diff --git "a/density of states/3.json" "b/density of states/3.json" new file mode 100644--- /dev/null +++ "b/density of states/3.json" @@ -0,0 +1 @@ +[ { "title": "2211.15016v1.Density_of_states_techniques_for_fermion_worldlines.pdf", "content": "Density of states techniques for fermion worldlines\nChristof Gattringer *\nFWF - Austrian Science Fund, 1090 Vienna, Austria\nE-mail: christof.gattringer@fwf.ac.at\nWorldline representations were established as a powerful tool for studying bosonic lattice field\ntheories at finite density. For fermions, however, the worldlines still may carry signs that originate\nfrom the Dirac algebra and from the Grassmann nature of the fermion fields. We show that a\ndensity of states approach can be set up to deal with this remaining sign problem, where finite\ndensity is implemented in a canonical approach by working with a fixed winding number of\nthe fermion worldlines. We discuss the approach in detail and show first results of a numerical\nimplementation in 2 dimensions.\n39th International Symposium on Lattice Field Theory - Lattice2022\nAugust 8 - 13, 2022\nBonn, Germany\n*Currently on leave of absence from Institute of Physics, University of Graz, 8010 Graz, Austria.\n© Copyright owned by the author(s) under the terms of the Creative Commons\nAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2211.15016v1 [hep-lat] 28 Nov 2022Density of states techniques for fermion worldlines\n1. Introductory comments\nEssentially all lattice field theories allow for representations where the matter degrees of freedom\nare represented by worldlines and the gauge degrees of freedom by worldsheets (see, e.g., [ 1] – [10]\nfor some key examples). In several cases these representations completely remove complex action\nproblems that originate from finite chemical potential or a topological term [ 11] – [14]. However,\nif the matter is fermionic then the corresponding worldlines still carry signs that emerge from the\nGrassmann nature of the fields as well as from the g-algebra. Thus Monte Carlo simulations of\nworldlines were so far restricted to bosonic theories, with the exception of massless staggered\nfermions in 2d [ 15,16,17], where the signs are known to be absent.\nHowever, worldlines are a very powerful and conceptually elegant framework, e.g., the net\nparticle number has the form of a topological invariant, such that it is worth exploring approaches\nthat may overcome the remaining sign problem of fermionic worldlines. Here we present a first\nexploratory study to address this sign problem with a suitable density of states (DoS) approach (see\nalso [ 18] for experiments in this direction). DoS techniques were initially introduced to lattice field\ntheory in [ 19,20], but saw a major revival based on a paper by Langfeld, Lucini and Rago [ 21],\nwhere a modern formulation with considerably improved accuracy was presented (see also [ 22] –\n[27]). Based on a variant [ 28] – [34] of these developments, the functional fit approach (FFA), we\nhere set up a version of DoS that is suitable for addressing the sign problem of fermion worldlines.\nThe basic idea is to define the density r(n)as a function of the number nof fermion loops\nwith a negative sign1, such that the partition function Zthen is given as Z=å¥\nn=0r(n)(\u00001)n. Two\nprerequisites are needed for this idea to work: 1) An update algorithm for the fermion loops that\nallows one to control the signs of the loops and at the same time is ergodic. 2) The density r(n)\nmust be fast decaying with n, such that the sum for Zconverges quickly and can be truncated after a\nreasonable number of terms. In this exploratory presentation we show that for the case of staggered\nfermions indeed both prerequisites can be fulfilled, i.e., we discuss suitable Monte Carlo steps that\nallow to control the loop signs and present preliminary numerical results that illustrate (at least in\n2d) exponential decay of r(n)for large n,\n2. Worldline representation for fermions and formulation of the DoS approach\nThe DoS method for fermion worldlines introduced here is a rather general approach, but for clarity\nof the presentation we discuss it for a specific model: The dynamical degrees of freedom are the 1-\ncomponent Grassmann-valued fermion fields yxandyxassigned to the sites xof ad-dimensional\nlattice. The boundary conditions are periodic for d\u00001 of the dimensions and anti-periodic for\ndimension dwhich is the euclidean time direction. The corresponding action is given by\nS=å\nx;ngx;nyxemdn;dyx+ˆn\u0000e\u0000mdn;dyx\u0000ˆn\n2+Må\nxyxyx\u0000J\n4å\nx;nyxyxyx+ˆnyx+ˆn: (2.1)\nIn the sums xruns over the sites of the lattice and nover the directions 1 ;2:::d, with ˆndenoting\nthe unit vector in direction n. The first sum in ( 2.1) is the kinetic term with the staggered sign\n1The fermion worldlines form closed loops such that from now on we will often use the term fermion loops .\n1Density of states techniques for fermion worldlines\n- 1+ 1+ 1\nFigure 1: Example of an admissible configuration of monomers (dots), dimers (double lines) and fermion\nloops (single lines with arrows) in 2d. The numbers next to the loops give the sign of the respective loop.\nfactors gx;n= (\u00001)x1+x2:::xd\u00001, the second sum the mass term and the last sum constitutes a quartic\nself-interaction with coupling J. A chemical potential mhas been introduced by weighting the\ntemporal ( n=d) hops in the kinetic term with e\u0006m. The partition sum of the system is given by\nintegrating the Boltzmann factor e\u0000Swith the product of Grassmann measuresR\nÕxdyxdyx.\nThe standard approach to a Monte Carlo simulation of the system would first introduce a Hub-\nbard Stratonovich (HS) field to break up the quartic interaction into a bilinear that couples to the\nHS field. In this form the fermions can be integrated out giving rise to a fermion determinant. The\npartition sum is then obtained by integrating the fermion determinant over all configurations of the\nHS field with a Gaussian weight. However, for m6=0 the fermion determinant is complex such\nthat it has no probability interpretation and only the case of vanishing chemical potential is acces-\nsible for Monte Carlo simulations. While for many bosonic systems the worldline representation\ncompletely solves the complex action problem, this is not the case for fermions, since the fermion\nworldlines still have signs. We will see, however, that the fermionic worldline picture allows for a\nnatural implementation of a DoS approach.\nIn the worldline representation the partition sum is exactly rewritten into a sum over configu-\nrations of monomers, dimers and loops. Monomers occupy a single site, dimers two neighboring\nsites, i.e., a link, and the fermion loops are oriented closed contours of links. The loops may not\ntouch or intersect. In addition monomers, dimers and loops obey a constraint: Each site of the\nlattice is either occupied by a monomer, is the endpoint of a dimer or is run through by a loop.\nFig.1shows an example of an admissible configuration in 2d, where monomers are shown as dots,\ndimers as double lines and loops as oriented single lines. The partition sum now is given by\nZ=1\n2Vå\nfm;d;lg(2M)#m(1+J)#dembWÕ\nlsign(l) =1\n2Vå\nfm;d;lg(2M)#m(1+J)#dembW(\u00001)N;(2.2)\nwhere the sum åfm;d;lgover the configurations of monomers m, dimers dand loops lis a restricted\nsum, where only admissible configurations are taken into account. Each configuration comes with\na weight factor where # mdenotes the number of monomers and # dthe number of dimers. bis the\ntemporal extent of the lattice, which corresponds to the inverse temperature in lattice units and W\nis the total net winding number of the loops around the compact time direction. The fact that W\nappears as factor of mbshows that Wcan be identified with the net particle number. From now on\nwe will switch to the canonical formalism, i.e., we work at a fixed net particle number, i.e., a fixed\nnet winding number Wand the factor embWwill no longer appear.\nIn Eq. ( 2.2) sign (l)denotes the sign of a loop l. This sign has several contributions: 1) The\nproduct of staggered sign factors gx;nalong the links of the loop l. 2) A factor of\u00001 for every\n2Density of states techniques for fermion worldlines\nlink that is run through by the loop in negative direction. 3) An overall sign originating from the\nreordering of the Grassmann variables along the loop. 4) A factor of \u00001 for every winding of the\nloop around compact time, which is due to the anti-periodic temporal boundary conditions. The\noverall sign of a configuration is then given by Õlsign(l), which in the second step of ( 2.2) was\nwritten as (\u00001)N, where Nis the number of loops with a negative sign.\nWe will show below, that in the canonical setting, i.e., at fixed winding number W, local\nupdate steps can be combined into an ergodic algorithm. In each of these local update steps one\ncan evaluate the change of the loop signs, such that one has an ergodic algorithm with full control\nof the total number Nof loops with negative sign. Thus it makes sense to consider a density of\nstates that is a function of the number nof negative sign loops. We define the density r(n)as\nr(n) =å\nfm;d;lg(2M)#m(1+J)#ddn;N; (2.3)\nwhere the sum over the dynamical degrees of freedom is now understood such that the net winding\nnumber Wremains fixed. The Kronecker delta dn;Nin (2.3) restricts the sum over the dynamical\ndegrees of freedom such that the number Nof negative sign loops is frozen to N=n. Obviously\nthe partition sum then is given by\nZ=¥\nå\nn=0r(n) (\u00001)n: (2.4)\nThe inclusion of observables will be discussed in an upcoming paper and we continue with intro-\nducing a suitable parameterization for the density r(n).\n3. Parameterization and evaluation of the density\nThe next step is to find a suitable parameterization of the density r(n)and a strategy for its de-\ntermination. Our parameterization is chosen piecewise constant, but it is convenient to write the\nconstants in exponential form, such that the parameterization in terms of real exponents anreads\nr(n) =r(n\u00001)e\u0000anwith a0=f(M;J),r(0) =e\u0000f(M;J): (3.1)\nThe overall normalization of the density can be chosen freely with some restrictions that become\nimportant when observables are introduced. The normalization obviously is tied to the first expo-\nnent a0and as a normalization we choose a0=f(M;J)where f(M;J)can be chosen as a function\nof the couplings. Of course all exponents anwill depend on the couplings implicitly, but the free-\ndom of normalization allows for one explicit choice, i.e., our a0=f(M;J).\nHaving parameterized the density in terms of the exponents anwe now need to describe how\nthese parameters can be evaluated. Once they are determined one can compute the partition sum\nusing ( 3.1) and ( 2.4). For computing the anwe introduce a restricted partition sum defined as\nZn(l) =å\nfm;d;lg(2M)#m(1+J)#delNQn(N)with Qn(N) =(\n1 for N2fn;n+1g\n0 otherwise.(3.2)\nThe restricted partition sum Zn(l)depends on the real control parameter lwhich in the sum over\nall configurations couples via elNto the number of negative sign loops N. This number Nis\n3Density of states techniques for fermion worldlines\nrestricted by the support function Qn(N)which admits only N=nandN=n+1. We have already\noutlined that our updates allow one to control the number N. Furthermore no more sign factors\nappear in ( 3.2) such that Zn(l)can be studied as function of lusing Monte Carlo simulations.\nUsing the definition ( 2.3) of the density, as well as its parameterization ( 3.1) we may express\nthe restricted partition sum Zn(l)as\nZn(l) =r(n)eln+r(n+1)el(n+1)=r(n)elnh\n1+el\u0000an+1i\n: (3.3)\nTaking the derivative of the logarithm of this partition sum with respect to the control parameter l\nwe obtain the restricted vacuum expectation value of the number Nof negative sign loops,\nhNin(l) =¶lnZn(l)\n¶l=1\nZn(l)å\nfm;d;lg(2M)#m(1+J)#dQn(N)elN=n+1\n2\u0014\n1+tanh\u0012l\u0000an+1\n2\u0013\u0015\n;\n(3.4)\nwhere the third expression is the form in terms of the path integral over the dynamical variables\nm;d;l, while the final expression is the form based on the density. As discussed for the restricted\npartition sum above, also the path integral form of hNin(l)can be evaluated with Monte Carlo\ntechniques. After a trivial normalization we find V(l)\u00112hNin(l)\u00002n\u00001=tanh\u0010\nl\u0000an+1\n2\u0011\n.\nThe Monte Carlo results for V(l)for different values of lcan be fit with the simple 1-parameter\nfunction tanh ((l\u0000an+1)=2)and the exponent anis obtained from this fit. Repeating this procedure\nfor different nwe obtain the exponents anand from those the density r(n).\n4. Monte Carlo simulation\nWe now come to presenting a set of local update steps that can be combined into an ergodic al-\ngorithm and have the property that at each update step we may control how the signs of the loops\nchange. This is necessary to take into account the support function Qn(N)in the simulation of\nhNin(l), which restricts the number Nof loops with negative signs to fn;n+1g. Of course our\nupdate steps also have to update the monomer and dimer degrees of freedom and the configurations\nhave to be admissible ones, i.e., each site has to be run through by one fermion worldline or is the\nendpoint of a dimer or is occupied by a monomer. Furthermore we need to take into account that\nour DoS approach uses a canonical setting, i.e., we work at a fixed temporal net winding number W\nof the fermion loops. In order to implement that we use as a starting configuration a configuration\nwith Wstraight loops in direction dthat close around compactified time. The sites not visited by\nthe fermion loops are then occupied with monomers. The update steps we discuss below do not\nchange the winding number, such that the simulation remains in the sector with net particle number\nW. This also implies that we can ignore the contribution of the anti-periodic boundary conditions\nto the loop signs. We finally remark that the update steps we discuss are illustrated in 2 dimensions,\nbut it is straightforward to see that they work also in higher dimensions.\nWe begin the discussion of the updates with two steps that only involve monomers and dimers.\nThey are depicted in the lhs. plots of Fig. 2. Obviously an admissible configuration remains ad-\nmissible if we exchange a dimer on some link by two monomers at the endpoints of the link. We\naccept such a step with the corresponding Metropolis probability (Fig. 2, lhs., top). Another step\nwe use is to identify a dimer with a monomer on a site neighboring one of its endpoints and then\n4Density of states techniques for fermion worldlines\nExchanging monopoles and dimers (Metropolis)Interchanging monopole and dimer positions (50/50)\nShrinking/expanding loops (heat bath)\nFlipping a loop corner (50/50)\nsign L - sign L\nFigure 2: Elementary update steps of our algorithm. Top left: Exchange of two monomers on neighboring\nsites with the corresponding dimer (accepted with a Metropolis step). Bottom left: Shift of a dimer and\na monomer on an adjacent site (accepted with probability 1/2). Top right: Expanding or shrinking a loop\naround three sides of a plaquette with a heat bath step. The sign of the loop does not change. Bottom right:\nFlipping the corner of a loop. This step is accepted with probability 1/2 and changes the sign of the loop.\nshift the dimer towards the monopole site and place the monomer at the now empty site (Fig. 2,\nlhs., bottom). Here the weights are not changed and we accept this move with probability 1/2. We\nremark that this step is equivalent to two of the previous steps, which, however, might have a poor\nacceptance probability at small mass.\nNext we come to a set of update steps that change the contour of a loop. The first one (top right\nof Fig. 2) changes a loop along three sides of a plaquette by either shrinking or expanding the loop\naround that plaquette and adding or removing either two monomers or a dimer. It is straightforward\nto show that the sign of the loop remains the same, since a minus sign from adding/removing a flux\nin negative direction is compensated by a minus sign from the staggered sign factors. The change\nof weight from adding/removing dimers or monomers is taken into account by a heat bath.\nAnother way of altering the loop is shown in the bottom right of Fig. 2, where we flip the\ncorner of a loop and move one monomer. Here the weight does not change, such that this step is\naccepted with probability 1/2. Here, however, the sign of the loop is flipped, due to the change of\nthe staggered sign factors. This implies that this update step can be admitted only if also after the\nstep the number Nof negative loops is still in fn;n+1gas required by the support function Qn(N).\nFinally we come to update steps that change the number of loops. The first one, illustrated\nin the top of Fig. 3inserts/removes an elementary loop around a single plaquette by remov-\ning/inserting a combination of dimers or monomers or both. The elementary loop has positive sign,\nand when inserting such a loop the orientation can be chosen randomly. The change of weight from\nadding/removing the dimers or monomers can again be taken into account by a heat bath.\nThe last step we discuss, is a re-routing of two antiparallel fluxes on opposite links of a pla-\nquette. As the bottom plot of Fig. 3shows, this either gives rise to splitting a single loop Linto two\nloops L1andL2or the inverse step by fusing two loops L1andL2into a single loop L. One finds\nthat the signs of the loops obey sign (L) =sign(L1)sign(L2), which can be seen from the property\nthat the product of the staggered sign factors over the links of a plaquette equals \u00001 and the fact\nthat each individual loop has an overall factor of \u00001. The weights do not change here, such that\nthe acceptance probability is 1/2, but of course the number of negative loops can change, implying\n5Density of states techniques for fermion worldlines\nExchanging elements on a plaquette (heat bath)\nJoining/splitting loops (50/50)\nx 2x 2x 4sign L = sign L1 sign L2LL1L2\nFigure 3: Update steps that change the number of loops. Top: Combinations of dimers and monomers that\nfill the sites and links of a plaquette can be exchanged with a loop around the plaquette (heat bath). Bottom:\nChanging two anti-parallel fluxes on a plaquette splits a single loop Linto two loops L1;L2or vice versa.\nThe step is accepted with probability 1/2 and the loop signs obey sign L= sign L1signL2.\nthat also this step can only be offered if the number of negative loops remains in fn;n+1g.\nFor a first test of the new approach we set up a simulation of our model in 2 dimensions. We\nwork on lattices of size L\u0002Land consider L=8;16;24 at vanishing winding number, i.e., we set\nW=0. The different update steps of our simulation strategy discussed in the previous paragraphs\nare combined into sweeps where a full sweep is defined as applying each update step once to all\nlinks or plaquettes it can act on. For our simulations we typically use 105sweeps for equilibration\nand a statistics of 105measurements separated by 10 sweeps for decorrelation. We determine\nhNin(l)for typically 50 values of land fit the corresponding V(l)with tanh ((l\u0000an+1)=2)to\ndetermine the exponent an+1. An example of this step is shown in the lhs. plot of Fig. 4. Obviously\nthe data (symbols) are described by the fit function (continuous curve) extremely well.\nFrom the exponents anone can determine the density r(n)using Eq. ( 3.1). To get a first\nimpression of its properties, in the rhs. plot of Fig. 4we show r(n)as a function of nfor different\nvolumes, using a normalization such that r(0) =1, i.e., a0=0. The plot is for couplings M=0:5\nandJ=0:0. Here the main question is to understand whether the density is fast decreasing for large\nn. Obviously this is the case (note the logarithmic scale on the vertical axis). However, it is obvious\nthat for the larger two volumes a maximum of r(n)appears for some nmax>0, which implies that\nfor increasing volume the most likely number of negative sign loops is at this nmax>0. Only for\nn>nmaxthe exponential decrease of the density sets in. One naturally expects that a negative\nsign loop has a characteristic coupling-dependent size2, such that with increasing volume more\nnegative sign loops of the characteristic size fit on the lattice. Thus one expects, that in leading\norder nmaxscales proportional to the volume, such that also the range of values nwhere r(n)needs\nto be evaluated until r(n)becomes sufficiently small scales with the volume. This expectation that\nthe range where the density has to be computed scales with the volume is in agreement with the\nexperience from other DoS applications.\nThe first numerical tests presented here show that indeed the density r(n)is fast decreasing for\nsufficiently large n, and it is plausible that the range of nwhere r(n)needs to be evaluated scales\n2Note that the smallest possible negative sign loop is around a 3 \u00023 square.\n6Density of states techniques for fermion worldlines\n-2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 λ-1.0-0.50.00.51.0 V(λ) fit\n MC data\n0 1 2 3 4 5 6 7 8 9 n10-910-810-710-610-510-410-310-210-11 ρ(n)24 x 24\n16 x 16\n 8 x 8\nFigure 4: Left: Example for the fit of the data for V(l)with tanh ((l\u0000an+1)=2). Right: The density r(n)\nas a function of nfor different lattice volumes.\nlinearly with the system size. Currently we are working on the implementation of observables and\nprepare a systematic comparison of the DoS results for free fermions to the corresponding exact\ncalculation, in order to assess whether the necessary accuracy of r(n)can be achieved, such that\nan efficient use of the new method is feasible.\nReferences\n[1] M. G. Endres, PoS LAT2006 , 133 (2006) doi:10.22323/1.032.0133 [arXiv:hep-lat/0609037]\n[2] M. G. Endres, Phys. Rev. D 75, 065012 (2007) doi:10.1103/PhysRevD.75.065012\n[arXiv:hep-lat/0610029]\n[3] U. Wolff, Nucl. Phys. B 832, 520-537 (2010) doi:10.1016/j.nuclphysb.2010.02.005 [arXiv:1001.2231]\n[4] T. Korzec, U. Wolff, PoS LATTICE2010 , 029 (2010) doi:10.22323/1.105.0029 [arXiv:1011.1359]\n[5] C. Gattringer, T. Kloiber, Nucl. Phys. B 869, 56-73 (2013) doi:10.1016/j.nuclphysb.2012.12.005\n[arXiv:1206.2954]\n[6] Y . Delgado Mercado, C. Gattringer, A. Schmidt, Phys. Rev. Lett. 111141601 (2013)\ndoi:10.1103/PhysRevLett.111.141601 [arXiv:1307.6120]\n[7] C. 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Orasch, PoS LATTICE2021 , 158 (2022) doi:10.22323/1.396.0158,\n[arXiv:2111.09535]\n8" }, { "title": "2301.06225v1.Non_phononic_density_of_states_of_two_dimensional_glasses_revealed_by_random_pinning.pdf", "content": "Non-phononic density of states of two-dimensional glasses revealed by random pinning\nKumpei Shiraishi,1,\u0003Hideyuki Mizuno,1and Atsushi Ikeda1, 2\n1Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan\n2Research Center for Complex Systems Biology, Universal Biology Institute,\nUniversity of Tokyo, Komaba, Tokyo 153-8902, Japan\n(Dated: January 18, 2023)\nThe vibrational density of states of glasses is considerably di\u000berent from that of crystals. In\nparticular, there exist spatially localized vibrational modes in glasses. The density of states of these\nnon-phononic modes has been observed to follow g(!)/!4, where!is the frequency. However, in\ntwo-dimensional systems, the abundance of phonons makes it di\u000ecult to accurately determine this\nnon-phononic density of states because they are strongly coupled to non-phononic modes and yield\nstrong system-size and preparation-protocol dependencies. In this article, we utilize the random\npinning method to suppress phonons and disentangle their coupling with non-phononic modes and\nsuccessfully calculate their density of states as g(!)/!4. We also study their localization properties\nand con\frm that low-frequency non-phononic modes in pinned systems are truly localized without\nfar-\feld contributions. We \fnally discuss the excess density of states over the Debye value that\nresults from the hybridization of phonons and non-phononic modes.\nI. INTRODUCTION\nLow-frequency vibrational states of glasses have been\nattracting considerable attention in recent years. Unlike\ncrystals [1], their low-frequency vibrational modes are not\ndescribed by phonons alone; there exist spatially local-\nized vibrations. The vibrational density of states of these\nnon-phononic localized modes follows g(!)/!4[2, 3].\nThe localized modes are widely observed in various sys-\ntems regardless of interaction potentials [4], details of\nconstituents [5], asphericity of particles [6], and stability\nof con\fgurations [7].\nTheoretical backgrounds of the non-phononic vibra-\ntional density of states have been studied. Mean-\feld\ntheories predict that glasses exhibit the non-Debye scal-\ning law of g(!)/!2at low frequencies by both the\nreplica theory [8] and the e\u000bective medium theory [9], and\nnumerical simulations of high-dimensional packings con-\n\frm this behavior [10, 11]. Recently, replica theories of\ninteracting anharmonic oscillators [12, 13] and the e\u000bec-\ntive medium theory [14{16] have also successfully derived\nthe!4scaling of glasses. Of these mean-\feld theories,\nthe e\u000bective medium theory [9, 14{16] naturally deals\nwith phonon modes together with non-phononic modes,\nwhereas the other theories focus on non-phononic modes\nwithout particular attention on phonon modes.\nMeanwhile, phonons do exist even in the amorphous\nsolids, which strongly hybridize with the non-phononic\nlocalized modes. In this case, the scaling of g(!) is\ndescribed by the framework of the generalized Debye\nmodel [17{20] that predicts that the exponent should be\nconsistent with that of the Rayleigh scattering \u0000 /\nd+1\nof acoustic attenuation (\u0000 is attenuation rate, \n is propa-\ngation frequency, and dis the spatial dimension). There-\nfore,g(!) is predicted to scale with !d+1. Numerical\n\u0003kumpeishiraishi@g.ecc.u-tokyo.ac.jpsimulations of three-dimensional glasses ( d= 3) show\nthat the phonon attenuation rate follows \u0000 /\n4[19, 21{\n24]. This behavior of Rayleigh scattering has also been\nobserved in recent experimental studies [25{28]. Simu-\nlation study also reveals that the vibrational density of\nstates follows g(!)/!4[3]. Thus, in three-dimensional\nsystems, acoustic attenuation and the vibrational den-\nsity of states both exhibit the exponent of d+ 1 = 4,\nconsistent with the generalized Debye theory.\nHowever, in two-dimensional glasses, con\ricting results\nhave been reported. In acoustic attenuation simulations\nin two-dimensional glasses ( d= 2), the Rayleigh scatter-\ning of \u0000/\n3is indeed observed [23, 29{31]. In sim-\nulations of direct measurements of g(!) in two dimen-\nsions, Mizuno et al. performed the vibrational analysis\nof glass con\fgurations of large system sizes and revealed\nthat localized vibrations were too few to determine the\nnon-phononic scaling of g(!) [3]. Afterward, Kapteijns\net al. reported that the !4scaling holds even for two-\ndimensional glasses by performing simulations of systems\nwith small system sizes for a large ensemble of con\fgura-\ntions to extract the su\u000ecient number of modes below the\n\frst phonon frequency [32]. However, a recent study by\nWang et al. reported a contradictory result of g(!)/!3:5\nfrom simulations of small systems [33]. More recently,\nLerner and Bouchbinder suggested that the exponent de-\npends on the glass formation protocol and system size\nand claimed the exponent to be 4 in the thermodynamic\nlimit even in two dimensions [34]. In contrast, a recent\nwork by Wang et al. claimed that there are no system-size\ne\u000bects and the exponent remains as 3.5 [35].\nThe above con\ricting results could be due to the emer-\ngence of phonons and their coupling with the localized\nmodes, making it di\u000ecult to accurately determine the\nnon-phononic vibrational density of states. Here, we uti-\nlize the random pinning method to resolve this prob-\nlem. Originally, this method is used to realize equilib-\nrium glass states [36{38]. Angelani et al. showed that\nthis method can be used to suppress phononic modes andarXiv:2301.06225v1 [cond-mat.soft] 16 Jan 20232\nprobe the non-phononic density of states [39]. Recently,\nwe revealed that low-frequency localized modes of pinned\nglasses are disentangled with phonons by numerical sim-\nulations of three-dimensional glasses [40]. By performing\nvibrational analysis in two-dimensional pinned glasses,\nwe can expect to put an end to the controversial results\nof the non-phononic density of states.\nIn this paper, we report the properties of low-frequency\nlocalized modes in two-dimensional glasses induced by\nrandomly pinned particles. First, we study the partic-\nipation ratio of each mode and show the low-frequency\nmodes of pinned glasses indeed have a localized character.\nSecond, we also study their localization properties by cal-\nculating the decay pro\fle. Those modes show exponen-\ntially decaying pro\fles, that is, they are truly localized.\nFinally, we evaluate the vibrational density of states of\nlocalized modes and observe the scaling of g(!)/!4in\ntwo-dimensional glasses with pinned particles. Our re-\nsults elucidate the bare nature of low-frequency localized\nmodes of glasses by obliterating harmful phononic modes\nusing the random pinning operation.\nII. METHODS\nWe perform vibrational mode analyses on the ran-\ndomly pinned Kob-Andersen system in two-dimensional\nspace [41], which is identical to a model studied by Wang\net al. [33]. We consider a system of Nparticles with iden-\ntical masses of menclosed in a square box with periodic\nboundary conditions. The linear size Lof the box is de-\ntermined by the number density of \u001a= 1:204. Particles A\nand B are mixed in a ratio of 65:35 to avoid crystalliza-\ntion [41]. The particles interact via the Lennard-Jones\npotential\nV(rij) =\u001e(rij)\u0000\u001e(rcut\nij)\u0000\u001e0(rcut\nij)(rij\u0000rcut\nij);(1)\nwith\n\u001e(rij) = 4\u000fijh\n(\u001bij=rij)12\u0000(\u001bij=rij)6i\n; (2)\nwhererijdenotes the distance between interacting par-\nticles, and the cut-o\u000b distance is set to rcut\nij= 2:5\u001bij.\nThe interaction parameters are chosen as follows: \u001bAA=\n1:0; \u001bAB= 0:8; \u001bBB= 0:88; \u000fAA= 1:0; \u000fAB=\n1:5; \u000fBB= 0:5. Lengths, energies, and time are mea-\nsured in units of \u001bAA,\u000fAA, and\u0000\nm\u001b2\nAA=\u000fAA\u00011=2, respec-\ntively. The Boltzmann constant kBis set to unity when\nmeasuring the temperature T.\nTo prepare the randomly pinned system, we \frst run\nmolecular dynamics simulations in the NVT ensemble\nto equilibrate the system in the normal liquid state at\nT= 5:0 for the time of t= 2:0\u0002102, which is su\u000e-\nciently longer than the structural relaxation time. Af-\nter the equilibration, we randomly choose particles and\nfreeze their positions. The FIRE algorithm [42] is ap-\nplied to the system to minimize energy (stop condition\n106\n105\n104\n103\n102\n101\n100pk\n(a)N=40K\nN=80K\nN=200K\nN=400K\nN=1000K\nN=2000K\n101\n100101ωk106\n105\n104\n103\n102\n101\n100pk\n(b)N=16K\nN=40K\nN=80K\nN=200K\nN=400K\nN=1000K\nN=2000KFIG. 1. Participation ratio pkversus mode frequencies !k.\nThe \fgures show the data of the lowest-frequency region. The\nfractionscof pinned particles are (a) 0.03 and (b) 0.20.\nis maxiFi<3:0\u000210\u000010), which produces the glass-solid\nstate at zero temperature, T= 0. We denote the frac-\ntion of pinned particles as c(0\u0014c\u00141) and the number\nof unpinned (vibrating) particles as Nup= (1\u0000c)N.\nThen, we perform the vibrational mode analysis on the\nrandomly pinned system and obtain the eigenvalues \u0015k\nand eigenvectors ek=\u0010\ne1\nk;e2\nk;:::; eNup\nk\u0011\n, wherek=\n1;2;:::; 2Nup[43]. Note that two zero-frequency modes\ncorresponding to global translations do not appear in\npinned systems because the existence of pinned parti-\ncles breaks translational invariance [39, 40]. For details\nof the random pinning procedure and vibrational mode\nanalysis, please refer to Ref. 40.\nIII. RESULTS\nA. Participation ratio\nFirst, we study the participation ratio\npk=1\nNupPNup\ni=1\f\fei\nk\f\f4; (3)3\n100101102r108\n107\n106\n105\n104\n103\n102\n101\nd(r)\nFIG. 2. Decay pro\fle d(r) of a low-frequency mode of the\nsystem with N= 2;000;000 andc= 0:20. The mode has\nthe eigenfrequency of !k= 0:5968 and participation ratio of\npk= 1:9\u000210\u00006. The dashed line indicates the power-law\nbehavior of d(r)/r\u00001.\nwhich quanti\fes the fraction of particles that participate\nin the mode k[44, 45]. Figure 1 shows pkversus eigenfre-\nquencies!k=p\u0015kforc= 0:03 andc= 0:20. The num-\nber of particles in the systems ranges from N= 16;000\ntoN= 2;000;000.\nAs we can easily recognize from Fig. 1, pinned systems\nhave numerous localized modes with low pkin the low-\nfrequency region. This result is strikingly di\u000berent from\nthe unpinned system, where most low-frequency modes\nare spatially extended phonons and localized modes are\nhard to observe in two dimensions [3]. When pinned\nparticles are introduced, these phonon modes are sup-\npressed because translational invariance is violated, and\nlow-frequency localized modes emerge, like in three-\ndimensional glasses [39, 40]. Comparing the cases of\nc= 0:03 and 0:20 shows that pkis lower in c= 0:20.\nIn particular, when c= 0:20, there are modes whose par-\nticipation ratio is near pk= 1=Nup, indicating that only\none particle out of Nupparticles vibrates in the mode k.\nAs mentioned in the Introduction, the di\u000eculty\nof studying low-frequency localized modes in two-\ndimensional glasses originates from the abundance of\nlow-frequency phonons [34]. As demonstrated in Fig. 1,\nphonon modes are well-suppressed in two-dimensional\npinned glasses, and only low-frequency localized modes\nremain. Our data of pkclearly shows that random\npinning excludes phonons and resolves this di\u000eculty\nfor analysis on non-phononic modes in two-dimensional\nglasses.\nB. Decay pro\fle\nNext, we scrutinize the spatial structure of a low-\nfrequency mode by calculating the decay pro\fle d(r) as\n100101ω105\n104\n103\n102\n101\n100C(ω)\nc=0.03\nc=0.20FIG. 3. Cumulative density of states C(!) of systems with\nc= 0:03 andc= 0:20. The dashed lines indicate C(!)/!5.\nin Ref. 2, which is de\fned as\nd(r) =\f\fei\nk\f\f\nmaxi\f\fei\nk\f\f: (4)\nWhen calculating d(r), we take the median of each con-\ntribution\f\fei\nk\f\ffrom particles inside a shell with radius\nrfrom the most vibrating particle imax= argmaxi\f\fei\nk\f\f.\nFigure 2 presents the decay pro\fle d(r) of a low-frequency\nmode of a con\fguration with N= 2;000;000 andc= 0:20\n(Nup= 1;600;000).\nAs in Fig. 2, the decay pro\fle of pinned glasses devi-\nates from the power-law behavior of d(r)/r\u00001[32]. In-\nstead,d(r) shows an exponential decay, consistent with\nthe behavior in three-dimensional pinned glasses [40].\nThis result indicates that the spatial structures of low-\nfrequency localized modes are signi\fcantly di\u000berent from\nthose of unpinned glasses [2, 32, 46]. The power-law de-\ncay ofd(r)/r\u00001, which is missing in Fig. 2, is a conse-\nquence of the absence of hybridization with phonons [2].\nTherefore, we conclude that the random pinning method\nprevents non-phononic localized modes from hybridiz-\ning with phonon modes, as in the three-dimensional sys-\ntem [40].\nC. Vibrational density of states\nFinally, we study the vibrational density of states\nin the low-frequency regime of randomly pinned two-\ndimensional glasses. The vibrational density of states\nis calculated as\ng(!) =1\nNmodeX\nk\u000e(!\u0000!k); (5)\nwhereNmode = 2Nupis the number of all eigenmodes\nand\u000e(x) is the Dirac delta function. However, the value\nofg(!) is sensitive to the binning setups used for the4\ncalculation. To determine the density of states without\nthe arbitrariness of binning, we present the cumulative\ndensity of states:\nC(!) =Z!\n0g(!0)d!0: (6)\nFigure 3 presents C(!) forc= 0:03 andc= 0:20. When\ngenerating Fig. 3, we averaged C(!) of di\u000berent system\nsizes presented in Fig. 1. The results from these di\u000ber-\nent system sizes provide information for the very low-\nfrequency regime [3]. We recall that each of these sys-\ntems has the fraction cof pinned particles; therefore, the\nnumber of vibrating particles Nupis smaller than N.\nAs shown in Fig. 3, C(!) obeys!5scaling, that is,\nthe vibrational density of states obeys g(!)/!4in the\nlow-frequency regime, which is the main result of this\nwork. This behavior is consistent with various reports\nin three-dimensional glasses [2, 3, 39, 40]. Our result\nis also consistent with a report by Kapteijns et al. [32]\nwho studied two-dimensional unpinned glasses of small\nsystems.\nHere, we emphasize that the random pinning method\nsuppresses phononic modes and enables us to directly\nprobe the non-phononic density of states without gener-\nating a large ensemble of small systems. Furthermore,\nbecause the low-frequency modes of pinned glasses do\nnot hybridize with phonons, the investigation of g(!)\nwith random pinning is free of \fnite-size e\u000bects [34, 47]\nor glass-formation-protocol dependence [34] appearing in\ng(!).\nIV. DISCUSSIONS\nIn summary, we report the properties of low-frequency\nvibrations of two-dimensional glasses with randomly\npinned particles. While there exist a large number of\nphonon modes in two-dimensional glasses, which cause\nhybridization with non-phononic modes, the random pin-\nning operation can well suppress phonon modes to dis-\nentangle their hybridization. We con\frm the disentan-\nglement numerically by observing the participation ra-\ntio and decay pro\fle and conclude that non-phononic\nmodes are truly localized modes that are not coupled to\nphonon modes. Therefore, we can easily intrude on the\nnon-phononic density of states of localized modes at low\nfrequencies. Then, our main result is that the cumula-\ntive vibrational density of states of non-phononic modes\nobeysC(!)/!5, that is, the vibrational density of states\nfollowsg(!)/!4in two-dimensional glasses. This result\nprovides a sound basis for the controversial vibrational\ndensity of states of two-dimensional glasses and could re-\nsolve the con\ricting reports of the exponent [32{35]. Our\nwork also demonstrates the bene\ft of the random pinning\nmethod, not only for the glass transition studies but also\nfor the material properties of amorphous solids.\nOur present analysis of randomly pinned two-\ndimensional glasses reveals the vibrational density ofTABLE I. Dimensional dependence of the exponent of the\nexcess density of states g(!)/!\f(with phonons) with corre-\nsponding values of the non-phononic density of states (with-\nout phonons). Note that \fhas not yet been measured for the\nexcess density of states (with phonons) and d\u00154; however,\nwe might expect \f= 4 (see the main text).\nDimension With phonons Without phonons\nd= 2\f= 3 [23] \f= 4 (this paper)\nd= 3\f= 4 [23] \f= 4 [40]\nd\u00154\f= 4 (expected) \f= 4 [32]\nstates of non-phononic modes that are completely free\nfrom hybridization with phonons. On the other hand, in\nthe following, we discuss the \\excess\" density of states\nover the Debye value in a situation where abundant\nphonon modes exist and hybridize with non-phononic\nmodes. Here, we use the phrase \\excess density of states\"\nbecause we generally cannot distinguish non-phonon\nmodes from phonon modes when they are strongly hy-\nbridized. In this situation, we can apply the generalized\nDebye theory [17{20, 23] to measure the \\excess\" density\nof states, where the exponent \fofg(!)/!\fis provided\nby the exponent \rof the acoustic attenuation \u0000 /\n\r.\nWe refer to previous studies and summarize the val-\nues of\fin Table I for the spatial dimensions of d= 2,\nd= 3, andd\u00154. In the table, we also present the\ncorresponding values of \fof the non-phononic density\nof states without phonon modes for comparison. For\nthe case without phonons, where truly-localized modes\nare realized, the non-phononic density of states follows\ng(!)/!4ford= 2 andd= 3, as con\frmed in the\npresent (d= 2) and previous study [40] ( d= 3) using\nthe random pinning method. The previous numerical\nwork [32] also provided the value of \f= 4 ford= 2\ntod= 4. In addition, the mean-\feld theories predicted\ng(!)/!4[12, 13], which validates the value of \f= 4 for\nd\u00154.\nIn contrast, for the case with phonons where non-\nphononic modes hybridize with phonons to become quasi-\nlocalized, the Rayleigh scattering behavior of \u0000 /\n\r=\n\nd+1is observed in d= 2 andd= 3, leading to the value\nof\f=d+ 1 [23]. For the larger dimensions of d\u00154,\nthere are no numerical results so far; however, we spec-\nulate the following. The non-phononic density of states\nwithout phonons, g(!)/!4, shows larger orders of val-\nues thang(!)/!d+1(in the low-frequency regime) since\n4s t;(2)where\u001bsands0are the width and the mean of the\nlognormal portion, respectively, and \u000bis the slope of\nthe power-law portion.\nAs discussed in Section 1, the density PDF is a key\ncomponent of many analytic star formation models, in-\ncluding sub-grid models for the SFR. In these models,\nthe density PDF is used to quantify the gas mass frac-\ntion that can form stars (because it is dense enough to\ncollapse), and, weighted by a free-fall time factor, pre-\ndict the SFR. The density PDF therefore quanti\fes how\nmuch of the gas is primarily turbulent versus how much\ngas is collapsing.\nThe SFR can be calculated by integrating over the\ndensity PDF above a critical density for star forma-\ntion and multiplying by the appropriate timescales and\ndensities (e.g., see Krumholz & McKee 2005; Padoan &\nNordlund 2011a; Hennebelle & Chabrier 2011; Federrath\n& Klessen 2012; Burkhart 2018). The critical density\n(scrit) can be de\fned in a number of ways; Federrath &\nKlessen (2012) give a thorough overview of several com-\nmonly used critical densities (see also Burkhart 2018,\nand references therein). Ultimately, each de\fnition of\nthe critical density \fnds a way to characterize the den-\nsity at which gravity becomes dynamically important.\nFor example, Krumholz & McKee (2005) directly com-\npares the scales at which gravity and turbulence become\nequal, i.e., the sonic scale (Federrath et al. 2021). Al-\nternatively, Burkhart (2018) suggests that the critical\ndensities discussed in Federrath & Klessen (2012) are ef-\nfectively traced by the transition density ( st) where the\nlognormal density PDF changes to a power-law PDF.\nThe transition between these regimes serves as evidence\nwithin the density PDF of the fact that gravity be-\ncomes dynamically dominant (Burkhart 2018; Burkhart\n& Mocz 2019).\nIn our analysis, we explore how di\u000berent physical pro-\ncesses impact the gas dynamics and how these dynamics\nare re\rected in the density PDF. Thus, we are particu-\nlarly interested in investigating the density at which self-\ngravity dominates over other physical processes and how\nthe gas density PDF relates to the SFR. Hence, we will\ninvestigate if the transition density, st, from Burkhart\n(2018) is the density at which gravity becomes dynami-\ncally dominant and sets the stage for star formation.\n3.2. Calculating the Compression Rate\nThe density PDF describes the density distribution\nof the gas at a single point in time, but it can be con-\nnected to the underlying gas dynamics with the conti-\nnuity equation. We quantify the rate of change of the\ndensity using the Lagrangian formulation of the conti-The Density PDF and the Gas Compression and Expansion Rates 7\nnuity equation, which is given by\nD\u001a\nDt+\u001a(r\u0001~ v) = 0; (3)\nwhere we use D=Dt\u0011@=@t +~ v\u0001ras a shorthand for the\nLagrangian derivative (see Appendix A for derivation).\nTo compare this expression to the density PDF, which\nwe have calculated in terms of the natural logarithm of\nthe normalized density,\ns= ln(\u001a=\u001a0); (4)\nwe rearrange Eq. 3 to be expressed in terms of s(see\nAppendix A for details):\nDs\nDt\u0011\u0000(r\u0001~ v): (5)\nThus, we have a connection between the time evolution\nof the natural logarithm of the normalized density (the\nDs=Dt term) and the gas dynamics (as represented by\nthe velocity vector, ~ v).\nBecausesis dimensionless, the quantity Ds=Dt has\ndimensions of inverse time, i.e., it is the rate of change of\nthe logarithmic density contrast s. This quantity repre-\nsents the \row rate of gas into or out of a particular region\nof the simulation. For the purpose of this study, we re-\nfer to positive values of Ds=Dt as compression rates and\nnegative values of Ds=Dt as expansion rates. Regions\nwith a positive rate ( Ds=Dt > 0) have net compress-\ning gas, i.e., converging \rows ( r\u0001~ v <0), which means\nthat more gas is entering that region than is leaving it.\nConversely, regions with a negative rate ( Ds=Dt < 0)\nhave net expanding gas, i.e., diverging \rows ( r\u0001~ v>0),\nwhich means that more gas is leaving that region than\nis entering it.\nWe compute the compression rate, Ds=Dt , for every\nsimulation cell using the ytpackage (Turk et al. 2011).\nWe calculate the gradient of each component of the ve-\nlocity \feld, which we then use to construct the diver-\ngence of the velocity \feld\nDs\nDt=\u0000(r\u0001~ v) =\u0000(@xvx+@yvy+@zvz):(6)\nThus, for every cell in the simulation, we now have both\na density value and a compression or expansion rate, and\nwe can investigate how the compression and expansion\nrates behave as functions of time and how they compare\nto the density PDF.\n4.RESULTS\n4.1. Star Formation throughout Each Run\nWe \frst compare the impact of the di\u000berent physical\nprocesses in each simulation by examining the integrated\n0246810SFE (%)G\nGT\nGTM\nGTMJ\nGTMJR\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5\nt/tff106\n105\n104\nM (M/yr)\nFigure 2. The integrated star formation e\u000eciency (SFE;\ntop panel) and the star formation rate (SFR; bottom panel)\nas a function of time for each of the simulations described\nin Table 2. The t0= 0 point is de\fned to be the point at\nwhich gravity is turned on for each simulation. As additional\nphysical processes are included, the SFR drops and it takes\nlonger for each simulation to reach a similar integrated SFE.\nSFE and the SFR for each simulation. In the upper\npanel of Fig. 2, we show the SFE as a function of time\nfor each of the simulations, where the integrated SFE is\nde\fned as\nSFE =M\u0003\nMinit; (7)\nwhereM\u0003is the total stellar mass formed and Minitis\nthe initial cloud mass. Fig. 2 demonstrates that the\nSFE evolves at very di\u000berent rates for each simulation\ndue to the di\u000berent physical properties that are included\nin each run. To account for this di\u000berence when the\nvarious runs are compared, we use SFE to characterize\nthe evolutionary stage of the simulation in addition to\nthe simulation time. For example, Fig. 1 shows density\nprojections of snapshots at SFE \u00195%.\nThe evolution of the SFR { which directly relates to\nthe slope of the SFE as a function of time { is plotted\nin the lower panel of Fig. 2. Although there is a lot\nof variation over time in the SFR for all of the simula-8 Appel et al.\n0.2 0.4 0.6\nt/tff101\n100101Sink Masses (M)\nG\n0.75 1.00 1.25\nt/tffGT\n0.5 1.0 1.5\nt/tffGTM\n1 2\nt/tffGTMJ\n1 2 3\nt/tffGTMJR\nFigure 3. The evolution of the masses of individual sink particles as a function of time for each of the simulations described\nin Table 2. Each sink particle follows a single line on the plot. The t0= 0 point is de\fned to be the point at which gravity is\nturned on for each simulation. Each run takes a di\u000berent amount of time to form the \frst sink particle. Thus, note that the\nx-axis range varies for each panel.\ntions, there is a clear overall decrease in the SFR as more\nphysical processes are included. This is also evident in\nthe mean SFR values shown in Table 2. This decrease\nin the SFR with the inclusion of additional physics has\nbeen seen and quanti\fed in previous studies (see e.g.,\nFederrath 2015).\nAs an additional diagnostic of the variations between\neach simulation, we consider the masses of the individual\nsink particles as a function of time. In Fig. 3, we plot the\nmass growth of each individual sink particle as a func-\ntion of time and Table 2 reports the \fnal number of sink\nparticles (Nsinks) for each run. We see that run G(the\nfar left panel) forms a large number of sink particles, but\nonly forms relatively low mass ( <4 M\f) sink particles\nthroughout the simulation. Run GTforms fewer, higher\nmass sink particles compared to run G. The inclusion of\nmagnetic \felds results in yet less fragmentation due to\nadditional magnetic support and run GTM forms only\nfour sink particles, including two that reach up to \u001814\nM\f. The inclusion of jet feedback, however, increases\nfragmentation (Federrath et al. 2014; Guszejnov et al.\n2020; Mathew & Federrath 2021) and runs GTMJ and\nGTMJR both produce a greater number of sink parti-\ncles that are all 5 :8 M\for less.\nFrom both Fig. 2 and Fig. 3 we can see that there\nis very little di\u000berence between runs GTMJ andGT-\nMJR . The inclusion of radiative feedback in run GT-\nMJR slightly slows down star formation relative to run\nGTMJ , which only includes protostellar jet feedback.\nSimilarly, the inclusion of radiative feedback does not\nsubstantially change the amount of fragmentation, as\nseen in Fig. 3 and Table 2. Thus, we choose to omit run\nGTMJ in our subsequent analysis, because it is very\nsimilar to run GTMJR . We include some further dis-\ncussion of this run in Appendix B.\n4.2. Expansion and Compression Rates as Functions\nof DensityIn Fig. 4, we compare the density PDF to the com-\npression and expansion rates for a single snapshot of run\nGTwhere the SFE\u0018=0:5%. Given the value of Ds=Dt\nfor every cell in the simulation, we can split up all of the\ngas in the simulation into expanding and compressing\ngas and consider the density PDF for each component\nof the gas. We plot the compression and expansion rates\nin separate panels because the collapsing and expanding\ngas trace di\u000berent density regimes. In particular, the\ncollapsing gas traces higher densities and includes higher\nrate values than the expanding gas. Thus, the top pan-\nels of Fig. 4 shows the separate volume-weighted den-\nsity PDFs for the expanding gas (solid line; left panel)\nand the compressing gas (solid line; right panel). For\ncomparison, the overall density PDF (dotted lines) is\nalso shown. In the bottom panels of Fig. 4, we show\nthe compression and expansion rates, Ds=Dt , given by\nEq. 6 in units of Myr\u00001versuss= ln(\u001a=\u001a0) as a 2D\nhistogram (the heat maps in the bottom two panels of\nFig. 4). The left panel shows the 2D histogram of the ex-\npansion rate as a function of sof the expanding gas (gas\nwith a negative Ds=Dt ) and the right panel shows the\ncorresponding histogram for the compression rate of the\ncompressing gas (gas with a positive Ds=Dt ). We also\nshow the median compression and expansion rates as a\nfunction of density with the interquartile range shown\nas a shaded region. For comparison, we show the free-\nfall rate (i.e., the reciprocal of the gravitational free-fall\ntime) as a function of density, as given by\n1\nt\u000b=\u00123\u0019\n32G\u001a\u0013\u00001=2\n: (8)\nFrom this plot, we can see that the expanding gas\ncontributes more to the overall PDF at low densities as\ncompared to the compressing gas. Conversely, at high\ndensities, the overall density PDF is primarily composed\nof compressing gas. Indeed, at densities s&5, the com-\npressing gas PDF and the overall PDF are essentiallyThe Density PDF and the Gas Compression and Expansion Rates 9\n107\n106\n105\n104\n103\n102\n101\np=(1/V) dV/dst= 2.854 Myr\nNsinks= 3\nSFE=0.501%Expanding Gas (Ds/Dt<0)\nFull PDF\nExpanding PDFCompressing Gas (Ds/Dt>0)\nFull PDF\nCompressing PDF\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5 10.0\ns=ln(/0)\n102\n101\n100101102103|Ds/Dt v| (1/Myr)\n1/tff(s)\nst\nssink\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5 10.0\ns=ln(/0)\n104910511053\nVol. (cm3)\nFigure 4. The density PDFs and histograms of the compression and expansion rates for a single snapshot of run GTwhere\nthe SFE \u00190:5%.Top: The volume-weighted density PDF for all of the gas in the simulation region (dotted line) and the\nvolume-weighted density PDF for only the expanding gas ( Ds=Dt < 0; left column) or only the compressing gas ( Ds=Dt > 0;\nright column) within the simulation region. Bottom: A 2D, volume-weighted histogram of the expansion rate as a function of\ndensity for only the expanding gas ( Ds=Dt < 0; left column) or of the compression rate as a function of density for only the\ncompressing gas ( Ds=Dt > 0; right column). The corresponding median rate as a function of density (with the 25th to 75th\npercentiles) is shown as a blue line (and shaded region). The free-fall rate as a function of density is overplotted. All: For all\npanels, the transition density ( st) from Appel et al. (2022) is overplotted as a vertical dashed line. The sink formation density\nthreshold ( ssink) for the simulation is also shown as a dotted vertical line. Key information (time, number of sink particles, and\nintegrated SFE) for the particular snapshot shown here is indicated on the plot.\nidentical and the expansion rate dramatically drops o\u000b,\nmeaning that compression dominates in this portion of\nthe PDF. As we will see in the subsequent analysis, this\ndensity is the point at which the gas mass \rux equals the\nSFR in our simulations. This is also the density range\nwhere we see the development of a second power-law tail\nin the PDF, in agreement with previous work (Khullar\net al. 2021).In Fig. 4, we show the density PDFs for only a sin-\ngle snapshot of a single simulation. However, the same\ntrend (e.g., the overall PDF matching the expanding\ngas PDF at low densities and the compressing gas PDF\nat high densities) is apparent for all physics cases and\nthroughout the run of the simulations. Appendix C\nshows examples of the density PDF for several physics\ncases and multiple points in time.10 Appel et al.\nComparison of top and bottom panels shows that\nsome aspects of the distribution of the heat map cor-\nrespond to the shape of the density PDF. In particular,\nthere are many fewer counts and less spread in the com-\npressing gas at high densities where the density PDF\nhas a lower value. Indeed, there is very little gas at all\nat high densities in the expanding gas heat map. Mean-\nwhile, there is a higher concentration of gas at average\nto low densities in the expanding gas heat map than in\nthe compressing gas heat map. The heat maps empha-\nsize that for both expanding and compressing gas there\nis a lot of spread in the rate values, with gas at a wide\nvariety of rates at most densities. However, the median\nline and the interquartile range indicate that the major-\nity of the gas is clustered around a particular rate for a\ngiven density. For example, the median expansion rate\ndrops at high densities, while the median compression\nrate increases at high densities.\n4.3. The E\u000bect of Di\u000berent Physics on Gas\nCompression and Expansion Rates\nFigure 5 shows the rate-density plots for our four main\nsimulations (runs G,GT,GTM , and GTMJR ). All\nfour physics cases are shown at an approximate mid-\npoint of each simulation, corresponding to SFE \u0018=5%\n(the corresponding times are shown on the plot). Again,\nwe show the expanding gas for a given snapshot in the\nleft-hand panels and the compressing gas in the right-\nhand panels.\nProgressing from the top of the \fgure to the bottom,\nwe can see how the rate-density plot changes as di\u000ber-\nent physical processes are added. The top row of Fig. 5\nshows run G, which only includes self-gravity. The rate-\ndensity plot for this case shows a few distinctive features.\nFirst, the median line of the compressing gas follows or\nis slightly above the free-fall rate at all densities. In con-\ntrast, the expanding gas has a relatively \rat distribution\naround the mean density and only increases toward the\nfree-fall rate at higher densities.\nHowever, in the second row of Fig. 5, we see that the\nmedian rate for run GThas an elevated and approxi-\nmately \rat distribution for both the expanding and com-\npressing gas at all densities below the transition den-\nsity (vertical dashed line), suggesting that turbulence\nreduces the density dependence of the compression and\nexpansion rates below st. Abovest, the compressing\ngas rises faster than the free-fall rate while the expand-\ning gas remains \rat before turning slightly down. This\nsuggests that for gas below st, turbulence increases both\nthe rate of convergence and the rate of divergence of the\ngas relative to the free-fall rate. Above st, the behavior\nof the compressing gas is mostly determined by gravity.In the third row of Fig. 5, we see that the inclusion\nof magnetic \felds has a less prominent e\u000bect than the\ndi\u000berence between run Gand run GT. However, it still\nproduces some di\u000berences in the rate-density plot. In\nparticular, the median rate of the compressing gas near\nthe mean density becomes lower than in run GT. This\nsuggests that magnetic \felds serve to dampen the ef-\nfects of turbulence on compressing gas. Furthermore,\nin regions where magnetic pressure dominates over self-\ngravity it will take longer for gas to move to higher den-\nsity, which may explain the larger spread in the rates\nof low density gas relative to run GT. At high densi-\nties, the expanding gas follows the free-fall rate closely,\nsuggesting that magnetic pressure is sub-dominant to\nself-gravity above st.\nIn the \fnal row of Fig. 5, we see that the inclusion\nof protostellar out\rows has an impact on the lowest-\ndensity gas. This is in agreement with the \fndings of\nAppel et al. (2022), which show that protostellar out-\n\rows produce an excess of low-density gas. The plots\nthat include protostellar out\rows show the same evi-\ndence of the mean \rattening at densities near the mean\ndensity as in run GTM . However, run GTMJR also\nshows a large upturn in the rate of the very lowest-\ndensity gas { for both compressing and expanding gas.\nThis suggests that protostellar out\rows are producing\nboth rapidly expanding and rapidly compressing low-\ndensity gas. The compressing low-density material is\nlikely associated to the bow shocks produced by the jets\nas they propagate through and entrain low-density gas.\n4.4. Net Compression and Expansion Rates\nIn Fig. 6 we show the median compression and expan-\nsion rates as in Fig. 5, but now averaged over time. We\ntake the median in time for snapshots between SFE= 2%\nand SFE= 10% in order to avoid \ructuations in the com-\npression and expansion rates that we see at early times.\nThese \ructuations at early times likely re\rect the fact\nthat it takes a while for the simulations to settle into\nan approximate steady state, and we exclude this tran-\nsient period from our analysis. All four simulations are\nshown with the 1-sigma variations shown as shaded re-\ngions around the median values. The leftmost panel\nshows the median rate for the expanding gas and the\ncentral panel shows the median rate for the compressing\ngas. The rightmost panel of Fig. 6 shows the net rate,\nwhich we calculate by taking the volume-weighted aver-\nage of all of the expanding gas rates and the compressing\ngas rates for a single density bin for a single snapshot.\nWe then plot the median in time of this net rate in the\nrightmost panel of Fig. 6.The Density PDF and the Gas Compression and Expansion Rates 11\n102\n101\n100101102103|Ds/Dt|G t= 2.527 MyrExpanding Gas (Ds/Dt<0)\nCompressing Gas (Ds/Dt>0)\n102\n101\n100101102103|Ds/Dt|GT t= 3.267 Myr\n102\n101\n100101102103|Ds/Dt|GTM t= 3.139 Myr\n5\n 0 5 10\ns=ln(/0)\n102\n101\n100101102103|Ds/Dt|GTMJR t= 4.624 Myr\n5\n 0 5 10\ns=ln(/0)\nst\nssink\n1/tff(s)\n104910511053\nVol. (cm3)\nFigure 5. Each row is the same as the bottom panels of Fig. 4 and shows the 2D, volume-weighted histograms of the expansion\nand compression rates as a function of density. Each row shows a single snapshot of a di\u000berent simulation. The snapshots\nchosen for this plot all have SFE \u00195% and the time since gravity was turned on is shown on the plot for each snapshot.\nSimilar to the median lines for individual snapshots\nseen in Fig. 5, we see that distributions of the median\nrates in Fig. 6 vary with the inclusion of additional phys-\nical processes. The gravity-only simulation (run G) has\na very low and \rat distribution of expanding gas for den-sities around the mean density, in accordance with the\nlack of any turbulent velocities. The net rate for run G\nincreases with density, close to the free-fall rate for all\ndensities, especially at low densities.12 Appel et al.\n5\n 0 5 10\ns=ln(/0)\n102\n101\n100101102103|Ds/Dt u| (1/Myr)\nMin SFE=2.0%\nMax SFE=10.0%Expanding Gas (Ds/Dt<0)\nG\nGT\nGTM\nGTMJR\n1/tff(s)\nst\nssink\n5\n 0 5 10\ns=ln(/0)\n102\n101\n100101102103|Ds/Dt u| (1/Myr)\nCompressing Gas (Ds/Dt>0)\n5\n 0 5 10\ns=ln(/0)\n20\n020406080Net Rate (1/Myr)Net Rate\nFigure 6. The median in time ( \u00061-sigma variation in time) of the median rates for the expanding gas ( Ds=Dt < 0; left panel)\nand for the compressing gas ( Ds=Dt > 0; center panel). The right-most panel shows the median in time ( \u0006one sigma variation\nin time) of the net rate on a linear scale. The free-fall rate as a function of density is overplotted in each panel, as is the\ntransition density from Appel et al. (2022) (vertical dashed line). The sink formation density ( ssink) for each of the simulations\nis also shown as a dotted vertical line.\nThe other simulations, which all include turbulent ve-\nlocities and driving, have generally \ratter (i.e., almost\nindependent of density) distributions of rates around\nthe mean density. Importantly, the turbulence-driven\nrates of gas expansion and compression at these densi-\nties (s < 5; see Figs. 5 and 6) have comparable mag-\nnitudes and are much higher than the free-fall rate at\nthe same densities (dashed-dotted line). This statisti-\ncal equilibrium between compression and expansion is\nindicative of continuous gas cycling between the self-\ngravitating, high-density gas in the power-law tail, out of\nwhich stars form, and the low-density, non-star-forming\ngas that corresponds to the lognormal portion of the\ndensity PDF (Appel et al. 2022, see also Semenov et al.\n2017, 2018 for analogous processes in the galactic con-\ntext).