of freedom, even if they are there in the physical space-time, they are inaccessible to an external observer. The only thing that this observer can ‘see’ are the degrees of freedom at the horizon, and these degrees of freedom are thus responsible for the entropy associated to the horizon.
In the last part of this article, we have focussed our attention on some recent results pertaining to the counting of states for Planck size horizons. As we have shown, even when these black holes lie outside the original domain of validity of the isolated horizon formalism (tailored for large black holes), the counting of such states has shed some light on such important issues as the BI parameter, responsible for the asymptotic behavior, and the first order, logarithmic, correction to entropy. We have also found, furthermore, that there is a rich structure underlying the area spectrum and the number of black hole states that could not have been anticipated by only looking at the large area limit. In particular, it has been found that the apparent periodicity in the entropy [*vs*]{} area relation yields an approximate ‘quantization´ of the entropy that makes contact with Bekenstein’s heuristic considerations [@CDF-2], and is independent on the choice of relevant states, and its associated counting. Details will be published elsewhere [@CDF-4].
Acknowledgments {#acknowledgments .unnumbered}
===============
AC would like to thank the organizers of the NEB XII International Conference in Nafplio, for the kind invitation to deliver a talk on which this contribution is based. This work was in part supported by CONACyT U47857-F, ESP2005-07714-C03-01 and FIS2005-02761 (MEC) grants, by NSF PHY04-56913, the Eberly Research Funds of Penn State and by the AMC-FUMEC exchange program. J.D. thanks MEC for a FPU fellowship.
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[^1]: The end result is that one should not regard $E[S,f]$ as phase space functions subject to the ordinary Poisson bracket relations, but rather should be viewed as arising from vector fields $X^\alpha$ on ${\cal A}$. The non-trivial bracket is then due to the non-commutative nature of the corresponding vector fields. This was shown in [@ACZ] where details can be found
[^2]: this has to be contrasted with the ordinary Fock representation where such objects do [*not*]{} give raise to well defined operators on Fock space. This implies that the loop quantum theory is qualitatively different from the standard quantization of gauge fields.
[^3]: The appearance of the parameter $\delta$ is standard in statistical mechanics. It has to be much smaller than the macroscopic parameters of the system but larger than level spacings in the spectrum of the operator under consideration.
[^4]: Note that there may be a large number –possibly infinite– of bulk states which are compatible with a given ‘surface data’ in this sense. This number does not matter because the bulk states are ‘traced out’ in calculating the entropy of the horizon. What matters for the entropy calculation is only the dimensionality of $S_{(p_I, j_I)}$.
[^5]: The under-counting was noticed in [@Dom:Lew], and a new counting was there proposed and carried out in [@meiss]. However the choice of relevant states to be counted there is slightly different from our case. Details of the comparison between two methods will be reported elsewhere [@CDF-3].
|
---
abstract: 'We consider collision of two particles near the horizon of a nonextremal static black hole. At least one of them is accelerated. We show that the energy in the center of mass can become unbounded in spite of the fact that a black hole is neither rotating nor electrically charged. In particular, this happens even for the Schwarzschild black hole. The key ingredient that makes it possible is the presence of acceleration. This acceleration can be caused by an external force in the case of particles or some engine in the case of a macroscopic body (“rocket”).'
author:
- 'O. B. Zaslavskii'
title: Schwarzschild black hole as accelerator of accelerated particles
---
Introduction
============
During last decade, much attention was focused on high energy collisions near black holes. It was stimulated by the observation that, under certain conditions, collisions of free falling particles near the horizon can lead to the unbounded energy $E_{c.m.}$ in the center of mass frame. This is the Bañados-Silk-West (BSW) effect [@ban] that occurs due to the existence of special fine-tuned (so-called critical) trajectories. For rotating black holes the corresponding relation between the energy at infinity $E$ and angular momentum $L$ reads $E=\omega _{H}L$, where $\omega $ is the metric coefficient responsible for rotation and proportional to the angular momentum of the black hole, subscript “H” refers to the horizon. The counterpart of the BSW effect can happen also near static but electrically charged black holes [@jl], with the difference that instead of geodesics, particles move under the action of the electrostatic field. Then, the critical trajectories are characterized by equality $E=\frac{qQ}{r_{+}}$, where $q$ and $Q$ are the charges of a particle and black hole, respectively, $r_{+}$ being the horizon radius. In other words, for the BSW effect to occur, a black hole should be rotating or/and charged. If one puts $\omega =0$, $Q=0$, this kills the effect in both cases. Correspondingly, the Schwarzschild black hole was unable to serve as particle accelerator to unbounded $E_{c.m.}$ (We put aside the case when colliding particles are spinning.). More precisely, if two particles of equal mass $m$ collide in the Schwarzschild background, $E_{c.m.}$ $\leq 2\sqrt{5}m$ [@baush].
It turns out, however, that modification of the set-up changes the situation drastically, if instead of geodesic motion, we consider motion under the action of some finite force. In contrast to motion of a particle in the Reissner-Nordström space-time, this force can have external source. We will see that for the motion under such a force the analogue of the BSW effect is possible even in the Schwarzschild background, i.e. for a nonrotating and electrically neutral black hole.
The role of a finite force was discussed earlier [@rad], [@ne] but just for rotating and/or charged black holes. It was implied in the aforementioned papers that the effect under discussion existed without this force. The question was whether or not the force spoils the effect. It was found that (under some additional weak restrictions on the behavior of the force) the BSW effect survives. Meanwhile, now the existence of such a force is not a potential obstacle but, by contrary, it is the only cause of the effect.
In what follows, we use the geometric system of units in which fundamental constants $G=c=1$.
Basic formulas
==============
Let us consider the black hole metric$$ds^{2}=-fdt^{2}+f^{-1}dt^{2}+r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})%
\text{,}$$where the horizon is located at $r=r_{+}$, so $f(r_{+})=0$. We will consider pure radial motion with the four-velocity $u^{\mu }$ and four-acceleration $%
a^{\mu }$ with $$a_{\mu }a^{\mu }\equiv a^{2}, \label{a2}$$where by definition $a\geq 0$. Then, it is convenient to use the expression for the acceleration in the simple form (see e.g. eqs. 16, 17 in [@sym] and [@pk]): $$a^{t}=f^{-1}u^{r}\frac{d}{dr}(fu^{t})\text{,} \label{at}$$$$a^{r}=fu^{t}\frac{d}{dr}(fu^{t})\text{.} \label{ar}$$In [@sym], [@pk], eqs. (\[at\]), (\[ar\]) were exploited for investigation of trajectories with the constant acceleration in the Schwarzschild space-time but they are valid in a more general case as well.
By substitution into (\[a2\]), one finds for a particle having the mass $m$:$$mu^{t}=\frac{X}{f}\text{,}$$$$mu^{r}=\sigma P\text{, }P=\sqrt{X^{2}-m^{2}f}\text{,} \label{ur}$$$\sigma =\pm 1$.$$X=m\beta (r)+E\text{.}$$Here, $E$ is a constant of integration, $$\beta =\delta \int^{r}dr^{\prime }a(r^{\prime })\text{,}$$$\delta =\pm 1$. The forward-in-time condition $u^{t}>0$ requires $X\geq 0$. It follows from (\[ar\]) that $$a^{r}=\frac{aX}{m}\delta \text{,}$$so $\delta =signa^{r}$.
If $a(r)$ tends to zero at infinity rapidly enough, it is convenient to choose the limit of integration in such a way that$$\beta =-\delta \int_{r}^{\infty }dr^{\prime }a(r^{\prime })\text{,}$$so $sign\beta =-\delta $. If $a=const$ we can choose $\beta =\delta ar$ and we return to the trajectory considered in [@pk]. Then, $sign\beta
=+\delta $. If, additionally, $f=1$, we obtain motion along the Rindler trajectory with $r=\frac{\cosh a\tau -\frac{E}{m}}{a}$, $t=\frac{\sinh a\tau
}{a},$ where we put $\delta =+1$ to have $r>0$ for large $\tau $. For the Reissner-Nordström (RN) case $a=\frac{\left\vert qQ\right\vert }{mr^{2}}$, where $q$ is the particle charge, $Q$ is that of a black hole. The BSW effect exists in the RN background if for trajectories with $E>0$ we take $%
\delta =+1$, $\beta <0$, $qQ>0$ [@jl]. Then, $$a^{r}>0, \label{pos}$$so we have repulsion between a particle and a black hole.
Let particles 1 and 2 move from infinity and collide in some point $r_{0}$. The energy in the center of mass frame$$E_{c.m.}^{2}=-(m_{1}u_{1}^{\mu }+m_{2}u_{2}^{\mu })(m_{1}u_{1\mu
}+m_{2}u_{2\mu })=m_{1}^{2}+m_{2}^{2}+2m_{1}m_{2}\gamma \text{,}$$where $\gamma =-u_{1\mu }u^{2\mu }$ is the Lorentz factor of relative motion. It follows from the above equations that$$\gamma =\frac{X_{1}X_{2}-\sigma _{1}\sigma _{2}P_{1}P_{2}}{m_{1}m_{2}f}.
\label{lor}$$
High energy collisions
======================
Now, the standard classification applies. A particle is called usual if $%
X_{H}>0$ if it is separated from zero and it is critical if $X_{H}=0$. This is possible if $\delta =+1$, so $\beta <0$ for a particle coming from infinity, when $E>0$. Then, for the critical particle$$E=\left\vert \beta (r_{+})\right\vert \text{.} \label{cr}$$Assuming at infinity $f=1$, we require $E>0$ where it has the meaning of the Killing energy. We also assume that both particles move towards the horizon, $\sigma _{1}=\sigma _{2}=-1$.
Near $r_{+}$, one has $X\approx X_{H}$ for a usual particle. Near the horizon, using the main term in the Taylor expansion, we have for the critical particle, $$X(r)\approx b(r-r_{+})\text{, }b>0.$$
Then, for collisions of the critical particle and a usual one we obtain that near the horizon the expression inside the square root in (\[ur\]) becomes negative. This is manifestation of the known fact that the critical particle cannot reach the horizon in the nonextremal case. At first, this property was considered as an obstacle again the BSW effect [@berti], [@ted]. However, the situation changes if instead of exactly critical particle, we consider a near-critical one with small but nonzero $X_{H}$. (For the first time, this idea was realized for the Kerr black hole in [@gp]). For such a particle $$X\approx X_{H}+b(r-r_{+})\text{.} \label{x}$$Let $X_{H}(r_{0})\approx d\sqrt{r_{0}-r_{+}}$, where $d$ is come constant.
Then, it is the first term in (\[x\]) which dominates$.$ For the metric function,$$f(r_{0})\approx 2\kappa (r_{0}-r_{+})\text{,}$$where $\kappa $ is the surface gravity. We assume that near-critical particle 1 collides with a usual particle 2. As a result, we find from ([lor]{}) that
$$\gamma \approx \frac{D}{\sqrt{(r_{0}-r_{+})}}\text{, }D=\frac{\left(
X_{2}\right) _{H}(d-\sqrt{d^{2}-2\kappa })}{2\kappa m_{2}}\text{,}
\label{ga}$$
where we also assumed that $d>\sqrt{2\kappa }$. Then, taking $r_{0}$ as close to $r_{+}$ as one likes, we obtain the unbounded growth of $\gamma $ and $E_{c.m.}^{2}$ Eq. (\[ga\]) can be thought of as a counterpart of eq. (19) from [@gp] where the Kerr metric was considered. Thus there is a close analogy between our case and the BSW effect near nonextremal black holes. In particular, now the same difficulties persist that forbid arrival of the near-extremal particle from infinity because of the potential barrier typical of any nonextremal black hole (for the Kerr metric, see Figure and accompanying discussion in [@gp42]). Therefore, either such a particle is supposed to be created already in the vicinity of the horizon from the very beginning or one is led to exploiting scenarios of multiple scattering [@gp].
Discussion and conclusions
==========================
What is especially interesting is that the effect under discussion is valid for the Schwarzschild black hole. It is also worth mentioning that for the extremal case, both terms inside the square root in (\[ur\]) have the same order and the result is similar to that for the RN black hole [@jl]. However, now the acceleration may be caused not by interaction between the particle charge and that of a black hole but by some external force or the engine on a rocket in the macroscopic case. The results are valid both for $%
a=const$ and for $a\rightarrow 0$ at infinity.
It is worth mentioning that there is one more but a quite different situation when high energy collisions also occur in the Schwarzschild background. This happens if a black hole is immersed in a magnetic field. In doing so, a particle orbiting around a black hole collides with another one coming from infinity. The magnetic field is supposed to be rather big to affect equations of particle motion and ensure the innermost stable circular orbit to be located near the horizon [@fr]. Meanwhile, the mechanism considered by us works for radial motion and any *finite* $a$ and has quite universal character.
Usually, the factor connected with additional forces (like gravitational radiation) are referred to as obstacles to gaining large $E_{c.m.}$[berti]{}. To the extent that such influence can be modeled by some force, backreaction does not spoil the effect [@rad], [@ne]. Meanwhile, as we saw now, in our context the presence of the force not only is compatible with the BSW effect but it can be its origin! It works even in the simplest case of radial motion in the Schwarzschild background.
This work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.
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---
abstract: 'Fundamental studies of the effect of specific ions on hydrophobic interactions are driven by the need to understand phenomena such as hydrophobically driven self-assembly or protein folding. Using $\beta$-peptide-inspired nano-rods, we investigate the effects of both free ions (dissolved salts) and proximally immobilized ions on hydrophobic interactions. We find that the free ion effect is correlated with the water density fluctuation near a non-polar molecular surface, showing that such fluctuation can be an indicator of hydrophobic interactions in the case of solution additives. In the case of immobilized ion, our results demonstrate that hydrophobic interactions can be switched on and off by choosing different spatial arrangements of proximal ions on a nano-rod. For globally amphiphilic nano-rods, we find that the magnitude of the interaction can be further tuned using proximal ions with varying ionic sizes. In general, univalent proximal anions are found to weaken hydrophobic interactions. This is in contrast to the effect of free ions, which according to our simulations strengthen hydrophobic interactions. In addition, immobilized anions of increasing ionic size do not follow the same ordering (Hofmeister-like ranking) as free ions when it comes to their impact on hydrophobic interactions. The immobilized ion effect is not simply correlated with the water density fluctuation near the non-polar side of the amphiphilic nano-rod. We propose a molecular picture that explains the contrasting effects of immobilized versus free ions.'
author:
- Kai Huang
- Sebastian Gast
- 'C. Derek Ma'
- 'Nicholas L. Abbott'
- Izabela Szlufarska
bibliography:
- 'reference.bib'
title: 'Comparison between Free and Immobilized Ion Effects on Hydrophobic Interactions: A Molecular Dynamics Study'
---
Introduction
============
Hydrophobic interactions have been recognized as a key driving force for water-mediated self-assembly processes [@ss1; @ss2; @ss3; @ss4; @ss6; @ss8] such as protein folding and micelle formation. Because of the ubiquitous nature of ions in biological environments, much research has been dedicated to understanding the effects of specific ions on hydrophobic interactions [@beyond]. While previous studies were mostly focused on the effects of soluble salts, a less explored effect (although of similar importance) is that of proximally immobilized ions. Immobilized charged or polar residues are often present on surfaces of macromolecules, where they are distributed within or are adjacent to nonpolar domains. The impact of these residues on hydrophobicity of the neighboring domains is key to understanding hydrophobic interactions in complex biological environments. Recent measurements based on the atomic force microscopy (AFM) [@Acevedo; @Derek] have revealed that the strength of hydrophobic interactions can be modulated by the presence of proximally immobilized ions. This effect was found to be sensitive to the three dimensional nano-patterning of the charged and non-polar groups [@Pomerantz4] and the specific charge (ion) type. Interestingly, the measurable effects of proximally immobilized ions were interpreted to extend over one nanometer in distance, i.e., they are long-range. This finding is in contrast to the recently reported [@Funkner; @Stirnemann] short-range nature (i.e., limited to the first ionic hydration shell) of the specific ion effects of soluble salts. The above AFM experiments raise a number of new questions related to specific ions effects. Do the specific proximally immobilized ions follow the same ranking as a function of ion size (Hofmeister order) [@Hofmeister] as free ions? For the same type of ion, how does its influence change when it is transformed from a free ion to a proximally immobilized ion? Without the freedom for ion segregation or depletion from the non-polar domain, what is the molecular origin of the effects of proximally immobilized ions on hydrophobic interactions?
As a first step towards providing insights into these questions, we report MD simulations of hydrophobic interactions between a non-polar surface and a non-polar or amphiphilic molecule in the presence of proximal charges or free ions. Inspired by the experimental studies that use oligo $\beta$-peptide [@beta1; @beta2; @beta3] that exhibits a well-defined helical conformation, we perform our simulations with a model nano-rod that has a number of key features in common with the experimental system. Specifically, our nano-rod has a well-defined shape and side groups that can be arranged to mimic globally amphiphilic (GA) and non-gobally amphiphilic ([*iso*]{}-GA) oligo-$\beta$-peptides. In addition, we construct a reference hydrophobic (HP) nano-rod with all side groups being non-polar. There is no corresponding purely hydrophobic $\beta$-peptide studied in experiment as it would be impractical to purify such molecules due to their physical properties. Nevertheless, such HP nano-rods in simulation serve as a useful reference system, allowing us to evaluate the effects of proximally immobilized ions and soluble salts.
In the remainder of this paper, we present potential of mean force (PMF) calculations of the model nano-rods near a non-polar plate. We first explore the interaction between an HP nano-rod and the extended non-polar plate as a reference system. We discuss the structure of the PMFs and thermodynamics of the hydrophobic interaction. We then investigate the effect of free ions (dissolved salts) by adding alkali halide salts to modify the hydrophobic interaction involving the HP nano-rod. We analyze the water structure and dynamics near the nano-rod, in search for a potential descriptor of the specific free ion effect. We revisit the anomalous effect of lithium and use controlled simulations to investigate different hypotheses for molecular origin of this anomaly. To study the effect of immobilized ions, we replace some of the non-polar side groups of the HP nano-rod by ionic groups. We construct both $iso$-GA and GA nano-rods to explore the effect of surface nano-patterns. For the GA nano-rods, we further study the specific effects of immobilized ions which are long-ranged in nature. Finally, we compare the immobilized and the free ion effects and discuss the origins of their differences.
Molecular Model and Simulation Methodology
==========================================
All MD simulations are performed using the GROMACS software package [@gromacs] with explicit solvent since the details of water structuring are key to accurate modeling of hydrophobic interactions. We use the SPC/E force field [@spce] to model water. Long-range electrostatic interactions are calculated using the particle mesh Ewald (PME) method. All hydrogen bonds are constrained through the LINCS [@lincs] algorithm to enable a simulation time step of 2 fs. The non-polar surface is modeled by 31 Lennard-Jones (LJ) particles arranged into a flat plate. The particles are arranged in a triangular lattice with a lattice constant of 0.32 nm. The same LJ particles (except for their different arrangement) are used to represent the non-polar groups of the nano-rod, whereas the ionic groups are modeled as monoatomic ions. Each of the nine side groups of the nano-rod is bonded to one of the three backbone residues of the molecule and the side group-backbone-side group angles are 120 degree. The backbone residues do not interact with water since they are buried inside the peptide, but their presence allows us to arrange the functional groups in a controlled manner, resembling the rigid helical structure of $\beta$-peptide [@QC].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Coarse-graining of a $\beta$-peptide into a nano-rod with three surface nano-patterns: hydrophobic (HP), globally amphiphilic (GA) and non-globally amphiphilic ([*iso*]{}-GA). The nano-rod is immersed in water and interacts with an extended non-polar surface. Coarse-grained sites of the non-polar surface are shown in cyan. Non-polar side groups of the nano-rod are shown in blue, backbone in white, and the immobilized ions in yellow. The backbone refers to the coarse-grained residues to which the functional groups are attached.[]{data-label="system"}](system1 "fig:") ![Coarse-graining of a $\beta$-peptide into a nano-rod with three surface nano-patterns: hydrophobic (HP), globally amphiphilic (GA) and non-globally amphiphilic ([*iso*]{}-GA). The nano-rod is immersed in water and interacts with an extended non-polar surface. Coarse-grained sites of the non-polar surface are shown in cyan. Non-polar side groups of the nano-rod are shown in blue, backbone in white, and the immobilized ions in yellow. The backbone refers to the coarse-grained residues to which the functional groups are attached.[]{data-label="system"}](structure2 "fig:")
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
As shown in Fig. \[system\], we construct nano-rods with three different nano-patterns: HP, GA and [*iso*]{}-GA. Each nano-rod resembles a triangular prism that is approximately 1 nm long. Each edge of the triangular side has a length of approximately 0.5 nm. Similar coarse-grained models have been used in earlier simulations of the self-assembly of $\beta$-peptides [@sbeta2], but in that case implicit solvent was used. To model the proximal ion effects, we use hypothetical halide ions as ionic side groups tethered to the coarse-grained backbone of the nano-rod. These immobilized halide ions have the same charge and size as the free halide ions to allow a better comparison between immobilized and free ion effects. We model both immobilized and free ions with the OPLS force field (parameters of this force field can be found in Ref [@opls]). Lorentz-Berthelot mixing rules are applied to generate LJ parameters between different types of atoms. Other LJ parameters used in our study are listed in Table \[t.lj\]. The cut-off distance for the LJ interaction and for short-range Coulomb interaction is 1.2 nm. The strength of the hydrophobic interaction between the nano-rod and the extended surface is largely determined by the parameter $\epsilon$ of the LJ potential for interactions between the non-polar site (NS) of the nano-rod and the oxygen atom of water, as well as between the extended nonpolar surface (ES) and the oxygen atom of water. We chose $\epsilon$ for NS-O and ES-O to be 0.6 kJ/mol, which represents a typical hydrophobicity of hydrocarbon molecules [@zangi]. The nano-rod and non-polar surface are solvated in a 2.5 nm$\times$2.5 nm$\times$6 nm water box (around 1100 water molecules). By carrying out two sets of simulations (with three counter-ions added or absent from the solutions) we found that such counter-ions have a negligible effect on the calculated free energy, indicating that the effect of highly dilute counter-ions and their interaction with immobilized ions can be reasonably neglected (see the Supporting Information). When investigating free ion effects, 1 molar concentration (approximately 80 ions) of various alkali halides is added to the solution. Such concentration has been used in previous simulation work to study the free ion effect and in past simulations the Hofmeister ordering of free ions was found to be independent of the concentration of ions for neutral non-polar hydrophobes [@Thomas; @Schwierz].
Atom types $\epsilon$ (kJ/mol) $\sigma$(Å)
------------ --------------------- -------------
H-\* 0.0 0.0
BB-\* 0.0 0.0
NS-O 0.6 3.52
ES-O 0.6 3.52
NS-ES 0.1 4.0
PC-ES 0.1 4.0
: Parameters of the Lennard-Jones potential. H: hydrogen atom of water, O: oxygen atom of water, BB: coarse-grained backbone atom of nano-rod, NS: coarse-grained non-polar site of nano-rod, PC: coarse-grained immobilized ionic group of nano-rod, ES: coarse-grained hydrophobic atom of the extended surface. \* symbol refers to atom types that are different from the ones before the hyphen symbol[]{data-label="tbl:example"}
\[t.lj\]
We use the umbrella sampling method to calculate the PMF between each nano-rod and non-polar surface. We define the reaction coordinate to be the $z$ projection of the distance between the second backbone atom (in the middle of the nano-rod) and the non-polar surface, where $z$ direction is perpendicular to the surface. The separation between neighboring sampling windows is 0.05 nm and the spring constant of the external harmonic constraint is 4000 kJ/mol$\cdot\mathrm{nm}^2$. The PMF reported in the main text corresponds to nano-rods that can freely rotate. In addition we have carried out controlled simulations where the rotational degree of freedom is forbidden (details in Supporting Information). We found that the rotation of the molecules does not impact the qualitative conclusions reported in this study. We have calculated uncertainties in PMF using the bootstrapping method [@bootstrap]. Other methods for calculating uncertainties in PMF have also been reported in literature [@zhu]. A 20 ns sampling time for each PMF calculation allowed us to reach an uncertainty smaller than 0.1 kcal/mol. In our simulations, the system is first relaxed for 1 ns at 300 K and 1 atmosphere using constant pressure constant temperature ensemble with coupling constants $\tau=1$ ps for both the thermostat and the barostat. The velocity rescaling thermostat is used for temperature coupling and the Berendsen barostat with the compressibility of $4.5\times10^5~\mathrm{bar}^{-1}$ is used for pressure coupling.
Results and Discussion
======================
Hydrophobic interaction between HP nano-rod and non-polar plate
----------------------------------------------------------------
Before exploring the specific ion effects, we first study the hydrophobic interaction between a HP nano-rod and an extended non-polar surface as a reference system. Free ion effect can be studied by adding salts to the reference system, whereas proximal charge effect can be investigated by replacing some of the non-polar sites with ionic groups. The PMFs of the hydrophobic interactions for the reference system at different temperatures are plotted in Fig. \[HP\]. It is interesting that the overall free energy landscape shown here is more complicated than energy landscapes reported for the interactions between two non-polar solutes of simple shapes [@Thomas; @zangi]. While there is only one contact minimum in the attractive part of the PMF in the case of simple solutes, our PMF has two minima (except at temperature higher than 325 K). Specifically, in addition to the primary contact minimum near 0.5 nm, we have a weak secondary minimum near 0.76 nm. The two minima correspond to two energetically favorable configurations of the nano-rod/plate contacts as shown in the insets of Fig. \[HP\]. Since there is no electrostatic interaction between the nano-rod and the non-polar surface and the assigned van der Waals interaction is very weak (see Table \[t.lj\]), the attractive part of the PMF is largely due to the water-mediated hydrophobic interaction. Therefore, these two contact minima are both hydrophobic in nature.
It is known that the thermodynamics of the hydrophobic hydration is size dependent [@Chandler1; @Blokzijl]. Since the hydrophobic interaction in our system is between a small molecule (nano-rod) and a large surface (non-polar plate), it is interesting to ask what is the thermodynamic driving force of such interaction. According to the partition of the free energy, $\Delta G=\Delta H-T\Delta S$ (where $H$ is enthalpy, $T$ is temperature and $S$ is entropy), the temperature dependence in Fig. \[HP\] is indicative of a positive entropy change during the process of association between the nano-rod and the non-polar wall. At $T=300$ K we estimate (based on the free energy difference between $T=275$ K and $T=325$ K) the contribution to the free energy ($-T\Delta S$) to be $-5.4\pm0.9$ kcal/mol, which is more than half of the total free energy change $\Delta G=-8.8$ kcal/mol. The remainder of the free energy change is due to a reduction of enthalpy ($\Delta H=-3.4$ kcal/mol) associated with the interaction. Therefore, the interaction between the HP nano-rod and the non-polar plate is favorable both in entropy and enthalpy. Recall that the association of two macroscopic non-polar surfaces is purely enthalpy driven and association of two small hydrophobes is purely driven by entropy. Our results demonstrate that both of these driving forces (entropy increase and enthalpy reduction) can be present in a mixed system comprised of a small molecule and a large surface. Additional information on the thermodynamic driving force will be presented when discussing Fig. \[iso\]. It is worth noting that even for the interaction between small hydrophobes, the thermodynamics can be strongly affected by a nearby flat non-polar surface [@patel2].
Free ion effects
----------------
We now investigate the specific ion effects on the interaction between the nano-rods and the non-polar plate, starting with free ions. The ordering of free ions with respect to their effect on hydrophobic interactions (the so-called Hofmeister series), have been extensively studied in literature [@Thomas; @Schwierz] and can be used to test of our simulations. Simulations of free ion effects also allows a direct comparison with the immobilized ion effects that will be discussed later.
Here we investigate the free ion (dissolved salt) effect by carrying out simulations of the interactions between the HP nano-rod and the non-polar plate in the presence of alkali halide salts (NaF, NaCl, NaI, LiCl, CsCl) at modest molar concentration (1M). Before showing the results, we shall however emphasize that the ordering of the Hofmeister series is not trivially determined by the properties of the ions, but also depends on the solute surface [@Schwierz]. For example, the ordering of ions that change the solubility of proteins with a net positive charge can be reversed if the protein becomes negatively charged. As in this paper we are dealing with hydrophobic interaction between charge neutral surfaces, we will use the term Hofmeister series to refer to ordering of ions in the presence of charge neutral hydrophobes. Such Hofmeister series for halide anions with respect to their salting-out ability is $\mathrm{I}^-<\mathrm{Br}^-<\mathrm{Cl}^-<\mathrm{F}^-$ [@Schwierz]. For alkali cations, the corresponding ranking is $\mathrm{Cs}^+<\mathrm{Li}^+<\mathrm{K}^+<\mathrm{Na}^+$ [@Schwierz].
Figure \[free\] shows the PMFs of the interaction between the nano-rod and the hydrophobic plate in the presence of different salts. As ${\text{Na}^+}$ and ${\text{Cl}^-}$ take positions in the middle of the Hofmeister series and the long-range cooperative ion effects are believed to be small at low and moderate concentrations [@Funkner; @Stirnemann], it is reasonable to assume that the effect of sodium halide (alkali chloride) mainly reflects the specific effect of the halide anions (alkali cations). Comparisons within the sodium halide series and alkali chloride series in Fig. \[free\] show that the strength of hydrophobic interaction follows the orders of NaI<NaCl<NaF and LiCl$\approx$CsCl <NaCl. Therefore we can rank halide anions and alkali cations as ${\text{I}^-}$<${\text{Cl}^-}$<${\text{F}^-}$ and ${\text{Li}^+}$$\approx$${\text{Cs}^+}$<${\text{Na}^+}$ in their salting-out effects. Such orderings agree with previous reports of Hofmeister series for non-polar solutes (${\text{I}^-}$<${\text{Br}^-}$<${\text{Cl}^-}$<${\text{F}^-}$, ${\text{Cs}^+}$<${\text{Li}^+}$<${\text{K}^+}$<${\text{Na}^+}$) very well [@Schwierz].
-- --
-- --
To shed more light on the molecular origin of the Hofmester ranking of free ions, and inspired by the work of Garde and coworkers [@Jamadagni], we examine the structure and dynamics of the hydrating water around the HP nano-rod by analyzing the water radial distribution function (RDF) and its time-dependent fluctuation in the presence of different salts (see Supporting Information). Here, the fluctuation of the water density is not merely a measure of the uncertainty of density, but more importantly it is an indicator of the dynamics of water in the hydration shells. For all systems examined in our simulations, we found the RDF of water around the HP nano-rod to be insensitive to the identity of the salts. In contrast, the fluctuation of the RDF is strongly affected by salts. More interestingly, the ion-modulated RDF fluctuation is well correlated with the ion-modulated strength of the hydrophobic interaction ($|\Delta G|$) with an approximate linear relationship as shown in Fig. \[linear\].
The only significant deviation from this linearity comes from the LiCl salt (see the point labeled LiCl(clustered) in Fig. \[linear\]). Anomalous behavior of Li$^+$ has been previously noted in literature. Specifically, while the positions of ions in the Hofmeister series can often be correlated with ion size, lithium ion was found to be an exception to this trend [@Thomas; @Schwierz]. Two different ways to rationalize the anomaly of Li$^+$ have been proposed. Thomas and Elcock [@Thomas] postulated that this anomaly is primarily due to clustering between lithium atoms and counter-ions. On the other hand Schwierz [*et al*]{} [@Schwierz] argued that the anomaly of lithium can be explained by its large effective size if one considers the rigid first hydration shell to be part of the ion. To determine the primary mechanism responsible for the lithium anomaly in our simulations, we carried out an additional calculation of $|\Delta G|$ for LiCl salt solution with a constraint on lithium cations such that Li$^+$ cannot form clusters with chloride anions. The result of this calculation is labeled in Fig. \[linear\] as LiCl(dissolved). Interestingly, the salting-out effect (quantified by $|\Delta G|$) of dissociated LiCl is weaker than that of NaCl and NaF and it is stronger than the effect of NaI and CsCl. In other words, even for fully dissolved Li$^+$, its effect on hydrophobic interaction does not correlate with its bare ionic size. Instead, as shown in Fig. \[linear\] the effect of dissociated LiC salt follows the same linear trend as other salts, demonstrating that the strength of hydrophobic interactions $|\Delta G|$ is correlated with RDF fluctuations of water around the nano-rod. These results reveal that while the clustering can contribute to the anomaly of lithium, the primary reason for this anomaly is the large effective size of this ion [@Schwierz]. It has been shown in recent literature [@Laage1; @Rezus; @Tielrooij; @Tielrooij2; @Funkner; @Stirnemann] that large ions (with low charge densities) tend to accelerate the reorientation of water. In this light, our simulation observation of the large RDF fluctuation of water in the presence of Li$^+$ is consistent with the hypothesis that Li$^+$ has a large effective ionic size.
Our conclusion that water fluctuations are a better indicator of hydrophobicity align with the prior results reported by Garde and coworkers [@Jamadagni] for flat surfaces. We demonstrate that these conclusions apply to curved surfaces and more complex molecular structures.
Immobilized ion effects
-----------------------
Now we turn to the effects of immobilized ions. It has been reported that GA and [*iso*]{}-GA $\beta$-peptides self-assemble into different structures [@Pomerantz4], which reflects interactions mediated by the different arrangements of functional groups on the two types of molecules [@Pomerantz1; @Pomerantz2; @Pomerantz3; @sbeta1; @sbeta2; @sbeta3]. Recent AFM single-molecule force spectroscopy measurements [@Acevedo] revealed that the adhesive forces between $\beta$-peptides and non-polar surfaces depend on the nature of the nano-patterns formed by functional groups on the $\beta$-peptides. Inspired by these experiments, our first goal here is to understand how nano-scale chemical patterns affect intermolecular interactions and how these interactions can be modulated by immobilized ions. To this end, we calculate the PMF between a non-polar plate and nano-rods with different nanoscale patterns. For amphiphilic nano-rods, we use hypothetical $\mathrm{Cl}^-$ ions (see method section) as the proximally immobilized ions in both GA and [*iso*]{}-GA patterns.
The PMF profiles for these two types of nano-rods are shown in Fig. \[iso\]a together with a reference PMF (without immobilized ions). All PMF curves are shifted so that the reference state (corresponding to the nano-rod not interacting with the surfaces) has zero free energy. The GA nano-rod interacts with the surface in such a way that six of the nano-rod’s non-polar sites face the surface and three of its ionic sites point away from the surface, similarly to the contact type $\text{I}$ of the HP nano-rod shown in Fig. \[HP\]. This arrangement minimizes the exposure of non-polar sites to water and thereby maximizes hydrophobic interaction. For nano-rod with the GA pattern, the primary contact minimum is still pronounced, but it is reduced as compared to the contact minimum for the HP nano-rod. The second minimum, which was observed for the HP nano-rod, disappears in the case of the GA pattern. In contrast, the PMF of the [*iso*]{}-GA pattern has a very shallow contact minimum of type $\text{II}$ and is largely repulsive in the regime where the primary minimum (corresponding to configuration $\text{I}$) would occur. Although it is possible for the [*iso*]{}-GA nano-rod to align itself so that up to four of its non-polar sites face the surface, it is clear that the strength of hydrophobic interaction does not simply scale with the number of non-polar sites that face the surface. The striking difference in the PMF of the GA and the [*iso*]{}-GA nano-rods reflects the distinct hydration status of the two molecules in water. More specifically, we find that hydrophobic interaction can be effectively destroyed by adding proximally immobilized ions between non-polar sites. Consequently, our results demonstrate that the hydrophobic interaction is a result of the collective behavior of hydrating water molecules and it is not a simple function of the surface area of non-polar domains. This conclusion is generally consistent with the AFM measurement by Acevedo [*et al*]{} [@Acevedo], although in the experiments a non-vanishing pull-off force persisted in the case of [*iso*]{}-GA $\beta$-peptide. This force was established to be electrostatic in nature due to the non-polar surface accumulating an excess negative charge when immersed in water. The immobilized ion effect in the [*iso*]{}-GA nano-rod is similar to the one recently reported in MD simulations by Acharya [*et al*]{} [@Acharya] where it was shown that polar groups in the middle of a flat non-polar domain can substantially modulate hydrophobic interactions. Our results extend Acharya [*et al*]{}’s conclusion for polar groups to immobilized ions, and from a flat geometry to a nano-rod.
-- --
-- --
To determine the thermodynamic origin of the hydrophobic interaction predicted by our MD simulations for nano-rods with the GA nano-pattern, we calculate the PMFs for hydrophobic interactions at a number of different temperatures as shown in the inset of Fig. \[iso\]a. Based on this temperature dependence, we partition the free energy into the entropy and enthalpy contributions. As shown in the Fig. \[iso\]b, around room temperature, both the entropy and enthalpy driving forces are present in the interactions between the non-polar plate and the nano-rods (with and without immobilized ions). In other words, both the HP and GA nano-rods have a negative hydration entropy associated with their non-polar surfaces. The entropy driving force decreases with temperature as the hydrating water becomes more disordered.
Having verified that the hydrophobic interaction involving amphiphilic molecules can be turned on and off by choosing either the GA or [*iso*]{}-GA nano-pattern of the nano-rod, we now test the possibility of tuning the strength of the hydrophobic interactions by varying the size of the proximally immobilized ions of the GA nano-rod. We also investigate whether the effects of proximally immobilized ion follows the same Hofmeister order as the dissolved free ions, a question that is of practical importance for the rational design of hydrophobically driven self-assembly of materials [@ss1; @ss2; @ss3; @ss4; @ss6; @ss8].
To shed light on the question of the effect of the ionic size of a proximally immobilized ion on hydrophobic interaction, we calculate the PMF for the nano-rods with the proximally immobilized ions being halide anions. We choose halide anions for our test because their specific ion effects are known to be stronger than alkali cations, and their Hofmeister order in the case of free ions correlates well with the ionic sizes or, equivalently, charge densities. The PMF results are presented in Fig. \[size\]a. Interestingly, we find that the strength of hydrophobic interaction between the nano-rod and the non-polar surface does not depend monotonically on the size of the proximally immobilized ion. Instead, the ranking of the interaction, from weak to strong, with different proximally immobilized ions follows $\mathrm{I}^-<\mathrm{F}^-<\mathrm{Cl}^-\approx\mathrm{Br}^-$. To ensure that the ordering of ions is not affected by the possibility of the nano-rod to reorient itself during simulations, we carried out additional simulations where the orientational freedom of the nano-rod is frozen out. In these constrained simulations, all nano-rods have their non-polar surface lying parallel to the plate during the sampling so that the reaction paths are identical for all ions. The results shown in the Supporting Information show the same ordering of ions with the interaction strength, confirming that the rotational freedom of nano-rod (or its absence) does not alter the order of the specific immobilized ion effects. In addition, we have calculated uncertainty of the PMF calculations (as explained in the Section Molecular Model and Simulation Methodology) and it was found to be smaller than 0.1 kcal/mol.
-- --
-- --
Two types of interactions can hypothetically contribute to the observed effects of immobilized ions on hydrophobic attraction and they are both illustrated by arrows in Fig. \[size\]b. One contribution comes from the electrostatic interaction between the immobilized ion and water molecules residing at the nano-rod/non-polar surface interface. This interaction occurs across the diameter of the GA nano-rod. The second contribution comes from long-range perturbation of hydration shells by the immobilized ions. Both of these contributions can affect free energy of the interfacial water (between the nano-rod and the non-polar surface). The electrostatic (charge-dipole) contribution to interaction is sensitive to the charge of the ion and not to its size. On the other hand, the contribution from long-range perturbation of hydration shell can affect the entropy of interfacial water and it is sensitive to the size of the immobilized ion. The overall specific immobilized ion effect is expected to be a balance of these two mechanisms. It is worth pointing out that the water density fluctuation that correlates with the free ion effect is no longer an indicator of the long-range immobilized ion effect. Specifically, analysis (not shown here) of the water density and its fluctuation near the non-polar side of the GA nano-rods shows that these quantities are not simply correlated with the specific immobilized ion effect. The lack of correlation here suggests that perturbation of the free energy of the water beyond the first ionic hydration shell is complicated in nature and further studies are needed to better understand the underlying molecular details.
Although many differences exist between the recent experiment studies using $\beta$-peptides and the models used in our simulations, our simulations do support the experimental finding that the immobilized ions can have a long-range effect on hydrophobic interactions.
Comparison between the effects of proximally immobilized ion and soluble salts
------------------------------------------------------------------------------
It is instructive to compare the effects from the same ion in different states, soluble and immobilized. In Fig. \[compare\] we summarize the effects of immobilized ions and free ions. We choose the reference (a dashed line in Fig. \[compare\]) for both cases to be the hydrophobic interaction strength between the HP nano-rod and the extended non-polar surface without any ions. Inspection of Fig. \[compare\] reveals that the immobilized ions tend to weaken the hydrophobic interaction while the free ones tend to strengthen it. This difference is related to the distinct spatial distributions of immobilized and free ions. One can think of the hydrophobic interaction as a dehydrating reaction accompanied by the release of water molecules from the surface of the solute to the bulk region, which can be symbolically written as $$\text{Nano-rod}\cdot \mathrm{H_2O}+\text{Surface}\cdot \mathrm{H_2O} \xrightleftharpoons[\text{hydrate}] {\text{dehydrate}}\text{Nano-rod|surface}+\mathrm{H_2O}(\text{bulk}),
\label{reaction}$$ where symbol $\cdot$ means hydrating and symbol $|$ means in contact. The strength of hydrophobic interaction is determined by $$\begin{split}
\Delta{G}_{\text{dehydrate}}&=G_{\mathrm{H_2O}(\text{bulk})}-[G_{\text{Nano-rod}\cdot\mathrm{H_2O}}+G_{\text{Surface}\cdot\mathrm{H_2O}}-G_{\text{Nano-rod|surface}}]\\
&=G_{\mathrm{H_2O}(\text{bulk})}-G_{\mathrm{H_2O}(\text{surface})}.
\label{G}
\end{split}$$ Here, we defined $G_{\text{H2O(surface)}}$, which qualitatively can be understood as the free energy of water near the non-polar surfaces (the non-polar domain of GA nano-rod and the flat non-polar surface) when they are far from each other. The specific ion effect on hydrophobic interaction can therefore be generally understood as the difference in the manners the ion modifies the free energy of hydrating water and of bulk water $$\Delta(\Delta{G}_{\text{dehydrate}})=\Delta G_{\mathrm{H_2O}(\text{bulk})}-\Delta G_{\mathrm{H_2O}(\text{surface})}.
\label{ion}$$ The sign of $\Delta(\Delta{G}_{\text{dehydrate}})$ determines whether the ion increases or decreases the strength of hydrophobic interaction. As we are interested in the immobilized ion effect, we here focus on the free energy of water hydrating the nano-rod. The free energy of water hydrating the non-polar plate is not sensitive to the proximally immobilized ions and can be treated as a constant reference because the plate is well isolated from the nano-rod in the non-interacting reference state. The distinct effects from an immobilized ion and a free ion result from their different spatial distributions with respect to the nano-rod, i.e., immobilized ions are immobilized near the nano-rod and absent in the bulk water whereas free ions are excluded from the nano-rod and remain in the bulk. Therefore in Eq \[ion\], the $\Delta G_{\mathrm{H_2O}(\text{bulk})}$ term is important for free ions but it is negligible for immobilized ions. This means that an immobilized ion weakens the hydrophobic interaction solely by lowering the free energy of water hydrating the nano-rod. In contrast, the free ion effect is a result of the changes in both the hydrating free energy and bulk free energy. In our case, as the nano-rod is small in size and highly convex in curvature, free ions are generally excluded from it (see Supporting Information) and mainly lower the free energy of bulk water. The exclusion of ions, including ${\text{I}^-}$, from small hydrophobes have been recently reported in experiments [@Rankin1; @Rankin2]. For bulkier complex ion such as guanidinium, it has been shown that the dispersive interaction between the ion and the non-polar solute can favor the dissolution of the latter [@godawat; @breslow]. It is interesting to note that regardless of whether ions are immobilized or dissolved, the same ions have different rankings with respect to their modulations of strength of hydrophobic interactions.
Concluding remarks
==================
We have used MD simulations to study the effects of proximally immobilized ions on hydrophobic interactions between a $\beta$-peptide inspired nano-rod and an extended non-polar surface. By comparing the effect of immobilized ions with different nano-patterns and ionic radii, we demonstrated that hydrophobic interaction can be largely eliminated by [*iso*]{}-GA patterning and can be modulated by proximally immobilized ions in the globally amphiphilic (GA) nano-rod. Several trends observed in our simulations agree with recent single-molecule AFM measurements [@Acevedo; @Derek] and we provide a molecular mechanistic understanding of these trends in the context of the models used in the MD simulation. In the broader context of specific-ion effects, we have shown that the immobilized ions do not necessarily follow the same ordering as the free ions. Our analysis of the structure and dynamics of water near the hydrophobic nano-rod shows that dynamics is a better indicator of the specific ion effect than the static water structure. This result extends prior results of Garde and coworkers [@Jamadagni]. Our results provide new insights into specific ion effects that may, in the long term, guide the rational design of hydrophobic interactions and self-assembly process driven by these interactions.
Associated content
==================
Supporting Information
Figure S1. Radial distribution function (RDF) of free ions around the nano-rod. Figure S2. Radial distribution function (RDF) and its fluctuation for water oxygen around nano-rod. Figure S3. Anomaly of the lithium effect. Figure S4. Comparison between simulations with and without counter-ions. Figure S5. Proximally immobilized ion effect in constrained simulation. This material is available free of charge via the Internet at http://pubs.acs.org.
The authors gratefully acknowledge support from NSF-NSEC at UW-Madison DMR-0832760, from NSF grant CMMI-0747661, and from the ARO (W911NF-14-1-0140).
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---
abstract: 'The chemical and photometric evolution of star forming disk galaxies is investigated. Numerical simulations of the complex gasdynamical flows are based on our own coding of the Chemo - Dynamical Smoothed Particle Hydrodynamical (CD - SPH) approach, which incorporates the effects of star formation. As a first application, the model is used to describe the chemical and photometric evolution of a disk galaxy like the Milky Way.'
author:
- Peter Berczik
title: 'Chemo - Photometric evolution of star forming disk galaxy'
---
Introduction
============
Galaxy formation is a highly complex subject requiring many different approaches of investigation. Recent advances in computer technology and numerical methods have allowed detailed modeling of baryonic matter dynamics in a universe dominated by collisionless dark matter and, therefore, the detailed gravitational and hydrodynamical description of galaxy formation and evolution. The most sophisticated models include radiative processes, star formation and supernova feedback ([@K92; @SM94; @FB95]).
The results of numerical simulations are fundamentally affected by the star formation algorithm incorporated into modeling techniques. Yet star formation and related processes are still not well understood on either small or large spatial scales. Therefore the star formation algorithm by which gas is converted into stars can only be based on simple theoretical assumptions or on empirical observations of nearby galaxies.
Among the numerous methods developed for modeling complex three dimensional hydrodynamical phenomena, Smoothed Particle Hydrodynamics (SPH) is one of the most popular ([@M92]). Its Lagrangian nature allows easy combination with fast N - body algorithms, making possible the simultaneous description of complex gas-stellar dynamical systems ([@FB95]). As an example of such a combination, an TREE - SPH code ([@HK89; @NW93]) was successfully applied to the detailed modeling of disk galaxy mergers ([@MH96]) and of galaxy formation and evolution ([@K92]). A second good example is an GRAPE - SPH code ([@SM94; @SM95]) which was successfully used to model the evolution of disk galaxy structure and kinematics.
Model
=====
The hydrodynamical simulations are based on our own coding of the Chemo - Dynamical Smoothed Particle Hydrodynamics (CD - SPH) approach, including feedback through star formation (SF). The dynamics of the “star” component is treated in the frame of a standard N - body approach. Thus, the galaxy consists of “gas” and “star” particles. For a detailed description of the CD - SPH code (the star formation algorithm, the SNII, SNIa and PN production, the chemical enrichment and the initial conditions) the reader is referred to [-@BerK96; -@Ber99]. Here we briefly decsribe the basic features of our algorithm.
We modify the standard SPH SF algorithm ([@K92; @SM94; @SM95]), taking into account the presence of chaotic motion in the gaseous environment and the time lag between the initial development of suitable conditions for SF, and SF itself.
Inside a “gas” particle, the SF can start if the absolute value of the “gas” particle gravitational energy exceeds the sum of its thermal energy and energy of chaotic motions:
E\_i\^[gr]{} > E\_i\^[th]{} + E\_i\^[ch]{}.
The chosen “gas” particle produces stars only if the above condition holds over the time interval exceeding its free - fall time:
t\_[ff]{} = .
We also check that the “gas” particles remain cool, $ t_{cool} < t_{ff} $. We rewrite these conditions following [-@NW93]:
\_i > \_[crit]{}.
We set the value of $\rho_{crit} = 0.03$ cm$^{-3}$.
When the collapsing particle $ i $ is defined, we create the new “star” particle with mass $ m^{star} $ and update the “gas” particle $ m_i $ using these simple equations:
{
[lll]{} m\^[star]{} = m\_i,\
\
m\_i = (1 - ) m\_i.\
.
In the Galaxy, on the scale of giant molecular clouds, the typical values for SF efficiency are in the range $ \epsilon \approx 0.01
\div 0.4 $ ([@DIL82; @WL83]).
We did not fixe this value but rather also derived $ \epsilon $ from the “energetics” condition:
= 1 - .
At the moment of birth, the positions and velocities of new “star” particles are set equal to those of parent “gas” particles. Thereafter these “star” particles interact with other “gas” and “star” or “dark matter” particles only by gravity.
For the thermal budget of the ISM, SNIIs play the main role. Following to [-@K92; -@FB95], we assume that the energy from the explosion is converted totally to thermal energy. The total energy released by SNII explosions ($ 10^{44} \; J $ per SNII) within “star” particles is calculated at each time step and distributed uniformly between the surrounding “gas” particles ([@RVN96]).
In our SF scheme, every new “star” particle represents a separate, gravitationally bound, star formation macro region (like a globular clusters). The “star” particle has its own time of birth $ t_{begSF} $ which is set equal to the moment the particle is formed. After the formation, these particles return the chemically enriched gas into surrounding “gas” particles due to SNII, SNIa and PN events.
We concentrate our treatment only on the production of $^{16}$O and $^{56}$Fe, yet attempt to describe the full galactic time evolution of these elements, from the beginning up to present time ($
t_{evol} \approx 15.0 $ Gyr).
The code also includes the photometric evolution of each “star” particle, based on the idea of the Single Stellar Population (SSP) ([@BCF94; @TCBF96]).
At each time - step, absolute magnitudes: M$_U$, M$_B$, M$_V$, M$_R$, M$_I$, M$_K$, M$_M$ and M$_{bol}$ are defined separately for each “star” particle. The SSP integrated colours (UBVRIKM) are taken from [-@TCBF96]. The spectro - photometric evolution of the overall ensemble of “star” particles forms the Spectral Energy Distribution (SED) of the galaxy.
We do not model the energy distribution in spectral lines nor the scattered light by dust. However according to [-@TCBF96] our approximation is reasonable, especially in the UBV spectral brand.
Results
=======
The model presented descibes well the time evolution of the basic chemical and photometric parameters of a disk galaxy similar to the Milky Way. The metallicity, luminosity and colors obtained are typical of such disk galaxies.
- Figure \[lum-t\]. Luminosity evolution of the model galaxy.
- Figure \[ubvk-t\]. Photometric evolution of the model galaxy.
- Figure \[ci-t\]. Color index evolution of the model galaxy.
- Figure \[bv-v\]. Evolution of the model galaxy in the (B-V) vs. V plane.
- Figure \[ub-bv\]. Evolution of the model galaxy in the (U-B) vs. (B-V) plane.
- Figure \[ub-vk\]. Evolution of the model galaxy in the (U-B) vs. (V-K) plane.
- Figure \[fe-v\]. Evolution of the model galaxy in the \[Fe/H\] vs. V plane.
- Figure \[ml-t\]. The “real” M$_{star}$/L$_{V}$ evolution of the model galaxy.
The author is grateful to Yu.I. Izotov and L.S. Pilyugin for fruitful discussions during the process of preparing this work.
The work was supported by the German Science Foundation (DFG) under grants No. 436 UKR 18/2/99, 436 UKR 17/11/99 and partially supported by NATO grant NIG 974675.
Special thanks for hospitality to the Astronomisches Rechen-Institute (ARI) Heidelberg, where part of this work has been done.
It is a pleasure to thank Christian Boily for comments on an earlier version of this work.
Berczik P., Kravchuk S.G., 1996, 27
Berczik P., 1999, å[348]{} 371, ([astro-ph/9907375]{})
Bressan A., Chiosi C., Fagotto F., 1994, 63
Duerr R., Imhoff C.L. & Lada C.J., 1982, 135
Friedli D., Benz W., 1995, å[301]{} 649
Hernquist L., Katz N., 1989, 419
Katz N., 1992, 502
Mihos J.C., Hernquist L., 1996, 641
Monaghan J.J., 1992, 543
Navarro J.F., White S.D.M., 1993, 271
Raiteri C.M., Villata M., Navarro J.F., 1996, å[315]{} 105
Steinmetz M., Muller E., 1994, å[281]{} L97
Steinmetz M., Muller E., 1995, 549
Tantalo R., Chiosi C., Bressan A., Fagotto F., 1996, å[311]{} 361
Wilking B.A. & Lada C.J., 1983, 698
|
$$$$
[BCK-algebras arising from block codes]{}
$$$$
Cristina FLAUT
$$$$
**Abstract.** [In this paper, we will provide an algorithm which allows us to find a BCK-algebra starting from a given binary block code.]{}
**Keywords:** BCK-algebras; Block codes.
**AMS Classification. **06F35$$$$
**0. Introduction**
$$$$
BCK-algebras were first introduced in mathematics in 1966 by Y. Imai and K. Iseki, through the paper \[4\], as a generalization of the concept of set-theoretic difference and propositional calculi. The class of BCK-algebras is a proper subclass of the class of BCI-algebras and there exist several generalizations of BCK-algebras as for example: generalized BCK-algebras \[3\], dual BCK-algebras \[9\] , BE-algebras \[1\], \[8\]. These algebras form an important class of logical algebras and have many applications to various domains of mathematics, such as: group theory, functional analysis, fuzzy sets theory, probability theory, topology, etc. For other details about BCK-algebras and about some new applications of them, the reader is referred to \[2\], \[5\], \[6\], \[10\], \[11\], \[12\], \[13\] .
One of the recent applications of BCK-algebras was given in the Coding Theory. In Coding Theory, a block code is an error-correcting code which encode data in blocks. In the paper \[7\], the authors constructed a finite binary block-codes associated to a finite BCK-algebra. At the end of the paper, they put the question if the converse of this statement is also true.
In the present paper, we will prove that, in some circumstances, the converse of the above statement is also true.
$$$$
**1. Preliminaries**$$$$
**Definition 1.1.** An algebra $(X,\ast ,\theta )$ of type $(2,0)$ is called a *BCI-algebra* if the following conditions are fulfilled:
$1)~((x\ast y)\ast (x\ast z))\ast (z\ast y)=\theta ,$ for all $x,y,z\in X;$
$2)~(x\ast (x\ast y))\ast y=\theta ,$ for all $x,y\in X;$
$3)~x\ast x=\theta ,$ for all $x\in X$;
$4)$ For all $x,y,z\in X$ such that $x\ast y=\theta ,y\ast x=\theta ,$ it results $x=y$.
If a BCI-algebra $X$ satisfies the following identity:
$5)$ $\theta \ast x=\theta ,~$for all $x\in X,$ then $X$ is called a *BCK-algebra*.
A BCK-algebra $X$ is called *commutative* if $x\ast (x\ast y)=y\ast
(y\ast x),$ for all $x,y\in X$ and *implicative* if $x\ast (y\ast
x)=x,$ for all $x,y\in X.$
The partial order relation on a BCK-algebra is defined such that $x\leq y$ if and only if $x\ast y=\theta .$
If $(X,\ast ,\theta )$ and $(Y,\circ ,\theta )$ are two BCK-algebras, a map $f:X\rightarrow Y$ with the property $f\left( x\ast y\right) =f\left(
x\right) \circ f\left( y\right) ,$ for all $x,y\in X,$ is called a *BCK-algebras morphism*$.$ If $f$ is a bijective map, then $f$ is an *isomorphism* of BCK-algebras.
In the following, we will use some notations and results given in the paper \[7\] .
From now on, in whole this paper, all considered BCK-algebras are finite.
Let $A$ be a nonempty set and let $X$ be a BCK-algebra.
**Definition 1.2.** A mapping $f:A\rightarrow X$ is called a *BCK-function* on $A.$ A *cut function* *of* $f$ is a map $f_{r}:A\rightarrow \{0,1\},r\in X,$ such that $$f_{r}\left( x\right) =1,\text{ if and only if \ }r\ast f\left( x\right)
=\theta ,\forall x\in A.$$A *cut subset* of $A$ is the following subset of $A$
$$A_{r}=\{x\in A:r\ast f\left( x\right) =\theta \}.$$
**Remark 1.3.** Let $f:A\rightarrow X$ be a BCK-function on $A.$ We define on $X$ the following binary relation $$\forall r,s\in X,r\sim s~~\text{if~and~only~if~~}A_{r}=A_{s}.$$This relation is an equivalence relation on $X$ and we denote with $\widetilde{r}$ the equivalence class of the element $r\in X.\medskip $
**Remark 1.4.** (\[7\] ) Let $A$ be a set with $n$ elements. We consider $A=\{1,2,...,n\}$ and let $X$ be a BCK-algebra. For each BCK-function $f:A\rightarrow X,$ we can define a binary block-code of length $n.$ For this purpose, to each equivalence class $\widetilde{x},x\in X,$ will correspond the codeword $w_{x}=x_{1}x_{2}...x_{n}$ with $x_{i}=j,$ if and only if $f_{x}\left( i\right) =j,i\in A,j\in \{0,1\}.$We denote this code with $V_{X}. $
Let $V$ be a binary block-code and $w_{x}=x_{1}x_{2}...x_{n}\in V,$ $w_{y}=y_{1}y_{2}...y_{n}\in V$ be two codewords$.$ On $V$ we can define the following partial order relation: $$w_{x}\preceq w_{y}\text{ if and only if }y_{i}\leq x_{i},i\in \{1,2,...,n\}.
\tag{1.1.}$$
In the paper \[7\], the authors constructed binary block-codes generated by BCK-functions. At the end of the paper they put the following question: *for each binary block-code* $V$, *there is a BCK-function which determines* $V$? The answer of this question is partial affirmative, as we can see in Theorem 2.2 and Theorem 2.9. $$$$
**2. Main results**$$$$
Let $(X,\leq )$ be a finite partial ordered set with the minimum element $\theta $ . We define the following binary relation $"\ast "~$on $X:$$$\left\{
\begin{array}{c}
\theta \ast x=\theta \text{ and }x\ast x=\theta ,\forall x\in X; \\
x\ast y=\theta ,\text{ if \ }x\leq y,\ \ \ x,y\in X; \\
x\ast y=x,\text{ if }y<x,~\ x,y\in X; \\
x\ast y=y,\text{ if }x\in X\text{ ~and~ }y\in X\text{ ~can't be compared.}\end{array}\right. \tag{2.1.}$$
**Proposition 2.1.** *With the above notations, the algebra* $\left( X,\ast ,\theta \right) $ *is a non-commutative and non-implicative BCK-algebra.* $\Box \medskip $
If the above BCK-algebra has $n$ elements, we will denote it with $\mathcal{C}_{n}.\medskip $
Let $V$ be a binary block-code with $n$ codewords of length $n.$ We consider the matrix $M_{V}=\left( m_{i,j}\right) _{i,j\in \{1,2,...,n\}}\in \mathcal{M}_{n}(\{0,1\})$ with the rows consisting of the codewords of $V.$ This matrix is called *the matrix associated to the code* $V.\medskip $
**Theorem 2.2.** *With the above notations, if the codeword* $\underset{n-\text{time}}{\underbrace{11...1}}$ *is in* $V$ *and the matrix* $M_{V}$ *is upper triangular with* $m_{ii}=1,$ *for all* $i\in \{1,2,...,n\}$, *there are a set* $A$ *with* $n$ *elements, a BCK-algebra* $X$ *and a BCK-function* $f:A\rightarrow X$ *such that* $f$ *determines* $V.\medskip $
**Proof.** We consider on $V$ the lexicographic order, denoted by $\leq _{lex}$. It results that $(V,\leq _{lex})$ is a totally ordered set. Let $V=\{w_{1},w_{2},...,w_{n}\},$ with $w_{1}\geq _{lex}w_{2}\geq
_{lex}...\geq _{lex}w_{n}.$ From here, we obtain that $w_{1}=\underset{n-\text{time}}{\underbrace{11...1}}$ and $w_{n}=\underset{(n-1)-\text{time}}{\underbrace{00...0}1}.$ On $V$ we define a partial order $\preceq $ as in Remark 1.4. Now, $\left( V,\preceq \right) $ is a partial ordered set with $\ w_{1}\preceq w_{i},i\in \{1,2,...,n\}.$ We remark that $w_{1}=\theta \ $ is the “zero” in $\left( V,\preceq \right) $ and $w_{n}$ is a maximal element in $\left( V,\preceq \right) .$ We define on $\left( V,\preceq
\right) $ a binary relation $"\ast "$ as in Proposition 2.1. It results that $X=\left( V,\ast ,w_{1}\right) $ becomes a BCK-algebra and $V$ is isomorphic to $\mathcal{C}_{n}$ as BCK-algebras. We consider $A=V$ and the identity map $f:A\rightarrow V,f\left( w\right) =w$ as a BCK-function. The decomposition of $f$ provides a family of maps $V_{\mathcal{C}_{n}}=\{f_{r}:A\rightarrow \{0,1\}~/~$ $f_{r}\left( x\right) =1,$ if and only if $r\ast f\left( x\right) =\theta ,\forall x\in A,r\in X\}.$ This family is the binary block-code $V$ relative to the order relation $\preceq
. $ Indeed, let $w_{k}\in V,1<k<n,$ $w_{k}=\underset{k-1}{\underbrace{00...0}}x_{i_{k}}...x_{i_{n}},~~x_{i_{k}}...x_{i_{n}}\in \{0,1\}.$ If $x_{i_{j}}=0,$ it results that $w_{k}\preceq w_{i_{j}}$ and $w_{k}\ast w_{i_{j}}=\theta .$ If $x_{i_{j}}=1,$ we obtain that $w_{i_{j}}\preceq w_{k}$ or $w_{i_{j}}$ and $w_{k}$ can’t be compared, therefore $w_{k}\ast w_{i_{j}}=w_{k}.\medskip
\Box $
**Remark 2.3.** Using technique developed in \[7\], we remark that a BCK-algebra determines a unique binary block-code, but a binary block-code as in Theorem 2.2 can be determined by two or more algebras(see Example 3.1). If two BCK-algebras, $A_{1},A_{2}$ determine the same binary block-code, we call them *code-similar algebras,* denoted by $A_{1}\thicksim A_{2}$. We denote by $\mathfrak{C}_{n}$ the set of the binary block-codes of the form given in the Theorem 2.2.$\medskip $
**Remark 2.4.** If we consider $\mathfrak{B}_{n}$, the set of all finite BCK-algebras with* *$\ n$ elements, then the relation code-similar is an equivalence relation on $\mathfrak{B}_{n}$*.* Let $\mathfrak{Q}_{n}$ be the quotient set. For $V\in \mathfrak{C}_{n}$, an equivalent class in $\mathfrak{Q}_{n}$ is $\widehat{V}=\{B\in \mathfrak{B}_{n}$ $/$ $B$ determines the binary block-code $V\}.\medskip $
**Proposition 2.5.** *The quotient set* $\mathfrak{Q}_{n}$ *has* $2^{\frac{\left( n-1\right) \left( n-2\right) }{2}}$ *elements, the same cardinal as the set* $\mathfrak{C}_{n}$.
**Proof.** We will compute the cardinal of the set $\mathfrak{C}_{n}.$ For $V\in \mathfrak{C}_{n},$ let $M_{V}$ be its associated matrix. This matrix is upper triangular with $m_{ii}=1,$ for all $i\in \{1,2,...,n\}.$ We calculate in how many different ways the rows of such a matrix can be written. The second row of the matrix $M_{V}$ has the form $\left(
0,1,a_{3},...,a_{n}\right) ,$ where $a_{3},...,a_{n}\in \{0,1\}.$ Therefore, the number of different rows of this type is $2^{n-2}$ and it is equal with the number of functions from a set with $n-2$ elements to the set $\{0,1\}.$ The third row of the matrix $M_{V}$ has the form $\left(
0,0,1,a_{4},...,a_{n}\right) ,$ where $a_{4},...,a_{n}\in \{0,1\}.$ In the same way, it results that the number of different rows of this type is $2^{n-3}.$ Finally, we get that the cardinal of the set $\mathfrak{C}_{n}$ is $2^{n-2}2^{n-3}...2=$ $2^{\frac{\left( n-1\right) \left( n-2\right) }{2}}$.$\Box \medskip $
**Remark 2.6.** If $\mathfrak{N}_{n}$ is the number of all finite non-isomorphic BCK-algebras with $\ n$ elements, then* *$\mathfrak{N}_{n}\geq 2^{\frac{\left( n-1\right) \left( n-2\right) }{2}}.\medskip $
**Remark 2.7.** 1) Let $V_{1},V_{2}\in \mathfrak{C}_{n}$ and $M_{V_{1}},M_{V_{2}}$ be the associated matrices. We denote by $r_{j}^{V_{i}}
$ a row in the matrix $M_{V_{i}},i\in \{1,2\},~j\in \{1,2,...,n\}.$ On $\mathfrak{C}_{n},~$we define the following totally order relation $$V_{1}\succeq _{lex}V_{2}~\text{if~there~is~}i\in \{2,3,...,n\}~\text{such~that~}r_{1}^{V_{1}}=r_{1}^{V_{2}},...,r_{i-1}^{V_{1}}=r_{i-1}^{V_{2}}~\text{and~~}r_{i}^{V_{1}}\geq _{lex}r_{i}^{V_{2}},$$where $\geq _{lex}$ is the lexicographic order$.$
2\) Let $V_{1},V_{2}\in \mathfrak{C}_{n}$ and $M_{V_{1}},M_{V_{2}}$ be the associated matrices. We define a partially order on $\mathfrak{C}_{n}$ $$V_{1}\ll V_{2}~\text{if~there~is~}i\in \{2,3,...,n\}\ \text{such\ that\ }r_{1}^{V_{1}}=r_{1}^{V_{2}},...,r_{i-1}^{V_{1}}=r_{i-1}^{V_{2}}\ \text{and\ }r_{i}^{V_{1}}\preceq r_{i}^{V_{2}},$$ where $\preceq $ is the order relation given by the relation $\left(
1.1\right) .$
3\) Let $\Theta =\left( \theta _{ij}\right) _{i,j\in \{1,2,...,n\}}\in
\mathcal{M}\left( \{0,1\}\right) $ be a matrix such that$~\theta _{ij}=1,$ $i\leq j,$ for all $i,j\in \{1,2,...,n\}$ and $\theta _{ij}=0$ in the rest. It results that the code $\Omega ,$ such that $M_{\Omega }=$ $\Theta ,$ is the minimum element in the partial ordered set $\left( \mathfrak{C}_{n},\text{ }\ll \right) ,$ where elements in $\mathfrak{C}_{n}$ are descending ordered relative to $\succeq _{lex}$ defined in 1). Using the multiplication $"\ast "$ given in relation $\left( 2.1\right) $ and Proposition 2.1, we obtain that $\left( \mathfrak{C}_{n},\ast ,\Omega \right) $ is a non-commutative and non-implicative BCK-algebra*. *Due to the above remarks and relation $\left( 2.1\right) ,~$this BCK-algebra determines a binary block-code $V_{\mathfrak{C}_{n}}\ $of length $2^{\frac{\left( n-1\right) \left( n-2\right) }{2}}.$ Obviously, $\widehat{V}_{\mathfrak{C}_{n}}\in \mathfrak{C}_{2^{\frac{\left( n-1\right) \left(
n-2\right) }{2}}}.\medskip $
**Proposition 2.8.** *Let* $A=\left( a_{i,j}\right)
_{\substack{ i\in \{1,2,...,n\} \\ j\in \{1,2,...,m\}}}\in \mathcal{M}_{n,m}(\{0,1\})$ *be a matrix with rows lexicographic ordered in the descending sense. Starting from this matrix, we can find a matrix* $B=\left( b_{i,j}\right) _{i,j\in \{1,2,...,q\}}\in \mathcal{M}_{q}(\{0,1\}),$ $q=n+m,$ *such that* $B$ *is an upper triangular matrix, with* $b_{ii}=1,\forall i\in \{1,2,...,q\}$ *and* $A$ *becomes a submatrix of the matrix* $B.\medskip $
**Proof.** We insert in the left side of the matrix $A$ ( from the right to the left) the following $n$ new columns of the form $\underset{n}{\underbrace{00...01}},\underset{n}{\underbrace{00...10}},...,\underset{n}{\underbrace{10...00}}.$ It results a new matrix $D$ with $n$ rows and $n+m$ columns. Now, we insert in the bottom of the matrix $D$ the following $m$ rows: $\underset{n}{\underbrace{00...0}}\underset{m}{\underbrace{10...00}}$ , $\underset{n+1}{\underbrace{00...0}}\underset{m-1}{\underbrace{01...00}},...,\underset{n+m-1}{\underbrace{000}}1.$ We obtained the asked matrix $B.\Box \medskip $
**Theorem 2.9.** *With the above notations, we consider* $V$ * a binary block-code with* $n$ *codewords of length* $m,n\neq m,$ *or a block-code with* $n$ *codewords of length* $n\,\ $*such that* *the codeword* $\underset{n-\text{time}}{\underbrace{11...1}}$ *is not in* $V,$* or a block-code with* $n$ *codewords of length* $n\,\ $*such that the matrix* $M_{V}$ *is not upper triangular*$.$ *There are a natural number* $q\geq \max \{m,n\}$, *a set* $A$ *with* $m$ *elements and a BCK-function* $f:A\rightarrow \mathcal{C}_{q}$ *such that the obtained block-code* $V_{\mathcal{C}_{n}}$ *contains the block-code* $V$ *as a subset.*
**Proof.** Let $V$ be a binary block-code, $V=\{w_{1,}w_{2},...,w_{n}\}, $ with codewords of length $m.$ We consider the codewords $w_{1,}w_{2},...,w_{n}$ lexicographic ordered, $w_{1}\geq _{lex}w_{2}\geq
_{lex}...\geq _{lex}w_{n}.$ Let $M\in \mathcal{M}_{n,m}(\{0,1\})$ be the associated matrix with the rows $w_{1},...,w_{n}$ in this order. Using Proposition 2.8, we can extend the matrix $M$ to a square matrix $M^{\prime
}\in \mathcal{M}_{q}(\{0,1\}),q=m+n,$ such that $M^{\prime }=\left(
m_{i,j}^{\prime }\right) _{i,j\in \{1,2,...,q\}}$ is an upper triangular matrix with $m_{ii}=1,$ for all $i\in \{1,2,...,q\}.$ If the first line of the matrix $M^{\prime }$ is not $\underset{q}{\underbrace{11...1}},$ then we insert the row $\underset{q+1}{\underbrace{11...1}}$ as a first row and the column $1\underset{q}{\underbrace{0...0}}~$as a first column $.$ Applying Theorem 2.2 for the matrix $M^{\prime },$ we obtain a BCK-algebra $\mathcal{C}_{q}=\{x_{1},...,x_{q}\},$with $x_{1}=\theta $ the zero of the algebra $\mathcal{C}_{q}$ and a binary block-code $V_{\mathcal{C}_{q}}.\ $Assuming that the initial columns of the matrix $M$ have in the new matrix $M^{\prime
}$ positions $i_{j_{1}},i_{j_{2}},...,i_{j_{m}}\in \{1,2,...,q\},$ let $A=\{x_{j_{1}},x_{j_{2}},...,x_{j_{m}}\}\subseteq \mathcal{C}_{q}.$ The BCK-function $f:A\rightarrow \mathcal{C}_{q},f\left( x_{j_{i}}\right) =$ $x_{j_{i}},$ $i\in \{1,2,...,m\},$ determines the binary block-code $V_{\mathcal{C}_{q}}$ such that $V\subseteq V_{\mathcal{C}_{q}}.\Box $
$$$$
**3. Examples**
$$$$
**Example 3.1.** Let $V=\{0110,0010,1111,0001\}$ be a binary block code. Using the lexicographic order, the code $V$ can be written $V=\{1111,0110,0010,0001\}=\{w_{1},w_{2},w_{3},w_{4}\}.$ From Theorem 2.2, defining the partial order $\preceq $ on $V,$ we remark that $w_{1}\preceq
w_{i},i\in \{2,3,4\},w_{2}\preceq w_{3},w_{2}$ can’t be compared with $w_{4}$ and $w_{3}$ can’t be compared with $w_{4}$. The operation $"\ast "$ on $V$ is given in the following table:
$\ast $ $w_{1}$ $w_{2}$ $w_{3}$ $w_{4}$
--------- --------- --------- --------- ---------
$w_{1}$ $w_{1}$ $w_{1}$ $w_{1}$ $w_{1}$
$w_{2}$ $w_{2}$ $w_{1}$ $w_{1}$ $w_{2}$
$w_{3}$ $w_{3}$ $w_{3}$ $w_{1}$ $w_{3}$
$w_{4}$ $w_{4}$ $w_{4}$ $w_{4}$ $w_{1}$
.
Obviously, $V$ with the operation $"\ast "$ is a BCK-algebra.
We remark that the same binary block code $V$ can be obtained from the BCK-algebra $(A,\circ ,\theta )$
$\circ $ $\theta $ $a$ $b$ $c$
----------- ----------- ----------- ----------- -----------
$\theta $ $\theta $ $\theta $ $\theta $ $\theta $
$a$ $a$ $\theta $ $\theta $ $a$
$b$ $b$ $a$ $\theta $ $b$
$c$ $c$ $c$ $c$ $\theta $
with BCK-function, $f:V\rightarrow V,f(x)=x.$(see \[7\] , Example 4.2). From the associated Cayley multiplication tables, it is obvious that the algebras $(A,\circ ,\theta )$ and $(V,\ast ,w_{1})$ are not isomorphic. From here, we obtain that BCK-algebra associated to a binary block-code as in Theorem 2.2 is not unique up to an isomorphism. We remark that the BCK-algebra $(A,\circ
,\theta )$ is commutative and non implicative and BCK-algebra $(V,\ast
,w_{1})$ is non commutative and non implicative. Therefore, if we start from commutative BCK-algebra $(A,\circ ,\theta )$ to obtain the code $V,$ as in \[7\], and then we construct the BCK-algebra $(V,\ast ,w_{1}),$ as in Theorem 2.2, the last obtained algebra lost the commutative property even that these two algebras are code-similar.
**Example 3.2.** Let $X$ be a non empty set and $\mathfrak{F}=\{f:X\rightarrow \{0,1\}~/$ $f$ function$\}.$ On $\mathfrak{F}$ is defined the following multiplication$$(f\circ g)\left( x\right) =f\left( x\right) -\text{\textit{min}}\{f\left(
x\right) ,g\left( x\right) \},\forall x\in X.$$
$\left( \mathfrak{F},\circ ,\mathbf{0}\right) $, where $\mathbf{0}\left(
x\right) =0,\forall x\in X,$ is an implicative BCK-algebra(\[12\], Theorem 3.3 and Example 1).
If $X$ is a set with three elements, we can consider $\mathfrak{F}=\{000,001,010,011,100,101,110,111\}$ the set of binary block-codes of length $3.$ We have the following multiplication table.
$\circ $ $000$ $001$ $010$ $011$ $100$ $101$ $110$ $111$ [The obtained binary code-words]{}
---------- ------- ------- ------- ------- ------- ------- ------- ------- ------------------------------------
$000$ 000 000 000 000 000 000 000 11111111
001 000 001 000 001 000 001 000 01010101
010 010 000 000 010 010 000 000 00110011
011 010 001 000 011 010 001 000 00010001
100 100 100 100 000 000 000 000 00001111
101 100 101 100 001 000 001 000 00000101
110 110 100 100 010 010 000 000 00000011
111 110 101 100 011 010 001 000 00000001
.
We obtain the following binary block-code$V=\{11111111,01010101,00110011,00010001,$$00001111,00000101,00000011,00000001\},$ with the elements lexicographic ordered in the descending sense$.$ From Theorem 2.2, defining the partial order $\preceq $ on $V$ and the multiplication $"\ast ",$ we have that $\left( V,\ast ,11111111\right) $ is a non-implicative BCK-algebra and the algebras $\left( V,\ast ,11111111\right) $ and $\left( \mathfrak{F},\circ ,\mathbf{0}\right) $ are code-similar.
**Example 3.3.** Let $V=\{11110,10010,10011,00000\}$ be a binary block code. Using the lexicographic order, the code $V$ can be written $V=\{11110,10011,10010,00000\}=\{w_{1},w_{2},w_{3},w_{4}\}.$ Let $M_{V}\in
\mathcal{M}_{4,5}\left( \{0,1\}\right) $ be the associated matrix$,$ $M_{V}=\left(
\begin{tabular}{lllll}
$1$ & $1$ & $1$ & $1$ & $0$ \\
$1$ & $0$ & $0$ & $1$ & $1$ \\
$1$ & $0$ & $0$ & $1$ & $0$ \\
$0$ & $0$ & $0$ & $0$ & $0$\end{tabular}\right) .$ Using Proposition 2.8, we construct an upper triangular matrix, starting from the matrix $M_{V}.$ It results the following matrices: $D=\left(
\begin{tabular}{lllllllll}
$1$ & $0$ & $0$ & $0$ & $$ & $$ & $$ & $$ & $$ \\
$0$ & $1$ & $0$ & $0$ & $$ & $$ & $$ & $$ & $$ \\
$0$ & $0$ & $1$ & $0$ & $$ & $$ & $$ & $$ & $$ \\
$0$ & $0$ & $0$ & $1$ & $$ & $$ & $$ & $$ & $$\end{tabular}\right) $ and $B=\left(
\begin{array}{ccccccccc}
1 & 0 & 0 & 0 & \mathbf{1} & \mathbf{1} & \mathbf{1} & \mathbf{1} & \mathbf{0} \\
0 & 1 & 0 & 0 & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{1} \\
0 & 0 & 1 & 0 & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\
0 & 0 & 0 & 1 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right) .$
Since the first row is not $\underset{9}{\underbrace{11...1}},$ using Theorem 2.8, we insert a new row $\underset{10}{\underbrace{11...1}}$ as a first row and a new column $\underset{10}{\underbrace{10...0}}$ as a first column. We obtain the following matrix: $B^{\prime }=\left(
\begin{array}{cccccccccc}
1 & 1 & 1 & 1 & 1 & \mathbf{1} & \mathbf{1} & \mathbf{1} & \mathbf{1} &
\mathbf{1} \\
0 & 1 & 0 & 0 & 0 & \mathbf{1} & \mathbf{1} & \mathbf{1} & \mathbf{1} &
\mathbf{0} \\
0 & 0 & 1 & 0 & 0 & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{1} &
\mathbf{1} \\
0 & 0 & 0 & 1 & 0 & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{1} &
\mathbf{0} \\
0 & 0 & 0 & 0 & 1 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} &
\mathbf{0} \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right) .$The binary block-code $W=\{w_{1},...,w_{10}\},$ whose codewords are the rows of the matrix $B^{\prime },$ determines a BCK-algebra $(X,\ast ,w_{1}).$ Let $A=\{w_{6},w_{7},w_{8},w_{9},w_{10}\}$ and $f:A\rightarrow X,f\left(
w_{i}\right) =w_{i},i\in \{6,7,8,9,10\}$ be a BCK-function which determines the binary block-code$U=\{11111,11110,10011,10010,00000,10000,01000,00100,00010,00001\}.$ The code $V$ is a subset of the code $U.\medskip $
**Conclusions.** In this paper, we proved that to each binary block-code $V$ we can associate a BCK-algebra $X$ such that the binary block-code generated by $X,V_{X},$ contains the code $V$ as a subset. In some particular case, we have $V_{X}=V.$
From Example 3.1 and 3.2, we remark that two code-similar BCK-algebras can’t have the same properties. For example, some algebras from the same equivalence class can be commutative and other non-commutative or some algebras from the same equivalence class can be implicative and other non-implicative. As a further research, will be very interesting to study what common properties can have two code-similar BCK-algebras.
Due to this connection of BCK-algebras with Coding Theory, we can consider the above results as a starting point in the study of new applications of these algebras in the Coding Theory. $$$$**Acknowledgements**
I want thank to anonymous referees for their comments, suggestions and ideas which helped me to improve this paper. The author also thanks Professor Arsham Borumand Saeid for having brought \[7\] to my attention.$$$$**References**$$$$
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\[13\] X. Xin, Y. Fu, *Some results of convex fuzzy sublattices*, J. Intell. Fuzzy Syst., **27(1)(2014)**, 287-298.$$$$Cristina FLAUT
[Faculty of Mathematics and Computer Science, Ovidius University,]{}
[Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA]{}
[http://cristinaflaut.wikispaces.com/; http://www.univ-ovidius.ro/math/]{}
[e-mail: [email protected]; cristina\[email protected]]{}
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abstract: 'Near-term studies of Venus-like atmospheres with JWST promise to advance our knowledge of terrestrial planet evolution. However, the remote study of Venus in the Solar System and the ongoing efforts to characterize gaseous exoplanets both suggest that high altitude aerosols could limit observational studies of lower atmospheres, and potentially make it challenging to recognize exoplanets as “Venus-like”. To support practical approaches for exo-Venus characterization with JWST, we use Venus-like atmospheric models with self-consistent cloud formation of the seven TRAPPIST-1 exoplanets to investigate the atmospheric depth that can be probed using both transmission and emission spectroscopy. We find that JWST/MIRI LRS secondary eclipse emission spectroscopy in the 6 $\mu$m opacity window could probe at least an order of magnitude deeper pressures than transmission spectroscopy, potentially allowing access to the subcloud atmosphere for the two hot innermost TRAPPIST-1 planets. In addition, we identify two confounding effects of sulfuric acid aerosols that may carry strong implications for the characterization of terrestrial exoplanets with transmission spectroscopy: (1) there exists an ambiguity between cloud-top and solid surface in producing the observed spectral continuum; and (2) the cloud-forming region drops in altitude with semi-major axis, causing an increase in the observable cloud-top pressure with decreasing stellar insolation. Taken together, these effects could produce a trend of thicker atmospheres observed at lower stellar insolation—a convincing false positive for atmospheric escape and an empirical “cosmic shoreline”. However, developing observational and theoretical techniques to identify Venus-like exoplanets and discriminate them from stellar windswept worlds will enable advances in the emerging field of terrestrial comparative planetology.'
author:
- 'Jacob Lustig-Yaeger'
- 'Victoria S. Meadows'
- 'Andrew P. Lincowski'
bibliography:
- 'bib.bib'
title: |
A mirage of the cosmic shoreline:\
Venus-like clouds as a statistical false positive for exoplanet atmospheric erosion
---
Introduction\[sec:intro\]
=========================
Venus-like exoplanets pose unique opportunities and challenges for the near-term characterization of terrestrial exoplanet atmospheres [@Arney2018b]. Exo-Venuses are key near-term observational targets due to the transit bias in favor of finding and characterizing planets at short orbital period (e.g. high transit probability, high transit frequency, high equilibrium temperature), particularly in the *TESS* era [@Ostberg2019]. Planets at similar insolation and with similar bulk properties to Venus are also favorable laboratories to empirically test runaway greenhouse theory, identify the location of the inner edge of the habitable zone (HZ), and probe the impact of atmospheric escape on an ensemble of terrestrial planets [@Kane2014b]. Additionally, the comparative study of Venus and Venus-like exoplanets are mutually beneficial research avenues [@Arney2018b]. Within the exoplanet population, if exo-Venuses are found to be common, they would point to a common end-state of terrestrial exoplanet evolution that Venus exemplifies. However, if true Venus analogs are rare, that may point to a more specific origin for Venus. Within the solar system, future orbiters and descent probes could provide detailed, in situ measurements to help answer outstanding questions about evolutionary processes and the current state of Venus, which will provide crucial context for the population of exo-Venuses [see recent white papers: @Kane2018; @Kane2019; @Wilson2019]. However, these exciting opportunities are contingent upon our ability to properly recognize and accurately characterize a Venus-like exoplanet when we see one.
Remote sensing observations have been used to understand and probe beneath the optically thick and global sulfuric acid clouds and hazes, which extend from 48 to 90 km altitude, and obscure the lower atmosphere and surface of Venus at most wavelengths. Although clouds were suspected early on due to Venus’s high albedo and UV markings [@Hunten1983book], their composition was unknown until optical phase curves ruled out water clouds [@Arking1968; @Hansen1971a], multi-band polarization phase curves matched the real index of refraction for a concentrated solution of sulfuric acid [@Hansen1971b], and NIR absorption features confirmed [@Pollack1974]. These clouds thoroughly obscure the hot lower atmosphere at visible wavelengths, but the first clue to the extremely hot nature of the surface environment was a radio brightness temperature measurement of ${\sim} 560$ K at 3.15 cm by @Mayer1958, which was later confirmed by spacecraft observations [e.g. @Barath1963] and descent probes [e.g. @Marov1973]. Despite these challenges, peering beneath the clouds into the hot lower atmosphere has been possible with spectroscopy targeting near-infrared windows on the Venus night side through which thermal emission from below the clouds escapes [e.g. @Allen1984; @Allen1987; @Carlson1991; @Crisp1991], enabling remote studies of the Venus lower atmosphere and surface [e.g. @Drossart1993; @deBergh1995; @Meadows1996; @Barstow2012; @Arney2014].
Extending the lessons learned from Venus remote sensing to the characterization of potential exo-Venuses may be challenging as the historically most informative Venus observations lack feasible exoplanet analogs, either because they were made from orbiters or descent probes, or used radio brightness or precise optical and polarization phase curves.
After launch, the *James Webb Space Telescope* (JWST) will likely be used to attempt characterization of Venus-like exoplanets [@Barstow2016; @Morley2017; @Lincowski2018], but these observations may be limited by how transmission and emission spectra are both significantly impacted by Venus-like clouds [@Lustig-Yaeger2019]. It has been well established by theory and observation that transmission spectroscopy is sensitive to obscuration by high altitude aerosols [e.g. @Fortney2005; @Berta2012; @Ehrenreich2014; @Knutson2014; @Kreidberg2014; @Nikolov2015; @Morley2013; @Charnay2015b; @Charnay2015a]. Additionally, although dayside thermal emission, which is sensitive to the cloud deck temperatures, may be observed for exoplanets via secondary eclipse, nightside NIR thermal emission windows, which are sensitive to the lower atmosphere, may be significantly more challenging to observe for exoplanets when the far brighter dayside portion is included in the disk average. Despite these challenges, modeling efforts in advance of JWST indicate that the presence of cloudy Venus-like atmospheres could be detected for all seven planets in the TRAPPIST-1 system using JWST transmission spectroscopy to identify absorption features in the thin atmosphere above the clouds [@Lustig-Yaeger2019].
Another potential complication in identifying and interpreting spectra of Venus-like planets comes from the behavior of sulfuric acid cloud formation as a function of semi-major axis. The super-luminous pre-main-sequence phase of late M dwarfs, like TRAPPIST-1, could produce a string of Venuses, extending from interior to, through, and beyond the HZ [@Lincowski2018]. During the pre-main-sequence phase, each of the TRAPPIST-1 planets may have been subjected to runaway greenhouse driven water loss and subsequent buildup [@Luger2015; @Bolmont2017; @Lincowski2018], even for planets well beyond the HZ. The subsequent sequestration of and the outgassing of volatiles over time [@Schaefer2016; @Garcia-Sage2017] may have allowed high- Venus-like atmospheres to develop.
@Lincowski2018 conducted a systematic study of the seven TRAPPIST-1 planets assuming they possess Venus-like atmospheres using a self-consistent 1D photochemical and climate model, which included sulfuric acid cloud formation. Interestingly, these models demonstrated that sulfuric acid clouds form high in the atmospheres of hot Venus-like planets, but drop to lower altitudes for cooler Venus-like planets at lower incident stellar fluxes [@Lincowski2018]. Since high altitude clouds can obscure molecular features in a transmission spectrum, the hottest cloudy exo-Venus atmospheres may actually be more difficult to detect than cooler cloudy exo-Venuses [@Lustig-Yaeger2019]—a practical manifestation of only probing the atmosphere above the clouds. However, if only the upper, above-cloud, region of the atmosphere is readily probed, we may remain ignorant to the existence of a lower atmosphere, unable to distinguish cloud-top from solid surface. Furthermore, the predicted increases in cloud top pressure with semi-major axis occur across a stellar insolation range that could also completely erode planetary atmospheres [@Dong2018]. Thus, observing a trend of thicker cloud-truncated atmospheres at lower stellar insolation may produce a statistical false positive for atmospheric escape across a population of terrestrial exoplanets [e.g. @Bean2017; @Checlair2019] and a mirage of the “cosmic shoreline”—an empirical dividing line between planets with and without atmospheres [@Zahnle2017].
In this letter we explore two fundamental questions on the characterization of Venus-like exoplanets and their potential contribution to our understanding of terrestrial exoplanet atmospheric evolution: *(1) how do we infer the presence of and study sub-cloud atmospheres, and (2) what consequences and misinterpretations may arise if we cannot?* In particular, we demonstrate how the presence of sulfuric acid clouds in *thick* Venus-like atmospheres can mimic *thin* cloud-free atmospheres in a transmission spectrum. We then explore how an observed decrease in cloud top altitude as a function of orbital distance could be misinterpreted as a surface pressure trend. Furthermore, such a trend with incident stellar flux could arise due to (1) atmospheric erosion via photoevaporation/thermal escape if the spectral continuum is assumed to be a solid surface, or (2) cloud top altitude variations due to condensation temperature if the continuum is assumed to be a cloud top. Finally, we offer observational and theoretical research avenues that may help to resolve this potential statistical false positive.
In Section \[sec:methods\] we describe the TRAPPIST-1 Venus-like atmospheric models used in this paper. In Section \[sec:results\] we investigate the atmospheric regions probed by the transmission and emission spectra of Venus-like exoplanets applicable to JWST observations. In Section \[sec:discussion\] we discuss the optimal paths towards inferring the presence of lower atmospheres for Venus-like exoplanets and we also expand on the hypothesis that, if we cannot detect lower atmospheres, atmospheric erosion could be invoked to explain mistakenly thin atmospheres, particularly in statistical characterization populations if Venus-like exoplanets are intrinsically common. We conclude in Section \[sec:conclusion\].
Methods {#sec:methods}
=======
We use the clear and cloudy Venus-like TRAPPIST-1 planet atmospheric models from @Lincowski2018 as a foundation for the investigations in this paper. Briefly, @Lincowski2018 used the VPL Climate model, a 1D radiative-convective equilibrium climate model applicable to terrestrial planet atmospheres [@Meadows2018; @Robinson2018b]. The climate model is coupled to a 1D atmospheric photochemistry model originally developed by @Kasting1979 and significantly improved upon by @Zahnle2006; this code is described in detail in @Meadows2018 and has been used extensively for terrestrial exoplanet photochemical modeling across a broad range of redox states [e.g. @Segura2005; @Arney2016; @Arney2017; @Schwieterman2016; @Arney2019]. In particular, the photochemical code was specifically updated and validated for modeling Venus-like atmospheres [@Lincowski2018].
@Lincowski2018 used radiatively-active, photochemically-self-consistent sulfuric acid aerosols in their climate and spectral calculations. These calculations considered the photochemical production of vapor given the forcing from the late M dwarf SED, the temperature-dependent condensation of , the sedimentation of condensates, and their thermal decomposition at high temperatures in the lower atmosphere. Together these effects determined the aerosol effective radii and concentration in each layer. @Lincowski2018 used refractive indices for sulfuric acid solutions from @Palmer1975 ranging in concentration from 25-100%, which were calculated for each atmosphere using the vapor pressure equilibrium between the and gases and the condensed solution. Mass-conserving log-normal aerosol particle distributions were computed (with geometric standard deviation equal to 0.25, as used in @Crisp1986 for Venus), from which mie-scattering phase functions and optical depths were calculated. We note that @Lincowski2018 found that sulfuric acid clouds did not form for a Venus-like TRAPPIST-1 b because the atmosphere was too hot for them to condense, so TRAPPIST-1 b is omitted from our cloudy Venus-like cases. We discuss the observational implications of this result in Section \[discussion:mitigation\].
Transmission and emission spectra of the TRAPPIST-1 Venus-like planets were produced in @Lincowski2018 using the Spectral Mapping Atmospheric Radiative Transfer (SMART) code [developed by D. Crisp; @Meadows1996]. SMART is a line-by-line, multi-stream, multi-scattering radiative transfer code that includes layer dependent gaseous and aerosol absorption and scattering, and treats both stellar and thermal source functions. SMART also calculates transmission spectra for transiting exoplanets using a ray tracing algorithm that includes the refraction of stellar light passing through the atmosphere [@Misra2014a; @Robinson2018]. Gaseous rotational-vibrational line absorption coefficients were calculated using the LBLABC model [@Meadows1996] with the HITEMP2010 and HITRAN2012 line lists [@Rothman2010; @Rothman2013]. To understand the maximum possible depth that can be probed into clear and cloudy Venus-like atmospheres, we processed the atmospheric structure and spectra of the Venus-like TRAPPIST-1 models from @Lincowski2018 to reveal the average pressure into each atmosphere that can be probed with transmission and emission spectroscopy. For transmission spectroscopy, we used the relationship between altitude and pressure for atmospheres in hydrostatic equilibrium to interpolate the effective transit height to an effective transit pressure. Although this so called “transit pressure” is not a direct observable, it is tied to the observable transit depth, $(R_p/R_s)^2$, and approximates the depth into the atmosphere at which it becomes optically thick in the slant transit geometry. While this is similar to the effective transit height, the transit height increases radially out of the atmosphere and must assume a zero-point altitude that presumes knowledge of the planet’s solid body radius—a key point of interest—but which is unknown *a priori*. Alternatively, the transit pressure increases into the atmosphere from space—a known zero point pressure boundary condition—to the maximum pressure probed, and is therefore a good measure of how deep into the atmosphere the transmission spectrum probes. The maximum transit pressure across a given wavelength range also provides forward modeling insight into the cloud top pressure or reference pressure that would be inferred by atmospheric retrievals [e.g. @Benneke2013; @Kreidberg2015; @Line2016; @Benneke2015].
To assess the depth probed into Venus-like atmospheres with emission spectroscopy, we recomputed the radiative transfer with SMART for the @Lincowski2018 Venus-like models to solve for the atmospheric pressure at which the total optical depth is unity at normal incidence. SMART calculates the pressure of optical depth unity in terms of Rayleigh scattering, gaseous absorption, and aerosol extinction. We combine these three terms into a single “emission pressure” or “brightness pressure” spectrum, which reflects the dominating process at each wavelength and approximates the pressure from which thermal emission emerges the atmosphere. Note that unlike the transit pressure, which resembles the observable transmission spectrum $(R_p/R_s)^2$, the “emission pressure” is significantly different from the observable eclipse depths of an emission spectrum ($F_p / F_s$), which have a strong wavelength dependence in accordance with the planet and star fluxes. Rather, the emission pressure more closely resembles the transit pressure, allowing for qualitative and quantitative comparisons between the two observational techniques.
Results {#sec:results}
=======
Transmission Spectroscopy
-------------------------
{width="98.00000%"} {width="98.00000%"}
To explore the impact of clouds on the range of pressures probed by transmission spectroscopy, Figure \[fig:transit\_pressure\] shows model transmission spectra for each TRAPPIST-1 planet with and without clouds in the upper and lower panels, respectively. The transmission spectrum is shown between about 1-5 $\mu$m—the range of the JWST/NIRSpec Prism instrument, which is optimal for detecting the atmospheres of the TRAPPIST-1 planets [@Batalha2018; @Lustig-Yaeger2019]—in units of the pressure into the atmosphere that is probed. The right panels of Fig. \[fig:transit\_pressure\] show the thermal structure of each TRAPPIST-1 planet atmosphere on the same pressure y-axis as the transmission spectrum. Thicker line styles indicate the vertical region of the atmosphere that the transmission spectrum is sensitive to, and the circular points denote the maximum atmospheric pressure that is probed. The top-right panel also shows the cumulative vertical optical depth for the aerosols on the top x-axis, which increases going down into the atmosphere.
Although the clear and cloudy transmission spectra appear similar due to the common presence of absorption bands, their respective continua vary by up to 3 orders of magnitude in pressure, which affects the strength and detectability of absorption [see @Lustig-Yaeger2019] and the depth into the atmosphere that may be probed by the spectrum. In units of pressure, the transmission spectra of different planets with atmospheres of similar compositions look quite similar, despite having different radii, masses, and temperatures. At wavelengths where the atmosphere is optically thick due to strong absorption (e.g. 2.7, 4.3, and 15.0 $\mu$m), the opacity is sufficiently high above the clouds that the peak absorption in the bands occurs at the same pressure in the upper atmosphere (1-10 Pa) for all of the TRAPPIST-1 planets, uninfluenced by the clouds and hazes at higher pressures below. At wavelengths where the atmosphere is optically thin, the presence of aerosols significantly raises the spectral continuum altitude to lower atmospheric pressures. For instance, the continuum pressure at 2.5 $\mu$m is $10^5$ Pa and $10^2$ Pa for clear and cloudy TRAPPIST-1d models, respectively. These cloudy results show both a significant departure from the clear atmosphere cases for each TRAPPIST-1 planet (${>} 100 {\times}$ lower continuum pressures for the cloudy inner planets compared to clearsky) and a significant variance in the pressure of the spectral continuum from one cloudy planet to the next (${\sim} 100 {\times}$ lower continuum pressure for the cloudy inner planets compared to the cloudy outer planets).
{width="48.00000%"} {width="48.00000%"}
The left panel of Figure \[fig:transit\_probed\] shows the maximum pressure probed by the transmission spectra (shown in Fig. \[fig:transit\_pressure\]) as a function of semi-major axes for the TRAPPIST-1 planets. Clear Venus-like atmospheres (blue lines) have their highest pressures accessible in the 1-3 $\mu$m range between the bands (e.g. at about 1.7, 2.4, and 3.1 $\mu$m), while Venus-like atmospheres with clouds (red lines) have their highest pressures accessible in the 2.4-2.6 $\mu$m range where aerosol scattering is weakest and just short of the 2.7 $\mu$m band, which has notably pressure broadened wings when not obscured by aerosols. These wavelengths offer the best opportunity to probe deepest into Venus-like atmospheres in transmission, and offer observation leverage for retrieving a cloud-top or surface pressure.
There are no wavelengths at which the transmission spectra of Venus-like TRAPPIST-1 planets access their lower atmospheres. For the inner TRAPPIST-1 planets, if they have Venus-like atmospheres, then the transmission spectrum will only probe down to about the Martian surface pressure (610 Pa). If they are not cloudy, then they may be probed down to about the surface pressure of Earth (101 kPa). For the outer TRAPPIST-1 planets, the presence of clouds minimally affects the maximum transit pressure, and yet they still cannot be probed to higher pressures than about $10^{4}$ Pa. At least two, and up to five, orders of magnitude in pressure exist between the maximum transit pressure and the unseen Venusian surface pressure, which these models share.
Despite the inaccessibility of lower atmospheres, clear and cloudy atmospheres exhibit distinctly opposing trends in the depth into their atmospheres that may be probed as a function of semi-major axis. Clear atmospheres gently slope from higher pressures accessible for the inner planets to lower pressures for the outer planets. However, Venus-like atmospheres with clouds generally increase from lower pressures accessible for the inner planets to higher pressures accessible for the outer planets.
The divergent scaling with semi-major axis seen between clear and cloudy atmospheres is a result of the underlying physics that controls the transmission spectrum continuum. For clear atmospheres, refraction places a fundamental limit on the depth into the atmosphere that can be accessed by a transmission spectrum [@Betremieux2014; @Misra2014a]. Clear atmospheres with larger semi-major axes cannot be probed as deeply as those closer to the star due to the dependence of the critical refraction pressure on (1) the angular size of the host star as seen from the transiting planet and (2) the scale height of the planetary atmosphere, both of which decrease with semi-major axis for similar composition planets within the same planetary system.
For cloudy atmospheres, the optically thick cloud deck limits the depth into the atmosphere that can be probed by the transmission spectrum. However, since the fixed temperature of sulfuric acid cloud condensation occurs lower in the atmospheres (at higher pressures) for cooler planets, @Lincowski2018 demonstrate a steady transition from high altitude clouds at low pressures for the inner TRAPPIST-1 planets to low altitude clouds at higher pressures for the outer TRAPPIST-1 planets. This cloud formation trend is manifested in the observable transmission spectrum continuum, and enables increasingly higher pressures to be accessed for planets with increasing distance from their parent star. The clear and cloudy trends in Figure \[fig:transit\_probed\] converge for planets at sufficiently large orbital separations (e.g. TRAPPIST-1 f, g and h) as the cloud tops drop below the critical refraction pressure.
{width="98.00000%"} {width="98.00000%"}
Emission Spectroscopy
---------------------
We also consider the limits of atmospheric study for Venus-like planets using emission spectroscopy. Figure \[fig:emission\_pressure\] shows the depth into the atmosphere that is probed by a secondary eclipse thermal emission spectrum for Venus-like models of the TRAPPIST-1 planets with and without clouds. Similar to Figure \[fig:transit\_pressure\], the left panels of Figure \[fig:emission\_pressure\] show the “emission pressure”, or the average pressure level in the atmosphere from which most of the thermal emission emerges at normal incidence. We show the emission pressure over a wavelength range that it is applicable to JWST’s Mid-IR Instrument (MIRI) Low Resolution Spectrometer (LRS), which is optimal for observing thermal emission from the TRAPPIST-1 and similar exoplanets during secondary eclipse [@Lustig-Yaeger2019]. The right panels of Figure \[fig:emission\_pressure\] show the atmospheric thermal structure on the same pressure y-axis as the emission pressure for comparison, with line thickness highlighting the thermally emitting region of the atmosphere.
Numerous transparent windows in the near- and mid-IR offer glimpses into the deeper atmosphere of Venus-like planets. For instance, at 6 $\mu$m there is a prominent window where thermal emission can be seen coming from pressures of about $10^5 - 10^6$ Pa. There are also windows in the NIR that probe even deeper into the atmosphere, but which are not shown here due to the insensitivity of JWST to thermal emission in the NIR. Note, however, that these NIR windows have been used extensively to study the surface and near-surface of Venus [see @Meadows1996; @deBergh2006].
The right panel of Figure \[fig:transit\_probed\] shows the maximum pressure reached by an emission spectrum over the MIRI LRS bandpass as a function of semi-major axis for the TRAPPIST-1 planets. In all cases considered here, emission spectroscopy probes higher pressures than transmission spectroscopy. In general, cloudy atmospheres emit from higher altitudes and lower pressures than clear atmospheres. However, even thermal emission from the clear atmospheres is coming from over an order of magnitude lower pressures than the surface.
Whereas transmission spectroscopy is more sensitive to the location of the cloud top due to the slant optical depth [@Fortney2005], emission spectroscopy is more sensitive to the total optical depth of the clouds in the atmospheric column. The maximum emission pressure trend with semi-major axis for the cloudy Venus-like models notably tracks the total extinction optical depth of the aerosols, seen in Figure 7[^1] of @Lincowski2018. TRAPPIST-1 c and d effectively bracket the small particle haze and thick cloud regimes, respectively, which both exist in the Venus atmosphere [@Crisp1986]. That is, TRAPPIST-1 c’s total aerosol extinction optical depth is of order unity, which is does not substantially modify the thermal emission spectrum from the clear sky case. However, TRAPPIST-1 d’s aerosol total optical depth peaks among the TRAPPIST-1 planets at $\tau {\sim} 30$, due to the strong formation rate, cooler temperature, and lower gravity, which allows larger particles to be sustained and suspended. The resulting extended haze and cloud layer significantly mutes spectral features in the thermal emission spectrum and restricts remote sensitivity to the lower atmosphere. Beyond TRAPPIST-1 d, the total aerosol optical depth decreases with semi-major axis, revealing higher atmospheric pressures and explaining the convergence of the clear and cloudy lines in the right panel of Figure \[fig:transit\_probed\]. However, observing thermal emission spectra from cool terrestrial exoplanets is not feasible with JWST, which we discuss next.
{width="98.00000%"} {width="98.00000%"}
Figure \[fig:emission\_spectra\] shows simulated secondary eclipse spectra for our Venus-like TRAPPIST-1 models. At short wavelengths each eclipse spectrum is dominated by reflected light, while at long wavelengths they are dominated by thermal emission. Both radiative source functions decrease with semi-major axis, making eclipse spectroscopy of temperate and cool planets require at least an order of magnitude higher precision observations. Note that thick clouds can mute the thermal emission spectrum features, as best exemplified by TRAPPIST-1 d in our models. The aforementioned 6 $\mu$m spectral window shows enhanced flux of thermal radiation, particularly in the clear sky models, extending nearly 100 ppm above the thermal continuum for TRAPPIST-1 b.
Discussion {#sec:discussion}
==========
We used the TRAPPIST-1 planets to demonstrate the difficulty intrinsic to identifying and studying the lower atmospheres of Venus-like exoplanets with transmission and emission spectroscopy, which is applicable to near-term efforts with JWST. In the case of true Venus analog exoplanets with sulfuric acid clouds and 92 bar surface pressures, transmission spectroscopy will only be sensitive to pressures between $10^2$ - $10^4$ Pa (0.001 - 0.1 bar). Although the tenuous above-cloud atmosphere could still be detected for all of the TRAPPIST-1 planets with JWST [@Lustig-Yaeger2019], inferring the presence of the lower atmosphere—which perhaps best defines the very nature of Venus—will be a significant challenge, likely exceeding the scope of transmission spectroscopy. In the following discussion we will present observational approaches that may best constrain lower atmospheres for Venus-like exoplanets (§\[discussion:lower\_atmospheres\]). We then discuss how when lower atmospheres cannot be observationally constrained, there exists a potential ambiguity between cloud-top and solid surface (§\[discussion:cloud\_surfaces\]) that may pose clouds as a false positive for atmospheric erosion (§\[discussion:false\_positive\]). We will finish with a discussion of possible strategies and opportunities to mitigate these challenges (§\[discussion:mitigation\]).
Accessing the lower atmospheres of Venus-like exoplanets {#discussion:lower_atmospheres}
--------------------------------------------------------
Although emission spectroscopy is able to probe deeper than transmission, optically thick clouds may still impede lower atmosphere studies. Additionally, the NIR transparent windows that are used to probe down to the surface of Venus for spatially-resolved remote-sensing studies in our Solar System, are out of observational reach for exoplanets in secondary eclipse because dayside reflected light will overwhelm photons emerging from the lower atmosphere (see Fig. \[fig:emission\_spectra\]), analogous to how the illuminated crescent of Venus must be spatially avoided when observing nightside thermal windows [@Meadows1996]. Instead, thermal emission measurements that are sensitive to the lower atmosphere must push to longer wavelengths where the reflected stellar SED is naturally dimmed in the Rayleigh-Jeans tail. In particular, there is a 6 $\mu$m opacity window that is optimally located for observations with MIRI LRS. However, this MIR window does not probe within an order of magnitude of the surface pressure and falls short of the surface temperature by over $200$ K in our TRAPPIST-1 models. Additionally, water vapor has the potential to close the 6 $\mu$m opacity window, so atmospheres with more atmospheric water than the @Lincowski2018 Venus-like models may not have this observable window into the lower atmosphere.
Promising observational approaches have been proposed to efficiently identify the presence, or lack, of hot terrestrial atmospheres using photometry, and in some cases these methods may immediately favor the existence of thick atmospheres. Thermal phase curves with large day-night contrasts can rule out thick atmospheres that would otherwise redistribute heat to the nightside [@Seager2009; @Selsis2011; @Kreidberg2016; @Koll2016; @Kreidberg2019], while an offset hot spot from the substellar point could favor a thick atmosphere [@Demory2016]. Similarly, secondary eclipse photometry could indicate a low dayside temperature due to atmospheric heat redistribution [@Koll2019] or a high bond albedo due to subsolar clouds [@Mansfield2019]. Although promising for warm to hot planets ($T_{eq} = 300-880$ K), constraints from secondary eclipses, and thermal studies in general, become quickly infeasible with JWST for cooler planets into and beyond the HZ [@Lustig-Yaeger2019; @Koll2019]. For these cooler planets, transmission spectroscopy is especially favorable because the strength of spectral features scales with the planet’s atmospheric scale height, $H = k T / \mu g \propto T_{eq}$, rather than the planet’s thermal emission which scales much more strongly with temperature for temperate planets not in the Rayleigh-Jeans limit[^2] [@Winn2010], as shown for the TRAPPIST-1 planets in Fig. \[fig:emission\_spectra\].
An ambiguity between cloud-top and solid surface {#discussion:cloud_surfaces}
------------------------------------------------
However, because of a lack of thermal emission data possible for the temperate and cooler planets, the interpretation of their transmission spectra is paramount in the era of JWST, but it may be complicated by an ambiguity between cloud-top and solid surface.
For any single exoplanet the presence of aerosols may be quite difficult to diagnose with transmission spectroscopy. In principle, scattering slopes and/or absorption features from aerosols may be used to break the cloud-surface degeneracy. However, high S/N observations would be needed to detect these features as they are 10-20 ppm in Venus-like TRAPPIST-1 models [@Lincowski2018], which is much smaller than any of the potentially detectable spectral features with JWST [@Lustig-Yaeger2019]. As a result, this cloud-top–solid-surface ambiguity is more likely to emerge for spectra with low S/N either due to prohibitively long exposure times or observing strategies that seek a large sample of spectra at low to moderate S/N for statistical comparative planetology [e.g. @Bean2017; @Checlair2019].
Clouds as a statistical false positive for atmospheric loss {#discussion:false_positive}
-----------------------------------------------------------
Across a population of exoplanets—either within a single planetary system, as in the case of TRAPPIST-1, or for an ensemble of planets from many systems—inferred trends in cloud-top pressure with stellar irradiation (as seen in Figure \[fig:transit\_probed\]) for similar composition atmospheres may erroneously appear as a surface pressure trend due to atmospheric loss processes. Specifically, the left panel of Figure \[fig:transit\_probed\] clearly shows that higher pressures are probed for cloudy exo-Venuses with larger semi-major axes. Without our prior knowledge on the inclusion of clouds in our models, and under the veil of the cloud-surface ambiguity, these trends could readily appear as trends in surface pressure. That is, *are we seeing thicker atmospheres as stellar irradiation decreases, or lower cloud decks, or both?*
This potential statistical false positive may be particularly nefarious because atmospheric loss is predicted to play a major role in sculpting the atmospheres of small rocky planets orbiting late M dwarfs. Models indicate that the TRAPPIST-1, and similar close-in, planets may have had their atmospheres completely eroded by x-ray and extreme ultraviolet radiation [XUV; @Airapetian2017; @Garcia-Sage2017; @Roettenbacher2017; @Zahnle2017; @Dong2018; @Fleming2019], although sufficient volatile outgassing could help maintain atmospheres [@Bolmont2017; @Garcia-Sage2017; @Bourrier2017b]. Furthermore, @Dong2018 found that the outer TRAPPIST-1 planets are capable of retaining their atmospheres over billions of years, while the inner planets may not be able to. Thus, observing a trend of thin atmospheres for the inner TRAPPIST-1 planets to thick atmospheres for the outer TRAPPIST-1 planets may appear consistent with a “cosmic shoreline”—an empirical division between planets with and without atmospheres based on the relationship between total incident stellar radiation and planetary escape velocities [@Zahnle2013; @Zahnle2017]. Testing the cosmic shoreline hypothesis on exoplanet data will require a statistical comparative planetology approach, as outlined in @Bean2017 and @Checlair2019, but care must be taken to understand and mitigate degenerate exoplanet population trends.
Mitigation strategies & opportunities {#discussion:mitigation}
-------------------------------------
Ambiguous trends in the maximum pressure seen across a population of planetary transmission spectra can also be used to implicate clouds and potentially expose their composition. First, @Lincowski2018 found that sulfuric acid clouds did not condense in the Venus-like model atmospheres of TRAPPIST-1 b, and so our analysis did not include a cloudy TRAPPIST-1 b. However, as noted by @Lincowski2019, detecting the atmosphere of the innermost planet in multi-planet systems could strongly increase the likelihood that similar size planets at longer orbits have atmospheres, because the loss of volatiles due to escape over a planet’s history is expected to decrease with increasing distance from the star. Second, by understanding what physical and chemical conditions may produce continuum pressure trends in terrestrial transmission spectra, it may be possible to rule out false positives scenarios, in the same way that false positive biosignatures may be identified and mitigated using additional environmental context from the atmosphere and stellar environment [@Meadows2017; @Meadows2018b; @Catling2018]. If distinct population trends seen with stellar insolation are consistent with predictions from cloud condensation modeling, then clouds could potentially be revealed by statistical characterization where they were unidentifiable in any single planet spectrum. This is a terrestrial exoplanet analog to cloud condensation trends observed in brown dwarf atmospheres across the L/T transition [e.g. @Ackerman2001; @Morley2012].
Furthermore, the distinctly opposing trends that we found between clear and cloudy atmospheres as a function of semi-major axis (see Figure \[fig:transit\_probed\]) highlights the potential to use the transmission spectrum continuum pressure to group similar populations of terrestrial exoplanets. The maximum pressure probed in clear atmospheres is set by the critical refraction pressure which scales with the angular size of the star as seen from the planet [@Betremieux2014; @Misra2014a], and prevents access to higher pressures at larger semi-major axes (for planets orbiting similar sized stars, e.g. late M dwarfs). Conversely, sulfuric acid cloud condensation allows access to higher pressures at larger semi-major axes. These contrasting trends could be detected by retrieving cloud top or reference pressures for multiple planets within the same system and may enable thick clear and cloudy atmospheres to be distinguished.
These distinguishing characteristics extend into the terrestrial domain the concepts presented in @Sing2016 of an observable distinction between clear and cloudy atmospheres in an ensemble of exoplanet spectra. @Sing2016 found that the strength of water absorption features in the spectra of hot Jupiter exoplanets is correlated with cloud and haze scattering slopes in the spectra, indicating that clouds/hazes may be obscuring the water column, rather than seeing an intrinsic trend in water vapor abundance. Similarly, we have explored how the strength of gaseous absorption features relative to the spectral continuum could potentially be used to discriminate between populations of thin, thick/clear, and thick/cloudy atmospheres, even if the transmission spectra are individually difficult to diagnose.
The transition between planets with and without atmospheres and the transition between terrestrial and gaseous planets are two bookends of the high mean molecular weight, terrestrial atmosphere regime. The emerging paucity of planets with radii ${\sim} 1.6$ R$_\oplus$ orbiting Sun-like stars [@Rogers2015; @Fulton2017] likely constrains the presence of terrestrial atmospheres on the large planet boundary. JWST will offer a first opportunity to investigate this boundary on the small planet end, as we continue to explore the effects of atmospheric escape [e.g. @Lehmer2017], and attempt to map the cosmic shoreline.
Conclusions {#sec:conclusion}
===========
The lower atmospheres of Venus-like exoplanets may elude our characterization efforts with JWST due to the presence of sulfuric acid clouds, which both dictate remote studies of Venus and constitute a potential terrestrial exoplanet analog to the high altitude clouds and hazes that currently limit the charaterization of gaseous exoplanets with transmission spectroscopy. For hot exo-Venuses, MIR opacity windows observed during secondary eclipse may offer glimpses of thermal emission from the atmosphere just below the clouds, potentially allowing for high-pressure, greenhouse-heated lower atmospheres to be directly inferred.
However, for temperate to cold exo-Venuses observed with transmission spectroscopy, a sulfuric acid cloud deck may appear indistinguishable from a solid surface at low to moderate S/N as both cause the spectral continuum to be flat. In these cases, Venus-like atmospheres should still be detectable via absorption features, but appear like tenuous low pressure atmospheres due to a lack of observational constraint from the lower atmosphere. For Venus-like atmospheric models of the TRAPPIST-1 planets, we demonstrated that the sulfuric acid clouds drop in altitude to higher pressures with semi-major axis. This effect has the potential to be misinterpreted as a trend of increasing surface pressure with decreasing stellar insolation and may appear suspiciously consistent with atmospheric escape. Looking ahead, the prospect of different populations of terrestrial exoplanets—cloudy exo-Venuses and stellar windswept worlds—presenting similar observables motivates the need for additional climate, photochemical, cloud formation, and atmospheric escape modeling to uncover observable characteristics that effectively discriminate between different populations of exoplanets, and observing strategies tailored to test these hypotheses.
This work was supported by NASA’s NExSS Virtual Planetary Laboratory funded by the NASA Astrobiology Program under grant 80NSSC18K0829. This work made use of the advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington.
[^1]: [Link to figure on the Astronomy Image Explorer](http://www.astroexplorer.org/details/apjaae36af7)
[^2]: For instance, the blackbody flux scales approximately as $T^8$ near 300 K and at 15 $\mu$m (for $< 1\%$ errors incurred by a Taylor series expansion of the exponential).
|
---
abstract: 'Surveys of diffuse $\gamma$-ray in the interstellar medium (ISM) can be used to probe hydrogen-antihydrogen oscillations, by detecting the $\gamma$-ray emission from antihydrogen annihilation. A bound on the oscillation parameter $\delta$ was originally derived by Feinberg, Goldhaber and Steigman (1978). In this paper, we re-visit the original derivation by performing a more detailed analysis that (1) incorporates suppression effects from additional elastic and inelastic processes, (2) treats the ISM as a multi-phase medium, and (3) utilises more recent $\gamma$-ray data from the *Fermi* Large Area Telescope. We find that suppression from elastic scattering plays a more important role than previously thought, while the multi-phase nature of the ISM affects how the $\gamma$-ray data should be utilised. We derive a more accurate bound on the oscillation period that is about an order of magnitude weaker than the older bound.'
author:
- Yuval Grossman
- Wee Hao Ng
- Shamayita Ray
bibliography:
- 'paper.bib'
title: 'Re-visiting the bounds on hydrogen-antihydrogen oscillations from diffuse $\gamma$-ray surveys'
---
Introduction
============
At the classical level, baryon ($B$) and lepton ($L$) numbers are conserved quantities in the Standard Model (SM). One of Sakharov’s condition [@Sakharov:1967dj] for a dynamical explanation of the baryon asymmetry in the universe requires that $B$ conservation be violated. Mechanisms like electroweak baryogenesis [@Kuzmin:1985mm] or leptogenesis [@Fukugita:1986hr] achieve this through sphaleron processes that makes use of $B+L$ violation in the SM at the quantum level, while mechanisms like baryogenesis in the Grand Unified Theories (GUT) [@Nanopoulos:1979gx] introduce processes that directly violate $B$ at the classical level. However, proton decay imposes strong constraints on models that directly allow $\Delta B = \Delta L = 1$ processes. One intriguing possibility is to consider models [@Mohapatra:1982aj; @FileviezPerez:2011dg] where proton decay is forbidden/suppressed, but yet allow processes with $\Delta B = 2$ or $\Delta B = \Delta L = 2$ to occur. In these cases, processes such as neutron-antineutron oscillations [@Phillips:2014fgb], $pp \to e^+e^+$ annihilations [@Bramante:2014uda] or hydrogen-antihydrogen (H-[$\bar{\text{H}}$]{}) oscillation may become more important probes of $B$ violation. In this paper we concentrate on H-[$\bar{\text{H}}$]{}oscillation.
One way to detect H-[$\bar{\text{H}}$]{}oscillations is through $\gamma$-rays from the annihilation of [$\bar{\text{H}}$]{}with other particles in its vicinity (henceforth called “oscillation-induced $\gamma$-rays”). A good place to look for this is the interstellar medium (ISM), first because of the immense amount of atomic hydrogen present, and second because the low density allow a larger oscillation amplitude and hence a larger proportion of [$\bar{\text{H}}$]{}to exist than in terrestrial sources. These $\gamma$-rays then show up in diffuse $\gamma$-ray surveys on top of other $\gamma$-ray emitting processes, such as cosmic ray (CR) interaction with matter. This idea is not new and a bound on the oscillation was first derived in [@Feinberg:1978sd]. The goal of the present paper is to revisit the bounds, for the following reasons.
1. In the original derivation, the amplitude of oscillation was assumed to be limited by H-[$\bar{\text{H}}$]{}annihilation. However, we do not know *a priori* how this compares to the effects of other processes such as elastic scattering.
2. We now have a better understanding of the phases of the ISM, $\gamma$-ray production within the ISM, as well as updated $\gamma$-ray survey results from the *Fermi* Large Area Telescope (LAT).
3. Finally, many steps are involved in deriving the experimental bounds on the oscillations. While we are only interested in an order-of-magnitude estimate, we want to reduce the uncertainty in each step as much as possible to avoid having the cumulative errors become too large. Therefore, besides improving on the oscillation and ISM model, we also want to utilise updated parameter values from literature rather than just rely on crude estimates.
This paper is structured as follows. In Sec. \[sec:formalism\], we present a model that describes H-[$\bar{\text{H}}$]{}oscillations in a medium, and use the model to derive a formula for the oscillation-induced $\gamma$-ray emissivity. In Sec. \[sec:ism-and-parameter\], we use this formula, together with available data for various elastic and inelastic processes, to calculate the emissivities of the relevant phases of the ISM. It then allows us in Sec. \[sec:bounds\] to obtain a bound on the oscillation parameter $\delta$ based on the *Fermi* LAT data presented in Ref. [@Abdo:2009ka]. We conclude in Sec. \[sec:discussion\] with a comparison of our bound with that from other $\Delta B=\Delta L =\pm 2$ processes. To keep the text focused, most technical details have been placed in the appendices.
Model of H-[$\bar{\text{H}}$]{}oscillation {#sec:formalism}
==========================================
To infer the oscillation-induced $\gamma$-ray emissivity, we need to know the probability of an H atom in the ISM becoming a [$\bar{\text{H}}$]{}. This in turn can be derived from a single-atom model of H-[$\bar{\text{H}}$]{}oscillation. The vacuum formalism is very straightforward; however the main issue here is to account for interactions with the environment. Some of the effects are well-understood: for example, forward scattering gives rise to coherent matter effects known from neutrino oscillations, while inelastic processes such as [$\bar{\text{H}}$]{}annihilation cause the state to leave the Hilbert space of interest and hence their effects are analogous to decays in meson oscillations. Both of these effects can be taken care of by modifications to the effective Hamiltonian.
Less well-recognised are effects that require going beyond the effective Hamiltonian, and require a density matrix formalism. First, say H and [$\bar{\text{H}}$]{}have different elastic scattering amplitudes off the same target, i.e. $f(\theta) \ne \bar{f}(\theta)$, where $\theta$ is the angle of scattering. Then non-forward scattering cause the identity of the atom (H or [$\bar{\text{H}}$]{}) to become entangled with its momentum and hence a two-level pure state formalism does not work if we want to incorporate elastic scattering beyond just forward scattering. Also, since the scattering environment is usually random, even a pure state formalism incorporating both identity and momentum degrees of freedom is insufficient. Second, chemical reactions such as recombination generate new “unoscillated” H atoms to replenish those lost to inelastic processes. Since these reactions should be treated as classical source terms, again a density matrix formalism is required. The model we adopt is similar to the original Feinberg-Weinberg model [@Feinberg:1961zza] that was also used in [@Feinberg:1978sd]. We then extend it to take into account more general sources of suppression. We also highlight the differences between our work and that of [@Feinberg:1978sd].
Model description {#sec:model-description}
-----------------
We regard H and [$\bar{\text{H}}$]{}as basis states of a two-level system (Hilbert space $\mathcal{H}_A$). In principle, there are other degrees of freedom such as momentum, atomic level and spin (Hilbert space $\mathcal{H}_B$), but since we are only interested in finding the probability of being [$\bar{\text{H}}$]{}, we trace them out in the full density matrix $\rho_{\text{full}}(t)$ to obtain a reduced $2 \times 2$ density matrix $\rho(t)$. The quantum kinetic equation of $\rho(t)$ will then depend on the moments of the other degrees of freedom, e.g. $\text{Tr}_B[p^2\rho_{\text{full}}(t)]$, and is hence not closed. To close this equation, we replace, say, the example above by $\langle p^2(t)\rangle \rho(t)$, and assume that $\langle p^2(t)\rangle$ is just given by the present-day value (since we are only interested in a quasi-steady solution). Also, since most of the atoms in the ISM phases of interest are in the $1S$ state, any average involving atomic level and spin is equivalent to a $1S$ hyperfine average.
### Elastic scattering
First, we take into account elastic scattering of the atom with other particles (targets). Let $i$ denote the target species. Then $\rho(t)$ satisfies the kinetic equation [@Feinberg:1961zza] $$\partial_t \rho(t) = -i [H \rho(t) - \rho(t)H^\dagger] + \sum_i\left[ n_i v_i \int d\Omega F_i(\theta) \rho F_i^\dagger(\theta)\right],
\label{eq:formalism-elastic}$$ where $$H \equiv \begin{pmatrix}
E - \sum_i\left[\frac{2\pi n_i v_i}{p_i} f_{i,p_i}(0)\right] & \frac{\delta}{2} \\
\frac{\delta^*}{2} & E - \sum_i\left[\frac{2\pi n_i v_i}{p_i} \bar{f}_{i,p_i}(0)\right]
\end{pmatrix}, \quad
F_i(\theta) \equiv \begin{pmatrix} f_{i,p_i}(\theta) & 0 \\ 0 & \bar{f}_{i,p_i}(\theta) \end{pmatrix},$$ and the symbols used here are defined as follows:
- $E$: the mean energy of an atom in vacuum (equal for H and [$\bar{\text{H}}$]{}by CPT) in the ISM rest frame,
- $n_i$: the number density of species $i$,
- $v_i$: the r.m.s. speed of approach between atom and a species $i$ particle,
- $p_i$: the r.m.s. momentum in centre-of-mass frame of the atom and a species $i$ particle,
- $f_{i,p_i}(\theta)$ ($\bar{f}_{i,p_i}(\theta)$): scattering amplitude of H ([$\bar{\text{H}}$]{}) off a species $i$ particle with momentum $p_i$ in centre-of-mass frame, and
- $\frac{\delta}{2}$: off-diagonal matrix element generated by $\Delta B=\Delta L = \pm 2$ operators.
The assumptions involved are presented in App. \[app:formalism\]. We just explain a few features of Eq. (\[eq:formalism-elastic\]) here. The first term describes the usual time-evolution with an effective non-Hermitian Hamiltonian $H$, comprising the energy $E$ of the atom in vacuum, the oscillation term $\delta$, and coherent forward scattering $f_{i,p_i}(0)$ and $\bar{f}_{i,p_i}(0)$, summed over all target species $i$. Differences in $f_{i,p_i}(0)$ and $\bar{f}_{i,p_i}(0)$ can suppress the oscillations, just like coherent matter effects in neutrino oscillations. The optical theorem ensures that even for elastic scattering $f_{i,p_i}(0)$ and $\bar{f}_{i,p_i}(0)$ are complex quantities, with the imaginary parts related to the total scattering rate. As a result, time evolution under the first term alone cause the total probability represented by $\text{Tr}(\rho)$ to decrease. This decrease is analogous to the effects of the “out” collision term in Boltzmann transport equation. Probability conservation is restored by the second term, analogous to the “in” collision term.
### Inelastic and production processes {#sec:inelastic}
To complete the picture, we want to include inelastic processes as well. We argue in App. \[app:inelastic\] that among all the inelastic processes, only those where the H/[$\bar{\text{H}}$]{}atom “disappears” are potentially important. This includes ionisation, chemical reactions as well as [$\bar{\text{H}}$]{}annihilation. Since these processes take the state out of the Hilbert space $\mathcal{H}_A$, they can be represented by imaginary contributions $i\omega_I/2$ and $i\bar{\omega}_I/2$ to the diagonal elements of $H$, where $\omega_I$ ($\bar{\omega}_I$) denotes the total rate of these processes per H ([$\bar{\text{H}}$]{}) atom.
However, just as H/[$\bar{\text{H}}$]{}atoms can “disappear”, they can also “reappear” through production process such as recombination and [$\text{H}_2$]{}dissociation. These processes correspond to source terms for the $\rho_{11}$ matrix element, which we introduce as $\omega_P \rho_{11}$ in Eq. (\[eq\_inelastic\]). $\omega_P$ can be interpreted as the rate of H production per unit volume, normalised by the number density of H. Furthermore, if we assume that the ISM is in a quasi-steady state (approximate ionisation balance, chemical equilibrium, etc.), then this source term can be approximated as $\omega_P \simeq \omega_I$ up to a small difference of order the quasi-steady rate of change. In principle, we can also include a source term for $\rho_{22}$, e.g. from re-combination of CR positrons and antiprotons to form [$\bar{\text{H}}$]{}. However, based on measurements of the CR antiproton flux [@Aguilar:2016kjl], this contribution is expected to be negligible compared to [$\bar{\text{H}}$]{}production from oscillations at the upper bound of $|\delta|$.
The time-evolution equation is then given by $$\partial_t \rho = -i [H\rho - \rho H^\dagger] + \sum_i \left[ n_i v_i \int d\Omega F_i(\theta) \rho F_i^\dagger(\theta) \right] +
\begin{pmatrix}
\omega_P \rho_{11} & 0 \\
0 & 0 \end{pmatrix}$$ with a modified effective Hamiltonian $$H \equiv \begin{pmatrix}
E - \sum_i\left[\frac{2\pi n_i v_i}{p_i} f_{i,p_i}(0)\right] - \frac{i}{2}\omega_I & \frac{\delta}{2} \\
\frac{\delta^*}{2}
& E - \sum_i\left[\frac{2\pi n_i v_i}{p_i} \bar{f}_{i,p_i}(0)\right] - \frac{i}{2} \bar{\omega}_I
\end{pmatrix}.
\label{eq_inelastic}$$
### Reformulating the model
It is instructive to rewrite $\rho(t)$ as a column vector $\rho(t) \equiv (\rho_{11}, \rho_{12}, \rho_{21}, \rho_{22})^T$ [@Feinberg:1978sd]. The time evolution equation then becomes $$\partial_t \rho(t) = M \rho,
\label{eq:formalism-column}$$ where $$M \equiv \begin{pmatrix} \omega_P -\omega_I & i\frac{\delta^*}{2} & -i\frac{\delta}{2} & 0 \\
i\frac{\delta}{2} & \epsilon' & 0 & -i\frac{\delta}{2}\\
-i\frac{\delta^*}{2} & 0 & \epsilon^{\prime *}
& i\frac{\delta^*}{2} \\
0 & -i\frac{\delta^*}{2} & i\frac{\delta}{2} & - \bar{\omega}_I
\end{pmatrix},$$ $$\epsilon' \equiv i \sum_i n_i v_i \left[ \Delta_i + \int d\Omega \text{Im}(\bar{f}_{i,p_i}^*f_{i,p_i}) \right] - \left[\frac{\omega_I + \bar{\omega}_I}{2} + \sum_i \frac{n_i v_i}{2} \int d\Omega |f_{i,p_i} - \bar{f}_{i,p_i}|^2 \right],
\label{eq:epsilonprime}$$ $$\Delta_i \equiv \frac{2\pi}{p_i} \text{Re}[f_{i,p_i}(0)-\bar{f}_{i,p_i}(0)].$$
Some observations:
- If $f_{i,p_i} = \bar{f}_{i,p_i}$, then all instances of $f_{i,p_i}$ and $\bar{f}_{i,p_i}$ vanish from $M$. In other words, elastic scattering does not suppress oscillations unless it can differentiate between H and [$\bar{\text{H}}$]{}amplitude-wise. This means, for example, that we can ignore elastic scattering with photons.
- If $\omega_I = \bar{\omega}_I$, then their combined contributions to $M$ is just proportional to the identity, so they only lead to an overall decay factor. Therefore, inelastic processes also do not suppress oscillations unless they can differentiate between H and [$\bar{\text{H}}$]{}rate-wise.
- Oscillations are also suppressed by the source term $\omega_P \rho_{11}$, although the physical mechanism is somewhat indirect. Here new H atoms that have yet to oscillate are being added to the system. This suppression is why despite our previous comment, we still need to consider inelastic processes such as photo-ionisation that have the same rate for H and [$\bar{\text{H}}$]{}, since $\omega_I$ informs us about $\omega_P$ in the quasi-steady state.
Note that our formalism here is similar to the one used in [@Feinberg:1978sd] (see Eq. (2.4) there). However, they did not include a source term $\omega_P$, and they also assumed that the only important process is H-[$\bar{\text{H}}$]{}annihilation. As a result, they have $\bar{\omega}_I \gg \omega_I$ (since it is much easier for a [$\bar{\text{H}}$]{}to find a H to annihilate with, than vice versa) and $|\epsilon'| \simeq \bar{\omega}_I/2$. In contrast, we do not make the same assumptions but instead consider a wide range of elastic and inelastic processes.
Formula for $\gamma$-ray emissivity
-----------------------------------
We want to use our model to derive a formula for the $\gamma$-ray emissivity. To do so, we need to find the solution to Eq. (\[eq:formalism-column\]) that best describes a H/[$\bar{\text{H}}$]{}atom in the ISM, from which we can then obtain the [$\bar{\text{H}}$]{}number density and hence the emissivity.
Most of the parameters in $M$ depend on the number densities of atomic hydrogen and other species in the ISM, so Eq. (\[eq:formalism-column\]) is actually much harder to solve than it seems. However, since we are only interested in the quasi-steady solution, it is actually self-consistent to assume these parameters as constants, at least for timescales short compared to the quasi-steady rate of change. Even though the quasi-steady solution based on this assumption may become inaccurate at longer times, it does not matter since we are using *present-day* parameter values. In other words, the reference starting time is actually the present, so we read off the present-day [$\bar{\text{H}}$]{}probability $\rho_{22}$ from the solution at $t=0$.
With this assumption, among the four eigenvectors of $M$, three have eigenvalues with negative real parts of order $|\epsilon'|$ or $\bar{\omega}_I$, while the fourth is given by $$\lambda = \omega_P - \omega_I + \mathcal{O}(\epsilon^2|\epsilon'|, \epsilon^2|\bar{\omega}_I|)$$ where $\epsilon \equiv \text{Max}\left\lbrace \left\vert \frac{\delta}{\epsilon'}\right\vert, \left\vert \frac{\delta}{\bar{\omega}_I}\right\vert \right\rbrace$ is a small parameter. The first three solutions correspond to transients that decay rapidly (although the actual decay rate may be somewhat different since these solutions are not consistent with the assumption about the parameters being constant), while the fourth solution does indeed change at the quasi-steady rate $|\omega_P-\omega_I|$ and is thus the one we want. The corresponding eigenvector is given by $$v = \begin{pmatrix}
1 + \mathcal{O}(\epsilon^2)\\
-\frac{i\delta}{2(\epsilon'+\omega_I-\omega_P)} + \mathcal{O}(\epsilon^3)\\
\left[-\frac{i\delta}{2(\epsilon'+\omega_I-\omega_P)} + \mathcal{O}(\epsilon^3)\right]^*\\
-\left\vert \frac{\delta}{\epsilon' + \omega_I - \omega_P} \right\vert^2 \frac{\text{Re}(\epsilon' + \omega_I - \omega_P)}{2(\bar{\omega}_I - \omega_I + \omega_P)} + \mathcal{O}(\epsilon^4)
\end{pmatrix}.$$ We observe that of the four components, $v_1 \simeq 1$, $v_2 = v_3^* \sim \mathcal{O}(\epsilon)$, and $v_4 \sim \mathcal{O}(\epsilon^2)$.
Since $\tfrac{v_1}{v_1+v_4}$ and $\tfrac{v_4}{v_1 + v_4}$ correspond to the probability of being H and [$\bar{\text{H}}$]{}, we can estimate the rate of [$\bar{\text{H}}$]{}annihilation per unit volume as $$\frac{v_4}{v_1} n_{{\text{H}}}n_i \langle \sigma_i v_i\rangle
\simeq -\left\vert \frac{\delta}{\epsilon'} \right\vert^2 \frac{\text{Re}(\epsilon')}{2\bar{\omega}_I}
\bar{\omega}_{\text{ann}}$$ where $\bar{\omega}_{\text{ann}}$ is the annihilation rate per [$\bar{\text{H}}$]{}(we allow it to differ from $\bar{\omega}_I$ in case there are other more important [$\bar{\text{H}}$]{}“disappearance” processes), and we have dropped the much smaller quasi-steady rate $|\omega_P - \omega_I|$ relative to $\bar{\omega}_I$ and $\epsilon'$. This is a positive quantity since $\text{Re}(\epsilon') < 0$. Note that $\omega_P$ has disappeared completely (it is not present in $\epsilon'$) since its main role is to cancel $\omega_I$ at certain places to give a much smaller quasi-steady rate that can then be neglected.
For comparison with $\gamma$-ray data later, it is useful to convert the previous rate per unit volume into an oscillation-induced emissivity per H atom, which gives $$\epsilon_{\gamma} = -\frac{g_{\gamma}}{4\pi}\left\vert \frac{\delta}{\epsilon'} \right\vert^2 \frac{\text{Re}(\epsilon')}{2\bar{\omega}_I}
\bar{\omega}_{\text{ann}} \text{ photons } \text{sr}^{-1},
\label{eq:emissivity}$$ where $g_{\gamma}$ is the average number of $\gamma$-ray photons emitted in the annihilation. We discuss its value below for specific situations.
Calculating the emissivities {#sec:ism-and-parameter}
============================
In the previous section, we derived a formula for the oscillation-induced $\gamma$-ray emissivity per H atom, Eq. (\[eq:emissivity\]). To make further progress, we need numerical values of the parameters in this formula, except for the unknown $|\delta|$ that we want to constrain. We begin this section by identifying phases of the ISM that are expected to be the dominant sources of these $\gamma$-rays. Using available data for a wide variety of elastic and inelastic processes, we then calculate the parameter values and hence the emissivity for each phase. We adopt the standard astronomical notation of [H<span style="font-variant:small-caps;">i</span>]{}and [H<span style="font-variant:small-caps;">ii</span>]{}for atomic and ionised hydrogen.
Phases of the ISM {#sec:ism}
-----------------
The *Fermi* LAT data presented in Abdo *et al.* [@Abdo:2009ka] focuses on $\gamma$-ray emission from [H<span style="font-variant:small-caps;">i</span>]{}and is hence of particular relevance to our work. We want to consider the same sector of the ISM, bounded by Galactic longitude $200^\circ < l < 260^\circ$, and latitude $22^\circ < |b| < 60^\circ$. Even within this sector, the ISM is not homogeneous and has a number of phases, each with a different [H<span style="font-variant:small-caps;">i</span>]{}density and presenting a different environment for H-[$\bar{\text{H}}$]{}oscillations.
In App. \[app:ism\], we describe these phases and explain why we expect most of the oscillation-induced $\gamma$-rays to come from three of them, namely the cold neutral medium (CNM), warm neutral medium (WNM) and warm ionised medium (WIM). Here we present a short description of these three phases, as well as the nominal values we assume for their physical properties [@Ferriere:2001rg; @cox2005three; @tielens2005physics; @draine2010physics]. $T$ here represents the phase temperature, and $x$ the ionisation fraction.
- CNM: Comprises clumps of cold [H<span style="font-variant:small-caps;">i</span>]{}clouds.\
$n_{{\text{H}}} \simeq 50 \, \text{cm}^{-3}$, $T \simeq 80 \, \text{K}$, $x=0.001$.
- WNM: Intercloud region containing warm diffuse [H<span style="font-variant:small-caps;">i</span>]{}.\
$n_{{\text{H}}} \simeq 0.5 \, \text{cm}^{-3}$, $T \simeq 8000 \, \text{K}$, $x=0.05$.
- WIM: Intercloud region containing warm diffuse [H<span style="font-variant:small-caps;">ii</span>]{}.\
$n_{{\text{H}^{\text{+}}}} \simeq 0.3 \, \text{cm}^{-3}$, $T \simeq 8000 \, \text{K}$, $x=0.9$.
The uncertainties in these nominal values, in particular the ionisation fraction, is a significant source of error in our analysis. Henceforth, most values that we present should only be interpreted as *order-of-magnitude estimates*.
Emissivities of the CNM, WNM and WIM
------------------------------------
We now want to determine the oscillation-induced emissivities of the three phases. To do so, we first need the values of $\epsilon'$, $\bar{\omega}_I$ and $\bar{\omega}_{\text{ann}}$ used in the emissivity formula Eq. (\[eq:emissivity\]). The values we present below incorporate a wide range of elastic targets as well as inelastic processes, using available data on scattering phase shifts, cross-sections and reaction rate constants [@schwartz1961electron; @Armstead:1968zz; @bhatia1971generalized; @Morgan:1973zz; @bhatia1974rigorous; @kolos1975hydrogen; @register1975algebraic; @fon1978elastic; @morgan1989atomic; @mitroy1993close; @krstic1999atomic; @sinha2004total; @chakraborty2007cold] (more details can be found in App. \[app:parameter\]):
- CNM:\
$\epsilon' \simeq (-1 \pm i) \times 10^{-7} \, \text{s}^{-1}$, mostly from elastic scattering with H.\
$\bar{\omega}_I \simeq \bar{\omega}_{\text{ann}} \simeq 6 \times 10^{-8} \, \text{s}^{-1}$, mostly from H-[$\bar{\text{H}}$]{}annihilation.
- WNM:\
$\epsilon' \simeq (-5 \pm 5i) \times 10^{-9} \, \text{s}^{-1}$, mostly from elastic scattering with H.\
$\bar{\omega}_I \simeq \bar{\omega}_{\text{ann}} \simeq 8 \times 10^{-10} \, \text{s}^{-1}$, mostly from H-[$\bar{\text{H}}$]{}annihilation.
- WIM:\
$\epsilon' \simeq (-2 - i) \times 10^{-8} \, \text{s}^{-1}$, mostly from elastic scattering with [$\text{e}^{\text{-}}$]{}.\
$\bar{\omega}_I \simeq \bar{\omega}_{\text{ann}} \simeq 7 \times 10^{-10} \, \text{s}^{-1}$, mostly from [$\text{H}^{\text{+}}$]{}-[$\bar{\text{H}}$]{}annihilation.
Our estimate for $\epsilon'$ are a few orders of magnitude larger than in [@Feinberg:1978sd], where it was assumed that $2|\epsilon'| \simeq \bar{\omega}_I \simeq 10^{-10} \, \text{s}^{-1}$. This discrepancy is mainly due to contributions from elastic scattering that they have neglected. Hence, their assumption that H-[$\bar{\text{H}}$]{}oscillations are mainly suppressed by [$\bar{\text{H}}$]{}annihilation is not justified.
With these values, we can finally obtain the following oscillation-induced $\gamma$-ray emissivities per H atom.
- CNM: $\epsilon_{\gamma} \simeq 2g_\gamma|\delta|^2 \times 10^5 \, \text{s} \, \text{srad}^{-1}$.
- WNM: $\epsilon_{\gamma} \simeq 4g_\gamma|\delta|^2 \times 10^6 \, \text{s} \, \text{srad}^{-1}$.
- WIM: $\epsilon_{\gamma} \simeq g_\gamma|\delta|^2 \times 10^6 \, \text{s} \, \text{srad}^{-1}$.
Since the $\gamma$-ray data in [@Abdo:2009ka] starts at $100 \, \text{MeV}$, using the experimental and simulation results in [@Backenstoss:1983gu], we estimate the average number of photons from [$\bar{\text{H}}$]{}annihilation above this threshold to be $g_\gamma \simeq 2.7$.
Deriving bound on $|\delta|$ using *Fermi* LAT data {#sec:bounds}
===================================================
In this section, we explain how we derive a bound on the oscillation parameter $|\delta|$ using *Fermi* LAT data. The main idea is to compare the results of $\gamma$ ray measurements with predictions from astrophysical models. The difference between them can then be used to constrain additional oscillation-induced emissivity and hence $|\delta|$.
More specifically, one can perform a linear regression of the observed $\gamma$ ray intensity against the [H<span style="font-variant:small-caps;">i</span>]{}column density. The slope corresponds to the emissivity per H atom, and the offset (intercept) a spatially homogeneous source of emissivity. The observed slope can be compared with independent astrophysical predictions to constrain $|\delta|$, and this was indeed what was done in [@Feinberg:1978sd]. However, we argue that the oscillation-induced emissivity should really show up in the offset rather than the slope, which lacks an independent prediction. Therefore, the whole measured offset is used to constrain $|\delta|$. We explain these points in more details below.
Review of relevant $\gamma$-ray data
------------------------------------
In this section we review the analysis and results in [@Abdo:2009ka]. One of their goals was to determine the [H<span style="font-variant:small-caps;">i</span>]{}$\gamma$-ray emissivity, and compare it with predictions based on CR interaction with matter. The authors used *Fermi* LAT $\gamma$-ray data from the sector we previously described, in the energy range $100 \, \text{MeV}-9.05 \, \text{GeV}$. This sector is known to be free of large molecular clouds. In this region, [H<span style="font-variant:small-caps;">ii</span>]{}column-density is relatively smooth and is in the range $(1-2) \times 10^{20} \, \text{cm}^{-2}$, while [H<span style="font-variant:small-caps;">i</span>]{}distribution is more clumpy with a column density in the range $(1-18) \times 10^{20} \, \text{cm}^{-2}$.
Known background such as point sources and inverse Compton scattering of soft photons with CR electrons were subtracted, leaving only data that are expected to come from CR interaction with matter as well as an isotropic extragalactic diffuse background. By comparing the post-subtraction $\gamma$-ray intensity map (Fig. 1 of [@Abdo:2009ka]) with a [H<span style="font-variant:small-caps;">i</span>]{}column density map derived from $21\, \text{cm}$ radio surveys (Fig. 3 of [@Abdo:2009ka]), the authors found a linear relationship between the $\gamma$-ray intensity $I_\gamma$ and the [H<span style="font-variant:small-caps;">i</span>]{}column density $N(\textsc{Hi})$ for each energy bin, which we index by $i$ (Fig. 4 of [@Abdo:2009ka]) I\_[,i]{} S\_i N(<span style="font-variant:small-caps;">Hi</span>) + O\_i where the slope $S_i$ represents the [H<span style="font-variant:small-caps;">i</span>]{}emissivity per atom, and the offset $O_i$ the contributions from residual particles and the extragalactic background. The authors found good agreement between the slope-derived [H<span style="font-variant:small-caps;">i</span>]{}emissivity and the predictions based on CR interaction with matter. Summing the results in Tab. 1 of [@Abdo:2009ka] over the bins in the energy range $100 - 1130 \, \text{MeV}$ (relevant for [$\bar{\text{H}}$]{}annihilation), we find that the [H<span style="font-variant:small-caps;">i</span>]{}emissivity given by the combined slopes is $$S = 1.5 \times 10^{-26} \text{ photons } \, \text{s}^{-1} \, \text{sr}^{-1} \text{~per H atom}.$$ and the combined offset is $$\label{eq:val-of-O}
O= 1.4 \times 10^{-5} \text{ photons } \text{cm}^{-2} \, \text{s}^{-1} \, \text{sr}^{-1}.$$
Bounds on $|\delta|$ {#sec:scenario}
--------------------
Let us now consider what happens if there are extra oscillation-induced $\gamma$-rays on top of the known sources. Distribution-wise, both the WIM and WNM have relatively low volume densities and large volume filling factors, so their contributions to the [H<span style="font-variant:small-caps;">i</span>]{}column density should be relatively uniform over the column density map. In contrast, the CNM is clumpy with much higher density and smaller filling factor, so the small regions in the map with high column densities probably correspond to lines of sight which pass through the CNM. In other words, lines of sight with more H from the CNM provide the high leverage points that determine the slope in the linear regression of emissivity against column density. On the other hand, as we have seen, the extra emissivity per H atom varies among the three phases of ISM, with the WNM and WIM values being one order of magnitude higher than the CNM. Together, this suggests that the extra $\gamma$-ray intensity is more likely to show up in Fig. 4 of [@Abdo:2009ka] as a contribution to the offset rather than the slope.
We perform a simple calculation to show that this is indeed the case. The WNM and WIM are assumed to be layers parallel to the galactic disk. Therefore, their contributions to the [H<span style="font-variant:small-caps;">i</span>]{}column density are constant, except for a $\tfrac{1}{\sin|b|}$ latitudinal variation since a more “glancing” line of sight travels a longer distance through the layer. Using Eq. (\[eq:cnm-wnm\]) and (\[eq:wim\]) and the nominal ionisation fraction, this corresponds to a contribution of $\tfrac{1.7}{\sin|b|} \times 10^{20} \, \text{cm}^{-2}$ from the WNM and $\tfrac{0.08}{\sin|b|} \times 10^{20} \, \text{cm}^{-2}$ from the WIM. On top of that, the CNM is assumed to add a random contribution that ranges from 0 to $\tfrac{10}{\sin|b|} \times 10^{20} \, \text{cm}^{-2}$. For each line of sight within the latitudinal range of interest, we calculate the total [H<span style="font-variant:small-caps;">i</span>]{}column density and oscillation-induced $\gamma$-ray intensity, repeated many times over different random CNM contributions. Fig. \[fig:scenario\] shows a plot of intensity against column density, with the horizontal errorbars indicating the bin intervals, and the vertical errorbars the intensity range of the corresponding bins. The plot is mostly horizontal, indicating that the extra intensity is indeed more likely to show up in the offset, with a contribution of roughly O\_ 4||\^2 10\^[27]{} \^[-2]{} \^[-1]{} \^[-1]{}. \[eq:val-of-O-ind\]
![Results of a simple calculation showing how the oscillation-induced $\gamma$-ray intensity varies with the [H<span style="font-variant:small-caps;">i</span>]{}column density.[]{data-label="fig:scenario"}](fig-scenario.pdf){width="0.6\linewidth"}
To obtain a bound on $|\delta|$, we identify this extra offset with the entire experimental offset value, which we found earlier to be around $1.4 \times 10^{-5} \text{ photons } \text{cm}^{-2} \, \text{s}^{-1} \, \text{sr}^{-1}$. In principle, we could have performed further background subtraction from this experimental value before making the identification. Possible background includes CR interaction with smoothly-distributed residual particles such as [H<span style="font-variant:small-caps;">ii</span>]{}, incomplete earlier subtraction of inverse Compton scattering due to model uncertainties, as well as extragalactic sources. However, these contributions are either not well-quantified, or turn out to be small compared to the experimental value, so the subtraction is unlikely to have made a big difference. Comparing $O$ and $O_{\text{osc.}}$ from Eqs. (\[eq:val-of-O\]) and (\[eq:val-of-O-ind\]), we find that $$|\delta| \lesssim 6 \times 10^{-17} \, \text{s}^{-1}.$$ This is about one order of magnitude weaker than the bound derived in [@Feinberg:1978sd]. In other words, the earlier bound may have been too stringent. We also note that [@Feinberg:1978sd] used the slope (from older $\gamma$-ray data [@Fichtel:1977]) instead of the offset to derive the bound, so it did not account for the most likely scenario in which the CNM is mainly responsible for the variation in [H<span style="font-variant:small-caps;">i</span>]{}column density from which the slope is derived, whereas the WNM dominates the extra oscillation-induced intensity.
Discussion and conclusions {#sec:discussion}
==========================
The bounds we have derived on $|\delta|$ can be translated to a bound on four-fermion contact operators involving protons and electrons. For instance, [@Feinberg:1978sd] considered the operator $$\mathcal{O}_1 = \frac{1}{\Lambda^2} [\bar{p}^c \gamma_\mu (1+\gamma_5) e][\bar{p}^c \gamma^\mu (1+\gamma_5) e] + \text{h.c.},$$ and found that $\delta$ is related to $\Lambda$ via $$\delta = \frac{16}{\Lambda^2 \pi a^3}, \label{eq:delta-from-cutoff}$$ where $a$ is the Bohr radius.
On the other hand, $ppee$ operators can also be constrained by other processes such as $pp\to ee$. For instance, results from Super-Kamiokande can be used to set an upper bound on the proton annihilation rate in oxygen nuclei. For a benchmark operator $$\mathcal{O}_2 = \frac{1}{\Lambda^2} (i\bar{p}^c \gamma_5 p)(i \bar{e}^c \gamma_5 e) + \text{h.c.},$$ this translates to a bound of $\Lambda > 7 \times 10^{14} \,\text{GeV}$ [@Bramante:2014uda]. If we now assume that the same cutoff scale can be used in Eq. (\[eq:delta-from-cutoff\]) to estimate a bound on $|\delta|$, we find that $$|\delta| \lesssim 10^{-21} \, \text{s}^{-1},$$ which is actually four orders of magnitude more stringent than the bound that we have obtained from $\gamma$-ray observations.
It is unlikely that choosing a different region for $\gamma$-ray observations can give an improved bound on $|\delta|$ that is just as competitive, so it is worth speculating whether a terrestrial laboratory-based oscillation experiment might do better. For instance, if a falling H atom oscillates partially into an [$\bar{\text{H}}$]{}, the experiment can attempt to detect $\gamma$-rays from annihilation when this atom comes into contact with a solid surface. Compared to measurements based on the ISM, the advantages are that annihilation no longer relies on chance encounters with other atoms, and that the $\gamma$-rays background can potentially be controlled. If there are $N$ H atoms each with a characteristic flight time $t$ before reaching a solid surface, then the absence of $\gamma$-rays indicate a crude bound of $(|\delta| t)^2 \lesssim \tfrac{1}{N}$. Unfortunately, even obtaining a bound close to that from the ISM is unlikely to be feasible. For instance, a bound of $|\delta| \lesssim 10^{-16} \, \text{s}^{-1}$, assuming a flight time of $t = 1 \, \text{s}$, will require about $10^8 \, \text{mol}$ of atomic hydrogen, a very large number. In addition, there are practical concerns about how rarefied the H atoms should be so that they do not start to interact, and the cryogenics required so that thermal motion does not substantially reduce the flight time.
To conclude, we have updated the bounds on H-[$\bar{\text{H}}$]{}oscillations based on oscillation-induced $\gamma$-ray emission in the ISM. Suppression from elastic collisions turn out to be more significant than assumed in previous work, and using a multi-phase ISM model as well as updated parameter values and $\gamma$-ray data, we show that the upper bound on $|\delta|$ is about $6 \times 10^{-17} \, \text{s}^{-1}$, one order of magnitude weaker than previously thought.
We thank Lorenzo Calibbi, Shmuel Nussinov and Chelsea Sharon for helpful discussions. Work of YG is supported in part by the NSF grant PHY1316222. Work of SR is supported by the Department of Science and Technology, Government of India through INSPIRE Faculty Fellowship (Grant no. IFA12-PH-41).
More details about the H-[$\bar{\text{H}}$]{}oscillation model
==============================================================
Elastic scattering {#app:formalism}
------------------
The model we used in this work was originally derived in [@Feinberg:1961zza] somewhat heuristically based on the notion of a classical sum over different “histories”, where in each infinitesimal time interval $\delta t$, the atom may undergo either elastic scattering or quantum time evolution. We have been able to re-derive the model on a more rigourous basis as follows.
The atom is originally described by a density matrix in the product space $\mathcal{H}_A \otimes \mathcal{H}_B$, where $\mathcal{H}_A$ is associated with the atom’s identity, and $\mathcal{H}_B$ with momentum degrees of freedom (for simplicity we neglect atomic level and spin; including them simply increases the number of Wigner functions). We then extend the impurity-scattering formalism described in [@rammer2004quantum] to derive quantum kinetic equations for the $2 \times 2$ Wigner functions. By making a number of assumptions before and after integrating over momentum space (equivalent to tracing out $\mathcal{H}_B$), we finally obtain the same kinetic equation for the reduced $2\times 2$ density matrix $\rho(t)$ as [@Feinberg:1961zza].
We now examine the various assumptions made in this derivation.
- The derivation of the Wigner function kinetic equations assumed that the mean free path be much larger than the de Broglie wavelength, and that quantum degeneracy as well as two-body correlation between atom and target can be ignored. These are probably reasonable assumptions for an atom in the ISM.
- In further reducing these kinetic equations to the one for $\rho(t)$, two further assumptions are made. First, we take the classical limit of the scattering terms, which requires that memory effects be neglected, again a reasonable assumption given that the momentum relaxation time of an atom is much shorter than our timescale of interest (the quasi-steady rate of change). Second, as mentioned in Sec. \[sec:model-description\], in order to close the kinetic equation for $\rho(t)$, we assume that moments in momentum and other degrees of freedom can be replaced by products of $\rho(t)$ with the relevant expectation values. While some errors are introduced in doing so, they are not expected to be very significant.
- The impurity-scattering formalism assumes that the targets are immobile, certainly not true for real targets in the ISM. Nonetheless, this can be addressed by replacing $v$ and $p$, not by the r.m.s. values in the lab frame, but rather the r.m.s. values evaluated in the two-particle centre-of-mass frame comprising the atom and a target particle (hence this also involves averaging over the target velocity distribution). Only $E$ should still be the lab frame value.
- Finally, the impurity-scattering formalism assumes that the atom and target are distinguishable particles. This is clearly violated if we consider scattering with other H atoms. Both $f(\theta)$ and $f(\pi - \theta)$ will then contribute to the same H-H scattering process, and one must also be careful not to double-count the phase space. This is probably the biggest source of error (possibly up to a factor of 2) in the model, at least for the CNM and WNM. However, there is not much point in trying to derive a more accurate treatment due to the lack of accurate scattering data.
Inelastic processes {#app:inelastic}
-------------------
In Sec. \[sec:inelastic\], we only considered inelastic processes where the H/[$\bar{\text{H}}$]{}atom “disappears”, e.g. [$\text{H}_2$]{}formation or [$\bar{\text{H}}$]{}annihilation. These processes cause the state to leave the Hilbert space $\mathcal{H}_A$ and can hence be represented by imaginary diagonal contributions to the effective Hamiltonian. However, there are other processes where the atom does not disappear but are nonetheless inelastic. We now explain why they can be neglected.
First, we consider processes like ${\text{H}}/{\bar{\text{H}}}(1S) + X \to {\text{H}}/{\bar{\text{H}}}(1S) + Y$, where the H/[$\bar{\text{H}}$]{}atom remains in the $1S$ state but the target is collisionally excited/ionised/dissociated. As far as the H/[$\bar{\text{H}}$]{}atom is concerned, these processes are not very different from elastic scattering, and so enters the model in a similar manner (except without a forward scattering contribution). However, we expect them to be less important than elastic scattering off the same target $X$ since the rates are usually Boltzmann-suppressed in comparison, even in the warm phases.
Next, we consider collisional and photo-excitations of H/[$\bar{\text{H}}$]{}to $n \ge 2$ atomic states. These processes (together with collisional and radiative decays) are responsible for maintaining the quasi-steady distribution of atomic levels. However, if the transition amplitudes for H and [$\bar{\text{H}}$]{}are different, then one also needs to examine how they might directly affect the oscillations. Collisional excitations can again be neglected since they are Boltzmann-suppressed compared to elastic scattering. For photo-excitations, the electric dipole transition amplitudes for H and [$\bar{\text{H}}$]{}do indeed differ by a sign; however, there is hardly any time for the $\mathcal{H}_A$ part of the state to evolve (except by an overall phase) before the atom undergoes radiative decay that undoes the sign change. Therefore, the net direct effects are also unimportant.
The arguments above do not apply to $1S$ hyperfine transitions. In particular, collisional excitations to the higher-energy hyperfine state are not Boltzmann-suppressed. However, since these processes involve electron spin flips, they are either magnetic in nature and hence have smaller cross-sections, or rely on electron exchange (e.g. when the target is [$\text{e}^{\text{-}}$]{}or other H atoms) and hence already included in conventional elastic scattering data. Photo-excitations can also occur via dipole transition to $nP$ states followed by decays to the higher $1S$ hyperfine state, but as explained above the net direct effects are unimportant due to sign cancellation.
Phases of the ISM {#app:ism}
=================
The ISM comprises a number of phases that accounts for most of its mass and volume. Parameter values are taken from [@Ferriere:2001rg; @cox2005three; @tielens2005physics; @draine2010physics].
- Neutral atomic gases: There are two phases that contain predominantly [H<span style="font-variant:small-caps;">i</span>]{}. The CNM comprises [H<span style="font-variant:small-caps;">i</span>]{}clouds typically of size $\mathcal{O}(10) \, \text{pc}$, number density $20 - 50 \, \text{cm}^{-3}$, temperature $50 - 100 \, \text{K}$ and volume filling factor $\mathcal{O}(0.01)$. The WNM comprises diffuse intercloud [H<span style="font-variant:small-caps;">i</span>]{}, typically with a lower number density $0.2 - 0.6 \, \text{cm}^{-3}$, and higher temperature $5000 - 10000 \, \text{K}$ and filling factor $0.3-0.4$. Locally, a simple model for the vertical [H<span style="font-variant:small-caps;">i</span>]{}distribution (filling factor incorporated) is given by $$n_{{\text{H}}}(z) / \text{cm}^{-3} = 0.40 e^{-\left(\frac{z}{127 \, \text{pc}}\right)^2} + 0.10 e^{-\left(\frac{z}{318 \, \text{pc}}\right)^2} + 0.063 e^{-\frac{|z|}{403 \, \text{pc}}},
\label{eq:cnm-wnm}$$ where the first term corresponds to the CNM, and the second and third terms the WNM.
- Warm ionised gases: Radiation from O and B stars cause almost-complete ionisation of nearby clouds, so most of the hydrogen are in the ionised form [H<span style="font-variant:small-caps;">ii</span>]{}. These [H<span style="font-variant:small-caps;">ii</span>]{}regions, typically of size $\mathcal{O}(1) \, \text{pc}$, are generally very dense and hot, with number densities up to $\mathcal{O}(10^5) \, \text{cm}^{-3}$, temperatures $8000 - 10000 \, \text{K}$, and negligibly small filling factors. Besides these dense regions, there also exists a diffuse warm ionised phase called the WIM. This phase has comparable temperature, but much lower number density $\sim 0.1 - 0.5 \, \text{cm}^{-3}$, and much higher filling factor $0.05 - 0.25$. A simple “two-disk” model for the vertical [H<span style="font-variant:small-caps;">ii</span>]{}distribution is given by $$n_{e}(z) / \text{cm}^{-3} = 0.015 e^{-\frac{|z|}{70 \, \text{pc}}} + 0.025 e^{-\frac{|z|}{900 \, \text{pc}}},
\label{eq:wim}$$ where the first term represents the collection of localised [H<span style="font-variant:small-caps;">ii</span>]{}regions as a “thin-disk”, and the second term the WIM as a “thick disk”.
- Coronal gases: Besides the WIM, there is another diffuse ionised phase referred to as coronal gases, because the temperature and ionisation state are believed to be similar to that of the solar corona. This phase is much hotter and rarefied, with temperature $\mathcal{O}(10^5 - 10^6) \, \text{K}$, number density $0.003 - 0.007 \, \text{cm}^{-3}$, and filling factor $0.2 - 0.5$. The vertical profile depends on the measurements used (e.g. choice of spectral lines) but usually fits a large scale height of 3 kpc (assuming exponential distribution) or above.
- Molecular clouds: These comprise gravitationally-bound clouds, typically of size $\mathcal{O}(10) \, \text{pc}$ with [$\text{H}_2$]{}as the dominant species. They are typically very cold and dense, with temperature $10 - 20 \, \text{K}$, number density up to $\mathcal{O}(10^6) \, \text{cm}^{-3}$, and negligible filling factor. Vertically, they tend to be concentrated near the galactic disk, with a Gaussian scale height around $70 - 80 \, \text{pc}$.
While the main constituents in these phases are H, [$\text{H}_2$]{}, [$\text{H}^{\text{+}}$]{}and [$\text{e}^{\text{-}}$]{}, also present are other gaseous elements and dust.
- Other gaseous elements: From photospheric and meteoritic measurements, the cosmic composition in terms of number density are as follows: He $10\%$, C $0.03\%$, O $0.05\%$, and all other species individually each below $0.01\%$ (combined $\sim 0.03\%$). There is also evidence that a significant fraction of these elements might have been locked up in dust and hence depleted in the gaseous form.
- Dust: Dust grains are generally well-mixed with the gases in the ISM, with a dust-to-gas mass ratio believed to be around $\mathcal{O}(0.01)$. The dust grains are primarily composed of heavier elements like C, N, O, Mg, Si and Fe, with a typical specific density of $3 \, \text{g} \, \text{cm}^{-3}$. A popular model for the grain-size distribution (based on the extinction curve) is the Mathis-Rumpl-Nordsieck model. In the model, the dust grains are assumed to graphite and silicates, and the distribution given by $$n_i(a) da = A_i n_{{\text{H}}} a^{-3.5} da,$$ where $a$ is the grain size, and $A_i$ is $7.8 \times 10^{-26}$ and $6.9 \times 10^{-26} \, \text{cm}^{2.5}$ for silicates and graphite respectively. This relation holds over the range $50 \, \text{\AA} < a < 2500 \, \text{\AA}$. Besides large dust grains, it is also believed that there exists a population of large polycyclic aromatic hydrocarbon molecules, with an relative abundance of $\mathcal{O}(10^{-5})\%$.
Having described the phases of the ISM, we now argue that we only need to consider oscillation-induced $\gamma$-ray contributions from the CNM, WNM and WIM. For instance, consider the dense molecular clouds. Looking at Eq. (\[eq:emissivity\]), since most contributions to $\epsilon'$, $\bar{\omega}_I$ and $\bar{\omega}_{\text{ann}}$ scale roughly with the gas density, this means that the emissivity per H atom is much smaller than in the more rarefied phases. While the gas column density may be very high along lines of sight passing through the clouds, only a tiny fraction of the gas is [H<span style="font-variant:small-caps;">i</span>]{}, so this is unlikely to compensate for the lower emissivity per H atom. In addition, [@Abdo:2009ka] specifically mentions that large molecular clouds are known to be absent in the sector of interest. Similar types of arguments can also be made for the dense [H<span style="font-variant:small-caps;">ii</span>]{}regions and the coronal gases to explain why they can be neglected.
Parameter values {#app:parameter}
================
We present here a summary of the contributions from both elastic and inelastic processes to the parameters $\epsilon'$, $\bar{\omega}_I$ and $\bar{\omega}_{\text{ann}}$. Properties of the three phases are assumed to follow the nominal values given in Sec. \[sec:ism\].
Elastic scattering {#elastic-scattering-1}
------------------
From Eq. (\[eq:epsilonprime\]), recall that the contribution of elastic scattering to $\epsilon'$ from target species $i$ is given by $$\Delta \epsilon' = n_i v_i \left\lbrace - \int d\Omega \tfrac{|f_{i,p_i} - \bar{f}_{i,p_i}|^2}{2} + i\left[ \tfrac{2\pi\text{Re}[f_{i,p_i}(0)-\bar{f}_{i,p_i}(0)]}{p_i} + \int d\Omega \text{Im}(\bar{f}_{i,p_i}^*f_{i,p_i}) \right]\right\rbrace.$$ We now calculate this contribution for different target species.
### [$\text{e}^{\text{-}}$]{}as targets
It is useful to begin with elastic (H/[$\bar{\text{H}}$]{})-[$\text{e}^{\text{-}}$]{}scattering for the WNM and WIM (we neglect the CNM due to its extremely low ionisation fraction). First, amplitude data are available for both H and [$\bar{\text{H}}$]{}. Second, [$\text{e}^{\text{-}}$]{}may potentially be the dominant target species, since the much lower reduced mass (around $m_{\text{e}}$) implies a higher speed of approach $v$ and smaller centre-of-mass momentum $p$, hence boosting $\Delta\epsilon'$.
For H-[$\text{e}^{\text{-}}$]{}partial wave phase shifts, we use [@schwartz1961electron; @Armstead:1968zz; @fon1978elastic], while for [$\bar{\text{H}}$]{}-[$\text{e}^{\text{-}}$]{}phase shifts, we use [@bhatia1971generalized; @bhatia1974rigorous; @register1975algebraic; @mitroy1993close]. At the warm phase temperature (about $1 \, \text{eV}$), we find that $$\begin{aligned}
\tfrac{1}{4} \int d\Omega \tfrac{|f_s - \bar{f}|^2}{2} + \tfrac{3}{4} \int d\Omega \tfrac{|f_t - \bar{f}|^2}{2} &\simeq 13 \, \text{\AA}^2,\\
\tfrac{1}{4} \tfrac{2\pi\text{Re}[f_s(0)-\bar{f}(0)] }{p} + \tfrac{3}{4} \tfrac{2\pi\text{Re}[f_t(0)-\bar{f}(0)]}{p} &\simeq -11 \, \text{\AA}^2,\\
\tfrac{1}{4} \int d\Omega \text{Im}(f_s \bar{f}) + \tfrac{3}{4} \int d\Omega \text{Im}(f_t \bar{f}) &\simeq 3.8 \, \text{\AA}^2,
\end{aligned}$$ where $f_s$ and $f_t$ and are the electronic singlet and triplet H-[$\text{e}^{\text{-}}$]{}amplitudes. To check that the first value makes sense, we note that the elastic H-[$\text{e}^{\text{-}}$]{}singlet and triplet cross-sections ($39 \, \text{\AA}^2$ and $15 \, \text{\AA}^2$) are much larger than the [$\bar{\text{H}}$]{}-[$\text{e}^{\text{-}}$]{}cross-section ($1.6 \, \text{\AA}^2$). This suggests that $f_s, f_t \gg \bar{f}$, in which case the first value should be approximately half the spin-averaged H-[$\text{e}^{\text{-}}$]{}cross-section. This gives a reasonably close value of $11\, \text{\AA}^2$.
For an r.m.s. speed of approach $v = \sqrt{\tfrac{3kT}{m_{\text{e}}}} \simeq 6 \times 10^7 \, \text{cm} \, \text{s}^{-1}$, we obtain
- WNM: $\Delta \epsilon' \simeq (-2 - i) \times 10^{-9} \, \text{s}^{-1}$.
- WIM: $\Delta \epsilon' \simeq (-2 - i) \times 10^{-8} \, \text{s}^{-1}$.
### [$\text{H}^{\text{+}}$]{}as targets
Next, we consider elastic (H/[$\bar{\text{H}}$]{})-[$\text{H}^{\text{+}}$]{}scattering, again for the WNM and WIM. Here, a number of issues arise. First, a much larger number of partial waves are required to accurately reconstruct the scattering amplitudes, since the centre-of-mass momentum $p$ is now much higher. For H-[$\text{H}^{\text{+}}$]{}scattering, while phase shifts for nearly 200 partial waves are available [@hunter1980scattering], we found that they are nonetheless insufficient for the forward scattering amplitude[^1]. Second, we have not been able to find scattering data for [$\text{H}^{\text{+}}$]{}-[$\bar{\text{H}}$]{}scattering. Therefore, unlike the previous case, here an accurate calculation is not possible. The approach we adopt is as follows.
[@morgan1989atomic] claims that the elastic H-$\bar{\text{p}}$ (charge-conjugate of [$\bar{\text{H}}$]{}-[$\text{H}^{\text{+}}$]{}) cross-section is comparable to the re-arrangement cross-section ($11 \, \text{\AA}^2$ from [@Morgan:1973zz]). Should this indeed be the case, this implies that the elastic [$\bar{\text{H}}$]{}-[$\text{H}^{\text{+}}$]{}cross-section is much smaller than that of H-[$\text{H}^{\text{+}}$]{}($160 \, \text{\AA}^2$ from [@krstic1999atomic] after nuclear-spin averaging). If we then assume that $\bar{f} \ll f$, we can drop $\bar{f}$ in the expression for $\Delta \epsilon'$, giving $$\Delta \epsilon' \simeq n v \left\lbrace - \int d\Omega \tfrac{|f|^2}{2} + i \tfrac{2\pi\text{Re}[f(0)]}{p}\right\rbrace,$$ so only H-[$\text{H}^{\text{+}}$]{}data is required. The first term requires the nuclear-spin averaged cross-section, and the second term the averaged forward scattering amplitude.
Instead of the phase shifts from [@hunter1980scattering], we mostly rely on the averaged differential and total cross-sections from [@krstic1999atomic], since the latter is more recent and includes a larger number of partial waves (more than 500). To extract the averaged $\text{Re}[f(0)]$, we first note that the nuclear singlet and triplet amplitudes are given by $f_{s,t}(\theta) = f_d(\theta) \pm f_e(\pi - \theta)$, where $f_d$ and $f_e$ are the “direct” and “charge exchange” amplitudes had the nuclei been distinguishable [@krstic1999atomic]. At energies $\gtrsim 1\, \text{eV}$, both $f_d(\theta)$ and $f_e(\theta)$ become so forward-distributed that $f_s(0) \simeq f_t(0) \simeq f_d(0)$, while the overlap between $f_d(\theta)$ and $f_e(\pi - \theta)$ become so small that the singlet and triplet total cross-sections become identical. We then use the optical theorem to estimate $\text{Im}[f_d(0)]$ from the spin-averaged cross-section, which in turn can be used to estimate $|\text{Re}[f_d(0)]|$ from the spin-averaged differential cross-section at $\theta\simeq 0$. We only use the phase shifts from [@hunter1980scattering] to fix the sign of $\text{Re}[f_d(0)]$ and to check the validity of the assumptions above. We find that $$\begin{aligned}
\int d\Omega \tfrac{|f|^2}{2} &\simeq 81 \, \text{\AA}^2,\\
\tfrac{2\pi\text{Re}[f(0)]}{p} &\simeq 74 \, \text{\AA}^2,\end{aligned}$$ from which we obtain
- WNM: $\Delta \epsilon' \simeq (-4 + 4i) \times 10^{-10} \, \text{s}^{-1}$.
- WIM: $\Delta \epsilon' \simeq (-5 + 4i) \times 10^{-9} \, \text{s}^{-1}$.
These $\Delta \epsilon'$ values are smaller than that of (H/[$\bar{\text{H}}$]{})-[$\text{e}^{\text{-}}$]{}scattering, mostly due to the much smaller speed of approach $v$.
### H as targets
Finally, we consider elastic (H/[$\bar{\text{H}}$]{})-H scattering for the CNM and WNM (we neglect the WIM due to its high ionisation fraction). We have not been able to find amplitude-level data, and even differential cross-section data is only limited to the WNM. Therefore, we will only perform a crude estimate of $\Delta \epsilon'$ using total cross-section data. We use [@krstic1999atomic] and [@chakraborty2007cold] for H-H and [@sinha2004total] for H-[$\bar{\text{H}}$]{}cross-sections. Actually [@sinha2004total] only covers up to $0.27 \,\text{eV}$, a few times lower than the WNM temperature. However, since the cross-section appears relatively constant near $0.27 \, \text{eV}$, the cross-section should not differ significantly between $0.27 \, \text{eV}$ and $1\, \text{eV}$.
For H-H scattering, the CNM electronic singlet and triplet cross-sections are around $130 \, \text{\AA}^2$ and $60 \, \text{\AA}^2$, and the WNM spin-averaged cross-section $50 \, \text{\AA}^2$. For H-[$\bar{\text{H}}$]{}scattering, the CNM cross-section is $90 \, \text{\AA}^2$, and the WNM $60 \, \text{\AA}^2$. Based on these cross-sections, we now assume that $-\text{Re}(\Delta \epsilon') \simeq |\text{Im}(\Delta\epsilon')| \simeq n v (100 \, \text{\AA}^2)$ for the CNM, and $ n v (50 \, \text{\AA}^2)$ for the WNM. We then obtain
- CNM: $\Delta \epsilon' \simeq (-1 \pm i) \times 10^{-7} \, \text{s}^{-1}$.
- WNM: $\Delta \epsilon' \simeq (-5 \pm 5i) \times 10^{-9} \, \text{s}^{-1}$.
### Other targets
While other neutral targets such as He and [$\text{H}_2$]{}may offer slightly larger cross-sections than H, nonetheless their much lower abundances mean that their contributions to $\epsilon'$ can be ignored. The same can be said for other charged targets compared to [$\text{H}^{\text{+}}$]{}or [$\text{e}^{\text{-}}$]{}.
Inelastic processes {#inelastic-processes}
-------------------
For inelastic processes, we consider [$\bar{\text{H}}$]{}annihilation, ionisation of H/[$\bar{\text{H}}$]{}, as well as chemical reactions involving H. Keep in mind that $\omega_I$ only enters Eq. (\[eq:emissivity\]) as $\omega_I + \bar{\omega}_I$, so even the dominant contribution to $\omega_I$ can be ignored if it turns out to be much smaller than $\bar{\omega}_I$.
### [$\bar{\text{H}}$]{}annihilation with H
We use the semi-classical calculations of the rearrangement cross-section from [@kolos1975hydrogen]. Note that while there are fully-quantum calculations of the annihilation cross-section that include both rearrangement and annihilation-in-flight [@voronin1998antiproton; @jonsell2001stability; @armour2002calculation; @armour2005inclusion], they only include the $s$-wave component and hence give values that are much smaller. We now discuss each phase in turn.
- CNM: The cross-section is $\sigma \simeq 60 \, \text{\AA}^2$, corresponding to a rate coefficient of $\langle \sigma v \rangle \simeq 10^{-9} \, \text{cm}^3 \, \text{s}^{-1}$. The contribution to $\bar{\omega}_I$ is given by $n_{{\text{H}}}\langle \sigma v\rangle \simeq 6 \times 10^{-8} \, \text{s}^{-1}$.
- WNM: The cross-section is $\sigma \simeq 8 \, \text{\AA}^2$, corresponding to a rate coefficient of $\langle \sigma v \rangle \simeq 2 \times 10^{-9} \, \text{cm}^3 \, \text{s}^{-1}$. The contribution to $\bar{\omega}_I$ is given by $n_{{\text{H}}}\langle \sigma v\rangle \simeq 8 \times 10^{-10} \, \text{s}^{-1}$.
We ignore this for the WIM due to the high ionisation fraction.
### [$\bar{\text{H}}$]{}annihilation with [$\text{H}^{\text{+}}$]{}
We again use semi-classical calculations from [@Morgan:1973zz], since more updated cross-sections are either again for $s$-waves [@voronin1998antiproton], or do not fully cover our energy range of interest [@sakimoto2001protonium; @sakimoto2001full]. (In any case, we note that discrepancies between [@Morgan:1973zz] and [@sakimoto2001protonium; @sakimoto2001full] where they do overlap are rather small.)
We ignore this for the CNM due to the extremely low ionisation fraction. For the WNM and WIM, we find a cross-section of $\sigma = 10 \, \text{\AA}^2$, corresponding to a rate coefficient of $\langle \sigma v \rangle \simeq 2 \times 10^{-9} \, \text{cm}^3 \, \text{s}^{-1}$. Hence we obtain the following results.
- WNM: The contribution to $\bar{\omega}_I$ is $n_{{\text{H}^{\text{+}}}}\langle \sigma v\rangle \simeq 6 \times 10^{-11} \, \text{s}^{-1}$.
- WIM: The contribution to $\bar{\omega}_I$ is $n_{{\text{H}^{\text{+}}}}\langle \sigma v\rangle \simeq 7 \times 10^{-10} \, \text{s}^{-1}$.
### Other [$\bar{\text{H}}$]{}annihilation processes
One might expect [$\text{e}^{\text{-}}$]{}-[$\bar{\text{H}}$]{}annihilation to be important (especially in the WIM) since the relative speed $v$ is much higher. However, the annihilation cross-section turns out to be much smaller, due to the $6.8 \,\text{eV}$ energy threshold for re-arrangement, and that direct annihilation-in-flight in this case involves the electromagnetic interaction as opposed to the strong interaction [@armour2005inclusion].
Finally, annihilation of [$\bar{\text{H}}$]{}with any other neutral or charged species is expected to be less important than with H or [$\text{H}^{\text{+}}$]{}, due to their much lower abundances.
### Ionisation
Ionisation in the [H<span style="font-variant:small-caps;">i</span>]{}phases proceeds mainly via CR ionisation, at a rate per atom of order $10^{-16} \, \text{s}^{-1}$ [@tielens2005physics; @draine2010physics]. For the WIM, photo-ionisation plays the more important role [@tielens2005physics]. A reasonable ionisation rate per atom in the WIM is $\mathcal{O}(10^{-13} - 10^{-12}) \, \text{s}^{-1}$, consistent with the degree of ionisation given typical recombination rates, as well as estimates of the ionisation parameter based on spectral measurements. Nonetheless, we see that in all three phases, the ionisation rates are much smaller than the contributions to $\bar{\omega}_I$ from [$\bar{\text{H}}$]{}annihilation.
### Chemical reactions
Many chemical reactions involve H and may contribute to $\omega_I$. However, all the rates are much smaller than $\bar{\omega}_I$, either because they involve species with very low abundances, or that they have very small rate coefficients. We discuss a number of examples here. The rate coefficients are taken from [@tielens2005physics].
- Neutral reaction ${\text{H}}+ \text{CH} \to \text{C} + {\text{H}_2}$ has a rate coefficient $k = 1.2 \times 10^{-9} \left(\tfrac{T}{300 \, \text{K}}\right)^{0.5} e^{-\tfrac{2200 \, \text{K}}{T}}$. Even in the warm phases where the exponential suppression (from the activation barrier) becomes insignificant, the rate per H atom remains small due to the low abundance of CH.
- [$\text{H}_2$]{}formation through ${\text{H}}+ \text{H}^- \to {\text{H}_2}+ {\text{e}^{\text{-}}}$ has a high rate coefficient $k = 1.3 \times 10^{-9} \, \text{cm}^3 \, \text{s}^{-1}$, but the [$\text{H}^{\text{-}}$]{}abundance is very low.
- Radiative association ${\text{H}}+ {\text{e}^{\text{-}}}\to {\text{H}^{\text{-}}}+ \gamma$ has a very low rate coefficient $k = 10^{-18} \tfrac{T}{1 \, \text{K}} \, \text{cm}^3 \, \text{s}^{-1}$.
- Radiative association ${\text{H}}+ {\text{H}}\to {\text{H}_2}+ \gamma$ has a very low rate coefficient $k \lesssim 10^{-23} \, \text{cm}^3 \, \text{s}^{-1}$.
- Accretion of H on dust grain surface (an important catalytic reaction for [$\text{H}_2$]{}formation) occurs at a very low rate of roughly $10^{-17} \left(\tfrac{T}{10 \, \text{K}}\right)^{0.5} n_{{\text{H}}} \, \text{s}^{-1}$ per atom. (The $n_{{\text{H}}}$ dependence comes from the assumption of a constant dust-to-gas mass ratio.)
[^1]: Recall that for partial wave amplitudes $a_l$, $f(0)$ involves a summation of $(2l+1)a_l$ as opposed to $(2l+1)|a_l|^2$ for the total cross-section, hence implying a slower convergence.
|
---
abstract: 'We study the minimal wave speed and the asymptotics of the traveling wave solutions of a competitive Lotka Volterra system. The existence of the traveling wave solutions is derived by monotone iteration. The asymptotic behaviors of the wave solutions are derived by comparison argument and the exponential dichotomy, which seems to be the key to understand the geometry and the stability of the wave solutions. Also the uniqueness and the monotonicity of the waves are investigated via a generalized sliding domain method.'
address: 'Department of Mathematics and Statistics, University of North Carolina-Wilmington, Wilmington, NC 28409'
author:
- xiaojie hou
title: on the minimal speed and asymptotics of the wave solutions for the lotka volterra system
---
**Introduction and the Main result\[sec:1\]**
=============================================
We study the minimal wave speed and the asymptotic behaviors of the traveling wave solutions of the following classical Lotka-Volterra competition system
$$\left\{ \begin{array}{l}
u_{t}=u_{xx}+u(1-u-a_{1}v),\\
\\v_{t}=v_{xx}+rv(1-a_{2}u-v)\end{array}\quad\quad(x,t)\in\mathbb{R}\times\mathbb{R}^{+}\right.\label{eq:1.01}$$
where $u=u(x,t),v=v(x,t)$ and $a_{1},\: a_{2},\: r$ are positive constants.
In the wave coordinates $\xi=x+ct$, is changed into
$$\left\{ \begin{array}{l}
u_{\xi\xi}-cu_{\xi}+u(1-u-a_{1}v)=0,\\
\\v_{\xi\xi}-cv_{\xi}+rv(1-a_{2}u-v)=0\end{array}\quad\quad\xi\in\mathbb{R}.\right.\label{eq:1.02}$$
Fei and Carr [@Fei] investigated the traveling wave solutions and their minimal wave speed of system under the assumptions:
**[\[]{}H1[\]]{}**.\[\[H1\]\] $0<a_{1}<1<a_{2}$,
**[\[]{}H2[\]]{}**.\[\[H2\]\] $1-a_{1}\leq r(a_{2}-1)$.
Requiring further $r(a_{2}-1)\leq1$, they showed that for each speed $c\geq2\sqrt{r(a_{2}-1)}$ system admits monotonic traveling waves $(u(\xi),v(\xi))^{T}$ satisfying the following boundary conditions:
$$\left(\begin{array}{l}
u\\
v\end{array}\right)(-\infty)=\left(\begin{array}{l}
0\\
1\end{array}\right),\quad\left(\begin{array}{l}
u\\
v\end{array}\right)(+\infty)=\left(\begin{array}{l}
1\\
0\end{array}\right).\label{eq:1.03}$$
Under additional assumptions $r=1,$ $a_{1}+a_{2}=2$ or $r(a_{2}-1)=1-a_{1}$ they also showed that system has monotonic traveling wave solutions satisfying for $c\geq2\sqrt{1-a_{1}}$. However, the question of the minimal wave speed for the wave solutions of - remains unanswered.
We will prove that the minimal wave speed for is indeed $2\sqrt{1-a_{1}}$ if the following additional assumption
**[\[]{}H3[\]]{}.\[\[H3\]\]** $\qquad r(a_{2}-1)<(1-a_{1})(2-a_{1}+r)$.
is imposed. Noting that if $r=1$ the condition **[\[]{}H3[\]]{}** includes Fei and Carr’s additional condition.
System has three non-negative equilibria $(0,0)$, $(0,1)$ and $(1,0)$, with $(0,0)$ and $(0,1)$ unstable and $(1,0)$ stable ([@Fei]). For the convenience of later use, we introduce the transformation $\hat{v}=1-v$ to change system into local monotone. Upon dropping the hat on the function t $v$ system is changed into $$\left\{ \begin{array}{l}
u_{\xi\xi}-cu_{\xi}+u(1-a_{1}-u+a_{1}v)=0,\\
\\v_{\xi\xi}-cv_{\xi}+r(1-v)(a_{2}u-v)=0\end{array}\quad\xi\in\mathbb{R},\right.\label{eq:1.04}$$ and the boundary conditions are changed into$$\left(\begin{array}{l}
u\\
v\end{array}\right)(-\infty)=\left(\begin{array}{l}
0\\
0\end{array}\right),\quad\left(\begin{array}{l}
u\\
v\end{array}\right)(+\infty)=\left(\begin{array}{l}
1\\
1\end{array}\right).\label{eq:1.05}$$ We have,
\[thm:1\]Assuming conditions **[\[]{}H1[\]]{}-[\[]{}H3[\]]{}**, then $c^{*}=2\sqrt{1-a_{1}}$ is the minimal wave speed for system -, namely, corresponding to each fixed $c\geq c^{*}$ - has a unique traveling wave solution $(u(\xi),v(\xi))^{T}$; while for $0<c<c^{*}$, - does not have any monotonic traveling wave solutions. Furthermore, the traveling wave solution has the following asymptotic behaviors:
1\. Corresponding to each wave speed $c>2\sqrt{1-a_{1}}$, the traveling wave solution $(u(\xi),v(\xi))^{T}$ satisfies, as $\xi\rightarrow-\infty$; $$\left(\begin{array}{c}
u(\xi)\\
\\v(\xi)\end{array}\right)=\left(\begin{array}{c}
A_{1}\\
\\A_{2}\end{array}\right)e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}+o(e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}),\label{eq:1.06}$$ While as $\xi\rightarrow+\infty$ we have two cases to deal with: if $r(a_{2}-1)\leq1$, then $$\left(\begin{array}{c}
u(\xi)\\
\\v(\xi)\end{array}\right)=\left(\begin{array}{c}
1\\
\\{\displaystyle 1}\end{array}\right)-\left(\begin{array}{c}
\bar{A}_{1}\\
\\\bar{A}_{2}\end{array}\right)e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}+o(e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}),\label{eq:1.07}$$ and if $r(a_{2}-1)>1$, then $$\left(\begin{array}{c}
u(\xi)\\
\\v(\xi)\end{array}\right)=\left(\begin{array}{c}
1\\
\\{\displaystyle 1}\end{array}\right)-\left(\begin{array}{c}
\hat{A}_{1}e^{\frac{c-\sqrt{c^{2}+4}}{2}\xi}+\hat{A}_{2}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}\\
\\\hat{A}_{3}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}\end{array}\right)+\left(\begin{array}{c}
o(e^{\frac{c-\sqrt{c^{2}+4}}{2}\xi})\\
\\o(e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi})\end{array}\right)\label{eq:1.08}$$ where $A_{1}$, $A_{2}$, $\bar{A}_{1}$, $\bar{A}_{2}$, are positive constants, and $\hat{A}_{2}$ is a real number.
2\. For the traveling wave with the critical speed $c_{\mbox{ }}^{*}=2\sqrt{1-a_{1}}$, $(u(\xi),v(\xi))^{T}$ satisfies $$\left(\begin{array}{c}
u(\xi)\\
\\v(\xi)\end{array}\right)=\left(\begin{array}{c}
A_{11c}+A_{12c}\xi\\
\\A_{21c}+A_{22c}\xi\end{array}\right)e^{\sqrt{1-a_{1}}\xi}+o(\xi e^{\sqrt{1-a_{1}}\xi})\label{eq:1.09}$$ as $\xi\rightarrow-\infty$, and if $r(a_{2}-1)\leq1$, we have$$\begin{array}{cc}
\left(\begin{array}{c}
u(\xi)\\
\\v(\xi)\end{array}\right)= & \left(\begin{array}{c}
1\\
\\{\displaystyle 1}\end{array}\right)-\left(\begin{array}{c}
\bar{A}_{11}\\
\\\bar{A}_{22}\end{array}\right)e^{(\sqrt{1-a_{1}}-\sqrt{1-a_{1}+r(a_{2}-1)})\xi}\\
\\ & +o(e^{(\sqrt{1-a_{1}}-\sqrt{1-a_{1}+r(a_{2}-1)})\xi});\end{array}\label{eq:1.10}$$
while if $r(a_{2}-1)>1$, we have$$\begin{array}{ccc}
\left(\begin{array}{c}
u(\xi)\\
\\v(\xi)\end{array}\right) & = & \left(\begin{array}{c}
1\\
\\{\displaystyle 1}\end{array}\right)-\left(\begin{array}{c}
\hat{A}_{11}e^{(\sqrt{1-a_{1}}-\sqrt{2-a_{1}})\xi}+\hat{A}_{12}e^{(\sqrt{1-a_{1}}-\sqrt{1-a_{1}+r(a_{2}-1)})\xi}\\
\\\hat{A}_{22}e^{(\sqrt{1-a_{1}}-\sqrt{1-a_{1}+r(a_{2}-1)})\xi}\end{array}\right)\\
\\ & & +\left(\begin{array}{c}
o(e^{(\sqrt{1-a_{1}}-\sqrt{2-a_{1}})\xi})\\
\\o(e^{(\sqrt{1-a_{1}}-\sqrt{1-a_{1}+r(a_{2}-1)})\xi})\end{array}\right)\end{array}\label{eq:1.11}$$ as $\xi\rightarrow+\infty$, where $A_{12c},\, A_{22c}<0$, $A_{11c}$, $A_{21c}\in\mathbb{R}$ and $\bar{A}_{11},$$\bar{A}_{22}$, $\hat{A}_{12}$, $\hat{A}_{22}>0$, $\hat{A}_{11}$ is a real number.
In the next section we prove the theorem. The proof uses monotone iteration of a pair of upper and lower solutions, which is different from that of [\[]{}FeiCarr[\]]{}. In fact, we fully explore properties of the wave solutions of the classical K.P.P ([@Sattinger]) equation and the monotonic structure of system . For the existence of the traveling wave solutions to the Lotka Volterra systems with different assumptions on parameters, we refer to [@Hosono; @Kanel; @KanelZhou; @Kan-on; @TangFife] and the references therein. Noting in the above mentioned results little attention has been paid to the asymptotics of the wave solutions. However, such information is the key to the understanding of the other properties of the traveling wave solutions such as the strict monotonicity, the uniqueness as well as the stability. As a final remark we point out that the existence and stability of the traveling wave solutions for is investigated in [@LeungHou] under conditions H1 and H2 (with the inequality reversed).
\[sec:2\]**The proof**
======================
The proof of the Theorem is devided into several parts.
\[sub:2.1\]The existence.
-------------------------
We show the existence of the traveling wave solutions by monotone iteration method given by [@WuZou]. Such method reduces the existence of the wave solutions to the finding of an ordered pair of upper and lower solutions. The construction of the upper and lower solutions seems to be new, see also [@LeungHou].
\[def:2\]A continuous and essentially bounded function $(\bar{u}(\xi),\bar{v}(\xi))$, $\xi\in\mathbb{R}$ is an upper solution of if it satisfies $$\left\{ \begin{array}{l}
u''-cu'+u(1-u-a_{1}+a_{1}v)\leq0,\\
\\v''-cv'+r(1-v)(a_{1}u-v)\leq0,\end{array}\right.\mbox{ for }\xi\in\mathbb{R}/\left\{ y_{1,}y_{2},...y_{m}\right\} \label{eq:2.01}$$ and the boundary conditions $$\left(\begin{array}{c}
u\\
v\end{array}\right)(-\infty)\geq\left(\begin{array}{c}
0\\
0\end{array}\right),\,\;\left(\begin{array}{c}
u\\
v\end{array}\right)(+\infty)\geq\left(\begin{array}{c}
1\\
1\end{array}\right).\label{eq:2.02}$$ while at $y_{i}$, $i=1,2..m$, $m\in\mathbb{N}$,
$$(\bar{u}'(y_{i}-0),\bar{v}'(y_{i}-0))^{T}\geq(\bar{u}'(y_{i}+0),\bar{v}'(y_{i}+0))^{T}.\label{eq:2.02-1}$$
A lower solution of is defined similarly by reversing the above inequalities in , and .
First recall the following classical result ([@Sattinger]) on the traveling wave solutions of K.P.P equation $$\left\{ \begin{array}{l}
w''-cw'+f(w)=0,\\
\\w(-\infty)=0,\quad w(+\infty)=b.\end{array}\right.\label{eq:2.03}$$ where $f\in C^{2}([0,\, b])$ and $f>0$ on the open interval $(0,b)$ with $f(0)=f(b)=0$, $f'(0)=d_{1}>0$ and $f'(b)=-d_{2}<0$. The Lemma below describes the properties of the wave solurtions of .
\[lem:2\]Corresponding to every fixed wave speed $c\geq2\sqrt{d_{1}}$, system has a unique (up to a translation of the origin) strictly monotonically increasing traveling wave solution $w(\xi)$ for $\xi\in\mathbb{R}$. The traveling wave solution $w$ has the following asymptotic behaviors:
For the wave solution with non-critical speed $c>2\sqrt{d_{1}}$, we have $$w(\xi)=a_{w}e^{\frac{c-\sqrt{c^{2}-4d_{1}}}{2}\xi}+o(e^{\frac{c-\sqrt{c^{2}-4d_{1}}}{2}\xi})\mbox{ as }\xi\rightarrow-\infty,\label{eq:2.04}$$ $$w(\xi)=b-b_{w}e^{\frac{c-\sqrt{c^{2}+4d_{2}}}{2}\xi}+o(e^{\frac{c-\sqrt{c^{2}+4d_{2}}}{2}\xi})\mbox{ as }\xi\rightarrow+\infty,\label{eq:2.05}$$ where $a_{w}$ and $b_{w}$ are positive constants.
For the wave with critical speed $c=2\sqrt{d_{1}}$, we have$$w(\xi)=(a_{c}+d_{c}\xi)e^{\sqrt{d_{1}}\xi}+o(\xi e^{\sqrt{d_{1}}\xi})\mbox{ as }\xi\rightarrow-\infty,\label{eq:2.06}$$ $$w(\xi)=b-b_{c}e^{(\sqrt{d_{1}}-\sqrt{d_{1}+d_{2}})\xi}+o(e^{(\sqrt{d_{1}}-\sqrt{d_{1}+d_{2}})\xi})\mbox{ as }\xi\rightarrow+\infty,\label{eq:2.07}$$ where the constants $d_{c}$ is negative, $b_{c}$ is positive and $a_{c}\in\mathbb{R}$.
According to Lemma \[lem:2\], we let $c\geq2\sqrt{1-a_{1}}$ be fixed and $\underline{u}(\xi)$, $\xi\in\mathbb{R}$ be a solution of the following form of K.P.P equation
$$\left\{ \begin{array}{l}
g''(\xi)-g'(\xi)+(1-a_{1})g(\xi)(1-g(\xi))=0,\\
\\g(-\infty)=0,\, g(+\infty)=1,\end{array}\quad\quad\xi\in\mathbb{R}\right.\label{eq:2.08}$$
and for the same $c$ let $l$ be a number such that $\frac{r(a_{2}-1)-(1-a_{1})}{1-a_{1}+r}\leq l<1-a_{1}$ and $\bar{\bar{u}}(\xi)$, $\xi\in\mathbb{R}$ be a solution of a K.P.P equation with the form $$\left\{ \begin{array}{l}
h''(\xi)-ch'(\xi)+(1-a_{1})h(\xi)(1-\frac{1-a_{1}-l}{1-a_{1}}h(\xi))=0,\\
\\h(-\infty)=0,\quad h(+\infty)=\frac{1-a_{1}}{1-a_{1}-l},\end{array}\quad\quad\xi\in\mathbb{R}.\right.\label{eq:2.09}$$
Define
$$\left(\begin{array}{l}
\bar{u}(\xi)\\
\\\bar{v}(\xi)\end{array}\right)=\left(\begin{array}{l}
\min_{\xi\in\mathbb{R}}\{\bar{\bar{u}}(\xi),1\}\\
\\\min_{\xi\in\mathbb{R}}\{(1+l)\bar{\bar{u}}(\xi),1\}\end{array}\right),\quad\left(\begin{array}{l}
\underline{u}(\xi)\\
\\\underline{v}(\xi)\end{array}\right)=\left(\begin{array}{l}
\underline{u}(\xi)\\
\\\underline{u}(\xi)\end{array}\right).\label{eq:2.10}$$
We have the following
\[lem:3\]For every fixed $c\geq2\sqrt{1-a_{1}}$, $(\bar{u}(\xi),\bar{v}(\xi))^{T}$ and $(u(\xi),\underbar{v}(\xi))^{T}$ in define respectively an upper and lower solutions for -.
The verification for $(\underline{u},\underline{v}$) being a lower solution for - is straightforward, so we skip it. According to Lemma \[lem:2\] $\bar{\bar{u}}(\xi)$ is strictly monotonically increasing for $\xi\in\mathbb{R}$, there exist $N_{1},N_{2}\in\mathbb{R}$ such that $\bar{\bar{u}}(N_{1})=\frac{1}{1+l}$ and $\bar{\bar{u}}(N_{2})=1$. We can therefore rewrite $(\bar{u},\bar{v})$ as follows
$$\left(\begin{array}{l}
\bar{u}(\xi)\\
\\\bar{v}(\xi)\end{array}\right)=\left\{ \begin{array}{l}
(\bar{\bar{u}}(\xi),\bar{\bar{v}}(\xi))^{T},\quad\mbox{for }-\infty<\xi\leq N_{1};\\
\\(\bar{\bar{u}}(\xi),1)^{T},\quad\mbox{for }N_{1}<\xi\leq N_{2};\\
\\(1,1)^{T},\quad\mbox{for }N_{2}<\xi<+\infty.\end{array}\right.\label{eq:2.11-1}$$
For $N_{2}<\xi<+\infty$, $(\bar{u}(\xi),\bar{v}(\xi))^{T}=(1,1)^{T}$ is obviously a solution of and satisfying the inequality on the boundary, we only need to verify that $(\bar{u}(\xi),\bar{v}(\xi))^{T}$ satisfies the inequality on the intervals $(-\infty,N_{1}]$ and $(N_{1},N_{2})$ respectively. For the pair $(\bar{u}(\xi),\bar{v}(\xi))^{T}=(\bar{\bar{u}}(\xi),(1+l)\bar{\bar{u}}(\xi))^{T}$ on $-\infty<\xi\leq N_{1}$, we have
$$\begin{array}{ll}
& \bar{\bar{u}}''-c\bar{\bar{u}}'+\bar{\bar{u}}(1-a_{1}-\bar{\bar{u}}+a_{1}\bar{\bar{v}})\\
\\= & -(1-a_{1})\bar{\bar{u}}(1-\frac{1-a_{1}-l}{1-a_{1}}\bar{\bar{u}})+\bar{\bar{u}}(1-a_{1}-\bar{\bar{u}}+a_{1}(1+l)\bar{\bar{u}})\\
\\= & -(1-a_{1})\bar{\bar{u}}(1-\frac{1-a_{1}-l}{1-a_{1}}\bar{\bar{u}}-1+\frac{1-a_{1}(1+l)}{1-a_{1}}\bar{\bar{u}})\\
\\= & -(1-a_{1})l\bar{\bar{u}}^{2}\leq0,\end{array}$$ and$$\begin{array}{ll}
& \bar{\bar{v}}''-c\bar{\bar{v}}'+r(1-\bar{\bar{v}})(a_{2}\bar{\bar{u}}-\bar{\bar{v}})\\
\\= & (1+l)(\bar{\bar{u}}''-c\bar{\bar{u}}'+\frac{r}{1+l}(1-(1+l)\bar{\bar{u}})(a_{2}\bar{\bar{u}}-(1+l)\bar{\bar{u}}))\\
\\= & (1+l)[\bar{\bar{u}}''-c\bar{\bar{u}}'+(1-a_{1})\bar{\bar{u}}(1-\bar{\bar{u}}+\frac{l}{1-a_{1}}\bar{\bar{u}})\\
\\ & -(1-a_{1})\bar{\bar{u}}(1-\bar{\bar{u}}+\frac{l}{1-a_{1}}\bar{\bar{u}})+\frac{r}{1+l}(1-(1+l)\bar{\bar{u}})\bar{\bar{u}}(a_{2}-(1+l))]\\
\\= & (1+l)\bar{\bar{u}}[\frac{r}{1+l}(1-(1+l)\bar{\bar{u}})(a_{2}-(1+l))-(1-a_{1})(1-\bar{\bar{u}}+\frac{l}{1-a_{1}}\bar{\bar{u}})]\\
\\= & (1+l)\bar{\bar{u}}[r(\frac{a_{2}}{1+l}-1)-(1-a_{1})-\bar{\bar{u}}(r(a_{2}-1-l)-(1-a_{1}-l)]\\
\\\leq & 0.\end{array}$$ The last inequality is true because of [\[]{}**H3**[\]]{} and the choice of $l$.
For $N_{1}<\xi\leq N_{2}$, we verify that $(\bar{\bar{u}},1)^{T}$ satisfies the inequality . We only verify for the first component since the one for $v=1$ is trivial.
$$\begin{array}{ll}
& \bar{\bar{u}}''-c\bar{\bar{u}}'+\bar{\bar{u}}(1-a_{1}-\bar{\bar{u}}+a_{1}v)\\
\\= & \bar{\bar{u}}''-c\bar{\bar{u}}'+\bar{\bar{u}}(1-\bar{\bar{u}})\\
\\ & +(1-a_{1})\bar{\bar{u}}(1-\frac{1-a_{1}-1}{1-a_{1}}\bar{\bar{u}})-(1-a_{1})\bar{\bar{u}}(1-\frac{1-a_{1}-1}{1-a_{1}}\bar{\bar{u}})\\
\\= & \bar{\bar{u}}(1-\bar{\bar{u}}-(1-a_{1})+(1-a_{1}-l)\bar{\bar{u}})\\
\\\leq & 0\end{array}$$
Therefore, we have the conclusion of the Lemma.
To show the orderliness of the upper and lower solution pairs, we first introduce a sliding domain method which applies to a sightly more general system than . Noting that no monotonicity requirements are imposed on the upper and lower sloutions.
\[lem:4\]Let the $C^{2}$ vector functions $\bar{U}(\xi)=(\bar{u}_{1}(\xi),\bar{u}_{2}(\xi),...,\bar{u}_{n}(\xi))^{T}$ and $\underline{U}(\xi)=(\underline{u}_{1}(\xi),\underline{u}_{2}(\xi),...,\underline{u}_{n}(\xi))^{T}$ be $C^{2}$ and satisfy the following inequalities
$$D\bar{U}''-c\bar{U}'+F(U)\leq0\leq D\underline{U}''-c\underline{U}'+F(\underline{U})\quad\mbox{for\,}\,\xi\in[-N,N]\label{eq:2.11}$$
and
$$\underline{U}(-N)<\bar{U}(\xi)\quad\mbox{for\,}\,\xi\in(-N,N],\label{eq:2.12}$$
$$\underline{U}(\xi)<\bar{U}(N)\quad\mbox{for\,}\,\xi\in[-N,N),\label{eq:2.13}$$
where $D$ is a diagonal matrix with positive entries $D_{i}$, $i=1,2...n$, $F(U)=(F_{1}(U),...,F_{n}(U))^{T}$ is $C^{1}$ with respect to its components and $\frac{\partial F_{i}}{\partial u_{j}}\geq0$ for $i\neq j$, $i,j=1,2...n$, then
$$\underline{U}(\xi)\leq\bar{U}(\xi),\qquad\xi\in[-N,N].\label{eq:2.14}$$
We adapt the proof of [@Berestycki] . Shift $\bar{U}(\xi)$ to the left, for $0\leq\mu\leq2N$, consider $\bar{U}^{\mu}(\xi):=\bar{U}(\xi+\mu)$ on the interval $(-N-\mu,N-\mu)$. On both ends of the interval, by and , we have
$$\underline{U}(\xi)<\bar{U}^{\mu}(\xi).\label{eq:2.15}$$
Starting from $\mu=2N$, decreasing $\mu$, for every $\mu$ in $0<\mu<2N$, the inequality is true on the end points of the respective interval. On decreasing $\mu$, suppose that there is a first $\mu$ with $0<\mu<2N$ such that
$$\underline{U}(\xi)\leq\bar{U}^{\mu}(\xi)\quad\xi\in(-N-\mu,N-\mu)$$ and there is one component, for example the $i-th$, such that the equality holds on a point $\xi_{1}$ inside the interval. Let $W(\xi)=(w_{1}(\xi),w_{2}(\xi),...,w_{n}(\xi))^{T}=\bar{U}^{r}(\xi)-\underline{U}(\xi)$, then $w_{i}(\xi)$, $i=1,2,...,n$ satisfies
$$\left\{ \begin{array}{l}
D_{i}w_{i}''-cw_{i}'+\frac{\partial F_{i}}{\partial u_{i}}w_{i}\leq D_{i}w_{i}''-cw_{i}'+\Sigma_{j=1}^{n}\frac{\partial F_{i}}{\partial u_{j}}w_{j}\leq0,\\
\\w_{i}(\xi_{1})=0,\; w_{j}(\xi)\geq0\:\mbox{\, f\mbox{or}}\:\xi\in[-N-\mu,N-\mu],\end{array}\right.$$ the Maximum principle further implies that $w_{i}\equiv0$ for $\xi\in[-N-\mu,N-\mu]$, but this is in contradiction with on the boundary points $\xi=-N-\mu$ and $\xi=N-\mu$. So we can decrease $\mu$ all the way to zero. This proves the Lemma.
\[lem:5\]There exists a $\nu\geq0$ such that $(\bar{u},\bar{v})^{T}(\xi+\nu)\geq(\underbar{u},\underbar{v})^{T}(\xi)\:\mbox{\, f\mbox{or}}\:\xi\in\mathbb{R}.$
We only prove for the wave speed $c>2\sqrt{1-a_{1}}$ and $r(a_{2}-1)\leq1$ since it is similar to show the other cases. We first derive the asymptotic behaviors of the upper- and lower-solutions at infinities. By Lemma \[lem:2\], we have the following asymptotics for the upper and lower solutions $$\left(\begin{array}{c}
\bar{u}\\
\\\bar{v}\end{array}\right)(\xi)=\left(\begin{array}{c}
A_{1}\\
\\(1+l)A_{1}\end{array}\right)e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}+o(e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi})\label{eq:2.16}$$ and
$$\left(\begin{array}{c}
\underline{u}\\
\\\underline{v}\end{array}\right)(\xi)=\left(\begin{array}{c}
B_{1}\\
\\B_{1}\end{array}\right)e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}+o(e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi})\label{eq:2.17}$$
as $\xi\rightarrow-\infty$;
$$\left(\begin{array}{c}
\bar{u}\\
\\\bar{v}\end{array}\right)(\xi)\equiv\left(\begin{array}{c}
1\\
\\1\end{array}\right)\label{eq:2.18}$$
and
$$\left(\begin{array}{c}
\underline{u}\\
\\\underline{v}\end{array}\right)(\xi)=\left(\begin{array}{c}
1\\
\\1\end{array}\right)-\left(\begin{array}{c}
B_{2}\\
\\B_{2}\end{array}\right)e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}+o(e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi})\label{eq:2.19}$$
as $\xi\rightarrow+\infty$, where $A_{1}$, $A_{2}$, $B_{1}$, **$B_{2}$** are positive constants.
Since (\[eq:2.06\]) is translation invariant, $\bar{v}^{\tilde{r}}(\xi)\equiv\bar{v}(\xi+\tilde{r})$ is also a solution of (\[eq:2.06\]) for any $\tilde{r}\in\mathbb{R}$. It then follows that $(\bar{u}^{\tilde{r}},\bar{v}^{\tilde{r}})^{T}(\xi)$ is also an upper-solution pair for system -. For the asymptotic behavior of $(\bar{u},\bar{v})^{\tilde{r}}(\xi)$ at $-\infty$, we can simply replace $(A_{1},\:(1+l)A_{1})^{T}$ by $(A_{1},\:(1+l)A_{1})^{T}e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\,\tilde{r}}$ in . Now we choose $\tilde{r}>0$ large enough such that
$$e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\,\tilde{r}}>1.$$ Then there exists a sufficiently large $N_{1}>0$ such that
$$\left(\begin{array}{c}
\bar{u}^{\tilde{r}}(\xi)\\
\\\bar{v}^{\tilde{r}}(\xi)\end{array}\right)>\left(\begin{array}{c}
\underline{u}(\xi)\\
\\\underline{v}(\xi)\end{array}\right)\quad{\normalcolor \mbox{for}\;}\xi\in(-\infty,-N_{1}].\label{eq:2.20}$$
On the other hand, the boundary conditions of the upper- and lower-solutions at $+\infty$ also imply, on increasing $N_{1}$ if necessary, that
$$\left(\begin{array}{c}
\bar{u}^{\tilde{r}}(\xi)\\
\\\bar{v}^{\tilde{r}}(\xi)\end{array}\right)>\left(\begin{array}{c}
\underline{u}(\xi)\\
\\\underline{v}(\xi)\end{array}\right)\quad{\normalcolor \mbox{for}\;}\xi\in[N_{1},\:+\infty).\label{eq:2.21}$$
On the interval $[-N_{1},N_{1}],$ the strict monotonicity of the upper and lower solutions $(\bar{u}^{\tilde{r}},\bar{v}^{\tilde{r}})^{T}$ and $(\underline{u},\underline{v})^{T}$, and the inequalities - imply that
$$(\underline{u},\underline{v})^{T}(-N_{1})<(\bar{u}^{\tilde{r}},\bar{v}^{\tilde{r}})^{T}(\xi)\quad\mbox{for}\,\xi\in(-N_{1},N_{1}],$$ and
$$(\underline{u},\underline{v})^{T}(\xi)<(\bar{u}^{\tilde{r}},\bar{v}^{\tilde{r}})^{T}(N_{1})\quad\mbox{for}\,\xi\in[-N_{1},N_{1}).$$ Therefore, by Lemma \[lem:4\] we have
$$(\underline{u},\underline{v})^{T}(\xi)\leq\bar{(\bar{u}^{\tilde{r}},\bar{v}^{\tilde{r}})^{T}(\xi)}\quad\mbox{for}\,\xi\in[-N_{1},N_{1}].\label{eq:2.22}$$
Inequality along with , show the validity of the Lemma.
**Proof of the Existence:** We still use $(\bar{u},\bar{v})^{T}(\xi)$ to denote the shifted upper-solution as given in lemma \[lem:4\]. Applying the monotone iteration method given in [@WuZou] to the upper and lower solutions defined in , we then have the existence of the traveling wave solutions for $c\geq2\sqrt{a_{1}-1}$. The boundary conditions that the upper and lower solutions satisfying lead to the boundary conditions for traveling waves.
\[sub:2.2\]The asymptotics and the monotonicity.
------------------------------------------------
To derive the asymptotic decay rate of the traveling wave solution at $\pm\infty$, we let $c\geq2\sqrt{1-a_{1}}$ and $$U(\xi):=(u(\xi),v(\xi))^{T}\quad\mbox{for}\:-\infty<\xi<\infty\label{eq:2.23}$$ be the corresponding traveling wave solution of - resulted from the monotone iteration. Lemma \[lem:2\] implies that the upper- and the lower-solutions as derived in Lemma \[lem:3\] have the same asymptotic rates at $-\infty$. and then follow from Lemma \[lem:5\]. We differentiate with respect to $\xi$, and note that $(U(\xi))':=(w_{1},w_{2})^{T}(\xi)$ satisfies $$(w_{1})_{\xi\xi}-c(w_{1})_{\xi}+A_{11}(u,v)w_{1}+A_{12}(u,v)w_{2}=0,\label{eq:2.24}$$ $$(w_{2})_{\xi\xi}-c(w_{2})_{\xi}+A_{21}(u,v)w_{1}+A_{22}(u,v)w_{2}=0,\label{eq:2.25}$$ where
$$\begin{array}{ll}
A_{11}(u,v)=1-a_{1}-2u+a_{1}v,\quad & A_{12}(u,v)=a_{1}u\\
\\A_{21}(u,v)=a_{2}r(1-v), & A_{22}(u,v)=-r(a_{2}u+1-2v)\end{array}$$
We next study the exponential decay rates of the traveling wave solution $U(\xi)$ at $+\infty$. The asymptotic system of (\[eq:2.24\]) and (\[eq:2.25\]) as $\xi\rightarrow+\infty$ is $$\left\{ \begin{array}{l}
(\psi_{1})''-c(\psi_{1})'-\psi_{1}+a_{1}\psi_{2}=0,\\
\\(\psi_{2})''-c(\psi_{2})'-r(a_{2}-1)\psi_{2}=0.\end{array}\right.\label{eq:2.26}$$
It is easy to see that the system admits exponential dichotomy. Since the traveling wave solution $(u(\xi),v(\xi))^{T}$ converges monotonically to a constant limit as $\xi\rightarrow\pm\infty$, the derivative of the traveling wave solution satisfies $(w_{1}(\pm\infty),w_{2}(\pm\infty))=(0,0)$ ([@WuZou], p$658$ Lemma $3.2$). Hence we are only interested in finding exponentially decaying solutions of at $+\infty$.
One can write the the general solution of the second equation of as $$\psi_{2}=A^{1}e^{\frac{c+\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}+B^{1}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}$$ for some constants $A^{1}$ and $B^{1}$. Since $w_{2}\rightarrow0$ as $\xi\rightarrow+\infty$, one immediately has $A^{1}=0$.
We then study the solution of the second equation of (\[eq:2.26\]), rewriting the equation as $$(\psi_{1})''-c(\psi_{1})'-\psi_{1}=-a_{1}\psi_{2},\label{eq:2.27}$$ we have the following expression for the solution of , $$\psi_{1}=\bar{B}_{1}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}+\bar{B}_{2}e^{\frac{c-\sqrt{c^{2}+4}}{2}\xi}+\bar{B}_{3}e^{\frac{c+\sqrt{c^{2}+4}}{2}\xi}.\label{eq:2.28}$$
Since $w_{2}(\xi)\rightarrow0$ as $\xi\rightarrow+\infty$, then $\bar{B}_{3}=0.$ Also noticing that (\[eq:2.27\]) is non-homogeneous, we have $\bar{B}_{1}\neq0$. By roughness of the exponential dichotomy (levinson) and integration, we obtain the asymptotic decay rate of the traveling wave solutions at $+\infty$ given in (\[eq:1.10\]).
We next show the monotonicity of the traveling wave solutions. By the monotone iteration process [@WuZou], the traveling wave solution $U(\xi)$ is increasing for $\xi\in\mathbb{R}$, it then follows that $W(\xi)=U'(\xi)\geq0$ satisfying and and
$$w_{1},w_{2}\geq0,\;(w_{1},w_{2})^{T}(\pm\infty)=0.\label{eq:2.31}$$
The Maximum Principle implies that $(w_{1},w_{2})^{T}(\xi)>0$ for $\xi\in\mathbb{R}$. This concludes that the traveling wave solution is strictly increasing on $\mathbb{R}$.
The Uniqueness.
---------------
On the uniqueness of the traveling wave solution for every $c\geq2\sqrt{1-a_{1}}$, we only prove the conclusion for traveling wave solutions with asymptotic behaviors and , since other case can be proved similarly. Let $U_{1}(\xi)$ and $U_{2}(\xi)$ be two traveling wave solutions of system - with the same speed $c>2\sqrt{1-a_{1}}$. There exist positive constants $A_{i}$, $B_{i}$, $i=1,2,3,4$ and a large number $N>0$ such that for $\xi<-N$,$$U_{1}(\xi)=\left(\begin{array}{c}
(A_{1}+o(1))e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}\\
\\(A_{2}+o(1))e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}\end{array}\right)\label{eq:2.32}$$ $$U_{2}(\xi)=\left(\begin{array}{c}
(A_{3}+o(1))e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}\\
\\(A_{4}+o(1))e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}\end{array}\right);\label{eq:2.33}$$ and for $\xi>N$,$$U_{1}(\xi)=\left(\begin{array}{c}
{\displaystyle 1-(B_{1}+o(1))e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}}\\
\\{\displaystyle 1-(B_{2}+o(1))e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}}\end{array}\right)\label{eq:2.34}$$ $$U_{2}(\xi)=\left(\begin{array}{c}
{\displaystyle 1-(B_{3}+o(1))e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}}\\
\\{\displaystyle 1-(B_{4}+o(1))e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}}\end{array}\right)\label{eq:2.35}$$ The traveling wave solutions of system are translation invariant, thus for any $\theta>0$, $U_{1}^{\theta}(\xi):=U_{1}(\xi+\theta)$ is also a traveling wave solution of and . By and , the solution $U_{1}(\xi+\theta)$ has the following asymptotic behaviors:$$U_{1}^{\theta}(\xi)=\left(\begin{array}{c}
(A_{1}+o(1))e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\theta}e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}\\
\\(A_{2}+o(1))e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\theta}e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}\end{array}\right)\label{eq:2.36}$$ for $\xi\leq-N$;$$U_{1}^{\theta}(\xi)=\left(\begin{array}{c}
{\displaystyle 1-(B_{1}+o(1))e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\theta}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}}\\
\\{\displaystyle 1-(B_{2}+o(1))e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\theta}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\xi}}\end{array}\right)\label{eq:2.37}$$ for $\xi\geq N$.
It is clear that for $\theta$ large enough, we have $$A_{1}e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\theta}>A_{3},\label{eq:2.38}$$ $$A_{2}e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\theta}>A_{4},\label{eq:2.39}$$ $$B_{1}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\theta}<B_{3},\label{eq:2.40}$$ $$B_{2}e^{\frac{c-\sqrt{c^{2}+4r(a_{2}-1)}}{2}\theta}<B_{4}.\label{eq:2.41}$$ Inequalities - imply that for $\theta$ large enough, $$U_{1}^{\theta}(\xi)>U_{2}(\xi)\label{eq:2.42}$$ for $\xi\in(-\infty,-N]$$\cup$$[N+\infty).$ We now consider $U_{1}^{\theta}(\xi)$ and $U_{2}(\xi)$ on $[-N,+N]$.
On the interval $[-N_{1},N_{1}],$ the strict monotonicity of $U_{1}$ and $U_{2}$ and the inquality imply that
$$U_{2}(-N_{1})<U_{1}^{\theta}(\xi)\quad\mbox{for}\,\xi\in(-N_{1},N_{1}],$$ and
$$U_{2}(\xi)<U_{1}^{\theta}(N_{1})\quad\mbox{for}\,\xi\in[-N_{1},N_{1}).$$ Therefore, by Lemma \[lem:4\] we have
$$U_{1}^{\theta}(\xi)\geq U_{2}(\xi)\quad\mbox{for}\,\xi\in[-N_{1},N_{1}].\label{eq:2.43}$$
Inequalities along with show the validity of the Lemma.
The range of the wave speed.
----------------------------
The next Theorem shows that the lower bound $2\sqrt{1-a_{1}}$ for the wave speed is optimal, hence $c=2\sqrt{\alpha}$ is the critical wave speed.
\[lem:6-1\]There is no monotone traveling wave solution of - for any $0<c<2\sqrt{\alpha}$.
Suppose there is a constant $c$ with $0<c<2\sqrt{1-a_{1}}$ and a solution $V(\xi)=(v_{1},v_{2})^{T}(\xi)$ of (\[eq:1.04\]-\[eq:1.05\]) corresponding to it. Similar to \[sub:2.2\], the asymptotic behaviors of $V(\xi)$ at $-\infty$ are described by $$\left(\begin{array}{c}
v_{1}(\xi)\\
\\v_{2}(\xi)\end{array}\right)=\left(\begin{array}{c}
A_{s}\\
\\B_{s}\end{array}\right)e^{\frac{c-\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}+\left(\begin{array}{c}
\bar{A_{s}}\\
\\\bar{B_{s}}\end{array}\right)e^{\frac{c+\sqrt{c^{2}-4(1-a_{1})}}{2}\xi}+h.o.t,$$ where $(A_{s},B_{s})^{T}$and $(\bar{A_{s},}\bar{B_{s}})$ are not both zero. The condition $0<c<2\sqrt{1-a_{1}}$ implies that $V(\xi)$ is oscillating. This concludes that any solution of - with $c<2\sqrt{1-a_{1}}$ is not strictly monotonic.
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J. I Kanel, Li Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Analysis, Theory, Methods & Applications, 27, No. 5, (1996) 579-587.
Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Analysis 44 (2001) 239-246.
Y. Kan-on, Fisher wave fronts for the lotka-volterra competition model with diffusion, Nonlinear Analysis, Theory, methods & Applications, 28 No. 1, (1997) 145-164.
A. Leung, X. Hou and W. Feng, Traveling wave solutions for the Lotka Volterra system revisited. Submitted.
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M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rat. Mech. Anal., (73) 1980, pp 69-77.
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---
abstract: 'The electrical transport properties of corner-shaped Al superconducting microstrips have been investigated. We demonstrate that the sharp turns lead to asymmetric vortex dynamics, allowing for easier penetration from the inner concave angle than from the outer convex angle. This effect is evidenced by a strong rectification of the voltage signal otherwise absent in straight superconducting strips. At low magnetic fields, an enhancement of the critical current with increasing magnetic field is observed for a particular combination of field and current polarity, confirming a recently theoretically predicted competing interplay of superconducting screening currents and applied currents at the inner side of the turn.'
author:
- 'O.-A. Adami[^1]'
- 'D. Cerbu$^{*}$'
- 'D. Cabosart'
- 'M. Motta'
- 'J. Cuppens'
- 'W.A. Ortiz'
- 'V. V. Moshchalkov'
- 'B. Hackens'
- 'R. Delamare'
- 'J. Van de Vondel'
- 'A. V. Silhanek'
title: 'Superconducting properties of corner-shaped Al microstrips'
---
The ability of superconductors to carry electricity without resistance holds in a restricted current density range $j<j_{max}$. Several physical mechanisms can be identified as responsible for limiting $j_{max}$ such as the motion of vortices, the formation of phase slip centers or eventually when the pair-breaking current, $j_{pb}$, is reached.
In principle, it is possible to attain the ultimate limit $j_{max}$=$j_{pb}$ by properly choosing the dimensions of the superconducting strip. Indeed, if the width $w$ of the strip is such that $w<4.4\xi$, where $\xi$ is the coherence length, vortices cannot fit into the sample[@likharev] and therefore $j_{max}$ cannot be limited by a vortex depinning process. In 1980 Kupriyanov and Lukichev [@kupriyanov] were able to determine theoretically $j_{pb}$ for all temperatures, by solving the Eilenberger equations, and only two years later their predictions were experimentally confirmed by Romijn *et al.*[@romijn] using *straight* Al strips. These works focused on the case where $w\ll\Lambda$, with $\Lambda=2\lambda^{2}/d$ the Pearl length[@pearl], $\lambda$ the London penetration depth, and $d$ the thickness of the superconductor.
Recently a renewed interest for understanding the limiting factors of $j_{max}$ in *non-straight* strips has arisen, partially motivated by the ubiquitous presence of sharp turns in more realistic architectures as those used in the superconducting meanders for single photon and single electron detectors [@bulaevski; @victor].
Early theoretical calculations by Hagerdorn and Hall[@hall] showed that a sharp bend in a superconducting wire leads to current crowding effects at the inner corner of the the bend, which in turn reduces the total critical current when compared to a straight wire. Not only sharp angles along the superconducting bridge, but any sudden change in the cross section of the wire, can lead to a reduction of the critical current. For instance, it has been pointed out in Ref.[\[]{} that a sudden increase in the cross section of a transport bridge leads to severe modifications of the voltage-current characteristics rendering unreliable those measurements performed in cross-shaped geometries. More recently, Clem and Berggren[@clem-berggren] have theoretically demonstrated that sudden increases in the cross section of a transport bridge, as those caused by voltage leads, also produce current crowding effects and the consequent detriment of the critical current, similarly to right-angle bends. These predictions have been independently confirmed experimentally by Hortensius *et al.*[@hortensius] and by Henrich *et al.*[@henrich] in submicron scale samples of NbTiN and NbN, respectively, and found to be also relevant in larger samples[@vestgarden].
The effect of a magnetic field applied perpendicularly to the plane containing the superconducting wire with a sharp turn has been discussed in Ref.[\[]{} and Ref.[\[]{}. Strikingly, in Ref.[\[]{} it is theoretically predicted that due to compensation effects between the field induced stream-lines and the externally applied current at the current crowding point, the critical current of thin and narrow superconducting bridges ($\xi\ll w\ll\Lambda$) should *increase* with field for small fields values and for a particular polarity of the applied field.
In this work we provide experimental confirmation of the theoretical predictions of Ref.[\[]{} and show that current crowding leads also to a clearly distinct superconducting response for positive and negative fields (or currents), making these asymmetric superconducting nanocircuits potentially efficient voltage rectifiers.
The samples investigated were all co-fabricated on the same chip and consist of electron-beam lithographically defined Al structures of thickness $d$ = 67 $\pm$ 2 nm, deposited by rf sputtering on top of a Si/SiO$_{2}$ substrate. We focus on two different geometries. Sample S90 consist of a 3.3 $\mu$m wide transport bridge with a 90$^{\circ}$ corner equidistant from two voltage probes separated 9.6 $\mu$m from the inner angle of the sharp bend. Similarly, S180 is a conventional straight transport bridge 3.7 $\mu$m wide and with voltage probes separated by 20.9 $\mu$m. These dimensions depart from the nominal values and were obtained via atomic force microscopy as shown in Figure 1(a)-(b).
The field dependence of the superconducting-to-normal metal transitions, $T_{c}$($H$), determined as 0.95$R_{N}$, where $R_{N}$ is the normal state resistance, and using an ac-current[@electronics] of 1 $\mu$A, is basically the same for the two samples studied (see Figure 1(c)). This similarity of the phase boundaries allows us to make reliable and direct comparisons between the two samples without the necessity to work with reduced temperatures or field units. The critical temperature at zero field is $T_{c0}=1.320\pm0.008$ K and the superconducting coherence length obtained from the Ginzburg-Landau approximation is $\xi(0)=121\pm3$ nm. The BCS coherence length for Al of similar characteristics [@romijn] ($T_{c0}$ and $d$) as the one used here is $\xi_{0}=1320$ nm, indicating that our Al falls in the dirty limit $\ell\ll\xi_{0}$, with $\ell$ the electronic mean free path. Using the relation $\xi(0)=0.855\sqrt{(}\xi_{0}\ell)$ we deduce $\ell\sim15$ nm. An independent estimation of $\ell\sim17$ nm can be obtained from the normal state resistivity $\rho=2.0\pm0.1^{.}10^{-8}$ $\Omega$m, and taking[@romijn] $\rho\ell=4^{.}10^{-16}$ $\Omega$m$^{2}$. In the dirty limit the magnetic penetration depth is given by $\lambda(0)=\lambda_{L}(0)\sqrt{\xi_{0}/\ell}\approx$ 145 nm, where $\lambda_{L}(0)=$16 nm is the London penetration depth. For thin film geometry with a perpendicular external field we need to use the Pearl length[@pearl] $\Lambda=2\lambda^{2}/d$. In the considered samples $\Lambda>2w$ for $T>$1.19 K.
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Let us now concentrate on the current-voltage characteristics, $V$($I$), of the considered systems. At zero external field, the $V$($I$) curves and, in particular the critical current, $I_{c}$, should be uniquely defined, irrespective of the direction of the applied current. This independence on the direction of the current persists at all fields for the S180 sample, but does not hold for the S90 sample. Indeed, on the one hand, the outer angle of the sharp corner has a larger surface nucleation critical field $H_{c3}$ (a factor $\sim1.16$ higher for the S90) when compared to the critical field at the inner corner[@schweigert] thus making the outer corner a point of enhanced superconductivity[@victor]. On the other hand, stream-lines of the applied current tend to conglomerate at the inner corner[@hall], depleting the order parameter at that place. Notice that both effects, larger surface nucleation field and lower applied current density at the sharper corner, share the same origin in the impossibility of both, screening or applied currents, to reach the tip of the bend.
The fact that current crowding at the inner corner leads to local depletion of the superconducting order parameter implies automatically a reduction of the surface barrier for vortex penetration[@clem-peeters] as long as the applied current is such that the Lorentz force pushes vortices from the inner towards the outer corner. However, if the current is reversed, vortices will not penetrate from the outer corner (where total current is nearly zero) but rather symmetrically from the straight legs of the bridge[@clem-peeters]. As a consequence of this different nucleation position and nucleation condition for the two opposite current directions, it is predicted that such a simple corner shape wire will give rise to asymmetric $V$($I$) characteristics and therefore to a vortex ratchet effect.
In order to demonstrate the existence of vortex motion rectification we submitted the samples to an ac current excitation of zero mean, $I_{ac}$, while measuring simultaneously the dc drop of voltage $V_{dc}$. The results of these measurements $V_{dc}$($I_{ac}$) are presented in Figure 2 for both samples. The chosen temperature $T=1.22$ K is such that $4.4\xi=1.9\mu$m $<$ $w=3\mu$m $<$ $\Lambda=8.3\mu$m ensuring the existence of vortices within the superconductor. There are several points that deserve to be highlighted here, (i) rectification effects are almost completely absent in the S180 sample, (ii) there is a very strong ratchet signal for the S90 sample, (iii) the ratchet signal changes polarity at zero field. Ideally, we expect no ratchet effect at all from the S180 sample, however, the fact that both voltage contacts are on the same side of the strip already impose a weak asymmetry in the system which can lead to asymmetric vortex penetration[@vodolazov-peeters; @cerbu]. In any case, the rectification signal obtained in the S180 sample is negligible in comparison to that observed in the sample with the sharp turn. The fact that the rectification signal is positive at positive fields for the S90 sample, and according to the sign convention depicted in Fig. 1(a), we conclude that the easy direction of vortex flow is from the inner corner towards the outer corner, in agreement with the theoretical findings [@clem-peeters]. In Fig.3 we show how the ratchet signal progressively disappears as the temperature approaches 1.280 K. For temperatures above this value vortices cannot fit anymore in the bridge and consequently the difference between the two corners vanishes. Similar ratchet effects due to surface barrier asymmetry, have been recently reported[@kajino] in high-Tc superconducting asymmetric nanobridges, with one side straight and the other having a constriction with an angle of 90$^{\circ}$.
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Notice that the ratchet effect here described results from the crowding of the applied current at the inner corner, and it would exist even if no screening currents were present. Let us now consider the additional effect of the screening currents. As it has been pointed out in Ref.[\[]{} based on both, London and Ginzburg-Landau theories, for a given direction of the applied current (as indicated in Fig.1(a)) a positive magnetic field will reinforce the total current (i.e. applied plus screening) at the inner corner and therefore the critical current will decrease as the field intensity increases. On the contrary, a negative applied magnetic field will induce a screening current which partially compensates the applied current at the inner corner and a field dependent increase of the critical current is expected [@clem-peeters]. We have experimentally confirmed this prediction by measuring the critical current using a voltage criterion of 1 $\mu$V as a function of field and current orientation. The results are presented in Figure 4(a) for three different temperatures and for the case where $\xi<w<\Lambda$. For positive current and field (as defined in Fig.1(a)), we observe a monotonous decrease of $I_{c}$. In contrast to that, for positive current and negative field, a clear enhancement of $I_{c}$ with field is observed for $H<H_{max}$, whereas for $H>H_{max}$ a monotonous decrease of $I_{c}$ is recovered as a consequence of antivortices induced by the magnetic field[@clem-peeters] that start to penetrate the sample. Reversing the applied current should lead to the opposite behavior, as indeed observed in Fig.4(a). This double test for all polarities of current and field also permits us to accurately determine the value of zero external field at the point where both curves cross each other. This has been convincingly confirmed by independent measurement of the remanent field in the S180 sample. For the sake of comparison, in the inset of Fig.4(b) we show the critical current for the S180 sample as a function of field. Notice that for this sample, the peak of maximum critical current is located at $H=0$, in contrast to the behavior observed in sample S90. It is important to point out that in Ref.\[13\] the theoretical prediction of the curves in Fig.4(a) corresponds to a sharp inverted-V shape according to the London model, whereas the Ginzburg-Landau calculations yield a rounded top, which becomes sharper the smaller the ratio of $\xi$ to $w$. This effect appears to be confirmed, at least qualitatively, in Fig. 4(a), in which the peaks become more rounded as the temperature increases and $\xi$ increases.
The compensation field $H_{max}$ is expected to depend on temperature since it is determined by the screening currents. In Fig.4(b) we plot the temperature dependence of $H_{max}/H_{c2}(T)$ where it can be noticed that this compensation field $H_{max}$ is a small fraction of the upper critical field $H_{c2}(T)$ in agreement with the theoretical calculations[@clem-peeters].
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To summarize, the superconducting properties of corner-shaped Al microstrips have been investigated. We show that sharp 90 degrees turns lead to asymmetric vortex penetration, being easier for vortices to penetrate from the inner side than from the outer side of the angle. We provide experimental confirmation of the predicted[@clem-peeters] competing interplay of superconducting screening currents and applied currents at the inner side of the turn. We prove that current crowding leads to a distinctly different superconducting responses for positive and negative fields (or currents). These effects are evidenced also by a field dependent critical current enhancement and also by a strong rectification of the voltage signal, thus making these asymmetric superconducting nanocircuits efficient voltage rectifiers. Complementary measurements done in samples with 30$^{\circ}$ and 60$^{\circ}$ corners (not shown) reproduce the results presented here, i.e. ratchet signal and field-induced increase of critical current.
This work was partially supported by the Fonds de la Recherche Scientifique - FNRS, FRFC grant no. 2.4503.12, the Methusalem Funding of the Flemish Government, the Fund for Scientific Research-Flanders (FWO-Vlaanderen), the Brazilian funding agencies FAPESP and CNPq, and the program for scientific cooperation F.R.S.-FNRS-CNPq. J. V.d.V. acknowledges support from FWO-Vl. The authors acknowledge useful discussions with V. Gladilin.
[10]{} K. K. Likharev, Rev. Mod. Phys. **51**, 101 (1979)
M. Yu. Kupriyanov and V. F. Lukichev, Sov. J. Low Temp. Phys. **6**, 210 (1980)
J. Romijn, T. M. Klapwijk, M. J. Renne, and J. E. Mooij, Phys. Rev. B **26**, 3648 (1982)
J. Pearl, Appl. Phys. Lett. 5, 65 (1964)
see L. N. Bulaevskii, M. J. Graf, and V. G. Kogan, Phys. Rev. B **85**, 014505 (2012) and references therein.
Notice that the nucleation of superconductivity itself is also influenced strongly by the presence of the corners, see, for example, GL treatment of this problem in a wedge in F. Brosens, V. M. Fomin, J. T. Devreese, V. V. Moshchalkov, Solid State Commun. 144, 494(2007) and in S.N. Klimin, V.M. Fomin, J.T. Devreese, V.V. Moshchalkov, Solid State Commun. 111, 589-593 (1999)
F. B. Hagedorn and P. M. Hall, J. Appl. Phys. **34**, 128 (1963)
A. V. Silhanek, J. Van de Vondel, V. V. Moshchalkov, A. Leo, V. Metlushko, B. Ilic, V. R. Misko, and F. M. Peeters, Appl. Phys. Lett. **92**, 176101 (2008)
J. R. Clem and K. K. Berggren, Phys. Rev. B **84**, 174510 (2011)
H. L. Hortensius, E. F. C. Driessen, T. M. Klapwijk, K. K. Berggren, J. R. Clem, Appl. Phys. Lett. **100**, 182602 (2012)
D. Henrich, P. Reichensperger, M. Hofherr, J. M. Meckbach, K. Ilâin, M. Siegel, A. Semenov, A. Zotova, D. Yu. Vodolazov, arXiv:1204.0616v2
J. I. Vestgarden and T.H. Johansen, Supercond. Sci. Technol. **25**, 104001 (2012)
J. R. Clem, Y. Mawatari, G. R. Berdiyorov, F. M. Peeters, Phys. Rev. B **85**, 144511 (2012)
The transport measurements have been done with the sample immersed in superfluid $^{4}$He for minimizing heating effects. Special care has been taken to avoid the high frequency noise signal (above $\sim$ 1 MHz ) by using a pi-filter.
V. A. Schweigert and F. M. Peeters, Phys. Rev. B **60**, 3084 (1999)
D. Y. Vodolazov and F. M. Peeters, Phys. Rev. B **72**, 172508 (2005)
D. Cerbu *et al.* unpublished
K. Kajino, K. Fujita, B. An, M. Inoue, A. Fujimaki, Jpn. J. Appl. Phys. **51**, 053101 (2012)
[^1]: These authors contributed equally to this work.
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abstract: |
We study the partition function of two versions of the continuum directed polymer in $1+1$ dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in ${\ensuremath{\mathbb{R}}}$, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in ${\ensuremath{\mathbb{R}}}_{-}$. The partition functions solves the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar-Parisi-Zhang equation.
We derive exact formulas for the Laplace transforms of the partition functions. In the full-space this is expressed as a Fredholm determinant while in the half-space this is expressed as a Fredholm Pfaffian. Taking long-time asymptotics we show that the limiting free energy fluctuations scale with exponent $1/3$ and are given by the GUE and GSE Tracy-Widom distributions. These formulas come from summing divergent moment generating functions, hence are not mathematically justified.
The primary purpose of this work is to present a mathematical perspective on the polymer replica method which is used to derive these results. In contrast to other replica method work, we do not appeal directly to the Bethe ansatz for the Lieb-Liniger model but rather utilize nested contour integral formulas for moments as well as their residue expansions.
address:
- 'A. Borodin, Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA, and Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow 127994, Russia'
- 'A. Bufetov, Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA, and International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, Moscow, Russia'
- 'I. Corwin, Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA, and Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, Providence, RI 02903, USA'
author:
- Alexei Borodin
- Alexey Bufetov
- Ivan Corwin
title: Directed random polymers via nested contour integrals
---
ß
\[section\] \[section\] \[theorem\][Conjecture]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Claim]{} \[theorem\][Critical point derivation]{} \[theorem\][Experimental Result]{}
\#1 \#1[[\#1]{}]{} \#1
\[theorem\][Remark]{}
\[theorem\][Example]{}
\[theorem\][Definition]{}
\[theorem\][Definitions]{}
Introduction
============
The replica method for studying directed polymers, pioneered by Kardar in 1987 [@K], has received a great deal of attention recently (cf. [@Dot; @CDR; @CDprl; @ProS1; @ProS2; @ProSpoComp; @ImSa; @ImSaKPZ; @CDlong; @ImSaKPZ2; @Dot2; @Dot3; @Dot4; @Dot5; @ISS]). In this paper we present a version of this method which ultimately leads to both the GUE and GSE Tracy-Widom distributions (for full-space and half-space polymers, respectively). Our aim is primarily pedagogical. We strive for mathematical clarity and do away with certain assumptions used in previous works (such as completeness of the Bethe ansatz and the evaluation of eigenfunction norms).
In this introduction we briefly highlight the central feature of our version of the polymer replica method – the use of nested contour integrals and their residue expansions (rather than direct application of the Bethe ansatz) – and briefly explain its connection to earlier work. The final result of our computations provides formulas characterizing the distributions of the partition function of full-space and half-space continuum directed random polymers.
We have attempted to provide computations which are as mathematically sound as possible (without getting into too many technicalities). Thus, up to a few technical assumptions on uniqueness of solutions to the delta Bose gas, our computations of the polymer partition function moments are mathematically rigorous. In the course of this computation all stated lemmas and propositions are accompanied by proofs.
Unfortunately, the moments of the partition function [*do not*]{} uniquely characterize the one-point distribution. Therefore, our final step of recovering the partition function’s Laplace transform from the (divergent) moment generating function is, at its heart, unjustifiable. In fact, this issue plagues all of the aforementioned replica method works. We proceed nevertheless and by an illegal application of a Mellin-Barnes summation trick we convert this divergent generating function into a convergent series of integrals. In the case of the full-space polymer this series can be matched to the formula derived independently and in parallel in [@ACQ; @SaSp; @Dot; @CDR] and proved in [@ACQ] (see also [@BCF] for a second proof). In the half-space polymer there are no corresponding rigorous results yet with which to compare the answer. The replica method work of [@LD] (see also [@LDexpand]) also deals with the half-space polymer in the limit as our parameter $a$ characterizing the interaction of the polymer with the wall goes to infinity (the wall becomes absorbing). We do not take this limit and hence do not compare our results to those derived in [@LD]. However, let us remark that in [@LD], the authors recover the predicted GSE Tracy-Widom distribution.
We believe that the most convincing argument for why this unjustified procedure produces the correct answer is that it is a shadow of a totally parallel procedure which can be performed rigorously on a suitable $q$-deformed regularization of our present model. For the full-space case this has been done in [@BorCor; @BCS; @BorCordiscrete; @CorPetpush; @Corhahn; @CSS; @CorPetdual; @BarCorHahn; @BarCorBeta]. Taking a suitable $q\to 1$ limit of the final formulas from these works provides a rigorous derivation of the Laplace transform formula we non-rigorously derive herein. A parallel treatment of the half-space case has not yet been performed (see Section \[mathrig\] for more on rigorous mathematical work related to this paper).
Nested contour integrals
------------------------
The polymer replica method relies upon the fact that the joint moments (at a fixed time $t$ and different spatial locations $x_1,\ldots, x_k$) of the partition function for the directed polymer (introduced in Section \[modelssec\]) satisfy certain closed systems of evolution equations (see Section \[mappingsec\]). These systems go by the name of the delta Bose gas or the Lieb-Liniger model with two-body delta interaction (we will use both names interchangeably), and variants of them hold in relation to both the full-space and half-space polymers. They are known to be integrable, which means that solving them can be reduced to solving a system of $k$ free one-body evolution equations subject to $k-1$ two-body boundary conditions (see Definitions \[Akmbs\] and \[Bkmbs\] for these systems).
The typical approach employed to solve the free evolution equation with $k-1$ two-body boundary conditions is to try to diagonalize the system (for instance, via the Bethe ansatz). The Bethe ansatz produces eigenfunctions, but does not a priori provide the knowledge of the relevant subspace of eigenfunctions on which to decompose the initial data as well as the knowledge of the norms of the eigenfunctions. Though we remark more on this approach below in Section \[relationBAsec\], it is not the route we follow. Instead, for delta function initial data we directly solve the system via a single $k$-fold nested contour integral (see Lemmas \[solnlemma2\] and \[solnlemma\] in Section \[solnsec\]). This solution is easily checked to satisfy the desired system by simple residue calculus.
The type of result one hopes to get from the spectral approach is recovered by deforming the nested contours in these formulas to all coincide with the same contour ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. The eigenfunctions, their norms and the relevant subspace all come immediately out of the residue expansion coming from the poles crossed during these contour deformations (see Section \[relationBAsec\] for more on this). In Section \[modelssec\] we introduce the full-space continuum directed random polymer partition function $Z(t,x)$ as well as the half-space analog $Z^a(t,x)$ where $a$ determines an energetic cost/rewards for the polymer paths reflected at the origin ($a>0$ corresponds to an energetic cost or repulsive interaction and $a<0$ corresponds to an energetic reward or attractive interaction). We define the joint moments for these partition functions as $$\bar{Z}(t;\vec{x}) :={\ensuremath{\mathbb{E}}}\big[Z(t,x_1)\cdots Z(t,x_k)\big], \qquad\qquad \bar{Z}^a(t;\vec{x}) :={\ensuremath{\mathbb{E}}}\big[Z^a(t,x_1)\cdots Z^a(t,x_k)\big]$$ where $\vec{x}= (x_1,\ldots, x_k)$ is assumed to be ordered as $x_1\leq \cdots \leq x_k$ for the full-space case and as $x_1\leq \cdots \leq x_k\leq 0$ for the half-space case.
We record equations (\[Aknci\]) and (\[Bknci\]) which are nested contour integral formulas for these joint moments. In the full-space case we show that $$\label{Aknciintro}
\bar{Z}(t;\vec{x}) = \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-1} \, \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2 + x_jz_j},$$ where we assume that $\alpha_1>\alpha_2 + 1 > \alpha_3 + 2> \cdots > \alpha_k + (k-1)$. And in the half-space case we show that $$\label{Bknciintro}
\bar{Z}^{a}(t;\vec{x}) = 2^k\int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-1}\, \frac{z_A+z_B}{z_A+z_B-1} \, \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2 + x_jz_j}\frac{z_j}{z_j+a},$$ where we assume that $\alpha_1>\alpha_2 + 1 > \alpha_3 + 2> \cdots > \alpha_k + (k-1)$ and $\alpha_k=\max(-a+{\epsilon},0)$ for ${\epsilon}>0$ arbitrary.
We call the above expressions nested contour integrals since the contours respect a certain infinite version of nesting (so as to avoid poles coming from the denominator). In the full-space case this formula seems to have first appeared in 1985 work of Yudson [@Yudson]. As solutions of the delta Bose gas (or Yang’s system) with general type root systems (the above formulas correspond with type $A$ and type $BC$ root systems, respectively) such formulas appeared in 1997 work of Heckman-Opdam [@HO].
Using Cauchy’s theorem and the residue theorem we may deform the contours in both expressions until they all coincide with ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. In the course of these deformations we encounter first-order poles coming from the terms $z_A-z_B-1$ (and in the half-space case, also $z_A+z_B-1$ and $z_j+a$ participate) in the denominator. It is this expansion into residue subspaces which replaces the spectral decomposition (see Section \[relationBAsec\]). For the half-space case, when $a=0$ the faction $\frac{z}{z+a} = 1$ which considerably simplifies the analysis versus $a\neq 0$. We do not presently attempt to work out the $a\neq 0$ residue expansion (see Remark \[nota\]).
In order to develop the expansion of our nested contour formulas (in both the full-space and half-space cases, as well as in various $q$-deformed cases – cf. Proposition \[321\]) there are three parallel steps:
[**Step 1:**]{} We identify the residual subspaces which arise in such an expansion. In terms of a meta-formula (letting NCI represent the words “Nested contour integral”): $$\textrm{NCI} = \sum_{\substack{I\in \textrm{Residual}\\\textrm{subspaces}}} \int_{{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}\cdots\int_{{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} {\underset{{I}}{\mathbf{Res}}}\big(\textrm{integrand}\big),$$ where ${\underset{{I}}{\mathbf{Res}}}$ represents taking the residue along the residual subspace indexed by $I$, and where the integrals on the right-hand side are over the variables which remain after computing the residues. As an example, consider (\[Aknciintro\]) for $k=2$. Then if we choose $\alpha_2=0$ from the start, we must deform the $z_1$ contour to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. Regarding $z_2$ as fixed along ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, as we deform the $z_1$ contour we necessarily encounter a pole at $z_1=z_2+c$ and thus the nested contour integral is expanded into two terms – one in which the integrals of $z_1$ and $z_2$ are both along ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ and the second in which only the $z_2$ integral remains and the integrand is replaced by the residue at $z_1=z_2+c$.
In general, due to the Vandermonde term in the numerator of the integrands we consider, the residual subspaces we encounter are indexed by certain strings of residues, as well as partitions $\lambda\vdash k$ which identify the sizes of the strings.
[**Step 2:**]{} We show that via the action of the symmetric group (in the full-space case) or the hyperoctahedral group (in the half-space case) we can transform our residual subspaces into a canonical form, only indexed by a partition $\lambda\vdash k$. Moreover, even though only certain elements of these groups arise from such transformations, we readily check that all other group elements lead to zero residue contribution. Thus, we can rewrite our sum over residual subspaces as a sum over $\lambda\vdash k$ and the symmetric or hyperoctahedral group. Using $G$ to denote either of these groups, we arrive at our second meta-formula (we have suppressed certain constants arising from group symmetries) $$\textrm{NCI} = \sum_{\lambda\vdash k} \,\sum_{\sigma\in G}\, \int_{{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}\cdots\int_{{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} {\underset{{\lambda}}{\mathbf{Res}}}\big(\sigma(\textrm{integrand})\big).$$ Now ${\underset{{\lambda}}{\mathbf{Res}}}$ represents taking the residue along the canonical form residual subspace corresponding with $\lambda$.
[**Step 3:**]{} Due to the form of the integrand we may rewrite it as a $G$-invariant function, times a remainder function which does not contain any of the poles presently relevant. This allows us to reach our final meta-formula: $$\label{astar}
\textrm{NCI} = \sum_{\lambda\vdash k} \,\int_{{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}\cdots\int_{{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} {\underset{{\lambda}}{\mathbf{Res}}}\big(\textrm{$G$-invariant part}\big)\, {\underset{{\lambda}}{\mathbf{Sub}}}\big(\sum_{\sigma\in G} \sigma(\textrm{remainder part})\big).$$ Here ${\underset{{\lambda}}{\mathbf{Sub}}}$ represents substitution or restriction of a function to the canonical form residual subspace corresponding with $\lambda$. In the cases we deal with in this paper, the residue term above can be explicitly evaluated, and the substitution term is generally left as is, though simplified considerably in certain cases (such as when the locations $x_i\equiv 0$).
In the full-space case, Proposition \[Akcprop\] and its proof substantiate the above outlined steps. This type of residue expansion for the delta-Bose gas goes back at least to [@HO], and this particular proposition is already present in [@BorCor Proposition 3.2.1] and [@BCPS1 Lemma 7.3]. In the half-space case, we provide Conjecture \[Bkcprop\] which explains what we believe to be the manner through which this expansion works. We additionally provide some evidence for the conjecture in Section \[sec:one-string\]. This conjecture involves some rather subtle cancelations of residues which at first appear to complicate the situation.
Relation to Bethe ansatz, Plancherel theory, and Macdonald symmetric functions {#relationBAsec}
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The Bethe ansatz goes back to Bethe’s 1931 solution (i.e. diagonalization) of the spin $1/2$ Heisenberg XXX spin chain [@Bethe]. Lieb-Liniger diagaonlized their eponymous model (in the repulsive case) in 1963 utilizing this eigenfunction ansatz [@LL] (it appears that this was the first application of the ansatz after Bethe’s original work and the approach was essentially rediscovered by Lieb-Liniger). Soon after, McGuire formulated the string hypotheses for the attractive case of the Lieb-Liniger model [@McGuire]. A great deal of work on this model (and its half-space variant) ensued, most notably in [@Gaudin; @Yang1; @Yang2] (see [@GutkinSuther] for a review of this early work).
The (nested) contour integrals for solutions of the Lieb-Liniger model with arbitrary initial conditions play a prominent role in [@HO][^1]. Their form can be traced back to the classical works of Harish-Chandra in the 1950’s on harmonic analysis on Riemannian symmetric spaces (see [@Hel] and references therein). In [@HO] they are used to prove the completeness of the Bethe ansatz eigenfunctions. The space of functions in which they work does not contain the $\delta$ initial condition with which this paper is concerned. However, their type of contour shifting arguments can be extended to this case well, and this extension is central for the present paper. In [@BCPS1; @BCPS2], $q$-deformed versions of the nested contour integrals are utilized to prove Plancherel theories for more general classes of eigenfunctions (related to the $q$-Boson and higher-spin six vertex models).
The connection to the Bethe ansatz can also be seen in the formulas above: the ${\underset{{\lambda}}{\mathbf{Sub}}}$ part of (\[astar\]) (denoted $E_{\vec{x}}^c$ later in the text) turns out to be an eigenfunction of the rewritten Lieb-Liniger model Hamiltonian (see Definitions \[Akmbs\] and \[Bkmbs\]), while the ${\underset{{\lambda}}{\mathbf{Res}}}$ term incorporates the inverse squared norm of the eigenfunction and its pairing with the initial condition. Our choice of the $\delta$ initial condition yields simple expressions that eventually allow us to access the asymptotics we need.
In earlier work on Macdonald processes [@BorCor], $(q,t)$-deformed versions of these nested contour integrals arose and independently led to the formulas above (in the full-space case). In [@BorCor] these formulas encoded the application of the Macdonald (first) difference operators to multiplicative functions. The reasons why iterating Macdonald difference operators is related to the same nested integrals as arise in [@HO] are still unclear to us (see however some exploration into this in [@BorCordiscrete Lemma 6.1].
Recent rigorous mathematical work on positive temperature directed polymers {#mathrig}
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Even though the replica method for the continuum directed polymer falls short on mathematical rigor, it can be put on a solid mathematical ground by appealing to certain discretizations which preserve the model’s Bethe ansatz solvability. Let us work for the moment just with full-space systems. Presently, the most general class of such discretizations are known of as the [*higher-spin vertex models*]{} (see [@BorodinR; @CorPetdual] for more details). These stochastic interacting particle systems are solved in [@CorPetdual] by an analog of the replica method known of as [*Markov dualities*]{} (see [@BorodinPetrovhigher] for a different approach than duality to study these systems). There are many degenerations of these higher-spin vertex models and the duality / replica method applies to all of them. These degenerations include interacting particle systems (such as the stochastic six vertex model [@BCG], ASEP [@IS; @BCS], Brownian motions with skew reflection [@SaSpBM], $q$-Hahn TASEP [@Pov; @Corhahn; @BarCorHahn], discrete time $q$-TASEP [@BorCordiscrete], $q$-TASEP [@BCS], $q$-pushASEP [@CorPetpush]) and directed polymer models (such as the Beta polymer [@BarCorBeta], inverse Beta polymer [@LeDoussalThierry2], log-gamma polymer [@LeDoussalThierry1], strict-weak polymer [@CSS], semi-discrete Brownian polymer [@BCS]). In some cases the moments do determine the distribution and the duality / replica method can be rigorously performed, while other cases suffer a similar fate in that higher moments growth too fast, or even become infinite. However, once the distribution (or Laplace transform) is computed rigorously, if the model converges to the continuum polymer (as shown in various cases [@BG; @ACQ; @AKQ; @CorTsai]) then taking a limit of the distribution function or Laplace transform provides a rigorous derivation of the continuum formula.
The approach taken in [@TW1; @TW2; @TW3] by Tracy and Widom in studying ASEP involves using Bethe ansatz to directly study transition probabilities and eventually extra marginal distributions from these formulas. While this approach bares some similarity to the duality / replica method, it does not have a clear degeneration to, for instance, the level of the continuum directed polymer. This work did, however, provide the first means to rigorously study the continuum directed polymer [@ACQ]. Besides the duality / replica method and Tracy and Widom’s approach, the Macdonald process [@BorCor] approach has also proved quite fruitful in providing rigorous results on exact formulas for directed polymer models – for further references, see the review [@ICICM] and reference therein. There exist other probabilistic methods which provide scaling exponents (though not exact distributions) for directed polymer models – see for example, [@SeppLog] in the context of the log-gamma polymer.
For half-space systems, much less has been done. Notably, for ASEP with finitely many particles, transition probabilities have been computed [@TWhalfspace], but no one-point marginal distribution formulas have been extracted. For the half-space log-gamma polymer, an analysis (analogous to the full-space case results of [@OCon; @COSZ]) has been undertaken in [@OSZ]. A Laplace transform formula is conjectured therein, though does not seem to be readily accessible to any asymptotics. For zero-temperature polymers (i.e. last passage percolation) there are considerably more results as explained in Section \[LPP\].
Conventions {#partsec}
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We write ${\ensuremath{\mathbb{Z}}}_{\geq 0} = \{0,1,\ldots\}$ and ${\ensuremath{\mathbb{Z}}}_{>0}=\{1,2,\ldots\}$. When we perform integrals along vertical complex contours (such as ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$) we will always assume that the contour is slightly to the right of the specified real part (i.e. ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}+{\epsilon}$) so as to avoid any poles on said contour.
A [*partition*]{} is a sequence $\lambda=(\lambda_1,\lambda_2,\ldots)$ of non-negative integers with finitely many nonzero entries, such that $\lambda_1\geq\lambda_2\geq \cdots\geq 0$. The [*length*]{} $\ell(\lambda)$ is the number of non-zero $\lambda_i$ in $\lambda$ and the weight $|\lambda| = \lambda_1+\lambda_2+\cdots$. If $|\lambda|=k$ then we say that $\lambda$ [*partitions*]{} $k$, which is written $\lambda\vdash k$. An alternative notation for $\lambda$ is $\lambda= 1^{m_1}2^{m_2}\cdots$ where $m_i$ represents the multiplicity of $i$ in the partition $\lambda$.
Let us also define the $q$-Pochhammer symbol (cf. Chapter 10 of [@AAR]) as $(a;q)_n = (1-a)(1-qa) \cdots (1-q^{n-1}a)$ and the Pochhammer symbol (or rising factorial[^2]) as $(a)_{n} = a(a+1)\cdots (a+n-1)$.
Outline
-------
In Section \[modres\] we introduce our polymer models (along with some background) and state our main results. In Section \[mappingsec\] we explain how the problem of computing the joint moments of the polymer partition function is mapped to a many body system. In Section \[solnsec\] we solve that many body system via nested contour integral formulas. In Section \[expsec\] we show how these formulas expand into residues when the nested contours are taken together to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ (this is a conjecture in the half-space case). In Section \[mgfsec\] we use the expansion of the moment formulas from earlier to compute the Laplace transform of the full-space and half-space polymer partition functions. We also perform long-time asymptotics and in the half-space case explain how we recognize the GSE Tracy-Widom distribution. Finally, in Section \[proofsec\] we provide proofs of the residue expansion formulas, with the half-space conjecture being proved modulo Claim \[claim:Res-are-zero\]. In Section \[partsec\] we provide notions concerning partitions and in Section \[Asymsec\], we provide background converning the symmetric and hyperoctahedral groups. Evidence for Claim \[claim:Res-are-zero\] is subsequently provided in Section \[sec:one-string\].
Acknowledgements
----------------
The authors have benefited from discussions with Pierre Le Doussal, Herbert Spohn and Jeremy Quastel during the Simons Symposium on the KPZ equation. We especially thank Le Doussal for further discussions regarding the Bethe ansatz bound states. A. Borodin was partially supported by the NSF grant DMS-1056390. A. Bufetov was partially supported by the RFBR grant 13-01-12449, and by the Government of the Russian Federation within the framework of the implementation of the 5-100 Programme Roadmap of the National Research University Higher School of Economics. I. Corwin was partially supported by the NSF through grant DMS-1208998 as well as by Microsoft Research through the Schramm Memorial Fellowship, by the Clay Mathematics Institute through the Clay Research Fellowship, by the Institute Henri Poincare through the Poincare Chair, and by the Packard Foundation through a Packard Fellowship in Science and Engineering.
Models and results {#modres}
==================
Models of directed polymers in disordered media in $1+1$ dimensions provide a unified framework for studying a variety of physical and mathematical systems as well as serve as a paradigm in the general study of disordered systems. Physically they provide models of domain walls of Ising type models with impurities [@HuHe; @LFC], vortices in superconductors [@BFG], roughness of crack interfaces [@HHR], Burgers turbulence [@FNS], and interfaces in competing bacterial colonies [@HHRN] (see also the reviews [@HHZ] or [@FH] for more applications). Mathematically, they are closely related to stochastic (partial) differential equations [@BQS; @ACQ], stochastic optimization problems (including important problems in bio-statistics [@SM; @HL; @MMN; @SAY] and operations research [@BBSSS]), branching Markov processes in random environments [@Bra], as well as certain aspects of integrable systems and combinatorics (which will be discussed below). The free energy of the polymer partition function is related to the Kardar-Parisi-Zhang (KPZ) equation [@KPZ] whose spatial derivative is the stochastic Burgers equation [@FNS]. These are representatives of a large universality class of growth models and interacting particle systems (see the review [@ICreview]).
The physical motivation of the present work is to study the probability distribution of the free energy (logarithm of the partition function) of the continuum directed polymer in the full-space, as well as in the presence of a hard reflecting / absorbing wall. We compute the Laplace transform (hence also the probability distribution) of the partition function in both geometries (though for the half-space case we only work with $a=0$). Taking a large time limit we show that in both geometries the free energy fluctuations scale with exponent $1/3$, though they display different limiting probability distributions – the GUE (written $F_{{\rm GUE}}$ or $F_2$) versus GSE (written $F_{{\rm GSE}}$ or $F_4$) Tracy-Widom distributions. In the context of last passage percolation with full and half-space geometries, these scalings and distributions first arose in the work of Johansson [@KJ] and Baik-Rains [@BaikRains].
The models {#modelssec}
----------
Let $\xi(t,x)$ represent a Gaussian space-time white noise in $1+1$ dimension with covariance $\left\langle \xi(s,y)\xi(t,x)\right\rangle = \delta(s-t)\delta(y-x)$ (this is understood in the sense of a generalized function – see [@ACQ; @W] for more details) and let ${\ensuremath{\mathbb{E}}}$ represent the expectation with respect to this random noise. We define the full-space and half-space polymer partition functions via the expectation of the exponential of path integrals through this noise field. This definition is properly made sense of either through smoothing the noise or through a chaos series expansion. To indicate either one of these (equivalent) renormalization procedures we use the notation of a Wick exponential $:\exp:$. The polymer partition function may equivalently be defined as the solution to a stochastic heat equation with suitable symmetry constraint in the half-space case[^3].
### Full-space polymer
Let $\mathcal{E}$ represent the expectation of a one dimensional Brownian motion $b(\cdot)$ with $b(0)=0$. Define the [*full-space continuum directed random polymer partition function*]{} as $$Z(t,x) = \mathcal{E}\left[:\exp:\left\{\int_0^t \xi(s,b(s))ds \right\} \delta(b(t)=x) \right].$$ See Section \[fspsec\] or [@ACQ] for details on how to make mathematical sense of this definition.
Equivalently, $Z(t,x)$ may be defined as the solution to the (well-posed) stochastic heat equation on ${\ensuremath{\mathbb{R}}}$ with multiplicative noise and delta initial data $$\frac{d}{dt} Z(t,x) = \frac{1}{2} \frac{d^2}{dx^2} Z(t,x) + \xi(t,x) Z(t,x), \qquad Z(0,x) = \delta_{x=0}.$$
### Half-space polymer
Let $\mathcal{E}^{R}$ represent the expectation of a one dimensional Brownian motion $b(\cdot)$ reflected so as to stay on the left of the origin and started from $b(0)=0$. Then for all $a\in {\ensuremath{\mathbb{R}}}$, define the [*half-space continuum directed random polymer partition function with parameter $a$*]{} as $$Z^{a}(t,x) = \mathcal{E}^R\left[:\exp:\left\{\int_0^t \big(\xi(s,b(s)) - a \delta_{b(s)=0}\big)ds \right\} \delta(b(t)=x)\right].$$ The integral of $\delta_{b(s)=0}$ is the local time of the Brownian motion at the origin.
Equivalently, $Z^a(t,x)$ may be defined as the solution to the (well-posed) stochastic heat equation on $(-\infty,0)$ with Robin (i.e. mixed Dirichlet and von Neumann) boundary condition at the origin $$\frac{d}{dt} Z^a(t,x) = \frac{1}{2} \frac{d^2}{dx^2} Z^a(t,x) + \xi(t,x) Z^a(t,x), \qquad \left(\frac{d}{dx}+a\right) Z^a(t,x)\big\vert_{x\to 0^-} = 0, \qquad Z^a(0,x) = \delta_{x=0}.$$ The initial data above means that for continuous bounded test functions $f:(-\infty,0)\to {\ensuremath{\mathbb{R}}}$, $$\lim_{t\to 0} \int_{-\infty}^{0} f(x) Z^a(t,x) dx = f(0).$$
The behavior of this partition function should depend on the sign of $a$. When $a$ is positive, the polymer measure will favor paths which tend to avoid touching the origin – something which can be done at relatively minor entropic cost. However, when $a$ is negative, the polymer paths will be rewarded for staying near the origin. Another way to see this difference is through a Feynman-Kac representation (cf. [@Mol; @BCLyapunovpaper]) in which when $a$ is positive, the Brownian paths are killed at a rate given by the local time multiplied by $a$, whereas when $a$ is negative, the Brownian paths duplicate at rate given by the local time multiplied by $|a|$.
This difference in path behavior should be reflected in the asymptotic behavior of the partition function. For $a\geq 0$, it is reasonable to expect that the asymptotic behavior of $\log Z^a$ should be independent of the magnitude of $a$. On the other hand, when $a<0$, one expects a localization transition in which the asymptotic growth behavior of $\log Z^a$ changes from that of $\log Z^0$.
In this article we derive moment formulas for $Z^a$ valid for all $a\in {\ensuremath{\mathbb{R}}}$, however we restrict our attention to the Laplace transform of the $a=0$ partition function only as $a\neq 0$ introduces some additional complications (see Remark \[nota\]).
Laplace transform formulas {#Laplacesec}
--------------------------
The output of the calculations we present are the following Laplace transform formulas. Though our method is not mathematically rigorous, the full-space formula matches the result proved in [@ACQ; @BCF]. No such rigorous derivation exists for the half-space formula as of yet. The partition function in question are positive random variables[^4], hence their Laplace transforms uniquely characterize their distributions.
### Full-space polymer {#fullspacelapsec}
We summarize our calculation with the following result. For $\zeta\in {\ensuremath{\mathbb{C}}}$ with ${\ensuremath{\mathrm{Re}}}(\zeta)<0$ $$\label{fullspacedet}
{\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = 1 + \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_{0}^{\infty} dx_1\cdots \int_{0}^{\infty} dx_L \det\left[K_{\zeta}(x_i,x_j)\right]_{i,j=1}^{L}$$ where the kernel is given by $$K_{\zeta}(x,x') =\int_{-\infty}^{\infty} dr \frac{1}{1 + \exp\left\{(\frac{t}{2})^{1/3}(r + u)\right\}} \Ai(x-r)\, \Ai(x'-r),$$ and where $\zeta$ and $u$ are related according to $$\log(-\zeta) = -\left(\frac{t}{2}\right)^{1/3} u + \frac{t}{24}.$$
The right-hand side of (\[fullspacedet\]) may be written in terms of a Fredholm determinant as $$\det(I-K_{\zeta})_{L^2([0,\infty))}$$ where the integral operator $K_{\zeta}$ acts on $L^2([0,\infty))$ via its kernel $K_{\zeta}(x,x')$ given above.
As pointed out in [@ACQ], by invariance of space-time white noise under affine shifts, the marginal distribution of $Z(t,x)$ is equal to the marginal distribution of $p(t,x) Z(t,0)$. Here $p(t,x) = (2\pi)^{-1/2} e^{-x^2/2t}$ is the standard heat kernel.
The distribution of $Z(t,0)$ was first discovered independently and in parallel in [@ACQ; @SaSp; @Dot; @CDR] with a mathematically rigorous proof of the formula given in [@ACQ] (and later a different proof given in [@BCF]). The formula above agrees with the rigorously proved result.
### Half-space polymer {#halfspacelapsec}
Our half-space polymer calculation for $a=0$ similarly leads to the following result. (It should be noted that along the way, we utilize a residue expansion formula of which we do not have a complete proof. This expansion formula is stated as Conjecture \[Bkcprop\].) For $\zeta\in {\ensuremath{\mathbb{C}}}$ with ${\ensuremath{\mathrm{Re}}}(\zeta)<0$,
$$\label{halfspacedet}
{\ensuremath{\mathbb{E}}}\left[e^{\frac{\zeta}{4} Z^0(t,0)}\right] = 1 + \sum_{L=1}^{\infty}\frac{(-1)^L}{L!} \int_{-\infty}^{\infty} dr_1\cdots \int_{-\infty}^{\infty} dr_L \prod_{k=1}^{L} \frac{\zeta}{e^{r_j}-\zeta} \Pf\left[ K(r_i,r_j)\right]_{i,j=1}^{L}$$
where the above Pfaffian is of a $2L\times 2L$ sized matrix composed of $2\times 2$ blocks $K^{(t)}(r,r')$ with components $$\begin{aligned}
K^{(t)}_{11}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_2}{2\pi {\ensuremath{\mathbf{i}}}} \frac{-w_1+w_2}{w_1 w_2} \frac{1}{F^{(t)}(w_1) F^{(t)}(w_2)} e^{-r w_1-r'w_2} e^{xw_1+xw_2},\\
K^{(t)}_{12}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{-w-s}{w} \frac{F^{(t)}(s)}{F^{(t)}(w)} e^{-r w+r's} e^{xw-xs},\\
K^{(t)}_{22}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_1}{2\pi {\ensuremath{\mathbf{i}}}}\int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_2}{2\pi {\ensuremath{\mathbf{i}}}} (s_1-s_2) F^{(t)}(s_1)F^{(t)}(s_2) e^{r s_1+r's_2} e^{-xs_1-xs_2},\end{aligned}$$ and $K^{(t)}_{21}(r,r') = - K^{(t)}_{12}(r',r)$. In the above we have used $$F^{(t)}(w) =\frac{\Gamma(w)}{\Gamma(w+\frac{1}{2})} e^{\frac{t}{2} \left(\frac{s^3}{3} - \frac{s}{12}\right)}.$$
This formula can be manipulated further to appear closer to that of (\[fullspacedet\]). However, unlike in the full-space case, there is no simple transformation which maps this $x=0$ result to a general $x$ result. Though almost all of our work applied to the general $x$ case, we presently do not have a suitable analog of the identity (\[Bkfacsym\]) for $x\neq 0$.
Long-time asymptotic distributions {#asydistsec}
----------------------------------
### Full-space polymer {#full-space-polymer-1}
We may rewrite (\[fullspacedet\]) in a suggestive manner for taking the $t\to \infty$ asymptotics:
$${\ensuremath{\mathbb{E}}}\left[ \exp\left\{-\exp\left\{ \left(\frac{t}{2}\right)^{1/3}\left[\frac{\log Z(t,0) + \frac{t}{24}}{\left(\frac{t}{2}\right)^{1/3}} \, -u \right]\right\}\right\} \right] = \det(I-K_{u})_{L^2([0,\infty))}$$
with kernel $$K_{u}(x,x') =\int_{-\infty}^{\infty} dr \frac{1}{1 + \exp\left\{(\frac{t}{2})^{1/3}(r + u)\right\}} \Ai(x-r)\, \Ai(x'-r).$$ Since $e^{-e^{\lambda x}} \to \mathbf{1}_{x<0}$ as $\lambda\to +\infty$ we see that as $t\to \infty$, the left-hand side becomes the expectation of an indicator function (hence a probability), while the right-hand side also has a clear limit since as $t \to \infty$, $$\frac{1}{1 + \exp\left\{(\frac{t}{2})^{1/3}(r + u)\right\}} \to \mathbf{1}_{r+u<0}.$$
The output of these observations is the following limit distribution result[^5]: $$\lim_{t\to \infty} {\ensuremath{\mathbb{P}}}\left(\frac{\log Z(t,0) + \frac{t}{24}}{\left(\frac{t}{2}\right)^{1/3}} \leq u \right) = \det(I-K_{{\rm Ai}})_{L^2((u,\infty])} = F_{{\rm GUE}}(u)$$ where $F_{{\rm GUE}}(u)$ is the GUE Tracy-Widom distribution, given in terms of the above Fredholm determinant with $$K_{{\rm Ai}}(x,x') = \int_{0}^{\infty} dr \Ai(x+r)\Ai(x'+r).$$
### Half-space polymer {#hsplim}
Just as in the full-space Laplace transform, we may rewrite the left-hand side of (\[halfspacedet\]) in a suggestive manner for taking the $t\to \infty$ asymptotics: $${\ensuremath{\mathbb{E}}}\left[ \exp\left\{-\frac{1}{4}\exp\left\{ \left(\frac{t}{2}\right)^{1/3}\left[\frac{\log Z^0(t,0) + \frac{t}{24}}{\left(\frac{t}{2}\right)^{1/3}} \, -u \right]\right\}\right\} \right].$$ The above expression converges to the limiting probability distribution of $\log Z^0(t,0)$ (centered by $t/24$ and scaled by $(t/2)^{1/3}$) as $t\to \infty$. The calculation of Section \[RHSder\] shows that the right-hand side of (\[halfspacedet\]) has a clear limit as well. Combining these facts we find that $$\label{almostGSE}
\lim_{t\to \infty} {\ensuremath{\mathbb{P}}}\left(\frac{\log Z^0(t,0) + \frac{t}{24}}{\left(\frac{t}{2}\right)^{1/3}} \leq u \right) = 1 + \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_{u}^{\infty} dr_1 \cdots \int_{u}^{\infty} dr_{L} \Pf\left[K^{\infty}(r_i,r_j)\right]_{i,j=1}^{L}$$ where the above Pfaffian is of a $2L\times 2L$ sized matrix composed of $2\times 2$ blocks $K^{\infty}(r,r')$ with components $$\begin{aligned}
\label{Kkern}
\nonumber K^{\infty}_{11}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_2}{2\pi {\ensuremath{\mathbf{i}}}} (w_2-w_1) e^{-\frac{w_1^3}{3} + w_1(r+x)} e^{-\frac{w_2^3}{3}+w_2(r'+x)},\\
K^{\infty}_{12}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{w+s}{s} e^{-\frac{w^3}{3} + w(r+x)} e^{\frac{s^3}{3} - s(r'+x)},\\
\nonumber K^{\infty}_{22}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_1}{2\pi {\ensuremath{\mathbf{i}}}}\int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_2}{2\pi {\ensuremath{\mathbf{i}}}} \frac{s_1-s_2}{s_1s_2} e^{\frac{s_1^3}{3}-s_1(r+x)} e^{\frac{s_2^3}{3} - s_2(r'+x)},\end{aligned}$$ and $K^{\infty}_{21}(r,r') = - K^{\infty}_{12}(r',r)$. It is shown in Section \[RHSdist\] that this Fredholm Pfaffian is equal to the GSE Tracy-Widom distribution. Hence we conclude that $$\lim_{t\to \infty} {\ensuremath{\mathbb{P}}}\left(\frac{\log Z^0(t,0) + \frac{t}{24}}{\left(\frac{t}{2}\right)^{1/3}} \leq u \right) = F_{{\rm GSE}}(u).$$
We expect that as long as the interaction at the origin is repulsive (i.e., $a\geq 0$), its strength does not effect the asymptotic free energy fluctuations. We do not, however, include an analysis of the general $a>0$ case herein due to extra complications which arise in its analysis (see Remark \[nota\]).
Comparison to Johansson and Baik-Rains last passage percolation work {#LPP}
--------------------------------------------------------------------
Last passage percolation (LPP) is a discrete time and space zero temperature polymer. For geometric weight distributions, the model is exactly solvable via methods of determinantal and Pfaffian point processes. We recover a full-space result due to Johansson [@KJ] and a corresponding half-space result due to Baik-Rains [@BaikRains].
For the full-space set up, let $w_{n,x}$ (with $n\in {\ensuremath{\mathbb{Z}}}_{\geq 0}$ and $x\in {\ensuremath{\mathbb{Z}}}$) be independent geometric random variables with parameter $p\in (0,1)$. Let $\pi(\cdot)$ represent the trajectory of a simple symmetric random walk started from $\pi(0)=0$. Then define the full-space last passage time $$L(N) = \max_{\pi:\pi(2N)=0} \left(\sum_{n=0}^{2N} w_{n,\pi(n)}\right).$$ For the half-space set up, let $w_{n,x}$ (with $n\in {\ensuremath{\mathbb{Z}}}_{\geq 0}$ and $x\in {\ensuremath{\mathbb{Z}}}_{< 0}$) be independent geometric random variables with parameter $p\in (0,1)$ and let $w_{n,0}$ be independent geometric random variables with parameter $\alpha \sqrt{p}\in (0,1)$. Then define the half-space last passage time with parameter $\alpha$ as $$L(N) = \max_{\pi:\pi(2N)=0, \pi(\cdot)\leq 0} \left(\sum_{n=0}^{2N} w_{n,\pi(n)}\right).$$
The following theorem is due to Johansson [@KJ] in the full-space case and Baik-Rains [@BaikRains] in the half-space case.
Setting $$\eta(p) = \frac{2\sqrt{p}}{1-\sqrt{p}}, \qquad \rho(p) = \frac{p^{1/6} (1+\sqrt{p})^{1/3}}{1-\sqrt{p}},$$ then in the full-space case $$\lim_{N\to \infty} {\ensuremath{\mathbb{P}}}\left(\frac{L(N)-\eta(p)N}{\rho(p) N^{1/3}}\leq u\right) = F_{{\rm GUE}}(u)$$ and in the half space case $$\lim_{N\to \infty} {\ensuremath{\mathbb{P}}}\left(\frac{L(N)-\eta(p)N}{\rho(p) N^{1/3}}\leq u\right) =
\begin{cases}
F_{{\rm GSE}}(u) & 0\leq \alpha<1\\
F(u;w) & \alpha = 1 - \frac{2 w}{\rho(p) N^{1/3}}\\
0 & \alpha>1.
\end{cases}$$
The definition of the distribution $F(u;w)$ above is given in [@BaikRains Definition 2].
The full-space result agrees with our positive temperature analog. The half-space result likewise agrees, as we are in the regime of a repulsive origin (here presented by $\alpha<\sqrt{p}$). There is a regime of $\sqrt{p}\leq \alpha\leq 1$ when the origin provides extra reward over the other weights, but the limiting statistics remain unchanged (this corresponds with the path not being sufficiently rewarded for the entropic cost of staying localized near the origin), and a regime of $\alpha = 1 - \frac{2 w}{\rho(p) N^{1/3}}$ when the origin strength becomes critical. Beyond this, the maximizing path visits the origin a macroscopic proportion of the time and the reason for the 0 limit above is because the law of large number centering by $\eta(p)$ becomes insufficient. In this regime the law of large numbers increases and the fluctuations are Gaussian.
We expect the same sort of transition should occur in the half-space continuum polymer we consider herein. In order to see the critical behavior we must tune $a<0$ appropriately. Since we do not presently deal with asymptotics for this $a<0$ regime, we cannot yet access this behavior.
Mapping to many body system {#mappingsec}
===========================
The basic observation explained in this section is that joint moments of the polymer partition function solve certain nice closed evolution equations. This at least goes back to the independent work of Kardar [@K] and Molchanov [@Mol].
Full-space polymer {#fspsec}
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Consider $\vec{x} = (x_1,\ldots,x_k) \in {\ensuremath{\mathbb{R}}}^k$ and define $$\bar{Z}(t;\vec{x}) = {\ensuremath{\mathbb{E}}}\left[\prod_{i=1}^{k} Z(t,x_i)\right],$$ where recall that $Z(t;x)$ is the full-space polymer partition function and ${\ensuremath{\mathbb{E}}}$ represents the expectation with respect to the white noise $\xi$. Then the basic observation one makes is that $$\frac{d}{dt} \bar{Z}(t;\vec{x}) = H^1 \bar{Z}(t;\vec{x}), \qquad \bar{Z}(0;\vec{x}) = \prod_{i=1}^{k} \delta(x_i=0)$$ where $H^c$, for general $c\in {\ensuremath{\mathbb{R}}}$, is given by $$H^c = \frac{1}{2} \sum_{i=1}^{k} \frac{d^2}{dx_i^2} + \frac{c}{2} \sum_{i\neq j} \delta(x_i-x_j=0).$$ Such a statement also holds for solutions of the stochastic heat equation with generic initial data, with $\bar{Z}(0;\vec{x})$ modified accordingly.
The above system is the imaginary time Lieb-Liniger model [@LL] with two body delta interaction and coupling constant $-c$. It is also called a delta Bose gas, as one restricts to Bosonic solutions which are symmetric in relabeling the indices of $\vec{x}$. When the coupling constant $-c$ is negative (i.e. $c>0$) the model is called attractive, whereas when it is positive the model is called repulsive. The partition function moments correspond with the attractive case.
The fact that these moments satisfy the above system can be shown through smoothing the noise $\xi$ in space, replicating the path integrals defining the partition function (which now make sense), interchanging the $k$-fold replica expectation with the Gaussian expectation (which can now be evaluated in terms of local times) and finally taking away the smoothing. This procedure is performed in a mathematically rigorous manner in Bertini-Cancrini [@BC Proposition 2.3] where they show that
$$\bar{Z}(t;\vec{x}) = \prod_{i=1}^{k} p(t,x_i) \, \mathcal{E}^{k} \left[ \exp\left\{\frac{1}{2} \sum_{i\neq j} L_t(b_i-b_j)\right\}\right]$$ where $p(t,x)$ is the transition probability for a Brownian motion to go from 0 to $x$ in time $t$, $\mathcal{E}^k$ is the expectation over $k$ independent Brownian bridges $b_1(\cdot),\ldots, b_k(\cdot)$ which start at 0 at time 0 and end at $x_1,\ldots, x_k$ (respectively), and $L_t(b_i-b_j)$ is the intersection local time of $b_i-b_j=0$ over the time interval $[0,t]$. From this formula and the Feynman-Kac representation, one readily checks that $\bar{Z}(t;\vec{x})$ satisfies the desired evolution equation above[^6].
Notice that $\bar{Z}(t;\vec{x}) = \bar{Z}(t;\sigma\vec{x})$ for any permutation $\sigma$ hence it suffices to compute $\bar{Z}(t;\vec{x})$ for $\vec{x} \in W(A_k)$. Here $W(A_k)$ is the type $A_k$ Weyl chamber $x_1\leq x_2\leq \cdots \leq x_k$ (see Section \[Asymsec\]).
\[Akmbs\] We say that $u^c:{\ensuremath{\mathbb{R}}}_{\geq 0} \times {\ensuremath{\mathbb{R}}}^k\to {\ensuremath{\mathbb{R}}}$ solves the [*type $A_k$ free evolution equation with $k-1$ boundary conditions and coupling constant $c$*]{} if
1. For all $t>0$ and $\vec{x}\in {\ensuremath{\mathbb{R}}}^k$, $$\frac{d}{dt} u^c(t;\vec{x}) = \frac{1}{2} \sum_{i=1}^{k} \frac{d^2}{dx_i^2} u^c(t;\vec{x});$$
2. For all $t>0$ and $\vec{x}\in W(A_k)$ such that $x_i=x_{i+1}$, $$\left(\frac{d}{dx_i} - \frac{d}{dx_{i+1}} -c\right) u^c(t;\vec{x}) = 0;$$
3. For all continuous bounded $L^2$ functions $f:W(A_k)\to {\ensuremath{\mathbb{R}}}$, $$\lim_{t\to 0} k!\int_{W(A_k)}f(\vec{x}) u^c(t;\vec{x}) =f(0).$$
Using the local time representation for $\bar{Z}(t;\vec{x})$ one finds that restricted to $\vec{x}\in W(A_k)$, $$\bar{Z}(t;\vec{x})=u^1(t,\vec{x}).$$ While there is little doubt of the above equality, in fact (to our knowledge) this observation has not been made in a rigorous manner and also relies upon an assumption of uniqueness of solutions for the above system.
This reduction to the form of a free evolution equation with $k-1$ boundary conditions is a hallmark of integrability and will be the starting point for our analysis. In fact, all replica method works relies on this rewriting of the delta Bose gas. It is worth noting that if the two body delta interaction term is replaced by a smoothed version (as corresponds to the case of spatially smoothed noise) there is no reduction to such an integrable form. However, there does exist discrete space versions of the polymer as well as a $q$-deformed discrete version of the polymer for which this integrability persists (see, for example, [@BCS]). For these discretizations, the questions of uniqueness are easily overcome.
Half-space polymer {#half-space-polymer-1}
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Consider $\vec{x} = (x_1,\ldots,x_k) \in {\ensuremath{\mathbb{R}}}^k$ and define $$\bar{Z}^a(t;\vec{x}) = {\ensuremath{\mathbb{E}}}\left[\prod_{i=1}^{k} Z^a(t,x_i)\right],$$ where recall that $Z^a(t;x)$ is the half-space polymer partition function with parameter $a$. Though this is only defined for $x_i< 0$, we can extend it to $x_i\in {\ensuremath{\mathbb{R}}}$ by setting declaring $Z^a(t,x)=Z^a(t,-x)$. Of course, this extended version of $Z^a(t,x)$ solves a stochastic heat equation on ${\ensuremath{\mathbb{R}}}$ (with the Robin jump condition for the derivative at 0), but now with doubled initial data $\prod_{i=1}^{k} 2\delta(x_i=0)$. In fact, (up to the factor of two) this corresponds with considering a full-space polymer with noise $\xi$ which is symmetric through the origin (i.e., $\xi(t,x)=\xi(t,-x)$) and which has an energetic contribution from the polymer path local time at the origin.
Just as in the full-space polymer, these moments solve nice closed evolution equations: $$\frac{d}{dt} \bar{Z}^a(t;\vec{x}) = H^{1,a} \bar{Z}(t;\vec{x}), \qquad \bar{Z}^a(0;\vec{x}) = \prod_{i=1}^{k} 2 \delta(x_i=0),$$ where $H^{c,a}$, for general $c,a\in {\ensuremath{\mathbb{R}}}$, is given by $$H^{c,a} = \frac{1}{2} \sum_{i=1}^{k} \frac{d^2}{dx_i^2} + \frac{c}{2} \sum_{i\neq j} \delta(x_i-x_j=0) - a \sum_{i=1}^{k} \delta(x_i=0).$$ One restricts to solutions which are invariant with respect to the action of hyperoctahedral group $BC_k$ (see Section \[Asymsec\]). Such an extension seems to have been first considered by Gaudin [@Gaudin] (see [@GutkinSuther; @HO] for further developments).
The fact that these moments satisfy the above system can be shown in a similar way as for the full-space polymer. Doing this one finds that $$\bar{Z}(t;\vec{x}) = \prod_{i=1}^{k} p^R(t,x_i) \, \mathcal{E}^{k} \left[ \exp\left\{\frac{1}{2} \sum_{i\neq j} L_t(b_i-b_j)- a \sum_{i=1}^{k} L_t(b_i)\right\}\right]$$ where $p^R(t,x)$ is the transition probability for a reflected Brownian motion to go from 0 to $x$ in time $t$, $\mathcal{E}^k$ is the expectation over $k$ independent reflected Brownian bridges $b_1(\cdot),\ldots, b_k(\cdot)$ which start at 0 at time 0 and end at $x_1,\ldots, x_k$ (respectively), $L_t(b_i-b_j)$ is the intersection local time of $b_i(\cdot)-b_j(\cdot)=0$ over the time interval $[0,t]$, and $L_t(b_i)$ is the intersection local time of $b_i(\cdot)=0$ over the time interval $[0,t]$. From this formula and the Feynman-Kac representation, one readily checks that $\bar{Z}^a(t;\vec{x})$ satisfies the desired evolution equation above[^7].
Notice that $\bar{Z}^a(t;\vec{x}) = \bar{Z}(t;\sigma\vec{x})$ for any $\sigma\in BC_k$. Therefore it suffices to compute $\bar{Z}^a(t;\vec{x})$ for $\vec{x} \in W(BC_k)$. Here $W(BC_k)$ is the type $BC_k$ Weyl chamber $x_1\leq x_2\leq \cdots \leq x_k\leq 0$ (see Section \[Asymsec\]).
\[Bkmbs\] We say that $u^{c,a}:{\ensuremath{\mathbb{R}}}_{\geq 0} \times {\ensuremath{\mathbb{R}}}^k\to {\ensuremath{\mathbb{R}}}$ solves the [*type $BC_k$ free evolution equation with $k-1$ boundary conditions and coupling constants $c$ and $a$*]{} if
1. For all $t>0$ and $\vec{x}\in {\ensuremath{\mathbb{R}}}^k$, $$\frac{d}{dt} u^{c,a}(t;\vec{x}) = \frac{1}{2} \sum_{i=1}^{k} \frac{d^2}{dx_i^2} u^{c,a}(t;\vec{x});$$
2. For all $t>0$ and $\vec{x}\in W(BC_k)$ such that $x_i=x_{i+1}$, $$\left(\frac{d}{dx_i} - \frac{d}{dx_{i+1}} -c\right) u^{c,a}(t;\vec{x}) = 0;$$
3. For all $t>0$ and $\vec{x}\in W(BC_k)$ such that $x_k=0$, $$\left(\frac{d}{dx_k} + a \right) u^{c,a}(t;\vec{x}) = 0;$$
4. For all continuous bounded $L^2$ functions $f:W(BC_k)\to {\ensuremath{\mathbb{R}}}$, $$\lim_{t\to 0} k!\int_{W(BC_k)}f(\vec{x}) u^{c,a}(t;\vec{x}) =f(0).$$
The above many body system is called Yang’s system in the work of Heckman-Opdam [@HO] due to Yang’s work in the late 1960’s [@Yang1; @Yang2].
Just as in the full-space case, we are not aware of a proof of uniqueness results for the above system. However, assuming this and using the local time representation for $\bar{Z}^a(t;\vec{x})$ one finds (again, we are not aware of a rigorous proof of this) that restricted to $\vec{x}\in W(BC_k)$, $$\bar{Z}^a(t;\vec{x})=u^{1,a}(t,\vec{x}).$$
Unlike the type $A$ case, we do not presently know of any discrete space or $q$-deformed discrete space versions of half-space polymers which display a similar degree of integrability as the continuum version. The work of [@TWhalfspace] and [@OSZ] are, however, suggestive of such a possibility.
Solution via nested contour integral ansatz {#solnsec}
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We solve the many body systems given above without appealing to the eigenfunction expansion (such as provided by the Bethe Ansatz). Even though we are presently only interested in the case $c=1$ and $a\geq 0$, the below formula applies for all $c,a\in {\ensuremath{\mathbb{R}}}$.
Full-space polymer {#full-space-polymer-2}
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\[solnlemma2\] The type $A_k$ free evolution equation with $k-1$ boundary conditions and coupling constant $c\in {\ensuremath{\mathbb{R}}}$ (see Definition \[Akmbs\]) is solved by $$u^c(t;\vec{x}) = \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} dz_1 \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} dz_k \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c}\, \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2 + x_jz_j},$$ where for $c\geq 0$ we assume that $\alpha_1>\alpha_2 + c > \alpha_3 + 2c> \cdots > \alpha_k + (k-1)c$, and for $c<0$, all $\alpha_i\equiv 0$.
Thus (up to assuming uniqueness of solutions to the many body system of Definition \[Akmbs\]) we may conclude that for $x_1\leq x_2\leq \cdots \leq x_k$, $$\label{Aknci}
\bar{Z}(t;\vec{x}) = \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-1} \, \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2 + x_jz_j},$$ for any $\alpha_1>\alpha_2 + 1 > \alpha_3 + 2> \cdots > \alpha_k + (k-1)$.
A proof of this result can be found in [@BorCor], Proposition 6.2.3. As this proof is a straightforward modification of the proof of Lemma \[solnlemma\] given below, we do not reproduce it presently.
Half-space polymer {#half-space-polymer-2}
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\[solnlemma\] The type $BC_k$ free evolution equation with $k-1$ boundary conditions and coupling constant $c, a \in {\ensuremath{\mathbb{R}}}$ (see Definition \[Bkmbs\]) is solved by $$u^{c,a}(t;\vec{x}) = 2^k \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c}\, \frac{z_A+z_B}{z_A+z_B-c} \, \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2 + x_jz_j} \frac{z_j}{z_j+a},$$ where for $c\geq 0$ we assume that $\alpha_1>\alpha_2 + c > \alpha_3 + 2c> \cdots > \alpha_k + (k-1)c$ and $\alpha_k=\max(-a+{\epsilon},0)$ for ${\epsilon}>0$ arbitrary, and for $c<0$, all $\alpha_i\equiv \max(-a+{\epsilon},0)$ for ${\epsilon}>0$ arbitrary.
Thus (up to assuming uniqueness of solutions to the many body system of Definition \[Bkmbs\]) we may conclude that for $x_1\leq x_2\leq \cdots \leq x_k\leq 0$, $$\label{Bknci}
\bar{Z}^{a}(t;\vec{x}) =2^k\int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-1}\, \frac{z_A+z_B}{z_A+z_B-1} \, \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2 + x_jz_j}\frac{z_j}{z_j+a},$$ where we assume that $\alpha_1>\alpha_2 + 1 > \alpha_3 + 2> \cdots > \alpha_k + (k-1)$ and $\alpha_k=\max(-a+{\epsilon},0)$ for ${\epsilon}>0$ arbitrary.
We sequentially check that $u^{c,a}(t;\vec{x})$ given in the statement of the lemma satisfies conditions (1), (2), (3) and (4) of the many body system in Definition \[Bkmbs\].
(1): This follows immediately from Leibniz’s rule and the fact that $$\frac{d}{dt} e^{\frac{t}{2} z^2 +zx} = \frac{1}{2} \frac{d^2}{dx^2} e^{\frac{t}{2} z^2 +zx}.$$(2): Applying $\left(\frac{d}{dx_i} - \frac{d}{dx_{i+1}} -c\right)$ to the integrand of $u^{c,a}(t;\vec{x})$ changes the integrand by simply bringing out a factor of $(z_i-z_{i+1}-c)$. This cancels the corresponding term in the denominator from the product over $1\leq A<B\leq k$. As a result, the integrand no longer has a pole corresponding to $z_i -z_{i+1} = c$ which means that without crossing any poles, we can deform the $z_{i+1}$ contour to coincide with the $z_{i}$ contour (the Gaussian decay at infinity justifies the deformation of these infinite contours, and the fact that all of the contours lie to the right of $-a$ implies no pole from the denominator $z_j+a$ is crossed). Note that if $c<0$ such a deformation is not necessary.
Since $x_i=x_{i+1}$, the integrand can now be rewritten in the form $$\int dz_i \int dz_{i+1} (z_i-z_{i+1}) G(z_i,z_{i+1})$$ where $G$ is symmetric in its two variables and all of the other integrations have been absorbed into its definition. As the contours of integrals coincide, it follows immediately that this integral is 0.(3): For $\vec{x}$ with $x_k=0$, we find that $$\begin{aligned}
\left(\frac{d}{dx_k}+a\right) u^{c,a}(t;\vec{x}) \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad& \\
=2^k\int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} dz_1 \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} dz_k \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c}\, \cdot \frac{z_A+z_B}{z_A+z_B-c} \, \prod_{j=1}^{k-1} e^{\frac{t}{2} z_j^2 + x_jz_j} \frac{z_j}{z_j+a} \, e^{\frac{t}{2} z_k^2} z_k. &\end{aligned}$$ We are now free to deform the $z_k$ contour from $\alpha_k+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ without crossing any poles. The above integrand is invariant under the change of variables $z_k \mapsto -z_k$ as is the contour ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ (up to a factor of $-1$ coming from the change in orientation of ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$). This, however, implies that the integral above is equal to the integral with the orientation of the $z_k$ contour reversed, which, in turn, implies that the integral is 0.(4): First note that $$\frac{z_A-z_B}{z_A-z_B-c} = 1 + \frac{c}{z_A-z_B-c}, \qquad \textrm{and} \qquad \frac{z_A+z_B}{z_A+z_B-c} = 1 + \frac{c}{z_A+z_B-c}.$$ We can write $u^{c,a}(t;\vec{x}) = \sum u^{c,a}_{K_1,K_2}(t;\vec{x})$, where the sum is over all $K_1, K_2\subset \left\{(A,B):1\leq A<B\leq k\right\}$, and where $$\begin{aligned}
u^{c,a}_{K_1,K_2}(t;\vec{x}) \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad &\\
= 2^k \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{(A,B)\in K_1} \frac{c}{z_A-z_B-c} \prod_{(A,B)\in K_2} \frac{c}{z_A+z_B-c} \,\prod_{j=1}^{k} e^{\tfrac{t}{2} z_j^2 + z_j x_j}\frac{z_j}{z_j+a}.&\end{aligned}$$ For use later, let $\ell = |K_1|+|K_2|$.
Define the unit vector in the direction of $\vec{x}$ as $v = \vec{x} / ||\vec{x}||$ and note that for $x_1< x_2< \cdots< x_k< 0$, it follows that $-v$ is strictly positive. On account of this fact, we can deform the $z_j$ contours to lie along $-t^{1/2} v_j + {\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ (this is because of the ordering of the elements of $\vec{x}$ and the Gaussian decay which justifies the deformation of contours). Let $w_j = t^{1/2} z_j$, then $$\begin{aligned}
u^{c,a}_{K_1,K_2}(t;\vec{x}) &=&2^k c^\ell t^{\ell/2} t^{-k/2} \int_{-v_1-{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}^{-v_1+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}}\cdots \int_{-v_k-{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}^{-v_k+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} \frac{dw_k}{2\pi {\ensuremath{\mathbf{i}}}}\\
& & \times \, \prod_{(A,B)\in K_1} \frac{1}{w_A-w_B-ct^{1/2}} \prod_{(A,B)\in K_2} \frac{1}{w_A+w_B-ct^{1/2}} e^{t^{-1/2} \vec{x}\cdot \vec{w}}\, \prod_{j=1}^{k} e^{\tfrac{1}{2} w_j^2}\frac{w_j}{w_j+t^{1/2}a}.\end{aligned}$$ Due to the choice of contours, ${\ensuremath{\mathrm{Re}}}(\vec{x}\cdot \vec{w}) = - ||\vec{x}||$. Taking absolute values we find that $$\begin{aligned}
\big\vert u^{c,a}_{K_1,K_2}(t;\vec{x})\big\vert &=& 2^k c^\ell t^{\ell/2} t^{-k/2} e^{-t^{-1/2} ||x||} \int_{-v_1-{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}^{-v_1+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}}\cdots \int_{-v_k-{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}^{-v_k+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} \frac{dw_k}{2\pi {\ensuremath{\mathbf{i}}}}\\
& &\times\,\left|\prod_{(A,B)\in K_1} \frac{1}{w_A-w_B-ct^{1/2}} \prod_{(A,B)\in K_2} \frac{1}{w_A+w_B-ct^{1/2}}\, \prod_{j=1}^{k} e^{\tfrac{1}{2} w_j^2} \frac{w_j}{w_j+t^{1/2}a} \right|.\end{aligned}$$ The integral above is now clearly convergent, independent of $x$, and bounded by a constant $C$ which is uniform in $t<t_0$. Thus we have established that $$\label{ubound}
\big\vert u^{c,a}_{K_1,K_2}(t;\vec{x})\big\vert \leq C' t^{\ell/2} t^{-k/2} e^{-t^{-1/2} ||x||}$$ for some $C'>0$.
For any bounded continuous function $f$ we may compute the integral against $u^{c,a}_{K_1,K_2}$. The boundedness of $f$ means that $|f(x)|<M$ for some constant $M$ as $x$ varies. By the triangle inequality $$\left|\int_{W(BC_k)} u^{c,a}_{K_1,K_2}(t;\vec{x}) f(\vec{x})d\vec{x}\right| \leq \int_{W(BC_k)} \left|u^{c,a}_{K_1,K_2}(t,\vec{x})\right| M d\vec{x} \leq C'' t^{\ell/2}.$$ The second inequality follows by substituting (\[ubound\]) and performing the change of variables $y_i=t^{-1/2} x_i$ in order to bound the resulting integral (this change of variables results in the cancelation of the $t^{-k/2}$ term). Observe that $\ell\geq 1$ for all choices of $K_1$ and $K_2$ which are not simultaneously empty, and as $t\to 0$ these terms disappear.
The only term in the expansion of $u^{c,a}$ which does not contribute negligibly as $t\to 0$ corresponds to when $K_1=K_2=\emptyset$ are empty. In that case the desired result can be seen in the following manner. We showed that $$k! \int_{W(BC_k)} f(x) u^{c,a}(t;\vec{x}) d\vec{x} = o(1) + 2^k k! \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \, \tilde{f}(\vec{z}) \prod_{j=1}^{k} e^{\frac{t}{2}z_j^2} \frac{z_j}{z_j+a},$$ where $$\tilde{f}(\vec{z}) = \int_{W(BC_k)} f(x) \prod_{j=1}^{k} e^{x_j z_j},$$ and where the $o(1)$ term comes from all other terms in the expansion of $u^{c,a}$ and goes to $0$ as $t\to 0$. Writing $\frac{z}{z+a}$ as $1+ \frac{a}{z+a}$ and using decay of $\tilde{f}(\vec{z})$ as the real part of $z_j$ goes to infinity, it is possible to show that $$\lim_{t\to 0} 2^k k! \int_{W(BC_k)} f(x) u^{c,a}(t;\vec{x}) d\vec{x} = 2^k k! \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \, \tilde{f}(\vec{z}).$$ Observe that $\tilde{f}(\vec{z})$ is the Fourier transform of a function which is $f(\vec{x})$ for $\vec{x}\in W(BC_k)$ and zero outside, we readily see that $$2^k k! \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \, \tilde{f}(\vec{z}) = f(0),$$ which is exactly as desired.
Expansion of nested contour integrals into residues {#expsec}
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At this point we have seen two types of nested contour integral formulas – those related to type $A$ (the full-space polymer) and those related to type $BC$ (the half-space polymer). In this section we present formulas for each type which shows the effect of deforming the nested contours to all lie upon the imaginary axis ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. This amounts to computing the residue expansion coming from the various poles crossed during this deformation.
The general scheme followed in these computations are quite similar between the two types (though in the BC case we are presently unable to complete the proof, thus the result remains a conjecture). The first step is to identify the residual subspaces resulting from the deformation. The second step is to show that (in both cases) the residual subspaces can be brought (via the respective action of the symmetric or hyperoctahedral groups) into a certain canonical form indexed only by partition $\lambda\vdash k$. And, the third step is to compute the residue corresponding to a given partition $\lambda$. This splits into computing the residue of the portion of the integrand containing poles and computing the substitution of the residual subspace relations into the remaining portion of the integrand. The observation of the first step, once made, is relatively easy to confirm, and computing residues and substitutions is straightforward as well (if not slightly technical). In the BC case, there is a subtlety involving contour deformations which we encounter in step 2. Essentially, in order to bring our expressions into a canonical form we end up getting contours of integration which need to be shifted back to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. During this shifting we encounter a number of poles which we believe, but have not been able to prove, all cancel. This is the content of Claim \[claim:Res-are-zero\]. The basic structure of the computation behind these results comes from the proof of Theorem 3.13 of [@HO], see also Section \[expsec\] for its non-technical description. Proposition \[Akcprop\] and Conjecture \[Bkcprop\] are stated below, though their proofs are held off until Section \[proofsec\]. Immediately after presenting these residue expansions we show how they are applied to the full-space and half-space polymer partition function joint moment formulas. In the case of the half-space polymer, Conjecture \[Bkcprop\] is only applied for $a\geq 0$ (repulsive interaction at the origin). The source of this restriction has to do with the occurrence of additional poles in the contour deformations when $a<0$.
In the statements and use of these propositions we use some notation related to partitions, the symmetric group $S_k$, the hyperoctahedral group $BC_k$, as well as Pochhammer symbols $(a)_{n}$, which all can be found in Section \[partsec\] or \[Asymsec\].
Type $A_k$ expansion and full-space polymer application
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We state Proposition \[Akcprop\] and then immediately apply it to expand the nested contour integral formula for $\bar{Z}(t;\vec{x})$ calculated earlier in equation (\[Aknci\]). The proof of the below proposition can be found in Section \[proofsec\].
\[Akcprop\] Fix $k\geq 1$ and $c\in (0,\infty)$. Given a set of real numbers $\alpha_1,\ldots,\alpha_k$ and a function $F(z_1,\ldots ,z_k)$ which satisfy
1. For all $1\leq j \leq k-1$, $\alpha_j>\alpha_{j+1}+c$;
2. For all $1\leq j \leq k$ and $z_1,\ldots,z_k$ such that $z_i\in \alpha_i+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ for $1\leq i<j$ and $z_i\in \alpha_k+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ for $j<i\leq k$, the function $z_j\mapsto F(z_1,\ldots ,z_j,\ldots, z_k)$ is analytic in the complex domain $\{z: \alpha_k\leq {\ensuremath{\mathrm{Re}}}(z) \leq \alpha_j\}$ and is bounded in modulus on that domain by ${\ensuremath{\textrm{const}}}\, {\ensuremath{\mathrm{Im}}}(z_j)^{-1-\delta}$ for some constants ${\ensuremath{\textrm{const}}},\delta>0$ (depending on $z_1,\ldots,z_{j-1},z_{j+1},\ldots,z_k$ but not $z_j$).
Then we have the following residue expansion identity: $$\begin{aligned}
\int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c} F(\vec{z})
\qquad\qquad\qquad\qquad\qquad\qquad&\\
=c^k \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{1}{m_1! m_2!\cdots} \, \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}}\, \det\left[\frac{1}{w_i +c\lambda_i -w_j}\right]_{i,j=1}^{\ell(\lambda)}&\\
\times\,E^c(w_1,w_1+c,\ldots, w_1 + c(\lambda_1-1), \ldots, w_{\lambda_{\ell(\lambda)}},w_{\lambda_{\ell(\lambda)}}+c, \ldots, w_{\lambda_{\ell(\lambda)}}+c(\lambda_{\ell(\lambda)}-1)),&\end{aligned}$$ where $$E^c(z_1,\ldots, z_k) := \sum_{\sigma\in S_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-z_{\sigma(B)}-c}{z_{\sigma(A)}-z_{\sigma(B)}} F(\sigma(\vec{z})).$$
We may apply this proposition to the formula for $\bar{Z}(t,\vec{x})$ provided in equation (\[Aknci\]). The hypotheses of the proposition are easily checked and the outcome is the following expansion: $$\begin{aligned}
\bar{Z}(t;\vec{x}) = \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{1}{m_1! m_2!\cdots} \, \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}} \det\left[\frac{1}{w_i +\lambda_i -w_j}\right]_{i,j=1}^{\ell(\lambda)}&\\
\times\,E^1_{\vec{x}}(w_1,w_1+1,\ldots, w_1 + (\lambda_1-1), \ldots, w_{\lambda_{\ell(\lambda)}},w_{\lambda_{\ell(\lambda)}}+1,\ldots, w_{\lambda_{\ell(\lambda)}}+(\lambda_{\ell(\lambda)}-1)),&\end{aligned}$$ where $$E^1_{\vec{x}}(z_1,\ldots, z_k) := \sum_{\sigma\in S_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-z_{\sigma(B)}-1}{z_{\sigma(A)}-z_{\sigma(B)}} \prod_{j=1}^{k} e^{\frac{t}{2} z_{\sigma(j)}^2 + x_j z_{\sigma(j)}}.$$
As we are interested in ${\ensuremath{\mathbb{E}}}[Z(t,0)^k]$ it suffices to consider all $x_i\equiv 0$. Using the symmetrization identity[^8] given in equation (\[Akfacsym\]) we find that $$\begin{aligned}
E^1_{\vec{0}}(w_1,w_1+1,\ldots, w_1 +(\lambda_1-1), \ldots, w_{\lambda_{\ell(\lambda)}},w_{\lambda_{\ell(\lambda)}}+1,\ldots, w_{\lambda_{\ell(\lambda)}}+(\lambda_{\ell(\lambda)}-1)) &\\
= k! \prod_{j=1}^{\ell(\lambda)} e^{\frac{t}{2}\big(w_j^2 + (w_j+1)^2+\cdots (w_j+\lambda_j-1)^2\big)} = k! \prod_{j=1}^{\ell(\lambda)} \frac{e^{\frac{t}{2} G(w_j+\lambda_j)}}{e^{\frac{t}{2} G(w_j)}},\qquad\qquad\qquad\qquad&\end{aligned}$$ where $$\label{Gdef}
G(w)=\frac{w^3}{3}-\frac{w^2}{2}+\frac{w}{6}.$$ In this last line we have used the fact that $$\label{Grat}
\frac{G(w+\ell)}{G(w)} = w^2 + (w+1)^2 + \cdots (w+\ell-1)^2.$$
Using this, we conclude that $$\begin{aligned}
\label{Asupexp}
{\ensuremath{\mathbb{E}}}[Z(t,0)^k] = k! \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{1}{m_1! m_2!\cdots} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&\\
\nonumber \times\, \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}} \det\left[\frac{1}{w_i +\lambda_i -w_j}\right]_{i,j=1}^{\ell(\lambda)} \prod_{j=1}^{\ell(\lambda)} \frac{e^{\frac{t}{2} G(w_j+\lambda_j)}}{e^{\frac{t}{2} G(w_j)}}.\end{aligned}$$
Type $BC_k$ expansion and half-space polymer application {#bkcase}
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We present an analogous result to Proposition \[Akcprop\] above, which we then apply to $\bar{Z}^0(t;\vec{x})$. We explain below in Remark \[nota\] why the residue expansion for the general case of $a\neq 0$ presents some additional complexities (and possibly additional terms). We actually give a complete proof of the below conjecture modulo Claim \[claim:Res-are-zero\] which deals with the cancelation of the sum of certain residues arising from contour deformations in the course of our computations. Partial evidence why Claim \[claim:Res-are-zero\] is correct is given in Section \[sec:one-string\], where cancelation of some of the arising residues is shown.
\[Bkcprop\] Fix $k\geq 1$ and $c\in (0,\infty)$. Given a set of real numbers $\alpha_1,\ldots,\alpha_k$ and a function $F(z_1,\ldots ,z_k)$ which satisfy
1. $\alpha_k=0$;
2. For all $1\leq j \leq k-1$, $\alpha_j>\alpha_{j+1}+c$;
3. For $z_1,\ldots, z_k$ in the complex domain $\{x+{\ensuremath{\mathbf{i}}}y: -\alpha_1\leq x\leq \alpha_1, y\in {\ensuremath{\mathbb{R}}}\}$, for all $1\leq j\leq k$, the function $z_j\mapsto F(z_1,\ldots ,z_j,\ldots, z_k)$ is analytic and is bounded by a constant (depending on $z_1,\ldots,z_{j-1},z_{j+1},\ldots,z_k$ but not $z_j$) times ${\ensuremath{\mathrm{Im}}}(z_j)^{-1-\delta}$ for some $\delta>0$.
Then we have the following residue expansion identity: $$\begin{aligned}
\label{mainform}
&\int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty}\frac{dz_1}{2 \pi{\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_k}{2 \pi{\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c} \,\frac{z_A+z_B}{z_A+z_B-c} F(\vec{z})\\
\nonumber &= c^k \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{(-1)^{\ell(\lambda)}}{m_1! m_2!\cdots} \, \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2 \pi{\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_k-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_k+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2 \pi{\ensuremath{\mathbf{i}}}} \prod_{j=1}^{\ell(\lambda)} \frac{1}{4 c}\, \frac{\left(\tfrac{2w_j+c}{2c}\right)_{\lambda_j-1}}{\left(\tfrac{2w_j}{2c}\right)_{\lambda_j}} \Pf\left[\frac{u_i-u_j}{u_i+u_j}\right]_{i,j=1}^{2\ell(\lambda)} \\
\nonumber & E^{c}(w_1,w_1+c,\ldots, w_1+c(\lambda_1-1),\ldots, w_{\ell(\lambda)},w_{\ell(\lambda)}+c,\ldots, w_{\ell(\lambda)} + c(\lambda_{\ell(\lambda)}-1))\end{aligned}$$ where $$(u_1,\ldots, u_{2\ell(\lambda)}) :=\big(-w_1+\tfrac{c}{2}, w_1-\tfrac{c}{2}+\lambda_1 c, -w_2+\tfrac{c}{2},w_2-\tfrac{c}{2}+\lambda_2 c, \ldots, -w_{\ell(\lambda)}+\tfrac{c}{2}, w_{\ell(\lambda)}-\tfrac{c}{2} + \lambda_{\ell(\lambda)} c\big)$$ and where $$E^{c}(z_1,\ldots, z_k) := \sum_{\sigma\in BC_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-z_{\sigma(B)}-c}{z_{\sigma(A)}-z_{\sigma(B)}}\frac{z_{\sigma(A)}+z_{\sigma(B)}-c}{z_{\sigma(A)}+z_{\sigma(B)}} F(\sigma(\vec{z})).$$
We apply this conjecture to the formula for $\bar{Z}^0(t,\vec{x})$ provided in equation (\[Bknci\]). Observe that the formula in (\[Bknci\]) can be written in the form of the left-hand side of (\[mainform\]) by taking $c=1$ and $$\label{eq:F-apply}
F(\vec{z}) = \prod_{j=1}^{k} e^{\frac{t}{2}z_j^2 + x_j z_j}.$$ Notice that the contours satisfy the first two hypotheses of Conjecture \[Bkcprop\] and that the function $F$ satisfies the analyticity and decay conditions of the third hypothesis.
\[nota\] The reason we have restricted our attention at this point to the case of $a=0$ is because of the following: For general $a\neq 0$ the function $F(\vec{z})$ contains an extra term $$\prod_{j=1}^{k} \frac{z_j}{z_j+a}.$$ When $a<0$ the contours from equation (\[Bknci\]) are such that $\alpha_k=-a+\epsilon>0$. If we want to shift to $\alpha_k=0$ then we cross poles, which contribute additional terms which are not accounted for in the above result. For $a>0$ we have $\alpha_k=0$, however, if $a<(k-1)$, then the third hypothesis is violated since the pole coming from the denominator of the above fraction will be included in the complex domain in which analyticity is desired (see Remark \[aexception\] for an explanation of how this hypothesis is utilized).