\nAt high densities, we see that the rate of the com-\npressing gas for all simulations increases with density\nfaster than the free-fall rate at densities well above st\n(e.g.,s&5). The fact that all of the simulations con-\nverge on the same behavior as run Gsuggests that the\ncompressing gas at high densities is strongly in\ruenced\nby gravity in all cases. The faster-than-free-fall collapse\nis likely a consequence of the details of the density dis-\ntribution and dynamics of the gas that is accreting onto\nthe sink particles. In particular, the faster-than-free-fall\ncollapse may be a consequence of plotting the rate as\na function of the svalue of individual cells; since sis\na cell-by-cell quantity, it may not re\rect how the gas is\nactually distributed around the sinks and the resulting\ngravitational potential. Alternatively, this may indicatethe in\ruence of accretion processes. In particular, pre-\nvious work suggests that s\u00185 (which is where the net\nrate increases above free-fall) is where the accretion disk\nforms around the sinks (i.e., Khullar et al. 2021). This\nis also the density where the second power law forms in\nthe density PDF (Federrath & Klessen 2013; Burkhart\n2018; Khullar et al. 2021).\nIn the net rate panel, we also see that for densities\nabove the mean density, all of the physics cases are dom-\ninated by compressing gas. Only at densities below the\nmean does the net rate take on a negative value for some\nof the physics cases, corresponding to the gas being dom-\ninated by expansion. This e\u000bect is most evident in the\nnet rate of the run with feedback (run GTMJR ), which\ntakes on large negative values for the lowest density gas\n(e.g.,s.\u00005) as a result of the inclusion of protostel-\nlar jet feedback. Similarly, at the lowest densities, we\nsee that both the expanding and compressing gas rises\nsigni\fcantly for run GTMJR . This agrees with our un-\nderstanding that jets drive gas out of dense regions and\ninto low-density, rapidly expanding gas (see also Appel\net al. 2022).\n4.5. The Gas Mass Flux\nIn the previous sections, we considered the compres-\nsion and expansion rates, which have dimensions of in-\nverse time. By converting to a mass-weighted distri-\nbution of the rates versus density, we can calculate a\ngas mass \rux in units of M \f/yr that we can compare\ndirectly to the SFR.The Density PDF and the Gas Compression and Expansion Rates 13\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5 10.0\ns=ln(/0)\n107\n106\n105\n104\nGas Mass Flux (M/yr)\nMin SFE=2.0%\nMax SFE=10.0%Expanding Gas (Ds/Dt<0)\nG\nGT\nGTM\nGTMJR\nst\nssink\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5 10.0\ns=ln(/0)\nCompressing Gas (Ds/Dt>0)\nFigure 7. The median in time ( \u00061-sigma variation in time) of the gas mass \rux in units of solar masses per year for the\nexpanding gas ( Ds=Dt < 0; left panel) and for the compressing gas ( Ds=Dt > 0; right panel). The average SFR of each run\nis overplotted as a horizontal line. As in Fig. 6, the transition density from Appel et al. (2022) and the sink formation density\n(ssink) for each of the simulations are also shown.\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5 10.0\ns=ln(/0)\n0.00000.00010.00020.00030.0004Gas Mass Flux (M/yr)\nMin SFE=2.0%\nMax SFE=10.0%G\nGT\nGTM\nGTMJR\nst\nssink\nFigure 8. The median in time ( \u00061-sigma variation in time)\nof the net gas mass \rux. The average SFR of each run is over-\nplotted as a horizontal line. The values of s\u0003, the densities\nat which the net gas mass \rux meets the SFR, for each of\nthe runs except run Gare shown as black stars. The hori-\nzontal, thin, grey line shows the SFR = 0 line. As in Fig. 6,\nthe transition density from Appel et al. (2022) and the sink\nformation density ( ssink) for each of the simulations are also\nshown.\nLetHmbe the mass-weighted version of the heatmaps\nin Figs. 4 and 5, or equivalently, the amount of mass at\na given compression (or expansion) rate bin, \u0001 R, anddensity bin, \u0001 s. Then, the net gas mass \rux (in M \f/yr)\nfor a given density bin is the product of Hmand the\ncompression rate, summed over every rate:\nF\u0001s=X\n\u0001RHm\u0001R: (9)\nThis gives us a net gas mass \rux as a function of density,\nor a metric of how much gas is expanding or compressing\nin M\f/yr as a function of density. As with the net rate\nin Section 4.4, a negative net gas mass \rux corresponds\nto net expanding gas and, conversely, a positive net gas\nmass \rux corresponds to net compressing gas.\nTo get the gas mass \rux for only the compressing gas,\nwe sum over all of the gas with positive rates. Similarly,\nto get the gas mass \rux for only the expanding gas, we\nsum over all of the gas with negative rates.\nWe show the median in time of the gas mass \rux for\nthe expanding gas (left panel) and the compressing gas\n(right panel) in Fig. 7, where the shaded regions show\nthe 1-sigma variations in time. The net gas mass \rux\nis shown in Fig. 8. For both Figs. 7 and 8, we again\nhave only considered snapshots between SFE=2% and\nSFE=10%. We overplot the mean SFR (also for the\nsnapshots between SFE=2% and SFE=10%) for each\nsimulation as a horizontal line in both \fgures.\n4.5.1. The Compressing and Expanding Gas Mass Flux\nThe \ruxes of both compressing and expanding gas\nshow prominent peaks near the mean density, s= 0.\nThis is because there is much greater gas mass near14 Appel et al.\ns= 0, where the mass-weighted density PDF peaks,\nand the \ruxes quickly fall o\u000b at higher and lower den-\nsities where the density PDF also falls o\u000b. Compres-\nsion clearly dominates near st, which corresponds to the\npostshock density where mass piles up due to shocks\n(Federrath 2016b). At even higher densities, the expan-\nsion rate exponentially drops o\u000b. At s\u00196 the compress-\ning gas mass \rux \rattens out before rising again at yet\nhigher densities. As Fig. 8 shows, this is also near the\ndensity at which the net gas mass \rux is roughly equal\nto the SFR (see a more detailed discussion below).\nThe compressing gas mass \rux continues to increase\nabove this density ( s\u00196) and peaks at ssink. We inves-\ntigated the formation of this second peak and found that\nit develops as sink particles start forming. In particular,\nright before the \frst sink particle is formed, no peak is\npresent atssink, and the plateau in the compressing gas\nmass \rux extends all the way to ssink(except for run G\nwhich exhibits very di\u000berent behavior, as discussed be-\nlow). Therefore, this second peak is likely a numerical\nartifact resulting from the e\u000bects of limited resolution\nand of the sink particle model on the local gas dynam-\nics.\nRunGexhibits rather di\u000berent behavior as a function\nof density and over time than the other runs. The me-\ndian line shown in Figs. 7 and 8 for run Ghas roughly\nconstant compressing gas mass \rux with increasing den-\nsity abovestup untils\u00196, after which the compress-\ning (and net) gas mass \rux increases and peaks at Ssink.\nUnlike the other runs, however, before the peak at ssink\nforms, there is not a plateau in the compressing gas mass\n\rux ats\u00196 for run G; instead the compressing gas\nmass \rux continues to decrease with increasing density\nabovestuntil well after the \frst sink forms. Indeed, the\npeak in the compressing gas mass \rux at ssinkfor run G\ndevelops slowly over many snapshots after the \frst sink\nis formed, suggesting this increase in compressing gas\nmass \rux is due to the presence of sink particles. Fur-\nthermore, the lack of plateau before the formation of\nthe \frst sink particle con\frms that this plateau in the\ncompressing gas mass \rux is a consequence of physical\nprocesses beyond gravity.\nWe also note that the density range at which the com-\npressing gas mass \rux begins to increase again for all of\nthe runs (s\u00196) corresponds to approximately the den-\nsity at which the second power-law tail is expected to\nform in the density PDF, due to accretion disks begin-\nning to form (see e.g., Khullar et al. 2021). Thus, the\nincrease in \rux just below the sink threshold density\nmay also be in\ruenced by the process of accretion onto\nsink particles.The gas mass \rux does fall o\u000b above ssink, but as there\nis only a very small amount of gas at these densities and\nthis is, by de\fnition, above the density at which sinks\nform, it is unclear how much we can trust any metrics\nof the behavior of the gas at these densities.\n4.5.2. The Net Gas Mass Flux\nLooking at Fig. 8, we see that the net gas mass \rux\nis very close to zero at the lowest densities, correspond-\ning to equal expanding and compressing gas mass \rux,\nas expected for driven turbulence. At around the mean\ndensity, the net gas mass \rux transitions rapidly to a\npositive value, indicating that gas at and above the mean\ndensity is net compressing. The transition to net com-\npressing gas at approximately the mean density matches\nthe behavior of the net rate in Fig. 6, and is due to shock\ncompression which piles up gas from the mean density\nto over-densities, up to the isothermal jump-condition\nofM2\ns(see e.g., Padoan & Nordlund 2011b; Federrath\n2016b, and references therein). The result of this shock\ncompression is that the net gas mass \rux peaks at the\ntransition density, st, indicating a gas pileup and the\nformation of \flamentary structures in the cloud.\nSimilar to the behavior of the compressing gas mass\n\rux in Fig. 7, the net gas mass \rux decreases in all runs\nat densities greater than st, except run G(the purple\nline in Fig. 8). As discussed, for run G, gravity is the\nonly dynamical process controlling the dynamics of the\ngas paststuntil sinks form and therefore the gas is mov-\ning with constant acceleration, hence the \rat or increas-\ning net gas mass \rux between s=stands\u00195). For\nthe other simulations, the addition of turbulence dra-\nmatically changes the way gas moves around and past\nthe post-shock density (i.e., the gas in the power-law\nportion of the density PDF). With driven supersonic\nturbulence, a strong peak at stin the gas mass \rux con-\n\frms this density traces the post-shock density and the\nformation of \flamentary features in the simulations (see\ne.g., Padoan & Nordlund 2011b; Federrath 2016b). At\ns > s t, gravity begins to dominate the dynamics, the\ndensity PDF forms the \frst power-law tail, and the gas\nmass \rux falls o\u000b until it reaches a plateau at around\ns= 5\u00006 (marked with a star symbol in Figure 8).\nFor ease of reference, we refer to the density at which\nthe net gas mass \rux \frst matches the mean SFR (after\nrising above the SFR at st) ass\u0003. The values of s\u0003for\neach of the runs except run G(which exhibits a very\ndi\u000berent behavior between standssink, as discussed)\nare reported in Table 2 and are shown as black stars in\nFig. 8. The values in Table 2 and Fig. 8 are calculated\nfor the mean SFR and median gas mass \rux between\nSFE=2% and SFE=10%.The Density PDF and the Gas Compression and Expansion Rates 15\n106\n105\n104\nMass Flux (M/yr)\nGTSFR\nAt st\nAt s*\n106\n105\n104\nMass Flux (M/yr)\nGTM\n1.0 1.5 2.0 2.5 3.0\nt/tf106\n105\n104\nMass Flux (M/yr)\nGTMJR\nFigure 9. The net gas mass \rux at two di\u000berent \fxed den-\nsities ( s=stands=s\u0003) are plotted in comparison to the\nsmoothed SFR as a function of time for each simulation. The\nnet gas mass \rux values are the time dependent counterparts\nto the values in Fig. 8 and are measured at a single density\nbin with a center equal to or just above the corresponding\ndensity.\nThe fall o\u000b in the gas mass \rux between stands\u0003\nis due to the fact that the acceleration (i.e., derivative\nof the rate) is smaller than free-fall collapse would sug-\ngest due to magnetic pressure and turbulent support,\nthereby slowing down collapse. The acceleration picks\nup at around s= 5\u00007, where an increase in the slope\nof the compression rate, shown in the middle panel of\nFig. 6, can be observed. At this point, the plateau de-\nvelops, signaling a constant gas mass \rux. Interestingly,\nthe value of the gas mass \rux at this plateau matches\nthe mean SFR. At the highest densities, a strong peak\ndevelops around the sink threshold density, as mass is\nrapidly funneled into the sink particles.\n4.5.3. Connecting the Gas Mass Flux and the SFRWe further investigate the relationship between s\u0003,\nthe gas mass \rux, and the SFR in Fig. 9, where\nwe plot the SFR as a function of time for runs GT,\nGTM , and GTMJR . The SFR is smoothed using\nscipy 'sgaussian_filter1d function with a sigma of\n\u00180:02 Myr. We plot the net gas mass \rux value (sam-\npled at every \ffth snapshot to reduce noise) at two key\ndensity bins: at stand ats\u0003. The values plotted in\nFig. 9 are the value of the net gas mass \rux for a single\nsnapshot at the density bin at or just above the density\nofstors\u0003. We note that we use the same value of s\u0003\n(the value in Table 2) for every snapshot, and, since this\nvalue is calculated based on where the median net gas\nmass \rux meets the mean SFR, may not re\rect exactly\nwhere the net gas mass \rux of an individual snapshot\nmeets the SFR of that snapshot.\nWe \fnd that the net gas mass \rux at the transition\ndensity,st, is higher than the SFR by about 40%. This\nis consistent with Fig. 8 which shows that the median\ngas mass \rux near stis higher than the mean SFR for\nall three cases. This discrepancy likely corresponds to\nthe core mass e\u000eciency factor for star formation models\nwhich take all the gas in the power-law tail portion of\nthe density PDF as star-forming (Burkhart 2018). The\ncore mass e\u000eciency factor accounts for the fact that the\ninclusion of turbulence and magnetic \felds makes the\nprocess of forming stars take much longer than free-fall\nand that the inclusion of out\rows cycles dense gas back\nto densities below stbefore it forms stars. This slow\ndown of the formation of stars from dense, star-forming\ngas is clearly evident in the discrepancy between the net\ngas mass \rux at stand the smoothed SFR. Further-\nmore, although the di\u000berence between the net gas mass\n\rux atstand the SFR is largest (at most times) for\nrunGTMJR , there is a signi\fcant discrepancy for all\nthree runs shown in Fig. 9, con\frming that both tur-\nbulence and magnetic \felds, and not only out\rows, are\ncontributing to preventing gas above the transition den-\nsity from forming sink particles.\nHowever, for all three runs, the net gas mass \rux at\ns\u0003as a function of time closely matches the value of the\nSFR. Time variable features of the SFR are also found in\nthe net gas mass \rux at s\u0003. For example, there is a large\njump in the SFR for run GTMJR aroundt=t\u000b\u00182:2\nthat is echoed by a similar jump in the net gas mass\n\rux ats\u0003, but not in the gas mass \rux at st. The\nsimilarity between the value of the net gas mass \rux at\ns\u0003and the SFR indicates that the gas dynamics around\ns\u0003play a key role in setting the SFR. We explore some\nof the implications of this connection in the Discussion\nsection below.16 Appel et al.\n5.DISCUSSION\n5.1. The Compression and Expansion Rates\nOur work investigates the ways in which di\u000berent\nphysical processes a\u000bect the gas dynamics of star-\nforming regions as a function of density and how they are\nre\rected in the density PDF shape. In density regimes\nwhere the in\ruence of gravity is dynamically dominant\n(e.g., above the transition density st), the gas dynamics\nof the simulations that include turbulence do not com-\npletely match the behavior of the gravity-only run until\ndensities above s&5 (i.e., at densities where self-gravity\ndominates). Only at the highest densities of compressing\ngas does the behavior of the compression rate become\nvery similar for all of the simulations presented in this\nwork. This suggests that turbulence, magnetic \felds,\nand feedback act to signi\fcantly alter how much gas\nreaches the highest densities and in\ruence the structure\nof collapsing regions.\nSimilarly, we see that turbulence acts to increase the\nrate of both the compressing and expanding gas at den-\nsities below the transition density. Below the transition\ndensity turbulence dominates, and the compressing and\nexpanding motions balance out, resulting in a net rate\nthat is near zero. Above the transition density, gravity\ndominates and the net rate rises rapidly. We also \fnd\nthat the net gas mass \rux peaks at st, suggesting that\nthe density where the \frst power-law tail forms is an ex-\ncellent tracer of the post-shock density (see e.g., Padoan\n& Nordlund 2011b; Federrath 2016b).\nIn contrast to the e\u000bect of turbulence, magnetic \felds\nact to decrease the rate of both the compressing gas and\nthe expanding gas at most densities below the transition\ndensity, relative to the run with only turbulence and\ngravity. This suggests that magnetic pressure acts to\ndampen the increased motion from turbulence.\nFinally, we see that the inclusion of protostellar out-\n\rows slightly increases the median rates of both the com-\npressing gas and the expanding gas, relative to the run\nwith magnetic \felds (as seen in Fig. 6), in agreement\nwith the expectation that the inclusion of protostellar\nout\rows will increase the kinetic energy of the gas (Ap-\npel et al. 2022). The most dramatic e\u000bect of proto-\nstellar out\rows is on the lowest density gas, where we\nsee that protostellar out\rows produce rapidly expand-\ning and compressing low-density gas. In addition, the\nnet rate (rightmost panel in Fig. 6) has a lot more vari-\nation in time for the cases with protostellar out\rows at\nlow densities, suggesting that the inclusion of protostel-\nlar out\rows introduces signi\fcant time variation in the\ncompression and expansion rates of the low-density gas\ncarved out by out\rows.In Fig. 6, we \fnd that the compression rate increases\nwith density faster than the free-fall rate for all physics\ncases, once the density exceeds approximately s\u00185.\nAs discussed above, this may be a consequence of plot-\nting the rate as a function of s, whensis a cell-by-cell\nquantity and may not re\rect how the gas is actually\ndistributed or the resulting gravitational potential.\nThis density also heralds the formation of the sec-\nond power-law tail in these simulations, as studied by\nKhullar et al. (2021) and which roughly corresponds to\nthe formation of accretion disks. This process of ac-\ncretion onto the sinks may also contribute to the faster\nthan free-fall collapse seen in Fig. 6, although our runs\nlikely do not fully resolve accretion disks, making this\nconnection uncertain. Regardless, the rate at which the\ngas passes through this density range appears to play\na role in setting the star formation rate in our simula-\ntions (see Figs. 8 and 9, and discussed below). Future\nwork will determine how the accretion disk forms and\nhow the gas mass \rux and compression and expansion\nrates depend on the sonic Mach number, Alfv\u0013 enic Mach\nnumber, and virial parameters.\nIn Section 4.2 we also explored the relationship be-\ntween the compression and expansion rates and the den-\nsity PDF. We see that the high-density end of the den-\nsity PDF is most closely matched by the PDF of the\ncompressing gas and the low-density end of the density\nPDF is most closely matched by the PDF of the expand-\ning gas. This agrees with our understanding that the\ncompressing gas is mostly at higher densities (that are\ndominated by gravity) and the expanding gas is mostly\nat lower densities where gravity is subdominant.\n5.2. The Gas Mass Flux\nThe gas mass \rux (Fig. 7) combines information from\nthe compression and expansion rates (Fig. 6) and the\noverall density PDF. At low densities, the gas mass \rux\nis low due to both low rates and small quantities of gas.\nNear the transition density from lognormal to power-law\ndistributions, the gas mass \rux peaks due to the forma-\ntion of shocks (Federrath 2016b). The drop-o\u000b in the net\ngas mass \rux above the transition density is evidence of\nvarious processes actively preventing collapse of the gas\nsince the acceleration of the gas is stalled relative to free-\nfall acceleration. The net gas mass \rux declines until it\nreaches a constant value (i.e., it plateaus), analogous to\na terminal velocity where the resistive forces are mag-\nnetic \felds and turbulent motions. The net gas mass\n\rux at this plateau matches the SFR at a density ( s\u0003)\nthat is well above the transition density. This behav-\nior of the gas mass \rux matches the fact that analytical\nmodels of the SFR that integrate over all densities aboveThe Density PDF and the Gas Compression and Expansion Rates 17\nthe critical density require an additional e\u000eciency fac-\ntor, implying that not all of the gas above the critical\ndensity ends up in a star. This agrees with the fact that\nthe gas mass \rux at stis higher than the SFR before\ndropping o\u000b { there are processes preventing and delay-\ning much of this gas from actually forming stars.\nKhullar et al. (2021) demonstrate the existence of a\nsecond power-law in the density PDF that begins at\ndensities greater than s\u00185. Their model suggests\nthat the lognormal portion of the density PDF is tur-\nbulence dominated, the \frst power-law is gravity domi-\nnated, and the second power-law (corresponding to the\nhighest density gas) is disk or rotation dominated. This\nsuggests a possible interpretation for the behavior of the\nnet gas mass \rux at high densities. In particular, the\npoint where the net gas mass \rux matches the SFR ( s\u0003)\nmay correspond to the beginning of this second power-\nlaw and the increase in the net gas mass \rux above s\u0003\nmay be due to the in\ruence of disk rotation. Although,\nagain, our simulations do not fully resolve the disk accre-\ntion, meaning that further work is needed to verify this\nconnection. As discussed above, however, the \rux at\nwhich the gas passes through this density range appears\nto set the star formation rate in our simulations. This\nrate is highest in run GTand lower in the runs with\nfeedback from protostellar out\rows; hence the SFR is\nlower when out\row feedback is included. Further work\nis needed to con\frm this connection and to compare the\nsdvalue from Khullar et al. (2021) to the value of s\u0003\nfound here.\n5.3. Other Implications and Future Work\nOur work may have important implications for sub-\ngrid models for isolated GMC simulations, or even\ngalaxy formation simulations. We demonstrated that\nthe SFR is set by the gas mass \rux at s\u0003. This can\nact as a minimum resolvable density required to set the\nSFR in simulations. However, measuring the gas mass\n\rux peak and \ftting a curve to higher densities could\nresult in an empirical sub-grid SFR model that could\nbe used by simulations to \\resolve\" protostellar core\nphysics. Doing so would require measuring r\u0001~ vand\ndetermining where this is negative (i.e., where gas is\ncompressing). Turning this into a gas mass \rux (Fig. 7)\ncould then yield similar curves which could be extrapo-\nlated to higher-than-resolved densities (i.e., protoplane-\ntary disk densities) where the star formation rate is then\nset.\nFuture work will explore how this s\u0003density depends\non the cloud mass, the virial parameter, the sonic Mach\nnumber, the Alfv\u0013 enic Mach number, and the magnetic\n\feld properties. We would also like to study caseswithout driven turbulence and with and without self-\nconsistent feedback driven turbulence. Future studies\ncould also explore how the gas dynamics, and s\u0003in par-\nticular, changes with the inclusion of more realistic ther-\nmal physics, such as that associated with an ambient\nFUV \feld and comsic rays (e.g., similar to the thermal\nphysics set up in Wu et al. 2017). As discussed, further\nwork is also needed to understand the potential con-\nnection between the values of s\u0003and thesdvalue from\nKhullar et al. (2021), as well as the role of disk rotation\nand accretion.\n6.CONCLUSIONS\nPrevious work has presented evidence for the cy-\ncling of gas between di\u000berent parts of gas density PDF\nwithin molecular regions: the high-density power-law\ntail, out of which stars form, and the non-star-forming\nlog-normal portion at average and low densities (Appel\net al. 2022). In this paper, we build on this analysis and\nfurther investigate the gas dynamics within star-forming\nregions using metrics such as the compression and ex-\npansion rates of the gas as a function of the gas density,\nand the gas mass \rux through di\u000berent portions of the\ndensity PDF.\nWe \fnd that:\n•The overall gas dynamics are dominated by com-\npressing gas at densities above the mean density\n(corresponding to the power-law part of the den-\nsity PDF), in agreement with the fact that the sim-\nulations are undergoing net gravitational collapse\nat high densities. In particular, at the highest den-\nsities, the net rate of all of our runs matches the\nnet rate of the run with only gravity, suggesting\nthat processes other than gravity have little e\u000bect\nat these densities.\n•At average to low densities (corresponding to the\nlognormal part of the density PDF), turbulence\nproduces both compression and expansion, and re-\nsults in a relatively constant rate, independent of\ngas density. This rate is signi\fcantly higher than\nthe free-fall rate at these low densities.\n•We \fnd that the net gas mass \rux peaks at the\ntransition between the lognormal and power-law\nforms of the density probability distribution func-\ntion. This is consistent with the transition density\ntracking the post-shock density, which promotes\nan enhancement of mass at this density (i.e., shock\ncompression and \flament formation).\n•The inclusion of stellar feedback in the form of\nprotostellar out\rows has a signi\fcant e\u000bect on the18 Appel et al.\ngas dynamics at low densities where protostellar\nout\rows result in very rapidly expanding and com-\npressing gas.\n•For simulations that include turbulent velocities,\nthe net gas mass \rux above the transition density\ndeclines until it reaches a constant value (i.e., it\nplateaus). The net gas mass \rux becomes constant\nat a density within the power-law tail, which we\ndenote ass\u0003. The gas mass \rux at s\u0003closely traces\nthe SFR, despite it being a far lower density than\nthe sink threshold. This suggests that the gas dy-\nnamics at this density, s\u0003, play an important role\nin setting the SFR. We \fnd that s\u0003varies slightly\nwith the inclusion of di\u000berent physics.ACKNOWLEDGMENTS\nS.M.A. & B.B. acknowledge support from NSF\ngrant AST-2009679. B.B. is grateful for generous sup-\nport by the David and Lucile Packard Foundation and\nAlfred P. Sloan Foundation. Support for V.S. was\nprovided by NASA through the NASA Hubble Fel-\nlowship grant HST-HF2-51445.001-A awarded by the\nSpace Telescope Science Institute, which is operated\nby the Association of Universities for Research in As-\ntronomy, Inc., for NASA, under contract NAS5-26555,\nand by Harvard University through the Institute for\nTheory and Computation Fellowship. C.F. acknowl-\nedges funding by the Australian Research Council (Fu-\nture Fellowship FT180100495 and Discovery Projects\nDP230102280), and the Australia-Germany Joint Re-\nsearch Cooperation Scheme (UA-DAAD). A.L.R. ac-\nknowledges support from the National Science Foun-\ndation (NSF) Astronomy and Astrophysics Postdoc-\ntoral Fellowship under award AST-2202249. J.C.T. ac-\nknowledges support from NSF grant AST-2009674. The\nanalysis and simulations were performed using comput-\ning resources provided by the Flatiron Institute and\nthe NCI Gadi cluster. We further acknowledge high-\nperformance computing resources provided by the Leib-\nniz Rechenzentrum and the Gauss Centre for Super-\ncomputing (grants pr32lo, pn73\f, and GCS Large-scale\nproject 22542), and the Australian National Computa-\ntional Infrastructure (grant ek9) in the framework of the\nNational Computational Merit Allocation Scheme and\nthe ANU Merit Allocation Scheme. The software used\nin this work was developed in part by the DOE NNSA\n- and DOE O\u000ece of Science - supported Flash Center\nfor Computational Science at the University of Chicago\nand the University of Rochester. The authors further\nacknowledge the use of the following software: yt (Turk\net al. 2011), \rash (Fryxell et al. 2000), SciPy (Virtanen\net al. 2020), scikit-learn (Pedregosa et al. 2011), Mat-\nplotlib (Hunter 2007), astropy (Astropy Collaboration\net al. 2013).\nAPPENDIX\nA.EULERIAN AND LAGRANGIAN CONTINUITY\nEQUATION\nIn Section 3.2, we use the Lagrangian formulation of\nthe continuity equation. Here, we brie\ry show how to\nderive the Lagrangian formation of the continuity equa-\ntion.\nFirst, let us consider a \ruid element of volume Vand\ndensity\u001a. The mass of this element remains constantin time, even as the density and volume may change.\nThus,\nD\nDt(\u001aV) = 0()VD\u001a\nDt+\u001aDV\nDt= 0 (A1)\nwhere we use D=Dt as a reminder that we are using\nthe Lagrangian derivative. Considering only the secondThe Density PDF and the Gas Compression and Expansion Rates 19\nterm of the latter expression:\nDV\nDt=I\nA\u0010\n~ v\u0001~A\u0011\nds\n=Z\nV(r\u0001~ v)dV\n= (r\u0001~ v)V (A2)\nwhere~ vis the velocity \feld, ~Ais the normal vector of\nthe surface, and we have used the Divergence theorem\nto go from the \frst to the second line. We can then\nrewrite Eq. A1 as:\nVD\u001a\nDt+\u001a[(r\u0001~ v)V] = 0\nD\u001a\nDt+\u001a(r\u0001~ v) = 0 (A3)\nwhich gives us the familiar Lagrangian formulation of\nthe continuity equation in terms of \u001a, shown in Eq. 3.\nSince we wish to compare this expression to the den-\nsity PDF, which we have calculated in terms of s=\nln(\u001a=\u001a0), we rearrange Eq. A3 in terms of s. First, we\nrewrite\u001a=\u001a0es. Then,\nD\nDt(\u001a0es) + (\u001a0es) (r\u0001~ v) = 0\n(\u001a0es)Ds\nDt+ (\u001a0es) (r\u0001~ v) = 0\nDs\nDt+ (r\u0001~ v) = 0 (A4)\nThus,\nD(s)\nDt\u0011\u0000(r\u0001~ v) (A5)\nwheres= ln(\u001a=\u001a0). Thus, we have a connection be-\ntween the time evolution of the density (the Ds=Dt\nterm) and the gas dynamics (as represented by the ve-\nlocity vector, ~ v).\nHowever, we can also derive the Lagrangian continu-\nity equation from the Eulerian formulation. From the\nstandard Eulerian continuity equation, we \fnd\n@\u001a\n@t+r\u0001(\u001a~ v) = 0\n()@\u001a\n@t+ (~ v\u0001r)\u001a=\u0000\u001ar\u0001~ v\n()1\n\u001aD\u001a\nDt=\u0000r\u0001~ v\n()Ds\nDt=\u0000r\u0001~ v; (A6)where the Lagrangian (co-moving) derivative D=Dt =\n@=@t +(~ v\u0001r) ands= ln(\u001a=\u001a0) were used in the last two\nsteps.\nB.ANALYSIS OF RUN GTMJ\nIn Sections 3 through 5, we focused on run GTMJR\nand did not show the results for run GTMJ since the\ndi\u000berences between these runs for the purposes of our\nanalysis are minimal. We found that including both\nlines in our \fgures signi\fcantly cluttered our plots with-\nout substantially enhancing the understanding of our re-\nsults. However, for completeness, we use this appendix\nto present Figs. 6, 7, and 8 with run GTMJ also shown.\nFigure 10 reproduces Fig. 6 with the addition of run\nGTMJ . Very little di\u000berence can be found between\nrunsGTMJ andGTMJR , although there seems to be\nslightly more time variation in run GTMJR for the net\nrate at the lowest densities.\nFigure 11 reproduces Fig. 7 with the addition of run\nGTMJ . Again, there is very little di\u000berence between\nrunsGTMJ andGTMJR . In fact, the mean SFRs are\nalmost identical. The inclusion of radiative heating in\nrunGTMJR appears to slightly lower the median value\nof the compressing gas mass \rux relative to run GTMJ\nnears\u00186:5, however the di\u000berence is small and well\nwithin the 1-sigma time variation of both runs.\nFigure 12 reproduces Fig. 8 with the addition of run\nGTMJ . Again, there is very little di\u000berence between\nrunsGTMJ andGTMJR . Indeed, the value of s\u0003is\nvery similar for the two runs, as can be seen in Table 2.\nC.CHECK TIME VARIATION OF THE DENSITY\nPDF\nSimilar to the upper panels of Fig. 4, Fig. 13 shows\na comparison between the overall PDF and the expand-\ning or compressing PDF for all four physics cases (we\ndo not include run GTMJ here). We show three dif-\nferent points in time for each simulation, corresponding\nto just before the formation of the \frst sink particle\n(SFE = 0%), the approximate mid-point of each sim-\nulation (SFE = 5%), and the end of each simulation\n(SFE = 10%). For all four physics cases and all three\npoints in time, the same trend is apparent. At high\ndensities, the overall density PDF is well matched by\nthe compressing gas PDF but is much higher than the\nexpanding gas PDF. However, at low densities, the over-\nall PDF is well matched by the expanding gas PDF but\ndiverges from the compressing gas PDF. This con\frms\nthat most of the expanding gas is at low densities while\nthe compressing gas is predominantly at high densities.\nThe transition between these regimes is continuous and\nfairly gradual.20 Appel et al.\n5\n 0 5 10\ns=ln(/0)\n102\n101\n100101102103|Ds/Dt u| (1/Myr)\nMin SFE=2.0%\nMax SFE=10.0%Expanding Gas (Ds/Dt<0)\nG\nGT\nGTM\nGTMJ\nGTMJR\n1/tff(s)\nst\nssink\n5\n 0 5 10\ns=ln(/0)\n102\n101\n100101102103|Ds/Dt u| (1/Myr)\nCompressing Gas (Ds/Dt>0)\n5\n 0 5 10\ns=ln(/0)\n20\n020406080Net Rate (1/Myr)Net Rate\nFigure 10. Figure 6 with run GTMJ .\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5 10.0\ns=ln(/0)\n107\n106\n105\n104\nGas Mass Flux (M/yr)\nMin SFE=2.0%\nMax SFE=10.0%Expanding Gas (Ds/Dt<0)\nG\nGT\nGTM\nGTMJ\nGTMJR\nst\nssink\n7.5\n 5.0\n 2.5\n 0.0 2.5 5.0 7.5 10.0\ns=ln(/0)\nCompressing Gas (Ds/Dt>0)\nFigure 11. Figure 7 with run GTMJ .\nREFERENCES\nAppel, S. M., Burkhart, B., Semenov, V. A., Federrath, C.,\n& Rosen, A. 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S. 2011, ApJ, 731, 62,\ndoi: 10.1088/0004-637X/731/1/62\nFryxell, B., Olson, K., Ricker, P., et al. 2000, ApJS, 131,\n273, doi: 10.1086/317361\n106\n104\n102\np=(1/V) dV/ds\nGExpanding Gas (Ds/Dt<0)Compressing Gas (Ds/Dt>0)\nFull PDF\nExp./Comp. PDF\n106\n104\n102\np=(1/V) dV/ds\nGT\n106\n104\n102\np=(1/V) dV/ds\nGTMSFE = 0%\nSFE = 5%\nSFE = 10%\n5\n 0 5 10\ns=ln(/0)\n106\n104\n102\np=(1/V) dV/ds\nGTMJR\n5\n 0 5 10\ns=ln(/0)\nst\nssinkFigure 13. Each panel shows the volume-weighted density\nPDF for all of the gas in the simulation region (dotted line)\nand the volume-weighted density PDF for only the expand-\ning gas ( Ds=Dt < 0; left column) or only the compressing\ngas (Ds=Dt > 0; right column) within the simulation region.\nEach row shows a di\u000berent simulation and the color corre-\nsponds to three di\u000berent points in time (SFE = 0 ;5;10%).\nAs in Fig. 4, the transition density ( st) from Appel et al.\n(2022) is overplotted. 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C., Christie, D., et al. 2017, ApJ, 841, 88,\ndoi: 10.3847/1538-4357/aa6\u000ba" }, { "title": "2301.08197v1.Stochastic_entropy_production_associated_with_quantum_measurement_in_a_framework_of_Markovian_quantum_state_diffusion.pdf", "content": "Stochastic entropy production associated with quantum measurement in a framework\nof Markovian quantum state diffusion\nClaudia L. Clarke and Ian J. Ford\nDepartment of Physics and Astronomy and London Centre for Nanotechnology,\nUniversity College London, Gower Street, London WC1E 6BT, U.K.\nThe reduced density matrix that characterises the state of an open quantum system is a projection\nfrom the full density matrix of the quantum system and its environment, and there are many full\ndensity matrices consistent with a given reduced version. Without a specification of relevant details\nof the environment, the evolution of a reduced density matrix is therefore typically unpredictable,\neven if the dynamics are deterministic. With this in mind, we investigate a two level open quantum\nsystem using a framework of quantum state diffusion. We consider the pseudorandom evolution of\nits reduced density matrix when subjected to an environment-driven process of continuous quantum\nmeasurement of a system observable, using dynamics that asymptotically send the system to an\neigenstate. The unpredictability is characterised by a stochastic entropy production, the average\nof which corresponds to an increase in the subjective uncertainty of the quantum state adopted\nby the system and environment, given the underspecified dynamics. This differs from a change in\nvon Neumann entropy, and can continue indefinitely as the system is guided towards an eigenstate.\nAs one would expect, the simultaneous measurement of two non-commuting observables within the\nsame framework does not send the system to an eigenstate. Instead, the probability density func-\ntion describing the reduced density matrix of the system becomes stationary over a continuum of\npure states, a situation characterised by zero further stochastic entropy production. Transitions\nbetween such stationary states, brought about by changes in the relative strengths of the two mea-\nsurement processes, give rise to finite positive mean stochastic entropy production. The framework\ninvestigated can offer useful perspectives on both the dynamics and irreversible thermodynamics of\nmeasurement in quantum systems.\nI. INTRODUCTION\nIn classical mechanics, entropy quantifies subjective\nuncertainty in the adopted configuration of a system\nwhen only partial detail is available concerning the coor-\ndinates of the component particles. Predictability of fu-\nture behaviour when such a system is coupled to an sim-\nilarly underspecified environment is limited and knowl-\nedge of the state worsens with time, even if the dynam-\nics are entirely deterministic. The total entropy of the\nsystem and environment increases as a consequence. In\nmany situations such evolution can be associated with\nthe dissipation of potential energy into heat as the world\nprogresses into the future, and this underpins the role\nplayed by entropy in the (19th century) second law of\nthermodynamics [1–3].\nThe 21st century concept of entropy production, how-\never, is based on dynamical consideration of the proba-\nbilities of forward and backward sequences of events gov-\nerned by an effective stochastic dynamics. In this frame-\nwork of ‘stochastic thermodynamics’, entropy change is\nthe expectation value of a ‘stochastic entropy produc-\ntion’, and this has clarified a number of long standing\nconceptual issues [4–8].\nThe central aim of this paper is to employ entropy as a\ndescriptionofuncertaintyofadoptedconfigurationatthe\nlevel of a reduced density matrix in quantum mechanics.\nIn the absence of quantum measurement, the full density\nmatrix of a system together with its environment (the\n‘world’) evolves deterministically according to the uni-\ntary dynamics of the von Neumann equation. This cangive rise to a non-unitary evolution of the reduced den-\nsity matrix describing the system, corresponding to ther-\nmalisation for example [9–11]. But the trajectory that\na reduced density matrix follows will be unpredictable\nif the complete initial state of the world is not speci-\nfied. This intrinsic unpredictability holds whether or not\nwe impose traditional ideas of randomness arising from\nquantum mechanical measurement. It is natural, there-\nfore, to consider the idea of an effective Brownian motion\nof the reduced density matrix, with associated entropy\nincrease. The concept is illustrated in Fig. 1.\nIn developing this idea, we regard the reduced density\nmatrix as an analogue of classical system coordinates and\nhence as a physical description of the quantum state, not\nmerely as a vehicle for specifying probabilities of pro-\njective measurements or a representation of a state of\nknowledge. There is a ‘real’ evolution trajectory that\nneeds to be modelled, and if this is effectively stochastic,\nthen there is also a subjective uncertainty in the actual\nstate adopted by the world over time. But coordinates\nthat describe a real state of a system ought not to change\ndiscontinuously, which would seem to raise difficulties in\nconnection with the instantaneous jumps normally con-\nsidered to arise from quantum measurement. A realist\nviewpoint therefore obliges us to describe quantum mea-\nsurement in a fashion that avoids jumps.\nWe can use the description of ‘weak’ or continuous\nmeasurement processes in quantum mechanics to achieve\nthis [12–14], though we employ the ideas in a slightly\nunconventional fashion. Instead of regarding weak mea-\nsurement as a consequence of projective measurementsarXiv:2301.08197v1 [quant-ph] 19 Jan 20232\nFigure 1. The box and the grey area represent the phase\nspaces of the density matrix of the world \u001aworldand of the re-\nduceddensitymatrix \u001aofaconstituentopenquantumsystem,\nrespectively. Deterministic trajectories \u001aworld(t)that start at\nt= 0from a macrostate subspace (shown as a red line) char-\nacterised by a given initial value \u001a(0)of the reduced den-\nsity matrix, are manifested as pseudorandom trajectories \u001a(t)\nwhen projected into the reduced phase space.\nof remote parts of the environment coupled to the sys-\ntem, we imagine that complex dynamical interactions ex-\nist between the system and environment that can guide\nthe system towards eigenstates of observables under cer-\ntain conditions [15]. In other words, we imagine a situ-\nation where quantum measurement is just an aspect of\nthe unitary dynamics of the world, its stochasticity be-\ning a consequence of a failure to specify the degrees of\nfreedom of the environment, or more precisely those of a\nmeasuring device. This is reminiscent of ideas employed\nin classical statistical mechanics.\nThe implications of such a point of view can be cap-\ntured by a continuous, Markovian, stochastic evolution\nof the reduced density matrix according to a framework\nknown as quantum state diffusion [16–20], a broad cate-\ngory of dynamics that includes weak measurement. More\nelaborate schemes are also possible, for example involv-\ning non-Markovian dynamics. Such modelling is consis-\ntent with strong projective measurements as a limiting\nbehaviour and can be made compatible with the Born\nrule. Measurement is then a process driven by specific\nsystem-environment coupling terms in the Hamiltonian\nand takes place without discontinuities [21–23]. This is\na quantum dynamics that resembles classical dynamics,\nbut where the dynamical variables are the elements of\na reduced density matrix. It combines both aspects of\nquantum evolution: determinism of the von Neumann\nequation together with effective stochasticity represent-\ning measurement or more general environmental effects\n[24]. It is not without its controversies [25–28].\nIt is nevertheless highly advantageous to employ an\nevolution of the reduced density matrix that avoids dis-\ncontinuities, because then the concept of stochastic en-\ntropy production can be implemented in quantum me-chanics in a straightforward way [29–38]. If the dy-\nnamical variables evolve according to Markovian stochas-\ntic differential equations (SDEs), or Itô processes [39],\nthen it is possible to derive a related Itô process for the\nstochastic entropy production [7]. We can then compute\na stochastic entropy production associated with individ-\nual Brownian trajectories taken by the reduced density\nmatrix of a system. This can include situations where\nthe system is guided towards an eigenstate of an observ-\nable, henceenablingustocomputethestochasticentropy\nproduction characterising a process of measurement.\nA positive expectation value of stochastic entropy pro-\nduction represents increasing subjective uncertainty in\nthe quantum state of the world. Growth in uncertainty\nis natural since we model the evolution using stochastic\nmethods starting from an incompletely specified initial\nstate. The state of the system can become lessuncer-\ntain, a necessary consequence of the performance of mea-\nsurement, but subjective uncertainty regarding the state\nof the rest of the world increases by a greater amount,\nthereby allowing the second law of thermodynamics to\nbe satisfied. It should be noted that stochastic entropy\nproduction does not correspond to a change in von Neu-\nmann entropy, which instead describes the uncertainty of\noutcomes when a system is subjected to projective mea-\nsurement in a specific basis.\nIn Section II we develop these ideas in the context of\nthe measurement of a single observable in a two level\nquantum system starting in a mixed state [40]. Mean\nstochastic entropy production is found to be positive and\nwithout limit as the system is guided, asymptotically in\ntime, into one or other of the two eigenstates. We go\non in Section III to consider the simultaneous measure-\nment of two non-commuting observables and show how\nthe stochastic entropy production is finite, a consequence\noftheinabilityofthedynamics, inthissituation, toguide\nthe system asymptotically into a definite eigenstate of ei-\nther observable. We interpret the results in Section IV\nand summarise our conclusions in Section V, suggesting\nthat dynamics based on quantum state diffusion, with an\ninterpretation of the reduced density matrix as a set of\nphysical properties of a state, together with the use of\nstochastic entropy production to monitor the process of\neigenstate selection, can provide some conceptual clarifi-\ncation of the quantum measurement problem [41].\nII. MEASUREMENT OF \u001bz\nA. Dynamics\nThe two level system will be described by a reduced\ndensity matrix (hereafter, simply a density matrix \u001a) de-\nfined in a basis of eigenstates j\u00061iof the\u001bzoperator.\nPure states denoting occupation of each of the two levels\ncorrespond to \u001ae\n\u0006=j\u00061ih\u00061j. Starting in the mixed\nstate\u001a=1\n2\u0000\na1\u001ae\n++a\u00001\u001ae\n\u0000\u0001\n, wherea\u00061are amplitudes,\nwe use a quantum state diffusion approach to model the3\nstochastic evolution of the system into one or other of\nthe levels in accordance with the Born rule.\nWe consider a minimal scheme [13] employing a rule\nfor stochastic transitions given by\n\u001a!S\u0006(\u001a) =\u001a0\u0006=M\u0006\u001aMy\n\u0006\nTr\u0010\nM\u0006\u001aMy\n\u0006\u0011;(1)\nusing two Kraus operators:\nM\u0006=1p\n2\u0012\nI\u00001\n2cycdt\u0006cp\ndt\u0013\n; (2)\nwherec=\u000bz\u001bz, withrealscalarparameter \u000bzdesignated\nasthestrengthofmeasurement. Theprobabilitiesforthe\nselection of one of the two possible outcomes \u001a0\u0006after an\ninfinitesimal timestep dtare\np\u0006(\u001a) = Tr\u0010\nM\u0006\u001aMy\n\u0006\u0011\n=1\n2\u0010\n1\u0006Cp\ndt\u0011\n;(3)\nwhereC= Tr\u0000\n\u001a\u0000\nc+cy\u0001\u0001\n. The quantum map in Eq. (1)\npreserves the trace of \u001a. Furthermore, since the Kraus\noperators in Eq. (2) differ incrementally from (a multi-\nple of) the identity, the positive definiteness of \u001ais main-\ntained [24]. The operator identity My\n+M++My\n\u0000M\u0000=I\nis also satisfied. This scheme defines a stochastic dynam-\nics representing the effect of a coupled measuring device\non the two level system, whereby the eigenstates of \u001bz\nare stationary, i.e. p+(\u001ae\n+) =p\u0000(\u001ae\n\u0000) = 1,p\u0000(\u001ae\n+) =\np+(\u001ae\n\u0000) = 0, andS+(\u001ae\n+) =\u001ae\n+,S\u0000(\u001ae\n\u0000) =\u001ae\n\u0000.\nThe two possible increments d\u001a\u0006=\u001a0\u0006\u0000\u001aavailable\nin the timestep dtunder the dynamics are\nd\u001a\u0006=\u0012\nc\u001acy\u00001\n2\u001acyc\u00001\n2cyc\u001a\u0013\ndt\u0000\u0000\n\u001acy+c\u001a\u0000C\u001a\u0001\nCdt\n\u0006\u0000\n\u001acy+c\u001a\u0000C\u001a\u0001p\ndt; (4)\nand by evaluating the mean and variance of this incre-\nment in\u001ait may be shown that the evolution can also\nbe represented by the Itô process\nd\u001a=\u0012\nc\u001acy\u00001\n2\u001acyc\u00001\n2cyc\u001a\u0013\ndt+\u0000\n\u001acy+c\u001a\u0000C\u001a\u0001\ndW;\n(5)\nwheredWis a Wiener increment with mean hdWi= 0\nand variancehdW2i=dt, with the brackets represent-\ning an average over the stochasticity. Note that terms of\nhigher order than linear in dtwill be neglected through-\nout.\nSuch a process of averaging over the stochasticity then\nleads to the standard Lindblad equation [42]:\nd\u0016\u001a\ndt=c\u0016\u001acy\u00001\n2\u0016\u001acyc\u00001\n2cyc\u0016\u001a; (6)\nwith \u0016\u001a=h\u001ai, suggestingthatsucha(deterministic)equa-\ntion describes the average dynamical behaviour of an en-\nsemble of density matrices. The actual trajectory fol-\nlowed by a system as it responds to external interactions,\n0.0 0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.0Figure 2. Four stochastic trajectories rz(t)derived from Eq.\n(7) with strength of measurement \u000bz= 1. Starting at rz(0) =\n0, they evolve towards eigenstates of the \u001bzobservable at\nrz=\u00061.\nhowever, is specified by the stochastic Lindblad equation\n(5) [43, 44]. The environment disturbs the system in\na manner represented by one of the transformations or\nmoves given in (1), selected at random with probabilities\n(3) that arise from the underspecification of the environ-\nmental state and hence of \u001aworld.\nFurthermore, if we represent the density matrix in the\nform\u001a=1\n2(I+rz\u001bz), it may be shown that the dynam-\nics of Eq. (5) correspond to the evolution of the real\nstochastic variable rz(t)according to [13]\ndrz= 2\u000bz\u0000\n1\u0000r2\nz\u0001\ndW: (7)\nExamplerealisationsofsuchbehaviour, startingfromthe\nfully mixed state at rz(0) = 0, are shown in Fig. 2.\nNotice that rzevolves asymptotically towards \u00061, cor-\nresponding to density matrices \u001ae\n\u0006, and note also that\nthe average increment hdrziover the ensemble satisfies\nhdrzi=dhrzi= 2\u000bz\u0000\n1\u0000hr2\nzi\u0001\nhdWi= 0, implying that\nhrziis time independent and that h\u001aiis as well. A similar\nconclusion can be reached simply by evaluating the right\nhand side of Eq. (6).\nThe standard Lindblad equation cannot capture sys-\ntem ‘collapse’ to an eigenstate, but instead describes the\naverage behaviour of an ensemble of collapsing systems.\nFor a closer consideration of the dynamics and thermo-\ndynamics of collapse, we need to ‘unravel’ the standard\nLindblad equation into its stochastic version (5), using it\nto generate an ensemble of trajectories that model pos-\nsible physical evolutions of the open quantum system.\nUsing Itï¿œ’s lemma, it can be shown that the purity\nof the state, P= Tr\u001a2=1\n2\u0000\n1 +r2\nz\u0001\n, evolves according to\ndP= 8\u000b2\nz(1\u0000P)2dt+ 4\u000bzrz(1\u0000P)dW:(8)\nThe dynamics take the purity asymptotically towards a\nfixed point at P= 1, or the density matrix towards one\nof\u001ae\n\u0006, which is clearly a necessary consequence of the\nprocess of measurement.4\nFigure 3. A probability density function p(rz;t), evolving\naccording to the Fokker-Planck equation (9), describing the\nevolution of an ensemble of density matrices under measure-\nment of\u001bz. A gaussian centred initially at the origin accu-\nmulates asymptotically at rz=\u00061. This complements the\ndirect computation of trajectories rz(t)illustrated in Fig. 2.\nThe Fokker-Planck equation describing the evolution\nof the probability density function (pdf) p(rz;t)for the\nsystem variable rzis\n@p\n@t=@2\n@r2z\u0010\n2\u000b2\nz\u0000\n1\u0000r2\nz\u00012p\u0011\n; (9)\nand this provides further insight into the dynamics. Fig-\nure3illustratesthedevelopmentstartingfromagaussian\npdf centred on the maximally mixed state at rz= 0. The\nensemble of density matrices is separated by the dynam-\nics into equal size groups that evolve asymptotically to-\nwards the eigenstates of \u001bzatrz=\u00061. The preservation\nof the ensemble average of rzis apparent.\nB. Stochastic entropy production\nThe (total) stochastic entropy production associated\nwith the evolution of a stochastic variable in a certain\ntime interval is defined in terms of probabilities for the\ngeneration of a ‘forward’ set of moves in the phase space\nand the corresponding ‘backward’ set [4]. For the coor-\ndinaterz, and the time interval dt, we need to consider\nthe quantity\nd\u0001stot(rz;t!rz+drz;t+dt) (10)\n= ln (Prob(forward) =Prob(backward))\n= lnp(rz;t)\u0001rz(rz)T(rz!rz+drz)\np(rz+drz;t+dt)\u0001rz(rz+drz)T(rz+drz!rz);\nwhere theTare conditional probabilities for the tran-\nsitions indicated. For stochastic variables that are odd\nunder time reversal symmetry, additional features have\nto be included in this definition, but since rzis even we\ncan ignore such complications [7, 45].It may be shown that the expectation or mean of\nd\u0001stotis never negative, which ultimately provides an\nunderpinning for the second law of thermodynamics [4].\nWe shall discuss the contributions to d\u0001stotinvolv-\ning the pdf p(rz;t)and the volume increment \u0001rz(rz)\nshortly, but first let us consider the ratio of conditional\nprobabilities. The two choices of forward move \u001a!\u001a0\u0006\nin Eqs. (1) and (2) are selected with probabilities\np\u0006=1\n2\u0010\n1\u00062\u000bzrzp\ndt\u0011\n: (11)\nThe corresponding backward moves \u001a0\u0006!\u001aare de-\nscribed by the quantum maps\n\u001a=~M\u0007\u001a0\u0006~My\n\u0007\nTr\u0010\n~M\u0007\u001a0\u0006~My\n\u0007\u0011; (12)\nin terms of reverse Kraus operators ~M\u0007that can be iden-\ntified from the condition that the initial density matrix\nis recovered. Inserting Eq. (1) into Eq. (12) we have\n\u001a=~M\u0007M\u0006\u001aMy\n\u0006~My\n\u0007\nTr\u0010\n~M\u0007M\u0006\u001aMy\n\u0006~My\n\u0007\u0011; (13)\nwhich requires ~M\u0007M\u0006to be proportional to the identity,\nup to linear order in dt. Forc=cythis can be achieved\nusing\n~M\u0007=1p\n2\u0012\nI\u00001\n2c2dt\u0007cp\ndt\u0013\n=M\u0007;(14)\nand specifically for c=\u000bz\u001bzwe have\n~M\u0007M\u0006=1\n2\u0000\n1\u00002\u000b2\nzdt\u0001\nI: (15)\nHence the probabilities for the backward moves are\np0\n\u0007= Tr\u0010\n~M\u0007\u001a0\u0006~My\n\u0007\u0011\n=Tr\u0010\nM\u0007M\u0006\u001aMy\n\u0006My\n\u0007\u0011\nTr\u0010\nM\u0006\u001aMy\n\u0006\u0011;(16)\nleading to\np0\n\u0007=\u0000\n1\u00004\u000b2\nzdt\u0001\n2\u0010\n1\u00062\u000bzrzp\ndt\u0011: (17)\nThe ratio of conditional probabilities T(rz!rz+\ndr\u0006\nz)=T(rz+dr\u0006\nz!rz)is then\np\u0006\np0\n\u0007= 1\u00064\u000bzrzp\ndt+ 4\u000b2\nz\u0000\n1 +r2\nz\u0001\ndt:(18)\nThe two possible increments in rzare\ndr\u0006\nz= Tr\u0000\n\u001a0\u0006\u001bz\u0001\n\u0000rz\n=\u00004\u000b2\nzrz\u0000\n1\u0000r2\nz\u0001\ndt\u00062\u000bz\u0000\n1\u0000r2\nz\u0001p\ndt;(19)5\nand we note that the mean and variance over the two\npossibilities are\nhdrzi=p+dr+\nz+p\u0000dr\u0000\nz= 0; (20)\nand\n\u001b2\nrz=p+\u0000\ndr+\nz\u0000hdrzi\u00012+p\u0000\u0000\ndr\u0000\nz\u0000hdrzi\u00012\n= 4\u000b2\nz\u0000\n1\u0000r2\nz\u00012dt: (21)\nconfirming that the evolution is consistent with the SDE\nforrzin Eq. (7). The moves and their probabilities are\nillustrated in Fig. 4.\nWe now write\nd\u0001s\u0006\ntot=d\u0001s\u0006\nA+d\u0001s\u0006\nB; (22)\nwhere\nd\u0001s\u0006\nA= ln\u0012T(rz!rz+dr\u0006\nz)\nT(rz+dr\u0006z!rz)\u0013\n= ln\u0012p\u0006\np0\n\u0007\u0013\n;(23)\nand\nd\u0001s\u0006\nB= ln\u0012p(rz;t)\u0001rz(rz)\np(rz+dr\u0006z;t+dt)\u0001rz(rz+dr\u0006z)\u0013\n:(24)\nInserting Eq. (18) we have\nd\u0001s\u0006\nA=\u00064\u000bzrzp\ndt+ 4\u000b2\nz\u0000\n1\u0000r2\nz\u0001\ndt;(25)\nwhichprovidestwochoicesofincrementalcontributionto\nthe stochastic entropy production in the forward move.\nWe can compute the mean of d\u0001s\u0006\nA:\nhd\u0001sAi=p+d\u0001s+\nA+p\u0000d\u0001s\u0000\nA\n= (p+\u0000p\u0000) 4\u000bzrzp\ndt+ (p++p\u0000) 4\u000b2\nz\u0000\n1\u0000r2\nz\u0001\ndt\n= 4\u000b2\nz\u0000\n1 +r2\nz\u0001\ndt; (26)\nand the variance:\n\u001b2\nA=p+\u0000\nd\u0001s+\nA\u0000hd\u0001sAi\u00012+p\u0000\u0000\nd\u0001s\u0000\nA\u0000hd\u0001sAi\u00012\n= 16\u000b2\nzr2\nzdt; (27)\nfrom which we conclude that the evolution can be repre-\nsented by an Itô process for a stochastic variable \u0001sA:\nd\u0001sA= 4\u000b2\nz\u0000\n1 +r2\nz\u0001\ndt+ 4\u000bzrzdW: (28)\nWenowconsiderthecontribution d\u0001s\u0006\nBtothestochas-\ntic entropy production given in Eq. (24). The volume\n\u0001rz(rz)is the region bounded by increments1\n2dr\u0006\nzstart-\ning fromrz. It is the patch of phase space associated\nwith coordinate rz, as illustrated in Fig. 4. We write\n\u0001rz=1\n2(dr+\nz\u0000dr\u0000\nz) = 2\u000bz\u0000\n1\u0000r2\nz\u0001p\ndtand then\nd\u0001s\u0006\nB=\u0000dlnp\u0006+d\u0001s\u0006\nC; (29)\nrz\nFigure 4. Available moves on a discrete set of locations on the\nrzaxis according to the stochastic dynamics of measurement\nof\u001bz, illustrating Eqs. (11), (17) and (19). The size of the\ncircles represents the local probability density p(rz;t). The\nshaded rectangle represents the volume \u0001rz=1\n2\u0000\ndr+\nz\u0000dr\u0000\nz\u0001\nof the continuum phase space associated with a given location\nrz.\nwheredlnp\u0006= lnp(rz+dr\u0006\nz;t+dt)\u0000lnp(rz;t)and\nd\u0001s\u0006\nC= ln\u0012\u0001rz(rz)\n\u0001rz(rz+dr\u0006z)\u0013\n= 4\u000b2\nz\u0000\n1\u0000r2\nz\u0001\ndt\u00064\u000bzrzp\ndt:(30)\nThe mean of d\u0001s\u0006\nCis\nhd\u0001sCi=p+d\u0001s+\nC+p\u0000d\u0001s\u0000\nC\n= 4\u000b2\nz\u0000\n1 +r2\nz\u0001\ndt; (31)\nand the variance is\n\u001b2\nC=p+\u0000\nd\u0001s+\nC\u0000hd\u0001sCi\u00012+p\u0000\u0000\nd\u0001s\u0000\nC\u0000hd\u0001sCi\u00012\n= 16\u000b2\nzr2\nzdt; (32)\nso the Itô process for this component of stochastic en-\ntropy production is\nd\u0001sC= 4\u000b2\nz\u0000\n1 +r2\nz\u0001\ndt+ 4\u000bzrzdW: (33)\nSimilarly, it may be shown that the term \u0000dlnp\u0006in\nEq. (29) makes a contribution of \u0000dlnpto the Itô pro-\ncess ford\u0001stot. Combining this with Eqs. (22), (28),\n(29) and (33), the stochastic entropy production can be\nshown to evolve according to the Itô process\nd\u0001stot=\u0000dlnp(rz;t) + 8\u000b2\nz\u0000\n1 +r2\nz\u0001\ndt+ 8\u000bzrzdW:\n(34)\nNote that the term \u0000dlnp(rz;t)is usually referred\nto as the stochastic entropy production of the system,\nd\u0001ssys. The remaining terms are then regarded as\nstochastic entropy production in the environment (in\nthis case the measuring device), and denoted d\u0001senvor\nd\u0001smeas:Note that the evolution of the stochastic en-\ntropy production in Eq. (34), with a system contribution\nthat depends on the pdf p(rz;t)over the phase space of\nthe density matrix, is continuous. This is in contrast\nto implementations of stochastic entropy production in\nquantum mechanics that involve the probability distri-\nbution over eigenstates of the measured operator in the\nformalism, or that invoke projective measurements caus-\ning discontinuities that are potentially infinite in magni-\ntude [33].6\nC. Derivation of d\u0001stotfrom the dynamics\nThe derivation of d\u0001stotin the previous section is in-\ntricate, but there is an alternative approach that is much\nmore straightforward [6, 7] and does not require the iden-\ntification of reverse Kraus operators [46]. Let us consider\nan Itô process for a stochastic variable xin the form\ndx=\u0000\nArev(x;t) +Airr(x;t)\u0001\ndt+B(x;t)dW;(35)\nwhere the terms proportional to ArevandAirrrepresent\nmodes of deterministic dynamics that satisfy and violate\ntime reversal symmetry, respectively. Then the stochas-\ntic entropy production is given by\nd\u0001stot=\u0000dlnp(x;t) +Airr\nDdx\u0000ArevAirr\nDdt+@Airr\n@xdt\n\u0000@Arev\n@xdt\u00001\nD@D\n@xdx+(Arev\u0000Airr)\nD@D\n@xdt\n\u0000@2D\n@x2dt+1\nD\u0012@D\n@x\u00132\ndt; (36)\nwhereD(x;t) =1\n2B(x;t)2. This may not seem very\nintuitive, but for dynamics that possess a stationary\nstate with zero probability current, characterised by a\npdfpst(x), Eq. (36) reduces to the simpler expression\nd\u0001stot=\u0000dln (p(x;t)=pst(x)).\nForthedynamicsof rzgivenbyEq. (7)wehave Arev=\nAirr= 0andB= 2\u000bz\u0000\n1\u0000r2\nz\u0001\n. HenceD= 2\u000b2\nz(1\u0000\nr2\nz)2, leading to dD=drz=\u00008\u000b2\nzrz(1\u0000r2\nz),d2D=dr2\nz=\n\u00008\u000b2\nz(1\u00003r2\nz), and\nd\u0001stot=\u0000dlnp\u00001\nDdD\ndrzdrz\u0000d2D\ndr2zdt+1\nD\u0012dD\ndrz\u00132\ndt\n=\u0000dlnp+ 8\u000b2\nz\u0000\n1 +r2\nz\u0001\ndt+ 8\u000bzrzdW: (37)\nThis is the same as Eq. (34), but the derivation is much\nmoredirect. ExtensiontosetsofcoupledItôprocessesfor\nseveral stochastic variables fxigis straightforward, and\nwe shall encounter an example of such a generalisation in\nSection III.\nD. Results\nLet us now consider the character of the stochastic\nentropy production described by Eq. (37). It is straight-\nforward to evaluate \u0001stot(t)numerically, employing solu-\ntions to the Fokker-Planck equation (9) and the Itô pro-\ncess forrz(t). Example evolutions of \u0001stot(t)associated\nwith trajectories rz(t)are shown in Fig. 5, for \u000bz= 1.\nThe mean stochastic entropy production over a sample of\ntrajectories appears to rise linearly in time. The increase\nreflects the fact that the pdf p(rz;t)does not reach a sta-\ntionary state, but instead progressively sharpens towards\ntwo\u000e-function peaks at rz=\u00061. The system approaches\none of the eigenstates but never quite reaches it. A sys-\ntem that continues to evolve in response to time reversal\n1.0 1.2 1.4 1.6 1.8 2.0024681012Figure 5. Four trajectories illustrating the stochastic entropy\nproduction \u0001stot(t)forthedynamicsofEq. (7)intheinterval\n1\u0014t\u00142, starting from a gaussian pdf centred on rz= 0at\nt= 0, and with \u000bz= 1. The mean over a sample of 40\ntrajectories is consistent with an asymptotic average rate of\nproduction equal to 8\u000b2\nz, as suggested in Eq. (47).\nasymmetric dynamics (which includes the noise term as\nwellasthedeterministiccontributionproportionalto Airr\nin Eq. (35)) is characterised by never-ending stochastic\nentropy production.\nThe calculations of \u0001stotin Fig. 5 were actually ob-\ntainedafterperformingatransformationofthestochastic\nvariable to avoid difficulties arising from the singulari-\nties inp(rz;t)ast!1. It is possible to do this since\nthe stochastic entropy production is invariant under a\ncoordinate transformation. Consider, then, the variable\ny= tanh\u00001rz, which evolves in time according to\ndy= 4\u000b2\nztanhydt+ 2\u000bzdW; (38)\nusing Itï¿œ’s lemma. The phase space \u00001\u0014rz\u00141maps\nto\u00001\u0014y\u00141. We identify Arev(y) = 0,Airr(y) =\n4\u000b2\nztanhyandD(y) = 2\u000b2\nzand write\nd\u0001stot=\u0000dlnp(y;t) +Airr\nDdy+dAirr\ndydt\n=\u0000dlnp(y;t) + 4\u000b2\nz\u0000\n1 + tanh2y\u0001\ndt+ 4\u000bztanhydW;\n(39)\nwhere the pdf for ysatisfies the Fokker-Planck equation\n@p\n@t=\u00004\u000b2\nz@\n@y(tanhyp) + 2\u000b2\nz@2p\n@y2:(40)\nSolving Eqs. (38), (39) and (40) numerically produces\nthe trajectories in Fig. 5.\nWe can perform an analysis of the evolution at late\ntimes, where rzis close to 1 or\u00001such thatjyjis large.\nThe dynamics are then approximated by\ndy=\u00064\u000b2\nzdt+ 2\u000bzdW; (41)7\nemployingtheplussignif y>0andthenegativeif y<0.\nThe Fokker-Planck equation is\n@p\n@t=\u00004\u000b2\nzsgn(y)@p\n@y+ 2\u000b2\nz@2p\n@y2; (42)\nwhich has an approximate asymptotic solution:\np(y;t)/1\nt1=2\u0014\nexp\u0014\n\u0000(y\u00004\u000b2\nzt)2\n8\u000b2zt\u0015\n+ exp\u0014\n\u0000(y+ 4\u000b2\nzt)2\n8\u000b2zt\u0015\u0015\n;\n(43)\nconsisting of two gaussians in the yphase space, moving\nwith equal and opposite drift velocities towards \u00061and\nsimultaneously broadening.\nFromEq. (39)weobtainstochasticentropyproduction\nfor a trajectory with y\u001d0of\nd\u0001stot\u0019\u0000dlnp+(y;t) + 8\u000b2\nzdt+ 4\u000bzdW; (44)\nwith\np+/1\nt1=2exp\u0012\n\u0000(y\u00004\u000b2\nzt)2\n8\u000b2zt\u0013\n; (45)\nand hence\nd\u0001stot\u0019d\u0012(y\u00004\u000b2\nzt)2\n8\u000b2zt\u0013\n+1\n2dlnt+ 8\u000b2\nzdt+ 4\u000bzdW;\n(46)\nthe average of which is\ndh\u0001stoti\u00191\ntdt\u0000h(y\u00004\u000b2\nzt)2i\n8\u000b2zt2dt+ 8\u000b2\nzdt\n=1\ntdt\u00004\u000b2t\n8\u000b2t2dt+ 8\u000b2\nzdt; (47)\nwhich reduces to 8\u000b2\nzdtast!1. A similar conclusion\ncan be reached if y\u001c0, so we expect mean stochastic\nentropy production at a constant rate 8\u000b2\nzast!1,\nconfirming the behaviour seen in Fig. 5.\nE. Contrast with von Neumann entropy\nAt this point we should consider whether stochastic\nentropy production is related to a change in the von\nNeumann entropy SvN=\u0000Tr\u001aln\u001a, a more commonly\nemployed expression for entropy in quantum mechanics.\nThe mean stochastic entropy production is the change\nin subjective uncertainty with regard to the quantum\nstate adopted by the world. We are unable to make exact\npredictions when the dynamical influence of the environ-\nmentonthesystemisnotspecifiedindetail. Thedynam-\nics then become effectively stochastic and our knowledge\nof the adopted state is reduced with time.\nIn contrast, the von Neumann entropy is the uncer-\ntainty presented by a quantum state with regard to the\noutcomes of projective measurement in a basis in which\nthe density matrix is diagonal. It is a Shannon entropy\n\u0000P\niPilnPiwherePiis the probability of projectioninto eigenstate iof the observable. For a two level system\nthe number of such outcomes is two and so the von Neu-\nmann entropy has an upper limit of ln 2. In contrast, the\nupper limit of the mean stochastic entropy production,\nrepresenting the change in uncertainty in the adopted\nquantum state of the world, is infinite, since there is a\ncontinuum of possible states that could be taken. The\ncontinued mean production of stochastic entropy associ-\nated with measurement, discussed in previous sections,\nrepresents this progressively greater uncertainty.\nAlso note that the stochastic entropy production we\nhave been considering has no connection with heat trans-\nfer or work. The two level system under consideration\ndoes not possess a Hamiltonian Hand the selection of\none or other level through measurement does not in-\nvolve a change in system energy: specifically TrH\u001a= 0\nthroughout. Stochastic entropy production is not neces-\nsarily associated with the dissipation of potential energy\ninto heat. Indeed it need not be in classical mechanics,\nfor example in the free expansion of an ideal gas. In both\nclassical and quantum settings the purpose of entropy is\nto specify the degree of configurational uncertainty. In\nclassical mechanics the configurations are described by\nsets of classical coordinates: in quantum mechanics they\nare specified by collections of (reduced) density matrix\nelements.\nVon Neumann entropy does play a role in computing\nthethermodynamicentropyofaquantumsysteminasit-\nuation where it is subjected to projective measurement\nand thereafter regarded as occupying one of the eigen-\nstates. However, it is not straightforward to involve von\nNeumann entropy in discussions of the second law and\nthe arrow of time. This is evident if we consider that the\nvonNeumannentropy \u0000Tr\u0016\u001aln \u0016\u001aoftheensembleaveraged\ndensity matrix \u0016\u001aremains constant under the measure-\nment dynamics employed here (because \u0016\u001aremains con-\nstant). And the von Neumann entropy of a typical mem-\nber of the considered ensemble of density matrices falls\nto zero under the dynamics. This is illustrated in Fig.\n6 for the two level system where \u001aevolves towards one\nof the\u001ae\n\u0006: the latter are pure states with SvN= 0. The\nmean von Neumann entropy change \u0000\u0001Trh\u001aln\u001aiassoci-\nated with the measurement process is negative. Neither\nof these outcomes makes it easy to argue that the von\nNeumann entropy has a role to play in the second law:\nthe stochastic entropy production is a better candidate.\nIII. SIMULTANEOUS MEASUREMENT OF \u001bz\nAND\u001bx\nA. Evolution towards purity\nNow we turn our attention to a slightly more compli-\ncated case of stochastic entropy production associated\nwith the dynamics of an open quantum system. We con-\ntinue to use the framework of quantum state diffusion,\ninvolving transformations according to Eq. (1), but we8\nFigure 6. Evolution of the von Neumann entropy of the re-\nduced density matrix of the two level system, for 10 stochastic\ntrajectories governed by the dynamics of Eq. (7) with \u000bz= 1.\nMean behaviour is also shown. Asymptotic values of zero im-\nply that the system is purified under measurement.\nnow represent the stochasticinfluence of theenvironment\non the system using twopairs of Kraus operators, given\nby\nM1\u0006=1\n2\u0012\nI\u00001\n2cy\n1c1dt\u0006c1p\ndt\u0013\nM2\u0006=1\n2\u0012\nI\u00001\n2cy\n2c2dt\u0006c2p\ndt\u0013\n;(48)\nwithc1=\u000bz\u001bzandc2=\u000bx\u001bx. The first and second pair\ndescribe the dynamics of continuous measurement of ob-\nservables\u001bzand\u001bx, respectively, and together therefore\nrepresent an attempt to perform simultaneous measure-\nment. Since \u001bzand\u001bxdo not commute, we expect this\nnot to succeed, and quantum state diffusion provides an\ninteresting illustration of what this means.\nProbabilities of stochastic changes in the reduced den-\nsity matrix of the system, brought about by interactions\nwith the environment, may be deduced for these opera-\ntors, and a stochastic Lindblad equation for its evolution\nmay be derived:\nd\u001a=X\ni=1;2\u0012\nci\u001acy\ni\u00001\n2\u001acy\nici\u00001\n2cy\nici\u001a\u0013\ndt\n+\u0010\n\u001acy\ni+ci\u001a\u0000Ci\u001a\u0011\ndWi; (49)\nwithCi= Tr\u0010\n(ci+cy\ni)\u001a\u0011\n. Upon inserting the repre-\nsentation\u001a=1\n2(I+rz\u001bz+rx\u001bx), the dynamics can be\nexpressed as\ndrz= 2\u000bz\u0000\n1\u0000r2\nz\u0001\ndWz\u00002\u000b2\nxrzdt\u00002\u000bxrzrxdWx\ndrx= 2\u000bx\u0000\n1\u0000r2\nx\u0001\ndWx\u00002\u000b2\nzrxdt\u00002\u000bzrxrzdWz;\n(50)\nwheredWxanddWzare independent Wiener increments.\nExample stochastic trajectories starting from the maxi-\nmally mixed state at rx=rz= 0are shown in Fig. 7.\n1.0 0.5Figure 7. Two trajectories of the density matrix coordinates\n(rx(t);rz(t)) generated by the dynamics of simultaneous mea-\nsurement of \u001bxand\u001bz, Eq. (50), starting from the maximally\nmixed state at the origin and for equal strengths of measure-\nment\u000bxand\u000bz. The black circle represents a condition of\npurity, towards which the system evolves. Eigenstates of \u001bx\nand\u001bzlie at\u0012=\u0006\u0019=2and\u0012= 0;\u0019on the circle, respectively.\nFigure 8. Evolution of purity for the system trajectories in\nFig. 7.\nThe purity P= Tr\u001a2=1\n2\u0000\n1 +r2\u0001\n, wherer2=r2\nx+r2\nz,\nevolves according to\ndP= 4\u0000\n\u000b2\nx\u0000\n1\u0000r2\nx\u0001\n+\u000b2\nz\u0000\n1\u0000r2\nz\u0001\u0001\n(1\u0000P)dt\n+ 4\u000bxrx(1\u0000P)dWx+ 4\u000bzrz(1\u0000P)dWz;(51)\nsuch thatP= 1is a fixed point that is reached asymp-\ntotically in time. Examples of such system purification\nare shown in Fig. 8.\nThe dynamics can be recast in terms of Y= tanh\u00001r2,\nwhich tends to 1asr!1, and an angle \u0012=9\ntan\u00001(rx=rz). For\u000bx=\u000bz=\u000bthe SDEs are\ndY=4\u000b2\n(1 + tanhY)2\u0000\n2 + tanhY+ 3 tanh2Y\u0001\ndt\n+4\u000bp\ntanhY\n1 + tanhYdWY\nd\u0012= 2\u000bdW\u0012=p\ntanhY; (52)\nwheredWY=r\u00001(rzdWz+rxdWx)anddW\u0012=\nr\u00001(\u0000rxdWz+rzdWx)are independent Wiener incre-\nments. As t!1, Eq. (51) implies that r2!1and\nhence tanhY!1, in which case we can write\ndY\u00196\u000b2dt+ 2\u000bdWY; (53)\nand for late times we have Y\u00196\u000b2t+ 2\u000bWY+ const.\nThe SDE for \u0012in this limit is d\u0012= 2\u000bdW\u0012, such that\nthe pdf becomes uniform over \u0012at late times. We write\np(Y;\u0012;t )!(2\u0019)\u00001F(Y;t), in terms of a travelling and\nbroadening gaussian in Y:\nF(Y;t) =1\n(8\u0019\u000b2t)1=2exp\u0014\n\u0000(Y\u00006\u000b2t)2\n8\u000b2t\u0015\n:(54)\nThe stochastic entropy production can now be com-\nputed using the framework of Yand\u0012coordinates. We\nshall do so first for late times where Y!1and the\ndynamical equations (52) become independent. We can\nidentify coefficients Airr\nY= 6\u000b2,Arev\nY= 0,DY= 2\u000b2, and\nAirr\n\u0012= 0,Arev\n\u0012= 0,D\u0012= 2\u000b2anduseEq. (36)toidentify\ncontributions to the stochastic entropy production. The\nsystem stochastic entropy production can be computed\nusing the pdf in Eq. (54). After some manipulation we\nfind that\nd\u0001stot\u001918\u000b2dt+ 6\u000bdWY; (55)\nand thus the stochastic entropy production increases at a\nmeanrateof 18\u000b2. Thisismorethantwicethemeanrate\nof production in Eq. (46) for the case of measurement of\n\u001bzalone. The continued increase is once again a conse-\nquence of the non-stationary character of the evolution:\nthe dynamics have the effect of purifying the system, but\nonly ast!1:\nFor the more general situation, without taking tto be\nlarge, it is possible to compute the stochastic entropy\nproduction numerically, based on the more elaborate co-\nefficients of the SDEs in Eqs. (52), and a general solution\nto the associated Fokker-Planck equation. Mean stochas-\nticentropyproductionoveranensembleof10trajectories\nis given in Fig. 9, separating h\u0001stotiinto contributions\nh\u0001ssysi=\u0000\u0001hlnpiandh\u0001smeasi=h\u0001stoti\u0000h \u0001ssysi.\nThe significance of this separation is that\n\u0000\u0001hlnpi=\u0000Z\np(Y;\u0012;t ) lnp(Y;\u0012;t )dYd\u0012\n+Z\np(Y;\u0012;0) lnp(Y;\u0012;0)dYd\u0012; (56)\nFigure 9. Mean stochastic entropy production h\u0001stotifor si-\nmultaneous measurement of observables \u001bxand\u001bz, separated\ninto contributions associated with the system and measuring\ndevice,h\u0001ssysiandh\u0001smeasi, respectively. The strengths of\nmeasurement \u000bxand\u000bzare both set to unity and the numer-\nically generated ensemble consisted of ten trajectories. The\nmean stochastic entropy production is consistent with the es-\ntimate in Eq. (55).\nis the change in Gibbs entropy \u0001SGof the system when\ndescribed using the pdf in Y;\u0012coordinates. Note that\nthe Gibbs entropy is coordinate frame dependent and is\ntherefore a measure of the uncertainty of adopted coordi-\nnatesinaspecificframe. Incontrast, themeanstochastic\nentropy production is independent of coordinate frame.\nB. Measurement of two non-commuting\nobservables for a pure state\nSimultaneous measurement of \u001bzand\u001bxleads asymp-\ntotically to a pure state located on a circle of radius\nr=p\nr2x+r2z= 1in the (rx;rz)coordinate space. It\nis of interest now to consider how the pdf of this pure\nstate over the angle \u0012(shown in Fig. 7) depends on the\nratio of the strengths of measurement of the two observ-\nables, and to compute the stochastic entropy production\narising from changes in this ratio.\nWe therefore return to Eq. (50), set rx= sin\u0012,rz=\ncos\u0012and derive an SDE for \u0012in the form\nd\u0012=\u0000\n\u000b2\nx\u0000\u000b2\nz\u0001\nsin 2\u0012dt+ 2\u000bxcos\u0012dWx\u00002\u000bzsin\u0012dWz\n=\u0000\n\u000b2\nx\u0000\u000b2\nz\u0001\nsin 2\u0012dt+ 2\u0000\n\u000b2\nxcos2\u0012+\u000b2\nzsin2\u0012\u00011=2dW;\n(57)\nwhich depends on the two measurement strengths \u000bxand\n\u000bz, and where dWis a Wiener increment. The Fokker-\nPlanck equation for the pdf p(\u0012;t)reads\n@p(\u0012;t)\n@t=\u0000@\n@\u0012h\u0000\n\u000b2\nx\u0000\u000b2\nz\u0001\nsin 2\u0012p(\u0012;t) (58)\n\u00002@\n@\u0012\u0000\n\u000b2\nxcos2\u0012+\u000b2\nzsin2\u0012\u0001\np(\u0012;t)i\n;(59)10\nFigure 10. Stationary pdfs pst(\u0012)for simultaneous measure-\nment of\u001bxand\u001bzwith strengths \u000bxand\u000bz, respectively,\nwith strength ratio \u0016=\u000bx=\u000bz.\nand this has stationary solutions (with zero probability\ncurrent) given by\npst(\u0012) =p\n2\u00162\u0000\n1 +\u00162\u0000\u0000\n1\u0000\u00162\u0001\ncos 2\u0012\u0001\u00003=2\nE(1\u0000\u00162) +\u0016E(1\u0000\u0016\u00002);(60)\nwhereE(x) =R\u0019=2\n0\u0000\n1\u0000xsin2\u001e\u00011=2d\u001eis the complete\nelliptical integral of the second kind and \u0016=\u000bx=\u000bzis\nthe ratio of the two measurement strengths. Examples\nof stationary pdfs for various values of \u0016are shown in\nFig. 10. Clearly a greater strength of measurement of\nobservable\u001bxproduces higher probability density in the\nvicinity of the eigenstates of \u001bxat\u0012=\u0006\u0019=2than in the\nvicinity of the eigenstates of \u001bzat\u0012= 0and\u0019, and vice\nversa.\nNotethataformofHeisenberguncertaintyisexhibited\nby the stationary pdf. In quantum state diffusion, rx=\nTr(\u001a\u001bx)andrz= Tr(\u001a\u001bz)are real physical properties of\nthe quantum state that are correlated in their evolution.\nThe expectation value of each in the stationary state is\nzero:\nhrzi=Z\u0019\n\u0000\u0019cos\u0012pst(\u0012)d\u0012= 0\nhrxi=Z\u0019\n\u0000\u0019sin\u0012pst(\u0012)d\u0012= 0; (61)\nwhile the variances hr2\nzi\u0000hrzi2=R\u0019\n\u0000\u0019cos2\u0012pst(\u0012)d\u0012and\nhr2\nxi\u0000hrxi2=R\u0019\n\u0000\u0019sin2\u0012pst(\u0012)d\u0012sum to unity. A higher\nmeasurement strength for one observable drives up the\nvariance of the associated variable (namely the adopted\nvalues lie close to either 1 or \u00001) while driving down\nthe variance of the other variable (the value lies close to\nzero).\nThe stochastic entropy production associated with\nthe dynamics of \u0012is specified by Arev\n\u0012= 0,Airr\n\u0012=\u0000\n\u000b2\nx\u0000\u000b2\nz\u0001\nsin 2\u0012, andD\u0012= 2\u0000\n\u000b2\nxcos2\u0012+\u000b2\nzsin2\u0012\u0001\nFigure 11. The asymptotic mean stochastic entropy produc-\ntion brought about by an abrupt change in the ratio \u0016=\n\u000bx=\u000bz, starting from equal measurement strengths \u000bx=\u000bz.\nThe initial and final stationary pdfs for \u00162= 0:2, 2 and 5,\nfrom Fig. 10, together with arrows indicating the change in\nshape in the process, are shown in the insets.\nwhich leads to\nd\u0001stot= \n6\u0000\n\u000b2\nx\u0000\u000b2\nz\u0001\ncos 2\u0012+9\u0000\n\u000b2\nx\u0000\u000b2\nz\u00012sin22\u0012\n2\u0000\n\u000b2xcos2\u0012+\u000b2zsin2\u0012\u0001!\ndt\n+3\u0000\n\u000b2\nx\u0000\u000b2\nz\u0001\nsin 2\u0012\n\u0000\n\u000b2xcos2\u0012+\u000b2zsin2\u0012\u00011=2dW\u0000dlnp(\u0012;t):\n(62)\nThe dynamic and entropic consequences of changing the\nratio of measurement strengths, for an initially pure\nstate, can be established by solving Eqs. (57), (58) and\n(62) for a given protocol. But we instead focus attention\non a case with an analytic result. The asymptotic mean\nproduction of stochastic entropy for a transition from\na uniform stationary pdf over \u0012, at equal measurement\nstrengths\u000bi\nx=\u000bi\nz, to a final stationary state brought\nabout by an abrupt change in measurement strengths to\n\u000bf\nx=\u0016\u000bf\nzatt= 0, takes the form of a Kullback-Leibler\ndivergence:\nh\u0001stoti1=Z\npi\nst(\u0012) ln\u0010\npi\nst(\u0012)=pf\nst(\u0012)\u0011\nd\u0012;(63)\nwhere thepi;f\nst(\u0012)correspond to Eq. (60) with the in-\nsertion of\u000bi;f\nxand\u000bi;f\nz. This can be derived by not-\ning thatd\u0001stot=\u0000dln (p(\u0012;t)=pst(\u0012))in this case. We\nploth\u0001stoti1for various ratios of final measurement\nstrengths\u0016in Fig. 11. Note that elevation of the mea-\nsurementstrengthofoneoftheobservablesrelativetothe\nother leads to positive mean stochastic entropy produc-\ntion, inaccordancewiththesecondlaw, andtheeffectfor\nenhanced measurement of \u001bxrelative to\u001bzis the same as\nfor similarly enhanced measurement of \u001bz, i.e. the same\nproduction emerges for ratios \u0016and1=\u0016.11\nIV. INTERPRETATION\nWe return now to the physical interpretation of\nstochastic entropy production in open quantum systems.\nBy analogy with situations in classical dynamics, the av-\nerage of the stochastic entropy production \u0001stotthat ac-\ncompanies the evolution expresses change in subjective\nuncertainty concerning the details of the quantum state\noftheworld. Wehavearguedthatthisuncertaintyisgen-\nerated in the same way as in classical physics. Namely\nthat the dynamical evolution of the world is determinis-\ntic, but chaotic, and that we do not or cannot attempt\nto solve the equations of motion for the coordinates ex-\nactly. We instead coarse-grain aspects of the description\nand employ a set of stochastic equations that capture the\nresulting unpredictability in evolution, again just as in a\nclassicalsituation. Suchmodellingmethodscanonlypro-\nvide statistical predictions, and hence are characterised\nbyanincreaseinentropyof(ourperceptionof)theworld.\nThis is not a physical effect, but merely a measure of the\nabsence of subjective knowledge, again just as in classi-\ncal thermodynamics. The key point is that we take the\nquantumstatevectoroftheworld, andhencethereduced\ndensity matrix of an open system, to be the appropriate\nfundamental physical description, analogous to classical\nphase space coordinates.\nIt is possible to derive such a stochastic model from an\nunderlying Hamiltonian describing the system and envi-\nronment [38], but here we have adopted a more direct\napproach, using a framework of quantum state diffusion\nto represent the environmental disturbances. The result-\ning Markovian stochastic rules of evolution, specified by\nKraus operators, are designed to drive a system contin-\nuously and (pseudo)randomly towards one of its eigen-\nstates. This is our conception of the process of quantum\nmeasurement, instead of instantaneous projection. The\nresulting evolution of the reduced density matrix resem-\nbles a path taken by a Brownian particle, and it can be\ndescribed using a Fokker-Planck equation for a pdf over\na suitable phase space, or an Itô process that specifies a\nstochastic trajectory.\nButwhatisthepointofstochasticentropyproduction?\nIts main purpose, in both classical and quantum systems,\nis in providing a measure of the apparent irreversibility of\nevolution and hence an arrow of time. Both of these de-\npendonthescaleofthecoarse-graining. Thedefinitionin\nEq. (10) involves a comparison between the likelihoods,\ncomputed according to the stochastic model employed,\nof forward and backward sequences of events. A depar-\nture of \u0001stotfrom zero indicates that the model dynam-\nics generate one of these sequences preferentially; that\nthe dynamics are effectively irreversible in the sense of\nbreaking time reversal symmetry. And since the stochas-\ntic model is intended to capture the effects of chaos in the\nunderlyingdynamics, thepreferredsequenceswillexhibit\neffects such as dispersion rather than assembly.\nNevertheless, parts of the world can become better de-\nfined as time evolves according to these models. Entropyproduction in a quantum framework can be used to char-\nacterise the approach of an open system towards a sta-\ntionary state as well as the selection of an eigenstate un-\nder measurement. The latter is not so very different from\nclassical measurement, which can also be shown to have\nan entropic cost in simple models [47]. Furthermore, we\ncan conceive of quantum processes that are reversible, in\nthe sense that the average of \u0001stotis zero. This would\narise, as in classical circumstances, when the driving of\nthe system, for example the rate of change of coupling\nto a measuring device, becomes quasistatic. Hence quan-\ntum measurement need not be irreversible, neither in the\ndynamic nor in the entropic sense.\nV. CONCLUSIONS\nEntropy production represents increasing subjective\nuncertainty of microscopic configuration brought about\nby employing stochastic models of the dynamics instead\nof the underlying deterministic equations of motion that\nare responsible for typical chaotic, dispersive behaviour.\nThese ideas can apply to quantum systems, where we re-\ngard the reduced density matrix as a physical property\nanalogous to a set of physical coordinates of a classi-\ncal system. The reduced density matrix evolves pseu-\ndorandomly through interactions with an underspecified\nenvironment, which we represent in a minimal fashion\nusing Kraus operators and a framework of Markovian\nquantum state diffusion. We concern ourselves with the\nuncertainty regarding the reduced density matrix that\nis actually adopted by the system. Stochastic entropy\nproduction can then be computed using analysis of the\nrelative probabilities of forward and backward Brownian\ntrajectories of the reduced density matrix.\nThe usual features of quantum mechanics are captured\nby the dynamics, in particular the stochastic selection of\nan eigenstate according to the Born rule. A further fea-\nture has been explored, for a simple two level system,\nwhere the simultaneous measurement of two observables\nrepresented by non-commuting operators can be consid-\nered. The system is prevented from selecting an eigen-\nstate of either operator, as expected, and instead adopts\na pure state with correlated stationary uncertainty with\nrespect to the two observables.\nThe models of measurement used here have the effect\nofpurifyingthesystem, i.e. eliminatinganyinitialentan-\nglement between the system and its environment. Such\nentanglement is often considered to be the outcome of\nmeasurement, so this is perhaps an unusual viewpoint,\nthough perfectly in line with the idea that a system is\nleft in an eigenstate after the process of measurement.\nThe final state of the environment (the measuring de-\nvice) is correlated with the final state of the system even\nin the absence of entanglement, and hence is able to con-\nvey information about the system and preserve a record\nof the measurement.\nWe suggest that the reduced density matrix typically12\nused to describe an open quantum system is an aver-\nage over an ensemble of adoptable states; pure as well as\nthose entangled with the environment. Moreover, the en-\nsemble average is not a suitable description for eigenstate\nselection. This problem is usually overcome by introduc-\ning a process of projective measurement that takes place\noutside the regular dynamics, but such a difficulty is not\npresent in quantum state diffusion.\nThe dynamics therefore conceptualise quantum me-\nchanics as the evolution of real physical properties that\nbehave in a complex but relatively unmysterious fashion.\nThe quantum state is more than a provider of informa-\ntion about probabilities of projective measurement out-\ncomes. The reduced density matrix, and by implication\nthe quantum state vector of the world, are treated as\nphysical coordinates and not merely bearers of informa-\ntion.\nHowever, the main purpose of this paper has been\nto use such a conceptual framework to provide explicit\nexamples of stochastic entropy production for a simple\nopen quantum system, and to suggest that this quan-\ntity is the most appropriate extension into the quantumregime of the modern concept of entropy production. 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Ford, Maxwell’s demon and the management of\nignorance in stochastic thermodynamics, Contemporary\nPhysics 57, 309 (2016)." }, { "title": "2301.12190v1.Density_of_states_for_the_Unitary_Fermi_gas_and_the_Schwarzschild_black_hole.pdf", "content": "arXiv:2301.12190v1 [cond-mat.quant-gas] 28 Jan 2023Density of states for the Unitary Fermi gas and the Schwarzsc hild black hole\nLuca Salasnich1,2,3\n1Dipartimento di Fisica e Astronomia “Galileo Galilei” and P adua QTech,\nUniversit` a di Padova, Via Marzolo 8, 35131 Padova, Italy\n2Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR),\nVia Nello Carrara 1, 50019 Sesto Fiorentino, Italy\n3Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Pa dova, Via Marzolo 8, 35131 Padova, Italy\nThe density of states of a quantum system can be calculated fr om its definition but, in some\ncases, this approach is quite cumbersome. Alternatively, t he density of states can be deduced\nfrom the microcanonical entropy or from the canonical parti tion function. After discussing the\nrelationship among these procedures, we suggest a simple nu merical method, which is equivalent\nin the thermodynamic limit to perform a Legendre transforma tion, to obtain the density of states\nfrom the Helmholtz free energy. We apply this method to deter mine the many-body density of\nstates of the unitary Fermi gas, a very dilute system of ident ical fermions interacting with divergent\nscattering length. The unitary Fermi gas is highy symmetric due to the absence of any internal scale\nexcept for the average distance between two particles and, f or this reason, its equation of state is\ncalled universal. In the last part of the paper , by using the same thermodynamical techniques, we\nreview some properties of the density of states of a Schwarzschild black hole, which sh ares with the\nunitary Fermi gas the problem of finding the density of states directly from its definition.\nI. INTRODUCTION\nThe density of states appears in many contexts of statistical mec hanics [1] and quantum physics [2]. The density\nof states, which tells you how many quantum states exist in a given ra nge of energy (or momentum), is extremely\nuseful in the experimental and theoretical determination of seve ral physical quantities [1, 2]. In some cases one deals\nwith the single-particle density of states, namely the density of sta tes of a single quantum particle in the presence of\nan external potential. The determination of this single-particle den sity of states is often quite simple. Instead, the\ncalculation of the many-body density of states, i.e. the density of s tates of system composed by many interacting\nquantum particles, is usually a difficult task. Indeed, although the de nsity of states of a quantum system can be,\nin principle, derived from its definition, this approach is not always str aightforward, in particular for many-body\nproblems. As an alternative, the density of states can be deduced from the microcanonical entropy or the canonical\npartition function. As well know, in the appropriate thermodynamic limit, microcanonical observables can be related\nto the corresponding canonical ones by means of a Legendre tran sformation [1].\nIn this paper we suggest a straightforward technique for deriving the density of states from the Helmholtz free\nenergy. This procedure isnothing else than a Legendre transformation of the entropy from the canonical ensemble\nto the microcanonical ensemble. We apply this method to calculate th e many-body density of states of the unitary\nFermi gas, characterized by an interaction potential with a diverg ent s-wave scattering length [3]. When the s-wave\nscattering length becomes very large the attractive Fermi gas of fermionic pairs with opposite spins is made of weakly\nbound dimers. Strictly speaking, the unitary Fermi gas is made of dim ers with zero binding energy [3]. This system\nis very peculiar due to the absence of any intrinsic parameter, exce pt for the number density. As a consequence,\nits equation of state is called universal [3]. In addition, for the unita ry Fermi gas, the conformal invariance plays\nan important role, as discussed by Son and Wingate [4]. We show that a direct microcanonical evaluation of the\nmany-body density of states of the unitary Fermi gas gives rise to a formula which seems intractable. Then, we\nexhibit an elegant derivation of the many-body density of states of the unitary Fermi gas starting from the canonical\nensemble and applying a Legendre transformation. We believe that t his canonical approach could be applied to other\nmany-body systems, for instance atomic nuclei, Bose-Einstein con densates, and superconductors.\nTaking into account that the density of states is currently an very hot topic in the physics of black holes also for the\ncondensed matter theoreticians [5], for the sake of entertainment , in the last part of the paper we reviewthe density\nof states and other thermodynamical quantities of the Schwarzs child black hole. Also for this astrophysical system\nwe derive the density of states starting from the canonical ensem ble and applying a Legendre transformation. In this\ncase, however, the procedure is quite simple .\nII. GENERAL PROPERTIES OF THE DENSITY OF STATES\nLet us consider a quantum system with microscopic Hamiltonian ˆHand macroscopic internal energy between E\nandE+∆, with ∆ ≪E[1]. Following the Boltzmann’s idea [6], in the microcanonical ensemble the entropyS(E)2\nof the system can be written as\nS(E) =kBln(W(E)), (1)\nwherekBis the Boltzmann constant and W(E) is the number of accessible microstates between EandE+∆, that\nwe call adimensional density of states, given by [1]\nW(E) =N(E+∆)−N(E), (2)\nwhere\nN(E) = Tr[Θ( E−ˆH)] (3)\nis the number of states up to the energy E, with Θ( x) the Heaviside step function [1]. If ∆ is sufficiently small one\nhas\nW(E)≃D(E) ∆, (4)\nwhere\nD(E) = Tr[δ(E−ˆH)] (5)\nis the density of states, with Tr the trace on the Hilbert space of qu antum states and δ(x) the Dirac delta function\n[1]. In the thermodynamic limit one often writes\nS(E)≃kBln(D(E)Es), (6)\nwithEsa characteristic energy scale of the system (for instance Es=/planckover2pi12n2/3/mforNidentical particles of mass min\na volume Vand number density n=N/V), because the intensive quantity ln(∆ /Es) becomes negligible with respect\nto the extensive quantity ln( D(E)Es) [1].\nKnowing the Hamiltonian ˆHone can calculate D(E) by using Eq. (5). Alternatively, knowing the microcanical\nentropyS(E), one easily derives the adimensional density of states W(E) from the entropy S(E) as\nW(E) =eS(E)/kB. (7)\nThe third principle of thermodynamics [1] states that S(Egs) = 0 with Egsthe ground-state energy of the system.\nConsequently, from Eq. (7) we obtain W(Egs) = 1.\nIn the canonical ensemble, the Helmholtz free energy F(T) of the system at temperature Tis given by [1]\nF(T) =−kBTln(Z(T)), (8)\nwhereZ(T) is the partition function, defined as\nZ(T) = Tr[e−ˆH/(kBT)]. (9)\nIt is not difficult to show that the partition function Z(T) is directly related to the density of states D(E). In fact,\nTr[e−ˆH/(kBT)] = Tr[/integraldisplay\ndE δ(E−ˆH)e−E/(kBT)] =/integraldisplay\ndETr[δ(E−ˆH)]e−E/(kBT)(10)\nand consequently\nZ(T) =/integraldisplay\ndE D(E)e−E/(kBT). (11)\nInverting this formula one gets the density of states D(E) as a function of the partition function Z(T), and then\nalsoD(E) as a function of the Helmholtz free energy F(T). However, this procedure is quite cumbersome because it\ninvolves the calculation of an anti-Laplace transformation.\nIn this paper we suggest a much simpler procedure to obtain the adim ensional density of states W(E) from the\nHelmholtz free energy F(T). In the canonical ensemble, the entropy Sas a function of the temperature T, namely\nS(T), is given by\nS(T) =−/parenleftbigg∂F(T)\n∂T/parenrightbigg\nN,V, (12)3\nthat is the partial derivative of the Helmholtz free energy F(T) with respect to the temperature Tat fixed number\nNof particles and volume V. Moreover, the internal energy E(T) reads\nE(T) =F(T)+T S(T). (13)\nBothS(T) andE(T)depend on the temperature T. This means that Tcan be considered as a dummy variable to get,\nor analytically or numerically, the parametric curve SvsE, i.e.S=S(E), which could be a multivalued function.\nIn this way we are actually performing, in the thermodynamic limit, a Le gendre transformation of the entropy from\nthe canonical ensemble to the microcanonical ensemble. Having this result, one can then use Eq. (7) to find the\nadimensional density of states W(E).\nIII. UNITARY FERMI GAS\nIn 2004 the crossover from the Bardeen-Cooper-Schrieffer (BC S) state of weakly-correlated pairs of fermions to\nthe Bose-Einstein condensation (BEC) of diatomic molecules was obs erved with ultracold gases of two-component\nfermionic40K or6Li atoms [8–10]. This BCS-BEC crossover is obtained by using a Fano- Feshbach resonance to\nchange the strength of the effective inter-atomic attraction and , consequently, the 3D s-wave scattering length a[3, 7].\nGiven a gas of Natomic fermions in a volume Vwith two equally-populated spin components, i.e. N↑=N↓=N/2,\nthe system is dilute if the characteristic range reof the inter-atomic potential is much smaller than the average\ninterparticle separation d=n−1/3withn=N/Vthe total number density, namely\nre≪d . (14)\nThe system is strongly-interacting if the scattering length aof the inter-atomic potential greatly exceeds the average\ninterparticle separation d=n−1/3, i.e.\nd≪ |a|. (15)\nThe unitarity regime [7] is characterized by both these conditions:\nre≪d≪ |a|. (16)\nUnder these conditions the dilute but strongly-interacting Fermi g as is called unitary Fermi gas.\nIdeally, the unitarity limit corresponds to\nre= 0 and a=±∞. (17)\nIn a uniform configuration and at zero temperature, the only lengt h characterizingthe Fermi gas in the unitarity limit\nis the average interparticle distance d=n−1/3.\nIn this case, simply for dimensional reasons, the ground-state en ergy must be [3]\nEgs=ξ3\n5/planckover2pi12\n2m(3π2)2/3n2/3N=ξ3\n5ǫFN (18)\nwithǫF=/planckover2pi12(3π2)2/3n2/3/(2m) Fermi energy of the ideal gas and ξa universal unknown parameter: the Bertsch\nparameter. Monte Carlo calculations and experimental data with dilu te and ultracold atoms suggest that, at zero\ntemperature, the unitary Fermi gas is a superfuid with ξ≃0.4 [3].\nWe model [11–13] the many-body quantum Hamiltonian ˆHof the uniform unitary Fermi gas with the simple\neffective Hamiltonian\nˆH=Egs+/summationdisplay\nσ=↑,↓/summationdisplay\nkǫsp(k) ˆc†\nkσˆckσ+/summationdisplay\nqǫcol(q)ˆb†\nqˆbq, (19)\nwhere ˆc†\nkσis the creation operator and ˆ ckσthe annihilation operator of fermionic single-particle excitations cha rac-\nterized by energy ǫsp(k), spinσ, and wavevector k. Similarly, ˆb†\nqis the creation operator and ˆbqthe annihilation\noperator of bosonic collective excitations with energy ǫcol(q) and wavevector q.\nThe energy of the the BCS-like excitations can be written as\nǫsp(k) =/radicalBigg/parenleftbigg/planckover2pi12k2\n2m−ζǫF/parenrightbigg2\n+∆2\n0 (20)4\nwhereζ= 0.9 takes into account many-body effects on the Fermi surface [14]. Instead, ∆ 0=γǫFis the energy gap\nwithγ= 0.45 [15].\nThe energy of collective elementary excitations is instead assumed t o be given by\nǫcol(q) =/radicalBigg\n/planckover2pi12q2\n2m/parenleftbigg\n2mc2\nB+λ/planckover2pi12q2\n2m/parenrightbigg\n, (21)\nwherecB=/radicalbig\nξ/3vFwithvF=/radicalbig\n2ǫF/m.In Ref. [12] we used the value λ= 0.25, which is consistent with a\nmacroscopic time-dependent nonlinear Schr¨ odinger equation app roach without the inclusion of spurious terms [16].\nIn a recent paper [13] we used instead λ= 0.08, which is the value obtained [17] from the beyond-mean-field GPF\ntheory [18] at unitarity.\nA. Attempt of direct evaluation of the many-body density of s tates\nWe try to write the density of states D(E) of the unitary Fermi gas by using Eq. (5) with Eq. (19). We immediat ely\nfind\nD(E) =/summationdisplay\n{nkσ}/summationdisplay\n{nq}δ/parenleftBig\nE−Egs−/summationdisplay\nσ=↑,↓/summationdisplay\nkǫsp(k)nkσ−/summationdisplay\nqǫcol(q)nq/parenrightBig\n, (22)\nwherenkσandnqare the occupation numbers of single-particleand collectiveexcitat ions, respectively. It is important\ntoremarkthatEq. (22)isthemany-bodydensityofstatesofthe systemandnotthemuchmorefamiliarsingle-particle\ndensity of states.\nTaking into account the Fourier representation of the Dirac delta f unctionδ(x) we have\nD(E) =/summationdisplay\n{nkσ}/summationdisplay\n{nq}1\n2π/integraldisplay\ndξ eiξ/parenleftbig\nE−Egs−/summationtext\nσ=↑,↓/summationtext\nkǫsp(k)nkσ−/summationtext\nqǫcol(q)nq/parenrightbig\n=1\n2π/integraldisplay\ndξ eiξ(E−Egs)/summationdisplay\n{nkσ}e−iξ/summationtext\nσ=↑,↓/summationtext\nkǫsp(k)nkσ/summationdisplay\n{nq}e−iξ/summationtext\nqǫcol(q)nq\n=1\n2π/integraldisplay\ndξ eiξ(E−Egs)/productdisplay\nσ=↑,↓/productdisplay\nk/summationdisplay\nnkσ=0,1e−iξǫsp(k)nkσ/productdisplay\nq+∞/summationdisplay\nnq=0e−iξǫcol(q)nq\n=1\n2π/integraldisplay\ndξ eiξ(E−Egs)/productdisplay\nσ=↑,↓/productdisplay\nk(1+e−iξǫsp(k))/productdisplay\nq1\n1−e−iξǫcol(q). (23)\nUnfortunately, Eq. (23) does not help very much to obtain a tract able expression of the density of states. For this\nreason, in the next Section, we analyze the same system in the cano nical ensemble, where we will find a similar,\nbut more manageable, formula for the partition function. The reas on is that working in the canonical ensemble\nthe statistical independence of non-interacting macroscopic sub systems is ensured by the canonical density operator\ne−ˆH/(kBT), which is an exponential operator [1].\nB. Canonical ensemble\nAs previously discussed, in the canonical ensemble the Helmholtz fre e energy F(T) of the system is obtained from\nthe partition function Z(T) adopting Eqs. (8) and (9) [1]. In particular, by using Eqs. (9) and ( 19) we find\nZ(T) =/summationdisplay\n{nkσ}/summationdisplay\n{nq}e−/parenleftbig\nEgs+/summationtext\nσ=↑,↓/summationtext\nkǫsp(k)nkσ+/summationtext\nqǫcol(q)nq/parenrightbig\n/(kBT)\n=e−Egs/(kBT)/summationdisplay\n{nkσ}e−/parenleftbig/summationtext\nσ=↑,↓/summationtext\nkǫsp(k)nkσ/parenrightbig\n/(kBT)/summationdisplay\n{nq}e−/parenleftbig/summationtext\nqǫcol(q)nq/parenrightbig\n/(kBT)\n=e−Egs/(kBT)/productdisplay\nσ=↑,↓/productdisplay\nk/summationdisplay\nnkσ=0,1e−(ǫsp(k)/(kBT))nkσ/productdisplay\nq+∞/summationdisplay\nnq=0e−(ǫcol(q)/(kBT))nq(24)5\nThus, we can write\nZ(T) =Zgs(T)Zsp(T)Zcol(T), (25)\nwhere\nZgs(T) =e−Egs/(kBT)(26)\nZsp(T) =/productdisplay\nσ=↑,↓/productdisplay\nk(1+e−ǫsp(k)/(kBT)) (27)\nZcol(T) =/productdisplay\nq1\n1−e−ǫcol(q)/(kBT). (28)\nWith the help of Eq. (8) the corresponding Helmholtz free energy re ads\nF(T) =Fgs+Fsp(T)+Fcol(T), (29)\nwhereFgs=Egs,\nFsp(T) =−2kBT/summationdisplay\nkln[1+e−ǫsp(k)/(kBT)], (30)\nand\nFcol(T) =−kBT/summationdisplay\nqln[1−e−ǫcol(q)/(kBT)]. (31)\nQuite remarkably, the total free energy F(T) can be written [12] in a compact form as\nF(T) =NǫFΦ(T\nTF), (32)\nwhere Φ( x) depends on the scaled temperature x≡T/TFonly, with TF=ǫF/kBthe Fermi emperature. In particular,\nwe have\nΦ(x) =3\n5ξ−3x/integraldisplay+∞\n0ln/bracketleftBig\n1+e−˜ǫsp(u)/x/bracketrightBig\nu2du\n+3\n2x/integraldisplay+∞\n0ln/bracketleftBig\n1−e−˜ǫcol(u)/x/bracketrightBig\nu2du . (33)\nHere the integrals replace the summations. For example,/summationtext\nk→V/integraltextd3k/(2π)3. Moreover, we set ˜ ǫcol(u) =/radicalbig\nu2(4ξ/3+λu2) and ˜ǫsp(u) =/radicalbig\n(u2−ζ)2+γ2.\nWe can now calculate the entropy S(T) and the internal energy E(T) by using Eqs. (12) and (13). In particular,\ntaking into account Eq. (33) we find for the entropy\nS(T) =−NkBΦ′(T\nTF). (34)\nwhere Φ′(x) =dΦ(x)/dx. Furthermore, for the internal energy Ewe obtain the expression\nE(T) =NǫF/bracketleftbigg\nΦ(T\nTF)−T\nTFΦ′(T\nTF)/bracketrightbigg\n. (35)\nIn Fig. 1 we plot the free energy F(T), the entropy S(T), and the internal energy E(T) of the unitary Fermi gas by\nusing Eqs. (32), (34), and (35). We choose the following values for the parameters of our model: ξ= 0.42,λ= 0.25,\nζ= 0.9, andγ= 0.45. The figure shows that, by increasing the temperature T, the Helmholtz free energy F(T)\nmonotonically decreases, while both the internal energy E(T) and the entropy S(T) are monotonic growing functions.\nIt is important to stress that our model for the low-temperature thermodyamics of the unitary Fermi gas seems to\nbe in quite good agreement with both Monte Carlo simulations [19] and experimental data [20]. In particular, in Fig.\n2 we compare our internal energy E(T) (solid line) with Monte Carlo calculations (filled circles) and experiment al\nresults (filled squares). Indeed, the agreement among these diffe rent datasets is impressive. We stress that Eq. (19)\nis a low-temperature Hamiltonian because we are not taking into acco unt the fact that, in general, the elementary\nexcitations ǫsp(k) andǫcol(q) depend on the temperature T.6\n00.050.10.150.20.250.30.350.4\nT/TF-0.500.511.52\nF(T)/(NεF)\nS(T)/(NkB)\nE(T)/(NεF)\nFIG. 1: Unitary Fermi gas: Scaled free energy F(T)/(NǫF), scaled entropy S(T)/(NkB), and scaled internal energy\nE(T)/(NǫF) deduced from our model, as a function of the scaled temperat ureT/TFwithTF=ǫF/kBthe Fermi temper-\nature.\n00.050.10.150.20.250.30.350.4\nT/TF00.20.40.60.81E(T)/(NεF)Our theory\nMonte Carlo simulations\nExperimental data\nFIG. 2: Unitary Fermi gas: Scaled internal energy E(T)/(NǫF), as a function of the scaled temperature T/TF. Solid line is\nobtained with our model. Filled circles: Monte Carlo simula tions [19]. Squares with error bars: experimental data [20] .\nC. Numerical calculation of the many-body density of states\nAs previously discussed, having S(T) andE(T), we can immediately get the curve S=S(E) by using Tas a\ndummy variable. This is a Legendre transformation from S(T) toS(E). In Fig. 3 we plot this curve (dashed line)\nand also the curve (solid line) of the adimensional many-body density of states W(E), which is obtained from Eq.\n(7).\nFig. 3 shows that, by increasing the internal energy E, there is a monotonic growth of both entropy S(E) and\nadimensional density of states W(E). This is a consequence of the monotonic behavior as a function of Tof bothS\nandE. Clearly, S(Egs) = 0,W(Egs) = 1, and W(E) is an exponental function of the internal energy E.\nWe remark that the entropy (34) and the internal energy (35) ar e additive, i.e.\nS(T) =Sgs+Ssp(T)+Scol(T) (36)\nE(T) =Egs+Esp(T)+Ecol(T) (37)\nwithSsg= 0 and Egs= (3/5)NǫFξ. Moreover, the adimensional density of states W(E) satisfies, in the thermody-\nnamic limit, the equation\nW(E) =Wgs(Egs)Wsp(Esp)Wcol(Ecol), (38)\nwhereWgs(Egs) =eSgs(Egs)/kB= 1,Wsp(Esp) =eSsp(Esp)/kB, andWcol(Ecol) =eScol(Ecol)/kB. In Fig. 4 we plot the\nentropy and the adimensional density of states of both collective a nd single-particle elementary excitations.7\n00.5 11.5 22.5 33.5 4\nE/(NεF)10-210-1100101102103104\nS(E)/(N kB)\nW(E)\nFIG. 3: Unitary Fermi gas: The dashed line is the scaled entro pyS(E)/(NkB), as a function of the scaled internal energy\nE/(NǫF). The solid line is the adimensional many-body density of st atesW(E), as a function of the scaled internal energy\nE/(NǫF).\n0 0.2 0.4 0.6 0.8 1\nEcol/(NεF)10-1100101\nScol(Ecol)/(N εF)\nWcol(Ecol)\n0 0.2 0.4 0.6 0.8 1\nEsp/(NεF)10-1100101\nSsp(Eps)/(N kB) \nWsp(Esp)\nFIG. 4: Unitary Fermi gas. Upper panel: The dashed line is the scaled entropy Scol(Ecol)/(NkB) of bosonic collective\nelementary excitations, as a function of the scaled interna l energy Ecol/(NǫF) of the collective elementary excitations. The\nsolid line is the adimensional density of states Wcol(Ecol) of collective elementary excitations, as a function of the scaled internal\nenergyEcol/(NǫF) of collective elementary excitations. Lower panel: The da shed line is the scaled entropy Ssp(Esp)/(NkB)\nof fermionic single-particle excitations, as a function of the scaled internal energy Esp/(NǫF) of single-particle elementary\nexcitations. The solid line is the adimensional density of s tatesWsp(Esp) of single-particle excitations, as a function of the\nscaled internal energy Esp/(NǫF) of single-particle excitations.\nIV. SCHWARZSCHILD BLACK HOLE\nIn this section we derive the adimensional density of states of a Sch warzschild black hole [5, 21] from the micro-\ncanonical entropy and also from the canonical Helmholtz free ener gy. These are known results but they are however\nhighly non trivial. We remark that the derivation of the density of sta tes of a black hole directly from its definition\nis quite controversial because a fully consistent quantum Hamiltonia nˆHof the black hole is not yet available [21].\nFor aSchwarzschildblackhole ofmass M, which does not rotateand hasno electric charge, the Bekenstein -Hawking8\n0 0.2 0.4 0.6 0.8 1 1.2 1.4\nE/EP10-610-31001031061091012\nS(E)/kB\nW(E)\nFIG. 5: Schwarzschild black hole: The dashed line is the scal ed entropy S(E)/kB, as a function of the scaled internal energy\nE/EP, withEP=/radicalbig\n/planckover2pi1c5/Gthe Planck energy. The solid line is the adimensional densit y of states W(E), as a function of the\nscaled internal energy E/EP.\nentropy [22, 23] is given by\nS(M) =4πkB\n/planckover2pi1GM2\nc, (39)\nwhereGisthe gravitationalconstant, /planckover2pi1is thereducedPlanckconstantand cisthe speedoflightin vacuum. Assuming\nthat the internal energy Eof the system is [24]\nE=Mc2, (40)\nwe obtain immediately the microcanonical entropy\nS(E) =4πkBG\n/planckover2pi1c5E2(41)\nand also, by using Eq. (7), the adimensional density of states\nW(E) =e4πG\n/planckover2pi1c5E2. (42)\nFrom Eq. (40) we have Egs= 0 and, as expected, from Eq. (42) it follows W(Egs) =W(0) = 1.\nIn Fig. 5 we plot the curves of the scaled entropy S(E)/kB(dashed line) and of the adimensional density of states\nW(E), obtained with Eqs. (41) and (42). The vertical axis is in a log scale t o contain the huge increase of W(E)\nwithE.\nNotice that, in the microcanonical ensemble, the temperature Tof the system is defined as\n1\nT=∂S(E)\n∂E. (43)\nFor the Schwarzschild black hole, using Eq. (41) we find\n1\nT=8πkBG\n/planckover2pi1c5E (44)\nor, equivalently, using also Eq. (40) we get\nT=/planckover2pi1c3\n8MπkBG, (45)\nthat is the so-called Hawking temperature [25].\nLet us now consider the Schwarzschild black hole within the framewor k of the canonical ensemble. Because the\nquantum Hamiltonian ˆHof a black hole is somehow unknown, Gibbons and Hawking in their appro ach [26] did not9\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8 2\nT/TP10-310-210-1100101102103\nF(T)/EP\nS(T)/kB\nE(T)/EP\nFIG. 6: Schwarzschild black hole: Scaled free energy F(T)/EP, scaled entropy S(T)kB, and scaled internal energy E/EPas a\nfunction of the scaled temperature T/TPwithTP=EP/kBthe Planck temperature and and EP=/radicalbig\n/planckover2pi1c5/Gthe Planck energy.\nuse Eq. (9). Instead, they derived the canonical partition funct ionZ(T) of the Schwarzschild black hole from the\npath integral formula\nZ(T) =/integraldisplay\nD[gµν(x)]e−1\n/planckover2pi1/integraltextd3x/integraltext/planckover2pi1/(kBT)\n0dτ√gL(gµν(x)), (46)\nwheregµ(x) is the metric tensor, x= (cτ,x) is the space-time coordinate with τthe imaginary time, gis the\ndeterminant of the metric tensor, and L(gµν(x)) is the Euclidean Lagrangian density of the Einstein-Hilbert action\n[27, 28]. Taking into account the Schwarzschild solution of metric ten sor [29] generated by a spherical object of mass\nMand using a semiclassical approximation of Eq. (46) with the inclusion o f appropriate boundary terms, Gibbons\nand Hawking [26] basically found\nZ(T) =e−c5/planckover2pi1/(16πGk2\nBT2). (47)\nIt is important to observe that Eq. (47) was obtained by Gibbons an d Hawking by using also Eq. (45), which is a\ncrucial constraint derived imposing the regularity of the Euclidean S chwarzschild metric at the Schwarzschild radius\nrs= 2GM/c2.\nFrom Eqs. (8) and (47) we then get the Helmholtz free energy\nF(T) =c5/planckover2pi1\n16πGkBT. (48)\nWe now calculate the entropy S(T) and the internal energy E(T) by using Eqs. (12) and (13). We find\nS(T) =c5/planckover2pi1\n16πGkBT2(49)\nand\nE(T) =c5/planckover2pi1\n8πGkBT. (50)\nFor the sake of completeness, in Fig. 6 we report the free energy F(T), the entropy S(T), and the internal energy\nE(T), obtained with Eqs. (48), (49), and (50). The figure shows the v ery unusual behavior of these quantities by\nincreasing the temperature T: they are all monotonically decreasing.\nAs previously discussed, both the canonical entropy S(T) and the canonical internal energy E(T) are functions of\nthe absolute temperature T, which can be considered as a dummy variable to get the parametric c urveSvsE. In\nthis case we can directly find the inverse of Eq. (50), which is exactly Eq. (44). Inserting this formula into Eq. (49),\ni.e. performing analytically a Legendre transformation, we obtain th e microcanonical entropy S(E) of Eq. (41) and\nfinally the adimensional density of states W(E) given, again, by Eq. (42).10\nV. CONCLUSIONS\nIn conclusion, we stress again that the knowledge of the number of quantum states in a given range of energy is a\ncrucial quantity in many context of Physics. Although thedensity of states can be determined from its microcanonical\ndefinition, this method is not always simple. As an alternative, the den sity of states can be inferred from the\nmicrocanonical entropy or the canonical partition. After discuss ing how these processes relate to one another, we\nhave offered a straightforward technique, based on the Legendr e transformation, for deriving the density of states\nfrom the Helmholtz free energy. As an enlightening example, the unit ary Fermi gas, an extremely dilute system of\nidentical fermions interacting with divergent scattering length, ha s been studied to determine the many-body density\nof states. In Section IIIA we have found that the computation of the density of states from its definition is an hard\ntask, while in Section IIIC we have obtained it quite esily working in the c anonical ensemble. Finally, we have used\nthe same thermodynamical framework to reviewthe adimensional density of states of a Schwarzschild black hole\nwith the Gibbson-Hawking formalism. Also in this case, a calculation of t he density of states from its definition is\nhighly demanding. However, for the Schwarzschild black hole, one ge ts quite easily the density of state fromthe\nmicrocanonical entropy ( as discussed in several textbooks ) orfromthe canonical free energy.\nThe author thanks Fulvio Baldovin, Davide Cassani , Pieralberto Marchetti, Fabio Sattin, Antonio Trovato, and\nAlexander Yakimenko for useful discussions. This work is partially su pported by BIRD Project ”Ultracold atoms\nin curved geometries” of the University of Padova and by Iniziativa S pecifica “Quantum” of INFN. The author\nacknowledges funding from the European Union-NextGenerationE U, within the National Center for HPC, Big Data\nand Quantum Computing (Project No. CN00000013, CN1 Spoke 1: “ Quantum Computing”).\n[1] K-Huang, Statistical Mechanics (Wiley, 2008).\n[2] L.D. Landau, E.M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, vol. 9, Statistical Physics: Theor y of the\nCondensed State (Butterworth-Heinemann, Oxford, 1980).\n[3] W. 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Hilbert, Die Grundlagen der Physik, in Nachrichten v on der Gesellschaft der Wissenschaften zu G¨ ottingen -\nMathematisch-Physikalische Klasse 3, 395 (1915).\n[29] K.Schwarzschild, SitzungsberichtederKoniglich Pre ussischen AkademiederWissenschaften zuBerlin, Phys.-Ma th. Klasse,\n189 (1916)." }, { "title": "2302.02796v1.Density_of_photonic_states_in_aperiodic_structures.pdf", "content": "Density of photonic states in aperiodic structures\nVladislav A. Chistyakov1,\u0003Mikhail S. Sidorenko1, Andrey D. Sayanskiy1, and Mikhail V. Rybin1;2\n1School of Physics and Engineering, ITMO University, St Petersburg 197101, Russia and\n2Io\u000be Institute, St Petersburg 194021, Russia\n(Dated: February 7, 2023)\nPeriodicity is usually assumed to be the necessary and su\u000ecient condition for the formation of\nbandgaps, i.e. energy bands with a suppressed density of states. Here, we check this premise\nby analyzing the bandgap properties of three structures that di\u000ber in the degree of periodicity\nand ordering. We consider a photonic crystal, disordered lattice and ordered but non-periodic\nquasicrystalline structure. A real-space metric allows us to compare the degree of periodicity of\nthese di\u000berent structures. Using this metric, we reveal that the disordered lattice and the ordered\nquasi-crystal can be attributed to the same group of material structures. We examine the density of\ntheir photonic states both theoretically and experimentally. The analysis reveals that despite their\ndramatically di\u000berent degrees of periodicity, the photonic crystal and the quasicrystalline structure\ndemonstrate an almost similar suppression of the density of states. Our results give a new insight\ninto the physical mechanisms resulting in the formation of bandgaps.\nI. INTRODUCTION\nModern condensed matter physics have stemmed from\ntranslational symmetry analysis, which determines the\ntransport properties of waves according to the band\nstructure, following the Bloch theorem, which is appli-\ncable to any periodic system. Thus, bandgaps related to\nthe periodicity of dielectric index give photonic crystals a\nunique degree of control over electromagnetic waves. For\nexample, photonic crystals enable localization of light in-\nside small volume cavities, design of waveguides with spe-\nci\fc dispersion properties, and modi\fcation of the light\nemission rate1{4. These features make photonic crystals\nuseful in optical applications; for instance, they are used\nas resonators5, optical sensor6, and topological insulators\nfor photons7. However, certain restrictions on the choice\nof material and symmetry for photonic crystals limit the\nfreedom in their design. In particular, an omnidirectional\nband gap opens up only for a su\u000eciently high contrast\nof the refractive index and high symmetry of a structure.\nReverse engineering method with a numerical optimiza-\ntion can be used to overcome this problem, but it often\nyields overly complex structures8{11.\nAnother strategy to obtain bandgaps is to consider\nnon-periodic systems. For example, disordered hyper-\nuniform structures have the demanded bandgap proper-\nties despite their random structure12{15. Quasicrystalline\nstructures are remarkable among other non-periodic sys-\ntems, because, despite the lack of translational symme-\ntry, they are ordered. Recent advances in 3D nanomanu-\nfacturing enabled the fabrication of these structures with\na \fne precision allowing them to operate even in the vis-\nible light range16{22.\nQuasiperiodic photonic structures are a special class\nof photonic structures with a high degree of order in re-\nciprocal space. The opportunity to design structures in\nthe reciprocal space and then obtain their structure in\nreal space by inverse Fourier transform attracts consid-\nerable attention20. Real-space methods allow engineer-\ning photonic structures with the desired properties, forexample, complex moir\u0013 e patterns23. Recently, quasicrys-\ntalline structures based on multiple one-dimensional (1D)\ngratings oriented in all directions were reported24. Even\na weak dielectric contrast in case of polymer materials\nallows such structures to exhibit an unprecedented sup-\npression of radiation due to the bandgaps created in 2D\nand 3D cases.\nTranslational symmetry allows classi\fcation of solu-\ntions by their wave vector: such solutions are either a\npropagating or evanescent wave4. In the latter case, the\nwave vector has a non-zero imaginary part and for a cer-\ntain direction, there is a band gap. In disordered lattices,\nthere is another source of evanescent solutions related to\nthe Anderson localization25,26. Figure 1 shows an or-\ndered photonic crystal with a periodic structure, a non-\nperiodic ordered quasicrystal, and a disordered lattice.\nThe quasicrystalline structure lacks both conditions for\nevanescent waves, because it has neither periodicity nor\ndisorder.\nIn this paper, we study the suppression of light in a\nquasicrystalline structure and compare it with similar\nproperties obtained in a photonic crystal with a honey-\ncomb lattice, and in a disordered lattice structure. By\nusing a real-space metric, we determine the amount of\nperiodicity that allows us to numerically compare the\nquasiperiodic structure with the disordered lattice with\na given degree of disorder. Numerical results reveal a\nstrong suppression of wave propagation in the quasicrys-\ntalline structure comparable to its photonic crystal coun-\nterpart; in contrast, for the disordered lattice with the\nsame degree of periodicity, the suppression is weaker.\nThe predicted properties are farther con\frmed by mi-\ncrowave transmission measurements.\nII. DESIGN OF PHOTONIC STRUCTURES\nHere, we describe photonic structures with a spatial\ndistribution of permittivity modulated in two dimen-\nsions. We start with rigorously ordered quasiperiodicarXiv:2302.02796v1 [physics.class-ph] 27 Jan 20232\n(a) Regular crystal\n(c) Quasiperiodic structure(b) Disordered lattices\n100% Periodicity degree\nFIG. 1. Landscape of ordered and disordered structures with\ndi\u000berent degrees of periodicity. (a) Honeycomb photonic crys-\ntal, an ordered periodic structure. (b) Disordered lattice with\nrods randomly shifted with respect to the rod positions in\nthe honeycomb lattice, a disordered and non-periodic struc-\nture. (c) Quasiperiodic structure, an ordered but non-periodic\nstructure.\nstructures with no translational symmetry, then we study\na photonic crystal with perfect ordering and periodicity,\nand then we consider a disordered non-periodic lattice.\nFollowing Ref. 24, we generate the quasiperiodic struc-\nture as a superposition of several one-dimensional grat-\nings with a sine-type modulated refractive index. The\ngrating orientations are uniformly distributed over the\npolar angle, and the continuous spatial distribution of\nthe refractive index is:\n\u0001nc(r) =NoptX\ni=1\u0001nisin (bi\u0001r+\u001ei); (1)\nwhereienumerates the gratings, Noptis the optimal\nnumber of gratings, \u0001 niis the modulation amplitude of\nthe refractive index in the i-th grating, biis the vector\ndetermining the grating period and orientation, and \u001ei\nis a random phase. For such quasicrystalline structures,\nthe optimal number of gratings is Nopt\u00192:36 (n=\u0001n)2=3,\nwherenis the mean refractive index and \u0001 nis the am-\nplitude of the refractive index deviation from the average\nvaluen, i.e.n1=n+ \u0001nandn2=n\u0000\u0001n.\nOptical properties of the structure can be analyzed in\nthe reciprocal space strictly connected to Fourier trans-\nform. The Fourier transform of the sine function corre-\nsponds to two Bragg maxima (Dirac delta functions) with\nthe direction\u0006biin reciprocal space. Each lattice has the\nsame period a, so the lattice vector length is de\fned by\nb=jbij= 2\u0019=a. The uniform angular distribution of the\ngratings leads to the superposition in (1), which results\nin dense distribution of \frst-order Bragg maxima around\na circle in reciprocal space, shown in Fig. 2a. Because\nr(a) (b)\n(c) (d)n+∆n–n−∆n–\n (mm) x (mm) y\n (mm) x (mm) y\n (mm) x (mm) yib\ni−b\n (mm-1)xb (mm-1)yb\naFIG. 2. (a) Schematics of fourteen Bragg maxima located\nalong the circle in the reciprocal space corresponding to the\nseven gratings merged into a single structure, which is used\nfor generating a quasiperiodic structure with an omnidirec-\ntional Bragg bandgap. (b) Fragment of the quasiperiodic\nstructure with a binary distribution of two materials. The\nstructure corresponds to seven gratings with the distribution\nof the maxima in reciprocal space shown in panel (a). (c)\nOrdered honeycomb photonic crystal composed of dielectric\nrods in air. (d) Disordered lattice of dielectric rods in air. The\nrod positions are randomly shifted in both xandydirections\nwith respect to the nodes of the honeycomb lattice.\nof randomly chosen phases and su\u000ecient number of grat-\nings, the local perturbation of the refractive index has\na homogeneous distribution with no remarkable features\nin any speci\fc direction (see Fig. 2b). Notably, accord-\ning to the recent report24, such a quasiperiodic structure\nhas a complete photonic band gap for all directions of\npropagation of the electromagnetic wave.\nTo obtain a structure that comprises only two dielectric\nmaterials, a binarization procedure27{31is applied to the\nrefractive index sum \u0001 nc(r)\nnb(r) =n+ \u0001nsign (\u0001nc(r)): (2)\nAfter binarization, each grating contributes to the\nlight scattering with an e\u000bective amplitude \u0001 nb;i=\n2\u0001n=p\n\u0019Nopt. The binarization procedure introduces\nadditional noise in the reciprocal space, and its integral\nbackground value is 36% of the Bragg maxima intensity.\nHowever, for a su\u000eciently high grating number, these\nmaxima are so densely distributed that they are almost\nindistinguishable in a \fnite-size structure32(see Fig. 2a).\nTherefore, a proper choice of the grating number provides\na su\u000ecient density of the maxima along the circle.3\naN\nWN\nl1000 \n 500 \n 1\nmkn\n1 400(a) (b) (c)\n1 500 1000400\n 1\n 1\n 0.7\n 0.4 WNaNk\nn\nFIG. 3. Illustration of calculation of a discrete convolution\nfor the ordered photonic crystal. (a) Choosing a \fnite area\nWin a complete structure F. (b) Calculation of the discrete\nconvolution values. (c) Convolution of the ordered photonic\ncrystal.\nWe chose a photonic crystal with a honeycomb lat-\ntice as a periodic and ordered structure (Fig. 2c). Each\nhexagonal unit cell contains six dielectric rods with a ra-\ndius ofr= 0:26asurrounded by air. The \flling fraction\nof the rod material matches that of the quasiperiodic\nstructure. Both structures have the same lattice con-\nstanta. The disordered lattices with no translational\nsymmetry33{36are based on the honeycomb photonic\ncrystal, but the position of each rod was randomly shifted\naccording to x=x0;j+\u000exjandy=y0;j+\u000eyj. Here,\u000exj\nand\u000eyjare random values distributed uniformly in the\ninterval [0:::d r]. Fig. 2d displays an example of such a\ndisordered lattice.\nIII. METRIC FOR PERIODICITY DEGREE\nSince we aim to reveal how periodicity a\u000bects the struc-\ntures' properties, a proper metric is required for quanti-\ntative analysis. Such a metric would allow us to compare\nordered quasiperiodic structures with photonic crystals\nwith a certain value of disorder \u001b. Currently, there exists\nno standard metric with general qualitative and quanti-\ntative properties for an arbitrary case. Various metrics\nwere developed and applied to a wide range of disordered\nsystems37. Most of the proposed metrics use the charac-\nteristics of the disorder spectrum in reciprocal space38{41.