The result of applying the above conjecture for $a=0$ is that (from here on out, all half-space formulas will assume the validity of Conjecture \[Bkcprop\]) $$\begin{aligned}
\bar{Z}^0(t;\vec{x}) = 2^k \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{(-1)^{\ell(\lambda)}}{m_1! m_2!\cdots} \, \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}}
\prod_{j=1}^{\ell(\lambda)} \frac{1}{4}\, \frac{\left(w_j+\frac{1}{2}\right)_{\lambda_j-1}}{(w_j)_{\lambda_j}} \Pf\left[\frac{u_i-u_j}{u_i+u_j}\right]_{i,j=1}^{2\ell(\lambda)}&\\
\times\,E^{1}_{\vec{x}}(w_1,w_1+1,\ldots, w_1 + (\lambda_1-1), \ldots, w_{\lambda_{\ell(\lambda)}},w_{\lambda_{\ell(\lambda)}}+1,\ldots, w_{\lambda_{\ell(\lambda)}}+(\lambda_{\ell(\lambda)}-1)),&\end{aligned}$$ where $$\label{uabove}
(u_1,\ldots, u_{2\ell(\lambda)}) :=\big(-w_1+\tfrac{1}{2}, w_1-\tfrac{1}{2}+\lambda_1 , -w_2+\tfrac{1}{2},w_2-\tfrac{1}{2}+\lambda_2 , \ldots, -w_{\ell(\lambda)}+\tfrac{1}{2}, w_{\ell(\lambda)}-\tfrac{1}{2} + \lambda_{\ell(\lambda)} \big)$$ and $$E^{1}_{\vec{x}}(z_1,\ldots, z_k) := \sum_{\sigma\in BC_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-z_{\sigma(B)}+c}{z_{\sigma(A)}-z_{\sigma(B)}}\frac{z_{\sigma(A)}+z_{\sigma(B)}+c}{z_{\sigma(A)}+z_{\sigma(B)}} \, \prod_{j=1}^{k} e^{\frac{t}{2} z_{\sigma(j)}^2 + x_j z_{\sigma(j)}}.$$
As we are interested in ${\ensuremath{\mathbb{E}}}[Z^0(t,0)^k]$ it suffices to consider all $x_i\equiv 0$. Using the symmetrization identity[^9] given in equation (\[Bkfacsym\]) we find that $$\begin{aligned}
E^{1}_{\vec{0}} (z_1,\ldots, z_k) &=& \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2} \sum_{\sigma\in BC_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-z_{\sigma(B)}+c}{z_{\sigma(A)}-z_{\sigma(B)}}\frac{z_{\sigma(A)}+z_{\sigma(B)}+c}{z_{\sigma(A)}+z_{\sigma(B)}}\\
&=& 2^k k! \prod_{j=1}^{k} e^{\frac{t}{2} z_j^2}.\end{aligned}$$ Note that the identity was used in going from the first to second line. The equality in the first line came from the $BC_k$ invariance of the product which is factored out of the summation over $BC_k$.
Owing to equation (\[Grat\]) we may rewrite $$\begin{aligned}
E^{1}_{\vec{0}}(w_1,w_1+1,\ldots, w_1 +(\lambda_1-1), \ldots, w_{\lambda_{\ell(\lambda)}},w_{\lambda_{\ell(\lambda)}}+1,\ldots, w_{\lambda_{\ell(\lambda)}}+(\lambda_{\ell(\lambda)}-1)) &\\
= 2^k k! \, \prod_{j=1}^{\ell(\lambda)} \frac{e^{\frac{t}{2} G(w_j+\lambda_j)}}{e^{\frac{t}{2} G(w_j)}}. &\end{aligned}$$
Using this, we conclude that $$\begin{aligned}
\label{Bsupexp}
{\ensuremath{\mathbb{E}}}[Z^0(t,0)^k] = 4^k k! \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{(-1)^{\ell(\lambda)}}{m_1! m_2!\cdots} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&\\
\nonumber \times\,\int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}}
\prod_{j=1}^{\ell(\lambda)} \frac{1}{4}\, \frac{\left(w_j+\frac{1}{2}\right)_{\lambda_j-1}}{(w_j)_{\lambda_j}} \frac{e^{\frac{t}{2} G(w_j+\lambda_j)}}{e^{\frac{t}{2} G(w_j)}} \Pf\left[\frac{u_i-u_j}{u_i+u_j}\right]_{i,j=1}^{2\ell(\lambda)},&\end{aligned}$$ with the $u$ variables are defined in terms of the $w$ variables in (\[uabove\]).
Summing the moment generating function {#mgfsec}
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We seek now to derive concise formulas for the Laplace transforms of $Z(t,0)$ and $Z^a(t,0)$ for $a\geq 0$. Knowledge of the Laplace transform of a positive random variable uniquely identifies the random variable’s distribution and can easily be inverted. Unfortunately, one readily checks that the moments of these random variables grow too rapidly (in $k$) to characterize their Laplace transforms and distributions[^10]. As such, we proceed formally (i.e. in a mathematically unjustified manner) and still attempt to use the moments to recover the Laplace transform. This means that we write $$\label{mgf}
{\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = \sum_{k=0}^{\infty} \frac{\zeta^k}{k!} {\ensuremath{\mathbb{E}}}[Z(t,0)^k].$$ The left-hand side above is clearly finite for all $\zeta$ with ${\ensuremath{\mathrm{Re}}}(\zeta)<0$. On the other hand, since ${\ensuremath{\mathbb{E}}}[Z(t,0)^k]$ grows super-exponentially (as can be seen from analysis of (\[Asupexp\])) the right-hand side is a divergent series for all $\zeta$, and does not converge in any sense. Of course the source of this apparent inconsistency is in the interchange of expectations and infinite summation which is not mathematically justifiable – hence the divergent right-hand side. Our derivation of the Laplace transform formula, therefore, involves “summing” this divergent moment generating function series. This is done via an illegal application of a Mellin-Barnes trick which says that for functions $g(z)$ with suitable decay and analyticity properties (cf. Lemma 3.2.13 of [@BorCor]) we have (using the fact that the residue of $\pi/\sin(-\pi z)$ at $z=n$ is $(-1)^{n+1}$) that $$\label{Mellin}
\sum_{n=1}^{\infty} g(n) \zeta^n = \int \frac{dz}{2\pi {\ensuremath{\mathbf{i}}}} \frac{\pi}{\sin(-\pi z)} g(z) (-\zeta)^z.$$ The negatively oriented $z$ contour should encircle the positive integers and no poles of $g(z)$, as well as satisfy certain properties related to the growth of $g(z)$ at infinity. Our illegal application neglects consideration of the growth condition (which clearly is not satisfied). However, the result after the application of this trick becomes perfectly well-defined and convergent. The result of this is given below and summarized in Section \[Laplacesec\] of the introduction.
Despite this divergent summation, in the full-space case the final answer we arrive at for the Laplace transform of $Z(t,0)$ can be compared to the formula proved in [@ACQ; @BCF] and it is in agreement. Presently we have no alternative proof of the analogous final result for the half-space case. It is quite reasonable to wonder why the formal manipulations of divergent series that we perform (at least in the full-space case) actually lead to the correct Laplace transform formula. This can be seen as a consequence of the fact that there exist $q$-deformed discrete versions of the polymers (both $q$-TASEP and ASEP work) for which every step below can be performed in an analogous, yet totally rigorous manner (see [@BorCor; @BCS]). Thus, from this particular perspective, the manipulations below can be viewed as shadows of rigorous manipulations at a $q$-deformed level. Presently we do not have any analogous $q$-deformed results for the half-space polymer.
Full-space polymer {#full-space-polymer-3}
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We proceed formally – mimicking the steps in the mathematically rigorous work of [@BorCor; @BCS]. We seek to sum the moment generating function (\[mgf\]). Using (\[Asupexp\]) we find that $$\frac{\zeta^k}{k!} {\ensuremath{\mathbb{E}}}[Z(t,0)^k] = \sum_{L=1}^{\infty} \frac{1}{L!} \sum_{\lambda_1=1}^{\infty} \cdots \sum_{\lambda_L=1}^{\infty} \mathbf{1}_{\lambda_1+\cdots + \lambda_L = k} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} I_{L}(\lambda;w;\zeta)$$ where $$I_{L}(\lambda;w;\zeta)= \det\left[\frac{1}{w_i+\lambda_i -w_j}\right]_{i,j=1}^{L} \prod_{j=1}^{L} \zeta^{\lambda_j} \frac{e^{\frac{t}{2} G(w_j+\lambda_j)}}{e^{\frac{t}{2} G(w_j)}}.$$ Notice that we have replaced the summation over partitions with a summation over $L$ and then arbitrary (unordered) $\lambda_1,\ldots, \lambda_L$. This requires insertion of the term $\mathbf{1}_{\lambda_1+\cdots + \lambda_L = k}$ to enforce that the $\lambda_j$ sum to $k$, as well as division by $$\frac{L!}{m_1!m_2!\cdots}$$ from the change between ordered and unordered $\lambda_j$’s. This factor exactly cancels the terms in (\[Asupexp\]) and yields the above claimed formula.
By summing over $k$ (and performing infinite range reordering within our already divergent series) we find that $$\label{almostfreddet}
{\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = 1+ \sum_{L=1}^{\infty} \frac{1}{L!} \sum_{\lambda_1=1}^{\infty} \cdots \sum_{\lambda_L=1}^{\infty} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} I_{L}(\lambda;w;\zeta).$$ The indicator function disappeared as a result of summing over $k$. The above equality is only formal since the right-hand side is still divergent at this point.
### The Mellin-Barnes trick
Notice that for $w_1,\ldots, w_L$ on ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, the function $I_{L}(\lambda;w;\zeta)$ is analytic in each $\lambda_1,\ldots,\lambda_L$ as long as their real parts stay positive. We use this fact in order to apply the Mellin-Barnes trick explained above in (\[Mellin\]) to replace the summations of $\lambda_1,\ldots, \lambda_L$ by an $L$-fold integral: $$\label{sumdiv}
\sum_{\lambda_1=1}^{\infty} \cdots \sum_{\lambda_L=1}^\infty I_{L}(\lambda_1,\ldots, \lambda_L ;w;\zeta) = \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \, \prod_{j=1}^{L} \frac{\pi}{\sin(-\pi \lambda_j)}\, I_{L}(\lambda;w;-\zeta).$$ Note that $3/4$ in the integral limits can be replaced by any real number in $(1/2,1)$. The upper bound is from the requirement (in the Mellin-Barnes trick application) that the $\lambda$ contour contain the positive integers including 1, and the lower bound comes later when, after changing variables by subtracting $1/2$, we want to apply the integral representation for the Airy function in equation \[Airyintrep\]. That application requires being along a positive real part contour, hence the lower bound.
Of course, this equality is only formal (in the $\zeta$ variable) in terms of a residue expansion and does not make mathematical sense due to the growth of the integrand near infinity.
It is, however, via this illegal application of the Mellin-Barnes trick that our divergent series has been “summed” and replaced by a convergent integral: $$\label{almostfreddet2}
{\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = 1+ \sum_{L=1}^{\infty} \frac{1}{L!} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \, \prod_{j=1}^{L} \frac{\pi}{\sin(-\pi \lambda_j)}\, I_{L}(\lambda;w;-\zeta).$$ In fact, the above formula (though arrived at through mathematically non-rigorous means) is equivalent to the rigorously proved formulas in [@ACQ; @BCF]. To put this in the same form (so as to compare) will require a few more manipulations – all of which now are mathematically justifiable.
### Manipulating the formula
We employ two identities (the first valid for ${\ensuremath{\mathrm{Re}}}(\zeta)<0$ and the second valid for ${\ensuremath{\mathrm{Re}}}(w+\lambda-w')>0$) $$\frac{\pi}{\sin(-\pi \lambda)}\, (-\zeta)^{\lambda} = \int_{-\infty}^{\infty}dr \frac{\zeta e^{\lambda r}}{e^{r}-\zeta}, \qquad \qquad \frac{1}{w+\lambda-w'} = \int_0^{\infty} dx e^{-x(w+\lambda-w')},$$ perform the change of variables $w_j+\lambda_j = s_j$, and use the linearity properties of the determinant to rewrite $$\label{freddet}
{\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = 1+ \sum_{L=1}^{\infty} \frac{1}{L!} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} \det\left[K^1_{\zeta}(w_i,w_j)\right]_{i,j=1}^{L} = \det(I+K^1_{\zeta})_{L^2({\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}},\frac{d\mu}{2\pi {\ensuremath{\mathbf{i}}}})}.$$ Here $\det(I+K^1_{\zeta})_{L^2({\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}},\frac{d\mu}{2\pi {\ensuremath{\mathbf{i}}}})}$ is the Fredholm determinant of the operator $K^1_{\zeta}$ which acts on functions in $L^2({\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}},\frac{d\mu}{2\pi {\ensuremath{\mathbf{i}}}})$ (with $\frac{d\mu}{2\pi {\ensuremath{\mathbf{i}}}}$ representing the measure $\frac{dw}{2\pi {\ensuremath{\mathbf{i}}}}$ along $w\in {\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$). The integral operator $K^1_{\zeta}$ is specified by its kernel $$K^1_{\zeta}(w,w') =\int_0^{\infty} dx \int_{-\infty}^{\infty} dr \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{\zeta e^{r(s-w)}}{e^r-\zeta} e^{-x(s-w')} \frac{e^{\frac{t}{2}G(s)}}{e^{\frac{t}{2}G(w)}}.$$ The change of variables $w \mapsto w+1/2$ and $s\mapsto s+1/2$ yields $${\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = \det(I+K^2_{\zeta})_{L^2(-\frac{1}{2}+ {\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}},\frac{d\mu}{2\pi {\ensuremath{\mathbf{i}}}})}$$ with $$K^2_{\zeta}(w,w') =\int_0^{\infty} dx \int_{-\infty}^{\infty} dr \int_{\frac{1}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{1}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{\zeta e^{r(s-w)}}{e^r-\zeta} e^{-x(s-w')} \frac{e^{\frac{t}{2}G(s+\frac{1}{2})}}{e^{\frac{t}{2}G(w+\frac{1}{2})}}.$$ We may factor the operator $K^2_{\zeta}(w,w') = \int_0^{\infty} dx A(w,x)B(x,w')$ where $$\begin{aligned}
A(w,x) &=& \int_{-\infty}^{\infty} dr \int_{\frac{1}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{1}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{\zeta}{e^r-\zeta} e^{-xs} \frac{e^{\frac{t}{2}G(s+\frac{1}{2})}}{e^{\frac{t}{2}G(w+\frac{1}{2})}},\\
B(x,w) &=& e^{xw}.\end{aligned}$$ Determinants satisfy a fundamental identity (see [@Deift] for details and uses) that $\det(I+AB) = \det(I+BA)$. This enables us to rewrite $${\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = \det(I+K^3_{\zeta})_{L^2([0,\infty))}$$ where $$K^3_{\zeta}(x,x') =\int_{-\infty}^{\infty} dr \frac{\zeta}{e^r-\zeta} \int_{-\frac{1}{2}-{\ensuremath{\mathbf{i}}}\infty}^{-\frac{1}{2} + {\ensuremath{\mathbf{i}}}\infty} \frac{dw}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\frac{1}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{1}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{F_x(s)}{F_{x'}(w)}$$ and $$F_x(s) = \exp\left\{ \frac{t}{2}\,\frac{s^3}{3} + s\left(r-x-\frac{t}{24}\right)\right\}.$$
Finally, we perform the change of variables $$\begin{aligned}
\label{changes1}
s \mapsto \left(\frac{t}{2}\right)^{-1/3} s, \qquad w \mapsto \left(\frac{t}{2}\right)^{-1/3} w, \qquad x \mapsto \left(\frac{t}{2}\right)^{1/3} x,\qquad x' \mapsto \left(\frac{t}{2}\right)^{1/3} x', \\
\nonumber r\mapsto \left(\frac{t}{2}\right)^{1/3} r + \frac{t}{24}, \qquad \log(-\zeta) \mapsto -\left(\frac{t}{2}\right)^{1/3} u + \frac{t}{24}.\end{aligned}$$ Noting that $$\frac{\zeta}{e^r-\zeta} = \frac{-1}{1+ e^{r-\log(-\zeta)}}$$ and taking into account the Jacobian contribution from the change of variables (and rescaling the contours), we arrive at $${\ensuremath{\mathbb{E}}}\left[e^{\zeta Z(t,0)}\right] = \det(I-K^4_{\zeta})_{L^2([0,\infty))}$$ where $$\begin{aligned}
K^4_{\zeta}(x,x') =\int_{-\infty}^{\infty} d r \frac{1}{1 + \exp\left\{(\frac{t}{2})^{1/3}(r + u)\right\}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&\\
\times\,\int_{\frac{1}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{1}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d s}{2\pi {\ensuremath{\mathbf{i}}}} \exp\left\{\frac{s^3}{3}+ s( r-x)\right\} \int_{-\frac{1}{2}-{\ensuremath{\mathbf{i}}}\infty}^{-\frac{1}{2} + {\ensuremath{\mathbf{i}}}\infty} \frac{d w}{2\pi {\ensuremath{\mathbf{i}}}} \exp\left\{-\frac{ w^3}{3}- w( r-x')\right\},&\end{aligned}$$ and $u$ is related to $\zeta$ as in (\[changes1\]).
For any $\delta>0$, the Airy function has the following contour integral representation $$\label{Airyintrep}
\Ai(v) = \int_{\delta -{\ensuremath{\mathbf{i}}}\infty}^{\delta+{\ensuremath{\mathbf{i}}}\infty} \frac{dz}{2\pi {\ensuremath{\mathbf{i}}}} \exp\left\{\frac{z^3}{3} - vz\right\},$$ from which it follows that $$K^4_{\zeta}( x, x') =\int_{-\infty}^{\infty} d r \frac{1}{1 + \exp\left\{(\frac{t}{2})^{1/3}( r + u)\right\}} \Ai( x-r)\, \Ai( x'- r).$$
At this point we have reached the final form of our formula, cf. Section \[fullspacelapsec\] above.
Half-space polymer {#half-space-polymer-3}
------------------
We seek here to compute the Laplace transform of $Z^0(t,0)$. We do so via formally “summing” the divergent moment generating function (\[mgf\]) in a similar way as worked for the full-space polymer. For convenience of our formulas, we replace $\zeta$ in that formula with $\zeta/4$. Using (\[Bsupexp\]) we find that $$\frac{\zeta^k}{4^k k!} {\ensuremath{\mathbb{E}}}[Z^0(t,0)^k] = \sum_{L=1}^{\infty} \frac{(-1)^{L}}{L!} \sum_{\lambda_1=1}^{\infty} \cdots \sum_{\lambda_L=1}^{\infty} \mathbf{1}_{\lambda_1+\cdots + \lambda_L = k} \int_{-{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}}^{{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} I^0_{L}(\lambda;w;\zeta)$$ where $$I^0_{L}(\lambda;w;\zeta)= \Pf\left[\frac{u_i-u_j}{u_i+u_j}\right]_{i,j=1}^{2L} \prod_{j=1}^{L} \frac{\zeta^{\lambda_j}}{4} \frac{1}{w_j-\tfrac{1}{2}} \frac{\tilde F(w_j+\lambda_j)}{\tilde F(w_j)}$$ where the $u$ variables are written in terms of the $w$ variables as in (\[uabove\]) and $$\tilde F(w) =\frac{\Gamma(w-\frac{1}{2})}{\Gamma(w)} e^{\frac{t}{2} G(w)}$$ with $G(w)$ as in (\[Gdef\]). In addition to the same considerations as in the full-space case, presently this step also relies upon the fact that the Pochhammer symbol $(x)_i$ can be expressed through Gamma functions as $\Gamma(x+i)/\Gamma(x)$.
By summing over $k$ (and performing infinite range reordering within our already divergent series) we find that $$\label{Balmostfreddet}
{\ensuremath{\mathbb{E}}}\left[e^{\frac{\zeta}{4} Z^0(t,0)}\right] = 1+ \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \sum_{\lambda_1=1}^{\infty} \cdots \sum_{\lambda_L=1}^{\infty} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} I^0_{L}(\lambda;w;\zeta).$$ The indicator function disappeared as a result of summing over $k$ and the above formula is still only formal, the right-hand side is divergent.
### The Mellin-Barnes trick
Notice that for $w_1,\ldots, w_L$ on ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, the function $I^0_{L}(\lambda;w;\zeta)$ is analytic in each $\lambda_1,\ldots,\lambda_L$ as long as their real part stays positive. We use this fact in order to apply the same Mellin-Barnes trick as in the full-space case in order to replace the summations of $\lambda_1,\ldots, \lambda_L$ by an $L$-fold integral: $$\label{Bsumdiv}
\sum_{\lambda_1=1}^{\infty} \cdots \sum_{\lambda_L=1}^\infty I^0_{L}(\lambda_1,\ldots, \lambda_L ;w;\zeta) = \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \, \prod_{j=1}^{L} \frac{\pi}{\sin(-\pi \lambda_j)}\, I^0_{L}(\lambda;w;-\zeta).$$ As before this is only a formal (in the $\zeta$ variable) residue expansion due to growth of the integrand near infinity.
Also as before, this Mellin-Barnes trick has turned our divergent series into a convergent integral: $$\label{Balmostfreddet2}
{\ensuremath{\mathbb{E}}}\left[e^{\frac{\zeta}{4} Z^0(t,0)}\right] = 1+ \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\frac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\frac{3}{4} + {\ensuremath{\mathbf{i}}}\infty} \frac{d\lambda_1}{2\pi {\ensuremath{\mathbf{i}}}} \, \prod_{j=1}^{L} \frac{\pi}{\sin(-\pi \lambda_j)}\, I^0_{L}(\lambda;w;-\zeta).$$ Both the left-hand and right-hand sides of the above formula make sense now, though we have only shown it in a formal manner. As before, it is useful to perform a few additional manipulations to the right-hand side in order to prepare the formula for large time asymptotics.
### Manipulating the formula
We employ essentially the same two identities as in the full-space (the first valid for ${\ensuremath{\mathrm{Re}}}(\zeta)<0$ and the second valid for ${\ensuremath{\mathrm{Re}}}(u_i+u_j)>0$) $$\frac{\pi}{\sin(-\pi \lambda)}\, (-\zeta)^{\lambda} = \int_{-\infty}^{\infty}dr \frac{\zeta e^{\lambda r}}{e^{r}-\zeta}, \qquad \qquad \frac{u_i-u_j}{u_i+u_j} = (u_i-u_j) \int_0^{\infty} dx e^{-x(u_i+u_j)},$$ and perform the change of variables $w_j+\lambda_j = s_j$ to rewrite $$\begin{aligned}
\label{Bfreddet}
\nonumber {\ensuremath{\mathbb{E}}}\left[e^{\frac{\zeta}{4} Z^0(t,0)}\right] &=& 1+ \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_{-\infty}^{\infty} dr_1 \cdots \int_{-\infty}^{\infty} dr_L \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_L}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\tfrac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{3}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\tfrac{3}{4}-{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{3}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_L}{2\pi {\ensuremath{\mathbf{i}}}} \\
&&\times\,\prod_{j=1}^{L} \frac{ \zeta e^{r_j(s_j-w_j)}}{e^{r_j}-\zeta}\, \frac{1}{4} \frac{1}{w_j-\tfrac{1}{2}} \frac{\tilde F(s_j)}{\tilde F(w_j)} \Pf\left[(u_i-u_j)\int_0^{\infty} dx e^{-x(u_i+u_j)}\right]_{i,j=1}^{2L}.\end{aligned}$$ Performing the change of variables $w \mapsto w+1/2$ and $s\mapsto s+1/2$ as well as using properties of the Pfaffian to bring the $s$ and $w$ integrals inside the Pfaffian, we find that this can be rewritten as $$\label{preasybform}
{\ensuremath{\mathbb{E}}}\left[e^{\frac{\zeta}{4} Z^0(t,0)}\right] = 1 + \sum_{L=1}^{\infty} \frac{(-1)^L}{L!}\int_{-\infty}^{\infty} dr_1\cdots \int_{-\infty}^{\infty} dr_L \prod_{L=1}^{k} \frac{\zeta}{e^{r_j}-\zeta} \Pf\left[ K^{(t)}(r_i,r_j)\right]_{i,j=1}^{L}$$ where $K^{(t)}$ is a $2\times 2$ matrix with components $$\begin{aligned}
K^{(t)}_{11}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_2}{2\pi {\ensuremath{\mathbf{i}}}} \frac{-w_1+w_2}{w_1 w_2} \frac{1}{F^{(t)}(w_1) F^{(t)}(w_2)} e^{-r w_1-r'w_2} e^{xw_1+xw_2},\\
K^{(t)}_{12}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{-w-s}{w} \frac{F^{(t)}(s)}{F^{(t)}(w)} e^{-r w+r's} e^{xw-xs},\\
K^{(t)}_{22}(r,r') &=& \frac{1}{4} \int_0^{\infty} dx \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_1}{2\pi {\ensuremath{\mathbf{i}}}}\int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_2}{2\pi {\ensuremath{\mathbf{i}}}} (s_1-s_2) F^{(t)}(s_1)F^{(t)}(s_2) e^{r s_1+r's_2} e^{-xs_1-xs_2},\end{aligned}$$ and $K^{(t)}_{21}(r,r') = - K^{(t)}_{12}(r',r)$. In the above we have used $$F^{(t)}(w) =\frac{\Gamma(w)}{\Gamma(w+\frac{1}{2})} e^{\frac{t}{2} \left(\frac{w^3}{3} - \frac{w}{12}\right)}.$$
This is the final form of our Laplace transform formula, cf. Section \[halfspacelapsec\] above.
### Long-time asymptotics {#RHSder}
We perform the necessary asymptotics in order to show the results claimed in Section \[hsplim\]. In that section we rewrote the half-space Laplace transform in a suggestive manner for taking the $t\to \infty$ asymptotics: $${\ensuremath{\mathbb{E}}}\left[ \exp\left\{-\frac{1}{4}\exp\left\{ \left(\frac{t}{2}\right)^{1/3}\left[\frac{\log Z^0(t,0) + \frac{t}{24}}{\left(\frac{t}{2}\right)^{1/3}} \, -u \right]\right\}\right\} \right].$$ This amounts to taking $\zeta$ in (\[preasybform\]) and replacing it by $\log(-\zeta) \mapsto -\left(\frac{t}{2}\right)^{1/3} u + \frac{t}{24}$. Without changing the value of the right-hand side of (\[preasybform\]) we may perform the following change of variables as well $$\begin{aligned}
s \mapsto \left(\frac{t}{2}\right)^{-1/3} s, \qquad w \mapsto \left(\frac{t}{2}\right)^{-1/3} w, \qquad r\mapsto \left(\frac{t}{2}\right)^{1/3} r + \frac{t}{24},\\
x \mapsto \left(\frac{t}{2}\right)^{1/3} x, \qquad x' \mapsto \left(\frac{t}{2}\right)^{1/3} x'.\end{aligned}$$ We may (using Cauchy’s theorem and the decay of the integrand) deform our rescaled contours back to their original locations without crossing any poles. Notice that under the change of variables, $$F^{(t)}(w)e^{(r-x)w} \mapsto \frac{\Gamma\left(\left(\frac{t}{2}\right)^{-1/3} w\right)}{\Gamma\left(\left(\frac{t}{2}\right)^{-1/3} w + \frac{1}{2} \right)} e^{\frac{w^3}{3}+(r-x)w}.$$ The decay of the exponential term along the vertical contours of integration implies that it suffices to consider the termwise limits of these Gamma function factors as $t\to\infty$. The function above becomes behaves (for large $t$) like $$\frac{\alpha}{\left(\frac{t}{2}\right)^{-1/3} w} e^{\frac{w^3}{3}+(r-x)w}, \qquad \alpha=\frac{1}{\Gamma(\tfrac{1}{2})}=\pi^{-1/2}.$$ We may combine this limiting behavior with the Jacobian contribution coming from the change of variables to find that
$$\lim_{t\to \infty} {\ensuremath{\mathbb{E}}}\left[e^{\frac{\zeta}{4} Z^0(t,0)}\right] = 1 + \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_{-\infty}^{-u} dr_1 \cdots \int_{-\infty}^{-u} dr_{L} \Pf\left[\tilde K^{\infty}(r_i,r_j)\right]_{i,j=1}^{L}$$
where $\tilde K^{\infty}$ is a $2\times 2$ matrix with components $$\begin{aligned}
\tilde K^{\infty}_{11}(r,r') &=&-\alpha^{-2} \frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw_2}{2\pi {\ensuremath{\mathbf{i}}}} (-w_1+w_2) e^{-\frac{w_1^3}{3} - w_1(r-x)} e^{-\frac{w_2^3}{3}-w_2(r'-x)},\\
\tilde K^{\infty}_{12}(r,r') &=& -\frac{1}{4} \int_0^{\infty} dx \int_{-\tfrac{1}{2} -{\ensuremath{\mathbf{i}}}\infty}^{-\tfrac{1}{2}+{\ensuremath{\mathbf{i}}}\infty} \frac{dw}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds}{2\pi {\ensuremath{\mathbf{i}}}} \frac{-w-s}{s} e^{-\frac{w^3}{3} - w(r-x)} e^{\frac{s^3}{3} + s(r'-x)},\\
\tilde K^{\infty}_{22}(r,r') &=& -\alpha^2 \frac{1}{4} \int_0^{\infty} dx \int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_1}{2\pi {\ensuremath{\mathbf{i}}}}\int_{\tfrac{1}{4} -{\ensuremath{\mathbf{i}}}\infty}^{\tfrac{1}{4}+{\ensuremath{\mathbf{i}}}\infty} \frac{ds_2}{2\pi {\ensuremath{\mathbf{i}}}} \frac{s_1-s_2}{s_1s_2} e^{\frac{s_1^3}{3}+s_1(r-x)} e^{\frac{s_2^3}{3} +s_2(r'-x)},\end{aligned}$$ and $\tilde K^{\infty}_{21}(r,r') = - \tilde K^{\infty}_{12}(r',r)$. Note that $\alpha$ is defined above. However, using the expansion to matrix elements for the Pfaffian, it becomes clear that the Pfaffian does not depend on the value of $\alpha$. It will be convenient (in order to match this expression to the known form of the GSE Tracy-Widom distribution) to define a final kernel $K^{\infty}$ in which $\alpha=-1$ and we have taken $r\mapsto -r$. This leads to the kernel $K^{\infty}$ in (\[Kkern\]) as well as the result recorded in (\[almostGSE\]).
### Recognizing the GSE Tracy-Widom distribution {#RHSdist}
The GSE Tracy-Widom distribution $F_{{\rm GSE}}(u)$ can be expressed via the following formula (e.g., page 124 of [@DeiftG]): $$F_{{\rm GSE}}(u) = \sqrt{\det(I-K^{(4)})_{L^2(u,\infty)}}$$ where the determinant above is given by the following infinite series $$\det(I-K^{(4)})_{L^2(u,\infty)} = 1+ \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_u^{\infty} dr_1\cdots \int_r^{\infty} dr_L \det\left(K^{(4)}(r_i,r_j)\right)_{i,j=1}^{L}$$ with $K^{(4)}(r,r')$ a $2\times 2$ matrix kernel with entries $$\begin{aligned}
K_{11}^{(4)} (r,r') &=& K_{22}^{(4)} (r',r) = \frac{1}{2} K_{\Ai}(r,r') - \frac{1}{4} \Ai(r)\, \int_{r'}^{\infty} \Ai(t) dt\\
K_{12}^{(4)} (r,r') &=& -\frac{1}{2} \frac{d}{dr'} K_{\Ai}(r,r') - \frac{1}{4} \Ai(r) \Ai(r')\\
K_{21}^{(4)} (r,r') &=& -\frac{1}{2} \int_{r}^{\infty} K_{\Ai}(t,r')dt + \frac{1}{4} \int_{r}^{\infty} \Ai(t) dt \, \int_{r'}^{\infty} \Ai(t)dt,\end{aligned}$$ and $K_{22}^{(4)} (r,r')= K_{11}^{(4)} (r',r)$. Here $K_{\Ai}$ is the Airy kernel $$K_{\Ai}(r,r') = \int_{0}^{\infty} dx \Ai(x+r)\Ai(x+r').$$ We note two facts which can be easily checked and which will be useful very soon. The first is that $$\label{kdiff}
\left(\frac{d}{dr}+\frac{d}{dr'}\right)K_{\Ai}(r,r') = -\Ai(r)\Ai(r'),$$ and the second is that the different elements of the kernel for $K^{(4)}$ are related via $$\label{k4rel}
K^{(4)}_{12}(r,r') = - \frac{d}{dr'} K^{(4)}_{11}(r,r'),\qquad K^{(4)}_{21}(r,r') = -\int_{r}^{\infty} K^{(4)}_{11}(t,r')dt.$$
Recall that if $M$ is a $2n\times 2n$ dimensional skew symmetric matrix, then $\sqrt{\det(M)} = \Pf(M)$. Letting $$J = \left(
\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}
\right)$$ and noting that $\det(J) =1$ and that $K^{(4)}J$ is skew symmetric, let us perform the following formal manipulations: $$F_{{\rm GSE}}(u)= \sqrt{\det(I-K^{(4)})_{L^2(u,\infty)}} = \sqrt{\det(IJ-K^{(4)}J)_{L^2(u,\infty)}} = \Pf(IJ-K^{(4)}J)_{L^2(u,\infty)}.$$ What is meant by this last expression is the following infinite series expansion $$1+ \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_u^{\infty}dr_1\cdots \int_u^{\infty}dr_L \Pf\left(K^{(4)}J(r_i,r_j) \right)_{i,j=1}^{L}.$$ Though the above manipulation was just formal, it is possible to prove that $$F_{{\rm GSE}}(u)=1+ \sum_{L=1}^{\infty} \frac{(-1)^L}{L!} \int_u^{\infty}dr_1\cdots \int_u^{\infty}dr_L \Pf\left(K^{(4)}J(r_i,r_j) \right)_{i,j=1}^{L}$$ by observing that the right-hand side is the inclusion exclusion formula for the distribution of the top particle of a point process with correlation functions given by the individual Pfaffian expressions. Since these correlation functions correspond with the edge scaling limit of the GSE (cf. [@Forrester Chapter 9.7]), we find that the right-hand side is exactly the limit distribution of the edge of the GSE – that is to say, it is $F_{{\rm GSE}}(u)$.
At this point, compare the above formula for $F_{{\rm GSE}}(u)$ to the formula derived in (\[almostGSE\]). We would like to show that the $2\times 2$ matrix kernel $K^{\infty}(r,r')$ in (\[almostGSE\]) exactly match the kernel $K^{(4)}J(r,r')$ above. In other words, we seek to check the following four equalities: $$\begin{aligned}
\label{4equals}
K^{\infty}_{11}(r,r') = -K^{(4)}_{12}(r,r'),\qquad K^{\infty}_{12}(r,r') = K^{(4)}_{11}(r,r'),\\
\nonumber K^{\infty}_{21}(r,r') = -K^{(4)}_{22}(r,r'), \qquad K^{\infty}_{22}(r,r') = K^{(4)}_{21}(r,r').\end{aligned}$$ The middle two are equivalent. We show these by explicitly checking the first equality, and then checking that the relations in (\[k4rel\]) hold for $K^{\infty}$ in the same way as for $K^{(4)}$.
From the integral representation of the Airy function given in (\[Airyintrep\]) we see that $$\begin{aligned}
K^{\infty}_{11}(r,r') &=& \frac{1}{4} \left(\frac{d}{dr'} - \frac{d}{dr}\right) K_{\Ai}(r,r')\\
&=& \frac{1}{4} \left(2\frac{d}{dr'} K_{\Ai}(r,r') + \Ai(r)\Ai(r')\right)\\
&=& -K^{(4)}_{12}(r,r').\end{aligned}$$ In going from the first line to the second line we utilized (\[kdiff\]) and the second to third line was by the definition of $K^{(4)}_{12}(r,r')$ and the symmetry of $K_{\Ai}(r,r')$. It is readily confirmed from (\[Kkern\]) that $$\frac{d}{dr'} K^{\infty}_{12}(r,r') = K^{\infty}_{11}(r,r'),\qquad -\int_r^{\infty} K^{\infty}_{12}(t,r')dt = K^{\infty}_{22}(r,r').$$ These are the same relations as those satisfied by the elements of $K^{(4)}$, cf. (\[k4rel\]). This implies[^11] that all of the equalities in (\[4equals\]) are satisfied and hence proves the desired equality of (\[almostGSE\]) with $F_{{\rm GSE}}(u)$.
Proofs of residue expansion results {#proofsec}
===================================
In this section we use residue calculus (briefly reviewed in Section \[resreview\] along with necessary notation for the proofs) to prove Proposition \[Akcprop\] and provide a partial proof of Conjecture \[Bkcprop\], modulo \[claim:Res-are-zero\]. A similar result as given in Proposition \[Akcprop\] was proved in [@BorCor Proposition 3.2.1], though by a considerably more involved inductive argument. The proofs given below are relatively elementary and in the full-space case (this type of proof was essentially given previously in [@BCPS1 Lemma 7.3]). A similar structure to the proof exists in both cases so we include the full-space proof mostly to help motivate the considerably harder half-space case. The proof also readily extends to a suitable $q$-deformed versions of the results (in fact, [@BorCor Proposition 3.2.1] and [@BCPS1 Lemma 7.3] were proved for the full-space case at this $q$-deformed level). To illustrate this, in the type $A$ (full-space) case we prove the $q$-deformed proposition and conclude via a limit as $q\to 1$ the desired Proposition \[Akcprop\]. In the type $BC$ (half-space) case, we prove Conjecture \[Bkcprop\], modulo Claim \[claim:Res-are-zero\], directly (though the $q$-deformed argument works quite similarly).
Residue calculus review and notation {#resreview}
------------------------------------
As almost all of the formulas within this paper deal with complex contour integrals, it is no surprise that the Cauchy and residue theorems play prominent roles in our calculations. A reader entirely unfamiliar with these tools of complex analysis is referred to [@Alhfors Chapters 2 and 4]. We briefly recall some consequences of these two theorems (so as to also fix notation for later applications).
For the present, all curves we consider in ${\ensuremath{\mathbb{C}}}$ are positively oriented, simple, smooth and closed (e.g. circles) or a infinite straight lines. In many of our applications we consider horizontal translates of the infinite contour ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ and in order to justify contour deformations as below it is necessary to bound the contribution of the integral near $\pm {\ensuremath{\mathbf{i}}}\infty$ as negligible and use Cauchy’s theorem on the finite approximation to the contours, before repairing them to be infinite (with a similar tail bound). As this is fairly standard, we do not dwell on it.
Given a deformation $D$ of a curve $\gamma$ to another curve $\gamma'$ and a function $f(z)$ which is analytic in a neighborhood the area swept out by the deformation $D$, Cauchy’s theorem implies that $$\int_{\gamma}\frac{dz}{2\pi {\ensuremath{\mathbf{i}}}} f(z) = \int_{\gamma'}\frac{dz}{2\pi {\ensuremath{\mathbf{i}}}} f(z).$$ This means that we can freely deform the contour $\gamma$ to $\gamma'$ without changing the value of the integral of $f(z)$, so long as no points of non-analyticity of $f(z)$ are crossed in the process.
For concreteness, let us now fix that $\gamma$ is a circle and $\gamma'$ is a second circle which is contained within $\gamma$. Consider a function $f(z)$ which is analytic in a neighborhood of the region between these circles, except at a finite collection of singularities (i.e. poles) $a_j$. The residue theorem along with the Cauchy theorem implies that $$\int_{\gamma}\frac{dz}{2\pi {\ensuremath{\mathbf{i}}}} f(z)= \int_{\gamma'}\frac{dz}{2\pi {\ensuremath{\mathbf{i}}}} f(z)+ \sum_j {\underset{{z=a_j}}{\mathbf{Res}}}\, f(z) ,$$ which is to say that we [*expand*]{} our integral into an integral on the smaller contour and residue terms from crossing the poles. Here ${\underset{{z=a}}{\mathbf{Res}}}f(z)$ is the residue of $f(z)$ at $a$. In general, this is defined as the unique complex number $R$ for which $f(z)-R/(z-a)$ is the derivative of a single valued analytic function in some annulus $0<|z-a|<\delta$. For a simple pole $a$, the residue can be computed by $${\underset{{z=a}}{\mathbf{Res}}}\, f(z) = \lim_{z\to a} (z-a) f(z).$$ In particular, if $f(z) = g(z)/(z-a)$ where $g(z)$ is analytic at $a$, we find that $$\label{ressubeqn}
{\underset{{z=a}}{\mathbf{Res}}}\, f(z) = {\underset{{z=a}}{\mathbf{Res}}}\, \frac{1}{z-a} \,\, {\underset{{z=a}}{\mathbf{Sub}}}\, g(z) = g(a),$$ where we have now also introduced the notation of [*substitution*]{} of a value into a function.
In our applications we will primarily be interested in functions of several complex variables $z_1,\ldots, z_k$. However, it will be easier to think of all but one of the variables of these functions as being fixed, and then apply the Cauchy / residue theorems to the remaining variable. In this way, we can justify various contour deformations as well as residue expansions. Let us consider the following:
\[ex1\] Fix $q\in (0,1)$ and consider the integral $$\int_{\gamma_1} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\gamma_2} \frac{dz_2}{2\pi {\ensuremath{\mathbf{i}}}} \,\frac{z_1-z_2}{z_1-qz_2} f(z_1,z_2)$$ where the $\gamma_2$ contour is along a small circle around $1$ and the $\gamma_1$ contour is another circle around $1$, with radius large enough so as to contain the image of $q$ times the $\gamma_2$ contour. Assume that for every $z_2\in \gamma_2$, the function $z_1\mapsto f(z_1,z_2)$ is analytic in the area between $\gamma_1$ and $\gamma_2$. Notice that for each fixed $z_2\in \gamma_2$ as we deform the $\gamma_1$ contour towards $\gamma_2$, the contour crosses a simple pole at $z_1= qz_2$ (coming from the denominator of the fraction in the integrand). To take this into account, we apply the residue theorem in the $z_1$ variable and find that $$\begin{aligned}
\int_{\gamma_1} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\gamma_2} \frac{dz_2}{2\pi {\ensuremath{\mathbf{i}}}} \frac{z_1-z_2}{z_1-qz_2} f(z_1,z_2) &=& \int_{\gamma_2} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\gamma_2} \frac{dz_2}{2\pi {\ensuremath{\mathbf{i}}}} \frac{z_1-z_2}{z_1-qz_2} f(z_1,z_2) + \int_{\gamma_2} \frac{dz_2}{2\pi {\ensuremath{\mathbf{i}}}} {\underset{{z_1=qz_2}}{\mathbf{Res}}} \left(\frac{z_1-z_2}{z_1-qz_2} f(z_1,z_2)\right)\\
&=& \int_{\gamma_2} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \int_{\gamma_2} \frac{dz_2}{2\pi {\ensuremath{\mathbf{i}}}} \frac{z_1-z_2}{z_1-qz_2} f(z_1,z_2) + \int_{\gamma_2} \frac{dz_2}{2\pi {\ensuremath{\mathbf{i}}}} (qz_2-z_2) f(qz_2,z_2).\end{aligned}$$
In what follows we will deal with slightly more involved residue calculations, so it is important to introduce reasonable notation. We will consider residues which occur according to multiplicative strings with parameter $q$ as well as additive strings with parameter $c$. Fix $k\geq 1$ and a partition $\lambda\vdash k$. Then, for a function $f(y_1,\ldots,y_k)$ we define ${\underset{{\lambda}}{\mathbf{Res}^q}}f(y_1,\ldots, y_k)$ to be the residue of $f(y_1,\ldots, y_k)$ at $$\begin{aligned}
\label{resqvalues}
\nonumber &y_{\lambda_{1}}=qy_{\lambda_{1}-1},\quad y_{\lambda_{1}-1}=qy_{\lambda_{1}-2}, \quad \ldots, \quad y_{2}=qy_{1}\\
&y_{\lambda_{1}+\lambda_2}=qy_{\lambda_1+\lambda_2-1},\quad y_{\lambda_{1}+\lambda_2-1}=qy_{\lambda_1+\lambda_2-2}, \quad \ldots, \quad y_{\lambda_{1}+2}=qy_{\lambda_1+1}\\
\nonumber &\vdots &\end{aligned}$$ with the output regarded as a function of the terminal variables $\big(y_1,y_{\lambda_1+1},\ldots, y_{\lambda_1+\cdots+\lambda_{\ell(\lambda)-1}-1}\big)$. We call each sequence of identifications of variables a [*string*]{}. As all poles which we encounter in what follows are simple, the above residue evaluation amounts to $$\label{amts}
{\underset{{\lambda}}{\mathbf{Res}^q}}f(y_1,\ldots, y_k) = \lim_{\substack{y_2\to q y_1\\ \cdots\\y_{\lambda_1}\to q y_{\lambda_1-1}}}\prod_{i=2}^{\lambda_1}(y_i-qy_{i-1}) \quad \lim_{\substack{y_{\lambda_1+2}\to q y_{\lambda_1+1}\\ \cdots \\ y_{\lambda_1+\lambda_2}\to q y_{\lambda_1+\lambda_2-1}}}\prod_{i=\lambda_1+2}^{\lambda_1+\lambda_2}(y_i-qy_{i-1})\quad \cdots \,\, f(y_1,\ldots, y_k).$$
It is also convenient to define ${\underset{{\lambda}}{\mathbf{Sub}^q}}f(y_1,\ldots, y_k)$ as the function of $\big(y_1,y_{\lambda_1+1},\ldots, y_{\lambda_1+\cdots+\lambda_{\ell(\lambda)-1}-1}\big)$, which is the result of substituting the relations of (\[resqvalues\]) into $f(y_1,\ldots, y_k)$.
Under the change of variables $q\mapsto e^{-{\epsilon}c}$ and $y\mapsto e^{-{\epsilon}y}$, as ${\epsilon}\to 0$ the above relations in (\[resqvalues\]) become $$\begin{aligned}
\label{rescvalues}
\nonumber &y_{\lambda_1} = y_{\lambda_1 - 1} +c, \quad y_{\lambda_1-1} = y_{\lambda_1 - 2} +c, \quad \ldots, \quad y_{2} = y_{1} +c &\\
&y_{\lambda_1+\lambda_2} = y_{\lambda_1+\lambda_2 - 1} +c, \quad y_{\lambda_1+\lambda_2-1} = y_{\lambda_1+\lambda_2 - 2} +c ,\quad \ldots, \quad y_{\lambda_1+2} = y_{\lambda_1+1} +c&\\
\nonumber &\vdots &\end{aligned}$$ In the same way as for the $q$ case, we define ${\underset{{\lambda}}{\mathbf{Res}^c}}$ and ${\underset{{\lambda}}{\mathbf{Sub}^c}}$ as the residue and substitution operators with respect to the above strings of relations, which take a function $f(y_1,\ldots, y_k)$ and output a function of $\big(y_1,y_{\lambda_1+1},\ldots, y_{\lambda_1+\cdots+\lambda_{\ell(\lambda)-1}-1}\big)$.
Proof of Proposition \[Akcprop\] (type $A$ expansion) {#akproof}
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Proposition \[Akcprop\] can be proved by either taking a limit as ${\epsilon}\to 0$ of Proposition \[321\] (stated and proved below) with the change of variables $q\mapsto e^{-{\epsilon}c}$, $z\mapsto e^{-{\epsilon}z}$, $w\mapsto e^{-{\epsilon}w}$, or it can be proved directly by mimicking the proof of Proposition \[321\]. The two hypotheses of Proposition \[321\] play analogous roles to those of Proposition \[Akcprop\]. Additionally, the decay bound of the second hypothesis of Proposition \[Akcprop\] is necessary because contours are infinite, and in order to apply the Cauchy / residue theorems it is necessary to have suitable decay near infinity. Otherwise, the proof goes through without any significant changes.
Thus, in this section we state and prove the above mentioned $q$-deformed analog of Proposition \[Akcprop\]. The following proposition is a generalization of Proposition 3.2.1 of [@BorCor], however the proof which we present here is considerably simpler than the inductive one presented therein. This proof also exposes the generality of the result as stated below.
\[321\] Fix $k\geq 1$ and $q\in (0,1)$. Given a set of positively oriented, closed contours $\gamma_1,\ldots,\gamma_k$ and a function $F(z_1,\ldots, z_k)$ which satisfy
1. For all $1\leq A<B\leq k$, the interior of $\gamma_A$ contains the image of $\gamma_B$ multiplied by $q$;
2. For all $1\leq j\leq k$, there exist deformations $D_j$ of $\gamma_j$ to $\gamma_k$ so that for all $z_1,\ldots, z_{j-1},z_j,\ldots, z_k$ such with $z_i\in \gamma_i$ for $1\leq i<j$, and $z_i\in \gamma_k$ for $j<i\leq k$, the function $z_j\mapsto F(z_1,\ldots ,z_j,\ldots, z_k)$ is analytic in a neighborhood of the area swept out by the deformation $D_j$.
Then we have the following residue expansion identity: $$\begin{aligned}
\label{qAK}
(-1)^k q^{\frac{k(k-1)}{2}} \oint_{\gamma_1} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \oint_{\gamma_k} \frac{dz_k}{2\pi {\ensuremath{\mathbf{i}}}} \prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-q z_B} F(z_1,\ldots, z_k) \qquad\qquad\qquad\qquad\qquad\qquad&\\
\nonumber = \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{(1-q)^{k}}{m_1! m_2!\cdots} \oint_{\gamma_k} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}}\cdots \oint_{\gamma_k} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}} \det\left[\frac{1}{w_i q^{\lambda_i} -w_j}\right]_{i,j=1}^{\ell(\lambda)}\qquad\qquad\qquad\qquad&\\
\nonumber \times\, \prod_{j=1}^{\ell(\lambda)} w_j^{\lambda_j} q^{\frac{\lambda_j(\lambda_j-1)}{2}} \, E^q(w_1,qw_1,\ldots, q^{\lambda_1-1}w_1,\ldots, w_{\lambda_{\ell(\lambda)}},q w_{\lambda_{\ell(\lambda)}},\ldots, q^{\lambda_{\ell(\lambda)}-1} w_{\lambda_{\ell(\lambda)}}),&\end{aligned}$$ where $$E^q(z_1,\ldots, z_k) = \sum_{\sigma\in S_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-qz_{\sigma(B)}}{z_{\sigma(A)}-z_{\sigma(B)}} F(\sigma(\vec{z})).$$
The meaning of the various terms and notation used below can be found in Section \[resreview\]. First notice that for $k=1$ the result follows immediately. Hence, in what follows we assume $k\geq 2$.
We proceed sequentially and deform (using the deformation $D_{k-1}$ afforded from the hypotheses of the theorem) the $\gamma_{k-1}$ contour to $\gamma_k$, and then deform (using the deformation $D_{k-2}$ afforded from the hypotheses of the theorem) $\gamma_{k-2}$ contour to $\gamma_k$, and so on until all contours have been deformed to $\gamma_k$. However, due to the $z_A-qz_B$ terms in the denominator of the integrand, during the deformation of $\gamma_A$ we may encounter simple poles at the points $z_A=q z_B$, for $B>A$. The residue theorem (cf. Section \[resreview\]) implies that the integral on the left-hand side of (\[qAK\]) can be expanded into a summation over integrals of possibly few variables (all along $\gamma_k$) whose integrands correspond to the various possible residue subspaces coming from these poles.
Our proof splits into three basic steps. First, we identify the residual subspaces upon which our integral is expanded via residues. This brings us to equation (\[RHS2\]). Second, we show that these subspaces can be brought to a canonical form via the action of some $\sigma\in S_k$. This enables us to simplify the summation over the residual subspaces to a summation over partitions $\lambda\vdash k$ and certain subsets of $\sigma\in S_k$. Inspection of those terms corresponding to $\sigma\in S_k$ not in these subsets shows that they have zero residue contribution and hence the summation can be completed to all of $S_k$. This brings us to equation (\[above3\]). And third, we rewrite the function whose residue we are computing as the product of an $S_k$ invariant function (which contains all of the poles related to the residual subspace) and a remainder function. We use Lemma \[reslemma\] to evaluate the residue of the $S_k$ invariant function and we identify the summation over $\sigma\in S_k$ of the substitution into the remainder function as exactly giving the $E^q$ function in the statement of the proposition we are presently proving.
[**Step 1:**]{} It is worthwhile to start with an example. The case $k=2$ is effectively worked out in Example \[ex1\] so we start with $k=3$. Figure \[qcontourshifting\] accompanies the example and illustrates the deformations and locations of poles.
![The expansion of the $k=3$ nested contour integral as explained in Example \[ex2\]. On the left-hand side, the $\gamma_2$ contour is deformed to the $\gamma_3$ contour and a pole is crossed along $q\gamma_3$ at the point $qz_3$ (here the point $z_3$ is drawn as a small filled disk and its location is on the solid line circle; and the point $q z_3$ is drawn as a small unfilled disk and its location is on the dotted line circle). On the upper right-hand side the effect of picking the integral is shown. The $\gamma_3$ contour is represented as a (doubled) solid line circle since both $z_3$ and $z_2$ are integrated along it (these correspond to the two filled disks). The unfilled disks are along $q\gamma_3$ (the dotted line circle) and represent $qz_3$ and $qz_2$. As the $\gamma_1$ contour is deformed to $\gamma_3$ these residues must be taken. On the lower right-hand side the effect of picking the initial residue at $z_2=qz_3$ is shown. As the $\gamma_1$ contour is deformed to $\gamma_3$ a pole is encountered along $q^2\gamma_3$ at the location $q^2 z_3$ (as before $z_3$ is the filled disk and $q^2 z_3$ is the unfilled disk).[]{data-label="qcontourshifting"}](qcontourshifting.eps)
\[ex2\] When $k=3$ the integrand on the left-hand side of (\[qAK\]) contains the fractions $$\label{cross}
\frac{z_1-z_2}{z_1-qz_2}\, \frac{z_1-z_3}{z_1-qz_3}\,\frac{z_2-z_3}{z_2-qz_3},$$ times the function $F(z_1,z_2,z_3)$ which (by the hypotheses of the proposition) does not have poles between $\gamma_j$ and $\gamma_k$ (for $j=1,2,3$). Thinking of $z_3$ as fixed along the contour $\gamma_3$, we begin by deforming the $\gamma_2$ contour to $\gamma_3$. As we proceed, we necessarily encounter a single simple pole at $z_2=qz_3$. The residue theorem implies that our initial integral equals the sum of (A) the integral where $z_2$ is along $\gamma_3$, and (B) the integral with only $z_1$ and $z_3$ remaining and integrand given by taking the residue of the initial integrand at $z_2=qz_3$.
Let us consider separately these two pieces. For (A) we now think of $z_2$ and $z_3$ as fixed along $\gamma_3$ and deform the $\gamma_1$ contour to $\gamma_3$, encountering two simple poles at $z_1=qz_2$ and $z_1=q z_3$. Thus (A) is expanded into a sum of three terms: the integral with $z_1, z_2$ and $z_3$ along $\gamma_3$; and the integral with only $z_2$ and $z_3$ remaining (along $\gamma_3$) and the residue taken at either $z_1=q z_2$ or $z_1=q z_3$.
For (B) we now think of $z_3$ as fixed along $\gamma_3$ and deform the $\gamma_1$ contour to $\gamma_3$, encountering a simple pole at $z_1=q^2 z_3$. This is because the residue of (\[cross\]) at $z_2=q z_3$ equals $$\frac{z_1-z_3}{z_1-q^2 z_3}\,(qz_3-z_3).$$ Thus (B) is expanded into a sum of two terms: the integral with $z_1$ and $z_3$ (along $\gamma_3$ and with the above expression in the integrand); and the integral with only $z_3$ remaining (along $\gamma_3$) and the residue of the above term taken at $z_1=q^2 z_3$.
Gathering the various terms in this expansion, we see that the residue subspaces we sum over are indexed by partitions of $k$ (here $k=3$) and take the form of geometric strings with the parameter $q$. For example, $\lambda = (1,1,1)$ corresponds to the term in the residue expansion in which all three variables $z_1,z_2$ and $z_3$ are still integrated, but along $\gamma_3$. On the other hand, $\lambda = (3)$ corresponds to the term in which the residue is taken at $z_1=qz_2=q^2z_3$ and the only variable which remains to be integrated along $\gamma_3$ is $z_3$. The partition $\lambda=(2,1)$ corresponds to the three remaining terms in the above expansion in which two integration variables remain. In general, $\ell(\lambda)$ corresponds to the number of variables which remain to be integrated in each term of the expansion.
Let us now turn to the general $k$ case. By the hypotheses of the theorem, the function $F$ has no poles which are encountered during contour deformations – hence it plays no role in the residue analysis. As we deform sequentially the contours in the left-hand side of (\[qAK\]) to $\gamma_k$ we find that the resulting terms in the residue expansion can be indexed by partitions $\lambda\vdash k$ along with a list (ordered set) of disjoint ordered subsets of $\{1,\ldots, k\}$ (whose union is all of $\{1,\ldots, k\}$) $$\begin{aligned}
\label{ijvar}
\nonumber &i_{1}< i_{2} < \cdots < i_{\lambda_1}&\\
&j_{1}< j_{2} < \cdots < j_{\lambda_2}&\\
\nonumber & \vdots&
$$ Let us call such a list $I$. For a given partition $\lambda$ call $S(\lambda)$ the collection of all such lists $I$ corresponding to $\lambda$. For $k=3$ and $\lambda=(2,1)$, in Example \[ex2\] we saw there are three such lists which correspond with $$S(\lambda) = \Big\{ \big\{1<2 , 3\big\}, \big\{1<3 ,2\big\}, \big\{2<3 , 1\big\}\Big\}.$$
For such a lists $I$, we write ${\underset{{I}}{\mathbf{Res}}} f(z_1,\ldots, z_k)$ as the residue of the function $f$ at $$\begin{aligned}
&z_{i_{1}}=qz_{i_{2}}, z_{i_2} = q z_{i_3}, \quad \ldots, \quad z_{i_{\lambda_1-1}}=q z_{i_{\lambda_1}}&\\
&z_{j_{1}}=qz_{j_{2}}, z_{j_2} = q z_{j_3}, \quad \ldots, \quad z_{j_{\lambda_2-1}}=q z_{j_{\lambda_2}}&\\
&\vdots&
$$ and regard the output as a function of the terminal variables $(z_{i_{\lambda_1}}, z_{j_{\lambda_2}}, \ldots)$. There are $\ell(\lambda)$ such strings and consequently that many remaining variables (though we have only written the first two strings above).
With the above notation in place, we may write the expansion of the integral on the left-hand side of (\[qAK\]) as $$\begin{aligned}
\label{RHS2}
{\rm LHS} (\ref{qAK}) = (-1)^k q^{\frac{k(k-1)}{2}} \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{1}{m_1! m_2!\cdots} \sum_{I\in S(\lambda)}\oint_{\gamma_k} \frac{dz_{i_{\lambda_1}}}{2\pi {\ensuremath{\mathbf{i}}}} \oint_{\gamma_k} \frac{dz_{j_{\lambda_{2}}}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \\
\nonumber \times\,{\underset{{I}}{\mathbf{Res}}} \left(\prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-q z_B} F(\vec{z}) \right).\end{aligned}$$ The factor of $\frac{1}{m_1! m_2!\cdots}$ arose from multiple counting of terms in the residue expansion due to symmetries of $\lambda$. For example, for the partition $\lambda=(2,2,1)$, each $I\in S(\lambda)$, corresponds uniquely with a different $I'\in S(\lambda)$ in which the $i$ and $j$ variables in (\[ijvar\]) are switched. Since these correspond with the same term in the residue expansion, this constitutes double counting and hence the sum should be divided by $2!$. The reason why our residual subspace expansion only corresponds with strings is because if we took a residue which was not of the form of a string, then for some $A\neq A'$ we would be evaluating the residue at $z_A=qz_B$ and $z_{A'}=qz_B$. However, the Vandermonde determinant in the numerator of our integrand would then necessarily evaluate to zero. Therefore, such possible non-string residues (coming from the denominator) in fact have zero contribution.
[**Step 2:**]{} For each $I\in S(\lambda)$ relabel the $z$ variables as $$\begin{aligned}
&(z_{i_{1}}, z_{i_{2}}, \ldots, z_{i_{\lambda_1}}) \mapsto (y_{\lambda_1},y_{\lambda_1-1},\ldots, y_1)&\\
&(z_{j_{1}}, z_{j_{2}}, \ldots, z_{j_{\lambda_2}}) \mapsto (y_{\lambda_1+\lambda_2}, \ldots, y_{\lambda_1+1})&\\
&\vdots&
$$ and observe that there exists a unique permutation $\sigma\in S_k$ for which $(z_1,\ldots, z_k) = (y_{\sigma(1)},\ldots, y_{\sigma(k)})$. Let us also call $w_j = y_{\lambda_1+\cdots \lambda_{j-1}+1}$, for $1\leq j\leq \ell(\lambda)$.
We rewrite the term corresponding to a partition $\lambda$ in the right-hand side of (\[RHS2\]) as $$\label{above3}
\frac{1}{m_1! m_2!\cdots} \sum_{\sigma\in S_k} \oint_{\gamma_k} \frac{dw_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \oint_{\gamma_k} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}}
{\underset{{\lambda}}{\mathbf{Res}^q}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-q y_{\sigma(B)}} F(\sigma(\vec{y})) \right),$$ where ${\underset{{\lambda}}{\mathbf{Res}^q}}$ is defined in equation (\[resqvalues\]). Note that the output of the residue operation is a function of the variables $(y_1,y_{\lambda_1+1},\ldots) = (w_1,w_2,\ldots)$.
One should observe that the above expression includes the summation over all $\sigma\in S_k$, and not just those which arise from an $I\in S(\lambda)$ as above. This, however, is explained by the fact that if $\sigma$ does not arise from some $I\in S(\lambda)$, then the residue necessarily evaluates to zero (hence adding these terms is allowed). To see this, observe that in the renumbering of variables discussed above, only permutations with $$\sigma^{-1}(1)>\sigma^{-1}(2)\cdots>\sigma^{-1}(\lambda_1), \qquad \sigma^{-1}(\lambda_1+1)>\sigma^{-1}(\lambda_1+2)\cdots>\sigma^{-1}(\lambda_1+\lambda_2),\qquad \ldots$$ participated. Any other $\sigma$ must violate one of these strings of conditions. Consider, for example, some $\sigma$ with $\sigma(\lambda_1-1) < \sigma(\lambda_1)$. This implies that the term $y_{\lambda_1-1}-qy_{\lambda_1}$ shows up in the denominator of (\[above3\]), as opposed to the term $y_{\lambda_1}-qy_{\lambda_1-1}$. Residues can be taken in any order, and if we first take the residue at $y_{\lambda_1} = qy_{\lambda_1-1}$, we find that the above denominator does not have a pole (nor do any other parts of (\[above3\])) and hence the residue is zero. Similar reasoning works in general.
[**Step 3:**]{} All that remains is to compute the residues in (\[above3\]) and identify the result (after summing over all $\lambda\vdash k$) with the right-hand side of (\[qAK\]) as necessary to prove the theorem.
It is convenient to rewrite the following expression as an $S_k$ invariant function, times a function which is analytic at the points in which the residue is being taken: $$\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-q y_{\sigma(B)}} = \prod_{1\leq A\neq B\leq k} \frac{y_A-y_B}{y_A-qy_B} \prod_{1\leq B<A\leq k} \frac{y_{\sigma(A)} - qy_{\sigma(B)}}{y_{\sigma(A)} - y_{\sigma(B)}}.$$ Since it is only the $S_k$ invariant function which contains the poles with which we are concerned, it allows us to rewrite (\[above3\]) as $$\begin{aligned}
\frac{1}{m_1! m_2!\cdots} \oint \frac{w_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \oint \frac{w_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}} {\underset{{\lambda}}{\mathbf{Res}^q}} \left( \prod_{1\leq A\neq B\leq k} \frac{y_A-y_B}{y_A-qy_B} \right)\\
\nonumber \times\,{\underset{{\lambda}}{\mathbf{Sub}^q}} \left( \sum_{\sigma\in S_k} \prod_{1\leq B<A\leq k} \frac{y_{\sigma(A)} - qy_{\sigma(B)}}{y_{\sigma(A)} - y_{\sigma(B)}} F(\sigma(\vec{y})) \right),\end{aligned}$$ where ${\underset{{\lambda}}{\mathbf{Sub}^q}}$ is the substitution operator defined in Section \[subsec\]. We use Lemma \[reslemma\] to evaluate the above residue, and we easily identify the substitution on the second line with $E^q(w_1,q w_1,\ldots)$ as in the statement of the proposition. Combining these two expressions and summing the resulting expression over $\lambda\vdash k$ we arrive at the desired residue expansion claimed in the statement of the proposition.
Proof of Conjecture \[Bkcprop\] (type $BC$ expansion) modulo Claim \[claim:Res-are-zero\] {#bcproof}
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We turn now to Conjecture \[Bkcprop\]. Note that when $k=1$ the proof follows immediately, hence we assume $k\geq 2$.
The argument follows a similar line to that of Proposition \[321\]. The type $BC$ symmetry makes the problem more difficult and we must assume at some point a non-trivial residue cancelation in the form of Claim \[claim:Res-are-zero\]. We do not prove this claim presently, though give a proof of a special case of this in Section \[sec:one-string\]. It would, therefore, be advised that the reader first reviews the proof of Proposition \[321\], before studying the below argument.
We proceed sequentially and shift the $\alpha_{k-1}+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ contour to $\alpha_k+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ which, since $\alpha_k=0$, is just ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. Then we repeat this procedure for $\alpha_{k-2}$ through $\alpha_1$. Despite these contours being of infinite length, the deformations are easily justified by virtue of the decay to zero of the function $F$ ensured by the hypotheses. We do, however, need to take care in keeping track of the residue contributions, which result from the simple poles when $z_A \pm z_B -c =0$. The residue theorem (see Section \[resreview\]) implies that the integral on the left-hand side of (\[mainform\]) can be expanded into a summation over integrals of possibly few variables (all along ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$) whose integrands correspond to the various possible residue subspaces coming from these poles.
Our proof splits into the same three steps as the proof of Proposition \[321\], but with the role of $S_k$ replaced by $BC_k$ and with a more complicated collection of residual subspaces. In the first step, equation (\[RHS2\]) from the proof of Proposition \[321\] is replaced by equation (\[firstclaim\]); in the second step equation (\[above3\]) is replaced by equation (\[ref31prime\]) – showing this we appeal to Claim \[claim:Res-are-zero\]; and in the third step, Lemma \[reslemma\] is replaced by Lemma \[fourthlemma\] and the $E^q$ function is replaced by $E^c$ as defined in the statement of Conjecture \[Bkcprop\].
[**Step 1:**]{} We start with the example of $k=3$ so as to introduce the basic idea and notation for counting these subspaces. Figure \[contourshifting\] accompanies the example and illustrates the deformations and locations of poles.
![The expansion of the $k=3$ nested contour integral as explained in Example \[ex3\]. Recall that $\alpha_3=0$. On the left-hand side, the contour $\alpha_2+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ is deformed to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ and two poles are crossed at $z_2=\pm z_3+c$ (here the point $z_3$ is drawn as a small filled disk and its location is on the solid line; the points $\pm z_3+c$ are drawn as small unfilled disks and their locations are on the dotted line). On the upper right-hand side the effect of picking the integral is shown (this is case (A) in the example). The ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ contour on which $z_2$ and $z_3$ are integrated is represented as a doubled solid line and $z_2$ and $z_3$ are represented by the two filled disks. The four unfilled disks are along $c+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ and represent $\pm z_3+c $ and $\pm z_2+c$. As the $z_1$ contour is deformed from $\alpha_1+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ these residues must be taken. On the lower right-hand side the effect of picking the initial residues at $z_2=z_3+c$ and then at $z_2=-z_3+c$ are shown (these are cases (B) and (C) from the example, respectively). In case (B) as the $z_1$ contour is deformed, poles are encountered at $z_1= z_3 + 2c$ and $z_1=-z_3 +c$, whereas in case (C) poles are encountered at $z_1=-z_3+2c$ and $z_1=z_3+c$.[]{data-label="contourshifting"}](contourshifting.eps)
\[ex3\] For $k=3$, the right-hand side of (\[mainform\]) contains the fractions $$\label{asabovefrac}
\frac{z_1-z_2}{z_1-z_2-c}\, \frac{z_1+z_2}{z_1+z_2-c}\, \frac{z_1-z_3}{z_1-z_3-c}\, \frac{z_1+z_3}{z_1+z_3-c}\, \frac{z_2-z_3}{z_2-z_3-c}\, \frac{z_2+z_3}{z_2+z_3-c},$$ times the function $F(z_1,z_2,z_3)$ which (by the hypotheses of the proposition) does not have poles in the area in which we are deforming contours. Thinking of $z_3$ as fixed along ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, we begin by deforming the $z_2$ contour, $\alpha_2+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. As we proceed, we necessarily encounter two simple poles at $z_2 = \pm z_3+c$. The residue theorem implies that our initial integral equals the sum of (A) the integral where $z_2$ is along ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, (B) the integral with only $z_1$ and $z_3$ remaining and integrand given by taking the residue of the initial integrand at $z_2=z_3+c$, and (C) the integral with only $z_1$ and $z_3$ remaining and integrand given by taking the residue of the initial integrand at $z_2=-z_3+c$.
Let us consider separately these three pieces. For (A) the fraction in the integrand remains the same as above in (\[asabovefrac\]). Thus, as we deform the $z_1$ contour, $\alpha_1+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, we encounter four simple poles at $z_1= \pm z_2 + c$ and $z_1 = \pm z_3+c$. Thus (A) is expanded into five terms (one when none of the residues are picked, and four for each of the residue contributions).
For (B) the fraction in the resulting integrand (after taking the residue at $z_2=z_3+c$ and canceling terms) is given by $$c\,\frac{z_1-z_3}{z_1-z_3-2c}\, \frac{z_1+z_3+c}{z_1+z_3-c}\,\frac{2z_3+c}{2z_3}.$$ Thus, as we deform the $z_1$ contour, $\alpha_1+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, we encounter two simple poles at $z_1= z_3+2c$ and $z_1 = -z_3+c$. Since one should think of $z_2=z_3+c$, the first pole is really at $z_1=z_2+c$. Thus (B) is expanded into three terms.
For (C) the fraction in the resulting integrand (after taking the residue at $z_2=-z_3+c$ and canceling terms) is given by $$c\,\frac{z_1+z_3}{z_1+z_3-2c}\, \frac{z_1-z_3+c}{z_1-z_3-c}\,\frac{-2z_3+c}{-2z_3}.$$ Thus, as we deform the $z_1$ contour, $\alpha_1+{\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, we encounter two simple poles at $z_1= z_3+c$ and $z_1 = -z_3+2c$. Since one should think of $z_2=-z_3+c$, the second pole is really at $z_1=z_2+c$. Thus (C) is expanded into three terms.
We will be well served by introducing good notation to keep track of the various terms in such a residue expansion. Let us give the example of the notation for the $k=3$ case considered above, and then explain what it means in terms of residues and integrals.
The five terms in (A) correspond with the following diagrams $$\substack{\displaystyle 1\\ \\ \displaystyle 2\\ \\ \displaystyle 3} \qquad\qquad
\substack{\displaystyle 1\rightp 2 \\ \\ \displaystyle 3\qquad\,\,\,\,\\ \,\\}\qquad\qquad
\substack{\displaystyle 1\rightm 2 \\ \\ \displaystyle 3\qquad\,\,\,\,\\ \,\\}\qquad\qquad
\substack{\displaystyle 1\rightp 3 \\ \\ \displaystyle 2\qquad\,\,\,\,\\ \,\\}\qquad\qquad
\substack{\displaystyle 1\rightm 3 \\ \\ \displaystyle 2\qquad\,\,\,\,\\ \,\\},$$ while the three terms in (B) correspond with the following diagrams $$\substack{\displaystyle 2\rightp 3 \\ \\ \displaystyle 1\qquad\,\,\,\,\\ }\qquad\qquad
\substack{\displaystyle 1\rightp 2 \rightp 3 \\ \\ \\ }\qquad\qquad
\substack{\displaystyle 1\rightm 3 \leftp 2 \\ \\ \\ },$$ while the three terms in (C) correspond with the following diagrams $$\substack{\displaystyle 2\rightm 3 \\ \\ \displaystyle 1\qquad\,\,\,\,\\ }\qquad\qquad
\substack{\displaystyle 1\rightp 2 \rightm 3 \\ \\ \\ }\qquad\qquad
\substack{\displaystyle 1\rightp 3 \leftm 2 \\ \\ \\ }.$$
In the above notation, each separate line of a diagram represents a single integral (over the $z_j$ variable where $j$ is the largest number in the line) and each directed arrow from $i$ to $j$ (with $i<j$) represents a residue taken at $z_i =\pm z_j +c$, where the choice of plus or minus is indicated by the label above the arrow. For instance, when $k=4$, the term corresponding with the diagram $$\substack{\displaystyle 2\rightm 3\leftp 1 \\ \\ \displaystyle 4\,\,\,\,\,\,\,\qquad\qquad\\ }$$ in the residue expansion of the left-hand side of (\[mainform\]) is $$\int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_3}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_4}{2\pi {\ensuremath{\mathbf{i}}}}\, {\underset{{\substack{z_2 = -z_3+c \\ z_1=z_3+c}}}{\mathbf{Res}}} \left(\prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c} \,\frac{z_A+z_B}{z_A+z_B-c} F(\vec{z})\right).$$
Let us now turn to the general $k$ case. Using the notation introduced in the above example, let us consider a general set of diagrams. Since it will be useful in proving the residue expansion formula which follows, consider $p\in \{0,\ldots, k\}$. (This number will correspond to the number of contours that have not been deformed to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ yet.) Then for each partition $\lambda\vdash k-p$ we consider diagrams of the form $$\substack{\displaystyle i_1\rightp i_2\rightp \cdots \rightp i_{\mu_1-1}\rightpm i_{\mu_1} \leftmp i_{\mu_1+1}\leftp \cdots \leftp i_{\lambda_1}\\ \\
\displaystyle j_1\rightp j_2\rightp \cdots \rightp j_{\mu_2-1}\rightpm j_{\mu_2} \leftmp j_{\mu_2+1}\leftp \cdots \leftp j_{\lambda_2}\\ \\ \displaystyle \vdots}$$ where the following conditions hold: The numbers on each line are pairwise disjoint and the union of the numbers over all lines equals the set $\{p+1,\ldots ,k\}$ (when $p=k$ this is the empty set); the arrows change from pointing right to pointing left only once (call the number at which this change occurs $i_{\mu_1},j_{\mu_2},\ldots$); the directed arrows are always from smaller numbers to larger numbers; within each line all arrows have $+$ above them except for the two (or possibly one if all arrows point in the same direction) surrounding $i_{\mu_1},j_{\mu_2},\ldots$ for which one sign is $+$ and the other is $-$ (this choice may change line to line). We write $S(\lambda)$ for the collection of all such diagrams for the partition $\lambda$, and we write $I$ for one such diagram. For such a list $I$ as above, we write ${\underset{{I}}{\mathbf{Res}}} f(z_1,\ldots,z_k)$ as the residue of the function $f$ at $$\substack{\displaystyle z_{i_1}= z_{i_2}+c, z_{i_2} = z_{i_3} +c, \ldots, z_{i_{\mu_1}-1} = \pm z_{i_{\mu_1}} +c,\qquad z_{i_{\lambda_1}} = \mp z_{i_{\lambda_1}-1} +c, \ldots z_{i_{\mu_1}+1} = z_{i_{\mu_1}} +c \\ \\
\displaystyle z_{j_1}= z_{j_2}+c, z_{j_2} = z_{j_3} +c, \ldots, z_{j_{\mu_2}-1} = \pm z_{j_{\mu_2}} +c,\qquad z_{j_{\lambda_2}} = \mp z_{j_{\lambda_2}-1} +c, \ldots z_{j_{\mu_2}+1} = z_{j_{\mu_2}} +c}$$ $$\label{replacements}
\vdots$$ and regard the output as a function of the variables $z_1,\ldots, z_p, z_{i_{\mu_1}},z_{j_{\mu_2}},\ldots$. Note that the $\pm$ choices for each line above correspond with the choices in the diagram $I$, as do the strings of equalities. We will thus identify a diagram $I$ with the string of equalities given above.
We claim that for any $p\in \{0,\ldots, k\}$ $$\begin{aligned}
\label{firstclaim}
{\rm LHS} (\ref{mainform}) &=& \sum_{\substack{\lambda\vdash k-p\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{1}{m_1! m_2!\cdots} \frac{1}{2^{m_2+m_3+\cdots}} \sum_{I\in S(\lambda)} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_{i_{\mu_1}}}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_{j_{\mu_2}}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \\
\nonumber &&\times\, \int_{\alpha_1-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_1+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_1}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{\alpha_p-{\ensuremath{\mathbf{i}}}\infty}^{\alpha_p+{\ensuremath{\mathbf{i}}}\infty} \frac{dz_p}{2\pi {\ensuremath{\mathbf{i}}}}\, {\underset{{I}}{\mathbf{Res}}} \left(\prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c}\frac{z_A+z_B}{z_A+z_B-c} F(\vec{z}) \right).\end{aligned}$$ Notice that when $p=k$ the partition $\lambda$ is empty and hence this reduces back to the exact expression on the left-hand side of (\[mainform\]). To see that this holds for general $p$ we proceed inductively (decreasing $p$). As we deform the $z_m$ contour to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$ we should consider whether there are poles at any of the points of the form $z_m = \pm z_B +c$ for $B>p$. We claim that the only points for which there are actually poles are those which correspond to augmenting the diagram $I$ into another diagram $I'$ (with $p$ replaced by $p-1$) by inserting $p$ on the left or right of any line of $I$, or into a new line. Any arrow direction and sign which leads to $I'$ satisfying the conditions listed earlier may arise. This fact follows from the Vandermonde determinant factor in the numerator, which necessarily evaluates to zero for any other augmentation of $I$ to $I'$. Another way to see this is to observe that diagrams which contain the following two snippets have zero residue: $$A\rightm B\rightpm C \qquad C\leftpm B\leftm A \qquad A\rightpm B\leftpm C.$$ Any other augmentation than described above would necessary include one of the subdiagrams.
There are two symmetries which lead to the combinatorial factors above. The first (which also was present in the proof of Proposition \[321\]) has to do with the symmetry of the parts of $\lambda$. In particular, there are $m_1!m_2!\cdots$ permutations which interchange lines in $I$ with the same length. The other symmetry is new to the type $BC$ diagrams $I$. For any line of $I$ which has at least one arrow (i.e. correspond with $\lambda_i\geq 2$) it is possible to reflect around the number $\mu_i$ and arrive at a new diagram $I'$. However, the residual subspace associated with $I$ and $I'$ is the same. Hence, for all $i$ with $\lambda_i$ we have a multiplicity of $2$. As there are $m_2+m_3+\cdots$ such $i$, we find the extra factor $2^{m_2+m_3+\cdots}$. Combining these two considerations yields the formula above. [**Step 2:**]{} Now focus on the above formula with $p=0$. We claim that for any $\lambda=1^{m_1}2^{m_2}\cdots \vdash k$ $$\begin{aligned}
\label{ref31prime}
&&\sum_{I\in S(\lambda)} \, \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_{i_{\mu_1}}}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_{j_{\mu_2}}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots {\underset{{I}}{\mathbf{Res}}} \left(\prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c}\frac{z_A+z_B}{z_A+z_B-c} F(\vec{z})) \right) \\
\nonumber &&=\frac{1}{2^{m_1}} \sum_{\sigma\in BC_k} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{1}}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{2}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c} F(\sigma(\vec{y})) \right),\end{aligned}$$ where $w_i = y_{\lambda_1+\cdots +\lambda_{i-1}+1}$. This can be seen in two pieces (in the first piece we appeal to the as of yet unproved Claim \[claim:Res-are-zero\]). First, we consider $I\in S(\lambda)$ and replace the $z$ variables as follows. Start with the first line of $I$ (the other lines work similarly). There are two possible forms that this first line may take. The first form is $$i_1\rightp i_2\rightp \cdots \rightp i_{\mu_1-1}\rightp i_{\mu_1} \leftm i_{\mu_1+1}\leftp \cdots \leftp i_{\lambda_1},$$ where the arrows neighboring $i_{\mu_1}$ have (respectively) plus and then minus signs. The second form is $$i_1\rightp i_2\rightp \cdots \rightp i_{\mu_1-1}\rightm i_{\mu_1} \leftp i_{\mu_1+1}\leftp \cdots \leftp i_{\lambda_1},$$ where the arrows neighboring $i_{\mu_1}$ have (respectively) minus and then plus signs.
If the first line of $I$ is in the first form, then make the following change of variables $$(z_{i_1},\ldots, z_{i_{\mu_1}-1},z_{i_{\mu_1}},z_{i_{\mu_1}+1},\ldots, z_{i_{\lambda_1}}) \mapsto (y_{\lambda_1}, \ldots, y_{\lambda_1 - \mu_1 +2},y_{\lambda_1 - \mu_1 +1},-y_{\lambda_1 - \mu_1},\ldots, -y_1);$$ whereas if the first line of $I$ is in the second form, then make the following change of variables $$(z_{i_1},\ldots, z_{i_{\mu_1}-1},z_{i_{\mu_1}},z_{i_{\mu_1}+1},\ldots, z_{i_{\lambda_1}}) \mapsto (y_{\lambda_1}, \ldots, y_{\lambda_1 - \mu_1 +2},-y_{\lambda_1 - \mu_1 +1},-y_{\lambda_1 - \mu_1},\ldots, -y_1).$$
Similarly for the other lines of $I$, make the analogous change of variables, where for the $i^{th}$ line, the participating $z$-variables are replaced by the variables $y_{\lambda_1+\cdots +\lambda_{i-1}+1}, \ldots, y_{\lambda_1+\cdots +\lambda_i}$.
The above procedure almost uniquely specifies how to change from $z$ to $y$ variables. The exception is for singleton lines. For example, if the first line of $I$ is a singleton $z_{i_1}$, then we may either make the change of variables $z_{i_1} \mapsto y_{1}$ or $z_{i_1}\mapsto -y_1$.
One quickly sees that this change of variables procedure turns ${\underset{{I}}{\mathbf{Res}^c}}$ into ${\underset{{\lambda}}{\mathbf{Res}^c}}$, and that for a given $I$, there are exactly $2^{m_1}$ different changes of variables which accomplishes this (here $m_1$ is the number of singletons in $I$).
Note that since the variables $z_{i_{\mu_1}},z_{j_{\mu_2}},\ldots$ were being integrated along ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$, the variables $w_i = y_{\lambda_1+\cdots +\lambda_{i-1}+1}$ may end up being integrated along a shift of ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. We need to deform these contours back to ${\ensuremath{\mathbf{i}}}{\ensuremath{\mathbb{R}}}$. However, during this deformation we meet poles of the integrand.
\[claim:Res-are-zero\] For any $\lambda$ the total sum over $S(\lambda)$ of all residues which we need to encounter during this deformation equals $0$.
We do not give a complete proof of Claim \[claim:Res-are-zero\] in this paper. In Section \[sec:one-string\] we give a proof of a partial result towards this claim: We prove that all residues which come from elements of $S(\lambda)$ consisting of one string give zero contribution (that is $\lambda=(k)$).
\[aexception\] When $a\neq 0$ (recall $a$ is the parameter controlling the behavior of the half-space polymer at the origin), there exist poles coming from the term $\frac{z}{z+a}$ which may block this deformation. We do not attempt to keep track of this possibly more complicated residue expansion presently.
For a given change of variables from $z$’s to $y$’s we can associate an element $\sigma\in BC_k$ so that $(z_1,\ldots, z_k) \mapsto (y_{\sigma(1)},\ldots, y_{\sigma(k)})$. Here an element $\sigma\in BC_k$ acts on a vector $\vec{y}$ in the manner described in Section \[Asymsec\]. Thus, we have almost shown (\[ref31prime\]), except that on the right-hand side of that equation, the summation is presently only over those $\sigma$ which arise in the manner described above.
The second piece to showing the claimed equality in (\[ref31prime\]) is to recognize those $\sigma$ which do not arise from the above replacement scheme, correspond to terms in the right-hand side of (\[ref31prime\]) which have zero residue. Hence the partial summation over $BC_k$ can be completed to the full sum, as claimed. This second piece follows in the same manner as in step 2 of the proof of Proposition \[321\] by observing that any $\sigma$ which does not arise as above, leads to the evaluation of a residue at a point with no pole.
Summarizing what we have learned so far: $$\begin{aligned}
\label{31prime}
{\rm LHS} (\ref{mainform}) &=& \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{1}{m_1! m_2!\cdots} \frac{1}{2^{\ell(\lambda)}} \sum_{\sigma\in BC_k} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{1}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}} \\
\nonumber && \times\, {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c} F(\sigma(\vec{y})) \right).\end{aligned}$$
[**Step 3:**]{} It remains to compute the residues and identify the result with the right-hand side of (\[mainform\]). It is convenient to rewrite the following expression as: $$\begin{aligned}
\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad &\\
= \prod_{1\leq A< B \leq k} \frac{y_{A}-y_{B}}{y_{A}-y_{B}-c}\frac{y_{A}+y_{B}}{y_{A}+y_{B}-c} \frac{y_{A}-y_{B}}{y_{A}-y_{B}+c}\frac{y_{A}+y_{B}}{y_{A}+y_{B}+c}\qquad\qquad&\\
\times\,\qquad \prod_{1\leq B<A\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}-c}{y_{\sigma(A)}-y_{\sigma(B)}}\frac{y_{\sigma(A)}+y_{\sigma(B)}-c}{y_{\sigma(A)}+y_{\sigma(B)}}.\qquad\qquad\qquad&\end{aligned}$$ On the right hand side above there are two products. The first is easily seen to be invariant under the action of $BC_k$, while the second product has no residue at the poles we are considering. These considerations allow us to rewrite (\[31prime\]) as $$\begin{aligned}
\label{31prime2}
{\rm LHS} (\ref{mainform}) &=& \sum_{\substack{\lambda\vdash k\\ \lambda= 1^{m_1}2^{m_2}\cdots}} \frac{1}{m_1! m_2!\cdots} \frac{1}{2^{\ell(\lambda)}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{1}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{\ell(\lambda)}}{2\pi {\ensuremath{\mathbf{i}}}} \\
&&\nonumber\times\, {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A< B \leq k} \frac{y_{A}-y_{B}}{y_{A}-y_{B}-c}\frac{y_{A}+y_{B}}{y_{A}+y_{B}-c} \frac{y_{A}-y_{B}}{y_{A}-y_{B}+c}\frac{y_{A}+y_{B}}{y_{A}+y_{B}+c} \right)\\
&&\nonumber\times\, {\underset{{\lambda}}{\mathbf{Sub}^c}} \left(\sum_{\sigma\in BC_k} \prod_{1\leq B<A\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}-c}{y_{\sigma(A)}-y_{\sigma(B)}}\frac{y_{\sigma(A)}+y_{\sigma(B)}-c}{y_{\sigma(A)}+y_{\sigma(B)}} F(\sigma(\vec{y}))\right).\end{aligned}$$
We use Lemma \[fourthlemma\] to evaluate the above residue, and we easily identify the substitution with $E^c(w_1,w_1+c,\ldots)$ as in the statement of the proposition. Combining these two expressions yields the desired residue expansion of the proposition.
Evaluation of certain residues and substitutions {#subsec}
------------------------------------------------
The purpose of this section is to evaluate certain residues and substitutions which arise in the proofs of Propositions \[Akcprop\], \[321\], and Conjecture \[Bkcprop\]. We first state our lemmas in the $q$-deformed setting and then take $q\to 1$. These calculations are straightforward but require some care as they involve large products.
For a partition $\lambda\vdash k$ and $q\in (0,1)$, recall the definitions of ${\underset{{\lambda}}{\mathbf{Res}^q}}$ and ${\underset{{\lambda}}{\mathbf{Sub}^q}}$ from Section \[resreview\], and recall that the outcome of these residue or substitution operators on functions $f(y_1,\ldots, y_k)$ are functions of the terminal variables $\big(y_1,y_{\lambda_1+1},\ldots, y_{\lambda_1+\cdots+\lambda_{\ell(\lambda)-1}-1}\big)$. We rename these remaining variables as $w_j = y_{\lambda_1+\cdots \lambda_{j-1}+1}$, for $1\leq j\leq \ell(\lambda)$.
\[reslemma\] For all $k\geq 1$, $\lambda\vdash k$ and $q\in (0,1)$, we have that $$\begin{aligned}
\label{reslemmaeqn}
{\underset{{\lambda}}{\mathbf{Res}^q}} \left(\prod_{1\leq i\neq j\leq k} \frac{y_i-y_j}{y_i-qy_j} \right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&\\
\nonumber =(-1)^k (1-q)^k q^{-\frac{k^2}{2}} \prod_{j=1}^{\ell(\lambda)} w_j^{\lambda_j} q^{\frac{\lambda_j^2}{2}} \frac{\prod_{1\leq i<j\leq \ell(\lambda)}(w_j-w_i)(w_i q^{\lambda_i} - w_j q^{\lambda_j})}{\prod_{i,j=1}^{\ell(\lambda)} (w_i q^{\lambda_i} - w_j)} \qquad\qquad\qquad\,\,&\\
\nonumber =(-1)^k (1-q)^k q^{-\frac{k^2}{2}} \prod_{j=1}^{\ell(\lambda)} w_j^{\lambda_j} q^{\frac{\lambda_j^2}{2}} \det\left[\frac{1}{w_i q^{\lambda_i}-w_j}\right]_{i,j=1}^{\ell(\lambda)}.\qquad\qquad\qquad\qquad\qquad\qquad&\end{aligned}$$
The second equality of the lemma follows immediately from the Cauchy determinant. Though straightforward, proving the first equality of lemma does require some care in keeping track of large products. The product on the left-hand side of (\[reslemmaeqn\]) involves terms in which $i$ and $j$ are in the same string of variables in (\[resqvalues\]) as well as terms in which they are in different strings. We need to compute the residue of the same string terms and multiply it by the substitution of variables into the different string terms.
Let us first evaluate same string residues. Consider variables $y_1,\ldots, y_{\ell}$, $\ell\geq 2$, and observe that $$\begin{aligned}
{\underset{{\substack{y_2=qy_1\\y_3=qy_2\\\cdots\\y_{\ell}= qy_{\ell-1}}}}{\mathbf{Res}}} \left( \prod_{1\leq i\neq j\leq \ell} \frac{y_i-y_j}{y_i-qy_j}\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&\\
\nonumber = \frac{\prod\limits_{1\leq i\neq j\leq \ell} (q^{i-1}y_1 - q^{j-1} y_1)}{\prod\limits_{\substack{1\leq i\neq j\leq \ell\\i\neq j+1}} (q^{i-1}y_1-q^j y_1)} =
\frac{\prod\limits_{1\leq i\neq j\leq \ell} (q^{i-1}y_1 - q^{j-1} y_1)}{\prod\limits_{\substack{1\leq i \leq \ell \\2\leq j'\leq \ell+1\\i\neq j'-1,\, i\neq j'}} (q^{i-1}y_1-q^{j'-1} y_1)} \qquad\qquad\qquad\qquad\qquad&\\
\nonumber = (-1)^{\ell-1} y_1^{\ell-1} \frac{(q-1)\cdots (q^{\ell-1}-1) (q-1)^{\ell-1} q^{\frac{(\ell-1)(\ell-2)}{2}}}{(1-q^{\ell})(q-q^{\ell})\cdots (q^{\ell-2}-q^{\ell})} = (-1)^{\ell-1} y_1^{\ell-1} \frac{(1-q)^{\ell}}{1-q^{\ell}}. \,\,\,&\end{aligned}$$
Now turn to the cross term between two strings of variables. Consider one set of variables $y_1,\ldots, y_{\ell}$ with $\ell\geq 2$ and a second set of variables $y'_1\ldots, y'_{\ell'}$ with $\ell'\geq 2$. Then $$\begin{aligned}
{\underset{{\substack{y_2=qy_1\\y_3=qy_2\\\cdots\\y_{\ell}= qy_{\ell-1}}}}{\mathbf{Sub}}} {\underset{{\substack{y'_2=qy'_1\\y'_3=qy'_2\\\cdots\\y'_{\ell'}= qy'_{\ell'-1}}}}{\mathbf{Sub}}} \left( \prod_{i=1}^{\ell}\prod_{j=1}^{\ell'} \frac{y_i-y_j}{y_i-qy_j}\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad & \\
\nonumber = \prod_{i=1}^{\ell}\prod_{j=1}^{\ell'} \frac{y_1 q^{i-1}- y'_1 q^{j-1}}{(y_1 q^{i-1} -y'_1 q^j)} = \prod_{i=1}^{\ell} \frac{\prod\limits_{j=1}^{\ell'} (y_1 q^{i-1}- y'_1 q^{j-1})}{\prod\limits_{j'=2}^{\ell'} y_1 q^{i-1} y'_1 q^{j'-1}} = \prod_{i=1}^{\ell} \frac{y_1 q^{i-1}-y'_1}{y q^{i-1} - y'_1 q^{\ell'}}. \qquad\qquad&\end{aligned}$$
Since the strings of variables also come interchanged, we should multiply the above expression by the same term with $(y_1,\ell)$ and $(y'_1,\ell')$ interchanged. This gives $$\prod_{i=1}^{\ell} \prod_{j=1}^{\ell'} \frac{(y_1 q^{i-1} - y'_1 q^{j-1})(y'_1 q^{j-1} -y_1 q^{i-1})}{(y_1 q^{i-1} - y'_1 q^{j})(y'_1 q^{j} - y_1 q^{i})} = q^{-\ell \ell'} \frac{ (y_1-y'_1) (y_1 q^{\ell}-y'_1 q^{\ell'})}{(y_1 q^{\ell}-y'_1)(y_1 - y'_1 q^{\ell'})}.$$
Returning to the statement of the lemma, we see that we can evaluate the desired residue by multiplying the same string terms over all strings in (\[resqvalues\]) as well as multiplying all terms corresponding to pairs of different string. Using the above calculations we obtain $${\underset{{\lambda}}{\mathbf{Res}^q}} \left(\prod_{1\leq i\neq j\leq k} \frac{y_i-y_j}{y_i-qy_j} \right) = \prod_{j=1}^{\ell(\lambda)} \frac{(1-q)^{\lambda_j}}{(1-q^{\lambda_j})} (-1)^{\lambda_j-1} w_{j}^{\lambda_j-1} \, \prod_{1\leq i<j\leq \ell(\lambda)} q^{-\lambda_i\lambda_j} \frac{(w_i-w_j)(w_i q^{\lambda_i} - w_j q^{\lambda_j})}{(w_i q^{\lambda_i} - w_j)(w_i -w_j q^{\lambda_j})}.\qquad\qquad$$ It is easy now to rewrite the above expression so as to produce the first equality of the lemma, as desired.
\[sublemma\] For all $k\geq 1$, $\lambda\vdash k$ and $q\in (0,1)$, we have $$\begin{aligned}
\label{sublemmaeqn}
{\underset{{\lambda}}{\mathbf{Sub}^q}} \left(\prod_{1\leq i<j\leq k} \frac{1-y_i y_j}{q-y_i y_j} \frac{1- y_i y_j}{1-q y_i y_j}\right) \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&\\
\nonumber = q^{-\frac{k^2}{2}} q^{\frac{k}{2}} \prod_{j=1}^{\ell(\lambda)} \frac{1-q^{\lambda_j} \tfrac{w_j^2}{q}}{1-\tfrac{w_j^2}{q}} \, \frac{(\tfrac{w_j^2}{q};q^2)_{\lambda_j}}{(w_j^2;q^2)_{\lambda_j}}
\prod_{1\leq i<j\leq \ell(\lambda)} \frac{1-q^{\lambda_i} \tfrac{w_i w_j}{q}}{1- \tfrac{w_i w_j}{q}}\,
\frac{1-q^{\lambda_j} \tfrac{w_i w_j}{q}}{1- q^{\lambda_i+\lambda_j}\tfrac{w_i w_j}{q}}.&\end{aligned}$$
The same considerations as in the proof of Lemma \[reslemma\] apply here. Let us first evaluate same string residues. Consider variables $y_1,\ldots, y_{\ell}$, $\ell\geq 2$, and observe that
$$\begin{aligned}
{\underset{{\substack{y_2=qy_1\\y_3=qy_2\\\cdots\\y_{\ell}= qy_{\ell-1}}}}{\mathbf{Sub}}} \left( \prod_{1\leq i<j\leq \ell} \frac{1-y_i y_j}{q-y_i y_j} \frac{1- y_i y_j}{1-q y_i y_j}\right) &=& \frac{ \prod\limits_{1\leq i<j\leq \ell} (1-q^{i-1+j-1} y_1^2)^2}{\prod\limits_{1\leq i<j\leq \ell} q(1-q^{i-1+j-1-1}y_1^2)(1-q^{i-1+j-1+1}y_1^2)} \\
\nonumber &=& q^{-\frac{\ell(\ell-1)}{2}} \frac{ \prod\limits_{1=j<j\leq \ell} (1-q^{i+j-2} y_1^2)\, \prod\limits_{1<j=i+1\leq \ell} (1-q^{i+j-2}y_1^2)}{\prod\limits_{\substack{i=0\\1<j\leq \ell}} (1-q^{i+j-2} y_1^2) \, \prod\limits_{1<i=j\leq \ell} (1-q^{i+j-2} y_1^2)}\\
\nonumber &=& q^{-\frac{\ell(\ell-1)}{2}} \frac{\prod\limits_{2\leq j\leq \ell} (1-q^{j-1}y_1^2) \, \prod\limits_{2\leq j\leq \ell} (1-q^{2j-3} y_1^2)}
{\prod\limits_{2\leq j\leq \ell} (1-q^{j-2}y_1^2) \, \prod\limits_{2\leq j\leq \ell} (1-q^{2j-2} y_1^2)}\\
\nonumber &=& q^{-\frac{\ell(\ell-1)}{2}} \frac{1-q^{\ell-1}y_1^2}{1-y_1^2}\, \frac{(1-q y_1^2)(1-q^3 y_1^2) \cdots (1-q^{2\ell-3}y_1^2)}{(1-q^2 y_1^2)(1-q^4 y_1^2) \cdots (1-q^{2\ell-2}y_1^2)}.\end{aligned}$$
Now turn to the cross term between two strings of variables. Consider one set of variables $y_1,\ldots, y_{\ell}$ with $\ell\geq 2$ and a second set of variables $y'_1\ldots, y'_{\ell'}$ with $\ell'\geq 2$. We also multiply by the same term with the string interchanged. Thus, $$\begin{aligned}
&{\underset{{\substack{y_2=qy_1\\y_3=qy_2\\\cdots\\y_{\ell}= qy_{\ell-1}}}}{\mathbf{Sub}}} {\underset{{\substack{y'_2=qy'_1\\y'_3=qy'_2\\\cdots\\y'_{\ell'}= qy'_{\ell'-1}}}}{\mathbf{Sub}}}\left(\prod_{1\leq i<j\leq k} \frac{1-y_i y_j}{q-y_i y_j} \frac{1- y_i y_j}{1-q y_i y_j}\, \frac{1-y_j y_i}{q-y_j y_i} \frac{1- y_j y_i}{1-q y_j y_i}\right) \\
&\nonumber = \prod_{i=1}^{\ell} \prod_{j=1}^{\ell'} \frac{(1-q^{i-1+j-1}y_1y'_1)^2}{q(1-q^{i-1+j-1-1}y_1y'_1)(1-q^{i-1+j-1+1}y_1y'_1)} = q^{-\ell \ell'} \prod_{j=1}^{\ell'} \frac{(1-q^{j-1} y_1 y'_1) (1-q^{\ell -1 +j-1} y_1 y'_1)}{(1-q^{j-2} y_1 y'_1)(1-q^{\ell +j-1} y_1 y'_1)}\\
&\nonumber = q^{-\ell \ell'}\frac{1-q^{\ell'-1}y_1 y'_1}{1-q^{-1} y_1 y'_1} \, \frac{1-q^{\ell-1} y_1 y'_1}{1-q^{\ell+\ell'-1} y_1 y'_1}.\end{aligned}$$
Returning to the statement of the lemma, we see that we can evaluate the desired residue by multiplying the same string terms over all strings in (\[resqvalues\]) as well as multiplying all terms corresponding to pairs of different string. Using the above calculations and the replacement $w_j= y_{\lambda_1+\cdots \lambda_{j-1}+1}$ we obtain
$$\begin{aligned}
{\underset{{\lambda}}{\mathbf{Sub}^q}} \left(\prod_{1\leq i<j\leq k} \frac{1-y_i y_j}{q-y_i y_j} \frac{1- y_i y_j}{1-q y_i y_j}\right) \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&\\
\nonumber = \prod_{i=1}^{\ell(\lambda)} q^{-\frac{\lambda_i(\lambda_i-1)}{2}} \frac{1-q^{\lambda_i} \frac{w_i^2}{q}}{1-\frac{w_i^2}{q}} \, \frac{\big(\frac{w_i^2}{q};q^2\big)_{\lambda_i}}{\big(w_i^2;q^2\big)_{\lambda_i}} \, \prod_{1\leq i<j\leq \ell(\lambda)} q^{-\lambda_i\lambda_j} \frac{(1-q^{\lambda_i} \frac{w_i w_j}{q})(1-q^{\lambda_j }\frac{w_i w_j}{q})}{(1-\frac{w_i w_j}{q})(1-q^{\lambda_i+\lambda_j}\frac{w_i w_j}{q})}. &\end{aligned}$$
The powers of $q$ can be simplified further using ($\lambda_1+\lambda_2+\cdots + \lambda_{\ell(\lambda)}=k$) $$\prod_{i=1}^{\ell(\lambda)} q^{-\frac{\lambda_i(\lambda_i-1)}{2}} \prod_{1\leq i<j\leq \ell(\lambda)} q^{-\lambda_i\lambda_j} = q^{-\frac{k^2}{2}} q^{\frac{k}{2}}.$$
Using this simplification, the above formula reduces to the right-hand side of (\[sublemmaeqn\]), thus proving the lemma.
Combining these two lemmas along with (\[ressubeqn\]) immediately yields
\[thirdlem\] For all $k\geq 1$, $\lambda\vdash k$ and $q\in (0,1)$, we have that $$\begin{aligned}
{\underset{{\lambda}}{\mathbf{Res}^q}} \left(\prod_{1\leq i<j\leq k} \frac{y_i-y_j}{y_i-qy_j} \frac{y_j-y_i}{y_j-qy_i} \frac{1-y_i y_j}{q-y_i y_j} \frac{1- y_i y_j}{1-q y_i y_j}\right)\qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad\qquad &\\
\nonumber = (-1)^k (1-q)^{k} q^{-k^2} \prod_{j=1}^{\ell(\lambda)} w_j^{\lambda_j} q^{\frac{\lambda_j(\lambda_j+1)}{2}} \frac{1-q^{\lambda_j} \tfrac{w_j^2}{q}}{1-\tfrac{w_j^2}{q}} \, \frac{(\tfrac{w_j^2}{q};q^2)_{\lambda_j}}{(w_j^2;q^2)_{\lambda_j}}\qquad \qquad \qquad\qquad \qquad &\\
\nonumber \times\,\prod_{1\leq i<j\leq \ell(\lambda)} \frac{(w_j-w_i)(w_i q^{\lambda_i} - w_j q^{\lambda_j}) (1-q^{\lambda_i} \tfrac{w_i w_j}{q})(1-q^{\lambda_j} \tfrac{w_i w_j}{q})}{(1- \tfrac{w_i w_j}{q})(1- q^{\lambda_i+\lambda_j}\tfrac{w_i w_j}{q})} \prod_{i,j=1}^{\ell(\lambda)} \frac{1}{w_i q^{\lambda_i} - w_j}.\end{aligned}$$
Recall the definitions of ${\underset{{\lambda}}{\mathbf{Res}^c}}$ and ${\underset{{\lambda}}{\mathbf{Sub}^c}}$ from Section \[resreview\]. By taking a limit as ${\epsilon}\to 0$ of the above results under the change of variables $q\mapsto e^{-{\epsilon}c}$, $y\mapsto e^{-{\epsilon}y}$ and $w\mapsto e^{-{\epsilon}w}$ we find the following:
\[fourthlemma\] For all $k\geq 1$, $\lambda\vdash k$ and $c\in (0,\infty)$, we have the following $$\begin{aligned}
{\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{ y_A-y_B}{y_A-y_B-c} \frac{y_A-y_B}{y_A-y_B+c} \frac{y_A+y_B}{y_A+y_B+c} \frac{y_A+y_B}{y_A+y_B-c}\right) && \\
= c^k \prod_{j=1}^{\ell(\lambda)} \frac{-1}{2c}\, \frac{\left(\tfrac{2w_j+c}{2c}\right)_{\lambda_j-1}}{\left(\tfrac{2w_j}{2c}\right)_{\lambda_j}} \Pf\left[\frac{u_i-u_j}{u_i+u_j}\right]_{i,j=1}^{2\ell(\lambda)},&&\end{aligned}$$ where $$(u_1,\ldots, u_{2\ell(\lambda)}) =\big(-w_1+\tfrac{c}{2}, w_1-\tfrac{c}{2}+c\lambda_1 , -w_2+\tfrac{c}{2},w_2-\tfrac{c}{2}+c\lambda_2, \ldots, -w_{\ell(\lambda)}+\tfrac{c}{2}, w_{\ell(\lambda)}-\tfrac{c}{2} + c\lambda_{\ell(\lambda)} \big).$$
The immediate consequence of taking the ${\epsilon}\to 0$ limit of Lemma \[thirdlem\] (with the change of variables $q\mapsto e^{-{\epsilon}c}$, $y\mapsto e^{-{\epsilon}y}$ and $w\mapsto e^{-{\epsilon}w}$) is that $$\begin{aligned}
\label{almostthere}
&&{\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{ y_A-y_B}{y_A-y_B-c} \frac{y_A-y_B}{y_A-y_B+c} \frac{y_A+y_B}{y_A+y_B+c} \frac{y_A+y_B}{y_A+y_B-c}\right)\\
&&\nonumber = (-1)^{k-\ell(\lambda)}(-c)^k \prod_{j=1}^{\ell(\lambda)} \frac{ -2 w_i + c - c\lambda_i}{-2 w_i +c} \, \frac{(2w_i-c)(2w_i-c +2c) \cdots (2 w_i -c +2c(\lambda_i-1))}{(2w_i)(2w_i +2c) \cdots (2 w_i +2c(\lambda_i-1))} \\
&&\nonumber \times\,\prod_{1\leq i<j\leq \ell(\lambda)} \frac{ (w_j-w_i) (-w_i-c\lambda_i +w_j + c\lambda_j)(-w_i-w_j+c-c\lambda_i)(-w_i-w_j+c-c\lambda_j)}{(-w_i-w_j+c) (-w_i-w_j+c-c(\lambda_i+\lambda_j))(-w_i-c\lambda_i +w_j)(-w_i+w_j+c\lambda_j)}\\
&& \times\,\prod_{j=1}^{\ell(\lambda)} \frac{1}{-c\lambda_j}.\end{aligned}$$ Note that the $(-1)^{k-\ell(\lambda)}$ factor arose from the limit of ${\underset{{\lambda}}{\mathbf{Res}^q}}$ to ${\underset{{\lambda}}{\mathbf{Res}^c}}$. We can recognize a Pfaffian in this expression. Take $u$ as in the statement of the lemma and use the Pfaffian identity (cf. [@M III.8]) $$\Pf\left[\frac{u_i-u_j}{u_i+u_j}\right]_{i,j=1}^{2\ell(\lambda)} = \prod_{1\leq i<j\leq \ell(\lambda)} \frac{u_i-u_j}{u_i+u_j}.$$ Then right-hand side of the above expression evaluates to $$\prod_{1\leq i<j\leq \ell(\lambda)} \frac{(-w_i+w_j) (-w_i-w_j +c - c\lambda_j) ( w_i+w_j -c + c\lambda_i) (w_i -w_j +c\lambda_i-c\lambda_j)}{(-w_i-w_j +c) (-w_i+w_j + c\lambda_j) (w_i -w_j +c\lambda_i) (w_i+w_j -c +c(\lambda_i+\lambda_j))} \, \prod_{i=1}^{\ell(\lambda)} \frac{ -2w_i +c -c\lambda_i}{c \lambda_i}$$ which, compared with (\[almostthere\]), yields the desired result.
Type $A$ and $BC$ symmetry {#Asymsec}
--------------------------
The symmetry group of type $A_k$ is identified with the permutation group on $k$ elements, written $S_k$. We write $\sigma\in S_k$ to identify a group element as a permutation. A permutation $\sigma\in S_k$ acts on a vector $\vec{z}=(z_1,\ldots, z_k)$ by permutating indices, yielding $\sigma(\vec{z}) = (z_{\sigma(1)},\ldots, z_{\sigma(k)})$.
A fundamental domain for the action of $S_k$ is a type $A_k$ Weyl chamber, defined here as $$W(A_k) = \left\{\vec{x} = (x_1\leq x_2\leq \cdots \leq x_k)\right\}.$$
An example of an $S_k$ invariant function (which is utilized in this work) is $$\prod_{1\leq i\neq j\leq k} \frac{z_A-z_B}{z_A-q z_B}.$$
The following symmetrization identity is proved in [@M III.1] as a special case of computing the Hall-Littlewood polynomial normalization: $$\sum_{\sigma\in S_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-tz_{\sigma(B)}}{z_{\sigma(A)}-z_{\sigma(B)}} = \prod_{j=1}^{k} \frac{1-t^j}{1-t}.$$
Changing variables as $t\mapsto e^{-{\epsilon}c}$ and $z \mapsto e^{-{\epsilon}z}$ and taking the limit ${\epsilon}\to 0$, we find that this becomes $$\label{Akfacsym}
\sum_{\sigma\in S_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-z_{\sigma(B)}-c}{z_{\sigma(A)}-z_{\sigma(B)}} = k!.$$
The symmetry group of type $BC_k$ is called the [*hyperoctahedral group*]{} and is identified with the [*signed*]{} permutation group on $k$ elements (we also denote this group by $BC_k$). The group is sometimes written as $S_k\ltimes {\ensuremath{\mathbb{Z}}}_2^k$ as it is a wreath product of $S_k$ and ${\ensuremath{\mathbb{Z}}}_2^k$. An element $\sigma\in BC_k$ maps $\{\pm 1,\ldots, \pm k\}$ onto itself with the condition that $\sigma(-i) = -\sigma(i)$. As such, it suffices to specify the images of $\{1,\ldots ,k\}$ under $\sigma$. For example for $k=3$, $\sigma\in BC_3$ could map $1\mapsto \sigma(1)=-2$, $2\mapsto \sigma(2)=3$ and $3\mapsto\sigma(3)=-1$ (and hence $-1\mapsto 2$, $-2\mapsto -3$ and $-3\mapsto 1$). A signed permutation $\sigma\in BC_k$ can act on a vector $\vec{z}=(z_1,\ldots,z_k)$ either [*multiplicatively*]{} or [*additively*]{}. If $\sigma$ acts multiplicatively, then $$\sigma(\vec{z}) = (z_{|\sigma(1)|}^{\sgn(\sigma(1))},\ldots,z_{|\sigma(k)|}^{\sgn(\sigma(k))})$$ and if $\sigma$ acts additively, then $$\sigma(\vec{z}) = (\sgn(\sigma(1)) z_{|\sigma(1)|},\ldots,\sgn(\sigma(k)) z_{|\sigma(k)|}).$$
A fundamental domain for the additive action of $BC_k$ is a type $BC_k$ Weyl chamber, defined here as $$W(BC_k) = \left\{ \vec{x} = (x_1\leq x_2\leq \cdots \leq x_k \leq 0)\right\}.$$
The following identity is proved in [@Vidya] as a special case of computing the $q=0$ Koornwinder polynomial normalization (type $BC$ analogs of Hall-Littlewood polynomials). In particular by combining equations (1), (5) and Theorem 2.6 of [@Vidya] and choosing $\lambda=\emptyset$ we have that for arbitrary $a$, $b$, $t$, $$\sum_{\sigma\in BC_k} \prod_{1\leq A<B\leq k} \frac{1-t\frac{z_{\sigma(B)}}{z_{\sigma(A)}}}{1-\frac{z_{\sigma(B)}}{z_{\sigma(A)}}}\frac{1-t\frac{1}{z_{\sigma(A)}z_{\sigma(B)}}}{1-\frac{1}{z_{\sigma(A)}z_{\sigma(B)}}} \prod_{j=1}^{k} \frac{\left(1-a \frac{1}{z_{\sigma(j)}}\right)\left(1-b \frac{1}{z_{\sigma(j)}}\right)}{1-\frac{1}{z_{\sigma(j)}^2}} = \prod_{j=1}^{k} \frac{1-t^j}{1-t} (1-ab t^{j-1}).$$ In the above, the action of $BC_k$ is taken to be multiplicative. Note also that the product over $1\leq A<B\leq k$ above can also be taken over $1\leq B<A\leq k$, which simply amounts to renaming the variables $z_j\mapsto z_{k+1-j}$, $1\leq j\leq k$.
Changing variables as $a\mapsto e^{-{\epsilon}a}$, $t\mapsto e^{-{\epsilon}c}$, $z_j\mapsto e^{-{\epsilon}z_j}$, and setting $b\equiv 0$, we find that the ${\epsilon}\to 0$ limit of the above identity yields (now with additive action of $BC_k$) $$\label{Bkfacsym}
\sum_{\sigma\in BC_k} \prod_{1\leq B<A\leq k} \frac{z_{\sigma(A)}-z_{\sigma(B)}-c}{z_{\sigma(A)}-z_{\sigma(B)}}\frac{z_{\sigma(A)}+z_{\sigma(B)}-c}{z_{\sigma(A)}+z_{\sigma(B)}} \prod_{j=1}^{k} \frac{z_{\sigma(j)}-a}{z_{\sigma(j)}} = 2^k k!.$$
The case of one string in Claim \[claim:Res-are-zero\] {#sec:one-string}
======================================================
The goal of this section is to present a proof of a partial result towards Claim \[claim:Res-are-zero\]. It is given by equation below. We deal with the case when the element of $I$ consists of one string only. Let us denote by $BC^{I(\lambda)}_k$ the set of elements of $BC_k$ which come from the diagrams with the partition type $\lambda$. Equation can be written as $$\begin{aligned}
\label{ref31prime-2}
&&\sum_{I\in S(\lambda)} \, \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_{i_{\mu_1}}}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dz_{j_{\mu_2}}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots {\underset{{I}}{\mathbf{Res}}} \left(\prod_{1\leq A<B\leq k} \frac{z_A-z_B}{z_A-z_B-c}\frac{z_A+z_B}{z_A+z_B-c} F(\vec{z})) \right) =\frac{1}{2^{m_1}} \\
\nonumber && \times \sum_{\sigma\in BC_k^{I(\lambda)}} \int_{ e_1^\sigma c -{\ensuremath{\mathbf{i}}}\infty}^{e_1^\sigma c + {\ensuremath{\mathbf{i}}}\infty} \frac{dw_{1}}{2\pi {\ensuremath{\mathbf{i}}}} \int_{e_2^\sigma c -{\ensuremath{\mathbf{i}}}\infty}^{e_2^\sigma c + {\ensuremath{\mathbf{i}}}\infty} \frac{dw_{2}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c} F(\sigma(\vec{y})) \right),\end{aligned}$$ where $w_i = y_{\lambda_1+\cdots +\lambda_{i-1}+1}$, and $e_1(\sigma), e_2(\sigma), \dots$ determine the contours of integration after the change of variables; they are certain parameters of $\sigma$.
Claim \[claim:Res-are-zero\] asserts that $$\begin{aligned}
&& \sum_{\sigma\in BC_k^{I(\lambda)}} \int_{ e_1^\sigma c -{\ensuremath{\mathbf{i}}}\infty}^{e_1^\sigma c + {\ensuremath{\mathbf{i}}}\infty} \frac{dw_{1}}{2\pi {\ensuremath{\mathbf{i}}}} \int_{e_2^\sigma c -{\ensuremath{\mathbf{i}}}\infty}^{e_2^\sigma c + {\ensuremath{\mathbf{i}}}\infty} \frac{dw_{2}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c} F(\sigma(\vec{y})) \right) \\
\nonumber && = \sum_{\sigma\in BC_k^{I(\lambda)}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dw_{1}}{2\pi {\ensuremath{\mathbf{i}}}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{ {\ensuremath{\mathbf{i}}}\infty} \frac{dw_{2}}{2\pi {\ensuremath{\mathbf{i}}}} \cdots {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c} F(\sigma(\vec{y})) \right),\end{aligned}$$ that is, that the contours can be deformed back to the imaginary axis, and the total contribution of arising residues is 0.
Let $BC^{I(k)}_k$ be the set of elements of $BC_k$ which correspond to one-string diagrams. Our aim is to prove the following equality $$\begin{aligned}
\label{12}
&& \sum_{\sigma\in BC_k^{I(k)}} \int_{ e_1^\sigma c -{\ensuremath{\mathbf{i}}}\infty}^{e_1^\sigma c + {\ensuremath{\mathbf{i}}}\infty} \frac{dy_{1}}{2\pi {\ensuremath{\mathbf{i}}}} {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c} F(\sigma(\vec{y})) \right) \\
\nonumber && = \sum_{\sigma\in BC_k^{I(k)}} \int_{-{\ensuremath{\mathbf{i}}}\infty}^{{\ensuremath{\mathbf{i}}}\infty} \frac{dy_{1}}{2\pi {\ensuremath{\mathbf{i}}}} {\underset{{\lambda}}{\mathbf{Res}^c}} \left(\prod_{1\leq A<B\leq k} \frac{y_{\sigma(A)}-y_{\sigma(B)}}{y_{\sigma(A)}-y_{\sigma(B)}-c}\frac{y_{\sigma(A)}+y_{\sigma(B)}}{y_{\sigma(A)}+y_{\sigma(B)}-c} F(\sigma(\vec{y})) \right).\end{aligned}$$
Initial considerations
----------------------
We start with an analysis which residues we need to sum up.
Recall that we have a sum over diagrams. We will consider the case when only one string of arrows is present in a diagram. Thus, we will assume that the variables in our string have indices from $1$ through $k$. We will denote the set of all diagrams with one string of arrows and with a structure explained in Step 1 of Section \[bcproof\] by ${\mathcal S}_1$. After Step 1, for each such diagram we have an integral over the contour ${\ensuremath{\mathrm{Re}}}(z_k)= {\epsilon}>0$, for ${\epsilon}\ll 1$ (not precisely 0 due to a possible pole at 0). In Step 2, we need to make a change of variables of the form $y_1 := z_k- \mathbf{a} c$ or $y_1 := - z_k- \mathbf{b} c$, where $\mathbf a, \mathbf b \in \mathbb N$ are certain parameters of our diagram. After this change of variables the contour of integration will be close to ${\ensuremath{\mathrm{Re}}}(y_1) = - \mathbf{a} c$ (or $- \mathbf{b} c$), and we need to deform it back to the contour $\Re(y_1)={\epsilon}>0$. During this deformation we will pick up some residues which we are interested in.
The residues depend on the function $$R(z_1, \dots, z_k) = \prod_{i<j} \frac{(z_i-z_j) (z_i+z_j)}{(z_i-z_j-c) (z_i+z_j-c)},$$ the analytic function $F(z_1, \dots, z_k)$, and a diagram $s \in {\mathcal S}_1$ under consideration.
Each diagram $s\in {\mathcal S}_1$ is of the form $$\qquad i_1 {\xrightarrow[]{+}}i_2 {\xrightarrow[]{+}}\dots i_{\mu-1} {\xrightarrow[]{+}}k {\xleftarrow[]{-}}i_{\mu+1} {\xleftarrow[]{+}}\dots {\xleftarrow[]{+}}i_k,$$ and encodes the substitution of variables $z_{i_1} = z_{i_2} +c$, $\dots$, $z_{i_{\mu}-1} = z_k+ c$, $z_{i_{\mu+1}} = c - z_k$, $\dots$, $z_{i_{k}}= z_{i_{k-1}}+c$, or, if the arrow of minus-type comes to $k$ from the another direction then the substitution has a similar form, see equation . We refer to such a substitution as the substitution prescribed by $s$.
Let us introduce notations $$R_s (z_k):= {\underset{{s}}{\mathbf{Res}}}\, R(z_1,z_2, \dots, z_k), \qquad F_s (z_k) := {\underset{{s}}{\mathbf{Sub}}}\, F(z_1,\dots, z_k).$$ We obtain the function $R_s (z_k)$ by taking residues of the function $R$ as specified by $s$. This amounts to removing the factors in the denominator of $R (z_1, \dots, z_k)$ which correspond to the poles at which we took residues, and and making the substitution prescribed by $s$ for what remains. The function $F_s (z_k)$ is merely a substitution, because it does not have poles. Our main concern will be the function $R_s (z_k)$, while $F_s (z_k)$ will not produce any difficulties for our analysis.
Assume that $k=6$ and $s$ is given by the string $$1 {\xrightarrow[]{+}}3 {\xrightarrow[]{+}}4 {\xrightarrow[]{+}}5 {\xrightarrow[]{-}}6 {\xleftarrow[]{+}}2.$$ During the deformation of contours we use factors $(z_5+z_6 -c)$, $(z_4-z_5-c)$, $(z_3-z_4-c)$, $(z_2-z_6-c)$, and $(z_1-z_3-c)$; we omit these factors from the denominator of $R(z_1, \dots, z_k)$. After this we make a substitution of variables $z_5= - z_6+c$, $z_4 = -z_6 +2c$, $z_3 = -z_6 +3c$, $z_2 = z_6 +c$, $z_1 = -z_6 +4c$ (which is prescribed by $s$) into all other factors and obtain the function $$R_s (z_6) = \frac{15c^5 (-2 z_6+5c)(-2 z_6+7c) (2 z_6+c)}{z_6 (-z_6+c)^2}.$$ For this diagram above we make a change of variables $y_1 := - z_6 - c$. This comes from the change of variables from $z$’s to $y$’s in step 2 along with the fact that the $z_6$ contour has real part ${\epsilon}$ and the relation between $y_1$ and $z_6$.
If instead we were working with a diagram $$2 {\xrightarrow[]{+}}6 {\xrightarrow[]{-}}5 {\xleftarrow[]{+}}4 {\xleftarrow[]{+}}3 {\xleftarrow[]{+}}1,$$ we need to set $y_1 := z_6 - 4c$, while the substitution and the function $R_s (z_6)$ are exactly the same.
As we already noticed, for a diagram $s$ we need to make a change of variables of the form $y_1 := z_k- \mathbf{a} c$ or $y_1 := - z_k- \mathbf{b} c$, where $ \mathbf{a} = \mathbf{a} (s) $ and $ \mathbf{b} = \mathbf{b} (s)$ are certain parameters of the diagram $s$. In more detail, if a diagram $s$ has a form (recall that $k$ is the largest index) $$q_{\mathbf{a}} {\xrightarrow[]{+}}q_{ \mathbf{a}-1} {\xrightarrow[]{+}}\dots q_{1} {\xrightarrow[]{-}}k {\xleftarrow[]{+}}p_1 {\xleftarrow[]{+}}\dots {\xleftarrow[]{+}}p_{\mathbf{b}},$$ then we need to make a change of variables $y_1 := -z_k - \mathbf{b} c$. Let $\tilde R_s (y_1)$ be the resulting function. Let $\mathcal W (s)$ be the sum of residues which we obtain in the process of moving the contour of integration of $\tilde R_s (y_1)$ in the left-hand side of from ${\ensuremath{\mathrm{Re}}}(y_1) = - \mathbf{b} c - {\epsilon}$ to ${\ensuremath{\mathrm{Re}}}(y_1) = {\epsilon}$.
If the diagram $s$ has instead the form $$p_{\mathbf{b}} {\xrightarrow[]{+}}p_{ \mathbf{b}-1} {\xrightarrow[]{+}}\dots p_{1} {\xrightarrow[]{+}}k {\xleftarrow[]{-}}q_1 {\xleftarrow[]{+}}\dots {\xleftarrow[]{+}}q_{\mathbf{a}},$$ then we need to make a change of variables $y_1 := z_k - \mathbf{a} c$. Let $ \mathcal W(s)$ be the contribution of poles which we obtain in the process of moving the integral of $\tilde R_s (y_1)$ in the left-hand side of from the contour ${\ensuremath{\mathrm{Re}}}(y_1) = - \mathbf{a} c + {\epsilon}$ to the contour ${\ensuremath{\mathrm{Re}}}(y_1) = {\epsilon}$.
Let $$\mathcal W := \sum_{s \in {\mathcal S}_1} \mathcal W(s)$$ be the total contribution coming from all our diagrams from ${\mathcal S}$. Our goal is to prove the following proposition.
\[theorem-main-oneLine\] We have $\mathcal W=0$.
By definition, this proposition is equivalent to equation .
We will consider a pairing on the set ${\mathcal S}_1$. For a diagram $s$ we can read the whole line in the opposite direction; let us denote such a diagram by $s^{\prime}$ (an example of such a diagram is given in the example above). It is clear that $R_s(z_k) = R_{s^{\prime}} (z_k)$, because the operations with multivariate integrals are the same for these diagrams. However, we need to make different changes of variables for them: In one case, we should move the $z_k$ variable to the right, and in the other case — to the left.
Let us denote the set of all diagrams in which $k-1$ is lying to the left of $k$ by the symbol ${\mathcal S}$. This set includes a half of all diagrams $s$ — another half can be obtained by taking $s^{\prime}$.
For an analytic function $G(z)$ we shall denote by ${\mathcal P}_G (x_1,x_2)$ the set of poles of this function lying in the real interval $(x_1,x_2)$.
\[initial\] We have $$\mathcal W = \sum_{s \in {\mathcal S}} \left( - \sum_{z^{*} \in {\mathcal P}_{R_s} ({\epsilon};\mathbf{a}(s) c+{\epsilon})} F_s (z^*) {\underset{{z_k = z^*}}{\mathbf{Res}}} R_s (z_k) + \sum_{z^* \in {\mathcal P}_{R_s} (-\mathbf{b}(s) c-{\epsilon};{\epsilon})} F_s (z^*) {\underset{{z_k = z^*}}{\mathbf{Res}}} R_s (z_k) \right)$$
The two sums correspond with the two types of change of variables relating $y_1$ to $z_k$. Each $s\in {\mathcal S}$ corresponds to one type of change of variable, and $s^{\prime}$ corresponds to the other type.
In the case when $y_1 := z_k- \mathbf{a}(s) c$ we pick up all poles of $\tilde R_s (y_1) = R_s (z_k- \mathbf{a}(s) c)$ between $- \mathbf{a}(s) c + {\epsilon}$ and ${\epsilon}$. This is the same as picking up all poles of $R_s (z_k)$ between ${\epsilon}$ and $ \mathbf{a}(s) c +{\epsilon}$. In the case $y_1 := -z_k - \mathbf{b}(s) c$ we obtain a sign from the Jacobian; but we also get a sign from the direction of integration over contours. These signs cancel out. Thus, we need to pick poles in the movement of the contour in $\int \tilde R_1 (y_1) = \int R_s (-z_k- \mathbf{b}(s) c)$ from ${\ensuremath{\mathrm{Re}}}(y_1) = - \mathbf{b}(s) c - {\epsilon}$ to ${\ensuremath{\mathrm{Re}}}(y_1) = {\epsilon}$. Equivalently, we need to pick poles in the movement of the contour in $\int R_s (z_k)$ from ${\ensuremath{\mathrm{Re}}}(z_k) = {\epsilon}$ to ${\ensuremath{\mathrm{Re}}}(z_k) = - \mathbf{b}(s) c -{\epsilon}$. The statement of the proposition readily follows. The difference of signs is because we move through these poles in different directions.
Pole at 0
---------
We know that for each diagram there exists no more than one arrow of minus-type; this arrow must end in $k$. It can connect $k-1$ and $k$ (the first line below), or a number $x$ and $k$; in the latter case $k-1$ and $k$ are connected by an arrow of plus-type (the second line below). $$\begin{aligned}
\label{second_pairing}
\dots k-1 {\xrightarrow[]{-}}k {\xleftarrow[]{+}}x \dots \\
\dots k-1 {\xrightarrow[]{+}}k {\xleftarrow[]{-}}x \dots.\end{aligned}$$
Let us introduce a new pairing of ${\mathcal S}$; diagrams in each pair are the same except for the change of plus-type and minus-type arrows leading to $k$ (as shown in \[second\_pairing\]). Note that when $k$ is at the end of the string, the pairing is between the diagrams $$\begin{aligned}
1 {\xrightarrow[]{+}}\dots (k-1) {\xrightarrow[]{+}}k \\
1 {\xrightarrow[]{+}}\dots (k-1) {\xrightarrow[]{-}}k\end{aligned}$$ Thus, we partition ${\mathcal S}$ into disjoint pairs. For $s \in {\mathcal S}$ we shall denote $\bar s \in {\mathcal S}$ the paired element through this plus/minus arrow switching.
Diagrams $$s = 1 {\xrightarrow[]{+}}3 {\xrightarrow[]{+}}4 {\xrightarrow[]{+}}5 {\xrightarrow[]{-}}6 {\xleftarrow[]{+}}2$$ and $$\bar s = 1 {\xrightarrow[]{+}}3 {\xrightarrow[]{+}}4 {\xrightarrow[]{+}}5 {\xrightarrow[]{+}}6 {\xleftarrow[]{-}}2$$ form a pair of the described pairing of ${\mathcal S}$. We have $$R_s (z_6) = \frac{15c^5 (-2 z_6+5c)(-2 z_6+7c) (2 z_6+c)}{z_6 (-z_6+c)^2}, \qquad R_{\bar s} (z_6) = \frac{15c^5 (2 z_6+5c)(2 z_6+7c) (-2 z_6+c)}{-z_6 (z_6+c)^2}.$$
\[codeTranspose\] For any $s \in {\mathcal S}$ we have $$R_s (z_k) = R_{\bar s} (- z_k), \qquad F_s (0) = F_{\bar s} (0).$$
For two diagrams $$\begin{gathered}
i_1 {\xrightarrow[]{+}}\dots {\xrightarrow[]{+}}k-1 {\xrightarrow[]{-}}k {\xleftarrow[]{+}}x \dots {\xleftarrow[]{+}}i_2 \\
i_1 {\xrightarrow[]{+}}\dots {\xrightarrow[]{+}}k-1 {\xrightarrow[]{+}}k {\xleftarrow[]{-}}x \dots {\xleftarrow[]{+}}i_2,\end{gathered}$$ the substitutions have the form $$\begin{gathered}
z_{i_1} = \mathbf{a} c -z_k;\,\, \dots \,\,z_{k-1} = c-z_k;\,\, z_k = z_k;\,\, z_x = z_k+ c;\,\, \dots\,\,z_{i_2} = z_k+ \mathbf{b} c; \\
z_{i_1} = z_k + \mathbf{a} c;\,\, \dots \,\,z_{k-1} = z_k+ c;\,\, z_k = z_k;\,\, z_x = c - z_k;\,\, \dots \,\,z_{i_2} = \mathbf{b} c -z_k.\end{gathered}$$ Note that our initial function $R(z_1, \dots, z_k)$ are stable under the transform $z_k \to (-z_k)$ (due to the fact that the index $k$ is the largest one). The statements of the lemma are thus clearly visible.
\[prop2\] We have $$\mathcal W = \sum_{s \in {\mathcal S}} \left( - \sum_{z^* \in {\mathcal P}_{F_s} ({\epsilon};\mathbf{a}(s)c+{\epsilon})} F_s (z^*) {\underset{{z_k = z^*}}{\mathbf{Res}}} F_s (z_k) + \sum_{z^* \in {\mathcal P}_{F_s} (- \mathbf{b}(s)c-{\epsilon};-{\epsilon})} F_s (z^*) {\underset{{z_k=z^*}}{\mathbf{Res}}} F_s (z_k) \right)$$
Note that the only change from Proposition \[initial\] is that the residues at 0 do not enter this summation since the interval $[-\mathbf{b}(s)c,{\epsilon}]$ is now replaced by $[-\mathbf{b}(s)c,-{\epsilon}]$.
Let us consider two diagrams from the same pairing of ${\mathcal S}$. Lemma \[codeTranspose\] shows that the poles of $R_s(z_k)$ and $R_{\bar s} (z_k)$ are closely related: If $\{e_1, \dots, e_M \}$ are the poles of $R_s(z_k)$, then $\{-e_1, \dots, -e_M\}$ are the poles of $R_{\bar s} (z_k)$. It is clear that ${\underset{{z_k = e_i}}{\mathbf{Res}}} R_s (z_k) = - {\underset{{z_k = -e_i}}{\mathbf{Res}}} R_{\bar s} (z_k)$. In particular, ${\underset{{z_k = e_i}}{\mathbf{Res}}} R_s (0) = - {\underset{{z_k = -e_i}}{\mathbf{Res}}} R_{\bar s} (0)$. Therefore, Lemma \[codeTranspose\] implies the statement of the proposition.
We need to prove that $\mathcal W$ is equal to 0. We shall prove a stronger theorem — in fact, cancellations already happen when we fix a complex number as a pole.
\[main\] For any complex number $z^* \ne 0$ we have $$\sum_{s \in {\mathcal S}} F_s (z^*) {\underset{{z_k = z^*}}{\mathbf{Res}}} R_s (z_k) = 0.$$
This theorem is nontrivial for points of the form $nc/2$, $n \in \mathbb N \backslash \{0\}$, only. We shall prove Theorem \[main\] in next sections. Due to Proposition \[prop2\], this will imply Theorem \[theorem-main-oneLine\].
Structure of poles {#structure}
------------------
For a fixed diagram $s \in {\mathcal S}$ we need to understand the structure of the function $R_s (z_k)$. We shall need some notation.
A diagram $s \in {\mathcal S}$ gives rise to two disjoint sets $A(s), B(s) \in \{1,2, \dots, k\}$, $A(s) \sqcup B(s) = \{1, \dots, k\}$; the elements of $A(s)$ are the indices of variables such that the substitution prescribed by $s$ has a form $\mathbf{q} c - z_k$ for these variables, and the elements of $B(s)$ are $k$ and the indices of variables such that the prescribed substitution has a form $\mathbf{q} c + z_k$, for some $\mathbf{q} \in \mathbb N$.
For two numbers $x_1,x_2$ in $\{1,\ldots,k\}$ we shall write ${\mathsf{set}}(x_1,x_2)=1$ if they both belong to $A(s)$ or both belong to $B(s)$, and we shall write ${\mathsf{set}}(x_1,x_2)=-1$ otherwise. For any number $x$ we shall denote by ${\mathsf{prev}}(x)$ the number such that the plus-type arrow goes from ${\mathsf{prev}}(x)$ to $x$. In the following, if for some $x$ the number ${\mathsf{prev}}(x)$ does not exist, we assume that all statements about ${\mathsf{prev}}(x)$ are true.
\[examples\] For a diagram $s = 1 {\xrightarrow[]{+}}3 {\xrightarrow[]{+}}4 {\xrightarrow[]{+}}5 {\xrightarrow[]{-}}6 {\xleftarrow[]{+}}2$ we have $A(s) = \{1,3,4,5\}$ and $B(s)=\{2,6\}$. We have ${\mathsf{set}}(1,5)=1$, ${\mathsf{set}}(2,6)=1$, ${\mathsf{set}}(2,3)=-1$, ${\mathsf{prev}}(3)=1$, ${\mathsf{prev}}(6)=2$.
Let us fix a number $n \in \mathbb Z \backslash \{0\}$. We shall consider some combinatorial quantities which are determined by fixed parameters $n, A(s), B(s)$. We seek to figure out what order of pole the function $R_s(z_k)$ has at the point $z_k = \frac{n c}{2}$. We omit the dependence on $s$ in notation in the rest of this section because $s$ will be fixed.
\[def:func-v\] Let $\tilde a$ be the index such that there is an arrow of minus-type from $z_{\tilde a}$ to $z_k$. Let us define a function $v: \{1,\dots, k\} \to \mathbb Z/2$ as follows: Let $v(k) := n/2$, and $v(\tilde a) := 1 - n/2$. All other values are defined inductively: If we have $v(x)=l$, then $v( {\mathsf{prev}}(x)) := l+1$. We call a pair of numbers $x_1, x_2$ such that $v(x_1) + v(x_2) =0$ a *plus-zero*. We call a pair of numbers $x_1, x_2$ such that ${\mathsf{set}}(x_1,x_2) =1$ and $v(x_1) + v(x_2) =1$ a *plus-pole*.
For the diagram $s$ as in Example \[examples\] and $n/2=1$ we have $v(6) = 1$, $v(5)= 0$, $v(4)=1$, $v(3)=2$, $v(2)=2$, $v(1)=3$. We have no plus-zeros, $(4,5)$ is a plus-pole (but $(5,6)$ is not a plus-pole).
\[plus\] If $n/2$ is a noninteger then the number of plus-poles is equal to the number of plus-zeros. If $n/2$ is an integer then the number of plus-poles is equal to the number of plus-zeros plus one.
By definition, we have $v(\tilde a) + v(k) =1$. Therefore, plus-poles and plus-zeros can appear only in one of the sets $A$ and $B$ (if $v(\tilde a) < v(k)$ then in $A$, if $v(\tilde a) > v(k)$, then in $B$). For $l \in \mathbb N \cup \{0 \}$ note that in the set $\{-l, -(l-1), \dots, 0, \dots, l, l+1\}$ the number of pairs with sum 1 is greater than the number of pairs with sum 0 by 1; also note that in the set $\{-l+1/2, \dots, 1/2, \dots, l+1/2 \}$ the number of pairs with sum 1 is equal to the number of pairs with sum 0. The statement of the lemma follows from these observations.
We call a pair of numbers $x_1 < x_2$ such that $v(x_1) - v(x_2) =0$ a *minus-zero*. We call a pair of numbers $x_1 < x_2$ such that ${\mathsf{set}}(x_1,x_2) = -1$ and $v(x_1) - v(x_2) = 1$ a *minus-pole*. Let $N_0^-$ and $N_p^-$ denote the number of minus-zeros and minus-poles, respectively.
Consider a pair of elements $x_1, x_2$ such that ${\mathsf{set}}(x_1,x_2)=-1$, $v(x_1)= v(x_2)$ and ${\mathsf{prev}}(x_1) <x_2$, ${\mathsf{prev}}(x_2) <x_1$ (recall that if some of elements ${\mathsf{prev}}$ do not exist, corresponding inequalities are assumed to be true); we call such a pair a *pivot pair*. We shall denote by the symbol $L$ the number of pivot pairs.
For $s$ and $n$ as above $(2,4)$, $(3,6)$, $(2,1)$ are minus-poles (but $(5,6)$ is not a minus-pole); $(4,6)$ and $(2,3)$ are minus-zeros; also both of these minus-zeros are pivot pairs. Therefore, $L= 2$.
\[minus\] Assume that $N_p^- \ge 1$. For any such diagram and $n$ we have $$N_0^- +L -1 \ge N_p^-.$$
We shall consider the set ${\mathcal Q}_r := \{ x \in \{1, \dots, k\} : v(x) \le r \}$. Let us prove the statement of the lemma by induction in $r\geq \min\big(v(k), v(\tilde a)\big)$; on each step we consider only numbers which belong to ${\mathcal Q}_r$.
While increasing $r$ we have the following situations (note that $v^{-1}(r)$ is the preimage under the map $v$ of $r$):
1. $v^{-1} (r)$ and $v^{-1} (r-1)$ (i.e. their union) consists of one or zero elements. At this point in the induction no minus-poles and minus-zeros exist in ${\mathcal Q}_r$, and inequality clearly holds.
2. $v^{-1} (r)$ consists of two elements while $v^{-1} (r-1)$ consists of one element. The second element in $v^{-1} (r)$ must be $k$ or $\tilde a$. We know that $v(k)+v(\tilde a)=1$; therefore, the second element can give a minus-pole only if it is $\tilde a$ and $v(\tilde a) = 1$, $v(k)=0$. But we exclude the case $v(k) = n/2 =0$ from the consideration. Therefore, on this step minus-poles cannot appear. However, one minus-zero appears for sure because we have two elements in the set $v^{-1} (r)$. Thus, our inequality holds.
3. $v^{-1} (r)$ and $v^{-1} (r-1)$ each consist of two elements. On this step one minus-zero is added. Also it is easy to see that one new minus-pole appears always, and two minus-poles appear if and only if the pair of numbers forming $v^{-1} (r-1)$ is pivot. Thus, our inequality still holds.
4. $v^{-1} (r)$ consists of one element and $v^{-1} (r-1)$ consists of two elements. No minus-zero appears on this step, and one minus-pole appears if and only if the pair of numbers forming $v^{-1} (r-1)$ is pivot. Also note that if $v^{-1} (r)$ consists of zero elements, and $v^{-1} (r-1)$ consists of two elements, then no new minus-zeros and minus-poles added on this step, while $v^{-1} (r-1)$ is a pivot pair. Therefore, in this case we obtain a stronger inequality $N_0^- +L -2 \ge N_p^-$.
5. $v^{-1} (r)$ and $v^{-1} (r-1)$ consist of one element. On these steps new minus-poles and minus-zeros do not appear.
In light of having checked all of these case, we verify the statement of the lemma holds.
Assume that a diagram $s \in {\mathcal S}$ has $L= L(s)$ pivot pairs.
- If $n/2$ is a noninteger, then $R_s(z_k)$ has a pole of order not greater than $L-1$ at the point $z_k = z^* = n \frac{c}{2}$.
- If $n/2$ is an integer, then $R_s(z_k)$ has a pole of order not greater than $L$ at the point $z_k = z^* = n \frac{c}{2}$.
During the transformation of $$R(z_1, \dots, z_k) = \prod_{i<j} \frac{(z_i-z_j) (z_i+z_j)}{(z_i-z_j- c) (z_i+ z_j -c )}$$ into one-dimensional integral over $z_k$ encoded by a diagram $s \in {\mathcal S}$ some poles disappear. In more detail, in this transformation we use the poles $z_i - z_j-c$, where $i<j$ and ${\mathsf{set}}(i,j) = 1$; also we use the pole in $z_{k-1} + z_k -c$. Therefore, the poles and zeros of the function $R_s (z_k)$ at $z^*$ should come from expressions of the other form. It is easy to see that the factor $(z_i-z_j-c)$ gives rise to a pole of $R_s (z_k)$ at $z^*=nc/2$ if and only if $(i,j)$ is a minus-pole in our terminology; in the similar vein, the same is true for factors of the form $(z_i+z_j-c)$ and plus-poles, factors $(z_i-z_j)$ and minus-zeros, factors $(z_i+z_j)$ and plus-zeros. Combining Lemmas \[plus\] and \[minus\] we obtain the statement of the proposition.
\[emptyCaseRemark\] If $s$ and $n$ are such that there are no new indices involved in steps 4) and 5) in the proof of Lemma \[minus\], then the order of the pole also does not exceed $L-1$, see step 4) of the proof of Lemma \[minus\].
Proof of Theorem \[main\]
-------------------------
Let us fix $z^* = nc/2$, $n \in \mathbb Z \backslash \{0\}$. Our goal is to prove that the function $$\sum_{s \in {\mathcal S}} R_s (z) F_s (z)$$ has no pole at $z^*$. We shall give a partition of ${\mathcal S}$ into several disjoint sets such that the sum of functions over each set has no pole at $z^*$. Note that these sets depend on a fixed $z^*$. Moreover, the value $F_s( z^*)$ will be the same for diagrams $s$ from the same set, so our difficulties will come from the sum of the functions $R_s(z)$.
Let us choose $s \in {\mathcal S}$ and let us describe which diagrams are in the same set with $s$. Recall that $L(s)$ is the number of pivot pairs in $s$, and that the order of the pole at $z^*$ of the function $R_s(z_k)$ does not exceed $L(s)$.
\[def:pivot-pairs\] For the set $A(s)$ let us consider all elements of pivot pairs in this set: $w_1 < w_2 < \dots < w_{L(s)}$. For $1 \le i \le L(s)-1$ we consider the set $\mathcal A_i = \{ {\mathsf{prev}}(w_i), {\mathsf{prev}}({\mathsf{prev}}(w_i)), \dots, w_{i+1} \}$; we call $\mathcal A_i$ a *block*. We also give the same definition for blocks $\mathcal B_i$ in $B(s)$. Note that for each $i$, $\mathcal A_i$ and $\mathcal B_i$ start and end with the elements of the same pivot pairs and have the same values of the function $v_s$ on its elements (see the definition of the function $v = v_s$ in Definition \[def:func-v\]).
We obtain the function $R_s (z_k)$ after certain substitution of variables prescribed by the diagram $s$. After this all variables $z_i$ are expressed through the variable $z_k$; let us denote by $z_i^s$ the value of variable $z_i$ after the substitution prescribed by $s$.
Let us also make a change of variables ${\epsilon}_k := z_k - nc/2$. It is easy to see that other variables can be written in the following way: if $i \in A(s)$, then $z_i^s = v(i) c -{\epsilon}_k$; if $i \in B(s)$, then $z_i^s = v(i) c + {\epsilon}_k$.
Our key transformation is a *“swap”* of blocks $\mathcal A_i$ and $\mathcal B_i$. For a diagram $s \in {\mathcal S}$ we can consider the sets $\hat A_i := (A \backslash \mathcal A_i) \cup \mathcal B_i$ and $\hat B_i := (B \backslash \mathcal B_i) \cup \mathcal A_i$. Note that for any $x \in \{\mathcal A_i, \mathcal B_i\}$ and $y \in \{1, \dots, k\} \backslash \{\mathcal A_i, \mathcal B_i\}$ inequalities $x<y$ and $v(x) > v(y)$ are true or false simultaneously; this follows from the definition of pivot pairs and the fact that our blocks start and end with elements of the same pivot pairs. We obtain that the sets $\hat A_i$ and $\hat B_i$ give rise to the new diagram $s^i$.
For a diagram $$1 {\xrightarrow[]{+}}2 {\xrightarrow[]{+}}3 {\xrightarrow[]{+}}6 {\xrightarrow[]{+}}7 {\xrightarrow[]{-}}8 {\xleftarrow[]{+}}5 {\xleftarrow[]{+}}4$$ and the pole at $z_k=c$ we have $n=2$, $z_8= c + {\epsilon}_k$, $z_7 = 0 - {\epsilon}_k$, $z_6 = c-{\epsilon}_k$, $z_5= 2c + {\epsilon}_k$, $z_4= 3c + {\epsilon}_k$, $z_3= 2c - {\epsilon}_k$, $z_2= 3c - {\epsilon}_k$, $z_1= 4c - {\epsilon}_k$. It is convenient to depict this diagram and $n$ as a diagram $$\begin{gathered}
7 \ \ \ 6 \ \ \ 3 \ \ \ 2 \ \ \ 1 \\
8 \ \ \ 5 \ \ \ 4\end{gathered}$$ In such a diagram the numbers with the same value of $v$-function are on the same vertical.
The pairs $(6,8)$ and $(2,4)$ are pivot pairs, and $(3,5)$ is not a pivot pair because $4$ is to the right of $3$ on the diagram above. $\mathcal A_1$ is $\{3,2\}$, and $\mathcal B_1$ is $\{5,4\}$.
The diagram $s^1$ is $$1 {\xrightarrow[]{+}}4 {\xrightarrow[]{+}}5 {\xrightarrow[]{+}}6 {\xrightarrow[]{+}}7 {\xrightarrow[]{-}}8 {\xleftarrow[]{+}}3 {\xleftarrow[]{+}}2.$$
It turns out that we can control how the functions $R_s (z_k)$ and $R_{s^i} (z_k)$ are related. Let us describe this relation.
Recall that the function $R(z_1, \dots, z_k)$ is obtained as a product of factors $\frac{(z_i-z_j) (z_i+z_j)}{(z_i-z_j-c) (z_i+z_j-c)}$ over all pairs of integers; when we consider a one-dimensional integral some factors disappear, and we should substitute $z_r^s = v_s (r) + {\epsilon}_k$ into all other factors.
Note that the values of variables from the blocks $\mathcal A_i$ and $\mathcal B_i$ change as $v_s (z_j)+ {\epsilon}_k \to v_s (z_j) - {\epsilon}_k$ and vice versa in $s$ and $s^i$. All other variables remain the same.
Let $I_i$ be the subset of all pairs of numbers from $1$ to $k$ such that both elements of the pair belong to $\mathcal A_i \cup \mathcal B_i$. Let $${\mathcal R}_{I_i} (z_1, \dots, z_k) := \prod_{(a,b) \in I_i} \frac{z_a - z_b}{z_a - z_b -c},$$ and let $R_{s,i} ({\epsilon}_k)$ be the function obtained from ${\mathcal R}_{I_i}$ after all substitutions prescribed by $s$. Note that we can write $$R_s \left(\frac{nc}{2}+{\epsilon}_k\right) = R_{s,0} \left(\frac{nc}{2} + {\epsilon}_k\right) R_{s,i} \left(\frac{nc}{2}+{\epsilon}_k\right),$$ where the function $R_{s,0}$ comes from the product of all other factors.
\[transpose\] We have $$R_{s^i} \left( \frac{nc}{2} + {\epsilon}_k \right) = R_{s,0} \left( \frac{nc}{2} + {\epsilon}_k \right) R_{s,i} \left( \frac{nc}{2} - {\epsilon}_k \right).$$ Therefore, in order to obtain the function $R_{s^i} ({\epsilon}_k)$ from $R_s ({\epsilon}_k)$ we need to change the sign of the variable ${\epsilon}_k$ in some part of the expression.
Note that the whole set of variables takes exactly the same values in the cases of $s$ and $s^i$. Therefore, the product of factors of the form $(z_i+ z_j)$ and $(z_i + z_j -c)$ over all pairs gives exactly the same in both cases.
If both indices are inside $\mathcal A_i \cup \mathcal B_i$, then the value of $(z_i- z_j)$ can be obtained by the change ${\epsilon}_i \to (-{\epsilon}_i)$, because in both variables this change happens.
Let us consider the case when one variable ${\mathbf j}$ is outside of blocks, while the other variable is inside blocks. Assume that ${\mathbf j}$ is less than all numbers inside the blocks (the opposite case, when ${\mathbf j}$ is greater than all numbers inside the blocks, can be considered in the same way). We can group all these factors into pairs $$\frac{(z_{{\mathbf j}} - z_a) (z_{{\mathbf j}} - z_{\bar a})}{(z_{{\mathbf j}} - z_a - c) (z_{{\mathbf j}} - z_{\bar a} - c)},$$ where $(a,\bar a)$ are such that $a \in \mathcal A_i$, $\bar a \in \mathcal B_i$, and $v_s(a) = v_s (\bar a)$. Note that in both cases ($s$ and $s^i$) the expression above is equal to $$\frac{(z_{{\mathbf j}}^s - v_s(a) - {\epsilon}_k) (z_{{\mathbf j}}^s - v_s(a)+ {\epsilon}_k)}{(z_{{\mathbf j}}^s - v_s(a) - c - {\epsilon}_k) (z_{{\mathbf j}}^s - v_s(a) + {\epsilon}_k - c)},$$ therefore, the product over the pairs with fixed index ${\mathbf j}$ outside the blocks and another index from the blocks gives the same expression for the diagrams $s$ and $s^i$.
Now we are able to prove Theorem \[main\] in the case of non-integer $n/2$. We know that in this case the order of the pole does not exceed $L(s)-1$. We also know that the diagram $s$ contains $L(s)-1$ pairs of blocks $(\mathcal A_1, \mathcal B_1)$, $\dots$, $(\mathcal A_{L-1}, \mathcal B_{L-1})$. Let us “swap” these blocks in all possible ways. We obtain $2^{L(s)-1}$ different diagrams; denote this set by $T_s$.
It is enough to prove that the sum of $R_s (z_k)$ over $T_s$ does not have a pole at $z^*$. Indeed, the value of $F_s (z_k)$ is the same for any diagram $s$ from $T_s$, because all variables take the same values for diagrams from $T_s$.
Note that the sets of pairs of indices in which the transformation ${\epsilon}_k \to (-{\epsilon}_k)$ happens do not intersect for different parameters $i$ of blocks $(\mathcal A_i$, $\mathcal B_i)$. Therefore, one can write $$R_s (nc/2+{\epsilon}_k) = \hat R_{s,0} (nc/2 + {\epsilon}_k) R_{s,1} (nc/2 + {\epsilon}_k) \dots R_{s, L(s)-1} (nc/2 + {\epsilon}_k),$$ where $\hat R_{s,0}$ does not change under our “swaps”, and $R_{s,i}$ corresponds to the “swap” of $(\mathcal A_i, \mathcal B_i)$. With the use of Lemma \[transpose\] we see that the sum of all functions corresponding to these $2^{L(s)} -1$ diagrams can be written as $$\hat R_0 \left( \frac{nc}{2} + {\epsilon}_k \right) \sum_{s \in T_s} R_1 \left( \frac{nc}{2} \pm {\epsilon}_k \right) \dots R_{L-1} \left( \frac{nc}{2} \pm {\epsilon}_k \right).$$ Now we need to consider a pole of this sum at ${\epsilon}_k=0$. Consider the pairs of indices such that one of them belongs to $\mathcal A_i$ and the other one belongs to $\mathcal B_i$. It is easy to see that there are exactly $|\mathcal A_i|$ minus-zeroes among these pairs, and exactly $|\mathcal A_i|-1$ minus-poles. Therefore, there are exactly $2 |\mathcal A_i|-1$ changes of signs in the factors which give ${\epsilon}_k$ or ${\epsilon}_k^{-1}$. Thus, the sum in the previous formula can be written as
$$\frac{\hat R_0 \left( \frac{nc}{2} + {\epsilon}_k \right)}{{\epsilon}_k^{deg}} \sum_{s \in T_s} (-1)^{\mbox{number of minuses}} \tilde R_1 \left( \frac{nc}{2} \pm {\epsilon}_k \right) \dots \tilde R_{L-1} \left( \frac{nc}{2} \pm {\epsilon}_k \right),$$
where $\hat R_0$, $\tilde R_i$ has no singularities at ${\epsilon}_k =0$ and the degree $deg$ does not exceed $L(s)-1$. But one can write such a sum in the form $$\frac{\hat R_0 \left( \frac{nc}{2} + {\epsilon}_k \right)}{{\epsilon}_k^{deg}} \left( \tilde R_1 \left( \frac{nc}{2} + {\epsilon}_k \right) - \tilde R_1 \left( \frac{nc}{2} - {\epsilon}_k \right) \right) \dots \left( \tilde R_{L-1} \left( \frac{nc}{2} + {\epsilon}_k \right) - \tilde R_{L-1} \left( \frac{nc}{2} - {\epsilon}_k \right) \right).$$ It is clear that such an expression does not have a pole at ${\epsilon}_k =0$.
One can readily see that all diagrams from ${\mathcal S}$ can be split into such disjoint groups $T_s$. This completes the proof of Theorem \[main\] in the case of non-integer $n/2$.
Now we need to consider the case when $n/2$ is an integer. In this case the order of the pole at ${\epsilon}_k=0$ can be equal to $L(s)$. The proof uses the same mechanism, but we need to consider one more allowed transform of the diagram $s$ (in addition to “swaps” of blocks $\mathcal A_i$ and $\mathcal B_i$ for $1 \le i \le L(s)-1$). Let us define new blocks $\mathcal A_L = \{ x \in A(s) : x< w_L \}$ and $\mathcal B_L = \{ x \in B(s) : x < w'_L \}$ (see the definition of $w_{L}, w'_{L}$ in Definition \[def:pivot-pairs\]).
First, let us consider the case when both new blocks are empty. Then the results of Section \[structure\] (see Remark \[emptyCaseRemark\]) show that the order of the pole at ${\epsilon}_k=0$ does not exceed $L(s)-1$; therefore, it is enough to consider “swaps” of previously defined blocks in order to obtain a cancelation of residues.
In the general case, the difference of this pair of blocks from the previous ones is that in this pair the blocks can have different number of elements. However, we still can “swap” these blocks as we did above and obtain a new diagram which we denote $s^L$ (again, we obtain a correctly defined diagram due to the definition of pivot pairs).
Let us describe how the functions $R_s (z_k)$ and $R_{s^L} (z_k)$ are related.
We call the elements of $\mathcal A_L \cup \mathcal B_L$ which belong to some minus-zero *regular*, and we call other numbers from $\mathcal A_L \cup \mathcal B_L$ *irregular*. Let us denote by $\mathcal P$ the elements from $\{1,\dots,k\} \backslash (\mathcal A_L \cup \mathcal B_L)$ which belong to some minus-zero; the other elements from $\{1,\dots,k\} \backslash (\mathcal A_L \cup \mathcal B_L)$ we denote by $\mathcal{NP}$.
Consider a diagram with $n/2=-1$ represented by the diagram $$\begin{gathered}
15 \ \ \ 13 \ \ \ 12 \ \ \ 10 \ \ \ \ 9 \ \ \ \ 8 \ \ \ 6 \ \ \ 5 \ \ \ 3 \ \ \ 1 \\
\ \ \ \ \ \ \ \ \ 14 \ \ \ 11 \ \ \ 7 \ \ \ 4 \ \ \ 2\end{gathered}$$ For this diagram we have 1 plus-zero $(12,15)$, 2 plus-poles $(10,15)$ and $(12,13)$, 5 minus-zeros, 6 minus-poles, two pivot pairs $(7,8)$ and $(9,11)$ (other minus-zeros are not pivot pairs), $L(s)=2$. We have $\mathcal A_1=\{7 \}$ and $\mathcal B_1 = \{8\}$, $\mathcal A_2 = \{2,4 \}$ and $\mathcal B_2 = \{1,3,5,6\}$. The indices $\{1,3\}$ are irregular, the indices $\{2,5,4,6\}$ are regular. The set $\mathcal{NP}$ is $\{12,13,15\}$ and the set $\mathcal P$ is $\{7,8,9,11,10,14\}$.
Let $J_1$ be the set of pairs of indices such that one of the indices is irregular and another belongs to $\mathcal{NP}$. Let $J_2$ be the set of pairs of indices such that one of the indices is irregular and another belongs to $\mathcal{P}$. Let $J_3$ be the set of pairs of indices such that both indices belong to $\mathcal A_L \cup \mathcal B_L$.
Let $$R_{J_1,L} (z_1, \dots, z_k) := \prod_{(i,j) \in J_1} \frac{z_i+z_j}{z_i+z_j-c} \frac{z_i-z_j}{z_i-z_j-c},$$
$$R_{J_2,L} (z_1, \dots, z_k) := \prod_{(i,j) \in J_2} \frac{z_i+z_j}{z_i+z_j-c} \frac{z_i-z_j}{z_i-z_j-c},$$
$$R_{J_3,L} (z_1, \dots, z_k) := \prod_{(i,j) \in J_3} \frac{z_i+z_j}{z_i+z_j-c} \frac{z_i-z_j}{z_i-z_j-c},$$ let $R^1_s (nc/2+ {\epsilon}_k)$, $R^2_s (nc/2+ {\epsilon}_k)$, and $R^3_s (nc/2+ {\epsilon}_k)$ be the functions which are obtained from the functions above after the substitution of variables prescribed by $s$, and let $$R_{s,L} \left( \frac{nc}{2} + {\epsilon}_k \right) := R^1_s \left( \frac{nc}{2} + {\epsilon}_k \right) R^2_s \left( \frac{nc}{2} + {\epsilon}_k \right) R^3_s \left( \frac{nc}{2} + {\epsilon}_k \right).$$ Note that we can write $$R_s \left( \frac{nc}{2} + {\epsilon}_k \right) = \hat R_{s,0} \left( \frac{nc}{2} + {\epsilon}_k \right) R_{s,L} \left( \frac{nc}{2} + {\epsilon}_k \right),$$ where $\hat R_0$ comes as a product over other pairs of indices.
We have for $j=1,2,3$ $$R^j_{s^L} \left( \frac{nc}{2}+ {\epsilon}_k \right) = R^j_s \left( \frac{nc}{2} - {\epsilon}_k \right).$$ We also have the equality $$R_{s^L} \left( \frac{nc}{2}+ {\epsilon}_k \right) = \hat R_0 \left( \frac{nc}{2} + {\epsilon}_k \right) R_L \left( \frac{nc}{2} - {\epsilon}_k \right),$$ similar to the case of other blocks.
Let $i$ be an irregular index; it is less than indices from $\mathcal P$ and $\mathcal{NP}$ due to the definition of a pivot pair. Then $z^{s^L}_i = v_s(i) c \mp {\epsilon}_k$.
Let $N := \max (n/2, - n/2)$; after the substitution prescribed by $s$ the variables from $\mathcal{NP}$ take values $\{-Nc+{\epsilon}_k, \dots, 0+{\epsilon}_k, \dots, Nc+{\epsilon}_k\}$ or the same expressions with $(-{\epsilon}_k)$ instead of $(+{\epsilon}_k)$. Assume that the $+$ sign appears here; the opposite case can be considered in the same way. For an index $j \in \mathcal{NP}$ with $z^s_j = v_s (j) c + {\epsilon}_k$ let $\bar j$ denote the index such that $z^s_{\bar j} = - v_s(j) c +{\epsilon}_k$. For a fixed irregular index $i$ and for any $j \in \mathcal{NP}$ one can directly verify that the sets $\{ z_i^s - z_j^s, z_i^s+z_j^s\}$ and $\{z_i^{s^L} - z_{\bar j}^{s^L}, z_i^{s^L} + z_{\bar j}^{s^L} \}$ are obtained from each other by ${\epsilon}_k \to (-{\epsilon}_k)$. Also the same is true for the sets $\{ z_i^{s^L} - z_j^{s^L}, z_i^{s^L}+z_j^{s^L} \}$ and $\{z_i^{s} - z_{\bar j}^{s}, z_i^{s} + z_{\bar j}^{s} \}$. The statement of the lemma about the function $R^1_{s^L}$ follows from this.
For any $a \in \mathcal P$ denote by $b$ the index such that $(a,b)$ is a minus-zero. We have $z_a^s = v_s(a) + {\epsilon}_k$ and $z_b^s = v_s(a) - {\epsilon}_k$ (or vice versa). One can directly verify that the sets $\{ z_i^s - z_a^s, z_i^s- z_b^s\}$ and $\{z_i^{s^L} - z_a^{s^L}, z_i^{s^L} - z_b^{s^L} \}$ are obtained from each other by ${\epsilon}_k \to (-{\epsilon}_k)$. Also the same is true for the sets $\{ z_i^s + z_a^s, z_i^s + z_b^s\}$ and $\{z_i^{s^L} + z_a^{s^L}, z_i^{s^L} + z_b^{s^L} \}$. The statement of the lemma about the function $R^2_{s^L}$ follows from this.
If both indices belong to $\mathcal A_{L(s)} \cup \mathcal B_{L(s)}$, then in both indices the change ${\epsilon}_k \to (-{\epsilon}_k)$ happens. The statement of the lemma about the function $R^3_{s^L}$ follows from this.
If both indices do not belong to $\mathcal A_{L(s)} \cup \mathcal B_{L(s)}$, then obviously corresponding factors do not change.
The only remaining case is that one index is regular, and the other is from $\{1, \dots, k\} \backslash \left( \mathcal A_L \cup \mathcal B_L \right)$; the product over corresponding factors does not change — this can be shown in the same way as in the proof of Lemma \[transpose\].
Note that the sets of pairs of indices which give rise to the change ${\epsilon}_k \to (-{\epsilon}_k)$ in corresponding factors do not intersect for the pairs of blocks $(\mathcal A_1, \mathcal B_1)$, $\dots$, $(\mathcal A_L, \mathcal B_L)$. Let us swap these blocks in all possible ways, obtaining $2^{L}$ different diagrams. Recall that the order of the pole at ${\epsilon}_k=0$ does not exceed $L$. In exactly the same way as before we can show that the sum over these diagrams does not have a pole at ${\epsilon}_k=0$ which implies Theorem \[main\] in the general case.
Consider the diagram $s$ $$1 {\xrightarrow[]{+}}2 {\xrightarrow[]{+}}3 {\xrightarrow[]{+}}6 {\xrightarrow[]{+}}7 {\xrightarrow[]{-}}8 {\xleftarrow[]{+}}5 {\xleftarrow[]{+}}4.$$ One can compute that $$R_s (z_8) = \frac{315(-2 z_8+5c)(2 z_8+c) c^7 (-2 z_8+9c)(-2 z_8+7c)(-4z_8^2+9c^2)}{(-2z_8+c)^2 (-z8+c)^2 z_8^2}.$$ and the residue at point $z_8= c$ is equal to $-4110750 c^8$. In this case our blocks are $\mathcal A_1 = \{2,3\}$, $\mathcal B_1 = \{4,5\}$, $\mathcal A_2 = \{1\}$, and the block $\mathcal B_2$ is empty. Transposing these blocks in all possible ways we obtain three new diagrams: $$\begin{aligned}
1 {\xrightarrow[]{+}}4 {\xrightarrow[]{+}}5 {\xrightarrow[]{+}}6 {\xrightarrow[]{+}}7 {\xrightarrow[]{-}}8 {\xleftarrow[]{+}}3 {\xleftarrow[]{+}}2, \\
2 {\xrightarrow[]{+}}3 {\xrightarrow[]{+}}6 {\xrightarrow[]{+}}7 {\xrightarrow[]{-}}8 {\xleftarrow[]{+}}5 {\xleftarrow[]{+}}4 {\xleftarrow[]{+}}1, \\
4 {\xrightarrow[]{+}}5 {\xrightarrow[]{+}}6 {\xrightarrow[]{+}}7 {\xrightarrow[]{-}}8 {\xleftarrow[]{+}}3 {\xleftarrow[]{+}}2 {\xleftarrow[]{+}}1.\end{aligned}$$ Computations of residues of functions corresponding to these diagrams at the point $z_8=c$ give numbers $-850500 c^8$, $3827250 c^8$, and $1134000 c^8$; we have $$-4110750 -850500 + 3827250 + 1134000 =0.$$
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[^1]: Specifically, the Plancherel formula for the repulsive case is given therein as Theorem 1.3. Page 14 then explains why this formula holds in both repulsive and attractive cases, with nested contours in the attractive case. Theorem 3.13 is the residue computation when the contours come together, but it is not complete as the constants are not computed exactly (Heckman-Opdam did not need the exact values).
[^2]: While this is the standard notation, we note that Wikipedia records $(a)_n$ as the falling factorial.
[^3]: From a mathematical perspective, the equivalence of these two formulations has only been shown in the full-space case [@BC], however, one expects the same methods to apply in the half-space case.
[^4]: This fact is proved in [@M] and more recently in [@Gregorio] for the full-space case. A proof does not seem to exist presently for the half-space case, though one expects the methods should extend without much difficulty.
[^5]: These asymptotic calculations can be performed in an entirely mathematically rigorous manner – see [@ACQ; @BCF].
[^6]: We note, however, that we are not aware of a uniqueness result for this delta Bose gas which includes the delta initial data we consider.
[^7]: We note, however, that we are not aware of a uniqueness result for this delta Bose gas which includes the delta initial data we consider.
[^8]: As long as all $x_i$ are equal, the identity in equation (\[Akfacsym\]) can be applied. However, when the $x_i$ differ it is not clear how to simplify $E_{\vec{x}}$ in a similar manner.
[^9]: We are presently only aware of a suitable identity to deal with the case $x_i\equiv 0$ (even $x_i\equiv x$ for $x\neq 0$ is unclear).
[^10]: One shows that $k^{th}$ moment grows as $\exp({\ensuremath{\textrm{const}}}\cdot k^3)$ for some ${\ensuremath{\textrm{const}}}>0$ and $k$ and $t$ large, see e.g. [@BC], [@BCLyapunovpaper].
[^11]: Strictly speaking the above only shows that $\frac{d}{dr'}K^{\infty}_{12}(r,r') = \frac{d}{dr'} K^{(4)}_{11}(r,r')$; hence the two functions of $r'$ could (in principle) differ by a constant. However, it is easy to compare the two functions as $r'$ goes to infinity and see that they are equal there, hence the constant is zero and the equality holds.
|
---
abstract: 'Fine-grained visual recognition is challenging because it highly relies on the modeling of various semantic parts and fine-grained feature learning. Bilinear pooling based models have been shown to be effective at fine-grained recognition, while most previous approaches neglect the fact that inter-layer part feature interaction and fine-grained feature learning are mutually correlated and can reinforce each other. In this paper, we present a novel model to address these issues. First, a cross-layer bilinear pooling approach is proposed to capture the inter-layer part feature relations, which results in superior performance compared with other bilinear pooling based approaches. Second, we propose a novel hierarchical bilinear pooling framework to integrate multiple cross-layer bilinear features to enhance their representation capability. Our formulation is intuitive, efficient and achieves state-of-the-art results on the widely used fine-grained recognition datasets.'
author:
- Chaojian Yu
- Xinyi Zhao
- Qi Zheng
- Peng Zhang
- 'Xinge You$^{(}$$^{)}$'
bibliography:
- 'samplepaper.bib'
title: |
Hierarchical Bilinear Pooling\
for Fine-Grained Visual Recognition
---
=1
Introduction
============
With the development of artificial intelligence, increasing demand appears to recognize subcategories of objects under the same basic-level category, e.g., brand identification for businessman, plant recognition for botanist. Thus recent years have witnessed great progress in fine-grained visual recognition, which has been widely used in applications such as automatic driving [@sochor2016boxcars], expert-level image recognition [@krause2016unreasonable], etc. Different from general image classification task (e.g., ImageNet classification [@russakovsky2015imagenet]) that is to distinguish basic-level categories, fine-grained visual recognition is very challenging as subcategories tend to own small variance in object appearance and thus can only be recognized by some subtle or local differences. For example, we discriminate breeds of birds depending on the color of their back or the shape of their beak.
Motivated by the observation that local parts of object usually act a role of importance in differentiating subcategories, many methods [@zhang2016spda; @branson2014bird; @simon2015neural; @zhang2016picking] for fine-grained classification were developed by exploiting the parts, namely part-based approaches. They mainly consist of two steps: firstly localize the foreground object or object parts, e.g., by utilizing available bounding boxes or part annotations, and then extract discriminative features for further classification. However, these approaches suffer from two essential limitations. First, it is difficult to ensure the manually defined parts are optimal or suitable for the final fine-grained classification task. Second, detailed part annotations incline to be time consuming and labor intensive, which is not feasible in practice. Therefore, some other approaches employ unsupervised techniques to detect possible object regions. For example, Simon and Rodner [@simon2015neural] proposed a constellation model to localize parts of objects, leveraging convolutional neural network (CNN) to find the constellations of neural activation patterns. Zhang *et al.* [@zhang2016picking] proposed an automatic fine-grained image classification method, incorporating deep convolutional filters for both selection and description related to parts. These models regard CNN as part detector and obtain great improvement in fine-grained recognition. Unlike part-based methods, we treat activations from different convolution layers as responses to different part properties instead of localizing object parts explicitly, leveraging cross-layer bilinear pooling to capture inter-layer interaction of part attributes, which is proved to be useful for fine-grained recognition.
Alternatively, some researches [@cai2017higher; @gao2016compact; @lin2015bilinear; @kong2017low] introduced bilinear pooling frameworks to model local parts of object. Although promising results have been reported, further improvement suffers from the following limitations. First, most existing bilinear pooling based models only take activations of the last convolution layer as representation of an image, which is insufficient to describe various semantic parts of object. Second, they neglect intermediate convolution activations, resulting in a loss of discriminative information of fine-grained categories which is significant for fine-grained visual recognition.
In this work, we present new methods to address the above challenges. We find that inter-layer part feature interaction and fine-grained feature learning are mutually correlated and can reinforce each other. To better capture the inter-layer feature relations, we propose a cross-layer bilinear pooling approach. The proposed method is efficient and powerful. It takes into account the inter-layer feature interactions while avoiding introducing extra training parameters. In contrast to other bilinear pooling based works which only utilize feature from one single convolution layer, our architecture exploits the interaction of part features from multiple layers, which is useful for fine-grained feature learning. Besides, our framework is highly consistent with the human coarse-to-fine perception, the visual hierarchy segregates local and global features in cortical areas V4 based on spatial differences and builds a temporal dissociation of the neural activity [@lu2018revealing]. We find that our cross-layer bilinear model is closer to the unique architecture of cortical areas V4 for processing spatial information.
It is well known that information loss exists in the propagation of CNNs. In order to minimize the loss of information that is useful for fine-grained recognition, we propose a novel hierarchical bilinear pooling framework to integrate multiple cross-layer bilinear features to enhance their representation power. To make full use of the intermediate convolution layer activations, all cross-layer bilinear features are concatenated before the final classification. Note that the features from different convolution layer are complementary, they contribute to discriminative feature learning. Thus the proposed network benefits from the mutual reinforcement between inter-layer feature interaction and fine-grained feature learning. Our contributions are summarized as follows:
- We develop a simple but effective cross-layer bilinear pooling technique that simultaneously enables the inter-layer interaction of features and the learning of fine-grained representation in a mutually reinforced way.
- Based on cross-layer bilinear pooling, we propose a hierarchical bilinear pooling framework to integrate multiple cross-layer bilinear modules to obtain the complementary information from intermediate convolution layers for performance boost.
- We conduct comprehensive experiments on three challenging datasets (CUB Birds, Stanford Cars, FGVC-Aircraft), and the results demonstrate the superiority of our method.
The rest of this paper is organized as follows. Section \[sec:re\_w\] reviews the related work. Section \[sec:model\] presents the proposed method. Section \[sec:exp\] provides experiments as well as result analysis, followed by conclusion in Section \[sec:conc\].
Related Work {#sec:re_w}
============
In the following, we briefly review previous works from the two viewpoints of interest due to their relevance to our work, including fine-grained feature learning and feature fusion in CNNs.
Fine-Grained Feature Learning
-----------------------------
Feature learning plays an important and fundamental role in fine-grained recognition. Since the differences between subcategories are subtle and local, capturing global semantic information with merely fully connected layers limits the representation capacity of a framework, and hence restricts further promotion of final recognition [@babenko2015aggregating]. To better model subtle difference for fine-grained categories, Lin *et al.* [@lin2015bilinear] proposed a bilinear structure to aggregate the pairwise feature interactions by two independent CNNs, which adopted outer product of feature vectors to produce a very high-dimensional feature for quadratic expansion. Gao *et al.* [@gao2016compact] applied Tensor Sketch [@pham2013fast] to approximate the second-order statistics and to reduce feature dimension. Kong *et al.* [@kong2017low] adopted low-rank approximation to the covariance matrix and further reduced the computational complexity. Yin *et al.* [@cui2017kernel] aggregated higher-order statistics by iteratively applying the Tensor Sketch compression to the features. The work in [@moghimi2016boosted] utilized bilinear convolutional neural network as baseline model and adopted an ensemble learning method to incorporate boosting weights. In [@lin2017improved], matrix square-root normalization was proposed and proved to be complementary to existing normalization. However, these approaches only consider the feature from single convolution layer, which is insufficient to capture various discriminative parts of object and model the subtle differences among subcategories. The method we propose overcome this limitation via integrating inter-layer feature interaction and fine-grained feature learning in a mutually reinforced manner and is therefore more effective.
Feature Fusion in CNNs
----------------------
Due to the success of deep learning, CNNs have emerged as general-purpose feature extractors for a wide range of visual recognition tasks. While feature maps from single convolution layer are insufficient for finer-grained tasks, thus some recent works [@cai2017higher; @hariharan2015hypercolumns; @long2015fully; @xie2015holistically] attempt to investigate the effectiveness of exploiting feature from different convolution layers within a CNN. For example, Hariharan *et al.* [@hariharan2015hypercolumns] considered the feature maps from all convolution layers, allowing finer grained resolution for localization tasks. Long *et al.* [@long2015fully] combined the finer-level and higher-level semantic feature from different convolution layers for better segmentation. Xie *et al.* [@xie2015holistically] proposed a holistically-nested framework where the side outputs are added after lower convolution layers to provide deep supervision for edge detection. The very recent work [@cai2017higher] concatenated the activation maps from multiple convolution layers to model the interaction of part features for fine-grained recognition. However, simply cascading the feature map introduces lots of training parameters and even fails to capture inter-layer feature relations when incorporating with more intermediate convolution layers. Instead, our network treats each convolution layer as attribute extractor for different object parts and models their interactions in an intuitive and effective way.
Hierarchical Bilinear Model {#sec:model}
===========================
In this section, we develop a hierarchical bilinear model to overcome those limitations mentioned above. Before presenting our hierarchical bilinear model, we first introduce the general formulation of factorized bilinear pooling for fine-grained image recognition in Sect. \[sec:model\_fbp\]. Based on this, we propose a cross-layer bilinear pooling technique to jointly learn the activations from different convolution layers in Sect. \[sec:model\_cbp\], which captures the cross-layer interaction of information and leads to better representation capability. Finally, our hierarchical bilinear model combining multiple cross-layer bilinear modules generates finer part description for better fine-grained recognition in Sect. \[sec:model\_hbm\].
Factorized Bilinear Pooling {#sec:model_fbp}
---------------------------
Factorized bilinear pooling has been applied to visual question answer task, Kim *et al.* [@kim2016hadamard] proposed factorized bilinear pooling using Hadamard product for an efficient attention mechanism of multimodal learning. Here we introduce the basic formulation of factorized bilinear pooling technique for the task of fine-grained image recognition. Suppose an image $I$ is filtered by a CNN and the output feature map of a convolution layer is $X \in \mathbb{R}^{h \times w \times c}$ with height $h$, width $w$ and channels $c$, we denote a $c$ dimensional descriptor at a spatial location on $X$ as $\mathbf{x} = [x_1,x_2,\cdots,x_c]^T$. Then the full bilinear model is defined by $$\begin{aligned}
\label{eq:eq1}
z_i=\mathbf{x}^TW_i\mathbf{x}\end{aligned}$$ Where $W_i \in \mathbb{R}^{c \times c}$ is a projection matrix, $z_i$ is the output of the bilinear model. We need to learn $\mathbf{W}=[W_1,W_2,\cdots,W_o] \in \mathbb{R}^{c \times c \times o}$ to obtain a $o$ dimensional output $\mathbf{z}$. According to matrix factorization in [@rendle2010factorization], the projection matrix $W_i$ in Eq. (\[eq:eq1\]) can be factorized into two one-rank vectors $$\begin{aligned}
\label{eq:eq2}
z_i=\mathbf{x}^TW_i\mathbf{x}=\mathbf{x}^TU_iV_i^T\mathbf{x}=U_i^T\mathbf{x} \circ V_i^T\mathbf{x}\end{aligned}$$ where $U_i \in \mathbb{R}^c$ and $V_i \in \mathbb{R}^c$. Thus the output feature $\mathbf{z} \in \mathbb{R}^o$ is given by $$\begin{aligned}
\label{eq:eq3}
\mathbf{z}=P^T(U^T\mathbf{x} \circ V^T\mathbf{x})\end{aligned}$$ where $U \in \mathbb{R}^{c \times d}$ and $V \in \mathbb{R}^{c \times d}$ are projection matrices, $P \in \mathbb{R}^{d \times o}$ is the classification matrix, $\circ$ is the Hadamard product and $d$ is a hyperparameter deciding the dimension of joint embeddings.
Cross-Layer Bilinear Pooling {#sec:model_cbp}
----------------------------
Fine-grained subcategories tend to share similar appearances and can only be discriminated by subtle differences in the attributes of local part, such as color, shape, or length of beak for birds. Bilinear pooling, which captures the pairwise feature relations, is an important technique for fine-grained recognition. However, most bilinear models only focus on learning the features from single convolution layer while completely ignoring the cross-layer interaction of information. Activations of individual convolution layer are incomplete since there are multiple attributes in each object part which can be crucial in differentiating subcategories.
Actually in most cases, we need to simultaneously consider multi-factor of part feature to determine the category for a given image. Therefore, to capture finer grained part feature, we develop a cross-layer bilinear pooling approach that treats each convolution layer in a CNN as part attributes extractor. After that the features from different convolution layers are integrated by element-wise multiplication to model the inter-layer interaction of part attributes. Accordingly, Eq. (\[eq:eq3\]) can be rewritten as $$\begin{aligned}
\mathbf{z}=P^T(U^T\mathbf{x} \circ V^T\mathbf{y})\end{aligned}$$ where $\mathbf{x}$ and $\mathbf{y}$ represent local descriptors from different convolution layers at the same spatial location.
It is worth noting that the features from different convolution layers are expanded into high-dimensional space by independent linear mappings. It is expected that the convolution activations and project activations encode global and local feature of object respectively, as shown in Fig. \[fig:fig3\]. It is highly consistent with the human coarse-to-fine perception: human and non-human primates often see the global “gist” of an object, or a scene, before discerning local detailed features [@lu2018revealing]. For example, neurons in macaque inferotemporal cortex that are active during face perception encode the global facial category is earlier than they begin to encode finer information such as identity or expression.
![Illustration of our Hierarchical Bilinear Pooling (HBP) network architecture for fine-grained recognition. The bottom image is the input, and above it are the feature maps of different layers in the CNN. First the features from different layers are expanded into a high-dimensional space via independent linear mapping to capture attributes of different object parts and then integrated by element-wise multiplication to model the inter-layer interaction of part attributes. After that sum pooling is performed to squeeze the high-dimensional features into compact ones. Note that we obtain the visual activation maps above by computing the response of sum-pooled feature vector on every single spatial location. []{data-label="fig:fig1"}](./images/figure1.pdf){height="6.5cm"}
Hierarchical Bilinear Pooling {#sec:model_hbm}
-----------------------------
Cross-layer bilinear pooling proposed in Sect. \[sec:model\_cbp\] is intuitive and effective, as it has superior representation capacity than traditional bilinear pooling models without increasing training parameters. This inspires us that exploiting the inter-layer feature interactions among different convolution layers is beneficial for capturing the discriminative part properties between fine-grained subcategories. Therefore, we extend the cross-layer bilinear pooling to integrate more intermediate convolution layers, which further enhances the representation capacity of features. In this section, we propose a generalized Hierarchical Bilinear Pooling (HBP) framework to incorporate more convolutional layer features by cascading multiple cross-layer bilinear pooling modules.
Specifically, we divide the cross-layer bilinear pooling module into interaction stage and classification stage, which formulates as follows $$\begin{aligned}
\mathbf{z}_{int}=U^T\mathbf{x} \circ V^T\mathbf{y}\\
\mathbf{z}=P^T\mathbf{z}_{int} \in \mathbb{R}^o\end{aligned}$$ To better model inter-layer feature interactions, the interaction feature of the HBP model is obtained by concatenating multiple $\mathbf{z}_{int}$ of the cross-layer bilinear pooling modules. Thus we can derive final output of the HBP model by $$\begin{aligned}
\mathbf{z}_{HBP}&=HBP(\mathbf{x},~\mathbf{y},~\mathbf{z},~\cdots)=P^T\mathbf{z}_{int}\\
&=P^Tconcat(U^T\mathbf{x} \circ V^T\mathbf{y},~U^T\mathbf{x} \circ S^T\mathbf{z},~V^T\mathbf{y} \circ S^T\mathbf{z},~\cdots)\end{aligned}$$ where $P$ is the classification matrix, $U,V,S,\ldots$ are the projection matrices of convolution layer feature $\mathbf{x},\mathbf{y},\mathbf{z},\ldots$ respectively. The overall flowchart of the HBP framework is illustrated in Fig. \[fig:fig1\].
Experiments {#sec:exp}
===========
In this section, we evaluate the performance of HBP model for fine-grained recognition. The datasets and implementation details of HBP are firstly introduced in Sect. \[sec:dataset\]. Model configuration studies are performed to investigate the effectiveness of each component in Sect. \[sec:config\]. Comparison with state-of-the-art methods is provided in Sect. \[sec:comp\]. Finally in Sect. \[sec:qual\], qualitative visualization is present to intuitively explain our model.
Datasets and Implementation Details {#sec:dataset}
-----------------------------------
### Datasets:
We conduct experiments on three widely used datasets for fine-grained image recognition, including Caltech-UCSD Birds (CUB-200-2011) [@wah2011caltech], Stanford Cars [@krause20133d] and FGVC-Aircraft [@maji2013fine]. The detailed statistics with category numbers and data splits are summarized in Table \[table:table1\]. Note that we only use category labels in our experiments.
Datasets \#Category \#Training \#Testing
-------------------------------- ------------ ------------ -----------
CUB-200-2011 [@wah2011caltech] 200 5994 5794
Stanford Cars [@krause20133d] 196 8144 8041
FGVC-Aircraft [@maji2013fine] 100 6667 3333
: Summary statistics of datasets[]{data-label="table:table1"}
### Implementation Detail:
For fair comparison with other state-of-the-art methods, we evaluate our HBP with VGG-16 [@simonyan2014very] baseline model pretrained on ImageNet classification dataset [@russakovsky2015imagenet], removing the last three fully-connected layers and inserting all the components in our framework. It is worth noting that our HBP can be also applied to other network structures, such as Inception [@szegedy2015going] and ResNet [@he2016deep]. The size of input image is $448 \times 448$. Our data augmentation follows the commonly used practice, i.e., random sampling (crop $448 \times 448$ from $512 \times S$ where $S$ is the largest image side) and horizontal flipping are utilized during training, and only center cropping is involved during inference.
We initially train only the classifiers by logistic regression, and then fine-tune the whole network using stochastic gradient descent with a batch size of 16, momentum of 0.9, weight decay of $5 \times 10^{-4}$ and a learning rate of $10^{-3}$, periodically annealed by 0.5. All experiments are implemented with the Caffe toolbox [@jia2014caffe] and performed on a server with Titan X GPUs. The source code and trained model will be made available at <https://github.com/ChaojianYu/Hierarchical-Bilinear-Pooling>
Configurations of Hierarchical Bilinear Pooling {#sec:config}
-----------------------------------------------
Cross-layer bilinear pooling (CBP) has a user-define projection dimension $d$. To investigate the impact of $d$ and to validate the effectiveness of the proposed framework, we conduct extensive experiments on the CUB-200-2011 [@wah2011caltech] dataset, with results summarized in Fig. \[fig:fig2\]. Note that we utilize $relu5\_3$ in FBP, $relu5\_2$ and $relu5\_3$ in CBP, $relu5\_1,~relu5\_2$ and $relu5\_3$ in HBP to obtain the results in Fig. \[fig:fig2\] and we also provide quantitative experiments about the choice of layers in the following. We focus on $relu5\_1,~relu5\_2$ and $relu5\_3$ in VGG-16 [@simonyan2014very] as they contain more part semantic information compared with shallower layers.
![Classification accuracy on the CUB dataset. Comparison of general Factorized Bilinear Pooling (FBP), Cross-layer Bilinear Pooling (CBP) and Hierarchical Bilinear Pooling (HBP) with various projection dimensions.[]{data-label="fig:fig2"}](./images/figure2.pdf){height="6.5cm"}
In Fig. \[fig:fig2\], we compare the performance of CBP with the general factorized bilinear pooling model, namely FBP. Futhermore, we explore HBP with combination of multiple layers. Finally, we analyze the impact factors of hyperparameter $d$. We can draw the following significant conclusions from Fig. \[fig:fig2\]
- First, under the same $d$, our CBP significantly outperforms FBP, which indicates that the discriminative power can be enhanced by the inter-layer interaction of features.
- Second, HBP further outperforms CBP, which demonstrates the efficacy of activations from intermediate convolution layers for fine-grained recognition. This can be explained by the fact that information loss exists in the propagation of CNNs, thus discriminative features crucial for fine-grained recognition may be lost in intermediate convolution layers. In contrast to CBP, our HBP takes more feature interactions of intermediate convolution layers into consideration and is therefore more robust, since HBP has presented the best performance. In the following experiments, HBP is used to compare with other state-of-the-art methods.
- Third, when $d$ varies from 512 to 8192, increasing $d$ leads to higher accuracy for all models and HBP is saturated with $d=8192$. Therefore, $d=8192$ is used for HBP in our following experiments in consideration of feature dimension, computational complexity as well as accuracy.
We then provide quantitative experiments on the CUB-200-2011 [@wah2011caltech] dataset to analyze the impact factor of layers. The accuracies in Table \[table:table11\] are obtained under the same embedding dimension ($d=8192$). We consider the combination of different layers for CBP and HBP. The results demonstrate that the performance gain of our framework comes mainly from the inter-layer interaction and multiple layers combination. As the HBP-3 already presents the best performance, thus we utilize $relu5\_1,~relu5\_2$ and $relu5\_3$ in all the experiments in Sect. \[sec:comp\].
---------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
FBP-1[^1] CBP-1[^2] CBP-2[^3] CBP-3[^4] HBP-1[^5] HBP-2[^6] HBP-3[^7]
Accuracy 85.70 86.75 86.85 86.67 86.78 86.91 87.15
---------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: Quantitative analysis results on CUB-200-2011 dataset
\[table:table11\]
We also compare our cross-layer integration with hypercolumn [@cai2017higher] based feature fusion. For fair comparison, we re-implement hypercolumn as the feature concatenation of $relu5\_3$ and $relu5\_2$, followed by factorized bilinear pooling (denoted as HyperBP) under the same experimental settings. Table \[table:table33333\] shows that our CBP obtains slightly better result than HyperBP with nearly 1/2 parameters, which again indicates that our integration framework is more effective in capturing inter-layer feature relations. This is not surprising since our CBP is consistent with human perception to some extent. On the contrary of the HyperBP, which obtains even worse result when integrating more convolution layer activations [@cai2017higher], our HBP is able to capture the complementary information within intermediate convolution layers and achieves an obvious improvement in recognition accuracy.
Method Accuracy Model Size
--------- ------------------ ------------
HyperBP 86.60 18.4M
CBP 86.75 10.0M
HBP $\mathbf{87.15}$ 17.5M
: Classification accuracy on the CUB dataset and model sizes of different feature integrations[]{data-label="table:table33333"}
Comparison with State-of-the-art {#sec:comp}
--------------------------------
### Results on CUB-200-2011.
CUB dataset provides ground-truth annotations of bounding boxes and parts of birds. The only supervised information we use is the image level class label. The classification accuracy on CUB-200-2011 is summarized in Table \[table:table2\]. The table is split into three parts over the rows: the first summarizes the annotation-based methods (using object bounding boxes or part annotations); the second includes the unsupervised part-based methods; the last illustrates the results of pooling-based methods.
Method Anno. Accuracy
--------------------------------------- --------- -----------------
SPDA-CNN [@zhang2016spda] $\surd$ 85.1
B-CNN [@lin2015bilinear] $\surd$ 85.1
PN-CNN [@branson2014bird] $\surd$ 85.4
STN [@jaderberg2015spatial] 84.1
RA-CNN [@fu2017look] 85.3
MA-CNN [@zheng2017learning] 86.5
B-CNN [@lin2015bilinear] 84.0
CBP [@gao2016compact] 84.0
LRBP [@kong2017low] 84.2
HIHCA [@cai2017higher] 85.3
Improved B-CNN [@lin2017improved] 85.8
BoostCNN [@moghimi2016boosted] 86.2
KP [@cui2017kernel] 86.2
FBP($relu5\_3$) 85.7
CBP($relu5\_3 + relu5\_2$) 86.7
HBP($relu5\_3 + relu5\_2 + relu5\_1$) $\mathbf{87.1}$
: Comparison results on CUB-200-2011 dataset. Anno. represents using bounding box or part annotation[]{data-label="table:table2"}
From results in Table \[table:table2\], we can see that PN-CNN [@branson2014bird] uses strong supervision of both human-defined bounding box and ground truth parts. SPDA-CNN [@zhang2016spda] uses ground truth parts and B-CNN [@lin2015bilinear] uses bounding box with very high-dimensional feature representation (250K dimensions). The proposed HBP($relu5\_3+relu5\_2+relu5\_1$) achieves better result compared with PN-CNN [@branson2014bird], SPDA-CNN [@zhang2016spda] and B-CNN [@lin2015bilinear] even without bbox and part annotation, which demonstrates the effectiveness of our model. Compared with STN [@jaderberg2015spatial] which uses stronger inception network as baseline model, we obtain a relative accuracy gain with 3.6% by our HBP($relu5\_3+relu5\_2+relu5\_1$). We even surpass RA-CNN [@fu2017look] and MA-CNN [@zheng2017learning], which are the recently-proposed state-of-the-art unsupervised part-based methods, with 2.1% and 0.7% relative accuracy gains, respectively. Compared with the baselines of pooling-based model B-CNN [@lin2015bilinear], CBP [@gao2016compact] and LRBP [@kong2017low], the superior result that we achieve mainly benefits from the inter-layer interaction of feature and the integration of multiple layers. We also surpass BoostCNN [@moghimi2016boosted] which boosts multiple bilinear networks trained at multiple scales. Although HIHCA [@cai2017higher] proposes similar ideas to model feature interaction for fine-grained recognition, our model can achieve higher accuracy because of the mutual reinforcement framework for inter-layer feature interaction and discriminative feature learning. Note that HBP($relu5\_3+relu5\_2+relu5\_1$) outperforms CBP($relu5\_3+relu5\_2$) and FBP($relu5\_3$), which indicates that our model can capture the complementary information among layers.
### Results on Stanford Cars.
The classification accuracy on Stanford Cars is summarized in Table \[table:table3\]. Different car parts are discriminative and complementary, thus object and part localization may play a significant role here [@yang2015large]. Although our HBP has no explicit part detection, we achieve the best result among state-of-the-art methods. Relying on inter-layer feature interaction learning, we even surpass PA-CNN [@krause2015fine] by 1.2% relative accuracy gains, which uses human-defined bounding box. We can observe significant improvement compared with unsupervised part-based method MA-CNN [@zheng2017learning]. Our HBP is also better than pooling-based methods BoostCNN [@moghimi2016boosted] and KP [@cui2017kernel].
Method Anno. Accuracy
----------------------------------- --------- -----------------
FCAN [@liu2016fully] $\surd$ 91.3
PA-CNN [@krause2015fine] $\surd$ 92.6
FCAN [@liu2016fully] 89.1
RA-CNN [@fu2017look] 92.5
MA-CNN [@zheng2017learning] 92.8
B-CNN [@lin2015bilinear] 90.6
LRBP [@kong2017low] 90.9
HIHCA [@cai2017higher] 91.7
Improved B-CNN [@lin2017improved] 92.0
BoostCNN [@moghimi2016boosted] 92.1
KP [@cui2017kernel] 92.4
HBP $\mathbf{93.7}$
: Comparison results on Stanford Cars dataset. Anno. represents using bounding box[]{data-label="table:table3"}
### Results on FGVC-Aircraft.
Different aircraft models are difficult to be recognized, due to subtle differences, e.g., one may be able to distinguish them by counting the number of windows in the model. The classification accuracy on FGVC-Aircraft is summarized in Table \[table:table4\]. Still, our model achieves the highest classification accuracy among all the methods. We can observe stable improvement compared with annotation-based method MDTP [@wang2016mining], part learning-based method MA-CNN [@zheng2017learning], and pooling-based BoostCNN [@moghimi2016boosted], which highlights the efficacy and robustness of the proposed HBP model.
Method Anno. Accuracy
----------------------------------- --------- -----------------
MG-CNN [@wang2015multiple] $\surd$ 86.6
MDTP [@wang2016mining] $\surd$ 88.4
RA-CNN [@fu2017look] 88.2
MA-CNN [@zheng2017learning] 89.9
B-CNN [@lin2015bilinear] 86.9
KP [@cui2017kernel] 86.9
LRBP [@kong2017low] 87.3
HIHCA [@cai2017higher] 88.3
Improved B-CNN [@lin2017improved] 88.5
BoostCNN [@moghimi2016boosted] 88.5
HBP $\mathbf{90.3}$
: Comparison results on FGVC-Aircraft dataset. Anno. represents using bounding box[]{data-label="table:table4"}
Qualitative Visualization {#sec:qual}
-------------------------
[ccccccc]{}
![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_1_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_1_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_1_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_1_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_1_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_1_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_1_proj_3.jpg "fig:"){width="14.00000%"}\
\[-2pt\] ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_2_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_2_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_2_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_2_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_2_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_2_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_2_proj_3.jpg "fig:"){width="14.00000%"}\
\[-2pt\] ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_3_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_3_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_3_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_3_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_3_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_3_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/cub_3_proj_3.jpg "fig:"){width="14.00000%"}\
\[5pt\]
![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_1_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_1_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_1_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_1_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_1_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_1_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_1_proj_3.jpg "fig:"){width="14.00000%"}\
\[-2pt\] ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_2_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_2_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_2_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_2_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_2_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_2_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_2_proj_3.jpg "fig:"){width="14.00000%"}\
\[-2pt\] ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_3_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_3_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_3_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_3_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_3_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_3_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/car_3_proj_3.jpg "fig:"){width="14.00000%"}\
\[5pt\]
![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_1_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_1_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_1_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_1_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_1_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_1_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_1_proj_3.jpg "fig:"){width="14.00000%"}\
\[-2pt\] ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_2_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_2_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_2_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_2_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_2_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_2_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_2_proj_3.jpg "fig:"){width="14.00000%"}\
\[-2pt\] ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_3_origin.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_3_relu5_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_3_relu5_2.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_3_relu5_3.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_3_proj_1.jpg "fig:"){width="14.00000%"} & ![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_3_proj_2.jpg "fig:"){width="14.00000%"} &![Visualization of model response of different layers on the CUB, Cars and Aircraft datasets. It can be seen that our model tend to ignore features in the cluttered background and focus on the most discriminative parts of object.[]{data-label="fig:fig3"}](./images/air_3_proj_3.jpg "fig:"){width="14.00000%"}\
\[5pt\]
$Original$ & $relu5\_1$ & $relu5\_2$ & $relu5\_3$ & $project5\_1$ & $project5\_2$ & $project5\_3$
To better understand our model, we visualize the model response of different layers in our fine-tuned network on different datasets. We obtain the activation maps by computing the magnitude of feature activations averaged across channel. In Fig. \[fig:fig3\], we show some randomly selected images from three different datasets and their corresponding visualizations.
The visualizations all suggest that the proposed model is capable of ignoring cluttered backgrounds and tends to activate strongly on highly specific semantic parts. The highlighted activation regions in $project5\_1,~project5\_2$ and $project5\_3$ are strongly related to semantic parts, such as heads, wings and breast in CUB; front bumpers, wheels and lights in Cars; cockpit, tail stabilizers and engine in Aircraft. These parts are crucial to distinguish the category. Moreover, our model is highly consistent with the human perception that resolve the fine details when perceive scenes or objects. In Fig. \[fig:fig3\], we can see that the convolution layers ($relu5\_1,relu5\_2,relu5\_3$) provide a rough localization of target object. Based on this, the projection layers ($project5\_1,project5\_2,project5\_3$) further determine essential parts of the object, which distinguish its category by successive interaction and integration of different part features. The process is consistent with the coarse-to-fine nature of human perception [@lu2018revealing] inspired by the Gestalt dictum that the “whole” is prior to the “parts” and it also provides an intuitive explanation as to why our framework can model subtle and local differences between subcategories without explicit part detection.
Conclusions {#sec:conc}
===========
In this paper, we propose a hierarchical bilinear pooling approach to fuse multi-layer features for fine-grained recognition, which combines inter-layer interactions and discriminative feature learning in a mutually-reinforced way. The proposed network requires no bounding box/part annotations and can be trained end-to-end. Extensive experiments on birds, cars and aircrafts demonstrate the effectiveness of our framework. In the future, we will conduct extended research on two directions, i.e., how to effectively fuse more layer features to obtain part representation at multiple scales, and how to merge effective methods for parts localization to learn better fine-grained representation.
### Acknowledgements. {#acknowledgements. .unnumbered}
This work was supported in part by the National Natural Science Foundation of China (No.61772220, 61571205), in part by National Key Technology Research and Development Program of Ministry of Science and Technology of China (No.2015BAK36B00), in part by the Technology Innovation Program of Hubei Province (No.2017AAA017), in part by the Key Program for International S&T Cooperation Projects of China (No.2016YFE0121200).
[^1]: $relu5\_3\ast relu5\_3$.
[^2]: $relu5\_3\ast relu5\_2$.
[^3]: $relu5\_3\ast relu5\_1$.
[^4]: $relu5\_3\ast relu4\_3$.
[^5]: $relu5\_3\ast relu5\_2+relu5\_3\ast relu5\_1$.
[^6]: $relu5\_3\ast relu5\_2+relu5\_3\ast relu5\_1+relu5\_3\ast relu4\_3$.
[^7]: $relu5\_3\ast relu5\_2+relu5\_3\ast relu5\_1+relu5\_2\ast relu5\_1$.
|
\#1\#2[0=1=to 0[\#2]{}\#1-201]{}
Introduction: State Measurement {#intro}
===============================
The question of how to measure the quantum state of a single mode of the electromagnetic field in a cavity has recently attracted a great deal of attention [@kn:spec], as it determines the feasibility of the [*measurement*]{} of the state of a genuine quantum system [@kn:disc]. Several proposals have been made in the last few years to answer this question. Among them, quantum-state endoscopy [@kn:bard] and quantum-optical homodyne state-tomography [@kn:tomo] are two notable examples. The former proposal makes use of a beam of two-level atoms—sent with controlled speed through the cavity—to infer the properties of the field state inside the cavity. On the other hand, optical homodyne tomography is a method for obtaining the Wigner function (or, more generally, the matrix elements of the density operator in some representation) of the electromagnetic field. It therefore consists of an ensemble of repeated measurements of one quadrature operator for different phases relative to the local oscillator of the homodyne detector. The major drawback of the first method is the low detection efficiency for atoms, whereas in the second one one has to couple the field out of the resonator.
In the present paper we propose to couple the field via a quantum-non-demolition (QND) interaction [@kn:sum] to a meter field on which we then perform tomography using a balanced homodyne detector. In this way we combine the idea of probing, that is doing endoscopy on the field without taking it out of the cavity, and the tool of tomography and arrive at the method of endoscopic quantum state tomography. In contrast to the method of quantum state tomography [@kn:tomo] based on homodyne detection, the present technique does not couple the signal field out of the resonator.
Our Model {#hami}
=========
Let us assume that the electromagnetic field we want to probe (the signal mode) is in a pure quantum state and neglect dissipation. We note, however, that our method applies as well to a field described by a density operator. In order to measure the signal field we couple it to a meter field. Both fields are then coupled to a pump field. This coupling leads us to a quantum-non-demolition Hamiltonian describing the interaction between the signal and the meter mode.
Our model starts from the Hamiltonian $$\hat {H} \equiv i \hbar \chi [\hat {a}_s^\dagger \hat {a}_m^\dagger \hat
{a}_p - \hat {a}_s \hat {a}_m \hat {a}_p^\dagger] + i\hbar \sigma [\hat {a}_s
^\dagger \hat {a}_m - \hat {a}_s \hat {a}_m^\dagger]\;,
\label{eq:hami}$$ where $\hat{a}_s (\hat{a}_s^\dagger)$, $\hat {a}_m (\hat{a}_m^\dagger)$, and $\hat{a}_p (\hat{a}_p^\dagger)$ represent the annihilation (creation) operators of the signal, meter, and pump field, respectively. The parameters $\chi$ and $\sigma$ measure the coupling between the three fields, and the meter and signal field, respectively. When the pump field is highly excited we can describe it by a coherent state of amplitude $\alpha$ and phase $2\phi$, that is $$\hat{a}_p \simeq \alpha e^{2i\phi}\;.
\label{eq:coh}$$ If we substitute the coherent state approximation, Eq. (\[eq:coh\]), into the Hamiltonian, Eq. (\[eq:hami\]), after some algebra we obtain $$\hat{H} = 2\hbar \sigma \hat X_s (\phi + \pi/2) \cdot \hat{X}_m (\phi)\;,
\label{eq:cohami}$$ where we have arranged the strength $\alpha$ of the pump field such that $\chi\alpha = \sigma$, and the quadrature operators are given by $$\hat{X}_j (\theta) \equiv \frac{1}{\sqrt{2}} \left(\hat{a}_j e^{-i\theta} +
\hat{a}_j^\dagger e ^{i\theta}\right)$$ of the signal $(j=s)$ and the meter $(j=m)$ mode at the angle $\theta$.
It is particularly interesting to note that due to the special choice $\chi\alpha = \sigma$ we have been able to obtain an interaction between the signal and the meter which couples the quadrature operator $\hat{X}_m (\phi)$ of the meter at phase angle $\phi$ to the out-of-phase quadrature operator $\hat{X}_s(\phi + \pi/2)$ of the signal [@kn:als; @kn:schl]. The present Hamiltonian is a genuine QND Hamiltonian. In the next sections we shall see how such a Hamiltonian can be used to measure the quantum state of the signal field. We note that according to the QND Hamiltonian Eq. (\[eq:cohami\]) a (homodyne) measurement of the meter at a fixed phase $\phi$ of the pump field yields information about the signal in the orthogonal quadrature. By varying the phase $\phi$ of the pump field we can probe all quadratures of the signal.
Entangled State {#enta}
===============
The aim of the present section is to calculate the combined state $|\Psi\rangle$ of signal and meter achieved after some interaction time according to the QND interaction Hamiltonian, Eq. (\[eq:cohami\]).
When we couple for an interaction time $t$ the signal and meter mode prepared initially in the states $|\psi_s\rangle$ and $|\psi_m\rangle$ we find the quantum state $$\begin{aligned}
|\Psi(t) \rangle &=& \exp (-i\hat{H}t/\hbar)|\psi_m\rangle|\psi_s\rangle
\nonumber \\
& = & \exp[-2i\sigma t\hat{X}_s(\phi + \pi/2)
\hat{X}_m(\phi)]|\psi_m\rangle|\psi_s\rangle
\label{eq:evol}\end{aligned}$$ for the combined system.
We expand the initial signal state in quadrature states $|X_s(\phi + \pi/2)\rangle$, $$|\psi_s\rangle = \int\limits_{-\infty}^\infty dX_s\, \psi_s(X_s; \phi + \pi/2)
|X_s (\phi + \pi/2)\rangle\;.
\label{eq:exp}$$ We stress that this representation and, in particular, the wave function $\psi_s(X_s; \phi + \pi/2) \equiv \langle X_s(\phi + \pi/2)|\psi_s\rangle$ depend crucially on the angle $\theta_s$.
Combining Eqs. (\[eq:evol\]) and (\[eq:exp\]), we may rewrite the combined state as $$\begin{aligned}
|\Psi(t) \rangle = \int\limits_{-\infty}^\infty & dX_s & \psi_s (X_s; \phi +
\pi/2)|X_s(\phi + \pi/2)\rangle
\nonumber \\
& \times & \exp[-2i\sigma t X_s \hat{X}_m (\phi)]|\psi_m\rangle\;.
\label{eq:comb}\end{aligned}$$
Expanding $|\psi_m\rangle$ in quadrature states $|X_m(\theta)\rangle$ of the meter at the angle $\theta$, that is $$|\psi_m\rangle = \int\limits_{-\infty}^\infty dX_m\, \psi_m (X_m; \theta)
|X_m (\theta)\rangle\;,
\label{eq:expan}$$ where $\psi_m (X_m;\theta)\equiv \langle X_m(\theta)|\psi_m\rangle$ denotes the wave function of the meter state at the angle $\theta$, it is straightforward to find [@kn:lou] $$\begin{aligned}
& \exp &[-i(2\sigma tX_s)\hat X_m(\phi)]|\psi_m \rangle
\nonumber \\
& = & \int\limits_{-\infty}^\infty dX_m\exp[-i\gamma(X_s,X_m;\theta-\phi)]
\label{eq:action} \\
& \times & \psi_m[X_m - 2\sigma tX_s\sin (\theta-\phi);\theta]
|X_m(\theta)\rangle\;,
\nonumber\end{aligned}$$ where $$\begin{aligned}
\gamma(X_s, X_m; \theta-\phi) & \equiv & (\sigma t X_s)^2\sin[2(\theta-\phi)]
\nonumber \\
& & + 2\sigma t X_s X_m \cos(\theta-\phi)\;.
\label{eq:gamma}\end{aligned}$$
Hence the combined quantum state reads $$\begin{aligned}
|\Psi(t)\rangle &=& \int\limits_{-\infty}^\infty dX_s
\int\limits_{-\infty}^\infty dX_m \psi_s(X_s; \phi + \pi/2)
\nonumber\\
& & \quad \quad \times \psi_m [X_m - 2\sigma tX_s \sin (\theta - \phi); \theta]
\nonumber\\
& & \quad \quad \times \exp[-i\gamma(X_s,X_m; \theta-\phi)]
\nonumber\\
& & \qquad \times |X_s (\phi + \pi/2)\rangle| X_m (\theta) \rangle\;.
\label{eq:combstate}\end{aligned}$$ We note that due to the coupling between the meter and the signal via the Hamiltonian Eq. (\[eq:cohami\]), the meter wave function $\psi_m (X_m;\theta)$ at the angle $\theta$ gets shifted by an amount $\delta X_m \equiv 2\sigma tX_s \sin (\theta - \phi)$.
Effect of the Meter Measurement on the Signal State {#cond}
===================================================
In the present section we shall show how a measurement of the meter influences the state of the signal. Let us first consider an arbitrary quadrature state of phase angle $\theta$.
According to Eq. (\[eq:combstate\]) the conditioned state $$|\psi_s^{(c)}\rangle=\frac{1}{\sqrt{W(X_m)}}\langle X_m(\theta)|\Psi(t)\rangle
\label{eq:condstate}$$ of the signal given that our quadrature measurement at angle $\theta$ has provided the value $X_m$ reads $$|\psi_s^{(c)} \rangle = \int\limits_{-\infty}^\infty\, dX_s \psi_s (X_s; \phi
+ \pi/2) f (X_s|X_m)| X_s (\phi + \pi/2)\rangle\;,
\label{eq:estra}$$ where the “filter function” $f$ is given by $$\begin{aligned}
f (X_s| X_m) &=& \frac{1}{\sqrt{W(X_m)}} \psi_m [X_m - 2\sigma tX_s
\sin (\theta- \phi); \theta]
\nonumber\\
& \times & \exp[-i \gamma (X_s, X_m; \theta - \phi)]\;.
\label{eq:filt}\end{aligned}$$ The normalization condition directly yields he probability $W(X_m)$ of finding the meter variable $X_m$, that is $$\begin{aligned}
W (X_m) = \int\limits_{-\infty}^\infty & dX_s &|\psi_s (X_s; \phi + \pi/2)|^2
\nonumber\\
& \times &|\psi_m[X_m - 2\sigma tX_s \sin (\theta - \phi); \theta]|^2\;.
\label{eq:dostar}\end{aligned}$$ Equation (\[eq:estra\]) clearly shows how the measurement of the meter influences the quantum state of the signal: The filter function determined by the wave function of the meter selects those parts of the signal wave function that are entangled with the corresponding parts in the meter. To study this in more detail we now calculate the Wigner function $$\begin{aligned}
W_s^{(c)} (X_s, P_s| X_m) &=& \frac{1}{2\pi} \int\limits_{-\infty}^\infty dY
\, e^{iP_sY}\langle X_s - Y/2| \psi_s^{(c)}\rangle
\nonumber\\
& & \qquad \quad \times \langle \psi_s^{(c)}|X_s + Y/2 \rangle
\label{eq:wigsigcon}\end{aligned}$$ of the signal state conditioned on the measured meter value $X_m$. Substituting the state $|\psi_s^{(c)}\rangle$, Eq. (\[eq:estra\]), into this expression we arrive at $$\begin{aligned}
&W&_s^{(c)}(X_s,P_s| X_m)=\frac{1}{2\pi} \int\limits_{-\infty}^\infty
dY \, e^{iP_sY} \psi_s (X_s -Y/2)
\nonumber \\
& \times & \psi_s^\ast (X_s + Y/2) f(X_s -Y/2| X_m) f^\ast(X_s + Y/2| X_m)\;.
\label{eq:wigpar}\end{aligned}$$ The integral may be expressed as the convolution $$\begin{aligned}
& & W_s^{(c)} (X_s, P_s| X_m) =
\nonumber \\
& & \qquad \int\limits_{-\infty}^\infty dP'\, W_s (X_s, P_s - P')
W_f (X_s, P'| X_m)
\label{eq:conv}\end{aligned}$$ between the Wigner function of the original signal state $$\begin{aligned}
& & W_s (X_s, P_s) =
\nonumber \\
& & \qquad \frac{1}{2\pi} \int\limits_{-\infty}^\infty dY \, e^{iP_sY}
\psi_s (X_s -Y/2) \psi_s^\ast (X_s + Y/2)\;,
\label{eq:wig1}\end{aligned}$$ and the Wigner function $$\begin{aligned}
& & W_f (X_s, P_s| X_m) =
\nonumber \\
& & \frac{1}{2\pi} \int\limits_{-\infty}^\infty dy \, e^{iP_sY}
f(X_s - Y/2| X_m) f^\ast (X_s + Y/2| X_m)
\label{eq:wig2}\end{aligned}$$ of the “filter” provided by the measurement on the meter.
Special Examples {#exa}
================
In phase measurement {#inphase}
--------------------
If we take the angle $\theta$ equal to $\phi$, the state $|\Psi \rangle$, Eq. (\[eq:combstate\]), of the complete system reduces to $$\begin{aligned}
|\Psi(t)\rangle &=& \int\limits_{-\infty}^\infty\, dX_s
\int\limits_{-\infty}^\infty dX_m\psi_s(X_s;\phi+\pi/2)\psi_m(X_m;\phi)
\nonumber \\
& & \times \exp(-i2\sigma tX_s X_m)|X_s(\phi + \pi/2)\rangle|X_m(\phi)
\rangle\;.
\label{eq:redstate}\end{aligned}$$ In this case the meter wave function is not shifted. Nevertheless, the states of signal and meter are still entangled. Since the shift $\delta X_m$ vanishes, the probability $$W(X_m) = |\psi_m|^2 \int\limits_{-\infty}^\infty
dX_s|\psi_s(X_s)|^2= |\psi_m|^2 \;,
\label{eq:prob}$$ of finding the meter variable $X_m$ following from Eq. (\[eq:dostar\]) for $\theta = \phi$ is identical to the initial probability of the meter, that is $$W(X_m)=|\psi_m(X_m)|^2\;.
\label{eq:inprob}$$ Hence, up to an overall phase $\mu_m$ determined by the meter wave function $\psi(X_m)=|\psi(X_m)|\exp[i\mu(X_m)]$, we find from Eq. (\[eq:filt\]) the filter function $f(X_s|X_m)=\exp(-i2\sigma t X_s X_m)$, and from Eq. (\[eq:estra\]) the conditioned signal state $$|\psi_s^{(c)}\rangle = \int\limits_{-\infty}^{\infty}\, dX_s\psi_s(X_s)\exp
(-i2\sigma tX_s X_m)|X_s (\phi + \pi/2)\rangle\;.
\label{eq:stern}$$ Note that the measurement of the meter has indeed changed the [*state*]{} of the system but did not alter the probability $$\begin{aligned}
W(X_s) &=& |\langle X_s|\psi_s^{(c)}\rangle|^2
= |\psi_s(X_s)\exp(-i2\sigma t X_s X_m)|^2
\nonumber\\
&=& |\psi_s(X_s)|^2
\label{eq:probab}\end{aligned}$$ of finding the signal variable $X_s$. Hence, the measurement has left untouched the shape of the original state (in the Wigner function representation) but has moved it along the momentum axis by an amount of $2\sigma tX_m$. Consequently, the measurement did not change the probability distribution in the conjugate variable, namely the $X_s$ variable. We note, however, that in this way we cannot gain information about the signal since according to Eqs. (\[eq:prob\]) and (\[eq:inprob\]) the probability distribution $W(X_m)$ of measuring the variable $X_m$ is identical to the original distribution.
This finding is actually a rather general result. In fact, it can be rigorously shown [@kn:noi] that a (QND) measurement which does not change the probability density of the observable which is being measured on a single quantum system gives no information about the measured observable.
Out of phase measurement {#outofphase}
------------------------
Turning now to the case of $\theta = \phi + \pi/2$, we see that the shift $\delta X_m = 2\sigma tX_s$ in the meter wave function is maximal and according to Eq. (\[eq:gamma\]) the phase $\gamma$ vanishes. Hence, the combined state $$\begin{aligned}
&\null&|\Psi(t)\rangle = \int\limits_{-\infty}^\infty dX_s
\int\limits_{-\infty}^\infty dX_m \psi_s(X_s; \phi + \pi/2)
\label{eq:costate} \\
&\times&\!\psi_m (X_m-2\sigma tX_s;\phi + \pi/2)\,
|X_s(\phi + \pi/2)\rangle\,|X_m(\phi + \pi/2)\rangle
\nonumber\end{aligned}$$ is an entangled state in which the entanglement between the meter and signal is due to the shift of the meter. In contrast to the discussion of Sec. \[inphase\] we can now deduce properties of the signal from the shift of the meter wave function. Unfortunately, we cannot simultaneously keep the probability distribution $W(X_s)=|\psi_s(X_s)|^2$ of the original signal state invariant, in accordance with the discussion at the end of Sec. \[inphase\]. Indeed, we find from Eqs. (\[eq:estra\]) or (\[eq:stern\]) the conditional state $$|\tilde{\psi}_s ^{(c)} \rangle = \frac{1}{\sqrt{\tilde{W}(X_m)}} \; \int
\limits_{-\infty}^\infty \!\!dX_s \, \psi_s (X_s) \psi_m (X_m - 2 \sigma t X_s)
\,|X_s \rangle
\label{eq:contilde}$$ of the system given the meter measurement at phase $\phi+\pi/2$ has provided the value $X_m$. The probability $$\tilde{W}(X_m) = \int\limits_{-\infty}^\infty dX_s |\psi_s (X_s) |^2|
\psi_m (X_m - 2 \sigma tX_s)|^2
\label{eq:wtilde}$$ of finding the meter value $X_m$ following from Eq. (\[eq:dostar\]) is now a convolution of the system and the meter function.
Special Measurements {#meter}
====================
Weak measurements {#weak}
-----------------
If $\psi_m$ is broad compared to $\psi_s$ we can evaluate $\psi_m$ at some characteristic value of $X_s$, such as $\langle X_s \rangle$. As a consequence the conditional state, Eq. (\[eq:contilde\]), is simply given by $$|\psi_s^{(c)} \rangle \equiv \int\limits_{-\infty}^{\infty}
dX_s\, \psi_s (X_s)| X_s \rangle\;.
\label{eq:weakcon}$$ The probability $$\tilde{W}(X_m) \equiv |\psi_m (X_m+2\sigma t\langle X_s \rangle)|^2
\label{eq:weakprob}$$ reduces to the original meter probability shifted by an amount $2\sigma t\langle X_s\rangle$. Hence, when this shift $2\sigma t\langle X_s \rangle$ is larger than the width of $W_m (X_m) = |\psi_m (X_m)|^2$, we can learn about $\langle X_s \rangle$. As seen from Eq. (\[eq:weakcon\]), in this case the state of the signal mode does not change appreciably.
Tomographic measurements {#tomo}
------------------------
In the present section we show that it is possible to perform tomography on the meter mode to obtain information about the signal state. To this end, we rewrite Eq. (\[eq:wtilde\]) $$\tilde{W}(X_m) = \int\limits_{-\infty}^\infty dX_s |\psi_s (X_s) |^2|
\psi_m (X_m - 2 \sigma tX_s)|^2\;,
\label{eq:tildew}$$ which gives the marginal distribution of the meter (probability distribution of the results of the measurements of $\hat{X}_m$) in the case of out of phase measurements. Let us assume that the meter wave function is extremely narrow, that is the meter is initially in a highly squeezed state, for example a squeezed vacuum $|0,r\rangle$, where $r$ is the (real) squeezing parameter. Then, according to Eq. (\[eq:tildew\]), the marginal distribution $\tilde{W}(X_m)$ is given by a convolution of the modulus square of the signal wave function with a narrow Gaussian $$\begin{aligned}
|\psi_m(X_m&-&2\sigma tX_s)|^2=\frac{1}{\protect\sqrt{\pi}\cosh
r(1-\tanh r)}
\label{eq:psivac} \\
&\times&\exp\left\{-\left[\frac{1+\tanh r}{1-\tanh r}\right]
(X_m-2\sigma tX_s)^2\right\}\;.
\nonumber\end{aligned}$$
Now, if the squeezing parameter $r$ is large enough, the Gaussian (\[eq:psivac\]) approaches a delta function in the meter and signal variables $$|\psi_m(X_m-2\sigma tX_s)|^2\longrightarrow \frac{1}{|2\sigma t|}
\delta\left(X_s-\frac{X_m}{2\sigma t}\right)\;,
\label{eq:appdel}$$ and Eq. (\[eq:tildew\]) reduces to
\[eq:double\] $$\begin{aligned}
\tilde{W}(X_m)&\cong&\frac{1}{|2\sigma t|}\int\limits_{-\infty}^{\infty}
dX_s\, |\psi_s(X_s)|^2\delta\left(X_s-\frac{X_m}{2\sigma t}\right)
\label{eq:doublea} \\
&=& \frac{1}{2\sigma t} \left|\psi_s\left(\frac{X_m}{2\sigma t}\right)\right|^2
=\frac{1}{2\sigma t} W\left(\frac{X_m}{2\sigma t}\right)\;.
\label{eq:doubleb}\end{aligned}$$
Hence, by measuring the probability distribution $\tilde{W}(X_m)$ of the outcomes of the meter variable $X_m$ (for example via balanced homodyne detection performed on the meter field) we indirectly obtain the probability distribution $W(X_s)$, up to a rescaling given by the factor $2\sigma t$. However, from Eq. (\[eq:contilde\]) it is clear that in this case the signal wave function is changed, and therefore we need to prepare the signal field again in the same state after each measurement. This is what is usually done in quantum optical tomography [@kn:tomo].
The advantage of the present scheme is that we perform an indirect measurement: We do not detect the signal mode outside the cavity (that is, we do not have to take the signal field outside the cavity), but we couple it to a meter field which is successively detected, thus overcoming the smearing effect introduced by the direct detection of the signal [@kn:tomo]. Moreover, there is no need of a smoothing procedure, since we are interested in the marginal probability distribution $W(X_s)$ which is directly related to $\tilde{W}(X_m)$ through Eq. (\[eq:double\]). In order to probe the full state of the signal field, however, we would need to measure the probability distribution $\tilde{W}(X_m)$ for various values of the phase [@kn:tomo].
Conclusions {#conclu}
===========
To summarize, we have suggested a method to measure the quadrature probability distribution (or, more generally, the full quantum state) of a single mode of the electromagnetic field inside a cavity. It is based on indirect homodyne measurements performed on a meter field which is coupled to the signal field via a QND interaction Hamiltonian. We have named this procedure “endoscopic tomography” because (i) it does not require (in contrast to Ref. [@kn:tomo]) to take the field out of the cavity, just as in “quantum state endoscopy” [@kn:bard], where a beam of two-level atoms is used as a probe; (ii) tomographic measurements performed (by balanced homodyne detection) on the meter mode allow us to reconstruct the marginal probability distribution of the signal variable or even the full quantum state.
It is a pleasure for us to acknowledge several enlightening discussions with O. Alter, F. Harrison, J.H. Kimble and K. Mölmer. We thank the European Union (through the TMR Programme), the Deutsche Forschungsgemeinschaft, the Land Baden-Württemberg, and INFM (through the 1997 PRA-CAT) for partial support.
Corresponding author. E-mail: [email protected] See the Special Issue on [*Quantum State Preparation and Measurement*]{}, edited by M.G. Raymer and W.P. Schleich: J. Mod. Opt. [**44**]{}, No. 11/12 (1997). See also the long-standing discussion among the scientific community about the measurability of the state of a single quantum system: A. Royer, Phys. Rev. Lett. [**73**]{}, 913 (1994); A. Royer, Phys. Rev. Lett. [**74**]{}, 1040 (1995). Y. Aharonov, J. Anandan, and L. Vaidman, Phys. Rev. A [**47**]{}, 4616 (1993). O. Alter and Y. Yamamoto, Phys. Rev. Lett. [**74**]{}, 4106 (1995). O. Alter and Y. Yamamoto in: [*Fundamental Problems in Quantum Theory*]{}, ed. by D. Greenberger and A. Zeilinger, Vol. 755 (New York Academy of Sciences, New York, 1995). O. Alter and Y. Yamamoto, Phys. Rev. A [**53**]{}, R2911 (1996). Y. Aharonov and L. Vaidman, Phys. Rev. A [**56**]{}, 1055 (1997); O. Alter and Y. Yamamoto, [*ibid.*]{}, 1057 (1997). O. Alter And Y. Yamamoto, in [*Quantum Physics, Chaos Theory and Cosmology*]{}, edited by M. Namiki, I. Ohba, K. Maeda and Y. Aizawa (The American Institute of Physics, New York, 1996), pp. 151-172. O. Alter and Y. Yamamoto, in [*Quantum Coherence and Decoherence*]{}, edited by K. Fujikawa and Y.A. Ono (Elsevier, Amsterdam, 1996), pp. 31-34. G.M. D’Ariano and H.P. Yuen Phys. Rev. Lett. [**76**]{}, (1996). P. J. Bardroff, E. Mayr, and W. P. Schleich, Phys. Rev. A [**51**]{}, 4963 (1995); P. J. Bardroff, E. Mayr, W. P. Schleich, P. Domokos, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. A [**53**]{}, 2736 (1996). The method of quantum state tomography has been first proposed by K. Vogel and H. Risken, Phys. Rev. A [**40**]{}, 2847 (1989). For experimental implementations see D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. [**70**]{}, 1244 (1993); M. Beck, D.T. Smithey, and M.G. Raymer, Phys. Rev. A [**48**]{}, R890 (1993); D.T. Smithey, M. Beck, J. Cooper, and M.G. Raymer, [*ibid.*]{} [**48**]{}, 3159 (1993). The method has been further developed by G.M. D’Ariano, C. Macchiavello, and M.G.A. Paris, Phys. Rev. A [**50**]{}, 4298 (1994); G.M. D’Ariano, U. Leonhardt, and H. Paul, Phys. Rev. A [**52**]{}, R1801 (1995). S. Schiller, G. Breitenbach, S.F. Pereira, T. Müller, and J. Mlynek, Phys. Rev. Lett. [**77**]{}, 2933 (1996). V.B. Braginsky and F.Y. Khalili [*Quantum Measurement*]{} (Cambridge University Press, Cambridge, 1992). P. Alsing, G. J. Milburn and D. F. Walls, Phys. Rev. A [**37**]{}, 2970 (1988). B. Yurke, W. Schleich, and D. F. Walls, Phys. Rev. A [**42**]{}, 1703 (1990). W. H. Louisell, [*Quantum Statistical Properties of Radiation*]{} (Wiley, New York, 1973). M. Fortunato, P. Tombesi, and W.P. Schleich (submitted to Phys. Rev. A).
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abstract: 'The HfV$_2$Ga$_4$ compound was recently reported to exhibit unusual bulk superconducting properties, with the possibility of multiband behavior. To gain insight into its properties, we performed ab-initio electronic structure calculations based on the Density Functional Theory (DFT). Our results show that the density of states at the Fermi energy is mainly composed by V–$d$ states. The McMillan formula predicts a superconducting critical temperature ($T_{c}$) of approximately 3.9K, in excellent agreement with the experimental value at 4.1K, indicating that superconductivity in this new compound may be explained by the electron-phonon mechanism. Calculated valence charge density maps clearly show directional bonding between Hf and V atoms with 1D highly populated V-chains, and some ionic character between Hf–Ga and V–Ga bonds. Finally, we have shown that there are electrons occupying two distinct bands at the Fermi level, with different characters, which supports experimental indications of possible multiband superconductivity. Based on the results, we propose the study of a related compound, ScV$_2$Ga$_4$, showing that it has similar electronic properties, but probably with a higher $T_c$ than HfV$_2$Ga$_4$.'
author:
- 'P. P. Ferreira'
- 'F. B. Santos'
- 'A. J. S. Machado'
- 'H. M. Petrilli'
- 'L. T. F. Eleno'
title: 'Insights on unconventional superconductivity in HfV$_2$Ga$_4$ and ScV$_2$Ga$_4$ from first principles electronic structure calculations'
---
Introduction
============
Although superconductivity has attracted the attention of the scientific community for a long time, the understanding of the phenomenon, which started with the model proposed by Bardeen, Cooper and Schrieffer (BCS) [@Bardeen1957], is still very challenging. The BCS theory, although useful for a large class of superconducting materials, fails to correctly explain other experimentally observed superconducting elements or compounds [@Luders2005] and a plethora of different behaviors demands new approaches.
First-principles electronic structure calculations, within the framework of the Density Functional Theory (DFT), has proven to be an important tool to study superconducting materials. Although strongly correlated systems are beyond the scope of the Kohn-Sham scheme of the DFT, many successful attempts have been made to deal with the description of superconducting materials. In particular, some specific properties of the normal state, e.g. electronic band dispersions and electronic density of states, are very useful to elucidate aspects of the superconducting mechanism and to predict relevant parameters, such as the critical temperature $T_c$ and the electron-phonon coupling constant $\lambda$. In the last few years, an increasing number of studies appeared using this methodology, either as support for experimental discoveries [@sefat2008; @chang2016; @Machado2017] or fully theoretical investigations [@singh2008; @subedi2013; @tian2016; @heil2017].
Superconductivity was recently [experimentally]{} reported, by some of the present authors, for the HfV$_2$Ga$_4$ compound, with a critical temperature ($T_c$) of 4.1K [@Santos2018]. The investigators observed some deviations from the more conventional BCS theory signatures, such as an unusual inflection near $T_c$ in lower and upper critical field as a function of reduced temperature, and a second jump in the specific heat vs. temperature curve. The authors speculated that the experimental results could be either due to sample inhomogeneity or to the presence of more than one superconducting gap at the Fermi surface, resulting in a two-band superconductivity [@zehetmayer2013].
These recent experimental results for the bulk HfV$_2$Ga$_4$ point to a new promising class of materials to study unconventional superconducting behavior. Motivated by these results, here we perform ab-initio electronic structure calculations for HfV$_2$Ga$_4$. We focus our attention on the analysis of the possible mechanisms behind the superconducting properties. [The theoretical study was extended to a new (possibly) bulk superconducting compound with the same prototype structure, ScV$_2$Ga$_4$]{}, as a way to manipulate the electronic structure aiming at enhancing the superconducting transition temperature.
Computational Methods
=====================
The ab-initio electronic structure calculations were performed in the framework of the Kohn-Sham scheme [@kohn1965] within Density Functional Theory (DFT) using the Full Potential – Linearized Augmented Plane Wave plus local orbitals (FP-LAPW+lo) method [@Singh2006], as implemented in the WIEN2k computational code [@blaha2001]. The Exchange and Correlation (XC) functional was described by the Generalized Gradient Approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) [formulation]{} [@perdew1996], taking relativistic corrections and spin-orbit coupling (SOC) effects into account. We used muffin-tin spheres with radius $R_\text{MT}=2.0\,a_0$ (Bohr’s radius) for all atoms, with $R_\text{MT}\,K_\text{max}$ = 9.0, where $K_\text{max}$ is related to the basis set size [[@blaha2001]]{}. Self-consistent-field (SCF) calculations were carried out with a $32 \times 32 \times 32$ Monkhorst-Pack [@monkhorst1976] shifted k-point mesh discretization in the first Brillouin zone. All lattice parameters and internal degrees of freedom were relaxed in order to guarantee a ground state convergence to about 10$^{-5}$Ry in the total energy, 10$^{-4}$*e* for [electron]{} density and 0.5mRy/$a_0$ for forces acting on the nuclei. The Birch-Murnaghan equation of state [@birch1947] was used to fit the total energy as a function of the unit cell volume (keeping $c/a$ constant) at several $c$ values in order to obtain the ground state lattice constants and bulk modulus.
Finally, six different lattice distortions, with 15 intensities for each one (a total of 90 different structures), were used to provide data for the determination of the elastic properties with the ElaStic code [@elastic], using [Quantum Espresso]{} [@Giannozzi2009] for DFT calculations of deformed structures. The [Quantum Espresso]{} calculations were performed using PBE SG15 Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials [@Schlipf2015], with a cutoff energy of 220Ry and 1728 $k$-points in the first Brillouin zone. Anderson’s simplified method [@anderson1963] was then employed for the calculation of the Debye temperature.
Results and Discussion
======================
HfV$_2$Ga$_4$ electronic structure calculations
-----------------------------------------------
HfV$_2$Ga$_4$ crystallize in the YbMo$_2$Al$_4$ prototype (space group *I4/mmm*, Pearson symbol $tI14$), a body-centered tetragonal structure composed by a cage-like structure, where Hf sites, at 2a (0, 0, 0) Wyckoff positions, are surrounded by V and Ga sites at 4d (0, 1/2, 1/4) and 8h (0.303, 0.303, 0), respectively [@fornasini1976], as schematically shown in Figure \[fig:YbMo2Al4\].
![HfV$_2$Ga$_4$ body-centered tetragonal unit cell (conventional setting). Hf (gray), V (blue) and Ga (red) sites are at the 2a (0, 0, 0), 4d (0, 1/2, 1/4) and 8h (0.303, 0.303, 0) Wyckoff positions, respectively.[]{data-label="fig:YbMo2Al4"}](HfV2Ga4.png){width=".6\columnwidth"}
The calculated optimized lattice parameters are in excellent agreement with the experimental data reported in the literature [@Grin1980], as seen in Table \[tab:lat-par\]. There is a slight difference of at most 1% with respect to the experimental values, which is commonly related to the inherent imprecision of the approximations required by the computational method [@Palumbo2014; @Lejaeghere2014; @Lejaeghere2016]. The calculated bulk modulus [is]{} 134.75GPa, with a Poisson ratio of 0.24, resulting in 416.3K for [the]{} Debye temperature ($\Theta_D$). Our ab initio calculations for $\Theta_D$ reproduce with great accuracy the 418.97K [value]{} obtained through experimental measurements [@Santos2018].
calc. exp.
-------------- ------------------- -------------------
$a$, $b$ (Å) 6.459 6.45
$c$ (Å) 5.197 5.20
$8h$ (Ga) (0.303, 0.303, 0) (0.303, 0.303, 0)
: Calculated lattice parameters and optimized $8h$ (Ga) atomic position for the HfV$_{2}$Ga$_{4}$ tetragonal compound, compared to experimental values [@Grin1980].[]{data-label="tab:lat-par"}
The total density of states (DOS), as well as the site and orbital projected density of states (PDOS), are shown in Figures \[fig:dos\_HfV2Ga4\]a-d. Both occupied and unoccupied states involve considerable hybridization, as seen in Figure \[fig:dos\_HfV2Ga4\]a. In the lowest energy region Ga orbitals are dominant, with some contribution from V; in the region around the Fermi level (from $-2.5$eV to 3eV), V states are prevailing, mainly due to V-$d$ character (notice the different PDOS scales on Figures \[fig:dos\_HfV2Ga4\]b-d), with some Hf-$d$ and Ga-$p$ contributions; in the higher (above 3eV), unoccupied energy region, Hf and V states contribute equally. Almost half of the total DOS at the Fermi level is due to V, although these states are extended along the whole studied energy region.
\
The calculated total DOS at the Fermi level is $ N(E_F) = 2.29$states/eV. This quantity is related to the linear coefficient of the electronic specific heat $\gamma$, known as Sommerfeld coefficient, given by $$\gamma = \dfrac{\pi^2}{3}k_{B}^{2}N(E_F)\,,
\label{eq:gamma}$$ where $k_{B}$ is the Botzmann constant.The calculated $N(E_F)$ leads to a value of 5.41mJmol$^{-1}$K$^{-2}$ for the theoretical $\gamma_\text{calc}$. From the value of the Sommerfeld coefficient $\gamma_\text{calc}$ resulting from the ab-initio calculations and the experimentally measured value ($\gamma_\text{exp}=8.263\,$mJmol$^{-1}$K$^{-2}$) [@Santos2018], we can estimate reasonably well the electron-phonon coupling constant $\lambda$ using the well-known approximation [@dugdale2011; @ram2012] $$\lambda = \dfrac{\gamma_\text{exp}}{\gamma_\text{calc}} - 1\,,
\label{eq:lambda}$$ which stems from the fact that the calculations give static (0K) results. Following Eq. (\[eq:lambda\]), we arrive at $\lambda = 0.53$. This value can be used to calculate the superconducting transition temperature $T_{c}$ using the empirical McMillan formula [@mcmillan1968], $$T_c = \dfrac{\Theta_D}{1.45}\exp\left[-\frac{1.04(1+\lambda)}{\lambda - \mu^{*}(1 + 0.62\lambda)}\right],
\label{eq:mcmillan}$$ where $\mu^{*}$ is the Coulomb pseudopotential, which measures the strength of the electron-electron Coulomb repulsion [@McMillan1965]. A typical value of $\mu^{*}$ is 0.12, as used in many previous works [@ram2012; @dugdale2011; @subedi2008]. For the HfV$_2$Ga$_4$ compound, using the calculated $\Theta_D$ and the above values for $\lambda$ and $\mu^*$, we arrive at an estimated critical temperature $T_c=3.9\,$K, in excellent agreement with the experimental (4.1K) value. This indicates that the electron-phonon interaction may be the mechanism behind superconductivity in HfV$_2$Ga$_4$. The V states dominate the $N(E_F)$ and therefore have the major contribution for the pairing.
\
\
The nature of the atomic bonding can be further elucidated with the help of valence electron density plots such as those shown in Figure \[fig:density\_HfV2Ga4\], in which the electron density is plotted, with an appropriate logarithmic scale in a (001) plane, passing through the center of Hf and Ga nuclei within a unit cell (Figure \[fig:density\_HfV2Ga4\]a), and a (100) plane, passing through the center of Hf and V nuclei (Figure \[fig:density\_HfV2Ga4\]b). It should be noted that, in Figure \[fig:density\_HfV2Ga4\]a, the non-labelled high-density regions are V nuclei not centered on the (100) plane. The plots clearly show a directional shared bond between Hf and V atoms, evidenced by the density contours in the (100) and (001) planes. This reveal that Hf atoms, which are “locked” in the center of a cage-like structure, are not simply passive electron donors: they stabilize the charge transfer to the V atoms (as also observed in Figure \[fig:dos\_HfV2Ga4\]) that, in turn, commands the electronic properties.
Furthermore, it is interesting to note that the charge density gives rise to a kind of electron sharing channel in the lattice, composed by directional, strongly-bonded, highly populated V chains in the (100) and (010) crystallographic planes. The Hf nuclei are weakly bonded with the two V atoms within adjacent unit cells in these 1D chains. As a consequence of these V chains that concentrate most of the electronic states that will give rise to Cooper pairs, this electronic configuration may lead to a high anisotropy that could be identified via transport measurements. Finally, despite the small difference in electronegativity between the atomic species, Hf–Ga and V–Ga bonds exhibit some ionic character. In Figure \[fig:density\_HfV2Ga4\](a) we can clearly observe isolated clusters containing four Ga atoms within a unit cell, forming weak bonds with adjacent Hf atoms.
Figure \[fig:band\_HfV2Ga4\] shows the resulting band character plots along high symmetry points in the first Brillouin zone, not including (Fig. \[fig:band\_HfV2Ga4\]a) and including (Fig. \[fig:band\_HfV2Ga4\]b) spin-orbit coupling (SOC) effects in the calculations. In the band character plots, stronger colors mean stronger character due to the respective orbital projection. Indeed, the cage-like symmetry of the lattice gives rise to complex dispersive metallic bands in the vicinity of the Fermi level. There are two bands crossing the Fermi energy, with very different Hf and V characters. The fact that there are electrons occupying two distinct bands in disconnected sheets of the Fermi surface (corresponding to the two bands crossing the Fermi level) supports the experimental evidence of a possible two-gap superconductivity [@Santos2018]. These results open a promising scenario for a possible multiband behaviour, so a superconducting gap calculation [@Koretsune2017; @Giustino2017] would be a very interesting test for that hypothesis.
In the band character plots for HfV$_2$Ga$_4$ shown in Figure \[fig:band\_HfV2Ga4\], we can see that a hole pocket develops in the M point, with a maximum at $\approx\,$0.4eV, originated mainly from the Hf-$d$ states containing some mixing with the V-$d$ states. In particular, the electron band crossing the P point just below $E_F$ is made up mostly by V-$d$ states. Notice that, near the Fermi level, the band plot unveils dispersive cones with zero gap at M and also along the M–$\Gamma$ direction, as well as one such feature at P. However, when SOC effects are considered, these features are gapped. Indeed, SOC leads to a visible lifting of some band degeneracies, mainly at M and M–$\Gamma$, and less-pronounced at P (just a few meV). Moreover, although these compounds are metallic, SOC broken degeneracy creates a continuous pseudo-gap around the Fermi energy, although the gap almost closes at P (not visible in the scale of Figure \[fig:band\_HfV2Ga4\]b). This kind of signature also occurs in a few nontrivial topological materials like Bi$_{14}$Rh$_3$In$_9$, PbTaSe$_2$ and Cu$_x$ZrTe$_{2-y}$ [@rasche2013; @ali2014; @Machado2017]. Therefore, more detailed experimental and theoretical studies about the possibility of nontrivial topological effects in HfV$_2$Ga$_4$ could be an interesting subject for futures investigations.
Theoretical predictions for ScV$_2$Ga$_4$
-----------------------------------------
Several compounds that crystallize in the same body-centered tetragonal prototype YbMo$_2$Al$_4$, such as RTi$_2$Ga$_4$ (R = Ho, Er, Dy) and RV$_2$Ga$_4$ (R = Sc, Zr, Hf), have been reported in the literature. These compounds are poorly investigated, most efforts having been focused exclusively on magnetic properties in rare-earth compounds [@ghosh1993; @lofland1994].
The results reported above for HfV$_2$Ga$_4$ led us to consider an effective way to manipulate the electronic structure of such compounds, aiming at enhancing superconducting properties. In Figure \[fig:dos\_HfV2Ga4\] we can observe that the Fermi level is situated down a deep valley in the total DOS. As a consequence, the density of states at $E_F$ is extremely sensitive. So, considering a rigid band model, it is a reasonable to assume that an element with a different valence configuration in the $2a$ site of HfV$_2$Ga$_4$ could shift the Fermi level to higher states. Based on what has been presented, we also have carried out first principles electronic structure calculations for ScV$_2$Ga$_4$, to test this hypothesis.
Table \[tab:lat-par\_ScV2Ga4\] shows the relaxed calculated lattice parameters, together with experimental reported values for ScV$_2$Ga$_4$. Following the same methodology applied in the previous section, we reached $\Theta_D = 447.8\,$K for ScV$_2$Ga$_4$. Unfortunately, in this case, there are no experimental data for comparison.
calc. exp.
-------------- --------------------- -------------------
$a$, $b$ (Å) 6.497 6.432
$c$ (Å) 5.200 5.216
$8h$ (Ga) (0.3004, 0.3004, 0) (0.303, 0.303, 0)
: Calculated lattice parameters and optimized $8h$ (Ga) atomic position for the ScV$_{2}$Ga$_{4}$, compared to experimental values [@Grin1980].[]{data-label="tab:lat-par_ScV2Ga4"}
The nature of atomic bonding is the same showed for the HfV$_2$Ga$_4$ in Figure \[fig:density\_HfV2Ga4\], with high populated 1D covalent V-chains and Sc atoms acting to stabilize the transfer of charge to the V atoms. Therefore, the DOS overall appearance for HfV$_2$Ga$_4$ is qualitatively identical to ScV$_2$Ga$_4$, as can be verified in Figure \[fig:ScV2Ga4\](a). Hence, the contribution of each orbital in the density of states for ScV$_2$Ga$_4$ is also very similar to HfV$_2$Ga$_4$, with a higher contribution due to Sc-$d$ states in the unoccupied bands. The calculated value of $N(E_F)$ is 3.62states/eV, that leads to $\gamma_\text{calc} = 8.53$mJmol$^{-1}$K$^{-2}$ using Eq. (\[eq:gamma\]). Confirming our hypothesis, the presence of Sc atoms instead of Hf in the $2a$ sites causes the Fermi level to shift to a higher DOS value, an increase of about 60%, escaping from the bottom of the well.
\
In Figures \[fig:ScV2Ga4\](b) and \[fig:ScV2Ga4\](c) we show the calculated band structure without and with SOC effects, respectively. It may be seen that the band structure is related to that presented for HfV$_2$Ga$_4$ (Fig. \[fig:band\_HfV2Ga4\]), with similar features in the vicinity of the Fermi energy. Fermi bands in ScV$_2$Ga$_4$ are well described as coming from hybridization between mainly V-$d$ and some Sc-$d$ states. However, SOC in ScV$_2$Ga$_4$ plays only a marginal role, making nontrivial topological effects unlikely. Nevertheless, the important point here resides on the fact that, similar to HfV$_2$Ga$_4$, there are two bands crossing the Fermi level, opening again the possibility for a multiband scenario.
The large contribution of V-d state electrons and the higher DOS value at the Fermi level, attached to the fact that there are electrons originated from two distinct bands in the Fermi surface, strongly suggest that ScV$_2$Ga$_4$ could be a new example of two-band electron-phonon superconducting material with a considerable higher critical temperature than the one reported for the HfV$_2$Ga$_4$ compound.
Conclusions
===========
In this work we presented ab-initio calculations for the bulk superconductor HfV$_2$Ga$_4$. The McMillan formula predicts a $T_c$ of 3.9K, in excellent agreement with experimental reported values (4.1K), indicating that superconductivity can be readily explained in an electron-phonon framework. From the signature of the DOS in the vicinity of the Fermi energy, we have proposed to improve the superconducting critical temperature by investigating the ScV$_2$Ga$_4$ compound. Theoretically, we have shown that the presence of Sc instead of Hf in the crystal structure causes the Fermi level to shift to a higher DOS value. The band structure around the Fermi level, which comes mainly from V-$d$ states, and the DOS overall appearance, are qualitatively very similar for both compounds. Valence electron density plots unveil Hf(Sc)-V shared bonding and 1D highly populated V-chains, while Hf(Sc)-Ga and V-Ga bonds have a partially ionic character. It was found that there are electrons derived from two distinct bands in disconnected sheets of the Fermi surface for both compounds, in agreement with the experimental evidence [@Santos2018] of a possible two-gap superconductivity for HfV$_2$Ga$_4$. Finally, we argue that ScV$_2$Ga$_4$ is presumably a new candidate for two-band electron-phonon superconductivity with a higher $T_c$ than HfV$_2$Ga$_4$, a result that should be confirmed experimentally.
We gratefully acknowledge the financial support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico (Cnpq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes), and the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), under Procs. 2017/11023-2, 2016/11774-5, and 2016/11565-7.
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---
abstract: |
The spin-Seebeck effect (SSE) in platinum (Pt) and tantalum (Ta) on yttrium iron garnet (YIG) has been investigated by both externally heating the sample (using an on-chip Pt heater on top of the device) as well as by current-induced heating. For SSE measurements, external heating is the most common method to obtain clear signals. Here we show that also by current-induced heating it is possible to directly observe the SSE, separate from the also present spin-Hall magnetoresistance (SMR) signal, by using a lock-in detection technique. Using this measurement technique, the presence of additional 2$^\textrm{nd}$ order signals at low applied magnetic fields and high heating currents is revealed. These signals are caused by current-induced magnetic fields (Oersted fields) generated by the used AC-current, resulting in dynamic SMR signals.
PACS numbers
: 72.25.Mk, 72.80.Sk, 75.70.Tj, 75.76.+j
author:
- 'N. Vlietstra'
- 'M. Isasa'
- 'J. Shan'
- 'F. Casanova'
- 'J. Ben Youssef'
- 'B. J. van Wees'
bibliography:
- 'YIGPt.bib'
title: 'Simultaneous detection of the spin-Hall magnetoresistance and the spin-Seebeck effect in Platinum and Tantalum on Yttrium Iron Garnet'
---
Introduction
============
For the investigation of pure spin transport phenomena, yttrium iron garnet (YIG) is shown to be a very suitable candidate. YIG is a ferrimagnetic insulating material having a low magnetization damping as well as a very low coercive field. In combination with a high spin-orbit coupling material such as platinum (Pt), many different experiments have been performed, showing spin-pumping[@HillebrandsSPmagnons; @SpinPump; @SaitohFreqDep; @CastelPRB], spin transport[@travellingSW; @Kajiwara2010nature] and spin-wave manipulation[@AzevedoAPL2013; @Kurebayashi2011nmat; @AzevedoPRL2011] as well as the recently discovered spin-Hall magnetoresistance (SMR).[@BauerTheorySMR; @AlthammerSMR; @VlietstraSMR; @BauerSMR; @VlietstraSMR2; @JamalTaPt]
Recently also experiments were performed showing the spin-Seebeck effect[@SSESaitoh; @SSETheory; @SSETheory2; @reportSSE] (SSE) as well as the spin-Peltier effect[@PeltierYIG] in YIG/Pt systems. The SSE is observed when a temperature gradient is present over a ferromagnetic/non-magnetic interface. In a YIG/Pt system, this temperature gradient causes the creation of thermal magnons, resulting in transfer of angular momentum at the YIG/Pt interface, generating a pure spin-current into the Pt.[@reportSSE] This spin-current can then be detected electrically via the inverse spin-Hall effect (ISHE). So far, most experiments on the SSE are performed using external heating sources to create a temperature gradient over the device. Interestingly, Schreier et al.[@SSEGoennenwein] showed that a clear SSE signal can also be extracted from more easily performed current-induced heating experiments. In their experiments a temperature gradient is created by sending a charge current through the detection strip. A disadvantage of their measurement method is the presence of a much larger signal originated from the SMR, which should be subtracted to reveal the SSE signal.
In this paper we investigate both the SSE and SMR in a YIG-based device, showing the possibility to simultaneously, but separately, detect the SSE and SMR by using a lock-in detection technique. Whereas Schreier et al. only performed their measurements applying high magnetic fields, fully saturating the magnetization of the YIG, we show that when lowering the applied magnetic field, dynamic behavior of the magnetization of the YIG can be picked up as additional 2$^\textrm{nd}$ order signal. Only by using a lock-in detection technique these signals can be separately detected and analyzed. Having platinum (Pt) or tantalum (Ta) as detection layer, we investigate the evolution of the SSE and the SMR signal as a function of the magnitude and the direction of the applied field, focusing especially on their low-field behavior.
The first experiments described in this paper show SSE measurements where a temperature gradient is generated by externally heating the sample using a second Pt strip on top of the device. By using devices consisting of both Pt and Ta on YIG, we confirm the opposite sign of the spin-Hall angle for Ta and Pt.[@SHE-Ta; @SHE-Metals] In the secondly shown experiments, the samples are heated by current-induced heating through the metal detection strip, such that both the SSE and SMR are present. Additionally detected 2$^\textrm{nd}$ harmonic signals for low applied fields and high heating currents are discussed and ascribed to dynamic behavior of the magnetization of the YIG, caused by the applied AC-current. Finally, we derive a dynamic SMR term, which is used to explain the observed features.
The same kind of experiments could as well be used for detection of spin-transfer torque effects on the magnetization of the YIG, like the generation of spin-torque ferromagnetic resonance as formulated by Chiba et al.[@ChibaCurrent]. However, as will be shown in this paper, when applying low magnetic fields and high currents, the detected magnetization behavior is dominated by current-induced magnetic fields (like the Oersted field). So, to be able to detect any effect of the spin-transfer torque, its contribution should be increased, for example by decreasing the YIG thickness.
Sample characteristics
======================
For the experiments shown in this paper, two Hall-bar shaped devices have been used, one consisting of a 5nm-thick Pt layer and the other of a 10nm-thick Ta layer. The Hall-bars have a length of 500$\upmu$m and a width of 50$\upmu$m, with side contacts of 10$\upmu$m width. Both Hall-bars are deposited on top of a 4x4mm$^2$ YIG sample, by dc sputtering.
The used sample consists of a 200nm thick layer of YIG, grown by liquid phase epitaxy on a single crystal (111)Gd$_3$Ga$_4$O$_{12}$ (GGG) substrate. The YIG magnetization shows isotropic behavior of the magnetization in the film plane, with a low coercive field of only 0.06mT.[@CastelPRB; @VlietstraSMR] For external heating experiments, a Ti/Pt bar of 5/40nm thick is deposited on top of both Hall-bars, separated from the main channel by a 80nm-thick insulating Al$_2$O$_3$ layer. The size of the heater is 400x25$\upmu$m$^2$. Finally, both Hall-bars and Pt heaters are contacted by thick Ti/Au pads \[5/150nm\]. All structures are patterned using electron-beam lithography. Before each fabrication step the sample has been cleaned by rinsing it in acetone, no further surface treatment has been carried out. A microscope image of the device is shown in Fig. \[fig:Fig1\](a).
Measurement methods
===================
To observe the SSE, two measurement methods have been investigated. At first, to generate a clear SSE signal, a temperature gradient is created using an external heating source to heat one side of the sample. In our case, we have a Ti/Pt strip on top of the Hall-bar, electrically insulated from the detection channel, which can be used as an external heater. By sending a large current (up to 10mA) through the heater, the strip will be heated by Joule heating. Hereby, a temperature gradient will be formed over the YIG/Pt(Ta) stack, giving rise to the SSE.
A second method to generate the SSE, is the generation of a temperature gradient by current-induced heating through the detection strip. In this case a charge current is sent through the Hall-bar itself, which also leads to Joule heating, resulting in a temperature gradient over the YIG/Pt(Ta) stack. As the Hall-bar is directly in contact with the YIG, also the SMR will be present when using this heating method.
To separately detect the SSE and the SMR signals, a lock-in detection technique is used. Using up to three Stanford SR-830 Lock-in amplifiers, the 1$^\textrm{st}$, 2$^\textrm{nd}$ and higher harmonic voltage responses of the system are separately measured. As SMR scales linearly with the applied current, its contribution will be picked up as a 1$^\textrm{st}$ harmonic signal. Similarly, the SSE scales quadratically with current, so its contribution will be detected as a 2$^\textrm{nd}$ harmonic signal. For lock-in detection an AC-current is used with a frequency of 17Hz. The magnitude of the applied AC-currents is defined by their rms values.
![\[fig:Fig1\] (a) Microscope image of the device structure, consisting of a Pt or Ta Hall-bar detector (bottom layer) and a Pt heater (top layer), separated by an insulating Al$_2$O$_3$ layer. Ti/Au pads are used for contacting the device. For external heating experiments, the device is contacted as marked. The applied field direction is given by $\alpha_0$, as defined in the figure. (b) 2$^\textrm{nd}$ harmonic voltage signal generated by the SSE for a fixed magnetic field direction of $\alpha_0=90^\circ$ and (c) angular dependence of the SSE signal applying a magnetic field of 50mT, in both Pt and Ta. ](Fig1SSEheater){width="8.5cm"}
{width="18cm"}
Evaluating the working mechanism of the lock-in detection technique in more detail (see appendix), shows that in order to obtain the linear response signal of the system, both the measured 1$^\textrm{st}$ harmonic signal as well as the 3$^\textrm{rd}$ harmonic signal have to be taken into account. Including both harmonic signals following the analysis explained in the appendix, we find the shown 1$^\textrm{st}$ order response. Note that the measured 2$^\textrm{nd}$ harmonic signal is directly plotted, without any corrections.
In all experiments, an external magnetic field is applied to define the direction of the magnetization M of the YIG. The direction of the applied field is defined by $\alpha_0$, which is the in-plane angle between the current direction (along $x$) and the applied field direction, as it is marked in Fig. \[fig:Fig1\](a). Not only experiments at high saturation magnetic fields are performed, also the low field behavior is investigated. The applied magnetic field strength was measured by a LakeShore Gaussmeter (model 421) using a transverse Hall probe, to correct the set magnetic field for any present remnant field. All measurements are carried out at room temperature.
Results and Discussion
======================
Spin-Seebeck effect by external heating
---------------------------------------
For the external heating experiment an AC-current is sent through the top Pt strip as marked in Fig. \[fig:Fig1\](a). By measuring the 2$^\textrm{nd}$ harmonic voltage signals along the Hall-bar, the SSE is detected via the ISHE in Pt and Ta. Fig. \[fig:Fig1\](b) shows the typical SSE signals for both YIG/Pt and YIG/Ta samples for an applied field perpendicular to the longitudinal direction of the Hall-bar ($\alpha_0=90^\circ$). Changing the sign of B (and thus M) changes the sign of the signal, as the spin-polarization direction of the pumped spin-current is reversed. Due to the low coercive field of YIG almost no hysteresis is observed for the reversed field sweep. For the YIG/Pt and YIG/Ta sample opposite magnetic field dependence is observed, proving the opposite sign of the spin-Hall angle for Pt versus Ta.
As the spin-polarization direction of the generated spin-current is dependent on the direction of the YIG magnetization, the SSE/ISHE voltage shows a sine shaped angular dependence with a period of 360$^\circ$. By rotating the sample in a constant applied magnetic field of 50mT, this angular dependence is detected as is shown in Fig. \[fig:Fig1\](c). Also here the effect of the opposite sign of the spin-Hall angle, for Ta compared to Pt, is clearly visible.
From Fig. \[fig:Fig1\] it is observed that the SSE signal for the YIG/Ta sample is almost a factor 10 smaller than for the YIG/Pt sample ($V_{SSE,Pt}/V_{SSE,Ta}=-9.8$). To compare, we calculate the expected ratio from the theoretical description of the SSE voltage, as reported by Schreier et al.[@SSETheory]: $$\label{eq:SSE}
V_{SSE}=C_{YIG} \cdot \Delta T_{me} G_r \Theta_{SH} \rho l \eta \frac{\lambda}{t} \tanh{\left( \frac{t}{2\lambda}\right) }$$ where $C_{YIG}$ contains all parameters describing properties of YIG, including some physical constants (defined in ref.[@SSETheory]), so $C_{YIG}$ is constant for both the YIG/Pt and the YIG/Ta sample. $\Delta T_{me}$ is the temperature difference between the magnons and electrons at the YIG/metal interface. $\rho$, $\lambda$, $t$ and $l$ are the resistivity, spin-diffusion length, thickness of the normal metal layer (Pt/Ta) and the distance between the voltage contacts, respectively. $\eta$ is the backflow correction factor, defined as $$\label{eq:backflow}
\eta = \left[ 1 + G_r \rho \lambda \coth{\left(\frac{t}{\lambda}\right)} \right]^{-1}$$
Previously, in ref.[@VlietstraSMR2], we have determined the real part of the spin-mixing conductance at the YIG/Pt interface ($G_r=4.4\times10^{14} \Omega^{-1} \textrm{m}^{-2}$), the spin-Hall angle ($\Theta_{SH,Pt}=0.08$) and the spin-diffusion length ($\lambda_{Pt}=1.2$nm) of Pt. For the YIG/Ta sample we take the magnitude of these system parameters as reported by Hahn et al.[@JamalTaPt] ($G_r=2\times10^{13} \Omega^{-1} \textrm{m}^{-2}$, $\Theta_{SH,Ta}=-0.02$ and $\lambda_{Ta}=1.8$nm). As a check, we also used these parameter-values to calculate the 1$^\textrm{st}$ order SMR signals for Ta, and found good agreement with the measured signals (not shown).
To get an estimate for $V_{SSE,Pt}/V_{SSE,Ta}$ we assume $\Delta T_{me}$ to be constant for both samples. By inserting the values of the mentioned parameters, the dimensions of the Hall-bars and the measured resistivity of the Pt and Ta layers ($\rho_{Pt}=3.4\times10^{-7}\Omega$m and $\rho_{Ta}=3.5\times 10^{-6} \Omega$m, respectively), we find $V_{SSE,Pt}/V_{SSE,Ta}=-10.6$, which is close to the experimentally observed ratio.
Current-induced spin-Seebeck effect
-----------------------------------
The second method used to detect the SSE is by current-induced heating through the metal detection strip itself, as recently was reported by Schreier et al.[@SSEGoennenwein]. In this section we show that we can achieve more directly similar results, by using a lock-in detection technique. By this technique, the SSE signals can directly be detected as a 2$^\textrm{nd}$ harmonic signal, fully separated from the SMR signal, which shows up in the 1$^\textrm{st}$ harmonic response. Furthermore, the lock-in detection technique enables us to reveal and investigate additional signals appearing when applying low magnetic fields.
The inset of Fig. \[fig:Fig2\](c) shows a microscope image of the sample, marking the position of the current and voltage probes for the current-induced heating experiments. The magnetic field direction is again defined by $\alpha_0$. This measurement configuration is similar to the method used to detect transverse SMR[@VlietstraSMR; @VlietstraSMR2] and therefore we expect to observe SMR in the 1$^\textrm{st}$ order signal, as is shown in Figs. \[fig:Fig2\](a) and (b). In Fig. \[fig:Fig2\](b) it is observed that down to very low applied magnetic fields, the average magnetization direction of the YIG nicely follows the applied field direction, resulting in the $\textrm{sin}(2\alpha_0)$ angular dependence of the SMR.[@AlthammerSMR] Only for the lowest applied field of 0.9mT a small deviation of the signal around $\alpha_0=\pm90^\circ$ is observed, showing this field strength is not sufficient to assume M being (on average) fully along the applied field direction. Similar to the external heating experiment, the SSE signal shows up in the 2$^\textrm{nd}$ harmonic signal. Figs. \[fig:Fig2\](c) and (d) show the magnetic field dependence and angular dependence of the detected 2$^\textrm{nd}$ harmonic signal, respectively. Comparing the shape of the 2$^\textrm{nd}$ harmonic data of the external heating experiments (Fig. \[fig:Fig1\](b)) to the current-induced heating experiments (Fig. \[fig:Fig2\](c)), an enhanced signal is observed in Fig. \[fig:Fig2\](c) for fields of a few mT. This additional signal cannot be explained by the angular dependence of the SSE, neither by the rotation of M in the plane towards B (by which the 1$^\textrm{st}$ harmonic SMR peaks in Fig. \[fig:Fig2\](a) are explained[@VlietstraSMR]).
The angular dependence of this additional signal, as presented in Fig. \[fig:Fig2\](d), shows that besides an increased amplitude of the SSE signal (black symbols in Figs. \[fig:Fig2\](c) and (d)), also at $\alpha_0=\pm90^\circ$ additional peaks appear for low applied fields. By increasing the applied magnetic field, all extra signals disappear, leaving the expected SSE signal showing a 360$^\circ$ periodic angular dependence.
![\[fig:Fig3\] AC-current dependence of the 2$^\textrm{nd}$ harmonic voltage for an applied field of (a) 0.9mT and (b) 50mT. For the shown experiments, the transverse current-induced heating measurement configuration has been used. The vertical dashed lines mark the data plotted in (c), which shows the AC-current dependence of the magnitude of the signal at $\alpha_0=0^\circ$ for B=0.9mT (red dots) and B=50mT (black squares) and the average magnitude of the peaks (peak to peak) around $\alpha_0=\pm90^\circ$ for B=0.9mT (blue triangles). The dashed lines are a guide for the eye. ](Fig3CurrentDep){width="8.5cm"}
To further characterize the additionally observed features at low applied magnetic fields, also their AC-current dependence has been measured and these results are shown in Fig. \[fig:Fig3\]. It can be seen that the current dependence is very similar to the shown magnetic field dependence, giving maximal additional signals for low applied fields and high applied AC-currents. From Figs. \[fig:Fig3\](a) and (b), the magnitude of the signal at $\alpha_0=0^\circ$ is extracted and plotted separately in Fig. \[fig:Fig3\](c). As can be seen from this figure, for both the applied magnetic field of 0.9mT (Fig. \[fig:Fig3\](a)) and 50mT (Fig. \[fig:Fig3\](b)), the amplitude of the signal quadratically scales with the applied AC-current. The magnitude of the peaks around $\alpha_0=\pm90^\circ$, plotted in blue in Fig. \[fig:Fig3\](c), increases faster than quadratically, pointing to the presence of higher order effects.
To fully exclude the SSE being the origin of the additionally detected signals, the current-induced heating measurements were repeated on the YIG/Ta sample. Results of those measurements are shown in Fig. \[fig:Fig4\]. The applied current in those experiments is only 1.9mA, limited by the high resistance of the Ta bar ($\rho_{Ta}=3.5\times 10^{-6} \Omega \textrm{m}$). In both Fig. \[fig:Fig4\](a) and (b) it can be seen that the high-field signal nicely changes sign compared to the YIG/Pt data, as predicted for the SSE/ISHE, because of the opposite sign of the spin-Hall angle of Ta compared to Pt. Contrary, the low-field peak in the magnetic field sweep (Fig. \[fig:Fig4\](a)) keeps the same sign as in the YIG/Pt sample (Fig. \[fig:Fig2\](c)), showing the SSE cannot be the origin of this phenomenon. Furthermore, this result also excludes the observed feature being originated by any other effect linearly related to the spin-Hall angle of a material. So, possible deviations of M caused by spin-transfer torque, due to a spin-current created via the SHE, cannot directly be used to explain the observed features. Note that the SMR signal depends quadratically on the spin-Hall angle,[@BauerTheorySMR; @VlietstraSMR2] which makes any effect related to the SMR a likely candidate for explaining the observed features.
![\[fig:Fig4\] 2$^\textrm{nd}$ harmonic response of the transverse voltage for the YIG/Ta sample. (a) Magnetic field sweep for $\alpha_0=0^\circ$ and (b) angular dependence for two different applied magnetic fields (0.9mT and 50mT). The SSE signal (=signal at high field) has opposite sign compared to the YIG/Pt sample, whereas the low-field behavior is similar for both samples. The black symbols in the figures point out the equal measurement conditions comparing both figures. ](Fig4PtTa){width="8.5cm"}
Summarizing, the current-induced heating experiments show that when applying a sufficiently high magnetic field ($>$10mT), the SMR and SSE can be simultaneously, but separately, detected using an AC-current combined with a lock-in detection technique. By this method the SSE can thus be very easily and directly detected, without being interfered with the SMR signal. Furthermore, it is observed that for low magnetic fields, and/or high heating currents, additional signals appear on top of the SSE. The origin of these additional signals might be related to the SMR-effect, which will be discussed in more detail in the next section.
Dynamic Spin-Hall Magnetoresistance
-----------------------------------
During the measurements, large AC-currents are sent through the Hall-bar structure, which can generate magnetic fields. One source of these magnetic fields will be Oersted fields (B$_{oe}$) generated around the Hall-bar, perpendicular to the current direction. As the Hall-bar structure is very thin compared to its lateral dimensions, the Oersted field above/below the center of the bar can be estimated using the infinite plane approximation: $B_{oe}=\frac{\mu_0I}{2w}$, where $\mu_0$ is the permeability in vacuum, $I$ is the applied current and $w$ is the width of the Pt bar. Note that the generated field in this case is independent of the distance from the plane, so the full thickness of the YIG below the Hall-bar will be exposed to this field.[^1] For example, for an applied current of 8mA, an Oersted field of 0.1mT will be generated, which is significant compared to an applied magnetic field of 0.9mT. Therefore, for low applied fields, the magnetization direction of the YIG will also be affected by the generated Oersted field.
As we are dealing with AC-currents, the generated Oersted field (and any other current-induced magnetic field) will continuously change sign, which might cause M to oscillate around the applied field direction. In this case, $\alpha$ (defining the direction of M) is current-dependent,[@SOTnat] giving rise to dynamic equations for both the SMR and the SSE. The current-dependent behavior of the SMR signal is derived starting from the equation for transverse SMR[@BauerTheorySMR; @VlietstraSMR2] $$\label{eq:SMR}
V_{T,SMR}=IR_{T,SMR}=\Delta R_1Im_xm_y+\Delta R_2Im_z %=\Delta R_1sin(\alpha)cos(\alpha)+\Delta R_2sin(\beta)$$ where $\Delta R_1$ and $\Delta R_2$ are resistance changes dependent on the spin-diffusion length, spin-Hall angle and spin-mixing conductance of the system, as defined in refs.[@BauerTheorySMR; @VlietstraSMR2]. $m_{x,y,z}$ are the components of M pointing in respectively the x-, y-, and z-direction (where z is the out-of-plane direction). $m_x$ and $m_y$ can be expressed as $\sin(\alpha)$ and $\cos(\alpha)$, respectively. As the applied magnetic field is in-plane, as well as the generated Oersted field, combined with the large demagnetization field of YIG for out-of-plane directions, $m_z$ will be small and therefore neglected in further derivations.
Small oscillations of M, due to the presence of AC-current generated magnetic fields, result in a current-dependent SMR signal, which can be expressed in first order as $$\label{eq:1storder}
V_{T,SMR}(I) \approx V_{T,SMR} (\alpha_0)+I \left.\frac{\textrm{d}V_{T,SMR}}{\textrm{d}I} \right|_{\alpha_0}$$ where $\alpha_0$ gives the equilibrium direction of M around which it is oscillating (assuming it to be equal to the applied field direction, as concluded from the measured 1$^\textrm{st}$ harmonic SMR response). Calculating the derivative in Eq.(\[eq:1storder\]), using Eq.(\[eq:SMR\]), neglecting $m_z$ and keeping in mind that $\alpha$ is dependent on $I$, gives $$\label{eq:SMRder}
\left.\frac{\textrm{d}V_{T,SMR}}{\textrm{d}I}\right|_{\alpha_0}=\Delta R_1\sin(\alpha_0)\cos(\alpha_0)+\Delta R_1I\cos(2\alpha_0)\frac{\textrm{d}\alpha}{\textrm{d}I}$$ The first term on the right side of Eq.(\[eq:SMRder\]) describes the 1$^\textrm{st}$ order response (linear with $I$), showing the expected transverse SMR behavior (Eq.(\[eq:SMR\])). The second term is a 2$^\textrm{nd}$ order response ($R_{2,SMR}$) and will therefore show up in addition to the expected SSE signal. $\frac{\textrm{d}\alpha}{\textrm{d}I}$ is the term which includes the deviation of M due to current-induced magnetic fields, and its magnitude is dependent on both $\alpha_0$ and the magnitude of the total magnetic field (applied field, coercive field and the current-induced fields). For large applied magnetic fields, the current-induced magnetic fields will have a negligible effect on M, so $\frac{\textrm{d}\alpha}{\textrm{d}I}$ goes to zero, leaving only the SSE signal in the 2$^\textrm{nd}$ harmonic signal (as is observed). Note that also in the described external heating experiments Oersted fields are generated, influencing M, but there the dynamic SMR signal will not be detected, as the SMR itself is not present.
To find an expression for $\frac{\textrm{d}\alpha}{\textrm{d}I}$, first the direction of M (given by $\alpha$) is defined, taking into account the Oersted fields (causing $\Delta\alpha$): $$\label{eq:Mdir}
\alpha=\alpha_0+\Delta\alpha=\alpha_0+\textrm{atan}\left(\frac{B_{oe}}{B_{ex}}\right)\cos(\alpha_0)$$ where $B_{ex}$ is the applied magnetic field and $\textrm{atan}(\frac{B_{oe}}{B_{ex}})$ is the maximum deviation of M from $\alpha_0$, which is the case for $\alpha_0=0^\circ$ ($B_{ex}$ perpendicular to $B_{oe}$, neglecting any other field contributions). From Eq.(\[eq:Mdir\]) now $\frac{\textrm{d}\alpha}{\textrm{d}I}$ ($=\frac{\textrm{d}\alpha}{\textrm{d}B_{oe}}\frac{\textrm{d}B_{oe}}{\textrm{d}I}$) can be derived, finding $$\label{eq:dalpha}
\frac{\textrm{d}\alpha}{\textrm{d}I}=\frac{\mu_0}{2w}\frac{B_{ex}}{B_{ex}^2+B_{oe}^2}\cos(\alpha_0)$$ Substituting the derived equation for $\frac{\textrm{d}\alpha}{\textrm{d}I}$ in Eq.(\[eq:SMRder\]), we calculate the expected 2$^\textrm{nd}$ harmonic SMR signal due to dynamic behavior of M, caused by the current-induced Oersted field as: $$\label{eq:R2ndHarm}
R_{2,SMR}=\Delta R_1\frac{\mu_0}{2w}\frac{B_{ex}}{B_{ex}^2+B_{oe}^2}\cos(2\alpha_0)\cos(\alpha_0)$$
Additional to $R_{2,SMR}$, also the SSE will be present as a 2$^\textrm{nd}$ harmonic signal, showing $\cos(\alpha_0)$ behavior with an amplitude independent from the applied magnetic field strength. The amplitude of the SSE signal can be derived from the high-field measurements shown in Fig. \[fig:Fig3\](b) and Fig. \[fig:Fig4\](b).
Figures \[fig:Fig5\](a) and (b) show the total calculated 2$^\textrm{nd}$ harmonic voltage signal for the YIG/Pt sample, taking into account both the SSE, extracted from the measurements, and the dynamic SMR term, given by Eq.(\[eq:R2ndHarm\]) (where $V_{2,SMR}=(I^2/\sqrt{2})R_{2,SMR}$). For the calculation of $\Delta R_1$ system parameters from ref.[@VlietstraSMR2] are used. Both the calculated current dependence (Fig. \[fig:Fig5\](a)) as well as the calculated magnetic field dependence (Fig. \[fig:Fig5\](b)) show similar features as the measurements, only its magnitude and exact shape do not fully coincide. Following the explanation of the lock-in detection method as described in the appendix, we find that these discrepancies are mainly caused by the presence of a non-negligible 4$^\textrm{th}$ harmonic signal (and possibly even higher harmonics). When determining the 2$^\textrm{nd}$ order response of the system, taking into account the measured 2$^\textrm{nd}$ and 4$^\textrm{th}$ harmonic signals, the peaks observed around $\alpha_0=\pm90^\circ$ get wider and smoother, more closely reproducing the calculated signals as presented in Fig. \[fig:Fig5\].
The amplitude of the overall calculated signal is slightly larger than the measurements, up to a factor 1.6 (even after taking into account the 4$^\textrm{th}$ harmonic signal). One reason for this discrepancy might be that, for the calculations, only the applied magnetic field and the generated Oersted field are taken into account, neglecting any other present fields. For example, the coercive field of the YIG is assumed to be absent, as well as the effect of spin-torque and the presence of non-uniform magnetic fields. These additional fields will influence the amplitude of the oscillations of M, giving a different value for $\frac{\textrm{d}\alpha}{\textrm{d}I}$ than the assumed deviation from only $B_{oe}$ and $B_{ex}$. Furthermore, the assumed perfect $\cos(\alpha_0)$ behavior of $\frac{d\alpha}{dI}$ might also be disturbed by the presence of these other fields. Secondly, in the calculations it is assumed that M is fully aligned with the total magnetic field, which might not always be the case, as we investigate the system applying magnetic fields approaching the coercive field of the YIG. Therefore, the definition of $\alpha_0$ can be slightly off from the assumed ideal case. Full characterization of the magnetization dynamics of the system and the magnetic field distribution would be needed to be able to give a more complete theoretical analysis of the observed features.
![\[fig:Fig5\] (a) Current dependence and (b) magnetic field dependence of the calculated 2$^\textrm{nd}$ harmonic response, including the dynamic SMR (from Eq.(\[eq:R2ndHarm\])) and the SSE for the YIG/Pt system. (c) Calculated 2$^\textrm{nd}$ harmonic response for the YIG/Ta sample using a scaling factor for $\frac{\textrm{d}\alpha}{\textrm{d}I}$ of 0.5. For all calculations the SSE signal is extracted from the measurements. ](Fig5Calculated){width="8.5cm"}
The same calculations have been repeated for the YIG/Ta system. The system parameters needed to calculate $\Delta R_1$ are taken from ref.[@JamalTaPt], as given in section IV A. As a check, these system parameters firstly were used to calculate the 1$^\textrm{st}$ order SMR signal, finding good agreement with the measured signals (not shown). For the calculated 2$^\textrm{nd}$ harmonic signal again it is found that the amplitude of $\frac{\textrm{d}\alpha}{\textrm{d}I}$ has to be lowered to be able to reproduce the measured behavior. When lowering the calculated $\frac{\textrm{d}\alpha}{\textrm{d}I}$ by a chosen scaling factor of 0.5, results as shown in Fig. \[fig:Fig5\](c) are obtained. Comparing the calculated angular dependence to the measurement as shown in Fig. \[fig:Fig4\](b), good agreement is found in the observed behavior. Note that for the YIG/Ta system the contribution of the 4$^\textrm{th}$ harmonic term is much less pronounced, as the applied current is only 1.9mA (compared to 8mA for the YIG/Pt experiments). Following the derivation of the dynamic SMR as explained above, also dynamic SSE signals can be expected. As the SSE is a 2$^\textrm{nd}$ order effect, any dynamic SSE signals are expected to appear as a 3$^\textrm{rd}$ order signal. Measurements of the 3$^\textrm{rd}$ harmonic signal indeed show additionally appearing signals at low applied magnetic fields, but these additional signals are one order of magnitude too large to be explained by the derived possible dynamic SSE signals. This shows that other higher harmonic effects are present, which makes it at this moment impossible to exclusively extract any contribution of possibly present dynamic SSE signals.
Concluding this section, the dynamic SMR is a good candidate for explaining the observed low-field 2$^\textrm{nd}$ harmonic behavior. For both the YIG/Pt and YIG/Ta sample, the features observed in the experiments can be well reproduced by the dynamic SMR model. However, one has to keep in mind that more non-linear effects are present, such that higher harmonic signals need to be taken into account. Furthermore, the derived model is not sufficient to fully be able to reproduce the measured data. Further analysis of the magnetization dynamics in the YIG at low applied fields and high applied currents is necessary to be able to derive a more complete model.
Summary
=======
We have shown the detection of the SSE in YIG/Pt and YIG/Ta samples by both external heating and current-induced heating. The external heating experiments directly show the SSE and clearly show the effect of the opposite spin-Hall angle for Ta compared to Pt. For the current-induced measurements, besides the SSE, the SMR is also present. By using a lock-in detection technique we are able to simultaneously, but separately, measure the SSE and SMR signals. Investigation of the low-field behavior of the SMR and SSE, reveals an additional 2$^\textrm{nd}$ harmonic signal. This additional signal is explained by the presence of a dynamic SMR signal, caused by alternating Oersted fields. Calculations show reproducibility of the observed 2$^\textrm{nd}$ harmonic features, however further analysis of the magnetization dynamics in the YIG is needed to derive a more complete model of the system behavior.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to acknowledge M. de Roosz, H. Adema and J. G. Holstein for technical assistance. This work is supported by NanoNextNL, a micro and nanotechnology consortium of the Government of the Netherlands and 130 partners, by the Marie Curie Actions (Grant 256470-ITAMOSCINOM), the Basque Government (PhD fellowship BFI-2011-106), by NanoLab NL and by the Zernike Institute for Advanced Materials (Dieptestrategie program).
Appendix {#appendix .unnumbered}
========
Lock-in detection
-----------------
All measurements shown in the main text are performed using a lock-in detection technique. By this technique, 1$^\textrm{st}$, 2$^\textrm{nd}$ and higher order responses of a system on an applied AC-current can be determined. In general, any generated voltage can be written as the sum of 1$^\textrm{st}$, 2$^\textrm{nd}$ and higher order current-responses as: $$\label{eq:Lockin1}
V(t) = R_1I(t)+R_2I^2(t)+R_3I^3(t)+R_4I^4(t)+...$$ where R$_n$ is the n$^\textrm{th}$ order response of the measured system to an applied current $I(t)$. By applying an AC-current $I(t)=\sqrt2I_0\textrm{sin}(\omega t)$, with angular frequency $\omega$ and rms value $I_0$, a lock-in amplifier can be used to detect individual harmonic voltage responses of the investigated system, making use of the orthogonality of sinusoidal functions. To extract the separate harmonic responses, the output signal and the reference input signal (a sine wave function) are multiplied and integrated over a set time. When both signals have different frequencies, the integration over many periods will result in zero signal, whereas integration of two sine wave functions with the same frequency and no phase shift will result in a non-zero signal. Besides being able to separately extract the different harmonic responses of the system, the lock-in detection technique also reduces the noise in the signal, compared to dc voltage measurements, as the measurement is only sensitive to a very narrow frequency spectrum.
The detected $n$-th harmonic signal of a lock-in amplifier at a set phase $\phi$ is defined as $$\label{eq:Lockin2}
V_{n}(t) = \frac{\sqrt2}{T} \int\limits_{t-T}^{t} \sin(n\omega s + \phi)V_{in}(s)ds$$ By evaluating Eq.(\[eq:Lockin2\]) for a given input voltage V$_{in}$, one can obtain the different harmonic voltage signals that can be measured by the lock-in amplifier ($V_{n}$). Assuming a voltage response up till the 4$^\textrm{th}$ order, the following lock-in voltages are calculated: $$\label{eq:V1}
V_{1} = R_1 I_0 + \frac{3}{2} R_3 I_0^3
\;\;\;\;\;\; \textrm{for} \; \phi=0^\circ$$ $$\label{eq:V2}
V_{2} = \frac{1}{\sqrt2} ( R_2 I_0^2 + 2 R_4 I_0^4) \;\;\;\;\;\; \textrm{for}\; \phi=-90^\circ$$ $$\label{eq:V3}
V_{3} = -\frac{1}{2} R_3 I_0^3
\;\;\;\;\;\; \textrm{for} \; \phi=0^\circ$$ $$\label{eq:V4}
V_{4} = -\frac{1}{2\sqrt2} R_4 I_0^4
\;\;\;\;\;\; \textrm{for} \; \phi=-90^\circ$$ So, using different lock-in amplifiers to measure the 1$^\textrm{st}$, 2$^\textrm{nd}$, 3$^\textrm{rd}$ and 4$^\textrm{th}$ harmonic voltage responses, $R_n$ can be deduced from Eqs.(\[eq:V1\])-(\[eq:V4\]). To detect the 2$^\textrm{nd}$ and 4$^\textrm{th}$ harmonic response, the phase of the lock-in amplifier should be set to $\phi=-90^\circ$.
Note that $V_1$ ($V_2$) does not purely scale linearly (quadratically) with $I_0$. A 3$^\textrm{rd}$ (4$^\textrm{th}$) order current dependence is also present in the measured voltage response. Thus, to obtain the 1$^\textrm{st}$ order response ($R_1$) of the measured system, not only the measured 1$^\textrm{st}$ harmonic signal ($V_1$) has to be taken into account, also the 3$^\textrm{rd}$ harmonic signal ($V_3$) has to be included: $$\label{eq:R1}
R_1 = \frac{1}{I_0}(V_{1} + 3V_{3})$$ Similarly, the 2$^\textrm{nd}$ order response is calculated as $$\label{eq:R2}
R_2 = \frac{\sqrt2}{I_0^2}(V_{2} + 4V_{4})$$
For the current-induced SSE and SMR measurements described in the main text, Fig. \[fig:Fig6\] shows the effect of including the higher harmonic responses of the system (up to the 4$^\textrm{th}$ harmonic), compared to assuming them to be negligible. For this comparison Eq.(\[eq:R1\]) and Eq.(\[eq:R2\]) are used. In Fig. \[fig:Fig6\](a) and (c) $V_3$ and $V_4$ are assumed to be zero, whereas in Fig. \[fig:Fig6\](b) and (d) the measured 3$^\textrm{rd}$ and 4$^\textrm{th}$ harmonic response have been taken into account, respectively. From these figures it can be concluded that in our low-field measurements the higher harmonic signals are not negligibly small, and thus should be accounted for.
In Fig. \[fig:Fig2\](a) and (b) of the main text, the 1$^\textrm{st}$ order response of the system is plotted, taking into account the measured 3$^\textrm{rd}$ harmonic voltage response as derived in Eq.(\[eq:R1\]) and shown in Fig. \[fig:Fig6\](b). For the other figures, regarding the 2$^\textrm{nd}$ order response of the system, the measured lock-in voltage is directly plotted ($V_2$). To find the 2$^\textrm{nd}$ order response of the system Eq.(\[eq:R2\]) should be evaluated, including both $V_2$ and $V_4$. Taking this correction into account, the observed features of $V_2$ slightly change, as is shown in Fig. \[fig:Fig6\](d), more closely following the expected behavior from the calculated dynamic SMR signal.
![\[fig:Fig6\] Evaluation of Eq.(\[eq:R1\]) for a selected set of measurements: (a) assuming $V_3=0$ and (b) including the measured values of $V_3$. Similarly for Eq.(\[eq:R2\]): (c) assuming $V_4=0$ and (d) including the measured values of $V_4$. ](Fig6AppendixHarmonics){width="8.5cm"}
[^1]: This approximation was checked by modeling the system using COMSOL Multiphysics, showing indeed a nearly constant Oersted field at least several hundreds of nanometers above/below the plane
|
---
abstract: 'The connection between geodesics on the modular surface $\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb{H}}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma\backslash {\mathbb{H}}$ and odd and grotesque continued fractions, where $\Gamma\cong {\mathbb Z}_3 \ast {\mathbb Z}_3$ is the index two subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by the free elements of order three $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix} \right)$, and having an ideal quadrilateral as fundamental domain. A similar connection between geodesics on $\Theta\backslash {\mathbb{H}}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)$.'
address:
- 'Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801'
- 'E-mail: [email protected]'
- 'E-mail: [email protected]'
author:
- 'Florin P. Boca'
- Claire Merriman
title: Coding of geodesics on some modular surfaces and applications to odd and even continued fractions
---
Introduction
============
The connection between geodesics on the modular surface ${\mathcal M}=\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb{H}}$ and regular continued fractions (RCF), originating in the seminal work of Artin [@Art], has generated a significant amount of interest. Inspired by earlier work of Moeckel [@Moe], Series [@Ser] established explicit connections between the geodesic flow on ${\mathcal M}$, geodesic coding, and RCF dynamics. The way ideal triangles of the Farey tessellation ${\mathbb{F}}$ cut oriented geodesics $\gamma$ on the upper half-plane ${\mathbb{H}}$ play a central role in this approach. The geodesic crosses two sides of a triangle, and the geodesic arc is labeled $L$ or $R$ according to whether the vertex shared by the sides is to the left or right, respectively, of the geodesic. This labeling is invariant under $\Gamma (1)=\operatorname{PSL}(2,{\mathbb Z})$, and hence under any of its subgroups. These geodesics $\gamma$ are lifts of geodesics $\bar{\gamma}$ on ${\mathcal M}$, which are uniquely determined by infinite two-sided cutting sequences $ \ldots L^{n_{-1}} R^{n_0} L^{n_1} \ldots$. The sequence of positive integers $(n_i)_{i=-\infty}^\infty$ is intimately related with regular continued fractions, since $$\gamma_{-\infty}=\frac{-1}{n_0+\frac{1}{n_{-1}+\frac{1}{n_{-2}+\cdots}}} =
-[ n_0,n_{-1},n_{-2},\ldots ]\ \ \mbox{\rm and} \ \
\gamma_{\infty}=n_1 +\frac{1}{n_2+\frac{1}{n_3 + \cdots}} =
[ n_1;n_2,n_3,\ldots ]$$ are the negative and positive endpoints of some lift $\gamma$ of $\bar{\gamma}$ to ${\mathbb{H}}$. Shifting along the cutting sequences is related to the Gauss map $T$ on $[0,1)$ and its natural extension ${\bar{T}}$ on $[0,1)^2$, defined by $$T\big( [n_1,n_2,n_3,\ldots]\big) =[n_2,n_3,n_4,\ldots ],$$ $${\bar{T}}\big( [n_1,n_2,n_3,\ldots ],[n_0,n_{-1},n_{-2},\ldots ]\big) =
\big( [n_2,n_3,n_4,\ldots ],[n_1,n_0,n_{-1},\ldots ]\big) .$$ Some different approaches for coding the geodesic flow on ${\mathcal M}$ were considered by Arnoux in [@Arn] and by Katok and Ugarcovici [@KU1].
A large class of continued fractions has been studied in the context of the geodesic flow and symbolic dynamics. A non-exhaustive list includes backward continued fractions [@AF1; @AF2], even continued fractions [@AaD; @BL; @Cel; @KU1], Rosen continued fractions [@MS], $(a,b)$-continued fractions [@KU2], Nakada $\alpha$-continued fractions and $\alpha$-Rosen continued fractions [@AS], or other classes of complex or Heisenberg continued fractions [@LV]. In different directions, the symbolic dynamics associated with the billiard flow on modular surfaces of uniform triangle groups, and with the geodesic flow on two-dimensional hyperbolic good orbifolds, have been thoroughly investigated by Fried [@Fr] and Pohl [@Po].
This note describes codings of geodesics on the modular surfaces ${\mathcal M}_o$ and ${\mathcal M}_e$, associated with subgroups of index two and three in $\Gamma (1)$, respectively. The coding of ${\mathcal M}_o$ is hereby connected, in the spirit of [@Ser], to the dynamics of odd and grotesque continued fractions (OCF and GCF respectively), and the coding of ${\mathcal M}_e$ is connected to the even (ECF) and extended even continued fractions. These continued fractions were first investigated by Rieger [@Rie] and by Schweiger [@Sch1; @Sch2].
We consider the modular surface ${\mathcal M}_o =\Gamma \backslash {\mathbb{H}}=\pi_o ({\mathbb{H}})$, where $\Gamma$ is the index two subgroup of $\Gamma (1)=\operatorname{PSL}(2,{\mathbb Z})$ generated by the M" obius transformations $S(z)=\frac{-1}{z+1}$ and $T(z)=z+2$ acting on ${\mathbb{H}}$. Equivalently, $\Gamma$ is generated by the order three matrices $S=\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix}\right)$ and $ST^{-1}=\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix} \right)$. The corresponding Dirichlet fundamental domain with respect to $i$ is the quadrilateral ${\mathfrak F}$ bounded by the geodesic arcs $[0,\omega]$, $[\omega,\infty]$, $[0,\omega^2]$ and $[\omega^2,\infty]$, where $\omega=\frac{1}{2} (1+i\sqrt{3})$ (see Figure \[Figure1\]). The transformation $S$ fixes $\omega^2=\omega-1$ and cyclically permutes the points $\infty, 0, -1$, and respectively $i,\frac{-1+i}{2},-1+$. On the other hand $ST^{-1}$ fixes $\omega$ and permutes $\infty, 0, 1$, and $1,\frac{1+i}{2},1+i$ (see Figures \[Figure1\] and \[Figure3\]). As shown by Lemma \[lemma7\], the point $\pi_o(\infty)$ is the only cusp of ${\mathcal M}_o$, while $\pi_o(i)$ is a regular point for ${\mathcal M}_o$ since we deal with a two-fold cover ramified at $i$ which makes the singularity disappear.
The parts of the fundamental region ${\mathfrak F}$ on either side of the imaginary axis are considered separately. First, we consider the triangle $\omega^2,0,\infty$. The union of the images of this triangle under $I, S,$ and $S^2$ gives the ideal triangle $-1,0,\infty$. Similarly, the triangle with vertices $\omega, 0, \infty$, under $I, {ST^{-1}}$ and $({ST^{-1}})^2$ is the ideal triangle $1,0,\infty$. Together, these regions form the ideal quadrilateral $\Delta$ with vertices $-1,0,1$ and $\infty$. The images of $\Delta$ under $\Gamma$ form the Farey tessellation. That is, two rational numbers $\frac{p}{q},\frac{p'}{q'}$ are joined by a side of the Farey tessellation precisely when $pq'-p'q=\pm 1$.
![The fundamental region ${\mathfrak F}$ and the modular surface ${\mathcal M}_o=\Gamma \backslash {\mathbb{H}}=\pi_o ({\mathbb{H}})$[]{data-label="Figure1"}](fundamental_domain.eps "fig:") ![The fundamental region ${\mathfrak F}$ and the modular surface ${\mathcal M}_o=\Gamma \backslash {\mathbb{H}}=\pi_o ({\mathbb{H}})$[]{data-label="Figure1"}](modsurface1.eps "fig:")
![The checkered Farey tessellation[]{data-label="Figure2"}](tessellation.eps){width="12cm"}
This is the same tessellation considered by Series [@Ser], but we add a checkerboard coloring, as shown in Figure \[Figure2\]. The triangle $-1, 0, \infty$ is light, while $1,0,\infty$ is dark, then continue in a checkerboard pattern, so that each of the three neighboring Farey cells of a light cell are dark, and vice versa. We code oriented geodesics by including the shade of the Farey cell. Concretely, a light $L$ is denoted by ${\mathbb{L}}$, a dark $L$ by ${\mathbf{L}}$, a light $R$ by ${\mathbb{R}}$, and a dark $R$ by ${\mathbf{R}}$. This way, every geodesic in ${\mathbb{H}}$ with irrational endpoints is assigned an infinite two-sided sequence of symbols ${\mathbb{L}}$, ${\mathbf{L}}$, ${\mathbb{R}}$ and ${\mathbf{R}}$. We also require that a light letter ${\mathbb{L}}$ or ${\mathbb{R}}$ can only be succeeded and preceded by a dark one, and vice versa.
The coding of geodesics on ${\mathcal M}_o$ is described by concatenating words of the following type: $$\label{string_types_o}
({\mathbb{L}}{\mathbf{L}})^{k-1} {\mathbb{L}}{\mathbf{R}},\quad
({\mathbb{L}}{\mathbf{L}})^{k-1} {\mathbb{R}},\quad
({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{L}},\quad
({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{R}}{\mathbb{L}},\qquad k\geqslant 1.$$ Single letter strings are allowed. We require colors of individual letters to alternate. Thus, strings of the first and third type must be succeeded by those of the first or second. A powerful tool in the study of endomorphisms in ergodic theory is provided by the natural extension, an invertible transformation that dilates the given endomorphism and preserves many of its ergodic properties, such as ergodicity or mixing [@CFS]. Here, we describe the natural extension of the Gauss type OCF map (or, by reversing order, of the Gauss type GCF map) as a factor of a certain explicit cross-section of the geodesic flow on ${\mathcal M}_o$. In particular, this provides a direct geometric proof for the ergodicity of this map and allows us to recapture the invariant measures for the Gauss type OCF and GCF maps. Other applications include a characterization of quadratic surds in terms of their OCF expansion and their conjugate GCF expansion, as well as a tail-equivalence type description of the orbits of the action of $\Gamma$ on the real line.
We also observe that similar results can be obtained for even continued fractions, using the Farey tessellation without coloring and the modular surface ${\mathcal M}_e=\Theta \backslash {\mathbb{H}}$, where $\Theta$ denotes the index three Theta subgroup in $\Gamma(1)$ generated by the transformations $S(z)=-\frac{1}{z}$ and $T(z)=z+2$.
Odd, grotesque, and even continued fractions {#fractiontypes}
============================================
In this section we review some properties and dynamics of odd, grotesque and even continued fractions. Odd and even continued fractions are part of the broader class of $D$-continued fractions introduced in [@Kra] (see also [@HK; @Mas; @DHKM]), while the GCF is the dual algorithm of the odd continued fractions. Instead of giving a compact presentation, we chose to be more repetitive, in order to clarify notation and the difference between these three classes of continued fractions.
Odd continued fractions {#ocf}
-----------------------
The OCF expansion of a number $x\in[0,1]\setminus {\mathbb Q}$ is given by $$\label{eq2}
x=[\![(a_1,{\epsilon}_1), (a_2, {\epsilon}_2), (a_3, {\epsilon}_3), ...]\!]_o = \cfrac{1}{a_1+\cfrac{{\epsilon}_1}{a_2+\cfrac{{\epsilon}_2}{a_3+\cdots}}},$$ where ${\epsilon}_i ={\epsilon}_i(x)\in \{\pm1\}$, $a_i =a_i(x)\in 2{\mathbb N}-1$ and $a_i + {\epsilon}_i \geqslant 2$. Such an expansion is unique. We also consider $$x=[\![(a_0,{\epsilon}_0);(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),\ldots ]\!]_o :=a_0+{\epsilon}_0 [\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),\ldots ]\!]_o \in [1,\infty),$$ with ${\epsilon}_0\in \{ \pm 1\}$, $a_0 \in 2{\mathbb N}-1$ and $a_0+{\epsilon}_0 \geqslant 2$, so that $x\in (a_0,a_0+1)$ if ${\epsilon}_0=1$ and $x\in (a_0-1,a_0)$ if ${\epsilon}_0=-1$.
The odd Gauss map $T_o$ acts on $[0,1]$ by $$T_o(x)= {\epsilon}\bigg( \frac{1}{x}-2k+1\bigg)
\quad
\mbox{\rm if} \quad
x\in B({\epsilon},k) :=\begin{cases} \big( \frac{1}{2k},\frac{1}{2k-1}\big)
& \mbox{\rm if ${\epsilon}=1$, $k\geqslant 1$} \\
\big( \frac{1}{2k-1},\frac{1}{2k-2}\big) & \mbox{\rm if ${\epsilon}=-1$, $k\geqslant 2.$}
\end{cases}$$ Symbolically, $T_o$ acts on the OCF representation by removing the leading digit $(a_1,{\epsilon}_1)$ of $x$, i.e. $$T_o \big( [\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots]\!]_o\big) =[\![ (a_2,{\epsilon}_2),(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_o .$$ The probability measure $d\mu_o (x)=\frac{1}{3\log G} ( \frac{1}{G-1+x}+\frac{1}{G+1-x}) dx$ on $[0,1]$ is $T_o$-invariant (see [@Rie; @Sch1]), where we denote $$G=\tfrac{1}{2}(\sqrt{5}+1).$$
Grotesque continued fractions {#gcf}
-----------------------------
Rieger’s GCF representation of an irrational $y\in I_G:=(G-2,G)$ is given by $$\label{eq3}
y={\langle\! \langle}(b_0,{\epsilon}_0), (b_1, {\epsilon}_1), (b_2, {\epsilon}_2), ...{\rangle\! \rangle}_o =
\cfrac{{\epsilon}_0}{b_0+\cfrac{{\epsilon}_1}{b_1+\cfrac{{\epsilon}_2}{b_2+\cdots}}},$$ where ${\epsilon}_i \in \{\pm1\}$, $b_i=b_i(y) \in 2{\mathbb N}-1$ and $b_i + {\epsilon}_i \geqslant 2$. Every irrational number $y\in I_G$ can be uniquely represented as above, by taking ${\epsilon}_0={\epsilon}_0(y)=\operatorname{sign} y$ and $b_0=b_0(y)$ the unique odd positive integer with $\frac{1}{\lvert y\rvert}-G \leqslant b_0(y) \leqslant
\frac{1}{\lvert y\rvert}-G+2$. The corresponding Gauss map $\tau_o$ acts on $I_G$ by $$\tau_o (y)=\frac{{\epsilon}_0(y)}{y}-b_0(y)=\frac{1}{\lvert y\rvert}-b_0(y) ,$$ or on the symbolic representation by $$\tau_o \left( {\langle\! \langle}(b_0,{\epsilon}_0),(b_1,{\epsilon}_1),(b_2,{\epsilon}_2),\ldots {\rangle\! \rangle}_o \right)
={\langle\! \langle}(b_1,{\epsilon}_1),(b_2,{\epsilon}_2),(b_3,{\epsilon}_3),\ldots {\rangle\! \rangle}_o ,$$ and $d\nu_o (y)=\frac{1}{3\log G} \cdot \frac{dy}{y+1}$ provides a $\tau_o$-invariant probability measure (see [@Rie; @Seb]).
Consider $\Omega_o =(0,1)\times I_G$. The natural extension of $T_o$ can be realized as the invertible map $${\bar{T}}_o :\Omega_o \rightarrow \Omega_o , \qquad {\bar{T}}_o (x,y) =\bigg( T_o (x),\frac{{\epsilon}_1(x)}{a_1(x)+y}\bigg),$$ and it acts on $\Omega_o \cap ({\mathbb R}\setminus{\mathbb Q})^2$ as a two-sided shift as follows: $$\begin{split}
{\bar{T}}_o & \big([\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o ,{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),\ldots {\rangle\! \rangle}_o \big)
\\ & = \big([\![(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_o ,{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o \big).
\end{split}$$
It is convenient to consider the extension ${{\widetilde{T}}}_o$ of ${\bar{T}}_o$ to ${{\widetilde{\Omega}}}_o :=\Omega_o \times \{ -1,1\}$ defined by $$\label{eq4}
{{\widetilde{T}}}_o (x,y,{\epsilon}) := \big( {\bar{T}}_o (x,y), -{\epsilon}_1(x) {\epsilon}\big) ,$$ with inverse $$\label{eq2.4}
\begin{split}
{\widetilde{T}}^{-1}_o & \big([\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o ,
{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),\ldots {\rangle\! \rangle}_o ,{\epsilon}\big)
\\ & = \big([\![(a_0,{\epsilon}_0),(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),\ldots ]\!]_o ,
{\langle\! \langle}(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),(a_{-3},{\epsilon}_{-3}),\ldots {\rangle\! \rangle}_o ,-{\epsilon}_0 {\epsilon}\big).
\end{split}$$
Even continued fractions {#ecf}
------------------------
The ECF expansion in $[0,1]\setminus{\mathbb Q}$ is given by $$x=[\![(a_1,{\epsilon}_1), (a_2, {\epsilon}_2), (a_3, {\epsilon}_3), ...]\!]_e = \cfrac{1}{a_1+\cfrac{{\epsilon}_1}{a_2+\cfrac{{\epsilon}_2}{a_3+\cdots}}},$$ where ${\epsilon}_i ={\epsilon}_i(x)\in \{\pm1\}$ and $a_i =a_i(x)\in 2{\mathbb N}$. Such an expansion is unique. Every number $x\in [1,\infty) \setminus {\mathbb Q}$ has a unique infinite ECF expansion $$x=[\![(a_0,{\epsilon}_0);(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),\ldots ]\!]_e :=a_0+{\epsilon}_0 [\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),\ldots ]\!]_e \in [1,\infty),$$ with ${\epsilon}_0\in \{ \pm 1\}$ and $a_0 \in 2{\mathbb N}$.
The even Gauss map $T_e$ acts on $[0,1]$ by $$T_e (x)= {\epsilon}\bigg( \frac{1}{x}-2k\bigg)
\quad
\mbox{\rm if} \quad
x\in B({\epsilon},k) =\begin{cases} \big( \frac{1}{2k+1},\frac{1}{2k}\big)
& \mbox{\rm if ${\epsilon}=1$, $k\geqslant 1$} \\
\big( \frac{1}{2k},\frac{1}{2k-1}\big) & \mbox{\rm if ${\epsilon}=-1$, $k\geqslant 1.$}
\end{cases}$$ Symbolically, $T_e$ acts on the ECF representation by removing the leading digit $(a_1,{\epsilon}_1)$ of $x$, i.e. $$T_e \big( [\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots]\!]_e \big) =[\![ (a_2,{\epsilon}_2),(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_e .$$ Here the infinite measure $d\mu_e (x)=\big( \frac{1}{1+x}+\frac{1}{1-x}\big) dx$ is $T_e$-invariant (see [@BL; @Sch1; @Sch2]).
The even continued fraction equivalent of the grotesque continued fractions are the extended even continued fractions. Given ${\epsilon}_i \in \{ \pm 1\}$ and even positive integers $b_i$, we denote $$y={\langle\! \langle}(b_0,{\epsilon}_0),(b_1,{\epsilon}_1),\ldots {\rangle\! \rangle}_e :=
{\epsilon}_0 [\![(b_0,{\epsilon}_1),(b_1,{\epsilon}_2),(b_2,{\epsilon}_3),\ldots ]\!]_e = \cfrac{{\epsilon}_0}{b_0+\cfrac{{\epsilon}_1}{b_1+\cdots}}
\in (-1,1).$$ The corresponding shift $\tau_e$ acts on $(-1,1)$ by $$\tau_e (y)=\frac{{\epsilon}_0(y)}{y}-b_0(y),$$ or in the symbolic representation by $$\tau_e \big( {\langle\! \langle}(b_0,{\epsilon}_0),(b_1,{\epsilon}_1),(b_2,{\epsilon}_2),\ldots {\rangle\! \rangle}_e \big)
={\langle\! \langle}(b_1,{\epsilon}_1),(b_2,{\epsilon}_2),(b_3,{\epsilon}_3),\ldots {\rangle\! \rangle}_e .$$
Consider $\Omega_e =(0,1)\times (-1,1)$ and the natural extension of $T_e$, realized as the map [@Sch2] $${\bar{T}}_e :\Omega_e \rightarrow \Omega_e , \qquad {\bar{T}}_e (x,y) =\bigg( T_e (x),\frac{{\epsilon}_1(x)}{a_1(x)+y}\bigg).$$ Equivalently, ${\bar{T}}_e$ is acting on $\Omega_e \cap ({\mathbb R}\setminus{\mathbb Q})^2$ as a two-sided shift: $$\begin{split}
{\bar{T}}_e & \big([\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_e ,{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),\ldots {\rangle\! \rangle}_e \big)
\\ & = \big([\![(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),(a_4,{\epsilon}_4)\ldots ]\!]_e ,{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_e \big).
\end{split}$$
It is also convenient to consider the extension ${{\widetilde{T}}}_e$ of ${\bar{T}}_e$ to ${{\widetilde{\Omega}}}_e =\Omega_e \times \{ -1,1\}$ defined by $${{\widetilde{T}}}_e (x,y,\epsilon) = \big( {\bar{T}}_e (x,y), -{\epsilon}_1(x) {\epsilon}\big) ,$$ with inverse $$\label{eq2.6}
{{\widetilde{T}}}_e^{-1} (x,y,{\epsilon})= \big( {\bar{T}}_e^{-1}(x,y), -{\epsilon}_0(y) {\epsilon}\big) .$$
Cutting sequences for geodesics on ${\mathcal M}_o$ {#oddseq}
===================================================
The group $\Gamma$ and the modular surface ${\mathcal M}_o=\Gamma\backslash {\mathbb{H}}$ {#oddsurface}
-----------------------------------------------------------------------------------------
The fundamental Dirichlet region corresponding to the point $i$ and the order three generators $S$ and ${ST^{-1}}$ is the quadrilateral $${\mathfrak F} =\{ z\in {\mathbb{H}}: \lvert {\operatorname{Re}}z \rvert \leqslant \tfrac{1}{2} , \
\lvert z-1\rvert \geqslant 1,\ \lvert z+1\rvert \geqslant 1\} ,$$ with edges $[\omega,\infty]$, $[0,\omega]$ identified by ${ST^{-1}}=
\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix}\right)$, and with $[0,\omega^2]$, $[\omega^2,\infty]$ identified by $S$. The resulting quotient space ${\mathcal M}_o$ is homeomorphic to the union of two cones glued along their basis $[0,\infty]$, with vertices corresponding to $\omega$ and $\omega^2$, cuts along the geodesic arcs $[\omega,\infty]$ and $[\omega^2,\infty]$, and a cusp at $\pi_o (\infty)$ (see Figure \[Figure1\]).
\[lemma1\] The group $\Gamma$ is an index two subgroup of $\Gamma (1)$ and coincides with $$\Gamma_o = \left\{ M \in \Gamma (1):
M\equiv \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) ,\
\left( \begin{matrix} 0 & 1 \\ 1& 1 \end{matrix}\right) \
\mbox{or}\
\left( \begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix} \right) \pmod{2} \right\} .$$
The generators $S$ and $T$ are contained in the group $\Gamma_o$. Since $\lvert \operatorname{SL}(2,{\mathbb Z}_2)\rvert =6$, we have $[ \Gamma (1):\Gamma_o ]=2$. Finally, we notice that ${{\mathfrak F}={\mathcal F} \cup
\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right) {\mathcal F}}$, where ${\mathcal F}=\big\{ z\in {\mathbb{H}}: \lvert z\rvert \geqslant 1, \lvert
\operatorname{Re} z \rvert \leqslant \frac{1}{2} \}$ is the standard fundamental domain for $\Gamma(1) \backslash {\mathbb{H}}$. This yields $[\Gamma (1):\Gamma ]=2$ and $\Gamma =\Gamma_o$.
The lift of the group $\Gamma$ to $\operatorname{GL}(2,{\mathbb Z})$ played an important role in the analysis of the renewal time for odd continued fractions, pursued by Vandehey and one of the authors in [@BV].
![The Farey tessellation in the disk model and the rotations $S$ and $ST^{-1}$[]{data-label="Figure3"}](disk_tessellation_1.eps){width="9cm"}
Let ${\mathcal{A}}_o$ denote the set of geodesics $\gamma$ in ${\mathbb{H}}$ with endpoints satisfying $$({\gamma_{\infty}},{\gamma_{-\infty}}) \in
{\mathcal{S}}_o :=\left( (1,\infty) \times (-I_G) \right) \cup \left( (-\infty,-1) \times I_G\right).$$
\[Lemma2\] Every geodesic $\bar{\gamma}$ on ${\mathcal M}_o$ lifts on ${\mathbb{H}}$ to a geodesic $\gamma \in {\mathcal{A}}_o$.
Without loss of generality, we can take $\bar{\gamma}$ to be a positively oriented geodesic arc in ${\mathfrak F}$ connecting $[\omega^2,\infty]$ to $[\omega,\infty]$, $[0,\omega^2]$ to $[\omega,\infty]$, $[\omega^2,\infty]$ to $[0,\omega]$, or $[0,\omega^2]$ to $[0,\omega]$. Note that $[0,1]\subset I_G =(G-2,G)$. There are four cases to consider. First, $\gamma_{-\infty} <-1<1<\gamma_\infty$, and an appropriate $2{\mathbb Z}$-translation gives $\gamma_{-\infty} \in -I_G$ and $\gamma_\infty >1$. Second, $\gamma$ connects the arc $[-1,0]$ to $[\omega,\infty]$, hence $-1<\gamma_{-\infty} <0<1<\gamma_\infty$. Third, we have $\gamma_{-\infty} <-1<0<\gamma_\infty<1$ so $\gamma\in{\mathcal{A}}_o$.
Finally, in the fourth case $TS^{-1}$ maps $\bar{\gamma}$ to a geodesic arc connecting $[\omega+1,\infty]$ to $[\omega,\infty]$, hence $\gamma_{-\infty} >2>0>\gamma_\infty$. An appropriate $2{\mathbb Z}$-translation ensures that $\gamma_{-\infty} \in I_G$ and $\gamma_\infty <-1$.
Cutting sequences and odd/grotesque continued fraction expansions {#oddcut}
-----------------------------------------------------------------
As described in the introduction, our coding of geodesics on ${\mathcal M}_o$ refines the Series coding. An oriented geodesic $\gamma$ in ${\mathbb{H}}$ is cut into segments as it crosses triangles in the Farey tessellation ${\mathbb{F}}$. Each segment of the geodesic crosses two sides of a triangle in the tessellation. If the vertex where the two sides meet is on the left, we label the segment $L$, if it is on the right we label it $R$. We use ${\mathbb{L}}$ and ${\mathbb{R}}$ when the geodesic is in a light cell and ${\mathbf{L}}$ or ${\mathbf{R}}$ for a dark cell. This way, we assign to every geodesic in ${\mathbb{H}}$ with irrational endpoints an infinite two-sided sequence of symbols ${\mathbb{L}}$, ${\mathbf{L}}$, ${\mathbb{R}}$ and ${\mathbf{R}}$, with alternating shades.
Next, we analyze in detail the connection between the GCF expansion of $\gamma_{-\infty}$, the OCF expansion of $\gamma_\infty$, and the strings in . For every Möbius transformation $\bar{\rho}$ leaving ${\mathcal{S}}_o$ invariant, we still denote by $\bar{\rho}$ the product map $\bar{\rho}\times \bar{\rho}$, viewed as a transformation of ${\mathcal{S}}_o$. To every geodesic $\gamma \in {\mathcal{A}}_o$ we associate the positively oriented geodesic arc $[\xi_\gamma,\eta_\gamma]$, where $$\xi_\gamma:=\begin{cases}
\gamma \cap [1,\infty] & \mbox{\rm if $\gamma_\infty >1$} \\
\gamma \cap [-1,\infty] & \mbox{\rm if $\gamma_\infty <-1$}
\end{cases}
\quad \mbox{\rm and} \quad
\eta_\gamma:=\begin{cases}
\gamma \cap [a_1,a_1+{\epsilon}_1] & \mbox{\rm if $\gamma_\infty >1$} \\
\gamma \cap [-a_1,-a_1-{\epsilon}_1] & \mbox{\rm if $\gamma_\infty <-1,$}
\end{cases}$$ with $(a_1,{\epsilon}_1)=(a_1(\gamma_\infty),{\epsilon}_1(\gamma_\infty))$.
When ${\gamma_{\infty}}>1$, we write ${\gamma_{\infty}}=[\![(a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\!]_o$, ${\gamma_{-\infty}}=-{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}), \ldots {\rangle\! \rangle}_o \in -I_G$. When ${\gamma_{\infty}}<-1$, ${\gamma_{\infty}}=-[\![(a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\!]_o$, ${\gamma_{-\infty}}={\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}), \ldots {\rangle\! \rangle}_o \in I_G$. Four cases will occur:
![The strings $\xi_\gamma ({\mathbb{L}}{\mathbf{L}})^{k-1} {\mathbb{L}}{\mathbf{R}}\eta_\gamma$ and $\xi_\gamma ({\mathbb{L}}{\mathbf{L}})^{k-1} {\mathbb{R}}\eta_\gamma$ []{data-label="Figure4"}](I_1_odd.eps "fig:"){width="8cm"} ![The strings $\xi_\gamma ({\mathbb{L}}{\mathbf{L}})^{k-1} {\mathbb{L}}{\mathbf{R}}\eta_\gamma$ and $\xi_\gamma ({\mathbb{L}}{\mathbf{L}})^{k-1} {\mathbb{R}}\eta_\gamma$ []{data-label="Figure4"}](I_2_odd.eps "fig:"){width="7.5cm"}
\(A) $\gamma_\infty \in(2k,2k+1)$, ${\epsilon}_1=-1$, $a_1=2k+1$. The M" obius transformation $
\rho_o (x)=\frac{1}{a_1-x}
$ belongs to $\Gamma$ and $$\label{eq3.1}
\begin{split}
\rho_o \big( [\![(a_1,{\epsilon}_1); & (a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o , -{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}), \ldots {\rangle\! \rangle}_o \big) \\ & =
\big( [\![(a_2,{\epsilon}_2);(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),,\ldots ]\!]_o ,
-{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o \big) .
\end{split}$$ In this situation $\rho_o$ transforms the arc $[\xi_\gamma,\eta_\gamma]$ of $\gamma$ connecting the geodesics $[1,\infty]$ and $[a_1-1,a_1]$ into an arc connecting $[0,\frac{1}{a_1-1}]$ with $[1,\infty]$. Following the orientation of $\gamma$, we assign the string $\xi_\gamma ({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{L}}{\mathbf{R}}\eta_\gamma$ to the arc $[\xi_\gamma,\eta_\gamma]$ (see Figure \[Figure4\]).
\(B) $\gamma_\infty \in(2k-1,2k),
{\epsilon}_1=+1$, $a_1=2k-1$. The M" obius transformation $
\rho_o (x)=\frac{1}{a_1-x}
$ belongs to $\Gamma$ and $$\label{eq3.2}
\begin{split}
\rho_o \big( [\![(a_1,{\epsilon}_1); & (a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o , -{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}), \ldots {\rangle\! \rangle}_o \big) \\ & =
\big( -[\![(a_2,{\epsilon}_2);(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_o ,
{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o \big) .
\end{split}$$ In this situation $\rho_o$ transforms the arc $[\xi_\gamma,\eta_\gamma]$ of $\gamma$ connecting the geodesics $[1,\infty]$ and $[a_1,a_1+1]$ into an arc connecting $[0,\frac{1}{a_1-1}]$ with $[-1,\infty]$. Following the orientation of $\gamma$, to the arc $[\xi_\gamma,\eta_\gamma]$ we assign the string $\xi_\gamma ({\mathbb{L}}{\mathbf{L}})^{k-1} {\mathbb{R}}\eta_\gamma$ (see Figure \[Figure4\]).
\(C) $\gamma_\infty \in(-2k-1,-2k), {\epsilon}_1=-1$, $a_1=2k+1$. The M" obius transformation $
\rho_o (x)=\frac{1}{-a_1-x}
$ belongs to $\Gamma$ and $$\label{eq3.3}
\begin{split}
\rho_o \big( -[\![(a_1,{\epsilon}_1); & (a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o , {\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}), \ldots {\rangle\! \rangle}_o \big) \\ & =
\big( -[\![(a_2,{\epsilon}_2);(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_o ,
{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o \big) .
\end{split}$$ In this situation $\rho_o$ transforms the arc $[\xi_\gamma,\eta_\gamma]$ of the geodesic $\gamma$ connecting the geodesics $[-1,\infty]$ and $[-a_1,-a_1+1]$ into an arc connecting $[\frac{-1}{a_1-1},0]$ with $[-1,\infty]$. Following the orientation of $\gamma$, to the arc $[\xi_\gamma,\eta_\gamma]$ we assign the string $\xi_\gamma ({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{R}}{\mathbb{L}}\eta_\gamma$ (see Figure \[Figure6\]).
![The strings $\xi_\gamma ({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{R}}{\mathbb{L}}\eta_\gamma$ and $\xi_\gamma ({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{L}}\eta_\gamma$ []{data-label="Figure6"}](II_1_odd.eps "fig:"){width="8cm"} ![The strings $\xi_\gamma ({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{R}}{\mathbb{L}}\eta_\gamma$ and $\xi_\gamma ({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{L}}\eta_\gamma$ []{data-label="Figure6"}](II_2_odd.eps "fig:"){width="7.7cm"}
\(D) $\gamma_\infty \in (-2k,-2k+1), {\epsilon}_1=+1$, $a_1=2k-1$. The M" obius transformation $
\rho_o (x)=\frac{1}{-a_1-x}
$ belongs to $\Gamma$ and $$\label{eq3.4}
\begin{split}
\rho_o \big( -[\![(a_1,{\epsilon}_1); & (a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o , {\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}), \ldots {\rangle\! \rangle}_o \big) \\ & =
\big( [\![(a_2,{\epsilon}_2);(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_o ,
-{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o \big) .
\end{split}$$ In this situation $\rho_o$ transforms the arc $[\xi_\gamma,\eta_\gamma]$ of the geodesic $\gamma$ connecting the geodesics $[-1,\infty]$ and $[-a_1-1,-a_1]$ into an arc connecting $[\frac{-1}{a_1-1},0]$ with $[1,\infty]$. Following the orientation of $\gamma$, to the arc $[\xi_\gamma,\eta_\gamma]$ we assign the string $\xi_\gamma ({\mathbf{R}}{\mathbb{R}})^{k-1} {\mathbf{L}}\eta_\gamma$ (see Figure \[Figure6\]).
Summarizing -, we obtain the following general formula for the action of $\rho_o$ on ${\mathcal{S}}_o$, where $\epsilon =+1$ in cases A and B, and $\epsilon =-1$ in cases C and D: $$\label{eq3.5}
\begin{split}
\rho_o & \big( {\epsilon}[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o, -{\epsilon}{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),\ldots {\rangle\! \rangle}_o) \\
& = (-{\epsilon}_1) \big( {\epsilon}[\![ (a_2,{\epsilon}_2);(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_o,
-{\epsilon}{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o \big),
\end{split}$$ with inverse $$\begin{split}
\rho_o^{-1} & \big( {\epsilon}[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o, -{\epsilon}{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),\ldots {\rangle\! \rangle}_o) \\
& = (-{\epsilon}_0) \big( {\epsilon}[\![ (a_0,{\epsilon}_0);(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),\ldots ]\!]_o,
-{\epsilon}{\langle\! \langle}(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),(a_{-3},{\epsilon}_{-3}),\ldots {\rangle\! \rangle}_o \big).
\end{split}$$ This also gives that $\rho_o$ reflects across the imaginary axis exactly when ${\epsilon}_1=+1$, so that $\rho_o$ agrees with the transformation $\rho$ from [@Ser] exactly where the RCF and OCF agree. We also get that cases A and C are followed by A or B, and cases B and D are followed by cases C or D.
\[Lcommdiagodd\] The map $\rho_o :{\mathcal{S}}_o \rightarrow {\mathcal{S}}_o$ is invertible, and the diagram $$\xymatrix{ \mbox{${\mathcal{S}}_o$} \ar@{->} [r]^(.5){\rho_o}
\ar@{->} [d]_(.4) {J_o} & \mbox{${\mathcal{S}}_o$} \ar@{->} [d]^(.4) {J_o} \\
\mbox{${{\widetilde{\Omega}}}_o$} \ar@{->} [r] ^(.5){{{\widetilde{T}}}_o} & {{\widetilde{\Omega}}}_o }$$ commutes, where $J_o:{\mathcal{S}}_o \rightarrow {{\widetilde{\Omega}}}_o$ is the invertible map defined by $$J_o (x,y):=\operatorname{sign} x (1/x,-y,1) = \begin{cases}
(1/x,-y,1) & \mbox{\rm if $x> 1$, $y\in -I_G$} \\
(-1/x,y,-1) & \mbox{\rm if $x< -1$, $y\in I_G.$}
\end{cases}$$
Set $x= {\epsilon}[\![ (a_1,{\epsilon}_1); (a_2,{\epsilon}_2),\ldots ]\!]_o$, $y=-{\epsilon}{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o$ with ${\epsilon}=\pm 1$, so that $(x,y)\in {\mathcal{S}}_o$. Using formulas and , we get that $$\begin{split}
J_o \rho_o (x,y) & = J_o \big( -{\epsilon}_1 {\epsilon}[\![(a_2,{\epsilon}_2); (a_3,{\epsilon}_3),\ldots ]\!]_o,
{\epsilon}_1 {\epsilon}{\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),\ldots {\rangle\! \rangle}_o \big) \\
& = \big( [\![(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o, {\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),\ldots {\rangle\! \rangle}_o ,-{\epsilon}_0 {\epsilon}\big) \\
& = {{\widetilde{T}}}_o \big( [\![(a_1,{\epsilon}_1),(a_2,{\epsilon}_2),\ldots ]\!]_o,
{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o ,{\epsilon}\big)
= {{\widetilde{T}}}_o J_o (x,y). \qedhere
\end{split}$$
\[cor4\] If $\alpha =[\![\overline{(a_1,{\epsilon}_1);(a_2,{\epsilon}_2), \ldots,(a_r,{\epsilon}_r)}]\!]_o>1$ and $\beta=-{\langle\! \langle}\overline{((a_r,{\epsilon}_r),\ldots,(a_1,{\epsilon}_1)} {\rangle\! \rangle}_o\in -I_G$, then
- $\ \rho_o^r (\alpha,\beta)=(-{\epsilon}_1)\cdots (-{\epsilon}_r) (\alpha,\beta)$.
- $\ \rho_o^{2r} (\alpha,\beta) =(\alpha,\beta)$.
To check (i), we use Proposition \[Lcommdiagodd\] and compute $$\begin{split}
J_0^{-1} {{\widetilde{T}}}_o^{r} J_0 (\alpha,\beta) & =J_0^{-1} {{\widetilde{T}}}_o^{r} (1/\alpha,-\beta,1) =
J_0^{-1} \big( 1/\alpha,-\beta,(-{\epsilon}_1)\cdots (-{\epsilon}_r)\big) =(-{\epsilon}_1)\cdots (-{\epsilon}_r) (\alpha,\beta) .
\end{split}$$
\(ii) We consider the case $(-{\epsilon}_1)\cdots (-{\epsilon}_r)=-1$, when we use Proposition \[Lcommdiagodd\] and $$\begin{split}
J_o^{-1} {{\widetilde{T}}}_o^{r} J_o (-\alpha,-\beta) & =J_o^{-1} {{\widetilde{T}}}_o^{r} (1/\alpha,-\beta,-1)
=J_o^{-1} \big( 1/\alpha,-\beta,-(-{\epsilon}_1)\cdots (-{\epsilon}_r) \big) \\
& =J_o^{-1} (1/\alpha,-\beta,1) =(\alpha,\beta). \qedhere
\end{split}$$
Finally, we notice the equality $$\label{3.6}
{\bar{T}}_o^{-1} (u,v)=\bigg( -\frac{\operatorname{sign}(v)}{\rho_o (1/u)},\operatorname{sign} (v) \rho_o (-v)\bigg),
\quad \forall (u,v)\in\Omega_o \cap ({\mathbb R}\setminus {\mathbb Q})^2 .$$
Connection with cutting sequence and RCF {#cut}
========================================
Odd continued fractions {#cutodd}
-----------------------
We now explore the connection between the cutting sequences of the regular continued fractions and the odd continued fractions. Here, we use $x$ to mark the imaginary axis, as in [@Ser], and $\xi_{\gamma}=\gamma\cap \pm [1,\infty]$ as defined above.
In cases $A$ and $B$, we get the cutting sequence $\ldots xL^{n_1}R^{n_2}L^{n_3}\ldots$ with regular continued fraction expansion $[n_1;n_2,n_3,\ldots]$. Without coloring, this corresponds to $\ldots L {\xi_\gamma}L^{n_1-1}R^{n_2}L^{n_3}\eta_\gamma\ldots$. We have two cases to consider for the first digit of the odd continued fraction expansion.
(A)
: $n_1=2k$ is even, and ${\gamma_{\infty}}\in(2k,2k+1)$. This gives the cutting sequence $\ldots {\xi_\gamma}({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{L}}{\mathbf{R}}\eta_\gamma$, and we get $(2k+1,-1)=(a_1,{\epsilon}_1)$. The next digit is represented by $L^n R$ for $n\geqslant 0$. When $n_2>1$, the next digit is $L^0{\mathbb R}$, corresponding to $(1,+1)$. This gives the cutting sequence $\ldots {\xi_\gamma}({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{L}}{\mathbf{R}}\eta_\gamma{\mathbb{R}}\ldots$. This corresponds to $$2k+\cfrac{1}{n_2+z}=2k+1-\cfrac{1}{1+\cfrac{1}{n_2-1+z}}.$$ When $n_2=1$, we proceed with $\ldots {\xi_\gamma}({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{L}}{\mathbf{R}}\eta_\gamma {\mathbb{L}}\ldots$. This corresponds to $$2k+\cfrac{1}{1+\frac{1}{n_3+z}}=2k+1+\frac{-1}{n_3+1+z}.$$ These equalities correspond to the singularization and insertion algorithm introduced by Kraaikamp in [@Kra] and explicitly computed for the OCF in Masarotto’s master’s thesis [@Mas] (see also [@HK]). This algorithm is based on the identity $$a+\cfrac{{\epsilon}}{1+\frac{1}{b+z}}=a+{\epsilon}+\frac{-{\epsilon}}{b+1+z}
\quad \mbox{\rm where $\epsilon \in \{ \pm 1\}$.}$$
(B)
: $n_1=2k-1$ is odd, and ${\gamma_{\infty}}\in(2k-1,2k)$. This gives the cutting sequence $\ldots {\xi_\gamma}({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{R}}\eta_\gamma$, and we get $(2k-1,+1)=(a_1,{\epsilon}_1)$. The next digit is represented by $R^nL$, where $n\geqslant 0$. If $n_2=1$, we have $ \ldots {\xi_\gamma}({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb R}\eta_\gamma {\mathbf{L}}\ldots$, and the next digit corresponds to $R^0{\mathbf{L}}$, which gives $(1,+1)$.
Strings starting with ${\mathbf{R}}$ are treating similarly, with $({\mathbf{R}}{\mathbb{R}})^{k-1}{\mathbf{L}}$ corresponding to $(2k-1, +1)$ and $({\mathbf{R}}{\mathbb{R}})^{k-1}{\mathbf{R}}{\mathbb{L}}$ to $(2k+1, -1)$.
Geodesics of Type C and D can be classified similarly. In this case, we get the cutting sequence $\ldots xR^{n_1}L^{n_2}R^{n_3}\ldots$, where ${\gamma_{\infty}}=-[n_1;n_2,n_3,\ldots]$. This gives, without coloring, the cutting sequence $\ldots R {\xi_\gamma}R^{n_1-1}L^{n_2}R^{n_3}\ldots$, and we interpret the strings in the same way as above.
Grotesque continued fractions {#grot}
-----------------------------
The grotesque continued fractions are the dual continued fraction expansion of the odd continued fractions, which changes the restriction on the digits. That is, for ${\epsilon}_1 /(a_1+ {\epsilon}_2 /\ldots)$, the odd continued fractions require $a_i+{\epsilon}_{i+1}\geqslant 2$, and the grotesque require $a_i+{\epsilon}_{i}\geqslant 2$. This means we need a different insertion and singularization algorithm to convert regular continued fractions to grotesque continued fractions that the one used for odd continued fractions. As with Series’ description of the regular continued fractions [@Ser], the forward endpoints are read from left to right, but the backwards endpoints are read from right to left. Thus, for the odd continued fractions, we can read the strings one at a time, but for grotesque continued fractions, we must also consider whether the preceding string would be valid.
We consider cases A, B, C, and D similar to those above. To stay consistent with how we normally read, we say that the string ends in the letter on the right. The preceding string is the string to the left of the one we are considering.
(A)
: ${\gamma_{\infty}}>1, {\gamma_{-\infty}}\in(0,2-G)$, and ${\epsilon}_0=-1$. We get the cutting sequence $\ldots ({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{L}}{\mathbf{R}}{\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k+1,-1)$ as in Figure \[Figure4\]. The preceding string must end in ${\mathbf{R}}$ or ${\mathbf{L}}$, as in case A or B.
(B)
: ${\gamma_{\infty}}>1, {\gamma_{-\infty}}\in(-G,0),$ and ${\epsilon}_0=+1$. We get the cutting sequence $\ldots ({\mathbf{R}}{\mathbb{R}})^{k-1}{\mathbf{L}}{\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k-1,+1)$, as in Figure \[Figure4\]. Note that we cannot have $a_0=1$. The preceding string must end in ${\mathbb R}$ or ${\mathbb{L}}$, as in case C or D.
(C)
: ${\gamma_{\infty}}<-1,{\gamma_{-\infty}}\in(G-2,0),$ and ${\epsilon}_0=-1$. We get the cutting sequence $\ldots ({\mathbf{R}}{\mathbb{R}})^{k-1}{\mathbf{R}}{\mathbb{L}}{\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k+1,-1)$ as in the image on the left of Figure \[Figure6\]. The preceding string must end in ${\mathbb R}$ or ${\mathbb{L}}$, as in case C or D.
(D)
: ${\gamma_{\infty}}<-1, {\gamma_{-\infty}}\in(0, G),$ and ${\epsilon}_0=+1$. We get the cutting sequence $\ldots ({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{R}}{\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k-1,+1)$ as in the image on the right of Figure \[Figure6\]. The preceding string must end in ${\mathbf{R}}$ or ${\mathbf{L}}$, as in case A or B.
To illustrate the difference between odd and grotesque continued fraction expansions, we consider two example strings. First, note that for case C, we have the RCF cutting sequence $\ldots L^{n_{-2}} R^{n_{-1}}L^{n_0}x R\ldots$ for the RCF $[n_0,n_{-1},n_{-2},\ldots]$, where $x=\gamma\cap [0,\infty]$. If we ignore coloring, this corresponds to the string $\ldots L^{n_{-2}} R^{n_{-1}}L^{n_0-1}R {\xi_\gamma}\ldots$. We compare the GCF expansions for $[2,1,2,\ldots]$ and $[2,2,2,\ldots]$, as well as the OCF expansion of $[2;2,2,\ldots]$.
In the first case, we have the string $\ldots{\mathbb{R}}{\mathbf{L}}{\mathbb{L}}{\mathbf{R}}{\mathbb{L}}{\mathbf{L}}{\mathbb{R}}{\xi_\gamma}\ldots$. Following the above rules, we get the grouping $\ldots{\mathbb{R}}{\mathbf{L}}({\mathbb{L}}{\mathbf{R}})({\mathbb{L}}{\mathbf{L}}{\mathbb{R}}){\xi_\gamma}\ldots$, and $[2,1,2,\ldots]={\langle\! \langle}(3,+1),(3,-1),\ldots{\rangle\! \rangle}_o$.
In the second case, we have $\ldots{\mathbf{R}}{\mathbb{L}}{\mathbf{L}}{\mathbb{R}}{\mathbf{R}}{\mathbb{L}}{\mathbf{L}}{\mathbb{R}}{\xi_\gamma}\ldots$, and $[2,2,2,\ldots]={\langle\! \langle}(1,+1),(1,+1),(3,-1),\ldots{\rangle\! \rangle}_o$. We need to change the grouping for the first digit to make the second digit an allowable string, since ${\mathbf{R}}$ must be preceded by $({\mathbb{L}}{\mathbf{L}})^{k-1}{\mathbb{L}}$. Thus, we get $\dots{\mathbf{R}}{\mathbb{L}}{\mathbf{L}}{\mathbb{R}}({\mathbf{R}}{\mathbb{L}})({\mathbf{L}})({\mathbb{R}}){\xi_\gamma}\dots$, where the final grouping is determined based on whether the previous letter is ${\mathbb{L}}$ or ${\mathbb{R}}$.
Finally, we contrast this with the regular continued fraction $\frac{1}{{\gamma_{-\infty}}}=[2;2,2,\ldots]$. The corresponding cutting sequence $\ldots{\xi_\gamma}({\mathbb{L}}{\mathbf{R}})({\mathbb{R}})( {\mathbf{L}}){\mathbb{L}}{\mathbf{R}}\ldots$ gives the OCF. We can see the connection between the RCF, GCF, and OCF by noting that $$2+\cfrac{1}{2+z}=1+\cfrac{1}{1-\cfrac{1}{3+z}}=3-\cfrac{1}{1+\cfrac{1}{1+z}}.$$
Applications
============
Invariant measures {#invmeas}
------------------
Section \[oddseq\] provides a cross-section ${\mathcal X}$ for the geodesic flow on $T_1 ({\mathcal M}_o)$ consisting of those elements $(\xi_\gamma,u_\gamma)$ with base point $\xi_\gamma$ on $\pi_o (\pm 1 +i{\mathbb R})$ such that the cutting sequence of $\pi_o(\gamma)$ is broken into segments as in , and the unit vector $u_\gamma$ points along the geodesic. Proposition \[Lcommdiagodd\] shows that the first return map of the geodesic flow to ${\mathcal X}$ corresponds to the transformation ${{\widetilde{T}}}_o$ on ${\widetilde{\Omega}}_o$.
As in [@Ser], it is convenient to express $(x,u)\in T_1({\mathbb{H}})$ in coordinates $(\alpha,\beta,t)$ given by the endpoints $\alpha,\beta$ of the geodesic $\gamma(u)$ through $u$ and the distance $t$ from the midpoint of $\gamma(u)$ to $u$. It is possible to transform the usual measure Haar measure $y^{-2} dx dy d\theta$ on ${\mathbb{H}}\times S^1 \cong T_1 ({\mathbb{H}}) \cong \operatorname{PSL}(2,{\mathbb R})$ to $(\alpha-\beta)^{-2} d\alpha d\beta dt$. Section 3 shows that ${\mathcal{S}}_o$ and ${\mathcal X}$ are naturally identified, up to a null-set, by mapping $\gamma\in{\mathcal{S}}_o$ with irrational endpoints $(\gamma_\infty,\gamma_{-\infty})\in {\mathcal{S}}_o$ to $(\xi_\gamma,u_\gamma)\in {\mathcal X}_o$ as above. The first return map to ${\mathcal X}$ is subsequently identified with $\rho_o$, acting on ${\mathcal{S}}_o$ with corresponding invariant measure $\mu=(\alpha-\beta)^{-2} d\alpha d\beta$. We define $\pi_o,\pi_1,\pi_2$ on ${\widetilde{\Omega}}_o$ by $\pi_o (x,y,\epsilon)=(x,y), \pi_1 (x,y,\epsilon)=x$, and $\pi_2 (x,y,\epsilon)=y$. The push-forward of $\mu$ under $\pi_o\circ J_o$ provides a $\bar{T}_o$-invariant measure $\bar{\mu}_o$ on $\Omega_o$. The push-forward of $\mu$ under $\pi_1 \circ J_o$ provides a $T_o$-invariant measure $\mu_o$ on $(0,1)$. Finally, the push-forward of $\mu$ under $\pi_2 \circ J_o$ provides a $\tau_o$-invariant measure $\nu_o$ on $I_G$. For every rectangle $E=[a,b]\times [c,d] \subset \Omega_o$, we have $$(\pi_o \circ J_o)^{-1}(E) =[b^{-1},a^{-1}] \times [-d,-c] \cup
[-a^{-1},-b^{-1}] \times [c,d] \qquad \mbox{\rm and}$$ $$\bar{\mu}_o (E) =
\int_{1/b}^{1/a} \int_{-d}^{-c} \frac{d\alpha\, d\beta}{(\alpha-\beta)^2} +
\int_{-1/a}^{-1/b} \int_c^d \frac{d\alpha\, d\beta}{(\alpha-\beta)^2}
= 2\iint_E \frac{dx\, dy}{(1+xy)^2} ,$$ showing that $\bar{\mu}_o =(1+xy)^{-2}dx dy$ is a finite $\bar{T}_o$-invariant measure. Using $\frac{1}{G}=G-1$ and $\frac{1}{2-G}=G+1$, we see that for every interval $[a,b]\subset (0,1)$, $$\begin{split}
\mu_o ([a,b]) & = \bar{\mu}_o \big( [1/b,1/a] \times (-I_G) \cup [-1/a,-1/b] \times I_G \big) \\
& = \int_{1/b}^{1/a} \int_{-G}^{2-G} \frac{d\alpha\, d\beta}{(\alpha-\beta)^2} +
\int_{-1/a}^{-1/b} \int_{G-2}^G \frac{d\alpha\, d\beta}{(\alpha-\beta)^2} \\
& = \int_a^b \int_{-G}^{2-G} \frac{1}{u^2}\cdot \frac{du\, dv}{(1/u-v)^2} +
\int_a^b \int_{G-2}^G \frac{1}{u^2} \cdot \frac{du\, dv}{(-1/u-v)^2} \\
& =2 \int_a^b \left( \frac{1}{u+G-1}-\frac{1}{u-G-1}\right) du.
\end{split}$$ This gives that $\mu_o =( \frac{1}{u+G-1}-\frac{1}{u-G-1}) du$ is a finite $T_o$-invariant measure. Finally, for every interval $[c,d] \subset I_G$ we have $$\begin{split}
\nu_o ([c,d]) & = \bar{\mu}_o \big( [1,\infty) \times [-d,-c] \cup (-\infty,-1] \times [c,d]\big)
\\ & = \int_1^\infty \int_{-d}^{-c} \frac{d\alpha\, d\beta}{(\alpha-\beta)^2} +
\int_{-\infty}^{-1} \int_c^d \frac{d\alpha\, d\beta}{(\alpha-\beta)^2} \\ &
= 2 \int_0^1 \int_c^d \frac{1}{u^2}\cdot \frac{du\, dv}{(1/u+v)^2} =
2\int_c^d \frac{dv}{1+v} ,
\end{split}$$ showing that $\nu_o=\frac{dv}{1+v}$ is a finite $\tau_o$-invariant measure.
Quadratic surds and their conjugates
------------------------------------
\[cor5\] A real number $\alpha >1$ has a purely periodic OCF expansion if and only if $\alpha$ is a quadratic surd with $-G < {\bar{\alpha}}<2-G$. Furthermore, if $$\label{eq5.1}
\alpha =[\![ \, \overline{(a_1,{\epsilon}_1); (a_2,{\epsilon}_2), \ldots ,(a_{r},{\epsilon}_{r})} \,]\!]_o ,$$ then $$\label{eq5.2}
{\bar{\alpha}}=-{\langle\! \langle}\,\overline{(a_{r},{\epsilon}_{r}), \ldots, (a_1,{\epsilon}_1)} \,{\rangle\! \rangle}_o .$$
In one direction, suppose that $\alpha$ is given by . Consider the geodesic $\gamma \in {\mathcal{A}}_o$ with endpoints at $\gamma_\infty=\alpha$ and $\gamma_{-\infty}=\beta =- {\langle\! \langle}\,\overline{(a_{r},{\epsilon}_{r}), \ldots, (a_1,{\epsilon}_1)} \,{\rangle\! \rangle}_o
\in -I_G$. Corollary \[cor4\] shows that the geodesic $\gamma$ is fixed by $\rho_o^{2r}$, so it is fixed by some $M\in\Gamma$, $M\neq I$. Hence both $\alpha$ and $\beta$ are fixed by $M$; in particular, $\beta={\bar{\alpha}}$.
In the opposite direction, suppose that $A\alpha^2+B\alpha+C=0$ with $\operatorname{gcd} (A,B,C)=1$, $A\geqslant 1$, and ${\bar{\alpha}}\in -I_G$. The quadratic surds $\alpha$, ${\bar{\alpha}}$, $-\alpha$, $\overline{-\alpha}=-{\bar{\alpha}}$, and $M\alpha=\frac{a\alpha+b}{c\alpha+d}$ with $M\in \Gamma \subset \Gamma(1)$ have the same discriminant. From $\alpha-{\bar{\alpha}}= \frac{\sqrt{\Delta}}{A} > G-1$ and $-2AG < 2A{\bar{\alpha}}=
-B-\sqrt{\Delta} < 2A(2-G)$, we infer that the number of quadratic surds $\alpha$ with fixed discriminant $\Delta=B^2-4AC$ must be finite. Employing equality , it follows that both components of ${\bar{T}}_o^k (\frac{1}{\alpha},-{\bar{\alpha}})$ are quadratic surds with discriminant $\Delta$ for every $k\geqslant 0$. Since they satisfy the same kind of restrictions as $\alpha$ above, there exist $k,k^\prime \geqslant 0$, $k\neq k^\prime$ such that ${\bar{T}}^k_o (\frac{1}{\alpha},-{\bar{\alpha}})={\bar{T}}^{k^\prime}_o (\frac{1}{\alpha},-{\bar{\alpha}})$. The map ${\bar{T}}_o$ is invertible, hence there exists $r\geqslant 1$ such that ${\bar{T}}^r_o (\frac{1}{\alpha},-{\bar{\alpha}})=(\frac{1}{\alpha},-{\bar{\alpha}})$, showing that $\alpha$ must be of the form and ${\bar{\alpha}}$ of the form .
Action of $\Gamma$ on ${\mathbb R}$ and continued fractions
-----------------------------------------------------------
Define the $m$-tail of an irrational number $\alpha =[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\!]_o >1$ by $$t_m (\alpha):= (-{\epsilon}_1)\cdots (-{\epsilon}_m) [\![ (a_{m+1},{\epsilon}_{m+1}), (a_{m+2},{\epsilon}_{m+2}), \ldots ]\!]_o .$$
\[cor6\] Two irrationals $\alpha,\beta >1$ are $\Gamma$-equivalent if and only if there exist $r,s\geqslant 0$ such that $$\label{eq5.3}
t_r(\alpha)=t_s (\beta) .$$
The proof follows closely the outline of statement 3.3.3 in [@Ser]. In one direction, if holds, then $\alpha$ and $\beta$ are $\Gamma$-equivalent because $-a_1-\frac{1}{\alpha} =t_1(\alpha)$.
Conversely, suppose that $g\alpha=\beta$ for some $g\in\Gamma$. Fix $\delta \in -I_G$ and consider the geodesics $\gamma,\gamma^\prime \in {\mathcal A}_o$ with $\gamma_{-\infty} =\gamma_{-\infty}^\prime =\delta$, $\gamma_\infty=\alpha$ and $\gamma^\prime_\infty=\beta$. Their cutting sequences are $\ldots \xi_\gamma A_1 A_2 \ldots$ and $\ldots\xi_\gamma B_1 B_2 \ldots$, respectively, with $A_i,B_i$ strings of type $A$, $B$, $C$ or $D$. The geodesics $\gamma^{\prime\prime} =g\gamma$ and $\gamma^\prime$ have the same endpoint $\beta$. Since their $\Gamma(1)$-cutting sequences in $L$ and $R$ coincide (cf. [@Ser Lemma 3.3.1]), their cutting sequences also coincide, implying that the cutting sequence of $\gamma^{\prime\prime}$ is of the form $\xi_{\gamma^{\prime\prime}} \ldots B_k B_{k+1}\ldots$ for some $k\geqslant 1$. As $\gamma$ and $\gamma^{\prime\prime}$ are $\Gamma$-equivalent geodesics, their cutting sequences (after equivalent initial points) will coincide, implying that the cutting sequences of $\gamma$ and $\gamma^\prime$ are of the form $\ldots \xi_\gamma A_1 \ldots A_r D_1 D_2 \ldots$ and $\ldots \xi_\gamma B_1 \ldots B_s D_1 D_2 \ldots$ respectively. Upon this implies .
Two rational numbers are $\Gamma$-equivalent, as shown by the following elementary
\[lemma7\] $\ \Gamma \infty ={\mathbb Q}$.
The problem of characterizing $\Gamma$-equivalence classes for a broad class of subgroups of $\Gamma(1)$ has been recently investigated, with a different approach, in [@Pan].
\[cor8\] The OCF expansion of an irrational $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic surd.
If the OCF tail of $\alpha$ is eventually periodic, then $g\alpha ={\epsilon}[\![ \overline{(a_1,{\epsilon}_1),\ldots ,(a_r,{\epsilon}_r)} ]\!]_o$ for some $g\in\Gamma$ and ${\epsilon}=\pm 1$. Proposition \[cor5\] gives that $g\alpha$ is a quadratic surd, hence $\alpha$ is a quadratic surd.
Conversely, assume that $\alpha$ is a quadratic surd and $\gamma$ is the geodesic connecting $\alpha$ to its conjugate root $\overline{\alpha}$. Let $g\in\Gamma$ such that the geodesic $g\gamma$ is a lift of $\pi_o (\gamma)$ to ${\mathbb{H}}$ with $g\gamma\in {\mathcal A}_o$. We can assume that $g\alpha >1$, reversing $\alpha$ and $\overline{\alpha}$ if necessary. Proposition \[cor5\] shows that $g\alpha$ has purely periodic OCF expansion, and Proposition \[cor6\] shows that the OCF expansion of $\alpha$ is eventually periodic.
Proposition \[cor8\] is known in more general situations, for instance it holds for all $D$-continued fractions (see, e.g., [@Mas]).
Closed geodesics on ${\mathcal M}_o$
------------------------------------
Employing Proposition \[cor6\] and standard arguments, one can prove
\[cor9\] A geodesic $\bar{\gamma}$ on ${\mathcal M}_0$ is closed if and only if it has a lift $\gamma\in {\mathcal{A}}_o$ with purely periodic endpoints $$\gamma_\infty ={\epsilon}[\! [ \overline{(a_1\,e_1);(a_2,{\epsilon}_2),\ldots, (a_r,{\epsilon}_r)} ]\!]_o \quad \mbox{and}
\quad \gamma_{-\infty} =-{\epsilon}{\langle\! \langle}\overline{(a_r,{\epsilon}_r),\ldots ,(a_2,{\epsilon}_2),(a_1,{\epsilon}_1)} {\rangle\! \rangle}_o$$ for some ${\epsilon}=\pm 1$ and $(-{\epsilon}_1)\cdots (-{\epsilon}_r)=1$.
The roof function and length of closed geodesics on ${\mathcal M}_o$
--------------------------------------------------------------------
Our construction describes the geodesic flow on $T_1 ({\mathcal M}_o)$ as a suspension flow over the measure preserving transformation $({{\widetilde{\Omega}}}_o, {{\widetilde{T}}}_o,{\widetilde{\mu}}_o)$ identified with $({\mathcal{S}}_o, \rho_o,(\alpha-\beta)^{-2} d\alpha d\beta)$. The roof function is given by the hyperbolic distance between two consecutive return points to ${\mathcal{S}}_o$: $$r_o (\xi_\gamma )=d (\xi_\gamma,\eta_\gamma),$$ with $\xi_\gamma,\eta_\gamma\in {\mathbb{H}}$ as in Subsection \[oddcut\]. The points $\xi_\gamma$ and $\eta_\gamma$ can also be identified with elements in $T_1 ({\mathcal M}_o)$ and represent two consecutive changes in type for the cutting sequence of the geodesic $\gamma$. It is convenient to replace the geodesic arc $[\xi_\gamma,\eta_\gamma]$ with $[\xi,\eta]$, where $\xi=\rho_o(\xi_\gamma)$, $\eta=\rho_o (\eta_\gamma)$. Considering $\gamma_- =\rho(\gamma_{-\infty})$, $\gamma_+=\rho_o(\gamma_\infty)$, we employ as in [@Ser] the formula $$d(\xi_\gamma,\eta_\gamma)= d(\xi,\eta) =\log \bigg|
\frac{\gamma_{-}-\eta}{\gamma_{-}-\xi} \cdot
\frac{\gamma_{+}-\xi}{\gamma_{+}-\eta} \bigg| .$$
![The first return length[]{data-label="Figure8"}](length1.eps){width="8cm"}
Assume first that $\gamma$ is in case A, which means $\rho_o(x)=\frac{1}{a_1-x}$ and $\xi=x+iy =[0,\beta] \cap [\gamma_-,\gamma_+]$ where $\beta=\frac{1}{a_1-1}$ and $\eta =[1,\infty] \cap [\gamma_-,\gamma_+]$. Trigonometry of right triangles with vertices $\gamma_-$, $\eta$, $\gamma_+$ and $\gamma_{-},\xi,\gamma_+$ provides $$\frac{\lvert \gamma_{-}-\eta\rvert}{\lvert \gamma_+ -\eta \rvert} =
\sqrt{\frac{1-\gamma_{-}}{\gamma_+ -1}} \qquad \mbox{\rm and} \qquad
\frac{\lvert \gamma_+ -\xi \rvert}{\lvert \gamma_{-} -\xi \rvert} =
\sqrt{\frac{\gamma_+ -x}{x-\gamma_{-}}} .$$ The equalities $\lvert x+iy-\frac{\beta}{2}\rvert =\frac{\beta}{2}$ and $\lvert x+iy -\frac{1}{2} (\gamma_{-}+\gamma_+) \rvert =\frac{1}{2}(\gamma_+ -\gamma_{-})$ lead to $$x=\operatorname{Re} \xi =
\frac{\gamma_{-} \gamma_+}{\gamma_{-}+\gamma_+ -\beta} ,$$ and thus $$d_A (\xi_\gamma,\eta_\gamma) =\frac{1}{2} \log \bigg(
\frac{\gamma_+ -x}{\gamma_+-1} \cdot \frac{\gamma_- -1}{\gamma_- -x} \bigg)
=\frac{1}{2} \log \bigg( \frac{\gamma_+^2 (\beta^{-1} -\gamma_+^{-1})}{\gamma_+-1} \cdot
\frac{\gamma_- -1}{\gamma_-^2 (\beta^{-1} -\gamma_-^{-1})} \bigg) .$$ Employing $\frac{1}{\beta}-\rho_o(x)=x-1$, we gather $$d_A (\xi_\gamma , \eta_\gamma) =\frac{1}{2} \log \bigg(
\frac{F_A (\gamma_\infty)}{F_A (\gamma_{-\infty})} \bigg),
\quad \mbox{\rm where} \quad
F_A (x)= \rho_o (x)^2 \, \frac{x-1}{\rho_o(x)-1} .$$
Similar computations in each of the cases B, C, D provide $$F_B (x)= \rho_o(x)^2\, \frac{x-1}{\rho_o(x)+1} ,\qquad
F_C (x)= \rho_o(x)^2\, \frac{x+1}{\rho_o (x)+1},\qquad
F_D (x)= \rho_o(x)^2\, \frac{x+ 1}{\rho_o(x)-1} ,$$ and actually we get the general formula $$\label{eq5.4}
d(\xi_\gamma, \eta_\gamma) =\frac{1}{2} \log \bigg(
\frac{F (\gamma_\infty)}{F (\gamma_{-\infty})}\bigg) \quad
\mbox{\rm with} \quad
F(x)= F_\gamma (x)=\rho_o(x)^2\, \frac{x-\operatorname{sign}(\gamma_\infty)}{\rho_0 (x)-\operatorname{sign}(\rho_o(\gamma_\infty))} .$$
If $(\alpha,\beta)\in {\mathcal{S}}_o$, then there exists ${\epsilon}\in \{\pm 1\}$ such that $({\epsilon}\alpha,{\epsilon}\beta)=(\alpha^\prime,\beta^\prime)$ with $$\alpha^\prime =[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\!]_o >1,\quad\beta^\prime =- {\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o
\in -I_G.$$ Eqs. and then provide $$\label{eq5.5}
\begin{split}
F (\alpha) & =\rho_o(\alpha)^2 (-{\epsilon}_1) \,\frac{[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),(a_3,{\epsilon}_3),\ldots ]\!]_o -1}{[\![
(a_2,{\epsilon}_2);(a_3,{\epsilon}_3),(a_4,{\epsilon}_4),\ldots ]\!]_o -1} =\rho_o(\alpha)^2 F_+ (\alpha) , \\
F (\beta) & =
\rho_o( \beta)^2 (-{\epsilon}_1) \,\frac{{\langle\! \langle}(a_{0},{\epsilon}_{0}),(a_{-1},{\epsilon}_{-1}),(a_{-2},{\epsilon}_{-2}),\ldots {\rangle\! \rangle}_o +1}{{\langle\! \langle}(a_{1},{\epsilon}_{1}),(a_{0},{\epsilon}_{0}),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_o +1} =\rho_o (\beta)^2 F_- (\beta) .
\end{split}$$
When $\bar{\gamma}$ is a closed geodesic on ${\mathcal M}_o$ with endpoints as in Proposition \[cor9\], the contribution of each of the factors $F_+$ and $F_-$ to the length of $\bar{\gamma}$ is one. Using $(a_{n+r},{\epsilon}_{n+r})=(a_n,{\epsilon}_n)$, we find $$\label{eq5.6}
\operatorname{length}(\bar{\gamma}) =
\log \Bigg( \prod\limits_{k=1}^{r}
\frac{[\![ \overline{(a_{k+1},{\epsilon}_{k+1});(a_{k+2},{\epsilon}_{k+2}),\ldots ,(a_{k+r},{\epsilon}_{k+r})} ]\!]_o^2}{
{\langle\! \langle}\overline{(a_{k},{\epsilon}_{k}),(a_{k-1},{\epsilon}_{k-1}),\ldots, (a_{k+r-1},{\epsilon}_{k+r-1})}{\rangle\! \rangle}_o^2}\Bigg)
=\log \Bigg( \frac{(\rho_o^r)^\prime (\gamma_\infty)}{(\rho_o^r)^\prime (\gamma_{-\infty})} \Bigg) .$$
Geodesic coding and even continued fractions
============================================
The group $\Theta$ and the modular surface ${\mathcal M}_e=\Theta\backslash{\mathbb{H}}$
----------------------------------------------------------------------------------------
The Theta group $$\Theta :=\left\{ M\in \Gamma (1) : M \equiv I_2 \ \mbox{\rm or}\ \left( \begin{matrix}
0 & 1 \\ 1 & 0 \end{matrix}\right) \pmod{2} \right\} ,$$ is the index three subgroup of $\Gamma(1)$ generated by $S=\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix} \right)$ and $T=\left( \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)$. The image of the left half of the standard Dirichlet region $\{ \lvert \operatorname{Re} z \rvert < 1, \lvert z\rvert >1 \}$ of $\Theta \backslash {\mathbb{H}}$ under the transformation $S$ coincides with the region $\{ \operatorname{Re} z >0, \lvert z\rvert <1,\lvert z-\frac{1}{2}\rvert >\frac{1}{2}\}$. As a result, the standard Farey cell $\{ 0< \operatorname{Re} z <1, \lvert z-\frac{1}{2}\rvert >\frac{1}{2}\}$ provides a fundamental domain for ${\mathcal M}_e=\Theta \backslash {\mathbb{H}}=\pi_e ({\mathbb{H}})$. We find that ${\mathcal M}_e$ is homeomorphic to a sphere with a conic point at $\pi_e(i)$ and cusps at $\pi_e (\infty)$ and $\pi_e(1)$.
The edges of the Farey tessellation project to the line running from $\pi_e(1)$ to $\pi_e (\infty)$. Bauer and Lopes [@BL] realized the ECF natural extension ${\bar{T}}_e$ as a section of the billiard flow on ${\mathcal M}_e$. Here we describe the extension ${\widetilde{T}}_e$ of $\bar{T}_e$ as a section of the geodesic flow on $T_1({\mathcal M}_e)$.
The coding of geodesics on $\Theta \backslash {\mathbb{H}}$ is analogous to the coding for $\Gamma \backslash {\mathbb{H}}$ described in Sections \[oddseq\] and \[cut\]. However, in this case we do not use a checkerboard coloring. Here ${\mathcal{A}}_e$ is the set of geodesics in ${\mathbb{H}}$ with endpoints $$({\gamma_{\infty}},{\gamma_{-\infty}}) \in {\mathcal{S}}_e := \big( (-\infty ,-1) \cup (1,\infty)\big) \times (-1,1) ,$$ while $\xi_\gamma$ and $\eta_\gamma$ are defined by $$\xi_\gamma =\begin{cases} \gamma \cap [1,\infty] & \mbox{\rm if $\gamma_\infty >1$} \\
\gamma \cap [-1,\infty] & \mbox{\rm if $\gamma_\infty < -1$,} \end{cases}
\qquad
\eta_\gamma =\begin{cases} \gamma \cap [a_1,a_1+{\epsilon}_1] & \mbox{\rm if $\gamma_\infty >1$} \\
\gamma \cap [-a_1,-a_1 -{\epsilon}_1] & \mbox{\rm if $\gamma_\infty < -1$,} \end{cases}$$ as in the odd continued fraction case.
Every geodesic $\bar{\gamma}$ on ${\mathcal M}_e$ lifts to a geodesic $\gamma \in{\mathcal{A}}_e$. Let ${\mathcal X}_e$ be the set of elements $(\xi_\gamma,u_\gamma)\in T_1 ({\mathcal M}_e)$ with base point $\xi_\gamma \in \pi_e (\pm 1 +i{\mathbb R})$ and unit tangent vector $u_\gamma$ pointing along the geodesic $\pi_e(\gamma)$ such that $\pi_e(\eta)$ gives the base point of the first return of $\pi_e (\gamma)$ to ${\mathcal X}_e$. The base point $\xi_\gamma$ breaks the cutting sequence of $\pi_e(\gamma)$ into strings $L^{2k-2}R$, $L^{2k-1}R$, $R^{2k-2}L$, $R^{2k-1}L$ that are concatenating according to the rules that will be described in Subsection \[evencut\].
As in Subsection \[oddcut\] four cases can occur, depicted in Figures \[Figure9\] and \[Figure10\]. Again, we consider cases A, B, C, and D. Here, case A corresponds to ${\gamma_{\infty}}\in(2k-1,2k)$ and ${\epsilon}_1=-1$, case B to ${\gamma_{\infty}}\in(2k,2k+1)$ and ${\epsilon}_1=+1$, case C to ${\gamma_{\infty}}\in(-2k,-2k+1)$ and ${\epsilon}_1=-1$, and case D to ${\gamma_{\infty}}\in(-2k-1,-2k)$ and ${\epsilon}_1=+1$.
In cases A and B, we write $\gamma_\infty =[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\!]_e$ and $\gamma_{-\infty} =- {\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_e$. For C and D, we write $\gamma_\infty =-[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\!]_e$ and $\gamma_{-\infty} = {\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_e$.
![ECF cases A ($\gamma_\infty >1,{\epsilon}_1=-1,\xi_\gamma L^{2k-2}R\eta_\gamma$) and B ($\gamma_\infty >1,{\epsilon}_1=+1,\xi_\gamma L^{2k-1}R\eta_\gamma$)[]{data-label="Figure9"}](I_1_even.eps)
![ECF cases C ($\gamma_\infty <-1, {\epsilon}_1=-1,\xi_\gamma R^{2k-2}L\eta_\gamma$) and D ($\gamma_\infty <-1, {\epsilon}_1=+1,\xi_\gamma R^{2k-1}L\eta_\gamma$)[]{data-label="Figure10"}](II_1_even.eps)
When ${\gamma_{\infty}}>1$, the M" obius transformation $\rho_e(x)=\frac{1}{a_1-x}$ belongs to $\Theta$ and we have $$\rho_e (\gamma_{-\infty}) ={\epsilon}_1 {\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),\ldots {\rangle\! \rangle}_e \in (0,1) \quad
\mbox{\rm and} \quad \rho_e (\gamma_\infty) = (-{\epsilon}_1) [\![ (a_2,{\epsilon}_2); (a_3,{\epsilon}_3),\ldots ]\!]_e.$$
When $\gamma_\infty <-1$ and we have the M" obius transformation $\rho_e(x)=\frac{1}{-a_1-x}$ belongs to $\Theta$ and $$\rho_e (\gamma_{-\infty}) =-{\epsilon}_1 {\langle\! \langle}(a_1,{\epsilon}_1),(a_0,{\epsilon}_0),\ldots {\rangle\! \rangle}_e \in (-1,0) \quad
\mbox{\rm and} \quad \rho_e (\gamma_\infty) = {\epsilon}_1 [\![ (a_2,{\epsilon}_2); (a_3,{\epsilon}_3),\ldots ]\!]_e.$$ In all cases we have $$\label{eq6.1}
\begin{split}
\rho_e & \big( {\epsilon}[\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\!]_e , -{\epsilon}{\langle\! \langle}(a_0,{\epsilon}_0),(a_{-1},{\epsilon}_{-1}),\ldots {\rangle\! \rangle}_e) \\
& = -{\epsilon}_1 {\epsilon}\big( [\![ (a_2,{\epsilon}_2);(a_3,{\epsilon}_3),\ldots ]\!]_e,
- {\langle\! \langle}(a_{1},{\epsilon}_{1}),(a_{0},{\epsilon}_{0}),\ldots {\rangle\! \rangle}_e \big) ,
\end{split}$$ where ${\epsilon}=+1$ in cases A and B, and ${\epsilon}=-1$ in cases C and D. Again, we get that when ${\epsilon}_1=+1$, $\rho_e({\gamma_{\infty}})$ and ${\gamma_{\infty}}$ are on opposite sides of the imaginary axis, and $\rho_e$ agrees with the transformation $\rho$ for the RCF in [@Ser]. As in the OCF case, this is exactly where the RCF and ECF agree, cases A and C are followed by A or B, and cases B and D are followed by cases C or D.
The map $J_e:{\mathcal{S}}_e \rightarrow {{\widetilde{\Omega}}}_e$, $J_e (x,y)=\operatorname{sign} (x)
(\frac{1}{x},-y,1)$ is invertible. Direct verification reveals $$\label{eq6.2}
J_e \rho_e J_e^{-1} = {{\widetilde{T}}}_e .$$
As in Subsection \[invmeas\], the push-forwards of the measure $(\alpha-\beta)^{-2}d\alpha d\beta$ on ${\mathcal{S}}_e$ under the maps $\pi\circ J_e$, $\pi_1 \circ J_e$, and $\pi_2 \circ J_e$ are $\bar{T}_e$-invariant, $T_e$-invariant, and $\tau_e$-invariant, respectively. For intervals $[a,b]\subset (0,1)$, $[c,d]\subset (-1,1)$ and $E=[a,b]\times [c,d]$, we find $$\begin{split}
\bar{\mu}_e (E) & =2\iint_E \frac{dxdy}{(1+xy)^2} ,
\qquad \nu_e ([c,d]) = 2\int_c^d \frac{dv}{1+v} , \\
\mu_e ([a,b]) & =\int_{1/b}^{1/a} \int_{-1}^1 \frac{d\alpha d\beta}{(\alpha-\beta)^2} +
\int_{-1/a}^{-1/b} \int_{-1}^1 \frac{d\alpha d\beta}{(\alpha-\beta)^2} =
\int_a^b \Big( \frac{1}{1+u}+\frac{1}{1-u}\Big) du ,
\end{split}$$ which coincide with the invariant measures from [@Sch2].
The analogues of Propositions \[cor5\], \[cor6\], \[cor8\] and \[cor9\] come from changing the subscript $o$ to $e$, since the same equalities hold.
\[evencors\] [*(i)*]{} A real number $\alpha >1$ has a purely periodic ECF expansion if and only if $\alpha$ is a quadratic surd with $-1 < {\bar{\alpha}}<1$. Furthermore, if $$\alpha =[\![ \, \overline{(a_1,{\epsilon}_1); (a_2,{\epsilon}_2), \ldots ,(a_{r},{\epsilon}_{r})} \,]\!]_e ,$$ then $${\bar{\alpha}}=-{\langle\! \langle}\,\overline{(a_{r},{\epsilon}_{r}), \ldots, (a_1,{\epsilon}_1)} \,{\rangle\! \rangle}_e .$$
[*(ii)*]{} Two irrational numbers $\alpha,\beta >1$ are $\Theta$-equivalent if and only if there are $r,s\geqslant 0$ such that $$\label{eventail}
t_r(\alpha)=t_s (\beta) ,$$ where $t_m ([\![ (a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots ]\! ]_e) :=(-{\epsilon}_1)\cdots (-{\epsilon}_m)
[\![ (a_{m+1},{\epsilon}_{m+1});(a_{m+2},{\epsilon}_{m+2}),\ldots ]\!]_e$.
[*(iii)*]{} The ECF expansion of an irrational $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic surd.
[*(iv)*]{} A geodesic $\bar{\gamma}$ on ${\mathcal M}_e$ is closed if and only if it has a lift $\gamma\in {\mathcal{A}}_e$ with purely periodic endpoints $$\label{eq6.4}
\gamma_\infty ={\epsilon}[\! [ \overline{(a_1,{\epsilon}_1);(a_2,{\epsilon}_2),\ldots, (a_r,{\epsilon}_r)} ]\!]_e \quad \mbox{and}
\quad \gamma_{-\infty} =-{\epsilon}{\langle\! \langle}\overline{(a_r,{\epsilon}_r),\ldots ,(a_2,{\epsilon}_2),(a_1,{\epsilon}_1)} {\rangle\! \rangle}_e$$ for some ${\epsilon}\in \{\pm 1\}$ and $(-{\epsilon}_1)\cdots (-{\epsilon}_r)=1$.
The second point should be compared with Theorem 1 in [@KL]. It seems that the definition of the tail $t_n(x)$ in [@KL Eq.(1.3)] should be changed to $t_n(x)=(-e_1)\cdots (-e_n) [0;e_{n+1}/a_{n+1},e_{n+2}/a_{n+2}, \ldots]$ for that statement to hold.
Analogous formulas for the roof function and for the length of a closed geodesic also hold, as in and . Since ${\langle\! \langle}(b_0,{\epsilon}_0),(b_1,{\epsilon}_1),\ldots {\rangle\! \rangle}_e^{-1}={\epsilon}_0
[\![ (b_0,{\epsilon}_1);(b_1,{\epsilon}_2),(b_2,{\epsilon}_3),\ldots]\!]_e$, the analogue of shows that the length of a closed geodesic $\bar{\gamma}$ on ${\mathcal M}_e$ with endpoints as in is given by $$\label{eq6.5}
\log \bigg( \prod\limits_{k=1}^r
[\![ \overline{(a_{k+1},{\epsilon}_{k+1});\ldots,(a_{k+r},{\epsilon}_{k+r})} ]\!]_e^2
[\![ \overline{(a_k,{\epsilon}_{k-1});(a_{k-1},{\epsilon}_{k-2}),\ldots,(a_{k-r+1},{\epsilon}_{k-r})} ]\!]_e^2 \bigg).$$
In this case ${\mathcal M}_e$ has two cusps, $\pi_e (\infty)$ and $\pi_e (1)$, and the group $\Theta$ splits the rationals in two equivalence classes, as shown by the following elementary
\[lemma10\] $\ \Theta \infty=\big\{ \frac{a}{c}\in {\mathbb Q}:\mbox{$a$ or $c$ is even}\big\}$ and $\ \Theta 1=\big\{ \frac{m}{n}\in{\mathbb Q}: \mbox{$m$ and $n$ are odd}\big\}$.
Connection with cutting sequence {#evencut}
--------------------------------
As before, cases A and B give the cutting sequence $\ldots xL^{n_1}R^{n_2}L^{n_3}\ldots$, where $x$ indicates $\gamma\cap [0,\infty]$ and $[n_1;n_2,n_3\ldots]$ is the regular continued fraction expansion of ${\gamma_{\infty}}$. This again corresponds to $\ldots L {\xi_\gamma}L^{n_1-1}R^{n_2}L^{n_3}\ldots$. We have two cases to consider for the first digit of the even continued fraction expansion.
(A)
: $n_1=2k-1$ is odd, and ${\gamma_{\infty}}\in(2k-1,2k)$. This gives the cutting sequence $\ldots{\xi_\gamma}L^{2k-2}R\eta_\gamma$ and we get $(2k,-1)=(a_1,{\epsilon}_1)$. The next digit is represented by $L^nR$. In the case $n_2>1$ the next digit is $L^0R$, corresponding to $(2,-1)$. This again corresponds to the insertion and singularization algorithm [@BL], as follows: $$\begin{aligned}
&2k-1+\cfrac{1}{1+\frac{1}{n_3+\ldots}}=[\![(2k,-1);(n_3+1,+1),\ldots]\!]_e, \\
&2k-1+\frac{1}{n_2+\ldots}=[\![(2k,-1);(2,-1)^{n_2-1},(n_3+1,+1),\ldots]\!]_e
\quad \mbox{\rm if $n_2 >1$,}\end{aligned}$$ where $(2,-1)^{t}$ means the digit $(2,-1)\ t$-times.
(B)
: $n_1=2k$ is even, and ${\gamma_{\infty}}\in(2k,2k+1)$. This gives the cutting sequence $\ldots {\xi_\gamma}L^{2k-1}R\eta_\gamma$, and we get $(2k,+1)=(a_1,{\epsilon}_1)$. The next digit is represented by $R^nL$. In the first case, we needed to consider what happened when $n_2>1,$ here we need to look at $n_2=1$. Then, we have $\ldots {\xi_\gamma}L^{2k-1}R\eta_\gamma L^{n_3}\ldots$, and the next digit corresponds to $R^0L$, which is again $(2,-1)$.
Strings starting with $R$ are treating similarly, with the roles of $L$ and $R$ switched.
Cases C and D are classified similarly, with roles of $L$ and $R$ exchanged. In this case, we get the cutting sequence $\ldots {\xi_\gamma}R^{n_1-1}L \eta_\gamma L^{n_2-1}R^{n_3}\ldots$, where ${\gamma_{\infty}}=-[n_1;n_2,n_3,\ldots]$.
Extended even continued fractions
---------------------------------
For the end point ${\gamma_{-\infty}}$, we must consider the extended even continued fractions. Unlike the odd and grotesque continued fractions, the extended even continued fractions correspond to reindexing of the even continued fractions.
We consider cases A, B, C, and D similar to the grotesque continued fraction case.
(A)
: ${\gamma_{\infty}}>1, {\gamma_{-\infty}}\in(0,1)$, and ${\epsilon}_0=-1$ gives the cutting sequence $\ldots L^{2k-2}R {\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k,+1)$.
(B)
: ${\gamma_{\infty}}>1, {\gamma_{-\infty}}\in(-1,0),$ and ${\epsilon}_0=+1$ gives the cutting sequence $\ldots R^{2k-1}L {\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k,-1)$.
(C)
: ${\gamma_{\infty}}<-1, {\gamma_{-\infty}}\in(0, 1),$ and ${\epsilon}_0=+1$ gives the cutting sequence $\ldots L^{2k-1}R {\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k,+1)$.
(D)
: ${\gamma_{\infty}}<-1,{\gamma_{-\infty}}\in(-1,0),$ and ${\epsilon}_0=-1$ gives the cutting sequence $\ldots R^{2k-2}L {\xi_\gamma}\ldots$ and $(a_0,{\epsilon}_0)=(2k,-1)$.
That is, we interpret the strings the same way as in the even continued fractions. However, strings of type A and C must be preceded by those of type A or B, and strings of type B and D must be preceded by C or D. This is similar to the restrictions on the grotesque continued fractions from Section \[grot\].
Acknowledgments
===============
We are grateful to Pierre Arnoux, Byron Heersink, and to the anonymous referee for constructive comments and suggestions.
The first author acknowledges partial support during his visits to IMAR Bucharest by a grant from Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project PN-III-P4-ID-PCE-2016-0823, within PNCDI III.
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abstract: 'We consider overdetermined problems of Serrin’s type in convex cones for (possibly) degenerate operators in the Euclidean space as well as for a suitable generalization to space forms. We prove rigidity results by showing that the existence of a solution implies that the domain is a spherical sector.'
address:
- 'Giulio Ciraolo, Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123, Palermo, Italy'
- 'Alberto Roncoroni, Dipartimento di Matematica F. Casorati, Università degli Studi di Pavia, Via Ferrata 5, 27100 Pavia, Italy'
author:
- Giulio Ciraolo and Alberto Roncoroni
title: 'Serrin’s type overdetermined problems in convex cones'
---
[^1]
Introduction
============
Given a bounded domain $E \subset \mathbb{R}^N$, $N\geq 2$, the classical Serrin’s overdetermined problem [@Serrin] asserts that there exists a solution to $$\label{pb_serrin}
\begin{cases}
\Delta u = -1 & \textmd{ in } E \,, \\
u=0 & \textmd{ on } \partial E \,, \\
\partial_{\nu}u = -c & \textmd{ on } \partial E \,,
\end{cases}$$ for some constant $c>0$, if and only if $E=B_R(x_0)$ is a ball of radius $R$ centered at some point $x_0$. Moreover, the solution $u$ is radial and it is given by $$\label{u_radial}
u(x) = \frac{R^2 - |x-x_0|^2}{2N}\,,$$ with $R=Nc$. Here, $\nu$ denotes the outward normal to $\partial \Omega$.
The starting observation of this manuscript is the following. Let $\Sigma$ be an open cone in $\mathbb{R}^N$ with vertex at the origin $O$, i.e. $$\Sigma=\lbrace tx \, : \, x\in\omega, \, t\in(0,+\infty)\rbrace$$ for some open domain $\omega\subset \mathbb{S}^{N-1}$. We notice that if $x_0$ is chosen appropriately then $u$ given by is still the solution to $$\begin{cases}
\Delta u = -1 & \textmd{ in } B_R(x_0) \cap \Sigma \,, \\
u=0 \text{ and } \partial_\nu u = -c & \textmd{ on } \partial B_R(x_0) \setminus \overline{\Sigma}\,, \\
\partial_{\nu}u = 0 & \textmd{ on } B_R(x_0) \cap \partial \Sigma \,.
\end{cases}$$ More precisely, $x_0$ may coincide with $O$ or it may be just a point of $\partial \Sigma \setminus \{O\}$ and, in this case, $B_R(x_0) \cap \Sigma$ is half a sphere lying over a flat portion of $\partial \Sigma$. Hence, it is natural to look for a characterization of symmetry in this direction, as done in [@Pacella-Tralli] (see below for a more detailed description).
In order to properly describe the results, we introduce some notation. Given an open cone $\Sigma$ such that $\partial\Sigma\setminus\lbrace O\rbrace$ is smooth, we consider a bounded domain $\Omega\subset\Sigma$ and denote by $\Gamma_0$ its relative boundary, i.e. $$\Gamma_0 = \partial \Omega \cap \Sigma \,,$$ and we set $$\Gamma_1=\partial\Omega\setminus\bar{\Gamma}_0 \,.$$ We assume that $\mathcal{H}_{N-1}(\Gamma_1)>0$, $\mathcal{H}_{N-1}(\Gamma_0)>0$ and that $\Gamma_0$ is a smooth $(N-1)$-dimensional manifold, while $\partial\Gamma_0=\partial\Gamma_1 \subset\partial\Omega\setminus\lbrace O\rbrace$ is a smooth $(N-2)$-dimensional manifold. Following [@Pacella-Tralli], such a domain $\Omega$ is called a *sector-like domain*. In the following, we shall write $\nu=\nu_x$ to denote the exterior unit normal to $\partial\Omega$ wherever is defined (that is for $x\in\Gamma_0\cup\Gamma_1\setminus\lbrace O\rbrace$).
Under the assumption that $\Sigma$ is a convex cone, in [@Pacella-Tralli] it is proved that if $\Omega$ is a sector-like domain and there exists a classical solution $u \in C^2(\Omega) \cap C^1(\Gamma_0 \cup \Gamma_1 \setminus \{O\})$ to $$\label{pb_serrin_cone_PT}
\begin{cases}
\Delta u = -1 & \textmd{ in } \Omega \,, \\
u=0 \textmd{ and } \partial_\nu u = -c & \textmd{ on } \Gamma_0\,, \\
\partial_{\nu}u = 0 & \textmd{ on } \Gamma_1 \setminus \{O\} \,,
\end{cases}$$ and such that $$u \in W^{1,\infty}(\Omega) \cap W^{2,2}(\Omega) \,,$$ then $$\Omega = B_R(x_0) \cap \Sigma$$ for some $x_0 \in \mathbb{R}^N$ and $u$ is given by . Differently from the original paper of Serrin [@Serrin], the method of moving planes is not helpful (at least when applied in a standard way) and the rigidity result in [@Pacella-Tralli] is proved by using two alternative approaches. One is based on integral identities and it is inspired from [@BNST], the other one uses a $P$-function approach as in [@Weinberger].
In this paper, we generalize the rigidity result for Serrin’s problem in [@Pacella-Tralli] in two directions. The former is by considering more general operators than the Laplacian in the Euclidean space, where the operators may be of degenerate type. Here, the generalization is not trivial due to the lack of regularity of the solution (the operator may be degenerate) as well as to other technical details which are not present in the linear case.
The latter is by considering an analogous problem in space forms, i.e. the hyperbolic space and the (hemi)sphere. The operator that we consider is linear and it is interesting since it has been shown that it is a helpful generalization of the torsion problem to space forms ([@CV1], [@QuiXia], [@QX]).
[**More general operators in the Euclidean space.**]{} Let $\Omega$ be a sector like domain in $\mathbb{R}^N$ and let $f:[0,+\infty) \to [0,+\infty)$ be such that $$\label{f_HP}
\begin{aligned}
f \in C([0,\infty))\cap C^3(0,\infty) & \textmd{ with } f(0)=f'(0)=0, \ f''(s)>0 \text{ for } s>0 \\
& \text{ and } \lim_{s\rightarrow +\infty }\dfrac{f(s)}{s}=+ \infty \,.
\end{aligned}$$ We consider the following mixed boundary value problem $$\label{pb cono}
\begin{cases}
L_f u=-1 &\mbox{in } \Omega, \\ u=0 &\mbox{on } \Gamma_0 \\ \partial_{\nu}u=0 \, &\mbox{on } \Gamma_1\setminus\lbrace O\rbrace,
\end{cases}$$ where the operator $L_f$ is given by $$\label{operator}
L_f u={\mathrm{div}}\left(f'(|\nabla u|)\dfrac{\nabla u}{|\nabla u|}\right),$$ and the equation $L_f u = -1$ is understood in the sense of distributions $$\int_\Omega\dfrac{f'(|\nabla u|)}{|\nabla u|}\nabla u\cdot\nabla\varphi\, dx=\int_{\Omega}\varphi\, dx$$ for any $$\varphi\in T(\Omega):=\lbrace \varphi\in C^1(\Omega) \, : \, \varphi\equiv 0 \, \textit{ on } \Gamma_0\rbrace.$$ Notice that the operator $L_f$ may be of degenerate type.
We notice that the solution to $L_f u =-1$ in $B_R(x_0)$ (a ball of radius $R$ centered at $x_0$) such that $u= 0$ on $\partial B_R(x_0)$ is radial and it is given by $$\label{u_radial_Lf}
u(x)=\int_{|x-x_0|}^{R}g'\left(\dfrac{s}{N}\right)\, ds \,,$$ where $g$ denotes the Fenchel conjugate of $f$ (see for instance [@Crasta] or [@FGK]), i.e. $$g=\sup\lbrace st-f(s) \, : \, s\geq 0\rbrace$$ (hence for us $g'$ is the inverse function of $f'$). Our first main result is the following.
\[teo 1 cono\] Let $f$ satisfy . Let $\Sigma$ be a convex cone such that $\Sigma\setminus\lbrace O\rbrace$ is smooth and let $\Omega$ be a sector-like domain in $\Sigma$. If there exists a solution $u\in C^1(\Omega\cup\Gamma_0\cup\Gamma_1\setminus\lbrace O\rbrace)\cap W^{1,\infty}(\Omega)$ to such that $$\label{overdetermined cond}
\partial_{\nu}u=-c \, \textit{ on } \, \Gamma_0$$ for some constant $c$, and satisfying $$\label{key property}
\dfrac{f'(|\nabla u|)}{|\nabla u|}\nabla u \in W^{1,2}(\Omega,\mathbb{R}^N) \,,$$ then there exists $x_0 \in \mathbb{R}^N$ such that $\Omega=\Sigma\cap B_{R}(x_0)$ with $c=g'(|\Omega|/|\Gamma_0|)$, $R=N|\Omega|/|\Gamma_0|$. Moreover $u$ is given by , where $x_0$ is the origin or, if $\partial \Sigma$ contains flat regions, it is a point on $\partial \Sigma$.
When $L_f = \Delta$ (i.e. $f(t)=t^2/2$), Theorem \[teo 1 cono\] is essentially Theorem 1.1 in [@Pacella-Tralli]. Condition holds (at least locally in $\Omega$) for uniformly elliptic operators, such as the mean curvature operator ($f(t) = \sqrt{1+t^2}$), and also for degenerate operators such as the $p-$Laplace operator ($f(t)=t^p/p$), see [@Mingione] and [@CiMa]. We stress that the validity of up to the boundary is more subtle, since it depends strongly on how $\Gamma_0$ and $\Gamma_1$ intersect.
We observe that the overdetermined problem with the condition can be seen as a partially overdetermined problem (see for instance [@FV1] and [@FV2]), since we impose both Dirichlet and Neumann conditions only on a part of the boundary, namely $\Gamma_0$, while a sole homogeneous Neumann boundary condition is assigned on $\Gamma_1$ (where, however, there is the strong assumption that it is contained in the boundary of a cone).
We notice that the proof of Theorem \[teo 1 cono\] still works when $\Gamma_1 = \emptyset$ (hence $\partial\Omega=\Gamma_0$). In this case we obtain the celebrated result of Serrin [@Serrin] for the operator $L_f$ (see also [@BC], [@BNST], [@CMS], [@FGK], [@FarinaKawhol], [@Garofalo-Lewis], [@Roncoroni], [@Weinberger]). Moreover, the proof is also suitable to be adapted to the anisotropic counterpart of the overdetermined problem and by following the approach used in this manuscript and in [@BC] (see also [@CS] and [@WX]). We also mention that rigidity theorems in cones are related to the study of relative isoperimetric and Sobolev inequalities in cones, and we refer to [@Pacella-Tralli] for a more detailed discussion (see also [@Baer-Figalli; @Cabre-RosOton-Serra; @Figalli-Indrei; @Grossi-Pacella; @Lions-Pacella-Tricarico; @Lions-Pacella]).
[**Serrin’s problem in cones in space forms.**]{} A space form is a complete simply-connected Riemannian manifold $(M,g)$ with constant sectional curvature $K$. Up to homotheties we may assume $K=0$,$1$,$-1$: the case $K=0$ corresponds to the Euclidean space $\mathbb{R}^N$, $K=-1$ is the hyperbolic space $\mathbb{H}^N$ and $K=1$ is the unitary sphere with the round metric $\mathbb{S}^N$. More precisely, in the case $K=1$ we consider the hemisphere $\mathbb{S}^N_+$. These three models can be described as warped product spaces $M=I\times \mathbb S^{N-1}$ equipped with the rotationally symmetric metric $$g=dr^2+h(r)^2\,g_{\mathbb S^{N-1}},$$ where $g_{\mathbb S^{N-1}}$ is the round metric on the $(N-1)$-dimensional sphere $\mathbb S^{N-1}$ and
1. $h(r)=r$ in the Euclidean case ($K=0$), with $I=[0,\infty)$;
2. $h(r)=\sinh(r)$ in the hyperbolic case ($K=-1$), with $I=[0,\infty)$;
3. $h(r)=\sin(r)$ in the spherical case ($K=1$), with $I=[0,\pi/2)$ for $\mathbb{S}^N_+$.
By using the warping structure of the manifold, we denote by $O$ the pole of the model and it is natural to define a *cone $\Sigma$ with vertex at $\{O\}$* as the set $$\Sigma=\lbrace tx \, : \, x\in\omega, \, t\in I \rbrace$$ for some open domain $\omega\subset \mathbb{S}^{N-1}$. Moreover, we say that $\Sigma$ is a *convex cone* if the second fundamental form $\mathrm {II}$ is nonnegative defined at every $p \in \partial \Sigma$.
Serrin’s overdetermined problem for semilinear equations $\Delta u + f(u) =0$ in space forms has been studied in [@Kumaresan_Prajapat] and [@Mol] by using the method of moving planes. If one considers the corresponding problem for sector-like domains in space forms, the method of moving planes can not be used and one has to look for alternative approaches. As already mentioned, in the Euclidean space these approaches typically use integral identities and $P$-functions (see [@BNST; @Weinberger]) and have the common feature that at a crucial step of the proof they use the fact that the radial solution attains the equality sign in a Cauchy-Schwartz inequality, which implies that the Hessian matrix ${\nabla^2}u $ is proportional to the identity. Since the equivalent crucial step in space forms is to prove that the Hessian matrix of the solution is proportional to the metric, then the equation $\Delta u =-1$ is no more suitable (one can easily verify that in the radial case the Hessian matrix of the solution is not proportional to the metric) and a suitable equation to be considered is $$\label{eq_space_forms}
\Delta u+NKu=-1$$ as done in [@CV1] and [@QuiXia], [@QX]. It is clear that for $K=0$, i.e. in the Euclidean case, the equation reduces to $\Delta u = -1$. For this reason, we believe that, in this setting, is the natural generalization of the Euclidean $\Delta u=-1$ to space forms.
A Serrin’s type rigidity result for can be proved following Weinberger’s approach by using a suitable $P$-function associated to (see [@CV1] and [@QX]). This approach is helpful for proving the following Serrin’s type rigidity result for convex cones in space forms, which is the second main result of this paper.
\[teo 2 cono\] Let $(M,g)$ be the Euclidean space, hyperbolic space or the hemisphere. Let $\Sigma \subset M$ be a convex cone such that $\Sigma\setminus\lbrace O\rbrace$ is smooth and let $\Omega$ be a sector-like domain in $\Sigma$. Assume that there exists a solution $u\in C^1(\Omega\cup\Gamma_0\cup\Gamma_1\setminus\lbrace O\rbrace)\cap W^{1,\infty}(\Omega) \cap W^{2,2}(\Omega)$ to $$\label{pb cono sf}
\begin{cases}
\Delta u+NKu=-1 &\mbox{in } \Omega, \\ u=0 &\mbox{on } \Gamma_0 \\ \partial_{\nu}u=0 \, &\mbox{on } \Gamma_1\setminus\lbrace O\rbrace,
\end{cases}$$ such that $$\label{overdetermined cond spaceforms}
\partial_{\nu}u=-c \, \textit{ on } \, \Gamma_0$$ for some constant $c$. Then $\Omega=\Sigma\cap B_{R}(x_0)$ where $B_R(x_0)$ is a geodesic ball of radius $R$ centered at $x_0$ and $u$ is given by $$u(x)=\frac{H(R)-H(d(x,x_0))}{n \dot h(R)} \,,$$ with $$H(r)=\int_0^r h(s) ds
$$ and where $d(x,x_0)$ denotes the distance between $x$ and $x_0$.
[**Organization of the paper.**]{} The paper is organized as follows: in Section \[section\_preliminary\] we introduce some notation, we recall some basic facts about elementary symmetric function of a matrix and prove some preliminary result needed to prove Theorem \[teo 1 cono\]. Theorems \[teo 1 cono\] and \[teo 2 cono\] are proved in Sections \[section\_proofthm1\] and \[section\_spaceforms\], respectively.
Preliminary results for Theorem \[teo 1 cono\] {#section_preliminary}
==============================================
In this section we collect some preliminary results which are needed in the proof of Theorem \[teo 1 cono\]. Let $f$ satisfy and consider problem $$\begin{cases}
L_f u=-1 &\mbox{in } \Omega, \\ u=0 &\mbox{on } \Gamma_0 \\ \partial_{\nu}u=0 \, &\mbox{on } \Gamma_1\setminus\lbrace O\rbrace,
\end{cases}$$ where the operator $L_f$ is given by $$L_f u={\mathrm{div}}\left(f'(|\nabla u|)\dfrac{\nabla u}{|\nabla u|}\right) \,.$$
$u\in C^1(\Omega\cup\Gamma_0\cup\Gamma_1\setminus\lbrace O\rbrace)$ is a solution to Problem if $$\label{weak sol cono}
\int_\Omega\dfrac{f'(|\nabla u|)}{|\nabla u|}\nabla u\cdot\nabla\varphi\, dx=\int_{\Omega}\varphi\, dx$$ for any $$\varphi\in T(\Omega):=\lbrace \varphi\in C^1(\Omega) \, : \, \varphi\equiv 0 \, \textit{ on } \Gamma_0\rbrace.$$
We observe some facts that will be useful in the following. Since the outward normal $\nu$ to $\Gamma_0$ is given by $$\label{normale}
\nu=-\frac{\nabla u}{|\nabla u|}|_{\Gamma_0} \,,$$ then implies that $$\label{gradiente cost}
|\nabla u|=c \quad \text{ on } \quad \Gamma_0.$$ Moreover we observe that the constant $c$ in the statement is given by $$\label{valore_c}
c=g'\left(\frac{|\Omega|}{|\Gamma_0|}\right),$$ as it follows by integrating the equation $L_fu=-1$, by using the divergence theorem, formula and the fact that $\partial_{\nu}u=0$ on $\Gamma_1\setminus\{ O\}$. We also notice that $$x\cdot\nu=0 \quad \text{ on } \quad \Gamma_1.$$
It will be useful to write the operator $L_f$ as the trace of a matrix. Let $V:{\mathbb{R}}^N\rightarrow{\mathbb{R}}$ be given by $$\label{definizione di V}
V(\xi)=f(|\xi|) \quad \text{for } \xi \in {\mathbb{R}}^N,$$ and notice that $$\label{derivatediV}
V_{\xi_i}(\xi)=f'(|\xi|)\dfrac{\xi_i}{|\xi|} \quad \text{ and } \quad V_{\xi_i\xi_j}(\xi)=f''(|\xi|)\dfrac{\xi_i\xi_j}{|\xi|^2}-f'(|\xi|)\dfrac{\xi_i\xi_j}{|\xi|^3}+f'(|\xi|)\dfrac{\delta_{ij}}{|\xi|}.$$ Hence, by setting $$W=(w_{ij})_{i,j=1,\dots,N}$$ where $$\label{Wdef}
w_{ij}(x)=\partial_{j} V_{\xi_i}(\nabla u(x)) \,,$$ we have $$\label{Lfu_trW}
L_f(u) = {\mathrm{Tr}}(W).$$ Notice that at regular points, where $\nabla u \neq 0$, it holds that $$\label{Wdef_II}
W=\nabla^2_{\xi}V(\nabla u)\nabla^2 u \,.$$ Our approach to prove Theorem \[teo 1 cono\] is to write several integral identities and just one pointwise inequality, involving the matrix $W$. Writing the operator $L_f$ as trace of $W$ has the advantage that we can use the generalization of the so-called Newton’s inequalities, as explained in the following subsection.
Elementary symmetric functions of a matrix
------------------------------------------
Given a matrix $A=(a_{ij})\in{\mathbb{R}}^{N\times N}$, for any $k=1,\dots,N$ we denote by $S_k(A)$ the sum of all the principal minors of $A$ of order $k$. In particular, $S_1(A)={\mathrm{Tr}}(A)$ is the trace of $A$, and $S_n(A)=\mathrm{det}(A)$ is the determinant of $A$. We consider the case $k=2$. By setting $$\label{def_S^2}
S^2_{ij}(A)=-a_{ji}+\delta_{ij}{\mathrm{Tr}}(A),$$ we can write $$\label{defS_2}
S_2(A)=\frac{1}{2}\sum_{i,j}S^2_{ij}(A)a_{ij}=\frac{1}{2}(({\mathrm{Tr}}(A)^2-{\mathrm{Tr}}(A^2)) \,.$$ The elementary symmetric functions of a symmetric matrix $A$ satisfy the so called Newton’s inequalities. In particular, $S_1$ and $S_2$ are related by $$\label{CS per S_2}
S_2(A)\leq \frac{N-1}{2N}(S_1(A))^2 \,.$$ When the matrix $A=W$, with $W$ given by , we have $$\label{defS_ij}
S^2_{ij}(W)=-V_{\xi_j\xi_k}(\nabla u)u_{ki}+\delta_{ij}L_f u \,,$$ and $S_{ij}^2(W)$ is divergence free in the following (weak) sense $$\label{derivata di S nulla}
\dfrac{\partial}{\partial x_j}S_{ij}^2(W)=0 \,.$$ We will need a generalization of to not necessarily symmetric matrices, which is given by the following lemma.
\[lemma matrici generiche\] Let $B$ and $C$ be symmetric matrices in ${\mathbb{R}}^{N\times N}$, and let $B$ be positive semidefinite. Set $A=BC$. Then the following inequality holds: $$\label{matrices}
S_2(A)\leq\dfrac{N-1}{2N}{\mathrm{Tr}}(A)^2.$$ Moreover, if ${\mathrm{Tr}}(A)\neq 0$ and equality holds in , then $$A=\dfrac{{\mathrm{Tr}}(A)}{N}I,$$ and $B$ is, in fact, positive definite.
Some properties of solutions to
--------------------------------
In this subsection we collect some properties of the solutions to . We assume that the solution is of class $C^1(\Omega) \cap W^{1,\infty}(\Omega)$. From standard regularity elliptic estimates one has that $u$ is of class $C^{2,\alpha}$ where $\nabla u \neq 0$. If one has more information about the degeneracy at zero of $f$ (see [@Mingione] and [@CiMa]), then one may conclude that $u \in C^{1,\alpha} ( \Omega)$ as well as that $$\frac{f'(|\nabla u|)}{|\nabla u|} \nabla u \in W^{1,2}_{loc}(\Omega, \mathbb{{\mathbb{R}}}^N) \,.$$ The regularity up to the boundary is more difficult to be understood, and it strongly depends on how $\Gamma_0$ and $\Gamma_1$ intersect. This will be one of the major points of the proof of Theorem \[teo 1 cono\].
In the following two lemmas we show that $u>0$ in $\Omega\cup\Gamma_1$ and we prove a Pohozaev-type identity.
\[prop 2\] Let $f$ satisfy and let $u$ be a solution of . Then $$u>0 \quad \textit{ in } \quad \Omega\cup\Gamma_1.$$
We write $u=u^+-u^-$ and use $\varphi = u^-$ as test function in : $$0 \geq -\int_{\Omega \cap \{u<0\}} \dfrac{f'(|\nabla u|)}{|\nabla u|}\, |\nabla u^-|^2\, dx= \int_{\Omega \cap \{u<0\}} u^- \, dx \geq 0\,,$$ which implies that $u\geq 0$ in $\Omega$. Moreover, if one assumes that $u(x_0)=0$ at some point $x_0 \in \Omega \cup \Gamma_1$, then $\nabla u(x_0)=0$, which leads to a contradiction by using the comparison principle between the solution $u$ and the radial solution in a suitable ball.
\[Pohozaev identity\] Let $\Omega$ be a sector-like domain and assume that $f$ satisfies . Let $u\in W^{1,\infty}(\Omega)$ be a solution to . Then the following integral identity $$\label{Pohozaev}
\int_\Omega[(N+1) u-N f(|\nabla u|)]\, dx=\int_{\Gamma_0}[f'(|\nabla u|)|\nabla u|-f(|\nabla u|)]x\cdot\nu\, d\sigma$$ holds.
We argue by approximation. For $t\geq 0$ and $\varepsilon\in(0,1)$, let $$f_{\varepsilon}(t)=f(\sqrt{\varepsilon^2+t^2})-f(\varepsilon) \,.$$ We define $F(t)=f'(t)t$ and $F_\varepsilon(t)=f_\varepsilon'(t)t$. From a standard argument (see for instance [@CFV Lemma 4.2]) we have that $$\label{convergenze}
f_\varepsilon\rightarrow f \quad \text{and} \quad F_\varepsilon\rightarrow F \quad \text{uniformly on compact sets of $[0,+\infty)$.}$$ We recall that $V(\xi)=f(|\xi|)$ (see ) for $\xi\in\mathbb{R}^N$, and we define $V^\varepsilon:\mathbb{R}^N\rightarrow\mathbb{R}$ as $$V^\varepsilon(\xi):=f_\varepsilon(|\xi|).$$ We approximate $\Omega$ by domains $\Omega_\delta$ obtained by chopping off a $\delta$-tubular neighborhood of $\partial\Gamma_0$ and a $\delta$-neighborhood of $O$. For $n \in \mathbb{N}$, we consider $u^n_{\delta}\in C^\infty(\Omega_\delta)\cap C^1(\bar \Omega_\delta)$ such that $$u^n_{\delta} \rightarrow u \, \text{ in } \, C^1(\overline{\Omega}_\delta),$$ as $n$ goes to infinity (see for instance [@Burenkov Section 2.6]). $\\ $ Since $${\mathrm{div}}\left(x\cdot\nabla u^n_{\delta}\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\right)=x\cdot\nabla u^n_{\delta}{\mathrm{div}}\left(\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\right)+\nabla(x\cdot\nabla u^n_{\delta})\cdot\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})$$ and from $$\begin{aligned}
\nabla(x\cdot\nabla u^n_{\delta})\cdot\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})=&\nabla u^n_{\delta}\cdot\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})+x\nabla^2(u^n_{\delta})\cdot\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\\
=&{\mathrm{div}}\left(u^n_{\delta}\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\right)-u^n_{\delta}{\mathrm{div}}\left(\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\right) \\
&+{\mathrm{div}}(xV^\varepsilon(\nabla u^n_{\delta}))-NV^\varepsilon(\nabla u^n_{\delta}) \,,
\end{aligned}$$ we obtain $$\label{div_smart}
{\mathrm{div}}\left(\varphi_n\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})-xV^\varepsilon(\nabla u^n_{\delta})\right)=\varphi_n{\mathrm{div}}\left(\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\right)-NV^\varepsilon(\nabla u^n_{\delta}) \,,$$ where $$\varphi_n(x)=x\cdot\nabla u^n_{\delta}(x)-u^n_{\delta}(x) \,.$$ Moreover, from the divergence theorem we have $$\label{angela}
\int_{\Omega_\delta} \nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\cdot\nabla\varphi_n\, dx=-\int_{\Omega_\delta} \varphi_n{\mathrm{div}}\left( \nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\right)\, dx + \int_{\partial\Omega_\delta}\varphi_n\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\cdot\nu\, d\sigma \,.$$ We are going to apply the divergence theorem in $\Omega_\delta$; to this end we set $$\Gamma_{0,\delta}=\Gamma_0\cap\partial\Omega_\delta\, , \quad \Gamma_{1,\delta}=\Gamma_1\cap\partial\Omega_\delta \quad \text{and} \quad \Gamma_\delta=\partial\Omega_\delta\setminus( \Gamma_{0,\delta}\cup \Gamma_{1,\delta})\, .$$ From and by integrating in $\Omega_\delta$ we obtain $$\begin{aligned}
\int_{\Omega_\delta} \nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\cdot\nabla\varphi_n\, dx=&-N\int_{\Omega_\delta} V^\varepsilon(\nabla u^n_{\delta})\, dx -\int_{\Omega_\delta}{\mathrm{div}}\left(\varphi_n\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\right)\, dx \\
&+\int_{\Omega_\delta}{\mathrm{div}}\left(xV^\varepsilon(\nabla u^n_{\delta})\right)\, dx \,, \\
\end{aligned}$$ and from $x\cdot\nu=0$ on $\Gamma_{1,\delta}$, we find $$\begin{aligned}
\int_{\Omega_\delta} \nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\cdot\nabla\varphi_n\, dx
=&-N\int_{\Omega_\delta} V^\varepsilon(\nabla u^n_{\delta})\, dx-\int_{\Gamma_{0,\delta}\cup\Gamma_{1,\delta}}\varphi_n\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})\cdot\nu\, d\sigma \\
&+\int_{\Gamma_{0,\delta}}V^\varepsilon(\nabla u^n_{\delta})x\cdot\nu\, d\sigma \\
&-\int_{\Gamma_\delta}[\varphi_n\nabla_\xi V^\varepsilon(\nabla u^n_{\delta})-xV^\varepsilon(\nabla u^n_{\delta})]\cdot\nu\, d\sigma \, .
\end{aligned}$$ By taking the limit as $\varepsilon\rightarrow 0$ and then as $n\rightarrow\infty$, using that $\nabla u\cdot\nu=0$ on $\Gamma_{1,\delta}$ (since $\partial_{\nu}u=0$ on $\Gamma_1$), we obtain $$\label{ryan}
\begin{aligned}
\int_{\Omega_\delta} \nabla_\xi V(\nabla u)\cdot\nabla\varphi\, dx=&-N\int_{\Omega_\delta} V(\nabla u)\, dx -\int_{\Gamma_{0,\delta}}\varphi\nabla_\xi V(\nabla u)\cdot\nu\, d\sigma +\int_{\Gamma_{0,\delta}}V(\nabla u)x\cdot\nu\, d\sigma \\
&-\int_{\Gamma_\delta}[\varphi\nabla_\xi V(\nabla u)-xV(\nabla u)]\cdot\nu\, d\sigma
\end{aligned}$$ where we let $$\label{phi_def_def}
\varphi(x)=x\cdot\nabla u(x)-u(x) \,.$$ Now, we take the limit as $\delta\rightarrow 0$. Since $u\in W^{1,\infty}(\Omega)$ and $\mathcal{H}_{N-1}(\Gamma_\delta)$ goes to $0$ as $\delta \to 0$, we have that the last term in vanishes and we obtain $$\int_{\Omega} \nabla_\xi V(\nabla u)\cdot\nabla\varphi\, dx=-N\int_{\Omega} V(\nabla u)\, dx -\int_{\Gamma_0}\varphi\nabla_\xi V(\nabla u)\cdot\nu\, d\sigma +\int_{\Gamma_{0}}V(\nabla u)x\cdot\nu\, d\sigma \,, \\$$ i.e. (in terms of $f$) $$\int_\Omega \frac{f'(|\nabla u|)}{|\nabla u|}\nabla u\cdot\nabla\varphi\, dx=-N\int_{\Omega}f(|\nabla u|)\, dx - \int_{\Gamma_0}\varphi\frac{f'(|\nabla u|)}{|\nabla u|}\partial_{\nu}u\, d\sigma +\int_{\Gamma_0}f(|\nabla u|)x\cdot\nu\, d\sigma.$$ Since $u$ satisfies , we get $$\label{stato_sociale}
\int_\Omega \varphi\, dx=-N\int_{\Omega}f(|\nabla u|)\, dx - \int_{\Gamma_0}\varphi\frac{f'(|\nabla u|)}{|\nabla u|}\partial_{\nu}u\, d\sigma +\int_{\Gamma_0}f(|\nabla u|)x\cdot\nu\, d\sigma.$$ From and since $u=0$ on $\Gamma_0$ and $\partial_\nu u=0$ on $\Gamma_1$, we have $$\int_\Omega \varphi\, dx=-(N+1)\int_{\Omega}u\, dx$$ and $$\label{annamo}
\int_{\Gamma_0}\varphi\frac{f'(|\nabla u|)}{|\nabla u|}\partial_{\nu}u\, d\sigma=\int_{\Gamma_0}f'(|\nabla u|)|\nabla u|x\cdot\nu\, d\sigma \,,$$ where we used the expresion of the unit exterior normal on $\Gamma_0$ given by . From and we obtain $$-(N+1)\int_{\Omega}u\, dx+N\int_{\Omega}f(|\nabla u|)\, dx=-\int_{\Gamma_0}f'(|\nabla u|)|\nabla u|x\cdot\nu\, d\sigma+\int_{\Gamma_0}f(|\nabla u|)x\cdot\nu\, d\sigma.$$ which is , and the proof is complete.
We conclude this subsection by exploiting the boundary condition $\partial_\nu u=0$ on $\Gamma_1$. Before doing this, we need to recall some notation from differential geometry (see also [@ecker Appendix A]). We denote by $D$ the standard Levi-Civita connection. Recall that, given an $(N-1)$-dimensional smooth orientable submanifold $M$ of $\mathbb{R}^N$ we define the *tangential gradient* of a smooth function $f:M\rightarrow\mathbb{R}$ with respect to $M$ as $$\nabla^Tf(x)=\nabla f(x)-\nu\cdot\nabla f(x)\nu$$ for $x\in M$, where $\nabla f$ denotes the usual gradient of $f$ in $\mathbb{R}^N$ and $\nu$ is the outward unit normal at $x$ to $M$. Moreover, we recall that the *second fundamental form* of $M$ is the bilinear and symmetric form defined on $TM\times TM$ as $$\mathrm{II}(v,w)=D\nu(v) w \cdot \nu\, ;$$ a submanifold is called *convex* if the second fundamental form is non-negative definite.
\[lemma\_tangential\] Let $u$ be the solution to . Then $$\label{piove}
\nabla_\xi V(\nabla u)\cdot\nu=0 \quad \text{on} \quad \Gamma_1 \,,$$ and $$\label{grad_0}
\nabla (\nabla_\xi V(\nabla u)\cdot\nu)\cdot\nabla u=0 \quad \text{on} \quad \Gamma_1 \, .$$
Since $\partial_\nu u=0$ on $\Gamma_1$, we immediately find . By taking the tangential derivative in we get $$0=\nabla^T (\nabla_\xi V(\nabla u)\cdot\nu)
=\nabla (\nabla_\xi V(\nabla u)\cdot\nu)-\nu\cdot \nabla(\nabla_\xi V(\nabla u)\cdot\nu)\nu \quad \text{on} \quad \Gamma_1\, .$$ By taking the scalar product with $\nabla u$ we obtain $$0=\nabla (\nabla_\xi V(\nabla u)\cdot\nu)\cdot\nabla u-\nu\cdot \nabla(\nabla_\xi V(\nabla u)\cdot\nu)\partial_\nu u
\, ,$$ and since $u_\nu=0$ on $\Gamma_1$, we find .
Integral Identities for $S_2$
-----------------------------
In this Subsection we prove some integral inequalities involving $S_2(W)$ and the solution to problem .
\[formula\_1\] Let $\Omega \subset \mathbb{R}^N$ be a sector-like domain and assume that $f$ satisfies . Let $u\in W^{1,\infty}(\Omega)$ be a solution of such that holds. Then the following inequality $$\label{integral_inequality_1}
2\int_{\Omega}S_2(W)u \, dx\geq -\int_{\Omega}S^2_{ij}(W)V_{\xi_i}(\nabla u)u_j\, dx$$ holds. Moreover the equality sign holds in if and only if $\mathrm{II}(\nabla^T u,\nabla^T u)=0$ on $\Gamma_1$.
We split the proof in two steps.
*Step 1: the following identity* $$\label{Bianchini_Ciraolo}
2\int_{\Omega}S_2(W)\phi \, dx=-\int_{\Omega}S^2_{ij}(W)V_{\xi_i}(\nabla u)\phi_j\, dx$$ *holds for every* $\phi\in C^{1}_0(\Omega)$.
For $t>0$ we set $\Omega_t=\lbrace x\in\Omega \, : \, {\mathrm{dist}}(x,\partial\Omega)>t\rbrace$. Let $\phi\in C^1_0(\Omega)$ be a test function and let $\varepsilon_0>0$ be such that $\Omega_{\varepsilon_0}\subset\Omega$ and $supp(\phi)\subset\Omega_{\varepsilon_0}$. For $\varepsilon<\varepsilon_0$ sufficiently small, we set $$a^i(x)=V_{\xi_i}(\nabla u(x)) \quad \text{ for every } \, i=1,\dots,N, \, x\in\Omega.$$ From we have that $a^i \in W^{1,2}(\Omega)$, $i=1,\dots,N$. With this notation, the elements $w_{ij}=\partial_{j} V_{\xi_i}(\nabla u)$ of the matrix $W$ are given by $$w_{ij}=\partial_j a^i \,.$$ Let $\lbrace\rho_\varepsilon\rbrace$ be a family of mollifiers and define $a^i_\varepsilon=a^i\ast\rho_\varepsilon$. Let $W^\varepsilon=(w^{\varepsilon}_{ij})_{i,j=1,\dots,N}$ where $w^{\varepsilon}_{ij}=\partial_j a_\varepsilon^i$, and notice that $$a^i_\varepsilon\rightarrow a^i \quad \text{ in } W^{1,2}(\Omega_{\varepsilon_0}) \quad \text{ and } \quad W^\varepsilon\rightarrow W \quad \text{ in } L^{2}(\Omega_{\varepsilon_0})\, ,$$ as $\varepsilon \to 0$. Moreover $$\label{ug_traccia}
{\mathrm{Tr}}W^\varepsilon={\mathrm{Tr}}W=-1$$ for every $x\in\Omega_\varepsilon$.
Let $i,j=1,\dots,N$ be fixed. We have $$\begin{aligned}
w^{\varepsilon}_{ji}w^{\varepsilon}_{ij}&=\partial_j(a^i_\varepsilon\partial_ia^j_\varepsilon)-a^i_\varepsilon\partial_j\partial_i a^j_\varepsilon \\
&=\partial_j(a^i_\varepsilon\partial_i a^j_\varepsilon)- a^i_\varepsilon\partial_i\partial_j a^j_\varepsilon \\
&=\partial_j(a^i_\varepsilon\partial_i a^j_\varepsilon)- a^i_\varepsilon\partial_i w^{\varepsilon}_{jj} \, ,
\end{aligned}$$ for every $x\in\Omega_\varepsilon$, and by summing up over $j=1,\dots,N$, using (hence $\partial_i\sum_j w^\varepsilon_{jj}=0$), we obtain $$\begin{aligned}
\sum_{j} w^{\varepsilon}_{ji} w^{\varepsilon}_{ij}&=\sum_{j}\partial_j(a^i_\varepsilon\partial_i a^j_\varepsilon)\\
&=w^{\varepsilon}_{ii}{\mathrm{Tr}}W^\varepsilon-\sum_j \partial_j(S^2_{ij}(W^\varepsilon)a^i_\varepsilon), \quad x\in\Omega_\varepsilon.
\end{aligned}$$ By summing over $i=1,\dots,N$, from we have $$\label{gesso}
2S_2(W^\varepsilon)=\sum_{i,j}\partial_j(S^2_{ij}(W^\varepsilon) a^i_\varepsilon), \quad x\in\Omega_\varepsilon.$$ Since $$\int_{\Omega_{\varepsilon_0}}\partial_j(S^2_{ij}(W^\varepsilon) a^i_\varepsilon)\phi\, dx+\int_{\Omega_{\varepsilon_0}}S^2_{ij}(W^\varepsilon)a^i_ \varepsilon\phi_j\, dx =\int_{\partial\Omega_{\varepsilon_0}}S^2_{ij}(W^\varepsilon)a^i_\varepsilon\nu_j \phi \, d\sigma=0 \,,$$ from and by letting $\varepsilon$ to zero, we obtain .
*Step 2.* Let $\delta>0$ and consider a cut-off funtion $\eta^\delta\in C^\infty_c(\Omega)$ such that $\eta^\delta=1$ in $\Omega_\delta$ and $|\nabla\eta^\delta|\leq\frac{C}{\delta}$ in $\Omega\setminus\Omega_\delta$ for some constant $C$ not depending on $\delta$. By taking $\phi(x)=u(x)\eta^\delta(x)$ for $x\in\Omega$ in we obtain $$\label{anderson}
2\int_{\Omega}S_2(W)u\eta^\delta \, dx=-\int_{\Omega}S^2_{ij}(W)V_{\xi_i}(\nabla u)u_j\eta^\delta\, dx -\int_{\Omega}S^2_{ij}(W)V_{\xi_i}(\nabla u)u\eta^{\delta}_j\, dx \, .$$ From we have that $W \in L^2 (\Omega)$ and the dominated convergence theorem yields $$\label{anderson_bis}
2\int_{\Omega}S_2(W)u\eta^\delta \, dx\rightarrow 2\int_{\Omega}S_2(W)u \, dx$$ as $\delta \to 0$. Analogously, $$\label{anderson_chi}
\int_{\Omega}S^2_{ij}(W)V_{\xi_i}(\nabla u) u_j\eta^\delta \, dx \rightarrow \int_{\Omega}S^2_{ij}(W)V_{\xi_i}(\nabla u) u_j \, dx$$ as $\delta \to 0$.
Now, we consider the last term in . We write $\Omega$ in the following way: $$\label{divisione_omega}
\Omega=A_0^\delta \cup A_1^\delta\,,$$ where $$A_0^\delta=\lbrace x\in\Omega \, : \, {\mathrm{dist}}(x,\Gamma_0)\leq\delta\rbrace \quad \text{ and } \quad A_1^\delta=\Omega \setminus A_0^\delta.$$ Since $u=0$ on $\Gamma_0$, we get that $$u(x)\leq ||u||_{W^{1,\infty}(\Omega)}\, {\mathrm{dist}}(x,\Gamma_0)\leq ||u||_{W^{1,\infty}(\Omega)}\, \delta$$ for every $x\in A_0^\delta$ and we obtain $$\left|\int_{A_0^\delta}S^2_{ij}(W)V_{\xi_i}(\nabla u) u\eta^{\delta}_j\, dx\right|\leq C_1 |A_0^\delta| \,,$$ where $C_1$ is a constant depending on $||u||_{W^{1,\infty}(\Omega)}$ and $\|W\|_{L^2(\Omega)}$, which implies that $$\label{anderson_tris}
\lim_{\delta \to 0} \int_{A_0^\delta}S^2_{ij}(W)V_{\xi_i}(\nabla u) u\eta^{\delta}_j\, dx = 0\, .$$
Now we show that $$\label{adesso}
\lim_{\delta \to 0} \int_{A_1^\delta}S^2_{ij}(W(x))V_{\xi_i}(\nabla u(x)) u(x)\eta^{\delta}_j(x)\, dx\geq 0\, .$$ By choosing $\delta$ small enough, a point $x\in A_1^\delta$ can be written in the following way: $x=\bar{x}+t\nu(\bar{x})$ where $\bar{x}=\bar{x}(x)\in\Gamma_1$ and $t=|x-\bar x|$ with $0<t<\delta$. Moreover, by using a standard approximation argument, $\eta^\delta$ can be chosen in such a way that $\eta^\delta(x)=\frac{1}{\delta}{\mathrm{dist}}(x,\Gamma_1)$ for any $x\in A_1^\delta$, so that $$\label{gradiente_eta_delta}
\nabla\eta^\delta(x)=-\frac{1}{\delta}\nu(\bar x) \,,$$ for every $x\in A_1^\delta \setminus \Omega_\delta$. For simplicity of notation we set $F=(F_1, \ldots, F_N)$, where $$\label{Fj}
F_j(x)=u(x) S^2_{ij}(W(x))V_{\xi_i}(\nabla u(x))$$ for $j=1,\ldots,N$, and hence $$\label{presidenza}
\int_{A_1^\delta}S^2_{ij}(W)V_{\xi_i}(\nabla u) u\eta^{\delta}_j\, dx= \int_{A_1^\delta} F(x)\cdot\nabla\eta^\delta(x)\, dx\, .$$ Since $\nabla \eta^\delta = 0 $ in $\Omega_\delta$ and $\nabla\eta^\delta(x)=-\frac{1}{\delta}\nu(\bar x)$, for every $x\in A_1^\delta \setminus \Omega_\delta$, we have $$\begin{aligned}
\int_{A_1^\delta}F(x)\cdot\nabla\eta^\delta(x)\, dx &=-\dfrac{1}{\delta}\int_{A_1^\delta \setminus \Omega_\delta} F(x)\cdot\nu(\bar x)\, dx \\
&=-\dfrac{1}{\delta}\int_{0}^{\delta}\, dt \int_{\lbrace x\in A_1^\delta \, : \, {\mathrm{dist}}(x,\Gamma_1)=t\rbrace}F(x)\cdot\nu(\bar{x})\, d\sigma \\
\end{aligned}$$ where we used coarea formula. Since we are in a *small* $\delta$-tubular neighborhood of (part of) $\Gamma_1$, we can parametrize $A_1^\delta \setminus \Omega_\delta$ over (part of) $\Gamma_1$ as from [@GT Formula 14.98] we obtain that $$\label{co-area}
\int_{A_1^\delta}F(x)\cdot\nabla\eta^\delta(x)\, dx =-\dfrac{1}{\delta}\int_{0}^{\delta}\, dt \int_{\Gamma_1}F(\bar{x}+t\nu(\bar{x}))\cdot\nu(\bar{x})|\det(Dg)|\, d\sigma \,.$$ We notice that, by using this notation, proving is equivalent to prove $$\label{aimstep2}
\lim_{\delta \to 0} \int_{A_1^\delta}F(x)\cdot\nabla\eta^\delta(x)\, dx \geq 0 \,,$$ for $\delta>0$ sufficiently small.
From , and the definition of $S^2_{ij}$ , we have $$\begin{aligned}
F(x) \cdot \nu (\bar x) & = -\delta_{ij}V_{\xi_i}(\nabla u(x)) u(x)\nu_j(\bar{x}) - w_{ji}(x) V_{\xi_i}(\nabla u(x)) u(x)\nu_j(\bar{x}) \\
& = -\left\lbrace\delta_{ij}V_{\xi_i}(\nabla u(x)) u(x)\nu_j(\bar{x}) + u(x)\frac{f'(|\nabla u(x)|)}{|\nabla u(x)|}w_{ji}(x) u_i(x)\nu_j(\bar{x})\right\rbrace
\end{aligned}$$ for almost every $x=\bar{x}+t\nu(\bar{x})\in A_1^\delta \setminus \Omega_\delta$, with $0 \leq t \leq \delta$. Since $$w_{ij}\nu_i u_j=\partial_j(V_{\xi_i}(\nabla u)\nu_i)u_j-V_{\xi_i}(\nabla u)\partial_j \nu_i u_j\, ,$$ we have $$\label{air_cond}
\begin{aligned}
F(x) & \cdot \nu (\bar x) = -u(x) \nabla_{\xi} V(\nabla u(x))\cdot\nu(\bar x) \\
& -u(x)\frac{f'(|\nabla u(x)|)}{|\nabla u(x)|}\left\lbrace \nabla (\nabla_\xi V(\nabla u(x))\cdot\nu(\bar x))\cdot\nabla u(x)-\frac{f'(|\nabla u(x)|)}{|\nabla u(x)|} \partial_j\nu_i(\bar x)u_j(x)u_i(x) \right\rbrace
\end{aligned}$$ for almost every $x=\bar{x}+t\nu(\bar{x})\in A_1^\delta \setminus \Omega_\delta$, with $0 \leq t \leq \delta$. Let $$\Gamma_1^{\delta,t} = \lbrace x\in A_1^\delta \, : \, {\mathrm{dist}}(x,\Gamma_1)=t\rbrace \,.$$ We notice that if $x\in \Gamma_1^{\delta,t}$ then $\nu(\bar x)=\nu^t(x)$ where $\nu^t(x)$ is the outward normal to $\Gamma_1^{\delta,t}$ at $x$. Hence $$\label{moduli}
\partial_j\nu_i(\bar x)u_j(x)u_i(x)=\mathrm{II}_x^{\delta,t}(\nabla^T u(x),\nabla^T u(x))$$ where $\mathrm{II}_x^{\delta,t}$ is the second fundamental form of $\Gamma_{1}^{\delta,t}$ at $x$. Since $\Sigma$ is a convex cone then the second fundamental form of $\Gamma_1 \setminus \{O\}$ is non-negative definite. This implies that the second fundamental form of $\Gamma_1^{\delta,t}$ is non-negative definite for $t$ sufficiently small [@GT Appendix 14.6] and hence $$\label{sole}
\partial_j\nu_i(\bar x)u_j(x)u_i(x)\geq 0 \,.$$ From and we obtain $$\label{lunga_lunga}
F(x) \cdot \nu (\bar x) \geq -u(x) \nabla_{\xi} V(\nabla u(x))\cdot\nu(\bar x) -u(x)\frac{f'(|\nabla u(x)|)}{|\nabla u(x)|} \nabla (\nabla_\xi V(\nabla u(x))\cdot\nu(\bar x))\cdot\nabla u(x)$$ for almost every $x=\bar{x}+t\nu(\bar{x})\in A_1^\delta \setminus \Omega_\delta$, with $0 \leq t \leq \delta$. We use in the right-hand side of and, by taking the limit as $\delta \to 0$, we obtain $$\lim_{\delta \to 0} \int_{A_1^\delta}F(x)\cdot\nabla\eta^\delta(x)\, dx \geq -\int_{\Gamma_1}u \left( \nabla_{\xi} V(\nabla u)\cdot\nu + \frac{f'(|\nabla u|)}{|\nabla u|}\nabla (\nabla_\xi V(\nabla u)\cdot\nu)\cdot\nabla u \right) d\sigma\,.$$ From and we find , and hence . From , , , , and , we obtain .
Proof of Theorem \[teo 1 cono\] {#section_proofthm1}
===============================
We divide the proof in two steps. We first show that $$\label{Uguaglianza}
W=-\dfrac{1}{N}Id \quad \text{ a.e. in } \Omega.$$ and $$\label{step1_II}
\mathrm{II}(\nabla^T u, \nabla^T u) = 0 \quad \text{ on } \Gamma_1\,,$$ and then we exploit in order to prove that $u$ is indeed radial.
*Step 1.* Let $g$ be the Fenchel conjugate of $f$ (in our case $g'=(f')^{-1}$), using we get that $$\begin{aligned}
{\mathrm{div}}\left(g(|\nabla_{\xi}V(\nabla u)|)\nabla_{\xi}V(\nabla u)\right)&= g'(|\nabla_{\xi}V(\nabla u)|)\nabla|\nabla_{\xi}V(\nabla u)|V_{\xi_j}(\nabla u) + g(|\nabla_{\xi}V(\nabla u)|){\mathrm{Tr}}(W)\\
&= g'(f'(|\nabla u|))\dfrac{V_{\xi_i}(\nabla u)}{|\nabla_{\xi}V(\nabla u)|}\partial_j(V_{\xi_i}(\nabla u))V_{\xi_j}(\nabla u) + g(f'(|\nabla u|)){\mathrm{Tr}}(W) \,,
\end{aligned}$$ a.e. in $\Omega$, where we used . Since $\partial_j V_{\xi_i}(\nabla u) = w_{ij}$ and $g'=(f')^{-1}$, we obtain $${\mathrm{div}}\left(g(|\nabla_{\xi}V(\nabla u)|)\nabla_{\xi}V(\nabla u)\right) = u_iw_{ij}V_{\xi_j}(\nabla u)+g(f'(|\nabla u|)){\mathrm{Tr}}(W)$$ a.e. in $\Omega$, and using again we find $${\mathrm{div}}\left(g(|\nabla_{\xi}V(\nabla u)|)\nabla_{\xi}V(\nabla u)\right) =\dfrac{f'(|\nabla u|)}{|\nabla u|}u_i w_{ij}u_j+g(f'(|\nabla u|)){\mathrm{Tr}}(W)$$ a.e. in $\Omega$. Since $$\label{gfprimo}
g(f'(t))=tf'(t)-f(t)$$ and ${\mathrm{Tr}}(W)=-1$, we obtain $$\label{nuova}
{\mathrm{div}}\left(g(|\nabla_{\xi}V(\nabla u)|)\nabla_{\xi}V(\nabla u)\right)= \dfrac{f'(|\nabla u|)}{|\nabla u|}u_i w_{ij}u_j+f(|\nabla u|) - |\nabla u|f'(|\nabla u|)$$ a.e. in $\Omega$.
Since , and yield $$-S^2_{ij}(W)V_{\xi_i}(\nabla u)u_j=\dfrac{f'(|\nabla u|)}{|\nabla u|} w_{ji}u_iu_j+ f'(|\nabla u|)|\nabla u|\, ,$$ a.e. in $\Omega$, from we obtain $$\label{nuovissima}
-S^2_{ij}(W)V_{\xi_i}(\nabla u)u_j= {\mathrm{div}}\left(g(|\nabla_{\xi}V(\nabla u)|)\nabla_{\xi}V(\nabla u)\right) +2 f'(|\nabla u|)|\nabla u| - f(|\nabla u|) \,,$$ a.e. in $\Omega$.
From Lemma \[formula\_1\] and , we obtain $$\begin{aligned}
2\int_{\Omega}S_2(W)u\, dx\geq&- \int_{\Omega}S^2_{ij}(W)V_{\xi_i}(\nabla u)u_j\, dx \\
=&\int_{\partial\Omega}g(|\nabla_{\xi}V(\nabla u)|)\nabla_{\xi}V(\nabla u)\cdot\nu\, d\sigma+\int_{\Omega} \left[2f'(|\nabla u|)|\nabla u| - f(|\nabla u|) \right]\, dx \,.
\end{aligned}$$ From and we find $$2\int_{\Omega}S_2(W)u\, dx\geq \int_{\Gamma_0}g(|\nabla_{\xi}V(\nabla u)|)\dfrac{f'(|\nabla u|)}{|\nabla u|}\partial_\nu u\, d\sigma+\int_{\Omega} \left[2f'(|\nabla u|)|\nabla u| - f(|\nabla u|) \right]\, dx \,.$$ From and we have $$2\int_{\Omega}S_2(W)u\, dx\geq -g(f'(c))f'(c)|\Gamma_0|+\int_{\Omega} \left[2f'(|\nabla u|)|\nabla u| - f(|\nabla u|) \right]\, dx$$ and from we obtain $$\label{eq1}
2\int_{\Omega}S_2(W)u\, dx\geq-[cf'(c)-f(c)]f'(c)|\Gamma_0|+\int_{\Omega} \left[2f'(|\nabla u|)|\nabla u| - f(|\nabla u|) \right]\, dx \,.$$ From the Pohozaev identity and we get $$(N+1)\int_\Omega u\, dx-N\int_\Omega f(|\nabla u|)\, dx=(f'(c)c-f(c))N|\Omega|\, ;$$ which we use in to obtain $$\label{eq1_bis}
2\int_{\Omega}S_2(W)u\, dx\geq-\dfrac{f'(c)|\Gamma_0|}{N|\Omega|}\int_\Omega \left[(N+1) u-Nf(|\nabla u|) \right] dx+\int_{\Omega} \left[2 f'(|\nabla u|)|\nabla u| - f(|\nabla u|) \right]\, dx \,.$$ We notice that from we have $$|\Omega|=f'(c)|\Gamma_0|,$$ and from we obtain $$\label{eq1_1}
2\int_{\Omega}S_2(W)u\, dx\geq- \dfrac{N+1}{N} \int_\Omega u \, dx+2\int_{\Omega}f'(|\nabla u|)|\nabla u|\, dx\,.$$ By using $u$ as a test function in we have that $$\int_\Omega u \, dx = \int_{\Omega}f'(|\nabla u|)|\nabla u| \, dx\,,$$ and from we find $$\label{Huisken}
2\int_{\Omega}S_2(W)u\, dx\geq\dfrac{N-1}{N}\int_\Omega u\, dx \,.$$ From and using the fact that ${\mathrm{Tr}}(W)=L_f u=-1$, we get that also the reverse inequality $$\label{legame tra u e S_2}
\dfrac{N-1}{N}\int_\Omega u\, dx\geq\int_\Omega 2S_2(W)u\, dx$$ holds. From and , we conclude that the equality sign must hold in and . From Lemma \[lemma matrici generiche\] we have that $$W=\frac{{\mathrm{Tr}}(W)}{N} Id$$ a.e. in $\Omega$, and since ${\mathrm{Tr}}(W)=-1$ we obtain . Moreover, Lemma \[formula\_1\] yields .
*Step 2: $u$ is a radial function.* From we have that $$-\dfrac{1}{N}\delta_{ij}=\partial_{j} V_{\xi_i}(\nabla u(x)) \, ,$$ for every $i, j=1,\dots, N$, which implies that there exists $x_0\in\mathbb{R}^N$ such that $$\nabla_{\xi} V(\nabla u(x))=-\dfrac{1}{N}(x-x_0),$$ i.e. according to $$\dfrac{f'(|\nabla u(x)|)}{|\nabla u(x)|}\nabla u(x)=-\dfrac{1}{N}(x-x_0) \,.$$ Hence $$\nabla u(x)=- g'\left(\dfrac{|x-x_0|}{N}\right) \frac{x-x_0}{|x-x_0|} \quad \text{ in } \Omega \,.$$ Since $u=0$ on $\Gamma_0$, we obtain and in particular $u$ is radial with respect to $x_0$. Moreover, from we find that $x_0$ must be the origin or, if $\partial \Sigma$ contains flat regions, a point on $\partial \Sigma$.
Cones in space forms: proof of Theorem \[teo 2 cono\] {#section_spaceforms}
=====================================================
The goal of this section is to give an easily readable proof of Theorem \[teo 2 cono\]. More precisely we assume more regularity on the solution than the one actually assumed in Theorem \[teo 2 cono\] in order to give a coincise and clear idea of the proof in this setting, and we omit the technical details which are, in fact, needed. A rigorous treatment of the argument described below can be done by adapting the (technical) details in Section \[section\_proofthm1\] and in [@Pacella-Tralli].
Before starting the proof we declare some notations we use in the statement of Theorem \[teo 2 cono\] and we are going to adopt in the following. Given a $N$-dimensional Riemannian manifold $(M,g)$, we denote by $D$ the Levi-Civita connection of $g$. Moreover given a $C^2$-map $u:M\rightarrow\mathbb{R}$, we denote by $\nabla u$ the gradient of $u$, i.e. the dual field of the differential of $u$ with respect to $g$, and by $\nabla^2u=Ddu$ the Hessian of $u$. We denote by $\Delta$ the Laplace-Betrami operator induced by $g$; $\Delta u$ can be defined as the trace of $\nabla^2u$ with respect to $g$. Given a vector field $X$ on an oriented Riemannian manifold $(M, g)$, we denote by ${\mathrm{div}}X$ the divergence of $X$ with respect to $g$. If $\lbrace e_k\rbrace$ is a local orthonormal frame on $(M, g)$, then $${\mathrm{div}}X=\sum_{k=1}^N g(D_{e_k}X,e_k)\, ;$$ notice that, if $u$ is a $C^1$-map and if $X$ is a $C^1$ vector field on $M$, we have the following *integration by parts* formula $$\int_\Omega g(\nabla u,\nu)\, dx=-\int_\Omega u{\mathrm{div}}X\, dx + \int_{\partial\Omega}ug(X,\nu)\, d\sigma\, ,$$ where $\nu$ is the outward normal to $\partial\Omega$ and $\Omega$ is a bounded domain which is regular enough. Here and in the following, $dx$ and $d\sigma$ denote the volume form of $g$ and the induced $(N-1)$-dimensional Hausdorff measure, respectively.
We divide the proof in four steps.
*Step 1: the $P$-function.* Let $u$ be the solution to problem and, as in [@CV1], we consider the $P$-function defined by $$P(u)=|\nabla u|^2+\dfrac{2}{N}u+Ku^2 \,.$$ Following [@CV1 Lemma 2.1], $P(u)$ is a subharmonic function and, since $u=0$ on $\Gamma_0$ and from , we have that $P(u)=c^2$ on $\Gamma_0$. Moreover, $$\label{posto29}
\nabla P(u)=2{\nabla^2}u\nabla u+\dfrac{2}{n}\nabla u+2Ku\nabla u \,.$$ From the convexity assumption of the cone $\Sigma$, we have that $$\label{posto29E}
g({\nabla^2}u \nabla u,\nu) \leq 0 \,.$$ Indeed, since $u_\nu = 0$ on $\Gamma_1$ and by arguing as done for , we obtain that $$0= g( \nabla u_\nu , \nabla u)= g({\nabla^2}u \nabla u,\nu) + \mathrm{II}(\nabla u , \nabla u) \geq g({\nabla^2}u \nabla u,\nu) \quad \text{on} \quad \Gamma_1 \, ,$$ which is . From and we obtain $$\partial_{\nu}P(u)=2 g({\nabla^2}u \nabla u,\nu) + \dfrac{2}{n}\partial_\nu u+2Ku\partial_\nu u\leq 0 \quad \text{ in } \, \Gamma_1\setminus\lbrace O\rbrace \,.$$ Hence, the function $P$ satisfies: $$\begin{cases}
\Delta P(u)\geq 0 &\mbox{in } \Omega, \\ P(u)=c^2 &\mbox{on } \Gamma_0 \\ \partial_{\nu}P(u)\leq 0 \, &\mbox{on } \Gamma_1\setminus\lbrace O\rbrace \,.
\end{cases}$$ Moreover, again from [@CV1 Lemma 2.1], we have that $$\label{uguaglianza}
\Delta P(u) = 0 \quad \text{ if and only if } \quad {\nabla^2}u = \left(-\frac{1}{N}-Ku\right) g \,.$$
*Step 2: we have* $$P(u)\leq c^2 \quad \textit{ in } \, \Omega.$$ Indeed, we multiply $\Delta P(u) \geq 0$ by $(P(u)-c^2)^+$ and by integrating by parts we obtain $$0 \geq \int_{\Omega \cap \{P>c^2\}} |\nabla P|^2\, dx - \int_{\partial \Omega} (P(u)-c^2)^+ \partial_\nu P\, d\sigma \,.$$ Since $P(u)=c^2$ on $\Gamma_0$ and $\partial_{\nu}P(u)\leq 0 $ on $\Gamma_1$ we obtain that $$0 \geq \int_{\Omega \cap \{P>c^2\}} |\nabla P|^2\, dx \geq 0$$ and hence $P(u) \leq c^2$.
*Step 3: $P(u)=c^2$*. By contradiction, we assume that $P(u)<c^2$ in $\Omega$. Since $\dot h>0$, we have $$c^2 \int_\Omega \dot h\, dx > \int_\Omega \dot h |\nabla u|^2\, dx + \frac{2}{n} \int_\Omega \dot h u\, dx + K \int_\Omega \dot h u^2\, dx \, .$$ Since $${\mathrm{div}}(\dot h u \nabla u)= \dot h |\nabla u|^2 + \dot h u \Delta u + \ddot h u \partial_r u$$ and $$\ddot h = -K h \,,$$ and from $u=0$ on $\Gamma_0$ and $\partial_{\nu}u=0$ on $\Gamma_1\setminus\{ O\}$, we have that $$\begin{split}
c^2 \int_\Omega \dot h\, dx & > - \int_\Omega \dot h u \Delta u\, dx - \int_\Omega \ddot h u \partial_r u\, dx + \frac{2}{n} \int_\Omega \dot h u\, dx + K \int_\Omega \dot h u^2\, dx \\ & = (n+1)K \int_\Omega \dot h u^2\, dx + \left(1 + \frac 2n \right) \int_\Omega \dot h u\, dx + K \int_\Omega h u \partial_r u\, dx \,.
\end{split}$$ From ${\mathrm{div}}(h\partial_r) = n \dot h$ we have $${\mathrm{div}}(u^2 h \partial_r) = n \dot h u^2 + 2 h u \partial_r u \,,$$ and from $u=0$ on $\Gamma_0$ and $\partial_{\nu}u=0$ on $\Gamma_1\setminus\{ O\}$ we obtain $$\label{minore_stretto}
c^2 \int_\Omega \dot h\, dx > \left(1 + \frac 2n \right)\left( \int_\Omega \dot h u\, dx - K \int_\Omega h u \partial_r u \, dx\right) \,.$$ Now we show that if $u$ is a solution of satisfying then the equality sign holds in . Indeed, let $X=h\partial_r$ be the radial vector field and, by integrating formula (2.8) in [@CV1], we get $$- \frac{c^2}{n} \int_{\partial \Omega} g(X,\nu)\, d\sigma
+ \frac{n+2}{n} \int_\Omega \dot h u\, dx
-( n-2) K \int_\Omega \dot h u^2\, dx + \left(\frac{2}{n}-3\right) K \int_\Omega u g(X,\nabla u)\, dx = 0\,.$$ Since ${\mathrm{div}}X=n \dot h$ we obtain $$c^2 \int_\Omega \dot h\, dx = \frac{n+2}{n} \int_\Omega \dot h u \, dx
-( n-2) K \int_\Omega \dot h u^2\, dx + \left(\frac{2}{n}-3\right) K \int_\Omega u g(X,\nabla u) \, dx \,,$$ i.e. $$c^2 \int_\Omega \dot h \, dx = \left(1 + \frac 2n \right)\left( \int_\Omega \dot h u\, dx - K \int_\Omega h u \partial_r u\, dx \right) \,,$$ where we used that $u=0$ on $\Gamma_0$, $\partial_{\nu}u=0$ on $\Gamma_1\setminus\{ O\}$ and $g(X,\nu)=0$ on $\Gamma_1$. From we find a contradiction and hence $P(u)\equiv c^2$ in $\Omega$.
*Step 4: $u$ is radial*. Since $P(u)$ is constant, then $\Delta P(u) = 0$ and from we find that $u$ satisfies the following Obata-type problem $$\label{luigifava}
\begin{cases}
{\nabla^2}u = (-\frac{1}{N}-Ku) g & \text{in } \Omega \,,\\
u=0 & \text{on } \Gamma_0 \,, \\
\partial_{\nu}u=0 & \text{on } \Gamma_1\setminus\{O\} \,.
\end{cases}$$ We notice that the maximum and the minimum of $u$ can not be both achieved on $\Gamma_0$ since otherwise we would have that $u\equiv 0$. Hence, at least one between the maximum and the minimum of $u$ is achieved at a point $p\in\Omega\cup\Gamma_1$. Let $\gamma:I\rightarrow M$ be a unit speed maximal geodesic satisfying $\gamma(0)=p$ and let $f(s)=u(\gamma(s))$. From the first equation of it follows $$f''(s)=-\dfrac{1}{N}-Kf(s) \,.$$ Moreover, the definition of $f$ and the fact that $\nabla u(p)=0$ yield $$f'(0)=0 \quad \text{ and } \quad f(0)=u(p),$$ and therefore $$f(s)=\left(u(p)-\dfrac{1}{N}\right)H(s)-\dfrac{1}{N}.$$ This implies that $u$ has the same expression along any geodesic strating from $p$, and hence $u$ depends only on the distance from $p$. This means that $\Omega=\Sigma\cap B_{R}$ where $B_R$ is a geodesic ball and $u$ depends only on the distance from the center of $B_R$.
[**Acknowledgement.**]{} The authors wish to thank Luigi Vezzoni for suggesting useful remarks regarding Section \[section\_spaceforms\]. The authors have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMPA) of the “Istituto Nazionale di Alta Matematica” (INdAM, Italy).
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[^1]:
|
---
abstract: 'Ram pressure stripping may be the dominant mechanisms driving the evolution of galaxy colors in groups and clusters. In this paper, an analytic model of ram pressure stripping is confronted with observations of galaxy colors and star formation rates in groups using a group catalog drawn from the Sloan Digital Sky Survey. An observed increase in the fraction of galaxies residing on the red sequence, the red fraction, with both increasing group mass, $M_{gr}$, and decreasing satellite luminosity, $L_{sat}$, is predicted by the model. The size of the differences in the red fraction can be understood in terms of the effect of the scatter in satellite and cluster morphologies and satellite orbits on the relationship between $M_{gr}$ and $L_{sat}$ and the stripped gas fraction. Observations of the group galaxies’ H$\delta$ and 4000Å break spectral measures and a comparison of the distribution of $SFR/M_{\ast}$ for star forming galaxies in the groups and in isolation both indicate that the color differences observed in the groups are the result of slowly declining SFRs, as expected if the color change is driven by stripping of the outer H <span style="font-variant:small-caps;">i</span> disk.'
author:
- 'J. A. Hester'
title: 'Ram Pressure Stripping in Groups: Comparing Theory and Observations'
---
INTRODUCTION {#intro}
============
Galaxies in clusters have earlier type morphologies, redder colors, and lower star formation rates (SFRs) than galaxies in the field of similar luminosity [@Gomez; @et; @al; @Goto; @et; @al; @Dressler]. Galaxies in groups and clusters formed both earlier and in denser environments than their field counterparts. Cluster galaxies may therefore naturally appear older than field galaxies. In addition, group and cluster galaxies are more likely to have undergone major mergers [@GKK] and are susceptible to harassment [@MLK] and interactions with the ICM [@Gunn; @Gott]. These interactions with the environment work to transform blue, late-type galaxies into red, early-type galaxies.
Several recent observational studies have explored the relationship between galaxy properties and environment. @Blanton [@et; @al.; @05b] and @Hogg [@et; @al.; @03] study the dependence of environment on galaxy properties and find that bright, red, and concentrated galaxies prefer to reside in environments with higher densities. Large galaxy surveys have revealed strong bimodalities in both stellar properties; color, SFR, 4000Å break; and structural properties; concentration, central surface brightness, $\mu_0$, [@Baldry; @et; @al.; @04; @Blanton; @et; @al.; @03a; @Kauffmann; @et; @al.; @03a; @Li; @et; @al.; @06; @Strateva; @et; @al]. As the local density increases the fraction of galaxies found in the red and high concentration modes increases [@Kauffmann; @et; @al.; @04], and galaxies in the red or high concentration modes are more clustered than galaxies in the blue or low concentration modes [@Li; @et; @al.; @06].
Galaxy properties are strongly related to each other as well as to the local environment. Bivariate distributions of $M_i$, $g-r$ color, central i-band surface brightness, $\mu_i$, and the i-band Sersic index, $n_i$, are shown in @Blanton [@et; @al.; @05b]. Galaxies tend to be either blue with low concentrations or red with high concentrations. In addition, the average 4000Å break, g-r color, concentration, and $\mu_0$ all monotonically increase with stellar mass [@Kauffmann; @et; @al.; @03b]. Understanding the effect of environment on galaxy evolution requires studying the effect of environment on both the distribution of galaxy properties and on the relationships between galaxy properties.
Three separate studies show that the relationship between color and environment is stronger than the relationship between concentration and environment. @Blanton [@et; @al.; @05b] demonstrate that the observed relationship between $n_i$ and environment is reproduced when galaxies are assigned new local densities based on their $g-r$ colors and i-band magnitudes, $M_i$. However, the observed relationship between color and environment cannot be reproduced by assigning local densities based on $n_i$ and $M_i$. @Kauffmann [@et; @al.; @03b] show that the relationship between concentration and stellar mass depends on the local environment only weakly and only at low stellar masses, $M_{\ast}<3\times10^{10}M_{\odot}$. In contrast the relationship between the 4000Å break or the SFR and the stellar mass shows a strong dependence on the local environment. @Li [@et; @al.; @06] study the correlation function, in bins of luminosity, for galaxies belonging to the different galaxy modes. Galaxies that are red, have large 4000Å breaks, high concentrations, or bright $\mu$ all have enhanced correlation functions on scales less than 5Mpc. However, for galaxies that are red and have large breaks, the enhancement is larger and extends to larger scales.
If environment causes galaxy colors to evolve, then the timescale on which evolution occurs is a clue to the processes responsible. @Kauffmann [@et; @al.; @04] and [@Balogh; @et; @al.; @04] both study the timescale over which galaxies’ colors evolve and come to different conclusions. @Kauffmann [@et; @al.; @04] compare Sloan Digital Sky Survey (SDSS) spectra to stellar evolution models and observe that the absence of galaxies with strong H$\delta$ for their 4000Å break strengths argues against the predominance of any process that truncates star formation on timescales less than a Gyr. In contrast, @Balogh [@et; @al.; @04] use observations of $H\alpha$ equivalent widths to argue against slowly evolving SFRs. While they observe an increase in the fraction of non-star forming galaxies in high density environments, they do not see a downward shift in the $W(H\alpha)$ distributions for either actively star forming galaxies or galaxies with blue colors. These observations indicate that galaxies pass quickly from the star forming to the non-star forming populations.
Ram pressure stripping of galaxies in groups and clusters is one process that can truncate star formation and redden galaxies. @P1 (Hereafter, Paper 1) presents a model of ram pressure stripping that predicts the fraction of the gas that is stripped from the outer H <span style="font-variant:small-caps;">i</span> disk and the hot galactic halo. This fraction depends primarily on the ratio of the galaxy’s mass to the mass of the group or cluster in which it orbits and secondly on several descriptive model parameters.
In this paper, the importance of ram pressure stripping is examined by combining the predictions of the model presented in Paper 1 with observations of galaxy groups in the SDSS. The group catalog used here was assembled by A. Berlind and is presented in @Berlind [@et; @al.]. It is reviewed briefly in § \[SDSS\]. The predictions of the model for the galaxy and group masses of interest are presented in § \[Model\]. In § \[stripping and sfr\], the timescale on which the SFR declines after a galaxy is stripped of the gas outside its inner star forming disk is discussed. In § \[r and d\], observations of the groups are compared to past observations and to the model of ram pressure stripping. The timescale of the decline in the SFR in the groups and the role ram pressure stripping may play in determining general trends in galaxy properties are also examined. § \[conclusion\] concludes.
THE GROUP CATALOG {#SDSS}
=================
The Sloan Digital Sky Survey (SDSS) [@York] is conducting an imaging and photometric survey of $\Pi$ $sr$ in the northern hemisphere as well as three thin slices in the southern hemisphere. Observing is done using a dedicated 2.5m telescope in Apache Point, NM. The telescope operates in drift scan mode and observes in five bandpasses [@Fukigita; @et; @al.]. Magnitude calibration is carried out using a network of standard stars [@Smith]. Three sets of spectroscopic targets are selected automatically, the main galaxy sample, the luminous red galaxy sample, and the quasar sample [@Straus; @et; @al.; @Eisenstein; @et; @al.; @Richards; @et; @al.]. Objects in the main galaxy sample have Petrosian magnitudes $r'<17.7$ and are classified as extended. Magnitudes are corrected for galactic extinction using the reddening maps of @Schlegel [@Finkbeiner; @Davis] prior to selection. Spectroscopy is taken using a pair of fiber-red spectrographs, and targets are assigned to fibers using an adaptive tiling algorithm [@Blanton; @et; @al.; @03d]. Data reduction for the SDSS is done using a series of automated pipelines [@Hogg; @et; @al.; @01; @Ivezic; @et; @al.; @Lupton; @et; @al.; @Pier; @et; @al.; @Smith].
The catalog used here is a volume limited sample drawn from the NYU Value Added Galaxy Catalog (NYU-VAGC) [@Blanton; @et; @al.; @05a]. The sample goes down to $M_{r}<-19.0$ and has a redshift range of 0.015-0.068. The group finding algorithm is described in detail in @Berlind [@et; @al.]. The group finder is a friends of friends algorithm with two linking lengths, one for projected distances and one in red shift space. Linking lengths are chosen such that the multiplicity function, richness, and projected size of recovered groups from simulations projected into redshift space are unbiased measures. The linking lengths are also chosen to maximize the number of groups recovered and minimize the number of spurious groups. The velocity dispersions of the recovered groups are systematically low because the group finder does not link the fastest moving group members to the group. However, as the fastest group members are also those most likely to be stripped, not including these galaxies biases against observing the signature of ram pressure stripping.
Virial masses and radii for the groups are determined by assuming a monotonic relationship between group mass and richness and matching a $\Lambda$CDM mass function to the measured group luminosity function [@Berlind; @et; @al.]. This assumes no scatter in the relationship between mass and richness. In this paper, galaxies are grouped by virial mass, and the assumption that there is no scatter should not be important. The absolute magnitudes used here are k-corrected and corrected for passive evolution to $z=0.1$ [@Blanton; @et; @al.; @03b; @Blanton; @et; @al.; @03c]. Membership in the red sequence is defined using the $M_r$ dependent color cut presented in @Li [@et; @al.; @06]; $g-r>-0.788-0.078M_r$. The i-band Sersic index, $n_i$, is defined as $I_i(r)=A\exp\left[-(r/r_0)^{(1/n_i)}\right]$ and its measurement is discussed in [@Blanton; @et; @al.; @03c]. The $H\delta$ and the 4000Å break are those measured by @Kauffmann [@et; @al.; @03a]. The measurements of the SFR are from @Brinchmann [@et; @al] and are normalized to the galaxies’ stellar masses. The affect of the changing physical size of the spectroscopic fiber’s 3 arcsec diameter is corrected for in $SFR/M_{\ast}$ by assuming that the relationship between color and $SFR/M_{\ast}$ is constant throughout the disk. This correction may bias against observing galaxies with blue colors and low $SFR/M_{\ast}$ and is important for galaxies at the low redshifts of this sample.[^1]
RAM PRESSURE STRIPPING
======================
@Gunn [@Gott] proposed ram pressure striping to explain the observed absence of gas rich galaxies in clusters. Galaxies in clusters feel an intracluster medium (ICM) wind that can overcome the gravitational attraction between the stellar and gas disks and strip the gas disk. They introduced the following condition to estimate when this occurs; $$\label{eqn rps}
\rho_{ICM}v_{sat}^{2} < 2 \pi G\sigma_{\ast}\sigma_{gas}.$$ The left-hand side is a ram pressure, where $\rho_{ICM}$ is the density of the ICM and $v_{sat}$ is the orbital speed of the satellite. The right-hand side is a gravitational restoring pressure where $\sigma_{\ast}$ and $\sigma_{gas}$ are the surface densities of the stellar and gas disks respectively. Using this condition they concluded that spirals should lose their gas disks when they pass through the centers of clusters.
Galaxies in nearby clusters are observed to be deficient in H <span style="font-variant:small-caps;">i</span> and to have truncated gas disks when compared to field galaxies of similar morphology and optical size [@Bravo-Alfaro; @et; @al; @2000; @Cayatte; @et; @al.; @Giovanelli; @and; @Haynes; @1983; @Solanes; @et; @al; @2001]. In addition, asymmetric extra-planar gas that appears to have been pushed out of the disk is observed in several Virgo spirals [@Kenney; @et; @al; @2004; @Kenney; @and; @Koopmann; @Kenney; @van; @Gorkom; @Vollmer; @2004]. These observations can be explained by ram pressure stripping, and observing an undisturbed stellar disks accompanied by extra-planar gas is a strong indication that the gas disk is interacting with the ICM. Ram pressure stripping has also been repeatedly observed in simulations of disk galaxies in an ICM wind [@Abadi; @Moore; @Bower; @Marcolini; @Brighenti; @D'Ercole; @Quilis; @Moore; @Bower; @Roediger; @and; @Hensler; @Schulz; @and; @Struck; @2001].
Paper 1 uses an analytical model of ram pressure stripping to explore the range of environments in which stripping can occur and the galaxy masses that are susceptible to stripping in each environment. It focuses on the H <span style="font-variant:small-caps;">i</span> disk beyond the stellar disk, and stripping of gas from within the stellar disk is not modeled. The gravitational restoring pressure is found by placing a flat H <span style="font-variant:small-caps;">i</span> disk in a gravitational potential consisting of a dark matter halo, a stellar disk, and a stellar bulge. The ram pressure is determined by letting the satellite orbit in an NFW potential through a $\beta$ profile ICM. The gas fraction that a galaxy is striped of is found to depend on the ratio of the satellite mass to the group mass, $M_{sat}/M_{gr}$, and the values of several descriptive parameters. Observations of stripped spirals in clusters compare well with the model’s predictions for large $M_{gr}$ and $M_{sat}$. Paper 1 concludes that many galaxies, particularly low-mass galaxies, can be stripped of a substantial fraction of their outer H <span style="font-variant:small-caps;">i</span> disks in a wide range of environments. The specific model predictions for the SDSS group catalog are given below and the possible effects of stripping the outer gas are discussed.
Model Predictions {#Model}
-----------------
In this section, the analytic model developed in Paper 1 is used to predict trends for galaxy colors in the SDSS groups.
The model predicts that the extent of stripping depends on the ratio $M_{sat}/M_{gr}$. However, two separate trends should in fact be observed, one with $M_{gr}$ at fixed $M_r$ and one with $M_r$ at fixed $M_{gr}$. This is mainly because the average ICM density of groups decreases as $M_{gr}$ decreases, and the average gas fraction in the disk increases as $M_{sat}$ decreases, which shifts the values of $M_{sat}/M_{gr}$ for which stripping occurs [@Sanderson; @Ponman; @Swaters; @et.; @al.]. In addition there is a strong color magnitude relation seen in both groups and the field. Therefore, it is necessary to in fact observe the change in the difference between the red fraction in the groups and in isolation with $M_r$ rather than the red fraction itself.
Observing a color or SFR trend due to ram pressure stripping requires a group catalog with both an appropriate range of group and galaxy masses and a sufficient number of galaxies and groups. A useful group catalog is one in which the effectiveness of stripping varies so that trends in $f_r$ with $M_{gr}$ and $M_r$ are expected. The groups in the catalog have masses between $10^{12}$ and $10^{15}M_{\odot}$ and the galaxies have $-19>M_r>-22$. In Tables \[tbl50\] - \[tbl90\] the model’s predictions for the satellite mass, $M_{sat}$, and absolute magnitude, $M_r$, at which a galaxy is stripped of $\approx
50,~80,~90\%$ of its H <span style="font-variant:small-caps;">i</span> disk are given for groups of masses $M_{gr} = 10^{13, 13.5, 14,14.5} M_{\odot}$. The model galaxy parameters are those from Paper 1 for a large spiral. The ICM parameters for $M_{gr}=10^{13}M_{\odot}$ are from the low-mass group model. The ICM parameters for $M_{gr}=10^{13.5, 14,14.5}
M_{\odot}$ are from the middle-mass cluster model. The values in Tables 1-3 assume a galaxy orbiting inclined to the ICM wind that is stripped of an intermediate gas fraction between those expected for a galaxy traveling face-on or edge-on to the wind. Total mass to light ratios in the r-band of 40, 65, and 90 are used to convert $M_{sat}$ to $M_r$. These are a factor 10-15 higher than the stellar mass to light ratios of 4-6 found for the R-band by @Maraston. In Tables 1-3 galaxies with $M_r<-19$ are highlighted. The predictions of the model are not exact, but they do demonstrate that the range of $M_{gr}$ and $M_r$ in the group catalog is appropriate for this project. The high-mass groups should contain many stripped galaxies, but the low-mass groups are capable of stripping few of the galaxies in the sample. Though the range in $M_{sat}$ is not as great as the range in $M_{gr}$, in higher mass groups, galaxies with $M_r=-21$ and $M_r=-19$ are stripped of different gas fractions.
[ccccc]{}
13.0 & 11.2 & **-19.8** & **-20.3** & **-19.4**\
13.5 & 11.7 & **-21.1** & **-21.6** & **-20.7**\
14.0 & 12.2 & **-22.3** & **-22.8** & **-22.0**\
14.5 & 13.0 & **-23.3** & **-23.8** & **-23.0**\
\[tbl50\]
[ccccc]{}
13.0 & 10.9 & -18.2 & -18.7 & -17.8\
13.5 & 11.4 & **-19.5** & **-20.0** & **-19.1**\
14.0 & 11.9 & **-20.7** & **-21.2** & **-20.4**\
14.5 & 12.6 & **-21.7** & **-22.2** & **-21.3**\
\[tbl80\]
[ccccc]{}
13.0 & 10.2 & -17.1 & -17.7 & -16.8\
13.5 & 10.7 & -18.4 & **-18.9** & -18.1\
14.0 & 11.2 & **-19.6** & **-20.2** & **-19.3**\
14.5 & 12.0 & **-20.6** & **-21.1** & **-20.3**\
\[tbl90\]
The model also predicts the extent of stripping of the hot galactic halo. The galactic halo is modeled by placing gas at the virial temperature of the galaxy’s dark matter halo in hydrostatic equilibrium with an NFW potential. It is assumed that the galactic halo is stripped down to the radius at which the ram pressure equals the thermal pressure. The galaxies in this sample can only maintain $\le0.5$% of their mass in the galactic halo without the gas cooling rapidly. Using this mass fraction, the model predicts that the galaxies in the group sample have more than 70% of this gas stripped in a $10^{13.5}M_{\odot}$ group and the entire halo is stripped in any group with a mass above $10^{14}M_{\odot}$. This is an upper limit on the hot halo gas these galaxies can retain in the absence of fresh in-falling gas. Therefore, it is assumed for the rest of the paper that the galactic halo is stripped for all galaxies in the sample, and that any observed trends in $f_r$ are due to ram pressure stripping of the H <span style="font-variant:small-caps;">i</span> disk.
The size of the galaxy catalog needed to detect stripping is determined by the scatter in the relationship between the fraction of the H <span style="font-variant:small-caps;">i</span> disk mass that is stripped, $M_{str}$, and the satellite galaxy mass, $M_{sat}$. The scatter can be thought of in terms of a distribution of effective masses, $M_{eff}$, at each physical satellite mass $M_{sat}$. The effective mass is defined such that all satellites with the same effective mass are stripped of the same fraction of their H <span style="font-variant:small-caps;">i</span> disk. This scatter is substantial. As shown in Paper 1, a $10^{11}M_{\odot}$ galaxy can be stripped as much as a $10^{10}M_{\odot}$ galaxy or as little as a $10^{12}M_{\odot}$ galaxy. The scatter is mainly due to differences in the galaxies’ orbits, stellar and H <span style="font-variant:small-caps;">i</span> disk scale lengths, and in the density and extent of the groups’ ICM. Paper 1 discusses the number of groups and galaxies a sample must contain in order to observe systematic changes in $M_{str}$ with $M_{sat}$ and $M_{gr}$. However, the relationship between $M_{str}$ and color is not simple. Therefore, because the connection between $M_{str}$ and residing on the red sequence is probably more direct, the red fraction will be studied rather than average galaxy colors. Galaxies that have been stripped of more than a critical gas fraction, $M_c$, should eventually join the red sequence. In this picture, the fraction of blue disk galaxies entering groups that join the red sequence is the fraction with $M_{str}>M_{c}$, and the fraction of galaxies with a given $M_{sat}$ that join the red sequence is intimately related to the distribution of $M_{eff}$ at $M_{sat}$. If the scatter in $M_{eff}$ is large compared to the difference in $M_{sat}$ or $M_{gr}$ between two sub-samples, the difference in the red fraction between the sub-samples should be small. For this sample, the scatter in $M_{eff}$ is similar range in $M_{sat}$, but smaller than the range in $M_{gr}$.
To get a rough estimate of the necessary catalog size, the number of galaxies needed to observe a difference in the red fraction of 0.1 is determined here. As errors in the colors are small in comparison, it is assumed that the scatter in the observed red fraction is given by $\sigma_{fr}^2 = f_{rt}(1-f_{rt})/N$, where $f_{rt}$ is the true red fraction. In the case that two $f_{rt}$ are close to 0.5, approximately 400 galaxies are needed in each sub-sample to observe a difference in the observed red fractions between them of 0.1 with 3$\sigma$ confidence. The scatter in $M_{eff}$ is due to both variations in orbital and satellite parameter values, which vary within a single group, and to variations in the ICM parameter values, which vary across groups. Therefore, this estimate is only valid for a sample that includes a large number of groups. The SDSS group catalog contains 2700 groups, 15400 group galaxies, and 22500 isolated galaxies. It is large enough to place 400 galaxies into reasonable bins in $M_r$ and $M_{gr}$ and has $\langle N_{gal}/N_{gr}\rangle\approx 5.5$. In other words, the catalog is large enough to detect differences in the red fraction as small as 0.1 with reasonable significance. This demonstrates why observations of colors and SFRs in the SDSS groups are being used to search for the signal of ram pressure stripping rather than the more obvious approach of using H <span style="font-variant:small-caps;">i</span> observations. Potentially observing a trend due to ram pressure stripping requires observing thousands of galaxies, preferably across a wide range of satellite and group masses, with uniform determinations of galaxy mass or luminosity and group mass.
The catalog of SDSS groups used for this project covers a range of $M_{gr}$ and $M_r$ in which stripping should occur and across which the degree of stripping should vary. The fraction of galaxies that belong to the red sequence will be focused on rather than average galaxy colors. This is mainly because residence in the red sequence and ram pressure stripping should be related in a more straight forward manner. It also simplifies comparisons between this project and others that use the galaxy color bimodality. The group catalog should be large enough to observe the signal of ram pressure stripping. The range in $M_{gr}$ is larger than that in $M_{sat}$. Therefore, the change in $f_r$ across the $M_{gr}$ range should be larger.
Gas Loss and the SFR {#stripping and sfr}
--------------------
The model of ram pressure stripping presented in Paper 1 addresses the stripping of the outer H <span style="font-variant:small-caps;">i</span> disk. Galaxies can be stripped of their inner gas disk. However, this probably only happens to the smallest galaxies or in the highest density environments. As is seen in Paper 1 and reviewed above, ram pressure stripping of the outer gas disk occurs for a wide range of satellite masses and environments. In this paper, the colors of galaxies, rather than H <span style="font-variant:small-caps;">i</span> observations, are used to confront the predictions of the disk stripping model. It is therefore assumed that stripping of the outer H <span style="font-variant:small-caps;">i</span> disk affects galaxies’ future SFRs. This section discusses the likelihood that the outer gas disk fuels future star formation and the time scale on which the SFR should decline if this gas is removed.
While disk galaxies are currently forming stars from the gas in their inner disks, they cannot continue to form stars at their current rate for longer than a few Gyr unless this gas is replenished. Outside of groups star formation can be sustained by in-fall of new gas into and the continual cooling of the hot galactic halo. However, in almost any group environment, most galaxies are stripped of their hot galactic halo gas and experience no new in-fall. If the loss of this gas were responsible for the majority of the reddening seen in groups, then all but the brightest galaxies would be uniformly reddened in any group with an ICM. This is seen for semi-analytic models of galaxy colors which include galactic halo stripping and ignore the extended H <span style="font-variant:small-caps;">i</span> disk [@Weinmann06]. In this scenario the SFR declines slowly as the galaxy consumes its inner gas disk.
Star formation may also be fed by the inflow of gas from the outer H <span style="font-variant:small-caps;">i</span> disk. Inflow is expected if there is any viscosity in the disk. In-falling gas should often join the outer H <span style="font-variant:small-caps;">i</span> disk, and feeding star formation with a viscous disk can be used to form exponential stellar profiles [@Bell; @Lin; @Pringle; @a; @Lin; @Pringle; @b]. Observationally, @Gavazzi find that late-type galaxies that are moderately deficient in H <span style="font-variant:small-caps;">i</span> as compared to galaxies with the same morphology and optical size also have low H$\alpha$ equivalent widths for their morphology. If star formation is fed by gas in the outer gas disk, then galaxies in groups that retain this gas will be able to continue forming stars while galaxies that do not will experience a decline in their SFRs. In this case galaxies’ SFRs and colors should depend the effectiveness of ram pressure stripping.
When the gas disk is stripped down to the radius at which star formation is occurring, the time scale over which star formation declines is determined by the rate at which star formation consumes the gas disk. The timescale for the decline in the star formation rate can be defined as $t_c\equiv-\Sigma_{SFR}/\dot{\Sigma}_{SFR}$, where $\Sigma_{SFR}$ is the surface density of star formation. If $\Sigma_{SFR}$ is related to the gas surface density, $\Sigma_H$, by a local Schmidt law, $\Sigma_{SFR}\propto\Sigma_{H}^{\alpha}$, then $t_c\approx\Sigma_{H,0}/(\alpha\Sigma_{SFR,0})$. @Kennicutt found that $\Sigma_{SFR}=2.5\times10^{-4}(\Sigma_H/1M_{\odot}pc^{-2})^{1.4}~M_{\odot}~yr^{-1}~kpc^{-2}$. For $\Sigma_H$ of order a few $M_{\odot}~pc^{-2}$, this relation gives $t_c$ of order a Gyr. If star formation is fed by inflow in the H <span style="font-variant:small-caps;">i</span> disk, galaxies in groups that retain a significant portion of their H <span style="font-variant:small-caps;">i</span> disk can continue to form stars. Galaxies that are stripped of their gas disks will experience a slow decline in their SFRs with a timescale of order a Gyr.
RESULTS AND DISCUSSION {#r and d}
======================
In this section the properties of the group galaxies are compared to past observations of the relationship between galaxy properties and environment and to the model of ram pressure stripping presented in Paper 1. The timescale of the star formation decline occurring in the groups is also examined. In addition, the role of ram pressure stripping in determining more general trends in galaxy properties is discussed.
Comparison with Other Observation and with the Model
----------------------------------------------------
In Figures \[red fraction\]a-c, the fraction of galaxies that belong to the red sequence, $f_r$, is plotted against the r-band absolute magnitude, $M_r$. In Figures \[red fraction\]d-e the fraction of galaxies with $n_i>2.5$, $f_{n2.5}$, is plotted. Figures \[red fraction\]a,d include all galaxies, \[red fraction\]b(c) includes only galaxies with $g-r<0.7$($g-r>0.7$), and \[red fraction\]e(f) includes galaxies with $n_i<2.5$($n_i>2.5$). Figure \[red fraction difference\] consists of the same set of plots, only for the difference between $f_r$ or $f_{n2.5}$ in the groups and in isolation, $f_{gr}-f_{is}$. In Figure \[break fraction\], similar plots are made for the fraction of galaxies with large 4000Å breaks. The four lines correspond to isolated galaxies and galaxies in groups of $log(M_{gr}(M_{\odot}))<~13$, 13 - 14, and 13.5 - 14.5. Isolated galaxies are defined as galaxies that are either in singlets or pairs. The $M_r$ bins for Figures 1-4 are chosen such that the number of galaxies in the middle $M_{gr}$ bin with blue colors is the same in each bin. This results in uniform errors across the $M_r$ bins. The plotted errors in Figures \[red fraction\] and \[break fraction\] are given by $\sigma^2=f(1-f)/N$. These errors are propagated to determine the errors for Figures \[red fraction difference\] and \[alpha\].
The observations presented in Figures \[red fraction\] and \[red fraction difference\] agree well with the previous observations discussed in § \[intro\]. Figures \[red fraction\]a,c and \[red fraction difference\]a,c show that both $f_r$ and $f_{n2.5}$ increase with $M_{gr}$. As seen most clearly in Figure \[red fraction difference\], the excess of galaxies with $n_i>2.5$ is smaller than the excess of red galaxies. Figure \[break fraction\] demonstrates that the 4000Å break behaves similarly to the $g-r$ color. These relations are the group-based counterpart of the observations of @Kauffmann [@et; @al.; @03b], @Kauffmann [@et; @al.; @04], @Blanton [@et; @al.; @05b], and @Hogg [@et; @al.; @03] correlating these properties with local over-density. The large $f_r$ and $f_{n2.5}$ in the groups will also increase the correlation function due to the higher clustering of groups compared to isolated galaxies. The increase in $f_r$ with $M_{gr}$ still appears Figures \[red fraction\]b and \[red fraction difference\]b, which include only galaxies with $n_i<2.5$. However, in Figures \[red fraction\]e and \[red fraction difference\]e, which include only blue galaxies, a difference in $f_{n2.5}$ between groups of different mass is not seen. From this it can be concluded that the majority of galaxies responsible for the excess of $n_i>2.5$ galaxies in the groups are red, but the galaxies responsible for the excess of red galaxies do not all have $n_i>2.5$. This asymmetry is reflected in the observation by @Blanton [@et; @al.; @05b] that color is a better predictor of environment that Sersic index.
The analytic model of ram pressure stripping presented in Paper 1 predicts that $f_r$ should increase as $M_{gr}$ increases and as $M_{sat}$ decreases. An increase in $f_r$ as $M_{gr}$ increases is clearly seen in Figures \[red fraction\] and \[red fraction difference\]. In Figure \[red fraction difference\]a, a decrease in $f_{r,gr}-f_{r,i}$ as $L_{sat}$ increases can also be seen. While the difference between consecutive $M_r$ bins is small, for the two higher $M_{gr}$ samples, a clear trend is seen across the range of $M_r$ and the difference in $f_{r,gr}-f_{r,i}$ between the brightest and dimmest bins is several times the errors. For galaxies with $n_i<2.5$, $f_{r,gr}-f_{r,i}$ increases with $M_{gr}$ and, for the highest $M_{gr}$ bin, with $M_r$. The trend in $f_{r,gr}-f_{r,i}$ with $M_r$ for all galaxies is shallower in the middle $M_{gr}$ bin and is not observed in Figure \[red fraction difference\]b. Possible reasons for this are given below and in § \[d2\].
The observed differences in $f_r$ in the $n_i<2.5$ sub-population may not reflect true differences. Both the measurement errors for $n_i$ and the intrinsic scatter in the two $n_i$ modes are substantial. Galaxies that scatter out of the $n_i>2.5$ populations into the $n_i<2.5$ population have a higher red fraction than those scattering in the opposite direction, which tends to make the measured $f_{r}(n_i<2.5)$, $f_{rm<}$, greater than the true value, $f_{rt<}$. A consequence of this is that a change in $f_{n2.5}$ can cause a change in $f_{rm<}$ while $f_{rt<}$ remains unchanged. In order to examine this effect and to determine the significance of the observed changes in $f_{r}(n_i<2.5)$ the following definitions are made: let $n$ be the percentage of the true $n_i>2.5$ galaxies that scatter into the observed $n_i<2.5$ population, $t$ be the percentage of the true $n_i<2.5$ galaxies that scatter in the opposite direction, and $f_{rt<(>)}$ be the true $f_r$ in the $n_i<(>)2.5$ population. At fixed $M_r$, $n$ and $t$ should not change with $M_{gr}$. With these definitions $$\label{eqn1}
f_{rm<}=\frac{N_{rm}}{N_{m}} =
\frac{f_{rt<}(1-f_{n2.5})(1-t)+f_{rt>}f_{n2.5}n}{(1-f_{n2.5})(1-t)+f_{n2.5}n},$$ and for small $\Delta f_{n2.5}$, $$\label{eqn2}
\Delta f_{rm<} = \Delta f_{n2.5}
\frac{(f_{rt>}-f_{rt<})}{(1-f_{n2.5})^2} \frac{n(1-t)}{(1-t+n
f_{n2.5}/(1-f_{n2.5})) ^2}.$$ A fortunate property of equation (\[eqn2\]) is that for $0<n<1$ and $0<t<1$, $n(1-t)/(1-t+n)^2<0.25$, which is a useful bound when $f_{n2.5}\sim0.5$. To make a conservative estimate of the size of $\Delta f_{rm<}$ that can be caused by the increase in $f_{n2.5}$ with $M_{gr}$ in the dimmest $M_{r}$ bin, let $f_{rt>}=1$, $f_{rt<}=0.4$, $f_{n2.5}=0.45$, $\Delta f_{n2.5} = 0.1$, and $n(1-t)/(1-t+n)^2=0.25$ . With these values, $\Delta
f_{rm<}\leq0.05$, which is significantly smaller than the observed change in $f_{r}(n_i<2.5)$ between the low and high $M_{gr}$ bins. Estimating the size of the induced change in $f_{rm<}$ with $M_r$ is less straight forward because $f_{n2.5}$, $n$, and $t$ all change. For the highest $M_{gr}$ bin, $f_{n2.5}$ increases from $\sim0.5$ to $\sim0.8$ across the $M_r$ range. Again assuming that $f_{rt>}=1$, the induced change in $f_{rm<}$ is $$\Delta f_{rm<} = \frac{f_{rt<}(1-t_2)+4n_2}{1-t_2+4n_2}
-\frac{f_{rt<}(1-t_1)+n_1}{1-t_1+n_1}.$$ In general, scatter between the populations tends to make $f_{rm<}$ behave similarly to $f_{n2.5}$. Contrary to this expectation, while $f_{n2.5}$ increases, $f_{rm<}$ decreases from $\sim0.6$ to $\sim0.5$. If the scatter between the two populations decreases as $f_{n2}$ increases, $f_{rm<}$ may decrease. However, producing the $f_{rm<}$ and $f_{n2.5}$ observed without a decrease in $f_{rt<}$ requires a somewhat unrealistic decrease in $n$. Assuming that both $n$ and $t$ are between 0.05 and 0.5, $n$ would need to decrease by at least 70% (for example, $n_1=0.5$ and $n_2=0.15$ or $n_1=0.25$ and $n_2=0.05$ can reproduce the observations with appropriate values of $t_1$ and $t_2$). It is therefore probable that both of the observed trends in $f_{r}(n_i<2.5)$ are real. The decrease in $f_{n2.5}$ with $M_r$ may contribute to erasing the shallow slope of the correlation of $f_{r,gr}-f_{r,i}$ with $M_r$ between Figures \[red fraction difference\]a and \[red fraction difference\]b.
The model’s predictions refer specifically to the fraction of the in-falling blue disk galaxies that eventually join the red sequence rather than to the difference in $f_r$ between the group and isolated galaxies. In Figure \[alpha\] the difference between $f_r$ in the groups and in isolation is normalized by the blue fraction in isolation. This normalized difference is an estimate of the fraction of the in-falling blue galaxies that join the red sequence. It is a rough estimate because the properties of the isolated galaxies may not reflect those of the galaxies that actually joined the groups. The normalized difference increases with $M_{gr}$ and for the higher $M_{gr}$ bins decreases with $L_{sat}$, and the increase with $M_{gr}$ is smaller than that with $L_{sat}$. According to the model of ram pressure stripping, the small size of the change in the normalized $f_r$ with $M_r$ results from the comparable sizes of the scatter in $M_{eff}$ and the range of $M_{sat}$ in the sample. Rather than an indication that ram pressure stripping is not occurring, the existence of a small change with $M_r$ is expected if stripping is occurring.
Timescale of the SFR Decline
----------------------------
As discussed in § \[stripping and sfr\], the timescale over which a galaxy’s SFR should decline after the outer H <span style="font-variant:small-caps;">i</span> disk is stripped is of order a Gyr. Evolution of the SFRs on this timescale is consistent with the galaxies’ spectral indices remaining on the standard track in $H\delta$ versus the 4000Å break, as seen in @Kauffmann [@et; @al.; @04]. However, it is inconsistent with lack of an observable shift in the SFR distribution for star forming galaxies seen in @Balogh [@et; @al.; @04]. Both of these tests are repeated here to determine whether color evolution in these groups is fast or slow.
Figure \[h delta vs break\] shows $H\delta$ versus the 4000Å break for $13>log(M_{gr}(M_{\odot}))<14.5$ for all galaxies and for galaxies with $g-r<0.7$. Compared to Figure 8 in @Kauffmann [@et; @al.; @04], there is no excess of galaxies residing above the standard track, where a galaxy that has had its star formation abruptly truncated should lie. In Figure \[SFR hist\] distributions of $SFR/M_{\ast}$ are plotted. Figure \[SFR hist\]a shows the distributions for star-forming galaxies, $log(SFR/M_{\ast})>-11$, and Figure \[SFR hist\]b shows the distributions for galaxies with $g-r<0.7$. For the medium and large $M_{gr}$ bins, a clear shift in the distribution of $SFR/M_{\ast}$ is seen for the star-forming galaxies. A KS test gives a less than 0.01% probability that the $SFR/M_{\ast}$ distributions in the groups and in isolation are drawn from the same sample. Though less visually apparent, a slight shift also exists for the blue galaxies. The average $SFR/M_{\ast}$ is slightly lower in the groups than in isolation, and a KS test gives a 0.5% probability that they are drawn from the same distribution. These observations are consistent with a decline in the star formation rate on timescales of order a Gyr.
In Figure \[SFR hist\], the $SFR/M_{\ast}$ values from @Brinchmann [@et; @al] are used rather than the $H\alpha$ equivalent widths, $W_0(H\alpha)$, which are used in @Balogh [@et; @al.; @04]. The $SFR/M_{\ast}$ are likely a better measure of the relative star formation rate in part because they measure $SFR/M_{\ast}$ rather than $SFR/L$. The $SFR$ distributions presented here also differ from the @Balogh [@et; @al.; @04] distributions in that group membership is used for all comparisons. When the $W_0(H\alpha)$ distributions for all isolated and group galaxies with $W_0(H\alpha)>4$Å are compared, $\langle W_0(H\alpha)\rangle$ is lower for the group galaxies and a KS test rules out the possibility that the two populations are drawn from the same distribution. This results disappears when only galaxies with $W_0(H\alpha)>10$Å are considered. However, if the galaxies’ $W_0(H\alpha)$ are divided by their $M_{\ast}/L_z$ ratios, as measured by [@Kauffmann; @et; @al.; @03a], then the difference in the $W_0(H\alpha)$ distributions remain regardless of the cut used. In both instances a slight difference is also seen in the blue population, and a KS test gives a few percent chance that the blue group and isolated galaxies are drawn from the same population.
The size of any shift in the distribution of $SFR/M_{\ast}$ in the blue galaxy population is likely to be small. As can be seen in Figure \[alpha\], approximately half of the in-falling blue galaxies are unaffected by reddening. The $SFR/M_{\ast}$ distribution for blue galaxies is therefore a superposition of the unaffected blue galaxy population and a transiting population of blue galaxies with low SFRs that have not yet joined the red sequence. In addition, the difference in $SFR/M_{\ast}$ between an average in-falling blue galaxy, $\log(SFR/M_{\ast})\approx-10$, and the $SFR/M_{\ast}$ at which galaxies appear to leave the blue population, $\log(SFR/M_{\ast})\approx-10.4$, is small. Furthermore, the use of color to estimate the effect of the finite fiber size on $SFR/M_{\ast}$ biases against seeing a shift in $SFR/M_{\ast}$ in the blue galaxy population. The change in the average $\log(SFR/M_{\ast})$ in the middle(high) $M_{gr}$ bin is 0.04(0.06). If the decay in the SFR is exponential, then placing 20%(30%) of the blue galaxies halfway along a journey from their incoming $SFR/M_{\ast}$ to joining the red sequence will result in this shift. Therefore, a significantly larger shift in the $SFR/M_{\ast}$ than that observed for the blue group galaxies isn’t expected.
The location of the galaxies in this sample in the H$\delta$ versus 4000Å break plane and the relative distributions of $SFR/M_{\ast}$ for star-forming and blue galaxies in the groups versus in isolation both indicate that galaxies that transition between the blue and red populations experience a decline in their SFRs on long timescales. This is consistent with color evolution that is driven by the ram pressure stripping of gas from the outer H <span style="font-variant:small-caps;">i</span> disks of blue disk galaxies.
The Larger Picture {#d2}
------------------
The differences observed in color and structure between the group and isolated galaxy populations can be summarized as follows. For both galaxies with $n_i>2.5$ and $n_i<2.5$, the red fraction is larger in the groups than in isolation. This can be seen in Figures \[red fraction\]b,c and \[red fraction difference\]b,c. In addition, color selected galaxy populations have identical $n_i$ distributions both inside and outside groups (Fig. \[red fraction\]e,f & \[red fraction difference\]e,f; [@Quintero]). In this section, the role that ram pressure stripping may play in this larger picture is discussed.
In isolation, $f_{n2.5}$ for red galaxies and $f_r$ for galaxies with $n_i<2.5$ are both large. Therefore, for the color selected $n_i$ distributions in the groups to match those in isolation, the normalized difference in the red fraction between group and isolated galaxies, that is $(f_{r,gr}-f_{r,i})/(1-f_{r,i})$, must be larger for galaxies with $n_i>2.5$ than for galaxies with $n_i<2.5$. In addition, though the $n_i$ distributions within color selected subsamples are identical in the group and isolated galaxy populations, the larger red fraction in the groups is accompanied by a larger fraction of galaxies with $n_i>2.5$. Ram pressure stripping can create an excess of red galaxies in groups in both the $n_i>2.5$ and $n_i<2.5$ populations. In particular, because ram pressure stripping does not affect galaxies’ structures, it can transform a blue disk galaxy into a red disk. Furthermore, a greater fraction of the in-falling spherical galaxies that have gas will be stripped than of the in-falling disk galaxies. However, ram pressure stripping cannot account for the structural differences that are observed.
It is worth pointing out that, while in the most massive groups in this sample almost no mergers should occur, the merger rates in the low and medium-mass groups may be quite high [@Mamon]. In addition, N-body simulations show that a greater fraction of the halos that reside in clusters today underwent major mergers in their pasts [@GKK]. Therefore, the high-mass group galaxy population may have experienced an enhanced merger rate in the past while the low- and middle-mass group galaxy populations may be experiencing enhanced merger rates currently. In this light, Figure \[red fraction difference\]b becomes interesting. The slope of the correlation between $f_{r,gr}-f_{r,i}$ and $M_r$ is shallower in the middle $M_{gr}$ bin than in the high $M_{gr}$ bin. Mergers may tend to wash out the dependence of $f_r$ on $M_{sat}$ in the middle-mass groups while the dependence is preserved in the high-mass groups. A larger merger remnant fraction in the groups may, of course, also account for some of the structural differences observed in the group galaxy populations.
CONCLUSIONS {#conclusion}
===========
Observations of the red fraction, $f_r$, the fraction of galaxies with a large 4000Å break, $f_{4000}$, and the fraction of galaxies with $n_i>2.5$, $f_{n2.5}$, in the SDSS groups are presented in Figures \[red fraction\] and \[red fraction difference\]. These are the group-based analog of previous observations and are consistent with the correlations between galaxy properties and local environment in @Kauffmann [@et; @al.; @03b; @Kauffmann; @et; @al.; @04], @Hogg [@et; @al.; @03], and @Blanton [@et; @al.; @05b]. In addition, the observed excess of red galaxies in the groups regardless of $n_i$ makes color a better predictor of environment, as observed in @Blanton [@et; @al.; @05b].
In § \[Model\] several predictions are made for the group sample using an analytical model of ram pressure stripping presented in @P1. In general, the model predicts that the red fraction should increase as either the group mass, $M_{gr}$, increases or the satellite mass, $M_{sat}$, decreases. For the SDSS group sample in particular, it predicts that mild stripping should occur in the low-mass groups while moderate to severe stripping should occur in the middle- and high-mass groups. Therefore, differences in $f_r$ with both $M_{gr}$ and $M_r$ are expected if ram pressure stripping is driving color change. The expected scatter in an effective mass for stripping at fixed $M_{sat}$ is smaller than the range of $M_{gr}$ present in the sample, but comparable to the range of $M_{sat}$. Therefore, in the stripping scenario, observed changes in $f_r$ with $M_r$ should be small while the change with $M_{gr}$ should be larger. In addition, if stripping of the outer H <span style="font-variant:small-caps;">i</span> disk, along with the hot galactic halo, is causing the decline in SFRs in the groups, then the timescale of the decline should be of order a Gyr.
The predicted behaviors are observed in the group sample. As seen in Figures \[red fraction\]ab, \[red fraction difference\]ab, and \[alpha\], the red fraction increases with both $M_{gr}$ and, for the higher mass groups, with $M_{r}$ for both all galaxies and galaxies with $n_i<2.5$. This is despite a decrease in $f_{n2.5}$ with $M_r$. In the lowest mass groups, where only mild disk stripping is expected, the increase in $f_r$ is slight and may be due to galactic halo stripping alone. In Figure \[alpha\], an estimate of the fraction of the in-falling blue galaxies that join in the red sequence after entering a group is presented. This estimate is the best test of the model’s predictions for the sizes of the change in $f_r$. As predicted, the change in $f_r$ across the $M_r$ range is quite small while the change across the $M_{gr}$ range is significantly larger. As shown in Figures \[h delta vs break\] and \[SFR hist\], the positions of the group members in the H$\delta$ versus 4000Å break plane and the distributions of $SFR/M_{\ast}$ for the star forming and blue galaxies both demonstrate that SFRs in the groups are declining on timescales greater than a Gyr. These observations all indicate that the observed color differences in the groups are the result of the ram pressure stripping of the outer H <span style="font-variant:small-caps;">i</span> disk followed by a gradual decline in the galaxies’ SFRs.
One alternate scenario for color evolution in groups postulates that the two dominate processes in groups are galactic halo stripping and mergers. The observed trends in Figures \[red fraction\] and \[red fraction difference\] do combine color change for galaxies at all values of $n_i$ and structural differences which are accompanied by color differences. If a galaxy’s structure is altered, for instance by a major merger, while it orbits in a group, then any remaining gas should be quickly stripped and the SFR should decline. In addition, if star formation in disk galaxies is fed only by the cooling of the galactic halo directly onto the star-forming disk, then the stripping of this halo will result in red galaxies at all $M_r$ and $M_{gr}$. Similar uniform reddening should be seen for spherical galaxies.
The observed trends in $f_r$ in Figures \[red fraction\]ab, \[red fraction difference\]ab, and \[alpha\] are an excellent match to predictions based on the ram pressure stripping model, which is in itself a strong indication that disk stripping plays a role in galaxy evolution in groups. However, the observation that the red fraction increases with group mass is particularly strong and can be used to differentiate between the scenario presented in the previous paragraph and the disk stripping scenario. In § \[Model\], the model presented in Paper 1 is combined with an estimate of the maximum gas density that can be sustained in these galaxies in order to place an upper limit on the fraction of the hot halo gas that the galaxies can retain. All galaxies in both the middle- and high-mass groups should be stripped of the majority of their galactic halo gas. Considering the small mass of the remaining gas and the absence of any new in-falling gas, the remaining galactic halo cannot continue to feed star-formation in any of these galaxies. This would lead to the expectation that $f_r$ should not depend on either $M_{gr}$ or $M_{r}$ in these groups. This loose theoretical expectation is affirmed by simulations. In N-body based semi-analytic models in which star formation is shut off when satellite galaxies enter a group, the red fraction in groups is independent of $M_{gr}$ [@Weinmann06]. However, in Figure \[red fraction\]b a substantial difference in $f_r$ is seen for groups of different $M_{gr}$. This difference, which is unexpected and must be rationalized in the strangulation scenario, is easily understood in the disk stripping scenario.
The observations presented in this paper favor ram pressure stripping of the outer H <span style="font-variant:small-caps;">i</span> disk as the dominant driver of color change for disk galaxies in groups. Despite this, other processes must occur at some level. Galaxies in groups are expected to undergo mergers and some merging galaxies will have retained their H <span style="font-variant:small-caps;">i</span> disks. In the dense inner regions of clusters stripping of gas from within the stellar disk should occur on occasion. Finally, even galaxies that retain their disks are stripped of their galactic halo and will eventually experience a decline in the SFRs. However, for the low redshift groups studied here, ram pressure stripping appears to be driving the relationships between SFR, color, $M_r$, and environment.
Sorting out which processes are dominant for different $M_{sat}$ and at different redshifts will require combining further modeling and observations. To determine the importance of ram pressure stripping it is most important to understand how galaxies fuel their star formation. Understanding trends like those shown in Figures \[red fraction\] and \[red fraction difference\] will also require understanding the processes that can alter structure.
ACKNOWLEDGEMENTS
================
This project was advised by D. N. Spergel and funded by NASA Grant Award \#NNG04GK55G. I’d like to thank A. Berlind, M. Blanton, and D. Hogg for the use of the SDSS group catalog and for useful discussions.
Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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[^1]: H$\delta$, 4000Å break, $M_{\ast}$, and $SFR/M_{\ast}$ are available from http://www.mpa-garching.mpg.de/SDSS/
|
---
abstract: 'We investigate dipole oscillations of ultracold Fermi gases along the BEC-BCS crossover through disordered potentials. We observe a disorder-induced damping of oscillations as well as a change of the fundamental Kohn-mode frequency. The measurement results are compared to numerical density matrix renormalization group calculations as well as to a three-dimensional simulation of non-interacting fermions. Experimentally, we find a disorder-dependent damping, which grows approximately with the second power of the disorder strength. Moreover, we observe experimentally a change of oscillation frequency which deviates from the expected behavior of a damped harmonic oscillator on a percent level. While this behavior is qualitatively expected from the theoretical models used, quantitatively the experimental observations show a significantly stronger effect than predicted by theory. Furthermore, while the frequency shift seems to scale differently with interaction strength in the BEC versus BCS regime, the damping coefficient apparently decreases with the strength of interaction, but not with the sign, which changes for BEC and BCS type Fermi gases. This is surprising, as the dominant damping mechanisms are expected to be different in the two regimes.'
bibliography:
- 'bibliography.bib'
title: |
Dipole oscillations of fermionic superfluids\
along the BEC-BCS crossover in disordered potentials
---
Introduction
============
Ultracold, dilute gases allow to experimentally probe fundamental properties of quantum fluids [@Anderson95; @Davis95; @OHara02]. An important question concerns the transport properties of quantum fluids in disordered media, which have been subject of intense studies since the observation of superfluid flow of helium [@Chan1988]. This has sparked the investigation of ultracold quantum gases in disordered media [@Lye05; @Clement08; @Dries10; @Fort05]. In disordered potentials, the phenomenon of Anderson localization [@Anderson58], i.e. interference-induced absence of diffusion, has been observed in various physical realizations of quasi-noninteracting gases [@Billy08; @Roati08; @Kondov11; @Jendrzejewski2012; @white2019].
A fascinating feature of cold gases is the ability to additionally control the interaction strength via magnetic Feshbach resonances [@Inouye98]. For ultracold Fermi gases, this has opened the door to experimentally access the crossover from a molecular Bose-Einstein condensate (mBEC) via a resonantly interacting Fermi gas, to a BCS-type superfluid [@Regal04; @Bartenstein04; @Zwierlein04; @Kinast04; @Chin04].
The fundamental oscillation mode in a harmonic trap, the so-called Kohn mode, is not affected by interactions [@Kohn1961; @Dobson1994]. For an additional external potential, however, the frequency and damping of the Kohn mode can sensitively indicate interactions with the environment. Dipole oscillations of a quantum gas have revealed, for example, the damping of an oscillating BEC in weak disorder [@Dries10] or the mutual influence of oscillating Bose- and Fermi quantum gases [@Delehaye15]. Moreover, it was shown that weak disorder is expected to introduce a shift of the oscillation frequency [@Falco07]. Furthermore, for numerical simulations of the interacting Gross-Pitaevskii equation of one-dimensional gases in disorder [@Hsueh18], signatures for localization of weakly interacting gases were found in the thermalization of dipole oscillations.
Here, we study the dipole oscillation of an interacting $^6$Li Fermi gas across the BEC-BCS crossover. We focus on probing the oscillation frequency and damping of the quantum gas for different interaction scenarios along the BEC-BCS crossover. We compare our findings to one-dimensional density matrix renormalization group (DMRG) calculations as well as to a three-dimensional simulation of non-interacting fermions.
Experimental setup
==================
![Schematic illustration of experimental setup and measurement sequence. (a) Experimental setup. The sample (red ellipsoid) is trapped in a superposition of an optical dipole trap (blue tube) and a magnetic saddle potential (yellow surface). The beams used for absorption imaging (red arrow) and speckle potential (green volume) propagate along the same axis in opposing directions. The inset shows a $\SI{15}{\micro\meter}\times \SI{15}{\micro\meter}$ section of the speckle intensity distribution in the $x$-$y$-plane. (b) Measurement sequence. Orange: magnetic field setting the scattering length $a$, blue: magnetic field gradient for cloud displacement, green: disorder potential strength $\avg{V}$. After a variable hold time $\xi$, the column-density distribution in the $x$-$y$-plane is recorded using absorption imaging (red line).[]{data-label="fig:figure1"}](figure1.pdf)
Experimentally, we prepare quantum gases of $N \simeq 10^6$ fermionic $^6$Li atoms by forced evaporative cooling in an equal mixture of the two lowest-lying Zeeman substates of the electronic ground state $^2\mathrm{S}_{1/2}$. Evaporation takes place in a hybrid magnetic-optical trap at a magnetic field of $\SI{840}{\gauss}$ close to a Feshbach resonance centered at [@Zuern2013], for details of setup and sequence, see [@Gaenger2018]. After evaporation, the sample is held at a constant trap depth of $\SI{250}{\nano\kelvin}\times\boltzmann$ for to ensure thermal equilibrium before the magnetic field is ramped to its final value during ((b)), setting the required value of the $s$-wave scattering length $a$ and, thus, the interaction parameter $\intp$ [@Grimm2007], with $k_\mathrm{F}$ the Fermi wave vector. The trapping frequencies are , yielding the Fermi energy , where $\hbar$ is the reduced Planck constant. The precise value of $\Omega_y$ depends on the magnetic field [@supps].
Depending on the magnitude and sign of the interaction parameter, the gas is in the BEC , unitary or BCS regime [@Grimm2007]. In the BEC regime, fermions of opposite spin form bosonic molecules.
We characterize the gas in the BEC regime at a magnetic field of (), where it is possible to measure the absolute temperature by fitting the characteristic bimodal density profile [@Naraschewski1998]. From this, we infer the temperature and corresponding reduced temperature , where is the Fermi temperature and $\boltzmann$ the Boltzmann constant. According to [@Chen2005], the reduced temperature $\ttf$ deep in the BEC regime is an upper bound for $\ttf$ in the strongly-interacting and BCS regime, provided that $\intp$ changes adiabatically during magnetic field ramps.
The speckle potential is created by passing a laser beam of wavelength through a diffusive plate and focusing the light using an objective with numerical aperture onto the atoms. They experience a repulsive and spatially random (but temporally constant) dipole potential $V$, which we characterize by its average $\avg{V}$ at the focal point of the objective. The typical grain size of the speckle is given by the Gaussian-shaped autocorrelation function of the potential with $1/e$ widths, i.e. correlation lengths [@Kuhn2007], transversely to and along the beam propagation direction. As the speckle beam profile has a Gaussian envelope with waist , the disorder potential is slightly inhomogeneous with less than variation of $\avg{V}$ across the typical cloud size and oscillation trajectory. Importantly, molecules experience double the disorder strength as they possess twice the polarizability of unbound atoms.
In order to initiate oscillations, we displace the cloud by [@supps] along its long axis ($y$ in (a)) by application of a magnetic gradient field, which is increased during a ramp. Subsequently, the speckle disorder potential is introduced during a linear ramp of intensity in order to minimize excitation of the gas. We release the cloud by suddenly extinguishing the magnetic gradient field and, therefore, almost instantaneously shifting the trap center to its initial position, see (a). The shift amplitude $A$ sets the initial potential energy which drives the dipole oscillation in the combined potential of the magnetic-optical trap and the disorder. Here, $m$ is the atomic mass for gases in the BCS and unitary regime and the molecular mass $2m_\mathrm{Li}$ in the BEC regime. After variable hold times $\xi$ of up to one second, we record the atomic density distribution using resonant high-intensity absorption imaging [@Reinaudi2007]. The center of mass position of the cloud is extracted by fitting a 2D Thomas-Fermi profile to the measured density distribution.
shows time series of cloud oscillations for disorder strength $\avg{V}/\epot=\num{0.06}$ and all three explored interaction parameters. In all cases, we observe a damped harmonic oscillation of the center of mass position.
![(a) Schematic illustration of experimental sequence. (a) and (b) and (c) Oscillation trajectories of density profiles for and . Colored lines depict the center of mass position and white lines mark the trap center. (b) BEC regime (). (c) Unitary regime (). (d) BCS regime ().[]{data-label="fig:figure2"}](figure2.pdf)
Numerical DMRG calculations
===========================
In order to theoretically simulate the dipole oscillations as a function of time in a quantum many-body system we consider a one-dimensional (1D) version of the corresponding setup based on the Hamiltonian $$\begin{aligned}
\label{FermiGas}
\hat{H}=& &
\sum_{\sigma=\uparrow,\downarrow}
\int \mathrm{d}x
\Psi_\sigma^\dagger(x)
\left( -\frac{\hbar^2}{2m} \partial_x^2 + \frac{1}{2} m \Omega^2 x^2 \right)
\Psi_\sigma(x)
\end{aligned}$$ modeling Fermions with spin . In the time-dependent DMRG [@vidal2004; @daley2004; @white2004] simulations we implement the model on a lattice including also disorder and interactions
$$\begin{aligned}
\label{FermiHubbard}
\hat{H}= &&\sum_{i} \sum_{\sigma=\uparrow,\downarrow} \left[ -J\left( c_{i,\sigma}^\dagger c_{i+1,\sigma}^{\phantom{\dagger} }+ \text{H.c.} \right) \right. \nonumber\\
&& +\left. \left(V_i + \alpha x_i
^2\right) n_{i,\sigma} + \frac{U}{2} n_{i,\sigma} n_{i,\overline{\sigma}}\right],
\end{aligned}$$
in the limit of flat traps and small densities, where the parameters are related by
$$\begin{aligned}
m=\frac{\hbar^2}{2Jd^2}, \phantom{abcd} \Omega = \frac{2 d \sqrt{\alpha J}}{\hbar}, \phantom{abcd} x_i = id,
\end{aligned}$$
and $U$ is the on-site interactions using the local densities $n_{i,\sigma}= c_{i,\sigma}^\dagger c_{i,\sigma}^{\phantom{\dagger} }$ and lattice spacing $d$. The uncorrelated disorder is modeled in the form $V_i=\delta\, r_i$ with a disorder strength $\delta$ and a random value taken from the continuous uniform distribution $r_i \in [0,1]$ for each site. Specifically, we time-evolve an oscillating wave packet of $N=6$ particles with equal number of particles in each spin component in a trap with $\alpha = 0.0015~J/d^2$ after shifting by $A=8d$. The $1/e$ radius of the cloud at $U=0$ is about 10 sites. Using a time step of $\tau=\frac{1}{10}\frac{1}{J}$ we managed to resolve about $2.5$ full oscillations with disorder averaging over 8 realizations, keeping up to $M=800$ states.
It is known that for negative $U$ the 1D system crosses over from a weakly bound BCS-like state for $U=0^-$ to a BEC-like state for $U \to - \infty$ [@Astra04] which mimics the corresponding behavior of the 3D system. For repulsive interactions $U>0$ the 1D system is described by dimer excitations [@Astra04; @tokatly2004; @fuchs2004], which do not have a simple correspondence in the 3D experiment and will not be considered here. The relation between $U$ and scattering length $a$ was determined by considering a simple 1D scattering problem on a lattice to be . The effective Fermi wave vector $k_\mathrm{F}=\sqrt{2m E_\mathrm{F}}/\hbar$ is given by the energy $E_\mathrm{F} = (N+1) \hbar \Omega/2 $ of the highest occupied state in the trap, where we have neglected the interaction dependence [@soeffing11]. Therefore, we find $$\frac{1}{a k_\mathrm{F}} = - \frac{U}{(N+1)^{1/2}d^{1/2} \alpha^{1/4} J^{3/4}}$$ which we have used in the following to compare the simulations to experimental data.
Numerical simulation of non-interacting fermions
================================================
Additionally, we model the experiment considering non-interacting fermions in a three dimensional harmonic trap which are subjected to a speckle potential. We investigate the evolution of the atomic cloud after exciting its dipole mode by rapidly shifting the center of the trap, following the experimental sequence. In order to simulate this system, we consider the time-dependent Schrödinger equation $i\hbar \frac{\partial}{\partial t }|\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle$ with Hamiltonian $$\begin{aligned}
\label{eq:ham}
\hat{H}=&-&\frac{\hbar^2}{2m}\nabla^2+ \frac{1}{2}m \left(\Omega_x^2 x^2+\Omega_y^2 (y-A)^2+\Omega_z^2 z^2\right)\nonumber\\
&+& V(x,y,z)
\end{aligned}$$ where the first term is the kinetic energy of particles with mass , the second term describes the harmonic trapping potential, with frequencies as in the experiment, centered at position $(0, A, 0)$, and the last term is the speckle potential $V$ with the spatially averaged value $\avg{V}$. In the experiment, the correlation length of the disorder $\sigma_z$ in $z$-direction is large, therefore we assume the speckle potential to be constant in this direction . We also assume that , which enables us to separate the Schrödinger equation in the three different dimensions. We diagonalize the discrete form of the Hamiltonian (Eq. \[eq:ham\]) in each direction with discrete sizes and calculate the density of particles in time.
We consider fermions at zero temperature in the harmonic trapping potential shifted to position $A=\SI{71}{\micro\meter}$ along the $y$-direction. At time , we quench the position of the trap to and calculate the evolution of the density of atoms up to time $t=\SI{1}{\second}$ with time step . We simulate this system for three different discrete sizes of $\Delta y = \SI{52.5}{\nano\meter}$, and where the results converge for $\Delta y \leq \SI{105}{\nano\meter}$. In the following we choose for a system of size $\SI{525}{\micro\meter} \times \SI{126}{\micro\meter} \times \SI{126}{\micro\meter}$ where the number of discrete points along different directions are $L_y= 5000$ and $L_x=L_z=1200$. We calculate the center of mass evolution in time for different disorder strengths $\avg{V}$ as considered in the experiment. For each disorder strength, the results are averaged over different realizations of disorder. When the speckle potential is off (), the center of mass shows pure dipole oscillations.
Results
=======
Experimentally as well as for both numerical approaches, we extract the center of mass oscillation for identical relative disorder strengths $\avg{V}/\epot$. For the DMRG calculations, also similar interaction strengths are considered. All oscillation trajectories are fitted by the classical equation of an underdamped harmonic oscillator $$y(t) = A_\mathrm{fit} \exp\!\left(-\gamma t\right)\sin (\omega_\mathrm{fit} t + \phi) + y_0,$$ with amplitude $A_\mathrm{fit}$, fitted oscillation frequency $\omega_\mathrm{fit}$, damping coefficient $\gamma$, phase $\phi$ and offset $y_0$. For a classical harmonic oscillator we expect $$\omega_\mathrm{cl} = \sqrt{\Omega_y^2 - \gamma^2},$$ and we are interested in the deviations from this expectation due to the disorder potential, as well as in the disorder-induced damping. For the experimental data, we correct for the finite curvature of the speckle envelope by introducing the corresponding trapping frequency and writing $\omega_\mathrm{cl}=\sqrt{\Omega_y^2-\gamma^2-\omega_\mathrm{s}^2}$. As for all explored disorder strengths, the maximum relative change in $\omega_\mathrm{cl}$ is below $\SI{0.1}{\percent}$.
![Relative deviation of dipole-oscillation frequency from the classical case. Dots indicate experimental data at the color-coded interaction parameter, crosses indicate results from the 1D DMRG calculation, and squares show results of the simulation of non-interacting fermions. The diamonds indicate the frequency shift predicted in [@Falco07].[]{data-label="fig:figure3"}](figure3.pdf)
shows the results for the deviation of the oscillation frequency. We use the relative deviation from the expectation, i.e. $(\omega_\mathrm{fit}-\omega_\mathrm{cl}) / \omega_\mathrm{cl}$. Experimentally, we find that the frequency of oscillation shifts to smaller values at a percent level. Moreover, while the unitary gas shows the strongest shift, the oscillation frequency in the BCS regime is less affected than the BEC despite stronger interactions.
A quantitative comparison to theoretical predictions from the models described above, and to a prediction for a BEC with healing length larger than the disorder correlation length [@Falco07], shows a qualitatively similar behavior. But while the theoretical predictions all lie within the same range of oscillation shifts in the sub-percent range, the experiment shows a much stronger effect. We attribute this to a combined effect of strong interactions, superfluid flow and three dimensions of the problem, which in this combination are not captured by the models we use. Importantly, for disorder strengths $\avg{V}/\epot \gtrsim 0.2$, the strong damping (see ) obstructs the determination of an oscillation frequency from experimental data, because the system approaches the overdamped regime.
![Damping $\gamma$ of the oscillation through disorder. Dots indicate experimental data at the color-coded interaction parameter, crosses indicate results from the 1D DMRG calculation, and squares show results of the simulation of non-interacting fermions. The dotted line marks the critical damping . (a) Experimental results. The inset shows the same data in log-log scale with the solid lines being quadratic fits to the data points. (b) Comparison of experimental results to models.[]{data-label="fig:figure4"}](figure4.pdf)
The results for the damping are depicted in , where we show the fitted coefficients normalized by the disorder-free oscillation frequency $\gamma / \Omega_y$. Experimentally, we find that the damping grows approximately quadratically with disorder strength in all three interaction regimes ((a)). This is consistent with the experimental outcomes for a BEC oscillating in disorder in [@Dries10]. Notably, the damping is considerably weaker for the resonantly interacting superfluid and BCS gas as for the BEC, which is in accordance with the observation of superfluidity with relatively large critical velocity in the unitary regime [@Combescot2006; @Miller2007; @Weimer2015]. Moreover, it seems that the general scaling of the damping only depends on the absolute value of $\intp$ (see inset of (a)), despite the fact that dephasing and wave-like excitation of the quantum fluid is expected to prevail in the BEC, while pair-breaking might occur in the BCS regime.
The theoretical models produce qualitatively the same result as the experiment, i.e. the damping increases monotonously with disorder strength. The numerical DMRG calculations show that the damping significantly increases with attractive interactions ($U<0$) towards the BEC regime, which is also seen in experiment. The magnitude of this effect, however, is in all cases more than one order of magnitude lower as in the measurements. In particular, numerical simulations show weakly damped oscillations for all disorder strengths considered, while experimentally we find the crossover to the overdamped regime for a disorder strength which corresponds to only of the Fermi energy.
We attribute this relatively strong damping to strong interactions in the cloud. At the same time, the quantitative comparison with theory shows again a much stronger experimental effect than predicted by theory. Also here, the combination of strong interactions and three spatial dimensions might explain this strongly damped dynamics.
Conclusion
==========
The following physical picture is suggested by our investigations. For the oscillating cloud, strong interactions allow the quantum gas to react on small length scales to the disordered potential. This bending is associated with a large kinetic energy. As the quantum fluid flows, a strong energy change on small length scales will lead to excitations in the quantum gas, causing damping. At the same time, it facilitates retarding the dynamics of the gas leading to stronger reduction of the oscillation frequency with stronger interactions. The concrete quantitative description of the frequency shift and damping will be the focus of future studies, in particular the scaling with interaction parameter $\intp$ and the microscopic mechanisms underlying the transport of a BEC wave function versus the gas in the BCS regime.
In the future it will be interesting to observe the transition from the weakly damped to the overdamped regime in order to investigate the connection between superfluid dynamics and diffusion in disorder. Moreover, the quantum phase transition to a quantum fluid is predicted to be affected by disorder [@Orso07; @Han10], and dipole oscillations have shown to be a sensitive tool for probing superfluid properties. Furthermore, for the blue-detuned speckle potential used experimentally, the classical percolation threshold, i.e. the energy threshold below which particles cannot explore the full potential, is around $10^{-4}\times\avg{V}$ [@Pilati2010]. Thus, the system is highly sensitive to reveal even the smallest fraction of localized particles if a localization transition occurs.
We thank B. Gänger and J. Phieler for experimental work in the early stages of the project. We acknowledge helpful discussions with C. Kollath and A. Pelster. This work is funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the Collaborative Research Center SFB/TR185 (Project No. 277625399).
|
---
abstract: 'We modeled observations of the C$_2$ $\mathrm{d} ^3\Pi_g - \mathrm{a} ^3\Pi_u$ (Swan) $\Delta \nu = 0$ sequence observed in spectra of comet 122P/de Vico obtained with the 2.7m Harlan J. Smith Telescope and Tull Coude spectrograph of McDonald observatory on 10/03/1995 and 10/04/1995. The data used spanned 4986-5169Å at R=$\lambda/\Delta\lambda$=60,000. We used the PGOPHER molecular spectra model to generate and fit synthetic spectra with the d$^3\Pi_g$ having one and two rotational temperatures. We found the excited state had a two component rotational temperature, similar to that found for comet Halley. The modeled spectrum was sufficiently high quality that local perturbations were important to include. The large perturbation, b$^3\Sigma_g^-(\nu=10)$, was added to our fits and some new estimates on its molecular constants were found.'
author:
- Tyler Nelson
- 'Anita L. Cochran'
- Colin Western
title: 'Rotational Temperature Modeling of the Swan $\Delta\nu = 0$ Band Sequence in Comet 122P/de Vico'
---
Introduction
============
Comets and asteroids represent the leftovers from the epoch of planet formation within the solar system. Comets differ from the rocky asteroids by the presence of ices, which are a direct indication that they formed in a region beyond Jupiter. Since comets spend most of their orbit in cold regions and have insufficient mass to differentiate or undergo structure-altering processes, they contain the least altered material from the primordial solar system. Thus, studies of comets offer constraints on conditions in the early solar nebula.
Comets are mixtures of ices and dust, with the mass being split approximately equally between those two substances. The cometary ice is about 80% H$_2$O, the remainder having contributions from other ices such as CO, CO$_2$, CH$_4$, NH$_3$, etc. As the comet approaches the Sun, the ices are sublimated and flow out from the nucleus carrying dust along with it. The dust quickly disentrains from the gas, flowing outwards more slowly than the gas. The gas travels outwards from the nucleus at a velocity that is dependent on heliocentric distance, being approximately 0.85 kmsec$^{-1}$ at 1[au]{}. Beyond a short distance from the nucleus (500 – 1000km depending on how active the comet is) the density of the gas is too low for collisions. Thus, the spectra of comets are marked by resonance fluorescence of the molecules excited by sunlight.
We observe only fragment species in the optical region of the spectrum. One of the most prominent set of molecular emissions observed are the $\mathrm{d} ^3\Pi_g - \mathrm{a} ^3\Pi_u$ Swan bands of C$_{2}$. The Swan bands are the dominant features in the green, orange and red part of the spectrum. Cometary spectra allow investigation of C$_2$ with longer path lengths and better vacuum than lab conditions. Thus, we are using observations of a comet to probe the structure of the C$_{2}$ spectrum and to determine the production pathways.
Outside the collisional zone, C$_2$ can reach high J levels because it has no pure rotational spectrum and only $\Delta\text{J}\pm1$, are allowed for electronic transitions (see Ref. ). The rotational temperature characterizes the distribution of J levels. @Lambert1990[@Lambert1990] demonstrated that the observed C$_2$ Swan (0-0) band of comet 1P/Halley had two rotational temperatures, whereas their observations of an acetylene torch spectrum had only one. These two populations in Halley had temperatures, T$_\text{low} = 600-700$K and T$_\text{high} \approx 3200$K. Previous rotational temperature calculations in comets for C$_2$ by @Lambert1983[@Lambert1983], @Ahearn1978[@Ahearn1978], and @Danylewych1978[@Danylewych1978] only were modeled with a single temperature which was similar to the high temperature component of Halley.
In this paper, we present evidence for two rotational temperatures in comet 122P/de Vico using modern spectroscopy modeling software and specifying the $\mathrm{d} ^3\Pi_g$ population function. This approach has several advantages, including simplicity, uniform treatment of uncontaminated blended lines, and the ability to treat perturbations. We also give estimates on some molecular constants for $\mathrm{b} ^3\Sigma^-_g (\nu = 10)$, which perturbs the $\mathrm{d} ^3\Pi_g \ (\nu = 0)$ state for N = 47.
Observations
============
We observed comet 122P/de Vico using the 2.7m Harlan J. Smith Telescope of The University of Texas at Austin’s McDonald Observatory. We used the high-spectral resolution Tull 2Dcoude spectrograph [@TullCoude]. The spectra have a resolving power, R=$\lambda/\Delta\lambda$=60,000 using a slit that was 1.2arcsec wide and 8.2arcsec long. Table \[log\] is a log of the observations.
The comet was relatively bright, resulting in high signal/noise (S/N). In addition, this comet has an extremely high gas-to-dust ratio, resulting in very little continuum from solar photons reflected off the dust. With so little dust, we neglected any solar spectrum removal since it plays such a small part in these observations. The spectral dispersion of the observations was computed using separate spectra of a ThAr lamp and had an rms error of the wavelength of $\sim24$mÅ. The spectra were Doppler shifted to the laboratory rest frame using the geocentric radial velocity of the comet.
The $\Delta \nu=0$ band sequence of C$_{2}$ is spread over many echelle orders and each order has slightly different sensitivity. We combined the orders together by correcting for the sensitivity variations using a separate solar spectrum obtained with the same instrument. This let us model the $\Delta\nu=0$ sequence from the (0-0) bandhead at 5165Å to the (1-0) bandhead at 4737Å with few strong contaminants. We chose to exclude the 4737-4783Å section in the following analysis because it added only a few lines while adding many more noisy data points, thereby degrading the quality of the fit from imperfect noise removal.
Methods
=======
We modeled the de Vico data using the PGOPHER[@Colin2017] molecular modeling code. PGOPHER is a code to model rotational, vibrational and electronic spectra of molecules. It can be used to fit a spectrum of line wavelengths and intensities or, as we used it, to input molecular constants to simulate a molecular spectrum. With its high degree of precision and customization, PGOPHER was well suited for our analysis. Serendipitously, the $\Delta\nu = 0$ band sequence up to (4-4) was already included in a sample constants file making it easy to start our analysis. @Brooke2013[@Brooke2013] modeled up to (9-9), so we added in their constants to those that came with PGOPHER and have included vibrational levels (0-0) through (9-9) in our models.
Unlike in a laboratory, where a single type of molecule is being studied, the comet spectrum consists of many different molecular emissions (see ). Thus, the spectral region of the Swan bands can contain contaminants from other species, such as NH$_2$. In order that the contaminating lines not influence the quality of the fit, strong contaminating lines were masked out. Many features that were likely perturbed were also masked out. We further excluded wavelengths less than 4986.2Å because of persistent offsets between the peaks of the model and data of 0.05 to 0.1Å. This left 4107 data points covering the wavelength region 4986-5169Å to model.
@Lambert1990[@Lambert1990] concluded that the Swan band population of comet Halley represented two different rotational temperatures. It is not clear how universal is the need in cometary spectra for populations at two temperatures. So we tested this by modeling our de Vico spectrum both with a single temperature and with two temperatures. The single temperature version used the built-in simulation temperature. The two-temperature distribution, $f$, had the form $$f(a,T_1,T_2) = e^{(-E/(k_BT_1))} + ae^{(-E/(k_BT_2))}$$ where $a$ is the ratio of the contribution of each population, and $T_1, T_2$ are the two temperatures. In addition, a data scaling factor, a$_\text{spec}$, was used for both fits since our data are not flux calibrated. a$_\text{spec}$ does not change the profile of the modeled spectrum. For both of the rotational temperature distributions, we also included models with and without a vibrational temperature (T$_\text{vib}$).
The intensity error is dominated by Poisson noise from the CCD, which is proportional to $\sqrt{\text{Counts}}$ measured, so we use this as a first order weighting scheme. To include these weights in our fit, we made minor modifications to PGOPHER that have been incorporated into the current development version. By including weights, the influence of noise and weak features is reduced.
We also made a preliminary model for the N = 47 perturbation of d$^3\Pi_g(\nu=0)$ by b$^3\Sigma^-_g(\nu=10)$. While this perturbation has been suspected since [-@Callomon1963][@Callomon1963], we could not find the molecular constants required to model it. If unaccounted for, perturbations were a large source of error in our residuals. We used constants from @Chen2015[@Chen2015] as a starting point for b$^3\Sigma^-_g (\nu = 10)$ and adjusted constants until the model achieved a reasonable match with our observations and the identifications given in @Tanabashi2007[@Tanabashi2007]
Results and Discussion
======================
The best fit values with 1$\sigma$ uncertainties and reduced chi-square ($\chi_r^2$) for the one- and two-temperature populations are given in Tables \[tab:1\] and \[tab:2\] respectively. The instrumental broadening dominates the line width with a Gaussian $\text{FWHM} = 0.102$Å. We convolved the models with this broadening to match the data. A model including all relevant perturbations is not available, so a minimum model involving just the states necessary was developed starting with the available constants[@Chen2015][@Tanabashi2007] and adjusting as required. The constants used for the perturbing state are given in Table \[tab:3\]. Visual comparisons of the one- and two-temperature models with the observations in a few important sections are given in Figures \[fig:2\] and \[fig:3\]. The correlation matrix for the two-temperature fit is given in Table \[tab:4\]. We also used a Markov chain Monte Carlo (MCMC) model[@emceeDan] to examine the posterior distributions of the fitted parameters. The a$_\text{spec}$ parameter was found to be almost identical across all the fits, with a value of 5.26 adopted throughout. The posterior probability distributions for the two-temperature fit are given in Figure \[fig:4\]. The effect of the perturbation is illustrated in Figure \[fig:1\]. While the $\chi_r^2$ values are large for all of the populations used, there is an obvious preference for the two-temperature population. It is also clear that the contribution from the vibrational temperature cannot be excluded, as it improves models with both one and two rotational temperatures. Comparing the residual panels of Figures \[fig:2\] and \[fig:3\], it is apparent that the two-temperature model reproduces the (0-0) bandhead better than the single temperature. The one- and two-temperature models were similar for the (1-1) and (2-2) bandheads. We did not see bandheads for (3-3) or higher $\Delta\nu=0$ transitions. This is not surprising since the (2-2) bandhead is almost washed out by the (1-1) transition, and the (3-3) strength is less than half that of the (2-2).
We investigated the possibility of a three-rotational-temperature population. The fit with three temperatures was worse than that with two temperatures and actually converged toward the two-temperature solution by driving the coefficient of the third component to zero.
The high value for $\chi^2_\text{r}$ that we derive probably comes from either errors in the reduction of the de Vico data or unaccounted physics in the model that we have employed. However, inspection of the figures shows that the fits are quite good in general. Since de Vico has high S/N and low dust, the most likely source of error in the data reduction is order de-tilting. The tilting is most pronounced toward the blue end of the spectrum, which is where the Swan band lines are the weakest and have the least bearing on the quality of the fit. Therefore, we conclude that which physical processes are included/excluded drive our “poor” fit. Small differences in wavelength, usually less than 0.05Å, between the peaks in the model and data were also seen. All models gave similar residuals around these mismatches so we conclude the temperatures are largely insensitive to this effect. The offsets could result from the peaks falling between two pixels on the CCD.
Inclusion of the d$^3\Pi_g \ \text{N}=47$ perturbation reduced the $\chi^2_r$ by more than 16% in both models. This seems to be the strongest perturbation but is unlikely to be the only one. For example, the P branch of (1-1) N = 62 seems offset, as shown in the right panel of Figure \[fig:5\]. @Lambert1990[@Lambert1990] saw intensity distributions of some P and R features that did not match the acetylene flame. If this mismatch is a consequence of the different environmental conditions then some deviation would be built in unless the different environments were dealt with explicitly. We removed blends with known contaminants to the best of our ability, but these frequently occur near or within C$_2$ lines, so a compromise must be reached so that not too much C$_2$ data is removed. These blends, especially for lines that were completely blended with C$_2$, will obviously enhance the observed intensity of the data with respect to the predictions. With many unidentified lines in the our bandpass, the possibility of undiscovered lines that are buried in strong C$_2$ emission could also exist. We also did not include the Swings[@Swings1941] effect, in which some lines that coincide with strong absorptions in the solar spectrum are suppressed since the C$_2$ band is produced via resonance fluorescence.
Our result agrees with previous work [@Lambert1990]. We also found that the two-temperature populations extend to the (1-1) and (2-2) bandheads. There are two explanations for this bimodal temperature. @Jackson1996[@Jackson1996] found this as an outcome of the photolysis of C$_2$H, arguing that either the sudden or phase-space models of dissociation can apply. The sudden model is a classical treatment of the break up, where there is a maximum attainable rotational energy which corresponds to max J. In comparison, the phase-space model proceeds more slowly, allowing all J levels to be accessed. The sudden model gives a low J level distribution because the J values are constrained by classical conservation models, whereas the phase-space model allows for the high J level distribution. Which model applies depends on the trajectory of the H atom along the C$_2$H potential surface during photodissociation. Communication from Jackson to Lambert in 1989 (see ) also indicated that a similar bimodal population can result from other parents/grandparents. The other explanation outlined by @Lambert1990[@Lambert1990] is driven by intercombinational cooling from c$^3\Sigma^+_u - \text{X} ^1\Sigma^+_g$ and a$^3\Pi_u - \text{X}^1\Sigma^+_g$ transitions. Since the triplet-singlet transitions occur more readily than the singlet-triplet ones there is a net loss of energy from the triplet system. Thus a-X and c-X cool the Swan system. Simulation results by @Gredel1989[@Gredel1989] and @KSwamy1997[@KSwamy1997] offer support for this.
Quantifying the importance of the formation pathways versus the intercombinational transitions depends on our ability to distinguish or isolate them. A first attempt could be examining the spectrum as a function of cometocentric distance, assuming that the formation influence diminishes far away from the nucleus. This relies on the time for a C$_2$ molecule to establish fluorescence equilibrium with the Sun, $\tau_\text{eq}$. @Lambert1983[@Lambert1983] and @Odell1988[@Odell1988] estimated this value as $\tau_\text{eq} < 500$ seconds. The lifetime against photodisocciation of C$_{2}$ at de Vico’s heliocentric distance can be computed using the cometary scale lengths of Cochran[@Cochran1985] of $5.7 \times 10^4$ km and an outflow velocity of 1 kmsec$^{-1}$. Thus the expected lifetime of C$_{2}$ is of order $5.7 \times 10^4$ sec. Once appreciable amounts of new C$_{2}$ are no longer made, the contribution from formation should be small.
@Jackson1996[@Jackson1996] produced bimodal temperatures for the X$^1\Sigma^+_g$, A$^1\Pi_u$, and B’$^1\Sigma^+_g$ singlet states in the laboratory. They found each electronic state has significantly different temperatures. @KSwamy1997[@KSwamy1997] predicted that the Mulliken and Phillips systems should also have bimodal temperatures. We intend to investigate whether there are two temperatures in the Phillips system for de Vico observations in a later analysis.
@KSwamy1997[@KSwamy1997] seemingly reproduced the two-temperature distribution observed in comet Halley by including Phillips, Mulliken, Ballik-Ramsay, Swan, and the a$^3\Pi_u - \text{X}^1\Sigma^+_g$ systems. He assumed all molecules start in the $\nu = \text{J} = 0$ ground state and modeled them by exposure to a smoothed solar spectrum. He did not discuss why the photolysis of C$_2$H would produce this population. Applying either of the aforementioned photodissociation mechanisms implies some initial J distribution. What results from populations with other rotational, vibrational, and electronic levels that emulate cometary conditions is unknown. Krishna Swamy recovered a bimodal distribution without including the c-X transition. @Gredel1989[@Gredel1989] found the rotational temperature strongly depends on both a-X and c-X strengths since both move energy into the singlet system. A simulation, testing different source functions, pressures, and initial population distributions would be of great interest to constrain what the bimodal temperature depends on.
Conclusions
===========
In this paper, we have applied new models of the C$_2$ Swan band in comet 122P/de Vico. These models fit most of the $\Delta \nu = 0$ band well by incorporating a two-temperature source population as well as including some perturbations between states. We showed that this two-temperature model fits the data better than either a single rotational temperature or three rotational temperatures. The reason for the two temperature populations is not entirely clear, though we quote several papers with potential models. We plan to add to our understanding of the processes that produce the C$_2$ spectrum by using additional observations of comets. We will explore how universal is the two-temperature population with heliocentric distance and cometary orbital dynamical type in future work.
This work was performed under NASA grant NNX17A186G. TN was supported by the Dean’s Excellence Fellowship.
@ifundefined
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Herzberg, G. *Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules*; R.E. Krieger Publishing Company, 1989 , D. L.; [Sheffer]{}, Y.; [Danks]{}, A. C.; [Arpigny]{}, C.; [Magain]{}, P. [High-Resolution Spectroscopy of the C$_2$ Swan 0-0 Band from Comet P/Halley]{}. *Astrophys. J.* **1990**, *353*, 640–653 , D. L.; [Danks]{}, A. C. [High-Resolution Spectra of C$_2$ Swan Bands from Comet West 1976 VI]{}. *Astrophys. J.* **1983**, *268*, 428–446 , M. F. [Synthetic Spectra of C$_2$ in Comets]{}. *Astrophys. J.* **1978**, *219*, 768–772 , L. L.; [Nicholls]{}, R. W.; [Neff]{}, J. S.; [Tatum]{}, J. B. [Absolute Spectrophotometry of Comets 1973XII and 1975IX. II - Profiles of the Swan Bands]{}. *Icarus* **1978**, *35*, 112–120 , R. G.; [MacQueen]{}, P. J.; [Sneden]{}, C.; [Lambert]{}, D. L. [The High-Resolution Cross-dispersed Echelle White-pupil Spectrometer of the McDonald Observatory 2.7-m Telescope]{}. *Pub. Astron. Soc. Pac.* **1995**, *107*, 251–264 Western, C. M. PGOPHER: A Program for Simulating Rotational, Vibrational and Electronic Spectra. *J. Quant. Spectrosc. Radiat. Transfer* **2017**, *186*, 221 – 242, Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting, October 2015 Brooke, J. S.; Bernath, P. F.; Schmidt, T. W.; Bacskay, G. B. Line Strengths and Updated Molecular Constants for the C$_2$ Swan System. *J. Quant. Spectrosc. Radiat. Transfer* **2013**, *124*, 11 – 20 , A. L.; [Cochran]{}, W. D. [A High Spectral Resolution Atlas of Comet 122P/de Vico]{}. *Icarus* **2002**, *157*, 297–308 Callomon, J. H.; Gilby, A. C. New Observations on the Swan Bands of C$_2$. *Can. J. Phys.* **1963**, *41*, 995–1004 Chen, W.; Kawaguchi, K.; Bernath, P. F.; Tang, J. Simultaneous Analysis of the Ballik-Ramsay and Phillips Systems of C$_2$ and Observation of Forbidden Transitions between Singlet and Triplet States. *J. Chem. Phys.* **2015**, *142*, 064317 , A.; [Hirao]{}, T.; [Amano]{}, T.; [Bernath]{}, P. F. [The Swan System of C$_{2}$: A Global Analysis of Fourier Transform Emission Spectra]{}. *Astrophys. J., Suppl. Ser.* **2007**, *169*, 472–484 , D.; [Hogg]{}, D. W.; [Lang]{}, D.; [Goodman]{}, J. [emcee: The MCMC Hammer]{}. *Pub. Astron. Soc. Pac.* **2013**, *125*, 306 , P.; [Elvey]{}, C. T.; [Babcock]{}, H. W. [The Spectrum of Comet Cunningham, 1940C.]{} *Astrophys. J.* **1941**, *94*, 320 , W. M.; [Blunt]{}, V.; [Lin]{}, H.; [Green]{}, M.; [Olivera]{}, G.; [Fink]{}, W. H.; [Bao]{}, Y.; [Urdahl]{}, R. S.; [Mohammad]{}, F.; [Zahedi]{}, M. [Non-Adiabatic Interactions in Excited C$_{2}$H Molecules and their Relationship to C$_{2}$ Formation in Comets]{}. *Astrophys. Space Sci.* **1996**, *236*, 29–47 , R.; [van Dishoeck]{}, E. F.; [Black]{}, J. H. [Fluorescent Vibration-Rotation Excitation of Cometary C$_2$]{}. *Astrophys. J.* **1989**, *338*, 1047–1070 , K. S. [On the Rotational Population Distribution of C$_{2}$ in Comets]{}. *Astrophys. J.* **1997**, *481*, 1004–1006 , C. R.; [Robinson]{}, R. R.; [Krishna Swamy]{}, K. S.; [McCarthy]{}, P. J.; [Spinrad]{}, H. [C$_2$ in Comet Halley - Evidence for its being Third Generation and Resolution of the Vibrational Population Discrepancy]{}. *Astrophys. J.* **1988**, *334*, 476–488 , A. L. [A Re-evaluation of the Haser Model Scale Lengths for Comets]{}. *Astron. J.* **1985**, *90*, 2609–2614
[l | c | c]{}
& Heliocentric & Geocentric\
& Distance (AU) & Distance (AU)\
03 Oct 1995 & 0.66 & 1.00\
04 Oct 1995 & 0.66 & 0.99\
------------------- -------------------------- ---------------------- ---------------------
Without Perturbation
Without T$_{\text{vib}}$ With T$_\text{vib}$ With T$_\text{vib}$
T$_\text{rot}$(K) $5075\pm17$ $4031\pm25$ $3880\pm30$
T$_\text{vib}$(K) - $5860\pm29$ $6100\pm34$
$\chi_r^2$ $560$ $406$ $492$
------------------- -------------------------- ---------------------- ---------------------
------------------- -------------------------- ---------------------- ---------------------
Without Perturbation
Without T$_{\text{vib}}$ With T$_\text{vib}$ With T$_\text{vib}$
a $1.249\pm0.034$ $1.112\pm0.037$ $1.313\pm0.054$
T$_1$(K) $6130\pm45$ $5432\pm76$ $5663\pm103$
T$_2$(K) $1179\pm43$ $931\pm43$ $1095\pm50$
T$_\text{vib}$(K) - $5519\pm25$ $5674\pm30$
$\chi_r^2$ $313$ $280$ $360$
------------------- -------------------------- ---------------------- ---------------------
[c | c]{} Origin & $19836.25468$\
B & $1.5418643$\
$\lambda$ & $0.3593$\
o & $-1.498$^*a*^\
$\gamma \times 10^4$ & $3.929$\
p$\times 10^5$ & $2.1$^*a*^\
D$\times 10^6$ & $6.417$\
H$\times 10^{12}$ & $4.88$^*a*^\
Strength & $0.03392826$\
\
a T$_1$ T$_2$ T$_\text{vib}$
---------------- -------- -------- -------- ----------------
a 1.0 0.779 0.552 -0.290
T$_1$ 0.779 1.0 0.724 -0.478
T$_2$ 0.552 0.724 1.0 -0.224
T$_\text{vib}$ -0.290 -0.478 -0.224 1.0
|
---
abstract: 'Recent observations suggest that magnetic flux cancellation may play a crucial role in heating the Sun’s upper atmosphere (chromosphere, transition region, corona). Here, we intended to validate an analytic model for magnetic reconnection and consequent coronal heating, driven by a pair of converging and cancelling magnetic flux sources of opposite polarities. For this test, we analyzed photospheric magnetic field and multi-wavelength UV/EUV observations of a small-scale flux cancellation event in a quiet-Sun internetwork region over a target interval of 5.2 hr. The observed cancellation event exhibits a converging motion of two opposite-polarity magnetic patches on the photosphere and red-shifted Doppler velocities (downflows) therein consistently over the target interval, with a decrease in magnetic flux of both polarities at a rate of 10$^{15}$Mxs$^{-1}$. Several impulsive EUV brightenings, with differential emission measure values peaked at 1.6–2.0MK, are also observed in the shape of arcades with their two footpoints anchored in the two patches. The rate of magnetic energy released as heat at the flux cancellation region is estimated to be in the range of (0.2–1)$\times$10$^{24}$ergs$^{-1}$ over the target interval, which can satisfy the requirement of previously reported heating rates for the quiet-Sun corona. Finally, both short-term (a few to several tens of minutes) variations and long-term (a few hours) trends in the magnetic energy release rate are clearly shown in the estimated rate of radiative energy loss of electrons at temperatures above 2.0MK. All these observational findings support the validity of the investigated reconnection model for plasma heating in the upper solar atmosphere by flux cancellation.'
author:
- 'Sung-Hong Park'
title: An Observational Test of Solar Plasma Heating by Magnetic Flux Cancellation
---
Introduction {#sec:intro}
============
Reconnection is a physical process in a magnetized plasma, in which magnetic field lines with antiparallel components are brought together in a current sheet or at a magnetic null point, break up, and then reconnect to establish a new, lower-energy configuration [@2000mare.book.....P]. Through this reconnection process, free energy stored in stressed magnetic fields is liberated in the forms of plasma thermal and kinetic energy, and hence radiation, all of which may be relevant for solar activity including explosive, transient and/or eruptive events. In order to understand how the reconnection process facilitates these solar eruptive events, there have been many recent developments in modelling reconnection in kinetic, hybrid and magnetohydrodynamic (MHD) frameworks with a variety of different current sheet structures in two or three dimensions [for reviews, see @1995GMS....90..139L; @1999SoPh..190....1P; @2010RvMP...82..603Y; @2011AdSpR..47.1508P; @2012RSPTA.370.3169P; @2016RSPSA.47260479Z and references therein]. In addition, many observed signatures have been reported, mainly interpreting large-scale morphological structures, plasma motions, heating or acceleration as signatures suggested in reconnection models. Examples include hot, cusp-shaped flare loops in soft X-ray [@1992PASJ...44L..63T], loop-top hard X-ray sources [@1994Natur.371..495M; @2003ApJ...596L.251S], current sheet-like structures [@2003JGRA..108.1440W; @2010ApJ...723L..28L; @2016NatCo...711837X; @2018ApJ...853L..18Y], plasma inflows [@2001ApJ...546L..69Y; @2009ApJ...703..877L; @2013NatPh...9..489S; @2015NatCo...6.7598S; @2015ApJ...798L..11Y] and outflows in the vicinity of current sheet-like structures or coronal magnetic null points [@2004ASPC..325..361A; @2010ApJ...722..329S; @2013ApJ...767..168L; @2016ApJ...818L..27C], hot plasmoids ejected from flare sites [@1995ApJ...451L..83S; @2010ApJ...711.1062N; @2012ApJ...745L...6T], inverse Y-shaped jet-like features [@1999ApJ...513L..75C; @2007Sci...318.1591S; @2014Sci...346A.315T], interaction between coronal loops [@1996SSRv...77....1S; @2006ApJ...646..605S; @2018ApJ...854..178N], and drifting pulsating structures observed in radio dynamic spectra . However, more direct observational evidence that quantitatively characterizes magnetic reconnection in the solar atmosphere, particularly in the context of energy release, has been rarely reported.
Photospheric magnetic flux cancellation has been thought to appear in consequence of (or as a cause of) magnetic reconnection; it is observed as the mutual disappearance of converging photospheric magnetic patches of opposite polarities [@1985AuJPh..38..855L; @1985AuJPh..38..929M]. Two scenarios are often employed, the emergence of U-shaped loops and the submergence of $\Omega$-shaped loops . In these scenarios, magnetic patches of opposite polarities are brought closer to each other, make a field-line connection via the process of magnetic reconnection [@1993SoPh..143..119W], and then the newly formed U-shaped/$\Omega$-shaped loops continue to emerge/submerge respectively. Such convergence and gradual disappearance of the opposite-polarity magnetic patches are found in photospheric magnetogram time sequences, sometimes accompanied by consistently blue-shifted or red-shifted Doppler velocities, on average, within the patches in the course of cancellation when observed near the solar disk center . Flux cancellation events are also frequently observed with transient brightenings in many spectral lines in the vicinity of cancellation regions .
Recent observations of solar magnetic fields with unprecedented spatial and temporal resolutions have revealed the presence of a complex, mixed polarity magnetic field distribution, as well as a significant, rapid decrease of magnetic flux, at the photospheric base of active region coronal loops as well as in both quiet-Sun internetwork and network regions . For example, @2017ApJS..229....4C detected the cancellation of photospheric magnetic flux at a rate of 10$^{15}$Mxs$^{-1}$ at the footpoints of bright coronal loops in extreme ultraviolet (EUV), using line-of-sight (LOS) magnetic field data from the Imaging Magnetograph eXperiment [IMaX; @2011SoPh..268...57M] on the SUNRISE balloon-borne observatory [@2010ApJ...723L.127S; @2017ApJS..229....2S]. At the loop footpoints, they also found small-scale chromospheric jets during the flux cancellation which may supply mass and energy to overlying coronal loops.
Motivated by such high-resolution observations of magnetic flux cancellation, @2018ApJ...862L..24P proposed an analytic model of magnetic reconnection for solar chromospheric and coronal heating. The model allows for calculating the rate of magnetic energy converted to heat at a reconnecting current sheet, based on a few key parameters which may be derived from photospheric magnetic field observations. In order to examine the model validity, two-dimensional (2D) and three-dimensional (3D) resistive MHD simulations of two cancelling magnetic sources of opposite polarities in the presence of an overlying, background horizontal magnetic field and a stratified atmosphere have been constructed [@2019ApJ...872...32S; @2020ApJ...891...52S]. It has been found from these simulations that the analytic model’s estimate for the magnetic energy release rate is in good agreement with that explicitly calculated from the values assigned to the associated simulation parameters.
In this study we analyze photospheric magnetic field and multi-wavelength UV/EUV observations of a small-scale flux cancellation event in a quiet-Sun internetwork region. We intend to (1) estimate the rate of magnetic energy released as heat as proposed in the @2018ApJ...862L..24P reconnection model and (2) quantitatively validate how the model performs comparing the estimated magnetic energy release rate with the rate of energy loss by radiation. The model is described in §\[sec:model\], the data analysis techniques are presented in §\[sec:data\_analysis\], and analysis results in §\[sec:results\]. Finally, in §\[sec:discussion\], we summarize our main findings.
![Reproduction of Figure 3 of @2018ApJ...862L..24P, showing a cartoon of magnetic field structures for two opposite-polarity sources (A and B) of magnetic flux $\pm F$ situated in an overlying uniform horizontal magnetic field $B_{0}$ when (a) $d$$>$$d_{0}$, (b) $d$$=$$d_{0}$, and (c) $d$$<$$d_{0}$, where $d$ is the half-separation between the two flux sources and $d_{0}$ is the flux interaction distance. Separatrix magnetic field lines are marked by dashed lines, other magnetic field lines by solid lines, a null point (N) by a filled circle, and a separator (S) by an open circle.[]{data-label="fig:priest_model"}](priest_cartoon.eps){width="75.00000%"}
Reconnection Model for Heating {#sec:model}
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An analytic model for magnetic reconnection driven by converging photospheric magnetic patches of opposite polarities was proposed by @2018ApJ...862L..24P, in which two opposite-polarity photospheric sources (A and B as marked in Figure \[fig:priest\_model\]) with magnetic flux $\pm F$ situated in an overlying uniform horizontal magnetic field $B_{0}$. The two sources are initially separated from each other by a distance $d$, and start to reconnect as they approach each other. Figure \[fig:priest\_model\] schematically describes (a) the initial configuration of the model magnetic field lines and separatrix magnetic field lines, and (b–c) their reconfiguration over the course of reconnection. The configuration of the model magnetic field can be described by the value of a key parameter, called the flux interaction distance $d_{0}$ [@1998ApJ...507..433L], which for 3D sources, is written as $$d_{0} = \sqrt{\frac{F}{\pi B_{0}}}.
\label{eq:d0}$$ When $d$$>$$d_{0}$ as shown in Figure \[fig:priest\_model\][a]{}, there is no magnetic field connecting the two sources and two first-order null points lie on the photosphere between the sources. In the case of $d$$=$$d_{0}$, there is a local bifurcation in which the nulls combine to form a high-order null at the origin (see Figure \[fig:priest\_model\][b]{}). Following @2018ApJ...862L..24P, as the sources approach closer to each other, such that $d$$<$$d_{0}$ (Figure \[fig:priest\_model\][c]{}), reconnection is driven and the location of a semicircular separator rises in the upper atmosphere to the height $z_{s}$ given by $$z_{s} = \sqrt{{d_{}}^{2/3}\,{d_{0}}^{4/3} - {d_{}}^{2}}.
\label{eq:zs}$$ In the case of fast reconnection, the total rate of magnetic energy released as heat is $$\frac{\mathrm{d}W}{\mathrm{d}t} = 0.4\,S_{i} = 0.8\,\frac{v_{i}{B_{i}}^{2}}{\mu} L L_{s} = \frac{1.6\pi}{3} \frac{v_{0}{B_{0}}^{2}}{\mu} {d_{0}}^{2} \frac{M_{A0}}{\alpha} \frac{1-(d/d_{0})^{4/3}}{(d/d_{0})^{2/3}},
\label{eq:dwdt}$$ where $S_{i}$ is the Poynting flux flowing into both sides of the current sheet region of given length $L$ and depth $L_{s}$$=$$\pi z_{S}$ at the separator. $v_{i}$ and $B_{i}$ are the reconnection inflow speed and the strength of the magnetic field drawn into the current sheet, respectively. $\mathrm{d}W/\mathrm{d}t$ is derived based on the assumption that 40% of the Poynting influx is converted to heat during fast reconnection [@2014masu.book.....P]. The physical quantities of $v_{i}$, $B_{i}$, $L$ and $L_{s}$, which are associated with the reconnection current sheet and magnetized plasma therein, can be basically derived as functions of $d$, $d_{0}$, $B_{0}$ and the flux source speed on the photosphere $v_{0}$$=$$\dot{d}$. An Alfvén Mach number $\alpha$ is defined as $v_{i}/v_{Ai}$ (where $v_{Ai}$$=$$B_{i}/\sqrt{\mu \rho_{i}}$ and $\rho_{i}$ is the density of the inflowing plasma), and a hybrid Alfvén Mach number $M_{A0}$ as $v_{0}/v_{A0}$ (where $v_{A0}$$=$$B_{0}/\sqrt{\mu \rho_{i}}$). To estimate $\mathrm{d}W/\mathrm{d}t$ for a flux cancellation event in a quiet-Sun region, we adopt here values of $\alpha$$=$0.1 and $M_{A0}$$=$0.1 [@2014masu.book.....P; @2018ApJ...862L..24P].
Observations and Data Analysis {#sec:data_analysis}
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An explosive, small-scale flux cancellation event was observed on 2017 November 7 in a quiet-Sun internetwork region (located at heliographic coordinates N05E30) by the *Solar Dynamics Observatory* [*SDO*, @2012SoPh..275....3P] and the *Interface Region Imaging Spectrograph* [*IRIS*, @2014SoPh..289.2733D]. In this study, we analyze the following dataset over an interval of 5.2 hr from 2017 November 07 00:00 TAI (hereafter referred to as the “target interval”), comprising: (1) photospheric line-of-sight magnetograms with a spatial resolution of 1 (pixel size of 0.5) obtained with a 45 s cadence by the *SDO*/Helioseismic and Magnetic Imager [HMI, @2012SoPh..275..207S], (2) multi-wavelength solar images (with 0.6/pixel) at 12 s cadence in seven EUV channels and at 24 s cadence in one UV channel by the *SDO*/Atmospheric Imaging Assembly [AIA, @2012SoPh..275...17L], and (3) *IRIS* slit-jaw images (SJIs) at the Si IV 1400[Å]{} with a 40[Å]{} bandpass at 36 s cadence. Note that *IRIS* SJIs at 1400[Å]{} are only available from 04:50 TAI on 2017 November 07. The observation dataset is summarized in Table \[tbl:obs\_data\].
[l|l|l]{}\[b!\] \[tbl:obs\_data\] *SDO*/HMI & Line-of-sight magnetograms & Photospheric magnetic flux\
*SDO*/AIA & EUV images at 94, 131, 171, 193, 211, 304 and 335[Å]{} & $\log_{10}$T$=$$4.9 \-- 6.85$\[K\]\
*SDO*/AIA & UV images at 1600[Å]{} & $\log_{10}$T$=$4\[K\]\
*IRIS* & Slit-jaw images at 1400[Å]{} & $\log_{10}$T$=$$3.7 \-- 5.2 $\[K\]\
The radial component $B_\mathrm{r}$ of the magnetic field to the solar surface is derived from the LOS component $B_\mathrm{LOS}$, assuming the magnetic field is radial at the photosphere: i.e., $B_\mathrm{r}$=$B_\mathrm{LOS}$/cos($\theta$), where $\theta$ is the heliocentric angle of the given observed location (i.e., the angle between the LOS and the local normal to the surface) - here $\cos(\theta)$$=$0.85–0.89. This assumption is consistent with plage field structure as observed by HMI, and more appropriate than other correction options given the nature of the field and the location [refer to @2017SoPh..292...36L].
![A full-disk photospheric LOS magnetogram observed by *SDO*/HMI on 2017 November 7 00:00 TAI is shown in panel (a). The white box in panel (a) indicates a quiet-Sun internetwork region shown in panel (b). Our region of interest (ROI, panel (c)) is marked with the white box in panel (b), in which a pair of opposite-polarity magnetic flux patches show a converging motion as well as a decrease in magnetic flux of both polarities (i.e., a so-called “photospheric magnetic flux cancellation event”). The black box in panel (b) is a weaker-field region compared to the nearby ROI, which is chosen to calculate the average strength of the overlying horizontal magnetic field.[]{data-label="fig:target_region"}](target_region.eps){width="98.00000%"}
As shown in Figure \[fig:target\_region\], the flux cancellation event is located in a quiet-Sun internetwork region denoted by the white box in the full-disk LOS magnetogram in panel (a). The white box in panel (b) outlines the region of interest (ROI, panel (c)) in which a pair of opposite-polarity and similar-sized magnetic flux patches in close proximity converge steadily over the target interval. The patches eventually show a concurrant decrease in magnetic flux in both the positive and negative polarities for the last 1.5 hr of the target interval. During the cancellation event, we find transient brightenings with lifetimes of a few to several tens of minutes from all UV and EUV wavelength channels on top of a long-term (a few hours) gradual increase and then decrease of the average UV/EUV intensities. A detailed description of the flux cancellation event is presented in Section\[sec:results\].
In order to determine the parameters of $d$, $d_{0}$ and $v_{0}$ in Equation (\[eq:dwdt\]) for the pair of opposite-polarity magnetic patches in the ROI, using a sequence of co-aligned $B_\mathrm{r}$ images, we develop an algorithm for automatic identification and tracking of magnetic patches. First, patch identification is carried out for a given $B_\mathrm{r}$ image of the ROI at a single point in time, based on the following steps: (1) making a bitmap based on a given threshold value $B_\mathrm{th}$ of $|B_\mathrm{r}|$, containing one of three values, i.e., $+$1 for $B_\mathrm{r}$$\geq$$B_\mathrm{th}$, 0 for $-B_\mathrm{th}$$<$$B_\mathrm{r}$$<$$B_\mathrm{th}$, and $-$1 for $B_\mathrm{r}$$\leq$$-B_\mathrm{th}$, (2) finding and labeling positive polarity patches, each of which is defined as a group of pixels signed with $+$1 and also located next to each other; repeating the same for negative polarity patches (i.e., signed with $-$1), (3) grouping patches if the shortest distance between the patches is less than a grouping distance $d_\mathrm{gp}$, and (4) removing too small-sized patches (i.e., the total number of pixels belonging to a given patch$<$$N_\mathrm{min}$) and patch pixels located within a distance $d_\mathrm{sd}$ from all four sides of the image, if any. Next, all patches identified and labeled therein are tracked in time from two consecutive $B_\mathrm{r}$ images, considering four possible cases of moving, merging, splitting and newly emerging patches as follows. (1) Moving: if any single pixel of a patch $\mathcal{A}$ at $\mathrm{T_{0}}$ is matched with pixels of a patch $\mathcal{B}$ at $\mathrm{T_{1}}$=$\mathrm{T_{0}}$$+$45 s with the same polarity as $\mathcal{A}$, then $\mathcal{A}$ and $\mathcal{B}$ are considered as the same labeled patch. Note that when comparing the pixels between $\mathcal{A}$ and $\mathcal{B}$, a parameter $d_\mathrm{ds}$ for the maximum possible displacement in both $\pm$x and $\pm$y directions is used to take into account patch motions over 45 s. (2) Merging: if pixels of a patch $\mathcal{C}$ at $\mathrm{T_{1}}$ are matched with pixels of two or more patches at $\mathrm{T_{0}}$ even considering their motions by $\pm d_\mathrm{ds}$, then those patches at $\mathrm{T_{0}}$ are regarded as merging into one patch $\mathcal{C}$. (3) Splitting: if pixels of a patch $\mathcal{D}$ at $\mathrm{T_{0}}$ are matched with pixels of two or more patches (apart from each other with a separation distance greater than $d_\mathrm{sp}$), then we decide that $\mathcal{D}$ splits into two or more. The largest patch at $\mathrm{T_{1}}$ continues to be labeled $\mathcal{D}$ while the smaller patches are newly labeled at $\mathrm{T_{1}}$. (4) Newly emerging: if any single pixel of a patch $\mathcal{E}$ at $\mathrm{T_{1}}$ is not matched with pixels of all patches identified at $\mathrm{T_{0}}$ as well as their displacement by $\pm d_\mathrm{ds}$, then $\mathcal{E}$ is considered as a newly emerging patch. Here, values for all input parameters of the patch identification and tracking algorithm are set as follows: $B_\mathrm{th}$=20G, $d_\mathrm{gp}$=1.5, $N_\mathrm{min}$=5 pixels, $d_\mathrm{ed}$=2.5, $d_\mathrm{ds}$=0.5, and $d_\mathrm{sp}$=4. After an across-the-board survey with different sets of the input parameter values, the above values have been chosen in two respects: first, both the main positive and negative patches (as shown in Figure \[fig:target\_region\](a)) of the flux cancellation event are persistently tracked, and second, the center-of-mass (CoM) separation between the two patches (i.e., the distance between the flux-weighted centroid positions of the two patches) smoothly changes with a minimum of fluctuation over the target interval.
![Parameters of (a) CoM separation $d$ between the main positive and negative magnetic patches on the photosphere, (b) magnetic flux $F$, (c) average overlying horizontal magnetic field strength $B_{0}$ (solid line) with its standard deviation $\sigma(B_{0})$ (error bars) and (d) flux interaction distance $d_{0}$, derived from a sequence of co-aligned $B_\mathrm{r}$ images with the magnetic patch identification and tracking algorithm developed in this study. In panel (a), the solid and dashed lines represent, respectively, the 40-point (corresponding to 0.5 hr) running average of $d$ (raw data marked by cross symbols) and the least squares regression line of the running average. In panel (c), the dotted line indicates the angle $\alpha$ between the average overlying horizontal field vector and the line connecting the CoM of the positive patch to that of the negative patch.[]{data-label="fig:patch_outputs"}](patch_outputs.eps){width="100.00000%"}
With the outputs of the algorithm for identification and tracking of magnetic patches (i.e., pixel positions of tracked patches), we determine the CoM separation (Figure \[fig:patch\_outputs\][(a)]{}) between the two main patches of opposite polarities, as a function of time, which is assigned as $d$ in Equation \[eq:dwdt\]. The converging speed $v_{0}$ of $\sim$0.1 km/s is then derived from the slope of the least squares regression line of the 40-point (i.e., 0.5 hr) running average of $d$ over the target interval, in order to avoid some unreliable, sudden, large fluctuations mainly due to the fact that each of the main positive and negative patches occasionally but instantaneously merges with nearby patches and/or splits into smaller patches over the course of tracking. The adjusted R-squared value, widely used goodness-of-fit measure, is 0.95, indicating that the linear least squares regression method provides an adequate fit to the time series of $d$. The magnetic flux $F$ (which is the same between the positive and negative polarities in the proposed model by @2018ApJ...862L..24P) is defined here by the total unsigned flux of the two main positive and negative patches divided by a factor of 2 (see Figure \[fig:patch\_outputs\][(b)]{}). To estimate the overlying horizontal magnetic field strength ($B_{0}$, solid line in Figure \[fig:patch\_outputs\][(c)]{}), we first construct a potential magnetic field from a given photospheric $B_\mathrm{r}$ image of the full field of view in panel (b) of Figure \[fig:target\_region\]. $B_{0}$ is then computed by averaging the horizontal component of the potential field, at heights ranging from 7 to 15Mm, over a weaker-field region (marked with the black box in Figure \[fig:target\_region\][(b)]{}) compared to the ROI. The weaker-field region considered here to calculate $B_{0}$ was selected to avoid the closed coronal magnetic field connecting the opposite-polarity patches, but to include a highly inclined field (on average, $>$75) with respect to the radial direction. The specified height range of 7–15Mm was chosen, of which the lower limit is defined as the maximum height of $z_{s}$ over the target interval (see Figure \[fig:time\_profiles\][(b)]{}) and the upper limit as the height where the average strength of the horizontal potential field becomes smaller than that of the vertical field. The selected weaker-field region is found to have the following characteristics: (1) it is located close to the ROI; (2) the horizontal potential field in the selected region is, on average, stronger than the vertical field by a factor of $\sim$2 at heights of 7–15Mm; (3) the average angle of the overlying horizontal potential field vector relative to the line connecting the CoM of the positive patch to that of the negative patch ranges from 130 to 180 (refer to Figure \[fig:patch\_outputs\][(c)]{}). The flux interaction distance $d_{0}$ (plotted in Figure \[fig:patch\_outputs\][(d)]{}) is calculated from $F$ and $B_{0}$ as defined in Equation \[eq:d0\].
The differential emission measure (DEM) is an estimate of the total number of electrons squared along the observed LOS (similar to a column mass) at a given temperature. It has been extensively used as a diagnostic tool to characterize temporal variations and spatial distributions of electron density and temperature in the solar atmosphere. The regularized inversion code developed by is used here to produce DEM maps at 48 s cadence from *SDO*/AIA co-aligned images in six EUV channels centered at 94, 131, 171, 193, 211 and 335[Å]{}, which have strong responses to logarithmic temperatures of $\log_{10}$T$=$6.85, 5.75, 5.95, 6.20, 6.25 and 5.35\[K\], respectively, from the latest Version 9 [@2019ApJS..241...22D] of CHIANTI. The temperature bins, as the inputs to the regularization algorithm, are chosen to be a total of 24 bins in the range of $\log_{10}$T$=$5.3–7.7\[K\], equally spaced on a logarithmic scale. In , it was found that this regularized inversion method is able to successfully recover the expected DEM from simulated data of a variety of model DEMs. Note that the inversion code provides uncertainties in both the DEM and temperatures, which allows us to estimate the accuracy of the regularized DEM solution.
Results {#sec:results}
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We first examine the morphological structure and overall dynamics of the two main magnetic patches of opposite polarities over the target interval, based on a sequence of $B_\mathrm{r}$ images (row 1 in Figure \[fig:img\_seq\]). At the beginning, the CoM of the leading negative patch is 3.2Mm apart from that of the following positive patch. The separation distance between the CoM positions of the leading and following patches continues to decrease throughout the target interval; i.e., the two patches consistently show a converging motion. We also find that at first the line connecting the two CoM positions is tilted $\sim$0 with respect to the east-–west direction. The tilt angle then gradually increases counterclockwise as the leading/following patches on the photosphere consistently move northward/southward, respectively. This rotation motion between the two opposite-polarity patches along the north–south direction is not thought to be caused by the differential rotation of the solar surface, because their directions are perpendicular to each other. The observed tilt angles disobey Joy’s law; in particular, for the latter half of the target interval they differ from that expected by Joy’s law angle by more than 90.
![A sequence of images for the target interval of 5.2 hr that show the flux cancellation event: (from top to bottom) HMI $B_\mathrm{r}$, HMI $V_\mathrm{LOS}$, AIA 1600Å, *IRIS* SJI 1400Å, all AIA EUV channels (304, 131, 171, 193, 211, 335, and 94Å).[]{data-label="fig:img_seq"}](img_seq.eps){width="100.00000%"}
![A sequence of different emission measure (DEM) maps at different temperature ranges.[]{data-label="fig:dem"}](dem.eps){width="100.00000%"}
In the time-sequence images of $B_\mathrm{r}$ (row 1 in Figure \[fig:img\_seq\]), it is shown that the pair of the opposite-polarity patches experience a large decrease of magnetic flux in both the positive and negative polarities over the last 1.5 hr of the target interval (i.e., at a rate of 10$^{15}$Mxs$^{-1}$ as shown in Figure \[fig:patch\_outputs\][(b)]{}). At the same time, the patches decrease in size and eventually disappear. Meanwhile, an emergence of negative-polarity magnetic flux occurs in the middle of the observation (i.e., between $\sim$02:00 to 02:40 TAI; refer to panel (b) of Figure \[fig:patch\_outputs\]) at the location ($x$,$y$)$=$(8,4)Mm. A smaller-sized, weaker, positive-polarity counterpart of the emerging negative flux first appears at ($x$,$y$)$=$(13,4)Mm, which is sufficiently far from the main positive patch (green contour, row 1 in Figure \[fig:img\_seq\]), so that it is not assigned to the main positive patch until they get close enough at 02:57 TAI to be considered as being merged by the tracking algorithm (cf. Section \[sec:data\_analysis\]). From HMI LOS Doppler velocity images (row 2, Figure \[fig:img\_seq\]), signatures of red-shifted Doppler velocities (downflows) are found in most areas of the two patches over the target interval. In multi-wavelength UV and EUV images (rows 3 to 11, from low- to high-temperature channels), we see a couple of impulsive and explosive brightenings localized around the two patches of flux cancellation in the shape of arcades whose two ends are anchored in the main positive and negative magnetic patches on the photosphere (e.g., see the AIA EUV images at 02:47:11 TAI in Figure \[fig:img\_seq\]). These transient, bright arcade-shaped structures (located around the center of the EUV images) tend to appear as small as $\sim$1–2Mm while the total unsigned flux of the two patches, as well as their CoM separation, decreases during the second half of the target interval. In Figure \[fig:dem\], similar arcade-shaped structures are found in DEM maps over the wide range of T$=$0.5–20MK. They have relatively large DEM values, compared to those in the background quiet-Sun corona, peaking at T$=$1.6–2.0MK. Those multi-temperature, arcade structures comprise two distinct populations of electrons at different temperatures: i.e., one in the T$=$0.5–2.5MK range and the other in the higher temperature range of T$=$8.0–20MK. Note that the mean coronal temperature is found to be in the range of $\sim$1.4–1.8MK for quiet-Sun regions for years 2010 to 2017 [refer to @2017SciA....3E2056M].
In Figure \[fig:time\_profiles\], we present how key parameters in the examined reconnection model vary over the target interval of 5.2 hr, specifically in relation to variations in the average EUV intensity over the ROI. Three AIA EUV channels (94, 171 and 304[Å]{}) are selected here to calculate the average EUV intensity for reference. In the ROI, we first find a long-term (a few hours) variation of a gradual increase and then decrease in the normalized average EUV intensity profiles ($I_\mathrm{avg}$, colored lines in panels (a–c)) from all three different channels over the target interval. On top of the long-term variation, there are numerous impulsive brightenings with two significant peaks in particular at 02:45 and 05:00 TAI. It is also found that the ratio of the separation distance to the interaction distance ($d/d_{0}$, black solid line in panel (a)) shows declining phases (marked with gray bars) at nearly constant rates about 0.5–1 hr prior to the onset of these EUV brightenings. Moreover, $d/d_{0}$ tends to show relatively large fluctuations and/or persistent increases during the brightenings, compared to the declining phases. The height of the reconnecting magnetic separator ($z_{s}$, panel (b)) is estimated from Equation (\[eq:zs\]) to be steadily located at about 6Mm until 03:30, but then it decreases until its final estimated height at $\sim$2.5Mm.
![Time profiles of (a) the ratio of the separation distance to the interaction distance ($d/d_{0}$, black line) with the normalized average EUV intensities at 94, 171 and 304[Å]{} ($I_\mathrm{avg}$, colored lines), (b) the height of the reconnecting magnetic separator ($z_{s}$, black line), (c) the total rate of magnetic energy released as heat through reconnection ($\mathrm{d}W/\mathrm{d}t$, gray bars).The vertical error bars on top of the $\mathrm{d}W/\mathrm{d}t$ bar plot indicate upper and lower bounds of $\mathrm{d}W/\mathrm{d}t$. In panels (d–f), $\mathrm{d}W/\mathrm{d}t$ is plotted with the average electron number densities ($n_\mathrm{e}\mathrm{(T)}$), the total electron counts ($N_\mathrm{e}\mathrm{(T)}$), and the radiative energy loss rates ($\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$) at different temperature bins, respectively. The black line with cross symbols in panel (f) represents the sum of the estimated radiative loss rates for electrons at T$>$1.6MK.[]{data-label="fig:time_profiles"}](time_profiles.eps){width="100.00000%"}
The total rate of magnetic energy released as heat through reconnection ($\mathrm{d}W/\mathrm{d}t$, gray bars in panel (c) of Figure \[fig:time\_profiles\]) is estimated to be several $10^{23}$ergs$^{-1}$, or $10^{5}$ergcm$^{-2}$s$^{-1}$ in the ROI. Upper and lower bounds of $\mathrm{d}W/\mathrm{d}t$ are also derived applying $B_{0}$$\pm$$\sigma (B_{0})$ to Equation (\[eq:dwdt\]), respectively. We find that the estimated magnitude of $\mathrm{d}W/\mathrm{d}t$ corresponds to the energy flux required to heat the quiet-Sun corona . In panel (c), we find that $\mathrm{d}W/\mathrm{d}t$ shows similar variations on both short-term (a few to several tens of minutes) and long-term (a few hours) scales as seen in the EUV intensity profiles, sometimes with a time lag of $\sim$5–10 minutes. This time lag between the magnetic energy release and EUV intensity can be explained by the heating scenario of chromospheric evaporation [first proposed by @1968ApJ...153L..59N] in which the rapid downward acceleration of non-thermal particles produced by reconnection, as well as the onset of the consequent evaporation, precedes thermal (soft X-ray and EUV) coronal emissions typically by several minutes .
We finally explore how the estimated magnetic energy released as heat by flux cancellation is converted into (or contributes to) radiative energy losses of electrons in the ROI as a function of temperature. For this, the average electron number density ($n_\mathrm{e}\mathrm{(T)}$, shown in panel (d) of Figure \[fig:time\_profiles\]) over the ROI is first derived at the different ranges of temperature used in the DEM estimations, assuming that the integration distance $d_\mathrm{LOS}$ contributing to the DEM along the LOS is 1Mm: i.e., $n_\mathrm{e}\mathrm{(T)}$$=$$\sqrt{\mathrm{DEM(T)}/d_\mathrm{LOS}}$. The value of $d_\mathrm{LOS}$ used here was chosen, because, in the case of $d_\mathrm{LOS}$$=$1Mm, we can achieve a typical value of $n_\mathrm{e}$(T$=$1.3–1.6MK)=$10^{9}$cm$^{-3}$ as reported in the quiet-Sun corona. In this calculation of $n_\mathrm{e}\mathrm{(T)}$, we consider only the pixels with the signal-to-noise ratios of the estimated DEM values greater than 5. The total number of electrons ($N_\mathrm{e}\mathrm{(T)}$, panel (e)) is also estimated in the volume of interest $V$$=$$0.5 \pi S z_{s}$, where $S$ is the total area of both the main positive and negative patches on the photospheric surface. This volume of interest is estimated based on a half-torus geometry with a major (outer) radius of $z_{s}$ and a minor (inner) radius of $\sqrt{0.5 S / \pi}$. It is found that in general temporal variations of the total electron counts at T$\geq$7.9MK follow along with those of $\mathrm{d}W/\mathrm{d}t$, as well as shown in the EUV intensity profiles. The time profiles of $N_\mathrm{e}$(T$=$1.3–2.5MK) show relatively large values for the first 2 hr of the target interval, when compared to those for the rest interval. The two prominent peaks in the time profiles of $N_\mathrm{e}\mathrm{(T)}$ at 02:45 and 05:00 TAI, are clearly seen only at T$\geq$7.9MK. The total rate of radiative energy loss ($\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$, panel (f) of Figure \[fig:time\_profiles\]) for a given temperature bin is $$\frac{\mathrm{d}L\mathrm{(T)}}{\mathrm{d}t} = \int_{V} \tilde{\Lambda}\mathrm{(T)} \,n_\mathrm{e}^{2}\mathrm{(T)}\,\mathrm{d}V,
\label{eq:rad_loss}$$ where $\tilde{\Lambda}\mathrm{(T)}$ is the average of the radiative loss function (i.e., emissivity per unit emission measure, adopted from Version 9 of CHIANTI) over the given temperature bin. The short-term dynamic changes in $\mathrm{d}W/\mathrm{d}t$ are shown in variations in $\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$ of electrons at most temperature bins over the target interval, as in $N_\mathrm{e}\mathrm{(T)}$. We also find that values of $\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$ at T$=$1.0–2.5MK are of the same order of magnitude as $\mathrm{d}W/\mathrm{d}t$, of which overall values are in the range of about $10^{23}$–$10^{24}$ergs$^{-1}$. In addition, $\mathrm{d}W/\mathrm{d}t$ is well correlated with the sum (black line in Figure \[fig:time\_profiles\][(f)]{}) of $\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$ for electrons at higher temperatures (i.e., T$\geq$2.0MK) compared to the mean quiet-Sun coronal temperature of $\sim$1.8MK in early 2017 reported by @2017SciA....3E2056M. This supports the validity of our estimate for $\mathrm{d}W/\mathrm{d}t$, in the respect that it can be used to reproduce the observed coronal heating and radiative energy loss over the target interval.
Summary and Conclusions {#sec:discussion}
=======================
In this paper we have investigated photospheric magnetic field and multi-wavelength UV/EUV observations of a small-scale magnetic flux cancellation event in a quiet-Sun internetwork region of interest (ROI) over a target interval of 5.2 hr. Specifically, we focused on how much the total rate of magnetic energy released as heat ($\mathrm{d}W/\mathrm{d}t$) at the ROI can contribute to the heat requirement of the coronal plasma therein, and also how well temporal variations in $\mathrm{d}W/\mathrm{d}t$ are correlated with those in the total rate of radiative energy loss ($\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$) of electrons at different temperature ranges. To answer the questions, $\mathrm{d}W/\mathrm{d}t$ was estimated using the analytic reconnection model of @2018ApJ...862L..24P in which key parameters can be derived from the observed photospheric magnetic field of the cancellation event. The regularized inversion method by was applied to produce the differential emission measure (DEM) maps and eventually $\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$ at the ROI, as a function of temperature, from the multi-channel EUV observations. Our main findings are summarized as follows:
1. The observed flux cancellation event involves a pair of opposite-polarity magnetic flux patches on the photosphere that show a converging motion and red-shifted Doppler velocities (downflows) over the target interval, together with an accompanying decrease in magnetic flux of both polarities at a rate of 10$^{15}$Mxs$^{-1}$ for the last 1.5 hr of the target interval.
2. Several impulsive EUV brightenings are observed in the form of arcades with their two footpoints anchored in these two magnetic patches.
3. The EUV bright arcade-shaped structures are visible in the DEM maps with relatively large DEM values, compared to the background quiet-Sun corona, peaking at T$=$1.6–2.0MK but with contribution over the broad range of T$=$0.5–20MK.
4. The estimated $\mathrm{d}W/\mathrm{d}t$ at the ROI ranges between $2\times10^{23}$ to $1\times10^{24}$ergs$^{-1}$ over the target interval. Based on the total area of the opposite-polarity patches, the energy flux is determined to be $\sim$10$^{5}$ergcm$^{-2}$s$^{-1}$, which corresponds to the same order of magnitude as previously reported in observations of the quiet-Sun corona.
5. Both short-term dynamic variations and long-term gradual trends in the 5.2 hr profile of $\mathrm{d}W/\mathrm{d}t$ are clearly shown in $\mathrm{d}L\mathrm{(T)}/\mathrm{d}t$ of electrons at T$\geq$2.0MK.
All these observational findings lend support to the @2018ApJ...862L..24P model in which reconnection driven by converging and cancelling magnetic patches of opposite polarities on the photosphere can provide sufficient energy to heat the Sun’s upper atmosphere (i.e., chromosphere, transition region, corona). Recent high-resolution observations of quiet-Sun magnetic fields are also favorable to the scenario for coronal heating by magnetic flux cancellation, in the context that a large amount of flux cancellation occurs frequently (i.e., at a rate of 10$^{15}$Mxs$^{-1}$) between dynamic, small-scale magnetic patches on the photosphere [e.g., @2017ApJS..229....4C]. The short-term variations seen in the 5.2 hr profile of $\mathrm{d}W/\mathrm{d}t$ have timescales of a few to several tens of minutes, comparable to the waiting time between successive nanoflares as suggested by hydrodynamic simulations of impulsive coronal heating [e.g., @2014ApJ...784...49C; @2016ApJ...833..217B]. It should be noted, however, that flux cancellation events such as the one examined here should be distributed all over the solar surface with a variety of temporal and spatial scales, and they ought to occur at suitable rates to contribute the required energy dissipation rate, or a significant portion thereof, in order to maintain the coronal temperature of the quiet Sun. In simulations by @2019ApJ...872...32S [@2020ApJ...891...52S], distinct outflows/jets were found to occur at different temperatures at reconnection sites during flux cancellation. However, in the small-scale flux cancellation event of this study, no clear morphological or kinematic signature of outflows/jets was found over the target interval amongst the AIA UV/EUV images, or the *IRIS* 1400[Å]{} SJIs. We note that such narrow sub-arcsecond features as implied by the simulations are difficult to detect at the spatial resolution of AIA and *IRIS*. *IRIS* spectroscopic observations of small-scale flux cancellation events may help identify cancellation-driven outflows/jets and their evolution through temporal analysis of spectral properties, but this is beyond the scope of the present study.
The analytic reconnection model of @2018ApJ...862L..24P presented in Section \[sec:model\] is constructed for a particular case of two equal flux sources of opposite polarities aligned with an overlying uniform horizontal magnetic field. In reality, however, the quiet-Sun magnetic field is more complicated so that the model may need to be improved considering various magnetic field structures and their evolution. The cancellation-based heating model also needs to be further examined with more flux cancellation events observed in both quiet-Sun and active regions as well as in data-driven (meaning realistic) simulations of the solar atmosphere. Moreover, statistical properties of flux cancellation events, such as a space-frequency distribution of estimated magnetic energy released as heat at flux cancellation regions, will help us to examine whether this cancellation-based heating scenario accounts for several aspects of the long-standing coronal heating problem, and also to evaluate how significant the coronal heating by flux cancellation is compared to the other competing heating mechanisms.
The author would like to thank an anonymous referee for thoughtful comments, and K. D. Leka, Kanya Kusano, Graham Barnes, Karin Dissauer, and Manolis K. Georgoulis for their valuable comments and suggestions. The data used in this work are courtesy of the NASA/*SDO* and the AIA and HMI science teams, as well as the NASA/*IRIS* team. *IRIS* is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre. This research has made extensive use of the NASA’s Astrophysics Data System (ADS) as well as the computer system of the Center for Integrated Data Science (CIDAS), Institute for Space-Earth Environmental Research (ISEE), Nagoya University. This work was partially supported by MEXT/JSPS KAKENHI Grant No. JP15H05814.
|
---
author:
- Ulrich Görtz
date: October 2000
title: |
On the flatness of local models\
for the symplectic group
---
Introduction
============
In order to study arithmetic properties of a variety over an algebraic number field, it is desirable to have a model over the ring of integers. We are interested in the particular case of a Shimura variety (of PEL type), and ask for a model over $O_E$, where $E$ is the completion of the reflex field at a prime of residue characteristic $p>0$. Since a Shimura variety of PEL type is a moduli space of abelian varieties with certain additional structure, it is natural to define a model by posing the moduli problem over $O_E$; in the case of a parahoric level structure at $p$, which is the case we are interested in, such models have been defined by Rapoport and Zink [@RZ].
These models are almost never smooth, and it is interesting to study the singularities occuring in the special fibre. It even is not obvious whether the models are flat over $O_E$, although flatness certainly belongs to the minimum requirements for a reasonable model. Rapoport and Zink have conjectured that their models are indeed flat. In the case of a unitary group that splits over an unramified extension of ${{\mathbb Q}}_p$, flatness has been proved in [@G]. In this article we will extend this result to the case of the symplectic group, i.e. we will show the flatness for groups of the form ${\rm Res}_{F/{{\mathbb Q}}_p} GSp_{2r}$, where $F$ is a finite [*unramified*]{} extension of ${{\mathbb Q}}_p$. In the case where the underlying group does not split over an unramified extension, the flatness conjecture is not true as it stands, as has been pointed out by Pappas [@P]; see also [@PR] for ideas on how to proceed in this case.
Several special cases of our main theorem have been known before, by the work of Chai and Norman [@CN], de Jong [@dJ] and Deligne and Pappas [@DP].
To study local questions such as flatness, it is useful to work with the so called local model which étale locally around each point of the special fibre coincides with the model of the Shimura variety, but which can be defined in terms of linear algebra and is thus easier to handle (cf. [@dJ], [@RZ]).
The proof of the flatness conjecture in the symplectic case then starts with the same approach as in the unitary case. Namely, we embed the special fibre of the local model into the affine flag variety, and identify it with the intersection of certain Schubert varieties, coming from simpler local models. Via the theory of Frobenius splittings we can conclude that this intersection is reduced. To make the proof work some additional ingredients are needed, most notably Faltings’ result on the normality of Schubert varieties in the affine flag variety (see [@F] and section \[normality\] below). Furthermore, to analyse the ’simplest’ singular local models, we use a theorem of de Concini about so-called doubly symplectic standard tableaux (see [@dC] and section \[deConcini\]). More precisely, an open neighborhood of their worst singularity in the special fibre is essentially isomorphic to ${\mathop{\rm Spec}}R$, where $$R= {{\mathbb F}}_p[c_{\mu \nu}, \mu, \nu = 1, \dots 2i] /
(CJ_{2i}C^t, C^tJ_{2i}C).$$ Here $C = (c_{\mu\nu})_{\mu,\nu}$ and $J_{2i} =\left( \begin{array}{cc} & {\mathbf J} \\ -{\mathbf J} &
\end{array} \right)$, where ${\mathbf J}$ is the $i \times i$-matrix with 1’s on the second diagonal and 0’s elsewhere. In loc. cit., de Concini gives a ${{\mathbb F}}_p$-basis of $R$, and this will enable us to show that $R$ is reduced (see proposition \[Rred\]). Note that the same singularity has also been studied by Faltings [@F1].
The case treated by Chai and Norman respectively by Deligne and Pappas corresponds to the equations $$CD = DC = p,\quad C = C^t,\quad D = D^t,$$ where $C$ and $D$ are $r \times r$-matrices of indeterminates over ${{\mathbb Z}}_p$. They show that these singularities are Cohen-Macaulay, and that the special fibre of the corresponding scheme is reduced. The proofs rely on the theory of algebras with straightening laws. Our approach gives a new proof for this result and it may be interesting to note that for this special case we do not need to invoke de Concini’s result. See remark i) after theorem \[mainthm\].
The precise statement of the main theorem as well as a sketch of its proof can be found in section \[statement\]. The proof proper of the main theorem is actually rather short. However, to make the paper more accessible, we have included an exposition of some of the results of others that we use. In section \[normality\] we give an exposition of Faltings’ proof of the normality result mentioned above in the case of interest to us, and in section \[deConcini\] we explain the result of de Concini and the crucial consequence of it that we need. The final section \[proof\] brings everything together and contains the proof of the main results.
Finally it is a pleasure to express my gratitude to those who have helped me with this work. First of all, I am much obliged to M. Rapoport for many helpful discussions and for his steady interest in my work. I am grateful to S. Orlik and T. Wedhorn for several useful conversations; T. Wedhorn also made several useful remarks on this text. Furthermore I would like to thank G. Faltings for briefly explaining to me his proof of the theorem on the normality of affine Schubert varieties. This manuscript has been finished during my stay at the Institute for Advanced Study. I am grateful to the Institute for providing a great working environment and to the Deutsche Forschungsgemeinschaft and the National Science Foundation (grant DMS 97-29992) for their support.
The standard local model for the symplectic group {#statement}
=================================================
First of all, let us give the definition of the [*standard local model*]{} which is the object of our studies. Let $O$ be a complete discrete valuation ring with perfect residue class field of characteristic $p > 0$. Let $\pi$ be a uniformizer of $O$ and let $k$ be an algebraic closure of the residue class field of $O$.
Fix a positive integer $r$, and let $n = 2r$.
Denote the quotient field of $O$ by $K$. Let $e_1, \dots, e_n$ be the canonical basis of $K^n$. We endow $K^n$ with the standard symplectic form, given by the matrix $J_{2r} =\left( \begin{array}{cc} & {\mathbf J} \\ -{\mathbf J} &
\end{array} \right)$, where ${\mathbf J} = (\delta_{r-i+1, j})_{ij}$ is the $r \times r$-matrix with 1’s on the second diagonal and 0’s elsewhere.
Let $\Lambda_i$, $0 \le i \le n-1$, be the free $O$-module of rank $n$ with basis $\pi^{-1} e_1, \dots, \pi^{-1} e_{i}, e_{i+1},
\dots, e_n$. This gives rise to a complete lattice chain
& & \_0 & & \_1 & & & & \_[n-1]{} & & \^[-1]{}\_0 &
Furthermore, the lattice chain is selfdual, i.e. for each $\Lambda$ occuring in the lattice chain, the dual $$\Lambda^\ast = \{ x \in K^n; \langle x, y \rangle \in O \text{ for all }
y \in \Lambda \}$$ appears as well. More precisely, $\Lambda_0^\ast = \Lambda_0$, $\Lambda_r^\ast = \pi \Lambda_r$, and $\Lambda_i^\ast = \pi \Lambda_{2r-1}$, $i = 1, \dots, r-1$.
Finally, choose a subset $I = \{ i_0 < \cdots < i_{m-1} \}$, such that for each $i \in I$, $1 \le i \le 2r-1$, also $2r-i \in I$.
The standard local model ${{\rm\bf M}^{\rm loc}}_I$, as defined by Rapoport and Zink, is the $O$-scheme representing the following functor (cf. [@RZ], definition 3.27):
For every $O$-scheme $S$, ${{\rm\bf M}^{\rm loc}}_I(S)$ is the set of isomorphism classes of commutative diagrams
\_[i\_0, S]{} & & \_[i\_1,S]{} & & & & \_[i\_[m-1]{},S]{} & \^& \_[i\_0,S]{}\
& & & & &&& &\
[[F]{}]{}\_0 & & [[F]{}]{}\_1 & & & & [[F]{}]{}\_[m-1]{} && [[F]{}]{}\_0
where $\Lambda_{i,S}$ is $\Lambda_i {\otimes}_{O} {{\cal O}}_S$, and where the ${{\cal F}}_\kappa$ are locally free ${{\cal O}}_S$-submodules of rank $r$ which Zariski-locally on $S$ are direct summands of $\Lambda_{i_\kappa,S}$.
Furthermore, the ${{\cal F}}_i$ have to satisfy the following duality condition: for each $i \in I$, the map $${{\cal F}}_i {\longrightarrow}\Lambda_{i,R} \cong \widehat{\Lambda}_{2r-i,R} {\longrightarrow}\widehat{{{\cal F}}}_{2r-i}$$ is the zero map.
It is clear that this functor is indeed representable. In fact, ${{\rm\bf M}^{\rm loc}}_I$ is a closed subscheme of a product of Grassmannians. More precisely, the local model for the symplectic group $GSp_{2n}$ is a closed subscheme of the local model associated to $GL_{2n}$ (for the same subset $I$); for its definition see [@RZ] or [@G].
We write ${{\rm\bf M}^{\rm loc}}:= {{\rm\bf M}^{\rm loc}}_{\{0, \dots, 2r-1\}}$. Furthermore, we will often write ${{\rm\bf M}^{\rm loc}}_{i_0, \dots, i_{m-1}}$ instead of ${{\rm\bf M}^{\rm loc}}_{\{i_0, \dots, i_{m-1} \}}$.
The aim of this article is to prove the following theorem:
\[mainthm\] The local model ${{\rm\bf M}^{\rm loc}}_I$ is flat over $O$, and its special fibre is reduced. The irreducible components of the special fibre are normal with rational singularities, so in particular they are Cohen-Macaulay.
Sketch of proof (in the Iwahori case $I = \{ 0, \dots, 2r-1 \}$):
We embed the special fibre ${\overline{\rm\bf M}^{\rm loc}}$ of the local model into the affine flag variety ${{\cal F}}= Sp_{2r}(k{(\!( t )\!)})/B$. Furthermore, we denote by ${\widetilde{\rm \bf M}^{\rm loc}}_I$ the inverse image of ${\overline{\rm\bf M}^{\rm loc}}_I \subset Sp_{2r}(k{(\!( t )\!)})/P^I$ under the projection ${{\cal F}}{\longrightarrow}Sp_{2r}(k{(\!( t )\!)})/P^I$.
Then we have $${\overline{\rm\bf M}^{\rm loc}}= {\widetilde{\rm \bf M}^{\rm loc}}_0 \cap {\widetilde{\rm \bf M}^{\rm loc}}_r \cap
\bigcap_{i=1}^{r-1} {\widetilde{\rm \bf M}^{\rm loc}}_{i, 2r-i}.$$
Now the proof of the flatness theorem can be divided into the following three steps.
- The special fibres of the local models ${{\rm\bf M}^{\rm loc}}_0$, ${{\rm\bf M}^{\rm loc}}_r$, ${{\rm\bf M}^{\rm loc}}_{i, 2r-i}$, $ i= 1, \dots, r-1$, are reduced (and irreducible) and thus are Schubert varieties (see section \[simplest\]). Here we use a result of de Concini, which is explained in section \[deConcini\].
- Schubert varieties in the affine flag variety for the symplectic group are normal and Frobenius split, as was shown by Faltings (see section \[normality\]). This implies that intersections of Schubert varieties are reduced. In particular we see that ${\overline{\rm\bf M}^{\rm loc}}$ is reduced (see section \[inters\]).
- The generic points of the irreducible components of the special fibre ${\overline{\rm\bf M}^{\rm loc}}$ can be lifted to the generic fibre (see section \[liftability\]). Since ${\overline{\rm\bf M}^{\rm loc}}$ is reduced, this implies that ${{\rm\bf M}^{\rm loc}}$ is flat over $O$.
[**Remarks.**]{} i) Choosing $I = \{ 0, r \}$, we get the result of Chai and Norman [@CN] respectively Deligne and Pappas [@DP]. Since in this case we have $${\widetilde{\rm \bf M}^{\rm loc}}_I = {\widetilde{\rm \bf M}^{\rm loc}}_0 \cap {\widetilde{\rm \bf M}^{\rm loc}}_r,$$ and ${\widetilde{\rm \bf M}^{\rm loc}}_0$ and ${\widetilde{\rm \bf M}^{\rm loc}}_r$ are smooth, we can conclude that ${\widetilde{\rm \bf M}^{\rm loc}}_I$ (and thus ${\overline{\rm\bf M}^{\rm loc}}_I$) is reduced without using de Concini’s result. Thus in this case our approach gives a proof that does not use the theory of algebras with straightening laws, unlike those of Chai/Norman and Deligne/Pappas.
ii\) While the theorem as it stands deals with the case of $GSp_{2r}$ over ${{\mathbb Q}}_p$, it can easily be generalized to the case of groups of the form ${\rm Res}_{F/{{\mathbb Q}}_p} GSp_{2r}$, where $F$ is an unramified extension of ${{\mathbb Q}}_p$. See [@RZ], chapter 3, for the definition of the local model in this case; see [@G], section 4.6, for details on how to obtain the generalization.
iii\) On the other hand, by some kind of continuity argument, it can be shown that the theorem holds as well if the characteristic of the residue class field of $O$ is 0. Confer [@G], section 4.5.
iv\) It is an interesting question, if ${{\rm\bf M}^{\rm loc}}$ itself is Cohen-Macaulay. In view of the flatness, this is equivalent to ${\overline{\rm\bf M}^{\rm loc}}$ being Cohen-Macaulay. This question can be reduced to a purely combinatorial problem in the affine Weyl group; unfortunately the combinatorics involved seems to be rather difficult.
The affine flag variety for the symplectic group {#normality}
================================================
Definition
----------
In this section we will give the definition of the affine flag variety and reproduce Faltings’ proof for the normality of Schubert varieties contained in it. We will mostly follow Faltings’ paper [@F], and no claim for originality is made. Furthermore, we restrict ourselves to the case of the symplectic group.
Let $G= Sp_{2r}$ (over ${{\mathbb Z}}$). All of the following obviously works for $SL_n$ as well, and in fact even works for any split, simply connected, semi-simple and simple group. We denote by $T$ the torus of diagonal matrices in $G$, and by $B$ the Borel subgroup of upper triangular matrices.
The loop group $LG$ is the ind-scheme (over ${{\mathbb Z}}$) such that for all rings $R$ $$LG(R) = G(R{(\!( t )\!)}).$$ Furthermore, we have the (infinite dimensional) schemes $L^+G$ and $L^{++}G$, $$\begin{aligned}
L^+G(R) & = & G(R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }) \\
L^{++}G & = & \text{kernel of the reduction map } L^+G {\longrightarrow}G,\end{aligned}$$ and the ind-schemes $ L^-G$ and $L^{--}G$, given by $$\begin{aligned}
L^-G(R) & = & G(R[t^{-1}]) \\
L^{--}G & = & \text{kernel of the reduction map } L^-G {\longrightarrow}G.\end{aligned}$$ Inside $L^+G$, we have the standard Iwahori subgroup ${{\cal B}}$, which is the inverse image of the standard Borel subgroup of $G$ under the reduction map $L^+G {\longrightarrow}G$.
The affine flag variety ${{\cal F}}$ is, by definition, the quotient $LG / {{\cal B}}$ in the sense of $fpqc$-sheaves.
Just as the usual flag variety can be interpreted as the space of flags in a vector space, the affine flag variety can be seen as a space of lattice chains. Let us make this more explicit.
Let $R$ be a ring. A lattice in $R{(\!( t )\!)}^n$ is a sub-$R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }$-module ${{\scr L}}$ of $R{(\!( t )\!)}^n$ which is locally free of rank $n$, and such that ${{\scr L}}{\otimes}_{R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }} R{(\!( t )\!)}= R{(\!( t )\!)}^n$.
A sequence ${{\scr L}}_0 \subset {{\scr L}}_1 \subset \dots \subset
{{\scr L}}_{n-1} \subset t^{-1} {{\scr L}}_0$ of lattices in $R{(\!( t )\!)}^n$ is called a complete lattice chain, if ${{\scr L}}_{i+1}/{{\scr L}}_i$ is a locally free $R$-module of rank 1 for all $i$.
By adding all $t^N {{\scr L}}_i$, $N \in {{\mathbb Z}}$, to a lattice chain $({{\scr L}}_i)_i$, we get a complete [*periodic*]{} lattice chain.
We endow $R{(\!( t )\!)}^n$ with the alternating pairing $\langle \cdot , \cdot \rangle$ given by the matrix $J_{2r} = \left( \begin{array}{cc} & {\mathbf J} \\ -{\mathbf J} &
\end{array} \right)$, where ${\mathbf J} = (\delta_{r-i+1, j})_{ij}$ is the $r \times r$-matrix with 1’s on the second diagonal and 0’s elsewhere.
Then we call a lattice chain selfdual, if for all lattices ${{\scr L}}$ in the lattice chain the dual $${{\scr L}}^\ast = \{ x \in K^n; \langle x, y \rangle \in R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }\text{ for all }
y \in {{\scr L}}\}$$ also occurs in the periodic lattice chain.
For example, the standard lattice chain $$\lambda_i = t^{-1} R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^i \oplus R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^{n-i},\quad i=0, \dots, n-1,$$ is selfdual; more precisely we have $\lambda_0^\ast = \lambda_0$, $\lambda_r^\ast = t \lambda_r$, and $\lambda_i^\ast = t \lambda_{2r-1}$, $i = 1, \dots, r-1$.
Finally, we call a lattice chain $({{\scr L}}_i)_i$ special, if $\bigwedge^n {{\scr L}}_0 = R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }$ inside $\bigwedge^n R{(\!( t )\!)}^n = R{(\!( t )\!)}$. For a selfdual lattice chain, this is equivalent to the condition that ${{\scr L}}_0$ is a selfdual lattice.
\[F\_latticechains\] We have a functorial isomorphism
[[F]{}]{}(R) & \^& { ([[L]{}]{}\_i)\_i R[(( t ))]{}\^n }
In particular, ${{\cal F}}$ is an ind-scheme again.
[*Proof.*]{} The map is of course the map induced by $$g \mapsto g \cdot (\lambda_i)_i.$$ It can be seen that the resulting morphism is surjective by proving that special selfdual complete lattice chains locally on $R$ admit a normal form. See the appendix to chapter 3 in [@RZ]. [$\square$]{}
We get a variant of the proposition by considering not special lattice chains, but $r$-special lattice chains, i. e. lattice chains $({{\scr L}}_i)_i$ such that $\bigwedge^n {{\scr L}}_0 = t^r R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }$. This will be useful for embedding the special fibre of the local model into ${{\cal F}}$.
The affine Weyl group $W_a$ for the symplectic group is the semidirect product $$W_r \ltimes \{ (x_1, \dots, x_n, -x_n, \dots, -x_1);
x_i \in {{\mathbb Z}}\},$$ where $W_r = {\mathfrak S}_r \ltimes \{ \pm 1 \}^r$ is the finite Weyl group of the symplectic group. The affine Weyl group $W_a$ for $Sp_{2r}$ is a subgroup of the affine Weyl group for $SL_{2r}$ in a natural way.
Denote the simple reflections in $W$ by $s_1, \dots, s_r$; these together with the affine simple reflection $s_0$ generate $W_a$.
Furthermore, let ${{\cal P}}_i$ be the parahoric subgroup of $LG$ generated by ${{\cal B}}$ and $s_i$. More generally, for $I \subseteq
\{ 0,\dots, r \}$, we write ${{\cal P}}_I$ for the parahoric subgroup generated by ${{\cal B}}$ and $s_i$, $i \in I$, and ${{\cal P}}^I = {{\cal P}}_{ \{ 0,\dots, r \} - I }$.
Corresponding to $s_i$, $i = 0, \dots, r$, we have a subgroup $SL_2$ inside ${{\cal P}}_i$. Concretely, the subgroup $SL_2(R)$ in $LG(R)$ corresponding to $s_i$, $i\in \{1, \dots, r-1\}$ consists of the matrices $${\mathop{\rm diag}}(1^{i-1}, \left( \begin{array}{cc} a & b \\ c & d
\end{array}\right), 1^{2r-2i}, \left( \begin{array}{cc} a & -b \\ -c & d
\end{array}\right), 1^{i-1}), \quad
\left( \begin{array}{cc} a & b \\ c & d
\end{array}\right) \in SL_2(R),$$ and the subgroup corresponding to $s_r$ is $${\mathop{\rm diag}}(1^{r-1}, \left( \begin{array}{cc} a & b \\ c & d
\end{array}\right), 1^{r-1}), \quad
\left( \begin{array}{cc} a & b \\ c & d
\end{array}\right) \in SL_2(R).$$ The subgroup $SL_2$ corresponding to the affine simple reflection $s_0$ is $$\left( \begin{array}{ccccc}
a & & & & t^{-1}b \\
& 1 & & & \\
& & \ddots & & \\
& & & 1 & \\
tc & & & & d
\end{array} \right), \quad
\left( \begin{array}{cc} a & b \\ c & d
\end{array}\right) \in SL_2(R).$$
The unipotent subgroup $\left\{ \left( \begin{array}{cc} 1 & b \\ 0 & 1
\end{array} \right) \right\} $ of $SL_2$ gives us, for each affine simple root $\alpha$, a subgroup ${{\mathbb G}}_a \cong U_\alpha \subset LG$, and by conjugation with elements of the affine Weyl group we get a subgroup $U_\alpha$ for every affine root $\alpha$.
Now let $k$ be an algebraically closed field. We have the Bruhat decomposition $${{\cal F}}(k) = \bigcup_{w \in W_{a}} {{\cal B}}w {{\cal B}}/ {{\cal B}}.$$ The closure $C_w$ of ${{\cal B}}w {{\cal B}}/{{\cal B}}$ is called an (affine) Schubert variety. It is a projective algebraic variety. We have $C_{w'} \subseteq C_w$ if and only if $w' \le w$ with respect to the Bruhat order.
In the case of characteristic 0, the normality of Schubert varieties has been known for some time. Indeed Faltings has proved (see [@BL], [@LS]) that in characteristic 0 the affine flag variety defined above coincides with the one arising in the theory of Kac-Moody algebras. This ’Kac-Moody affine flag variety’ has been investigated by Kumar [@Ku], Littelmann [@Li] and Mathieu [@M]. In particular, it has been shown that in this case Schubert varieties are normal and have rational singularities (see, for example, [@Ku], theorems 2.16 and 2.23).
Now consider the affine flag variety over an algebraically closed field of characteristic $p>0$. We want to show:
[**(Faltings)**]{} All Schubert varieties in ${{\cal F}}$ are normal, with rational singularities. Further, each Schubert variety $X\subset {{\cal F}}$ is Frobenius split, and all Schubert subvarieties of $X$ are simultaneously compatibly split.
For the notion of Frobenius splittings, which was introduced by Mehta and Ramanathan, we refer to their articles [@MR] and [@Ram]. The facts that we will need can also be found in [@G].
The strategy of Faltings’ proof is the following:
The Schubert varieties admit a resolution of singularities $ D(w) {\longrightarrow}C_w$ by the so-called Demazure varieties, which of course factors over the normalization $\widetilde{C}_w$ of $C_w$. One shows that the Demazure varieties are Frobenius split, compatibly with their Demazure subvarieties, and that this gives a Frobenius splitting of $\widetilde{C}_w$. Furthermore for $w' \le w$ we get a closed immersion $\widetilde{C}_w' \subseteq \widetilde{C}_w$. This allows us to define the ind-scheme $\widetilde{{{\cal C}}}$ of the normalizations $\widetilde{C}_w$, and this ind-scheme can even be defined over ${{\mathbb Z}}$. Now the key point is that the action of the $SL_2(R) \subseteq G(R{(\!( t )\!)})$ associated to the simple affine roots can be lifted to an action on $\widetilde{{{\cal C}}}$. The subgroup ${{\cal N}}^-$, which is the inverse image of the unipotent radical of $B^-$ under the projection $L^-G {\longrightarrow}G$, maps isomorphically onto an open subset of ${{\cal F}}$. Let ${\widehat{\cal N}^-}$ be the formal completion of ${{\cal N}}^-$ at the origin. We will see that ${\widehat{\cal N}^-}(R)$ is contained in the subgroup generated by the $SL_2(R)$’s associated to the simple affine roots. As a consequence we get a section of the natural map between the completions of the local rings at the origin of $\widetilde{{{\cal C}}}$ and ${{\cal F}}$. By exploiting the fact that the normality of Schubert varieties is already known in characteristic 0, one shows that this section is indeed an isomorphism; this implies the normality result.
Demazure varieties
------------------
### Definitions
In this section we define the Demazure varieties, which yield a resolution of singularities of Schubert varieties. We continue to work over an algebraically closed field $k$ and we proceed exactly as in the case of the special linear group; cf. [@M], [@G]. Therefore the proofs are omitted.
Let $w \in W_a$ and let $ \tilde{w} = s_{i_1} \cdots s_{i_\ell}$ be a reduced expression for $w$. (We sometimes write $\tilde{w}$ instead of $w$ to indicate that the following definitions really depend on the choice of a reduced decomposition.)
The variety $$D(\tilde{w}) := {{\cal P}}_{i_1} \times^{{\cal B}}\cdots \times^{{\cal B}}{{\cal P}}_{i_\ell}/{{\cal B}}.$$ is called the Demazure variety corresponding to $\tilde{w}$.
If $\tilde{u} = s_{i_1} \cdots \widehat{s_{i_k}} \cdots s_{i_\ell}$ is reduced, we have a closed immersion $$\label{closedimm}
\sigma \colon D(\tilde{u}) {\longrightarrow}D(\tilde{w}).$$
If $\tilde{w} = \tilde{u} \tilde{v}$, i. e. $\tilde{u} = s_{i_1} \cdots s_{i_k}$, $\tilde{v} = s_{i_{k+1}} \cdots s_{i_\ell}$, for some $k$, we get a canonical projection morphism $D(\tilde{w}) {\longrightarrow}D(\tilde{u})$, which is a locally trivial fibre bundle with fibre $D(\tilde{v})$. In particular, let $\tilde{u} = s_{i_1} \cdots s_{i_{\ell-1}}$, $\tilde{v} = s_{i_\ell}$. Then we get a ${{\mathbb P}}^1$-fibration $$\label{p1fibr}
\pi \colon D(\tilde{w}) {\longrightarrow}D(\tilde{u}).$$ The closed immersion $D(\tilde{u}) {\longrightarrow}D(\tilde{w})$ defined above is a section of this fibration. Multiplication gives us a morphism $\Psi_{\tilde{w}} : D(\tilde{w}) {\longrightarrow}X_w$.
\[demvar\_compatible\] i) The Demazure variety $D(\tilde{w})$ is smooth and proper over $k$, and has dimension $l(w)$.
ii\) The morphism $\Psi_{\tilde{w}}:D(\tilde{w}) {\longrightarrow}X_w$ is proper and birational. If $\tilde{u} = s_{i_1} \cdots \widehat{s_{i_k}} \cdots s_{i_\ell}$ is reduced, these morphisms together with the closed immersion (\[closedimm\]) yield a commutative diagram
D() & & X\_u\
& &\
D() & & X\_w
In the Demazure variety $D(\tilde{w})$, we have $l(w)$ divisors $Z_1^{\tilde{w}} ,\dots, Z_{l(w)}^{\tilde{w}} $. These are defined inductively on the length of $w$, as follows.
Write $\tilde{w} = \tilde{u} s_{i_\ell}$. We have the map $\pi: D(\tilde{w}) {\longrightarrow}D(\tilde{u})$, which is a ${{\mathbb P}}^1$-fibration, and we also have the section $\sigma: D(\tilde{u}) {\longrightarrow}D(\tilde{w})$ of $\pi$, as defined above.
We define $$\label{def_divisors}
\begin{array}{rcl}
Z_i^{\tilde{w}} & := & \pi^{-1}(Z_i^{\tilde{u}}), \quad i = 1,\dots, l(w)-1
\\[.2cm]
Z_{l(w)}^{\tilde{w}} & := & \sigma(D(\tilde{u})).
\end{array}$$
We denote by $Z^{\tilde{w}}$ the sum of the divisors $Z_i^{\tilde{w}}$.
\[prop\_zi\] \[codim1var\_appear\] i) The subvarieties $Z^{\tilde{w}}_i$ are smooth of codimension 1 in $D(\tilde{w})$.
ii\) The scheme-theoretic intersection $P^{\tilde{w}}:= \bigcap_i Z_i^{\tilde{w}}$ is just a point.
iii\) If $\tilde{v} < \tilde{w}$, $l(v) = l(w)-1$, then $D(\tilde{v})$ (considered as a closed subscheme of $D(\tilde{w})$ by the embedding defined above) is one of the $Z_i^{\tilde{w}}$. [$\square$]{}
### The canonical bundle
Let $w \in W_a$ and choose a reduced decomposition $\tilde{w}$. We want to describe the canonical bundle of the Demazure variety $D(\tilde{w})$. As above, we identify the affine flag variety with the space of special complete selfdual lattice chains. Again we denote by $(\lambda_i)_i$ the standard lattice chain.
The Schubert variety $X_w$ consists of certain lattice chains $({{\scr L}}_i)_i$. We can find $N > 0$, such that all lattices occuring here lie between $t^{-N} k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n$ and $t^N k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n$.
Let $n_i = \dim_k \lambda_i/t^N k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n$, $i= 0, \dots, r$. We get maps $$\varphi_i \colon X_w {\longrightarrow}{\mathop{\rm Grass}\nolimits}(t^{-N}k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n/t^N k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n, n_i), \
({{\scr L}}_i)_i \mapsto {{\scr L}}_i/t^N k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n.$$ These maps yield a closed embedding $$\label{emb_in_grass}
\begin{diagram}
\varphi\colon X_w & \rInto & \prod_{i=0}^{r}
{\mathop{\rm Grass}\nolimits}(t^{-N}k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n/t^N k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n, n_i).
\end{diagram}$$
Now let $L_i$ be the very ample generator of the Picard group of ${\mathop{\rm Grass}\nolimits}(t^{-N}k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n/t^N k{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^n, n_i)$, and define $$L_w := \varphi^\ast \bigotimes_{i=0}^{n-1} L_i.$$ This line bundle does not depend on $N$.
Let us remark that we could also obtain this line bundle as the pull-back of a certain line bundle on the affine flag variety; see [@F], section 2.
Denote by $L_{\tilde{w}}$ the pull back of $L_w$ along the morphism $\Psi_{\tilde{w}}: D(\tilde{w}) {\longrightarrow}X_w$. As in [@G], one proves
\[deg\_along\_fibres\]
i\) Let $\tilde{u} < \tilde{w}$, $l(\tilde{u}) = \tilde{w} - 1$. The pull back of $L_{\tilde{w}}$ along the embedding $\sigma: D(\tilde{u}) {\longrightarrow}D(\tilde{w})$ is $L_{\tilde{u}}$.
ii\) The line bundle $L_{\tilde{w}}$ does not have a base point.
iii\) Write $\tilde{w} = \tilde{u} s_{i_\ell}$. The degree of $L_{\tilde{w}}$ along the fibres of $\pi: D(\tilde{w}) {\longrightarrow}D(\tilde{u})$ is $1$.
Now the following description of the canonical bundle of $D(\tilde{w})$ is a purely formal consequence of the above (cf. [@M], [@G]).
\[canbundle\] The canonical bundle of $D(\tilde{w})$ is $$\omega_{D(\tilde{w})} = {{\cal O}}(-Z^{\tilde{w}}) {\otimes}L_{\tilde{w}}^{-1}.$$
### Demazure varieties are Frobenius split
Assume that our algebraically closed field $k$ has characteristic $p > 0$. Now that we have computed the canonical bundle, we can apply the criterion of Mehta and Ramanathan for Frobenius split varieties, and get
\[demvarsplit\] The Demazure variety $D(\tilde{w})$ admits a Frobenius splitting which compatibly splits all the divisors $Z_i^{\tilde{w}}$.
The ind-scheme $\widetilde{{{\cal C}}}$
---------------------------------------
At the moment, we still work over an algebraically closed field of characteristic $p > 0$.
Denote by ${\widetilde{C}}_w$ the normalization of $C_w$. The resolution $\pi_w \colon D(w) {\longrightarrow}C_w$ factors as
D(w) & & \_w & \^[\_w]{} & C\_w.
If $w' \le w$, the map $D(w') {\longrightarrow}D(w)$ induces a map ${\widetilde{C}}_{w'} {\longrightarrow}{\widetilde{C}}_w$, which is independent of the chosen resolution of $w$.
By push forward, the Frobenius splitting of $D(w)$ gives us a splitting of ${\widetilde{C}}_w$; all ${\widetilde{C}}_{w'}$, $w'<w$ are simultaneously compatibly split.
The map $\pi_w$ has geometrically connected fibres as can be seen by induction on the length of $w$ and by factoring $\pi_w$ as
D(w) = [[P]{}]{}\_i \^[[[B]{}]{}]{} D(w’) & & [[P]{}]{}\_i \^[[[B]{}]{}]{} C\_[w’]{} & & C\_w,
where $w = s_i w'$; indeed, the first map has geometrically connected fibres by induction, and the fibres of the second map are points respectively projective lines.
This implies that the normalization map $\psi_w$ is a universal homeomorphism.
The natural maps $\widetilde{C}_{w'} {\longrightarrow}\widetilde{C}_{w}$, $w'\le w$, are closed immersions.
[*Proof.*]{} Since $\psi_w$ is a universal homeomorphism, the maps $\widetilde{C}_{w'} {\longrightarrow}\widetilde{C}_{w}$ are universally injective.
Denote by $C'$ the image of $\widetilde{C}_{w'}$ in $\widetilde{C}_{w}$. Then the extension of the function fields of $\widetilde{C}_{w'}$ and $C'$ is purely inseparable.
On the other hand, these varieties are compatibly Frobenius split, i.e. we have a commutative diagram of ${{\cal O}}_{C'}$-modules
[[O]{}]{}\_[C’]{} & & [[O]{}]{}\_[\_[w’]{}]{}\
\^& & \^\
[[O]{}]{}\_[C’]{} & & [[O]{}]{}\_[\_[w’]{}]{}
where $\varphi$ resp. $\psi$ are sections to the Frobenius map $x \mapsto x^p$. From this we get an analogous diagram for the function fields, and this implies that they are indeed isomorphic. [$\square$]{}
\[rat\_sing\] The higher direct images $R^i \pi_{w,\ast} {{\cal O}}_{D(w)}$, $i>0$, vanish. Furthermore $\widetilde{C}_w$ has rational singularities and in particular is Cohen-Macaulay.
[*Proof.*]{} To show the first part of the lemma, write $w = s_i u$, $l(u) = l(w)-1$. Let us factor $\pi_w$ as $$D(w) = {{\cal P}}_i \times^{{\cal B}}D(u)
{\longrightarrow}{{\cal P}}_i \times^{{\cal B}}\widetilde{C}_u {\longrightarrow}\widetilde{C}_w.$$ By induction, we may assume that the higher direct images of ${{\cal O}}_{D(w)}$ under the first morphism vanish. To deal with the second part, observe that its fibre over $x \in \widetilde{C}_w$ is either a point or a ${{\mathbb P}}^1$. Thus we can apply the following lemma (see [@MS] for a proof) and are done.
[**(Mehta - Srinivas)**]{} Let $f : X {\longrightarrow}Y$ be a projective birational morphism of algebraic varieties over $k$, such that $X$ is Frobenius split, and for all $y \in Y$ we have $H^i(X_y, {{\cal O}}_{X_y})
=0$ for all $i > 0$. Then $R^i f_\ast {{\cal O}}_X = 0$ for all $i > 0$.[$\square$]{}
Finally, to show that $\pi_w$ is a rational resolution, it remains to show that the higher direct images of the canonical bundle vanish as well. This follows from the Grauert-Riemenschneider theorem for Frobenius split varieties, see [@MvK]. [$\square$]{}
Now we define the ind-scheme $\widetilde{{{\cal C}}}$ as the inductive limit of the normalizations ${\widetilde{C}}_w$, $w \in W_{a}$.
We even can do this over ${{\mathbb Z}}$; to do so, we will use the following proposition which is a nice generalized version of the well-known cohomology and base change theorem.
[**(Faltings)**]{} \[coh\_bc\] Let $S$ be a noetherian scheme, let $X$ and $Y$ be noetherian $S$-schemes, and let $f \colon X {\longrightarrow}Y$ be a proper morphism of $S$-schemes. Furthermore, assume that $F$ is a coherent sheaf on $X$ which is flat over $S$. If $R^1 f_{s,\ast} F_s =0$ for all $ s\in S$, then $f_\ast F$ is flat over $S$ and is compatible with base change $S' {\longrightarrow}S$.
[*Proof.*]{} It is clear that $f_\ast F$ is flat over $S$. To show the compatibility with base change, we more or less follow section III.12 in Hartshorne’s book [@H].
Without loss of generality, we may assume that $S$ and $Y$ are affine, say $S={\mathop{\rm Spec}}A$, $Y={\mathop{\rm Spec}}B$.
Now consider the (covariant) functor $$T^i(M) = H^i(X, F \otimes_A M)$$ on the category of $A$-modules. The functor $T^i$ is additive and exact in the middle. The family $(T^i)_{i \ge 0}$ is a $\delta$-functor.
(see [**[@H] III 12.5**]{}) For any $M$, we have a natural map $\varphi: T^i(A) \otimes M {\longrightarrow}T^i(M)$. Furthermore, the following conditions are equivalent:
i\) $T^i$ is right exact
ii\) $\varphi$ is an isomorphism for all $M$
iii\) $\varphi$ is surjective for all $M$.
[*Proof.*]{} The proof in loc. cit. carries over literally. [$\square$]{}
Let us first show that $T^1(M)=0$ for every $A$-module $M$. By a flat base change, we may assume that $A$ is local, with maximal ideal ${{\mathfrak m}}$. We denote the closed point of its spectrum by $s$. Furthermore, it is enough to show the claim for finitely generated $A$-modules, since $T^i$ commutes with direct limits.
First consider the case of a module of finite length. By our assumption $T^1(k(s))=0$, i. e. $T^1$ vanishes on the $A$-module of length one. Since $T^1$ is exact in the middle, this implies that $T^1(M)=0$ for all $A$-modules $M$ of finite length.
Now let $M$ be an arbitrary $A$-module of finite type. Assume that $T^1(M)$ does not vanish, and let $y \in {\mathop{\rm Spec}}B$ be a point in the support of $T^1(M)$. By further localizing $A$ if necessary, we may assume that $y$ maps to the closed point $s \in {\mathop{\rm Spec}}A$.
We know already that $T^1(M/{{\mathfrak m}}^n M)=0$ for all $n$. So by the theorem on formal functions, applied to $f:X{\longrightarrow}Y$, we get $$0= \lim_{\longleftarrow} T^1(M/{{\mathfrak m}}^n M) = H^1(X, F\otimes M)^\wedge,$$ where $H^1(X, F\otimes M)^\wedge$ denotes the completion of the $B$-module $H^1(X, F \otimes M)$ with respect to the ideal ${{\mathfrak m}}B$. But since ${{\mathfrak m}}B \subseteq {{\mathfrak p}}_y$, this implies that the stalk $T^1(M)_y$ vanishes which is a contradiction.
So we have proved that $T^1$ vanishes, and thus that $T^0$ is right exact. By the lemma above this yields that $T^0(M) \cong T^0(A) \otimes M$ for all $A$-modules $M$, which is what we needed to prove. [$\square$]{}
Now we can define the ind-scheme of normalizations of Schubert varieties over the integers. The Demazure varieties $D(w)$ are obviously defined over ${{\mathbb Z}}$. Now we let ${\widetilde{C}}_w = {\mathop{\rm Spec}}\pi_\ast({{\cal O}}_{D(w)})$, where $\pi$ is the map $D(w) {\longrightarrow}{{\cal F}}$ induced by multiplication. We get flat ${{\mathbb Z}}$-schemes that are independent of the chosen reduced decomposition of $w$ (since ${\mathop{\rm Spec}}\pi_\ast({{\cal O}}_{D(w)})$ is the normalization of the scheme-theoretic image of $D(w)$ in ${{\cal F}}$). This construction is compatible with base change by lemma \[rat\_sing\] and proposition \[coh\_bc\]. Furthermore, ${\widetilde{C}}_{w'}$ is a closed subscheme of ${\widetilde{C}}_w$ for $w' \le w$, and we obtain an ind-scheme $$\widetilde{{{\cal C}}} = \lim_{{\longrightarrow}} {\widetilde{C}}_w$$ over ${{\mathbb Z}}$, which maps to ${{\cal F}}$.
The action of the $SL_2$’s associated to the simple affine roots lifts to $\widetilde{{{\cal C}}}$.
[*Proof.*]{} Fix $i \in \{0, \dots, r \} $. The subgroup $SL_2$ associated to a simple affine root $\alpha_i$ acts on $C_w$ if $l(s_iw) = l(w)-1$. If $l(s_iw) = l(w)-1$, we can even find a reduced decomposition of $w$ that starts with $s_i$. Then $SL_2$ acts on the Demazure variety corresponding to this decomposition (since it acts on the parahoric subgroup ${{\cal P}}_i$). Thus we also get an $SL_2$-action on ${\widetilde{C}}_w$.
Since there is a cofinal system of elements $w$ with $l(s_iw) = l(w)-1$ in $W_{a}$, the action lifts to $\widetilde{{{\cal C}}}$. [$\square$]{}
The open cell
-------------
We continue to work over the integers. Let us for a moment consider the special linear group $SL_n$. Again, we denote by $B$ the standard Borel subgroup, by $B^-$ the opposite Borel subgroup, and by ${{\cal B}}$ the Iwahori subgroup of $L\,SL_n$ corresponding to $B$.
Denote by ${{\cal N}}^- = {{\cal N}}^-_{SL_n}$ the inverse image of the unipotent radical of $B^-$ under the projection $L^- SL_n = SL_n(k[t^{-1}]) {\longrightarrow}SL_n$.
The canonical map ${{\cal N}}^- {\longrightarrow}L\,SL_n {\longrightarrow}L\,SL_n/{{\cal B}}$ is an open immersion.
Before we prove the proposition, let us first look at the situation in the affine Grassmannian, which is similar. Faltings ([@F], lemma 2) shows that the natural map $L^{--}SL_n {\longrightarrow}L\, SL_n / L^+ SL_n$ is an open immersion, the image of which consists of those lattices ${{\scr L}}\subset R{(\!( t )\!)}^n$, such that $${{\scr L}}\oplus t^{-1} R[t^{-1}]^n = R{(\!( t )\!)}^n.$$ If ${{\scr L}}$ satisfies this condition, we get the corresponding matrix in $L^{--}SL_n$ in the following way: Denote by $e_1, \dots, e_n$ the standard basis of $R{(\!( t )\!)}^n$, and write $$e_i = u_i + m_i, \quad u_i \in {{\scr L}},\ m_i \in t^{-1} R[t^{-1}]^n.$$ Then it can be shown that the $u_i$ are a basis for ${{\scr L}}$, and the matrix $(u_1, \dots, u_n) = 1- (m_1, \dots, m_n)$ lies in $L^{--}SL_n$.
Now let us generalize this observation to the affine flag variety, and thus prove the proposition.
[*Proof.*]{} Let $R$ be a ${{\mathbb Z}}$-algebra. Denote the standard lattice chain by $(\lambda_i)_i$. Furthermore, write ${{\scr M}}_i = t^{-2}R[t^{-1}]^i \oplus t^{-1}R[t^{-1}]^{n-i}$. We will show that the image of ${{\cal N}}^-(R)$ is the set of lattice chains $({{\scr L}}_i)_i$ such that $$\label{cond}
{{\scr L}}_i \oplus {{\scr M}}_i = R{(\!( t )\!)}, \qquad\text{ for all } i.$$ This is an open condition; more precisely, we see that ${{\cal N}}^-$ is identified with the intersection of the inverse images of the open cells of the affine Grassmannians.
It is clear that the condition (\[cond\]) is satisfied by lattice chains of the form $h((\lambda_i)_i)$, $h\in {{\cal N}}^-(R)$. Now take a lattice chain $({{\scr L}}_i)_i \in LSL_n/{{\cal B}}$ that satisfies (\[cond\]).
For each $i \in \{0,\dots,n-1\}$ write $$\begin{aligned}
t^{-1} e_j & = & u^i_j + m^i_j, \quad u^i_j \in {{\scr L}}_i,
\ m^i_j \in {{\scr M}}_i,\ j=1, \dots, i \\
e_j & = & u^i_j + m^i_j, \quad u^i_j \in {{\scr L}}_i,
\ m^i_j \in {{\scr M}}_i,\ j=i+1, \dots, n\end{aligned}$$ where again $e_1, \dots, e_n$ denotes the standard basis of $R{(\!( t )\!)}^n$. By what we have said above, the $u^i_j$, $j=1,\dots, n$, are a basis of ${{\scr L}}_i$.
Claim: The matrix $h:=(tu^1_1, tu^2_2, \dots, tu^{n-1}_{n-1}, u^0_n)$ lies in ${{\cal N}}^-(R)$, and $h(\lambda_i)_i = ({{\scr L}}_i)_i$.
By the definition of the $u^i_j$, it is clear that $h\in {{\cal N}}^-(R)$. So it remains to show that $h(\lambda_i) = {{\scr L}}_i$, for all $i$. This is done by an easy computation, and to keep the notations simple, we will only treat the case $i=0$. We have, for $1 \le j < n$, $$\begin{aligned}
e_j & = & t(u^j_j + m^j_j) \\
& = & t u^j_j + \sum_{k=j+1}^n a_k e_k
+ m^0, \quad a_k \in R,\ m^0 \in {{\scr M}}_0, \\
& = & t u^j_j + \sum_{k=j+1}^n a_k u^0_k + m'_0, \quad m'_0 \in {{\scr M}}_0,\end{aligned}$$ so we have $$u^0_j = t u^j_j + \sum_{k=j+1}^n a_k u^0_k,$$ and this proves, by descending induction on $j$, that indeed $tu^1_1, \dots, tu^{n-1}_{n-1}, u^0_n$ is a basis of ${{\scr L}}_0$. [$\square$]{}
Now we switch back to the symplectic group $G = Sp_{2r}$. Inside $L^-G$, we have the subgroup ${{\cal N}}^-_{Sp_{2r}}$, which is by definition the inverse image of the unipotent radical $N^-$ of the opposite Borel subgroup $B^-$, under the projection $L^-G {\longrightarrow}G$. In fact, ${{\cal N}}^-_{Sp_{2r}}$ is just the intersection (inside $L\,SL_n$) of ${{\cal N}}^-_{SL_{2r}}$ with $LG$. From now on, we will simply write ${{\cal N}}^-$ instead of ${{\cal N}}^-_{Sp_{2r}}$. As before, we denote the affine flag variety $L\,Sp_{2r}/{{\cal B}}$ by ${{\cal F}}$. The proposition gives us the following corollary.
The natural map ${{\cal N}}^- {\longrightarrow}{{\cal F}}$ is an open immersion. [$\square$]{}
We denote by ${\widehat{\cal N}^-}$ the formal completion of ${{\cal N}}^-$ at the identity. In other words, $$\begin{aligned}
{\widehat{\cal N}^-}(R) = \{ g \in G(R[t^{-1}]); && g \equiv 1 \text{ modulo
a nilpotent ideal in $R$,} \\
&& \text{ and the constant term of $g$
lies in $N^-$} \}. \end{aligned}$$ It is clear (think of the elements of ${\widehat{\cal N}^-}$ as matrices) that ${\widehat{\cal N}^-}$ is ind-represented by a power series ring ${{\cal R}}:= {{\mathbb Z}}[\hspace{-.12em}[ x_i, i\in {{\mathbb N}}]\hspace{-.12em} ]$ in infinitely many variables, i. e. $${\widehat{\cal N}^-}= \lim_{{\longrightarrow}} {\mathop{\rm Spec}}{{\cal R}}/I,$$ where the limit runs over those ideals $I$ which contain almost all $x_i$, and contain some power of each $x_i$. These ideals $I$ will be called open. It is clear that we can just as well take the limit over those ${\mathop{\rm Spec}}{{\cal R}}/I$, such that additionally ${{\cal R}}/I$ is ${{\mathbb Z}}$-torsion free.
Let $g \in {\widehat{\cal N}^-}(R)$. Then the image of $g$ in $G(R{(\!( t )\!)})$ lies in the subgroup $K$ generated by the subgroups $SL_2(R)$ associated to the simple affine roots.
[*Proof.*]{} Of course, the subgroups $U_\alpha$ associated to the simple affine roots $\alpha$ are contained in $K$. On the other hand, we get all elements of the affine Weyl group, since the simple reflections are contained in the $SL_2(R)$’s. Thus all $U_\alpha$ lie in $K$.
The entries on the diagonal of $g$ have the form $1 + t^{-1} a$, $a \in {\rm nil}(R)$, hence are units in $R[t^{-1}]$. Thus we can write $g$ as a product of elements of $L^{--}T$ and of certain $U_\alpha$’s.
So it only remains to show that $L^{--}T$ is contained in $K$. But obviously $L^{--}T$ is generated by the subgroups ${{\mathbb G}}_m(R[t^{-1}])
\cong \{ {\mathop{\rm diag}}(1^i, a, 1^{2r-2i-2}, a^{-1}, 1^i) \}$ associated to the (finite) simple roots. To show that these ${{\mathbb G}}_m(R[t^{-1}])$ are contained in $K$, it suffices to treat the case of $SL_2$. In this case the assertion follows from the formula ($a\in R[t^{-1}]^\times$) $$\left(\begin{array}{cc}
a & 0 \\
0 & a^{-1}
\end{array} \right)
=
\left(\begin{array}{cc}
1 & a \\
0 & 1
\end{array} \right)
\left(\begin{array}{cc}
1 & 0 \\
-a^{-1} & 1
\end{array} \right)
\left(\begin{array}{cc}
1 & a \\
0 & 1
\end{array} \right)
\left(\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array} \right).$$ [$\square$]{}
Proof of the normality theorem
------------------------------
Denote by ${{\cal A}}$ the projective limit of the local rings of the formal completions of ${\widetilde{C}}_w$ at the origin. Since $\widetilde{{{\cal C}}}$ maps to ${{\cal F}}$, we have a natural map ${{\cal R}}{\longrightarrow}{{\cal A}}$. We will first define a section of this map, and then show that it is an isomorphism.
Let $I \subseteq {{\cal R}}$ be an open ideal such that $R:= {{\cal R}}/I$ is torsion free over ${{\mathbb Z}}$, and let $g \in {\widehat{\cal N}^-}(R)$. By the lemma in the previous section, $g$ lies in the subgroup of $LG$ generated by $SL_2(R)$’s, so we can lift the corresponding point $z \in {{\cal F}}$ to a point $\tilde{z} \in \widetilde{{{\cal C}}}$, say $\tilde{z} \in {\widetilde{C}}_{w}$, which again lies in the formal completion at the origin.
On the level of rings, this means the following: The map ${{\cal R}}{\longrightarrow}R$ corresponding to $g$ lifts to a continous map ${{\cal A}}{\longrightarrow}R$. This lift is unique because $R$ is ${{\mathbb Z}}$-torsion free and ${\widetilde{C}}_{w,{{\mathbb Q}}} {\longrightarrow}C_{w,{{\mathbb Q}}}$ is an isomorphism. By taking the limit, we get a section ${{\cal A}}{\longrightarrow}{{\cal R}}$ of the natural map ${{\cal R}}{\longrightarrow}{{\cal A}}$.
We want to show that these maps actually are isomorphisms, i. e. that the composition ${{\cal A}}{\longrightarrow}{{\cal R}}{\longrightarrow}{{\cal A}}$ is the identity map as well.
Let $w \in W_{a}$, and denote by $A$ the ring of the formal completion of ${\widetilde{C}}_w$ at the origin. Choose a lift of the origin to a ${{\mathbb Z}}$-valued point of the Demazure variety $D(w)$, and denote the ring of the formal completion at this point by $B$. The ring $B$ is simply a ring of formal power series over ${{\mathbb Z}}$. By $B_n$ we will denote the quotient of $B$ modulo some power of its defining ideal.
We have an inclusion $A \subseteq B$, and it is enough to show that the two endomorphisms of ${{\cal A}}$, namely the composition ${{\cal A}}{\longrightarrow}{{\cal R}}{\longrightarrow}{{\cal A}}$ and the identity map, become equal after composing them with ${{\cal A}}{\longrightarrow}A {\longrightarrow}B {\longrightarrow}B_n$. Now these two compositions correspond to $B_n$-valued points of ${\widetilde{C}}_w$ which become equal in ${{\cal F}}$. Since $B_n$ is torsion-free and ${\widetilde{C}}_{w,{{\mathbb Q}}} \cong C_{w,{{\mathbb Q}}}$, these points are actually equal.
So in fact ${{\cal R}}\cong {{\cal A}}$, and this implies that the map ${\widetilde{C}}_w {\longrightarrow}{{\cal F}}$ is a closed immersion near the origin, because it induces a surjection on the local rings of the formal completions.
Now consider the situation over an algebraically closed field $k$ again. What we have shown implies that $C_w$ is normal at the origin. On the other hand, the locus where $C_w$ is not normal is closed and ${{\cal B}}$-stable, hence, if it was not empty, it would have to contain the origin, which is the unique closed ${{\cal B}}$-orbit.
Doubly symplectic standard tableaux {#deConcini}
===================================
Let $r \ge 1$ be an integer, $n=2r$, let $k$ be a field and $$R= k[c_{\mu \nu}, \mu, \nu = 1, \dots 2r] / (CJ_{2r}C^t, C^tJ_{2r}C).$$ Here $C=(c_{\mu\nu})_{\mu,\nu}$ and $J_{2r}$ again denotes the matrix $\left( \begin{array}{cc} & {\mathbf J} \\ -{\mathbf J} &
\end{array} \right)$, where ${\mathbf J}$ is the $r \times r$-matrix with 1’s on the second diagonal and 0’s elsewhere.
We will see later that this singularity appears in the ’simplest’ singular local models, and it will be important for us to know that $R$ is reduced. To prove reducedness, we make use of a theorem of de Concini [@dC] which we explain now.
The theorem will give a $k$-basis of $R$ in terms of doubly symplectic standard tableaux; to give the definition of these standard tableaux, we need several notations.
First of all, if $I=\{i_1, \dots, i_k \}$ and $J=\{ j_1, \dots, j_\ell \}$ are subsets of $\{ 1, \dots, r\}$, we say $ I \le J$ if $ k \ge \ell $ and $i_\mu \le j_\mu$, $\mu = 1, \dots, \ell$.
Now if $i_1, \dots, i_s, j_1, \dots, j_s \in \{ 1, \dots, r\}$, we denote by $(i_s, \dots, i_1 | j_1, \dots, j_s)$ the minor of $C$ consisting of the rows $i_1, \dots, i_s$ and columns $j_1, \dots, j_s$.
Since we are in a sense dealing with the symplectic group, it will be useful to consider the rows (resp. columns) $1, \dots, r$ and $r+1, \dots, 2r$ separately. Thus we make the following definition:
Let $s \le k \le r$, $I = \{ i_1, \dots, i_s \}, J = \{ j_1, \dots, j_{k-s} \}
\subseteq \{ 1, \dots, r \}$. Let $\Gamma = I \cap J$, and write $I = \tilde{I} \stackrel{\cdot}{\cup} \Gamma$, $J = \tilde{J} \stackrel{\cdot}{\cup} \Gamma$. Finally, let $\tilde{I} = \{ \tilde{i}_1 < \dots < \tilde{i}_{s-\lambda} \}$, $\tilde{J} = \{ \tilde{j}_1 < \dots < \tilde{j}_{k-s-\lambda} \}$, $\Gamma = \{ \gamma_1 < \dots < \gamma_\lambda \}$. Then we define $$\begin{aligned}
(J, I | h_1, \dots, h_k ) & = &
(n-\tilde{j}_1+1, \dots, n-\tilde{j}_{k-s-\lambda}+1,
\tilde{i}_{s-\lambda}, \dots, \tilde{i}_1, \\
&& n-\gamma_\lambda+1, \gamma_\lambda, \dots, n-\gamma_1+1, \gamma_1
| h_1, \dots, h_k ). \end{aligned}$$
Next we have to define the notion of admissible minor. So let $I$, $J$ and $\Gamma = I\cap J$ be as above.
The minor $P = (J, I | h_1, \dots h_k)$ is called admissible, if there exists a subset $T \subseteq \{1, \dots, r \} - ( I \cup J)$, such that $ |T| = |\Gamma|$ and $T \ge \Gamma$.
In this case we will write $$\begin{aligned}
P & = & \left(\left. \begin{array}{c} {J', I} \\ {J, I'} \end{array}
\right| h_1, \dots, h_k \right) \\
& = & \left(\left. \begin{array}{c} n-j_1'+1, \dots, n-j_{k-s}'+1,
i_s, \dots, i_1 \\ n-j_1+1, \dots, n-j_{k-s}+1,
i_s', \dots, i_1' \end{array}
\right| h_1, \dots, h_k \right),\end{aligned}$$ where $I = \tilde{I} \cup \Lambda = \{ i_1' < \dots < i_s' \}$, $J' = \tilde{J} \cup \Lambda = \{ j_1' < \dots < j_{k-s}' \} $, and where $\Lambda \subseteq \{1, \dots, r \} - ( I \cup J)$ is the smallest subset with the above properties.
[*Example.*]{} The minor $(r, \dots, 1 | 1, \dots, r) =
(\emptyset, \{1, \dots, r\} | 1, \dots, r)$ is obviously admissible. We have $\Gamma = \Lambda = \emptyset$, and $$(\emptyset, \{1, \dots, r\} | 1, \dots, r)
= \left(\left. \begin{array}{c} {r, \dots, 1 } \\
{ r, \dots, 1 } \end{array}
\right| 1, \dots, r \right).$$
The minor $P$ will be called doubly admissible, if the admissibility condition holds not only for the rows, but also for the columns. In this case, we can write $$\begin{aligned}
P & = &
\left(\left. \begin{array}{c} J', I \\ J, I' \end{array}
\right| \begin{array}{c} H, K' \\ H', K \end{array} \right) \\
& = & \left(\left. \begin{array}{c} t_k, \dots, t_1 \\
v_k, \dots, v_1 \end{array}
\right| \begin{array}{c} s_1, \dots, s_k \\
u_1, \dots, u_k \end{array} \right) .\end{aligned}$$
We have some kind of partial order on the set of doubly admissible minors, that is defined in the following way. Let $$P = \left(\left. \begin{array}{c} t_k, \dots, t_1 \\
v_k, \dots, v_1 \end{array}
\right| \begin{array}{c} s_1, \dots, s_k \\
u_1, \dots, u_k \end{array} \right),
\quad
P' = \left(\left. \begin{array}{c} t'_{k'}, \dots, t'_1 \\
v'_{k'}, \dots, v'_1 \end{array}
\right| \begin{array}{c} s'_1, \dots, s'_{k'} \\
u'_1, \dots, u'_{k'} \end{array} \right)$$ be doubly admissible minors. Then we say that $P \le P'$ if $$\{ v_1, \dots, v_k \} \le \{ t'_1, \dots t'_{k'}\}
\text{ and }
\{ u_1, \dots, u_k \} \le \{ s'_1, \dots s'_{k'}\}.$$ Note that, in general, it is not true that $P \le P$.