\nHowever, they are ine\u000ecient for the description of sam-\nples that are small in real space. Although several studies\nproposed non-trivial real-space metrics that allow distin-\nguishing structures in terms of pseudodisorder33,34, they\ncannot be generalized for other classes of photonic crys-\ntals, including quasiperiodic structures. Thus, an ad hoc\napproach is required.\nIn this study, we exploit a metric based on self-\nconvolution in real space. A reliable \fgure of merit is\nobtained by numerical convolution of the sample with\nitself42, which means calculating the structure numeri-\ncally after discretizing the real-space domain into \fnitesquare samples\nC2D(n;k) =NWX\ni=1NWX\nj=1W(i;j)F(n+i\u00001;k+j\u00001);(3)\nwheren= 1:::N F\u0000NW+ 1,k= 1:::N F\u0000NW+ 1 are\nthe elements of the output convolution matrix, F(l;m)\nis anNFbyNFbinary matrix describing the complete\nstructure,W(i;j) is anNWbyNWarea of the complete\nmatrix of the structure for ( NF=Na\u00004) periods with the\nlattice constant Na=a=\u0001, and \u0001 = 0 :1 is the sam-\npling step. Fig. 3 shows such a 2D convolution for the\nordered photonic crystal. At each location, we calculate\nthe product of each element of the square area and each\nelement of the complete matrix, i.e. we \fnd their over-\nlap, and then we sum the results to obtain the output\nin the current location43. Figs. 5a and 5b show the con-\nvolution maps for the considered disordered lattice and\nquasiperiodic structure, respectively. The maxima in the\nmaps re\rect the degree of periodicity of a structure with\na speci\fc lattice constant a. To quantify the periodic-\nity degree, we consider the nearest maxima located at\na distance from the center equal to the lattice constant\na. We notice here that for the quasiperiodic sample, the\nmaxima merge and form a ring, which indicates an om-\nnidirectional band gap. The non-periodic distribution of\nstructures is manifested in the broadening of the maxima\non the map along with the decrease in their intensity.\nAt the next step, we consider a one-dimensional nor-\nmalized convolution C1D(x) for the selected maxima of\nboth structures, shown in Fig. 5c with solid curves. These\ndata are described adequately by using a second convo-\nlution of Gaussian G(x) and rectangular rect( x) func-\ntions, and the disorder is quanti\fed with respect to the\nstandard deviation of the Gaussian function. The corre-\nsponding equation read\nCrG(x) =X\nx=1rect(x)G(k\u0000x+ 1);(4)\nC0\n1D(x) =X\nx=1CrG(x)CrG(k\u0000x+ 1)\u0019C1D(x);(5)\nwhere rect( x) is a unit-height rectangular pulse of length\nC1D(x),G(x) = exp(\u0000x2=(2\u001b2))=(\u001bp\n2\u0019), and\u001bis the\nstandard deviation. By applying the one-dimensional\nconvolution to our structures, we \fnd the standard de-\nviation\u001bfor the Gaussian distribution when C0\n1D(x)\u0019\nC1D(x). The larger the structure and the square area\nare, the more stable the standard deviation value is. For\nour ordered photonic crystal, the dispersion \u001b= 0, which\nis proven by the fact that the corresponding convolution\nfunction is triangular, see the green curve in Fig. 5c. The\ndashed curve in Fig. 5c shows the metric function C0\n1D(x)\nfor the disordered lattice. This function describes the\noriginal maximum in the map, therefore, we can apply\nthis metric to describe the degree of periodicity in ran-\ndom photonic crystals as well.\nWe \fnd the minimum of the RMSE function44of the\ndi\u000berence in the convolution maps of the quasiperiodic4\n(a) (b) (c)\nFIG. 4. Photographs of samples printed on a 3D printer. (a) Ordered honeycomb photonic crystal of dielectric rods in air. (b)\nDisordered lattice of dielectric rods in air. The rod positions are randomly shifted away from the honeycomb lattice positions\nin bothxandydirections (the disorder degree is dr= 2 mm). (c) Quasiperiodic structure with a binary distribution of two\nmaterials. The structure corresponds to seven gratings merged into a single binary structure. All the structures are 5 mm high.\nStructures in (b) and (c) have the same degree of periodicity \u001b= 4:78.\nstructure and disordered lattice. By this criterion, the\nquasiperiodic structure equivalent to the disordered lat-\ntice with a degree of disorder of dr= 2 mm, as can be\nseen in the inset in Fig. 5c. In this case, the described\nmetric reveals a standard deviation of \u001b= 4:78 for both\nstructures. Thus, these two structures have the same de-\ngree of periodicity, which is the parameter responsible for\nthe formation of photonic band gaps.\nIV. PROBING OF BAND GAP PROPERTIES\nNow we compare the electromagnetic properties of the\nquasiperiodic structure and the disordered lattice with\nthe same degree of periodicity according to the metric\ndescribed above. First, we carried out full-wave simu-\nlation by using the time-domain solver of CST Studio\nSuite software. A linear dipole emitter was located at\nthe center of each structure under consideration. We also\nanalyzed the results after averaging the data for 15 di\u000ber-\nent dipole positions in the central unit cell. To facilitate\nfurther experiments, we chose a dipole source with TM\npolarization (in this case, electric \feld oscillates along\nthe vertical zdirection). The top and bottom bound-\naries of the structure were chosen to be a perfect electric\nconductor; the vertical boundaries of the structure in the\nx-yplane were a perfectly matched layer to simulate open\nboundary conditions.\nTo verify our theoretical predictions in experiment, we\nused polymer samples fabricated with a 3D printer. A\npair of aluminum plates formed a plane-parallel waveg-\nuide for TM polarization, and the samples were placed\nbetween the plates. A small dipole antenna was located\nin the center of the structures. We measured the param-\neters of the structures in the frequency range from 10.8\nto 15 GHz. For each structure, the re\rection coe\u000ecient\n(S11parameter) was measured using a vector network\nanalyzer, and the real part of S11allowed estimating the\nemitted power P. To measure the reference radiation\npower in vacuum P0, we placed the dipole antenna in theplane-parallel waveguide without the structure.\nFor the structures both in the simulation and in the\nexperiment, we chose polylactide (PLA) plastic with a\npermittivity of about \"1= 2:2 (n1\u00191:5) and low losses\n(tan\u000e\u00191\u000110\u00002) in the microwave range45. The dielectric\nstructures were surrounded by air ( n2\u00191). The optimal\nnumber of gratings for the quasiperiodic structure with\nthe refractive contrast n1=n2= 1:5 wasNopt= 7. The\nscale of the structures was chosen for the lattice constant\nto have the same value, a= 10 mm, at di\u000berent fre-\nquencies of the band gap of the quasiperiodic structure\nand the photonic crystals. For the photonic crystals, the\nradius of the dielectric rods was r= 0:26a. These struc-\ntures are shown in Fig. 4. The structures had the dimen-\nsions of 250\u0002250\u00025 mm3(25a\u000225a\u00020:5a), and the\ntotal volume \flling fraction was about 50% for all the\nsamples.\nFirst, we simulated the radiation e\u000eciency and stud-\nied the radiation suppression due to the modi\fcation of\nthe local density of photonic states. The emitted power of\nthe dipolePwas estimated by real part of the impedance\nnormalized to that of the dipole located in free space\nP018,46,47. For convenience, we use dimensionless fre-\nquencya=\u0015for representation of the emission spectra of\nthe dipole in the structures under study (the dashed blue\ncurve in Fig. 6). The photonic crystal structures exhibit\na suppression band around 0 :46a=\u0015(Fig. 6a). A pair of\nemission suppression gaps is clearly seen around 0 :38a=\u0015\nand 0:41a=\u0015in the power spectrum of the quasiperiodic\nstructure (Fig. 6c). We notice that the spectrum of dis-\nordered lattice is quite similar for di\u000berent realizations of\nthe structure. Thus, we do not average the spectrum over\nthe ensemble. The suppression of the radiation related\nto the local density of photonic states decreases strongly\nwithin the band gaps. Noteworthy, the \fnite sizes of\nthe structures result in non-zero local density of photonic\nstates in the band gap. Macroscopic interference e\u000bects\ncause oscillation fringes that are observed in all the spec-\ntra at the frequencies outside the bandgaps, but these\nfeatures are neglected in the further discussion. An im-5\na\n a(a) (b) (c)\n (mm) x (mm) y4.78σ=\n (mm)rdRMSE1 (PC)DС\n1 (DL)DС\n1 (QPS)DС'\n1DС (mm) y\n (mm) x (mm) x\nFIG. 5. Convolution maps of the disordered lattice (a) and the quasiperiodic structure (b) shown in Figs. 4b and 4c,\nrespectively. The vectors of translation are shown with white arrows. (c) Comparison of the cut of convolution along the x-axis\nfor the honeycomb photonic crystal (green curves), the quasiperiodic structure (red curve), the disordered lattice (blue curve)\nand the metric function for the disordered lattice (blue dashed curve). Inset: RMSE di\u000berence between the convolutions of the\nquasiperiodic structure and the disordered lattice as a function of the degree of disorder of the disordered lattice.\n(a) (b) (c)\nQPC PC DL0/PP\n/aλ /aλ /aλSimulation\nExperiment\nFIG. 6. Normalized radiation power P=P 0of a dipole placed inside the honeycomb photonic crystal (a), the disordered lattice\n(b) and the quasiperiodic structure (c). The plots correspond to the theory (dashed blue curve) and the experiment (solid red\ncurve). The gray dashed line shows the level of radiation suppression up to 0.2.\nportant observation is that the both regular structures\ndemonstrate almost the same suppression of radiation\ndown to the baselevel of 20% despite the rather distinct\ndegree of periodicity. The disordered lattice with the\nsame degree of periodicity as the quasiperiodic structure\ndemonstrates a signi\fcantly weaker suppression, about\n40% (Fig. 6b).\nThe normalized TM power emission spectra of the\ndipole obtained in experiment for each structure are\nshown by the solid red curve in Fig. 6. In contrast to\nthe simulation, high-frequency \ructuation fringes are ob-\nserved in the spectra for all the structures. These fringes\nare likely to result from the spatial gaps between the sam-\nple and metal planes in the experimental setup. These\ngaps create additional waveguide modes, which are ab-\nsent in the simulations. However, such oscillations do not\na\u000bect the analysis of the bandgap suppression features.\nFor all the three structures, the spectra exhibit\nbandgaps around 0 :46a=\u0015(photonic crystal samples in\nFig. 6a) and 0 :38a=\u0015and 0:41a=\u0015(the quasiperiodic sam-\nple in Fig. 6c), which is in excellent agreement with theones obtained in simulations. Moreover, the suppression\nof the electromagnetic radiation due to the Bragg band\ngaps is stronger for the ordered structures (down to the\nbaseline at 20%). In the experiment, the disordered lat-\ntice (Fig. 6b) with the same degree of periodicity as that\nof the quasiperiodic structure demonstrates weaker ef-\nfects related to the emission suppression (with a baseline\nat about 40%). Thus, the experimental data completely\ncon\frms the theoretical predictions.\nLet us discuss the parameters related to the modi\fca-\ntion of the local density of photonic states. The main\nparameters contributing to this e\u000bect are the following:\nthe size of the structure, dielectric contrast, stop bands\noverlapping for all spatial directions, and the degree of\nperiodicity. The translational symmetry of a crystal lat-\ntice limits possible rotational symmetries, which are re-\nsponsible for the overlap of the stop bands in all direc-\ntions. If the dielectric contrast is low, there is an addi-\ntional leakage of radiation in photonic crystals, and non-\nperiodicity introduces another leakage channel. In con-\ntrast, quasiperiodic structures have no strict restrictions6\non the overlapping of stop bands, but a \fxed degree of\nperiodicity opposes the suppression of the local density of\nstates. These competing e\u000bects create opportunities for\ndesigning polymer-based structures with a local density\nof states due to an additional degree of freedom provided\nby rotational symmetry.\nV. CONCLUSION\nIn this work, we have analyzed the opportunities pro-\nvided by quasiperiodic structures for advanced manip-\nulation of electromagnetic radiation. Our study focuses\non the suppression of local density of photonic states and\nits interplay with the degree of periodicity of the struc-\ntures. We have proposed a real-space metric to compare\nthe photonic properties of quasiperiodic structures and\ndisordered lattices. We have found that quasiperiodic\nstructures made of available plastic materials achieve the\nsame suppression of local density of photonic states asphotonic crystal with a perfect translational symmetry\ndoes. We have carried out both theoretical and exper-\nimental research, and the results are in excellent agree-\nment. Surprisingly, we have revealed that the lack of\nperiodicity is bene\fcial for the suppression of density of\nphotonic states, and non-periodic structures have inher-\nent advantages over the ordered ones in this regard. Our\n\fndings pave the way for engineering photonic structures\nmade of various low-index materials with an additional\ndegree of freedom enabled by quasicrystal design. We an-\nticipate that polymer-based structures empowered with\na quasiperiodic topology will broaden the possible appli-\ncations of photonic structures.\nACKNOWLEDGMENTS\nWe thank Alexander Petrov for fruitful discussions.\nAuthors acknowledge a support from the Russian Science\nFoundation (Grant No. 20-79-10316).\n\u0003v.chistyakov@metalab.ifmo.ru\n1E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).\n2J. Sajeev, Phys. Rev. Lett. 58, 2486 (1987).\n3J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature\n386, 143 (1997).\n4J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D.\nMeade, Photonic Crystals: Molding the Flow of Light , 2nd\ned. (Princeton: Princeton University Press, 2008).\n5R. Merlin and S. M. Young, Optics Express 22, 18579\n(2014).\n6S. O. Konorov, A. M. Zheltikov, and M. 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Rev.\nLett.117, 085503 (2016)." }, { "title": "2302.08001v1.Learning_Density_Based_Correlated_Equilibria_for_Markov_Games.pdf", "content": "Learning Density-Based Correlated Equilibria for Markov\nGames\nLibo Zhang1,2,∗, Yang Chen2,∗,†, Toru Takisaka1, Bakh Khoussainov1, Michael Witbrock2,\nand Jiamou Liu2,†\n1University of Electronic Science and Technology of China\nChina\n{takisaka,bmk}@uestc.edu.cn\n2The University of Auckland\nNew Zealand\nlzha797@aucklanduni.ac.nz,{yang.chen,jiamou.liu,m.witbrock}@auckland.ac.nz\nABSTRACT\nCorrelated Equilibrium (CE) is a well-established solution concept\nthat captures coordination among agents and enjoys good algorith-\nmic properties. In real-world multi-agent systems, in addition to\nbeing in an equilibrium, agents’ policies are often expected to meet\nrequirements with respect to safety, and fairness. Such additional\nrequirements can often be expressed in terms of the state density\nwhich measures the state-visitation frequencies during the course\nof a game. However, existing CE notions or CE-finding approaches\ncannot explicitly specify a CE with particular properties concerning\nstate density; they do so implicitly by either modifying reward func-\ntions or using value functions as the selection criteria. The resulting\nCE may thus not fully fulfil the state-density requirements. In this\npaper, we propose Density-Based Correlated Equilibria (DBCE), a\nnew notion of CE that explicitly takes state density as selection\ncriterion. Concretely, we instantiate DBCE by specifying different\nstate-density requirements motivated by real-world applications.\nTo compute DBCE, we put forward the Density Based Correlated\nPolicy Iteration algorithm for the underlying control problem. We\nperform experiments on various games where results demonstrate\nthe advantage of our CE-finding approach over existing methods\nin scenarios with state-density concerns.\nKEYWORDS\nCorrelated Equilibrium; State Density; Markov Games\nACM Reference Format:\nLibo Zhang1,2,∗, Yang Chen2,∗,†, Toru Takisaka1, Bakh Khoussainov1, Michael\nWitbrock2,, and Jiamou Liu2,†. 2023. Learning Density-Based Correlated\nEquilibria for Markov Games. In Proc. of the 22nd International Conference on\nAutonomous Agents and Multiagent Systems (AAMAS 2023), London, United\nKingdom, May 29 – June 2, 2023 , IFAAMAS, 11 pages.\n1 INTRODUCTION\nA central question in the study of multi-agent systems is finding\npolicies for rational game players to reach a particular form of\nequilibrium. A more recent trend in the investigation of this ques-\ntion is to incorporate policies’ side effects [ 23]. Indeed, in many\n* Equal contributions\n†Corresponding author\nProc. of the 22nd International Conference on Autonomous Agents and Multiagent Sys-\ntems (AAMAS 2023), A. Ricci, W. Yeoh, N. Agmon, B. An (eds.), May 29 – June 2, 2023,\nLondon, United Kingdom .©2023 International Foundation for Autonomous Agents\nand Multiagent Systems (www.ifaamas.org). All rights reserved.real-world scenarios, it is difficult to define a reward function that\ncaptures all aspects of desired outputs of the agents. For example, a\nrobotic system may gain a high reward by performing a specific\nrisky manoeuvre that is less-than-desirable or engaging in actions\nthat are seen as unethical [ 31]. When finding policies for agents, it\nis therefore not optimal to simply enable agents to achieve the high-\nest possible rewards. Still, more importantly, the procedures must\nalso satisfy other desirable properties, such as safety and fairness,\nthat are not reflected by rewards.\nFor simplicity, we formulate this type of problem as taking an\n𝑁-player Markov game as input while asking for policies of agents\nthat satisfy two types of requirements:\n(1)Reward Requirement: First, we expect that the agents, being\nrational, will not unilaterally deviate from their policies due to\nutility concerns, and\n(2)Non-reward Requirements: Then, the policies must satisfy\ncertain non-utility-based requirements that confine the runs of\nthe multi-agent system.\nIn this paper, we study an instance of the general problem above.\n(1) For the reward requirement, we specify a solution concept that\nfactors into the possible coordination among agents. More specif-\nically, we adopt correlated equilibrium (CE) [ 17] as the solution\nconcept. Compared to Nash equilibrium (NE), widely adopted in\nthis field [ 19], CE does not require independence among agents\nand is suitable for a wider range of practical scenarios. Moreover,\nthe set of CEs constitutes a convex polytope. Therefore, it is easy\nto compute via linear programming. Many adaptive procedures\nare shown to converge to CE rather than the more restricted NE\n[13,16]. (2) For the non-reward requirements, we consider essential\nproperties which can be loosely translated to, e.g., “certain situation\nshould not take place” ,“certain situation should happen with a pre-\nscribed frequency” and“two situations should happen with the same\nfrequency” . These properties can be captured by examining a run,\ni.e., the sequence of states the agents are in during the game. More\nprecisely, they are state-distribution requirements that are defined\nin terms of the visitations to states in the game:\n•Safety requirements. These conditions demand that certain bad\nstates should not be visited . Many industrial applications involve\ndangerous states that should never happen. Take, e.g., the low-\npower status of a robot system [ 29]. Similar concerns can happen\nfrom an ethical perspective as well [31].arXiv:2302.08001v1 [cs.AI] 16 Feb 2023•Frequency requirements. These conditions demand that certain\nstates should be visited with a fixed frequency . To generalise safety,\nthe system may be expected to visit certain states with a certain\nproportion in the long term. For example, we may want a robotic\nsystem to run in high-efficiency mode 30% of the time, and the\nrest 70% time in normal mode.\n•Fairness requirements. These conditions demand that two states\nshould be visited with the same frequency . One may also wish to\nbalance the visitation frequency of two different states of the\nsystem for the sake of, e.g., system stability. For instance, if there\nare two charging stations for a team of uncrewed aerial vehicles,\none may wish to balance their use rate to avoid unnecessary\nqueuing. Or at a crossroads, traffic lights in two directions should\nbe green with equal frequency.\nThestate density function may be employed to measure the state\nvisitation frequency when navigating the environment using a\npolicy [ 29]. The function can express the aforementioned state-\ndistribution requirements. However, so far, no work on CE or CE-\nfinding algorithms has explicitly incorporated requirements defined\nby the state density function. On the other hand, methods have been\nintroduced to implicitly express these non-reward requirements\nby imposing additional constraints on rewards. Yet, these methods\nmay not be sufficient to meet these desired requirements. In detail,\nthe existing techniques fall into two categories:\n(1)Risk-sensitive reward modification. This method adds additional\nterms that tweak the reward structure, such as negative rewards\nfor undesired states or imposing variance as risk-terms [ 5,12,25,\n32]. This method has been preferred when the target is simple,\nand a tweaking strategy can be efficiently designed. However, it\nrequires parameter fine-tuning as the optimal policy is sensitive\nto reward settings. When computing CE, this method changes\nthe shape of the CE set of the original game. As a result, the\noptimal policy found in a modified game may not be a CE to the\noriginal game. Moreover, designing a reward modification for\ncomplex targets such as value-targeting requirement requires\nexpert domain knowledge, which is challenging in real-world\napplications.\n(2)Constrained methods. This method directly takes the subset\nof policies by adding explicit constraints such as constrained\nMarkov games [3]. However, it requires parameter fine-tuning\nbecause the threshold for constraints can directly impact the\ngame’s performance and feasibility. Before solving the game,\nthe optimal solution is invisible to the designers so setting a\ncorrect threshold is challenging. Additionally, when computing\nCE, the introduced constraints may reduce the size of the CE\nset of the original game.\nBoth methods above may change the shape or size of the feasible\nCE set of the original game. Such changes are illustrated in Fig. 1\nIn this paper, we introduce Density-Based Correlated Equilibria\n(DBCE) in the context of Markov games. By using density functions\nas a selection criterion, DBCE explicitly integrates state-distribution\nrequirements (non-reward requirements) and reward requirements\nto a novel CE concept without suffering the issues above in the\nexisting CE notions or CE-finding approaches.\nHowever, having an equilibrium concept does not necessarily\nimply an effective way to find it. Directly computing a DBCE is\nRisk-sensitive Reward ModificationConstrained MethodsOriginal Feasible SetFigure 1: This diagram shows the changes to the feasible\nCE sets by the two existing methods. A constrained method\nleads to a size-reduced feasible set, which can be empty if ad-\nditional constraints are infeasible. The risk-sensitive reward\nmodification generates a new game; thereby, the feasible set\nmay be shifted. Moreover, an improper reward modification\nmay cause an empty intersection between the feasible sets\nfor the modified game and the original game.\nintractable due to the inconsistency between the measurements of\na policy’s state density and cumulative rewards, preventing us from\noptimising the two in the same space. To settle this challenge, we\nemploy the notion of occupancy measure , i.e., the cumulative state-\naction visitation frequency, in terms of which both the state density\nand cumulative rewards can be represented; it thus allows us to\noptimise the two in a unified fashion. This machinery gives rise to\nour proposed algorithm for computing DBCE named Density Based\nCorrelated Policy Iteration (DBCPI). More specifically, DBCPI runs\nin such an iterative manner that alternates between the update\nof agents’ policies and the occupancy measure: the policies are\nupdated by finding a CE that is induced by the current occupancy\nmeasure and satisfies the non-reward requirements; the current\noccupancy is subsequently updated in accordance with the updated\npolicy. Moreover, we provide a theoretical justification for DBCPI\nwhere the convergence conditions are given.\nOur primary contributions are summarised as follows:\n(1)We propose a new CE concept for Markov games– Density-\nBased Correlated Equilibria (DBCE) – which exploits the state\ndensity function to explicitly capture non-reward requirements\nwithout changing the set of all feasible CEs.\n(2)To compute DBCE, we come up with Density-Based Correlated\nPolicy Iteration (DBCPI). We show that under certain assump-\ntions, this mechanism converges to a valid DBCE.\n(3)We test DBCPI against existing approaches on different simu-\nlated scenarios motivated by real-world applications. Experi-\nmental results demonstrate our machinery’s advantage in find-\ning CE with those mentioned above, additional non-reward\nrequirements, i.e., safety, frequency and fairness.\n2 RELATED WORKS\nEquilibrium Concepts. In multi-player games, especially non- coop-\nerative games, solving a game amounts to finding an equilibrium.\nNash Q-learning [ 20] extended the canonical Q-learning to general-\nsum Markov games to find Nash equilibrium. As for finding CEs,\nsome work [ 7,26] attempted to calculate the whole set of CEs\nby determining or approximating the boundary of the resultingexpected-reward space, which has been shown to be a convex poly-\ntope. Alternatively, Greenwald et al . [14] proposed a Q-learning-like\nalgorithm to find an instance of CE in a Markov game rather than\nthe whole set. Another line of work focuses on exploiting the ap-\nplication value of CEs in real-world scenarios, such as Yu et al . [37]\nand Han et al . [15] used CEs to coordinate equipment in industrial\nscenarios; Jin et al . [22] used CEs as the solution to a outsource\ntask pricing problem. A series of work captures some particular\nproperties by selecting a special subclass of CEs from the entire set.\nFor instance, Ortiz et al . [28] and Ziebart et al . [38] choose the CE\nwith the maximum policy entropy to ensure the uniqueness of the\nsolution to a game.\nNon-reward Requirements. In real-world applications, the non-reward\nrequirements are inevitable. Some work adopts an implicit way to\nsatisfy these requirements by modifying the reward functions [ 9].\nOne popular method is to augment the reward function with risk-\nsensitive terms such as variance [ 24] and exponential utility func-\ntion [ 6]. Rather than implicit reward tweaking, logic instruction [ 18]\nthat explicitly describes the goal is also considered as one method\nto modify the reward function. Recently, reinforcement learning\nalgorithms with different risk-sensitive factors have been studied\nin various aspects [ 5,12,25,32]. Some work along this line adds\nconstraints to the learned policy in order to capture safety concerns:\nAltman [2]studied constrained MDP, and subsequently, Q-learning\nwas extended to constrained MDP by [ 10]. Constrained method was\nlater further extended to Markov Games in the multi-agent setting\n[1,3,11,21]. Among these constraints, the state density stands\nout as a particular one. Typical work includes [ 12] that directly\nspecified the unwanted states to avoid getting in, and Qin et al .\n[29] proposed to use density functions as constraints to guide the\nfinding of an optimal policy in reinforcement learning.\n3 PRELIMINARIES\nThe set of all natural numbers, reals, and non-negative reals are\ndenoted by N,R, andR≥0, respectively. For a natural number 𝑁,\nthe set{1,...,𝑁}is denoted by[𝑁].\nMarkov Games. Markov games, also known as stochastic games,\nare extensions of Markov decision processes to the multi-agent\nsetting, where a set of agents act in a stochastic environment, each\naiming to maximise its cumulative rewards.\nDefinition 1. An𝑁-agent Markov game is a tuple\n(S,{A𝑖}𝑁\n𝑖=1,𝑃,{𝑟𝑖}𝑁\n𝑖=1,𝜂,𝛾),where\n•S is the set of states ,\n•A𝑖is the set of actions for the𝑖th agent,\n•𝑃:S×A→Δ(S)is the transition function that specifies\nthe transition probability between two states given a joint action\n𝒂=(𝑎1,...,𝑎𝑛), whereA=×𝑁\n𝑖=1A𝑖is the space of joint actions\nandΔ(S)denotes the set of probability distributions over S,\n•𝑟𝑖:S×A→Ris areward function that determines agent 𝑖’s\nimmediate reward of a joint action in a state,\n•𝜂∈Δ(S)is the initial distribution of states,\n•𝛾∈(0,1)is adiscount factor .\nThroughout, we use bold variables without subscripts to repre-\nsent the concatenation of the corresponding variables for all agentsand use the subscript −𝑖to denote all agents other than 𝑖, e.g.,\n𝒂=(𝑎1,...,𝑎𝑛)=(𝑎𝑖,𝒂−𝑖)denotes a joint action of all agents.\nDefinition 2. The agents’ (stationary) joint policy is a function\n𝝅:S→Δ(A)\nwhich specifies agents’ probabilistic choice of actions according to the\ncurrent state. The set of all joint policies is denoted by Π.\nEach agent aims to find a policy to maximise its own cumulative\nrewards during the whole course of a game:Í∞\n𝑡=0𝛾𝑡𝑟𝑖(𝑠𝑡,𝒂𝑡).For\neach agent𝑖, the expected return of a state-joint action pair under a\njoint policy 𝝅is defined as:\n𝑄𝝅\n𝑖(𝑠,𝒂)≜E\"∞∑︁\n𝑡=0𝛾𝑡𝑟𝑖(𝑠𝑡,𝒂𝑡)\f\f\f\f𝑠0=𝑠,𝒂0=𝒂,𝑃,𝝅#\n.\nCorrelated Equilibria. A solution to a Markov is called an equilib-\nrium that amounts to a joint policy where no agent has an incentive\nto unilaterally deviate to gain rewards. Two canonical equilibrium\nconcepts stand out concerning assumptions on different degrees of\nthe independence among agents’ policies. The well-known Nash\nequilibrium (NE) [ 8] requires independence among the agents, i.e.,\n𝝅=×𝑁\n𝑖=1𝜋𝑖where𝜋𝑖:S→Δ(A𝑖)denotes the policy of an individ-\nual agent. In comparison, correlated equilibrium (CE) [ 4] generalises\nNE by capturing the coordination among agents, which is more suit-\nable for multi-agent systems where agents coordinate their actions.\nConceptually, agents are coordinated by a correlation device that\nrecommends an action 𝑎𝑖∈A𝑖to each agent 𝑖, who is aware of all\nother agents’ conditional distribution 𝝅−𝑖(𝒂−𝑖|𝑠,𝑎𝑖). To be in a CE,\neach agent has no incentive to disobey the recommendation, i.e.,\nselecting an alternate action 𝑎′\n𝑖∈A𝑖, called the deviation action .\nDefinition 3. Acorrelated equilibrium (CE) for a Markov game\nis a joint policy 𝝅that satisfies:\n∀𝑖∈[𝑁],𝑠∈S,𝑎𝑖,𝑎′\n𝑖∈A𝑖,reg𝝅(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖)≤0. (1)\nHere, the regret reg𝝅(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖)embodies the expected reward gain\nof shifting to a deviation action:\nreg𝝅(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖)≜E𝒂−𝑖∼𝝅−𝑖(·|𝑠,𝑎𝑖)\u0002\n𝑄𝝅\n𝑖(𝑠,𝑎′\n𝑖,𝒂−𝑖)−𝑄𝝅\n𝑖(𝑠,𝑎𝑖,𝒂−𝑖)\u0003\n.\nThe general existence of NE [ 8] implies the existence of CE. CE\nhas nicer mathematical properties than NE in the sense that the\nconstraints in Eq. (1)define an𝑁-dimension polytope in agent’s\nexpected returns while the set of NE consists of isolated points\n[27] in the polytope. Consequently, the set of CEs for normal-form\ngames (equivalent to one-shot Markov games) can be derived using\nlinear programming as Eq. (1)is a system of linear inequalities. Still,\nexactly computing CE for Markov games is generally intractable\ndue to two reasons: (i) the constraints of CE turn to non-linear\ninequalities because both 𝑄and𝝅are unknown in Eq. (1); (ii) the\nnumber of corners of the CE polytope grows exponentially with\nthe horizon increases [38].\nDensity Functions. Adensity function [30]𝜌:S→R≥0measures\nthe visitation frequency of states when navigating the environment\nwith a policy. Formally, for an infinite-horizon Markov game with\nits initial distribution 𝜂, discounted factor 𝛾, the density function\nunder a joint policy 𝝅is defined as𝜌𝝅(𝑠)≜∞∑︁\n𝑡=0𝛾𝑡Pr(𝑠𝑡=𝑠|𝝅,𝑠0∼𝜂).\nNotice that the density function can also be written in a recursive\nform as\n𝜌𝝅(𝑠)=𝜂(𝑠)+𝝅(𝑠,𝑎)𝛾∑︁\n𝑠′∈S∑︁\n𝒂∈A𝑃(𝑠|𝑠′,𝒂)𝜌𝝅(𝑠′).\nOccupancy Measure. Similar to the density function, the occupancy\nmeasure𝜌(𝑠,𝑎):S×A→R≥0measures the visitation frequency\nof state-action pairs given a stationary policy. Formally, the occu-\npancy measure 𝜌𝝅under 𝝅is defined as\n𝜌𝝅(𝑠,𝒂)≜∞∑︁\n𝑡=0𝛾𝑡Pr(𝑠𝑡=𝑠,𝒂𝑡=𝒂|𝝅,𝑠0∼𝜂).\nWe can calculate the density of a state by an equation 𝜌𝝅(𝑠)=Í\n𝒂∈A𝜌𝝅(𝑠,𝒂). Occupancy measure also has several properties\nuseful in policy synthesis via optimisation [ 33]. First, a function\n𝑓:S×A→Ris the occupancy measure under some stationary\npolicy if and only if it satisfies the following Bellman flow (BF)\nconstraints :\nBFError𝑓(𝑠)=0,∀𝑠∈S AND𝑓(𝑠,𝒂)≥0,∀𝑠∈S,𝒂∈A,(2)\nwhere BFError𝑓(𝑠)denotes the Bellman residual with respect to the\nstate-action visitation frequency:\nBFError𝑓(𝑠)=∑︁\n𝒂∈A𝑓(𝑠,𝒂)−𝜂(𝑠)−𝛾∑︁\n𝑠′∈S∑︁\n𝒂∈A𝑃(𝑠|𝑠′,𝒂)𝑓(𝑠′,𝒂).\nOn the other direction, for an 𝑓satisfying BF constraints, there is\na unique stationary policy 𝝅∈Πassociated with 𝑓such that𝑓is\nthe occupancy measure under 𝝅(i.e.,𝜌𝝅=𝑓); furthermore, such a\npolicy can be constructed by\n𝝅(𝑠,𝒂)=𝑓(𝑠,𝒂).∑︁\n𝒂′∈A𝑓(𝑠,𝒂′). (3)\nNon-reward Requirements. In addition to reward requirements cap-\ntured by equilibrium concepts, non-reward requirements have also\ndrawn attention. Here, we consider the three typical non-reward\nrequirements: safety, frequency and fairness requirements. These\nrequirements can be measured as the counts of occurrences of cer-\ntain states in a game trajectory , i.e., a sequence of states generated\nby a policy in a Markov game. A finite trajectory with length 𝑛+1is\nwritten as𝜏≜𝑠0,𝑠1,...,𝑠𝑛. We can formalise the above-mentioned\nthree types of non-reward requirements in a trajectory-centric way:\n•Safety: For a set of undesired states 𝑆∗, we expect the count\nof undesired states in the trajectory equals zero,Í\n𝑖∈[0,𝑛]I(𝑠𝑖∈\n𝑆∗)=0where Iis the indicator function;\n•Frequency: For a set of specific states 𝑆∗, we expect the count\nof such states occur in trajectory with a certain proportion 𝑐,Í\n𝑖∈[0,𝑛]I(𝑠𝑖∈𝑆∗)/(𝑛+1)=𝑐\n•Fairness: For 2 sets of states 𝑆∗\n1,𝑆∗\n2, we expect the counts of such\nstates from 2 sets to be equal in trajectory,Í\n𝑖∈[0,𝑛]I(𝑠𝑖∈𝑆∗\n1)=Í\n𝑖∈[0,𝑛]I(𝑠𝑖∈𝑆∗\n2).\nIntuitively, we demonstrate the three types of non-reward re-\nquirements in Fig. 2 on a Markov game with two different states.\nSafety Avoid visiting unsafe (black) statesFrequency Black states are wished to be visited exactly at a 25% frequencyFairnessBlack and white states are wished to be visited at the same frequencyFigure 2: Demonstrations of three types of non-reward re-\nquirements. Sequences of circles represent the trajectory,\nand black and white colors represent two different states.\n4 DENSITY-BASED CORRELATED\nEQUILIBRIA\nIn this section, we first propose the general definition of Density-\nBased Correlated Equilibria (DBCE). We then instantiate it by spec-\nifying the selection criterion as above mentioned three types of\nnon-reward requirements.\nRecall that the motivation of DBCE is to find an equilibrium that\ncan capture both agents’ coordination and policies’ side effects that\ncannot be simply represented in terms of rewards but can instead\nbe interpreted using the density function. To this end, we formalise\nDBCE by taking one or a set of density functions as the selection\ncriterion to identify the subset of CEs that satisfy the desired non-\nreward requirements. In such a way, DBCE is defined as a solution\nto a constrained optimisation problem, where the density functions\nserve as the objective and the constraints enforce the conditions of\nbeing a CE.\nDefinition 4. Let the following be given:\n•A Markov Game(S,{A𝑖}𝑁\n𝑖=1,𝑃,{𝑟𝑖}𝑁\n𝑖=1,𝜂,𝛾);\n•A subset of statesS∗={𝑠1,...,𝑠𝑚}⊆S ;\n•A real-valued function 𝐹:R𝑚→R;\n•A function𝜑(𝝅)=𝐹(𝜌𝝅(𝑠1),...,𝜌𝝅(𝑠𝑚)), which we call the\ndensity error of𝝅.\nA joint policy 𝝅is called an (𝐹-specified) density-based correlated\nequilibria (DBCE) if it is a solution to the following constrained opti-\nmisation problem:\nmin\n𝝅∈Π𝜑(𝝅)subject to\nreg𝝅(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖)≤0,∀𝑖∈[𝑁],𝑠∈S,𝑎𝑖,𝑎′\n𝑖∈A𝑖.(4)\nWe can use the value of the density error 𝜑(𝝅)to indicate the\nquality of 𝝅in terms of the state density. By choosing a suitable\nfunction𝐹, we can instantiate the DBCE that captures a specific\nnon-reward requirements. Here, we introduce the following specific\nDBCEs with respect to safety, frequency and fairness requirements:\na DBCE is called a\n•Minimum Density CE (MDCE) when we have\n𝜑(𝝅)=Í\n𝑠∈S∗𝜌𝝅(𝑠);\n•Frequency Matching CE (FMCE) when we have\n𝜑(𝝅)=|Í\n𝑠∈S∗𝜌𝝅(𝑠)−𝑐|for some𝑐∈R≥0;•Minimum Density Gap CE (MDGCE) when we have\n𝜑(𝝅)=|Í\n𝑠∈S1𝜌𝝅(𝑠)−Í\n𝑠∈𝑆2𝜌𝝅(𝑠)|for𝑆1,𝑆2⊆S∗.\nIntuitively, we use MDCE, FMCE and MDGCE to represent CEs\nwith requirements concerning safety, frequency, and fairness re-\nquirements, respectively. These instantiations indicate the general\nability of DBCE to characterise the equilibria with some density-\nrelated properties, which are not yet able to be represented by other\nequilibrium notions.\n5 DENSITY-BASED CORRELATED\nEQUILIBRIA FINDING\nThis section is devoted to the introduction of a policy iteration\nalgorithm to compute DBCE and the proof of its convergence under\ncertain assumptions. For simplicity, our analysis centers around the\nMinimum Density CE (MDCE); but it applies to any other instance\nof DBCE.\n5.1 Density-Based Correlated Policy Iteration\nRecall that computing an MDCE requires solving the constrained\noptimisation problem defined in Eq. (4)where𝜑(𝝅)=Í\n𝑠∈S∗𝜌𝝅(𝑠).\nHowever, directly solving it is intractable because the density func-\ntions in the objective and the expected return in the constraints\nare defined in two different spaces; this prevents us from optimis-\ning the density-related objective whilst satisfying rewards-related\nconstraints in a unified fashion. We thus ask for a way to unify\nthe representations of the state density and expected return. Fortu-\nnately, we observe that both the state density and expected return\ncan be rewritten in terms of the occupancy measure introduced in\nSec. 3. We can thereby simultaneously control the two by maintain-\ning a single variable, rather than in two separate spaces.\nWe next show how to derive an equivalent yet tractable form of\nthe original constrained optimisation problem. We first rewrite the\nconstraints of Eq. 4 by occupancy measure, reg′\n𝑓(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖), which\nis defined as follows:\n∑︁\n𝒂−𝑖𝑓(𝑠,𝑎𝑖,𝒂−𝑖)\u0002\n𝑄𝝅\n𝑖(𝑠,𝑎′\n𝑖,𝒂−𝑖)−𝑄𝝅\n𝑖(𝑠,𝑎𝑖,𝒂−𝑖)\u0003\n.\nThis is equivalent to reg′\n𝜌𝝅(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖), shown via Eq. (3).\nWith the relationship between the occupancy measure and the\ndensity function, the objective function 𝜑(𝝅)can also be rewritten\nas𝜑′(𝑓), where\n𝜑′(𝑓)≜𝐹\u0000∑︁\n𝒂𝑓(𝑠1,𝒂),...,∑︁\n𝒂𝑓(𝑠𝑚,𝒂)\u0001. (5)\nRecall that the density function can be rewritten as the sum of\noccupancy measures within one state: 𝜌𝝅(𝑠)=Í\n𝒂∈A𝜌𝝅(𝑠,𝒂), the\nnew objective function becomes equivalent to the original objective\nfunction. So far, we achieve the consistency between non-reward\nrequirements and reward requirements by occupancy measure. By\nthe properties of occupancy measure discussed in the previous\nsection, we recast the problem as follows:Problem 1. min\n𝑓:S×A→R∑︁\n𝑠∈𝑆∗∑︁\n𝒂∈A𝑓(𝑠,𝒂)subject to\nreg′\n𝑓(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖)≤0,∀𝑖∈[𝑁],𝑠∈S,𝑎𝑖,𝑎′\n𝑖∈A𝑖;(6)\nBFError𝑓(𝑠)=0,∀𝑠∈S; (7)\n𝑓(𝑠,𝒂)≥0,∀𝑠∈S,𝒂∈A. (8)\nHere, the Bellman flow constraints (7) and (8) enforce 𝑓to be\nthe occupancy measure under some 𝝅∈Π. For such an 𝑓, the new\nobjective functionÍ\n𝒂∈A𝑓(𝑠∗,𝒂)is equal to𝜌𝝅(𝑠∗), and (6) en-\nforces 𝝅to be a CE. Due to the one-to-one correspondence between\noccupancy measures and stationary policies, a solution of Prob. 1\nis the occupancy measure under a solution to the original problem.\nHowever, Prob. 1 is still difficult to solve directly because both\n𝑓and𝑄(involved in reg′\n𝑓) are unknown. We introduce an itera-\ntive approach to handle the problem that we call Density-Based\nCorrelated Policy Iteration (DBCPI). It alternates between: (i) policy\nevaluation : estimating 𝑄values according to the current policy;\nand (ii) policy improvement : computing a DBCE under the current\n𝑄function. More formally, let 𝑡denote the index of iterations. At\neach iteration, Q𝑡={𝑄𝑡\n𝑖}𝑖∈[𝑁]defines a stage game with constant\n𝑄values. Define reg𝑡\n𝑓(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖)as follows:\n∑︁\n𝒂−𝑖𝑓(𝑠,𝑎𝑖,𝒂−𝑖)\u0002\n𝑄𝑡\n𝑖(𝑠,𝑎′\n𝑖,𝒂−𝑖)−𝑄𝑡\n𝑖(𝑠,𝑎𝑖,𝒂−𝑖)\u0003\n.\nBy substituting reg′\n𝑓in Prob. 1 with reg𝑡\n𝑓, the stage game Q𝑡is now\ntractable to solve using linear programming. After deriving the\nDBCE 𝝅𝑡of the current stage game Q𝑡, we head back to update 𝑄\nfunctions and derive Q𝑡+1. The pseudocode is presented in Alg. 1.\nAlgorithm 1 Density-Based Correlated Policy Iteration\n1:Input: A Markov game(S,A,𝑃,{𝑟𝑖}𝑁\n𝑖=1,𝜂,𝛾).\n2:Initialisation: 𝑄𝑖for each𝑖∈[𝑁], learning rate 𝛼\n3:𝝅(𝑠,𝒂)←𝑓(𝑠,𝒂)/Í\n𝒂′∈A𝑓(𝑠,𝒂′)\n4:foreach iteration do\n5:𝑓←(solution to Prob. 1 with {𝑄𝑖}𝑖∈[𝑁])\n6: 𝝅(𝑠,𝒂)←𝑓(𝑠,𝒂)/Í\n𝒂′∈A𝑓(𝑠,𝒂′)\n7:while Not converge do\n8: Initialise state 𝑠∈S\n9: Observe transition (𝑠,𝒂,𝒓,𝑠′)\n10: foreach𝑖∈[𝑁]do\n11:𝑉𝑖(𝑠′)←Í\n𝒂′∈A𝝅(𝑠′,𝒂′)𝑄𝑖(𝑠′,𝒂′)\n12:𝑄𝑖(𝑠,𝒂)←( 1−𝛼)𝑄𝑖(𝑠,𝒂)+𝛼(𝑟𝑖+𝛾𝑉𝑖(𝑠′))\n13: end for\n14: Decay𝛼\n15: end while\n16:end for\n17:Output: A joint policy 𝝅, and𝜑′(𝑓)as the error of 𝝅.\n5.2 Convergence Analysis\nWe next prove that Q𝑡converges to the 𝑄values under a DBCE\nas Alg. 1 is applied. We begin by introducing the following useful\ntechnical assumptions.\nAssumption 1. Each state𝑠∈ S and action𝑎𝑖∈ A𝑖for all\n𝑖∈[𝑁]are visited infinitely often.Assumption 2. The reward is bounded by some constant.\nAssumption 3. The learning rate 𝛼𝑡satisfies the following condi-\ntions: 0≤𝛼��<1∀𝑡,Í\n𝑡𝛼𝑡=∞andÍ\n𝑡𝛼2\n𝑡<∞.\nWe also need the following lemma that guarantees the policy\nestimation procedure after solving each stage game can converge\nto𝑄values under 𝝅𝑡.\nLemma 1 ([ 34]).LetQbe the space of all 𝑄functions. Under\nAssumption 1-3, the iteration defined by the following converges to\n𝑄𝝅with probability 1:\n𝑄𝑡+1(𝑠,𝒂)=(1−𝛼)𝑄𝑡(𝑠,𝒂)\n+𝛼𝑡\u0010\n𝑟(𝑠′,𝒂)+𝛾∑︁\n𝒂𝝅(𝒂|𝑠′)𝑄𝑡(𝑠′,𝒂)\u0011\n.\nNow, we are ready to present our main theorem which shows\nthat DBCPI converges to a DBCE 𝑄function under assumptions.\nTheorem 1. Under Assumption 1-3, the 𝑄function iteratively\nupdated in Alg. 1 will converge to the one under a DBCE if for all 𝑡,\n𝑠∈S, and𝑖∈[𝑁], the policy 𝝅𝑡is recognised as the global optimum\nexpressed as:\n∀𝝅′∈Π,E𝒂∼𝝅𝑡[𝑄𝑡\n𝑖(𝑠,𝒂)]≥E𝒂∼𝝅′[𝑄𝑡\n𝑖(𝑠,𝒂)].\nProof sketch. The basic idea is to show that the policy guided\nby DBCPI monotonically improves in terms of rewards. By Lemma 1,\nfor all𝑡, after sufficient rounds of updates, we derive 𝑄functions\nQ𝑡+1={𝑄𝑡+1\n𝑖}𝑖∈[𝑁]under 𝝅𝑡. By assumption, there always exists\na globally optimal policy for at each encountered stage game Q𝑡.\nAs a result, every iteration the policy monotonically improves as\nthe iteration progresses:\nE𝒂∼𝝅𝑡+1[𝑄𝑡+1\n𝑖(𝑠,𝒂)]≥E𝒂∼𝝅𝑡[𝑄𝑡+1\n𝑖(𝑠,𝒂)],\nfor all𝑡,𝑠∈ S, and𝑖∈ [𝑁]. This implies that after sufficient\nnumber of iterations, the policy converges to a globally optimal\none, so does the 𝑄function. By solving each stage game Q𝑡using\nlinear programming, all constraints in Prob. 1 can be satisfied. At\nconvergence, the policy is thus a feasible DBCE. □\nAlthough Thm. 1 tells us that the convergence holds true under\nstrong constraints on every stage game, in experiments we find the\nconstraint is not necessary for DBCPI to converge. This fact is in\naccordance with the empirical analysis in [20, 36].\n6 EXPERIMENTS\nWe seek to answer the following questions via experiments:\nQ1.Does our algorithm find a CE better than other approaches?\nWe evaluate this by checking the following values upon the termi-\nnation of Alg. 1 after 𝐾iterations:\nMaxReg ≜max\n𝑠,𝑖,𝑎 𝑖,𝑎′\n𝑖reg𝐾+1\n𝑓(𝑠,𝑖,𝑎𝑖,𝑎′\n𝑖),\nMaxBF ≜max𝑠|BFError𝑓(𝑠)|.\nMaxReg can be seen as a “distance” between 𝝅and the CE-set:\nthe larger the value is, the larger incentive there exists for some\nagent to deviate from 𝝅. In particular, 𝝅is a CE when this value is\nnon-positive. MaxBF evaluates the soundness of the computation\nin line 5: the larger the value is, the further 𝑓deviates from 𝜌𝝅, i.e.,\n(a) FairGamble\n (b) Hunters\n (c) CaE\nFigure 3: Screenshots of games.\nthe occupancy measure of 𝝅. Such a deviation of 𝑓implies that the\nvalue𝜑′(𝑓)is unreliable as the error of 𝝅.\nTo evaluate the algorithms in this question, we focus on the\nMaxReg andMaxBF values in all cases. The smaller the values are,\nthe better the algorithms are. 9 extra MDCE tasks are carried out\nto compare the MaxReg in modified games and original games for\nrisk-sensitive reward modification.\nQ2.Does our policy generate desired trajectories?\nWe evaluate this by examining the patterns of individual trajectories\nunder the policy computed by Alg. 1 on all three requirements.\nQ3.What is the accuracy of our DBCPI compared to existing ones?\nTo make the comparison in accuracy, we compare the errors (i.e.,\nthe value𝜑′(𝑓)in the output of Alg. 1), and select a few instances\nfrom our data and illustrate them by plots.\nQ4.What is the convergence performance of our DBCPI on learning?\nWe evaluate this by performing the iteration error plots of our\nDBCPI in the experiments.\n6.1 Experiment Setup\n6.1.1 Game environments and tasks. We consider three game mod-\nels (Fair Gamble, Hunters, Collect and Explore); for each of them,\nwe impose three different state-distribution requirements, which\nmake nine instances of the input to Alg. 1 in total. We have anony-\nmously published animated demonstrations of our algorithm on\nthese games, which are available at https://github.com/nanaralala/\nDensity-based-Correlated-Equilibrium/. The screenshots are pre-\nsented in Fig. 3. The descriptions of each game is shown below.\n(1)Fair Gamble . In this game, two gamblers play games with each\nother, and they choose from 3 different games. 3 games are\nextremely fair, so no matter what they do, the reward will be\ngiven randomly. Game 1 gives 0 rewards fairly; Game 2 gives 0.5\nto a gambler and -0.5 to another; Game 3 gives 1 to a gambler\nand -1 to another.\nIn each round, the gamblers choose a number from 0,1,2 and\nwe compare the number to select which game they play. See\nFig. 3a for the explanation.\nWe consider three state-related requirements, namely\n(a) a safety requirement for gamblers to avoid game 3.\n(b)a frequency matching requirement for gamblers to choose\ngame 3 in 10% of the time.\n(c)a fairness requirement that demands game 1 and game 2 to\nhave equal frequency.\n(2)Hunters . In this non-cooperative game, 3 hunters live in one\nvillage. In each round, they are inside the village or outside the\nvillage, and they can choose between going hunting or guardingthe village against the animals, see Fig. 3b. If one hunter goes\nhunting from the village, the hunter will get a high reward (1)\nand the rest of the hunters will get a low reward (0.1). If one\nhunter guards the village, all hunters will get the same mid-level\nreward (0.5). If one hunter is outside of the village and still stays\nhunting outside, we consider the behaviour is not safe enough,\nso the reward he gains becomes smaller (0.5), and the others\nwill get a punishment reward (-0.5). Additionally, if there is less\nor equal than 1 hunter guarding the village, they will receive\nhigh punishment reward (-3).\nWe consider three state-related requirements, namely\n(a)a safety requirement that demands at least 2 hunters to stay\nin the village.\n(b)a frequency matching requirement that requires less or equal\nto one agent guarding the village 10% of the time.\n(c)a fairness requirement that demands that the frequency of\nhunter 1 goes hunting and the sum of hunter 2 and 3 go\nhunting to be equal.\n(3)Collect and Explore (CaE). In this cooperative game, 3 agents are\ntrapped in a forest, see Fig. 3c. They can choose to explore the\nenvironment or collect some food nearby their accommodation.\nIf more than one agent chooses to go out, we randomly choose\none of them to go out, and the others will get no rewards. In\neach round, if existing agents go exploration, we add 1 to the\nreward; if any agent collects foods nearby, we add 0.3 to the\nreward. Since it’s a cooperative environment, we set the same\nreward for all agents.\nWe consider three state-related requirements, namely\n(a) a safety requirement for agent 1 to not go out.\n(b)a frequency matching requirement that confines agent 1 to\ngo out 10% of the time.\n(c)a fairness requirement that demands that the frequency of\nagent 1 go explore be equal to the sum of the frequencies that\nagent 2 and agent 3 go explore.\n6.1.2 Baselines. We implement utilitarian CE-Q [14] with risk-\nsensitive reward modification and constrained methods. The detail\nis shown as follows:\n(1)Risk-sensitive Reward modification (RM). A negative constant\n𝑝<0is added to the reward at 𝑠∈𝑆∗. We write RM- 𝑝to denote\nthe algorithm with a specific 𝑝.\n(2)Constrained method (CM). An additional constraint 𝜑′(𝑓)≤𝑏\nis added (cf. eq. (5)). We write CM- 𝑏to denote the algorithm\nwith a specific 𝑏.\nRM is used as a baseline for the safety requirement only; we are not\naware if there exists a canonical way to do that for frequency match-\ning and fairness requirements. CM is used for all state requirements\nwe consider in the experiment.\n6.1.3 Implementation details. The iteration number of Alg. 1 is\nset to 250. Parameters in Alg. 1 are set to 𝛾=0.99and𝛼decays\nfrom 0.3 to 0.001. We run the algorithm 3 times for each experiment\nenvironment, and the results are taken as the mean of 3 runs. In the\nprogram, the optimisation problem in line 5 of Alg. 1 was solved\nusing an optimizer in [35].Table 1: Comparisons on capabilities of found CE.\nGame Metric MethodRequirement\nSafety Fairness Freq-10\nFairGambleErrorDBCE 1.225 32.967 8.683\nCM-0.05 8.331 0.466 6.053\nCM-5 9.65 8.814 5.668\nRM-1.5 17.725 — —\nMaxBFDBCE 0.464 0.13 0.521\nCM-0.05 16.496 8.16 11.612\nCM-5 18.054 7.781 6.623\nRM-1.5 0.032 — —\nMaxRegDBCE 0.164 0.034 0.08\nCM-0.05 0.174 0.172 0.22\nCM-5 0.391 0.107 0.231\nRM-1.5 0.11 — —\nHuntersErrorDBCE 14.129 4.442 3.643\nCM-0.05 2.313 2.283 0.05\nCM-5 2.124 2.174 5\nRM-1.5 0.828 — —\nMaxBFDBCE 0 0 0.035\nCM-0.05 0.005 0.001 0\nCM-5 0.004 0.032 0\nRM-1.5 0.001 — —\nMaxRegDBCE 0.044 0.061 0.037\nCM-0.05 9.171 0.52 0.225\nCM-5 0.479 1.18 0.18\nRM-1.5 0.842 — —\nCaEErrorDBCE 0.419 0.002 7.608\nCM-0.05 0.242 0.05 1.174\nCM-25 16.802 9.688 11.854\nRM-0.5 23.838 — —\nMaxBFDBCE 0 0 0\nCM-0.05 0.023 0 0\nCM-25 0.004 1.151 0\nRM-0.5 0.023 — —\nMaxRegDBCE 0.002 0.001 0.003\nCM-0.05 1.266 0.002 0.174\nCM-25 0.419 6.695 0.001\nRM-0.5 1.434 — —\n6.2 Experiment Results\nThe following results and discussions answer questions asked at\nthe beginning of this section.\nQ1.The results for this question are found in the MaxBF and\nMaxReg rows in Tab. 1. DBCE has the smallest MaxBF value in\n7 of 9 cases, which means DBCE performs better in finding oc-\ncupancy measures and mapping them to policies. DBCE also has\nthe smallest MaxReg value in 7 of 9 cases, which means DBCE\nperforms better in finding policies in the CE-set. In conclusion of\nthis observation, our selection criteria perform better in finding CE\npolicies in these experiments. In 9 extra MDCE runs, 7 of 9 runs\nshow that the distance to CE-set in the original game is longer thanthe distance to CE-set in the modified game, which indicates the\ndisadvantage of the RM method.\nQ2.The result for this question is found in Fig. 4a for MDCE, Fig. 4b\nfor FMCE and Fig. 4c for MGDCE. Each plot includes a few trajec-\ntories generated by our algorithm, we estimate the gap between\nthe trajectories and the expected visitation counts, so the closer to\n0 the value is, the better the trajectory is. In Fig. 4a, we observe\nthe trajectories have the expected property, which is the count of\nvisitations to undesired states is low. In Fig. 4b, we observe the\ntrajectories generated by our algorithm gradually converge to the\nexpected proportion of visitation. Few trajectories have a larger\ndeviation from the desired value, which indicates a larger deviation,\nbut others perform well. In Fig. 4c, all trajectories perform well in\nbalancing visitation in 30 steps (a similar pattern appears in the\nrest 220 steps).\nQ3.The selected result for this question is shown in Fig. 2. There,\nwe picked up the cases with comparable MaxReg andMaxBF val-\nues, which means we drop the cases with large MaxReg orMaxBF\nvalues; notice that these numbers should be (close to) zero to claim\nthat the algorithm found a CE with the computed error. The ex-\nhaustive result is found in Tab. 1.\nIn detail, DBCE has the smallest MaxReg andMaxBF values in\nmost cases. In CaE-MinGap case, DBCE and Cons005 have almost\nthe same performance in MaxReg andMaxBF , but DBCE has a\nsmaller error.\nQ4.The results for this question are found in Fig. 5. We can observe\nthe error-step line gradually gets stable. Fluctuations in the early\nstages vanish along with learning. We believe this early-stage fluc-\ntuation can be caused by the updating of Q-functions, and risk gets\nstable after the Q-functions get stable, which is reasonable since\nit’s a policy-iteration process.\nTable 2: Explaintion on Comparable Instances\nTask Method MaxReg MaxBF Error\nHunters-MinGapDBCE 0.061 0 4.442\nCons005 0.52 0.001 2.283\nCons5 1.18 0.032 2.174\nFairGamble-MDCEDBCE 0.164 0.464 1.225\nRewMod 0.11 0.032 17.725\nCaE-MinGapDBCE 0.001 0 0.002\nCons005 0.002 0 0.05\nCons5 6.695 1.151 9.688\nCaE-MDCEDBCE 0.002 0 0.419\nCons005 1.266 0.023 0.242\nCons25 0.419 0.004 16.802\nRewMod 1.434 0.023 23.838\n7 CONCLUSION AND FUTURE WORKS\nIn this paper, we propose a new concept of the correlated equilib-\nrium, the Density-Based correlate equilibrium (DBCE). It enables\nus to find joint policies that satisfy both reward requirements, i.e.,equilibrium, and non-reward requirements characterised by state\ndensity functions. Different from existing methods, DBCE neither\nmodifies the shape or size of feasible CE-set to a game nor suffers\nthe parameter tuning problem. We connect density and reward by\noccupancy measure, and design Density-Based Correlated Policy\nIteration (DBCPI) to compute DBCE. Experiments on various games\nprove the advantage of our method in finding desired CEs. In fu-\nture works, one may be interested in implementing parameterised\nversion of DBCPI to solve more complex games with continuous\nstate-action space games. Additionally, one density-based objective\nmay lead to multiple points in the CE space, so further selection\namong those candidates can also be the next step.\nREFERENCES\n[1]E. Altaian, K. Avrachenkov, Nicolas Bonneau, mérouane Debbah, Rachid El-\nAzouzi, and Daniel Menasché. 2007. Constrained Stochastic Games in Wireless\nNetworks. GLOBECOM - IEEE Global Telecommunications Conference , 315 – 320.\nhttps://doi.org/10.1109/GLOCOM.2007.66\n[2]Eitan Altman. 1993. 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Here we show experimental results on 20 runs, to provide stronger statistical information.\nMethod TaskError\nMeanError\nStdMaxBF\nMeanMaxBF\nStdMaxReg\nMeanMaxReg\nStdRunTime\nMeanRunTime\nStd\ncons005 CaEMinGap 0.07 0.06 0.00 0.01 0.27 0.81 161.20 28.49\ncons25 CaEMinGap 12.04 10.22 0.00 0.01 1.31 3.37 194.49 37.35\nDBCE CaEMinGap 4.27 8.39 0.05 0.13 0.31 0.68 255.07 42.78\ncons005 CaEMDCE 0.48 0.66 0.05 0.08 3.49 7.28 256.63 33.06\ncons25 CaEMDCE 13.93 10.78 0.70 3.02 0.58 1.09 171.41 27.09\nDBCE CaEMDCE 2.23 5.99 0.00 0.01 0.22 0.67 166.01 23.90\nModRew CaEMDCE 20.70 16.81 0.42 1.75 0.07 0.13 208.10 21.92\ncons005 CaEFreq-10 1.30 2.50 0.03 0.10 0.26 0.47 265.13 88.26\ncons25 CaEFreq-10 10.84 5.91 0.00 0.01 0.07 0.21 211.06 33.89\nDBCE CaEFreq-10 11.35 4.78 0.04 0.14 0.03 0.05 203.83 44.49\ncons005 FairGambleMinGap 3.07 4.34 10.84 8.03 0.18 0.17 150.41 20.72\ncons5 FairGambleMinGap 8.50 6.00 6.75 3.85 0.15 0.07 177.02 53.26\nDBCE FairGambleMinGap 23.47 16.63 0.46 0.78 0.12 0.12 266.40 31.82\ncons005 FairGambleMDCE 5.69 5.93 11.25 11.79 0.19 0.21 347.57 182.65\ncons5 FairGambleMDCE 6.54 5.86 9.73 12.22 0.20 0.21 297.36 128.42\nDBCE FairGambleMDCE 8.05 14.31 0.19 0.34 0.15 0.34 370.30 36.37\nModRew FairGambleMDCE 8.72 14.34 0.24 0.32 0.18 0.35 338.67 46.01\ncons005 FairGambleFreq-10 5.68 1.57 13.47 10.93 0.34 0.30 130.20 38.12\ncons5 FairGambleFreq-10 6.17 1.52 8.04 9.80 0.22 0.19 195.69 67.60\nDBCE FairGambleFreq-10 9.63 7.32 0.42 0.94 0.07 0.07 322.61 43.02\ncons005 HuntMinGap 1.30 1.89 0.00 0.01 1.69 1.07 153.83 8.26\ncons5 HuntMinGap 1.04 1.57 0.01 0.04 1.32 1.31 154.30 12.01\nDBCE HuntMinGap 2.76 4.35 0.90 2.94 0.14 0.35 397.15 666.03\ncons005 HuntMDCE 1.20 0.73 0.04 0.13 4.33 1.78 525.69 131.39\ncons5 HuntMDCE 2.29 2.44 0.02 0.07 1.05 1.50 162.71 8.36\nDBCE HuntMDCE 11.49 7.07 0.01 0.03 0.03 0.05 206.20 40.72\nModRew HuntMDCE 1.67 1.62 4.31 18.43 2.81 1.75 200.18 14.53\ncons005 HuntFreq-10 2.59 4.67 0.02 0.05 0.91 1.20 435.08 37.31\ncons5 HuntFreq-10 5.41 1.24 0.14 0.47 0.70 1.40 257.46 27.31\nDBCE HuntFreq-10 7.69 5.26 0.00 0.00 0.07 0.09 298.59 39.70\nBelow we show the error plots of 2 baselines in Fig. 6, which can be understood in a similar way as Fig. 5.\n(a) FairGamble-MDCE\n (b) CaE-MinGap\nFigure 6: The error plot of constrained-25 and ModRew8 DETAILS OF GAME MODELS\nThe details of models are as follows.\n(1)Fair Gamble . In this game, two gamblers play games with each other, and they choose from 3 different games. 3 games are extremely\nfair, so no matter what they do, the reward will be given randomly. Game 1 gives 0 rewards fairly; Game 2 gives 0.5 to a gambler and\n-0.5 to another; Game 3 gives 1 to a gambler and -1 to another.\nEach round, the gamblers choose a number from 0,1,2 and we compare the number to select which game they play. See figure3a for\nthe explanation.\n(2)Hunters . In this game, 3 hunters live in one village. In each round, they are inside the village or outside the village, and they can\nchoose between going hunting or guarding the village from the animals, see Figure3b. If one hunter goes hunting from the village, the\nhunter will get a high reward (1) and the rest of the hunters will get a low reward (0.1). If one hunter guards the village, all hunters\nwill get the same mid-level reward (0.5). If one hunter is outside of the village and still hunts outside, the behaviour is not safe enough,\nso the reward he gains is smaller (0.5), while the others will get a punishment reward (-0.5). Additionally, if there is less or equal than\n1 hunter guarding the village, they will receive high punishment reward (-3).\n(3)Collect and Explore . This is a cooperative game. In this environment, 3 agents are trapped in a forest, see Figure 3c. They can choose to\nexplore the environment or collect some foods nearby their accommodation. In each round, only one agent can go out to explore\nthe environment. If more than one agent chooses to go out, we randomly choose one of them to go out, and the others will get no\nrewards. In each round, if existing agents go exploration, we add 1 to the reward; if any agent collects foods nearby, we add 0.3 to the\nreward. Since it’s a cooperative environment, we set the same reward for all agents." }, { "title": "2302.09171v2.Light_front_synchronization_and_rest_frame_densities_of_the_proton__Electromagnetic_densities.pdf", "content": "NT@UW-23-03\nLight front synchronization and rest frame densities of the proton: Electromagnetic densities\nAdam Freese1,∗and Gerald A. Miller1,†\n1Department of Physics, University of Washington, Seattle, WA 98195, USA\nWe clarify the physical origin and meaning of the two-dimensional relativistic densities of the light front\nformalism. The densities are shown to originate entirely from the use of light front time instead of instant\nform time, which physically corresponds to using an alternative synchronization convention. This is shown by\nusing tilted light front coordinates, which consist of light front time and ordinary spatial coordinates, and which\nare also used to show that the obtained densities describe a system at rest rather than at infinite momentum.\nThese coordinates allow all four components of the electromagnetic current density to be given clear physical\nmeanings. We explicate the formalism for spin-half targets, obtaining charge and current densities of the proton\nand neutron using empirical form factor parametrizations, as well as up and down quark densities and currents.\nAngular modulations in the densities of transversely-polarized states are explained as originating from redshifts\nand blueshifts due to quarks moving in different longitudinal directions.\nI. INTRODUCTION\nThere has recently been renewed interest and debate about the proper manner of understanding spatial densities of hadrons,\nespecially regarding their application to the energy-momentum tensor [1–4]. This interest is especially pertinent to the upcoming\nElectron Ion Collider [5–7], since spatial tomography of partons in hadrons is one of the facility’s central focuses. It is vital to\nthe success of this program that an accurate formalism be employed to obtain spatial distributions of partons within hadrons.\nFor the longest time, most authors have calculated relativistic densities using three-dimensional Fourier transforms of form\nfactors, obtaining results referred to as Breit frame densities or Sachs densities [8]. For a long time, too, this approach has been\ncriticized as relativistically inexact and as not following from elementary field-theoretic definitions [9–11]. While most authors\ncontinue to use Breit frame densities uncritically, a variety of criticisms and defenses of the Breit frame densities and alternative\nformalisms have been proposed (see Refs. [2, 9–15] for a variety of perspectives).\nThe light front formalism has stood out in these debates as providing relativistically exact two-dimensional densities [16, 17]\nthat can be obtained from elementary field-theoretic definitions [3, 10] in a wave-packet-independent way [15]. Additionally, the\nlight front densities are related to generalized parton distributions [11, 16, 18], which simultaneously encode electromagnetic\nand gravitational form factors through their polynomiality property [19, 20], and which extend light front densities to three-\ndimensional distributions over the transverse plane and the light front momentum fraction x.\nDespite the success of the light front formalism, misgivings about light front densities persist. It is widely believed that light\nfront densities describe the apparent structure of a system moving at the speed of light, and thus contain kinematic distortions\ncaused by boosting to infinite momentum. Another criticism—given in Ref. [14] for instance—is that the “minus” components\nof four-vector and tensor densities do not have clear meanings, and indeed existing works on light front densities merely ignore\nthese components.\nBoth of these misgivings can be addressed with a minor change to the light front coordinates. By using light front time\nx+=t+zwith ordinary spatial coordinates (x, y, z )—an idea first explored in Ref. [21], where these are called tilted light front\ncoordinates —all existing results for light front densities are reproduced, but the previously-neglected components of vectors and\ntensors obtain clear physical meanings. Moreover, by using ordinary spatial coordinates, it can unambiguously be shown that the\npreviously-known and newly-found light front densities alike describe a system moving at any velocity, even a system at rest.\nThe use of tilted light front coordinates has a straightforward conceptual interpretation: using light front time x+instead of\ninstant form time tsimply means changing the rules for how spatially distant clocks are synchronized. In this work, we will show\nhow the entire formalism of light front densities follows from changing the time synchronization convention. Due to the lengthy\nexposition required for the coordinate system, this paper will focus entirely on electromagnetic densities of spin-half systems;\nthe energy-momentum tensor will be explored in a sequel work.\nThis work is organized as follows. Tilted coordinates are defined in Sec. II, which also contains a detailed exposition of their\nproperties: general properties are explored in Sec. II A; Lorentz transforms in Sec. II B; conserved currents and identification\nof the “charge” component in Sec. II C. The wave packet and spin density matrix are constructed in Sec. III. The formulas for\nintrinsic electromagnetic densities of spin-half systems are derived in Sec. IV. In Sec. V, these formulas are applied to empirical\nproton and neutron form factors to obtain charge and current densities, as well as up and down quark densities. We provide\nconcluding remarks and an outlook in Sec. VI. A uniqueness proof for light front synchronization is given in Appendix B.\n∗afreese@uw.edu\n†miller@uw.eduarXiv:2302.09171v2 [hep-ph] 25 Jul 20232\nt\nzAlice BobtAtB\nτ=tA+tB\n2Einstein synchronization\n˜τ\nzAlice BobtAτ=tB tBLight front synchronization\nFIG. 1. Pictorial representations of time synchronization conventions. In these pictures, Alice sends a light signal to Bob, who promptly\nsends a return signal. The time ∆t=tB−tAelapsed throughout the exchange from Alice’s perspective is a local observable, independent\nof synchronization convention. In the Einstein convention (left panel), the local time measured by Bob’s clock when the signal is reflected is\nhalfway between the start and end times. In light front synchronization (right panel), the clocks are synchronized under the convention that the\nreturn signal travels instantaneously, so Bob’s clock reads tBwhen the signal is reflected. The invariance of ∆trequires that the signal to Bob\ntravel at c/2in this convention.\nII. TILTED COORDINATES AND LIGHT FRONT SYNCHRONIZATION\nTilted light front coordinates (hereafter “tilted coordinates”) were first defined by Blunden, Burkardt and Miller (BBM) in\nRef. [21] 1:\n˜τ= ˜x0≡x+=t+z (1a)\n˜x1≡x (1b)\n˜x2≡y (1c)\n˜x3≡z . (1d)\nWe use tildes for quantities in tilted coordinates for clarity. The three spatial coordinates are the usual Cartesian coordinates of\nEuclidean space. Crucially, this means that a system at rest in tilted coordinates is really at rest as understood by common sense.\nThe tilted time variable ˜τis the same as light front time, and can be given an operational meaning. In fact, the familiar time tof\nMinkowski coordinates is also defined operationally, as first clarified by Einstein [22]. The local time τat which an event occurs\nis determined by its simultaneity with the reading of a clock in the local vicinity of the event. Einstein postulates by definition\n(“durch Definition ”—his emphasis) that distant clocks be synchronized by light signals under the convention that the time delay\nof the signal is equal in both directions. This definition has come to be called the Einstein synchronization convention, and using\nit gives us the Einstein (or instant form) time coordinate τ=x0=t. A pictorial representation of this convention is given in the\nleft panel of Fig. 1.\nThe use of Einstein’s convention is ultimately arbitrary—a point which has been discussed for a long time by both physicists\n(see Refs. [23, 24] and references therein) and philosophers [25, 26] 2. The use of an anisotropic synchronization convention\nnotoriously results in the same empirical predictions as Einstein’s convention, since mathematically this merely corresponds to\na coordinate transformation [24].\nLight front time (and thus tilted time) is defined using an anisotropic synchronization convention, which we call light\nfront synchronization. Observers aligned in the zdirection synchronize their clocks under the convention that light travels\ninstantaneously in the −zdirection. For instance, if Alice is located downstream of Bob ( zAlice< z Bob), she synchronizes her\nclock to read the time she currently sees on Bob’s clock. This is depicted in the right panel of Fig. 1. The local time elapsed\nduring the signal’s round trip is convention-independent, meaning the two-way speed of light is given by the invariant constant\nc[23, 25]. Under light front synchronization, this means light must travel at c/2in the +zdirection, as depicted in the right\npanel of Fig. 1. Clocks at the same zcoordinate but displaced in the transverse xyplane are synchronized using the Einstein\nconvention. Following these rules for clock synchronization gives us the light front time τ= ˜x0=x+.\n1BBM use ζ=−zinstead of ˜z=zas a coordinate, but this is a minor difference.\n2This feature of special relativity has also recently been popularized by the science education channel Veritasium [27].3\nAs noted above, the empirical predictions of a relativistic theory are independent of the synchronization convention. The\nuse of a particular convention is ultimately arbitrary, but different conventions may be more conducive to solving particular\nproblems. The motivation for using light front synchronization—and thus tilted coordinates—is that light front time is invariant\nunder a Galilean subgroup of the Poincar ´e group [16, 28]. Ultimately, this allows physical densities at fixed ˜τto be separated\ninto state-independent internal structures and state-dependent smearing functions in the manner explored in Ref. [15].\nA. Properties of tilted coordinates\nLet us consider some of the properties of tilted coordinates, especially since they differ from the more familiar light front\ncoordinates. Some of these properties were reported previously by BBM [21], but some of these relationships are new. Firstly,\nthe metric and inverse metric are given by:\n˜gµν=∂xα\n∂˜xµ∂xβ\n∂˜xνgαβ=\n1 0 0 −1\n0−1 0 0\n0 0 −1 0\n−1 0 0 0\n (2a)\n˜gµν=∂˜xµ\n∂xα∂˜xν\n∂xβgαβ=\n0 0 0 −1\n0−1 0 0\n0 0 −1 0\n−1 0 0 −1\n (2b)\nwhere gµν=gµν= diag(+1 ,−1,−1,−1)is the usual Minkowski metric 3. This means that the invariant proper time element\nis given by:\nds2= ˜gµνd˜xµd˜xν= d˜τ2−2 d˜τd˜z−d˜x2\n⊥, (3)\nwhere ˜x⊥= (˜x,˜y)are the transverse coordinates, while the d’ Alembertian is given by:\n∂2= ˜gµν˜∂µ˜∂ν=−2˜∂z˜∂τ−˜∇2, (4)\nwhere ˜∇= (˜∂x,˜∂y,˜∂z)is the three-dimensional gradient. The latter was found previously by BBM [21].\nCovariant components of a four-vector ˜Aµ(lower index) are related to contravariant components ˜Aµ(upper index) through\nthe metric:\n˜Aµ= ˜gµν˜Aν ˜Aµ= ˜gµν˜Aν, (5a)\nwhich in terms of individual components gives:\n˜A0=˜A0−˜A3 ˜A0=−˜A3 (5b)\n˜A1=−˜A1 ˜A1=−˜A1 (5c)\n˜A2=−˜A2 ˜A2=−˜A2 (5d)\n˜A3=−˜A0 ˜A3=−˜A0−˜A3. (5e)\nComponents of four-momentum are defined as generators of spacetime translations:\ni[ˆPµ,ˆO(x)] =∂µˆO(x), (6)\nand accordingly the energy and vector momentum are identified using the covariant vector ˜pµ:\n˜pµ≡(˜E;−˜px,−˜py,−˜pz). (7)\nNote that when we use x,yorzas a subscript, we are indicating a component of the three-vector ˜prather than components of\nthe four-vector pµ:\n˜p= (˜px,˜py,˜pz) = (−˜p1,−˜p2,−˜p3). (8)\n3Note that, while using tilted coordinates, we use numerical indices 0,1,2,3rather than +,1,2,3to label components of tensors.4\nWith this identification, the tilted energy and vector momentum become time and space translation generators as usual:\ni[˜E,ˆO(x)] =˜∂0ˆO(x) (9a)\n−i[˜p,ˆO(x)] = ˜∇ˆO(x). (9b)\nThe scalar product between four-momentum and a spacetime coordinate is:\n˜p·˜x= ˜pµ˜xµ=˜E˜τ−˜p·˜x, (10)\nwhich is a point of central importance to the density formalism. The mass-shell relation can be written:\nm2= ˜gµν˜pµ˜pν= 2˜pz˜E−˜p2(11a)\n˜E=m2+˜p2\n2˜pz, (11b)\nwhich was found previously by BBM [21].\nThe relationships between the tilted and Minkowski four-momenta can be found using:\n˜pµ=∂xν\n∂˜xµpν, (12)\nwhich entails:\n˜E=E (13a)\n˜px=px (13b)\n˜py=py (13c)\n˜pz=E+pz=p+. (13d)\nTwo remarkable things have happened here. Firstly, the tilted energy is equal to the instant form energy. This means that a\nmass decomposition in tilted coordinates will be the same as a mass decomposition in Minkowski coordinates (in contrast to the\nstandard light front energy p−[29]), and a spatial density of ˜Eis a density of the familiar instant form energy E. Secondly, the\nconjugate momentum to the zcoordinate is p+, which is the momentum variable used to define momentum fractions in parton\ndistribution functions and generalized parton distributions.\nLastly, we discuss particle velocities and the conditions for a particle being at rest. Classically, the velocity of a free particle\nis obtained from Hamilton’s equations:\n˜va=∂˜E\n∂˜pa, (14a)\n(where a=x, y, z ) which, upon using Eq. (11b) gives\n˜vx=˜px\n˜pz(14b)\n˜vy=˜py\n˜pz(14c)\n˜vz= 1−˜E\n˜pz(14d)\nfor the individual components of the velocity. All three of these can be summarized with the help of Eq. (5) as:\n˜vi=˜pi\n˜pz. (14e)\nNote the upper index in the components of the momentum four-vector here. This relationship can be made quantum mechanical\nsimply by promoting it to an operator relationship. The free particle is at rest when ˜v= 0, which occurs when:\n˜px= 0 (15a)\n˜py= 0 (15b)\n˜pz=m . (15c)\nThe relationship between longitudinal velocity and longitudinal momentum is strange and counter-intuitive, but follows from the\nelementary definition of momentum as the generator of spatial translations at a fixed time (which is fixed light front time here).\nRegardless, Eq. (15) allows us to clearly identify if a particle is at rest—and since the spatial coordinates ˜x=xare the usual\nCartesian coordinates, a system at rest with respect to these coordinates is unambiguously at rest as understood by common sense.5\nB. Lorentz transforms and the Galilean subgroup\nWe next explore the algebra of the Lorentz group and the effects of finite boosts in tilted coordinates. Since the goal of\nthis work is to obtain rest frame densities, special focus will be given to properties that are invariant under boosts, since these\nproperties remain unaltered when boosting the system to rest.\nThe standard generator basis for the Poincar ´e group in Minkowski coordinates consists of four translation generators Pµand\nsix Lorentz transform generators Jµν=−Jνµ, with the latter consisting of three rotation generators J= (J23, J31, J12)and\nthree boost generators K= (J10, J20, J30). We have already discussed the translation generators in tilted coordinates above.\nThe tilted rotation generators are the same as in instant form coordinates, ˜J=J, since the same spatial coordinates are used in\nboth systems. The boosts in tilted coordinates remain to be analyzed.\nUnder an infinitesimal passive transverse instant form boost, with velocities dβxanddβyin the xandydirections, the instant\nform coordinates transform as:\nt′=t−dβxx−dβyy (16a)\nx′=x−dβxt (16b)\ny′=y−dβyt (16c)\nz′=z . (16d)\nIn terms of tilted coordinates, this same infinitesimal transformation can be written:\n˜τ′= ˜τ−dβx˜x−dβy˜y (17a)\n˜x′= ˜x−dβx˜τ+ dβx˜z (17b)\n˜y′= ˜y−dβy˜τ+ dβy˜z (17c)\n˜z′= ˜z . (17d)\nRemarkably, this transformation is not merely a boost in tilted coordinates, but a combined boost and rotation. In fact, this is the\norigin of Terrell-Penrose rotations [30, 31]: what is merely boost according to a spatially extended reference frame with clocks\nsynchronized by the Einstein convention looks like a boost and a rotation, and in fact isa boost and a rotation under the light front\nsynchronization convention. Because of this, to define a mere boost in tilted coordinates, we must counteract the Terrell-Penrose\nrotation. The appropriate boost generators are:\n˜B1=K1−J2 (18a)\n˜B2=K2+J1. (18b)\nThe remaining boost generator remains unmodified:\n˜B3=K3. (18c)\nAll three of these boosts modify the ˜zcoordinate in unintuitive ways. Under a finite light front transverse boost by a velocity v\nalong the xdirection, the tilted coordinates transform as:\n˜τ′= ˜τ (19a)\n˜x′= ˜x−v˜τ (19b)\n˜y′= ˜y (19c)\n˜z′= ˜z+v˜x−v2\n2˜τ , (19d)\nwhile under a finite longitudinal boost of rapidity 4ηthey transform as:\n˜τ′= e−η˜τ (20a)\n˜x′= ˜x (20b)\n˜y′= ˜y (20c)\n˜z′= eη˜z−sinh(η)˜τ . (20d)\n4Recall that in instant form coordinates, the rapidity of a boost is given by η=1\n2log\u0010\nc+vIF\nc−vIF\u0011\n. In tilted coordinates, Eq. (21) instead gives the relationship\nbetween rapidity and longitudinal velocity.6\nAn object that is stationary in the original frame moves with a velocity\nvz=−eηsinh(η) =1\n2\u0000\n1−e2η\u0001\n(21)\nin the new frame, which is bounded to (−∞,1/2)—the bounds being the one-way speed of light in each direction along the z\naxis.\nThe longitudinal boost is worth understanding conceptually. Let ∆˜τbe a time interval measured by a clock in the original\nframe; the same time interval according to the transformed frame will be ∆˜τ′= e−η∆˜τ. Since the boost was a passive\ntransformation, the now-moving clock will measure the interval ∆˜τ= eη∆˜τ′while a stationary clock in the new frame measures\n∆˜τ′. Thus, when η >0, time measured by the moving clock is dilated, while it is instead quickened when η <0.\nNext, let ∆˜zbe the length between two comoving bodies as measured in the original frame; the length will be ∆˜z′= eη∆˜z\nin the transformed frame. If η >0, the length will appear dilated to the observer in the new frame, while when η <0the length\nappears contracted.\nThese transformation properties make sense when we recall the meaning behind light front synchronization discussed in\nSec. II. To an observer downstream of (with a lesser ˜zcoordinate than) the system under observation, η >0means the system\nis moving away from the observer (in the +zdirection), and η <0means the system is moving towards them. For η >0, the\ndilation of time and length of a moving system corresponds to a redshift while the time quickening and length contraction of a\nsystem moving with η <0corresponds to a blueshift. These are exactly the usual relativistic redshift and blueshift.\nThe Poincar ´e group has long been known to have a (2 + 1) -dimensional Galilean subgroup [16, 28, 32–34]. This subgroup is\ngenerated by the basis {˜E,˜Pi\n⊥,˜Pz,˜J3,˜Bi\n⊥}, where the ⊥subscript means ionly runs over 1and2. Under this group, the time ˜τ\nremains invariant. The algebra of this subgroup is:\n[˜J3,˜Bi\n⊥] = iϵij3˜Bj\n⊥(22a)\n[˜Bi\n⊥,˜Bj\n⊥] = 0 (22b)\n[˜J3,˜Pi\n⊥] = iϵij3˜Pj\n⊥(22c)\n[˜Bi\n⊥,˜Pj\n⊥] = iδij˜Pz (22d)\n[˜Bi\n⊥,˜E] = i˜Pi\n⊥, (22e)\nwith all other commutators zero. Notably, ˜Pzcommutates with all other generators in the group. It is accordingly considered a\ncentral charge of the group, and this observation is identical to the well-known fact that P+is a central charge of the Galilean\nsubgroup in light front coordinates [16, 28]. Physically, this means that the transformations in the Galilean subgroup (including\nall translations, transverse boosts, and rotations around the zaxis) will leave ˜Pzunaltered.\nThe Galilean subgroup has another central charge, which is related to the Pauli-Lubanski axial four-vector [35, 36]:\nWµ=−1\n2ϵµνρσJνρPσ, (23)\ndefined with the convention that ϵ0123= +1 . For its expression in tilted coordinates, one need only put tildes over each of the\ntensors involved. Just as with momentum, we define individual components of the associated three-vector using the covariant\ncomponents of the Pauli-Lubanski vector: ˜Wµ= (˜W0;−˜Wx,−˜Wy,−˜Wz). Through some algebra, one can show that:\n˜Wz=˜Pz˜J3−(˜B⊥טP⊥)·ˆz , (24)\nand additionally one can show that ˜Wzcommutes with all the generators of the Galilean subgroup, making it another central\ncharge of this group. Additionally,\nˆλ=˜P−1\nz˜Wz=˜J3−˜P−1\nz(˜B⊥טP⊥)·ˆz (25)\nis invariant under the Galilean subgroup, and is equal to the zcomponent of angular momentum for a system at transverse\nrest. The operator ˆλis the standard light front helicity operator [28], and characterizes the zcomponent of intrinsic spin in a\nGalilean-invariant manner. In particular—as pointed out in Ref. [28]—this operator characterizes with spin projection along the\nzaxis when the system is at rest.\nOne can show that, under longitudinal boosts:\n[˜B3,˜Pz] =i˜Pz (26a)\n[˜B3,˜Wz] =i˜Wz (26b)\n[˜B3,ˆλ] = 0 . (26c)7\nt\nzτ0 τΩnµ\nV\nFIG. 2. A finite spacetime region Ωbounded by two hypersurfaces of equal light front time τ0andτ. Each slice of fixed light front time\ncontains the same spatial region V. The future-directed normal nµto the equal-light-front-time hypersurfaces is also indicated in this diagram.\nThe light front helicity operator ˆλis invariant under longitudinal boosts. This makes light front helicity an especially useful label\nfor characterizing the spin states of hadrons: a rest frame state with definite λcan be transformed to an arbitrary momentum state\nwith the same λby performing a longitudinal boost followed by a transverse boost. We will therefore use light front helicity to\nlabel the spin content of hadrons throughout the remainder of this work.\nC. Conserved currents and densities at fixed light front time\nBefore proceeding to the construction of internal hadronic densities, it is important to understand how conserved currents\nshould be understood in terms of light front time, especially for correctly identifying which component should be identified as\nthe static density (e.g., as the charge density or as the energy density). A conserved current obeys the differential continuity\nequation:\n∂µjµ(x) =˜∂µ˜jµ(x) = 0 , (27)\nwhich is the same in Minkowski and tilted coordinates. We will first proceed using Minkowski coordinates. The continuity\nequation can be cast into integral form by integrating it over a spacetime region Ω:\nZ\nΩd4x ∂µjµ(x) = 0 . (28)\nThe specific region of spacetime considered consists of a spatial volume Vbounded by two equal-light-front-time hypersurfaces\natx+=τ0andx+=τ, and is depicted in Fig. 2. Using the divergence theorem, the integral of the continuity equation over Ω\ncan be rewritten as an integral over its surface. If we take nµto be the future-pointing normal vector to the equal-light-front-time\nsurfaces, then:\nZ\nVd3xnµjµ(x)\f\f\f\f\nx+=τ−Z\nVd3xnµjµ(x)\f\f\f\f\nx+=τ0+Zτ\nτ0dτ′Z\n∂VdS·j(x)\f\f\f\f\nx+=τ′= 0, (29)\nand then differentiating this with respect to τgives:\nd\ndτ\u0014Z\nVd3xnµjµ(x)\f\f\f\f\nx+=τ\u0015\n=−Z\n∂VdS·j(x)\f\f\f\f\nx+=τ. (30)\nThis form has a clear interpretation as the integral form of a continuity equation. The left-hand side quantifies how the amount\nof “charge” in a spatial region Vis changing with time, and the right-hand side quantifies how much current is flowing through\nthe boundary of this spatial region at a fixed light front time τ. The right-hand side becomes zero if Vis all of space and jµ\nis localized in space (as is expected of physically realistic currents), and in this case the equation is just a statement of charge\nconservation. Since this relation must hold for any spatial region V, the static charge density is given by\nρ(x) =nµjµ(x) =j+(x) (31)8\nwhen expressed in Minkowski coordinates, which is identical to the standard light front charge density. This is the logical result\nof using light front time synchronization, and not an arbitrary choice.\nThe same derivation can be done using tilted instead of Minkowski coordinates. Fixed-time surfaces are given by fixed ˜τ, and\nthe future-pointing normal to equal light front time surfaces gives\nρ(x) = ˜nµ˜jµ(x) =˜j0(x). (32)\nComparing to Eq. (1), one can see that ˜j0(x) =j+(x), so these coordinate systems give consistent results if the same equal-\nlight-front-time surface is used in both. The vector current is given by ˜j(x) =j(x)regardless of synchronization convention.\nIII. WA VE PACKETS AND PROBABILITY CURRENTS IN TILTED COORDINATES\nTo isolate purely internal properties of composite spin-half systems, we require a formulation for the wave function describing\nthe location of the system itself. We shall construct that formulation here, along with four-currents for the probability and spin\nprojections, which will appear in Sec. IV as smearing functions.\nWe assume that it is possible to prepare a state with a single particle of the species under consideration; we consider a proton\nfor concreteness. Single-proton states are characterized by the three independent components of their four-momentum, namely\n˜p, as well as their light front helicity λ. As noted by BBM [21], light front quantization can be used with tilted coordinates, since\nequal-time surfaces are defined the same way in both systems. The creation and annihilation operators for the proton field obey\nthe canonical anticommutation relations:\n{b(˜p, λ), b†(˜p′, λ′)}={d(˜p, λ), d†(˜p′, λ′)}= 2˜pz(2π)3δ(3)(˜p−˜p′)δλλ′, (33)\nand all other anticommutators are zero. Here, bandb†annihilate and create protons, while dandd†annihilate and create\nanti-protons. Single-proton states of definite momentum are given by:\n|˜p, λ⟩=b†(˜p, λ)|0⟩, (34)\nand the projection operator for identifying single-proton states is:\nΛ1p=X\nλZd3˜p\n2˜pz(2π)3|˜p, λ⟩⟨˜p, λ|. (35)\nLet|Ψ⟩signify a normalized single-proton state, meaning:\n⟨Ψ|Ψ⟩= 1 (36a)\n|Ψ⟩=Λ1p|Ψ⟩. (36b)\nWhen dealing with such a state, we can treat Λ1pas an identity operator and use Eq. (35) as a completeness relation. The\nmomentum-representation wave function can be defined as ⟨˜p, λ|Ψ⟩, which obeys the normalization rule:\nX\nλZd3˜p\n2˜pz(2π)3\f\f⟨˜p, λ|Ψ⟩\f\f2= 1. (37)\nThe normal mode expansion of the proton field operator is given by:\nˆψ(x) =X\nλZd3˜p\n2˜pz(2π)3n\nu(˜p, λ) e−i˜p·˜xb(˜p, λ) +v(˜p, λ) e+ i˜p·˜xd†(˜p, λ)o\n, (38)\nwhere the spinors uandvare normalized such that ¯u(˜p, λ′)u(˜p, λ) = 2 mδλλ′and¯v(˜p, λ′)v(˜p, λ) =−2mδλλ′5. Explicit\nexpressions for the spinors can be found in Appendix A. A fully covariant coordinate-representation wave function for a single-\nproton state can be defined through the field operator as:\nΨcov(x) =⟨0|ˆψ(x)|Ψ⟩, (39)\n5We remark for clarity that ¯u(˜p, λ) =u†(˜p, λ)γ0and¯v(˜p, λ) =v†(˜p, λ)γ0as usual, and that ˜γ0is not used to define the barred spinors. In the context of\ndefining barred spinors, γ0plays the role of swapping left-handed and right-handed components, rather than as the 0th component of a four-vector. Put another\nway,¯u(˜p′, λ′)u(˜p, λ)is a scalar, and therefore should not change with the transformation from Minkowski to tilted coordinates, meaning the definition of\n¯u(˜p, λ)should not change to use ˜γ0.9\nand is related to the momentum-representation wave function through a Fourier transform:\nΨcov(x) =X\nλZd3˜p\n2˜pz(2π)3u(˜p, λ)⟨˜p, λ|Ψ⟩e−i˜p·˜x. (40)\nThrough this relation, it can be shown that Ψ(˜x,˜τ)obeys the Dirac equation:\n(i/∂−m)Ψcov(x) =X\nλZd3˜p\n2˜pz(2π)3(/p−m)u(˜p, λ)⟨˜p, λ|Ψ⟩e−i˜p·˜x= 0, (41)\nwhere (/p−m)u(˜p, λ) = 0 was used.\nA. Helicity-component wave functions\nIt is helpful to consider helicity components of the proton wave function. These are defined:\nΨλ(˜x,˜τ) =¯u(i˜∇, λ)\n2mΨcov(x) =Zd3˜p\n2˜pz(2π)3⟨˜p, λ|Ψ⟩e−i˜p·˜x, (42)\nand are given by a Fourier transform of the momentum-representation wave function. From inverting Eq. (42) and placing it into\nEq. (37), the following normalization relation can be derived for the helicity components:\nX\nλZ\nd3˜xΨ∗\nλ(˜x,˜τ) i← →˜∂0Ψλ(˜x,˜τ) = 1 , (43)\nwhere ˜∂0≡˜g0µ˜∂µand where the two-sided derivative is defined as f← →∂µg=f(∂µg)−(∂µf)g. Notably, this appears to be the\n0th component of a four-current:\nPµ(˜x,˜τ) =X\nλΨ∗\nλ(˜x,˜τ) i← →˜∂µΨλ(˜x,˜τ), (44)\nand it can be confirmed through the help of Eq. (42) that:\n˜∂µPµ(˜x,˜τ) = 0 . (45)\nThusPµ(˜x,˜τ)is a conserved current. As discussed in Sec. II C, P0(˜x,˜τ)is thus the density of the conserved “charge”\nassociated with this current. Its normalization to unity confers it an interpretation as a probability density, and thus Pµ(˜x,˜τ)\ncan be interpreted as the probability four-current.\nB. Spin density matrices\nLastly, we define spin-projection densities and currents. In this formulation, we emphasize properties that are invariant under\nlight front boosts and use linear combinations of definite- λstates to define spin densities. In particular, we follow the conventional\ndefinitions in the light front literature [37]:\n|sz=±1/2⟩=|λ=±1/2⟩ (46a)\n|sx=±1/2⟩=|λ= +1 /2⟩ ± |λ=−1/2⟩√\n2(46b)\n|sy=±1/2⟩=|λ= +1 /2⟩ ±i|λ=−1/2⟩√\n2. (46c)\nIt should be remarked that for systems with non-zero velocity, these are not actually eigenstates of the appropriate components\nof the Pauli-Lubanski vector. However, for a system at rest, these do become true spin eigenstates. Moreover, these states are\ninvariant under light front boosts, and accordingly can be identified as states with definite spin in their own rest frame.10\nStates with definite spin along the aaxis (where a=x, y, z ) are eigenstates of the Pauli matrix σa, and accordingly a spin\ndensity can be defined:\nSa(˜x,˜τ) =X\nλλ′Ψ∗\nλ′(˜x,˜τ) i← →˜∂0(σa)λ′λΨλ(˜x,˜τ). (47)\nThis is essentially a density for the expectation value of σa. Just as with the probability density, this is the 0th component of a\nfour-current that is conserved.\nSince spin, defined in this manner, is boost-invariant, it is meaningful to consider it an intrinsic property of the system under\nconsideration. It is thus helpful to consider a spin density matrix:\nPµ\nλ′λ(˜x,˜τ) =Ψ∗\nλ′(˜x,˜τ) i← →˜∂µΨλ(˜x,˜τ), (48)\nthe trace of which gives the probability four-current of Eq. (44). Spin four-currents can similarly be obtained from multiplying\nby(σa)λ′λand summing over λ, λ′, with the 0th component of this operation reproducing the spin-projection density in Eq. (47).\nIV. DERIVATION OF ELECTROMAGNETIC DENSITIES\nIn this section, we derive the internal electromagnetic densities of the proton when light front time synchronization is used.\nWe closely follow the methodology of Ref. [15]: the physical density is identified as the expectation value of the electromagnetic\ncurrent operator ˆjµ(x)within a physical state, which is decomposed into a sum of wave-packet-independent internal densities\nconvolved with packet-dependent smearing functions.\nA. General formula for the physical electromagnetic current\nThe physical electromagnetic current is the expectation value of the electromagnetic current operator for a physical state:\n⟨˜jµ(˜x)⟩=⟨Ψ|ˆjµ(˜x)|Ψ⟩. (49)\nUsing ˆjµ(˜x) = eiˆP·˜xˆjµ(0) e−iˆP·˜xand the one-particle projection operator, we can write:\n⟨˜jµ(˜x)⟩=X\nλ,λ′Zd3˜p\n2˜pz(2π)3Zd3˜p′\n2˜p′z(2π)3Ψ∗(˜p′, λ′)⟨˜p′, λ′|ˆjµ(0)|˜p, λ⟩Ψ(˜p, λ) ei˜∆·˜x, (50)\nwhere ˜∆= ˜p′−˜p. Using the inversion of Eq. (42) here gives:\n⟨˜jµ(˜x)⟩=X\nλ,λ′Z\nd3˜RZ\nd3˜R′Zd3˜P\n(2π)3Zd3˜∆\n(2π)3Ψ∗\nλ′(˜R′,˜τ)⟨˜p′, λ′|ˆjµ(0)|˜p, λ⟩Ψλ(˜R,˜τ) e−i˜∆·(˜x−˜R+˜R′\n2)e+ i˜P·(˜R′−˜R),\n(51)\nwhere ˜P=1\n2\u0000\n˜p+ ˜p′\u0001\n. As observed in Refs. [13, 15], the integral over ˜Pcan be performed, producing a delta function that sets\n˜R′=˜R, if the substitutions\n˜P7→i\n2← →˜∂ (52)\nare implicitly made within the matrix element ⟨˜p′, λ′|ˆjµ(0)|˜p, λ⟩. The result is:\n⟨˜jµ(˜x)⟩=X\nλ,λ′Z\nd3˜RZd3˜∆\n(2π)3Ψ∗\nλ′(˜R,˜τ)⟨˜p′, λ′|ˆjµ(0)|˜p, λ⟩Ψλ(˜R,˜τ) e−i˜∆·(˜x−˜R). (53)\nThe matrix element appearing here can be decomposed into form factors as:\n⟨˜p′, λ′|ˆjµ(0)|˜p, λ⟩= ¯u(˜p′, λ′)(\n˜γµF1(∆2) +i˜σµν˜∆ν\n2mF2(∆2))\nu(˜p, λ), (54)11\nwhere we use the normalization convention that F2(0) = κgives the anomalous magnetic moment. The Lorentz-invariant\nmomentum transfer in tilted coordinates is:\n∆2= 2˜∆z˜∆0−˜∆2= 2˜∆z\u0000\n˜p′\n0−˜p0\u0001\n−˜∆2= 2˜∆z\nm2+\u0010\n˜P+1\n2˜∆\u00112\n2˜Pz+˜∆z−m2+\u0010\n˜P−1\n2˜∆\u00112\n2˜Pz−˜∆z\n−˜∆2, (55)\nwhich clearly depends on ˜P. This ˜Pdependence causes the argument of the form factors to depend on the wave packet, which\nprovides an obstacle to separating internal structure from wave packet dependence. As an alternative to Eq. (55), one can\ntransform ∆2into a d’ Alembertian acting on the operator as a whole (as is explored in Appendix B), but the presence of a time\nderivative in the d’ Alembertian still impedes eliminating wave packet dependence from the form factor. (See Appendix B for\ndetails on this approach.)\nAs previously observed in Ref. [15], this undesired ˜Pdependence can be eliminated by integrating out ˜zin the physical\ndensity. Doing so turns e−i˜∆z˜zinto a Dirac delta function that sets ˜∆z= 0. We are left with a two-dimensional density:\n⟨˜jµ(˜x⊥,˜τ)⟩2D≡Z\nd˜z⟨˜jµ(˜x)⟩=X\nλ,λ′Z\nd3˜RZd2˜∆⊥\n(2π)2Ψ∗\nλ′(˜R,˜τ)⟨˜p′, λ′|ˆjµ(0)|˜p, λ⟩Ψλ(˜R,˜τ) e−i˜∆⊥·(˜x−˜R), (56)\nwhere the substitution of Eq. (52) is still implicitly being applied. With ˜∆z= 0, the Lorentz-invariant momentum transfer is\njust:\n∆2=−˜∆2\n⊥. (57)\nAt this point, we can use the explicit spinor elements found in Appendix A to write:\n⟨˜p′, λ′|ˆjµ(0)|˜p, λ⟩= 2˜Pµ \n(σ0)λ′λF1(−˜∆2\n⊥)−iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ\n2mF2(−˜∆2\n⊥)!\n−iϵµνρσ˜nν˜Pρ˜∆σ\n(˜P·˜n)(σ3)λ′λGM(−˜∆2\n⊥) +m˜nµ\n(˜P·˜n)iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λGM(−˜∆2\n⊥),(58)\nwhere GM(−˜∆2\n⊥) =F1(−˜∆2\n⊥)+F2(−˜∆2\n⊥)is the Sachs magnetic form factor, and where ˜nµ= (1; 0 ,0,0)and˜¯nµ= (1; 0 ,0,0)\nare four-vectors that project out the time component of ˜xµand the energy component of ˜pµrespectively—and in this sense are\nanalogous to the nand¯nvectors used in standard light front coordinates. Noting the substitution rule of Eq. (52), wave packet\ndependence is indicated by ˜Pdependence. The wave packet dependence in the first line of Eq. (58) can be identified as the spin\ndensity matrix defined in Eq. (48). For the second line, additional smearing functions must be defined:\nJν,λ′λ(˜R,˜τ) = 2mΨ∗\nλ′(˜R,˜τ)−˜Pν\n(˜P·˜n)Ψλ(˜R,˜τ) (59a)\nKλ′λ(˜R,˜τ) = 2mΨ∗\nλ′(˜R,˜τ)m\n(˜P·˜n)Ψλ(˜R,˜τ), (59b)\nwhere factors of 2mwere introduced so that these functions have the same units as the spin density matrix. If we additionally\ndefine the three following internal densities:\nρλ′λ(˜b⊥) =Zd2˜∆⊥\n(2π)2 \n(σ0)λ′λF1(−˜∆2\n⊥)−iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ\n2mF2(−˜∆2\n⊥)!\ne−i˜∆⊥·˜b⊥(60a)\njµν\n⊥,λ′λ(˜b⊥) =−Zd2˜∆⊥\n(2π)2iϵµνρσ˜nρ˜∆σ\n2m(σ3)λ′λGM(−˜∆2\n⊥) e−i˜∆⊥·˜b⊥(60b)\njµ\n∥,λ′λ(˜b⊥) =Zd2˜∆⊥\n(2π)2iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ\n2m˜nµGM(−˜∆2\n⊥) e−i˜∆⊥·˜b⊥, (60c)\nthen the physical density can be written as:\n⟨˜jµ(˜x⊥,˜τ)⟩2D=X\nλ,λ′Z\nd3˜R(\nPµ\nλ′λ(˜R,˜τ)ρλ′λ(˜x⊥−˜R⊥)\n+Jν,λ′λ(˜R,˜τ)jµν\n⊥,λ′λ(˜x⊥−˜R⊥) +Kλ′λ(˜R,˜τ)jµ\n∥,λ′λ(˜x⊥−˜R⊥))\n.(61)\nWith the general expression now in hand, we must now investigate the meaning of the internal densities in Eq. (60).12\nB. Interpretation of internal densities\nLet us now consider the interpretation of the internal densities in Eq. (60), with a special focus on the spatial structure of the\nproton in its rest frame. Although there is no rest state of the proton in Hilbert space—since wave functions must necessarily\nbe square integrable—we will show that the internal densities are boost-invariant quantities, and that artifacts of the proton’s\nmotion have all been absorbed into the smearing functions in Eq. (61). Moreover, these interpretations can be further validated\nby utilizing the rest condition P= (0,0, m)found in Eq. (15).\nWe shall consider states with definite light front spin1\n2ˆs—as described in Sec. III B—since these correspond to a proton with\ndefinite spin in its own rest frame. For these states, one can substitute:\n(σ0)λ′λ7→1 (62a)\n(σi)λ′λ7→ˆsi (62b)\nin the formulas of Eqs. (48), (59) and (60) to obtain spin-state densities, which will now be functions of ˆsinstead of λandλ′.\n1. Internal charge density\nWe consider the charge density first. As discussed in Sec. II C, the charge density can be identified by contracting the\nelectromagnetic current with ˜nµ, giving the 0th component. Since j0ν\n⊥(˜b⊥,ˆs) = 0 andj0\n∥(˜b⊥,ˆs) = 0 are both zero—a result\nthat is frame-independent, since boosts do not mix spatial components into light front time (see Sec. II B)—the internal charge\ndensity is given entirely by:\nρ(˜b⊥,ˆs) =Zd2˜∆⊥\n(2π)2 \nF1(−˜∆2\n⊥) +(ˆs×i˜∆⊥)·ˆz\n2mF2(−˜∆2\n⊥)!\ne−i˜∆⊥·˜b⊥, (63)\nwhere Eq. (A10) was used to rewrite the terms multiplying F2(−˜∆2\n⊥). Notably, this reproduces the results of Refs. [16, 38, 39] for\nlongitudinally-polarized nucleons, and (up to a sign discrepancy) the results of Carlson and Vanderhaegen [37] for transversely-\npolarized nucleons 6. This internal charge density is smeared over space by the probability current, as expected and as previously\nobserved for spin zero targets [15]. This probability current transforms as a Lorentz four-vector, and effectively absorbs all of\nthe boost dependence of the charge density contribution to the physical electromagnetic four-current. Since the spin degrees\nof freedom are characterized by light front helicity, and since ˜b⊥=˜x⊥−˜R⊥is given by a difference between transverse\nspatial coordinates, the internal density ρ(˜b⊥,ˆs)is fully invariant under both transverse and longitudinal boosts, and therefore\ncharacterizes the internal charge density of the proton in every reference frame—including its rest frame.\n2. Transverse current density\nWe next consider the density jµν\n⊥(˜b⊥,ˆs). Like the internal charge density, it is boost-invariant, with boost dependence having\nbeen absorbed into the smearing function Jν(˜R,˜τ,ˆs). The presence of ˜nρin its definition ensures that the µ= 0 andν= 0\ncomponents are all zero. The index µsignifies which component of the physical four-current this density contributes to, and ν\nsignifies which component of the total four-momentum this density is to be contracted with. With the help of Eq. (A10), we can\nwrite out the non-zero components as:\nj23\n⊥(˜b⊥,ˆs) =−j32\n⊥(˜b⊥,ˆs) = + ˆs·ˆzZd2˜∆⊥\n(2π)2i˜∆x\n2mGM(−˜∆2\n⊥) e−i˜∆⊥·˜b⊥(64a)\nj13\n⊥(˜b⊥,ˆs) =−j31\n⊥(˜b⊥,ˆs) =−ˆs·ˆzZd2˜∆⊥\n(2π)2i˜∆y\n2mGM(−˜∆2\n��) e−i˜∆⊥·˜b⊥. (64b)\nThese entail, first of all, of a transverse current density:\nj⊥(˜b⊥,ˆs) = ( ˆs·ˆz)Zd2˜∆⊥\n(2π)2(ˆz×i˜∆⊥)\n2mGM(−˜∆2\n⊥) e−i˜∆⊥·˜b⊥, (65)\n6The sign discrepancy is due to a mistake in Eq. (1) of Ref. [37], where the complex exponential should be e−iq⊥·b⊥instead of eiq⊥·b⊥. The sign mistake\noriginated in Ref. [38], where only helicity states were considered so that this sign flip had no effect on its results.13\nwhich, considering Eq. (59), is smeared over the transverse plane by the square of the wave function. This smearing function\nis boost-invariant, meaning the contribution of j⊥(˜b⊥,ˆs)to the physical transverse current density is frame-independent. This\ndensity can therefore be identified as the intrinsic transverse current density of the proton, applicable to all frames including the\nrest frame of the proton.\nIn addition, the µ= 3components of jµν\n⊥(˜b⊥,ˆs)entail a longitudinal current density:\nj3\nind.,i(˜b⊥,ˆs) = ( ˆs·ˆz)Zd2˜∆⊥\n(2π)2(i˜∆⊥׈z)i\n2mGM(−˜∆2\n⊥) e−i˜∆⊥·˜b⊥, (66)\nwhich—considering Eq. (59)—must be contracted with the expectation value of the transverse velocity (˜P⊥/˜Pz)ito give a\ncontribution to the physical longitudinal current. The smearing function associated with this contribution is not boost invariant,\nand in fact vanishes in a frame where the expectation value of the transverse velocity is zero. In fact, an intrinsic transverse\ncurrent can induce a longitudinal current under transverse boosts, since transverse boosts mix transverse spatial coordinates into\nthezcoordinate—see Eq. (19). Since j3\nind.,i(˜b⊥,ˆs)vanishes in a frame where the expected value of the transverse velocity is\nzero, we call it a boost-induced longitudinal current, in contrast to an intrinsic longitudinal current. We shall address the intrinsic\nlongitudinal current density next.\n3. Longitudinal current density\nThe last intrinsic density defined in Eq. (60) is jµ\n∥. Considering ˜nµ= (0; 0 ,0,−1), the only component is µ= 3, which for\ndefinite-spin states can be written (with the help of Eq. (A10)):\nj3\n∥(˜b⊥,ˆs) =Zd2˜∆⊥\n(2π)2(ˆs×i˜∆⊥)·ˆz\n2mGM(−˜∆2\n⊥) e−i˜∆⊥·˜b⊥. (67)\nThis is smeared over the transverse plane by the function H(˜R,˜τ,ˆs)defined in Eq. (59), which is invariant under the Galilean\nsubgroup (including transverse boosts) but not under longitudinal boosts. Under an active boost of rapidity η, the smearing\nfunction scales as e−η, meaning the longitudinal current is redshifted (slowed down) for η >0and blueshifted (sped up) for\nη <0. We remind the reader than η >0increases ˜Pzand has the system move away from the observer, while η <0decreases ˜Pz\nand has the system move towards the observer, assuming it is located upstream from (at a larger ˜zcoordinate than) the observer.\nAlthough the intrinsic transverse current and intrinsic longitudinal current are separated in our formulation—the separation\nbeing necessary by virtue of the different transformation properties of their smearing functions under boosts—we may choose to\nformally combine Eqs. (65) and (67) into a single expression for the intrinsic electric current:\n˜j(˜b⊥,ˆs) =Zd2˜∆⊥\n(2π)2ˆs×i˜∆⊥\n2mGM(−˜∆2\n⊥) e−i˜∆⊥·˜b⊥. (68)\nV. EMPIRICAL REST FRAME CHARGE AND CURRENT DENSITIES\nUsing the results from Sec. IV, we proceed to present numerical results for the internal rest frame densities of the proton and\nthe neutron. For F1(∆2)andF2(∆2), we use the parametrizations fit to empirical data by Kelly [40], combined with improved\nmeasurements of the neutron electric form factor by Riordan et al. [41].\nWe first present results for the rest frame charge densities in Fig. 3, which can all be calculated using Eq. (63). The neutron\ncharge density for helicity states reproduces the finding of Miller [38], in which the neutron has a negatively-charged core\nsurrounded by a diffuse cloud of positive charge. The charge densities for transversely-polarized states reproduce the previous\nresults of Carlson and Vanderhaegen [37] up to the previously noted sign discrepancy in the sinϕmodulations.\nIt is worth emphasizing that we have reproduced and affirmed previously-known results for the transverse charge density in\nthe standard light front formalism. These previous results were found to be entirely frame-independent, with the spin degree of\nfreedom specifically quantifying the nucleon’s spin in its own rest frame. The limit of infinite momentum was never considered\nor taken. It is mistaken to attribute features of these densities to kinematic effects from boosting to infinite momentum. The\ncause of angular modulations in the densities of transversely-polarized states must lie elsewhere, and in fact originates from the\ntime synchronization convention. We shall comment on this further in Sec. V B.\nFor now, we further remark that the angular modulations in the charge densities of transversely-polarized states produce an\neffective electric dipole moment, which can be calculated from Eq. (63) as follows:\npeff≡Z\nd2˜b⊥˜b⊥ρrest(˜b⊥,ˆs) =ˆz׈s\n2mF2(0) = (ˆ z׈s)κ\n2m, (69)14\nFIG. 3. Rest frame charge densities of the proton (top row) and neutron (bottom row). The left column depicts positive-helicity states (spin-up\nalong the zaxis), while the right column depicts transversely-polarized states with spin up along the xaxis.\nwhere κis the anomalous magnetic moment of the nucleon. Since it is an effect of using light front synchronization, we call this\na synchronization-induced electric dipole moment. From the empirical values of the masses and anomalous magnetic moments,\nthe induced dipole moments of the proton and neutron are:\npp= 0.19fm (70a)\npn=−0.20fm. (70b)\nWe next present in Fig. 4 numerical results for the internal electric current distributions of the proton and neutron. It should\nbe noted that we use a right-handed coordinate system in these plots in which the xaxis is oriented vertically and the yaxis is\noriented horizontally, so that the zaxis points into the page. This choice is made so that the distributions reflect what an observer\nlocated downstream from the target (i.e., at a lesser zcoordinate) actually sees. The clockwise revolution of the electric current\nfor a proton with spin up along the zaxis (top left panel) thus entails a positive magnetic moment, while the counterclockwise\nrevolution of a neutron in the same state (bottom left panel) entails a negative magnetic moment. The helicity-state results in the\nleft column reproduce the findings of Chen and Lorc ´e [14].\nFor a transversely polarized nucleon, the only non-zero component of its internal current density—as defined in Eq. (68)—is\n˜j3(˜b⊥). This is depicted in the right column of Fig. 4. That the zaxis points into the page is worth emphasizing when interpreting\nthese plots. For a proton that is spin-up along the xaxis, the current is moving towards the observer at by<0and away from\nthe observer at by>0. Since the observer looks in the +zdirection, a current towards the observer is negative. Using the15\nFIG. 4. Rest frame electric current distributions in the proton (top row) and neutron (bottom row). The left column depicts positive-helicity\nstates (spin-up along the zaxis), while the right column depicts transversely-polarized states with spin up along the xaxis. These figures use a\nright-handed coordinate system with a vertical xaxis and horizontal yaxis, so the zaxis points into the page.\nright-hand rule, the transversely-polarized proton (top-right panel) has an electric current that co-revolves with its spin, while\nthe transversely-polarized neutron (bottom-right panel) has an electric current that counter-rotates against its spin. These are\nconsistent with the picture suggested by the helicity-state nucleons.\nIt is worth remarking that the ˜j3(˜b⊥)results for transversely-polarized states are new, and could be obtained and meaningfully\ninterpreted by virtue of using tilted coordinates instead of standard light front coordinates.\nA. Quark densities and currents\nInformation about the densities and currents of up quarks and down quarks in the proton can be found by invoking charge\nsymmetry. The uandddensities in the proton are assumed equal to the dandudensities in the neutron. Thus, for any density\nρ(˜b⊥),\nρp(˜b⊥) =euρu(˜b⊥) +edρd(˜b⊥) =2\n3ρu(˜b⊥)−1\n3ρd(˜b⊥) (71a)\nρn(˜b⊥) =euρd(˜b⊥) +edρu(˜b⊥) =2\n3ρd(˜b⊥)−1\n3ρu(˜b⊥), (71b)16\nFIG. 5. Rest frame quark densities in a transversely-polarized proton. Rest frame electric densities of the proton (top row) and neutron (bottom\nrow). The proton has spin-up along the xaxis. The left panel is the up quark density and the right panel is the down quark density.\nnund⟨(˜b2\n⊥)u⟩ ⟨ (˜b2\n⊥)d⟩ κu κd\n2 1\u0000\n0.65fm\u00012\u0000\n0.68fm\u000121.68−2.03\nTABLE I. Static quantities associated with up and down quarks in the proton.\nwhich invert to:\nρu(˜b⊥) = 2ρp(˜b⊥) +ρn(˜b⊥) (72a)\nρd(˜b⊥) = 2ρn(˜b⊥) +ρp(˜b⊥). (72b)\nThe up and down quark densities in a helicity-state proton were found previously by Miller [38]. We are now equipped to\nadditionally provide quark densities in transversely-polarized protons, as well as the quark current densities.\nUp and down quark densities in a transversely-polarized proton at rest are presented in Fig. 5. These densities are not weighted\nby charge, so they are positive-definite number densities, normalized to nu= 2 andnd= 1 respectively. Both densities are\noff-center, but the down quark density is much further displaced from the center. These displacements are in opposite directions,\nand since the up and down quarks have opposite charge, they make constructive contributions to the synchronization-induced\nelectric dipole moment.\nUp and down quark current distributions in a proton are presented in Fig. 6. These are not weighted by charge, so the direction\nof the current is the genuine direction of motion of the respective quark flavor. For both the longitudinally and transversely-\npolarized state, the up quark current is in the direction of proton spin, while the down quark current is in the opposite direction.\nThis seems to conflict with model calculations [42, 43] and lattice QCD results [44] finding that up quarks carry negative orbital\nangular momentum (OAM) and down quarks carry positive OAM, at least at renormalization scales of a few GeV2or higher.\nHowever, the OAM contains sea quark contributions—hence its renormalization scale dependence—while the quark currents\ndescribe only valence quark contributions. This conflict is resolved if the majority of OAM is carried by sea quarks.\nWe remark, in comparing Fig. 5 to Fig. 6, that each quark density appears to be enhanced where the quarks are moving away\nfrom the observer (on the right side of the transversely-polarized plots of up quark distributions, or the left side for down quark\ndistributions), and conversely suppressed where the quarks are moving towards the observer. This is a natural consequence of\nusing light front time synchronization, and will be explored further in Sec. V B.\nIn Table I, we present individual quark flavors’ contributions to the static quantities associated with the proton’s charge and\ncurrent distributions. The quark number, quark radius, and quark contribution to the anomalous magnetic moment (without17\nFIG. 6. Rest frame quark current distributions in the proton. Up distributions (top row) and down distributions (bottom row) are shown. The\nleft column depicts positive-helicity states (spin-up along the zaxis), while the right column depicts transversely-polarized states with spin up\nalong the xaxis. These figures use a right-handed coordinate system with a vertical xaxis and horizontal yaxis, so the zaxis points into the\npage.\ncharge weight) are respectively:\nnq=F1q(0) (73a)\n⟨(˜b2\n⊥)q⟩=4\nF1q(0)dF1q(t)\ndt\f\f\f\f\nt=0(73b)\nκq=F2q(0). (73c)\nB. Discussion and further interpretation\nWe now further discuss the interpretation of the internal densities we have obtained, with a special focus on the angular\nmodulations in the charge densities of transversely-polarized nucleons (right column of Fig. 3). Since these densities are frame-\ninvariant and therefore describe a proton even at rest, they cannot be attributed to kinematic effects from boosting to infinite\nmomentum.\nModulations such as these can appear for spinning targets at rest by virtue of using light front synchronization. What a\ndownstream observer sees at any fixed light front time ˜τis everything that a single light front encounters while traveling in the18\n0 10 20 30 40 50\nt(ys)−1.0−0.50.00.51.0Coordinate (fm)Toy model: Einstein time\nz(t)\ny(t)\n0 10 20 30 40 50\n˜τ(ys)−1.0−0.50.00.51.0Coordinate (fm)Toy model: light front time\nz(˜τ)\ny(˜τ)\nFIG. 7. yandzcoordinates of the toy revolver model as functions of Einstein time (left panel) and light front time (right panel), for R= 1fm\nandω= 250 ZHh.\n−zdirection. For an extended rotating body, the light front takes a non-zero amount of Einstein time tto cross the extent of the\nbody, and particles within it will advance in their motion while the light front is crossing it. Particles are more likely to encounter\nthe light front while they’re moving against it (in the +zdirection), so the particle density will be skewed in the ˆz׈sdirection.\nThis can explain the direction of the modulations in of Figs. 3 and 5, as well as the sign of the synchronization-induced electric\ndipole moment in Eq. (69).\nThis same phenomenon can also be rephrased in the language of light front time ˜τ. Suppose that a particle revolves around\na transversely-polarized body, taking the same amount of time from its own perspective to reach the lowest- zpoint of its orbit\nand the highest- zpoint of its orbit. Since its time is dilated when moving in the +zdirection and quickened when moving in the\n−zdirection from the perspective of a stationary observer (see discussion in Sec. II B), it will spend less time moving towards\nthe observer and more time moving away from the observer from the observer’s perspective. This makes it more likely that the\nobserver will see the particle when it’s in the ˆz׈sdirection, again skewing the particle density in that direction.\nThis can be explicitly illustrated in a simple classical toy model. Suppose we have a charged body revolving around a central\npoint with angular velocity ω. The revolution is around the x-axis, and the trajectory in terms of Einstein time tis:\nr(t) =R\u0010\nˆycos(ωt) + ˆzsin(ωt)\u0011\n. (74)\nAt any Einstein time, the zcoordinate is:\nz(t) = ˆz·r(t) =Rsin(ωt). (75)\nThe light front time ˜τand Einstein time are related by:\n˜τ=t+z(t)\nc≡t+˜z(˜τ)\nc, (76)\nso that the zcoordinate as a function of the light front time ˜τis given by the implicit formula:\n˜z(˜τ) =Rsin\u0000\nω[˜τ−˜z(˜τ)]\u0001\n. (77)\nThe function ˜z(˜τ)can be solved for numerically. Additionally, from this one can find the ycoordinate as a function of light front\ntime from this:\n˜y(˜τ) =y(t) =Rcos\u0000\nω[˜τ−˜z(˜τ)]\u0001\n. (78)\nNumerical results for the revolver coordinates as a function of light front time ˜τare presented in the right panel of Fig. 7, with\nthe Einstein time trajectory given in the left panel for contrast. In terms of Einstein time t, the coordinates are simple sinusoids,\nas expected. In terms of light front time, however, the trajectory appears highly distorted. The revolver spends most of its time\nmoving away from the observer with a low velocity, and then quickly swings back towards the observer at a high velocity in19\na short span of light front time. Because of this, the revolver is at a positive ycoordinate for longer than it is at a negative y\ncoordinate, and the light-front-time-averaged ycoordinate is non-zero. In fact, this average can be calculated analytically:\n⟨˜y(˜τ)⟩LF≡lim\nT→∞1\n2˜τ(T)Z+˜τ(T)\n−˜τ(T)d˜τ′˜y(˜τ′) = lim\nT→∞1\n2T+ 2R\ncsin(ωT)Z+T\n−Tdtd˜τ(t)\ndty(t) =R2ω\n2c. (79)\nWe can use this model to give an extremely rough estimate of how quickly the valence up and down quarks revolve around\nthe proton. These estimates will be rough because quarks do not orbit the proton in well-defined circular trajectories, but they\nshould give an idea of the order of magnitude of the angular frequency of quark revolutions. If nq⟨˜y(˜τ)⟩LFis interpreted as the\nsynchronization-induced dipole moment of the quark distribution, then using this with Eq. (69) gives:\nωq=cκq\nmpnqR2q. (80)\nFor the sake of these estimates, the mean-squared radii given in Table I can be used in place of R2\nq. Doing this gives the following\nrough estimates:\nωu≈0.417c/fm= 125 ZHz (81a)\nωd≈ −0.922c/fm=−276ZHz, (81b)\nwhere the minus sign for the down quark’s angular frequency indicates revolution in the opposite direction. These exact numbers\ndepended on a simplistic toy model and should be taken only to give a ballpark estimate of the angular velocity of up and down\nquarks’ orbits around the proton.\nVI. CONCLUSIONS AND OUTLOOK\nIn this work, we used tilted light front coordinates—light front time x+and ordinary spatial coordinates (x, y, z )—to\ndemonstrate that the formalism for light front densities follows from use of a peculiar time synchronization convention, rather\nthan from boosting to infinite momentum. Internal densities of electric charge and currents, as well as up and down quark\ndensities and currents, were obtained for a proton. These densities are boost-invariant, and thus describe a proton in any state\nof motion, including at rest. The densities and transverse currents we obtained reproduced prior results, while the longitudinal\ncurrent densities were newly-obtained by virtue of using tilted coordinates.\nThis formalism is equally applicable to the energy-momentum tensor (EMT), allowing all sixteen of its components to be\ngiven a clear physical meaning. This will become the subject of a sequel paper. Care will need to be taken about upper and lower\nindices; since the energy and momentum are identified as covariant components of ˜pµ(as discussed in Sec. II A), and since the\nEMT is conserved with respect to its first index, the tilted energy density for instance will be given by:\n˜T0\n0=˜T00−˜T03=T00+T30, (82)\nwhich involves off-diagonal components of the EMT. For spin-half targets in particular, the energy density may differ depending\non whether the EMT contains the antisymmetric piece discussed in Refs. [2, 45]. Tilted light front coordinates will be helpful\nnot only in defining relativistically exact densities for all components of the EMT—including an energy density that integrates to\nthe usual instant form energy E—but may also provide additional insight into the question of whether the physical EMT contains\nan antisymmetric piece.\nACKNOWLEDGMENTS\nThe authors would like to thank Ian Clo ¨et, Wim Cosyn, and Yang Li for helpful discussions that contributed to this work.\nThis work was supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number\nDE-FG02-97ER-41014.\nAppendix A: Explicit light front helicity spinors\nTo evaluate internal proton densities, we will need to evaluate explicit spinor matrix elements. To this end, it is helpful to\nhave explicit expressions for light front helicity spinors. Explicit expressions for these spinors were found already by Kogut and\nSoper [46]—the Kogut-Soper spinors are in fact eigenstates of the light front helicity operator ˆλdefined in Eq. (25).20\nIn terms of tilted momentum components, the Kogut-Soper spinors can be written:\nu\u0012\n˜p,+1\n2\u0013\n=1√˜pz\n˜pz\n˜px+ i˜py\nm\n0\n, u\u0012\n˜p,−1\n2\u0013\n=1√˜pz\n0\nm\n−˜px+ i˜py\n˜pz\n. (A1)\nIn this basis, the tilted gamma matrices can be written in block matrix form as:\n˜γ0=\u0014\n0 1−σ3\n1+σ3 0\u0015\n, ˜γi=\u0014\n0−σi\nσi0\u0015\n, (A2)\nwhere 1is a2×2identity matrix and σiare the usual 2×2Pauli matrices.\nThe barred spinors are defined as usual by:\n¯u(˜p, λ) =u†(˜p, λ)γ0(A3a)\n¯v(˜p, λ) =v†(˜p, λ)γ0(A3b)\nwhere the untilted γ0matrix is given by:\nγ0=\u0014\n0 1\n10\u0015\n. (A4)\nThe reason that γ0rather than ˜γ0is used is that γ0plays the role here of swapping left-handed and right-handed components\nof the spinor, rather than of isolating a component of a four-vector. Indeed, since ¯u(˜p′, λ′)u(˜p, λ) =u†(˜p′, λ′)γ0u(˜p, λ)is a\nscalar, it should be invariant under the transformation from Minkowski to tilted coordinates.\nIt is helpful to have explicit matrix elements of spinors, in particular when momentum in the zdirection remains unchanged.\nLet˜p=˜P−1\n2˜∆and˜p′=˜P+1\n2˜∆, and ˜∆z= 0. For equal-helicity spinors, we find the following matrix elements:\n¯u(˜p′, λ)u(˜p, λ) = 2m (A5a)\n¯u(˜p′, λ)˜γ0u(˜p, λ) = 2 ˜Pz (A5b)\n¯u(˜p′, λ)˜γi\n⊥u(˜p, λ) = 2 ˜Pi\n⊥−2λiϵij3˜∆j(A5c)\n¯u(˜p′, λ)˜γ3u(˜p, λ) = 2 ˜Pz \n1−˜E+˜E′\n2˜Pz!\n+2λi(˜P⊥ט∆⊥)·ˆz\n˜Pz, (A5d)\nwhere ˜Eand˜E′are the mass-shell energies of ˜pand˜p′, respectively. The gamma matrix elements in particular can be summarized\nin the following manifestly-covariant form:\n¯u(˜p′, λ)˜γµu(˜p, λ) = 2 ˜Pµ−2λiϵµνρσ˜nν˜Pρ˜∆σ\n(˜P·˜n), (A6)\nwhere ˜nµ= (1; 0 ,0,0)projects out the “time” component of contravariant four-vectors (e.g., ˜nµ˜xµ= ˜x0= ˜τ).\nLet us consider the helicity-flip case next. For λ′̸=λ, let∆λ=λ′−λ. We find:\n¯u(˜p′, λ′)u(˜p, λ) =\u0000\n∆λ˜∆x−i˜∆y\u0001\n(A7a)\n¯u(˜p′, λ′)˜γ0u(˜p, λ) = 0 (A7b)\n¯u(˜p′, λ′)˜γi\n⊥u(˜p, λ) = 0 (A7c)\n¯u(˜p′, λ′)˜γ3u(˜p, λ) =−m\n˜Pz\u0000\n∆λ˜∆x−i˜∆y\u0001\n. (A7d)\nWriting these in a manifestly-covariant manner is trickier, but can be done by utilizing the four-vector ˜¯nµ= (1; 0 ,0,0)which\nprojects out the “time” component of covariant four-vectors (e.g., ˜¯nµ˜pµ= ˜p0=˜E). Using Eq. (5), we note that ˜nµ= (0; 0 ,0,−1)\nand˜¯nµ= (1; ,0,0,−1). With this in mind, we can write:\n¯u(˜p′, λ′)u(˜p, λ) = iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ (A8a)\n¯u(˜p′, λ′)˜γµu(˜p, λ) =m˜nµ\n(˜P·˜n)iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ (A8b)21\nforλ′̸=λ, where σδ= ( 1;σ1, σ2, σ3)is a covariant four-vector of Pauli matrices.\nThe definite-helicity and helicity-flip cases can now be combined:\n¯u(˜p′, λ′)u(˜p, λ) = 2m(σ0)λ′λ+ iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ (A9a)\n¯u(˜p′, λ′)˜γµu(˜p, λ) = 2 ˜Pµ(σ0)λ′λ−iϵµνρσ˜nν˜Pρ˜∆σ\n(˜P·˜n)(σ3)λ′λ+m˜nµ\n(˜P·˜n)iϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ. (A9b)\nUsing Gordon decomposition, we can also write:\n¯u(˜p′, λ′)iσµν˜∆ν\n2mu(˜p, λ) =−iϵµνρσ˜nν˜Pρ˜∆σ\n(˜P·˜n)(σ3)λ′λ+ \nm˜nµ\n(˜P·˜n)−˜Pµ\nm!\niϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ. (A9c)\nFor the sake of explicit evaluations, it is helpful to note:\n−iϵ0νρσ˜nν˜Pρ˜∆σ\n(˜P·˜n)= 0 (A10a)\n−iϵiνρσ˜nν˜Pρ˜∆σ\n(˜P·˜n)= i(ˆzט∆⊥)i :i= 1,2 (A10b)\n−iϵ3νρσ˜nν˜Pρ˜∆σ\n(˜P·˜n)=i(˜Pט∆⊥)·ˆz\n˜Pz(A10c)\niϵαβγδ˜nα˜¯nβ˜∆γ(σδ)λ′λ=−i(σλ′λט∆⊥)·ˆz . (A10d)\nAppendix B: Uniqueness of light front synchronization\nWe prove two statements in this appendix. Firstly, for any coordinate system utilizing Cartesian spatial coordinates and a locally\ntime-independent but otherwise arbitrary synchronization convention, it is impossible to express the physical electromagnetic\ncurrent density of a spin-zero target as a three-dimensional internal density smeared by the probability current. In other words,\nthere is no function ρ(b)that is independent of the state |Ψ⟩such that:\n⟨Ψ|ˆjµ(x)|Ψ⟩=Z\nd3RΨ∗(R, t) i← →∂µΨ(R, t)ρ(x−R) (B1)\nfor every state |Ψ⟩. Secondly, the only possible coordinate systems which allow a two-dimensional reduction of the physical\ncurrent density to be expressed as a two-dimensional internal density smeared by the probability current is a tilted light front\ncoordinate system with an arbitrary unit vector ndefining the longitudinal direction.\nOne clarification is in order before proceeding. We mean that a time synchronization convention is locally time-independent if\nthe proper time element dτ=p\ngµνdxµdxνmeasured by a stationary clock at any location is always dt. This means we must\nhave˜g00= 1in the transformed coordinate system. In other words, we choose by convention [25] that the local time coordinate\nshould match the physical time measured by a local stationary clock.\nThe coordinate system for the hypothetical time synchronization convention can be written:\n˜x0=x0+S(x) (B2a)\n˜x1=x1(B2b)\n˜x2=x2(B2c)\n˜x3=x3, (B2d)\nwhere we use tildes for the transformed coordinates, and where we call S(x)the synchronization function. The requirement of\nlocal time independence imposes:\n��g00=∂xµ\n∂˜x0∂xν\n∂˜x0gµν=\u0012\n1 +∂S(x)\n∂x0\u00132\n= 1, (B3)\nand thus:\nS(x) =S(x). (B4)\nThe synchronization function is therefore a function only of space.22\n1. The non-existence of three-dimensional densities\nTo proceed, require an expression for the physical density ⟨Ψ|ˆjµ(˜x)|Ψ⟩in terms of the transformed coordinates. To avoid\nany complications related to trying to quantize the theory on a curvilinear equal-time surface, we proceed within instant form\nquantization and Minkowski coordinates, and obtain a formula to which the coordinates can be transformed according to Eq. (B2)\ndirectly.\nSince matrix elements for momentum kets can be written:\n⟨p′|ˆjµ(0)|p⟩= 2PµF(∆2), (B5)\nthe physical electromagnetic four-current density is:\n⟨jµ(x)⟩ ≡ ⟨Ψ|ˆjµ(x)|Ψ⟩=Zd3p\n2Ep(2π)3Zd3p′\n2Ep′(2π)3⟨Ψ|p′⟩eip′·x2PµF(∆2)⟨p|Ψ⟩e−ip·x. (B6)\nSimilar to the procedure in Sec. IV A, one can substitute 2Pµ7→i← →∂µhere. Additionally, the substitution\n∆7→ − i\u0000− →∂+← −∂\u0001\ncan be made. If this substitution is formally made within the form factor, we obtain:\n⟨jµ(x)⟩=F(−∂2)\u0014\u0012Zd3p′\n2Ep′(2π)3⟨Ψ|p′⟩eip′·x\u0013\ni← →∂µ\u0012Zd3p\n2Ep(2π)3⟨p|Ψ⟩e−ip·x\u0013\u0015\n. (B7)\nThe position-representation wave function can be defined in a covariant way in terms of the field operator ˆϕ(x):\nΨ(x, t) =⟨0|ˆϕ(x)|Ψ⟩, (B8)\nwhich is related to the momentum-representation wave function by:\nΨ(x, t) =Zd3p\n2Ep(2π)3⟨p|Ψ⟩e−ip·x. (B9)\nThus, the physical electromagnetic current density can be written:\n⟨jµ(x)⟩=F(−∂2)h\nΨ∗(x, t) i← →∂µΨ(x, t)i\n≡F(−∂2)Pµ(x). (B10)\nEach piece of this expression can be rewritten in terms of the alternative coordinate system by employing Eq. (B2):\n⟨˜jµ(˜x)⟩=F(−˜∂2)\u0014\nΨ∗(˜x,˜τ) i← →˜∂µΨ(˜x,˜τ)\u0015\n≡F(−˜∂2)˜Pµ(˜x). (B11)\nTo proceed, we need the d’ Alembertian in the transformed coordinate system. In a general coordinate system [47]:\n˜∂2f(˜x) =1√−˜g˜∂µhp\n−˜g˜gµν˜∂νf(˜x)i\n, (B12)\nwhere ˜g= det(˜ gµν). The inverse metric in this coordinate system is:\n˜gµν=∂˜xµ\n∂xα∂˜xν\n∂xβgαβ=\n1−(∇S)2−∇xS−∇yS−∇zS\n−∇xS −1 0 0\n−∇yS 0−1 0\n−∇zS 0 0 −1\n, (B13)\nand its determinant is −1, meaning the determinant of ˜gµνis also −1. Thus, the d’ Alembertian in the transformed coordinates\ncan be written:\n˜∂2=\u0000\n1−(∇S)2\u0001˜∂2\n0−(∇2S)˜∂0−(∇S)·˜∇˜∂0−˜∇2. (B14)23\nIf a distribution ρ(˜b)defining the three-dimensional state-independent electric charge density exists, then the following relation\nshould be satisfied for any state |Ψ⟩:\n⟨˜jµ(˜x)⟩=Z\nd3˜R˜Pµ(˜R,˜τ)ρ(˜x−˜R). (B15)\nWe shall proceed to show that no such distribution exists, by means of a proof by contradiction.\nTo start, we note that for Eq. (B15) to hold, the integral on the right-hand side must converge, which places restrictions on the\ndistribution ρ(˜b)which depend on the restrictions in place on the probability four-current ˜Pµ(˜R,˜τ). A reasonable restriction on\nthe probability current is that ˜Pµ(˜R,˜τ)be a Schwartz function, meaning that it and all its derivatives vanish at spatial infinity\nfaster than any power of |˜R|. In this case, ρ(˜b)is a tempered distribution, a broad space of generalized functions which includes\nthe Dirac delta distribution and its derivatives. (See Chapter 9 of Folland [48] for more details.) Using Eqs. (B11) and (B14),\nthis means:\nF\u0010\n−\u0000\n1−(∇S)2\u0001˜∂2\n0+ (∇2S)˜∂0+ (∇S)·˜∇˜∂0+˜∇2\u0011\n˜Pµ(˜x) =Z\nd3˜R˜Pµ(˜R,˜τ)ρ(˜x−˜R). (B16)\nBecause the Fourier transform of a tempered distribution can always be defined [48], we can (assuming Eq. (B15) is true)\nintroduce φ(k)as the Fourier transform of the charge density:\nρ(˜b) =Zd3k\n(2π)3φ(k)e−ik·˜b, (B17)\nit follows that:\nZ\nd3˜R˜Pµ(˜R,˜τ)ρ(˜x−˜R) =φ(i˜∇)˜Pµ(˜x,˜τ), (B18)\nand Eq. (B16) becomes:\nF\u0010\n−\u0000\n1−(∇S)2\u0001˜∂2\n0+ (∇2S)˜∂0+ (∇S)·˜∇˜∂0+˜∇2\u0011\n˜Pµ(˜x) =φ(i˜∇)˜Pµ(˜x,˜τ). (B19)\nThe requirement that this holds for any state imposes:\nF\u0010\n−\u0000\n1−(∇S)2\u0001˜∂2\n0+ (∇2S)˜∂0+ (∇S)·˜∇˜∂0+˜∇2\u0011\n=φ(i˜∇), (B20)\nin which the right-hand side depends only on spatial derivatives. This equality can only hold if the left-hand side does not depend\non˜∂0, which imposes the following three constraints that cannot simultaneously hold:\n∇2S= 0 (B21a)\n(∇S)2= 1 (B21b)\n∇S= 0. (B21c)\nIn particular, the second and third conditions contradict each other. We therefore conclude that Eq. (B15) is false and thus that\nno state-independent three-dimensional internal charge density exists, regardless of the synchronization convention used.