A tuple $(P_1, \dots, P_\ell)$ of doubly admissible minors, such that $P_i \le P_{i+1}$ for all $i$, is called a doubly symplectic standard tableaux. We denote the set of all doubly symplectic standard tableaux by ${{\cal T}}$.
We have a map $\varphi \colon {{\cal T}}{\longrightarrow}R$, which maps an element $(P_1, \dots, P_\ell)$ of ${{\cal T}}$ to the product $P_1 \cdot \dots P_\ell$ of the minors $P_i$.
With these notations, de Concini ([@dC], theorem 6.1) proved the following
[**(de Concini)**]{} The elements $\varphi(P)$, $P \in {{\cal T}}$, form a $k$-basis of $R$.
We will in fact only need the following corollary to de Concini’s theorem.
\[f\_NNT\] The element $f = (r, \dots, 1 | 1, \dots, r)$ is not a zero divisor in $R$.
[*Proof.*]{} Obviously, $f$ is a doubly admissible minor; see the example above. Furthermore, the set $\{1, \dots, r\}$ is the smallest element among all subsets of $\{1, \dots, r\}$ with respect to the partial order defined above. Thus, if $T$ is an arbitrary doubly symplectic standard tableau, $fT$ is again a doubly symplectic standard tableau. In view of the theorem this clearly implies that $f$ is not a zero divisor.
[**Remark.**]{} What we really need to know, and what we will derive from this corollary in the course of the proof of the flatness theorem, is that $R$ is reduced; see proposition \[Rred\]. It might well be that this result is - at least implicitly - already contained in de Concini’s paper. For example, it would follow if one knew that $R$ equipped with the basis of doubly symplectic standard tableaux becomes an algebra with straightening law. I do not know if this is true.
Proof of the flatness theorem {#proof}
=============================
We embed the special fibre ${\overline{\rm\bf M}^{\rm loc}}$ of the local model into the affine flag variety ${{\cal F}}= LG/{{\cal B}}$. More precisely, we identify ${{\cal F}}$ with the space of $r$-special selfdual complete lattice chains, as explained in proposition \[F\_latticechains\] and the remark following it. Let $R$ be a $k$-algebra. As before, we denote by $(\lambda_i)_i$ the standard lattice chain: $$\lambda_i = t^{-1} R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^i \oplus R{[\hspace{-.12em}[ t ]\hspace{-.12em} ] }^{n-i}.$$ Then we map an element $({{\cal F}}_i)_i \in {\overline{\rm\bf M}^{\rm loc}}(R)$ to the lattice chain $({{\scr L}}_i)_i$, where ${{\scr L}}_i$ is the inverse image of ${{\cal F}}_i$, considered as a subspace of $\Lambda_{i,R} \cong R^n$, under the projection $\lambda_i {\longrightarrow}\lambda_i/t \lambda_i \cong R^n$.
It is easily checked that in this way we get an element of ${{\cal F}}(R)$, and that the resulting map is a closed immersion. Via this embedding, we can identify ${\overline{\rm\bf M}^{\rm loc}}$ with the set $$\label{descrMloc}
{\overline{\rm\bf M}^{\rm loc}}\cong \{ ({{\scr L}}_i)_i \in {{\cal F}}; \ t \lambda_i \subseteq {{\scr L}}_i \subseteq \lambda_i
\text{ for all } i \}.$$
We can do the same thing for the parahoric local model associated to a subset $I \subseteq \{ 1, \dots, n \}$ (such that $2r-i \in I$ for each $i\in I$, $i\ne 0$). Denote by ${{\cal P}}^I \subseteq LG$ the stabilizer of the partial lattice chain $(\lambda_i)_{i\in I}$. We get an embedding ${\overline{\rm\bf M}^{\rm loc}}_I \subset LG/{{\cal P}}^I$. Denote by ${\widetilde{\rm \bf M}^{\rm loc}}_I$ the inverse image of ${\overline{\rm\bf M}^{\rm loc}}_I \subset Sp_{2r}(k{(\!( t )\!)})/P^I$ under the projection ${{\cal F}}{\longrightarrow}Sp_{2r}(k{(\!( t )\!)})/P^I$. It can be identified with the following set of lattice chains: $${\widetilde{\rm \bf M}^{\rm loc}}_I = \{ ({{\scr L}}_i)_i \in {{\cal F}}; \ t \lambda_i \subseteq
{{\scr L}}_i \subseteq \lambda_i \text{ for all } i \in I \}.$$
Then we have (scheme-theoretic intersection) $${\overline{\rm\bf M}^{\rm loc}}= {\widetilde{\rm \bf M}^{\rm loc}}_0 \cap {\widetilde{\rm \bf M}^{\rm loc}}_r \cap
\bigcap_{i=1}^{r-1} {\widetilde{\rm \bf M}^{\rm loc}}_{i, 2r-i}.$$
All the ${\widetilde{\rm \bf M}^{\rm loc}}_I$ and also ${\overline{\rm\bf M}^{\rm loc}}$ are, set-theoretically, unions of Schubert varieties in ${{\cal F}}$, since they are invariant under the action of ${{\cal B}}$.
The simplest local models {#simplest}
-------------------------
The local models ${{\rm\bf M}^{\rm loc}}_0$ and ${{\rm\bf M}^{\rm loc}}_{r}$ are isomorphic to the Grassmannian of totally isotropic subspaces of $\Lambda_0$ resp. $\Lambda_r$. So they are smooth over $O$.
The case of ${{\rm\bf M}^{\rm loc}}_{i,2r-i}$, $i \in \{ 1, \dots, r-1 \}$, is more complicated. First we determine the equations of the local model in this case. For simplicity we assume that $r \le 2i$ (the case $r > 2i$ is completely analogous).
In an open neighborhood of the worst singularity, we can represent the subspaces ${{\cal F}}_i, {{\cal F}}_{2r-i}$ with $({{\cal F}}_i, {{\cal F}}_{2r-i}) \in {{\rm\bf M}^{\rm loc}}_{i,2r-i}$ by matrices of the form
$${{\cal F}}_i \hat{=}
\left(
\begin{array}{ccc}
a_{r-i+1,1} & \cdots & a_{r-i+1,r} \\
\vdots & & \vdots \\
a_{r1} & \cdots & a_{rr} \\
1 & & \\
& \ddots & \\
& & 1 \\
a_{11} & \cdots & a_{1r} \\
\vdots & & \vdots \\
a_{r-i,1} & \cdots & a_{r-i,r}
\end{array}
\right), \qquad
{{\cal F}}_{2r-i} \hat{=}
\left(
\begin{array}{cccccc}
& & & 1 & & \\
& & & & \ddots & \\
& & & & & 1 \\
b_{11} & \cdots & b_{1i} & b_{1,i+1} & \cdots & b_{1r} \\
\vdots & & \vdots & & \vdots & \vdots \\
b_{r1} & \cdots & b_{ri} & b_{r,i+1} & \cdots & b_{rr} \\
1 & & & & & \\
& \ddots & & & & \\
& & 1 & & &
\end{array}
\right).$$
The duality condition implies that $$b_{\mu \nu} = \varepsilon_{\mu \nu} a_{r-\nu+1, r-\mu+1},$$ where $$\varepsilon_{\mu \nu} =
\left\{ \begin{array}{rl} 1 & \mu,\nu \le i \text{ or }
\mu, \nu \ge i+1, \\ -1 & \text{otherwise.} \end{array}\right.$$
To write down in terms of the $a_{\mu \nu}$ what it means that these subspaces are mapped into one another, we divide the matrices $ A = (a_{\mu \nu})_{\mu, \nu} $ and $B = (b_{\mu \nu})_{\mu, \nu}$ into blocks: $$\begin{array}{rcr}
A = &
\left(
\begin{array}{cc}
A_4 & A_3 \\
A_1 & A_2
\end{array}
\right) & \begin{array}{c} \mbox{\tiny $2(r-i)$} \\ \mbox{\tiny $2i-r$}
\end{array} \\
& \begin{array}{cc} \mbox{\tiny $2(r-i)$} & \mbox{{\tiny $2i -r$}}
\end{array} &
\end{array}, \qquad
\begin{array}{rcr}
B = &
\left(
\begin{array}{cc}
B_2 & B_3 \\
B_1 & B_4
\end{array}
\right) & \begin{array}{c} \mbox{\tiny $ 2i-r$} \\ \mbox{\tiny $2(r-i)$}
\end{array} \\
& \begin{array}{cc} \mbox{\tiny $2i-r$} & \mbox{{\tiny $ 2(r-i)$}}
\end{array} &
\end{array}.$$ Now we get the following equations as condition that ${{\cal F}}_i$ is mapped into ${{\cal F}}_{2r-i}$: $$B_4 A_4 = \pi, \quad B_2 = A_2 - B_3 A_3, \quad A_1 = B_3 A_4, \quad
B_1 = - B_4 A_3.$$ The condition that ${{\cal F}}_{2r-i}$ is mapped into ${{\cal F}}_i$ is expressed by the following equations: $$A_4 B_4 = \pi, \quad \pi B_2 = A_1 B_1 + \pi A_2, \quad
\pi B_3 = A_1 B_4, \quad \pi A_3 = -A_4 B_1.$$ Altogether, this gives us the following: $$B_4 A_4 = A_4 B_4 = \pi, \quad B_2 = A_2 - B_3 A_3, \quad A_1 = B_3
A_4.$$ All the other equations follow from these four equations.
It is easy to see that we can choose all the coefficients of $A_3$ and those of $A_2$ that lie on or above the secondary diagonal. Then the other coefficients of $A_2$ are determined, as well as the coefficients of $A_1$. Of course, $B_1$, $B_2$ and $B_3$ are then determined, too.
Furthermore note that $B_4$ is just the adjoint matrix of $A_4$ with respect to the standard symplectic form.
So our open subset is isomorphic to $$\begin{aligned}
&& {\mathop{\rm Spec}}O[c_{\mu \nu}, \mu, \nu = 1, \dots 2(r-i)] /
(C C^{ad} = C^{ad} C = \pi) \\
&&\times
{{\mathbb A}}_O^{2(i-r)(2i-r)+ \frac{1}{2}(2i-r)(2i-r+1)}.\end{aligned}$$
Thus to show that ${\overline{\rm\bf M}^{\rm loc}}_{i,2r-i}$ is reduced, we may obviously restrict ourselves to the case where $r = 2i$. Since $C^{ad} = -J_{2i} C^t J_{2i}$, where $J_{2i}$ is the matrix describing the standard symplectic form as above, in this case an open neighborhood of the worst singularity in the special fibre is isomorphic to ${\mathop{\rm Spec}}R$, where $$R= k[c_{\mu \nu}, \mu, \nu = 1, \dots r] /
(CJ_{2i}C^t, C^tJ_{2i}C),$$ and we have to show
\[Rred\] The ring $R$ is reduced.
[*Proof.*]{} We denote by $f\in R$ the $i \times i$-minor of $C$ consisting of the rows $1, \dots, i$ and columns $1, \dots, i$, and by $R_f$ the localization of $R$ with respect to $f$.
The scheme ${\mathop{\rm Spec}}R_f$ is smooth; more precisely, it is an open subscheme of ${{\mathbb A}}_k^{r(r+1)/2}$.
[*Proof.*]{} Consider the open subset $U \subset {\overline{\rm\bf M}^{\rm loc}}_{i,2r-i}$, where we can represent the subspaces as $${{\cal F}}_i =
\left(
\begin{array}{ccc}
a_{11} & \cdots & a_{1r} \\
\vdots & & \vdots \\
a_{r1} & \cdots & a_{rr} \\
1 & & \\
& \ddots & \\
& & 1
\end{array}
\right), \qquad
{{\cal F}}_{2r-i} =
\left(
\begin{array}{ccc}
b_{11} & \cdots & b_{1r} \\
\vdots & & \vdots \\
b_{r1} & \cdots & b_{rr} \\
1 & & \\
& \ddots & \\
& & 1
\end{array}
\right).$$ The duality condition implies that $b_{\mu \nu} = a_{r-\nu+1,, r-\mu+1}$.
It is easy to see by a direct computation that this open subset is isomorphic to ${{\mathbb A}}_k^{r(r+1)/2}$. We could also derive this by using the corresponding fact for $GL_n$; cf. section \[liftability\]. Since ${\mathop{\rm Spec}}R_f$ obviously is an open subscheme of $U$, the lemma follows. [$\square$]{}
Furthermore, corollary \[f\_NNT\] shows that $f$ is not a zero divisor.
Since ${\mathop{\rm Spec}}R_f$ is irreducible and $f$ is not a zero divisor, we see that ${{\rm\bf M}^{\rm loc}}_{i, 2r-i}$ is irreducible. This can also be proved by analysing the affine Weyl group (cf. the article [@KR] of Kottwitz and Rapoport, sections 4 and 10).
Now we can conclude that $R$ is reduced: obviously this is true generically, so all we have to show is that $R$ has no embedded associated prime ideals. This is clear for the localization $R_f$, and on the other hand all primes in the complement of ${\mathop{\rm Spec}}R_f$ contain the non-zero divisor $f$, and thus cannot be associated prime ideals. [$\square$]{}
Intersections of Schubert varieties {#inters}
-----------------------------------
From Faltings’ theorem we get
Let $Y_1 , \dots Y_k \subset {{\cal F}}$ be unions of Schubert varieties. Then the scheme-theoretic intersection $\bigcap_i Y_i$ is reduced.
[*Proof.*]{} Embed all the $Y_i$ in a sufficiently big Schubert variety $X$, and use the fact that $X$ is Frobenius split, compatibly with all its Schubert subvarieties. [$\square$]{}
The corollary shows that ${\overline{\rm\bf M}^{\rm loc}}$ is a union of Schubert varieties in the affine flag variety even scheme-theoretically. In particular, it is reduced and its irreducible components are normal with rational singularities.
The final ingredient for the flatness theorem is that we can lift the generic points of the special fibre to the generic fibre. This point is dealt with in the next section.
Lifting the generic points of the special fibre {#liftability}
-----------------------------------------------
We want to show that the generic points of the special fibre of a local model ${{\rm\bf M}^{\rm loc}}_I$ can be lifted to the generic fibre. To keep the notation a little bit simpler, we deal only with ${{\rm\bf M}^{\rm loc}}$. See [@G] for a detailed account on this question in the case of $GL_n$.
As we have seen, ${\overline{\rm\bf M}^{\rm loc}}$ is the union of certain Schubert varieties, that correspond to certain elements of the affine Weyl group, or in other words to certain alcoves (in the standard apartment of the Bruhat-Tits building) for $Sp_{2r}$.
The question which Schubert varieties occur has a natural answer in terms of the notions of admissible and permissible alcoves introduced by Kottwitz and Rapoport [@KR], which we will use freely in the following.
It is clear that the set of alcoves contributing to ${\overline{\rm\bf M}^{\rm loc}}$ is precisely the set of minuscule or $\mu$-permissible alcoves, where $\mu$ is the minuscule dominant coweight $(1^r, 0^r)$. As Kottwitz and Rapoport have shown, this set coincides with the set of $\mu$-admissible alcoves, which consists of all alcoves that are smaller than some conjugate of $\mu$ under the finite Weyl group.
Thus the ’extreme’ elements, which correspond to the irreducible components of ${\overline{\rm\bf M}^{\rm loc}}$, are the conjugates of $\mu$ under $W$.
Let $x = (x_0, \dots, x_{2r-1})$ be an extreme minuscule alcove for $GSp_{2r}$ (cf. [@KR]). Then $x_i = x_0 + (1^i, 0^{2r-i})$, and $\{ x_0(i), x_0(2r-i+1) \} = \{ 0, 1 \}$, $i = 1, \dots, r$.
Let $I = \{ i_1 < \cdots < i_r \} = \{ i; x_0(i) = 0 \}$, $J = \{ j_1 < \cdots < j_r \} = \{ j; x_0(j) = 1 \}$.
We denote by $U_x$ the open subset of ${{\rm\bf M}^{\rm loc}}$, where each subspace ${{\cal F}}_i$ can be represented by a matrix with the unit matrix in rows $j_1, \dots, j_r$. Note that for each ${{\cal F}}_i$ the unit matrix is in the same place. The special fibre $\overline{U}_x$ of $U_x$ contains the Schubert cell corresponding to $x$ as an open subscheme.
We have $U_x \cong {{\mathbb A}}_O^{r(r+1)/2}$.
[*Proof.*]{} We can consider $U_x$ as a closed subscheme of the corresponding extreme stratum $U_x(GL_{2r})$ in the local model for $GL_{2r}$, and we know that $U_x(GL_{2r}) \cong {{\mathbb A}}^{r^2}$ (cf. [@G]).
Recall how this isomorphism was constructed. We denote the coefficients (in the rows $i_1, \dots, i_r$) in the matrix describing ${{\cal F}}_i$ by $a^i_{\lambda \mu}$.
Now fix $\lambda$ and $\mu$. Then for $i=0, \dots, 2r-1$ we get three types of equations:
[*First case.*]{} $\pi a^i_{\lambda \mu} = a^{i+1}_{\lambda \mu}.$
[*Second case.*]{} $a^i_{\lambda \mu} = \pi a^{i+1}_{\lambda \mu}.$
[*Third case.*]{} $a^i_{\lambda \mu} = a^{i+1}_{\lambda \mu}.$
The first and second case occur precisely once, namely when $i = i_\lambda - 1$ resp. $i = j_\mu$. So we see that we can choose $a^{i_\lambda-1}_{\lambda \mu}$ arbitrarily, and that all other $a^i_{\lambda \mu}$ are then uniquely determined; more precisely we have $$a^i_{\lambda \mu} =
\left\{
\begin{array}{rl}
a^{i_\lambda-1}_{\lambda \mu} & \quad i \in [ j_\mu, i_\lambda ) \\
\pi a^{i_\lambda-1}_{\lambda \mu} & \quad\text{otherwise}
\end{array}
\right. ,$$ where we consider $[ j_\mu, i_\lambda )$ as an interval in ${{\mathbb Z}}/2r{{\mathbb Z}}$.
Now we have to investigate the effect of the duality condition that describes $U_x$ inside $U_x(GL_{2r})$. It says that for all $i, \lambda, \mu$, we must have $a^i_{\lambda \mu} = a^{2r-i}_{r-\mu+1, r-\lambda+1}$. We want to get our isomorphism $U_x \cong {{\mathbb A}}^{r(r+1)/2}$ by choosing just one of these two, and the following lemma shows that we can do so.
For all $\lambda, \mu$, we have $$i\in [j_\mu, i_\lambda )\quad \Longleftrightarrow \quad
2r-i \in [j_{r-\lambda+1}, i_{r-\mu+1}).$$
[*Proof.*]{} Since we started with an extreme minuscule alcove [*for the symplectic group*]{}, we have $$i_\lambda = 2r - j_{r-\lambda+1} +1, \quad
j_\mu = 2r- i_{r-\mu+1} +1,$$ and the lemma follows immediately. [$\square$]{}
Since an open subset of the irreducible component of ${\overline{\rm\bf M}^{\rm loc}}$ corresponding to $x$ is contained in $U_x$ as an open subscheme, the proposition implies that the generic point of this irreducible component can be lifted to the generic fibre. This concludes the proof of the flatness theorem.
[MvK]{}
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Ulrich Görtz\
Mathematisches Institut\
der Universität zu Köln\
Weyertal 86–90
50931 Köln (Germany)
[email protected]
.5cm\
School of Mathematics\
Institute for Advanced Study\
Einstein Drive
Princeton, NJ 08540 (USA)
[email protected]
|
---
abstract: 'We study the interplay of spin-orbit coupling (SOC) and strong p-wave interaction to the scattering property of spin-1/2 ultracold Fermi gases. Based on a two-channel square-well potential generating p-wave resonance, we show that the presence of an isotropic SOC, even for its length much longer than the potential range, can greatly modify the p-wave short-range boundary condition(BC). As a result, the conventional p-wave BC cannot predict the induced molecules near p-wave resonance, which can be fully destroyed to vanish due to strong interference between s- and p-wave channels. By analyzing the intrinsic reasons for the breakdown of conventional BC, we propose a new p-wave BC that can excellently reproduce the exact molecule solutions and also equally apply for a wide class of single-particle potentials besides SOC. This work reveals the significant effect of SOC to both the short- and long-range properties of fermions near p-wave resonance, paving the way for future exploring interesting few- and many-body physics in such system.'
author:
- Xiaoling Cui
title: ' Multichannel Molecular State and Rectified Short-range Boundary Condition for Spin-orbit Coupled Ultracold Fermions Near p-wave Resonances'
---
The interplay of spin-orbit coupling (SOC) and interaction has generated tremendous research interests in condensed matter physics[@review_1; @review_2], while ultracold atomic gases offer an ideal platform for its study giving successful realizations of synthetic SOC using Raman lasers[@gauge2exp; @fermisocexp1; @fermisocexp2; @2dsoczhang1; @Wu2015] and highly tunable interactions via Feshbach resonances[@Chin]. Nevertheless, before studying the complex many-body physics the very first question to address is how to model the fundamental two-body interactions. A crucial factor here is the asymptotic behavior of two-body wave function in the short-range limit, called the short-range boundary condition(BC), which is the basis for constructing the Huang-Yang pseudo-potentials[@LHY; @Stock05] and also equivalent to the use of renormalized contact models[@review_Fermi_gas; @Gurarie]. In the presence of SOC, studies have shown that the usual s-wave short-range BC, giving the conventional s-wave models, is hardly modified near s-wave resonances given the typical length of realistic SOC much longer than the potential range[@Cui; @Yu; @Zhang]. Despite the negligible short-range consequence, SOC can greatly change the long-range (low-energy) scattering properties from near the threshold[@Cui; @Yu; @Blume] to intermediate energy regime[@Blume; @You; @Greene]. Moreover, with conventional s-wave models it has been found that SOC can induce shallow molecules[@Vijay] and universal trimers[@trimer_soc; @borromean_soc] more easily, and lead to various fascinating many-body phenomena in both bosons and fermions atomic systems[@socreview1; @socreview2; @socreview4; @socreview5; @socreview6].
Besides s-wave, the p-wave interacting atomic gases have also attracted great attention in recent years[@Chin; @thesis], in particular, in view of the very recent explorations of universal properties near p-wave resonance[@Toronto; @Ueda; @Yu_p; @Zhou]. In this work, we study the interplay of SOC and strong p-wave interaction to the short-range and long-range two-body physics. Specifically, we ask the question how would SOC affect the p-wave short-range BC and induce shallow molecules? There have been a few related discussions in literature without full answers[@Cui; @Yu]. Addressing this problem will be fundamentally important for future exploring a new set of few- and many-body physics due to the interplay of SOC and high partial-wave scatterings.
We consider two spin-1/2 fermions subject to an isotropic SOC, which is hopefully realizable in future considering a number of proposals[@proposal1; @proposal2]. By adopting a two-channel square-well potential generating the p-wave Feshbach resonance[@Chin] (see Fig.1a), we can exactly pin down the binding energy of induced molecule as well as its multi-channel structures and short-range behaviors. We find that even for the SOC length much longer than the potential range, it can still greatly modify the p-wave short-range BC near p-wave resonances, on contrary to previous findings in the s-wave channel. Consequently, the conventional p-wave BC cannot predict the induced molecules near p-wave resonance, which instead can be fully destroyed to vanish within a visible range of interaction strength. Its breakdown is partly due to the intrinsic sensitivity of p-wave scattering parameters to short-range details, and partly due to the strong interference between s- and p-wave channels via SOC. Based on these observations, we finally propose a new p-wave BC, which excellently reproduces the molecule solution in this case and also equally applies for a wide class of single-particle potentials besides SOC.
![(Color online) (a) Schematic plot of s- and p-wave square-well potentials with range $r_0$(see text). The shaded area is the region where single-particle SOC is applied. (b) p-wave scattering volume ($v_p$) and effective range ($r_p$) as a function of closed-channel molecule $\epsilon_c$ with fixed $v_{bg}=1.2r_0^3$ and $\Omega/|V_{p,c}-V_{p,o}|$ around $0.12$ across the whole region. The red arrow marks the location of p-wave resonance. []{data-label="fig1"}](fig1.pdf){width="9cm" height="4cm"}
We start from a square-well model potential, as depicted in Fig.1a, for two spin-1/2 fermions with relative distance ${\bf r}$: $$V(\cp r)=V_p P_{11;00} + V_s \theta(r_0-r)P_{00;00}. \label{V}$$ Here $P_{L_rS;m_lm_s}$ is the projection operator to two-particle state with relative orbital angular momenta $\{L_rm_l\}$ and total spin angular momenta $\{Sm_s\}$. Explicitly, $|Sm_s\rangle$ can be expanded as: $|10\rangle = \frac{|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle}{\sqrt{2}}, \ |00\rangle = \frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt{2}}$. For the p-wave interaction ($L_r=1$), we have selected out $m_l=0$ channel whose resonance can be well separated from the other channels[@Chin; @thesis]; to mimic the realistic p-wave Feshbach resonance, we adopt a two-channel potential[@Chin] $$V_p = \left(\begin{array}{cc} V_{p,o} & \Omega \\ \Omega & V_{p,c}\end{array}\right)\theta(r_0-r)+\left(\begin{array}{cc} 0& 0 \\ 0 & \infty \end{array}\right)\theta(r-r_0).$$ Here $V_{p,o}$ and $V_{p,c}$ are respectively the open- and closed-channel potentials within the range $r_0$. $V_{p,o}$ provides the background p-wave volume $v_{bg}=-j_2(q_or_0)/(3j_0(q_or_0))$ ($q_o=\sqrt{-mV_{p,o}}$), and $V_{p,c}$ gives a closed-channel molecule at $\epsilon_c$ satisfying $j_1(\sqrt{m(\epsilon_c-V_{p,c})}r_0)=0$. Given $\epsilon_c$ close to the scattering threshold and an inter-channel coupling $\Omega$, a p-wave resonance can be induced in the open-channel with scattering volume $v_p\rightarrow\infty$ and a finite range $r_p$. Explicitly, $v_p$ and $r_p$ are defined through the phase shift expansion in low-energy limit ($k\ll 1/r_0$) as $k^3\cot\delta_p=-1/v_p+r_pk^2/2$, where the phase shift $\delta_p$ appears in the open-channel wave function $\psi_{p,o}=j_1(kr)-\tan\delta_p n_1(kr)$ and can be determined by requiring $\psi_{p,c}(r_0)=0$ and the continuity of $\psi'_{p,o}/\psi_{p,o}$ at $r_0$ in the current model. In Fig.1b, we plot $v_p$ and $r_p$ as a function of $\epsilon_c$ for a weak $v_{bg}=1.2r_0^3$ and $\Omega\ll |V_{p,c}-V_{p,o}|$. A resonance of $v_p$ occurs at $\epsilon_c=-0.37$ with a finite range $r_p=-7/r_o$.
Importantly, the scattering parameters $v_p$ and $r_p$ also appear in the asymptotic behavior of $\psi_{p,o}\equiv \psi_p$ in the short-range regime $r_0\ll r \ll 1/k$: $$\psi_{p}(r)\rightarrow \frac{1}{r^2} + \frac{k^2}{2} +\frac{r}{3}(-\frac{1}{v_p}+\frac{r_pk^2}{2})+o(r^3). \label{short_range}
%\psi_{p,o}(r)\rightarrow (\frac{1}{r^2}-\frac{r}{3v_p}) + k^2(\frac{1}{2} +\frac{r_p r}{2}).$$ Thus $v_p, r_p$ determine the ratio between the coefficients of $ r$ and most singular $1/r^2$ terms, which sets the conventional short-range BC in p-wave channel: $$\frac{(r^2\psi_p)'''}{r^2\psi_p} = %-\frac{r^2}{v_p}+k^2 r\left( 1+\frac{r^3}{v_p} +r_p r (\frac{1}{2}+\frac{r^3}{3v_p})\right)
2\left( -\frac{1}{v_p}+ \frac{r_pk^2}{2}\right) +o(r^2), \label{old_BC}$$ here the superscript $'''$ denotes the third derivative in terms of $r$. In fact, the pseudo-potential method[@LHY; @Stock05] and p-wave contact model[@Gurarie] all correspond to guaranteeing above short-range BC, regardless of the presence of any external or internal single-particle potentials.
Note that Eq.\[V\] also includes a weak potential $V_s$ in s-wave channel at $r<r_0$, giving the scattering length $a_s/r_0=1-\tan(q_sr_0)/(q_sr_0)$ ($q_s=\sqrt{-mV_s}$) and effective range $r_s$. The reason for its inclusion will be explained later.
Now we consider the SOC part. Different from previous studies[@Cui; @Yu], in this work we consider the single-particle SOC applied only to the region outside the interaction potential, as shown by shaded area in Fig.1a, since the laser-generated SOC in experiments[@gauge2exp; @fermisocexp1; @fermisocexp2; @2dsoczhang1; @Wu2015] can hardly reach the very short-range regime given so deep interaction potentials therein. Thus SOC will not modify the scattering inside the potential ($r<r_0$). Nevertheless, we will show below that it does greatly modify the p-wave short-range BC in the regime $r_0\ll r\ll 1/k$.
Under the isotropic SOC ($(\lambda/m) \cp{k}\cdot \boldsymbol\sigma$ with $ \boldsymbol\sigma$ the Pauli matrix), the single-particle eigen-state at momentum $\cp k$ has two orthogonal branches $|\cp k^{(\pm)}\rangle$ with eigen-energies $\epsilon^{(\pm)}_{\cp k}=(k\pm\lambda)^2/(2m)\ (k\equiv|\mathbf{k}|)$. In single-particle level, the isotropic SOC enables the conservation of total angular momentum $\mathbf{j}=\mathbf{l}+\mathbf{s}\ (\mathbf{s}=\frac{1}{2}\boldsymbol\sigma)$. Consequently in two-body level, the total angular momentum $\mathbf{J}=\mathbf{L}+\mathbf{S}$ is also conserved, and the orbital $\mathbf{L}$ can be reduced to the relative component $\mathbf{L}_r$ when we consider the ground state scattering with zero center-of-mass momentum[@Cui]. Given the interaction potential (\[V\]), we expect the relevant $J$ can be 0 and 2, which can be composed by $S=0,\ L_r=0,2$ or $S=1,\ L_r=1,3$. Such multi-channel structure is revealed clearly in the scattered wave function studied below.
Based on the Lippmann-Schwinger equation, the scattered wave function ($r>r_0$) at energy $E$ reads: $$\begin{aligned}
\Psi^{sc}(\cp r)=\int d\mathbf{r}' \langle \mathbf{r}|G|\mathbf{r}' \rangle \langle\mathbf{r}'|V |\Psi\rangle. \label{psi}\end{aligned}$$ where the Green function can be expanded as $$\langle\mathbf{r}|G|\mathbf{r}'\rangle=\frac{1}{2}\sum_{\sigma\sigma'=\pm} \frac{\langle \cp r|\cp{k}^{(\sigma)},-\cp{k}^{(\sigma')}\rangle \langle \cp{k}^{(\sigma)},-\cp{k}^{(\sigma')} |\cp r'\rangle}{E-\epsilon_{\cp k}^{(\sigma)}-\epsilon_{-\cp k}^{(\sigma')} + i0^{+}}; \label{G}$$ and the interaction part can be parametrized as $$\begin{aligned}
\langle \mathbf{r}|V|\Psi \rangle=F_p(r) Y_{10}(\Omega_r) |10\rangle+ F_s(r) Y_{00}(\Omega_r) |00\rangle, \ (r<r_0)\label{U_P}\end{aligned}$$ with $F_s,\ F_p$ respectively the scattering amplitudes in s- and p-wave channel based on Eq.(\[V\]).
Lengthy but straightforward calculation of Eq.\[psi\] indeed results in various $|J;L_rS\rangle$ states. These states include $|2;11\rangle$, $|2;31\rangle$, $|2;20\rangle$, $|0;11\rangle$, $|0;00\rangle$ generated by p-wave interaction ($F_p$ part in Eq.\[U\_P\]), and $|0;11\rangle$, $|0;00\rangle$ generated by s-wave interaction ($F_s$ part in Eq.\[U\_P\]). By analyzing their coefficients, it is found that the dominated $1/r^{L_r+1}$ singularity only occur in $L_r=0,1$ channels[@note_divergence], which reads $\psi_p Y_{10}(\Omega_r) |10\rangle + \psi_sY_{00}(\Omega_r) |00\rangle$ with $$\psi_p=f_p A - f_s B;\ \ \ \ \psi_s= f_s C+f_p D. \label{psi_sp}$$ Here $f_s=\int_0^{r_0} dr r^2F_s(r)$, $f_p=\int_0^{r_0} dr r^3F_s(r)$, and $A,B,C,D$ are the functions of $r$ and $E^{+}\equiv E+i0^+$:$$\begin{aligned}
A&=&\frac{2}{3\pi} \int dk k^3 j_1(kr)\left( \sum_{\sigma=\pm} \frac{3/10}{E^+-2\epsilon_{\cp k}^{(\sigma)}} + \frac{2/5}{E^+-\epsilon_{\cp k}^{(+)}-\epsilon_{-{\cp k}}^{(-)}}\right); \nonumber\\
B&=&\frac{i}{\sqrt{3}\pi} \int dk k^2 j_1(kr)\left( \frac{1}{E^+-2\epsilon_{\cp k}^{(+)}} - \frac{1}{E^+-2\epsilon_{\cp k}^{(-)}}\right); \nonumber\\
C&=&\frac{1}{\pi} \int dk k^2 j_0(kr)\left( \frac{1}{E^+-2\epsilon_{\cp k}^{(+)}} + \frac{1}{E^+-2\epsilon_{\cp k}^{(-)}}\right); \nonumber\\
D&=&\frac{i}{3\sqrt{3}\pi} \int dk k^3 j_0(kr)\left( \frac{1}{E^+-2\epsilon_{\cp k}^{(+)}} - \frac{1}{E^+-2\epsilon_{\cp k}^{(-)}}\right). \nonumber\end{aligned}$$ Note that we have used $\epsilon_{\cp k}^{(\sigma)}=\epsilon_{-\cp k}^{(\sigma)}$ to simplify above equations. In small $r$ limit, $A\sim 1/r^2$, and $C, D\sim 1/r$. Here $D\sim 1/r$ shows an interesting fact that the p-wave interaction alone ($f_p\neq0,\ f_s=0$) can cause short-range singularity in the s-wave channel[@Cui]. Thus to ensure a physical $\psi_s$ near $r=0$, an s-wave potential has to be activated at short-range, as shown in Fig.1a. In other words, one has to explicitly include s-wave interaction while considering the physics near p-wave resonance. Note that this is in sharp difference to the pure s-wave interaction ($f_p=0,\ f_s\neq0$), which does not induce any singularity in p-wave channel[@Cui; @Vijay] and a single s-wave model is adequate to describe the physics near s-wave resonance.
Next we study the bound state solution with $E=-\kappa^2/m$, whose wave function is fully given by $\Psi^{sc}$ (Eq.\[psi\]) for $r>r_0$. Specifically, $\kappa$ and the ratio $f_{sp}\equiv-if_s/f_p$ can be exactly determined by requiring the continuity of both $\psi_s'/\psi_s$ and $\psi_p'/\psi_p$ at the potential boundary $r=r_0$. These quantities also determine the asymptotic behaviors of $\psi_s$ and $\psi_p$ (Eq.\[psi\_sp\]) in short-range regime ($r_0\ll r \ll 1/\kappa$): $$\begin{aligned}
\psi_p &\rightarrow& \frac{1}{r^2} +(-\frac{\kappa^2}{2}-\sqrt{3}f_{sp}\lambda+0.7\lambda^2)+\eta_p r; \label{short_psip}\\
\eta_p &=& \frac{f_{sp}}{\sqrt{3}\kappa}\lambda(3\kappa^2-\lambda^2)+\frac{\lambda^4-6\lambda^2\kappa^2+\kappa^4}{5\kappa} +\frac{2(\lambda^2+\kappa^2)^{3/2}}{15}, \nonumber\\
\psi_s &\rightarrow& \frac{1}{r}+\eta_s; \\
\eta_s&=&\frac{3\sqrt{3}f_{sp}(\lambda^2-\kappa^2)-\lambda(\lambda^2-3\kappa^2)}{(3\sqrt{3}f_{sp}-2\lambda)\kappa}. \nonumber\end{aligned}$$ One can check that in $\lambda=0$ limit, these equations well reproduce the free-space results $\eta_s=-\kappa$ and $\eta_p=\kappa^3/3$. When turn on SOC ($\lambda\neq 0$), under certain limit of $f_{sp}$ they can reproduce the results from individual s- or p-wave BC. Namely, the individual s-wave BC corresponds to equating $\eta_s$ with $1/a_s-r_s\kappa^2$ in $f_{sp}\rightarrow\infty$ limit[@Vijay; @Cui]; while the individual p-wave BC corresponds to equating $\eta_p$ with $-\frac{1}{3}(\frac{1}{v_p}+\frac{r_p}{2}\kappa^2)$ in $f_{sp}\rightarrow0$ limit[@Cui]. Of course one can further improve the theory by relaxing $f_{sp}$ and imposing the s- and p-wave BC simultaneously. We will show below that none of these theories predict correctly the bound states near p-wave resonance.
![(Color online) Bound state solution across p-wave resonance. $\lambda r_0=0.1$, $a_s=-4.5 r_0$. Square dots show exact solutions by solving the square-well potential in Fig.1a (i.e., by requiring the continuity of $\psi_s'/\psi_s$ and $\psi_p'/\psi_p$ at $r=r_0$). For comparison, we also show results from individual s/p-wave BC (gray/orange short-dashed lines), from the combined s-wave and conventional p-wave BC (blue dashed; by requiring the continuity of $\psi_s'/\psi_s$ at $r_0$ and matching $\eta_p$ to $-\frac{1}{3}(\frac{1}{v_p}+\frac{r_p}{2}\kappa^2)$, see Eq.\[old\_BC\]), and from the combined s-wave and new p-wave BC (red dashed-dot; by requiring the continuity of $\psi_s'/\psi_s$ at $r_0$ and matching $\psi_p$ in Eq.\[short\_psip\] to Eq.\[new\] at $r_0$). []{data-label="fig2"}](fig2.pdf){width="8.3cm"}
In Fig.2, we plot the exact solution of $\kappa/\lambda$ across the p-wave resonance (square dots), taking a small SOC strength $\lambda r_0=0.1$ and a weak s-wave interaction with $a_s=-4.5 r_0$. It shows two branches of solutions. Far away from p-wave resonance, the two branches follow the predictions from individual s-wave and individual p-wave BC(short-dashed lines), respectively giving the s-wave and p-wave dominated molecules. However, when close to p-wave resonance with $1/(\lambda v_p^{1/3})\sim[-3, 3]$, the exact solutions no longer follow the predictions from individual s-wave or individual p-wave BC, nor from the combined s-wave and conventional p-wave BC (Eq.\[old\_BC\]) (blue dashed). Instead, no molecule solution is found in this regime, which can be attributed to the enhanced interference between s- and p-wave channels due to the presence of SOC and strong p-wave interaction.
![(Color online) $f_{sp}$ and its associated weight $W_{sp}$ in $\eta_p$ for one branch of molecule solution in Fig.2. The black and red arrows show the change of $f_{sp}$ and $W_{sp}$ following the trajectory also arrow-marked in Fig.2. []{data-label="fig3"}](fig3.pdf){width="7.5cm"}
In the following, we show that the SOC-induced s-p interference (resulting in a finite $f_{sp}$) greatly modify the p-wave asymptotic parameter $\eta_p$, such that the usual p-wave BC breaks down even near the p-wave resonance. To see this clearly, in Fig.3 we plot $f_{sp}$ and its associated weight in $\eta_p$ ($W_{sp}$, i.e., the weight of the first term in $\eta_p$) for one branch of solution shown in Fig.2. We see that as $v_p$ is tuned to approach resonance, $|f_{sp}|$ gradually increases from zero to around $0.5$, and $W_{sp}$ increases from zero to nearly 1, suggesting the molecule evolves from the p-wave dominated to s-p strongly interfered regime. In the latter regime, the s-p interference play an essential role in determining the actual $\eta_p$, which can be very different from the predictions of usual p-wave BC (Eq.\[short\_range\]).
In comparison to the robust s-wave BC near s-wave resonance under SOC[@Cui; @Yu; @Zhang], here the fragile p-wave BC near the p-wave resonance has its intrinsic reasons. This is because the usual p-wave BC (Eq.\[old\_BC\]) is characterized by the ratio between very singular $1/r^2$ and very weak $r$ terms as shown in Eq.\[short\_range\]. Such ratio can be very easily destroyed by perturbations outside the potential, since it is related to the third derivative of $\psi_p$ (see Eq.\[old\_BC\]) while any realistic BC can only guarantee the continuity of $\psi_p$ and its first derivative across the boundary. This is quite contrary to the s-wave case where the short-range behavior is characterized by the ratio between $1/r$ and constant terms. In this case the continuity of $\psi_s$ and its first derivative is adequate to guarantee the continuity of this ratio. Therefore the short-range behavior of $\psi_s$ outside the potential ($r>r_0$) is universally given by the physics inside the potential ($r<r_0$), which is hardly modified by small perturbations such as SOC.
Motivated by above observations, we can set up a new p-wave BC following the persistent continuity of $\psi_p'/\psi_p$ at the short-range boundary. Given the wave function inside the potential is not modified by SOC, the new BC can be formulated using the original scattering parameters $v_p$ and $r_p$ in free space (see Eq.\[short\_range\]). Namely, at the potential boundary where SOC just starts to take effect, we have ($r\ll k^{-1},\ |v_p|^{1/3}$) $$\frac{(r^2\psi_p)'}{r^2\psi_p} = %-\frac{r^2}{v_p}+k^2 r\left( 1+\frac{r^3}{v_p} +r_p r (\frac{1}{2}+\frac{r^3}{3v_p})\right)
\left( -\frac{1}{v_p}+ \frac{r_pk^2}{2}\right) r^2 +k^2 r +o(r^4). \label{new}$$ Note that this rectified BC reflects the whole structure of $\psi_p$ in short-range limit (Eq.\[short\_range\]), including the constant term which has been omitted by usual p-wave BC (Eq.\[old\_BC\]). Because this constant is more dominated than $r$ term in short-range limit, and because the SOC greatly modifies such constant (Eq.\[short\_psip\]), it is important for its effect to be taken into account. In Fig.2, we plot the re-calculated $\kappa$ (red dashed-dot) using Eq.\[new\] to replace the usual p-wave BC for $\psi_p$ at $r=r_0$. The obtained results are excellently consistent with the exact solutions across p-wave resonance. We also check its robustness by choosing other boundaries $r(>r_0)$ in matching Eq.\[new\], and find good consistence with exact solutions for $r$ up to a few times of $r_0$. This justifies Eq.\[new\] as the correct p-wave BC in the presence of SOC. In fact, since Eq.\[new\] only relies on the wave function continuity near the potential boundary, it will be generally applicable for any type of SOC or other single-particle potentials, as long as they do not modify the physics inside the short-range potential.
In summary, we have studied the interplay of an isotropic SOC and strong p-wave interaction to both the short-range physics and shallow molecules of two interacting fermions. We find that even for SOC length much longer than the range of interaction potential, it can still induce strong interference between s- and p-wave channels, which leads to the vanishing of molecules near p-wave resonance and the breakdown of usual p-wave short-range BC. The proposed new p-wave BC, which applies for a wide class of single-particle potentials including SOC, will hopefully play more roles in dealing with few- and many-body problems near p-wave resonance.
Finally, since our scheme to solve the two-body problem and the conclusion of s-p interference do not rely on the specific type of SOC, our results will shed light on the molecule formation in current experiments with one or two-dimensional SOC[@gauge2exp; @fermisocexp1; @fermisocexp2; @2dsoczhang1; @Wu2015] and near p-wave resonance. In particular, it would be practical to realize SOC in quite a number of two-species fermion systems with p-wave Feshbach resonance, such as the $F=9/2$ $^{40}$K Fermi gas with two hyperfine states $|m_{F1}=-9/2; m_{F2}=-5/2\rangle$ at $B_0=215$G, $|-3/2;-1/2\rangle$ at $338$G, and $|9/2;5/2\rangle$ at $44$G[@thesis]. It is interesting to explore in future the multichannel molecules in these systems. [**Acknowledgment.**]{} I like to thank Doerte Blume, Qingze Guan and Peng Zhang for useful discussions on the few-body physics with spin-orbit coupling during the KITP program “Universality in Few-Body Systems” in the winter of 2016. This project is supported by the National Natural Science Foundation of China (No.11374177, No.11626436, No. 11421092, No. 11534014), and the National Key Research and Development Program of China (2016YFA0300603).
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We also find a [*weak*]{} singularity as $ \lambda/r$ in $|2;20\rangle$ state, while the dominated singularity $1/r^3$ is absent. Therefore we consider it a negligible short-range consequence to d-wave channel and discard it in this work.
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abstract: 'We developed a hybrid associations model () to generate sequential recommendations using two factors: 1) users’ long-term preferences and 2) sequential, both high-order and low-order association patterns in the users’ most recent purchases/ratings. uses simplistic pooling to represent a set of items in the associations. We compare with three the most recent, state-of-the-art methods on six public benchmark datasets in three different experimental settings. Our experimental results demonstrate that significantly outperforms the state of the art in all the experimental settings, with an improvement as high as 27.90%.'
author:
- Bo Peng$^1$
- Zhiyun Ren$^2$
- |
Srinivasan Parthasarathy$^{1,2}$Xia Ning$^{1,2}$[^1]\
$^1$Department of Computer Science and Engineering, The Ohio State University, Columbus, USA\
$^2$Department of Biomedical Informatics, The Ohio State University, Columbus, USA\
[email protected], {ren.685,ning.104}@osu.edu, [email protected]
bibliography:
- 'paper.bib'
title: ': Hybrid Associations Model with Pooling for Sequential Recommendation'
---
Introduction {#sec:intro}
============
Sequential recommendation aims to identify and recommend the next few items for a user that the user is most likely to purchase/review, given the user’s purchase/rating history. It becomes an effective tool to chronologically help users select favorite items from a variety of options. A key challenge in sequential recommendation is to identify, learn or represent the patterns and dynamics in users’ purchase/rating sequences that are most pertinent to inform their future interactions with other items, and also to capture the relations between such patterns and future interactions. With the prosperity of deep learning, many deep models, particularly based on recurrent neural networks [@hidasi2015session; @hidasi2018recurrent] and with attention mechanisms, have been developed for sequential recommendation purposes. These methods typically model users’ sequential behaviors and their long-term/short-term preferences, and have significantly improved recommendation performance.
However, given the notoriously sparse nature of most recommendation data without any side information on the items or users, a question on these deep learning methods, particularly those with attention mechanisms, is that whether the sparse recommendation data is sufficient to enable well-learned attention weights that play effective roles in identifying important information leading to accurate recommendations. Recent studies [@dacrema2019we; @ludewig2020] bring such concerns on recommendation algorithms in general, demonstrating that complicated deep recommendation methods may not always outperform simple ones.
We propose a hybrid associations model () with a simplistic pooling mechanism to better model users historical purchase/rating sequences. generates recommendations for the next items using two factors: 1) users’ general/long-term preferences and 2) sequential, association patterns in the users’ most recent purchases/ratings. The users’ general preferences are learned by leveraging their all historical purchases/ratings and represented in user embeddings. The item associated patterns used in include both high-order associations (i.e., more items induce the next, a few items) and low-order associations (i.e., fewer items induce the next, a few items). We use simplistic pooling to represent a set of items in the associations, and recommend the next items based on their recommendation scores aggregated from users’ general preferences and item association patterns. We compare the models with 3 the most recent, state-of-the-art methods on 6 public benchmark datasets in 3 different experimental settings. Our experimental results show that models significantly outperform the state-of-the-art methods in all the experimental settings, with a best improvement 27.90%. [^2] The major contributions in this paper are as follows:
- To the best of our knowledge, is the first method that explicitly models both high-order and low-order sequential associations among items for sequential recommendation. significantly outperforms the state-of-the-art methods.
- uses simplistic pooling instead of learned attentions to represent a set of items.
- We investigated the attention weights learned in benchmark datasets and studied the potential scenarios in which pooling could outperform attention mechanisms in sequential recommendation.
- We studied various experimental settings in which sequential recommendation performance is evaluated, and discussed the potential issues in the most widely used experimental setting in literature.
Literature Review {#sec:review}
=================
Numerous sequential recommendation methods have been developed, particularly using Markov Chains (MCs), Recurrent Neural Networks (RNNs), Convolutional Neural Networks (CNNs), attention and gating mechanisms, etc. Specifically, MCs-based methods, such as factorized personalized Markov chains (FPMC) [@rendle2010factorizing], use MCs to capture pairwise item-item transition relations to recommend the next item for each user. Based on FPMC, He [[@he2016fusing]]{} developed a factorized sequential prediction model (fossil), which uses high-order MCs to capture the impact from all historical purchases/ratings on the next item. Recently, many RNN-based methods have been developed to model the sequential patterns in users’ purchase/rating sequences. For instance, Hidasi [@hidasi2015session; @hidasi2018recurrent] used gated recurrent units (GRUs) to capture the users’ short-term preferences. Kang [@kang2018self] developed a self-attention based sequential model () to capture a few, most informative items in users’ purchase/rating sequences to generate recommendations. Recent work also adapts CNNs for sequential recommendation. For example, Tang [[@tang2018personalized]]{} developed a convolutional sequence embedding recommendation model (), which uses multiple convolutional filters on the most recent purchases/ratings to extract sequential features from a set of recent items. Ma [@ma2019hierarchical] developed a hierarchical gating network (), which captures item-item transition relations and user long-term preferences, and uses gating mechanisms to identify important items and their latent features from users’ historical purchases/ratings. has been demonstrated as the state of the art, and outperforms an extensive set of existing methods including and .
Methods {#sec:methods}
=======
Figure \[fig:method\] presents the architecture and Table \[tbl:notations\] presents the key notations. In this paper, the historical purchases or ratings of user $i$ in chronological order are represented as a sequence $\sequences_i$=$\{s_i(1), s_i(2), \cdots\}$, where $s_i(t)$ is the $t$-th purchased/rated item. A length-$l$ subsequence of $S_i$ starting at the $t$-th purchase/rating is denoted as $S_i(t, l)$, that is, $S_i(t, l) = \{s_i(t), s_i(t$$+$$1), \cdots, s_i(t$$+$$l$$-$$1)\}$. When no ambiguity arises, we will eliminate $i$ in $S_i$/$S_i(t, l)$.
generates recommendations for the next items for each user using two factors: 1) the user’s general/long-term preferences, and 2) the sequential, association patterns in the user’s most recent purchases/ratings. These two factors will be used to calculate a recommendation score for each item candidate in order to prioritize and recommend the next items.
Modeling Users’ General Preferences {#sec:method:general}
-----------------------------------
It has been shown that users’ general preferences play an important role in their purchases/ratings [@he2016fusing]. Therefore, in , we learn users’ general preferences using an embedding matrix $\U \in \mathbb{R}^{m\times d}$. serves as a lookup table, in which the $i$-th row, denoted as $\mathbf{u}_i$, represents the general preferences of user $i$.
Modeling Sequential Hybrid Associations {#sec:method:sequential}
---------------------------------------
Previous study [@tang2018personalized] has shown the existence of sequential associations in recommendation data. Here, we denote the sequential association at time $t$ from its previous $\LL$ purchases/ratings to the next $\T$ subsequent purchases/ratings as $S(t$$-$$\LL, \LL)$$\rightarrow$$S(t, \T)$, and the number of the involved $\LL$$+$$\T$ items as the order of the association. It has also been shown [@tang2018personalized] that the sequential associations among items in benchmark recommendation datasets (used in experiments as in Section \[sec:experiments:datasets\]) have different orders. For example, about 50% significant associations have $\LL$$=$$2$ and $\T$$\leq$$2$ (i.e., previous 2 purchases/ratings have immediate effects on the next 1 or 2 purchases/ratings), and about 15% significant associations have $\LL$$=$$4$ and $\T$$\leq$$2$.
We propose to explicitly model the item associations of different orders in such that the aggregated information from the previous different numbers of purchases/ratings will contribute to the recommendation scores of all the subsequent item candidates. Particularly, we include a high-order association $S(t$$-$$\LL, \LL)$$\rightarrow$$S(t, \T)$ and a low-order association $S(t$$-$$\M, \M)$$\rightarrow$$S(t, \T)$ ($\M$$<$$\LL$, $S(t$$-$$\M, \M) \subset S(t$$-$$\LL, \LL)$) to recommend the next $\T$ items. During training, we split the historical purchase/rating sequence of each user into multiple subsequences (training instances) of length $\LL$+$\T$ (these subsequences will overlap), and learn the individual item embeddings, and thus the embeddings of $\LL$/$\M$ successive items, from all the length-($\LL$+$\T$) subsequences. Note that can be a general framework, in which arbitrary numbers of various-order associations can be incorporated.
#### Mean and Max Pooling from Previous Items
In order to represent the information from the previous $\LL$/$\M$ purchased/rated items as a whole, we use pooling mechanisms, and learn one embedding $\mathbf{h}$/$\mathbf{o}$ for the $\LL$/$\M$ successive purchases/ratings using 1) mean pooling and 2) max pooling, respectively, from individual item embeddings of the $\LL$/$\M$ purchased/rated items. Given the sparse nature of the recommendation data, it might be difficult to learn and differentiate the contributions of different items. The mean pooling is a simplistic solution to average the effects from each individual item. with mean pooling is denoted as . The hypothesis of max pooling is that the purchased/rated items contribute in various dimensions in determining the next $\T$ purchases/ratings. with max pooling is denoted as .
Heterogeneous Item Embeddings
-----------------------------
It has also been shown [@rendle2010factorizing; @kang2018self] that the item transitions may be asymmetric: item $j$ might be frequently purchased/rated after item $k$, but not vise versa. To model such asymmetry in , we follow the ideas of heterogeneous item embeddings [@kang2018self] and learn two item embedding matrices, denoted as $\EM \in \mathbb{R}^{n \times d}$ and $\QM \in \mathbb{R}^{n \times d}$, respectively. and are lookup tables, in which the $j$-th row, denoted as $\E_j$ and $\Q_j$, respectively, represent item $j$. If item $j$ is used to recommend the next items, it is represented by $\E_j$. If item $j$ is a candidate to be recommended, it is represented by $\Q_j$.
Recommendation Scores {#sec:method:prediction}
---------------------
The recommendation scores are calculated from the user embedding $\mathbf{u}$, the embedding of previous $\LL$ purchases/ratings $\mathbf{h}$ and the embedding of previous $\M$ purchases/ratings . For user $i$, given the subsequence $S_i(t$$-$$\LL,\LL)$, the estimated recommendation score of user $i$ on item candidate $j$, denoted as $\rs^t_{ij}$, is calculated as follows: $$\vspace{-3pt}
\label{eqn:prediction}
\rs^t_{ij} = ~~~\underbrace{\mathbf{u}_i \Q_j^\top}_{\mathclap{\text{user's general preferences}}}
~~~~~~+
\underbrace{\Esub \Q_j^\top}_{\text{high-order association}} +
\underbrace{\Esubsub \Q_j^\top}_{\text{low-order association}},
\vspace{-3pt}$$ where $\Q_j$ is embedding of item $j$, is the embedding for $S_i(t$$-$$\LL,\LL)$ and is the embedding for $S_i(t$$-$$\M,\M)$; $\mathbf{u}_i \mathbf{w}_j^\top$ measures how user $i$’s general preferences match item candidate $j$; $\Esub \mathbf{w}_j^\top$ measures how $S_i(t$$-$$\LL,\LL)$ induces item candidate $j$, and $\Esubsub \mathbf{w}_j^\top$ measures how $S_i(t$$-$$\M,\M)$ induces item candidate $j$. For each user, we recommend the items of [top-$k$]{} largest recommendation scores. Note that we do not explicitly weight the three factors, as their weights can be learned as part of the user/sequence/item embeddings.
Objective Function {#sec:method:obj}
------------------
We adapt the Bayesian personalized ranking objective [@rendle2012bpr] and minimize the loss that occurs when the truly purchased/rated items are ranked below those not purchased/rated. The objective function is as follows: $$\label{eqn:obj}
\min\limits_{\boldsymbol{\Theta}} \sum^m_{i=1}
\sum_{\scriptsize{\substack{\sequences_i(t,\T) \\ \subset \sequences_i}}}~~~~
\sum_{\mathclap{\substack{\scriptsize{j \in \sequences_i(t, \T)}\\ \scriptsize{k \notin \sequences_i(t, \T)}}}}
-\log\sigma(\rs_{ij} - \rs_{ik}) + \lambda(\|\boldsymbol{\Theta}\|^2),
\vspace{-3pt}$$ where $\boldsymbol{\Theta} = \{\U, \EM, \QM\}$ is the set of the parameters, $\sigma$ is the sigmoid function, $\sequences_i(t, \T)$ is a sequence of $\T$ items in $\sequences_i$, $j$ denotes an item in $\sequences_i(t,\T)$, and $k$ denotes an item not in $\sequences_i(t,\T)$. Given the huge number of items not in $\sequences_i(t,\T)$, following the ideas in literature [@tang2018personalized; @ma2019hierarchical], we randomly sample a non-purchased/rated item $k$ for each purchased/rated item $j$. Please note that, in , the recommendation scores of purchased/rated item are not necessarily close to their ground-truth ratings, as long as the scores of purchased/rated items are higher than scores of those not purchased/rated. Also note that each of the items in $\sequences_i(t,\T)$ will be recommended and evaluated independently from $S_i(t$$-$$\LL,\LL)$, following the literature [@ma2019hierarchical].
Materials {#sec:materials}
=========
Baseline Methods {#sec:experiments:baseline}
----------------
We compare with following state-of-the-art methods. **Caser** [@tang2018personalized] uses multiple convolutional filters on the most recent purchases/ratings of a user to extract the user’s sequential features and items’ group features. These two features and the users’ long-term preferences are used to calculate item recommendation scores. **** [@kang2018self] uses self-attention mechanisms to capture the most informative items in users’ purchase/rating sequences to recommend the next item. **** [@ma2019hierarchical] uses gating mechanisms to identify important items and their latent features from users’ historical purchases/ratings to recommend next items. has been compared with a comprehensive set of other methods and has been demonstrated as the state of the art. Thus, we compare with instead of the methods that outperforms.
Datasets {#sec:experiments:datasets}
--------
We evaluate the methods on 6 public benchmark datasets: Amazon-Books (Books) and Amazon-CDs (CDs) [@he2016ups], Goodreads-Children (Children) and Goodreads-Comics (Comics) [@wan2018item], and MovieLens-1M (ML-1M) and MovieLens-20M (ML-20M) [@harper2016movielens]. The Books and CDs datasets are from Amazon reviews [@amazon], which contain users’ 1-5 star ratings and reviews on books and CDs, respectively. The Children and Comics datasets are from goodreads website [@goodreads]. These two datasets contain users’ implicit feedback (i.e., if a user has read the book or not), explicit feedback (i.e., ratings) and reviews on children and comics books. The ML-1M and ML-20M datasets are from the MovieLens website [@movielens] with user-movie ratings. Following the data preprocessing protocol in [@ma2019hierarchical], among the 6 datasets, we only kept the users with at least 10 ratings, and items with at least 5 ratings. We converted the rating values into binary values by setting rating 4 and 5 to value 1, and the lower ratings to value 0. Table \[tbl:dataset\] presents the statistics of the 6 datasets after the preprocessing.
Experimental Settings {#sec:experiments:setting}
---------------------
We use the following three experimental settings to evaluate the methods. Figure \[fig:setting\] presents the three settings.
**80-20-cut-off setting ()**: We extract the first 70% of each user’s sequence as training set, the next 10% as validation set for parameter tuning, and the remaining 20% as testing set [^3]. is the most widely used experimental setting in sequential recommendation literature [@yuan2014graph; @zhao2016stellar; @tang2018personalized; @ma2019hierarchical]. **80-3-cut-off setting ()**: We use the same training and validation set as in , but only the next 3 items after the validation set as the testing set. Compared to , recommends the immediate next few items, not potentially many items that might be only purchased/rated much later (e.g., in , 20% of a long user sequence may have many items).
**Leave-out setting ():** We use only the last 3 items in each user sequence for testing and all the previous items for training and validation. The validation set contains only the 3 items before the testing items. Thus, maximizes the data for training and recommends the immediate next few items.
Evaluation Metrics {#sec:experiments:metrics}
------------------
Following the literature [@ma2019hierarchical], we use [Recall@$k$]{} and [NDCG@$k$]{} to evaluate the different methods. For each user, [Recall@$k$]{} measures the proportion of all the ground-truth purchased/rated items in the testing set that are correctly recommended. The overall [Recall@$k$]{} value is calculated as the average over all the users. Higher [Recall@$k$]{} indicates better recommendation performance. [NDCG@$k$]{} is the normalized discounted cumulative gain for among top-$k$ ranking, in which $\text{gain} \in\{0,1\}$, indicating whether a ground-truth purchased/rated item has been recommended (i.e., 1) or not (i.e., 0). Thus, [NDCG@$k$]{} measures the positions of the correctly recommended items among the top-$k$ recommendations. Higher [NDCG@$k$]{} indicates better performance.
Experimental Results {#sec:results}
====================
Overall Performance in Setting {#sec:results:lof}
------------------------------
Table \[tbl:performance\] presents the results on all the six datasets in . Note that is the setting used in and . In , for , we used the parameters reported by its authors on CDs, Books, Children, Comics and ML-20M; on ML-1M we tuned the parameters using grid search. For , and other baseline methods, we also tuned their parameters using grid search and report the best results. For , we achieved similar results as reported; for other baseline methods, the results are slightly better than those reported [@ma2019hierarchical].
Table \[tbl:performance\] shows that in terms of [Recall@$10$]{}, achieves the best performance on 5 out of 6 datasets, and the second best performance on the rest ML-1M dataset; achieves the second best performance on 3 out of 6 datasets. In terms of [NDCG@$10$]{}, achieves the best performance on 4 out of 6 datasets, and the second best performance on the ML-20M dataset; achieves the second best performance on 2 of 6 datasets. On average, achieves 27.90%, 18.90% and 7.10% improvement in terms of [Recall@$10$]{} , and 14.08%, 12.27% and 4.31% improvement in terms of [NDCG@$10$]{} over all 6 datasets compared to , and , respectively. This indicates that outperforms the state of the art on most benchmark datasets with significant improvement.
Table \[tbl:performance\] also shows that outperforms on all the datasets. The difference between and is that uses mean pooling over the last $\LL$ items and last $\M$ items to recommend the next items, whereas uses gating mechanisms over the last items and also over their latent features to differentiate the importance of such items and features. However, as Table \[tbl:dataset\] shows, each user typically has only a few items (compared to all the possible items), and each item is typically only purchased/rated by a few users (compared to all the possible users). Therefore, the data sparsity issue may lead to less meaningful gating weights learned by parameterized gating mechanisms, whereas equal weights from mean pooling would suffice. Similarly, , which uses max pooling to capture item difference, could also be effective, demonstrated by its relatively similar performance as in Table \[tbl:performance\]. In addition, combines both high-order and low-order sequential patterns, conforming to the discovery in [@tang2018personalized], which may also contribute to the superior performance. In addition, outperforms on all the datasets in terms of Recall@$10$, and in terms of NDCG@$10$ on 5 out 6 datasets except on ML-1M. A key difference between and is that leverages item associations in the most recent purchases/ratings, whereas only uses the long-term user preferences. As demonstrated in literature [@zhou2019deep], user preferences may shift over time and thus preferences from the most recent purchases/ratings might provide more pertinent information for the next recommendations. Moreover, outperforms across all datasets consistently. This might be due to a similar reason as for , that is, the sparse data does not substantially enable well-learned difference among latent item features. Table [\[tbl:performance\]]{} also shows that outperforms except on the relatively dense datasets ML-1M and ML-20M, which may exhibit strong local patterns in the latent item features that learns from sufficient data and utilizes for recommendation.
Overall, outperforms the best baseline methods with very high percentage improvement when the dataset is very sparse (e.g., 14.16% on CDs on Recall@$10$), and significant improvement when the datasets is moderately sparse (e.g., 4.67% on Comics on Recall@$10$). When the datasets are dense (e.g., ML-1M as in Table \[tbl:dataset\]), could be slightly worse than the other baseline methods. However, most of the recommendation problems always have very sparse datasets, on which will be effective. More detailed analysis on the data sparsity aspect is available later in Section \[sec:results:weight\].
Overall Performance in Setting {#sec:results:lot}
------------------------------
Table \[tbl:near\] presents the results in . In , for , and all the baseline methods, the parameters are tuned by grid search on the validation sets, and the best results are reported. Overall, the performance comparison between and the baseline methods has a similar trend as in . Particularly, achieves overall the best performance compared to the other methods. In terms of NDCG@$10$, although is slightly worse than on Children and Comics, it still achieves the second best performance. Table \[tbl:performance\] and \[tbl:near\] together show that in terms of Recall@$10$, all the methods have in general better performance in (evaluating on the immediate next few items) than in (evaluating on the rest 20% items). Note that and have the same training sets but different testing sets. The results in the two settings correspond to our intuition that the historical purchase/rating sequences are most informative for the immediately next few items compared to the items purchased/rated much later. In terms of NDCG@$10$, however, the results in are better. It may be due to that although the recall rate is low in , the number of accurately recommended items is larger, which increases NDCG@$10$.
Overall Performance in Setting {#sec:results:loo}
------------------------------
Table \[tbl:performancelevel\] presents the results in . Overall, outperforms the other methods, achieving the best performance in terms of both Recall@$10$ and NDCG@$10$ on 5 datasets, and the second best performance on the rest dataset. Table \[tbl:near\] and \[tbl:performancelevel\] together show that in terms of Recall@$10$ and NDCG@$10$, all the methods have in general better performance in (i.e., the next 3 items after the validation set of each user are used for testing and the first 80% sequence are used for training and validation) than in (i.e., the last 3 items of each user are used for testing and all the previous items are used for training and validation). Compared to setting, the training sets in setting contain more early purchases/ratings (i.e., purchases/ratings that occurred long time ago before the testing items). These early purchases/ratings may not accurately represent users’ preferences at the time of the testing items as such preferences may shift [@zhou2019deep].
Attention Weight Analysis {#sec:results:weight}
-------------------------
[0.23]{}
[0.23]{}
[0.23]{}
[0.23]{}
It has been shown that the learned attention weights may not always be meaningful [@jain2019attention; @serrano2019attention]. Therefore, we further investigate the attention weights in to interpret their significance and to understand why instead the simplistic mean pooling in would suffice. We use datasets CDs, Comics, ML-1M and ML-10M in the investigation, because these datasets, as Table \[tbl:dataset\] shows, represent different data sparsities (i.e., CDs is highly sparse, Comics is moderately sparse, and ML-1M and ML-10M are dense). Figure \[fig:total\_distribution\] (the x-axis in the figure is logarithmized item frequencies and then normalized into \[0,1\]) shows that most of the items in CDs and Comics are very infrequent, whereas in ML-1M infrequent items are fewer; in ML-20M, the infrequent items (compared to other frequent items in ML-20M) still have many purchases/ratings (Table \[tbl:dataset\]).
Figure \[fig:cd\_weights\], \[fig:comics\_weights\], \[fig:ml1m\_weights\] and \[fig:ml20m\_weights\] present the distributions of attention weights from the best performing models on the CDs, Comics, ML-1M and ML-20M, respectively. Note that a same item can have different weights in different user sequences; we use all the weights of a same item from all the users in the figures. The distributions for CDs and Comics datasets show very similar pattens: for very infrequent items, their attention weights are highly centered around 0.5 (i.e., the initialization value); for the most frequent items, their attention weights are slightly off 0.5 and have some different values than 0.5. Given that as Figure \[fig:total\_distribution\] shows, most of the items in CDs and Comics are infrequent, the weight distribution over infrequent items indicates that the weights might not be well learned to differentiate the importance of infrequent items. The weights on frequent items might be relatively better learned. However, unfortunately, frequent items are not many and their weights may not substantially affect recommendations. The distributions for ML-1M and ML-20M datasets also show similar pattens: the weights for both infrequent and frequent items are closely centered at 0.5, indicating that such weights might not well differentiate item importance. Such weight distributions from both sparse and dense datasets indicate that the learned weights may not play an effective role in recommendation. Instead, a special case of weights, that is, equal weights as we have in , should also achieve comparable performance as . As a matter of fact, equal weights on better learned item representations as in actually improve the recommendation performance.
Parameter and Ablation Study {#sec:results:parameter}
----------------------------
Table [\[tbl:parameter\]]{} presents the parameter study on in the on two sparse datasets CDs and Comics. Recall that $\LL$/$\M$ and $\T$ are the number of items in high-order/low-order associations, and the number of items to be recommended used for training. Table \[tbl:parameter\] shows that as more items are used to learn high-order item associations (i.e., larger $\LL$) and more items are recommended during training (i.e., larger $\T$), the recommendation performance over the remaining 20% items is better. Still, the best performance is achieved when $\LL$ and $\T$ are small ($\LL$ is 4 or 5; $\T$ is 2 to 4), indicating that the most recent associations among a few items are effective in recommending next items.
Table \[tbl:ablution\] presents the results when low-order associations among items are not included in (i.e., $\M = 0$) in . Table \[tbl:ablution\] shows that when only a small number of items are used to learn item associations (e.g., $\LL$=3) and a small number of items are recommended during training (e.g., $\T$=2), without low-order item associations achieves its best performance. Comparing Table \[tbl:parameter\] and Table \[tbl:ablution\], it is noticed that when low-order associations are not used, the recommendation performance (Table \[tbl:ablution\]) is significantly worse than that when low-order associations are used (Table \[tbl:parameter\]). It indicates that explicitly modeling the low-order associations, together with high-order associations, enables to better capture the hybrid impact from the previous, different number of items, and thus to improve the performance.
Table \[tbl:parameterT\] and \[tbl:parameterO\] present the parameter study of in and , respectively. Similarly as in Table \[tbl:parameter\], a small number of items in high-order associations (e.g., $\LL$ = 4) is sufficient for the best performance on sparse datasets.
Discussions on Experimental Settings {#sec:discussions}
====================================
(i.e., to evaluate the last 20% items) and NDCG@$k$ (e.g., $k$=10) are the most commonly used experimental setting and evaluation metric. NDCG@$k$ in could over-estimate the sequential recommendation performance, particularly for long sequences in which the last 20% items include many. In such long sequences, NDCG@$k$ could be high when items that are purchased/rated very late are recommended on top. However, such recommendations would have limited use scenarios. Meanwhile, many testing items will increase the chances that the recommended $k$ items will be included in the many items, and thus inflate the NDCG@$k$ values. and mitigate the over-estimation issue because always a same number of items will be tested and NDCG@$k$ calculated over a same number of testing items will not be affected by the number of testing items.
[^1]: Contact Author
[^2]: we will publish the source code once this paper is accepted.
[^3]: We used the data splits from <https://github.com/allenjack/HGN>.
|
---
abstract: 'During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules $S^\lambda$ for the symmetric group $\Sigma_d$, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the $p$-rank of the defect group of its block. The Georgia group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of $\lambda$ is built out of $p \times p$ blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the $p$-weight of partitions and branching.'
address: |
Department of Mathematics\
University at Buffalo, SUNY\
244 Mathematics Building\
Buffalo, NY 14260, USA
author:
- 'David J. Hemmer'
bibliography:
- 'references0808.bib'
date: November 2008
title: '**The complexity of certain Specht modules for the symmetric group**'
---
[^1]
Introduction {#sec: Introduction}
============
In 2004 the VIGRE algebra research group at the University of Georgia computed some examples of the complexity of Specht modules $S^\lambda$ for the symmetric group $\Sigma_d$. Their data led them to focus on partitions $\lambda$ of a curious form, specifically:
\[defin: pxpdef\] A partition $\lambda \vdash d$ is $p \times p$ if $\lambda=(\lambda_1^{a_1}, \lambda_2^{a_2}, \ldots, \lambda_s^{a_s})$ where $p\mid \lambda_i$ and $p \mid a_i$ for all $i$.
Such $\lambda$ can exist only if $p^2 \mid d$. Equivalently, $\lambda$ is [$p \times p$ ]{}if both $\lambda$ and its transpose $\lambda'$ are of the form $p\tau$. Also equivalently, the Young diagram of $\lambda$ is built from $p \times p$ blocks.
Now suppose $S^\lambda$ is in a block $B(\lambda)$ of weight $w$ corresponding to a $p$-core $\tilde{\lambda} \vdash d-pw$. Then the defect group of $B(\lambda)$ is isomorphic to a Sylow $p$-subgroup of $\Sigma_{pw}$ and has $p$-rank $w$. In particular, the maximum complexity of any module in the block $B(\lambda)$ is $w$. The VIGRE group made the following conjecture:
\[conj: Vigre\] Let $S^\lambda$ be in a block $B$ of weight $w$. Then the complexity of $S^\lambda$ is $w$ if and only if $\lambda$ is not [$p \times p$ ]{}.
Conjecture \[conj: Vigre\] implies that almost every Specht module has maximal complexity among modules in its block. Indeed it would imply that if $p^2\not\,\mid d$, then all the Specht modules for $\Sigma_d$ have this property. As far as we know the condition we call [$p \times p$ ]{}has not appeared anywhere in the literature, and it seems quite mysterious. However we will prove that it arises quite naturally from considering the weights of $\Sigma_d$ blocks and the branching theorems. Specifically we prove:
\[thm: main\] Suppose $\lambda \vdash d$ has $p$-weight $w$. Then $\lambda$ is [$p \times p$ ]{}if and only if $w(\lambda_A) \leq w-2$ for each removable node $A$ of $\lambda$. In this case, $w(\lambda_A)$ is always equal to $w-2$.
Theorem \[thm: main\] give one direction of Conjecture \[conj: Vigre\] as a fairly immediate corollary.
\[cor: onewayofconjecture\] Suppose $\lambda \vdash p^2d$ is [$p \times p$ ]{}, and hence of weight $w=pd$. Then the complexity of $S^\lambda$ is less than $w$.
\[rmk: d<p\^2\]When $d<p^2$ Conjecture \[conj: Vigre\] can be deduced immediately from the dimensions of the corresponding Specht modules, as was noticed by the VIGRE group and proven by Lim [@LimVarietySpechtpreprint Thm. 4.1].
\[rmk: Lim\] The conjecture was verified for $d=p^2$ by Lim in [@LimVarietySpechtpreprint Thm. 3.1], where he showed $S^{(p^p)}$ has complexity $p-1$ even though $(p^p)$ has weight $p$. Lim’s proof uses the fact that ${\operatorname{Res}}_{\Sigma_{p^2-1}}(S^{(p^p)}) \cong S^{(p^{p-1},p-1)}$ and $(p^{p-1},p-1)$ has weight $p-2$. Our proof is essentially a large generalization of this observation about the branching theorems.
Blocks and abaci
================
In this section we review the description of the blocks of $k\Sigma_d$ and the representation of partitions of $d$ on the abacus. For further details on this, and as good sources for the representation theory of the symmetric group, see the books [@JamesKerberbook] and [@Jamesbook], where definitions of basic terms not defined here, like removable node, addable node, residue of a node, etc.., can be found.
Let $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_t) \vdash d$ with $\lambda_t>0$. Choose an integer $r \geq t$ and define a *sequence of beta-numbers* by: $$\beta_i=\lambda_i -i+r$$ for $1 \leq i \leq r$. For $r=s$ the $\beta$-numbers are just the first-column hook lengths in the Young diagram of $\lambda$. It is easily seen that $\lambda$ can be recovered uniquely from a corresponding sequence of beta numbers.
Now take an abacus with $p$ runners, labeled $0, 1, \ldots, p-1$ from left to right. Label the positions on the abacus so that row $s$ (counting down from the top) runner $i$ is labeled $i+(s-1)p$. To represent $\lambda$ on the abacus with $r$ beads, simply place a bead at position $\beta_i$ for each $1 \leq i \leq r$.
Sliding all the beads in the abacus display as far up as possible, one obtains an abacus display for the $p$-core $\tilde{\lambda}$. Moving a bead up a single spot corresponds to removing a rim $p$-hook from the Young diagram of $\lambda$. Thus, if $w$ is the total number of such moves, then $\tilde{\lambda} \vdash d-pw$. The number $w$ is called the $p$-weight of $\lambda$. The following proposition collects some well-known facts, all found in the book [@JamesKerberbook].
\[prop:abacusproperties\] Let $\lambda \vdash d$ have $p$-weight $w$, $p$-core $\tilde{\lambda} \vdash d-pw$, and exactly $t$ parts. Let $\mu \vdash d$. Then:
- The Specht modules $S^\lambda$ and $S^\mu$ lie in the same $p$-block if and only if $\tilde{\lambda}=\tilde{\mu}.$
- Suppose $B(\lambda)$ is the block containing $S^\lambda$. Then a defect group of $B(\lambda)$ is isomorphic to a Sylow $p$-subgroup of the symmetric group $\Sigma_{pw}$.
- Removing a removable node from $\lambda$ corresponds to sliding a bead in the abacus display of $\lambda$ from position $l$ to (unoccupied) position $l-1$. Adding an addable node corresponds to sliding a bead from position $l$ to (unoccupied) position $l+1$.
- If $\lambda$ is represented on an abacus with a multiple of $p$ beads then removable nodes which correspond to beads on runner $i$ have $p$-residue $i$.
- Let $\{A_1, A_2, \ldots, A_m\}$ denote the removable nodes of $\lambda$ and let $\lambda_A \vdash d-1$ denote $\lambda$ with removable node $A$ removed. Then ${\operatorname{Res}}_{\Sigma_{d-1}}(S^\lambda)$ has a filtration by Specht modules where each $S^{\lambda_{A_i}}$ occurs once.
Motivated by Proposition \[prop:abacusproperties\](d), we assume henceforth all our abacus displays have a multiple of $p$ beads. Given such an abacus display for $\lambda$, let $\lambda^{[i]}$ be the number of beads on runner $i$. Knowing these numbers is equivalent to knowing the $p$-core $\tilde{\lambda}$. For each runner we define a partition which captures how far each bead moves when sliding the beads of $\lambda$ all the way up. Define $\lambda_j^i$ as the number of empty spots above the $j$th bead from the top on runner $i$, i.e. the number of times this bead is moved up to obtain the $p$-core. Then $\lambda(i)=(\lambda_1^i, \lambda_2^i, \ldots )$ is a partition and the sequence of partitions $(\lambda(0), \lambda(1), \ldots, \lambda(p-1))$ is called the *$p$-quotient* of $\lambda$. It should be clear that $\lambda$ is uniquely determined by its $p$-core and $p$-quotient, and that the parts of the $p$-quotient are partitions of integers which sum to $w$.
The main theorem {#sec: The main theorem}
================
In this section we will prove Theorem \[thm: main\]. Our first observation is that the $p \times p$ condition is easily described in terms of the abacus display:
\[prop: pxpintermsofabacus\] Let $\lambda \vdash d$ have $p$-quotient $(\lambda(0), \lambda(1), \ldots, \lambda(p-1))$. Then $\lambda$ is [$p \times p$ ]{}if and only if $\tilde{\lambda}=\emptyset$ and $\lambda(0)=\lambda(1)= \cdots =\lambda(p-1)$.
This is a straightforward exercise. Suppose $\lambda$ is [$p \times p$ ]{}and write $$\lambda=(p\tau_1^{pa_1}, p\tau_2^{pa_2}, \ldots, p\tau_r^{pa_r}).$$ Then $\lambda$ has $\sum_{i=1}^r pa_i$ parts and $p$-weight $\sum_{i=1}^r pa_i\tau_i$. Representing $\lambda$ on an abacus with $\sum_{i=1}^r pa_i$ beads, one easily verifies that each $\lambda(i)$ in the $p$-quotient is just $(\tau_1^{a_1}, \tau_2^{a_2}, \ldots, \tau_r^{a_r})$. Conversely given a partition with constant $p$-quotient $\lambda(i)=\tau$ and empty $p$ core, one immediately observes that $\lambda$ is [$p \times p$ ]{}.
\[rmk: abacusform\] The condition on $\lambda$ in Proposition \[prop: pxpintermsofabacus\] is equivalent to each row of the abacus display being either empty or completely full. For example when $p=5$ the partition $\lambda=(20^{10},10^5,5^5)$ has abacus display shown in Figure \[fig: abacusdisplay\], where the corresponding $\tau$ is $(4,4,2,1)$.
(10,14)(0,0)
(2,0)[(0,1)[14]{}]{} (4,0)[(0,1)[14]{}]{} (6,0)[(0,1)[14]{}]{} (8,0)[(0,1)[14]{}]{} (10,0)[(0,1)[14]{}]{} (1.8,0)[(1,0)[0.4]{}]{}(1.8,2)[(1,0)[0.4]{}]{}(1.8,4)[(1,0)[0.4]{}]{} (1.8,6)[(1,0)[0.4]{}]{}(1.8,8)[(1,0)[0.4]{}]{}(1.8,10)[(1,0)[0.4]{}]{} (1.8,12)[(1,0)[0.4]{}]{}(1.8,14)[(1,0)[0.4]{}]{}
(3.8,0)[(1,0)[0.4]{}]{}(3.8,2)[(1,0)[0.4]{}]{}(3.8,4)[(1,0)[0.4]{}]{} (3.8,6)[(1,0)[0.4]{}]{}(3.8,8)[(1,0)[0.4]{}]{}(3.8,10)[(1,0)[0.4]{}]{} (3.8,12)[(1,0)[0.4]{}]{}(3.8,14)[(1,0)[0.4]{}]{}
(5.8,0)[(1,0)[0.4]{}]{}(5.8,2)[(1,0)[0.4]{}]{}(5.8,4)[(1,0)[0.4]{}]{} (5.8,6)[(1,0)[0.4]{}]{}(5.8,8)[(1,0)[0.4]{}]{}(5.8,10)[(1,0)[0.4]{}]{} (5.8,12)[(1,0)[0.4]{}]{}(5.8,14)[(1,0)[0.4]{}]{}
(7.8,0)[(1,0)[0.4]{}]{}(7.8,2)[(1,0)[0.4]{}]{}(7.8,4)[(1,0)[0.4]{}]{} (7.8,6)[(1,0)[0.4]{}]{}(7.8,8)[(1,0)[0.4]{}]{}(7.8,10)[(1,0)[0.4]{}]{} (7.8,12)[(1,0)[0.4]{}]{}(7.8,14)[(1,0)[0.4]{}]{}
(9.8,0)[(1,0)[0.4]{}]{}(9.8,2)[(1,0)[0.4]{}]{}(9.8,4)[(1,0)[0.4]{}]{} (9.8,6)[(1,0)[0.4]{}]{}(9.8,8)[(1,0)[0.4]{}]{}(9.8,10)[(1,0)[0.4]{}]{} (9.8,12)[(1,0)[0.4]{}]{}(9.8,14)[(1,0)[0.4]{}]{}
(2,12)(4,12)(6,12) (8,12)(10,12)
(2,8)(4,8)(6,8) (8,8)(10,8)
(2,2)(4,2)(6,2) (8,2)(10,2)
(2,0)(4,0)(6,0) (8,0)(10,0)
Suppose for the remainder of this section that $\lambda \vdash d$ has $p$-core $\tilde{\lambda} \vdash c$ and weight $w$, so $d=c+pw$. Suppose further that for every removable node $A$, the partition $\lambda_A$ which results from removing the node $A$ satisfies $w(\lambda_A) \leq w-2$. We will show that $\lambda$ has an abacus display as described in Remark \[rmk: abacusform\], and thus that $\lambda$ is [$p \times p$ ]{}. Recall the notation $\lambda^{[i]}$ for the number of beads on runner $i$, thus $\lambda(i) \vdash \lambda^{[i]}$.
\[lem: removablerunner12etc\] Suppose $\lambda$ has a removable node $A$ of residue $i> 0$ Then $\lambda^{[i-1]} \geq \lambda^{[i]}+1$.
Removing a node corresponds to sliding a bead one spot to the left on the abacus. Suppose the bead corresponding to $A$ is in row $s$ of the abacus display, so there is no bead on runner $i-1$ in row $s$. Further, suppose runner $i-1$ has $y$ beads in the first $s-1$ rows and $\lambda^{[i-1]}-y$ beads in rows $>s$. Assume runner $i$ has $x$ beads in the first $s-1$ rows and $\lambda^{[i]}-x-1$ beads in rows $>s$. Runners $i-1$ and $i$ in the abacus display for $\lambda$ are illustrated in Figure \[fig: columni\].
(15,18)(0,0)
(5,-1)[(0,1)[17]{}]{} (10,-1)[(0,1)[17]{}]{} (4.5,2)[(1,0)[1]{}]{}(4.5,4)[(1,0)[1]{}]{} (4.5,6)[(1,0)[1]{}]{}(4.5,8)[(1,0)[1]{}]{} (4.5,10)[(1,0)[1]{}]{}(4.5,12)[(1,0)[1]{}]{}(4.5,14)[(1,0)[1]{}]{} (4.5,0)[(1,0)[1]{}]{} (9.5,2)[(1,0)[1]{}]{}(9.5,4)[(1,0)[1]{}]{} (9.5,6)[(1,0)[1]{}]{}(9.5,8)[(1,0)[1]{}]{} (9.5,10)[(1,0)[1]{}]{}(9.5,12)[(1,0)[1]{}]{}(9.5,14)[(1,0)[1]{}]{}
(10,8) (10,12) (10,2) (10,0) (5,6) (5,10) (5,2) (5,14) (10,0)
(4.2,16.5)[$i-1$]{} (9.8,16.5)[$i$]{} (-3,7.5)[row $s$]{} (1,8)[(1,0)[2]{}]{}
(3,13)[(2,6)\[l\]]{} (1.5,13)[(1,0)[0.5]{}]{} (-4,12.5)[$y$ beads]{}
(3,16)[(1,0)[0.5]{}]{} (3,10)[(1,0)[0.5]{}]{} (3,6)[(1,0)[0.5]{}]{} (3,0)[(1,0)[0.5]{}]{} (12,16)[(-1,0)[0.5]{}]{} (12,10)[(-1,0)[0.5]{}]{} (12,6)[(-1,0)[0.5]{}]{} (12,0)[(-1,0)[0.5]{}]{}
(12,13)[(2,6)\[r\]]{} (13,13)[(1,0)[0.5]{}]{} (14,12.5)[$x$ beads]{}
(14,7.5)[bead of node $A$]{} (13,8)[(-1,0)[2]{}]{} (3,3)[(2,6)\[l\]]{} (1.5,3)[(1,0)[0.5]{}]{} (-9,2.5)[$\lambda^{[i-1]}-y$ beads]{}
(12,3)[(2,6)\[r\]]{} (13,3)[(1,0)[0.5]{}]{} (14,2.5)[$\lambda^{[i]}-x-1$ beads]{}
Then the weight of bead $A$ changes from $s-x-1$ to $s-y-1$. In $\lambda_A$, the bottom $\lambda^{[i-1]}-y$ beads on runner $i-1$ are now weight one less and the bottom $\lambda^{[i]}-x-1$ beads on runner $i$ are weight one more. Thus the change in weight between $\lambda$ and $\lambda_A$ is:
$$\begin{aligned}
\label{eq:changeinweight}
w(\lambda_A)-w(\lambda) &=& s-y-1+\lambda^{[i]}-x-1-(s-x-1+\lambda^{[i-1]}-l)\\
&=& \lambda^{[i]}-\lambda^{[i-1]}-1. \nonumber\end{aligned}$$
Our assumption on $\lambda_A$ gives $\lambda^{[i]}-\lambda^{[i-1]}-1 \leq -2$, and thus $\lambda^{[i-1]} \geq \lambda^{[i]}+1$.
\[prop: nonincreonpcore\]The number of beads on each runner in the abacus display of $\lambda$ is nonincreasing from left to right, i.e. $\lambda^{[i-1]} \geq \lambda^{[i]}.$
If $\lambda^{[i-1]} < \lambda^{[i]}$ then the extra beads on runner $i$ ensure $\lambda$ has a removable node of residue $i$, which then contradicts Lemma \[lem: removablerunner12etc\].
Next we consider the situation of a removable node of residue zero.
\[lem: removableresidue0\]Suppose $\lambda$ has a removable node of residue zero. Then $\lambda^{[0]}=\lambda^{[p-1]}.$
Suppose $\lambda$ has a removable node of residue zero, so the abacus display has a bead in row $s+1$ of runner zero, and no bead in row $s$ of runner $p-1$. We let the reader draw the corresponding picture to Figure \[fig: columni\], and observe that $$\label{eq: residuezero}
w(\lambda_A)-w(\lambda)=\lambda^{[0]}-\lambda^{[p-1]}-2.$$ By Prop. \[prop: nonincreonpcore\] plus our assumption that $w(\lambda_A)-w(\lambda) \leq -2$, we obtain the result.
If any of the inequalities in Proposition \[prop: nonincreonpcore\] were strict, then we would have $\lambda^{[0]}>\lambda^{[p-1]}$, which would contradict Lemma \[lem: removableresidue0\]. Thus we have proven that:
$$\label{eq: all lambaeq}
\lambda^{[0]}=\lambda^{[1]} \cdots =\lambda^{[p-1]}.$$
Comparing equations and , we discover that all removable nodes of $\lambda$ have weight zero, since removable nodes of any other residue decrease the weight of the partition by only one. So the abacus display of $\lambda$ has the same number of beads on each runner, and all the removable nodes have residue zero. This forces the abacus display to be as described in Remark \[rmk: abacusform\], and completes the proof of Theorem \[thm: main\]. Notice that and implies $w(\lambda_A)$ is always exactly $w-2$, although our assumption was only that it is $\leq w-2$.
Complexity
==========
Finally we prove Corollary \[cor: onewayofconjecture\], which gives one direction of Conjecture \[conj: Vigre\]. A module has complexity $c$ if the dimensions in a minimal projective resolution of that module are bounded by a polynomial of degree $c-1$. The complexity of the trivial module is known to be the maximal rank of an elementary abelian subgroup, which for $\Sigma_{pd}$ is just $d$. In general the complexity of a module for the symmetric group in a block of weight $w$ is at most $w$. Good for this theory are [@Bensonrepteoryvolume2] or [@EvensCohobook].
So suppose $\lambda \vdash p^2d$ is [$p \times p$ ]{}and so has weight $w=pd$. By Proposition \[prop:abacusproperties\](e) and Theorem \[thm: main\], all summands of ${\operatorname{Res}}_{\Sigma_{p^2d-1}}S^\lambda$ lie in blocks of weight $w-2$. By [@HNsupportvariety 4.2.1d], ${\operatorname{Res}}_{\Sigma_{p^2d-1}}S^\lambda$ has complexity at most $w-2$. An elementary argument using support varieties (see [@LimVarietySpechtpreprint] for example) implies that $S^\lambda$ cannot have complexity $w$. If it did, then for $E$ a maximal elementary abelian subgroup of rank $w$, the support variety $V_E(S^\lambda)$ would have dimension $w$, which would force the complexity of ${\operatorname{Res}}_{\Sigma_{p^2d-1}}S^\lambda$ to be $w-1.$
For the case $\lambda=p^p$, dimension considerations immediately give the complexity of $S^{(p^{p-1},p-1)}$, which lets Lim deduce [@LimVarietySpechtpreprint] that the complexity of $S^{(p^p)}$ is exactly $p-1$. This part of the argument does not appear to generalize, i.e. in general it large powers of $p$ can divide the dimension of $S^{\lambda_A}$ when $\lambda$ is [$p \times p$ ]{}.
Problems
========
There are several obvious problems left unsolved.
Resolve the other direction of Conjecture \[conj: Vigre\].
Suppose $\lambda$ is [$p \times p$ ]{}of weight $w$. Is the complexity of $S^\lambda$ equal to $w-1$, or can it drop further?
One can generalize the definition of [$p \times p$ ]{}. For example the first obvious generalization would be to require $\lambda$ be [$p \times p$ ]{}and each $\lambda(i)$ be [$p \times p$ ]{}. (And recursively for higher generalizations.) Can one say anything interesting about these situations? Perhaps the complexity drops by even more in this case?
[^1]: Research of the author was supported in part by NSF grant DMS-0556260
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---
abstract: 'Recently, heat manipulation has gained the attention of scientific community due to its several applications. In this letter, based on transformation thermodynamic (TT) methodology, a novel material, which is called thermal-null medium (TNM), is proposed that enables us to design various thermal functionalities such as thermal bending devices, arbitrary shape heat concentrators and omnidirectional thermal cloaks. In contrary to the conventional TT-based conductivities, which are inhomogeneous and anisotropic, TNMs are homogeneous and easy to realize. In addition, the attained TNMs are independent of the device shape. That is if the geometry of the desired device is changed, there is no need to recalculate the necessitating conductivities. This feature of TNM will make it suitable for scenarios where re-configurability is of utmost importance. Several numerical simulations are carried out to demonstrate the TNM capability and its applications in directional bending devices, heat concentrators and thermal cloaks. The proposed TNM could open a new avenue for potential applications in solar thermal panels and thermal-electric devices.'
author:
- Hooman Barati Sedeh
- Mohammad Hosein Fakheri
- Ali Abdolali
- Fei Sun
bibliography:
- 'achemso-demo.bib'
title: 'Thermal-null medium (TNM) : a novel material to achieve feasible thermodynamics devices beyond conventional challenges'
---
1. Introduction
===============
In the times of energy shortage, the recycling of heat energy and heat manipulation becomes an important topic. Therefore, how to handle the heat dissipation, the heat storage and the control of heat flux becomes a significant subject of debates among the scientific community. Various methods have been proposed for this aim which among all of them, transformation thermodynamics (TT) demonstrates a high flexibility to manipulate heat in an unprecedented manner [@guenneau2012transformation; @schittny2013experiments]. The main idea of TT is derived from its electromagnetic (EM) counterpart which was proposed by Pendry *et al.* and named as transformation optics (TO) [@pendry2006controlling]. As it was analyzed by Pendry’s group, an equivalence between Maxwell’s equations described in an initial coordinate system (i.e. virtual domain) and their counterparts in another arbitrary transformed coordinate system (i.e. physical domain), will result in a direct link between permittivity and permeability of the occupying material and the transformed space metric tensor, which contains the desired EM properties. Soon after the introduction of TO, many novel devices which deemed impossible to be achieved with natural materials such as EM cloaks [@liu2009broadband; @xu2015conformal; @fakheri2017carpet], multi emission lenses [@tichit2014spiral; @zhang2016experimental; @jiang2012broadband; @ashrafian2019space; @barati2019exploiting], EM wave concentrator [@rahm2008design; @yang2009metamaterial; @sadeghi2015transformation; @zhao2018feasible; @yang2019arbitrarily; @zhou2018perfect; @abdolali2019geometry] and beam splitters [@barati2018experimental; @sedeh2019advanced; @kwon2008polarization] have been proposed. The same as TO methodology, TT has gained lots of attention due to the intrinsic degree of freedom that offers. Although TT could paves the way towards designing various devices such as heat cloak [@guenneau2012transformation], heat flux concentrators [@yu2011design] and heat transferring devices [@hu2016directional], it has some serious drawbacks which restrict the usage of this approach for real-life scenarios. The main problem of TT is that the attained conductivities through this approach are inhomogeneous and anisotropic which result in difficulties for their fabrications [@guenneau2012transformation]. Although thermal metamaterials have received lots of attention in recent years, it is still a challenging task to design a thermal cell which possess both inhomogenity and anisotropy [@farhat2016transformation]. In addition to the inhomogenity problem, the TT-based conductivities are extremely depended on the device input shape. That is if the geometry of the device is changed, one must not only perform tedious mathematical calculations to attain desired conductivities but also redesign them, which is time consuming and not applicable for scenarios where re-configurability is important. Therefore, the above-mentioned reasons cause this question to be asked that whether is there any alternative way of designing heat devices that its necessitating conductivities are homogeneous and independent of the input geometry?
In this paper, we propose a new method for achieving various thermal devices based on new material called thermal-null medium (TNM) [@sun2019thermal; @liu2018fast]. In contrary to the conventional TT-based conductivities, the attained TNMs through our approach are homogenous and geometry free. That is if the desired device shape is changed, there is no need to recalculate the necessitating conductivities. This will make the TNM a good alternative for being used in scenarios where re-configurability is of utmost importance. For instance, as the first application of TNM, a directional heat bending device is designed that regardless of its deflection angle, only one material is used and perfect functionality will be achieved. In addition, for the first time we have used the TNM as a mean to obtain arbitrary shape heat concentrators with homogenous materials. It is shown that by utilizing a TNM, one can freely change the geometry of the concentrator while using the same material. Finally, a thermal omnidirectional cloak is also proposed that is capable of guiding the thermal distributions from any incident direction in a manner that they become undetectable from an outside observer.
The rest of this paper is organized as follows. First, the fundamental theoretical formulation of the method is given, then several numerical simulations are performed to demonstrate the capability of propounded approach for achieving different practical functionalities in section 3. Finally, the paper will end with a summary of the obtained results in the conclusion section.
2. Theoretical formulations
===========================
The derivation of TNM is based on the null-space mapping, which is a kind of transformation that will result in optical/thermal materials with large anisotropic parameters. Here, we will briefly discussed the derivation approach, while for more comprehensive study one can refer to [@sun2019thermal; @liu2018fast]. Assume a slab with width of $W_v$ is mapped to another slab with width of $W_p$ as shown in Fig.1.
![Schematic diagram of null-space transformation to obtain TNM. a) A thin slab in the virtual space is mapped to b) a slab with finite thickness. c) Schematic of a typical shape designed by bending a slab in (b).[]{data-label="fgr:fig1"}](fig1.jpg){width="0.6\linewidth"}
The transformation function of such a mapping could be written as $$\begin{aligned}
x^\prime =
\ \left\{
\begin{array}{lll}
x & x^\prime \in (-\infty, 0]\\
\frac{W_p}{W_v} x & x^\prime \in [0, W_p] \\
x-W_v+W_p & x^\prime \in [W_p,\infty)
\end{array}
\right. \\end{aligned}$$ When the thickness of the slab in the virtual space approaches zero (i.e., $W_v \rightarrow 0$), its interfaces become very close to each other (see $S_1$ and $S_2$ in Fig.1 (a)). After a null-space transformation (i.e., Eq. (1) when $W_v \rightarrow 0$), the thickness of this slab is greatly expanded. In other words, while surface $S_1$ is fixed, surface $S_2$ is stretched by $W_p/W_v$ times in the physical space as shown in Fig.1 (b). For a steady state case, the thermal conduction equation without a source can be written as $\nabla\cdot(\kappa \nabla T)=0$, where $\kappa$ is thermal conductivity and $T$ is the temperature. On the basis of the invariance of heat conduction equation under coordinate transformations [@guenneau2012transformation], the thermal conduction equation in the transformed space can also be written as $\nabla^\prime \cdot(\kappa^\prime \nabla^\prime T^\prime)=0$, through which we can obtain $$\kappa^\prime=\frac{\Lambda \times \kappa \times \Lambda^T}{det(\Lambda)}$$ where $\Lambda=\partial(x^\prime,y^\prime,z^\prime)/\partial(x,y,z)$ is the Jacobian matrix which relates the metrics of virtual space ,$(x,y,z)$, to the ones of physical space, $(x^\prime,y^\prime,z^\prime)$. By substituting Eq. (1) into Eq. (2) the corresponding heat conductivity will be achieved as $$\frac{\kappa^\prime}{\kappa}=
\begin{bmatrix}
\frac{W_p}{W_v} &
0&
0 \\
0 &
\frac{W_v}{W_p} &
0\\
0 &
0 &
\frac{W_v}{W_p}
\end{bmatrix}$$ However, since $W_v \rightarrow 0$ (null-space transformation), the obtained conductivity will be changed to $$\kappa^\prime=
\begin{bmatrix}
\infty &
0&
0 \\
0 &
0 &
0\\
0 &
0 &
0
\end{bmatrix}$$ The obtained material of Eq.(4) is named as thermal null medium (TNM), which was designed by null-space transformation. Since thermal resistance is inversely proportional to the thermal conductivity, it could be concluded that $R_\parallel \rightarrow 0$ and $R_\perp \rightarrow \infty$, which leads to a super-transferring performance along the “stretching” direction without dissipation. In other words, the function of TNM is to guide heat very fast along its main stretching direction (i.e., $\hat{x}$) and extremely slow in other directions (i.e., $\hat{y}$ and $\hat{z}$ ). In addition, as it was comprehensively discussed recently in [@sun2019thermal; @liu2018fast], TNMs are not restricted to the cartesian coordinate system. In fact, TNMs with any other shapes could be obtained by bending the slab of Fig.1(b) and still guide the thermal field directionally without any dissipation as shown in Fig.1 (c).
3. Numerical simulations
========================
To verify the correctness of the proposed TNM, heat directional bending devices, are analyzed by performing simulations that were carried out using COMSOL Multiphysics finite element solver. Fig. 2 shows the primary goal of our design. As it can be seen, there is a need to change the direction of a heat flow to the angle of $\alpha$.
![The schematic transformation for describing heat directional bending device []{data-label="fgr:fig4"}](fig2.jpg){width="0.5\linewidth"}
As it is shown in Fig.2, the hot interface is separated from the cold one with the deflection angle of $\alpha$. The output surface (that has higher temperature $T_H$) and input surface (that has lower temperature $T_C$) are labeled by $S_1$ and $S_2$, respectively. It is clear from Fig.2 that a common plane of $S$ (shown with red color) can be obtained, if one extends the edges of surfaces $S_1$ and $S_2$ (indicated by black dashed lines). According to the above-mentioned discussion, filling the region between surfaces $S_1$ and $S$ with a TNM with its main axis along the $\hat{ x}$ direction (i.e., $\kappa_x \rightarrow \infty$ and $\kappa_y \rightarrow 0$) and the area between surfaces $S$ and $S_2$ with a TNM with its main axis along along the *u* direction (which is rotated to the angle of $\alpha$ with respect to the $\hat{x}$ direction), will result in point-to-point mapping and in turn will give rise to bending the heat distributions directionally from $S_1$ to $S_2$. In other words, surfaces of $S_1$, $S_2$ and $S$ are all equivalent surfaces, in a manner that heat distributions on $S_1$ will first be projected onto $S$ along the $\hat{x}$ direction and then projected onto $S_2$ along the $\alpha$ direction. It is notable to mention that the entire design process is general, without any mathematical calculations and is valid for any arbitrary deflection angle of $\alpha$. To demonstrate such a capability, several thermal bending devices with different bending angles and structures are simulated and their results are depicted in Fig.3. As can be seen from this figure, the obtained TNM is capable of connecting two thermal surfaces with any desired temperature without any deviation in their thermal contours. This will make it a suitable material for being used in scenarios where reconfigurability is of utmost importance or there is a prerequisite to manipulate heat distributions directionally in an arbitrary path. Furthermore, according to Fig.3 (d), the fictitious surface $S$ could also be located in cylindrical coordinate system (i.e., azimuthal direction ($\hat{\phi}$) in Fig.3 (d)) rather than cartesian coordinate and remains its functionality.
![Thermal distributions of a heat bending device with different bending angles of (a) $\alpha=30^\circ$,(b) $\alpha=45^\circ$ and with TNM in (c) cartesian, (d) cylindrical coordinate system[]{data-label="fgr:fig7"}](fig3.jpg){width="0.8\linewidth"}
In addition to the thermal directional bending devices, over the last few decades, much more attention has been paid to renewable energy since the fossil fuels are being depleted. Among all the proposed methods, solar energy has been widely exploited as one of the most important renewable energy for converting the sun energy to electricity. In general, the conversion of solar energy into electricity is performed via two approaches of direct solar-electrical energy conversion and indirect conversion. In the former, the solar cell devices are used while in the latter thermal energy is used as a mediator by means of a device named as solar thermal collector (STC). Heat flux concentrator is an example of the second group (i.e., STC) that received a lot of attention in the past decade. Recently, TT method has also been extended to manipulate heat current and localize thermal energies with heat flux concentrators. However, the conventional challenges of TT-based conductivities (i.e., inhomogenity and shape dependent) are still the main drawback of TT-based concentrators and restrict their applicability for being used in practical situations. However, by extending the idea of null-space transformation, one can exploit TNMs as a mean to collect thermal energies in any arbitrary region of interest.To this aim, the schematic diagram of Fig.4 is used as the space transformation, which three cylinders with arbitrary cross sections of $R_1(\phi)= \tau_1 R(\phi)$, $R_2(\phi)=\tau_2 R(\phi)$ and $R_3(\phi)=\tau_3 R(\phi) $ divide the space (i.e., virtual space) into three different regions. It should be noted that $\tau_1$, $\tau_2$ and $\tau_3$ are constant coefficients which satisfy the condition of $\tau_1 < \tau_2 < \tau_3$ and $R(\phi)$ is an arbitrary continuous function with period of $2\pi$ that is specified with Fourier series as $$R(\phi) = a_{0}+ \sum_{n=1}^{\infty} \{ a_{n} \cdot cos (n\phi) +b_{n} \cdot cos (n\phi) \}$$
where $a_n$ and $b_n$ are constant coefficients that specify the contour shape.
![ The schematic of coordinate transformation for achieving arbitrary shape concentrators.[]{data-label="fig:fig8"}](fig4.jpg){width="\linewidth"}
To concentrate the thermal energy in a predefined region of $R_1(\phi)$, one must collect the energy that was originally located in $\rho<R_2(\phi)$ into the region of $\rho^{\prime} < R_1 (\phi^\prime)$ as shown in Fig.4. To this aim, as explained in [@guenneau2012transformation; @farhat2016transformation], the region of $\rho\in[0, R_2 (\phi)]$ must be compressed into the region $\rho^\prime\in[0, R_1 (\phi^\prime)]$, while at the same time $\rho\in[R_2 (\phi), R_3 (\phi)]$ is stretched into the region of $\rho^\prime\in[R_1 (\phi^\prime), R_3 (\phi^\prime)]$. Since these two steps occur simultaneously, all the energy, which previously located in $\rho < R_2 (\phi)$ is now localized in the region of $\rho^{\prime} < R_1 (\phi^\prime)$ and as a result, the thermal intensity will be increased in the mentioned domain. The function which is competent to perform such a transformation could be expressed as $$\ \left\{
\begin{array}{ll}
f_c(\rho,\phi)= \frac{\tau_1}{\tau_2}\rho& \rho'\in [0, R_1(\phi))\vspace{0.5 cm} \\
f_s(\rho,\phi)= \Omega \times \rho + \Psi \times R(\phi) & \rho'\in[R_1 (\phi), R_3 (\phi)]
\end{array}
\right. $$ where subscripts of $c$ and $s$ represent the compressed and stretched region, respectively and $\Omega=(\tau_3-\tau_1)/(\tau_3-\tau_2)$, $\Psi= [(\tau_1-\tau_2)/(\tau_3-\tau_2)]\tau_3$. By substituting Eq. (6) into Eq. (2) the necessitating materials for each region will be achieved as $$\frac{\kappa'_{c}}{\kappa_0}=
\begin{bmatrix}
1 &
0&
0 \\
0 &
1 &
0\\
0 &
0 &
(\tau_2/\tau_1)^2
\end{bmatrix} ,
\frac{\kappa'_{s}}{\kappa_0}=
\begin{bmatrix}
m_{11} &
m_{12}&
0 \\
m_{21} &
m_{22}&
0\\
0 &
0 &
m_{33}
\end{bmatrix}$$ where the coefficients of $m_{ij}$ are $$\begin{aligned}
&m_{11} = \frac{(\tau_3 -\tau_2)\rho^\prime - \tau_3(\tau_1-\tau_2)R(\phi)}{(\tau_3-\tau_2)\rho^\prime}+
\frac{\tau^2_3(\tau_1-\tau_2)^2(dR(\phi)/d\phi)^2 }{[(\tau_3-\tau_2)^2 r^\prime - \tau_3 (\tau_1-\tau_2)(\tau_3-\tau_2)R(\phi)]\rho^\prime} \\ \nonumber
&m_{12} = m_{21}= \frac{(\tau_3(\tau_1-\tau_2)(dR/d\phi))}{(\tau_3-\tau_2)\rho^\prime- \tau_3 (\tau_1 -\tau_2) R(\phi)},m_{22} = \frac{(\tau_3-\tau_2)r^\prime}{(\tau_3-\tau_2)\rho^\prime-\tau_3(\tau_1-\tau_2)R(\phi) } \\ \nonumber
&m_{33}=\frac{(\tau_3-\tau_2)^2 \rho^\prime -\tau_3(\tau_1-\tau_2)(\tau_3-\tau_2)R(\phi)}{(\tau_3-\tau_1)^2 \rho^\prime}\end{aligned}$$
As can be seen from Eq.(7) and Eq.(8), the obtained materials of the stretched region are inhomogeneous and anisotropic with off-diagonal components of $m_{12} / m_{21}$ which cause serious difficulties in their realization process. In fact, the reason why there is no experimental verification of arbitrary shape thermal concentrators are yet proposed is due to the existence of these inhomogeneous and off-diagonal components.
In addition to the inhomogenity that the existence of $R(\phi)$ in the components of $m_{ij}$ dictates, it also demonstrate the dependency of obtained materials to the structure geometry. In other words, an alternation in the coefficients of Eq.(5),which results in new geometry, leads to new materials which must be recalculated. However, since $ R_2(\phi)=\tau_2 \times R(\phi)$ is a fictitious region, $\tau_2$ can achieve any arbitrary value. This will give us a degree of freedom to arbitrarily select the value of $\tau_2$ in a manner that it will eradicate the effect of the off-diagonal components of $m_{12}$ (also $m_{21}$). Therefore, without the loss of generality one can assume that $\tau_2 \rightarrow \tau_3$(null-space transformation). By setting such a value for $\tau_2$ the coefficients of Equation (8) will be attained as $$\ \left\{
\begin{array}{lll}
m_{11}=\frac{1}{\Delta}
\vspace{0.5 cm} \\
m_{12}=m_{21}=-\kappa_0\frac{dR(\phi)/d\phi}{R(\phi)}
\vspace{0.5 cm} \\
m_{22}=m_{33}=\Delta
\end{array}
\right. $$ whence $\Delta \rightarrow 0$. As can be seen from Equation (9), the obtained conductivities are still suffer from anisotropy and inhomogeneity problem due to the existence of $m_{12}$ (also $m_{21}$ ) and their dependence to the contour shape of $R(\phi)$. However, it is known that in the absence of a heat source the heat diffusion equation of steady state is governed by the Laplace equation as $$\begin{aligned}
&\nabla \cdot(\bar{\bar{\kappa}}\nabla T)=\sum_{j,k=1}^{n}\frac{\partial ^{j,k}}{\partial x^j}\kappa\frac{\partial T}{\partial x^k}=0\end{aligned}$$ Therefore, after some simple calculation the Laplace equation in cylindrical coordinate will be achieved as $$\begin{aligned}
&\frac{1}{\rho} m_{11} \partial T/\partial \rho + m_{11} \partial^2 T/\partial \rho^2 + \frac{2}{\rho} m_{12} \partial^2 T/\partial \rho \partial \phi +
\frac{1}{\rho} \frac{\partial m_{12}}{\partial \phi} \partial T/\partial \rho +\frac{1}{\rho^2}m_{22} \partial^2 T/\partial \phi^2 \\ \nonumber
& +m_{33} \partial^2 T/\partial z^2 =0 \end{aligned}$$ By substituting Equation (9) into Equation (11), one can easily obtain $$\begin{aligned}
&\frac{1}{\rho} \partial T/\partial \rho + \partial^2 T/\partial \rho^2 + \frac{2}{\rho} m_{12} \times \Delta \times \partial^2 T/\partial \rho \partial \phi +
\frac{1}{\rho} \frac{\partial m_{12}}{\partial \phi} \times \Delta \times \partial T/\partial \rho \\ \nonumber & +\frac{1}{\rho^2}\Delta^2 \partial^2 T/\partial \phi^2
+\Delta^2 \partial^2 T/\partial z^2 =0 \end{aligned}$$ Since $\Delta \rightarrow 0$ and $m_{12}$ has a finite value according to Eq. (9), the exact value of $m_{12}$ is not important. This is because only the products of these values play crucial role as shown in the Laplace equation (i.e. Eq. (12)) not each of them individually. Therefore, one can assume any desirable finite value for the off-diagonal components. Here we assumed $m_{12}=0$, this assumption would eradicate the off-diagonal components of Eq. (9). Therefore, the final conductivities, which describe the performance of an arbitrary shape thermal concentrator, will be achieved as $$\frac{\kappa'_{c}}{\kappa_0}=
\begin{bmatrix}
1 &
0&
0 \\
0 &
1&
0\\
0 &
0 &
(\tau_2/ \tau_1)^2
\end{bmatrix} ,
\frac{\kappa'_{s}}{\kappa_0}=
\begin{bmatrix}
\infty &
0&
0 \\
0 &
0&
0\\
0 &
0 &
0
\end{bmatrix}$$ To validate the concept, several arbitrary shape concentrators were simulated. The solving area is consisted of a square which at the position of $x=-0.3 m$ , there is a planar metallic plate with the temperature of $T=315 $ K, and the metallic plate at $x=+0.3 m$ has the temperature $T=275$ K. For all the performing simulations, it is assumed that the values of $\tau_1=0.5$, $\tau_2=0.99$ and $\tau_3=1$ are constant while $R(\phi)$ is changed for each new case. Therefore, the necessitating conductivies will be attained according to Eq. (13) as $$\frac{\kappa'_{c}}{\kappa_0}=
\begin{bmatrix}
1 &
0&
0 \\
0 &
1&
0\\
0 &
0 &
3.92
\end{bmatrix} ,
\frac{\kappa'_{s}}{\kappa_0}=
\begin{bmatrix}
\infty &
0&
0 \\
0 &
0&
0\\
0 &
0 &
0
\end{bmatrix}$$ The first example is dedicated to circular and elliptical cross-section concentrators and their results are illustrated in Figure 5.
![ The temperature distribution results of the heat flux concentrator with different cross sections of (a),(c) circular and elliptical. (b),(d) their corresponding thermal flow.[]{data-label="fig:fig9"}](fig5.jpg){width="0.8\linewidth"}
As can be seen from Fig. 5(a) and Fig.5(c), the thermal concentrator does not change the thermal distribution which is well abide with the theoretical investigations. In addition, since in the background medium( in a specific location) the heat flux is equal to $350 w/m^2$, it is expected that in this particular location, the heat flux be enhanced by the ratio of $\tau_2/\tau_1=1.98$. This has been verified by the numerical simulations, as shown in Fig.5(b) and Fig.5(d). It is evident that in the compressed region the heat flux is increased from $350 w/m^2$ to $693 w/m^2$ which is in good agreement with the theoretical predictions.
To have full control of heat flux it might be necessary in some cases that the thermal energy is localized into a certain domain having an arbitrary cross-section. To date, no systematic work has been proposed for achieving arbitrary shape thermal concentrator. As it was mentioned, the presented method in this article will result in anisotropic conductivities that are independent to the geometry of the concentrator. That is the attained conductivites of Eq.(14) are sufficient for any desired geometry. This will in turn make the proposed approach a good candidate for scenarios where reconfigurability is of utmost importance. To show this, assume that the contour coefficients of $R(\phi)$ in Eq. (5) are given in a way that an arbitrary shape concentrator is generated. By utilizing the obtained conductivities (i.e., Eq. (14)) for each compressed and stretched regions, the results of the thermal flux and temperature distributions will be achieved as shown in Figure 6.
![ (a),(c) The temperature distribution results of heat flux concentrator with arbitrary cross-section. (b),(d) their corresponding thermal flow.[]{data-label="fig:fig10"}](fig6.jpg){width="0.8\linewidth"}
The same as the previous case, in the new scenario the thermal flow is increased from $350 W/m^2$ to $693 W/m^2$ in the compressed region without any distortion in its temperature distributions. However, it is noteworthy to mention that changing the ratio of $\tau_2/\tau_1$ will result in different values for the thermal energy inside the compressed domain.
As the final example, a square shape heat flux cloak is proposed via utilization of TNM. To date, no systematic work has been proposed that yield to omnidirectional heat cloak. However, in this paper for the first time we have proposed a square shape cloak that is capable of guiding the thermal distributions in a manner that the object becomes undetectable from an outside observer. Since heat will be diffused from a higher temperature region to a lower one, if an obstacle is located in the path of the thermal flux, a distortion in the isothermal lines will occur and in turn give rise to degradation of efficiency. However, when a square shape heat cloak is exploited, it will allow the heat flux to pass smoothly around the cloaked region without creating any distortion as shown schematically in Fig.7 (a).
![ (a) The schematic of the heat flux path when the TM-based cloak is utilized.(b) The demanding mapping function used for designing a square shape cloak[]{data-label="fig:fig11"}](fig7.jpg){width="0.8\linewidth"}
To design such a cloak based on coordinate transformation, the transformation function of Fig.7 (b) is used. In contrary to the previous reported cloaks which mapped a point in the virtual space into a circle in the physical space, here the cloak region has divided into different regions and for each domain a linear transformation function is exploited. The space between the square with side length of $L$ (i.e., *$ A B C D$*) in the virtual space is transformed to the same square with the same length in physical space(i.e., *$A^\prime B^\prime C^\prime D^\prime$*), while at the same time inner square with the side of *$l_1$* (i.e., *$E F G H$*)in the virtual space is mapped to a larger square with the side of *$l_2$* (i.e., *$E^\prime F^\prime G^\prime H^\prime$*) in the physical space.
Without the loss of generality, one can assume that the inner square side length is approaching to zero (i.e., $l_1 \rightarrow 0$ ). Therefore, under this assumption, the triangles $\triangle OAB$, $\triangle OBC$, $\triangle OCD$ and $\triangle ODA$ in the virtual space (Fig.7 (b)) are transformed to triangles $\triangle F^\prime A^\prime B^\prime $, $\triangle G^\prime B^\prime C^\prime$, $\triangle H^\prime C^\prime D^\prime$ and $\triangle E^\prime D^\prime A^\prime$ in the physical space respectively. Meanwhile, the lines $AO$, $BO$, $CO$ and $DO$ in the virtual space must also be transformed to triangles $\triangle A^\prime E^\prime F^\prime$, $\triangle B^\prime F^\prime G^\prime$, $\triangle C^\prime G^\prime H^\prime$ and $\triangle D^\prime H^\prime E^\prime$ in the physical space, respectively. Consequently, the object which was supposed to be cloaked will not affect the heat distributions and will be invisible from the outside detector. By introducing a local coordinate system for each of the pink triangular region ($\triangle D^\prime H^\prime E^\prime$ and $\triangle B^\prime F^\prime G^\prime$ are indicated as I and III, respectively while $\triangle A^\prime E^\prime F^\prime$ and $\triangle C^\prime G^\prime H^\prime$ are shown by II and IV ), the demanding conductivity for each of these regions is given by $$\frac{\kappa^\prime_{I,III}}{\kappa_0}=
\begin{bmatrix}
0&
0&
0 \\
0 &
\infty &
0\\
0 &
0 &
0
\end{bmatrix},
\frac{\kappa^\prime_{II, IV}}{\kappa_0}=
\begin{bmatrix}
\infty &
0&
0 \\
0 &
0 &
0\\
0 &
0 &
0
\end{bmatrix}$$ In addition, the conductivities of the remaining regions (shown by $R$ subscript) in their local coordinate system (i.e.,$ (u,v,z)$) will also be attained as $$\frac{\kappa^\prime_R}{\kappa_0}=
\begin{bmatrix}
\frac{1}{\Gamma} &
0&
0 \\
0 &
\Gamma &
0\\
0 &
0 &
\frac{1}{\Gamma}
\end{bmatrix},$$ where $\Gamma=1-(\sqrt{2}l_2/L)$. Compared with the previous cloak prototypes with inhomogeneous and anisotropic extreme material parameters, the square cloak is simplified to one with only two homogeneous materials. The latter is a simple diagonal anisotropic mass density tensor which could be easily implemented via thermal metamaterials [@farhat2016transformation; @yang2014experimental] and the former is the TNM which is introduced in this paper. To demonstrate the effectiveness of the proposed cloak, we have simulated it with the conductivities given in Eq. (15) and Eq. (16) for each of the corresponding regions under two different angles of $\theta_{inc}=0^\circ$ and $\theta_{inc}=15^\circ$ and their results are shown in Fig.8.
![ The heat flux distributions when the designed square shape cloak is used under different incident angles of a) $\theta=0^\circ$ and b) $\theta=15^\circ$[]{data-label="fig:fig12"}](fig8.jpg){width="\linewidth"}
As can be seen from Fig.8 (a), the isothermal contours will smoothly pass the object without any distortion. However, in contrary with previous cloaks, when the incident angle is changed, the functionality of the designed cloak will remain unchanged as shown in Fig.8 (b). In other words, the functionality of the proposed cloak is not restricted to any specific incident angle and we used $\theta=15^\circ$ as the demonstration of the concept. This feature of the propounded omnidirectional cloak will consequently give rise to making the object be cloaked from an outside detector no matter where the location of the detector is, which is of utmost importance in many practical applications.
4. Conclusion
=============
In conclusion, in this paper we have presented a new material based on null-space transformation which is capable of obviating the conventional challenges of TT-based materials. The attained material which is called TNM, is a homogeneous and anisotropic conductivity that is shape independent. In other words, when the geometry of the structure (e.g., the deflection angle of the directional heat devices) is changed, the same material could be used again without the demand of recalculating or re-fabricating it. This will in turn make the proposed material a good candidate to be used for more practical and re-configurable scenarios. several numerical simulations are proposed which corroborate the validity and effectiveness of the propounded approach. As the first example, a directional heat bending device was designed that despite of its deflection angle, only a constant TNM can be used. In addition, we have used the TNM as a mean to obtain arbitrary shape heat concentrators with homogenous materials. It is shown that by utilizing a TNM, one can freely change the geometry of the concentrator and still use the same material. Finally, a thermal omnidirectional cloak is also proposed that is capable of guiding the thermal distributions in a manner that they become undetectable from an outside observer.
|
---
abstract: 'Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the $z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points $z=\pm i/n$ are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed.'
author:
- |
José Luis López\
Departamento de Ingenieria Matématica e Informática,\
Universidad Pública de Navarra, 31006-Pamplona, Spain\
- |
Nico M. Temme\
Centrum Wiskunde & Informatica,\
Science Park 123, 1098 XG Amsterdam, The Netherlands.\
[ e-mail: [ [email protected], [email protected]]{}]{}
title: Large Degree Asymptotics of Generalized Bessel Polynomials
---
0.8cm
2000 Mathematics Subject Classification: 30E10, 33C10, 33C15, 41A60.
Keywords & Phrases: generalized Bessel polynomials, asymptotic expansions, Bessel functions, Kummer functions.
Introduction {#sec:intro}
============
Generalized Bessel polynomials of degree $n$, complex order $\mu$ and complex argument $z$, denoted by $Y_n^\mu(z)$, have been introduced in [@Krall:1949:ANC], and can be defined by their generating function. We have [@Grosswald:1978:BPS]: $$\label{Intro1}
\left(\frac2{1+\sqrt{{1-2zw}}}\right)^\mu\,\frac{e^{2w/(1+\sqrt{{1-2zw}})}}{\sqrt{{1-2zw}}}=
\sum_{n=0}^\infty \frac{Y_n^{\mu}(z)}{ n!}w^n,
\quad \vert 2zw\vert<1,$$ with special values $Y_n^{\mu}(0)=1$, $n=0,1,2,\ldots$.
The generalized Bessel polynomials are important in certain problems of mathematical physics. For a historical survey and discussion of many interesting properties, we refer to [@Grosswald:1978:BPS].
In [@Dunster:2001:UAE] and [@Wong:1997:AEG] detailed contributions on asymptotic approximations are given on the generalized Bessel polynomials. Wong and Zhang use integral representations and Dunster’s approach is based on a differential equation. Our approach also uses integrals as starting point.
Our approach is different from that of Wong and Zhang [@Wong:1997:AEG]. We give expansions that are similar to those of the modified Bessel functions, and the expansions reduce to these expansions when $\mu=0$. For the expansions in terms of elementary functions we give a simple description for the domains of validity.
In §\[sec:Besalt\] we start with a simple expansion valid outside a fixed neighborhood of the origin. In §\[sec:elem\] we give expansions that hold uniformly with respect to $z$ inside the sectors $\vert\phase\,\pm z\vert\le \frac12\pi-\delta$, where $\delta$ is a small positive number. At the end of this section we compare our results with those in [@Dunster:2001:UAE] and [@Wong:1997:AEG]. For complementary sectors (and extensions of these) we give an expansion in terms of the modified Bessel function $K_{n+\frac12}(z)$ and its derivative. In fact, this is an expansion in terms of $Y_n^{0}(z)$, and this expansion holds for all $z$. In the Appendix §\[sec:modBes\] we summarize the expansions of the modified Bessel functions. These expansions play a role when comparing the results for the generalized Bessel polynomials with those for $\mu=0$.
Representations and relations with Bessel and Kummer functions {#sec:Rels}
==============================================================
For $\mu=0$ the generalized Bessel polynomials become well-known polynomials that occur in representations of Bessel functions of fractional order. We have in terms of the modified $K-$Bessel function [@Olver:2010:BFS][^1]: $$\label{ybes}
Y_n^{0}(z)=\sqrt{\frac{2}{\pi z}}e^{1/z}K_{n+\frac12}(1/z)=
\sum_{k=0}^n \binomial{n}{k}\,(n+1)_k\,\left(\tfrac12 z\right)^k,$$ where $(p)_k$ is the Pochhammer symbol defined by $$\label{i4}
(p)_0=1,\quad (p)_k=\frac{\Gamma(p+k)}{\Gamma(p)},\quad k\ge1.$$ For $Y_n^{\mu}(z)$ an explicit formula reads [@Krall:1949:ANC] $$\label{ydef}
Y_n^{\mu}(z)=\sum_{k=0}^n\binomial{n}{ k}\,(n+\mu+1)_k\,\left(\tfrac12z\right)^k.$$
The simple integral representation $$\label{yint}
Y_n^{\mu}(z)=\frac1{\Gamma(n+\mu+1)}\int_0^\infty t^{n+\mu}\left(1+\tfrac12zt\right)^n e^{-t}\,dt$$ gives the representation in by expanding $(1+\frac12zt)^n$ in powers of $z$.
From the generating function in we have the Cauchy integral representation $$\label{Besint}
Y_n^{\mu}(z)=\frac{n!}{2\pi i}\int_{\calC}\left(\frac2{1+\sqrt{{1-2zw}}}\right)^\mu\,\frac{e^{2w/(1+\sqrt{{1-2zw}})}}{\sqrt{{1-2zw}}}\,\frac{dw}{w^{n+1}},$$ where $\calC$ is a circle with radius smaller than $1/\vert 2z\vert$.
When $z=0$ all polynomials reduce to unity, the first few polynomials being
$$\label{y012}
\begin{array}{l}
\dsp{Y_0^{\mu}(z)=1,\quad Y_1^{\mu}(z)=1+\tfrac12(\mu+2)z,}\\
\dsp{Y_2^{\mu}(z)=1+(\mu+3)z+\tfrac14(\mu+3)(\mu+4)z^2.}
\end{array}$$
More values can be obtained from the recurrence relation $$\label{yrecn}
A_nY_{n+2}^{\mu}(z)=B_nY_{n+1}^{\mu}(z)+C_nY_{n}^{\mu}(z),$$ where
$$\label{yrecc}
\begin{array}{l}
\dsp{A_n=2(2n+\mu+2)(n+\mu+2),}\\
\dsp{B_n=(2n+\mu+3)(2\mu+z(2n+\mu+4)(2n+\mu+2)),}\\
\dsp{C_n=2(n+1)(2n+\mu+4).}
\end{array}$$
There is also a recursion with respect to $\mu$: $$\label{yrecmu}
(n+\mu+2)Y_{n}^{\mu+2}(z)=(2n+\mu+2-2/z)Y_{n}^{\mu+1}(z)+(2/z)Y_{n}^{\mu}(z),$$ and for the derivative we have $$\label{ydiff}
\frac{d}{dz}Y_{n}^{\mu}(z)=\tfrac12n(n+\mu+1)Y_{n-1}^{\mu+2}(z)=
\frac{n+\mu+1}{z}\left(Y_{n}^{\mu+1}(z)-Y_{n}^{\mu}(z)\right).$$ These relations are special cases of known analytic continuation formula for the Kummer functions. See, for example, [@Olde:2010:CHF] and [@Slater:1960:CHF p. 13]. They follow also from by integrating by parts.
The relations with the Kummer functions are
$$\label{ykum}
\begin{array}{@{}r@{\;}c@{\;}l@{}}
Y_n^{\mu}(1/z)&=&\dsp{(2z)^{n+\mu+1}U(n+\mu+1,2n+\mu+2,2z),}\\
&=&\dsp{(2z)^{-n}\frac{\Gamma(2n+\mu+1)}{\Gamma(n+\mu+1)}{}_1F_1(-n;-2n-\mu;2z).}
\end{array}$$
For $\Re z<0$ it is convenient to have the representation $$\label{ykumneg}
Y_n^{\mu}(-1/z)=F_n^{\mu}(1/z)+U_n^{\mu}(1/z),$$ where $$\label{ykumnegFU}
\begin{array}{l}
F_n^{\mu}(1/z)=\dsp{\frac{n!\,(2z)^{n+\mu+1}e^{-2z}}{\Gamma(2n+\mu+2)}{}_1F_1(n+1;2n+\mu+2;2z),}\\
U_n^{\mu}(1/z)=\dsp{\frac{(-1)^n n!\,(2z)^{n+\mu+1}e^{-2z}}{\Gamma(n+\mu+1)}U(n+1,2n+\mu+2,2z).}
\end{array}$$
For $\mu=0$ we have $$\label{zn06}
F_n^{0}(z)=\sqrt{2\pi z}e^{-z} I_{n+\frac12}(1/z), \quad
U_n^{0}(z)=(-1)^n\sqrt{\frac{2z}{\pi }}e^{-z}K_{n+\frac12}(z),$$ and this corresponds to the relation $$\label{zn05}
Y_n^{0}(-z)=\sqrt{\frac{2}{\pi z}}e^{-1/z}\left((-1)^nK_{n+\frac12}(1/z)+\pi I_{n+\frac12}(1/z)\right).$$
A simple expansion {#sec:Besalt}
==================
The interesting region in the $z$ plane for uniform asymptotic expansions is a neighborhood of size $\bigO(1/n)$ of the origin, where the zeros appear. For $z$ outside a fixed neighborhood a simple expansion will be derived.
First we mention $$\label{Ba01}
\begin{array}{l}
\dsp{Y_n^{\mu}(z)=2^{\mu+\frac12}\left(\frac{2nz}{e}\right)^n\,e^{1/z} \ \times}\\
\quad\quad\quad\quad
\dsp{\left(1-\frac{1+6\mu(\mu+1+2z^{-1})+6z^{-2}}{24n}+\bigO\left(1/n^{2}\right)\right).}
\end{array}$$ This is derived in [@Docev:1962:OGB] and mentioned in [@Grosswald:1978:BPS p. 124] and [@Wong:1997:AEG]. More terms in this expansion can be obtained, for example by using the Cauchy integral given in .
In this section we derive a simple asymptotic expansion related to the result in by expanding part of the integrand in in powers of $W=\sqrt{1-2zw}$. First we notice that the main asymptotic contributions from the contour integral in come from the singular point $w=1/(2z)$, and when $w\sim1/(2z)$ the quantity $W$ is small.
We have $$\label{Ba02}
\left(\frac2{1+\sqrt{{1-2zw}}}\right)^\mu\,e^{2w/(1+\sqrt{1-2zw})}=2^\mu e^{1/z}(1+W)^{-\mu}e^{-W/z}$$ and we expand for $m=0,1,2,\ldots\,$ $$\label{Ba03}
(1+W)^{-\mu}e^{-W/z}=\sum_{k=0}^{m-1} L_k^{-\mu-k}(1/z)W^k +W^m U_m(W).$$ The appearance of the Laguerre polynomials becomes clear when expanding both the exponential and binomial and by comparing the coefficients with the representation $$\label{Ba04}
L_n^{\alpha}(x)=\sum_{m=0}^n \binomial{n+\alpha}{n-m}\frac{(-x)^m}{m!}.$$ Introducing this expansion in we find $$\label{Ba05}
Y_n^{\mu}(z)= n!\,2^\mu e^{1/z} \sum_{k=0}^{m-1} L_k^{-\mu-k}(1/z)\Phi_k^{(n)}+n!\,R_m(n),$$ where $$\label{Ba06}
\Phi_k^{(n)}=\frac{1}{2\pi i}\int_{\calC}(1-2zw)^{(k-1)/2}\frac{dw}{w^{n+1}}=(2z)^n\frac{\left(\tfrac12-\tfrac12k\right)_n}{n!},$$ and $$\label{Ba07}
R_m(n)=\frac{2^\mu e^{1/z}}{ 2\pi i}\int_{\calC}(1-2wz)^{(m-1)/2}U_m(W)\,\frac{dw}{ w^{n+1}},$$ with $\calC$ a circle with radius less than $1/\vert 2z\vert$.
After the change of variable $w=(1-t/n)/(2z)$ we have $$\label{Ba08}
R_m(n)=\frac{(2z)^n2^\mu e^{1/z}}{ 2\pi i\,n^{(m+1)/2}}\int_{\calC}t^{(m-1)/2}U_m\left(\sqrt{t/n}\right)\,\frac{dt}{ (1-t/n)^{n+1}}.$$ The function $U_m(W)$ is analytic in $\vert W\vert<1$ and ${\cal O}(1)$ as $W\to 0$. This means that $\vert U_m(W)\vert<C_m$ with $C_m$ a positive constant, on and inside the path $\calC$ of integration in . Hence, $\vert U_m(\sqrt{t/n})\vert<C_m$ on $\calC$ (indeed, the path $\calC$ has been modified after the change of variable, but we can set it equal to the previous path). Also, $(1-t/n)^{-(n+1)}$ is bounded on $\calC$ (and converges to $e^t$ for large $n$). Therefore, the above integral is $\bigO(1)$ as $n\to \infty$. Hence, $$\label{Ba09}
R_m(n)={\bigO}\left(\frac{(2z)^n}{ n^{(m+1)/2}}\right),\quad n\to \infty, \ z\ne0,$$ which is comparable with the large $n$ behavior of $\Phi_m^{(n)}$.
From the integral in it easily follows that $\Phi_{2k+1}^{(n)}=0$ for $k=0, 1, 2, \ldots, n$, and we see that, when $n$ is large, only the even terms in the series give contributions. We notice that $\Phi_{2k}^{(n)}$ constitute an asymptotic sequence. This follows from $$\label{Ba10}
\frac{\Phi_{2k+2}^{(n)}}{\Phi_{2k}^{(n)}}=\frac{\Gamma(-\frac12-k+n)}{\Gamma(\frac12-k+n)}=\frac{1}{-\frac12-k+n}=\bigO\left(n^{-1}\right),\quad n\to\infty.$$
We can collect the results of the section as follows.
\[thm1\] For $n\to \infty$ we have the asymptotic expansion $$\label{Ba11}
Y_n^{\mu}(z)\sim(2z)^n\,2^\mu e^{1/z} \sum_{k=0}^\infty L_k^{-\mu-k}(1/z)\left(\tfrac12-\tfrac12k\right)_n,$$ which holds uniformly for $\vert z\vert \ge z_0$, where $z_0$ is a positive constant.
$z=10^j$ $z=-10^j$
--------- ------- ---------------- -------------- -- ----------- ---------------- -------------- -- --
$j\ $ $Y_n^{\mu}(z)$ [$\delta$]{} [$j\ $]{} $Y_n^{\mu}(z)$ [$\delta$]{}
$ n=50$ -1 0.4232e034 0.16e-03 -1 0.1961e026 0.26e-11
0 0.1211e081 0.38e-07 0 0.1778e080 0.62e-08
1 0.5131e130 0.17e-07 1 0.4235e130 0.14e-07
2 0.4707e180 0.16e-07 2 0.4617e180 0.15e-07
3 0.4666e230 0.15e-07 3 0.4657e230 0.15e-07
$n=100$ -1 0.1681e093 0.30e-07 -1 0.5251e084 0.68e-15
0 0.3190e189 0.10e-10 0 0.4501e188 0.18e-11
1 0.1325e289 0.47e-11 1 0.1089e289 0.39e-11
2 0.1213e389 0.43e-11 2 0.1189e389 0.42e-11
3 0.1202e489 0.43e-11 3 0.1200e489 0.42e-11
: Relative errors $\delta$ in the asymptotic expansion in with terms up and including $k=20$ for $\mu=17/4$ and several values of $z$ and $n$. \[Atab\]
In Table \[Atab\] we give the relative errors $\delta$ when we use the expansion in with terms up and including $k=20$, for $\mu=17/4$ and several values of $z$ and $n$. We see a quite uniform error with respect to $z$, except when $z=\pm\frac{1}{10}$.
\[Barem1\] [If we wish we can expand the Pochhammer symbols in for large $n$ and rearrange the series. In that way we can obtain an expansion of $Y_n^{\mu}(z)$ in negative powers of $n$, and this expansion is comparable with an expansion of which the first terms are given in . ]{}
\[Barem2\] [In we expand the generalized Bessel polynomials $Y_n^\mu(z)$ in terms of another set of polynomials, the generalized Laguerre polynomials. Because the degree of these polynomials does not depend on the large parameter, they can be evaluated much easier than the polynomials $Y_n^\mu(z)$. In fact, to compute the Laguerre polynomials we can use a recurrence relation, which follows from differentiating with respect to $W$. Let $c_k=L_k^{-\mu-k}(1/z)$, then $$\label{Ba12}
z(k+1)c_{k+1}=-(\mu z+kz+1)c_k-c_{k-1}, \quad k=1, 2, 3,\ldots,$$ with initial values $c_0=1$, $c_1=-(\mu z+1)/z$. ]{}
\[Barem3\] [For general values of $\mu$ the expansion in is not convergent, but for $\mu=0,-1,-2,\ldots$ it is. For example, for $\mu=0$ a relation for the $K-$Bessel function should arise. We have from $L_k^{-k}(1/z)=(-1)^k/(k!\,z^k)$, which gives the convergent expansion $$\label{Ba13}
Y_n^{0}(z)=(2z)^n e^{1/z} \sum_{k=0}^\infty\frac{(-1)^k}{k!\,z^k}\left(\tfrac12-\tfrac12k\right)_n.$$ Summing the series, separating the terms with even and odd $k$, we obtain $$\label{Ba14}
Y_n^{0}(z)=(-1)^ne^{1/z} \sqrt{\frac{\pi}{2z}}\left(I_{-n-\frac12}(1/z)-I_{n+\frac12}(1/z)\right),$$ and by using a well-known relation between the modified Bessel functions the representation in arises. ]{}
\[Barem4\] [The expansion in is simpler than those of the following sections: it is easier to obtain and the coefficients are easily computed. When $z$ is small the expansion in breaks down. The Laguerre polynomials ($k\ge1$) are not bounded, although the factor $(2z)^n$ in front of the expansion has some control. But the main concern is the exponential factor $e^{1/z}$, which has an essential singularity at $z=0$. Recall that the polynomials $Y_n^{\mu}(z)$ all tend to unity when $z\to0$. As mentioned in Theorem \[thm1\], for the expansion in we have to exclude a fixed neighborhood of the point $z=0$. ]{}
Expansions in terms of elementary functions {#sec:elem}
===========================================
By using saddle point methods we obtain expansions that hold uniformly inside sectors $\vert\phase\,\pm z\vert\le\frac12\pi-\delta$, where $\delta$ is a fixed small positive number.
For $Y_n^{\mu}(z)$ we take $\nu=n+\frac12$ as the large parameter. This gives a suitable identification of the results with those for the Bessel function $K_{n+\frac12}(z)$ when $\mu=0$; see also . In addition we replace the argument $z$ of $Y_n^{\mu}(z)$ by $1/(\nu z)$ (observe that in [@Dunster:2001:UAE] $\nu$ is also the large parameter, and the Bessel polynomial is considered with reversed argument).
Because for both cases $\vert\phase\,\pm z\vert\le\frac12\pi-\delta$ the derivation of the asymptotic expansion is very similar we first summarize the results in the following two subsections, and in §\[sec:deriv\] we give the details of the analysis.
Expansion holding for {#sec:Beszpos}
----------------------
\[thm2\] For large values of $n$ we have the expansion $$\label{Ynp15}
Y_n^{\mu}(\zeta)\sim\frac{\left(1-z+\sqrt{1+z^2}\right)^{\mu}\sqrt{z}}{(1+z^2)^{\frac14}}e^{\nu z-\nu \eta}
\sum_{k=0}^\infty \frac{A_{k}(\mu,z)} {\nu^k},$$ and the expansion holds uniformly inside the sector $\vert\phase\,z\vert\le\frac12\pi-\delta$. Here, $\delta$ is a small positive constant, $\nu=n+\frac12$, $\zeta=1/(\nu z)$, $A_0(\mu,z)=1$, $$\label{Ynp23}
A_1(\mu,z)=\frac{t(5t^2-3)}{24}-\frac{\mu t^2(z+1)}{4}+\frac{\mu^2(tz-1)}{4},$$ and $$\label{etat0}
t=\frac{1}{\sqrt{1+z^2}}, \qquad
\eta=\sqrt{1+z^2}+\log\frac{z}{1+\sqrt{1+z^2}}.$$
For $\mu=0$ the coefficients $A_{k}(\mu,z)$ reduce to those in the expansion in , that is, $A_{k}(0,z)=(-1)^k u_k(t)$, and $$\label{Ynp17}
Y_n^{0}(\zeta)=\sqrt{\frac{2\nu z}{\pi}}e^{\nu z}K_{\nu}(\nu z)\sim\frac{\sqrt{z}}{(1+z^2)^{\frac14}}e^{\nu z-\nu \eta}
\sum_{k=0}^\infty \frac{(-1)^k u_k(t)} {\nu^k},$$ which indeed gives the expansion in
For $\zeta=0$ all Bessel polynomials $Y_n^{\mu}(\zeta)$ reduce to unity. We have as $\zeta\to0$: $$\label{Ynp24}
z\to\infty, \ t\to0, \ zt\to1, \ z-\eta\to0, .$$ As a consequence, $A_1(\mu,z)\to0$ as $\zeta\to0$. In fact all coefficients $A_k(\mu,z)$ with $k\ge1$ vanish as $\zeta\to0$, and both sides of reduce to unity.
Recall that the simple expansion in §\[sec:Besalt\] is no longer valid when the argument of the Bessel polynomials approaches the origin.
Expansions holding for {#sec:Beszneg}
-----------------------
In this case we write (see and ) $$\label{zn02n}
Y_n^{\mu}(-1/z)=F_n^{\mu}(1/z)+U_n^{\mu}(1/z).$$
We have the following results.
\[thm3\] For large values of $n$ we have the expansions $$\label{zn13}
U_n^{\mu}(\zeta)\sim(-1)^n\frac{\left(1+z+\sqrt{1+z^2}\right)^{\mu}\sqrt{z}}{(1+z^2)^{\frac14}}e^{-\nu z-\nu \eta}
\sum_{k=0}^\infty \frac{B_{k}(\mu,z)} {\nu^k},$$ $$\label{zn27}
F_n^{\mu}(\zeta)\sim
\frac{\left(1+z-\sqrt{1+z^2}\right)^\mu\sqrt{z}}{(1+z^2)^{1/4}}e^{-\nu z +\nu\eta}\sum_{k=0}^\infty\frac{C_k(\mu,z)}{\nu^k},$$ and the expansions hold uniformly inside the sector $\vert\phase\,z\vert\le\frac12\pi-\delta$. Here, $B_{0}(\mu,z)=1$, $C_0(\mu,z)=1$, $$\label{zn15}
B_1(\mu,z)=\frac{t(5t^2-3)}{24}+\frac{\mu t^2(z-1)}{4}-\frac{\mu^2(zt+1)}{4},$$ and $$\label{zn29}
C_1(\mu,\zeta)=-\frac{t(5t^2-3)}{24}+\frac{\mu t^2(z-1)}{4}+\frac{\mu^2(zt-1)}{4}.$$ The quantities $t$, $\zeta$, $\nu$ and $\eta$ are as in Theorem \[thm2\].
For $\mu=0$ the expansions reduce to those for the modified Bessel functions mentioned in .
Integral representations {#sec:intrep}
------------------------
For deriving the asymptotic expansions we introduce the integrals
$$\label{intrep1}
\begin{array}{l}
\dsp{P_\nu^\mu(z)=\int_0^\infty p_\mu(s) e^{-\nu\phi(s)}\,ds,}\\
\dsp{Q_\nu^\mu(z)=\frac{1}{2\pi i}\int_\calL q_\mu(s) e^{\nu\phi(s)}\,ds,}
\end{array}
\renewcommand{{1.75}}{1}$$
where $\nu>0$ and $$\label{intrep2}
\phi(s)=2zs-\ln\,s-\ln(1+s).$$ When $z>0$ the contour $\cal L$ is a vertical line with $\Re s>0$; when $z$ is complex we can deform the contour in order to keep convergence. For the same purpose we can rotate the path of integration for $P_\nu^\mu(z)$ in .
For certain choices of $p_\mu(s)$ and $q_\mu(s)$ these integrals give representations of the functions $Y_n^{\mu}(z)$, $F_n^{\mu}(z)$, and $U_n^{\mu}(z)$. We have $$\label{intrep3}
Y_n^{\mu}(\zeta)=\frac{(2\nu z)^{n+\mu+1}}{\Gamma(n+\mu+1)} P_\nu^\mu(z), \quad p_\mu(s)=\frac{s^{\mu}}{\sqrt{s(1+s)}},$$ $$\label{intrep4}
U_n^{\mu}(\zeta)=\frac{(-1)^n(2\nu z)^{n+\mu+1}e^{-2\nu z}}{\Gamma(n+\mu+1)}P_\nu^\mu(z), \quad p_\mu(s)=\frac{(1+s)^\mu}{\sqrt{s(1+s)}},$$ $$\label{intrep5}
F_n^{\mu}(\zeta)=\frac{n!}{(2\nu z)^{n}} Q_\nu^\mu(z), \quad q_\mu(s)=\frac{(1+s)^{-\mu}}{\sqrt{s(1+s)}}.$$ The multi-valued functions in $\phi(s)$, $p_\mu(s)$, and $q_\mu(s)$ have there principal branches and are real for $s>0$.
The representations in and follow from the well-known integral $$\label{intrep6}
U(a,c,z)=\frac{1}{\Gamma(a)}\int_0^\infty t^{a-1}(1+t)^{c-a-1} e^{-zt}\,dt, \quad \Re\,a,z>0,$$ the first line in , and the second line in . For we refer to the first line in and the integral representation (see [@Slater:1960:CHF p. 46]) $$\label{intrep7}
\frac{1}{\Gamma(c)}{}_1F_1(a;c;z)=\frac{z^{1-c-}e^z}{2\pi i}\int_\calL e^{zs}(1+s)^{a-c}s^{-a} \,ds,$$ where $\calL$ is a vertical line with $\Re s>0$.
Construction of the expansions {#sec:deriv}
------------------------------
We use the saddle point method for obtaining asymptotic expansions of the integrals in . The saddle points follow from the equation $\phi^\prime(s)=0$, where $$\label{der1}
\phi^\prime(s)=\frac{2z s^2+2(z-1)s-1}{s(1+s)},$$ and are given by $$\label{der2}
{s_{+}=\frac{1-z+\sqrt{1+z^2}}{2z},} \quad
{s_{-}=\frac{1-z-\sqrt{1+z^2}}{2z}.}$$ When $z>0$ the saddle points are well-separated, with $-1<s_-< -\frac12$ and $s_+>0$. We have the following limits: $\lim_{z\to0} s_{+}=+\infty$ and $\lim_{z\to\infty}s_{+}=0$.
Also, $$\label{der3}
s_{+}(1+s_{+})=\frac{1+\sqrt{1+z^2}}{2z^2},$$ and $$\label{der4}
\phi(s_+)=1-z+\ln(2z)+\eta, \quad
\phi^{\prime\prime}(s_+)=\frac{4z^2\sqrt{1+z^2}}{1+\sqrt{1+z^2}},$$ with $\eta$ defined in .
We use Laplace’s method with the transformation $$\label{der5}
\phi(s)-\phi(s_+)=\tfrac12\phi^{\prime\prime}(s_+) w^2, \quad \sign(w)=\sign(s-s_+).$$ We have $s=w+\bigO(w^2)$ as $w\to0$. The integrals in become
$$\label{der6}
\begin{array}{l}
\dsp{P_\nu^\mu(z)=e^{-\nu\phi(s_+)}\int_{-\infty}^\infty f(w)\, e^{-\tfrac12\nu\phi^{\prime\prime}(s_+) w^2}\,dw,\quad f(w)=p_\mu(s)\frac{ds}{dw},}\\
\dsp{Q_\nu^\mu(z)=\frac{e^{\nu\phi(s_+)}}{2\pi i}\int_{-i\infty}^{i\infty} g(w)\, e^{\tfrac12\nu\phi^{\prime\prime}(s_+) w^2}\,dw,\quad g(w)=q_\mu(s)\frac{ds}{dw}.}
\end{array}
\renewcommand{{1.75}}{1}$$
By expanding $f(w)=\sum_{k=0}^\infty f_k w^k$ and $g(w)=\sum_{k=0}^\infty g_k w^k$ we obtain the asymptotic expansions
$$\label{der7}
\begin{array}{l}
\dsp{P_\nu^\mu(z)\sim f_0 e^{-\nu\phi(s_+)}\sqrt{\frac{2\pi }{\nu \phi^{\prime\prime}(s_+)}}
\sum_{k=0}^\infty \frac{F_{k}(\mu,z)} {\nu^k},}\\
\dsp{Q_\nu^\mu(z)\sim g_0 \frac{e^{\nu\phi(s_+)}}{2\pi}\sqrt{\frac{2\pi }{\nu \phi^{\prime\prime}(s_+)}}
\sum_{k=0}^\infty (-1)^k \frac{G_{k}(\mu,z)} {\nu^k},}
\end{array}
\renewcommand{{1.75}}{1}$$
where (see also )
$$\label{der8}
\begin{array}{l}
\dsp{F_k(\mu,z)= \frac{(\frac12)_k 2^k }{(\phi^{\prime\prime}(s_+))^k}\frac{f_{2k}}{f_0}}, \quad
\dsp{f_0=p_\mu(s_+)},\\
\dsp{G_k(\mu,z)= \frac{(\frac12)_k 2^k }{(\phi^{\prime\prime}(s_+))^k}\frac{g_{2k}}{g_0}}, \quad
\dsp{g_0=q_\mu(s_+)},
\end{array}
\renewcommand{{1.75}}{1}$$
because $ds/dw=1$ at $w=0$.
By using – and – it follows that $$\label{der9}
Y_n^{\mu}(\zeta)\sim\frac{\left(1-z+\sqrt{1+z^2}\right)^{\mu}\sqrt{z}}{(1+z^2)^{\frac14}}
\frac{e^{\nu z-\nu \eta}}{\Gamma^*(\nu+\mu+\frac12)}
\sum_{k=0}^\infty \frac{F_{k}^{(1)}(\mu,z)} {\nu^k},$$ $$\label{der10}
U_n^{\mu}(\zeta)\sim(-1)^n\frac{\left(1+z+\sqrt{1+z^2}\right)^{\mu}\sqrt{z}}{(1+z^2)^{\frac14}}
\frac{e^{-\nu z-\nu \eta}}{\Gamma^*(\nu+\mu+\frac12)}
\sum_{k=0}^\infty \frac{F_{k}^{(2)}(\mu,z)} {\nu^k},$$ $$\label{der11}
F_n^{\mu}(\zeta)\sim\Gamma^*\left(\nu+\tfrac12\right)\frac{\left(1+z-\sqrt{1+z^2}\right)^\mu\sqrt{z}}{(1+z^2)^{1/4}}e^{-\nu z +\nu\eta}\sum_{k=0}^\infty(-1)^k\frac{G_k(\mu,z)}{\nu^k}.$$ The coefficients $F_{k}^{(1)}(\mu,z)$ are obtained from and the function $f(w)$ of with the function $p_\mu(s)$ as given in , and $F_{k}^{(2)}(\mu,z)$ follow from taking the function $p_\mu(s)$ as given in . The function $\Gamma^*$ is the slowly varying part of the corresponding gamma function. That is, $$\label{der12}
\Gamma^*(\nu+\alpha)=\frac{\Gamma(\nu+\alpha)}{\sqrt{2\pi}\,\nu^{\nu+\alpha-\frac12}e^{-\nu}}\sim
\sum_{k=0}^{\infty}\frac{\gamma_k(\alpha)}{\nu^k}, \quad \gamma_0(\alpha)=1, \quad \nu\to\infty.$$ The coefficients $\gamma_k(\alpha)$ follow from standard methods for the gamma function; see §\[subsec:coeffY\].
The final form of the expansion of $Y_n^{\mu}(\zeta)$ given in in Theorem \[thm2\] can be obtained by dividing the expansion in by the expansion of $\Gamma^*(\nu+\mu+\frac12)$ given in , and similar for the other expansions. This gives for $k=0,1,2,\ldots$ $$\label{der13}
A_{k}(\mu,z)=F_{k}^{(1)}(\mu,z)-\sum_{j=0}^{k-1} A_{j}(\mu,z)\gamma_{k-j}\left(\mu+\tfrac12\right),$$ $$\label{der14}
B_{k}(\mu,z)=F_{k}^{(2)}(\mu,z)-\sum_{j=0}^{k-1} B_{j}(\mu,z)\gamma_{k-j}\left(\mu+\tfrac12\right),$$ $$\label{der15}
C_{k}(\mu,z)=\sum_{j=0}^k (-1)^jG_{j}(\mu,z)\gamma_{k-j}\left(\tfrac12\right).$$
Computation of the coefficients {#subsec:coeffY}
-------------------------------
To compute the coefficients $F_k(\mu,z)$ and $A_k(\mu,z)$ we need the coefficients in the expansion $$\label{comp1}
s=s_++\sum_{k=1}^\infty s_kw^k,$$ which follow from . We write, as in , $$\label{comp2}
t=\frac{1}{\sqrt{1+z^2}}$$ and obtain
$$\label{comp3}
\begin{array}{l}
\dsp{s_1=1,\quad s_2=\frac{2-t}{6}, \quad s_3=\frac{(1-t)(5t^3-6t^2+2)}{18t^2},} \\
\dsp{s_4=-\frac{z(1-t)(40t^4-65t^3+24t^2-2t+4)}{135t^2}.}
\end{array}$$
With these coefficients we can compute the coefficients $f(w)$ and $g(w)$ of by choosing the appropriate $p_\mu(s)$ and $q_\mu(s)$.
To obtain the coefficients in – we first compute $\gamma_k(\mu+\frac12)$ that appear in . We have
$$\label{comp4}
\begin{array}{l}
\dsp{\gamma_0(\mu+\tfrac12)=1, \quad \gamma_1(\mu+\tfrac12)=\tfrac{1}{24}\left(-1+12\mu^2\right),}\\
\dsp{\gamma_2=\tfrac{1}{1152}\left(1+48\mu-24\mu^2-192\mu^3+144\mu^4\right),}\\
\dsp{\gamma_3=\tfrac{1}{414720}\left(1003-720\mu-17100\mu^2+11520\mu^3+32400\mu^4-}
\right.\\
\quad\quad\quad\quad
\left.
\dsp{34560\mu^5+8640\mu^6\right),}\\
\dsp{\gamma_4=\tfrac{1}{39813120}\left(-4027-288864\mu+151824\mu^2+1618560\mu^3-}
\right.\\
\quad
\left.
\dsp{1239840\mu^4-1645056\mu^5+2177280\mu^6-829440\mu^7+103680\mu^8\right).}
\end{array}$$
Extending the result to complex values of {#subsec:extz}
------------------------------------------
From [@Olver:1997:ASF p. 378] it follows that the expansions in and hold for large values of $\nu$ and are uniformly valid for complex values of $z$ inside the sector $\vert\phase\,z\vert\le \frac12\pi-\delta$ with $\delta$ a small positive number. As can be seen from the front factor and the coefficients, it becomes invalid when $z$ approaches $\pm i$. In that case the singularities of the functions $f$ and $g$ in approach the origin.
The singularities come from those of the mapping in . This mapping does not depend on $\mu$ and, hence, Laplace’s method remains applicable for all fixed values of $\mu$, and also is uniformly valid for complex values of $z$ inside the sector $\vert\phase\,z\vert\le \frac12\pi-\delta$.
For complex $z$ inside the sector $\vert\phase\,z\vert< \frac12\pi$ the saddle points given in move into the complex plane, and it is for all these values of $z$ possible to find a single saddle point contour from $0$ through $s_+$ such that $\phase(zs)=0$ at infinity. If $\vert\phase\,z\vert\le \frac12\pi-\delta$ the singular points of the mapping in and of the function $f$ and $g$ in stay away from the origin.
In Figure \[BP.fig1\] we show the saddle points $s_+$ (black balls) and the corresponding saddle point contours of the first integral in for $z= e^{i\theta}$, $\theta=k\pi/10$, $k=1,2,3,4,5$. The black balls at the left are the saddle points $s_-$. When $z=i$ the saddle points $s_+$ and $s_-$ coincide at $s=-\frac12(1+i)$. The saddle point contours of the second integral in are the paths of steepest ascent of the first integral.
Comparison with earlier expansions {#subsec:comp}
----------------------------------
In [@Dunster:2001:UAE] the expansions of $Y_n^\mu(z)$ are given for large $n$ with possibly large values of $\mu$ as well. This makes comparison with our expansions rather complicated. In fact in Dunster’s expansions given in [@Dunster:2001:UAE §§6-7] the expansions can be re-expanded for small values of a parameter $\alpha$ corresponding with (in our notation) $\mu/\nu$, and in this way our results of the present section may be obtained. Dunster has used Olver’s theory [@Olver:1997:ASF] for linear differential equations of second order, with bounds for the remainders in the expansions and a recursion formula for the coefficients.
Starting from an integral we show how to include $\mu$ as a second large parameter, leaving out the details. Let $\mu =\alpha\nu$ and write in the form $$\label{D1}
Y_n^{\mu}(\zeta)=\frac{(2\nu z)^{n+\mu+1}}{\Gamma(n+\mu+1)}
\int_0^\infty \frac{e^{-\nu\psi(s)}}{\sqrt{s(1+s)}}\,ds,$$ where $\zeta=1/(\nu z)$, $\nu=n+\frac12$, and $$\label{D2}
\psi(s)=2zs-(1+\alpha)\ln\,s-\ln(1+s).$$ Then the saddle point analysis for the result of §\[sec:Beszpos\] can be repeated, giving an expansion that holds again in the sector $\vert\phase\,z\vert\le \frac12\pi-\delta$ and $\alpha\ge-1+\varepsilon$, with $\delta, \varepsilon$ small positive numbers. In a similar way the expansions of §\[sec:Beszneg\] can be modified.
With respect to the results in terms of elementary functions given in [@Wong:1997:AEG] we observe the following points.
- A remarkable point is that for $Y_n^\mu(z)$ the point $z=0$ excluded, whereas in our results this point is accepted as long we approach it inside the sectors $\vert\phase(\pm z)\vert \le \frac12\pi-\delta$. In fact the results are essentially the same as our results, although the notation and scaling of the parameters is different. For example, the factor $e^{-1/z}$ at the left-hand side of [@Wong:1997:AEG (2.26)] can be combined with $e^{(n+1)f(\zeta_+,\alpha)}$ to give a regular expression as $z\to0$. Also, the domains of validity are different and are simpler in our case (just the sectors $\vert\phase(\pm z)\vert \le \frac12\pi-\delta$).
- The expansions are derived from the integral representation , after a transformation. A detailed discussion is given about the location of saddle points and paths of steepest descent and the domains for the expansions to be derived. Our start is from two integrals related to those for the Kummer functions.
- The reader has to become familiar with several aspects of the detailed description of the paths of integration, the domains of validity and the proper choices of the branches of some multi-valued quantities. In this sense, our approach is more accessible for the reader who wants to use the results and to construct more terms.
- Complex quantities arise in the coefficients and front factors (which, of course for real $z$ will provide real expansions). Our expansions show quantities that are real for real $z$.
- It is not indicated how the results reduce to the well-known expansions of the modified Bessel functions; relations with the Kummer functions are not given.
Expansions in terms of modified Bessel functions {#sec:gen}
================================================
The expansions for $Y_n^\mu(1/(\nu z))$ in the previous section §\[sec:elem\] become invalid when $z$ approaches the points $\pm i$, because in that case the saddle points coincide. As shown in [@Dunster:2001:UAE] and [@Wong:1997:AEG] it is possible to derive uniform expansions in terms of Airy functions, and these expansions are valid in large $z-$domains.
For the modified Bessel functions $I_\nu(\nu z)$ and $K_\nu(\nu z)$ similar asymptotic phenomena arise when $z$ approaches the points $\pm i$, and the expansion in terms of Airy functions is available in the literature. In fact expansions for the Hankel functions and the ordinary Bessel functions can be used. See [@Olver:2010:BFS] and [@Olver:1997:ASF Chapter 11].
Because the asymptotic phenomena of the generalized Bessel polynomials $Y_n^\mu(z)$ for large $n$ and fixed $\mu$ are the same as those of the polynomial $Y_n^0(z)$, we approach the problem for obtaining uniform expansions by expanding the generalized polynomials in terms of the modified Bessel functions $K_\nu(z)$ (with $\nu=n+\frac12$), which are the same as the reduced Bessel polynomials $Y_n^0(z)$ (see ). By using the existing results for the Bessel functions a complete description is available in this way.
We summarize the results of this section as follows.
\[thm4\] For $n\to \infty$ we have the asymptotic expansion $$\label{gen13}
Y_n^{\mu}(\zeta)\sim\frac{(2\nu z)^{\mu} n!\,e^{\nu z}}{\Gamma(n+\mu+1)}\sqrt{\frac{2\nu z}{\pi}}\,
\left(K_\nu(\nu z)\sum_{k=0}^{\infty}\frac{C_k}{\nu^k}+
K_\nu^\prime(\nu z)\sum_{k=0}^{\infty}\frac{D_k}{\nu^k}\right),$$ and the expansion holds uniformly with respect to all $z$. Here, $\zeta=1/(\nu z)$, $\nu=n+\frac12$,
$$\label{gen17}
C_0=\left(2^{-\frac12}e^{-\frac34\pi i}\right)^{\mu},\quad
C_1=\tfrac1{24}(1-i)\mu(\mu-1)(-2\mu+1+3i)C_0,$$
and $$\label{gen18}
D_0=\tfrac12(1-i)\mu C_0,\quad
D_1=-\tfrac1{24}i \mu^2 (\mu-1) (-\mu+2+3 i) C_0.$$
The construction of the expansion {#subsec:constr}
---------------------------------
To start the construction of the expansion we write in the form $$\label{Ynp2}
Y_n^{\mu}(\zeta)=\frac{(2\nu z)^{n+\mu+1}}{\Gamma(n+\mu+1)}
\int_0^\infty \frac{s^{\mu}}{\sqrt{s(1+s)}}e^{-\nu\phi(s)}\,ds,$$ where again $$\label{Ynp3}
\zeta=\frac{1}{\nu z},\quad \nu=n+\tfrac12, \quad \phi(s)=2zs-\ln\,s-\ln(1+s).$$ We write $$\label{gen01}
f_0(s)= s^\mu= A_0+B_0s +\phi^{\prime}(s)g_0(s),$$ and substitute $s=s_+$ and $s=s_-$ to obtain $$\label{gen02}
A_0=\frac{s_+f_0(s_-)-s_-f_0(s_+)}{s_+-s_-},\quad
B_0=\frac{f_0(s_+)-f_0(s_-)}{s_+-s_-}.$$ Putting into we obtain $$\label{gen03}
Y_n^{\mu}(\zeta)=A_0\Phi_0+B_0\Phi_1+ \frac{(2\nu z)^{n+\mu+1}}{\Gamma(n+\mu+1)}
\int_0^\infty \frac{\phi^{\prime}(s)g_0(s)}{\sqrt{s(1+s)}}e^{-\nu\phi(s)}\,ds,$$ where $$\label{gen04}
\Phi_0=\frac{(2\nu z)^{\mu} n!}{\Gamma(n+\mu+1)}Y_n^0(\zeta),\quad
\Phi_1=\frac{(2\nu z)^{\mu-1} (n+1)!}{\Gamma(n+\mu+1)}Y_n^1(\zeta).$$ By using and it follows that
$$\label{gen05}
\begin{array}{l}
\dsp{\Phi_0=\frac{(2\nu z)^{\mu} n!}{\Gamma(n+\mu+1)}\sqrt{\frac{2\nu z}{\pi}}\,
e^{\nu z}K_{\nu}(\nu z),} \\
\dsp{\Phi_1=\frac{(2\nu z)^{\mu} n!}{2\Gamma(n+\mu+1)}
\sqrt{\frac{2\nu z}{\pi}}\,e^{\nu z}
\left((1/z-1)K_\nu(\nu z)-K_\nu^\prime(\nu z)\right)}.
\end{array}$$
In the second line we can also write [@Temme:1996:SFI p. 234] $$\label{gen06}
(1/z-1)K_\nu(\nu z)-K_\nu^\prime(\nu z)=K_{\nu+1}(\nu z)-K_\nu(\nu z),$$ but we prefer the notation with the derivative, because the asymptotic expansions of $K_\nu(\nu z)$ and $K_\nu^\prime(\nu z)$ are quite related and usually presented together.
The next step is to use integration by parts in , and this gives $$\label{gen07}
Y_n^{\mu}(\zeta)=A_0\Phi_0+B_0\Phi_1+ \frac{(2\nu z)^{n+\mu+1}}{\nu\Gamma(n+\mu+1)}
\int_0^\infty\frac{ f_1(s)}{\sqrt{s(1+s)}}e^{-\nu\phi(s)}\,ds,$$ where $$\label{gen08}
f_1(s)=\sqrt{s(1+s)}\frac{d}{ds}\frac{g_0(s)}{\sqrt{s(1+s)}}.$$ Repeating this procedure by writing for $k\ge0$ $$\label{gen09}
f_k(s)= A_k+B_ks +\phi^{\prime}(s)g_k(s),\quad f_0(s)=s^\mu,$$ $$\label{gen10}
A_k=\frac{s_+f_k(s_-)-s_-f_k(s_+)}{s_+-s_-},\quad
B_k=\frac{f_k(s_+)-f_k(s_-)}{s_+-s_-},$$ $$\label{gen11}
f_{k+1}(s)=\sqrt{s(1+s)}\frac{d}{ds}\frac{g_k(s)}{\sqrt{s(1+s)}}
=g_k^\prime(s)-\frac{2s+1}{2s(s+1)}g_k(s),$$ we obtain for $K\ge0$
$$\label{gen12}
\begin{array}{l}
\dsp{Y_n^{\mu}(\zeta)=\Phi_0\sum_{k=0}^{K-1}\frac{A_k}{\nu^k}+
\Phi_1\sum_{k=0}^{K-1}\frac{B_k}{\nu^k}\,+}\\
\quad\quad\quad\quad\quad\quad
\dsp{ \frac{(2\nu z)^{n+\mu+1}}{\nu^M\Gamma(n+\mu+1)}
\int_0^\infty\frac{ f_K(s)}{\sqrt{s(1+s)}}e^{-\nu\phi(s)}\,ds.}
\end{array}$$
We rearrange the expansion by using and writing $$\label{gen14}
C_k= A_k+\frac{1-z}{2z}B_k, \quad D_k=-\tfrac12B_k, \quad k=0,1,2,\ldots\,$$ to obtain the expansion given in Theorem \[thm4\].
\[Genrem1\] [To compute the coefficients $A_k, B_k$ defined in and the functions $f_k(s)$, say, by using a computer algebra package, it is convenient to write the functions $f_k(s)$ in the form of two-point Taylor expansions at the saddle points $s_+$ and $s_-$. More details on this method can be found in [@Lopez:2002:TPT], [@Lopez:2004:MPT], [@Vidunas:2002:SEC]. ]{}
\[Genrem2\]
For integer values of $\mu$ we have the following simple cases.
1. For $\mu=0, 1,2,\ldots$ the expansion in has a finite number of terms which can also be obtained from the recursion in .
2. For $\mu=-1,-2,-3,\ldots$ we can also obtain an exact result. When $\mu=-1$ we have $$\label{gen19}
C_k=\frac{z-1}{2^k}, \quad D_k=-\frac{z}{2^k},\quad k=0,1,2,\ldots,$$ and we can sum the convergent series when $2\nu =2n+1>1$. This gives a result that corresponds to the relation in with $\mu=-1$.
Concluding remarks {#Conc}
==================
In §\[sec:Besalt\] we have given a new simple expansion of $Y_n^{\mu}(z)$ that is valid outside a compact neighborhood of the origin in the $z-$plane and new forms of expansions in terms of elementary functions valid in the sectors $\vert\phase(\pm z)\vert \le \frac12\pi-\delta$ not containing the turning points $z=\pm i/n$. To avoid mappings for obtaining expansions in terms of Airy functions we have given expansions in terms of modified Bessel functions. For these functions very detailed Airy-type expansions are available, which can be used to obtain similar expansions of $Y_n^{\mu}(z)$.
Appendix: Expansions of the modified Bessel functions {#sec:modBes}
=====================================================
Because we compare the expansions for the Bessel polynomials to those of the modified Bessel functions, we summarize a few details about the uniform expansions of the $K-$ and $I-$Bessel functions.
We have [@Olver:2010:BFS][^2], [@Olver:1997:ASF p. 378] $$\label{Knuzuniform}
K_{\nu}(\nu z)\sim \sqrt{\frac{\pi}{2\nu}}\frac{e^{-\nu\eta}}{(1+z^2)^{1/4}}
\sum_{k=0}^{\infty}(-1)^k\frac{u_k(t)}{\nu^k},$$ $$\label{Inuzuniform}
I_{\nu}(\nu z)\sim\frac{1}{ \sqrt{2\pi\nu}}\frac{e^{\nu\eta}}{(1+z^2)^{1/4}}
\sum_{k=0}^{\infty}\frac{u_k(t)}{\nu^k},$$ which hold when $\nu\to\infty$, uniformly with respect to $z$ such that $\vert\phase\, z\vert\le\frac12\pi-\delta$, $\delta$ being an arbitrary positive number in $(0,\frac12\pi)$. Here, $$\label{etat}
t=\frac{1}{\sqrt{1+z^2}}, \qquad
\eta=\sqrt{1+z^2}+\log\frac{z}{1+\sqrt{1+z^2}}.$$
The first coefficients $u_k(t)$ are $$\label{ukt}
u_0(t)=1, \qquad u_1(t)=\frac{3t-5t^3}{24},
\qquad u_2(t)=\frac{81t^2-462t^4+385t^6}{1152},$$ and other coefficients can be obtained by applying the formula $$\label{recuk}
u_{k+1}(t)=\tfrac 12 t^2(1-t^2)u_k'(t)+\tfrac 18 \int_0^t (1-5s^2)u_k(s) ds,
\quad k=0,1,2,\ldots.$$
For the derivatives we have $$\label{Knuzduniform}
K_{\nu}^\prime(\nu z)\sim - \sqrt{\frac{\pi}{2\nu}}\frac{(1+z^2)^{1/4}}{z}e^{-\nu\eta}
\sum_{k=0}^{\infty}(-1)^k\frac{v_k(t)}{\nu^k},$$ $$\label{Inuzduniform}
I_{\nu}^\prime(\nu z)\sim\frac{1}{ \sqrt{2\pi\nu}}\frac{(1+z^2)^{1/4}}{z}e^{\nu\eta}
\sum_{k=0}^{\infty}\frac{u_k(t)}{\nu^k},$$ where $$\label{vkt}
v_0(t)=1, \qquad v_1(t)=\frac{-9t+7t^3}{24},
\qquad v_2(t)=\frac{-135t^2+594t^4-455t^6}{1152},$$ and other coefficients can be obtained by applying the formula $$\label{recvk}
v_{k}(t)=u_{k}(t)+t(t^2-1)\left(\tfrac12u_{k-1}(t)+tu_{k-1}^\prime(t)\right),
\quad k=0,1,2,\ldots.$$
The expansions in and become invalid when $z$ approaches the turning points $\pm i$. In that case expansions are available in terms of Airy functions. First the functions $I_{\nu}(z)$ and $K_{\nu}(z)$ should be written in terms of ordinary Bessel functions, and then the results for these functions can be used; see [@Olver:1997:ASF p. 419–426].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank the referee for helpful comments on the first version of the paper. The authors acknowledge financial support from [*Gobierno of Navarra*]{}, Res. 07/05/2008; NMT acknowledges support from [*Ministerio de Ciencia e Innovación*]{}, project MTM2009–11686.
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[^1]: http://dlmf.nist.gov/10.49.E12
[^2]: http://dlmf.nist.gov/10.41
|
---
abstract: 'Occlusion relationship reasoning demands closed contour to express the object, and orientation of each contour pixel to describe the order relationship between objects. Current CNN-based methods neglect two critical issues of the task: (1) simultaneous existence of the relevance and distinction for the two elements, i.e, occlusion edge and occlusion orientation; and (2) inadequate exploration to the orientation features. For the reasons above, we propose the Occlusion-shared and Feature-separated Network (OFNet). On one hand, considering the relevance between edge and orientation, two sub-networks are designed to share the occlusion cue. On the other hand, the whole network is split into two paths to learn the high-level semantic features separately. Moreover, a contextual feature for orientation prediction is extracted, which represents the bilateral cue of the foreground and background areas. The bilateral cue is then fused with the occlusion cue to precisely locate the object regions. Finally, a stripe convolution is designed to further aggregate features from surrounding scenes of the occlusion edge. The proposed OFNet remarkably advances the state-of-the-art approaches on PIOD and BSDS ownership dataset. The source code is available at <https://github.com/buptlr/OFNet>.'
author:
- |
Rui Lu$^{1}$Feng Xue$^{1}$Menghan Zhou$^{1,3}$Anlong Ming$^{1}$Yu Zhou$^{2,*}$\
[$^{1}$Beijing University of Posts and Telecommunications, Beijing, China]{}\
[$^{2}$Huazhong University of Science and Technology, Wuhan, China]{}\
[$^{3}$Lenovo Research, Beijing, China]{}\
[ [email protected] [email protected]]{}
bibliography:
- 'ICCV2019\_arxiv.bib'
title: |
Occlusion-shared and Feature-separated Network\
for Occlusion Relationship Reasoning
---
Introduction {#sec:intro}
============
Reasoning the occlusion relationship of objects from monocular image is fundamental in computer vision and mobile robot applications, such as [@Jacobson2012An; @Ayvaci2011Detachable; @Sargin2009Probabilistic; @Marshall1996Occlusion; @Stein2009Occlusion]. Furthermore, it can be regarded as crucial elements for scene understanding and visual perception[@Yu2016Similarity; @Yu2016Human; @Zhou2014ONLINE; @NIPS2012_4792; @Ming2016Monocular], such as object detection, image segmentation and 3D reconstruction [@Gao2011A; @Alper2012Detachable; @Zhang2015Monocular; @He2017Mask; @Shen2015DeepContour; @Xue2019A]. From the perspective of the observer, occlusion relationship reflects relative depth difference between objects in the scene.
![ (a) visualization result of DOC-HED, (b) visualization result of DOOBNet, (c) visualization result of ours, (d) the occlusion cue, (e) the bilateral feature, (f) visualization result of ground truth. Occlusion relationship (the red arrows) is represented by orientation $\theta\in(-\pi,\pi]$ (tangent direction of the edge), using the “left” rule where the left side of the arrow means foreground area. Notably, “red” pixels with arrows: correctly labeled occlusion boundaries; “cyan”: correctly labeled boundaries but mislabeled occlusion; “green”: false negative boundaries; “orange”: false positive boundaries (Best viewed in color). []{data-label="Fig:demo"}](demo.pdf){width="1\linewidth"}
{width="0.9\linewidth"}
Previously, a number of influential studies infer the occlusion relationship by designing hand-crafted features, e.g. [@Hoiem2007Recovering; @Ren2006Figure; @Saxena2005Learning; @Ma2017Object; @inproceedings; @Zhou2018Learning]. Recently, driven by Convolutional Neural Networks (CNN), several deep learning based approaches outperform traditional methods at a large margin. DOC [@Peng2016DOC] specifies a new representation for occlusion relationship, which decomposes the task into the occlusion edge classification and the occlusion orientation regression. And it utilizes two networks for these two sub-tasks, respectively. DOOBNet [@Wang2018DOOBNet] employs an encoder-decoder structure to obtain multi-scale and multi-level features. It shares backbone features with two sub-networks and simultaneously acquires the predictions.
In occlusion relationship reasoning, *the closed contour is employed to express the object, and the orientation values of the contour pixels are employed to describe the order relationship between the foreground and background objects.* We observe that two critical issues have rarely been discussed. Firstly, the two elements, i.e, occlusion edge and occlusion orientation, have the relevance and distinction simultaneously. They both need the occlusion cue, which describes the location of the occluded background, as shown in Fig.\[Fig:demo\] (d). Secondly, the high-level features for orientation prediction are not fully revealed. It needs additional cues from foreground and background areas (shown in Fig.\[Fig:demo\] (e)). Consequently, existing methods are limited in reasoning accuracy. Compared with our approach (shown in Fig.\[Fig:demo\] (c)), previous works [@Peng2016DOC; @Wang2018DOOBNet] (shown in Fig.\[Fig:demo\] (a)(b)) exist false positive and false negative detection of edge, as well as false positive prediction of orientation. Aiming to address the two issues above, and boost the occlusion relationship reasoning, a novel Occlusion-shared and Feature-separated Network (**OFNet**) is proposed. As shown in Fig.\[Fig:netdemo\] (c), considering the relevance and distinction between edge and orientation, our network is different from the other works (shown in Fig.\[Fig:netdemo\] (a)(b)). Two separate network paths share the occlusion cue and encode different high-level features. Furthermore, a contextual feature for orientation prediction is extracted, which is called the bilateral feature. To learn the bilateral feature, a Multi-rate Context Learner (**MCL**) is proposed. The learner has different scales of receptive field so that it can fully sense the two objects, i.e, the foreground and background objects, fundamentally assisting the occlusion relationship reasoning. To extract the feature more accurately, the Bilateral Response Fusion (**BRF**) is proposed to fuse the occlusion cue with the bilateral feature from **MCL**, which can precisely locate the areas of foreground and background. To effectively infer the occlusion relationship by the special orientation features, a stripe convolution is designed to replace the traditional plain convolution, which elaborately integrates the bilateral feature to distinguish the foreground and background areas. Experiments prove that we achieve SOTA performance on both PIOD [@Peng2016DOC] and BSDS ownership [@Ren2006Figure] dataset.
The main contributions of our approach lie in:
- The relevance and distinction between occlusion edge and occlusion orientation are re-interpreted. The two sub-tasks share the occlusion cues, but separate the contextual features.
- The bilateral feature is proposed, and two particular modules are designed to obtain the specific features, i.e, Multi-rate Context Learner (**MCL**) and Bilateral Response Fusion (**BRF**).
- To elaborately infer the occlusion relationship, a stripe convolution is designed to further aggregate the feature from surrounding scenes of the contour.
Related Work {#sec:related}
============
[**[Contextual Learning]{}**]{} plays an important role in scene understanding and perception [@DBLP:journals/corr/ChenPSA17; @Wang2018Understanding]. At first, Mostajabi et al. [@Mostajabi2015Feedforward] utilize multi-level, zoom-out features to promote feedforward semantic labeling of superpixels. Meanwhile, Liu et al. [@Liu2015ParseNet] propose a simple FCN architecture to add the global context for semantic segmentation. Afterwards, Chen et al. [@Chen2018DeepLab] apply the *Atrous Spatial Pyramid Pooling* to extract dense features and encode image context at multi-scale.
[**[Multi-level Features]{}**]{} are extracted from different layers, which are widely used in image detection [@Long2014Fully; @Ronneberger2015U; @Wei2016Object; @Wang2018DeepFlux]. Peng et al. [@Peng2017Large] fuse feature maps from multi-layer with refined details. Shrivastava et al. [@DBLP:journals/corr/ShrivastavaSMG16] adopt lateral connections to leverage top-down context and bottom-up details. [**[Occlusion Relationship Representation]{}**]{} has evolved overtime from triple points and junctions representation [@Hoiem2011Recovering; @Ren2006Figure] to pixel-based representation [@Teo2015Fast; @Peng2016DOC]. The latest representation [@Peng2016DOC] applies a binary edge classifier to determine whether the pixel belongs to an occluded edge, and a continuous-valued orientation variable is proposed to indicate the occlusion relationship by the left-hand rule [@Peng2016DOC].
![ Illustration of our proposed network architecture. The length of the block expresses the map resolution and the thickness of the block indicates the channel number. []{data-label="Fig:network"}](network8.pdf){width="1\linewidth"}
OFNet {#sec:network}
=====
Two elements of occlusion relationship reasoning, i.e, edge and orientation, are in common of necessity for the occlusion cue while differing in the utilization of specific contextual features. In this section, a novel Occlusion-shared and Feature-separated Network (**OFNet**) is proposed. Fig.\[Fig:network\] illustrates the pipeline of the proposed OFNet, which consists of a single stream backbone and two parallel paths, i.e, edge path and orientation path. Specifically, for the edge path (see Sec.\[ssec:edgepath\]), a structure similar to [@Lu2019Context] is employed to extract consistent and accurate occlusion edge, which is fundamental for occlusion reasoning. For the orientation path (see Sec.\[ssec:oripath\]), to learn more sufficient cues near the boundary for occlusion reasoning, the high-level bilateral feature is obtained, and a Multi-rate Context Learner (**MCL**) is proposed to extract the feature (see Sec.\[ssec:learner\]). To enable the learner to locate the foreground and background areas precisely, a Bilateral Response Fusion module (**BRF**) is proposed to fuse the bilateral feature and the occlusion cue (see Sec.\[ssec:fusion\]). Furthermore, a stripe convolution is proposed to infer the occlusion relationship elaborately (see Sec.\[ssec:inference\]).
Edge Path {#ssec:edgepath}
---------
The occlusion edge expresses the position of objects, and defines the boundary location between the bilateral regions. It requires reserved resolution of the original image to provide the accurate location and large receptive field to perceive the mutual constraint of pixels on the boundary.
We adopt the module proposed in [@Lu2019Context], which has a high capability to capture accurate location cue and sensitive perception of the entire object. In [@Lu2019Context], the low-level cue from the first three side-outputs preserves the original size of the input image and encodes abundant spatial information. Without losing resolution, the large receptive field is achieved via dilated convolution [@Yu2015Multi] after *res50* [@He_2016_CVPR]. The Bilateral Response Fusion (**BRF**) shown in Fig.\[Fig:network\] is presented to compensate the precise position for high-level features and suppress the clutter of non-occluded pixels for low-level features. Different from [@Lu2019Context], we employ an additional convolution block to refine the contour, and integrate specific task features provided by diverse channels. Besides, this well-designed convolution block eliminates the gridding artifacts [@Yu2017Dilated] caused by the dilated convolution in high-level layers. The resulting edge map embodies low-level and high-level features, which guarantees the consistency and accuracy of the occlusion edge. Specifically, the edge path provides complete and continuous contour, which makes up the object region. The object region is delineated by a set of occlusion edges.
Orientation Path {#ssec:oripath}
----------------
For the orientation path, we innovatively introduce the bilateral feature, which is conducive to describe the order relationship. Specifically, the bilateral feature represents information of surrounding scenes, which includes sufficient ambient context to deduce whether it belongs to the foreground or background areas.
### Multi-rate Context Learner {#ssec:learner}
Bilateral feature characterizes the relationship between the foreground and background areas. To infer the occlusion relationship between objects, the sufficient receptive field for the objects with different sizes is essential.
To perceive the object with various ranges and learn the bilateral feature, the Multi-rate Context Learner (**MCL**) is designed, which consists of three components, as shown in Fig.\[Fig:module1\]. Firstly, the high-level semantic cue is convolved by multiple dilated convolutions, which allows the pixels on the edge to perceive the foreground and background objects as completely as possible. The dilated convolutions have kernel size of *3$\times$3* with various dilation rates. With various dilated rates of the dilated convolutions, the learner is able to perceive the scene cue at different scales from the foreground and background areas, which is beneficial to deduce which side of the region is in front. Secondly, an element-wise convolution module, i.e., *1$\times$1* conv, is used to integrate the scene cue between various channels and promote cross-channel bilateral feature aggregation at the same location. Compared to dilated convolution, the element-wise convolution module retains the local cues near the contour. Besides, it greatly clarifies occlusion cue and bilateral cue in occlusion reasoning. The function $Dilated()$ represents the dilated convolution and the function $Conv_1()$ represents the *1$\times$1* conv. $X_i$ is the input of convolution. $W_i$ is the convolution layer parameters to be learned. Thirdly, the *1$\times$1* conv is once again applied to normalize the values nearby the contour, where the bilateral cue is further enhanced and other irrelevant cues are suppressed. The **MCL** learns the cues of the foreground and background objects. The feature map of bilateral cue, $i.e.,\{B\}$, is denoted as: $$\label{Eq:MCL}
\{B\} = Conv_1(\sum_{i=1}^{3}Dilated(X_i,\{W_i\}) + Conv_1(X_4,\{W_4\}))$$
![ Illustration of our proposed Multi-rate Context Learner (**MCL**). The **MCL** module includes 3 dilated convolutions with kernel size of *3$\times$3* and dilation rate of 6, 12 and 18, respectively. []{data-label="Fig:module1"}](module1.pdf){width="0.9\linewidth"}
![ Illustration of our proposed Bilateral Response Fusion (**BRF**). []{data-label="Fig:module2"}](module2.pdf){width="0.9\linewidth"}
[**[Difference with **ASPP**:]{}**]{} Notably, our **MCL** module is inspired by the “Atrous Spatial Pyramid Pooling” (ASPP) [@Chen2018DeepLab], but there exists several differences. Firstly, we add a parallel element-wise convolution module, which additionally gains local cues of the specific region. It compensates for the deficiency that dilated convolution is not sensitive to nearby information. Secondly, the convolution blocks after each branch remove the gridding artifacts [@Yu2017Dilated] caused by the dilated convolution. Thirdly, the *1$\times$1* conv can adjust channel numbers and explore relevance between channels.
### Bilateral Response Fusion {#ssec:fusion}
Respectively, the bilateral cue obtained by the method in Sec.\[ssec:learner\] discriminates which side of the contour belongs to foreground area, the occlusion cue obtained through the decoder represents the location information of the boundary. As shown in Fig.\[Fig:aspp\], after the bilinear upsampling, bilateral feature is hard to locate the exact location of the contour. Hence, to sufficiently learn the feature for occlusion relationship reasoning, more precise location of object region is demanded, which is provided by the occlusion cue from the decoder. Thus, it is necessary to introduce clear contour to describe the areas of the foreground and background objects, thereby extracting the object features more accurately. The Bilateral Response Fusion (**BRF**), shown in Fig.\[Fig:module2\], is proposed to fuse these two disparate streams of features, i.e. the bilateral map $\{B\}$ and occlusion map $\{D\}$. The unified orientation fused map of ample bilateral response and emphatic occlusion is formed, which is denoted as $\{F\}$, where $Conv_3()$ represents the *3$\times$3* conv: $$\label{Eq:BRF}
\{F\} = Conv_3(Conv_3(Concat(\{B\}, \{D\})))$$ $\{\emph{F}\}$ denotes the feature map generated by **BRF** module, and each element of the set is a feature map. Subsequently, $\{\emph{F}\}$ has *224$\times$224* spatial resolution and is taken as the input of the **Occlusion Relationship Reasoning** module (Sec.\[ssec:inference\]), as shown in Fig.\[Fig:aspp\]. Through **BRF**, the occlusion feature is effectively combined with the bilateral feature. For occlusion relationship reasoning, the fused orientation map not only possesses the boundary location between two objects with occlusion relationship, but also own contextual information of each object. The **BRF** module provides adequate cues for the following feature learning module to infer the foreground and background relationship. Besides, by integrating bilateral feature, the scene cue near the contour is enhanced.
![ The test demo of the generation of orientation fused map. We acquire the fused map by adopting **BRF** to complement bilateral feature with occlusion feature. []{data-label="Fig:aspp"}](aspp1.pdf){width="0.9\linewidth"}
### Occlusion Relationship Reasoning {#ssec:inference}
By utilizing the **MCL** and **BRF**, the bilateral feature is learned and fused, an inference module is necessary to determine the order of the foreground and background areas, which makes full use of this feature. Existing method [@Wang2018DOOBNet] utilizes *3$\times$3* conv to learn the features. This small convolution kernel only extracts the cues at the local pixel patch, which is not suitable to infer the occlusion relationship. The reason is that the tiny perceptive field is unable to perceive the learned object cue. Thus, a large convolution kernel is necessary for utilizing the bilateral feature, which is able to perceive surrounding regions near the contour.
Nevertheless, large convolution kernels are computation demanding and memory consuming. Instead, two stripe convolutions are proposed, which are orthogonal to each other. Compared to the *3$\times$3* conv, which captures only nine pixels around the center (shown in Fig.\[Fig:stripe\](a)), the vertical and horizontal stripe convolutions have *11$\times$3* and *3$\times$11* receptive field, as shown in Fig.\[Fig:stripe\](b). Specifically, for a contour pixel with arbitrary orientation, its tangent direction can be decomposed into vertical and horizontal directions. Contexts along orthogonal directions make varied amount of contributions in expressing the orientation representation. Thus, tendency of the extended contour and occluded relationship of bilateral scenes are recognized.
In addition, two main advantages are achieved. First, the large receptive field aggregates contextual information of object to determine the depth order, which is without large memory consuming. Second, although the slope of the edge is not exactly perpendicular or parallel to the ground, one of the stripe convolutions can successfully perceive the foreground and background objects. After the concatenation of the two orthogonal convolution modules, we apply the *3$\times$3* conv to refine the features.
Loss Function
-------------
[**[Occlusion Edge:]{}**]{} Occlusion edge characterizes depth discontinuity between regions, reflecting as the boundary between objects. Given a set of training images $\Psi = \{I_1,I_2,\dots,I_N\}$, the corresponding ground truth edge of the $k$-th input image at pixel $p$ is $E_k(p\mid I)\in \{0,1\}$ and we denote $\overline{E}_k(p\mid I,W)\in [0,1]$ as its network output, indicating the computed edge probability.
[**[Occlusion Orientation:]{}**]{} Occlusion orientation indicates the tangent direction of the edge using the left rule (i.e. the foreground area is on the left side of the background area). Following the mathematical definition above, for the $k$-th input image, its orientation ground truth at pixel $p$ is $O_k(p\mid I)\in (-\pi,\pi]$. The regression prediction result of orientation path is $\overline{O}_k(p\mid I,W)\in (-\pi,\pi]$.
[**[Occlusion Relationship:]{}**]{} During the testing phase, we first refine the $\overline{E}_k$ by conducting non-maximum suppression $\hat{E}_k = NMS(\overline{E}_k)$. The nonzero pixels of sharpened $\hat{E}_k$ form the binary matrix $M_k = sign(\hat{E}_k)$. We then perform element-wise product of $M_k$ and orientation map $O_k$, obtain refined orientation map $\hat{O}_k = M_k\circ O_k$. Finally, following \[11\], we adjust the $\hat{O}_k$ to the tangent direction of ${E}_k$ and gain the final occlusion edge map.
[**[Loss Function:]{}**]{} Following [@Wang2018DOOBNet], we use the following loss function to supervise the training of our network. $$\label{Eq:loss_all}
l(W)=\frac{1}{M}(\sum\limits_{j}{\sum\limits_{i}{AL({{y}_{i}},{{{e}}_{i}})+\lambda \sum\limits_{j}{\sum\limits_{i}{SL(f({\overline{a}_{i}},{{{a}}_{i}})}}}})$$
The parameters include: collection of all standard network layer parameters ($W$), predicted edge value at pixel $i$ ($y_i\in [0,1]$), mini-batch size ($M$), image serial number in a mini-batch ($j$),the Attention Loss ($AL$), the Smooth $L_1$ Loss ($SL$) [@Wang2018DOOBNet].
![ The schematic illustration of how orientation information propagates in the feature learning phase. (a) the plain convolution. (b) the stripe convolution. []{data-label="Fig:stripe"}](stripe.pdf){width="0.9\linewidth"}
Experiments {#sec:Experiments}
===========
In this section, abundant experiments are demonstrated to validate the performance of the proposed OFNet. Further, we present some ablation analyses for discussions of the network design choices.
Implementation Details {#ssec:details}
----------------------
[**[Dataset:]{}**]{} Our method is evaluated on two challenging datasets: PIOD [@Peng2016DOC] and BSDS ownership [@Ren2006Figure]. The PIOD dataset is composed of 9,175 training images and 925 testing images. Each image is annotated with ground truth object instance edge map and its corresponding orientation map. The BSDS ownership dataset includes 100 training images and 100 testing images of natural scenes. Following [@Wang2018DOOBNet], all images in the two datasets are randomly cropped to *320$\times$320* during training while retaining their original sizes during testing. [**[Initialization:]{}**]{} Our network is implemented in Caffe [@Jia2014Caffe] and finetuned from an initial pretrained *Res50* model. All convolution layers added are initialized with $``$msra$"$ [@He2015Delving].
[**[Evaluation Criteria:]{}**]{} Following [@Peng2016DOC], we compute precision and recall of the estimated occlusion edge maps (i.e.OPR) by performing three standard evaluation metrics: fixed contour threshold (ODS), best threshold of each image (OIS) and average precision (AP). Notably, the orientation recall is only calculated at the correctly detected edge pixels. Besides, the above three metrics are also used to evaluate the edge map after *NMS*.
** ODS OIS AP FPS ODS OIS AP FPS
---- ------------------------------- ------------------------------- ------------------------------- --------------- ------------------------------- ------------------------------- ------------------------------- ---------------
$.268$ $.286$ $.152$ $0.018$ $.419$ $.448$ $.337$ $0.018$
$.460$ $.479$ $.405$ $18.3\dagger$ $.522$ $.545$ $.428$ $19.6\dagger$
$.601$ $.611$ $.585$ $18.9\dagger$ $.463$ $.491$ $.369$ $21.1\dagger$
$\textbf{{\color{blue}.702}}$ $\textbf{{\color{blue}.712}}$ $\textbf{{\color{blue}.683}}$ $26.7\dagger$ $\textbf{{\color{blue}.555}}$ $\textbf{{\color{blue}.570}}$ $\textbf{{\color{blue}.440}}$ $25.8\dagger$
$\textbf{{\color{red}.718}}$ $\textbf{{\color{red}.728}}$ $\textbf{{\color{red}.729}}$ $28.3\dagger$ $\textbf{{\color{red}.583}}$ $\textbf{{\color{red}.607}}$ $\textbf{{\color{red}.501}}$ $27.2\dagger$
: OPR results on PIOD (left) and BSDS ownership dataset (right). - represent SRF-OCC [@Teo2015Fast], DOC-HED [@Peng2016DOC], DOC-DMLFOV [@Peng2016DOC], DOOBNet [@Wang2018DOOBNet] and ours, respectively. $\dagger$ refers to GPU running time. Red bold type indicates the best performance, blue bold type indicates the second best performance (the same below).
\[tab:OPR\]
** ODS OIS AP ODS OIS AP
---- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
$.345$ $.369$ $.207$ $.511$ $.544$ $.442$
$.509$ $.532$ $.468$ $\textbf{{\color{blue}.658}}$ $\textbf{{\color{blue}.685}}$ $\textbf{{\color{red}.602}}$
$.669$ $.684$ $.677$ $.579$ $.609$ $.519$
$\textbf{{\color{blue}.736}}$ $\textbf{{\color{blue}.746}}$ $\textbf{{\color{blue}.723}}$ $.647$ $.668$ $.539$
$\textbf{{\color{red}.751}}$ $\textbf{{\color{red}.762}}$ $\textbf{{\color{red}.773}}$ $\textbf{{\color{red}.662}}$ $\textbf{{\color{red}.689}}$ $\textbf{{\color{blue}.585}}$
: EPR results on PIOD (left) and BSDS ownership dataset (right).
\[tab:EPR\]
Evaluation Results {#ssec:eva}
------------------
[**[Quantitative Performance:]{}**]{} We evaluate our approach with comparisons to the state-of-the-art algorithms including SRF-OCC [@Teo2015Fast], DOC-HED [@Peng2016DOC], DOC-DMLFOV [@Peng2016DOC] and DOOBNet [@Wang2018DOOBNet].
![ OPR results on two datasets. []{data-label="Fig:OPR"}](OPR.pdf){width="1\linewidth"}
![ EPR results on two datasets. []{data-label="Fig:EPR"}](EPR.pdf){width="1\linewidth"}
{width="0.9\linewidth"}
As shown in Table.\[tab:OPR\] and Fig.\[Fig:OPR\], our approach outperforms all other state-of-the-art methods for OPR results. Specifically, in terms of the PIOD dataset, our method performs the best, outperforming the baseline DOOBNet of 4.6% AP. This is due to the efficiency of extracting high semantic features of the two paths separately. Edge path succeeds in enhancing contour response and orientation path manages to perceive foreground and background relationship. Splitting these two tasks into two paths enables the promotion of the previous algorithms. For the BSDS ownership dataset, which is difficult to train due to the small training samples, the proposed OFNet obtains the gains of 2.8% ODS, 3.7% OIS and 6.1% AP compared with the baseline DOOBNet. Specifically, our approach increases bilateral cue between the foreground and background objects, and fuses them with high semantic features to introduce clear contour, which describes the areas of the foreground and background better. Besides, stripe convolution in our network plays an important role in harvesting the surrounding scenes of the contour. The improvement in orientation proves the effectiveness of the module.
EPR results are presented in Table.\[tab:EPR\] and Fig.\[Fig:EPR\]. For the PIOD dataset, our approach superiorly performs against the other evaluated methods, surpassing DOOBNet by 5.0% AP. We take the distinction between edge and orientation into consideration, and extract specific features for sub-networks, respectively. For edge path, by utilizing the contextual features, which reflect pixels constraint on the occlusion edge, our network outputs edge maps with augmented contour and less noise around. With location cue extracted from low-level layers, the predicted edge in our method fits the contour better, thus avoiding false positive detections compared to others. For the BSDS ownership dataset, our approach achieves the highest ODS as well.
[**[Qualitative Performance:]{}**]{} Fig.\[Fig:quality\] shows the qualitative results on the two datasets. The top four rows show the results of the PIOD dataset [@Peng2016DOC], and the bottom four rows represent the BSDS ownership dataset [@Ren2006Figure]. The first column to the sixth column show the original RGB image from datasets, ground truth, the result predicted by DOOBNet [@Wang2018DOOBNet], the result predicted by the proposed OFNet, the detected occlusion edge and the predicted orientation, respectively. In the resulting image, the right side of the arrow direction is the background, and the left side corresponds to the foreground area.
In detail, the two occluded buses in the first row have similar appearances. Thus, it is hard to detect the dividing line between them, just as our baseline DOOBNet fails. However, our method detects the occlusion edge consistently. In the second row, the occlusion relationship between the wall and the sofa is easy to predict failure. Instead of the small receptive field, which is difficult to perceive objects with large-area pure color, our method with sufficient receptive field correctly predicts the relationship. The third scene is similar to the second row. Compared with the baseline, our method predicts the relationship between the sofa and the ground correctly. In the fourth row, the color of the cruise ship is similar to the hill behind, which is not detected by the baseline. By using the low-level edge cues, our method accurately locates the contour of the ship. The fifth row shows people under the wall, and the orientation cannot be correctly detected due to the low-level features in the textureless areas. Our method correctly infers the relationship by using the high-level bilateral feature. The last three scenes have the same problem as the third row, i.e, the object with a large region of pure color. Our method outperforms others in this situation by a large margin, which proves the effectiveness of our designed modules.
Ablation Analysis {#ssec:ablation}
-----------------
[**[One-branch or Multi-branch Sub-networks:]{}**]{} To evaluate that our method provides different high-level features for different sub-tasks, an existing method [@Wang2018DOOBNet], which adopts a single flow architecture by sharing high-level features, is used to be compared with our method. As shown in Table.\[tab:ablation1\], the high-level features for two paths promote the correctness of occlusion relationship. In addition, each path is individually trained for comparison, validating the help of occlusion cue for orientation prediction in our method.
*Methods* ODS OIS AP ODS OIS AP
------------------------- ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------
Baseline $.736$ $.746$ $.723$ $.702$ $.712$ $.683$
Baseline(split decoder) $.720$ $.735$ $.694$ $.702$ $.712$ $.683$
Single edge stream $.739$ $.750$ $.685$ $-$ $-$ $-$
Single ori stream $-$ $-$ $-$ $.705$ $.716$ $.674$
Ours $\textbf{{\color{red}.751}}$ $\textbf{{\color{red}.762}}$ $\textbf{{\color{red}.773}}$ $\textbf{{\color{red}.718}}$ $\textbf{{\color{red}.728}}$ $\textbf{{\color{red}.729}}$
: Experimental results of baseline DOOBNet [@Wang2018DOOBNet], baseline with split decoder, baseline with single stream sub-network and our approach. The experiments are conducted on the PIOD dataset (the same below).
\[tab:ablation1\]
*Methods* ODS OIS AP ODS OIS AP
-------------------------- ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------
Ours(w/o low-cues) $.746$ $.758$ $.764$ $.715$ $.722$ $.715$
Ours(w/o edge high-cues) $.742$ $.753$ $.758$ $.709$ $.717$ $.698$
Ours(w/o ori high-cues) $.743$ $.756$ $.759$ $.711$ $.719$ $.703$
Ours $\textbf{{\color{red}.751}}$ $\textbf{{\color{red}.762}}$ $\textbf{{\color{red}.773}}$ $\textbf{{\color{red}.718}}$ $\textbf{{\color{red}.728}}$ $\textbf{{\color{red}.729}}$
: Experimental results of our model without low-cues, without edge high-cues, without orientation high-cues and our model.
\[tab:ablation2\]
[**[Necessity for Each Feature:]{}**]{} In order to verify the role of various low-level and high-level features, each feature is removed to construct an independent variant for evaluation, as shown in Table.\[tab:ablation2\]. Intuitively, if the low-level features for edge path are removed, the occlusion edge is difficult to be accurately located. If the high-level features for edge path are removed, the occlusion edge is failed to be detected consistently. Furthermore, if the high-level features for orientation path are removed, although the occlusion edge could be detected accurately and consistently, the ability to reason occlusion relationship reduces sharply. The intrinsic reason is that the **MCL** perceives the bilateral cue around the contour, and affirms the foreground and background relationship. The bilateral feature plays an important role in occlusion relationship reasoning.
[**[Proportion of Bilateral and Contour Features:]{}**]{} The bilateral feature provides relative depth edgewise, and occlusion cue supplies the location of the boundary. We fuse them with various channel ratios to best refine the range of the foreground and background. The proportion of bilateral and occlusion features determines the effectiveness of the fusion. Table.\[tab:ablation3\] reveals various experimental results with different proportions of two features. Experiments prove that fusing bilateral feature and occlusion feature with *64:16* channel ratio in the **BRF** outperforms others. It reveals that bilateral feature plays a more important role in the fusion operation. Occlusion cue mainly plays an auxiliary role, which distinguishes the region of foreground and background. However, when the bilateral feature occupies an excess proportion, the boundary will be ambiguous, blurring the boundary between foreground and background, which causes a negative impact on the effect.
*Scale* ODS OIS AP ODS OIS AP
----------------- ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------
scale = *16:16* $.742$ $.752$ $.749$ $.710$ $.719$ $.703$
scale = *32:16* $.741$ $.754$ $.759$ $.712$ $.722$ $.709$
scale = *48:16* $.744$ $.758$ $.765$ $.715$ $.726$ $.717$
scale = *64:16* $\textbf{{\color{red}.751}}$ $\textbf{{\color{red}.762}}$ $\textbf{{\color{red}.773}}$ $\textbf{{\color{red}.718}}$ $\textbf{{\color{red}.728}}$ $\textbf{{\color{red}.729}}$
scale = *80:16* $.747$ $.757$ $.764$ $.715$ $.726$ $.722$
: Experimental results of bilateral feature and occlusion feature with various fusion ratio in **BRF** module.
\[tab:ablation3\]
*Scale* ODS OIS AP ODS OIS AP
---------------------- ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------
conv = *3$\times$3* $.746$ $.753$ $.754$ $.712$ $.719$ $.694$
conv = *3$\times$5* $.747$ $.755$ $.760$ $.712$ $.720$ $.696$
conv = *3$\times$7* $.747$ $.754$ $.758$ $.713$ $.721$ $.699$
conv = *3$\times$9* $.750$ $.759$ $.767$ $.716$ $.723$ $.712$
conv = *3$\times$11* $\textbf{{\color{red}.751}}$ $\textbf{{\color{red}.762}}$ $\textbf{{\color{red}.773}}$ $\textbf{{\color{red}.718}}$ $\textbf{{\color{red}.728}}$ $\textbf{{\color{red}.729}}$
: Experimental results of stripe convolutions with different aspect ratios.
\[tab:ablation4\]
[**[Plain or Stripe Convolution:]{}**]{} To evaluate the effect of stripe convolution for occlusion relationship reasoning, the stripe-based convolution variants with different aspect ratios are employed to make comparisons. As shown in Table.\[tab:ablation4\], intuitively, even if the slope of the edge is not in a horizontal or vertical direction, the convolution kernels possess large receptive field and tend to learn the cues of both directions, respectively. Nevertheless, the larger convolution layer takes up too much computation cost, which increases the number of parameters. Consequently, the stripe convolutions in orthogonal directions extract the tendency of edges and bilateral cue around the contour.
Conclusion
==========
In this paper, we present a novel OFNet, which shares the occlusion cue from the decoder and separately acquires the contextual features for specific tasks. Our algorithm builds on top of the encoder-decoder structure and side-output utilization. For learning the bilateral feature, an **MSL** is proposed. Besides, a **BRF** module is designed to apply the occlusion cue to precisely locate the object regions. In addition, we utilize a stripe convolution to further aggregate features from surrounding scenes of the contour. Significant improvement of the state-of-the-art through numerous experiments on PIOD and BSDS ownership dataset demonstrates the effectiveness of our network.
[**[Acknowledgement.]{}**]{} This work was supported by the National Natural Science Foundation of China Nos. 61703049, 61876022, 61872047, and the Fundamental Research Funds for the Central Universities No. 2019kfyRCPY001.
Appendix
========
{width="0.85\linewidth"}
In this appendix material, we provide full qualitative analysis for the ablation study. The experiments are conducted on the PIOD dataset.
One-branch or Multi-branch Sub-networks
---------------------------------------
Previous approach DOOBNet [@Wang2018DOOBNet] adopts a single flow architecture by sharing decoder features that represent high-level features. The shared decoder features reflect the contour cues, which are necessary for both edge and orientation estimations. Besides, edge detection and orientation detection are different in the choice of feature extraction, especially in the case of high semantic layers. We innovatively split the features produced by side-outputs and share decoder features to fit both tasks, respectively. Fig.\[Fig:ablation1\] reveals the effectiveness of our design.
Necessity for Each Feature
--------------------------
To verify the role of various low-level and high-level features, each feature is removed to construct an independent variant for evaluation. If the low-level features for edge path are removed, the occlusion edge is difficult to be accurately located, leading to decrease in the accuracy of occlusion relationship reasoning (shown in Fig.\[Fig:ablation2\](*w/o low-cues*)). If the high-level feature for edge path is removed, the occlusion edge is failed to be detected consistently, which decreases the accuracy at a large margin (shown in Fig.\[Fig:ablation2\](*w/o high-cues*)). By capturing spatial and contextual cues from the side-outputs respectively, the network is able to explore specific features for individual predictions.
{width="0.85\linewidth"}
Proportion of Bilateral-Contour Features
----------------------------------------
Previous works utilize inappropriate feature maps to predict the orientation, which reflects as the characteristic of the edge outline. The features of orientation on both sides of the contour are filtered gradually which are adversely affected by the edge prediction. We take advantage of an **MCL** to perceive the bilateral cues around the contours, and affirm the foreground and background relationship. As shown in Fig.\[Fig:ablation3\], fusing bilateral feature and occlusion feature with *64:16* channel ratio in the **BRF** outperforms others.
{width="1\linewidth"}
Plain or Stripe Convolution
---------------------------
Plain convolutions perceive information about surrounding small areas. To extract the tendency of edges to extend and bilateral cues around contours, we employ stripe convolutions in orthogonal directions. The convolution kernels possess large receptive field and tend to learn the cues of both directions, respectively. We test stripe convolution kernels with different aspect ratios, which are exhibited in Fig.\[Fig:ablation3\]. The larger convolution layer takes up too much computation cost, which increases the number of parameters. We evaluate the performance of the model with *11$\times$11 conv* on PIOD and BSDS datasets, the *EPR* (left) and *OPR* (right) are reported in Table.\[tab:abl\]. Compared with *3$\times$11 conv*, the model with *11$\times$11 conv* achieves limited improvement, while it increases about *50$\%$* gpu memory usage (*10031MB* to *14931MB*).
{width="1\linewidth"}
**Dataset** **Scale** **ODS** **OIS** **AP** **ODS** **OIS** **AP**
-------------- ----------------------- --------- --------- -------- --------- --------- --------
\*[*PIOD*]{} *conv = 3$\times$11* $.751$ $.762$ $.773$ $.718$ $.728$ $.729$
*conv = 11$\times$11* $.753$ $.764$ $.776$ $.719$ $.730$ $.732$
\*[*BSDS*]{} *conv = 3$\times$11* $.662$ $.689$ $.585$ $.583$ $.607$ $.501$
*conv = 11$\times$11* $.663$ $.690$ $.587$ $.585$ $.608$ $.503$
: Results of our model with different conv kernel sizes.
\[tab:abl\]
|
---
abstract: 'Clustering of charged particle tracks along the beam axis is the first step in reconstructing the positions of hadronic interactions, also known as primary vertices, at hadron collider experiments. We use a 2036 qubit D-Wave quantum annealer to perform track clustering in a limited capacity on artificial events where the positions of primary vertices and tracks resemble those measured by the Compact Muon Solenoid experiment at the Large Hadron Collider. The algorithm, which is not a classical-quantum hybrid but relies entirely on quantum annealing, is tested on a variety of event topologies from 2 primary vertices and 10 tracks up to 5 primary vertices and 15 tracks. It is benchmarked against simulated annealing executed on a commercial CPU constrained to the same processor time per anneal as time in the physical annealer, and performance is found to be comparable with an intriguing quantum advantage noted for 2 vertices and 16 tracks.'
address: 'Department of Physics and Astronomy, Purdue University'
author:
- 'S. Das'
- 'A. J. Wildridge'
- 'S. B. Vaidya'
- 'A. Jung'
bibliography:
- 'TrackClusteringDWave\_NIM.bib'
title: Track clustering with a quantum annealer for primary vertex reconstruction at hadron colliders
---
“Hadron collider" “Particle tracking" “Track vertexing" “Quantum annealing" “LHC" “D-Wave"
\[sec:Introduction\]Introduction
================================
{width="0.8\linewidth"}
Hadron colliders circulate counter-rotating beams of hadrons in closely packed bunches that cross at designated interaction points. These interaction points are instrumented with experiments that detect particles produced at hadron-hadron collisions when the bunches cross. Reconstructing the positions of these collisions within a bunch crossing, also known as primary vertices, from the trajectories of charged particles detected by the apparatuses is of paramount importance for physics analyses. The Large Hadron Collider (LHC) is a high luminosity collider that produces an average of 35 proton-proton (p-p) collisions at each bunch crossing, distributed in one dimension along the beam axis. At one of the LHC interaction points, the Compact Muon Solenoid experiment (CMS) reconstructs the paths of charged particles from p-p collisions as tracks detected by its silicon tracker [@Chatrchyan:2008aa]. Track reconstruction uncertainties obscure which tracks originated together at a primary vertex. Thus, primary vertex reconstruction begins with a one-dimensional clustering of tracks by their positions along the beam axis where they approach it most closely, also known as the tracks’ $z_0$. In this paper, we demonstrate a method of performing this clustering on a D-Wave quantum annealer and report preliminary results benchmarked against simulated annealing on a classical computer.
The D-Wave 2000Q quantum computer, available from D-Wave Systems Inc., performs computations through quantum annealing [@Finnila:1994; @Kadowaki:1998; @Santoro:2002]. The quantum processing unit (QPU) has 2036 RF-SQUID flux qubits implemented as superconducting niobium loops [@Harris:2010]. Each qubit has a programmable external magnetic field to bias it. The network of qubits is not fully connected and programmable couplings have been implemented between 5967 pairs of qubits. A computational problem is defined by setting the biases ($h_i$) and couplings ($J_{ij}$) such that the ground state of the qubits’ Hamiltonian corresponds to the solution. We call this the “problem Hamiltonian" ($H_p$) $$H_p = \sum_{i} h_i \sigma_z^i + \sum_{i} \sum_{j > i} J_{ij} \sigma_z^i \sigma_z^j,
\label{eq:ProblemHamiltonian}$$ where $\sigma_z^i$ is a spin projection observable of the $i^\mathrm{th}$ qubit with eigenvalues +1 and -1. (This $z$ direction is not related to the beam axis at CMS.) It may be trivially mapped to a bit observable $q_i$ with eigenvalues 0 and 1 through the shift $2 q_i = \sigma_z^i + I$, where $I$ is the identity matrix. The problem Hamiltonian may then be expressed for quadratic unconstrained binary optimization (QUBO) as $$H_p = \sum_{i} a_i q_i + \sum_{i} \sum_{j > i} b_{ij} q_i q_j,
\label{eq:ProblemHamiltonianQUBO}$$ notwithstanding energy offsets that are irrelevant for optimization. The D-Wave 2000Q programming model allows us to specify a problem in QUBO form by specifying $a_i$ and $b_{ij}$.
At the beginning of a typical annealing cycle in the QPU, a driver Hamiltonian puts all qubits in a superposition of the computational basis states by introducing a global energy bias in the transverse $x-$direction. Annealing proceeds by lowering this driver Hamiltonian while simultaneously increasing the problem Hamiltonian as $$H = A(s) \sum_i \sigma_x^i + B(s) H_p,
\label{eq:Annealing}$$ where $A$ is a monotonically decreasing function and $B$ is a monotonically increasing function defined on $s \in [0, 1]$. $A$ and $B$ have units of energy. The adiabatic theorem of quantum mechanics guarantees that the qubits will land in the ground state of $H_p$ if this change is sufficiently gradual, the ground state is unique with a non-zero energy separation from other states, and the initial state of the qubits is the ground state of the initial field [@BornFock:1928; @Kato:1950; @AvronElgart:1999]. These conditions are difficult to achieve experimentally. We therefore anneal within tens of microseconds during which quantum tunneling leaves the system in a low energy configuration at the end of the annealing process [@ThermallyAssistedQuantumAnnealing]. We measure the final state of the qubits as a solution, and repeat several times. The lowest energy solution is then taken as the best one.
We map track clustering to finding the ground state of a problem Hamiltonian. The Hamiltonian, which may be thought of as an objective function to minimize, is a measure of distances between the tracks given an association matrix $p_{ik}$ between the $i^\mathrm{th}$ track’s $z_0$ {$z_i$}, and candidate vertices labeled by integer $k$. We discuss the case where the matrix element $p_{ik}$ is 0 or 1 and hence expressed by one bit. Given {$z_i$} and the number of expected vertices, the quantum annealer solves for $p_{ik}$. A generic version of this algorithm has been described by V. Kumar, et al [@BAHPaper]. Further, we distort the Hamiltonian to encourage annealing to the ground state given the clustering characteristics of tracks observed at CMS. Given current technological limitations, we cannot cluster thousands of tracks into 35 or more primary vertices as expected at the LHC. We show how our method scales with event complexity by testing it on a variety of event topologies from 2 primary vertices and 10 tracks to 5 primary vertices and 15 tracks, where the positions of vertices and tracks are drawn from measured distributions at CMS. Our algorithm is not a quantum-classical hybrid, and relies entirely on thermally assisted quantum tunneling. This allows us to benchmark it clearly against simulated annealing running on a commercial CPU in the same period of time as a sampling cycle on the QPU.
\[sec:Formulation\]Formulation
==============================
For track clustering, we seek the ground state of the problem Hamiltonian $$\begin{split}
H_p = \sum_k^{n_V} \sum_i^{n_T} \sum_{j > i}^{n_T} p_{ik}p_{jk} g(D(i, j); m) \\
+ \lambda \sum_i^{n_T} \left( 1 - \sum_k^{n_V} p_{ik} \right)^2,
\label{eq:ObjectiveFunction}
\end{split}$$ where $n_T$ is the number of tracks, $n_V$ is the number of vertices, $p_{ik} \in \left[0, 1\right]$ is the probability of the $i^\mathrm{th}$ track to be associated with the $k^\mathrm{th}$ vertex, and $D(i, j)$ is a measure of distance between the reconstructed $z_0$ parameters of the $i^\mathrm{th}$ and $j^\mathrm{th}$ tracks.
As in any clustering algorithm, a density threshold of tracks must be set that determines $n_V$. For the study presented in this paper, we order the tracks in $z_0$ and count the number of gaps greater than a threshold of 5 mm. $n_V$ is set to this number plus one.
For $D(i, j)$, we find the absolute distance between $z_i$ and $z_j$ divided by the quadrature sum of the measurement uncertainties $\delta z_i$ and $\delta z_j$ to be an effective measure: $$D(i, j) = \frac{|z_i - z_j|}{\sqrt{\delta z_i^2 + \delta z_j^2}}.
\label{eq:Dij}$$ CMS distributions of track $z_0$ around vertices tend to cluster $D(i, j)$ near zero. This results in a cluster of energy levels near the ground state. To distribute the energy levels more uniformly, we use a distortion function on $D(i, j)$ $$g(x; m) = 1 - e^{-mx},
\label{eq:ExponentialSqueezing}$$ where $m$ is the distortion parameter. This reduces the spread of energy drops between intermediate states as the system anneals and is seen to improve convergence efficiency. $m$ is set to 5 for event topologies considered here.
$\lambda$ is a penalty parameter chosen to discourage $p_{ik}$ for each track to add up to anything other than 1. While it should be large enough to discourage the probability of a single track to be assigned to multiple vertices, it must not drown out the energy scale of $D(i, j)$. We tried several values of $\lambda$ from 1.0 to 2.0 times the maximum of $D(i, j)$ and settled on 1.2 times the maximum of $D(i, j)$ for optimal performance. Not all solutions from the QPU have $p_{ik}$ add up to 1 for all tracks; these are checked offline by a CPU and marked as invalid.
If each $p_{ik}$ is represented by one logical qubit in the QPU, Eq. \[eq:ObjectiveFunction\] is already in the QUBO form of the problem Hamiltonian described in Eq. \[eq:ProblemHamiltonianQUBO\]. Therefore, it can be directly programmed into a D-Wave QPU. A logical qubit is one physical qubit or a set of strongly coupled physical qubits created to compensate for the limited connectivity of a single physical qubit and to mitigate bit flips from thermal fluctuations. The graph embedding used to map the network of logical qubits to the network of physical qubits is found using default D-Wave algorithms [@CaiMacready; @VickyChoi1; @VickyChoi2] and can be re-used for multiple events. We need to program $n_V n_T$ logical qubits and $n_V n_T (n_V + n_T - 2) /2 $ couplings between them to encode $H_p$. Customizing embedding algorithms, optimizing the chain lengths and weights of logical qubits, modifying the overall annealing schedule [@ReverseAnnealing1; @ReverseAnnealing2] and the annealing schedule per chain [@Lanting2017] are all exercises at the frontier of quantum annealing and may be pursued in the future. We did not study the influence of QPU parameters such as the annealing time and the re-thermalization delay on convergence efficiency, and these may also be tuned for this class of problems.
\[sec:Results\]Results
======================
To test the algorithm, we generate artificial events with vertex positions in one dimension sampled from a simulated distribution of p-p bunch crossings at the LHC interaction point within CMS. A Gaussian with a width of 35 mm is a good representation of this distribution. The $z_0$ parameter of toy tracks are sampled from Gaussians centered around the generated vertices with widths corresponding to track resolutions measured in CMS [@CMSTracking]. These widths range from 0.1 to 0.7 mm depending on the momentum of the tracks, which are also sampled from measured track momentum distributions in CMS [@CMSTracking].
\[sec:OneEvent\]Primary vertexing one event
-------------------------------------------
![Biases and couplings between the logical qubits of the QPU as coefficients of the QUBO form used to solve a particular event with 3 primary vertices and 15 tracks. The diagonal terms are biases corresponding to the $\lambda$ term in Eq. \[eq:ObjectiveFunction\].[]{data-label="fig:c_qubo"}](c_qubo.png){width="\linewidth"}
![Energy spectrum of solutions for one event with 3 primary vertices and 15 tracks explored by the QPU with 10,000 samples. Energies corresponding to valid solutions, where the $p_{ik}$ add up to 1 for every track, are plotted with solid lines while all solutions are plotted with dashed lines. Error bars correspond to statistical uncertainties. The best and next-to-best valid solutions are indicated as Solutions 1 and 2, respectively. For clarity, the histogram is binned by 1 GHz below 10 GHz, by 10 GHz for 10 – 100 GHz, and by 100 GHz above 100 GHz. Events in 10 (100) GHz bins are normalized by 10 (100).[]{data-label="fig:c_energy_3V15T_PAPER"}](c_energy_3V15T_PAPER.png){width="\linewidth"}
To illustrate the algorithm, we generate an event with 3 vertices where 5 tracks emanate from each vertex. This requires 45 logical qubits to encode. The biases and couplings between them are obtained from Eq. \[eq:ObjectiveFunction\] in QUBO form and displayed in Fig. \[fig:c\_qubo\].
It takes 8 ms to program the biases and couplings into the QPU. Sampling the QPU for solutions consists of annealing as described in Eq. \[eq:Annealing\], readout of the qubits, and a delay for re-thermalizing the qubits to mitigate inter-sample correlations. Annealing is allowed for 20 $\mu$s by default. Readout takes 123 $\mu$s, and the delay is set to 21 $\mu$s. Thus, a single sample takes 164 $\mu$s. This period of time is used for benchmarking studies described in Section \[sec:Benchmark\].
We sample the QPU 10,000 times to evaluate the efficiency of finding the correct solution. The energy spectrum of the solutions, of which 6,825 are valid (where $p_{ik}$ add up to 1 for every track) is shown in Fig. \[fig:c\_energy\_3V15T\_PAPER\]. The energy scale is set by $B(1)$ as defined in Eq. \[eq:Annealing\], which is $6h$ GHz in the QPU used for this study. Of the valid solutions, 6,615 have landed on the lowest energy solution, marked as “Solution 1" in the figure. On investigating the qubit states, we find that the lowest energy solution corresponds to the correct clustering of the tracks with their respective vertices. Thus, the efficiency of finding the correct solution is noted as 66%. A small number of valid solutions correspond to “Solution 2" where one track has been mis-associated with a vertex. Further mis-associations result in higher energy valid solutions in the spectrum.
\[sec:Ensemble\]Performance on an ensemble of events
----------------------------------------------------
![In black is a histogram of QPU convergence efficiency to the correct solution for the case of 3 primary vertices and 15 tracks using 100 independent events. In red is the convergence efficiency for the same set of 100 events solved using simulated annealing on a CPU where the processor time for an annealing sample is set equal to the sampling time on the QPU. Error bars correspond to statistical uncertainties.[]{data-label="fig:c_ConvergenceEfficiency_3V15T_PAPER"}](c_ConvergenceEfficiency_3V15T_PAPER.png){width="\linewidth"}
![The dependence of QPU (black) and simulated annealing (red) convergence efficiency on the Dunn index, a measure of event clumpiness as defined in Eq. \[eq:Dunn\]. Events with higher Dunn indices have higher convergence efficiencies that asymptote to different values for the QPU and simulated annealing performed in comparable time as described in Section \[sec:Benchmark\]. The asymptotic values are marked with flat lines.[]{data-label="fig:c_Dunn_eff_3V15T_PAPER"}](c_Dunn_eff_3V15T_PAPER.png){width="\linewidth"}
Having studied the performance of the quantum annealer on one particular event, we consider an ensemble of 100 such events with 3 primary vertices and 15 tracks thrown from measured CMS distributions. Events with vertices spaced closely together compared to the spread of their tracks are difficult for the QPU to solve correctly and result in lower convergence efficiencies than events where vertices are widely separated. This results in a distribution of efficiencies shown in Fig. \[fig:c\_ConvergenceEfficiency\_3V15T\_PAPER\] with a mean of 42% and a standard deviation of 25%.
Since we generate the events, we can characterize the separation of vertices with respect to the intra-vertex spread of tracks, or the event “clumpiness", with the Dunn index $$\mathrm{Dunn} = \frac{\mathrm{min}(d(z_k^V, z_m^V))}{\mathrm{max}(d(z_i^T, z_j^T))}.
\label{eq:Dunn}$$ The numerator of the Dunn index measures the minimum distance between all pairs of primary vertex positions $(z_k^V, z_m^V)$. The denominator measures the maximum distance between all pairs of tracks from one vertex $(z_i^T, z_j^T)$, scanning over all vertices. Thus, a higher Dunn index corresponds to higher clumpiness. Events with higher Dunn indices should be easier for the quantum annealer to reconstruct. This is what we observe, as the convergence efficiency rises with the Dunn index and asymptotes as shown in Fig. \[fig:c\_Dunn\_eff\_3V15T\_PAPER\]. The sharp spikes and large variance in the QPU convergence efficiency may be attributed to hysteresis between sampling in the D-Wave 2000Q processor that is difficult to exactly reproduce.
\[sec:Scaling\]Scaling with event complexity
--------------------------------------------
To characterize how convergence efficiency scales with event complexity, we repeat our investigation for five other event topologies: 2 vertices and 10 tracks, 2 vertices 16 tracks, 4 vertices 12 tracks, 4 vertices 16 tracks, and 5 vertices 15 tracks. The distributions of convergence efficiency and its dependence on the Dunn index for each event topology is shown in Figs. \[fig:c\_ConvergenceEfficiency\_2V10T\_PAPER\] – \[fig:c\_ConvergenceEfficiency\_5V15T\_PAPER\]. We note the maximum asymptotic efficiency versus the Dunn index for each event topology, and plot it against the topology’s complexity measured by the number of logical qubits needed to express the problem in Fig. \[fig:c\_MaxConvEff\_qubits\_error\_PAPER\]. The central values correspond to the mean efficiency in the asymptotic region and the uncertainty corresponds to the uncertainty in the mean.
![The dependence of QPU (black) and simulated annealing with CPU (red) convergence efficiency on track clustering problem complexity. The trend is shown by complexity measured in the number of logical qubits needed. The processor time spent on the CPU per sample is equal to that spent by the QPU per sample, for all topologies. Central values correspond to the mean asymptotic efficiency indicated in Fig. \[fig:c\_ConvergenceEfficiency\_2V10T\_PAPER\] – \[fig:c\_ConvergenceEfficiency\_5V15T\_PAPER\], and the error bars correspond to uncertainty in the mean. Dotted lines are drawn to guide the eye.[]{data-label="fig:c_MaxConvEff_qubits_error_PAPER"}](c_MaxConvEff_qubits_error_PAPER.png){width="\linewidth"}
\[sec:Benchmark\]Benchmarking against Simulated Annealing
---------------------------------------------------------
![Illustration of the constraint on simulated annealing to use the same time as the quantum annealer per sample. For the D-Wave 2000Q quantum annealer, time per sample consists of annealing time, readout time and a re-thermalization delay as described in Section \[sec:OneEvent\]. The time / sweep of the simulated annealer is measured as shown in Fig. \[fig:c\_TimePerSweep\].[]{data-label="fig:Benchmarking"}](Benchmarking.png){width="\linewidth"}
![Process time of simulated annealing on a 3.1 GHz Intel Core i7-5557U CPU as a function of number of sweeps for various event topologies. The slopes indicate time per sweep for each topology and is used to benchmark against the D-Wave 2000Q QPU.[]{data-label="fig:c_TimePerSweep"}](c_TimePerSweep.png){width="\linewidth"}
We benchmark the D-Wave QPU’s performance on this problem against a time-optimized implementation of Simulated Annealing (SA) on a 3.1 GHz Intel Core i7-5557U CPU. The SA algorithm is written for a problem expressed generally in QUBO form for a vector of bits of size $N$ as shown in Eq. \[eq:ProblemHamiltonianQUBO\]. The algorithm increments an inverse-temperature parameter $\beta$ linearly from $\beta_i$ to $\beta_f$ in $n_{S}$ number of steps. $\beta_i$ and $\beta_f$ correspond to the hottest and coldest temperatures relevant to the problem, respectively. We set $\beta_i$ to the order of magnitude of the inverse-temperature at which the most strongly coupled bit has a 50% probability to flip as shown in Eq. \[eq:beta\_i\]. This requires us to compute the energy difference due to the flip, $\Delta E_{max}$, if all bits connected to it are 1. By studying all event topologies, we find $\beta_i$ may be reasonably set to 0.1. Similarly, $\beta_f$ is set to the inverse-temperature at which the smallest coupling in the QUBO, $\Delta E_{min}$, has a 1% chance of a bit flip across it as shown in Eq. \[eq:beta\_f\]. After studying all topologies, we set $\beta_f$ to 10. $$\beta_i = -\log(0.5) / \Delta E_{max}
\label{eq:beta_i}$$ $$\beta_f = -\log(0.01) / \Delta E_{min}
\label{eq:beta_f}$$ At each increment in $\beta$, we “sweep" through the $N$ bits to flip each one at a time. If flipping a bit results in a lower energy, the flip is accepted. If flipping it results in a state with energy $\Delta E$ higher, we accept the flip with probability $e^{-\beta\Delta E}$. Else, the flip is reversed.
We compile the algorithm in `C++` enabling `-O2` optimization. To time-optimize the implementation, we construct a sorted `std::map` from the QUBO where keys are bit indices $q_i$ and values are `std::vector<std::pair <unsigned int, double>>` where the first element of the pair is another bit $q_j$ it couples to, and the second element is the $b_{ij}$ coupling between them. As a bit flip requires us to only compute the energy difference due to it, this organization streamlines the lookup of the relevant bits and couplings.
Since the D-Wave QPU takes 164 $\mu$s per sample, we first measure how many SA sweeps ($n_S$) can be accomplished in that time as illustrated in Fig. \[fig:Benchmarking\]. To that end, we graph the process time on the CPU against $n_S$ equal to 10, 50, 100, and 150 for each event toplogy as shown in Fig. \[fig:c\_TimePerSweep\]. Thus, we eliminate overhead times and extract the time per sweep. We find the time per sweep is linear in problem size defined by $n_V n_T$ as expected from the memory organization. From Fig. \[fig:c\_TimePerSweep\], we infer that 42, 21, 15, 15, 10 and 8 sweeps can fit within 164 $\mu$s for 2 vertices 10 tracks, 2 vertices 16 tracks, 3 vertices 15 tracks, 4 vertices 12 tracks, 4 vertices 16 tracks, and 5 vertices 15 tracks event topologies, respectively. Therefore, we set $n_S$ equal to these numbers of sweeps for each event topology run on the CPU to *constrain the effective working time to be equal between the CPU and the QPU*, and then we compare their convergence efficiencies. In Figs. \[fig:c\_ConvergenceEfficiency\_3V15T\_PAPER\], \[fig:c\_Dunn\_eff\_3V15T\_PAPER\], and \[fig:c\_ConvergenceEfficiency\_2V10T\_PAPER\] – \[fig:c\_ConvergenceEfficiency\_5V15T\_PAPER\], we overlay the convergence efficiency of the CPU over the QPU for each event topology. We note that while performances are comparable for the 2 vertices 10 tracks topology, quantum annealing is twice as efficient as simulated annealing for 2 vertices 16 tracks, and this advantage persists for the 3 vertices 15 tracks topology to some extent.
In Fig. \[fig:c\_MaxConvEff\_qubits\_error\_PAPER\], we overlay the maximum asymptotic convergence efficiency of simulated annealing on that of quantum annealing for various event topologies. A decreasing trend in efficiency with increasing problem complexity is observed.
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\[sec:Conclusions\]Conclusions and outlook
==========================================
With noisy intermediate-scale quantum computers on the horizon, we find that a D-Wave 2000Q quantum annealer can already be used to solve track clustering for primary vertex reconstruction at LHC experiments like CMS in a limited capacity. The convergence efficiency for finding the correct solution is similar to simulated annealing on a commercial CPU that is constrained to the same sampling time as the quantum annealer. The quantum annealer is found to possess an intriguing advantage over simulated annealing for the 2 vertex 16 tracks event topology.
Extended to two dimensions and used for hierarchical clustering, the method described in this paper can be used for other high energy physics applications like clustering of energy deposits in calorimeters to identify particle showers. Research in this direction may accelerate both quantum information science and high energy physics instrumentation.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge useful discussions with Joel Gottlieb, Mark Johnson, and Alexander Condello at D-Wave Systems Inc., and are grateful for the opportunity to work with the 2000Q quantum annealer. We also acknowledge helpful discussions with Jan-Frederik Schulte, Jason Thieman, Amandeep Singh Bakshi, and Giulia Negro. The authors acknowledge support from the Department of Energy under the grant $\mathrm{DE-SC0007884}$, and from the Purdue Research Foundation.
References {#references .unnumbered}
==========
|
---
abstract: 'We investigate a numerical method for studying resonances in quantum mechanics. We prove rigorously that this method yields accurate approximations to resonance energies and widths for shape resonances in the semiclassical limit.'
address:
- ' Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061–0123, U.S.A.'
- ' Facultad de Física, P.U. Católica de Chile, Casillo 306, Santiago 22, Chile'
author:
- 'George A. Hagedorn and Bernhard Meller'
title: Resonances In a Box
---
Introduction
============
In this paper we rigorously analyze the validity of a numerical technique for studying resonances in quantum mechanics. The technique is called, “A spherical box approach to resonances,” by its inventors, Maier *et al* [@MaiCedDom80a]. We prove that the technique yields correct energies and lifetimes for shape resonances in the semiclassical limit.
The technique is an “$L^{2}$ method,” in contrast to time–independent scattering theory methods, such as the calculation of phase shifts near energies where a resonance is expected. These $L^2$ methods are surveyed, [[*e.g.$\;$*]{}]{}, in [@KukulinEtAl].
The basic physical idea underlying all $L^{2}$ methods is that a resonance wavefunction is a state that is concentrated mainly in the interaction region. In contrast, states associated with the rest of the continuous spectrum are not concentrated in any bounded interval. As a consequence, when the system is confined to a box that is large compared to the interaction region and the size of the box is varied, the resonance wavefunction is much less influenced than the states from the rest of the continuous spectrum. This should be visible in the spectrum, and is the basis of the technique we study.
To make this precise, we consider the Schrödinger operator $$\label{eq:Hamiltonian}
H:=D^{2} +V\,, \qquad D:= \frac{\hbar}{i}\frac{d}{dx}$$ with a resonance producing potential $ V $ that is defined on all of ${I\!\!R}$. We restrict the system to the interval $(-\ell,\,\ell)$ with Dirichlet boundary conditions at $ x=\pm \ell $, and plot the eigenvalues of the resulting operator $H(\ell)$ as a function of $ \ell $.
Figure \[fig:lines\] presents the results obtained by doing this for the potential $ V $ that is depicted in Figure \[fig:pot\].
In this figure, one can clearly distinguish between eigenvalues that depend strongly on $\ell$ and others that seem to be almost independent of $\ell$. Furthermore, there are avoided eigenvalue crossings when a strongly dependent eigenvalue is close to an eigenvalue that is almost independent of $\ell$. Note that in our example, eigenvalues are not expected to cross [@WigVNeu27a], since the potential has no apparent symmetry properties.
In addition to relating the almost constant eigenvalues to resonance energies, Maier *et al* [@MaiCedDom80a] also relate the sizes of the gaps in the avoided crossings to the imaginary part (or width, or inverse lifetime) of the resonance. In [@MaiCedDom80a], spherically symmetric potentials are treated. After the reduction to an angular momentum subspace, the particle can escape to infinity in only one way, by increasing the radial coordinate $r$. In the model we consider, the particle can escape toward either plus or minus infinity. Since the probablilities for going in the two directions can be different, we observe two different size gaps for each given resonance. This is obviously the case in Figure \[fig:lines\]. For our model, the resonance width is related to the larger of the two gaps.
In this paper, we provide rigorous justification of these results in the semiclassical limit. As a first step, we adopt a standard definition of a resonance that is presented in [@AguCom71; @Reed-SimonIV]. This definition identifies a resonance with a complex eigenvalue of a suitably constructed analytic family of operators obtained from the original Hamiltonian (\[eq:Hamiltonian\]).
In many instances, as in the case of shape resonances, such a complex eigenvalue can be viewed as arising from the perturbation of an eigenvalue embedded in the continuous spectrum. We take this viewpoint and employ the framework of, “The Shape Resonance,” [@ComDucKleSei87a] by Combes *et al*. We temporarily impose supplementary Dirichlet boundary conditions at points $\omega_{\pm}$ to decouple the interaction region from the rest of ${I\!\!R}$. This yields an unperturbed operator on all of ${I\!\!R}$ that has embedded eigenvalues whose eigenfunctions are supported in the interaction region. Removal of these Dirichlet conditions perturbs the embedded eigenvalues to produce the resonances (that are realized as complex eigenvalues of certain non-self-adjoint operators). The perturbation calculations are facilitated by the use of Krein’s formula [@ComDucKleSei87a].
To relate the resonances of $H$ defined on $L^2({I\!\!R})$ to the almost $\ell$–independent eigenvalues of $H(\ell)$, we show that the techniques of [@ComDucKleSei87a] can also be applied in a box to study $H(\ell)$. We then employ the following strategy: For small values of $\hbar$, resonances of $H$ are very close to embedded eigenvalues of $H$ with supplementary Dirichlet conditions at $\omega_{\pm}$. For $\ell >\max\,\{|\omega_+|,\,|\omega_-|\}$, these embedded eigenvalues are also eigenvalues of $H(\ell)$ with supplementary Dirichlet conditions at $\omega_{\pm}$. For large $\ell$ and small $\hbar$, removal of these supplementary Dirichlet conditions perturbs these eigenvalues only slightly. Thus, the resonances of $H$ are near eigenvalues of $H(\ell)$. These results are made precise in Theorem \[theo:1\].
This approach also allows us to prove rigorously that the gap in the avoided crossing is on the order of the square root of the resonance width, in accordance with [@MaiCedDom80a]. We accomplish this by relating both the gap and the resonance width to the thickness of the potential barrier as measured by the Agmon distance [@Agmon]. The relationship between resonance widths and Agmon distances is already established in [@ComDucKleSei87a], so we need only examine the relationship between the Agmon distances and the gaps in the avoided crossings. This is done in Theorem \[theo:2\].
Hypothesis and Results
======================
For simplicity, we assume the potential $V$ to be bounded. We wish to study resonances that are produced by a single well and to avoid asymptotically degenerate eigenvalues with an exponentially small separation in $\hbar$. Furthermore, we want the bottom $ v_{0} $ of the well to be above the scattering threshold. We force this situation by imposing a hypothesis that can be expressed nicely with the help of the notion of *the classical forbidden region at energy $E$*. This is defined as $$J(E):=\{x\in{I\!\!R}:\ V(x)>E\}\,.$$ Our precise hypothesis is the following:
- $V\in C^{1}({I\!\!R})$ is bounded and has a local minimum $v_{0}$ at $ x_{0}$, such that $\overline{J(v_{0})}$ is connected, and $\limsup_{|x|{\rightarrow}\infty}V(x)<v_{0}$.
By translating the origin if necessary, we choose an interior region $$\Omega_{i}:=(\omega_{-},\,\omega_{+}),\quad\mbox{ with }
\omega_{-}<0\mbox{ and }\omega_{+}>0,\mbox{ such that }\quad
\overline{\Omega}_{i}\setminus\{x_{0}\}\subset J(v_{0}).\,$$ We define the exterior region to be $\Omega_{e}:={I\!\!R}\setminus\overline{\Omega}_{i}$, and let $\Omega_{e}^-=(-\infty,\,\omega_-)$ and $\Omega_{e}^+=(\omega_+,\,\infty)$. We define the decoupled comparison operator $ H^{d} $ as having the same symbol as $ H $, but with supplementary Dirichlet conditions at $\omega_-$ and $\omega_+$. This operator decomposes into $$H^{d}=H^{i}\oplus H^{e}
\quad\mbox{with}\quad {\cal D}(H^{\alpha})=
{\cal H}^{1}_{0}\cap{\cal H}^{2}(\Omega_{\alpha}),\quad
\mbox{where}\quad\alpha\in\{i,\,e\}\,.$$
Since we want to focus on shape resonances, we impose a hypothesis that prevents resonances from being produced in the exterior region for energies near $ v_{0} $. We phrase this hypothesis in terms of a non-trapping condition [@BriComDuc87a]: We say *the potential $V$ is non-trapping in $\Omega_{e}$ at energy $E$ (abbreviated $E$ is NT)*, if the following condition is satisfied for $ \alpha\in\{-,\,+\} $: $$\exists S>0,\ \forall x\in\Omega_{e}^{\alpha}\setminus J(E), \quad
\frac{x\!-\!\omega_{\alpha}}{x}\big( 2(V(x)\!-\!E)+xV'(x)\big)<-S
\,.
\label{eq:VirialCDKS}$$ We assume:
- $v_{0}$ is NT.
Note that formula (\[eq:VirialCDKS\]) is equivalent to the more standard virial condition $$\exists \tilde{S}>0,\ \forall x\in\Omega_{e}\setminus J(E), \quad
2(V(x)-E)+xV'(x)<-\tilde{S}\,.$$ Furthermore (\[eq:VirialCDKS\]) implies the “exterior” virial condition $$\exists S>0,\ \forall x\in\Omega_{e}^{\alpha}\setminus J(E), \quad
\big( 2(V(x)-E)+(x-\omega_{\alpha})V'(x)\big)<-S,\
\alpha\in\{-,\,+\}\,.$$ Our third hypothesis concerns analyticity under exterior dilation. For $\theta\in{I\!\!R}$, we define
[2]{} U\_: L\^2([IR]{}) &L\^2([IR]{}) &&\
U\_: &r\_ & r\_(x) &:= {
[cl]{} \_-+e\^(x-\_-),& x<\_-\
x,& x(\_-,\_+) .\
\_+ +e\^(x-\_+),& x>\_+
.
We then assume:
- $V_\theta:=U_\theta V U_\theta^{-1}$ defined initially for $\theta\in{I\!\!R}$, has an analytic continuation as a bounded operator to the strip $\{ \theta\in{I\!\!\!\!C}: |{{\rm Im}}\,\theta | < \beta\}$, for some $\beta\in (0,\,\pi/4)$.
For $\theta\in{I\!\!R}$ we also define the operators $ H_\theta := U_\theta H U_\theta^{-1}$ and $H^d_\theta := U_\theta H^d U_\theta^{-1} $. It is a straightforward calculation to obtain the associated symbol $$\begin{aligned}
U_\theta (D^2+V)U_\theta^{-1} =\, &{r_\theta '}^{-2} D^{2}+V\circ
r_\theta, \\
\mbox{where\hspace{4cm}}[&{r_\theta '}^{-2} D^{2}u](x)=\left\{
\begin{aligned}
-\hbar^2 u''(x),&\quad x\in(\omega_-,\,\omega_+)\\
-\hbar^2 e^{-2\theta} u''(x),&\quad x{\mbox{$\hskip 6pt /\hskip -10pt\in\,$}}[\omega_-,\,\omega_+]\,.
\end{aligned}
\right.\end{aligned}$$ Since $U_\theta$ is a unitary operator on $L^2({I\!\!R})$ for $\theta\in{I\!\!R}$, we easily compute the domains for the operators $H^d_\theta $ and $H_\theta$, for $\theta\in {I\!\!R}$: $$\begin{aligned}
\notag
{\cal D}(H^d_\theta)&={\cal D}(H^i)\oplus {\cal D}(H^e),\\
\label{eq:Htheta}
{\cal D}(H_\theta) &=
\{u_i\oplus u_e\in {\cal H}^2(\Omega_i)\oplus {\cal H}^2(\Omega_e) :
u_e(\omega_\pm)\!=\! e^{\frac{\theta}{2}}u_i(\omega_\pm),\, u_e'(\omega_\pm)\!=\! e^{\frac{3\theta}{2}}u_i'(\omega_\pm)\}.
$$ We define the restrictions of these operators to the box $(-\ell,\,\ell)$ to be $$\begin{aligned}
\notag
H^d_\theta(\ell)&:={r_\theta '}^{-2} D^{2}+V\circ r_\theta
\quad\mbox{on}\quad {\cal D}(H^d_\theta) \cap {\cal H}^1_0((-\ell,\ell)),
\quad\mbox{and}\\
H_\theta(\ell)&:={r_\theta '}^{-2} D^{2}+V\circ r_\theta
\quad\mbox{on}\quad {\cal D}(H_\theta) \cap {\cal H}^1_0((-\ell,\ell))\,.
\label{eq:Hthetaell}\end{aligned}$$ For $\theta=0$, $H_{\theta=0}(\ell)$ is simply the Schrödinger operator $H(\ell)$ described in the introduction that is used to produce plots, such as Figure 1.
The following lemma describes the analytic continuations of these families of operators to complex values of $\theta$:
Hypotheses **(H1)**–**(H3)** imply the following two conclusions:
1. $\{H^d_{\theta},\, |{{\rm Im}}\theta|<\beta\}$ and $\{H^d_{\theta}(\ell),\, |{{\rm Im}}\theta|<\beta\}$ are self-adjoint analytic families of Type (A) of m-sectorial operators.
2. $\{H_{\theta},\, |{{\rm Im}}\theta|<\beta\}$ and $\{H_{\theta}(\ell),\, |{{\rm Im}}\theta|<\beta\}$ are self-adjoint analytic families of operators.
[[*Proof:*]{}]{} These conclusions for the families $H^d_\theta$ and $H_\theta$ are proved in [@ComDucKleSei87a]. The same proofs apply for the families $H^d_\theta(\ell)$ and $H_\theta(\ell)$ since the proofs in [@ComDucKleSei87a] make no use of the (un)boundedness of $\Omega_e$.
We next recall the Agmon distance [@Agmon], that we denote by the symbol $ d_{E} $. It is the distance associated to the pseudo-metric $ ds^{2}:=\max\{0,V(x)-E\}dx^{2}\,$. We introduce the abbreviations
[3]{} d\_[v\_[0]{}]{}\^&:= d\_[v\_[0]{}]{}(x\_[0]{},), {-,+},& && d\^&:= {d\_[v\_[0]{}]{}\^[-]{},d\_[v\_[0]{}]{}\^[+]{}} .
The following theorem gives precise information about the resonance on the line and the “resonance in the box.” Its first conclusion follows from [@ComDucKleSei87a].
\[theo:1\] Assume **(H1)**–**(H3)** and that $E^{d}$ is the $n^{th}$ eigenvalue of $H^{i}$.
- For any $ \vartheta\in(0,1) $ and sufficiently small $\hbar$, there exists $\beta_0\in(0,\beta)$, such that $H_{i\beta_0}$ has a (complex) eigenvalue $E$ close to $ E^{d} $ that satisfies: $$\label{eq:tunnel exp}
E=E^{d}+\sum_{n\geq 1}\frac{t^{n}\sigma_{n}}{n!},
\quad\mbox{with}\quad
t=o(e^{-2\vartheta d^{\star}/\hbar}),\quad \mbox{where}\quad
\sigma_{n}=o(1),\ \forall n\geq 1.$$
- The same is true for the operator $H_{i\beta_0}(\ell)$. Furthermore its eigenvalue is stable in the sense of Kato , as the box size $\ell$ tends to infinity. As $\ell$ tends to infinity, this eigenvalue converges to the corresponding eigenvalue of $H_{i\beta_0}$.
- For sufficiently small $ \hbar $ and those values of $ \ell $, for which there exist positive constants $c$ and $ N $, such that $\mbox{\em dist}(E^{d}, \sigma (H^{d}(\ell))\setminus \{E^{d}\})
\geq c\hbar^{N}$, there exists a real eigenvalue of $ H(\ell) $ close to $ E^{d} $ that satisfies the same type of expansion as above.
**Remark:** (a) Note that in [@ComDucKleSei87a], the theorem is stated with $ d^{\star} $ replaced by $ d_{v_{0}}(x_{0},\partial\Omega_{i}) $. Due to the possible choices of $ \omega_{\pm} $, the difference between the two quantities can be made arbitrarily small and can be absorbed into $ \vartheta $. But then, how small $ \hbar $ must be chosen, depends on $ \vartheta $.
\(b) In the third conclusion of this theorem, one cannot expect uniform results in $\ell$ and $ \hbar $. The eigenvalues of the exterior operator $H^e(\ell)$ have different dependence on $\ell$ and $ \hbar $ than the eigenvalues of the interior operator $H^i$. The condition [dist]{}$(E^{d}, \sigma (H^{d}(\ell))\setminus \{E^{d}\}) \geq c\hbar^{N}$ is technical; we cannot handle exponentially closely spaced eigenvalues. It is well known that under our hypotheses, the eigenvalues of $H^i$ near the bottom of the well (close to $v_0$) cannot be spaced more closely than ${\cal O}(\hbar^\gamma)$. Here, the constant $\gamma$ is strictly smaller than 2. Its value depends on how flat the bottom of the well is. In order to prove that eigenvalues from $H^e(\ell)$ do not cause [dist]{}$(E^{d}, \sigma (H^{d}(\ell))\setminus \{E^{d}\}) \geq c\hbar^{N}$ to be violated for all $\ell$, we would need an additional assumption on the decay of the potential. For example, together with dilation analyticity, it would be enough to assume that $V$ tends to a limit at infinity like $|x|^{-\epsilon}$ for any $\epsilon>0$. We now turn our attention to the gaps in the avoided crossings that occur in graphs of the eigenvalues of $H(\ell)$. For this part of our analysis, we replace hypotheses **(H2)** and **(H3)** by the following:
- $V\in C^3({I\!\!R})$. For $x\in\Omega_e\setminus \overline{J(v_0)}$, the potential obeys $V(x)< v_0$ and there exist two constants $v_\pm<v_0$, such that $V-v_\pm = {\cal O}(|x|^{-\epsilon})$ as $x$ tends to $\pm\infty$. Furthermore for $n=1,2$, we have $V^{(n)}= {\cal O}(|x|^{-\epsilon-1})$ as $x$ tends to $\pm\infty$.
This hypothesis allows us to use WKB estimates to analyze the behavior of eigenvalues of $H^e(\ell)$. We note that $H^e(\ell)$ decomposes into the direct sum of $ H^{e}_{-}(\ell) $ and $ H^{e}_{+}(\ell) $, where $ H^{e}_{-}(\ell) $ acts on $ L^{2}((-\ell,\,\omega_{-})) $ and $ H^{e}_{+}(\ell) $ acts on $ L^{2}((\omega_{+},\,\ell)) $.
We have the following result on the gaps:
\[theo:2\] Assume **(H1)** and **(H4)**. Suppose $ E^{d}$ is an eigenvalue of $ H^i $ and of $H^e_\alpha(\ell_0)$, but not of $H^e_{-\alpha}(\ell_0)$. Assume it satisfies $\mbox{\em dist}(E^{d}, \sigma (H^{e}_{-\alpha}(\ell_0)))
\geq c\hbar^{N}$, for some positive constants $ c$ and $ N$ and $\alpha\in\{-,\,+\}$. Then we have the following: For fixed values of $ \hbar $ that are sufficiently small, there exists a neighborhood $ {\cal V}(\ell_{0}) $ of $ \ell_{0} $, such that for all $\ell$ in $ {\cal V}(\ell_{0}) $, $ H(\ell) $ has two eigenvalues $E_+$ and $E_{-}$ that are exponentially close to $ E^{d} $. These two eigenvalues are separated by a gap that satisfies $$\min_{\ell\in{\cal V}(\ell_{0})}\{ |E_{+}-E_{-}|\}
=\left|\sum_{n\geq 1}\frac{(t_{1}+t_{2})^{n}\sigma_{n}}{n!}
\right|\,,
\text{ where}\quad \sigma_{n}=o(1),\
\forall n\geq 1.$$ In this estimate, $t_1$ and $t_2$ satisfy the following for any $\vartheta\in(0,1) $: $$\begin{aligned}
{3} &t_{1}&=t_{2}=o(\exp({-\vartheta d^{-}_{v_0}/\hbar}))\quad
&\text{if}\quad&
E^{d}&\in\sigma(H^{i})\cap\sigma(H^{e}_{-}(\ell_{0}))\,,
\\
&t_{1}&=t_{2}=o(\exp({-\vartheta d^{+}_{v_0}/\hbar}))\quad
&\text{if}\quad&
E^{d}&\in\sigma(H^{i})\cap\sigma(H^{e}_{+}(\ell_{0}))\,.
\end{aligned}$$
**Remark:** (a) Here, $ \hbar $ does not depend on $ \vartheta $.
\(b) The width of the resonance is given by the tunneling parameter $t$ according to Theorem \[theo:1\]. We do not know whether the resonance is going to tunnel to the left or right, so we only obtain the estimate $ t= o(\exp({-\vartheta 2 d^{\star}/\hbar}))$. In Theorem \[theo:2\] we know to which side the resonance escapes, and the result is more precise. We also obtain estimates for both of the gap sizes that can occur in the avoided crossings for a given resonance. Since $d^{\star}=\min\{d_{v_{0}}^{-},\,d_{v_{0}}^{+}\}$, Theorem \[theo:2\] shows that the larger gap is of the same order as the square root of the resonance width. We again note that in [@MaiCedDom80a], a radial symmetric situation is studied, so that there is only one way for the resonance to escape, and hence only one gap size.
\(c) The eigenvalues of $ H^{i} $ are obviously independent of $ \ell $, but not of $ \omega_{\pm} $. Thus, it might seem that the condition of having a double eigenvalue is crucially dependent on the choice of $ \omega_{\pm} $. This is not case: From Theorem \[theo:1\] (iii) we see that the eigenvalues of $ H^{i} $ vary only by an exponentially small quantity in $ \hbar $ when the $ \omega_{\pm} $ are varied. For the eigenvalues of $ H^{e}_{\pm}(\ell) $, we show in \[app:2\], that **(H4)** implies that eigenvalues $ E\in \sigma(H^{e}_{\alpha}(\ell))$ that belong to an interval $ (v_{0},v_{0}+\delta) $ are related to $ \hbar $, $ \ell $, and a quantum number $ m $ by the asymptotic formula $$E=v_{\alpha}+\left((m+\frac{3}{4})\frac{\pi\hbar}{\ell}\right)^{2}\left(1+
{\cal O}(\hbar) +{\cal O}(\ell^{-\epsilon})\right)\,,
\quad\alpha\in\{-,\,+\}
\,.$$ We thus have the following consequence: Suppose, for example, that the $n$-th eigenvalue $ E^{d} $ of $ H^{i} $ coincides with an eigenvalue of $ H^{e}_{+}(\ell_{0}) $ for some choice of $ \omega_{\pm} $, and that $ E^{d} $ is at least a distance of $ {\cal O}(\hbar^{N}) $ from the spectrum of $ H^{e}_{-}(\ell_0) $. Then for any other choice of $ \omega_{\pm} $, there exists an $ \ell $ in a neighborhood of $ \ell_{0} $, such that $ E^{d} $ is an eigenvalue of $ H^{e}_{+}(\ell) $, and the distance from $ E^{d} $ to the spectrum of $ H^{e}_{-}(\ell) $ is still at least $ {\cal O}(\hbar^{N}) $.
The Proofs
==========
Inspection of the proofs of [@ComDucKleSei87a] for Theorem \[theo:1\] (i) shows that they are valid whether or not $\Omega_e$ is bounded. Furthermore, these proofs can be separated into two parts: The first is a formal algebraic part that shows the stability of the eigenvalue of $ H^{i} $ for the whole operator and constructs the asymptotic expansion of the perturbed eigenvalue in powers of the tunneling parameter $ t $. It is quite simple and short. The second part is the justification of these algebraic formulas with the corresponding estimates. This part is more complicated and involves estimation of the operators involved in Krein’s formula.
We present the formal algebraic part, which is needed in all of the situations treated in Theorems \[theo:1\] (ii), \[theo:1\] (iii), and Theorem \[theo:2\]. We do this in Section \[sec:S\] in the context of Theorem \[theo:1\] (iii). In Section \[sec:S2\], we treat the stability of the resonance eigenvalue of $H_{i\beta_0}(\ell)$ as the box size $\ell$ tends to infinity. Finally, in Section \[sec:S3\] we prove Theorem \[theo:2\]. In the Appendix, we recall Krein’s formula and present the more technical estimates, including the WKB estimates.
We omit the estimates required to prove the existence and the series expansion of the eigenvalue of $H_{i\beta_0}(\ell)$ because they are identical to those in [@ComDucKleSei87a].
Stability and Tunneling Expansion for The Box {#sec:S}
---------------------------------------------
We view $ H(\ell) $ as a perturbation of $ H^{d}(\ell) $. This perturbation involves two Dirichlet conditions. It is most easily approached by way of Krein’s formula, that exhibits the difference of the resolvents of $ H(\ell) $ and $ H^{d}(\ell) $ as a rank two operator.
The norm of this rank two operator is not small. However, because the Dirichlet conditions are imposed inside the classically forbidden region, its norm does not explode in proportion to the inverse of the distance from the spectrum to the spectral parameter in the resolvents. This allows us to choose the parameters in such a way that the resolvent of the resolvent of $ H^{d}(\ell) $ is small in norm, and we can still use perturbation theory.
The tunneling expansion is based upon a Feshbach type reduction of the eigenvalue equation with respect to the unperturbed eigenprojection. This leads to an implicit equation that we solve by using the Lagrange inversion formula.
### Stability
To simplify the notation, we suppress the $\ell$ dependence in many of the formulas. We define $$R^{d}(z):=(H^{d}(\ell)-z)^{-1}\quad\mbox{and}
\quad R(z):= (H(\ell)-z)^{-1}\,.$$ We choose a contour $\Gamma$ that lies in the resolvent set of $ H^{d}(\ell) $ and encloses only $ E^{d} $ in $ \sigma(H^{d}(\ell)) $. We then choose a point $z_0$ in the intersection of the resolvent sets of $ H(\ell) $ and $ H^{d}(\ell) $, but outside of $\Gamma$. By using the identity $$\left(R^{d}(z_{0})-\frac{1}{z-z_{0}}\right)^{-1}
= -(z-z_{0})-(z-z_{0})^{2}R^{d}(z),
\label{eq:identity}$$ we obtain the following expression for the eigenprojection $
P^{d}\equiv P^{d}(\ell) $ associated to $ E^{d} $: $$P^{d}\,=\,-\,\frac{1}{2\pi i}\,\int_{\Gamma}\,R^{d}(z)\,dz
\,=\,-\,\frac{1}{2\pi i}\,\int_{\widetilde{\Gamma}}\,
\left(R^{d}(z_{0})-\widetilde{z}\right)^{-1}\!d\widetilde{z}\,,$$ where $\{\,\widetilde{\Gamma}:=\widetilde{z}\in{I\!\!\!\!C}:\,
\widetilde{z} =\frac{1}{z-z_{0}},\,z\in\Gamma\,\}$. By defining $$\pi(z_0):= (H(\ell)-z_0)^{-1}-(H^{d}(\ell)-z_{0})^{-1}\,,$$ we can formally write the eigenprojection $ P\equiv P(\ell) $ associated to the perturbed eigenvalue $ E $ as $$P \,=\,-\,\frac{1}{2\pi i}\,\int_{\widetilde{\Gamma}}\,
\left(R^{d}(z_{0})-\widetilde{z}\right)^{-1}
\left(1+\pi(z_{0})\left(R^{d}(z_{0})-
\widetilde{z}\right)^{-1}\right)^{-1}
d\widetilde{z}\,.$$ If we can choose $\Gamma $ and $z_0$, such that $\left\|\,\pi(z_{0})\left(R^{d}(z_{0})
-\widetilde{z}\right)^{-1}\,\right\|<1$, then the inverse term in the integral for $P$ can be computed by geometric series. Then the eigenprojection is well defined, and by standard arguments, we can deduce the stability of the eigenvalue for $ H(\ell) $.
To see that we can do this, fix any $ n\in{I\!\!N}$. Let $$\label{eq:defi delta}
\Delta:= \mbox{dist}
(E^{d}, \sigma (H^{d}(\ell))\setminus \{E^{d}\}),
\quad \mbox{and fix}\quad
r\in[\min\{\hbar^{n},\,\frac{1}{2}\Delta\},\,\frac{1}{2}\Delta ]\,.$$ Note that by hypothesis, $ \Delta\geq c\hbar^{N} $, for some $ N\in{I\!\!N}$, and that we can choose $ r $ to be as small as any power of $ \hbar $.
We define $ \Gamma:=\{z\in{I\!\!\!\!C}:|z-E^{d}| =r\} $ and $ z_{0}= E^{d}+2ir $. Then, formula (\[eq:identity\]) implies $ \left(R^{d}(z_{0})-\widetilde{z}\right)^{-1}={\cal O}(r) $. Thus, the stability follows from the following proposition that we prove in \[app:1.2\]:
\[prop:pi\] $ \pi(z_{0})={\cal O}(1) $.
### Tunneling expansion {#sec:TE}
Since we have proven the stability of the eigenvalue and constructed $P(\ell)$, we can write the eigenvalue equation as $$R(z_{0})P(\ell)= \frac{1}{E-z_{0}}P(\ell)\,.$$ We perform a Feshbach type reduction to this equation, with respect to the projections $ P^{d} $ and $ Q^{d}=1-P^{d} $. We define the “reduced” resolvent $$\widehat{R}(z;\,z_{0}):=Q^{d}\,
{\left(Q^{d}(R(z_{0})-z)Q^{d}\right)}^{-1}\,Q^{d}
\,.$$ It satisfies the following estimate:
\[prop:hatR\] For any $ z $ in the disc delimited by $ \Gamma $, one has $ \widehat{R}(\frac{1}{z-z_{0}};\,z_{0}) ={\cal O}(r) $.
[[*Proof:*]{}]{} If we replace the $ R(z_{0}) $ by $ R^{d}(z_{0}) $ in the definition of $ \widehat{R}(z;\,z_{0}) $, we obtain a trivial result. The conclusion to the proposition is obtained by applying perturbation theory to this trivial result.For $ ({E\!-\!z_{0}})^{-1} $ the reduction yields the implicit equation $$(\frac{1}{E\!-\!z_{0}}-\frac{1}{E^{d}\!-\!z_{0}})P^{d}P =
P^{d}\left(\pi(z_{0}) -\pi(z_{0})\,
\widehat{R}(\frac{1}{E\!-\!z_{0}};\,z_{0})\, \pi(z_{0}) \right)P^{d}P\,.$$ Using properties of the trace and the factorization $ \pi(z)=\hbar A^{\star}(\bar z)B(z) $, [[*c.f.$\;$*]{}]{}\[app:1.2\], we obtain $$\begin{aligned}
\notag
\frac{1}{E\!-\!z_{0}} -\frac{1}{E^{d}\!-\!z_{0}}\,&=\,
\hbar\,\mbox{Tr} \left(B(z_{0}) P^{d} A^{\star}(z_{0})
\Big(1-\hbar B(z_{0})\,\widehat{R}(\frac{1}{E\!-\!z_{0}};\,z_{0})
\,A^{\star}(\bar{z}_{0})\Big)\right)\,,\\
\intertext{or equivalently}
\label{eq:implicit F}
\frac{1}{E\!-\!z_{0}} -\frac{1}{E^{d}\!-\!z_{0}}\,&=\,
t\, s\,(\frac{1}{E\!-\!z_{0}})\,,\end{aligned}$$ where (suppressing $ z_{0}$ in $ A $ and $ B $) $$t:=\hbar\,|\mbox{Tr}(B P^{d}A^{\star})|\quad\mbox{and}\quad
s(z):= \frac{1}{t}\,\mbox{Tr} \left(\hbar B
P^{d}A^{\star}(1-\hbar B\widehat{R}(z;z_{0})A^{\star})\right)\,.
\label{eq:tands}$$ For any $z$ in the disc delimited by $\Gamma$ and $\tilde{z}=\frac{1}{z-z_{0}}$, we have the following estimate on $ s(\tilde{z}) $: $$|s(\tilde{z})|\,\leq\,\|1-\hbar B\widehat{R}(\tilde{z};z_{0})A^{\star}\|
\,=\,1+{\cal O}(r).$$ This follows from Proposition \[prop:hatR\] and the bound on $ \pi $, [[*c.f.$\;$*]{}]{}\[app:1.2\]. If we can establish the estimate $ t=o(e^{-2\vartheta d^{\star}/\hbar}) $ of Theorem \[theo:1\], then equation can be solved with Lagrange’s inversion formula [@Dieudonne p.250] $$\frac{1}{E\!-\!z_{0}}=\frac{1}{E^{d}\!-\!z_{0}}+\sum_{n\geq
1}\frac{t^{n}}{n!}\left[\frac{d^{n-1}}{dz^{n-1}}s^{n}
\right](\frac{1}{E^{d}\!-\!z_{0}})
=:\frac{1}{E^{d}\!-\!z_{0}}+\sum_{n\geq
1}\frac{t^{n}}{n!}\widetilde{\sigma}_{n}
\,.$$ Multiplying by $ (E\!-\!z_{0})(E^{d}\!-\!z_{0}) $ and rearranging, we obtain $$E = E^{d}- (E\!-\!z_{0})(E^{d}\!-\!z_{0})
\sum_{n\geq 1}\frac{t^{n}}{n!}\widetilde{\sigma}_{n}
= E^{d}-\sum_{k\geq 1} (z_{0}\!-\!E^{d})^{k+1}
\biggl(\sum_{n\geq
1}\frac{t^{n}}{n!}\widetilde{\sigma}_{n}\biggr)^{k}\,.$$ We estimate the coefficients $ \widetilde{\sigma}_{n} $ by using the Cauchy formula $$\widetilde{\sigma}_{n} = \frac{(n-1)!}{2\pi i}\,
\int_{\widetilde{\Gamma}}\,
\frac{s(\widetilde{z})^{n}}{(\frac{1}{E^{d}\!-\!z_{0}}-\widetilde{z})^{n}}
\,d\widetilde{z},\quad\text{and}\quad
(\frac{1}{E^{d}\!-\!z_{0}}-\widetilde{z})^{-1}= {\cal O}(r)\,.$$ We define $ \sigma_{n}:= (E\!-\!z_{0})(E^{d}\!-\!z_{0})\widetilde{\sigma}_{n} $ and easily obtain the estimate $ \sigma_{n}=o(1) $ of Theorem \[theo:1\].
### The tunneling parameter
The above calculation relies on the estimate $ t=o(e^{-2\vartheta d^{\star}/\hbar}) $. To prove this, we note that if $ \phi_{d} $ denotes the eigenfunction associated to $
E^{d} $, then using the definitions and estimations of \[app:1.2\], $$\begin{aligned}
t\,&\leq\,
\hbar\,\|B\phi_{d}\|\,\|A\phi_{d}\|
\,\leq\,\hbar^{2}\,\|TRT^{\star}\|\, \|B\phi_{d}\|^{2} \\
&=\,\frac{\hbar^{2}\,\|TRT^{\star}\|}{|E^{d}\!-\!z_{0}|^{2}}\,
\|T^{d}D\phi_{d}\|^{2}
\,\leq\,\frac{c\hbar^{3}}{4r^{2}}\,
(|\phi_{d}'(\omega_{-})|^{2}+|\phi_{d}'(\omega_{+})|^{2})\,.\end{aligned}$$ For each part of Theorem \[theo:1\], we can estimate the expression $|\phi_{d}'(\omega_{-})|^{2}+|\phi_{d}'(\omega_{+})|^{2}$ by the well known decay estimates of Agmon [@Agmon]. This implies the results of Theorem \[theo:1\].
Stability as The Box Size Tends to Infinity {#sec:S2}
-------------------------------------------
We consider the operator $$H^{D}_{\theta}(\ell):= H_{\theta}(\ell)\oplus
H^{ee}_{\theta}(\ell)\,,
\label{eq:HD}$$ where $ H_{\theta}(\ell) $ is the operator defined in , and $$H^{ee}_{\theta}(\ell):= e^{-2\theta}D^{2}+ V\circ r_{\theta}
\quad\mbox{on} \quad
{\cal H}^{1}_{0}\cap{\cal H}^{2} ({I\!\!R}\setminus [-\ell,\ell])
\,.$$ It is easy to see that $ H^{ee}_{\theta}(\ell) $ is an analytic family of Type (A) in $ \theta $, and that we have the following resolvent estimate:
\[prop:Hee\] Assume **(H1)**–**(H3)** and let $S$ denote the constant in the non-trapping condition **(H2)**. Let $\nu=\{z\in{I\!\!\!\!C}: |{{\rm Re}}\,z -v_0|< S/4,\,{{\rm Im}}\,z > -S/4\}$. Then $$\forall\, z\in\nu,
\qquad \|R^{ee}_{i\beta}(z)\|\,\leq\,
\frac{4}{|\beta | S}\,(1+{\cal O}(\beta)).$$
[[*Proof:*]{}]{} $ H^{ee}_{i\beta}(\ell) $ decomposes into a direct sum of operators that act on $L^2((-\infty,\,-\ell))$ and $L^2((\ell,\,\infty))$. We consider only the term associated to the interval $ (\ell,\infty) $; analogous formulas hold for the other term. We mimic arguments of [@BriComDuc87a]. For $ u\in{\cal H}^{1}_{0}\cap{\cal H}^{2} ((\ell,\infty))$ and any $ v\in L^{2}((\ell,\infty)) $, we have $$\begin{aligned}
\|v\|\,\|(H^{ee}_{i\beta}(\ell)-z)u\| &
\,\geq\,{{\rm Re}}\,((H^{ee}_{i\beta}(\ell)-z)u,\,v)\,.\end{aligned}$$ For $ \beta>0 $ we use this with $ v= -ie^{-i2\beta}u$ to obtain $$\begin{aligned}
{{\rm Re}}\,((H^{ee}_{i\beta}(\ell)-z)u,v)\,&=\,
-\,{{\rm Im}}\,(e^{2i\beta}(e^{-2i\beta}D^{2}+V\circ r_{i\beta}-z)u,\,u)\\
&=\,-\,{{\rm Im}}\,(e^{2i\beta}(V\circ r_{i\beta}-z)u,\,u)\\
&=\,-\,((\beta\,( 2(V-{{\rm Re}}\,z)+(x\!-\!\omega_{+})V' -{{\rm Im}}\,z ) +
{\cal O}(\beta^{2}))u,\,u)\\
&>\,(\beta\,(S-2(v_{0}\!-\! {{\rm Re}}\,z)+{{\rm Im}}\,z)+{\cal O}(\beta^{2}))\,
\|u\|^{2} \\
&>\,\left(\,\frac{\beta\,S}{4} +{\cal O}(\beta^{2})\,\right)\,
\|u\|^{2}\,.\end{aligned}$$ For negative $ \beta $ we repeat this calculation with $ v\!=\! ie^{-i2\beta}u $. This proves the proposition.We now fix $ \theta=i\beta_{0} $ as in [@ComDucKleSei87a]. With the definitions of $ z_{0}$ and $ \widetilde{\Gamma} $ as in Section \[sec:S\], we define $$P_{i\beta_0}(\ell) \,=\,-\,\frac{1}{2\pi i}\,\int_{\widetilde{\Gamma}}\,
\left((H_{i\beta_0}(\ell)\!-\!z_{0})^{-1}-\widetilde{z}\right)^{-1}
\oplus
\left((H^{ee}_{i\beta_0}(\ell)\!-\!z_{0})^{-1}-\widetilde{z}\right)^{-1}
d\widetilde{z}\,.$$ Here, $ P_{i\beta_0}(\ell) $ projects onto the eigenspace for the eigenvalue $ E \in \sigma( H_{i\beta_0}(\ell)) $, but does so in the space $ L^{2}({I\!\!R}) $. To prove stability of the eigenvalue in the generalized sense ([[*c.f.$\;$*]{}]{}Kato, [@Kato Sec. VIII.1.4]), we must show that $ P_{i\beta_0}(\ell)\stackrel{s}{{\longrightarrow}} P_{i\beta_0} $ as $ \ell $ tends to $ \infty $, where $$P_{i\beta_0} \,=\,-\,\frac{1}{2\pi i}\,\int_{\widetilde{\Gamma}}\,
\left((H_{i\beta_0}\!-\!z_{0})^{-1}-\widetilde{z}\right)^{-1}
d\widetilde{z}\,.$$ It is shown in [@ComDucKleSei87a] that for sufficiently small $ \hbar $, $ \left((H_{i\beta_0}\!-\!z_{0})^{-1}-\widetilde{z}\right)^{-1} =
{\cal O}(r) $, uniformly on $ \widetilde{\Gamma} $. The estimates of [@ComDucKleSei87a] are also valid for $ \left((H_{i\beta_0}(\ell)\!-\!z_{0})^{-1}-\widetilde{z}\right)^{-1} $. So, from Proposition \[prop:Hee\] and identity , we see that, $$\left((H^{D}_{i\beta_0}(\ell)\!-\!z_{0})^{-1}-\widetilde{z}\right)^{-1} =
{\cal O}(r)\,,$$ uniformly on $ \widetilde{\Gamma} $. Thus, we need only show that for any $ u\in L^{2}({I\!\!R}) $, $$\lim_{\ell{\rightarrow}\infty}\left\|\left( (H_{i\beta_0}-z_{0})^{-1}-
(H^{D}_{i\beta_0}(\ell)-z_{0})^{-1}\right) u\right\| =0\,,$$ uniformly in $ \hbar $. This is shown in \[app:1.1\]
Proof of Theorem 3 {#sec:S3}
------------------
In the degenerate case, we must solve for two eigenvalues. So, we cannot *a priori* use the Lagrange inversion formula to solve equation in the disc delimited by $ \Gamma $.
However, we could use the formula if one of the solutions were known to be $ \frac{1}{E^{d}\!-\!z_{0}} $. This would happen if $ \pi $ were a rank one operator. In that case, the spectra of $ H^{d} $ and $ H $ would intertwine, and as a consequence, at the crossing of two eigenvalues of $ H^{d} $ there would have to be an eigenvalue of $ H $.
In our situation such a scenario can be realized by lifting the two Dirichlet conditions one after the other.
It suffices to consider the case where $ E^{d}\in\sigma(H^{i})\cap\sigma(H^{e}_{+}(\ell_{0}))$. In the first step, we consider the operators $$H^{d}_{-}(\ell) := H^{e}_{-}(\ell)\oplus H^{i}
\quad\text{and}\quad
H_{-}(\ell) := D^{2} +V \quad\text{on}\quad
L^{2}((-\ell,\omega_{+}))\,.$$ By hypothesis, $ \hbar $ is small and fixed, and $ H^{i} $ has the eigenvalue $ E^{d} $, which for $ \ell=\ell_{0} $ is a distance of $ {\cal O}(\hbar^{N}) $ from the rest of the spectrum of $ H^{d}_{-}(\ell_{0}) $, [[*i.e.$\;$*]{}]{}$ E^{d} $ is a simple, conveniently isolated eigenvalue of $ H^{d}_{-}(\ell_{0}) $. Thus, the analog Theorem \[theo:1\] (iii) is valid:
\[lem:cutbox\] Assume the hypotheses of Theorem \[theo:2\] with $ E^{d}\in\sigma(H^{i})\cap\sigma(H^{e}_{+}(\ell_{0}))$. Then there exists a neighborhood of $ {\cal V}(\ell_{0}) $, of size $ c\hbar^{N} $, such that for each $ \ell\in{\cal V}(\ell_{0}) $, the operator $ H_{-}(\ell) $ has an eigenvalue $ E_{-} $ close to $ E^{d} $ that satisfies the following for any $\vartheta\in(0,1) $ $$E_{-}=E^{d}+\sum_{n\geq 1}\frac{t^{n}\sigma_{n}}{n!}
\quad\text{with}\quad
t= o(e^{-2\vartheta d_{v_0}(\omega_{-},x_{0})/\hbar})
\quad\text{and}\quad
\sigma_{n}=o(1),\,\forall n\geq 1.$$
[[*Proof:*]{}]{} We first note that as we vary $ \ell $, with the restriction that $ |\ell-\ell_{0}|\leq c\hbar^{N} $, $ E^{d} $ remains isolated from the rest of the spectrum by a distance of size $ c\hbar^{N} $. Thus, we can prove the lemma by mimicking the proof of Theorem \[theo:1\] (iii).For the second step, we note that due to the behavior of $ E^{e}_{+}(\ell) $ there exists an $ \ell_{1}\in {\cal V}(\ell_{0}) $, such that $$E_{-}= E^{e}_{+}(\ell_{1}) \,.$$ We now use the intertwining of the spectra of $ H_{-}(\ell_{1})\oplus H^{e}_{+}(\ell_{1}) $ and $ H(\ell_{1}) $. We obtain the following lemma by using the techniques we used for Lemma \[lem:cutbox\] and noting that the eigenfunction $ \phi_{d} $ associated to $ E_{-} $ has the form $ \phi_{d}=\phi_{-}\oplus\phi_{+} $, where $ H_{-}(\ell_{1})\phi_{-}=E_{-}\phi_{-} $ and $ H^{e}_{+}(\ell_{1})\phi_{+}=E_{-}\phi_{+} $:
Assume **H(1)** and **(H4)** and that $ E_{-} $ is a double eigenvalue of $H_{-}(\ell_{1})\oplus
H^{e}_{+}(\ell_{1})$ as constructed above. Then the operator $ H(\ell_{1}) $ has two eigenvalues $E_{-}$ and $ E_{+} $ that satisfy $$E_{+}=E_{-}
+\sum_{n\geq 1}\frac{(t_{1}+t_{2})^{n}\sigma_{n}}{n!}
\quad\text{with}\quad
\sigma_{n}=o(1),\ \forall n\geq 1\,,$$ where, for any $\vartheta\in(0,1) $,
[3]{} t\_[1]{}&=o(e\^[-2d\_[v\_[0]{}]{}(x\_[0]{},\_[+]{})/]{})& && t\_[2]{}&=o(e\^[-2d\_[v\_[0]{}]{}(\_[+]{},\_[1]{})/]{}).
The last step in the proof of Theorem \[theo:2\] is to note that the first two steps can be done for any admissible $ \omega_{+} $. The $ n $-th eigenvalue $ E^{d} $ of $ H^{i} $ changes by only an exponentially small amount in $ \hbar $ when $ \omega_{+} $ is varied, so it remains properly isolated from $ \sigma( H^{e}_{-}(\ell_{0})) $. Furthermore, by the behavior of the exterior eigenvalues, there exists an $ \ell_{2} $ in a neighborhood of $ \ell_{0} $, such that the new $ E_{-} $ is also an eigenvalue of $ H^{e}_{+}(\ell_{2}) $. The optimal estimate is obtained when $ t_{1}=t_{2} $, in which case we have $ t_{1}=t_{2}= o(e^{-\vartheta d^{+}_{v_{0}}/\hbar})$.
Krein’s Formula
===============
Since we need Krein’s formula for one and two supplementary Dirichlet boundary conditions, taken at different points depending on the situation, we wish to present the formula in a general setting. On the other hand, for simplicity, we leave out the exterior dilation. We deal with this only when necessary.
Suppose $n\geq 2$, and $ -\infty\leq x_{0}<x_{1}<\ldots<x_{n}\leq \infty $ are specified. Let $ \Omega:=(x_{0},x_{n}) $ and $ \Omega_{k}:=(x_{k-1},x_{k}) $ for $ k=1,\ldots,n $. Let $ H:=D^{2}+V $ be a Schrödinger operator on $ \Omega $, with self-adjoint boundary conditions at $ x_{0} $ and $ x_{n} $, and let $ H^{d} $ be the corresponding decoupled operator with supplementary Dirichlet conditions at $ x_{1},\,x_{2},\ldots ,\,x_{n-1} $. Denote their resolvents by $ R $ and $ R^{d} $, respectively.
Let $ z\in \rho(H)\cap \rho(H^{d}) $ and $ u,v\in L^{2}({I\!\!R}) $. Define $ \hat{u}:=R^{d}(z)u $ and $ \hat{v}:=R({z})^{\star}v $. Clearly, $ \hat{u}\in {\cal D}(H^{d})$, and thus, $ \hat{u}=\oplus_{k=1}^{n}\hat{u}_{k} $ with $ \hat{u}(x_{k})=0,\,k=1,\ldots,n-1$. We have $$\begin{aligned}
\left( (R(z)-R^{d}(z))u,v\right)\,&=\,
(u,\hat{v})-(\hat u,v)
\\
&=\,\sum_{k=1}^{n}\,(D^{2}\hat{u}_{k},\hat{v})_{\Omega_{k}} -
(\hat{u}_{k},D^{2}\hat{v})_{\Omega_{k}}\\
&=\,- \hbar^{2}\,\sum_{k=1}^{n}\,
\hat{u}'_{k}\,\overline{\hat{v}}\Big|_{\partial\Omega_{k}}
\\ &=\,\hbar^{2}\,\sum_{k=1}^{n-1}\,
\left(\,\hat{u}'_{k+1}\!-\!\hat{u}'_{k}\,\right)\,
\overline{\hat{v}}\big|_{x_{k}}\,.\end{aligned}$$ We use standard Sobolev space notation and define functionals $ T^{j}_{x_{k}} $ by the following relations, where $ f\in\oplus_{k=1}^{n}{\cal H}^{1}(\Omega_{k}) $: $$T^{j}_{x_{k}}:{\cal H}^{1}(\Omega_{j}){\longrightarrow}{I\!\!\!\!C},\quad
T^{j}_{x_{k}}f:=\lim_{y{\rightarrow}x,\,y\in\Omega_{j}} f(y),\quad\mbox{for}\quad
j=k,\,k+1,\quad k=1,\ldots,n-1.$$ If $ T^{k}_{x_{k}}f= T^{k+1}_{x_{k}}f $ for all $f$, we simply write $ T_{x_{k}} $. It is well known that $ T^{j}_{x_{k}} $ is compact, and consequently, $ (T^{j}_{x_{k}})^{\star}:{I\!\!\!\!C}^{2}{\rightarrow}{\cal H}^{-1}(\Omega_{j}) $ is continuous. Furthermore, Lemma 4 of Section III of [@ComDucKleSei87a] shows that whenever $ \chi\in C^{\infty}_{0}({I\!\!R}) $ satisfies $ \chi(x_{k})= 1 $ for $ k=1,\ldots,n-1 $, $$\left\|T^{j}_{x_{k}}u\right\|^{2}\,\leq\,
2 \hbar^{-1}\left\|\chi u_{j}\right\|
\left\|D \chi u_{j}\right\|
\,\leq\,c\hbar^{-1}\left\|\chi u_{j}\right\|_{{\cal H}^{1}}\,,
\quad\mbox{for}\quad j =k,\,k+1.
\label{eq:trace}$$ Finally, we define $$T^{-}:=\begin{pmatrix} T^{1}_{x_{1}} \\ \vdots\\ T^{n-1}_{x_{n-1}}\end{pmatrix},
\quad
T^{+}:=\begin{pmatrix} T^{2}_{x_{1}} \\ \vdots\\ T^{n}_{x_{n-1}}\end{pmatrix},
\quad
T^{d}:= -\,T^{-}\oplus T^{+}
\quad\mbox{and}\quad
T:=\begin{pmatrix} T_{x_{1}} \\ \vdots\\ T_{x_{n-1}}\end{pmatrix}.$$ With these definitions, we have the following formula, $$\left((R(z)-R^{d}(z))u,\,v\right)
\,=\,\hbar\,\big(R(z) T^{\star}iT^{d}DR^{d}(z)u,\,v\big)\,,$$ where all the multiplications are understood to be matrix multiplications.
Applying Krein’s Formula for Theorem \[theo:1\] (iii) {#app:1.2}
-----------------------------------------------------
In the proof of Theorem \[theo:1\] (iii), we have $x_{0}=-\ell$, $x_{1}=-\omega_{-}$, $x_{2}=\omega_{+}$, $x_{3}=\ell$, $$R^{d}(z)=(H^{d}(\ell)-z)^{-1},\quad\mbox{and}
\quad R(z)= (H(\ell)-z)^{-1}\,.$$ Following [@ComDucKleSei87a], we define $$B(z):= i T^{d} D R^{d}(z) \quad\mbox{and}\quad
A(z):= T R(z)\,.$$ Since $ H $ is self-adjoint, we can write $$\pi(z)=R(z)-R^{d}(z)=\hbar\,A^{\star}(\overline{z}) B(z).$$ Furthermore, since $ TR^{d}(z)=0 $, we have $$T \pi(z) = TR(z)= A(z)=
\hbar\,T R(z) T^{\star} B(z) \,.
$$ We combine the two formulas to obtain $$\pi(z)= \hbar^{2}B^{\star}(\overline{z}) \,
TR(z)T^{\star}\,B(z)\,.$$ Proposition \[prop:pi\] now follows from
\[lem:esti B\] Let $ z_{0}= E^{d}+2ir $. Fix any $ N\in {I\!\!N}$. Then for sufficiently small $ \hbar $ and any $ r\in[\min\{\hbar^{N},\,\frac{1}{2}\Delta\},\,\frac{1}{2}\Delta ]$, $$B(z_{0})={\cal O}(\hbar^{-1/2})\qquad\mbox{and}\qquad
TR(z_{0})T^{\star}={\cal O}(\hbar^{-1})\,.$$
*Proof:* The assertion on $ TR(z_{0})T^{\star} $ is proved in step 5 of the proof of Theorem III.3 of [@ComDucKleSei87a]. As for $ B(z_{0}) $ we have $$\|B(z_{0})\|^{2}= \|T^{1}_{\omega_{-}}DR^{e}(z_{0})\|^{2}
+\|T^{2}_{\omega_{-}}DR^{i}(z_{0})\|^{2}
+\|T^{2}_{\omega_{+}}DR^{i}(z_{0})\|^{2}
+\|T^{3}_{\omega_{-}}DR^{e}(z_{0})\|^{2}\,,$$ where $ R^{i}(z_{0}) := (H^{i}-z_{0})^{-1}$ and $ R^{e}(z_{0}) := (H^{e}(\ell)-z_{0})^{-1}$. Let $ \chi $ be a $ C^{\infty}_{0} $ function supported around $\omega_{\pm}$ such that $ \chi(\omega_{\pm})=1 $. Using the estimate , it suffices to find a uniform bound on the expressions $$\chi DR^{i}(z_{0}),\quad D\chi DR^{i}(z_{0}),
\qquad\mbox{and}\qquad
\chi DR^{e}(z_{0}),\quad D\chi DR^{e}(z_{0})\,.$$ We choose $ \chi $, such that $ V(x)-v_{0}\geq\varepsilon>0 $ for $ x\in \mbox{supp}\,\chi $. Then steps 1 and 2 of the proof of Theorem III.3 of [@ComDucKleSei87a] show that all theses terms are uniformly bounded.
Applying Krein’s Formula for Theorem \[theo:1\] (ii) {#app:1.1}
----------------------------------------------------
Here we consider the difference of the resolvents of the operators $ H_{i\beta_0} $ defined by formula of the introduction and $ H^{D}_{i\beta_0}(\ell) $ defined by formula . In this case, $n=3$, $x_{0}=-\infty$, $x_{1}=-\ell$, $x_{2}=\ell$, and $x_{3}=\infty $. The difference of the resolvents is $$R_{i\beta_0}(z_{0})-R_{i\beta_0}^{d}(z_{0})
= \hbar R_{i\beta_0}(z_{0}) T^{\star}
ie^{-2i\beta_0}T^{d}DR^{d}_{i\beta_0}(z_{0})\,.$$ Let $ \chi $ be a $ C^{\infty}_{0} $ function supported around $\pm\ell$, with $ \chi(\pm\ell)=1 $. To show that $T^{d} D R^{D}_{i\beta_0}(z_{0})$ and $ T R_{-i\beta}(\overline{z}_{0}) $ are uniformly bounded operators we use the estimate . Thus, it suffices to show that $\chi D R^{D}_{i\beta_0}(z_{0})$ and $ \chi R_{-i\beta}(\overline{z}_{0}) $ are uniformly bounded operators from $ L^{2} $ to $ {\cal H}^{1}$. If that is true, then for $ u\in L^{2}({I\!\!R}) $, we have $$\begin{aligned}
\left\|\left( R_{i\beta_0}(z_{0})-R_{i\beta_0}^{d}(z_{0})\right) u\right\|
&\leq
\left\|TR_{-i\beta}(\overline{z}_{0}) \right\|
(|\hat{u}(-\ell)|+|\hat{u}(\ell)|) \\
&\leq c(|\hat{u}(-\ell)|+|\hat{u}(\ell)|)
\stackrel{\ell{\rightarrow}\infty}{{\longrightarrow}}0 \,,\end{aligned}$$ uniformly in $ \hbar $, since $ \hat u =\chi DR^{D}_{i\beta_0}(z_{0})u \in {\cal H}^{1}({I\!\!R}) $.
We now address the required uniform bounds. Commuting $ \chi $ and $ D $, we need only consider $ \chi DR^{D}_{i\beta_0}(z_{0}) $ and $ \chi D^{2}R^{D}_{i\beta_0}(z_{0}) $. The expressions for $ T R_{-i\beta}(\overline{z}_{0}) $ are analogous and can be treated the same way. The formula $$\|\chi DR^{D}_{i\beta_0}(z_{0})\|^{2}=
{{\rm Re}}\,R^{D}_{i\beta_0}(z_{0})^{\star} \chi^{2}
\left(D^{2}-2\hbar^{2}(\chi^{2})''\right) R^{D}_{i\beta_0}(z_{0})$$ shows that it is sufficient to bound $ \chi D^{2}R^{D}_{i\beta_0}(z_{0}) $ and $ \chi R^{D}_{i\beta_0}(z_{0}) $. We have $$\begin{aligned}
\chi D^{2}R^{D}_{i\beta_0}(z_{0}) &=
e^{2i\beta}\chi \left(H^{D}_{i\beta_0}-z_{0}-(V\circ r_{i\beta_0}-z_{0})
\right)R^{D}_{i\beta_0}(z_{0})\\
&=e^{2i\beta}\left(1-(V\circ r_{i\beta_0}-z_{0})
\chi R^{D}_{i\beta_0}(z_{0})\right)\end{aligned}$$ For $ \chi R^{D}_{i\beta_0}(z_{0}) $, we set $ \pi_{i\beta_0}(z_{0}):=(H_{i\beta_0}(\ell)\!-\!z)^{-1}
-(H^{d}_{i\beta_0}(\ell)\!-\!z)^{-1} $, and then write $$\begin{aligned}
\chi R^{D}_{i\beta_0}(z_{0}) &=
\chi \left(R^{i}(z_{0})\oplus (H^{e}_{i\beta_0}(\ell)-z_{0})^{-1}
+\pi_{i\beta_0}(z_{0})
\right) \oplus R^{ee}_{i\beta_0}(z_{0})\\
&=\chi\left((H^{e}_{i\beta_0}(\ell)-z_{0})^{-1}
+\pi_{i\beta_0}(z_{0})\right) \oplus
R^{ee}_{i\beta_0}(z_{0})\,.\end{aligned}$$ The right hand side is uniformly bounded in $ \hbar $ and $ \ell $ by Propostion \[prop:Hee\] and Lemma II.3 and Theorem III.3 of [@ComDucKleSei87a], which are also valid for $ (H^{e}_{i\beta_0}(\ell)-z_{0})^{-1} $ and $ \pi_{i\beta_0}(z_{0})
$, respectively.
WKB Estimates {#app:2}
=============
For these estimates, we follow Olver [@Olver Ch. 11]. The goal is to find approximate solutions to the differential equation $$-\hbar^{2}w'' +(V-E)w=0\,
\label{eq:Schroedinger}$$ in $ \Omega_{e} $ with $ v_{0}+\delta>E>v_{0} $ for some positive $ \delta $. Due to either the non-trapping condition or the explicit assumption **(H4)**, there is at most one turning point in each of the intervals $ (\omega_{+},\,\infty) $ and $(-\infty,\,\omega_{-}) $. There is exactly one, if $ \delta $ is sufficiently small. We assume $\delta$ has been chosen so that this is the case.
It suffices to consider the interval $ (\omega_{+},\infty) $, and we denote the turning point by $ x_{t} $. We define a new independent variable $ \xi:= s(x) $ by $$s(x)\,s'(x)^{2}= E-V(x)\,, \quad s(x_{t})=0\,,\quad s'(x_{t})>0\,.$$ By integration, we obtain $$\xi= {{\rm sgn\,}}(x-x_{t})\biggl(\frac{3}{2}{\cal S}(x)\biggr)^{2/3}
\quad\mbox{where}\quad
{\cal S}(x):= \int_{\min\{x,x_{t}\}}^{\max\{x,x_{t}\}}
\sqrt{|V(t)-E|}\,dt
\,.$$ Note that $ {{\rm sgn\,}}( V(x)-E) = {{\rm sgn\,}}(x_{t}-x) $. It is easy to check that under our conditions, Theorem 3.1 of [@Olver Ch. 11] shows that equation has two $ C^{2} $ solutions $ w_{1} $ and $ w_{2} $ in $ (\omega_{+},\infty) $, such that $$\label{eq:WKB1tp sol}
\begin{split}
w_{1}(x;\hbar) & =
s'(x)^{-1/2}\biggl(
\operatorname{Bi}(-\xi/\hbar^{2/3})+{\cal O}(\hbar\operatorname{Bi}(-\xi/\hbar^{2/3}))\biggr)\,,\\
w_{2}(x;\hbar) & =
s'(x)^{-1/2}\biggl(
\operatorname{Ai}(-\xi/\hbar^{2/3})+{\cal O}(\hbar\operatorname{Ai}(-\xi/\hbar^{2/3}))\biggr)\,.
\end{split}
$$ Higher order approximations are also known, [[*c.f.$\;$*]{}]{}[@Olver Sec. 11.7].
The Dirichlet boundary conditions imply the quantization condition $$w_{1}(\omega^{+};\hbar) w_{2}(\ell;\hbar)-
w_{2}(\omega^{+};\hbar) w_{1}(\ell;\hbar) =0\,.$$ Factoring the error in in the classically forbidden region, using the asymptotic expansions of the Airy functions [@Olver p.392/3], and substituting all this into the quantization condition yields $$e^{{\cal S}(\omega^{+})/\hbar}\biggl(
\cos(\frac{{\cal S}(\ell)}{\hbar}-\frac{\pi}{4}) + {\cal O}(\hbar)
\biggr) + \frac{1}{2}
e^{-{\cal S}(\omega^{+})/\hbar}\biggl(
\sin(\frac{{\cal S}(\ell)}{\hbar}-\frac{\pi}{4}) + {\cal O}(\hbar)
\biggr)=0\,.$$ If this equation is satisfied, then necessarily, $ \cos(\frac{{\cal S}(\ell)}{\hbar}-\frac{\pi}{4}) = {\cal O}(\hbar)
$. This implies $
\frac{{\cal S}(\ell)}{\hbar}-\frac{\pi}{4} =
\frac{2n+1}{2}\pi + {\cal O}(\hbar)
$, or equivalently $$\int_{x_{t}}^{\ell}\sqrt{E-V(t)}\,dt =
(n+\frac{3}{4})\pi\hbar +{\cal O}(\hbar^{2})\,.$$ Now using **(H4)**, we have $$\begin{aligned}
\int_{x_{t}}^{\ell}\sqrt{E-V(t)}\,dt\,&=\,
\int_{x_{t}}^{\ell}\sqrt{E-v_{+}}\,dt +
\int_{x_{t}}^{\ell}\,(\sqrt{E-V(t)}-\sqrt{E-v_{+}})\,dt
\\
&=\,\sqrt{E-v_{+}}\,(\ell-x_{t})+
\int_{x_{t}}^{\ell}\,\frac{v_{+}-V(t)}{\sqrt{E-V(t)}+\sqrt{E-v_{+}}}\,dt
\\
&=\,\ell\,\sqrt{E-v_{+}}\,(1+{\cal O}(\ell^{-\epsilon}))\end{aligned}$$ From this, it follows that $$E=v_{+}+\left(\frac{(n+3/4)\,\pi\hbar}{\ell}\right)^{2}\left(1+
{\cal O}(\hbar) +{\cal O}(\ell^{-\epsilon})\right)\,.$$
[1]{}
S. Agmon. . Number 29 in Princeton Mathematical Notes. Princeton University Press, 1982.
J. Aguilar and J.-M. Combes. A class of analytic perturbations for one-body schr[ö]{}dinger hamiltonians. , 22:269–279, 1971.
Philippe Briet, Jean-Michel Combes, and Pierre Duclos. On the location of resonances for [Schr[ö]{}dinger]{} operators in the semiclassical limit [I]{}: [Resonance]{} free domains. , 125:90–99, 1987.
Jean-Michel Combes, Pierre Duclos, Markus Klein, and Ruedi Seiler. The shape resonance. , 110:215–236, 1987.
Jean [Dieudonné]{}. . Collection Méthodes. Hermann, Paris 1968.
V.I. Kukulin *et al*. . Reidl Texts in the Mathematical Sciences. Kluwer Academic Publishers, 1989.
Tosio Kato. . Number 132 in Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 3rd edition, 1980.
C.H. Maier, L.S. Cederbaum, and W. Domcke. A spherical-box approach to resonances. , 13:L119–L124, 1980.
F. W. J. Olver. . Computer Science and Applied Mathemetics. Academic Press, New York and London, 1974.
Michael Reed and Barry Simon. , volume IV of [*Methods of Modern Mathematical Physics*]{}. Academic Press, San Diego, 1978.
Eugene P. Wigner and von Neumann. . , 1927.
|
---
abstract: 'These lectures provide a concise introduction to the so-called “Beyond the Standard Model” physics, with particular emphasis on the problem of the microscopic origin of the Higgs mass term and of the Electro-Weak symmetry breaking scale in connection with Naturalness. The standard scenarios of Supersymmetry and Composite Higgs are shortly reviewed. An attempt is made to summarise the implications of the LHC run-$1$ results on what we expect to lie beyond (or behind) the Standard Model.'
author:
- Andrea Wulzer
title: Behind the Standard Model
---
Beyond the Standard Model; Supersymmetry; Composite Higgs; Naturalness.
BSM: What For? {#wf}
==============
Physics is the continuous effort towards a deeper understanding of the laws of Nature. The Standard Model (SM) theory summarises the state-of-the-art of this understanding, providing the correct description of all known fundamental particles and interactions (including Gravity) at the energy scales we have been capable to explore experimentally so far. “Beyond the SM” (BSM) physics aims to the next step of this understanding, namely to unveil the microscopic origin of the SM itself, of its field content, Lagrangian and parameters. From this viewpoint, the acronym “BSM” should better be read as “Behind” rather than “Beyond” the SM, from which the unconventional title (see however [@RR]) I gave to these lectures. The main focus is indeed not on new physics (beyond what predicted by the SM) per se, but on the solution of some of the mysteries associated with the microscopic theory that lie behind the SM itself. In this respect, a lack of discovery, namely a non-trivial confirmation of the SM that closes the door to BSM physics potentially associated with one of these mysteries, might be as informative as the observation of new physics.
The one described above is only one of the possible approaches to forefront research in fundamental physics. A valid alternative is to start from observations rather than from theory, in particular from those observations that cannot be accounted for by the SM, signalling the existence of new physics. What I have in mind are of course neutrino masses and oscillations and evidences of Dark Matter, Inflation and Baryogenesis. Dedicated lectures were given at this School on these topics [@pklect] [@dslect]. Even within the context of high-energy physics research, where no BSM discovery crossed our horizon yet [^1], new physics searches driven by data rather than by theory are highly desirable and complementary to the study of specific signal topologies dictated by theoretical BSM scenarios. Also, we should not discard the possibility of performing theory-unbiased new physics searches in final states that appear promising because of their simplicity, of their low SM background and/or of their experimental purity. Notice however that a fully “unbiased” approach to new physics searches is virtually impossible. A certain degree of theory bias is unavoidably needed in order to limit the infinite variety of possible channels (or of experiments) one could search in. Even the very fact that TeV-scale reactions at the LHC are promising places to look at is in itself a theory bias, though dictated by extremely general and robust BSM considerations. Theory-unbiased or theory-driven new physics searches thus just correspond to a different gradation of BSM bias we decide to apply.
No-Lose Theorems
----------------
Sometimes, the quest for the microscopic origin of known particles and interactions has extremely powerful implications, leading to absolute [*guarantees*]{} of new physics discoveries. A mathematical argument based on currently established laws of Nature, which ensures future discoveries provided the experimental conditions become favourable enough (i.e., high enough energy in the examples that follow), is what we call a “No-Lose Theorem”. Though exceptional in the long history of science, several No-Lose Theorem could be formulated (and exploited, resulting in a number of discoveries) in the context of fundamental interaction physics over the last several decades. So many No-Lose Theorem existed, and for so long, that we got used to them, somehow forgetting their importance and their absolutely exceptional nature. They deserve a review now, after the discovery of the Higgs which prevents the formulation of new No-Lose Theorems marking the end of the age of guaranteed discoveries.
The simplest No-Lose Theorem is the one that guarantees the existence of new physics beyond (and behind) the Fermi Theory of Weak interactions. To appreciate the value of this theorem we must go back to the times when the Fermi Theory was the only experimentally established, potentially “fundamental”, description of Weak interactions. At that times, our knowledge of the Weak force was entirely encapsulated in a four-fermions operator of energy dimension $d=6$, the Fermi interaction, with its $d=-2$ coefficient, the Fermi constant $G_F$.[^2] The question of whether the Fermi theory can be truly fundamental or not, and correspondingly whether or not $G_F$ can be a fundamental constant of Nature, has a very sharp negative answer, schematically summarised below\
{width="80.00000%"}
[The]{} point is that the four-fermions scattering amplitude grows with the square of the center-of-mass energy “$E$” of the reaction, a fact that trivially follows from dimensional analysis (since the amplitude is dimensionless and proportional to the $d=-2$ coupling constant $G_F$) and is intrinsically linked with the non-renormalizable nature of the Fermi Theory. But the Weak scattering amplitude becoming too large, overcoming the critical value of $16\pi^2$, means that the Weak force gets too strong to be treated as a small perturbation of the free-fields dynamics and the perturbative treatment of the theory breaks down. Of course there is nothing conceptually wrong in the Weak force entering a non-perturbative regime, the problem is that this regime cannot be described by the Fermi Theory, which is intrinsically defined in perturbation theory. Namely, the Fermi Theory does not give trustable predictions and becomes internally inconsistent as soon as the non-perturbative regime is approached. Therefore a new theory, i.e. new physics, is absolutely needed. Either in order to modify the energy behaviour of the amplitude before it reaches the non-perturbative threshold, keeping the Weak force perturbative, or to describe the new non-perturbative regime. In all cases this new, more fundamental, theory will account for the microscopic origin of the Fermi interaction and of its coupling strength $G_F$ as a low-energy effective description of the Weak force. According to the theorem, the microscopic theory must show up at an energy scale below $4\pi/\sqrt{G_F}\simeq 4\pi v$, having expressed $G_F=1/\sqrt{2}v^2$ in terms of the ElectroWeak Symmetry Breaking (EWSB) scale $v\simeq246$ GeV. We now know that the new physics beyond the Fermi Theory is the Intermediate Vector Boson (IVB) theory, which was confirmed by discovering the $W$ boson at the scale $m_W\simeq80$ GeV, far below $4\pi v$ compatibly with the theorem.
As everyone knows, well before the discovery of the $W$ discovery we already had strong indirect indications on the validity of the IVB theory and a rather precise estimate of the $W$ boson mass. These indications came from fortunate theoretical speculations and from the measurement of the Weak angle through neutrino scattering processes, and are completely unrelated with the No-Lose Theorem outlined above. Indeed, the theorem makes no assumption on, and gives no indication about, the details of the microscopic physics that lies behind the Fermi Theory. Namely, the theorem guarantees that something would have been discovered in fermion-fermion scattering, possibly not the $W$ and possibly not at a scale as low as $m_W$, even if all the theoretical speculations about the IVB theory had turned out to be radically wrong. This means in particular that if the UA$1$ and UA$2$ experiments at the CERN SPS collider had not discovered the $W$, we would have for sure continued searching for it, or for whatever new physics lies behind the Fermi theory, by the construction of higher energy machines.
A situation like the one described above was indeed encountered in the search for the top quark, which according to a widespread belief was expected to be much lighter than $m_t\simeq 173$ GeV, where it was eventually observed. Consequently, the top discovery was expected at several lower-energy colliders, constructed before the Tevatron, which instead produced a number of negative results. However we never got discouraged and we never even considered the possibility of giving up searching for the top quark, or for some other new physics related with the bottom quark, because of a second No-Loose Theorem:\
{width="80.00000%"}
[The]{} theorem relies on the validity of the IVB theory and on the existence of the bottom quark with its neutral current interactions, which we consider here as experimentally established facts at the times when the top was not yet found. The observation is that the amplitude for longitudinally polarised $W$ bosons production from a $b$ $\overline{b}$ pair grows quadratically with the energy if the top quark is absent or if it is too heavy to be relevant. It is indeed the t-channel contribution from the top exchange that makes the amplitude constant at high energies in the complete SM. Perturbativity thus requires new physics at a scale below $4\pi m_W/g_W\simeq 4\pi v$, having used the relation $m_W=g_W v/2$. When interpreted in the SM, the upper bound on the new physics scale translates in the familiar perturbativity bound on the top mass, however the Theorem does not rely on the SM and on the existence of the top quark. It states that the top, or something else, must exist beyond the bottom quark in order to moderate the growth with the energy of the scattering amplitude. More physically, the Theorem says that the microscopic origin of the bottom quark (e.g., the fact that its left-handed component lives in a doublet together with the top) must reveal itself below $4\pi v$.
Another particle whose discovery was significantly “delayed” with respect to the expectations is the Higgs boson, which also comes with its own No-Loose Theorem:\
{width="80.00000%"}
[The]{} growth with the energy of the longitudinally polarised $W$ bosons scattering amplitude in the IVB theory requires the presence of new particles and/or interactions, once again below the critical threshold of $4\pi v\sim3$ TeV. Given that the TeV scale is within the reach of the LHC collider, the Theorem above offered absolute guarantee of new physics discoveries at the LHC and was heavily used to motivate its construction. Now the Higgs has been found, with couplings compatible with the SM expectations, we know that it is indeed the Higgs particle the agent responsible for cancelling (at least partially, given the limited accuracy of the Higgs couplings measurements) the quadratic term in the scattering amplitude. This leaves us, as I will better explain below, with no No-Loose Theorem and thus with no guaranteed discovery to organise our future efforts in the investigation of fundamental interactions.
Each of the No-Lose theorems discussed above emerges because of the anomalous power-like growth with the energy of some scattering amplitude, a behaviour which unmistakably signals that a non-renormalizable interaction operator of energy dimension $d>4$ is present in the theory. This being the case is completely obvious for the Fermi theory, a bit less so in the two other examples. In the latter cases it requires, to be understood, somewhat technical considerations related with the Goldstone boson Equivalence Theorem [@Horejsi:1995jj] which go beyond the purpose of the present lectures. It suffices here to say that one given $d=6$ non-renormalizable operator, responsible for the $E^2$ growth of the scattering amplitude, can be identified for each of the $3$ No-Lose theorems above. When each theorem was “exploited” by discovering the associated new physics we “got rid” of the corresponding operator by replacing it with a more fundamental theory that explains its origin as a low-energy effective description. Having exploited all the theorems, we got rid of all the non-renormalizable operators and we are left, for the first time, with an experimentally verified renormalizable theory of electroweak and strong interactions. No new No-Lose theorems can be thus formulated in this theory, at least not as simple and powerful ones as the ones listed above.
![The RG running of the most relevant couplings of the SM, namely the three gauge couplings $g_{1,2,3}$, the top and bottom Yukawa’s $y_{t,b}$ and the Higgs quartic coupling $\lambda$. See Ref. [@Buttazzo:2013uya] and references therein for more details. \[SM\_RG\]](Figures/SM_RG){width="40.00000%"}
However the SM is not only a theory of electroweak and strong interactions. It can be (and it must be, to account for observations) extended to incorporate Gravity and the only sensible way to do so is by introducing and quantising the Einstein-Hilbert action. This produces a number of non-renormalizable interaction operators involving gravitons, giving rise to another well-known No-Lose theorem\
{width="80.00000%"}
[where]{} $M_{\textrm{P}}\simeq10^{19}$ GeV is the Planck scale. What the theorem says is that the SM is for sure not the “final theory” of Nature, because it does not provide a complete description of Gravity at the quantum level. It does incorporate a description of quantum gravity that is valid and predictive at low energy but breaks down at a finite scale $\Lambda_{\textrm{{SM}}}$, which we call the “SM cutoff”. BSM particles and interactions are present at that scale, which however can be as high as $10^{19}$ GeV. Given our technical inability to test such an enormous scale, it is unlikely that we might ever exploit this last No-Lose Theorem as a guide towards a concrete new physics discovery.
The second aspect to be discussed is that even in a renormalizable theory the scattering amplitudes can actually grow with the energy. Not with a power-law, but logarithmically, through the Renormalisation Group (RG) running of the dimensionless coupling constants of the theory. The RG evolution can make some of the couplings grow with the energy until they violate the perturbativity bound, producing a new No-Lose Theorem. Obviously this No-Lose Theorem would most likely be not as powerful as those obtainable in non-renormalizable theories because the RG evolution is logarithmically slow and thus the perturbativity violation scale is exponentially high, but still it is interesting to ask if one such a theorem exists for the SM and at which scale it points to. The answer is that perturbativity violation does not occur in the SM below the Planck mass scale, at which new physics is anyhow needed to account for gravity, as shown in fig. \[SM\_RG\]. The only coupling that grows significantly with the energy is the one associated with the gauge group, $g_1$, which however is still well below the perturbativity bound at the Planck scale. Notice that the result crucially depends on the initial conditions of the running, namely on the values of the SM parameters measured at the $100$ GeV scale. The result would have been different, and an additional No-Lose Theorem would have been produced, if that values were radically different than what we actually observed.
The vacuum stability problem [@Krive:1976sg] is yet another potential source of high-energy inconsistencies (and thus of No-Lose Theorems) in renormalizable theories that display, like the SM, a non-trivial structure of the vacuum state. The problem is again due to RG evolution effects, which modify the form of the Higgs potential at very high values of the Higgs field and potentially make it develop a second minimum. If the energy of this second minimum is lower than the first one, transitions can occur via quantum tunnelling from the ordinary EWSB vacuum where $v\simeq246$ GeV to an inhospitable minimum characterised by a very large vacuum expectation value (VEV) of the Higgs field. Whether this actually happens or not depends, once again, on the measured value of the SM parameters and in particular on the Higgs boson and top quark masses as displayed in fig. \[STAB\]. We see that our vacuum is not stable and thus it is fated to decay provided we wait long enough. However it falls in the “meta-stability” region of the diagram, which is where the vacuum lifetime is longer than the age of the Universe. Therefore the decay of our vacuum might not have had enough time to occur. Some people find disturbing that we live in a meta-stable vacuum. Some others [@Buttazzo:2013uya] find intriguing the fact that we live close (see the right panel of fig. \[STAB\]) to the boundary between the stability and meta-stability regions and suggest that we should measure $m_t$ better in order to be sure of how close we actually are. Anyhow what is sure (and what matters for our discussion) is that the analysis of the vacuum stability does not reveal any concrete inconsistency of the SM at high energy. Consequently, no new No-Lose Theorem is found.
The “SM-only” Option {#SMONLY}
--------------------
![Stability, instability, metastability and non-perturbativity regions for the SM in the plane of the Higgs and top masses. A zoom on the experimentally viable region is displayed in the right plot, with the $1$, $2$ and $3\sigma$ regions allowed by $m_H$ and $m_t$ uncertainties. From Ref. [@Buttazzo:2013uya]. \[STAB\]](Figures/STAB){width="80.00000%"}
Two extremely important (and in some sense contradictory) facts emerge from the previous considerations. On one hand, we know that BSM physics exists at a finite energy scale $\Lambda_{\textrm{SM}}$. This makes that the SM is necessarily an approximate low-energy description of a more fundamental theory, i.e. an Effective Field Theory (EFT) with a finite cutoff $\Lambda_{\textrm{SM}}$. On the other hand, the only upper bound on the cutoff scale is provided by the Planck mass, which is to a very good approximation equal to infinity compared with the much lower scales we are able to explore experimentally today and in any foreseeable future. We are thus led to consider the “SM-only” option for high-energy physics. Namely the possibility that the SM cutoff $\Lambda_{\textrm{SM}}$ (i.e., the scale of new physics) is extremely high, much above the TeV as depicted in fig. \[SMEFT\]. Values as high as $\Lambda_{\textrm{SM}}\sim M_{\textrm{P}}$ and $\Lambda_{\textrm{SM}}\sim10^{15}$ GeV $\equiv{M}_{\textrm{GUT}}$ can be considered.
The SM-only option is not just a logical possibility. On the contrary, it is a predictive and phenomenologically successful scenario for high-energy physics. To appreciate its value, we look again at fig. \[SMEFT\], starting from the high energy (UV) region and we ask ourselves how the SM theory emerges in the IR. As pictorially represented in the figure, we have no idea of how the theory in the UV looks like. It might be a string theory, a GUT model (for a review, see for instance Refs. [@Langacker:1980js; @Raby:2006sk]), or something completely different we have not yet thought about. All what we know about the UV theory is that, by assumption, its particle content reduces to the one of the SM at $\Lambda_{\textrm{SM}}$, all BSM particles being at or above that scale.[^3] Below $\Lambda_{\textrm{SM}}$ the UV theory thus necessarily reduces, after integrating out the heavy states, to a low-energy EFT which only describes the light SM degrees of freedom. A technically consistent description of the force carriers (gluon and EW bosons) requires invariance under the gauge group, but apart from being gauge (and Lorentz) invariant there is not much we can tell a priori on how the SM effective Lagrangian will look like. It will consist of an infinite series of local gauge- and Lorentz-invariant operators with arbitrary energy dimension “$d$”, constructed with the SM Matter, Gauge and Higgs fields as in fig. \[SMEFT\]. The coefficient of the operators must be proportional to $1/\Lambda_{\textrm{SM}}^{d-4}$ by dimensional analysis, given that $[{\mathcal{L}}]=E^4$ and $\Lambda_{\textrm{SM}}$ is the only relevant scale. This simple observation lies at the heart of the phenomenological virtues of the SM-only scenario but also, as we will see, of its main limitation.