\n2. The uniqueness of light front synchronization\nAlthough no three-dimensional internal density exists, we already know that light front synchronization allows an internal two-\ndimensional density to be found. We next inquire whether other synchronization conventions permit alternative two-dimensional\ndensities to be obtained. (For instance, is there a synchronization convention that is rotationally symmetric and allows an angular\ndensity to be obtained by integrating out the radial coordinate?)\nIn the formulation used in this appendix, light front synchronization works by eliminating derivatives with respect to the spatial\ncoordinate that has been integrated out. Thus, the term (∇S)·˜∇˜∂0in the d’ Alembertian can be made to vanish without requiring24\nthat∇S= 0. The way that is works with S(x) =zas an example is:\nZ∞\n−∞d˜z F\u0010\n˜∂z˜∂0+˜∇2\u0011\n˜Pµ(˜x) =∞X\nn=0F(n)(0)\nn!Z∞\n−∞d˜z\u0010\n˜∂z˜∂0+˜∇2\u0011n˜Pµ(˜x)\n=∞X\nn=0nX\nk=0n!\nk!(n−k)!F(n)(0)\nn!Z∞\n−∞d˜z˜∂n−k\nz˜∂n−k\n0(˜∇2)k˜Pµ(˜x)\n=∞X\nn=0F(n)(0)\nn!Z∞\n−∞d˜z(˜∇2)n˜Pµ(˜x) =Z∞\n−∞d˜z F(˜∇2)˜Pµ(˜x). (B22)\nIn the last line, we have dropped all total derivatives with respect to zthat appeared in the second line, since the associated\nsurface terms at infinity vanish. Since the mixed derivative in the d’ Alembertian has been eliminated by integration, the third\ncondition in Eq. (B21) is no longer necessary to eliminate the time derivative dependence from the form factor. Thus, to obtain\ntwo-dimensional densities, only the first two conditions of Eq. (B21) are required to hold.\nThe most general solution to ∇2S= 0without a singularity at the origin is:\nS(r) =∞X\nl=0rllX\nm=−lClmYlm(θ, ϕ). (B23)\nFor this general solution:\n(∇S)2=X\nl,l′rl+l′−2lX\nm=−ll′X\nm′=−l′ClmC∗\nl′m′\u0012\nll′Ylm(θ, ϕ)Y∗\nl′m′(θ, ϕ)\n+∂Ylm(θ, ϕ)\n∂θ∂Y∗\nl′m′(θ, ϕ)\n∂θ+1\nsin2θ∂Ylm(θ, ϕ)\n∂ϕ∂Y∗\nl′m′(θ, ϕ)\n∂ϕ\u0013\n.(B24)\nTo obtain (∇S)2= 1, it is necessary that every power of rgreater than 0vanish in this expression. Thus, for any n >0:\nn+2X\nl=0n+2X\nl′=0δl′,n+2−llX\nm=−ll′X\nm′=−l′ClmC∗\nl′m′\u0012\nll′Ylm(θ, ϕ)Y∗\nl′m′(θ, ϕ)\n+∂Ylm(θ, ϕ)\n∂θ∂Y∗\nl′m′(θ, ϕ)\n∂θ+1\nsin2θ∂Ylm(θ, ϕ)\n∂ϕ∂Y∗\nl′m′(θ, ϕ)\n∂ϕ\u0013\n= 0.(B25)\nIntegrating this over the surface of a unit sphere, and using integration by parts, gives:\nn+2X\nl=0n+2X\nl′=0δl′,n+2−llX\nm=−ll′X\nm′=−l′ClmC∗\nl′m′Zπ\n0dθsinθZ2π\n0dϕ Y∗\nl′m′(θ, ϕ)\u0012\nll′Ylm(θ, ϕ)+\n−1\nsinθ∂\n∂θ\u0014\nsinθ∂Ylm(θ, ϕ)\n∂θ\u0015\n−1\nsin2θ∂2Ylm(θ, ϕ)\n∂ϕ2\u0013\n= 0.(B26)\nUsing the differential equation defining the spherical harmonics, and their orthogonality, gives:\nn+2X\nl=0n+2X\nl′=0δl′,n+2−llX\nm=−ll′X\nm′=−l′ClmC∗\nl′m′δll′δmm′l(l′+l+ 1) = 0 . (B27)\nThe Kronecker delta product δl′,n+2−lδll′will give 0automatically if nis odd. For even n, using the Kronecker deltas to\neliminate most the sums gives:\nlX\nm=−l|Clm|2\f\f\f\f\nl=n/2+1= 0. (B28)\nSince this must be true for even n >0, it must be true for any l >1. Thus Clm= 0for all l >1. 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A\npseudo-density matrix is a hermitian matrix of unit trace whose marginals are density ma-\ntrices, and in this work, we make use a factorization system for quantum channels to asso-\nciate a pseudo-density matrix with a quantum system which is to evolve according to a finite\nsequence of quantum channels. We then view such a pseudo-density matrix as the quantum\nanalog of a local patch of spacetime, and we make an in-depth mathematical analysis of\nsuch quantum dynamical pseudo-density matrices and the properties they satisfy. We also\nshow how to explicitly extract quantum dynamics from a given pseudo-density matrix, thus\nsolving an open problem posed in the literature.\nContents\n1 Introduction 2\n2 Preliminaries 5\n3 Classical probability as CPTP dynamics 9\n4 In quantum bloom 14\n5 Blooming 2-chains 19\n6 Blooming n-chains 25\n7 States over time for n-step processes 32\n8 The Y-function 34\n9 Extracting dynamics from a pseudo-density matrix 40\nBibliography 42\n1arXiv:2304.03954v3 [quant-ph] 2 Sep 20231 Introduction\nIn 1907 Hermann Minkowski showed that the work of Maxwell, Lorentz and Einstein could\nbe viewed geometrically as a 4-dimensional theory of spacetime [15], after which he boldly pre-\ndicted that space by itself, and time by itself, were then \"doomed to fade away in mere shad-\nows\". Minkowski’s prediction was indeed correct, and ever since we have only furthered our\nunderstanding of the inextricable connection between space and time. By 1927 quantum theory\nhad been established through the revolutionary work Bohr, Heisenberg, Schrödinger, Born and\nothers, where the fundamental view was that at the sub-atomic realm, reality was appropriately\ndescribed by \"quantum states\" evolving in time. While debates have raged through the ages\nregarding the ontological status of a quantum state, the one thing that most can agree upon is\nthat at a fundamental level a quantum state consists of information , and there is an emerging\nviewpoint — famously coined by Wheeler as \"it from bit\" — that such quantum information is\nthe basis of our physical reality. But while our macroscopic view of the world has blossomed\nfrom Minkowski’s unification of space and time, for the most part there has not been an analo-\ngous unification of information and time in quantum theory, where we seemed to have left the\ninsights of Minkowski behind. In particular, while in relativity theory space evolves in time to\nform a single mathematical object called spacetime, quantum theory lacks a standard notion of\nhow to encapsulate the global evolution of a quantum state over successive instances of time\ninto a single entity, or rather, as a \"state over time\". In this letter, we then provide an answer to\nthe \"?\" in the following analogy:\nGeneral Relativity :space + time =spacetime\nQuantum Theory :state + time =?\nMoreover, as spacetime is fundamental to our understanding of gravity, it seems only natu-\nral that a mathematically precise formulation of the above analogy will yield insights into the\nnature of quantum gravity.\nTo summarize our approach to such an analogy, consider a local patch Sof spacetime, which\nwe may view as a fibration φ:S→[t0,t1]of spatial slices over an interval of time [t0,t1].\nIn such a case, we may view Sas a cobordism , i.e., as representing the process of the spatial\nsliceS0=φ−1(t0)evolving over time into the spatial slice S1=φ−1(t1). As such, in general\nrelativity, the objects encoding evolution over time are of the same class of entity as the objects\nwhich are evolving over time, as S,S0andS1are all manifolds. However, a crucial distinction\nbetween the process manifold Sand the spatial slices S0andS1, is that while S0andS1are\nof euclidean signature, the signature of the process manifold Spicks up a negative sign as it\nextends over time.\nFor our quantum analogy of a local patch of spacetime, suppose there is a quantum system\nin state ρ∈A0at time t=t0, which is to evolve according to a quantum channel E:A0→A1,\nso that E(ρ)∈A1is then viewed as the state of the system at time t=t1. We also assume that\n∆t=t1−t0is on the order of the Planck time, so that the interval [t0,t1]may be viewed as a\nsingle discrete time-step. Then in accordance with our spacetime analogy, we wish to associate\n2a \"state over time\"\nψ(ρ,E)∈A0⊗A1\nencoding the dynamical evolution of ρaccording to the channel E:A0→A1. Moreover, in\nanalogy with the spatial slices S0andS1being the source and target of the process manifold\nS, the states ρandE(ρ)should be the reduced density matrices of the state over time ψ(ρ,E)\nwith respect to tracing out A1andA0respectively. In the case that A0andA1admit large tensor\nfactorizations\nA0=mO\ni=1Mi & A1=nO\nj=1Mj\n(where Mkdenotes the matrix algebra of k×kmatrices with complex entries), each tensor\nfactorMiandMjmay be identified with a region of space at times t=t0andt=t1respec-\ntively, and in such a case, we view the state over time ψ(ρ,E)as a quantum mechanical unit of\nspacetime.\nWhile various formulations of dynamical quantum states have appeared in the literature,\nincluding the pseudo-density operator (PDO) formalism for systems of qubits first appearing\nin the work of Fitzsimons, Jones and Vedral [3, 13, 14, 22, 23], the causal states of Leifer and\nSpekkens [10–12], the right bloom construction of Parzygnat and Russo [18, 19], the super-\ndensity operators of Cotler et alia [1], the compound states of Ohya [16], the generalized condi-\ntional expectations of Tsang [20], the Wigner function approach of Wooters [21] and the process\nstates of Huang and Guo [7], in this work we focus mainly on the state over time ψ(ρ,E)given\nby\nψ(ρ,E) =1\n2{(ρ⊗1)J[E]}, (1.1)\nwhere {∗,∗}denotes the anti-commutator, and\nJ[E] =X\ni,j|i⟩⟨j|⊗E(|j⟩⟨i|)\nis the Jamiołkowski matrix1associated with the channel E:A0→A1. The state over time given\nby (1.1) was shown in [5, 17] to satisfy a list of desiderata for states over time put forth in [6],\nand at present it is the only known state over time construction to satisfy such properties. We\nnote however that while the state over time given by (1.1) is hermitian and of unit trace, it\nis not positive in general, and as such, is often referred to as a pseudo-density matrix . In our\nspacetime analogy, the negative eigenvalues of a state over time are analogous to the negative\nsign appearing in the time-component of the spacetime metric, and further justification for such\na state over time to have negative eigenvalues appears in [3], where it is argued that negative\neigenvalues serve as a witness to temporal correlations encoded in ψ(ρ,E).\nOne desideratum for states over time is a property often referred to as \"associativity\", as it\nallows one to unambiguously associate a \"2-step state over time\"\nψ(ρ,E,F)∈A0⊗A1⊗A2\n1The Jamiołkowski matrix is not to be confused with the Choi matrix C[E] =P\ni,j|i⟩⟨j|⊗E(|i⟩⟨j|).\n3associated with the evolution of a state ρaccording to a 2-chain A0E→A1F→A2of quantum\nchannels. The potential ambiguity stems from the fact that there are two parenthezations (A0⊗\nA1)⊗A2andA0⊗(A1⊗A2)of the matrix algebra A0⊗A1⊗A2, which lead to 2 fundamentally\ndifferent constructions of a 2-step state over time ψ(ρ,E,F). The associativity condition then\nensures that these two distinct constructions of 2-step states over time are in fact equal, leading\nto a well-defined notion of a 2-step state over time.\nIn this work, we consider the case of a quantum state ρevolving according to an n-chain of\nquantum channels\nA0E1− →A1− → ··· − → An−1En− →An, (1.2)\nwith which we wish to associate an \" n-step\" state over time\nψ(ρ,E1, ...,En)∈A0⊗ ··· ⊗ An.\nSimilar to how the ambiguity for 2-step states over time arises from the two different paren-\nthezations of the matrix algebra A0⊗A1⊗A2, there is an ambiguity for n-step states over time\ncorresponding to the nth Catalan number cn=1\nn+1\u00002n\nn\u0001\nof ways to parenthesize the matrix al-\ngebra A0⊗ ··· ⊗ An, leading to cnfundamentally distinct constructions of an n-step state over\ntime. We then prove that the cndifferent constructions of such an n-step state over time are\nall in fact equal (see item ii of Theorem 7.4), leading to a well-defined notion of an n-step state\nover time. In accordance with our spacetime analogy, we then view an n-step state over time\nψ(ρ,E1, ...,En)as a quantum analog of a local patch of spacetime fibered over an interval of time\n[t0,tn] = [t0,t1]∪ ··· ∪ [tn−1,tn],\nwhere ∆t=ti−ti−1is on the order Planck time, so that [ti−1,ti]is viewed as a discrete time-step\ncorresponding to the evolution of the quantum system according to the channel Ei:Ai−1→Ai.\nAs the n-step extension of the state over time given by (1.1) is the only known construction\nwhich always yields a hermitian state over time, we denote the associated state over time by\nY(ρ,E1, ...,En), and refer to the mapping\n(ρ,E1, ...,En)7− →Y(ρ,E1, ...,En)\nas theY-function . In Example 8.20 we show that in the case of dynamically evolving systems of\nqubits, the output of the Y-function coincides with the coarse-grained pseudo-density operator\n(PDO) associated with the system, which was introduced in [3] to treat temporal and spatial\ncorrelations in quantum theory on equal footing. As such, the Y-function may be viewed as\na generalization to arbitrary finite-dimensional quantum systems of the pseudo-density oper-\nator formalism for systems of qubits. We also prove that given a unit-trace hermitian element\nτ∈A0⊗ ··· ⊗ Ansatisfying some technical conditions, one may explicitly extract a dynami-\ncal process (ρ,E1, ...,En)such that Y(ρ,E1, ...,En) =τ(see Theorem 9.5). This solves an open\nquestion recently posed in [9], where it is stated \"Another interesting and closely relevant open\nquestion is, for a given PDO (pseudo-density operator), how to find a quantum process to real-\nize it.\".\n4In what follows, we provide the necessary definitions and notation which will be used\nthroughout in Section 2. In Section 3, we motivate the definition of quantum state over time by\nfirst analyzing the classical case of a random variable Xstochastically evolving into a random\nvariable Y. In particular, we show that classical joint distribution P(x,y)associated with such\nstochastic evolution arises from a unique factorization of the associated classical channel, and\nmoreover, that P(x,y)may be re-written in a way that is valid for any 1-step quantum process\n(ρ,E). In Section 4 we then define a factorization system for quantum channels in terms of\naquantum bloom map , and then we use such a factorization system to define quantum states\nover time associated with 1-step processes (ρ,E). In Section 5 we recall the associativity prop-\nerty for quantum bloom maps which allow states over time for 1-step processes to extend to\nwell-defined states over time for 2-step processes (ρ,E,F), and prove some general results re-\ngarding 2-step states over time. In Section 6 we then show how the associativity property for\nquantum bloom maps yields uniquely-defined extensions to quantum bloom maps for n-chains\nof quantum channels. In Section 7 we then use the quantum bloom maps for n-chains to de-\nfine states over time ψ(ρ,E1, ...,En)associated with n-step processes (ρ,E1, ...,En)for arbitrary\nn > 0. In Section 8 we then focus on the n-step states over time associated with the Y-function,\nand we prove some general properties of the Y-function. We also two distinct formulas for\nY(ρ,E1, ...,En)in terms of ρand the states J[E1], ...,J[En]which are Jamiołkowski-isomorphic\nto the channels E1, ...,En. In Section 9 we then show that the Y-function yields a bijection when\nrestricted to a suitably nice set of n-step processes, and we prove an explicit formula for its\ninverse in such a case.\nAcknowledgements. We thank Arthur J. Parzygnat and Franceso Buscemi for many useful\ndiscussions. This work is supported in part by the Blaumann Foundation.\n2 Preliminaries\nIn this section we provide the basic definitions, notation and terminology which will be used\nthroughout.\nDefinition 2.1. LetXbe a finite set. A function p:X→Rwill be referred to as a quasi-\nprobability distribution if and only ifP\nx∈Xp(x) =1. In such a case, p(x)∈Rwill be denoted\nbypxfor all x∈X. Ifpx∈[0, 1]for all x∈X, then pwill be referred to as a probability\ndistribution .\nDefinition 2.2. LetXandYbe finite sets. A stochastic mapf:X Yconsists of the assignment\nof a probability distribution fx:Y→[0, 1]for every x∈X. In such a case, (fx)ywill be denoted\nbyfyxfor all x∈Xandy∈Y, which is interpreted as the probability of ygiven x. A stochastic\nmap f:X Ytogether with a prior distribution pon its set of inputs Xwill be denoted by\n(p,f). The set of stochastic maps from XtoYwill be denoted by Stoch (X,Y).\nRemark 2.3. A stochastic map f:X Yis also commonly referred to as a Markov kernel , or a\ndiscrete memoryless channel .\n5Notation and Terminology 2.4. Given a natural number m∈N, the set of m×mmatrices with\ncomplex entries will be denoted by Mm, and will be referred to as a matrix algebra . As the\nmatrix algebra M1is simply the complex numbers, it will be denoted by C. The matrix units in\nMmwill be denoted by E(m)\nij(or simply Eijifmis clear from the context), and for every ρ∈Mm,\nρ†∈Mmdenotes the conjugate-transpose of ρ. Given a finite set X, a direct sumL\nx∈XMmx\nwill be referred to as a multi-matrix algebra , whose multiplication and addition are defined\ncomponent-wise. If AandBare multi-matrix algebras, then the vector space of all linear maps\nfromAtoBwill be denoted by Hom (A,B). The trace of an element A=L\nx∈XAx∈L\nx∈XMmx\nis the complex number tr (A)given by tr (A) =P\nx∈Xtr(Ax)(where tr (∗)is the usual trace\non matrices), and the dagger ofAis the element A†∈L\nx∈XMmxgiven by A†=L\nx∈XA†\nx.\nGivenE∈Hom(A,B)withAandBmulti-matrix algebras, we let E∗∈Hom(B,A)denote the\nHilbert–Schmidt dual (oradjoint ) ofE, which is uniquely determined by the condition\ntr\u0010\nE(A)†B\u0011\n=tr\u0010\nA†E∗(B)\u0011\nfor all A∈AandB∈B. The identity map between algebras will be denoted by id, while the\nunit element in an algebra will be denoted by 1(subscripts such as id Aand1Awill be used if\ndeemed necessary).\nRemark 2.5. Every finite-dimensional C∗-algebra is isomorphic to a multi-matrix algebra [2].\nNotation 2.6. Given a finite set X, the multi-matrix algebraL\nx∈XCis canonically isomorphic to\nalgebra CXof complex-valued functions on X. As such, we will write an element ρ∈L\nx∈XC∼=\nCXas\nρ=X\nx∈Xρxδx∈CX,\nwhere δx∈CXis the Dirac-delta ofx, which is the function taking the value 1 at xand 0\notherwise. In such a case, ρx∈Cwill be referred to as the x-component ofρfor all x∈X.\nDefinition 2.7. LetXbe a finite set and let A=L\nx∈XMmxbe a multi-matrix algebra. An\nelement A=L\nx∈XAx∈Ais said to be\n•self-adjoint if and only if A†\nx=Axfor all x∈X.\n•positive if and only if Ax∈Mmxis self-adjoint and has non-negative eigenvalues for all\nx∈X.\n• a state if and only if Ais positive and of unit trace. If Xcontains only one element, then\na state Awill often be referred to as a density matrix . The set of all states on Awill be\ndenoted by S(A).\nRemark 2.8. IfXis a finite set and A=CX, then a state ρ∈S(CX)is of the form ρ=P\nx∈Xρxδx\nwith ρx⩾0 andP\nx∈Xρx=1. As such, any state on CXmay be identified with a probability\ndistribution on X.\n6Definition 2.9. LetA=L\nx∈XMmxandB=L\ny∈YMny. The tensor product ofAandBis the\nmulti-matrix algebra A⊗Bgiven by\nA⊗B=M\n(x,y)∈X×YMmx⊗Mny,\nwhereMmx⊗Mnyis the usual tensor product of matrix algebras. Given elementsL\nx∈XAx∈A\nandL\ny∈YBy∈B, the element (L\nx∈XAx)⊗(L\ny∈YBy)∈A⊗Bis the multi-matrix given by\n M\nx∈XAx!\n⊗\nM\ny∈YBy\n=M\n(x,y)∈X×YAx⊗By,\nwhere Ax⊗Byis the usual tensor product of matrices. Given maps E∈Hom(A,A′)and\nF∈Hom(B,B′), thenE⊗F∈Hom(A⊗B,A′⊗B′)is the map corresponding to the linear\nextension of the assignment (E⊗F)(A⊗B) =E(A)⊗F(B).\nDefinition 2.10. Given multi-matrix algebras A=L\nx∈XMmxandB=L\ny∈YMmy, an element\nE∈Hom(A,B)consists of elements Eyx∈Hom(Mmx,Mny)such that\nE M\nx∈Xρx!\n=M\ny∈Y X\nx∈XEyx(ρx)!\n.\nIn such a case, the Eyxwill be referred to as the component functions of E.\nDefinition 2.11. LetAandBbe multi-matrix algebras. A map E∈Hom(A,B)is said to be\n•†-preserving if and only if E(A)†=E(A†)for all A∈A.\n•trace-preserving if and only if tr (E(A)) = tr(A)for all A∈A.\n•positive if and only if E(A)is positive whenever A∈Ais positive.\n•completely positive if and only if E⊗idC:A⊗C→B⊗Cis positive for every multi-\nmatrix algebra C.\nNotation 2.12. Let(A,B)be pair of multi-matrix algebras. The subset of Hom (A,B)consisting\nof completely positive, trace-preserving maps will be denoted by CPTP (A,B), while the subset\nof Hom (A,B)consisting of trace-preserving maps will be denoted by TP(A,B). An element of\nCPTP (A,B)withAandBmatrix algebras will often be referred to as a quantum channel .\nDefinition 2.13. LetXandYbe finite sets. An element E∈CPTP (CX,CY)will be referred to as\naclassical channel . In such a case, we let Eyx∈[0, 1]denote the elements such that\nE(δx) =X\ny∈YEyxδy.\nIn such a case, the elements Eyxwill be referred to as the conditional probabilities associated\nwithE.\n7Definition 2.14. Given a pair (A,B)of multi-matrix algebras, an element (ρ,E)∈S(A)×\nCPTP (A,B)will be referred to as a process , and the set of processes S(A)×CPTP (A,B)will\nbe denoted by P(A,B). The subset of P(A,B)consisting of processes (ρ,E)with ρinvert-\nible will be denoted by P+(A,B). When A=CXandB=CYfor finite sets XandY, then\n(ρ,E)∈P(CX,CY)will be referred to as a classical process .\nDefinition 2.15. Let(A,B)be a pair of multi-matrix algebras, and let E∈Hom(A,B). The\nchannel state ofEis the element J[E]∈A⊗Bgiven by\nJ[E] = ( idA⊗E)(µ∗(1)),\nwhere µ∗:A→A⊗Ais the Hilbert-Schmidt dual of the multiplication map µ:A⊗A→A.\nRemark 2.16. Given a pair (A,B)of multi-matrix algebras, the map Hom (A,B)− →A⊗B\ngiven by E7− →J[E]is a linear isomorphism, which we refer to as the Jamiołkowski iso-\nmorphism [8]. In this work we will also make heavy use of the the inverse J−1:A⊗B− →\nHom(A,B)of the Jamiołowski isomorphism, which is given by\nJ−1(τ)(ρ) =trA((ρ⊗1)τ).\nDefinition 2.17. IfA,BandCare matrix algebras, there exists an associator isomorphism A⊗\n(B⊗C)− →(A⊗B)⊗Cwhich then allows one to define tensor products of of a finite number\nof matrix algebras iteratively. We can then extend such a constriction to multi-matrix algebras\nas follows. Let {Ax}x∈Xbe a collection of multi-matrix algebras indexed by a finite set X, where\nAx=L\nλx∈ΛxAλx. The tensor product of the algebras Axis the multi-matrix algebraN\nx∈XAx\nindexed by the set Γ=Q\nx∈XΛxgiven by\nO\nx∈XAx=M\n(λx)x∈X∈Γ O\nx∈XAλx!\n.\nIf the index set Xis of the form X={0, 1, ..., n}, then we will writeN\nx∈XAxasA0⊗ ··· ⊗ An.\nDefinition 2.18. LetA=A0⊗ ··· ⊗ Anbe a tensor product of multi-matrix algebras. The ith\npartial trace fori∈{0, ...,n}is the map tr i:A− →Aigiven by the linear extension of the\nassignment\ntri(A0⊗ ··· ⊗ An) =tr\u0010\nA0⊗ ··· ⊗ cAi⊗ ··· ⊗ An\u0011\nAi∈Ai,\nwhere cAidenotes the empty matrix for all i∈{0, ...,n}.\nNotation 2.19. IfC=A⊗B, then the partial trace maps tr 1:A⊗B− →Aand tr 2:A⊗B− →B\nwill be denoted by tr Band tr Arespectively. If A=CXandB=CY, then the partial trace\nmaps trCX:CX⊗CY− →CYand trCY:CX⊗CY− →CXwill be denoted by tr Xand tr Y. If\nA=A0⊗ ··· ⊗ An, then we will frequently make use of the nth partial trace tr n:A− →An,\nand as such, we will often denote tr nsimply by tr.\n8Definition 2.20. LetXbe a finite set and let A=L\nx∈XMmxbe a multi-matrix algebra. A self-\nadjoint element τ∈Ais said to be a pseudo-density operator with respect to the factorization\nA=A0⊗ ··· ⊗ Anif and only if for all i∈{0, ...,n}we have\ntri(τ)∈S(Ai).\nIfAis a matrix algebra (i.e., when Xconsists of a single element), then a pseudo-density op-\nerator τ∈Awith respect to the factorization A=A0⊗ ··· ⊗ Anwill be referred to as a\npseudo-density matrix . The set of all pseudo-density operators with respect to the factorization\nA=A0⊗ ··· ⊗ Anwill be denoted by T(A0⊗ ··· ⊗ An).\nRemark 2.21. A pseudo-density operator with respect to the trivial factorization A=A0is\nsimply a state, so that T(A0) =S(A0).\nRemark 2.22. Since all of the marginals of a pseudo-density operator are of unit trace, it follows\nthat a pseudo-density operator is necessarily of unit trace as well.\n3 Classical probability as CPTP dynamics\nIn this section we recall how classical probability may be recast in terms of CPTP maps between\nmulti-matrix algebras. In particular, we will show how the joint distribution associated with a\nclassical channel E:CX− →CYarises from a unique factorization of the channel of the form\nE=trX◦!\nE, where!\nE:CX− →CX×Y∼=CX⊗CYis a map referred to as the bloom ofE. Ifρ∈CX\nis a state, then the element!\nE(ρ)is the joint distribution associated with the classical process\n(ρ,E), which in a more dynamical language we refer to as the state over time associated with\n(ρ,E). We will then re-write the classical state over time!\nE(ρ)in a way that is valid for any\nquantum process (ρ,E)∈P(A,B)withAandBarbitrary multi-matrix algebras, thus paving\nthe way for a quantum generalization of the classical state over time!\nE(ρ).\nThe following proposition shows how classical stochastic maps may be recast in terms of\nCPTP dynamics.\nProposition 3.1. Let(X,Y)be a pair of finite sets, and let Q:Stoch (X,Y)− → CPTP (CX,CY)be the\nmap given by\nQ(f)(δx) =X\ny∈Yfyxδy.\nThen Qis a continuous bijection.\nProof. This follows form the fact that stochastic maps f:X Yand CPTP maps E:CX− →CY\nare completely determined by their associated conditional probabilities fyxandEyx. ■\nThe next proposition shows that factorizations of classical channels are essentially unique,\nwhich we will use to characterize joint distributions associated with stochastic dynamics.\n9Proposition 3.2. Let(X,Y)be a pair of finite sets, let E∈CPTP (CX,CY), and let!\nE∈\nCPTP (CX,CX⊗CY)be such that E=trX◦!\nEand idCX=trY◦!\nE. Then!\nEis the linear map\ngiven by!\nE X\nx∈Xρxδx!\n=X\n(x,y)∈X×YρxEyx(δx⊗δy), (3.3)\nwhereEyxare the conditional probabilities associated with the map E.\nProof. We show that given x0∈X, we have!\nE(δx0) =X\n(x,y)∈X×Yδxx0Eyx(δx⊗δy).\nThe result then follows as!\nEis assumed to be linear and Dirac deltas form a basis of CX. So let\nx0∈Xbe arbitrary, and let eE(x,y)x0be the conditional probabilities associated with the map!\nE,\nso that!\nE(δx0) =X\n(x,y)∈X×YeE(x,y)x0(δx⊗δy).\nFrom the conditions E=trX◦!\nEand idCX=trY◦!\nEwe then have\nX\ny∈YEyx0δy=E(δx0) =trX(!\nE(δx0)) =X\ny∈Y X\nx∈XeE(x,y)x0!\nδy\nand\nX\nx∈Xδxx0δx=δx0=trY\u0000!\nE(δx0)\u0001\n=X\nx∈X\nX\ny∈YeE(x,y)x0\nδx,\nwhere δxx0is the Kronecker delta. For all y∈Ywe then have\nX\nx∈XeE(x,y)x0=Eyx0,\nand for all x∈Xwe haveX\ny∈YeE(x,y)x0=δxx0.\nIt then follows that for all (x,y)∈X×Ywe have eE(x,y)x0=δxx0Eyx, thus!\nE(δx0) =X\n(x,y)∈X×YeE(x,y)x0(δx⊗δy) =X\n(x,y)∈X×Yδxx0Eyx(δx⊗δy),\nas desired. ■\nDefinition 3.4. Let(X,Y)be a pair of finite sets and let (ρ,E)∈P(CX,CY)be a classical process.\n• The map!\nE∈CPTP (CX,CX⊗CY)defined by (3.3) will be referred to as the bloom ofE.\n10Figure 1: The bloom map \"opens up\" the classical channel E.\n• The element!\nE(ρ)∈S(CX⊗CY)will be referred to as the state over time associated with\nthe classical process (ρ,E).\n• The map on classical processes given by (ρ,E)7− →!\nE(ρ)will be referred to as the classical\nstate over time function .\nRemark 3.5. Given a pair of finite sets (X,Y)and a classical process (ρ,E)∈P(CX,CY), if we\nidentify Eyxwith conditional probabilities P(y|x),ρ∈S(CX)with a probability distribution\nP(x)onX, and!\nE(ρ)with a joint probability distribution P(x,y)onX×Y, then the equation!\nE(ρ)(x,y)=ρxEyx\nmay be re-written as\nP(x,y) =P(x)P(y|x), (3.6)\nwhich is the classical equation relating a joint distribution with a conditional distribution and\na prior. As such, Proposition 3.2 may be interpreted as saying that the joint distribution P(x,y)\nassociated with a classical process (ρ,E)∈P(CX,CY)arises as an intermediary step in a canon-\nical factorization E=trX◦!\nEof the associated channel. This observation will be crucial for\ngeneralizing joint distributions to the quantum domain.\nNext we show that in the classical domain, joint distributions are in a bijective correspon-\ndence with classical processes.\nNotation 3.7. Given a pair (X,Y)of finite sets, let SX\n+(CX⊗CY)⊂S(CX⊗CY)denote the subset\ngiven by\nSX\n+(CX⊗CY) =\nτ∈S(CX⊗CY)|trY(τ)is invertible\u000b\n.\nProposition 3.8. Let(X,Y)be a pair of finite sets, and let\nS :P+(CX,CY)− →SX\n+(CX⊗CY)\n11be the map given by S(ρ,E) =!\nE(ρ). Then Sis a bijection, whose inverse is determined by the condition\nS−1(τ) = (ρ,E)=⇒Eyx=τ(x,y)\nρx∀(x,y)∈X×Y.\nProof. Let(X,Y)be a pair of finite sets, and let C :SX\n+(CX⊗CY)− →P+(CX,CY)be the map\nuniquely determined by the condition\nC(τ) = (ρ,E)=⇒Eyx=τ(x,y)\nρx∀(x,y)∈X×Y.\nWe will show C◦S = id and S◦C = id. For the former case, let (ρ,E)∈P+(CX,CY), and let\n(eρ,eE) = C (S(ρ,E)). By the definition of Cwe have\nC(τ) = (ρ,E)=⇒Eyx=τ(x,y)\nρx∀(x,y)∈X×Y\n=⇒X\ny∈Yτ(x,y)=X\ny∈YρxEyx=ρx∀x∈X\n=⇒ trY(τ) =ρ,\nthus\neρ=trY(S(ρ,E))=trY(!\nE(ρ))=ρ.\nMoreover, by definition of C, for all (x,y)∈X×Ywe have\neEyx=\n!\nE(ρ)(x,y)\neρx=ρxEyx\nρx=Eyx=⇒eE=E,\nthusC◦S = id.\nNow let τ∈SX\n+(CX⊗CY), let(ρ,E) = C( τ), and let eτ= S(C( τ)). Then for all (x,y)∈X×Y\nwe have\neτ(x,y)= S(ρ,E)(x,y)=ρxEyx=ρx\u0012τ(x,y)\nρx\u0013\n=τ(x,y)=⇒eτ=τ,\nthusS◦C = id, as desired. ■\nRemark 3.9. In light of Proposition 3.8, it follows that a state τ∈SX\n+(CX⊗CY)may either be\nviewed as a joint distribution associated with two space-like separated random variables Xand\nYoccurring in parallel, or as a state over time!\nE(ρ)associated with the process (ρ,E) = C( τ)∈\nP+(CX,CY). These two equivalent viewpoints are what we refer to as the static-dynamic dual-\nityfor classical joint states in SX\n+(CX⊗CY). Another way to think of the static-dynamic duality\nfor classical joint states, is that classical probability does not distinguish between temporal cor-\nrelations and spatial correlations, revealing a certain symmetry between space and time for\nclassical random variables. For quantum systems however it turns out that such a symmetry\nbetween space and time does not exist, as there exists quantum processes for which temporal\ncorrelations may not be viewed as spatial correlations and vice-versa [3]. As such, the static-\ndynamic duality for classical joint states does not generalize to the quantum domain, and we\nwill see that this is manifested in the fact that quantum states over time are not positive in\ngeneral. This fact will play a crucial role in the general theory of states over time for quantum\nprocesses.\n12In the next proposition, we show how the state over time!\nE(ρ)associated with a classical\nprocess (ρ,E)∈P(CX,CY)may be re-written in a way that is valid for all quantum processes,\nwhich we will use in the next section for a quantum generalization of states over time.\nProposition 3.10. Let(X,Y)be a pair of finite sets, and let E∈CPTP (CX,CY). Then!\nE(ρ) = (ρ⊗1)J[E] (3.11)\nfor all ρ∈CX. In particular, if ρ∈S(CX), then (ρ⊗1)J[E]is the state over time associated with the\nclassical process (ρ,E).\nProof. Letµ:CX⊗CX− →CXdenote the multiplication map. From the definition of the Hilbert-\nSchmidt dual we have\nµ∗(1) =X\nx∈Xδx⊗δx,\nthus\nJ[E] = (idCX⊗E)(µ∗(1)) =(idCX⊗E) X\nx∈Xδx⊗δx!\n=X\nx∈Xδx⊗E(δx)\n=X\nx∈Xδx⊗\nX\ny∈YEyxδy\n=X\n(x,y)∈X×YEyx(δx⊗δy).\nAnd since for all ρ∈CXwe have\nρ⊗1= X\nx∈Xρxδx!\n⊗\nX\ny∈Yδy\n=X\n(x,y)∈X×Yρx(δx⊗δy),\nit follows that\n(ρ⊗1)J[E] =\nX\n(x,y)∈X×Yρx(δx⊗δy)\n\nX\n(x,y)∈X×YEyx(δx⊗δy)\n\n=X\n(x,y)∈X×YρxEyx(δx⊗δy)\n=!\nE(ρ),\nas desired. ■\nRemark 3.12. While up to now the left-hand side of equation (3.11) is only defined for classical\nprocesses, the right-hand side is defined for anyquantum process (ρ,E). Moreover, since for\nclassical processes we have [J[E],(ρ⊗1)]=0, the right-hand side of equation (3.11) coincides\nwith\nλ(ρ⊗1)J[E] + (1−λ)J[E](ρ⊗1) (3.13)\nfor all λ∈C, which is also well-defined for any quantum process (ρ,E). However when\n[(ρ⊗1),J[E]]̸=0 and λ̸=1 the expression (3.13) is distinct from (ρ⊗1)J[E], thus in the\n13quantum domain, (3.13) yields a parametric family of states over time associated with a general\nquantum process (ρ,E)which generalizes the classical state over time!\nE(ρ). Another crucial\npoint, is that while the multi-matrix (3.13) is of unit trace for all λ∈C, in the quantum domain\nit is rarely positive (even for λ∈R), and is only guaranteed to be self-adjoint for λ=1/2. As\nsuch, when generalizing states over time to the quantum domain, we will relax the positivity\ncondition and only require states over time to be self-adjoint elements of unit trace. In fact, as\nfirst pointed out in [3], it is the negative eigenvalues of a quantum state over time which act as\na witness to causal correlations which exist in the associated process, and are a feature of the\ntheory rather than a defect.\n4 In quantum bloom\nThe classical bloom map!\nEfrom Proposition 3.2 may be viewed as the output of a mapping\nthat associates every pair (X,Y)of finite sets with a function!\n(∗):CPTP (CX,CY)− →CPTP (CX,CX⊗CY) (4.1)\nsuch that tr X◦!\nE=Eand tr Y◦!\nE=idCXfor allE∈CPTP (CX,CY). Moreover, the bloom map!\n(∗)naturally extends to a map on all of Hom (CX,CY), which together with the partial trace pro-\nvides a factorization system on Hom (CX,CY)which yields the state over time!\nE(ρ)associated\nwith every classical process (ρ,E)∈P(CX,CY). In this section, we extend the classical bloom\nmap (4.1) to the quantum domain. Though there are various such extensions, the extension\nreferred to as the symmetric bloom yields a state over time which is always self-adjoint, and will\nbe the focus of our work.\nDefinition 4.2. Abloom map associates every pair (A,B)of multi-matrix algebras with a map!:TP(A,B)− →TP(A,A⊗B)such that tr A◦!(E) =Eand tr B◦!(E) =idAfor allE∈TP(A,B).\nDefinition 4.3. If(ρ,E)∈P(A,B)is a quantum process, and!is a bloom map, then!(E)(ρ)\nwill be referred to as the state over time associated with the process (ρ,E)and the bloom map!, and will be denoted by!(ρ,E). The map on quantum processes given by\n(ρ,E)7− →!(ρ,E)\nwill then be referred to as the state over time function associated with!.\nDefinition 4.4. Given a bloom map!, the factorization E=trA◦!(E)will be referred to as the\nbloom-shriek factorization ofE∈TP(A,B).\nRemark 4.5. An analogue of bloom-shriek factorization for free gs-monoidal categories appears\nin [4] under the name bloom-circuitry factorization .\nExample 4.6. Let(A,B)be a pair of multi-matrix algebras, and let!:TP(A,B)− →TP(A,A⊗B)\n14be the map given by!(E)(ρ) = (ρ⊗1)J[E]. When Ais a matrix algebra it was shown in [19]\nthatµ∗\nA(1) =P\ni,jEij⊗Eji, where Eijare the matrix units in A. We then have!(E)(ρ) = ( ρ⊗1)J[E] = (ρ⊗1)(idA⊗E)(µ∗\nA(1))\n= (ρ⊗1)(idA⊗E)\nX\ni,jEij⊗Eji\n= (ρ⊗1)\nX\ni,jEij⊗E(Eji)\n\n=X\ni,jρEij⊗E(Eji),\nfrom which it follows that\ntrA(!(E)(ρ))=X\ni,jtr(ρEij)E(Eji) =X\ni,jρjiE(Eji) =E(ρ),\nand\ntrB(!(E)(ρ))=X\ni,jtr(E(Eji))ρEij=X\ni,jδjiρEij=X\niρEii=ρ,\nthus!defines a bloom map in this case. When A=L\nx∈XMmxandB=L\ny∈YMnyare both\nmulti-matrix algebras, then\nρ=M\nx∈Xρx&J[E] =M\n(x,y)∈X×YJ[Eyx],\nwhereEyxare the component functions of E∈Hom(A,B). We then have!(E)(ρ) = (ρ⊗1)J[E] =M\n(x,y)∈X×Y(ρx⊗1)J[Eyx],\nthus the above arguments in the case when Ais a matrix algebra may be applied component-\nwise to deduce tr A◦!(E) =Eand tr B◦!(E) = idA, thus!defines a bloom map. Moreover, it\nfollows from Proposition 3.10 that this bloom map recovers the classical state over time!\nE(ρ)\nassociated with classical processes (ρ,E)∈P(CX,CY).\nIn light of the previous example, the following definition provides three examples of bloom\nmaps.\nDefinition 4.7. Let(A,B)be a pair of multi-matrix algebras.\n• The right bloom is the map!\nR:TP(A,B)− →TP(A,A⊗B)given by E7− →!\nR(E), where!\nR(E)(ρ) = (ρ⊗1)J[E]\nfor all ρ∈A.\n• The left bloom is the map!\nL:TP(A,B)− →TP(A,A⊗B)given by E7− →!\nL(E), where!\nL(E)(ρ) =J[E](ρ⊗1)\nfor all ρ∈A.\n15• The symmetric bloom is the map Y:TP(A,B)− → TP(A,A⊗B)given by E7− →Y(E),\nwhere\nY(E)(ρ) =1\n2((ρ⊗1)J[E] +J[E](ρ⊗1))\nfor all ρ∈A.\nRemark 4.8. The symmetric bloom was introduced in [5], where it was proved that the associ-\nated state over time function (ρ,E)7− →Y(ρ,E) :=Y(E)(ρ)satisfies a list of axioms set forth in\n[6]. The list of axioms include hermiticity of Y(ρ,E), bilinearity of Y(∗,∗), a classical limit ax-\niom, and an associativity axiom which ensures that Yextends in a well-defined way to n-step\nprocesses (ρ,E1, ...,En). In a later work we will prove that the symmetric bloom is in fact the\nonly state over time function satisfying the aforementioned list of axioms.\nThe following statement yields an important property of the symmetric bloom which will\nbe useful for our purposes.\nProposition 4.9. The(A,B)be a pair of multi-matrix algebras, and let E∈TP(A,B). Then the\nfollowing statements are equivalent.\ni.Eis†-preserving.\nii.J[E]is self-adjoint.\niii.Y(E)is†-preserving.\nProof. We prove the statement in the case of matrix algebras.\ni=⇒ii: Letµ:A⊗A− →Adenote the multiplication map. It follows from [19] that\nµ∗(1) =X\ni,jEij⊗Eji,\nthus\nJ[E] =(idA⊗E)µ∗(1) =(idA⊗E)\nX\ni,jEij⊗Eji\n=X\ni,jEij⊗E(Eji).\nWe then have\nJ[E]†=\nX\ni,jEij⊗E(Eji)\n†\n=X\ni,jE†\nij⊗E(Eji)†\n=X\ni,jEji⊗E(E†\nji) =X\ni,jEji⊗E(Eij)\n=J[E],\nas desired.\n16ii=⇒iii: Letρ∈A. Then\nY(E)(ρ)†=1\n2((ρ⊗1)J[E] +J[E](ρ⊗1))†\n=1\n2\u0010\nJ[E]†(ρ⊗1)†+ (ρ⊗1)†J[E]†\u0011\n=1\n2\u0010\nJ[E](ρ†⊗1) + (ρ†⊗1)J[E]\u0011\n=Y(E)(ρ†),\nthusY(E)is†-preserving.\niii=⇒i: Since\ntrA(A⊗B)†=(tr(A)B)†=tr(A)B†=tr(A†)B†=trA(A†⊗B†) =trA\u0010\n(A⊗B)†\u0011\n,\nit follows that tr Ais†-preserving. For all ρ∈Awe then have\nE(ρ)†= (trA◦Y(E))(ρ)†=trA(Y(E)(ρ))†=trA\u0010\nY(E)(ρ)†\u0011\n=trA\u0010\nY(E)(ρ†)\u0011\n=E(ρ†),\nas desired. ■\nDefinition 4.10. A bloom map!is said to be\n•classically reducible if and only if [(ρ⊗1),J[E]]=0=⇒!(E)(ρ) = (ρ⊗1)J[E].\n•hermitian if and only if!(E)is†-preserving whenever Eis†-preserving.\nRemark 4.11. By Proposition 4.9 it follows that the symmetric bloom is hermitian, and it follows\ndirectly from its definition that the symmetric bloom is classically reducible. These are two key\nproperties one should expect from a quantum bloom map if it is to be viewed as generalization\nof the classical bloom. While we have reason to believe that the symmetric bloom is the only\nbloom map which is both hermitian and classically reducible, we have yet to prove such a\nresult.\nWe now generalize Proposition 3.8 using the symmetric bloom. In particular, given a\npseudo-density operator on τ∈A⊗B, we derive an explicit formula for a process (ρ,E)such\nthatτ=Y(ρ,E). For this, we recall that T(A⊗B)denotes the set of pseudo-density operators\nwith respect to the tensor factorization A0=A⊗B(see Definition 2.20).\nLemma 4.12. Letτ∈T(A⊗B)be such that trB(τ)is invertible. Then there exists a unique element\nXτ∈A⊗Bsuch that\n(trB(τ)⊗1)Xτ+Xτ(trB(τ)⊗1) =2τ. (4.13)\nProof. We prove the statement in the case of matrix algebras, where we will use the fact from\nlinear algebra that if A,B,C∈Mn, then the Sylvester equation AX+XB=Chas a unique\nsolution X∈Mnif and only if Aand−Bhave disjoint spectrums. So let τ∈T(A⊗B)be such\n17that tr B(τ)is invertible. Since tr B(τ)is a state which is invertible, it follows that the eigenvalues\nof trB(τ)are strictly positive, thus the spectrums of tr B(τ)⊗1and−(trB(τ)⊗1)are disjoint. It\nthen follows that the Sylvester equation\n(trB(τ)⊗1)X+X(trB(τ)⊗1) =2τ\nhas a unique solution Xτ∈A⊗B, thus proving the statement. ■\nNotation 4.14. Given a pair (A,B)of multi-matrix algebras, we let\nT∗(A⊗B)⊂TA\n+(A⊗B)⊂T(A⊗B) (4.15)\ndenote the subsets given by\nTA\n+(A⊗B) ={τ∈T(A⊗B)|trB(τ)is invertible },\nand\nT∗(A⊗B) =\nτ∈TA\n+(A⊗B)|J−1(Xτ)∈CPTP (A,B)\u000b\n,\nwhere Xτ∈A⊗Bis the unique element satisfying (4.13).\nLemma 4.16. Let(A,B)be a pair of multi-matrix algebras. Then Y(ρ,E)∈T∗(A⊗B)for all (ρ,E)∈\nP+(A,B).\nProof. Let(ρ,E)∈P+(A,B)and let τ=Y(ρ,E). To show τ∈T∗(A⊗B)we need to show that\nτis a pseudo-density operator such that tr B(τ)is invertible and J−1(Xτ)is CPTP . To show τis\na pseuso-density operator we first show τis self-adjoint. Now since Eis CPTP it is †-preserving\nthusY(E)is†-preserving by Proposition 4.9. We then have\nτ†=Y(ρ,E)†=Y(E)(ρ)†=Y(E)(ρ†) =Y(E)(ρ) =τ,\nthus τis self-adjoint. And since the marginals of τareρandE(ρ)with ρinvertible (since\n(ρ,E)∈P+(A,B)), it follows that τ∈TA\n+(A⊗B). Moreover, since ρ=trB(τ)and\nτ=1\n2((ρ⊗1)J[E] +J[E](ρ⊗1)),\nit follows that Xτ=J[E]andJ−1(Xτ) =E∈CPTP (A,B), thus τ∈T∗(A⊗B), as desired. ■\nTheorem 4.17. Let(A,B)be a pair of multi-matrix algebras, and let S :P+(A,B)− →T∗(A⊗B)\nbe the map given by S(ρ,E) =Y(ρ,E). Then Sis a bijection, whose inverse is given by\nS−1(τ) =\u0010\ntrB(τ),J−1(Xτ)\u0011\n, (4.18)\nwhere Xτ∈A⊗Bis the unique element satisfying (4.13) .\n18Proof. LetC :T∗(A⊗B)− →P+(A,B)be the map given by C(τ) = ( trB(τ),J−1(Xτ)). We now\nshow C◦S = id and S◦C = id. So let (ρ,E)∈P+(A,B)and let τ= S(ρ,E), so that ρ=trB(τ)\nand\nτ=1\n2((ρ⊗1)J[E] +J[E](ρ⊗1))=⇒J[E] =Xτ=⇒E=J−1(Xτ).\nWe then have\n(C◦S)(ρ,E) = C( τ) = ( trB(τ),J−1(Xτ)) = ( ρ,E),\nthusC◦S = id. Now let τ∈T∗(A⊗B), so that\n1\n2(trB(τ)⊗1)Xτ+Xτ(trB(τ)⊗1)=τ.\nWe then have\n(S◦C)(τ) = S (C(τ))\n= S( trB(τ),J−1(Xτ))\n=1\n2\u0010\n(trB(τ)⊗1)J[J−1(Xτ)] +J[J−1(Xτ)](trB(τ)⊗1)\u0011\n=1\n2(trB(τ)⊗1)Xτ+Xτ(trB(τ)⊗1)\n=τ,\nthusS◦C = id, as desired. ■\n5 Blooming 2-chains\nOur next goal is to extend bloom maps to n-chains, but first we need to consider the case\nn=2. In this section we will show is that if a bloom map satisfies an associativity condition\non 2-chains, then it naturally extends to a unique bloom map on 2-chains. We then show that\nthe right bloom, left bloom and symmetric bloom are all associative, and as such, yield well-\ndefined state over time functions for 2-step processes. In such a case we also prove an explicit\nformula for the symmetric bloom state over time function, and show that is satisfies a certain\ncompositionality property. This will provide us with the mathematical foundation for using\nbloom maps to define explicit states over time associated with the dynamics of n-step quantum\nprocesses for all n > 0.\nDefinition 5.1. Given a triple (A,B,C)of multi-matrix algebras, a 2 -chain consists of a pair\n(E,F)∈TP(A,B)×TP(B,C). The set of all such 2-chains will be denoted by TP(A,B,C).\nExample 5.2 (Blooming 2-chains) .LetAE− →BF− →Cbe a 2-chain, let!be a bloom map, and\nconsider the following diagram.\nA B CA⊗B B ⊗C\nE//\nF//\n!DD\ntr\n\u001a\u001a\n!DD\ntr\n\u001a\u001a(5.3)\n19Using only maps in the above diagram, there are two maps one may construct in TP(A,A⊗B⊗\nC), namely,!(!(F)◦E)and!(F◦tr)◦!(E). As the map!(!(F)◦E)has codomain A⊗(B⊗C)\nand the map!(F◦tr)◦!(E)has codomain (A⊗B)⊗C, there is a bijection between 2-blooms\nconstructed from the above diagram and parenthezations of the multi-matrix algebra A⊗B⊗C.\nIt is then a natural to question whether or not the two maps!(!(F)◦E)and!(F◦tr)◦!(E)are\nequal. If the two maps are in fact equal, then the bloom map!naturally extends to a unique\nbloom map on 2-chains. Such considerations then motivate the following definition.\nDefinition 5.4. A bloom map!is said to be associative if and only if for every 2-chain\nAE− →BF− →C\nwe have!(!(F)◦E)=!(F◦tr)◦!(E). (5.5)\nIt turns out that the right bloom, left bloom and symmetric bloom are indeed associative,\nbut before giving the proof, we will first prove the following lemma, which will also be useful\nlater on.\nLemma 5.6. Let(A,B,C)be a triple of matrix algebras, and let (E,F)∈TP(A,B,C)be a 2-chain.\nThen the following statements hold.\ni.J[!\nR(F)◦E] =(J[E]⊗1) (1⊗J[F])\nii.J[!\nL(F)◦E] =(1⊗J[F]) (J[E]⊗1)\niii.J[F◦tr] =1⊗J[F]\niv.!\nL(!\nR(F)◦E) =!\nR(F◦tr)◦!\nL(E)\nv.!\nR(!\nL(F)◦E) =!\nL(F◦tr)◦!\nR(E)\nProof. Item i : Indeed, for all ρ∈Awe have\nJ[!\nR(F)◦E] =X\ni,jEA\nij⊗(!\nR(F)◦E)(EA\nji)\n=X\ni,jEA\nij⊗!\nR(F)\u0010\nE(EA\nji)\u0011\n=X\ni,jEA\nij⊗\u0010\n(E(EA\nji)⊗1)J[F]\u0011\n=X\ni,jEA\nij⊗\n(E(EA\nji)⊗1)\nX\nk,lEB\nkl⊗F(EB\nlk)\n\n\n=X\ni,j,k,lEA\nij⊗E(EA\nji)EB\nkl⊗F(EB\nlk)\n=\nX\ni,jEA\nij⊗E(EA\nji)⊗1\n\nX\nk,l1⊗EB\nkl⊗F(EB\nlk)\n\n=(J[E]⊗1) (1⊗J[F]),\n20as desired.\nItem ii : The proof is similar to that of item i.\nItem iii : Indeed,\nJ[F◦tr] =X\nr,s,k,l\u0010\nEA\nrs⊗EB\nkl\u0011\n⊗\u0010\n(F◦tr)(EA\nsr⊗EB\nlk)\u0011\n=X\nr,s,k,l\u0010\nEA\nrs⊗EB\nkl\u0011\n⊗\u0010\nδsrF(EB\nlk)\u0011\n=X\nr,k,l\u0010\nEA\nrr⊗EB\nkl\u0011\n⊗F(EB\nlk)\n=X\nk,l\u0010\n1⊗EB\nkl\u0011\n⊗F(EB\nlk)\n=1⊗J[F],\nas desired.\nItem iv : Indeed, for all ρ∈Awe have\n(!\nR(F◦tr)◦!\nL(E))(ρ) =!\nR(F◦tr)((!\nL(E)(ρ))\n=(!\nL(E)(ρ)⊗1)J[F◦tr]\n=(J[E]⊗1) (ρ⊗1⊗1) (1⊗J[F])\n=(J[E]⊗1) (1⊗J[F]) (ρ⊗1⊗1)\n=J[!\nR(F)◦E](ρ⊗1)\n=!\nL(!\nR(F)◦E)(ρ),\nwhere the third and fifth equalities follows from items iii and i. It then follows that!\nL(!\nR(F)◦\nE) =!\nR(F◦tr)◦!\nL(E), as desired.\nItem v : The proof is similar to that of item iv. ■\nProposition 5.7. The right bloom, left bloom and symmetric bloom are all associative.\nProof. Right bloom : Indeed, for all ρ∈Awe have\n(!\nR(F◦tr)◦!\nR(E))(ρ) =!\nR(F◦tr)((!\nR(E)(ρ))\n=(!\nR(E)(ρ)⊗1)J[F◦tr]\n= (ρ⊗1)(J[E]⊗1) (1⊗J[F])\n= (ρ⊗1)J[!\nR(F)◦E]\n=!\nR(!\nR(F)◦E)(ρ),\nwhere the third and fourth equalities follow from items iii and i of Lemma 5.6. It then follows\nthat!\nR(!\nR(F)◦E)=!\nR(F◦tr)◦!\nR(E), as desired.\nLeft bloom : The proof is similar to that for the right bloom.\n21Symmetric bloom : We first compute\nY(Y(F)◦E) =1\n2\u0012!\nR\u00121\n2(!\nR(F) +!\nL(F))◦E\u0013\n+!\nL\u00121\n2(!\nR(F) +!\nL(F))◦E\u0013\u0013\n=1\n4(!\nR(!\nR(F)◦E)+!\nL(!\nL(F)◦E)+!\nR(!\nL(F)◦E)+!\nL(!\nR(F)◦E)).\nNow let ℵ=Y(F◦tr)◦Y(E). Then\nℵ=Y(F◦tr)◦Y(E)\n=1\n2(!\nR(F◦tr) +!\nL(F◦tr))◦1\n2(!\nR(E) +!\nL(E))\n=1\n4(!\nR(F◦tr)◦!\nR(E) +!\nL(F◦tr)◦!\nL(E) +!\nR(F◦tr)◦!\nL(E) +!\nL(F◦tr)◦!\nR(E))\n=1\n4(!\nR(!\nR(F)◦E)+!\nL(!\nL(F)◦E)+!\nR(!\nL(F)◦E)+!\nL(!\nR(F)◦E))\n=Y(Y(F)◦E),\nwhere the second-to-last equality follows from the associativity of!\nR, the associativity of!\nL, and\nitems v and iv of Lemma 5.6. ■\nAssociative bloom maps yield well-defined state over time functions on 2-step processes,\nas we now show.\nDefinition 5.8. Given a triple (A,B,C)of multi-matrix algebras, an element (ρ,E,F)∈S(A)×\nCPTP (A,B)×CPTP (B,C)will be referred to as a 2-step process , and the set of 2-step processes\nS(A)×CPTP (A,B)×CPTP (B,C)will be denoted by P(A,B,C). The subset of P(A,B,C)\nconsisting of processes (ρ,E,F)with ρinvertible will be denoted by P+(A,B,C).\nDefinition 5.9. Suppose!is associative. If (ρ,E,F)∈P(A,B,C)is a 2-step process, then!(!(F)◦E)(ρ) =(!(F◦tr)◦!(E))(ρ)∈A⊗B⊗C\nwill be referred to as the state over time associated with the 2-step process (ρ,E,F)and the\nbloom map!, and will be denoted by!(ρ,E,F). The map on 2-step quantum processes given\nby\n(ρ,E,F)7− →!(ρ,E,F)\nwill then be referred to as the 2-step state over time function associated with!.\nThe next result shows that the state over time!(ρ,E,F)has the expected marginals.\nProposition 5.10. Suppose!is associative, and let (ρ,E,F)∈P(A,B,C)be a 2-step process. Then\nthe following statements hold.\ni.trC(!(ρ,E,F))=!(ρ,E)\nii.trA(!(ρ,E,F))=!(E(ρ),F)\n22iii.trB⊗C(!(ρ,E,F))=ρ\niv.trA⊗C(!(ρ,E,F))=E(ρ)\nv.trA⊗B(!(ρ,E,F))=F(E(ρ))\nProof. Associativity is used in the proofs by identifying!(ρ,E,F)with either (!(F◦tr)◦!(E))(ρ)\nor!(!(F)◦E)(ρ).\nItem i :\ntrC(!(ρ,E,F))=trC(!(F◦tr)(!(E)(ρ)))\n=!(E)(ρ)\n=!(ρ,E),\nas desired.\nItem ii :\ntrA(!(ρ,E,F))=trA(!(!(F)◦E)(ρ))\n= (!(F)◦E)(ρ)\n=!(F)(E(ρ))\n=!(E(ρ),F),\nas desired.\nItem iii : Indeed,\ntrB⊗C(!(ρ,E,F))=trB(trC(!(ρ,E,F)))\n=trB(!(ρ,E))\n=ρ,\nwhere the second equality follows from item i.\nItem iv : Indeed,\ntrA⊗C(!(ρ,E,F))=trA(trC(!(ρ,E,F)))\n=trA(!(ρ,E))\n=E(ρ),\nwhere the second equality follows from item i.\nItem v : Indeed,\ntrA⊗B(!(ρ,E,F))=trB(trA(!(ρ,E,F)))\n=trB(!(E(ρ),F))\n= (F◦E)(ρ),\nwhere the second equality follows from item ii. ■\n23We now derive a formula for the symmetric bloom state over time Y(ρ,E,F)in terms of ρ,\nJ[E]andJ[F].\nDefinition 5.11. LetAbe a multi-matrix algebra. The normalized Jordan product is the map\nJor:A×A− →Agiven by\nJor(A,B)=1\n2(AB+BA).\nLemma 5.12. Let(E,F)∈TP(A,B,C)be a 2-chain. Then\nJ[Y(F)◦E] =Jor(J[E]⊗1,1⊗J[F]).\nProof. Indeed,\nJ[Y(F)◦E] =J\u00141\n2(!\nR(F) +!\nL(F))◦E\u0015\n=1\n2J[!\nR(F)◦E+!\nL(F)◦E]\n=1\n2(J[!\nR(F)◦E] +J[!\nL(F)◦E])\n=1\n2((J[E]⊗1) (1⊗J[F])+(1⊗J[F]) (J[E]⊗1))\n=Jor(J[E]⊗1,1⊗J[F]),\nwhere the second to last equation follows from items i and ii of Lemma 5.6. ■\nProposition 5.13. Let(ρ,E,F)∈P(A,B,C)be a 2-step process. Then\nY(ρ,E,F) =Jor(ρ⊗1,Jor(J[E]⊗1,1⊗J[F])). (5.14)\nProof. Indeed,\nJ[Y(F)◦E]) =J\u00141\n2(!\nR(F) +!\nL(F))◦E\u0015\n=1\n2(J[!\nR(F)◦E] +!\nL(F)◦E])\n=1\n2((J[E]⊗1)(1⊗J[F]) + (1⊗J[F])(J[E]⊗1))\n=Jor(J[E]⊗1,1⊗J[F]),\nwhere the third equality follows from items iv and v Lemma 5.6. We then have\nY(ρ,E,F) =Y(Y(F)◦E)(ρ)\n=Jor(ρ⊗1,J[Y(F)◦E])\n=Jor(ρ⊗1,Jor(J[E]⊗1,1⊗J[F])),\nas desired. ■\n24To conclude the section, we prove a result which will be useful later on.\nProposition 5.15 (Compositionality) .Let(ρ,E,F)∈P(A,B,C)be a two-step process. Then\ntrB(Y(ρ,E,F))=Y(ρ,F◦E). (5.16)\nProof. It follows by direct computation that\nJor(J[E]⊗1,1⊗J[F])=X\ni,j,k,lEA\nij⊗Jor\u0010\nE(EA\nij),EB\nkl\u0011\n⊗F(EB\nlk),\nthus\ntrB((ρ⊗1)Jor(J[E]⊗1,1⊗J[F]))=trB\nX\ni,j,k,lρEA\nij⊗Jor\u0010\nE(EA\nij),EB\nkl\u0011\n⊗F(EB\nlk)\n\n=X\ni,j,k,lρEA\nij⊗\u0010\nE(EA\nij)EB\nklF(EB\nlk)\u0011\n=X\ni,j,k,lρEA\nij⊗\u0010\nE(EA\nij)lkF(EB\nlk)\u0011\n= (ρ⊗1)X\ni,jEA\nij⊗(F◦E)(EA\nji)\n= (ρ⊗1)J[F◦E].\nSimilarly, we have\ntrB(Jor(J[E]⊗1,1⊗J[F])(ρ⊗1))=J[F◦E](ρ⊗1),\nthus\ntrB(Y(ρ,E,F))(5.14)= trB(Jor(ρ⊗1,Jor(J[E]⊗1,1⊗J[F])))\n= Jor(ρ⊗1,J[F◦E])\n=Y(ρ,F◦E),\nas desired. ■\n6 Blooming n-chains\nIn this section we show how associative bloom maps naturally extend to bloom maps for arbi-\ntrary n-chains.\nDefinition 6.1. Given an (n+1)-tuple (A0, ...,An)of multi-matrix algebras, an n-chain consists\nof an-tuple\n(E1, ...,En)∈TP(A0,A1)× ··· × TP(An−1,An).\nThe set of all such n-chains will be denoted by TP(A0, ...,An).\n25Definition 6.2. Ann-bloom associates every (n+1)-tuple (A0, ...,An)of multi-matrix algebras\nwith a map!:TP(A0, ...,An)− →TP(A0,A0⊗ ··· ⊗ An)such that\ntri◦!(E1, ...,En) =\u000e\nEi◦ ··· ◦E1 fori∈{1, ...,n}\nidA0fori=0(6.3)\nfor all (E1, ...,En)∈TP(A0, ...,An).\nExample 6.4 (Blooming 3-chains) .Let!be a bloom map, and let\nA0E1− →A1E2− →A2E3− →A3\nbe a 3-chain, which after incorporating bloom-shriek factorizations yields the following dia-\ngram:\nA0 A1 A2 A3A0⊗A1 A1⊗A2 A2⊗A3\nE1//\nE2//\nE2//\n!DD\ntr\n\u001a\u001a\n!DD\ntr\n\u001a\u001a\n!DD\ntr\n\u001a\u001a(6.5)\nSimilar to the case of 2-chains, there are then five 3-blooms in TP(A0,A0⊗A1⊗A2⊗A3)one\nmay construct from the above diagram (6.5), each of which corresponds to a parenthezation of\nA0⊗A1⊗A2⊗A3.\nFor example, the 3-bloom associated with the parenthezation A0⊗(A1⊗(A2⊗A3))is con-\nstructed as follows. We first associate a map with what is inside the outermost parantheses of\nA0⊗(A1⊗(A2⊗A3)), namely, A1⊗(A2⊗A3), which is a parenthezation of the multi-matrix\nalgebra A1⊗A2⊗A3. As we’ve already seen in the case of 2-chains, the 2-bloom associated\nwithA1⊗(A2⊗A3)is then!(!(E3)◦E2):A1− →A1⊗(A2⊗A3),\nand then post-composing with E1yields!(!(E3)◦E2)◦E1:A0− →A1⊗(A2⊗A3),\nso that taking a final bloom yields!(!(!(E3)◦E2)◦E1):A0− →A0⊗(A1⊗(A2⊗A3)),\nwhich is the desired 3-bloom. The 3-blooms associated with the other 4 parenthezations of A0⊗\nA1⊗A2⊗A3may then be obtained by considering the pentagon where each vertex corresponds\nto one of the 5 parenthezations of A0⊗A1⊗A2⊗A3, and a directed edge is drawn between two\nvertices if and only if they are related by a single application of the associator transformation\n26A⊗(B⊗C)7− →(A⊗B)⊗C:\nA0⊗(A1⊗(A2⊗A3))\n(A0⊗A1)⊗(A2⊗A3)\n((A0⊗A1)⊗A2)⊗A3 (A0⊗(A1⊗A2))⊗A3A0⊗((A1⊗A2)⊗A3)vv\n\u001a\u001a\noo\u0004\u0004((\nThe associated 3-blooms for the other 4 parenthezations of A0⊗A1⊗A2⊗A3can then be ob-\ntained from!(!(!(E3)◦E2)◦E1)by applying corresponding associator transformations of maps!(!(F)◦E)7− →!(F◦tr)◦!(E)along the pentagon:!(!(!(E3)◦E2)◦E1)!(!(E3)◦E2◦tr)◦!(E1)!(E3◦tr)◦!(E2◦tr)◦!(E1)!(E3◦tr)◦!(!(E2)◦E1)\n!(!(E3◦tr)◦!(E2)◦E1)vv\n\u001a\u001a\noo\u0004\u0004((\nThis procedure then yields the following 3-blooms associated with each parenthezation of\nA0⊗A1⊗A2⊗A3:\ni.A0⊗(A1⊗(A2⊗A3)):!(!(!(E3)◦E2)◦E1)\nii.(A0⊗A1)⊗(A2⊗A3):!(!(E3)◦E2◦tr)◦!(E1)\niii.((A0⊗A1)⊗A2)⊗A3:!(E3◦tr)◦!(E2◦tr)◦!(E1)\niv.A0⊗((A1⊗A2)⊗A3):!(!(E3◦tr)◦!(E2)◦E1)\nv.(A0⊗(A1⊗A2))⊗A3:!(E3◦tr)◦!(!(E2)◦E1)\nIf!is assumed to be associative, then this implies that these five 3-blooms are all equal:\n27i=ii: The statement follows from (5.5) with E=E1andF=!(E3)◦E2.\nii=iii: The statement follows from (5.5) with E=E2◦tr andF=E3.\ni=iv: The statement follows from (5.5) with E=E2andF=E3.\niv=v: The statement follows from (5.5) with E=!(E2)◦E1andF=E3◦tr.\nRemark 6.6. Note that when showing iv =v we have set tr ◦tr=tr as we are eliding the domains\nand codomains of the partial trace, which we will continue to do throughout.\nWe now generalize the previous two examples to n-chains for all n > 1. Let!be a bloom\nmap, let n > 1, and let\nA0E1− →A1− → ··· − → An−1En− →An (6.7)\nbe an n-chain. With every parenthezation ϑ(A0, ...,An)of the multi-matrix algebra A0⊗···⊗ An\nwe associate a map!\nϑ(E1, ...,En) :A0− →ϑ(A0, ...,An)\nconstructed from the bloom map!, the maps Eifori=1, ...,nand the partial trace tr. For the\ncasen=2, we know from Example 5.2 that there are two parenthezations of A0⊗A1⊗A2,\nnamely,\nϑ(A0,A1,A2) = (A0⊗A1)⊗A3&eϑ(A0,A1,A2) =A0⊗(A1⊗A2).\nand we let!\nϑ(E1,E2) =!(E2◦tr)◦!(E1)&!\neϑ(E1,E2) =!(!(E2)◦E1).\nIn such a case we have that!\nϑ(E1,E2)may be obtained from!\neϑ(E1,E2)by a transformation of the\nform!(!(F)◦E)7− →!(F◦tr)◦!(E)\nwithF=E2andE=E1. For general n > 2 we will build up recursively from this case.\nSo now let n > 2 and let ϑ(A0, ...,An)be an arbitrary parenthezation of A0⊗ ··· ⊗ An. It\nthen follows that ϑ(A0, ...,An)is in one of the three forms:\nI:ϑ(A0, ...,An) = A0⊗(ω(A1, ...,An))\nII:ϑ(A0, ...,An) = (ω(A0, ...,An−1))⊗An\nIII:ϑ(A0, ...,An) = (ω(A0, ...,Ak−1))⊗(χ(Ak, ...,An))\nwhere 1 < k < n andω(A1, ...,An),ω(A0, ...,An−1),ω(A0, ...,Ak−1)andχ(Ak, ...,An)are the\nparenthezations of the multi-matrix algebras A1⊗ ··· ⊗ An,A0⊗ ··· ⊗ An−1,A0⊗ ··· ⊗ Ak−1\nandAk⊗ ··· ⊗ Anfor which the above equations hold. In each of the three cases we define the\nassociated n-blooms recursively as follows.\nI:!\nϑ(E1, ...,En) =!(!\nω(E2, ...,En)◦E1)\nII:!\nϑ(E1, ...,En) =!(En◦tr)◦!\nω(E1, ...,En−1)\nIII:!\nϑ(E1, ...,En) =!(!\nχ(Ek+1, ...,En)◦Ek◦tr)◦!\nω(E1, ...,Ek−1)\nThe next proposition shows that!\nϑ, when viewed as a map on n-chains, is an actual n-bloom,\ni.e., that!\nϑsatisfies the condition (6.3) in Definition 6.2.\n28Proposition 6.8. Letn > 1, let (A0, ...,An)be an (n+1)-tuple of multi-matrix algebras, let\nϑ(A0, ...,An)be a parenthezation of A0⊗ ··· ⊗ An, and let!be a bloom map. Then!\nϑis an n-bloom,\ni.e.,\ntri◦!\nϑ(E1, ...,En) =\u000e\nEi◦ ··· ◦E1 fori∈{1, ...,n}\nidA0fori=0(6.9)\nfor all (E1, ...,En)∈TP(A0, ...,An).\nProof. We use induction on n. For n=2 the statement follows from Proposition 5.10. Now\nsuppose the result holds for n=m−1>2, and let ϑ(A0, ...,Am)be a parenthezation of A0⊗\n··· ⊗Am. We then consider the three cases as for ϑ(A0, ...,Am)as defined above.\nCase I: In this case we have ϑ(A0, ...,Am) =A0⊗(ω(A1, ...,Am))and!\nϑ(E1, ...,Em) =!(!\nω(E2, ...,Em)◦E1)\nfor all m-chains (E1, ...,Em). For i=0 we then have\ntr0◦!\nϑ(E1, ...,En) =tr0◦!(!\nω(E2, ...,Em)◦E1)=idA0,\nwhere the last equality follows from the definition of a bloom map. As for 0 < i⩽m, we have\ntri◦!\nϑ(E1, ...,En) = trA0⊗···⊗cAi⊗···⊗Am◦!(!\nω(E2, ...,Em)◦E1)\n= trA1⊗···⊗cAi⊗···⊗Am◦trA0◦!(!\nω(E2, ...,Em)◦E1)\n= trA1⊗···⊗cAi⊗···⊗Am◦!\nω(E2, ...,Em)◦E1\ninduction= ( Ei◦ ··· ◦E2)◦E1\n=Ei◦ ··· ◦E1,\nthus!\nϑis an m-bloom.\nCase II: In this case we have ϑ(A0, ...,Am) =(ω(A0, ...,Am−1))⊗Amand!\nϑ(E1, ...,Em) =!(Em◦tr)◦!\nω(E1, ...,Em−1)\nfor all m-chains (E1, ...,Em), where tr =trA0⊗···⊗Am−2:A0⊗ ··· ⊗ Am−1− →Am−1. For i=mwe\nthen have\ntrm◦!\nϑ(E1, ...,En) = trA0⊗···⊗Am−1◦!(Em◦tr)◦!\nω(E1, ...,Em−1)\n=Em◦(tr◦!\nω(E1, ...,Em−1))\ninduction=Em◦(Em−1◦ ··· ◦E1)\n=Em◦ ··· ◦E1.\n29As for 0 ⩽i < m , we have\ntri◦!\nϑ(E1, ...,En) = trA0⊗···⊗cAi⊗···⊗Am−1◦(trAm◦!(Em◦tr))◦!\nω(E1, ...,Em−1)\n= trA0⊗···⊗cAi⊗···⊗Am−1◦idA0⊗···⊗Am−1◦!\nω(E1, ...,Em−1)\n= trA0⊗···⊗cAi⊗···⊗Am−1◦!\nω(E1, ...,Em−1)\ninduction=\u000e\nEi◦ ··· ◦E1 fori∈{1, ...,m−1}\nidA0fori=0,\nthus!\nϑis an m-bloom.\nCase III: In this case we have ϑ(A0, ...,Am) =(ω(A0, ...,Ak−1))⊗(χ(Ak, ...,Am))for some\n1< k < m and!\nϑ(E1, ...,Em) =!(!\nχ(Ek+1, ...,Em)◦Ek◦tr)◦!\nω(E1, ...,Ek−1)\nfor all m-chains (E1, ...,Em), where tr =trA0⊗···⊗Ak−2:A0⊗ ··· ⊗ Ak−1− →Ak−1. Now let\nℵ=tri◦!\nϑ(E1, ...,En). Then for 0 ⩽i < k we have\nℵ = tri◦!\nϑ(E1, ...,En)\n= trA0⊗···⊗cAi⊗···⊗Am◦!(!\nχ(Ek+1, ...,Em)◦Ek◦tr)◦!\nω(E1, ...,Ek−1)\n= trA0⊗···⊗cAi⊗···⊗Ak−1◦\u0000\ntrAk⊗···Am◦!(!\nχ(Ek+1, ...,Em))◦Ek◦tr\u0001\n◦!\nω(E1, ...,Ek−1)\n= trA0⊗···⊗cAi⊗···⊗Ak−1◦idA0⊗···⊗Ak−1◦!\nω(E1, ...,Ek−1)\n= trA0⊗···⊗cAi⊗···⊗Ak−1◦!\nω(E1, ...,Ek−1)\ninduction=\u000e\nEi◦ ··· ◦E1 fori∈{1, ...,k−1}\nidA0fori=0.\nAs for k⩽i⩽m, we have\nℵ = tri◦!\nϑ(E1, ...,En)\n= trA0⊗···⊗cAi⊗···⊗Am◦!(!\nχ(Ek+1, ...,Em)◦Ek◦tr)◦!\nω(E1, ...,Ek−1)\n= trAk⊗···⊗cAi⊗···⊗Am◦\u0000\ntrA0⊗···⊗Ak−1◦!(!\nχ(Ek+1, ...,Em)◦Ek◦tr)\u0001\n◦!\nω(E1, ...,Ek−1)\n= trAk⊗···⊗cAi⊗···⊗Am◦(!\nχ(Ek+1, ...,Em)◦Ek◦tr)◦!\nω(E1, ...,Ek−1)\n=\u0010\ntrAk⊗···⊗cAi⊗···⊗Am◦!\nχ(Ek+1, ...,Em)\u0011\n◦Ek◦\u0000\ntrA0⊗···⊗Ak−2◦!\nω(E1, ...,Ek−1)\u0001\ninduction= \u000e\nEi◦ ··· ◦Ek+1 fori∈{k+1, ...,m}\nidAkfori=k!\n◦Ek◦Ek−1◦ ··· ◦E1\n=Ei◦ ··· ◦E1,\nthus!\nϑis an m-bloom, as desired. ■\nRemark 6.10. Ifϑ(A0, ...,An)andeϑ(A0, ...,An)are two parenthezations of A0⊗ ··· ⊗ Ansuch\nthateϑ(A0, ...,An)is obtained from ϑ(A0, ...,An)by an associator transformation of the form\nA⊗(B⊗C)7− →(A⊗B)⊗C, then there exists a 2-chain\nAE− →BF− →C\n30such that!\neϑis obtained from!\nϑby an associator transformation at the level of maps associated\nwith the 2-chain (E,F), i.e.,!\nϑ=!(!(F)◦E)7− →!(F◦tr)◦!(E) =!\neϑ.\nThe previous remark motivates the following definition.\nDefinition 6.11. Letn > 1 and let 0 < k < l ⩽n. The associator relation on the set of maps\nTP(Al−k,Al−k⊗ ··· ⊗ Al)is the subset\nA⊂TP(Al−k,Al−k⊗ ··· ⊗ Al)×TP(Al−k,Al−k⊗ ··· ⊗ Al)\ngiven by (ψ,φ)∈Aif and only if φis obtained from ψfrom successive transformations of the\nform!(!(F)◦E)7− →!(F◦tr)◦!(E).\nIn such a case, we use the notation ψ↘φto denote the fact that (ψ,φ)∈A.\nLemma 6.12. Letk >0. Then!(!(F1)◦ ··· ◦!(Fk)◦E)↘!(F1◦tr)◦ ···!(Fk◦tr)◦!(E).\nProof. We use induction on k. The case k=1 holds by definition of the associator relation, so\nnow assume the result holds for k=m−1. Then!(!(F1)◦ ··· ◦!(Fm−1)◦!(Fm)◦E)induction\n↘!(F1◦tr)◦ ···!(Fm−1◦tr)◦!(!(Fm)◦E)\nassociator\n↘!(F1◦tr)◦ ···!(Fm−1◦tr)◦!(Fm◦tr)◦!(E),\nthus the result holds for k=m, as desired. ■\nProposition 6.13. Letn > 1, and let ϑ(A0, ...,An)be a parenthezation of A0⊗ ··· ⊗ An. Then!\nϑ(E1, ...,En)↘!(En◦tr)◦!(En−1◦tr)◦ ··· ◦!(E2◦tr)◦!(E1).\nProof. We will consider each of the three cases given above for the definition of!\nϑ, and use\nstrong induction on n. The case n=2 follows by definition, so now assume the result holds for\nallmwith 1 < m < n .\nCase I : Indeed,!\nϑ(E1, ...,En) =!(!\nω(E2, ...,En)◦E1)\ninduction\n↘!(!(En◦tr)◦ ··· ◦!(E3���tr)◦!(E2)◦E1)\nLemma 6.12\n↘!(En◦tr)◦!(En−1◦tr)◦ ··· ◦!(E2◦tr)◦!(E1),\nwhere in the last line we note that we have repeatedly used the elision tr ◦tr=tr, thus the\nresult holds.\n31Case II : Indeed,!\nϑ(E1, ...,En) =!(En◦tr)◦!\nω(E1, ...,En−1)\ninduction\n↘!(En◦tr)◦!(En−1◦tr)◦ ··· ◦!(E2◦tr)◦!(E1),\nas desired.\nCase III : Letℵ=!\nϑ(E1, ...,En). Then\nℵ =!\nϑ(E1, ...,En)\n=!(!\nχ(Ek+1, ...,En)◦Ek◦tr)◦!\nω(E1, ...,Ek−1)\ninduction\n↘!(!(En◦tr)◦ ··· ◦!(Ek+1)◦Ek◦tr)◦!(Ek−1◦tr)◦ ··· ◦!(E2◦tr)◦!(E1)\nLemma 6.12\n↘!(En◦tr)◦ ··· ◦!(Ek+1◦tr)◦!(Ek◦tr)◦!(Ek−1◦tr)◦ ··· ◦!(E2◦tr)◦!(E1),\nwhere again in the last line we note that we have repeatedly used the elision tr ◦tr=tr, thus\nthe result holds. ■\nThe following theorem follows directly from Proposition 6.13, and will form the basis for a\ndefinition of states over time associated with n-step processes, as we will see in the next section.\nTheorem 6.14. If!is associative, then!\nϑ(E1, ...,En) =!(En◦tr)◦!(En−1◦tr)◦ ··· ◦!(E2◦tr)◦!(E1)\nfor every parenthezation ϑ(A0, ...,An)ofA0⊗ ··· ⊗ An.\n7 States over time for n-step processes\nIn this section we use bloom maps for n-chains to define states over time for arbitrary finite-step\nquantum processes.\nDefinition 7.1. Given an (n+1)-tuple (A0, ...,An)of multi-matrix algebras, an element\n(ρ,E1, ...,En)∈S(A0)×CPTP (A0,A1)× ··· × CPTP (An−1,An)\nwill be referred to as an n-step process , and the set S(A0)×CPTP (A0,A1)×···× CPTP (An−1,An)\nof all such n-step processes will be denoted by P(A0, ...,An). Given an n-step process\n(ρ,E1, ...,En), we let ρ0=ρandρi= (Ei◦ ··· ◦E1)(ρ)for all i∈{1, ...,n}.\nDefinition 7.2. Ann-step state over time function associates every (n+1)-tuple (A0, . . . ,An)\nof multi-matrix algebras with a map\nψ:P(A0, ...,An)− →T(A0⊗ ··· ⊗ An)\n32such that\ntri(ψ(ρ,E1, ...,En))=ρi, (7.3)\nfor all i∈{0, ...,n}. In such a case, the pseudo-density operator ψ(ρ,E1, ...,En)will be referred\nto as the state over time associated with the n-step process (ρ,E1, ...,En). For n=1 ann-step\nstate over time function will be referred to simply as a state over time function .\nTheorem 7.4. Let!be a bloom map which is hermitian, let n > 0, and let ψbe the map on n-step\nprocesses given by\nψ(ρ,E1, ...,En) = (!(En◦tr)◦ ··· ◦!(E2◦tr)◦!(E1))(ρ). (7.5)\nThen the following statements hold.\ni. The map ψis ann-step state over time function.\nii. If!is also associative and (ρ,E1, ...,En)∈P(A0, ...,An), then ψ(ρ,E1, ...,En) =!\nϑ(E1, ...,En)(ρ)\nfor every parenthezation ϑ(A0, ...,An)ofA0⊗ ··· ⊗ An.\nProof. Item i : Let (A0, ...,An)be an (n+1)-tuple of multi-matrix algebras, let (ρ,E1, ...,En)∈\nP(A0, ...,An)be an n-step process. Then for all i∈{0, ...,n}we have\ntri(ψ(ρ,E1, ...,En))= trA0⊗···⊗cAi⊗···⊗An((!(En◦tr)◦ ··· ◦!(E2◦tr)◦!(E1))(ρ))\n(6.9)=\u000e\n(Ei◦ ··· ◦E1)(ρ) fori∈{1, ...,n}\nidA0(ρ) fori=0\n=ρi,\nthusψsatisfies condition (7.3) of being a state over time function.\nTo conclude the proof, we must show that ψ(ρ,E1, ...,En)is self-adjoint, for which we use\ninduction on n. For n=1 we have ψ(ρ,E1) =!(E1)(ρ), and since E1is CPTP it is †-preserving,\nthus!(E1)is†-preserving by the hermitian assumption on!. We then have!(E1)(ρ)†=!(E1)(ρ†) =!(E1)(ρ),\nwhere the last equality follows from the fact that ρis a state (and so self-adjoint), thus!(E1)(ρ)\nis self-adjoint. Now assume the result holds for n=m−1>1, and let σ= (!(Em−1◦tr)◦ ··· ◦!(E2◦tr)◦!(E1))(ρ), so that\nψ(ρ,E1, ...,Em) =!(Em◦tr)(σ).\nNow since Emand tr are both completely positive, it follows that Em◦tr is†-preserving, thus\nby the hermitian assumption on!it follows that!(Em◦tr)is†-preserving as well. We then have\nψ(ρ,E1, ...,Em)†=!(Em◦tr)(σ)†=!(Em◦tr)(σ†) =!(Em◦tr)(σ) =ψ(ρ,E1, ...,Em),\nwhere the last equality follows from the fact that σis self-adjoint by the inductive hypothesis,\nthusψ(ρ,E1, ...,Em)is self-adjoint.\nItem ii : The statement follows from Theorem 6.14. ■\n338 The Y-function\nSince the symmetric bloom Yis hermitian and associative, it follows from Theorem 7.4 that Y\nextends to a uniquely defined n-step state over time function for all n∈N, which we refer\nto as the Y-function . In this section we will prove various properties the Y-function satisfies,\nincluding multi-linearity, classical reducibility and a multi-marginal property that will be used in\nthe next section to solve an open problem from the literature. We also prove a general formula\nfor the Y-function evaluated on an n-step process (ρ,E1, ...,En)in terms of ρand the channel\nstatesJ[E1], ...,J[En]. To conclude the section, we show that in the case of a closed system\nof dynamically evolving qubits, the Y-function recovers the pseudo-density matrix formalism\nfirst introduced by Fitzsimons, Jones and Vedral [3].\nDefinition 8.1. TheY-function is the function on all finite-step processes given by\nY(ρ,E1, ...,En) = (Y(En◦tr)◦ ··· ◦Y(E2◦tr)◦Y(E1))(ρ). (8.2)\nRemark 8.3. Note that the Y-function as given by (8.2) is well-defined even if ρis not a state\nand even if Eiis not CPTP for any i∈{1, ...,n}. As such, we may view the Y-function as being\ndefined on the set\nA0×Hom(A0, ...,An)\nfor any (n+1)-tuple of multi-matrix algebras (A0, ...,An). However, the element Y(ρ,E1, ...,En)\nwill only be viewed as a state over time when (ρ,E1, ...,En)∈P(A0, ...,An).\nWe now prove various properties the Y-function satisfies, but first we introduce some no-\ntation.\nNotation 8.4. LetA=A0⊗ ··· ⊗ An. Given 0 ⩽i1<···< im⩽n, we let\ntri1···im:A− →Ai1⊗ ··· ⊗ Aim\ndenote the partial trace map.\nTheorem 8.5. TheY-function satisfies the following properties.\ni.(Hermiticity) Y(ρ,E1, ...,En)is self-adjoint of unit trace for all finite step processes (ρ,E1, ...,En).\nii.(Multi-Linearity) TheY-function is multi-linear, i.e.,\nY(λρ+ (1−λ)σ,E1, ...,En) =λY(ρ,E1, ...,En) + ( 1−λ)Y(σ,E1, ...,En) (8.6)\nand\nY(ρ,E1, ...,λEi+ (1−λ)Fi, ...,En) =λY(ρ,E1, ...,Ei, ...,En) + ( 1−λ)Y(ρ,En, ...,Fi, ...,En)\n(8.7)\nfor all i∈{1, ...,n}and for all λ∈C.\niii.(Reduction) Y(ρ,E1, ...,En) =Y(ρ,E1, ...,Y(Ei+1)◦Ei,Ei+2◦tr,Ei+3, ...,En)for all i∈\n{1, ...,n−1}.\n34iv.(The Multi-Marginal Property) Let0⩽i1<···< im⩽n. Then\ntri1···im(Y(ρ,E1, ...,En))=Y(ρi1,Ei2◦ ··· ◦Ei1+1, ...,Eim◦ ··· ◦Eim−1+1) (8.8)\nv.(Classical Reducibility) If(ρ,E1, ...,En)∈P(CX0, ...,CXn)is a classical process, then\nY(ρ,E1, ...,En) =M\n(x0,...,xn)∈X0×···× XnP(xn|xn−1)···P(x1|x0)P(x0),\nwhereP(x0)is the classical distribution associated with ρandP(xi|xi−1)are conditional probabil-\nities associated with J[Ei].\nProof. Item i : The statement follows from item i of Theorem 7.4.\nItem ii : Equation (8.6) follows from the fact that Y(En◦tr)◦···◦Y(E1)is a linear map, while\nequation (8.7) follows from the fact that\nY((λEi+ (1−λ)Fi)◦tr) =λY(Ei◦tr) + ( 1−λ)Y(Fi◦tr)\nfor all i∈{1, ...,n}and all λ∈C.\nItem iii : Leti∈{1, ...,n−1}, and let ℵ=Y(ρ,E1, ...,Y(Ei+1)◦Ei,Ei+2◦tr,Ei+3, ...,En). Then\nℵ=Y(ρ,E1, ...,Y(Ei+1)◦Ei,Ei+2◦tr,Ei+3, ...,En)\n=(Y(En◦tr)◦ ··· ◦Y(Y(Ei+1)◦Ei◦tr)◦ ··· ◦Y(E1))(ρ)\n=(Y(En◦tr)◦ ··· ◦Y(Ei+1◦tr)◦Y(Ei◦tr)◦ ··· ◦Y(E1))(ρ)\n=Y(ρ,E1, ...,En),\nwhere the third equality follows from the associativity of the symmetric bloom.\nItem iv : The proof relies on two lemmas, which we will prove after we finish proving the\ntheorem. The case m=1 follows from item i of Theorem 7.4, so now suppose m > 1. The\npartial trace map tr i1···immay be written as\ntri1···im=\u0010\ntrAi1+1⊗···⊗Ai2−1◦ ··· ◦ trAim−1+1⊗···⊗Aim−1\u0011\n◦trA0⊗···⊗Ai1−1⊗Aim+1⊗···⊗An,\nwhere if ij−ij−1=1, we set tr Aij−1+1⊗···⊗Aij−1=id. Now by items i and ii Lemma 8.9 we have\ntrA0⊗···⊗Ai1−1⊗Aim+1⊗···⊗An(Y(ρ,E1, ...,En))=Y(ρi1,Ei1+1, ...,Eim)∈Ai1⊗Ai1+1⊗ ··· ⊗ Aim,\nthus\ntri1···im(Y(ρ,E1, ...,En))=trAi1+1⊗···⊗Ai2−1◦ ··· ◦ trAim−1+1⊗···⊗Aim−1\u0000\nY(ρi1,Ei1+1, ...,Eim)\u0001\n=Y(ρi1,Ei2◦ ··· ◦Ei1+1, ...,Eim◦ ··· ◦Eim−1+1),\nwhere the last equality follows from Lemma 8.10.\nItem v : In the proof of Proposition 3.10 we established that if E:CX− →CYis a classical\nchannel, then\nJ[E] =X\n(x,y)∈X×YEyx(δx⊗δy),\nwhere we recall Eyxare the conditional probabilities associated with the classical channel E.\nThe statement then follows directly from the definition of the symmetric bloom. ■\n35We now prove the lemmas needed for the proof of the multi-marginal property of the Y-\nfunction (item iv of Theorem 8.5).\nLemma 8.9. Let(ρ,E1, ...,En)∈P(A0, ...,An)be an n-step process. Then the following statements\nhold.\ni.trA0⊗···⊗Ai−1(Y(ρ,E1, ...,En)) =Y(ρi,Ei+1, ...,En)for all i∈{1, ...,n}.\nii.trAi+1⊗···⊗An(Y(ρ,E1, ...,En)) =Y(ρ,E1, ...,Ei)for all i∈{0, ...,n−1}.\nProof. Item i : We use induction on i. For i=0 we have\ntrA0(Y(ρ,E1, ...,En))=trA0((Y(En◦tr)◦ ··· ◦Y(E2◦tr)◦Y(E1))(ρ))\n=trA0(Y(Y(En◦tr)◦ ··· ◦Y(E2)◦E1)(ρ))\n=(Y(En◦tr)◦ ··· ◦Y(E2))(E1(ρ))\n=Y(ρ1,E2, ...,En),\nwhere the second equality follows from Lemma 6.12 and the third equality follows from the\ndefinition of a bloom map, thus the result holds for i=0. Now suppose the result holds for\ni=m−1 with m−1< n. Then\ntrA0⊗···⊗Am−1(Y(ρ,E1, ...,En))=trAm−1\u0000\ntrA0⊗···⊗Am−2(Y(ρ,E1, ...,En))\u0001\n=trAm−1(Y(ρm−1,Em, ...En))\n=Y(ρm,Em+1, ...En),\nwhere the last equality follows from a calculation similar to the i=0 case. It then follows that\nthe result holds for i=m, as desired.\nItem ii : We prove the equivalent statement tr An−i⊗···⊗An(Y(ρ,E1, ...,En)) =Y(ρ,E1, ...,En−i−1)\nfor all i∈{0, ...,n−1}by induction on i. For i=0 we have\ntrAn(Y(ρ,E1, ...,En))=trAn(Y(En◦tr)(Y(ρ,E1, ...,En−1)))=Y(ρ,E1, ...,En−1),\nwhere the last equality follows from the definition of bloom map, thus the result holds for i=0.\nNow suppose the result holds for i=m−1 with m−1< n−1. Then\ntrAn−m⊗···⊗An(Y(ρ,E1, ...,En)) = trAn−m\u0000\ntrAn−m+1⊗···⊗An(Y(ρ,E1, ...,En))\u0001\n=trAn−m\u0000\nY(ρ,E1, ...,En−(m−1)−1)\u0001\n=trAn−m(Y(ρ,E1, ...,En−m))\n=Y(ρ,E1, ...,En−m−1),\nwhere the last equality follows from a calculation similar to the i=0 case. It then follows that\nthe result holds for i=m, as desired. ■\nLemma 8.10. Let(E1, ...,En)∈TP(A0, ...,An)be an n-chain, let 0=i1<···< im=n, letρ∈Ai1,\nand let ℵ∈Ai1⊗Ai2⊗ ··· ⊗ Aimbe the element given by\nℵ=trAim−1+1⊗···⊗Aim−1◦ ··· ◦ trAi1+1⊗···⊗Ai2−1\u0000\nY(ρ,Ei1+1, ...,Eim)\u0001\n,\n36where we set trAij−1+1⊗···⊗Aij−1=idifij−ij−1=1. Then\nℵ=Y\u0000\nρ,Ei2◦ ··· ◦Ei1+1, ...,Eim◦ ··· ◦Eim−1+1\u0001\n. (8.11)\nProof. We use induction on m. The case m=1 follows from item ii of Lemma 8.9. So assume\nthe result holds for m=k−1>1, and let ℵkbe the element given by\nℵk=trAik−1+1⊗···⊗Aik−1◦ ··· ◦ trAi1+1⊗···⊗Ai2−1\u0000\nY(ρ,Ei1+1, ...,Eik)\u0001\n.\nThe result then follows once we show\nℵk=Y\u0000\nρ,Ei2◦ ··· ◦Ei1+1, ...,Eik◦ ··· ◦Eik−1+1\u0001\n.\nFor this, we first consider the 2-chain\nAi1E− →Ai1+1⊗ ··· ⊗ Ai2−1F− →Ai2⊗Ai2+1⊗ ··· ⊗ Aim,\nwhereE=Y(Ei2−1◦tr)◦···◦Y(Ei1+2)◦Ei1+1andF=Y(Eik◦tr)◦···◦Y(Ei2+1)◦Ei2◦tr. Then\nafter letting χ=Y(ρ,E,F), we have\nχ=Y(ρ,E,F)\n=(Y(F◦tr)◦Y(E))(ρ)\n=\u0000\nY(Y(Eik◦tr)◦ ··· ◦Y(Ei2+1)◦Ei2◦tr)◦Y\u0000\nY(Ei2−1◦tr)◦ ··· ◦Y(Ei1+2)◦Ei1+1\u0001\u0001\n(ρ)\n=\u0000\nY(Ei2◦tr)◦ ··· ◦Y(Ei1+2◦tr)◦Y(Ei1+1)\u0001\n(ρ)\n=Y(ρ,Ei1+1, ...,Eik),\nwhere the fourth equality follows from the associativity of the symmetric bloom. Then after\nletting ξ=trAi1+1⊗···⊗A12−1\u0000\nY(ρ,Ei1+1, ...,Eik)\u0001\n, we then have\nξ= trAi1+1⊗···⊗A12−1\u0000\nY(ρ,Ei1+1, ...,Eik)\u0001\n= trAi+1⊗···⊗Aj−1(Y(ρ,E,F))\n(5.16)=Y(ρ,F◦E)\n=Y\u0000\nρ,Y(Eik◦tr)◦ ··· ◦Y(Ei2+1)◦Ei2◦tr◦Y(Ei2−1◦tr)◦ ··· ◦Y(Ei1+2)◦Ei1+1\u0001\n=Y\u0000\nρ,Y(Eik◦tr)◦ ··· ◦Y(Ei2+1)◦Ei2◦Ei2−1◦ ··· ◦Ei1+1\u0001\n=Y\u0000\nY(Eik◦tr)◦ ··· ◦Y(Ei2+1)◦Ei2◦Ei2−1◦ ··· ◦Ei1+1\u0001\n(ρ)\n=\u0000\nY(Eik◦tr)◦ ··· ◦Y(Ei2+1◦tr)◦Y\u0000\nEi2◦Ei2−1◦ ··· ◦Ei1+1\u0001\u0001\n(ρ)\n=Y(ρ,Ei2◦Ei2−1◦ ··· ◦Ei1+1,Ei2+1, ...,Eik),\nwhere the fifth equality follows from the fact that Ei2◦tr◦Y(Ei2−1◦tr)◦ ··· ◦Y(Ei1+2)◦Ei1+1=\nEi2◦Ei2−1◦ ··· ◦Ei1+1, and the seventh equality follows from the associativity of symmetric\nbloom. Now let σ=Y(ρ,Ei2◦Ei2−1◦ ··· ◦Ei1+1), so that\nY(ρ,Ei2◦Ei2−1◦ ··· ◦Ei1+1,Ei2+1, ...,Eik) =\u0000\nY(Eik◦tr)◦ ··· ◦Y\u0000\nEi2◦Ei2−1◦ ··· ◦Ei1+1\u0001\u0001\n(ρ)\n=\u0000\nY(Eik◦tr)◦ ··· ◦Y\u0000\nEi2+1◦tr\u0001\u0001\n(σ)\n=Y(σ,Ei2+1◦tr,Ei2+2, ...,Eik).\n37We then have\nℵk = trAik−1+1⊗···⊗Aik−1◦ ··· ◦ trAi1+1⊗···⊗Ai2−1\u0000\nY(ρ,Ei1+1, ...,Eik)\u0001\n= trAik−1+1⊗···⊗Aik−1◦ ··· ◦ trAi2+1⊗···⊗Ai3−1\u0010\ntrAi1+1⊗···⊗A12−1\u0000\nY(ρ,Ei1+1, ...,Eik)\u0001\u0011\n= trAik−1+1⊗···⊗Aik−1◦ ··· ◦ trAi2+1⊗···⊗Ai3−1\u0000\nY(ρ,Ei2◦Ei2−1◦ ··· ◦Ei1+1,Ei2+1, ...,Eik)\u0001\n= trAik−1+1⊗···⊗Aik−1◦ ··· ◦ trAi2+1⊗···⊗Ai3−1\u0000\nY(σ,Ei2+1◦tr,Ei2+2, ...,Eik)\u0001\ninduction=Y\u0000\nσ,Ei3◦ ··· ◦Ei2+1◦tr, ...,Eik◦ ··· ◦Eik−1+1\u0001\n=Y\u0000\nY(ρ,Ei2◦Ei2−1◦ ··· ◦Ei1+1),Ei3◦ ··· ◦Ei2+1◦tr, ...,Eik◦ ··· ◦Eik−1+1\u0001\n=Y\u0000\nρ,Ei2◦ ··· ◦Ei1+1, ...,Eik◦ ··· ◦Eik−1+1\u0001\n,\nwhere the final equality follows from item iii of Theorem ii, thus concluding the proof. ■\nWe now prove a formula for Y(ρ,E1, ...,En)in terms of ρand the channel states J[E1], ...,\nJ[En].\nNotation 8.12. Let[n] ={1, ...,n}for every natural number n∈N. For every subset {i1, ...,im}⊂\n[n], the subscripts ijare chosen so that i1<···< im. In such a case, we denote the complement\n[n]\\ {i1, ...,im}by{i1, ...,in−m}, with the same convention that i1<···< in−m.\nNotation 8.13. Given an n-chain (E1, ...,En)∈TP(A0, ...,An)and an element j∈[n], we let\nfJ[Ej]∈A0⊗ ··· ⊗ Anbe the element given by\nfJ[Ej] =\n\nJ[E1]⊗1 ifj=1\n1⊗J[Ej]⊗1if 1< j < n\n1⊗J[En] ifj=n.\nTheorem 8.14. Let(ρ,E1, ...,En)∈P(A0, ...,An)be an n-step process. Then\nY(ρ,E1, ...,En) =1\n2nX\n{i1,...,im}⊂[n]fJ[Eim]···fJ[Ei1](ρ⊗1)fJ[Ei1]···fJ[Ein−m] (8.15)\nProof. We use induction on n. For n=1 the result holds by definition of the Y-function. So\nnow suppose the result holds for n=k−1>1. Then after letting ℵ=Y(ρ,E1, ...,Ek), we have\nℵ=Y(ρ,E1, ...,Ek)\n= (Y(Ek◦tr)◦ ··· ◦Y(E2◦tr)◦Y(E1))(ρ)\n=Y(Ek◦tr)(Y(ρ,E1, ...,Ek−1))\n=Jor(Y(ρ,E1, ...,Ek−1)⊗1,J[Ek◦tr])\n=Jor(Y(ρ,E1, ...,Ek−1)⊗1,1⊗J[Ek])\n=Jor\n\n1\n2k−1X\n{i1,...,im}⊂[k−1]fJ[Eim]···fJ[Ei1](ρ⊗1)fJ[Ei1]···fJ[Eik−1−m]\n⊗1,fJ[Ek]\n\n=1\n2kX\n{i1,...,im}⊂[k]fJ[Eim]···fJ[Ei1](ρ⊗1)fJ[Ei1]···fJ[Eik−m],\n38where the second to last equality follows from the inductive assumption. It then follows that\nthe result holds for n=k, as desired. ■\nWe now extend the normalized Jordan product to an n-ary operation for all n > 0, which\nwe will then use to derive a formula for the Y-function which may be viewed as the n-step\ngeneralization of the 2-step formula (5.14).\nDefinition 8.16. Give a multi-matrix algebra A, the extended Jordan product is the map\nJor:A× ··· × A− →A\ngiven by\nJor(A1, ...,An) =Jor(A0,Jor(A1,Jor(···,Jor(An−1,An)))). (8.17)\nTheorem 8.18. Let(ρ,E1, ...,En)∈P(A0, ...,An)be an n-step process. Then\nY(ρ,E1, ...,En) =Jor\u0000\nρ⊗1,J[E1]⊗1, ...,1⊗J[Ej]⊗1, ...,1⊗J[En]\u0001\n(8.19)\nProof. The statement follows from Theorem 8.14 together with the fact that (ρ⊗1)commutes\nwithfJ[Ej]for all j∈[n]with j >1. ■\nTo conclude this section, we show how the Y-function recovers the pseudo-density matrix\nformalism for qubits.\nExample 8.20 (The pseudo-density matrix formalism for systems of qubits) .Letρ∈Nk\ni=1M2\nbe the initial state of an k-qubit system, let {σ0,σ1,σ2,σ3}be the Pauli basis of M2, and let\n(ρ,E1, ...,En)be an n-step process with initial state ρ. In the formalism introduced by Fitzsi-\nmons, Jones and Vedral [3], the pseudo-density matrix associated with (ρ,E1, ...,En)is the ma-\ntrixRn∈Nn\nj=0M⊗k\n2given by\nRn=1\n2k(n+1)4kX\ni0=1···4kX\nin=1\n(eσiα)n\nα=0\u000bnO\nα=0eσiα,\nwhere eσiα∈{σ0, ...,σ3}⊗kand\n{eσiα}n\nα=0\u000b\ndenotes the expectation value associated with the\nobservable (eσiα)n\nα=0. In [13], it was recently proved that\nR1=Jor(ρ⊗1,J[E1])≡Y(ρ,E1)&Rn=Jor(Rn−1⊗1,1⊗J[En]),\nand since\nY(ρ,E1, ...,En) = ( Y(En◦tr)◦ ··· ◦Y(E2◦tr)◦Y(E1))(ρ)\n=Y(En◦tr)(Y(ρ,E1, ...,En−1))\n=Jor(Y(ρ,E1, ...,En−1)⊗1,J[En◦tr])\n=Jor(Y(ρ,E1, ...,En−1)⊗1,1⊗J[En]),\nit follows via induction that Y(ρ,E1, ...,En) =Rn(the final equality follows from item iii of\nLemma 5.6). As such, the Y-function recovers the pseudo-density matrix formalism for systems\nof qubits first introduced in [3], and moreover, Theorems 8.14 and 8.18 yield new formulas for\nRn.\n399 Extracting dynamics from a pseudo-density matrix\nIn this section, we derive an explicit expression for the inverse of the Y-function restricted to\na suitably nice subset of n-step processes. In particular, given a pseudo-density matrix τ, we\nshow how to identify τwith an n-step process (ρ,E1, ...,En)such that Y(ρ,E1, ...,En) =τ. This\nsolves an open problem from [9], where it is stated \"Another interesting and closely relevant\nopen question is, for a given PDO (pseudo-density matrix), how to find a quantum process to\nrealize it.\".\nNotation 9.1. Given an element τ∈A0⊗ ··· ⊗ Anwith tr 0(τ)∈A0invertible, we let Xτ∈\nA0⊗ ··· ⊗ Anbe the unique element such that τ=Jor((tr(τ0)⊗1),Xτ).\nNotation 9.2. Given an (n+1)-tuple (A0, ...,An)of multi-matrix algebras, we let P+(A0, ...,An)⊂\nP(A0, ...,An)denote the subset given by\nP+(A0, ...,An) ={(ρ,E1, ...,En)∈P(A0, ...,An)|ρiis invertible for i=0, ...,n−1},\nand we let T∗(A0⊗ ··· ⊗ An)⊂T(A0⊗ ··· ⊗ An)be the subset consisting of elements τ∈\nT(A0⊗ ··· ⊗ An)such that\n•τi∈T∗(Ai−1⊗Ai)fori=1, ...,n, where T∗(Ai−1⊗Ai)is as defined by (4.15).\n•Xτ=Jor(Xτ1⊗1, ...,1⊗Xτn)\nwhere τi=tri−1,i(τ)and tr i−1,i:A0⊗ ··· ⊗ An− →Ai−1⊗Aiis the partial trace map.\nRemark 9.3. We have reason to believe that the two conditions defining T∗(A0⊗ ··· ⊗ An)are\nin fact equivalent, but we have yet to find a proof.\nWe now are going to prove that the Y-function restricted to P+(A0, ...,An)defines a bijec-\ntion onto T∗(A0⊗ ··· ⊗ An), and we will also give a precise formula for the inverse. Before\ndoing so however, we first prove that the image of the Y-function restricted to P+(A0, ...,An)\nindeed lies in T∗(A0⊗ ··· ⊗ An).\nProposition 9.4. Let(ρ,E1, ...,En)∈P+(A0, ...,An). ThenY(ρ,E1, ...,En)∈T∗(A0⊗ ··· ⊗ An).\nProof. Let(ρ,E1, ...,En)∈P+(A0, ...,An), letτ=Y(ρ,E1, ...,En), and let\nτi=tri−1,i(Y(ρ,E1, ...,En))∈Ai−1⊗Ai\nfor all i∈[n]. Then it follows from the multi-marginal property of the Y-function that\nτi=Y(ρi−1,Ei), and since ρi−1is invertible by the definition of P+(A0, ...,An), it follows that\n(ρi−1,Ei)∈P+(Ai−1,Ai), thus τi∈T∗(Ai−1⊗Ai)fori=1, ...,nby Lemma 4.16, showing that\nτsatisfies the first condition for being an element of T∗(A0⊗ ··· ⊗ An). Moreover, it follows\n40from Theorem 4.17 that Ei=J−1(Xτi)for all i∈[n], thus\nτ=Y(ρ,E1, ...,En)\n=Y(tr0(τ),J−1(Xτ1), ...,J−1(Xτn))\n(8.19)= Jor\u0010\ntr0(τ)⊗1,J\u0010\nJ−1(Xτ1)\u0011\n⊗1, ...,1⊗J\u0010\nJ−1(Xτn)\u0011\u0011\n= Jor(tr0(τ)⊗1,Xτ1⊗1, ...,1⊗Xτn)\n= Jor((tr0(τ)⊗1),Jor(Xτ1⊗1, ...,1⊗Xτn))\n=⇒Xτ=Jor(Xτ1⊗1, ...,1⊗Xτn),\nthusτsatisfies the second condition for being an element of T∗(A0⊗ ··· ⊗ An). It then follows\nthatτ=Y(ρ,E1, ...,En)∈T∗(A0⊗ ··· ⊗ An), as desired.\n■\nTheorem 9.5. Let(A0, ...,An)be an (n+1)-tuple of multi-matrix algebras, let\nS :P+(A0, ...,An)− →T∗(A0⊗ ··· ⊗ An)\nbe the restriction of the Y-function to P+(A0, ...,An), and let\nC :T∗(A0⊗ ··· ⊗ An)− →P+(A0, ...,An)\nbe the map given by\nC(τ) =\u0010\ntr0(τ),J−1(Xτ1), ...,J−1(Xτn)\u0011\n, (9.6)\nwhere τi=tri−1,i(τ)andXτi∈Ai−1⊗Aiis defined via (4.13) for all i∈{1, ...,n}. Then Sis a\nbijection, and C = S−1.\nProof. Let(ρ,E1, ...,En)∈P+(A0, ...,An), letτ= S(ρ,E1, ...,E1)≡Y(ρ,E1, ...,En), and let\nτi=tri−1,i(S(ρ,E1, ...,En))∈Ai−1⊗Ai\nfor all i∈{1, ...,n}. Then we have seen in the proof of Proposition 9.4 that J−1(Xτi) =Ei, thus\n(C◦S)(ρ,E1, ...,En) = C (S(ρ,E1, ...,En))\n= C(Y(ρ,E1, ...,En))\n=\u0010\ntr0(Y(ρ,E1, ...,En)),J−1(Xτ1), ...,J−1(Xτn)\u0011\n= (ρ,E1, ...,En).\nIt then follows that C◦S = id, thus Sis injective.\nNow let τ∈T∗(A0⊗ ··· ⊗ An), so that\nC(τ) =\u0010\ntr0(τ),J−1(Xτ1), ...,J−1(Xτn)\u0011\n.\n41From the definition of T∗(A0⊗ ··· ⊗ An)we have τi∈T∗(Ai−1,Ai)for all i∈[n], from which\nit follows that J−1(Xτi)is CPTP for all i∈[n]and also that tr 0(τ)is invertible, thus C(τ)∈\nP+(A0, ...,An). We then have\n(S◦C)(τ) = Y\u0010\ntr0(τ),J−1(Xτ1), ...,J−1(Xτn)\u0011\n(8.19)= Jor\u0010\ntr0(τ)⊗1,J\u0010\nJ−1(Xτ1)\u0011\n⊗1, ...,1⊗J\u0010\nJ−1(Xτn)\u0011\u0011\n= Jor(tr0(τ)⊗1,Xτ1⊗1, ...,1⊗Xτn)\n= Jor((tr0(τ)⊗1),Jor(Xτ1⊗1, ...,1⊗Xτn))\n= Jor((tr0(τ)⊗1),Xτ)\n=τ,\nwhere the second to last equality follows from the fact that τ∈T∗(A0⊗ ··· ⊗ An). It then\nfollows that S◦C = id, thus concluding the proof. ■\nRemark 9.7. As the inverse J−1:A⊗B− → Hom(A,B)of the Jamiołowski isomorphism\nadmits the explicit formula\nJ−1(τ)(ρ) =trA((ρ⊗1)τ),\nand since there are algorithms to compute Xτwith τ∈T∗(A0⊗ ··· ⊗ An), the formula (9.6) for\nthe inverse of the Y-function is easily implementable in symbolic computing software.\nReferences\n[1] Jordan Cotler, Chao-Ming Jian, Xiao-Liang Qi, and Frank Wilczek, Superdensity operators for spacetime quantum\nmechanics , Journal of High Energy Physics 9(2018). ↑3\n[2] Douglas R. 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A 98(2018), 052312, available at 1711.05955 .↑3\n43" }, { "title": "2304.13829v2.Controlled_density_transport_using_Perron_Frobenius_generators.pdf", "content": "Controlled density transport using Perron Frobenius generators\nJake Buzhardt and Phanindra Tallapragada\nAbstract — We consider the problem of the transport of a den-\nsity of states from an initial state distribution to a desired final\nstate distribution through a dynamical system with actuation.\nIn particular, we consider the case where the control signal is\na function of time, but not space; that is, the same actuation\nis applied at every point in the state space. This is motivated\nby several problems in fluid mechanics, such as mixing and\nmanipulation of a collection of particles by a global control\ninput such as a uniform magnetic field, as well as by more\ngeneral control problems where a density function describes an\nuncertainty distribution or a distribution of agents in a multi-\nagent system. We formulate this problem using the generators\nof the Perron-Frobenius operator associated with the drift\nand control vector fields of the system. By considering finite-\ndimensional approximations of these operators, the density\ntransport problem can be expressed as a control problem for\na bilinear system in a high-dimensional, lifted state. With this\nsystem, we frame the density control problem as a problem of\ndriving moments of the density function to the moments of a\ndesired density function, where the moments of the density can\nbe expressed as an output which is linear in the lifted state. This\noutput tracking problem for the lifted bilinear system is then\nsolved using differential dynamic programming, an iterative\ntrajectory optimization scheme.\nI. I NTRODUCTION\nIn this paper, we consider the problem of controlled\ndensity transport, where given an initial distribution of states\nspecified by a density function, we seek to determine a\ncontrol sequence to drive this initial distribution to a desired\nfinal distribution. We consider the case where a common\ncontrol signal is applied to the entire distribution of states.\nThis differs from the usual formulation of swarm control\nand optimal transport problems, where typically each agent\ncan select a control input independently, making the control\nsignal a function of the states and time. This problem of\ndensity transport is motivated by problems of manipulation\nof a large collection of agents using a uniform control\nsignal [1], [2]. The transport of density also has relevance\nto the propagation of an uncertainty distribution arising due\nto uncertainty in the initial state or of a model parameter\nthrough an otherwise deterministic control system (see, e.g.\n[3]–[5]). We formulate and solve this problem using an\noperator theoretic approach, specifically using the generator\nof the Perron-Frobenius operator.\nIn recent years the operator theoretic approach to dy-\nnamical systems and control has gained significant research\nattention [6]–[8]. A dynamical system can be framed in\nterms of such an operator either by considering the evolution\nof observable functions of the state using the Koopman\nThe authors are with the Department of Mechanical Engineering, Clem-\nson University, Clemson, SC, USA. jbuzhar@g.clemson.edu,\nptallap@clemson.eduoperator or by considering the evolution of densities of states\nusing the Perron-Frobenius operator [9]. The interest in these\napproaches is primarily due to the fact that these operators\nallow for a linear, although typically infinite dimensional\nrepresentation of a nonlinear system. The linearity of these\noperators is useful from an analytical perspective, as it\nallows for the use of linear systems techniques such as the\nanalysis of eigenvalues and eigenfunctions, but also from a\ncomputational perspective, as in many cases a useful approx-\nimation for these operators can be found by considering a\nfinite dimensional approximation in which the operator is\nrepresented as a matrix acting on coordinates corresponding\nto a finite set of a set of dictionary functions [10]–[12].\nIn applications in control systems, much of the recent work\nhas been on developing methods involving the Koopman\noperator [8], [13], as the transformation to a space of\nobservable functions can be viewed as a nonlinear change of\ncoordinates which maps the system to a higher dimensional\nspace where the dynamics are (approximately) linear [14].\nThis makes the numerical approximation of the operator\nparticularly amenable to linear control methods, such as\nthe linear quadratic regulator (LQR) and model predictive\ncontrol (MPC) [14]–[16]. On the other hand, the Perron-\nFrobenius operator propagates densities of states forward in\ntime along trajectories of the system, which can have mul-\ntiple interpretations in the controlled setting. For example,\nthe Perron-Frobenius operator and the Liouville equation,\nthe related PDE formulation, have been used to determine\ncontrols for agents in an ensemble or swarm formulation\n[17], [18]. Such formulations are closely related to optimal\ntransport problems which also involve driving an initial\ndistribution to a desired final distribution (see, e.g., [18]–\n[20]). Formulations involving the Perron-Frobenius operator\nhave also been used in the context of fluid flows to study\nthe transport of distributions of fluid particles and to detect\ninvariant or almost invariant sets [21], [22].\nOur approach involves first obtaining a finite dimensional\napproximation of the Perron-Frobenius generators associated\nwith the drift and control vector fields of the system, which\nallow us to represent the density transport dynamics as a\nbilinear system in a lifted state. With this system, we frame\nthe density control problem as a problem of driving moments\nof the density function to the moments of a desired density\nfunction, where the moments of the density can be expressed\nas an output which is linear in the lifted state. This output\ntracking problem for the lifted bilinear system is then solved\nusing differential dynamic programming (DDP), an iterative\ntrajectory optimization scheme.arXiv:2304.13829v2 [eess.SY] 25 Sep 2023II. P RELIMINARIES\nConsider first the autonomous dynamical system on a\nmeasure space (X⊂Rn,A, µ)with a σ-algebra AonX\nandµa measure on (X,A),\n˙x=f(x) (1)\nand denote the associated time- tflow from an initial state x0\nasΦt(x0), where x∈Xis the state. The Perron-Frobenius\noperator Pt:L1(X)7→L1(X)associated with the flow map\nΦtis defined asZ\nA\u0002\nPtρ\u0003\n(x)dx=Z\n(Φt)−1(A)ρ(x)dx (2)\nfor any A∈ A , assuming that the relevant measure µis\nabsolutely continuous with respect to the Lebesgue measure\nand can thus be expressed in terms of a density ρ(i.e.,\ndµ(x) =µ(dx) =ρ(x)dx). It can be shown that the family\nof these operators {Pt}t≥0form a semigroup, (see [9]). The\ngenerator of this semigroup is known as the Liouville opera-\ntor, denoted L, or Perron-Frobenius generator and expresses\nthe deformation of the density ρunder infinitesimal action\nof the operator Pt[9], [23]. That is,\ndρ\ndt=Lρ=−∇x·(ρf) (3)\nAlternatively, the action of the generator can be written in\nterms of the Perron-Frobenius operator as\nLρ= lim\nt→0Ptρ−ρ\nt= lim\nt→0\u0012Pt− I\nt\u0013\nρ (4)\nwhere Iis the identity operator.\nLemma 1: Suppose the Liouville operator associated with\na vector field f1:X7→Rnis denoted by L1and the\nLiouville operator associated with the vector field f2:X7→\nRnbyL2, then the Liouville operator associated with the\nvector field f(x) =f1(x) +f2(x), isL=L1+L2.\nProof: The proof is a direct consequence of Eq. 3.\nSuppose f(x) =f1(x) +f2(x). Then Lρ=−∇x·(ρ(f1+\nf2)) =−∇x·(ρf1)− ∇ x·(ρf2) = (L1+L2)ρ.\nThe Koopman operator Kt:L∞(X)7→L∞(X)propa-\ngates observable functions forward in time along trajectories\nof the system and is defined as\n[Kth](x) = [h◦Φt](x) (5)\nwhere h(x)is an observable. The Koopman and Perron-\nFrobenius operators are adjoint to one another,\nZ\nX[Kth](x)ρ(x)dx=Z\nXh(x)[Ptρ](x)dx . (6)\nIII. N UMERICAL APPROXIMATION OF THE\nPERRON -FROBENIUS OPERATOR AND GENERATOR\nHere, we implement extended dynamic mode decompo-\nsition (EDMD) [11] for the computation of the Perron-\nFrobenius operator, which we outline below, largely follow-\ning [12].\nWe begin by selecting a dictionary Dofkscalar-valued\nbasis functions, D={ψ1, ψ2, . . . , ψ k}, where ψi:X7→Rfori= 1, . . . , k , and denote by Vthe function space spanned\nby the elements of D. We then collect trajectory data with\nfixed timestep, ∆t, arranged into snapshot matrices as\nX=\u0002\nx1,···, x m\u0003\n(7)\nY=\u0002\nx+\n1,···, x+\nm\u0003\n(8)\nwhere the subscript i= 1, . . . , m is a measurement index\nandx+\ni= Φ∆t(xi).\nWe then approximate the observable function hand den-\nsityρin Eq. 6 by their projections onto V. That is,\nh(x)≈ˆhTΨ(x) (9)\nρ(x)≈ΨT(x)ˆρ (10)\nwhere ˆh,ˆρ∈Rkare column vectors containing the projec-\ntion coefficients and Ψ :X7→Rkis a column-vector valued\nfunction where the elements are given by [Ψ(x)]i=ψi(x).\nSubstituting these expansions into Eq. 6, we have\nZ\nXK∆t[ˆhTΨ]ΨTˆρ dx=Z\nXˆhTΨP∆t[ΨTˆρ]dx . (11)\nThen replacing [K∆tΨ](x) = Ψ( x+)and assuming that\nP∆tcan be approximated by a matrix Poperating on the\ncoordinates ˆρgives a least-squares problem for the matrix P\nmin\nP∥ΨYΨT\nX−ΨXΨT\nXP∥2\n2 (12)\nwhere ΨX,ΨY∈Rk×mare matrices with columns formed\nby evaluating Ψon the columns of XandYrespectively.\nThe analytical solution of this least squares problem is\nP=\u0000\nΨXΨT\nX\u0001†ΨYΨT\nX (13)\nwhere (·)†is the Moore-Penrose pseudoinverse.\nGiven this matrix approximation of the operator, P, if\nthe timestep ∆tchosen in the data collection is sufficiently\nsmall, the corresponding matrix approximation Lof the\nPerron Frobenius generator can be approximated based on\nthe limit definition of the generator in Eq. 4. as\nL≈P−Ik\n∆t(14)\nwhere Ikis the k×kidentity matrix. Just as the matrix\noperator Papproximates the propagation of a density func-\ntionρby advancing the projection coordinates ˆρforward\nfor a finite time, the approximation of the generator allows\nus to approximate the infinitesimal action of the operator\nPtby approximating the time derivative of the projection\ncoordinatesdˆρ\ndt=Lˆρ . (15)\nA. Extension to controlled systems\nIn the context of applying the Koopman operator to control\nsystems, several recent works have noted the usefulness of\nformulating the problem in terms of the Koopman generator,\nrather than the Koopman operator itself (see [24]–[26] and\nothers), which typically results in a lifted system that is\nbilinear in the control and lifted state. This approach allowsFig. 1: Moment propagation of proposed method for a Duffing oscillator with sinusoidal forcing. Red points show trajectories\nfrom initial conditions sampled from the initial density, ρ(x(0))∼ N([−0.5; 1],0.05I). Red circle and red ellipse show the\nsample mean and 2 σsample covariance ellipse, respectively. The black circle and black ellipse are the predicted mean and\n2σcovariance ellipse.\nfor a better approximation of the effects of control, especially\nfor systems in control-affine form\n˙x=f(x) +ncX\ni=1gi(x)ui (16)\nas it expresses the effect of the control vector fields giin a\nway that is also dependent on the lifted state. Here we apply a\nsimilar approach to the density transport problem, expressed\nin terms of the Perron-Frobenius generator. As shown in\nRef. [26] for the Koopman generator, by the property of\nthe Perron-Frobenius generator given in Lemma 1, if the\ndynamics are control-affine, then the generators are also\ncontrol affine, as can be seen by application of Eq. 3. This\nleads to density transport dynamics of the following form\nd\ndtρ(x) = (L0ρ)(x) +ncX\ni=1ui(Biρ)(x) (17)\nwhere L0is the Perron Frobenius generator associated with\nthe vector field f(x)and similarly, the Biare the Perron\nFrobenius generators associated with the control vector fields\ngi(x). Therefore, given the finite dimensional approximation\nof these generators, we can approximate the density transport\ndynamics as\ndˆρ\ndt=L0ˆρ+ncX\ni=1uiBiˆρ (18)\nwhere the matrices L0andBiare the matrix approximations\nof the operators in Eq. 17.\nB. Propagation of moments\nIn order to make the problem of controlling a density\nfunction using a finite dimensional control input well-posed,\nwe formulate the problem as a control problem of a finite\nnumber of outputs. In particular, we will describe the density\nfunction, in terms of a finite number of its moments. For a\nscalar x, recall that the ithraw moment, miis defined as\nmi=R\nXxiρ(x)dxand the ithcentral moment, µiabout the\nmean m1isµi=R\nX(x−m1)iρ(x)dx.\nGiven a projection of the density as in Eq. 10, the mean\nis approximated as\nmi\n1=Z\nxiρ(x)dx= ˆρTZ\nxiΨ(x)dx (19)which simply indicates that the mean of the density can\nbe written as a summation of the means of the dictionary\nfunctions, weighted by the projection coefficients. For higher\nmoments, if the central moment is considered, it will be poly-\nnomial in the projection coefficients due to its dependence\non the mean, whereas the raw moments remain linear in the\nprojection coefficients. For this reason, we choose to work\nwith the raw moments, as the central moments can also be\nexpressed in terms of the raw moments. Similarly, the second\nraw moment can be written as\nmij\n2=Z\nxixjρ(x)dx= ˆρTZ\nxixjΨ(x)dx (20)\nwhere, again, superscripts iandjare coordinate indices.\nC. Numerical example\nTo illustrate the ability of the proposed framework to\npropagate density functions forward in time, we consider\nthe propagation of an initial density for a forced Duffing\noscillator system, given by\nd\ndt\u0012x1\nx2\u0013\n=\u0012x2\nx1−x3\n1+u\u0013\n(21)\nwhere uis the control input. For the purpose of this sim-\nulation, we set u(t) = sin(4 πt)and the prediction results\nare shown in Figs. 1 and 3. For the generator calculation, a\ndictionary of Gaussian radial basis functions is used where\nthe centers lie on an evenly spaced 30×30grid ranging\nfrom−2.5to2.5inx1andx2. The operators are approxi-\nmated using data collected from short time trajectories with\n∆t= 0.005 for a 50×50grid of initial conditions on\nthe same region. The predicted moment is compared to the\nsample moment obtained from 1000 trajectories from initial\nconditions sampled according to the initial density. We see\nthat the moment propagation of the proposed method is good\nfor at least 3 seconds, which motivates the use of this method\nin a control formulation, as detailed in the following sections.\nAlso shown for comparison in Fig. 3 is a linear predic-\ntion, which is computed by propagating the initial Gaussian\nthrough a linearization of Eq. 21, where the linearization is\nre-computed at each timestep about the predicted mean, as is\ncommonly done in the a priori prediction step of an extended\nKalman filter.Fig. 2: Controlled transport through the Duffing system from an initial density ρ(x(0))∼ N([0; 1] ,0.025I)\ntoward the equilibrium point at (0,1)(green circle). Red circle and red ellipse show the sample mean and 2 σsample\ncovariance ellipse, respectively. The black circle and black ellipse are the predicted mean and 2σcovariance ellipse.\n- 101m1\n1\n- 101m2\n1\n0 1 2 3 4 5\nt- 101u012m1 1\n2S a m p l e P F P r e d i c t i o n L i n e a r P r e d i c t i o n\n012m2 2\n2\n0 1 2 3 4 5\nt- 202m1 2\n2\nFig. 3: Moment propagation for a forced Duffing oscillator.\nLeft, top: First raw moment (mean). Left, bottom: sinusoidal\ncontrol signal. Right: 2nd raw moment.\nIV. C ONTROL FORMULATION\nWe have shown in Sec. III-A and III-B that the problem\nof steering a density ρto a desired density can be expressed\nas an output tracking problem on a lifted, bilinear system\ngiven by Eq. 18, where the projection coefficients ˆρcan be\ninterpreted as the lifted state. Then, if the raw moments are\ntaken to be the relevant output, the output, yis linear in the\nlifted state, y=Cˆρ, where the elements of the output matrix\nCare given by rewriting Eqs. 19, 20 in matrix form.\nFor the optimal output tracking problem, we consider a\ndiscrete time optimal control problem\nmin\nu1,u2,...,u H−1H−1X\nt=1l(ˆρt, ut) +lH(ˆρH) (22a)\ns.t. ˆρt+1=F(ˆρt, ut) (22b)\nyt=Cˆρt (22c)\nwhere His the number of timesteps in the time horizon and\nEq. 22b represents the discrete time version of Eq. 18.\nIn particular, for output tracking, we consider a quadraticcost of the form\nl(ˆρt, ut) = (yt−yref\nt)TS(yt−yref\nt) +uT\ntRut (23)\nlH(ˆρH) = (yH−yref\nH)TSH(yH−yref\nH) (24)\nwhere SandRare weighting matrices which define the rela-\ntive penalty on tracking error and control effort, respectively.\nThis cost is quadratic in ˆρt(with a linear term).\nIt is well known that for optimal control problems on\nbilinear systems with quadratic cost, an effective way of\nsolving the problem is by iteratively linearizing and solving\na finite time linear quadratic regulator (LQR) problem about\na nominal trajectory, utilizing the Ricatti formulation of\nthat problem [27]. For this reason, we solve the optimal\ncontrol problem using differential dynamic programming\n(DDP) [28], [29], which is closely related to the method\nof iterative LQR.\nV. V ALIDATION AND RESULTS\nHere we consider two examples of the density control\nformulation using Perron Frobenius generators.\nA. Example 1: Forced Duffing oscillator\nAs a first example, we consider the forced Duffing oscil-\nlator of Eq. 21, and we use DDP to determine a control\nsequence to steer a Gaussian initial density ρ(x(0))∼\nN([1; 1] ,0.025I)toward the equilibrium point at (1,0)over\na time horizon of 2s. The generators are approximated using\nthe same data as described in Sec. III-C and the same\ntimestep of 0.005s is used for DDP. In differential dynamic\nprogramming, the in-horizon cost weights on the reference\nerror of the first two moments and control effort are all set\nto unity. The terminal cost on the reference error in the\nmoments is set to 1000 , and only the first two moments\nare considered. The target raw second moment is computed\nfrom the desired mean, with the desired variance taken to be\nzero. The results of this computation are shown in Fig. 2-5.\nIn Fig. 6, we show a comparison of the performance of the\nproposed controller, labelled ‘PF-DDP’ with a standard DDP\ncontroller, which computes a control on the Duffing system\ndirectly (rather than a lifted state), with the initial condition\nbeing the mean. That is, the standard DDP controller acts on\nthe mean as if it were a deterministic initial condition of the\nsystem. The comparison shown is the mean and one standardFig. 4: Controlled transport of fluid particles, driven by two rotlets, or micro-rotors, in a Stokes flow from an initial density,\nρ(x(0))∼ N([1; 1] ,0.025I)toward a target mean (green circle) at (−1,−1). The rotors are located at (−1,0)and(0,1),\nas indicated by the black circle-cross. Red circle and red ellipse show the sample mean and 2 σsample covariance ellipse,\nrespectively. The black circle and black ellipse are the predicted mean and 2σcovariance ellipse. Gray streamlines indicate\nthe flow field produced by the rotlets at the instant shown.\nFig. 5: Control of raw moments for the forced Duffing\nsystem. The black line indicates the target (reference). Left,\nbottom: control from differential dynamic programming for\nthe generator system. Right: 2nd raw moment. Predicted\nvalues are from the Perron-Frobenius generator computation.\nTrue values are given by the sample moment.\n0 0 . 5 1 1 . 5 2\nt- 0 . 500 . 511 . 5x 1\nP F - D D P\nD D P\nR e f e r e n c e\n0 0 . 5 1 1 . 5 2\nt- 0 . 500 . 511 . 5x 2\nFig. 6: Comparison of the proposed Perron Frobenius gener-\nator DDP with standard DDP for the forced Duffing system.\nShaded region indicates one standard deviation\ndeviation region from trajectories from a set of 500 initial\nconditions sampled from the initial distribution, with each of\nthe respective controllers applied. We see that the proposed\ncontroller moves the mean of the distribution closer to the\nreference, while maintaining nearly the same variance as the\nstandard DDP controller.\nB. Example 2: Rotor-driven Stokes flow\nAs a second example, we consider the problem of steering\na distribution of fluid particles in a Stokes flow, where theflow is produced by two micro-rotors. The micro-rotors are\nmodelled as rotlets, the Stokes flow singularity associated\nwith a point torque in the fluid. For a collection of Nrrotlets,\nthis flow is given by\ndx\ndt=NrX\ni=1Ti×(x−¯ xi)\n∥x−¯ xi∥3(25)\nwhere ¯ xiandTidenote the location and torque of the i-th\nrotor, respectively. We consider the case where there are two\nsuch rotors lying in the x1-x2plane, located at ¯ x1= (−1,0)\nand¯ x2= (1,0)and the controls for the problem are taken\nto be torques u1=T1,u2=T2, where the direction of\nthese torques is taken to be normal to the x1-x2plane (in\nthe positive x3direction.\nFor this example, the task is to drive a Gaussian initial\ndensity ρ(x(0))∼ N ([1; 1] ,0.025I)toward a target mean\nat(−1,−1)over a time horizon of 2s. Again, we consider\na timestep of ∆t= 0.005s for both the computation of the\ngenerators and for DDP. The in horizon costs weights for\nthe mean error are set to 2, while the weights for the second\nmoment error and control effort are unity. The terminal cost\nweight on the mean error is 1000 and the terminal cost\nweight on the second moment error is 500. Results from\nthis computation are shown in Figs. 4 and 7.\nWe see in Fig. 7 that DDP yields a control which drives\nthe mean to the target by first giving a significant counter-\nclockwise torque on the right rotor to drive the density to\na point between the rotors, at which point the left rotor is\ninitiated to drive the flow with a clockwise torque, pulling\nthe density mean near to the target.\nThis example demonstrates an alternative physical in-\nterpretation of the density transport problem, where the\ndensity represents a distribution of fluid particles. This also\ndemonstrates the effectiveness of the proposed method on\na system which is linear in the controls, but in which the\ncontrol vector fields are nonlinear.\nVI. C ONCLUSION\nIn this work, we have studied the problem of transporting\ndensity functions of states through a controlled dynamicalFig. 7: Control of raw moments for the rotlet-driven Stokes\nflow. The black line indicates the target (reference). Left, top:\n1st raw moment. Left, bottom: control inputs uLanduRfor\nthe left and right rotor. Right: 2nd raw moment. Predicted\nvalues are from the Perron-Frobenius generator computation.\nTrue values are given by the sample moment.\nsystem. This problem formulation has applications both in\nfluid mechanics and in the control of uncertain systems. Our\napproach is based on approximations of the Perron-Frobenius\noperator, whereby we show that approximations of this\noperator and its generator can be used to model the density\ntransport dynamics as a high-dimensional system which is\nbilinear in the lifted state and the control. We demonstrated\nthis approach on two examples, a forced Duffing system,\nin which the density can have the interpretation as an\nuncertainty in the initial state and on a rotor driven Stokes\nflow, in which the density formulation takes on the fluid\nmechanics interpretation of describing a distribution of fluid\nparticles. Future work in these areas could include extending\nthe proposed control formulation for use in a constrained\nmodel predictive control framework for uncertain systems or\nby studying the fluid transport by more realistic biological\nmicroswimmers or artificial microrobots [30].\nREFERENCES\n[1] A. Becker, G. Habibi, J. Werfel, M. Rubenstein, and J. McLurkin,\n“Massive uniform manipulation: Controlling large populations of\nsimple robots with a common input signal,” in 2013 IEEE/RSJ\ninternational conference on intelligent robots and systems , pp. 520–\n527, IEEE, 2013.\n[2] J. Buzhardt and P. 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Tibken, “An iterative method for the finite-time\nbilinear-quadratic control problem,” Journal of optimization Theory\nand applications , vol. 57, pp. 411–427, 1988.\n[28] Y . Tassa, T. Erez, and E. Todorov, “Synthesis and stabilization of\ncomplex behaviors through online trajectory optimization,” in 2012\nIEEE/RSJ International Conference on Intelligent Robots and Systems ,\npp. 4906–4913, IEEE, 2012.\n[29] S. Yakowitz and B. Rutherford, “Computational aspects of discrete-\ntime optimal control,” Applied Mathematics and Computation , vol. 15,\nno. 1, pp. 29–45, 1984.\n[30] J. Buzhardt and P. Tallapragada, “Optimal trajectory tracking for\na magnetically driven microswimmer,” in 2020 American Control\nConference (ACC) , pp. 3211–3216, IEEE, 2020." }, { "title": "2305.00869v1.Estimating_the_Density_Ratio_between_Distributions_with_High_Discrepancy_using_Multinomial_Logistic_Regression.pdf", "content": "Published in Transactions on Machine Learning Research (03/2023)\nEstimating the Density Ratio between Distributions with\nHigh Discrepancy using Multinomial Logistic Regression\nAkash Srivastava∗akashsri@mit.edu\nMIT-IBM Watson AI Lab and IBM Research\nSeungwook Han∗swhan@mit.edu\nMIT\nKai Xu xuk@amazon.com\nAmazon\nBenjamin Rhodes ben.rhodes@ed.ac.uk\nSchool of Informatics, University of Edinburgh\nMichael U. Gutmann∗michael.gutmann@ed.ac.uk\nSchool of Informatics, University of Edinburgh\nReviewed on OpenReview: https: // openreview. net/ forum? id= jM8nzUzBWr\nAbstract\nFunctions of the ratio of the densities p/qare widely used in machine learning to quantify\nthe discrepancy between the two distributions pandq. For high-dimensional distributions,\nbinary classification-based density ratio estimators have shown great promise. However,\nwhen densities are well separated , estimating the density ratio with a binary classifier is\nchallenging. In this work, we show that the state-of-the-art density ratio estimators per-\nform poorly on well separated cases and demonstrate that this is due to distribution shifts\nbetween training and evaluation time. We present an alternative method that leverages\nmulti-class classification for density ratio estimation and does not suffer from distribution\nshift issues. The method uses a set of auxiliary densities {mk}K\nk=1and trains a multi-class\nlogistic regression to classify the samples from p,qand{mk}K\nk=1intoK+ 2classes. We\nshow that if these auxiliary densities are constructed such that they overlap with pandq,\nthen a multi-class logistic regression allows for estimating logp/qon the domain of any of\ntheK+ 2distributions and resolves the distribution shift problems of the current state-of-\nthe-art methods. We compare our method to state-of-the-art density ratio estimators on\nboth synthetic and real datasets and demonstrate its superior performance on the tasks of\ndensity ratio estimation, mutual information estimation, and representation learning. Code:\nhttps://www.blackswhan.com/mdre/\n1 Introduction\nQuantification of the discrepancy between two distributions underpins a large number of machine learning\ntechniques. For instance, distribution discrepancy measures known as f-divergences (Csiszár, 1964), which\nare defined as expectations of convex functions of the ratio of two densities, are ubiquitous in many domains\nof supervised and unsupervised machine learning. Hence, density ratio estimation is often a central task\nin generative modeling, mutual information and divergence estimation, as well as representation learning\n(Sugiyamaetal.,2012;Gutmann&Hyvärinen,2010;Goodfellowetal.,2014;Nowozinetal.,2016;Srivastava\net al., 2017; Belghazi et al., 2018; Oord et al., 2018; Srivastava et al., 2020). However, in most problems\n∗equal contribution\n1arXiv:2305.00869v1 [stat.ML] 1 May 2023Published in Transactions on Machine Learning Research (03/2023)\nof interest, estimating the density ratio by modeling each of the densities separately is significantly more\nchallenging than directly estimating their ratio for high dimensional densities (Sugiyama et al., 2012). Hence,\ndirect density ratio estimators are often employed in practice.\nOne of the most commonly used density ratio estimators (DRE) utilizes binary classification via logistic\nregression (BDRE). Once trained to discriminate between the samples from the two densities, BDREs have\nbeen shown to estimate the ground truth density ratio between the two densities (e.g. Gutmann & Hyväri-\nnen, 2010; Gutmann & Hirayama, 2011; Sugiyama et al., 2012; Menon & Ong, 2016). BDREs have been\ntremendously successful in problems involving the minimization of the density-ratio based estimators of dis-\ncrepancy between the data and the model distributions even in high-dimensional settings (Nowozin et al.,\n2016; Radford et al., 2015). However, they do not fare as well when applied to the task of estimating the dis-\ncrepancy between two distributions that are far apart or easily separable from each other . This issue has been\ncharacterized recently as the density-chasm problem by Rhodes et al. (2020). We demonstrate this in Figure\n1 where we employ a BDRE to estimate the density ratio between two 1-D distributions, p=N(−1,0.1)\nandq=N(1,0.2)shown in panel (a). Since pandqare considerably far apart from each other, solving the\nclassification problem is relatively simple as illustrated by the visualization of the decision boundary of the\nBDRE. However, as shown in panel (b), even in this simple setup, BDRE completely fails to estimate the\nratio. Kato & Teshima (2021) have also confirmed that most DREs, especially those implemented with deep\nneural networks, tend to overfit to the training data in some way when faced with the density-chasm prob-\nlem. Since BDRE-based plug-in estimators are often used in many high-dimensional tasks such as mutual\ninformation estimation, representation learning, energy-based modeling, co-variate-shift resolution, and im-\nportance sampling (Rhodes et al., 2020; Choi et al., 2021b;a; Sugiyama et al., 2012), resolving density-chasm\nis an important problem of high practical relevance.\nA recently introduced solution to the density-chasm problem, telescopic density-ratio estimation (TRE;\nRhodes et al., 2020), tackles it by replacing the easier-to-classify, original logistic regression problem, by a set\nof harder-to-classify logistic regression problems. In short, TRE constructs a set of Kauxiliary distributions\n({mk}K\nk=1) to bridge the two target distributions ( p=:m0andq=:mK+1) of interest and then trains a\nset ofK+ 1BDREs on every pair of consecutive distributions (mk−1andmkfork= 1,...,K), which are\nassumed to be close enough (i.e. not easily separable) for BDREs to work well. After that, an overall density\nratio estimate is obtained by taking the cumulative (telescopic) product of all individual estimates.\nIn this work, we argue that the aforementioned solution to the density chasm problem has an inherent issue\nofdistribution shift that can lead to significant inaccuracies in the final density ratio estimation. Notice that\nthei-th BDRE in the chain of BDREs that TRE constructs is only trained on the samples from distributions\nmiandmi+1. However, post-training, it is typically evaluated on regions where the distributions from the\noriginal density ratio estimation problem (i.e. pandq) have non-negligible mass. If the high-probability\nregions ofp,qand the auxiliary distributions mido not overlap, the training and evaluation distributions\nforthei-thBDREaredifferent. Becauseofthisdistributionshiftbetweentrainingandevaluation, theoverall\ndensity ratio estimation can end up being inaccurate (see Figure 2 and Section 2.1 for further details). We\nhere provide another solution to the density-chasm problem that avoids this distribution shift.\nWe present Multinomial Logistic Regression based Density Ratio Estimator (MDRE), a novel\nmethodfordensityratioestimationthatsolvesthedensity-chasmproblemwithoutsufferingfromdistribution\nshift. This is done by using auxiliary distributions and multi-class classification .MDREreplaces the easy\nbinary classification problem with a singleharder multi-class classification problem. MDREfirst constructs\na set ofKauxiliary distributions {mk}K\nk=1that overlap with pandqand then uses multi-class logistic\nregression on the K+ 2distributions to obtain a density ratio estimator of logp/q. We will show that the\nmulti-class classification formulation avoids the distribution shift issue of TRE.\nThe key contributions of this work are as follows:\n1. We study the state-of-the-art solution to the density-chasm problem (TRE; Rhodes et al., 2020)\nand identify its limitations arising from distribution shift. We illustrate that this inherent issue can\nsignificantly degrade its density ratio estimation performance.\n2Published in Transactions on Machine Learning Research (03/2023)probability\n0 20 40 60 80 100\nIteration0246810Train Loss\n10\n 8\n 6\n 4\n 2\n 0 2 4\nSamples500\n400\n300\n200\n100\n0100200Log RatioTrue p/q\nBDRE p/q\n0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nIteration0246810Test Loss\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n1.5\n 1.0\n 0.5\n 0.0 0.5 1.0 1.5\nSamples012345Probp\nq\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n(a) BDRE class probability\nlog-ratio\n0 20 40 60 80 100\nIteration0246810Train Loss\n10\n 8\n 6\n 4\n 2\n 0 2 4\nSamples500\n400\n300\n200\n100\n0100200Log RatioTrue p/q\nBDRE p/q\n0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nIteration0246810Test Loss\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n1.5\n 1.0\n 0.5\n 0.0 0.5 1.0 1.5\nSamples012345Probp\nq\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\n(b) BDRE log density-ratio\nprobability\n0 5000 10000 15000 20000 25000 30000 35000 40000\nIteration0246810Train Loss\n10\n 8\n 6\n 4\n 2\n 0 2 4\nSamples500\n400\n300\n200\n100\n0100200Log RatioTrue p/q\nCoB p/q\n0 100 200 300 400 500 600 700 800\nIteration0246810Test Loss\n0 10000 20000 30000 40000\nIteration0246810Train Loss\n1.5\n 1.0\n 0.5\n 0.0 0.5 1.0 1.5\nSamples012345Probp\nm\nq\n0 10000 20000 30000 40000\nIteration0246810Test Loss\n0 10000 20000 30000 40000\nIteration0246810Train Loss\n6\n 4\n 2\n 0 2 4 6 8 10\nSamples1000\n010002000300040005000Log RatioTrue p/q\nTRE p/q\n0 10000 20000 30000 40000\nIteration0246810Test Loss\n(c)MDREclass probability\nlog-ratio\n0 5000 10000 15000 20000 25000 30000 35000\nIteration0246810Train Loss\n10\n 8\n 6\n 4\n 2\n 0 2 4\nSamples500\n400\n300\n200\n100\n0100200Log RatioTrue p/q\nMDRE p/q\n0 100 200 300 400 500 600 700\nIteration0246810Test Loss\n0 10000 20000 30000 40000\nIteration0246810Train Loss\n1.5\n 1.0\n 0.5\n 0.0 0.5 1.0 1.5\nSamples012345Probp\nm\nq\n0 10000 20000 30000 40000\nIteration0246810Test Loss\n0 10000 20000 30000 40000\nIteration0246810Train Loss\n6\n 4\n 2\n 0 2 4 6 8 10\nSamples1000\n010002000300040005000Log RatioTrue p/q\nTRE p/q\n0 10000 20000 30000 40000\nIteration0246810Test Loss\n(d)MDRElog density-ratio\nFigure 1: BDRE vs proposed MDRE on estimation of logdensity ratio where p=N(−1,0.1)andq=\nN(1,0.2). For MDRE, the auxillary distribution mis CauchyC(0,1). Plots (a) and (c) show the class\nprobabilities P(Y|x)learned for BDRE and MDRE respectively overlayed on the plots of p,qandm.\nPlots (b) and (d) show the estimated log density-ratio by BDRE and MDRErespectively. Using auxiliary\ndistribution mallows MDREto better estimate the log density-ratio.\n2. We formally establish the link between multinomial logistic regression and density ratio estimation\nand propose a novel method ( MDRE) that uses auxiliary distributions to train a multi-class clas-\nsifier for density ratio estimation. MDREresolves the aforementioned distribution shift issue by\nconstruction and effectively tackles the density chasm problem.\n3. We construct a comprehensive evaluation protocol that significantly extends on benchmarks used in\nprior works. We conduct a systematic empirical evaluation of the proposed approach and demon-\nstrate the superior performance of our method on a number of synthetic and real datasets. Our\nresults show that MDREis often markedly better than the current state-of-the-art of density ratio\nestimation on tasks such as f-divergence estimation, mutual information estimation, and represen-\ntation learning in high-dimensional settings.\n2 Related Work\nTelescopic density-ratio estimation (TRE, Rhodes et al., 2020) uses a two step, divide-and-conquer strategy\nto tackle the density-chasm problem. In the first step, they construct Kwaymark distributions{mk}K\nk=1\nby gradually transporting samples from ptowards samples from q. Then, they train KBDREs, one for\neach consecutive pair of distributions. This allows for estimating the ratio rp/qas the product of K+ 1\nBDREs,rp/q:=p\nq=p\nm1×···×mK\nq. Rhodes et al. (2020) introduced two schemes for creating waymark\ndistributions that ensure that consecutive pairs of distributions are packed closely enough so that none of\ntheK+ 1BDREs suffer from the density-chasm issue. Hence, TRE addresses the density-chasm issue by\nreplacing the ratio between pandqwith a product of K+ 1intermediate density ratios that, by design of\nthe waymark distribution, should not suffer from the density-chasm problem. In a new work, Choi et al.\n(2021b) introduced DRE- ∞, a method that takes the number of waymark distributions in TRE to infinity\nand derives a limiting objective that leads to a more scalable version of TRE.\nF-DRE is other interesting related work that comes from Choi et al. (2021a). F-DRE uses a FLOW-\nbased model (Rezende & Mohamed, 2015) which is trained to project samples from a mixture of the two\ndistributions onto a standard Gaussian. They then train a BDRE. It is easy to show that any bijective\nmap will preserve the original density ratio rp/qin the feature space as the Jacobian correction term simply\ncancels out. However, due to the bijectivity of the FLOW map, such a method cannot bring the projected\ndistributions any closer than the discrepancy between the original distributions. At best, the method can\nshift the discrepancy between the original distributions along different moments after projection. Due to\nthis issue, we found that F-DRE did not work well for the problems we considered (see experimental results\nin Section 4). Recently, Liu et al. (2021) introduced an optimization-based solution to the density-chasm\nproblem in exponential family distributions by using (a) normalized gradient descent and (b) replacing the\nlogistic loss with an exponential loss. Finally, while BDRE remains the dominant method of density ratio\n3Published in Transactions on Machine Learning Research (03/2023)\nestimation in recent literature, prior works, such as Bickel et al. (2008) and Nock et al. (2016), have studied\nmulti-class classifier-based density ratio estimation for estimating ratios between a set of densities against a\ncommon reference distribution and its applications in multi-task learning.\n2.1 TRE’s performance can degrade due to training-evaluation distribution shifts\nIn supervised learning, distribution shift (Quiñonero-Candela et al., 2009) occurs when the training data\n(x,y)∼ptrainand the test data (x,y)∼ptestcome from two different distributions, i.e. ptrain/negationslash=ptest.\nCommon training methods, such as those used in BDRE, only guarantee that the model performs well on\nunseen data that comes from the same distribution as ptrain. Thus, in the case of distribution shift at test\ntime, themodel’sperformance degrades proportionatelyto theshift. Wenowshow thatasimilardistribution\nshift can occur in TRE when distributions pandqare sufficiently different. Recall that in TRE, we use\nBDREs to estimate K+ 1density ratios p/m 1,m2/m1,...,mK/qthat are combined in a telescopic product\nto form the overall ratio p/q. Let us denote the estimates of the K+ 1ratios by ˆη1,..., ˆηK+1.\nGiven the theoretical properties of BDRE, for any i∈{1,...,K +1},ˆηiestimatesrmi−1/miover the support\nofmi(Sugiyama et al., 2012; Gutmann & Hyvärinen, 2010; Menon & Ong, 2016). However, in TRE, when we\nevaluate the target ratio p/qon the supports of pandq, we evaluate the individual ˆηion domains for which\nwe lack guarantees that they perform well. Since the overall estimator for p/q≈ˆη1∗···∗ ˆηK+1combines\nmultiple ratio estimators, it suffers from the distribution shift issue if anyof the individual estimators’\nperformance deteriorates. Thus, if the supports of {mi}K\ni=1,p, andqare different, or when the samples from\n{mi}K\ni=1,p, andqdo not overlap well enough, the training and evaluation domains of the ˆηiare different\nand we expect the ratio estimate ˆηiand, in turn, the overall estimator for p/qto be poor. We now illustrate\nthis with a toy example.\nWe consider estimating the density ratio between p=N(−1,0.1)andq=N(1,0.2). Since,pandqare well\nseparated, we introduce three auxiliary distributions m1,m2,m3to bridge them, providing the waymarks\nthat TRE needs. The auxiliary distributions m1,m2,m3are constructed with the linear-mixing strategy\nthat will be described in Section 3.2. This setup is shown in the top-left panel of Figure 2. We train 4\nBDREs ˆη1,ˆη2,ˆη3,ˆη4to estimate ratios p/m 1,m1/m2,m2/m3andm3/qrespectively. We begin by showing\nthat each of the trained BDREs estimates their corresponding density ratio accurately on their corresponding\ntraining distributions. To show this, in panels 2-5 in the first row of Figure 2, we evaluate ˆη1,ˆη2,ˆη3,ˆη4on\nsamples from their respective denominator densities m1,m2,m3,qand plot them via a scatter plot where\nthe x-axis is labeled with the distribution that we draw the samples from and the y-axis is the log-density\nratio (red). We plot the true density ratio in blue for comparison. As evident, red and blue scatter plots\noverlap significantly, indicating the individual ratio estimators are accurate on their respective denominator\n(training) distributions.\nNext, we evaluate the BDREs ˆη1,ˆη2,ˆη3,ˆη4, on samples from pandqinstead of their corresponding training\ndistributions as before. Distributions pandqare shown in panel 1 of the second row in Figure 2. In the rest\nof the panels (2-5) in the second row, estimators ˆη1,ˆη2,ˆη3,ˆη4are compared to the ground-truth log-density\nratios (blue) p/m 1,m1/m2,m2/m3andm3/qthat are also evaluated on samples from pandq. Unlike in\nrow 1, the estimated log-density ratios do not match the ground-truth. This reflects the training-evaluation\ndistribution-shift issues pointed out above. We show now that this deterioration in accuracy on the level\nof the individual BDREs results in an deterioration of the overall performance of TRE. To this end, we\nfirst recover the TRE estimator by chaining the individually trained BDREs via a telescoping product, i.e.\nˆη1∗ˆη2∗ˆη3∗ˆη4and then evaluate it on samples from all the 5 distributions p,m 1,m2,m3,q. The results are\nshown in panels 1-5 of the third row. The estimated log-density ratios (red) do not match the corresponding\nground-truth log-density ratios (blue), which demonstrates that the distribution-shift in the training and\nevaluation distributions of the individual BDREs significantly degrades the overall estimation accuracy of\nTRE. Additional issues occur when both pandqdo not have full support, as discussed in Appendix G.\n4Published in Transactions on Machine Learning Research (03/2023)\nFigure 2: TRE for p=N(−1,0.1)andq=N(1,0.2)from Figure 1. In all scatter plots, x-axis denotes\nthe sampling distribution and y-axis denotes the log-density-ratio. The density plot in the first row shows\np,qand the 3 waymarks; the density plot in the second row shows pandqonly. The scatter plots in the\nfirst row show individual density ratio estimators evaluated on samples from their corresponding training\ndata (denominator density), demonstrating accurate estimation on the training set. The scatter plots in the\nsecond show individual density ratio estimators evaluated on samples from pandq. The estimation accuracy\nhas degraded notably due to the train-eval distribution shift. The last row shows the performance of the\noverall density ratio estimator on samples from p,m 1,m2,m3,q. We see that the overall ratio estimate is\nsignificantly affected by the deterioration of the individual ratio estimates, illustrating the sensitivity of TRE\nto distribution shift problems in case of well-separated distributions.\n3 Density Ratio Estimation using Multinominal Logistic Regression\nWe propose Multinomial Logistic Regression based Density Ratio Estimator (MDRE) to tackle\nthe density-chasm problem while avoiding the distribution shift issues of TRE. As in TRE, we introduce a\nset ofK≥1auxiliary distributions {mk}K\nk=1. But, in constrast to TRE, we then formulate the problem\nof estimating logp/qas a multi-class classification problem rather than a sequence of binary classification\nproblems. We show that this change leads to an estimator that is accurate on the domain of all K+ 2\ndistributions and, therefore, does not suffer from distribution shift.\n3.1 Loss function\nWehereestablishaformallinkbetweendensityratioestimationandmultinomiallogisticregression. Consider\na set ofCdistributions{pc}C\nc=1and letpx(x) =/summationtextC\nc=1πcpc(x)be their mixture distribution, with prior\nclass probabilities πc.1The multi-class classification problem then consists of predicting the correct class\nY∈{1,...,C}from a sample from the mixture px. For this purpose, we consider the model\nP(Y=c|x;θ) =πcexp(hc\nθ(x))/summationtextC\nk=1πkexp(hk\nθ(x)), (1)\n1In our simulations, we will use a uniform prior over the classes.