![Pictorial view of the SM as an effective field theory, with its Lagrangian generated at the scale $\Lambda_{\textrm{SM}}$.[]{data-label="SMEFT"}](Figures/SM_as_EFT.png){width="75.00000%"}
We now classify the SM effective operators by their energy dimension and discuss their implications, starting from those with $d=4$. They describe almost all what we have seen in Nature, namely EW and strong interactions, quarks and charged leptons masses. They define a renormalizable theory and thus, together with the $d=2$ operator we will introduce later, they are present in the textbook SM Lagrangian formulated in the old times when renormalizability was taken as a fundamental principle.
Several books have been written (see for instance Refs. [@OKUN; @Cheng:1985bj; @Schwartz:2013pla]) on the extraordinary phenomenological success of the renormalizable SM Lagrangian in describing the enormous set of experimental data [@Agashe:2014kda] collected in the past decades. In a nutshell, as emphasized in Ref. [@10lectures], most of this success is due to symmetries, namely to “accidental” symmetries. We call “accidental” a symmetry that arises by accident at a given order in the operator classification, without being imposed as a principle in the construction of the theory. The renormalizable ($d\leq4$) SM Lagrangian enjoys exact (or perturbatively exact) accidental symmetries, namely baryon and lepton family number, and approximate ones such as the flavour group and custodial symmetry. For brevity, we focus here on the former symmetries, which have the most striking implications. Baryon number makes the proton absolutely stable, in accordance with the experimental limit $\Gamma_p/m_p\lesssim10^{-64}$ on the proton width over mass ratio. It is hard to imagine how we could have accounted for the proton being such a narrow resonance in the absence of a symmetry. Similarly lepton family number forbids exotic lepton decays such as $\mu\rightarrow e\gamma$, whose branching ratio is experimentally bounded at the $10^{-12}$ level. From neutrino oscillations we know that the lepton family number is actually violated, in a way that however nicely fits in the SM picture as we will see below. Clearly this is connected with the neutrino masses, which exactly vanish at $d=4$ because of the absence, in what we call here “the SM”, of right-handed neutrino fields.
We now turn to non-renormalizable operators with $d>4$. Their coefficient is proportional to $1/\Lambda_{\textrm{SM}}^n$, with $n=d-4>0$, thus their contribution to low-energy observables is suppressed by $(E/\Lambda_{\textrm{SM}})^n$ with respect to renormalizable terms. Given that current observations are at and below the EW scale, $E\lesssim m_{\textrm{EW}}\simeq100$ GeV, their effect is extremely suppressed in the SM-only scenario where $\Lambda_{\textrm{SM}}\gg$ TeV. This could be the reason why Nature is so well described by a renormalizable theory, without renormalizability being a principle.
Non-renormalizable operators violate the $d=4$ accidental symmetries. Lepton number stops being accidental already at $d=5$ because of the Weinberg operator [@Weinberg:1979sa] \[WOP\] (\_L H\^c)(\_L\^c H\^c), where $\ell_L$ denotes the lepton doublet, $\ell_L^c$ its charge conjugate, while $H$ is the Higgs doublet and $H^c = i \sigma^2 H^*$. The indices are contracted within the parentheses and the spinor index between the two terms. A generic lepton flavour structure of the coefficient, leading to the breaking of lepton family number, is understood. Surprisingly enough, the Weinberg operator is the unique $d=5$ term in the SM Lagrangian. When the Higgs is set to its VEV, the Weinberg operator reduces to a Majorana mass term for the neutrinos, $m_\nu\sim{c}\,v^2/\Lambda_{\textrm{SM}}$. For $\Lambda_{\textrm{SM}}\simeq10^{14}$ GeV and order one coefficient “$c$” it generates neutrino masses of the correct magnitude ($m_\nu\sim0.1$ eV) and neutrino mixings that can perfectly account for all observed neutrino oscillation phenomena. Baryon number is instead still accidental at $d=5$ and its violation is postponed to $d=6$. We thus perfectly understand, qualitatively, why lepton family violation effects are “larger”, thus easier to discover, while baryon number violation like proton decay is still unobserved. At a more quantitative level we should actually remark that the bounds on proton decay from the $d=6$ operators, with order one numerical coefficients, set a limit $\Lambda_{\textrm{SM}}\gtrsim10^{15}$ GeV that is in slight tension with what required by neutrino masses. However few orders of magnitude are not a concern here, given that there is no reason why the operator coefficient should be of order one. A suppression of the proton decay operators is actually even expected because they involve the first family quarks and leptons, whose couplings are reduced already at the renormalizable level. Namely, it is plausible that the same mechanism that makes the first-family Yukawa couplings small also reduces proton decay, while less suppression is expected in the third family entries of the Weinberg operator coefficient that might drive the generation of the heaviest neutrino mass.
The considerations above make the SM-only option a plausible picture, which becomes particularly appealing if we set $\Lambda_{\textrm{SM}}\sim{M}_{\textrm{GUT}}$. This choice happens to coincide with the gauge coupling unification scale, but this doesn’t mean that the new physics at the cutoff is necessarily a Grand Unified Theory. On the contrary, the physics at the cutoff can be very generic in this picture, the compatibility with low-energy observations being ensured by the large value of the $\Lambda_{\textrm{SM}}$ scale and not by the details of the UV theory. New physics is virtually impossible to discover directly in this scenario, but this doesn’t make it completely untestable. Purely Majorana neutrino masses would be a strong indication of its validity while observing a large Dirac component would make it less appealing.
Having discussed the virtues of the SM-only scenario, we turn now to its limitations. One of those, which was already mentioned, is the hierarchy among the Yukawa couplings of the various quark and lepton flavours, which span few orders of magnitude. This tells us that the new physics at $\Lambda_{\textrm{SM}}$ cannot actually be completely generic, given that it must be capable of generating such a hierarchy in its prediction for the Yukawa’s. This limits the set of theories allowed at the cutoff but is definitely not a strong constraint. Whatever mechanism we might imagine to generate flavour hierarchies at $\Lambda_{\textrm{SM}}\sim M_{\textrm{GUT}}$, it will typically not be in contrast with observations given that the bounds on generic flavour-violating operators are “just” at the $10^{8}$ GeV scale. Incorporating dark matter also requires some modification of the SM-only picture, but there are several ways in which this could be done without changing the situation dramatically. Perhaps the most appealing solution from this viewpoint is “minimal dark matter” [@Cirelli:2005uq], a theory in which all the symmetries that are needed for phenomenological consistence are accidental. This includes not only the SM accidental symmetries, but also the additional $\Zdouble_2$ symmetry needed to keep the dark matter particle cosmologically stable. Similar considerations hold for the strong CP problem, for inflation and all other cosmological shortcomings of the SM. The latter could be addressed by light and extremely weakly-coupled new particles or by very heavy ones above the cutoff. In conclusion, none of the above-mentioned issues is powerful enough to put the basic idea of very heavy new physics scale in troubles. The only one that is capable to do so is the Naturalness (or Hierarchy) problem discussed below.[^4]
We have not yet encountered the Naturalness problem in our discussion merely because we voluntarily ignored, in our classification, the operators with $d<4$. The only such operator in the SM is the Higgs mass term, with $d=2$.[^5] When studying the $d>4$ operators we concluded that their coefficient is suppressed by $1/\Lambda_{\textrm{SM}}^{d-4}$. Now we have $d=2$ and we are obliged to conclude that the operator is [*enhanced*]{} by $\Lambda_{\textrm{SM}}^2$, [*[i.e.]{}*]{} that the Higgs mass term reads \[mhUV\] c\_\^2 H\^H, with “$c$” a numerical coefficient. In the SM the Higgs mass term sets the scale of EWSB and it directly controls the Higgs boson mass. Today we know that $m_H=125$ GeV and thus the mass term is $\mu^2=m_H^2/2=(89\,{\textrm{GeV}})^2$. But if $\Lambda_{\textrm{SM}}\sim M_{\textrm{GUT}}$, what is the reason for this enormous hierarchy? Namely \~10\^[-28]{}1 This is the essence of the Naturalness problem.
Further considerations on the Naturalness problem and implications are postponed to the next section. However, we can already appreciate here how radically it changes our expectations on high energy physics. The SM-only picture gets sharply contradicted by the Naturalness argument since the problem is based on the same logic ([*[i.e.]{}*]{}, dimensional analysis) by which its phenomenological virtues ([*[i.e.]{}*]{}, the suppression of $d>4$ operators) were established. The new picture is that $\Lambda_{\textrm{SM}}$ is low, in the $100$ GeV to few TeV range, such that a light enough Higgs is obtained “Naturally”, [*[i.e.]{}*]{} in accordance with the estimate in eq. (\[mhUV\]). The new physics at the cutoff must now be highly non-generic, given that it cannot rely any longer on a large scale suppression of the BSM effects. To start with, baryon and lepton family number violating operators must come with a highly suppressed coefficient, which in turn requires baryon and lepton number being imposed as symmetries rather than emerging by accident. In concrete, the BSM sector must now respect these symmetries. This can occur either because it inherits them from an even more fundamental theory or because they are accidental in the BSM theory itself. Similarly, if $\Lambda_{\textrm{SM}}\sim$ TeV flavour violation cannot be generic. Some special structure must be advocated on the BSM theory, Minimal flavour Violation (MFV) [@Glashow:1976nt; @D'Ambrosio:2002ex] being one popular and plausible option. The limits from EW Precision Tests (EWPT) come next; they also need to be carefully addressed for TeV scale new physics. On one hand this makes Natural new physics at the TeV scale very constrained. On the other hand it gives us plenty of indications on how it should, or it should not, look like.
The Naturalness Argument {#natarg}
------------------------
The reader might be unsatisfied with the formulation of the Naturalness problem we gave so far. All what eq. (\[mhUV\]) tells us is that the numerical coefficient “$c$” that controls the actual value of the mass term beyond dimensional analysis should be extremely small, namely $c\sim10^{-28}$ for GUT scale new physics. Rather than pushing $\Lambda_{\textrm{SM}}$ down to the TeV scale, where all the above-mentioned constraints apply, one could consider keeping $\Lambda_{\textrm{SM}}$ high and try to invent some mechanism to explain why $c$ is small. After all, we saw that there are other coefficients that require a suppression in the SM Lagrangian, namely the light flavours Yukawa couplings. One might argue that it is hard to find a sensible theory where $c$ is small, while this is much simpler for the Yukawa’s. Or that $28$ orders of magnitude are by far much more than the reduction needed in the Yukawa sector. But this would not be fully convincing and would not make full justice to the importance of the Naturalness problem.
In order to better understand Naturalness we go back to the essential message of the previous section. The SM is a low-energy effective field theory and thus the coefficients of its operators, which we regard today as fundamental input parameters, should actually be derived phenomenological parameters, to be computed one day in a more fundamental BSM theory. Things should work just like for the Fermi theory of weak interactions, where the Fermi constant $G_F$ is a fundamental input parameter that sets the strength of the weak force. We know however that the true microscopic description of the weak interactions is the IVB theory. The reason why we are sure about this is that it allows us to predict $G_F$ in terms of its microscopic parameters $g_W$ and $m_W$, in a way that agrees with the low-energy determination. What we have in mind here is merely the standard textbook formula G\_F=, that allows us to carry on, operatively, the following program. Measure the microscopic parameters $g_W$ and $m_W$ at high energy; compute $G_F$; compare it with low-energy observations.[^6] Since this program succeeds we can claim that the microscopic origin of weak interaction is well-understood in terms of the IVB theory. We will now see that the Naturalness problem is an obstruction to repeating the same program for the Higgs mass and in turn for the EWSB scale.
Imagine knowing the fundamental, “true” theory of EWSB. It will predict the Higgs mass term $\mu^2$ or, which is the same, the physical Higgs mass $m_H^2=2\mu^2$, in terms of its own input parameters “$p_{\textrm{true}}$”, by a formula that in full generality reads \[mHtrue\] m\_H\^2=\_0\^ dE(E;p\_). The integral over energy stands for the contributions to $m_H^2$ from all the energy scales and it extends up to infinity, or up to the very high cutoff of the “true” theory itself. The integrand could be localized around some specific scale or even sharply localized by a delta-function at the mass of some specific particle, corresponding to a tree-level contribution to $m_H^2$. Examples of theories with tree-level contributions are GUT [@Langacker:1980js; @Raby:2006sk] and Supersymmetric (SUSY) models, where $m_H$ emerges from the mass terms of extended scalar sectors. The formula straightforwardly takes into account radiative contributions, which are the only ones present in the composite Higgs scenario (see sect. \[CH\]). Also in SUSY, as discussed in sect. \[SUSY\], radiative terms have a significant impact given that the bounds on the scalar (SUSY and soft) masses that contribute at the tree-level are much milder than those on the coloured stops and gluinos that contribute radiatively. In the language of old-fashioned perturbation theory [@Weinberg:1995mt], “$E$” should be regarded as the energy of the virtual particles that run into the diagrams through which $m_H^2$ is computed.
![Some representative top, gauge and Higgs boson loop diagrams that contribute to the Higgs mass.[]{data-label="HP"}](Figures/HP.png){width="100.00000%"}
Consider now splitting the integral in two regions defined by an intermediate scale that we take just a bit below the SM cutoff. We have \[splitint\] m\_H\^2&&=\_0\^ dE(E;p\_)+\_\^ dE(E;p\_)\
&&=\_m\_H\^2+\_m\_H\^2, where $\delta_{\textrm{BSM}}m_H^2$ is a completely unknown contribution, resulting from energies at and above $\Lambda_{\textrm{SM}}$, while $\delta_{\textrm{SM}}m_H^2$ comes from virtual quanta below the cutoff, whose dynamics is by assumption well described by the SM. While there is nothing we can tell about $\delta_{\textrm{BSM}}m_H^2$ before we know what the BSM theory is, we can easily estimate $\delta_{\textrm{SM}}m_H^2$ by the diagrams in Figure \[HP\], obtaining \[deltamh\] \_m\_H\^2= \_\^2 - (14+1[8\^2\_W]{})\_\^2 -\_\^2, from, respectively, the top quark, EW bosons and Higgs loops. The idea is that we know that the BSM theory must reduce to the SM for $E<\Lambda_{\textrm{SM}}$. Therefore no matter what the physics at $\Lambda_{\textrm{SM}}$ is, its prediction for $m_H^2$ must contain the diagrams in fig \[HP\] and thus the terms in eq. (\[deltamh\]). These terms are obtained by computing $dm_H^2/dE$ from the SM diagrams and integrating it up to $\Lambda_{\textrm{SM}}$, which effectively acts as a hard momentum cutoff. The most relevant contributions come from the quadratic divergences of the diagrams, thus eq. (\[deltamh\]) can be poorly viewed as the “calculation” of quadratic divergences. Obviously quadratic divergences are unphysical in quantum field theory. They are canceled by renormalization and they are even absent in certain regularizations schemes such as dimensional regularization. However the calculation makes sense, in the spirit above, as an estimate of the low-energy contributions to $m_H^2$.
The true nature of the Naturalness problem starts now to show up. The full finite formula for $m_H^2$ obtained in the “true” theory receives two contributions that are completely unrelated since they emerge from separate energy scales. At least one of those, $\delta_{\textrm{SM}}m_H^2$, is for sure very large if $\Lambda_{\textrm{SM}}$ is large. The other one is thus obliged to be large as well, almost equal and with opposite sign in order to reproduce the light Higgs mass we observe. A cancellation is taking place between the two terms, which we quantify by a fine-tuning $\Delta$ of at least \[deltatuning\] =()\^2()\^2. Only the top loop term in eq. (\[deltamh\]) has been retained for the estimate since the top dominates because of its large Yukawa coupling and because of color multiplicity. Notice that the one above is just a lower bound on the total amount of cancellation $\Delta$ needed to adjust $m_H$ in the true theory. The high energy contribution $\delta_{\textrm{BSM}}m_H^2$, on which we have no control, might itself be the result of a cancellation, needed to arrange for $\delta_{\textrm{BSM}}m_H^2\simeq -\delta_{\textrm{SM}}m_H^2$. Examples of this situation exist both in SUSY and in composite Higgs.
The problem is now clear. Even if we were able to write down a theory that formally predicts the Higgs mass, and even if this theory turned out to be correct we will never be able to really predict $m_H$ if $\Lambda_{\textrm{SM}}$ is much above the TeV scale, because of the cancellation. For $\Lambda_{\textrm{SM}} = M_{\textrm{GUT}}$, for instance, we have $\Delta\gtrsim10^{24}$. This means that in the “true” theory formula for $m_H$ a $24$ digits cancellation is taking place between two a priori unrelated terms. Each of these terms must thus be known with at least $24$ digits accuracy even if we content ourselves with an order one estimate of $m_H$. We will never achieve such an accuracy, neither in the experimental determination of the $p_{\textrm{true}}$ “true” theory parameters $m_H$ depends on, nor in the theoretical calculation of the Higgs mass formula. Therefore, we will never be able to repeat for $m_H$ the program we carried on for $G_F$ and we will never be able to claim we understand its microscopic origin and in turn the microscopic origin of the EWSB scale. A BSM theory with $\Lambda_{\textrm{SM}} = M_{\textrm{GUT}}$ has, in practice, the same predictive power on $m_H$ as the SM itself, where eq. (\[mHtrue\]) is replaced by the much simpler formula m\_H\^2=m\_H\^2. Namely if such an high-scale BSM theory was realized in Nature $m_H$ will remain forever an input parameter like in the SM. The microscopic origin of $m_H$, if any, must necessarily come from new physics at the TeV scale, for which the fine-tuning $\Delta$ in eq. (\[deltatuning\]) can be reasonably small.
The Higgs mass term is the only parameter of the SM for which such an argument can be made. Consider for instance writing down the analog of eq. (\[mHtrue\]) for the Yukawa couplings and splitting the integral as in eq. (\[splitint\]). The SM contribution to the Yukawa’s is small even for $\Lambda_{\textrm{SM}} = M_{\textrm{GUT}}$, because of two reasons. First, the Yukawa’s are dimensionless and thus, given that there are no couplings in the SM with negative energy dimension, they do not receive quadratically divergent contributions. The quadratic divergence is replaced by a logarithmic one, with a much milder dependence on $\Lambda_{\textrm{SM}}$. Second, the Yukawa’s break the flavour group of the SM. Therefore there exist selection rules (namely those of MFV) that make radiative corrections proportional to the Yukawa matrix itself. The Yukawa’s, and the hierarchies among them, are thus “radiatively stable” in the SM (see sect. \[tale\] for more details). This marks the essential difference with the Higgs mass term and implies that their microscopic origin and the prediction of their values could come at any scale, even at a very high one. The same holds for all the SM parameters apart from $m_H$.
The formulation in terms of fine-tuning (\[deltatuning\]) turns the Naturalness problem from a vague aesthetic issue to a concrete semiquantitative question. Depending on the actual value of $\Delta$ the Higgs mass can be operatively harder or easier to predict, making the problem more or less severe. If for instance $\Delta\sim10$, we will not have much troubles in overcoming a one digit cancellation once we will know and we will have experimental access to the “true” theory. After some work, sufficiently accurate predictions and measurements will become available and the program of predicting $m_H$ will succeed. The occurrence of a one digit cancellation will at most be reported as a curiosity in next generation particle physics books and we will eventually forget about it. A larger tuning $\Delta=1000$ will instead be impossible to overcome. The experimental exploration of the high energy frontier will tell us, through eq. (\[deltatuning\]), what to expect about $\Delta$. Either by discovering new physics that addresses the Naturalness problem or by pushing $\Lambda_{\textrm{SM}}$ higher and higher until no hope is left to understand the origin of the EWSB scale in the sense specified above. One way or another, a fundamental result will be obtained.
What if Un-Natural?
-------------------
I argued above that searching for Naturalness at the LHC is relevant regardless of the actual outcome of the experiment. Such a bold statement needs to be more extensively defended. The case of a discovery is so easy that it would not even be worth discussing. If new particles are found at the TeV scale, with properties that resemble what predicted by a Natural BSM theory such as the ones described in the following sections, Naturalness would have guided us towards the discovery of new physics. Moreover, it will provide the theoretical framework for the interpretation of the discoveries, by which the new particles will eventually find their place in a concrete BSM model. If instead nothing related with Naturalness will be found, strong limits will be set on $\Delta$ and we will be pushed towards the idea that the $m_H^2$ parameter does not have a canonical “microscopic” origin as previously explained. This would still qualify as a discovery: the discovery of “Un-Naturalness”.[^7] The profound implications of this potential discovery are discussed below.
If Un-Naturalness will be discovered, other options will have to be considered to explain the origin of the Higgs mass term. The two known possibilities are that $m_H^2$ has an “environmental” or a “dynamical” origin rather than a “microscopic” one, as previously assumed. A well-known parameter with environmental origin is the Gravity of Earth $g=9.8\,m/s^2$. It is the input parameter of Ballistics, a theory of great historical relevance which in Galileo’s times might have been conceivably thought to be a fundamental theory of Nature. The origin of $g$ is obviously dictated by the environment in which the theory is formulated, namely by the fact that Ballistics applies to processes that occur close to the surface of Earth. Its value depends on the Earth’s mass and radius and it cannot be inferred just based on the knowledge of the “truly fundamental” theory of Gravity (Newton’s law) and of its parameters (Newton’s constant). This is not the case for those parameters, such as $G_F$, with a purely microscopic origin. The dependence on the environment can help explaining the size of an environmental parameter by the so-called “Anthropic” argument. In fact, the value of $g=9.8\,m/s^2$ is rather peculiar. It is much larger than the one we would observe in interstellar space and much smaller than the one on the surface of a neutron star, very much like $m_H$ is much smaller than $M_{\textrm{P}}$ or $M_{\textrm{GUT}}$. However we do perfectly understand the magnitude of $g$, for the very simple reason that no ancient physicist might have lived in empty space or on a neutron star. The magnitude of $g$ must be compatible with what is needed for the development of intelligent life, otherwise no physicist would have existed and nobody would have measured it.
The Weinberg prediction of the cosmological constant [@Weinberg:1987dv] proceeds along similar lines. The cosmological constant operator suffers of exactly the same Naturalness problem as the Higgs mass. Provided we claim we understand gravity well enough to estimate them, radiative corrections push the cosmological constant to very high values, tens of orders of magnitude above what we knew it had to be (and was subsequently observed) in order for galaxies being able to form in the early universe. Weinberg pointed out that the most plausible value for the cosmological constant should thus be close to the maximal allowed value for the formation of galaxies because galaxies are essential for the development of intelligent life. The idea is that if many ground state configurations (a landscape of vacua) are possible in the fundamental theory, typically characterised by a very large cosmological constant but with a tail in the distribution that extends up to zero, the largest possible value compatible with galaxies formation, and thus with the very existence of the observer, will be actually observed. A similar argument can be made for the Higgs mass (see for instance Ref. [@Hall:2007ja]), however it is harder in the SM to identify sharply the boundary of the anthropically allowed region of the parameter space.
I tried here to vulgarise the mechanism of anthropic vacua selection by the example of Gravity of Earth, however the analogy is imperfect under several respects. Perhaps the most important difference is that the landscape of vacua cannot be viewed as a set of physical regions (like the interstellar space or the neutron star) separated in space, where $m_H$ or the cosmological constant assume different values. Or at least, since the other vacua live in space-time regions that are causally disconnected from us, it will be impossible to have access to them and check directly that the mechanism works.
The possibility of a “dynamical” origin of the Higgs mass term is quite new [@Graham:2015cka] and not much studied.[^8] The idea, first proposed in [@Abbott:1984qf] as an unsuccessful attempt to solve the cosmological constant problem, is that $m_H$ might be set by the expectation value of a new scalar field, whose value evolves during cosmological Inflation. This field is called “relaxion” in [@Graham:2015cka] because it is similar to the QCD axion needed to address the strong-CP problem and because it sets the value of $m_H$ by a dynamical relaxation mechanism. At the beginning of Inflation, the relaxion VEV is such that the Higgs mass term is large and positive, but it evolves in the course of time making the Higgs mass term decrease and eventually cross zero so that EWSB can take place. The structure of the theory is such that once a non-vanishing Higgs VEV is generated, a barrier develops in the relaxion potential and makes it stop evolving. The Higgs mass term gets thus frozen to the value which is just sufficient for an high enough barrier to form. If the theory is special enough (but not necessarily complicate), this value can be small and the Hierarchy problem can be solved.
You might find these speculations extremely interesting. Or you might believe that they have no chance to be true. Anyhow, their vey existence demonstrates how radically the discovery of Un-Naturalness would change our perspective on the physics of fundamental interactions. They show the capital importance of searching for Naturalness or Un-Naturalness at the LHC and, perhaps, at future colliders.
Composite Higgs {#CH}
===============
One aspect of the Naturalness problem which has not yet emerged is the fact that addressing it requires BSM physics of rather specific nature at $\Lambda_{\textrm{{SM}}}\lesssim$ TeV. Namely, it is true that any BSM scenario that Naturally explains the origin of $m_H$ is obliged to show up at the TeV by eq. (\[deltatuning\]), but this does not mean that the presence of generic new particles at the TeV scale would solve the Naturalness problem. Conversely, it is not true that any BSM particle we might happen not to discover at the TeV scale would signal that the theory is fine-tuned as a naive application of eq. (\[deltatuning\]) would suggest. Natural BSM physics would show up through new particles (and/or, indirect effects on SM processes) of specific nature and it is only the non-discovery of these particles the one that matters for the tuning $\Delta$. Addressing this point requires studying concrete BSM solutions to the Naturalness problem.
Among the various scenarios which have been proposed to address the Naturalness problem I decided to focus on two of them: Supersymmetry and Composite Higgs. The reason for this choice is that they are representative of the only two known mechanisms which truly address the problem of the microscopic origin of $m_H$ by a well-defined high-energy picture. Alternative Natural models are often reformulations or deformations of these basic scenarios, or a combination of the two.[^9] You are referred to Ref. [@Pomarol:2012sb] for a comprehensive overview.
The Basic Idea {#TBI}
--------------
![\[CH\_drawing\]Pictorial representation of the Composite Higgs solution to the Naturalness problem.](Figures/CH_drawing){width="70.00000%"}
The composite Higgs scenario offers a simple solution to the problem of Naturalness. Suppose that the Higgs, rather than being a point-like particle as in the SM, is instead an extended object with a finite geometric size $l_H$. We will make it so by assuming that it is the bound state of a new strong force characterised by a confinement scale $m_*=1/l_H$ of TeV order. In this new theory the $dm_H^2/dE$ integrand in the Higgs mass formula (\[mHtrue\]), which stands for the contribution of virtual quanta with a given energy, behaves as shown in fig. \[CH\_drawing\]. Low energy quanta have too a large wavelength to resolve the Higgs size $l_H$. Therefore the Higgs behaves like an elementary particle and the integrand grows linearly with $E$ like in the SM, resulting in a quadratic sensitivity to the upper integration limit. However this growth gets canceled by the finite size effects that start becoming visible when $E$ approaches and eventually overcomes $m_*$. Exactly like the proton when hit by a virtual photon of wavelength below the proton radius, the composite Higgs is transparent to high-energy quanta and the integrand decreases. The linear SM behaviour is thus replaced by a peak at $E\sim m_*$ followed by a steep fall. The Higgs mass generation phenomenon gets localised at $m_*=1/l_H$ and $m_H$ is insensitive to much higher energies. This latter fact is also obvious from the fact that no Higgs particle is present much above $m_*$. Therefore there exist no Higgs field and no $d=2$ Higgs mass term to worry about.
Implementing this idea in practice requires a theory with the structure in fig. \[elcomp\]. The three basic elements are a “Composite Sector” (CS), an “Elementary Sector” (ES) and a set of interactions “${\mathcal{L}}_{\textrm{int}}$” connecting the two. The Composite Sector contains the new particles and interactions that form the Higgs as a bound state and it should be viewed as analogous to the QCD theory of quarks and gluons. The CS plays the main role for the composite Higgs solution to the Naturalness problem as it gives physical origin to the Higgs compositeness scale $m_*$. In the analogy with QCD, $m_*$ corresponds to the QCD confinement scale $\Lambda_{\textrm{QCD}}$ and it is generated, again like in QCD, by the mechanism of dimensional transmutation. Thanks to this mechanism it is insensitive to other much larger scales which are present in the problem. For instance the microscopic origin of the CS itself might well be placed at $\Lambda_{\textrm{UV}}\sim{M}_{\textrm{GUT}}$, but still $m_*$ could be Naturally of TeV order, very much like $\Lambda_{\textrm{QCD}}\sim300$ MeV$\,\ll{m}_{\textrm{EW}}$ is perfectly Natural within the SM.
The Elementary Sector contains all the particles we know, by phenomenology, cannot be composite at the TeV scale.[^10] Those are basically all the SM gauge and fermion fields with the possible exception of the right-handed component of the top quark. The most relevant operators in the ES Lagrangian, namely those that are not suppressed by $1/\Lambda_{\textrm{UV}}^n$, are thus just the ordinary $d=4$ SM gauge and fermion kinetic terms and gauge interactions. Since there is no Higgs, no dangerous $d=2$ operator is present in the ES and thus the theory is perfectly Natural. Obviously the lack of a Higgs also forbids Yukawa couplings and a different mechanism will have to be in place to generate fermion masses and mixings.
![The basic structure of the composite Higgs scenario.[]{data-label="elcomp"}](Figures/elementary_composite.png){width="75.00000%"}
The Elementary-Composite interactions ${\mathcal{L}}_{\textrm{int}}$ consist of two classes of terms: those involving the elementary gauge fields and those involving the elementary fermionic field. The latter are responsible for fermion masses and will be discussed later. The former are instead sharply dictated by gauge invariance and read \_\^=\_[i=1,2,3]{}g\_i A\^\_i J\_\^i, \[gaugeint\] where $i$ runs over the three irreducible factors of the SM gauge group and $g_i$ denotes the corresponding gauge coupling. In the equation, $J_\mu^i$ represents the global current operators of the Composite Sector, namely the Noether currents associate with each of the three irreducible factors of the SM group. Notice that for this to make sense the CS must be invariant under the SM symmetries, therefore the complete global symmetry group of the CS, denoted by “${\mathpzc{G}}$” in fig. \[elcomp\], must at least contain the SM one as a subgroup. Good reasons to make ${\mathpzc{G}}$ larger will be discussed shortly. Pushing forward the analogy with low-energy QCD and hadron physics, the ES sector is analogous to the photon plus light leptons system, whose coupling to the CS proceed through the electromagnetic gauge interaction precisely as in eq. (\[gaugeint\]).
The generic framework described until now has an important pitfall, which is overcame in what we nowadays properly call the “Composite Higgs” scenario [^11] by the fact that the Higgs is a pseudo Nambu–Goldstone Boson (pNGB). The pitfall is that if the Higgs is a generic bound state of the CS dynamics one generically expects its mass to be of the order of the CS confinement scale $m_*$, namely $m_H\sim{m}_*$. In a sense, the point is that the mechanism of fig. \[CH\_drawing\] does indeed solve the Naturalness problem by making the shape of $dm_H^2/dE$ localised at $m_*$ but tells us nothing about the normalisation of the $dm_H^2/dE$ function. In the absence of a special mechanism one can estimate $dm_H^2/dE\sim{m}_*$ at $E\sim{m_*}$ and the result of the integral is $m_H^2\sim{m}_*^2$. One can reach the same conclusion heuristically by exploiting the analogy with QCD and browsing one of the many PDG [@Agashe:2014kda] summary tables devoted to the properties of hadrons. By picking one generic (random) hadron in the list one would find that its mass is around the QCD confinement scale $\Lambda_{\textrm{QCD}}$ and that it is surrounded by many other hadrons (a bit heavier or lighter) with similar properties. The Higgs particle is instead alone in the spectrum, or at least we are pretty sure that we would have seen (directly and/or indirectly) at least some of the other particles that would come with it if $m_*$ was around $m_H\sim100$ GeV. Therefore $m_*$ must be of the TeV or multi-TeV order and some mechanism must be in place to explain why $m_H\ll{m}_*$. The problem is actually even more severe than that because the Higgs, on top of being light, is a narrow weakly coupled particle and furthermore its couplings are measured to agree with what predicted by the SM at the $10$ or $20\%$ level.[^12] The existence of a CS resonance obeying these non-trivial properties by accident for no special underlying reason, appears extremely unlikely. The explanation of all these facts might be that the Higgs is a pNGB, namely a special CS hadron associated with the spontaneous breaking of the CS’s global symmetry group ${\mathpzc{G}}$. The Higgs is said a “pseudo” NGB (pNGB) because ${\mathpzc{G}}$ is not an exact but an approximate symmetry. This is precisely what happens in QCD, where the $\pi$ mesons are light because they are pNGB’s associated with the spontaneous breaking of the chiral group. The Higgs might be analogous to a pion, rather than to a random hadron in the PDG list.
The theory of Nambu–Goldstone Bosons works as follows. If the CS is endowed by the global group of symmetry ${\mathpzc{G}}$, it is generically expected that this group will be broken spontaneously to a subgroup ${\mathpzc{H}}\subset{\mathpzc{G}}$ by CS confinement. If this happens, the Goldstone Theorem guarantees that a set of scalar particles, exactly massless as long as ${\mathpzc{G}}$ is an exact symmetry, are present in the spectrum. The theorem says that one such massless NGB particle arises for each of the symmetry generators that are broken in the ${\mathpzc{G}}\rightarrow {\mathpzc{H}}$ pattern, namely one for each generator in ${\mathpzc{G}}$ which is not part of the unbroken ${\mathpzc{H}}$. The broken generators and the corresponding NGB’s are collected in what is called the “${\mathpzc{G}}/{\mathpzc{H}}$ coset”. If the Higgs emerges as one of those particle, which we can achieve by a judicious choices of the coset as discussed in the next section, it will be Naturally light given that its mass cannot be generated from the CS alone, which is exactly invariant under ${\mathpzc{G}}$. A non-vanishing Higgs mass requires the interplay with the ES that breaks the ${\mathpzc{G}}$ symmetry and communicates the breaking to the CS trough ${\mathcal{L}}_{\textrm{int}}$ as in fig. \[elcomp\]. Given that the Elementary/Composite interactions are weak and perturbative, such as the gauge couplings in eq. (\[gaugeint\]), a considerable gap between $m_H$ and $m_*$ is Naturally expected.
It is important to remark that the pNGB nature of the Higgs can also explain why its couplings are close to the SM expectations. This comes from a general mechanism called “vacuum misalignment” discovered in Refs. [@Kaplan:1983fs; @Kaplan:1983sm; @Dugan:1984hq]. I will illustrate how it works in the next section through an example. The picture according to which the Higgs might be the lightest state of the CS, and thus the first one in being discovered, because it is a pNGB, turns out to be rather plausible.
The Minimal Composite Higgs Couplings {#MCHC}
-------------------------------------
A rigorous and complete description of the Composite Higgs (CH) scenario goes beyond the purpose of these lectures, the interested reader is referred to the extensive reviews in [@Contino:2010rs; @Panico:2015jxa]. However most of the relevant features of CH can be illustrated by performing a specific calculation in a specific CH model, namely by computing the couplings of the Higgs to SM particles in the so-called Minimal CH Model (MCHM). Studying Higgs couplings and their possible departures from the SM expectations is one of the ways in which CH models have been and are being searched for at the LHC. Therefore the relevance of the calculation goes beyond its pedagogical value.
![The composite Higgs setup. The elementary SM gauge fields are the three $W$’s, the hypercharge boson $B$ and the eight QCD gluons. The elementary fermionic quark and lepton fields are collectively denoted as $\psi_L$ and $\psi_R$. The Higgs is labeled as “$\Pi$” (see the main text) and $\sigma$, $\Psi$ represent Composite Sector resonances.[]{data-label="ecomp"}](Figures/elem_comp){width="70.00000%"}
The MCHM [@Agashe:2004rs] is based on the choice ${\mathpzc{G}}={\textrm{SO}}(5)$ and ${\mathpzc{H}}={\textrm{SO}}(4)$, which delivers NGB’s in the so-called “minimal coset” ${\textrm{SO}}(5)/{\textrm{SO}}(4)$. According to the Goldstone theorem, the number of real NGB scalar fields in this theory is $4=10-6$, equal to the number of generators in ${\textrm{SO}}(5)$ minus those in ${\textrm{SO}}(4)$. Four real scalars are just sufficient to account for the two complex components of one Higgs doublet. Therefore the ${\textrm{SO}}(5)/{\textrm{SO}}(4)$ coset delivers a single doublet, rather than an extended Higgs sector as it would be the case if larger ${\mathpzc{G}}$ and ${\mathpzc{H}}$ groups are considered. This is why it is called the minimal coset. The Goldstones, i.e. the Higgs, are the lightest particles of the CS, as shown in fig. \[ecomp\]. Therefore they can be studied independently of the other hadrons of the CS (called “resonances”) at all energies below the resonance mass scale $m_*\sim\;$ TeV. On-shell Higgs couplings are low-energy observables in this context, thus they can be computed independently of the detailed knowledge of the resonance dynamics.
A simple model for Goldstone bosons is defined as follows. Be $\ve\Phi$ a five-components vector of real fields, on which the ${\textrm{SO}}(5)$ group acts as rotations in five dimensions, and impose on it the condition \[constr\] =f\^2. The constant parameter $f$ is called the “Higgs decay constant” because it plays in CH the same role of the pion decay constant $f_\pi$ in the low-energy theory of QCD pions. It has the dimensionality of energy and it represents the scale of ${\mathpzc{G}}\rightarrow{\mathpzc{H}}$ spontaneous breaking. The $4$ Goldstone bosons $\Pi_i$, $i=1,\ldots,4$ are introduced as the fields that parameterise the solutions to the constraint (\[constr\]), namely =f, \[fred5\] where $\Pi=\sqrt{\vet\Pi\cdot\ve\Pi}$. Geometrically (see fig. \[vmis\]), $\ve\Phi$ lives on a sphere in the five-dimensional space and $\ve\Pi$ are the four angular variables which are needed to parametrise the sphere. Notice that the constraint (\[constr\]) is invariant under ${\textrm{SO}}(5)$ rotations of $\ve\Phi$, therefore the theory of Goldstone Bosons we will construct out of it will respect the ${\textrm{SO}}(5)$ symmetry. A controlled and perturbative breaking of the symmetry will emerge from the coupling with SM gauge fields and fermions.
The four $\Pi$’s are the Higgs, but this is not yet apparent because the Higgs field is typically represented as a two-components complex doublet $H=(h_u,h_d)^T$ rather than a real quadruplet. The conversion between the two notations is provided by $$\label{dh}
\vec{\Pi}=\left[\begin{matrix}\Pi_1\\ \Pi_2\\ \Pi_3\\ \Pi_4\end{matrix}\right]=
\frac1{\sqrt{2}}\left[\begin{matrix}-i\,(h_u-h_u^\dagger)\\
h_u+h_u^\dagger
\\ i\,(h_d-h_d^\dagger)\\
h_d+h_d^\dagger
\end{matrix}\right]\,.$$ The deep meaning of this equation is that the unbroken group is actually equivalent to the product of two groups, , where is the habitual SM one and is a generalisation of the SM Hypercharge .[^13] Namely, contains the Hypercharge, which is identified with its third generator, $Y=T_R^3$. The Higgs quadruplet $\ve\Pi$ is a $\mathbf{4}$ of , or equivalently a $\mathbf{(2,2)}$ of . The $\mathbf{(2,2)}$ transforms as a $\mathbf{2_{1/2}}$ Higgs doublet under the SM subgroup. The conversion formula in eq. (\[dh\]) does depend on the convention chosen for the generators. I thus report them for completeness T\_[LR]{}\^=, (t\^\_[LR]{})\_[ij]{} = -. In the equation, capital $T_{L{\textrm{/}}R}^\alpha$ ($\alpha=1,2,3$) denote the $5\times5$ generators of seen as a subgroup of , small $t_{L{\textrm{/}}R}^\alpha$ are the habitual generators written as $4\times4$ matrices.
The Lagrangian for $\ve\Phi$, out of which the one of the Goldstones will be straightforwardly extracted, simply reads \[CHlag0\] =12 D\_D\^, D\_=(\_-i g W\_\^ T\_L\^- i g’B\_T\_R\^3). Notice that the couplings with the SM gauge fields $W^\alpha$ and $B$ come from the covariant derivative and they are completely determined by the requirement of gauge invariance. This is exactly what happens when we construct the SM through the habitual gauging procedure and follows from the fact that we decided, in eq. (\[gaugeint\]), to introduce the SM $W$ and $B$ as gauge fields. As a result of this fact, a very sharp prediction will be obtained for the Higgs couplings to the SM vector bosons. To compute the couplings of the physical Higgs we go to the unitary gauge \[ugauge\] H=, and eq. (\[CHlag0\]) becomes =12(\_h)\^2 + 4 f\^2 \^2 (|W|\^2+1[2c\_w\^2]{}Z\^2), \[LUN\] where $W$ and $Z$ denote the ordinary SM mass and charge eigenstate fields, $c_w$ is the cosine of the weak mixing angle defined as usual by . The parameter $V$ denotes the VEV of the Higgs field, induced by a yet unspecified potential.
![A geometrical illustration of EWSB through vacuum misalignment, in the case of the spatial rotations group with . The breaking from vacuum misalignment is proportional to the projection of $\vec{F}$ on the plane, $v=f\sin\langle\theta\rangle$.[]{data-label="vmis"}](Figures/vmis_1){width="32.00000%"}
We can learn a lot on CH by looking at eq. (\[LUN\]). First of all, we can read the mass of the SM vector bosons m\_W=c\_w m\_Z=12 g f 12 g v, \[mwz\] and, by comparing with the corresponding SM formulas, extract the definition of the physical EWSB scale $v\simeq 246$ GeV. We see that $v$, unlike in the SM, is not directly provided by the composite Higgs VEV, but rather it is given by \[xval\] v=f. The geometrical reason for this equation is illustrated in fig. \[vmis\]. According to eq. (\[fred5\]), the vacuum configuration assumed by $\ve\Phi$ when the Higgs takes a VEV, call it $\langle\ve\Phi\rangle$, is a vector of norm $f$ that forms an angle $\langle\theta\rangle=\langle\Pi\rangle/f=V/f$ with the reference vector $\ve{F}=(0,0,0,0,f)^T$. The reference vector is the vacuum configuration $\ve\Phi$ would assume if the Higgs had vanishing VEV and the angle $\langle\theta\rangle$ measures how far the true VEV is from the reference vector. If $\langle\ve\Phi\rangle=\ve{F}$, the vacuum would be invariant under , and thus in particular under the SM group which is part of . The amount of breaking of the EW symmetry is thus measured by the transverse component of $\langle\ve\Phi\rangle$ with respect to $\ve{F}$ because it is only this component the one that makes the vacuum configuration non-invariant under the SM group. From this observation, eq. (\[xval\]) follows. An important property of eq. (\[mwz\]) that I should not forget to outline is that the $W$ and $Z$ boson masses are related by the familiar SM tree-level condition $m_W=c_w m_Z$, which is accurately established experimentally. This property is due to the unbroken group and it furnishes one example of the ability of this “custodial” symmetry to suppress BSM effects as mentioned in footnote \[cuss\].
Next, we can Taylor-expand eq. (\[LUN\]) in powers of the physical Higgs field $h(x)$ and notice that it provides an infinite set of local interactions involving two gauge and an arbitrary number of Higgs fields. The first few terms in the expansion are \[CHC\] (|W|\^2+1[2c\_w\^2]{}Z\^2), where we traded the parameters $V$ and $f$ for the physical EWSB scale $v$ and for the parameter \[xiset\] ==\^21. $\xi$ measures how smaller the scale of EWSB scale is with respect to the scale of breaking or, equivalently, the magnitude of the misalignment angle $\langle\theta\rangle$. The capital importance of the $\xi$ parameter in CH models will become apparent by the discussion that follows. Eq. (\[CHC\]) contains single- and double-Higgs vertices similar to those which arise in the SM, but with modified couplings k\_V=,=1-2. \[hvc\] Also, it contains higher-dimensional vertices with more Higgs field insertions which are absent for the SM Higgs. By measuring Higgs couplings and/or (if possible) by searching for these higher-dimensional vertices we can thus test experimentally the possible composite nature of the Higgs boson.
One peculiarity of eq. (\[hvc\]) that you might have noticed already is that both formulas approach $1$ in the limit $\xi\rightarrow0$, meaning that both the $hVV$ and the $hhVV$ couplings reduce to the values predicted by the SM in this limit. Moreover the coupling strength of the higher-dimensional vertices in eq. (\[CHC\]) are proportional to $\xi$ so that they disappear for $\xi\rightarrow0$ and the same happens to all other interactions of even higher order in the Taylor series. In summary, the complete Lagrangian for the Higgs and the EW boson collapses to the one of the SM for $\xi\rightarrow0$ so that the Composite Higgs becomes effectively indistinguishable from the elementary SM Higgs in this limit. The reason for this is that the $\xi\rightarrow0$ limit is taken at fixed $v$ by sending $f\rightarrow\infty$, and $f$ is related with the typical energy scale of the Composite Sector. For $f\gg{v}$ the CS decouples from the EWSB scale while the Higgs stays light because it is a NGB. The only way in which the theory can account for this large scale separation is by turning itself, spontaneously, into the SM. Of course $\xi$ is not zero, but provided it is sufficiently small this phenomenon explains why the measured couplings of the Higgs boson are close to the SM predictions, which is a priori not trivial at all as discussed in section \[TBI\]. The very existence of the parameter $\xi$ and the possibility of adjusting it in order to mimic the SM predictions with arbitrary accuracy marks the essential difference between the modern CH construction and the old idea of Technicolor [@Weinberg:1975gm; @Weinberg:1979bn; @Susskind:1978ms] (see Ref. [@Lane:2002wv] for a review). Not only in Technicolor, unlike in CH, there is no structural reason to expect the presence of a light Higgs boson. There is not even a reason why this scalar, if accidentally present in the spectrum, should have couplings which are similar to the SM ones. Notice however that taking $\xi$ very small, as we will be obliged to do if the agreement with the SM will survive more precise measurement, does not come for free in CH models. I will come back to this point in the next section.
Let us now turn to the calculation of the Higgs couplings to fermions. In order to proceed we first need to specify the structure of the fermionic part of the interaction that connects the elementary and the composite sector as in fig. \[ecomp\]. This is taken to be similar to the gauge part in eq. (\[gaugeint\]), namely \[ferint\] \_\^\~, where $\psi$ is one of the SM fermion fields in the elementary sector, ${\mathcal{O}}$ is a composite sector local operator and $\lambda$ is a free parameter that sets the strength of the interaction. One such operator is present for each of the SM chiral fermions, each with its own coupling strength $\lambda$. Below we will mostly focus on the top quark sector, in which case the relevant SM fields are the $\psi=q_L$ doublet and the $\psi=t_R$ singlet. The similarity with eq. (\[gaugeint\]) consists in the fact that $\psi$ is an elementary sector field just like $A_\mu$, which is coupled linearly to an operator ${\mathcal{O}}$ made of composite sector constituents very much like $A_\mu$ couples to the composite sector current operator $J_\mu$. Linear fermion couplings of the type (\[ferint\]) were first introduced in Ref. [@Kaplan:1991dc] and are said to have the “Partial Compositeness” structure for a reason that I will explain in the next section.
An important difference between gauge (\[gaugeint\]) and fermion (\[ferint\]) interactions is that in the former case we do know perfectly what the CS operator $J$ is, while in the latter one we have to deal with an operator ${\mathcal{O}}$ of yet unspecified properties. What we know is that ${\mathcal{O}}$ must be a spin $1/2$ fermionic operator in order for equation (\[ferint\]) to comply with Lorentz invariance and that it must be a triplet of QCD colour to respect the symmetry. This latter property will have important phenomenological implications in that it obliges the CS to carry QCD colour and thus to produce coloured resonances which are easy to produce at the LHC. We also know that ${\mathcal{O}}$ must be in some multiplet of the CS global group ${\mathpzc{G}}$ but we don’t know in which one. The only constraint is that the representation in which ${\mathcal{O}}$ lives must contain the SM group representation of the corresponding $\psi$ fermion, in order for eq. (\[ferint\]) not to break the EW group. Few options (focussing on reasonably small multiplets) exist to solve this constraint and for each option the calculation of Higgs couplings might produce a different result. Unlike those with gauge bosons, Higgs couplings to fermions are thus not uniquely predicted in terms of $\xi$.
One simple option is to make ${\mathcal{O}}$ be in the ${\mathbf{5}}$, in which case eq. (\[ferint\]) becomes \[pcrew\] \_\^=\_[L]{}(\_[L]{})\^I\_I+\_[R]{}(\_R)\^I\_I. The index $I$ runs from $1$ to $5$ and it transforms in the ${\mathbf{5}}$ of ${\mathpzc{G}}={\textrm{SO}}(5)$. The capital $Q$ and $T$ fields are two quintuplets that contain the elementary $q_L=(t_L,b_L)$ and $t_R$ fermions. Explicitly, they are \_[L]{}=1(-ib\_L,-b\_L,-it\_L,t\_L,0)\^T,\_R=(0,0,0,0,t\_R)\^T. Their form is chosen in such a way that $(t_L,b_L)$ and $t_R$ appear precisely in those components of the $Q_{L}$ and $T_R$ quintuplets that display the transformation properties of a ${\mathbf{2}}_{1/6}$ and of a ${\mathbf{1}}_{2/3}$ of the SM subgroup. In short, the form of the embeddings is fixed by the requirement that eq. (\[pcrew\]) must respect the SM gauge symmetry.[^14]
Top Bottom
---------------------------------- ------------------------------------------------------------ -----------------------------------------
\[-10pt\] ${\mathbf{5\oplus5}}$ $\ds k_t=\frac{1-2\,\xi}{\sqrt{1-\xi}}\;\;\;\;\;c_2=-2\xi$ $\ds k_b=\frac{1-2\,\xi}{\sqrt{1-\xi}}$
\[8pt\]
\[-10pt\] ${\mathbf{4\oplus4}}$ $\ds k_t=\sqrt{1-\xi}\;\;\;\;\;c_2=-\frac{\xi}2$ $k_b={\sqrt{1-\xi}}$
\[8pt\]
\[-10pt\] ${\mathbf{14\oplus1}}$ $\ds k_t=\frac{1-2\,\xi}{\sqrt{1-\xi}}\;\;\;\;\;c_2=-2\xi$ $\ds k_b=\frac{1-2\,\xi}{\sqrt{1-\xi}}$
\[8pt\]
: Kappa factor and anomalous $c_2$ coupling predictions in the top and bottom quark sector for different choices of the fermionic operators representations under the group. \[tabC\]
Once the representation is chosen, Higgs couplings are determined by symmetries. There is indeed a unique ${\mathpzc{G}}$-invariant operator we can form with $\ve\Phi$ (i.e., the Higgs), the embeddings and no derivatives. Furthermore the coefficient of this operator is fixed by the fact that the correct top mass must be reproduced when the Higgs is set to its VEV. The operator is \[yukup\] \_\^[t]{}&&=-\_I\_[L]{}\^I[T]{}\_R=- t\
&&=-m\_t t - k\_t h t - c\_2 h\^2 t+…. It produces the top quark mass plus, after Taylor-expanding, a set of interactions of the physical Higgs with $t{\overline{t}}$. The first interaction is an $h$-$t{\overline{t}}$ vertex like the one we have in the SM. The second one is an exotic $hh$-$t{\overline{t}}$ coupling which is absent in the SM and could be tested in the double-Higgs production process [@Grober:2010yv; @Contino:2012xk]. The modified single-Higgs coupling and the double-Higgs vertex read k\_t\^=,c\_2\^=-2, \[htc5\] where the $^{\mathbf{5}}$ superscript reminds us that the prediction depends on the choice of the representation (the ${\mathbf{5}}$) for the fermionic operator ${\mathcal{O}}$.
One proceeds in exactly the same way to generate the mass and the Yukawa coupling for the bottom quark, obtaining the bottom coupling modification $k_b$ and an anomalous $hh$-$b{\overline{b}}$ vertex, which however is weighted by the bottom mass and thus it is too small to be phenomenologically relevant. Also for the bottom, the ${\mathbf{5}}$ could be a valid representation for the corresponding ${\mathcal{O}}$ operator. Other choices like the ${\mathbf{4}}$ could be considered both for the bottom and for the top, with the results reported in table \[tabC\]. In the table, the notation “${\mathbf{5\oplus5}}$” means that the fermionic operators that couples to the left-handed doublet $q_L$ and the one that couples to the right-handed singlet ($t_R$ or $b_R$) are in the same representation, i.e. the ${\mathbf{5}}$, while their are both in the ${\mathbf{4}}$ in the “${\mathbf{4\oplus4}}$” case. However the two representations might be different, in spite of the fact that a single name was given for shortness to the two ${\mathcal{O}}$ operators in eq. (\[pcrew\]). A reasonable option is to take the doublet mixed with a ${\mathbf{14}}$ and the singlet mixed with a singlet operator. This is denoted as the “${\mathbf{14\oplus1}}$” case in the table. Up to caveats which is not worth discussing here, table \[tabC\] exhausts what are considered to be the “most reasonable” options for the fermionic operator representations and the corresponding predictions of Higgs couplings. Other patterns which could be worth studying are in Appendix B of Ref. [@Pomarol:2012qf].
Composite Higgs Signatures
--------------------------
Now that the basic structure of the CH scenario has been introduced, I can start illustrating its phenomenology. Additional structural aspects that were left out from the previous discussion will be introduced when needed. The signatures of CH that have been searched for at the $8$ TeV LHC run (run-$1$) and we will keep studying at run-$2$ and possibly at future colliders are Higgs couplings modifications, vector resonances and top partners.
### Higgs Couplings Modifications {#higgs-couplings-modifications .unnumbered}
The current status of our field is that we are not sure of which kind of new physics we are looking for. This is much different from what it used to be the case when the Higgs still had to be discovered. In searching for the Higgs one could rely on one single full-fledged model (the SM) with only one at that time unknown parameter (the Higgs mass). Searching for the Higgs boson was basically equivalent to searching for the SM theory, which was capable to provide detailed and specific predictions for the expected signal to be searched for in the data. We are not anymore in this situation. Even if we focus on one given BSM hypothesis (CH, in the present case, but the same applies to SUSY, WIMP DM or whatever else), this hypothesis is not at all equivalent to a single specific model. This is why in BSM searches so much importance is given to model-independence. Namely to the fact that we should not organise our efforts around specific signatures of specific benchmark models, but rather on generic model-independent features of the scenario we aim to investigate, ideally on those features that are unmistakably present in all the models that provide specific realisations of the generic scenario.
![Fit of the Higgs coupling strength to the gauge bosons ($k_V$) and fermions ($k_F$) obtained by the ATLAS (red contours) and CMS collaborations (blue contours) from the combination of the $7$ and $8\ \mathrm{TeV}$ LHC data. Solid black lines show the CH predictions, depending on the fermionic operators representation, at different values of $\xi$.[]{data-label="fig:kvkf"}](Figures/kvkf){width=".6\textwidth"}
Model-independence is the first reason to be interested in coupling modifications in CH, given that we saw in the previous section how Higgs couplings can be universally predicted as a function of $\xi$. This prediction is independent of the detailed dynamics of the Composite Sector resonances, for which many different explicit models (with plenty of free parameters) can be written down (see e.g. [@Agashe:2004rs; @Panico:2011pw]). The Higgs couplings predictions in all these models are always (up to small corrections) those in eq. (\[hvc\]) and in table \[tabC\]. Higgs couplings have been measured at the LHC run-$1$ both by ATLAS [@Aad:2015gba] and CMS [@Khachatryan:2014jba], with the result reported in fig. \[fig:kvkf\] in the plane. $k_F$ is a common rescaling factor for the SM coupling to fermions, therefore the plot assumes $k_t=k_b=k_F$. The CH predictions are also reported on the plot for different values of $\xi$. The curve labeled “” follows the trajectory in the second line of table \[tabC\], while the “” one represents the first and the third lines. The resulting limit quoted by ATLAS in Ref. [@Aad:2015pla] is $\xi<0.12$ in the and $\xi<0.10$ in the at $95\%$ CL. ATLAS limit is stronger than the CMS one because the ATLAS central value is slightly away from the SM in the opposite direction than the one predicted by CH. The resulting limit is thus stronger than the expected one. Because of this stringent bound, it is unlikely that much progress will be made with the next runs of the LHC, given that the expected limit with the full luminosity of is of around $\xi<0.1$ [@CMS-NOTE-2012/006; @ATL-PHYS-PUB-2013-014; @Dawson:2013bba], very close to the present one. Of course if the central value will not sit on the SM the limit could improve, but we can definitely exclude the occurrence of the discovery of a non-vanishing $\xi$.
We saw that ATLAS and CMS are doing a rather good job in studying Higgs couplings modifications due to compositeness. The study is however not fully complete, and it could be generalised in three directions. First, one can easily construct models where $\kappa_t\neq{k}_b$. It is sufficient for instance to place the fermionic operators associated with the top quark in the $\mathbf{5}$ representation while assigning those for the bottom to a $\mathbf{4}$. In this case $k_t$ will follow the prediction in the first line of table \[tabC\], while $k_b$ will follow the second line. Studying this case is straightforward even if it requires going beyond the plane. No much improvement is however expected in the compatibility of the model since $k_V$ is still the one in eq. (\[hvc\]) and the ATLAS preference for $k_V>1$, independently of the fermion couplings, is already sufficient to produce a limit on $\xi$ not much above $0.1$. A second direction of improvement is to study not only the modification of the Higgs vertices that exist already in the SM, but also anomalous couplings such as $hh$-$t{\overline{t}}$ in eq. (\[yukup\]). The latter might be visible in double-Higgs production when enough luminosity will be collected. However existing studies (see e.g. [@Azatov:2015oxa]) suggest that even with the high-luminosity stage of the LHC (HL-LHC) it might be hard to reach a competitive accuracy. A third direction of improvement would be to generalise the analysis to non-minimal cosets, namely to go beyond the minimal example we discussed here. The problem is that non-minimal cosets produce an extended Higgs sector and thus the modification of the Higgs couplings emerge from the pile-up of two effects. One has the modifications due to compositeness, which are analogous to those in eq. (\[hvc\]) and table \[tabC\], plus further modifications due to the mixing of the Higgs boson with extra light scalar states. The former effect is easy to compute, while the latter one is hard to parametrise with a sufficient degree of generality as it depends on the properties of the extra scalars that mix with the Higgs. Furthermore, all this should be studied in correlation with the direct searches for extra scalars. A detailed phenomenological analysis of extended cosets is missing in the literature, in spite of the fact that extended cosets are not at all implausible from the view-point of model-building. The original CH model [@Kaplan:1983fs], for instance, was based on an coset, which delivers one complex and one real scalar triplet, plus one singlet, on top of the ordinary Higgs doublet.
The second reason to be interested in Higgs couplings modification is the (almost) direct connection between the parameter $\xi$, which couplings measurements are capable to probe, and the level of fine-tuning $\Delta$ of the theory. We discussed in the previous section that for $\xi\rightarrow0$ CH models reduce to the SM, which is an eminently Un-Natural theory. It is thus expected that taking $\xi$ small might be dangerous in terms of fine-tuning. In order to illustrate how this works, let us write down the structure of the Higgs potential, as it emerges in a certain class of models and under certain approximations.[^15] It reads \[hpot\] V\[H\]-f\^2\^2+f\^2\^4, where the coefficients $\alpha$ and $\beta$ can be computed within explicit models (see ...) and depend on some of their free parameters. By adjusting the free parameters one can set $\alpha$ and $\beta$ in such a way that the VEV $V$ of the Higgs field (i.e., the minimum of the potential) produces our favorite value of $\xi$ through eq. (\[xiset\]) and also to reproduce the observed Higgs boson mass. These two constraints read, respectively \[VEVM\] &=&2,\
m\_H\^2&=&8(1-). Both conditions might cost fine-tuning, let us however momentarily focus only on the first one. It tells us that the “expected” value of $\xi$ is proportional to $(\alpha)_{\textrm{expected}}/(\beta)_{\textrm{expected}}$, where by “expected” I mean the size of the $\alpha$ and $\beta$ coefficients that are generically encountered in the parameter space of the model. In all existing CH models, the expected magnitudes of $\alpha$ and $\beta$ either are comparable, or $\alpha$ is larger than $\beta$, making that having $\xi\ll1$ is never an expected structural feature of the model. In this situation, enforcing $\xi\ll1$ requires fine-tuning. Namely, a cancellation must take place in the prediction for $\alpha$, obtained by finely adjusting the parameters of the underlying model. This tuning is at least of order \[xitun\] =1[2]{}. The above equation displays the anticipated connection between $\xi$ and the level of Un-Naturalness of the theory. The current bound $\xi<0.1$ corresponds to a not fully Natural (but still acceptably so) theory with a level of tuning $\Delta>5$.
Actually, we are not sure of the connection between $\xi$ and $\Delta$ in a fully model-independent way. In principle, it would be sufficient to find a model where $\alpha$ is structurally smaller than $\beta$ in order to avoid the tuning in the Higgs VEV and to have $\xi$ Naturally small. The problem, as mentioned above, is that no such model currently exists, but this does not mean that one could not be invented in the future. Engineer a Naturally small $\xi$ is the purpose of the Little Higgs constructions [@Schmaltz:2005ky; @Perelstein:2005ka], however as of now I’m not aware of any convincing and realistic model of this class.
### Vector Resonances {#vector-resonances .unnumbered}
Searching for modified couplings of the Higgs boson is not the only way to test Higgs compositeness experimentally. Direct searches for new particles also play an important role, which will become the leading role at the LHC run-$2$ thanks to the improved collider energy. The new particles to be searched for are the resonances that emerge, together with the Higgs, from the Composite Sector of the theory (see fig. \[ecomp\]). Resonances at a scale $m_*\sim$ TeV are unmistakably present in CH, they are the “hadrons” of the new strong force we are obliged to postulate if we want the Higgs to be a composite object. If we are lucky and the CH scenario is realised in Nature, plenty of such resonances exist and a sort of new “Subatomic Zoo” is waiting to be discovered at the TeV scale.
Predicting the quantum numbers and the properties of the CS resonances is not completely straightforward. However a valid rule of thumb is that resonances are associated with the operators of the CS. Namely, for each resonance it should be possible to identify at least one CS operator that is capable to excite it from the vacuum. The first set of operators we encountered are the global currents “$J$” in eq. (\[gaugeint\]), associated to a set of resonances “$\rho$" through the equation \[op\_par\] |J|00. The currents are bosonic operators that transform as vectors of the Lorentz group, therefore we expect $\rho$ to be a spin-$1$ vector particle in order for eq. (\[op\_par\]) to comply with Lorentz symmetry.[^16] The analogous hadrons in QCD are the $\rho$ mesons, the $\omega$ and the $a_1$, each associated with one of the global currents of the chiral group. Eq. (\[op\_par\]) also tells us the quantum numbers of $\rho$ under the SM group. If for instance ${\mathpzc{G}}={\textrm{SO}}(5)$, the global current $J$ is in the Adjoint $\mathbf{10}$ representation of the group, which decomposes in a $\mathbf{3}_0$, plus a $\mathbf{1}_0$, plus a $\mathbf{1}_1$ and a $\mathbf{2}_{1/2}$ of the SM subgroup (i.e., a $\mathbf{(3,1)\oplus{{(1,3)}}\oplus{(2,2)}}$ of ). $\rho$ particles in all these representations are thus expected, plus one further $\mathbf{1}_0$ because ${\mathpzc{G}}={\textrm{SO}}(5)$ actually needs to be enlarged to ${\textrm{SO}}(5)\times{\textrm{U}}(1)_X$ (see Footnote \[u1x\]) in order to incorporate SM fermion masses into the theory. The existence of vectors with these quantum numbers is confirmed by explicit models. A first study of their phenomenology in the context of holographic realisations of the CH scenario was performed in Ref.s [@Agashe:2007ki; @Agashe:2008jb; @Agashe:2009bb]. Other interesting particles of this class are coloured spin-$1$ vectors, the so-called “KK-gluons” [@Agashe:2006hk]. KK-gluons emerge because the CS (see section \[MCHC\]) needs to carry QCD colour and thus it contains an extra group of symmetry on top of the “electroweak” ${\textrm{SO}}(5)\times{\textrm{U}}(1)_X$ factors. This produces extra global current operators and their corresponding particles in the octet of the QCD group.
All particles above are worth searching for, however here I will focus, for definiteness, on vector resonances in the $\mathbf{3}_0$ triplet, the so-called Heavy Vector Triplet (HVT) [@Pappadopulo:2014qza]. The reason for this choice is that HVT’s display a quite simple phenomenology, still varied enough and promising in terms of mass-reach. Furthermore, the $\mathbf{3}_0$ vectors are associated with the global currents of the SM subgroup of the CS symmetry group. The existence of such subgroup is absolutely unavoidable in CH models, independently of whether or not we stick to the minimal coset or even of whether the Higgs is a pNGB or not. HVT’s thus unmistakably emerge in all models where a strong dynamics is involved in the mechanism responsible for EWSB. This includes old-fashioned Technicolor, in which these particles are also present and are known as “techni-rho” mesons.
Characterising the HVT phenomenology requires a little digression on how we do expect, in general, Composite Sector particles to be coupled among themselves and with the gauge and fermionic fields in the Elementary Sector. This expectation can be encapsulated (see Ref. [@Giudice:2007fh] and Ch. 3 of [@Panico:2015jxa]) in a “power-counting rule”, namely a formula that tells us the expected size of the interaction vertices or, which is the same, of the interaction operators in the Lagrangian. The rule is based on the idea that the CS is characterised by one typical mass scale $m_*$ (the confinement scale) and by one typical coupling strength parameter “$g_*$”. It is thus said to be a “$1$ Scale $1$ Coupling” () power-counting. The parameter $g_*$ represent the typical magnitude of the interaction vertices involving CS particles, among which the Higgs. It can thus be expressed in terms of the Higgs decay constant $f$ and defined as \[gsf\] g\_\*=. The coupling $g_*$ can easily be very large, close to the absolute maximal value $g_*\sim4\pi$ a coupling strength parameter can assume. It is for instance very large in real-world QCD, where it can be identified with the $\rho$ meson coupling $g_\rho\simeq6$. It can however be smaller if the underlying strongly-interacting theory is characterised by a large number of colours $N_c$. For instance, $g_*\sim4\pi/\sqrt{N_c}\rightarrow0$ in the large-$N_c$ limit of QCD. We are thus entitled to consider values of $g_*$ anywhere from $0$ to $4\pi$, however basic phenomenological consistency of the CH scenario requires it to be above around $y_t\simeq1$. Therefore in what follows we will take $g_*\in[1,4\pi]$.
On top of $g_*$, the other couplings that are present in the theory are the SM gauge couplings “$g$” in eq. (\[gaugeint\]) and the fermionic interactions “$\lambda$” in eq. (\[pcrew\]). They control the strength of those interactions of the Elementary Sector fields (gauge and fermions, respectively) that are generated by the CS dynamics, such as for instance their couplings with the Higgs and with the CS resonances. The complete power-counting formula, which takes care both of CS particles self-couplings and of Elementary/Composite interactions, reads \[1s1c\] =, where ${\widehat{\mathcal{L}}}$ is a dimensionless polynomial function with order one coefficients. In the equation, $\sigma$ represents a bosonic CS resonance, such as a spin-$1$ particle like the $\rho$’s we aim to study, while $\Psi$ denotes a fermionic resonance such as the Top Partners we will discuss in the next section. The different power of $m_*$ in the denominator simply follows from the different energy dimensionality ($1$ and $3/2$) of bosonic and and fermionic fields. The fields $A_\mu$ and $\psi$ collectively denote the ES sector gauge and fermions, each entering in the power-counting formula with its own “$g$” and “$\lambda$” coupling. For instance $A_\mu=W_\mu^\alpha$ couples through the weak coupling $g$ while the QCD gluons, $A_\mu=G_\mu^a$, couples through the strong coupling $g_S$. Similarly the third family $q_L$ doublet couples through the $\lambda_L$ parameter in eq. (\[pcrew\]) and $t_R$ couples with strength $\lambda_R$. Notice that light generation quarks and leptons couple with their own strengths, which are typically much smaller than $\lambda_L$ and $\lambda_R$ because their role in the theory is to generate the light fermions Yukawa’s rather than the large top Yukawa coupling. An estimate of light generation couplings is postponed to the next section, since they will turn out to be very small we are entitled to neglect them in what follows.
Let us now turn to HVT phenomenology. Since $g_*$ is the largest coupling in the theory, the strongest vertices of $\rho$ are those that only involve CS particles and no ES degrees of freedom. Among those we have a coupling with the Higgs field \[C\_H\] g\_\*c\_H\_\^aiH\^\^a\^H, where $\rho_\mu^{a=1,2,3}$ denotes the components of the triplet, $\tau^a=\sigma^a/2$ are the ${\textrm{SU}}(2)_L$ generators and the double arrow denotes the covariant derivative acting on the right minus the one acting on the left. The coefficient of the operator has been estimated with eq. (\[1s1c\]) up to an unknown order one parameter $c_H$. The one in eq. (\[C\_H\]) is the unique gauge-invariant operator involving the $\rho$ and two Higgs fields that cannot be eliminated by the equations of motions. It produces couplings of $\rho$ with all the four real components of the Higgs doublet which correspond to the physical Higgs boson plus the three longitudinal polarisation components of the SM $W^\pm$ and $Z$ massive vector bosons.[^17] The operator thus mediates the decay of $\rho$ to different combinations of vector bosons and Higgs final states, with decay widths \[bdec\] \_[\_0W\^+W\^-]{}\_[\_0Zh]{}\_[\_W\^Z]{}\_[\_W\^h]{}. With obvious notation, $\rho_0$ and $\rho_\pm$ respectively denote the electrically neutral and charged $\rho$’s, obtained as linear combinations of the $\rho^a$ triplet components. Neutral and charged resonances are approximately degenerate in mass because of the ${\textrm{SU}}(2)_L$ symmetry. Their common mass is denoted as $m_\rho$.
The second term to be considered is the one responsible for the interaction of $\rho$ with light quarks and leptons. Notice that such an interaction cannot occur directly with an operator involving light elementary fermionic fields because we argued above that the insertion of such fields in the Lagrangian (\[1s1c\]) costs very small $\lambda$’s that make the resulting vertices negligible. However what we can do is to write, compatibly with gauge invariance, an operator that mixes $\rho$ with the elementary $W$ boson field. Since the $W$ couples to quarks and leptons just like in the SM, this $\rho$-$W$ mixing eventually generates the interaction we are looking for. In accordance with the power-counting (\[1s1c\]), the mixing and the resulting interaction reads \[C\_F\] c\_F W\_\^a D\^\^\_a, c\_F \^\_a J\_\^a, with $c_F$ an unknown order one parameter. In the equation, $J_\mu^a=\overline{f}_L\gamma_\mu\tau^af_L$ denotes the ordinary ${\textrm{SU}}(2)_L$ current, namely the one to which $W_\mu^a$ couples in the SM. Since the interaction emerges from the mixing with the $W$, this is precisely the structure we should have expected for the $\rho$ coupling. The scaling of the coefficient is also easily understood. The power-counting formula predicts a $g/g_*$ for the $\rho$ mixing with $W$, while the $W$ coupling with fermions gives an extra power of $g$. The result is the rather peculiar $g^2/g_*$ factor, which makes that the $\rho$ coupling with fermions decreases when $g_*$ increases and the CS becomes more and more strongly coupled. The opposite behaviour is observed for the coupling to bosons in eq. (\[C\_H\]). The translation between the mixing and the interaction operator reported in eq. (\[C\_F\]) is obtained by performing by a field redefinition, namely by shifting the $W$ field by an amount proportional to $\rho$ in such a way that the mixing cancels and the interaction is generated. However this technicality should not obscure the fact that the coupling physically emerges from the mixing with the $W$.
The mixing in eq. (\[C\_F\]) is responsible for $\rho$ decays to quarks and to leptons. Leptonic decays are particularly important because searches in $l^+l^-$ and $l\nu$ final states (with $l=e,\mu$) are extremely sensitive to the presence of resonances. These decays are controlled by one parameter only, $c_F$, therefore the processes $\rho_\pm\to l^\pm\nu$ and $\rho_0\to l^+l^-$ (and the decays to quarks as well) are universally related very much like we saw for the bosonic channels in eq. (\[bdec\]). The widths are \[ldec\] \_[\_l\^]{}2\_[\_0l\^+l\^-]{}()\^2. Notice the presence of $g_*^2$ in the denominator. Together with the $g_*^2$ factor in the numerator of the bosonic decay widths (\[bdec\]), it makes the relative branching fraction between leptons and bosons scales like $1/g_*^4$, which is a strong suppression in the large $g_*$ limit. In this limit, leptonic final states are suppressed and the $\rho$ is better seen in diboson channels in spite of the fact that the reach in terms of cross-section is much better for the leptonic than for the diboson searches. Eq. (\[C\_F\]) is also responsible for $\rho$ Drell-Yan production from a quark anti-quark pair.[^18] The relative magnitude of the $\rho_\pm$ and $\rho_0$ couplings to quarks are fixed and thus the $\rho_\pm$ and $\rho_0$ relative production rate is entirely determined by the parton luminosities. For $m_\rho\sim$ TeV, $\sigma(\rho_\pm)\simeq2\,\sigma(\rho_0)$ at the LHC. The absolute normalisation of the cross section is of course also easily computed, depending on the parameter $c_F$. Together with the partial widths (\[bdec\]) and (\[ldec\]) (plus the analogous formula for the decay to quarks), and assuming that no other decay channel is present, cross sections times branching ratios can be computed for all the channels of interest in terms of two free parameters only, $c_H$ and $c_F$. [^19] Or better, if not willing to assume a fixed $g_*$, in terms of the parameter combinations $c_Hg_*$ and $c_Fg^2/g_*$, which are those that appear in the vertices. This provides a synthetic approximate description of the HVT phenomenology which allows for a comprehensive experimental investigation of the HVT signal [@Pappadopulo:2014qza]. Notice that the production rate scales like $1/g_*^2$, again due to the $1/g_*$ suppression of the vertex (\[C\_F\]), and the HVT’s become more and more elusive in the strong coupling regime.
The left panel of figure \[HVT1\] gives an idea of current limits on HVT from the negative searches performed at the LHC run-$1$. The figure assumes $c_H=c_F=1$, which leaves $g_*$ as the only free parameter. The bound is thus simply reported as an excluded region in the mass versus coupling plane. The yellow region is excluded by resonance searches in leptonic final states (specifically, $l\nu$) while two diboson searches are reported in blue (see [@Pappadopulo:2014qza] for details). The behaviour is the expected one. Namely, the mass reach deteriorates at large $g_*$ because of the suppression of the production rate and the one in the leptonic channel deteriorates much faster than the diboson ones because of the suppression of the leptonic branching ratio. Diboson searches thus become competitive and overcome the leptonic sensitivity for $g_*\gtrsim3$. This behaviour is peculiar of HVT’s with a composite origin, as apparent from the right panel of the figure where the bounds are shown for an “elementary” HVT such as those encountered in $W'$ models. Elementary HVT’s are massive vector bosons emerging from an underlying gauge theory, therefore all their couplings emerge as gauge interactions and thus there is no way in which the coupling to vector bosons can scale differently with $g_*$ than the one to fermions. The branching ratios to leptons and bosons thus remain comparable even at large $g_*$ and the diboson channels never win in terms of mass-reach. Overall, we see that current limits are rather poor in the composite case. Resonance as low as $2$ or $3$ TeV, perfectly compatible with Naturalness and with EWPT limits (reported in black in the figure), are still allowed for a reasonable $g_*$ of order $3$. A priori $g_*$ could be even larger than that, making composite HVT’s virtually invisible, however a moderate value is suggested by other kind of considerations. The left panel of figure \[HVT2\] shows how much the next runs of the LHC could improve the limits, both in the high mass and in the high coupling directions. The plot is based on an approximate extrapolation of current bounds to the $14$ TeV LHC [@Thamm:2015zwa] and assumes a total luminosity of . The HL-LHC reach, with , is also reported, and the exercise is repeated in the right panel of the figure for an hypothetical future $100$ TeV collider. The dashed straight lines in the plot represent indirect limits from the Higgs coupling measurements described in the previous section. The logic is that the resonance mass $m_\rho$ is expectedly comparable with the CS confinement scale $m_*$. If we take them exactly equal we can use eq. (\[gsf\]) to compute $f$, and in turn $\xi$ (\[xiset\]), on the $(m_\rho,g_*)$ plane. Lines are shown for $\xi=0.1$, $\xi=0.08$, $\xi=0.01$ and $\xi=0.004$, corresponding to the reach of the LHC, of the HL-LHC, of ILC and TLEP/CLIC future colliders (see references in [@Thamm:2015zwa]). This shows the complementarity of direct and indirect searches of the Composite Higgs scenario.
![\[HVT1\]Run-$1$ limits on HVT’s from leptonic (yellow) and bosonic (blue) searches. HVT’s of the “composite” type, namely with properties that comply with the expectations of the CH scenario, are shown on the left panel. The case of an “elementary” model, namely the $W'$ of Ref. [@Barger:1980ix] is displayed on the right. Black curves are limits from EWPT. See [@Pappadopulo:2014qza] for details.](Figures/HVT1){width="100.00000%"}
![\[HVT2\]Expected exclusion limits on composite HVT’s compared with indirect constraints from Higgs coupling measurements. From Ref. [@Thamm:2015zwa].](Figures/gV-mV_14 "fig:"){width="35.00000%"} ![\[HVT2\]Expected exclusion limits on composite HVT’s compared with indirect constraints from Higgs coupling measurements. From Ref. [@Thamm:2015zwa].](Figures/gV-mV_100 "fig:"){width="35.00000%"}
### Top Partners {#top-partners .unnumbered}
Top partners are the Composite Sector resonances associated with the fermionic operator ${\mathcal{O}}$ introduced in eq. (\[ferint\]) to couple the third family $q_L=(t_L,b_L)$ doublet and the singlet $t_R$ with the CS. Similarly to what we saw for vectors in eq. (\[op\_par\]), top partners quantum numbers can be extracted from the relation \[op\_par\_TP\] ||00. Since ${\mathcal{O}}$ is a Lorentz Dirac spinor, $\Psi$ must be a spin $1/2$ particle in order to be excited by ${\mathcal{O}}$ from the vacuum. Also, ${\mathcal{O}}$ is in the triplet of the QCD colour group and thus $\Psi$ must also be coloured as I anticipated in section \[MCHC\]. Finally, top partners are CS resonances and as such their mass must be large, of order $m_*\sim$ TeV, barring special suppression mechanisms which we have no reason to expect a priori. The large top partners mass comes directly from the CS and it is unrelated with the occurrence of EWSB. Unlike quarks and leptons, top partners masses would be present in the theory even if the EW gauge symmetry was unbroken, meaning that top partners must be endowed with a perfectly gauge-invariant Dirac mass term. This requires top partners to be “vector-like” fermions, i.e. to come as complete Dirac fields with their left- and the right-handed components transforming in the same way under the gauge group. Coloured particles of this sort are said to be Vector-Like Quarks (VLQ’s). Top partners are VLQ’s of specific type and with specific properties.[^20]
Top partners gauge quantum numbers can also be extracted from eq. (\[op\_par\_TP\]). The result depends on the representation of ${\mathcal{O}}$ under the CS global group ${\textrm{SO}}(5)$, which is a priori ambiguous as I explained in section \[MCHC\]. However any valid representation of ${\textrm{SO}}(5)$, or actually any valid representation of any CS group ${\mathcal{G}}$ we might decide to deal with, going beyond the minimal choice ${\mathcal{G}}={\textrm{SO}}(5)$, must contain at least one SM doublet with $1/6$ Hypercharge and one singlet with Hypecharge $2/3$. The reason is of course that eq. (\[ferint\]) must comply with gauge invariance and thus some of the components of ${\mathcal{O}}$ must have the same gauge quantum numbers as those of the SM $q_L$ and $t_R$ fields. Top partners are thus at least one $(T,B)$ doublet and one ${\widetilde{T}}$ singlet, plus extra states that possibly emerge from the decomposition of ${\mathcal{O}}$. Among those, one extra doublet with exotic Hypercharge of $7/6$ is often present, producing one additional top partners doublet $(X_{2/3},X_{5/3})$ with electric charge $2/3$ and $5/3$, respectively. It is possible to show that all choices of the ${\mathcal{O}}$ representation for which the extra doublet is absent, such as the ${\bf{4}}$ we mentioned in section \[MCHC\], are typically in serious phenomenological troubles because of unacceptably large modifications of the $Zb{\overline{b}}$ coupling [@Contino:2006qr; @Agashe:2006at]. We thus have good reasons to expect the presence of the extra top partners doublet and thus good reasons to search for it.
Similarly to what we saw above for vector resonances, top partners phenomenology can be characterised by employing symmetry, which constrain the structure of their interactions, and power-counting (\[1s1c\]), which sets the expected strengths of the different couplings. The characterisation is slightly more complicate than the one for vectors, mainly because the whole symmetry structure of the theory must be taken into account and not just the SM gauge group. This includes the unbroken group of the CS and even the full non-linearly realised which takes care of the pNGB nature of the Higgs. The analysis produces relatively sharp predictions [@Matsedonskyi:2012ym; @DeSimone:2012fs] of the top partners mass spectrum, decay and production processes. As shown in figure \[TPS\], particles within the two doublets are essentially degenerate, but also the two doublets are quite close in mass, with a splitting between them of around $100$ GeV. The exotic Hypercharge doublet is always the lightest of the two. This spectrum is due to the fact that the two doublets emerge as a single quadruplet and by the peculiar way in which the symmetry is broken by the pNGB Higgs VEV. The ${\widetilde{T}}$ singlet can have any mass, significantly below or above (or close to) the two doublets. The top partners decay branching ratios are approximately universal, as shown in the right panel of the figure. This feature is not peculiar of top partners, it holds for any VLQ with a mass much above the EW scale and follows from considerations related with the Equivalence Theorem similar to those that led us to eq. (\[bdec\]) for vector resonances.
![\[TPS\]Typical top partners mass spectrum and decay branching ratios.](Figures/TPS){width="80.00000%"}
Top partners are colour triplets, thus they are produced in pair by QCD interactions at a fixed and predictable rate as a function of their mass. Since the branching ratios are also known, negative searches for top partners pair production allow to set sharp mass limits, of around $800$ or $900$ GeV at the LHC run-$1$. The run-$2$ reach in terms of exclusions is around $1.2$ or $1.5$ TeV, and it is unlikely it will ever overcome $1.7$ TeV even when the full luminosity of the HL-LHC will be available in many years from now (see [@Matsedonskyi:2014mna; @Matsedonskyi:2015dns] and references therein). The reach could however be extended up to around $2$ TeV by exploiting another sizeable production mechanism top partners are found to possess, namely single production (see figure \[TPSP\]) in association with a top or with a bottom plus a forward jet from the splitting of an EW boson out of a quark line. Single production emerges from a vertex with schematic form \[spv\] \_\~[\_[LR]{}]{}Hq\_[LR]{}, with $q=t$ or $q=b$. The vertex couples top partners with third family quarks and the Higgs, and its power-counting estimate (\[1s1c\]) is rather sizeable because it is controlled by third family $\lambda_{L{\textrm{/}}R}$ couplings. The Equivalence Theorem relates as usual the Higgs field components to longitudinally polarised EW bosons (see Footnote \[ET\]), therefore the operator produces single-production vertices like the one in figure \[TPSP\]. These vertices are of course also responsible for Top Partners decays. Single production cross-section, as the figure shows, is favoured at high mass by the steeply falling parton luminosities and readily starts to dominate over pair production. The mass-limit one can set for single production is not as sharp as the one from pair production because the reach crucially depends on the magnitude of the interaction vertex (\[spv\]), which is not fully predicted. The above-mentioned expected reach ($\sim2$ TeV [@Matsedonskyi:2014mna; @Matsedonskyi:2015dns]) is based on a conservative estimate of the single production coupling strength.
![\[TPSP\]Top partners production cross-section for typical values of the single-production coupling at the $14$ TeV LHC. Pair production is shown as a continuous red line.](Figures/TPSP){width="80.00000%"}
Top partners are arguably the most important CH signatures to be searched for in the forthcoming LHC runs, in spite of the fact that the mass-reach is not great if compared with the one on vectors that can easily overcome $3$ TeV by exploiting the complementarity between direct and indirect searches as in figure \[HVT2\]. The point is that a $3$ or even $5$ TeV bound on vectors would not be as problematic for the CH scenario as a $2$ TeV bound on top partners. Conversely, we don’t have a strong theoretical preference for vectors below $3$ or $5$ TeV, or at least not such a strong one as we have for top partners below $2$ TeV. Of course all resonance masses are set by the same scale, $m_*$, therefore we expect them to be comparable but a factor of two hierarchy between vectors and top partners is perfectly conceivable. What makes top partners special is that it is their mass the one that actually enters in the fine-tuning formula in eq. (\[deltatuning\]), not the mass of vectors or of other CS resonances. Namely, the statement which I will now justify is that the generic estimate of fine-tuning in eq. (\[deltatuning\]) specialises, in the case of the CH scenario, to $\Lambda_{\textrm{SM}}=M_\Psi$. Top partners at $2$ TeV would thus cost a tuning well above ten.
The connection between top partners and fine-tuning is due to the fact that top quark loops (see section \[HP\] and in particular figure \[natarg\]) are the dominant term in the low-energy contribution to the Higgs mass which is at the origin of the fine-tuning problem, and top partners are strongly coupled with the top quark. An example of such coupling is the single production operator in eq. (\[spv\]). Another relevant interaction is the top/top partners mixing of the form [^21] \[mix\] \_\~m\_\*t\_[L]{}+m\_\*t\_[R]{}, and analogously for the $b_L$ mixing with the $B$. In explicit model it is only the mixing term above which is actually generated (in the appropriate field basis) and all the other quarks interactions such as those in eq. (\[spv\]) emerge after diagonalization. The mixing can be used to construct loop diagrams like the one in the left panel of figure \[potyuk\], involving the exchange of a virtual top and a top partner. These diagrams generate a mass for the Higgs, of order \[mhtop\] m\_H\^2\~a\_LM\_\^2+a\_RM\_\^2, where the two terms stand respectively for the exchange of a virtual $t_L$ and $t_R$. The order one numerical coefficients $a_L$ and $a_R$ are calculable in explicit CH models (see e.g. [@Matsedonskyi:2012ym]) and, depending on the model’s microscopic parameters, can assume any sign. The estimate of $m_H$ has been performed by counting the powers of $\lambda$ and $g_*$, reported in figure \[potyuk\], multiplying by the loop factor $1/16\pi^2$ and by two powers of the top partners mass $M_\Psi$ because of dimensionality. This is quite right in spite of the fact that the diagram is still logarithmically divergent because the log only produces order one numerical coefficients which is not worth retaining in our rough estimate.[^22]
![\[potyuk\]Left panel: one representative diagram contributing to the Higgs mass. The Higgs-top partners vertex is a purely CS interaction and thus it has been estimated as $g_*$. The insertion of the mixing weights as in eq. (\[mix\]). Right panel: the generation of the top Yukawa coupling through mixing.](Figures/yuk "fig:"){width="20.00000%"}
Eq. (\[mhtop\]) requires some clarification. As I explained at length in the previous sections, the fact that the Higgs is a NGB prevents the generation of its mass as long as the Goldstone symmetry, i.e. the group ${\mathcal{G}}$, is an exact symmetry of the theory. Since the CS is exactly invariant under ${\mathcal{G}}$, no contribution to $m_H$ can come from the CS alone. In our language this contribution would be a tree-level Higgs mass-term, and the fact that it is absent is the reason why to estimate $m_H$ we had to go at the loop level as in figure \[natarg\]. The diagrams in the figure have the chance to produce a mass because they do feel ${\mathcal{G}}$ breaking through the insertion of the top/top partner mixing. Remember that are indeed the Composite/Elementary Sector interactions the ones responsible for ${\mathcal{G}}$ breaking (see figure \[elcomp\]) in our construction, and the mixing is one of those interaction. Moreover the mixing is the largest of those interaction because it is associated with the generation of the largest coupling of the Higgs boson, namely the top quark Yukawa $y_t$. Other Elementary/Composite interaction such as the gauge couplings also contribute to $m_H$, producing however only small corrections to eq. (\[mhtop\]). This is the reason why it is the top partners mass scale $M_\Psi$, and not for instance the mass of spin one resonances, the one that controls the size of the Higgs mass.
Mixed top/top partners loops generate not only a mass-term, but a full potential for the Higgs field. The potential has the form of eq. (\[hpot\]), with an $\alpha$ parameter \~a\_L\_L\^2 + a\_R\_R\^2 . This estimate is slightly more accurate than the one in eq. (\[mhtop\]), in particular it takes into account the number of colours $N_c=3$, but it scales in the same way with the parameters. The physical mass of the Higgs boson, obtained by combining the two lines of eq. (\[VEVM\]), thus reads m\_H\^2=4(1-)\~a\_L\_L\^2 + a\_R\_R\^2. If $M_\Psi$ is large, obtaining the correct Higgs mass $m_H=125$ GeV requires a cancellation between the two terms, obtained by choosing the fundamental parameters of the models such that $a_L$ is almost equal and opposite to $a_R$. This means a fine-tuning \[tuntpp\] = ()\^2\^2 ()\^2, having assumed $\lambda_L\simeq\lambda_R\equiv\lambda$, which is the configuration that minimises the required amount of tuning. The equation clearly illustrates that light top partners are needed for a Natural (low-tuning) CH model.
Our estimate closely resembles the general formula (\[deltatuning\]) with $\Lambda_{\textrm{SM}}=M_\Psi$, apart from the prefactor $\lambda^2$ that is replaced by $y_t^2$ in eq. (\[deltatuning\]). In order to see that the two formulas match we should relate $\lambda$ with the top Yukawa coupling, by proceeding as follows. The top/top partners mass-mixing (\[mix\]) makes that the two chirality components of the physical top quark, which is massless before EWSB is taken into account, are a quantum mechanical superimposition of Elementary and Composite degrees of freedom \[part\_comp\] |t\_L\^=\_L|t\_L\^+\_L|T\_L\^,\
|t\_R\^=\_R|t\_R\^+\_R|\_R\^, with $\sin\phi_L\simeq\lambda_L/g_*$ and $\sin\phi_R\simeq\lambda_R/g_*$. A similar formula holds for the $b_L$. This comes from diagonalising the mass-matrix of the top/top partners system, which consists of the mass-mixing (\[mix\]) plus the vector-like mass-term $M_\Psi$ for the partners. For the estimate we took $m_*=M_\Psi$ in eq. (\[mix\]), consistently with what implicitly done in the estimate of the $m_H$. Eq. (\[part\_comp\]) shows, in the first place, why we call “Partial Compositeness” [@Kaplan:1991dc] the mechanism (\[ferint\]) we are using to couple ES fermions with the CS: it is because it produces physical particles that are partially made of Composite degrees of freedom. Second, the formula allows us to estimate the top Yukawa generated by mixing as in the right panel of figure \[potyuk\], obtaining \[yukest\] y\_t=\_L\_Rg\_\*, =. But we said that $g_*$ has to be large, at least above $y_t\simeq1$, therefore the above equation tells us $\lambda>y_t$ and eq. (\[tuntpp\]) can be turned into a lower bound \[tpptf\] > ()\^2()\^2, identical to eq. (\[deltatuning\]).
Notice that the estimate of the Yukawa couplings can be carried on for the light quarks (including the bottom) and leptons in exactly the same way as for the top, producing expressions for the corresponding $\lambda$ parameters which are identical to eq. (\[yukest\]) aside from the fact that the light quarks and leptons Yukawas, rather than $y_t$, are involved. Light generation $\lambda$’s are thus very suppressed and this is why we could systematically ignore them. Correspondingly, light fermions compositeness fraction $\sin\phi\sim\lambda/g_*$ are very small. Light fermions are thus almost entirely elementary particles, with a tiny composite component which is however essential to generate their Yukawa’s and masses.[^23]
In summary, the importance of top partners stems from their connection with tuning in eq. (\[tpptf\]). Not finding them at the LHC below $2$ TeV would cost more tuning than what negative searches of Higgs couplings modifications (whose reach is $\xi<0.1$) would imply through eq. (\[xitun\]).[^24] Vector resonances are mildly connected with tuning, therefore even a multi-TeV bound on their mass would not be competitive in terms of fine-tuning reach. Top partners searches are so important because their capable to put the very idea of Naturalness in serious troubles, at least in the Composite Higgs framework. We will see in the next chapter that a similar role is played in Supersymmetry by the stops. It is also important to keep in mind that top partners might very well be discovered at the LHC run-$2$. Current bounds are below $1$ TeV and thus their impact on tuning is modest, well below ten and comparable with the one from coupling measurements. The interesting mass region is the one from $1$ to $2$ TeV in which we are about to enter.
Supersymmetry {#SUSY}
=============
Supersymmetry (SUSY) is probably the most intensively studied theoretical subject of the last $30$ or $40$ years. Its applications range from string theory and supergravity down to collider phenomenology, with digressions on correspondence and holography, dualities and scattering amplitude properties. I mention this to outline that the scope of SUSY is much broader than phenomenology and to explain why theorists care about SUSY a priori, independently of its applicability to the real world on a short timescale. Plenty of excellent reviews [@Sohnius:1985qm; @Derendinger:1990tj; @Drees:1996ca; @Martin:1997ns; @Peskin:2008nw], lecture notes[^25] and books [@WessBagger; @WeinbergSUSY] have been written about SUSY, just to mention some of those that are relevant in the (relatively narrow, as I mentioned) context of SUSY phenomenology. With all this literature available, it makes no sense trying to condense a self-contained introduction to SUSY in these few pages. I will thus keep introductory material to the minimum, focusing only on few basic concepts and results that are absolutely needed for the discussion. Next, in sections \[tale\] and \[after\], I will describe SUSY phenomenology building around two specific questions which I find particularly important to address in this particular moment.
Basics of SUSY
--------------
Symmetries are so much important in particle physics that Coleman and Mandula in ‘$67$ found interesting to ask themselves what is the largest symmetry content a relativistic theory of interacting particles can posses [@Coleman:1967ad]. Their answer was that the largest global symmetry group is Poincaré, generated by the $6$ $M^{\mu\nu}$ Lorentz generators plus the $4$ $P^\mu$’s associated with space-time translations, times a generic Lie group of symmetries generated by a set of charges ${\vec{Q}}_{\textrm{B}}$. Here “times” means direct product, namely their result was that all the internal symmetry generators have to commute with those of the Poincaré group \[comm0\] =0,=0. Remember that commutators are the way in which the symmetry generators act on the other operators. The first equation thus means that the ${\vec{Q}}_{\textrm{B}}$’s are invariant under translations and the second one means that the ${\vec{Q}}_{\textrm{B}}$’s are Lorentz scalars, namely that they stay the same in any reference frame. With a modern terminology we would say that what Coleman and Mandula had in mind were “bosonic” generators, this is why I labeled them with the subscript “B”. Concretely what they had in mind are generators that obey ordinary commutation relations among them, of the form $[{{Q}}_{\textrm{B}}^i,{{Q}}_{\textrm{B}}^j]=if^{ijk}{{Q}}_{\textrm{B}}^k$.
However Gol’fand and Likhtman proved that Coleman and Mandula were wrong, and in so doing they discovered SUSY [@Golfand:1971iw]. They pointed out that a set of $2$ symmetry generators $Q_\alpha$ ($\alpha=1,2$) exist which do not obey eq. (\[comm0\]), but instead \[comm1\] =0,=-(\^)\_\^Q\_. The $Q$’s are still invariant under translations, and in particular under time translations (which is of course obvious if they are conserved), but they are not anymore invariant under Lorentz transformations. Under Lorentz, i.e. under commutation with $M^{\mu\nu}$, the $Q$’s transform with the matrix $\sigma^{\mu\nu}$ which is a $2\times2$ representation of the Lorentz group called the (left-handed) Weyl representation. Therefore we will say that the SUSY generators (or charges) $Q_\alpha$ form a two-components Weyl spinor under the Lorentz group.
The reader unfamiliar with the formalism of Weyl spinors is referred to standard textbooks or to Ref. [@Derendinger:1990tj] (section 4.1 and Appendix A and B) for a concise introduction. The essential point is that Weyl spinor fields are the “building” blocks of the habitual Dirac fermions we normally employ to describe [spin-$1/2$]{} particles. Namely, one four-components Dirac spinor $\Psi$ can be decomposed as \[deco\]=
[c]{} {width="20.00000%"}\
{width="20.00000%"}
[in]{} terms of two two-components spinors $(\psi_1)_\alpha$ and $(\psi_2)_\alpha$ called “left-handed” Weyl spinors.[^26] As anticipated, the Lorentz generators acting on these objects are the $\sigma^{\mu\nu}$ matrices. Namely, under an infinitesimal Lorentz transformation (\_[1,2]{})\_=-2\_(\^)\_\^,\^=4(\^\^-\^\^), where $\sigma^\mu=(\Id,\vec\sigma)$ and $\overline\sigma^\mu=(\Id,-\vec\sigma)$, with $\vec\sigma$ the Pauli matrices. Notice that unlike $\psi_1$, $\psi_2$ does not enter the decomposition formula directly, but rather through the object $({\overline{\psi}}_2)^{\dot{\alpha}}$ which is related to $\psi_2$ by complex conjugation. Namely \[LR\] ()\^=\^\[(\_2)\_\]\^\*, where $\varepsilon$ is the antisymmetric Levi-Civita tensor in two dimensions and the sum over $\beta=1,2$ is understood. We sometimes call $({\overline{\psi}}_2)^{\dot{\alpha}}$ a “right-handed” Weyl spinor, however the previous formula shows that there is no actual distinction between left- and right-handed spinors because one can be turned into the other by complex conjugation. What is normally done in the SUSY literature is to use only left-handed spinors to describe fermions, an habit which can be confusing for beginners. For instance, because of this convention the SM right-handed top quark is represented by a left-handed spinor with electric charge equal to $-2/3$ rather than $+2/3$ because the correspondence between left and right spinors involves complex conjugation.
If the Dirac spinor $\Psi$ is massless, the two Weyl components are endowed with a very simple physical interpretation, pictorially reported in eq. (\[deco\]). $\psi_1$ corresponds to a massless fermion $f$ with helicity $h=-1/2$ plus its anti-particle $\overline{f}$ with $h=+1/2$, while $\psi_2$ is an $h=+1/2$ fermion plus an $h=-1/2$ anti-fermion. If instead the Dirac spinor is massive, namely if it is endowed with a Dirac mass term, there is no direct correspondence between Weyl spinors and physical particles because the Dirac mass mixes the two Weyl components and produces physical particles which are combinations of the two components. Still, a Weyl spinor can be in direct correspondence with a massive fermion, but only if it is a completely neutral particle, not endowed with any conserved charge or quantum number. In this case there is no way to distinguish particle from anti-particle, namely $f=\overline{f}$, and the two helicity states of each Weyl can be interpreted as the two helicity (or spin) eigenstates of a single massive fermion. A mass term given to a single Weyl spinor, which unlike the Dirac mass does not mix the two Weyl components, is called a “Majorana” mass. One Weyl $2$-component spinor can be equivalently representations as a $4$-component spinor called a “Majorana spinor”. There is no physical distinction between the two representations, thus a Weyl fermion with Majorana mass is often called a Majorana fermion.
After this interlude on Weyl spinors, we return to our historical introduction to SUSY. Gol’fand and Likhtman could find a counterexample to the Coleman–Mandula theorem because Coleman and Mandula made too restrictive assumptions in their proof. Namely they assumed bosonic internal symmetry generators, characterised by ordinary commutation relations as previously mentioned. The Gol’fand–Likhtman SUSY charges are instead fermionic generators, characterised by anti-commutation relations \[AC\]{Q\_,Q\_}=0,{Q\_,\_}=2(\^)\_P\_, where $\overline{Q}$ is the conjugate of the SUSY charge [^27] \_=\[Q\_\]\^\*. SUSY charges are thus very different from the ordinary generators of internal symmetries like baryon and lepton number, isospin, etc. Unlike the latter, they do not form an algebra, specified by commutation relations, but rather what is called a “super-algebra”, specified by relations that involve the anti-commutators. Moreover, and perhaps more importantly, SUSY generators do not commute with $M^{\mu\nu}$ (\[comm1\]) unlike the ordinary bosonic charges (\[comm0\]). The story ends with Haag, Lopuszański and Sohnius, who had the final word on the maximal symmetry content of a relativistic theory (with massive particles) [@Haag:1974qh]. They found that it consists of the Poincaré generators, plus bosonic bosonic charges, plus a set of replicas, $Q_\alpha^i$ with $i=1,\ldots,N$, of Gol’fand–Likhtman’s SUSY generators. Actually they also found other symmetries related with SUSY, called “$R$-charges”. Extended SUSY, namely $N\neq1$, does not play an important role in phenomenology, therefore in what follows we will stick to the minimal case $N=1$. A similar consideration holds for the continuos $R$-symmetry group, aside from a discrete subgroup of it called “$R$-parity” which is instead very relevant and will be discussed in the next section.
SUSY-invariant theories display a number of remarkable properties, some of which can be summarised by the famous rule \[b=f\] . The rule has several meanings, the simplest one being that SUSY requires bosonic and fermionic particles with the same mass. In order to see why it is so, consider the state $|h,p\rangle$ describing a single particle with helicity $h$ and four-momentum $p$. For definiteness, we will take the particle moving along the $z$-axis so that the helicity operator coincides with the third component of the angular momentum, i.e. the $1-2$ component of the Lorentz generator, $S_3=M^{12}$. Let us now act on the state with one of the SUSY charges, $Q_\alpha$. This produces a new single-particle state, $Q_\alpha|h,p\rangle$, with the following properties \[MH\] &&P\^|h,p=p\^|h,pP\^( Q\_|h,p)=(Q\_P\^+\[P\^,Q\_\])|h,p=p\^( Q\_|h,p),\
&&M\^[12]{} |h,p=h|h,pM\^[12]{}( Q\_|h,p)=(Q\_M\^[12]{}+\[M\^[12]{},Q\_\])|h,p=(h1/2)( Q\_|h,p),where the $-1/2$ is for $\alpha=1$ and the $+1/2$ for $\alpha=2$. The first equation tells us that the new particle has the same four-momentum as the original one, and thus in particular the same mass. It follows from the first relation in eq. (\[comm1\]), which states that the SUSY charges commute with the $P^\mu$ operator. This first result is of course not at all surprising. Any symmetry generator commutes with $P^\mu$ and connects among each other particles with the same mass. The second relation in eq. (\[MH\]) is instead peculiar of SUSY. Ordinary generators commute with $M^{12}$ and as such they connect particles with the same spin and the same helicity. The commutator of SUSY charges with $M^{12}$ is instead $[M^{12},Q_\alpha]=-1/2(\sigma^3)_\alpha^{\;\beta}Q_\beta$, as dictated by eq. (\[comm1\]), so that SUSY connects particles with helicity $h$ to particles with helicity $h\mp1/2$ as in eq. (\[MH\]). Given that it shifts the helicity by a semi-integer amount, SUSY relates bosons with fermions and thus it requires the existence of mass-degenerate multiplets containing at the same time bosonic and fermionic particles.
By proceeding along these lines, i.e. by repeatedly acting with $Q$ and $\overline{Q}$, one can classify the irreducible representations of $N=1$ SUSY. The relevant ones are those that contain particles of spin two at most, i.e. the chiral, vector and gravity multiplets, schematically represented in fig. \[SUSY\_mult\]. When constructing supersymmetric extensions of the SM, chiral multiplets are used to describe the SM chiral fermions (quarks and leptons), plus the corresponding SUSY particles (squarks and sleptons). The latter are complex scalars with the same quantum numbers of the corresponding SM fermions under the SM gauge group. A chiral multiplet (actually, two of them, as we will see) also describes the SM Higgs field, plus the “higgsinos” superpartners, which are $2$-components fermions. Vector multiplets describe the SM gauge field (photon, gluons, $W$ and $Z$) with their partners, which are again $2$-components fermions called photino, gluinos, wino and zino. Clearly the vector multiplets describe the $W$ and $Z$ bosons, plus their super-partners, before the breaking of the EW symmetry, when they are massless. The gauge fields becoming massive require extra components taken from the Higgs multiplet, like in the SM. The graviton is part of the gravity multiplet, together with a particle of spin $3/2$, the gravitino. A proper description of the gravity multiplet and thus of the gravitino requires a supersymmetric theory of gravity, i.e. a Supergravity model. This goes far beyond the purpose of the present lectures, we will thus not consider the gravity multiplet anymore in what follows.
Notice that each of the multiplets in fig. \[SUSY\_mult\] contains the exact same number ($2$) of bosonic and of fermionic degrees of freedom.[^28] If we combine them to form a SUSY theory we will thus obtain a model with the same number of bosonic and of fermionic degrees of freedom, in accordance with the general rule “bosons $=$ fermions” in eq. (\[b=f\]). Notice that if SUSY is spontaneously broken, bosons and fermions will not anymore form mass-degenerate multiplets according to fig. \[SUSY\_mult\], but still the total number of bosonic and of fermionic degrees of freedom in the theory will remain the same. It is interesting to remark that the validity of the “bosons $=$ fermions” rule crucially relies on the fact that the trivial representation, i.e. the singlet, does not exist in SUSY, unlike any other ordinary symmetry group. If it existed, it would be possible to add “SUSY-singlet” states (bosonic or fermionic) to the theory, violating in this way the equality of the number of bosonic and fermionic degrees of freedom. No SUSY-singlet particle exists because a singlet would be a state that is invariant under SUSY, which means that is must be annihilated both by $Q$ and by $\overline{Q}$. But since $\{Q,\overline{Q}\}\propto P_\mu$, this hypothetical SUSY singlet would be also annihilated by $P_\mu$ and thus it would have vanishing four-momentum and could not be interpreted as a particle. The only state with such properties, i.e. the only SUSY-singlet state, is a (SUSY-invariant) vacuum configuration.
![The $N=1$ SUSY multiplets that are relevant for phenomenology. \[SUSY\_mult\]](Figures/M1 "fig:"){width="30.00000%"} ![The $N=1$ SUSY multiplets that are relevant for phenomenology. \[SUSY\_mult\]](Figures/M2 "fig:"){width="30.00000%"} ![The $N=1$ SUSY multiplets that are relevant for phenomenology. \[SUSY\_mult\]](Figures/M3 "fig:"){width="30.00000%"}
Let us now turn to the problem of writing down SUSY-invariant theories. If SUSY was an ordinary (bosonic) global symmetry, this would be a trivial step to take, once the single-particle state multiplets are known. One would just introduce one field for each particle and construct a multiplet of fields that transform under the symmetry in the exact same way as the corresponding particle multiplets. Symmetric Lagrangians will eventually be obtained by constructing invariant combinations of the field multiplets. The situation is more complicate in SUSY. Constructing invariant Lagrangians requires the concept of “auxiliary fields” and the one of “super-fields”. The issue comes from the “bosons $=$ fermions” rule in eq. (\[b=f\]), which happens to hold not only for the states, but also for the fields. Namely, any set of fields that form a representation of SUSY must contain the same number of bosonic and of fermionic fields components. Consider for instance the chiral multiplet of particles. We describe its scalar degrees of freedom by one complex scalar field $\phi(x)$, which has $2$ real bosonic components, while to describe the $2$-components fermion we must use a Weyl spinor $\psi_\alpha(x)$, which amounts to $4$ real ($2$ complex) fermionic components. Purely in terms of fields, i.e. before we impose the Equations Of Motion (EOM) that reduce the number of fermionic degrees of freedom to $2$, there is a mismatch between the number of bosonic and fermionic components. This mismatch means that a SUSY multiplet cannot just contain the $\{\psi,\phi\}$ fields. One additional complex scalar field, the auxiliary field $F(x)$, is needed to match bosonic and fermionic components. The chiral multiplet is thus made of the set of fields $\{\psi,\phi,F\}$. The exact way in which the SUSY symmetry acts on this multiplet is not worth reporting here. What matters is that a consistent SUSY transformation exists and thus the problem of writing down a SUSY-invariant theory boils down, from this point on, to the one of combining these fields in order to form a SUSY-invariant Lagrangian. The super-field formalism turns out to be extremely effective for this purpose.
Before discussing super-fields, it is important to clarify the role of the auxiliary fields in the construction of SUSY theories. They are introduced in order to comply with the “bosons $=$ fermions” rule applied to the fields, but of course their presence cannot invalidate the rule at the particle level. Namely, auxiliary fields cannot produce extra propagating degrees of freedom, and for this being the case their Lagrangian must not contain a kinetic term. The simplest SUSY-invariant Lagrangian for a chiral multiplet indeed reads \[simp\]=i\^\_-2(+)+\_\^\^-m(F+\^F\^)+F\^F, and it is such that the dependency on the auxiliary field $F$ is purely polynomial. Consequently, the EOM for $F$ is polynomial and can be solved exactly, leading to F=m\^. A field whose EOM can be uniquely solved in terms of the other fields in the theory produces no physical particles, and furthermore it can be eliminated (or, “integrated out”) from the theory by plugging the solution into the Lagrangian. Auxiliary fields thus will not enter in the final expressions for our SUSY-invariant Lagrangian, in spite of the fact that their presence was needed in order to construct it (in the super-field formalism, at least). In the case of eq. (\[simp\]) we obtain =i\^\_-2(+)+\_\^\^-m\^2\^, which is simply the Lagrangian of a free complex scalar, with mass $m$, plus a Majorana fermion (i.e., a neutral Weyl spinor endowed with a Majorana mass-term) with the same mass.
![The chiral and vector superfields, together with the physical degrees of freedom they produce after the EOM are applied to get rid of the auxiliary fields $F$ and $D$. The variable $y$ that appears in the chiral super-field is defined as $y^\mu=x^\mu-i\theta\sigma^\mu\overline\theta$.[]{data-label="SF"}](Figures/SF){width="100.00000%"}
All fields (including the auxiliary ones) in a SUSY multiplet can be collected in a single object, called a super-field. Super-fields can be thought as fields in an extended coordinate space (the super-space), which contains four additional “fermionic” coordinates $\theta^\alpha$ and $\overline{\theta}_{\dot{\alpha}}$ on top of the ordinary “bosonic” space-time coordinates $x^\mu$. The idea is to treat SUSY charges in analogy with the $P_\mu$ momentum operator, which acts on ordinary fields $\mathcal{F}(x)$ as a shift $x\rightarrow x+\delta x$ of the coordinates. A super-field is a function $\mathcal{F}(x,\theta,\overline\theta)$ and the SUSY charges $Q$ and $\overline{Q}$ act on it (almost) as translations $\theta\rightarrow\theta+\delta\theta$ and $\overline\theta\rightarrow\overline\theta+\delta\overline\theta$. A SUSY invariant Lagrangian [^29] is thus constructed as a functional of the super-field that is translational-invariant in the super-space. The $\theta$ and $\overline\theta$ coordinates are however very different from the ordinary space-time ones. Rather than real numbers, they are “Grassmann variables”, namely the product of two of them anti-commutes rather than commuting. This has several bizarre implications, among which the fact that the square of one of the $\theta$ or $\overline\theta$ components just vanishes. The most general super-field thus is not an arbitrary function of $\theta$ and $\overline\theta$, but just a fourth order (corresponding to the total number of independent components) polynomial in $\theta$ and $\overline\theta$, whose coefficients are ordinary fields in the $x$ space. Namely \[gensf\](x,,)=a(x)+b(x)+c(x)+d(x)+e(x)+f\_(x)\^+g(x)+h(x)+i(x). The super-field is taken to be a bosonic object, therefore the fields in the decomposition that accompany even powers of $\theta$ and $\overline\theta$ (i.e., $a$, $d$, $e$, $f_\mu$ and $i$) are bosonic while the ones that come with odd powers ($b$, $c$, $g$ and $h$) are fermionic Weyl fields.
The generic super-field in eq. (\[gensf\]) (or, which is the same, the fields $a,\ldots,i$ it is made of) is a representation of the SUSY algebra, but it is a reducible one. Irreducible representations, corresponding the the chiral and to the vector multiplets, are restricted versions of the general super-field reported in fig. \[SF\]. We already discussed the auxiliary field $F$ appearing in the chiral field multiplet, we now see that it corresponds to the $\theta\theta$ component of the chiral super-field. This component is thus sometimes dubbed the “$F$-component”. A real auxiliary field $D$ is present in the vector multiplet, together with the gauge field $A_\mu$ and the Weyl gaugino fields $\lambda_\alpha$. The auxiliary $D$ is needed because the $A_\mu$ field is taken to be in the Feynman gauge, i.e. it is subject to the condition $\partial_\mu A^\mu=0$ that reduces to three its independent components. One extra real field is thus required in order to match the $4$ real components of the gaugino field. The $D$ field is the $\theta\theta\overline\theta\overline\theta$ component of the vector super-field, which is thus called “$D$-component”.
The rules to construct SUSY-invariant Lagrangians out of super-fields are rather simple. The first one is that (generic) super-fields, like ordinary fields, can be summed, multiplied and conjugated to produce other super-fields. Super-fields can also be derived with respect to the ordinary $x^\mu$ coordinates and also with respect to the SUSY coordinates $\theta$ and $\overline\theta$, by defining certain differential operators called “SUSY covariant derivatives”. I will not define SUSY covariant derivatives here, the reader is referred to the literature. Chiral super-fields can also be summed and multiplied producing other chiral super-fields, but they cannot be conjugated. The conjugate of a chiral super-field is still a super-field, but not a chiral one (it is called “anti-chiral”). The product of a chiral super-field with its conjugate is instead neither chiral nor anti-chiral. An important composite chiral super-field, which we will readily use to construct our SUSY Lagrangian, is the super-potential W()=a+12 m \^2 + 13 \^3. It is a cubic polynomial in the chiral super-field $\Phi$, with an obvious generalisation to the case in which several super-fields $\Phi_i$ are present. The super-potential is the SUSY generalisation of the ordinary scalar potential. However unlike the latter it cannot contain the conjugate of the chiral field, $\Phi^\dagger$, otherwise it would not be a chiral super-field as previously explained. A super-potential can actually contain higher power of $\Phi$. I stopped at the third order because higher term would produce non-renormalizable interactions in the Lagrangian.
The last set of rules tells us how to extract invariant Lagrangians out of functionals (sums, products and derivatives) of super-fields. All SUSY invariants happen to be either the $D$ component (i.e., $\theta\theta\overline\theta\overline\theta$) of a generic super-field or the $F$ component (i.e., $\theta\theta$) of a chiral super-field. The most general SUSY-invariant Lagrangian for a chiral super-field (with obvious generalisation to several super-fields) is thus &&\_F=i\^\_+ \_\^\^+F\^F,\
&&\_D+=.|\_F -12.|\_+. \[lesssimp\] We see that the simple SUSY-invariant Lagrangian in eq. (\[simp\]) is recovered for $a=\lambda=0$ in the super-potential. Also notice that even in the more general Lagrangian in eq. (\[lesssimp\]) the auxiliary field $F$ does not possess a kinetic term and it can be integrated out by solving its EOM, which is just $F^\dagger=-\partial W/\partial\Phi$. This results in a potential for the scalar component $\phi$ of the chiral super-field \[fterm\] V\_F()=|.|\_|\^2=|a+m+\^2|\^2, which is called “$F$-term potential”.
Similarly, one can write down the Lagrangian for the vector super-field and the interactions between the vector super-field and the chiral one. The vector super-field is the SUSY generalisation of the $A_\mu$ gauge field, therefore its interactions are dictated by gauge-invariance (plus SUSY), very much like the interaction of an ordinary gauge field. For a single vector super-field, corresponding to a gauge symmetry (the generalisation to non-abelian groups like the ones of the SM is rather straightforward) and a single chiral super-field with charge $q$ under the group, the Lagrangian consists of the two following terms &&14\_F+= -14 A\_ A\^+ i\^\_+12 D\^2,\[gaugelag\]\
&&\_D =D\_\^D\^+i\^D\_ +F\^F -iqg+iqg\^-gq\^D. The first one is simply the kinetic term for the gauge and for the gaugino fields, plus a quadratic (non-derivative, as it should) term for the auxiliary $D$.[^30] The second contains the kinetic terms of the scalar and Weyl fields in the chiral multiplet, with “$D_\mu$” denoting the ordinary covariant derivative with with charge $q$ and gauge coupling $g$, which produces the habitual gauge interactions. Interestingly enough, Yukawa couplings are also present involving $\phi$, $\psi$ and the gaugino $\lambda$. These are supersymmetric generalisations of the $A_\mu$ gauge interactions with $\psi$ and with $\phi$ and they emerge with a coupling strength, $\sqrt{2} gq$, which is completely fixed by gauge invariance. Also notice that the auxiliary field can, as usual, easily be integrated out producing another contribution to the scalar potential called “$D$-term potential”. It reads V\_D()=12q\^2g\^2||\^2. Once again, like the Yukawa’s previously mentioned, its coefficient is completely specified in terms of the representation of the gauge group in which the field lives (i.e., the charge $q$ in our example) and by the gauge coupling $g$ of the theory.
Why SUSY is Great: a Tale from the 80’s {#tale}
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The possibility of SUSY being the right tool to construct realistic extensions of the SM below or at the TeV scale (and not “just” a tool to build string theories of quantum gravity and to study deep theoretical aspects of Quantum Field Theory) is supported by a number of surprising phenomenological properties SUSY theories happen to possess (see [@Drees:1996ca; @Martin:1997ns] for a complete discussion). These properties were discovered in the early 80’s and produced enormous excitement in the theory community. The virtues of SUSY, which I will describe in the present section, of course are still there today. However they are now accompanied by a set of issues, related with negative searches of super-particles at different experimental facilities and with the determination of the Higgs boson mass, as I will explain in sect. \[after\]. None of these experimental issues were of course known in the 80’s, and thus the great excitement about SUSY was fully justified. The situation is different now. SUSY might still be waiting to be discovered at the TeV scale, but apparently not in the simple “vanilla” form theorists imagined in the 80’s.
The main reason why SUSY should be relevant for TeV scale physics is that SUSY models can solve the Naturalness Problem, as first pointed out by several authors in ’81, among which S. Dimopoulos, H. Georgi and E. Witten. In order to see how this works, let us recall the Naturalness Argument, as formulated in sect. \[natarg\], for the Higgs mass parameter $m_H^2$. The problem has to do with a contribution that comes (in whatever new physics model is ultimately responsible for the microscopic origin of the Higgs mass) from low-energies, below the SM cutoff $\Lambda_{\textrm{SM}}$, i.e. at energies where physics is known and is provided by the SM. We focus on the largest contribution, the one from the top quark loop in eq. (\[deltamh\]) \[dmh\] \_m\_H\^2= \_\^2. It can be interpreted, poorly speaking, as a divergent contribution to the Higgs mass. The Naturalness Problem is that this term becomes much larger than the actual value of $m_H^2$, obliging us to a cancellation, if $\Lambda_{\textrm{SM}}$ is much above the TeV, as eq. (\[deltatuning\]) shows.
The problem emerges because the Higgs mass has two properties, which have to be simultaneously verified for the Naturalness Problem to arise. These are the fact that the Higgs mass term is a parameter with positive energy dimension and the fact that it is not protected by any symmetry, namely no new symmetry emerges in the SM Lagrangian if $m_H^2$ is taken to vanish. Parameters that violate both conditions are instead, for instance, the SM Yukawa couplings. Take for simplicity one single fermion, with its Yukawa coupling $y_f$ to the Higgs field, and repeat for $y_f$ the considerations that led us to the Naturalness Problem in sect. \[natarg\]. We can still split the integral expression for it as in eq. (\[splitint\]), but now if we compute the $E<\Lambda_{\textrm{SM}}$ contribution we find \[yukNNP\] \_y\_f\~(\_/M\_). In the expression, $M_{\textrm{EW}}$ denotes the EW scale and $g$ is one SM coupling which, depending on the diagram, could be either $y_f$ itself or one of the gauge couplings. The result contains no power-like divergence, for the very simple reasons is that all the SM couplings are dimensionless. It is thus impossible to have a polynomial divergence on a dimensionless quantity, only logarithmic divergencies are allowed. Clearly a logarithm is much less dangerous. Even for $\Lambda_{\textrm{SM}}=M_P$, the log is around $40$ and hardly compensates the $g^2/16\pi^2$ loop factor. The contribution to $y_f$ is thus of order $y_f$ or smaller and no cancellation is required. The fact that $\delta_{\textrm{SM}}y_f$ contains at least one power of $y_f$ is instead less trivial and has to do with the fact that a symmetry (chiral symmetry, i.e. two independent phase transformations acting on the two chirality components) is recovered at $y_f=0$. Then if $y_f$ was really vanishing, loop corrections could not generate it. A diagram contributing to it must thus contain at least one insertion of the $y_f$ vertex. As mentioned in sect. \[natarg\], this symmetry argument can be extended in order to deal with all the three SM families, ensuring that the correction of each Yukawa coupling is proportional to itself. This avoids, for instance, relatively large contribution to the Yukawa coupling of the up induced by the much larger coupling of the top quark.
A less simple case is the one of a massive fermion. Of course we don’t have one in the SM since no fermion masses (but only Yukawa’s) are present in the SM above the EW scale. Consider however a toy model in which a massive fermion is included in the theory, coupled through a set of dimensionless couplings “$g$” to the SM fields. Its mass $m_F$ has of course positive energy dimension like $m_H$, but still the low-energy contribution to it is only logarithmically divergent [^31] \_m\_F\~m\_F(/M\_). Unlike the Yukawa’s, $m_F$ has positive dimension but, exactly like the Yukawa’s, it is protected by the chiral symmetry which is recovered in the theory if $m_F=0$. Thus loop corrections are proportional to $m_F$ itself and are not large. As such, $m_F$ does not suffer of a Naturalness Problem.
![The ordinary Yukawa coupling (left) with its SUSY counterpart (right).\[FD\]](Figures/FD1 "fig:"){height="13.00000%"} ![The ordinary Yukawa coupling (left) with its SUSY counterpart (right).\[FD\]](Figures/FD2 "fig:"){height="13.00000%"}
We just discovered that a fermion, unlike a scalar boson, can be “Naturally” light, even if the cutoff $\Lambda$ of the theory it is part of is extremely large. It is thus now clear why SUSY, which obliges the mass of the scalar Higgs boson to be equal to the one of its fermionic higgsino partner, can help us with the Naturalness Problem. If the former is “Naturally” light, the latter must be “Natural” as well in a SUSY model. In order to illustrate how this works, let us only consider the Higgs boson, the top quark and the Yukawa interaction between them, which is responsible for the largest contribution to $m_H^2$ in eq. (\[dmh\]). I will even ignore the bottom quark, as well as the other components of the Higgs doublet, and I will just focus on the neutral Higgs field component $h$ coupled to $t_L$ and $t_R$ through the Yukawa coupling. In order to construct a supersymmetric version of this theory, three chiral super-fields need to be introduced: $\Phi_h$, $\Phi_{t_L}$ and $\Phi_{t_R}$. After integrating out the auxiliary fields, they lead respectively to the fields $\{h,\widetilde{h}\}$, $\{t_L,\widetilde{t}_L\}$ and $\{t_R^c,\widetilde{t}_R^\dagger\}$, where $\widetilde{h}$ is the higgsino, $\widetilde{t}_L$ is the left-handed stop and $\widetilde{t}_R$ is the right-handed stop. Notice that what appears in $\Phi_{t_R}$ is the conjugate of the right-handed top, $t_R^c$, which is a left-handed field and as such can appear in the chiral super-field. Correspondingly, the right-handed stop is defined with a conjugate such that it has the same quantum numbers of the (not conjugate) SM $t_R$. Introducing the SM Yukawa in the theory requires us to put a trilinear term in the super-potential W=\_h\_[t\_L]{}\_[t\_R]{} {
[l]{} - h t\_R,\
F -2h\^2\[|\_L|\^2+|\_R|\^2\].
. Therefore in SUSY the Yukawa coupling, diagrammatically represented on the left panel of fig. \[FD\], is necessarily associated with a quartic $h^2\widetilde{t}_{L,R}^2$ vertex with the stops, reported on the right. Both couplings must be included in the calculation of $\delta_{\rm{IR}}m_H^2$, and the stop loop cancel exactly the one of the top\
{width="50.00000%"}
The result is that the Naturalness Problem is solved, as expected, in a supersymmetric theory.
Obviously in oder to exploit the solution to the Naturalness Problem offered by supersymmetry we cannot just replace the SM with its SUSY version. This would be in sharp contrast with observations given that the particles we know about, their spectrum and interactions, do not respect SUSY. What one has to do is to first extend the SM to its (possibly minimal, but not necessarily so) SUSY version, and then include extra terms in the Lagrangian that break supersymmetry and reconcile the model with observations. Very importantly, it turns out that it is possible to do this without spoiling the SUSY solution to the Naturalness Problem, by introducing a special set of SUSY-breaking terms called “soft terms”. Equally importantly, explicit microscopic models exist where SUSY is exact at very high scale, gets spontaneously broken and produces only soft breaking terms at low energy. Soft SUSY-breaking terms, namely terms that break SUSY but preserve Naturalness, include (see e.g. [@Drees:1996ca; @Martin:1997ns]) mass, bilinear and trilinear terms for the scalar fields and gaugino mass terms. Including them in the Lagrangian happens to be sufficient to make all the SUSY partners of the SM particles heavy, explaining why we have not yet seen them. SUSY models addressing the Naturalness Problem can thus be made fully realistic, as a result of a fortunate series of “coincidences” related with a bunch of non-trivial properties of SUSY.
The SUSY picture of high-energy physics is thus the one of fig. \[SUSYPIC\]. Starting from above, the theory is exactly supersymmetric at very high energies, until the scale $M_\slashed{\rm{S}}$ where SUSY is broken producing a set of soft terms. The typical mass-scale $M_{\rm{soft}}$ of the soft terms generated by the breaking, among which we have the mass of the supersymmetric particles, needs however not to be of the order of $M_\slashed{\rm{S}}$. It can be of that size in specific SUSY breaking scenarios, but it can also easily be much smaller than that, $M_{\rm{soft}}\ll M_\slashed{\rm{S}}$, as in the framework of “gravity-mediated” SUSY breaking (which used to be very popular in the 80’s). Below $M_\slashed{\rm{S}}$, the theory reduces to a supersymmetric extensions of the SM containing both the SM particles and the SUSY partners as propagating degrees of freedom, the latter ones with a mass of order $M_{\rm{soft}}$, larger than the EW scale. Below $M_{\rm{soft}}$, SUSY partners decouple from the theory and one is left with the SM. Seen from below, $M_{\rm{soft}}$ is the scale at which BSM particles appear and thus it provides the SM cutoff $\Lambda_{\rm{SM}}$.
In view of the identification $M_{\rm{soft}}\sim\Lambda_{\rm{SM}}$, it is clear that the SUSY partners cannot be arbitrarily heavy if we really want to solve the Naturalness Problem, because of eq. (\[dmh\]). This is readily checked by giving a mass $M_{\widetilde{t}}\simeq M_{\rm{soft}}$ to the stops and repeating the calculation of $\delta_{\rm{IR}}m_H^2$. It is rather obvious by dimensional analysis that we are going to obtain \[dmhSUSY\] \_[[IR]{}]{}m\_H\^2=M\_(M\_/M\_). Up to the log, which can just worsen the situation, we get the same expression as in eq. (\[dmh\]) with $\Lambda_{\rm{SM}}$ replaced by $M_{\widetilde{t}}\simeq M_{\rm{soft}}$. Consequently we get a large fine-tuning $\Delta$, as in eq. (\[deltatuning\]), if SUSY particles are not at the TeV scale or below.
![The SUSY picture of high-energy physics.\[SUSYPIC\]](Figures/SUSYPIC){height="30.00000%"}
SUSY having to show up before the TeV was of course not at all an issue in the 80’, when this scale was far to be directly probed experimentally. It was actually a reason for excitement having all these new particles close enough to be discovered in the future. More reasons for excitement came from two more arguments, seemingly unrelated with SUSY: coupling unification and Dark Matter. Coupling unification (see [@Langacker:1980js; @Raby:2006sk] for a review) is the idea that the three SM gauge forces might have a common origin at very high scales, where they are all described by a single simple unified gauge group (e.g., or ), characterised by a single gauge coupling. This is supported, in the first place, by the fact that the SM matter fermion content fills, for no obvious reason, complete multiplets of the unified group (see [@Pomarol:2012sb] for a concise discussion). These multiplets contain at the same time quarks and leptons. GUT models are also supported by the fact that the running of the three SM gauge couplings makes them approach each other at high scale. As shown in fig. \[GUT\], this more or less happens (but not very accurately) in the SM at a scale $M_{\rm{GUT}}\sim10^{14}$ GeV. At this scale, the full unified theory should show up. In particular, new massive gauge bosons should appear, with interactions connecting leptons and quarks that sit in the same GUT multiplets as previously mentioned. These interactions make the proton decay at an unacceptably large rate if $M_{\rm{GUT}}\sim10^{14}$ GeV. The situation is much better in supersymmetric extensions of the SM, as shown in the right panel of fig. \[GUT\]. First, the couplings unify more accurately, simply due to the effect of the super-partners on the running, which happen to go in the right direction for no obvious reason. Second, unification is postponed to $M_{\rm{GUT}}\sim10^{16}$ GeV and proton decay experiments are not sensitive to such a high suppression scale. All this of course happens only provided $M_{\rm{soft}}$ is small enough for the super-partners starting to contribute to the running early enough, $M_{\rm{soft}}\sim100$ GeV is assumed in the plot. Clearly the running is logarithmically slow, so that $M_{\rm{soft}}=10$ TeV or even more would not change the situation radically. However it is clear that also coupling unification, as well as the Naturalness Argument, point towards low-energy supersymmetry. The positive interplay between low-energy SUSY and unification is a very strong argument in favour of SUSY and of unification as well.
![\[GUT\]The , and inverse structure constant ($\alpha_i^{-1}=4\pi/g_i^2$) renormalisation group running in the SM (left) and in its minimal supersymmetric extension, the MSSM (right).](Figures/GUT){width="100.00000%"}
The interplay between SUSY and DM is equally impressive. It originates from a serious phenomenological problem of SUSY and of its solution, which consists in imposing a discrete symmetry called “$R$-parity”. I stressed in sect. \[SMONLY\] the phenomenological importance of Baryon and Lepton number as accidental symmetry in the SM and how much non-trivial it is that these symmetries emerge at $d=4$ without being imposed in the construction of the theory. I also argued that BSM scenarios will in general not possess accidental Baryon and Lepton number and that those symmetries will have to be imposed in some way. This is the case also in SUSY. Indeed, when trying to construct the minimal supersymmetric extension of the SM (the Minimal Supersymmetric Standard Model, MSSM), one immediately encounters terms in the super-potential, allowed by the gauge symmetries, that violate both the Baryon and the Lepton number. For instance, Baryon number is violated by (see e.g. [@Martin:1997ns] for more details) \[dw\] W\_[B=1]{}=” \_\_[u\_R]{}\^\_[d\_R]{}\^\_[d\_R]{}\^, where $\alpha,\beta,\gamma$ are QCD color indices while the flavour indices are understood. Adding those terms in the super-potential produces SUSY-invariant $d=4$ interactions that violate Baryon and Lepton number, in sharp contrast with observations. However all these dangerous terms, and all the soft SUSY-breaking ones which also violate Baryon and Lepton number, are avoided by imposing only one discrete symmetry, $R$-parity. $R$-parity consists in the sign-flip $\theta\rightarrow-\theta$ and $\overline\theta\rightarrow-\overline\theta$ of the super-space coordinates, times an additional overall minus sign for all the matter fermions (quarks and leptons) super-fields. A quick look at fig. \[SF\] immediately reveals that with this assignment all the SM fields (quarks, leptons, gauge and Higgs) are even and all the super-partners (or s-particles) are odd. The super-potential in eq. (\[dw\]) is obviously odd under $R$-parity and it is thus forbidden, together with all the other Baryon and Lepton number-violating terms, if $R$-parity is imposed as a symmetry of the MSSM.
Since they are odd under $R$-parity, s-particles cannot decay to SM particles only, at least one s-particle must be present in the final state. In particular this means that the lightest of the s-particles (the LSP) cannot decay at all and it is absolutely stable. If it happens to be electrically and QCD neutral, it is potentially a viable DM canditate. Moreover, the LSP mass will be of the order of $M_{\rm{soft}}$, which we argued above to be likely of the $100$ GeV to TeV order. Furthermore, the LSP will typically couple to SM through EW gauge interactions. A particle with these properties is called a Weakly-Interacting Massive Particle (WIMP) and it can perfectly account for the observed DM component of the Universe through the mechanism of thermal freeze-out (see [@dslect] for a review). This is the so-called “WIMP miracle”, which automatically emerges as a byproduct of SUSY model-building.
SUSY after LEP, Tevatron and LHC run-$1$ {#after}
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Naturalness, coupling unification and Dark Matter are extremely strong arguments in favour of low-scale SUSY, and all the enthusiasm they triggered towards SUSY is perfectly justified. However this enthusiasm cooled considerably after 30 years of negative experimental s-particle searches. LEP was the collider at which SUSY had its first chance to be discovered, in spite of the fact that LEP energy was far below $450$ GeV (see eq. (\[deltatuning\])), which is what we nowadays consider to be threshold for “Natural” BSM physics. This is because in the fine-tuning estimate we should not forget the logarithmic term we found in eq. (\[dmhSUSY\]), and we should remember that a high SUSY-breaking scale was expected in the 80’s. With this expectation, taking for definiteness $M_\slashed{\rm{S}}=10^{15}$ GeV, the log is around $30$ and the Naturalness threshold moves down to $450/\sqrt{30}=82$ GeV. Even taking into account that s-particles must be produced in pairs because of $R$-parity, the LEP collider (in the LEP-II stage) could have had enough energy to produce them. Of course $M_\slashed{\rm{S}}$ needs not to be that high, viable SUSY-breaking scenarios exist where $M_\slashed{\rm{S}}$ is not far from the TeV scale and the log is small. Still, negative LEP search were the first evidence against the “vanilla” SUSY picture described in the previous section.
The search for s-particles continued at Tevatron and at the LHC run-$1$, with negative results.[^32] Current limits on certain SUSY particles (light squarks and gluinos) are as high as $1.7$ TeV signalling, if taken at face value, that SUSY is a quite “Un-Natural” theory. One should however be more careful, because the s-particles needs not to be all degenerate and a bound on few of them cannot be directly translated into a bound on $M_{\rm{soft}}$. Furthermore, not all the s-particles are equally important as far as fine-tuning is concerned because the way in which they contribute to the Higgs mass is very different. For instance, the stops are those that give the largest radiative contribution, in eq. (\[dmhSUSY\]), because their coupling to the Higgs is the largest one. The $450$ GeV threshold only applies to the stops, and the limit on their mass is only $700$ GeV or less, still compatible with Naturalness.[^33] The strong limit on the light squarks is instead irrelevant for Naturalness, given that the squark contribution to $m_H^2$ is extremely suppressed by the small Yukawa couplings. The partners of the EW gauge bosons (EWinos) give the second largest radiative contribution (see eq. (\[deltamh\])), which is proportional to the Weak coupling square rather than to $y_t$. The Naturalness threshold for the EWinos is thus around the TeV, much above the limits. The gluinos are also relevant for Naturalness. In spite of the fact that their contribution to $m_H^2$ arises at two loops, the strong QCD coupling and certain color multiplicity factors produce a Naturalness threshold for gluinos around the TeV, which is comparable with the run-$1$ limit. The overall picture that emerges from this kind of considerations (the so-called “Natural SUSY” approach) is that the LHC run-$1$ started probing the “Natural” parameter space of SUSY, but no conclusive statement can be made. For an extensive presentation of this viewpoint and a quantitative discussion of run-$1$ searches the reader is referred to the lecture notes in Ref. [@Craig:2013cxa].
The very last topic of these lectures is the structure of the Higgs potential in supersymmetry. This topic is relevant by itself, as it constitutes the starting point for SUSY Higgs phenomenology, extensively discussed in [@Djouadi:2005gj]. However it is also relevant in order to assess the current status of SUSY because it will allow us to understand and to qualify the often-heard statement that the LEP bound on the Higgs mass (and its measurement at the LHC) is problematic for SUSY. The first important point is that any SUSY extension of the SM requires us to introduce at least two Higgs chiral super-fields: $\Phi_{\rm{u}}$ and $\Phi_{\rm{d}}$. This follows from the fact in order to generate the Yukawa couplings in the up and in the down sector two Higgs doublets are needed, with respectively Hypercharge equal to $1/2$ and $-1/2$. Only one doublet is introduced in the SM because the other one can be obtained by complex conjugation, but this is impossible in SUSY since the conjugate of a chiral super-field cannot appear in the super-potential. Two chiral super-fields are thus needed,[^34] with super-potential terms W\_[[u]{}]{}=y\_[[u]{}]{}\_[q\_L]{}\_[[u]{}]{}\_[u\_R]{}, W\_[[d]{}]{}=y\_[[d]{}]{}\_[q\_L]{}\_[[d]{}]{}\_[d\_R]{}. Therefore two scalar Higgs doublets $H_{\rm{u}}$ and $H_{\rm{d}}$ are present in SUSY, coupled to up- and down-type quarks respectively. After EWSB, three of these $8$ real degrees of freedom are eaten by the EW bosons becoming massive, one provides the neutral SM Higgs boson and the remain four are extra scalars which are absent in the SM. The extra scalars in SUSY are one heavy neutral CP-even state $H_0$, one charged $H_\pm$ and a neutral CP-odd $A$. Searching for these particles directly, or indirectly by studying their effects (through mixing) on the couplings of the SM-like Higgs, is one way to test supersymmetry.
The scalar potential for the $H_{\rm{u}}$ and $H_{\rm{d}}$ doublets consists of three terms [^35] V(H\_[[u]{}]{},H\_[[d]{}]{})&=&\^2(|H\_[[u]{}]{}|\^2+|H\_[[d]{}]{}|\^2)\
&&+8(|H\_[[u]{}]{}|\^2-|H\_[[d]{}]{}|\^2)\^2+2 |H\_[[u]{}]{}\^|H\_[[d]{}]{}|\^2\
&&+ m\_[[u]{}]{}\^2 |H\_[[u]{}]{}|\^2+m\_[[d]{}]{}\^2 |H\_[[d]{}]{}|\^2 + B (H\_[[u]{}]{} H\_[[d]{}]{}+H\_[[u]{}]{}\^\* H\_[[d]{}]{}\^\*).\[Higgspot\] The one on the first line is an $F$-term, originating from the $\mu$-term $\mu\Phi_{\rm{u}}\Phi_{\rm{d}}$ in the super-potential. The one on the second line is a $D$-term, dictated by the gauge quantum numbers of the Higgs doublets. Notice that it is the only one that contains quartic couplings, which are thus completely fixed in terms of the SM gauge couplings $g$ and $g'$. The soft SUSY-breaking terms are displayed in the last line. The potential in eq. (\[Higgspot\]) allows, with the appropriate choice of its parameters, EWSB to occur. Also, it allows (or better, generically requires) both doublets to get a VEV |H\_[[u]{}]{}|\^2=2,|H\_[[d]{}]{}|\^2=2. The sum of the square of the two VEVs is fixed to $v_{\rm{u}}^2+v_{\rm{d}}^2=v^2$, where $v\simeq246$ GeV, but the ratio between them is a free parameter, which is typically traded for the tangent of the “$\beta$” angle . Notice that both Higgses taking VEV is necessary in order to generate quark masses since, as we discussed, the up- and down-type Yukawa couplings are only present for $H_{\rm{u}}$ and for $H_{\rm{d}}$, respectively.
With the knowledge of the 80’s, the potential in eq. (\[Higgspot\]) is quite successful. It produces realistic EWSB and fermion masses, and an extended Higgs sector which was perfectly plausible at that times, in which almost no experimental information was available on Higgs physics. After LEP could not discover the Higgs boson and set a lower bound $m_H>115$ GeV, the potential (\[Higgspot\]) started being in tension with observations. Indeed, it is possible to show that the structure of the potential is such that the Higgs mass is unavoidably smaller than the one of the $Z$ boson. More precisely, it turns out that for any choice of the free parameters one has \[mhSUSY\] m\_H|2| m\_Zm\_Z. The relation follows from the fact that the quartic terms in the potential are not free parameters, but instead they are uniquely dictated, through supersymmetry, by the gauge coupling. In order to see how this works, consider a simplified limit, the so-called “decoupling limit”, in which the soft mass of the $H_{\rm{d}}$, $m_{\rm{d}}^2$, is taken to be large. In the limit, $H_{\rm{d}}$ decouples and it can be just ignored (i.e., set to zero) in eq. (\[Higgspot\]), obtaining a SM-like potential V=\_[[SM]{}]{}\^2|H\_[[u]{}]{}|\^2+\_[[SM]{}]{}|H\_[[u]{}]{}|\^4, with $\mu_{\rm{SM}}^2=\mu^2+m_{\rm{u}}^2$ and $\lambda_{\rm{SM}}=(g^2+g^{\prime2})/8$. The habitual SM formula $m_H=\sqrt{2\lambda} v$ thus tells us that $m_H=m_Z$ in the decoupling limit. This matches eq. (\[mhSUSY\]) because in the decoupling limit one finds that $\tan\beta\rightarrow\infty$, i.e. $\beta\rightarrow\pi/2$.
Since the mass relation in eq. (\[mhSUSY\]) is violated experimentally, we might wander if it excludes SUSY as a realistic theory of Nature. Of course it does not, because of two reasons, but it has important implications. The first point is that eq. (\[mhSUSY\]) is only valid at the tree-level order, radiative corrections violate it. For instance top and stop loops contribute to the quartic by an amount \~, so that by making stops heavy one can get a large enough quartic and a large enough Higgs mass. Working in the decoupling limit for simplicity (and because it is the most favourable one), the shift we need on $\lambda$ is =0.06, M\_1.3 . That heavy stops cost quite a lot of fine-tuning, definitely above ten. More refined estimates [@Hall:2011aa], taking into account the need of a separation between $M_\slashed{\rm{S}}$ and the weak scale (i.e., of some log enhancement in the tuning), reveal that the tuning needed to accomodate the $125$ GeV Higgs mass is at least $100$ .
The second reason why eq. (\[mhSUSY\]) cannot disprove supersymmetry is that it only holds in the MSSM, thus it is not a robust property of SUSY models. It can be violated in SUSY scenarios like $\lambda$SUSY [@Barbieri:2006bg] (A.K.A. NMSSM), in which an extra singlet chiral super-field $\Phi_S$ is added to the theory with a $\lambda \Phi_S \Phi_{\rm{u}} \Phi_{\rm{d}}$ term in the super-potential. This contributes to the quartic Higgs coupling and leads to a heavy enough Higgs boson if $\lambda$ is sufficiently large. The main drawback of this construction is that $\lambda$ needs to be relatively big, therefore its RG-running is very fast and reaches a Landau pole not much above the $10$ TeV scale. The alternative to a fine-tuned scenario seems thus to be a model which cannot be extended far above the TeV scale. This clearly seems very different from the basic SUSY picture we had in mind in fig. \[SUSYPIC\]. However there might be caveats, new model-building ways out, or space for a “partially Un-Natural”, but still true, SUSY model at the TeV scale. Let us wait and see, the LHC run-$2$ will tell us more about SUSY.
Conclusions and Outlook
=======================
My purpose, when giving these lectures, was to outline that BSM physics is not (only) a collection of models, but rather a set of structural questions on fundamental physics and of possible answers to be checked against data. The microscopic origin of the Higgs mass, in connection with Naturalness (or Un-Naturalness), is only one of such questions. However it is the one about which decisive experimental progress will be made at the LHC, this is why I built the lectures around it. Several other relevant questions and ideas were encountered during the lectures, among which GUT, DM, neutrino masses and vacuum stability, each of which deserves a separate course. Some of these courses were given at this School [@pklect; @dslect]. For what is missing, the lectures in Ref. [@Pomarol:2012sb] are a valid starting point. The course aimed at providing a pedagogical introduction to BSM physics, for this reason basic material was presented and many recent developments were left out from the discussion. This should not obscure the fact that “Natural” BSM model-building is an active research area. Approaches related with the “Twin Higgs” mechanism [@Chacko:2005pe] are worth mentioning in this context.
Concerning the future of BSM physics, there is not much I can add to what discussed in sect. \[wf\]. There is not guarantee that the ongoing LHC program will produce a new physics discovery, but it is sure that it will improve our comprehension of fundamental interactions. This is more than enough to work on LHC physics to the best of our abilities. On a longer timescale, the future is impossible to predict. We will definitely keep asking structural questions on fundamental physics, however it is unclear if high-energy collider experiments will continue being the optimal investigation tools to search for answers. My viewpoint is well summarised by a famous sentence\
*“Learn from yesterday, live for today, hope for tomorrow.*
*The important thing is not to stop questioning.”*
– Albert Einstein
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[^1]: Still, the ongoing LHC program makes the direct exploration of the energy frontier the most promising tool of investigation we currently have to our disposal. Also, one should not forget the strong impact of Flavour physics [@sglect], because of its capability of indirectly exploring very high-energy scales, on BSM physics.
[^2]: Of course the Cabibbo angle was also needed in order to describe hadronic Weak processes.
[^3]: The presence of light feebly coupled BSM particles would not affect the considerations that follow.
[^4]: See Refs. [@Barbieri:2013vca] and [@Giudice:2008bi] for recents essays on the Naturalness problem. The problem was first formulated in Refs. [@thooftNat] and [@Dimopoulos:1979es; @Susskind:1978ms], however according to the latter references it was K.Wilson who first raised the issue.
[^5]: There is also the cosmological constant term, of $d=0$. It poses another Naturalness problem that I will mention later.
[^6]: Actually $G_F$ is taken as an input parameter in actual calculations because it is better measured than $g_W$ and $m_W$, but this doesn’t affect the conceptual point we are making.
[^7]: Deciding whether or not negative LHC results will have the last word on Naturalness is a matter of taste, to some extent, since it is unclear how much tuning we can tolerate. It also depends on how good we will be in searching for Natural new physics and consequently how strong and robust the limit on $\Delta$ will actually be. It is nevertheless undoubtable that negative LHC results will put the idea of Naturalness in serious troubles.
[^8]: The word “dynamical” is used here in its proper sense, related with evolution in the course of time. It has nothing to do with the generation of energy scales (e.g,, the QCD confinement scale) induced by an underlying strongly-coupled theory, which is also said to be a “dynamical” generation mechanism.
[^9]: For instance, certain Randall-Sundrum models are reformulations of the Composite Higgs scenario with or without the Higgs being a pseudo-Nambu–Goldstone Boson (pNGB). Little Higgs (see [@Schmaltz:2005ky; @Perelstein:2005ka] for a review) is a pNGB Higgs endowed with a special mechanism which could make it more Natural. Twin Higgs [@Chacko:2005pe] is an additional protection for $m_H$ which postpones the emergence of coloured particles in the spectrum. It can be applied both to the Composite Higgs and to the SUSY scenario.
[^10]: Those particles might be “partially composite”, a concept that we will introduce below.
[^11]: See [@Kaplan:1983fs; @Kaplan:1983sm; @Dugan:1984hq] for earlier references and [@Agashe:2004rs; @Giudice:2007fh] for more recent ones.
[^12]: We nowadays know this directly from the LHC Higgs couplings determinations. Indirect evidences of SM-like couplings for the Higgs boson could however already be extracted from precision LEP data.
[^13]: \[cuss\]This group is also called the “custodial” . It plays a major role in BSM physics as it suppresses certain BSM effects constrained by LEP and often helps the compatibility of BSM models with data.
[^14]: I’m being quite sloppy here. In order to make the thing work one needs to enlarge the global group of the CS promoting it to ${\mathpzc{G}}={\textrm{SO}}(5)\times{\textrm{U}}(1)_X$ and to change the definition of the SM Hypercharge into $Y=T_R^3+X$, with $X$ the charge under the newly introduced ${\textrm{U}}(1)_X$ group. It is only by giving an $X$ charge of $2/3$ to ${\mathcal{O}}$ and to $Q_L$ and $T_R$ that one finds a ${\mathbf{2}}_{1/6}$ and of a ${\mathbf{1}}_{2/3}$ in the decomposition and eq. (\[pcrew\]) truly complies with gauge invariance.\[u1x\]
[^15]: The one that follows is an approximate formula for the Higgs potential in models where the fermionic operators in the top quark sector are in the ${\mathbf{5\oplus5}}$ or in the ${\mathbf{14\oplus1}}$ configurations. The connection between the Higgs potential and the top quark sector will be explained later. Further details can be found in Chapter 3 of Ref. [@Panico:2015jxa].
[^16]: $\rho$ cannot have spin greater than $1$ because a Lorentz vector operator cannot have a non-vanishing matrix element between the vacuum and a high-spin particle. Massless scalars can instead be excited from the vacuum by a conserved current if it is associated with a spontaneously broken generator. These scalars are nothing but the NGB’s of the theory we already discussed extensively.
[^17]: \[ET\]The correspondence between longitudinally polarised vector bosons and the so-called “unphysical” components of the Higgs field (i.e., the charged $h_u$ and the imaginary part of the neutral $h_d$ component of the doublet) is ensured by the Equivalence Theorem [@Horejsi:1995jj]. It holds at energies much above the vector boson masses, which is an excellent approximation for our purposes. In practice the theorem says that the Feynman amplitudes with longitudinal vector bosons on the external legs can be equivalently be computed as the amplitude for the corresponding scalar fields.
[^18]: The $\rho$ can also be produced in vector boson fusion (VBF) through the $c_H$ operator (\[C\_H\]), however the VBF rate is too small to be relevant, at least at the current stage of the LHC.
[^19]: This is not necessarily accurate for the channels involving third family quarks. The large $\lambda$ coupling of the third family produced extra contribution to the vertex that can easily overcome the one from the mixing in eq. (\[C\_F\]). This enhances $\rho_0\rightarrow{tt}$ and $\rho_\pm\rightarrow{tb}$ making them promising search channels [@Liu:2015hxi]. Composite HVT’s might also dominantly decay to other composite sector particles like the fermionic top partners [@Bini:2011zb], if kinematically allowed. These decays can also be searched for.
[^20]: VLQ’s are somehow similar to a fourth family of quarks, but they are also radically different in that their vector-like mass allows them to be at the TeV scale without need of huge Yukawa couplings. Unlike a fourth family, VLQ’s are not excluded by the measurement of the Higgs production rate from gluons. See [@Azatov:2011qy] for an analysis of the (moderate) impact of CH top partners on Higgs physics.
[^21]: Order one coefficients, which of course are be predicted by the power-counting formula (\[1s1c\]), are understood in both terms.
[^22]: This would not be the case if a parametrically large separation was present between $M_\Psi$ and the confinement scale $m_*$ at which the loop is naturally cut off. We assume a factor of a few separation at most, which does not qualifies as parametrically large and thus the estimate is correct.
[^23]: There are however exceptions to this rule. On one hand, it is possible to make largely composite one of the light quarks chirality components recovering the small Yukawa by giving very very small compositeness to the other one. This helps in evading flavour constraints [@Redi:2011zi; @Barbieri:2012uh] and produces interesting LHC signatures related with the fermionic partners of the light quarks [@Redi:2013eaa; @Delaunay:2013pwa]. On the other hand, it is possible to avoid partial compositeness altogether for the light fermions [@Matsedonskyi:2014iha; @Panico:2016ull] and obtain their mass by bilinear interactions.
[^24]: Eq. (\[tpptf\]) does not supersede eq. (\[xitun\]). The two equations estimate tuning from different sources, namely the one from the Higgs VEV and from the Higgs mass, respectively. Therefore the maximum of the two expressions should be taken for a complete estimate of $\Delta$.
[^25]: At least $22$ of them, counting only those produced by the CERN ESHEP school founded in $1993$.
[^26]: This assumes that the Weyl basis is chosen for the $\gamma$ matrices, otherwise the decomposition is more complicate.
[^27]: Weyl spinor indices can be raised or lowered by acting with $\varepsilon^{\alpha\beta}=\varepsilon^{\dot\alpha\dot\beta}=-\varepsilon_{\alpha\beta}=-\varepsilon_{\dot\alpha\dot\beta}$. With this convention the definition of ${\overline{Q}}_{\dot{\alpha}}$ reported below matches with eq. (\[LR\]).
[^28]: By “degree of freedom” we mean single-particles states of given helicity and quantum numbers.
[^29]: More precisely, an invariant Action since SUSY Lagrangian are often only invariant up to total derivatives
[^30]: The chiral super-fields ${\mathcal{W}}_\alpha$ are a SUSY generalisation of the field-strength in ordinary gauge theories. Their definition is not worth reporting here.
[^31]: From now on, since the low-energy (IR) theory we are considering to compute the low energy contribution is not anymore the SM, I will substitute “$\delta_{\textrm{SM}}$” with “$\delta_{\textrm{IR}}$” and “$\Lambda_{\textrm{SM}}$” with $\Lambda$, representing the cutoff scale of the IR theory.
[^32]: And at the LHC run-$2$, however here I stick to the run-$1$ results, the only ones that were available when I gave the lectures.
[^33]: Tree-level contributions to $m_H^2$ emerge from higgsinos, and thus the Naturalness threshold on these particles is extremely low. However there no tension with the experimental bounds, which are too weak.
[^34]: The cancellation of gauge anomalies also requires two Higgs super-fields.
[^35]: The contraction with the $\varepsilon_{ij}$ tensor is understood in last term of the equation that follows.
|
---
abstract: 'Recently, a new class of codes, called sparse superposition or sparse regression codes, has been proposed for communication over the AWGN channel. It has been proven that they achieve capacity using power allocation and various forms of iterative decoding. Empirical evidence has also strongly suggested that the codes achieve capacity when spatial coupling and approximate message passing decoding are used, without need of power allocation. In this note we prove that state evolution (which tracks message passing) indeed saturates the potential threshold of the underlying code ensemble, which approaches in a proper limit the optimal threshold. Our proof uses ideas developed in the theory of low-density parity-check codes and compressive sensing.'
author:
- '\'
bibliography:
- 'IEEEabrv.bib'
- 'bibliography.bib'
title: |
Proof of Threshold Saturation for\
Spatially Coupled Sparse Superposition Codes
---
[Threshold Saturation for Spatially Coupled Sparse Superposition Codes]{}
Introduction
============
Sparse superposition (SS) codes introduced by Barron and Joseph for reliable communication over the additive white Gaussian noise channel (AWGNC) have been proven to approach capacity using power allocation and various efficient decoders [@barron2010sparse; @JosephB14; @barron2012high]. An approximate message passing (AMP) decoder was introduced in [@barbier2014replica], and the recent analysis [@rush2015capacity] proves that this allows to reach capacity with the help of power allocation. Spatially coupled SS codes were introduced in [@barbierSchulkeKrzakala; @BarbierK15] and empirically shown to reach capacity under AMP [*without any need for power allocation*]{}. The empirical evidence shows that spatially coupled SS codes perform better than power allocated ones in the sense that they approach capacity faster in a proper limit [@BarbierK15]. Given this evidence, it is of interest to develop a rigorous theory for such coding constructions.
It is natural to address two conjectures. First, that spatially coupled SS codes allow to reach the so-called *potential threshold* of state evolution (SE). Second, that SE correctly tracks the AMP decoder. As we will argue, the potential threshold tends to capacity in a proper limit, so this would prove that the codes are capacity achieving.
The purpose of this note is to settle the first conjecture. We prove that for a general ensemble of spatially coupled coding matrices, the AMP threshold attains (in a suitable limit) the potential threshold of SE. This phenomenon is called [*threshold saturation*]{}. The precise statements of our main results are Theorem \[th:mainTheorem\] and Corollary \[cor:maincorollary\].
Threshold saturation was first established in the context of spatially coupled Low-Density Parity-Check codes for general binary input memoryless symmetric channels in [@Kudekar-Urbanke-Richardson-2013], and is recognized as the basic mechanism underpinning the excellent performance of such codes [@Zigangirov-Costello-2010]. It is interesting that essentially the same phenomenon can be established for a coding system operating on a channel with [*continuous inputs*]{}. This result is a stepping-stone towards establishing that spatially coupled SS codes achieve capacity on the AWGNC under AMP decoding. The remaining analysis to settle the second conjecture would require extending the one given for compressive sensing [@Montanari-Javanmard] to signals with correlated components in a spatially coupled system, as already done for the power allocated system [@rush2015capacity].
To establish threshold saturation, we use the *potential method* along the lines of [@YedlaJian12; @PfisterMacrisBMS] developed for LDPC and LDGM codes. Note that a similar (but different) potential to the one used here has been introduced in the context of scalar compressive sensing [@BayatiMontanari10; @6887298]. It is interesting that the potential method goes through for the present system involving a dense coding matrix and a fairly wide class of spatial couplings [@KrzakalaMezard12].
Coding constructions of the underlying and coupled ensembles are described in Sec. \[sec:codeens\]. Sec. \[sec:stateandpot\] reviews SE and potential formulations adapted to the present context. The AMP thresholds of underlying and coupled ensembles as well as the potential threshold are given precise definitions. Sec. \[sec:propCoupledSyst\] introduces a notion of degradation and settles monotonicity properties of SE. The essential steps for the proof of threshold saturation are presented in Sec. \[sec:proofsketch\].
Code ensembles {#sec:codeens}
==============
We first define the underlying and spatially coupled ensembles of SS codes for transmission over an AWGNC. We will often use the shorter notations $[a_1 : a_n]$ and $\{b_1 : b_n\}$ instead of $[a_1, \dots, a_n]$ and $\{b_1, \dots , b_n\}$ for $n$-tuples.
The underlying ensemble
-----------------------
In the framework of SS codes, the *information word* $\bs = [\bs_1 : \bs_L]$ is a vector made of $L$ *sections*. Each section $\bs_l$, $l\in\{1:L\}$, is a $B$-dimensional vector with one component equal to $1$ and $B-1$ components equal to $0$. For example if $(B=3,L=4)$, a valid information word is $\tbf s = [001,010,100,010]$. We call $B$ the *section size* (or alphabet size) and set $N=LB$. A *codeword* $\bF\bs \in \mathbb{R}^{M}$ is generated from a fixed *coding matrix* $\bF\in \mathbb{R}^{M \times N}$. We consider random codes generated by $\bF$ drawn from the ensemble of random matrices with i.i.d real Gaussian entries with distribution $\mathcal{N}(0, 1/L)$. The cardinality of the code is $B^L$, the block length is $M$, and the (design) rate is $R=L\log_2(B)/M = N\log_2(B)/(M B)$. The code is thus specified by the basic parameters $(M, R, B)$.
Codewords are transmitted through an AWGNC, i.e., the received signal is $\by= \bF\bs +\boldsymbol\xi$ where $\xi_\mu \sim\mathcal{N}(0, \sigma^2) \ \forall \ \mu$. Power is normalized thanks to the $1/L$ variance of the entries of $\bF$, so that the signal-to-noise ratio is ${\rm snr = 1/\sigma^2}$.
The decoding task is to infer the information word $\bs$ from channel observations $\by$. This is obviously very similar in spirit to compressive sensing, where one wants to infer a sparse signal from a certain number of measurements. For this reason, our language and techniques are sometimes borrowed from compressive sensing. In particular, the code rate can be linked to a “measurement rate” $\alpha \defeq M/N = \log_2(B)/ (BR)$.
The spatially coupled ensemble
------------------------------
![A spatially coupled coding matrix $\in\mathbb{R}^{M\times N}$ made of $\Gamma\times \Gamma$ blocks indexed by $(r,c)$, each with $N/\Gamma$ columns and $M/\Gamma =\alpha N/\Gamma$ rows. The i.i.d elements in block $(r,c)$ have distribution $\mathcal{N}(0,J_{r,c}/L)$. Away from the boundaries, in addition to the diagonal (in red), there are $w$ forward and backward coupling blocks. In this example, the design function $g_w$ enforces a stronger backward coupling. Blocks are darker at the boundaries because the variances are larger so as to enforce the *variance normalization* $\sum_{c=1}^\Gamma J_{r,c}/\Gamma = 1 \ \forall \ r$. The yellow shape emphasizes *variance symmetry* $\sum_{r=1}^\Gamma J_{r,k}/\Gamma=\sum_{c=1}^\Gamma J_{k,c}/\Gamma=1$ verified if $k\in \{2w+1:\Gamma-2w\}$.[]{data-label="fig:opSpCoupling"}](./figures/seededMat){width=".45\textwidth"}
The construction has similarities with [@CaltagironeZ14]. We consider spatially coupled codes based on coding matrices in $\mathbb{R}^{M\times N}$ as described in details in Fig. \[fig:opSpCoupling\]. This ensemble of matrices is parametrized by $(M,R,B,\Gamma,w,g_w)$, where $w$ is the *coupling window* and $g_w$ is the *design function*. This is any function verifying $g_w(x) = 0$ if $|x|>1$ and $g_w(x)\ge g_0>0$ else, which is Lipschitz continuous on its support with Lipschitz constant $g_*$. The constants $g_0$, $g_*$ are independent of $w$. Moreover, we impose the normalization $\sum_{k=-w}^{w} g_w(k/w)/(2w+1) = 1$.
From $g_w$, we construct the *variances* of the blocks: the i.i.d entries inside the block $(r,c)$ are distributed as $\mathcal{N}(0,J_{r,c}/L)$, where $J_{r,c} \defeq \gamma_r \Gamma g_w((r-c)/w)/(2w+1)$. Here $\gamma_r$ enforces the *variance normalization* $\sum_{c=1}^{\Gamma}J_{r,c}/\Gamma = 1 \ \forall \ r$. This implies (by the law of large numbers) that $[\bF \bs]_\mu^2 \to 1 \ \forall \ \mu$ as $L\to\infty$, i.e. the asymptotic power $\lim_{L\to \infty}\sum_{\mu=1}^M [\bF \bs]_\mu^2/M = 1$, and the ${\rm snr} = 1/\sigma^2$ is homogeneous.
The spatial coupling induces a natural decomposition of the signal into $\Gamma$ *blocks* (associated with the block-columns of the matrix), each made of $L/\Gamma$ sections. One key element of spatially coupled codes is the *seed* introduced at the boundaries. We assume the sections in the first and last $4w$ blocks of the information word ${\tbf s}$ to be known by the decoder. This boundary condition can be interpreted as perfect side measurements that propagate inward and boost the performance. The seed induces a rate loss in the *effective rate* of the code, $R_{\text{eff}} = R (1-\frac{8w}{\Gamma})$. However, this loss vanishes as $\Gamma \rightarrow \infty$.
State evolution and potential formulation {#sec:stateandpot}
=========================================
We now give the [*state evolution*]{} associated to the underlying and spatially coupled ensembles, which is conjectured to track the performance of the AMP decoder. We then define an appropriate [*potential function*]{} for each ensemble.
State evolution
---------------
The goal is to iteratively compute the mean square error (MSE) $\tilde E^{(t)} =\frac{1}{L}\mathbb{E}_{\bs, \by}[\sum_{j=1}^N
(\hat{s}_j^{(t)} - s_j)^2]$ of the AMP estimates $\{\hat{s}_j^{(t)}\}$ at iteration $t\in \mathbb{N}$. We need a few definitions to express the iterations. Let $\Sigma(E) \defeq \sqrt{R(\sigma^2 +E)}$. We define a [*denoiser*]{} $f_i(\Sigma(E))$ as the minimum mean square error (MMSE) estimator of the $i$-th component of a section sent through an [*effective*]{} AWGNC with a noise $\mathcal{N}(0, \Sigma(E)^2/\log_2(B))$, when the input signal is uniformly distributed. Explicitly, if $\bz = [z_1 : z_B]$ denote $B$ independent standardized Gaussian random variables (with zero mean and unit variance), and $p_0(\bx) = \frac{1}{B}\sum_{i=1}^B\delta_{x_i, 1}\prod_{k\neq i}^B\delta_{x_k, 0}$, we set for $i=1 : B$ $$\begin{aligned}
& f_i(\Sigma(E)) \defeq
\frac{\sum_{\bx} e^{-\frac{\|\bx - (\bs +\bz \Sigma(E)/\sqrt{\log_2(B)})\|_2^2}{2\Sigma(E)^2/\log_2(B)}} p_0(\bx) x_i}
{\sum_{\bx} e^{-\frac{\|\bx - (\bs +\bz \Sigma(E)/\sqrt{\log_2(B)}) \|_2^2}{2\Sigma(E)^2/\log_2(B)}} p_0(\bx)}
\nonumber \\ &
=
\Bigg[1+\sum_{k\neq i}^B
e^{(s_k- s_i)\log_2(B)/\Sigma(E)^2 + (z_k - z_i)\sqrt{\log_2(B)}/\Sigma(E)}\Bigg]^{-1}.
\nonumber\end{aligned}$$ Note that the denoiser also depends on $\bs, \bz$. Furthermore, we define the [*SE operator of the underlying system*]{} as $$\begin{aligned}
\label{equ:stateevopunderlying}
T_\text{u}(E) \defeq \mathbb{E}_{\bs, \bz}\biggl[\sum_{i=1}^B \big(f_i(\Sigma(E)) - s_i\big)^2\biggr].\end{aligned}$$ This is nothing else than the MSE associated to the MMSE estimator of the effective AWGNC with noise $\mathcal{N}(0, \Sigma(E)^2/\log_2(B))$. The SE iterations for the underlying system’s MSE can now be expressed as $$\begin{aligned}
\label{equ:stateevolutionunderlying}
\tilde E^{(t+1)} = T_\text{u}(\tilde E^{(t)}), \qquad t\geq 0.\end{aligned}$$ To track the performance of the AMP decoder the iterations are initialized with $\tilde E^{(0)}=1$. The monotonicity and boundedness of the iterations of SE, discussed in Sec. \[sec:propCoupledSyst\], ensure that actually all initial conditions reach a fixed point.
The MSE floor $E_0$ is defined as the fixed point reached from the initial condition $\tilde E^{(0)} = 0$. In other words $E_0 = T_\text{u}^{(\infty)}(0)$.
The basin of attraction of the MSE floor $E_0$ is $\mathcal{V}_0 \defeq \big\{ E \ \! |\ \! T_\text{u}^{(\infty )}(E) = E_0 \big\}$.
\[Threshold of underlying ensemble\] The AMP threshold is $R_{\text{u}} \defeq {\rm sup}\{R>0\ \! |\ \! T_{\text{u}}^{(\infty)}(1) = E_0\}$.
It can be shown that for the present system $T_{\text{u}}^{(\infty)}(0)$ and $T_{\text{u}}^{(\infty)}(1)$ are the only two possible fixed points. For $R<R_{\text{u}}$, there is only one fixed point, namely the “good” one $T_{\text{u}}^{(\infty)}(0)$, and for large section size $B$ the MSE floor and section error rate are small. Instead if $R>R_{\text{u}}$, the decoder is blocked by the “bad” fixed point $T_{\text{u}}^{(\infty)}(1)\neq E_0$ and AMP cannot decode.
We now turn our attention to SE for the spatially coupled system. The perfomance of AMP is described by an [*MSE profile*]{} $\{\tilde E_c \ \! |\ \! c= 1: \Gamma\}$ along the “spatial dimension”. Since we assume that the boundary sections are known to the decoder, we enforce the *pinning condition* $\tilde E_c =0$ for $c\in\{1: 4w\}\cup\{\Gamma-4w+1 : \Gamma\}$. For $c$ not in this set, by definition $\tilde E_c \defeq \frac{\Gamma}{L} \sum_{l\in c} \mathbb{E}_{\bs, \by}[\|\hat{\tbf s}_l - {\tbf s}_l\|_2^2]$, where the sum $l\in c$ is over the set of indices of the $L/\Gamma$ sections composing the $c$-th block of $\bs$. It is more convenient to work with a *smoothed error profile* (referred as the error profile) ${\tbf E}= \{E_r \ \! | \ \! r=1 : \Gamma\}$, $E_r \defeq \frac{1}{\Gamma} \sum_{c=1}^{\Gamma} J_{r,c} \tilde E_c$. Indeed, this change of variables makes the problem mathematically more tractable for spatially coupled codes. The pinning condition becomes $E_r=0$ for $r\in\mathcal{A}\defeq\{1: 3w\}\cup\{\Gamma-3w+1: \Gamma\}$.
We define an [*effective noise*]{} for block $c\in\{1: \Gamma$}, $$\label{equ:effectivenoisecoupled}
\Sigma_{c}({\tbf E} )\defeq \biggl[\frac{1}{\Gamma}\sum_{r=1}^{\Gamma} \frac{J_{r,c}}{R(\sigma^2 +E_r)}\biggr]^{-1/2}$$ and the [*coupled SE operator*]{} $$\label{equ:stateevopcoupled}
[T_{\rm c}({\tbf E})]_r = \frac{1}{\Gamma}
\sum_{c=1}^{\Gamma}J_{r, c}\mathbb{E}_{\bs, \bz}\biggl[\sum_{i=1}^{B}\big(f_i(\Sigma_{c}({\tbf E} )) - s_i\big)^2\biggr].$$ $T_{\rm c}({\tbf E})$ is vector valued and here we have written its $r$-th component. The SE iterations then read for $r\notin \mathcal{A}$ $$\label{equ:statevolutioncoupled}
E_r^{(t+1)} = [T_{\rm c}({\tbf E}^{(t)})]_r, \qquad t\geq 0.$$ For $r\in \mathcal{A}$, the pinning condition $E_r^{(t)} =0$ is enforced at all times. SE is initialized with $E_r^{(0)}=1$ for $r\notin \mathcal{A}$.
Let ${\tbf E}_0 \defeq [E_r=E_0, r=1: \Gamma]$ be the *MSE floor profile*.
\[Threshold of coupled ensemble\]\[def:AMPcoupled\] The AMP threshold of the coupled system is defined as $R_{\text c} \defeq {\liminf}_{\Gamma, w\to \infty} {\rm sup}\{R>0\ \! |\ \! T_{\text{c}}^{(\infty)}(\boldsymbol{1}) \prec \tbf E_0\}$ where $\boldsymbol{1}$ is the all ones vector. Here the ${\liminf}_{\Gamma, w \to \infty}$ is taken along sequences where [*first*]{} $\Gamma \to \infty$ and [*then*]{} $w\to\infty$.
Potential formulation {#subsec:potentials}
---------------------
The fixed point equations associated to SE iterations can be reformulated as stationary point equations for suitable [*potential functions*]{}. These are in general not unique. However, the “correct” guess (i.e. the one that allows to prove full threshold saturation) can be derived by the replica method of statistical physics [@barbier2014replica] . The potential of the underlying ensemble is $F_{\text{u}}( E) \defeq U_{\text{u}}(E) - S_{\text{u}}( \Sigma( E))$, with $$\begin{aligned}
\begin{cases}
U_{\text{u}}(E) \defeq \frac{1}{2R} \log_2\Big((\sigma^2 + E) e^{-\frac{ E}{\sigma^2 + E}}\Big), \\
%
S_{\text{u}}( \Sigma( {E})) \defeq \mathbb{E}_\bz\bigr[\log_B\big(1+\sum_{i=2}^B e_{i}(\Sigma( {E}))\big)\bigl],
\end{cases}\end{aligned}$$ where $e_{i}(x) \defeq \exp\big((z_i-z_1)\sqrt{\log_2(B)}/x-\log_2(B)/x^2 \big)$.
The potential of the spatially coupled ensemble is $F_{\text{c}}({\tbf E}) = U_{\text{c}}({\tbf E}) - S_{\text{c}}({\tbf E})$ where $$\begin{aligned}
\begin{cases}
U_{\text{c}}(\tbf{E}) \defeq \sum_{r=1}^{\Gamma} U_{\text{u}}( E_r), \\
%
S_{\text{c}}(\tbf{E}) \defeq\sum_{c=1}^{\Gamma} S_{\text{u}}( \Sigma_c({\tbf E})).
\end{cases}\end{aligned}$$
Let us pause for an instant and give the statistical physics interpretation of these formulas. The posterior distribution $p(\bs\vert \by) = \exp(-\|\by - \bF\bs\|_2^2/2\sigma^2) p_0(\bs) /Z$ ($Z$ the normalizing factor) can be interpreted as the Gibbs distribution of a disordered spin system ($\bs$ being the annealed “spin” degrees of freedom, $\by$ and $\bF$ the quenched “disorder”). The potential functions $F$ are “Bethe free energies” averaged over the disorder. They are equal to “energy” terms $U$ minus “entropy” terms $S$. One can prove that both terms are increasing in $\Sigma$. This is “physically” natural if the effective channel noise $\Sigma$ is interpreted as a kind of effective “temperature”.
The free energy gap is $\Delta F_{\text{u}} \defeq {\rm inf} _{E \notin \mathcal{V}_0 } (F_{\text{u}}( E) - F_{\text{u}}(E_0))$, with the convention that the infimum over the empty set is $\infty$ (this happens for $R < R_{\text{u}}$).
\[def:potThresh\] The potential threshold is $R_{\rm pot} \defeq {\rm sup}\{R>0\ \! |\ \! \Delta F_{\text{u}} > 0\}$.
The connection between these potentials and SE is given by the following Lemma.
\[lemma:fixedpointSE\_extPot\] If $\mathring{E}$ is a fixed point of , i.e., $T_{\text u}(\mathring E) = \mathring E$, then $[\frac{\partial F_{\text{u}}}{\partial E}]_{\mathring E} =0$. Similarly for the coupled system, if $\mathring{{\tbf E}}$ is a fixed point of then $[\frac{\partial F_{\text{c}}}{\partial E_r}]_{\mathring{{\tbf E}}} = 0 \ \forall \ r\in \{3w+1 : \Gamma-3w\}$.
The proof is technical and we skip it here. Let us just indicate that it proceeds by computing the derivatives of the potentials, uses Gaussian integration by parts formulas and channel symmetry. Similar results can be found in [@YedlaJian12; @PfisterMacrisBMS]. It is noteworthy that what is called a “Bethe entropy” in the statistical physics literature, has an information theoretic interpretation, and is actually a Shannon conditionnal entropy for an effective channel. Let $\bX$ be a $B$-dimensional random vector with distribution $p_0(\bx) = \frac{1}{B}\sum_{i=1}^B\delta_{x_i, 1}\prod_{k\neq i}^B\delta_{x_k, 0}$. Take the output ${\tbf Y}$ of an AWGNC with i.i.d noise $\mathcal{N}(0, \Sigma^2/\log_2(B))$ for each component, when the input is $\bX$. Then it is an exercise to check that $H(\bX \vert {\tbf Y}) = S_{\text{u}}(\Sigma) \log_2(B)$. This identification clearly shows that $S_{\text{u}}(\Sigma)$ must be an increasing function of $\Sigma$. Also, it allows to give information theoretic expressions for the potential functions. We note that such expressions have already been written down for [*scalar*]{} compressive sensing [@BayatiMontanari10; @6887298].
We also point out that there is another way to derive potential functions by integrating out the SE fixed point equations after premultiplication by an “integrating factor”. When the correct “integrating factor” is used, one recovers the information theoretic expressions of the potential functions. This method is discussed in [@6887298] for a wide range of problems including scalar compressive sensing, and one can check it extends to the present setting. A key point for this method is the well known relation between mutual information (or conditional entropy) and MMSE [@guo2011estimation]. Here this relation takes the form $\frac{d H(\bX| {\tbf Y})}{d (\Sigma^{-2})} = - \frac{1}{2} {\rm mmse}(\Sigma)$ (here ${\rm mmse} = T_{\rm u}$ where $T_{\rm u}$ is the r.h.s of viewed as a function of $\Sigma$).
Large $B$ analysis and connection with Shannon’s capacity {#sec:larg_B}
---------------------------------------------------------
Let us now emphasize the connection between the potential threshold $R_{\rm pot}$ and Shannon’s capacity $C= \frac{1}{2}\log_2(1+{\rm snr})$. The large section size limit of the potential of the underlying system becomes [@phdBarbier] $$\begin{aligned}
\lim_{B\to\infty} F_{\text{u}}(E) = U_{\text{u}}(E) - {\rm max}\Big(0,1 - \frac{1}{2\ln(2)\Sigma(E)^2}\Big),\end{aligned}$$ (where we recall $\Sigma(E)^2 \defeq R(\sigma^{2} +E)$). The analysis of this function of $E\in[0,1]$ shows the following. For $R<[(1+\sigma^2)2\ln(2)]^{-1}$ there is a unique minimum at $E=0$. For $[(1+\sigma^2)2\ln(2)]^{-1} < R < C$, $E=0$ is the global minimum but there exists a local minimum at $E=1$. When $R>C$ the global minimum is at $E=1$ and $E=0$ is a local minimum. Therefore we can identify $\lim_{B\to \infty} R_{\rm pot} = C$ and $\lim_{B\to \infty} R_{\rm u} = [(1+\sigma^2)2\ln(2)]^{-1}$.
Let us also point out that these are static properties of the system which are independent of the decoding algorithms. In a sense they confirm in an alternative way that the codes must achieve capacity under optimal decoding [@leastsquareBarron].
Properties of the coupled system {#sec:propCoupledSyst}
================================
Monotonicity properties of the SE operators $T_{\rm u}$ and $T_{\rm c}$ are key elements in the analysis. We start by defining a suitable notion of [*degradation*]{}.
A profile ${\tbf{E}}$ is degraded (resp. strictly degraded) with respect to another one ${\tbf{G}}$, denoted as ${\tbf{E}} \succeq {\tbf{G}}$ (resp. ${\tbf{E}} \succ {\tbf{G}}$), if $E_r \ge G_r \ \forall \ r$ (resp. if ${\tbf{E}} \succeq {\tbf{G}}$ and there exists some $r$ such that $E_r > G_r$).
The SE operator of the coupled system maintains degradation in space, i.e., if ${\tbf E} \succeq {\tbf G}$, then $T_{\txt{c}}({\tbf E}) \succeq T_{\txt{c}}({\tbf{G}})$. This property is verified by $T_{\txt{u}}$ for a scalar error as well. \[lemma:spaceDegrad\]
From it is immediately seen that ${\tbf E} \succeq {\tbf G}$ implies $\Sigma_c({\tbf E}) \geq \Sigma_c({\tbf G})\ \forall \ c$. Now, the SE operator can be interpreted as an average over the spatial dimension of local MMSE’s. The local MMSE’s are the ones of $B$-dimensional AWGN channels with effective noise $\mathcal{N}(0,\Sigma_c^2/\log_2(B))$. The MMSE’s are increasing functions of $\Sigma_c^2$: this is intuitively clear but we provide an explicit formula for the derivative below. Thus $[T_{\txt{c}}( {\tbf E})]_r \geq [T_{\txt{c}}( {\tbf G})]_r \ \forall \ r$, which means $T_{\txt{c}}({\tbf E}) \succeq T_{\txt{c}}({\tbf{G}})$.
The derivative of the MMSE of the Gaussian channel with i.i.d noise $\mathcal{N}(0, \Sigma^2)$ can be computed as $$\begin{aligned}
\frac{d \ \! {\rm mmse}(\Sigma)}{d(\Sigma^{-2})} = - 2 \mathbb{E}_{\bX, {\tbf Y}}\bigl[\|\bX - \mathbb{E}[\bX\vert {\tbf Y}]\|_2^2 {\rm Var}[\bX\vert {\tbf Y}]\bigl].\end{aligned}$$ This formula is valid for vector distributions $p_0(\bx)$, and in particular, for our $B$-dimensional sections. It explicitly confirms that $T_{\rm u}$ (resp. $[T_{\txt{c}}]_r$) is an increasing function of $\Sigma$ (resp. $\Sigma_c$).
![A profile $\tbf{E}^{*}$ (solid) and its associated *saturated* non decreasing profile $\tbf E$ (dashed). The former has a plateau at $0$ for all $r \le 3w$ and it increases until $r_{\rm max}$ where it reaches its maximum value $E_{{\rm max}}$. Then it decreases to $0$ at $\Gamma-3w+1$ and remains null after. The saturated profile starts with a plateau at $E_0$ for all $r \le r_*$, where $r_*$ is the position defined by $E^{*}_{r_*}=E_0$, and then matches $\tbf E^{*}$ for all $r\in\{r_*:r_{\rm max}\}$. It then saturates to $E_{{\rm max}}$ for all $r \ge r_{\rm max}$. By construction, ${\tbf E}$ is non decreasing and verifies ${\tbf E} \succ {\tbf E}^{*}$.[]{data-label="fig:errorProfile"}](./figures/profile){width=".5\textwidth"}
\[cor:timeDegrad\] The SE operator of the coupled system maintains degradation in time, i.e., $T_{\txt{c}}({\tbf E}^{(t)}) \preceq {\tbf E}^{(t)}$, implies $T_{\txt{c}}({\tbf E}^{(t+1)}) \preceq {\tbf E}^{(t+1)}$. Similarly $T_{\txt{c}}({\tbf E}^{(t)}) \succeq {\tbf E}^{(t)}$ implies $T_{\txt{c}}({\tbf E}^{(t+1)}) \succeq {\tbf E}^{(t+1)}$. Furthermore, the limiting error profile $ {\tbf E}^{(\infty)} \defeq T_{\txt{c}}^{(\infty)}({\tbf E}^{(0)})$ exists. This property is verified by $T_{\txt{u}}$ for the underlying system as well.
The degradation statements are a consequence of Lemma \[lemma:spaceDegrad\]. The existence of the limits follows from the monotonicity of the operator and boundedness of the MSE.
Proof of threshold saturation {#sec:proofsketch}
=============================
The pinning step together with the monotonicity properties of the coupled SE imply that the [*fixed point*]{} profile ${\tbf E}^{*}$ must adopt a shape similar to Fig. \[fig:errorProfile\] (note $E_{\rm max} \in [0,1]$). We associate to ${\tbf E}^{*}$ a [*saturated profile*]{} denoted $\tbf E$. Its construction is described in detail in Fig. \[fig:errorProfile\]. The saturated profile $\tbf E$ is strictly degraded with respect to ${\tbf E}^{*}$, thus $\tbf E$ serves as an upper bound in our proof.
Coupled potential change evaluation by Taylor expansion
-------------------------------------------------------
The *shift operator* is defined pointwise as $[\text{S}({\tbf E})]_1 \defeq E_0, \ [\text{S}({\tbf E})]_r \defeq E_{r-1}$.
\[leminterp\] Let ${\tbf E}$ be a saturated profile. If $\hat {\tbf E} \defeq (1-\hat{t}) {\tbf E} + \hat{t}\text{S}({\tbf E})$ and $\Delta {E}_{r} \defeq {E}_{r} -{E}_{r-1}$, then for a proper $\hat{t}\in[0,1]$, $$\begin{aligned}
F_{\text{c}}(\text{S}({\tbf E})) &-F_{\text{c}}({\tbf E}) = \frac{1}{2} \sum_{r,r'=1}^{\Gamma} \Delta {E}_r \Delta {E}_{r'} \left[\frac{\partial^2 F_{\text{c}}}{\partial E_{r}\partial E_{r'}}\right]_{\hat {\tbf E}}.\end{aligned}$$ \[lemma:Fdiff\_quadraticForm\]
$F_{\text{c}}(\text{S}({\tbf E})) - F_{\text{c}}({\tbf E})$ can be expressed as $$\begin{aligned}
\frac{1}{2}\sum_{r,r'=1}^{\Gamma} \Delta {E}_{r} \Delta {E}_{r'} \left[\frac{\partial^2 F_{\text{c}}}{\partial E_{r}\partial E_{r'}}\right]_{\hat {\tbf E}} - \sum_{r=1}^{\Gamma} \Delta {E}_{r} \left[\frac{\partial F_{\text{c}}}{\partial E_r}\right]_{{\tbf E}}. \label{eq:expFbsminusFb}\end{aligned}$$ By saturation of $\tbf E$, $\Delta E_r=0 \ \forall \ r \in\mathcal{B}\defeq\{1:r_*\}\cup \{r_{\rm max}+1:\Gamma\}$. Moreover for $r\notin\mathcal{B}$, $E_r=[T_c(\tbf E)]_r$, and thus by Lemma \[lemma:fixedpointSE\_extPot\] the potential derivative cancels at these positions. Hence the last sum in (\[eq:expFbsminusFb\]) cancels.
The saturated profile ${\tbf E}$ is *smooth*, i.e. $|\Delta {E}_{r} | < (g_* + \tilde{g}) /w$ where $\tilde{g}\defeq \max (1+ g_*, g_0 + 2g_*)$. Recall $g_0$ and $g_*$ are independent of $w$ and $\Gamma$. \[lemma:Evariesslowly\]
$\Delta E_r=0$ for all $r \in\mathcal{B}$. By construction of $\{J_{r,c}\}$ and using Lipschitz continuity of $g_w$, we have for all $r \notin\mathcal{B}$ that $|\Delta {E}_{r}| = \big|\sum_{c} (J_{r,c} - J_{r-1,c}) \tilde E_c\big|/\Gamma < (g_* + \tilde{g})/w$.
\[lemma:quadFormBounded\] The coupled potential verifies $$\begin{aligned}
\frac{1}{2}\Bigg|\sum_{r,r'=1}^{\Gamma} \Delta {E}_{r} \Delta {E}_{r'}\left[\frac{\partial^2 F_{\text{c}}}{\partial E_{r}\partial E_{r'}}\right]_{\hat{\tbf E}}\Bigg| < K/w, \label{eq:quadFormBounded}\end{aligned}$$ where $K$ is independent of $w$ and $\Gamma$.
The proof uses Lemma \[lemma:Evariesslowly\] and the monotonicity of $E_r$. Bounding the Hessian is rather long but does not present major difficulties.
Direct evaluation of the coupled potential change
-------------------------------------------------
Let ${\tbf E}$ be a saturated profile such that ${\tbf E}\succ {\tbf E}_0$. Then $ E_{\rm max} \notin \mathcal{V}_0$. \[lemma:outside\_basin\]
The non decreasing error profile and the assumption that $ {\tbf E} \succ {\tbf E}_0$ imply that $ E_{\rm max} > E_{0}$. Moreover, $E_{\rm max} \le [T_\text{c}({\tbf E})]_{r_{\rm max}} \le T_\text{s}(E_{\rm max})$ where the first inequality follows from $\tbf E\succ \tbf E^*$ and the monotonicity of $T_\text{c}$, while the second comes from the variance symmetry at $r_{\rm max}$ and the fact that $\tbf E$ is non decreasing. Combining these with the monotonicity of $T_\text{s}$ gives: $T_{\text s}(E_{\rm max}) \ge E_{\rm max} \Rightarrow T_{\text s}^{(\infty)}( E_{\rm max}) \ge E_{\rm max}> E_0 \Rightarrow E_{\rm max} \notin \mathcal {V}_0$.
Let ${\tbf E}$ be a saturated profile such that ${\tbf E}\succ {\tbf E}_0$. Then $F_{\text{c}}(\text{S}({\tbf E})) - F_{\text{c}}({\tbf E}) \leq -\Delta F_{\text{u}}.$ \[lemma:diffShited\_directEval\]
The contribution of the change in the energy term is a perfect telescoping sum: $$\begin{aligned}
U_{\text{c}}(\text{S}({\tbf E})) - U_{\text{c}}({\tbf E}) = U_{\text{u}}(E_0) - U_{\text{u}}(E_{\rm max}). \label{eq:DeltaU}\end{aligned}$$ We now deal with the contribution of the change in the entropy term. We first notice that, using the variance symmetry, $\Sigma_c({\tbf E}) = \Sigma_{c+1}(\text{S}({\tbf E})) \ \forall \ c \in \{2w + 1:\Gamma-2w-1\}$, which makes the sum telescoping in this set. Thus $$\begin{aligned}
&S_{\text{c}}({\tbf E})-S_{\text{c}}(\text{S}({\tbf E}))= S_{\text{u}}( \Sigma_{\Gamma-2w}({\tbf E}))-S_{\text{u}}( \Sigma_{2w+1}(\text{S}{(\tbf E)})) \nonumber \\
&- \sum_{c\in \mathcal{S}} [S_{\text{u}}( \Sigma_{c}(\text{S}{(\tbf E)})) - S_{\text{u}}( \Sigma_{c}({\tbf E}))], \label{eq:diffSB}
% \end{aligned}$$ where $\mathcal{S}\defeq\{1:2w\}\cup\{\Gamma-2w+1:\Gamma\}$. By saturation, $\tbf E$ possesses the following property: $[\text{S}( {\tbf E})]_r = [ {\tbf E}]_r \ \forall \ r \in \{1:r_*\}\cup\{r_{\rm max}+1:\Gamma\} \Rightarrow \Sigma_c(\text{S}( {\tbf E})) = \Sigma_c( {\tbf E}) \ \forall \ c \in \mathcal{S}$ and thus the sum in (\[eq:diffSB\]) cancels. Furthermore, one can show, using the saturation of $\tbf E$ and the variance symmetry, that $\Sigma_{2w+1}(\text{S}({\tbf E})) = \Sigma( E_{0})$. Using the same arguments, in addition to the fact that $\tbf E\succ \tbf{E}_0\Rightarrow r_{\rm max} \le \Gamma-3w$, we obtain $\Sigma_{\Gamma-2w}({\tbf E}) = \Sigma( E_{\rm max})$. Hence, (\[eq:diffSB\]) yields the following: $$\begin{aligned}
S_{\text{c}}({\tbf E}) - S_{\text{c}}(\text{S}({\tbf E})) = S_{\text{u}}(\Sigma( E_{\rm max})) - S_{\text{u}}(\Sigma( E_{0})).
\label{eq:DeltaS}\end{aligned}$$ Combining (\[eq:DeltaU\]) with (\[eq:DeltaS\]) and using Lemma \[lemma:outside\_basin\] gives $$\begin{aligned}
F_{\text{c}}(\text{S}({\tbf E})) - F_{\text{c}}({\tbf E}) = -F_{\text{u}}( E_{\rm max}) + F_{\text{u}}({E_0}) \leq -\Delta F_{\text{u}}.\end{aligned}$$
Assume a spatially coupled SS code ensemble is used for communication through an AWGNC. Fix $R<R_{\txt{pot}}$, $w>K/\Delta F_{\text{u}}$ ($K$ is independent of $w$ and $\Gamma$) and $\Gamma> 8w$ (such that the code is well defined). Then any fixed point error profile ${\tbf{E}^*}$ of the coupled SE satisfies ${\tbf{E}}^* \prec {\tbf{E}}_0$. \[th:mainTheorem\]
Assume that, under these hypotheses, there exists a saturated profile $\tbf E$ associated to $\tbf E^*$ such that $\tbf E\succ {\tbf{E}}_0$. By Lemma \[lemma:diffShited\_directEval\] and the positivity of $\Delta F_{\text{u}}$ as $R<R_{\txt{pot}}$, we have $|F_{\text{c}}({\tbf E}) - F_{\text{c}}(\text{S}({\tbf E}))| \ge \Delta F_{\text{u}}$. Therefore, by Lemmas \[lemma:Fdiff\_quadraticForm\] and \[lemma:quadFormBounded\] we get $K/w > \Delta F_{\text{u}} \Rightarrow w < K/\Delta F_{\text{u}}$, a contradiction. Hence, ${\tbf{E}} \preceq {\tbf{E}}_0$. Since $\tbf E \succ {\tbf E}^*$ we have ${\tbf E}^* \prec {\tbf{E}}_0$.
\[cor:maincorollary\] By first taking $\Gamma \to \infty$ and then $w\to\infty$, the AMP threshold of the coupled ensemble satisfies $R_{\text c}\geq R_{\text{pot}}$.
This result follows from Theorem \[th:mainTheorem\] and Definition \[def:AMPcoupled\]. It says that the AMP threshold for the coupled codes saturates the potential threshold. To prove it cannot surpass it requires a separate treatment. For our purposes this is not really needed because we have necessarily $R_{\text c} < C$ and we know (Sec. \[sec:larg\_B\]) that $\lim_{B\to \infty}R_{\text{pot}}=C$. Thus $\lim_{B\to \infty} R_{\text c} = C$.
Acknowledgments {#acknowledgments .unnumbered}
===============
J.B and M.D acknowledge funding from the Swiss National Science Foundation grant num. 200021-156672. We thank Marc Vuffray and Rüdiger Urbanke for discussions.
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---
abstract: 'Using McMahon pseudo-metrics, for any minimal semiflow admitting an invariant measure, we study the relationships between its equicontinuous structure relation, regionally proximal relation and Veech’s relations; and characterize its weak-mixing. We show its Veech Structure Theorem if it is almost automorphic.'
address: 'Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China'
author:
- Xiongping Dai
title: 'The McMahon pseudo-metrics of minimal semiflows with invariant measures'
---
Automorphy $\cdot$ equicontinuous structure $\cdot$ regional proximity $\cdot$ minimality $\cdot$ weak-mixing $\cdot$ $E$-semiflow $\cdot$ chaos
37A25 $\cdot$ 37B05 $\cdot$ 54H15
Introduction {#sec1}
============
As usual, a *semiflow* with phase space $X$ and with phase semigroup $T$ is understood as a pair $(T,X)$ where, unless specified otherwise, we assume that
- $X$ is a compact $\textrm{T}_2$-space, **not necessarily metrizable**, and $T$ a topological **semigroup** with the identity $e$; moreover, $T$ acts jointly continuously on $X$ by phase mapping $(t,x)\mapsto tx$ such that $ex=x$ and $(st)x=s(tx)$ for all $x\in X$ and $s,t\in T$.
When $T$ is a topological group here, $(T,X)$ will be called a *flow*.
Given $x\in X, U\subseteq X$, and $V\subseteq X$, for $(T,X)$, if no confusion, we write for convenience
1. $N_T(x,U)=\{t\,|\,tx\in U\}$ and $N_T(U,V)=\{t\,|\,U\cap t^{-1}V\not=\emptyset\}$.
We shall say that:
1. $(T,X)$ is *minimal* if and only if $Tx$ is dense in $X$: $\mathrm{cls}_XTx=X$, for all $x\in X$; An $x\in X$ is called a *minimal point* or an *a.p. point* of $(T,X)$ if $\mathrm{cls}_XTx$ is a minimal subset of $X$.
2. $(T,X)$ admits of an *invariant measure* $\mu$ if $\mu$ is a Borel probability measure on $X$ such that $\mu(B)=\mu(t^{-1}B)$ for all Borel set $B\subseteq X$ and each $t\in T$. If every non-empty open subset of $X$ is of positive $\mu$-measure, then we say $\mu$ is of *full support*.
3. $(T,X)$ is *surjective* if for each $t$ of $T$, $x\mapsto tx$ is an onto self-map of $X$.
4. $(T,X)$ is *invertible* if for each $t\in T$, $x\mapsto tx$ is a 1-1 onto self-map of $X$. In this case, by $\langle T\rangle$ we mean the smallest group of self-homeomorphisms of $X$ with $T\subseteq\langle T\rangle$.
5. $T$ is called *amenable* if any semiflow on a compact $\textrm{T}_2$-space with the phase semigroup $T$ admits an invariant measure. Particularly, each abelian semigroup is amenable.
Let $\varDelta_X=\{(x,x)\,|\,x\in X\}$ and $\mathscr{U}_X$ the compatible symmetric uniform structure of $X$. For $\varepsilon\in\mathscr{U}_X$ and $x\in X$, set
1. $\varepsilon[x]=\{y\in X\,|\,(x,y)\in\varepsilon\}$, which is a neighborhood of $x$ in $X$.
Recall that a set-valued map $f\colon X\rightsquigarrow X$ is said to be *continuous* at $x_0\in X$ if and only if given $\varepsilon\in\mathscr{U}_X$ there is a $\delta\in\mathscr{U}_X$ such that $x\in\delta[x_0]$ implies $f(x)\subseteq\varepsilon[f(x_0)]$ and $f(x_0)\subseteq\varepsilon[f(x)]$.
1. A surjective semiflow $(T,X)$ is called *bi-continuous* if for each $t\in T$, $x\rightsquigarrow t^{-1}x$ is continuous at every point of $X$.
Clearly, if $(T,X)$ is invertible, it is bi-continuous. If let $X=\{0,1\}^{\mathbb{Z}_+}$ with the usual topology and $\sigma\colon X\rightarrow X$ defined by $(x_i)_{i\in\mathbb{Z}_+}\mapsto(x_{i+1})_{i\in\mathbb{Z}_+}$, then the cascade $(\sigma,X)$ is bi-continuous. More generally, if $(T,X)$ is such that for each $t\in T$, $x\mapsto tx$ is an $N$-to-1 locally homeomorphism of $X$, then $(T,X)$ is a bi-continuous semiflow.
In this paper, we shall study the equicontinuous structure relation, weak-mixing and weak disjointness, and Veech’s relations by using the McMahon pseudo-metric of minimal (bi-continuous) semiflows with invariant measures.
Equicontinuous structure relation {#sec1.1}
---------------------------------
First of all we recall a basic notion—equicontinuity, which is valid for any semiflow $(T,X)$ and for which there are some equivalent conditions in [@AD].
\[def1.1\]$(T,X)$ is *equicontinuous* if and only if given $\varepsilon\in\mathscr{U}_X$, there exists some $\delta\in\mathscr{U}_X$ such that $T\delta\subseteq\varepsilon$, i.e., if $(x,x^\prime)\in\delta$ then $(tx,tx^\prime)\in\varepsilon\ \forall t\in T$.
As is proved in [@EG; @V77; @Aus], for $(T,X)$ there always exists on $X$ a least closed $T$-invariant equivalence relation, denoted $S_{\textit{eq}}(X)$, such that $(T,X/S_{\textit{eq}}(X))$, defined by $$\begin{gathered}
(t,S_\textit{eq}[x])\mapsto S_\textit{eq}[tx]\quad \forall (t,x)\in T\times X,\end{gathered}$$ is an equicontinuous semiflow. Then:
\[def1.2\]Based on a semiflow $(T,X)$:
1. $S_{\textit{eq}}(X)$ is called the *equicontinuous structure relation* of $(T,X)$. Simply write $X_{\textit{eq}}=X/S_{\textit{eq}}(X)$ if no confusion.
2. $(T,X_{\textit{eq}})$ is called the *maximal equicontinuous factor* of $(T,X)$.
3. Write $\pi\colon X\rightarrow X_{\textit{eq}}$, viz $x\mapsto S_{\textit{eq}}[x]$, for the canonical projection.
In a number of situations the equicontinuous structure relation of minimal flows is known explicitly (cf., e.g., [@V68; @Aus; @AEE; @Aus01; @AGN]). Particularly, if $Q$ is the “regionally proximal relation” (cf. Definition \[def1.3\] below), then the following two facts are well known:
1. **Theorem A** (cf. [@V77 Theorem 2.6.2]). [*Let $(T,X)$ be a minimal flow such that $(T,X\times X)$ has a dense set of minimal points; then $S_\textit{eq}(X)=Q(X)$.*]{} (This is due to R. Ellis and W.A. Veech independently.)
<!-- -->
1. **Theorem B** (cf. McMahon [@McM] and also see [@Aus Theorem 9.8]). [*If $(T,X)$ is a minimal flow admitting an invariant measure, then $S_{\textit{eq}}(X)=Q(X)$.*]{}
Based on the recent work [@DX; @AD; @Dai], we will generalize the above classical Theorem B to semiflows on compact $\textrm{T}_2$-spaces; see Theorem \[thm1.4\] below.
\[def1.3\]Let $(T,X)$ be a semiflow with phase semigroup $T$.
1. $(T,X)$ is called *distal* if given $x,x^\prime\in X$ with $x\not=x^\prime$, there is an $\varepsilon\in\mathscr{U}_X$ such that $t(x,x^\prime)\not\in\varepsilon$ for all $t\in T$. Then by [@AD Theorem 1.15], whenever $(T,X)$ is distal, then it is invertible.
2. **Theorem C** (cf. [@AD]). [*If $(T,X)$ is a (minimal) distal semiflow, then $(\langle T\rangle,X)$ is a (minimal) distal flow. Hence by Furstenberg’s theorem [@F63], $(T,X)$ admits invariant measures.*]{}
3. An $x\in X$ is called a *distal point* of $(T,X)$ if no point of $\textrm{cls}_XTx$ other than $x$ is proximal to $x$; i.e., if $y\in\textrm{cls}_XTx$ is such that $t_n(x,y)\to(z,z)$ for some $z\in X$ and some net $\{t_n\}$ in $T$ then $y=x$. It is easy to verify that:
4. **Lemma D**. [*$(T,X)$ is distal iff every point of $X$ is distal for $(T,X)$.*]{}
$(T,X)$ is called *point-distal* if there is a distal point whose orbit is dense in $X$.
5. We say $x\in X$ is *regionally proximal to* $x^\prime\in X$ for $(T,X)$, denoted by $(x,x^\prime)\in Q(X)$ or $x^\prime\in Q[x]$, if there are nets $\{x_n\}, \{x_n^\prime\}$ in $X$ and $\{t_n\}$ in $T$ with $x_n\to x, x_n^\prime\to x^\prime$ and $\lim_n t_nx_n=\lim_nt_nx_n^\prime$.
Then by the equality: $$\begin{gathered}
Q(X)={\bigcap}_{\alpha\in\mathscr{U}_X}\mathrm{cls}_{X\times X}{\bigcup}_{t\in T}t^{-1}\alpha\end{gathered}$$ $Q(X)$ is a closed reflexive symmetric relation. An example due to McMahon [@M76] shows it is not always an equivalence relation even in minimal flows.
If $(T,X)$ is a flow and $(x,x^\prime)\in Q(X)$, then for all $t\in T$, by $\lim_n(t_nt^{-1})(tx_n)=\lim_n(t_nt^{-1})(tx_n^\prime)$ and $t_nt^{-1}\in T$, it easily follows that $t(x,x^\prime)\in Q(X)$ and so $Q(X)$ is invariant.
In general, $Q(X)$ is neither invariant nor transitive in semiflows; yet if $(T,X)$ is minimal admitting an invariant measure, as we will see, $Q(X)$ is an invariant closed equivalence relation on $X$, and in fact
\[thm1.4\]Let $(T,X)$ be a minimal semiflow, which admits an invariant measure on $X$, then $S_{\textit{eq}}(X)=Q(X)$. Hence $(T,X_\textit{eq})$ is equicontinuous invertible.
It should be noted that the condition “admitting an invariant measure” is important for our statement above. For example, let $X=[0,1]$ the unit interval with the usual topology and for every $\alpha$ with $0<\alpha<1$, define two injective mappings of $X$ into itself: $$\begin{gathered}
f_\alpha\colon X\rightarrow X,\ x\mapsto\alpha x\quad {\textrm{and}}\quad
g_\alpha\colon X\rightarrow X,\ x\mapsto 1-\alpha(1-x).\end{gathered}$$ Now let $$T=\left\{\left.f_{\alpha_1}^{\epsilon_1}g_{\gamma_1}^{\delta_1}\dotsm f_{\alpha_n}^{\epsilon_n}g_{\gamma_n}^{\delta_n}\,\right|\,0<\alpha_i<1, 0<\gamma_i<1,\ \epsilon_i=0\textrm{ or } 1,\ \delta_i=0\textrm{ or }1,\ n=1,2,\dotsc\right\}$$ be the discrete semigroup generated by $f_\alpha,g_\alpha$, $0<\alpha<1$. It is easy to see that each $t\in T$ is injective and that $(T,X)$ is equicontinuous minimal so $S_{\textit{eq}}(X)=\varDelta_X$, but $Q(X)=X\times X$. See [@AD Example 1.2].
Thus, as consequences of Theorem \[thm1.4\], we can easily obtain the following two corollaries.
\[cor1.5\]Let $(T,X)$ be a minimal semiflow with $T$ amenable; then $S_{\textit{eq}}(X)=Q(X)$.
Thus if $f\colon X\rightarrow X$ is a minimal continuous transformation of $X$, then $S_{\textit{eq}}(X)=Q(X)$ associated to the natural $\mathbb{Z}_+$-action.
Next, replacing amenability of $T$ in Corollary \[cor1.5\] by distality of $(T,X)$, we can obtain the following result:
\[cor1.6\]Let $(T,X)$ be a minimal distal semiflow; then $S_{\textit{eq}}(X)=Q(X)$.
Since $(T,X)$ is distal, each $t\in T$ is a self-homeomorphism of $X$. Then by Theorem C, $(T,X)$ admits an invariant measure. Then Corollary \[cor1.6\] follows from Theorem \[thm1.4\].
$\S\S\ref{sec2}$ and \[sec3\] of this paper will be devoted to proving Theorem \[thm1.4\]. Notice that although we will see that $(T,X_\textit{eq})$ is invertible in the situation of Theorem \[thm1.4\], yet each $t\in T$ itself need not be invertible with respect to $(T,X)$. This will cause the difficulty.
Weak-mixing semiflows {#sec1.2}
---------------------
Let $(T,X)$ be a semiflow. Let $X^n=X\times\dotsm\times X$ ($n$-times), for any integer $n\ge2$. Based on $(T,X)$, $(T,X^n)$ is also a semiflow, which is defined by $t\colon (x_i)_{i=1}^n\mapsto(tx_i)_{i=1}^n$. Given any cardinality $\mathfrak{c}\ge1$, we can similarly define $(T,X^\mathfrak{c})$. By $\mathfrak{U}(X)$ we denote the collection of non-empty open subsets of $X$. Then we introduce some basic standard notions.
1. $(T,X)$ is called *topologically transitive* (T.T.) iff every $T$-invariant set with non-empty interior is dense in $X$ iff $N_T(U,V)\not=\emptyset$ for $U,V\in\mathfrak{U}(X)$.
2. $(T,X)$ is called *point-transitive* (P.T.) if there is some point $x\in X$ with $Tx$ dense in $X$. In this case, write $x\in \textrm{Tran}\,(T,X)$.
3. $(T,X)$ is called *weak-mixing* if $(T,X\times X)$ is a T.T. semiflow.
4. $(T,X)$ is called *discretely thickly T.T.* if given $U,V\in\mathfrak{U}(X)$, $N_T(U,V)$ is *discretely thick* (i.e. for any finite subset $F$ of $T$ there is $t\in T$ such that $Ft\subseteq N_T(U,V)$).
5. $A\subseteq T$ is called an *IP-set* of $T$ if there exists a sequence $\{t_n\}$ in $T$ such that $t_{n_1}t_{n_2}\dotsc t_{n_k}\in A$ for all $1\le n_1<n_2<\dotsm<n_k<\infty$ and $k\ge1$. $(T,X)$ is said to be *IP-T.T.* if $N_T(U,V)$ is an IP-set of $T$ for $U,V\in\mathfrak{U}(X)$.
6. Let $n\ge2$ be an integer. We say $(T,X)$ satisfies the *$n$-order Bronstein condition* if $X^n$ contains a dense set of a.p. points. We say $(T,X)$ and $(T,Y)$ satisfies the *Bronstein condition* if the a.p. points are dense in $X\times Y$.
In [@F67 p. 26], T.T. is called “ergodic”. It should be noted that in our situation, T.T. $\not\Rightarrow$ P.T., and weak-mixing $\not\Rightarrow$ P.T. as well. In $\S\ref{sec4}$, we will characterize “weak-mixing” of minimal semiflow with invariant measures by using the McMahon pseudo-metric; see Theorem \[thm4.12\].
It is a well-known important fact that (cf. [@F67 Proposition II.3] for cascades, [@G76 Theorem 1.11] for $T$ abelian groups, and [@DT Lemma 3.2] for abelian semigroups):
> **Theorem E**. *If $(T,X)$ is weak-mixing with $T$ abelian, then $(T,X^n)$ is T.T. for all integer $n\ge2$*.
In fact, this very important theorem can be extended to amenable semigroups as follows, which is new even for flows.
\[thm1.7\]Let $(T,X)$ be a minimal semiflow, which admits of an invariant measure $\mu$ $($e.g. $T$ is an amenable semigroup$)$. Then $(T,X)$ is weak-mixing iff $(T,X\times Y)$ is T.T. for all $(T,Y)$ which is T.T. and admits of a full ergodic center iff $Q(X)=X\times X$.
This theorem is comparable with the following theorem in which our ergodic condition “$\mu$” and “full ergodic center” is replaced by the topological condition—“Bronstein condition”:
> **Theorem F** (cf. Veech [@V77 Theorem 2.6.3 for the relativized version]). *Let $T$ be a group. Let $(T,X)$ be minimal which satisfies the 2-order Bronstein condition, and suppose $S_\textit{eq}(X)=X\times X$. If $(T,Z)$ is T.T. such that $(T,X)$ and $(T,Z)$ satisfies the Bronstein condition, then $(T,X\times Y)$ is a T.T. flow*.
Theorem F implies that
- [*Let $(T,X)$ be a minimal flow satisfying the 2-order Bronstein condition. Then $(T,X)$ is weak-mixing iff $Q(X)=X\times X$.*]{}
The “ergodic center” will be defined in Definition \[def4.3\]. This theorem will be proved in $\S$\[sec4.1\] under the guises of Corollary \[cor4.8\] for the first “if and only if” and Theorem \[thm4.12\] for the second “if and only if”. By induction based on Theorem \[thm1.7\] we can obtain the following
\[thm1.8\]Let $(T,X)$ be a minimal semiflow, which admits an invariant measure $($e.g. $T$ an amenable semigroup$)$. Then $(T,X)$ is weak-mixing iff $(T,X^\mathfrak{c})$ is T.T. for all cardinality $\mathfrak{c}\ge2$ iff $(T,X)$ is a discretely thickly T.T. semiflow.
Similar to Theorem \[thm1.7\], using the Bronstein condition instead of our ergodic condition we can obtain the following
> **Theorem G** (cf. Veech [@V77 Theorem 2.6.4 for the relativized version]). *Let $(T,X)$ be a minimal flow which satisfies the $n$-order Bronstein condition for each $n\ge2$. Then $(T,X^n)$ is T.T. for all $n\ge2$*.
Theorem \[thm1.8\] will be proved in $\S\ref{sec4.1}$ under the guise of Theorem \[thm4.11\]. It is not known whether these assertions still hold for a non-minimal semiflow with amenable phase semigroup.
An $x\in X$ is called *locally almost periodic* (l.a.p.) for $(T,X)$ if for each neighborhood $U$ of $x$ there are a neighborhood $V$ of $x$ and a discretely syndetic subset $A$ of $T$ such that $AV\subseteq U$.
Here by a “discretely syndetic” set $A$ we mean that there is a finite subset $K$ of $T$ such that $Kt\cap A\not=\emptyset\ \forall t\in T$. A subset of $T$ is discretely syndetic if and only if it intersects non-voidly each discretely thick subset of $T$ (cf., e.g., [@AD]).
\[cor1.10\]Let $(T,X)$ be a minimal semiflow with $T$ amenable. If $(T,X)$ is weak-mixing, then there is no l.a.p. point of $(T,X)$.
Suppose the contrary that there is an l.a.p. point $x_0$. Since $X$ is a non-singleton $\textrm{T}_2$-space, we can choose disjoint $U,U^\prime\in\mathfrak{U}(X)$ with $x_0\in U$. Then there are a discretely syndetic set $A\subseteq T$ and an open neighborhood $V$ of $x$ such that $AV\subseteq U$. But this contradicts that $N_T(V,U^\prime)$ is discretely thick in $T$ by Theorem \[thm1.8\]. This proves Corollary\[cor1.10\].
Recall that an $x\in X$ is referred to as an *IP$^*$-recurrent point* of $(T,X)$ if and only if given a neighborhood $U$ of $x$, $N_T(x,U)$ is an *IP$^*$-set* in the sense that it intersects non-voidly every IP-set of $T$ (cf. [@Fur; @DL]).
> **Theorem H** (cf. [@Fur Theorem 9.11] and [@DL Theorem 4]). [*Let $(T,X)$ be any semiflow and $x\in X$; then $x$ is a distal point of $(T,X)$ if and only if $x$ is an IP$^*$-recurrent point of $(T,X)$.*]{}
\[cor1.11\]If $(T,X)$ is a minimal weak-mixing semiflow with $T$ amenable, then there exists no distal point of $(T,X)$.
$\,$
1. See Veech [@V70] and Furstenberg [@Fur Theorem 9.12] for cascade $(f,X)$ with $X$ a compact metric space and Dai-Tang [@DT Proposition 3.5] for IP-T.T. semiflow $(T,X)$ with first countable phase space $X$. Here without any countability our proof is completely new.
2. If $T$ is a group in place of “$T$ ameanble”, the consequence of Corollary \[cor1.11\] still holds using Veech’s Structure Theorem of point-distal flows; see Theorem \[thm2.10\].
3. By Corollary \[cor1.11\], a point-distal semiflow is T.T. but it is never an IP-T.T. semiflow.
Assume for a contradiction that $(T,X)$ has a distal point, say $x_0\in X$. Let $W$ be a closed subset of $X$ with $x_0\not\in W$ and $\textrm{Int}_XW\not=\emptyset$. Let $\beta\in\mathscr{U}_X$ be such that $\beta[x_0]\cap W=\emptyset$. For all $\varepsilon\in\mathscr{U}_X$ with $\varepsilon\le\beta$, define a closed subset of $X$ as follows: $W_\varepsilon=\left\{x\in W\,|\,\exists\,t\in T\textit{ s.t. }tx_0,tx\in\textrm{cls}_X\varepsilon[x_0]\right\}$. Since $x_0$ is IP$^*$-recurrent by Theorem H, $N_T(x_0,\varepsilon[x_0])$ is an IP$^*$-set. On the other hand, by Theorem \[thm1.8\], $N_T(W,\varepsilon[x_0])$ is discretely thick and so $N_T(W,\varepsilon[x_0])$ is an IP-set (cf., e.g., [@Fur Lemma 9.1] and [@Dai (1.2a)]). Therefore, $W_\varepsilon\not=\emptyset$ for all $\varepsilon\in\mathscr{U}_X$. Given any $\varepsilon_1,\dotsc,\varepsilon_n\le\beta$, let $\varepsilon\le\varepsilon_1\cap\dotsm\cap\varepsilon_n$; then $W_{\varepsilon_1}\cap\dotsm\cap W_{\varepsilon_n}\supseteq W_\varepsilon\not=\emptyset$. Thus $\{W_\varepsilon\,|\,\varepsilon\le\beta\}$ has the finite intersection property. This shows that $\bigcap_{\varepsilon\le\beta}W_\varepsilon\not=\emptyset$. Now let $y\in\bigcap_{\varepsilon\le\beta}W_\varepsilon$ be any given. Clearly, $x_0\not=y$. Since for each $\varepsilon\le\beta$ there is some $t=t_\varepsilon\in T$ such that $tx_0, ty\in\textrm{cls}_X\varepsilon[x_0]$, hence $y$ is proximal to $x_0$. This is a contradiction to the assumption that $x_0$ is a distal point of $(T,X)$. This proves Corollary \[cor1.11\].
\[thm1.12\]Let $(T,X)$ be a point-distal non-trivial semiflow with $T$ amenable. Then $(T,X)$ has a non-trivial equicontinuous factor.
By Theorem \[thm1.4\], $S_\textit{eq}(X)=Q(X)$. Suppose the contrary that $(T,X)$ has no non-trivial equicontinuous factor, then $Q(X)=X\times X$ and so $(T,X)$ is weakly mixing by Theorem \[thm1.7\]. But this contradicts Corollary \[cor1.11\]. The proof is complete.
It should be noticed that “$T$ amenable” may be relaxed by “$(T,X)$ admitting of an invariant measure” in Corollaries \[cor1.10\] and \[cor1.11\] and Theorem \[thm1.12\].
As another application of Theorem \[thm1.8\] we will consider the chaotic dynamics of weak-mixing semiflows with amenable phase semigroups in $\S\ref{sec4.2}$; see Theorems \[thm4.17\] and \[thm4.19\] and Corollary \[cor4.21\].
Veech’s relations of surjective dynamics {#sec1.3}
----------------------------------------
\[def1.13\]Let $(T,X)$ be any semiflow.
1. An $x\in X$ is called *almost automorphic* (a.a.) for $(T,X)$, denoted $x\in P_{\!aa}(T,X)$, if and only if $t_nx\to y, x_n^\prime\to x^\prime, t_nx_n^\prime=y$ implies $x=x^\prime$ for every net $\{t_n\}$ in $T$.
2. If $x\in P_{\!aa}(T,X)$ and $\mathrm{cls}_XTx=X$, then $(T,X)$ is called an *a.a. semiflow*.
3. We say that $(x,x^\prime)$ is in Veech’s relation $V$ of $(T,X)$, denoted $(x,x^\prime)\in V(X)$ or $x^\prime\in V[x]$, if there exist a net $x_n^\prime\to x^\prime$, a point $y\in X$, and a net $\{t_i\}$ in $T$ such that $t_nx\to y$ and $t_nx_n^\prime=y$. Then:
- $V[x]=\{x\}$ if and only if $x\in P_{\!aa}(T,X)$.
4. Given $x\in X$, define $D[x]$ for $(T,X)$ by $$D[x]={\bigcap}_{\varepsilon\in\mathscr{U}_X}D(x,\varepsilon),\quad \textrm{where }D(x,\varepsilon)=\mathrm{cls}_XA^{-1}Ax\textrm{ and }A=N_T(x,\varepsilon[x]).$$ $D[x]$ is closed, and, of course, $x\in D[x]$. We will say $(x,y)\in D(X)$ if and only if $y\in D[x]$. Then
- $D[x]=\{y\,|\,\exists\{t_n\}, \{s_n\}\textrm{ in }T, \{y_n\}\textrm{ in }X\textrm{ s.t. }(t_nx,s_nx,y_n)\to(x,x,y), t_ny_n=s_nx\}$.
$D(x,\varepsilon)$ is originally defined as $D(x,\varepsilon)=\mathrm{cls}_XAA^{-1}x$ by Veech in [@V68]. However, since $T$ is an abelian group in [@V68], so our definition here agrees with Veech’s in flows with abelian phase groups.
Since an a.a. point is a distal point of any surjective $(T,X)$ (cf. [@Dai Lemma 1.8] and [@Fur Theorem 9.13]), hence an a.a. surjective semiflow is point-distal and so minimal. We shall consider another Veech relation $U(X)$ in $\S$\[sec3.3\].
\[def1.14\]Let $(T,X)$ and $(T,Y)$ be two any semiflows.
1. $\pi\colon X\rightarrow Y$ is called an *epimorphism* between $(T,X)$ and $(T,Y)$, denoted $(T,X)\xrightarrow{\pi}(T,Y)$, if $\pi$ is continuous surjective with $\pi(tx)=t\pi(x)$ for all $t\in T$ and $x\in X$. In this case, $(T,Y)$ is called a *factor* of $(T,X)$, and $(T,X)$ an *extension* of $(T,Y)$.
2. $(T,X)\xrightarrow{\pi}(T,Y)$ is said to be of *almost 1-1 type* if there exists a point $y\in Y$ such that $\pi^{-1}(y)$ is a singleton set.
It is clear that if $(T,X)$ is surjective (resp. minimal), then its factors are also surjective (resp. minimal). Of course, its factor need not be invertible if $(T,X)$ is invertible. Moreover, even if $(T,X)$ has an invertible non-trivial factor, $(T,X)$ itself need not be invertible. As mentioned before, we will be mainly concerned with the maximal equicontinuous factor $(T,X_\textit{eq})$ of a minimal semiflow $(T,X)$.
Using a variation of a theorem of Bogoliouboff and F[ø]{}lner [@V68 Theorem 4.1] that is valid only for discrete abelian groups, Veech proved the following theorem:
> **Theorem I** (cf. [@V68 Theorem 1.1]). [*If $(T,X)$ is a minimal flow with $T$ an abelian group, then $D(X)=S_\textit{eq}(X)$.*]{}
Although Veech’s proof of the above Theorem I does not work for non-abelain flows, yet using different approaches—McMahon pseudo-metrics, we can obtain the following generalization in $\S$\[sec3\] and $\S$\[sec5\].
\[thm1.15\]Let $(T,X)$ be a minimal bi-continuous semiflow, which admits an invariant measure. Then:
1. $D[x]=Q[x]$ for all $x\in X$.
2. When $(T,X)$ is invertible, then $Q[x]=D[x]=\overline{V[x]}$ for all $x\in X$.
3. $(T,X)$ is $\mathrm{a.a.}$ if and only if $\pi\colon(T,X)\rightarrow(T,X_\textit{eq})$ is of almost 1-1 type.
Hence if $X$ is compact metric and $(T,X)$ is a.a., then $P_{\!aa}(T,X)$ is a residual subset of $X$.
\[cor1.16\]Let $(T,X)$ be an invertible minimal semiflow with $T$ an amenable semigroup. Then:
1. $Q[x]=D[x]=\overline{V[x]}$ for all $x\in X$.
2. $(T,X)$ is a.a. if and only if $\pi\colon(T,X)\rightarrow(T,X_\textit{eq})$ is of almost 1-1 type.
Since $T$ is amenable, $(T,X)$ admits an invariant measure. Then Corollary \[cor1.16\] follows from Theorem \[thm1.15\].
\[cor1.17\]Let $(T,X)$ be a minimal bi-continuous semiflow, which admits an invariant measure. Then $(T,X)$ is equicontinuous iff all points are a.a. points.
This will be proved in $\S\ref{sec5}$. We note that Corollary \[cor1.17\] in the case that $T$ is an abelian group and $X$ a compact metric space is [@AM Corollary 21]. Moreover, the statement of Corollary \[cor1.17\] still holds without the condition “which admits an invariant measure” by using different approaches (cf. [@Dai Theorem 4.5]).
\[cor1.18\]Let $(T,X)$ be a minimal bi-continuous semiflow, which admits an invariant measure. If $(T,X)$ is a.a., then it has l.a.p. points.
By Theorem \[thm1.15\], $\pi\colon X\rightarrow X_\textit{eq}$ is of almost 1-1 type. Since $(T,X_\textit{eq})$ is equicontinuous invertible, it is l.a.p. and so $(T,X)$ has l.a.p. points. The proof is complete.
Theorem \[thm1.15\] will be proved in $\S$\[sec3\] and $\S$\[sec5\]. In fact, (1) of Theorem \[thm1.15\] follows from Theorem \[thm3.10\] and Corollary \[cor3.13\]; (2) of Theorem \[thm1.15\] is Theorem \[5.3\]; and (3) of Theorem \[thm1.15\] is Theorem \[thm5.5\]. In fact, since $(T,X)$ is bi-continuous, $(T,X)$ must be invertible when it is a.a by [@Dai Lemma 1.1].
Preliminary lemmas {#sec2}
==================
To prove our main results Theorem \[thm1.4\], Theorem \[thm1.7\] and Theorem \[thm1.15\], we will need some preliminary lemmas and theorems. Among them, Theorem \[thm2.7A\] asserts that an epimorphism can transfer $Q(X)$ of a minimal semiflow $(T,X)$ onto the regionally proximal relation of its invertible factor. And Theorem \[thm2.10\] is a supplement of Corollary \[cor1.11\].
Preliminary notions
-------------------
Let $(T,X)$ be a semiflow with phase semigroup $T$ and with compact $\textrm{T}_2$ phase space $X$. We have introduced some necessary notions in $\S\ref{sec1}$. Here we need to introduce another one.
\[def2.1\]We shall say that $(T,X)$ admits an *invariant quasi-regular Borel probability measure* $\mu$, provided that $\mu$ is an invariant Borel probability measure on $X$ such that
- $\mu$ is *quasi-regular* in the sense that for any Borel set $B$ and $\varepsilon>0$ one can find an open set $U$ with $B\subseteq U$ and $\mu(U-B)<\varepsilon$, and for any open set $U$ and $\varepsilon>0$ one can find a compact set $K$ with $K\subset U$ and $\mu(U-K)<\varepsilon$.
Since $X$ is compact $\textrm{T}_2$ here, a quasi-regular Borel probability measure must be regular. Moreover, it is well known that if $T$ is an amenable semigroup or $(T,X)$ is distal, then $(T,X)$ always admits an invariant Borel probability measure.
Recall that a point is called *minimal* if its orbit closure is a minimal subset. Since $X$ is compact, then by Zorn’s lemma, minimal points always exist. A minimal point is also called an “almost periodic point” or a “uniformly recurrent point” in some works like [@E69; @Fur; @Aus]. If $(T,X)$ is a minimal semiflow, then $\varDelta_X$ is a minimal set of $(T,X\times X)$.
If the set of minimal points is dense in $X$, then we will say $(T,X)$ has a *dense set of minimal points* or *with dense almost periodic points*.
Preliminary lemmas {#preliminary-lemmas}
------------------
Now we will introduce and establish in this subsection some preliminary lemmas needed in our later discussion.
\[lem2.2\]Let $(T,X)$ be a semiflow. Then:
1. If $(T,X)$ is distal, then it is invertible.
2. If $(T,X)$ is surjective equicontinuous, then it is distal.
3. If $(T,X)$ is invertible equicontinuous, then $(\langle T\rangle,X)$ is an equicontinuous flow.
\[lem2.3\]$(T,X)$ is an equicontinuous surjective semiflow if and only if $Q(X)=\varDelta_X$ if and only if $(\langle T\rangle,X)$ is an equicontinuous flow.
\[lem2.4A\]Let $(T,X)$ be a minimal semiflow. If $(T,X)$ admits an invariant measure, then $(T,X)$ is surjective.
\[lem2.5A\]Let $\pi\colon(T,X)\rightarrow(T,Y)$ be an epimorphism where $(T,X)$ contains a dense set of minimal points and $(T,Y)$ is minimal invertible, and let $\varepsilon\in\mathscr{U}_X$. Then $(\pi\times\pi)(T^{-1}\varepsilon)$ belongs to $\mathscr{U}_Y$.
Let $V\in\mathfrak{U}(X)$ such that $V\times V\subset\varepsilon$ and take a minimal point $x$ of $X$ with $x\in V$. Since $(T,Y)$ is minimal, hence $\pi(\textrm{cls}_XTx)=Y$. Thus there is some $\tau\in T$ such that $\textrm{Int}_Y\pi(\tau^{-1}V)\not=\emptyset$. Let $\eta=T^{-1}(\textrm{Int}_Y\pi(\tau^{-1}V)\times \textrm{Int}_Y\pi(\tau^{-1}V))$. Then $\eta$ is a non-empty $T^{-1}$-invariant open set which meets $\varDelta_Y$. Since $\varDelta_Y$ is minimal for $(T,Y\times Y)$ and $(Y\times Y)\setminus\eta$ is $T$-invariant, it follows that $\varDelta_Y\subset\eta$ and so $\eta\in\mathscr{U}_Y$. Then by $(\pi\times\pi)(T^{-1}\varepsilon)\supseteq T^{-1}(\pi(\tau^{-1}V)\times\pi(\tau^{-1}V))\supseteq\eta$, we can conclude that $(\pi\times\pi)(T^{-1}\varepsilon)\in\mathscr{U}_Y$. The proof is complete.
The following result is useful, which generalizes [@F63 Corollary to Theorem 8.1] and [@V70 Proposition 2.3] that are in minimal distal flows with compact metric phase spaces by different approaches.
\[cor2.6A\]Let $\pi\colon(T,X)\rightarrow(T,Y)$ be an epimorphism of semiflows where $(T,X)$ has a dense set of minimal points and $(T,Y)$ is minimal invertible. Assume $\{(y_n,y_n^\prime)\}$ is a net in $Y\times Y$ such that $\lim_n(y_n,y_n^\prime)\in\varDelta_Y$. Then there are nets $\{(x_i,x_i^\prime)\}$ in $X\times X$ and $\{t_i\}$ in $T$ such that $\lim_it_i(x_i,x_i^\prime)\in\varDelta_X$ and that $\{(\pi(x_i),\pi(x_i^\prime))\}$ is a subnet of $\{(y_n,y_n^\prime)\}$.
Given any $\varepsilon\in\mathscr{U}_X$, since $(\pi\times\pi)(T^{-1}\varepsilon)\in\mathscr{U}_Y$ by Lemma \[lem2.5A\] and $\lim_n(y_n,y_n^\prime)\in\varDelta_Y$, then we can take some $n_\varepsilon$ with $(y_{n_\varepsilon},y_{n_\varepsilon}^\prime)\in\{(y_n,y_n^\prime)\}$ such that $(y_{n_\varepsilon},y_{n_\varepsilon}^\prime)\in(\pi\times\pi)(T^{-1}\varepsilon)$. Let $t_\varepsilon\in T$ and $x_{n_\varepsilon},x_{n_\varepsilon}^\prime\in X$ such that $t_\varepsilon(x_{n_\varepsilon},x_{n_\varepsilon}^\prime)\in\varepsilon$ and $\pi(x_{n_\varepsilon})=y_{n_\varepsilon}$ and $\pi(x_{n_\varepsilon}^\prime)=y_{n_\varepsilon}^\prime$. This completes the proof of Corollary \[cor2.6A\].
Corollary \[cor2.6A\] is very useful for our later Theorem \[thm2.7A\]. Moreover, it should be noted that we are not permitted to use $(\pi\times\pi)(T\varepsilon)\in\mathscr{U}_Y$ instead of $(\pi\times\pi)(T^{-1}\varepsilon)\in\mathscr{U}_Y$, for $T\varepsilon$ is not necessarily open in our semiflow context.
Regional proximity relations of extensions and factors
------------------------------------------------------
Now we will be concerned with the relationship of $Q(X)$ with the same relation of its factors. The point of Theorem \[thm2.7A\] below is that $(T,X)$ need not be invertible.
\[thm2.7A\]Let $\pi\colon(T,X)\rightarrow(T,Y)$ be an epimorphism, where $(T,X)$ is surjective with a dense set of minimal points and $(T,Y)$ is invertible minimal. Then $(\pi\times\pi)Q(X)=Q(Y)$.
Clearly $(\pi\times\pi)Q(X)\subseteq Q(Y)$. For the other direction inclusion, let $(y,y^\prime)\in Q(Y)$. Then there are nets $\{(y_n,y_n^\prime)\}$ in $Y\times Y$ and $\{\tau_n\}$ in $T$ with $(y_n,y_n^\prime)\to(y,y^\prime)$ and $\lim_n\tau_n(y_n,y_n^\prime)\in\varDelta_Y$. By Corollary \[cor2.6A\], choose $(x_n,x_n^\prime)\in X\times X$ and $t_n\in T$ (passing to a subnet if necessary) with $\lim_nt_n(x_n,x_n^\prime)\in\varDelta_X$ and $\pi(x_n)=\tau_ny_n,\ \pi(x_n^\prime)=\tau_n y_n^\prime$. Pick $\tilde{x}_n,\tilde{x}_n^\prime\in X$ with $\tau_n(\tilde{x}_n,\tilde{x}_n^\prime)=(x_n,x_n^\prime)$. Then $\lim t_n\tau_n(\tilde{x}_n,\tilde{x}_n^\prime)\in\varDelta_X$ and $\pi(\tilde{x}_n)=y_n,\ \pi(\tilde{x}_n^\prime)=y_n^\prime$. By taking a subnet of $\{(\tilde{x}_n,\tilde{x}_n^\prime)\}$ if necessary, let $\tilde{x}_n\to x$ and $\tilde{x}_n^\prime\to x^\prime$. Thus $(x,x^\prime)\in Q(X)$ with $\pi(x)=y$ and $\pi(x^\prime)=y^\prime$. Thus $Q(Y)\subseteq(\pi\times\pi)Q(X)$. This proves Theorem \[thm2.7A\].
\[cor2.8A\]Let $\pi\colon(T,X)\rightarrow(T,Y)$ be an epimorphism such that $(T,X)$ is surjective with dense minimal points and (T,Y) is minimal invertible. Then $(T,Y)$ is equicontinuous if and only if $(\pi\times\pi)Q(X)=\varDelta_Y$.
The “only if” follows from Lemma \[lem2.3\] and Theorem \[thm2.7A\]. Conversely, by Theorem \[thm2.7A\], $Q(Y)=\varDelta_Y$ and so the “if” follows from Lemma \[lem2.3\].
Now applying Theorem \[thm2.7A\] with $\pi\colon(T,X)\rightarrow(T,X_\textit{eq})$, we can easily obtain the following result, which is very useful in surjective semiflows and which partially generalizes [@E69 Proposition 4.20].
\[cor2.9A\]Let $(T,X)$ be a surjective minimal semiflow. Then $S_{\textit{eq}}(X)$ is the smallest closed invariant equivalence relation containing $Q(X)$.
Since $(T,X_\textit{eq})$ is equicontinuous surjective, Lemma \[lem2.2\] follows that $(T,X_\textit{eq})$ is minimal invertible, where $X_\textit{eq}=X/S_\textit{eq}$. Then by Lemma \[lem2.3\], $(\pi\times\pi)Q(X)\subseteq Q(X_\textit{eq})=\varDelta_{X_\textit{eq}}$ so that $Q(X)\subseteq S_{\textit{eq}}$. To show $S_\textit{eq}(X)$ is the smallest one, let $R$ be a closed invariant equivalence relation on $X$ with $Q(X)\subseteq R$, $X\xrightarrow{\pi}X/R$ the canonical map and $E(X)\xrightarrow{\pi_*} E(X/R)$ the standard semigroup homomorphism between the Ellis enveloping semigroups induced by $\pi$. If $u^2=u\in E(X/R)$ then $\pi_*^{-1}(u)$ is a closed subsemigroup of $E(X)$. Hence there exists $v^2=v\in E(X)$ with $\pi_*(v)=u$. Since $(x,v(x))\in P(X)\subseteq Q(X)\subseteq R$, then $\pi(x)=\pi(v(x))=u\pi(x)$ for all $x\in X$. Thus $u=\textit{id}_{X/R}$ the identity of $X/R$. This implies that $(T,X/R)$ is distal and so invertible (cf. Lemma \[lem2.2\]). Then by Theorem \[thm2.7A\] and $Q(X)\subseteq R$, $\varDelta_{X/R}=(\pi\times \pi)Q(X)=Q(X/R)$. Thus $(T,X/R)$ is equicontinuous by Lemma \[lem2.3\]. Whence $S_{\textit{eq}}\subseteq R$. This proves Corollary \[cor2.9A\].
We note here that when $T$ is a group, the proof of Corollary \[cor2.9A\] may be simplified much as follows:
Since $\pi\colon X\rightarrow X_\textit{eq}$ is an epimorphism of minimal flows, by $(\pi\times\pi)Q(X)\subseteq Q(X_\textit{eq})=\varDelta$ it follows that $Q(X)\subseteq S_\textit{eq}$. On the other hand, if $R$ is an invariant closed equivalence relation on $X$ with $Q(X)\subseteq R$, then by Theorem \[thm2.7A\] we can conclude that $Q(X/R)=\varDelta$ so $(T,X/R)$ is equicontinuous. Thus $S_\textit{eq}\subseteq R$. The proof is complete.
Recall that Lemma \[lem2.3\] is proved in [@DX pp. 46-47] by using Lemma \[lem2.2\], a.a. points and Veech’s relation $V(X)$ in (1) of Definition \[def1.13\]. In fact, we can simply reprove it by only using Lemma \[lem2.2\] and Corollary \[cor2.9A\].
Let $(T,X)$ be equicontinuous surjective; then $\varDelta_X=P(X)=Q(X)$ by Lemma \[lem2.2\]. Conversely, assume $Q(X)=\varDelta_X$; then $(T,X)$ is invertible and pointwise minimal. By Corollary \[cor2.9A\], it follows that $S_\textit{eq}=\varDelta_X$ so $(T,X)$ is equicontinuous surjective.
\[thm2.10\]Let $(T,X)$ be a minimal weak-mixing flow with $X$ non-trivial. Then no point of $X$ is distal.
Since $(T,X)$ is weak-mixing, $Q(X)=X\times X$ and so $S_\textit{eq}=X\times X$ by Corollary \[cor2.9A\]. This shows that $(T,X)$ has no non-trivial equicontinuous factor. However, if $(T,X)$ had a distal point it were point-distal. So by Veech’s Structure Theorem (cf. [@V70 Theorem 6.1] for $X$ metrizable and [@MN Corollary 2.2 and Lemma 2.3] for general $X$), $(T,X)$ would have a non-trivial equicontinuous factor. Thus $(T,X)$ has no distal point. The proof is complete.
In addition, when $(T,X)$ is an invertible semiflow, Corollary \[cor1.11\] can be proved by using Corollary \[cor2.9A\] and Veech’s Structure Theorem ([@AD Corollary 4.6]) as follows:
First $S_\textit{eq}=X\times X$ by Corollary \[cor2.9A\] and so $(T,X)$ has no non-trivial equicontinuous factor. However, if $(T,X)$ has a distal point with $T$ amenable, then by [@AD Corollary 4.6] it follows that $(T,X)$ has a non-trivial equicontinuous factor. Thus $(T,X)$ has no distal point and the proof is complete.
Quasi-regular invariant measure
-------------------------------
Since $X$ is not necessarily metrizable, hence a Borel probability measure does not need be regular. However, next we will show that $(T,X)$ admits an invariant Borel probability measure if and only if it admits an invariant quasi-regular (regular) Borel probability measure. Let $C_c(X)$ be the space of continuous real-valued functions with compact support.
> **Riesz-Markov theorem**. [*Let $X$ be a locally compact $\textrm{T}_2$ space and ${\boldsymbol{I}}$ a positive linear functional on $C_c(X)$. Then there is a Borel measure $\mu$ on $X$ such that $${\boldsymbol{I}}(f)=\int_Xfd\mu\quad \forall f\in C_c(X).$$ The measure $\mu$ may be taken to be quasi-regular. In this case, it is then unique.*]{}
\[lem2.11\]$(T,X)$ admits an invariant measure if and only if it admits an invariant quasi-regular Borel probability measure.
The “if” part is trivial. So we now assume $(T,X)$ admits an invariant Borel probability measure $\nu$. Then by the Riesz-Markov theorem, we can obtain a positive linear functional $${\boldsymbol{I}}(f)=\int_Xfd\nu\quad \forall f\in C(X).$$ Since $\nu$ is $T$-invariant, ${\boldsymbol{I}}$ is also $T$-invariant in the sense that ${\boldsymbol{I}}(f)={\boldsymbol{I}}(ft)$ for all $f\in C(X)$ and $t\in T$. Further by the Riesz-Markov theorem again, we can find a unique quasi-regular Borel probability measure $\mu$ such that $${\boldsymbol{I}}(f)=\int_Xfd\mu\quad \forall f\in C(X).$$ Since ${\boldsymbol{I}}$ is $T$-invariant, so $\mu$ is also invariant. Therefore the “only if” part holds.
In view of Lemma \[lem2.11\], we will identify an invariant Borel probability measure with an invariant quasi-regular Borel probability measure in our later arguments if no confusion arises.
Furstenberg’s structure theorem {#sec2.5}
-------------------------------
Let $T$ be any discrete semigroup with identity $e$ and let $\theta\ge1$ be some ordinal. Following [@F63], a *projective system* of minimal semiflows with phase semigroup $T$ is a collection of minimal semiflows $(T,X_\lambda)$ on compact $\textrm{T}_2$ spaces $X_\lambda$ indexed by ordinal numbers $\lambda\le\theta$, and a family of epimorphisms, $\pi_{\lambda,\nu}\colon(T,X_\lambda)\rightarrow(T,X_\nu)$, for $0\le\nu<\lambda\le\theta$, satisfying:
1. If $0\le\nu<\lambda<\eta\le\theta$, then $\pi_{\eta,\nu}=\pi_{\lambda,\nu}\circ\pi_{\eta,\lambda}$.
2. If $\mu\le\theta$ is a limit ordinal, then $X_\mu$ is the minimal subset of the Cartesian product semiflows $\left(T,\times_{\lambda<\mu}X_\lambda\right)$ consisting of all $x=(x_\lambda)_{\lambda<\mu}\in\times_{\lambda<\mu}X_\lambda$ with $x_\nu=\pi_{\lambda,\nu}(x_\lambda)$ for all $\nu<\lambda<\mu$ and then for $\lambda<\mu$, $\pi_{\mu,\lambda}\colon X_\mu\rightarrow X_\lambda$ is just the projection map. In this case, we say that $(T,X_\mu)$ is the *projective limit* of the family of minimal semiflows $\{(T,X_\lambda)\,|\,\lambda<\mu\}$.
Let $\pi\colon(T,X)\rightarrow(T,Y)$ be an epimorphism of two semiflows. Then $\pi$ is called *relatively equicontinuous* if given $\varepsilon\in\mathscr{U}_X$ there is a $\delta\in\mathscr{U}_X$ such that whenever $(x,x^\prime)\in\delta$ with $\pi(x)=\pi(x^\prime)$, then $(tx,tx^\prime)\in\varepsilon$ for all $t\in T$ (cf. [@F63] and [@Aus p. 95]). In this case, $(T,X)$ is also call a *relatively equicontinuous extension* of $(T,Y)$.
Now based on these definitions, we are ready to state the Furstenberg structure theorem for minimal distal semiflows as follows:
> **Furstenberg’s Structure Theorem** (cf. [@AD Theorem 5.14]). [*Let $\pi\colon (T,X)\rightarrow(T,Y)$ be an epimorphism between distal minimal semiflows. Then there is a projective system of minimal semiflows $\{(T,X_\lambda)\,|\,\lambda\le\theta\}$, for some ordinal $\theta\ge1$, with $X_\theta=X$, $X_0=Y$ such that if $0\le\lambda<\theta$, then $(T,X_{\lambda+1})\xrightarrow{\pi_{\lambda+1,\lambda}}(T,X_\lambda)$ is a relatively equicontinuous extension.*]{}
In fact, we will only need the special case that $(T,Y)$ is the semiflow with $Y$ a singleton space. See Lemma \[lem4.1\] below.
McMahon pseudo-metric and $S_\textit{eq}=Q$ {#sec3}
===========================================
This section will be mainly devoted to proving Theorem \[thm1.4\] and considering another proximity relation of Veech. The McMahon pseudo-metric $D_J$ and the induced equivalence relation $R_J$ based on an invariant closed subset $J$ of $X\times X$ are useful techniques for our aim here.
McMahon pseudo-metric {#sec3.1}
---------------------
In this subsection, let $(T,X)$ be a minimal semiflow; and suppose $(T,Y)$ is any semiflow, which admits an invariant (quasi-regular Borel probability) measure $\mu$. If $J$ is a closed invariant subset of $X\times Y$ and $x\in X$, then the *section* $J_x$ is defined by $$J_x=\{y\in Y\,|\,(x,y)\in J\}.$$ Such $J$ is called a *joining* of $(T,X)$ and $(T,Y)$ in some works if we additionally require that $\pi_Y(J)=Y$ where $\pi_Y\colon(x,y)\mapsto y$.
\[lem3.1\]Let $J$ be a closed invariant subset of $X\times Y$. If $x,x^\prime\in X$, then $\mu(J_x)=\mu(J_{x^\prime})$.
Let $\varepsilon>0$ and let $V$ be an open set in $Y$ with $J_x\subset V$ and $\mu(V)<\mu(J_x)+\varepsilon$. Let $\{t_n\}$ be a net in $T$ for which $t_nx^\prime\to x$. It is easy to see that $J_{t_nx^\prime}\subset V$ for $n\ge n_0$, for some $n_0$. Hence $\mu(J_{x^\prime})\le\mu(J_{t_nx^\prime})\le\mu(V)\le\mu(J_x)+\varepsilon$. Letting $\varepsilon\to 0$, we have $\mu(J_{x^\prime})\le\mu(J_x)$. By symmetry, $\mu(J_x)\le\mu(J_{x^\prime})$ and the lemma is thus proved.
\[def3.2\]If $J$ is a closed invariant subset of $X\times Y$, we define the *McMahon pseudo-metric* $D_J$ on $X$ by $$D_J(x,x^\prime)=\mu(J_x\vartriangle J_{x^\prime})\quad \forall x,x^\prime\in X.$$ It is easy to check that $D_J$ is a pseudo-metric on $X$. Moreover, $D_J(x,x^\prime)=0$ if $J_x\subseteq J_{x^\prime}$ or if $J_x\supseteq J_{x^\prime}$ by Lemma \[lem3.1\].
\[lem3.3\]The pseudo-metric $D_J$ on $X$ is continuous and satisfies $D_J(x,x^\prime)=D_J(tx, tx^\prime)$ for all $t\in T$ and any $x,x^\prime\in X$.
Let $x,x^\prime\in X$ and $t\in T$. First note that $tJ_x\subseteq J_{tx}$ and $\mu(tJ_x)=\mu(J_{tx})$ by Lemma \[lem3.1\]. So $tJ_x=J_{tx}\pmod0$; that is, $\mu(tJ_x\vartriangle J_{tx})=0$. Similarly, $tJ_{x^\prime}\subseteq J_{tx^\prime}$ and so $tJ_{x^\prime}=J_{tx^\prime}\pmod0$. Thus $$t^{-1}(tJ_x)=J_x\pmod0\quad \textrm{and}\quad t^{-1}(tJ_{x^\prime})=J_{x^\prime}\pmod0.$$ This implies that $J_x\cap J_{x^\prime}= t^{-1}(tJ_x\cap tJ_{x^\prime})=t^{-1}(J_{tx}\cap J_{tx^\prime})\pmod0$. Thus by Lemma \[lem3.1\], it follows that $D_J(x,x^\prime)=\mu(J_x)+\mu(J_{x^\prime})-\mu(J_x\cap J_{x^\prime})=\mu(J_{tx})+\mu(J_{tx^\prime})-\mu(J_{tx}\cap J_{tx^\prime})=D_J(tx,tx^\prime)$.
To show that $D_J$ is continuous, we first note that if $U$ is an open set in $Y$, with $J_x\subset U$, and $\mu(U-J_x)<\epsilon$, then $\mu(U-J_{x^\prime})<\epsilon$, for $x^\prime$ sufficiently close to $x$. Now, let $\{x_n\}$ be a net in $X$ with $x_n\to x$. Let $\epsilon>0$, $U$ open in $Y$ with $J_x\subset U$ and $\mu(U-J_x)<\epsilon$. Then if $n\ge n_0$, $J_{x_n}\subset U$ and then $\mu(J_x-J_{x_n})<\epsilon$, so $D_J(x,x_n)<2\epsilon$. Thus, if $x_n\to x$, then $D_J(x,x_n)\to 0$. It follows immediately that $D_J$ is continuous. In fact, let $(x_n,x_n^\prime)\to (x,x^\prime)$, then $$\lim_n|D_J(x,x^\prime)-D_J(x_n,x_n^\prime)|\le\lim_n|D_J(x,x_n)+D_J(x_n,x_n^\prime)+D_J(x_n^\prime,x^\prime)-D_J(x_n,x_n^\prime)|=0.$$ The proof of Lemma \[lem3.3\] is therefore completed.
Let $R_J$ be the equivalence relation on $X$ defined by the pseudo-metric $D_J$ as follows: $(x,x^\prime)\in R_J$ if and only if $D_J(x,x^\prime)=0$.
\[lem3.5\]$R_J$ is a closed $T$-invariant equivalence relation on $X$ which contains $Q(X)$.
Lemma \[lem3.3\] implies that $R_J$ is a closed invariant equivalence relation. Now if $(x,x^\prime)$ belongs to $Q(X)$, let $x_n\to x, x_n^\prime\to x^\prime$ and $t_n(x_n,x_n^\prime)\to(z,z)$ for some $z\in X$, as in Definition \[def1.3\]. Then $D_J(x_n,x_n^\prime)\to D_J(x,x^\prime)$ and $D_J(x_n,x_n^\prime)=D_J(t_nx_n,t_nx_n^\prime)\to D_J(z,z)=0$. So $D_J(x,x^\prime)=0$ and thus $(x,x^\prime)\in R_J$. This shows $Q(X)\subseteq R_J$.
\[thm3.6\]If $J$ is a closed invariant subset of $(T,X\times Y)$ and if $(T,X)$ is surjective. Then $S_{\textit{eq}}(X)\subseteq R_J$.
This follows immediately from Lemma \[lem3.5\] and Corollary \[cor2.9A\].
It should be noticed that in Theorem \[thm3.6\], we do not require that $\mu$ is such that $\mu(V)>0$ for all non-empty open subset $V$ of $Y$. For example, if $\mu$ is exactly concentrated on a fixed point, then $R_J=X\times X$ and so $(T,X/R_J)$ is trivial.
$S_\textit{eq}(X)=Q(X)$ {#sec3.2}
-----------------------
Now we will specialize to the case $(T,Y)=(T,X)$, which admits an invariant measure $\mu$. There is no loss of generality in assuming $\mu$ is quasi-regular by Lemma \[lem2.11\].
Let $(T,X)$ be a minimal semiflow, which admits an invariant quasi-regular measure $\mu$ on $X$. Then $S_\textit{eq}(X)=Q(X)$.
First by Lemma \[lem2.4A\], $(T,X)$ is surjective. Thus $Q(X)\subseteq S_{\textit{eq}}(X)$ by Corollary \[cor2.9A\]. To prove Theorem \[thm1.4\], it is sufficient to show that $S_{\textit{eq}}(X)\subseteq Q(X)$.
Let $(x,y)\in S_{\textit{eq}}(X)$, and let $V$ be a neighborhood of $x$. Consider the closed invariant subset $J$ of $X\times X$ defined by $$J=\overline{{\bigcup}_{x^\prime\in V}T(y,x^\prime)}=\overline{{\bigcup}_{x^\prime\in V}\overline{T(y,x^\prime)}}.$$ Hence $$J=\mathrm{cls}_{X\times X}\left\{(z,w)\,|\,\exists\,x^\prime\in V\textrm{ and }\exists\,\{t_n\}\textrm{ in }T\textit{ s.t. }t_n(y,x^\prime)\to(z,w)\right\}.$$ Then $V\subset J_y$ (by taking $\{t_n\}=\{e\}$). Now since $(x,y)\in S_{\textit{eq}}(X)$, so $(x,y)\in R_J$ by Theorem \[thm3.6\], and it follows that $V\subseteq J_x\pmod 0$ and so $V\subseteq J_x$ for $\textrm{supp}\,(\mu)=X$.
Summarizing, if $(x,y)\in S_{\textit{eq}}(X)$, and $V$ is a neighborhood of $x$, then there is an $x^\prime\in V$ and a $\tau\in T$ such that $\tau x^\prime$ and $\tau y$ are in $V$. Since $V$ is arbitrary, it follows that $(x,y)\in Q(X)$. Moreover, $(x,x)\in J$ by $V\subseteq J_x$.
The proof of Theorem \[thm1.4\] is thus completed.
Note that the proof just given shows that in the definition of $Q(X)$ one can take one of the nets in $X$ to be constant. Precisely, we have:
\[lem3.7\]Under the hypotheses of Theorem \[thm1.4\], the following conditions are pairwise equivalent:
1. $(x,y)\in Q(X)$.
2. There are nets $\{x_n\}$ in $X$ and $\{t_n\}$ in $T$ with $x_n\to x$, $t_nx_n\to x$ and $t_ny\to x$.
3. There are nets $\{x_n\}$ in $X$ and $\{s_n\}$ in $T$ such that $x_n\to x, s_nx_n\to y$ and $s_ny\to y$.
\(1) $\Leftrightarrow$ (2) in flows is due to McMahon [@McM].
“(1) $\Leftrightarrow$ (2)” and “(3) $\Rightarrow$” (1) are obvious. For “(1) $\Rightarrow$ (3)”, as in the proof of Theorem \[thm1.4\], since $(x,x)\in J$ and $J$ is closed invariant, so by minimality of $(T,X)$, it follows that $(y,y)\in J$. Thus, we can find nets $\{x_n\}$ and $\{s_n\}$ such that $x_n\to x$, $s_nx_n\to y$ and $s_ny\to y$. This proves Lemma \[lem3.7\].
The (1) $\Leftrightarrow$ (2) of Lemma \[lem3.7\] implies the following (cf. [@Aus Corollary 9.10] for $T$ a group).
\[cor3.8\]Let $(T,X)$ be a minimal semiflow admitting an invariant measure. Then, $$\begin{gathered}
Q[y]={\bigcap}_{\alpha\in\mathscr{U}_X}\overline{{\bigcup}_{t\in T}(t^{-1}\alpha)[y]},\end{gathered}$$ for all $y\in X$.
First, if $x\in\bigcap_{\alpha\in\mathscr{U}_X}\overline{\bigcup_{t\in T}(t^{-1}\alpha)[y]}$, then for every $\alpha\in\mathscr{U}_X$, there are $x_\alpha\in\alpha[x]$ and $t_\alpha\in T$ with $(t_\alpha y,t_\alpha x_\alpha)\in\alpha$. This shows that $x\in Q[y]$. Conversely, if $x\in Q[y]$, then by Lemma \[lem3.7\], $x\in\overline{\bigcup_{t\in T}(t^{-1}\alpha)[y]}$ for all $\alpha\in\mathscr{U}_X$. Thus $x\in\bigcap_{\alpha\in\mathscr{U}_X}\overline{\bigcup_{t\in T}(t^{-1}\alpha)[y]}$.
The (1) $\Leftrightarrow$ (3) of Lemma \[lem3.7\] is very useful for proving Theorem \[thm1.15\] in $\S\ref{sec5}$. The relation in (3) of Lemma \[lem3.7\] was first introduced and studied by Veech in [@V77]. See $\S\ref{sec3.3}$ for the details.
Another proximity relation and the proof of Theorem \[thm1.15\](1) {#sec3.3}
------------------------------------------------------------------
Following Veech [@V77 p. 806 and p. 819], we introduce the following notation.
Let $(T,X)$ be a minimal semiflow.
1. We say $(T,X)$ satisfies the *Bronstein condition* if $(T,X\times X)$ contains a dense set of minimal points.
2. Given $x\in X$, define $U[x]$ to be the set of $z\in X$ for which there exist nets $t_n\in T$ and $z_n\in X$ such that $z_n\to z$, $t_nz_n\to x$ and $t_nx\to x$.
It is clear that $P[x]\subseteq U[x]\subseteq Q[x]$ for all $x\in X$. In [@V77] Veech proved the following theorem:
> **Theorem** (cf. [@V77 Theorem 2.7.6]). [*Let $(T,X)$ be a minimal flow satisfying the Bronstein condition. Then $S_\textit{eq}[x]=U[x]=Q[x]$ for all $x\in X$.*]{}
Therefore, it holds that
> **Corollary** (cf. [@V77 Theorem 2.7.5]). [*If $(T,X)$ is a minimal distal flow, then we have $S_\textit{eq}[x]=U[x]=Q[x]$ for all $x\in X$.*]{}
With an invariant measure instead of the Bronstein condition, by using Lemma \[lem3.7\] and Theorem \[thm1.4\] we can easily obtain the following.
\[thm3.10\]Let $(T,X)$ be a minimal semiflow, which admits an invariant measure. Then $U[x]=Q[x]=S_\textit{eq}[x]$ for all $x\in X$.
Since every distal semiflow always has an invariant measure by Furstenberg’s theorem, we can easily obtain the following corollary to Theorem \[thm3.10\].
\[cor3.11\]If $(T,X)$ be a minimal distal semiflow, then $S_\textit{eq}[x]=U[x]=Q[x]$ for all $x\in X$.
In fact, we have the following, where $D[x]$ is as in (4) of Definition \[def1.13\].
\[lem3.12\]If $(T,X)$ is a minimal bi-continuous semiflow, then we have $D[x]=U[x]$ for all $x\in X$.
\[cor3.13\]If $(T,X)$ be a minimal bi-continuous semiflow, then $D[x]\subseteq Q[x]$ for all $x\in X$.
This follows at once from Theorem \[thm3.10\] and Lemma \[lem3.12\].
Weak-mixing minimal semiflows {#sec4}
=============================
In this section, we will characterize a minimal weak-mixing semiflow by using the McMahon pseudo-metric $D_J$ introduced in $\S\ref{sec3.1}$. Moreover, we will consider the chaotic dynamics of minimal weak-mixing semiflow with amenable phase semigroup and with metrizable phase space.
Characterizations of minimal weak-mixing semiflows {#sec4.1}
--------------------------------------------------
Let $(T,X)$ be a surjective semiflow. Recall that $\mathfrak{U}(X)$ is the collection of non-empty open subsets of $X$. Then it is clear that the weak-mixing is “highly non-equicontinuous”. In fact, if $(T,X)$ is weak-mixing, then $Q(X)=X\times X$; and then further its factor $(T,X_{\textit{eq}})$ is trivial by Corollary \[cor2.9A\].
\[lem4.1\]Let $(T,X)$ be a minimal surjective semiflow. Then the following two conditions are equivalent:
1. $(T,X)$ has no non-trivial distal factor.
2. $(T,X)$ has no non-trivial equicontinuous factor; that is, $S_\textit{eq}(X)=X\times X$.
$(1)\Rightarrow(2)$ by Lemma \[lem2.2\]. And $(2)\Rightarrow (1)$ follows easily from Furstenberg’s structure theorem stated in $\S\ref{sec2.5}$.
Therefore, any minimal surjective semiflow is “highly non-equicontinuous” if and only if it is “highly non-distal.”
\[def4.2\]Let $(T,X)$ and $(T,Y)$ be two semiflows with compact $\textrm{T}_2$ phase spaces and with the same phase semigroup $T$. We will say $(T,X)$ is *weakly disjoint* from $(T,Y)$, denoted $X\perp^wY$, if $(T,X\times Y)$ is a T.T. semiflow.
It should be noted here that since our phase spaces need not be metrizable, Definition \[def4.2\] is not equivalent to requiring that $(T,X\times Y)$ is point-transitive even for $T$ in groups as in [@Aus p. 150].
Let $(f,X)$ be a cascade where $f$ is a homeomorphism on a compact metric space, and assume $(f,X)$ has no non-trivial equicontinuous factor. Then $(f,X\times X)$ is T.T. ([@KR]). Next we will consider an open question of Furstenberg.
Let $T$ be a semigroup; then the class of minimal semiflow with phase semigroup $T$ will be denoted by $\texttt{SF}_{\min}$, and we write $$\begin{gathered}
\texttt{SF}_{tt}=\{(T,X)\,|\,(T,X)\textrm{ is T.T.}\}\quad \textrm{and}\quad\texttt{SF}_{wm}=\{(T,X)\,|\, (T,X\times X)\textrm{ is T.T.}\}.\end{gathered}$$ Note that all the phase spaces are compact $\textrm{T}_2$ for our semiflows. Moreover, $\texttt{SF}_{\min}\subset\texttt{SF}_{tt}$ for all semigroup $T$.
Furstenberg’s [@F67 Proposition II.11] asserts that $\texttt{SF}_{wm}\times\texttt{SF}_{\min}\subset\texttt{SF}_{tt}$ if $T=\mathbb{Z}_+$. In view of this, he further asked the following problem:
> *Is it true for* $T=\mathbb{Z}_+$ *that* $\texttt{SF}_{wm}\times\texttt{SF}_{tt}\subset\texttt{SF}_{tt}$*?* (See [@F67 Problem F].)
This is false in general; however we will consider the following question:
> [*$\left(\texttt{SF}_{wm}\cap\texttt{SF}_{\min}\right)\times\left(\texttt{SF}_{tt}\cap \textit{\textbf{?}}\right)\subset\texttt{SF}_{tt}$*]{}
As to this, there is an important theorem due to Veech:
> **Theorem J** (Veech [@V77 Theorem 2.1.6]). [*Let $T$ be a group. Let $(T,X)$ be “incontractible” minimal and assume $(T,X)$ has no nontrivial equicontinuous factor. Let $(T,Y)$ be T.T. having a dense set of a.p. points. Then $(T,X\times Y)$ is a T.T. flow.*]{}
Next we will introduce a class of semiflows which are weaker than minimal semiflows with amenable phase semigroups.
\[def4.3\]Let $T$ be a discrete semigroup and let $X$ vary in the set of compact $\textrm{T}_2$ spaces.
1. By the *ergodic center* of $(T,X)$, denoted by $\mathscr{C}_{erg}(T,X)$, we mean the smallest closed invariant subset of $X$ of $\mu$-measure $1$, for all invariant measure $\mu$ of $(T,X)$. If $(T,X)$ has no invariant measure, then we shall say $\mathscr{C}_{erg}(T,X)=\emptyset$.
2. $(T,X)$ is called an *E-semiflow*, denoted $(T,X)\in\texttt{SF}_{e}$, if $(T,X)\in\texttt{SF}_{tt}$ and $\mathscr{C}_{erg}(T,X)=X$. Equivalently, $(T,X)\in\texttt{SF}_{e}$ if and only if it is T.T. with full ergodic center.
The following lemma is a simple observation and so we will omit its proof details.
\[lem4.4\]Let $T$ be an amenable semigroup. If $(T,X)$ is T.T. such that the set of minimal points is dense in $X$, then $(T,X)\in\texttt{SF}_{e}$.
$\texttt{SF}_{e}$ is an extension of the class of *E*-systems of Glasner and Weiss [@GW1; @GW2]. If $(T,X)\in\texttt{SF}_{e}$ with $T$ a countable discrete semigroup, then $N_T(U,V)$ is syndetic in $T$ for all $U,V\in \mathfrak{U}(X)$.
\[lem4.5\]If $(T,X)\in\texttt{SF}_{e}$ and $U\in\mathfrak{U}(X)$, then there exists an invariant measure $\mu$ of $(T,X)$ such that $\mu(U)>0$. Hence every $(T,X)$ in $\texttt{SF}_{e}$ is surjective.
First we note that $\mathscr{C}_{erg}(T,X)$ is just equal to the closure of the union of the supports of all invariant measures of $(T,X)$. Then there is some invariant measure $\mu$ such that $\textrm{supp}\,(\mu)\cap U\not=\emptyset$ so that $\mu(U)>0$. Finally, let $t\in T$. Since $X\setminus tX$ is open and $tX$ is Borel, so if $tX\not=X$ we have $1>\mu(tX)=\mu(t^{-1}tX)=\mu(X)=1$ a contradiction. Thus $(T,X)$ is surjective. The proof is complete.
\[thm4.6\]Let $(T,X)\in\texttt{SF}_{\min}$ be surjective. Then $(T,X)$ has no non-trivial distal factor if and only if $X\perp^wY$ for all $(T,Y)\in\texttt{SF}_{e}$.
The “only if” part: By Lemma \[lem4.1\] we may assume $(T,X)$ has no non-trivial equicontinuous factor $X_\textit{eq}$. Let $(T,Y)$ be TT with $\mathscr{C}_{erg}(T,Y)=Y$. If $(T,X\times Y)\not\in\texttt{SF}_{tt}$, then there would be an invariant closed subset $J$ of $X\times Y$ such that $\textrm{Int}_{X\times Y}J\not=\emptyset$ and that $J\not=X\times Y$. So there could be found a point $x\in X$ and an $U\in\mathfrak{U}(Y)$ such that $U\subset J_x$. Moreover, $J_x\not=Y$. Let $V=Y-J_x$. Then $V\not=\emptyset$. Since $(T,Y)$ is TT, so there is $\tau\in T$ such that $\tau U\cap V\not=\emptyset$.
Since $U\cap\tau^{-1}V\in\mathfrak{U}(Y)$, we can find an invariant quasi-regular Borel probability $\mu$ on $Y$ such that $\mu(U\cap\tau^{-1}V)>0$. Since $\tau U\cap V=\tau(U\cap\tau^{-1}V)$, hence $\mu(\tau J_x\cap V)>0$ and then $\mu(J_{\tau x}\cap V)>0$. Having let $D_J$ be the McMahon pseudo-metric as in Definition \[def3.2\] associated to $J$ and $\mu$, $D_J(x,\tau x)>0$. Let $R_J$ be the closed invariant equivalence relation defined by $D_J$ on $X$. Then $R_J\not=X\times X$, and by Theorem \[thm3.6\], $(T,X/R_J)$ is a non-trivial equicontinuous factor of $(T,X)$. This contradiction shows that $J=X\times Y$ and therefore $(T,X\times Y)$ must be TT and so $X\perp^wY$.
The “if" part: Let $X\perp^wY$ for all $(T,Y)\in\texttt{SF}_{e}$. To be contrary, assume $X_\textit{eq}$ is not a singleton set. Then $(T,X_\textit{eq})$ is minimal distal by Lemma \[lem2.2\]. So $(T,X_\textit{eq})\in\texttt{SF}_{e}$ and thus $X\perp^wX_\textit{eq}$. Therefore $X_\textit{eq}\perp^wX_\textit{eq}$ and $(T,X_\textit{eq})$ is weak-mixing. This is a contradiction to Lemma \[lem2.3\]. Thus $X_\textit{eq}$ must be a singleton set.
Therefore the proof of Theorem \[thm4.6\] is completed.
It should be noted that $(T,X)$ need not admit an invariant measure in the above Theorem \[thm4.6\].
\[lem4.7\]Let $(T,X)\in\texttt{SF}_{\min}$ admits an invariant measure. Then it has no non-trivial equicontinuous factor if and only if it is weak-mixing.
First $(T,X)$ is surjective by Lemma \[lem2.4A\], and moreover, $(T,X)\in\texttt{SF}_{e}$. If $(T,X)$ has no non-trivial equicontinuous factor, then it is weak-mixing from Theorem \[thm4.6\]. Conversely, if $(T,X)$ is weak-mixing, then $Q(X)=X\times X$ so that $(T,X)$ has no non-trivial equicontinuous factor by Corollary \[cor2.9A\]. The proof is complete.
The following Corollary \[cor4.8\] implies the first “if and only if” of Theorem \[thm1.7\]. In addition, if $T$ is a nilpotent group (then it is amenable), every minimal flow with phase group $T$ is incontractible; thus for this case the forgoing Theorem J of Veech may follow from Corollary \[cor4.8\] and Theorem \[thm4.12\] below.
\[cor4.8\]Let $(T,X)\in\texttt{SF}_{\min}$ admits an invariant measure. Then $(T,X)$ is weak-mixing iff $X\perp^wY$ for all $(T,Y)\in\texttt{SF}_{e}$.
\[cor4.9\]Let $T$ be an amenable semigroup and $(T,X)\in\texttt{SF}_{\min}$. Then $(T,X)$ is weak-mixing iff $X\perp^wY$ for all $(T,Y)\in\texttt{SF}_{\min}$ iff $X\perp^wY$ for all $(T,Y)\in\texttt{SF}_{e}$.
Comparing with [@Aus Corollary 11.8] here our phase spaces are not required to be a metric space and further ‘T.T.’ does not imply ‘point-transitive’ in our setting. Moreover, the minimality of $(T,X)$ is important for Corollary \[cor4.9\]. In fact, following [@AG; @HY] there are non-minimal non-weak-mixing cascade $(T,X)$, which are weakly disjoint every *E*-system.
\[cor4.10\]Let $(T,X)\in\texttt{SF}_{\min}$ be surjective. If $(T,X)$ is weak-mixing, then $X\perp^wY$ for all $(T,Y)\in\texttt{SF}_{e}$.
By Corollary \[cor2.9A\], $S_\textit{eq}(X)\supseteq Q(X)$. Then the statement follows from Theorem \[thm4.6\] and the fact that $Q(X)=X\times X$.
Now Theorem \[thm1.8\] stated in $\S\ref{sec1.2}$ easily follows from the following more general theorem. Our proof below is of interest because there exists no Furstenberg’s intersection lemma ([@F67 Proposition II.3] and [@DT Lemma 3.2]) in the literature for non-abelian phase semigroup $T$.
\[thm4.11\]Let $(T,X)$ be a minimal semiflow, which admits an invariant measure. Then the following conditions are pairwise equivalent:
1. $(T,X)$ is weak-mixing.
2. $(T,X^n)$ is T.T. for all $n\ge2$.
3. $N_T(U,V)$ is discretely thick in $T$ for all $U,V\in\mathfrak{U}(X)$.
4. Let $I$ be any set with $\textrm{card}\,I\ge2$ and $(T,X_i)=(T,X)$ for all $i\in I$; then $(T,\times_{i\in I}X_i)$ is weak-mixing.
$(1)\Rightarrow(2)$. We will proceed to show that $(T,X^n)$ is T.T. for all $n\ge2$ by induction on $n$. First, by definition of weak-mixing, $(T,X\times X)$ is T.T. so the case of $n=2$ holds. Now, assume the statement holds for all integer $n=k\ge2$. Let $Y=X^k$ and then $(T,Y)$ is a T.T. semiflow. Since $(T,X)$ is minimal with an invariant measure, then $(T,X)$ is surjective by Lemma \[lem2.4A\] and there is an invariant measure $\nu$ on $X$ with $\textrm{supp}\,(\nu)=X$. Define a Borel probability measure $\mu=\nu^k$ on $Y$ by $\mu(V_1\times\dotsm\times V_k)=\nu(V_1)\dotsm\nu(V_k)$ for all open subsets $V_1,\dotsc,V_k$ of $X$. Clearly, $\mu$ is $T$-invariant on $Y$ and moreover every open subset of $Y$ has positive $\mu$-measure. Then by Theorem \[thm4.6\], $(T,X\times Y)$ must be T.T. and thus $(T,X^{k+1})$ is a T.T. semiflow.
$(2)\Rightarrow(3)$. Let $U,V\in\mathfrak{U}(X)$ and $K=\{k_1,\dotsc,k_n\}$ a finite subset of $T$. Since by Lemma \[lem2.4A\] each $k_i^{-1}V\not=\emptyset$, $U\times\dotsm\times U$ and $k_1^{-1}V\times\dotsm\times k_n^{-1}V$ both are non-empty open subsets of $X^n$, then there is some $t\in T$ such that $U\times\dotsm\times U\cap t^{-1}(k_1^{-1}V\times\dotsm\times k_n^{-1}V)\not=\emptyset$. Thus $Kt\subseteq N_T(U,V)$ and so $N_T(U,V)$ is discretely thick in $T$.
$(3)\Rightarrow(1)$. Let $U,V,U^\prime, V^\prime\in\mathfrak{U}(X)$. Then $N_T(U,V)$ is discretely thick in $T$; moreover, $N_T(U^\prime, V^\prime)$ is discretely syndetic in $T$ by [@DT Lemma 2.3]. So $N_T(U,V)\cap N_T(U^\prime, V^\prime)\not=\emptyset$. This shows that $N_T(U\times U^\prime, V\times V^\prime)\not=\emptyset$ and thus $(T,X\times X)$ is a T.T. semiflow.
$(2)\Leftrightarrow(4)$. $(4)\Rightarrow(2)$ is obvious; and $(2)\Rightarrow(4)$ follows from the definition of the product topology of $\times_{i\in I}X_i$.
The proof of Theorem \[thm4.11\] is thus completed.
The following Theorem \[thm4.12\] implies that:
> If $(T,X)$ is minimal with $T$ an amenable semigroup, then $(T,X)$ is weak-mixing if and only if it has no non-trivial distal factor.
This would just answer a question of Petersen in [@P70 p. 280]. Moreover Theorem \[thm4.12\] is a semigroup version of Theorem \[thm2.10\] by different approaches.
\[thm4.12\] Let $(T,X)$ be a minimal semiflow, which admits of an invariant measure. Then, the following five statements are pairwise equivalent:
1. $(T,X)$ is weak-mixing.
2. $Q(X)=X\times X$.
3. $S_{\textit{eq}}(T,X)=X\times X$.
4. $(T,X)$ has no non-trivial distal factor.
5. $(T,X)$ has no non-trivial equicontinuous factor.
In particular, each of $(1)$ through $(5)$ implies that $(T,X)$ has no distal point.
First of all, $(T,X)$ is surjective by Lemma \[lem2.4A\]. $(1)\Rightarrow(2)$ and $(3)\Rightarrow(5)$ both are trivial; $(2)\Rightarrow(3)$ is by Corollary \[cor2.9A\] and $(4)\Leftrightarrow(5)$ follows easily from Lemma \[lem4.1\]. Finally $(5)\Rightarrow(1)$ follows from Lemma \[lem4.7\] at once.
Finally let any of $(1)$ through $(5)$ hold. By Theorem \[thm4.11\], $(T,X)$ is thickly transitive, i.e., for all $U,V\in\mathfrak{U}(X)$, $N_T(U,V)$ is a thick subset of $T$. Then no point of $X$ is distal for $(T,X)$ by Corollary \[cor1.11\]. Precisely speaking, this follows from a proof similar to that of Corollary \[cor1.11\] with Theorem \[thm4.11\] in place of Theorem \[thm1.8\].
The proof of Theorem \[thm4.12\] is thus completed.
Note here that comparing with [@Aus Theorem 9.13] the only new ingredients of Theorem \[thm4.12\] are 1): our phase space $X$ is not required to be metrizable; and 2): $T$ is not necessarily a group. The non-metrizable condition of $X$ that is expected by Petersen in [@P70] is just the point for Theorem \[thm1.12\].
In particular, let $T=\mathbb{Z}$ and $X$ be a compact metric space, and assume $(T,X)$ has no nontrivial equicontinuous factor; then $(T,X\times X)$ is T.T. (Keynes and Robertson [@KR]).
Chaos of minimal weak-mixing semiflow {#sec4.2}
-------------------------------------
Let $(T,X)$ be a semiflow on a uniform non-singleton space $(X,\mathscr{U}_X)$, where $\mathscr{U}_X$ is a compatible symmetric uniform structure on $X$. We first introduce the notion of sensitivity.
$(T,X)$ is said to be *sensitive to initial conditions* if there exists an $\varepsilon\in\mathscr{U}_X$ such that for all $x\in X$ and $\delta\in\mathscr{U}_X$ there are $y\in\delta[x]$ and $t\in T$ with $(tx,ty)\not\in\varepsilon$.
Given any $\varepsilon\in\mathscr{U}_X$, let $$\textrm{Equi}_\varepsilon(T,X)=\left\{x\in X\,|\,\exists\,\delta\in\mathscr{U}_X\textit{ s.t. }t(\delta[x])\subseteq\varepsilon[tx]\ \forall t\in T\right\}.$$ Then,
> $(T,X)$ is not sensitive to initial conditions iff $\textrm{Equi}_\varepsilon(T,X)\not=\emptyset$ for all $\varepsilon\in\mathscr{U}_X$.
It is already known that “weak-mixing is highly non-equicontinuous” (cf. Theorem \[thm4.12\]). In fact, weak-mixing is even imcompatible with “$\varepsilon$-equicontinuity” as follows:
\[lem4.14\]If $(T,X)$ is a weak-mixing semiflow on a uniform $T_2$-space $(X,\mathscr{U}_X)$, then $(T,X)$ is sensitive to initial conditions.
By contradiction, suppose $(T,X)$ were not sensitive to initial conditions; then it holds that $\textrm{Equi}_\varepsilon(T,X)\not=\emptyset$ for all $\varepsilon\in\mathscr{U}_X$. By the weak-mixing property, $(T,X\times X)$ is a T.T. semiflow. Since $X\times X\not=\varDelta_X$, there is an $\varepsilon\in\mathscr{U}_X$ such that $U:=X\times X\setminus\textrm{cls}_{X\times X}\varepsilon\not=\emptyset$. Take an $x_0\in\textrm{Equi}_{\varepsilon/3}(T,X)$ where $\varepsilon/3$ is an entourage in $\mathscr{U}_X$ such that $\varepsilon/3\circ\varepsilon/3\circ\varepsilon/3\subseteq\varepsilon$. Then there is a $\delta\in\mathscr{U}_X$ such that $t(\delta[x_0]\times\delta[x_0])\subseteq\varepsilon$ for all $t\in T$. Thus $N_T(\delta[x_0]\times\delta[x_0],U)=\emptyset$, which is a contradiction to that $(T,X\times X)$ is a T.T. semiflow. The proof of Lemma \[lem4.14\] is therefore completed.
Let $\mathscr{K}(T)$ be the collection of non-empty compact subsets of $T$. Given $\varepsilon\in\mathscr{U}_X$ and $x\in X$, we define the $\varepsilon$-stable set of $(T,X)$ at $x$ as follows: $$W_\varepsilon^s(T,X;x)=\{y\in X\,|\,\exists\,K\in\mathscr{K}(T)\textit{ s.t. }(tx,ty)\in\varepsilon\ \forall t\in T\setminus K\}.$$
Then there holds the following lemma.
\[lem4.15\]Let $(T,X)$ be a semiflow such that $T$ is $\sigma$-compact. If $(T,X)$ is sensitive to initial conditions, then there is an $\varepsilon\in\mathscr{U}_X$ such that $W_\varepsilon^s(T,X;x)$ is of the first category in $X$ for all $x\in X$.
The following notion is stronger than the above sensitivity to initial conditions.
Let $(T,X)$ be a semiflow and $\varepsilon,\delta\in\mathscr{U}_X$ with $\varepsilon>\delta$. For $x\in X$ define a set $$\textrm{LY}_{\varepsilon\textrm{-}\delta}[x]=\left\{y\in X\,|\,y\not\in W_\varepsilon^s(T,X;x)\textit{ and }\exists\,t\in T\textit{ s.t. }(tx,ty)\in\delta\right\}.$$ We say that $(T,X)$ is *$\varepsilon\textrm{-}\delta$ Li-Yorke sensitive* if $\textrm{LY}_{\varepsilon\textrm{-}\delta}[x]$ is of the second category for all $x\in X$.
\[thm4.17\]Let $(T,X)$ be a minimal semiflow with $T$ $\sigma$-compact, which admits of an invariant measure (e.g. $T$ is amenable). If $(T,X)$ is weak-mixing, then there exists an $\varepsilon\in\mathscr{U}_X$ such that for all $\delta\in\mathscr{U}_X$ with $\delta<\varepsilon$, $(T,X)$ is $\varepsilon\textrm{-}\delta$ Li-Yorke sensitive.
By Lemma \[lem4.14\], $(T,X)$ is sensitive to initial conditions. So by Lemma \[lem4.15\], it follows that there exists an $\varepsilon\in\mathscr{U}_X$ such that $W_\varepsilon^s(T,X;x)$, for all $x\in X$, is of the first category.
Let $\delta\in\mathscr{U}_X$ with $\delta<\varepsilon$ be any given. Given any $x\in X$, let $P[x]$ be the proximal cell at $x$; i.e., $y\in P[x]$ iff $\exists\,z\in X$ and $\{t_n\}\subseteq T$ s.t. $(t_nx,t_ny)\to(z,z)$. Then by a slight modification of the proof of Corollary \[cor1.11\] in $\S\ref{sec1.2}$, we can conclude that $P[x]$ is dense in $X$. In addition, $$P[x]=\bigcap_{\alpha\in\mathscr{U}_X}\bigcup_{t\in T}t^{-1}(\alpha[tx]).$$ Then $\bigcup_{t\in T}t^{-1}(\alpha[tx])$ is an open dense subset of $X$ and $\left(\bigcup_{t\in T}t^{-1}(\delta[tx])\right)\setminus W_\varepsilon^s(T,X;x)\subseteq\textrm{LY}_{\varepsilon\textrm{-}\delta}[x]$. Thus, $\textrm{LY}_{\varepsilon\textrm{-}\delta}[x]$ is of the second category.
This thus completes the proof of Theorem \[thm4.17\].
Let $(T,X)$ be a semiflow and $\varepsilon\in\mathscr{U}_X$. For $x\in X$ define a set $$\textrm{LY}_{\varepsilon\textrm{-}0}[x]=\left\{y\in X\,|\,y\not\in W_\varepsilon^s(T,X;x)\textit{ and }\exists\,\{t_n\}\in T\textit{ s.t. }{\lim}_n(t_nx,t_ny)\in\varDelta_X\right\}.$$ We say that $(T,X)$ is *$\varepsilon\textrm{-}0$ Li-Yorke sensitive* if $\textrm{LY}_{\varepsilon\textrm{-}0}[x]$ is of the second category for all $x\in X$.
\[thm4.19\]Let $(T,X)$ be a minimal semiflow with $T$ $\sigma$-compact and $X$ a compact metric space, which admits of an invariant measure (e.g. $T$ is amenable). If $(T,X)$ is weak-mixing, then there exists an $\varepsilon\in\mathscr{U}_X$ such that $(T,X)$ is $\varepsilon\textrm{-}0$ Li-Yorke sensitive.
This follows from the proof of Theorem \[thm4.17\] by noting that $P[x]$ is a dense $G_\delta$-set in $X$ when $X$ is a compact metric space.
Let $T$ be $\sigma$-compact with a sequence of compact subsets $F_1\subset F_2\subset F_3\subset\dotsm$ such that $T=\bigcup F_n$ and $\epsilon>0$. $(T,X)$ with $X$ a metric space is called *densely Li-Yorke $\epsilon$-chaotic* if there is a dense Cantor set $\Theta\subseteq X$ such that for all $x,y\in\Theta$ with $x\not=y$, there are $\{t_n\}$ and $\{s_n\}$ in $T$ with the properties:
- $\lim_{n\to\infty}d(t_nx,t_ny)=0$;
- $s_n\not\in F_n$ and $\lim_{n\to\infty}d(s_nx,s_ny)\geq\epsilon$.
Now we can easily obtain a Li-Yorke chaos result from Theorem \[thm4.19\].
\[cor4.21\]Let $(T,X)$ be a minimal semiflow with $T$ $\sigma$-compact and $X$ a compact metric space, which admits of an invariant measure (e.g. $T$ is amenable). If $(T,X)$ is weak-mixing, then $(T,X)$ is densely Li-Yorke $\epsilon$-chaotic for some $\epsilon>0$.
Veech’s structure theorem for $\mathrm{a.a.}$ semiflows {#sec5}
=======================================================
Let $V$ and $D$ be Veech’s relations defined as in Definition \[def1.13\] in $\S\ref{sec1.3}$. This section will be devoted to proving (2) and (3) of Theorem \[thm1.15\].
For that, we need to introduce two important lemmas. The first one is borrowed from [@Dai].
\[lem5.1\]Let $(T,X)$ be an invertible minimal semiflow. Then $V[x]$ is a dense subset of $D[x]$ for each $x\in X$. If $X$ is metrizable, then $V[x]=D[x]$ for all $x\in X$.
\[lem5.2\]Let $(T,X)$ be a bi-continuous semiflow and let $x_0\in P_{\!aa}(T,X)$. Then $V[x_0]$ is a dense subset of $D[x_0]$. If $X$ is metrizable, then $V[x_0]=D[x_0]$.
Let $\Sigma$ be the set of all continuous pseudo-metrics on the compact $\textrm{T}_2$ space $X$. Let $\rho\in\Sigma$ be arbitrarily fixed. First, for every $\delta>0$ and all $x,y\in X$, we shall say $(x,y)\in\delta$ iff $\rho(x,y)<\delta$. For closed subsets $A,B$ of $X$, we say $\rho_H(A,B)<\delta$ if and only if $\rho(a,b)<\delta$ for all $a\in A$ and $b\in B$. We will say $\rho(A,x)<\delta$ if and only if $\rho(a,x)<\delta$ for some $a\in A$. Write $$\begin{gathered}
\langle T^{-1}\circ T\rangle=\left\{\tau_{i_1}^{-1}\sigma_{i_1}\dotsm\tau_{i_n}^{-1}\sigma_{i_n}\,|\,\tau_{i_k}, \sigma_{i_k}\in T \textit{ for }k=1,\dotsc,n, 1\le n<\infty\right\}.\end{gathered}$$ Let $\{\delta_n\}_{n=1}^\infty$ be a sequence of positive numbers such that $\sum_n\delta_n<\infty$.
If $x^\prime\in D[x_0]$, then by definition there exist, for each $n$ and $\epsilon>0$, elements $\sigma,\tau\in N_T(x_0,\delta_n[x_0])$ such that $\rho(\tau^{-1}\sigma x_0,x^\prime)<\epsilon$. Moreover, if $F$ is any finite subset of $\langle T^{-1}\circ T\rangle$ it can also be arranged that $\rho_H(s\gamma x_0,sx_0)<\delta_n\ \forall s\in F$, for $\gamma=\sigma$ and $\tau$.
Since $x_0$ is a distal point so it is an a.p. point, then $N_T(x_0,\delta[x_0])$ is syndetic in $T$ for all $\delta>0$. We can select a sequence $(\sigma_1,\tau_1), (\sigma_2,\tau_2), \dotsc$ inductively as follows. First we can choose $\sigma_1,\tau_1\in N_T(x_0,\delta_1[x_0])$ and $F_0=\{e\}$ with $\rho(\tau_1^{-1}\sigma_1x_0,x^\prime)<\delta_1$. Having chosen $(\sigma_1,\tau_1), \dotsc, (\sigma_n,\tau_n)$ let $F_n$ be the finite set of elements of $\langle T^{-1}\circ T\rangle$ which are representable as $$t=\tau_1^{\epsilon_1}\sigma_1^{\epsilon_1^\prime}\dotsm\tau_n^{\epsilon_n}\sigma_n^{\epsilon_n^\prime}\quad \textrm{where }\epsilon_i=0\textrm{ or }-1\textrm{ and } \epsilon_i^\prime=0\textrm{ or }1.$$ Then choose $\sigma_{n+1},\tau_{n+1}\in N_T(x_0,\delta_{n+1}[x_0])$ in such a way that
1. $\rho_H(s\gamma x_0,sx_0)<\delta_{n+1}\ \forall s\in F_n$, where $\gamma\in\{\sigma_{n+1},\tau_{n+1}\}$; and
2. $\rho(\tau_{n+1}^{-1}\sigma_{n+1}x_0,x^\prime)<\delta_{n+1}$.
Based on the sequence $\{(\sigma_n,\tau_n)\}$, we define $\alpha_1=\tau_1, \alpha_2=\sigma_1\tau_2$ in $T$, and in general, $$\alpha_n=\sigma_1\dotsm\sigma_{n-1}\tau_n\in T\quad (n=2,3,\dotsc).$$ If $m<n$ we have from (a) that $$\rho(\alpha_mx_0,\alpha_nx_0)\le\sum_{j=0}^{n-m-1}\rho(\alpha_{m+j}x_0,\alpha_{m+j+1}x_0)
\le\sum_{j=0}^{n-m-1}(2\delta_{m+j}+\delta_{m+j+1})$$ tends to $0$ as $m\to\infty$. Thus $\rho\textrm{-}\lim_n\alpha_nx_0$ exists because $\sum_{n=1}^{\infty}\delta_n<\infty$. Letting $y$ be the $\rho$-limit, we now claim $x^\prime\in\rho\textrm{-}\lim_{m\to\infty}\alpha_m^{-1}y$ (that is, $\exists x_m^\prime\in\alpha_m^{-1}y$ s.t. $\rho$-$\lim x_m^\prime=x^\prime$). To see this we note that if $n>m$, then $$\alpha_m^{-1}\alpha_nx_0=\tau_m^{-1}\sigma_{m-1}^{-1}\dotsm\sigma_1^{-1}\sigma_1\dotsm\sigma_{n-1}\tau_nx_0=\tau_m^{-1}\sigma_m\dotsm\sigma_{n-1}\tau_nx_0$$ because $t^{-1}tx=x\ \forall t\in T$ for all $x\in P_{\!aa}(T,X)$ and $P_{\!aa}(T,X)$ is invariant; and therefore by argument as above, for $n>m$, $$\rho_H\left(\alpha_m^{-1}\alpha_nx_0,\tau_m^{-1}\sigma_mx_0\right)\le\sum_{k=m+1}^n\delta_k\to 0\quad \textrm{as }m\to\infty.$$ Thus by (b), $$\lim_{m\to\infty}\rho\left(\alpha_m^{-1}y,x^\prime\right)=\lim_{m\to\infty}\lim_{n\to\infty}\rho\left(\alpha_m^{-1}\alpha_n x,x^\prime\right)=0$$ and then we can choose $x_m^\prime\in\alpha_m^{-1}y$ such that $x_m^\prime\to x^\prime$. By choosing a subnet $\{\beta_i\}$ from the sequence $\{\alpha_n\}$ in $T$, there are points $z,x^{\prime\prime}\in X$ such that $\beta_ix_0\to z$ and $x^{\prime\prime}\in V[x_0]\cap\lim_j\lim_i\beta_j^{-1}\beta_ix_0$ in $X$; and moreover, $\rho(x^{\prime\prime},x^\prime)=0$. Of course, if $\rho$ is just a metric on $X$, then $x^{\prime\prime}=x^\prime$.
This shows that $V[x_0]$ is dense in $D[x_0]$ and $V[x_0]=D[x_0]$ when $X$ is metrizable. The proof of Lemma \[lem5.2\] is thus complete.
The following result is exactly (2) of Theorem \[thm1.15\], which generalizes Veech’s [@V68 Theorem 1.1] from abelian groups to invertible semiflows admitting invariant measures by using completely different approaches. In fact, our proof is much more simpler than Veech’s one presented in [@V68] that depends on harmonic analysis on locally compact abelian groups.
\[5.3\]Let $(T,X)$ be an invertible minimal semiflow admitting an invariant measure. Then $Q[x]=D[x]=\overline{V[x]}$ for all $x\in X$.
Given $x\in X$, $D[x]\subseteq Q[x]$ is obvious. In fact, $V[x]\subseteq Q[x]$ and $Q[x]$ is closed by Definitions \[def1.3\] and \[def1.13\]. So $D[x]=\overline{V[x]}\subseteq Q[x]$ by Lemma \[lem5.1\]. To show $Q[x]\subseteq D[x]$ for all $x\in X$, we fix an $x\in X$ and let $y\in Q[x]$. Then by (3) of Lemma \[lem3.7\], there are nets $\{y_n\}, \{z_n\}$ in $X$ and $\{t_n\}$ in $T$ such that $$\begin{gathered}
y_n\to y,\quad z_n\to x,\quad z_n=t_ny_n,\quad t_nx\to x.\end{gathered}$$ Then $Q[x]\subseteq D[x]$ by Lemma \[lem3.12\]. Therefore, we have concluded that $D[x]=Q[x]$ for all $x\in X$. This proves Theorem \[5.3\].
Using Lemma \[lem5.2\] in place of Lemma \[lem5.1\], we can obtain the following, which is important for (3) of Theorem \[thm1.15\] stated in $\S\ref{sec1.3}$.
\[5.4\]Let $(T,X)$ be a minimal bi-continuous semiflow admitting an invariant measure. If $x_0\in P_{\!aa}(T,X)$, then $\{x_0\}=V[x_0]=D[x_0]=Q[x_0]$.
At first, by [@AD Lemma 3.5] we see $(T,X)$ is surjective. By Lemma \[lem5.2\], it follows that $\{x_0\}=V[x_0]=D[x_0]\subseteq Q[x_0]$. Moreover, $Q[x_0]\subseteq D[x_0]$ by Lemma \[lem3.12\]. Thus we can conclude that $\{x_0\}=V[x_0]=D[x_0]=Q[x_0]$ and the proof is complete.
Based on Theorem \[thm1.4\] and Lemma \[5.4\] we can obtain the next result which is just the third part of Theorem \[thm1.15\] stated in $\S\ref{sec1.3}$.
\[thm5.5\]Let $(T,X)$ be a minimal bi-continuous semiflow admitting an invariant measure. Then $(T,X)$ is $\mathrm{a.a.}$ if and only if $\pi\colon(T,X)\rightarrow(T,X_\textit{eq})$ is of almost 1-1 type.
Since $(T,X)$ is a minimal semiflow admitting an invariant measure, by Theorem \[thm1.4\] we have $S_\textit{eq}(X)=Q(X)$.
(1). Let $(T,X)$ be a.a.; then there exists an $x_0\in X$ such that $V[x_0]=\{x_0\}$. Then $Q[x]=\{x_0\}$ by Lemma \[5.4\]. Thus, $\pi\colon(T,X)\rightarrow(T,X_\textit{eq})$ is of almost 1-1 type at $x_0$. This has concluded the “only if” part.
(2). Assume $\pi\colon (T,X)\rightarrow(T,X_\textit{eq})$ is of almost 1-1 type. Then we can take an $x\in X$ such that $\pi^{-1}\pi(x)=\{x\}$. Set $y=\pi(x)$. Since $(T,X_\textit{eq})$ is minimal equicontinuous invertible by Lemmas \[lem2.4A\] and \[lem2.2\], hence $V[y]=\{y\}$. This implies that $V[x]=\{x\}$ and so $(T,X)$ is an a.a. semiflow. This proves the “if” part. The proof is complete.
Let $(T,X)$ be a minimal bi-continuous semiflow, which admits an invariant measure. Then $(T,X)$ is equicontinuous iff all points are a.a. points.
\(1) The “only if” part. By Lemma \[lem2.3\], $P(X)=Q(X)=\varDelta_X$. Thus $(T,X)$ is pointwise a.a. by Theorem \[thm1.15\].
\(2) The “if” part. By Theorem \[thm1.15\], $Q(X)=\varDelta_X$. Then $(T,X)$ is equicontinuous by Lemma \[lem2.3\]. This proves Corollary \[cor1.17\].
**Acknowledgments** {#acknowledgments .unnumbered}
===================
The author would like to thank Professor Joe Auslander and Professor Eli Glasner for their many helpful comments on the first version of this paper. This work was partly supported by National Natural Science Foundation of China (Grant Nos. 11431012, 11271183) and PAPD of Jiangsu Higher Education Institutions.
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abstract: 'We discuss various modifications of separability, precompactness and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<{\omega}}(X)$.'
address:
- 'T.Banakh: Ivan Franko National University of Lviv (Ukraine), and Jan Kochanowski University in Kielce (Poland)'
- 'I.Guran: Ivan Franko National University of Lviv (Ukraine)'
- 'O.Ravsky: Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine'
author:
- 'Taras Banakh, Igor Guran, Alex Ravsky'
title: 'Generalizing separability, precompactness and narrowness in topological groups'
---
In this paper we define and analyze various properties related to separability and narrowness in topological groups and test these properties for the permutation groups $S(X)$ and $S_{<{\omega}}(X)$. All topological groups in this paper are Hausdorff.
For a set $X$ by $S(X)$ we denote the permutation group of $X$, and by $S_{<{\omega}}(X)$ the normal subgroup of $X$, consisting of permutations $f:X\to X$ having finite support ${\mathrm{supp}}(f)=\{x\in X:f(x)\ne x\}$. The groups $S(X)$ and $S_{<{\omega}}(X)$ carry the topology of pointwise convergence, i.e., the topology inherited from the Tychonoff power $X^X$ of the discrete space $X$.
For subsets $A,B$ of a group $G$ let $AB=\{ab:a\in A,\;b\in B\}$ be their pointwise product in $G$. For a topological group $G$ by $\tau_*$ we denote the family of open neighborhoods of the identity $1_G$ in $G$.
By ${\omega}$ and ${\omega}_1$ we denote the smallest infinite and uncountable cardinals, respectively. For a set $X$ and a cardinal $\kappa$, let $[X]^{<\kappa}=\{A\subseteq X:|A|<\kappa\}$. Therefore, $[X]^{<{\omega}}$ and $[X]^{<{\omega}_1}$ are the families of finite and countable subsets of $X$, respectively.
As a motivation of subsequent definitions, let us consider characterizations of separable and precompact topological groups.
A topological space $X$ is [*separable*]{} if it contains a countable dense subset of $X$. Separable topological groups admit the following (trivial) characterization.
\[t:sep\] For any topological group $G$ the following conditions are equivalent:
1. $G$ is separable;
2. $\exists S_1\in[G]^{<{\omega}_1}\;\forall U_2\in\tau_*$ such that $S_1U_2=G$;
3. $\exists S_1\in[G]^{<{\omega}_1}\;\forall U_2\in\tau_*$ such that $U_2S_1=G$;
4. $\exists S_1\in[G]^{<{\omega}_1}\;\forall U_2\in\tau_*$ such that $U_2S_1U_2=G$.
Following [@BGR], we define a topological group $G$ to be [*duoseparable*]{} if $G$ contains a countable subset $S$ such that $SUS=G$ for every neighborhood $U\subseteq G$ of the unit. By [@BGR], every topological group is a subgroup of a duoseparable topological group, which means that the duoseparability is a strictly weaker property than the separability.
Next, we discuss (Roelcke) precompact and (Roelcke) narrow topological groups. A topological group $G$ is [*precompact*]{} (resp. [*narrow*]{}) if for any neighborhood $U$ of the neutral element $1_G$ there is a finite (resp. countable) subset $S\subseteq G$ such that $SU=G=US$. It is well-known [@AT 3.7.17] that a topological group $G$ is precompact if and only if $G$ is a subgroup of a compact topological group. By [@Gur] (see also [@AT 3.4.23]), a topological group is narrow if and only if it is topologically isomorphic to a subgroup of the Tychonoff product of second-countable topological groups.
Precompact topological groups admit the following characterization (see [@Us 15.81] or [@BT 4.3]), which resembles the characterization of separability in Theorem \[t:sep\].
\[t:precomp\] For any topological group $G$ the following conditions are equivalent:
1. $G$ is precompact;
2. $\forall U_1\in \tau_*\;\exists S_2\in [G]^{<{\omega}}$ such that $U_1S_2=G$;
3. $\forall U_1\in \tau_*\;\exists S_2\in [G]^{<{\omega}}$ such that $S_2U_1=G$;
4. $\forall U_1\in\tau_*\;\exists S_2\in [G]^{<{\omega}}$ such that $S_2U_1S_2=G$.
A topological group $G$ is called [*Roelcke precompact*]{} (resp. [*Roelcke narrow*]{}) if for any neighborhood $U$ of the neutral element $1_G$ there is a finite (resp. countable) subset $S\subseteq G$ such that $USU=G$. By [@Us], every topological group is a subgroup of a Roelcke precompact group, which implies that the Roelcke precompactness is a strictly weaker property than the precompactness and the Roelcke narrowness is strictly weaker than the narrowness.
Theorems \[t:sep\] and \[t:precomp\] motivate the following definitions that fit into a general scheme. We start with properties that generalize (Roelcke) precompactness and (Roelcke) narrowness.
\[d:n\] Let $\kappa,\lambda$ be infinite cardinals. A topological group $G$ is called
- $\mathsf u_1\mathsf s_2^{\kappa}$ if $\forall U_1\in \tau_*\;\;\exists S_2\in [G]^{<\kappa}$ such that $U_1S_2=G$;
- $\mathsf s_2^{\kappa}\mathsf u_1$ if $\forall U_1\in \tau_*\;\;\exists S_2\in [G]^{<\kappa}$ such that $S_2U_1=G$;
- $\mathsf s_2^{\kappa}\mathsf u_1\mathsf s_2^{\kappa}$ if $\forall U_1\in\tau_*\;\;\exists S_2\in [G]^{<\kappa}$ such that $S_2U_1S_2=G$;
- $\mathsf u_1\mathsf s_2^{\kappa}\mathsf u_1$ if $\forall U_1\in \tau_*\;\;\exists S_2\in [G]^{<\kappa}$ such that $U_1S_2U_1=G$;
- $\mathsf u_1\mathsf s_2^{\kappa}\mathsf u_1\mathsf s_3^\lambda\mathsf u_1$ if $\forall U_1\in \tau_*\;\exists S_2\in [G]^{<\kappa}\;\exists S_3\in[G]^{<\lambda}$ such that $U_1S_2U_1S_3U_1=G$;
- $\mathsf u_1\mathsf s_2^{\kappa}\mathsf u_3$ if $\forall U_1\in \tau_*\;\exists S_2\in [G]^{<\kappa}\;\forall U_3\in\tau_*$ such that $U_1S_2U_3=G$;
- $\mathsf u_1\mathsf s_2^{\kappa}\mathsf u_3\mathsf s_4^{\lambda}$ if $\forall U_1\in \tau_*\;\exists S_2\in [G]^{<\kappa}\;\forall U_3\in\tau_*\;\exists S_4\in[G]^{<\lambda}$ such that $U_1S_2U_3S_4=G$;
- $\mathsf s^\kappa_2\mathsf u_1\mathsf s_3^{\lambda}\mathsf u_4$ if $\forall U_1\in \tau_*\;\exists S_2\in [G]^{<\kappa}\;\exists S_3\in[G]^{<\lambda}\;\forall U_4\in\tau_*$ such that $S_2U_1S_3U_4=G$.
Observe that the properties (2),(3),(4) of Theorem \[t:precomp\] coincide with the properties $\mathsf u_1\mathsf s_2^{{\omega}}$, $\mathsf s_2^{{\omega}}\mathsf u_1$, $\mathsf s_2^{{\omega}}\mathsf u_1\mathsf s_2^{{\omega}}$, respectively. The narrowness is equivalent to the conditions $\mathsf u_1\mathsf s_2^{{\omega}_1}$ and $\mathsf s_2^{{\omega}_1}\mathsf u_1$, but is strictly stronger than $\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_2^{{\omega}_1}$, see Theorem \[t:main\].
Next, we introduce some generalizations of the (duo)separability.
\[d:s\] Let $\kappa,\lambda$ be infinite cardinals. A topological group $G$ is called
- $\mathsf s_1^{\kappa}\mathsf u_2$ if $\exists S_1\in [G]^{<\kappa}\;\;\forall U_2\in \tau_*$ such that $S_1U_2=G$;
- $\mathsf u_2\mathsf s_1^{\kappa}$ if $\exists S_1\in [G]^{<\kappa}\;\;\forall U_2\in \tau_*$ such that $U_2S_1=G$;
- $\mathsf s_1^{\kappa}\mathsf u_2\mathsf s_1^{\kappa}$ if $\exists S_1\in [G]^{<\kappa}\;\;\forall U_2\in \tau_*$ such that $S_1U_2S_1=G$;
- $\mathsf s_1^{\kappa}\mathsf u_2\mathsf s_1^{\kappa}\mathsf u_2$ if $\exists S_1\in [G]^{<\kappa}\;\;\forall U_2\in \tau_*$ such that $S_1U_2S_1U_2=G$;
- $\mathsf u_2\mathsf s_1^{\kappa}\mathsf u_2$ if $\exists S_1\in [G]^{<\kappa}\;\;\forall U_2\in \tau_*$ such that $U_2S_1U_2=G$;
- $\mathsf s_1^{\kappa}\mathsf u_2\mathsf s_3^\lambda$ if $\exists S_1\in [G]^{<\kappa}\;\;\forall U_2\in \tau_*\;\;\exists S_3\in[G]^{<\lambda}$ such that $S_1U_2S_3=G$;
- $\mathsf s_1^{\kappa}\mathsf u_2\mathsf s_3^\lambda\mathsf u_4$ if $\exists S_1\in [G]^{<\kappa}\;\;\forall U_2\in \tau_*\;\;\exists S_3\in[G]^{<\lambda}\;\;\forall U_4\in \tau_*$ such that $S_1U_2S_3U_4=G$.
Observe that the conditions (2), (3), (4) of Theorem \[t:sep\] coincide with the properties $\mathsf s_1^{{\omega}_1}\mathsf u_2$, $\mathsf u_2\mathsf s_1^{{\omega}_1}$, $\mathsf u_2\mathsf s_1^{{\omega}_1}\mathsf u_2$, respectively. The duoseparability coincides with $\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_1^{{\omega}_1}$.
Following the general scheme we could introduce infinitely many properties extending those in Definitions \[d:n\] and \[d:s\]. But we wrote down only the properties that will appear in Theorem \[t:main\] and Example \[ex1\], which are the main results of this paper.
For any topological group we have the following implications (for the unique nontrivial implication $\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_3^{\omega}{\Rightarrow}\mathsf u_1\mathsf s_2^{{\omega}_1}$, see Lemma 3.31 in [@Pachl]). $$\xymatrix{
\mathsf s_1^{\omega}\mathsf u_2 \mathsf s_1^{\omega}\ar@{<=>}[r]\ar@{=>}[d]&\mathsf s_1^{\omega}\mathsf u_2\ar@{<=>}[r]&\mbox{finite}\ar@{=>}[d]\ar@{<=>}[r]&\mathsf u_2\mathsf s_1^{\omega}\ar@{<=>}[rr]&&\mathsf u_2\mathsf s_1^{{\omega}}\mathsf u_2\ar@{=>}[ddl]\\
\mathsf s_2^{\omega}\mathsf u_1\mathsf s_2^{\omega}\ar@{<=>}[r]&\mathsf s_2^{\omega}\mathsf u_1\ar@{<=>}[r]&
\mbox{precompact}\ar@{<=>}[r]\ar@{=>}@/_35pt/[dd]&\mathsf u_1\mathsf s_2^{\omega}\ar@{<=>}[r]\ar@{=>}@/_30pt/[ddd]&\mathsf u_1\mathsf s_2^{{\omega}}\mathsf u_3\ar@{=>}[r]&{\color{red}\mathsf u_1\mathsf s_2^{{\omega}}\mathsf u_1}\ar@{=>}[dd]
\\
\mathsf s_1^{{\omega}_1}\!\mathsf u_2\mathsf s_1^{{\omega}_1}\ar@{=>}[dd]\ar@{=>}@/_18pt/[rrdd]&\mathsf s_1^{{\omega}_1} \mathsf u_2 \ar@{=>}[l]&
\mbox{separable}\ar@{=>}[d]\ar@{<=>}[r]\ar@{<=>}[l]&\mathsf u_2\mathsf s_1^{{\omega}_1}\ar@{<=>}[r]&\mathsf u_2\mathsf s_1^{{\omega}_1}\mathsf u_2\ar@{=>}[ddl]&
\\
&\mathsf s_2^{{\omega}_1}\mathsf u_1\ar@{<=>}[r]&
\mbox{narrow}\ar@{<=>}[r]\ar@{=>}[d]&\mathsf u_1\mathsf s_2^{{\omega}_1}\ar@{<=>}[r]&\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_3\ar@{<=>}[d]\ar@{=>}[r]&{\color{red}\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_1}\ar@{=>}^{?}[dd]
\\
{\color{blue}\mathsf s_1^{{\omega}_1}\!\mathsf u_2\mathsf s_1^{{\omega}_1}\!\mathsf u_2}\ar@{=>}[d]&{\color{green}\mathsf s_2^{{\omega}_1} \mathsf u_1\mathsf s_2^{{\omega}_1}}&\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_3^{{\omega}_1}\ar@{=>}[l]\ar@{=>}[dll]\ar@{=>}[d]&\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_3^{\omega}\ar@{=>}[d]\ar@{=>}[r]\ar@{=>}[l] &\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_3^{\omega}\ar@{=>}[dr]&
\\
{\color{blue}\mathsf s_1^{{\omega}_1}\!\mathsf u_2\mathsf s_3^{{\omega}_1}\!\mathsf u_4}&{\color{red}\mathsf u_1\mathsf s_2^{{\omega}_1}\!\mathsf u_3\mathsf s_4^{{\omega}_1}}&{\color{blue}\mathsf u_2s_1^{{\omega}_1}\mathsf u_2\mathsf s_3^{{\omega}_1}}\ar@{=>}[l]&{\color{blue}\mathsf u_2s_1^{{\omega}_1}\mathsf u_2\mathsf s_3^{\omega}}\ar@{=>}[r]\ar@{=>}[l]&{\color{red}\mathsf u_1\mathsf s_2^{{\omega}_1}\!\mathsf u_3\mathsf s_4^{{\omega}}}\ar@{=>}@/^20pt/[lll]\ar@{=>}_(.45){?}[r]&{\color{red}\mathsf u_1\mathsf s_2^{{\omega}_1}\!\mathsf u_1\mathsf s_3^{{\omega}}\mathsf u_1}\\
}$$
The implications which are not labeled by the question mark cannot be reversed, which is either obvious or witnessed by the examples constructed in Theorem \[t:main\] and Examples \[ex0\], \[ex1\]. A property in the diagram is drawn with
- the color if it holds for both groups $S_{<{\omega}}({\omega}_1)$ and $S({\omega}_1)$;
- the color if it holds for $S_{<{\omega}}({\omega}_1)$ but not for $S({\omega}_1)$;
- the color if it holds for $S({\omega}_1)$ but not for $S_{<{\omega}}({\omega}_1)$;
- the [*black*]{} color if it does not hold neither for $S_{<{\omega}}({\omega}_1)$ nor for $S({\omega}_1)$.
\[t:main\] For a set $X$ of infinite cardinality $\kappa$, the topological group
1. $S_{<{\omega}}(X)$ is $\mathsf u_1\mathsf s_2^{\omega}\mathsf u_1$, $\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_2^{{\omega}_1}$, $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_3\mathsf s_4^{\omega}$, but is neither $\mathsf s_1^\kappa\mathsf u_2\mathsf s_3^\kappa\mathsf u_4$ nor $\mathsf u_2\mathsf s_1^\kappa\mathsf u_2\mathsf s_3^\kappa$;
2. $S(X)$ is $\mathsf u_1\mathsf s_2^{\omega}\mathsf u_1$, $\mathsf s_1^{{\omega}_1}\!\mathsf u_2\mathsf s_1^{{\omega}_1}\mathsf u_2$, $\mathsf u_2\mathsf s_1^{{\omega}_1}\!\mathsf u_2\mathsf s_3^{\omega}$, but not $\mathsf s_2^{\kappa}\mathsf u_1\mathsf s_2^\kappa$.
The proof of this theorem will be divided into a series of lemmas.
The topological groups $S_{<{\omega}}(X)$ and $S(X)$ are $\mathsf u_1\mathsf s_2^{\omega}\mathsf u_1$.
Let $G$ denote the group $S(X)$ or $S_{<{\omega}}(X)$.
Given any neighborhood $U_1\in\tau_*$ of the identity $1_G$, find a finite subset $A\subset X$ such that $U_1\supseteq V=\{f\in G:\forall a\in A\;\;f(a)=a\}$. Let $B\subset X\setminus A$ be any set with $|B|=|A|$. Let $S_2$ be any finite subset of $G$ such that for each injective function $g:A\to A\cup B$ there exists a permutation $f\in S_2$ such that $f{\restriction} A=g$. We claim that $U_1S_2U_1=G$. Given any $h\in G$, find a permutation $v\in V$ such that $v(h(A)\setminus A)\subseteq B$. Then $v\circ h(A)=v(h(A)\cap A)\cup v(h(A)\setminus A)\subseteq A\cup B$. The choice of $S_2$ yields a permutation $s\in S_2$ such that $s{\restriction}A=v\circ h{\restriction}A$. It follows that the permutation $u=s^{-1}\circ v\circ h$ belongs to the set $V$ and hence $h=v^{-1}\circ s\circ u\in U_1S_2U_1$, witnessing that $G$ is $\mathsf u_1\mathsf s_2^{\omega}\mathsf u_1$.
The topological group $G=S_{<{\omega}}(X)$ is $\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_2^{{\omega}_1}$.
Given any neighborhood $U\in\tau_*$ of the identity of $G$, find a finite set $A\subset X$ such that the subgroup $\{f\in G:\forall a\in A\;\;f(a)=a\}$ is contained in $U_1$. Choose a countable infinite set $S_2\subseteq G$ such that the family $(s(A))_{a\in S_2}$ is disjoint and $s=s^{-1}$ for every $s\in S_2$. We claim that $G=\bigcup_{s\in S_2}sU_1s^{-1}$. Indeed, given any permutation $h\in G$ find $s\in S_2$ such that $s(A)$ is disjoint with the finite set ${\mathrm{supp}}(h)$. Then the permutation $u=s^{-1}\circ h\circ s$ belongs to the set $U_1$ and hence $h\in sU_1s^{-1}=sUs\subseteq S_2U_1S_2$, which means that $G$ is $\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_2^{{\omega}_1}$.
\[l:usus\] The topological group $G=S_{<{\omega}}(X)$ is $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_3\mathsf s_4^{\omega}$.
Given any neighborhood $U_1\in\tau_*$ of the identity in $G$, find a finite set $A_1\subset X$ such that $U_1\supset V_1:=\{f\in G:\forall a\in A_1\;\;f(a)=a\}$. Let $\mathcal A_1$ be an arbitrary countably infinite family of mutually disjoint subsets of $X$ such that $|A|=|A_1|$ for each $A\in\mathcal A_1$. Let $S_2=S_2^{-1}\subset G$ be a countable set such that for each $A'_1\in\mathcal A_1$ and each bijection $f_1:A_1\to A'_1$ there exists a permutation $f_2\in S_2$ such that $f_2{\restriction}A_1=f_1$. Next, let $U_3\in \tau_*$ be any neighborhood of the identity in $G=S_{<{\omega}}(X)$. Then there exists a finite subset $A_3\subset X$ such that $U_3\supseteq V_3:=\{f\in S_{<{\omega}}(X):\forall a\in A_3\;\;f(a)=a\}$. Let $\mathcal A_3$ be a finite family of pairwise disjoint subsets of $X$ such that $|\mathcal A_3|>|A_1|$ and $|A|=|A_3|$ for each $A\in\mathcal A_3$. Finally choose a finite subset $S_4=S_4^{-1}$ of $S_{<{\omega}}(X)$ such that for each $A'_3\in\mathcal A_3$ there exists $f_4\in S_4$ such that $f_4(A_3)=A'_3$. We claim that $S_{<{\omega}}(X)=U_1S_2U_3S_4$. Indeed, let $h$ be any element of $G$. Since $|\mathcal A_3|>|A_1|$, there exists a set $A'_3\in\mathcal A_3$ such that $A'_3\cap h^{-1}(A_1)=\varnothing$. Pick $f_4\in S_4$ such that $f_4(A_3)=A'_3$. Then $f_4^{-1}h^{-1}(A_1)\cap A_3=\varnothing$. Choose a set $A'_1\in\mathcal A_1$ such that $A'_1\cap A_3=\varnothing$. Pick an arbitrary $v_3\in V_3$ such that $v_2f_4^{-1}h^{-1}(A_1)=A'_1$. There exists $f_2\in S_2$ such that $f_2(a)=v_3f_4^{-1}h^{-1}(a)$ for each $a\in A_1$. Then $f_2^{-1}v_3f_4^{-1}h^{-1}(a)=a$ for each $a\in A_1$ and hence the permutation $\mathsf u_1=f_2^{-1}v_3f_4^{-1}h^{-1}$ belongs to $V_1\subseteq U_1$. Then $h=u_1^{-1}f_2^{-1}v_3f_4^{-1}\in U_1S_2U_3S_4$ and hence $S_{<{\omega}}(X)$ is $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_3\mathsf s_4^{\omega}$.
The topological group $G=S(X)$ is $\mathsf u_2\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_3^{\omega}$.
Fix a permutation $h\in S(X)$ such that for every $x\in X$ the points $h^n(x)$, $n\in{\mathbb Z}$, are pairwise distinct. Let $S_1=\{h^n\}_{n\in{\mathbb Z}}$. Given any neighborhood $U_2\subseteq G$ of the identity, find a finite set $A\subset X$ such that $U_2\supseteq\{g\in G:\forall a\in A\;\;g(a)=a\}$. Choose a finite family $S_3\in[G]^{<{\omega}}$ such that $|S_3|=|A|+1$ and the family $(s(A))_{s\in S_3}$ is disjoint and $s=s^{-1}$ for every $s\in S_3$.
We claim that $U_2S_1U_2S_3=G$. Given any permutation $g\in G$, find $s_3\in S_3$ such that $s_3(A)\cap g^{-1}(A)=\emptyset$. Such permutation $s_3$ exists since the family $(s(A))_{s\in S_3}$ is disjoint and consists of $|g^{-1}(A)|+1$ many sets. Then $gs_3(A)\cap A=\emptyset$. The choice of the permutation $h$ ensures that $s_1(A)\cap A=\emptyset$ for some $s_1\in S_1$.
Since the set $gs_3(A)\cup s_1(A)$ is disjoint with the set $A$, we can find a permutation $u_2\in U_2$ such that $u_2s_1{\restriction}A=gs_3{\restriction}A$. Then the permutation $u_2'=s_1^{-1}u_2^{-1}gs_3$ belongs to the neighborhood $U_2$ and hence $g=u_2s_1u_2's_3^{-1}=u_2s_1u_2's_3\in U_2S_1U_2S_3$.
The topological group $G=S_{<{\omega}}(X)$ is not $\mathsf s_1^\kappa\mathsf u_2\mathsf s_3^\kappa\mathsf u_4$.
We should prove that $\forall S_1\in[G]^{<\kappa}\;\exists U_2\in\tau_*\;\forall S_3\in [G]^{<\kappa}\;\exists U_4\in\tau_*$ such that $S_1U_2S_3U_4\ne G$.
Given any $S_1\in[G]^{<\kappa}$ find a point $a\in X\setminus\bigcup_{s\in S_1}{\mathrm{supp}}(s)$ and consider the open neighborhood $U_2=\{f\in G:f(a)=a\}$. For any set $S_3\in[G]^{<\kappa}$, we can find a point $b\in X\setminus\{f^{-1}(a):f\in S_3\}$ and consider the neighborhood $U_4=\{f\in G:f(b)=b\}\in\tau_*$.
We claim that for any $f\in S_1U_2S_3U_4$, we have $f(b)\ne a$. Find $s_1\in S_1,u_2\in U_2,s_3\in S_3,u_4\in U_4$ such that $f=s_1u_2s_3u_4$. The choice of $a\notin{\mathrm{supp}}(s_1)$ ensures that $s_1u_2(a)=s_1(a)=a$ and the choise of $b$ guarantees that $b\ne s_3^{-1}(a)$ and hence $s_3(b)\ne a$. Then $f(b)=s_1u_2s_3(b)\ne a$ and $S_1U_2S_3U_4\ne G$.
The topological group $G=S_{<{\omega}}(X)$ is not $\mathsf u_2\mathsf s_1^\kappa\mathsf u_2\mathsf s_3^\kappa$.
Assuming that $G$ is $\mathsf u_2\mathsf s_1^\kappa\mathsf u_2\mathsf s_3^\kappa$, find a set $S_1\in[G]^{<\kappa}$ such that for any neighborhood $U_2\subseteq G$ of the identity there exists a set $S_3\in[G]^{<\kappa}$ such that $U_2S_1U_2S_3=G$. Choose any point $a\in X\setminus\bigcup_{s\in S_1}{\mathrm{supp}}(s)$ and consider the neighborhood $U_2=\{g\in G:g(a)=a\}$ of the identity in $G$. The choice of $S_1$ yields a set $S_3\in [G]^{<\kappa}$ such that $G=U_2S_1U_2S_3$. Choose any permutation $g\in G$ such that $g^{-1}(a)\notin \{s^{-1}(a):s\in S_3\}$. Since $g\in G=U_2S_2U_2S_3$, there are permutations $u_2,u_2'\in U_2$, $s_1\in S_1$, $s_3\in S_3$ such that $g=u_2s_1u_2's_3$. Then for the point $b=s_3^{-1}(a)$, we get $g(b)=u_2s_1u_2's_3(b)=u_2s_1u_2'(a)=a$ and hence $g^{-1}(a)=b\in\{s^{-1}(a):s\in S_3\}$, which contradicts the choice of $g$.
The topological group $S(X)$ is $\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_1^{{\omega}_1}\mathsf u_2$.
Choose a permutation $h\in S(X)$ such that for every $x\in X$ the points $h^n(x)$, $n\in{\mathbb Z}$, are pairwise distinct. Consider the countable subset $S_1=\{h^n:n\in{\mathbb Z}\}$ of the group $S(X)$. We claim that $S_{<{\omega}}(X)\subseteq S_1U_2S_1$ for every neighborhood $U_2\in\tau_*$ of the identity in $S(X)$.
Given any $U_2\in\tau_*$, find a finite set $A\subset X$ such that the subgroup $V_2=\{f\in S(X):\forall a\in A\; f(a)=a\}$ is contained in $U_2$. The choice of the permutation $h$ guarantees that for any permutation $f\in S_{<{\omega}}(X)$, there is $n\in{\mathbb Z}$ such that $h^n(A)\cap{\mathrm{supp}}(f)=\emptyset$. This implies $h^{-n}\circ f\circ h^n\in V_2$ and hence $f\in h^nV_2h^{-n}\subseteq S_1U_2S_1$. The density of the subgroup $S_{<{\omega}}(X)\subseteq S_1U_2S_1$ in $S(X)$ implies that $S_1U_2S_1U_2=S(X)$.
The topological group $G=S(X)$ is not $\mathsf s_2^\kappa\mathsf u_1\mathsf s_2^\kappa$.
Fix any point $a\in X$ and consider the neighborhood $U_1=\{f\in G:f(a)=a\}$. Assuming that $G$ is $\mathsf s_2^\kappa\mathsf u_1\mathsf s_2^\kappa$, we can find a set $S_2\in[G]^{<\kappa}$ such that $G=S_2U_1S_2$. Choose any permutation $g\in S(X)$ such that $g\big(\{t^{-1}(a):t\in S_2\})\cap \{s(a):s\in S_2\}=\varnothing$. Find permutatuons $s,t\in S_2$ and $u\in U_1$ such that $g=sut$ and observe that $gt^{-1}(a)=su(a)=s(a)$, which contradicts the choice of $g$. This contradiction shows that $G$ is not $\mathsf s_2^\kappa\mathsf u_1\mathsf s_2^\kappa$.
In the following examples by ${\mathbb Z}$ be denote the additive group of integers, endowed with the discrete topology.
\[ex0\] In the Tychonoff power ${\mathbb Z}^{{\omega}_1}$ of ${\mathbb Z}$, consider the dense subgroup $$G=\big\{(x_i)_{i\in{\omega}_1}\in{\mathbb Z}^{{\omega}_1}:|\{i\in{\omega}_1:x_i\ne 0\}|<{\omega}\big\}$$ and observe that $G$ is narrow but not $\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_1^{{\omega}_1}\mathsf u_2$ and not $\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_3^{\omega}$.
\[ex1\] Let $X$ be a nonseparable Banach space and $X\rtimes {\mathbb Z}$ be the product $X\times {\mathbb Z}$ endowed with the group operation $*$ defined by $$(x,n)*(y,m)=(x+2^ny,n+m).$$ The topological group $X\rtimes{\mathbb Z}$ is $\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_1^{{\omega}_1}$ but not $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_3^{{\omega}}\mathsf u_1$.
Identify $X$ and ${\mathbb Z}$ with the subgroups $X\times\{0\}$ and $\{0\}\times{\mathbb Z}$ of the semidirect product $G=X\rtimes{\mathbb Z}$. It is clear that the binary operation $*:G\times G\to G$ is continuous and so is the inversion $(\cdot)^{-1}:G\to G$, $(\cdot)^{-1}:(x,n)\mapsto (-2^{-n}x,-n)$. Therefore, $(G,*)$ is a topological group. It can be shown (see [@BGR]) that $G={\mathbb Z}*U*{\mathbb Z}$ for any neigborhood $U\subseteq X$ of zero, which means that $G$ is $\mathsf s_1^{{\omega}_1}\mathsf u_2\mathsf s_1^{{\omega}_1}$.
To see that $G$ is not $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_3^{{\omega}}\mathsf u_1$, let $U_1$ be the open unit ball $\{x\in X:\|x\|<1\}$ of the Banach space $X$. Assuming that $G$ is $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_3^{{\omega}}\mathsf u_1$, we can find sets $S_2\in[G]^{<{\omega}_1}$ and $S_3\in[G]^{<{\omega}}$ such that $U_1S_2U_1S_3U_1=X$. Find a separable Banach subspace $H$ of $X$ such that $S_2S_3\subseteq H\times{\mathbb Z}$. Since the Banach space $X$ is not separable, $X\ne H$. By the Hahn-Banach Theorem, there exists a linear continuous functional $f:X\to\mathbb R$ such that $f(H)=\{0\}$ and $\|f\|=1$. Find $m\in{\mathbb N}$ such that $S_3\subset X\times[-m,m]$ and choose a point $x\in X$ with $|f(x)|>2+2^m$. Find elements $u_1,u_1',u_1''\in U_1$, $s_2\in S_2$, $s_3\in S_3$ such that $x=u_1s_2u'_1s_3u_1''$. Then $x=u_1s_2s_3s_3^{-1}u_1's_3u_1''$. Write the elements $s_2s_3\in H\times {\mathbb Z}$ and $s_3\in X\times[-m,m]$ as $s_2s_3=(h_2,n_2)$, $s_3=(h_3,n_3)$ for some $h_2,h_3\in X$ and $n_2,n_3\in{\mathbb Z}$ with $h_2\in H$ and $n_3\in[-m,m]$. Then $$\begin{aligned}
(x,0)=\;&(u_1,0)*(h_2,n_2)*(h_3,n_3)^{-1}*(u_1',0)*(h_3,n_3)*(u_1'',0)=\\
&(u_1+h_2,n_2)*(-2^{-n_3}h_3,-n_3)*(u_1',0)*(h_3,n_3)*(u_1'',0)=\\
&(u_1+h_2,n_2)*(2^{-n_3}u_1',0)*(u_1'',0)=(u_1+h_2+2^{n_2-n_3}u_1'+2^{n_2}u_1'',n_2)
\end{aligned}$$ and hence $n_2=0$ and $x=u_1+h_2+2^{-n_3}u_1'+u_1''$. Then $$|f(x)|\le |f(u_1)|+|f(h_2)|+2^{-n_3}|f(u_1')|+|u_1''|<1+0+2^{-n_3}\cdot 1+1\le 2+2^m,$$ which contradicts the choice of $x$. This contradiction shows that $G$ is not $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_3^{{\omega}}\mathsf u_1$.
We finish this paper by a problem suggested by implications with question marks in the diagram.
Is there a $\mathsf u_1\mathsf s_2^{{\omega}_1}\!\mathsf u_1\mathsf s_3^{{\omega}}\mathsf u_1$ topological group which is neither $\mathsf u_1\mathsf s_2^{{\omega}_1}\mathsf u_1$ nor $\mathsf u_1\mathsf s_2^{{\omega}_1}\!\mathsf u_3\mathsf s_4^{{\omega}}$?
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors express their sincere thanks to Jan Pachl for his valuable comment on the implication $\mathsf s_2^{{\omega}_1}\mathsf u_1\mathsf s_3^{\omega}{\Rightarrow}\mathsf u_1\mathsf s_2^{{\omega}_1}$ (proved in Lemma 3.31 of his book [@Pachl]).
A. Arhangel’skii, M. Tkachenko, [*Topological groups and related structures*]{}, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
T. Banakh, I. Guran, A. Ravsky, [*Each topological group embeds into a duoseparable topological group*]{}, preprint ([arxiv.org/abs/2002.06232]{}).
A. Bouziad, J.-P. Troallic, [*A precompactness test for topological groups in the manner of Grothendieck*]{}, Topology Proc. [**31**]{}:1 (2007), 19–30.
I.I. Guran, [*Topological groups similar to Lindelöf groups*]{}, Dokl. Akad. Nauk SSSR [**256**]{}:6 (1981), 1305–1307.
J. Pachl, [*Uniform spaces and measures*]{}, Fields Institute Monographs, 30. Springer, 2013.
V.V. Uspenskij, [*On subgroups of minimal topological groups*]{}, Topology Appl. [**155**]{}:14 (2008), 1580–1606.
|
---
abstract: 'We report the discovery of TeV gamma-ray emission from the Type Ia supernova remnant (SNR) G120.1+1.4, known as Tycho’s supernova remnant. Observations performed in the period 2008-2010 with the VERITAS ground-based gamma-ray observatory reveal weak emission coming from the direction of the remnant, compatible with a point source located at $00^{\rm h} \ 25^{\rm m} \ 27.0^{\rm s},\ +64^{\circ} \ 10^{\prime} \ 50^{\prime\prime}$ (J2000). The TeV photon spectrum measured by VERITAS can be described with a power-law $dN/dE = C(E/3.42\;\textrm{TeV})^{-\Gamma}$ with $\Gamma = 1.95 \pm 0.51_{stat} \pm 0.30_{sys}$ and $C = (1.55 \pm 0.43_{stat} \pm 0.47_{sys}) \times 10^{-14}$ cm$^{-2}$s$^{-1}$TeV$^{-1}$. The integral flux above 1 TeV corresponds to $\sim 0.9\%$ percent of the steady Crab Nebula emission above the same energy, making it one of the weakest sources yet detected in TeV gamma rays. We present both leptonic and hadronic models which can describe the data. The lowest magnetic field allowed in these models is $\sim 80 \mu$G, which may be interpreted as evidence for magnetic field amplification.'
author:
- 'V. A. Acciari, E. Aliu, T. Arlen, T. Aune, M. Beilicke, W. Benbow, S. M. Bradbury, J. H. Buckley, V. Bugaev, K. Byrum, A. Cannon, A. Cesarini, L. Ciupik, E. Collins-Hughes, W. Cui, R. Dickherber, C. Duke, M. Errando, J. P. Finley, G. Finnegan, L. Fortson, A. Furniss, N. Galante, D. Gall, G. H. Gillanders, S. Godambe, S. Griffin, J. Grube, R. Guenette, G. Gyuk, D. Hanna, J. Holder, J. P. Hughes, C. M. Hui, T. B. Humensky, P. Kaaret, N. Karlsson, M. Kertzman, D. Kieda, H. Krawczynski, F. Krennrich, M. J. Lang, S. LeBohec, A. S Madhavan, G. Maier, P. Majumdar, S. McArthur, A. McCann, P. Moriarty, R. Mukherjee, R. A. Ong, M. Orr, A. N. Otte, D. Pandel, N. H. Park, J. S. Perkins, M. Pohl, J. Quinn, K. Ragan, L. C. Reyes, P. T. Reynolds, E. Roache, H. J. Rose, D. B. Saxon, M. Schroedter, G. H. Sembroski, G. Demet Senturk, P. Slane, A. W. Smith, G. Tešić, M. Theiling, S. Thibadeau, K. Tsurusaki, A. Varlotta, V. V. Vassiliev, S. Vincent, M. Vivier, S. P. Wakely, J. E. Ward, T. C. Weekes, A. Weinstein, T. Weisgarber, D. A. Williams, M. Wood, B. Zitzer'
title: 'Discovery of TeV Gamma Ray Emission from Tycho’s Supernova Remnant'
---
Introduction
============
The object G120.1+1.4, commonly called Tycho’s supernova remnant (SNR), is the historical relic of a supernova which was first observed in 1572. Recent spectral analysis of the light echo from the explosion [@2008Natur.456..617K] has confirmed the long-standing conjecture [see, e.g., @2004ApJ...612..357R] that the event was a Type Ia supernova.
At 438 years old, Tycho is young among Galactic SNRs and is well-studied at many wavelengths. It has a radio spectral index of $0.65$ and a flux density at 1.4 GHz of $40.5$ Jy . A hint of spectral curvature, consistent with nonlinear shock acceleration, and a slightly flatter radio spectrum (0.61) has also been reported [@1992ApJ...399L..75R]. Radio images show a clear shell-like morphology with enhanced emission along the northeastern edge of the remnant [@1991AJ....101.2151D; @2009ApJ...696.1864S].
X-ray images reveal strong non-thermal emission concentrated in the SNR rim [@2002JApA...23...81H; @2005ApJ...621..793B; @2005ApJ...634..376W; @2010ApJ...709.1387K]. Thin filamentary X-ray structures in this region have been interpreted as evidence for electron acceleration [@2002JApA...23...81H; @2005ApJ...621..793B; @2005ApJ...634..376W]. This is supported by radio spectral tomography studies [@2000ApJ...529..453K] and by the detection of X-rays up to energies of 30 keV [e.g., @2009PASJ...61S.167T], which implies the presence of electrons up to at least $\sim10$ TeV.
The global expansion rate of the remnant, measured at many wavelengths [@1978ApJ...224..851K; @1997ApJ...491..816R; @2010ApJ...709.1387K], is consistent with an object transitioning into the Sedov phase. However, many of these studies have noted that the northeast quadrant of the remnant is expanding at a lower rate than the rest of the object. This has been attributed by some to interactions with a nearby high-density cloud, which has been studied through its HI and CO emissions .
Distance estimates for Tycho have varied. Estimates which combine measurements of proper motion with shock velocities inferred from H$_{\alpha}$ line widths tend to favor distances near $\sim2.5$ kpc [@1980ApJ...235..186C; @1991ApJ...375..652S; @1992ApJ...384..665S]. On the other hand, the distance derived from the light echo spectrum [@2008Natur.456..617K] is $3.8^{+1.5}_{-1.1}$ kpc, and HI absorption studies yield $4.5 \pm 0.5$ kpc. A recent X-ray study combining ejecta velocities with proper motion studies yields $4.0 \pm 1.0$ kpc [@2010arXiv1009.6031H]. The apparent association with a molecular CO cloud has not produced an unambiguous distance estimate , due to the complex velocity field in the direction of the Perseus arm, where Tycho appears to be located. For the same reason, estimates based on the HI measurements may be considered controversial. Finally, derive a lower limit of 3.3 kpc, based on the previous non-detection of Tycho in TeV gamma rays.
It is worth noting that many of the shorter distance estimates rely on assumptions about the post-shock proton temperature’s relationship to the shock velocity in the case of adiabatic expansion [see also @1988MNRAS.230..331S]. These assumptions may not be valid in the presence of efficient particle acceleration, as discussed, for instance, in [@2009Sci...325..719H].
Claims of such efficient nuclear particle acceleration have indeed been made (e.g., @2005ApJ...634..376W, though see also @2010ApJ...715L.146L), based on detailed studies of the shock front and contact discontinuity locations. Drawing on this result and on their own detailed simulations, claim that a detection of gamma rays from Tycho would represent “incontrovertible" evidence of nuclear particle acceleration in SNRs. More generally, TeV gamma-ray observations probe the high-energy end of the underlying parent particle distributions and, especially in conjunction with measurements at other wavelengths, can help constrain the associated acceleration mechanics (see, e.g., @2009astro2010S.145K).
Tycho has been observed several times at gamma-ray energies, with no credible detections previously reported. In the GeV regime, the source does not appear in either the first Fermi LAT catalog [@2010ApJS..188..405A] or the 3rd EGRET catalog [@1999ApJS..123...79H]. Upper limits in the TeV range have been presented by the Whipple Collaboration , the HEGRA Collaboration and the MAGIC Collaboration [@2009arXiv0907.1009C]. This last limit is the most constraining, with a $3\sigma$ (where $\sigma$ indicates standard deviations) flux upper limit of about 1.7% of the steady Crab Nebula flux above 1 TeV. All of the TeV limits assume a gamma-ray source located at the center of the remnant.
In this Letter we report the discovery by VERITAS of TeV gamma-ray emission from Tycho’s supernova remnant.
VERITAS Instrument & Observations
=================================
VERITAS is an array of four 12 m imaging atmospheric Cherenkov telescopes located at the Fred Lawrence Whipple Observatory in southern Arizona (1.3 km a.s.l., N 31$^\circ$40$^\prime$, W 110$^\circ$57$^\prime$). Each of the telescopes has a camera comprising 499 photomultiplier tubes covering a $3.5^{\circ}$ total field of view. The array is sensitive to photon energies between 100 GeV and 30 TeV, with an energy resolution of $\sim15\%$ and an angular resolution per event of $\sim0.1^{\circ}$. The array has been fully functional since September 2007. In summer 2009, telescope T1 was relocated, improving the sensitivity and angular resolution of the array [@2009arXiv0912.3841P]. Prior to this reconfiguration, VERITAS was capable of detecting a point source with 1% of the flux of the Crab Nebula at the $5\sigma$ level in less than 50 hours. After moving T1, such a source can be detected in less than 30 hours.
VERITAS observations of Tycho spanned two epochs, from October 2008 to January 2009 and from September 2009 to January 2010. All observations were taken in “wobble mode”, in which the telescopes are pointed to a position $0.5^{\circ}$ offset from the source position. The offset direction is sequentially varied from run to run between the four cardinal directions. Approximately equal amounts of data were acquired from each offset position. This technique allows for the collection of data with a simultaneous estimate of the background. From the period between October 2008 and January 2009, a total of 21.9 hours of data were retained after quality cuts on weather conditions and operational stability. The mean zenith angle of these data is $35^{\circ}$. Between September 2009 and January 2010, an additional 44.7 hours of data were analyzed after quality cuts. The mean zenith angle of this data set is $39^{\circ}$. Note that the second season’s data set was taken with the new array configuration, and therefore benefits from the improved sensitivity.
Data Analysis and Results
=========================
The data were analyzed following standard procedures [see, e.g., @2008ApJ...679.1427A]. The optimum gamma-ray selection criteria (*cuts*) depend upon the source flux and spectrum, which are not known a priori. We therefore define and apply two sets of cuts, appropriate for a reasonable range of source properties, and account for the statistical trials incurred in this process. The results presented here required that at least 3 of the telescopes in the array recorded an image with more than 800 digital counts ($\sim150$ photo-electrons, corresponding to an energy threshold of 800 GeV). These images were then used to reconstruct the air shower properties. Gamma-ray events were selected through cuts on the *mean reduced scaled width* and *mean reduced scaled length* parameters [@2006APh....25..380K], which were required to fall between -1.2 and 0.5, and on the square of the angular distance from the test position to the reconstructed arrival direction of the shower, $\theta^2$. For the 2008-2009 data set, we required $\theta^2 < 0.015\,\textrm{deg}^2$, and for the 2009-2010 data set, we required $\theta^2 < 0.01\,\textrm{deg}^2$ because of the improved angular resolution of the array. The ring background model was used to estimate the background .
This analysis produced an excess which is significant at the $5.8\sigma$ level (pre-trials). This is the peak excess found in a blind search region with sides of length 0.26$^{\circ}$ - roughly twice the diameter of the radio remnant. A conservative a priori trials factor was determined by tiling this area with square 0.04$^{\circ}$ bins [@2006ApJ...636..777A], and additionally accounting for the two sets of applied cuts. This results in a $5.0\sigma$ post-trials significance.
![\[fig:skymap\]VERITAS TeV gamma-ray count map of the region around Tycho’s SNR. The color scale indicates the number of excess gamma-ray events from a region, using a squared integration radius of 0.01 deg$^2$ for the 2009/2010 data and 0.015 deg$^2$ for the 2008/2009 data. The centroid of the emission is indicated with a thick black cross. Overlaid on the image are X-ray contours from a Chandra ACIS exposure [thin black lines; @2002JApA...23...81H] and $^{12}$CO emission (J=1-0) from the high-resolution FCRAO Survey [magenta lines; @1998ApJS..115..241H]. The CO velocity selection is discussed in the text. The VERITAS count map has been smoothed with Gaussian kernel of size $0.06^\circ$. The point-spread function of the instrument (see text) is indicated by the white circle.](Tycho_Map.pdf){width="0.99\linewidth"}
Morphology
----------
The morphology of the source was investigated by binning the uncorrelated acceptance-corrected map of excess event counts. Bins of size $0.05^\circ$ were used to provide sufficient statistics for fits to source models. The map is compatible ($\chi^2=508;\textrm{ndf}=438;\textrm{Prob}=1.2\%$) with a point source located at $00^{\rm h} \ 25^{\rm m} \ 27.0^{\rm s},\ +64^{\circ} \ 10^{\prime} \ 50^{\prime\prime}$ (J2000) and hence we designate the object VER J0025+641. This position is derived from a simple symmetric Gaussian fit with a width fixed at the instrument point-spread function (i.e., the 68% containment radius for photons, $\theta_{68\%}$= 0.11$^{\circ}$). While other source functions (e.g., an offset asymmetric Gaussian) may provide a marginally better fit (perhaps hinting at a more complex underlying source morphology) a likelihood ratio test shows that the extra degrees of freedom are not statistically justified in this data set.
As shown in Figure 1, the center of the fit position is offset by 0.04$^{\circ}$ from the center of the remnant. The statistical uncertainty in this location is $0.023^{\circ}$, while the systematic uncertainty resulting from telescope pointing accuracy is $0.014^{\circ}$. We note that the derived centroid position depends also on the source shape assumed for the fit. Future observations will allow more detailed study of the source morphology.
Spectrum
--------
The differential photon spectrum between 1 and 10 TeV is shown in Figure \[fig:spectrum\]. This spectrum is generated from the complete data set after quality selection. The shape is consistent with a power law $dN/dE = C(E/3.42~\textrm{TeV})^{-\Gamma}$ with $\Gamma = 1.95 \pm 0.51_{stat} \pm 0.30_{sys}$ and $C = (1.55 \pm 0.43_{stat} \pm 0.47_{sys}) \times 10^{-14}$ cm$^{-2}$ s$^{-1}$ TeV$^{-1}$, where the systematic error on the flux is dominated by uncertainty in the energy scale. The $\chi^2$ of the fit is 0.6 for 1 degree of freedom. The integrated flux above 1 TeV is $(1.87 \pm 0.51_{stat}) \times 10^{-13}$ cm$^{-2}$ s$^{-1}$, about 0.9% that of the steady Crab Nebula flux above the same energy.
Discussion
==========
Figure \[fig:skymap\] shows the TeV gamma-ray image of Tycho’s SNR. The color scale indicates the number of excess gamma-ray events in a region, using a squared integration radius of 0.015 deg$^2$ for the 2008/2009 data and 0.01 deg$^2$ for the 2009/2010 data. The map has been smoothed with a Gaussian kernel of radius 0.06$^{\circ}$. Overlaid on the image are X-ray contours from a Chandra ACIS exposure [thin black lines; @2005ApJ...634..376W]. A contour map of 115 GHz line emission associated with the molecular $^{12}$CO (J=1-0) transition, from the 14 m telescope of the Five College Radio Astronomy Observatory (FCRAO), is shown in magenta [@1998ApJS..115..241H; @2003AJ....125.3145T]. Following the analysis of [@2004ApJ...605L.113L], we have integrated over the velocity range -68 km s$^{-1}$ to -50 km s$^{-1}$, revealing a cloud possibly interacting with the northeast quadrant of the remnant.
As can be seen from Figure \[fig:skymap\], the peak of the gamma-ray emission is displaced somewhat to the northeast of the center of the remnant, towards the CO cloud. While this is provocative in the context of hadronically-induced gamma-ray emission , the statistical significance of the displacement is weak. Nevertheless, this is the general morphology expected in the case of a shock/cloud interaction leading to gamma-ray emission, which is seen in several other remnants [see, e.g., @2009ApJ...706L.270H]. On the other hand, OH maser emission, a telltale sign of such interactions [@2002Sci...296.2350W] has not been detected from this remnant [@1996AJ....111.1651F]. A catalog search within our error box reveals no likely gamma-ray emission candidates at other wavelengths, so we tentatively associate the source with Tycho.
![\[fig:models\]Radio, (nonthermal) X-ray, and VHE gamma-ray emission from Tycho’s SNR, along with models for the emission. The upper panel shows a lepton-dominated model while the lower panel shows a model dominated by hadrons. In each, the IC emission corresponds to the (cyan) long-dashed curve while the pion-decay emission corresponds to the (magenta) short-dashed curve. The solid curve at high energies is the sum of these components; at lower energies it corresponds to the synchrotron emission. See text for discussion.](Tycho_Models.pdf){width="0.90\linewidth"}
In Figure \[fig:models\] we show simple model fits to the broadband spectrum of Tycho, generated assuming no influence from the molecular cloud. Two versions are considered - one in which the TeV emission is dominated by leptonic (inverse Compton (IC) scattering; upper panel) processes, and one in which it is dominated by hadronic (pion decay; lower panel) processes [@2010ApJ...720..266S]. Here we have assumed particle spectra of the form: $$\frac{dN}{dE} = A E^{-\alpha} e^{-(E/E_c)}$$ with the spectral index $\alpha$ being fixed to the same value for the electrons and protons, but with the cutoff energy $E_c$ being allowed to differ, as expected for loss-dominated distributions. We note the crucial point that, in both the leptonic and hadronic models, the total relativistic particle energy is dominated by hadrons, and it represents a significant fraction of the total supernova kinetic energy, estimated to be $1.2 \times 10^{51}$ ergs [@2006ApJ...645.1373B]. Indeed, in both cases, the energy density of hadronic cosmic rays is likely sufficient to modify the supernova shock dynamics, supporting the conclusions of [@2005ApJ...634..376W]. The radio spectrum requires $\alpha \approx 2.2$, and we have assumed an ambient density $n_0 = 0.2 {\rm\ cm}^{-3}$, based on upper limits from X-ray measurements [@2010ApJ...709.1387K]. The distance is set to 4 kpc.
The radio data in Figure \[fig:models\] are compiled from [@1992ApJ...399L..75R]. The X-ray data (shown in blue) represent the unfolded spectrum between $\sim 4$ and $ 10$ keV extracted from a deep [*Suzaku*]{} observation of Tycho. Features from Fe-K and weaker line emission have been removed. The TeV data points are from the VERITAS results reported here.
For the lepton-dominated scenario, we find the normalization and magnetic field required to reproduce the radio and X-ray emission by synchrotron radiation. The gamma-ray emission is then generated by IC scattering of cosmic microwave background photons. (The impact of adding an additional - even maximal - contribution from the dust-IR emission was investigated and found to make only a small difference in the overall IC spectrum.) The associated magnetic field is $\sim 80 \mu{\rm G}$, with a $\sim 15\%$ uncertainty. Assuming an electron-to-proton number ratio $\kappa_{ep} = 10^{-2}$, as observed in the local cosmic-ray population, the total particle energy (dominated by protons) is $1.8 \times 10^{50}{\rm\ erg}$, and the associated gamma-ray emission from $\pi^0$ decay is negligible in the TeV band, as shown in the upper panel of Figure \[fig:models\]. The magnetic field required for this model is somewhat lower than the conventionally accepted $\sim 200-300 \mu$G value that many derive , though it is within the range found by [@2005ApJ...634..376W].
For the hadron-dominated scenario, we adjust the normalization of the proton spectrum to reproduce the observed TeV emission through $\pi^0$-decay (Figure \[fig:models\], lower panel). The normalization of the electron spectrum is then reduced to a level at which the hadron-induced gamma-ray emission dominates, yielding a $\kappa_{ep} \approx 4 \times 10^{-4}$. The associated magnetic field required to reproduce the synchrotron emission is $\sim 230 \mu{\rm G}$. This value can be considered a lower limit, since a smaller $\kappa_{ep}$ will require a larger magnetic field.
For this model, we find a total particle energy of $\sim 8 \times 10^{50}{\rm\ erg}$. While this is perhaps uncomfortably large, it is based on emission involving only the mean gas density of the remnant. Any possible contribution of target material from the cloud would reduce the energy requirement. This could also be achieved if we loosen the assumption of a single power-law particle spectrum.
Our hadronic model is broadly consistent with that of , who employ a nonlinear kinetic particle acceleration model to derive a hadronically-induced flux as a function of distance. As shown in Figure \[fig:spectrum\], with a source distance of $4$ kpc, an ambient density of $\sim0.2$ cm$^{-3}$, and a magnetic field of $\sim350\mu{\rm G}$, their model provides a reasonable fit to our data. However, by scaling the flux to improve the fit, a distance estimate of $3.8$ kpc can be obtained.
Finally, it is worth noting that the lowest magnetic field allowable in either of our models is $\sim 80 \mu{\rm G}$. This value is not only well above the typical $\sim 3\mu{\rm G}$ fields of the interstellar medium, it is significantly higher than expectations from shock-compression of that field in moderate-density environments [see, e.g., @2008ApJ...673L..47E]. Hence, our detection of gamma rays from Tycho’s SNR represents additional evidence for magnetic field amplification in this remnant. While this conclusion is dependent on the validity of our one-zone emission approximation, it is not subject to large systematic uncertainties arising from the choice of electron spectrum, since, for a $\sim 80 \mu{\rm G}$ field, the *Suzaku* and VERITAS data probe the same underlying electron energies.
Conclusion
==========
VERITAS has detected weak TeV gamma-ray emission from the SNR G120.1+1.4, also known as Tycho’s supernova remnant. The total flux from the remnant above 1 TeV is $\sim0.9\%$ that of the Crab Nebula, making it one of the weakest sources yet detected in TeV gamma rays, and only the second confirmed Type Ia SNR gamma-ray emitter. SN1006, another Type Ia remnant, was observed by H.E.S.S. to have two distinct regions of weak ($\sim 1\%$ Crab) gamma-ray emission, one at the northeastern edge of the $\sim 0.5^\circ$ diameter remnant and one at the southwestern edge . Both of these regions are highly correlated with non-thermal X-ray emission and have spectra that are compatible with power-laws of index $\Gamma \sim 2.3$. For SN1006, neither a leptonic nor a hadronic origin for gamma-ray emission can be firmly eliminated.
The photon spectrum of Tycho can be described by a power-law with differential index $1.95 \pm 0.51_{stat} \pm 0.30_{sys}$ and is compatible with the hadronic model of , scaled for a distance of $3.8$ kpc. We present another hadronic model which can also reproduce the data, although it requires a large energy content of cosmic rays. A leptonic external IC model also provides a tolerable fit to the data, though it requires a magnetic field somewhat lower than generally is accepted for Tycho. Notably, the lowest magnetic field allowed in either model is $\sim 80 \mu$G, which may be interpreted as evidence for magnetic field amplification. The morphology of the emission is compatible with a point source, with the peak of the emission possibly offset from the center of the remnant. Detailed spectral and spatial studies will be possible with a deeper exposure.
This research is supported by grants from the U.S. Department of Energy, the U.S. National Science Foundation and the Smithsonian Institution, by the Natural Sciences and Engineering Research Council (NSERC) in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748) and by the Science and Technology Facilities Council in the UK. The research presented in this paper has used data from the Canadian Galactic Plane Survey, a Canadian project with international partners, supported by NSERC. D.B. Saxon acknowledges the NASA Delaware Space Grant Program for its support of this research. J.P. Hughes acknowledges support from NASA grant NNX08AZ86G to Rutgers University.
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|
---
abstract: 'The problem of matching two sets of features appears in various tasks of computer vision and can be often formalized as a problem of permutation estimation. We address this problem from a statistical point of view and provide a theoretical analysis of the accuracy of several natural estimators. To this end, the minimax rate of separation is investigated and its expression is obtained as a function of the sample size, noise level and dimension of the features. We consider the cases of homoscedastic and heteroscedastic noise and establish, in each case, tight upper bounds on the separation distance of several estimators. These upper bounds are shown to be unimprovable both in the homoscedastic and heteroscedastic settings. Interestingly, these bounds demonstrate that a phase transition occurs when the dimension $d$ of the features is of the order of the logarithm of the number of features $n$. For $d=O(\log n)$, the rate is dimension free and equals $\sigma (\log n)^{1/2}$, where $\sigma$ is the noise level. In contrast, when $d$ is larger than $c\log n$ for some constant $c>0$, the minimax rate increases with $d$ and is of the order of $\sigma(d\log n)^{1/4}$. We also discuss the computational aspects of the estimators and provide empirical evidence of their consistency on synthetic data. Finally, we show that our results extend to more general matching criteria.'
author:
- |
Olivier Collier [email protected]\
Imagine- LIGM\
Université Paris EST\
Marne-la-Vallée, FRANCE Arnak S. Dalalyan [email protected]\
Laboratoire de Statistique\
ENSAE - CREST\
Malakoff, FRANCE
bibliography:
- 'biblio\_permut\_est.bib'
title: Minimax rates in permutation estimation for feature matching
---
[permutation estimation]{}, [minimax rate of separation]{}, [feature matching]{}
Introduction
============
The Statistical Problem of Feature Matching
-------------------------------------------
In this paper, we present a rigorous statistical analysis of the problem of permutation estimation and multiple feature matching from noisy observations. More precisely, let $\{X_1,\ldots,X_n\}$ and $\{\Xdiese_1,\ldots,\Xdiese_m\}$ be two sets of vectors from $\RR^d$, hereafter referred to as noisy features, containing many matching elements. That is, for many $X_i$’s there is a $\Xdiese_j$ such that $X_i$ and $\Xdiese_j$ coincide up to an observation noise (or measurement error). Our goal is to estimate an application $\pi^* : \{1,\ldots,n\}\to \{1,\ldots,m\}$ for which each $X_i$ matches with $\Xdiese_{\pi^*(i)}$ and to provide tight conditions which make it possible to accurately recover $\pi^*$ from data.
In order to define a statistical framework allowing us to compare different estimators of $\pi^*$, we confine[^1] our attention to the case $n=m$, that is when the two sets of noisy features have equal sizes. Furthermore, we assume that there exists a unique permutation of $\{1,\ldots,n\}$, denoted $\pi^*$, leading to pairs of features $(X_i,\Xdiese_{\pi^*(i)})$ that match up to a measurement error. In such a situation, it is clearly impossible to recover the true permutation $\pi^*$ if some features within the set $\{X_1,\ldots,X_n\}$ are too close. Based on this observation, we propose to measure the quality of a procedure of permutation estimation by the minimal distance between pairs of different features for which the given procedure is still consistent. This quantity will be called *separation distance* and will be the main concept of interest in the present study. In this respect, the approach we adopted is close in spirit to the minimax theory of hypotheses testing (see, for instance, [@Spokoiny96; @Ingster]).
A Motivating Example: Feature Matching in Computer Vision
---------------------------------------------------------
Many tasks of computer vision, such as object recognition, motion tracking or structure from motion, are currently carried out using algorithms that contain a step of feature matching, cf. [@Szeliski; @Hartley_Zisserman]. The features are usually local descriptors that serve to summarize the images. The most famous examples of such features are perhaps SIFT [@Lowe2004] and SURF [@surf]. Once the features have been computed for each image, an algorithm is applied to match features of one image to those of another one. The matching pairs are then used for estimating the deformation of the object, for detecting the new position of the followed object, for creating a panorama, etc. In this paper, we are interested in simultaneous matching of a large number of features. The main focus is on the case when the two sets of features are extracted from the images that represent the same scene with a large overlap, and therefore the sets of features are (nearly) of the same size and every feature in the first image is also present in the second one. This problem is made more difficult by the presence of noise in the images, and thus in the features as well. Typically, due to the high resolution of most images, the number of features is large and their dimension is relatively large as well (128 for SIFT and 64 for SURF). It is therefore important to characterize the behavior of various matching procedures as a function of the number of features, the dimension and the noise level.
Main Contributions
------------------
We consider four procedures of permutation estimation that naturally arise in this context. The first one is a greedy procedure that sequentially assigns to each feature $X_i$ the closest feature $\Xdiese_j$ among those features that have not been assigned at an earlier step. The three other estimators are defined as minimizers of the (profiled-)log-likelihood under three different modeling assumptions. These three modeling assumptions are that the noise level is constant across all the features (homoscedastic noise), that the noise level is variable (heteroscedastic noise) but known and that the noise level is variable and unknown. The corresponding estimators are respectively called least sum of squares (LSS) estimator, least sum of normalized squares (LSNS) estimator and least sum of logarithms (LSL) estimator.
We first consider the homoscedastic setting and show that all the considered estimators are consistent under similar conditions on the minimal distance between distinct features $\k$. These conditions state that $\k$ is larger than some function of the noise level $\s$, the sample size $n$ and the dimension $d$. This function is the same for the four aforementioned procedures and is given, up to a multiplicative factor, by $$\begin{aligned}
\label{kappastar}
\k^*(\s,n,d) = \s \max((\log n)^{1/2}, (d\log n)^{1/4}).\end{aligned}$$ Then, we prove that this expression provides the optimal rate of the separation distance in the sense that for some absolute constant $c$ if $\k \le c\k^*(\s,n,d)$ then there is no procedure capable of consistently estimating $\pi^*$.
In the heteroscedastic case, we provide an upper bound on the identifiability threshold ensuring the consistency of the LSNS and LSL estimators. Up to a proper normalization by the noise level, this bound is of the same form as (\[kappastar\]) and, therefore, the ignorance of the noise level does not seriously affect the quality of estimation. Furthermore, the LSL estimator is easy to adapt to the case $n\not = m$ and is robust to the presence of outliers in the features. We carried out a small experimental evaluation that confirms that in the heteroscedastic setting the LSL estimator is as good as the LSNS (pseudo-) estimator and that they outperform the two other estimators: the greedy estimator and the least sum of squares. We also argue that the three estimators stemming from the maximum likelihood methodology are efficiently computable either by linear programming or by the Hungarian algorithm.
Note that different loss functions may be used for measuring the distance between an estimated permutation and the true one. Most results of this paper are established for the 0-1 loss, which equals one if the estimator and the true permutation differ at least at one location and equals 0 otherwise. However, it is of interest to analyze the situation with the Hamming distance as well, since it amounts to controlling the proportion of the mismatched features and, hence, offers a more graduated evaluation of the quality of estimation. We show that in the case of the Hamming distance, in the regime of moderately large dimension ($d\ge c\log n$ for some constant $c>0$) the rate of separation is exactly the same as in the case of the 0-1 distance. The picture is more complex in the regime of small dimensionality $d=o(\log(n))$, in which we get the same upper bound as for the 0-1 loss but the lower bound is expressed in terms of the logarithm of the packing number of an $\ell_2$ ball of the symmetric group. We conjecture that this quantity is of the order of $n\log(n)$ and check this conjecture for relatively small values of $n$. If this conjecture is correct, our lower bound coincides up to a multiplicative factor with the upper bound.
Finally, let us mention that some of the results of the present work have been presented in the AI-STATS 2013 conference and published in the proceedings [@jmlr_CD13].
Plan of the Paper
-----------------
We introduce in Section \[section:model\] a model for the problem of matching two sets of features and of estimation of a permutation. The estimating procedures analyzed in this work are presented in Section \[section:estimators\], whereas their performances in terms of rates of separation distance are described in Section \[section:performance\]. In Section \[section:computationalaspects\], computational aspects of the estimating procedures are discussed while Section \[section:extensions\] is devoted to the statement of some extensions of our results. We report in Section \[section:simulations\] the results of some numerical experiments. The proofs of the theorems and of the lemmas are postponed to Sections \[section:theorems\] and \[section:lemmas\], respectively.
Notation and Problem Formulation {#section:model}
================================
We begin with formalizing the problem of matching two sets of features $\{X_1,\ldots,X_n\}$ and $\{\Xdiese_1,\ldots,\Xdiese_m\}$ with $n,m\ge 2$. In what follows we assume that the observed features are randomly generated from the model $$\label{model}
\begin{cases}
X_i = \t_i + \s_i\xi_i \ , \\
\Xdiese_j = \tdiese_j + \sdiese_j\xidiese_j,
\end{cases}\quad i=1,\ldots,n \text{ and } j=1,\ldots,m$$ where
- $\btheta=\{\t_1,\ldots,\t_n\}$ and $\btdiese=\{\tdiese_1,\ldots,\tdiese_m\}$ are two collections of vectors from $\RR^d$, corresponding to the original features, which are unavailable,
- $\s_1,\ldots,\s_n,\sdiese_1,\ldots,\sdiese_m$ are positive real numbers corresponding to the levels of noise contaminating each feature,
- $\xi_1,\ldots,\xi_n$ and $\xidiese_1,\ldots,\xidiese_m$ are two independent sets of i.i.d. random vectors drawn from the Gaussian distribution with zero mean and identity covariance matrix.
The task of feature matching consists in finding a bijection $\pi^*$ between the largest possible subsets $S_1$ and $S_2$ of $\{1,\ldots,n\}$ and $\{1,\ldots,m\}$ respectively, such that $$\label{pistar}
\forall\ i\in S_2,\quad \tdiese_i \equiv \t_{\pi^*(i)},$$ where $\equiv$ is an equivalence relation that we call *matching criterion*. The features that do not belong to $S_1$ or $S_2$ are called *outliers*. To ease presentation, we mainly focus on the case where the matching criterion is the equality of two vectors. However, as discussed in Section \[subsection5.2\], most results carry over the equivalence corresponding to equality of two vectors transformed by a given linear transformation. Furthermore, it turns out that statistical inference for the matching problem is already quite involved when no outlier is present in the data. Therefore, we make the following assumption $$\label{m=n}
m=n \qquad\text{and}\qquad S_1=S_2=\{1,\ldots,n\}.$$ Note however that the procedures we consider below admit natural counterparts in the setting with outliers. We will also restrict ourselves to noise levels satisfying some constraints. The two types of constraints we consider, referred to as homoscedastic and heteroscedastic setting, correspond to the relations $\s_i=\sdiese_i=\s$, $\forall i=1,\ldots,n$, and $\s_{\pi^*(i)}=\sdiese_i$, $\forall i=1,\ldots,n$.
In this formulation, the data generating distribution is defined by the (unknown) parameters $\btheta$, $\bsigma = (\s_i,\ldots,\s_n)$ and $\pi^*$. In the problem of matching, we focus our attention on the problem of estimating the parameter $\pi^*$ only, considering $\btheta$ and $\bsigma$ as nuisance parameters. In what follows, we denote by $\prob_{\btheta,\bsigma,\pi^*}$ the probability distribution of the vector $(X_1,\ldots,X_n,\Xdiese_1,\ldots,\Xdiese_n)$ defined by (\[model\]) under the conditions (\[pistar\]) and (\[m=n\]). We write $\esp_{\btheta,\bsigma,\pi^*}$ for the expectation with respect to $\prob_{\btheta,\bsigma,\pi^*}$. The symmetric group, the set of all permutations of $\{1,\ldots,n\}$, will be denoted by $\Sn$.
We will use two measures of quality for quantifying the error of an estimator $\hat\pi$ of the permutation $\pi^*$. These errors are defined as the 0-1 distance and the normalized Hamming distance between $\hat\pi$ and $\pi^*$, given by $$\label{dist}
\delta_{\text{0\,-1}}(\hat\pi,\pi^*)\triangleq\fcar_{\{\hat\pi\not=\pi^*\}},\qquad
\delta_H(\hat\pi,\pi^*)\triangleq\frac1n\sum_{k=1}^n \fcar_{\{\hat\pi(k)\not=\pi^*(k)\}}.$$ Our ultimate goal is to design estimators that have an expected error smaller than a prescribed level $\alpha$ under the weakest possible conditions on the nuisance parameter $\btheta$. The estimation of the permutation or, equivalently, the problem of matching is more difficult when the features are hardly distinguishable. To quantify this phenomenon, we introduce the separation distance $\k(\btheta)$ and the relative separation distance $\bar\k(\btheta,\bsigma)$, which measure the minimal distance between distinct features and the minimal distance-to-noise ratio, respectively. The precise definitions are $$\k(\btheta) \triangleq \min_{i\neq j} \|\t_i-\t_j\|,\qquad
\bar\k(\btheta,\bsigma) \triangleq \min_{i\neq j} \frac{\|\t_i-\t_j\|}{(\s_i^2+\s_j^2)^{1/2}}. \label{kappa}$$ We will see that in the heteroscedastic case the last quantity is more suitable for characterizing the behavior of the estimators than the first one.
Clearly, if $\bar\k(\btheta,\bsigma)=0$, then the parameter $\pi^*$ is nonidentifiable, in the sense that there exist two different permutations $\pi^*_1$ and $\pi^*_2$ such that the distributions $\prob_{\btheta,\bsigma,\pi^*_1}$ and $\prob_{\btheta,\bsigma,\pi^*_2}$ coincide. Therefore, the condition $\bar\k(\btheta,\bsigma)>0$ is necessary for the existence of consistent estimators of $\pi^*$. Furthermore, good estimators are those consistently estimating $\pi^*$ even if $\bar\k(\btheta,\bsigma)$ is small. To give a precise sense to these considerations, let $\a\in(0,1)$ be a prescribed tolerance level and let us call *perceivable separation distance* of a given estimation procedure $\hat\pi$ the quantity $$\bar\k_\a(\hat\pi) \triangleq \inf\Big\{\k>0 \suchthat \max_{\pi\in\Sn} \sup_{\bar\k(\btheta,\bsigma)>\k} \prob_{\btheta,\bsigma,\pi} (\hat\pi\neq\pi)\le\a\Big\},$$ where we skip the dependence on $n$, $d$, $\bsigma$ and $\bsigmadiese$. Here, the perceivable separation distance is defined with respect to the 0-1 distance, the corresponding definition for the Hamming distance is obtained by replacing $\prob_{\btheta,\bsigma,\pi} (\hat\pi\neq\pi)$ by $\esp_{\btheta,\bsigma,\pi} [\delta_H(\hat\pi,\pi)]$. Finally, we call *minimax separation distance* the smallest possible [perceivable separation distance]{} achieved by an estimator $\hat\pi$, $$\label{mmx}
\bar\k_\a \triangleq \inf_{\hat\pi}~\bar\k_\a(\hat\pi),$$ where the infimum is taken over all possible estimators of $\pi^*$. In the following sections, we establish nonasymptotic upper and lower bounds on the minimax separation distance which coincide up to a multiplicative constant independent of $n$, $d$, $\bsigma$ and $\bsigmadiese$. We also show that a suitable version of the maximum profiled likelihood estimator is minimax-separation-rate-optimal both in the homoscedastic and heteroscedastic settings.
Estimation Procedures {#section:estimators}
=====================
As already mentioned, we will consider four estimators. The simplest one, called greedy algorithm and denoted by $\pi^{\rm gr}$ is defined as follows: $\pi^{\rm gr}(1) = \arg\min_{j\in\{1,\ldots,n\}} \|X_j-\Xdiese_1\|$ and, for every $i\in\{ 2,\ldots,n\}$, recursively define $$\begin{aligned}
\label{greedy}
\pi^{\rm gr}(i)\triangleq \arg\min_{j\not\in\{\pi^{\rm gr}(1),\ldots,\pi^{\rm gr}(i-1)\}} \|X_j-\Xdiese_i\|.\end{aligned}$$ A drawback of this estimator is that it is not symmetric: the resulting permutation depends on the initial numbering of the features. However, we will show that this estimator is minimax-separation-rate-optimal in the homoscedastic setting.
A common approach for avoiding incremental estimation and taking into consideration all the observations at the same time consists in defining the estimator $\hat\pi$ as a maximizer of the profiled likelihood. In the homoscedastic case, which is the first setting we studied, the computations lead to the estimator $$\pi^{\textrm{LSS}} \triangleq \arg\min_{\pi\in\Sn} \sum_{i=1}^n \|X_{\pi(i)}-\Xdiese_i\|^2. \label{LSS}$$ which will be referred to as the *Least Sum of Squares* (LSS) estimator.
The LSS estimator takes into account the distance between the observations irrespectively of the noise levels. The fact of neglecting the noise levels, while harmless in the homoscedastic setting, turns out to cause serious loss of efficiency in terms of the perceivable distance of separation in the setting of heteroscedastic noise. Yet, in the latter setting, the distance between the observations is not as relevant as the signal-to-noise ratio ${\|\t_i-\t_j\|^2}/({\s_i^2+\s_j^2})$. Indeed, when the noise levels are small, two noisy but distinct vectors are easier to distinguish than when the noise levels are large.
The computation of the maximum likelihood estimator in the heteroscedastic case with known noise levels also suggests that the signal-to-noise ratio should be taken into account. In this setting, the likelihood maximization leads to the *Least Sum of Normalized Squares* (LSNS) estimator $$\pi^{\textrm{LSNS}} \triangleq \arg\min_{\pi\in\Sn}\sum_{i=1}^n \frac{\|X_{\pi(i)}-\Xdiese_i\|^2}{\s_{\pi(i)}^2+{\sdiese_i}^2}. \label{LSNS}$$ We will often call the LSNS a pseudo-estimator, to underline the fact that it requires the knowledge of the noise levels $\s_i$ which are generally unavailable. In the general setting, when no information on the noise levels is available, the likelihood is maximized over all nuisance parameters (features and noise levels). But this problem is underconstrained, and the result of this maximization is $\pinf$. This can be circumvented by assuming a proper relation between the noise levels. As mentioned earlier, we chose the assumption $$\label{sigma_h}
\forall\ i\in\{1,\ldots,n\},\quad \sdiese_i = \s_{\pi^*(i)}.$$ The maximum likelihood estimator under this constraint is the *Least Sum of Logarithms* (LSL) defined as $$\pi^{\textrm{LSL}} \triangleq \arg\min_{\pi\in\Sn} \sum_{i=1}^n \log \|X_{\pi(i)}-\Xdiese_i\|^2. \label{LSL}$$ We will prove that this estimator is minimax-rate-optimal both in the homoscedastic and the heteroscedastic cases.
Performance of the Estimators {#section:performance}
=============================
The purpose of this section is to assess the quality of the aforementioned procedures. To this end, we present conditions for the consistency of these estimators in the form of upper bounds on their perceivable separation distance. Furthermore, to compare this bounds with the minimax separation distance, we establish lower bounds on the latter and prove that it coincides up to a constant factor with the perceivable separation distance of the LSL estimator.
Homoscedastic Setup
-------------------
We start by considering the homoscedastic case, in which upper and lower bounds matching up to a constant are obtained for all the estimators introduced in the previous section.
\[upperbound1\] Let $\a\in (0,1)$ be a tolerance level and $\s_j=\sdiese_j=\s$ for all $j\in\{1,\ldots,n\}$. If $\hat\pi$ denotes any one of the estimators (\[greedy\])-(\[LSL\]), then $$\bar\k_\a(\hat\pi)\le 4 \max\Big\{\Big(2\log \frac{8n^2}{\a}\Big)^{1/2}, \Big(d\log\frac{4n^2}{\a}\Big)^{1/4} \Big\}.$$
An equivalent way of stating this result is that if $$\k = 4\s \max\Big\{\Big(4\log \frac{8n^2}{\a}\Big)^{1/2}, \Big(4d\log\frac{4n^2}{\a}\Big)^{1/4} \Big\}$$ and $\T_\k$ is the set of all $\btheta\in\RR^{n\times d}$ such that $\k(\btheta)\ge \k$, then $$\max_{\pi^*\in\Sn}\sup_{\btheta\in\T_\k}\prob_{\btheta,\bsigma,\pi^*}(\hat\pi\neq\pi^*) \le \a$$ for all the estimators defined in Section \[section:estimators\]. Note that this result is nonasymptotic. Furthermore, it tells us that the perceivable separation distance of the procedures under consideration is at most of the order of $$\max\Big\{(\log n)^{1/2}, (d \log n)^{1/4}\Big\}.$$ It is interesting to observe that there are two regimes in this rate, the boundary of which corresponds to the case where $d$ is of the order of $\log n$. For dimensions that are significantly smaller than $\log n$, the perceivable distance of separation is dimension free. On the other hand, when $d$ is larger than $c\log n$ for some absolute constant $c>0$, the perceivable distance of separation deteriorates with increasing $d$ at the polynomial rate $d^{1/4}$. However, this result does not allow us to deduce any hierarchy between the four estimators, since it provides the same upper bound for all of them. Moreover, as stated in the next theorem, this bound is optimal up to a multiplicative constant.
\[lowerbound1\] Assume that $n\ge6$, $\T_\k$ is the set of all $\btheta\in\RR^{n\times d}$ such that $\k(\btheta)\ge \k$. Then there exist two absolute constants $c,C>0$ such that for $$\k=2^{-5/2}{\s}\max\Big\{(\log n)^{1/2}, (c d \log n)^{1/4}\Big\},$$ the following lower bound holds $$\inf_{\hat\pi} \max_{\pi^*\in\Sn} \sup_{\btheta\in\T_\k}\prob_{\btheta,\bsigma,\pi^*}\big(\hat\pi\neq\pi^*\big) > C,$$ where the infimum is taken over all permutation estimators.
An equivalent way of stating this result is to say that the minimax distance of separation satisfies the inequality $$\bar\k_\a \ge 2^{-5/2}\max\big\{(\log n)^{1/2}, (c d \log n)^{1/4}\big\}.$$ Combined with Theorem \[upperbound1\], this implies that the minimax rate of separation is given by the expression $\max\big\{(\log n)^{1/2}, (d \log n)^{1/4}\big\}$.
In order to avoid any possible confusion, we emphasize that the rate obtained in this and subsequent sections concerns the speed of decay of the separation distance and not the estimation risk measured by $\prob_{\btheta,\bsigma,\pi^*}\big(\hat\pi\neq\pi^*\big)$. For the latter, considering $\kappa$ as fixed, one readily derives from Theorem \[upperbound1\] that $$\begin{aligned}
\label{rem:a}
\max_{\pi^*\in\Sn} \sup_{\btheta\in\T_\k}\prob_{\btheta,\bsigma,\pi^*}\big(\hat\pi\neq\pi^*\big)
\le \max\Big\{8n^2\exp\Big(-\frac{\kappa^2}{2^{6}\sigma^2}\Big),4n^2\exp\Big(-\frac{\kappa^4}{2^{10}d\sigma^4}\Big)\Big\}\end{aligned}$$ for the four estimators $\hat\pi$ defined in the previous section by equations (\[greedy\])-(\[LSL\]). We do not know whether the right-hand side of this inequality is the correct rate for the minimax risk $R^{\rm mmx} =\inf_{\hat\pi} \sup_{(\btheta,\pi^*)\in\T_\k\times\Sn}
\prob_{\btheta,\bsigma,\pi^*}\big(\hat\pi\neq\pi^*\big)$. In fact, one can adapt the proof of Theorem \[lowerbound1\] to get a lower bound on $R^{\rm mmx}$ which is of the same form as the right-hand side in (\[rem:a\]), but with constants $2^6$ and $2^{10}$ replaced by smaller ones. The ratio of such a lower bound and the upper bound in (\[rem:a\]) tends to 0 and, therefore, does not provide the minimax rate of estimation, in the most common sense of the term. However, one may note that the minimax rate of separation established in this work is the analogue of the minimax rate of estimation of the Bahadur risk, see [@Bahadur; @Korostelev; @KorostelevSpok].
Heteroscedastic Setup
---------------------
We switch now to the heteroscedastic setting, which allows us to discriminate between the four procedures. Note that the greedy algorithm, the LSS and the LSL have a serious advantage over the LSNS since they can be computed without knowing the noise levels $\bsigma$.
\[upperbound2\] Let $\a\in(0,1)$ and condition (\[sigma\_h\]) be fulfilled. If $\hat\pi$ is either $\pi^{\textrm{LSNS}}$ (if the noise levels $\s_i,\sdiese_i$ are known) or $\pi^{\textrm{LSL}}$ (when the noise levels are unknown), then $$\bar\k_\a(\hat\pi) \leq 4\max\Big\{\Big(2\log \frac{8n^2}{\a}\Big)^{1/2}, \Big(d\log\frac{4n^2}{\a}\Big)^{1/4} \Big\}.$$
This result tells us that the performance of the LSNS and LSL estimators, measured in terms of the order of magnitude of the separation distance, is not affected by the heteroscedasticity of the noise levels. Two natural questions arise: 1) is this performance the best possible over all the estimators of $\pi^*$ and 2) is the performance of the LSS and the greedy estimator as good as that of the LSNS and LSL?
To answer the first question, we should start by adapting the notion of minimax separation distance to the case of unknown noise levels. Indeed, the definition (\[mmx\]) given in previous sections involves a minimum over all possible estimators $\hat\pi$ which are allowed to depend on $\bsigma$. On the one hand, considering $\bsigma$ as known limits considerably the scope of applications of the methods. On the other hand, from a theoretical point of view, knowing $\bsigma$ may lead to a substantially better minimax rate and even to a separation equal to zero, which correspond to an estimator that has no real practical interest. Indeed, under condition (\[sigma\_h\]), it is possible to estimate the permutation $\pi^*$ by only considering the noise levels, without any assumption on the distance between the features. For instance, we can define an estimator $\hat\pi$ as follows. We set $\hat\pi(1) = \arg\min_{i=1,\ldots,n} \big| \frac1{2d}{\|X_i-\Xdiese_1\|^2} - \s_i^2 \big|$ and recursively, for every $j\in\{ 2,\ldots,n\}$, $$\hat\pi(j) = \arg\min_{i\not\in\{\hat\pi(1),\ldots,\hat\pi(j-1)\}} \Big| \frac1{2d}{\|X_i-\Xdiese_j\|^2} - \s_i^2 \Big|.$$ This leads to an accurate estimator of $\pi^*$—for vectors $\{\theta_j\}$ that are very close—as soon as the noise levels are different enough from each other. In particular, the estimated permutation $\hat\pi$ coincides with the true permutation $\pi^*$ on the event $$\label{eq:14}
\forall i\in\{1,\ldots,n\},\qquad \Big|\frac1{2d}{\|X_{\pi^*(i)}-\Xdiese_i\|^2} - \s_{\pi^*(i)}^2 \Big| < \min_{j\neq \pi^*(i)} \Big| \frac{\|X_j-\Xdiese_i\|^2}{2d} - \s_j^2\Big|.$$ Using standard bounds on the tails of the $\chi^2$ distribution recalled in Lemma \[concentration\] of Section \[section:theorems\], in the case when all the vectors $\theta_j$ are equal, one can check that the left-hand side in (\[eq:14\]) is of the order of $\s_{\pi^*(i)}^2\sqrt{(\log n)/d}$ while the right-hand side is at least of the order of $\frac12\min_{j\not = \pi^*(i)} |\s_j^2-\s_{\pi^*(i)}^2|- C(\sigma_{\pi^*(i)}^2+\s_{j}^2)\sqrt{(\log n)/d}$. This implies that we can consistently identify the permutation when $$\forall (i,j)\in\{1,\ldots,n\}^2,
\quad i\not= j,\qquad \Big|\frac{\s_j^2}{\s_i^2}-1\Big| \ \gg \ \sqrt{\frac{\log n}{d}},$$ even if the separation distance is equal to zero. In order to discard such kind of estimators from the competition in the procedure of determining the minimax rates, we restrict our attention to the noise levels for which the values $|\frac{\s_j^2}{\s_i^2}-1|$ are not larger than $C\sqrt{(\log n)/d}$ for $j\not=i$.
\[lowerbound2\] Assume that $n\ge6$, $\bar\T_\k$ is the set of all $\btheta\in\RR^{n\times d}$ such that $\bar\k(\btheta,\bsigma)\ge \k$ and $$\frac{\max_i\s^2_i}{\min_i\s^2_i} -1 \le \frac14 \ \sqrt{\frac{\log n}{d}}.$$ Then there exist two constants $c,C>0$ such that $\k<(1/8) \max\{ (\log n)^{1/2}, (cd\log n)^{1/4} \}$, implies that $$\inf_{\hat\pi} \max_{\pi^*\in\Sn} \sup_{\btheta\in\bar\T_{\k}} \prob_{\btheta,\bsigma,\pi^*} (\hat\pi\neq\pi^*) > C,$$ where the infimum is taken over all permutation estimators.
It is clear that the constants $c$ and $C$ of the previous theorem are closely related. The inspection of the proof shows that, for instance, if $c\le 1/20$ then $C$ is larger than $17\%$.
Let us discuss now the second question raised earlier in this section and concerning the theoretical properties of the greedy algorithm and the LSS under heteroscedasticity. In fact, the perceivable distances of separation of these two procedures are significantly worse than those of the LSNS and the LSL especially for large dimensions $d$. We state the corresponding result for the greedy algorithm, a similar conclusion being true for the LSS as well. The superiority of the LSNS and LSL is also confirmed by the numerical simulations presented in Section \[section:simulations\] below. In the next theorem and in the sequel of the paper, we denote by $\id$ the identity permutation defined by $\id(i) = i$ for all $i\in\{1,\ldots,n\}$.
\[lowerbound3\] Assume that $d\ge 225\log 6$, $n=2$, $\s_1^2 =3$ and $\s_2^2=1$. Then the condition $\k < 0.1(2d)^{1/2}$ implies that $$\sup_{\btheta\in\bar\T_\k}\prob_{\btheta,\bsigma,\id}(\pi^{\textrm{gr}}\neq\id) \ge 1/2.$$
This theorem shows that if $d$ is large, the necessary condition for $\pi^{\textrm{gr}}$ to be consistent is much stronger than the one obtained for $\pi^{\textrm{LSL}}$ in Theorem \[upperbound2\]. Indeed, for the consistency of $\pi^{\textrm{gr}}$, $\k$ needs to be at least of the order of $d^{1/2}$, whereas $d^{1/4}$ is sufficient for the consistency of $\pi^{\textrm{LSL}}$. Hence, the maximum likelihood estimators LSNS and LSL that take into account noise heteroscedasticity are, as expected, more interesting than the simple greedy estimator[^2].
Computational Aspects {#section:computationalaspects}
=====================
At first sight, the computation of the estimators (\[LSS\])-(\[LSL\]) requires to perform an exhaustive search over the set of all possible permutations, the number of which, $n!$, is prohibitively large. This is in practice impossible to do on a standard PC as soon as $n \geq 20$. In this section, we show how to compute these (maximum likelihood) estimators in polynomial time using, for instance, algorithms of linear programming[^3].
To explain the argument, let us consider the LSS estimator $$\pi^{{\textrm{LSS}}} = \arg\min_{\pi\in\Sn} \sum_{i=1}^n \|X_{\pi(i)}-\Xdiese_i\|^2.$$ For every permutation $\pi$, we denote by $P^\pi$ the $n\times n$ permutation matrix with coefficients $P^\pi_{ij} = \fcar_{\{ j=\pi(i) \}}$. Then we can give the equivalent formulation $$\label{discrete}
\pi^{\textrm{LSS}} = \arg\min_{\pi \in\Sn} \trace \big(M P^\pi\big),$$ where $M$ is the matrix with coefficient $\|X_i-\Xdiese_j\|^2$ at the $i^{th}$ row and $j^{th}$ column. The cornerstone of our next argument is the Birkhoff-von Neumann theorem stated below, which can be found for example in [@BudishCheKojimaMilgrom2009].
\[birkhoff-von-neumann\] Assume that $\mathcal{P}$ is the set of all doubly stochastic matrices of size $n$, the matrices whose entries are nonnegative and sum up to $1$ in every row and every column. Then every matrix in $\mathcal{P}$ is a convex combination of matrices $\{P^\pi : \pi\in\Sn\}$. Furthermore, permutation matrices are the vertices of the simplex $\mathcal P$.
In view of this result, the combinatorial optimization problem (\[discrete\]) is equivalent to the following problem of continuous optimization: $$\label{cont}
P^{\textrm{LSS}} = \arg\min_{P \in\mathcal P} \trace \big(M P\big),$$ in the sense that $\pi$ is a solution to (\[discrete\]) if and only if $P^\pi$ is a solution to (\[cont\]). To prove this claim, let us remark that for every $P\in\mathcal{P}$, there exist coefficients $\a_1,\ldots,\a_{n!}\in[0,1]$ such that $P = \sum_{i=1}^{n!} \a_i P^{\pi_i}$ and $\sum_{i=1}^{n!} \a_i = 1$. Therefore, we have $\trace\big(MP\big) = \sum_{i=1}^{n!} \a_i \trace\big(MP^{\pi_i}\big) \ge \min_{\pi\in\Sn} \trace\big(MP^{\pi}\big)$ and $\trace\big(MP^{\textrm{LSS}}\big)\ge \trace\big(MP^{\pi^{\textrm{LSS}}}\big).$
The great advantage of (\[cont\]) is that it concerns the minimization of a linear function under linear constraints and, therefore, is a problem of linear programming that can be efficiently solved even for large values of $n$. The same arguments apply to the estimators $\pi^{\textrm{LSNS}}$ and $\pi^{\textrm{LSL}}$, only the matrix $M$ needs to be changed.
There is a second way to compute the estimators LSS, LSNS and LSL efficiently. Indeed, the computation of the aforementioned maximum likelihood estimators is a particular case of the assignment problem, which consists in finding a minimum weight matching in a weighted bipartite graph, where the matrix of the costs is the matrix $M$ from above. This means that the cost of assigning the $i^{\text{th}}$ feature of the first image to the $j^{\text{th}}$ feature of the second image is either
- the squared distance ${\|X_i-\Xdiese_j\|^2}$,
- or the normalized squared distance $\|X_i-\Xdiese_j\|^2/(\s_i^2+(\sdiese_j)^2)$,
- or the logarithm of the squared distance $\log \|X_i-\Xdiese_j\|^2$.
The so-called Hungarian algorithm presented in @Kuhn1955 solves the assignment problem in time $O(n^3)$.
Extensions {#section:extensions}
==========
In this section, we briefly discuss possible extensions of the foregoing results to other distances, more general matching criteria and to the estimation of an arrangement.
Minimax Rates for the Hamming Distance
--------------------------------------
In the previous sections, the minimax rates were obtained for the error of estimation measured by the risk $\prob_{\btheta,\bsigma\pi^*}(\hat\pi\neq\pi^*)
=\esp_{\btheta,\bsigma,\pi^*}[\delta_{\text{0\,-1}}(\hat\pi,\pi^*)]$, which may be considered as too restrictive. Indeed, one could find acceptable an estimate having a small number of mismatches, if it makes it possible to significantly reduce the perceivable distance of separation. These considerations lead to investigating the behavior of the estimators in terms of the Hamming loss, *i.e.*, to studying the risk $$\esp_{\btheta,\bsigma,\pi^*}[\delta_{H}(\hat\pi,\pi^*)]=\esp_{\btheta,\bsigma,\pi^*}
\Big[\frac1n \sum\limits_{i=1}^n \fcar_{\{\hat\pi(i)\neq\pi^*(i)\}}\Big]$$ corresponding to the expected average number of mismatched features. Another advantage of studying the Hamming loss instead of the 0-1 loss is that the former sharpens the difference between the performances of various estimators. Note, however, that thanks to the inequality $\delta_{H}(\hat\pi,\pi^*)\le \delta_{\text{0\,-1}}(\hat\pi,\pi^*)$, all the upper bounds established for the minimax rate of separation under the 0-1 loss directly carry over to the case of the Hamming loss. This translates into the following theorem.
\[upperbound4\] Let $\a\in(0,1)$ and condition (\[sigma\_h\]) be fulfilled. If $\hat\pi$ is either $\pi^{\textrm{LSNS}}$ or $\pi^{\textrm{LSL}}$, then $\bar\k_\a(\hat\pi) \leq 4\max\big\{\big(2\log \frac{8n^2}{\a}\big)^{1/2}, \big(d\log\frac{4n^2}{\a}\big)^{1/4} \big\}$. That is, if $$\kappa=4\max\Big\{\Big(2\log \frac{8n^2}{\a}\Big)^{1/2}, \Big(d\log\frac{4n^2}{\a}\Big)^{1/4} \Big\}$$ and $\bar\T_\k$ is the set of all $\btheta\in\RR^{n\times d}$ such that $\bar\k(\btheta,\sigma)\ge \k$, then $$\max_{\pi^*\in\Sn}\sup_{\btheta\in\bar\T_\k}\esp_{\btheta,\bsigma,\pi^*}[\delta_{H}(\hat\pi,\pi^*)] \le \a.$$
------------------------------- ------ ------ ------ ------ ------- ------- ------- ------- -------
$n=$ 4 5 6 7 8 9 10 11 12
$M_n \ge $ 19 57 179 594 1939 3441 11680 39520 86575
$\frac{\log M_n}{n\log n}\ge$ 0.53 0.50 0.48 0.47 0.455 0.412 0.407 0.401 0.381
$\frac{\log M_n}{n\log n}\le$ 0.53 0.50 0.48 0.47 0.455 0.445 0.436 0.427 0.420
------------------------------- ------ ------ ------ ------ ------- ------- ------- ------- -------
: The values of $M_n=\mathcal M(1/4, B_{2,n}(2),\delta_H)$ for $n\in\{4,\ldots,12\}$. The lower bound is just the cardinality of one $\epsilon$-packing, not necessarily the largest one. The upper bound is merely the cardinality of the $\ell_2$-ball. \[tab:1\]
While this upper bound is an immediate consequence of Theorem \[upperbound2\], getting lower bounds for the Hamming loss appears to be more difficult. To state the corresponding result, let is consider the case of homoscedastic noise and introduce some notation. We denote by $\delta_2(\cdot,\cdot)$ the normalized $\ell^2$-distance on the space of permutations $\Sn$: $\delta_2(\pi,\pi')^2=\frac1n\sum_{k=1}^n\big(\pi(k)-\pi'(k)\big)^2$. Let $B_{2,n}(R)$ be the ball of $(\Sn,\delta_2)$ with radius $R$ centered at $\id$. As usual, we denote by $\mathcal M(\e, B_{2,n}(R),\delta_H)$ the $\e$-packing number of the $\ell_2$-ball $B_{2,n}(R)$ in the metric $\delta_H$. This means that $\mathcal M(\e, B_{2,n}(R),\delta_H)$ is the largest integer $M$ such that there exist permutations $\pi_1,\ldots,\pi_M\in B_{2,n}(R)$ satisfying $\delta_H(\pi_i,\pi_j)\ge \e$ for every $i\not=j$. One can easily check that replacing $B_{2,n}(R)$ by any other ball of radius $R$ leaves the packing number $\mathcal M(\e, B_{2,n}(R),\delta_H)$ unchanged. We set $M_n=\mathcal M(1/4, B_{2,n}(2),\delta_H)$.
\[lowerbound4\] Let $\sigma_k=\sigma$ for all $k\in\{1,\ldots,n\}$ and $\bar\T_\k$ is the set of all $\btheta\in\RR^{n\times d}$ such that $\bar\k(\btheta,\bsigma)\ge \k$. Furthermore, assume that one of the following two conditions is fulfilled:
- $n\ge 3$ and $\k = (1/4)\big(\frac{\log M_n}{n}\big)^{1/2}$,
- $n\ge 26$, $d\ge 24\log n$ and $\k\le (1/8)(d\log n)^{1/4}$.
Then $\inf_{\hat\pi} \max_{\pi^*\in\Sn} \sup_{\btheta\in\bar\T_\k} \esp_{\btheta,\bsigma,\pi^*} [\delta_H(\hat\pi,\pi^*)] > 2.15\%$.
This result implies that in the regime of moderately large dimension, $d\ge 24\log n$, the minimax rate of the separation is the same as the one under the 0-1 loss and it is achieved by the LSL estimator. The picture is less clear in the regime of small dimensions, $d=o(\log n)$. If one proves that for some $c>0$, the inequality $\log M_n\ge cn\log n$ holds for every $n\ge 3$, then the lower bound of the last theorem matches the upper bound of Theorem \[upperbound4\] up to constant factors and leads to the minimax rate of separation $\max\{(\log n)^{1/2},(d\log n)^{1/4}\}$. Unfortunately, we were unable to find any result on the order of magnitude of $\log M_n$, therefore, we cannot claim that there is no gap between our lower bound and the upper one. However, we did a small experiment for evaluating $M_n$ for small values of $n$. The result is reported in Table \[tab:1\].
More General Matching Criteria {#subsection5.2}
------------------------------
In the previous sections, we were considering two vectors $\t_i$ and $\tdiese_j$ as matching if $\t_i\equiv\tdiese_j$, and $\equiv$ was the usual equality. In this part, we show that our results can be extended to more general matching criteria, defined as follows. Let $A$, $\Adiese$ be two known $p\times d$ matrices with some $p\in\mathbb N$ and $b$, $\bdiese$ be two known vectors from $\RR^{d}$. We write $\t\equiv_{A,b} \tdiese$, if $$\label{A,b}
A(\theta-b)=\Adiese(\tdiese -\bdiese).$$ Note that the case of equality studied in previous sections is obtained for $A =\Adiese=\mathbf I_d$ and $b=\bdiese=0$, where $\mathbf I_d$ is the identity matrix of size $d$. Let us first note that without loss of generality, by a simple transformation of the features, one can replace (\[A,b\]) by the simpler relation $$\label{A'}
\bar\theta=B \bartdiese,$$ where $\bar\t\in\RR^{d_1}$ for $d_1 = \text{rank}(A)$, $\bartdiese\in\RR^{d_2}$ for $d_2 = \text{rank}(\Adiese)$ and $B$ is a $d_1\times d_2$ known matrix. Indeed, let $A = U^\top\Lambda V$ (resp. $\Adiese = \tilde U^\top \tilde\Lambda\tilde V$) be the singular value decomposition of $A$ (resp. $\Adiese$), with orthogonal matrices $U\in\RR^{d_1\times p}$, $V\in\RR^{d_1\times d}$ and a diagonal matrix $\Lambda\in\RR^{d_1\times d_1}$ with positive entries. Then, one can deduce (\[A’\]) from (\[A,b\]) by setting $\bar\t = V(\theta-b)$, $\bartdiese=\tilde V(\tdiese-\bdiese)$ and $B = \Lambda^{-1}U\tilde U^\top\tilde\Lambda$. Of course, the same transformation should be applied to the observed noisy features, which leads to $\bar X_i= V(X_i-b)$ and $\barXdiese_i=\tilde V(\Xdiese-\bdiese)$. Since $V$ and $\tilde V$ are orthogonal matrices, *i.e.*, satisfy the relations $VV^\top = \mathbf I_{d_1}$ and $\tilde V\tilde V^\top = \mathbf I_{d_2}$, the noise component in the transformed noisy features is still white Gaussian.
All the four estimators introduced in Section \[section:estimators\] can be adapted to deal with such type of criterion. For example, denoting by $M$ the matrix $B(B^\top B)^+B^\top+BB^\top$ where $M^+$ is the Moore-Penrose pseudoinverse of the matrix $M$, the LSL estimator should be modified as follows $$\pi^{\textrm{LSL}} \triangleq \arg\min_{\pi\in\Sn} \sum_{i=1}^n \log \|M^+(\bar X_{\pi(i)}-B\barXdiese_i)\|^2. \label{LSL1}$$ All the results presented before can be readily extended to this case. In particular, if we assume that all the nonzero singular values of $B$ are bounded and bounded away from 0 and $\text{rank}(B)=q$, then the minimax rate of separation is given by $\max\big\{(\log n)^{1/2}, (q \log n)^{1/4}\big\}$. This rate is achieved, for instance, by the LSL estimator.
Let us briefly mention two situations in which this kind of general affine criterion may be useful. First, if each feature $\theta$ (resp.$\tdiese$) corresponds to a patch in an image $I$ (resp. $\Idiese$), then for detecting pairs of patches that match each other it is often useful to neglect the changes in illumination. This may be achieved by means of the criterion $A\theta=A\tdiese$ with $A = \mathbf I_d-\frac1{d}\mathbf 1_d\mathbf 1_d^\top$. Indeed, the multiplication of the vector $\theta$ by $A$ corresponds to removing from pixel intensities the mean pixel intensity of the patch. This makes the feature invariant by change of illumination. The method described above applies to this case and the corresponding rate is $\max\big\{(\log n)^{1/2}, ((d-1)\log n)^{1/4}\big\}$ since the matrix $A$ is of rank $d-1$. Second, consider the case when each feature combines the local descriptor of an image and the location in the image at which this descriptor is computed. If we have at our disposal an estimator of the transformation that links the two images and if this transformation is linear, then we are in the aforementioned framework. For instance, let each $\theta$ be composed of a local descriptor $\mathbf d\in\RR^{d-2}$ and its location $\mathbf x\in\RR^2$. Assume that the first image $I$ from which the features $\theta_i=[\mathbf d_i,\mathbf x_i]$ are extracted is obtained from the image $\Idiese$ by a rotation of the plane. Let $R$ be an estimator of this rotation and $\tdiese_i=[\bddiese_i,\bxdiese_i]$ be the features extracted from the image $\Idiese$. Then, the aim is to find the permutation $\pi$ such that $\bddiese_i=\mathbf d_{\pi(i)}$ and $\bxdiese_i=R\mathbf x_{\pi(i)}$ for every $i=1,\ldots,n$. This corresponds to taking in (\[A’\]) the matrix $B$ given by $$B =\begin{pmatrix}
\mathbf I_{d-2} & 0\\
0 & R
\end{pmatrix}.$$ This matrix $B$ being orthogonal, the resulting minimax rate of separation is exactly the same as when $B = \mathbf I_d$.
An interesting avenue for future research concerns the determination of the minimax rates in the case when the equivalence of two features is understood under some transformation $A$ which is not completely determined. For instance, one may consider that the features $\theta$ and $\theta'$ match if there is a matrix $A$ in a given parametric family $\{A_\tau:\tau\in\RR\}\subset \RR^{d\times d}$ for which $\theta = A\theta'$. In other terms, $\theta\equiv\theta'$ is understood as $\inf_\tau \|\theta-A_\tau\theta'\|=0$. Such a criterion of matching may be useful for handling various types of invariance (see [@CollierDalalyan2011; @Collier2012] for invariance by translation).
Estimation of an Arrangement {#sec:arrangement}
----------------------------
An interesting extension concerns the case of the estimation of a general arrangement, the case when $m$ and $n$ are not necessarily identical. In such a situation, without loss of generality, one can assume that $n\le m$ and look for an injective function $$\pi^*:\{1,\ldots,n\}\to\{1,\ldots,m\}.$$ All the estimators presented in Section \[section:estimators\] admit natural counterparts in this rectangular setting. Furthermore, the computational method using the Birkhoff-von Neumann theorem is still valid in this setting, and is justified by the extension of the Birkhoff-von Neumann theorem recently proved by @BudishCheKojimaMilgrom2009. In this case, the minimization should be carried out over the set of all matrices $P$ of size $(n,m)$ such that $P_{i,j}\ge 0,$ and $$\begin{cases}
\ \sum_{i=1}^n P_{i,j}\le 1 \\
\ \sum_{j=1}^m P_{i,j}= 1
\end{cases},\quad (i,j)\in\{1,\ldots,n\}\times\{1,\ldots,m\}.$$ From a practical point of view, it is also important to consider the issue of robustness with respect to the presence of outliers, when for some $i$ there is no $\Xdiese_j$ matching with $X_i$. The detailed exploration of this problem being out of scope of the present paper, let us just underline that the LSL-estimator seems to be well suited for such a situation because of the robustness of the logarithmic function. Indeed, the correct matches are strongly rewarded because $\log(0)=-\infty$ and the outliers do not interfere too much with the estimation of the arrangement thanks to the slow growth of $\log$ in $\pinf$.
Experimental Results {#section:simulations}
====================
We have implemented all the procedures in Matlab and carried out numerical experiments on synthetic data. To simplify, we have used the general-purpose solver SeDuMi [@SeDuMi] for solving linear programs. We believe that it is possible to speed-up the computations by using more adapted first-order optimization algorithms, such as coordinate gradient descent. However, even with this simple implementation, the running times are reasonable: for a problem with $n=500$ features, it takes about six seconds to compute a solution to (\[cont\]) on a standard PC.
![Average error rate of the four estimating procedures in the experiment with homoscedastic noise as a function of the minimal distance $\k$ between distinct features. One can observe that the LSS, LSNS and LSL procedures are indistinguishable and perform much better than the greedy algorithm.\[fig3:1\] ](plot3new.pdf){width="48.00000%"}
![Left: Average error rate of the four estimating procedures in the experiment with heteroscedastic noise as a function of the minimal distance $\k$ between distinct features. Right: zoom on the same plots. One can observe that the LSNS and LSL are almost indistinguishable and, as predicted by the theory, perform better than the LSS and the greedy algorithm.\[fig3:2\] ](plot1new.pdf "fig:"){width="48.00000%"}![Left: Average error rate of the four estimating procedures in the experiment with heteroscedastic noise as a function of the minimal distance $\k$ between distinct features. Right: zoom on the same plots. One can observe that the LSNS and LSL are almost indistinguishable and, as predicted by the theory, perform better than the LSS and the greedy algorithm.\[fig3:2\] ](plot2new.pdf "fig:"){width="48.00000%"}
#### Homoscedastic noise
We chose $n=d=200$ and randomly generated a $n\times d$ matrix $\btheta$ with i.i.d. entries uniformly distributed on $[0,\tau]$, with several values of $\tau$ varying between $1.4$ and $3.5$. Then, we randomly chose a permutation $\pi^*$ (uniformly from $\Sn$) and generated the sets $\{X_i\}$ and $\{\Xdiese_i\}$ according to (\[model\]) with $\sigma_i=\sdiese_i=1$. Using these sets as data, we computed the four estimators of $\pi^*$ and evaluated the average error rate $\frac1n\sum_{i=1}^n \fcar_{\{\hat\pi(i)\not=\pi^*(i)\}}$. The result, averaged over 500 independent trials, is plotted in Fig. \[fig3:1\].
Note that the three estimators originating from the maximum likelihood methodology lead to the same estimators, while the greedy algorithm provides an estimator which is much worse than the others when the parameter $\k$ is small.
#### Heteroscedastic noise
This experiment is similar to the previous one, but the noise level is not constant. We still chose $n=d=200$ and defined $\btheta= \tau I_d$, where $I_d$ is the identity matrix and $\tau$ varies between $4$ and $10$. Then, we randomly chose a permutation $\pi^*$ (uniformly from $\Sn$) and generated the sets $\{X_i\}$ and $\{\Xdiese_i\}$ according to (\[model\]) with $\sigma_{\pi^*(i)}=\sdiese_i=1$ for 10 randomly chosen values of $i$ and $\sigma_{\pi^*(i)}=\sdiese_i=0.5$ for the others. Using these sets as data, we computed the four estimators of $\pi^*$ and evaluated the average error rate $\frac1n\sum_{i=1}^n \fcar_{\{\hat\pi(i)\not=\pi^*(i)\}}$. The result, averaged over 500 independent trials, is plotted in Fig. \[fig3:2\].
Note that among the noise-level-adaptive estimators, LSL outperforms the two others and is as accurate as, and even slightly better than the LSNS pseudo-estimator. This confirms the theoretical findings presented in foregoing sections.
Conclusion and Future Work
==========================
Motivated by the problem of feature matching, we proposed a rigorous framework for studying the problem of permutation estimation from a minimax point of view. The key notion in our framework is the minimax rate of separation, which plays the same role as in the statistical hypotheses testing theory [@Ingster]. We established theoretical guarantees for several natural estimators and proved the optimality of some of them. The results appeared to be quite different in the homoscedastic and in the heteroscedastic cases. However, we have shown that the least sum of logarithms estimator outperforms the other procedures both theoretically and empirically.
Several avenues of future work have been already mentioned in previous sections. In particular, investigating the statistical properties of the arrangement estimation problem described in Section \[sec:arrangement\] and considering the case of unspecified transformation relating the features may have a significant impact on the practice of feature matching.
Another interesting question is to extend the statistical inference developed here for the problem of feature matching to the more general assignment problem. The latter aims at assigning $m$ tasks to $n$ agents such that the cost of assignment is as small as possible. Various settings of this problem have been considered in the literature [@Pentico] and many algorithms for solving the problem have been proposed [@Romeijn]. However, to the best of our knowledge, the statistical aspects of the problem in the case where the cost matrix is corrupted by noise have not been studied so far.
Proofs of the Theorems {#section:theorems}
======================
In this section we collect the proofs of the theorems. We start with the proof of Theorem \[upperbound2\], since it concerns the more general setting and the proof of Theorem \[upperbound1\] can be deduced from that of Theorem \[upperbound2\] by simple arguments. We then prove the other theorems in the usual order and postpone the proofs of some technical lemmas to the next section.
To ease notation and without loss of generality, we assume that $\pi^*$ is the identity permutation denoted by $\id$. Furthermore, since there is no risk of confusion, we write $\prob$ instead of $\prob_{\btheta,\bsigma,\pi^*}$. We wish to bound the probability of the event $\O = \{\hat\pi\neq \id\}$.
Let us first denote by $\hat\pi$ the maximum likelihood estimator $\pi^{\textrm{LSL}}$ defined by (\[LSL\]). We have $$\O \subset \bigcup_{\pi\not=\id} \O_\pi,$$ where $$\begin{aligned}
\O_\pi &
= \Big\{\sum_{i=1}^n \log \frac{\|X_i-\Xdiese_i\|^2}{\|X_{\pi(i)}-\Xdiese_i\|^2}\ge 0\Big\}
= \Big\{\sum_{i : \pi(i)\not= i} \log \frac{\|X_i-\Xdiese_i\|^2}{\|X_{\pi(i)}-\Xdiese_i\|^2}\ge 0\Big\}.\end{aligned}$$ On the one hand, for every permutation $\pi$, $$\begin{aligned}
\sum_{\pi(i)\not =i} \log\Big({\textstyle\frac{2\s_i^2}{\s_i^2+\s_{\pi(i)}^2}}\Big)
&= \sum_{i=1}^n \big(\log(2\s_i^2) - \log(\s_i^2+\s_{\pi(i)}^2) \big)\\
&= \sum_{i=1}^n \frac{\log(2\s_i^2)+\log(2\s_{\pi(i)}^2)}2-\log(\s_i^2+\s_{\pi(i)}^2)\end{aligned}$$ so, using the concavity of the logarithm, this quantity is nonpositive. Therefore, $$\begin{aligned}
\O_\pi
&\subset \Big\{\sum_{i : \pi(i)\not= i}\!\! \log \textstyle\frac{\|X_i-\Xdiese_i\|^2/(2\s_i^2)}
{\|X_{\pi(i)}-\Xdiese_i\|^2/(\s_i^2+\s_{\pi(i)}^2)}\ge 0\Big\}\\
&\subset \bigcup_{i=1}^n \bigcup_{j\not =i}\Big\{\frac{\|X_i-\Xdiese_i\|^2}{2\s_i^2}\ge\frac{\|X_{j}-\Xdiese_i\|^2}{\s_j^2+\s_{i}^2}\Big\}.\end{aligned}$$ This readily yields $\O\subset \bar\O$, where $$\begin{aligned}
\label{inclusion}
\bar\O =\bigcup_{i=1}^n \bigcup_{j\not =i}\Big\{\frac{\|X_i-\Xdiese_i\|^2}{2\s_i^2}\ge \frac{\|X_{j}-\Xdiese_i\|^2}{\s_j^2+\s_{i}^2}\Big\}.\end{aligned}$$ Furthermore, the same inclusion is true for the LSNS estimator as well. Therefore, the rest of the proof is common for the estimators LSNS and LSL.
We set $\sigma_{i,j}=(\sigma_i^2+\sigma_j^2)^{1/2}$ and $$\begin{aligned}
\z_1 &= \max_{i\not =j} \bigg|
\frac{(\t_{i}-\t_j)^\top(\s_i\xi_i-\s_j\xidiese_j)}{\|\t_i-\t_j\|\s_{i,j}}\bigg|, \quad
\z_2 = d^{-1/2} \max_{i,j } \bigg|\,\bigg\|\frac{\s_i\xi_i-\s_j\xidiese_j}{\s_{i,j}}\bigg\|^2 - d\bigg|.\end{aligned}$$ Since $\pi^*=\id$, it holds that for every $i\in\{1,\ldots,n\}$, $$\begin{aligned}
\|X_i-\Xdiese_i\|^2 &= \s_i^2 \|\xi_i-\xidiese_i\|^2 \le 2\s_i^2(d+\sqrt{d}\z_2).\end{aligned}$$ Similarly, for every $j\not =i$, $$\begin{aligned}
\|X_{j}-\Xdiese_i\|^2
&= \|\t_{j}-\t_i\|^2+ \|\s_{j}\xi_{j}-\s_i\xidiese_i\|^2+ 2(\t_{j}-\t_i)^\top(\s_{j}\xi_{j}-\s_i\xidiese_i).\end{aligned}$$ Therefore, $$\begin{aligned}
\|X_{j}-\Xdiese_i\|^2
&\ge \|\t_{j}-\t_i\|^2+\s_{i,j}^2 (d-\sqrt{d}\z_2) - 2\|\t_{j}-\t_i\|\s_{i,j}\z_1.\end{aligned}$$ This implies that on the event $\O_1 = \{\bar\k(\btheta,\bsigma)\ge \z_1\}$ it holds that $$\begin{aligned}
\frac{\|X_{j}-\Xdiese_i\|^2}{\s_{i,j}^2}
\ge \bar\k(\btheta,\bsigma)^2 - 2\bar\k(\btheta,\bsigma)\z_1+d-\sqrt{d}\z_2.\end{aligned}$$ Combining these bounds, we get that $$\O\cap \O_1\subset \Big\{d+\sqrt{d}\z_2\ge\bar\k(\btheta,\bsigma)^2 - 2\bar\k(\btheta,\bsigma)\z_1+d-\sqrt{d}\z_2\Big\},$$ which implies that $$\begin{aligned}
\prob(\O)
&\le \prob(\O_1^\complement) + \prob\big(\O\cap\O_1\big)\nonumber\\
&\le \prob\big(\z_1\ge \bar\k(\btheta,\bsigma)\big)
+\prob(2\sqrt{d}\z_2+2\bar\k(\btheta,\bsigma)\z_1\ge\bar\k(\btheta,\bsigma)^2)\nonumber\\
&\le 2\prob\big(\z_1\ge \textstyle\frac{\bar\k(\btheta,\bsigma)}4\big)+
\prob\big(\z_2\ge \textstyle\frac{\bar\k(\btheta,\bsigma)^2}{4\sqrt{d}}\big).\label{ineq:2}\end{aligned}$$ Finally, one easily checks that for suitably chosen random variables $\z_{i,j}$ drawn from the standard Gaussian distribution, it holds that $\z_1 = \max_{i\not=j}|\z_{i,j}|$. Therefore, using the well-known tail bound for the standard Gaussian distribution in conjunction with the union bound, we get $$\begin{aligned}
\prob\big(\z_1\ge \textstyle\frac14\bar\k(\btheta,\bsigma)\big)
&\le \sum_{i\not =j}\nolimits \prob\big(|\z_{i,j}|\ge \textstyle\frac14\bar\k(\btheta,\bsigma)\big)
\le 2 n^2 e^{-\frac1{32}\bar\k(\btheta,\bsigma)^2}.\label{ineq:3}\end{aligned}$$ To bound the large deviations of the random variable $\zeta_2$, we rely on the following result.
\[concentration\] If $Y$ is drawn from the chi-squared distribution $\chi^2(D)$, where $D\in\mathbb N^*$, then, for every $x>0$, $$\begin{cases}
\ \prob\big(Y-D \le -2\sqrt{Dx} \big) \le e^{-x}, \phantom{\Big()}\\
\ \prob\big(Y-D \ge 2\sqrt{Dx} + 2x \big) \le e^{-x}. \phantom{\Big()}
\end{cases}$$ As a consequence, $\forall y>0$, $\prob\big(D^{-1/2}|Y-D| \ge y\big) \le 2\exp\big\{-\textstyle\frac18 y(y\wedge \sqrt{D})\big\}$.
This inequality, combined with the union bound, yields $$\begin{aligned}
\prob\Big(\z_2 \ge \textstyle\frac{\bar\k(\btheta,\bsigma)^2}{4\sqrt{d}}\Big)
&\le 2n^2 \exp\Big\{-\frac{(\bar\k(\btheta,\bsigma)/16)^2}{d}(\bar\k^2(\btheta,\bsigma)\wedge 8d)\Big\}.\label{ineq:4}\end{aligned}$$ Combining inequalities (\[ineq:2\])-(\[ineq:4\]), we obtain that as soon as $$\begin{aligned}
\bar\k(\btheta,\bsigma)\ge 4\Big(\sqrt{2\log(8n^2/\a)} \vee \big(d\log(4n^2/\a)\big)^{1/4}\Big),\end{aligned}$$ we have $\prob(\hat\pi\not = \pi^*)=\prob(\O)\le \a$.
Without loss of generality, we assume $\pi =\id$. It holds that, on the event $$\mathcal A= \bigcap_{i=1}^n\bigcap_{j\not =\pi^*(i)}\Big\{ \|X_{\pi^*(i)}-\Xdiese_{i}\|< \|X_{j}-\Xdiese_{i}\|\Big\},$$ all the four estimators coincide with the true permutation $\pi^*$. Therefore, we have $$\{\hat \pi\not=\pi^*\} \subseteq \bigcup_{i=1}^n\bigcup_{j\not =i}\Big\{ \|X_{i}-\Xdiese_{i}\|\geq \|X_{j}-\Xdiese_{i}\|\Big\}.$$ The latter event is included in $\bar\O$ at the right-hand side of (\[inclusion\]), the probability of which has been already shown to be small in the previous proof.
We refer the reader to the proof of Theorem \[lowerbound2\] below, which concerns the more general situation. Indeed, when all the variances $\s_j$ are equal, Theorem \[lowerbound2\] boils down to Theorem \[lowerbound1\].
To establish lower bounds for various types of risks we will use the following lemma:
\[Tsybakov\] Assume that for some integer $M\ge 2$ there exist distinct permutations $\pi_0,\ldots,\pi_M\in\Sn$ and mutually absolutely continuous probability measures $\bQ_0,\ldots,\bQ_M$ defined on a common probability space $(\mathcal Z,\mathscr Z)$ such that $$\frac{1}{M} \sum_{j=1}^M K(\bQ_j,\bQ_0) \le \frac{1}{8} \log M.$$ Then, for every measurable mapping $\tilde\pi:\mathcal Z\to\Sn$, $$\max_{j=0,\ldots,M} \bQ_j(\tilde\pi\neq\pi_j) \ge \frac{\sqrt{M}}{\sqrt{M}+1} \Big( \frac{3}{4}-\frac{1}{2\sqrt{\log M}} \Big).$$
To prove Theorem \[lowerbound2\], we consider separately the two cases $$\begin{aligned}
&\textbf{Case 1:}\qquad \max\Big\{(\log n)^{1/2} , (c d \log n)^{1/4}\Big\} = (\log n)^{1/2},\\
&\textbf{Case 2:}\qquad \max\Big\{(\log n)^{1/2} , (c d \log n)^{1/4} \Big\} = (c d \log n)^{1/4}.\end{aligned}$$
#### **Case 1:** We assume that $\k \le \frac18 \sqrt{\log n}$
Denote $m$ the largest integer such that $2m\le n$. We assume without loss of generality that the noise levels are ranked in increasing order: $\s_1\leq\ldots\leq\s_n.$ Then, we construct a least favorable set of vectors for the estimation of the permutation. To ease notation, we set $\s_{i,j}=(\s_i^2+\s_j^2)^{1/2}$.
\[construction\_btheta\] Assume that $m$ is the largest integer such that $2m\le n$. Then there is a set of vectors $\btheta$ such that $$\frac{\|\t_1-\t_2\|}{\s_{1,2}} = \ldots = \frac{\|\t_{2m-1}-\t_{2m}\|}{\s_{2m-1,2m}} = \k,$$ and for every pair $\{i,j\}$ different from the pairs $\{1,2\},\ldots,\{2m-1,2m\} $ we have $$\frac{\|\t_i-\t_j\|}{\s_{i,j}} > \k\bigg(1+\frac{\max_{1\le\ell\le n} \sigma_\ell}{\min_{1\le\ell\le n} \sigma_\ell}\bigg).$$
Denote $\btheta^0$ the constructed set. We define for every $k\in\{1,\ldots,m\}$, the set $\btheta^k=\btheta^0 + (0,\ldots,\eta_k,\eta_k,\ldots,0),$ where only the $(2k-1)^\text{th}$ and $2k^\text{th}$ components are modified by adding the vector $\eta_k=(\s_{2k}^2-\s_{2k-1}^2)(\t_{2k-1}-\t_{2k})/(2\s_{2k}^2+2\s_{2k-1}^2)$. It follows from Lemma \[construction\_btheta\] that $\btheta^0,\ldots,\btheta^m \text{ belong to } \bar\T_\k,$ so that, denoting for every $k\in\{1,\ldots,m\}, \pi_k=(2k-1~2k)$ the transposition of $\Sn$ that only permutes $2k-1$ and $2k$, and $\pi_0=\id,$ we get the following lower bound for the risk: $$\inf_{\hat\pi} \sup_{(\pi,\btheta)\in\Sn\times\bar\T_\k} \prob_{\btheta,\pi} (\hat\pi\neq\pi) \ge \inf_{\hat\pi} \max_{k=0,\ldots,m} \prob_{\btheta^{k},\pi_k}(\hat\pi\neq\pi_k).$$ In order to use Lemma \[Tsybakov\] with $\bQ_j=\prob_{\btheta^j,\pi_j}$, we compute $\forall k\in\{1,\ldots,m\}$ $$\begin{aligned}
2K(\prob_{\btheta^k,\pi_k},\prob_{\btheta^0,\pi_0}) = &\frac{\|\eta_k\|^2}{2\s_{2k-1}^2}+
\frac{\|\t_{2k-1}+\eta_k-\t_{2k}\|^2}{\s^2_{2k}} + d\Big(\frac{\s^2_{2k-1}}{\s^2_{2k}}-1\Big) \\
& +\frac{\|\eta_k\|^2}{2\s_{2k}^2}+\frac{\|\t_{2k}+\eta_k-\t_{2k-1}\|^2}{\s^2_{2k-1}} + d\Big(\frac{\s^2_{2k}}{\s^2_{2k-1}}-1\Big).\end{aligned}$$ Using the definition of $\eta_k$, we get $$\begin{aligned}
2K(\prob_{\btheta^k,\pi_k},\prob_{\btheta^0,\pi_0})
&= \frac{\|\t_{2k}-\t_{2k-1}\|^2}{\s_{2k}^2+\s_{2k-1}^2}
\bigg(\frac{3\s_{2k-1}^2}{8\s_{2k}^2}+\frac{3\s_{2k}^2}{8\s_{2k-1}^2}+\frac{10}{8} \bigg)
+ d \frac{\s^2_{2k-1}}{\s^2_{2k}} \Big( 1 - \frac{\s^2_{2k}}{\s^2_{2k-1}} \Big)^2 \\
&\le \k^2\bigg(\frac{13}{8}+\frac{3\s_{2k}^2}{8\s_{2k-1}^2}\bigg) + \frac{\log n}{16}.\end{aligned}$$ Next, we apply the following result.
Let $a_1,a_2,\ldots,a_m$ be real numbers larger than one such that $\prod_{k=1}^m a_k\le A$. Then, $\sum_{k=1}^m a_k \le n+\log A\max_k a_k$.
We use the simple inequality $e^x\le 1+xe^x$ for all $x\ge 0$. Replacing $x$ by $\log a_k$ and summing over $k=1,\ldots,m$ we get $$\sum_{k=1}^m a_k \le \sum_{k=1}^m 1+ a_k\log a_k\le m+ \max_{k=1,\ldots,m} a_k \log \prod_{k=1}^m a_k \le m+ \max_{k=1,\ldots,m} a_k \log A.$$ This completes the proof of the lemma.
We apply this lemma to $a_k = \s_{2k}^2/\s_{2k-1}^2$. Since the variances are sorted in increasing order, we have $\prod_{k=1}^m a_k \le \prod_{i=1}^{n-1} {\s_{i+1}^2}/{\s_i^2} = \s_n^2/\s_1^2\le 1+\frac14(\frac{\log n}{d})^{1/2}$. In conjunction with the inequality $\log(1+x)\le x$, this entails that $\sum_{k=1}^m \s_{2k}^2/\s_{2k-1}^2\le m+ \frac14(1+\frac14 (\frac{\log n}{d})^{1/2})(\frac{\log n}{d})^{1/2}$. Then, since $\log n\le 2\log m$ for $n\ge 6$ and $ (\frac{\log n}{d})^{1/2}\le ({\log n})^{1/2}\le \frac34\log n$, we get $$\begin{aligned}
\frac1m\sum_{k=1}^m K(\prob_{\btheta^k,\pi_k},\prob_{\btheta^0,\pi_0})
&\le\k^2\bigg(\frac{13}{16}+\frac{3}{16m}\sum_{k=1}^m\frac{\s_{2k}^2}{\s_{2k-1}^2}\bigg) + \frac{\log m}{16}\\
&\le\k^2\bigg(1+\frac{3}{64m}\Big\{1+\frac14 \Big(\frac{\log n}{d}\Big)^{1/2}\Big\}\Big(\frac{\log n}{d}\Big)^{1/2}\bigg) + \frac{\log m}{16}\\
&\le\k^2\Big(1+\frac{3}{64m}\Big) + \frac{\log m}{16}.\end{aligned}$$ Finally, using the fact that $m\ge 3$, we get $\frac1m\sum_{k=1}^m K(\prob_{\btheta^k,\pi_k},\prob_{\btheta^0,\pi_0}) \le
\frac{65}{64}\k^2+\frac{\log m}{16} \le \frac{\log m}{8}$ since $\k^2\le \frac2{65}\log n \le \frac4{65}\log m$. We conclude by Lemma \[Tsybakov\] and by the monotonicity of the function $m\mapsto \frac{\sqrt{m}}{1+\sqrt{m}}(\frac34-\frac1{2\sqrt{\log m}})$ that $$\inf_{\hat\pi} \sup_{(\pi,\btheta)\in\Sn\times\bar\T_\k} \prob_{\btheta,\pi} (\hat\pi\neq\pi) \ge \frac{\sqrt{3}}{\sqrt{3}+1} \Big( \frac{3}{4} - \frac{1}{2\sqrt{\log3}} \Big)\ge 0.17.$$
#### **Case 2:** We assume that $\frac18 \sqrt{\log n}\le \k \le \frac{1}{8} (cd\log n)^{1/4}$
In this case, we have $d\ge\frac{1}{c}\log n$. To get the desired result, we use Lemma \[Tsybakov\] for a properly chosen family of probability measures described below.
\[kullback\_leibler\_divergence\_heteroscedastic\] Let $\e_1,\ldots,\e_n$ be real numbers defined by $$\e_{{k}} = \sqrt{2/d}\;\k\s_{{k}},\qquad \forall{k}\in\{1,\ldots,n\},$$ and let $\mu$ be the uniform distribution on $\mathcal E=\{\pm\e_1\}^d\times\ldots\times\{\pm\e_{n}\}^d$. We denote by $\prob_{\mu,\pi}$ the probability measure on $\RR^{d\times n}$ defined by $\prob_{\mu,\pi}(A) = \int_{\mathcal E} \prob_{\btheta,\pi}(A)\,\mu(d\btheta)$. Assume that $\s_1\le\ldots\le \s_n$. For two positive integers $k<k'\le n$, set $\gamma = \frac{\s_{k'}^2}{\s_{k}^2}$ and let $\pi=({k}~{k}')$ be the transposition that only permutes ${k}$ and ${k}'$. Then $$\begin{aligned}
K(\prob_{\mu,\pi},\prob_{\mu,\id}) &\le {4\k^2}\big(1-\gamma^{-1}\big)+\frac{8\k^4}{d}\big(2+\big(1+(2/d)\k^2\big)^2\gamma^{2}\big)+ \frac12\big(d+2\k^2\big)(\gamma-1)^2\end{aligned}$$ and $\mu(\mathcal E\backslash\bar\T_\k) \le ({n(n-1)}/{2})\ e^{-d/8}$.
The assumption on the noise levels entails that, for any integer $k\in\{1,\ldots,k'\}$, $1\le \frac{\s_{k'}^2}{\s_{k}^2} \le 1+\frac14(\frac{\log n}{d})^{1/2}$, and consequently, $(\gamma-1)^2=\big(\frac{\s_{k'}^2}{\s_{k}^2} -1\big)^2\le 4^{-2}\big(\frac{\log n}{d}\big)$. Furthermore, $\frac{\k^2}{d}\le \frac{c}{64}\le \frac1{64}$ provided that $c\le 1$. Finally, for the Kullback-Leibler divergence between $\prob_{\mu,\pi_{k,k'}}$ and $\prob_{\mu,\id}$, where $\pi_{k,k'}=(k~k')$ is the transposition from $\Sn$ permuting only $k$ and $k'$, it holds $$\begin{aligned}
K(\prob_{\mu,\pi_{k,k'}},\prob_{\mu,\id}) &\le {\k^2}\sqrt{\frac{\log n}{d}}+\frac{8\k^4}{d}\Big(2+\frac{33^2}{32^2}\big(1+0.25\big)^2\Big)+
\frac{33d}{64}\times\frac{\log n}{16d}\\
&\le \frac{\log n}{8}\le \frac{\log n(n-1)/2}{8},\end{aligned}$$ where we have used once again the facts that $c\le 1$ and $n\ge 3$. Applying Lemma \[Tsybakov\] with $M = n(n-1)/2$, $\bQ_0 = \prob_{\mu,\id}$ and $\{\bQ_j\}_{j=1,\ldots,M}=\{\prob_{\mu,\pi_{k,k'}}\}_{k\not= k'}$, we obtain $$\begin{aligned}
\max_{\pi^*\in\Sn}\sup_{\btheta\in\bar\T_\k}\prob_{\btheta,\pi^*}\big(\hat\pi\neq\pi^*\big) &\ge \max_{\pi^*\in\{\id\}\cup\{\pi_{k,k'}\}}
\int_{\bar\T_\k} \prob_{\btheta,\pi^*} \big( \hat\pi \neq \pi^* \big) \frac{\mu(d\btheta)}{\mu(\bar\T_\k)} \\
&\ge \max_{\pi^*\in\{\id\}\cup\{\pi_{k,k'}\}} \prob_{\mu,\pi^*} \big( \hat\pi\neq\pi^*\big) - \mu(\mathcal E\backslash\bar\T_\k) \\
&\ge \frac{\sqrt{15}}{\sqrt{15}+1} \Big( \frac{3}{4} - \frac{1}{2\sqrt{\log15}} \Big) - \frac{n(n-1)}2 e^{-d/8}.\end{aligned}$$ In view of the inequalities $d\ge (1/c)\log n$, $c\le 1/20$ and $n\ge 6$, we get the inequality $\max_{\pi^*\in\Sn}\sup_{\btheta\in\bar\T_\k}\prob_{\btheta,\pi^*}\big(\hat\pi\neq\pi^*\big)\ge 22.4\%$.
Since the event $\{\pi^{\textrm{gr}}\not =\id\}$ includes the event $$\O_2=\{\|X_{1}-\Xdiese_1\|^2>\|X_{2}-\Xdiese_1\|^2\},$$ it is sufficient to bound from below the probability of $\O_2$. To this end, we choose any $\btheta\in\RR^{n\times d}$ satisfying $\|\t_1-\t_2\|=2\k$. This readily implies that $\btheta$ belongs to $\bar\T_\k$. Furthermore, for suitably chosen random variables $\eta_1\sim \chi^2_d$, $\eta_2\sim \chi^2_d$ and $\eta_3\sim\mathcal N(0,1)$, it holds that $\|X_{1}-\Xdiese_1\|^2-\|X_{2}-\Xdiese_1\|^2 = 6\eta_1 - 4\k^2-8\k \eta_3 - 4\eta_2$. The random terms in the last sum can be controlled using Lemma \[concentration\]. More precisely, for every $x>0$, each one of the following three inequalities holds true with probability at least $1-e^{-x^2}$: $$\begin{aligned}
\eta_1 \ge d- 2\sqrt{d} x,\quad
\eta_2 \le d+ 2\sqrt{d} x + 2x^2,\quad
\eta_3 \le \sqrt{2} x.\end{aligned}$$ This implies that with probability at least $1-3e^{-x^2}$, we have $$\|X_{1}-\Xdiese_1\|^2-\|X_{2}-\Xdiese_1\|^2 \ge 2d-20\sqrt{d} x-4(\k + \sqrt2 x)^2.$$ If $x=\sqrt{\log 6}$, then the conditions imposed on $\k$ and $d$ ensure that the right-hand side of the last inequality is positive. Therefore, $\prob(\O_2)\ge 1-3e^{-x^2}=1/2$.
The proof is split into two parts. In the first part, we consider the case $\k\le \frac14\sqrt{\frac{\log M_n}{n}}$, while in the second part the case $\kappa\le \frac18\big(\frac{\log n}{d}\big)^{1/4}$ with $d\ge 24\log n$ and is analyzed. In both cases, the main tool we use is the following result.
\[Tsybakov2\] Assume that for some integer $M\ge 2$ there exist distinct permutations $\pi_0,\ldots,\pi_M\in\Sn$ and mutually absolutely continuous probability measures $\bQ_0,\ldots,\bQ_M$ defined on a common probability space $(\mathcal Z,\mathscr Z)$ such that $$\begin{cases}
\ \exists\ s>0,\ \forall\ i\neq j, \ \d(\pi_i,\pi_j)\ge 2s, \\
\ \frac{1}{M} \sum_{j=1}^M K(\bQ_j,\bQ_0) \le \frac{1}{8} \log M.
\end{cases}$$ Then, for every measurable mapping $\tilde\pi:\mathcal Z\to\Sn$, $$\max_{j=0,\ldots,M} \bQ_j\big(\d(\hat\pi,\pi_j)\ge s\big) \ge \frac{\sqrt{M}}{\sqrt{M}+1} \Big( \frac{3}{4}-\frac{1}{2\sqrt{\log M}} \Big).$$
We now have to choose $M$ and $\pi_0,\ldots,\pi_M$ in a suitable manner, which will be done differently according to the relationship between $n$ and $d$.
#### **Case 1:** We assume that $\k\le \frac14\sqrt{\frac{\log M_n}{n}}$
Let $M = M_n$ with $M_n= \mathcal M(1/4, B_{2,n}(2),\delta_H)$ and let $\btheta = (\theta_1,\ldots,\theta_n)$ be the set of vectors $\theta_k = k\k\s(1,0,\ldots,0)\in\RR^d$. Clearly, $\btheta$ belongs to $\bar\T_\k$. By definition of the packing number, there exist $\pi_1,\ldots,\pi_{M_n}$, permutations from $\Sn$, such that $$\delta_2(\pi_j,\id)\le 2,\qquad \delta_H(\pi_i,\pi_j)\ge \frac14;\qquad\forall i,j\in\{1,\ldots,M_n\}, \ i\not=j.$$ Defining $\bQ_j=\prob_{\btheta,\pi_j}$ for $j=1,\ldots,M_n$ and $\bQ_0=\prob_{\btheta,\id}$ we get $$\begin{aligned}
K(\bQ_j,\bQ_0)
&= \frac1{2\s^2} \sum_{k=1}^n \|\theta_{\pi_j(k)}-\theta_{k}\|^2 = \frac{\k^2}{2}\sum_{k=1}^n (\pi_j(k)-k)^2\\
&= \frac{n\k^2}{2}\;\delta_2(\pi_j,\id)^2\le 2n\k^2.\end{aligned}$$ Therefore, using Lemma \[Tsybakov2\] with $s=1/8$ we infer from $\k\le \frac14\big(\frac{\log M_n}{n}\big)^{1/2}$ that $$\min_{\hat\pi}\max_{j=0,\ldots,M_n} \prob_{\btheta,\pi_j}(\delta_H(\hat\pi,\pi_j)\ge 1/8) \ge \frac{\sqrt{3}}{\sqrt{3}+1}\Big(\frac34-\frac1{2\sqrt{\log 3}}\Big) \approx 17.31\%.$$ As a consequence, we obtain that $$\begin{aligned}
\min_{\hat\pi}\max_{(\pi,\btheta)\in\Sn\times\bar\T_\k} \esp_{\btheta,\pi}[\delta_H(\hat\pi,\pi)]
&\ge \min_{\hat\pi}\max_{j=0,\ldots,M} \esp_{\btheta,\pi}[\delta_H(\hat\pi,\pi)\fcar_{\{\delta_H(\hat\pi,\pi)\ge 1/8\}}]\\
&\ge \frac18 \min_{\hat\pi}\max_{j=0,\ldots,M} \esp_{\btheta,\pi}[\fcar_{\{\delta_H(\hat\pi,\pi)\ge 1/8\}}]\\
&\ge 2.15\%.\end{aligned}$$ This completes the proof of the first case.
#### **Case 2:** We assume that $d\ge 24\log n$ and $\k \le \frac18 (d\log n)^{1/4}$
Let $\mu$ be the uniform distribution on $ \{\pm\e\}^{m\times d}$ with $\e={\sqrt{2/d}\;\s\k}$, as in Lemma \[kullback\_leibler\_divergence\_heteroscedastic\]. For any set of permutations $\{\pi_0,\ldots,\pi_M\}\subset\Sn$, in view of Markov’s inequality, $$\begin{aligned}
\sup_{(\pi,\btheta)\in\Sn\times\bar\T_\k} \esp_{\btheta,\pi} \big[\delta_H(\hat\pi,\pi)\big] &\ge
\frac{3}{16} \Big( \max_{i=0,\ldots,M} \prob_{\mu,\pi_i} \Big(\delta_H(\hat\pi,\pi_i) \ge \frac{3}{16} \Big) - \mu(\bar\T_\k^\complement)\Big).\end{aligned}$$ We choose $M$ and $\pi_0,\ldots,\pi_M$ as in the following lemma.
\[existence2\] For any integer $n\ge 4$ there exist permutations $\pi_0,\ldots,\pi_M\in\Sn$ such that $$\pi_0=\id,\qquad\ M\ge(n/24)^{n/6},$$ each $\pi_i$ is a composition of at most $n/2$ transpositions with disjoint supports, and for every distinct pair of indices $i,j\in\{0,\ldots,M\}$ we have $$\delta_H(\pi_i, \pi_j) \ge {3}/{8}.$$
As $\pi_i$ is a product of transpositions, the Kullback-Leibler divergence between $\prob_{\mu,\pi_i}$ and $\prob_{\mu,\pi_0}$ can be computed by independence thanks to Lemma \[kullback\_leibler\_divergence\_heteroscedastic\]: $$\frac1M\sum_{i=1}^M K(\prob_{\mu,\pi_i},\prob_{\mu,\pi_0}) \le \frac{n}{2} \times \frac{8\k^4}{d}\Big(2+\Big[1+\frac{2\k^2}{d}\Big]^2\Big) = \frac{16n\k^4}{d},$$ where the last inequality follows from the bound $\k\le 0.45 d^{1/2}$. For $n\ge 26$, it holds that $M\ge 2$ and $$\log M\ge \frac{n(\log n -\log 24)}{6}\ge \frac{\log n}{512}.$$ Consequently, $\frac{16n\k^4}{d}\le \frac18\log M$ which allows us to apply Lemma \[Tsybakov2\]. This yields $$\begin{aligned}
\inf_{\hat\pi} \sup_{(\pi,\btheta)\in\Sn\times\bar\T_\k} \esp_{\btheta,\pi} [\delta_H(\hat\pi,\pi)]
&\ge \frac{3}{16}\Big[\frac{\sqrt{2}}{\sqrt{2}+1} \Big(\frac{3}{4}-\frac{1}{2\sqrt{\log 2}} \Big) - \frac{n^2}{2}e^{-d/8}\Big]\\
&\ge \frac{3}{16}\Big[0.077-\frac{n^2}{2}e^{-24\log(n)/8}\Big]\\
&\ge \frac{3}{16}\Big[0.077-\frac{1}{2n}\Big]\ge 5.81\%.\end{aligned}$$
Proofs of the Lemmas {#section:lemmas}
====================
Let us denote $r_\sigma =\max_{1\le\ell\le n} \sigma_\ell/\min_{1\le\ell\le n} \sigma_\ell$. It suffices to set $\theta_1=0\in\RR^d$, $$\begin{aligned}
\theta_{2k+1}&=\kappa\Big(\s_{1,2}+\ldots+\s_{2k-1,2k}+k\big(1+r_\s\big),0,\ldots,0\Big)\in\RR^d,\\
\theta_{2k}&=\kappa\Big(\s_{1,2}+\ldots+\s_{2k-1,2k}+(k-1)\big(1+r_\s\big),0,\ldots,0\Big)\in\RR^d\end{aligned}$$ for all $k=1,\ldots,m-1$. If $n$ is impair, one can set $\theta_n=\theta_{n-1}+\k \big(1+r_\s\big) (1,0,\ldots,0)$. One readily checks that these vectors satisfy the desired conditions.
Without loss of generality, we assume hereafter that $\pi\in\Sn$ is the transposition permuting $1$ and $2$. Recall that the uniform distribution on $\{\pm\e_1\}^d\times\ldots\times\{\pm\e_{n}\}^d$ can also be written as the product $\mu = \bigotimes_{\ell=1}^n\mu_\ell$, where $\mu_\ell$ is the uniform distribution on $\{\pm\e_{\ell}\}^d$. Let us introduce an auxiliary probability distribution $\tilde\mu$ on $\RR^{n\times d}$ defined as $\tilde\mu = \delta_{\mathbf 0}\otimes\delta_{\mathbf 0}
\otimes \mu_2\otimes\ldots\otimes\mu_m$ with $\delta_{\mathbf 0}$ being the Dirac delta measure at $\mathbf 0\in\RR^d$. We set $\prob_{\tilde\mu,\id}(\cdot)= \int_{\Theta} \prob_{\btheta,\id}(\cdot)\tilde\mu(d\btheta)$.
We first compute the density of $\prob_{\mu,\pi}$ w.r.t. $\prob_{\mu,\id}$, which can be written as $$\begin{aligned}
\frac{d\prob_{\mu,\pi}}{d\prob_{\mu,\id}}(\bX,\bXdiese) = \frac{d\prob_{\mu,\pi}}{d\prob_{\tilde\mu,\id}}(\bX,\bXdiese) \bigg/ \bigg(\frac{d\prob_{\mu,\id}}{d\prob_{\tilde\mu,\id}}(\bX,\bXdiese)\bigg),\quad \bX,\bXdiese\in \RR^{n\times d}.\end{aligned}$$ For every $\theta_i\in\RR^d$ we denote by $\prob_{\theta_i,\s_i}$ the probability distribution of $X_i$ from (\[model\]), given by $$\frac{d\prob_{\t_i,\s_i}}{d\prob_{0,\s_j}}(x) = \exp\Big\{-\frac{\|\t_i\|^2}{2\s_i^2} + \frac{1}{\s_i^2}(x,\t_i)-\frac{\|x\|^2}{2}(\sigma_i^{-2}-\s_j^{-2})\Big\},\quad \forall x\in\RR^d.$$ With this notation, we have $$\begin{aligned}
&\frac{d\prob_{\mu,\pi}}{d\prob_{\tilde\mu,\id}}(\bX,\bXdiese) = \esp_{\mu} \Bigg[\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_1}}(X_1)\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_2}}(X_2)\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_1}}(\Xdiese_1)\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_2}}(\Xdiese_2)\Bigg] \\
&= \esp_{\mu} \Bigg[\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_1}}(X_1)\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_2}}(\Xdiese_2)\Bigg] \times \esp_{\mu}\Bigg[\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_2}}(X_2)\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_1}}(\Xdiese_1)\Bigg]\\
&= \prod_{k=1}^d\cosh\Big(\frac{\e_1}{\s_1^2}(X_{1,k}+\Xdiese_{2,k})\Big)\cosh\Big(\frac{\e_2}{\s_2^2}(X_{2,k}+\Xdiese_{1,k})\Big)\\
&\qquad\times\exp\Big\{-\frac12(\|\Xdiese_{1}\|^2-\|\Xdiese_{2}\|^2)(\s_2^{-2}-\s_1^{-2})\Big\}.\end{aligned}$$ Similarly, $$\begin{aligned}
\frac{d\prob_{\mu,\id}}{d\prob_{\tilde\mu,\id}}&(\bX,\bXdiese) =\esp_{\mu} \Bigg[\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_1}}(X_1)\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_2}}(X_2)\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_1}}(\Xdiese_1)
\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_2}}(\Xdiese_2)\Bigg] \\
&= \esp_{\mu} \Bigg[\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_1}}(X_1)\frac{d\prob_{\t_1,\s_1}}{d\prob_{0,\s_1}}(\Xdiese_1)\Bigg] \times \esp_{\mu}\Bigg[\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_2}}(X_2)\frac{d\prob_{\t_2,\s_2}}{d\prob_{0,\s_2}}(\Xdiese_2)\Bigg]\\
&= \prod_{k=1}^d\cosh\Big(\frac{\e_1}{\s_1^2}(X_{1,k}+\Xdiese_{1,k})\Big)\cosh\Big(\frac{\e_2}{\s_2^2}(X_{2,k}+\Xdiese_{2,k})\Big).\end{aligned}$$ Thus, we get that $$\begin{aligned}
\frac{d\prob_{\mu,\pi}}{d\prob_{\mu,\id}}(\bX,\bXdiese) &= \prod_{k=1}^d \frac{\cosh\big(\frac{\e_1}{\s_1^2}(X_{1,k}+\Xdiese_{2,k})\big)}{\cosh\big(\frac{\e_1}{\s^2_1}(X_{1,k}+\Xdiese_{1,k})\big)}
\times \prod_{k=1}^d \frac{\cosh\big(\frac{\e_2}{\s_2^2}(X_{2,k}+\Xdiese_{1,k})\big)}{\cosh\big(\frac{\e_2}{\s^2_2}(X_{2,k}+\Xdiese_{2,k})\big)}\\
&\qquad\times\exp\Big\{-\frac12(\|\Xdiese_{1}\|^2-\|\Xdiese_{2}\|^2)(\s_2^{-2}-\s_1^{-2})\Big\}.\end{aligned}$$ Then, we compute the Kullback-Leibler divergence, $$\begin{aligned}
K&(\prob_{\mu,\pi},\prob_{\mu,\id})
= \int \log \Big(\frac{d\prob_{\mu,\pi}}{d\prob_{\mu,\id}}(\bX,\bXdiese)\Big) d\prob_{\mu,\pi}(\bX,\bXdiese)\nonumber\\
& = \sum_{k=1}^d\sum_{j=1}^2 \bigg\{\esp_{\mu} \bigg[ \int \log\cosh\Big[ \frac{\e_j}{\s^2_j}(2\t_{j,k}+\s_j\sqrt{2}x)\Big] \varphi(x)dx \bigg] \nonumber\\
& \qquad - \esp_{\mu}\bigg[ \int \log\cosh \Big[\frac{\e_j}{\s^2_j}(\t_{1,k}+\t_{2,k}+\s_{12}x)\Big]\varphi(x) dx \bigg]\bigg\}\nonumber\\
&\qquad +\frac{d}2\esp_{\mu}\bigg[ \int_\RR ((\t_{1,1}+\s_1 x)^2-(\t_{2,1}+\s_2 x)^2)\varphi(x)\,dx\bigg](\s_2^{-2}-\s_1^{-2}),\nonumber\end{aligned}$$ where $\varphi$ is the density function of the standard Gaussian distribution. We evaluate the first two terms of the last display using the following inequalities: $$\label{cosh}
\forall u\in\RR,\qquad
\frac{u^2}{2}-\frac{u^4}{12} \le \log\cosh(u) \le \frac{u^2}{2},$$ while for the third term the exact computation yields: $$\begin{aligned}
\esp_{\mu}\bigg[ \int_\RR ((\t_{1,1}+\s_1 x)^2-(\t_{2,1}+\s_2 x)^2)\varphi(x)\,dx\bigg]&=
\e_1^2+\s_1^2-\e_2^2-\s_2^2\\
&= (\s_1^2-\s_2^2)(1+(2/d)\k^2).\end{aligned}$$ In conjunction with the facts that $\e_1/\s_1=\e_2/\s_2$, $\s_1\le\s_2$ and $\e_1\le\e_2$, this leads to $$\begin{aligned}
&\esp_{\mu} \bigg[ \int \log\cosh\Big[ \frac{\e_j}{\s_j^2}(2\t_{j,k}+\s_j\sqrt{2}x)\Big] \varphi(x)dx \le
\frac{\e_j^2}{\s_j^2} + 2\frac{\e_j^4}{\s_j^4}=\frac{\e_2^2}{\s_2^2} + 2\frac{\e_2^4}{\s_2^4},\\
&\esp_{\mu}\bigg[ \int \log\cosh \Big[\frac{\e_j}{\s_j^2}(\t_{1,k}+\t_{2,k}+\s_{1,2}x)\Big]\varphi(x) dx \bigg]\\
&\qquad\ge\frac{\e^2_j}{2\s^4_j}\Big(\e_1^2+\e_2^2+\s_1^2+\s_2^2\Big)\\
&\qquad\qquad- \frac{\e^4_j}{12\s^8_j}\Big(\e_1^4+\e_2^4+3(\s_1^2+\s_2^2)^2+6\e_1^2\e_2^2+6(\s_1^2+\s_2^2)(\e_1^2+\e_2^2)\Big)\\
&\qquad\ge\frac{\e^2_2(\e_1^2+\s_1^2)}{\s^4_2}- \frac{\e^4_1(\e_2^2+\s_2^2)^2}{\s^8_1}.\end{aligned}$$ Thus, we get that $$\begin{aligned}
(1/d)K(\prob_{\mu,\pi},\prob_{\mu,\id}) &\le \frac{2\e_2^2}{\s_2^2} + \frac{4\e_2^4}{\s_2^4}-\frac{2\e^2_2(\e_1^2+\s_1^2)}{\s^4_2}+\frac{2\e^4_1(\e_2^2+\s_2^2)^2}{\s^8_1}\\
&\qquad+ \frac12\big(1+(2/d)\k^2)(\s_1^2-\s_2^2)(\s_2^{-2}-\s_1^{-2})\\
&\le \frac{4\k^2}{d}\Big(1-\frac{\s_1^2}{\s_2^2}\Big)+\frac{16\k^4}{d^2}+\frac{8\k^4}{d^2}\big(1+(2/d)\k^2\big)^2\frac{\s_2^4}{\s_1^4}\\
&\qquad+ \frac12\big(1+(2/d)\k^2\big)\Big(\frac{\s_2^2}{\s_1^2}-1\Big)^2.\end{aligned}$$ To complete the proof, we need to evaluate $\mu(\mathcal E\setminus\bar\T_\k)$. We note that in view of the union bound, $$\begin{aligned}
\mu(\mathcal E\setminus\bar\T_\k) & = \mu\Big(\bigcup_{k=1}^n\bigcup_{k'\not=k}\{\btheta : \|\theta_{k}-\theta_{k'}\| < \k\s_{k,k'} \}\Big)\\
&\le \frac{n(n-1)}{2} \max_{k\not=k'}\mu\big(\{\btheta : \|\theta_{k}-\theta_{k'}\|^2 < \k^2\s_{k,k'}\}\big)\\
&= \frac{n(n-1)}{2} \max_{k\not=k'}\prob\big(d\e_{k}^2+d\e_{k'}^2-2d\e_{k}\e_{k'}\bar\zeta < \k^2\s_{k,k'}^2 \big),\end{aligned}$$ where $\bar\zeta=\frac1d\sum_{j=1}^d \zeta_j$ with $\zeta_1,\ldots,\zeta_d$ being i.i.d. Rademacher random variables (*i.e.*, random variables taking the values $+1$ and $-1$ with probability $1/2$). One easily checks that $$\frac{d\e_{k}^2+d\e_{k'}^2-\k^2\s_{k,k'}^2}{2d\e_{k}\e_{k'}} = \frac{2\s_{k}^2+2\s_{k'}^2-(\s_{k}^2+\s_{k'}^2)}{4\s_{k}\s_{k'}}\ge \frac12.$$ Therefore, using the Hoeffding inequality, we get $\mu(\mathcal E\setminus\bar\T_\k) \le \frac12n(n-1) \prob\big(\bar\zeta > 1/2 \big)\le \frac12n(n-1) e^{-d/8}$.
We first prove an auxiliary result.
\[existence\] For any integer $n\ge 2$ there exist permutations $\pi_0,\pi_1,\ldots,\pi_M$ in $\Sn$ such that $$\pi_0=\id,\qquad\ M \ge (n/8)^{n/2}$$ and for any pair $i,j\in\{0,\ldots,M\}$ of distinct indices we have $\delta_H(\pi_i, \pi_j) \ge \frac{1}{2}$.
When $n\le 8$, the claim of this lemma is trivial since one can always find at least one permutation that differs from the identity at all the positions and thus $M\ge 1\ge (n/8)^{n/2}$. Let us consider the case $n>8$. For every $\pi\in\Sn$, denote $$E_\pi \triangleq \Big\{ \pi'\in\Sn \suchthat \frac1n \sum_{i=1}^n \fcar_{\{\pi(i)\neq \pi'(i)\}} \ge 1/2 \Big\}.$$ We first notice that for every $\pi\in\Sn$, there is a one-to-one correspondence between $E_{id}$ and $E_\pi$ through the bijection $${\begin{array}{l|rcl}
\phi: & E_{id} & \longrightarrow & E_\pi\phantom{\Big()} \\
& \pi' & \longmapsto & \pi\circ\pi' \end{array}},$$ so that $\# E_\pi = \# E_{id}$. The following lemma, proved later on in this section, gives a bound for this number.
\[cardinalEid\] Let $n\ge 2$ be an integer and $m$ be the smallest integer such that $2m\ge n$. Then $$\# E_{id}^\complement \le \frac{4n!}{m!}.$$
Now we denote $\pi_0=id$ and choose $\pi_1$ in $E_{id}$. Then, it is sufficient to choose $\pi_2$ as any element from $E_{id}\cap E_{\pi_1}$, the latter set being nonempty since $$\begin{aligned}
\#\big(E_{\pi_0}\cap E_{\pi_1}\big) &\ge \# \Sn - \# E_{\pi_0}^\complement - \# E_{\pi_1}^\complement \\
&\ge n!\times\Big(1-\frac{8}{m!}\Big) > 0.\end{aligned}$$ We can continue the construction until $\pi_i$ if $$1-\frac{4i}{m!} > 0\qquad \ssi\qquad i < \frac{m!}{4}.$$ To conclude, we observe that $$\frac{m!}{4} > \frac14 \Big( \frac{n}{2e} \Big)^{n/2} \ge \Big( \frac{n}{8} \Big)^{n/2}.$$
Let us denote by $m$ the largest integer such that $2m\le n$, and choose $$\tilde\pi_0,\tilde\pi_1,\ldots,\tilde\pi_M\in\mathfrak{S}_m \quad \text{ with } \quad M\ge(m/8)^{m/2}$$ as in Lemma \[existence\], so that for every $i\neq j\in\{0,\ldots,M\}$, $\delta_H(\tilde\pi_i,\tilde\pi_j) \ge \frac{1}{2}$. We use each permutation $\tilde\pi_i\in\mathfrak{S}_m$ to construct a permutation $\pi_i\in\Sn$. The idea of the construction is as follows: the permutation $\pi_i$ is a product of $m$ transpositions of distinct supports, and each transposition permutes an even integer with an odd one. We set $\pi_0=id$ and for every $i$ in $\{1,\ldots,M\}$, $$\pi_i = \big(1 \ 2\tilde\pi_i(1)\big) \circ \big(3 \ 2\tilde\pi_i(2)\big) \circ \ldots \circ \big(2m-1 \ 2\tilde\pi_i(m)\big) \in \Sn.$$ With these choices, the number of differences between $\pi_i$ and $\pi_j$ is exactly twice as much as the number of differences between $\tilde\pi_i$ and $\tilde\pi_j$. To sum up, for every pair of distinct indices $i,j\in\{0,\ldots,M\}$, $$\frac1n \sum_{k=1}^n \fcar_{\{\pi_i(k)\neq \pi_j(k)\}} = \frac{2m}{n} \times \frac1m \sum_{k=1}^m \fcar_{\{\tilde\pi_i(k)\neq \tilde\pi_j(k)\}} \ge \frac{m}{n}\ge\frac{3}{8},\qquad \forall n\ge 4.$$ To complete the proof, we note that $m\ge n/3$.
For every $\ell\in\{m,\ldots,n\}$, counting all the permutations $\pi$ such that $\sum_{k=1}^n \fcar_{\{\pi(k)\neq k\}}=\ell$, we get $$\# E_{id} = !n + !(n-1)\binom{n}{1} + \ldots + !(n-m)\binom{n}{m} ,$$ where $!\ell$ is the number of derangements, the permutations such that none of the elements appear in their original position, in $\mathfrak{S}_\ell$ for $\ell\ge 1$. We know that $$\forall \ \ell\ge1, \qquad !\ell = \ell! \times \sum_{j=0}^\ell \frac{(-1)^j}{j!},$$ which, using the alternating series test, yields $$\forall \ \ell\ge1, \qquad !\ell \ge \ell !\times\Big(e^{-1}-\frac{1}{(\ell+1)!}\Big).$$ It follows that $$\begin{aligned}
\# E_{id} &\ge n!\times\Big(e^{-1}-\frac{1}{(n-m+1)!}\Big)\times\Big( 1 + \frac{1}{1!} + \ldots + \frac{1}{m!} \Big) \\
&\ge n!\times\Big(e^{-1}-\frac{1}{(n-m+1)!}\Big)\times\Big(e-\frac{e}{(m+1)!}\Big)\\
&\ge n!\times\Big(1-\frac{e}{(n-m+1)!} - \frac{1}{(m+1)!} \Big).\end{aligned}$$ Therefore, $$\# E_{id}^\complement \le n!\times\Big( \frac{e}{(n-m+1)!} + \frac{1}{(m+1)!} \Big) \le \frac{4n!}{m!}.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the grants Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047) and CALLISTO. The authors thank the Reviewers for many valuable suggestions. Special thanks to an anonymous Referee for pointing a mistake in the proof of Theorem \[lowerbound2\].
[^1]: These assumptions are imposed for the purpose of getting transparent theoretical results and are in no way necessary for the validity of the considered estimation procedures, as discussed later in the paper.
[^2]: It should be noted that the conditions of Theorem \[lowerbound3\] are not compatible with those of Theorem \[lowerbound2\]. Hence, strictly speaking, the former does not imply that the greedy estimator is not minimax under the conditions of the latter.
[^3]: The idea of reducing the problem of permutation estimation to a linear program has been already used in the literature, however, without sufficient theoretical justification: see, for instance, @Jebara2003.
|
---
abstract: 'The optical emittance of a hole pair perforated in an opaque metal film is studied from first-principles using the coupled-mode method. The geometrical simplicity of this system helps to understand the fundamental role played by surface plasmon polaritons (SPPs) in its optical response. A SPP interference model without fitting parameters is developed from the rigorous solution of Maxwell’s equations. The calculations show that the interference pattern of the hole pair is determined by two scattering mechanisms: (i) re-illumination of the holes by the in-plane SPP radiation and (ii) an effective impedance depending on the single-hole response. The conditions for constructive and destructive interference only depend on the phase difference provided by each of the two scattering mechanisms.'
author:
- 'F. de León-Pérez'
- 'F.J. García-Vidal'
- 'L. Martín-Moreno'
title: The role of surface plasmon polaritons in the optical response of a hole pair
---
Introduction
============
The extraordinary transmission through nanohole arrays milled into metallic films [@EbbesenN98] is attributed to the resonant excitation of surface electromagnetic (EM) modes by the incident light [@LMMPRL01]. In the optical regime, these surface EM modes are surface plasmon polaritons (SPPs), modified by the metal corrugation. The light-SPP coupling is made possible by the additional grating momentum provided by the scattering of the incident light by the hole array. Nevertheless, interference of excited SPPs is set up even for two interacting holes [@SoenninchensenAPL00; @SchoutenPRL05; @LalannePRL05; @AlarverdyanNP07; @PacificiOE08]. Increasing the number of holes, the transmission is enhanced due to better-defined peaks of the structure factor, appearing at the reciprocal lattice vectors [@BravoPRL04]. It must also be noted that light-SPP interaction is not the single mechanism behind the extraordinary optical transmission (EOT). The EOT physical scenario is completed by the excitation of localized and Fabry-Perot modes [@WannemacherOC01; @AbajoOE02; @Degiron200461; @FJPRL05; @MolenPRB05; @PopovAO05; @FJPRB06; @RindzeviciusJPC07], which may also contribute the whole process (see [@FJRMP10] for a comprehensive review).
The aim of the present paper is to study the interference pattern of the simplest interacting system: a hole pair. Since the original proposition of the “nanogolf” effect by Sönninchsen [*et al.*]{} [@SoenninchensenAPL00], several groups have measured the optical interaction of two holes, see for example [@AlarverdyanNP07; @PacificiOE08; @AlegretNJP08]. These groups have used basic SPP resonant models in order to explain the characteristic optical transmittance of the hole dimer, which oscillates as function of the hole-hole distance, with period equal to the SPP wavelength. These approaches have in common that the relevant scattering channels are assumed [*ad hoc*]{}: only SPP scattering channels are included in the final optical response.
In this paper we make no such assumption and solve Maxwell’s equations from first-principles using a coupled-mode method (CMM) [@FJRMP10; @FdLPNJP08]. We shall consider the two possible radiative channels: freely propagating light radiated out-of-plane into the far field, and SPP power scattered along the metal plane. The out-of-plane power $P_{\rm rad}$, normalized to the power incident on the hole area, gives the far field transmittance $T$. This is the quantity commonly used to characterize EOT. However, to the best of our knowledge, the in-plane SPP power $P_{spp}$ has not yet been measured for a hole pair. We shall analyze the relevant scattering mechanisms for each radiative channel. Moreover, we shall derive, without fitting parameters, the conditions for constructive and destructive interference, hereafter conditions for interference (CI), that explain experimental interference patterns [@AlarverdyanNP07; @AlegretNJP08].
The paper is organized as follows. In the next section we briefly review the CMM and give the expressions for $P_{spp}$ and $P_{\rm rad}$. The assumptions behind the CMM and some cumbersome mathematical formula are reported in the Appendix. For the sake of completeness, section \[sec:singlehole\] summaries the emittance of a single hole. Section \[sec:holepair\] discusses the optical response of the hole pair. A sub-section is devoted to clarify the scattering mechanisms dominating the conditions for interference. At the end, we outline the main conclusions of the paper.
![(Color). Schematic representation of the hole-pair geometry.[]{data-label="fig:scheme"}](Fig1.eps){height="3cm"}
Theoretical framework {#sec:me}
=====================
Figure \[fig:scheme\] renders the hole-pair geometry studied in this paper. Two identical circular holes of radius $r_{h}$, separated by a distance $R$, are milled into an infinite metal film of thickness $h$ and dielectric function $\epsilon_{\rm m}$. In general, the metal film lays on a substrate with dielectric constant $\epsilon$, it is covered with a dielectric superstrate $\epsilon_1$, and the space inside the holes is characterized by a dielectric constant $\epsilon_2$. For the sake of simplicity, $\epsilon=\epsilon_1=\epsilon_2=1$ is used along this paper. We consider in what follows that the metal film is illuminated by a normal-incident p-polarized plane wave, oriented along the main axis of the hole pair, as shown in Fig. \[fig:scheme\]. We shall focus on the energy power radiated into the transmission region ($z>0$).
Maxwell’s equations are solved self-consistently using a convenient representation for the EM fields [@FJRMP10; @FdLPNJP08]. In both substrate and superstrate the fields are expanded into an infinite set of plane waves with both p- and s-polarizations. Inside the holes the most natural basis is a set of circular waveguide modes. Convergence is fast achieved with a small number of such modes [@baudrionOE08; @przybillaOE08]. In fact, we shall see that the fundamental waveguide mode is a good approximation for our problem. The assumptions behind this coupled-mode method, as well as its relevant constitutive quantities, are briefly review in the Appendix under the single mode approximation.
The flux power traversing the hole is distributed into two channels [@FdLPNJP08]: (i) out-of-plane radiation, freely propagating into the far-field, and (ii) SPP power, scattered along the metal plane. The calculation of these two quantities is straightforward within the CMM after we know the amplitude of the fundamental waveguide mode at the hole openings $E'_i$, where $i=1,2$ labels each hole. For a normal-incident plane wave, both holes receive the same illumination $I$ (\[eq:I\]), therefore $E'_1=E'_2 \equiv E'$ due to the symmetry of the system with respect to the central point of the hole pair; $E'$ hence reads $$\begin{aligned}
\label{eq:Ep}
E'=\frac{G_{\rm \nu} I}{\left[G_{\rm sh}+G_{hh}(R)-\Sigma\right]^2-G^2_\nu},\end{aligned}$$ where the hole-hole propagator $G_{hh}(R)$ (\[eq:green\]) represents the coupling of the two holes as a function of the hole-hole distance $R$. This interaction can be seen as a re-illumination of the hole $i$ by the magnetic field $G_{hh}E_j$ radiated from the other hole $j$. Notice that there is also a self-illumination term for each hole, $G_{\rm sh}$, which adds to the single-hole scattering mechanisms $\Sigma$ (\[eq:S\]) and $G_{\rm \nu}$ (\[eq:Gnu\]). Using Eq. (\[eq:Ep\]), the out-of-plane power emitted by the hole pair simplifies to $$\begin{aligned}
\label{eq:Prad}
P_{\rm rad}(R)=\vert E' \vert^2 g_{\rm rad}(R),\end{aligned}$$ where the propagator $g_{\rm rad}(R)=g^{\rm sh}_{\rm rad}+g^{\rm int}_{\rm rad}(R)$ provides the far field radiated from the hole pair, $g^{\rm sh}_{\rm rad}$ represents the contribution of each single hole, and $g^{\rm int}_{\rm rad}(R)$ (\[eq:greenR\]) is a term arising from the interference of the fields radiated by the two holes.
On the other hand, we can obtain the power radiated into SPPs by computing the contribution from the plasmon pole in the propagator [@FdLPNJP08]. The power of the scattered SPPs is first computed, at a point $r$ on the metal surface several SPP wavelengths away from the nearest edge of the hole pair, by integrating the in-plane radial component of the Poynting vector, defined with the SPP fields, on a cylindrical surface of radius $r$ and semi-infinity extension in $z>0$; the power in the plasmon wave is then calculated using the known decay length of the SPP. The integrated power reads $$\begin{aligned}
\label{eq:Pspp}
P_{spp}(R)= \vert E' \vert^2 g_{spp}(R),\end{aligned}$$ where the propagator $g_{spp}(R)=g^{\rm sh}_{spp}+g^{\rm int}_{spp}(R)$ provides the total SPP field radially scattered along all possible angular directions in the metal plane, $g^{\rm sh}_{spp}$ (\[eq:Gspp0\]) represents the contribution of each single hole, and $g^{\rm int}_{spp}(R)$ (\[eq:gspp\]) is the interference term
The conservation of the energy flux for a lossless metal (\[eq:efc\]), imposes a constrain to the real part of the full interaction propagator $G(R)=G_{\rm sh}+G_{hh}(R)$, which fulfills $\mbox{Re}[G(R)]=g_{\rm rad}+ g_{spp}$, see Appendix. We shall see that this relation is a good approximation for lossy metals at optical frequencies, and we shall use it in section \[sec:holepair\]. However, the hole-hole interaction can not be fully understood without a previous knowledge of the optical response of a single hole, which is briefly reviewed in the next section.
Single hole emittance {#sec:singlehole}
=====================
![Normalized-to-hole-area out-of-plane ($P_{\rm rad}$) and in-plane SPP ($P_{spp}$) emittance as function of the hole radius $r_{h}$ (in nm) for a single hole milled in a silver film, free standing on air ($h=250$ nm and $\lambda=700$ nm). Symbols and lines represent converged results and the single mode approximation, respectively. The inset show the cutoff wavelength, $\lambda_{\rm c}$ (in nm). []{data-label="fig:single_hole"}](Fig2.eps){height="6.5cm"}
The emittance spectrum of a single circular hole is described in this section for the sake of completeness, although this issue have been largely study, see for example [@SoenninchensenAPL00; @WannemacherOC01; @AbajoOE02; @PopovAO05; @FJPRB06; @AlegretNJP08; @FdLPNJP08; @baudrionOE08; @przybillaOE08; @ChangOE05] and references therein. The behavior of both $P_{\rm rad}$ and $P_{spp}$ is depicted in Fig. \[fig:single\_hole\] as function of the hole radius, for a free-standing Ag film with $h=250$ nm. The Ag dielectric function $\epsilon_{\rm m}(\lambda)$ is fitted to Palik’s data [@Palik]. It is equal to $\epsilon_{\rm m}=-19.9+1.15 \: i$ for the incident wavelength, $\lambda=700$ nm, which is kept constant along the paper. Both $P_{\rm rad}$ and $P_{spp}$ are normalized to the power incident on the hole area.
Fig. \[fig:single\_hole\] renders $P_{spp}$ for both the fundamental mode approximation (dashed line) and converged results (open circles). Both curves practically overlap, so the fundamental mode is enough to achieve converged results for this emittance channel. For the out-of-plane emittance the agreement between single mode (solid line) and full calculations (full circles) is slightly worse, but still the difference is less than 15% and tendencies are well captured in the parameter range considered.
As already stressed in Ref. [@baudrionOE08], $P_{spp}(r_{h})$ presents a broad peak with maximum at $r_{h}=190$ nm, close to the cutoff radius, $r_{\rm c}=168$ nm for $\lambda=700$ nm. The cutoff wavelength, $\lambda_{\rm c}$, is represented in the inset of Fig. \[fig:single\_hole\] as a function of $r_{h}$. The resonance appears in the field at the opening $|E'|$ (not shown), while the decay for $r_{h}>r_{\rm c}$ is due to that in the single hole SPP propagator $g^{\rm sh}_{spp}$ (\[eq:Gspp0\]). For $r_{h}>r_{\rm c}$ most of the the energy is radiated out of the plane. In this case, both $g^{\rm sh}_{\rm rad}$ (not shown) and $P_{\rm rad}$ reach a fast saturation with the hole radius.
Optical response of a hole pair {#sec:holepair}
===============================
We define the [*normalized*]{} hole pair emittance as the power radiated into each channel, out-of-plane (\[eq:Prad\]) or in-plane-SPP (\[eq:Pspp\]), divided by twice the corresponding emitted power of a single hole located at $R=0$, i.e. $$\begin{aligned}
\label{eq:Jrad}
\eta_{rad}(R)= \frac{ P_{\rm rad}(R)}{2 P^{\rm sh}_{\rm rad}}=|E'_N|^2 g^{\rm N}_{\rm rad}(R), \\
\label{eq:Jspp}
\eta_{spp}(R)=\frac{P_{spp}(R)}{ 2 P^{\rm sh}_{spp}}=|E'_N|^ 2 g^{\rm N}_{spp}(R), \end{aligned}$$ where $E'_N$ is the ratio of the electric field at the hole openings $$\begin{aligned}
\label{eq:rE}
E'_N (R) &=& \frac{E'(R)}{E'_{\rm sh}} = \frac{\left[G_{\rm sh}-\Sigma\right]^2-G^2_\nu}{\left[G(R)-\Sigma\right]^2-G^2_\nu},\end{aligned}$$ and we have used that the illumination of an isolated hole is equal to the illumination of each hole in the pair for a normal incident plane wave; while the ratios for the out-of-plane and SPP propagators are given by $$\begin{aligned}
\label{eq:rrad}
g^{\rm N}_{\rm rad}(R)= \frac{g_{\rm rad}(R)}{g^{\rm sh}_{\rm rad}}, \\
\label{eq:rspp}
g^{\rm N}_{spp}(R) = \frac{g_{spp}(R)}{g^{\rm sh}_{spp}} .\end{aligned}$$
![(Color). Normalized out-of-plane emittance $\eta_{rad}$ (a) and normalized in-plane SPP emittance $\eta_{spp}$ (b) as function of $R/\lambda_{spp}$ for increasing $r_{h}$; $r_{h}=100$ nm (blue dashed line), 150 nm (red solid line), and 250 nm (black short-dashed line). The holes are milled in a free standing Ag film of thickness=250 nm. The illumination wavelength is $\lambda$=700 nm. The CI reported in \[sec:IC\] are included in both (a) and (b) for $r_{h}=150$ nm: maxima (at $R/\lambda_{spp}=m-1/4$ for $\eta_{rad}$ and $R/\lambda_{spp} \approx m$ for $ \eta_{spp}$) and minima (at $R/\lambda_{spp}=m+1/4$ for $\eta_{rad}$ and $R/\lambda_{spp} \approx m-1/2$ for $ \eta_{spp}$) are represented with arrows, while vertical dashed lines are used for the condition $\eta=1$ (at $R/\lambda_{spp}=(m+1)/2$ for $\eta_{rad}$ and $R/\lambda_{spp} \approx (2m+1)/4$ for $ \eta_{spp}$); $m=1,2,3,\dots$[]{data-label="fig:Agh250r150"}](Fig3a.eps "fig:"){height="6.5cm"} ![(Color). Normalized out-of-plane emittance $\eta_{rad}$ (a) and normalized in-plane SPP emittance $\eta_{spp}$ (b) as function of $R/\lambda_{spp}$ for increasing $r_{h}$; $r_{h}=100$ nm (blue dashed line), 150 nm (red solid line), and 250 nm (black short-dashed line). The holes are milled in a free standing Ag film of thickness=250 nm. The illumination wavelength is $\lambda$=700 nm. The CI reported in \[sec:IC\] are included in both (a) and (b) for $r_{h}=150$ nm: maxima (at $R/\lambda_{spp}=m-1/4$ for $\eta_{rad}$ and $R/\lambda_{spp} \approx m$ for $ \eta_{spp}$) and minima (at $R/\lambda_{spp}=m+1/4$ for $\eta_{rad}$ and $R/\lambda_{spp} \approx m-1/2$ for $ \eta_{spp}$) are represented with arrows, while vertical dashed lines are used for the condition $\eta=1$ (at $R/\lambda_{spp}=(m+1)/2$ for $\eta_{rad}$ and $R/\lambda_{spp} \approx (2m+1)/4$ for $ \eta_{spp}$); $m=1,2,3,\dots$[]{data-label="fig:Agh250r150"}](Fig3b.eps "fig:"){height="6.5cm"}
The normalized emittances $\eta_{rad}$ and $ \eta_{spp}$ are depicted in Fig. \[fig:Agh250r150\] as a function of the hole-hole distance $R$ for increasing hole radius; $r_{h}=100$ nm (blue dashed line), 150 nm (red solid line), and 250 nm (black short-dashed line). $R$ is normalized to the SPP wavelength $\lambda_{spp}=2\pi/\mbox{Re}[k_{spp}]$, where $k_{spp}$ (\[kspp\]) is the SPP propagation constant in silver; $\lambda_{spp}=682.3$ nm for the chosen $\lambda=700$ nm.
In accordance with experimental works [@AlarverdyanNP07; @PacificiOE08], the computed powers $\eta_{rad}$ and $ \eta_{spp}$ oscillate with period $\lambda_{spp}$. However, $\eta_{rad}$ behaves different than $ \eta_{spp}$ as a function of the hole radius. The amplitude of $\eta_{rad}$ strongly oscillates with $r_{h}$. Indeed, increasing $r_{h}$ from 100 nm to 150 nm at fixed $R$ we can transform a maximum of $\eta_{rad}$ into a minimum. To the best our knowledge this dependence of $\eta_{rad}$ on $r_{h}$ has not been previously reported. Moreover, in the thin-film limit it has been found that the CI only depend on the edge-edge distance, and not on $r_{h}$. Further experimental work is needed to study the dependence on $r_{h}$ for opaque metal films and hole sizes larger that the metal skin depth (the region of the parameter space targeted in this paper). Nevertheless, it is worth stressing that the available experimental data [@AlarverdyanNP07; @PacificiOE08] report the same CI for very different systems [@note1]. In both cases $r_{h}$ is very small ($\sim \lambda/20$), but while Ref. [@AlarverdyanNP07] considers a thin gold layer ($h=20$ nm) on a glass substrate, Ref. [@PacificiOE08] uses an optically thick silver film, immersed in a medium with refractive index $n=1.45$. Both experimental CI are the same as for a third different system, the particular case $r_{h}=150$ nm in Fig. \[fig:Agh250r150\] (a), i.e. maxima are at $R/\lambda_{spp}=m-1/4$, minima at $R/\lambda_{spp}=m+1/4$ (both represented with arrows), and $\eta_{rad}=1$ at $R/\lambda_{spp}=(m+1)/2$ (represented with vertical dashed lines), where $m=1,2,3,\dots$
In contrast, the amplitude of $ \eta_{spp}$ shows a stronger dependence on $R$, but does not present such large variations with size of the holes. Maxima of $ \eta_{spp}$ occur close to the conditions for constructive interference of SPPs at the flat metal surface ($R=m\lambda_{spp}$), while minima appear close to conditions for destructive interference of SPPs between the holes ($R=(2m-1)\lambda_{spp}/2$). As the energy traversing the holes is distributed into the out-of-plane and in-plane channels (\[eq:efc\]), we find in Fig. \[fig:Agh250r150\] that $\eta_{rad}$ and $ \eta_{spp}$ behave as complementary scattering channels, with a relative contribution that changes as a function of both $R$ and $r_{h}$. Taking into account the interference pattern on the SPP channel we solve the apparent paradox put forward in Ref. [@PacificiOE08]: although $\eta_{rad}$ is described by an SPP interference model, there is neither a transmission enhancement nor suppression at the conditions for constructive interference of SPPs.
The out-of-plane radiation, $\eta_{rad}$ (\[eq:Jrad\]), is mainly determined by $E'_N$, i.e. by the change in the field at the hole due to the presence of the other hole. This is illustrated in Fig. \[fig:j\_vs\_R\](a), where $\eta_{rad}$ is compared with both $|E'_N|^2$ and $g^{\rm N}_{\rm rad}$. We observe that the interference between the radiative field of the two holes, given by $g^{\rm N}_{\rm rad}$, practically does not change the total transmission for hole-hole distances larger than $2\lambda_{spp}$. Conversely, the normalized in-plane propagator $g^{\rm N}_{spp}$ plays an important role setting up the CI for $ \eta_{spp}$ (\[eq:Jspp\]), see Fig. \[fig:j\_vs\_R\](b). Although the contribution of $|E'_N|^2$ can not be neglected, the interference pattern of $g^{\rm N}_{spp}$ resembles the behavior of $ \eta_{spp}$.
![(Color). (a) Normalized out-of-plane emittance $\eta_{rad}$ (red solid line), its constituent terms $|E'_N|^2$ (black short-dashed line) and $g^{\rm N}_{\rm rad}$ (dark-yellow dashed line), and the approximate expression for $\eta_{rad}$ (orange dash-dotted line) of Eq. (\[eq:Pradapprox\]). (b) Normalized SPP emittance $ \eta_{spp}$ (blue solid line), its constituent terms $|E'_N|^2$ (black short-dashed line) and $g^{\rm N}_{spp}$(magenta dashed line), and the approximate expression for $ \eta_{spp}$ (violet dash-dotted line) of Eq. (\[eq:Psppapprox\]). All these quantities are represented as function $R/\lambda_{spp}$. The hole radius is $r_{h}=150$ nm, the rest of parameters are the same as in Fig. \[fig:Agh250r150\].[]{data-label="fig:j_vs_R"}](Fig4a.eps "fig:"){height="6.5cm"} ![(Color). (a) Normalized out-of-plane emittance $\eta_{rad}$ (red solid line), its constituent terms $|E'_N|^2$ (black short-dashed line) and $g^{\rm N}_{\rm rad}$ (dark-yellow dashed line), and the approximate expression for $\eta_{rad}$ (orange dash-dotted line) of Eq. (\[eq:Pradapprox\]). (b) Normalized SPP emittance $ \eta_{spp}$ (blue solid line), its constituent terms $|E'_N|^2$ (black short-dashed line) and $g^{\rm N}_{spp}$(magenta dashed line), and the approximate expression for $ \eta_{spp}$ (violet dash-dotted line) of Eq. (\[eq:Psppapprox\]). All these quantities are represented as function $R/\lambda_{spp}$. The hole radius is $r_{h}=150$ nm, the rest of parameters are the same as in Fig. \[fig:Agh250r150\].[]{data-label="fig:j_vs_R"}](Fig4b.eps "fig:"){height="6.5cm"}
The CI developed in the next section strongly depend on the properties of the in-plane propagator $G_{hh}(R)$, which is behind the interference pattern of both radiative channels. We use the following decomposition of the in-plane propagator $$\begin{aligned}
\label{eq:Ghhdecomp}
G_{hh}(R)=G^{hh}_{\rm rad}(R)+G^{hh}_{spp}(R)+G^{hh}_{\rm ev}(R),\end{aligned}$$ where $G^{hh}_{\rm rad}(R)$ (\[eq:Gfp\]) represents the contribution of radiative modes, $G^{hh}_{spp}(R)$ (\[eq:Gspphh\]) designate the contribution of the plasmon pole to evanescent modes, and $G^{hh}_{\rm ev}(R)$ (\[eq:Grnp\]) denotes the contribution of the remaining evanescent modes. This decomposition is not only the most natural way of connecting $G_{hh}$ (\[eq:green\]), to the radiative propagators $g_{\rm rad}$ (\[eq:Prad\]) and $g_{spp}$ (\[eq:Pspp\]), as well as to recover previous results for the PEC [@BravoPRL04], it is also related to the decomposition proposed in Ref. [@LalanneNP06] in order to compare SPP with non-SPP mediated interaction.
The real and imaginary parts of $G^{hh}_{\rm rad}(R)$, $G^{hh}_{\rm ev}(R)$, and $G^{hh}_{spp}(R)$ are compared with $G_{hh}(R)$ in Fig. \[fig:green\] for the same parameters of Fig. \[fig:j\_vs\_R\]. The most relevant feature observed in Figs. \[fig:green\] (a) and (b) is that the main contribution to $G_{hh}(R)$ comes from the SPP propagator, $G^{hh}_{spp}(R)$, which has a simple analytical form (\[eq:Gspphh\]). This allows us to find analytical expression for CI that will be presented in the next section. Notice that the agreement between $G_{hh}(R)$ and $G^{hh}_{spp}(R)$ has been previously reported for 1D defects separated a distance larger that $2-3\lambda$ [@FLTPRB05]. Regarding non-SPP channels, $G^{hh}_{\rm rad}(R)$ decays faster than $G^{hh}_{spp}(R)$ being negligible small for $R$ equal to a few $\lambda_{spp}$. On other hand, the real part of $G^{hh}_{\rm ev}(R)$ is vanishing small (see Appendix), while its imaginary part is in anti-phase to $G^{hh}_{\rm rad}(R)$. It must be noted that, as expected, the relative contribution of the different propagators changes when we approach the PEC limit [@LalanneNP06; @FLTPRB05; @SondergaardPRB04; @LiuN08].
![(Color). Real (a) and imaginary (b) parts of the propagator $G_{hh}(R)$ (black solid line), as well as its constituent terms for radiative modes, $G^{hh}_{\rm rad}(R)$ (red dashed line), SPP modes, $G^{hh}_{spp}(R)$ (blue short-dashed line), and remaining evanescent modes $G^{hh}_{\rm ev}(R)$ (green dash-dotted line). We use the same geometrical parameters of Fig. \[fig:j\_vs\_R\].[]{data-label="fig:green"}](Fig5a.eps "fig:"){height="6.5cm"} ![(Color). Real (a) and imaginary (b) parts of the propagator $G_{hh}(R)$ (black solid line), as well as its constituent terms for radiative modes, $G^{hh}_{\rm rad}(R)$ (red dashed line), SPP modes, $G^{hh}_{spp}(R)$ (blue short-dashed line), and remaining evanescent modes $G^{hh}_{\rm ev}(R)$ (green dash-dotted line). We use the same geometrical parameters of Fig. \[fig:j\_vs\_R\].[]{data-label="fig:green"}](Fig5b.eps "fig:"){height="6.5cm"}
Conditions for Interference {#sec:IC}
---------------------------
In this section we compute the conditions for constructive and destructive interference of both out-of-plane and in-plane radiative powers. We start with the simpler of these two quantities, $\eta_{rad}$. Three simplifications help in finding the results for $\eta_{rad}$. First, its interference pattern is accurately described by the normalized square field amplitude, $\eta_{rad} \approx |E'_N|^{2}$, see \[fig:j\_vs\_R\](a). Second, $G_{hh}(R) \approx G^{hh}_{spp}(R)$, as we have learned from Fig. \[fig:green\]. Third, $G^{hh}_{spp}(R) \ll G_{\rm sh}$. This last approximation is valid for $R \gg \lambda_{spp}$, but we shall see it gives results that work surprisingly well even for $R \sim \lambda_{spp}$. Expanding $E'_N$ (\[eq:rE\]) into Mclaurin series of $G^{hh}_{spp}(R)/G_{\rm sh}$ and keeping only the leading term, we find $$\begin{aligned}
\label{eq:Enapprox}
E'_N =\frac{E'}{E'_{\rm sh}} \approx 1-2 \, Z_E \, G^{hh}_{spp}(R), \nonumber \end{aligned}$$ where $Z_E=E_{\rm sh}/I$ (\[eq:ZE\]) is the effective impedance of a single hole, which gives the modal amplitude at the hole opening as a function of the illumination. From the simplified expression for $E'$ we can deduce that the interference pattern of the hole pair is set up by both the single hole impedance and the re-illumination of one hole by the other.
The CI for $\eta_{rad}$ can be written in terms of the phase shift of both $Z_E$ and $G^{hh}_{spp}(R)$. We thus define the single-hole phase shift, $\phi_{ZE}$, from $Z_E=|Z_E| \exp(i\phi_{ZE})$, as well as the phase difference acquire by the SPP when traveling from one hole to the other, $\phi_{hh}$, from $G^{hh}_{spp}(R)=|G^{hh}_{spp}(R)|\exp(i\phi_{hh})$. An approximate expression for the $\phi_{hh}$ can be obtained replacing the Hankel function in $G^{hh}_{spp}$ (\[eq:Gspphh\]) by its asymptotic expression, $H'^{(1)}_1(x) \approx (\pi x/2)^{ -1/2}\exp[i (x-\pi/4)]$. We have then $\phi_{hh}=k_{spp}R-\pi/4$. Keeping again the leading term in the expansion of $|E'_N|^2$, we obtain $$\begin{aligned}
\label{eq:Pradapprox}
\eta_{rad}\approx 1-4\left|Z_E G^{hh}_{spp} \right| \cos(k_{spp} R+\phi_{ZE}-\pi/4). \end{aligned}$$ This equation clearly shows that the out-of-plane radiation depends both on the optical path traveled by the SPP when going to one hole to the other and the phase picked up by the field given the extra illumination provided by the SPP coming from the other hole. The approximate equation (\[eq:Pradapprox\]) is compared with full calculations in Fig. \[fig:j\_vs\_R\](a). We find that Eq. (\[eq:Pradapprox\]) slightly underestimates $|E'_N|^2$ for $R < \lambda_{spp}$, but the agreement is excellent for $R> \lambda_{spp}$. This nice agreement is related to the fact that non-SPP waves decays faster than SPP waves as a function of the distance, see Fig. \[fig:green\]. The leading role of SPP waves for large $R$ have been already stressed in [@LalanneNP06; @FLTPRB05; @SondergaardPRB04; @LiuN08]. Eq. (\[eq:Pradapprox\]) also agrees with the one proposed in [@PacificiOE08] following an intuitive interference plasmon model, which, in contrast to first-principles derivation of (\[eq:Pradapprox\]), contains fitting parameters.
It is straightforward to derive the CI of $\eta_{rad}$ from Eq. (\[eq:Pradapprox\]) assuming that the absolute value of $G^{hh}_{spp}$ changes smoothly with $R$, and that the dependence on $R$ mainly comes from its phase. Then we have that extrema of $\eta_{rad}$ appear at $$\begin{aligned}
\label{eq:rEext}
k_{spp}R-\frac{\pi}{4}+\phi_{ZE}=n \pi,\end{aligned}$$ where the integer value of $n$ is equal to $n=2m-1$ for maxima, $n=2m$ for minima, and $m=1,2,3,...$; while the condition for $\eta_{rad}=1$ is shifted in $\pi/2$ with respect to the previous expression, i.e. $$\begin{aligned}
\label{eq:rEeq1}
k_{spp}R-\frac{\pi}{4}+\phi_{ZE}=(m+\frac{1}{2}) \pi.\end{aligned}$$
The single-hole phase shift, $\phi_{ZE}$, is depicted in Fig. \[fig:phivsrh\](a) as function of the of the hole radius, $r_{h}$, and for increasing metal thickness; $h=100$ nm (blue dashed line), 150 nm (red solid line), and 250 nm (black short-dashed line). The large variation in $\phi_{ZE}$ as function of $r_{h}$ (up to $\pi/2$ for increasing $r_{h}$ from 50 nm to 250 nm) accounts for the oscillations in $\eta_{rad}$ observed in Fig. \[fig:Agh250r150\](a). In Fig. \[fig:Agh250r150\](a) we compare full calculations with the CI given by Eqs. (\[eq:rEext\]) and (\[eq:rEeq1\]) for the case $r_{h}=150$, $h=250$ nm, for which $\phi_{ZE}=-\pi/4$ (see Fig. \[fig:phivsrh\](a)). An excellent agreement is obtained even for small values of $R/\lambda_{spp}$. Notice that $\eta_{rad}$ is largely independent on the metal thickness $h$ (although it is computed for a given value of of $h$ in optically thick film) given that $\phi_{ZE}$ is practically independent on $h$, see Fig. \[fig:phivsrh\].
Notice that the CI represented by Eqs. (\[eq:rEext\]) and (\[eq:rEeq1\]), which are valid for a wide range of hole sizes (larger than the metal skin depth) and opaque metal films, are expressed in terms of the distance between the centers of the holes. A previous work [@AlarverdyanNP07] suggested that, for thin-metal films and small hole sizes, the CI are a function of the edge-edge distance, independently from the hole radius. In our notation, this could only occurs if $\phi_{ZE}+2r_{h}/\lambda_{spp}=0$ in Eq. (\[eq:Pradapprox\]). However, we observe in Fig. \[fig:phivsrh\](a) that $-2r_{h}/\lambda_{spp}$ (dash-dotted line) is equal to $\phi_{ZE}$ only for a small region of the parameter space. This novel behavior demands further experimental work on opaque metal films and hole sizes larger than the metal skin depth.
![(Color). (a) Single-hole phase shift for the out-of-plane emittance, $\phi_{ZE}$, as function of the hole radius, $r_{h}$, and for increasing metal thickness $h$; $h=100$ nm (blue dashed line), 150 nm (red solid line), and 250 nm (black short-dashed line). The dash-dotted line represents the hole diameter normalized by $\lambda_{spp}$. (b) SPP phase shift for the in-plane emittance of a single hole, $\phi_{ZP}$.[]{data-label="fig:phivsrh"}](Fig6a.eps "fig:"){height="6.5cm"} ![(Color). (a) Single-hole phase shift for the out-of-plane emittance, $\phi_{ZE}$, as function of the hole radius, $r_{h}$, and for increasing metal thickness $h$; $h=100$ nm (blue dashed line), 150 nm (red solid line), and 250 nm (black short-dashed line). The dash-dotted line represents the hole diameter normalized by $\lambda_{spp}$. (b) SPP phase shift for the in-plane emittance of a single hole, $\phi_{ZP}$.[]{data-label="fig:phivsrh"}](Fig6b.eps "fig:"){height="6.5cm"}
Similar CI can be developed for the in-plane scattered power $ \eta_{spp}$. As commented in the discussion of Fig. \[fig:j\_vs\_R\](b), both terms $|E'_N|^{2}$ and $g^{\rm N}_{spp}$ contribute to $\eta_{spp}$ in Eq. \[eq:Jspp\]. We take the approximate expression of $|E'_N|^{2}$ from Eq. (\[eq:Pradapprox\]) and use the asymptotic expression $g^{\rm N}_{spp}=1+2J'_{1}(k_{spp}R)$ found in the Appendix. We recall that the last relation is exact when absorption is neglected, but otherwise it is still a good approximation. Using again the asymptotic expansion of the Bessel function, the in-plane emittance is thus simplified to $$\begin{aligned}
\label{eq:Psppapprox}
\eta_{spp} \approx |E'|^2 \left[1+2\sqrt{\frac{2}{\pi k_{spp} R}}\cos(k_{spp}R-\pi/4) \right].\end{aligned}$$ This equation tells us that, given the normalized amplitude of the electric field at the hole opening $E'_N$, the interference pattern of the in-plane scattering power is determined by the SPP optical path between the two holes. However, in order to quantify the CI of $ \eta_{spp}$, we should include the modulation of the field given by $|E'_N|^2$ (\[eq:Pradapprox\]). Expanding the two terms in Eq. (\[eq:Psppapprox\]) up to the first order in $G^{hh}_{spp}/G_{\rm sh}$, $\eta_{spp}$ can be straightforwardly rewritten to $$\begin{aligned}
\label{eq:PsppIC}
\eta_{spp} \approx 1+4\left|Z_{spp} G^{hh}_{spp} \right| \cos(k_{spp} R+\phi_{ZP}-\pi/4), \end{aligned}$$ where the effective impedance for the SPP channel $Z_{spp}=Z_E-(4|G_{spp}^{sh}|)^{-1}$ takes into account both the excitation of the EM field inside the hole, characterized by $Z_E$, and the excitation of the SPP at the hole, given by $(4|G_{spp}^{sh}|)^{-1}$. Like for the out-of-plane channel, the approximate Eq. (\[eq:PsppIC\]) shows an excellent agreement with full calculations in Fig. \[fig:j\_vs\_R\](b). However, the behavior of $\phi_{ZP}$ (defined from $Z_{spp}=|Z_{spp}|e^{i\phi_{ZP}}$) differs from $\phi_{ZE}$. Fig. \[fig:phivsrh\](b) renders $\phi_{ZP}$ as a function of $r_{h}$, showing a characteristic peak centered near the cutoff radius $r_{\rm c}=168$ nm, cf. Fig. \[fig:single\_hole\]. The phase difference with respect to $\phi_{ZE}$ is about $\pi/2$ for $r_{h} \leqslant r_{\rm c}$, and decreases to zero for $r_{h}>r_{\rm c}$.
The extreme values of $ \eta_{spp}$ (\[eq:PsppIC\]) satisfy $$\begin{aligned}
\label{eq:rPext}
k_{spp}R-\frac{\pi}{4}+\phi_{ZP}=n \pi,\end{aligned}$$ where the integer value of $n$ is equal to $n=2m$ for maxima, $n=2m-1$ for minima, and $m=1,2,3,...$ Notice that the values of $n\pi$ for $ \eta_{spp}$ are shifted in $\pi$ with respect to the extreme values of $\eta_{rad}$ (maxima are replaced by minima, and vise versa). This shifting is determined by the fact that the power traversing the hole is radiated into two complementary channels: $\eta_{rad}$ and $ \eta_{spp}$. As for $\eta_{rad}$, the condition $ \eta_{spp}=1$ is shifted in $\pi/2$ with respect to the previous expression for extreme values, i.e. $$\begin{aligned}
\label{eq:rPeq1}
k_{spp}R-\frac{\pi}{4}+\phi_{ZP}=(n + \frac{1}{2})\pi.\end{aligned}$$ In \[fig:Agh250r150\](b) we compare full calculations with the CI for the case $r_{h}=150$, $h=250$ nm, for which $\phi_{ZP}=0.27 \; \mbox{rad} \approx \pi/4$ (see Fig. \[fig:phivsrh\](b)). As for $\phi_{ZE}$, an excellent agreement is obtained.
Conclusions
===========
We have studied the emission pattern of a hole pair, focusing our attention in the role played by SPP resonances. Starting from the rigorous solution of the problem, we have developed a SPP interference model that does not contain fitting parameters. This model provides simple analytical expressions for the interference pattern of both the out-of-plane and in-plane radiation channels, which nicely agree with full calculations for noble metals at optical frequencies.
In agreement with experimental reports, both radiated powers oscillate with period $\lambda_{spp}$. However, they show different trends as a function of the hole-hole distance and the hole radius. The amplitude of $\eta_{rad}$ strongly oscillates with the hole radius, while the amplitude of $\eta_{spp}$ has a stronger dependence on R, but does not present such large variations with the hole size.
Maxima of $\eta_{spp}$ occur close to the conditions for constructive interference of SPPs at the flat metal surface ($R=m\lambda_{spp}$), while minima appear close to conditions for destructive interference of SPPs between the holes ($R=(2m-1)\lambda_{spp}/2$). CI for $ \eta_{spp}$ are shifted in $\pi$ with respect to those of $\eta_{rad}$ (maxima are replaced by minima, and vise versa), because $\eta_{rad}$ and $ \eta_{spp}$ are two complementary channels. The power traversing the hole is distributed into these two channels.
We have also shown that two scattering mechanisms determine the interference pattern of the hole pair: (i) re-illumination by the in-plane SPP radiation and (ii) an effective impedance depending on the single-hole response. The conditions for interference only depend on the phase difference provided by each of the two scattering mechanisms. The large variation in the effective impedance of the single hole accounts for the oscillations of $\eta_{rad}$ as a function of the hole size.
The authors gratefully acknowledge financial support from the Spanish Ministry of Science and Innovation under grants MAT2009-06609-C02, CSD2007-046-NanoLight.es, and AT2009-0027.
Coupled-mode method {#app:me}
===================
In this section we briefly review the coupled-mode method for the optical transmission through holes, under the fundamental waveguide mode (TE$_{11}$) approximation. We refer to [@FdLPNJP08] for the expressions of the full multimode formalism and their derivation. Within the CMM, Maxwell’s equations are solved self-consistently using a convenient representation for the EM fields. In both substrate and superstrate (see Fig. \[fig:scheme\]), the fields are expanded into an infinite set of plane waves with both p- and s-polarizations. Inside the holes the most natural basis is a set of circular waveguide modes [@stratton]. The parallel components of the fields are matched at the metal/dielectric interface using surface impedance boundary conditions (SIBCs) [@jackson]. Although SIBCs neglect the tunneling of EM energy between the two metal surfaces, this effect is not relevant for a metal thickness larger than a few skin depths.
At the lateral walls of the holes we choose the PEC approximation for the sake of analytical simplicity. We are thus neglecting absorption losses at the walls. Nevertheless, we upgrade the PEC approximation introducing two phenomenological corrections. First, the propagation constant of the PEC fundamental mode is replaced by the one computed for a real metal. This improves the comparison between CMM and both experimental and FDTD results for both the spectral position of the peaks and the dependence of optical properties on the metal thickness. Second, enlarging the radius of the hole by one skin depth simulates the real penetration of in field into the metallic walls. This value for the enlargement provides the best agreement with FDTD simulations for an infinite periodic array of holes [@przybillaOE08].
After matching the fields at the interface we arrive to the following system of tight binding-like equations $$\begin{aligned}
[ G_{\rm sh}-\Sigma ] E_1(R)+G_{hh}(R) E_2(R)-G_{\rm \nu} E'_1(R) = I_1 , \nonumber \\
\left[ G_{\rm sh}-\Sigma \right] E'_1(R)+G_{hh}(R) E'_2(R)-G_{\rm \nu} E'_2(R) = 0 ,\nonumber\end{aligned}$$ where $E_i$ is the modal amplitude of the electric field at the input opening of the $i^{th}$ hole, $i=1,2$, and $E'_i$ is the same quantity but at the output opening. Two additional equivalent equations are needed for $E_2$ and $E'_2$. Other relevant quantity is the illumination provided by the normal-incident p-polarized plane wave, with wavenumber $k_{\lambda}=2\pi/\lambda$ and admittance $Y_0=\sqrt{\epsilon_1}$, onto the lowest-energy mode $$\label{eq:I}
I \equiv I_1=I_2=\frac{\sqrt{2 Y_0}}{1+z_{s} Y_0 }\frac{k_{\lambda} }{\sqrt{u^2-1}},$$ where $z_{s}=\epsilon^{-1/2}_m$ is the metal impedance. In order to obtain a transmittance normalized by the flux impinging on the area covered by the holes, the illumination term $I$ already contains a factor $(\pi r^2_h Y_0)^{-1/2}$. The constant $u$ satisfies $J'_1(u)=0$ [@stratton], where $J(x)$ is the Bessel function of order $1$, and the prime denotes derivation with respect to its argument
The quantities $\Sigma$ and $G_{\rm \nu}$ represent scattering mechanisms already present in single holes. $\Sigma$ is related to the bouncing back and forth of the waveguide fields inside the holes. Its value is $$\label{eq:S}
\Sigma= Y_w \frac{ f^+_w \e^{i k_zh}+f^-_w \e^{-i k_zh}}{ {f^+_w}^2 \e^{i k_zh}-{f^-_w}^2 \e^{-i k_zh}},$$ where $k_z$ is the propagation constant of the waveguide mode, $h$ is the metal thickness, $f^{\pm}_{\rm w} = 1 \pm z_{s} Y_w$, $Y_w=k_z/k_{\epsilon_2}$ is the admittance for the excited TE$_{11}$ mode, and $k_{\epsilon_2}=\sqrt{\epsilon_2}k_{\lambda}$. The quantity $$\label{eq:Gnu}
G_{\rm \nu}= 2 Y_w \left[
{f^+_w}^2 \e^{i k_z
h}-{f^-_w}^2\e^{-i k_zh} \right]^{-1}$$ reflects the coupling between EM fields at the two sides of a given hole [@FdLPNJP08].
The propagator $G_{hh}(R)$ represents the coupling of the two holes. It results from the projection of the Green’s dyadic onto the waveguide modes in the holes. For the TE$_{11}$ mode, $G_{hh}$ can be written as the following integral in the plane of the reciprocal space parallel to the metal surface $$\begin{aligned}
\label{eq:green}
G_{hh}(R)&=&G_0\int^{\infty}_0 \left( \frac{G_{p}(q)}{q_{z}+z'_{s} } + \frac{G_{s}(q)}{q_{z}^{-1} +z'_{s}} \right) q d q,\end{aligned}$$ where $G_0 = 4 k^2_{\epsilon} r^2_h \sqrt{\epsilon}/(u^2-1)$ and the two terms in the integrand represent the contribution of p- and s-polarized plane waves in the infinite semi-space in contact with the metal surface. The denominators of these two terms stand for the response of the metal plane. In particular, the p-term has a pole at the SPP wavevector. The numerators $G_{p}$ and $G_{s}$ account for both the single hole response, which is a function of the hole radius $r_{h}$, and the hole-hole interaction, a function of $R$. They read $$\begin{aligned}
\label{eq:Gp}
G_{p} (q,r_{h},R)= \frac{ J^2_1 (k_{\epsilon} q r_{h})}{k^2_{\epsilon} q^2 r^2_h} J'_1 (k_\epsilon q R), \\
\label{eq:Gs}
G_{s} (q,r_{h},R)= \frac{J^{' 2}_1 (k_{\epsilon} q r_{h})}{\left( 1-\frac{k^2_{\epsilon} q^2 r^2_h}{u^2}\right)^2 } \frac{J_1(k_{\epsilon} q R)}{k_{\epsilon} q R}.\end{aligned}$$ The integrand is written in adimensional units normalizing the wavevector by $k_{\epsilon}=k_{\lambda} \sqrt{\epsilon}$. Notice that the $R$-dependent Bessel functions are obtained after the angular integration in the $\mathbf{k}_\parallel=k_\epsilon q (\cos \theta,\sin \theta)$ plane, where $\theta$ defines the direction of the component of wavevector parallel to the metal plane, $\mathbf{k}_\parallel$. The dielectric constant $\epsilon$ characterizes the dielectric material in contact with the metal surface (see Fig. \[fig:scheme\], $\epsilon=\epsilon_1=1$ is used along this paper).
The self-interaction term $G_{\rm sh}$ is obtained after tacking the limit $R \rightarrow 0$ in $G_{hh}(R)$, i.e. using the identities $\lim_{x \rightarrow 0} J'_1(x)=\lim_{x \rightarrow 0} J_1(x)/x=1/2$.
Other relevant function is the effective impedance $Z_E$, which is determined by the the three scattering mechanisms of the single hole ($G_{\rm sh}$, $G_{\rm \nu}$, and $\Sigma$), $$\begin{aligned}
\label{eq:ZE}
Z_E=\frac{E_{\rm sh}}{I}=\frac{G_{\rm sh}-\Sigma}{(G_{\rm sh}-\Sigma)^2-G^2_\nu}. \end{aligned}$$
We compute $G_{hh}$ using the decomposition $G_{hh}=G^{hh}_{\rm rad}+G^{hh}_{spp}+G^{hh}_{\rm ev}$ (\[eq:Ghhdecomp\]), where $G^{hh}_{\rm rad}(R)$ represents the contribution of radiative modes, $G^{hh}_{spp}(R)$ designate the contribution of the plasmon pole to evanescent modes, and $G^{hh}_{\rm ev}(R)$ denotes the contribution of the remaining evanescent modes. The contribution of $G^{hh}_{\rm rad}(R)$ can be written in terms of the functions $g^{\rm int}_{\rm rad}(R)$ and $\Delta G^{hh}_{\rm rad}(R)$, which always take real values, $$\begin{aligned}
\label{eq:Gfp}
G^{hh}_{\rm rad}(R)=g^{\rm int}_{\rm rad}(R)+z'^*_s \Delta G^{hh}_{\rm rad}(R), \nonumber\end{aligned}$$ where $$\begin{aligned}
\label{eq:greenR}
g^{\rm int}_{\rm rad}(R)&=&G_0\int^1_0 d q \left( \frac{q q_{z} G_{p}}{\vert q_{z}+z'_{s} \vert^2} + \frac{q q_{z} G_{s}}{\vert 1+q_{z} z'_{s} \vert^2} \right) \end{aligned}$$ provides the interference term of the far field radiated from the holes (\[eq:Prad\]), while the term proportional to the metal impedance $z'_{s}$ reads $$\begin{aligned}
\Delta G^{hh}_{\rm rad}(R)&=&G_0 \int^1_0 q d q \left( \frac{G_{p}}{\vert q_{z}+z'_{s} \vert^2} + \frac{q^2_z G_{s}}{\vert 1+q_{z} z'_{s} \vert^2} \right). \nonumber\end{aligned}$$ Both integrals are computed for free-propagating states ($0 \leq q \leq 1$). The real part of $z'_{s}$ is very small for typical noble metals, making $\mbox{Re}[G^{hh}_{\rm rad}] \approx g^{\rm int}_{\rm rad}$ and $\mbox{Im}[G^{hh}_{\rm rad}] \approx |z_{s}| \Delta G^{hh}_{\rm rad}$ a good approximation for $G^{hh}_{\rm rad}$. The same relations hold for the single hole propagator, $G^{sh}_{rad}$.
For non-propagating states ($q>1$) the integrand in $G_{hh}(R)$ is prolonged into the complex $q$-plane, see [@FdLPNJP08] for details. The residue of the Cauchy integral gives the SPP wave confined to the metal/air interface $$\begin{aligned}
\label{eq:Gspphh}
G^{hh}_{spp}=\pi i z'_{s} G_0 \frac{ J^2_1 ( k_{spp}r_{h})}{k^2_{spp} r^2_h} H'^{(1)}_1( k_{spp} R), \end{aligned}$$ where $k_{spp}$ is the parallel component of the SPP wavevector $$\label{kspp}
k_{spp} =k_\epsilon \left( \frac{\epsilon \; \epsilon_{\rm m}(\lambda)}{\epsilon_{\rm m}(\lambda)+\epsilon}\right)^{1/2} .$$ Eq. (\[kspp\]) defines $k_{spp}$ for a real metal and rigorous boundary condition at the metal/dielectric interface. In order to improve the accuracy our model we use Eq. (\[kspp\]) instead of the approximate SPP wavevector for SIBCs, $k^{\rm sibc}_{spp} =k_\epsilon [ \epsilon( 1-\epsilon^{-1}_{\rm m}) ]^{1/2}$. Besides the coupling propagator $G^{hh}_{spp}$, we define the radiative propagator $g_{spp}$, which provides the total SPP field radially scattered along all possible angular directions in the metal plane (\[eq:Pspp\]). We have that $g_{spp}=g^{\rm sh}_{spp}+g^{\rm int}_{spp}$, where the single-hole contribution read $$\begin{aligned}
\label{eq:Gspp0}
g^{\rm sh}_{spp}&=& \frac{\pi \vert z'_{s}\vert a_{\rm l} G_0}{2} \left| \frac{ J_1 (k_{spp} r_{h})}{k_{spp} r_{h}} \right|^2, \end{aligned}$$ and the interference term is equal to $$\begin{aligned}
\label{eq:gspp}
g^{\rm int}_{spp}(R)= g^{\rm sh}_{spp} \left[ 2 \mbox{Re}\left[J'_1(k_{spp}R)\right]+ J'_1(2 i \mbox{Im}[k_{spp}R])-\frac{1}{2} \right] , \nonumber \\\end{aligned}$$ while $a_{\rm l} = |k_{\rm zp}| \mbox{Re}[k_{spp}] /(\mbox{Im}[k_{\rm zp}] |k_{spp}|)$, and $k_{\rm zp}=(k^2_{\rm \epsilon}-k^2_{spp})^{1/2}$. Notice that $g^{\rm N}_{spp}=g^{\rm int}_{spp}/g^{\rm sh}_{spp}$ (\[eq:rspp\]) is independent of $r_{h}$.
For the sake of convenience, the integral for non-SPP evanescent states is computed along the vertical contour $q=1+ih \equiv q_+$, $h\in[0,\infty)$, after the integral variable is changed from $q$ to $h$ $$\begin{aligned}
\label{eq:Grnp}
%\fl
G^{hh}_{\rm ev}=\frac{G_0}{2} \int^\infty_0 q_+ d h \left( \frac{G_{p}}{\kappa_{z}-i z'_{s}}-\frac{G^*_{p}}{\kappa^*_z-i z'_{s}} \right. \nonumber \\
\left. -\frac{\kappa_{z} G_{s}}{1+i z'_{s} \kappa_{z}}+ \frac{\kappa^*_z G^*_{s}}{1+i z'_{s} \kappa^*_z}\right),\end{aligned}$$ where $\kappa_{z}=\sqrt{2i h-h^2}$, and the Bessel function $J_1(x)$ in both $G_{p}$ (\[eq:Gp\]) and $G_{s}$ (\[eq:Gs\]) is replaced by a Hankel function of the first kind $H^{(1)}_1(x)$.
The propagator $G_{hh}$ is further simplified when the metal absorption is neglected, i.e. for $\mbox{Im}[\epsilon_{\rm m}]=\mbox{Re}[z'_{s}]=0$ and $\mbox{Im}[z'_{s}]=-|z'_{s}|$. We find for the radiative modes that $\mbox{Re}[G^{hh}_{\rm rad}] = g^{\rm int}_{\rm rad}$, and $\mbox{Im}[G^{hh}_{\rm rad}] = |z_{s}| \Delta G^{hh}_{\rm rad}$, while for SPP modes $\mbox{Re}[G_{spp}(R)]=g_{spp}(R)=g^{\rm sh}_{spp}[1+2J'_1(k_{spp}R)]$. For the remaining evanescent modes we obtain that $G^{hh}_{\rm ev}$ is a pure imaginary function. The same relations that hold for $G_{hh}$ are valid for $G_{\rm sh}$. Therefore, only the radiative and SPP terms contribute to the real part of in-plane propagator, $$\begin{aligned}
\label{eq:ReG}
\mbox{Re}[G_{\rm sh}+G_{hh}(R)]=g_{\rm rad}(R)+g_{spp}(R). \end{aligned}$$ Under the lossless metal approximation, the total power traversing the two holes simplifies to $P_{\rm hole}=\mbox{Re}[G_{\rm \nu} E E'^*]$ [@FdLPNJP08]. We rewrite it in terms of $E'$ with help of the relation $G_{\rm \nu} E=\left[G_{\rm sh}+G_{hh}(R)-\Sigma\right] E'$, i.e. $$\begin{aligned}
\label{eq:Phole}
P_{\rm hole}=\vert E' \vert^2 \mbox{Re}[G_{\rm sh}+G_{hh}(R)]. \nonumber\end{aligned}$$ As $\Sigma$ (\[eq:S\]) is purely imaginary for a lossless media, this term does not contribute to $P_{\rm hole}$. Using (\[eq:ReG\]), we then have $$\begin{aligned}
\label{eq:efc}
P_{\rm hole}= P_{\rm rad}+P_{spp}.\end{aligned}$$
This equality represents the conservation of the power flux traversing the hole. These results can be easily generalized to an arbitrary number of holes, waveguide modes, and non-cylindrical geometries. It is also worth to mention that including absorption the computed powers differ in less than 5% from the lossless case, even for a large number of defects [@FdLPNJP08; @baudrionOE08; @przybillaOE08].
Finally, we recall that PEC is a particular case of a lossless metal with $z'_{s}=0$. In this case $G^{hh}_{\rm rad}=g^{\rm int}_{\rm rad}$, while for non-propagating states of a PEC only $G^{hh}_{\rm ev}$ survives because $G^{hh}_{spp}=0$ [@BravoPRL04].
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---
abstract: 'We present the first high-resolution X-ray spectrum of the Seyfert 1.5 galaxy NGC 4151. Observations with the Chandra High Energy Transmission Grating Spectrometer reveal a spectrum dominated by narrow emission lines from a spatially resolved (1.6 kpc), highly ionized nebula. The X-ray narrow-line region is composite, consisting of both photoionized and collisionally ionized components. The X-ray emission lines have similar velocities, widths, and spatial extent to the optical emission lines, showing that they arise in the same region. The clouds in the narrow-line region must contain a large range of ionization states in order to explain both the optical and X-ray photoionized emission. Chandra data give the first direct evidence of X-ray line emission from a hot plasma ($T\sim10^7$ K) which may provide pressure confinement for the cooler ($T=3\times10^4$ K) photoionized clouds.'
author:
- 'Ogle, P. M., Marshall, H. L., Lee, J. C., and Canizares, C. R.'
title: |
Chandra Observations of the X-ray Narrow-Line Region in\
NGC 4151
---
Introduction
============
NGC 4151 (z=0.00332) is a well-studied Seyfert 1.5 galaxy. It is known to be extended in soft X-rays from Einstein and ROSAT observations [@ebh83; @mwe95]. The nature of the extended X-ray emission was postulated to be either thermal emission from a collisionally ionized plasma or scattered nuclear flux. @wyh94 modeled the ASCA spectrum by 3 components: an absorbed power law plus reprocessed emission from a warm absorber and an extra scattered component. Variability was observed in the column density of the warm absorber.
Spectral complexity from unresolved emission lines and ionized absorption edges has been seen in the residuals of power-law fits to the spectra of many Seyfert galaxies. Recent observations with Chandra have revealed narrow absorption and emission lines in the spectra of Seyfert 1 galaxies NGC 5548 [@kml00] and NGC 3783 [@kbn00]. These two sources are dominated by continuum emission from the central source, so absorption lines are more prominent than emission lines. On the other hand, the nucleus of NGC 4151 is highly absorbed, giving a good view of its narrow emission line region.
In this paper we present Chandra High Energy Transmission Grating Spectrometer (HETGS) observations of NGC 4151. We identify and measure a large number of narrow emission lines in the X-ray spectrum. The high spectral resolution provided by HETGS allows us to study the ionization, temperature, and kinematics of the X-ray emitting regions. We discuss the association of the extended X-ray emission with the narrow-line region, and interpretations of the multiple X-ray emitting components in the spectrum.
Observations
============
We observed NGC 4151 for 48 ks on 2000 March 5 with Chandra HETGS [@c00], at a roll angle of $153.2\arcdeg$. The dispersion directions of the High Energy Grating (HEG) and the Medium Energy Grating (MEG) made acute angles of $42.5\arcdeg$ and $37.5\arcdeg$, respectively, with the direction of the extended X-ray emission. Since the source has appreciable extent along the dispersion direction (see below), the spectral and spatial dimensions are mixed. Spectral extractions and reductions were performed using CIAO [^1] and IDL. We used extraction windows of $7\farcs9$ width in the cross-dispersion direction, which were centered on the nucleus of the 0-order image. The windows included most of the flux from the extended source. We then summed the plus and minus orders of the spectra. The flux calibration of HETGS is currently accurate to 10% for energies $E>2$ keV and 20% for $E<2$ keV.
The wavelength calibration was tied to an HETGS observation of Capella which was reduced in an identical manner to NGC 4151 (using CIAO, v. 2.0$\alpha$, 2000 August 6). In this way we determined that velocities must be corrected by $-210\pm20$ km s$^{-1}$ to remove a systematic wavelength calibration error that exists in standard-processed data. This error is currently attributed to an error in the CCD pixel scale, due to thermal contraction [^2]. Corrected velocities are reported in Table 1. Marginally redshifted X-ray narrow emission lines have been reported for NGC 5548 [@kml00 $270\pm100$ km s$^{-1}$] and NGC 3783 [@kbn00 $230\pm170$ km s$^{-1}$]. These redshifts may be owing to similar wavelength calibration errors.
NGC 4151 was in a low flux state during our observation, with $ 3\times10^{-3}$ photons s$^{-1}$ cm$^{-2}$ Å$^{-1}$ at 2 Å. We show the first order HEG and MEG spectra in Figure 1. Above 6 Å (below 2.1 keV), the spectrum is dominated by emission lines. Table 1 gives a list of the identified emission lines, their fluxes, and velocities relative to the galaxy rest frame.
We compare the HETGS 0-order X-ray (0.4-2.5 keV) image of NGC 4151 to an optical \[O [iii]{}\] 5007 image from the Hubble Space Telescope (HST) (Fig. 2). There is a good, but not exact correspondence between the structure of the optical and soft X-ray emission regions. The X-ray emission has a similar extent ($23\arcsec=1.6~h_{70}^{-1}$ kpc) and orientation (PA$=63\arcdeg$) to the optical narrow-line region (NLR). Most (70%) of the soft X-ray emission is resolved by Chandra; the rest comes from an FWHM$=0\farcs9$ nucleus. The hard X-ray emission (E$>$2.5 keV) is consistent with a nuclear point source.
The spatial profiles of the narrow emission lines are extended (Figure 3). We show three lines, from three different ionization states: O [vii]{} (f), O [viii]{} Ly$\alpha$, and Fe [i]{} K$\alpha$. All three lines have similar spatial profiles to the 0-order soft X-ray image, and can be traced out to a distance of $3\arcsec=210~h_{70}^{-1}$ pc from the nucleus.
HETGS is rather insensitive to relativistically broadened Fe K$\alpha$ emission. [@yew95] fit ASCA observations of Fe K$\alpha$ with broad ($\sigma=0.7$ keV) and narrow components having equivalent widths of $\sim 300$ eV and $\sim 100$ eV, respectively. We find an equivalent width of $160\pm20$ eV for narrow Fe [i]{} K$\alpha$ with the HEG. The narrow core of the line is unresolved, with FWHM$=1800\pm 200$ km/s.
Discussion
==========
Photoionized X-ray Line Emission
--------------------------------
The spectrum of NGC 4151 is hybrid, with line emission coming from both photoionized and collisionally ionized plasmas. There are several strong indications of a photoionized component. First, we see narrow radiative recombination continua (RRCs) from N [vii]{}, O [vii]{}, O [viii]{}, Ne [ix]{}, and Ne [x]{}. We estimate half widths (at 37% peak flux) for the RRCs which range from 1-4 eV, corresponding to temperatures of $2-4\times
10^4$ K. To have such a low temperature and contain such high ionization states, the plasma where the RRC originates must be dominated by photoionization.
The ratio of the O [vii]{} forbidden to resonance emission lines ($f/r=4.6\pm1.5$) is also consistent with photoionized plasma [@pd00]. In addition, relatively strong $n=3-1$ transitions of N [vi]{}, Ne [ix]{}, and Mg [xi]{} are indicative of photoionization [@bk00]. The strength of O, Ne, Mg, and Si K-shell emission lines and relative weakness of Fe L-shell lines are additional characteristics of a photoionized plasma [@l99].
Thermal X-ray Line Emission
---------------------------
Except for O [vii]{}, the He-like lines have a small $f/r$ ratio, which means that collisional ionization is important. The Ne [ix]{} lines have $f/r=1.7\pm0.4$. A ratio $f/r\sim3.4$ is expected for a photoionized plasma, and $f/r<0.8$ for a collisionally ionized plasma with $T>2\times10^6$ K. There is clearly a mixture of line emission from both types of plasma. We note that fluorescent excitation by the continuum and resonance scattering may lead to enhancement of resonance lines under special circumstances [@bka98; @klo96]. However, we would expect these processes to equally effect the O [vii]{} and other He-like resonance lines, which is not the case.
The Ly$\alpha$ lines of O [viii]{}, Ne [x]{}, and Mg [xii]{} also have a large contribution from collisionally ionized plasma. The O [viii]{} Ly$\alpha$ line is much stronger than the O [viii]{} RRC, while a ratio near unity is expected for a pure photoionized plasma. We find that $89\pm5\%$ of the O [viii]{} Ly$\alpha$ emission comes from the hot plasma component of the NLR. Similarly, $70\pm10\%$ of the Ne [x]{} Ly$\alpha$ emission comes from this component. A large fraction of the Mg [xii]{} Ly$\alpha$ flux must also have a collisional origin, since the Mg [xii]{} RRC (6.32 Å) is not even detected.
From the range of formation temperatures of the collisionally augmented lines, we deduce that there is plasma in the temperature range $T=0.3-1\times10^7$ K. The nondetection of S [xvi]{} emission (4.73 Å, $2\pm5\times10^{-6}$ photons cm$^{-2}$ s$^{-1}$) is partly owing to the strong continuum at that wavelength. However, the detection of Fe [xxv]{} ($T_f=8\times10^7$ K) suggests that hotter gas is present. A more thorough analysis will be necessary to derive the volume emission measure vs. temperature distribution of the hot plasma.
Continuum Emission
------------------
We characterize the continuum emission from NGC 4151 with the sum of two absorbed power laws. The hard continuum ($E>3$ keV) has a photon index $\Gamma=0.4\pm0.3$ and normalization $A = 2.3^{+1.6}_{-0.9}\times10^{-3}$ photons keV$^{-1}$ cm$^{-2}$ s$^{-1}$ at 1 keV, absorbed by a neutral column of $N_H=3.7^{+1.1}_{-0.9}\times10^{22}$ cm$^{-2}$. The observed 2-10 keV flux is $5.5 \times10^{-11}$ ergs cm$^{-2} s^{-1}$. The soft continuum ($E<1.3$ keV) has a photon index of $\Gamma=3.1\pm0.5$ and normalization $A = 1.0\pm0.1\times10^{-3}$ photons keV$^{-1}$ cm$^{-2}$ s$^{-1}$ at 1 keV, absorbed by a Galactic column of $N_H=2.0\times10^{20}$ cm$^{-2}$.
We deduce that there is strong absorption of the hard nuclear flux by a cloud which does not occult the bulk of the soft X-ray emitting NLR. @wyh94 observed a 2-10 keV flux of $11-22\times10^{-11}$ ergs cm$^{-2}
s^{-1}$, $\Gamma\sim1.5$, and $N_H=2-5\times10^{22}$ cm$^{-2}$ with ASCA. The hard continuum was harder and a factor of $2-4$ times fainter during our observation, but $N_H$ was similar. However, we note that poor statistics for energies above 7 keV in our data may bias the fit value for $\Gamma$. @wyh94 also found variations in $N_H$ with ASCA on a time scale of 6 months, which suggests that absorption is taking place on a spatial scale much smaller than the NLR.
The soft continuum of NGC 4151 has a much steeper photon index than the hard continuum (3.1 vs 0.4). This argues against Thomson scattered emission as the source of the soft continuum, unless the intrinsic nuclear continuum steepens drastically below 3 keV. Thermal bremsstrahlung and blended Fe L emission lines are therefore the most likely sources of the soft continuum.
Kinematics of the X-ray Line Region
-----------------------------------
The locations and velocity field of optically emitting clouds in the NLR of NGC 4151 have been mapped in detail by [@kbh00], and are consistent with a biconical outflow from the nucleus. Integrated long-slit spectroscopy of the central $2\arcsec$ of the NLR gives a blueshift of -90 km s$^{-1}$ and a line width (FWHM) of 460 km s$^{-1}$ for the \[O [iii]{}\] 5007 line [@hmb81]. The blueshifts and blue asymmetry of the optical lines are likely owing to obscuration of the far cone by dust.
The unblended X-ray emission lines have an (unweighted) mean blueshift of $-120\pm50$ km s$^{-1}$ with respect to the galaxy rest frame, as measured from its stellar spectrum. This is consistent with the velocity of the optical NLR. Since the X-ray line widths are affected by spatial broadening, we can only get an upper limit to their velocity widths from the widths of the low energy lines. O [vii]{} (f) has an apparent width (Gaussian FWHM) of $460$ km s$^{-1}$, which equals the widths of the optical lines.
It is interesting that the X-ray emission region is more extended in the front (SW) cone than the back cone (Fig. 2), suggesting that the soft X-rays from the back cone are obscured, like the optical emission. The similar spatial extent of the X-ray and optical emission line regions (Figs. 2,3) and their similar kinematics are strong evidence that they are closely linked. X-ray emission from the NLR demonstrates that high energy processes are important in this region, at distances of 70-800 pc from the galactic nucleus.
The Multiphase Nature of the NLR
--------------------------------
For the first time, there is direct evidence for at least 2 distinct gas phases in the extended NLR of a Seyfert galaxy. One of the phases is cool ($T=3\times10^4$ K) and photoionized, while the second is hot ($T\sim10^7$ K) and collisionally ionized. In addition, there is a large range in ionization of the photoionized region, as demonstrated by O [i]{}-O [iii]{} in the optical-UV spectrum and O [vii]{} in the X-ray spectrum. There are also strong K$\alpha$ lines from neutral species in the X-ray spectrum, including Fe [i]{}, Si [i]{}, and Mg [i]{}. Detailed modeling will be necessary to determine the volume emission measure vs. ionization distribution of the photoionized component. A wide range of ionization is derived from photoionization models of the NLR of the Circinus galaxy [@skp00], and might be explained by a radial gradient in the NLR density.
A large percentage ($65\pm9\%$) of the narrow Fe [i]{} K$\alpha$ emission comes from the extended NLR (Fig. 3). This is contrary to the popular idea that the narrow emission comes primarily from a parsec scale torus. The remaining unresolved emission, which has an equivalent width of $56\pm9$ eV, may very well come from a torus. Unlike NGC 5548 [@ygn00], there is no evidence for Doppler broadening of the narrow Fe [i]{} K$\alpha$ line in NGC 4151.
It has been suggested that a hot, inter-cloud medium is necessary for pressure confinement of the NLR clouds [@ebh83]. The hot plasma we detect in our X-ray spectrum may form this medium. We derive volume emission measures for the thermal contribution to the Ne [x]{} and O [viii]{} Ly$\alpha$ lines of $n_e^2V=2-4\times10^{63} h_{70}^{-2}$ cm$^{-3}$, assuming $T=1\times10^7 K$, and line emissivities for solar abundances from [@mgv85]. [@ckh00] derive a half-opening angle of $36\arcdeg$ from kinematic models of the optical NLR. If the thermal X-ray line emission comes from a uniformly filled ($f=1$) bicone in the central $4\arcsec$ of the galaxy, then the mean plasma density is $n_e=3$ cm$^{-3}$ and the pressure is $n_eT=3\times10^7$ cm$^{-3}$ K. Pressure equilibrium between the hot and cool phases of the X-ray NLR would then require a density of $n_e=10^3$ cm$^{-3}$ for the cool phase.
The optical NLR clouds at $r=6-20\arcsec$ have a density $n_e=220$ cm$^{-3}$ and a pressure of $n_eT=3\times10^6$ cm$^{-3}$ K [@pra90; @mwe95], much lower than what we find closer to the nucleus. However, we know that the spatial profile of the NLR flux is centrally peaked. A density profile of $r^{-2}$ would account for the larger pressure of the nucleus than the extended NLR, assuming constant temperature. Estimates of the pressure can be improved by modeling the density distribution to match the spatial profiles of the X-ray emission.
A possible alternative is that the hot plasma is confined to shocks between the NL clouds and the host galaxy ISM. In that case, its filling factor $f$ would be much lower, and its density and pressure much higher than calculated above. The X-ray emission we observe in the Chandra image has a clumpy morphology which follows the optically emitting clouds. This may be partly owing to emission from a shocked component. There is evidence for deceleration of the NL clouds at a radius of about $2\arcsec$ from the nucleus [@ckh00], perhaps from interaction with the ISM.
Conclusions
===========
The power of Chandra HETGS to elucidate the nature of the X-ray emission from active galactic nuclei is demonstrated by high resolution spectra and images of NGC 4151. Direct emission from the nucleus is highly absorbed. Strong, narrow X-ray emission lines are seen from the spatially resolved NLR of NGC 4151. The narrow radiative recombination continuum features and the O [vii]{} $f/r$ ratio are consistent with photoionization from the hard nuclear X-ray source. There is also narrow line emission from a hot, collisionally ionized component in the extended NLR. This may come from the long-sought intercloud medium or shocked plasma. In addition, we find strong narrow Fe [i]{} K$\alpha$ emission from the extended NLR. The composite nature of the NLR indicates that both photoionization and collisional heating are important. The extent and kinematics of both X-ray photoionized clouds and hot thermal plasma match those of the optical NLR, showing that they are closely connected. We also suggest that the narrow X-ray emission lines seen in some other Seyfert galaxies may come from the NLR, and may be interpreted in the light of these NGC 4151 observations.
We thank everyone whose hard work made Chandra and HETGS possible. We also thank the referee, Kim Weaver, for many helpful comments. This work was funded in part by contracts NAS8-38249 and SAO SV1-61010. Some of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
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[lcccl]{}
1.79 & 460$\pm$420 & 3.3E-05 & 2.3 & Fe [xxv]{}\
1.937 & 250$\pm$80 & 1.8E-04 &11.0 & Fe [i]{} K$\alpha$\
6.182 & -20$\pm$120 & 1.0E-05 & 4.1 & Si [xiv]{} Ly$\alpha$\
6.648 & -30$\pm$130 & 1.1E-05 & 4.4 & Si [xiii]{} r\
6.740 & -40$\pm$40 & 1.1E-05 & 5.0 & Si [xiii]{} f\
7.106 &-380$\pm$110 & 5.3E-06 & 3.5 & Mg [xii]{} Ly$\beta$\
7.130 &-380$\pm$40 & 1.3E-05 & 5.6 & Si [i]{} K$\alpha$\
7.851 &-560$\pm$80 & 2.5E-06 & 2.4 & Mg [xi]{} 1s3p-1s$^2$\
8.421 &-210$\pm$90 & 1.1E-05 & 7.4 & Mg [xii]{} Ly$\alpha$\
9.01 &-380$\pm$170 & 3.5E-06 & 3.3 & Fe [xxii]{}\
9.102 && 7.2E-06 & 3.2 & Ne [x]{} RRC\
9.169 & -20$\pm$160 & 9.3E-06 & 6.3 & Mg [xi]{} r\
9.314 &-406$\pm$140 & 1.2E-05 & 7.4 & Mg [xi]{} f\
9.887 &-330$\pm$50 & 3.2E-06 & 2.7 & Mg [i]{} K$\alpha$\
10.239 &-180$\pm$70 & 7.4E-06 & 4.5 & Ne [x]{} Ly$\beta$\
10.368 && 9.3E-06 & 4.2 & Ne [ix]{} RRC\
11.001 & 110$\pm$140 & 3.8E-06 & 2.2 & Ne [ix]{} 1s4p-1s$^2$\
11.547 &-290$\pm$70 & 5.9E-06 & 3.0 & Ne [ix]{} 1s3p-1s$^2$\
12.134 & -70$\pm$80 & 2.0E-05 & 6.4 & Ne [x]{} Ly$\alpha$\
13.447 &-150$\pm$40 & 2.0E-05 & 4.8 & Ne [ix]{} r\
13.553 &-210$\pm$160 & 9.9E-06 & 2.5 & Ne [ix]{} i\
13.698 & 10$\pm$40 & 3.3E-05 & 6.1 & Ne [ix]{} f\
14.228 && 1.1E-05 & 2.3 & O [viii]{} RRC\
15.176 &-170$\pm$50 & 1.1E-05 & 3.2 & O [viii]{} Ly$\gamma$\
16.006 &-250$\pm$20 & 1.8E-05 & 3.8 & O [viii]{} Ly$\beta$\
16.771 && 5.2E-05 & 4.7 & O [vii]{} RRC\
18.588 && 2.6E-05 & 2.6 & N [vii]{} RRC\
18.969 & -40$\pm$30 & 1.0E-04 & 7.0 & O [viii]{} Ly$\alpha$\
21.602 & 106$\pm$20 & 6.8E-05 & 4.1 & O [vii]{} r\
21.804 &-410$\pm$140 & 5.1E-05 & 3.3 & O [vii]{} i\
22.101 &-132$\pm$20 & 3.1E-04 & 8.3 & O [vii]{} f\
24.781 &-190$\pm$60 & 6.3E-05 & 4.0 & N [vii]{} Ly$\alpha$\
24.898 & -10$\pm$20 & 3.9E-05 & 3.0 & N [vi]{} 1s3p-1s$^2$\
**Figure Captions**
[**Figure 1**]{}\
Chandra first order HEG (top) and MEG (middle, bottom) spectra of NGC 4151, with line identifications. Above 6 Å, the spectrum is dominated by narrow emission lines from both thermal and photoionized plasmas. Note that the ratio of forbidden to resonance emission in the He-like O [vii]{} lines is large, while the ratio for Ne [ix]{}, Mg [xi]{}, and Si [xiii]{} is smaller, indicating a hybrid spectrum. The narrow radiative recombination continua (RRCs) of several ions attest to photoionization in a cool $T=3\times10^4 K$ plasma.
[**Figure 2**]{}\
Comparison of X-ray and optical emission from the extended NLR in NGC 4151. Chandra contours (0.4-2.5 keV band) are overlayed on an HST \[O [iii]{}\] 5007 image. A 36$\farcs$4 by 23$\farcs$5 region is shown, with N at top and E at left. Note that the SW (front) cone is more extended than the NE (back) cone in both bands. X-ray contours are separated by factors of 2 in surface brightness, with the lowest contour at 2.1 photons arcsec$^{-2}$.
[**Figure 3**]{}\
Comparison of NGC 4151 zero-order spatial profile and 1st order emission line spatial profiles. The gaussian profile of the unresolved nuclear component is shown for comparison. The spatial profiles of O [vii]{}, O [viii]{}, and Fe K$\alpha$ are all resolved, from a region at least $6\arcsec$ in extent.
[^1]: http://asc.harvard.edu/ciao
[^2]: http://asc.harvard.edu/cal
|
---
abstract: |
Let $W$ be a compact simply connected triangulated manifold with boundary and $K\subset W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of ${W\backslash K}$ out of a model of the map of pairs $(K,K \cap {\partial W})\hookrightarrow (W,{\partial W})$ under some high codimension hypothesis.
We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicits models of these configuration spaces for a large class of compact manifolds.
address:
- 'H.C.B. and P.L.: IRMP, Université catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium'
- |
D.S.: University of Regina, Department of Mathematics\
College West, Regina, CANADA
author:
- Hector Cordova Bulens
- Pascal Lambrechts
- Don Stanley
date: 18 May 2015
title: |
Rational models of the complement\
of a subpolyhedron in a manifold\
with boundary
---
[^1]
Introduction
============
Let $W$ be a compact and simply-connected manifold with boundary (in this paper all manifolds are triangulated). Let $f\colon K
\hookrightarrow W$ be the inclusion of a subpolyhedron. The first goal of this paper is to determine the rational homotopy type of the complement ${W\backslash K}$. We will then apply this to deduce the rational homotopy type of the configuration space of two points in a manifold with boundary under 2-connectedness hypotheses. Hence this paper extends the results of [@LaSt:PE] and [@LaSt:FM2] to the case of manifolds *with boundary*.
The main result of [@LaSt:PE] is an explicit description of the rational homotopy type of ${W\backslash K}$ when $W$ is a *closed* manifold and $K$ is a subpolyhedron of codimension $ \geq(\dim W)/2 +2$. This rational homotopy type depends only on the rational homotopy class of the inclusion $K \hookrightarrow W$ ([@LaSt:PE Theorem 1.2]).
The situation for manifolds with boundary is different. For example, let $W$ be an $n$-dimensional disk $D^n$ and $K$ be a point. If $K$ is embedded in the interior of $D^n$ then ${W\backslash K}\simeq S^{n-1}$. On the contrary, if $K$ is embedded in the boundary of $D^n$ then ${W\backslash K}\simeq *$. Hence the complements ${W\backslash K}$ have different rational homotopy types, although the two inclusions $K\hookrightarrow W$ are homotopic. These examples show that we need more information to determine the rational homotopy type of ${W\backslash K}$. Our main result is that the only extra information needed is related to the inclusion of ${\partial W}\cap K$ in ${\partial W}$. More precisely, we have the following result
\[T:intro1\] Let $W$ be a compact simply connected triangulated manifold with boundary and let $K$ be a subpolyhedron in $W$.
Assume that $$\label{e:codim_intro}
\dim W \geq 2\dim K + 3.$$ Then the rational homotopy type of ${W\backslash K}$ depends only on the rational homotopy type of the square of inclusions $$\label{d:intro}
\xymatrix{ (K\cap {\partial W}) \ar@{^{(}->}[r]\ar@{^{(}->}[d] & {\partial W}\ar@{^{(}->}[d] \\
K \ar@{^{(}->}[r] & W.}$$ Moreover a CDGA model of ${W\backslash K}$ (that is, an algebraic model in the sense of Sullivan of this rational homotopy type, see Section \[s:RHT\]) can be explicitely constructed out of any CDGA model of Diagram .
Actually we will see that the high codimension hypothesis can be weakened. Indeed we will establish a sharp *unknotting condition*, which is an inequality relating the connectivity of the inclusion maps and the dimensions of the manifold and the subpolyhedron (see in Corollary \[C:mod\_conndim\]), under which we still get a CDGA model of the complement.
There is an interesting application of this theorem to the study of configuration spaces of 2 points in $W$, $${\operatorname{Conf}}(W,2) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\{(x_1, x_2) \in W\times W \colon x_1 \neq x_2\}.$$ Indeed this configuration space is the complement $${\operatorname{Conf}}(W,2) = W\times W {\setminus}\Delta (W)$$ where $\Delta \colon W \hookrightarrow W\times W$ is the diagonal embedding. We will deduce from Theorem \[T:intro1\] the following result.
\[T:intro2\] Let $W$ be a $2$-connected compact manifold with a $2$-connected or empty boundary. The rational homotopy type of the configuration space ${\operatorname{Conf}}(W,2)$ depends only on the rational homotopy type of the pair $(W,{\partial W})$.
In [@CLS:pretty] we prove that a large class of compact manifolds with boundary admit CDGA models of a special form that we call *surjective pretty models*. This class contains in particular even-dimensional disk bundles over a closed manifold and complements of high codimensional polyhedra in closed manifolds. As a consequence, such manifolds admit a CDGA model of the form $P/I$ where $P$ is a Poincaré duality CDGA and $I$ is some differential ideal. Poincaré duality CDGAs come with a natural *diagonal class* $\Delta \in (P\otimes P)^n$. We then get the following elegant model for the configuration space (see Section \[S:ConfPretty\] for more details)
(Theorem \[T:modele\_FW2\]) \[T:introFW2\] Let $W$ be a compact manifold of dimension $n$ with boundary and assume that $W$ and ${\partial W}$ are $2$-connected. If $(W,{\partial W})$ admits a surjective pretty model in the sense of [@CLS:pretty] then a CDGA model of ${\operatorname{Conf}}(W, 2)$ is given by $$\left(P/I \otimes P/I \right)\oplus_{\overline{\Delta^!}} ss^{-n} P/I,$$ where $P$ is the Poincaré duality CDGA and $I$ the ideal associated to the pretty model, and $\overline{\Delta^!}$ is a map induced by multiplication by the diagonal class $\Delta\in (P\otimes P)^n$.
When $W$ is a closed manifold, we have $I=0$ and the model of Theorem \[T:introFW2\] is exactly that of [@LaSt:FM2].
In the paper [@CLS:ConfWk] in preparation we will show how to build a model (of dgmodules) of ${\operatorname{Conf}}(W,k)$, $k\geq2$, which enables to compute effectively the homology of the space of configurations of any number of points in a manifold with boundary. This model will be of the form $$\left(\frac{(P/I)^{\otimes k}\otimes\Lambda(g_{ij}:1\leq i<j\leq
k)}{\textrm{(Arnold and symmetry
relations)}}\,,\,d(g_{ij})=\pi_{ij}^*(\overline{\Delta})\right),$$ mimicking the model in [@LaSt:remFMk].
Here is the plan of the paper. Section \[s:map\_con\] contains a very short review on rational homotopy theory, the notion of truncation of a CDGA, a discussion on CDGA structures on mapping cones, and the notion of homotopy kernel. Section \[s:lef\_dual\] is a first step to the understanding of a dgmodule model of the complement $W\setminus K$ and in Section \[s:complement\] we establish a CDGA model of that complement. In Section \[S:ConfW2\] we apply the previous results to the model of the configuration space of $2$ points in compact manifolds, with some developments of the examples of configuration spaces on a disk bundle or in the complement of a polyhedron in a closed manifold.
Truncation of dgmodules and CDGA’s, and CDGA structures on mapping cones. {#s:map_con}
=========================================================================
This section contains a quick review on some classical topics that we will need with some special development. In particular in \[S:truncation\_cdga\] we explain some notion of truncation of a CDGA, and in \[s:semi\_trivial\] we show how to endow a mapping cone (or its truncation) with the structure of a CDGA.
Rational homotopy theory {#s:RHT}
-------------------------
In this paper we will use the standard tools and results of rational homotopy theory, following [@FHT:RHT]. Recall that ${A_{PL}}$ is the Sullivan-de Rham functor and that for a 1-connected space of finite type, $X$, ${A_{PL}}(X)$ is a commutative differential graded algebra (CDGA for short), which completely encodes the rational homotopy type of $X$. Any CDGA weakly equivalent to ${A_{PL}}(X)$ is called a *CDGA model of $X$*. All our dgmodules and CDGAs are over the field ${\mathbb{Q}}$.
Truncation of a dgmodule
------------------------
The classical truncation of a cochain complex, i.e. ${\mathbb{Q}}$-dgmodule, $C$, is classicaly defined by (see [@Wei:HA Section 1.2.7]) $$\label{Eq:tronWeibel}
(\hat\tau^{\leq N} C)^i =
\begin{cases}
C^i & \text{ if } i<N\\
C^N \cap \ker d & \text{ if } i=N \\
0& \text{ if } i >N
\end{cases}$$ This comes with an inclusion $$\iota \colon \hat\tau^{\leq N} C \hookrightarrow C$$ which induces isomorphisms $H^i(\iota)$, for $i\leq N$, and such that $H^{>N}$ $(\hat\tau^{\leq N} C) =0$.
When $R$ is an $A$-dgmodule, the truncation $\hat\tau^{\leq N} R$ is not necessarily an $A$-dgmodule. In that case a better replacement would be to take for the truncation a quotient $R/I$ where $I$ is a suitable $A$-dgsubmodule such that $I^i=R^i$ for $i>N$. In this paper we will use the following:
Let $R$ be an $A$-dgmodule and let $N$ be a positive integer. A *truncation below degree $N$ of $R$* is an $A$-dgmodule, ${\tau^{\leq N}}R$, and a morphism $\pi \colon R \to~{\tau^{\leq N}}R$ of $A$-dgmodules verifying the two following conditions:
1. $({\tau^{\leq N}}R)^{>N} = 0$ and $({\tau^{\leq N}}R)^{<N}\cong R^{<N}$, and
2. the morphism $\pi$ is a surjection of $A$-dgmodules such that $H^i(\pi)$ is an isomorphism for $0\leq i \leq N$.
Contrary to $\hat \tau^{\leq N}$ from , our truncation ${\tau^{\leq N}}R$ is not unique and is not a functorial construction.
Truncation of a CDGA {#s:truncation_cdga}
--------------------
\[S:truncation\_cdga\]
Let $A$ be a connected CDGA. *A CDGA truncation below degree $N$ of $A$* is a truncation of $A$-dgmodule $ ({\tau^{\leq N}}A, \pi)$ such that ${\tau^{\leq N}}A$ is a CDGA and $\pi:A\to \hat\tau^{\leq N} A$ is a CDGA morphism.
Equivalently a CDGA truncation can be seen as a projection $\pi \colon A \to A/I$ where $I$ is an ideal of $A$ such that $I^{<N}=0$, $I^{>N}=A^{>N}$ and $I^N\oplus (\ker d \cap A^N)= A^N$.
Any two CDGA truncations below degree $N$ of a given connected CDGA are weakly equivalent.
Let $A$ be a connected CDGA and $N\in \mathbb{N}$. It is easy to construct a relative Sullivan model $$\xymatrix{\iota \colon A \ar@{ >->}[r] & (A\otimes \Lambda V, D)}$$ such that $H^{\leq N} (\iota)$ is an isomorphism, $V=V^{\geq N}$ and $H^{>N} (A\otimes \Lambda V,D)=0$. Indeed, one builds inductively $V=V^{\geq N}$ by adding generators to eliminate all the homology in degrees $>N$. It is straightforward to check that any CDGA truncation $\pi \colon A \to {\tau^{\leq N}}A$ factors as follows $$\xymatrix{A \ar[rr]^{\pi} \ar@{>->}[dr]_{\iota} &&{\tau^{\leq N} A}\\
& (A\otimes \Lambda V, D) \ar[ur]_{m} &}$$ where $m(V)=0$. Since $H^{\leq N} (\iota)$ and $H^{\leq N} (\pi)$ are isomorphisms and $H^{>N}(A\otimes \Lambda V, D)= H^{>N}({\tau^{\leq N} A})=0$, we deduce that $m$ is a quasi-isomorphism. Therefore any two truncation of $A$ are quasi-isomorphic to $(A\otimes \Lambda V, D)$, and hence are weakly equivalent
Semi-trivial C(D)GA structures on mapping cones {#s:semi_trivial}
-----------------------------------------------
Let $A$ be a CDGA and let $R$ be an $A$-dgmodule. We will denote by $s^k R$ the k-th suspension of $R$, i.e. $(s^k R)^p= R^{k+p}$, and for a map of $A$-dgmodules, $f\colon R \to Q$, we denote by $s^k f$ the k-th suspension of $f$. Furthermore, we will use $\#$ to denote the linear dual of a vector space, $\#V=hom(V,\mathbb{Q})$, and $\#f$ to denote the linear dual of a map $f$.
If $f\colon Q \to R$ is an $A$-dgmodule morphism, *the mapping cone* of $f$ is the $A$-dgmodule $$C(f){\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}(R\oplus_f sQ, \delta)$$ defined by $R\oplus sQ$ as an $A$-module and with a differential $\delta$ such that $\delta (r,sq) = (d_R (r) +f(q), -sd_Q(q))$.
When $R=A$, the mapping cone $C(f\colon Q\to A)$ can be equipped with a unique commutative graded algebra (CGA) structure that extends the algebra structure on $A$, respects the $A$-dgmodule structure, and such that $(s q) \cdot (sq')=0$, for $q,q' \in Q$. We will call this structure the *semi-trivial CGA structure* on the mapping cone $A\oplus_fsQ$ (see [@LaSt:PE Section 4]). The following result is very useful to detect when this CGA structure is, in fact, a CDGA structure.
\[D:balanced\] Let $A$ be a CDGA. An $A$-dgmodule morphism $f\colon Q \to A$ is *balanced* if : $$f(x)y=xf(y) \ \ \text{ for all $x,y\in Q$.}
\label{Eq:anodyne}$$
The importance of this notion comes from the following proposition.
\[p:nagata\] Let $Q$ be an $A$-dgmodule and $f\colon Q \to A$ be an $A$-dgmodule morphism. If $f$ is balanced then the mapping cone $C(f)=A\oplus_f sQ$ endowed with the semi-trivial CGA structure is a CDGA.
The only non trivially verified condition for $C(f)$ being a CDGA is the Leibniz rule for the differential, which is a consequence of $(\ref{Eq:anodyne})$. See the proof of [@CLS:pretty Proposition 2.2] for more details.
\[p:truncMC\] Let $A$ be a connected CDGA and let $f\colon Q\to A$ be an $A$-dgmodule morphism. Let $p$ and $N$ be natural integers such that $Q^{<p}=0$ and $N\leq 2p-3$. Then the semi-trivial CGA structure on the mapping cone $C(f)$ induces a CDGA structure on $\tau^{\leq N} (C(f))$, and $$A\to \tau^{\leq N} (C(f))$$ is a CDGA morphism.
Analogous argument as for the proof of Proposition 2.5. See the proof of [@LaSt:PE lemme 4.5] for more details.
In the rest of this paper, when a mapping cone is equipped with a CDGA structure it will be understood that it comes from the semi-trivial structure.
Homotopy kernel {#S:hoker}
---------------
In this section we recall the notion of homotopy kernel and some of its properties.
Let $f \colon M \to N$ be a morphism of $A$-dgmodules. The *homotopy kernel* of $f$ is the $A$-dgmodule mapping cone $$\mathrm{hoker} \ f {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}s^{-1}N\oplus_{s^{-1}f} M,$$ which comes with an obvious map $$\mathrm{hoker} \ f \to M \ ; \ (s^{-1}n, m) \mapsto m.$$
The following result is a consequence of the five lemma and justifies the terminology “homotopy kernel”.
Let $f\colon M \to N$ be a surjective morphism of $A$-dgmodules. Then the morphism $$\begin{array}{cccc}
\varphi : &\ker f &{\stackrel{\simeq}\longrightarrow}& \mathrm{hoker} \ f \\
& m &\longmapsto & (0, m)\end{array}$$ is an $A$-dgmodule quasi-isomorphim.
Lefschetz duality for manifolds with boundary {#s:lef_dual}
=============================================
The aim of this section is to prove Proposition \[modele\_apl\_bord\] below, which is a first step towards the description of the rational homotopy type of the complement of a subpolyhedron in a manifold with boundary.
Let $W$ be a closed connected oriented triangulated manifold of dimension $n$ with boundary and let $f: K\hookrightarrow W$ be the inclusion of a connected subpolyhedron of dimension $k$ in $W$. Denote by ${\partial W}$ the boundary of $W$ and set $$\label{E:dWK}
\partial_W K := K\cap \partial W.$$ In this section we will construct a dgmodule model of $W\setminus K$, extending [@LaSt:remFMk Theorem 6.3] to manifolds with boundary. Consider the diagram $$\label{diag1}
\xymatrix{
W &\ar@{_{(}->}_{f}[l] K \\
{\partial W}\ar@{_{(}->}[u] & \ar@{_{(}->}^{\partial f}[l] {\partial_W K},\ar@{_{(}->}[u], }$$ which after applying the ${A_{PL}}$ functor gives $$\label{diag2}
\xymatrix{
{A_{PL}}(W) \ar[r]^-{{A_{PL}}(f)} \ar[d] & {A_{PL}}(K)\ar[d] \\
{A_{PL}}({\partial W})\ar[r] &{A_{PL}}({\partial_W K}). }$$ Recall that for a map of spaces $Y\to X$, we set $${A_{PL}}(X,Y)= \ker ({A_{PL}}(X) \to {A_{PL}}(Y)).$$ The inclusion of pairs $$i\colon(K,{\partial_W K})\hookrightarrow (W,{\partial W})$$ induces an ${A_{PL}}(W)-$dgmodule morphism $$\label{E:apli}
{A_{PL}}(i) \colon {A_{PL}}(W, {\partial W}) \to {A_{PL}}(K,{\partial_W K}).$$ Using our notation for mapping cones, suspension and linear duals from Section \[s:semi\_trivial\], consider the map $$s^{-n}\# {A_{PL}}(i) \colon {s^{-n}\#}{A_{PL}}(K, {\partial_W K}) \to {s^{-n}\#}{A_{PL}}(W,{\partial W})$$ and its mapping cone $$\label{E:MCs-nApli}
C\left(s^{-n}\#
{A_{PL}}(i)\right)\,\,=\,\,{s^{-n}\#}{A_{PL}}(W,{\partial W})\,\oplus_{s^{-n}\# {A_{PL}}(i)}\,s
{s^{-n}\#}{A_{PL}}(K, {\partial_W K})$$ with the inclusion $$\label{E:iotaCsapli}
\iota\colon
{s^{-n}\#}{A_{PL}}(W,{\partial W})
\hookrightarrow
C\left(s^{-n}\#
{A_{PL}}(i)\right).$$ Since $(W,{\partial W})$ is an oriented compact manifold of dimension $n$, Poincaré duality induces a quasi-isomorphism of ${A_{PL}}(W)-$dgmodules $$\label{E:PhiW1}
\Phi_W\colon{A_{PL}}(W){\stackrel{\simeq}\longrightarrow}s^{-n}\#{A_{PL}}(W,{\partial W})$$ (see $(\ref{E:PhiW})$ in the proof of Proposition \[modele\_apl\_bord\] below for an explicit description of $\Phi_W$.)
\[modele\_apl\_bord\] The map $${A_{PL}}(W)\to{A_{PL}}({W\backslash K})$$ is weakly equivalent in the category of ${A_{PL}}(W)$-dgmodules to the map $$\iota\circ\Phi_W\colon{A_{PL}}(W)\to C(s^{-n}\# {A_{PL}}(i))$$ where $C(s^{-n}\# {A_{PL}}(i))$ is the mapping cone , $\iota$ is from , and $\Phi_W$ is from .
First we review from [@LaSt:remFMk Section 4] a variation of the functor ${A_{PL}}$ defined on ordered simplicial complex and having an improved excision property. Recall from [@FHT:RHT Chapter 10] that ${A_{PL}}$ is actually defined first on simplicial sets. Consider the category, ${\mathcal K}$, of ordered simplicial complexes. To any ordered simplicial complex, $K$, we can associate naturally a simplicial set, $K_\bullet$, whose non-degenerate simplices are exactly the simplices of $K$ (see [@Cur:SHT p.108]). Define the functor $$\widehat{{A_{PL}}} \colon {\mathcal K} \to ADGC ; K \to {A_{PL}}(K_\bullet).$$ This functor verifies the two following properties (see [@LaSt:remFMk Section 4]):
1. ${A_{PL}}(| K|) \simeq\widehat{{A_{PL}}} (K)$ naturally for every ordered simplicial complex (where $|K|$ is the geometric realization).
2. : Let $(K,L)$ be a pair of ordered simplicial complexes. Let $K'\subset K$ a sub-complex and $L'= K'\cap L$. If $K'\cup L= K$ then the inclusion $j \colon (K', L') \hookrightarrow (K, L)$ induces an *isomorphism* $$\widehat{{A_{PL}}}(j) \colon \widehat{{A_{PL}}}(K,L)
\stackrel{\cong}\longrightarrow \widehat{{A_{PL}}}(K', L').$$ (Note that ${A_{PL}}(j)$ is a quasi-isomorphism by the classical excision property.)
Consider now the triangulated compact manifold $W$ and its subpolyhedron $K$. Replace those polyhedra $W$ and $K$ by their second barycentric subdivision. Denote by $T$ the star of $K$ in $W$, which is a regular neighborhood (see [@Hud:PLT chapters 1 and 2]), hence $T$ is a codimension 0 submanifold with boundary and it retracts by deformation onto $K$. It is clear that the topological closure $\overline{W{\setminus}T}$ of ${W{\setminus}T}$ is homotopy equivalent to ${W\backslash K}$. Set $$\begin{aligned}
{\partial_+ T}&=&{\partial T}\cap {\partial W}\\ {\partial_- T}&=&\overline{ ({\partial T}\cap (W{\setminus}{\partial W}))}=T\cap \overline{W{\setminus}T} \\ \partial_0 T &=& {\partial_+ T}\cap {\partial_- T},\end{aligned}$$ which gives a decomposition of the boundary of $T$, $${\partial T}= {\partial_+ T}\cup_{{\partial_0 T}} {\partial_- T}.$$
Our next goal is to set up Diagram below. Let us fix an arbitrary order on the vertices of the simplicial complex $W$ such that $W$ and the subpolyhedron $T$, ${\partial T}$, ${\partial_+ T}$, ${\partial_- T}$, ${\partial_0 T}$, $K$ and ${\partial_W K}$ turn into ordered simplicial complexes. We can apply to them the functor ${\widehat{{A_{PL}}}}$ which is naturally quasi-isomorphic to ${A_{PL}}$. To prove the result, it suffices to show that the mapping cone $C({s^{-n}\#}$ ${\widehat{{A_{PL}}}}(i))$ is a model of ${\widehat{{A_{PL}}}}(W)$-dgmodule of ${\widehat{{A_{PL}}}}({\overline{W\backslash T}})$. To ease notations, in the rest of this proof we will write ${A_{PL}}$ instead of ${\widehat{{A_{PL}}}}$.
The inclusion of the pair $$(T,{\partial T}) \hookrightarrow (W, \overline{W\backslash T} \cup {\partial W})$$ induces by the strong excision property above an isomorphism $${A_{PL}}(W,\overline{W\backslash T}\cup {\partial W}) \stackrel{\cong}\rightarrow {A_{PL}}(T,{\partial T}).$$ Denote by $n$ the dimension of $W$. By Poincaré duality of the pair $(W,{\partial W})$, there exists an orientation $$\epsilon_W : {A_{PL}}(W,{\partial W}) \to s^{-n}\mathbb{Q},$$ i.e. a morphism of cochain complexes that induces an isomorphism in cohomology in degree $n$. Using this morphism we can define a morphism of ${A_{PL}}(W)$-dgmodules $$\label{E:PhiW}
\begin{array}{cccc}\Phi_W \colon& {A_{PL}}(W) &\longrightarrow &{s^{-n}\#}{A_{PL}}(W,{\partial W})\\
& \alpha & \longmapsto & \left(\Phi_W(\alpha)\colon \beta \mapsto
\epsilon_W (\alpha \beta)\right), \end{array}$$ which is a quasi-isomorphism by Poincaré duality of the pair $(W,{\partial W})$. The composition $$\epsilon_T: {A_{PL}}(T,{\partial T})\cong{A_{PL}}(W, \overline{W\backslash T} \cup {\partial W})\stackrel{{A_{PL}}(incl)}\longrightarrow {A_{PL}}(W,{\partial W}) \stackrel{\epsilon_W}{\longrightarrow} s^{n}\mathbb{Q}$$ induces an isomorphism in cohomology in degree $n$. Define $$\begin{array}{cccc}\Phi_T \colon& {A_{PL}}(T) &\longrightarrow &{s^{-n}\#}{A_{PL}}(T,{\partial T})\\
& \alpha & \longmapsto & \left(\Phi_T(\alpha)\colon \beta \mapsto \epsilon_T (\alpha \beta)\right) \end{array}$$ which is a quasi-isomorphism of ${A_{PL}}(W)$-dgmodules by Poincaré duality of the pair $(T,{\partial T})$. Also, using the quasi-isomorphism above and the five lemma, it is not difficult to see that the morphism $$\begin{array}{cccc}\tilde\Phi_T \colon& {A_{PL}}(T,{\partial_- T}) &\longrightarrow &{s^{-n}\#}{A_{PL}}(T,{\partial_+ T})\\
& \alpha & \longmapsto & \left(\tilde\Phi_T(\alpha)\colon \beta \mapsto \epsilon_T (\alpha \beta)\right) \end{array}$$ is a quasi-isomorphism of ${A_{PL}}(T)$-dgmodules, hence of ${A_{PL}}(W)$-dgmodules.
The inclusion $$(K,{\partial_W K}) \hookrightarrow (T,{\partial_+ T})$$ is a homotopy equivalence and induces a weak equivalence of ${A_{PL}}(W)$-dgmodules $${A_{PL}}(T,{\partial_+ T}) {\stackrel{\simeq}\longrightarrow}{A_{PL}}(K,{\partial_W K}).$$ By the strong excision property, the inclusion $$(T,{\partial_- T}) \hookrightarrow (W, {\overline{W\backslash T}})$$ induces an isomorphism $${A_{PL}}(W, {\overline{W\backslash T}}) \stackrel{\cong}\rightarrow {A_{PL}}(T,{\partial_- T}).$$ Combining all these morphisms we get the following commutative diagram of ${A_{PL}}(W)$-dgmodules $$\label{D:big}
{\xymatrix{
0\ar[r]& 0 \ar[r]\ar[d]^{0}\ar@{}[rd]|{(\ast)} &{A_{PL}}(W) \ar@{=}[r]\ar@{=}[d] &{A_{PL}}(W)\ar[r]\ar[d]_{{A_{PL}}(j)}& 0 \\
0\ar[r]& {A_{PL}}(W,{\overline{W\backslash T}}) \ar[r]\ar[d]^{\cong \text{ exc}} &{A_{PL}}(W) \ar[r]_{{A_{PL}}(j)}\ar[dd]_{\simeq}^{\Phi_W} &{A_{PL}}({\overline{W\backslash T}})\ar[r]& 0 \\
&{A_{PL}}(T,{\partial_- T}) \ar[d]^{\simeq \tilde\Phi_T}& \\
&s^{-n}\#{A_{PL}}(T,{\partial_+ T})\ar[r]&s^{-n}\#{A_{PL}}(W,{\partial W})\ar@{=}[d]\\
&s^{-n}\#{A_{PL}}(K,{\partial_W K}) \ar[u]_{\simeq} \ar[r]_{s^{-n}\#{A_{PL}}(i)}&
s^{-n}\#{A_{PL}}(W,{\partial W})
}}
$$ and the two top lines are short exact sequences.
Properties of mapping cones and of short exact sequences imply that, in the category of ${A_{PL}}(W)$-dgmodules, the morphism $$\label{E:aplj}
{A_{PL}}(j)\colon{A_{PL}}(W)\to{A_{PL}}({\overline{W\backslash T}})$$ on the top right of is equivalent to the map induced between the mapping cones of the horizontal maps of the square $(\ast)$ in Diagram $(\ref{D:big})$, $$\label{E:aplMC}
{\operatorname{id}}_{{A_{PL}}(W)}\oplus s0\colon {A_{PL}}(W)\oplus s0\to{A_{PL}}(W)\oplus s{A_{PL}}(W,{\overline{W\backslash T}}).$$ Since the vertical maps below the second line of are quasi-isomorphisms, the morphism ${\operatorname{id}}_{{A_{PL}}(W)}$ $\oplus s0$ in is equivalent to $$\iota\circ\Phi_W\colon{A_{PL}}(W)\to C(s^{-n}\# {A_{PL}}(i)).$$ The morphism ${A_{PL}}(j)$ of is clearly equivalent to $${A_{PL}}(W)\to{A_{PL}}({W\backslash K}).$$ This finishes the proof.
Rational model of the complement of a subpolyhedron in a manifold with boundary {#s:complement}
===============================================================================
In this section we establish the CDGA model of the complement $W\setminus K$ under some unknotting condition, in particular when the codimension of the subpolyhedron is high (Theorem \[T:mod\_conndim\]). We also state a partial CDGA model without unknotting condition (Proposition \[modele\_truncation\].)
Consider the same setting as at the beginning of Section \[s:lef\_dual\], in particular Diagram . Suppose given a commutative diagram of CDGAs $$\label{diag3}
\xymatrix{
A \ar[d]_{\alpha} \ar[r]^{\varphi} & B \ar[d]^{\beta} \\
{\partial A}\ar[r]_{\partial \varphi} &{\partial B}}$$ that is a CDGA model of $$\label{diag4}
\xymatrix{
W &\ar@{_{(}->}_{f}[l] K \\
{\partial W}\ar@{_{(}->}[u] & \ar@{_{(}->}^{\partial f}[l] {\partial_W K},\ar@{_{(}->}[u], }$$ in other words Diagram is quasi-isomorphic to Diagram ($\ref{diag2}$). Note that in Diagram (\[diag3\]), $\partial A$ and $\partial B$ are just the names of some CDGAs.
The goal of this section is to construct from Diagram a CDGA model of ${A_{PL}}({W\backslash K})$.
Dgmodule model of the complement ${W\backslash K}$
--------------------------------------------------
Let $\hat{A}$ be a CDGA such that we have the following zig-zag of quasi-isomorphisms $$\label{rho_rho'}
\xymatrix{ A&\ar[l]_{\rho}^{\simeq} \hat{A} \ar[r]^-{\rho'}_-{\simeq}& {A_{PL}}(W). }$$ The morphism $\rho'$ induces a structure of $\hat{A}$-dgmodule on Diagram (\[diag2\]) and the morphism $\rho$ induces a structure of $\hat{A}$-dgmodule on Diagram (\[diag3\]). From Diagram (\[diag3\]) we deduce an $\hat{A}$-dgmodules morphism between the homotopy kernels of $\alpha$ and $\beta$ (see Section \[S:hoker\]) $$\label{barfi}\bar{\varphi}: \text{hoker }\alpha \to \text{hoker } \beta.$$ Note also that by Poincaré duality of the pair $(W,\partial W)$, we have a quasi-isomorphism of $A$-dgmodules $$\theta_A\colon A{\stackrel{\simeq}{\longrightarrow}}s^{-n}\#{\text{hoker }}\alpha.$$
\[dual\] An $\hat A$-dgmodule model of $${A_{PL}}(W)\to{A_{PL}}({W\backslash K})$$ is given by the composite $$\xymatrix{
A\ar[r]^-{\simeq}_-{\theta_A}&s^{-n}\#{\text{hoker }}\alpha\ar@{^(->}[r]_-{\iota}&C(s^{-n}\#\bar{\varphi}) }$$ where $C(s^{-n}\#\bar{\varphi}) $ is the mapping cone of the $\hat{A}$-dgmodules morphism $$s^{-n}\#\bar{\varphi}\colon s^{-n}\#{\text{hoker }}\beta \to s^{-n}\# {\text{hoker }}\alpha.$$
Since is a CDGA model of , ${\text{hoker }}\alpha$ is weakly equivalent as an ${\hat{A}}$-dgmodule to ${A_{PL}}(W,{\partial W})$ and ${\text{hoker }}\beta$ is weakly equivalent as an ${\hat{A}}$-dgmodule to ${A_{PL}}(K,{\partial_W K})$. Hence, the result is a direct consequence of Proposition \[modele\_apl\_bord\].
If the morphisms $\alpha$ and $\beta$ are surjective then we can work with the genuine kernel instead of the homotopy kernel.
The major flaw of the dgmodule model of $W\setminus K$ of Proposition \[dual\] is that there is no natural CDGA structure on it. The next proposition is a first step to endow this dgmodule model of $W\setminus K$ with the structure of a CDGA.
\[modele\_de\_dgmod\] Assume given an ${\hat{A}}$-dgmodule morphism $\varphi^! \colon Q \to A$ weakly equivalent to $${s^{-n}\#}\bar\varphi \colon {s^{-n}\#}{\text{hoker }}\beta \to {s^{-n}\#}{\text{hoker }}\alpha.$$ Then an $\hat A$-dgmodule of ${A_{PL}}(W)\to{A_{PL}}({W\backslash K})$ is given by $$A\,\hookrightarrow\,C(\varphi^!)$$ where $C(\varphi^!)$ is the mapping cone $A\oplus_{\varphi^!} sQ$.
This is a direct consequence of Proposition \[dual\].
The existence of such a morphism $\varphi^!$ is guaranteed if we take for $Q$ a cofibrant $\hat{A}$-dgmodule model of $s^{-n}\#$ hoker $\beta$.
This new dgmodule model $C(\varphi^!) =$ $A\oplus_{\varphi^!} sQ$ of $
{W\backslash K}$ has the advantage that $A$ is a CDGA and therefore, under some dimension hypotheses, the semi-trivial CGA structure on the mapping cone described in Section \[s:semi\_trivial\] makes it into a CDGA. We develop this in the next section.
CDGA model of the complement ${W\backslash K}$
----------------------------------------------
We work in the set-up of diagrams -. Remember also the notion of semi-trivial CDGA structure on a mapping cone from Section \[s:semi\_trivial\] and the notion of CDGA truncation from Section \[s:truncation\_cdga\]. Under some codimension and connectedness hypothesis for the inclusion $f\colon K \hookrightarrow W$ we can construct a CDGA model of ${W\backslash K}$. More precisely, we have:
\[T:mod\_conndim\] Let $W$ be a compact connected oriented triangulated manifold of dimension $n$ with boundary, and let $K\subset W$ be a subpolyhedron of dimension $k$. Consider Diagram and its CDGA model . Let $r$ be an integer such that the induced morphisms on homology $H_*(f;{\mathbb{Q}})$ and $H_*(\partial f; {\mathbb{Q}})$ are $r$-connected, that is $H_{\leq r}(W,K;{\mathbb{Q}})=0$ and $H_{\leq r}(\partial W,\partial_W K;{\mathbb{Q}})=0$.
Suppose given an $A$-dgmodule $Q$ weakly equivalent to ${s^{-n}\#}\mathrm{hoker} \ \beta$ such that $Q^{<n-k}=0$ and an $A$-dgmodules morphism $$\label{e:shriek_map}
\varphi^! \colon Q \to A$$ weakly equivalent to $${s^{-n}\#}\bar{\varphi} \colon {s^{-n}\#}\mathrm{hoker} \ \beta \to {s^{-n}\#}\mathrm{hoker} \ \alpha.$$ If $$\label{e:connectivity}
r\geq 2k-n+2,$$ then every truncation $\tau^{\leq n-r-1} (C(\varphi^!))$ of the mapping cone $C(\varphi^!)=A\oplus_{\varphi^!} sQ$ equipped with the semi trivial structure is a CDGA, and the morphism $$\xymatrix{A\ar[r] & \tau^{\leq n-r-1}(C(\varphi^!))}$$ is a CDGA model of the inclusion $${W\backslash K}\hookrightarrow W.$$
Moreover it is always possible to construct an $A$-dgmodule $Q$ and a morphism $\varphi^!$ as in .
This generalizes the main result of [@LaSt:PE Theorem 1.2] to manifolds with boundary. A first direct consequence of this theorem is the following corollary on the rational homotopy invariance of the complement under some connectedness-codimension hypotheses.
\[C:mod\_conndim\] Let $W$ be a compact triangulated manifold with boundary and $K\subset W$ be a subpolyhedron. Assume that $W$ and ${\partial W}$ are 1-connected and that the inclusions $$K \hookrightarrow W \text{ and } K\cap {\partial W}\hookrightarrow {\partial W}$$ are $r$-connected with $$\label{unknot2}
r \geq 2 (\dim K) - \dim W +2.$$ Then the rational homotopy type of ${W\backslash K}$ depends only on the rational homotopy type of the diagram $$\xymatrix{
{\partial_W K}\ar@{^{(}->}^{\partial f}[r] \ar[d] & {\partial W}\ar[d] \\
K \ar@{^{(}->}_{f}[r] &W. }$$
The hypotheses (or equivalently ) is called *the unknotting condition* and it cannot be removed as shown in [@LaSt:PE Section 9].
Let ${\hat{A}}$ be a CDGA such that we have a zig-zag of CDGA quasi-isomorphisms $$\xymatrix{{A_{PL}}(W) &\ar[l]^-{\rho'}_-{\simeq} \hat{A} \ar[r]_-{\rho}^-{\simeq} &A.}$$ Set $N {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}2(n-k)-3$. By Proposition \[p:truncMC\] (with $p=n-k$), ${\tau^{\leq N} C(\varphi^!)}$ admits the structure of a CDGA induced by the semi-trivial CGA structure on the mapping cone, and the composite $$\xymatrix{ A \ar[r]^{\iota \ \ \ } & C(\varphi^!) \ar[r] & {\tau^{\leq N} C(\varphi^!)}}$$ is a CDGA morphism.
We now prove that $H^{>N}(W\setminus K)=0$ where (co)homology of spaces is understood with coefficients in ${\mathbb{Q}}$. By excision and the connectedness hypotheses on $H(\partial f)$ and $H(f)$, $$H_{\leq r}(K\cup_{\partial_W K}\partial W\,,\,K)\,\cong\,H_{\leq
r}(\partial W\,,\,\partial_WK)\,=\,0$$ and $$H_{\leq
r}(W\,,\,K)\,=\,0.$$ Lefschetz duality and the long exact sequence of the triple\
$(W\,,\,K\cup_{\partial_WK}\partial W\,,\,K)$ give $$H^{\geq n-r}(W\setminus K)\,\cong\,H_{\leq
r}(W\,,\,K\cup_{\partial_WK}\partial W)\,=\,0.$$ The unknotting hypothesis ($\ref{e:connectivity}$) implies that $N\geq n-r-1$, therefore $H^{>N}(W\setminus K)=0$.
By Proposition \[modele\_de\_dgmod\], $A\to C(\varphi^!)$ is an ${\hat{A}}$-dgmodule model of ${A_{PL}}(W)\to{A_{PL}}({W\backslash K})$. This implies that $$H^{>N} (C(\varphi^!)) \cong H^{>N} ({W\backslash K})=0,$$ therefore $$\operatorname{proj}: C(\varphi^!) \longrightarrow \tau^{\leq N} (C(\varphi^!))$$ is a quasi-isomorphism.
Thus the CDGA morphism $$A\to \tau^{\leq N}\left(C(\varphi^!)\right)$$ is a model of $\hat A$-dgmodules of ${A_{PL}}(W)\to{A_{PL}}({W\backslash K})$. We will prove that it is actually a CDGA model.
Take a minimal relative Sullivan model (in the sense of [@FHT:RHT Chapter 14]) $$\label{E:mimodel}
\xymatrix{\hat{A} \ar@{->>}[r]^{\rho'}_{\simeq} \ar@{ >->}[dr] & {A_{PL}}(W)\ar@{->>}[r] & {A_{PL}}({W\backslash K}) \\
&(\hat{A}\otimes \Lambda V,D). \ar@{->>}[ur]^{\lambda'}_{\simeq}}$$ By Proposition \[modele\_de\_dgmod\], $\xymatrix{ {\hat{A}}\ar@{ >->}[r]
& {\hat{A}}\otimes \Lambda V}$ is an ${\hat{A}}$-dgmodule model of $A\to
C(\varphi^!)$. Since $(\hat A \otimes \Lambda V, D)$ is a cofibrant $\hat A$-dgmodule, we can construct a weak equivalence of ${\hat{A}}$-dgmodules $$\lambda\colon {\hat{A}}\otimes \Lambda V\to C(\varphi^!)$$ making commute the following diagram, where the upper part is of CDGA and the lower part is of $\hat A$-dgmodules, $$\label{D:diagmodel}
\xymatrix{{A_{PL}}(W)\ar[r]& {A_{PL}}({W\backslash K})& \\
{\hat{A}}\ar[d]^{\simeq}_{\rho} \ar[u]_{\simeq}^{\rho'} \ar@{>->}[r] &
{\hat{A}}\otimes \Lambda V
\ar[dr]_-{\simeq}^-{\bar\lambda={\operatorname{proj}}\circ\lambda}\ar[u]_{\simeq}^{\lambda'}
\ar@{-->}[d]^{\simeq}_{\lambda}& \\
A \ar[r] & C( \varphi^!) \ar[r]^-{\simeq}_-{{\operatorname{proj}}} & {\tau^{\leq N} C(\varphi^!)}. }$$
By Lefschetz duality and the hypothesis on the dimension of $K$ $$H^{<n-k} (W,{W\backslash K}) \cong H_{>k} (K,\partial_W K)=0.$$ By minimality of the Sullivan relative model , this implies that $V^{<n-k-1}=0$. Therefore $(\Lambda^{\geq 2}V)^{\leq N}=0$ and, since $(\tau^{\leq N}C(\varphi^!))^{>N}=0$, this implies that the composition \[rho\] $$\bar\lambda :{(\hat{A}\otimes \Lambda V,D)}\stackrel{\lambda}{\longrightarrow}C(\varphi^!) \stackrel{\textrm{proj}}{\longrightarrow} \tau^{\leq N} (C(\varphi^!))$$ is a morphism of CDGA. Thus all the solid arrows in Diagram $(\ref{D:diagmodel})$ are of CDGAs. This achieves to prove that $A\to
\tau^{\leq N} (C(\varphi^!))$ is a CDGA model of ${W\backslash K}\hookrightarrow W$, as claimed.
It remains to prove the existence of an $A$-dgmodule $Q$ and a morphism $\varphi^!$. Since $H^{>k}$(hoker $\beta$) $\cong H^{>k}(K,\partial_W K)=0$, we have $H^{<n-k} (s^{-n} \#$ hoker $\beta)=0$. Therefore there exists a cofibrant $A$-dgmodule model $Q$ of $s^{-n} \#$ hoker $\beta$ such that $Q^{<n-k}=0$. Since, by Poincaré duality, $s^{-n}\#$ hoker $\alpha \simeq A$, there exists an $A$-dgmodule morphism $$\varphi^! : Q \longrightarrow A$$ weakly equivalent to $s^{-n}\# \bar \varphi$.
Actually even when the unknotting condition of Theorem \[T:mod\_conndim\] is not satisfied, we still get a partial model of ${W\backslash K}$. More precisely we get a CDGA model of ${W\backslash K}$ up to some degree, i.e. a model of the truncation of ${A_{PL}}({W\backslash K})$. This is the content of the next proposition.
\[modele\_truncation\] Consider the same hypotheses as in Theorem \[T:mod\_conndim\] except that we do not assume the unknotting condition
Let $l \colon A(W) \to A ({W\backslash K})$ be a CDGA model of ${A_{PL}}(W) \to {A_{PL}}({W\backslash K})$ such that $A(W)$ and $A({W\backslash K})$ are connected. Set $N=2(n-k)-3$. Then the CDGA morphism $$A \to {\tau^{\leq N} C(\varphi^!)}$$ is a CDGA model of the composite $$\pi \circ l \colon A(W) \hookrightarrow A({W\backslash K}) \to \tau^{\leq N} A({W\backslash K}).$$
\[Proof of Proposition \[modele\_truncation\]\] The proof is very similar to that of Theorem \[T:mod\_conndim\]. The details to change are left to the reader.
We would have preferred in Proposition \[modele\_truncation\] to state that $A\to \tau^{\leq N} (C(\varphi^!))$ is a CDGA model of ${A_{PL}}{(W)}\to \tau^{\leq N} ({A_{PL}}({W\backslash K}))$. But the latter is not well defined because ${A_{PL}}({W\backslash K})$ is not connected and hence we cannot take its truncation. This is the reason for considering instead a model $l\colon A(W)\to A({W\backslash K})$ between *connected* CDGAs.
Note that $N=n-(2k-n+2)-1$ and therefore, under the unknotting condition $r \geq 2k-n+2$, we have that $N\geq n-r-1$. But, Poincaré duality and the $r$-connectedness implies that $H^{\geq n-r} ({W\backslash K})=0$. Hence Theorem \[T:mod\_conndim\] is actually a corollary of Proposition \[modele\_truncation\]
Rational model of the configuration space of two points in a manifold with boundary {#S:ConfW2}
===================================================================================
In this section we use the results of Section \[s:complement\] to describe the rational homotopy type of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. In particular we prove in Corollary \[c:rht\_fw2\] that the rational homotopy type of ${\operatorname{Conf}}(W,2)$ depends only on the rational homotopy type of the pair $(W,{\partial W})$ when $W$ and ${\partial W}$ are 2-connected. We also construct in Theorem \[t:modele\_adgc\_FW2\_1\] an explicit CDGA model of ${\operatorname{Conf}}(W,2)$. Moreover in Theorem \[T:modele\_FW2\] we describe an elegant CDGA model for ${\operatorname{Conf}}(W,2)$ when the pair $(W,\partial W)$ admits a pretty surjective model in the sense of [@CLS:pretty].
Fix a compact connected orientable manifold of dimension $n$, $W$, with boundary ${\partial W}$. Let $$\Delta : W \hookrightarrow W\times W \ ; x\mapsto (x,x)$$ be the diagonal embedding. The configuration space of two points in $W$ is the complementary space $${\operatorname{Conf}}(W,2):= (W\times W) \backslash \Delta(W)=\{ (x,y) \in W\times W | x\neq y\}.$$ Notice that the diagonal embedding $\Delta$ is such that $\Delta(\partial W) \cong \partial W$ and $\Delta^{-1} ({\partial W}\times
{\partial W}) = {\partial W}$. In other words, with the notation of , $$\partial_{W\times W}\left(\Delta(W)\right)=\Delta(\partial W) \cong \partial W.$$ Therefore, according to Corollary \[C:mod\_conndim\], if $W$ and $\partial W$ are connected enough, then the rational homotopy type of ${\operatorname{Conf}}(W,2)=W\times W \backslash \Delta (W)$ is determined by the square $(\ref{diagFW2_2})$ of Proposition $\ref{modele_carre_FW2}$ below. The goal of the next section is to compute a CDGA model of that square.
CDGA model of the diagonal embedding of the pair $(W,{\partial W})$ into $(W\times W, \partial (W\times W))$
-------------------------------------------------------------------------------------------------------------
The goal of this section is to prove the following proposition.
\[modele\_carre\_FW2\] Let $W$ be a compact connected orientable manifold with boundary ${\partial W}$. Suppose given a CDGA surjective model $\xymatrix{\beta \colon B \ar@{->>}[r]& {\partial B}}$ of the inclusion ${\partial W}\hookrightarrow W$. Then a CDGA model of the square $$\label{diagFW2_2}
\xymatrix{W\times W && \ar@{_{(}->}[ll]_-{\Delta} W \\
\partial(W\times W) \ar@{_{(}->}[u]& &\ar@{_{(}->}[ll]^-{\partial\Delta} {\partial W}. \ar@{_{(}->}[u]}$$ where $\Delta$ is the diagonal map and $\partial \Delta$ is the composition ${\partial W}\stackrel{\Delta}{\hookrightarrow} {\partial W}\times {\partial W}\hookrightarrow \partial (W \times W)$ is given by the CDGA square $$\label{diagFW2ADGC_2}\xymatrix{{B\otimes B}\ar[r]^{\mu}\ar[d]_{\alpha} & B \ar[d]^{\beta}\\
\frac{{B\otimes B}}{(\ker\beta \otimes \ker \beta)} \ar[r]_-{\tilde \mu} & {\partial B}}$$ where $\mu$ is the multiplication, $\alpha$ is the projection on the quotient, and $\tilde{\mu}$ the map induced by $\beta \circ \mu$.
The rest of this section is devoted to the proof of this result and for the rest of it we will use the notations introduced in the proposition. First, notice that, since $W$ is a manifold with boundary, $W\times W $ is also a manifold with boundary $$\partial(W\times W)= W\times {\partial W}\cup_{{\partial W}\times{\partial W}} {\partial W}\times W.$$ In other words we have a pushout (and homotopy pushout) $$\label{d:bound}
\xymatrix{
\partial(W\times W) \ar@{}[rd]|{\textrm{pushout}}&\ar@{_{(}->}[l] W \times (\partial W) \\
(\partial W)\times W\ar@{^{(}->}[u] & \ar@{_{(}->}[l]\ar@{^{(}->}[u] \partial W \times \partial W.}$$
The key argument to prove Proposition \[modele\_carre\_FW2\] is that Diagram is the right upper half of the following diagram $$\label{D2}
\xymatrix{ W\times W &&\ar@{_{(}->}[ll]_{\Delta} W \\
\ar@{^{(}->}[u]\partial(W\times W) \ar@{}[rd]|{\textrm{pushout}} & \ar@{_{(}->}[l]W\times (\partial W) \\
(\partial W) \times W \ar@{^{(}->}[u] & \ar@{_{(}->}[l]{\partial W}\times {\partial W}\ar@{^{(}->}[u] & \ar@{_{(}->}[l] \partial W\ar@{^{(}->}[uu]}$$ where the maps are the obvious inclusions and diagonals, and the small left lower square in is the homotopy pushout .
\[lem\_po4\] The following diagram is a CDGA model of diagram $$\label{po4}
\xymatrix{B\otimes B \ar[d]^{\alpha}\ar[rr]^-{\mu}&& B\ar@{->>}[dd]_-{\beta}\\
P \pullback\ar[r]\ar[d] & B\otimes {\partial B}\ar@{->>}[d]^{\beta\otimes {\operatorname{id}}} &\\
{\partial B}\otimes B \ar@{->>}[r]_{{\operatorname{id}}\otimes \beta} &{\partial B}\otimes {\partial B}\ar[r]^{\mu}& {\partial B}}$$ where $P$ is the pullback of the small square, $\alpha$ is the morphism given by the universal property, and $\mu$ are the multiplication morphisms.
Using the classical CDGA models for products and diagonal maps on spaces, the fact that ${A_{PL}}$ turns homotopy pushout of topological spaces into homotopy pullbacks of CDGAs, that a pullback of CDGA surjections is a homotopy pullback, and standard techniques in rational homotopy theory we get that a CDGA model of Diagram is given by the following diagram, where $P'$ denotes the pullback of the left bottom corner of the square, $$\label{po3}
\xymatrix{{A_{PL}}(W)\otimes {A_{PL}}(W) \ar[d]\ar[rr]^{mult} && {A_{PL}}(W)\ar[dd] \\
P' \pullback \ar[r] \ar[d]& {A_{PL}}(W)\otimes {A_{PL}}({\partial W})\ar@{->>}[d] & \\
{A_{PL}}({\partial W}) \otimes {A_{PL}}(W) \ar@{->>}[r]&{A_{PL}}({\partial W}) \otimes {A_{PL}}({\partial W}) \ar[r]& {A_{PL}}({\partial W}).}$$ This diagram is easily seen to be equivalent to Diagram .
The following lemma computes the small lower left pullback square in Diagram $(\ref{po4})$.
\[l:PBCDGA\] We have a pullback in CDGA $$\label{PBCDGA}\xymatrix{\frac{{B\otimes B}}{(\ker\beta \otimes \ker \beta)}
\ar[r]^{\overline{id_B \otimes \beta}}\ar[d]_{\overline{\beta\otimes
id_B}} &
B\otimes {\partial B}\ar[d]^{\beta\otimes id_{\partial B}}\\
{\partial B}\otimes B\ar[r]_-{id_{\partial B}\otimes \beta} & {\partial B}\otimes {\partial B}. }$$
Consider the following diagram of CDGA’s where the internal square is a pullback and $\alpha$ is the map induced by the universal property: $$\xymatrix{ {B\otimes B}\ar@{-->}[rd]^{\alpha} \ar[rrd]^{\beta \otimes id_B} \ar[rdd]_{id_\beta\otimes \beta} \\
& P\ar[r]\ar[d] & B\otimes \partial B\ar@{->>}[d]^{\beta \otimes id_{\partial B}} \\
&{\partial B}\otimes B \ar@{->>}[r]_{ id_{\partial B} \otimes \beta } & {\partial B}\otimes {\partial B}. }$$ It is straightforward to check that $\alpha$ is surjective and that $$\ker\alpha = \ker \beta \otimes \ker \beta.$$ Therefore we have an induced isomorphism $$\overline{\alpha} \colon \frac{B\otimes B}{\ker \beta \otimes \ker \beta} \stackrel{\cong}{\longrightarrow} P.$$
Diagram (\[diagFW2\_2\]) is the upper right part of Diagram (\[D2\]), therefore, by Lemma \[lem\_po4\], a CDGA model of (\[diagFW2\_2\]) is given by the upper right part of (\[po4\]). Using Lemma \[l:PBCDGA\] which computes the pullback $P$, we deduce that this CDGA model is $(\ref{diagFW2ADGC_2})$.
A first CDGA model of ${\operatorname{Conf}}(W,2)$
--------------------------------------------------
Let ${\xymatrix{\beta \colon B \ar@{->>}[r]& {\partial B}}}$ be a surjective CDGA model of $i \colon{\partial W}\hookrightarrow W$. Using the results of Section \[s:complement\], a CDGA model of ${\operatorname{Conf}}(W,2)=W\times W\backslash \Delta(W)$ can be obtained from a CDGA model of $$\label{diagFW2_3}
\xymatrix{W\times W && \ar@{_{(}->}[ll]_-{\Delta} W \\
\partial(W\times W) \ar@{_{(}->}[u]& &\ar@{_{(}->}[ll]^-{\partial\Delta} {\partial W}= \partial_{W\times W} W. \ar@{_{(}->}[u]}$$ which, by Proposition \[modele\_carre\_FW2\] is given by
$$\xymatrix{{B\otimes B}\ar[r]^{\mu}\ar[d]_{\alpha} & B \ar[d]^{\beta}\\
\frac{{B\otimes B}}{(\ker\beta \otimes \ker \beta)} \ar[r]_-{\tilde \mu} & {\partial B}. }$$
\[t:modele\_adgc\_FW2\_1\] Let $W$ be a compact triangulated manifold with boundary such that $W$ and ${\partial W}$ are $2$-connected. Let ${\xymatrix{\beta \colon B \ar@{->>}[r]& {\partial B}}}$ be a surjective CDGA model of ${\partial W}\hookrightarrow W$ and consider the map $$\bar \mu \colon \ker \beta \otimes \ker \beta \to \ker \beta$$ induced by the multiplication $\mu \colon B\otimes B \to B$. Suppose given a ${B\otimes B}$-dgmodule morphism $$\delta^! \colon D \to {B\otimes B}$$ weakly equivalent to $${s^{-2n}\#}\bar\mu \colon {s^{-2n}\#}\ker \beta \to {s^{-2n}\#}(\ker \beta \otimes \ker \beta)$$ and such that $D^{<n}=0$.
Then every truncation $\tau^{\leq 2n-3}C(\delta^!)$ of the mapping cone of $\delta^!$ admits a semi-trivial CDGA structure and $${B\otimes B}\to \tau^{\leq 2n-3}C(\delta^!)$$ is a CDGA model of ${A_{PL}}(W\times W) \to {A_{PL}}({\operatorname{Conf}}(W,2))$.
Since $W$ and ${\partial W}$ are 2-connected, we have that the morphisms $\Delta \colon W\hookrightarrow W \times W$ and $\partial \Delta \colon {\partial W}\hookrightarrow \partial (W\times W)$ are $2$-connected. So we are under the hypothesis of Theorem \[T:mod\_conndim\] with $r=2$, and the result is a direct consequence of it.
We deduce the rational homotopy invariance of ${\operatorname{Conf}}(W,2)$.
\[c:rht\_fw2\] Let $W$ be a compact manifold with boundary. If $W$ and ${\partial W}$ are $2$-connected then the rational homotopy type of ${\operatorname{Conf}}(W,2)$ depends only of the rational homotopy type of the pair $(W, {\partial W})$.
The rational homotopy invariance of Conf $(W,2)$ when $W$ is a closed 2-connected has been established in [@LaSt:FM2], and [@Cor:FM2-1conn] gives partial results in the 1-connected case. When $W$ is not simply-connected, [@LoSa:con] shows that there is no rational homotopy invariance.
\[R:BB’\] If we have a CDGA quasi-isomorphism $B {\stackrel{\simeq}\longrightarrow}B'$ and $\delta'^! \colon
D' \to B'\otimes B'$ a $B'\otimes B'$-dgmodule morphism which is weakly equivalent as a ${B\otimes B}-$dgmodule morphism to ${s^{-2n}\#}\bar\mu$ then it follows immediately from Theorem \[t:modele\_adgc\_FW2\_1\] that $$B'\otimes B' \to \tau^{\leq 2n-3}C(\delta'^!)$$ is also a CDGA model of ${A_{PL}}(W\times W) \to {A_{PL}}({\operatorname{Conf}}(W,2))$.
A CDGA model of ${\operatorname{Conf}}(W, 2)$ when $(W,{\partial W})$ admits a surjective pretty model {#S:ConfPretty}
------------------------------------------------------------------------------------------------------
Let $W$ be a compact manifold of dimension $n$ with boundary ${\partial W}$ such that both $W$ and ${\partial W}$ are 2-connected. In this section we will construct an elegant CDGA model of ${\operatorname{Conf}}(W,2)$ when the pair $(W,\partial W)$ admits a *surjective pretty model* in the sense of [@CLS:pretty Definition 3.1]. Let us recall what this means. Suppose given
- a connected Poincaré duality CDGA, $P$, in dimension $n$ ;
- a connected CDGA, $Q$;
- a CDGA morphism, $\varphi \colon P \to Q$.
Since $P$ is a Poincaré Duality CDGA there exists an isomorphism of $P$-dgmodules $$\label{Eq:thetaP}
\theta_P \colon P \stackrel{\cong}{\longrightarrow} {s^{-n}\#}P.$$ Consider the composite $$\label{eq:phishriek}
\xymatrix{\varphi^! \colon {s^{-n}\#}Q \ar[r]^{{s^{-n}\#}\varphi} & {s^{-n}\#}P \ar[r]^{\theta_P^{-1}}& P,}$$ which is a morphism of $P$-dgmodules. Assume that the morphism $$\varphi \varphi^! \colon {s^{-n}\#}Q \to Q$$ is balanced (see Definition \[D:balanced\]) and consider the CDGA morphism $$\label{Eq:wdf}
\varphi \oplus {\operatorname{id}}\colon P \oplus_{\varphi^!} s{s^{-n}\#}Q \to Q \oplus_{\varphi \varphi^!} s{s^{-n}\#}Q.$$ When is a CDGA model of the inclusion ${\partial W}\hookrightarrow W$ we say that it is a *pretty model* of the pair $(W,\partial W)$. If moreover $\varphi$ is surjective (and hence also ) we say that is is a surjective pretty model. Then if we consider the differential ideal $$\label{Eq:ideal}
I= \varphi^! ({s^{-n}\#}Q) \subset P,$$ [@CLS:pretty Corollary 3.3] states that the CDGA $P/I$ is a CDGA model of $W$. In [@CLS:pretty] we proved that many compact manifolds admit surjective pretty models as for examples even-dimensional disk bundles over closed manifolds, complements of high codimensional polyhedra in a closed manifold, as well as any compact manifold whose boundary retracts rationally on its half-skeleton (see [@CLS:pretty Definition 6.1].)
The objective in this section is to use this model, $P/I$, of $W$ to construct an elegant model for ${\operatorname{Conf}}(W,2)$, analogous to the one constructed in [@LaSt:FM2] for configuration spaces in closed manifolds.
Since $P$ is a Poincaré duality CDGA, for any homogeneous basis $\{a_i\}_{0\leq i \leq N}$ of $P$, there exists a Poincaré dual basis $\{a^\ast_i\}_{0\leq i \leq N}$ characterized by $\epsilon (a_ia^\ast_j)=\delta_{ij}$ where $\epsilon : P^n \to {\mathbb{Q}}$ is an orientation of $P$ and $\delta_{ij}$ is the Kronecker symbol. Let $\Delta \in (P\otimes P)^n$ be the diagonal class of $P\otimes P$ defined as $$\label{eq:diag}
\Delta = \sum_{i=0}^N (-1)^{|a_i|} a_i\otimes a_i^*.$$ Denote by $$\pi \colon P \to P/I$$ the projection. Taking the image of the diagonal $\Delta$ by the projection $\pi \otimes \pi \colon P\otimes P \to P/I \otimes P/I$ we get a *truncated diagonal class* $$\label{eq:trunc_diag}
\overline {\Delta}= (\pi \otimes \pi) (\Delta)\in (P/I \otimes P/I)^n.$$ Define the map $$\label{e:delta_shriek}
\overline{\Delta}^! \colon s^{-n}P/I \to P/I \otimes P/I \ ; \ s^{-n}x \mapsto \overline{\Delta} \cdot (1\otimes x).$$
\[l:trunc\_diag\_bob\] The map $\overline{\Delta}^! \colon s^{-n}P/I \to P/I\otimes P/I$ defined in is a $P/I \otimes P/I$-dgmodules morphism.
In [@LaSt:FM2 Lemma 5.1] it is shown that for $P$ a connected Poincaré duality CDGA, the morphism $\Delta^! \colon s^{-n} P \to P\otimes P \ ; \ s^{-n} x \mapsto \Delta (1\otimes x)$ is a $P\otimes P$-dgmodules morphism.
We have the following commutative diagram $$\xymatrix{
s^{-n}P \ar[d]_{s^{-n}\pi}\ar[r]^{\Delta^!} & P\otimes P
\ar[d]^{\pi\otimes \pi}\\
s^{-n}P/I \ar[r]_{\overline{\Delta^!}} & P/I \otimes P/I.
}$$ Since $P/I$ is a $P$-dgmodule generated by $1\in P/I$, this implies that $\overline \Delta ^! $ is a $P\otimes P$-dgmodules morphism, and the surjectivity of the morphism $\pi:P \to P/I$ implies that $\overline \Delta ^! $ is a $P/I\otimes P/I$-dgmodules morphism.
The main result of this section is the following theorem.
\[T:modele\_FW2\] Let $W$ be a $2$ connected compact manifold of dimension $n$ whose boundary is $2$-connected. Suppose that $(W,{\partial W})$ admits a surjective pretty model of the form $(\ref{Eq:wdf})$, and let $\overline{\Delta}^!$ be the $P/I\otimes P/I$-dgmodules morphism defined in . Then the mapping cone $$C (\overline{\Delta}^!)=\left( P/ I \otimes P/I \right) \oplus_{\overline{\Delta}^!} ss^{-n}P/I,$$ equipped with the semi-trivial structure is a CDGA model of ${\operatorname{Conf}}(W,2)$.
Before proving the theorem, let us fix some notation and prove a lemma. Set $$B = P\oplus_{\varphi^!} s{s^{-n}\#}Q \text{\quad and \quad} {\partial B}= Q\oplus s{s^{-n}\#}Q.$$ By hypothesis $$\label{Eq:beta}
\beta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\varphi \oplus {\operatorname{id}}\colon B \twoheadrightarrow {\partial B}$$ is a surjective CDGA model for the inclusion ${\partial W}\hookrightarrow W$. Also let $B':=P/I$ and notice that the obvious projection $\pi\oplus
0\colon B \stackrel{\simeq}{\longrightarrow}B'$ is a quasi-isomorphism of CDGA. According to Theorem \[t:modele\_adgc\_FW2\_1\] and Remark \[R:BB’\], we only need to show that $\overline{\Delta}^!$ is equivalent to $s^{-2n}\#\bar \mu$, which is the content of the following lemma.
\[lem\_PI\_mubar\] There exists a $B\otimes B$-dgmodules commutative square: $$\label{diag_PI_mubar}
\xymatrix{s^{-n}P/I \ar[rr]^{\overline{\Delta}^!}\ar[d]^{\cong}_{\overline{\theta}_P} && P/I\otimes P/I \ar[d]_{\cong}^{\overline{\theta}_{P\otimes P}} \\
{s^{-2n}\#}\ker\beta \ar[rr]_-{{s^{-n}\#}\bar{\mu}} && {s^{-2n}\#}\ker\beta \otimes \ker\beta.}$$
By Poincaré duality of the CDGA $P$, we have a $P$-dgmodules isomorphism $\theta_P: P \stackrel{\cong}{\longrightarrow} {s^{-n}\#}P
$. This morphism induces, by construction of the differential ideal $I\subset P$ (see and ), a $P$-dgmodules isomorphism $${\bar\theta}_P \colon P/I \stackrel{\cong}{\longrightarrow} {s^{-n}\#}\ker \varphi.$$ The morphism $$\beta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\varphi \oplus {\operatorname{id}}\colon B \twoheadrightarrow {\partial B}$$ is a surjective CDGA model of ${\partial W}\hookrightarrow W$. We have an obvious isomorphism $\ker\beta \cong \ker \varphi$ as $P$-dgmodules. So, we have a $P$-dgmodules isomorphism (that we will also denote ${\bar\theta}_P$) $${\bar\theta}_P \colon P/I \stackrel{\cong}{\longrightarrow} {s^{-n}\#}\ker \beta.$$An easy computation shows that for $(p,u) \in B=P\oplus s{s^{-n}\#}Q$ and $x\in P/I$, $${\bar\theta}_P ( (p,u) \cdot x) =(p,u) {\bar\theta}(x).$$ Thus ${\bar\theta}_P$ is a morphism of $B$-dgmodules and, via the multiplication $\mu \colon B\otimes B \to B$, it is a $B\otimes B$-dgmodules morphism. As a direct consequence we have the $B\otimes B$-dgmodules isomorphism $$\overline{\theta}_{P\otimes P} \colon P/I \otimes P/I \stackrel{\cong}{\longrightarrow} {s^{-n}\#}\ker \beta \otimes {s^{-n}\#}\ker\beta \cong {s^{-2n}\#}(\ker \beta \otimes \ker\beta)$$
By Lemma \[l:trunc\_diag\_bob\], the morphism $\overline{\Delta}^!$ is a $P/I \otimes P/I$-dgmodules morphism, and hence it is also a morphism of $B\otimes B$-dgmodules.
Consider the following diagram of $B\otimes B$-dgmodules $$\xymatrix{s^{-n}P/I \ar[rr]^{\overline{\Delta}^!}\ar[d]^{\cong}_{\overline{\theta}_P} && P/I\otimes P/I \ar[d]_{\cong}^{\overline{\theta}_{P}\otimes {\bar\theta}_P} \\
{s^{-2n}\#}\ker\beta \ar[rr]_-{{s^{-n}\#}\bar{\mu}} && {s^{-2n}\#}\ker\beta \otimes
\ker\beta,}$$ and let us show that it commutes. Since $P/I$ is a $B\otimes B$-dgmodule generated by the element $1\in
P/I$, it suffices to prove that $${\bar\theta}_P \otimes {\bar\theta}_P (\overline{\Delta}^! (s^{-n}1)) = {s^{-n}\#}\bar\mu ({\bar\theta}_P(s^{-n}1)).$$ A straightforward computation shows that this is the case.
Since $W$ and ${\partial W}$ are $2$-connected, Lemma \[lem\_PI\_mubar\], Remark \[R:BB’\] and Theorem \[t:modele\_adgc\_FW2\_1\] imply that $$P/I \otimes P/I \to \tau^{\leq 2n-3}C(\overline{\Delta}^{!})$$ is a CDGA model of ${\operatorname{Conf}}(W,2) \hookrightarrow W\times W$. Moreover, we can verify that the morphism $\overline{\Delta^!}$ is balanced, therefore $C(\overline{\Delta}^!)$ is also a CDGA when equipped with the semi-trivial structure. By the $2$-connectedness of the manifold $W$ and for degree reasons we have that $C(\overline{\Delta}^!){\stackrel{\simeq}\longrightarrow}\tau^{\leq 2n-3}C(\overline{\Delta}^{!})$ is a CDGA quasi-isomorphism.
A CDGA model for ${\operatorname{Conf}}(W,2)$ when $W$ is a disk bundle of even rank over a closed manifold {#S:ConfDxi}
-----------------------------------------------------------------------------------------------------------
We apply the model constructed in Section \[S:ConfPretty\] to disk bundles.
Let $\xi$ be a vector bundle of even rank, $2k$, for some $k\geq 2$, over some 2-connected closed manifold, $M$, of dimension $m$. Then the disk bundle $D\xi$ is a compact manifold of dimension $m+2k$ with boundary the sphere bundle $S\xi$.
Let $Q$ be a Poincaré duality CDGA model of $M$, let $$\Delta_Q \in (Q\otimes Q)^m$$ be a diagonal class for $Q$, and let $$e\in Q^{2k} \cap \ker (d_Q)$$ be a representative of the Euler class of $\xi$. Denote by $(\Delta_Q \cdot (e\otimes 1))^!$ the $Q\otimes Q-$dgmodule morphism $$(\Delta_Q \cdot (e\otimes 1))^!:s^{-(m+2k)} Q \longrightarrow Q\otimes Q, s^{-(m+2k)} q \longmapsto \Delta_Q \cdot (e\otimes q),$$ which is balanced. Consider the mapping cone $$Q \otimes Q \bigoplus_{(\Delta_Q \cdot (1\otimes e))^!} ss^{-(m+2k)}Q$$ which is a CDGA.
\[T:ConfDxi\] With the notation above, assume that the vector bundle $\xi$ is of even rank $2k\geq 4$, and that the base, $M$, is a 2-connected closed manifold. Then $$Q\otimes Q \bigoplus_{(\Delta_Q \cdot (1\otimes e))^!} ss^{-(m+2k)}Q$$ is a CDGA model of ${\operatorname{Conf}}(D\xi,2)$.
Before proving this theorem, let us first deduce the rational homotopy invariance of that configuration space.
\[C:invConfDxi\] The rational homotopy type of the configuration space of 2 points in a disk bundle of even rank $\geq 4$ over a 2-connected closed manifold depends only on the rational homotopy type of the base and on the Euler class.
By the main result of [@LaSt:PD], the base of the bundle admits a Poincaré duality CDGA model, $Q$. Let $e\in Q \cap \ker(d_Q)$ be a representative of the Euler class. By Theorem \[T:ConfDxi\], a CDGA model of the configuration space, and hence its rational homotopy type since it is simply connected, depends only on those data.
Denote by $\bar z$ a generator of degree $2k$ and define the CDGA $$P:=\left(\frac{Q\otimes \wedge \bar z}{(\bar z^2-e\bar z)}, D \bar z = 0\right)$$ which is a Poincaré CDGA in dimension $n=m+2k$. Define the CDGA morphism $$\varphi: P \longrightarrow Q$$ by $\varphi (q_1+q_2 \bar z) = q_1+q_2 \cdot e$, for $q_1,q_2\in Q$. Then [@CLS:pretty Theorem 4.1] and its proof establishes that the pair $(D\xi,S\xi)$ admits a surjective pretty model associated to $\varphi$. We will then use Theorem \[T:modele\_FW2\] to establish the model of ${\operatorname{Conf}}(D\xi,2)$.
Following the notation of [@CLS:pretty proof of Theorem 4.1], one computes that $$I=\varphi^! (s^{-n}\# Q)=\Phi^! (s^{-2k}Q)=\bar z\cdot Q.$$ We need to compute the truncated diagonal class $\overline \Delta \in
P/I \otimes P/I$. Let $\{q_i\}$ be a homogeneous basis of $Q$ and let $\{q^\ast_i\}$ be its Poincaré dual basis. Denote by $\omega \in Q^m$ the fundamental class of $Q$, so that we have $$q_i\cdot q^\ast_j = \delta_{ij} \cdot \omega \mod Q^{<m}.$$ Then $$\label{E:basisP}
\{q_i\} \cup \{q_i\cdot \bar z\}$$ is a homogeneous basis of $P$ and $-\omega\bar z$ is a fundamental class of $P$. Then the Poincaré dual basis of is given by $$\{q^\ast_i \cdot (e-\bar z)\} \cup \{-q^\ast_i\}$$ because of the four equations $$\begin{aligned}
q_i \cdot q^\ast_j (e-\bar z ) &= -\delta_{ij} \omega \bar z& \mod P^{<n},
\\
q_i\cdot (-q^\ast_j) &=0& \mod P^{<n},
\\
(q_i\bar z) \cdot (q^\ast_j (e-\bar z)) &= q_iq^\ast_j (\bar z e-\bar z^2) = 0& \mod P^{<n},
\\
(q_i \bar z) \cdot (- q^\ast_j) &= -q_i q^\ast_j \bar z = -\delta_{ij} \omega \bar z&\mod P^{<n}.
\end{aligned}$$ Therefore the diagonal class in $P$ is given by $$\Delta_P = \sum_i (-1)^{|q_i|} (q_i \otimes q^\ast_i (e-\bar z)-q_i\bar z \otimes q^\ast_i)\in P\otimes P,$$ and, since $I=Q\bar z$, the truncated diagonal class is $$\overline \Delta_P = \sum_i (-1)^{|q_i|} q_i \otimes q^\ast_i e \in P/I \otimes P/I.$$ The diagonal class of $Q$ is $$\Delta_Q = \sum_i (-1)^{|q_i|} q_i \otimes q^\ast_i \in Q\otimes Q,$$ therefore, using the canonical isomorphism $P/I \cong Q$, we have $$\overline \Delta_P = \Delta_Q \cdot (1\otimes e).$$ The theorem is then a direct consequence of Theorem \[T:modele\_FW2\].
Note that the total space, $E\xi$, of the vector bundle $\xi$ is homeomorphic to the interior of $D\xi$, and therefore ${\operatorname{Conf}}(E\xi,2)\simeq {\operatorname{Conf}}(D\xi,2)$. In particular, when the bundle is trivial the above gives a model of ${\operatorname{Conf}}(M\times {\mathbb{R}}^{2k},2)$. Hence we recover partially the result [@CoTa:BAMS Theorem1].
Interestingly enough we get different models when the bundle is not trivial. Consider for example the quaternionic Hopf line bundle, $\eta$, over $S^4$, of rank 4. In that case we can take $Q=({\mathbb{Q}}[x]/(x^2),d_Q=0)$, with $\deg(x)=4$, as a model for $S^4$ and the Euler class is represented by $e=x$. Using the model of Theorem \[T:ConfDxi\] one computes easily that the rational cohomology algebra of ${\operatorname{Conf}}(D\eta,2)$ is the same as $H^\ast (S^4 \vee S^4 \vee S^{11};{\mathbb{Q}})$, but ${\operatorname{Conf}}(D\eta,2)$ is *not* formal because it admits a non trivial Massey product in degree 11.
By contrast, for the trivial bundle of rank $4$ over $S^4$, $\epsilon=S^4\times{\mathbb{R}}^4$, one computes that ${\operatorname{Conf}}(D\epsilon,2)$ is formal and its rational cohomology algebra is given by $$H^\ast({\operatorname{Conf}}(S^4\times {\mathbb{R}}^4,2);{\mathbb{Q}}) \cong \frac{\wedge(x,x',u)}{(x^2,x^{'2},ux-ux')}.$$ with $\deg(x)=\deg(x')=4$ and $\deg(u)=7$.
Thus the two compact manifolds $D\eta$ and $D\epsilon$ of dimension $8$ are homotopy equivalent but their configuration spaces have different Poincaré series. This is because their boundaries, $\partial D\eta=S^7$ and $\partial D\epsilon=S^4\times S^3$, are not homotopy equivalent.
A CDGA model for ${\operatorname{Conf}}(W,2)$ when $W$ is the complement of a subpolyhedron in a closed manifold
----------------------------------------------------------------------------------------------------------------
Let $M$ be a 2-connected closed manifold of dimension $n$. Let $K\subset M$ be a 2-connected subpolyhedron such that $\dim M \geq 2 \dim (K)+3$. In this section we explain how to build a CDGA model of ${\operatorname{Conf}}(M\backslash K,2)$.
Let $T$ be a regular neighborhood of $K$ in $M$, in other words $T$ is a compact codimension $0$ submanifold of $M$ that retracts by deformation on $K$. Then let $W$ be the closure of $ M\backslash T$ in $M$, which is a compact manifold with boundary $\partial
W= \partial T$. The interior of $W$ is homeomorphic to $M\backslash
K$. Therefore ${\operatorname{Conf}}(M\backslash K,2)$ is homotopy equivalent to ${\operatorname{Conf}}(W,2)$.
Let us recall how to build a pretty surjective model of $(W,\partial W)$. By [@CLS:pretty Proposition 5.4] one can construct a surjective CDGA model $$\varphi : P \twoheadrightarrow Q$$ of $K \hookrightarrow M$, where $P$ is a Poincaré duality CDGA and $Q^{\geq n/2-1}=0$. As explained in [@CLS:pretty end of Section 3], the main result of [@LaSt:PE] implies that the pretty model associated to $\varphi$, $$\varphi \oplus {\operatorname{id}}: P \bigoplus_{\varphi^!} ss^{-m} \# Q \longrightarrow Q \bigoplus_{\varphi\varphi^!} ss^{-n} \# Q$$ is a CDGA model of $(W,\partial W)$. Therefore, by Theorem \[T:modele\_FW2\], a CDGA model of ${\operatorname{Conf}}(M\backslash K,2)$ is given by $$P/I \otimes P/I \bigoplus_{\overline \Delta^!} ss^{-n}(P/I).$$
Let us illustrate this for the configuration space of a punctured manifold. Let $M$ be a closed 2-connected manifold and set $W=M\backslash \{x_0\}$, with $x_0\in M$. Let $P$ be a Poincaré duality CDGA model of $M$ with fundamental class $$\omega \in P^n.$$ Pick a homogeneous basis $\{a_i\}_{0\leq i\leq N}$ of $P$ with $a_0=1$ and $a_N=\omega$. Let $\{a_i^*\}$ be the Poincaré dual basis. Then the diagonal class is $$\Delta = 1 \otimes \omega + (-1)^n \omega\otimes 1 + \sum^{N-1}_{i=1} a_i \otimes a^\ast_i$$ with $1\leq \deg (a_i) \leq n-1$, for $1\leq i\leq N-1$. In that case we can take $I={\mathbb{Q}}\cdot \omega$ and we get that $$\overline P = P/({\mathbb{Q}}\cdot \omega)$$ is a CDGA model of $M\backslash \{x_0\}$ and the truncated diagonal is $$\overline \Delta = \sum^{N-1}_{i=1} a_i \otimes a^\ast_i.$$ Thus $$\overline P \otimes \overline P \bigoplus_{\overline \Delta^!} ss^{-n} \overline P$$ is a CDGA model of ${\operatorname{Conf}}(M\backslash \{x_0\},2)$.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[10]{}
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Hector Cordova Bulens, Pascal Lambrechts, and Don Stanley, *Models for configuration spaces in manifolds with boundary*, In preparation.
, *Pretty rational models for poincaré duality pairs*, Available on arXiv.
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Yves F[é]{}lix, Stephen Halperin, and Jean-Claude Thomas, *Rational homotopy theory*, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001.
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, *Algebraic models of [P]{}oincaré embeddings*, Algebr. Geom. Topol. **5** (2005), 135–182 (electronic).
, *Poincaré duality and commutative differential graded algebras*, Ann. Sci. Éc. Norm. Supér. (4) **41** (2008), no. 4, 495–509.
, *A remarkable [DG]{}module model for configuration spaces*, Algebr. Geom. Topol. **8** (2008), no. 2, 1191–1222.
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[^1]: H.C.B. and P.L. are supported by the belgian federal fund PAI Dygest
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---
abstract: 'We define a new quantifier of classicality for a quantum state, the Roughness, which is given by the $\mathcal{L}^2 (\R^2)$ distance between Wigner and Husimi functions. We show that the Roughness is bounded and therefore it is a useful tool for comparison between different quantum states for single bosonic systems. The state classification via the Roughness is not binary, but rather it is continuous in the interval \[0,1\], being the state more classic as the Roughness approaches to zero, and more quantum when it is closer to the unity. The Roughness is maximum for Fock states when its number of photons is arbitrarily large, and also for squeezed states at the maximum compression limit. On the other hand, the Roughness reaches its minimum value for thermal states at infinite temperature and, more generally, for infinite entropy states. The Roughness of a coherent state is slightly below one half, so we may say that it is more a classical state than a quantum one. Another important result is that the Roughness performs well for discriminating both pure and mixed states. Since the Roughness measures the inherent quantumness of a state, we propose another function, the Dynamic Distance Measure (DDM), which is suitable for measure how much quantum is a dynamics. Using DDM, we studied the quartic oscillator, and we observed that there is a certain complementarity between dynamics and state, i.e. when dynamics becomes more quantum, the Roughness of the state decreases, while the Roughness grows as the dynamics becomes less quantum.'
author:
- 'Humberto C. F. Lemos'
- 'Alexandre C. L. Almeida'
- Barbara Amaral
- 'Adélcio C. Oliveira'
title: Roughness as Classicality Indicator of a Quantum State
---
Acknowledgments {#acknowledgments .unnumbered}
===============
ACO, HCLF and ACLA gratefully acknowledge the support of Brazilian agency Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) through grant No. APQ-01366-16. BA acknowledges financial support from the Brazilian ministries MEC and MCTIC, CNPq (Grant No. 444927/2014-0) and INCT.
|
---
author:
- 'John C. Brown'
- 'Procheta C.V. Mallik'
title: 'Non-thermal recombination - a neglected source of flare hard X-rays and fast electron diagnostic '
---
Introduction {#sect:intro}
============
Ever since their first detection (Arnoldy, Kane and Winckler 1968, Kane and Andersen 1970), flare hard X-ray (HXR) bursts (photon energies $\epsilon > 10$ keV or so) have been recognised as an important diagnostic of electron acceleration and propagation (e.g. Brown 1971, Lin and Schwartz 1987, Johns and Lin 1992). The large electron flux and power imply they play a substantial role in flare energy budgets and pose challenges for electron acceleration mechanisms (see recent reviews by, e.g. Vilmer [*et al.*]{} 2003, Brown 2005, MacKinnon 2006). Recent copious high resolution HXR spectral data from the RHESSI mission (Lin et al 2002) have created the possibility of detailed reconstruction of source electron spectra (following Brown 1971) offering important constraints on the electron energy budget and acceleration processes (Piana [*et al.*]{} 2003, Conway [*et al.*]{} 2003, Massone [*et al.*]{} 2004, Kontar [*et al.*]{} 2004, 2005, Brown [*et al.*]{} 2006).
In inferring electron flux spectra $F(E$), the HXR radiation mechanism has always been taken to be f-f collisional bremsstrahlung of fast electron impacts with atoms and ions, gyrosynchrotron and inverse Compton radiation being negligible at these energies for solar magnetic and radiation fields (Korchak 1971). Though included for thermal electrons in hot (a few keV) plasma, f-b recombination radiation from non-thermals seems to have been assumed negligible other than in a preliminary study by Landini, Monsignori and Pallavicini (1973). In view of the importance of details in the photon spectrum $J(\epsilon)$ (photons sec$^{-1}$kev$^{-1}$) for accurate reconstruction of $F(E$), we re-examine this assumption, and conclude (cf Mallik and Brown 2007) that it is not valid under some conditions, which quite commonly exist in some flare HXR source regions.
It is not the intention of this paper to analyse precisely the theoretical recombination radiation spectrum from fast electrons under conditions (e.g. ionisation structure) for specific flares which are typically both inhomogeneous and time dependent. Rather we give approximate theoretical estimates of how important it may be relative to bremsstrahlung under various limiting conditions. Specifically, we compare the two in the simplest, Kramers, cross-section approximations, for limiting cases of plasma ionisation. The recombination emission rate per electron is very sensitive to the ionic charge, being $\propto Z^4A_Z$ (Kramers 1923) per plasma proton for hydrogenic ions of charge $Ze$ and number abundance $A_Z$. Thus the emitted f-b flux and spectrum depend strongly on the ionisation state, hence the temperature, of the plasma where the fast electrons recombine. In practice this will involve several ionisation stages of several target plasma species (since $Z^4A_Z$ may be large even for small abundance $A_Z$), which will vary along the paths of the electrons and be time dependent.
The paper is organised as follows. In Section 2 we briefly discuss relevant processes and the cross-section approximations we use, and obtain expressions for the total continuum photon spectral contributions $j(\epsilon)$ expected from an electron flux spectrum $F(E)$ from f-f and from f-b processes. In Section 3, we compare these for a power-law $F(E)$ with low cut off at $E<E_c$ and for a shifted power-law, and discuss implications for flare electron spectra and energy budgets under several limiting plasma ionisation assumptions. In Section 4 we look at thermal and non-thermal components to show how the relative importance of each contribution depends on conditions in the flare by varying parameters around those for a specific real event. Section 5 discusses the effect of including the f-b contribution on inverse problem inference of $F(E)$ from $j(\epsilon)$ while Section 6 summarises our conclusions and suggests directions for future work. Details of some of the equations are given in Appendix A. In Appendix B we discuss the total emission spectra from extended volumes for thin target, collisional thick target and thermal cases.
Free-free and free-bound emissivity spectra
===========================================
General considerations
----------------------
In this section, we discuss only local emissivities $j(\epsilon)$ (photons cm$^{-3}$ sec$ ^{-1}$ per unit $\epsilon$ ). Relativistic and directivity effects are disregarded ($E,\epsilon \ll m_ec^2$) since the f-b/f-f ratio is largest at low $E$. Then, if target atom/ion type $t$ has density $n_t$ and the fast electron flux spectrum is $F(E)$ (electrons sec$^{-1}$ cm$^{-2}$ per unit $E$), $j(\epsilon)$ for a collisional radiation process is
$$j(\epsilon)=\Sigma_t j_t(\epsilon)=\Sigma_t n_t
\int_{E_{tmin(\epsilon)}}^\infty
F(E)\frac{dQ_t}{d\epsilon}(\epsilon,E)dE,$$
where $dQ_t/d\epsilon(\epsilon,E)$ is the relevant cross-section per unit $\epsilon$ for target species $t$ and the integral is over the range of electron energies relevant to species $t$.
Bremsstrahlung
--------------
In the case of f-f (bremsstrahlung), $dQ_t/d\epsilon(\epsilon,E)$ is essentially the same for any state of ionisation of an atomic species $Z$ (Koch and Motz 1959), and the $t$ summation in (1) need only be carried out over elements $Z$ to give, for element abundances $A_Z$ (by number relative to hydrogen), and total proton (p+H) density $n_p$,
$$j_B(\epsilon)=n_p\Sigma_Z A_Z
\int_\epsilon^\infty F(E)\frac{dQ_{BZ}}{d\epsilon}(\epsilon,E)dE,$$
where $dQ_{BZ}/d\epsilon(\epsilon,E)$ is the bremsstrahlung cross-section for element $Z$ and $E_{min}=\epsilon$ since any free-free transition can only yield a maximum $\epsilon=E$. The bremsstrahlung cross-section per nucleus $Z$ scales as $Z^2$ and can be written
$$\frac{dQ_{BZ}}{d\epsilon}=\frac{8\alpha r_e^2
Z^2}{3}\frac{m_ec^2}{\epsilon E}q(\epsilon,E)~, ~\epsilon\le E$$
(and zero for $\epsilon >E$). Here $\alpha = e^2/\hbar c$ is the fine structure constant and $r_e=e^2/m_ec^2$ the classical electron radius, while $q(\epsilon, E)$ is the ratio of the actual cross section to the Kramers cross section (Kramers 1923), which is the factor in front of $q$. While this is only a first approximation, not suitable for accurate absolute spectral inversion/reconstruction algorithms (Brown 2005), it will be adequate for the present purpose of comparing f-f with f-b emission, which we also treat in the Kramer’s approximation. Then (2) and (3) give, for bremsstrahlung,
$$j_B(\epsilon) =\frac{8\alpha
r_e^2}{3}\frac{m_ec^2}{\epsilon}\zeta_Bn_p\int_\epsilon^\infty
\frac{F(E)}{E}dE,$$
where
$$\zeta_B=\Sigma_Z \zeta_{BZ}=\Sigma_Z A_ZZ^2$$
is the heavy element correction for bremsstrahlung, with $\zeta_B\approx 1.6$ for the solar coronal abundances we use - see later.
Recombination Radiation
-----------------------
0.2cm
Element Z $A_z$ $A_zZ^2$ $A_zZ^4$ $V_z = Z^2\chi$ (keV)
--------- ---- ------------------ ----------------- ----------------- -----------------------
H 1 1 1 1 0.0136
He 2 0.096 0.384 1.536 0.0544
C 6 3.57 x $10^{-4}$ 0.013 0.463 0.490
O 8 8.57 x $10^{-4}$ 0.055 3.511 0.870
Ne 10 1.07 x $10^{-4}$ 0.011 1.071 1.360
Mg 12 1.33 x $10^{-4}$ 0.019 2.755 1.958
Si 14 1.27 x $10^{-4}$ 0.025 4.871 2.666
S 16 1.61 x $10^{-5}$ 0.0041 1.053 3.482
Ca 20 8.50 x $10^{-6}$ 0.0034 1.360 5.440
Fe 26 8.61 x $10^{-5}$ 0.058 39.336 9.914
Ni 28 6.95 x $10^{-6}$ 0.0054 4.27 10.662
$\Sigma = 1.58$ $\Sigma = 61.2$
0.2cm
Element $Z-z$ $Z_{eff}$ $\Phi_{Z_{eff}}$ $A_z$ $A_zZ_{eff}^4$ $V_z = Z_{eff}^2\chi$ (keV)
---------- ------- ----------- ------------------ ------------------ ---------------- -----------------------------
Fe XXII 21 21.98 0.05 0.43 x $10^{-5}$ 1.004 6.57
Fe XXIII 22 22.61 0.14 1.21 x $10^{-5}$ 3.152 6.95
Fe XXIV 23 23.20 0.25 2.15 x $10^{-5}$ 6.232 7.32
Fe XXV 24 23.77 0.56 4.82 x $10^{-5}$ 15.381 7.68
The situation here is more complicated. Firstly, 2-body radiative recombination (we neglect 3-body recombination) of a free electron of energy $E$ to a bound level $m$ of energy $-V(Z,i,m)$ in ionic stage $i$ yields a photon energy $\epsilon$, which, apart from quantum uncertainty, is unique, namely:
$$\epsilon=E +V(Z,i,m).$$
That is, when a fast electron does recombine, all of its kinetic energy $E$ [*plus*]{} $V$ goes into a photon of that energy, in contrast to bremsstrahlung where photons of all energies $\epsilon\le E$ are emitted.
Furthermore, for each element $Z$, there is a range of $Z+1$ distinct ion stages $i$ each with its own distinct set of energy levels ($m$) and a set of $Z,i,m$-dependent recombination cross-sections. Thus recombination collisions of a mono-energetic beam with a multi-species plasma gives rise to a set of delta-function-like spectral features at all energies (6) corresponding to elements $Z$, ionic stages $i$ and levels $m$ . For a continuous electron spectrum, this yields a continuum photon spectrum that is a sum of an infinite series of energy-shifted electron flux contributions. In contrast to bremsstrahlung it does not involve an integral over a continuum of electron energies.
For a general plasma the basic particle type $"t"$ onto which recombination occurs is level $m$ of ion stage $i$ of element $Z$ with recombination cross-section differential in $\epsilon$ for that $t$:
$$\frac{dQ_{Rt}}{d\epsilon}(\epsilon)=Q_{Rt}\delta(E-\epsilon+V_t),$$
where $Q_{Rt}$ is the total radiative recombination cross-section for species $t$ and $\delta(E')$ is the delta-function in energy such that $\int_{-\infty}^\infty \delta(E')dE'=1$. Then the total recombination emission spectrum for electron flux spectrum $F(E)$ is
$$\begin{aligned}
j_R(\epsilon) = & n_p\Sigma_t A_t\int_{E_{min}(\epsilon,t)}^\infty
Q_{Rt}(\epsilon,E)\delta(E-\epsilon +V_t)F(E)dE & \nonumber
\\ = & \Sigma_t A_t n_p Q_{Rt}(\epsilon,\epsilon-V_t)F(\epsilon-V_t), &\end{aligned}$$
where $A_t$ is the numerical abundance of species $t$ relative to $n_p$. The forms for $Q_{Rt}$, for general $t$, are complicated and have to be calculated numerically, as do the values of $A_t$ when individual ionisation states are considered. However, in the Kramers approximation (with unit Gaunt factors) there is an analytic expression for hydrogenic ions, which we will use to estimate $dj_R/d\epsilon$ compared with $dj_B/d\epsilon$, namely, for recombination onto level $m$ of the hydrogenic ion of element $Z$ (Kramers 1923, Andersen [*et al.*]{} 1992, Hahn 1997)
$$Q_{R}=\frac{32\pi}{3{\sqrt
3}\alpha}r_e^2\frac{Z^4\chi^2}{m^3\epsilon E},$$
where $\chi=m_e e^4/2\hbar^2$ is the hydrogen ionisation potential.
For an element in its highest purely hydrogenic ion state the emissivity spectrum would then be
$$j_{RZ}(\epsilon)= \frac{32\pi}{3{\sqrt
3}\alpha}\frac{r_e^2\chi^2Z^4n_z}{\epsilon}\Sigma_m\frac{1}{m^3}
\frac{F(\epsilon-Z^2\chi /m^2)}{\epsilon-Z^2\chi /m^2}$$
with the $m$ summation over $m\ge Z(\chi/\epsilon)^{1/2}$, since recombination to level $m$ yields only photons of $\epsilon \ge
Z^2\chi/m^2$. If the source were so hot that all atoms were almost fully ionised the total for all $Z$ would be, in this approximation,
$$j_{R}(\epsilon) =\frac{32\pi}{3{\sqrt
3}\alpha}\frac{r_e^2\chi^2}{\epsilon}n_P\Sigma_Z
Z^4A_Z\Sigma_m\frac{1}{m^3}
\frac{F(\epsilon-Z^2\chi/m^2)}{\epsilon-Z^2\chi/m^2}$$
for element abundances $A_Z$, with the same $m$ summation limits.
In reality even super-hot coronal flare temperatures are not high enough to equal the ultra-hot $T\gg 10^8$ K needed to almost fully ionise all elements into their hydrogenic states, especially Fe, which is crucial in having by far the highest value of $A_ZZ^4$ - see Table 1. Consequently, to deal accurately with $j_R$ for real flare data, we would have to take into account the actual ionisation state of the flare plasma, which varies with time and location (being radically different in loop tops from loop footpoints), and actual forms of $Q_R(Z),V_Z$ for non-hydrogenic ion stages.
For our purpose of making first estimates we make the following simplifying approximations:
- We treat [*all*]{} ions using hydrogenic Equations (9) - (11) but with suitably chosen $Z_{eff}$ so that $$V_Z= Z_{eff}^2 \chi ~~;~~ Q_{RZ}=\frac{32\pi}{3{\sqrt
3}\alpha}r_e^2\frac{Z_{eff}^4\chi^2}{m^3\epsilon E},$$ where $Z_{eff}$ makes allowance for screening and other non-hydrogenic effects. While this will be a rough estimate for some ions, such approximations are often quite satisfactory for suitable $Z_{eff}$ (e.g. Hahn and Krstic 1994, Erdas, Mezzorani and Quarati 1993). Here we adopt $Z_{eff}$ such that hydrogenic Equation (12) gives the correct value of $Q_{RZ}$ as given by exact calculations such as those of Arnaud and Raymond (1992) for Fe, which is the most important ion in our analysis. Typically, for an element of atomic number $Z$ in an ionic state with $z$ bound electrons left, $Z_{eff}$ is between $Z-z$ and $Z-z+1$
- Noting that $Q_R \propto 1/m^3$ we include here only recombination to $m=1$ (in the sense of the lowest empty level of the ion - hydrogenic with $Z=Z_{eff}$ - not of the atom). Higher $m$ contributions are weaker, being $\propto 1/m^3$ though extending to lower energies with edges at $Z_{eff}^2\chi/m^2$. These should be included in quantitative data fitting.
- We focus on situations where the emitting region is near isothermal and either quite cool, so that only low $V_Z$ element recombination matters, or very hot so that high $V_Z$ elements (mainly Fe) are dominant. The former are typically loop chromospheric footpoints (thick target) and the latter very hot coronal loops which are either at the limb with their footpoints occulted, or are so dense as to be coronal thick targets (Veronig and Brown 2004).
Under these conditions, Equation (11) becomes
$$j_{R}(\epsilon)=\frac{32\pi}{3{\sqrt
3}\alpha}\frac{r_e^2\chi^2}{\epsilon}n_p\Sigma_{Z_{eff}}
Z_{eff}^4A_{Z_{eff}}
\frac{F(\epsilon-Z_{eff}^2\chi)}{\epsilon-Z_{eff}^2\chi},$$
where $A_{Z_{eff}}=A_Z\Phi_{Z_{eff}}$ with $\Phi_{Z_{eff}}$ the fraction of atoms of element $Z$ in ionic state $Z_{eff}$.
Note that, since there is no integration over $E$ here, if $F(E)$ contains a sharp feature at an electron energy $E_*$, such as a low or high $E$ cut-off, this will appear in the recombination contribution to the photon spectrum $j(\epsilon)$ as a series of sharp features at photon energies $\epsilon(m,Z,E_*)=E_*+Z_{eff}^2\chi/m^2~;~m=1,\infty$ for every ion $Z$ present. The same is true for broad features like smooth bumps or dips. This is in contrast with the bremsstrahlung contribution, in which such features are smoothed out by integration over $E$. Thus, even if $j_R\ll j_B$, it may have an important effect in inferring $F(E)$ from $j(\epsilon)$ since this essentially involves differentiating $j(\epsilon)$ (Section 5).
Element parameters and flare plasma ionisation
----------------------------------------------
The heavy element correction for bremsstrahlung, $\zeta_B$, is almost independent of ionisation state (since the bremsstrahlung cross sections for atoms and ions of the same $Z$ are essentially the same), being $\zeta_B\approx 1.6$ for solar abundances. On the other hand $\zeta_{RZ_{eff}}=Z_{eff}^4A_{Z_{eff}}$ depends on the number of empty ion levels available for recombination. The importance of fast electron recombination radiation thus depends on the state of ionisation of the plasma in which the fast electrons are moving, which is primarily a function of plasma temperature $T$.
In Table 1 we show the values of $Z$, $Z^2A_Z=\zeta_{BZ}$, $Z^4A_Z=\zeta_{RZ}$, $V_Z$ for various elements/ions whose $\zeta_{RZ}=Z^4A_Z$ is large enough to be significant, if the element is sufficiently ionised. With $\zeta_{RZ}\approx 40$ for FeXXVI, Fe is by far the most important if conditions are such that it is highly ionised. The $kT$ where maximum ionisation of an ion stage is reached is typically of the order $0.1 Z_{eff}^2\chi$ to $Z_{eff}^2\chi$. In Table 2 we show more detailed values for several stages of ionisation of Fe (XXII-XXV, i.e. 21+ to 24+) with the appropriate $A_{Z_{eff}}=A_{Z_{eff}}\Phi_{Z_{eff}}$ for each of these Fe ionic states for the typical coronal flare case of $T=2\times 10^7$ K. These are taken from Arnaud and Raymond (1992) as are the actual ionisation fractions we adopt later (Section 4) for the temperatures of the real flare we consider.
The radiative recombination coefficients give $Z_{eff}$, which differ slightly from the $Z$ values, as mentioned in Section 2.3. For the 2002 April 14 event, to which we return later, the peak flare temperature was 19.6 MK, $\sim 5\%$ of the iron appearing as Fe XXII ($Fe^{21+}$), $\sim 14\%$ in the Fe XXIII ($Fe^{22+}$) state, $\sim25\%$ appearing as Fe XXIV and $\sim56\%$ as Fe XXV. The respective $Z_{eff}$ values are 21.98, 22.61, 23.20 and 23.77.
Broadly speaking in typical flare/micro-flare conditions we can consider the following $T$ regimes:
- At $T\le 10^4$ K (’cold’) even H and other low $V_{Z_{eff}}$ ions are neutral so $\zeta_{RZ}\approx 0$ for all $Z$. This would be typical of very dense cool chromospheric thick target footpoints relevant to deeply penetrating electrons.
- For $10^5 \le T\le 10^6$ K (’cool’) the predominant elements ionised are H, O, Mg, Si giving $\Sigma_Z \zeta_{RZ}\approx 15$. This is most relevant to upper chromospheric dense warm plasma reached by moderate energy thick target electrons.
- At $T\ge 10^7$ K (’hot’) Fe is well ionised up to about Fe XXV giving $\Sigma_Z \zeta_{RZ}\approx 50$. This is relevant to the hot ’coronal’ loop regime, hence either to (i) typical upper (SXR) flare loops of moderate density (thin target) whose HXR emission is seen in isolation either by HXR spectroscopic imaging or volume integrated but with the cool footpoints occulted because they are over the solar limb; or (ii) cases of coronal thick target loops (Veronig and Brown 2004) where the upper loop density suffices to stop the fast electrons collisionally.
Local (thin target) HXR spectra of f-f and f-b for power-law $F(E)$ with cut-off
================================================================================
Basic expressions for $j_B,j_R$
-------------------------------
To estimate how the fast electron recombination $j_R(\epsilon)$ compares with bremsstrahlung $j_R(\epsilon)$, we first consider the commonly studied case of a power-law with a low energy cut-off
$$\label{definePL}
F(E) = (\delta-1)\frac{F_c}{E_c}\left(\frac{E}{E_c}\right)^{-\delta}~~;~~
E\ge E_c,$$
where $F_c$ is the total electron flux at $E\ge E_c$. Then, from Equations (4) and (14), we obtain for f-f emission
$$\begin{aligned}
\nonumber j_B(\epsilon) = &
\frac{\delta-1}{\delta}\frac{8\alpha\zeta_B}{3}\frac{m_ec^2r_e^2}{\epsilon}
\frac{n_pF_c}{E_c} &\\
\nonumber & \times ~~\left[\frac{\epsilon}{E_c}\right]^{-\delta}; &
\epsilon \ge E_c \\ & \times ~~1; & \epsilon < E_c,\end{aligned}$$
while for f-b emission from an ion of effective charge $Z_{eff}$,
$$\begin{aligned}
\nonumber j_{RZ_{eff}}(\epsilon) =& (\delta-1)\frac{32\pi
\zeta_{RZ_{eff}}}{3^{3/2}\alpha}\frac{r_e^2
\chi^2}{\epsilon}\frac{n_pF_c}{E_c^2} & \\
\nonumber & \times ~~\left[\frac{\epsilon-Z_{eff}^2\chi}{E_c}
\right]^{-\delta-1}; & \epsilon \ge E_c+Z_{eff}^2\chi \\
& \times ~~0; & \epsilon < E_c+Z_{eff}^2\chi,\end{aligned}$$
where $$\zeta_{R_{Z_{eff}}} = A_{Z_{eff}} Z_{eff}^4.$$ So the total for all relevant $V_{Z_{eff}}$ is
$$j_{R}(\epsilon) = \Sigma_{Z_{eff}\ge [(\epsilon-E_c)/\chi]^{1/2}}
j_{RZ_{eff}}(\epsilon).$$
Ratio of $j_R$ to $j_B$
-----------------------
For this truncated power-law case, the ratio of f-b to f-f emissivity is
$$\begin{aligned}
\nonumber \Psi = \frac{j_R(\epsilon)}{j_B(\epsilon)}& \frac{2\pi\delta}{\sqrt 3} \frac{\chi}{\epsilon}\Sigma
_{Z_{eff}^2>(\epsilon-E_c)/\chi)}~~
\frac{\zeta_{RZ_{eff}}}{\zeta_B}\left[1-\frac{Z_{eff}^2\chi}{\epsilon}\right]^{-\delta-1}
& \\ & \approx \frac{0.25(\delta/5)}{\epsilon(keV)}\Sigma
_{Z_{eff}^2>(\epsilon-E_c)/\chi)}~~
\frac{\zeta_{RZ_{eff}}}{\zeta_B}\left[1-\frac{Z_{eff}^2\chi}{\epsilon}\right]^{-\delta-1},
&\end{aligned}$$
where each term in the summation is zero at $\epsilon <
E_c+Z_{eff}^2\chi$.
For $\epsilon \gg E_c,\Psi \rightarrow
0.25\Sigma_{Z_{eff}}A_{Z_{eff}} Z_{eff}^4/\epsilon$(keV). In pure ionised H ($\Sigma_Z\zeta_{RZ}=1)$ this is only $2.5 \%$ at 10 keV. This rather small value of $\Psi$ must be the origin of the conventional wisdom that f-b can be ignored compared to f-f emission at HXR energies. However, this notion neglects several crucial facts:
- At high coronal flare temperatures, where all elements are highly ionised, in plasmas of cosmic chemical abundances, heavy elements are the main contributors to the $A_ZZ^4$ sum. For the extreme ultra-hot case of near-total ionisation of all $Z$, and for modern solar coronal abundances the $\Sigma_Z$ factor is $\approx 61.2$, mainly due to Fe as discussed in Section 2.4 - see Tables 1 and 2. Note that Fe coronal abundance, for example, has been assumed to be $2.9$ times photospheric Fe abundance (Feldman [*et al.*]{} 1992). Even higher factors of about 4 have been suggested (Dennis, personal communication).
- At lower $\epsilon$ the contribution from each $Z_{eff}$ rises steeply to a sharp recombination edge at $\epsilon = E_c+V_Z$, where the flux can be large, especially if $E_c$ is small and $\delta$ large.
- At the edge, the \[ \] factor in Equation (19) goes to $[1+Z_{eff}^2\chi/E_c]^{\delta+1}$. This is because the flux of electrons emitting recombination photons of energy $\epsilon$ is not the flux of those at $E\ge\epsilon$, as for bremsstrahlung, but of those at $E=\epsilon-Z_{eff}^2\chi$. Consequently $\Psi$ is not negligible even at $\epsilon \gg E_c$. For fully ionised Fe alone, this factor is $\approx [1+10/E_c$(keV)$]^{\delta+1}$, which, for $\delta =5$ and at $\epsilon = 10$ keV, is 64, 11.4, 5.5 for $E_c=$ 10, 20, 30 keV respectively. Even for lower stage Fe ions (e.g. XXV), common in flare coronal loops, evidently recombination must be a significant contributor to the HXR emission in those parts of the flare.
Typical results in limiting regimes
-----------------------------------
N.B. All spectrum figures in this paper (except Figure 5) have been plotted for a bin-width of 1 keV to match RHESSI’s spectral resolution. However, in Figure 5 we use 0.01 keV resolution so as to compare it with Plot A of Figure 4 to see how f-b edges would look if they were observed at a higher resolution. The 1 keV binning smears out a lot of the edges of different elements that are clearly noticeable in Figure 5. Hence in Figures 3 and 4, the edges are not ’infinitely’ steep as they should be; this is evident in Figure 5 where they do look ’infinitely’ steep due to the finer resolution. Also important to note is that the features seen in Figures 3, 4 and 5 are recombination edges and not spectral lines. None of the figures in this paper includes spectral lines - leaving them out shows more clearly where f-b edges exist in the HXR continuum.
In Figure 1 we show for $\delta=3, 5$ the actual spectral shapes for $E_c=10, 25$ keV respectively in plasmas of normal solar coronal abundances, which are: ultra-hot ($T \gg 10^8$ K; Fe is nearly fully ionised), hot ($T = 2\times 10^7$ K; Fe well ionised up to Fe XXV) and cool ($T = 10^6$ K; elements up to Si are almost fully ionised). In Figure 2 we show the ratios $\Psi(\epsilon)$ for the ultra-hot, hot and ’cool’ cases, respectively. The following key features of the hot thin target situation are apparent from these Figures:
- The peak non-thermal f-b contribution, in each hot or ultra-hot case shown, adds at least 50% to the usual f-f one and in some cases ($\delta=5,E_c=10$ keV) is up to 10 times greater (1000% increase) even when only ions up to Fe XXV are present. This is essentially due to the high abundance of Fe - much higher than thought when recombination spectra were first discussed (Culhane 1969, Culhane and Acton 1970). In appendix A we evaluate the efficiency with which f-b yields HXRs compared to f-f, and also derive the ratio $\Psi$ for the case of a smooth $F(E)$ with no cut-off. This proves, that in a hot enough plasma, far less electrons and power are needed than is found when only f-f is included and that, for smooth $F(E)$, $\Psi$ is largest for large $\delta$ and low $E$ spectral roll-over.
- In the ’cool’ case ($T\approx 10^6$ K) of elements up to Si almost fully ionised, the f-b contribution is smaller but not in general negligible. For example, in the bottom left panel of Figure 1 ($\delta=5, E_c=10$ keV), f-b is about 30 % of f-f at 15 keV energies. This is amply large enough to have a major impact on inferring $F(E)$ by inversion or by forward fitting (Section 5).
- In hot plasma, Fe is by far the most important contributor of recombination radiation.
- The peak ratio of f-b to f-f increases as $\delta$ is increased and/or $E_c$ is decreased. This is because f-b photons of energy $\epsilon$ are emitted by electrons of energy $E-V$ which have flux $F(E-V)\propto (E-V)^{-\delta}$ which is greatest when the minimum $E=E_c$ is smallest, $V$ is largest and the steepness $\delta$ greatest.
- Recombination edges are apparent for the elements with the highest values of $A_{Z_{eff}}Z_{eff}^4$ - Fe, Si, Mg and O and at energies $\epsilon = Ec+Z_{eff}^2\chi$, thereby creating the possibility of finding the location of a low energy cut-off $E_c$ should one exist.
- The harder asymptotic $\gamma=\delta+1$ for f-f compared with $\gamma=\delta+2$ for f-b (Equations (15) and (16)) results in an upward ’knee’ in the total spectrum clearly visible in Figure 1 for $E_c=10$ keV but also present for higher $E_c$ outside the $\epsilon$ range of the Figure. This could be an important signature in data of a substantial f-b contribution.
While the edge locations and the spectral shape trends will be roughly right, our use of the hydrogenic and $Z_{eff}$ approximations, and adoption of unit Gaunt factors, mean that these curves/analytic forms can only be used for approximate quantitative fitting of real data. As far as we are aware (Kaastra, personal communication) the Gaunt factors, rates etc. have only ever been systematically evaluated for Maxwellian $F(E)$ and sometimes for forms which can be written as sums of these (such as pure power-laws with no cut-off), and some occasional consideration of specific non-thermal spectra (e.g. Landini, Monsignori Fossi and Pallavicini 1973). Comparison of our Maxwellian results, in the unit Gaunt factor Kramers approximation, with those of Culhane for the same parameters shows the necessary corrections in the Maxwellian case to be significant for quantitative comparison with real data. In addition, in real cases the non-thermal emission will always be superposed on thermal contributions (especially important for the very hot plasmas of special interest here) and also in many cases on a thick target non-thermal contribution (unless this is from occulted footpoints), from the flare volume as a whole. In Appendix B we derive the generalisation of the above equations to the various cases involved in real flares, viz. finite volume thin targets, Maxwellian plasmas and thick targets for use in Section 4, where we evaluate the sum of all these contributions for a specific case.
Some practical case study results derived from a real flare
===========================================================
We saw above and in the appendices that the most favourable conditions for a substantial recombination contribution are when the maximum possible amount of the observable HXR source is a hot plasma (e.g. loop) at SXR temperatures. High density maximises the emission measure but may make the source/loop collisionally thick and smear recombination edge spectral signatures of low energy cut offs. So an optimal case could be a loop which is just tenuous enough to be collisionally thin and for which the cool dense thick target footpoints are occulted. (Footpoint removal by imaging is limited by RHESSI’s dynamic range). Such sources will have a strong HXR source in the coronal loop. One such event was adopted as a basis for a case study, starting from the real event parameters. This was the 2002 April 14 event, which Veronig and Brown (2004) showed to be a hot, dense, collisionally thick loop with a strong coronal HXR source and no footpoints up to at least 60 keV. Thus the hot coronal source of non-thermal f-b emission was not diluted by cold footpoint thick target f-f emission though the f-b edges were smeared because the hot loop itself slowed the fast electrons to rest. In Figure 3 we show the theoretical spectrum from a hypothetical resolved part of the coronal loop for two $E_c$ values. We have evaluated the theoretical thermal, non-thermal and the whole volume hypothetical total $J_B(\epsilon),J_R(\epsilon)$ (from Sections 2-3 and Appendix B) for such a loop, based on our approximate Kramers expressions, in three loop parameter regimes (Figure 4):
- Plot A: With the actual hot thick target loop parameters found by Veronig and Brown, namely $\delta=6.7$; $T=19.6$ MK; $L=45 \times 10^8$ cm; $A=19.1 \times 10^{16}$ cm$^2$; $n_p=10^{11}$ cm$^{-3}$; $N=4.9\times 10^{20}$ cm$^{-2}$; ${\cal
F}_1=5\times 10^{35}$ sec$^{-1}$ above $E_1=$ 25 keV. The total $J$ is dominated by thermal f-b and f-f at low $\epsilon$ but thick-target f-b at medium $\epsilon$ and thick-target f-f at high $\epsilon$. Locally within the loop volume, if this were spatially resolved, the spectrum $j$ would be like those in Figure 3, where edges are clearly visible in positions corresponding to cut-off energies of 15 and 21 keV. At a higher resolution, these edges would look similar to the edges shown in Figure 5. Should such edges be found in data, they can diagnose the all-important $E_c$ parameter.
- Plot B: With the actual parameters found by Veronig and Brown except with $n_p$ reduced by a factor of 25 so that the loop is collisionally thin above about 10 keV but with the footpoints hidden (limb occulted) so there is no cold thick target contribution. In this case the thermal emission is also much reduced because $EM=2n_p^2AL$ is down by a factor of 625. Somewhere between this and the first case should be the optimum condition for seeing maximum f-b contribution.
- Plot C: The same as B but with the dominant cold footpoint thick target emission added to show its diluting effect.
- Plot D: The same as C but with a reduced injection rate and so the thermal is more dominant than in C and this alters the total spectral shape a little bit.
The upward ’knee’ apparent in Figures 4 A,B at around 40 keV due to the transition from a f-b to a f-f dominated spectrum (cf. Section 3 and Figure 3) is rarely seen in data but may be present in some events (Conway [*et al.*]{} (2003)). A statistical survey of a large sample of events should shed light on conditions where non-thermal f-b is important. Also note that an upward ’knee’ is present at the transition from a thermal- to a non-thermal-dominated spectrum. The position of this knee depends on the plasma temperature and may interfere with the f-b to f-f ’knee’, which depends mainly on the $E_c$ parameter. Hence, although for certain parametric conditions one may be able to notice two separate upward ’knees’, if $E_c$ is low and $T$ is high, the ’knees’ may occur at similar $\epsilon$ and may not be distinguishable in real data.
The inverse problem - effect of f-f on $F(E)$ inferred from data on $j(\epsilon)$
=================================================================================
We note again that, since even the thin target $j_B$ involves an integral over $E$ while $j_R$ does not, any sharp features in $F(E)$ would be smoothed out in the bremsstrahlung contribution to the photon spectrum but not in the recombination contribution. Consequently, an important way to study the effect of including f-b on the required properties of $F(E)$ is to consider it as an inverse problem (Craig and Brown 1986) to infer $F(E)$ from observed $j(\epsilon)$. Here we consider the following experiment for the thin target case. (Thick target and thermal cases always involve even greater error magnification - Brown and Emslie 1988). Generate the total $j(\epsilon)$ including f-b as well as f-f from a specified $F_1(E)$ and evaluate the $F_2(E)$ which would be erroneously inferred by solving the inverse problem ignoring the presence of the f-b term, as is currently done in all HXR data analysis, whether by inversion or forward fitting.
By (4) and (11) the total f-f + f-b emission spectrum $dJ/d\epsilon$ from a homogeneous volume $V$ can be written
$$H(\epsilon)= \int_\epsilon^\infty G(E)dE + D\Sigma_{Z_{eff}\le{\sqrt
{\epsilon/\chi}}}Z_{eff}^4A_{Z_{eff}}G(\epsilon-V_{Z_{eff}}),$$
where $$H(\epsilon)=\frac{3}{8\alpha
r_e^2}\frac{1}{\zeta_Bm_ec^2n_pV}\epsilon\frac{dJ}{d\epsilon}; \hspace{0.2cm}G(E)= F(E)/E$$ and $D$ is as given in Equation (A.2). If we ignore the second (recombination) term in Equation (20), as has always been done in the past, for the Kramers f-f term, the inverse is just (Brown and Emslie 1988)
$$G(\epsilon) = - H'(E).$$
The neglect of the second term can be thought of as an ’error’ $\Delta H$ in our data and if we apply inversion formula (22) to this ’data’, ignoring the recombination ’error’ we get a resulting error $\Delta G$ in the inferred $G$ given by $$\begin{aligned}
\Delta G(E) = \frac{F_2(E)-F_1(E)}{E}\\ \nonumber =-D\Sigma_{Z_{eff}\le{\sqrt
{\epsilon/\chi}}}Z_{eff}^4A_{Z_{eff}}G'(E-V_{Z_{eff}}).\end{aligned}$$
It is at once clear that any sharp change in $j(\epsilon)$ i.e. in $H(E)$, such as the presence of f-b edges, however small, can have a very large effect on the inferred $F_2(E)$. (If the inverse problem is addressed for more realistic smoother forms of f-f cross section than Kramers, the ’error magnification’ is in general even larger - Brown and Emslie 1988, Piana [*et al.*]{} 2000). For a power law $F$ with cut off around say 20 keV, analytically speaking this expression gives infinite negatives in $\Delta G(E)$ at the spectral edges around 30 keV (for Fe). However when smoothed over a few keV and added to the f-f term the result would be a ’wiggle’ in the $F(E)$ solution in the 30-40 keV range. This is just where enigmatic features have been reported in some RHESSI spectra and variously attributed to the effects of photospheric albedo (Kontar [*et al.*]{} 2006), possibly pulse pile up (Piana [*et al.*]{} 2003), or a high value of $E_c$ (Zhang and Huang 2004).
Another case providing insight is that of a smooth shifted power-law $G(E)=A(E+E_*)^{-\delta-1}$, which has no edges though the corresponding $F(E)$ has a smooth peak at $E=E_*/\delta$. In this case the fractional error in $G$ due to applying (22) ignoring the recombination term can be expressed as
$$\frac{\Delta G(E)}{G(E)} = (\delta+1)\frac{D}{E+E_*}\Sigma_{Z_{eff}}Z_{eff}^4A_{Z_{eff}}\left[\frac{1}{1-V_{Z_{eff}}/(E+E_*)}\right]^{\delta+2},$$
where each term in the $Z_{eff}$ sum is zero for $E<V_{Z_{eff}}=Z_{eff}^2\chi$.
In the case of recombination onto Fe XXV alone (hot plasma), this gives for $\delta =5$,
$$\frac{\Delta G}{G}\approx \frac{10 ~ keV}{E+E_*}\left[1-7 ~
keV/(E+E_*)\right]^{-7},$$
which is shown in Figure 6 for $E_*=5,10,20$ keV. Evidently errors due to neglect of recombination can be large at low $E$. The reason is that the $Z_{eff}$ recombination contribution to the bremsstrahlung solution for $G(E)$ at $E$ comes from the slope of $G$, and not just $G$ itself and at $E-V_{Z_{eff}}$ not at $E$. Figure 6 is similar to Figure A.2 because $F_2/F_1=G_2/G_1= 1+\Delta
G/G_1$.
This error has very serious consequences for past analyses of HXR flare spectra, at least in cases where a significant hot dense coronal loop is involved. For example, the f-b emission spectrum is most important at lower energies (5-30 keV or so), depending on the plasma temperature $T$ and low energy electron cut-off or roll-over $E_c, E_*$ and is steeper than the free-free. This will offset some of the spectral flattening caused around such energies by photospheric albedo (Alexander and Brown 2003, Kontar [*et al.*]{} 2005) resulting in underestimation of the albedo contribution and hence of the downward beaming of the fast electrons. This fact would weaken the finding of Kontar and Brown (2006) that the electrons are near isotropic, in contradiction of the usual thick target description, but for the fact that the flares they used had rather hard spectra and substantial footpoint emission - conditions where the f-b correction should be rather small. Nevertheless it illustrates that care is needed to ensure f-b emission is properly considered.
Finally, recognising the presence of the f-b contribution, one can in fact convert integral Equation (20) into a differential/functional equation for $F(E)$ by differentiation, namely
$$G(E) -
D\Sigma_{Z_{eff}\ge{E/\chi}^{1/2}}A_{Z_{eff}}Z_{eff}^4G'(E-Z_{eff}^2\chi)
= -H'(E),$$
which is a wholly new class of functional equation in need of exploration.
Discussion and Conclusions
===========================
It is clear from our findings that ignoring non-thermal f-b contribution as negligible, as has been done in the past, is erroneous. Even if we ignore coronal enhancement of element abundances, and use photospheric abundances, f-b contribution can be very significant. In certain flaring regions, especially in dense-hot coronal sources or occulted loop-top events, fast electron recombination can be of vital importance in analysing data properly and in inferring electron spectra and energy budgets. It can have a major influence on inferred electron spectra both as an inverse problem and also in forward fitting parameters, including the important potential to find and evaluate low-energy electron cut-offs, which are vital to flare energy budgets. While incorporating f-b into spectral fitting procedures will make it considerably more complicated, an advantage is that the f-b, unlike the f-f, contribution retains its $J(\epsilon)$ signatures of any sharp features in $F(E)$.
A major consequence of the low energy f-b contribution is that, to fit an actual photon spectrum, less electrons are needed, than in f-f only modelling, at the low $E$ end, which is where most of the power in $F(E)$ lies. For example, if we consider the case $\delta=5,E_c=10$ keV and ionisation up to Fe XXV, then we see from Figures 1 and 2 that inclusion of f-b increases $j$ by a factor of 2-10 in the 15-20 keV range for $\delta=$ 3-5. Thus, to get a prescribed $j$ in that range we need only $10-50\%$ as many electrons as inferred from f-f emission only.
We also note that the importance of non-thermal f-b emission is greatest when non-thermal electrons are present at low $E$ and with large $\delta$ such as in microflares with ’hard’ XRs in the few to ten KeV range (Krucker [*et al.*]{} 2002). Such low energy electrons have short collisional mfps and so are more likely to emit mainly in hot coronal regions, if accelerated there. Microflares are therefore important cases for inclusion of f-b.
Before we conduct any precise fitting of $F(E)$, involving the f-b contribution, to real data (e.g. from RHESSI) and include it in software packages it will be important to include, for both f-b and f-f, more accurate cross-sections with Gaunt factors etc. and ionisation fractions as functions of plasma temperature. By doing this, it will be possible to show, for certain events, how vital recombination is and to improve our understanding of electron spectra and their roles in flares. However, our Kramers results already bring out the fact that recombination should not be ignored in the future, and that it may be invaluable in some cases as a diagnostic of the presence or otherwise of electron spectral features.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by a PPARC Rolling Grant and UC Berkeley Visitor funds (JCB) and by a Dorothy Hodgkin’s Scholarship (PCVM). Helpful discussions with A. Caspi, H.S. Hudson, A.G. Emslie and J. Kasparova are much appreciated, as are the helpful suggestions of the referee (S. Krucker).
0.5cm
Efficiency and smooth $F(E)$
============================
Comparison of the efficiency of f-b versus f-f HXR yield
--------------------------------------------------------
In Section 3.1 and 3.2, we predicted the $j_B, j_R$ from a power-law $F(E)$ and found that the $j_R$ contribution could sometimes be more important than $j_B$. It is of interest therefore, to consider the following question. If one observes a power-law $j(\epsilon)\propto
\epsilon^{-\gamma}$ above some $\epsilon \ge V_Z$, what electron flux $F_R(E)$ would be needed to generate it in a plasma of solar abundances [*purely*]{} by non-thermal electron recombination on ion $Z_{eff}$ as compared with the $F_B(E)$ required to do so [*purely*]{} by f-f bremsstrahlung? If we write, from Equation (15), the latter as $F_B(E)=CE^{-\gamma+1}$ then the former has to be, by Equation (16), $$F_R(E)=C(E+V_{Z_{eff}})^{-\gamma}/DZ_{eff}^4A_{Z_{eff}},$$ where $$D=\frac{2\pi \chi}{{\sqrt 3}\zeta_B}\approx 0.04 keV$$ and the ratio measuring recombination efficiency relative to bremsstrahlung is $$\frac{F_B(E)}{F_R(E)}=\gamma
Z_{eff}^4A_{Z_{eff}}\frac{D}{E}[1+V_{Z_{eff}}/E]^{\gamma},$$ which we show in Figure A.1 for $\gamma=5$ in terms of each of the dominant f-b contributions from fully ionised O, Mg, Si and Fe respectively while the f-f is for all elements. Evidently non-thermal recombination could be dominant over bremsstrahlung up to many 10s of keV as the most efficient HXR source if the electrons are emitted entirely in a plasma hot enough ($T\approx 20$MK) for elements up to Fe 24+ to be ionised and is significant even at lower temperatures.
In terms of the total required electron fluxes $F_{R1},F_{B1}$ above energy $E_1$, the ratio is
$$\begin{aligned}
\nonumber \frac{F_{B1}}{F_{R1}}& =\frac{\gamma-1}{\gamma-2}
Z_{eff}^4
A_{Z_{eff}}\frac{D}{E_1}[1+V_{Z_{eff}}/E_1]^{\gamma-1}&\\
\approx & 0.02
Z_{eff}^4A_{Z_{eff}}\frac{10~keV}{E_1}[1+V_{Z_{eff}}/E_1]^{\gamma-1},&\end{aligned}$$
which is about 10 for Fe, 0.25 for Si and 0.1 for Mg and O at $E_1=10$ keV.
At higher electron energies ($E \ge \approx 17$ keV), O becomes more efficient than Mg, as can be seen in Figure A.1, because of the combined effects of the $A_ZZ^4$ factor and the term containing $V_Z$.
Ratio of $j_R$ to $j_B$ for an example of a smooth $F(E)$ with no cut-off
-------------------------------------------------------------------------
All of the above results are for $F(E)$ with a sharp cut off $E_c$. To illustrate how the appearance of $j(\epsilon)$ is modified by inclusion of f-b as well as f-f for a smooth $F(E)$, a simple case to evaluate is $F(E)\propto E(E+E_*)^{-\delta-1}$, which behaves as $E^{-\delta}$ at $E\gg E_*$ but has a smooth roll-over at $E_*/\delta$. It is simple to show that the resulting $j_B(\epsilon)\propto (E+E_*)^{-\delta}/\delta$ for f-f alone and that the ratio of f-b to f-f in this case is, for ion $Z_{eff}$ alone,
$$\nonumber \Psi_{smooth} = \frac{D\zeta_{Z_{eff}}}{\epsilon+E_*}\left[1-\frac{Z_{eff}^2}{\epsilon+E_*}\right]^{-\delta-1},$$
which is shown in Figure A.2 for $\delta=5$, $Z_{eff}=23.77$ and $E_*=5, 10, 20$ keV. We see again that $\Psi_{smooth}$ is largest for large $\delta$ and for small $E_*$.
0.5cm
Whole Flare Thin Target, Thermal, and Thick Target Expressions for f-f and f-b HXR Emission Spectra
===================================================================================================
Here we extend the above results on local emissivities $j(\epsilon)$ to estimate total spectral emission rate $J(\epsilon)$ (photons sec$^{-1}$ per unit $\epsilon$) from extended flare volumes as required for real flare data.
Thin Target Coronal Loop
------------------------
A thin target is one in which $F(E)$ is not significantly modified by energy losses or gains over the volume. For a loop of half length $L$, transverse area $A$, volume $2AL$ and density $n_p$, the total emission rate spectra contributions $J_{thin}(\epsilon)$ are for a power law $F(E)$ with a low energy cut-off, by Equation (15),
$$\begin{aligned}
\nonumber J_{B_{thin}}(\epsilon) = &
\frac{\delta-1}{\delta}\frac{8\alpha\zeta_B}{3}\frac{m_ec^2r_e^2}{\epsilon}
\frac{2n_pALF_c}{E_c}\times &\\
\nonumber & ~~\left[\frac{\epsilon}{E_c}\right]^{-\delta}; &
\epsilon \ge E_c \\ & ~~1; & \epsilon < E_c\end{aligned}$$ and by Equation (16), $$\begin{aligned}
\nonumber J_{R_{thin}}(\epsilon) =& (\delta-1)\frac{32\pi
\zeta_{RZ_{eff}}}{3^{1/2}\alpha}\frac{r_e^2
\chi}{\epsilon}\frac{2n_pALF_c}{E_c^2} \times \Sigma_{Z_{eff}}& \\
\nonumber & \times ~~\left[\frac{\epsilon-Z_{eff}^2\chi}{E_c}
\right]^{-\delta-1}; & \epsilon \ge E_c+Z_{eff}^2\chi \\
\nonumber & \times ~~0; & \epsilon < E_c+Z_{eff}^2\chi, \\&&\end{aligned}$$ where the summation is over all $Z_{eff} \le (\epsilon-E_c]^{1/2}$. These spectral shapes $J(\epsilon)$ are of course just the same as the thin target $j$ forms, scaled by the plasma volume.
Hot Coronal Loop Thermal Emission (in the Kramers approximation)
----------------------------------------------------------------
Both f-f and f-b emissions are included in the standard analyses (e.g. Mewe [*et al.*]{} 1987, Dere [*et al.*]{} 1996) of isothermal hot plasma contributions to flare spectra, using full cross sections and ionisation balance expressions. It is therefore surprising that f-b is omitted from calculations of non-thermal emission, especially at low $\epsilon$, where electrons of comparable energy are present in both thermal and non-thermal populations. In applying our study of the non-thermal f-b to real data we wish to include thermal emission as it is important at energies under about 20 keV and so dilutes the visibility of non-thermal contributions. In order to treat the thermal and non-thermal $j$ consistently and allow meaningful comparisons we use the expressions for the thermal $j$ relevant to the Kramers cross sections just as in the non-thermal case - but see remarks previously and below concerning Gaunt factors and absolute accuracy of our results.
For an isothermal plasma the local Maxwellian electron flux spectrum is
$$F_{therm}(E)=\left[\frac{8}{\pi m_e}\right]^{1/2}
\frac{E}{(kT)^{3/2}}n_p\exp(-E/kT),$$
which, by Equation (4), gives for the thermal bremsstrahlung emission from a uniform loop
$$J_{Btherm}(\epsilon) = \frac{16\alpha r_e^2}{3}\zeta_B m_ec^2 \times
\left[\frac{8}{\pi m_e}\right]^{1/2}\frac{2n_p^2AL
e^{-\epsilon/kT}}{\epsilon(kT)^{1/2}}$$
and for the recombination
$$\begin{aligned}
\label{} \nonumber J_{Rtherm}(\epsilon) &= \sqrt{\frac {2\pi}{27
m_e}}\frac {64r_e^2\chi^2}{\alpha} \frac {2n_p^2AL}{\epsilon
(kT)^{3/2}} \Sigma_{Z_{eff}} \zeta_{RZ_{eff}} & \times \\ &
\exp\left(\frac{Z_{eff}^2\chi - \epsilon}{kT}\right).&\\ \nonumber\end{aligned}$$
These results can be compared with those of Culhane (1969) and Culhane and Acton (1970) who were among the first to explicitly address the X-Ray spectrum from hot coronal plasmas. Using the Kramers cross sections is essentially equivalent to setting to unity all Gaunt factors in their expressions. When we do so, the $\epsilon, T$ dependences of our $ J_{Rtherm}, J_{Btherm}$ are identical to theirs - e.g. $ J_{Rtherm}/ J_{Btherm}$ is independent of $\epsilon$, the only difference being that our $J_{Rtherm}$ is much larger (in absolute value) than theirs, mainly because they used the very much lower value of $A_Z$ for Fe believed at that time. Examination of the $\epsilon, T$ dependences of Culhane’s Gaunt factors shows that they affect quite significantly both the f-f and the f-b spectra from a Maxwellian $F(E)$ and we should expect the same to be true for non-thermal $F(E)$ like power-laws. Thus, any accurate absolute comparison of predictions with data will require incorporation of appropriate $g,G$. However, these do not affect the absolute orders of magnitude of $J_{Rtherm}, J_{Btherm}$ nor the dependencies on $n_p,V,F_c$ etc., nor the locations of edges. So, for the present purpose of demonstrating the importance of f-b, the Kramers expressions will suffice.
Thick target (dense loop or footpoint) f-f and f-b emission spectra
-------------------------------------------------------------------
In the thick target case, $j$ evolves in space along with the energy losses of the electrons. To find $j$ locally one uses the continuity equation (Brown 1972) and then integrates over volume to get $J$. However, to get the whole volume $J$, it is actually simpler (Brown 1971) to start with the electron injection rate spectrum ${\cal
F}_o(E_o)$ electrons/sec per unit injection energy $E_o$ and use the expression
$$J_{thick}(\epsilon) = \int_{E_o} {\cal F}_o(E_o) \eta (\epsilon,
E_o)dE_o,$$
where $\eta (\epsilon, E_o)$ is the total number of photons per unit $\epsilon$ emitted by an electron of energy $E_o$ as it decays in energy. For purely collisional losses $dE/dN=-K/E$ with $K=2\pi e^4
\Lambda$, $e$ being the electronic charge and $\Lambda$ the Coulomb Logarithm. Then
$$\eta(\epsilon,E_o) =\frac{1}{K}\int_E E\frac{dQ}{d\epsilon} dE$$
for the relevant radiation cross section $dQ/d\epsilon$. Note that this assumes H to be uniformly and fully ionised along the electron path. For partially ionised H the energy loss constant $K$ is reduced but this situation is not relevant to our hot source situations.
For our Kramers $dQ/d\epsilon$ f-f and f-b expressions (3), (7) and (9), the resulting expressions, in the case where $A_{Z_{eff}}$ are uniform along the path, Equation (B.7) gives
$$\begin{aligned}
\nonumber \eta_B(\epsilon,E_o) =& \frac{8\alpha\zeta_B}{3}\frac{r_e^2m_ec^2}{K} & \times \\
\nonumber & \left[\frac{E_o}{\epsilon}-1\right]; & \epsilon \le E_o \\
& 0; & \epsilon > E_o\end{aligned}$$
and
$$\begin{aligned}
\nonumber \eta_{RZ}(\epsilon, E_o) & = \frac{32\pi
A_{Z_{eff}}Z_{eff}^4}{3^{3/2}\alpha}\frac{r_e^2\chi^2}{K\epsilon} &\times\\
\nonumber & 1; & E_o\ge\epsilon+Z_{eff}^2 \chi \\
& 0; & E_o <\epsilon+Z_{eff}^2.\end{aligned}$$
For a power-law injection rate spectrum of spectral index $\delta_o$, viz
$${\cal F}_o(E_o) = (\delta_o-1)\frac{{\cal
F}_{oc}}{E_{oc}}\left[\frac{E_o}{E_{oc}}\right]^{-\delta_o}; E_o \ge E_{oc},$$
where ${\cal F}_{oc}$ is the total rate above low energy cut-off $E_{oc}$, the expressions for the non-thermal emission spectra are then by Equation (B.6)
$$\begin{aligned}
\label{} \nonumber J_{Bthick}(\epsilon) =& \frac{8\alpha
r_e^2}{3}\frac{\zeta_B m_ec^2{\cal F}_{oc}}{(\delta_o - 1)(\delta_o
-2)K} & \times \\
\nonumber & \left(\frac
{\epsilon}{E_c}\right)^{-\delta_o +1}; & \epsilon \ge E_c \\
\nonumber & \left[(\delta_o-1)\frac{E_c}{\epsilon}-(\delta_o-2)\right]; &\epsilon < E_c\\\end{aligned}$$
and, for ion $Z_{eff}$,
$$\begin{aligned}
\label{}
\nonumber J_{RZ_{eff}thick}(\epsilon) = & \frac{32\pi
r_e^2
m_ec^2}{3^{3/2}\alpha}\zeta_{RZ_{eff}}\frac{\chi^2}{K\epsilon}\frac{{\cal
F}_{oc}}{E_{oc}} \times &\\
\nonumber &
\left[\frac{\epsilon-Z_{eff}^2\chi}{E_{oc}}\right]^{-\delta_o + 1}; &
\epsilon \ge E_{oc}+Z_{eff}^2\chi \\
\nonumber & \left[\frac{E_{oc}-Z_{eff}^2\chi}{E_{oc}}\right]^{-\delta_o + 1}; & Z_{eff}^2\chi<\epsilon <
E_{oc}+Z_{eff}^2\chi\\
& 0; & \epsilon < Z_{eff}^2\chi.\end{aligned}$$
For the case of a cold thick target footpoint the total $\zeta_R$ can be almost as small as 1 if only hydrogen and some low $\zeta_R$ elements are ionised and even zero if $T<8000$ K or so (there being almost no charged ions present). In these sources the f-b contribution is negligible or at most a very small correction. For a collisonally thick hot loop $\zeta_R$ is, however, very much higher.
The main distinction of these hot thick target spectra compared to hot thin targets is that the decay of all electrons to zero energy means that the signature of the cut off $E_{oc}$ in the injection spectrum appears not as a discontinuity in $J(\epsilon)$ but only in its gradient $J'(\epsilon)$. This gradient break is very noticeable in Figure B.1 at energy $\epsilon = E_c+V_{Fe}$. So, even in the thick target case, spectral diagnosis of any $E_{oc}$ present is possible. The recombination edges themselves appear at the relevant ionisation energies $\epsilon = V_{eff}$, these being from thick target electrons decelerated to zero $E$. These non-thermal recombination spectral edges are then down in the energy regime below 10 keV which is complicated by Fe lines etc., making the interpretation of ${\cal F}_o$ there, and of the lines, more difficult.
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|
---
abstract: 'The 1-2-3 Conjecture, posed in 2004 by Karoński, [Ł]{}uczak, and Thomason, states that one may weight the edges of any connected graph on at least 3 vertices from the set $\{1,2,3\}$ (call the weight function $w$) so that the function $f(v) = \sum_{u \in N(v)} w(uv)$ is a proper vertex colouring. This paper presents the current state of research on the 1-2-3 Conjecture and the many variants that have been proposed in its short but active history.'
author:
- Ben Seamone
bibliography:
- 'references.bib'
title: 'The 1-2-3 Conjecture and related problems: a survey'
---
\[section\] \[thm\][Question]{} \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Definition]{} \[thm\][Problem]{}
Introduction {#ch:intro:KLTintro}
============
Unless otherwise stated, a graph $G = (V,E)$ is simple, finite, and undirected. Standard graph theory notation ([@BM08], [@Diestel]) is used throughout.
In 2004, Karo[ń]{}ski, [Ł]{}uczak, and Thomason [@KLT04] made the following conjecture:
If $G$ is a graph with no component isomorphic to $K_2$, then the edges of $G$ may be assigned weights from the set $\{1,2,3\}$ so that, for any adjacent vertices $u,v \in V(G)$, the sum of weights of edges incident to $u$ differs from the sum of weights of edges incident to $v$.
The motivation for the 1-2-3 Conjecture comes from the study of graph irregularity strength. An edge weighting of a graph $G$ is an [**irregular assignment**]{} if, for any pair of vertices $u, v \in V(G)$, the sum of weights of edges incident to $u$ differs from the sum of weights of edges incident to $v$. The [**irregularity strength**]{} of a graph $G$ is the smallest value of $k$ such that $G$ has an irregular assignment from $[k]$.
With a few simple definitions, we may state the 1-2-3 Conjecture more succinctly. An [**edge $k$-weighting**]{} is a function $w:E(G) \to [k] := \{1,2, \ldots, k\}$. An edge $k$-weighting $w$ is a [**proper vertex colouring by sums**]{} if $\sum_{e \ni u}w(e) \neq \sum_{e \ni v}w(e)$ for every $uv \in E(G)$. Denote by ${\chi_\Sigma^e}(G)$ the smallest value of $k$ such that a graph $G$ has a edge $k$-weighting which is a proper vertex colouring by sums. A graph $G$ is [**nice**]{} if no connected component is isomorphic to $K_2$. The 1-2-3 Conjecture may now be stated as follows:
If $G$ is nice, then ${\chi_\Sigma^e}(G) \leq 3$.
This paper surveys what is currently known about the 1-2-3 Conjecture, as well as the interesting questions (and answers) that have arisen during the conjecture’s brief but active history. Since the introduction of the 1-2-3 Conjecture, a variety of variations have been considered. For instance, one may vary the set to be weighted or coloured (edges, vertices, or both), the operation by which one obtains a colour, or the set from which one weights the objects (a variation of interest is to study weightings from arbitrary lists rather than from a fixed set). Due to the number of different weighting-colouring parameters one could conceivably study (and since there is no standard notation that is consistent across the literature on the subject), we have introduced the notation above, which is a modification of notation proposed by Gy[ő]{}ri and Palmer [@GP09]. Note the three components of ${\chi_\Sigma^e}(G)$ – “$\chi$” indicates that we desire a proper [*vertex*]{} colouring, the superscript “$e$” indicates that the weighting of interest is of the [*edges*]{}, and the subscript “$\Sigma$” indicates that vertex colours are obtained by [*summing*]{} edge weights. As new parameters are defined for different combinations of weightings and colourings, we will develop notation consistent with this format.
Progress on the 1-2-3 Conjecture
================================
Early approaches to the 1-2-3 Conjecture focused on relating ${\chi_\Sigma^e}(G)$ to $\chi(G)$. One of the first such results appears in the paper which introduces the 1-2-3 Conjecture:
\[odd\] If $(\Gamma,+)$ is a finite abelian group of odd order and $G$ is a nice $|\Gamma|$-colourable graph, then there is a weighting of the edges of $G$ with the elements of $\Gamma$ such that the vertex colouring by sums is a proper vertex colouring.
In particular, if $G$ is nice and $k$-colourable for $k$ odd, then ${\chi_\Sigma^e}(G) \leq k$. This theorem was then extended to the following:
If $G$ is 2-connected and $\chi(G) \geq 3$, then ${\chi_\Sigma^e}(G) \leq \chi(G)$. In particular, for any integer $k \geq 3$ and nice graph $G$, the following hold:
1. if $G$ is $k$-colourable for $k$ odd, then ${\chi_\Sigma^e}(G) \leq k$;
2. if $G$ is $k$-colourable for $k \equiv 0 \pmod 4$, then ${\chi_\Sigma^e}(G) \leq k$;
3. if $\delta(G) \leq k-2$, then ${\chi_\Sigma^e}(G) \leq k$;
4. if $G$ is $2$-connected, $k$-colourable, and has $\delta(G) \geq k+1$ for $k \equiv 2 \pmod 4$, then ${\chi_\Sigma^e}(G) \leq k$.
It is also shown in [@DLY] that ${\chi_\Sigma^e}(G) \leq \chi(G)$ for a connected graph $G$ if $|V(G)|$ is odd or if there is a proper $\chi(G)$-colouring where one colour class has even size. The most significant progress toward solving the 1-2-3 Conjecture is the establishment and improvement of constant bounds on ${\chi_\Sigma^e}(G)$ for every nice graph $G$. The best known bound to date is as follows:
\[sum5\] If $G$ is a nice graph, then ${\chi_\Sigma^e}(G) \leq 5$.
The proof of this bound involves an algorithmic argument; the graph’s vertices are processed in a linear order and, at each step, some weights of edges incident to a vertex are adjusted so that there are no colouring conflicts with previously considered vertices. Addario-Berry, Dalal, McDiarmid, Reed, and Thomason [@ADMRT07] had previously shown that ${\chi_\Sigma^e}(G) \leq 30$ for any nice graph $G$, a bound which was then improved to ${\chi_\Sigma^e}(G) \leq 16$ by Addario-Berry, Dalal, and Reed [@ADR08], then to ${\chi_\Sigma^e}(G) \leq 13$ by Wang and Yu [@WY08], and then to ${\chi_\Sigma^e}(G) \leq 6$ by Kalkowski, Karo[ń]{}ski, and Pfender [@KKP2].
It is easily seen that there exist nice graphs for which two edge weights do not suffice to colour the vertices by sums (e.g. $K_3$, $C_6$), and hence the best possible constant bound for all nice graphs is the conjectured value of 3. However, it is known that if $G$ is a random graph chosen from $\mathcal{G}_{n,p}$ for any constant $p \in (0, 1)$, then asymptotically almost surely ${\chi_\Sigma^e}(G) \leq 2$ [@ADR08]. It is also known that $2$ edge weights suffice to properly colour vertices by sums for any digraph (in fact, a stronger result is known, which is stated in Theorem \[BGNdigraph\]).
Clearly, ${\chi_\Sigma^e}(G) = 1$ if and only if adjacent vertices of $G$ always have different degrees. A classification of graphs for which ${\chi_\Sigma^e}(G) = 2$ does not yet exist, though some partial results are known. Chang, Lu, Wu, and Yu [@CLWY] showed that ${\chi_\Sigma^e}(G) \leq 2$ if $G$ is bipartite and $d$-regular for $d \geq 3$. Lu, Yu, and Zhang [@LYZ] proved that if $G$ is a nice graph which is either $3$-connected and bipartite or has minimum degree $\delta(G) \geq 8\chi(G)$, then ${\chi_\Sigma^e}(G) \leq 2$. Davoodi and Omooni [@DO] claim to have recently proven that, for any two bipartite graphs $G$ and $H$, one has that ${\chi_\Sigma^e}(G \,\Box\, H) \leq 2$ if $G \,\Box\, H \neq K_2$. Their manuscript also states that, for any two graphs $G$ and $H$, ${\chi_\Sigma^e}(G \,\Box\, H) \leq \max\{{\chi_\Sigma^e}(G), {\chi_\Sigma^e}(H)\}$ (though not stated in their paper, one should assume that both $G$ and $H$ are nice, or adopt the convention that ${\chi_\Sigma^e}(K_2) := \infty$). Khatirinejad et al [@Ben1] show that ${\chi_\Sigma^e}(G) \leq 2$ if all cycles of $G$ have length divisible by $4$. A [**generalized theta graph**]{}, denoted $\Theta{_{(m_1,\ldots m_d)}}$ ($d \geq 3$) is a graph constructed from $d$ internally disjoint paths between distinct vertices, where the $i^{\rm th}$ path has of length $m_i$. Khatirinejad et al [@Ben1] and Lu, Yang, and Zhang [@LYZ] show that ${\chi_\Sigma^e}(\Theta{_{(m_1,\ldots m_d)}}) = 2$ if and only if $\{m_1,m_2, m_3, \ldots m_d)\} \neq \{1, 4k_2 + 1, 4k_3+1, \ldots, 4k_d + 1\}$ for some set of positive integers $\{k_i \geq 1 \mid i=2,\ldots,d\}$. In fact, the [*only*]{} bipartite graphs for which it is known that ${\chi_\Sigma^e}(G) > 2$ are cycles $C_{4k+2}$ ($k \geq 1$), generalized theta graphs $\Theta{_{(1, 4k_2 + 1,4k_3+1 \ldots, 4k_d + 1)}}$, and an infinite subfamily of a class of bipartite graphs called polygon trees (Davoodi and Omoomi [@DO]). The following problem remains open:
Characterize the set of graphs $\{G \mid {\chi_\Sigma^e}(G) \leq 2, \textrm{$G$ bipartite}\}$.
Recalling Theorem \[odd\], one may consider the minimum $s$ such that, for any abelian group $(\Gamma,+)$ of order $s$, a graph $G$ has an edge weighting from $\Gamma$ which properly colours $V(G)$ by sums. This parameter, called the [**group sum chromatic number**]{} and denoted $\chi^{\Sigma}_g(G)$, was introduced by Anholcer and Cichacz [@AC1], who prove the following theorem:
\[groupsum\] If $G$ is a graph with no component having fewer than 3 vertices, then $\chi(G) \leq \chi^{\Sigma}_g(G) \leq \chi(G)+2$.
Furthermore, the authors give a complete characterization of which graphs have group sum chromatic number $\chi(G)$, $\chi(G)+1$, and $\chi(G) + 2$.
We conclude this introduction to the 1-2-3 Conjecture with two related open problems.
\[unique3\] Does there exist a graph $G$ which has a unique edge $3$-weighting (up to isomorphism) which properly colours $V(G)$ by sums?
We say that $G$ is [**$S$-weight colourable**]{} if $G$ has an edge weighting from $S$ which properly colours $V(G)$ by sums, and $G$ is [**$k$-weight colourable**]{} if $G$ is $S$-weight colourable for every set $S \subset {\mathbb{R}}$ of order $k$. In [@Ben1], gadget graphs are constructed which are uniquely $S$-weight colourable for any set $S$ of order $2$. A positive answer to Question \[unique3\] may similarly provide a class of gadget graphs useful for disproving the 1-2-3 Conjecture (and, hence, a negative answer would provide further evidence [*for*]{} the conjecture).
In [@Ben1], it was also noted that it is unknown how difficult it is to decide if a given graph $G$ admits an edge $2$-weighting which properly colours $V(G)$ by sums (and, more generally, an edge $\{a,b\}$-weighting for general $a,b \in {\mathbb{R}}$).
Is it NP-complete to decide whether or not a given graph $G$ is $2$-weight colourable?
Havet, Paramaguru, and Sampathkumar [@HPS] have shown that it is NP-complete to decide if two edge weights suffice for a cubic graph $G$ if one requires distinct [*multisets*]{} of weights at adjacent vertices (more on such colouring variations can be found in the next section). Dudek and Wajc [@DW] have recently proven that it is NP-complete to determine whether or not a graph $G$ is $S$-weight colourable for $S = \{1, 2\}$ and $S = \{0, 1\}$. The general problem remains open, though they suggest that their methods may work for any set of two rational numbers.
Variation I: Colouring by products, multisets, sets, and sequences {#ch:intro:var1}
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As mentioned in the introduction, there are a variety of ways in which one may modify the way in which vertex colours are obtained from an edge weighting of a graph. The first variations we consider are those where addition of edge weights as the colouring method is replaced by another operation; in particular, we consider variations where colours are obtained by taking a product, multiset, set, or sequence of weights from edges incident to $v$ for each $v \in V(G)$. If such a colouring is proper, then the edge $k$-weighting of $G$ is a [**proper vertex colouring by products, multisets, sets, or sequences**]{}, respectively. Note that a colouring by sequences relies on a well-defined method for ordering the weights from edges incident to a vertex. The natural way to do this is to first order the elements of $E(G)$, and then order the edges incident to a vertex according to the global ordering of $E(G)$. The smallest $k$ for which such colourings exist for a graph $G$ are denoted ${\chi_\Pi^e}(G)$ (products), ${\chi_m^e}(G)$ (multisets), ${\chi_s^e}(G)$ (sets), ${\chi_{\sigma^*}^e}(G)$ (sequences, where one can choose any ordering of $E(G)$), and ${\chi_\sigma^e}(G)$ (sequences, where a weighting must exist for any ordering of $E(G)$).
Let us first consider colouring by multisets. Clearly, two multisets are distinct if the sums of their elements are distinct, and hence colouring by multisets is a natural relaxation of the problem of colouring by sums. The following simple observation is used frequently:
\[multibound\] If $\min\{{\chi_\Sigma^e}(G),{\chi_\Pi^e}(G),{\chi_s^e}(G)\} \leq k$, then ${\chi_m^e}(G) \leq k$.
It follows from the 1-2-3 Conjecture and Proposition \[multibound\] that the expected upper bound for ${\chi_m^e}(G)$ is 3; whether or not this upper bound holds remains open. The first bound on ${\chi_m^e}(G)$ was established in [@KLT04], where Karo[ń]{}ski, [Ł]{}uczak, and Thomason used a probabilistic argument to show that if $G$ is a nice graph, then ${\chi_m^e}(G) \leq 183$. By using the following vertex partitioning lemma, an improved bound of ${\chi_m^e}(G) \leq 4$ was proven in [@AADR05]:
\[vpart\] Let $G$ be a connected graph which is not $3$-colourable. There exists a partition of $V(G)$ into sets $V_0, V_1, V_2$ such that there exists a weighting $w:E(G) \rightarrow \{c_0, c_1, c_2, c^*\}$ with the following properties
1. for each $i \in \{0,1,2\}$, every $v \in V_i$ is incident to at least one edge weighted $c_i$;
2. for each $i \in \{0,1,2\}$, the vertices in $V_i$ are incident only to edges weighted $c_i, c_{i-1 \pmod 3}, c^*$, and
3. for each $i \in \{0,1,2\}$, if $u, v \in V_i$ are adjacent, then the number of edges incident to $u$ with weight $c_i$ is different from the number of edges incident to $v$ with weight $c_i$.
\[me4\] If $G$ is a nice graph, then ${\chi_m^e}(G) \leq 4$.
If $G$ is $3$-colourable, then ${\chi_m^e}(G) \leq {\chi_\Sigma^e}(G) \leq 3$ by Theorem \[odd\] and Proposition \[multibound\]. If $G$ is not $3$-colourable, then the edge-weighting guaranteed by Lemma \[vpart\] gives the desired vertex colouring by multisets.
Addario-Berry et al. [@AADR05] also use Lemma \[vpart\] to prove that ${\chi_m^e}(G) \leq 3$ if $G$ is nice and $\Delta(G) \geq 1000$. It is also shown in [@HPS] that ${\chi_m^e}(G) \leq 2$ if $G$ is cubic and bipartite.
Colouring vertices by sequences of edge weights is a natural further relaxation of the problem of colouring by multisets. Given an ordering of $E(G)$ and a weighting of $E(G)$, let each vertex be coloured by the sequence obtained by taking the multiset of weights from incident edges and ordering these weights according to the order in which their corresponding edges appear. Recall that ${\chi_{\sigma^*}^e}(G)$ (respectively, ${\chi_\sigma^e}(G)$) is the smallest $k$ so that, for some (resp., any) ordering of $E(G)$, an edge $k$-weighting exists which properly colours $V(G)$ by sequences in this way. Clearly, ${\chi_{\sigma^*}^e}(G) \leq {\chi_\sigma^e}(G) \leq {\chi_m^e}(G)$, and so $4$ edge weights suffice for both colouring by sequences variations. The following are the best known bounds for ${\chi_{\sigma^*}^e}(G)$ and ${\chi_\sigma^e}(G)$:
\[sequencebounds\] If $G$ is a nice graph, then
1. ${\chi_{\sigma^*}^e}(G) \leq 2$, and
2. ${\chi_\sigma^e}(G) \leq 3$ if $\delta(G) \in \Omega\left(\log{\Delta(G)}\right)$.
In fact, stronger statements than those of Theorem \[sequencebounds\] are proven in [@Ben-LLL]; see Theorem \[listsequencebounds\]. The constant suppressed by the $\Omega$ notation in the bound on $\delta(G)$ above can be improved if one imposes a girth condition on $G$. Similar results for multigraphs are also proven in [@Ben-LLL].
Turning our attention to colouring by products, we note that a constant bound on ${\chi_m^e}(G)$ implies a constant bound on ${\chi_\Pi^e}(G)$. The following corollary to the work of Addario-Berry et al. was first noted in [@SK08]:
If $G$ is a nice graph, then ${\chi_\Pi^e}(G) \leq 5$.
If $G$ is $3$-colourable, then ${\chi_m^e}(G) \leq 3$ by Theorem \[odd\] and Proposition \[multibound\], and edge weights $\{2,3,5\}$ suffice. If $G$ is not $3$-colourable, then let $w:E(G) \to \{c_0, c_1, c_2, c^*\}$ be the edge weighting guaranteed by Theorem \[vpart\], and let $c_0 = 2$, $c_1 = 3$, $c_2 = 5$ and $c^* = 1$; this edge $5$-weighting properly colours $V(G)$ by products.
While constant bounds exist for ${\chi_\Sigma^e}(G)$, ${\chi_\Pi^e}(G)$ and ${\chi_m^e}(G)$, this is not possible for the parameter ${\chi_s^e}(G)$. To see this, note that an edge $k$-weighting allows at most $2^k - 1$ possible vertex colours by sets and so the complete graph on $2^k$ vertices must have ${\chi_s^e}(G) > k$. In the study of colouring by sets, Gy[ő]{}ri and Palmer establish a link between ${\chi_s^e}(G)$ and a hypergraph induced by $G$. A hypergraph is said to have [**Property B**]{} if there exists a colouring $c:V(H) \rightarrow \{1,2\}$ such that every hyperedge contains vertices of both colours; this concept has its roots in set theory and is due to Bernstein [@Bern].
Let $G$ be a bipartite graph with bipartition of the vertices $V(G) = X \cup Y$. If $H$ is the hypergraph with $V(H) = Y$ and $E(H) = \{N_G(x) \,:\, x \in X\}$, then ${\chi_s^e}(G) = 2$ if and only if $H$ has Property B.
In [@GHPW08], Gy[ő]{}ri, Hornák, Palmer, and Woźniak show that if $G$ is nice, then ${\chi_s^e}(G) \leq 2\lceil{\log_2{\chi(G)}}\rceil + 1$, a bound which was subsequently refined.
\[setbound\] If $G$ is nice, then ${\chi_s^e}(G) = \lceil{\log_2{\chi(G)}}\rceil + 1$.
The parameter ${\chi_s^e}(G)$ is denoted $\textup{gndi}(G)$ in [@GHPW08] and called the general neighbourhood distinguishing index. The motivation for the parameter comes from the study of the [**neighbourhood distinguishing index**]{}\[ndidef\] of a graph $G$, denoted $\textup{ndi}(G)$, which is the smallest integer $k$ such that $G$ has a proper edge $k$-colouring such that adjacent vertices have distinct sets of colours on their incident edges. This parameter is also known as the [**adjacent vertex distinguishing chromatic index**]{}, and the type of colouring is called an [**adjacent strong edge-colouring**]{} or [**1-strong edge-colouring**]{}. The neighbourhood distinguishing index was first introduced by Liu, Wang and Zhang [@LWZ], who propose the following conjecture:
\[ndi\] If $G \notin \{K_2, C_5\}$ and $G$ is connected, then $\Delta(G) \leq \textup{ndi}(G) \leq \Delta(G) + 2$.
While the lower bound is trivially true, the upper bound appears to be difficult to prove. Here are a few known bounds for ${\rm ndi}(G)$:
Conjecture \[ndi\] holds if $G$ is bipartite or $\Delta(G) \leq 3$.
If a graph $G$ is nice, then $\textup{ndi}(G) \leq \Delta(G) + O(\log{\chi(G)})$.
If a graph $G$ is nice and $\Delta(G) > 10^{20}$, then $\textup{ndi}(G) \leq \Delta(G) + 300$.
Dong and Wang [@DW12] have recently introduced the study of [**neighbour sum distinguishing colourings**]{}, where a proper edge colouring is used to properly colour vertices by sums. The smallest $k$ for which such an edge-colouring exists is denoted $\textup{ndi}_\Sigma(G)$; it is shown that $\textup{ndi}_\Sigma(G) \leq \max\{2\Delta(G) + 1, 25\}$ if $G$ is a planar graph and $\textup{ndi}_\Sigma(G) \leq \max\{2\Delta(G), 19\}$ if $G$ is a graph such that $\textup{mad}(G) \leq 5$ (where $\textup{mad}(G)$ is the maximum average degree taken over all subgraphs of $G$).
Finally, we present one further variation of neighbour-distinguishing edge weightings, introduced by Baril and Togni [@BT]. A [**proper $k$-tuple edge-colouring**]{} of a graph $G$ assigns a set of $k$ colours to each edge such that adjacent edges have disjoint colour sets. The vertex version of such a colouring was introduced by Stahl [@Stahl]. Let $S(v)$ denote the union of all sets of colours assigned to edges incident to $v \in V(G)$. If $S(u) \neq S(v)$ for each $uv \in E(G)$, then the edge colouring is a [**$k$-tuple neighbour-distinguishing colouring**]{}. Baril and Togni determine the smallest $k$ for which various classes of graphs have $k$-tuple neighbour-distinguishing colourings. They also make the following conjecture, which extends Conjecture \[ndi\] to multigraphs:
\[multindi\] If $G$ is a connected multigraph, $G \neq C_5$, with edge multiplicity $\mu(G)$ and maximum degree $\Delta(G)$, then ${\rm ndi}(G) \leq \Delta(G) + \mu(G) + 1$.
Variation II: Total and vertex weightings {#ch:intro:var2}
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Total weightings
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A [**total weighting**]{} of a graph $G$ is an assignment of a real number weight to each $e \in E(G)$ and each $v \in V(G)$. The concept of proper colourings induced by total $k$-weightings was introduced by Przyby[ł]{}o and Wozniak [@PW10]. Given a total $k$-weighting of $G$, we consider vertex colourings obtained by taking either the sum, product, multiset, or set of weights taken from the edges incident to $v$ and from $v$ itself for each $v \in V(G)$. If such a colouring is proper, then the total $k$-weighting of $G$ is a [**proper vertex colouring by sums, products, multisets or sets**]{}, respectively. The smallest values of $k$ for which a proper colouring of each type exists for a graph $G$ are denoted ${\chi_\Sigma^t}(G)$, ${\chi_\Pi^t}(G)$, ${\chi_m^t}(G)$ and ${\chi_s^t}(G)$, respectively. Note that, while one must exclude graphs with connected components containing precisely one edge when considering edge weightings which properly colour vertices by sums, this exclusion is not necessary for total weightings.
We first observe that, for the colouring methods considered so far, any total weighting parameter is bounded above by its edge weighting counterpart:
\[totalboundbyedge\] If $G$ is any graph, then
1. ${\chi_\Sigma^t}(G) \leq {\chi_\Sigma^e}(G)$,
2. ${\chi_\Pi^t}(G) \leq {\chi_\Pi^e}(G)$,
3. ${\chi_m^t}(G) \leq {\chi_m^e}(G)$,
4. ${\chi_s^t}(G) \leq {\chi_s^e}(G) =\lceil{\log_2{\chi(G)}}\rceil + 1$.
The proofs, while not difficult, are omitted. In many cases, allowing vertex weights in addition to edge weights can strictly decrease the number of weights needed to obtain a proper vertex colouring of a graph. The following conjecture motivates the study of total weightings and vertex colouring by sums:
For every graph $G$, ${\chi_\Sigma^t}(G) \leq 2$.
Hulgan, Lehel, Ozeki, and Yoshimoto [@HLOY] studied a variation of ${\chi_\Sigma^t}(G)$ similar to that of the $S$-weight colourability variation of ${\chi_\Sigma^e}(G)$. They consider weighting vertices and edges from a set of two non-negative real numbers $\{a,b\}$ and colouring vertices by sums; for some general values of $a,b \in {\mathbb{R}}$, they show that a variety of classes of graphs admit total weightings from $\{a,b\}$ which colour vertices by sums.
More generally, one may consider weightings which assign edges values from one set and vertices values from another. For example, a [**total $(k,l)$-weighting**]{} $w$ if a graph $G$ assigns $w(v) \in [k]$ for each $v \in V(G)$ and $w(e) \in [l]$ for each $e \in E(G)$. The best known bound on ${\chi_\Sigma^t}(G)$, to date, comes from the following result on total $(k,l)$-weightings:
\[12bound\] Every graph has a total $(2,3)$-weighting which is a proper vertex colouring by sums. As a consequence, ${\chi_\Sigma^t}(G) \leq 3$ for any graph $G$.
This theorem improves previous results from [@PW10], where it is shown that ${\chi_\Sigma^t}(G) \leq \min\{ \lfloor \chi(G)/2 \rfloor + 1, 11\}$ for any graph $G$, and from [@Prz08], where it is shown that ${\chi_\Sigma^t}(G) \leq 7$ if $G$ is regular. It is shown in [@PW10] that the 1-2 Conjecture holds if $G$ is complete, $3$-colourable, or $4$-regular. Little else is known when one varies the colouring operation. Theorem \[12bound\] implies that ${\chi_m^t}(G) \leq 3$ for any graph $G$. The following conjecture is proposed on colouring by products from total $k$-weightings:
\[SKprod\] For every graph $G$, ${\chi_\Pi^t}(G) \leq 2$.
In [@SK08], Conjecture \[SKprod\] was verified for graphs which are 3-colourable or complete, and it was shown that ${\chi_\Pi^t}(G) \leq 3$ for every graph $G$.
On the topic of total weightings which properly colour vertices by sets, much more attention has been paid to weightings which are themselves proper total colourings. Recall that, in a proper total colouring of $G$, adjacent vertices, adjacent edges, and incident vertices and edges must all receive different colours. As an extension of the neighbourhood distinguishing index discussed in Section \[ch:intro:var1\] (see page ), Zhang et al [@CLL+] study proper total colourings which, considered as weightings, also properly colour $V(G)$ by sets (the set of weights consisting of the colour of the vertex itself as well as the colours on all incident edges). The fewest number of colours required for such a colouring to exist for a graph $G$ is called the [**adjacent vertex distinguishing total chromatic number**]{}, which is denoted $\chi'_{at}(G)$. Since $\chi'_{at}(G) \geq \chi''(G)$, it follows immediately that $\chi'_{at}(G) \geq \Delta(G)+1$ for any graph $G$. Conversely, it is conjectured that any graph can be so coloured using a constant number of colours greater than its maximum degree.
\[adjstrong\] For any graph $G$, $\Delta(G) + 1 \leq \chi'_{at}(G) \leq \Delta(G) + 3$.
This conjecture is reminiscent of Conjecture \[ndi\] on edge colourings and, more notably, the Total Colouring Conjecture.
For any graph $G$, $\chi''(G) \leq \Delta(G) + 2$.
The best known bound on a graph’s total chromatic number is due to Molloy and Reed [@MR], who show that $\chi''(G) \leq \Delta(G) + 10^{26}$. The Total Colouring Conjecture is notoriously difficult, which suggests that Conjecture \[adjstrong\] may also be very difficult to solve. However, Coker and Johannson [@CJ] have recently shown that there exists a universal constant $C$ such that $\chi'_{at}(G) \leq \Delta(G) + C$. Their proof relies on a probabilistic argument to show that $\chi'_{at}(G) \leq \chi''(G) + C'$ for some other constant $C'$, then invokes Molloy and Reed’s aforementioned bound on $\chi''(G)$.
Pilśniak and Woźniak [@PW+] study a similar parameter, where one requires that a proper total colouring of $G$ distinguishes adjacent vertices by sums. They too conjecture that $\Delta(G) + 3$ colours should suffice for any graph $G$; they verify their conjecture for complete graphs, cycles, bipartite graphs, cubic graphs and graphs with maximum degree at most three.
Vertex weightings {#vertexweightings}
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Having considered edge and total weightings, we now turn our attention to vertex weightings. A [**vertex $k$-weighting**]{} of a graph $G$ is a mapping $w:V(G) \to [k]$; we consider the usual four methods of obtaining a vertex colouring from a vertex $k$-weighting, taking either the sum, product, multiset, or set of weights from vertices adjacent to $v$ for each $v \in V(G)$. If such a colouring is proper, then the vertex $k$-weighting of $G$ is a [**proper vertex colouring by sums, products, multisets or sets**]{}, respectively. The smallest values of $k$ such that a colouring of each type exists for a graph $G$ are denoted ${\chi_\Sigma^v}(G)$, ${\chi_\Pi^v}(G)$, ${\chi_m^v}(G)$ and ${\chi_s^v}(G)$, respectively.
A vertex weighting of $G$ which is a proper vertex colouring by sums is also known as an [**additive colouring**]{} of $G$ (initially called a [**lucky labelling**]{} of $G$), studied in [@CGZ] by Czerwiński, Grytczuk, and Żelazny. The value ${\chi_\Sigma^v}(G)$ is called the [**additive colouring number**]{} of $G$ (denoted $\eta(G)$ in [@CGZ]). The following conjecture is proposed:
\[lucky\] For every graph $G$, ${\chi_\Sigma^v}(G) \leq \chi(G)$.
It is shown in [@ADKM] that it is NP-complete to determine if a graph $G$ has ${\chi_\Sigma^v}(G) = k$ for any $k \geq 2$, and that it is still NP-complete if one considers the subproblem of determining if a $3$-colourable planar graph $G$ has ${\chi_\Sigma^v}(G) = 2$.
As evidence for the Additive Colouring Conjecture, note the following bound on ${\chi_m^v}(G)$:
\[multilucky\] For any graph $G$, ${\chi_m^v}(G) \leq \chi(G)$.
Let $k = \chi(G)$ and let $c: V(G) \rightarrow [k]$ be a vertex weighting which is a proper vertex colouring. Let $m(u)$ denote the multiset of weights from the vertices in $N_G(u)$. If $uv \in E(G)$, then $c(u) \in m(v) \setminus m(u)$. Thus, $m(v) \neq m(u)$ for any $uv \in E(G)$ and so $c$ is a vertex weighting which properly colours $V(G)$ by sums.
Note that the proof above actually shows the following:
For any graph $G$, ${\chi_s^v}(G) \leq \chi(G)$.
Furthermore, it is shown in [@COZ] that, for any graph $G$, there exists a set $S_G$ of $\chi(G)$ integers such that there exists a vertex weighting $w: V(G) \rightarrow S_G$ which is an additive colouring. This result is further extended in [@COZ2], where it is shown that the size of the smallest set of reals such that $G$ has an additive colouring is exactly ${\chi_m^v}(G)$ for any connected graph $G$.
It is easily shown that no constant bound is possible for ${\chi_\Sigma^v}(G)$ by noting that ${\chi_\Sigma^v}(K_n) = n$ for any $n \geq 2$. In [@BBC], it is shown that ${\chi_\Sigma^v}(G) \leq 468$ if $G$ is planar, ${\chi_\Sigma^v}(G) \leq 36$ if $G$ is planar and $3$-colourable, and ${\chi_\Sigma^v}(G) \leq 4$ if $G$ is planar and has girth at least $13$. Czerwiński et al. [@CGZ] prove a general upper bound on ${\chi_\Sigma^v}(G)$ in terms of the acyclic chromatic number of $G$, denoted $A(G)$.
\[luckybound\] For every graph $G$, ${\chi_\Sigma^v}(G) \leq p_1\cdots p_r$ where $p_i$ denotes the $i^{\textup{th}}$ odd prime number and $r = {A(G) \choose 2}$.
Unfortunately, the bound given by Theorem \[luckybound\] can grow large with $\Delta(G)$. For instance, it is shown in [@AMR] that there exist graphs with maximum degree $\Delta(G) = \Delta$ for which $A(G) \in \Omega\left(\frac{\Delta^{4/3}}{(\log\Delta)^{1/3}}\right)$, and hence the bound in Theorem \[luckybound\] is the product of $r \gg \Omega(\Delta^2)$ primes. Much improved bounds in the more general setting of list-weightings are presented in Section \[additivelistsection\]; in particular, we will see that ${\chi_\Sigma^v}(G) \in O(\Delta(G)^2)$ for every graph $G$.
As with the previously considered variations on the 1-2-3 Conjecture, one may vary the operation used to obtain a vertex colouring from a vertex weighting. In [@COSZ], Chartrand, Okamoto, Salehi, and Zhang study vertex weightings which properly colour vertices by multisets. Aside from establishing the easy upper bound of $\chi(G)$ for ${\chi_m^v}(G)$, exact values of ${\chi_m^v}(G)$ are determined for bipartite graphs, complete multipartite graphs, and powers of cycles.
Colouring by sets are considered in [@CORZ] by Chartrand, Okamoto, Rasmussen, and Zhang, who show that ${\chi_s^v}(G)$ is bounded below by $\left\lceil \log_2(\chi(G)+1) \right\rceil$ and by $1 + \left\lceil \log_2 \omega(G) \right\rceil$. Particular values are determined for some graph classes, and the effects of vertex and edge deletions are considered. This parameter is closely related to a graph’s [**locally identifying chromatic number**]{}, which is the fewest number of colours needed to properly colour $V(G)$ such that the set of colours in $N_G[u]$ differs from the set of colours in $N_G[v]$ for any adjacent vertices $u, v$ with $N_G[u] \neq N_G[v]$. This concept was introduced in [@Esp+] by Esperet et al., who show that there exist graphs which require $\Omega(\Delta(G)^2)$ colours. They ask whether or not $O(\Delta(G)^2)$ colours suffices for every admissible graph; this was answered in the affirmative by Foucould et al. in [@Fou+].
Variation III: List weightings {#ch:intro:var3}
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A popular variation of many colouring problems is to colour elements from independently assigned lists rather than from one universal colour set. Research on list-colouring problems often provides great insight into classical colouring problems. For instance, Thomassen’s proof that ${\textup{ch}}(G) \leq 5$ for any planar graph $G$ [@CTho] is arguably the most elegant proof of the classical result that every planar graph $G$ has $\chi(G) \leq 5$. Similarly, we hope to gain insight on the motivating conjectures presented thus far – the 1-2-3 Conjecture, the 1-2 Conjecture, and the Additive Colouring Conjecture – by considering colourings that come from list-weightings.
Let $G$ be a graph and $k, r, s \in {\mathbb{Z}}^{+}$. Assign to each edge $e \in E(G)$ a list of weights $L_e$ and to each vertex $v$ a list of weights $L_v$. Let $\mathcal{E} = \cup_{e \in E(G)} L_e$, $\mathcal{V} = \cup_{v \in V(G)} L_v$, and $\mathcal{L} = \mathcal{E} \cup \mathcal{V}$. An [**edge list-weighting**]{} of $G$ is a function $w: E(G) \rightarrow \mathcal{E}$ such that $w(e) \in L_e$ for each $e \in E(G)$, a [**vertex list-weighting**]{} of $G$ is a function $w: V(G) \rightarrow \mathcal{V}$ such that $w(v) \in L_v$ for each $v \in V(G)$, and a [**total list-weighting**]{} of $G$ is a function $w: E(G) \cup V(G) \rightarrow \mathcal{L}$ such that $w(e) \in L_e$ for each $e \in E(G)$ and $w(v) \in L_v$ for each $v \in V(G)$. If the size of each list is at most $k$, then each weighting is referred to as an [**edge $k$-list-weighting**]{}, [**vertex $k$-list-weighting**]{}, or [**total $k$-list-weighting**]{}, respectively. If a total list-weighting has $|L_v| \leq r$ and $|L_e| \leq s$ for each $L_v$ and $L_e$, then the weighting is called a [**total $(r,s)$-list-weighting**]{}.
Let ${{\textup{ch}}_\Sigma^e}(G)$ denote the smallest value of $k$ for which assigning a list of size $k$ of permissible weights, called a [**$k$-list assignment**]{}, to each edge of a graph permits an edge $k$-list-weighting which is a [**proper vertex colouring by sums**]{}; each of the parameters given thus far generalizes similarly.
Note that we refer to lists of “permissible” weights begin assigned to edges, vertices, or both in the preceding paragraph. For most colouring operations, these lists may be chosen freely from ${\mathbb{R}}$. However, if one were to colour vertices by products from an edge weighting, it is clear that one would never choose an edge weight to be $0$ as this would colour each of its ends $0$. Hence, we exclude the edge weight $0$ when colouring by products from edge list-weightings (for total and vertex list-weightings, $0$ [*is*]{} permissible for vertex lists).
Edge list-weightings {#edgelist}
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We will be primarily motivated by the following conjecture, a strengthening of the 1-2-3 Conjecture:
For every nice graph $G$, ${{\textup{ch}}_\Sigma^e}(G) \leq 3$.
We first note that it is not true that ${{\textup{ch}}_\Sigma^e}(G) = {\chi_\Sigma^e}(G)$ for every nice graph $G$. For example, let $G$ be a path on $4k+2$ vertices for some positive integer $k$ and assign to each edge the list $\{0,a\}$ for some arbitrary $a \in {\mathbb{R}}^+$; one can easily check that no edge list-weighting exists which properly colours $V(G)$ by sums.
The authors of the List 1-2-3 Conjecture have developed an approach to the problem which makes use of Alon’s Combinatorial Nullstellensatz [@A99]. Let $G = (V,E)$ be a graph, with $E(G) = \{e_1, \ldots, e_m\}$. Associate with each $e_i$ the variable $x_i$, and let $X_{v_j} = \sum_{e_i \ni v_j} \, x_i$. For an an orientation $D$ of $G$, define the following polynomial: $$\begin{aligned}
P_D(x_1, \ldots, x_m) = \prod_{(u,v) \in E(D)}(X_v - X_u) \\
\end{aligned}$$ Let $w$ be an edge weighting of $G$. By letting $x_i = w(e_i)$ for $1 \leq i \leq m$, $w$ is a proper vertex colouring by sums if and only if $P_D(w(e_1), \ldots, w(e_m)) \neq 0$. We can analyze the polynomial $P_D$ by using the Combinatorial Nullstellsatz:
Let ${\mathbb{F}}$ be an arbitrary field, and let $f=f(x_1,\ldots,x_n)$ be a polynomial in ${\mathbb{F}}[x_1,\ldots,x_n]$. Suppose the total degree of $f$ is $\sum_{i=1}^{n}t_i$, where each $t_i$ is a nonnegative integer, and suppose the coefficient of $\prod_{i=1}^{n}x_i^{t_i}$ in $f$ is nonzero. If $S_1,\ldots,S_n$ are subsets of ${\mathbb{F}}$ with $|S_i|>t_i$, then there are $s_1\in S_1, s_2\in S_2,\ldots,s_n\in S_n$ so that $$f(s_1,\ldots,s_n)\neq 0.$$
It can be shown that a term in $P_D$ has nonzero coefficient if and only if a related matrix has nonzero permanent (see [@BGN09; @PW11; @Ben-CN] for details). The following theorem, which is proven using this approach, gives the best known bound on ${{\textup{ch}}_\Sigma^e}(G)$:
\[chSebound\] If $G$ is a nice graph, then ${{\textup{ch}}_\Sigma^e}(G) \leq 2\Delta(G)+1$.
The Combinatorial Nullstellensatz is also used in [@BGN09] to show that ${{\textup{ch}}_\Sigma^e}(G) \leq 3$ if $G$ is complete, complete bipartite, or a tree. Upper bounds which do not rely on $\Delta(G)$ are proven in [@mythesis; @Ben-CN] for classes of Cartesian products, though these upper bounds are not constant.
While no improved bounds are known for ${{\textup{ch}}_m^e}(G)$, the Lovász Local Lemma can be used to prove the following results for edge-list-weightings which properly colour vertices by sequences.
\[listsequencebounds\] If $G$ is a nice graph, then
1. ${{\textup{ch}}_{\sigma^*}^e}(G) \leq 2$, and
2. ${{\textup{ch}}_\sigma^e}(G) \leq 3$ if $\delta(G) > \log_3(2\Delta(G)^2 - 2\Delta(G) + 1) + 2$.
The best known bound on products is a direct corollary of Theorem \[chSebound\]; one takes a fixed logarithm of the absolute value of every element in each list, and applies the theorem.
\[chPebound\] If $G$ is a nice graph, then ${{\textup{ch}}_\Sigma^e}(G) \leq 4\Delta(G)+2$.
One may easily extend the concept of vertex colouring edge weightings of graphs to the realm of digraphs. For an arc-weighting of a digraph $D$, the natural way to define a vertex colouring by sums is to define the colour of a vertex as the sum of the weights of its incident incoming arcs less the sum of the weights of its incident outgoing arcs. Such vertex colouring edge weightings are completely understood.
\[BGNdigraph\] For any digraph $D$, ${{\textup{ch}}_\Sigma^e}(D) \leq 2$.
The first set of authors cited in Theorem \[BGNdigraph\] proved this result by a constructive method; the second set used an adaptation of the Combinatorial Nullstellensatz method described above.
Total list-weightings
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Just as the 1-2-3 Conjecture has a natural list generalization, so too does the 1-2 Conjecture.
For every graph $G$, ${{\textup{ch}}_\Sigma^t}(G) \leq 2$.
In [@WZ], Wong and Zhu study [**$(k,l)$-total list-assignments**]{}, which are assignments of lists of size $k$ to the vertices of a graph and lists of size $l$ to the edges. If any $(k,l)$-total list-assignment of $G$ permits a total weighting which is a proper vertex colouring by sums, then $G$ is [**$(k,l)$-weight choosable**]{}. Obviously, if a graph $G$ is $(k,l)$-weight choosable, then ${{\textup{ch}}_\Sigma^t}(G) \leq \max\{k,l\}$. Wong and Zhu make the following two conjectures which, if true, would be stronger than the List 1-2-3 and List 1-2 Conjectures:
Every graph is $(2,2)$-weight choosable. Every nice graph is $(1,3)$-weight choosable.
It is easy to prove that every graph $G$ is $({\textup{ch}}(G),1)$-weight choosable, and hence we have the “trivial” upper bound of ${{\textup{ch}}_\Sigma^t}(G) \leq {\textup{ch}}(G) \leq \Delta(G)+1$. The polynomial method outlined in Section \[edgelist\] may be adapted to the problem of total weightings (see [@PW11] for details). This approach is used to prove the following improvement on the trivial upper bound on ${{\textup{ch}}_\Sigma^t}(G)$:
\[chStbound\] Any graph $G$ is $(\lceil\frac{2}{3}\Delta(G)\rceil + 1, \lceil\frac{2}{3}\Delta(G)\rceil + 1)$-weight choosable, and hence ${{\textup{ch}}_\Sigma^t}(G) \leq \lceil\frac{2}{3}\Delta(G)\rceil + 1$.
The other known results on $(k,l)$-weight choosability focus on graph classes which are $(k,l)$-weight choosable for small values of $k$ and $l$ (see [@BGN09], [@PY], [@PW11], [@WZ], [@WWZ], [@WYZ]). Table \[wctable\] on page summarizes these results, most of which are also proven using the Combinatorial Nullstellensatz approach described in Section \[edgelist\].
Type of graph $(k,l)$-weight choosability
------------------------------------- -----------------------------
$K_2$ $(2,1)$
$K_n$, $n \geq 3$ $(1,3)$, $(2,2)$
$K_{n,m}$, $n \geq 2$ $(1,2)$
$K_{m,n,1,\ldots,1}$ $(2,2)$
trees $(1,3)$, $(2,2)$
unicyclic graphs $(2,2)$
generalized theta graphs $(1,3)$, $(2,2)$
wheels $(2,2)$, $(1,3)$
hypercubes $Q_d$, $d$ even or $d=3$ $(1,3)$
planar $(1,10)$
outerplanar $(1,4)$
$s$-degenerate graphs $s \geq 2$ $(1,2s)$
: $(k,l)$-weight choosability of classes of graphs[]{data-label="wctable"}
Hornák and Woźniak [@HW] consider the list variation of ${\chi_s^e}(G)$ (or $\textup{gndi}(G)$). They denote this parameter $\textup{glndi}(G)$, while we denote it ${{\textup{ch}}_s^e}(G)$. They show that ${{\textup{ch}}_s^e}(G) = {\chi_s^e}(G) = \lceil{\log_2{\chi(G)}}\rceil + 1$ in the case where $G$ is a path or cycle, and that ${{\textup{ch}}_s^e}(T) \leq 3$ for any tree $T$. If the edge weighting is required to give a proper edge colouring as well, they denote the corresponding parameter $\textup{lndi}(G)$ in correspondence with the parameter $\textup{ndi}(G)$ for the non-list version of the problem. They show that $\textup{lndi}(G) = \textup{ndi}(G)$ if $G$ is a cycle or a tree and conjecture that $\textup{lndi}(G) = \textup{ndi}(G)$ for every graph $G$, a conjecture reminiscent of the List Colouring Conjecture (see [@JT] for a history of the List Colouring Conjecture).
We have already noted that ${{\textup{ch}}_\Sigma^t}(G) \leq {\textup{ch}}(G)$ for any graph $G$. This upper bound also holds for colourings by multisets and by products. It follows that the following graphs, which all have low choosability number, have small values of ${{\textup{ch}}_\Sigma^t}(G)$, ${{\textup{ch}}_ \Pi ^t}(G)$ and ${{\textup{ch}}_m^t}(G)$ (references stated are for the bound on ${\textup{ch}}(G)$ used to obtain the result on weighting parameters):
Let $n$ be a positive integer.
1. If $G$ is a graph whose core[^1] is $K_1$, $C_{2n+2}$, or $\theta_{2,2,2n}$, then ${{\textup{ch}}_\Sigma^t}(G), {{\textup{ch}}_ \Pi ^t}(G),$ and ${{\textup{ch}}_m^t}(G)$ are all at most $2$.
2. If all cycles of $G$ have length divisible by an integer $k \geq 3$, then ${{\textup{ch}}_\Sigma^t}(G), {{\textup{ch}}_ \Pi ^t}(G),$ and ${{\textup{ch}}_m^t}(G)$ are all at most $3$.
3. If $G$ is planar, then ${{\textup{ch}}_\Sigma^t}(G), {{\textup{ch}}_ \Pi ^t}(G),$ and ${{\textup{ch}}_m^t}(G)$ are all at most $5$.
4. If $G$ is bipartite and planar, then ${{\textup{ch}}_\Sigma^t}(G), {{\textup{ch}}_ \Pi ^t}(G),$ and ${{\textup{ch}}_m^t}(G)$ are all at most $3$.
Furthermore, every graph $G$ has the following two upper bounds on ${{\textup{ch}}_\Sigma^t}(G)$, ${{\textup{ch}}_ \Pi ^t}(G)$, and ${{\textup{ch}}_m^t}(G)$:
\[chbound\] If $G$ is a graph on $n$ vertices, then
1. each of ${{\textup{ch}}_ \Pi ^t}(G)$ and ${{\textup{ch}}_m^t}(G)$ is at most $\Delta(G) + 1$, and
2. each of ${{\textup{ch}}_\Sigma^t}(G), {{\textup{ch}}_ \Pi ^t}(G),$ and ${{\textup{ch}}_m^t}(G)\}$ is at most $\chi(G)\log{n}$.
Note that the bound ${{\textup{ch}}_\Sigma^t}(G) \leq \Delta(G) + 1$ is omitted since a stronger result is given in Theorem \[chStbound\].
We conclude with a brief consideration of colouring vertices by products from list-weightings. By excluding $0$ from our ground set of potential edge weights, it remains possible that ${{\textup{ch}}_ \Pi ^e}(G) \leq 3$ (were we not to exclude $0$, the lists $\{-1,0,1\}$ would not permit a vertex colouring for a nice non-bipartite graph). However, by assigning every vertex and edge the list $\{1,-1\}$, one sees that graphs which are not bipartite cannot be totally weighted from these lists in such a way as to properly colour vertices by products. So, while the List 1-2 Conjecture posits that ${{\textup{ch}}_\Sigma^t}(G) \leq 2$ for any graph $G$, a similar bound for total list-weightings is not possible for all graphs when colouring by products.
Vertex list-weightings {#additivelistsection}
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Denote by ${{\textup{ch}}_\Sigma^v}(G)$ the smallest $k$ such that $G$ has an additive colouring from any assignment of lists of size $k$ to the vertices of $G$, and call ${{\textup{ch}}_\Sigma^v}(G)$ the [**additive choosability number**]{} of $G$. Such list-weightings were considered by Czerwiński et al. [@CGZ], who show that if $T$ is a tree, then ${{\textup{ch}}_\Sigma^v}(T) \leq 2$, and if $G$ is a bipartite planar graph, then ${{\textup{ch}}_\Sigma^v}(G) \leq 3$.
The following conjecture follows in the spirit of the List 1-2-3 Conjecture:
For any graph $G$, ${{\textup{ch}}_\Sigma^v}(G) = {\chi_\Sigma^v}(G)$.
While there is no known bound on ${{\textup{ch}}_\Sigma^v}(G)$ in terms of ${\chi_\Sigma^v}(G)$ (a bound in terms of ${\textup{ch}}(G)$ or $\chi(G)$ would also be of interest), one may bound ${{\textup{ch}}_\Sigma^v}(G)$ by a function of the maximum degree of $G$:
\[listluckybound\] For any graph $G$, ${{\textup{ch}}_\Sigma^v}(G) \leq \Delta(G)^2 - \Delta(G) + 1$.
The following general bound can be proven with a straightforward inductive argument:
\[additivechoosability\] If $G$ is a $d$-degenerate graph, then ${{\textup{ch}}_\Sigma^v}(G) \leq d\Delta(G) + 1$.
It the easily follows that ${{\textup{ch}}_\Sigma^v}(G) \leq \Delta(G)\left(\chi(G) - 1\right) + 1$ if $G$ is chordal, ${{\textup{ch}}_\Sigma^v}(G) \leq 2\Delta+1$ if $G$ is $K_4$-minor free, and ${{\textup{ch}}_\Sigma^v}(G) \leq 5\Delta(G)+1$ if $G$ is planar. It is also shown in [@mythesis; @Ben-CN] that ${{\textup{ch}}_\Sigma^v}(G) \leq \chi(G)$ when $G$ is a complete multipartite graph (in fact, a slightly stronger statement is proven, one which generalizes a result of Chartrand, Okamoto, and Zhang [@COZ] who considered ${\chi_\Sigma^v}(G)$ for regular, complete multipartite graphs).
Variation IV: Edge colourings {#ch:intro:var4}
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Each section so far has dealt with the problem of properly colouring the [*vertices*]{} of a graph given an edge weighting, vertex weighting, or total weighting. We now define how one might similarly define edge colourings and total colourings from a weighting.
Given an edge $k$-weighting, we colour an edge $e \in E(G)$ with the sum, product, multiset, or set of weights from edges adjacent to $e$. If the resulting edge colouring is proper, then the $k$-edge weighting of $G$ is a [**proper edge colouring by sums, products, multisets or sets**]{}, respectively. The smallest $k$ for which such colourings exist for a graph $G$ are denoted ${{\chi'}_\Sigma^e}(G)$, ${{\chi'}_\Pi^e}(G)$, ${{\chi'}_m^e}(G)$ and ${{\chi'}_s^e}(G)$, respectively.
These parameters do not appear in the present literature, but a few bounds are easily obtained by noting that each is a particular case of its vertex-weighting vertex-colouring analog, e.g., ${{\chi'}_\Sigma^e}(G) = {\chi_\Sigma^v}(L(G))$, ${{{\textup{ch}}'}_\Sigma^e}(G) = {{\textup{ch}}_\Sigma^v}(L(G))$, etc. In the particular case of ${{\chi'}_\Sigma^e}(G)$, we have a few easy to prove upper bounds. Theorem \[luckybound\] bounds ${{\chi'}_\Sigma^e}(G)$ in terms of the acyclic chromatic number of $L(G)$. The case of proper colouring by multisets also has a bound which follows from previous work.
For any graph $G$, ${{\chi'}_m^e}(G) \leq \chi'(G)$.
By Proposition \[multilucky\], ${\chi_m^v}(L(G)) \leq \chi(L(G))$. Since ${{\chi'}_m^e}(G) = {\chi_m^v}(L(G))$, the result follows.
One may also colour edges from a vertex or total weighting, or by list weightings of each type; such scenarios have yet to be studied.
A total $k$-weighting of a graph $G$, $w: V(G) \cup E(G) \rightarrow [k]$ is a [**proper edge colouring by sums**]{} if the edge colouring function $c(uv) = w(u) + w(v) + w(uv)$ is proper. The smallest value of $k$ such that there exists a total $k$-weighting which is a proper edge colouring by sums is denoted ${{\chi'}_\Sigma^t}(G)$. While one may easily define similar parameters for proper colourings by products, multisets, or sets, only ${{\chi'}_\Sigma^t}(G)$ has been explicitly studied in published literature.
If a graph $G$ has maximum degree $\Delta(G) = \Delta$, then ${{\chi'}_\Sigma^t}(G) \leq \frac{\Delta}{2} + O(\sqrt{\Delta\log{\Delta}})$.
Brandt et al. also establish bounds for trees and complete graphs, and give necessary and sufficient conditions for a cubic graph $G$ to have ${{\chi'}_\Sigma^t}(G) = 2$. The topic of weightings and list-weightings which are proper total colourings via binary operations remains unstudied.
Distinguishing colourings and irregularity strength {#global}
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As mentioned, one motivation for the 1-2-3 Conjecture is the study of graph irregularity strength, which is is the smallest positive integer $k$ for which there exists an edge weighting $w: E(G) \to [k]$ such that $\sum_{e \ni u} w(e) \neq \sum_{e \ni v} w(e)$ for all $u,v \in V(G)$. Irregularity strength and its variations have generated a significant amount of interest since the topic’s inception in 1986. We will only highlight a few of the more significant results in the field here; the reader is referred to [@survey], [@Leh91], and [@West_IS] for a more complete survey of results on graph irregularity strength.
The original notation for the irregularity strength of a graph $G$, given by Chartrand et al. [@CJL+], is $\textup{s}(G)$; to maintain consistency with the notation developed so far, we will denote it $\textup{s}_{\Sigma}^e(G)$. There is a natural “global" variation of each of the “local” parameters defined so far. For the local parameters $\chi_x^y(G)$, ${\chi'}_x^y(G)$, and ${\chi''}_x^y(G)$, where $x \in \left\{\sum, \prod, m, s, \sigma, \sigma^* \right\}$ and $y \in \left\{e,v,t\right\}$, let $\textup{s}_x^y(G)$, ${\textup{s}'}_x^y(G)$, and ${\textup{s}''}_x^y(G)$ denote their respective “global" counterparts. Similarly, let $\textup{ls}_x^y(G)$, ${\textup{ls}'}_x^y(G)$, and ${\textup{ls}''}_x^y(G)$ denote the global versions of ${\textup{ch}}_x^y(G)$, ${{\textup{ch}}'}_x^y(G)$, and ${{\textup{ch}}''}_x^y(G)$, respectively.
A major conjecture on the topic of irregularity strength was one of Aigner and Triesch [@AT90], who asked if $\textup{s}_{\Sigma}^e(G) \leq |V(G)|-1$ for every nice graph. This conjecture was answered in the affirmative by Nierhoff [@Nie]. The following is the best known irregularity strength bound:
For any nice graph $G$, $\textup{s}_{\Sigma}^e(G) \leq \lceil 6n/\delta(G) \rceil$.
Recently, Anholcer and Cichacz proved a group analog of Aigner and Triesch’s conjecture. They define the [**group irregularity strength**]{} of a graph $G$, $s_g(G)$, to be the smallest value of $s$ such that, for any abelian group $(\Gamma,+)$ of order $s$, there exists an edge weighting of $G$ from $\Gamma$ such that distinct vertices receive distinct sums of edge weights.
For any connected graph of order $n \geq 3$, $n \leq s_g(G) \leq n+2$.
As with their result on group sum chromatic numbers (Theorem \[groupsum\]), Anholcer and Cichacz characterize which graphs have group irregularity strength $n$, $n+1$, and $n+2$.
Note that irregular weightings are closely related to antimagic labelings. A graph $G$ with $m$ edges is called [**antimagic**]{} if there exists a bijection between $E(G)$ and $\{1, \ldots, m\}$ such that every vertex has a distinct sum of edge weights from its adjacent edges. Such a bijection is called an [**antimagic labeling**]{}. Antimagic labelings and their variations are widely studied; a thorough survey may be found in Chapter 6 of [@survey].
Aigner et al. [@ATT] introduce a version of irregularity strength where each vertex is assigned the multiset of edge weights from incident edges rather than the sum; the corresponding parameter is $\textup{s}_{m}^e(G)$ in our notation and is called the [**multiset irregularity strength**]{} of $G$. The following result is proven using probabilistic methods:
\[ATT\] If $G$ is a $d$-regular graph, then $\textup{ls}_{m}^e(G) \in \Theta(n^{1/d})$.
Note that the original result published in [@ATT] does not refer explicitly to weighting from lists, however their proofs all directly translate to this setting. The constant in the upper bound suppressed by the $\Theta$ notation can be improved if one relaxes the colouring method to sequences given any ordering of the edges [@Ben-LLL] ($\textup{ls}_{\sigma}^e(G)$ is the [**general sequence irregularity strength**]{} of $G$; see Section \[ch:intro:var1\] for more on colouring by sequences). More generally, it is shown in [@Ben-LLL] that if $G$ is a graph on $n$ vertices with minimum degree $\delta(G) > c\log{n}$ for large enough $c=c(k)$, then $\textup{ls}_\sigma^e(G) \leq k$. Similar results are also given for total list-weightings.
Anholcer [@Anh] studied the [**product irregularity strength**]{} of a graph, or $\textup{s}_{\Pi}^e(G)$. He establishes bounds on $\textup{s}_{\Pi}^e(G)$ when $G$ is a cycle, path or grid. Pikhurko [@Pik] has shown that if a graph is sufficiently large, then $\textup{s}_{\Pi}^e(G) \leq |E(G)|$ (in antimagic labelling terminology, he shows that sufficiently large graphs are product antimagic).
A [**vertex-distinguishing edge-colouring**]{} of a graph $G$ is a colouring of the edges such that for any two vertices $u$ and $v$, the set of colours assigned to the set of edges incident to $u$ differs from the set of colours assigned to the set of edges incident to $v$. Harary and Plantholt [@HP85] introduced the study of vertex-distinguishing [*proper*]{} edge-colourings in 1985; they use $\chi_0(G)$ to denote the smallest value of $k$ such that a graph $G$ has such an edge-colouring, and they call the parameter the [**point-distinguishing chromatic index**]{} of $G$. Note that $\chi_0(G)$ is a global version of ${\chi_s^e}(G)$ with the added constraint that the edge weighting is a proper edge-colouring, and hence is an upper bound on $\textup{s}_s^e(G)$.
\[setirrstrength\] If $n_i$ denotes the number of vertices of degree $i$ in a graph $G$, then for any nice graph $G$, $\chi_0(G) \leq C\max\{n_i^{1/i} \,:\, 1 \leq i \leq \Delta(G)\}$, where $C$ is a constant relying only on $\Delta(G)$.
Burris and Schelp also make the following two conjectures:
1. Let $G$ be a simple graph with no more than one isolated vertex and no connected component isomorphic to $K_2$. If $k$ is the minimum integer such that ${k \choose d} \geq n_d$ for all $\delta(G) \leq d \leq \Delta(G)$, then $\chi_0(G) \in \{k, k+1\}$.
2. If $G$ is a simple graph with no more than one isolated vertex and no connected component isomorphic to $K_2$, then $\chi_0(G) \leq |V(G)| + 1$.
The second conjecture was answered in the affirmative by Bazgan et al. [@Baz+].
Bača, Jendrol’, Miller, and Ryan introduced the [**total edge irregularity strength**]{} of a graph $G$ in [@BJMR], which is defined to be the smallest $k$ for which there is a total $k$-weighting $w$ such that $w(u) + w(v) + w(uv) \neq w(x) + w(y) + w(xy)$ for any distinct $uv, xy \in E(G)$. In our notation, this parameter is ${\textup{s}'}_\Sigma^t(G)$. They show that if $G$ has $m$ edges, then $\left\lceil\frac{m+2}{3}\right\rceil \leq {\textup{s}'}_\Sigma^t(G) \leq m$. They also make the following conjecture:
If $G \ncong K_5$, then $${\textup{s}'}_\Sigma^t(G) = \max\left\{ \left\lceil \dfrac{\Delta(G)+1}{2} \right\rceil, \left\lceil\dfrac{|E(G)|+2}{3}\right\rceil \right\}.$$
This conjecture was recently confirmed in [@BMR] for graphs with $|E(G)| > 111\,000\Delta(G)$ and for a random graph from $\mathcal{G}(n,p)$ for any $p = p(n)$. It is also shown in [@BMR] that ${\textup{s}'}_\Sigma^t(G) \leq \left\lceil \frac{|E(G)|}{2}\right\rceil$ for any graph $G$ which is not a star. The conjecture is also verified in [@BMRRR] for any graph $G$ having $|E(G)| \leq \frac{3}{2}|V(G)| - 1$ and $\Delta(G) \leq \left\lceil\frac{|E(G)|+2}{3}\right\rceil$.
Bača et al. [@BJMR] also introduce the [**total vertex irregularity strength**]{} of a graph $G$, denoted ${\textup{s}}_\Sigma^t(G)$, which is the smallest $k$ such that $G$ has a total $k$-weighting $w$ such that $w(u) + \sum_{e \ni u}w(e) \neq w(v) + \sum_{e \ni v} w(e)$ for any distinct $u,v \in V(G)$. The best known bound on ${\textup{s}}_\Sigma^t(G)$ was given by Anholcer, Kalkowski, and Przyby[ł]{}o [@AKP], who showed that ${\textup{s}}_\Sigma^t(G) \leq 3\left\lceil n/\delta \right\rceil + 1$ for a graph on $n$ vertices with minimum degree $\delta(G) = \delta$. For a survey of results on total edge irregularity strength and total vertex irregularity strength, see Section 7.17 in [@survey].
Chartrand, Lesniak, VanderJagt, and Zhang [@CLVZ] have very recently considered vertex weightings for which the multiset of colours of the vertices in $N(u)$ differs from the multiset of colours of the vertices of $N(v)$ for any distinct vertices $u$ and $v$. The smallest $k$ such that a graph $G$ has such a vertex $k$-weighting is denoted ${\textup{s}}_m^v(G)$. Such weightings are also known as [**recognizable colourings**]{} and ${\textup{s}}_m^v(G)$ is the [**recognition number**]{} of $G$. Recognition numbers are determined in [@CLVZ] for a number of classes of graphs.
An [**edge-distinguishing vertex weighting**]{} of a graph $G$ is a weighting $w:V(G) \to [k]$ such that $\{w(u),w(v)\} \neq \{w(x),w(y)\}$ for any two distinct edges $uv, xy \in E(G)$. Such weightings were introduced by Frank, Harary and Plantholt [@FHP], where they were called [**line-distinguishing vertex colourings**]{}. The smallest value of $k$ for which a graph $G$ has an edge-distinguishing vertex-weighting is called the [**line-distinguishing chromatic number**]{}, and is denoted ${\textup{s}'}_{s}^v(G)$ (note that $\{w(u),w(v)\} \neq \{w(x),w(y)\}$ as sets if and only if they are distinct as multisets, and hence we could also use ${\textup{s}'}_{m}^v(G)$). Harary and Plantholt [@HP83] conjectured that ${\textup{s}'}_{m}^v(G) \geq \chi'(G)$ for every graph $G$, which was proven by Salvi [@Salvi].
Summary
=======
Although the ultimate goal of showing that ${\chi_\Sigma^e}(G) \leq 3$ remains out of reach for the moment, a good deal of progress has been made in many of the natural generalizations. A summary of the major results contained in the thesis are found in Table \[summaryofbounds\]. In each case, assume that $G$ is a nice graph if necessary, and that $D$ is a digraph. Not every bound is explicitly stated; for instance, since ${\chi_\sigma^e}(G) \leq {\chi_m^e}(G)$ and no bound for ${\chi_\sigma^e}(G)$ is known other than the best known bound on ${\chi_m^e}(G)$, ${\chi_\sigma^e}(G)$ is omitted from the table.\
Parameter Conjectured upper bound Known upper bound Reference
----------------------------------- --------------------------------- ------------------------------------------------------------ ----------------------
${\chi_\Sigma^e}(G)$ 3 5 [@KKP1]
${\chi_m^e}(G)$ 3 4 [@AADR05]
${\chi_\Pi^e}(G)$ 3 5 [@SK08]
${\chi_s^e}(G)$ – $\left\lceil \log_2\chi(G)\right\rceil + 1 $ [@GP09]
${\chi_\Sigma^t}(G)$ 2 3 [@Kal]
${\chi_\Pi^t}(G)$ 2 3 [@SK08]
${{\textup{ch}}_\Sigma^e}(G)$ 3 $2\Delta(G)+1$ [@mythesis; @Ben-CN]
${{\textup{ch}}_\Sigma^t}(G)$ 2 $\left\lceil \frac{2}{3}\Delta(G) \right\rceil + 1$ [@mythesis; @Ben-CN]
${{\textup{ch}}_ \Pi ^e}(G)$ – $4\Delta(G)+2$ [@mythesis; @Ben-CN]
${{\textup{ch}}_{\sigma^*}^e}(G)$ 2 2 [@Ben-LLL]
${{\textup{ch}}_\Sigma^e}(D)$ 2 2 [@BGN09], [@Ben2]
${{\textup{ch}}_\Sigma^v}(G)$ ${\chi_\Sigma^v}(G)$, $\chi(G)$ $\Delta(G)^2 + 1$ [@mythesis; @Ben-CN]
${{\chi'}_\Sigma^t}(G)$ – $\frac{\Delta(G)}{2} + O(\sqrt{\Delta(G)\log{\Delta(G)}})$ [@BBRS]
: Summary of derived colouring parameter values[]{data-label="summaryofbounds"}
Acknowledgements
================
Much of this work was compiled during the writing of the author’s doctoral thesis. As such, a great debt is owed to Brett Stevens for his supervisory role, Carleton University for their institutional support, and to the Natural Sciences and Engineering Research Council of Canada and the Fonds de recherche Québec - Nature et technologies for their financial support.
[^1]: The core of a graph is the subgraph obtained by repeated deletion of vertices of degree 1 until none remain. This is one of two standard definitions for the core of a graph, and should not be confused with the definition from the study of graph homomorphisms.
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abstract: 'We consider two finite index endomorphisms $\rho$, $\sigma $ of any AFD factor $M$. We characterize the condition for there being a sequence $\{ u_n\}$ of unitaries of the factor $M$ with $\mathrm{Ad}u_n \circ \rho \to \sigma $. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi–Sutherland–Takesaki and Masuda–Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.'
address: 'Department of Mathematical Sciences University of Tokyo, Komaba, Tokyo, 153-8914, Japan'
author:
- Koichi Shimada
title: Approximate unitary equivalence of finite index endomorphisms of the AFD factors
---
Introduction
============
In this paper, we characterize the approximate innerness of the difference of two finite index endomorphisms of the AFD factors of type III (Theorem \[main\]). More precisely, for two finite index endomorphisms $\rho$ ,$ \sigma $ of any AFD factor $M$, we give a good necessary and sufficient condition for there being a sequence $\{ u_n\}$ of unitaries of $M$ with $\mathrm{Ad}u_n \circ \rho \to \sigma $ as $n\to \infty $ in the sense of Masuda–Tomatsu [@MT1]. First of all, we explain the reason why we are interested in this topic. The reason is that this result should be useful for classifying group actions. It has been important to classify group actions on von Neumann algebras up to cocycle conjugacy. Since a remarkable work of Connes [@C], classification of group actions on the AFD factors has greatly been developed by many researchers. In particular, the actions of discrete amenable groups on the AFD factors are completely classified (See Jones [@J], Ocneanu [@O], Sutherland–Takesaki [@ST], Kawahigashi–Takesaki–Sutherland [@KwST] and Katayama–Sutherland–Takesaki [@KtST]). It is interesting to note that although there are many different actions up to conjugacy, they are clearly classified when we ignore the difference of cocycle conjugacy. One of the next problems is to classify actions of continuous groups. Among them, classification of actions of compact groups is considered to be relatively easy because the dual of a compact group is discrete. In fact, actions of compact abelian groups on the AFD factors have completely been classified by using this observation (See Jones–Takesaki [@JT] and Kawahigashi–Takesaki [@KwT]). However, when it comes to classifying actions of non-abelian compact groups, the problem is much more difficult. One of the reasons is that the dual action of an action of a non-abelian compact group is a collection of endomorphisms, not of automorphisms. Hence in order to proceed with classifying actions, it is important to understand the properties of endomorphisms. In the proof of classification theorems of group actions, whether the difference of two actions is approximated by inner automorphisms or not is very important. Hence we should characterize the approximate innerness of the difference of two endomorphisms of the AFD factors.
In this paper, we characterize the approximate innerness of the difference of two endomorphisms in the sense of Masuda–Tomatsu [@MT1] (Theorem \[main\]). In Masuda–Tomatsu [@MT3], they propose a conjecture of the complete invariant for actions of discrete Kac algebras on the AFD factors (Conjecture 8.2 of [@MT2]). The dual of minimal actions of compact groups are ones of them. Our main theorem implies that if two actions of discrete Kac algebras on the AFD factors of type III have the same invariants, the difference of these two actions is approximately inner (See Problem 8.3 and the preceding argument to that problem of [@MT3]). Our main theorem characterizes when one endomorphism transits to another endomorphism. Hence the theorem may also be seen as a kind of endomorphism counterpart of the main theorem of Haagerup–Størmer [@HS], which characterizes when one normal state of a von Neumann algebra transits to another normal state. It is important to note that our main theorem is a generalization of Theorem 1 (1) of Kawahigashi–Sutherland–Takesaki [@KwST] and Theorem 3.15 of Masuda–Tomatsu [@MT1]. The proof of our theorem is based on recent development on the Rohlin property of flows on von Neumann algebras, which does not depend on the types of the AFD factors. Our method is also applicable to the characterization of the central triviality of automorphisms (Theorem 1. (2) of Kawahigashi–Sutherland–Takesaki [@KwST]). In appendix, we give another proof of the characterization of the central triviality, which does not depend on the types of the AFD factors.
**Acknowledgment** The author thanks Professor Reiji Tomatsu for introducing him to this topic and for giving him useful comments and Professor Toshihiko Masuda for pointing out a mistake in the first version of this paper. The author is also thankful to Professor Yasuyuki Kawahigashi, who is his adviser, for his useful comments on the presentation of this work. The author is supported by Research Fellowships of the Japanese Society for the Promotion of Science for Young Scientists No.26-6590. This work is also supported by the Program for Leading Graduate Schools, MEXT, Japan.
Preliminaries
=============
Notations
---------
Let $M$ be a von Neumann algebra. For a normal positive linear functional $\psi $ of $M$ and $x\in M$, set $$\| x\| _{\psi}:=\sqrt{\psi (x^*x)},$$ $$\| x\| _{\psi}^\sharp :=\sqrt{\frac{\psi (x^*x)+\psi (xx^*)}{2}}.$$
\[inequality\] Let $\lambda $ be a $\sigma$-weakly continuous linear functional of a von Neumann algebra $M$ and $\lambda =\psi v$ be its polar decomposition. Then we have
$$\| \lambda a\| \leq \psi (vaa^*v^*)^{1/2}\| \lambda \|^{1/2} ,$$ $$\| a\lambda \| \leq \psi (a^*a)^{1/2}\| \lambda \|^{1/2}$$ for any $a\in M$.
By Cauchy–Schwarz’s inequality, for $x\in M$, we have $$\begin{aligned}
|\lambda a(x)|&=|\psi (vax)| \\
&\leq \psi (vaa^*v^*)^{1/2}\psi (x^*x)^{1/2} \\
&\leq \psi (vaa^*v^*)^{1/2} \| \lambda \| ^{1/2} \| x\| . \end{aligned}$$ The latter inequality is shown in a similar way.
A topology of semigroups of endomorphisms
-----------------------------------------
Let $M$ be a factor of type $\mathrm{III}$. Let $\mathrm{End}(M)_0$ be the set of all finite index endomorphisms $\rho$ of $M$. Let $d(\rho ) $ be the square root of the minimal index of $M\supset \rho (M)$ and $E_\rho $ be the minimal expectation from $M$ to $\rho (M)$. Set $\phi _\rho :=\rho ^{-1} \circ E_\rho$. In Masuda–Tomatsu [@MT1], a topology of $\mathrm{End} (M)_0$ is introduced in the following way. We have $$\rho _i \to \rho$$ if, by definition, $\| \psi \circ \phi _{\rho _i} -\psi \circ \phi _\rho \| \to 0$ for any $\psi \in M_*$.
Canonical extension of endomorphisms
------------------------------------
Let $\varphi $ be a normal faithful semifinite weight of $M$ and $\sigma ^\varphi$ be the group of modular automorphisms of $\varphi$. In Izumi [@I], an extension $\tilde{\rho}$ of $\rho \in \mathrm{End}(M)_0$ on the continuous core $\tilde{M}:=M\rtimes _{\sigma ^\varphi} \mathbf{R}$ is introduced in the following way. We have $$\tilde {\rho} (x\lambda _t^{\sigma ^\varphi})=d(\rho )^{it}\rho (x) [D\varphi \circ \phi _{\rho} :D\varphi ]_t \lambda _t^{\sigma^\phi}$$ for $t\in \mathbf{R}$, $x\in M$, where $[D\varphi \circ \phi _{\rho}:D\varphi ]_t$ is the Connes cocycle between $\varphi \circ \phi _{\rho}$ and $\varphi$. This extension does not depend on the choice of $\varphi$ under a specific identification (See Theorem 2.4 of Izumi [@I]). The extension $\tilde{\rho}$ is said to be the canonical extension of $\rho$.
In Lemma 3.5 of Masuda–Tomatsu [@MT1], it is shown that there exists a left inverse $\phi _{\tilde{\rho}}$ of $\tilde{\rho}$ satisfying $$\phi _{\tilde{\rho}} (x\lambda ^\varphi _t )=d(\rho )^{-it} \phi _\rho (x[D\phi :D\phi \circ \phi _\rho ]_t )\lambda _t^\varphi$$ for $x\in M$, $t\in \mathbf{R}$.
The main theorem
================
The main theorem of this paper is the following.
\[main\] Let $\rho$ , $\sigma $ be endomorphisms of an AFD factor $M$ of type $\mathrm{III}$ with $d(\rho ), d(\sigma )<\infty$. Then the following two conditions are equivalent.
We have $\phi _{\tilde{\rho}}\circ \theta _{-\log (d(\rho) /d(\sigma ))}|_{\mathcal{Z}(\tilde{M})}=\phi _{\tilde{\sigma }}|_{\mathcal{Z}(\tilde{M})}$.
There exists a sequence $\{ u_n\}$ of unitaries of $M$ with $\mathrm{Ad}u_n \circ \rho \to \sigma $ as $n\to \infty$.
As a Corollary, we have the following result.
Let $M$ be an AFD factor and $R_0$ be the AFD factor of type $\mathrm{II}_1$. Take endomorphisms $\rho _1$, $\rho _2\in \mathrm{End}(M)_0$ and $\sigma _1$, $\sigma _2\in ^mathrm{End}(R_0)_0$. Then the following two conditions are equivalent.
There exists a sequence of unitaries $\{u_n\}$ of $M \otimes R_0$ with $\mathrm{Ad}u_n \circ (\rho _1\otimes \sigma _1) \to \rho _2\otimes \sigma _2$ as $n\to \infty$.
There exists a sequence of unitaries $\{v_n\}$ of $M$ with $\mathrm{Ad}v_n \circ \rho _1 \to \rho _2$ as $n \to \infty$.
Since $\sigma _1$ and $\sigma _2 $ are approximately inner, we may assume that $\sigma _1=\sigma _2=\mathrm{id}_{R_0}$. By the identification $\mathcal{Z}((M\otimes R_0)\rtimes _{\sigma ^\varphi \otimes \mathrm{id}_{R_0}}\mathbf{R})\cong \mathcal{Z}((M\rtimes _{\sigma ^\varphi}\mathbf{R})\otimes R_0) \cong \mathcal{Z}(M\rtimes _{\sigma ^\varphi }\mathbf{R})$ by $$(x\otimes y)\lambda _t^{\sigma ^\varphi \otimes \mathrm{id}_{R_0}}\mapsto (x\lambda _t^{\sigma ^\varphi})\otimes y,$$ we have $\phi _{\rho _i \otimes \mathrm{id}_{R_0}}=\phi _{\rho_i}$ for $i=1,2$. We also have $\d(\rho _i \otimes \mathrm{id}_{R_0})=d(\rho _i)$. Hence by Theorem \[main\], conditions (1) and (2) are equivalent.
Note that this corollary would be quite difficult to show without Theorem \[main\] (See also Section 3 of Connes [@C2]).
Theorem \[main\] should also be useful for classifying actions of compact groups on the AFD factors of type $\mathrm{III}$. Popa–Wassermann [@PW] and Masuda–Tomatsu [@MT2] showed that any compact group has only one minimal action on the AFD factor of type $\mathrm{II}_1$, up to conjugacy. One of the next problems is to classify actions of compact groups on the AFD factors of type III. In Masuda–Tomatsus [@MT3] and [@MT4], they are trying to solve this problem, and some partial answers to this problem are obtained (Theorems A, B of [@MT3] and Theorem 2.4 of [@MT4]). However, still the problem has not been solved completely. In Masuda–Tomatsu [@MT3], a conjecture about this classification problem is proposed (Conjecture 8.2). Our main theorem implies that if two actions of discrete Kac algebras on the AFD factors of type III have the same invariants, the difference of these two actions is approximately inner (See Problem 8.3 and the preceding argument to that problem of Masuda–Tomatsu [@MT3]). In order to classify group actions, whether the difference of two actions is approximately inner or not is very important. Kawahigashi–Sutherland–Takesaki [@KwST] and Masuda–Tomatsu [@MT1] characterize the approximate innerness of endomorphisms under such a motivation. Theorem \[main\] is a generalization of their results.
In the following, we will show Theorem \[main\]. Implication $\Rightarrow $ is shown easily by using known results.
*Proof of implication $\Rightarrow$ of Theorem \[main\].* This is shown by the same argument as that of the proof of implication (1) $\Rightarrow $ (2) of Theorem 3.15 of [@MT1]. Assume that we have $\mathrm{Ad}u_n \circ \rho \to \sigma $ as $n \to \infty$. Then by the continuity of normalized canonical extension (Theorem 3.8 of Masuda–Tomatsu [@MT1]), we have $$\phi _{\tilde{\rho}}\circ \theta _{-\log d(\rho)} \circ \mathrm{Ad}u_n^*(x) \to \phi _{\tilde{\sigma }} \circ \theta _{-\log d(\sigma )} (x)$$ in the strong\* topology for any $x\in \tilde{M}$. Hence we have $$\phi _{\tilde{\rho}}\circ \theta _{-\log (d(\rho )/d(\sigma ))}|_{\mathcal{Z}(\tilde{M})} =\phi _{\tilde{\sigma }}|_{\mathcal{Z}(\tilde{M})}.$$
In the following, we will show the opposite implication. Our strategy is to reduce the problem to that of endomorphisms on semifinite von Neumann algebras. In order to achieve this, in Kawahigashi–Sutherland–Takesaki [@KwST] and Masuda–Tomatsu [@MT1], they have used discrete decomposition theorems (See Connes [@C]). However, in our situation, the centers of the images of caonical extensions may not coincide with that of $\tilde{M}$. This makes the problem difficult. It seems that Corollary 4.4 of Izumi [@I] means that it is difficult to show Theorem \[main\] by the same strategy as those in them. Instead, we will use continuous decomposition. We also note that our method gives a proof of Theorem (1) of Kawahigashi–Sutherland–Takesaki [@KwST] which does not depend on the types of the AFD factors.
Approximation on the continuous core
====================================
In order to prove implication $\Rightarrow$ of Theorem \[main\], we need to prepare some lemmas. We first show the implication when $\phi _{\tilde{\rho}}=\phi _{\tilde{\sigma}}$. Until the end of the proof of Lemma \[22.5\], we always assume that $d(\rho)=d(\sigma)$ and $\phi _{\tilde{\rho}}=\phi _{\tilde{\sigma}}$. Choose a dominant weight $\varphi $ of $M$ (For the definition of dominant weights, see Definition II.1.2. and Theorem II.1.3. of Connes–Takesaki [@CT]). Then by Lemma 2.3 (3) of Izumi [@I], it is possible to choose unitaries $u$ and $ v$ of $M$ so that $ (\varphi , \mathrm{Ad} u\circ \rho )$ and $(\varphi , \mathrm{Ad}v \circ \sigma )$ are invariant pairs (See Definition 2.2 of Izumi [@I]). More precisely, we have $$\varphi \circ \mathrm{Ad}u \circ \rho =d(\rho ) \varphi , \ \varphi \circ E_{\mathrm{Ad}u\circ \rho} =\varphi ,$$ $$\varphi \circ \mathrm{Ad} v \circ \sigma =d(\sigma ) \varphi , \ \varphi \circ E_{\mathrm{Ad}v\circ \sigma} =\varphi .$$ By replacing $\rho$ by $\mathrm{Ad} u \circ \rho $ and $\sigma $ by $\mathrm{Ad}v\circ \sigma $ respectively, we may assume that $(\varphi , \rho)$ and $(\varphi , \sigma )$ are invariant pairs. In the rest of this paper, we identify $\tilde{M}$ with $M\rtimes _{\sigma ^\varphi}\mathbf{R}$. Let $h$ be a positive self-adjoint operator affiliated to $\tilde{M}$ satisfying $h^{-it}=\lambda _t^\varphi$. Let $\tau$ be a trace of $\tilde{M}$ defined by $\hat{\varphi }(h\cdot)$.
\[expectation\] For $\rho \in \mathrm{End}(M)_0$, we have $\phi _{\tilde{\rho}}=\tilde{\rho}^{-1} \circ E_{\tilde{\rho}}$, where $E_{\tilde{\rho}}$ is the conditional expectation with respect to $\tau$.
For $x\in M$ and $t\in \mathbf{R}$, we have $$\begin{aligned}
\tilde{\rho} \circ \phi _{\tilde{\rho}} (x\lambda _t^\varphi ) &= \tilde{\rho } (d(\rho )^{-it} \phi _{\rho} (x[D\varphi : D\varphi \circ \phi _\rho ]_t ) \lambda _t^\varphi ) \\
&=d(\rho )^{it} d(\rho )^{-it} \rho (\phi _\rho (x[D\varphi :D\varphi \circ \phi _\rho ]_t ))[D\varphi \circ \phi _\rho : D\varphi ]_t \lambda _t^\varphi \\
&=E_\rho (x[D\varphi : D\varphi \circ \phi _\rho ]_t ) [D\varphi \circ \phi _\rho :D\varphi ]_t \lambda _t^\varphi\end{aligned}$$ Since $(\varphi , \rho )$ is an invariant pair, we have $$[D\varphi \circ \phi _\rho :D\phi ]_t =d(\rho )^{-it}.$$ Hence we have $$E_{\rho } (x[D\varphi :D\varphi \circ \phi _\rho ]_t )[D\varphi \circ \circ \phi _\rho :D\varphi ]_t \lambda _t^\varphi = E_{\rho }(x) d(\rho ) ^{it} d(\rho ) ^{-it} \lambda _t^\varphi =E_\rho (x) \lambda _t^\varphi .$$ Hence by an argument of p.226 of Longo [@L], it is shown that $\tilde{\rho}\circ \phi _{\tilde{\rho}}$ is the expectation with respect to $\tau$.
\[trace\] For $\rho \in \mathrm{End}(M)_0$, we have $\tau \circ \phi _{\tilde{\rho}} =d(\rho ) ^{-1} \tau$.
By Lemma \[expectation\], we have $\phi _{\tilde{\rho}}=\tilde{\rho}^{-1} \circ E_{\tilde{\rho}}$. On the other hand, by Proposition 2.5 (4) of Izumi [@I], we have $\tau \circ \tilde{\rho } =d(\rho ) \tau$. Hence we have $$\begin{aligned}
\tau \circ \phi _{\tilde{\rho}} &=d(\rho) ^{-1} \tau \circ \tilde{\rho }\circ \phi _{\tilde{\rho}} \\
&=d(\rho ) ^{-1} \tau \circ \tilde{\rho} \circ \tilde{\rho} ^{-1} \circ E_{\tilde{\rho }} \\
&= d(\rho ) ^{-1} \tau \circ E_{\tilde{\rho}} \\
&= d(\rho ) ^{-1} \tau .\end{aligned}$$
In the following, we identify $\mathcal{Z}(\tilde{M})$ with $L^\infty (X, \mu )$. Let $$\tau =\int _X^\oplus \tau _x \ d\mu (x)$$ be the direct integral decomposition of $\tau$.
\[kkey\] Let $\rho , \sigma $ be elements of $\mathrm{End}(M)_0$. Assume that $\phi _{\tilde{\rho}}|_{\mathcal{Z}(\tilde{M})}=\phi _{\tilde{\sigma }}|_{\mathcal{Z}(\tilde{M})}$ and $d(\rho )=d(\sigma )$. For $a\in \tilde{M}_+$ with $\tau (a) <\infty$, set $$b:=\tilde{\rho} (a)=\int _X^\oplus b_x \ d\mu (x),$$ $$c:=\tilde{\sigma } (a) =\int _X^\oplus c_x \ d\mu (x).$$ Then we have $$\tau _x (b_x)=\tau _x(c_x)$$ for almost every $x\in X$.
Take an arbitrary positive element $z$ of $\mathcal{Z}(\tilde{M})_+$. Then we have $$\begin{aligned}
\tau (bz) &= \int _X \tau _x(b_xz_x) \ d\mu (x) \\
&=\int _X \tau _x(b_x)z_x \ d\mu (x).\end{aligned}$$ Similarly, we have $$\tau (cz) =\int _X\tau _x(c_x)z_x \ d\mu (x).$$ On the other hand, by Lemma \[trace\], we have $$\begin{aligned}
\tau (bz)&= d(\rho ) \tau \circ \phi _{\tilde{\rho}}(bz) \\
&=d(\rho ) \tau \circ \phi _{\tilde{\rho}}(\tilde{\rho}(a)z) \\
&=d(\rho ) \tau \circ \tilde{\rho }^{-1} \circ E_{\tilde{\rho}} (\tilde{\rho }(a)z) \\
&=d(\rho ) \tau \circ \tilde{\rho } ^{-1} (\tilde{\rho} (a) E_{\tilde{\rho}}(z)) \\
&=d(\rho ) \tau (a \phi _{\tilde{\rho}}(z)).\end{aligned}$$ Since we assume $d(\rho )=d(\sigma ) $ and $\phi _{\tilde{\rho}}|_{\mathcal{Z}(\tilde{M})}=\phi _{\tilde{\sigma}}|_{\mathcal{Z}(\tilde{M})}$, the last number of the above equality is $d(\sigma ) \tau (a\phi _{\tilde{\sigma }}(z))$, which is shown to be $\tau (cz)$ in a similar way. Hence we have $$\int _X \tau _x(b_x)z_x \ d\mu (x)=\int _X \tau _x(c_x) z_x \ d\mu (x).$$ Since the maps $x\mapsto \tau _x(b_x)$ and $x\mapsto \tau _x(c_x)$ are integrable functions and $z\in L^\infty (X, \mu )=L^1(X, \mu )^*$ is arbitrary, we have $\tau _x(b_x)=\tau _x(c_x)$ for almost every $x\in X$.
Note that we have never used the assumption that $M$ is approximately finite up to this point. However, in order to show the following lemma, we need to assume that $M$ is approximately finite. Let $$\tilde{M}=\int ^\oplus _X\tilde{M}_x \ d\mu (x)$$ be the direct integral decomposition.
\[convergence in strong\] Let $M$ be an AFD factor of type $\mathrm{III}$ and $\rho$, $\sigma $ be as in Lemma \[kkey\]. Then for almost every $x\in X$, there exist a factor $B_x$ of type $I_\infty$, a unitary $u$ of $\tilde{M}_x$ and a sequence $\{ u_n\}$ of unitaries of $\tilde{M}_x$ with the following properties.
The relative commutant $B_x'\cap \tilde{M}_x$ is finite.
There exists a sequence of unitaries $\{ v_n\}$ of $B_x'\cap \tilde{M}_x$ with $u_n =(v_n \otimes 1)u$, where we identify $\tilde{M}_x$ with $(B_x'\cap \tilde{M}_x)\otimes B_x$.
For almost every $x\in X$ and for any $a\in \tilde{M}$, we have $\mathrm{Ad}u_n ((\tilde{\rho} (a))_x) \to (\tilde{\sigma }(a))_x $ in the strong \* topology.
We have $B_x\subset u(\tilde{\rho}(\tilde{M}))_xu^*\cap (\tilde{\sigma }(\tilde{M}))_x$.
Let $B_0 \subset \tilde{\rho} (\tilde{M})$ be a factor of type $\mathrm{I}_\infty$ with $Q:=\tilde{\rho} (\tilde{M}) \cap B_0'$ finite. Let $\{ f_{ij}^0 \}$ be a matrix unit generating $B_0$. We may assume that $\tau (f_{ii}^0)<\infty$. Then since $(\tau \circ E_{\tilde{\rho}})_x((f_{11}^0)_x)<\infty$ for almost every $x\in X$, $P:=\tilde{M} \cap B_0'$ is also finite. Then by Lemma \[kkey\], there exists a partial isometry $v$ of $\tilde{M}$ with $v^*v=\tilde{\rho }(f_{11}^0)$, $vv^*=\tilde{\sigma}(f_{11}^0)$. Set $$u:=\sum _{j=1}^\infty \tilde{\sigma}(f_{j1}^0)v\tilde{\rho}(f_{1j}^0).$$ Then $u$ is a unitary of $\tilde{M}$ with $u\tilde{\sigma}(f_{ij}^0)u^*=\tilde{\rho}(f_{ij}^0)$. Set $$B:=\tilde{\sigma }(B_0)(=u\tilde{\rho}(B_0)u^*),$$ $$f_{ij}:=\tilde{\sigma }(f_{ij}^0)(=u\tilde{\rho}(f_{ij}^0)u^*).$$ By replacing $\tilde{\rho} $ by $\mathrm{Ad}u \circ \tilde {\rho}$, we may assume that $\tilde{\rho}(f_{ij})=\tilde{\sigma }(f_{ij})$. In the following, we identify $\tilde{M}$ with $P \otimes B$ and $P $ with $R\otimes \mathcal{Z}(\tilde{M})$, where $R$ is the AFD factor of type $\mathrm{II}_1$. By the approximate finiteness of $R$ and $\mathcal{Z}(\tilde{M})$, there exists a sequence $\{ \{ e_{ij}^n\otimes a_k^n \}_{i,j,k}\}_{n=1}^\infty$ of systems of partial isometries of $P$ with the following properties.
\(1) For each $n$, the system $\{ e_{ij}^n\}_{i,j}$ is a matrix unit of $R$.
\(2) For each $n$, the system $\{ a_k^n\}_k$ is a partition of unity in $\mathcal{Z}(\tilde{M})$.
\(3) For each $n$, $\{ e_{ij}^{n+1}\}_{i,j} $ is a refinement of $\{ e_{ij}^n\}_{i,j}$.
\(4) For each $n$, $\{ a_k^{n+1}\}_k$ is a refinement of $\{ a_k^n\}_k$.
\(5) We have $\bigvee _{n=1}^\infty \{ e^n_{ij} \otimes a^n_k\}_{i,j,k}'' =P$.
Fix a natural number $n$. Then by Lemma \[kkey\], we have $$\tau _x ((\tilde{\rho}(e^n_{11} \otimes a_k^n\otimes f_{11}))_x)=\tau _x((\tilde{\sigma }(e^n_{11} \otimes a_k^n \otimes f_{11}))_x)$$ for almost every $x\in X$. Hence for almost every $x\in X$, there exists a partial isometry $v_k^n$ of $P_x=(\tilde{\rho}(f_{11})\tilde{M}\tilde{\rho}(f_{11}))_x$ with $${v_k^n}^*v_k^n =\tilde{\rho} (e^n_{11} \otimes a_k^n \otimes f_{11})_x, \ v_k^n {v_k^n }^* =\tilde{\sigma } (e^n_{11} \otimes a_k^n \otimes f_{11})_x.$$ Set $$v_n :=\sum _{k,j} \tilde{\sigma }(e_{j1}\otimes a_k^n \otimes f_{11})_xv_k^n \tilde{\rho }(e_{1j} \otimes a_k^n \otimes f_{11})_x.$$ Then $v_n$ is a unitary of $ \tilde{\rho} (f_{11})_x \tilde{M}_x \tilde{\rho} (f_{11})_x$ with $$v_n \tilde{\rho}(e_{ij}^n\otimes a_k^n \otimes f_{11} )_xv_n^* =\tilde{\sigma }(e_{ij}^n \otimes a_k^n \otimes f_{11})_x.$$ Hence for almost every $x\in X$, there exists a sequence $\{ v_n\} $ of unitaries of $P_x$ with $$\mathrm{Ad}(v_n \otimes 1) (\tilde{\rho}(a)_x )\to \tilde{\sigma }(a)_x$$ for any $a\in \tilde{M}$.
\[convergence in strong globaly\] Let $M$, $\rho$ and $\sigma $ be as in Lemma \[convergence in strong\]. Then there exist a unital subfactor $B$ of $\tilde{M}$, a unitary $u$ of $\tilde{M}$ and a sequence $\{ u_n\}$ of unitaries of $\tilde{M}$ with the following properties.
The factor $B$ is of type $I_\infty$.
The relative commutant $B'\cap \tilde{M}$ is finite.
There exists a sequence of unitaries $\{ v_n\}$ of $B'\cap \tilde{M}$ with $u_n =(v_n \otimes 1)u$, where we identify $\tilde{M}$ with $(B'\cap \tilde{M})\otimes B$.
For any $a\in \tilde{M}$, we have $\mathrm{Ad}u_n \circ \tilde{\rho} (a) \to \tilde{\sigma }(a) $ in the strong \* topology.
We have $B\subset u\tilde{\rho}(\tilde{M})u^*\cap \tilde{\sigma }(\tilde{M})$.
This is shown by “directly integrating” the above lemma.
The conclusion of Lemma \[convergence in strong globaly\] means that $\mathrm{Ad}u_n \circ \tilde{\rho}$ converges to $\tilde{\sigma }$ point \*strongly. However, this convergence is slightly weaker than the topology we consider. We need to fill this gap. In order to achieve this, the following criterion is very useful.
\[criterion\] Let $\rho$ and $\rho _n$, $n\in \mathbf{N}$ be be endomorphisms of a von Neumann algebra $N$ with left inverses $\Phi$ and $\Phi _n$, $n\in \mathbf{N}$, respectively. Fix a normal faithful state $\phi $ of $N$. Then the following two conditions are equivalent.
We have $\lim _{n \to \infty } \| \psi \circ \Phi _n -\psi \circ \Phi \| =0 $ for all $\psi \in N_*$.
We have $\lim _{n\to \infty } \| \phi \circ \Phi _n -\phi \circ \Phi \| =0$ and $\lim _{n\to \infty} \rho _n (a) =\rho (a) $ for all $a \in N$.
Hence what we need to do is to find a normal faithful state of $\tilde{M}$ satisfying condition (2) of Lemma \[criterion\].
\[key\] Let $M$, $\rho$, $\sigma $ be as in Lemma \[convergence in strong\]. Then there exists a sequence of unitaries $u_n$ of $\tilde{M}$ with $\mathrm{Ad}u_n \circ \tilde{\rho} \to \tilde{\sigma} $.
Take a subfactor $B$ of $\tilde{M}$, a unitary $u$ of $\tilde{M}$ and a sequence $\{ v_n\}$ of unitaries of $\tilde{M}$ as in Lemma \[convergence in strong globaly\]. By condition (5) in Lemma \[convergence in strong globaly\], we have $u^*Bu \subset \tilde{\rho}(\tilde{M})$. Set $$F:=\tilde{\rho}^{-1}(u^*Bu).$$ Then we have $$\tilde{\rho}^{-1}\circ \mathrm{Ad}u^*(B)=F,$$ $$\tilde{\rho}^{-1}\circ \mathrm{Ad}u^*(B'\cap \mathrm{Ad}u\circ \tilde{\rho}(\tilde{M}))=F'\cap \tilde{M}.$$ We also have $$\mathrm{Ad}u\circ E_{\tilde{\rho}}\circ \mathrm{Ad}u^*|_B=\mathrm{id}_B,$$ $$\mathrm{Ad}u \circ E_{\tilde{\rho}}\circ \mathrm{Ad}u^*(B'\cap \tilde{M})=B'\cap \mathrm{Ad}u \circ \tilde{\rho}(\tilde{M}).$$ Let $\{ f_{ij}\}$ be a matrix unit generating $B$. Set $$\overline{\tau} (a):=\tau (a\tilde{\rho}^{-1}(u^*f_{11}u))$$ for $a\in F'\cap \tilde{M}$, which is a faithful normal finite trace of $F'\cap \tilde{M}$. Let $\varphi$ be a normal faithful state of $F$. Let $\Psi _F: \tilde{M}\to (F'\cap \tilde{M})\otimes F$ is the natural identification map. Then by the above observation, for $a\in B'\cap \tilde{M}$ and $i$, $j$, we have $$\begin{aligned}
\ &(\overline{\tau} \otimes \varphi )\circ \Psi _F\circ \phi _{\tilde{\rho}} \circ \mathrm{Ad}u^*(af_{ij}) \\
&=(\overline{\tau} \otimes \varphi )\circ \Psi _F \circ (\tilde{\rho}^{-1}\circ \mathrm{Ad}u^*)\circ (\mathrm{Ad}u \circ E_{\tilde{\rho}}\circ \mathrm{Ad}u^*)(af_{ij}) \\
&=(\overline{\tau}\otimes \varphi ) \circ \Psi _F\circ (\tilde{\rho}^{-1}\circ \mathrm{Ad}u^*)((\mathrm{Ad}u \circ E_{\tilde{\rho}}\circ \mathrm{Ad}u^*|_{B'\cap \tilde{M}})(a)f_{ij}) \\
&=(\overline{\tau} \circ \phi _{\tilde{\rho}} \circ \mathrm{Ad}u^*) (a) (\varphi \circ \phi _{\tilde{\rho}} \circ \mathrm{Ad}u^*) (f_{ij}).\end{aligned}$$ Since $B \subset \tilde{\sigma }(\tilde{M})\cap \mathrm{Ad}u \circ \tilde{\rho}(\tilde{M})$, we have $$E_{\tilde{\sigma}}(af_{ij})=E_{\tilde{\sigma}}(a)f_{ij},$$ $$\mathrm{Ad}u \circ E_{\tilde{\rho}}\circ \mathrm{Ad}u^*(af_{ij})=\mathrm{Ad}u \circ E_{\tilde{\rho}} \circ \mathrm{Ad}u^*(a)f_{ij}$$ for $a\in B'\cap \tilde{M}$. Notice that $\tilde{\sigma }^{-1}(f_{ij})=\tilde{\rho}^{-1}(u^*f_{ij}u)$ by condition (3) of Lemma \[convergence in strong globaly\]. Then for any $a\in B'\cap \tilde{M}$, we have $$\begin{aligned}
\ &( \overline{\tau}\otimes \varphi )\circ \Psi _F \circ \phi _{\tilde{\rho}}\circ \mathrm{Ad}u^* (v_n^*\otimes 1)(af_{ij}) \\
&= (\overline{\tau}\otimes \varphi ) \circ \Psi _F \circ \phi _{\tilde{\rho}}((u^*(v_n^*av_n)u)(u^*f_{ij}u)) \\
&= \overline{\tau} \circ \phi _{\tilde{\rho}}(u^*(v_n^*av_n)u)\varphi (\tilde{\rho}^{-1}(u^*f_{ij}u)) \\
&=\tau (\phi _{\tilde{\rho}} (u^*(v_n^*av_n )u)\tilde{\rho}^{-1}(u^*f_{11}u))\varphi (\tilde{\rho}^{-1}(u^*f_{ij}u)) \\
&=\tau \circ \phi _{\tilde{\rho}}(u^*(v_n^*av_n)f_{11}u)\varphi (\tilde{\rho}^{-1}(u^*f_{ij}u)) \\
&=d(\rho ) \tau (u^*(v_n^*av_n)f_{11}u)\varphi (\tilde{\rho}^{-1}(u^*f_{ij}u)) \\
&=d(\sigma ) \tau (af_{11}) \varphi (\tilde{\sigma }^{-1}(f_{ij})) \\
&=\tau (\phi _{\tilde{\sigma }}(a)\tilde{\sigma }^{-1}(f_{11})) \varphi (\tilde{\sigma }^{-1}(f_{ij})) \\
&=\tau (\phi _{\tilde{\sigma }}(a)\tilde{\rho}^{-1}(u^*f_{11}u)) \varphi (\tilde{\sigma }^{-1}(f_{ij})) \\
&= (\overline{\tau}\otimes \varphi ) \circ \Psi _F \circ \phi _{\tilde{\sigma}}(af_{ij}).\end{aligned}$$ Hence we have $( \overline{\tau}\otimes \varphi )\circ \Psi _F \circ \phi _{\tilde{\rho}} \circ \mathrm{Ad}(u^*(v_n \otimes 1)^*)=(\overline{\tau}\otimes \varphi )\circ \Psi _F\circ \phi _{\tilde{\sigma }}$ for any $n$. Hence by Lemma \[convergence in strong globaly\] and Lemma \[criterion\], we have $\mathrm{Ad}((v_n\otimes 1)u)\circ \tilde{\rho} \to \tilde{\sigma}$.
Averaging by the trace-scaling action
=====================================
In this section, we always assume that $M$ is an AFD factor of type III. Let $\varphi $ be a dominant weight of $M$ and $\rho, \sigma \in \mathrm{End}(M)_0$ be finite index endomorphisms with $(\varphi , \rho)$ and $(\varphi , \sigma )$ invariant pairs. Set $$\tilde{M}:=M\rtimes _{\sigma ^{\varphi}} \mathbf{R}.$$ Let $\psi _0$ be a normal faithful state of $\tilde{M}$ and $\{ \psi _i\}_{i=1}^\infty $ be a norm dense sequence of the unit ball of $\tilde{M}_*$. Let $\theta $ be the dual action on $\tilde{M} $ of $\sigma ^\varphi$. We will replace the sequence $\{ u_n\}$ chosen in the previous section so that it is almost invariant by $\theta $. In order to achieve this, we use a property of $\theta $ which is said to be the Rohlin property. In order to explain this property, we first need to explain related things. Let $\omega $ be an ultrafilter of $\mathbf{N}$. A sequence $\{ [-1,1] \ni t \mapsto x_{n,t} \in \tilde{M}\}_{n=1}^\infty $ of maps from $[-1,1]$ to $\tilde{M}$ is said to be $\omega $-equicontinuous if for any $\epsilon >0$, there exist an element $U\subset \mathbf{N}$ of $\omega $ and $\delta >0$ with $\| x_{n,t} -x_{n,s}\| <\epsilon $ for any $s,t\in [-1,1]$ with $|s-t|<\delta $, $n\in U$. Set $$\mathcal{C}:=\{ (x_n)\in l^\infty (\tilde{M}) \mid \| x_n \psi -\psi x_n \| \to 0 \ \mathrm{for} \ \mathrm{any} \ \psi \in \tilde{M}.\} ,$$ $$\mathcal{C}_{\theta , \omega } :=\{ (x_n) \in \mathcal{C}_\omega \mid \mathrm{ the} \ \mathrm{maps} \ \{ t\mapsto \theta _t(x_n)\}_{n=1}^\infty \ \mathrm{are} \ \omega \ \mathrm{equicontinuous}.\} ,$$ $$\mathcal{I}_\omega :=\{ (x_n)\in l^\infty (\tilde{M}) \mid x_n \to 0 \ \mathrm{in } \ \mathrm{the } \ \mathrm{*strong} \ \mathrm{topology}.\} .$$ Then $\mathcal{I}_\omega $ is a (norm) closed ideal of $\mathcal{C}_{\theta , \omega }$, and the quotient $\tilde{M}_{\theta , \omega }:=\mathcal{C}_{\theta , \omega }/\mathcal{I}_\omega $ is a von Neumann algebra. As mentioned in Masuda–Tomtasu [@MT5], the action $\theta $ has the Rohlin property, that is, for any $R>0$, there exists a unitary $v$ of $\tilde{M}_{\theta ,\omega} $ with $$\theta _t(v)=e^{-iRt}v$$ for any $t\in \mathbf{R}$ (See Section 4 of Masuda–Tomatsu [@MT5]). Choose arbitrary numbers $r>0$ and $0<\epsilon <1$. Then since $M$ is of type III, there exists a real number $R$ which is not of the discrete spectrum of $\theta |_{\mathcal{Z}(\tilde{M})}$ and which satisfies $r/R<\epsilon ^2$. Then as shown in Theorem 5.2 of Masuda–Tomatsu [@MT5], there exists a normal injective \*-homomorphism $\Theta$ from $\tilde{M}\otimes L^\infty ([-R,R])$ to $\tilde{M}^\omega $ satisfying $x\otimes f\mapsto xf(v)$ for any $x\in \tilde{M}$, $f \in L^\infty([-R,R])$. For each $t\in \mathbf{R}$, set $$\gamma _t:L^\infty ([-R,R]) \ni f \mapsto f(\cdot -t)\in L^\infty ([-R,R]),$$ where we identify $[-R,R]$ with $\mathbf{R}/2R\mathbf{Z}$ as measured spaces. Then the \*-homomorphisms $\Theta $ and $\gamma _t$ satisfy $$\Theta \circ (\theta _t\otimes \gamma _t ) =\theta _t\circ \Theta$$ (See Theorem 5.2 of Masuda–Tomatsu [@MT5]).
\[10\] For $\psi \in \tilde{M}_*$ and $x\otimes f\in \tilde{M}\otimes L^\infty ([-R,R])$, we have $$\psi ^\omega \circ \Theta =\psi \otimes \tau _{L^\infty },$$ where $\tau _{L^\infty}$ is the trace coming from the normalized Haar measure of $L^\infty ([-R,R])$.
Let $\{ v_n\}$ be a representing sequence of $v$. Then we have $$\begin{aligned}
\psi ^\omega \circ \Theta (x\otimes f) &= \psi ^\omega (xf(v)) \\
&=\lim _{n\to \omega } \psi (xf(v_n)) \\
&=\psi (x) \lim _{n \to \omega }f(v_n) \\
&=\psi (x)\tau _{L^\infty }(f) \\
&=(\psi \otimes \tau _{L^\infty } )(x\otimes f).\end{aligned}$$
Since the maps $$[-R,R]\ni t\mapsto \psi _i \circ \phi _{\tilde{\rho}}\circ \theta _t \in (\tilde{M})_*,$$ $$[-R,R]\ni t\mapsto \psi _i \circ \phi _{\tilde{\sigma }}\circ \theta _t \in (\tilde{M})_*$$ are norm continuous, the union of their images $$\{ \psi _i \circ \phi _{\tilde{\rho}} \circ \theta _t \mid t\in [-R,R]\} \cup \{ \psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _t\mid t\in [-R,R]\}$$ is compact. Hence there exists a finite set $-R=t_0 <\cdots <t_J =R$ of $[-R,R]$ such that $$\| \psi _i \circ \phi _{\tilde{\rho}} \circ \theta _{t_j}-\psi _i \circ \phi _{\tilde{\rho}}\circ \theta _t \| <\epsilon ,$$ $$\| \psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _{t_j} -\psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _t\| <\epsilon$$ for any $i=1, \cdots , n$, $j=0, \cdots , J-1$ and $t\in [t_j, t_{j+1}]$. We may assume that $t_j=0$ for some $j$. Then by Lemma \[key\], there exists a unitary $u$ of $\tilde{M}$ with $$\| \psi _i \circ \phi _{\tilde{\rho}}\circ \theta _{t_j}\circ \mathrm{Ad}u -\psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _{t_j}\| <\epsilon$$ for any $j=0, \cdots , J-1$, $i=1, \cdots ,n$ (Notice that we used the fact that we have $\phi _{\tilde{\rho}}\circ \theta _{t_j}=\theta _{t_j} \circ \phi _{\tilde{\rho}}$ and that we have $\phi _{\tilde{\sigma }}\circ \theta _{t_j}=\theta _{t_j}\circ \phi _{\tilde{\sigma }}$ for any $j=0, \cdots , J-1$). Set $$U:[-R,R]\ni t\mapsto \theta _t(u) \in \tilde{M},$$ which is a unitary of $\tilde{M}\otimes L^\infty ([-R,R])$.
\[11\] We have $$\| (\psi _i \circ \phi _{\tilde{\rho}})^\omega \circ \mathrm{Ad} \Theta (U)|_{\mathrm{Im}\Theta } -(\psi _i \circ \phi _{\tilde{\sigma}})^\omega | _{\mathrm{Im}\Theta } \| <3\epsilon .$$
Let $m$ be the normalized Haar measure of $[-R,R]$. By Lemma \[10\], we have $$\begin{aligned}
\ &\| (\psi _i \circ \phi _{\tilde{\rho}})^\omega \circ \mathrm{Ad}\Theta (U) |_{\mathrm{Im}\Theta } - (\psi _i \circ \phi _{\tilde{\sigma }})^\omega |_{\mathrm{Im}\Theta }\| \\
&=\| ((\psi _i \circ \phi _{\tilde{\rho}})\otimes \tau _{L^\infty}) \circ \mathrm{Ad}U -(\psi _i \circ \phi _{\tilde{\sigma }} ) \otimes \tau _{L^\infty } \| \\
&= \int _{[-R,R]} \| (\psi _i \circ \phi _{\tilde{\rho}} ) \circ \mathrm{Ad}\theta _t(u) -\psi _i \circ \phi _{\tilde{\sigma }} \| \ dm(t) \\
&=\int _{[-R,R]} \| (\psi _i \circ \phi _{\tilde{\rho}}) \circ \theta _t\circ \mathrm{Ad}u -\psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _t \| \ dm(t) \\
&\leq \sum _{j=0}^{J-1} \int _{[t_j, t_{j+1}]} ( \| (\psi _i \circ \phi _{\tilde{\rho}} )\circ \theta _t -(\psi _i \circ \phi _{\tilde{\rho}}) \circ \theta _{t_j} \| \\
&+ \| \psi _i \circ \phi _{\tilde{\rho}}\circ \theta _{t_j} \circ \mathrm{Ad}u-\psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _{t_j}\| \\
&+ \| \psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _{t_j}-\psi _i \circ \phi _{\tilde{\sigma }} \circ \theta _{t} \| ) \ dm(t) \\
&\leq \sum _{j=0}^{J-1} \int _{[t_j, t_{j+1}]} (\epsilon + \epsilon +\epsilon ) \ dm(t) \\
&=3\epsilon .\end{aligned}$$
\[12\] We have $$\| \theta _s(\Theta (U))-\Theta (U) \| _{\psi _0^\omega }^\sharp <2\epsilon$$ for $|s|\leq r$.
Notice that we have $$(\theta _s\otimes \gamma _s) (U):t\mapsto \theta _s(U_{t-s}),$$ where $U_t$ denotes the evaluation of the function $U$ at the point $t$. Hence by the definition of $U$, we have $$(\theta _s\otimes \gamma _s)(U) _t=\theta _t(u)$$ for any $t\in [-R+r, R-r]$, where the left hand side is the evaluation of the function $(\theta _s\otimes \gamma _s) (U)$ at the point $t$. Hence by Lemma \[10\], we have $$\begin{aligned}
\ &\| \theta _s(\Theta (U))-\Theta (U) \| _{\psi _0^\omega }^\sharp \\
&= \| (\theta _s\otimes \gamma _s )(U) -U \| _{\psi _0 \otimes \tau _{L^\infty}}^\sharp \\
&=(\int _{[-R,R]} (\| ((\theta _s \otimes \gamma _s )(U))_t -U_t \| _{\psi _0}^\sharp )^2 \ dm(t) )^{1/2} \\
&\leq (\int _{[-R,-R+r]\cup [R-r,R]} 4 \ dm(t) ) ^{1/2} \\
&\leq (4\epsilon ^2 )^{1/2} \\
&=2\epsilon .\end{aligned}$$
Let $$\psi _i \circ \phi _{\tilde{\sigma }} =| \psi _i \circ \phi _{\tilde{\sigma }}|v_i$$ be the polar decompositions of $\psi _i \circ \phi _{\tilde{\sigma }}$ for $i=1, \cdots , n$.
\[13\] There exists a finite subset $-R=s_0< \cdots < s_K=R$ of $[-R,R]$ with the following properties.
We have $$\| (U-\sum _{k=0}^{K-1} \theta _{s_k}(u)e_k )v_i^* \| _{|\psi _i \circ \phi _{\tilde{\sigma }}|\otimes \tau _{L^\infty}} ^\sharp <\epsilon$$ for any $i=1, \cdots , n$, where $e_k:=\chi _{[s_k,s_{k+1}]}\in L^\infty ([-R,R])$.
We have $$\| U-\sum _{k=0}^{K-1} \theta _{s_k}(u) e_k \| _{|\psi _i \circ \phi _{\tilde{\rho}}|\otimes \tau _{L^\infty}}^\sharp <\epsilon$$ for any $i=1, \cdots , n$.
We have $$\| U-\sum _{k=0}^{K-1} \theta _{s_k} (u)e_k\| _{(\psi _0 \circ \theta _{t_j}) \otimes \tau _{L^\infty}}^\sharp <\epsilon$$ for any $i=1, \cdots , n$ and $j=0, \cdots , J-1$.
Since the map $t\mapsto \theta _t(u)$ is continuous in the strong \* topology, there exists a finite set $-R=s_0< \cdots < s_K=R$ of $[-R,R]$ with $$\| (\theta _t(u)-\theta _{s_k}(u) )v_i^* \| _{|\psi _i \circ \phi _{\tilde{\sigma }}| }^\sharp <\epsilon$$ for $i=1, \cdots , n$, $k=0, \cdots , K-1$ and $t\in [s_k, s_{k+1}]$, $$\| \theta _t(u)-\theta _{s_k}(u) \| _{|\psi _i \circ \phi _{\tilde{\rho}}|} ^\sharp <\epsilon$$ for $i=1, \cdots ,n$, $k=0, \cdots , K-1$ and $t\in [s_k, s_{k+1}]$, $$\| \theta _t(u)-\theta _{s_k}(u)\| _{\psi _0\circ \theta _{t_j}}^\sharp <\epsilon$$ for $j=0, \cdots , J-1$, $k=0, \cdots ,K-1$ and $t\in [s_k, s_{k+1}]$. Then we have $$\begin{aligned}
\ &\| (U-\sum _{k=0}^{K-1}\theta _{s_k}(u)e_k )v_i^* \| _{|\psi _i \circ \phi _{\tilde{\sigma }}|\otimes \tau _{L^\infty} }^\sharp \\
&=(\sum _{k=0}^{K-1} \int _{[s_k, s_{k+1})} (\| (\theta _t(u) -\theta _{s_k}(u))v_i^* \| _{|\psi _i \circ \phi _{\tilde{\sigma }}|}^\sharp )^2 \ dm(t) )^{1/2} \\
&< (\sum _{k=0}^{K-1} \int _{[s_k,s_{k+1})} \epsilon ^2 \ dm(t ))^{1/2} \\
&=\epsilon . \end{aligned}$$ The other inequalities are shown in a similar way.
Set $$V:=\sum _{k=0}^{K-1} \theta _{s_k}(u)e_k.$$ Take a representing sequence $\{ e_k^n\}_{n=1}^\infty$ of $\Theta (e_k)$ so that $\{ e_k^n\}_{k=0}^{K-1}$ is a partition of unity in $\tilde{M}$ by projections for each $n$. Set $$v_n:=\sum _{k=0}^{K-1} \theta _{s_k}(u)e_k^n,$$ which is a unitary. The sequence $\{ v_n\}_{n=1}^\infty$ represents the unitary $\Theta (V)$. Let $\{ u_n\}_{n=1}^\infty $ be a representing sequence of $\Theta (U)$.
\[P\] We have $$\lim _{n\to \omega } \| \theta _t (v_n)-v_n \| _{\psi _0}^\sharp <6\sqrt{\epsilon} .$$ for $t\in [-r,r]$.
Note that we have $$\begin{aligned}
\ &(\| \theta _t(a)\| _{\psi _0}^\sharp )^2 \\
&=\frac{1}{2}\psi _0 \circ \theta _t(a^*a+aa^*) \\
&=\frac{1}{2}(\psi _0 \circ \theta _{t_j}(a^*a+aa^*)) -\frac{1}{2}((\psi _0\circ \theta _{t_j}-\psi _0\circ \theta _t)(a^*a+aa^*)) \\
&\leq (\| a\| _{\psi _0\circ \theta _{t_j}}^\sharp )^2 +\| a\| ^2 \| \psi _0\circ \theta _{t_j}-\psi _0\circ \theta _t\|\end{aligned}$$ for any $a\in \tilde{M}$. Hence for $t\in [t_j, t_{j+1}]\cap [-r,r]$, we have $$\begin{aligned}
\ &\| \theta _t(v_n) -v_n \| _{\psi _0}^\sharp \\
&\leq \| \theta _t(v_n-u_n) \| _{\psi _0}^\sharp +\| \theta _t(u_n)-u_n\| _{\psi _0}^\sharp +\| u_n-v_n \| _{\psi _0}^\sharp \\
&\leq (4\| \psi _0 \circ \theta _{t_j}-\psi _0 \circ \theta _t\| +(\| v_n -u_n\| _{\psi _0\circ \theta _{t_j}}^\sharp )^2)^{1/2} \\
&+\| \theta _t(u_n)-u_n \| _{\psi _0}^\sharp +\| u_n -v_n \| _{\psi _0}^\sharp \\
&< (4\epsilon +(\| v_n -u_n\| _{\psi _0\circ \theta _{t_j}}^\sharp )^2 )^{1/2} \\
&+\| \theta _t(u_n)-u_n \| _{\psi _0}^\sharp + \| u_n -v_n \| _{\psi _0}^\sharp .\end{aligned}$$ Hence by Lemmas \[12\] and \[13\] (3), we have $$\begin{aligned}
\ &\lim _{n\to \omega} \| \theta _t(v_n) -v_n \| _{\psi _0}^\sharp \\
&\leq (4\epsilon +(\| V-U\| _{(\psi _0 \circ \theta _{t_j})\otimes \tau _{L^\infty}}^\sharp )^2)^{1/2} \\
&+\| \theta _t(U)-U\| _{\psi _0\otimes \tau _{L^\infty}}^\sharp +\| U-V\|_{\psi _0\otimes \tau _{L^\infty}}^\sharp \\
&<(4\epsilon +\epsilon ^2)^{1/2} +2\epsilon +\epsilon \\
&<6\sqrt{\epsilon} .\end{aligned}$$
\[Q\] We have $$\lim _{n\to \omega } \| v_n ^*\psi _i \circ \phi _{\tilde{\rho}}-\psi _i \circ \phi _{\tilde{\sigma }}v_n^* \| \leq 7\epsilon$$ for any $i=1, \cdots , n$.
Notice that we have $$\begin{aligned}
\ & \| u_n^* (\psi _i \circ \phi _{\tilde{\rho}})-(\psi _i \circ\phi _{\tilde{\sigma }})u_n ^*\| \\
& \| \Theta (U)^* (\psi _i \circ \phi_{\tilde{\rho}})^\omega |_M -(\psi _i \circ \phi _{\tilde{\sigma }})^\omega \Theta (U)^*|_M \| \\
&\leq \| (\psi _i \circ \phi _{\tilde{\rho}} )^\omega \circ \mathrm{Ad}\Theta (U)|_{\mathrm{Im}\Theta}-(\psi _i \circ \phi _{\tilde{\sigma }})^\omega |_{\mathrm{Im}\Theta}\| .
\end{aligned}$$ Hence by Lemmas \[11\] and \[13\] (1) (2), we have $$\begin{aligned}
\ &\lim _{n\to \omega } \| v_n^*\psi _i \circ \phi _{\tilde{\rho}}-\psi _i \circ \phi _{\tilde{\sigma }}v_n^* \| \\
&\leq \lim _{n\to \omega }(\| (v_n^*-u_n^*)\psi _i \circ \phi _{\tilde{\rho}} \| \\
&+\| u_n^*\psi _i \circ \phi _{\tilde{\rho}}-\psi _i \circ \phi _{\tilde{\sigma}}u_n^*\| +\| \psi _i \circ \phi _{\tilde{\sigma }}(u_n^*-v_n^*)\| ) \\
&\leq \lim _{n\to \omega }(\| (v_n-u_n)^*\| _{|\psi _i \circ \phi _{\tilde{\rho}}|} \\
&+\| (\psi _i \circ \phi _{\tilde{\rho}})^\omega \circ \mathrm{Ad}\Theta (U) |_{\mathrm{Im}\Theta} -(\psi _i \circ \phi _{\tilde{\sigma }})^\omega |_{\mathrm{Im}\Theta}\| \\
&+\| (v_n-u_n)v_i^*\| _{|\psi _i \circ \phi _{\tilde{\sigma }}|} )\\
&=\| V-U\| _{|\psi _i \circ \phi _{\tilde{\rho}}|\otimes \tau _{L^\infty}} \\
&+\| (\psi _i \circ \phi _{\tilde{\rho}})^\omega \circ \mathrm{Ad}\Theta (U) |_{\mathrm{Im}\Theta} -(\psi _i \circ \phi _{\tilde{\sigma }})^\omega |_{\mathrm{Im}\Theta}\| +\| (V-U)v_i^*\|_{|\psi _i \circ \phi _{\tilde{\sigma}}|\otimes \tau _{L^\infty}} \\
&\leq \epsilon +3\epsilon +\epsilon \\
&=5\epsilon .\end{aligned}$$ Note that in order to show the second inequality, we used Lemma \[inequality\].
By Lemmas \[P\] and \[Q\], we have the following proposition.
\[R\] There exists a sequence $\{ v_n\}_{n=1}^\infty $ of unitaries of $\tilde{M}$ with $$\lim _{n\to \infty }\| \theta _t(v_n)-v_n\| _{\psi _0}^\sharp =0,$$ $$\lim _{n\to \infty}\| v_n^*\psi _i \circ \phi _{\tilde{\rho}}-\psi _i \circ \phi _{\tilde{\sigma }}v_n^* \| =0$$ for any $i=1, 2, \cdots $.
Approximation on $\tilde{M}\rtimes _\theta \mathbf{R}$. {#lifting}
=======================================================
Set $$n_{\tau}:=\{ x\in \tilde{M}\mid \tau (x^*x)<\infty \}.$$
\[33\] Let $L^2(\tilde{M})$ be the standard Hilbert space of $\tilde{M}$ and $\Lambda :n_\tau \to L^2(\tilde{M})$ be the canonical injection. For each $x\in n_\tau$, set $V _{\tilde{\rho}}(\Lambda (x)):=\sqrt{d(\rho )}^{-1}\Lambda (\tilde{\rho}(x))$. Then $V_{\tilde{\rho}}$ defines an isometry of $L^2(\tilde{M})$ satisfying $$V_{\tilde{\rho}}^* xV_{\tilde{\rho}}=\phi _{\tilde{\rho}} (x)$$ for any $x\in \tilde{M}$.
Take $x\in n_\tau$. Then by Lemma 2.5 (4) of Izumi [@I], we have $$\begin{aligned}
\| V_{\tilde{\rho}} \Lambda (x)\| ^2 &=d(\rho )^{-1} \tau (\tilde{\rho }(x^*x) ) \\
&=\tau (x^*x) =\| \Lambda (x)\| ^2.\end{aligned}$$ Hence $V_{\tilde{\rho}}$ defines an isometry of $L^2(\tilde{M})$. Next, we show the latter statement. We have $V_{\tilde{\rho}}^* (\Lambda (x))=\sqrt{d(\rho )}\Lambda (\phi _{\tilde{\rho}}(x))$ because $$\begin{aligned}
\langle V_{\tilde{\rho}}^* \Lambda (x), \Lambda (y)\rangle &=\langle \Lambda (x), \sqrt{d(\rho )}^{-1} \Lambda (\tilde{\rho}(y))\rangle \\
&=\sqrt{d(\rho )}^{-1} \tau (\tilde{\rho}(y)^*x) \\
&=\sqrt{d(\rho )}\tau (y^*\phi _{\tilde{\rho}}(x)) \\
&=\langle \sqrt{d(\rho )} \Lambda (\phi _{\tilde{\rho}}(x)), \Lambda (y)\rangle\end{aligned}$$ for any $x,y \in n_\tau$. In order to show the third equality of the above, we used Lemma \[trace\]. Hence for any $x\in \tilde{M}$ and $y\in n_\tau$, we have $$\begin{aligned}
V_{\tilde{\rho}}^*xV_{\tilde{\rho}}\Lambda (y)&= \sqrt{d(\rho )}^{-1}V_{\tilde{\rho}}^*\Lambda (x\tilde{\rho}(y)) \\
&= \Lambda (\phi _{\tilde{\rho}}(x\tilde{\rho}(y))) \\
&=\phi _{\tilde{\rho}}(x) \Lambda (y).\end{aligned}$$
Let $\rho $ be an endomorphism of a von Neumann algebra $M$. Then since its canonical extension $\tilde{\rho}$ satisfies $\tau \circ \tilde{\rho}=d(\rho )\tau$, the endomorphism $\tilde{\rho}$ extends to $\tilde{M}\rtimes _\theta \mathbf{R}$ by $\lambda _t^\theta \mapsto \lambda _t^\theta $ for any $t\in \mathbf{R}$. We denote this extension by $\tilde{\tilde{\rho}}$.
\[18\] Let $\alpha $ and $\sigma $ be finite index endomorphisms of a separable infinite factor $M$ and $\varphi $ be a dominant weight of $M$. Assume that there exists a sequence $\{ u_n\}$ of unitaries of $\tilde{M} \rtimes _\theta \mathbf{R}$ with $\mathrm{Ad}u_n \circ \tilde{\tilde{\rho}} \to \tilde{\tilde{\sigma}}$ as $n \to \infty$. Then there exists a sequence $\{ v_n \}$ of unitaries of $M$ with $\mathrm{Ad}v_n \circ \rho \to \sigma $.
Since $(\varphi , \rho )$ and $(\varphi , \sigma )$ are invariant pairs, it is possible to identify $\tilde{\tilde{\rho}}$ with $\rho \otimes \mathrm{id}_{B(L^2\mathbf{R})}$ and $\tilde{\tilde{\sigma }}$ with $\sigma \otimes \mathrm{id}_{B(L^2\mathbf{R})}$ through Takesaki duality, respectively (It is possible to choose the same identification between $M\otimes B(L^2\mathbf{R}) $ and $\tilde{M}\rtimes _\theta \mathbf{R}$ for $\tilde{\tilde{\rho}}$ and $\tilde{\tilde{\sigma }}$. See the argument preceding to Lemma 3.10 of Masuda–Tomatsu [@MT1]). Then by (the proof of) Lemma 3.11 of Masuda–Tomatsu [@MT1], there exist an isomorphism $\pi $ from $M\otimes B(L^2\mathbf{R})$ to $M$ and unitaries $u_\rho$, $u_\sigma $ of $M$ satisfying $$\pi \circ (\rho \otimes \mathrm{id}) \circ \pi ^{-1} =\mathrm{Ad}u_\rho \circ \rho ,$$ $$\pi \circ (\sigma \otimes \mathrm{id} ) \circ \pi ^{-1} =\mathrm{Ad}u_\sigma \circ \sigma$$ (Although in the statement of Lemma 3.11 of Masuda–TOmatsu [@MT1], the isomorphism $\pi $ depends on the choice of $\rho$, $\pi $ turns out to be independent of $\rho $ by its proof). Then we have $$\begin{aligned}
\ & \mathrm{Ad}(u_\sigma ^*\pi (u_n) u_\rho ) \circ \rho \\
&= \mathrm{Ad}(u_\sigma ^*\pi (u_n ) )\circ \pi \circ (\rho\otimes \mathrm{id}_{B(L^2\mathbf{R})}) \circ \pi ^{-1} \\
&=\mathrm{Ad}u_\sigma ^* \circ \pi \circ (\mathrm{Ad}u_n \circ (\rho \otimes \mathrm{id}_{B(L^2\mathbf{R})}) ) \circ \pi ^{-1} \\
&\to \mathrm{Ad}u_\sigma ^* \circ \pi \circ (\sigma \otimes \mathrm{id}_{B(L^2\mathbf{R})} )\circ \pi ^{-1} \\
&=\mathrm{Ad}u_\sigma ^* \circ (\mathrm{Ad}u_\sigma \circ \sigma ) \\
&=\sigma .\end{aligned}$$
\[19\] Let $\rho $ be an endomorphism with finite index and with $(\varphi , \rho )$ an invariant pair. Let $E_{\tilde{\tilde{\rho}}}$ be the minimal expectation from $\tilde{\tilde{M}}$ to $\tilde{\tilde{\rho}}(\tilde{\tilde{M}})$. Then we have the following.
For each $x\in \tilde{M}$, we have $E_{\tilde{\tilde{\rho}}}(x)=E_{\tilde{\rho}}(x)$.
For any $s\in \mathbf{R}$, we have $E_{\tilde{\tilde{\rho}}}(\lambda _t^\theta )=\lambda _t^\theta$.
This is shown in the proof of Theorem 4.1 of Longo [@L].
\[36\] For $\xi \in L^2(\mathbf{R}, \tilde{M})$, set $$V_{\tilde{\tilde{\rho}}}(\xi )(s):=V_{\tilde{\rho}}(\xi (s)).$$ Then $V_{\tilde{\tilde{\rho}}}$ is an isometry of $L^2(\mathrm{R}, \tilde{M})$ satisfying $$V_{\tilde{\tilde{\rho}}}^*xV_{\tilde{\tilde{\rho}}}=\phi _{\tilde{\tilde{\rho}}}(x)$$ for any $x\in M$, where $\phi _{\tilde{\tilde{\rho}}}=\tilde{\tilde{\rho}}^{-1}\circ E_{\tilde{\tilde{\rho}}}$.
The first statement is shown by the following computation. $$\begin{aligned}
\| V_{\tilde{\tilde{\rho}}}(\xi )\| ^2&=\int _{\mathbf{R}}\| V_{\tilde{\rho}}(\xi (s))\| ^2 \ d\mu (s) \\
&=\int _{\mathbf{R}}\| \xi (s)\| ^2\ d\mu (s) \\
&=\| \xi \| ^2\end{aligned}$$ for $\xi \in L^2(\mathbf{R}, \tilde{M})$. Next, we show the latter statement. Choose $x\in M$ and $\xi \in L^2(\mathbf{R}, \tilde{M})$. Then we have $$\begin{aligned}
V_{\tilde{\tilde{\rho}}}^* \circ \pi _\theta (x) \circ V_{\tilde{\tilde{\rho}}}(\xi ) &= V_{\tilde{\tilde{\rho}} }^* \pi _\theta (x)(s\mapsto V_{\tilde{\rho}}(\xi (s))) \\
&=V_{\tilde{\tilde{\rho}}} ^* (s\mapsto \theta _{-s}(x)\circ V_{\tilde{\rho}}(\xi (s))) \\
&=(s\mapsto V_{\tilde{\rho}}^* \circ \theta _{-s}(x) \circ V_{\tilde{\rho}}(\xi (s))) \\
&=(s\mapsto \phi _{\tilde{\rho}}(\theta _{-s}(x))(\xi (s))) \\
&=(s\mapsto \theta _{-s}( \phi _{\tilde{\rho}}(x))(\xi (s))) \\
&=\pi _\theta (\phi _{\tilde{\rho}}(x))(\xi ) \\
&=\phi _{\tilde{\tilde{\rho}}} (\pi _\theta (x))(\xi ).\end{aligned}$$ In order to show the fourth equality of the above, we used Lemma \[33\]. The last equality of the above follows from Lemma \[19\]. For $t\in \mathbf{R}$ and $\xi \in L^2(\mathbf{R},\tilde{M})$, we have $$\begin{aligned}
V_{\tilde{\tilde{\rho}}}^*\lambda _t^\theta V_{\tilde{\tilde{\rho}}}\xi &= V_{\tilde{\tilde{\rho}}} ^*(s\mapsto V _{\tilde{\rho}}(\xi (s-t)) \\
&= s\mapsto V_{\tilde{\rho}}^*V_{\tilde{\rho}}(\xi (s-t)) \\
&= \lambda _t^\theta (\xi ).\end{aligned}$$
Thus we are done.
\[20\] Let $N$ be a von Neumann algebra and $\{ V_n\}_{n=0}^\infty $ be a sequence of isometries on the standard Hilbert space $L^2(N)$ such that for each $n$, the map $\Phi _n:N\ni x\mapsto V_n^*xV_n$ is a left inverse of an endomorphism of $N$. Then the following two conditions are equivalent.
The sequence of operators $\{ V_n\}_{n=1}^\infty $ converges to $V_0$ strongly.
We have $\|\psi \circ \Phi _n -\psi \circ \Phi _0\| \to 0$ for any $\psi \in N_*$.
This is shown by the same argument as that of the proof of Lemma 3.3 of Masuda–Tomatsu [@MT1].
\[22.5\] Let $\{ u_n\} $ be a sequence of unitaries of $\tilde{M}$ satisfying the following conditions.
We have $\mathrm{Ad}u_n \circ \tilde{\rho} \to \tilde{\sigma }$ as $n \to \infty$.
For any compact subset $F$ of $\mathbf{R}$, we have $\theta _t(u_n )-u_n \to 0$ uniformly for $t\in F$.
Then we have $\mathrm{Ad} u_n \circ \rho \to \sigma$.
By Lemmas \[18\], \[36\] and \[20\], it is enough to show that $V_{\tilde{\tilde{\rho}}}u_ n^*\to V_{\tilde{\tilde{\sigma }}}$. Notice that we have $$\begin{aligned}
\ &V_{\tilde{\tilde{\rho}} }u_n^*(\xi \otimes f) \\
&=(s\mapsto V_{\tilde{\rho}}(\theta _{-s}(u_n^*)(\xi ) )f(s))\end{aligned}$$ for any $\xi \in L^2(M)$ and $f\in L^2\mathbf{R}$. Hence we have $$\begin{aligned}
\ &\| V_{\tilde{\tilde{\rho}}}u_n^*(\xi \otimes f) -V_{\tilde{\tilde{ \sigma}} } (\xi \otimes f)\| ^2 \\
&=\int _{\mathbf{R}}\| V_{\tilde{\rho}}(\theta _{-s}(u_n^*)(\xi ))-V_{\tilde{\sigma}}(\xi ) \| ^2|f(s)|^2\ ds \\
&\leq \int _{\mathbf{R}} \| (V_{\tilde{\rho}}((\theta _{-s}(u_n^*) -u_n^*) (\xi ))\| ^2 |f(s)|^2 \ ds +\int _{\mathbf{R}} \| V_{\tilde{\rho}} (u_n^*(\xi )) -V_{\tilde{\sigma }}(\xi ) \| ^2 |f(s)|^2 \ ds \\
&\to 0\end{aligned}$$ by the Lebesgue dominant convergence theorem. Note that in order to show the last convergence, we use Lemmas \[R\], \[33\] and \[20\].
The proof of the main theorem
=============================
\[40\] Let $M$ be an AFD factor and $\sigma $ be a finite index endomorphism of $M$ with $d(\sigma )=d$. Then there exists an endomorphism $\lambda$ with the following properties.
The endomorphism $\lambda $ is approximately inner.
We have $d(\lambda )=d$.
The endomorphism $\lambda$ has Connes–Takesaki module and it is $\theta _{-\log d}|_{\mathcal{Z}(\tilde{M})}$.
By the proof of Theorem 3 of Kosaki–Longo [@KL], there exists an endomorphism $\lambda _0$ of the AFD factor of type $\mathrm{II}_1 $ with $d(\lambda _0)=d$. Then $\mathrm{id}_M \otimes \lambda _0$ is an endomorphism of $M$ with $d(\mathrm{id}\otimes \lambda _0)=d$ and with $\mathrm{mod}(\mathrm{id} \otimes \lambda _0)$ trivial. Hence by the existence of a right inverse of the Connes–Takesaki module of automorphisms (See Sutherland–Takesaki [@ST3]), there exists an automorphism $\alpha $ of $M$ with $\mathrm{mod}(\alpha \circ \lambda _0)=\theta _{-\log (d)}$. By Theorem 3.15 of Masuda–Tomatsu (or by the same argument of our paper), it is shown that $\lambda :=\alpha \circ \lambda _0$ is approximately inner.
Now, we return to the proof of the main theorem.
*Proof of implication $\Rightarrow $ of Theorem \[main\].* Let $\rho , \sigma $ be endomorphisms of $\mathrm{End}(M)_0$ with the first condition of Theorem \[main\]. Then by Lemma \[40\], there exist endomorphisms $\lambda , \mu \in \mathrm{End}(M)_0$ with the following properties.
\(1) We have $d(\lambda )=d(\sigma)$, $d(\mu ) =d(\rho )$.
\(2) We have $\tilde{\lambda}|_{\mathcal{Z}(\tilde{M})}=\theta _{-\log (d(\sigma ))}|_{\mathcal{Z}(\tilde{M})}$ and $\tilde{\mu}|_{\mathcal{Z}(\tilde{M})}=\theta _{-\log (d(\rho ))}|_{\mathcal{Z}(\tilde{M})}$.
\(3) The endomorphisms $\lambda $ and $\mu $ are approximately inner.
By the second condition, we have $$\begin{aligned}
\phi _{\tilde{\rho}}\circ \phi _{\tilde{\lambda}}|_{\mathcal{Z}(\tilde{M})} &=\phi _{\tilde{\rho}} \circ \theta _{\log d(\sigma)} |_{\mathcal{Z}(\tilde{M})} \\
&=\phi _{\tilde{\sigma }}\circ \theta _{-\log (d(\sigma )/d(\rho ))} \circ \theta _{\log d(\sigma)}|_{\mathcal{Z}(\tilde{M})} \\
&=\phi _{\tilde{\sigma}}\circ \theta _{\log (d(\rho))}|_{\mathcal{Z}(\tilde{M})} \\
&=\phi _{\tilde{\sigma }}\circ \phi _{\tilde{\mu }}|_{\mathcal{Z}(\tilde{M})}.\end{aligned}$$
Hence by replacing $\rho $ by $\lambda \circ \rho$ and $\sigma $ by $\mu \circ \sigma $ respectively, we may assume that $d(\rho)=d(\lambda )$ and $\phi _{\tilde{\rho}} |_{\mathcal{Z}(M)}=\phi _{\tilde{\sigma}} |_{\mathcal{Z}(M)}$. By Proposition \[R\], there exists a sequence $\{ u_n\}$ of unitaries of $\tilde{M}$ satisfying the assumptions of Lemma \[22.5\]. Hence by Lemma \[22.5\], we have $\mathrm{Ad}u_n \circ \rho \to \sigma $.
Appendix (A proof of the characterization of central triviality of automorphisms of the AFD factors)
====================================================================================================
In this section, we will see that it is possible to give a proof of a characterization theorem of central triviality of automorphisms of the AFD factors by a similar strategy to the proof of Theorem \[main\], which is independent of the types of the AFD factors.
Let $M$ be an AFD factor of type III. Let $\alpha $ be an automorphism of $M$ and $\tilde{\alpha}$ be its canonical extension. Set $$p:=\mathrm{min}\{ q\in \mathbf{N}\mid \tilde{\alpha }^q \ \mathrm{is} \ \mathrm{centrally} \ \mathrm{trivial}\} ,$$ $$G:=\mathbf{Z}/p\mathbf{Z}.$$
The action $\{ \tilde{\alpha }_n \circ \theta _t\}_{(n,t)\in G\times \mathbf{R}}$ of $G\times \mathbf{R}$ on $\tilde{M}_{\omega , \theta }$ is faithful.
We will show this lemma by contradiction. Let $\varphi $ be a normal faithful state of $\tilde{M}$ and $\{ \psi _j\}_{j=1}^\infty$ be a norm dense sequence of the unit ball of $\tilde{M}_*$. Assume that there existed a pair $(n ,t)\in (G\times \mathbf{R})\setminus \{(0,0)\} $ satisfying $\tilde{\alpha }_n \circ \theta _{-t}(a)=a$ for any $a\in \tilde{M}_{\omega , \theta }$. Then the automorphism $\tilde{\alpha }_n \circ \theta _{-t}$ would be centrally non-trivial because $\tilde{\alpha }_n \circ \theta _{-t}$ is trace-scaling if $t\not =0$. Hence there would exist an $x$ of $\tilde{M}_\omega $, which can never be of $\tilde{M}_{\omega, \theta }$, with $\tilde{\alpha} _n (x)\not =\theta _t(x)$ and with $\| x\|\leq 1$. Take a representing sequence $\{ x_k\}$ of $x$ with $\| x_k\| \leq 1 $ for any $k$. Then we would have $$\begin{aligned}
\ & \lim _{k\to \omega }\| \tilde{\alpha} _n (x_k) -\theta _t(x_k)\| _{\varphi \circ \theta _s}^\sharp \\
&=\mathrm{weak}\lim _{k \to \omega }\frac{1}{2} ( |\tilde{\alpha } _n (x_k) -\theta _t(x_k )|^2 +|(\tilde{\alpha }_n (x_k)-\theta _t(x_k))^*|^2 ) \\
& =2\delta >0\end{aligned}$$ for some $\delta >0$ (The constant $\delta $ does not depend on the choice of $s\in \mathbf{R}$). Then for each natural number $L$, there would exist $k \in \mathbf{N}$ satisfying the following three conditions.
\(1) We have $\| x_k\| \leq 1$.
\(2) We have $$\| \theta _t(x_k)\psi _j -\psi _j \theta _t(x_k)\| (=\| x_k (\psi _j\circ \theta _{t})-(\psi _j \circ \theta _{t})x_k\| )<\frac{1}{L}$$ for $j=1, \cdots , L$, $| t| \leq L$ (Use the compactness of $\{ \psi _j \circ \theta _t\mid t\in L\}$. See also the argument just after Lemma \[10\]).
\(3) We have $$\| \tilde{\alpha }_n (x_k)-\theta _t(x_k) \| _\varphi ^\sharp >\delta .$$
Let $\Theta :L^\infty ([-L,L], dm(t))\otimes (\tilde{M}, \varphi) \to (\tilde{M}_{\omega , \theta }, \varphi ^\omega )$ be the inclusion mentioned in Section 5 (an inclusion coming from the Rohlin property of $\theta$), where $dm(t)$ is the normalized Haar measure of $[-L,L]$. Set $$\tilde{y}:=([-L,L]\ni s\mapsto \theta _s(x_k)) \in L^\infty ([-L,L], dm(s))\otimes \tilde{M},$$ $$y:=\Theta (\tilde{y}).$$ Since we would have $\tilde{\alpha }_n\circ \theta _{-t}$ is trivial on $\tilde{M}_{\omega, \theta }$, we would have $$(\tilde{\alpha }_n (\Theta (f)))_s=\tilde{\alpha }_n (\Theta (f)_{s-t})$$ for $f\in L^\infty ([-L,L])\otimes \tilde{M}$ and $s\in [-L+t, L-t]$, where $f_s$ is the evaluation of the function $f$ at $s\in [-L,L]$. Hence we would have $$\begin{aligned}
\| \tilde{\alpha }_n (y) -y \| _{\varphi ^\omega }^\sharp &\geq (\int _{[-L+t,L-t]}(\| \tilde{\alpha }_n (\theta _{s-t}(x_k)) -\theta _{s}(x_k) \| _\varphi ^\sharp )^2 \ ds \\
&-\int _{[-L,-L+t]\cup [L-t,L]} 2^2 \ ds)^{1/2} \\
&\geq (\int _{[-L,L]}\delta ^2 \ ds -\frac{4t}{L})^{1/2} \\
&=(\delta ^2-\frac{4t}{L})^{1/2} .\end{aligned}$$ Since we have $$(\theta _r(y))_s=\theta _s(y)$$ for any $0<r<1$, $s\in [-L+r,L-r]$, we have $$\begin{aligned}
\| \theta _r(y)-y\| _{\varphi ^\omega}^\sharp &=(\int _{[-L,L]}(\| (\theta _r(y))_s-y_s \| _{\varphi }^\sharp )^2\ ds )^{1/2} \\
&\leq (\int _{[-L, -L+1]\cup [L-1,L]}2^2\ ds )^{1/2} \\
&=\frac{2}{\sqrt{L}}
\end{aligned}$$ for $| r|\leq 1$. We also have $$\begin{aligned}
\| [y,\psi _j ]\| &=\| [y, \psi _j]| _{\Theta (\mathbf{C}\otimes \tilde{M})} \| \\
&\leq \| [y, \psi _i ]|_{\Theta ( L^\infty ([-L,L])\otimes \tilde{M})} \| \\
&=\int _{[-L,L]} \| [ \tilde{y}_s , \psi _i ]\| \ ds \\
&= \int _{[-L,L]} \| [\theta _s(x_k), \psi _j]\| \ ds \\
&<\int _{[-L,L]} \frac{1}{L} \ ds \\
&=\frac{1}{L}
\end{aligned}$$ for $j=1, \cdots , L$. Hence there would exist a sequence $\{ y_l\}$ of $\tilde{M}$ with the following properties.
\(1) We have $\| y_l\| \leq 1$.
\(2) We have $\| [y_l, \psi _j]\| \to 0$ for any $j=1, 2, \cdots $.
\(3) For any $j=1,2, \cdots $, we have $\| \theta _r(y_l)-y_l\| _{\varphi }^\sharp \to 0$ uniformly for $s\in [-1,1]$.
\(4) We have $\| \tilde{\alpha }_n(y_l)-\theta _t(y_l)\| _{\varphi }^\sharp \geq \delta /2$ for any $l$.
This would contradict the assumption that $\tilde{\alpha }_n \circ \theta _{-t}$ were trivial on $\tilde{M}_{\omega , \theta }$.
For each $p\in \hat{(G\times \mathbf{R})}=\hat{G}\otimes \mathbf{R}$, there exists a unitary $u$ of $\tilde{M}_{\omega , \theta }$ with $\tilde{\alpha }_n \circ \theta _t(u)=\langle (n,t),p\rangle u$ for any $(n,t)\in G\times \mathbf{R}$.
The proofs of Theorems 4.10 and 7.7 of Masuda–Tomatsu [@MT5] works in our case.
There exist a non-zero projection $e$ of $(\tilde{M}_{\omega , \theta})^\theta $ with $\tilde{\alpha}(e)$ orthogonal to $e$.
By the previous lemma, for each natural number $l$, there exists a unitary $u$ of $\tilde{M}_{\omega , \theta }$ with $\tilde{\alpha }(u)=e^{2\pi i/p}u$ and with $\theta _t(u)=e^{-it/l}u$ for any $t$. Then there exists a spectral projection $e$ of $u$ with $\tilde{\alpha }(e)\leq 1-e$, $\tau ^\omega (e)=1/p$ and with $\tau ^\omega (e-\theta _t(e))\leq 1/(2l)$ for $|t|\leq 1$. By the usual diagonal argument, it is possible to choose a desired projection.
For an automorphism $\alpha $ of $M$, $\alpha $ is centrally trivial if and only if its canonical extension is inner.
First, assume that $\tilde{\alpha }$ is not centrally trivial. Then by the previous lemma, neither is $\tilde{\tilde{\alpha }}$. Hence neither is $\alpha $ centrally trivial (See, for example, Lemmas 5.11 and 5.12 of Sutherland–Takesaki [@ST]). The above argument means that if $\alpha $ is centrally trivial, then $\tilde{\alpha}$ is centrally trivial. Since $\tilde{M}$ is of type II, any centrally trivial automorphism of $\tilde{M}$ is inner. The opposite direction is trivial by the central triviality of a modular endomorphism group.
Finally, we remark that by our results and the result of Masuda [@M], if we admit that the AFD factors are completely classified by their flows of weights, it is possible to classify the actions of discrete amenable groups on the AFD factors without separating cases by the types of the factors.
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abstract: 'The CAIRNS (Cluster And Infall Region Nearby Survey) project is a large spectroscopic survey of the infall regions surrounding nine nearby rich clusters of galaxies. I describe the survey and use the kinematics of galaxies in the infall regions to estimate the cluster mass profiles. At small radii, these mass profiles are consistent with independent mass estimates from X-ray observations and Jeans analysis. I demonstrate the dependence of mass-to-light ratios on environment by combining these mass profiles with Two-Micron All-Sky Survey (2MASS) photometry. Near-infrared light is more extended than mass in these clusters, suggesting that dense cluster cores are less efficient at forming galaxies and/or more efficient at disrupting them. At large radii, galaxy populations in cluster infall regions closely resemble those in the field. The mass-to-light ratio at these radii should therefore be a good probe of the global mass-to-light ratio. The mass-to-light ratio in the infall region yields a surprisingly low estimate of $\Omega_m \sim 0.1$.'
date: '?? and in revised form ??'
title: The Distribution of Mass and Light in Cluster Infall Regions
---
Introduction
============
The relative distribution of matter and light in the universe is one of the outstanding problems in astrophysics. Clusters of galaxies, the largest gravitationally relaxed objects in the universe, are important probes of the distribution of mass and light. [@zwicky1933] first computed the mass-to-light ratio of the Coma cluster using the virial theorem and found that dark matter dominates the cluster mass. Recent determinations using the virial theorem yield mass-to-light ratios of $M/L_{B_j}\sim 250
{h M_\odot/L_\odot}$ (Girardi et al. 2000 and references therein). Equating the mass-to-light ratio in clusters to the global value provides an estimate of the mass density of the universe; this estimate is subject to significant systematic error introduced by differences in galaxy populations between cluster cores and lower density regions ([@cye97; @g2000]). Indeed, some numerical simulations suggest that cluster mass-to-light ratios exceed the universal value ([@diaferio1999; @kk1999; @bahcall2000]).
Determining the global matter density from cluster mass-to-light ratios therefore requires knowledge of the dependence of mass-to-light ratios on environment. Bahcall et al. (1995) show that mass-to-light ratios increase with scale from galaxies to groups to clusters. Ellipticals have larger overall values of $M/L_B$ than spirals, presumably a result of younger, bluer stellar populations in spirals. At the scale of cluster virial radii, mass-to-light ratios appear to reach a maximum value. Some estimates of the mass-to-light ratio on very large scales ($>$10${h^{-1}{\rm Mpc}}$) are available ([@bld95]), but the systematic uncertainties are large.
There are few estimates of mass-to-light ratios on scales between cluster virial radii and scales of 10${h^{-1}{\rm Mpc}}$ (Rines et al. 2000, Rines et al. 2001a, Biviano & Girardi 2003, Katgert et al. 2003, Kneib et al. 2003, Rines et al. 2004, Tully 2004 and references therein). On these scales, many galaxies near clusters are bound to the cluster but not yet in equilibrium ([@gunngott]). These cluster infall regions have received relatively little scrutiny because they are mildly nonlinear, making their properties very difficult to predict analytically. However, these scales are exactly the ones in which galaxy properties change dramatically (e.g., Ellingson et al. 2001, Lewis et al. 2002, Gomez et al. 2003, Treu et al. 2003, Balogh et al. 2004, Gray et al. 2004). Variations in the mass-to-light ratio with environment could have important physical implications; they could be produced either by a varying dark matter fraction or by variations in the efficiency of star formation with environment. In blue light, however, higher star formation rates in field galaxies could produce lower mass-to-light ratios outside cluster cores resulting only from the different contributions of young and old stars to the total luminosity ([@bahcall2000; @tully]).
Galaxies in cluster infall regions produce sharp features in redshift surveys. Early investigations of this infall pattern focused on its use as a direct indicator of the global matter density $\Omega_m$. Unfortunately, random motions caused by galaxy-galaxy interactions and substructure within the infall region smear out this cosmological signal (Diaferio & Geller 1997, Vedel & Hartwick 1998). Instead of sharp peaks in redshift space, infall regions around real clusters typically display a well-defined envelope in redshift space which is significantly denser than the surrounding environment ([@cairnsi], hereafter Paper I, and references therein).
Diaferio & Geller (1997) and Diaferio (1999) analyzed the dynamics of infall regions with numerical simulations and found that in the outskirts of clusters, random motions due to substructure and non-radial motions make a substantial contribution to the amplitude of the caustics which delineate the infall regions. Diaferio & Geller (1997) showed that the amplitude of the caustics is a measure of the escape velocity from the cluster; identification of the caustics therefore allows a determination of the mass profile of the cluster on scales $\lesssim 10{h^{-1}{\rm Mpc}}$.
Diaferio & Geller (1997) and Diaferio (1999) show that nonparametric measurements of caustics yield cluster mass profiles accurate to $\sim$50% on scales of up to 10 $h^{-1}$ Mpc. This method assumes only that galaxies trace the velocity field. Indeed, simulations suggest that little or no velocity bias exists on linear and mildly non-linear scales ([@kauffmann1999a; @kauffmann1999b]). The caustic method has been applied to systems as large as the Shapley Supercluster ([@rqcm]) and as small as the Fornax cluster ([@drink]) as well as to many nearby clusters (Paper I). Biviano & Girardi (2003) applied the caustic technique to an ensemble cluster created by stacking redshifts around 43 clusters from the 2dF Galaxy Redshift Survey. Rines et al. (2000) found an enclosed mass-to-light ratio of $M/L_R \sim 300 h$ within 4$~{h^{-1}{\rm Mpc}}$ of A576. Rines et al. (2001) used 2MASS photometry and the mass profile from Geller et al. (1999) to compute the mass-to-light profile of Coma in the K-band. They found a roughly flat profile with a possible decrease in $M/L_K$ with radius by no more than a factor of 3. Biviano & Girardi (2003) find a decreasing ratio of mass density to total galaxy number density. For early-type galaxies only, the number density profile is consistent with a constant mass-to-light (actually mass-to-number) ratio.
Here, we calculate the infrared mass-to-light profile within the turnaround radius for the CAIRNS clusters (Paper I), a sample of nine nearby rich, X-ray luminous clusters. We use photometry from 2MASS, the Two Micron All Sky Survey ([@twomass]) and add several new redshifts to obtain complete or nearly complete surveys of galaxies up to 1-2 magnitudes fainter than $M^*_{K_s}$ (as determined by Cole et al. 2001 and Kochanek et al. 2001). Infrared light is a better tracer of stellar mass than optical light; it is relatively insensitive to dust extinction and recent star formation. Despite these advantages, there are very few measurements of infrared mass-to-light ratios in clusters ([@tustin; @rines01a; @lin03; @cairnsii]).
Observations {#sec:observations}
============
The CAIRNS clusters are 8 of the 14 Abell clusters that are: nearby ($cz<15,000~{{\rm km~s}^{-1}}$), Abell richness class $R\geq1$, X-ray luminous ($L_X>2.5 \times 10^{43} h^{-2}$erg s$^{-1}$) galaxy clusters with declination $\delta>-15^\circ$. Between 1997 and 2003, we collected 5607 redshifts (both new and remeasured) in large areas around nearby clusters with the FAST spectrograph on the FLWO 1.5-m telescope in Arizona. We targeted galaxies within a projected radius of $\sim$10${h^{-1}{\rm Mpc}}$ of the clusters, selecting targets first from POSSII 103aE plates and later from 2MASS when it became available. The redshift catalogs in Rines et al. (2004) complete $K_s$ band magnitude-limited samples extending 1-2 magnitudes fainter than the characteristic magnitude $M_{K_s}^*$. These samples include $\sim$60-85% of the total light in the clusters and their infall regions.
Mass Profiles {#sec:massprof}
=============
Figure \[fig:caustics\] shows the redshifts of galaxies surrounding nearby clusters as a function of projected radius (normalized to $r_{200}$, the radius within which the average density is 200 times the critical density). The caustic pattern (a dense envelope in phase space with well-defined edges) is evident in the combined cluster as well as in each of the 9 clusters. Using the phase space distribution of galaxies in cluster infall regions, we apply the kinematic mass estimator of D99 to these clusters. The resulting mass profiles agree well with NFW (Navarro et al. 1997) and Hernquist (1990) models, but exclude an isothermal sphere.
These mass profiles agree with X-ray masses at small radius as well as virial masses (after correction for the surface pressure term) at slightly larger radii ([@rines02], Paper I). The latter is primarily a consistency check as the caustic technique utilizes the same kinematic data as the virial theorem. This consistency can be further demonstrated with the velocity dispersion profiles (see Paper I and Rines et al. 2004).
Near-Infrared Luminosity Functions {#sec:klf}
==================================
When using the mass-to-light ratio in clusters to estimate $\Omega_m$, one must determine whether the luminosity functions of field and cluster galaxies differ significantly. Given the well-known morphology-density relation, it is possible that the two LFs differ significantly. We use 2MASS photometry to determine the near-infrared luminosity functions for the CAIRNS clusters and infall regions ([@cairnsii]). The cluster and infall region LFs are very similar to each other (Figure \[fig:klfncomp\]) and to the field LF for relatively bright galaxies ($M_{K_s}\lesssim M_{K_s}^*+2$). Because of this similarity, we can correct for the luminosity in faint galaxies using the field LF. Our redshift surveys include 60–85% of the total light.
Near-Infrared Mass-to-Light Profiles {#sec:mlprof}
====================================
Both the surface number density and surface luminosity density profiles of cluster/infall region members are more extended than the mass profiles of Paper I. Figure \[fig:mlkprof\] shows the mass-to-light profiles of the CAIRNS clusters. The mass-to-light profiles are either flat or show a decreasing $M/L_K$ with increasing radius. The mean mass-to-light ratio within $r_{200}$ is a factor of $1.8\pm0.3$ larger than the mean value outside $r_{200}$.
The decreasing mass-to-light profiles could be caused by gradients in stellar populations with radius; such effects have been invoked to account for a similar result for B band mass-to-light profiles in simulations ([@bahcall2000]). In K band, however, changes in stellar populations with radius are expected to change the mean stellar mass-to-light ratio by $\lesssim 20\%$. Thus, the CAIRNS mass-to-light profiles provide tentative evidence for variations in the efficiency of galaxy formation and/or disruption. Environments with higher virial temperatures (like cluster centers) are more efficient at disrupting galaxies and/or less efficient at forming them.
A related trend has been noted by Lin et al. (2003, 2004), who show that K-band mass-to-light ratios within $r_{500}$ increase with increasing cluster mass. This result indicates that more massive clusters (with larger virial temperatures) have less efficient galaxy formation and/or more efficient galaxy disruption. We confirm this trend in Rines et al. (2004). Because cluster infall regions should be composed of galaxies inhabiting less massive systems ([@rines01b; @rines02]) and/or regions with lower virial temperatures, the above trend predicts that cluster infall regions should have smaller mass-to-light ratios than virial regions, consistent with the CAIRNS results.
Figure \[fig:combomlk\] shows the mass-to-light profile of the combined CAIRNS cluster (normalized to unity at $r_{200}$). The mass-to-light profile clearly decreases with radius. The combined cluster should be less susceptible to substructure than the individual clusters. Figure \[fig:combomlk\] shows that the mass-to-light ratio in shells decreases by about a factor of 2 from the virial region to the infall region, consistent with the results for individual clusters. Taking the mass-to-light ratio in cluster infall regions as an estimate of the global value and the SDSS luminosity density extrapolated to $K_s$ band ([@blanton03]), we estimate $\Omega_m=0.10\pm0.02$ (statistical). This estimate is somewhat lower if we use the $K_s$ band luminosity density of either Cole et al. (2001) or Kochanek et al. (2001). We discuss potential systematic effects in Rines et al. (2004). Most of these effects would flatten the observed mass-to-light profiles relative to the true profiles, suggesting that the observed decrease in mass-to-light ratio with radius is real.
Conclusions {#sec:conclusions}
===========
Cluster infall regions contain more galaxies than their virial regions. If currently popular cosmological models are correct, the mass in infall regions will eventually accrete onto the parent clusters, and their final masses will increase by a factor of about 2.
Near-infrared luminosity functions depend only weakly on environment, at least at the bright end. We show that near-infrared light is more extended than mass in cluster infall regions, suggesting environmental dependence of the efficiency of galaxy formation and/or disruption. If more efficient galaxy disruption is responsible, intracluster stars might be a significant component of stellar mass in clusters. The mass-to-light ratios in infall regions suggest a low $\Omega_m\sim
0.1$. Future work is needed to determine the significance of the conflict of this result with the currently favored $\Omega_m\sim 0.3$.
I would like to thank Margaret Geller and Antonaldo Diaferio for their many contributions as my primary collaborators in this work. I would also like to thank Tom Jarrett, Michael Kurtz, Joe Mohr, Gary Wegner, and John Huchra for their contributions to the CAIRNS project. Perry Berlind, Mike Calkins, and Susan Tokarz collected and reduced most of the spectroscopic data. I thank the Smithsonian Institution for support.
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|
---
abstract: 'We present 10 and 20$\mu$m images of IRAS 19500-1709 taken with the mid-infrared camera, OSCIR, mounted on the Gemini North Telescope. We use a 2-D dust radiation transport code to fit the spectral energy distribution from UV to sub-mm wavelengths and to simulate the images.'
author:
- 'K.L. Clube, T.M. Gledhill'
title: 'Mid-Infrared Imaging and Modelling of the Dust Shell around Post-AGB star HD 187885 (IRAS 19500-1709)'
---
Introduction
============
The circumstellar envelopes (CSEs) of post-AGB stars are cool and dust rich and hence radiate most of their energy at mid-infrared wavelengths. IRAS 19500-1709 is associated with the high galactic latitude F2-3I (Parthasarathy, Pottasch & Wamsteker 1988) post-AGB star HD 187885 and has the double-peaked spectral energy distribution (SED) typical of post-AGB stars with detached CSEs (Hrivnak, Kwok & Volk 1989). The expansion velocity of the envelope, based on CO-line emission, is 11 km s$^{-1}$ with wings up to 30 km s$^{-1}$ (Likkel et al. 1987). The envelope is carbon rich, showing no OH or H$_{2}$O maser emission (Likkel 1989). It has a broad emission feature from 10-13$\mu$m peaking at 12$\mu$m which could be attributed to a polycyclic aromatic hydrocarbon (PAH), such as chrysene (Justtanont et al. 1996). It has a weak 21$\mu$m feature (Justtanont et al. 1996) and a broad feature around 30$\mu$m which has been modelled with non-spherical magnesium sulphide (MgS) dust grains (Hony, Waters & Tielens 2002). Imaging polarimetry at near-infrared wavelengths shows that IRAS 19500-1709 has a bipolar structure in scattered light (Gledhill et al. 2001). We present mid-infrared imaging of IRAS 19500-1709 using the OSCIR camera mounted on the 8.1-m Gemini North Telescope which provides information on the inner part of the CSE. We fit the spectral energy distribution (SED) from UV to sub-mm wavelengths and simulate the images using a 2-D dust radiation transport (RT) code in order to derive the physical and chemical properties of the dust in the CSE.
Observations
============
Using the OSCIR camera on Gemini North, we obtained 10 and 20$\mu$m (N and Q3 band) images of IRAS 19500-1709 with spatial resolution of 0.5 arcsecs (N band) to gain information on the inner part of the CSE (Fig.1). The object is extended relative to the point spread function (PSF).
The N band suggests the object is elongated in a N-S direction. The outer contours appear rounder in the Q band image but there is evidence of elongation in the centre in a N-S direction.. Also the brightness peak in the N-band image appears approximately 0.3 arcsecs off centre to the east, relative to the outer contours.
Modelling of the Dust Shell
===========================
We modelled the images and SED of IRAS 19500-1709 using a 2-D axisymmetric RT dust code (Efstathiou & Rowan-Robinson 1990) to derive the physical and chemical properties of dust in the envelope. The SED is constrained by the available observational data, for IRAS 19500-1709, taken from the literature which includes a 7.6-23.6$\mu$m UKIRT spectrum (CGS3) provided by K.Justannont, optical and near-IR photometry (Hrivnak, Kwok & Volk 1989), 8.5, 10 and 12.2$\mu$m flux (Meixner et al. 1997), IRAS fluxes, IUE data[^1], ISO SWS[^2] and LWS[^3] spectra and SCUBA flux (Van der Veen et al. 1994). Fig. 2 shows our best fit to date.
In this model we use a spherically symmetric shell with dust composed of 0.01-2$\mu$m amorphous carbon (amC), silicon carbide (SiC) and MgS grains. The amC dust fits the continuum of the SED quite well. We include SiC to account for the presence of the infrared emission feature from 10-13$\mu$m. We attempted to fit this feature using PAHs but found that they produced emission features in the 16-25$\mu$m region of the spectrum which are not seen in the data. The MgS is included as being the possible carrier of the 30$\mu$m band, although it is possible that this feature is due to an unknown carbonaceous component. We used optical constants for Mg$_{x}$Fe$_{(1-x)}$S from Begemann et al ((1994) with x=0.9 as an approximation to MgS (also see Hony et al. 2003). These data do not extend below 10$\mu$m however, so it was necessary to approximate the optical and UV extinction cross-sections to ensure that the MgS grains absorb sufficient radiation at short wavelengths to radiate in the mid-IR. This was done by adopting the optical constants of graphite below 4$\mu$m and interpolating between 4 and 10$\mu$m to produce a smooth extinction curve. We assume all dust grains are spherical with a size distribution of n(a)$\alpha$a$^{-q}$ with q=5. The dust is composed of 30% amC (temp. 87-167K), 40% SiC (temp. 79-150K) and 30% MgS (temp. 59-121K). The central star is assumed to be a black body with T=7500K. The ratio of the stellar radius to the inner radius of the shell is determined to be 9x10$^{-5}$ in our model and A$_{v}$=1.5. The ratio of the inner shell radius to the outer shell radius is 0.2 which produces sufficient far-IR emission to fit the data longward of 100$\mu$m. Extending the outer boundary any further than this produces too much far-IR emission.
In Fig.3a we show a raw model image corresponding to the best fit envelope parameters. In Fig.3b the model image is smoothed using a gaussian filter to simulate the OSCIR N-band image. The degree of smoothing (FWHM of the Gaussian) is determined by matching profiles between the model and OSCIR images. A gaussian smoothing function with a FWHM of 40 pixels provides the best match. This corresponds to a model pixel size of 0.01 arcsec. Using this procedure we estimate that the inner radius of the detached shell is 0.2 arcsec.
Future Work
===========
There appears to be excess emission at 7-10$\mu$m as well as the broad feature around 30$\mu$m. We have attempted to fit this but the MgS peak is too narrow. This may be due to the MgS grains being too cold or because we have used spherical grains. Variations in grain shape can influence the emission profile with elliptical grains causing a wider peak (Hony, Waters & Tielens 2002). We will be running axisymmetric models to determine whether we can reproduce the elongation seen in the OSCIR images.
Begemann, B. et al. 1994, , 423, L71 Efstathiou, A; & Rowan-Robinson, M. 1990, , 245, 275 Gledhill, T. M. et al. 2001, , 322, 321 Hony, S; Waters, L. B. F. M; & Tielens, A. G. G. M. 2002, , 390, 533 Hony, S. et al. 2003, , 402, 211 Hrivnak, B. J; Kwok, S; & Volk, K. M. 1989, , 346, 265 Justtanont, K. et al. 1996, , 309, 612 Likkel, L. et al. 1987, , 173, L11 Likkel, L. 1989, , 344, 350 Meixner, M. et al. 1997, , 482, 897 Parthasarathy, M; Pottasch, S. R; & Wamsteker, W. 1988, , 203, 117 Van der Veen, W. E. C. J. et al. 1994, , 285, 551
[^1]: Based on INES data from the IUE satellite
[^2]: ISO SWS06 observing program rszczerba-PPN30
[^3]: ISO LWS01 observing program mbarlow-dust 3
|
---
abstract: 'Predicating macroscopic influences of drugs on human body, like efficacy and toxicity, is a central problem of small-molecule based drug discovery. Molecules can be represented as an undirected graph, and we can utilize graph convolution networks to predication molecular properties. However, graph convolutional networks and other graph neural networks all focus on learning node-level representation rather than graph-level representation. Previous works simply sum all feature vectors for all nodes in the graph to obtain the graph feature vector for drug predication. In this paper, we introduce a dummy super node that is connected with all nodes in the graph by a directed edge as the representation of the graph and modify the graph operation to help the dummy super node learn graph-level feature. Thus, we can handle graph-level classification and regression in the same way as node-level classification and regression. In addition, we apply focal loss to address class imbalance in drug datasets. The experiments on MoleculeNet show that our method can effectively improve the performance of molecular properties predication.'
author:
- |
Junying Li, Deng Cai, Xiaofei He\
State Key Laboratory of CAD&CG, College of Computer Science and Technology\
Zhejiang University\
[email protected], [email protected], [email protected]\
bibliography:
- 'graph-level.bib'
title: 'Learning Graph-Level Representation for Drug Discovery'
---
Introduction
============
Reducing the high attrition rates in drug development continues to be a key challenge for the pharmaceutical industry. When biological researches yield evidence that a particular molecule could modulate essential pathways to achieve therapeutic activity, the discovered molecule often fails as a potential drug for a number of reasons including toxicity, low activity, and low solubility[@waring2015analysis]. The primary problem of small-molecule based drug discovery is finding analogue molecules with increased efficacy and reduced safe risks to the patient to optimize the candidate molecule. Machine learning is a powerful tool of drugs virtual screening that can help drug developer eliminate unqualified candidate molecules quickly.
A molecule can be of arbitrary size and shape. However, most machine learning methods can only handle inputs of a fixed size. A wide-used conventional proposal is to utilize hand-crafted feature like ECFP[@rogers2010extended], Coulomb Matrix[@rupp2012fast] as input and feed it into conventional classifier like random forest and multi-layer perceptron. In the last few years, convolutional neural network clearly surpassed the conventional methods that use hand-crafted feature and SVM in object classification tasks[@krizhevsky2012imagenet]. It indicates that end-to-end learning with reasonable differentiable feature extractor may surpass the conventional two stage classifier. Following this idea, [@duvenaud2015convolutional] model a molecule as a graph, the nodes of which stand for atoms and the edges of which represent chemical bonds linking some of the atoms together, and propose graph convolutional network which directly takes the graph as input to learn the representations of molecules. Previous studies show that graph convolutional network broadly offers the best performance in most of the datasets in MoleculeNet[@wu2017moleculenet].
Molecular properties of drugs are almost macroscopic influences on human body, like efficacy and toxicity. In fact, unlike common graph-based application like web classification, citation network and knowledge graph, predicating drugs properties is graph-level classification and regression rather than node-level classification and regression. However, present graph neural networks, including graph convolutional network[@duvenaud2015convolutional], similar graph networks that take the original molecule as input[@altae2017low; @yao2017intrinsic; @schutt2017quantum; @gomes2017atomic], and other general graph neural networks designed for all types of graph[@li2015gated; @bruna2013spectral; @defferrard2016convolutional; @kipf2016semi; @niepert2016learning], all focus on learning node-level representation rather than graph-level representation. Previous works [@altae2017low; @wu2017moleculenet] simply sums all feature vectors for all nodes in the graph to obtain the graph representation for molecule properties classification and regression.
To learn better graph-level representation and handle graph-level classification and regression, we introduce a dummy super node that is connected with all nodes in the graph by a directed edge, as the representation of the graph. Thus, we can handle graph-level classification and regression in the same way as node-level classification and regression. The idea of taking the feature of one specific node as the feature of the graph has been proposed before[@scarselli2009graph], but earlier work simply assign the first atom in the atom-bond description as the specific node. In our work, the dummy super node is initialized to zero and get updated through graph convolutional networks simultaneously as the genuine nodes do. We modify the graph operations[@altae2017low] and apply node-level batch normalization to learn better features. We take the feature of the dummy super node as the feature of the graph and feed it into classifier.
Another problem of molecular properties predication is that the datasets of molecules are often unbalanced. That is, the positive samples may be only a small part of the total samples. Take HIV dataset[@hiv] as instance, there are only 1487 chemical compounds that are HIV active in the dataset of totally 41913 chemical compounds. Previous works[@altae2017low; @wu2017moleculenet] did not pay much attention to the fact of unbalanced data. To address the unbalanced data, we adopt focal loss[@lin2017focal] and show that it can effectively improve the classification performance in unbalanced molecular dataset.
We evaluated our model on several datasets of toxity, biological activities and solubility in MoleculeNet, including Tox21 ToxCast, PCBA, MUV, HIV, FreeSolv. The results show that our method could effectively improve the performance on molecular properties predication.
Related Work
============
Our work draws inspiration both from the field of molecular machine learning and graph neural network. In what follows, we provide a brief overview of previous works in both fields.
Molecule Mchine Learning
------------------------
Encoding molecules into fixed-length strings or vectors is a core challenge for molecular machine learning. [@rupp2012fast] introduce Coulomb Matrix as feature of molecules and apply a machine learning model to predict atomization energies of a diverse set of organic molecules. [@rogers2010extended] propose Extended-connectivity fingerprints (ECFP), a novel class of topological fingerprints for molecular characterization.
[@duvenaud2015convolutional] propose graph convolutional network that operates directly on graph, which could learn better representations than conventional method like ECFP. [@altae2017low] combine graph convolutional network with residual LSTM embedding for one-shot learning on drug discovery. [@yao2017intrinsic; @schutt2017quantum] apply neural network to predicate the molecular energy and reduced the predication error to 1 kcal/mol. [@gomes2017atomic] propose atomic neural network to predicate the binding free energy of a subset of protein-ligand complexes found in the PDBBind dataset.[@ramsundar2015massively] applied Massively multitask neural architectures to synthesize information from many distinct biological sources. [@wu2017moleculenet] introduce MoleculeNet, a large scale benchmark for molecular machine learning, which contains multiple public datasets, and establish metrics for evaluation and high quality open-source implementations.
Graph neural network
--------------------
Graph-based Neural networks have previously introduced in[@gori2005new; @scarselli2009graph] as a form of recurrent neural network. [@li2015gated] modify the graph neural network by using gated recurrent units and modern optimization techniques and then extend to output sequences. Spectral graph convolutional neural networks are introduced by [@bruna2013spectral] and later extended by [@defferrard2016convolutional] with fast localized convolutions. [@kipf2016semi] introduced a number of simplifications to spectral graph convolutinal neural network and improve scalibility and classification performance in large-scale networks. [@cao2016deep] propose a novel model for learning graph representations, which generates a low-dimensional vector representation for each vertex by capturing the graph structural information. [@niepert2016learning] propose a framework for learning convolutional neural networks for arbitrary graphs and applied it to molecule classification.
All above work focus on learning node-level representation in graph rather than graph-level representation. However, predicating molecular properties is in fact a problem of graph-level classification and regression. In this paper, we introduce a dummy super node that is connected to all other nodes by a directed edge to learn graph-level representation rather than simply using the sum of the vectors of all nodes as the graph-level representation[@duvenaud2015convolutional; @altae2017low].
Learning Graph-Level Representation
===================================
For most macroscopic molecular properties predication tasks, like efficacy and toxicity, we could neglect the detailed edge information and take a molecule as an undirected graph[@wu2017moleculenet]. We apply graph convolutional networks[@duvenaud2015convolutional] to learn representation of the atoms(nodes) in the molecule. To learn the representation of the molecule(graph), we introduce a dummy super node that is connected with all nodes in the graph by a directed edge. We modify the graph operation to help the dummy super node learn graph-level feature, and utilize neural network as a classifier for graph-level properties predication.
Graph Convolutional Network
---------------------------
\[fig::graph\]
{width="80.00000%"}
(0,0) (73,0)[GraphConv]{} (205,0)[GraphConv(Super Node)]{} (390,0)[GraphPool]{}
(90,28)[$\boldsymbol{v}$]{} (405,28)[$\boldsymbol{v}$]{} (237,60)[$\boldsymbol{S}$]{} (237,135)[$\boldsymbol{S}$]{} (396,60)[$\boldsymbol{S}$]{} (396,135)[$\boldsymbol{S}$]{}
(56,160) (54,150) (217,160) (212,150) (368,155)
A molecule could be modeled as a graph, the nodes of which stand for atoms and the edges of which represent chemical bonds linking some of the atoms together. In graph convolution network, for a specific node, we separately feed it and its neighbours into two dense layer, and calculate the sum as the new features of the node. The weights are shared between dense layers that operate on different node with the same degree. Formally, for a specific node $\boldsymbol{v}$, that have totally $d$ neighbours $\boldsymbol{n_i}(1 \leq i \leq d)$, the new feature of the node $\boldsymbol{v'}$ of graph convolution operation is formulated as $$\label{equ:node}
\boldsymbol{v'} = \boldsymbol{W_{self}^d v} + \sum_{1 \leq i \leq d}\boldsymbol{W_{nb}^d n_i} + \boldsymbol{b_d}$$
where $\boldsymbol{W_{self}^d}$ is the weight for self node, $\boldsymbol{W_{nb}^d}$ is the weight for neighbour node, $boldsymbol{b_d}$ is the bias. These weights vary from different degree $d$ of the specific node, for that the nodes with different number of neighbours are affected by neighbours in different ways. Since the output of graph convolution layer is also a graph, we could continually apply graph convolution layers to the graph. Thus, the receptive field of the network would be larger, and the nodes would be affected by further neighbours. The left thumbnail in figure \[fig::graph\] illustrates the graph convolution operation. The yellow arrows indicate the dense layer with weight $\boldsymbol{W_{nb}^d}$ for neighbours, representing the neighbours’ effect on the specific node. The orange arrows indicate the dense layer with weight $\boldsymbol{W_{self}^d}$ for the specific node self.
In analogy to pooling layers in convolutional neural networks, Graph Pooling is introduced by [@altae2017low]. Graph Polling is a simple operation that returns the maximum activation across the node and its neighbours. Simple illustration is shown in Figure \[fig::graph\]. With graph pooling, we could easily enlarge the receptive field without adding extra weights. However, graph pooling does not change the size of the graph as the pooling of CNN does to the features of images.
Although there is no real convolution operation in graph convolutional network, graph convolutional network is design for similar reason to convolutional neural network, the ’local geometry’ and reasonable weight sharing. Convolutional networks[@lecun1998gradient] apply convolutions to exploit the translation equivariance of image features. And the feature learned form CNN is translation equivariant to some extent. Graph convolutional network also focus on learning local feature, through sharing weight with dense layers operate on nodes with the same number of degree. Thus, the feature learned by graph convolutional is determined by local neighbours rather the global position of nodes. Such features are equivariant to permutation of atomic group.
\[fig::model\]
{width="94.00000%"}
(0,0) (75,0)[GC: Graph Convolution]{} (186,0)[GP: Graph Pooling]{} (280,0)[BN: Node-Level Batch Normalization]{}
(37,44)[GC+RELU+BN]{} (137,44)[GP]{}
(185,44)[GC+RELU+BN]{} (285,44)[GP]{}
(332,44)[GC+RELU+BN]{} (438,50)[Two-Layer]{} (438,38)[Classifier]{}
Dummy Super Node
----------------
Learning graph-level is a central problem of molecule classification and regression. Modern graph convolution networks often contain two graph convolution layers and two graph polling layers, the receptive field of the such model is too small, especially compared with the size of most drug compounds. Simply summing the node-level features can not enlarge the receptive of the model in a real sense. A naive way to enlarge the receptive field is simply applying more graph convolutional layers. However, such method makes model more likely to overfit and perform worse in validation and test subset for the lack of enough data in drug datasets.
In order to learn better graph-level representations, we introduce a dummy super node that is connected with all nodes in the graph by a directed edge, as the representation of the graph. Since the dummy super node is directly connected with all nodes in graph, it can easily learn global feature through one graph convolution layer. The directed edge pointed to the dummy super node from other genuine nodes, indicates that the dummy super node could learn features from all other genuine nodes, while none of the genuine nodes would be affected by the dummy super node. Consequently, the dummy super node could learn the global feature while the genuine nodes keep learning local features that is equivariant to permutation of atomic group. Since the feature of molecules is likely to be more complex than that of atoms, we could use a longer vector as the feature of the dummy super node.
Formally, we could formulate the new feature of the dummy super node $\boldsymbol{S'}$ of graph convolution layer as $$\label{equ:dummy}
\boldsymbol{S'} = \boldsymbol{W_{self} S} + \sum_{1 \leq i \leq N}\boldsymbol{W_{nb} v_i} + \boldsymbol{b}$$ where $\boldsymbol{W_{self}}$ is the weight for the dummy super node, $\boldsymbol{W_{nb}}$ is the weight for genuine node, $N$ is the number of nodes, $\boldsymbol{v_i}$ is the feature of $i$th node, $\boldsymbol{b}$ is the bias. The dummy super node $\boldsymbol{S}$ is initialized to zero and get updated through graph convolutional networks simultaneously as the genuine nodes do.
The new features of genuine nodes are not affected by the dummy super node, and keep the same as Equation \[equ:node\]. Graph pooling is only applied to genuine nodes, and the dummy super node is not regarded as neighbours of other nodes when graph pooling is applied.
Network Structure
-----------------
In addition to graph operations, we also apply node-level batch normalization, relu activation, tanh activation and dense layer in the model. For some datasets with imbalance data, we replace common cross entropy loss with focal loss. The whole network structure is shown in Figure \[fig::model\]. We add the dummy super node and its edges to the original graph before sending it to the network. In our model, We apply three graph convolution blocks(GC + RELU + BN) and two graph pooling layers, then we feed the feature of the dummy super node to a two-layer classifier. The batch normalization used there is node-level batch normalization that would be discussed later. Only the feature of the dummy super node is calculated in the third graph convolution block, for the new feature of other nodes would not be fed into following classifier.
Unlike the ResNet, we place relu activation before batch normalization, for the experiment shows that it is hard to train when we place relu activation behind. We argue that the molecule feature is different from that of images. The noise in images usually does not affect the image label, while that in molecules not.
### Node-Level Batch Normalization
Since the number of atoms vary from molecules, standard batch normalization[@ioffe2015batch] can not be applied to graph convolution network directly. We extend standard batch normalization to node-level batch normalization. We normalize the feature of each node, make it zero mean and unit variance. Since the dummy super node is also a node, we simply apply identical normalization to the dummy super node as well.
### Focal Loss
Data in drug datasets is often unbalanced. Take HIV dataset as instance, there are only 1487 chemical compounds that are HIV active in the dataset of totally 41913 chemical compounds. However, previous work[@altae2017low; @wu2017moleculenet] do not pay much attention to the fact of unbalanced data in Drug dataset.
Downsample is a simple method to tackle with unbalanced data. However, the number of data in drug datasets is always not large for the difficulty of acquiring data and get them labeled, and simple downsample may result in serious overfitting of neural network for the lack of sufficient data. In addition, Boosting is an effective resample method that may address imbalance data. But it’s not easy to integrate boosting with deepchem[@wu2017moleculenet], the modern chemical machine learning framework.
Focal Loss is an elegant and brief proposal that introduced by [@lin2017focal] to address class imbalance. It shows that focal loss could effectively improve the performance of one-stage detector in object detection. In fact, Focal Loss is a kind of reshaped cross entropy loss that the weighs of well-classified examples are reduced. Formally, Focal Loss is defined as
$$FL(p)=-(y(1-p)^\gamma \log p+(1-y)p^\gamma \log (1-p)).$$
where $y \in \{0,1\}$ specifies the ground-truth class, $p \in [0,1]$ is the model’s estimated probability for the class with label $y=1$, $\gamma$ is tunable focusing parameter. When $\gamma = 0$, focal loss is equivalent to cross entropy loss, and as $\gamma$ is increased the effect of the modulating factor is likewise increased. Focal loss focuses on training on a sparse set of hard examples and prevents the vast number of easy negatives from overwhelming the classifier during training.
Experiment
==========
We evaluate our model on several datasets of MoleculeNet, ranging from drug activity, toxity, solvation. Our code is released in Github[^1].
0.15in
-- ---------- ----------- ----------- ----------- ----------- ----------- -----------
$R^2$ RMSE MAE $R^2$ RMSE MAE
Index 0.935 0.909 0.703 0.941 0.963 0.738
Random 0.928 **0.982** **0.644** 0.895 1.228 0.803
Scaffold 0.883 2.115 1.555 0.709 2.067 1.535
Index **0.952** **0.787** **0.566** **0.945** **0.933** **0.598**
Random **0.933** 1.010 0.652 **0.910** **1.112** **0.659**
Scaffold **0.884** **2.076** **1.404** **0.746** **1.939** **1.415**
-- ---------- ----------- ----------- ----------- ----------- ----------- -----------
-0.1in
MoleculeNet
-----------
MoleculeNet is a dataset collection built upon multiple public databases, covering various levels of molecular properties, ranging from atomic-level properties to macroscopic influences on human body.
For chemical data, random splitting datasets into train/valid/test collections that is widely used in machine learning, is often not correct[@sheridan2013time]. Consequently, MoleculeNet implements multiple different splittings(Index Split, Random Split, Scaffold Split) for each dataset. Scaffold split attempts to separate structurally molecules in the training/validation/test sets, thus the scaffold split offers a greater challenge and demands a higher level of generalization ability for learning algorithms than index split and random split.
For quite a few datasets in MoleculeNet, the number of the positive samples and the number of negative samples is not balanced. Thus the accuracy metrics widely used in machine learning classification tasks are not suitable here. In MoleculeNet, classification tasks are evaluated by area under curve (AUC) of the receiver operating characteristic (ROC) curve, and regression tasks are evaluated by squared Pearson correlation coefficient (R2).
We pick following datasets(Tox21, ToxCast, HIV, MUV, PCBA, FreeSolv) on macroscopic chemical influences on human body from MoleculeNet, and evaluate our model.
### Tox21
The “Toxicology in the 21st Century” (Tox21) initiative created a public database measuring toxicity of compounds, which has been used in the 2014 Tox21 Data Challenge[@tox21]. Tox21 contains qualitative toxicity measurements for 8014 compounds on 12 different targets, including stress response pathways and nuclear receptors.
### ToxCast
ToxCast is another data collection (from the same initiative as Tox21) providing toxicology data for a large library of compounds based on in virtual high-throughput screening[@richard2016toxcast]. The processed collection in MoleculeNet contains qualitative results of over 600 experiments on 8615 compounds.
### MUV
The MUV dataset[@rohrer2009maximum] contains 17 challenging tasks for around 90,000 compounds and is specifically designed for virtual screening techniques. The positives examples in these datasets are selected to be structurally distinct from one another.
### PCBA
PubChem BioAssay (PCBA) is a database that consists of biological activities of small molecules generated by high-throughput screening[@wang2011pubchem]. The processed collection in MoleculeNet is a subset that contains 128 bioassays measured over 400,000 compounds.
### HIV
The HIV dataset was introduced by the Drug Therapeutics Program (DTP) AIDS Antiviral Screen, which tested the ability to inhibit HIV replication for 41,913 compounds[@hiv]. Screening results were evaluated and placed into three categories: confirmed inactive (CI), confirmed active (CA) and confirmed moderately active (CM). In MoleculeNet, the latter two labels are combined, making it a classification task between inactive (CI) and active (CA and CM).
### FreeSolv
The Free Solvation Database (FreeSolv) provides experimental and calculated hydration free energy of small molecules in water[@Mobley2014FreeSolv]. FreeSolv contains 643 compounds, a subset of which are also used in the SAMPL blind prediction challenge[@Mobley2014Blind].
Settings and Performance Comparisons
------------------------------------
We apply the identical network to all above datasets, except for the output channel number of the last layer that is determined by the number of tasks. For different datasets, we only tune the hyperparameter of training. We compare our model with logistic regression with ECFP feature as input and standard graph convolution network. To have a fair comparison, we reimplement graph convolution network, and most of the performance of our implements are much better than that reported by [@wu2017moleculenet].
### Classification Tasks
For classification tasks(Tox21, ToxCast, MUV, PCBA), we utilize the area under curve (AUC) of the ROC curve as metric(larger is better). The results are shown in Table \[tbl::tox21\], Table \[tbl::toxcast\], Table \[tbl::muv\] and Table \[tbl::pcba\]. Our model generally surpasses standard graph convolution network and conventional logistic regression in both small datasets on toxity(Tox21, ToxCast) and larger datasets on bioactivities(MUV, PCBA). It indicates that our method could generally improve the performance of graph convolution network. Compared with standard graph convolution network, in average, our model achieves an improvement of about 1.5% in the test subset of the four datasets.
0.15in
Model Split Method Train Valid Test
------- -------------- ----------- ----------- -----------
Index 0.903 0.704 0.738
Random 0.901 0.742 0.755
Scaffold 0.905 0.651 0.697
Index 0.945 0.829 0.820
Random 0.938 0.833 0.846
Scaffold 0.955 0.778 0.752
Index **0.965** **0.839** **0.848**
Random **0.964** **0.842** **0.854**
Scaffold **0.971** **0.788** **0.759**
: **The area under curve (AUC) of the ROC curve of various models in Tox21 dataset.** []{data-label="tbl::tox21"}
-0.1in
0.15in
Model Split Method Train Valid Test
------- -------------- ----------- ----------- -----------
Index 0.727 0.578 0.464
Random 0.713 0.538 0.557
Scaffold 0.717 0.496 0.496
Index 0.904 0.723 0.708
Random 0.901 0.734 0.754
Scaffold 0.914 0.662 0.640
Index **0.927** **0.747** **0.734**
Random **0.924** **0.746** **0.768**
Scaffold **0.929** **0.696** **0.657**
: **The area under curve (AUC) of the ROC curve of various models in ToxCast dataset.** []{data-label="tbl::toxcast"}
-0.1in
0.15in
Model Split Method Train Valid Test
------- -------------- ----------- ----------- -----------
Index 0.809 0.776 0.781
Random 0.808 0.772 0.773
Scaffold 0.811 0.746 0.757
Index 0.895 0.855 0.851
Random 0.896 0.854 0.855
Scaffold 0.900 0.829 0.829
Index **0.904** **0.869** **0.864**
Random **0.899** **0.863** **0.867**
Scaffold **0.907** **0.847** **0.845**
: **The area under curve (AUC) of the ROC curve of various models in PCBA dataset.** []{data-label="tbl::pcba"}
-0.1in
0.15in
Model Split Method Train Valid Test
------- -------------- ----------- ----------- -----------
Index 0.960 0.773 0.717
Random 0.954 0.780 0.740
Scaffold 0.956 0.702 0.712
Index 0.951 **0.816** 0.792
Random 0.949 0.787 0.836
Scaffold 0.979 0.779 0.735
Index **0.982** **0.816** **0.795**
Random **0.989** **0.800** **0.845**
Scaffold **0.990** **0.816** **0.762**
: **The area under curve (AUC) of the ROC curve of various models in MUV dataset.** []{data-label="tbl::muv"}
-0.1in
0.15in
Model Split Method Train Valid Test
------- -------------- ----------- ----------- -----------
Index 0.864 0.739 0.741
Random 0.860 0.806 0.809
Scaffold 0.858 0.798 0.738
Index 0.945 0.779 0.728
Random 0.939 0.835 0.822
Scaffold 0.938 0.795 0.769
Index 0.973 0.789 0.737
Random 0.951 0.842 0.830
Scaffold 0.967 0.813 0.763
Index **0.993** **0.793** **0.749**
Random **0.993** **0.843** **0.851**
Scaffold **0.992** **0.816** **0.776**
: **The area under curve (AUC) of the ROC curve of various models in HIV dataset.** With focal loss($\gamma = 2$), our method achieves further improvement.[]{data-label="tbl::hiv"}
-0.1in
### Regression Tasks
For regression tasks(FreeSolv), we utilize squared Pearson correlation coefficient($R^2$), RMSE, MAE as metrics. The experimental results are shown in Table \[tbl::freesolv\]. We only report the performance of graph convolution network and our model here. Our model has a better performance in general. We argue that our model failed to surpass graph convolution network in validation subset under random splitting for stochastic factors introduced by splitting method and too few numbers of compounds in FreeSolv.
It is interesting that how our method competes with classic ab-initio calculations. Hydration free energy that should be predicated in FreeSolv has been widely used as a test of computational chemistry methods. With energy values ranging from -25.5 to 3.4kcal/mol in the FreeSolv dataset, the RMSE of ab-initio calculations results reach up to 1.5kcal/mol[@Mobley2014Blind]. Our methods clearly outperform ab-initio calculations when we utilize index split and random split. However, our method fails to outperform ab-initio calculations when the dataset is split by scaffold split that separate molecules structurally. We argue that it is the insufficient data that results in weak generalization ability of our model. When fed with enough data, our model may overall surpass classic ab-initio calculations.
### imbalance Class
We apply focal loss in HIV dataset that have unbalanced class. The experimental results of Table \[tbl::hiv\] shown that focal loss could further improve the performance. It is interesting that our model overfits quickly in train dataset when we apply focal loss. We argue that focal loss helps our model handle hard example and fit better in train dataset.
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Conclusion
==========
In this paper, we point out that molecular properties predication demand graph-level representation. However, most of the previous works on graph neural network only focus on learning node-level representation. Such representation is clearly not sufficient for molecular properties predication. In order to learn graph-level representation, we propose the dummy super node that is connected with all nodes in the graph by a directed edge and modify the graph operation to help the dummy super node learn graph-level feature. Thus, we can handle graph-level classification and regression in the same way as node-level classification and regression. The experiment on MoleculeNet shows that our method could generally improve the performance of graph convolution network.
[^1]: <https://github.com/microljy/graph_level_drug_discovery>
|
---
abstract: 'Slotted ALOHA can benefit from physical-layer network coding (PNC) by decoding one or multiple linear combinations of the packets simultaneously transmitted in a timeslot, forming a system of linear equations. Different systems of linear equations are recovered in different timeslots. A message decoder then recovers the original packets of all the users by jointly solving multiple systems of linear equations obtained over different timeslots. We propose the batched BP decoding algorithm that combines belief propagation (BP) and local Gaussian elimination. Compared with pure Gaussian elimination decoding, our algorithm reduces the decoding complexity from cubic to linear function of the number of users. Compared with the ordinary BP decoding algorithm for low-density generator-matrix codes, our algorithm has better performance and the same order of computational complexity. We analyze the performance of the batched BP decoding algorithm by generalizing the tree-based approach and provide an approach to optimize the system performance.'
author:
- '[^1]'
title: 'Coding for Network-Coded Slotted ALOHA'
---
Introduction
============
In a wireless multiple-access network operated with the slotted ALOHA access protocol, a number of users transmit messages to a sink node through a common wireless medium. Time is divided into discrete slots and all transmissions start at the beginning of a timeslot. Collisions/interferences occur when more than one user transmits in the same timeslot [@roberts1975aloha]. Successive interference cancellation (SIC) was proposed for slotted ALOHA to resolve collisions so that signals contained in collisions can be leveraged to increase throughput [@casini2007contention]. In this approach, a number of timeslots are grouped together as a frame. Each user aims to deliver at most one packet per frame, but it can transmit copies of the same packet in different timeslots of the frame.
To see the essence, suppose we have two users and the first user transmits two copies of packet $v_1$ in timeslots $1$ and $2$ respectively, and the second user transmits one copy of packet $v_2$ in timeslots $2$. Since no collision occurs, $v_1$ can be correctly decoded by the sink node in timeslot $1$. A collision occurs in timeslot 2. In SIC, the sink node can use $v_1$ decoded from timeslot 1 to cancel the interference in timeslot 2. This approach can be applied iteratively to cancel more interference, in a manner similar to the belief propagation (BP) decoding of LT codes over erasure channels. Slotted ALOHA with SIC has been extensively studied based on the AND-OR-tree analysis, and optimal designs have been obtained [@liva2011graph; @paolini2011high; @stefanovic2012frameless; @narayanan2012iterative; @ghanbarinejad2013irregular].
*Physical-layer network coding (PNC)* [@Zhang06] (also known as *compute-and-forward* [@Nazer11]) is recently applied to wireless multiple-access network to improve the throughput [@lu14n; @ncma14; @cocco2014]. Such multiple-access schemes, called *network-coded multiple access (NCMA)*, employ both PNC and multiuser decoders at the physical layer to decode one or multiple linear combinations of the packets simultaneously transmitted in a timeslot. Specifically, Lu, You and Liew [@lu14n] demonstrated by a prototype that a PNC decoder may sometimes successfully recover linear combinations of the packets when the traditional multiuser decoder (MUD) [@verdu1998multiuser] that does not make use of PNC fails. In the existing works on PNC (or compute-and-forward), the decoding of the XOR of the packets of two users has been extensively investigated [@shengli09; @wubben10] (see also the overview [@liew2013physical]). The decoding of multiple linear combinations over a larger alphabet has been studied in [@Nazer11; @feng2013algebraic; @zhu2014isit]. In this paper, we consider slotted ALOHA employing PNC (and MUD) at the physical layer, called *network-coded slotted ALOHA (NCSA)*. We assume that the physical-layer decoder at the sink node can reliably recover one or multiple linear combinations of the packets transmitted simultaneously in one timeslot. Our work in this paper does not depend on a specific PNC scheme. Specifically, we consider a $K$-user NCSA system, where each user has one input packet to be delivered over a frame of timeslots. A packet is the smallest transmission unit, which cannot be further separated into multiple smaller transmission units. But it is allowed to send multiple copies of a packet in different timeslots. The number of copies, called the *degree*, is independently sampled from a *degree distribution*. The linear equations decoded by the physical layer in a timeslot form a system of linear equations. Different systems of linear equations are recovered in different timeslots. To recover the input packets of users, a message decoder is then required to jointly solve these systems of linear equations obtained over different timeslots. Though Gaussian elimination can be applied to solve the input packets, the computational complexity is $O(K^3+K^2T)$ finite-field operations, where $T$ the number of field symbols in a packet. In this paper, we study the design of NCSA employing an efficient message decoding algorithm.
With the possibility of decoding more than one linear combination of packets in a timeslot, the coding problem induced by NCSA becomes different from that of slotted ALOHA with SIC. We will show by an example that the ordinary BP decoding algorithm of LT codes over erasure channels is not optimal for NCSA. We instead propose a *batched BP decoding algorithm* for NCSA, where Gaussian elimination is applied locally to solve the linear system associated with each timeslot, and BP is applied between the linear systems obtained over different timeslots. The computational complexity of our algorithm is $O(KT)$ finite-field operations, which is of the same order as the ordinary BP decoding algorithm. We analyze the asymptotic performance of the batched BP decoding algorithm when $K$ is large by generalizing the tree-based approach in [@luby98]. We provide an approach to optimize the degree distribution based on our analytical results.
Though the batched BP decoding is similar to the one proposed for NCMA [@yang14l; @yang14c], we cannot apply the analysis therein. In NCMA, we assume that the number of users is fixed but the number of packets to be delivered by each user tends to infinity. In NCSA, each user has only one packet while the number of users can be large.
[[Similar schemes have been developed for random linear network coding over finite fields without explicitly considering the physical-layer effect, e.g., BATS codes and chunked codes (see [@yang14bats; @yang14d] and the references therein). Here the technique for NCSA is different from BATS (or chunked) codes in two aspects. First, in BATS codes the degree distribution of batches is the parameter to be optimized, while in NCSA the degree distribution of the input packets (variable nodes) is the parameter to be optimized. Second, the decoding of BATS codes only solves the associated linear system of a batch when it is uniquely solvable (and hence recovers all the input packets involved in a batch), while the decoding of NCSA processes the associated linear system of a batch even when it is not uniquely solvable.]{}]{}
In the remainder of this paper, Section \[sec:netw-coded-slott\] formally introduces NCSA and presents our main analytical result (Theorem \[the:1\]). An outline of the proof of the theorem is given in Section \[sec:performance-analysis\]. An example is provided in Section \[sec:example\] to demonstrate the degree distribution optimization and the numerical results.
Network-Coded Slotted ALOHA {#sec:netw-coded-slott}
===========================
In this section, we introduce the model of network-coded slotted ALOHA (NCSA), the message decoding algorithm and the performance analysis results.
Slotted Transmission
--------------------
Fix a *base field* $\mathbb{F}_q$ with $q$ elements and an integer $m>0$. Consider a wireless multiple-access network where $K$ source nodes (users) deliver information to a sink node through a common wireless channel. Each user has one input packet for transmission, formulated as a column vector of $T$ symbols in the extension field $\mathbb{F}_{q^m}$. All the users are synchronized to a *frame* consisting of $n$ timeslots of the same duration. The transmission of a packet starts at the beginning of a timeslot, and the timeslots are long enough for completing the transmission of a packet. Each user transmits a number of copies of its input packet within the frame. The number of copies transmitted by a user, called the degree of the packet, is picked independently according to a *degree distribution* $\Lambda=(\Lambda_1,\ldots,\Lambda_{D})$, where $D$ is the maximum degree. That is, with probability $\Lambda_d$, a user transmits $d$ copies of its input packet in $d$ different timeslots chosen uniformly at random in the frame. Let $\bar \Lambda = \sum_{i=1}^Di\Lambda_i$, $\Lambda(x) = \sum_{i} \Lambda_ix^i$ and $\Lambda'(x)=\sum_{i}i\Lambda_ix^{i-1}$. We also call $\Lambda(x)$ a degree distribution.
Denote by $v_i$ the input packet of the $i$-th user. Fix a timeslot. Let $\Theta$ be the set of indices of the users who transmit a packet in this timeslot. The elements in $\Theta$ are ordered by the natural order of integers. We assume that certain PNC scheme is applied, so that the physical-layer decoder of the sink node can decode multiple *output packets*, each being a linear combination of $v_s,
s\in\Theta$ with coefficients over the base field ${\mathbb{F}}_q$. Suppose that $B$ output packets are decoded ($B$ may vary from timeslot to timeslot). The collection of $B$ linear combinations can be expressed as $$\label{eq:batch}
[u_1,\ldots,u_B] = [v_s,s\in\Theta] {\mathbf{H}},$$ where ${\mathbf{H}}$ is a $|\Theta|\times B$ full-column-rank matrix over ${\mathbb{F}}_q$, called the *transfer matrix*, and $[v_s,s\in\Theta]$ is the matrix formed by juxtaposing the vectors $v_s$, where $v_{s'}$ comes before $v_{s''}$ whenever $s'<s''$.
Note that in , the algebraic operations are over the field ${\mathbb{F}}_{q^m}$. We call the set of packets $\{u_1,\ldots,u_B\}$ decoded in a timeslot a *batch*. The cardinality of $\Theta$ (the number of users transmitting in a timeslot) is called the *degree of the batch/timeslot*. [[ We call the ratio $K/n$ the *design rate* of NCSA.]{}]{}
\[lemma:1\] When $K/n \rightarrow R$ as $K\rightarrow \infty$, the degree of a timeslot converges to the Poisson distribution with parameter $\lambda = R\bar \Lambda$ as $K\rightarrow \infty$.
This is a special case of Lemma \[lemma:1a\] to be proved later in this paper.
Denote by $\mathcal{H}_{d}$ the collection of all the full-column-rank, $d$-row matrices over ${\mathbb{F}}_q$, [[where we assume that the empty matrix, representing the case that nothing is decoded, is an element of $\mathcal{H}_{d}$.]{}]{} For a timeslot of degree $d$, we suppose that the transfer matrix of the batch is ${\mathbf{H}}\in
\mathcal{H}_d$ with probability $g({\mathbf{H}}|d)$. Further, we consider all the users are symmetric so that for any $d\times d$ permutation matrix $\mathbf{P}$, $$\label{eq:4}
g({\mathbf{H}}|d) = g(\mathbf{P}{\mathbf{H}}|d).$$ The transfer matrices of all timeslots are independently generated given the degrees of the timeslots. Examples of the distribution $g$ will be given in Section \[sec:example\].
[[We say a rate $R$ is *achievable* by the NCSA system if for any $\epsilon>0$ and all sufficiently large $n$, at least $n(R-\epsilon)$ input packets are decoded correctly from the receptions of the $n$ timeslots with probability at least $1-\epsilon$.]{}]{}
Belief Propagation Decoding
---------------------------
For multiple access described above, the goal of the sink node is to decode as many input packets as possible during a frame. From the output packets of the $n$ timeslots decoded by the physical layer, the original input packets can be recovered by solving the linear equations of all the timeslots jointly. Gaussian elimination has a complexity $O(K^3+K^2T)$ finite-field operations when $n = O(K)$, which makes the decoding less efficient when $K$ is large.
The output packets of all the timeslots collectively can be regarded as a low-density generator matrix (LDGM) code. Similar to decoding an LT code, which is also a LDGM code, we can apply the (ordinary) BP algorithm to decode the output packets. In each step of the BP decoding algorithm, an output packet of degree one is found, the corresponding input packet is decoded, and the decoded input packet is substituted into the other output packets in which it is involved. The decoding stops when there are no more output packets of degree one. [[However, as we will show in the next example, the ordinary BP decoding cannot decode some types of batches efficiently. We can actually do better than the ordinary BP decoding with little increase of decoding complexity by exploiting the batch structure of the output packets.]{}]{}
For example, consider a batch of two packets $u_1$ and $u_2$ formed by $$\label{eq:1}
\begin{bmatrix}
u_1 & u_2
\end{bmatrix}
=
\begin{bmatrix}
v_1 & v_2 & v_3 & v_4
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
1 & 1 \\
1 & 1
\end{bmatrix}.$$ Suppose that we use the ordinary BP decoding algorithm, and when the BP decoding stops, $v_1$ is recovered [[by processing other batches]{}]{}, but $v_2$, $v_3$ and $v_4$ are not recovered. However, if we allow the decoder to solve the linear system , we can further recover $v_2=u_2-u_1+v_1$. The example shows that the BP decoding performance can be improved if the linear system associated with a timeslot can be solved locally.
Motivated by the above example, we propose the *batched BP decoder* for the output packets of the physical layer of NCSA. The decoder includes multiple iterations. In the $i$-th iteration of the decoding, $i=1,2,\ldots$ all the batches are processed individually by the following algorithm: Consider a batch given in . Let $S\subset \Theta$ be the set of indices $r$ such that $v_r$ is decoded in the previous iterations. When $i=0$, $S =
\emptyset$. Let ${\mathbf{i}}_\Theta:\Theta\rightarrow \{1,\ldots,|\Theta|\}$ be the one-to-one mapping preserving the order on $\Theta$, i.e., ${\mathbf{i}}_\Theta(s_1)<{\mathbf{i}}_\Theta(s_2)$ if and only if $s_1<s_2$. We also write ${\mathbf{i}}(s)$ when $\Theta$ is clear from the context. The algorithm first substitutes the values of $v_r, r\in
S$ into and obtain $$\label{eq:2}
[u_1,\ldots,u_B] - [v_r, r\in S] {\mathbf{H}}^{{\mathbf{i}}[S]} = [v_s,s\in\Theta\setminus S] {\mathbf{H}}^{{\mathbf{i}}[\Theta\setminus S]},$$ where ${\mathbf{H}}^{{\mathbf{i}}[S]}$ is the submatrix of ${\mathbf{H}}$ formed by the rows indexed by ${\mathbf{i}}[S]$. The algorithm then applies Gaussian (Gauss-Jordan) elimination on the above linear system so that ${\mathbf{H}}^{{\mathbf{i}}[\Theta\setminus S]}$ is transformed into the reduced column echelon form $\tilde {\mathbf{H}}$ and becomes $$\label{eq:3}
[\tilde u_1,\ldots, \tilde u_B] = [v_s,s\in\Theta\setminus S] \tilde {\mathbf{H}}.$$ Suppose that the $j$-th column of $\tilde {\mathbf{H}}$ has only one nonzero component (which should be one) at the row corresponding to user $s$. The value of $v_s$ is then $\tilde u_j$ and hene recovered. The algorithm returns the new recovered input packets by searching the columns of $\tilde {\mathbf{H}}$ with only one non-zero component.
For a batch with degree $d$, the complexity of the above decoding is $O(d^3+d^2T)$. Suppose that $K/n$ is a constant and the maximum degree $D$ does not change with $K$. Since the degree of a batch converge to the Poisson distribution with parameter $\frac{K}{n}\bar\Lambda$ (see Lemma \[lemma:1\]), the average complexity of decoding a batch is $O(T)$ finite-field operations. Hence the total decoding complexity is $O(KT)$ finite-field operations.
Decoding Performance
--------------------
For an integer $j$, denoted by $[j]$ the set of integers $\{1,\ldots,j\}$. When $j\leq 0$, $[j]=\emptyset$. For any $\mathbf{H}\in \mathcal{H}_{d}$, define $\gamma(\mathbf{H})$ as the collection of all subsets $V$ of $[d-1]$ such that in the linear system , $v_{{\mathbf{i}}^{-1}(d)}$ can be uniquely solved when the values of $v_r, r\in {\mathbf{i}}^{-1}[V]$ are known. Taking the transfer matrix in as an example, we have $$\gamma(\mathbf{H}) = \{ \{1, 3\}, \{2,3\}, \{1,2,3\}\}.$$ For a timeslot of degree one, the transfer matrix ${\mathbf{H}}$ is the one-by-one matrix with the unity. Then $\gamma({\mathbf{H}}) = \{ \emptyset\}$. For any intiger $k\geq 0$, define $$\Gamma_k(x) = \sum_{{\mathbf{H}}\in \mathcal{H}_{k+1}} g({\mathbf{H}}|k+1) \sum_{S\in
\gamma({\mathbf{H}})} x^{|S|}(1-x)^{k-|S|}.$$ [[In other words, $\Gamma_k(x)$ is the probability that when $k+1$ users transmitted in a timeslot, the input packets of the user with the largest index can be recovered if each of the other users’ packet is known with probability $x$.]{}]{}
We assume that the maximum degree $D$ is a contant that does not change with $K$. The following theorem, proved in the next section, tells us the decoding performance of $l$ iterations of the batched BP decoder when $K$ is sufficiently large. We apply the convention that $0^0=1$.
\[the:1\] Fix real numbers $R>0$, $\epsilon>0$ and an integer $l>0$. Consider a multiple-access system described above with $K$ users and $n =
\lceil K/R\rceil$ timeslots. Define $$z^*_l = 1 - \Lambda\left(1-\sum_k\frac{\lambda^k e^{-\lambda}}{k!}\Gamma_k(z_{l-1})\right),$$ where $z_0=0$ and for $1\leq i <l$ $$z_i = 1 - \Lambda'\left(1-\sum_k\frac{\lambda^k e^{-\lambda}}{k!}\Gamma_k(z_{i-1})\right)/\bar\Lambda,$$ where $\lambda = R \bar\Lambda$. Then for any sufficiently large $K$, $l$ iterations of the batch BP decoder will recover at least $K(z^*_l-\epsilon)$ input packets with probability at least $1-\exp(-c\epsilon^2K)$, where $c$ is a number independent of $K$ and $n$.
See Section \[sec:performance-analysis\].
\[lemma:gamma\] $\Gamma_k(x)$ is an increasing function of $x$.
This lemma can be proved by applying [@yang14c Lemma 13].
Degree Distribution Optimization {#sec:perf-eval}
--------------------------------
Theorem \[the:1\] induces a general approach to optimize the degree distribution $\Lambda$. Let $$f(x;\lambda) = 1 - \Lambda'\left(1-\sum_k\frac{\lambda^k
e^{-\lambda}}{k!}\Gamma_k(x)\right)/\bar\Lambda.$$ We have $z_i = f(z_{i-1};\lambda), i=1,\ldots,l-1$. Suppose that we allow $l\rightarrow \infty$. The sequence $\{z_i\}$ is increasing (implied by Lemma \[lemma:gamma\]) and converges to the first value $x>0$ such that $f(x;\lambda)=x$. For given value of $\lambda$, $0< \epsilon<1$ and $0<\eta\leq 1$, we can optimize the degree distribution $\Lambda$ by solving $$\label{eq:6}
\begin{IEEEeqnarraybox}[][c]{r.l}
\max & R \\
\text{s.t.} &
f(x;\lambda) \geq x(1+\epsilon), \quad \forall x\in (0,\eta], \\
& \sum_ii\Lambda_i = \lambda/R, \sum_i\Lambda_i=1, \Lambda_i\geq 0.
\end{IEEEeqnarraybox}$$
\[the:2l\] Denote by $R(\lambda,\epsilon)$ the optimal value of the above optimization. Then the rate $$R^*(\lambda,\epsilon) = R(\lambda,\epsilon)
\left(1-\Lambda\left(1-\sum_{k}\frac{\lambda^k e^{-\lambda}}{k!}
\Gamma_k(\eta)\right)\right)$$ packet per timeslot is achievable for the batched BP decoding algorithm.
For any $\delta >0$, let $R = R(\lambda,\epsilon) - \sqrt{\delta}$. We show that for sufficiently large $K$, there exists a degree distribution $\Lambda$ such that using $n\leq K/R$ timeslots, the batch BP decoding algorithm can recover at least $K(\eta^*-\sqrt{\delta})$ input packets with high probability, where $$\eta^* = \left(1-\Lambda\left(1-\sum_{k}\frac{\lambda^k e^{-\lambda}}{k!}
\Gamma_k(\eta)\right)\right).$$ That is the code has a rate at least $R^*(\lambda,\epsilon)-\delta$ packet per timeslot.
Let $n=\lceil K/R(\lambda,\epsilon)\rceil$. For the degree distribution $\Lambda$ achieving $R(\lambda,\epsilon)$ in , we know by Theorem \[the:1\] that at $K(z_l^*-\sqrt{\delta})$ input packets can be recovered with high probability. We know that the sequence $\{z_i\}$ converges to a value larger than $\eta$. Then there exists a sufficiently large $l$ such that $z_{l-1}\geq \eta$. Thus, $z_l^* \geq \eta^*$. The proof is completed.
Performance Analysis {#sec:performance-analysis}
====================
We generalize the tree-based approach [@luby98] to analyze the performance of the batched BP decoder and prove Theorem \[the:1\].
Decoding Graph
--------------
The relation between the input packets and the timeslots can be represented by a random Tanner graph $G$, where the input packets are represented by the variable nodes, and timeslots are represented by the check nodes. We henceforth equate a variable node with the corresponding input packet, and a check node with the corresponding timeslot. There exists an edge between a variable node and a check node if and only if the corresponding input packet is transmitted in the timeslot. Associated with each check node is a random transfer matrix $H$. For given degree $d$ of the timeslot, the distribution of $H$ is $g(\cdot|d)$.
The $l$-neighborhood of a variable node $v$, denoted by $G_l(v)$, is the subgraph of $G$ that includes all the nodes with distance less than or equal to $l$ from variable node $v$, as well as all the edges involved. Since $G_l(v)$ has the same distribution for all variable node $v$, we denote by $G_l$ the generic random graph with the same distribution as $G_l(v)$. After $l$ iterations of the batched BP decoding, whether or not a variable node $v$ is decoded is determined by its $2l$-neighborhood.
Motived by the tree-based approach, in the remainder of this section, we first analyze the decodable probability of the root node of a random tree, and then show that the decoding performance of $G_{2l}$ is similar to that of the tree. The proof of Theorem \[the:1\] is then completed by a martingale argument.
Tree Analysis
-------------
Fix two degree distributions $\alpha(x)$ and $\beta(x)$. Let $T_l$ be a tree of $l+1$ levels. The root of the tree is at level $0$ and the leaves are at level $l$. Each node at an even level is a variable node, and each node at an odd level is a check node. The probability that the root node has $i$ children is $\Lambda_i$. Except for the root node, all the other variable nodes have $i$ children with probability $\alpha_i$. All the check node has $i$ children with probability $\beta_i$.
An instance of $T_4$ is shown in Fig. \[fig:tree\].
Let $x_l^*$ be the probability that the root variable node is decodable by applying the batched BP decoding on $T_{2l}$. We have $$x_l^* = 1 - \Lambda\left(1-\textstyle \sum_{k} \beta_k \Gamma_k(x_{l-1}) \right),$$ where $x_0=0$ and for $1 \leq i < l$, $$x_i = 1 - \alpha\left(1-
\textstyle \sum_{k} \beta_k \Gamma_k(x_{i-1}) \right).$$
Denote by $y_i$ the probability that a check node at level $2(l-i)+1$ can recover its parent variable node by solving the associated linear system of this check node with possibly the knowledge of its children variable nodes. We have $x^*_l =
1-\Lambda(1-y_l)$. Suppose that a variable node at level $2(l-i)$ is decodable by at least one of its children check node with probability $\hat{x}_{i}$, $0\leq i < l$. We have $\hat{x}_i = 1 -
\alpha(1-y_{i})$ for $0<i<l$ and $\hat{x}_0=0$.
Fix a check node $c$ at level $2(l-i)+1$. With probability $g({\mathbf{H}}|k+1)\beta_k$, the check node has $k$ children and the associated linear system has ${\mathbf{H}}$ as the transfer matrix. We permutate the rows of ${\mathbf{H}}$ such that the *last* row of ${\mathbf{H}}$ corresponds to the parent variable node. By , the permutation does not change the distribution $g({\mathbf{H}}|k+1)$. Index the $k$ children by $1, \ldots, k$. Using Gaussian elimination in the batched BP decoder, the parent variable node of check node $c$ can be recovered if and only if for certain $S \in
\gamma({\mathbf{H}})$, all the children variable nodes indices by $S$ are decodable. Therefore, the probability that the parent variable node of $c$ is decodable is $\sum_{S\in \gamma({\mathbf{H}})}
\hat{x}_{i-1}^{|S|}(1-\hat{x}_{i-1})^{k-|S|}$ for transfer matrix ${\mathbf{H}}$. Considering all the possible transfer matrices, we have $y_i
= \sum_k\beta_k\Gamma_k(\hat{x}_{i-1})$. The proof is completed by $x_i=\hat{x}_i$.
at (0,0) child [ node\[cnode\] child [node\[vnode\]]{} child [node\[vnode\] child [node\[cnode\] child [node\[vnode\]]{} child [node\[vnode\]]{} ]{} child [node\[cnode\]]{} ]{} ]{} child [ node\[cnode\] ]{} child [ node\[cnode\] child [node\[vnode\] child [node\[cnode\] child [node\[vnode\]]{} child [node\[vnode\]]{} ]{} ]{} child [node\[vnode\] ]{} ]{};
at (3,-1.1) [ ;\
;\
;\
;\
;\
]{};
We prove the following stronger result than Lemma \[lemma:1\].
\[lemma:1a\] Suppose that $K/n \rightarrow R$ as $K\rightarrow \infty$. Fix a timeslot $t$ and an integer $k\geq 0$. Under the condition that a fixed set of $k$ users do not transmit at timeslot $t$, the degree of timeslot $t$ converges to the Poisson distribution with parameter $\lambda = R\bar \Lambda$ as $K\rightarrow \infty$.
Let $\Theta$ be the set of users that do not transmit at timeslot $t$. For each user that is not in $\Theta$, the probability that this user transmits a packet at timeslot $t$ is $\bar\Lambda/n$, when $n$ is larger than $D$. Therefore, the degree of timeslot $t$ follows a binomial distribution with parameter $(K-k,
\bar\Lambda/n)$, which converges to the Poisson distribution with parameter $R\bar\Lambda$ when $K\rightarrow \infty$.
For a positive integer $L$, let $\epsilon_L = 1 - \sum_{d=0}^L \frac{\lambda^de^{-\lambda}}{d!}.$ We are interested in the following instances of $\alpha$ and $\beta$
[r.r]{} \[eq:ab\] (x) = , & (x) = \_[k=0]{}\^L x\^k.
Let $\mathcal{G}_l(L)$ be the set of trees of $l+1$ levels where each check node has at most $L$ children and each variable node has at most $D$ children.
\[lemma:4\] When $K$ is sufficiently large, for any $\mathbf{G}_l\in\mathcal{G}_l(L)$, $$\Pr\{G_l = \mathbf{G}_l\} \geq \Pr\{T_l = \mathbf{G}_l\} - c_{l,L}\epsilon_L,$$ where $c_{l,L} = O(L^{\lfloor l/2\rfloor})$ and the degree distributions of $T_l$ are given in .
We show by induction that $$\label{eq:5}
\Pr\{G_l = \mathbf{G}_l\} \geq \Pr\{T_l =
\mathbf{G}_l\} - c_{l,L}\epsilon_L,$$ where $c_{l,L} = O(L^{\lfloor l/2\rfloor})$.
We prove the lemma by induction. When $l=1$, $G_1$ and $T_1$ follow the same distribution. For $l>1$, we have $$\Pr\{G_l = \mathbf{G}_l\} = \Pr\{G_l = \mathbf{G}_l| G_{l-1} =
\mathbf{G}_{l-1}\} \Pr\{G_{l-1} = \mathbf{G}_{l-1}\},$$ where $\mathbf{G}_{l-1}$ is the subgraph of $\mathbf{G}_l$ obtained by removing the leaf nodes. We assume that $$\Pr\{G_{l-1} = \mathbf{G}_{l-1}\} \geq \Pr\{T_{l-1} =
\mathbf{G}_{l-1}\} - c_{l-1,L}\epsilon_L,$$ for certain function $c_{l-1,L} = O(L^{\lfloor (l-1)/2 \rfloor})$. We then prove with $l>0$ for two cases:$l$ is even and $l$ is odd.
We first consider the case that $l$ is even. Suppose that $\mathbf{G}_{l-1}$ has $N$ leaf *check* nodes, which are at level $l-1$ of $\mathbf{G}_l$. Denote by $k_i$ the number of children variable nodes of the $i$-th check node at level $l-1$ in $\mathbf{G}_l$. Since $\mathbf{G}_l \in \mathcal{G}_l(L)$, we have $k_i \leq L$. By Lemma \[lemma:1a\], we have
[rCl]{} {G\_l = \_l| G\_[l-1]{} = \_[l-1]{}} \_[i=1]{}\^N ,K.
On the other hand, we have
[rCl]{} {T\_l = \_l| T\_[l-1]{} = \_[l-1]{}} = \_[i=1]{}\^N .
Therefore, for sufficiently large $K$,
[rCl]{}\
& & (1-\_L)\^N- 1 - \_L\
& & -(N+1)\_L.
Note that $N= O(L^{\lfloor l/2 \rfloor})$.
We then consider the case that $l$ is odd. Suppose that $\mathbf{G}_{l-1}$ has $N$ leaf *variable* nodes, which are at level $l-1$ of $\mathbf{G}_l$. Denote by $k_i$ the number of children check nodes of the $i$-th variable node at level $l-1$ of $\mathbf{G}_l$. We know that $k_i \leq D-1$. We then have
[rCl]{}\
& & \_[i=1]{}\^N\
& = & {T\_l = \_l| T\_[l-1]{} = \_[l-1]{}}.
Therefore, for sufficiently large $K$, $\Pr\{G_l = \mathbf{G}_l| G_{l-1} =
\mathbf{G}_{l-1}\} \geq \Pr\{T_l = \mathbf{G}_l| T_{l-1} =
\mathbf{G}_{l-1}\} - \epsilon_L$.
Proof of Theorem \[the:1\]
--------------------------
Now we are ready to prove Theorem \[the:1\]. We say $G_l$ or $T_l$ is decodable if its root is decodable by the batched BP decoding algorithm. Fix a sufficiently large $L$. We have
[rCl]{}\
& & \_[\_[2l]{}(L)]{} { } {G\_[2l]{} = }\
& & \_[\_[2l]{}(L)]{} { } ({[T]{}\_[2l]{} = } - )\
& & {[T]{}\_[2l]{} } - /4 = x\^\*\_l - /4 z\^\*\_l - /2,
where the second inequality follows from Lemma \[lemma:4\] and the last inequality follows that $x_l^*\rightarrow z_l^*$ when $L\rightarrow \infty$.
Let $A$ be the number of variable nodes $v$ with $G_{2l}(v) \in \mathcal{G}_{2l}(L)$ and decodable. We have $\operatorname{\mathbb{E}}[A] \geq (z^*_l-\epsilon/2)K$. For $i=1,\ldots, K$, denote $Z_i = G_1(v_i)$. Define $X_i = \operatorname{\mathbb{E}}[A|Z_1,\ldots,Z_i]$. By definition, $X_i$ is a Doob’s martingale with $X_0=\operatorname{\mathbb{E}}[A]$ and $X_K=A$. Since the exposure of a variable node will affect the degrees of a constant number of subgraphs $G_{2l}(v)$ with check node degree $\leq L+1$, we have $|X_i-X_{i-1}| \leq c'$, a constant does not depend on $K$. Applying the Azuma-Hoeffding Inequality, we have $$\Pr\{A \leq \operatorname{\mathbb{E}}[A] - \epsilon/2 K \} \leq \exp\left(- \frac{\epsilon^2K}{8c'^2} \right).$$ Hence $\Pr\{A > (z^*_l-\epsilon) K\} > 1 - \exp\left(-
\frac{\epsilon^2K}{8c'^2} \right)$. This completes the proof of Theorem \[the:1\].
An Example {#sec:example}
==========
In this section, we use an example to illustrate how the proposed NCSA scheme works. Here $q=2$ and $m=1$. Fix an integer $N\geq 2$. We consider the PNC scheme that has the following outputs: i) When one user transmits in a timeslot, the packet of the user is decoded; ii) When two to $N$ users transmit in a timeslot, one or two binary linear combinations of the input packets are decoded; and iii) When more than $N$ users transmit in a timeslot, nothing is decoded.
Taking $N=3$ as an example, when one user transmits in a timeslot, the transfer matrix is ${\mathbf{H}}_{1} = [1]$, and $g({\mathbf{H}}_1|1)=1$. When two users transmit in a timeslot, the possible transfer matrices are
[c]{} \_[21]{}=
1\
1
, \_[22]{}=
1 &\
& 1
, \_[23]{}=
& 1\
1 &
.
Since ${\mathbf{H}}_{22}$ and ${\mathbf{H}}_{23}$ have the same probability, $g({\mathbf{H}}_{21}|2) + 2g({\mathbf{H}}_{22}|2) = 1.$ Now consider that three users transmit in a timeslot. Define $${\mathbf{H}}_{31} =
\begin{bmatrix}
1 \\ 1 \\ 1
\end{bmatrix},
{\mathbf{H}}_{32} =
\begin{bmatrix}
1 & \\ 1 & \\ & 1
\end{bmatrix},
{\mathbf{H}}_{33} =
\begin{bmatrix}
1 & \\ 1 & 1 \\ & 1
\end{bmatrix}.$$ The possible transfer matrices are given by the row permutations of ${\mathbf{H}}_{3i}$, $i=1,2,3$. Note that for two transfer matrices that are permutation of each other, they have the same probability to occur. Thus we have $$g({\mathbf{H}}_{31}|3) + 3g({\mathbf{H}}_{32}|3) + 6g({\mathbf{H}}_{33}|3) = 1.$$
We then have
[rCl]{} \_0(x) & = & 1, \_1(x) = g(\_[21]{}|2)x+2g(\_[22]{}|2),\
\_2(x) & = & g(\_[31]{}|3)x\^2+g(\_[32]{}|3)(1+2x)\
& & +g(\_[33]{}|3)(8x-2x\^2).
In general, for a timeslot of $d$ users, we denote by ${\mathbf{H}}_{d1}$ the single column transfer matrix of all ones. For transfer matrices of two columns, there are three types of rows: $[0,1]$, $[1,0]$ and $[1,1]$. Denote by ${\mathbf{H}}_{d2}(a)$ a generic transfer matrix with $a$ rows of type $[0,1]$ and $d-a$ rows of type $[1,0]$. Here $0<a \leq \lfloor d/2 \rfloor$. All the row permutations of ${\mathbf{H}}_{d2}(a)$ are possible transfer matrices.
Denote by ${\mathbf{H}}_{d3}(a_1,a_2)$ a generic transfer matrix with $a_1$ rows of type $[0,1]$, $a_2$ rows of type $[1,0]$ and $d-a_1-a_2$ rows of type $[1,1]$. Here $a_2\geq a_1>0$ and $a_1+a_2<d$. All the row permutations of ${\mathbf{H}}_{d3}(a_1,a_2)$ are possible transfer matrices. Thus,
[rCl]{} 1 & = & g(\_[d1]{}|d) + \_[a=1]{}\^[d/2 ]{}g(\_[d2]{}(a)|d)\
& & + \_[a\_1=1]{}\^[d-2]{}\_[a\_2=a\_1]{}\^[d-a\_1-1]{} g(\_[d3]{}(a\_1,a\_2)|d).
We can then calculate that
[rCl]{}\
& + & \_[a=1]{}\^[d/2 ]{}g(\_[d2]{}(a)|d)\
& + & \_[a\_1=1]{}\^[d-2]{}\_[a\_2=a\_1]{}\^[d-a\_1-1]{} g(\_[d3]{}(a\_1,a\_2)|d) .
Given average degree $\lambda$ of a timeslot, the average number of output packets decoded in a timeslot converges to $$U(\lambda) = \sum_d \frac{\lambda^d e^{-\lambda}}{d!}
\sum_{{\mathbf{H}}\in \mathcal{H}_d} \operatorname{\text{rk}}({\mathbf{H}}) g({\mathbf{H}}|d),$$ when $K\rightarrow\infty$. The achievable rate of the NCSA system is upper bounded by $U(\lambda)$ packets per timeslot. In the case of this example, the achievable rate bound is given by
[rCl]{} U() & = & \_[d=1]{}\^N .
Note that the upper bound is in general not tight since the packets decoded in different timeslots can be the same.
We solve for the above example with the results in Fig. \[fig:1\], where we assume the uniform distribution for each possible transfer matrices. We also evaluate the corresponding upper bound $U(\lambda)$ for comparison.
table\[x=lambda,y=rate\] [ex2.txt]{}; table\[x=lambda,y=bound\] [ex2.txt]{};
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^1]: [[This work was partially supported by the National Natural Science Foundation of China under Grant 61471215. This work was partially supported by the General Research Funds (Project No. 414812) and the AoE grant (Project No. AoE/E-02/08), established under the University Grant Committee of the Hong Kong Special Administrative Region, China.]{}]{}
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---
abstract: 'We propose a simple algorithm to train stochastic neural networks to draw samples from given target distributions for probabilistic inference. Our method is based on iteratively adjusting the neural network parameters so that the output changes along a Stein variational gradient [@liu2016stein] that maximumly decreases the KL divergence with the target distribution. Our method works for any target distribution specified by their unnormalized density function, and can train any black-box architectures that are differentiable in terms of the parameters we want to adapt. As an application of our method, we propose an *amortized MLE* algorithm for training deep energy model, where a neural sampler is adaptively trained to approximate the likelihood function. Our method mimics an adversarial game between the deep energy model and the neural sampler, and obtains realistic-looking images competitive with the state-of-the-art results.'
author:
- |
Dilin Wang, Qiang Liu\
Department of Computer Science, Dartmouth College\
`{dilin.wang.gr, qiang.liu}@dartmouth.edu`
title: 'Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning'
---
Introduction
============
Modern machine learning increasingly relies on highly complex probabilistic models to reason about uncertainty. A key computational challenge is to develop efficient inference techniques to approximate, or draw samples from complex distributions. Currently, most inference methods, including MCMC and variational inference, are hand-designed by researchers or domain experts. This makes it difficult to fully optimize the choice of different methods and their parameters, and exploit the structures in the problems of interest in an automatic way. The hand-designed algorithm can also be inefficient when it requires to make fast inference repeatedly on a large number of different distributions with similar structures. This happens, for example, when we need to reason about a number of observed datasets in settings like online learning, or need fast inference as inner loops for other algorithms such as maximum likelihood training. Therefore, it is highly desirable to develop more intelligent probabilistic inference systems that can adaptively improve its own performance to fully the optimize computational efficiency, and generalize to new tasks with similar structures.
Specifically, denote by $p(x)$ a probability density of interest specified up to the normalization constant, which we want to draw sample from, or marginalize to estimate its normalization constant. We want to study the following problem:
\[pro:prob1\] Given a distribution with density $p(x)$ and a function $f(\eta;~\xi)$ with parameter $\eta$ and random input $\xi$, for which we only have assess to draws of the random input $\xi$ (without knowing its true distribution $q_0$), and the output values of $f(\eta;~\xi)$ and its derivative $\partial_\eta f(\eta;~\xi)$ given $\eta$ and $\xi$. We want to find an optimal parameter $\eta$ so that the density of the random output variable $x = f(\eta;~\xi)$ with $\xi\sim q_0$ closely matches the target density $p(x)$.
Because we have no assumption on the structure of $f(\eta;~\xi)$ and the distribution of random input, we can not directly calculate the actual distribution of the output random variable $x = f(\eta;~\xi)$; this makes it difficult to solve Problem \[pro:prob1\] using the traditional variational inference (VI) methods. Recall that traditional VI approximates $p(x)$ using simple proposal distributions $q_\eta(x)$ indexed by parameter $\eta$, and finds the optimal $\eta$ by minimizing KL divergence $\KL(q_\eta ~||~p) = \E_{q_\eta}[\log (q_\eta/p)]$, which requires to calculate the density $q_\eta(x)$ or its derivative that is not computable by our assumption (even when the Monte Carlo gradient estimation and the reparametrization trick [@kingma2013auto] are applied).
In fact, it is this requirement of calculating $q_\eta(x)$ that has been the major constraint for the designing of state-of-the-art variational inference methods with rich approximation families; the recent successful algorithms [e.g., @rezende2015variational; @tran2015variational; @ranganath2015hierarchical to name only a few] have to handcraft special variational families to ensure the computational tractability of $q_\eta(x)$ and simultaneously obtain high approximation accuracy, which require substantial mathematical insights and research effects. Methods that do not require to explicitly calculate $q_\eta(x)$ can significantly simplify the design and applications of VI methods, allowing practical users to focus more on choosing proposals that work best with their specific tasks. We will use the term *wild variational inference* to refer to new variants of variational methods that require no tractability $q_\eta(x)$, to distinguish with the *black-box variational inference* [@ranganath2013black] which refers to methods that work for generic target distributions $p(x)$ without significant model-by-model consideration (but still require to calculate the proposal density $q_\eta(x)$).
A similar problem also appears in importance sampling (IS), where it requires to calculate the IS proposal density $q(x)$ in order to calculate the importance weight $w(x) = p(x)/q(x)$. However, there exist methods that use no explicit information of $q(x)$, which, seemingly counter-intuitively, give better asymptotic variance or converge rates than the typical IS that uses the proposal information [e.g., @liu2016black; @briol2015probabilistic; @henmi2007importance; @delyon2014integral]. Discussions on this phenomenon dates back to @o1987monte, who argued that “Monte Carlo (that uses the proposal information) is fundamentally unsound” for violating the Likelihood Principle, and developed Bayesian Monte Carlo [@o1991bayes] as an example that uses no information on $q(x)$, yet gives better convergence rate than the typical Monte Carlo $\Od(n^{-1/2})$ rate [@briol2015probabilistic]. Despite the substantial difference between IS and VI, these results intuitively suggest the possibility of developing efficient variational inference without calculating $q(x)$ explicitly.
In this work, we propose a simple algorithm for Problem \[pro:prob1\] by iteratively adjusting the network parameter $\eta$ to make its output random variable changes along a Stein variational gradient direction (SVGD) [@liu2016stein] that optimally decreases its KL divergence with the target distribution. Critically, the SVGD gradient includes a repulsive term to ensure that the generated samples have the right amount of variability that matches $p(x).$ In this way, we “amortize SVGD” using a neural network, which makes it possible for our method to adaptively improve its own efficiency by leveraging fast experience, especially in cases when it needs to perform fast inference repeatedly on a large number of similar tasks. As an application, we use our method to amortize the MLE training of deep energy models, where a neural sampler is adaptively trained to approximate the likelihood function. Our method, which we call *SteinGAN*, mimics an adversarial game between the energy model and the neural sampler, and obtains realistic-looking images competitive with the state-of-the-art results produced by generative adversarial networks (GAN) [@goodfellow2014generative; @radford2015unsupervised].
#### Related Work
The idea of amortized inference [@gershman2014amortized] has been recently applied in various domains of probabilistic reasoning, including both amortized variational inference [e.g., @kingma2013auto; @jimenez2015variational], and data-driven proposals for (sequential) Monte Carlo methods [e.g., @paige2016inference], to name only a few. Most of these methods, however, require to explicitly calculate $q(x)$ (or its gradient). One exception is a very recent paper [@operator] that avoids calculating $q(x)$ using an idea related to Stein discrepancy [@gorham2015measuring; @liu2016kernelized; @oates2014control; @chwialkowski2016kernel]. There is also a raising interest recently on a similar problem of “learning to optimize” [e.g., @andrychowicz2016learning; @daniel2016learning; @li2016learning], which is technically easier than the more general problem of “learning to sample”. In fact, we show that our algorithm reduces to “learning to optimize” when only one particle is used in SVGD.
Generative adversarial network (GAN) and its variants have recently gained remarkable success on generating realistic-looking images [@goodfellow2014generative; @salimans2016improved; @radford2015unsupervised; @li2015generative; @dziugaite2015training; @nowozin2016f]. All these methods are set up to train latent variable models (the generator) under the assistant of the discriminator. Our SteinGAN instead performs traditional MLE training for a deep energy model, with the help of a neural sampler that learns to draw samples from the energy model to approximate the likelihood function; this admits an adversarial interpretation: we can view the neural sampler as a generator that attends to fool the deep energy model, which in turn serves as a discriminator that distinguishes the real samples and the simulated samples given by the neural sampler. This idea of training MLE with neural samplers was first discussed by @kim2016deep; one of the key differences is that the neural sampler in @kim2016deep is trained with the help of a heuristic diversity regularizer based on batch normalization, while SVGD enforces the diversity in a more principled way. Another method by @zhao2016energy also trains an energy score to distinguish real and simulated samples, but within a non-probabilistic framework (see Section \[sec:gan\] for more discussion). Other more traditional approaches for training energy-based models [e.g., @ngiam2011learning; @xie2016theory] are often based on variants of MCMC-MLE or contrastive divergence [@geyer1991markov; @hinton2002training; @tieleman2008training], and have difficulty generating realistic-looking images from scratch.
Stein Variational Gradient Descent (SVGD)
=========================================
Stein variational gradient descent (SVGD) [@liu2016stein] is a general purpose Bayesian inference algorithm motivated by Stein’s method [@stein1972; @barbour2005introduction] and kernelized Stein discrepancy [@liu2016kernelized; @chwialkowski2016kernel; @oates2014control]. It uses an efficient *deterministic* gradient-based update to iteratively evolve a set of particles $\{x_i\}_{i=1}^n$ to minimize the KL divergence with the target distribution. SVGD has a simple form that reduces to the typical gradient descent for maximizing $\log p$ when using only one particle $(n=1)$, and hence can be easily combined with the successful tricks for gradient optimization, including stochastic gradient, adaptive learning rates (such as adagrad), and momentum. To give a quick overview of the main idea of SVGD, let $p(x)$ be a positive density function on $\R^d$ which we want to approximate with a set of particles $\{ x_i\}_{i=1}^n$. SVGD initializes the particles by sampling from some simple distribution $q_0$, and updates the particles iteratively by $$\begin{aligned}
\label{equ:xxii}
x_i \gets x_i + \epsilon \ff(x_i), ~~~~ \forall i = 1, \ldots, n, \end{aligned}$$ where $\epsilon$ is a step size, and $\ff(x)$ is a “particle gradient direction” chosen to maximumly decrease the KL divergence between the distribution of particles and the target distribution, in the sense that $$\begin{aligned}
\label{equ:ff00}
\ff = \argmax_{\ff \in \F} \bigg\{ - \frac{d}{d\epsilon} \KL(q_{[\epsilon\ff]} ~|| ~ p) \big |_{\epsilon = 0} \bigg\},
$$ where $q_{[\epsilon \ff]}$ denotes the density of the updated particle $x^\prime = x + \epsilon \ff(x) $ when the density of the original particle $x$ is $q$, and $\F$ is the set of perturbation directions that we optimize over. We choose $\F$ to be the unit ball of a vector-valued reproducing kernel Hilbert space (RKHS) $\H^d = \H \times \cdots \times \H$ with each $\H$ associating with a positive definite kernel $k(x,x')$; note that $\H$ is dense in the space of continuous functions with universal kernels such as the Gaussian RBF kernel. Critically, the gradient of KL divergence in equals a simple linear functional of $\ff$, allowing us to obtain a closed form solution for the optimal $\ff$. @liu2016stein showed that $$\begin{aligned}
\label{equ:klstein00}
&- \frac{d}{d\epsilon} \KL(q_{[\epsilon\ff]} ~|| ~ p) \big |_{\epsilon = 0} = \E_{x\sim q}[\sumstein_p \ff(x)], \\[.5\baselineskip]
&~~~~\text{with}~~~~~ \sumstein_p \ff(x) = \nabla_x \log p(x) ^\top \ff (x)+ \nabla_x \cdot \ff(x), \end{aligned}$$ where $\sumstein_p$ is considered as a linear operator acting on function $\ff$ and is called the Stein operator in connection with Stein’s identity which shows that the RHS of equals zero if $p = q$: $$\begin{aligned}
\label{equ:steinid}
\E_{p}[\sumstein_p \ff] =\E_{p}[ \nabla_x \log p ^\top \ff + \nabla_x \cdot \ff] = 0. \end{aligned}$$ This is a result of integration by parts assuming the value of $p(x)\ff(x)$ vanishes on the boundary of the integration domain. Therefore, the optimization in reduces to $$\begin{aligned}
\label{equ:ksd}
\S(q ~||~ p) \overset{def}{=} \max_{\ff \in \H^d} \{ \E_{x\sim q} [\sumstein_p \ff(x)] ~~~s.t.~~~~ ||\ff ||_{\H^d} \leq 1\}, \end{aligned}$$ where $\S(q ~||~ p)$ is the kernelized Stein discrepancy defined in @liu2016kernelized, which equals zero if and only if $p = q$ under mild regularity conditions. Importantly, the optimal solution of yields a closed form $$\ff^*(x') \propto \E_{x\sim q}[\nabla_x \log p(x)k(x,x') + \nabla_x k(x,x')].$$ By approximating the expectation under $q$ with the empirical average of the current particles $
\{x_i\}_{i=1}^n$, SVGD admits a simple form of update: $$\begin{aligned}
\label{equ:update11}
&& &~~~~~~~~~~~ x_i ~ \gets ~ x_i ~ + ~ \epsilon \Delta x_i,
~~~~~~~~\forall i = 1, \ldots, n, \notag
\\
~
&&& \text{where~~~~~}\Delta x_i = \hat \E_{x\in \{x_i\}_{i=1}^n} [ \nabla_{x} \log p(x) k(x, x_i) + \nabla_{x} k(x, x_i) ],
$$ and $\hat\E_{x\sim \{x_i\}_{i=1}^n}[f(x)] = \sum_i f(x_i)/n$. The two terms in $\Delta x_i$ play two different roles: the term with the gradient $\nabla_x \log p(x)$ drives the particles toward the high probability regions of $p(x)$, while the term with $\nabla_x k(x,x_i)$ serves as a repulsive force to encourage diversity; to see this, consider a stationary kernel $k(x,x') = k(x-x')$, then the second term reduces to $\hat \E_x \nabla_{x} k(x,x_i) = - \hat \E_x \nabla_{x_i} k(x,x_i)$, which can be treated as the negative gradient for minimizing the average similarity $\hat \E_x k(x,x_i)$ in terms of $x_i$. Overall, this particle update produces diverse points for distributional approximation and uncertainty assessment, and also has an interesting “momentum” effect in which the particles move collaboratively to escape the local optima. It is easy to see from that $\Delta x_i$ reduces to the typical gradient $\nabla_x \log p(x_i)$ when there is only a single particle ($n=1$) and $\nabla_x k(x,x_i)$ when $x=x_i$, in which case SVGD reduces to the standard gradient ascent for maximizing $\log p(x)$ (i.e., maximum *a posteriori* (MAP)).
Set batch size $m$, step-size scheme $\{\epsilon_t\}$ and kernel $k(x,x')$. Initialize $\eta^0$. Draw random $\{\xi_i\}_{i=1}^m$, calculate $x_i = f(\eta^t;~\xi_i)$, and the Stein variational gradient $\Delta x_i$ in . Update parameter $\eta$ using , or .
Amortized SVGD: Towards an Automatic Neural Sampler {#sec:amortizedsvgd}
===================================================
SVGD and other particle-based methods become inefficient when we need to repeatedly infer a large number different target distributions for multiple tasks, including online learning or inner loops of other algorithms, because they can not improve based on the experience from the past tasks, and may require a large memory to restore a large number of particles. We propose to “amortize SVGD” by training a neural network $f(\eta;~\xi)$ to mimic the SVGD dynamics, yielding a solution for Problem \[pro:prob1\].
One straightforward way to achieve this is to run SVGD to convergence and train $f(\eta;~\xi)$ to fit the SVGD results. This, however, requires to run many epochs of fully converged SVGD and can be slow in practice. We instead propose an *incremental approach* in which $\eta$ is iteratively adjusted so that the network outputs $x = f(\eta;~\xi)$ changes along the Stein variational gradient direction in in order to decrease the KL divergence between the target and approximation distribution. To be specific, denote by $\eta^t$ the estimated parameter at the $t$-th iteration of our method; each iteration of our method draws a batch of random inputs $\{\xi_i\}_{i=1}^m$ and calculate their corresponding output $x_i = f(\eta;~\xi_i)$ based on $\eta^t$; here $m$ is a mini-batch size (e.g., $m=100$). The Stein variational gradient $\Delta x_i$ in would then ensure that $x'_i = x_i + \epsilon \Delta x_i$ forms a better approximation of the target distribution $p$. Therefore, we should adjust $\eta$ to make its output matches $\{x'_i\}$, that is, we want to update $\eta$ by $$\begin{aligned}
\label{equ:follow1}
\eta^{t+1} \gets \argmin_\eta \sum_{i=1}^m || f(\eta;~\xi_i) - x_i' ||_2^2, ~~~~~~ \text{where} ~~~~~~ x_i' = x_i + \epsilon \Delta x_i. \end{aligned}$$ See Algorithm \[alg:alg1\] for the summary of this procedure. If we assume $\epsilon$ is very small, then reduces to a least square optimization. To see this, note that $f(\eta;~\xi_i) \approx f(\eta^t;~\xi_i) + \partial_\eta f(\eta^t;~\xi_i) (\eta - \eta^t)$ by Taylor expansion. Since $x_i = f(\eta^t;~\xi_i)$, we have $$|| f(\eta;~\xi_i) - x_i' ||_2^2 \approx || \partial_\eta f(\eta^t;~\xi_i) (\eta - \eta^t) - \epsilon \Delta x_i ||_2^2.$$ As a result, reduces to the following least square optimization: $$\begin{aligned}
\label{equ:follow2}
\eta^{t+1} \gets \eta^t + \epsilon \Delta \eta^t,
\text{~~~~~where~~~~~}
\Delta \eta^t = \argmin_{\delta} \sum_{i=1}^m || \partial_\eta f(\eta^t;~\xi_i) \delta - \Delta x_i ||_2^2. \end{aligned}$$ Update can still be computationally expensive because of the matrix inversion. We can derive a further approximation by performing only one step of gradient descent of (or ), which gives $$\begin{aligned}
\label{equ:follow3}
\eta^{t+1} \gets \eta^t + \epsilon \sum_{i=1}^m \partial_\eta f(\eta^t;~\xi_i) \Delta x_i. \end{aligned}$$
Although update is derived as an approximation of -, it is computationally faster and we find it works very effectively in practice; this is because when $\epsilon$ is small, one step of gradient update can be sufficiently close to the optimum.
Update also has a simple and intuitive form: can be thought as *a “chain rule” that back-propagates the Stein variational gradient to the network parameter $\eta$*. This can be justified by considering the special case when we use only a single particle $(n=1)$ in which case $\Delta x_i$ in reduces to the typical gradient $\nabla_x \log p(x_i)$ of $\log p(x)$, and update reduces to the typical gradient ascent for maximizing $$\E_{\xi}[\log p(f(\eta;~\xi))],$$ in which case $f(\eta;~\xi)$ is trained to maximize $\log p(x)$ (that is, *learning to optimize*), instead of *learning to draw samples from $p$* for which it is crucial to use Stein variational gradient $\Delta x_i$ to diversify the network outputs. Update also has a close connection with the typical variational inference with the reparameterization trick [@kingma2013auto]. Let $q_\eta(x)$ be the density function of $x = f(\eta;~\xi)$, $\xi\sim q_0$. Using the reparameterization trick, the gradient of $\KL(q_\eta~||~p)$ w.r.t. $\eta$ can be shown to be $$\nabla_\eta\KL(q_{\eta}~||~p) = -\E_{\xi \sim q_0}[\partial_\eta f(\eta;~\xi)(\nabla_x \log p(x) - \nabla_x \log q_\eta(x))].$$ With $\{\xi_i\}$ i.i.d. drawn from $q_0$ and $x_i = f(\eta;~\xi_i), ~\forall i$, the standard stochastic gradient descent for minimizing the KL divergence is $$\begin{aligned}
\label{equ:rep}
\eta^{t+1} \gets \eta^t + \sum_i \partial_\eta f(\eta^t;~\xi_i) \tilde \Delta x_i, ~~~~ \text{where} ~~~~
\tilde \Delta x_i = \nabla_x \log p(x_i) - \nabla_x \log q_\eta(x_i). \end{aligned}$$ This is similar with , but replaces the Stein gradient $\Delta x_i$ defined in with $\tilde \Delta x_i$. The advantage of using $\Delta x_i$ is that it does not require to explicitly calculate $q_\eta$, and hence admits a solution to Problem 1 in which $q_\eta$ is not computable for complex network $f(\eta; ~\xi)$ and unknown input distribution $q_0$. Further insights can be obtained by noting that $$\begin{aligned}
\label{equ:tmp}
\Delta x_i
& \approx \E_{x\sim q}[\nabla_x \log p(x)k(x,x_i) + \nabla_xk(x,x_i)] \notag \\
& = \E_{x\sim q}[(\nabla_x \log p(x) - \nabla_x \log q(x))k(x,x_i)] \\
& = \E_{x\sim q} [( \tilde \Delta x) k(x, x_i)], \notag
$$ where is obtained by using Stein’s identity . Therefore, $\Delta x_i$ can be treated as a kernel smoothed version of $\tilde \Delta x_i $.
Amortized MLE for Generative Adversarial Training
=================================================
Our method allows us to design efficient approximate sampling methods adaptively and automatically, and enables a host of novel applications. In this paper, we apply it in an amortized MLE method for training deep generative models. Maximum likelihood estimator (MLE) provides a fundamental approach for learning probabilistic models from data, but can be computationally prohibitive on distributions for which drawing samples or computing likelihood is intractable due to the normalization constant. Traditional methods such as MCMC-MLE use hand-designed methods (e.g., MCMC) to approximate the intractable likelihood function but do not work efficiently in practice. We propose to adaptively train a generative neural network to draw samples from the distribution during MLE training, which not only provides computational advantage, and also allows us to generate realistic-looking images competitive with, or better than the state-of-the-art generative adversarial networks (GAN) [@goodfellow2014generative; @radford2015unsupervised] (see Figure \[fig:mnist\]-\[fig:facemore\]). To be specific, denote by $\{x_{i,obs}\}$ a set of observed data. We consider the maximum likelihood training of energy-based models of form $$p(x|\theta) = \exp(-\phi(x, \theta) - \Phi(\theta)), ~~~~~ \Phi(\theta) = \log \int \exp(-\phi(x,\theta))dx,$$ where $\phi(x; ~\theta)$ is an energy function for $x$ indexed by parameter $\theta$ and $\Phi(\theta)$ is the log-normalization constant. The log-likelihood function of $\theta$ is $$L(\theta) =\frac{1}{n}\sum_{i=1}^n\log p(x_{i,obs} | \theta),$$ whose gradient is $$\begin{aligned}
\nabla_\theta L(\theta) = - \hat\E_{obs} [\partial_\theta \phi (x; \theta) ] + \E_\theta [\partial_\theta \phi(x; \theta) ],
$$ where $\hat\E_{obs}[\cdot]$ and $\E_\theta[\cdot]$ denote the empirical average on the observed data $\{x_{i,obs}\}$ and the expectation under model $p(x |\theta)$, respectively. The key computational difficulty is to approximate the model expectation $\E_{\theta}[\cdot]$. To address this problem, we use a generative neural network $x = f(\eta;~\xi)$ trained by Algorithm \[alg:alg1\] to approximately sample from $p(x|\theta)$, yielding a gradient update for $\theta$ of form $$\begin{aligned}
\label{equ:updatetheta}
\theta \gets \theta + \epsilon \hat\nabla_\theta L(\theta), &\text{}&
\hat \nabla_\theta L(\theta) = - \hat\E_{obs} [\partial_\theta \phi (x; \theta) ] + \hat\E_{\eta} [\partial_\theta \phi(x; \theta) ],
$$ where $\hat \E_{\eta}$ denotes the empirical average on $\{x_i\}$ where $x_i = f(\eta;~\xi_i)$, $\{\xi_i\}\sim q_0$. As $\theta$ is updated by gradient ascent, $\eta$ is successively updated via Algorithm \[alg:alg1\] to *follow* $p(x|\theta)$. See Algorithm \[alg:gan\].
We call our method *SteinGAN*, because it can be intuitively interpreted as an adversarial game between the generative network $f(\eta;~\xi)$ and the energy model $p(x|\theta)$ which serves as a discriminator: The MLE gradient update of $p(x|\theta)$ effectively decreases the energy of the training data and increases the energy of the simulated data from $f(\eta;~\xi)$, while the SVGD update of $f(\eta;~\xi)$ decreases the energy of the simulated data to fit better with $p(x|\theta)$. Compared with the traditional methods based on MCMC-MLE or contrastive divergence, we *amortize the sampler as we train*, which gives much faster speed and simultaneously provides a high quality generative neural network that can generate realistic-looking images; see @kim2016deep for a similar idea and discussions.
MLE training for energy model $p(x|\theta) = \exp(-\phi(x,\theta) - \Phi(\theta))$. Initialize $\eta$ and $\theta$. Draw $\xi_i\sim q_0$, $x_i = f(\eta;~\xi_i)$; update $\eta$ using , or with $p(x)=p(x|\theta)$. Repeat several times when needed. Draw a mini-batch of observed data $\{x_{i,obs}\}$, and simulated data $x_i = f(\eta;~\xi_i)$, update $\theta$ by .
Conclusion
==========
We propose a new method to train neural samplers for given distributions, together with a new SteinGAN method for generative adversarial training. Future directions involve more applications and theoretical understandings for training neural samplers.
![More images generated by SteinGAN on CelebA.[]{data-label="fig:facemore"}](\dilinfig/faces/vgd_gan-20.pdf){width="90.00000%"}
|
---
abstract: 'We present an investigation of candidate Infrared Dark Cloud cores as identified by @2006ApJ...639..227S located within the SCUBA Legacy Catalogue. After applying a uniform noise cut to the Catalogue data we identify 154 Infrared Dark Cloud cores that were detected at 850$\umu$m and 51 cores that were not. We derive column densities for each core from their 8$\umu$m extinction and find that the IRDCs detected at 850$\umu$m have higher column densities (a mean of $1.7\times10^{22}$cm$^{-2}$) compared to those cores not detected at 850$\umu$m (a mean of $1.0\times10^{22}$cm$^{-2}$). Combined with sensitivity estimates, we suggest that the cores not detected at 850$\umu$m are low mass, low column density and low temperature cores that are below the sensitivity limit of SCUBA at 850$\umu$m. For a subsample of the cores detected at 850$\umu$m those contained within the MIPSGAL area) we find that two thirds are associated with 24$\umu$m sources. Cores not associated with 24$\umu$m emission are either “starless” IRDC cores that perhaps have yet to form stars, or contain low mass YSOs below the MIPSGAL detection limit. We see that those “starless” IRDC cores and the IRDC cores associated with 24$\umu$m emission are drawn from the same column density population and are of similar mass. If we then assume the cores without 24$\umu$m embedded sources are at an earlier evolutionary stage to cores with embedded objects we derive a statistical lifetime for the quiescent phase of a few 10$^{3}$–10$^{4}$years. Finally, we make conservative predictions for the number of observed IRDCs that will be observed by the Apex Telescope Galactic Plane Survey (ATLASGAL), the Herschel Infrared Galactic Plane Survey (Hi-GAL), the JCMT Galactic Plane Survey (JPS) and the SCUBA-2 “All Sky” Survey (SASSy).'
author:
- |
H. Parsons$^{1}$[^1], M. A. Thompson$^{1}$[^2] and A. Chrysostomou$^{1,2}$\
$^{1}$Centre for Astrophysics Research, Science & Technology Research Institute,\
University of Hertfordshire, College Lane, Hatfield, AL10 9AB, UK\
$^{2}$Joint Astronomy Centre, 660 North A’ohoku Place, University Park, Hilo, Hawaii 96720, U.S.A.
bibliography:
- 'bibliography.bib'
date: 'Accepted 2009 July 7. Received 2009 July 7; in original form 2009 March 13'
title: Infrared Dark Cloud Cores in the SCUBA Legacy Catalogue
---
\[firstpage\]
stars: formation – dust –infrared: ISM – submillimetre
Introduction
============
Infrared Dark Clouds (IRDCs) were first observed in the mid-1990s by the Infrared Space Observatory, ISO, and the Midcourse Space eXperiment, MSX, [@1998ApJ...494L.199E] as silhouettes against the bright mid-infrared Galactic background. Initially, [@1998ApJ...494L.199E] identified $\sim$2000 clouds by eye from the MSX Galactic Plane Survey images. A systematic study of the MSX data using an automated identification process by [@2006ApJ...639..227S] identified 10,931 candidate IRDCs within which a total of 12,774 compact IRDC core candidates were detected. By definition each IRDC contains at least one core.
An unsurprising consequence of this detection method is the fact that the Galactic distribution of IRDCs follows the mid-infrared background of the Galaxy. IRDCs are predominantly found in the first and fourth Galactic quadrants and near to the Galactic mid-plane [@2008ApJ...680..349J], precisely where the mid-infrared background is greatest. @2006ApJ...653.1325S used data from the Galactic Ring Survey (GRS) to obtain distance estimates for IRDCs found within the first quadrant of the Galaxy, using a morphological match with $^{13}$CO emission. This matching process identified distances to 313 candidate IRDCs. Further distance estimates have been obtained by @2008ApJ...680..349J for 316 IRDC candidates contained within the fourth quadrant and were obtained using single pointings of CS (J=2–1). The distances obtained from these large scale investigations show a peak in the radial galactocentric distribution of IRDCs corresponding to the location of the Scutum-Centaurus arm i.e. peak of R=5kpc in the first quadrant and R=6kpc in the fourth [@2008ApJ...680..349J].
In the decade since their discovery, our understanding of the physical properties of IRDCs has increased and it is now known that these objects are cold ($<$25K) dense (10$^{5}$cm$^{-3}$) regions, on scales of 1–10pc, with masses ranging between 10$^{2}$-10$^{5}$M$_{\odot}$ [@2006ApJ...641..389R]. Current theory suggests that cold dense starless cores found within IRDCs are the precursors to hot molecular cores (@2008ASPC..387...44J and references therein), indeed @2007ApJ...662.1082R report on a hot molecular core found within an IRDC. Other tracers of massive star formation such as HII regions and Class II methanol (CH$_{3}$OH) masers (@1998ApJ...508..721C; ; [@2009ApJS..181..360C]) have been found in association with a number of IRDCs. The association with high mass star formation is not exclusive: a number of individual studies find only low to intermediate mass young stellar objects embedded within IRDCs [@2008arXiv0807.3628V].
To date, with the exception of distance estimates, only small samples of the IRDCs originally published by @2006ApJ...639..227S have been investigated, with an observational bias towards the darkest high contrast clouds (@2006ApJ...641..389R; @2008ApJ...686..384D). The trends of global properties across a large sample of IRDCs have yet to be investigated, in particular the proportion of IRDCs that are associated with active star formation as opposed to IRDCs that are quiescent or starless.
This paper aims to address this issue by studying candidate IRDC cores originally identified by @2006ApJ...639..227S that are contained within the recently published SCUBA Legacy Catalogue by @2008ApJS..175..277D. In Section \[Section:CrossMatch\] we describe the cross matching method used on the two catalogues, obtain column density and mass estimates (or upper limits where applicable) and identify 24$\umu$m embedded sources associated with the the cores identified. The results of the cross matching procedure are presented in Section \[Section:Results\]. We discuss our findings in Section \[Section:Discussion\] with a mention of the impact on two of the forthcoming James Clerk Maxwell Telescope[^3] (JCMT) Legacy surveys. Finally we make some concluding remarks in Section \[Section:Conclusions\].
Cross-matching IRDCs in MSX, SCUBA and MIPSGAL {#Section:CrossMatch}
==============================================
Archival data
-------------
In [[email protected]], @2006ApJ...639..227S produced a catalogue of 10,931 candidate IRDCs using data from the MSX satellite that covered the entire Galactic plane from l=0–360$^{\circ}$ and $|$b$|$$\le$5$^{\circ}$. Candidate IRDCs were identified by modelling the Galactic background diffuse emission at 8$\umu$m, subtracting the 8$\umu$m MSX images from this model and then dividing by the background model to produce what is known as a “contrast image”. Regions of high extinction in the raw images appeared as positive objects with contrast values between 0 and 1 (1 for highly extincted objects) in the contrast images. IRDCs were then identified by looking for extended contrast sources, those with 36 or more continuous pixels with a contrast greater than 2$\sigma$. Cores within the clouds were identified by decomposing the clouds using two-dimensional elliptical Gausian fits [@2006ApJ...639..227S]. Although discovered by their mid-infrared absorption, it is at sub millimetre and far-infrared wavelengths that these objects have their peak emission.
@2008ApJS..175..277D present a comprehensive re-reduction of the entire 8 year sub millimetre continuum data set observed by SCUBA (Sub millimetre Common User Bolometer Array) on the JCMT in Hawaii. This data set is known as the SCUBA Legacy Catalogue and covers a total area of just over 29 square degrees at 850$\umu$m. A consequence of the varying weather conditions and method by which the data was collected, over the entire lifetime of SCUBA, is that the data within the SCUBA Legacy Catalogue is both non-uniform in noise and in its quality of opacity corrections. @2008ApJS..175..277D divided the SCUBA Legacy Catalogue into a Fundamental and an Extended Dataset. The former uses data for which there is well known atmospheric opacity calibration data (from both skydips and the CSO radiometer, @2008ApJS..175..277D) and the latter contains all observations regardless of the data quality. Coverage by the Extended data set is greater in area than the Fundamental by 9.7 square degrees. Discrete objects were identified within the SCUBA Legacy Catalogue (from both the Fundamental and Extended Dataset independently) using Clumpfind[^4] [@2008ApJS..175..277D]. This process provides information on the properties of each object such as flux (at the peak and integrated over its area) and apparent size.
MIPSGAL[^5] is a survey of the Galactic Plane from 10$<$l$<$65$^{\circ}$ and -10$>$l$>$-65$^{\circ}$ $|$b$|$$<$1$^{\circ}$ at 24 and 70$\umu$m, using the MIPS instrument on Spitzer[^6]. As a tracer of warm dust and with good resolution (6“ angular resolution as opposed to 20” and 14" for MSX and SCUBA respectively), MIPSGAL 24$\umu$m data are ideal for investigating warm embedded objects, such as Young Stellar Objects. Indeed [@2008arXiv0808.2053F] combined MIPSGAL data with IRAC (Infrared Camera on Spitzer), 2MASS (Two Micron All Sky Survey) and SCUBA data to identify Young Stellar Objects (YSOs) within IRDC MSXDC G048.65-00.29.
Cross identification {#subsection:crossidentification}
--------------------
In total, 428 MSX IRDC cores from [@2006ApJ...639..227S] were located in regions mapped by SCUBA (325 located in the Fundamental region and an additional 103 located in the Extended region). However the SCUBA Legacy Catalogue does not contain photometric measurements of objects located at the edges of maps (which may be subject to large scale background or noise fluctuations). In addition, regions of high noise persist within the catalogue due to the non-uniform way in which the data were taken. In order to make the catalogue more uniform we exclude data with an rms noise greater than 0.1Jypixel$^{-1}$, the pixel size for data in the SCUBA Legacy Catalogue is 6”. This 0.1Jypixel$^{-1}$ cut excludes the high noise Poisson tail present in the data [@2008ApJS..175..277D] and gives the remaining data approximately Gaussian noise statistics, with a mean and sigma of 0.05Jypixel$^{-1}$ and 0.024Jypixel$^{-1}$ respectively. Taking map edges into account and excluding regions with noise $>$0.1Jypixel$^{-1}$ leaves us with 251 MSX IRDC cores within the SCUBA Legacy Catalogue mapped region. The 251 MSX candidate IRDC cores were then cross matched to sources identified within the SCUBA Legacy Catalogue (as defined by Clumpfind). IRDC cores were matched against both the Fundamental and the Extended data Catalogues.
For the cross identification process, the locations of the MSX IRDC cores were positionally matched using TOPCAT[^7] to the locations of the Clumpfind SCUBA objects. The irregular morphology of the candidate objects in both the MSX 8$\umu$m contrast images and the SCUBA 850$\umu$m emission maps, meant that the task of cross matching cores between the two catalogues was non trivial. @2006ApJ...639..227S identified IRDCs within the MSX 8$\umu$m data as contiguous structures in the contrast images that were sufficently extended to be real clouds and not artifacts. The cores within the clouds were identified using two dimensional elliptical Gaussian fits to these contiguous structures. In contrast @2008ApJS..175..277D created the SCUBA Legacy Catalogue using Clumpfind which identifies irregular objects by following intensity contours. Due to this different approach in identification between the two catalogues the catalogued positions of IRDC cores and SCUBA clumps may differ by a considerable amount, even when the two are clearly morphologically associated with each other. A large positional matching radius was required to identify potential matches followed by further refinement by eye, checking that the individual IRDC cores were morphologically similar to the SCUBA 850$\umu$m emission. A matching radius of 1’ was chosen as @2006ApJ...639..227S quotes that typical core diameters lie between 0.75’ and 2’. For added confidence, those cores that were initially matched were additionally checked for 850$\umu$m emission at the location of the MSX core.
\[subsection:embedded\]
![Image of IRDC core: MSXDCG028.37+00.07 (d). Image is MSX 8$\umu$m with SCUBA contours overlaid.[]{data-label="pic:limitation"}](G028.37+00.07d.eps){width="40.00000%"}
On several occasions multiple matches were made to the same SCUBA sources where Clumpfind had only identified one object and vice versa. In these instances the closest positional Clumpfind match to the candidate MSX identified IRDC core was taken. A consequence of the differing techniques used to identify objects within the MSX and SCUBA catalogues was seen when classifying by eye those IRDC SCUBA detected candidates with and without embedded objects. Fig. \[pic:limitation\], shows IRDC core MSXDCG028.37+00.07 (d), Clumpfind identified two distinct objects in the 850$\umu$m data but the MSX identification process identified the dark complex as one object.
Of the 251 IRDC cores located within the SCUBA Legacy Catalogue mapped region a total of 46 core matches were manually excluded from the sample. In some cases, this was because the MSX identified IRDC cores were located on positions of extended 850$\umu$m emission which could not be morphologically matched to the compact candidate IRDC cores. Due to the large upper limit used for the matching radius a number of matches were also found to be inappropriate. In other cases, objects were found within 1’ but not coincident with 850$\umu$m emission. This may have occurred due to poor background modelling of the mid-infrared emission. The subtraction of a smoothed background model can potentially result in the creation of artifacts with high contrast values. Cores adjacent to bright extended 8$\umu$m infrared emission, were also amongst those excluded. These objects were removed due to concerns over the process of creating a contrast image in a complex enviroment.
Finally of the 205 remaining IRDC cores, a total of 154 IRDC cores were matched to SCUBA objects from the SCUBA Legacy Catalogue of @2008ApJS..175..277D. The other remaining 51 IRDC cores were identified as having no corresponding 850$\umu$m emission. These MSX identified candidate cores could be due to column densities and dust temperatures below the detection limit of SCUBA. Alternatively they could be a result of uncertainties in the MSX IRDC candidate identification process, we explore these possibilities further in section \[Section:Discussion1\].
Column densities and masses of the cores {#subsection:coldenmass}
----------------------------------------
We derive peak column densities for all IRDC cores within our sample, whether detected at 850$\umu$m or not, by applying the following extinction law to the MSX 8$\umu$m data: $$I_{i} = I_{b} e^{-\tau_{\lambda}}
\label{equation1}$$ where $I_{i}$ is the image intensity, $I_{b}$ is the background model intensity and $\tau_{\lambda}$ is the dust opacity which equals the cross sectional area, $\sigma_{\lambda}$, multiplied by the column density, $N(H_{2})$, i.e. $\tau_{\lambda}=\sigma_{\lambda}.N(H_{2})$. The peak contrast value $C$ is defined by [@2006ApJ...639..227S] as: $$C = \frac{I_{b} - I_{i}}{I_{b}} = 1 - e^{-\tau_{\lambda}}
\label{equation:C}$$ It is then possible, by substituting $\tau_{\lambda}$ from equation \[equation1\] into equation \[equation:C\], to derive: $$N(H_{2}) = \frac{-ln (1 - C)}{\sigma_{\lambda}}
\label{equation:N8(H2)}$$ We assume a value of $\sigma_{\lambda} = 2.3 \times 10^{-23}$cm$^{2}$ for the cross sectional area of the obscuring dust particles at 8.8$\umu$m [@2006ApJS..166..567R]. Column densities for the IRDCs derived by this method are contained in Tables \[table:classA\] and \[table:classC\]. Fig. \[Hist:ColDen\] shows the distribution of those cores detected and not detected at 850$\umu$m with peak contrast. We see that the median column density for the SCUBA detected candidates and the SCUBA non-detected candidates is $1.7\times 10^{22}$cm$^{-2}$ and $1.0\times 10^{22}$cm$^{-2}$ respectively.
For comparison with the values determined from the 8$\umu$m extinction we also calculated the peak column densities for cores detected at 850$\umu$m, using the SCUBA 850$\umu$m data to derive the mass (as defined by @1983QJRAS..24..267H [[email protected]]) and assuming spherical geometry:
$$N(H_{2}) = \frac{F_{\nu} C_{\nu}}{B_{\nu}(T) \pi (tan(B_{850}))^{2} 2 m_{H}}
\label{equation:N850(H2)}$$
where the mass coefficient $C_\nu$ is given by $$C_{\nu} = \frac{M_{g}}{M_{d} \kappa_{\nu}}$$ $F_{\nu}$ is the observed peak flux; $B_{\nu}(T)$ is the Planck function evaluated for dust temperature, $T$; $B_{850}$ is the radius of the beam at 850$\umu$m, which has a FWHM of 19” due to convolution during the data reduction [@2008ApJS..175..277D] and $m_{H}$ is the mass of a hydrogen atom. The value of C$_{\nu}=$ 50gcm$^{-2}$ at 850$\umu$m is taken from [@2001ApJ...552..601K], where $M_{g}$ is the gas mass and $M_{d}$ is the dust mass and $\kappa_{\nu}$ is the dust opacity, assuming a gas to dust ratio of 100 and an opacity gradient $\beta$ of 2.
When evaluating the Planck function, a temperature of 15K was assumed for all the cores as this is the midpoint of the observed range (8–25K) in IRDC temperatures observed by @1998ApJ...508..721C, and . Decreasing or increasing the temperature to 8 or 25K would increase or decrease these column density estimates by a factor of 3.5 and 2.2 respectively.
We compared the column densities derived by each method. In general the column density derived from the 8$\umu$m extinction agrees with that derived from the 850$\umu$m emission to within an order of magnitude. There is considerable scatter but the overall trend is that the 8$\umu$m column density underestimates the 850$\umu$m column density by roughly a factor of 2. This suggests that the average temperature for the IRDC cores may be closer to 10K than our assumption of 15K. However due to the large uncertainties in mass coefficients, the 8$\umu$m extinction law and contamination from foreground emission we do not expect close agreement between these two methods.
Masses were determined for the cores detected at 850$\umu$m using the method of @1983QJRAS..24..267H, i.e. $$M = \frac{d^{2} F_{\nu} C_{\nu}}{B_{\nu}(T)}
\label{mass}$$ A dust temperature of 15K was again assumed for all the cores. As before, decreasing or increasing the dust temperature to 8 or 25K would increase or decrease the masses derived by a factor of 3.5 and 2.2 respectively. $d$ is the distance to each core. Kinematic distances exist for 33 of our cores detected at 850$\umu$m from [@2006ApJ...639..227S], who derived distances by matching up the morphologies of candidate IRDC cores with CO morphologies from the GRS (Galactic Ring Survey). Our mass estimates for these 33 SCUBA detected cores, along with the distance estimates from [@2006ApJ...639..227S] can be found in Table \[table:mass\].
Embedded 24$\umu$m objects in the cores
---------------------------------------
Of the 154 IRDC cores detected at 850$\umu$m, 69 were located within the coverage area of MIPSGAL and 34 out of the 51 cores not detected at 850$\umu$m were also located within the MIPSGAL survey coverage area. The cores were visually inspected at 24$\umu$m and it was found that 48 of the cores detected at 850$\umu$m are positionally associated with one or more 24$\umu$m MIPSGAL sources (approximately half contain more than one 24$\umu$m source). None of the 34 cores not detected at 850$\umu$m are found to be associated with any MIPSGAL 24$\umu$m sources. Fig. \[Fig:cores1\] shows two cores that are seen at both 8$\umu$m and 24$\umu$m, one with an embedded object and one without.
Results {#Section:Results}
=======
MSX identified IRDCs in the SCUBA Legacy Catalogue
--------------------------------------------------
In total 205 candidate IRDC MSX cores were found to be within the SCUBA Legacy Catalogue coverage area. 154 candidate cores had detectable emission at 850$\umu$m, and 51 candidate cores did not. Of the 154 cores detected at 850$\umu$m, we find that they span a range of peak contrast values (0.11–0.62), column densities and masses, with the peak contrast distribution having a median of 0.32. 8$\umu$m column densities range from 0.56$\times 10^{22}$ to 4.21$\times 10^{22}$cm$^{-2}$ with a median of 1.7$\times 10^{22}$cm$^{-2}$. Mass estimates of these candidates range from 50 to 4,190M$_{\odot}$ with a median of 300M$_{\odot}$. Peak contrast values for cores detected and not detected at 850$\umu$m are given in Tables \[table:classA\] and \[table:classC\] respectively. The physical properties of those cores with distance information available (data taken from [@2006ApJ...639..227S] and [@2008ApJ...680..349J]) are seen in Table \[table:mass\].
Those cores not detected at 850$\umu$m are found predominantly at low contrast values ($\le$0.4), with a mean of 0.22 (seen in Fig. \[Hist:ColDen\]). This result is not surprising, SCUBA is naturally expected to detect high column density clouds (which would have high contrast values) and not detect low column density clouds (which would have low contrast values). Although the cores not detected at 850$\umu$m possess lower peak contrasts, we see no evidence for the existence of two separate populations. A Kolmogorov-Smirnoff (KS) two sample test, on the peak contrast values for cores with and without 850$\umu$m emission, reveals no significant difference (to 95%) that the cores originate from two separate populations. Inspection of the rms values for the 205 candidate IRDC cores, as with the 850$\umu$m emission, reveals no significant difference in values between those cores detected at 850$\umu$m and those not detected at 850$\umu$m as seen in Fig. \[Hist:error\]. Thus we are confident that the reason behind the cores not being detected at 850$\umu$m is not due to them simply lying in high noise regions of the SCUBA Legacy Catalogue. It is possible that some of the cores identified by @2006ApJ...639..227S that were not detected at 850$\umu$m are not true cores at all, rather voids in the mid-infrared background. This possibility is suggested by [@2006ApJ...653.1325S] and [@2008ApJ...680..349J], who state at low contrast values the number of mis-identified IRDCs is greater than at high contrasts.
--------------------- --------------------- ------------ --------------------- ------ ----------- ------------------ -------------------- -------------
MSX SCUBA RA Dec C F$_{850}$ N$_{8}$(H$_{2}$) N$_{850}$(H$_{2}$) MIPS
ID ID $\times 10^{22}$ $\times 10^{22}$ source [^8]
(MSXDC) (JCMTS[^9]) (hh:mm:ss) ( $^{\circ}$: ’: ”) (Jy/beam) (cm$^{-2}$) (cm$^{-2}$)
G000.06+00.21 (a) F\_J174457.0-284618 17:44:55.4 -28:46:38 0.22 1.09 1.08 7.91
G000.07+00.24 (a) F\_J174450.2-284449 17:44:50.7 -28:44:45 0.16 0.40 0.76 2.89
G000.08+00.19 (a) F\_J174503.4-284618 17:45:03.8 -28:46:09 0.15 0.76 0.77 5.51
G000.13$-$00.14 (a) F\_J174626.0-285437 17:46:26.8 -28:54:38 0.22 1.41 1.08 10.22
G000.20$-$00.25 (a) F\_J174706.6-285314 17:47:06.3 -28:53:26 0.19 0.57 0.92 4.13
G000.21+00.23 (a) F\_J174510.6-283812 17:45:13.8 -28:37:59 0.14 0.59 0.66 4.28
G000.25+00.02 (a) F\_J174607.0-284130 17:46:07.9 -28:41:35 0.44 6.62 2.52 47.95
G000.32$-$00.23 (a) F\_J174719.9-284656 17:47:19.3 -28:47:01 0.26 0.73 1.31 5.29
G000.32$-$00.18 (a) F\_J174707.2-284432 17:47:08.6 -28:45:19 0.33 0.46 1.74 3.33
G000.33+00.05 (a) F\_J174613.0-283654 17:46:12.7 -28:37:10 0.21 3.74 1.02 27.21
G000.35$-$00.24 (a) F\_J174725.4-284538 17:47:23.3 -28:45:47 0.16 0.34 0.76 2.46
G000.40$-$00.21 (a) F\_J174725.4-284156 17:47:25.3 -28:42:16 0.23 1.26 1.14 9.14
G000.40+00.04 (a) F\_J174624.4-283331 17:46:24.7 -28:33:47 0.45 6.76 2.60 49.03
G000.59+00.01 (e) F\_J174646.8-283155 17:46:47.5 -28:32:07 0.37 14.95 2.01 112.31
G000.59+00.01 (a) F\_J174643.2-283007 17:46:42.9 -28:30:27 0.47 7.83 2.76 56.80
G000.59+00.01 (c) F\_J174648.6-282944 17:46:49.2 -28:29:53 0.37 2.71 2.01 19.65
G000.59+00.01 (f) F\_J174653.2-282632 17:46:53.0 -28:26:40 0.34 2.18 1.81 15.81
G000.59+00.01 (d) F\_J174657.8-282332 17:46:59.7 -28:23:06 0.37 2.10 2.01 16.20
G000.59+00.01 (b) F\_J174709.6-282208 17:47:09.3 -28:21:54 0.37 3.38 2.01 24.41
G000.68$-$00.18 (a) F\_J174800.1-282714 17:48:01.2 -28:27:24 0.20 0.67 0.97 4.86
G000.72$-$00.08 (a) F\_J174740.1-282132 17:47:39.0 -28:21:39 0.16 3.49 0.76 25.27
G000.87$-$00.01 (a) E\_J174744.6-281150 17:47:45.4 -28:11:47 0.15 1.91 0.77 13.84
G000.90$-$00.02 (a) E\_J174751.4-281039 17:47:51.1 -28:10:46 0.13 1.87 0.61 13.56
G000.94+00.00 (a) E\_J174750.1-280803 17:47:49.5 -28:07:37 0.17 1.13 0.81 8.19
G000.97+00.04 (a) E\_J174744.6-280445 17:47:45.3 -28:04:45 0.22 1.18 1.08 8.55
G000.98+00.09 (a) E\_J174733.3-280302 17:47:32.7 -28:02:30 0.13 0.86 0.61 6.24
G000.97$-$00.06 (c) E\_J174831.4-280826 17:48:30.3 -28:08:25 0.22 1.39 1.08 10.09
G001.01+00.05 (b) E\_J174752.3-280321 17:47:53.0 -28:03:27 0.20 0.92 0.97 8.42
G000.97$-$00.06 (a) E\_J174814.6-280550 17:48:15.3 -28:06:05 0.26 1.55 1.31 11.23
G001.01+00.05 (a) E\_J174747.8-280157 17:47:49.1 -28:01:58 0.20 0.90 0.97 6.52
G001.02+00.02 (a) E\_J174755.5-280239 17:47:57.0 -28:02:48 0.11 0.67 0.57 4.86
G001.11$-$00.16 (a) F\_J174846.7-280338 17:48:47.9 -28:03:28 0.29 1.30 1.49 9.42
G001.11$-$00.16 (b) F\_J174851.3-280356 17:48:52.4 -28:04:01 0.24 0.59 1.19 4.28
G001.13$-$00.26 (a) F\_J174915.8-280531 17:49:16.5 -28:05:39 0.20 0.71 0.97 5.14
G001.26$-$00.23 (a) F\_J174930.3-275848 17:49:31.4 -27:59:01 0.14 0.36 0.66 2.61
G001.26+00.04 (a) F\_J174826.6-274949 17:48:25.3 -27:49:53 0.21 1.78 1.02 12.89
G001.29+00.03 (a) F\_J174833.4-274836 17:48:32.5 -27:48:45 0.17 1.62 0.81 11.75
G001.31$-$00.04 (a) F\_J174856.9-275047 17:48:57.0 -27:50:28 0.17 0.67 0.81 4.86
G001.34$-$00.08 (a) F\_J174906.9-274946 17:49:06.7 -27:49:48 0.13 0.42 0.61 3.05
G001.47$-$00.03 (a) F\_J174914.3-274211 17:49:13.2 -27:41:55 0.18 0.40 0.86 2.89
G001.46+00.03 (a) E\_J174855.8-273917 17:48:56.3 -27:39:24 0.29 2.31 1.49 16.74
G001.51$-$00.10 (a) F\_J174934.7-274212 17:49:36.7 -27:41:53 0.17 1.64 0.81 11.88
G001.53+00.14 (a) E\_J174842.2-273230 17:48:41.0 -27:33:04 0.13 0.82 0.61 5.94
G001.61$-$00.02 (b) E\_J174936.6-273248 17:49:36.5 -27:33:11 0.37 0.71 2.01 5.14
G001.61$-$00.02 (a) E\_J174944.3-273330 17:49:44.4 -27:33:11 0.41 2.67 2.29 19.35
G001.67$-$00.18 (b) E\_J175015.0-273437 17:50:16.0 -27:34:22 0.23 0.63 1.14 4.56
G002.51+00.18 (b) F\_J175045.8-263945 17:50:43.4 -26:40:21 0.38 2.77 2.08 20.09
G002.51+00.18 (a) F\_J175045.8-263945 17:50:48.5 -26:39:30 0.41 2.77 2.29 20.09
G008.83$-$00.05 (e) F\_J180525.3-211926 18:05:27.7 -21:20:17 0.27 4.22 1.37 30.67
G008.83$-$00.05 (a) F\_J180525.3-211926 18:05:26.3 -21:19:04 0.32 4.22 1.68 30.67
G009.84$-$00.03 (a) E\_J180733.9-202613 18:07:37.4 -20:26:20 0.43 0.65 2.44 4.71 yes
G010.71$-$00.16 (g) E\_J180944.4-194712 18:09:44.4 -19:47:02 0.35 0.46 1.87 3.33 yes
G010.71$-$00.16 (h) E\_J180953.3-194806 18:09:52.5 -19:47:40 0.34 0.65 1.81 4.71 yes
\[table:classA\]
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MSX SCUBA RA Dec C F$_{850}$ N$_{8}$(H$_{2}$) N$_{850}$(H$_{2}$) MIPS
ID ID $\times 10^{22}$ $\times 10^{22}$ source
(MSXDC) (JCMTS) (hh:mm:ss) ( $^{\circ}$: ’: ”) (Jy/beam) (cm$^{-2}$) (cm$^{-2}$)
G010.71$-$00.16 (b) E\_J180938.7-194512 18:09:38.7 -19:45:15 0.49 2.79 2.93 20.22 yes
G010.71$-$00.16 (d) E\_J180949.1-194442 18:09:48.9 -19:44:52 0.40 0.44 2.22 3.20 no
G010.71$-$00.16 (f) E\_J180940.8-194336 18:09:41.5 -19:43:42 0.36 2.35 1.94 17.04 no
G010.71$-$00.16 (a) E\_J180945.7-194206 18:09:45.5 -19:42:22 0.56 1.28 3.57 9.29 yes
G010.71$-$00.16 (e) E\_J181009.0-194507 18:10:10.0 -19:45:13 0.38 0.63 2.80 4.56 no
G010.71$-$00.16 (c) E\_J181003.1-194342 18:10:02.4 -19:43:28 0.44 0.84 2.52 6.09 yes
G010.99$-$00.07 (a) F\_J181007.1-192755 18:10:07.2 -19:27:60 0.55 1.39 3.47 10.09 yes
G011.11$-$00.11 (g) F\_J181013.5-192419 18:10:13.7 -19:24:36 0.30 0.67 1.55 4.86 no
G011.11$-$00.11 (b) F\_J181018.2-192431 18:10:18.7 -19:24:42 0.43 0.88 2.44 6.37 yes
G011.11$-$00.11 (a) F\_J181033.0-192201 18:10:32.0 -19:22:31 0.50 1.45 3.01 10.52 yes
G011.11$-$00.11 (f) F\_J181033.0-192201 18:10:35.4 -19:21:02 0.30 0.73 1.55 5.29 yes
G011.11$-$00.11 (d) F\_J181037.3-191820 18:10:37.7 -19:18:27 0.31 0.36 1.61 2.61 yes
G011.11$-$00.11 (c) F\_J181034.8-191702 18:10:35.3 -19:17:20 0.40 0.80 2.22 5.81 yes
G011.11$-$00.11 (e) F\_J181039.9-191132 18:10:40.6 -19:10:58 0.31 0.76 1.61 5.51 yes
G012.44$-$00.14 (b) F\_J181320.6-181220 18:13:21.6 -18:12:17 0.34 0.36 1.81 2.61 yes
G012.44$-$00.14 (a) F\_J181341.7-181239 18:13:41.5 -18:12:32 0.45 1.43 2.60 20.37 yes
G012.44$-$00.14 (d) F\_J181331.6-181115 18:13:32.5 -18:11:18 0.29 0.36 1.49 2.61 no
G012.88+00.53 (a) F\_J181145.3-173044 18:11:44.8 -17:31:17 0.36 1.11 1.94 8.06 yes
G013.02$-$00.83 (a) F\_J181700.1-180202 18:17:00.7 -18:02:18 0.47 0.27 2.76 1.97 yes
G013.68$-$00.60 (a) F\_J181725.7-172049 18:17:27.6 -17:20:59 0.24 0.42 1.19 3.05 no
G018.50$-$00.16 (c) E\_J182517.5-125526 18:25:17.5 -12:55:31 0.27 0.29 1.37 2.10 no
G018.50$-$00.16 (d) F\_J182523.2-125450 18:25:23.3 -12:54:55 0.25 0.59 1.25 4.28 no
G018.50$-$00.16 (b) F\_J182520.4-125014 18:25:21.2 -12:50:18 0.29 0.32 1.49 2.31 no
G018.58$-$00.08 (b) F\_J182507.3-124750 18:25:07.6 -12:48:00 0.32 0.46 1.68 3.33 yes
G018.58$-$00.08 (a) F\_J182508.5-124520 18:25:08.9 -12:45:20 0.37 1.91 2.01 13.84 yes
G019.27+00.07 (a) F\_J182552.1-120456 18:25:54.0 -12:04:56 0.50 1.03 3.01 7.47 yes
G022.35+00.41 (b) F\_J183029.6-091238 18:30:28.7 -09:12:31 0.37 0.38 2.01 2.76 no
G022.35+00.41 (a) F\_J183024.4-091038 18:30:24.7 -09:10:47 0.51 1.89 3.10 13.69 yes
G023.86$-$00.19 (a) E\_J183526.9-080814 18:35:26.6 -08:08:22 0.32 0.42 1.68 3.05 no
G024.00+00.15 (a) E\_J183428.8-075220 18:34:29.5 -07:52:23 0.20 1.11 0.97 8.06 yes
G024.36$-$00.16 (a) F\_J183618.3-074102 18:36:17.5 -07:41:27 0.41 0.88 2.29 6.37 yes
G024.37$-$00.21 (a) F\_J183630.0-074208 18:36:30.2 -07:42:16 0.34 0.34 1.81 2.46 yes
G024.60+00.08 (a) F\_J183540.1-071838 18:35:39.4 -07:18:51 0.49 2.02 2.93 14.64 yes
G024.68+00.17 (a) F\_J183540.1-071514 18:35:41.2 -07:15:22 0.20 0.25 0.97 1.81 yes
G025.04$-$00.20 (g) F\_J183712.0-071126 18:37:12.8 -07:11:23 0.36 0.76 1.94 5.51 yes
G025.04$-$00.20 (e) F\_J183719.2-071144 18:37:18.8 -07:11:49 0.41 1.13 2.29 8.19 yes
G025.04-00.20 (b) F\_J183734.6-070726 18:37:34.8 -07:07:39 0.44 0.38 2.52 2.76 yes
G025.04$-$00.20 (f) F\_J183738.2-070550 18:37:38.2 -07:06:00 0.38 0.29 2.08 2.10 no
G028.37+00.07 (a) F\_J184250.6-040314 18:42:50.6 -04:03:30 0.61 2.52 4.09 18.27 yes
G028.37+00.07 (d) F\_J184248.2-040133 18:42:48.6 -04:01:42 0.47 0.65 2.76 4.71 yes
G028.37+00.07 (b) F\_J184255.4-040150 18:42:55.5 -04:01:47 0.51 0.95 3.10 6.89 no
G028.37+00.07 (e) F\_J184300.2-040132 18:43:00.5 -04:01:36 0.45 0.69 2.60 5.01 no
G028.53$-$00.25 (g) F\_J184417.3-040208 18:44:17.0 -04:02:18 0.27 0.61 1.37 5.40 yes
G028.53$-$00.25 (b) F\_J184422.5-040150 18:44:23.7 -04:02:09 0.38 0.57 2.08 4.13 yes
G028.53$-$00.25 (c) F\_J184415.6-040056 18:44:16.6 -04:01:02 0.34 1.03 1.81 7.47 yes
G028.53$-$00.25 (a) F\_J184418.1-035938 18:44:17.1 -03:59:37 0.41 2.65 2.29 19.20 yes
G028.53$-$00.25 (e) F\_J184418.1-035938 18:44:17.7 -03:58:16 0.29 0.69 1.49 5.01 yes
G028.61$-$00.26 (a) F\_J184428.1-035750 18:44:29.0 -03:57:46 0.27 0.32 1.37 2.31 yes
G030.77+00.22 (a) F\_J184647.8-014856 18:46:47.1 -01:49:03 0.25 4.62 1.25 33.48 yes
G030.97$-$00.14 (a) E\_J184821.9-014832 18:48:24.2 -01:48:25 0.38 1.89 2.08 13.69 yes
G031.03+00.26 (b) F\_J184701.4-013438 18:47:01.5 -01:34:47 0.29 0.78 1.49 5.66 yes
G031.03+00.26 (c) F\_J184707.4-013432 18:47:07.7 -01:34:42 0.29 0.50 1.49 3.63 yes
G031.03+00.26 (a) F\_J184701.4-013314 18:47:01.2 -01:33:23 0.31 0.55 1.61 4.00 yes
G031.23+00.05 (a) E\_J184807.5-012844 18:48:08.3 -01:28:50 0.21 0.82 1.02 5.94 yes
G031.27+00.08 (a) E\_J184807.9-012626 18:48:08.0 -01:26:43 0.16 0.69 0.76 5.01 no
G031.38+00.29 (a) F\_J184732.6-011338 18:47:34.0 -01:13:59 0.38 1.53 2.08 11.10 no
G031.97+00.07 (b) F\_J184922.1-005038 18:49:22.2 -00:50:47 0.42 0.59 2.37 4.28 yes
G031.97+00.07 (c) F\_J184926.9-005002 18:49:27.0 -00:50:11 0.35 0.36 1.87 2.61 no
G033.69$-$00.01 (e) E\_J185248.6+003602 18:52:49.6 +00:35:55 0.32 0.63 1.68 4.56 no
G033.69$-$00.01 (b) E\_J185252.6+003832 18:52:53.1 +00:38:10 0.38 0.65 2.08 4.71 no
G033.69$-$00.01 (c) E\_J185253.8+004120 18:52:53.0 +00:40:44 0.37 0.59 2.01 4.28 no
\[table:classA2\]
--------------------- --------------------- ------------ --------------------- ------ ----------- ------------------ -------------------- --------
--------------------- --------------------- ------------ --------------------- ------ ----------- ------------------ -------------------- --------
MSX SCUBA RA Dec C F$_{850}$ N$_{8}$(H$_{2}$) N$_{850}$(H$_{2}$) MIPS
ID ID $\times 10^{22}$ $\times 10^{22}$ source
(MSXDC) (JCMTS) (hh:mm:ss) ( $^{\circ}$: ’: ”) (Jy/beam) (cm$^{-2}$) (cm$^{-2}$)
G033.69$-$00.01 (a) E\_J185257.0+004302 18:52:57.6 +00:42:59 0.38 1.60 2.08 11.60 yes
G034.43+00.24 (a) F\_J185318.9+012650 18:53:18.9 +01:26:39 0.34 0.38 1.81 2.76 yes
G038.95$-$00.47 (a) F\_J190407.5+050844 19:04:08.3 +05:08:49 0.53 1.37 3.28 9.94 yes
G048.52$-$00.47 (a) F\_J192207.4+133713 19:22:07.9 +13:36:58 0.38 0.29 2.08 2.10 no
G048.65$-$00.29 (a) F\_J192144.7+134925 19:21:45.3 +13:49:22 0.34 0.40 1.81 2.89 yes
G079.24+00.41 (b) E\_J203137.7+401935 20:31:38.1 +40:19:38 0.51 0.88 3.10 6.37
G079.24+00.41 (a) F\_J203157.6+401828 20:31:56.8 +40:18:12 0.52 1.99 3.19 14.43
G081.67+00.44 (a) F\_J203924.9+421555 20:39:21.0 +42:15:47 0.23 2.31 1.14 16.74
G081.73+00.59 (a) F\_J203859.3+422330 20:38:58.2 +42:23:55 0.27 8.19 1.37 59.39
G081.76+00.63 (a) F\_J203851.6+422717 20:38:52.6 +42:27:12 0.17 0.80 0.81 5.81
G351.52+00.69 (a) F\_J172056.6-354044 17:20:58.2 -35:40:28 0.39 2.48 2.15 17.97
G353.26$-$00.16 (a) F\_J172935.1-344316 17:29:33.6 -34:43:31 0.31 0.44 1.61 3.20
G353.37$-$00.33 (b) F\_J173012.1-344328 17:30:12.4 -34:43:45 0.38 1.47 2.08 10.65
G353.37$-$00.33 (a) F\_J173017.0-344217 17:30:18.8 -34:41:58 0.43 2.29 2.44 19.61
G353.90+00.25 (e) F\_J172902.5-335950 17:29:02.2 -34:00:12 0.46 1.13 2.68 8.19
G353.90+00.25 (a) F\_J172917.1-340017 17:29:12.8 -34:00:01 0.62 0.55 4.21 4.00
G353.90+00.25 (f) F\_J172917.1-340017 17:29:16.8 -34:00:25 0.46 0.61 2.68 5.40
G353.90+00.25 (c) F\_J172919.4-335550 17:29:19.1 -33:55:59 0.50 0.59 3.01 4.28
G353.90+00.25 (b) F\_J172928.5-335444 17:29:27.9 -33:55:06 0.59 0.50 3.88 3.63
G353.90+00.25 (d) F\_J172927.1-335302 17:29:25.0 -33:53:03 0.49 0.90 2.93 6.52
G359.05+00.00 (a) F\_J174321.6-294437 17:43:21.8 -29:44:43 0.27 0.63 1.37 4.56
G359.06$-$00.03 (a) F\_J174326.7-294531 17:43:29.7 -29:45:22 0.16 0.71 0.76 5.14
G359.08+00.04 (a) F\_J174314.8-294143 17:43:14.8 -29:42:02 0.20 0.50 0.97 3.63
G359.29$-$00.03 (a) F\_J174404.5-293302 17:44:03.5 -29:33:12 0.21 1.28 1.02 9.29
G359.37$-$00.28 (a) F\_J174514.5-293644 17:45:14.1 -29:37:27 0.12 0.27 0.56 19.57
G359.41+00.08 (a) F\_J174354.0-292314 17:43:53.9 -29:23:27 0.16 1.03 0.76 7.47
G359.46$-$00.03 (a) F\_J174428.9-292426 17:44:29.3 -29:24:18 0.28 2.42 1.43 19.54
G359.48$-$00.22 (a) F\_J174514.4-292902 17:45:15.1 -29:29:30 0.14 1.87 0.66 13.56
G359.59+00.02 (a) F\_J174436.3-291621 17:45:33.5 -29:24:26 0.31 1.87 1.61 15.56
G359.60$-$00.22 (b) F\_J174535.0-292456 17:44:32.0 -29:16:05 0.20 1.97 0.97 14.28
G359.60$-$00.22 (a) F\_J174535.0-292314 17:45:35.2 -29:23:15 0.36 5.50 1.94 39.96
G359.68$-$00.13 (a) F\_J174526.3-291608 17:45:25.3 -29:16:06 0.20 0.61 0.97 5.40
G359.80$-$00.13 (a) F\_J174539.0-291132 17:45:37.1 -29:11:25 0.32 1.62 1.68 11.75
G359.82+00.12 (a) F\_J174443.6-290127 17:44:43.1 -29:01:02 0.19 0.44 0.92 3.20
G359.83+00.18 (a) E\_J174429.3-285901 17:44:29.5 -28:59:13 0.21 0.38 1.02 2.76
G359.85+00.21 (a) E\_J174425.2-285643 17:44:25.3 -28:56:49 0.23 0.38 1.14 2.76
G359.87$-$00.09 (a) F\_J174544.0-290502 17:45:43.4 -29:05:22 0.25 5.44 1.25 3.95
G359.90$-$00.30 (a) F\_J174636.3-291011 17:46:35.8 -29:10:26 0.35 2.33 1.87 16.89
G359.91+00.17 (b) F\_J174444.4-285519 17:44:43.6 -28:55:28 0.45 0.90 2.60 6.52
G359.91+00.17 (a) F\_J174448.5-285349 17:44:47.6 -28:53:54 0.58 4.41 3.77 31.96
\[table:classA3\]
--------------------- --------------------- ------------ --------------------- ------ ----------- ------------------ -------------------- --------
--------------------- -------------- -------------- ------------ --------------------- ---------- ---------------------------
MSX ID l b RA Dec Peak N$_{8}$(H$_{2}$)
(MSXDC) ($^{\circ}$) ($^{\circ}$) (hh:mm:ss) ( $^{\circ}$: ’: ”) Contrast $\times 10^{22}$cm$^{-2}$
G000.13$-$00.14 (b) 0.163 -0.164 17:46:38.8 -28:52:56 0.19 0.92
G000.13$-$00.19 (a) 0.131 -0.196 17:46:41.8 -28:55:34 0.21 1.02
G000.36$-$00.21 (a) 0.364 -0.216 17:47:19.6 -28:44:14 0.20 0.97
G000.73$-$00.01 (a) 0.733 -0.014 17:47:24.6 -28:19:02 0.15 0.77
G001.62$-$00.08 (a) 1.631 -0.092 17:49:48.6 -27:35:17 0.16 0.76
G004.33$-$00.04 (a) 4.336 -0.052 17:55:48.0 -25:14:19 0.34 1.81
G004.33$-$00.04 (c) 4.358 -0.056 17:55:51.9 -25:13:18 0.29 1.49
G006.06-01.39 (a) 6.063 -1.397 18:04:43.5 -24:24:28 0.23 1.14
G006.09-01.39 (a) 6.094 -1.396 18:04:47.3 -24:22:49 0.24 1.19
G006.09-01.36 (a) 6.094 -1.367 18:04:40.6 -24:21:58 0.20 0.97
G010.57$-$00.30 (a) 10.588 -0.311 18:10:08.1 -19:55:35 0.33 1.74
G010.94$-$00.05 (a) 10.944 -0.059 18:09:55.7 -19:29:35 0.18 0.86
G012.37+00.50 (c) 12.431 0.496 18:10:54.4 -17:55:22 0.29 1.49
G012.44$-$00.20 (a) 12.449 -0.201 18:13:30.9 -18:14:31 0.15 0.77
G012.88+00.53 (c) 12.886 0.528 18:11:42.5 -17:30:31 0.29 1.49
G013.15+00.09 (a) 13.154 0.099 18:13:49.6 -17:28:46 0.25 1.25
G017.00+00.67 (a) 17.003 0.661 18:19:22.5 -13:49:34 0.33 1.74
G017.01+00.78 (a) 17.013 0.789 18:18:55.8 -13:45:24 0.13 0.61
G017.03+00.71 (a) 17.029 0.719 18:19:13.0 -13:46:33 0.14 0.66
G017.10+00.71 (a) 17.096 0.699 18:19:25.1 -13:43:34 0.31 1.61
G017.10+00.71 (b) 17.129 0.711 18:19:26.4 -13:41:29 0.31 1.61
G018.99$-$00.30 (b) 19.004 -0.306 18:26:44.4 -12:30:45 0.28 1.43
G024.60+00.08 (b) 24.659 0.163 18:35:40.9 -07:16:57 0.32 1.68
G024.60+00.08 (d) 24.596 0.131 18:35:40.7 -07:21:11 0.29 1.49
G025.12$-$00.16 (a) 25.126 -0.162 18:37:42.6 -07:01:01 0.12 0.56
G025.37$-$00.06 (a) 25.419 -0.104 18:38:02.7 -06:43:48 0.26 1.31
G025.42+00.10 (a) 25.426 0.109 18:37:17.7 -06:37:34 0.14 0.66
G027.93$-$00.34 (a) 27.924 -0.344 18:43:30.8 -04:36:47 0.19 0.92
G031.32+00.09 (a) 31.326 0.094 18:48:10.4 -01:23:11 0.17 0.81
G031.33+00.12 (a) 31.336 0.124 18:48:05.1 -01:21:49 0.16 0.76
G033.69$-$00.01 (d) 33.639 -0.056 18:52:55.6 +00:36:14 0.35 1.87
G034.94+00.37 (a) 34.941 0.384 18:53:44.2 +01:57:47 0.18 0.86
G042.75+00.01 (a) 42.751 0.019 19:09:25.2 +08:44:19 0.21 1.02
G042.75$-$00.19 (a) 42.753 -0.202 19:10:13.0 +08:38:18 0.22 1.08
G080.88$-$00.12 (a) 80.886 -0.131 20:39:16.7 +41:17:23 0.21 1.02
G081.49+00.13 (a) 81.504 0.129 20:40:11.5 +41:56:17 0.40 2.22
G081.49+00.13 (b) 81.498 0.161 20:40:02.2 +41:57:11 0.34 1.81
G081.56+00.57 (a) 81.564 0.581 20:38:27.4 +42:15:40 0.16 0.76
G081.57+00.50 (b) 81.576 0.523 20:38:44.7 +42:14:07 0.27 1.37
G081.60+00.58 (a) 81.603 0.586 20:38:33.7 +42:17:42 0.17 0.81
G081.69+00.71 (a) 81.699 0.708 20:38:21.2 +42:26:43 0.19 0.92
G351.50+00.66 (a) 351.509 0.661 17:21:05.1 -35:41:56 0.17 0.81
G353.26$-$00.16 (f) 353.281 -0.207 17:29:28.1 -34:43:09 0.24 1.19
G353.90+00.25 (g) 353.886 0.254 17:29:13.6 -33:57:36 0.40 2.22
G353.98+00.39 (a) 353.993 0.391 17:28:57.7 -33:47:43 0.20 0.97
G359.25+00.01 (a) 359.254 0.016 17:43:46.2 -29:33:50 0.15 0.77
G359.28+00.02 (a) 359.298 0.031 17:43:49.0 -29:31:06 0.38 2.08
G359.28+00.02 (b) 359.298 0.006 17:43:54.9 -29:31:54 0.34 1.81
G359.79$-$00.25 (b) 359.799 -0.267 17:46:11.1 -29:14:48 0.23 1.14
G359.81$-$00.29 (a) 359.814 -0.297 17:46:20.3 -29:14:58 0.17 0.81
G359.82$-$00.37 (b) 359.843 -0.367 17:46:40.9 -29:15:40 0.27 1.37
\[table:classC\]
--------------------- -------------- -------------- ------------ --------------------- ---------- ---------------------------
--------------------- --------------------- -------------- -------------- ---------- ------------ ---------- ------------- ----------- -- --
MSX ID SCUBA ID l b Peak Flux [^10] Distance Mass MIPS
(MSXDC) (JCMTS) ($^{\circ}$) ($^{\circ}$) Contrast Jy kpc M$_{\odot}$ detected?
G018.50$-$00.16 (b) F\_J182520.4-125014 18.558 -0.159 0.29 1.79 4.1 240 No
G018.58$-$00.08 (b) F\_J182507.3-124750 18.566 -0.092 0.32 0.97 3.8 110 Yes
G018.58$-$00.08 (a) F\_J182508.5-124520 18.608 -0.076 0.37 4.51 3.8 500 Yes
G019.27+00.07 (a) F\_J182552.1-120456 19.289 0.076 0.50 3.24 2.4 130 Yes
G022.35+00.41 (b) F\_J183029.6-091238 22.356 0.416 0.37 2.02 4.3 290 No
G022.35+00.41 (a) F\_J183024.4-091038 22.374 0.444 0.51 4.94 4.3 720 Yes
G023.86$-$00.19 (a) E\_J183526.9-080814 23.871 -0.179 0.32 1.92 4.0 240 No
G024.36$-$00.16 (a) F\_J183618.3-074102 24.366 -0.159 0.41 1.77 3.9 190 Yes
G024.37$-$00.21 (a) F\_J183630.0-074208 24.378 -0.212 0.34 0.46 3.9 50 Yes
G024.60+00.08 (a) F\_J183540.1-071838 24.628 0.154 0.49 5.22 3.7 560 Yes
G028.37+00.07 (a) F\_J184250.6-040314 28.341 0.058 0.61 10.8 5.0 1010 Yes
G028.37+00.07 (d) F\_J184248.2-040133 28.364 0.079 0.47 2.52 5.0 100 Yes
G028.37+00.07 (g) F\_J184239.7-040027 28.366 0.121 0.38 1.36 5.0 100 No
G028.37+00.07 (b) F\_J184255.4-040150 28.376 0.053 0.51 3.34 5.0 310 No
G028.37+00.07 (e) F\_J184300.2-040132 28.388 0.036 0.45 2.18 5.0 130 No
G028.37+00.07 (f) F\_J184252.3-035956 28.403 0.064 0.43 3.72 5.0 4160 No
G028.53$-$00.25 (a) F\_J184418.1-035938 28.563 -0.232 0.41 12.8 5.7 4190 Yes
G028.61$-$00.26 (a) F\_J184428.1-035750 28.613 -0.262 0.27 0.56 4.2 80 Yes
G030.97$-$00.14 (a) E\_J184821.9-014832 30.978 -0.149 0.38 6.30 5.1 1270 Yes
G031.03+00.26 (b) F\_J184701.4-013438 31.023 0.261 0.29 2.97 6.6 520 Yes
G031.03+00.26 (c) F\_J184707.4-013432 31.036 0.239 0.29 1.32 6.6 160 Yes
G031.03+00.26 (a) F\_J184701.4-013314 31.043 0.273 0.31 3.43 6.6 460 Yes
G031.38+00.29 (a) F\_J184732.6-011338 31.393 0.299 0.38 6.96 6.6 2460 No
G031.97+00.07 (b) F\_J184922.1-005038 31.943 0.074 0.42 0.88 6.9 420 Yes
G031.97+00.07 (c) F\_J184926.9-005002 31.961 0.061 0.35 0.82 6.9 490 No
G033.69$-$00.01 (e) E\_J185248.6+003602 33.623 -0.036 0.32 1.76 7.1 690 No
G033.69$-$00.01 (b) E\_J185252.6+003832 33.663 -0.032 0.38 1.60 7.1 620 No
G033.69$-$00.01 (c) E\_J185253.8+004120 33.701 -0.012 0.37 1.35 7.1 530 No
G033.69$-$00.01 (a) E\_J185257.0+004302 33.743 -0.012 0.38 7.05 7.1 2750 Yes
G034.43+00.24 (a) F\_J185318.9+012650 34.431 0.241 0.34 1.20 3.7 100 Yes
G038.95$-$00.47 (a) F\_J190407.5+050844 38.959 -0.469 0.53 4.58 2.7 290 Yes
G048.52$-$00.47 (a) F\_J192207.4+133713 48.519 -0.467 0.38 0.76 2.8 50 No
G048.65$-$00.29 (a) F\_J192144.7+134925 48.658 -0.289 0.34 1.97 2.5 100 Yes
\[table:mass\]
--------------------- --------------------- -------------- -------------- ---------- ------------ ---------- ------------- ----------- -- --
Detection rates
---------------
We investigated the fraction of cores detected at 850$\umu$m versus peak contrast to identify any trends in the detection fraction of IRDC cores. We calculated the detection fraction by dividing the number of cores with 850$\umu$m detections by the total number of MSX IRDC cores within the SCUBA Legacy Catalogue at specific contrast values. Overall the fraction of IRDC cores detected by SCUBA is found to be 75%. At low contrast values we detect over 60% of the IRDC cores, whereas all IRDC cores are detected when the contrast is high as seen in Fig. \[Hist:Reliability\]. The error bars in Fig. \[Hist:Reliability\] were calculated by assuming an uncertainty of $\sqrt{N}$ where $N$ is the number of cores detected by SCUBA per bin and using propagation of errors.
MIPSGAL 24$\umu$m sources associated with IRDCs detected at 850$\umu$m {#subsection:SCUBAdetectedIRDCs}
----------------------------------------------------------------------
We present a histogram of the 69 IRDC cores detected at 850$\umu$m within the MIPSGAL coverage area in Fig. \[Hist:embedded\]. 48 of the IRDC cores were found to be positionally associated with one or more 24$\umu$m MIPSGAL sources and 21 were found not to be associated with any MIPSGAL sources
To investigate if the cores with and without embedded 24$\umu$m objects were drawn from the same population, a KS test was once again performed on their peak contrast distribution and it was found that they are highly likely to originate from the same population, with no distinct differences in their peak contrast distribution to a level of significance $>$99%.
{width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"}
Discussion {#Section:Discussion}
==========
The reliability of the MSX IRDC catalogue {#subsection:Reliability}
-----------------------------------------
Originally, when the [@2006ApJ...639..227S] catalogue was published, an initial reliability of 82% was reported for IRDCs with contrast values $>$0.25. This initial reliability was estimated for the large high contrast clouds by comparison to other source lists from MSX and ISO data (@2006ApJ...639..227S and references there in). Later [@2008ApJ...680..349J] determined a reliability (against CS J=2–1 detections) for low contrast (0.2-0.4) objects of approximately 50% increasing to almost 100% at high contrasts ($>$0.6), with an overall reliability of $\sim$59%. However the [@2008ApJ...680..349J] sample lacked very low contrast objects, the selection criteria used were peak contrasts $>$0.32 and angular sizes $>$42”. Expanding this estimate to the whole MSX IRDC catalogue [@2008ApJ...680..349J] stated that it was $>$50% reliable for all contrasts. These estimates of reliability were obtained via molecular line spectroscopy of $^{13}$CO and CS data from [@2006ApJ...653.1325S] and [@2008ApJ...680..349J] respectively.
Our mean detection rate of IRDC cores with 850$\umu$m emission is 75% (this value varies over a range of peak contrast values as seen in Fig. \[Hist:Reliability\]) which is greater than the reliability of 50% stated by [@2008ApJ...680..349J]. However this detection rate does not take into account the number of inconclusive matches that make up 19% of the IRDC sample. The close correspondence of 850$\umu$m emission, CS 2–1 and $^{13}$CO detection rates for IRDCs places greater confidence in the high contrast [@2006ApJ...653.1325S] candidate IRDCs as true molecular clouds.
Cores not detected at 850$\umu$m {#Section:Discussion1}
--------------------------------
Of the 205 cores within our sample 51 cores were not detected at 850$\umu$m. Those cores detected at 850$\umu$m were found to have higher peak contrast vales and column densities than those cores not detected at 850$\umu$m. The difference in peak contrast values can be seen in Fig. \[Hist:ColDen\]. The median column density of the cores detected at 850$\umu$m is a factor of $\sim$1.6 times greater than the cores not detected at 850$\umu$m. A KS test of the peak contrast distribution could find no significant difference between the cores detected at 850$\umu$m and those that were not.
Although the mid-infrared contrast value for a particular cloud should vary with the intensity of the background, it is possible to determine an estimate of the peak contrast sensitivity of SCUBA as a function of temperature by substituting equation \[equation:N8(H2)\] into equation \[equation:N850(H2)\], using an 850$\umu$m flux limit of 3$\sigma$ (where $\sigma$ is the median rms sensitivity of the SCUBA Legacy Catalogue) for $F_{\nu}$ and rearranging for $C$: $$C = 1 - exp \left( -\frac{F_{\nu} C_{\nu} \sigma_{\lambda}}{B_{\nu}(T) \pi (tan(B_{850}))^{2} 2 m_{H}} \right)
\label{equation:sensitivity}$$ The median rms sensitivity value at 850$\umu$m in the SCUBA Legacy Catalogue after applying our noise cut is 50mJybeam$^{-1}$. We plot the limiting contrast of the SCUBA Legacy Catalogue as a function of temperature, derived using equation \[equation:sensitivity\], in Fig. \[Graph:Sensitivity\]. We also plot the forecast sensitivity of the SCUBA-2 Legacy Survey SASSy (the SCUBA-2 “All Sky” Survey; @2007arXiv0704.3202T) and JPS (the JCMT Plane Survey; @2005prpl.conf.8370M), which will be discussed further in Section \[section:SASSy\].
Fig. \[Graph:Sensitivity\] shows that the median sensitivity of the SCUBA Legacy Catalogue would be sufficient to detect the majority if the @2006ApJ...639..227S IRDC cores if they have temperatures greater than 10K. At this temperature we would detect IRDC cores with peak contrast greater than 0.15. This corresponds to the approximate completeness limit of the [@2006ApJ...639..227S] catalogue, where the turnover in peak contrast occurs. In our sample of 205 IRDCs within the SCUBA Legacy Catalogue 94% have peak contrast values greater than or equal to 0.15.
The IRDC cores that are not detected at 850$\umu$m are thus consistent with being low temperature, low column density cores but below the 0.1Jypixel$^{-1}$ noise cut we applied to the SCUBA Legacy Catalogue. Fig. \[Graph:Sensitivity\] shows that if they are true clouds they are likely to have temperatures less than 10K. Almost none were identified with 24$\umu$m MIPSGAL sources, corroborating our low temperature hypothesis and implying that they are either transient or potentially prestellar cores.
If we consider the cores not detected at 850$\umu$m, to be low temperature, low column density cores, this requires that they have a temperature lower than $\sim$10K, whereas the lowest contrast objects in the sample could have temperatures less than 14K. Typical temperatures for IRDCs range from 8–25K . However as we do not know how IRDC cores are arranged in temperature we cannot know if we ought to have detected the majority of these objects or not. We therefore cannot rule out the presence of a cold faint transient or prestellar population within the SCUBA non-detected sample, particularly at low contrast values where SCUBA is least sensitive. Recent theoretical models [@2007MNRAS.379.1390S] suggest that the temperature of prestellar cores may be lower than suspected ($\sim$5–10K), which would place the 850$\umu$m fluxes of the IRDC cores below our detection limit.
An alternative hypothesis is that a number of the cores not detected at 850$\umu$m are a result of the absence of background mid-infrared emission rather than its extinction by intervening cold dust in an IRDC. In this case some the cores not detected at 850$\umu$m would be localised “holes” or local minima in the mid-infrared background masquerading as IRDCs. This possibility is more likely for the higher contrast cores not detected at 850$\umu$m, as our temperature constraints for these objects mean that they are less likely to be prestellar. Artefacts may also be present as a result of the background subtraction process, which would result in false IRDC detections particularly in regions of complex emission [@2006ApJ...653.1325S]. Without deeper submillimetre continuum or molecular line data it is difficult to satisfactorily determine whether an IRDC core identified by @2006ApJ...639..227S and not detected at 850$\umu$m is a true cloud, void or artefact.
Cores detected at 850$\umu$m
----------------------------
We identified 154 cores detected at 850$\umu$m within the area covered by the SCUBA Legacy Catalogue. These cores have higher peak contrast values (as seen in Fig. \[Hist:ColDen\]) and column densities than those cores cores not detected at 850$\umu$m. Clearly as these objects are seen in submillimetre emission they are not voids or artefacts in the MSX contrast images. We determine estimates of the mass of the 33 cores in our sample with kinematic distances (see Section \[subsection:coldenmass\]). The median mass of cores within the sample is 300M$_{\odot}$, with a minimum mass of 50M$_{\odot}$ and a maximum mass of 4,190M$_{\odot}$ (assuming a dust temperature of 15K). Our results are consistent with those of @2006ApJ...641..389R, who observed the 38 highest contrast clouds from [@2006ApJ...653.1325S] at 1.2mm, taking into account differences in sample selection, assumed temperature and in the measurement of integrated fluxes (@2006ApJ...641..389R fit Gaussians to their sample whereas the SCUBA Legacy Catalogue uses Clumpfind).
Are these masses consistent with high mass star formation within the IRDCs? There is considerable uncertainty regarding the minimum mass core needed to form a high mass star, by considering the observed range in star formation efficiencies estimated that a core mass of at least 30–200M$_{\odot}$ would be required to form a 10M$_{\odot}$ star. Observed values for high mass star forming cores range from 720M$_{\odot}$ to 10$^{4}$M$_{\odot}$ . Our sample of SCUBA detected IRDC cores falls at the lower end of this observed range of masses and is largely consistent with the estimate of the mass required for high mass star formation. We thus conclude that the masses of the IRDC cores in our sample are sufficient to support intermediate to high mass star formation.
IRDC cores detected at 850$\umu$m without 24$\umu$m sources: could they be “starless” IRDCs? {#section:starless}
--------------------------------------------------------------------------------------------
Approximately two thirds of the IRDC cores detected at 850$\umu$m that are located within the MIPSGAL survey area are associated with embedded 24$\umu$m sources (48 cores or 69% of the sample), as shown in Section \[subsection:SCUBAdetectedIRDCs\]. We carried out KS tests of the peak contrast and column density distributions for SCUBA detected IRDC cores with and without associated 24$\umu$m sources. There is no evidence for the existence of two populations (see Fig. \[Hist:embedded\]), which implies that the IRDC cores detected at 850$\umu$m with and without associated 24$\umu$m sources originate from the same column density population. We searched for any signs of correlation for the limited sample of those cores with known kinematic distances (see Table \[table:mass\]) and did not find any correlation of the presence of an embedded object at 24$\umu$m with the mass of the IRDC core.
Given the similar properties of the cores with and without associated 24$\umu$m sources, the two types of core may be evolutionarily related. The cores that are without associated 24$\umu$m sources could represent an earlier “starless” evolutionary stage to the IRDC cores that have formed intermediate or high mass Young Stellar Objects and are associated with 24$\umu$m sources. A range of evolutionary stages have been observed in a handful of IRDCs [@2008arXiv0808.2973R], which supports this hypothesis. In this picture the “starless” IRDC cores (i.e. those without associated 24$\umu$m sources) represent the cold quiescent initial conditions for high mass star formation as suggested by @1998ApJ...508..721C and @2006ApJ...639..227S. The SCUBA detected candidate IRDC cores with associated MIPSGAL 24$\umu$m sources would then represent a star forming population of IRDCs with embedded (proto) stellar objects, and so we refer to these as star forming IRDCs.
Two alternative explanations are that the starless IRDC cores detected at 850$\umu$m are sterile, possibly unbound, condensations that may never go on to form stars, or that they are forming stars, but with luminosities too low to be detected by MIPSGAL. In order to address the likelihood of the former explanation in detail, we would need to determine the virial masses and gravitational stability of a large sample of the starless IRDC cores (via additional spectroscopy). However we note that the Jeans Mass for a core of similar temperature and number density to the starless cores ($\sim$15K and 10$^{4}$cm$^{-3}$) is 20M$_{\odot}$. Decreasing the temperature or increasing the number density decrease the Jeans mass. The minimum mass of our sample of cores detected at 850$\umu$m (with or without associated 24$\umu$m sources) is 50M$_{\odot}$. Allowing for uncertainties in our derivation of the mass we thus conclude that it is unlikely that many of the IRDC cores fall below this Jeans Mass and so the majority of the IRDC cores detected at 850$\umu$m ought to at least have the potential for star formation.
This approach implicitly assumes that the IRDC cores are single gravitationally bound objects. If instead they are composed of numerous smaller cores fragmented below the scale of the JCMT beam then this conclusion may not apply. Higher resolution interferometry would be needed in this case (e.g. @2008arXiv0808.2973R).
To assess the likelihood of the latter explanation that the starless IRDC cores are forming stars with luminosities below the detection limit of MIPSGAL we need to determine the sensitivity of MIPSGAL to YSOs as a function of YSO luminosity. Fortunately a series of studies carried out by the Spitzer c2d[^11] survey team on nearby star forming regions allows us to characterise the 24$\umu$m flux to total internal YSO luminosity fairly well for low mass YSOs. @dunham08 find an approximately linear relationship between the MIPS 24$\umu$m flux and total internal luminosity for low mass YSOs detected in the Spitzer c2d survey. This was found to be consistent with the predictions of radiative transfer models of low mass YSOs [@crapsi2008]. Unfortunately no such similar study exists for high mass YSOs and so we use the empirical mid to far-infrared flux relation of @lumsden02. We plot these relationships (corrected for an estimate of the average 24$\umu$m extinction of the IRDCs) for a range of YSO luminosities as a function of distance in Fig. \[Fig:dunhamlumsden\].
Uncertainties in Fig. \[Fig:dunhamlumsden\] result from uncertainties in extinction and in the relationships between flux and luminosity as taken from @lumsden02 and @dunham08. The main source of error in @dunham08 is the uncertainty in the relationship between flux and luminosity which were obtained from observations as well as theoretical models. This uncertaintly is depicted in Fig. \[Fig:dunhamlumsden\] by the shaded region in the plots on the left. The primary source of error in @lumsden02 is due to the large range in observed flux ratio. This uncertainty is depicted in the middle and right plots with the lower estimate in the middle plots and the upper estimates in the plots on the right. From Fig. \[Fig:dunhamlumsden\] we see that the relationships taken from @lumsden02 and @dunham08 are consistent with each other particularly the plots to the right and in the middle of Fig. \[Fig:dunhamlumsden\].
The quoted 5$\sigma$ point source sensitivity of MIPSGAL is 1.7mJy at 24$\umu$m [@MIPSdocumentation]. This limit will obviously vary from region to region depending upon the strength of the background emission and complexity of structures in the images. However the IRDCs presented here in general have low 24$\umu$m backgrounds and are relatively free from source crowding, hence we assume the given 5$\sigma$ limit is valid, which is show in Fig. \[Fig:dunhamlumsden\] by a dashed horizontal line.
The effect of extinction from the environments surrounding the cores found without associated 24$\umu$m emission and from the material contained within the cores themselves lowers any observed radiation emitted from within. With these starless IRDC cores having high column densities ($\sim2\times10^{22}$cm$^{-2}$ from 8$\umu$m and 850$\umu$m data) and hence high opacities there are three possible situations we should consider for the flux that is observed: *i)* there may be no extinction *ii)* there may be a medium amount of extinction (taking this from the lower of the quoted column densities derived from the 8$\umu$m data) *iii)* there may be high extinction (taking this from the higher of the quoted column densities as derived from the 850$\umu$m data). We derive values of visual extinction (A$_{v}$) for the two latter cases where extinction has an effect on the observed flux using the method given by @RiekeLebofsky85. The derived A$_{v}$ range between 11 and 94 which converts to a 24$\umu$m extinction (A$_{24}$) of 0.6 and 4.6 respectively. The effect of the differing extinction can be seen in Fig. \[Fig:dunhamlumsden\] in which the top graphs are for no extinction, the middle graphs take the medium extinction case (A$_{24}=0.6$), and the bottom graphs assume high extinction (A$_{24}=4.6$). Errors on these estimates are on the order of $\sim$40%, predominantly resulting from the uncertainty in the empirically derived conversion from N(H+H$_{2}$)/E(B–V) [@bohlin1978].
At the typical distance of IRDCs within our Galaxy, which is 3.8kpc, we look at the sensitivity of MIPSGAL. In Fig. \[Fig:dunhamlumsden\], we see that in the ‘worst case scenario’ MIPSGAL should be complete to embedded objects with luminosities above 100L$_{\odot}$ (when high mid-infrared extinction is considered; A$_{24}=4.6$, A$_{v}=94$). This completeness limit indicates the possibility of ruling out the presence of all but low mass YSOs. From @iben1965 we find that in the protostellar stages of evolution a star with a final main sequence mass $<$2M$_{\odot}$ will never reach luminosities greater than 100L$_{\odot}$. Greater constraints on the column densities of these objects are required to allow us to have a better handle on the potential luminosities of these cores and determine if indeed they are a low mass population of cores.
The lifetimes of starless and star forming IRDCs {#section:SF}
------------------------------------------------
From the previous section we have seen that the starless cores may be evolutionary related to those SCUBA detected cores associated with 24$\umu$m objects but with luminosities below the detection limit of MIPSGAL. Going one step further we may assume that the starless and star forming SCUBA detected IRDCs are at different evolutionary stages in the formation of high mass stars and so we can estimate the statistical lifetime of the starless quiescent phase. If each starless IRDC core evolves into a corresponding star forming IRDC core with one or more embedded 24$\umu$m source then the relative proportions of these objects in the sample should reflect the statistical life time of each type of object. In the sample of SCUBA detected IRDCs lying within the MIPSGAL survey area we find twice as many star forming IRDC cores with 24$\umu$m sources than starless IRDC cores. Thus if these two types of object do form an evolutionary sequence we would expect the starless phase to last half the lifetime of the star forming phase.
Estimates for the absolute lifetime of the embedded high mass star formation range from $10^{4}$–$10^{5}$ years for UCHII and embedded YSOs [@1989ApJ...340..265W; @1989ApJS...69..831W], a few 10$^{4}$ years for methanol masers [@2005MNRAS.360..153V], and 1.2–7.9$\times10^{4}$ years for embedded high mass protostars . Taking the upper and lower bounds of these estimates we conclude that the starless phase of IRDCs, as an upper limit due to our assumption on evolution, last a few $10^{3}$–$10^{4}$ years. The proportions of starless and star forming IRDCs that we see are consistent with the proportion of massive infrared quiet high mass protostars to the massive protostellar stage as found in Cygnus X by . The lifetime of the starless IRDC phase is comparable to that found for the infrared quiet protostellar phase by who calculated the statistical lifetimes based on the proportion of massive infrared quiet high mass protostars to the massive protostellar stage as found in Cygnus X. Our statistical lifetime estimate for the starless IRDC phase is also approximately one to two orders of magnitude less than the extended lifetime of the low mass Class 0 evolutionary phase recently calculated by @2008arXiv0811.1059E.
Caution must be applied to comparing the estimated lifetime of starless IRDCs to the estimated lifetime of the high mass pre-stellar phase. As shown by for Cygnus X there are no high mass starless cores, which implies an age of less than 10$^{3}$ years for this phase. The “starless” IRDCs that we identify from their mid-infrared quietness may yet display other signs of star formation such as molecular outflows or methanol masers which would imply that they have a proto-stellar nature. This may be supported by the fact that their statistical lifetime is similar to the high mass proto-stellar phase identified by . Future investigations of these clouds to search for identifiers of high mass star formation are needed to estimate the lifetimes of the pre-stellar and proto-stellar phases found within these clouds. A number of forthcoming Galactic Plane surveys have these aims, such as the Methanol Multi Beam Survey (MMB; @2008arXiv0810.5201G), the CORNISH[^12] 5GHz survey [@2008ASPC..387..389P], the Herschel Hi-Gal Survey [@2005prpl.conf.8163M], and the JCMT Legacy Surveys SASSy and JPS [@2007arXiv0704.3202T; @2005prpl.conf.8370M], see Section \[section:SASSy\]. However, regardless we have shown that the lifetime of a quiescent (before it shows evidence of activity in the mid-infrared) IRDC is approximately half of that spent in the embedded phase.
Predictions and implications for Galactic Plane surveys {#section:SASSy}
-------------------------------------------------------
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The astronomical comunity are planning a number of uniform and sensitive surveys of the Galactic Plane in the far infrared and sub millimetre that will detect a large number of the @2006ApJ...639..227S IRDC catalogue in emission. We use the results drawn form the SCUBA Legacy Catalogue to make predictions for the number of IRDCs that will be detected by four surveys in particular: SASSy, the SCUBA-2 “All Sky” Survey [@2007arXiv0704.3202T], JPS, the JCMT Galactic Plane Survey [@2005prpl.conf.8370M], Hi-GAL, the Herschel Infrared Galactic Plane Survey [@2005prpl.conf.8163M], and ATLASGAL the APEX Telescope Large Area Survey of the Galaxy [@ATLASGAL]. Each of these surveys will cover much larger regions of the plane than the SCUBA Legacy Cataogue and will be both deeper and more uniform, resulting in a much more unbiased survey of IRDCs that is free from the targeted and non-uniform nature of the SCUBA Legacy Catalogue. Fig \[pic:SASSy\] shows a region of the Galactic Plane with the coverage area of the SCUBA Legacy Catalogue and the positions of [@2006ApJ...639..227S] IRCDs, which clearly indicate the potential of these large area surveys to detect a large number of IRDCs.
As each of these surveys will detect IRDC cores by their emission rather than their extinction against the galactic mid-infrared background this means that they will also be sensitive to IRDCs located on the far side of the Galaxy that were not detected by @2006ApJ...639..227S. The forecast 1$\sigma$ sensitivities of HIGAL and JPS are 20mJybeam$^{-1}$ and 4mJybeam$^{-1}$ at 250$\umu$m and 850$\umu$m respectively, which are sufficient to detect cores of only a few tens of M$_{\odot}$ at 20kpc (assuming 20K dust with $\beta$=2 and a mass coefficient of 50gcm$^{-2}$). SASSy and ATLASGAL will have 1$\sigma$ sensitivities of 30mJybeam$^{-1}$ and 50-70mJybeam$^{-1}$ at 850$\umu$m which could detect cores of a few hundred M$_{\odot}$ out to 20kpc. Taking the masses of known IRDC cores into consideration each of these surveys has the potential to detect these objects at the far side of the Galaxy. In addition, as we have shown in Section \[Section:Discussion1\], the deeper surveys may find the low column density low temperature clouds that were not detected at 850$\umu$m in the SCUBA Legacy Catalogue. Thus as well as the increased number of detections resulting from surveying a larger area of the plane, we expect that the surveys will also detect a greater number of ‘IRDC cores’ on the far side of the Galaxy and the colder population that we have not detected with SCUBA.
Estimating an upper limit to the number of IRDC cores that could be detected by the surveys is difficult. For the IRDC cores located on the far side of the Galaxy that have foreground emission preventing them being detected by @2006ApJ...639..227S we may estimate their number by geometric means and considering the volume of the Galaxy probed by MSX. Following the argument presented by @2006ApJ...641..389R we estimate that the total number of IRDC cores in the Galaxy may be up to a factor of 3 greater than those detected by @2006ApJ...639..227S. To this number must be added an uncertain quantity of low column density cores whose intrinsic contrast falls below the [@2006ApJ...639..227S] criteria for detection but whose column density is great enough to be detected by the surveys (particularly Hi-GAL and JPS). We see from Fig. \[Graph:CumulativeMSX\] that the steep turnover of IRDC cores at low contrast values may indicate that the catalogue is incomplete at low contrasts. Without further information on the general temperature distribution of IRDCs it is currently not possible to place firm limits on the number of such cores and so whilst we note that the deeper surveys will detect this colder population (and Hi-GAL will determine the temperature distribution of IRDC cores) we do not include them in our estimate.
Currently no information exists on the temperature distribution of IRDCs in general, as by the nature of their detection the estimated temperature for each cloud is an upper limit. This means that we cannot take the column densities estimated from the MSX 8$\umu$m data (as contained in Table \[table:classA\]) and convert these into flux estimates, as the lack of temperature information renders these into rather loosely determined upper flux limits. In addition the large uncertainties in mass co-efficients, the 8$\umu$m extinction law and contamination from foreground emission introduce a considerable scatter between column densities derived from 8$\umu$m and 850$\umu$m (see Section \[subsection:coldenmass\]). We thus estimate lower limits for the detection rate of IRDC cores within the surveys by using the SCUBA detection fraction shown in Section \[subsection:Reliability\]. SASSY, JPS and Hi-GAL are deeper than the SCUBA Legacy Catalogue and so we expect these surveys to detect a greater fraction of IRDC cores, particularly at low contrast values where the surveys are more sensitive to low temperature low column density cores (see Fig. \[Graph:Sensitivity\]). Without knowing the temperature distribution of IRDC cores it is impossible to determine exactly what this fraction is, but given the greater sensitivities of these surveys they ought to detect at least the fraction of IRDC cores that SCUBA did. The depth of ATLASGAL is similar to the 0.1Jypixel$^{-1}$ noise cut that we applied to the SCUBA Legacy Catalogue and thus ATLASGAL should detect a similar fraction of IRDC cores from @2006ApJ...639..227S.
ATLASGAL will survey the inner third of the Galactic Plane ($|l|<60^{\circ}$ and $|b|<1.5^{\circ}$), within which there are 11,529 IRDC cores from the @2006ApJ...639..227S catalogue. Taking the SCUBA detection fraction of 75% we predict that ATLASGAL will detect at least 8,600 IRDC cores. We scale this number by the geometric argument of @2006ApJ...641..389R to estimate the number of cores that ATLASGAL will detect on the far side of the Galaxy and hence estimate that ATLASGAL may detect up to 26,000 IRDC cores. This is consistent with the preliminary results of the first 95 deg$^{2}$ of ATLASGAL which detects $\sim$6,000 sources, many of them infrared dark [@ATLASGAL]. The survey area of Hi-GAL again covers the inner third of the Galactic Plane but with a latitude range $|b|<1^{\circ}$. Of the 12,774 cores within the IRDC core catalogue by @2006ApJ...639..227S 10,644 IRDC cores are located within the Hi-GAL survey area. Scaling this to the detection fraction of SCUBA Hi-GAL will detect at least 8,000 IRDC cores. Again we use the geometric argument to estimate that within the entire Galaxy this number may increase to 24,000 cores. SASSy, covering $0^{\circ}\leq l \leq245^{\circ}$ and $|b|\leq 5^{\circ}$ of the Galactic plane has 6,160 IRDC cores from the [@2006ApJ...639..227S] within the coverage area. Taking the 75% detection fraction results in a lower estimate of 4,600 cores being obsevred. Again the number detected increases, when we consider geometric arguments, to 14,000. Finally JPS will survey two regions of the Galactic Plane, the GLIMPSE-N region ($10^{\circ}<l<65^{\circ}$ and $|b|\leq1^{\circ}$) and the FCRAO Outer Galaxy Survey region ($102.5^{\circ}<l<141.5^{\circ}$ and $|b|\leq1^{\circ}$). We see that 4,095 IRDC cores from @2006ApJ...639..227S are located within the coverage area. With a detection fraction of 75% we expect a lower limit of 3,000 IRDC cores to be detected. With geometric arguments this number may increase to 9,000 IRDC cores. Although by number we see that the predicted numbers of IRDC cores that ATLASGAL and Hi-GAL are expected to return higher source counts than SASSy and JPS, this is due to the larger area covered by these surveys. JPS and SASSy will however explore relatively unique parameter spaces. The high sensitivity JPS (1$\sigma$ $\sim$ 4mJybeam$^{-1}$ at 850 $\umu$m) will be ideal for identifying the most low temperature low column density IRDC cores. SASSy in contrast to the other surveys will have the benefit of observing greater latitudes of the Galactic Plane than any other survey and (as with JPS) will observe the outer Galaxy where low mid infrared backgrounds has restricted previous identifications of ‘IRDCs’ due to their very nature.
Summary and Conclusions {#Section:Conclusions}
=======================
From positional cross matching of the IRDC catalogue produced by @2006ApJ...639..227S with the coverage area of the SCUBA Legacy Catalogue (as published by @2008ApJS..175..277D) we have identified two populations of objects: candidate IRDC cores with and without associated 850$\umu$m emission. Column densities of these two populations were derived from the 8$\umu$m data by applying an extinction law to the peak contrast values (as defined by @2006ApJ...639..227S based upon observations at 8$\umu$m). For those cores that were associated with 850$\umu$m emission column densities were also derived assuming a spherical geometry and the assumption of @1983QJRAS..24..267H. From our findings outlined within this paper we make the following conclusions:
1. We find 154 cores with 850$\umu$m detected emission and 51 cores without 850$\umu$m emission. Those cores associated with detectable 850$\umu$m emission had a median peak contrast value of 0.32, a median column density of 1.7$\times10^{22}$cm$^{-2}$ and a median mass of 300M$_{\odot}$. We found that the overall detection fraction of IRDC cores with 850$\umu$m emission is 75%, as a lower limit which is in good agreement with the CS detection of @2008ApJ...680..349J.
2. Those cores without 850$\umu$m emission are found to have no significant difference in peak contrast distribution than those cores detected at 850$\umu$m. These cores are likely to be population of low temperature low column density transient or prestellar cores. However, a small number of these cores could also be “holes” in the background mid-infrared continuum emission or artefacts as a result of the identification procedure. Further observations of the cores not detected at 850$\umu$m, either deeper sub millimetre continuum data or molecular line data, are required to yield insight into the true nature of these objects.
3. On the nature of those cores detected at 850$\umu$m, we find that their range in masses ($50-4,190$M$_{\odot}$) are consistent with the lower mass end range observed in high mass star forming regions. 69% of those cores detected at 850$\umu$m lying within the MIPSGAL survey area are associated with an embedded object at 24$\umu$m. A KS test gave no indication for the existence of two populations. This could suggest these cores are related evolutionarily. Those cores detected at 850$\umu$m without 24$\umu$m sources could be “starless” IRDCs or they may be forming stars but with luminosities too low to be detected. An alternative explanation for their origins are that they are unbound condensations that may never go on to form stars. To make more detailed conclusions about the nature of the SCUBA detected cores and their embedded mid-IR sources requires a deeper understanding of their physical properties from follow up molecular line mapping.
4. Based on the assumption that the “starless” and star forming cores are related evolutionarily we derive an upper limit of $10^{3}-10^{4}$years for the lifetimes of starless IRDC cores. This lifetime is found to be comparable to the infrared quiet protostellar phase by and is approximately one to two orders of magnitude less than the extended lifetime of the low mass Class 0 evolutionary phase recently calculated by @2008arXiv0811.1059E.
5. Based on SCUBA detection rates found, we make a conservative prediction to a lower limit of the number of IRDC cores that the Galactic Plane surveys ATLASGAL, Hi-GAL, SASSy and JPS will potentially detect : 8,600, 8,000, 4,600 and 3,000 cores respectively. If we apply geometric arguments to these values to scale to the number of such cores in the far Galaxy [@2006ApJ...641..389R] we see that ATLASGAL, Hi-GAL, SASSy and JPS have the potential to observe up to 26,000, 24,000, 14,000 and 9,000 infrared dark cores respectively throughout the Galaxy.
We are now entering into an exciting time for sub millimetre and far infrared astronomy with the advent of Herschel and SCUBA-2. These two instruments will push the observational investigations of IRDCs, and in turn they will yield fresh insight into the role they may play in massive star formation
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the STFC and the University of Hertfordshire for support and an anonymous referee for a number of useful comments that substantially improved this work. We also thank James Fi Francesco for providing a file of all SCUBA Legacy Catalogue noise values. This research has made use of NASA’s Astrophysics Data System. We would also like to acknowledge the JCMT (The James Clerk Maxwell Telescope is operated by The Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the United Kingdom, the Netherlands Organisation for Scientific Research, and the National Research Council of Canada), MIPSGAL and Midcourse Space Experiment. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency.
\[lastpage\]
[^1]: E-mail: [email protected];
[^2]: E-mail: [email protected];
[^3]: The James Clerk Maxwell Telescope is operated by the Joint Astronomy Centre on behalf of the Scientific and Technology Facilities Council of the UK, the Netherlands Association for Scientific Research, and the National Research Council of Canada.
[^4]: The Clumpfind algorithm used to identify objects was adapted from [@1994ApJ...428..693W] [@2008ApJS..175..277D].
[^5]: MIPS (Multiband Imaging Photometer for Spitzer) Galactic Plane Survey. Data available from http://irsa.ipac.caltech.edu/data/SPITZER/MIPSGAL/images/
[^6]: The Spitzer Space Telescope is operated by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407.
[^7]: TOPCAT: Tool for OPerations on Catalogues And Tables. See http://www.star.bris.ac.uk/$\sim$mbt/topcat/
[^8]: cores with MIPS data avaliable are denoted by a yes (if an embedded 24$\umu$m source is present) or no (if an embedded 24$\umu$m is\
not present) in this column
[^9]: F stands for Fundamental catalogue and E stands for Extended catalogue
[^10]: Flux integrated over the area of the object as defined by Clumpfind [@2008ApJS..175..277D]
[^11]: The Spitzer Space Telescope Legacy program “From Molecular Cores to Planet-Forming Disks” [@Evans2003]
[^12]: The Co-Ordinated Radio ‘N’ Infrared Survey for High-mass star formation, @2008ASPC..387..389P
|
---
abstract: 'The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions, such as when there is some sparsity pattern of the optimal parameter. This work characterizes a certain strong convexity property of *general* exponential families, which allow their generalization ability to be quantified. In particular, we show how this property can be used to analyze generic exponential families under $L_1$ regularization.'
author:
- 'Sham M. Kakade'
- |
Sham M. Kakade\
Department of Statistics\
The Wharton School\
University of Pennsylvania, USA Ohad Shamir\
School of Computer Science and Engineering\
The Hebrew University of Jerusalem, Israel Karthik Sridharan\
Toyota Technological Institute\
Chicago, USA Ambuj Tewari\
Toyota Technological Institute\
Chicago, USA
bibliography:
- 'mybib.bib'
- 'bib.bib'
- 'sparse.bib'
title: ' Learning Exponential Families in High-Dimensions:\'
---
|
---
abstract: 'The *Chandra X-ray observatory* has proven to be a vital tool for studying high-energy emission processes in jets associated with Active Galactic Nuclei (AGN). We have compiled a sample of 27 AGN selected from the radio flux-limited MOJAVE (Monitoring of Jets in AGN with VLBA Experiments) sample of highly relativistically beamed jets to look for correlations between X-ray and radio emission on kiloparsec scales. The sample consists of all MOJAVE quasars which have over 100 mJy of extended radio emission at 1.4 GHz and a radio structure of at least 3$\arcsec$ in size. Previous *Chandra* observations have revealed X-ray jets in 11 of 14 members of the sample, and we have carried out new observations of the remaining 13 sources. Of the latter, 10 have X-ray jets, bringing the overall detection rate to $\sim$ 78$\%$. Our selection criteria, which is based on highly compact, relativistically beamed jet emission and large extended radio flux, thus provides an effective method of discovering new X-ray jets associated with AGN. The detected X-ray jet morphologies are generally well correlated with the radio emission, except for those displaying sharp bends in the radio band. The X-ray emission mechanism for these powerful FR II (Fanaroff-Riley type II) jets can be interpreted as inverse Compton scattering off of cosmic microwave background (IC/CMB) photons by the electrons in the relativistic jets. We derive viewing angles for the jets, assuming a non-bending, non-decelerating model, by using superluminal parsec scale speeds along with parameters derived from the inverse Compton X-ray model. We use these angles to calculate best fit Doppler and bulk Lorentz factors for the jets, as well as their possible ranges, which leads to extreme values for the bulk Lorentz factor in some cases. When both the non-bending and non-decelerating assumptions are relaxed the only constraints on the kpc scale jet from the Chandra and VLA observations are an upper limit on the viewing angle, and a lower limit on the bulk Lorentz factor.'
author:
- |
Brandon Hogan$^1$, Matthew Lister$^1$, Preeti Kharb$^2$,\
Herman Marshall$^3$, Nathan Cooper$^1$
title: 'Chandra Discovery of 10 New X-Ray Jets Associated With FR II Radio Core-Selected AGNs in the MOJAVE Sample'
---
Introduction
============
Blazar jets are generated in active galactic nuclei (AGN) as a result of accretion onto supermassive black holes, and can transport energy over large distances. These outflows tend to show apparent superluminal speeds, and are oriented at very shallow angles with respect to the line of sight [@AS80]. The blazar class encompasses flat spectrum radio quasars (FSRQs), which have Fanaroff-Riley type II jets (FR II; @FR74), and BL Lac objects, which are thought to have FR I type jets [@UP95]. The AGN outflows that we discuss here are of the powerful FR type II class which have well collimated jets and bright terminal hotspots (e.g., @PK08). In terms of X-ray production in the jet, the inverse Compton radiation process is suggested to be more important in FSRQs than in the less powerful BL Lac sources [@GT08; @HK06].
Prior to the launch of the *Chandra X-ray observatory*, there were very few AGN jet detections in the X-ray band. The only major X-ray imaging telescopes in use were *Einstein* and *ROSAT*. Only a few very bright, nearby X-ray jets were known, e.g., M87, 3C 273, Centaurus A, and a few lesser known sources (see [@RS04], [@HM05] and references therein). Since the launch of *Chandra* there have been approximately 50 new discoveries of X-ray jets that are spatially correlated to some extent with the radio emission. The excellent angular resolution of *Chandra* has revealed detailed structure in FR II jets, such as knots, lobes and hotspots [@HC02], and has opened up an entirely new subfield of AGN astronomy.
Many X-ray emitting jets were discovered in early surveys by [@RS04] & [@HM05]. The quasars in these surveys were selected mostly from radio imaging surveys of FSRQs, but the surveys were not statistically complete. For our study we have chosen to assemble a complete sample of beamed FR II jets according to well defined selection criteria. These jets generally have high Doppler factors and relativistic speeds. The MOJAVE Chandra Sample (MCS) is a complete subset of compact radio jets selected from the MOJAVE sample. The latter sample consists of all 135 known AGN with $\delta$ $>$ $-20^\circ$, $|$b$|$ $>$ 2.5$^\circ$ and VLBA 15 $\sim$ GHz correlated flux density exceeding 1.5 Jy at any epoch between 1994.0 and 2004.0 (2 Jy for AGN below $\delta = 0\arcdeg$) [@ML09a]. Since the long interferometric baselines of the VLBA are insensitive to large-scale unbeamed radio emission, the sample is heavily dominated by blazars. In Section 2, we describe how the MCS was selected using a set of criteria designed to maximize the chances of X-ray jet detection.
The goals of our study are threefold. First, we seek to identify new X-ray jets for future follow up with *Chandra*, *Spitzer*, and the *Hubble Space Telescope (HST)*. Because of the large redshift range of the MCS (0.033 $\leq$ z $\leq$ 2.099), we can examine the effects of proposed X-ray mechanisms such as inverse Compton scattering off of cosmic microwave background (IC/CMB) photons by relativistic electrons in the jets, which is highly dependent on redshift. Second, we wish to characterize the ratio of X-ray emission to radio emission for a large complete sample of jets. Such information is vital for determining the respective roles that deceleration and bending play in determining why jets associated with some AGN are strong X-ray emitters. Lastly, we can use the detailed viewing angle and speed information of the AGN jets on parsec (pc) scales provided by the MOJAVE program to better model the X-ray emission mechanism(s).
A total of 14 AGN in the MCS have previously been observed by *Chandra* [@HM05; @RS04; @CO02; @SJ06; @JM06; @WE07; @WA01]; here we present new 10 kilosecond exposures on the other 13 sources, in which we have detected 10 new X-ray jets. A detailed analysis of the full 27 source sample will be presented in subsequent papers.
Our paper is laid out as follows; we describe the MOJAVE Chandra Sample in Section 2, along with our data reduction method and selection criteria. In Section 3, we describe the jet observations for each specific source in which a jet was present in both the radio and X-ray images. In Section 4, we discuss overall source trends and provide additional ancillary information on selected sources. In Section 5, we discuss implications of the model with respect to the bulk Lorentz factor and viewing angle. We summarize our conclusions in Section 6. The limits for the derived Doppler factor and bulk Lorentz factors are given in the Appendix. Throughout this paper we use a standard cosmology with H$_0$ = 71 km s$^{-1}$ Mpc$^{-1}$, $\Omega$$_m$ = 0.27, and $\Omega$$_{\Lambda}$ = 0.73.
The *[MOJAVE Chandra]{} Sample*
===============================
Selection Criteria
------------------
In formulating the sample for our *Chandra* survey, we wished to focus on relativistic radio galaxy and blazar jets whose high Doppler factors would make them prime candidates for IC/CMB X-ray emission. We also decided to limit our survey to FR II radio galaxies and quasars, in order to avoid possible contamination by lower power (presumably FR-I type) BL Lac objects. The MOJAVE sample [@ML09b] provided a useful list of candidates for possible IC/CMB in this regard, since it comprises a complete set of compact radio jets in the northern sky. Its VLBA selection criteria favor highly Doppler-boosted blazar jets for which extensive pc scale kinematic information has been obtained [@ML09a]. Deep 1.4 GHz VLA A-configuration radio images are also available for the entire sample of 135 AGN [@NC07; @PK10]. In order to maximize the likelihood of X-ray jet detection, we considered all MOJAVE quasars and FR II radio galaxies having more than 100 mJy of extended kiloparsec (kpc) scale emission (where the extended emission is the total emission after the core emission has been removed) at 1.4 GHz and a radio structure that was at least 3$\arcsec$ in extent. This final list of 27 AGN (see Table \[table:mcs\]) comprises the MCS. A search of the *Chandra* archive revealed that 14 of these objects had been previously observed, where most of the sources had integration times $>$ 10 kiloseconds. During the period 2007, November to 2008, December, we obtained new *Chandra* 10 kilosecond ACIS images of the remaining 13 AGN.
Data Reduction and Analysis
---------------------------
In Figure \[fig:rxo\] we present X-ray-radio overlays for our new observations. We first obtained the 1.4 GHz VLA A-array data from the NRAO[^1] data archive and our own observations [@NC09; @PK10]. The observation dates and exposure times for the new *Chandra* targets are listed in Table \[table:obs\]. These were reduced following standard procedures in the Astronomical Image Processing System (AIPS). After the initial amplitude and phase calibration using the standard calibrators, the AIPS tasks CALIB and IMAGR were used iteratively to self-calibrate and image the sources. Self-calibration on both the phases (with solution intervals typically set to less than 0.5 mins in CALIB) and amplitude (with successively decreasing solution intervals) were performed until convergence in image flux and structure was achieved. The final radio maps had a typical rms noise of $\sim 0.2$ mJy beam$^{-1}$. The FWHM restoring beam of the radio images was adjusted to $\sim 1.4\arcsec$ on average, and the width of the *Chandra* FWHM was estimated to be $\sim$ 0.75$\arcsec$. The *Chandra* maps for the X-ray-radio overlays were created using the DS9 image tool. We started by loading the level 2 events files into CIAO for energy filtering. The event files were filtered to an energy range of 0.5 to 7 keV. After loading these event files into DS9, we used the analysis smoothing tool to smooth the image. This was set to use Gaussian smoothing with a kernel radius of three pixels, where a pixel size of 0.5 pixels per arcsecond are used. After that we adjusted the color scale so that the cores were oversaturated. This allowed for the jet emission to be detected easily by visual inspection. The radio contours were then superimposed. The images frames were aligned in DS9 using the WCS frame matching setting. Some of the core positions were slightly misaligned when overlayed. We registered the X-ray and radio images using the Fv program in the Ftools package provided by NASA[^2][@BB95]. The shifts were generally small, of order of 2 pixels or less.\
The centroid positions were also calculated via the method described by [@HM05]. The procedure involves determining the preliminary X-ray core centroid location by fitting Gaussians to the 1D histograms obtained from events within 30$\arcsec$ of the core region. This defines the rough centroid position. This process was then repeated using a region defined by a radius of 3$\arcsec$ from the previously calculated centroid position which allows for a more refined centroid position. This two step approach reduces the effect of the extended jet emission, which can bias the centroid. After calculating the centroid position, we used Poisson statistics to test for the existence of an X-ray jet. The radio profiles were first used to define the outer radius and position angle of the primary jet from visual inspection, which are listed in Table \[table:vla\]. The radii allowed us to create a box within which we could check for the existence of X-ray structures. The lengths of the radii varied but the width of each rectangle was fixed at 3$\arcsec$. The inner radius was fixed at 1.5$\arcsec$ to eliminate the core emission, with the exception of 1849+670 and 2345$-$167, which were fixed to values of 5$\arcsec$ and 2$\arcsec$ respectively, because of their elongated radio restoring beams (Table \[table:vla\]).
The detection algorithm assumes straight radio jets and that the region 90$^\circ$ to the primary jet is free of jet emission, which is a valid assumption for all of the jets in the sample except for 0119+115. This source has radio jet emission on both sides of the core perpendicular to the direction of the primary jet but lacks substantial X-ray emission, making the perpendicular radio jet emission irrelevant. We then produced profiles of the radio emission along the jet position angle, and at 90$^{\circ}$ to it (Figure \[fig:rp\]). These two quantities were subtracted from each other to eliminate core structure. The X-ray jet counts were then compared for the same sky region. For the X-ray profiles, we chose to use the jet axis region and the region 180$^\circ$ away from it (Figure \[fig:xp\]). We then compared the counts in these regions by using Poisson statistics. Sources with negative values in the count rate column in Table \[table:sjm\] have less X-ray emission compared to the area in the region opposite to it. Counter-jets in powerful AGN have rarely been seen in X-rays, presumably due to Doppler boosting effects [@WD09] and we found no evidence of any X-ray counterjets in our sample from visual inspection. We set the Poisson probability threshold for the detection of an X-ray jet to 0.0025 [@HM05]. This value yields a 5% chance of a false detection in one out of every 20 sources. The X-ray fluxes were computed from count rates using a conversion factor of 1 $\mu$Jy per count s$^{-1}$. This conversion is accurate to about 10% for typical jet power law spectra [@HM05]. Our analysis method indicated that there were X-ray jet detections in all of the sample sources except for 0119+115, 0224+671, and 2345$-$167. These sources did not show any appreciable X-ray emission above the background level except for their respective cores (Figure \[fig:rxo\]). This is despite the fact that their redshift and radio structure are comparable to the other sources in the sample. The relevant X-ray emission limits for these sources are listed in Tables \[table:sjm\] and \[table:bmp\].
Notes on Individual Sources
===========================
In this section we provide a general overview of the X-ray jet morphologies in our sample, and how they compare with the 1.4 GHz radio structure seen in the VLA A array radio images (Figure \[fig:rxo\]). We use the term “knot” for any excess of emission at a shock front that is not at the terminal point of the jet, and use the term “hotspot” for any knot-like structure or excess of emission located where the jet terminates in the radio band. Some of these images may show readout streaks which look similar to jet emission. We attempted to set the roll angle for each source that we viewed with *Chandra* so that the readout streak would not be aligned with the previously known jet emission in the radio band. We have labeled these in Figure \[fig:rxo\]. Figures \[fig:rp\] and \[fig:xp\] show the radio and X-ray jet profiles, respectively, and are described in section 2.2.
0106$+$013 (OC 12)
------------------
This blazar has a prominent X-ray jet which protrudes due south of the radio core. There is a strong correlation between the X-ray jet and the radio jet contours. Both have a hotspot-like structure 5$\arcsec$ from the core, which is the terminal point of their respective jets. There is also an excess of radio emission to the northeast that does not correlate with any X-ray emission.
0415$+$379 (3C 111)
-------------------
The VLA radio data on this powerful radio galaxy was obtained by [@LP84]. The image of the jet shows that there are 4 prominent radio knots present, and three of these show an excess of X-ray emission. The terminal knot, or hotspot, shows an excess of emission also, indicating an excellent correlation between the X-ray and radio emission in this jet. The jet lies at a reasonably small angle to the line of sight according to our IC/CMB calculations (see Section 4 and Table \[table:bmp\]). [@SJ05] give a value of the angle to the line of sight of 18.1 $\pm$ 5.0$^\circ$, which is somewhat larger than our value obtained via the IC/CMB method. This jet has a measured superluminal speed of 5.9c and therefore has a maximum value for the angle to the line of sight of roughly 19$^\circ$ [@ML09a]. The deprojected length from the core to the terminal hotspot for this source is 661 kpc for $\theta = 8^\circ$, 537 kpc for $\theta = 10^\circ$, and 302 kpc for $\theta = 18^\circ$, where the measured distance on the plane of the sky is $\sim$100$\arcsec$ from the core to the hotspot.
This object also shows only a single sided jet structure within 100$\arcsec$ of the core, which is likely to be the result of Doppler boosting and also shows a pc scale jet in approximately the same general direction as the kpc jet in the radio band. The terminal point of both the jet and counter jet are visible, with lobe-like radio emission at the respective terminal points, along with notable X-ray emission only at the primary jet terminal point. This object is one of only two radio galaxies, the other being Cygnus A, that met the selection criteria of the MCS and have X-ray jet structure [@WA01].
0529$+$075 (OG 050)
-------------------
This blazar has an X-ray jet which follows the initial radio jet position angle of $-$145$^\circ$. The radio jet terminates at $\sim$ 6$\arcsec$ from the core. There is also radio emission to the southeast of the core and X-ray emission more to the east of the core, but these two features are not coincident. The main jet emission seems to coincide with the X-ray jet in the south west direction. The south eastern emission could be coming from the counter-jet, since we see no X-ray emission there, which is what we might expect from the Doppler de-boosting of a counter jet, according to the IC/CMB model. The pc scale jet lies at a position angle of approximately $-$45$^\circ$, which implies significant bending between the pc and kpc scale jets.
1045$-$188
----------
The radio image of this blazar shows strong one sided jet emission and diffuse lobe emission from the counter jet. The main jet, which lies at a position angle of 125$^\circ$, makes a $\sim$ 90$^\circ$ bend at the bright radio knot at a distance of $\sim$ 8$\arcsec$ from the core. The X-ray emission follows the primary radio jet out to the bend but then terminates abruptly. There is no detectable X-ray emission from the counter jet or its associated lobe above the background level.
1334$-$127
----------
This blazar has an X-ray jet with a length of $\sim$ 6$\arcsec$ that follows the radio jet emission out to a 60$^{\circ}$ bend of the radio jet, then undergoes a drop in emission, but still terminates at the same point as the radio jet. Both jets initially follow a position angle of 135$^\circ$. The emission characteristics in the bend region are significantly different than the jet of 1045$-$188, which undergoes a sudden drop in X-ray emission after the bend.
1800$+$440
----------
The radio image of this jet shows a lobe to the northeast, and emission toward the southwest at a position angle of $-$130$^\circ$. The X-ray jet emission follows the radio jet emission to the southwest for $\sim$ 4$\arcsec$ and then terminates at the radio knot. The radio emission continues for another $\sim$ 3$\arcsec$ until it terminates in a hotspot. There also appears to be a shallow bend beyond the first radio knot. This is another example where the X-ray jet flux decreases beyond a radio knot located at a bend.
1849+670
--------
This quasar shows a short radio jet to the north with lobe structure to the south. The radio emission to the north shows a drop in emission at $\sim$ 15$\arcsec$, whereas the X-ray emission decreases drastically after a distance of about 9$\arcsec$ from the core. A visual inspection of the overlays (Figure \[fig:rxo\]), indicates no apparent correlation between the radio counter lobe emission and the X-ray emission in this source.
2155$-$152
----------
Although there is arcsecond scale radio emission on both sides of the core, the pc scale jet is oriented to the south, so we chose this direction to search for X-ray emission. The southern jet emission terminates at a distance of $\sim$ 8$\arcsec$ from the core, whereas the X-ray emission is seen up until a distance of $\sim$ 4$\arcsec$ south of the core where it terminates.
2201$+$315 (4C 31.63)
---------------------
The radio map of this object shows a $\sim$ 37$\arcsec$ long jet with a counter jet lobe located approximately the same distance away in the opposite direction. There is X-ray emission which correlates with the radio in the first knot structure as seen in Figures \[fig:rp\] and \[fig:xp\]. This is at a distance of $\sim$ 4$\arcsec$ from the core. Downstream from this knot, there is a significant decrease in X-ray flux. This detection is considered marginal and needs a longer exposure time to be able to produce a clear visual correlation in the overlay images.
2216$-$038
----------
The radio image shows a sharp knot-like structure at a distance of about 8$\arcsec$ from the core, along with an excess in the X-ray emission at the same location. The radio jet shows emission continuing out to a distance of $\sim$ 15$\arcsec$, but the overlayed image shows a sharp decrease in X-ray emission between the knot and the terminal hotspot. There is a counter jet lobe to the northeast in the radio portion of the overlay, but there is no X-ray emission in this region above the background emission.
X-ray Data Analysis
===================
Data Trends and Results
-----------------------
We have found that the extended flux densities, S$_{ext}$, are closely correlated with the detection rate of the X-ray emission. Interestingly, [@PK10] have reported a significant trend between pc scale (apparent) jet speeds and extended radio luminosity in the MOJAVE blazars. This could suggest a link between X-ray jet detection and jet speed. Although our selection criteria guaranteed S$_{ext}$ $>$ 100 mJy for all sources, we get a 100% X-ray jet detection fraction for S$_{ext}$ $>$ 300 mJy. Below that value, we find a significantly lower detection rate ($\sim$ 57%). Using an extended flux density threshold value as a selection criterion could prove to be a definitive way to predict X-ray jet detection in blazars and bright core radio galaxies from radio jet images alone.\
We also ran Kolmogorov-Smirnov tests on the MCS population for three different cases; the $\beta_{app}$ values with respect to the detection of sources, the $\beta_{app}$ values with respect the S$_{ext}$ threshold value (300 mJy), and the redshift value with respect to the detection of the sources. In all three cases, the *p* value does not reject the possibility that both populations could have the same parent population.
If we examine the X-ray jet morphologies of the MCS, as well as the sample of [@HM05], we find no visible instances of any X-ray counterjets. This is consistent with the predictions of the IC/CMB model, which suggest that the X-ray emission from the relativistic kpc scale jets must be highly beamed along the direction of the flow. Thus, any X-ray counterjet emission would not be visible to *Chandra*. Jet bending on the kpc scale can also limit the detection of X-ray jets that are bright in the radio band. This could occur if there is deceleration past the point of the bend, which may in turn limit the flow to mildly relativistic speeds. A secondary factor may be a significant change in the magnetic field strength downstream from the bend as in 3C 279 [@SJ04]. Other examples of jet bending where the X-ray flux decreases past the bend in the MCS are the blazars 0234$+$285, 1045$-$188, 1222$+$216, 1334$-$127, 1510$-$089, 1800$+$440, and 1928$+$738.
Emission Modeling
------------------
The MCS has improved the detection rate of X-ray jets when compared to previous surveys of the same kind. We have found that $\sim$ 78% (21 of 27) of the objects in the MCS have significant kpc scale X-ray jet emission in 10 kilosecond *Chandra* exposures, while the previous surveys have a $\sim$ 60% detection rate [@HM05; @RS04]. One difference between previous quasar surveys and ours is that the MCS is complete with respect to beamed synchrotron emission. Our sample is also more strongly biased toward blazars with superluminal jet speeds and higher Doppler factors. Our detection rate supports the hypothesis that relativistic beaming is indeed an important factor affecting large-scale X-ray jet emission.\
For our sample analysis of blazars and radio galaxies, we have chosen to use a method similar to the one used by [@HM05]. This model assumes that the magnetic fields in the kpc scale jets are in equipartition with the particle energies. Under this assumption, the magnetic field strength of the jet can be calculated. The photon energy density is then compared to the magnetic field energy density by using the previously calculated magnetic field strength and the CMB photon density. The parameter *K* (a function of $\beta$, the jet velocity in terms of the speed of light, and $\theta$, the angle to the line of sight), can be calculated by combining the magnetic field strength with the ratio of observed X-ray (inverse Compton) to radio (synchrotron) luminosities.
Since the single component synchrotron model has difficulties in explaining the X-ray emission in powerful blazar jets that have observed limits on their optical emission (e.g., @HM05), we have derived physical quantities for the X-ray emission using a standard IC/CMB model. We started with the same IC/CMB assumptions as [@HM05], which were obtained from [@HK02]. The first assumption is that the energy density of the CMB occurs at the peak of the blackbody distribution. The second assumption is that the jet frame equipartition holds between the particle energy densities and the magnetic field, with a filling factor ($\phi$) of 1. If relativistic protons contribute to the particle energy density, then this assumption will fail and beaming will become much more extreme. The third assumption is that the low energy spectral index for the synchrotron spectrum continues unchanged below the current range of the instruments used to measure them. The procedure involves first defining
$$B_{1} = \left[\frac{18.85C_{12}(1+k)L_{s}}{\Phi V}\right]^{2/7},$$
where *$B_{1}$* is the spatially averaged, minimum energy magnetic field of the jet in the case where there is no Doppler boosting, in Gauss, *$C_{12}$* is a weak function of the low frequency spectral index of the synchrotron spectrum ($\alpha_r$, where S$_\nu$ $\propto$ $\nu$$^{-\alpha_r}$), $\Phi$ is the filling factor, $L_{s}$ is the synchrotron luminosity (calculated from the radio flux and luminosity distance), *k* is the baryon energy fraction parameter, and *V* is the emitting volume [@AP70]. The values used for the constants are; *k*=0, *$C_{12}$*=5.7$\times$10$^7$, $\alpha_r$=0.8, and $\Phi$=1. The emitting volume is calculated using the length of the jet defined in Table \[table:sjm\] by taking the difference of the two radius values and then assuming a cylindrical cross section given by the width associated with the *Chandra* FWHM (0.75$\arcsec$). The 1.4 GHz (FWHM = 1.4$\arcsec$) radio data results in larger derived emitting volumes than the *Chandra* FWHM. This discrepancy causes the magnetic field value (*$B_{1}$*) to be considered a minimum value for all intents and purposes. One way to address this magnetic field discrepancy is to adjust the filling factor $\Phi$. If the $\Phi$ is decreased from unity by a factor of 10 the magnetic field quantity $B_{1}$ would only increase by roughly a factor of 2 [@HM05].
We next compute the X-ray to radio luminosity ratio (R) using
$$R = \frac{S_{x}(\nu/\nu_{x})^{-\alpha_r}}{S_{r}(\nu/\nu_{r})^{-\alpha_r}} = \frac{S_{x}\nu^{\alpha_r}_{x}}{S_{r}\nu^{\alpha_r}_{r}} = \left[\frac{\nu_{x}}{\nu_{r}}\right]^{\alpha_r-\alpha_{rx}},$$
where $\nu_{r}$ and $\nu_{x}$ are the radio and X-ray frequencies at which the flux densities $S_{r}$ and $S_{x}$ are observed, respectively. The luminosity distance and redshift are also important parameters, because they affect the derived synchrotron luminosity in the jet frame. Equation 2 is valid as long as both the X-ray and radio frequencies are far from the endpoints of the synchrotron and IC spectral breaks. We use $\nu$$_x$=2.42$\times$10$^{17}$ Hz for calculation purposes later in the paper, as well as the $\nu$$_r$ values listed in Table \[table:sjm\]. Based on previous IC/CMB modeling, and following the solution presented by [@HM05], we compute the quantity *K*, which is a function of constants or observed values only, given by
$$K=B_{1}(aR)^{1/(\alpha_r+1)}(1+z)^{-(\alpha_r+3)/(\alpha_r+1)}b^{(1-\alpha_r)/(\alpha_r+1)},$$
where *a*=9.947$\times$10$^{10}$ Gauss$^{-2}$ and *b*=3.808$\times$10$^{4}$ Gauss, as used by [@HK02]. The values for *a* and *b* are found by equating the expected and observed values of the ratio of X-ray to radio energy densities (*R*) under the equipartition assumption. This leads to *K* being a dimensionless number that is solely a function of intrinsic jet speed and viewing angle. [@HM05] showed that *K* is a simple function of the beaming parameters under the assumption that $\Gamma > 1.5$.
$$K=\frac{1-\beta+\mu-\beta\mu}{(1-\beta\mu)^{2}},
\label{eq:K_eqn}$$
which can then be solved for $\mu$ (where $\mu$ = cos $\theta$) for a given $\beta$ (see [@HM05] Eq. 5). The solution for $\mu$ has two roots and we have chosen, like [@HM05], to use the negative root of Equation \[eq:K\_eqn2\]. $$\mu=\frac{1-\beta+2K\beta\pm(1-2\beta+4K\beta+\beta^{2}-4K\beta^{3})^{1/2}}{2K\beta^{2}}
\label{eq:K_eqn2}$$
If the viewing angle is known, the following equations can be used to find $\delta$ and $\Gamma$:
$$\beta=\frac{{\beta_{app}}}{{\beta_{app}}\mu+\sqrt{1-\mu^2}},
\label{eq:beta_app_eqn}$$
$$\theta=\arctan\frac{2\beta_{app}}{\beta^2_{app}+\delta^2-1},$$
$$\Gamma=\frac{\beta^2_{app}+\delta^2+1}{2\delta}.$$
By examining the effect of each parameter on the K factor individually we have determined that the main source of observational error in *K* is the radio spectral index $\alpha_{r}$. Typical observed values for $\alpha_{r}$ in kpc scale jets are between $-$0.7 and $-$0.9. Since we do not have direct measurements of $\alpha_{r}$ for our jet sample, we carried out a Monte Carlo error analysis, using a Gaussian distribution of $\alpha_{r}$ with $\overline{\alpha_{r}}$ = $-$0.8 and $\sigma_{\alpha_{r}}$ = 0.1. We tabulate the resulting one sigma error values for *K* in Table \[table:bmp\].
Viewing Angle and Bulk Lorentz Factor Analysis
==============================================
Using VLBI observations, we can investigate the possible kpc-scale jet beaming parameters under an initial assumption that there is no deceleration or bending from pc to kpc scales. This was done by using the theoretical framework of [@HK02]. Given a pc-scale ${\beta_{app}}$ measurement, which is a function of $\theta$ and $\beta$, we can use the *K* and ${\beta_{app}}$ equations, along with the assumption that the value of ${\beta_{app}}$ is the same for the pc and kpc scale jets to solve for the viewing angle $\theta$. This then allows us to calculate the Doppler factor ($\delta$) and the bulk Lorentz factor ($\Gamma$) using the IC/CMB model (e.g., see @HK02). We discuss the possibilities of jet bending and deceleration on the pc to kpc scales in Sections 5.2 and 5.3.
IC/CMB model with No Jet Deceleration or Bending
------------------------------------------------
Equation 4 can be solved for $\theta$ as a function of *K* and $\beta$ (or $\Gamma$; @HM05) as shown in Figure \[fig:gt1\] (blue curve) for each source. Equation 5 defines a locus of allowed $\Gamma$ and $\theta$ values for a fixed ${\beta_{app}}$ observed in the pc scale jet (black curve). The intersection point of these curves yields the viewing angle and bulk Lorentz factor of the kpc scale jet, under the assumption that the X-ray emission is given by the IC/CMB model and that the jet directions and bulk Lorentz factors on pc and kpc scales are the same. The range of the error of the ${\beta_{app}}$ and *K* values defines a range for the value of $\Gamma$ described by the error curves associated with curves plotted in Figure \[fig:gt1\]. Note that in the cases of 0415+379 and 1800+440, as well as some other jets in the sample, the uncertainty in ${\beta_{app}}$ can translate into a large range of uncertainty on $\Gamma$ (Table \[table:bmp\]).
Our measured ranges of $\Gamma$ are consistent with previous investigations of beamed inverse Compton models for X-ray emission, which often require bulk Lorentz factors on the order of $\Gamma$$\approx$10 or greater. [@LM09], on the other hand, model a set of radio data using a Bayesian parameter-inference method, which provides $\Gamma$ values for a sample of FRII jets. These $\Gamma$ values range from 1.18 to 1.49, which are significantly lower than the values required by the inverse Compton model. The FRII jets in the [@LM09] sample, however, are selected on the basis of isotropic lobe emission, and thus their jets generally have large angles to the line of sight. They are therefore more representative of the general FRII population than our MCS sample, which is highly biased toward fast jets pointing nearly directly at us. As pointed out by [@LM97], unbiased orientation samples of radio jets are likely to have much lower bulk Lorentz factors than blazars, due to the relatively steep power law distribution of jet speeds in the parent population. [@NC10] produced a Monte Carlo simulation which describes the mean pc scale viewing angle distribution for a modeled MOJAVE sample. The sample is modeled by using 1000 trial populations of 135 sources which have their bulk Lorentz factor described by a power law ranging from 3 to 50 with an index of $-$1.5 and are based on the luminosity function for the MOJAVE parent population [@CL08]. This simulation produces a roughly Poisson distribution of pc scale angles for the sample which is peaked around 2$^{\circ}$. This non-uniform distribution for the pc scale viewing angle is produced because of the highly beamed nature of the MOJAVE sample. Since the MCS is a subsample of the MOJAVE sample we should expect to see a similar angle bias in it.
0106+013 and 1849+670 show extreme values for the bulk Lorentz factor (71 $<$ $\Gamma$ $<$ 133 and 97 $<$ $\Gamma$ $<$ 129 respectively) when compared to the rest of the MCS sample, as well as other samples of blazars. The largest value of $\Gamma$ on pc scales in the [@TH09] sample is 65 for 1730$-$130, which has an extremely large apparent velocity value ($\beta_{app}$ $\approx$ 35c). [@TH09] compare their sample to the [@PU92] sample, which has a maximum $\Gamma$ of about 40. The MOJAVE sample contains no known jets with superluminal speeds above 50c [@ML09b]. [@LM97] find that in large flux limited blazar samples the value for $\beta_{app,max}$ should always be very similar to the $\Gamma_{max}$ in the parent population. The MCS sources with extreme Lorentz factors have the smallest values of $\theta$ in the sample, with values less than 7$^{\circ}$, and also have the largest $\beta_{app}$ values in the sample.
Based on papers by [@CM93] & [@PM81] the small angle approximation with respect to the pc scale position angle ($\theta_{n}$) and the intrinsic misalignment angle between the pc and kpc scale jets ($\zeta$) for Equation 1 of [@CM93] is:
$$\tan(\eta)\approx\frac{\sin(\phi)}{\left(\frac{\theta_{n}}{\zeta}+\cos(\phi)\right)},$$
where $\eta$ is the change in the position angle on the plane of the sky between the pc scale to the kpc scale jets and $\phi$ is the azimuthal angle of the jet. For a small value of $\eta$, the value for $\zeta$ has to be small when compared to the value for $\theta_{n}$ for an arbitrary value of $\phi$. Thus, for the sources in our sample with a small $\eta$, misalignments are likely to be less than a degree, so any discrepancy between $\delta$ and $\Gamma$ is likely to require deceleration. Furthermore, to obtain a value for $\eta$ which is large, $\zeta$ must be larger than $\theta_{n}$. These large values for $\eta$ require that the value for $\zeta$ is comparable to or larger than ${\beta_{app}}^{-1}$. This is easily accomplished when the value for ${\beta_{app}}$ is large, as is true with most of the sources in this sample. [@PM81] states that it is quite likely for small values of $\theta_{max}$ to be attributed to large values of $\eta$, where $\theta_{max}$ is the largest value of $\theta_{n}$ which is likely to occur. Lastly, for sources with $\eta$ values which approach $90^{\circ}$ the value of $\zeta$ must be comparable to the value of $\theta_{n}$. Examining Figure \[fig:gt1\], it is evident that bending between pc and kpc scales cannot by itself resolve the high bulk Lorentz factor issue in these two extreme sources.
IC/CMB model with Jet Deceleration
----------------------------------
One way to reconcile the large $\Gamma$ values is to consider possible deceleration from the pc to kpc scale, where the deceleration is caused by a jet depositing its power into the surrounding medium in the form of kinetic energy [@GK04]. A one-zone model deceleration of jets can allow for a misalignment of knots and other jet structures between the radio and X-ray wavelengths, which we also find examples of in the MCS. Deceleration can also offer a way to reduce the unusually large values for $\Gamma$ in sources in the MCS by widening the beaming cone, assuming that the jets decelerate from ultra relativistic speeds near the base of the jet to mildly relativistic and even sub-relativistic speeds at the point of termination.
We now examine the application of the IC/CMB model allowing for possible deceleration, but no bending between the pc and kpc scales. The problem reduces to finding a possible family of horizontal lines in Figure \[fig:gt1\] that intersect both the pc (black) and kpc (blue) curves. In Figure \[fig:gt1\] we show a shaded region which represents this family of lines. The red dashed line corresponds to the best fit viewing angle in the non-bending/non-decelerating model of Section 5.1. Thus, if we relax our non-decelerating assumption, the kpc scale jet can lie on the low $\Gamma$ tail of the K curve, without the need to invoke any jet bending. We list the range of possible kpc scale $\Gamma$ values for this deceleration/non-bending scenario in column 10 of Table 5. These ranges are generally narrow. In the case of the two extreme blazars, for 0106+013 we have 1.89 $< \Gamma <$ 1.92 and for 1849+670, 2.21 $< \Gamma <$ 2.23. Thus, deceleration between pc and kpc scales offers a way to alleviate the need for unusually high bulk Lorentz factors in these blazar jets.
IC/CMB model with Deceleration and Jet Bending
----------------------------------------------
Many blazar jets display bent morphologies going from pc to kpc scales (e.g., @PK10), although their apparent magnitudes are often highly exaggerated by projections effects. For some of the jets in our sample, bending between the pc and kpc scales can lower the bulk Lorentz factor value required to reconcile the VLBI and X-ray observations, but can not change the requirement that $\Gamma \geq {\beta_{app}}$ on pc scales, which is derived from the superluminal motion of the radio pc scale jet. Allowing the possibility of deceleration, acceleration, and bending effectively renders the two curves in Figure \[fig:gt1\] independent, i.e., the kpc jet can now lie anywhere on the blue curve, and the pc jet anywhere on the black curve. There are still, however, limits that can be placed on the beaming parameters. For example, the expression for K (Equation \[eq:K\_eqn\]) sets an upper limit on $\theta_{kpc}$, which is a lengthy algebraic function of K [@HM05]. These limits range from 8$^{\circ}$ to 20$^{\circ}$ for the jets in our sample (Figure \[fig:gt1\]). Setting $\mu$ = 1 for an end-on jet in Equation \[eq:K\_eqn\] also yields a lower limit of
$$\Gamma_{min}=\frac{K}{2\sqrt{K-1}}$$
on the kpc scale. These are tabulated in column 11 of Table \[table:bmp\]. The IC/CMB model thus limits the kpc scale minimum bulk Lorentz factor to 1.6 $< \Gamma <$ 2.7 in most cases, although in two sources (0415+379 and 1334$-$127) the limit placed on the minimum bulk Lorentz factor must be at least 3.5.
Finally, the superluminal speed confines the Lorentz factor of the pc jet to $\geq {\beta_{app}}$, and its viewing angle to below 2 $\tan^{-1}({\beta_{app}}^{-1})$. If independent observations can further constrain the amount of intrinsic jet bending, then X-ray observations of blazars can provide useful limits on the amount of pc to kpc scale jet deceleration. It may be possible to pursue this method statistically using a larger sample.
Conclusions
===========
We have performed Chandra observations of a radio-core-selected sample of blazar jets. The selection criteria that we used to define our AGN sample has increased the overall fraction of correlations between X-ray and radio jets in radio selected AGN. Of the popular single zone models available (synchrotron or IC/CMB), we chose to apply the IC/CMB model to our sample, based on the earlier results of [@HM05]. The detected X-ray jets are generally well correlated spatially with the radio jet morphology, except for those radio jets that display sharp bends. The wide range of apparent X-ray to radio ratios among the jets suggests that no single overall emission model can explain all of the X-ray morphologies. We are currently analyzing follow-up *Chandra* and *HST* observations of selected AGN to obtain multiwavelength spectra of jet knots (Kharb et al. 2011, in prep.), which will allow us to investigate possible synchrotron and IC models for the emission beyond what we have discussed in this paper.
Our major findings are as follows:
- The selection criteria associated with the MCS has increased the detection rate from previous jet surveys [@RS04; @HM05] from a $\sim$ 60% detection rate to a $\sim$ 78% detection rate.
- We have found that the 1.4 GHz, VLA-A array extended radio jet flux density, S$_{ext}$, is a strong predictor of X-ray jet emission in a core-selected sample such as the MCS, which is related to the correlation of the extended luminosity and the pc scale jet (apparent) speed. Above a value of 300 mJy we find a 100% X-ray detection rate, with $\sim$ 57% detection rate for sources located below that threshold. This further reinforces the usefulness of our extended radio emission selection criteria for this sample.
- The IC/CMB assumptions can produce calculated values for the jet bulk Lorentz factor, $\Gamma$, which are larger than expected in some sources (eg. 0106+013 and 1849+670) under the assumption that the jet speed and direction are the same on both the pc and kpc scales.
- Bending alone can not reconcile the large $\Gamma$ values in these sources as it constrains the minimum $\Gamma$ value on the kpc scale to the minimum value on the pc scale. This can still be quite large as seen in sources such as 0106+013 and 1849+670.
- If we allow for the possibility of deceleration with out jet bending, the VLBI jet speeds and IC/CMB X-ray model can be reconciled, although jet bending is necessary in several cases. In this scenario the kpc scale relativistic jet bulk Lorentz factors typically range from $\sim$ 1.7 to 7.
- When both the non-bending and non-decelerating assumptions are relaxed the only constraints on the kpc scale jet from the Chandra and VLA observations are an upper limit on the viewing angle, and a lower limit on the bulk Lorentz factor. These typically range from $8^{\circ} < \theta < 20^{\circ}$ and 1.6 $< \Gamma_{min} <$ 3.5 for our sample.
This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team [@ML09a]. This work was supported by Chandra Award GO8-9113A. We would also like to thank, the referee, Dan Harris for the insightful comments he provided that helped to improve this paper.
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![X-ray images obtained from *Chandra* with VLA 1.4 GHz radio contours overlaid in black and white. The VLA contours are set at 5 times the rms noise level for the lowest contour, with the exception of 0415+379 and 1849+670, which had their starting values set to 10 and 2.5 times the rms noise respectively, and multiples of 2 greater than that for each successive level. The X-ray portion of each image has been energy filtered to a range of 0.5 to 7.0 keV in CIAO before being processed in DS9. The FWHM dimensions of the radio restoring beam are denoted by a cross in the bottom corner of each image and are also located in Table \[table:vla\].[]{data-label="fig:rxo"}](0106+013.ps)
![[]{data-label="fig:rp"}](radio1.ps)
![X-ray profiles for the sources in the sample. These are represented as histograms of the counts in 0.2$\arcsec$ bins. The solid lines give the profile along the position angle of the jet, as defined by the radio images. The dashed lines show the profile along the counter-jet direction, which is defined as 180$^\circ$ opposite to the jet.[]{data-label="fig:xp"}](xray1.ps)
![[]{data-label="fig:gt1"}](figure1.ps)
[lllllll]{} 0106$+$013& OC 12 & 2.099 & 0.53 & 26.5 $\pm$ 4.2 & This paper & 9281\
0119$+$115& & 0.57 & 0.11 & 17.1 $\pm$ 1.1 & This paper& 9290\
0224$+$671& 4C 67.05 & 0.523 & 0.15 & 11.6 $\pm$ 0.8 & This paper& 9288\
0234$+$285& CTD 20 & 1.207 & 0.10 & 12.3 $\pm$ 1.1 & Marshall et al. 2005 & 4898\
0415$+$379& 3C 111 & 0.0491 & 2.70 & 5.9 $\pm$ 0.3 & This paper& 9279\
0529$+$075& OG 050 & 1.254 & 0.13 & 12.7 $\pm$ 1.6 & This paper & 9289\
0605$-$085& & 0.872 & 0.12 & 19.8 $\pm$ 1.2 & Sambruna et al. 2004 & 2132\
1045$-$188& & 0.595 & 0.51 & 8.6 $\pm$ 0.8 & This paper & 9280\
1055$+$018& 4C 01.28 & 0.89 & 0.23 & 11.0 $\pm$ 1.2 & Sambruna et al. 2004& 2137\
1156$+$295& 4C 29.45 & 0.729 & 0.20 & 24.9 $\pm$ 2.3 & Coppi et al. & 0874\
1222$+$216& 4C 21.35 & 0.432 & 0.96 & 21.0 $\pm$ 2.2 & Jorstad & Marscher 2005 & 3049\
1226$+$023& 3C 273 & 0.158 & 17.67 & 13.4 $\pm$ 0.8 & Jester et al. 2006 & 4879\
1253$-$055& 3C 279 & 0.536 & 2.10 & 20.6 $\pm$ 1.4 & WEBT Collaboration 2007 & 6867\
1334$-$127& & 0.539 & 0.15 & 10.3 $\pm$ 1.1 & This paper& 9282\
1510$-$089& & 0.36 & 0.18 & 20.2 $\pm$ 4.9 & Sambruna et al. 2004& 2141\
1641$+$399& 3C 345 & 0.593 & 1.48 & 19.3 $\pm$ 1.2 & Sambruna et al. 2004& 2143\
1655$+$077& & 0.621 & 0.20 & 14.4 $\pm$ 1.4 & Marshall et al. 2005& 3122\
1800$+$440& S4 1800$-$44 & 0.663 & 0.25 & 15.4 $\pm$ 1.0 & This paper & 9286\
1828$+$487& 3C 380 & 0.692 & 5.43 & 13.7 $\pm$ 0.8 & Marshall et al. 2005& 3124\
1849$+$670& S4 1849$-$67 & 0.657 & 0.10 & 30.6 $\pm$ 2.2 & This paper & 9291\
1928$+$738& 4C 73.18 & 0.302 & 0.36 & 8.4 $\pm$ 0.6 & Sambruna et al. 2004 & 2145\
1957$+$405& Cygnus A & 0.0561 & 414.18 & 0.2 $\pm$ 0.1 & Wilson et al. 2001 & 1707\
2155$-$152& & 0.672 & 0.30 & 18.1 $\pm$ 2.0 & This paper & 9284\
2201$+$315& 4C 31.63 & 0.295 & 0.37 & 7.9 $\pm$ 0.6 & This paper & 9283\
2216$-$038& & 0.901 & 0.31 & 5.6 $\pm$ 0.6 & This paper & 9285\
2251$+$158& 3C 454.3 & 0.859 & 0.88 & 14.2 $\pm$ 1.1 & Marshall et al. 2005& 3127\
2345$-$167& & 0.576 & 0.14 & 13.5 $\pm$ 1.1 & This paper & 9328\
\[table:mcs\]
[ccccc]{}
0106$+$013 & 9.69 & 2007 Nov 21 & 1h8m38.771s & +1$^\circ$35$\arcmin$0.317$\arcsec$\
0119$+$115 & 9.95 & 2008 Oct 27 & 1h21m41.595s & +11$^\circ$49$\arcmin$50.413$\arcsec$\
0224$+$671 & 10.11 & 2008 Jun 27 & 2h28m50.051s & +67$^\circ$21$\arcmin$3.029$\arcsec$\
0415$+$379 & 10.14 & 2008 Dec 10 & 4h18m21.277s & +38$^\circ$1$\arcmin$35.800$\arcsec$\
0529$+$075 & 10.18 & 2007 Nov 16 & 5h32m38.998s & +7$^\circ$32$\arcmin$43.345$\arcsec$\
1045$-$188 & 10.18 & 2008 Apr 01 & 10h48m6.621s & $-$19$^\circ$9$\arcmin$35.727$\arcsec$\
1334$-$127 & 10.79 & 2008 Mar 09 & 13h37m39.783s& $-$12$^\circ$57$\arcmin$24.693$\arcsec$\
1800$+$440 & 10.19 & 2008 Jan 05 & 18h1m32.315s & +44$^\circ$4$\arcmin$21.900$\arcsec$\
1849$+$670 & 10.19 & 2008 Feb 27 & 18h49m16.072s & +67$^\circ$5$\arcmin$41.680$\arcsec$\
2155$-$152 & 10.19 & 2008 Jul 10 & 21h58m6.282s & $-$15$^\circ$1$\arcmin$9.328$\arcsec$\
2201$+$315 & 10.11 & 2008 Oct 12 & 22h3m14.976s & +31$^\circ$45$\arcmin$38.270$\arcsec$\
2216$-$038 & 10.16 & 2007 Dec 02 & 22h18m52.038s & $-$3$^\circ$35$\arcmin$36.879$\arcsec$\
2345$-$167 & 10.15 & 2008 Sep 01 & 23h48m2.609s & $-$16$^\circ$31$\arcmin$12.022$\arcsec$\
\[table:obs\]
[ccccccc]{}
0106$+$013 & 2004-09-19 & AL634 & 1.4$\times$10$^{-1}$ & 1.64 & 1.49 & 100\
0119$+$115 & 2004-09-19 & AL634 & 9.8$\times$10$^{-2}$ & 1.53 & 1.44 & 55\
0224$+$671 & 2004-09-19 & AL634 & 1.5$\times$10$^{-1}$ & 1.42 & 1.13 & 78\
0415$+$379 & 1982-06-14 & LINF & 1.9$\times$10$^{-1}$ & 1.60 & 1.47 & 168\
0529$+$075 & 2004-09-19 & AL634 & 4.6$\times$10$^{-2}$ & 1.70 & 1.35 & 63\
1045$-$188 & 2007-06-30 & AC874 & 3.4$\times$10$^{-1}$ & 1.00 & 1.00 & 90\
1334$-$127 & 1986-03-18 & AD176 & 7.1$\times$10$^{-2}$ & 1.73 & 1.22 & 89\
1800$+$440 & 1990-05-18 & AS396 & 1.8$\times$10$^{-1}$ & 2.54 & 1.02 & 7\
1849$+$670 & 2004-11-09 & AL634 & 2.0$\times$10$^{-1}$ & 2.77 & 1.06 & 146\
2155$-$152 & 2004-11-21 & AL634 & 2.0$\times$10$^{-1}$ & 1.90 & 1.26 & 106\
2201$+$315 & 2004-11-21 & AL634 & 1.1$\times$10$^{-1}$ & 1.57 & 1.43 & 164\
2216$-$038 & 2004-11-21 & AL634 & 1.8$\times$10$^{-1}$ & 1.58 & 1.34 & 113\
2345$-$167 & 2004-11-09 & AL634 & 1.6$\times$10$^{-1}$ & 1.88 & 1.22 & 102\
\[table:vla\]
[rrrrrrrrrrc]{}
0106$+$013 & $-$122 & 180 & 1.5 & 8.0 & 526.7 $\pm$ 0.4 & 1.40 & 9.90 $\pm$ 1.11 & 9.9 &$<$ 1$\times10^{-10}$ & Y\
0119$+$115 & 6 & 35 & 1.5 & 8.0 & 22.3 $\pm$ 0.3 & 1.40 & 0.00 $\pm$ 0.38 & $<$ 1.2 &5.54$\times10^{-1}$ & N\
0224$+$671 & $-$5 & $-$10 & 1.5 & 11.0 & 22.9 $\pm$ 0.8 & 1.40 & $-$0.55 $\pm$ 0.45 & $<$ 0.8 &9.62$\times10^{-1}$ & N\
0415$+$379 & 71 & 63 & 1.5 & 100.0& 50.6 $\pm$ 7.2 & 1.44 & 7.50 $\pm$ 2.49 & 7.5 &2.44$\times10^{-6}$ & Y\
0529$+$075 & $-$7 & $-$145 & 1.5 & 8.0 & 69.2 $\pm$ 0.3 & 1.40 & 1.52 $\pm$ 0.65 & 1.5 &1.99$\times10^{-4}$ & Y\
1045$-$188 & 146 & 125 & 1.5 & 10.0 & 167.8 $\pm$ 5.0 & 1.42 & 2.82 $\pm$ 0.82 & 2.8 &4.74$\times10^{-8}$ & Y\
1334$-$127 & 147 & 135 & 1.5 & 12.0 & 103.9 $\pm$ 0.3 & 1.49 & 17.07 $\pm$ 1.56 & 17.1 &$<$ 1$\times10^{-10}$ & Y\
1800$+$440 & $-$157 & $-$130 & 1.5 & 8.0 & 133.2 $\pm$ 0.5 & 1.51 & 6.28 $\pm$ 0.99 & 6.3 &$<$ 1$\times10^{-10}$ & Y\
1849$+$670 & $-$52 & 0 & 5.0 & 20.0 & 8.3 $\pm$ 0.8 & 1.40 & 1.08 $\pm$ 0.40 & 1.1 &1.36$\times10^{-6}$ & Y\
2155$-$152 & $-$146 & $-$170 & 1.5 & 12.0 & 231.4 $\pm$ 0.9 & 1.40 & 1.51 $\pm$ 0.85 & 1.5 &5.00$\times10^{-3}$ & Y\
2201$+$315 & $-$141 & $-$110 & 1.5 & 10.0 & 31.1 $\pm$ 0.7 & 1.40 & 1.96 $\pm$ 1.05 & 2.0 &1.54$\times10^{-3}$ & Y\
2216$-$038 & $-$170 & 135 & 1.5 & 15.5 & 164.3 $\pm$ 1.0 & 1.40 & 1.74 $\pm$ 0.78 & 1.7 &4.94$\times10^{-4}$ & Y\
2345$-$167 & 141 & $-$135 & 2.0 & 8.0 & 83.8 $\pm$ 0.4 & 1.40 & 0.65 $\pm$ 0.67 & $<$ 2.7 &8.92$\times10^{-2}$ & N\
\[table:sjm\]
[rrrrlrrcrrcc]{}
0106$+$013 & 0.0740 & 1.7$\times$10$^{3}$ & 146 & 13$\pm$2 & 0.94 $\pm$ 0.01 & 4.2 & 922 & 3.6 & $99^{+33}_{-29}$ & 1.89$-$1.92 & 1.9\
0119$+$115 & $<$ 0.2010 & 8.0$\times$10$^{2}$ & 29 & $<$ 19 & $>$ 0.88 & $>$ 6.3 & ... & 4.4 & 36 & ... & ...\
0224$+$671 & $<$ 0.1322 & 1.0$\times$10$^{3}$ & 26 & $<$ 14 & $>$ 0.91 & $>$ 8.9 & ... & 3.8 & 20 & ... & ...\
0415$+$379 & 0.4829 & 3.9$\times$10$^{1}$ & 19 & 49$\pm$12 & 0.84 $\pm$ 0.02 & 8.1 & 661 & 7.0 & $6^{+1}_{-1}$ & 4.72$-$7.22 & 3.5\
0529$+$075 & 0.0822 & 1.7$\times$10$^{3}$ & 58 & 11$\pm$2 & 0.93 $\pm$ 0.02 & 8.4 & 460 & 3.3 & $26^{+7}_{-6}$ & 1.81$-$1.86 & 1.7\
1045$-$188 & 0.0678 & 1.1$\times$10$^{3}$ & 48 & 17$\pm$3 & 0.94 $\pm$ 0.02 & 10.8 & 355 & 4.1 & $11^{+2}_{-2}$ & 2.44$-$2.71 & 2.1\
1334$-$127 & 0.6867 & 1.2$\times$10$^{3}$ & 38 & 52$\pm$14 & 0.82 $\pm$ 0.01 & 7.5 & 582 & 7.2 & $11^{+2}_{-2}$ & 4.68$-$7.43 & 3.6\
1800$+$440 & 0.1871 & 9.9$\times$10$^{2}$ & 51 & 29$\pm$6 & 0.89 $\pm$ 0.01 & 6.6 & 486 & 5.3 & $25^{+3}_{-3}$ & 3.00$-$3.10 &2.7\
1849$+$670 & 0.4859 & 2.2$\times$10$^{3}$ & 18 & 18$\pm$4 & 0.84 $\pm$ 0.02 & 3.7 & 2156 & 4.2 & $114^{+15}_{-17}$ & 2.21$-$2.23 & 2.2\
2155$-$152 & 0.0255 & 1.6$\times$10$^{3}$ & 52 & 10$\pm$2 & 0.99 $\pm$ 0.03 & 6.1 & 793 & 3.1 & $55^{+10}_{-13}$ & 1.69$-$1.72 & 1.7\
2201$+$315 & 0.2438 & 1.5$\times$10$^{3}$ & 27 & 29$\pm$7 & 0.87 $\pm$ 0.03 & 9.9 & 254 & 5.4 & $9^{+1}_{-2}$ & 3.47$-$5.67 & 2.7\
2216$-$038 & 0.0428 & 2.9$\times$10$^{3}$ & 49 & 9$\pm$1 & 0.97 $\pm$ 0.02 & 15.8 & 445 & 3.0 & $7^{+1}_{-2}$ & 1.84$-$2.06 & 1.6\
2345$-$167 & $<$ 0.1227 & 7.5$\times$10$^{2}$ & 44 & $<$ 21 & $>$ 0.91 & $>$ 7.5 & ... & 4.7 & 22 & ... & ...\
\[table:bmp\]
[^1]: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
[^2]: Information about Ftools can be found at http://heasarc.gsfc.nasa.gov/ftools/
|
---
abstract: 'We recently presented a constructive solution to the $N$-representability problem of the two-electron reduced density matrix (2-RDM)—a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic $N$-electron quantum system \[D. A. Mazziotti, Phys. Rev. Lett. [**108**]{}, 263002 (2012)\]. In this paper we provide additional details and derive further $N$-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the $(2,q)$-positivity conditions where the $q$ indicates their derivation from the nonnegativity of $q$-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) [*lifting conditions*]{}, that is conditions which arise from lifting lower $(2,q)$-positivity conditions to higher $(2,q+1)$-positivity conditions and (ii) [*pure conditions*]{}, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure $(2,q)$-positivity conditions for $q>3$ require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new $N$-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.'
author:
- 'David A. Mazziotti'
bibliography:
- 'SFDM5BIBR2.bib'
date: 'Submitted May 4, 2012; Published [*Phys. Rev. A*]{} [**85**]{}, 062507 (2012)'
title: 'Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of $N$-representability'
---
Introduction
============
Because electrons are indistinguishable with pairwise Coulomb interactions, the energies and properties of many-electron atoms and molecules can be evaluated from a knowledge of the two-electron reduced density matrix (2-RDM) [@RDM; @CY00; @M12a]. Minimizing the ground-state energy as a functional of the 2-RDM, however, requires non-trivial constraints on the 2-RDM to ensure that it represents an $N$-electron system ($N$-representability conditions) [@RDM; @CY00; @M12a; @C63; @GP64; @K67; @H78; @E78; @P78; @EJ00; @ME01; @N01; @M02; @M04; @P04; @C06; @M05; @M06; @GM07; @E07; @A09; @SI10; @M11; @BP12]. While advances in theory and computation enabled the accurate variational calculation of the 2-RDM for a variety of strongly correlated systems in chemistry and physics from polyaromatic hydrocarbons [@GM08; @PGG11] to quantum dots [@RM09], the known $N$-representability conditions for the 2-RDM, albeit rigorous, remained incomplete. Recently, we presented a constructive solution to the $N$-representability problem—a systematic approach to constructing complete $N$-representability conditions on the two-electron reduced density matrix (2-RDM)—as well as examples of new $N$-representability conditions [@M12b]. In the present paper we present additional details as well as further conditions on the 2-RDM that follow from the constructive solution.
The advantage of reduced variables such as the 2-RDM and the one-electron density is that, unlike the wavefunction expanded in terms of determinants, their degrees of freedom grow [*polynomially*]{} with the size of the quantum system [@M12a] even when the electrons are strongly correlated [@ZK05; @MA03]. Direct calculation of the reduced variables, however, requires that they and their functionals be consistent with a realistic $N$-electron quantum system; in other words, the reduced variables and functionals must be representable by the integration of an $N$-electron density matrix. Such consistency relations are known as the [*$N$-representability conditions*]{} [@RDM; @CY00; @M12a; @C63; @GP64; @K67; @H78; @E78; @P78; @EJ00; @ME01; @N01; @M02; @M04; @P04; @C06; @M05; @M06; @E07; @A09; @SI10; @M11; @BP12]. These conditions are particularly important to 2-RDM methods where they enable the direct calculation of the 2-RDM without the wavefunction, but they are also implicit in the design of realistic approximations to the density functional in density functional theory [@PY94; @CH08].
Minimizing the many-electron energy as a functional of the 2-RDM [*without*]{} $N$-representability conditions produces an energy that is [*much lower*]{} than the exact ground-state energy of the quantum system. The energy is too low because both the energy and the computed 2-RDM are not realistic—they are not $N$-representable. In the early 1960s the search for the set of necessary and sufficient $N$-representability conditions became known as the $N$-representability problem [@C63]. Three important constraints, known as the D, Q, and G (or 2-positivity) conditions, were developed by Coleman [@C63] and Garrod and Percus [@GP64]. The D, Q, and G conditions restrict the probability distributions of two electrons, two holes (where a [*hole*]{} is the absence of an electron), and an electron-hole pair to be nonnegative. Each condition can be expressed in the form of constraining a matrix to be positive semidefinite. A matrix is [*positive semidefinite*]{} if and only if its eigenvalues are nonnegative.
In 1978 Erdahl [@E78] discovered two additional semidefinite constraints on the 2-RDM known as the T1 and T2 (or partial 3-positivity) conditions [@P04; @M06; @P07; @M05], which are derivable from the nonnegativity of the three-electron probability distributions. Finally, Weinhold and Wilson [@WW67], Yoseloff and Kuhn [@YK69], McRae and Davidson [@MD72], and Erdahl [@E78] derived necessary conditions on the [*diagonal*]{} part of the 2-RDM. These diagonal conditions were shown, in the context of the Boole optimization problem [@Cuts], to be part of a complete set of [*classical $N$-representability conditions*]{} on the two-electron reduced density function which is the diagonal part of the 2-RDM in a coordinate representation [@KM08]. Despite the solution of the classical problem, the complete set of quantum $N$-representability conditions remained unknown except for the D, Q, G, T1, and generalized T2 conditions as well as unitary transformations of the classical $N$-representability conditions. In 2001 Mazziotti and Erdahl [@ME01] presented a systematic generalization of these constraints known as the [*$p$-positivity conditions*]{} and in 2002 Mazziotti [@M02; @HM05] introduced the [*lifting conditions*]{}; however, except for the conditions given above, the $p$-positivity conditions and the lifting conditions depend upon not only the 2-RDM but also higher-particle RDMs.
The constructive solution to the $N$-representability problem provides a systematic approach to building complete $N$-representability conditions on the two-electron reduced density matrix (2-RDM) [@M12b]. While an example of the new conditions was given previously, in the present paper we present further $N$-representability conditions on the 2-RDM that follow from the constructive solution. The conditions are in the form of a set of model Hamiltonians with pairwise interactions whose trace against the 2-RDM must be nonnegative. The resulting conditions can be classified into an increasing hierarchy of constraints, known as the $(2,q)$-positivity conditions where the first number $p$ in the name indicates the highest $p$-RDM required to evaluate the condition (the 2-RDM in our case) and the second number $q$ indicates the highest $q$-particle reduced density operators ($q$-RDOs) canceled by nonnegative linear combinations in the derivation of the condition. The $(p,p)$-positivity conditions are equivalent to the $p$-positivity conditions introduced earlier in Refs. [@EJ00; @ME01; @M02]. We will use the two conventions in nomenclature interchangeably.
In addition to the previously known T1 and T2 conditions [@E78; @P04; @M06; @P07; @M05], we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The conditions obtained can be divided into two general types: (i) [*lifting conditions*]{}, that is conditions which arise from lifting lower $(2,q)$-positivity conditions to higher $(2,q+1)$-positivity conditions and (ii) [*pure conditions*]{}, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure $(2,q)$-positivity conditions for $q>3$ require that the expansion coefficients in the model Hamiltonians be [*tensor decomposed*]{}. Subsets of the $N$-representability conditions can be employed with previously known conditions for polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems in chemistry and physics.
Theory
======
After the constructive solution of $N$-representability is reviewed in section \[sec:sol\], it is employed in sections \[sec:kcon\] and \[sec:ncon\] to derive known and new $N$-representability conditions, respectively. The new constraints are organized into sections on (2,3)-, (2,4)-, (2,5)-, and (2,6)-positivity conditions. Two algorithms for implementing the conditions in a variational 2-RDM calculation are briefly discussed in section \[sec:sd\].
Constructive solution {#sec:sol}
---------------------
The energy of an $N$-electron quantum system in a stationary state can be computed from the Hamiltonian traced against the state’s density matrix $$\label{eq:EN} E = {\rm Tr}({\hat H} \, {}^{N} D) ,$$ where the Hamiltonian operator is expressible in second quantization as $${\hat H} = \sum_{ijkl}{ {}^{2} K^{ij}_{kl} {\hat a}^{\dagger}_{i}
{\hat a}^{\dagger}_{j} {\hat a}_{l} {\hat a}_{k} }$$ in which the matrix ${}^{2} K$ is the reduced Hamiltonian operator in a finite one-electron basis set [@M98] and the indices label the members (orbitals) of the basis set. Because electrons are indistinguishable with pairwise interactions, the energy can also be universally written as a linear functional of only the 2-RDM $$\label{eq:E2} E = {\rm Tr}({\hat H} \, {}^{2} D) ,$$ where the 2-RDM can be formally defined from integration of the $N$-electron density matrix over all electrons save two $$\label{eq:D2} {}^{2} D = \frac{N(N-1)}{2} \int{ {}^{N} D \, d3 \dots
dN } .$$ The expression of the energy as a functional of the 2-RDM suggests the tantalizing possibility of computing the ground-state energy of any electronic system as a functional of only the 2-RDM [@M55; @RDM; @CY00]. Early calculations by Coleman [@C63], Tredgold [@T57], and others, however, showed that minimization of the energy as a 2-RDM functional produces unphysically low energies without additional constraints on the 2-RDM to ensure that it represents an $N$-electron density matrix. In 1963 Coleman called these constraints the $N$-representability conditions [@C63].
Building upon work by Garrod and Percus [@GP64], Kummer in 1967 showed by the bipolar theorem [@R71] that there exists a convex set (cone) of two-body operators $\{ {}^{2} {\hat O_{i}} \}$ whose trace against a potential 2-RDM will be nonnegative $${\rm Tr}({}^{2} {\hat O} \, {}^{2} D) \ge 0$$ if and only if the 2-RDM is $N$-representable [@K67]. Hence, the set of two-body operators $\{ {}^{2} {\hat O_{i}} \}$ defines the set $P^{2}_{N}$ of $N$-representable 2-RDMs. We say that the set $\{ {}^{2} {\hat O_{i}} \}$ is the polar of $P^{2}_{N}$ and denote it as ${P^{2}_{N}}^{*}$. Characterizing the set $P^{2}_{N}$ of $N$-representable 2-RDMs, therefore, would be complete if we could characterize its polar set ${P^{2}_{N}}^{*}$. Kummer’s original result demonstrates the existence of the set ${P^{2}_{N}}^{*}$, but it does not provide a prescription for constructing it.
Recently, a constructive solution to the $N$-representability problem has been derived through the complete characterization of the polar set ${P^{2}_{N}}^{*}$ [@M12b]. In Ref. [@M12b] it is proven that the second-quantized representation of the operators $\{ {}^{2} {\hat O_{i}} \}$ in ${P^{2}_{N}}^{*}$ can be explicitly constructed as follows $$\label{eq:O2} {}^{2} {\hat O} = \sum_{i}{ w_{i} {\hat C}_{i} {\hat
C}_{i}^{\dagger} }$$ where ${\hat C}_{i}$ are polynomials in the creation and/or annihilation operators of degree less than or equal to $r$ (the rank of the one-electron basis set) and $w_{i}$ are nonnegative integer weights. The proof relies on the fact that ${P^{2}_{N}}^{*}$ is [*contained within*]{} the set ${P^{r}_{N}}^{*}$ of operators of degree $\le 2r$ whose trace against an $N$-electron density matrix must be nonnegative. Because the extreme elements (rays) of the convex cone ${P^{r}_{N}}^{*}$ are readily expressed as [@H02] $${\hat C}_{i} {\hat C}_{i}^{\dagger} ,$$ the extreme elements (rays) of ${P^{2}_{N}}^{*}$ can be constructed from the [*conic combinations*]{} (or nonnegative linear combinations) given in Eq. (\[eq:O2\]). The conic combinations, if divided by $\sum_{i}{w_{i}}$, can be interpreted as [*convex combinations*]{}. Conic combinations are contained in ${P^{2}_{N}}^{*}$ if and only if they cancel all three- and higher-body operators, that is polynomials in creation and annihilation operators of degree greater than of equal to 6.
Practical implementation {#sec:sd}
------------------------
Before developing known and new $N$-representability conditions in sections \[sec:kcon\] and \[sec:ncon\] respectively, in this section we briefly indicate their practical applications by sketching two algorithms for computing the ground-state 2-RDM. Minimizing the ground-state energy as a function of the 2-RDM constrained by these conditions can be formulated as a linear program $$\begin{aligned}
\label{eq:Ex} {\rm minimize~~} E & = & {\rm Tr}({\hat H} \, {}^{2}
D) \\ {\rm such~that~~} {\rm Tr}({\hat O}_{j} \, {}^{2} D ) & \ge &
0~~~{\rm for~all}~j, \label{eq:lin}\end{aligned}$$ in which the necessary set of operators (model Hamiltonians) ${\hat
O}_{j}$, defining the boundary of the convex set of 2-RDMs, must be determined iteratively. Given an initial set of model-Hamiltonian constraints that bound the minimum energy, the three key steps of the algorithm are: (i) solving the linear program for the optimal 2-RDM, (ii) updating the set of model-Hamiltonian constraints in the linear program, and (iii) repeating steps (i) and (ii) until the 2-RDM is nonnegative in its trace with all model Hamiltonians explored in step (ii). In the second step, the trace of each model Hamiltonian with the 2-RDM is minimized by optimizing the Hamiltonian’s parameters (expansion coefficients), and if the final trace is negative, the model Hamiltonian with its optimized parameters is added to the constraints in Eq. (\[eq:lin\]). In practice, only a subset of model Hamiltonians from the constructive solution is employed.
Some of the $N$-representability constraints can be collected together as a single semidefinite constraint on the 2-RDM. The generalization of a linear program to include semidefinite constraints is known as a [*semidefinite program*]{}, and the solution of such a program is called [*semidefinite programming*]{} [@VB96; @HSV00]. Efficient large-scale semidefinite programming algorithms have been developed for the variational calculation of the 2-RDM [@M04; @P04; @C06; @FNY07; @M07; @A09; @M11; @BP12; @E79]. While the model Hamiltonians corresponding to previously known $N$-representability conditions in section \[sec:kcon\] can be expressed as semidefinite constraints, the model Hamiltonians corresponding to the new conditions in section \[sec:ncon\], which use tensor decompositions of the expansion coefficients in the ${\hat C}_{i}$ operators, cannot be written as traditional semidefinite constraints. In practice, however, we can add these non-standard constraints to a semidefinite program containing the standard semidefinite constraints by the three-step iterative procedure discussed above for the linear program. A main advantage of this second algorithm is that a large number of model Hamiltonians can be included by a single semidefinite constraint. A similar algorithm, to which we refer for further details, was proposed in Ref. [@JSM07] for imposing the T2 condition by recursively generated linear inequalities.
Known conditions {#sec:kcon}
----------------
All previously known $N$-representability conditions are generated by the constructive solution. The most important representability conditions on the 2-RDM, derived by Coleman [@C63] and Garrod and Percus [@GP64], are the D, Q, and G conditions—also, known as the 2-positivity conditions [@ME01]. These conditions restrict the two-particle RDM $^{2} D$, the two-hole RDM $^{2} Q$, and the particle-hole RDM $^{2} G$ to be positive semidefinite, that is $$\begin{aligned}
^{2} D & \succeq & 0 \\
^{2} Q & \succeq & 0 \\
^{2} G & \succeq & 0 ,\end{aligned}$$ where the elements of the RDMs are given by $$\begin{aligned}
^{2} D^{ij}_{kl} = \langle \Psi | {\hat a}^{\dagger}_{i}
{\hat a}^{\dagger}_{j} {\hat a}_{l} {\hat a}_{k} | \Psi \rangle \\
^{2} Q^{ij}_{kl} = \langle \Psi | {\hat a}_{i} {\hat
a}_{j} {\hat a}^{\dagger}_{l} {\hat a}^{\dagger}_{k} | \Psi \rangle \\
^{2} G^{ij}_{kl} = \langle \Psi | {\hat a}^{\dagger}_{i} {\hat
a}_{j} {\hat a}^{\dagger}_{l} {\hat a}_{k} | \Psi \rangle\end{aligned}$$ and $M\succeq 0$ indicates that the matrix $M$ is constrained to be positive semidefinite. Physically, these conditions correspond to constraining the probability distributions of two particles, two holes, as well as one particle and one hole to be nonnegative. The 2-positivity conditions are generated from the constructive solution by restricting the following three two-body operators from Eq. (\[eq:O2\]) to be nonnegative for all coefficients $b_{ij}$ $$\begin{aligned}
{}^{2} {\hat O}_{D} & = & {\hat C}_{D} {\hat C}_{D}^{\dagger} \label{eq:OD} \\
{}^{2} {\hat O}_{Q} & = & {\hat C}_{Q} {\hat C}_{Q}^{\dagger} \label{eq:OQ} \\
{}^{2} {\hat O}_{G} & = & {\hat C}_{G} {\hat C}_{G}^{\dagger}
\label{eq:OG}\end{aligned}$$ where the ${\hat C}_{D}$, ${\hat C}_{Q}$, and ${\hat C}_{G}$ cover all polynomials in creation and annihilation operators of degree two $$\begin{aligned}
{\hat C}_{D} & = & \sum_{ij}{ b_{ij} {\hat a}^{\dagger}_{i}
{\hat a}^{\dagger}_{j} } \\
{\hat C}_{Q} & = & \sum_{ij}{ b_{ij} {\hat a}_{i}
{\hat a}_{j} } \\
{\hat C}_{G} & = & \sum_{ij}{ b_{ij} {\hat a}^{\dagger}_{i} {\hat
a}_{j} } .\end{aligned}$$ Note that conic combinations are not present in these conditions because when the ${\hat C_{i}}$ operators are of degree 2, the expectation values of the ${\hat O_{i}}$ operators only involve the 2-RDM [@M12b].
The other previously known $N$-representability conditions—the T1 and T2 conditions [@E78; @P04; @M06; @P07; @M05]—are part of the (2,3)-conditions that follow from the constructive solution. These semidefinite conditions on the 2-RDM are obtainable from conic combinations of three-particle metric matrices that cancel their dependence on the 3-RDM [@M06; @M05] $$\begin{aligned}
T1 = {}^{3} D + {}^{3} Q & \succeq & 0 \\
T2 = {}^{3} E + {}^{3} F & \succeq & 0 \label{eq:T2}.\end{aligned}$$ where in second quantization the matrix elements of these metric matrices are definable as $$\begin{aligned}
{}^{3} D^{ijk}_{pqs} & = & \langle \Psi | {\hat a}^{\dagger}_{i}
{\hat a}^{\dagger}_{j} {\hat a}^{\dagger}_{k} {\hat a}_{s} {\hat
a}_{q} {\hat a}_{p} | \Psi \rangle \\
{}^{3} E^{ijk}_{pqs} & = & \langle \Psi | {\hat a}^{\dagger}_{i}
{\hat a}^{\dagger}_{j} {\hat a}_{k} {\hat a}^{\dagger}_{s} {\hat
a}_{q} {\hat a}_{p} | \Psi \rangle \\
{}^{3} F^{ijk}_{pqs} & = & \langle \Psi | {\hat a}_{p} {\hat a}_{q}
{\hat a}^{\dagger}_{s} {\hat a}_{k} {\hat a}^{\dagger}_{j}
{\hat a}^{\dagger}_{i} | \Psi \rangle \\
{}^{3} Q^{ijk}_{pqs} & = & \langle \Psi | {\hat a}_{p} {\hat a}_{q}
{\hat a}_{s} {\hat a}^{\dagger}_{k} {\hat a}^{\dagger}_{j} {\hat
a}^{\dagger}_{i} | \Psi \rangle\end{aligned}$$ The four metric matrices ${}^{3} D$, ${}^{3} E$, ${}^{3} F$, and ${}^{3} Q$ correspond to the probability distributions for three particles, two particles and a hole, one particle and two holes, and three holes, respectively [@ME01; @M02; @M06]. Restricting the ${}^{3} D$, ${}^{3} E$, ${}^{3} F$, and ${}^{3} Q$ matrices to be positive semidefinite generates the 3-positivity conditions [@ME01; @M06] which depend on the 3-RDM. While the T1 and T2 conditions are a subset of the 3-positivity conditions, they depend only upon the 2-RDM because the 3-particle parts of ${}^{3}
D$ and ${}^{3} Q$ (and ${}^{3} E$ and ${}^{3} F$) cancel upon addition [@M06; @M05]. For example, the matrix elements of $T1$ are given by $$T1^{ijk}_{pqs} = 6 \, {}^{3} I^{ijk}_{pqs} - 18 \, ^{1} D^{i}_{p}
\wedge {}^{2} I^{jk}_{qs} + 9 \, {}^{2} D^{ij}_{pq} \wedge
{}^{1}I^{k}_{s} ,$$ where $^{p} I$ is the $p$-particle identity matrix and $\wedge$ denotes the Grassmann wedge product [@S71; @M98].
While the T1 condition is unique, three distinct forms of the T2 condition can be generated from rearranging the second-quantized operators in the definition of the ${}^{3} F$ metric matrix relative to those in the ${}^{3} E$ metric matrix [@M06]. Consider the two variants of the ${}^{3} F$ matrix with the following matrix elements: $$\begin{aligned}
{}^{3} {\bar F}^{ijk}_{pqs} & = & \langle \Psi | {\hat a}_{p} {\hat
a}^{\dagger}_{s} {\hat a}_{q} {\hat a}^{\dagger}_{j}
{\hat a}_{k} {\hat a}^{\dagger}_{i} | \Psi \rangle \\
{}^{3} {\tilde F}^{ijk}_{pqs} & = & \langle \Psi | {\hat
a}^{\dagger}_{s} {\hat a}_{p} {\hat a}_{q} {\hat a}^{\dagger}_{j}
{\hat a}^{\dagger}_{i} {\hat a}_{k} | \Psi \rangle .\end{aligned}$$ The 3-positivity condition ${}^{3} F \succeq 0$ implies both ${}^{3}
{\bar F} \succeq 0$ and ${}^{3} {\tilde F} \succeq 0$ because reordering the creation and annihilation operators does not change the vector space covered by the metric matrix. Changing the ordering of the second-quantized operators in the ${}^{3} F$ matrix relative to those in the ${}^{3} E$ matrix, however, does generate two additional T2 conditions $$\begin{aligned}
{\bar T2} & = & {}^{3} E + {}^{3} {\bar F} \succeq 0 \label{eq:T2b}\\
{\tilde T2} & = & {}^{3} E + {}^{3} {\tilde F} \succeq 0
\label{eq:T2t}.\end{aligned}$$ It was the ${\tilde T2}$ form of the T2 condition that was originally implemented by Zhao [*et al.*]{} [@P04] and Mazziotti [@M05; @M06].
The three T2 conditions are generated in the constructive solution by keeping the following two-body operators from Eq. (\[eq:O2\]) nonnegative $$\begin{aligned}
^{2} {\hat O}_{T2} & = & {\hat C}_{E} \, {\hat
C}_{E}^{\dagger} + {\hat C}_{F} {\hat C}_{F}^{\dagger} \\
^{2} {\hat O}_{\bar T2} & = & {\hat C}_{E} \, {\hat
C}_{E}^{\dagger} + {\hat C}_{\bar F} {\hat C}_{\bar F}^{\dagger} \\
^{2} {\hat O}_{\tilde T2} & = & {\hat C}_{E} \, {\hat
C}_{E}^{\dagger} + {\hat C}_{\tilde F} {\hat C}_{\tilde
F}^{\dagger}\end{aligned}$$ where $$\begin{aligned}
{\hat C}_{E} & = & \sum_{ijk}{ b_{ijk} {\hat a}^{\dagger}_{i} {\hat
a}^{\dagger}_{j} {\hat a}_{k} } \\
{\hat C}_{F} & = & \sum_{ijk}{ b^{*}_{ijk} {\hat a}_{i} {\hat a}_{j}
{\hat a}^{\dagger}_{k} } \\
{\hat C}_{\bar F} & = & \sum_{ijk}{ b^{*}_{ijk} {\hat a}_{i} {\hat
a}^{\dagger}_{k} {\hat a}_{j} } \\
{\hat C}_{\tilde F} & = & \sum_{ijk}{ b^{*}_{ijk} {\hat
a}^{\dagger}_{k} {\hat a}_{i} {\hat a}_{j} } .\end{aligned}$$ The three T2 conditions can be combined into a single generalized T2 condition as shown in Refs. [@M06; @P07]. The T1 condition is also produced in the constructive solution by keeping the following two-body operator from Eq. (\[eq:O2\]) nonnegative $$^{2} {\hat O}_{T1} = {\hat C}_{D} \, {\hat C}_{D}^{\dagger} + {\hat
C}_{Q} {\hat C}_{Q}^{\dagger}$$ where $$\begin{aligned}
{\hat C}_{D} & = & \sum_{ijk}{ b_{ijk} {\hat a}^{\dagger}_{i} {\hat
a}^{\dagger}_{j} {\hat a}^{\dagger}_{k} } \label{eq:d3} \\
{\hat C}_{Q} & = & \sum_{ijk}{ b^{*}_{ijk} {\hat a}_{i} {\hat a}_{j}
{\hat a}_{k} } .\end{aligned}$$ Because the second-quantized operators in ${\hat C}_{D}$ and ${\hat
C}_{Q}$ are anticommutative, there is only one T1 condition. Unlike the D, Q, and G conditions, both T1 and T2 conditions arise from the conic combination of a pair of 3-positive operators that cancels their dependence on the 3-RDM.
New conditions {#sec:ncon}
--------------
The constructive solution also produces new $N$-representability conditions on the 2-RDM [@M12b]. In this section we will discuss the further conditions on the 2-RDM that emerge from conic combinations of three-, four-, five-, and six-particle operators in Eq. (\[eq:O2\]), which we denote as (2,3)-, (2,4)-, (2,5)-, and (2,6)-positivity conditions, respectively. All of the new $N$-representability conditions require a nonlinear factorization of the expansion coefficients to cancel the higher-particle operators.
### (2,3)-positivity conditions
---------------------------------------------------------------------------------------------
Class Type Representative Condition ${\hat C}$ Definition
------- -------------- ---------------------------------------------- -----------------------
1 Lifted (2,2) ${\rm Tr}( ({\hat O}(k,i,j) + {\hat O}({\bar Eq. (\[eq:23P2\])
k},i,j)) \, {}^{2} D)
\ge 0$
2 Pure (2,3) ${\rm Tr}( ({\hat O}(i,j,k) + {\hat O}({\bar Eq. (\[eq:d3\])
i},{\bar j},{\bar k})) \, {}^{2} D)
\ge 0$
---------------------------------------------------------------------------------------------
In addition to the T1 and T2 conditions there exists a second class of (2,3)-positivity conditions that can be generated from lifting the 2-positivity conditions to the three-particle space and then canceling the three-particle operators. Consider the pair of three-body operators $$\begin{aligned}
{\hat O}(i,j,k) & = & {\hat C}(i,j,k) {\hat C}(i,j,k)^{\dagger} \\
{\hat O}(i,j,{\bar k}) & = & {\hat C}(i,j,{\bar k}) {\hat C}(i,j,
{\bar k})^{\dagger}\end{aligned}$$ where $$\begin{aligned}
{\hat C}(i,j,k) & = & \sum_{ijk}{ b_{ij} d_{k} {\hat
a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} {\hat a}^{\dagger}_{k} } \label{eq:23L2} \\
{\hat C}(i,j,{\bar k}) & = & \sum_{ijk}{ b_{ij} d^{*}_{k} {\hat
a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} {\hat a}_{k} } .
\label{eq:23P}\end{aligned}$$ The notation for the operators ${\hat O}(i,j,k)$ and ${\hat
C}(i,j,k)$ includes their internal summation indices to indicate succinctly: (i) the ordering of the second-quantized operators with indices $i$, $j$, and $k$, and (ii) the type of second-quantized operator with $k$ denoting ${\hat a}^{\dagger}_{k}$ and ${\bar k}$ denoting ${\hat a}_{k}$. Note that the notation does not indicate the ordering of the indices on the tensor coefficients which is alphabetical in both ${\hat C}(i,j,k)$ in Eq. (\[eq:23P\]) and ${\hat C}(k,i,j)$ in Eq. (\[eq:23P2\]). Although the summation indices within ${\hat C}$ and its adjoint are distinct, we only show primes on the indices of the adjoint when the indices of the two operators appear in the same sum. Finally, for the $N$-representability conditions to be valid for real symmetric and general Hermitian RDMs, one-index tensors $d_k$ and $d_{\bar k}$ denote $d_{k}$ and $d_{k}^{*}$, respectively. For multi-index tensors we employ the convention that the first subscript determines conjugacy, that is $b_{ij..m} = b_{ij..m}$ and $b_{{\bar i}j..m} =
b^{*}_{ij..m}$.
The first operator ${\hat O}(i,j,k)$ arises from lifting the D condition through the insertion of a [*particle*]{} projection operator $$\sum_{k,k'}{ d_{k} d^{*}_{k'} {\hat a}^{\dagger}_{k} {\hat a}_{k'} }$$ while the second operator ${\hat O}(i,j,{\bar k})$ arises from lifting the D condition through the insertion of a [*hole*]{} projection operator $$\sum_{k,k'}{ d^{*}_{k} d_{k'} {\hat a}_{k} {\hat a}^{\dagger}_{k'} }
.$$ The nonnegativity of ${\hat O}(i,j,k)$ and ${\hat O}(i,j,{\bar k})$ generates a pair of [*lifting conditions*]{} discussed in Refs. [@M02; @HM05]. While these two conditions depend not just on the 2-RDM but on parts of the 3-RDM, the sum of these two three-body operators produces a two-body operator $$\label{eq:L1} ^{2} {\hat O}_{L1} = {\hat O}(i,j,k) + {\hat
O}(i,j,{\bar k}) .$$ Because the two-body operator $^{2} {\hat O}_{L1}$ simplifies to the two-body operator $^{2} {\hat O}_{D}$ in Eq. (\[eq:OD\]), its nonnegativity regenerates the D condition. With a generalization of this lifting process, however, we can generate (2,3)-positivity conditions that are distinct from the known conditions.
We can generalize the lifting process by inserting the creation operator and the annihilation operator responsible for lifting at non-adjacent positions. For example, consider the pair of three-body operators $$\begin{aligned}
{\hat O}(k,i,j) & = & {\hat C}(k,i,j) {\hat C}(k,i,j)^{\dagger} \\
{\hat O}({\bar k},i,j) & = & {\hat C}({\bar k},i,j) {\hat C}({\bar
k},i,j)^{\dagger}\end{aligned}$$ where $$\begin{aligned}
{\hat C}(k,i,j) & = & \sum_{ijk}{ b_{ij} d_{k} {\hat
a}^{\dagger}_{k}
{\hat a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} } \label{eq:23P2} \\
{\hat C}({\bar k},i,j) & = & \sum_{ijk}{ b_{ij} d^{*}_{k} {\hat
a}_{k} {\hat a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} } .\end{aligned}$$ In ${\hat O}(k,i,j)$ the creation operator ${\hat a}^{\dagger}_{k}$ in ${\hat C}$ and the annihilation operator ${\hat a}_{k'}$ in the adjoint of ${\hat C}$, which perform the lifting of the D condition, are separated from each other by four second-quantized operators; similarly, in ${\hat O}({\bar k},i,j)$ the creation and annihilation operators, ${\hat a}_{k}$ and ${\hat a}^{\dagger}_{k'}$ respectively, are separated from each other by four second-quantized operators. Because the components of the projectors are separated, the nonnegativity of ${\hat O}(k,i,j)$ and ${\hat O}({\bar k},i,j)$ generates a pair of generalized lifting conditions that extend those discussed in Refs. [@M02; @HM05].
While individually ${\hat O}(k,i,j)$ and ${\hat O}({\bar k},i,j)$ depend on three-particle operators, their sum generates a two-body operator $$\label{eq:L2} ^{2} {\hat O}_{L2} = {\hat O}(k,i,j) + {\hat O}({\bar
k},i,j) .$$ Unlike $^{2} {\hat O}_{L1}$, the nonnegativity of the lifted operator $^{2} {\hat O}_{L2}$ is not necessarily implied by the D, Q, G, T1, and T2 conditions. Importantly, $^{2} {\hat O}_{L2}$ does not simply rearrange to $^{2} {\hat O}_{L1}$ because the creation and annihilation operators are non-commutative. Based on the possible orderings of the fundamental second-quantized operators, there are nine distinct ways to lift the D condition while canceling the resulting three-particle operators and hence, nine distinct lifting conditions from the D condition. Similarly, there are nine distinct (2,3)-positivity conditions from lifting the Q condition and nine from lifting the G condition. Three of these 27 lifting conditions reduce to the D, Q, and G conditions, respectively while the other conditions are distinct because the second-quantized operators in quantum mechanics form a non-commutative algebra.
Table I summarizes the (2,3)-positivity conditions by giving a representative condition from each of the two classes: (i) the lifting conditions and (ii) the pure conditions. While the lifting conditions arise from lifting the 2-positivity conditions, the pure conditions cannot be obtained from lifting any of the lower conditions. Table I gives nonnegativity of $^{2} {\hat O}_{L2}$ and the T1 condition as representative conditions of the lifting and pure (2,3)-positivity conditions, respectively. All of the other (2,3)-conditions can be obtained from these representative conditions through two processes, (i) [*switching*]{} of the second-quantized operators in the ${\hat C}(i,j,k)$ between creators and annihilators and (ii) [*reordering*]{} of the second-quantized operators in the ${\hat C}(i,j,k)$.
---------------------------------------------------------------------------------------------------------------------
Class Type Representative Condition ${\hat C}$ Definition
------- -------------- ---------------------------------------------------------------------- -----------------------
1 Lifted (2,2) ${\rm Tr}( ({\hat O}(l,k,i,j) + {\hat O}(l,{\bar Eq. (\[eq:24L2\])
k},i,j) + {\hat O}({\bar l},k,i,j) + {\hat O}({\bar l},{\bar k},i,j)
) \, {}^{2} D)
\ge 0$
2 Lifted (2,3) ${\rm Tr}( ({\hat O}(l,i,j,k) + {\hat O}(l,{\bar Eq. (\[eq:24L3\])
i},{\bar j},{\bar k}) + {\hat O}({\bar l},{\bar i},{\bar j},{\bar
k}) + {\hat O}({\bar l},i,j,k) ) \, {}^{2} D)
\ge 0$
3 Pure (2,4) ${\rm Tr}( (3 {\hat O}(i,j,k,l) + {\hat Eq. (\[eq:24P\])
O}(i,j,k,{\bar l}) + {\hat O}(i,j,{\bar k},l) + {\hat O}(i,{\bar
j},k,l) + {\hat O}({\bar i},j,k,l) + {\hat O}({\bar i},{\bar
j},{\bar k},{\bar l})) \, {}^{2} D) \ge 0$
---------------------------------------------------------------------------------------------------------------------
Switching the second-quantized operators with index $j$ in the L2 condition of Eq. (\[eq:L2\]), for example, generates a lifted G condition $$^{2} {\hat O}_{L3} = {\hat O}(k,i,{\bar j}) + {\hat O}({\bar
k},i,{\bar j}) .$$ Note that switching the second-quantized operators with index $k$ in Eq. (\[eq:L2\]) simply regenerates the same condition while switching the second-quantized operators associated with indices $i$ and $j$ generates lifted G and Q conditions from the lifted D condition. Reordering of the second-quantized operators in the L2 condition of Eq. (\[eq:L2\]), by contrast, produces the other 9 lifted D conditions; for example, reordering L2 yields the L1 condition in Eq. (\[eq:L1\]). Similarly, for the pure (2,3)-positivity conditions switching of the second-quantized operators with index $k$ in T1 produces the T2 condition in Eq. (\[eq:T2\]). The other two distinct T2 conditions, ${\bar
T2}$ and ${\tilde T2}$, in Eqs. (\[eq:T2b\]) and (\[eq:T2t\]) are generated not by switching but by reordering the second-quantized operators in the T2 condition of Eq. (\[eq:T2\]).
### (2,4)-positivity conditions
The (2,4)-positivity conditions, arising from considering all ${\hat
C}_{i}$ operators of degree less than or equal to four in Eq. (\[eq:O2\]), consist of two classes of lifting conditions and one class of pure conditions, which are summarized in Table II. The two classes of lifting conditions are generated from lifting the two classes of (2,3)-positivity conditions. As in the previous section, the generalized lifting is performed by (i) inserting a creation operator into each ${\hat C}_{i}$ operator contributing to the condition, (ii) converting the inserted creation operator into an annihilation operator in the operator produced from step (i), and (iii) adding the two lifted operators from steps (i) and (ii) together to produce a two-particle operator. The nonnegativity of the resulting two-particle operator generates a lifting (2,4)-positivity condition. Representative lifting conditions for both classes are shown in Table II. The ${\hat C}$ operators in the first and second classes of lifted (2,3)-positivity conditions and the pure (2,4)-positivity condition are given by $$\begin{aligned}
{\hat C}(l,k,i,j) & = & \sum_{ijkl}{ b_{ij} d_{k} e_{l}
{\hat a}^{\dagger}_{l} {\hat a}^{\dagger}_{k} {\hat a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} } \label{eq:24L2} \\
{\hat C}(l,i,j,k) & = & \sum_{ijkl}{ b_{ijk} d_{l} {\hat
a}^{\dagger}_{l}
{\hat a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} {\hat a}^{\dagger}_{k} } \label{eq:24L3} \\
{\hat C}(i,j,k,l) & = & \sum_{ijkl}{ b_{i} d_{j} e_{k} f_{l} {\hat
a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} {\hat a}^{\dagger}_{k} {\hat
a}^{\dagger}_{l} } \label{eq:24P}\end{aligned}$$ where $b_{i}$, $d_{i}$, $e_{i}$, $f_{i}$, $b_{ij}$, $b_{ijk}$, and ${\hat a}^{\dagger}_{i}$ become $b_{i}^{*}$, $d_{i}^{*}$, $e_{i}^{*}$, $f_{i}^{*}$, $b_{ij}^{*}$, $b_{ijk}^{*}$, and ${\hat
a}_{i}$ when $i={\bar i}$. The rank of the largest tensor changes from three in Eq. (\[eq:24L2\]) to one in Eq. (\[eq:24P\]) to effect the cancelation of the 3- and 4-RDOs in the combinations of operators in Table II. Fusing the tensors $b_{i}$ and $d_{j}$ into a single rank-two tensor $b_{ij}$ in Eq. (\[eq:24P\]), for example, would cause the operator combinations in Table II to depend on the 3- and 4-RDOs. Additional (2,4)-positivity conditions can be generated from the representative conditions through a combination of switching and reordering of the creation and annihilation operators.
The pure (2,4)-positivity conditions, presented in Ref. [@M12b], depend upon only the 2-RDM through conic combinations that cancel the 3- and 4-RDMs. As in the (2,3)-conditions, the cancelations depend upon the conic combination of pairs of operators that differ from each other by an odd number of switchings—exchanges of creators and annihilators. Generating an extreme condition on the 2-RDM requires that we consider the minimum number of conic combinations that effect the cancelation of the higher RDMs. Each pure (2,4)-positivity condition involves the conic combination of eight four-particle operators by Eq. (\[eq:O2\]). These eight four-particle operators can be grouped into the four pairs that depend upon only three-particle operators: $$\begin{aligned}
{\hat O}({\bar i},j,k,l) & + & {\hat O}({\bar i},{\bar j},{\bar k},
{\bar l}) \\
{\hat O}(i,{\bar j},k,l) & + & {\hat O}(i,j,k,l) \\
{\hat O}(i,j,{\bar k},l) & + & {\hat O}(i,j,k,l) \\
{\hat O}(i,j,k,{\bar l}) & + & {\hat O}(i,j,k,l)\end{aligned}$$ The operators in the first pair differ from each other by the switching of three creation and annihilation operators while the operators in the other three pairs differ from each other by the switching of one creation operator and one annihilation operator. Rearranging the second-quantized operators in the four pairings into normal order with creators to the left of the annihilators generates expressions involving the sum of 9, 5, 3, and 1 3-RDOs, respectively. Upon summation the 9 3-RDOs from the one pairing with three switchings cancels with the 5, 3, and 1 3-RDOs from the three pairings with one switching, and hence the final operator depends upon only the 2-RDO.
------------------------------------------------------------------------------------------
Condition ${\hat C}$ Definition
------------------------------------------------------------------ -----------------------
$g_{1}({}^{2} D) = {\rm Tr}( (3 {\hat O}(i,j,k,l) + {\hat Eq. (\[eq:24P\])
O}(i,j,k,{\bar l}) + {\hat O}(i,j,{\bar k},l) + {\hat O}(i,{\bar
j},k,l) + {\hat O}({\bar i},j,k,l) + {\hat O}({\bar i},{\bar
j},{\bar k},{\bar l})) \, {}^{2} D)
\ge 0$
$g_{2}({}^{2} D) = {\rm Tr}( (3 {\hat O}(i,j,k,l) + {\hat Eq. (\[eq:24P\])
O}(i,j,k,{\bar l}) + {\hat O}(i,{\bar k},j,l) + {\hat O}(k,{\bar
j},i,l) + {\hat O}(j,{\bar i},k,l) + {\hat O}({\bar i},{\bar
j},{\bar k},{\bar l})) \, {}^{2} D) \ge 0$
$g_{3}({}^{2} D) = {\rm Tr}( (3 {\hat O}(i,j,k,l) + {\hat Eq. (\[eq:24P\])
O}(i,j,{\bar l},k) + {\hat O}(i,l,{\bar k},j) + {\hat O}(k,{\bar
j},i,l) + {\hat O}(j,{\bar i},k,l) + {\hat O}({\bar i},{\bar
j},{\bar k},{\bar l})) \, {}^{2} D) \ge 0$
$g_{4}({}^{2} D) = {\rm Tr}( (3 {\hat O}(i,j,k,l) + {\hat Eq. (\[eq:24P\])
O}(i,j,k,{\bar l}) + {\hat O}(i,{\bar k},j,l) + {\hat O}(k,{\bar
j},i,l) + {\hat O}(j,{\bar i},k,l) + {\hat O}({\bar i},{\bar
j},{\bar k},{\bar l})) \, {}^{2} D) \ge 0$
------------------------------------------------------------------------------------------
Other pure (2,4)-positivity conditions can be generated from the representative condition through switching and reordering of the second-quantized operators. To maintain the cancelation of the 3- and 4-RDOs, we must perform the same switching of creation and annihilation operators in each operator ${\hat C}(i,j,k,l)$ contributing to the condition. Because each fundamental second-quantized operator can be either a creation or an annihilation operator, there are $2^{4}$ or 16 conditions from switching. Eight of these conditions can be generated from the other 8 conditions by switching all creation and annihilation operators by particle-hole symmetry. In the limit that the expansion coefficients $b_{i}$, $d_{j}$, $e_{k}$, and $f_{l}$ become orthogonal unit vectors, these 16 conditions reduce to the 16 conditions in (2,4)-class of the classical (or diagonal) $N$-representability problem [@MD72; @E78; @Cuts]. The quantum mechanical formulation of these conditions, however, is much more general because the expansion coefficients need not be orthogonal. When the expansion coefficients are non-orthogonal, the creation and annihilation operators become non-commutative operators, and hence, the conditions depend upon their ordering.
In the quantum case additional conditions can be generated from each of the 16 conditions by reordering the creation and annihilation operators while preserving the cancelation of the 3- and 4-RDOs. These additional conditions are related to the original 16 conditions as the generalized T2 conditions are related to the T2 condition in the (2,3)-positivity conditions. Table III presents the representative pure (2,4)-positivity condition as well as three other conditions generated from its reordering. Each of these four conditions differs from the others by a few terms involving the 2-RDM. For example, the first and second conditions differ by only one term $$g_{2} = g_{1} + 4 \Re \left ( \alpha \beta \sum_{ik;i'l'}{ b_{i}
e_{k} \, {}^{2} D^{ik}_{i'l'} \, b^{*}_{i'} f^{*}_{l'} } \right )
\ge 0.$$ where $$\begin{aligned}
\alpha & = & \sum_{j}{d_{j} e^{*}_{j}} \\
\beta & = & \sum_{j}{f_{j} d^{*}_{j}}\end{aligned}$$ and $\Re$ selects the real part of the expression. When this term is negative, inequality $g_{2}$ is stronger than $g_{1}$, but when this term is positive, inequality $g_{1}$ is stronger than $g_{2}$. In the classical case, where the expansion coefficients are orthogonal, these two conditions are equivalent because both $\alpha$ and $\beta$ are zero, and hence, this additional term vanishes.
### (2,5)-positivity conditions
-----------------------------------------------------------------------------------------------------------------------------------------------------
Class Type Representative Condition ${\hat C}$ Definition
------- -------------- ------------------------------------------------------------------------------------------------------ -----------------------
1 Lifted (2,2) ${\rm Tr}( ({\hat O}(m,l,k,i,j) + {\hat Eq. (\[eq:25L2\])
O}(m,{\bar l},k,i,j) + {\hat O}({\bar m},l,k,i,j) + {\hat
O}({\bar m},{\bar l},k,i,j)$
$ ~~~~~~~~~~~~~~ + {\hat O}(m,l,{\bar k},i,j) +
{\hat O}(m,{\bar l},{\bar k},i,j) + {\hat O}({\bar m},l,{\bar
k},i,j) + {\hat O}({\bar m},{\bar l},{\bar k},i,j) ) \, {}^{2} D)
\ge 0$
2 Lifted (2,3) ${\rm Tr}( ({\hat O}(m,l,i,j,k) + {\hat Eq. (\[eq:25L3\])
O}(m,l,{\bar i},{\bar j},{\bar k}) + {\hat O}({\bar m},l,{\bar
i},{\bar j},{\bar k}) + {\hat O}({\bar m},l,i,j,k)$
$~~~~~~~~~~~~~~ + {\hat O}(m,{\bar l},i,j,k) +
{\hat O}(m,{\bar l},{\bar i},{\bar j},{\bar k}) + {\hat O}({\bar
m},{\bar l},{\bar i},{\bar j},{\bar k}) + {\hat O}({\bar m},{\bar
l},i,j,k) ) \, {}^{2} D)
\ge 0$
3 Lifted (2,4) ${\rm Tr}( (3 {\hat O}(m,i,j,k,l) + {\hat Eq. (\[eq:25L4\])
O}(m,i,j,k,{\bar l}) + {\hat O}(m,i,j,{\bar k},l)$
$~~~ + {\hat O}(m,i,{\bar j},k,l) + {\hat O}(m,{\bar i},j,k,l)
+ {\hat O}(m,{\bar i},{\bar j},{\bar k},{\bar l})$
$~+ 3 {\hat O}({\bar m},i,j,k,l) + {\hat
O}({\bar m},i,j,k,{\bar l}) + {\hat O}({\bar m},i,j,{\bar k},l)$
$~~~~~~~~~~~~~~~+ {\hat O}({\bar m},i,{\bar j},k,l) +
{\hat O}({\bar m},{\bar i},j,k,l) + {\hat O}({\bar m},{\bar i},{\bar j},{\bar k},{\bar l}))
\, {}^{2} D) \ge 0$
4 Pure (2,5) ${\rm Tr}( (3 {\hat O}(i,j,k,l,m) + {\hat Eq. (\[eq:25P\])
O}(i,j,k,l,{\bar m})
+ {\hat O}(i,j,k,{\bar l},m)$
$~~~ + {\hat O}(i,j,{\bar k},l,m) + {\hat O}(i,{\bar j},k,l,m)
+ {\hat O}({\bar i},j,k,l,m)$
$~+ 3 {\hat O}({\bar i},{\bar j},{\bar k},{\bar l},{\bar m}) +
{\hat O}({\bar i},{\bar j},{\bar k},{\bar l},m) + {\hat O}({\bar i},{\bar j},{\bar k},
l,{\bar m})$
$~~~~~~~~~~~~~~~ + {\hat O}({\bar i},{\bar j},k,{\bar l},{\bar m}) +
{\hat O}({\bar i},j,{\bar k},{\bar l},{\bar m}) + {\hat O}(i,{\bar j},{\bar k},{\bar l},{\bar m}))
\, {}^{2} D) \ge 0$
5 Pure (2,5) ${\rm Tr}( (6 {\hat O}(i,j,k,l,m) + 3 {\hat Eq. (\[eq:25P\])
O}(i,j,k,l,{\bar m}) + 3 {\hat O}(i,j,k,{\bar l},m)$
$~~ + 3 {\hat O}(i,j,{\bar k},l,m) + 3 {\hat O}(i,{\bar j},k,l,m)
+ 3 {\hat O}({\bar i},j,k,l,m)$
$~+ {\hat O}(i,j,k,{\bar l},{\bar m}) +
{\hat O}(i,j,{\bar k},l,{\bar m}) + {\hat O}(i,j,{\bar k},
{\bar l},m)$
$~+ {\hat O}(i,{\bar j},k,l,{\bar m}) +
{\hat O}(i,{\bar j},k,{\bar l},m) + {\hat O}(i,{\bar j},{\bar k},
l,m)$
$~+ {\hat O}({\bar i},j,k,l,{\bar m}) +
{\hat O}({\bar i},j,k,{\bar l},m) + {\hat O}({\bar i},j,{\bar k},
l,m)$
$ + {\hat O}({\bar i},{\bar j},k,l,m) +
{\hat O}({\bar i},{\bar j},{\bar k},{\bar l},{\bar m})) \, {}^{2} D)
\ge 0$
6 Pure (2,5) ${\rm Tr}( (6 {\hat O}(i,j,k,l,m) + 3 {\hat Eq. (\[eq:25P\])
O}(i,j,k,l,{\bar m}) + 3 {\hat O}(i,j,k,{\bar l},m)$
$~~ + 3 {\hat O}(i,j,{\bar k},l,m) + 3 {\hat O}(i,{\bar j},k,l,m)
+ {\hat O}({\bar i},j,k,l,m)$
$~+ {\hat O}(i,j,k,{\bar l},{\bar m}) +
{\hat O}(i,j,{\bar k},l,{\bar m}) + {\hat O}(i,j,{\bar k},
{\bar l},m)$
$~+ {\hat O}(i,{\bar j},k,l,{\bar m}) +
{\hat O}(i,{\bar j},k,{\bar l},m) + {\hat O}(i,{\bar j},{\bar k},
l,m)$
$~+ {\hat O}(i,j,k,l,{\bar m}) +
{\hat O}(i,j,k,{\bar l},m) + {\hat O}(i,j,{\bar k},
l,m)$
$ + {\hat O}(i,{\bar j},k,l,m) +
3 {\hat O}({\bar i},{\bar j},{\bar k},{\bar l},{\bar m})) \, {}^{2}
D) \ge 0$
-----------------------------------------------------------------------------------------------------------------------------------------------------
The (2,5)-positivity conditions are generated from considering all ${\hat C}_{i}$ operators of degree less than or equal to five in Eq. (\[eq:O2\]). These conditions consist of three classes of lifting conditions and three classes of pure conditions, which are given in Table IV. The lifting conditions arise from lifting the three different classes of (2,4)-positivity conditions. The ${\hat
C}$ operators of the first, second, and third classes of lifting conditions are given by $$\begin{aligned}
{\hat C}(m,l,k,i,j) & = & \sum_{ijklm}{ b_{ij} d_{k} e_{l} f_{m}
{\hat a}^{\dagger}_{m} {\hat a}^{\dagger}_{l} {\hat a}^{\dagger}_{k}
{\hat a}^{\dagger}_{i}
{\hat a}^{\dagger}_{j} } \label{eq:25L2} \\
{\hat C}(m,l,i,j,k) & = & \sum_{ijklm}{ b_{ijk} d_{l} e_{m} {\hat
a}^{\dagger}_{m} {\hat a}^{\dagger}_{l} {\hat a}^{\dagger}_{i} {\hat
a}^{\dagger}_{j} {\hat a}^{\dagger}_{k} } \label{eq:25L3} \\
{\hat C}(m,i,j,k,l) & = & \sum_{ijklm}{ b_{i} d_{j} e_{k} f_{l}
g_{m} {\hat a}^{\dagger}_{m} {\hat a}^{\dagger}_{i} {\hat
a}^{\dagger}_{j} {\hat a}^{\dagger}_{k} {\hat a}^{\dagger}_{l} },
\label{eq:25L4}\end{aligned}$$ respectively, and the ${\hat C}$ operators of the three classes of pure conditions are given by $${\hat C}(i,j,k,l,m) = \sum_{ijklm}{ b_{i} d_{j} e_{k} f_{l} g_{m}
{\hat a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} {\hat a}^{\dagger}_{k}
{\hat a}^{\dagger}_{l} {\hat a}^{\dagger}_{m} } \label{eq:25P}$$ where the $b_{i}$, $d_{i}$, $e_{i}$, $f_{i}$, $g_{i}$, $b_{ij}$, $b_{ijk}$, and ${\hat a}^{\dagger}_{i}$ become $b_{i}^{*}$, $d_{i}^{*}$, $e_{i}^{*}$, $f_{i}^{*}$, $g_{i}^{*}$, $b_{ij}^{*}$, $b_{ijk}^{*}$, and ${\hat a}_{i}$ when $i={\bar i}$. Note that the operators in Eqs. (\[eq:25L4\]) and (\[eq:25P\]) are not equivalent after switching. Switching of creators to annihilators in the ${\hat C}$ operators in the representative conditions in Table IV produces 16, 32, and 32 conditions in the pure classes 4, 5, and 6, respectively. Class 4 has fewer conditions because its conditions, unlike those in classes 5 and 6, possess particle-hole symmetry. Particle-hole symmetry is present in all of the pure (2,3)-positivity conditions and none of the pure (2,4)-conditions. Additional conditions can be generated from the representative conditions through reordering of the creation and annihilation operators. Like the (2,3)- and (2,4)-positivity conditions, the (2,5)-conditions generate all of the classical (diagonal) $N$-representability conditions when the expansion coefficients $b_{i}$, $d_{j}$, $e_{k}$, $f_{l}$, and $g_{m}$ are chosen to be orthogonal unit vectors.
### (2,6)-positivity conditions
[ccccccccccccccccccc]{}
&\
Operators & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18\
$\alpha {\hat O}(ijklmn)+ \beta {\hat O}({\bar i}{\bar j}{\bar
k}{\bar l}{\bar m}{\bar n})$ & 20/2 & 12/6 & 20/6 & 12/12 & 20/12 & 20/12 & 9/6 & 5/3 & 9/5 & 9/3 & 5/2 & 14/9 & 14/6 & 14/3 & 6/3 & 6/5 & 1/6 & 3/3\
$\alpha {\hat O}(ijklm{\bar n})+ \beta {\hat O}({\bar i}{\bar
j}{\bar k}{\bar l}{\bar m}n)$ & 12/0 & 6/2 & 12/2 & 6/6 & 12/6 & 12/6 & 5/3 & 2/1 & 3/1 & 5/1 & 3/1 & 9/5 & 9/3 & 9/1 & 5/2 & 5/3 & 0/3 & 2/1\
$\alpha {\hat O}(ijkl{\bar m}n)+ \beta {\hat O}({\bar i}{\bar
j}{\bar k}{\bar l}m{\bar n})$ & 12/0 & 6/2 & 12/2 & 6/6 & 12/6 & 12/6 & 6/3 & 3/1 & 6/3 & 6/1 & 3/1 & 9/5 & 9/3 & 9/1 & 3/1 & 3/3 & 0/3 & 1/1\
$\alpha {\hat O}(ijkl{\bar m}{\bar n})+ \beta {\hat O}({\bar i}{\bar
j}{\bar k}{\bar l}mn)$ & 6/0 & 2/0 & 6/0 & 2/2 & 6/2 & 6/2 & 3/1 & 1/0 & 1/0 & 3/0 & 1/0 & 5/2 & 5/1 & 5/0 & 3/1 & 2/1 & 0/1 & 1/0\
$\alpha {\hat O}(ijk{\bar l}mn)+ \beta {\hat O}({\bar i}{\bar
j}{\bar k}l{\bar m}{\bar n})$ & 12/0 & 6/2 & 12/2 & 6/6 & 12/6 & 12/6 & 5/3 & 2/1 & 5/2 & 5/1 & 3/1 & 9/5 & 9/3 & 9/1 & 3/1 & 3/2 & 0/3 & 3/2\
$\alpha {\hat O}(ijk{\bar l}m{\bar n})+ \beta {\hat O}({\bar i}{\bar
j}{\bar k}l{\bar m}n)$ & 6/0 & 2/0 & 6/0 & 2/2 & 6/2 & 6/2 & 2/1 & 0/0 & 1/0 & 2/0 & 1/0 & 5/2 & 5/1 & 5/0 & 2/0 & 3/1 & 0/1 & 3/1\
$\alpha {\hat O}(ijk{\bar l}{\bar m}n)+ \beta {\hat O}({\bar i}{\bar
j}{\bar k}lm{\bar n})$ & 6/0 & 2/0 & 6/0 & 2/2 & 6/2 & 6/2 & 3/1 & 1/0 & 3/1 & 3/0 & 2/1 & 5/2 & 5/1 & 5/0 & 1/0 & 1/1 & 0/1 & 2/1\
$\alpha {\hat O}(ijk{\bar l}{\bar m}{\bar n})+ \beta {\hat O}({\bar
i}{\bar j}{\bar k}lmn)$ & 2/2 & 0/0 & 2/0 & 0/0 & 2/0 & 2/0 & 1/0 & 0/0 & 0/0 & 1/0 & 0/0 & 2/0 & 2/0 & 2/0 & 1/0 & 1/0 & 1/0 & 3/1\
$\alpha {\hat O}(ij{\bar k}lmn)+ \beta {\hat O}({\bar i}{\bar
j}k{\bar l}{\bar m}{\bar n})$ & 12/0 & 6/2 & 12/2 & 6/6 & 12/6 & 12/6 & 3/2 & 3/2 & 6/3 & 6/2 & 3/1 & 6/3 & 9/3 & 6/0 & 3/1 & 3/2 & 1/5 & 2/3\
$\alpha {\hat O}(ij{\bar k}lm{\bar n})+ \beta {\hat O}({\bar i}{\bar
j}k{\bar l}{\bar m}n)$ & 6/0 & 2/0 & 6/0 & 2/2 & 6/2 & 6/2 & 1/1 & 1/1 & 2/1 & 3/1 & 2/1 & 3/1 & 5/1 & 3/0 & 3/1 & 3/1 & 0/2 & 1/1\
$\alpha {\hat O}(ij{\bar k}l{\bar m}n)+ \beta {\hat O}({\bar i}{\bar
j}k{\bar l}m{\bar n})$ & 6/0 & 2/0 & 6/0 & 2/2 & 6/2 & 6/2 & 1/0 & 1/0 & 3/1 & 3/0 & 1/0 & 3/1 & 5/1 & 3/0 & 1/0 & 1/1 & 0/2 & 0/1\
$\alpha {\hat O}(ij{\bar k}l{\bar m}{\bar n})+ \beta {\hat O}({\bar
i}{\bar j}k{\bar l}mn)$ & 2/2 & 0/0 & 2/0 & 0/0 & 2/0 & 2/0 & 0/0 & 0/0 & 0/0 & 1/0 & 0/0 & 1/0 & 2/0 & 1/1 & 2/1 & 1/0 & 0/0 & 0/0\
$\alpha {\hat O}(ij{\bar k}{\bar l}mn)+ \beta {\hat O}({\bar i}{\bar
j}kl{\bar m}{\bar n})$ & 6/0 & 2/0 & 6/0 & 2/2 & 6/2 & 6/2 & 1/1 & 1/1 & 3/1 & 3/1 & 2/1 & 3/1 & 5/1 & 3/0 & 1/0 & 1/0 & 1/3 & 1/1\
$\alpha {\hat O}(ij{\bar k}{\bar l}m{\bar n})+ \beta {\hat O}({\bar
i}{\bar j}kl{\bar m}n)$ & 2/2 & 0/0 & 2/0 & 0/0 & 2/0 & 2/0 & 0/1 & 0/1 & 1/1 & 1/1 & 1/1 & 1/0 & 2/0 & 1/1 & 1/0 & 2/0 & 1/1 & 1/0\
$\alpha {\hat O}(ij{\bar k}{\bar l}{\bar m}n)+ \beta {\hat O}({\bar
i}{\bar j}klm{\bar n})$ & 2/2 & 0/0 & 2/0 & 0/0 & 2/0 & 2/0 & 0/0 & 0/0 & 1/0 & 1/0 & 1/1 & 1/0 & 2/0 & 1/1 & 0/0 & 0/0 & 1/1 & 0/0\
$\alpha {\hat O}(i{\bar j}klmn)+ \beta {\hat O}({\bar i}j{\bar
k}{\bar l}{\bar m}{\bar n})$ & 12/0 & 6/2 & 12/2 & 6/6 & 6/2 & 12/6 & 3/1 & 3/1 & 5/2 & 5/1 & 2/0 & 3/1 & 3/0 & 9/1 & 1/0 & 1/1 & 1/5 & 1/1\
$\alpha {\hat O}(i{\bar j}klm{\bar n})+ \beta {\hat O}({\bar
i}j{\bar k}{\bar l}{\bar m}n)$ & 6/0 & 2/0 & 6/0 & 2/2 & 2/0 & 6/2 & 1/0 & 1/0 & 1/0 & 2/0 & 1/0 & 1/0 & 1/0 & 5/0 & 1/0 & 1/0 & 1/3 & 1/0\
$\alpha {\hat O}(i{\bar j}kl{\bar m}n)+ \beta {\hat O}({\bar
i}j{\bar k}{\bar l}m{\bar n})$ & 6/0 & 2/0 & 6/0 & 2/2 & 2/0 & 6/2 & 2/0 & 2/0 & 3/1 & 3/0 & 1/0 & 1/0 & 1/0 & 5/0 & 0/0 & 0/1 & 1/3 & 0/0\
$\alpha {\hat O}(i{\bar j}kl{\bar m}{\bar n})+ \beta {\hat O}({\bar
i}j{\bar k}{\bar l}mn)$ & 2/2 & 0/0 & 2/0 & 0/0 & 0/0 & 2/0 & 1/0 & 1/0 & 0/0 & 1/0 & 0/0 & 0/0 & 0/1 & 2/0 & 1/1 & 0/0 & 2/2 & 1/0\
$\alpha {\hat O}(i{\bar j}k{\bar l}mn)+ \beta {\hat O}({\bar
i}j{\bar k}l{\bar m}{\bar n})$ & 6/0 & 2/0 & 6/0 & 2/2 & 2/0 & 6/2 & 1/0 & 1/0 & 2/0 & 2/0 & 1/0 & 1/0 & 1/0 & 5/0 & 0/0 & 0/0 & 0/2 & 1/0\
$\alpha {\hat O}(i{\bar j}k{\bar l}m{\bar n})+ \beta {\hat O}({\bar
i}j{\bar k}l{\bar m}n)$ & 2/2 & 0/0 & 2/0 & 0/0 & 0/0 & 2/0 & 0/0 & 0/0 & 0/0 & 0/0 & 0/0 & 0/0 & 0/1 & 2/0 & 0/0 & 1/0 & 1/1 & 2/0\
$\alpha {\hat O}(i{\bar j}k{\bar l}{\bar m}n)+ \beta {\hat O}({\bar
i}j{\bar k}lm{\bar n})$ & 2/2 & 0/0 & 2/0 & 0/0 & 0/0 & 2/0 & 1/0 & 1/0 & 1/0 & 1/0 & 1/1 & 0/0 & 0/1 & 2/0 & 0/1 & 0/1 & 1/1 & 1/0\
$\alpha {\hat O}(i{\bar j}{\bar k}lmn)+ \beta {\hat O}({\bar
i}jk{\bar l}{\bar m}{\bar n})$ & 6/0 & 2/0 & 6/0 & 2/2 & 2/0 & 6/2 & 0/0 & 2/1 & 3/1 & 3/1 & 1/0 & 0/0 & 1/0 & 3/0 & 0/0 & 0/0 & 0/3 & 1/2\
$\alpha {\hat O}(i{\bar j}{\bar k}lm{\bar n})+ \beta {\hat O}({\bar
i}jk{\bar l}{\bar m}n)$ & 2/2 & 0/0 & 2/0 & 0/0 & 0/0 & 2/0 & 0/1 & 1/1 & 1/1 & 1/1 & 1/1 & 0/1 & 0/1 & 1/1 & 1/1 & 1/0 & 0/1 & 1/1\
$\alpha {\hat O}(i{\bar j}{\bar k}l{\bar m}n)+ \beta {\hat O}({\bar
i}jk{\bar l}m{\bar n})$ & 2/2 & 0/0 & 2/0 & 0/0 & 0/0 & 2/0 & 0/0 & 1/0 & 1/0 & 1/0 & 0/0 & 0/1 & 0/1 & 1/1 & 0/1 & 0/1 & 0/1 & 0/1\
$\alpha {\hat O}(i{\bar j}{\bar k}{\bar l}mn)+ \beta {\hat O}({\bar
i}jkl{\bar m}{\bar n})$ & 2/2 & 0/0 & 2/0 & 0/0 & 0/0 & 2/0 & 0/1 & 1/1 & 1/0 & 1/1 & 1/1 & 0/1 & 0/1 & 1/1 & 0/1 & 0/0 & 0/1 & 0/0\
$\alpha {\hat O}({\bar i}jklmn)+ \beta {\hat O}(i{\bar j}{\bar
k}{\bar l}{\bar m}{\bar n})$ & 12/0 & 6/2 & 6/0 & 2/2 & 6/2 & 2/0 & 5/3 & 3/2 & 3/1 & 3/0 & 3/1 & 6/3 & 6/1 & 6/0 & 5/3 & 3/3 & 0/3 & 1/2\
$\alpha {\hat O}({\bar i}jklm{\bar n})+ \beta {\hat O}(i{\bar
j}{\bar k}{\bar l}{\bar m}n)$ & 6/0 & 2/0 & 2/0 & 0/0 & 2/0 & 0/0 & 2/1 & 1/1 & 0/0 & 1/0 & 2/1 & 3/1 & 3/0 & 3/0 & 3/1 & 2/1 & 0/1 & 0/0\
$\alpha {\hat O}({\bar i}jkl{\bar m}n)+ \beta {\hat O}(i{\bar
j}{\bar k}{\bar l}m{\bar n})$ & 6/0 & 2/0 & 2/0 & 0/0 & 2/0 & 0/0 & 3/1 & 2/1 & 2/1 & 2/0 & 2/1 & 3/1 & 3/0 & 3/0 & 2/1 & 1/2 & 0/1 & 0/1\
$\alpha {\hat O}({\bar i}jk{\bar l}mn)+ \beta {\hat O}(i{\bar
j}{\bar k}l{\bar m}{\bar n})$ & 6/0 & 2/0 & 2/0 & 0/0 & 2/0 & 0/0 & 2/1 & 1/1 & 1/0 & 1/0 & 1/0 & 3/1 & 3/0 & 3/0 & 3/2 & 1/1 & 0/1 & 1/1\
$\alpha {\hat O}({\bar i}j{\bar k}lmn)+ \beta {\hat O}(i{\bar
j}k{\bar l}{\bar m}{\bar n})$ & 6/0 & 2/0 & 2/0 & 0/0 & 2/0 & 0/0 & 1/1 & 1/1 & 1/0 & 1/0 & 1/0 & 1/0 & 3/0 & 1/0 & 2/1 & 1/1 & 1/3 & 1/3\
$\alpha {\hat O}({\bar i}{\bar j}klmn)+ \beta {\hat O}(ij{\bar
k}{\bar l}{\bar m}{\bar n})$ & 6/0 & 2/0 & 2/0 & 0/0 & 0/0 & 0/0 & 1/0 & 1/0 & 1/0 & 1/0 & 1/0 & 0/0 & 0/0 & 3/0 & 1/1 & 0/1 & 0/2 & 0/1
As with the (2,$q$)-positivity conditions for $q \le 5$, the (2,6)-positivity conditions are generated from Eq. (\[eq:O2\]) by considering all ${\hat C}_{i}$ operators of degree less than or equal to six. Six classes of lifting (2,6)-positivity conditions arise from lifting the six classes of (2,5)-positivity conditions. While not shown explicitly, the representative conditions can be readily constructed from the conditions in Table IV. There are also 18 classes of pure (2,6)-positivity conditions. The ${\hat C}$ operators of these 18 conditions are given by $${\hat C}(ijklmn) = \sum_{ijklmn}{ b_{i} d_{j} e_{k} f_{l} g_{m}
h_{n} {\hat a}^{\dagger}_{i} {\hat a}^{\dagger}_{j} {\hat
a}^{\dagger}_{k} {\hat a}^{\dagger}_{l} {\hat a}^{\dagger}_{m} {\hat
a}^{\dagger}_{n}}$$ where the $b_{i}$, $d_{i}$, $e_{i}$, $f_{i}$, $g_{i}$, $h_{i}$, and ${\hat a}^{\dagger}_{i}$ become $b_{i}^{*}$, $d_{i}^{*}$, $e_{i}^{*}$, $f_{i}^{*}$, $g_{i}^{*}$, $h_{i}^{*}$, and $a_{i}$ when $i={\bar i}$. Table V provides a representative operator for each of the 18 classes. Each representative operator arises from the conic combination of potentially $2^{6}$ (or 64) six-particle operators, which are distinguished from each other by the switching between creation and annihilation operators. These 64 operators are grouped in 32 particle-hole pairs given in the rows of Table V. For each of the 18 representative conditions, the nonnegative integer weights $\alpha$ and $\beta$ of the operators in each pair are reported. The conic combination of all 32 pairs with the weights in the $x^{\rm th}$ column generates a representative operator for class $x$. The operator for each class depends only on the 2-RDO with the dependence on the 3-, 4-, 5-, and 6-RDOs canceling through the conic combination. The trace of each representative operator against the 2-RDM generates a representative condition on the 2-RDM. Additional (2,6)-positivity conditions can be generated from the representative conditions through a combination of switching and reordering of the creation and annihilation operators. From the particle-hole pairing it is easy to observe that only one class of the (2,6)-conditions—class 4—has particle-hole symmetry, that is $\alpha=\beta$ in all pairs.
The (2,6)-positivity conditions yield all classes of the classical (diagonal) $N$-representability conditions when the expansion coefficients $b_{i}$, $d_{j}$, $e_{k}$, $f_{l}$, $g_{m}$, and $h_{n}$ are chosen to be orthogonal unit vectors. Classically, all classes of (2,$q$)-conditions for $q \le 5$ are in the form of hypermetric inequalities [@Cuts; @MD72]. When $q=6$, however, new classes of classical $N$-representability conditions emerge [@E78; @MD72; @Cuts; @G89]. In the classical limit, the first 6 classes of pure (2,6)-positivity conditions in Table V reduce to hypermetric inequalities while the remaining 12 can be grouped into cycle, parachute, and Grishukhin inequalities [@G89].
Discussion and Conclusions
==========================
Both new and known $N$-representability conditions on the 2-RDM have been derived from the constructive solution to the $N$-representability problem [@M12b]. In addition to all of the previously known conditions, we generate new (2,3)-, (2,4)-, (2,5), and (2,6)-conditions where the first number $p$ in each pair indicates the highest $p$-RDM required to evaluate the condition (the 2-RDM in our case) and the second number $q$ indicates the highest RDMs canceled by conic (linear nonnegative) combinations in the derivation of the condition. There are two classes of (2,3)-conditions: (i) lifting conditions that are derivable from lifting the D, Q, and G (2-positivity) conditions to the three-particle space, and (ii) pure conditions that are not derivable from lifting and hence, are without precedent in the 2-positivity conditions. The (2,4)-conditions have two classes of lifting conditions and one class of pure conditions, the (2,5)-conditions have three classes of lifting conditions and three classes of pure conditions, and the (2,6)-conditions have six classes of lifting conditions and eighteen classes of pure conditions. A similar procedure of using conic combinations to cancel operators higher than two-body can be followed for deriving the $(2,q)$-conditions for $q>6$.
The classical (diagonal) $N$-representability conditions [@WW67; @YK69; @MD72; @E78; @Cuts] are constraints on the two-electron reduced density function, the diagonal part of the 2-RDM, to ensure that it represents an $N$-electron density function. A solution to the diagonal problem was developed in the context of both the Boole 0-1 programming problem and the maximum cut problem of graph theory [@Cuts; @P89]. The recent constructive solution of the $N$-representability problem for fermionic density matrices extends the classical solution to the more general quantum case. All of the quantum conditions can be cast in the form of restricting the trace of two-body operators (model Hamiltonians) against the 2-RDM to be nonnegative. In the limit that all tensors in the model Hamiltonians are decomposed into products of orthogonal rank-one (one-index) tensors, the quantum conditions reduce to the classical (diagonal) conditions for all unitary transformations of the one-electron basis set. The quantum (2,6)-conditions presented here reduce in the classical limit to the complete set of classical (2,6)-conditions [@MD72; @Cuts], which were shown to be complete by Grishukhin [@G89].
A significant difference between the classical and quantum conditions is the orthogonality (classical) or non-orthogonality (quantum) of the rank-one tensors. Consequently, in the classical case the creation and annihilation operators form a commutative algebra while in the quantum case they form a non-commutative algebra. The non-orthogonality leads to active $N$-representability conditions on the 2-RDM that lack a classical analogue. For example, all classes of lifting conditions that we presented are inactive in the classical limit. Because the creation and annihilation operators commute, each class of classical $(2,q)$-lifting conditions reduces to a class of classical $(2,p)$-pure conditions where $p<q$. Furthermore, typically more than one pure quantum condition reduces to each classical condition in the classical limit. Table III shows four pure (2,4)-conditions that reduce to the same classical condition. These quantum conditions differ only in the ordering of the creation and annihilation operators—a difference that disappears in the classical, commutative limit.
The conic combination of the extreme two-body operators in the $N$-representability conditions forms a convex set (cone) of model Hamiltonians for which the $N$-representability conditions are exact. From the perspective of quantum information the computational complexity of enforcing all $N$-representability conditions on the 2-RDM can be shown to be non-deterministic polynomial-time (NP) complete, meaning that in the worst-case scenario enforcing exact $N$-representability scales non-polynomially with system size. Despite this complexity, however, many realistic quantum systems are much more tractable than the worst-case scenario implies. For example, the 2-positivity conditions, particularly the G condition, are exact for pairing Hamiltonians whose ground states are antisymmetrized geminal power wavefunctions. Such pairing Hamiltonians have been employed to model the Cooper pairing and long-range order associated with superconductivity. For any strength of interaction the ground-state energy for this class of Hamiltonians can be computed in polynomial time.
More generally, for fixed $q$ the $(2,q)$-positivity conditions, which contain the lower positivity conditions, cover a large class of model Hamiltonians whose ground states are computable in polynomial time—in a time that scales polynomially with system size. Even when the Hamiltonian of interest is not rigorously contained in this class, the associated $N$-representability conditions, which intrinsically are not constrained by the approximations of perturbation theory, may produce an accurate lower bound on the ground-state energy. Computational experience with the variational calculation of the 2-RDM in atoms and molecules [@M12a; @M04; @P04; @M06; @GM08; @GM10; @SGM10; @PGG11] shows that sufficiently accurate lower-bound ground-state energies are often produced with $(2,q)$-positivity conditions where $q \le 3$.
The practical implementation of the variational 2-RDM method requires that the energy be minimized as a functional of the 2-RDM constrained by its $N$-representability conditions. Both the 2-positivity conditions and the T1 and T2 conditions can be expressed as positive semidefinite constraints (also known as linear matrix inequalities) in which metric matrices are constrained to be positive semidefinite. These constraints on the 2-RDM can be imposed during the minimization of the ground-state energy through a genre of constrained optimization known as semidefinite programming [@M04; @P04; @C06; @FNY07; @M07; @A09; @M11; @BP12; @E79]. The remaining $(2,q)$-positivity conditions, however, cannot be expressed as a traditional semidefinite constraint because the coefficients in the ${\hat C}_{i}$ operators must be tensor decomposed to remove the dependence of the constraints on the higher RDMs. Practically, as described in section \[sec:sd\], these constraints can be added to the semidefinite program through recursively generated linear inequalities, similar to those described in Ref. [@JSM07] for T2.
The constructive solution of $N$-representability establishes 2-RDM theory as a fundamental theory for many-particle quantum mechanics for particles with pairwise interactions. Lower bounds on the ground-state energy can be computed and improved systematically within the theory. While not all of the 2-RDM conditions will be imposed in practical calculations, a complete knowledge of the conditions—their form and function—can be invaluable in devising and testing approximate $N$-representability conditions for different types of quantum systems and interactions. Like Feynman diagrams the positivity conditions represent different physical interactions of the electrons. Adding positivity conditions to the 2-RDM calculation expands the class of exactly describable model Hamiltonians. Just as classes of Feynman diagrams differ in importance according to the nature of the interaction, for a given system some positivity conditions will be significantly more important than others. For example, both the G and T2 conditions have proven to be especially important in calculations of many-electron atoms and molecules [@M04; @P04; @M05] while the T1 condition has rarely been of any significance. Similar evaluations must be performed in a variety of many-electron quantum systems for the conditions resulting from the constructive solution.
Previous variational 2-RDM computations on metallic hydrogen chains [@SGM10], polyaromatic hydrocarbons [@GM08; @PGG11], and firefly luciferin [@GM10] show that they can capture strong, multi-reference correlation effects for which appropriate ans[ä]{}tze for the wavefunction are difficult to construct. With a suitable choice of $N$-representability conditions, therefore, strong electron correlation effects can be computed at a computational cost that scales polynomially with the system size. Although the exploration of the conditions following from the constructive solution is still in its earliest stages, a 2-RDM-based theory with systematically improvable accuracy promises fresh theoretical and computational possibilities for treating strong correlation in quantum many-electron systems.
The author thanks D. Herschbach, H. Rabitz, and A. Mazziotti for encouragement, and the NSF, ARO, Microsoft Corporation, Dreyfus Foundation, and David-Lucile Packard Foundation for support.
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abstract: 'We construct the representations of $\widehat{sl(2,\mathbb{R})}$ starting from the unitary representations of the loop $ax+b$-group. Our approach involves a combinatorial analysis of the correlation functions of the generators and renormalization of the appearing divergencies. We view our construction as a step towards a realization of the principal series representations of $\widehat{sl(2,\mathbb{R})}$.'
address:
- 'Department of Mathematics, Yale University,10 Hilhouse ave,New Haven, CT, [email protected]'
- ' Department of Mathematics,Columbia University, 2990 Broadway, New York,NY 10027, USA. [email protected]://math.columbia.edu/$\sim$zeitlin http://www.ipme.ru/zam.html '
author:
- 'Igor B. Frenkel'
- 'Anton M. Zeitlin'
title: 'On the continuous series for $\widehat{sl(2,\mathbb{R})}$ '
---
Introduction
============
The principal series representations of split reductive real Lie groups is one of the cornerstones of the classical representation theory. Recently a proper quantum analogue of the principal series in the case of the quantum group $U_q(sl(2,\mathbb{R}))$, or more precisely of its modular double [@faddeev],[@kls] has been found in [@teschner], [@teschner2], based on the previous work in [@schmudgen]. These unitary representations, which do not have a classical limit, but behave similarly to the finite-dimensional representations of the quantum group $U_q(su(2))$, in particular, they form a tensor category. A generalization of these representations to higher rank quantum groups has been also obtained in [@gkl],[@gerasimov],[@fip], [@ivan2], [@ivan3].
One of the fundamental results in the theory of quantum groups is the equivalence of braided tensor categories of finite-dimensional representations of quantum groups and of dominant highest weight representations of affine Lie algebras, associated with the same simple Lie algebra [@ms], [@KL]. By Drinfeld-Sokolov reduction, this equivalence can be extended to the so-called W-algebras, which in the simplest case is nothing but the Virasoro algebra. It was shown in [@teschner] that the braided tensor category of modular double representations of $U_q(sl(2,\mathbb{R}))$ is equivalent to a certain braided tensor category of unitary representations of the Virasoro algebra. Then the analogy with the compact case suggests that there should be the corresponding braided tensor category for $\widehat{sl(2,\mathbb{R})}$ and it must be composed of an affine counterpart of the principal series representations.
A generalization of the classical analytic construction of the principal series representations to the affine Lie algebra $\widehat{sl(2,\mathbb{R})}$ presents substantial analytic difficulties still unresolved even in $\widehat{su(2)}$ case, in which there exists only a heuristic physical construction, known as WZW model. In mathematical literature there exists a construction of a unitary representation of $\widehat{sl(2,\mathbb{R})}$, see [@ggv]. However, this construction has a trivial central extension and does not seem to be relevant for the problem of constructing an equivalent tensor category with the ones in [@teschner].
In this paper we combine an analytic and algebraic approaches to construction of representations of $\widehat{sl(2,\mathbb{R})}$, following the analogy with the quantum algebra case. The principal series representation of the modular double of $U_q(sl(2,\mathbb{R}))$ was realized by means of the algebra of the quantum plane, which is a quantum version of the $ax+b$-group [@ivan]. In our construction of $\widehat{sl(2,\mathbb{R})}$ representations we exploit the representations of the loop $ax+b$-group and the corresponding Lie algebra [@zeit]. The resulting formulas for generators at the first glance remind formulas for bosonization of the free field representation of highest weight modules [@gmm]. However, we encounter a problem of divergences in the correlation functions of the $\widehat{sl(2,\mathbb{R})}$ generators that endangers the foundations of our construction: this is the major issue which makes our construction different from highest weight case. One of the central results of our paper is the method of eliminating divergencies so that the resulting “renormalized” correlators still satisfy the correct commutation relations of $\widehat{sl(2,\mathbb{R})}$ Lie algebra. Our approach was inspired by the explicit combinatorial formula for the correlators of affine Lie algebra in a highest weight representation discovered in [@zhu]. The formula is expressed in terms of Feynman-like diagrams, where the connected ones had the form of cycles.
In our case, one can write down the expression for regularized correlators in terms of Feynman-like diagrams, where the connected diagrams involve tree and 1-loop graphs. We propose, that there is an explicit way to renormalize 1-loop graphs, so that the commutation relations and the Hermicity condition within the correlator remain intact. So, our construction produces representations for $\widehat{sl(2,\mathbb{R})}$ with a bilinear Hermitian form, which depends on an infinite family of renormalization parameters. It is an important problem to find a set of parameters for which the resulting representation, determined by renormalized correlators is unitary, i.e. the pairing is nondegenerate and positive-definite.
Following the analogy with the representation of the quantum group, one can generalize the construction of this article to the higher rank. In the quantum group case one had to use the representations of the multiple quantum planes [@gerasimov], [@kashaev], [@fip], [@ivan2], [@ivan3]. In our case it seems to be a similar technical problem: the role of quantum planes will be played by the loop $ax+b$-group.
The structure of the article is as follows. In Section 2 we recall the structure of principal series representations of $sl(2,\mathbb{R})$, we reformulate it algebraically, indicating its relation to the representations of \*-algebras $\mathcal{A}$, $\mathcal{K}$, the two versions of $ax+b$-algebra. In Section 3 we discuss the unitary and nonunitary representations of affinized versions of \*-algebras $\mathcal{A}$, $\mathcal{K}$, which we denote as $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$ correspondingly. Section 4 is devoted to our main results, namely a construction of the $\widehat{sl(2,\mathbb{R})}$ representations. First we give formal expressions for the generators of $\widehat{sl(2,\mathbb{R})}$, built from the generators of $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$ correspondingly, which satisfy the commutation relations of $\widehat{sl(2,\mathbb{R})}$ but which lead to divergent correlation functions for the $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$-representations. We show how to obtain the well-defined correlation functions of the generators, which determine the representation. At first we show how it works for nonunitary representations of $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$ as an example and then generalize the buildup to the unitary ones.\
[**Acknowledgements.**]{} We are grateful to H. Garland and A. Goncharov for illuminating discussions. A.M.Z. is indebted to the organizers of the Simons Workshop 2012, where this paper was partly written. I.B.F. is supported by NSF grant DMS-100163.
A realization of continuous series for $sl(2,\mathbb{R})$
=========================================================
[**2.1. $\mathcal{A}$ and $\mathcal{K}$ algebras and their unitary representations.**]{} In this section we present a construction of continuous series of $sl(2,\mathbb{R})$ in a special way, which will be convinient for generalizations to the loop case. We first construct certain versions of ax+b-algebras (affine transformations of a real line) which we call $\mathcal{A}$-algebra and $\mathcal{K}$-algebra: they have similar algebraic structure, but a different \*-structure. The $\mathcal{A}$-algebra is an algebra with three generators: $h,e^{\pm}$, so that $ih,ie^{+}$ generate the Lie algebra of ax+b-group, however we make the b-subgroup generator $e^{+}$ to be invertible, so that the inverse element is $e^{-}$. The commutation relations and the star structure are as follows: $$\begin{aligned}
[h, e^{\pm}]=\pm ie^{\pm}, \quad e^{\pm}e^{\mp}=1, \quad h^*=h, \quad e^{{\pm}^*}=e^{\pm}.\end{aligned}$$ The generators $h, \alpha^{\pm}$ of $\mathcal{K}$-algebra have similar commutation relations, but the star-structure is different: $$\begin{aligned}
[h, \alpha^{\pm}]=\mp \alpha^{\pm}, \quad \alpha^{\pm}\alpha^{\mp}=1, \quad h^*=h, \quad {\alpha^{\pm}}^*=\alpha^{\mp}.\end{aligned}$$ To construct the unitary representations of the above \*-algebras, one can realize the generators $h,e^{\pm}$ as unbounded self-adjoint operators $i\frac{d}{dx}, e^{\pm x}$ in Hilbert space of square-integrable functions on a real line $L_2(\mathbb{R})$ as well as generators $h, \alpha^{\pm}$ as the operators $i\frac{d}{d\phi}, e^{\pm i\phi}$ in Hilbert space of square integrable functions on a circle $L_2(S^1)$. These generators acting on a specific vector in the appropriate Hilbert space generate a dense set. More explicitly one has the following.\
[**Proposition 2.1.**]{} [*i) Let $D_A$ be the space spanned by the vectors $a\cdot v_0$, where $v_0=e^{-t x^2}\in L_2(\mathbb{R})$, $t>0$ and $a$ belongs to universal enveloping algebra of $\mathcal{A}$-algebra, such that the action of generators is realized as $h=i\frac{d}{dx}, e^{\pm}= e^{\pm x}$. Then $D_A$ is a dense set in $L_2(\mathbb{R})$.\
ii) Let $D_K$ be the space spanned by the vectors $a\cdot v_0$, where $v_0=1\in L_2(S^1)$ and $a$ belongs to universal enveloping algebra of $\mathcal{K}$-algebra, such that the action of generators is realized as $h=i\frac{d}{d\phi}, \alpha^{\pm}= e^{\pm i \phi}$, where $\phi\in [0,2\pi]\cong S^1$ . Then $D_K$ is a dense set in $L_2(S^1)$.*]{}\
[**Proof.**]{} i) Let us consider the space spanned by vectors $h^nv_0$. This way one can obtain all functions of the form $P(x)e^{-tx^2}$, where $P(x)$ is any polynomial function. It is well-known that this set is dense in $L_2(\mathbb{R})$. Therefore $D_A$ is dense in $L_2(\mathbb{R})$.\
ii) In this case let us consider the space spanned by vectors $(\alpha^+)^n(\alpha^-)^m v_0$. This way we obtain the space of all trigonometric polynomials. It is dense in $L_2(S^1)$. Therefore $D_K$ is a dense set in $L_2(S^1)$. $\blacksquare$.\
[**2.2. Continuous series of $sl(2,\mathbb{R})$ via $\mathcal{A}$ and $\mathcal{K}$ algebras.**]{} First of all, let us introduce some notations. We are interested in unitary representations of $sl(2,\mathbb{R})$ algebra, i.e. Lie algebra with generators $E,F, H$ such that $$\begin{aligned}
\label{sl2r}
&& [E,F]=H, \quad [H, E]=2E,\quad [H, F]=-2F\nonumber\\
&& E=-E^*, \quad F=-F^*, \quad H=-H^* .
\end{aligned}$$ We remind that there are two standard relaizations of continuous series of $sl(2,\mathbb{R})$, one is related to inducing the representations from the diagonal subgroup of $sl(2,\mathbb{R})$, corresponding to the generator $H$, the other one is related to the inducing representations from maximally compact subgroup generated by $J^3=E-F$. It is convenient to introduce the following change of generators (which provides the correspondence between $sl(2,\mathbb{R})$ and $su(1,1)$): $$\begin{aligned}
J^{\pm}=E+F\mp i H , \end{aligned}$$ so that the relations [(\[sl2r\])]{} can be rewritten in the following way: $$\begin{aligned}
&&[J^3, J^{\pm}]=\pm 2iJ^{\pm}, \quad
[J^+, J^-]=-iJ^3, \nonumber \\
&&{J^+}^*=-J^-, \quad {J^3}^*=-J^3.
\end{aligned}$$ Now we will give two realizations of $sl(2,\mathbb{R})$ via generators of $\mathcal{A}$- and $\mathcal{K}$-algebras. Using $\mathcal{A}$-algebra one can write down the expressions for $E, F, H$: $$\begin{aligned}
\label{real1}
&&E=\frac{i}{2}(e^+h+he^+)+i\lambda e^+,\nonumber\\
&&F=-\frac{i}{2}(e^-h +he^-)+i\lambda e^-,\\
&&H=-2ih,\nonumber
\end{aligned}$$ where $\lambda$ is a real parameter. Similarly, using $\mathcal{K}$-algebra, one can represent $J^3, J^{\pm}$ in a similar way: $$\begin{aligned}
\label{real2}
&&J^+=\frac{i}{2}(\alpha^+h+h\alpha^+)-\lambda \alpha^+\nonumber\\
&&J^-=\frac{i}{2}(\alpha^-h +h\alpha^-)+\lambda \alpha^-\\
&&J^3=2ih\nonumber
\end{aligned}$$
Therefore the following theorem is valid.\
[**Theorem 2.1.**]{}
*Let $\mathcal{D}_A$ be the space spanned by the vectors $a\cdot v_0$, where $v_0=e^{-t x^2}\in L_2(\mathbb{R})$ and $a$ belongs to universal enveloping algebra of $sl(2,\mathbb{R})$-algebra, such that the action of generators is realized as in [(\[real1\])]{}, so that $h=i\frac{d}{dx}, e^{\pm}= e^{\pm x}$. Then $\mathcal{D}_A$ is a dense set in $L_2(\mathbb{R})$ and it is a representation space for $sl(2,\mathbb{R})$.*
ii\) Let $\mathcal{D}_K$ be the space spanned by the vectors $a\cdot v_0$, where $v_0=1\in L_2(S^1)$ and $a$ belongs to universal enveloping algebra of $sl(2,\mathbb{R})$-algebra, such that the action of generators is realized as in [(\[real2\])]{}, so that $h=i\frac{d}{d\phi}, e^{\pm}= e^{\pm i \phi}$, where $\phi\in [0,2\pi]\cong S^1$. Then $\mathcal{D}_K$ is a dense set in $L_2(S^1)$ and it is representation space for $sl(2,\mathbb{R})$.
\
[**Proof.**]{} i) Let us consider the space spanned by vectors $H^nv_0$. This way one can obtain all functions of the form $P(x)e^{-tx^2}$, where $P(x)$ is any polynomial function. It is well-known that this set is dense in $L_2(\mathbb{R})$. Therefore $\mathcal{D}_A$ is dense in $L_2(\mathbb{R})$.
ii\) In this case let us consider the space spanned by vectors $(J^+)^n(J^-)^m v_0$. This way we obtain the space of all trigonometric polynomials. It is dense in $L_2(S^1)$. Therefore $\mathcal{D}_K$ is a dense set in $L_2(S^1)$. $\blacksquare$.\
Representations of $\widehat{\mathcal{A}}$ and $\widehat{\mathcal{K}}$
======================================================================
[**3.1. Definitions.**]{} In this section we consider the following \*-algebras, which we denote $\widehat{\mathcal{A}}$ and $\widehat{\mathcal{K}}$. The first algebra is close to the Lie algebra of a loop group associated with extended $\mathcal{A}$-group with properties, which is generated by $h_n$, $e^{\pm}_n$, $n\in \mathbb{Z}$, so that the generating “currents” are: $$\begin{aligned}
h(u)=\sum_{n\in\mathbb{Z}} h_{-n} e^{inu}, \quad e^{\pm}(u)=\sum_{n\in\mathbb{Z}} e^{\pm}_{-n}e^{inu} \end{aligned}$$ and obey the following commutation relations, expressed via generating functions: $$\begin{aligned}
\label{relcom}
&&[h(u), h(v)]=0, \quad [e^{\pm}(u), e^{\pm}(v)]=0, \quad [e^{\pm}(u), e^{\mp}(v)]=0,\nonumber\\
&&[h(u), e^{\pm}(v)]=\pm i\delta(u-v)e^{\pm}(v), \quad e^{\pm}(u)\cdot e^{\mp}(u)=1.\end{aligned}$$ The \*-structure is such that $h(u)$, $e^{\pm}(u)$ are Hermitian, i.e. $$\begin{aligned}
h(u)^*=h(u),\quad e^{\pm}(u)^*=e^{\pm}(u)\end{aligned}$$ The algebra $\widehat{\mathcal{K}}$ is generated by $h_n$, $\alpha^{\pm}_n$, $n\in \mathbb{Z}$, it has similar commutation relations, but a different \*-structure: $$\begin{aligned}
&&h(u)=\sum_n h_{-n} e^{inu}, \quad \alpha^{\pm}(u)=\sum_n \alpha^{\pm}_{-n}e^{inu},\nonumber\\
&&[h(u), h(v)]=0, \quad [\alpha^{\pm}(u), \alpha^{\pm}(v)]=0, \quad [\alpha^{\pm}(u), \alpha^{\mp}(v)]=0,\nonumber\\
&&[h(u), \alpha^{\pm}(v)]=\mp \delta(u-v)\alpha^{\pm}(v), \quad \alpha^{\pm}(u) \alpha^{\mp}(u)=1\nonumber\\
&&h(u)^*=h(u),\quad \alpha^{\pm}(u)^*=\alpha^{\mp}(u).\end{aligned}$$ After a short reminder of Gaussian integration, we will construct some representations of these algebras.\
[**3.2. Gaussian integration on Hilbert spaces.**]{} In this subsection we recall a few basic facts and formulas, for more information, see e.g. [@daprata], [@quo]. Suppose we have a real separable Hilbert space $H$ with the orthonormal basis $\{e^i\}$, $i\in \mathbb{N}$ and the pairing $\langle\cdot , \cdot\rangle$. Every element $x$ of this Hilbert space can be expressed as $x=\sum_i x_ie^i$. Let us introduce positive real numbers $\lambda_i$, $i\in \mathbb{N}$, so that $\sum_i\lambda_i<\infty$. Then one can say that numbers $\lambda_i$ define a diagonal trace class operator $A$ on our Hilbert space. Than it appears possible to define the sigma-additive measure $d\mu_A$ on $H$ and heuristically express it as follows: $$\begin{aligned}
d\mu_A(x)=(\sqrt{\det {2 \pi A}})^{-1}\cdot e^{-\frac{1}{2}\langle x,A^{-1}x \rangle}[dx],\end{aligned}$$ which can be thought as the infinite product of 1-dimensional Gaussian measures for each $i$: $d\mu_i=\sqrt{2\pi \lambda_i}^{-1}e^{-\lambda_i^{-1}x_i^2}$.
Since it is a sigma-additive measure, one can define the space of square-integrable functions $L_2(H, d\mu_A)$ with respect to it. One of the basic formulas is the translational shift in the measure. Namely, if $b\in Im A$, then $$\begin{aligned}
\int f(x) d\mu_A(x)=\int f(x+b) e^{{-\frac{1}{2}\langle b,A^{-1}b \rangle}-\langle x,A^{-1}b \rangle}d\mu_A(x).\end{aligned}$$ One can consider also an infinitesimal version of this formula. Making $b$ infinitesimal and parallel to $e_i$, we obtain that $$\begin{aligned}
\int D_i f(x)d\mu_A(x)=0, \quad D_i={\partial}_{x_i}-\lambda_i^{-1}x_i,\end{aligned}$$ if ${\partial}_{x_i} f(x)\in L_2(H, d\mu_A)$. It should be noted that the following monomials $$\begin{aligned}
\label{polexp}
(\prod^k_{i=1}\langle \alpha_i, x\rangle) e^{\langle \beta, x\rangle},\end{aligned}$$ where $\alpha_i, \beta$ are the elements of complexified Hilbert space, are always integrable with respect to $d\mu_A$, moreover, they belong to $L_2(H, d\mu_A)$. There is an explicit formula for the integral of the function [(\[polexp\])]{}, which can be derived from the simple result: $$\begin{aligned}
\label{gauss}
\int e^{\langle \beta, x\rangle}d\mu_A(x)=e^{\frac{1}{2}\langle \beta, A\beta\rangle}.\end{aligned}$$
[**3.3. Construction of representations.**]{} In order to construct representations of $\widehat{\mathcal{A}}$ and $\widehat{\mathcal{K}}$ one can use the Gaussian measure on a Hilbert space. Let us consider the Fourier series of a function from $L_2(S^1, \mathbb{R})$: $$\begin{aligned}
x(u)=\sum_{n\in \mathbb{Z}}x_{-n}e^{inu}, \quad x_0\in \mathbb{R}, \quad x_n^*=x_{-n}.\end{aligned}$$ Let us introduce two quadratic forms defining two types of trace-class operators on $L_2(S^1, \mathbb{R})$, which will determine the appropriate Gaussian measures: $$\begin{aligned}
\label{bil}
&& B_A(x,x)=\frac{1}{2}\sum_{n\ge 1}\xi_n^{-1}x_nx_{-n}+\xi_0^{-1}x_0^2,\nonumber\\
&& B_{K}(x,x)=\frac{1}{2}\sum_{n\ge 1}\xi_n^{-1}x_nx_{-n},\end{aligned}$$ where $\xi_n>0$ for all $n$ and $\sum_n\xi_n<\infty$. The Gaussian measures we are interested in, heuristically can be expressed as follows: $$\begin{aligned}
&& dw_{A}=(\sqrt{det(2\pi N_A)})^{-1}e^{-B_A(x,x)}dx_0\prod^{\infty}_{n=1}[\frac{i}{2}dx_n\wedge dx_{-n}],\nonumber \\
&& dw_{K}=(\sqrt{det(2 \pi N_{K})})^{-1}e^{-B_K(x,x)}d\phi\prod^{\infty}_{n=1}[\frac{i}{2}dx_n\wedge dx_{-n}],\end{aligned}$$ where $N_A, N_{K}$ are trace-class diagonal operators determined by the quadratics forms [(\[bil\])]{}. Here as before the range for $\phi$ is $[0,2\pi]$. Literally the difference between two measures is that in the second one we compactified the zero mode $x_0$ on a circle with parameter $\phi$ for $dw_{K}$. The Hilbert spaces of square-integrable functions with respect to these measures are denoted in the following as $\mathcal{H}_A$ and $\mathcal{H}_K$.
Let us construct the unitary representations of $\widehat{\mathcal{A}}$ and $\widehat{\mathcal{K}}$-algebras in the Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_K$, correspondingly.
We will explicitly define the operators, which will represent the generators. Let us start from $\widehat{\mathcal{A}}$-algebra. First of all, let us extend the index of $\xi_n$ to all integers, so that $\xi_{-n}=\xi_n$, $n\in \mathbb{Z}$. We need the following differential operators: $$\begin{aligned}
\label{ab}
b_{-n}=i({\partial}_n-\xi_n^{-1}x_{-n}),
\quad a_{-n}=i{\partial}_n,\end{aligned}$$ where ${\partial}_{n}=\frac{{\partial}}{{\partial}x_n}$. These operators are formally conjugate to each other, $$\begin{aligned}
a_n^*=b_{-n},\end{aligned}$$ considered on a certain dense set $D^l_A$, i.e. functions which are the sums of monomials $$\begin{aligned}
\prod^n_{k=1}\langle\mu_k,x\rangle\prod^m_{s=1} \langle \lambda_s, e^{\pm x}\rangle, \end{aligned}$$ where $\langle,\rangle$ is the standard $L_2(S^1, \mathbb{R})$ pairing and $\mu_k, \lambda_k$ are trigonometric polynomials. We define the operators $h_n$ as follows: $$\begin{aligned}
\label{ha}
h_{-n}=\frac{1}{2}(a_{-n}+b_{-n})=i({\partial}_n-\frac{1}{2}\xi_n^{-1}x_{-n}).\end{aligned}$$ Hence, on a dense set $h_n^*=h_{-n}$, so that the current $h(u)=\sum_{n\in \mathbb{Z}}h_{-n}e^{inu}$ is Hermitian. Also, note that Hermitian currents $h(u), x(v)$ generate infinite-dimensional Heisenberg algebra: $$\begin{aligned}
[h(u), x(v)]=i\delta(u-v).\end{aligned}$$ We also define the currents $e^{\pm}(u)$: $$\begin{aligned}
\label{ea}
e^{\pm}(u)=e^{\pm x(u)}.\end{aligned}$$ It also follows that they satisfy the commutation relations [(\[relcom\])]{}. One can show that $e^{\pm x(u)}$ for any $u$ are Hermitian operators, considered on a dense set $D^l_A$. Therefore, this gives a unitary representation of $\widehat{\mathcal{A}}$-algebra.\
[**Proposition 3.1.**]{} [*Let us consider the elements of the form $a\cdot v_0$, where $v_0\equiv 1\in \mathcal{H}_A$ and $a$ belongs to universal enveloping algebra of $\widehat{\mathcal{A}}$ and the action of the generators is given by the formulas [(\[ea\])]{}, [(\[ha\])]{}. Then such elements generate a dense set $D^l_A$ in $\mathcal{H}_A$, which is a unitary representation of $\widehat{\mathcal{A}}$-algebra.*]{}\
[**Proof.**]{} Idea of the proof is similar to the one from Proposition 2.1. Let us consider a linear span of the following vectors: $h^{k_1}_{n_1}....h^{k_p}_{n_p}v_0$ for $k_i\in \mathbb{Z}_{\ge 0}$, $n_i\in \mathbb{Z}$. The resulting space will contain all polynomials in $x_n$, which is a well-known dense set (see e.g. [@daprata]) in $\mathcal{H}_A$. Therefore $D^l_A$ is dense in $\mathcal{H}_A$. $\blacksquare$
Similarly one can construct the unitary representations of $\widehat{\mathcal{K}}$-algebra. Let us consider the dense set $D^l_A$ in $\mathcal{H}_K$ of the following form: $$\begin{aligned}
\prod^n_{k=1}\langle\mu_k,x\rangle\prod^m_{s=1} \langle \lambda_s, e^{\pm ix^c}\rangle, \end{aligned}$$ where $\lambda_i$ are trigonometric polynomials without the constant term and $x^c(u)=\phi+\sum_{n\neq 0}x_n e^{-inu}$. The currents $h(u), \alpha^{\pm}(u)$ are defined by the following formulas: $$\begin{aligned}
\label{ag}
&& h(u)=\sum_{n\neq 0} i({\partial}_n-\frac{1}{2}\xi_n^{-1}x_{-n})e^{inu}+i{\partial}_{\phi},\nonumber\\
&& \alpha^{\pm}(u)=e^{\pm ix^c(u)}.\end{aligned}$$ It is possible to introduce operators $a_n, b_n$ and define them by the same formulas as in [(\[ab\])]{} for all $n\neq 0$. For $n=0$ we put $a_0=b_0=i{\partial}_{\phi}$. This allows to formulate the following.\
[**Proposition 3.2.**]{} [*Let us consider the elements of the form $a\cdot v_0$, where $v_0\equiv 1\in \mathcal{H}_K$ and $a$ belongs to universal enveloping algebra of $\widehat{\mathcal{K}}$ and the action of the generators is given by the formulas [(\[ag\])]{}. Such elements generate a dense set $D^l_K$ in $\mathcal{H}_K$, which is a unitary representation of $\widehat{\mathcal{K}}$-algebra.*]{}\
[**Proof.**]{} The proof is ismilar to the one of Proposition 3.1. Let us consider a linear span of the following vectors: $h^{k_1}_{n_1}....h^{k_p}_{n_p}v_0$ for $k_i\in \mathbb{Z}_{\ge 0}$, $n_i\in \mathbb{Z}$. The resulting space will contain all polynomials of $x_n$, where $n\neq 0$. This would be a dense set in the Hilbert space with the Gaussian measure with out the $\phi$-variable. If one applies the action of $\alpha_0^{\pm}$ modes to elements of this space and considers a linear span then the resulting set is dense in the space of polynomials of $x_n$, $e^{\pm i\phi}$ where $n\neq 0$, which is a dense set (see e.g. [@daprata]) in $\mathcal{H}_K$. Therefore $D^l_K$ is dense in $\mathcal{H}_A$. $\blacksquare$\
[**3.4. Correlators and normal ordering.**]{} An important notion which is necessary for our construction is the correlator associated with the representation.
By the correlator of generators $T_1, ..., T_n$ of $\widehat{\mathcal{A}}$-algebra (resp. $\widehat{\mathcal{K}}$-algebra) we mean the following expression: $$\begin{aligned}
<T_1...T_n>\equiv \langle v_0, T_1...T_n v_0 \rangle,
\end{aligned}$$ where the pairing $\langle, \rangle $ is of the Hilbert space $\mathcal{H}_A$ (resp. $\mathcal{H}_K$), $v_0$ is the vector corresponding to the constant function $1$ and $T_1, ..., T_n$ are the generators $h_n, e^{\pm}_m$ (resp. $h_n, \alpha^{\pm}_m$).
We remind that $h_n=\frac{1}{2}(a_n+b_n)$. We note the following property which is the consequence of the properties of Gaussian integration.\
[**Proposition 3.3.**]{} [*The following correlators $$\begin{aligned}
\label{van}
<T_1...T_na_k>, \quad
<b_kT_1...T_n>\end{aligned}$$ vanish for any generators $T_1, ...,T_n$.*]{}\
[**Proof**]{}. The correlator of the first type $<T_1...T_na_k>= \langle v_0, T_1...T_n a_kv_0 \rangle$ vanishes, because $a_k v_0=0$ for any $k$. The correlator $<b_kT_1...T_n>$ vanishes because by complex conjugation it transforms in the correaltor of the first type.$\blacksquare$\
It is natural to call operators $a_n$ $annihilation$ operators and $b_n$ $creation$ operators. This allows us to define the normal ordering. Namely, when we write down the expression $$\begin{aligned}
:T_1...T_n:\end{aligned}$$ for the product of $n$ generators, we reorder them in such a way that creation operators will be to the left and annihilation to the right.
This procedure together with the vanishing of the correlators [(\[van\])]{} also gives an easy method to compute the correlators: by means of commutation relations of the generators, one can reduce the products $T_1, ...,T_n$ to the normally ordered expressions. Therefore, the result will reduce to the correlators of the generators $e^{\pm}_n$ or $\alpha^{\pm}_n$. We also note that in the case of $\widehat{\mathcal{K}}$-algebra the corresponding correlators are nonzero only if they have an equal number of $\alpha^+$ and $\alpha^-$ generators.\
In order to compute the correlators of these generators it is easier to consider the appropriate currents instead of modes and use the Gaussian integration.\
[**Proposition 3.4.**]{}[*One has the following expressions for correlation functions: $$\begin{aligned}
\label{core}
&&\langle e_+(u_1)...e_+(u_n)e_-(v_1)...e_-(v_m)\rangle=\nonumber\\
&&\exp(\sum^n_{i<j; i, j=1}N_A(u_i,u_j)+\sum^m_{r<s; r, s=1}N_A(v_r,v_s)-\sum^n_{k=1}\sum^m_{l=1}N_A(u_k,v_l)+\nonumber\\
&&\frac{n+m}{2}N_A(0,0)),\\
&&\label{corealpha}\langle \alpha_+(u_1)...\alpha_+(u_n)\alpha_-(v_1)...\alpha_-(v_m)\rangle=\\
&&\delta_{n, m}\exp(-\sum^n_{i<j; i, j=1}N_K(u_i,u_j)-\sum^n_{i<j; i, j=1}N_K(v_i,v_j)+\sum^n_{k,l=1}N_K(u_k,v_l)\nonumber\\
&&+nN_K(0,0)),\end{aligned}$$ where $$\begin{aligned}
&&N_A(u, v)=2\sum_{n\ge 0}cos(n(u-v))\xi_n, \nonumber\\
&&N_K(u, v)=2\sum_{n>0}cos(n(u-v))\xi_n.\end{aligned}$$*]{}
[**Proof.**]{} The proof of this result follows from the formula [(\[gauss\])]{}. Let us show that in the case of $e^{\pm}$-generators, the case of $\alpha^{\pm}$ ones is similar. We constructed $e^{pm}$ generators so that $e_+(u_1)...e_+(u_n)e_-(v_1)...e_-(v_m)=\exp{(\sum_i\delta_{u_i}+\sum_j\delta_{v_j}, x)}$, where $\delta_w$ stands for the delta function at the point $w$. Then one can see that if one treats $\sum_i\delta_{u_i}+\sum_j\delta_{v_j}$ as $\beta$ from [(\[gauss\])]{} one obtains the formula [(\[core\])]{}. To show that this is not only formally true, one has to consider $\delta_w$ as a weak limit of $L_2$-functions, the so-called $\delta$-like sequence, use that $\delta$-like approximation for $e^{\pm}$-generators, take a gaussian integral according to [(\[gauss\])]{} and then take a limit. For further details see e.g. [@daprata].$\blacksquare$.\
We note here that in the case of correlators of the currents involving not only $e^{\pm}, \alpha^{\pm}$, but also $h(u)$, the resulting expression will consist of monomials [(\[corealpha\])]{} multiplied on certain product of delta-functions, coming from commutation relations of $\widehat{\mathcal{K}}$, $\widehat{\mathcal{A}}$-algebras.\
[**3.5. Nonunitary representations.**]{} In the previous section we have shown that using Gaussian measure, one can construct unitary representations of $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$-algebras. However, it is possible to simplify those representations by making $a_k$ commute with $b_s$ for any $k$ and $s$. However, the resulting module will not be unitary, i.e. the pairing though nondegenerate will lose its positivity.
Let us at first give the explicit description of such module for $\widehat{\mathcal{K}}$-algebra. For this we consider the vacuum vector $v_0$, such that $v_0$ is annihilated by $a_k$: $$\begin{aligned}
a_kv_0=0.\end{aligned}$$ Then the module which we will refer to as $\mathcal{V}_A$ is spanned by the following vectors: $$\begin{aligned}
b_{m_1}...b_{m_s}e^{\pm}_{n_1}...e^{\pm}_{n_r}v_0,\end{aligned}$$ where $n_1, ..., n_r, m_1, ..., m_s\in \mathbb{Z}$. Let us begin to define the pairing with the postulation of the points: $e(u)=\sum_ne_ne^{inu}$ is a Hermitian current, $b(u)^*=a(u)$, so that $b(u)=\sum_n b_ne^{-inu}$, $a(u)=\sum_n a_ne^{-inu}$ and $$\begin{aligned}
[b(u), a(v)]=0.\end{aligned}$$ The pairing is uniquely defined by the correlator of $e^{\pm}$ currents is given by [(\[core\])]{}. Similarly one can define the module $\mathcal{V}_K$ for $\widehat{\mathcal{K}}$ algebra with the same conditions, just replacing $e^{\pm}$ with $\alpha^{\pm}$, as a result $\mathcal{V}_K$ is spanned by $$\begin{aligned}
b_{m_1}...b_{m_s}\alpha^{\pm}_{n_1}...\alpha^{\pm}_{n_r}v_0.\end{aligned}$$ One can define the pairing on $\mathcal{V}_K$ which is uniquely defined by the correlators of $\alpha^{\pm}$ currents [(\[corealpha\])]{}. Let us formulate this as a proposition.\
[**Proposition 3.5.**]{} [*The pairing defined above is Hermitean and nondegenerate. It gives a structure of nonunitary representations of the \*-algebras $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$ on the spaces $\mathcal{V}_A$, $\mathcal{V}_K$ correspondingly.*]{}\
We note here that clearly, these representations are nonunitary, because you can easily find vectors $v\in \mathcal{V}_A, \mathcal{V}_K$, such that $\langle v,v\rangle=0$.
Construction of representations for $\widehat{sl(2,\mathbb{R})}$
================================================================
[**4.1. Regularization and the commutator.**]{} In order to construct the $\widehat{sl(2,\mathbb{R})}$ representations via $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$, one has to find affine analogue of the formulas for $E,F, H$ and $J^3, J^{\pm}$. In the case of nontrivial central extension, the $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$ representations appear to be insufficient, one has to introduce the representation of the infinite-dimensional Heisenberg algebra, so that the generating current is $\rho(u)=\sum_{n\in \mathbb{Z}}\rho_ne^{-inu}$ and the commutation relations are: $$\begin{aligned}
[\rho_n, \rho_m]=2\kappa n\delta_{n,-m},\end{aligned}$$ where $\kappa\in \mathbb{R}_{>0}$. The irreducible module, so-called Fock module $F_{\kappa,p}$ of this algebra is defined as follows. We introduce a vector $vac_p$ with the property $\rho_n vac_p=0$, $p\in \mathbb{R}$, $n>0$ so that $$\begin{aligned}
F_{\kappa,p}=\{\rho_{-n_1}...\rho_{-n_k}vac_p; \quad n_1,..., n_k>0,\quad \rho_0 vac_p=p\cdot vac_p\}.\end{aligned}$$ The Hermitian pairing is defined so that $$\begin{aligned}
\langle vac_p, vac_p \rangle=1, \quad \rho^*_n=\rho_{-n}\end{aligned}$$ Another object required in this section is the regularized version of the $\rho$, $h$, $e^{\pm}$, $\alpha^{\pm}$ currents. Namely, for any $\varphi$, which stands for any of $\rho$, $h$, $x(u)$ or $x^c(u)$ we consider $$\begin{aligned}
\varphi(z,\bar z)=\sum_{n\ge 0}\varphi_n \bar z^n+\sum_{n> 0}\varphi_{-n} z^n,\end{aligned}$$ where $z=re^{iu}$, so that $0<r\le 1$. We denote $e^{\pm}(z,\b z)\equiv e^{\pm x(z,\b z)}$ and $\alpha^{\pm}(z,\b z)\equiv e^{\pm i x^c(z,\b z)}$. The Wick theorem implies that the correlators of the regularized Heisenberg currents $\rho(z_1, \bar z_1), ...., \rho(z_n,\bar z_n)$ are finite as long as $0<|z_i|< 1$, i.e. the expressions $$\begin{aligned}
\label{heicor}
\langle vac_p, \rho(z_1,\bar z_1)...\rho(z_n,\bar z_n) vac_p\rangle\end{aligned}$$ are finite. One can consider the limit $|z_i|\to1$ in the sense of distributions, so that [(\[heicor\])]{} is the sum of products of distributions, i.e. $$\begin{aligned}
\label{2pointf}
\langle vac_p, \rho(u_1)\rho(u_2)vac_p\rangle=\frac{2\kappa}{(1-e^{i(u_2-u_1-i0)})^2}+p^2.\end{aligned}$$ Next we consider the following sets of composite regularized currents $$\begin{aligned}
&&J^{\pm}(z,\bar z)=\\
&&\frac{i}{2}(b(z,\bar z)\alpha^{\pm}(z,\bar z)+\alpha^{\pm}(z,\bar z)a(z,\bar z))\pm\kappa{\partial}_u\alpha^{\pm}(z,\bar z)
\pm \rho(z,\bar z)\alpha^{\pm}(z,\bar z),\nonumber\\
&&J^3(z,\bar z)=2i h(z,\bar z)-2\kappa\alpha^-(z,\bar z){\partial}_u \alpha^+(z,\bar z),\nonumber\end{aligned}$$ and $$\begin{aligned}
&&E(z,\bar z)=\\
&&\frac{i}{2}(b(z,\bar z)e^{+}(z,\bar z)+e^{+}(z,\bar z)a(z,\bar z))+i\kappa{\partial}_ue^{+}(z,\bar z)
+ i\rho(z,\bar z)e^{+}(z,\bar z),\nonumber\\
&&F(z,\bar z)=\nonumber\\
&&-\frac{i}{2}(b(z,\bar z)e^{-}(z,\bar z)+e^{-}(z,\bar z)a(z,\bar z))+i\kappa{\partial}_ue^{-}(z,\bar z)
+i \rho(z,\bar z)e^{-}(z,\bar z)\nonumber\\
&&H(z,\bar z)=-2i h(z,\bar z)+2i\kappa e^-(z,\bar z){\partial}_u e^+(z,\bar z).\nonumber\end{aligned}$$ Moreover, we have the following Hermicity conditions: $$\begin{aligned}
&&E(z,\bar z)^*=-E(z,\bar z), \quad F(z,\bar z)^*=-F(z,\bar z), \quad H(z,\bar z)^*=-H(z,\bar z),\nonumber\\
&&J^3(z,\bar z)^*=-J^3(z,\bar z), \quad J^{\pm}(z,\bar z)^*=-J^{\mp}(z,\bar z).\end{aligned}$$ Then the following proposition is true.\
[**Proposition 4.1.**]{} [ *Let $\phi_k$ denote $E, F, H$ or $J^3, J^{\pm}$. Then the correlators $$\begin{aligned}
\label{gencor}
&&\langle \phi_{1}(z_1,\bar z_1)...\phi_{n}(z_n,\bar z_n)\rangle_p\equiv\nonumber\\
&&\langle v_0\otimes vac_p,\phi_{1}(z_1,\bar z_1)...\phi_{n}(z_n,\bar z_n)v_0\otimes vac_p\rangle,\end{aligned}$$ are well-defined for $0<|z_i|<1$. Moreover, if one of the currents is on the unit circle, i.e. considered at the point $|z|=1$, while all other are inside the unit circle, correlator [(\[gencor\])]{} is also well-defined.* ]{}\
[**Proof.**]{} Consider the case when all parameters are inside the unit circle. Since $|z_1|, ..., |z_n|<1$ after the normal ordering procedure $\phi_{1}(z_1,\bar z_1)...\phi_{n}(z_n,\bar z_n)$ will be represented as normally ordered products with coefficients which are continuous functions of $u_1,..., u_n$, where $z_i=r_ie^{u_i}$. Therefore, the expression $\langle \phi_{1}(z_1,\bar z_1)...\phi_{n}(z_n,\bar z_n)\rangle_p$ will be given by the sum of correlators of exponentials $\alpha^{\pm}$ or $e^{\pm}$ which are well-defined even on the unit circle as we know from the formulas [(\[core\])]{}. In the case if only one of the currents $\phi_k(z_k, \bar z_k)$ is such that $|z_k|=e^{iu_k}$, the situation will not change. This is because during normal ordering procedure, the commutators of $a$- and $b$- parts of this generator with other ones again produce continuous functions of $u_1,..., u_n$, since all other $z_i$ are inside the unit circle and the normal ordering wouldn’t produce any distributions[^1].$\blacksquare$\
Let us make sense of the commutator of two currents on the unit circle as follows. We begin by considering the difference of two correlators $$\begin{aligned}
&&\langle \phi_{1}(z_1,\bar z_1)...\xi(w_1,\bar w_1)\eta(w_2,\bar w_2)...\phi_{n}(z_n,\bar z_n)\rangle_p-\nonumber\\
&&\langle \phi_{1}(z_1,\bar z_1)...\eta(w_2,\bar w_2)\xi(w_1,\bar w_1)...\phi_{n}(z_n,\bar z_n)\rangle_p.\end{aligned}$$ It is clearly well-defined, however, we want $w_1, w_2$ to lie on the unit circle. Then we have the following.\
[**Proposition 4.2.**]{} [ *The following limit $$\begin{aligned}
\label{comm}
&&\lim_{r_1,r_2\to 1}\big(\langle \phi_{1}(z_1,\bar z_1)...\xi(w_1,\bar w_1)\eta(w_2,\bar w_2)...\phi_{n}(z_n,\bar z_n)\rangle_p-\nonumber\\
&&\langle \phi_{1}(z_1,\bar z_1)...\eta(w_2,\bar w_2)\xi(w_1,\bar w_1)...\phi_{n}(z_n,\bar z_n)\rangle_p\big ),\end{aligned}$$ exists in the sense of distributions, more specifically the answer will contain a linear combination of delta functions $\delta(u_1-u_2)$ and its derivatives. Here $\phi_k$ stands for $E, F, H$ or $J^3, J^{\pm}$, $w_i=r_ie^{iu_i}$, $\xi(u_1)\equiv \xi(e^{iu_1},e^{-iu_1})$, $\eta(u_2)\equiv \eta(e^{iu_2},e^{-iu_2})$.* ]{}\
[**Proof.**]{} The proof is based on the definition of the commutator [(\[comm\])]{} and the commuation relations of the Heisenberg algebra generated by $\rho(u)$ and $\widehat{\mathcal{A}}$, $\widehat{\mathcal{K}}$-algebras. We explicitly prove the $J^{\pm}, J^3$ commutation relations, for $E,F, H$-currents can be obtained in a similar fashion. Let $z=re^{iu}$, $w=te^{iu}$. In the notations below we drop the dependence on $\bar z$ variable in order to simplify the calculations. Let us start by computing the commutator of $J^+$ and $J^-$. Let us introduce the notation: $$\begin{aligned}
j^{\pm}(z,\bar z)\equiv\frac{i}{2}(b(z,\bar z)\alpha^{\pm}(z,\bar z)+\alpha^{\pm}(z,\bar z)a(z,\bar z)).\end{aligned}$$ Then $$\begin{aligned}
&& \lim_{r,t\to 1}[j^+(z,\bar z), j^-(\bar w, \bar w)]=-2h(u)\delta(u-v),\nonumber\\
&&\lim_{r,t\to 1}[j^+(z,\bar z), -\kappa{\partial}_v\alpha^-(w,\bar w)]=-i\kappa{\partial}_v(\delta(u-v)\alpha^-(v))\alpha^+(u),\nonumber\\
&&\lim_{r,t\to 1}[ \kappa{\partial}_u\alpha^+(z,\bar z), j^-(w,\bar w)]=
i\kappa{\partial}_u(\delta(u-v)\alpha^+(u))\alpha^-(v),\nonumber\\
&&-\lim_{r,t\to 1}[\rho(z)\alpha^+(z,\bar z), \rho(w)\alpha^-(w)]=2i\kappa{\partial}_u\delta(u-v)\alpha^+(u)\alpha^-(v).\end{aligned}$$ Summing all the terms and using the $\delta$-function properties we arrive to the desired commutation relation between $J^+$ and $J^-$ currents: $$\begin{aligned}
\label{jpm}
[J^+(u),J^-(v)]=i J^3(v)\delta(u-v)+4i\kappa\delta'(u-v).\end{aligned}$$ Similarly, $$\begin{aligned}
\lim_{r,t\to 1}[-2i h(z,\bar z),2\kappa\alpha^-(w,\bar w){\partial}_v \alpha^+(w,\bar w)]=-4i\kappa{\partial}_u\delta(u-v).\end{aligned}$$ This leads to commutation relations between $J^3$ currents, giving the proper central extension term: $$\begin{aligned}
\label{j3}
[J^3(u), J^3(v)]=-8i\kappa\delta'(u-v)\end{aligned}$$ Finally, the commutation relation formula for $J^3$ and $J^{\pm}$ follows from the formula $$\begin{aligned}
\lim_{r,t\to 1}[2\kappa\alpha^-(z,\bar z){\partial}_u \alpha^+(z,\bar z), j^{\pm}(w, \bar w)]=\pm2i\kappa{\partial}_v\delta(u-v)\alpha^{\pm}(v).\end{aligned}$$ Therefore, $$\begin{aligned}
\label{j3pm}
[J^3(u), J^{\pm}(v)]=\pm 2iJ^{\pm}(v)\delta(u-v).\end{aligned}$$ $\blacksquare$
We will denote the expression [(\[comm\])]{} as follows: $$\begin{aligned}
\langle \phi_{1}(z_1,\bar z_1)...[\xi(u_1),\eta(u_2)]...\phi_{n}(z_n,\bar z_n)\rangle_p. \end{aligned}$$ We discovered throught the proof of Proposition 4.2 (see [(\[jpm\])]{}, [(\[j3\])]{}, [(\[j3pm\])]{}) that the commutation relations between currents $J^3, J^{\pm}$ (similarly one can show that for $E,F, H$-currents), generate the $\widehat{sl(2,\mathbb{R})}$ algebra. However, it doesn’t mean that correlators [(\[gencor\])]{} define the representation of $\widehat{sl(2,\mathbb{R})}$, since the currents in [(\[gencor\])]{} will be often not well defined if two or more of the arguments are considered on the unit circle. Let us summarize what do we have so far in the following theorem.\
[**Theorem 4.1.**]{} [*The commutator, defined by the formula [(\[comm\])]{}, of the currents $E, F, H$ or $J^3, J^{\pm}$ exists and satisfies the commutation relations [^2] for $\widehat{sl(2,\mathbb{R})}$ algebra with the central charge $\kappa$: $$\begin{aligned}
&&[E(u),F(v)]=H(v)\delta(u-v)-4i\kappa\delta'(u-v), \quad [H(u), H(v)]=8i\kappa\delta'(u-v),\nonumber\\
&& [H(u), E(v)]=2E(v)\delta(u-v),\quad [H(u), F(v)]=-2F(v)\delta(u-v)\end{aligned}$$ and $$\begin{aligned}
&&[J^+(u),J^-(v)]=i J^3(v)\delta(u-v)+4i\kappa\delta'(u-v), \nonumber\\
&& [J^3(u), J^3(v)]=-8i\kappa\delta'(u-v),\nonumber\\
&&[J^3(u), J^{\pm}(v)]=\pm 2iJ^{\pm}(v)\delta(u-v).\end{aligned}$$*]{}
As we mentioned above, even though we managed to define the commutator, based on the regularized commutators, this definition doesn’t provide a representation, since the correlation functions of $E, F, H$ or $J^3, J^{\pm}$ do not exist, if more than one of the arguments lies on a circle. Then it is clear that the space from which we started, i.e. $\mathcal{H}_K\otimes F_p$ or $\mathcal{H}_A\otimes F_p$ are not suitable to be spaces for $\widehat{sl(2,\mathbb{R})}$-module. However, we still have the regularized correlators which obey commutation relations. If we manage to eliminate divergencies in such a way that commutation relations and Hermicity conditions would be preserved, then the correlators will determine representation with the Hermitian bilinear form. In the next subsection we show a method, how to get rid of the divergencies and redefine the correlator, so that it is well-defined when all the arguments are on a circle.\
[**4.2. Renormalization of correlators and construction of representation: nonunitary representations.**]{} In this section we will renormalize the correlators with currents on a circle for the cases of $\mathcal{V}_A$ and $\mathcal{V}_K$ representations. In both cases the description is very similar, so we focus on $\mathcal{V}_K$ case and the correlators of $J^3, J^{\pm}$ currents. Generalization to $\mathcal{V}_A$ and $E,F, H$ goes along the same path.
In order to renormalize, at first we have to understand what kind of divergencies we are dealing with. For that purpose it is convenient to write down the expression for correlator in the graphic form using Feynman-like diagrams.
Recall, that the easiest way to compute the correlator for $\widehat{\mathcal{K}}$ is to reduce the whole expression to the normal ordered form, i.e. all the creation operators $b(z,\bar z)$ are moved to the left and all the annihilation operators $a(z,\bar z)$ to the right.
Once we move the creation operator $b(z,\bar z)$, which is a part of the certain generator at the point to the left (or the annihilation operator $a(z,\bar z)$ to the right) it may produce the following terms arising from the commutation with the $\alpha^{\pm}$-generator of $J^{\pm}$ currents on the right (on the left in the case of $a(z,\bar z)$) of the given generator: $$\begin{aligned}
\label{lines}
&&[a(z,\bar z), \alpha^{\pm}(w,\bar w)]=\mp \alpha^{\pm}(w,\bar w)\delta(z,w),\nonumber\\
&&[\alpha^{\pm}(w,\bar w), b(z,\bar z)]= \pm \alpha^{\pm}(z,\bar z)\delta(z,w),\end{aligned}$$ where $\delta(z,w)=\sum_{n\ge 0} (z\bar w)^n+\sum_{n> 0} (\bar zw)^n$. If $z,w$ are on the circle, i.e. $z=e^{iu}, w=e^{iv}$, we get $\delta(z,w)=\delta(u-v)$. We will depict every term of the form [(\[lines\])]{}, which we obtain during the normal ordering procedure, as a line from one vertex to another:
$$\begin{aligned}
\begin{xy}
(20,0)*+{\bullet}="2";
(20,5)*+{}="0";
(30,5)*+{\delta(z,w)}="1";
(40,0)*+{\bullet}="3";
"2";"3" **\dir{-}; ?(.65)*\dir{>};
\end{xy}\end{aligned}$$
Here the initial vertex corresponds to the term containing creation/annihilation operator and the terminal vertex correspond to the term containing $\alpha^{\pm}$. The direction of the arrow (to the right or to the left) indicates whether it was annihilation or creation operator.
At the same time, each generator $J^{\pm}$ contributes terms like that $$\begin{aligned}
\alpha^{\pm}(z,\bar z)a(z,\bar z) , \quad b(z,\bar z)\alpha^{\pm}(z,\bar z).\end{aligned}$$ They enter the diagram as vertices with one outgoing line to the right or to the left and any amount of incoming lines: $$\begin{aligned}
\begin{xy}
(0,0)*+{}="1";
(20,0)*+{\bullet}="2";
(20,3)*+{-}="-";
(20,5)*+{+}="+";
(40,0)*+{}="3";
(3,6)*+{}="1t";
(10,12)*+{}="1tt";
(3,-6)*+{}="1b";
(10,-12)*+{}="1bb";
"1";"2" **\dir{-}; ?(.65)*\dir{>};
{\ar@{->} "2";"3"};
"1t";"2" **\dir{-}; ?(.65)*\dir{>};
"1tt";"2" **\dir{-}; ?(.65)*\dir{>};
"1b";"2" **\dir{-}; ?(.65)*\dir{>};
"1bb";"2" **\dir{-}; ?(.65)*\dir{>};
(60,0)*+{}="4";
(80,0)*+{\bullet}="5";
(80,3)*+{-}="-";
(80,5)*+{+}="+";
(100,0)*+{}="6";
(97,6)*+{}="6t";
(90,12)*+{}="6tt";
(97,-6)*+{}="6b";
(90,-12)*+{}="6bb";
"5";"6" **\dir{-}; ?(.35)*\dir{<};
{\ar@{->} "5";"4"};
"5";"6t" **\dir{-}; ?(.35)*\dir{<};
"5";"6tt" **\dir{-}; ?(.35)*\dir{<};
"5";"6b" **\dir{-}; ?(.35)*\dir{<};
"5";"6bb" **\dir{-}; ?(.35)*\dir{<};
\end{xy}\end{aligned}$$ Signs $\pm$ over the vertices correspond to $\alpha^{\pm}$, while we neglect for simplicity the dependence on $z, \bar z$ variables. The incoming line in the vertex forms when during the normal ordering procedure the creation/annihilation operators from other vertices leave commutator term [(\[lines\])]{} with $\alpha^{\pm}$ at a given vertex. The outgoing line forms when the creation/annihilation operator of a given vertex leaves a commutator term $\alpha^{\pm}$ from another vertex.
On the other hand, $J^{\pm}$ also have terms of the form $$\begin{aligned}
\label{term}
\kappa{\partial}_u\alpha^{\pm}(z,\bar z), \quad \rho(z,\bar z)\alpha^{\pm}(z,\bar z)\end{aligned}$$ According to the strategy formulated, we denote the first term from [(\[term\])]{} as a terminal vertex for graphs, because it contains only $\alpha^{\pm}$:
$$\begin{aligned}
\begin{xy}
(60,0)*+{}="4";
(80,0)*+{\bullet}="5";
(80,3)*+{-}="-";
(80,5)*+{+}="+";
(100,0)*+{}="6";
(97,6)*+{}="6t";
(90,12)*+{}="6tt";
(97,-6)*+{}="6b";
(90,-12)*+{}="6bb";
"5";"6" **\dir{-}; ?(.35)*\dir{<};
"5";"6t" **\dir{-}; ?(.35)*\dir{<};
"5";"6tt" **\dir{-}; ?(.35)*\dir{<};
"5";"6b" **\dir{-}; ?(.35)*\dir{<};
"5";"6bb" **\dir{-}; ?(.35)*\dir{<};
\end{xy}\end{aligned}$$
However the second term from [(\[term\])]{} is composite: it has contribution from $\alpha^{\pm}(z,\bar z)$ and $\rho(z,\bar z)$. We consider this term as a terminal vertex for the graphs coming from the normal ordering of $a,b$-operators, however, we should add one outgoing “wavy” line, corresponding to the normal ordering in the Fock space:
$$\begin{aligned}
\begin{xy}
(60,0)*+{}="4";
(80,0)*+{\bullet}="5";
(80,3)*+{-}="-";
(80,5)*+{+}="+";
(100,0)*+{}="6";
(97,6)*+{}="6t";
(90,12)*+{}="6tt";
(97,-6)*+{}="6b";
(90,-12)*+{}="6bb";
{\ar@{~} "5";"4"};
"5";"6" **\dir{-}; ?(.35)*\dir{<};
"5";"6t" **\dir{-}; ?(.35)*\dir{<};
"5";"6tt" **\dir{-}; ?(.35)*\dir{<};
"5";"6b" **\dir{-}; ?(.35)*\dir{<};
"5";"6bb" **\dir{-}; ?(.35)*\dir{<};
\end{xy}\end{aligned}$$
Each wavy line produces the term from 2-point correlator [(\[2pointf\])]{} and may appear only once in the connected graph: it is so, because of the combinatorial formula, expressing the Fock space n-point correlator via 2-point correlators.
Finally, there are terms coming from $J^3$-generator, namely $$\begin{aligned}
&&\label{ab2}\frac{1}{2}a(z,\bar z), \quad \frac{1}{2}b(z,\bar z),\\
&&\label{terminalj3}2\kappa\alpha^-(z,\bar z){\partial}_u \alpha^+(z,\bar z)\end{aligned}$$ According to our conventions, we denote first two elements [(\[ab2\])]{} as initial vertices with one outgoing line to the right and to the left correspondingly: $$\begin{aligned}
\begin{xy}
(20,0)*+{\bullet}="2";
(20,5)*+{0}="0";
(40,0)*+{}="3";
"2";"3" **\dir{-}; ?(.65)*\dir{>};
(60,0)*+{}="4";
(80,0)*+{\bullet}="5";
(80,5)*+{0}="0";
"4";"5" **\dir{-}; ?(.35)*\dir{<};
\end{xy}\end{aligned}$$ Zeros over the vertices represent the absence of $\alpha^{\pm}$ contribution from these terms. Finally, [(\[terminalj3\])]{} will correspond to the terminal vertex, because it has neither creation or annihilation operators: $$\begin{aligned}
\begin{xy}
(60,0)*+{}="4";
(80,0)*+{\bullet}="5";
(80,5)*+{0}="0";
(100,0)*+{}="6";
(97,6)*+{}="6t";
(90,12)*+{}="6tt";
(97,-6)*+{}="6b";
(90,-12)*+{}="6bb";
"5";"6" **\dir{-}; ?(.35)*\dir{<};
"5";"6t" **\dir{-}; ?(.35)*\dir{<};
"5";"6tt" **\dir{-}; ?(.35)*\dir{<};
"5";"6b" **\dir{-}; ?(.35)*\dir{<};
"5";"6bb" **\dir{-}; ?(.35)*\dir{<};
\end{xy}\end{aligned}$$ However, in this exceptional case we will consider every line as a sum of two terms contributing $\delta(z,w)$, because there are two $\alpha$’s in this vertex corresponding to the term $\alpha^-(z,\bar z){\partial}_u\alpha^+(z,\bar z)$, so we consider contribution of the commutator with each of them. Moreover, formally on the circle (if such a limit exists), there can be only one incoming line into this vertex, because of the commutation relations: $$\begin{aligned}
[a(u), \alpha^+(v){\partial}_v \alpha^-(v)]=[b(u), \alpha^+(v){\partial}_v \alpha^-(v)]=
-\delta'(u-v).\end{aligned}$$ Here are two sample diagrams
$$\begin{aligned}
\begin{xy}
(0,12)*+{+}="A+";
(20,12)*+{-}="B-";
(40,12)*+{-}="C-";
(60,12)*+{+}="D+";
(80,12)*+{-}="E-";
(0,10)*+{\bullet}="A";
(20,10)*+{\bullet}="B";
{\ar@{->} "A";"B"};
(40,10)*+{\bullet}="C";
(60,10)*+{\bullet}="D";
{\ar@{->} "C";"B"};
{\ar@/_2pc/ "B";"D"};
{\ar@{->} "D";"C"};
(80,10)*+{\bullet}="E";
{\ar@/_2pc/ "E";"C"};
(100,10)*+{\bullet}="F";
(100,12)*+{+}="F+";
{\ar@{->} "F";"E"}
\end{xy}\nonumber\end{aligned}$$
$$\begin{aligned}
\begin{xy}
(0,12)*+{+}="A+";
(20,12)*+{-}="B-";
(40,12)*+{-}="C-";
(60,12)*+{+}="D+";
(80,12)*+{-}="E-";
(0,10)*+{\bullet}="A";
(20,10)*+{\bullet}="B";
{\ar@{->} "A";"B"};
(40,10)*+{\bullet}="C";
(60,10)*+{\bullet}="D";
{\ar@{~} "C";"B"};
{\ar@{->} "D";"C"};
(80,10)*+{\bullet}="E";
(100,10)*+{\bullet}="F";
(100,12)*+{+}="F+";
{\ar@/_2pc/ "E";"B"}
{\ar@{->} "F";"E"}
\end{xy}\nonumber\end{aligned}$$
which belong to the expansion of the 6-point correlator $$\begin{aligned}
\langle J^+(z_1,\bar z_1)J^-(z_2,\bar z_2)J^-(z_3,\bar z_3)J^+(z_4,\bar z_4)J^-(z_5,\bar z_5)J^+(z_6,\bar z_6)\rangle\end{aligned}$$
After it was described how all terms from the generators fit into graphs describing the correlator, we are ready to formulate the first important statement about such graphical representation.\
[**Proposition 4.3.**]{} [*Every connected graph contributing to the correlator $$\begin{aligned}
\label{corgen}
\langle \phi_1(z_1,\bar z_1)....\phi_n(z_n,\bar z_n)\rangle_p,\end{aligned}$$ where $\phi_i=J^3, J^{\pm}$, has at most one loop.*]{}\
[**Proof.**]{} In order to see that, one has to analyze the structure of the vertices. Namely, each vertex have possibly many incoming lines, but at most one outgoing line. Let us consider the graph with one loop. It means that all outgoing lines of the vertices in this loop are included in the loop and all other lines are incoming. Therefore none of these vertices can participate in a different loop, since all outgoing lines are included in the first loop. Let us assume that there is another loop with different set of vertices in the same graph. However, since for all vertices participating all outgoing lines are “circulating” inside the loop there is no possibility to make a connected graph out of two loops. $\blacksquare$\
However, we see that each loop diagram, though producing a well-defined expression for currents inside the circle, in the limit $|z_i|\to 1$ produce a divergence of the type $\delta(0)$, since every line in the loop produces delta-function. It is easy to see, say in the case of correlator $$\begin{aligned}
\langle J^+(z_1,\bar z_1)J^-(z_2,\bar z_2)\rangle_p\end{aligned}$$ When the creation operator from $J^-$ contributes the commutator term [(\[lines\])]{} with the exponent in $J^+$ and at the same time, annihilation operator from $J^-$ contributes the commutator term [(\[lines\])]{} with the exponent in $J^+$, the following diagram is produced: $$\begin{aligned}
\xymatrix{
\bullet \ar@/_1pc/[r] &
\bullet \ar@/_1pc/[l]}\end{aligned}$$\
and leads to the divergent term $\delta(u_1-u_2)\cdot\delta(u_1-u_2)$ when considered on the circle.
The simplest way to regularize correlators to preserve commutation relations is to throw away all the loop diagrams. This allows to make sense of the correlators on a circle.\
[**Theorem 4.2.a.**]{} [*Let us consider the renormalized correlators $$\begin{aligned}
\label{rencor}
\langle \phi_1(z_1,\bar z_1)....\phi_n(z_n,\bar z_n)\rangle_p^R,\end{aligned}$$ where $\phi_i=J^3, J^{\pm}$, which contain only the tree graph contributions to [(\[corgen\])]{}. Then the limit ${r_k\to 1}$ (where $z_k=r_ke^{iu_k}$) of [(\[rencor\])]{} exists as a distribution. Moreover, the commutation relation between $J^3, J^{\pm}$ under the renormalized correlator reproduces $\widehat{sl(2,\mathbb{R})}$. Therefore, correlators [(\[rencor\])]{} considered on a circle define a module for $\widehat{sl(2,\mathbb{R})}$ with a Hermitian bilinear form.*]{}\
[**Proof.**]{} To prove this theorem, it is enough to look at relevant contibutions for the commutators. It appears that they all come from initial/terminal vertices and their closest neighbors. For example, let us consider diagram of this sort: $$\begin{aligned}
\label{diag1}
\begin{xy}
(0,0)*+{}="a";
(20,0)*+{\bullet}="b";
(80,0)*+{\bullet}="c";
(100,0)*+{}="d";
(15,3)*+{+}="+";
(75,3)*+{-}="-";
"a";"b" **\dir{-}; ?(.45)*\dir{<};
"b";"c" **\dir{-}; ?(.45)*\dir{<};
"c";"d" **\dir{-}; ?(.45)*\dir{<};
(14,16)*+{}="b1";
(20,16)*+{}="b2";
(26,16)*+{}="b3";
"b1";"b" **\dir{-}; ?(.75)*\dir{>};
"b2";"b" **\dir{-}; ?(.75)*\dir{>};
"b3";"b" **\dir{-}; ?(.75)*\dir{>};
(20,20)*+{A}="A";
(20,20)*\xycircle(10,5){};
(70,20)="e1";
(80,20)*+{B}="B";
(90,20)="e3";
(80,20)*\xycircle(10,5){};
(72,17)*+{}="c1";
(77,15)*+{}="c2";
(83,15)*+{}="c3";
(88,17)*+{}="c4";
"c1";"c" **\dir{-}; ?(.75)*\dir{>};
"c2";"c" **\dir{-}; ?(.72)*\dir{>};
"c3";"c" **\dir{-}; ?(.72)*\dir{>};
"c4";"c" **\dir{-}; ?(.75)*\dir{>};
\end{xy}\end{aligned}$$\
corresponding to the correlator $$\begin{aligned}
\label{diag2}
\langle\dots J^+(z_1,\bar z_1)J^-(z_2,\bar z_2)\dots\rangle_p.\end{aligned}$$ There is a diagram with equal contribution $$\begin{aligned}
\label{diag2'}
\begin{xy}
(0,0)*+{}="a";
(20,0)*+{\bullet}="b";
(80,0)*+{\bullet}="c";
(100,0)*+{}="d";
(15,3)*+{-}="-";
(75,3)*+{+}="+";
"a";"b" **\dir{-}; ?(.45)*\dir{<};
"b";"c" **\dir{-}; ?(.45)*\dir{<};
"c";"d" **\dir{-}; ?(.45)*\dir{<};
(12,17)*+{}="b1";
(17,15)*+{}="b2";
(23,15)*+{}="b3";
(28,17)*+{}="b4";
"b1";"b" **\dir{-}; ?(.75)*\dir{>};
"b2";"b" **\dir{-}; ?(.72)*\dir{>};
"b3";"b" **\dir{-}; ?(.72)*\dir{>};
"b4";"b" **\dir{-}; ?(.75)*\dir{>};
(20,20)*+{B}="B";
(20,20)*\xycircle(10,5){};
(70,20)="e1";
(80,20)*+{A}="A";
(90,20)="e3";
(80,20)*\xycircle(10,5){};
(74,16)*+{}="c1";
(80,16)*+{}="c2";
(86,16)*+{}="c3";
"c1";"c" **\dir{-}; ?(.75)*\dir{>};
"c2";"c" **\dir{-}; ?(.75)*\dir{>};
"c3";"c" **\dir{-}; ?(.75)*\dir{>}
\end{xy}\end{aligned}$$\
from the correlator $$\begin{aligned}
\langle\dots J^-(z_2,\bar z_2)J^+(z_1,\bar z_1)\dots\rangle_p.\end{aligned}$$ Therefore, they cancel each other when we take a commutator of $J^+(u_1), J^-(u_2)$, since the middle line produces $\delta(u_1-u_2)$. One can understand it in the following way: the commutator $[J^+(u_1), J^-(u_2)]$ involves $J^3$-term and $\delta'$-term, $J^3$-term contains only initial and terminal vertices, therefore the diagrams, which contribute to this commutator should have also $J^+$ or $J^-$ as initial or terminal vertices. As for the commutators of $J^3$ with $J^{\pm}$, the picture above indicates that only terminal/initial vertices and their closest neighbors contribute, because again, all terms in $J^3$ are depicted as initial/terminal vertices. Therefore, it is clear that one can cancel all loop contributions to the correlators and still the commutation relations of Theorem 4.1.will be valid. However, if all the loops are eliminated, one can consider the limit $|z_i|\to 1$, putting all currents on a circle, since all divergent graphs are gone. Therefore, we have the well-defined correlators of the generators of $\widehat{sl(2,\mathbb{R})}$ Lie algebra. One can see that the these correlators define a Hermitian bilinear form, because after the conjugation one can obtain one-to-one correspondence between tree graphs in conjugated correlators. $\blacksquare$\
It appears that the theorem above can be generalized: instead of eliminating loops completely, one can renormalize them, i.e. eliminate the divergence of the type $\delta(0)$. Namely, one can associate a real number $\mu_k$ with every loop which is the contribution of $k$ $J^{\pm}$-currents at the points $z_1,..., z_k$. This number $\mu_k$ enters the loop in the following way. Suppose currents $J^{\pm}(z_i)$ appear in the correlator in the given order from left to right. Our loop contains the following term $$\begin{aligned}
\label{delta1}
\delta (z_1,z_2)\cdot\delta(z_2,z_3)\dots\delta(z_{k-1}, z_k)\cdot \delta(z_{k}, z_1),\end{aligned}$$ where $\delta (z,w)=\sum_{n\ge 0}(z\bar w)^n+\sum_{n< 0}(w\bar z)^n$. In the limit $|z_k|\to 1$ we will substitute it with the expression $$\begin{aligned}
\label{delta2}
\mu_k\cdot \delta(u_1-u_2)\cdot\delta(u_2-u_3)\dots\delta(u_{k-1}-u_k),\end{aligned}$$ where we remind that $z_k=r_ke^{iu_k}$. This procedure again does affect neither commutation relations inside the correlator nor Hermicity condition for the resulting bilinear form, because loops do not participate in commutation relations. Therefore, we have a new version of the theorem above.\
[**Theorem 4.2.b.**]{} [*Let us consider renormalized correlators $$\begin{aligned}
\label{rencormu}
\langle \phi_1(z_1,\bar z_1)....\phi_n(z_n,\bar z_n)\rangle_p^{R, \{\mu_n\}},\end{aligned}$$ where $\phi_i=J^3, J^{\pm}$, such that the loop contributions to [(\[corgen\])]{} are replaced by their renormalized analogues with the family of arbitrary real parameters $\{\mu_n\}$. Then the limit ${r_k\to 1}$ (where $z_k=r_ke^{iu_k}$) of [(\[rencormu\])]{} exists and the commutation relations between $J^3, J^{\pm}$ under the renormalized correlator reproduce Lie algebra $\widehat{sl(2,\mathbb{R})}$, and correlators [(\[rencormu\])]{} considered on a circle define a module for $\widehat{sl(2,\mathbb{R})}$ with a Hermitian bilinear form.*]{}\
[**Proof.**]{} The proof goes along the same lines as in the part $a$ of the theorem. When we consider the loop contribution to the commutator, there will be two loop diagrams, each emerging from one of the terms precisely as in the pictures [(\[diag1\])]{}, [(\[diag2\])]{} which again will cancel each other, because of the delta function line between two vertices and because there always is one incoming line and outgoing line for each vertex associated with the currents, participating in the commutator in these two terms. Therefore, the commutation relations are unaffected in the presence of loops. It is easy to see that the same applies to the Hermicity condition. $\blacksquare$\
[**4.3. Renormalization of correlators and construction of representations: unitary representations.**]{} In the previous section, we have shown how to construct a module with a Hermitian pairing for $\widehat{sl(2,\mathbb{R})}$, starting from the nonunitary representations $\mathcal{V}_K$ of $\widehat{\mathcal{K}}$. In this section, we will do the same with the use of unitary module $D^l_K$, which is a dense subset in $H_K$. Needless to say, similar procedure will apply in the case of $\widehat{\mathcal{A}}$ and ${D^l_A}$. There is a major difference between the nonunitary and unitary case. The creation and annihilation operators $a_n$ and $b_n$ do not commute in this case, namely $$\begin{aligned}
[a(z,\bar z), b(w,\bar w)]=D(z, w), \quad D(z,w)=\sum_{n>0}\xi_n^{-1}(z^n\bar w^n+\bar z^n w^n).\end{aligned}$$ On a circle it is not a well-defined distribution, because of the behavior of $\xi_n$; however, it is well-defined on trigonometric polynomials and this is what we need to define the correlator of the modes of $J^{\pm}$. Inside the circle, however, we can choose regularization parameters $z,w$ in such a way that $D(z,w)$ converges. When we compute the correlator $$\begin{aligned}
\label{corgen2}
\langle \phi_1(z_1,\bar z_1)....\phi_n(z_n,\bar z_n)\rangle_p,\end{aligned}$$ which is now based on tensor product of Fock module and $D^l_K$, if $D(z,w)$ appears after we interchange $a, b$ operators, we indicate that with a dotted line. In other words, all the terms which contributed to vertices with solid outgoing lines in the case of nonunitary module, contribute also as vertices with outgoing dotted line and incoming solid lines. So, this is a set of new vertices for $J^{\pm}$ currents: $$\begin{aligned}
\label{dotted}
\begin{xy}
(0,0)*+{}="1";
(20,0)*+{\bullet}="2";
(20,3)*+{-}="-";
(20,5)*+{+}="+";
(40,0)*+{}="3";
(3,6)*+{}="1t";
(10,12)*+{}="1tt";
(3,-6)*+{}="1b";
(10,-12)*+{}="1bb";
"1";"2" **\dir{-}; ?(.65)*\dir{>};
{\ar@{--} "2";"3"};
"1t";"2" **\dir{-}; ?(.65)*\dir{>};
"1tt";"2" **\dir{-}; ?(.65)*\dir{>};
"1b";"2" **\dir{-}; ?(.65)*\dir{>};
"1bb";"2" **\dir{-}; ?(.65)*\dir{>};
(60,0)*+{}="4";
(80,0)*+{\bullet}="5";
(80,3)*+{-}="-";
(80,5)*+{+}="+";
(100,0)*+{}="6";
(97,6)*+{}="6t";
(90,12)*+{}="6tt";
(97,-6)*+{}="6b";
(90,-12)*+{}="6bb";
"5";"6" **\dir{-}; ?(.35)*\dir{<};
{\ar@{--} "5";"4"};
"5";"6t" **\dir{-}; ?(.35)*\dir{<};
"5";"6tt" **\dir{-}; ?(.35)*\dir{<};
"5";"6b" **\dir{-}; ?(.35)*\dir{<};
"5";"6bb" **\dir{-}; ?(.35)*\dir{<};
\end{xy}\end{aligned}$$ and $J^3$ current: $$\begin{aligned}
\begin{xy}
(20,0)*+{\bullet}="2";
(20,5)*+{0}="0";
(40,0)*+{}="3";
"2";"3" **@{--};
(60,0)*+{}="4";
(80,0)*+{\bullet}="5";
(80,5)*+{0}="0";
"4";"5" **@{--};
\end{xy}\end{aligned}$$ One can generalize the results of the Theorem 4.2 in the case of the unitary modules. The renormalization procedure remains the same with arbitrary parameters $\mu_n$, except that in this case we will have some extra tree graphs containing dotted lines, which may lead to divergencies, which we have to eliminate.\
[**Theorem 4.3.**]{} [*Let us consider renormalized correlators $$\begin{aligned}
\label{rencor2}
\langle \phi_1(z_1,\bar z_1)....\phi_n(z_n,\bar z_n)\rangle_p^{R, \{\mu_n\}}.\end{aligned}$$ Here $\phi_i=J^3, J^{\pm}$. The loop contributions to [(\[corgen2\])]{} are replaced by their renormalized analogues with a family of arbitrary real parameters $\{\mu_n\}$. The tree graphs, containing dotted lines are eliminated unless there is equal number of plus and minus vertices on one side of the dotted line [(\[dotted\])]{}. Then the limit ${r_k\to 1}$ (where $z_k=r_ke^{iu_k}$) of [(\[rencor\])]{} exists and the commutation relations between $J^3, J^{\pm}$ under the renormalized correlator reproduce Lie algebra $\widehat{sl(2,\mathbb{R})}$. The correlators [(\[rencor2\])]{} considered on a circle define the module for $\widehat{sl(2,\mathbb{R})}$ with the Hermitian bilinear form.*]{}\
[**Proof.** ]{}First, we make the following useful observation: there can be only one dotted line in the connected graph, and there is no loops in that graph. Alas, in most of cases these diagrams diverge on a circle because of the correlator between $\alpha^+,\alpha^-$, which, combined with $D(e^{iu},e^{iv})$ leads to divergence due to the product with $N_K(u,v)$. The basic example of such sort is the 2-point correlator between $J^+$ and $J^-$. Therefore, we will eliminate all of such graphs containing dotted lines so that the commutation relations remain intact.
However, there is still a small amount of these diagrams, which leads to finite expressions and which is important for keeping the commutation relations. In order to describe them, let us separate vertices into two groups: those connected through solid path with first vertex with outgoing dotted line and those connected with another one. Note that if the number of pluses and minuses on vertices in any of these two groups is equal, then the expression corresponding to such a graph is well defined. Actually, every solid line gives a $\delta$-function, and since there is an equal number of $\alpha^+, \alpha^-$ on one side, then there will be no divergence arising from $N_K(u,v)$ and $D(e^{iu},e^{iv})$ as described above. All these graphs contribute to commutation relation between $J^+$ and $J^-$, which leads to emergence of $J^3$ current and the resulting graph with terminal/initial dotted line, involving $J^3$.
In order to renormalize graphs with loops, one can follow the same route as in the previous subsection. In other words, one can renormalize all the loop diagrams in a similar way, using the formulas [(\[delta1\])]{}, [(\[delta2\])]{}. $\blacksquare$.
[25]{} G. Da Prato, [*An Introduction to Infinite-Dimensional Analysis*]{}, Springer, 2006. P.I. Etingof, I.B. Frenkel, A.A. Kirillov, Jr., [*Lectures on representation theory and Knizhnik-Zamolodchikov equations*]{}, Providence, USA, Am. Math. Soc. (1998). L. Faddeev, [*Modular Double of Quantum Group*]{}, Math.Phys.Stud.21:149-156, 2000. E. Frenkel, D. Ben-Zvi, [*Vertex algebras and algebraic Curves*]{}, Math. Surveys and Monographs [**88**]{}, AMS, 2004. I.B. Frenkel, I.C-H. Ip, [*Positive representations of split real quantum groups and future perspectives*]{}, arXiv:1111.1033. I.B. Frenkel, Y. Zhu, [*Vertex operator algebras associated to representations of affine and Virasoro algebras*]{}, Duke Math. J. 66 (1992) 123-168. I.M. Gelfand, M.I. Graev, A.M. Vershik, [*Representations of the group $SL(2,\mathbf{R})$, where $\mathbf{R}$ is a ring of functions*]{}, Russ. Math. Surv. [**28**]{} (1973) 87-132. A. Gerasimov, S. Kharchev, D. Lebedev, [*Representation theory and quantum integrability*]{}, Progress in Mathematics [**237**]{} (2005) 133-156. A. Gerasimov, S. Kharchev, D. Lebedev, S. Oblezin, [*On a Class of Representations of Quantum Groups*]{}, arXiv:math/0501473. A. Gerasimov, A. Marshakov, A. Morozov, [*Hamiltonian reduction of the Wess-Zumino-Witten theory from the point of view of bosonization*]{}, Phys.Lett. [**B**]{} 236 (1990) 269-272. I.C.H. Ip, [*The classical limit of representation theory of the quantum plane*]{}, arXiv:1012.4145. I.C.H. Ip, [*Positive Representations of Split Real Simply-laced Quantum Groups*]{}, arXiv:1203.2018. I.C.H. Ip, [*Positive representations of split real quantum groups of type $B_n$, $C_n$, $F_4$, and $G_2$* ]{}, arXiv:1205.2940. R. M. Kashaev, A. Yu. Volkov, [*From the Tetrahedron Equation to Universal R-Matrices*]{}, arXiv:math/9812017 D. Kazhdan, G. Lusztig, [*Tensor structures arising from affine Lie algebras I,II*]{}, J. Amer. Math. Soc. [**6**]{} (1993) 905-1011; [*Tensor structures arising from affine Lie algebras III,IV*]{}, J. Amer. Math. Soc. [**7**]{} (1994) 335-453. S. Kharchev, D. Lebedev, M. Semenov-Tian-Shansky, [*Unitary representation of $U_q(sl(2))$, the modular double and multiparticle q-deformed Toda chains*]{}, Commun. Math. Phys. [**225**]{} (2002) 573-609. H.-H. Kuo, [*Gaussian measures in Banach spaces*]{}, Springer-Verlag, 1975. B. Ponsot, J. Teschner, [*Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of $U_q(sl(2,\mathbb{R}))$*]{}, Commun.Math.Phys. [**224**]{} (2001) 613-655. B. Ponsot, J. Teschner, [*Liouville bootstrap via harmonic analysis on a noncompact quantum group*]{}, hep-th/9911110. G. Moore, N. Seiberg, [*Classical and quantum conformal field theory*]{}, Commun. Math. Phys. [**123**]{} (1989) 177-254. K. Schmudgen, [*Operator representations of $U_q(sl(2,\mathbb{R})$*]{}, Lett. Math. Physics [**37**]{} (1996) 211-222. J. Teschner, [*A lecture on the Liouville vertex operators*]{}, Int.J.Mod.Phys. A19S2 (2004) 436-458. A.M. Zeitlin, [*Unitary representations of a loop ax+b-group, Wiener measure and Gamma-function*]{}, J. Func. Anal. [**263**]{} (2012) 529-548, arXiv:1012.4826.
[^1]: As we will see below, in the case when two or more currents $\phi_i$ have their arguments lying on the unit circle the correlator diverges. Our aim will be to eliminate these divergencies consistently with the algebraic structure.
[^2]: Again, we draw attention that these formulas are valid and well-defined only under the correlator as in Proposition 4.2.
|
---
abstract: 'Under specific experimental circumstances, sputter erosion on semiconductor materials exhibits highly ordered hexagonal dot-like nanostructures. In a recent attempt to theoretically understand this pattern forming process, Facsko et al. \[Phys. Rev. B [**69**]{}, 153412 (2004)\] suggested a nonlocal, damped Kuramoto-Sivashinsky equation as a potential candidate for an adequate continuum model of this self-organizing process. In this study we theoretically investigate this proposal by (i) formally deriving such a nonlocal equation as minimal model from balance considerations, (ii) showing that it can be exactly mapped to a local, damped Kuramoto-Sivashinsky equation, and (iii) inspecting the consequences of the resulting non-stationary erosion dynamics.'
author:
- Sebastian Vogel
- 'Stefan J. Linz'
title: |
On continuum modeling of sputter erosion under normal incidence:\
interplay between nonlocality and nonlinearity
---
Introduction
============
Sputter erosion [@carter], the bombardment of solid target surfaces with ionized particles to remove or to detach target material, has a long tradition in physics as an experimental technique to clean, smooth, or appropriately prepare solid surfaces. The preparation of on the nanoscale perfectly flat surfaces, however, does not seem to be possible. This is because various surface roughening and smoothing processes compete during the sputtering process and can lead to self-organized pattern formation of the eroded surface morphology. Depending on the target material, temperature, ion beam energy, angle of incidence and various other parameters, one generically observes the development of rough surfaces, cellular patterns and, in particular under oblique incidence of the ion beam, the formation of more or less regular ripple patterns. For an overview on recent experimental results and theoretical approaches using continuum modeling we refer to Makeev et al. [@makeev] and Valbusa et al. [@valbusa].
During the last five years, however, spectacularly novel experimental results have been reported by Facsko et al [@facsko1; @facsko2; @facsko3; @facsko4] showing that GaSb and InSb semiconductor targets eroded by Ar$^+$ ions under normal incidence can develop into a rather well ordered surface morphology with basically hexagonally arranged dot structures. Similar results have been subsequently reported by Gago et al[@gago] for Si targets under normal incidence and, more generally, by Frost et al. [@frost1; @frost2; @frost3] for rotated InP, InSb and GaSb targets under oblique incidence (where as function of the inclination angle a variety of other patterns have also been observed). To theoretically explain the hexagonal ordering and taking advantage of earlier work by Elder et al. [@elder1; @elder2] (cf. also an even earlier study[@siva1] on this subject), Facsko et al. [@facsko-t] have recently suggested a (stochastically extended) stabilized or damped Kuramoto-Sivashinsky (KS) equation for the dynamics of the surface height $H$. This equation is given by $$\partial_t H= -v_0-\alpha H +\nu \nabla^2 H-D_{\rm eff}\nabla^4 H
+\frac{\lambda}{2}(\nabla H)^2+ \eta
\label{skse}$$ and might be considered as a useful continuum model for the ion-beam erosion under normal incidence since it can successfully reproduce the experimentally observed hexagonally ordered dot-type structures. The physical origin of the six terms on the rhs of Eq.(\[skse\]) is attributed to constant erosion velocity $v_0$, dissipation, effective surface tension (Bradley-Harper mechanism [@bradley]), thermal and erosion induced diffusion, tilt dependent sputtering yield, and some stochasticity of the erosion process, respectively (cf. the original study [@facsko-t] for details). Previous attempts[@facsko2; @kahng] based on the (standard) KS equation, i.e. without the term $-\alpha H$, have reproduced cellular patterns without convincing evidence of a regular hexagonal ordering. As noted by Facsko et al. at the end of their paper [@facsko-t], there are two basic problems with the applicability of Eq.(\[skse\]) to erosion processes since, as Eq.(\[skse\]) stands, it violates the translation invariance in the erosion direction and, moreover, the physical meaning of the dissipation term $-\alpha H$ is not directly obvious. Facsko et al. [@facsko-t] briefly argued that (i) translational invariance can be restored by assuming that $-\alpha H$ has to be replaced by $-\alpha (H-\overline{H})$ with $\overline{H}$ being the erosion depth averaged over the sample area and (ii) that the term $-\alpha (H-\overline{H})$ might be interpreted as an approximation of a newly suggested redeposition effect of the sputtered target particles. Due to the nonlocal character of this term, however, the systematic connection between this nonlocally extended damped KS equation and Eq.(\[skse\]) is far from being obvious and the main reason for our paper.
First, we argue on general grounds, by using balance considerations, symmetries and allowing for nonlocal dependencies, what the simplest functional form of the spatio-temporal dynamics of the morphology of ion sputtered surfaces under [*normal*]{} incidence of the ions might be, if nonlocal terms as suggested by Facsko et al. [@facsko-t] are taken into account. Second, we show that the resulting non-local model equation can be mapped to the damped KS equation by the use of a temporally nonlocal transformation. By that we show how the stabilized KS equation suggested by Facsko et al. [@facsko-t] systematically fits into the theoretical framework of a continuum description of eroded surfaces obeying all desired invariances. Third, we discuss in detail the subtle interplay between potentially nonlocal terms in the evolution equation for the eroding surfaces and the role of the KS-nonlinearity $(\nabla H)^2$. Finally, we also investigate some consequences on the general evolution dynamics in these equations.
Balance equation considerations
===============================
The starting point of our investigation is the general, stochastically extended balance equation for the spatio-temporal evolution of the eroded surface morphology $H({\bf x},t)$ measured perpendicularly to the initially flat target surface with coordinates ${\bf x}=(x,y)$. Assuming that the particle density at the surface of the target material can be basically considered as being constant [@linz1; @linz2], such a balance equation reads quite generally [@bs]
$$\partial_t H={\nabla}\cdot{\bf J}_H + F +\eta.
\label{eq1}$$
and expresses the fact that temporal changes of the erosion depth $H({\bf x},t)$ arise from two main contributions: (i) the detachment of target material leading to an in general inhomogenous and non-stationary, appropriately rescaled flux of eroded particles $F$ (given by the number of eroded particles per time and surface area divided by the particle density) and (ii) the local rearrangements of the particles at the surface leading to a relaxational current ${\bf J}_H$ along the surface. Note that this balance only accounts for the target particles. The underlying driving of the erosion process, the flux of eroding ions $I$, enters indirectly into such a description: all terms on the right hand side of (\[eq1\]) depend on $I$ in a way that they vanish if the sputtering process is turned off, $I=0$. Note that thermally activated processes that do not depend on $I$ and might lead to a further smoothing of the surface after the sputtering has been stopped are not taken into account. The spatio-temporal fluctuations $\eta=\eta({\bf x},t)$ entering in (\[eq1\]) mimic some stochasticity present in the erosion process and are usually assumed to be Gaussian white, i.e. having an average $\langle\eta({\bf x},t)\rangle_\eta=0$ and a covariance $\langle\eta({\bf x},t)\eta({\bf x^\prime},t^\prime)\rangle_\eta
=2 D \delta({\bf x}-{\bf x^\prime})\delta({t}-{t^\prime})$.
Considering periodic boundary conditions on an appropriately chosen, large enough sample area of the size $L^2$ and introducing the spatial average $\stackrel{-}{..}(t)=
(1/L^2)\int_{-L/2}^{L/2}dx \int_{-L/2}^{L/2} dy ...$ (being generally not equivalent to a stochastic average if taken on a finite area), the evolution of the mean erosion depth $\overline H(t)$ develops according to $\partial_t\overline H(t)=\overline F+\overline \eta $ which directly leads to $$\overline H(t)-\overline H(0)=\int_0^t\overline F\; dt^\prime
+\int_0^t\overline \eta\; dt^\prime$$ with $\overline H(0)$ being the spatial average of the initial condition $H({\bf x},0)$ (which is usually set to zero under the assumption of an initially flat target surface). Already at this stage it is clear that the mean erosion depth $\overline{H}(t)$ is, in general, not a linear function in $t$ implying a constant erosion velocity, but a rather complicated function that integrates over the history of the stochasticity and of the eroded flux.
In order to specify the admissible functional forms of the right hand side of Eq.(\[eq1\]), we reconsider the three fundamental symmetry requirements that are considered to be basic for the spatio-temporal evolution of surface morphologies [@bs]: (i) no dependence of (\[eq1\]) on the specific choice of the origin of time implying invariance of (\[eq1\]) under translation in time, (ii) no dependence of (\[eq1\]) on the specific choice of the origin of the coordinate system implying invariance of (\[eq1\]) under translation in the direction perpendicular to the erosion direction, and (iii) no dependence of (\[eq1\]) on the specific choice of the origin of the $H$-axis implying invariance of (\[eq1\]) under translation in growth direction. These symmetry requirements exclude any [*explicit*]{} dependence of ${\nabla}\cdot{\bf J}_H$ and $F$ on the time $t$, the spatial position ${\bf x}$ and the erosion depth $H$, respectively. Following Facsko et al.’s argument [@facsko-t], however, an implicite functional dependence on $H-\overline H$ and on the spatial derivatives of $H$ is still admissable. Consequently, the detachment contribution in (\[eq1\]) is quite generally given by $F[\nabla H, H-\overline H]$ where $[\nabla H, ..]$ is the short hand notation for any derivative or combination of derivatives of $H$ being compatible with the scalar character of $F$ or $H$.
To proceed we (i) additionally apply invariance under rotation and reflection in the plane perpendicular to the erosion direction (which is suggested by the experimentally observed amorphorization of the target surface due to the erosion), (ii) expand the height dependent term $f_\beta$ of $F$ in a power series in $H-\overline H$, i.e. $f_\beta=\sum_n \beta_n (H-\overline H)^n$, and (iii) perform a gradient expansion of the contribution $f_\alpha$ of $F$ that solely contains derivatives of $H$. The latter implies that the lowest order terms (up to forth order in $\nabla$ and second order in $H$) are given by $\nabla^2 H$, $(\nabla H)^2$, $\nabla^4 H$, $\nabla^2(\nabla^2 H) $, $(\nabla^2 H)^2$), and $\nabla \cdot [(\nabla H)(\nabla^2 H)]$. Keeping only the lowest order contributions, the erosion flux is determined by $$\begin{aligned}
F&=& F_0 \left[ 1 + \beta_1 (H-\overline H)+\alpha_1 \nabla^2 H+\alpha_2
(\nabla H)^2\right]\nonumber\\
&\;& +\, O\left(\nabla^4,H^2,(H-\overline H)^2\right)
\label{trafo1}\end{aligned}$$ with $F_0$ being a constant and negative since this part of the flux is antiparallel to the erosion direction. The lowest order possible mixing term between $\nabla H$ and $H-\overline H$ that could appear in (\[trafo1\]) is given by $(H-\overline H)\nabla^2 H$ and will be omitted since it can be considered as a higher order contribution to the term proportional to $\alpha_1$ or $\beta_1$ in (\[trafo1\]).
The functional form of the term representing the relaxational currents at the surface, ${\nabla}\cdot{\bf J}_H$ in (\[eq1\]) remains to be specified. Following Facsko et al. [@facsko-t], we suppose ${\nabla}\cdot{\bf J}_H=-D_{\rm eff}\nabla^4 H$. This mimics the tendency of surface particles to reach energetically more favorable positions with a positive curvature $\nabla^2 H>0$ and, therefore, leads to a current ${\bf J}_H\propto \nabla(\nabla^2 H)$.
Combining all ingredients and renaming the entering coefficients $b=\beta_1 F_0$, $a_1=\alpha_1 F_0$, $a_2=-D_{\rm eff}$, and $a_3=\alpha_2 F_0$, yields a minimal functional form for the evolution of the erosion depth under the afore-mentioned restrictions that is given by $$\begin{aligned}
\partial_t H&=& a_1 \nabla^2 H+a_2\nabla^4 H+a_3(\nabla H)^2\nonumber \\
&\,& + b (H-\overline H)+F_0+ \eta.
\label{trafo2}\end{aligned}$$ The functional form of Eq.(\[trafo2\]) looks like the equation suggested by Facsko et al[@facsko-t] that follows after their ad-hoc replacement of $-\alpha H$ by $- \alpha(H-\overline H)$ in the damped KS equation. Note however, that the spatially constant part of the flux $F$, i.e. $F_0$, is, in general, not given by the mean constant erosion velocity since the latter is not constant during the course of the evolution.
Next we transform Eq.(\[trafo2\]) into a coordinate system that moves with the mean erosion depth $\overline H(t)$, $$h({\bf x},t)=H({\bf x},t)-\overline H(t)
\label{trafo3}$$ with $h({\bf x},t)$ being the local erosion profile that obviously fulfills $\overline h=0$ and $\partial_t\overline h=0$ for all times $t$. The transformation (\[trafo3\]) also guarantees that the invariance under translation in erosion direction $H\rightarrow H+z$ with $z$ being an arbitrary constant (that only needs to hold in the coordinate system that is fixed in space) holds for [*any*]{} erosion profile $h({\bf x},t)$. As a consequence, the evolution dynamics for $h({\bf x},t)$ does not necessarily need to fulfill this requirement.
Using Eq.(\[trafo2\]) and the facts that $\partial_t H=\partial_t \overline H +\partial_t h$ and that any terms consisting only of spatial derivatives of $H$ have the same functional form in the comoving system when $H$ is substituted by $\overline H+h$, the evolution equation for the erosion process in the comoving frame reads explicitly $$\partial_t h= b h+a_1 \nabla^2 h+a_2\nabla^4 h+a_3(\nabla h)^2
+F_0-\partial_t\overline H+ \eta
\label{trafo4}$$ Eq.(\[trafo4\]) still constitutes a stochastic integro-partial differential equation by virtue of the nonlocal term $\partial_t\overline H$. Only if $\partial_t\overline H=F_0$ or, equivalently, the mean erosion depth $\overline H(t)$ were moving with a [*constant*]{} speed for all times, Eq.(\[trafo4\]) would reduce to the damped stochastic KS equation. In general, however, this cannot be invoked, in particular because of the presence of the KS-type nonlinearity $(\nabla h)^2$. By taking the spatial average of Eq.(\[trafo4\]), $a_3 \overline{(\nabla
h)^2}+F_0-\partial_t\overline H+\overline\eta=0$, one can directly connect the mean erosion depth $\overline H(t)$ to the dynamics of the local erosion profile $h({\bf x},t)$, $$\overline H(t)= F_0 t + \int_0^t dt^\prime \,\left[a_3\overline{(\nabla h)^2}
+\overline\eta\right]
\label{trafo5}$$ where $\overline H(0)=0$ has been assumed. Consider first the deterministic part of (\[trafo5\]), i.e. with $\overline\eta=0$. Since the integrand of the second term on the right hand side of Eq.(\[trafo5\]) is positive for all times, the mean erosion depth $\overline H(t)$ systematically deviates from a linear time evolution given by $F_0 t$. For $a_3$ being negative (positive) the mean erosion depth is therefore retarded (advanced) in comparison to $F_0 t$. Noteworthy, this fact is not a specific property triggered by the nonlocal term in Eq.(\[trafo2\]), it is already present in the standard KS equation and from related studies there[@linz1], it is known that the second term of the right hand side of (\[trafo5\]) significantly contributes to the time evolution of $\overline H(t)$. The additional impact of the term $\int_0^t dt^\prime\overline\eta$ is roughly that of a superposed Wiener process since $\eta$ has been assumed to be a Gaussian and white.
Transformation to a local equation
==================================
As explained in the previous section, it is not obvious how the nonlocal equations (\[trafo2\]) and (\[trafo4\]) are related to the damped KS equation. Here we show in a more general context how the nonlocal term can be elimated by an appropriate transformation. Specifically our statement is as follows: There is a transformation $h\rightarrow \hat h$ that maps the general form of a [*nonlocal*]{} stochastic evolution equation given by $$\partial_t h=b h + G[\nabla h]+ F_0- \partial_t \overline H +\eta
\label{eqta}$$ to a [*local*]{} evolution equation in the transformed variables $\hat h$ reading $$\partial_t \hat h=b \hat h + G[\nabla \hat h]+\eta
\label{eqtb}$$ where, quite generally, the functional $G$ in the Eqs.(\[eqta\]) and (\[eqtb\]) can contain any combination of derivatives of $h$ or $\hat h$, respectively, but no explicit dependence on $h$ or $\hat h$.
As subsequently useful observation, we note that Eq.(\[eqtb\]) is invariant under the transformation $\hat h\rightarrow \hat h+z\,\exp(bt)$ with $z$ being an arbitrary constant. Obviously, (\[eqta\]) contains the model equation (\[trafo4\]) as special case if we set $G[\nabla h]=a_1 \nabla^2 h+a_2\nabla^4 h+a_3(\nabla h)^2$.
To the constructive derivation of the statement: In order to find the desired transformation we use the ansatz $$h\ \rightarrow \hat h=h + \frac{F_0}{b} +f(t)
\label{eqtc}$$ where the time-dependent function $f(t)$ is so far arbitrary and will be subsequently determined. Since $\partial_t h= \partial_t \hat h-\partial_t f$ and spatial derivatives of $h$ transform without any change to spatial derivatives in $\hat h$, insertion of (\[eqtc\]) into Eq.(\[eqta\]) yields $$\partial_t \hat h=b \hat h + G[\nabla \hat h]+\eta +\partial_t f-bf-
\partial_t \overline H.
\label{eqtd}$$ In order to arrive from Eq.(\[eqta\]) at Eq.(\[eqtb\]), one has to demand that the last three terms on the right hand side of Eq.(\[eqtd\]) vanish at any time, $$\partial_t f=bf+\partial_t \overline H.
\label{eqte}$$ The linear nonautonomous ordinary differential equation (\[eqte\]) can be solved for $f(t)$ by standard means, e.g. by the method of variation of constants. Introducing $f(t)=c(t) \exp(bt)$ and solving the resulting equation $\partial_t c=\exp(-bt)\partial_t \overline H$ yields $$f(t)={\rm e}^{bt}\left[c(0)+\int_0^t dt^\prime\,{\rm e}^{-bt^\prime}
\partial_{t^\prime}
\overline H(t^\prime)\right].
\label{eqtf}$$ In general, the integration constant $c(0)$ in (\[eqtf\]) can be arbitrarily chosen. This is a consequence of the afore-mentioned invariance of Eq.(\[eqtb\]) with respect to $\hat h\rightarrow \hat h+\exp(bt)z$ ($z$ arbitrary and constant) which implies that there is actually a continuous number of transformations leading from Eq.(\[eqta\]) to Eq.(\[eqtb\]). A convenient choice, however, is to select as initial condition for $\hat h$ that $h({\bf x},t)$ and $\hat h({\bf x},t)$ initially coincide, i.e. $h({\bf x},0)=\hat h({\bf x},0)$. Consequently, $c(0)=-F_0/b$ holds and the transformation reads $$h({\bf x},t)=\hat h({\bf x},t)-\frac{F_0}{b}(1- {\rm e}^{bt})-{\rm
e}^{bt}\int_0^t dt^\prime\,{\rm e}^{-bt^\prime}\partial_{t^\prime}
\overline H(t^\prime).
\label{eqtg}$$ The transformation (\[eqtg\]) that reduces the nonlocal equation (\[eqta\]) in $h$-system to a local equation (\[eqtb\]) in $\hat h$-system has several specific properties. (i) It is a purely [*temporal*]{} (integral) transformation relating $h({\bf x},t)$ in the physically meaningful coordinate system comoving to the mean evolution of the erosion depth with the evolution of $\hat h({\bf x},t)$ in a temporally shifted system with no obvious physical significance. This shift $s(t)=h({\bf x},t)-\hat h({\bf x},t)$ being represented by the last two terms in (\[eqtg\]), takes over the nonlocal properties of Eq.(\[eqta\]) and is therefore nonlocal by virtue of the last term in (\[eqtg\]). Noteworthy, $s(t)$ integrates over the temporal history of the mean velocity of the erosion front via $\partial_{t^\prime}
\overline H(t^\prime)$. (ii) The stochastic part in Eq.(\[eqta\]) remains unchanged by the transformation. (iii) The transformation (\[eqtg\]) is highly useful because it allows for the direct applicability of theoretical and numerical results obtained from Eq.(\[eqtb\]) to Eq.(\[eqta\]). Note, however, that for a correct interpretation in the physical meaningful $H$- or $h$-system the full temporal information of the mean evolution of the erosion front $\overline H(t)$ needs to be separately determined. If the dynamics of $\hat h({\bf x},t)$ is known, this can be achieved by using $$\overline H(t)= F_0 t + \int_0^t dt^\prime \,\left[\overline{G[\nabla \hat h)]}
+\overline\eta(t^\prime)\right].
\label{trafo11}$$ (iv) The basic prerequisite for the transformation (\[eqtg\]) is that Eq.(\[eqta\]) is linear in $h$. Extensions to nonlinear dependences on $h$ in Eq.(\[eqta\]) do not seem to be generally feasible.
For the specific case under consideration, $G[\nabla h]=a_1 \nabla^2 h+a_2\nabla^4 h+a_3(\nabla h)^2$, the transformation (\[eqtg\]) can be further simplified. Using (\[trafo3\]) it follows that $$h({\bf x},t)=\hat h({\bf x},t)
-a_3\int_0^t dt^\prime\,{\rm e}^{-b(t^\prime-t)}
\overline{(\nabla h)^2}(t^\prime)
\label{eqtk}$$ and, consequently, $$\partial_t \hat h= b \hat h+a_1 \nabla^2 \hat h
+a_2\nabla^4 \hat h+a_3(\nabla \hat h)^2+\eta
\label{eqtx}$$ which constitutes the damped KS equation in the $\hat h$-system and possesses the obvious invariance under the transformation $\lbrace\hat h,a_3\rbrace
\rightarrow\lbrace -\hat h,-a_3\rbrace$. Eq.(\[eqtk\]) shows the importance and the genuine interrelation of the KS-type nonlinearity for the non-triviality of the transformation. If $a_3$ equals zero, then simply $h({\bf x},t)=\hat h({\bf x},t)$ follows.
This argument can be generalized: Separating the functional $G[\nabla h]$ in Eq.(\[eqta\]) in terms that can be rewritten as the divergence of a flux, $G_{\rm F}[\nabla h]=\nabla\cdot{\bf j}_{\rm F}$, and terms $G_{\rm NF}[\nabla h]$ that cannot, the corresponding transformation is determined by $h({\bf x},t)=\hat h({\bf x},t)
-\int_0^t dt^\prime\,{\rm e}^{-b(t^\prime-t)}
\overline{G_{\rm NF}[\nabla h]}(t^\prime)$. Consequently, any term being nonlinear and not originating from a flux leads to analogous complications as the term $\overline{(\nabla h)^2}$. Only if $G_{\rm NF}[\nabla h]=0$, then again $h({\bf x},t)=\hat h({\bf x},t)$ holds implying that $\overline{H}(t)=F_0 t$.
Some further results
====================
Role of the nonlocal term
-------------------------
The general idea behind model equations such as Eq.(\[trafo2\]) is that any individual term entering into the sum on the right hand side has its individual physical significance. As argued by Facsko et al. [@facsko-t], the nonlocal term in (\[trafo2\]) might be interpreted as a redepostion effect of eroded particles. To clarify the role of the nonlocal term in Eq.(\[trafo2\]), $b (H-\overline H)$, we disregard, for the moment, all terms in Eq.(\[trafo2\]) that depend on derivatives of $H$ and the stochastic fluctuations $\eta$. So, the equation $\partial_t H=F_0 +b (H-\overline H)$ remains to be solved. Obviously $\partial_t\overline H=F_0$ holds and, consequently, (i) the mean depth evolution increases linearly with time for all times $t$ according to $\overline H=F_0 t$ and (ii) $H-\overline H$ can be rewritten as $H-F_0 t$ showing that this term, individually considered, acts locally in the frame moving with $F_0 t$. The solution for the erosion profile $H$ is then simply given by $H({\bf x},t)=H({\bf x},0)\exp(bt)+F_0 t$. For negative $b$, the impact of the term $b (H-\overline H)$ is just an exponential diminishment of the initial target profile $H({\bf x},0)$ in such a way that the overall shape and, in particular, the maxima, minima, and inclination points of $H({\bf x},t)-F_0 t$ remain at same spatial positions. Only the amplitude of $H({\bf x},t)$ decays in time. Consequently, it is not obvious whether the term $b (H-\overline H)$ can be interpreted as a redeposition effect that should also lead to a lateral variation of the erosion profile.
Stability of a flat erosion front
---------------------------------
Taking advantage of the afore-mentioned arguments on the role of the nonlinearity in (\[trafo2\]), ignoring the stochastic fluctuations for the moment and linearizing Eq.(\[trafo2\]), i.e. omitting the term $(\nabla H)^2$, yields $\overline H(t)=F_0 t$. Therefore, a flat erosion front given by $H_{\rm FF}({\bf x},t)=F_0 t$ or, equivalently $h_{\rm FF}({\bf x},t)=0$ solves the deterministic limit of the linearized version of Eq.(\[trafo2\]) and determines its basic solution. Consequently, as far as the linear stability of the flat erosion front solution is concerned, the nonlocal term can be replaced by the local term $F_0 t$ in the linearized Eq.(\[trafo2\]). Then, standard techniques can be invoked for the stability analysis of $H_{\rm FF}$: Using an ansatz for a perturbation of $h_{\rm FF}({\bf x},t)=0$ of the form $h\propto \exp[i {\bf k}\cdot{\bf x}+\sigma t]$, yields a dispersion relation $\sigma=b-a_1 k^2+a_2 k^4$ ($k^2={\bf k}^2$) for the growth rate $\sigma(k)$ of perturbations with a wave vector ${\bf k}$. Therefore, $\sigma(k)$ has its maximum at $k^2_{\rm max}=a_1/2a_2$ with $\sigma(k_{\rm max})=b-a_1^2/4a_2$ and the flat erosion front $H_{\rm FF}$ is stable if $b\leq a_1^2/4 a_2$.
Time dependence of the mean erosion depth $\overline H$
-------------------------------------------------------
A crucial point of our discussion is the fact that variations of the mean erosion velocity about $F_0$, $\partial_t\overline H-F_0=a_3\overline{(\nabla H)^2} +\eta$, are generally non-zero and not even constant during the course of the erosion process. Here, we substantiate this by numerical simulations of Eq.(\[trafo2\]) for a representative set of parameter values that leads to hexagonal patterns and draw some further conclusions.
In Fig. 1 we show the evolution of the ensemble or stochastically averaged evolution of $\partial_t\overline H-F_0$ given by $\partial_t\langle\overline H\rangle-F_0=
a_3\langle\overline{(\nabla H)^2}\rangle$ since $\langle\overline\eta\rangle=\overline{\langle\eta\rangle}=0$. For this quantity, the stochasticity necessarily present in individual runs of the erosion process, as shown in Fig. 2, is leveled out. In both cases, the initial target profile has been kept fixed to a small Gaussian distribution of the initial amplitudes with a maximum amplitude of $\eta=0.01$ about the perfectly flat state.
The generic behavior as function of time consists of three parts: (i) For very small erosion times, $\partial_t\langle\overline H\rangle-F_0$ is very close to zero since here the nonlinearity in the evolution equation can be neglected and, therefore mainly the linear evolution of $H$ contributes. Since $\partial_t\overline H-F_0$ is proportional to $a_3\overline{(\nabla H)^2}$ this leads to a purely exponential increase for short times as depicted in the insets of Fig. 1. (ii) With increasing erosion time, the amplitude $H$ eventually reaches values where the nonlinear term in Eq.(\[trafo2\]) becomes comparable in size to linear terms. In this crossover time range, $\partial_t\langle\overline H\rangle-F_0$ increases very rapidly up to a point where the subsequent increase drastically slows down. (iii) For longer times, where the nonlinearity in Eq.(\[trafo2\]) is fully developed, the KS-nonlinearity $(\nabla H)^2$ mainly reduces the further increase of $\partial_t\langle\overline H\rangle-F_0$ almost to a constant. Our numerical simulations that went very far into the full nonlinear regime, however, do not indicate a full saturation, but a slight systematic increase with time that depends almost linearly on $t$ in the simulated time range. Similar behavior for individual erosion processes (albeit slightly modified by its intrinsic stochasticity) can also be read off from Fig. 2. An important consequence of this dynamical behavior of Eq.(\[trafo2\]) is the fact that the underlying surface morphology $H({\bf x},t)$ does not saturate in a steady state unless $\partial_t\overline H-F_0$ fully saturates into a constant value. To substantiate this statement, we show in the insets of Fig. 2 for two representative erosion times the corresponding morphology behavior that clearly indicates a non-steady evolution.
Finally, we like to draw attention to a challenging point for further experiments. Due to the non-monotonic behavior of $\partial_t\langle\overline H\rangle-F_0$ as function of the erosion time that is triggered by the KS-nonlinearity, a specific experimental measurement of this quantity can lead to experimental evidence for the existence and dominance of no-flux terms as discussed in section III.
Summary
=======
To conclude, we stress the most important points of our investigation: (i) The nonlocal, damped KS equation suggested by Facsko et al. [@facsko-t] can be obtained as minimal model from balance equation considerations if nonlocal terms compatible with the underlying invariances are allowed. Most importantly, such a nonlocal term being proportional to $H-\overline{H}$ leads in combination with the KS nonlinearity to a non-monotonic time evolution of the spatially averaged erosion depth and, connected with that, to a surface morphology that does not generally seem to approach a steady state (at least on the simulated time scales). (ii) The necessity for a non-local dependence arises from the fact that the KS nonlinearity cannot be rewritten in form of the divergence of a flux. If all terms entering into the evolution equation could be rewritten in form of such divergences, the non-local term could be replaced by a simple drift term $\overline{H}=F_0 t$. (iii) By using a specific transformation presented in section III, the nonlocal, damped KS equation can be exactly recast in form of a standard (non-local) damped KS equation. This property greatly simplifies the analysis of the evolution equations. Finally, we have also briefly proposed a simple experimental measurement in order to substantiate the pronounced effect of the KS nonlinearity or at least related non-flux nonlinearities during the erosion process. Building on the results presented here, we will discuss, in a subsequent publication, the rich pattern forming structure of an appropriately generalized non-local anisotropic damped KS equation for the case of oblique incidence.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank T. Allmers, M. Donath, and S. Facsko for interesting discussions.
[99]{} for recent reviews see e.g. G. Carter, J. Phys. D: Appl. Phys. [**34**]{}, R1 (2001); C. Teichert, Appl. Phys. A [**76**]{}, 653 (2003) M.A. Makeev, R. Cuerno, and A.-L. Barab[á]{}si, Nucl. Instr. and Meth. B [**197**]{}, 185 (2002) U. Valbusa, C. Boragno, and F. Buatier de Mongeot, J. Phys.: Condens. Matter [**14**]{}, 8153 (2002) S. Facsko, T. Dekorsy, C. Koerdt, C. Trappe, H. Kurz, A. Vogt, and H. Hartnagel, Science [**285**]{}, 1551 (1999); S. Facsko, T. Dekorsy, and H. Kurz, Phys. Rev. B [**63**]{}, 165329 (2001) S. Facsko, T. Bobek, H. Kurz, T. Dekorsy, S. Krysta and R. Cremer, Appl. Phys. Lett. [**80**]{}, 130 (2002) T. Bobek, S. Facsko, H. Kurz, T. Dekorsy, M. Xu, and C. Teichert, Phys. Rev. B [**68**]{}, 085324 (2003) R. Gago, L. V[á]{}zquez, R. Cuerno, M. Varela, C. Ballesteros, J.M. Albella, Appl. Phys. Lett. [**78**]{}, 3316 (2001) F. Frost, A. Schindler, and F. Bigl, Phys. Rev. Lett. [**85**]{}, 4116 (2000) F. Frost and B. Rauschenbach, Appl. Phys. A [**77**]{}, 1 (2003) F. Frost, B. Ziberi, T. Höche, and B. Rauschenbach, Nucl. Instr. and Meth. B [**216**]{}, 9 (2004) K.R. Elder, J.D. Gunton, and N. Goldenfeld, Phys. Rev. E [**56**]{}, 1631 (1997) M. Paniconi and K.R. Elder, Phys. Rev. E [**56**]{}, 2713 (1997) L. Shtilman and G. Sivashinsky, Can. J. Phys. [**68**]{}, 768 (1990) S. Facsko, T. Bobek, A. Stahl, H. Kurz, and T. Dekorsy, Phys. Rev. B [**69**]{}, 153412 (2004) R.M. Bradley and J.M.E. Harper, J. Vac. Sci. Technol. A [**6**]{},2390 (1988) B. Kahng, H. Jeong and A.-L. Barabási, Appl. Phys. Lett. [**78**]{}, 805 (2001); B. Kahng, H. Jeong and A.-L. Barabási, J. Korean Phys. Soc. [**39**]{}, 421 (2001) S.J. Linz, M. Raible and P. Hänggi, Lect. Notes Phys. [**41**]{}, 391 (2000) M. Raible, S.J. Linz, and P. Hänggi, Phys. Rev. E [**62**]{}, 1691 (2000); M. Raible, S.J. Linz, P. Hänggi, Phys. Rev. E [**64**]{}, 031506 (2001); S.J. Linz, M. Raible and P. Hänggi, Adv. Solid State Phys. [**41**]{}, 391 (2001) A.-L. Barabási and H.E. Stanley, [*Fractal concepts in surface growth*]{} (Cambridge University Press, 1995)
![Growth rate of the mean erosion depth calculated from Eq.(\[trafo2\]), averaged over 100 runs. Parameters are the same as in the paper by Facsko et al. , except for a spatial step size of $dx =1$ (mesh size $400\times 400$,$b=-0.24$, $a_1=-1$, $a_2=-1$, $a_3=0.0025$, white noise with a maximum amplitude of $\eta=0.01$). Solid line: $b=-0.22$, dotted line: $b=-0.24$. The inset shows a semilogarithmic plot of a portion of the data where linear terms determine the growth. A linear fit in this regime yields $\ln (\partial_t \langle\bar{H}\rangle- F_0)
\propto 0.018\cdot t$ for $b=-0.24$ and $\ln (\partial_t \langle\bar{H}\rangle- F_0)
\propto 0.057\cdot t$ for $b=-0.22$. This compares well to the result which can be obtained from the linearized version of Eq. (\[trafo2\]), i.e. $2\sigma_{max}=0.02$ for $b=-0.24$ and $2\sigma_{max}=0.06$ for $b=-0.22$.[]{data-label="dthgm"}](dkspaperbild1.ps){width=".4\textwidth"}
![Growth rate of the mean erosion depth calculated from Eq.(\[trafo2\]) for five different noise sequences. Parameters are the same as in Fig. with $b=-0.24$. Insets show a section of the surface morphology corresponding to the thick line in the figure at times $t=1\cdot 10^4$ and $t=2\cdot 10^4$.](dkspaperbild2klein.eps){width=".4\textwidth"}
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