\n5Published in Transactions on Machine Learning Research (03/2023)\nwhere thehc\nθ(x),c= 1,...,Care unnormalized log probabilities parametrized by θ. We estimate θby\nminimizing the negative multinomial log-likelihood (i.e. the softmax cross-entropy loss) L(θ)\nL(θ) =−C/summationdisplay\nc=1πcEx∼pc[logP(Y=c|x;θ)] =C/summationdisplay\nc=1πcEx∼pc/bracketleftbigg\n−logπc−hc\nθ+ logC/summationdisplay\nk=1πkexp(hk\nθ(x))/bracketrightbigg\n,(2)\nwhere, inpractice, theexpectationsarereplacedwithasampleaverage. Wedenotetheoptimalparametersby\nθ∗= arg minθL(θ). To ease the theoretical derivation, we consider the case where the hc\nθ(x)are parametrized\ninsuchaflexiblewaythatwecanconsidertheabovelossfunctiontobeafunctionalof Cfunctionsh1,...,hC,\nL(h1,...,hC) =C/summationdisplay\nc=1πcEx∼pc/bracketleftbigg\n−logπc−hc+ logC/summationdisplay\nk=1πkexp(hk(x))/bracketrightbigg\n. (3)\nThe following propositions shows that minimizing L(h1,...,hC)allows us to estimate the log ratios between\nany pair of the Cdistributions pc.\nProposition 3.1. Letˆh1,..., ˆhCbe the minimizers of L(h1,...,hC)in equation 3. Then the density ratio\nbetweenpi(x)andpj(x)for anyi,j≤Cis given by\nlogpi(x)\npj(x)=ˆhi(x)−ˆhj(x) (4)\nfor allxwherepx(x) =/summationtext\ncπcpc(x)>0.\nProof.Wefirstnotethatthesumofexpectations/summationtextC\nc=1πcEx∼pcinequation3isequivalenttotheexpectation\nwith respect to the mixture distribution px. Writing the expectation as an integral we obtain\nL(h1,...,hC) =C/summationdisplay\nc=1πcEx∼pc[−logπc−hc(x)] +/integraldisplay\npx(x)[logC/summationdisplay\nk=1πkexp(hk(x))]dx. (5)\nThe functional derivative of L(h1,...,hC)with respect to hi, i=1..., C, equals\nδL\nδhi=−πipi(x) +px(x)πiexp(hi(x))/summationtextC\nk=1πkexp(hk(x))(6)\nfor allxwherepx(x)>0. Setting the derivative to zero gives the necessary condition for an optimum\nπipi(x)\npx(x)=πiexp(hi(x))/summationtextC\nk=1πkexp(hk(x)), i = 1,...,C,and for allxwherepx(x)>0. (7)\nThe left-hand side of equation 7 equals the true conditional probability P∗(Y=i|x) =πipi(x)\npx(x). Hence, at\nthe critical point, ˆh1,..., ˆhCare such that P∗(Y|X)is correctly estimated. From equation 7, it follows that\nfor two arbitrary iandj, we have (πipi)/(πjpj) = (πiexp(ˆhi))/(πjexp(ˆhj))i.e.\nlogpi(x)\npj(x)=ˆhi(x)−ˆhj(x) (8)\nfor allxwherepx(x)>0, which concludes the proof.\nRemark 3.2 (Identifiability) .While we have Cunknownsh1,...,hCandCequations in equation 7, there is\na redundancy in the equations because\nC/summationdisplay\ni=1πipi(x)\npx(x)=C/summationdisplay\ni=1πiexp(hi(x))/summationtextC\nk=1πkexp(hk(x))=/summationtextC\ni=1πiexp(hi(x))\n/summationtextC\nk=1πkexp(hk(x))= 1\nThis means that we cannot uniquely identify all hiby minimising equation 3. However, the difference hi−hj,\nfori/negationslash=j, can be identified and is equal to the desired log ratio between piandpjper equation 8.\n6Published in Transactions on Machine Learning Research (03/2023)\nRemark 3.3 (Effect of parametrisation and finite sample size) .In practice, we only have a finite amount of\ntraining data and the parametrisation introduces constraints on the flexibility of the model. With additional\nassumptions, e.g. that the true density ratio logpi(x)−logpj(x)can be modeled by the difference of hi\nθ(x)\nandhj\nθ(x), we show in Appendix A that our ratio estimator is consistent. We here do not dive further\ninto the asymptotic properties of the estimator but focus on the practical applications of the key result in\nequation 8.\nImportantly, equation 8 allows us to estimate rp/qby formulating our ratio estimation problem as a multi-\nnomial nonlinear regression problem as summarized in the following corollary.\nCorollary 3.4. Let the distributions of the first two classes be pandq, respectively, i.e. p1≡p,p2≡q,\nand the remaining Kdistributions be equal to the auxiliary distributions mi, i.e.p3≡m1,...,pK+2≡mK.\nThen\nlog ˆrp/q(x) =ˆh1(x)−ˆh2(x). (9)\nRemark3.5 (Freefromdistributionshiftissues) .Sinceequation8holdsforall xwherethemixture px(x)>0,\nthe estimator ˆrp/q(x)in equation 9 is valid for all xin the union of the domain of p,q,m 1,...,mK. This\nmeans that MDREdoes not suffer from the distribution shift problems that occur when solving a sequence\nof binomial logistic regression problems as in TRE. We exemplify this in Section 3.3 after introducing three\nschemes to construct the auxiliary distributions m1,...,mK.\n3.2 Constructing the auxiliary distributions\nInMDRE, auxiliary distributions need to be constructed such that they have overlapping support with the\nempirical densities of pandq. This allows the multi-class classification probabilities to be better calibrated\nand leads to an accurate density ratio estimation. We demonstrate this in panel (c) of Figure 1, where\np=N(−1,0.1)andq=N(1,0.2)and the single auxiliary distribution mis set to be Cauchy C(0,1)that\nclearly overlaps with the other two distributions. The classification probabilities are shown as the scatter\nplot that is overlayed on the empirical densities of these distributions. Compared to the BDRE case in panel\n(a), which has high confidence in regions without any data, the multi-class classifier assigns, for pandq,\nhigh class probabilities only over the support of the data and not where there are barely any data points\nfrom these two distributions. Moreover, the auxiliary distribution well covers the space where pandqhave\nlow density, which provides the necessary training data to inform the values of ˆh1(x)andˆh2(x)in that area,\nwhich leads to an accurate estimate of the log-density ratio shown in panel (d). This is contrast to BDRE\nin panel (a) where the classifier, while constrained enough to get the classification right, is not learned well\nenough to also get the density ratio right (panel b). This subtle, yet important distinction between the\nusage of auxiliary distributions in MDREcompared to BDRE and TRE enables MDREto generalize on\nout-of-domain samples, as we will demonstrate in Section 4.1.\nNext, we briefly describe three schemes to construct auxiliary distributions for MDREand leave the details\nto Appendix B: (1) Overlapping Distribution Unlike TRE, the formulation of MDREdoes not require\n“gradually bridging” the two distributions pandq, hence, we introduce a novel approach to constructing\nauxiliary distributions. We define mkas any distribution whose samples overlap with both pandq, and\np<0. Hence the same arguments after equation 7 in the main text lead the\nparametric equivalent to equation 3.1, which we summarize in the following corollary.\nCorollary A.2. If the true conditional P∗(Y|x)is part of the parametric family {P(Y|x;θ)}θ, then\nhi\nˆθ(x)−hj\nˆθ(x) = logpi(x)\npj(x)(15)\nfor allxwherepx(x) =/summationtext\ncπcpc(x)>0.\nWe next derive conditions under which ˆθis the unique minimum, which is needed to prove consistency. For\nthat purpose, we perform a second-order Taylor expansion of L(θ)around ˆθ.\nLemma A.3.\nL(ˆθ+/epsilon1φ) =L(θ) +/epsilon12\n2φ/latticetopIφ (16)\nwhere/epsilon1 >0andI=−Epx(x)EP∗(Y|x)[H(Y,x)]. The matrix H(Y,x)contains the second derivatives of the\nlog-model, i.e. its (i,j)-th element is\n[H(Y,x)]ij=∂2\n∂θi∂θjlogP(Y|x;θ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nθ=ˆθ(17)\nwhereθiandθjare thei-th andj-th element of θ, respectively.\nProof.A second-order Taylor expansion around L(ˆθ)gives\nL(ˆθ+/epsilon1φ) =−Epx(x)/summationdisplay\ncP∗(Y=c|x) logP(Y=c|x,ˆθ+/epsilon1φ) (18)\n=L(ˆθ)−∇θL(θ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nθ=ˆθ−Epx(x)/summationdisplay\ncP∗(Y=c|x)/epsilon12\n2φ/latticetopH(Y=c,x)φ+O(/epsilon12) (19)\n=L(ˆθ)−/epsilon12\n2φ/latticetop/bracketleftBigg\nEpx(x)/summationdisplay\ncP∗(Y=c|x)H(Y=c,x)/bracketrightBigg\nφ+O(/epsilon12) (20)\nwhere we have used that the gradient of L(θ)is zero at a minimizer ˆθ. Since/summationtext\ncP∗(Y=c|x)H(Y=c,x) =\nEP∗(Y|x)H(Y,x), the result follows.\n15Published in Transactions on Machine Learning Research (03/2023)\nNote thatI(x) =−EP∗(Y|x)[H(Y,x)]is the conditional Fisher information matrix, and I=Epx(x)I(x)is its\nexpected value taken with respect to px(x).\nCorollary A.4. IfIis positive definite, then ˆθis the unique minimizer of L(θ).\nProof.IfIis positive definite, then φ/latticetopIφ > 0for all non-zero φand by Lemma A.3, L(ˆθ+/epsilon1φ)>L(ˆθ)\nwheneverφ/negationslash= 0.\nWe now consider the objective function Ln(θ)where the expectations in L(θ)are replaced by a sample\naverage over nsamples. Let ˆθn= arg minθLn(θ).\nProposition A.5. If (i)Iis positive definite and (ii) supθ|Ln(θ)−L(θ)|p− →0, then ˆθnp− →ˆθ.\nProof.By Corollary A.4, condition (i) ensures that ˆθ= arg minθL(θ)is a unique minimizer, and hence that\nchanging ˆθby a small amount will increase the cost function L(θ). Together with the technical condition\n(ii) on the uniform convergence of Ln(θ)toL(θ), this allows one to prove that ˆθnconverges in probability\ntoˆθas the sample size nincreases, following exactly the same reasoning as e.g. in proofs for consistency of\nmaximum likelihood estimation (Wasserman, 2004, Section 9.13) or noise-contrastive estimation (Gutmann\n& Hyvärinen, 2012, Appendix A.3.2).\nCorollary A.6. If (i)Iis positive definite, (ii) supθ|Ln(θ)−L(θ)|p− →0, and (iii) there is a parameter value\nθ∗such thatP∗(Y|x) =p(Y|x;θ∗), then ˆθnp− →θ∗\nProof.With Proposition A.5, condition (i) and (ii) ensure that ˆθnconverges to ˆθ= arg minθL(θ). With\nLemma A.1, ˆθis also minimizing Epx(x)KL(P∗(Y|x)||P(Y|x;θ)). Hence, if condition (iii) holds, ˆθ=θ∗,\nand the result follows.\nProposition A.7 (Consistency of the ratio estimator) .If (i)Iis positive definite, (ii) supθ|Ln(θ)−L(θ)|p− →\n0, (iii)P∗(Y|x) =p(Y|x;θ∗)for some parameter value θ∗, and (iv) the mapping from θtohc\nθis continuous,\nthen\nhi\nˆθn(x)−hj\nˆθn(x)p− →logpi(x)\npj(x)(21)\nfor allxwherepx(x) =/summationtext\ncπcpc(x)>0.\nProof.By Proposition A.5, condition (i) and (ii) ensure that ˆθnconverges to ˆθ= arg minθL(θ). By Corollary\nA.2, condition (iii) ensures that hi\nˆθ(x)−hj\nˆθ(x) = logpi(x)\npj(x)for allxwherepx(x) =/summationtext\ncπcpc(x)>0. Since\ncontinuous functions are closed under addition, the mapping from θtohi\nˆθ(x)−hj\nˆθ(x)is continuous if condition\n(iv) holds. We can then apply the continuous mapping theorem to conclude that hi\nˆθn(x)−hj\nˆθn(x)p− →\nhi\nˆθ(x)−hj\nˆθ(x) = logpi(x)\npj(x), which establishes the result.\nB Constructing M\nWe here elaborate on the three types of auxiliary distributions that we used in this work.\nOverlapping Distribution: TheMDREestimator, logp\nq= logp\nm−logm\nqis defined when p<95% for all classes), this implies that the classification\ntask is easy and therefore the DRE may suffer from the density chasm issue. On the other hand, if the\nclassification accuracy is too low ( <50% for all classes), then again, the DRE does not estimate well. We\nfound that targeting an accuracy curve as shown in Figure 10 (last panel) empirically leads to accurate\ndensity ratio estimation. This curve plots the test accuracy across all the classes and, empirically when it\nstays between the low and the high bounds of (50%,95%), the DRE estimates the ratios fairly well. The\nfirst panel shows that MDREestimates the ground truth ratio accurately across samples from all the K+ 2\ndistributions, the second panel shows that KL estimates of MDREis close to the ground truth KL and the\nthird panel shows that both test and training losses have converged. Figure 11 shows another example for\nthe case of randomized means. While MDREalso manages to get the ground truth KL correctly and most\nof the ratio estimates are also accurate, it does, however, slightly overestimate the log ratio for some of the\nsamples from p.\nF SpatialMultiOmniglot Experiment\nSpatialMultiOmniglot is a dataset of paired images uandv, whereuis an×ngrid of Omniglot characters\nfrom different Omniglot alphabets and vis an×ngrid containing the next characters of the corresponding\ncharacters in u. In this setup, we treat each grid of n×nas a collection of n2categorical random variables,\nnot the individual pixels. The mutual information I(u,v)can be computed as: I(u,v) =/summationtextn2\ni=1logli, where\nliis the alphabet size for the ithcharacter in u. This problem allows us to easily control the complexity of\nthe task since increasing nincreases the mutual information.\nFor the model, as in TRE, we use a separable architecture commonly used in MI-based representation\nlearning literature and model the unnormalized log-scores with functions of the form g(u)TWf(u), where\ngandfare 14-layer convolutional ResNets (He et al., 2015). We construct the auxiliary distributions via\ndimension-wise mixing—exactly the way that TRE does.\n21Published in Transactions on Machine Learning Research (03/2023)\n(a) Mutual information estimation\n (b) Representation learning accuracy\nFigure 12: SpatialMultiOmniglot representation learning results with same encoder for fandg.\nTo evaluate the representations after learning, we adopt a standard linear evaluation protocol to train a\nlinear classifier on the output of the frozen encoder g(u)to predict the alphabetic index of each character in\nthe gridu.\nAdditional Experiments\nIn addition to the experiments in the main text, we run an additional experiment with SpatialMultiOmniglot\nto test the effect of using the same encoder for gandf(i.e, modeling the unnormalized log-scores with the\nformg(u)TWg(v))instead of logp(u,v) =g(u)TWf(v)).\nSingle Encoder Design: We test the contribution of using two different encoders fandginstead of\none. As seen in Figure 12, in both cases of d= 4,9, the two models reach slightly different but similar MI\nestimates, but, interestingly, do not differ at all in the test classification accuracy. Empirically, we also found\nthat using one encoder helps the model converge to much faster. Overall, this experiment demonstrates that\nusing two different encoders does not necessarily work to our advantage.\nG TRE on Finite Support Distributions for K= 1\nForK= 1, TRE proposes the following telescoping: logp/q= logp/m + logm/q. As such, for TRE to be\nwell defined,dM\ndQi.e. the Radon-Nikodym Derivatives (RND) needs to exist. The consequence of this is that\nTRE is only defined when p<0Zω′\nc\nωn⟨Re[g(r,k, iωn)−1]⟩,(21)\nwhere the free-energy density is measured from its nor-\nmal value; F=FS− F N. In a homogeneous SC, the\nfree-energy density approaches to F(ρ)→ − ¯Fat low\ntemperature with ¯F=N0¯∆2/2 being the condensation\nenergy in the bulk.\nIn the numerical simulations, we fix the parameters:\nωc= 6πTc,T= 0.2Tc,δ= 0.01¯∆, and ω′\nc= 100 ¯∆ with\nξ0=ℏvF/2πTcbeing the coherence length.III. NUMERICAL RESULTS\nA. Pair potential and free-energy density\nWe first discuss the influences of the exchange poten-\ntial on the pair potential as shown in Fig. 2(a-c), where\nthe spatial variations of the pair potential are plotted for\nV0/¯∆ = 1 in (a), 2 in (b), and 3 in (c). The size of\nthe magnetic cluster is R0= 3ξ0. The pair potential is\nalmost homogeneous when the magnetization is compa-\nrable to (or smaller than) the superconducting gap (i.e.,\nV0<¯∆) as shown in Fig. 2(a). We refer to this state as\nthe zero-node state. For V0= 2¯∆ [Fig. 2(b)], the pair\npotential is suppressed and changes its sign once around\nρ= 2ξ0(one-node state). For V= 3¯∆ [Fig. 2(c)], the\npair potential changes the sign twice (two-node state):\nnegative approximately at 1 < ρ/ξ 0<2.1 and positive\natρ/ξ0<1. Namely, the FFLO-like superconducting\nstates are realized locally (i.e., only beneath the mag-\nnetic cluster).\nIn the presence of the Zeeman field, the condition\nfor the uniform superconducting state is given by V0<\n¯∆/√\n2.57,58The superconducting state for V0=¯∆ in\nFig. 2(a), however, goes beyond this limit. The pair po-\ntentials in Fig. 2(a-c) are self-consistently obtained as\nstable solutions of the Eilenberger equation. Such local\nFFLO states can be supported by the wide supercon-\nducting region outside the cluster. To confirm the valid-\nity of this argument, we calculate the free-energy density\nF(ρ) in Fig. 2(a-c). The vertical axis is normalized to\nthe condensation energy in an uniform superconductor\n¯F=¯∆2N0/2 at zero temperature. The free-energy den-\nsity outside the magnetic segment is negative (smaller\nthan the free-energy density in the normal state) and\napproaches to −¯Fforρ≫R0. Inside the magnetic seg-\nment, on the other hand, the free-energy density becomes\npositive locally. In particular, the free-energy density at\nV0=¯∆ in Fig. 2(a) is always positive at ρ < R 0. How-\never, the total free-energy FTot=R\nFdris always nega-\ntive because of the massive superconducting region out-\nside the cluster. Figure 2(b) and 2(c) show that introduc-\ning the nodes in the pair potential reduces the free-energy\ndensity at ρ < R 0drastically. Even so, the free energy at\nthe magnetic segment Fin=R\nr R 0.\nWhen the exchange potential increases to V0= 2¯∆ in\n(e), the LDOS spectra show a complicated profile due\nto the spatial variation of the pair potential. The sharp\nsubgap peak around E= 0.7¯∆ exist for 0 < ρ < 1.8ξ0\nandρ > 2.3ξ0. At the node of the pair potential ρ=\n2ξ0, the subgap peaks around E= 0.7¯∆ are drastically\nsuppressed and the LDOS shows almost flat spectra as it\ndoes in the normal state.\nThe same tendency can be also seen in the results for\nV0= 3¯∆ in Fig. 2(f). The spectra of LDOS for V0= 3¯∆\nin (f) become more inhomogeneous and complicated than\nthose in Figs. 2(d) and Fig. 2(e). At the outer node at\nρ= 2.2ξ0, the spectra are totally flat and LDOS does\nnot have large peaks below the gap. At the inner node\natρ=ξ0, however, the LDOS has a peak at zero energy.\nIn Sec. IV, we will discuss the LDOS spectra are very5\nsensitive to V0using the one-dimensional SF structure.\nC. Pairing correlations\nWe display the pairing correlation functions in\nFigs. 2(g-i) for V0=¯∆, 2 ¯∆, and 3 ¯∆, respectively. The\nresults are calculated for the lowest frequency ω0=πT.\nThe spin-singlet s-wave component for V0=¯∆ is al-\nways larger than the other components and almost flat\nas shown in Figs. 2(g). Only this component has a fi-\nnite amplitude far from the magnetic cluster. The two\np-wave components show a broad peak at the boundary\n(ρ=R0) as a result of the local inversion symmetry\nbreaking. The odd-frequency triplet s-wave component\nhas a relatively large amplitude than the induced p-wave\ncomponents around the center.\nIt is possible to derive the Eilenberger equation for\ncorresponding four coherence functions: γSW\n0,γPW\n0,γSW\n3,\nγPW\n3. The detailed results are displayed in the Appendix.\nThe equation following the first row of Eq. (A.5),\nvFkxd\ndxγSW\n0+ 2ωnγPW\n0+ 2V γPW\n3= 0, (22)\nindicates that the spatial variation of the pair poten-\ntial (spin-singlet s-wave superconductor) generates two\np-wave components γPW\n0andγPW\n3. In addition, the equa-\ntion following the second row of Eq. (A.5),\nvFkxd\ndxγPW\n0+ 2ωnγSW\n0+ 2V γSW\n3= ∆,(23)\nexplains the appearance of the spin-triplet s-wave com-\nponent γSW\n3even in a uniform pair potential. Here we\nsummarize our knowledge of the relation between the fre-\nquency symmetry of a Cooper pair and their influence on\nthe free energy and on the quasiparticle LDOS.\nI Usual even-frequency pairs indicate the diamagnetic\nresponse to magnetic fields and favor the spatially\nuniform superconducting phase at the ground state.\nOn the other hand, odd-frequency Cooper pairs are\nparamagnetic.59Therefore, odd-frequency pairs in-\ncrease the free energy of uniform ground state60\nand favor the spatial gradient of superconducting\nphase.37\nII The LDOS has a gapped energy spectrum in the\npresence of even-frequency pairs, whereas it tends\nto have peaks below the gap in the presence of odd-\nfrequency pairs.61,62\nThese properties qualitatively explain the characteris-\ntic behavior in the free-energy density and those in the\nLDOS. The LDOS for V0=¯∆ in Fig. 2(d) shows the\ngap-like energy spectra because even-frequency compo-\nnent fSW\n0is dominant everywhere. When the exchange\npotential increases to V0= 2¯∆, a node appears in the\npair potential. As a consequence, the spatial profile of\nFIG. 3. (a) Cluster-size dependence of pair potentials at\nV0= 2¯∆. (b,c) Pair potential at the center of the magnetic\ncluster ( ρ= 0) as a function of the exchange potential V0.\nThe radius of the cluster varies from R=ξ0to 6ξ0byξ0.\nThe results are plotted with the offset by ( R0/ξ0−1)¯∆ The\nhorizontal broken lines indicate zeros. The temperature is set\nto (a,b) T= 0.2Tcand (c) 0 .1Tc.\nthe pairing correlations drastically changes as shown in\nFig. 2(h). The two p-wave components have peaks in-\nside the boundary because the pair potential and the\nexchange potential break the inversion symmetry locally.\nThe amplitudes of the two p-wave components are larger\nthan those in Fig. 2(g). The two spin-triplet components\nhave large amplitudes around the node of the pair poten-\ntial. The sign change of the pair potential is equivalent to\nthe local π-phase shift in the pair potential. According to\nthe property I, odd-frequency spin-triplet s-wave compo-\nnents fSW\n3appear to decrease the free-energy density at\nthe node of the pair potential. The triplet p-wave compo-\nnentfPW\n3is the most dominant at the node ρ= 2ξ0. As a\nconsequence, LDOS at the node in Fig. 2(e) does not have\nlarge peaks below the gap. The two odd-frequency com-\nponents are the source of the subgap peak at E= 0.7¯∆\nin Fig. 2(e) for 0 < ρ < 1.8ξ0. Outside the magnetic seg-\nment, fSW\n0is a source of the coherence peak at E=¯∆\nandfPW\n0assists the subgap peak at E= 0.7¯∆. The\ntwo components fSW\n0andfPW\n3seem to affect the LDOS\nindependently at ρ > R 0.\nIn a two-node state at V0= 3¯∆, the pairing corre-6\nFIG. 4. Hysteresis loop of ∆ c. The degree of the hysteresis\nis more prominent for larger magnetic clusters and at lower\ntemperatures. The arrows indicate how the exchange poten-\ntial is changed in the numerical simulation. The radius of the\nisland is set to R0= 3ξ0.\nlation functions oscillate in the ferromagnetic segment\nmore rapidly as shown in in Fig. 2(i). At the outer\nnode ρ= 2.2ξ0, the two spin-triplet components ( fPW\n3\nandfSW\n3) have peaks. The spectra of LDOS show the\nflat structure because the amplitude of fPW\n3and that of\nfSW\n3are almost the same at the outer node. Around\nthe inner node at ρ=ξ0, the two odd-frequency pairing\ncorrelations have larger amplitudes than the two even-\nfrequency pairing correlations. As a consequence, LDOS\nhas a peak at zero energy for 0 .7ξ0< ρ < 2ξ0. Therefore,\nthe relative amplitudes among the four correlation func-\ntions govern the subgap spectra in the LDOS. When the\nodd-frequency (even-frequency) pairing correlations are\ndominant, the LDOS tend to have peak (gap) at E < ¯∆.\nAt the end of this subsection, we briefly summarize\nthe two properties of the local FFLO states. First, zero-\nenergy peaks appear at the edge of the ferromagnetic\nsegment ρ=R0in Figs. 2(e) and 2(f). When V0is close\nto a transition point between the n-node and n±1-node\nstates, the LDOS tends to have a zero-energy peak at the\nboundary. However, this zero-energy peak is not univer-\nsal and depends on the amplitude of the magnetization.\nWe will further explore this issue in Sec. IV.\nSecondly, Eq. (22) implies that the spatial variation\nof the singlet s-wave component (linked to the pair po-\ntential) generates two p-wave components. Simultane-\nously, we can state that the p-wave pairing correlations\ndrive the spatial variation of the pair potential. There-\nfore, p-wave pairing correlations are indispensable to re-\nalizing the FFLO states. This insight consistent with\na fact that the FFLO states are fragile againt impurity\nscatterings.63–65To our knowledge, a spin-singlet s-wave\nsuperconducting state is fragile when it contains odd-\nfrequency pairing correlations in the clean limit.66,67\nFIG. 5. Evolutions of pair amplitudes over magnetization.\n(a) Even-frequency spin-singlet s-wave, (b) Odd-frequency\ntriplet s-wave, (c) Odd-frequency singlet p-wave, and (d)\nEven-frequency triplet p-wave are shown. The spatial pro-\nfile of the pair potential is qualitatively the same as those in\n(a); the principal component. The magnetization varies from\nV0/¯∆ = 0 .5 to 4 .0 by 0 .1. The radius of the island and the\ntemperature are set to R0= 3ξ0andT= 0.1Tc. The even-\nfrequency components (a,d) exhibit discrete behavior, while\nthe odd-frequency components (b,c) vary gradually. The val-\nues of V0are given in the legend only for the thick lines.\nD. Discontinuous transition\nThe spatial profiles of the pair potential with several\nradii of the magnetic cluster are shown in Fig. 3(a), where\nwe choose R0/ξ0= 3, 5, and 7 and V0= 2¯∆. The results\nindicate that, if the radius of the magnetic cluster is large\nenough, the multi-node states appear even with a weak\nexchange potential. In Figs. 3(b) and 3(c), we plot the\npair potential at the center of the ferromagnetic segment\n∆c≡∆(ρ= 0) as a function of V0,68where the radius\nvaries from R0=ξ0to 6ξ0byξ0with the corresponding\noffset and the temperature is set to (b) T/T c= 0.2 and\n(c) 0.1. The pair potential keeps the homogeneous profile\n(i.e., ∆ c≈¯∆ without a node) until V0reaches to a crit-\nical value, which we define V1. At V0=V1, ∆cchanges\nthe sign abruptly, meaning that a node appears in the\npair potential. Each time ∆ cchanges the sign in Fig. 3,\nthe number of nodes in the pair potential changes by\none. The results show a relation V1≈¯∆ holds R0>2ξ0\nwhich limits the validity of the theoretical model for a\ntopologically nontrivial superconducting nanowire.69,70\nWe also find jumps in ∆ catV0=V1shows the hystere-\nsis between the two processes in the numerical simulation:\nincreasing and decreasing V0. The results for R0= 3ξ0\nare displayed in Fig. 4, where we choose (a) T= 0.2Tc\nand (b) 0 .1Tc. The hysteresis loop is more prominent at\na lower temperature. We have confirmed that the hys-\nteresis loop appears also between the one- and two-node\nstates if the temperature is low enough.\nComparing Figs. 3(b) and 3(c), we see the discontin-\nuous behavior is the more remarkable at the lower tem-7\nperature. The first term of the Eilenberger equation (7)\nis [∂x+(2ξT)−1]ˆγalong the xcoordinate, where we focus\non the lowest Matsubara frequency and ξT=ℏvF/2πT\nis the thermal coherence length in the clean limit. In\nthis case, the length scale of the spatial variation of ˆ γis\napproximately given by ξT. In other words, the spatial\nvariation in the coherence functions are correlated within\nthe range with πξ2\nT. At T= 0.1Tc, the thermal coher-\nence length becomes ξT= 10ξ0which covers the whole\nferromagnetic segment. In this case, ∆ prefers a homo-\ngeneous profile as much as possible until V0exceeds a\ncritical value because the influence of a node in the pair\npotential spreads over a wide region of approximately\nπξ2\nT. Therefore, at low temperature, the jump in ∆ cis\nmore abrupt. Figure 3 also indicates that the disconti-\nnuity between the zero-node and the one-node states is\nmore remarkable for larger clusters. At present, however,\nwe cannot think of reasons for this tendency.\nIn Fig. 5, we display the pairing correlations func-\ntions by changing the exchange potential gradually for\nR0= 3ξ0. The spatial profile of the spin-singlet s-wave\ncomponent in the n-node state is qualitatively different\nfrom those in n±1-node states [Fig. 5(a)]. Roughly\nspeaking, the spatial profiles of fSW\n0is insensitive to V0\nas long as the number of nodes in the pair potential re-\nmains the same value. As a result, the calculated results\nforfSW\n0are bundled for each state. The same discontinu-\nous behavior appears in the even-frequency spin-triplet p-\nwave component in Fig. 5(d). Such discontinuous behav-\nior in the pairing correlation functions is responsible for\nthe jump of the pair potential between the n-node state\nandn+1-node state. The spatial profile of the two even-\nfrequency pairing components is governed mainly by the\nnumber of nodes. The two odd-frequency components\nin Fig. 5(b) and 5(c), on the other hand, changes grad-\nually with increasing V0. Consequently, odd-frequency\npairs relax the effects of discontinuous change in the pair\npotential at the transition point. This might be a role\nof odd-frequency pairs in the FFLO states. The gradual\nchange of the odd-frequency pairing correlations causes\nthe gradual change of subgap spectra in the LDOS. We\nwill discuss this issue in Sec. IV.\nIV. ANALYSIS IN ONE-CHANNEL MODEL\nThe one-channel model is represented by putting ky=\n0 in all the equations in Sec. II and describes a one-\ndimensional superconducting structure including the ex-\nchange potential of V(x) =V0Θ(L0− |x|). Although the\none-channel model is not realistic, the characteristic be-\nhaviors in the LDOS in the one-channel model are simpler\nthan those in the two-dimension.\nThe pair potential at the center of the system (i.e.,\n∆c= ∆|x=0is shown in Fig. 6(a) where L0= 3ξ0and\nT= 0.2Tc. We focus on the several characteristic ex-\nchange potentials indicated by the arrows: V0/¯∆ = 0 .8\n(zero-node), 1 .1 (just before the transition), 1 .2 (just af-\nFIG. 6. Results for the one-channel model. The length of\nthe ferromagnet and the temperature are set to L0= 3ξ0and\nT= 0.2Tc, respectively. (a) Pair potential at the center of\nthe system. (b) Spatial profiles of the pair potential. (c-h)\nLocal density of states. The magnetization is set to the char-\nacteristic values indicated by the arrows in (a): V0/¯∆ = 0 .8\n(zero-node), 1 .1 (before the transition), 1 .2 (after the transi-\ntion), 1 .9 (one-node state), 2 .9 (after the second transition),\nand 3 .8 (two-node state). The vertical broken lines in (c-h)\nindicate the position of the nodes.\nter the transition), 1 .9 (one-node state), 2 .9 (just after\nthe second transition), and 3 .8 (two-node state). The\nspatial profile of the pair potentials in Fig. 6(b) is qual-\nitatively the same as those in the 2D case [See Fig. 2(a-\nc)]. We have also confirmed that the transition between\nthe one-node and two-node states becomes discontinu-\nous at a low temperature (The results are not shown).\nThe LDOS for the characteristic exchange potentials are\nshown in Fig. 6(c-h). The LDOS indicate that the quasi-\nparticle spectra are not simply determined by the number\nof nodes but depend sensitively on the exchange poten-\ntial. In the zero-node state in Fig. 6(c), the LDOS at\nthe interface ( x= 3ξ0) has peaks around |E|= 0.6 and\nthat at the center of the magnetic segment has peaks\naround |E|= 0.9. Just below the transition point to the\none-node state, the LDOS at the boundary ( x= 3ξ0)\nhas a peak at zero energy as shown in Fig. 6(d). The\nzero-energy peak at the boundary can be seen also just8\nabove the transition point in Fig. 6(e) and 6(g). At\nV0/¯∆ = 1 .9, the one-node state is stable because the ex-\nchange potential is far from the two transition points of\nV0/¯∆ = 1 .15 and 2 .85 [see Fig. 6(a)]. The corresponding\nLDOS [Fig. 6(f)] shows that the spectra are flat N≈N0\nat the node of the pair potential and gapped at the\nboundary. The same behavior appears also for the sta-\nble two-node states at V0/¯∆ = 3 .8 [Fig. 6(h)]. Although\nthe pair potentials for V0/¯∆ = 1 .2 and 1 .9 have the sim-\nilar profiles, the subgap spectra [Figs. 6(e) and 6(f)] are\nqualitatively different. As already discussed in Fig. 5, the\nspatial profiles of the odd-frequency components for these\nstates deviate from each other. As a result, the subgap\nspectra in (e) and (f) are totally different from each other\naccording to property II. The spatial profiles of the odd-\nfrequency components are very sensitive to V0and those\nfor these two states are qualitatively different (the results\nare not shown but similar to those in Fig. 5). Therefore,\nthe subgap spectra shown in Figs. 6(e) and 6(f) exhibit\ndistinct differences, which can be attributed to property\nII. This discussion can be applied also to the LDOS in\nthe two two-node states shown in Figs. 6(g) and 6(h).\nThus, the gradual changes of the odd-frequency pairing\ncorrelations are responsible for the gradual changes in the\nLDOS.\nV. CONCLUSION\nWe have theoretically studied the property of the\nFulde-Ferrell-Larkin-Ovchinnikov (FFLO) state which\nappears in a superconducting thin film attached to a\ncircular-shaped magnetic cluster. By solving the quasi-\nclassical Eilenberger equation, we calculate the pair po-\ntential, free-energy density, the pairing correlation func-\ntions, and the local density of states. The FFLO states\nare locally realized beneath the ferromagnetic cluster as\na stable solution of the self-consistent gap equation. The\nfree-energy density shows that the local FFLO states\nare supported by superconducting condensate surround-\ning the magnetic cluster. As the exchange potential in-\ncreases, the number of nodes in the pair potential in-\ncreases one by one.\nThe spatial profiles of the even-frequency pairing cor-\nrelations are not sensitive to the exchange potential but\nare determined mainly by the number of nodes in the\npair potential. On the other hand, the odd-frequency\npairing correlations show a gradual change with increas-\ning the exchange potential. The local density of states\nis inhomogeneous in the local FFLO state. In addition,\nthe subgap spectra depend sensitively on the exchangepotential because the odd-frequency pairs coexist with\nsubgap quasiparticles.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI (No.\nJP20H01857). S.-I. S. acknowledges Overseas Research\nFellowships by JSPS and the hospitality at the University\nof Twente. T. S. is supported in part by the establish-\nment of university fellowships towards the creation of sci-\nence technology innovation from the Ministry of Educa-\ntion, Culture, Sports, Science, and Technology (MEXT)\nof Japan.\nAppendix: Analysis in linearized Eilenberger\nequation\nIt is possible to derive the Eilenberger equation for the\ncorresponding four coherence functions defined by\nγSW\n0=1\n2{γ0(k) +γ0(−k)}, (A.1)\nγPW\n0=1\n2{γ0(k)−γ0(−k)}, (A.2)\nγSW\n3=1\n2i{γ3(k) +γ3(−k)}, (A.3)\nγPW\n3=1\n2i{γ3(k)−γ3(−k)}. (A.4)\nHere we assume that an s-wave ( p-wave) component is\nthe most dominant in an even-parity (odd-parity) coher-\nence function. The Eilenberger equation for these com-\nponents results in\nvFk· ∇\nγSW\n0\nγPW\n0\nγSW\n3\nγPW\n3\n+ 2\n0ω0V\nω 0V0\n0−V0ω\n−V0ω0\n\nγSW\n0\nγPW\n0\nγSW\n3\nγPW\n3\n\n+\n2(γSW\n0γPW\n0−γSW\n3γPW\n3)\n(γSW\n0)2+ (γPW\n0)2−(γSW\n3)2−(γPW\n3)2\n2(γSW\n0γPW\n3+γPW\n0γSW\n3)\n2(γSW\n0γSW\n3−γPW\n0γPW\n3)\n\n=\n0\n∆\n0\n0\n. 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Lett.\n105, 177002 (2010).\n70R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.\nLett. 105, 077001 (2010)." }, { "title": "2305.07601v2.Density_of_states_of_a_2D_system_of_soft__sphere_fermions_by_path_integral_Monte_Carlo_simulations.pdf", "content": "Density of states of a 2D system of soft–sphere\nfermions by path integral Monte Carlo simulations\nV. Filinov1, P. Levashov1,2, A. Larkin1\n1Joint Institute for High Temperatures, Russian Academy of Sciences,\nIzhorskaya 13 bldg 2, Moscow 125412, Russia\n2Moscow Institute of Physics and Technology, 9 Institutskiy per.,\nDolgoprudny, Moscow Region, 141700, Russia\nE-mail: vladimir filinov@mail.ru\nApril 2023\nAbstract. The Wigner formulation of quantum mechanics is used to derive a new\npath integral representation of quantum density of states. A path integral Monte Carlo\napproach is developed for the numerical investigation of density of states, internal\nenergy and spin–resolved radial distribution functions for a 2D system of strongly\ncorrelated soft–sphere fermions. The peculiarities of the density of states and internal\nenergy distributions depending on the hardness of the soft–sphere potential and particle\ndensity are investigated and explained. In particular, at high enough densities the\ndensity of states rapidly tends to a constant value, as for an ideal system of 2D fermions.\nKeywords: Density of states, Wigner representation, Path integral Monte Carlo\nSubmitted to: J. Phys. A: Math. Gen.\n1. Introduction\nDensity of states (DOS) is a key factor in condensed matter physics determining many\nproperties of matter [1]. The DOS is proportional to the fraction of states per unit\nvolume that have a certain energy. The product of the DOS and the probability\ndistribution function gives the fraction of occupied states at a given energy for a\nsystem in thermal equilibrium. Computing DOS is of fundamental importance and\nmany works for different systems are devoted to this problem. Popular approaches are\nbased on generalized ensembles and reweighting techniques. One of the most prominent\napproaches is the Wang–Landau (WL) algorithm, which is a well-known Monte Carlo\ntechnique for computing the DOS of classical systems [2–9].\nDOS is much more important for determining the properties of quantum systems of\nparticles [10]. For instance, researchers working in the solid–state and condensed matter\nphysics usually apply quantum mechanical approaches, such as density functional theoryarXiv:2305.07601v2 [physics.comp-ph] 11 Sep 2023Density of states of a 2Dsystem 2\n(DFT) [11, 12]. However, DFT is approximate and in some cases leads to a severe\ncomputational workload [13].\nThus, there are many attempts to develop fast and high–accuracy methods to\npredict the DOS and internal energy distribution of materials [14–16]. An interesting\napproach involving path integrals was suggested in the article [17], in which the entropic\nsampling [18] was applied within the Wang–Landau algorithm to calculate the DOS for a\n3D quantum system of harmonic oscillators at a finite temperature. In the path integral\nformalism quantum particles are presented as “ring polymers” consisting of a lot of\n“beads” connected by harmonic-like bonds (springs) [17]. In Ref. [17] the exact data for\nthe energy and canonical distribution were reproduced for a wide range of temperatures.\nIn this paper we propose a new path integral representation of DOS in the Wigner\nformulation of quantum mechanics and the path integral Monte Carlo method (WPIMC)\nfor its calculation. We hope that our approach will be a compromise between the\naccuracy and speed of calculations. The suggested approach is applicable for predicting\nDOS not only for bulk structures (3D) but also for surfaces (2D) in multi-component\nsystems. To illustrate the basic ideas we make use of a simple model of strongly coupled\nsoft-sphere fermions that is useful in statistical mechanics and capable of grasping some\nphysical properties of complex systems. This model includes also the one-component\nplasma (OCP), which is of great astrophysical importance being an excellent model for\ndescribing many features of superdense, completely ionized matter [19, 20]. Moreover,\ntheoretical studies of strongly interacting particles obeying the Fermi–Dirac statistics is\na subject of general interest in many fields of physics.\nAt strong interparticle interaction perturbative methods cannot be applied, so\ndirect computer simulations have to be used. At non-zero temperatures the most\nwidespread numerical method in quantum statistics is a Monte–Carlo (MC) method\nusually based on the representation of a quantum partition function in the form of path\nintegrals in the coordinate representation [21,22]. A direct computer simulation allows\none to calculate the thermodynamic properties of dense noble gases, dense hydrogen,\nelectron-hole and quark-gluon plasmas, etc. [23–29].\nThe main difficulty of the path integral Monte-Carlo (PIMC) method for Fermi\nsystems is the “fermionic sign problem” arising due to the antisymmetrization of a\nfermion density matrix [21]. For this reason thermodynamic quantities become small\ndifferences of large numbers associated with even and odd permutations. To overcome\nthis issue a lot of approaches have been developed. In Ref. [30, 31], to avoid the\n“fermionic sign problem”, a restricted fixed–node path–integral Monte Carlo (RPIMC)\napproach was proposed. In the RPIMC only positive permutations are taken into\naccount, so the accuracy of the results is unknown. More consistent approaches are\nthe permutation blocking path integral Monte Carlo (PB-PIMC) and the configuration\npath integral Monte Carlo (CPIMC) methods [25]. In the CPIMC the density matrix\nis presented as a path integral in the space of occupation numbers. However it turns\nout that both methods also exhibit the “sign problem” worsening the accuracy of PIMC\nsimulations.Density of states of a 2Dsystem 3\nIn [32, 33] an alternative approach based on the Wigner formulation of quantum\nmechanics in the phase space [34,35] was used to avoid the antisymmetrization of matrix\nelements and hence the “sign problem”. This approach allows to realize the Pauli\nblocking of fermions and is able to calculate quantum momentum distribution functions\nas well as transport properties [23,24].\nHere we propose a path integral representation of DOS in the Wigner phase space.\nThis approach also allows to reduce the “sign problem” as the exchange interaction is\nexpressed through a positive semidefinite Gram determinant [36].\nIn section II we consider the path integral description of quantum DOS in the\nWigner formulation of quantum mechanics. In section III we derive a pseudopotential\nfor soft spheres accounting for quantum effects in the interparticle interaction. In section\nIV we present the results of our simulations. For a 2D quantum system of strongly\ncorrelated soft–sphere fermions we present the DOS, internal energy distributions and\nspin – resolved radial distribution functions obtained by the new path integral Monte\nCarlo method (WPIMC) (see Supplemental Material) for the hardness of the soft–sphere\npotential smaller or of the order of unity. In section V we summarize the basic results.\n2. Path integral representation of the density of state\nWe consider a 2D system of Nsoft–sphere particles obeying the Fermi–Dirac statistics.\nThe Hamiltonian of the system ˆH=ˆK+ˆUcontains the kinetic ˆKand interaction\nenergy ˆUcontributions taken as the sum of pair interactions ϕ(r) =ϵ(σ/r)n, where r\nis the interparticle distance, σcharacterizes the effective particle size, ϵsets the energy\nscale and nis a parameter determining the potential hardness.\nDensity of state (DOS) is a fundamental function of a system and can be defined\nas Ω( E) = Tr {δ(EˆI−ˆH)}[37]. Ω( E)dEdetermines the number of states between E\nandE+ dEper unit volume [38] ( ˆI is the unit operator, while δis the delta function).\nThe DOS can be used to compute important thermodynamic properties such as, for\nexample, internal energy, entropy and heat capacity.\nIn our approach we are going to rewrite Ω( E) in an identical form using the property\nof the delta function\nΩ(E) = Tr {δ(EˆI−ˆH)ˆI}= Tr{δ(EˆI−ˆH) exp( EˆI−ˆH)}\n=1\n2πZ\ndωTr{exp iω\u0000\nEˆI−ˆH\u0001\nexp(E ˆI−ˆH)}=1\n2πZ\ndωTr{expκ(ω)\u0000\nEˆI−ˆH\u0001\n}\n=1\n2πZ\ndωZ\ndq1D\nq1\f\f\fexpκ(ω)\u0000\nEˆI−ˆH\u0001\f\f\fq1E\n,(1)\nwhere κ(ω) = 1+i ω, angular brackets ⟨q|˜q⟩mean the scalar products of the eigenvectors\n|q⟩and|˜q⟩of the coordinate operator ˆ q(⟨ˆq|q⟩=q|q⟩,⟨q|˜q⟩=δ(q−˜q)),ˆI =R\n|q⟩dq⟨q|\nis the unit operator, ψ(q) =⟨q|ψ⟩is the wave function [35] ), the angular brackets\nin expression ⟨q1|A|q⟩mean the scalar products of vectors |q1⟩and|ˆA|q⟩arising after\nthe action of operator ˆAon vector |q⟩, i is the imaginary unit. Further in the text, itDensity of states of a 2Dsystem 4\nis convenient to imply that energy is expressed in units of kBT(kBis the Boltzmann\nconstant, Tis the temperature of the system) and q1is a 2 N-dimensional vector of the\nparticle coordinates.\nSince the operators of kinetic and potential energy do not commute, an exact\nexplicit analytical expression for the DOS is unknown but can be constructed using a\npath integral approach [21,22,39] based on the operator identity exp\u0000\nκ(ω)\u0000\nEˆI−ˆH\u0001\u0001\n=\nexp\u0000\nϵ(ω)\u0000\nEˆI−ˆH\u0001\u0001\n× ··· × exp\u0000\nϵ(ω)\u0000\nEˆI−ˆH\u0001\u0001\nwith ϵ(ω) =κ(ω)/M, where Mis a\nlarge positive integer. So the DOS can be rewritten in the coordinate representation as\nΩ(E) =1\n2πZ\ndωZ\ndq1D\nq1\f\f\fexpκ(ω)\u0000\nEˆI−ˆH\u0001\f\f\fq1E\n=1\n2πZ\ndωMY\nj=1Z\ndqjd˜qj\n×\u001c\nq1\f\f\f\fexpiω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜q1\u001d\u001c\n˜q1\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜q2\u001d\n×\u001c\n˜q2\f\f\f\fexpiω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq2\u001d\u001c\nq2\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq3\u001d\n×\u001c\nq3\f\f\f\fexpiω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜q3\u001d\u001c\n˜q3\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜q4\u001d\n. . .\n×\u001c\n˜qM\f\f\f\fexpiω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fqM\u001d\u001c\nqM\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq1\u001d\n,(2)\nwhere we have used the coordinate representation of the unit operator ˆI =R\n|q⟩dq⟨q|[35].\nTo present the DOS in the Wigner representation of quantum mechanics let us\nconsider the Weyl symbol of an operator. For example, for the operator ˆHthe\ncorresponding Weyl symbol is the Hamiltonian function H(pq) [34,35]\nH(pq) =Z\ndξexp(i⟨p|ξ⟩)D\nq−ξ/2\f\f\fˆH\f\f\fq+ξ/2E\n, (3)\nwhere the vectors ξand momentum pare 2N–dimensional vectors.\nThe inverse Fourier transform allows to express matrix elements of operators\nthrough their Weyl symbols. So for large Mwith the error of the order of (1 /M)2\nrequired for the path integral approach [21,22] we have\n\u001c\nQj−ξj/2\f\f\f\fexpiω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fQj+ξj/2\u001d\n≈\u001c\nQj−ξj/2\f\f\f\fˆI +iω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fQj+ξj/2\u001d\n+ O\u00121\nM\u00132\n=\u00121\n2π\u0013(2N)Z\ndPjexp(−i⟨Pj|ξj⟩)\u0012\n1 +iω\nM\u0000\nE−H(Pj, Qj)\u0001\u0013\n≈\u00121\n2π\u0013(2N)Z\ndPjexp (−i⟨Pj|ξj⟩) exp\u0012iω\nM\u0000\nE−H(Pj, Qj)\u0001\u0013\n+ O\u00121\nM\u00132\n,(4)\nwhere new variable Qjandξjare defined by equations: Qj= (˜qj+qj/2),ξj= (˜qj−qj) for\nj= 1, . . . , M (qj=Qj−ξj/2, ˜qj=Qj+ξj/2) and H(Pj, Qj) =⟨Pj|Pj⟩/2m+U(Qj) areDensity of states of a 2Dsystem 5\nthe sums of the Hamilton functions for Nparicles at a given j. Further for convenience\nwe will use both set of variable ( Q,ξ) and ( q, ˜q). The final expression for the product\nis\nMY\nj=1\u001c\nqj\f\f\f\fexpiω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜qj\u001d\n≈\u00121\n2π\u0013(2NM)MY\nj=1Z\ndPjexp(−i⟨Pj|ξj⟩) expiω\nM\u0000\nE−H(Pj, Qj)\u0001\n.\n(5)\nThen the DOS is presented as\nΩ(E) =\u00121\n2π\u0013(2NM)1\n2πZ\ndQdPZ\ndωexp\u0000\niω\u0000\nE−H(P, Q)\u0001Z\ndξexp(−i⟨P|ξ⟩)\n×\u001c\n˜q1\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜q2\u001d\u001c\nq2\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq3\u001d\n···\n×\u001c\nqM\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq1\u001d\n,(6)\nwhere H(P, Q) =PM\nj=1H(Pj, Qj)/M,Q={Q1, . . . , Q M}andP={P1, . . . , P M},\nξ={ξ1, . . . , ξ M}are 2NM–dimensional vectors andQM\nj=1dqjd˜qj= dQdξ.\nThe final expression for the DOS in the Wigner approach to quantum mechanics\ncan be written as:\nΩ(E) = exp( E)Z\ndQdPδ(E−H(P, Q)W(P, Q), (7)\nwhere δ(E−H(P, Q)) is the path integral analogue of the Weyl symbol of the operator\nδ(EˆI−ˆH) [34,35]\nδ(E−H(P, Q)) =1\n2πZ\ndωexp iω\u0000\nE−H(P, Q)\u0001\n≈1\n2πZ\ndωexp(i⟨P|ξ⟩MY\nj=1\u001c\nqj\f\f\f\fexpiω\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜ qj\u001d\n.(8)\nSo the generalization of the Wigner function W(P, Q) is defined as\nW(P, Q) = (1\n2π)(2NM)exp(−E)Z\ndξexp(−i⟨P|ξ⟩)\n×\u001c\n˜q1\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜q2\u001d\u001c\nq2\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq3\u001d\n···\n×\u001c\nqM\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq1\u001d\n,(9)\nHerein we have assumed that the operators ˆHdo not depend on the spin variables.\nHowever, the spin variables σand the Fermi statistics can be taken into account by theDensity of states of a 2Dsystem 6\nfollowing redefinition of W(P, Q) in the canonical ensemble with temperature T\nW(P, Q) =1\nZ(β)N!λ2Nexp(−E)X\nσX\nP(−1)κˆPmS(σ,Pσ′)\f\f\nσ′=σZ\ndξexp(−i⟨P|ξ⟩)\n×\u001c\n˜q1\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\f˜q2\u001d\u001c\nq2\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq3\u001d\n···\u001c\nqM\f\f\f\fexp1\nM\u0000\nEˆI−ˆH\u0001\f\f\f\fq1\u001d\n=1\nZ(β)N!λ2NZ\ndξexp(−i⟨P|ξ⟩)ρ(1). . . ρ(M−1)\n×X\nσX\nP(−1)κPS(σ,Pσ′)\f\f\nσ′=σPρ(M)\f\f\nq(M+1)=q1(10)\nwhere the sum is taken over all permutations Pwith the parity κP, index jlabels the\noff–diagonal high–temperature density matrices ρ(j)≡ ⟨Qj±ξj/2|e−1\nMˆH|Q(j+1)±ξj/2⟩.\nWith the error of the order of 1 /M2each high–temperature factor can be presented\nin the form ρ(j)=⟨Qj±ξj/2|e−1\nMˆH|Q(j+1)±ξ(j+1)/2⟩ ≈ e−1\nMˆU(Qj±ξj/2)ρ(j)\n0with\nρ(j)\n0=⟨Qj±ξj/2|e−1\nMˆK|Q(j+1)±ξ(j+ 1)/2⟩, arising from neglecting the commutator\n[K, U]/(2M2) and further terms of the expansion. In the limit M→ ∞ the error of the\nwhole product of high temperature factors tends to zero ( ∝1/M) and we have an exact\npath integral representation of the Wigner functions.\nThe partition function Zfor a given temperature Tand volume Vcan be similarly\nexpressed as\nZ(β) =1\nN!λ2NX\nσZ\nVdQ1ρ(Q1, σ;β), (11)\nwhere ρ(Q1, σ;β) denotes the diagonal matrix elements of the density operator ˆ ρ=e−ˆH\nandλ=q\n2πℏ2β\nmis the thermal wavelength and β= 1/kBT. The integral in Eq. (11)\ncan be rewritten as\nX\nσZ\ndQ1ρ(Q1, σ;β)\n=Z\ndQ1. . .dQMρ(1). . . ρ(M−1)X\nσX\nP(−1)κPS(σ,Pσ′)Pρ(M)\f\f\nQ(M+1)=Q1,σ′=σ\n≈Z\ndQ1. . .dQMexp(\n−M−1X\nj=1\u0014\nπ\f\fQj−Q(j+1)\f\f2+1\nMU(Qj)\u0015)\ndet∥Ψ(QM, Q1∥,(12)\nwhere we imply that momentum and coordinate are dimensionless variables ˜pλ/ℏand\nq/˜λrelated to a temperature MT(˜λ=p\n2πℏβ/(mM)). Spin gives rise to the standard\nspin part of the density matrix S(σ,Pσ′) =QN\nk=1δ(σk, σPk), (δ(σk, σt) is the Kronecker\nsymbol) with exchange effects accounted for by the permutation operator Pacting on\ncoordinates of particles ˜ q(M+1)and spin projections σ′.\nIn the thermodynamic limit the main contribution in the sum over spin variables\ncomes from the term related to the equal numbers ( N/2) of fermions with the sameDensity of states of a 2Dsystem 7\nspin projection [23,24]. The sum over permutations gives the product of determinants\ndet∥Ψ(QM, Q1∥= det\r\re−π\f\f\fQ(k)\nM−Qt\n1\f\f\f2\r\rN/2\n1det\r\re−π\f\f\fQ(k)\nM−Qt\n1\f\f\f2\r\rN\n(N/2+1).\nIn general the complex-valued integral over ξin the definition of the Wigner function\n(10) can not be calculated analytically and is inconvenient for Monte Carlo simulations.\nThe second disadvantage is that Eqs. (10), (12) contain the sign–altering determinant\ndet∥Ψ(QM, Q1∥, which is the reason of the “sign problem” worsening the accuracy of\nPIMC simulations. To overcome these problems let us replace the variables of integration\nQjbyζjfor any given permutation Pusing the substitution [33,40]\nQj= (˜PQ (M+1)−Q1)j−1\nM+Q1+ζj, (13)\nwhere ˜Pis the matrix representing the operator of permutation Pand equal to the\nunit matrix Ewith appropriately transposed columns. This replacement presents each\ntrajectory Qjas the sum of the “straight line” ( ˜PQ (M+1)−Q1)j−1\nM+Q1and the deviation\nζjfrom it (here we assume Q(M+1)=Q1,j= 1,···, M+ 1).\nAs a consequence the matrix elements of the density matrix can be rewritten in the\nform of a path integral over “closed” trajectories {ζ1, . . . , ζ (M+1)}with ζ1=ζ(M+!)= 0\n(‘ring polymers”). By making use the approximation for potential Uarising from the\nTaylor expansion up to the first order in the ξ, and after the integration over ξ[33,40]\nand some additional transformations (see [36,41,42] for details) the Wigner function can\nbe written in the form containing the Maxwell distribution with quantum corrections\nW(P, Q)≈˜C(M)\nZ(β)N!exph\n−MX\nj=1\u0012\nπ|ηj|2+1\nMU\u0012\nQ1+ζj\u0013\u0013i\n×exp(\nM\n4πMX\nj=1*\niPj+ (−1)(j−1)1\n2M∂U(Q1+ζj)\n∂Q1\f\f\f\f\fiPj+ (−1)(j−1)1\n2M∂U(Q1+ζj)\n∂Q1+)\n×det∥˜ϕkt\r\rN/2\n1det\r\r˜ϕkt∥Ne\n(N/2+1),(14)\nwhere\n˜ϕkt= exp{−π|rkt|2/M}exp(\n−1\n2MMX\nj=1\u0012\nϕ\u0010\f\f\frtk2j\nM+rkt+(ζk\nj−ζt\nj)\f\f\f\u0011\n−ϕ\u0010\f\f\frkt+(ζk\nj−ζt\nj)\f\f\f\u0011\u0013)\n,\nηj≡ζj−ζ(j+1),rkt≡(Qk\n1−Qt\n1), (k, t= 1, . . . , N ). The constant ˜C(M) is canceled in\nMonte Carlo calculations.\nLet us stress that approximation (14) have the correct limits to the cases of\nweakly and strongly degenerate fermionic systems. Indeed, in the classical limit\nthe main contribution comes from the diagonal matrix elements due to the factor\nexp{−π|rkt|2/M}and the differences of potential energies in the exponents are equal to\nzero (identical permutation). At the same time, when the thermal wavelength is of the\norder of the average interparticle distance and the trajectories are highly entangled theDensity of states of a 2Dsystem 8\nterm rtk2j\nM(breaking ‘ring polymers”) in the potential energy ϕ\u0010\f\f\frtk2j\nM+rkt+(ζk\nj−ζt\nj)\f\f\f\u0011\ncan be neglected and the differences of potential energies in the exponents tend to\nzero [36,42].\nThus, the problem is reduced to calculating the matrix elements of the density\nmatrix ρ= exp ( −βˆH), which is similar to the simulation of thermodynamic properties\nand, according to Eq. (7), the problem of DOS calculation is reduced to considering the\ninternal–energy histogram in the canonical ensemble multiplied by exp( E).\n3. Quantum pseudopotential for soft–sphere fermions\nThe high–temperature density matrix ρ(j)=⟨r(j)|e−ϵˆH|r(j+1)⟩can be expressed as a\nproduct of two–particle density matrices [23]\nρ(rl, r′\nl, rt, r′\nt;ϵ) =1\n˜λ6exp\u0014\n−π\n˜λ2|rl−r′\nl|2\u0015\nexp\u0014\n−π\n˜λ2|rt−r′\nt|2\u0015\nexp[−ϵΦOD\nlt]. (15)\nThis formula results from the factorization of the density matrix into the kinetic and\npotential parts, ρ≈ρK\n0ρU. The off–diagonal density matrix element (15) involves an\neffective pair interaction by a pseudopotential, which can be expressed approximately\nvia the diagonal elements, ΦOD\nlt(rl, r′\nl, rt, r′\nt;ϵ)≈[Φlt(rl−rt;ϵ) + Φ lt(r′\nl−r′\nt;ϵ)]/2.\n/s48 /s49 /s50 /s51 /s52/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s32/s49\n/s32/s50/s44/s32\n/s114/s47\n/s32/s32\nFigure 1. (Color online) The soft–sphere potential ϕ(line 1) at n= 1/2 and respective\npseudopotential Φ (line 2) defined by Eq. (17) in conditional units.\nTo estimate Φ( r) for each high-temperature density matrix we use the Kelbg\nfunctional [43, 44] allowing to take into account quantum effects in interparticle\ninteraction. The peudopotential Φ( r) is defined by the Fourier transform v(t) of the\npotential ϕ(r). This transform can be found at n <3 for the corresponding Yukawa–\nlike potential exp( −κr)/rnin the limit of “zero screening” ( κ→0 )\nv(t) =4πtnΓ(2−n) sin(nπ/2)\nt3, (16)Density of states of a 2Dsystem 9\nwhere Γ is the gamma function. The resulting quantum pseudopotential has the\nfollowing form\nΦ(r) =√π\n8π3Z∞\n0v(t) exp(−(˜λt)2/4)sin(tr)erfi( ˜λt/2)\n(tr)˜λt4πt2dt, (17)\nwhere erfi(z) = i efr(iz), erf(z) is the error function [43]. This pseudopotential is finite\nat zero interparticle distance, Φ(0) = λ−nΓ(1−n/2), and decreases according to the\npower law ( λ/r)nfor distances larger than the thermal wavelength (see Figure 1).\nFor more accurate accounting for quantum effects the “potential energy”\nU(q(j), q(j+1)) in (10) and (12) has to be taken as the sum of pair interactions given\nby ΦODwith Φ( r). The pseudopotential Φ was also used in the Hamilton function\nH(pq) in the Weyl’s symbol of the operator δ(EˆI−ˆH).\nHowever, if the effective hardness of the pseudopotential Φ is less than 3 the\ncorresponding energyPM−1\nm=0ϵU(x(j)) may be divergent in the thermodynamic limit.\nTo overcome this deficiency let us modify the pseudopotential Φ according to the\ntransformation considered in [45]\n˜Φ(r) = [Φ( r)−1\nVZ\nVd3yΦ(r+y)] =Z\nd3yΦ(r+y)\u0012\nδ(y)−1\nV\u0013\n. (18)\nHere the uniformly “charged” background is introduced to compensate the possible\ndivergence of U(q(j)) similar to the case of one-component Coulomb plasma.\nLet us note that the pseudopotential Φ corresponding to the Coulomb potential\nwith hardness n= 1 was often used in PIMC simulations of one– and two–component\nplasma media in [23, 24, 44, 46–49] showing good agreement with the data available in\nthe literature.\n4. Results of simulations\nIn this section we investigate the dependence of a radial distribution function (RDF) and\nDOS on the hardness of the soft–sphere potential. Here, as an interesting example, we\npresent RDFs, internal–energy distributions (histograms) and DOS for the 2D system of\nFermi particles strongly interacting via the soft sphere potential with hardness nequal\nto 0.2, 0.6, 1, and 1.4. The density of soft spheres is characterized by the parameter\nrs=a/σ, defined as the ratio of the mean distance between the particles a= [1/(π˜ρ)]1/2\ntoσ(˜ρis the 2D particle density, rs= 1/√\nπnσ2is the 2D Brucker parameters). For\nexample, the results presented below have been obtained for the following physical\nparameters used in [29] for PIMC simulations of helium-3: ϵ= 26.7K,σ= 5.19aB(aB\nis the Bohr radius), ma= 3.016 (the soft–sphere mass in atomic units) and rsis of the\norder of 2.\nThe RDF [50, 51], internal–energy distribution functions (IED) and DOS can beDensity of states of a 2Dsystem 10\nwritten in the form\ngab(r) =Z\ndPdQ δ(|q1,a−q1,b| −r)W(P, Q),\nW(E) =Z\ndPdQ δ(E−H(P, Q)W(P, Q),\nΩ(E) = exp( E)W(E), (19)\nwhere EandH(P, Q) are energy per particle, aandblabel the spin value of a fermion.\nThe RDF gabis proportional to the probability density to find a pair of particles of types\naandbat a certain distance rfrom each other. In an isotropic system the RDF depends\nonly on the difference of coordinates because of the translational invariance. The DOS\ngives the number of states in the phase space per an infinitesimal range of internal\nenergy. In a non-interacting 2D system Ω = const [37] and gab≡1, whereas interaction\nand quantum statistics result in the redistribution of particles and non-constant DOS.\n/s48 /s49 /s50/s48/s49/s50/s51\n/s97/s41\n/s32/s49\n/s32/s50\n/s32/s51\n/s114/s47 /s115/s103\n/s48 /s49 /s50/s48/s44/s48/s48/s44/s53/s49/s44/s48\n/s98/s41\n/s32/s49\n/s32/s50\n/s32/s51\n/s114/s47 /s115/s103\nFigure 2. (Color online) The RDFs for the system of soft–sphere fermions at a fixed\ndensity rs= 2.2 and temperature T= 60 K.\nPanel a) — fermions with the same spin projections, panel b) — fermions with the\nopposite spin projections. Lines: 1—ideal system; 2— n= 0.6; 3— n= 1. Small\noscillations indicate the Monte-Carlo statistical error.\nFigure 2 presents the results of our WPIMC calculations for the spin–resolved\nRDFs for a fixed density and temperature but at different hardnesses of the soft–sphere\npotential. Let us discuss the difference revealed between the RDFs with the same and\nopposite spin projections. At small interparticle distances all RDFs tend to zero due\nto the repulsion nature of the soft–sphere potential. Additional contribution to the\nrepulsion of fermions with the same spin projection at distances of the order of the\nthermal wavelength (lines 1,2,3) is caused by the Fermi statistics effect described by the\nexchange determinant in (14) that accounts for the interference effects of the exchange\nand interparticle interactions. This additional repulsion leads to the formation of cavities\n(usually called exchange–correlation holes) for fermions with the same spin projection\nand results in the formation of high peaks on the corresponding RDFs due to the strong\nexcluded volume effect [52]. The RDFs for fermions with the same spin projection show\nthat the characteristic “size” of an exchange–correlation cavity with corresponding peaks\nis of the order of the soft–sphere thermal wavelength ( λ/σ∼0.5) that is here less thanDensity of states of a 2Dsystem 11\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s69/s45/s54/s49/s69/s45/s52/s48/s44/s48/s49/s97/s41/s87/s40/s69/s41\n/s69/s32/s49\n/s32/s50\n/s32/s51\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s44/s49/s49/s49/s48/s49/s48/s48\n/s98/s41/s87 /s40/s69/s41\n/s32/s49\n/s32/s50\n/s32/s51\n/s69\nFigure 3. (Color online) The energy distribution W(E) (panel a) and DOS (panel\nb) for the system of soft-sphere fermions at a fixed density rs= 2.2 and temperature\nT= 60 K (R\nW(E)dE= 1, Ω( E) in conditional units). Lines: 1—ideal system; 2—\nn= 0.6; 3— n= 1. Small oscillations indicate the Monte-Carlo statistical error.\nthe average interparticle distance rs= 2.2. Let us stress that the strong excluded volume\neffect was also observed in the classical systems of repulsive particles (system of hard\nspheres) seventy years ago in [53] and was derived analytically for 1D case in [51].\nNote that for fermions with the opposite spin projections the interparticle\ninteraction is not enough to form any peaks on the RDF. At large interparticle distances\nthe RDFs decay monotonically to unity due to the short–range repulsion of the potential.\nPanels a) and b) of Figure 3 present the results of WPIMC calculations for the\ninternal–energy distributions W(E) and DOS Ω( E) for the same parameters as in\nFigure 2. Here all W(E) are normalized to unity (R\ndE W(E) = 1).\nIn an ideal system the internal energy of chaotic particle configurations is defined\nby the Maxwell distribution. Soft–sphere repulsive interaction increases the energy of\nany given phase space configuration in comparison with the same configuration of the\nideal system. As the energy distribution is proportional to the fraction of phase space\nstates with the energy equal to E, then this fraction ( W(E)) have to be shifted to\na greater energy, which we can see in panel a) of Fig. 3. The characteristic value of\nthis energy shift is of the order of the product of characteristic values of the RDF and\npseudopotential at small interparticle distances as the main contribution to the shift is\ndefined by the region where g(r)−1 is nonzero. Let us remind that both Φ( r) and Φ(0)\nincrease with the hardness for small interparticle distances ( Φ(0) = λ−nΓ(1−n/2),\nΓ is gamma function), while the RDFs according to panel a) of Fig. 2 decrease. This\neffect is the physical reason of the nontrivial behavior of the DOS in panel b) of Fig. 3\n(lines 2, 3) in comparison with the DOS of ideal system, which is identically equal to a\nconstant (line 1) [37].\nPanel a) in Fig. 4 shows the RDFs at the same temperature but for a slightly\nhigher density ( rs= 2.1 instead of rs= 2.2) and an extended range of hardness in\ncomparison with panel b) of Fig. 3. As in the previous case the heights of the RDF\npeaks demonstrate non-monotonic behavior and reach the maximum values at n= 0.6,Density of states of a 2Dsystem 12\nwhile the widths of the peaks are practically the same being of the order of the fermion\nthermal wavelength. Generally speaking the changes in the DOS behavior are non-\nmonotonic and show interesting peculiarities at n= 0.6.\n/s48 /s49 /s50 /s51/s48/s49/s50/s51\n/s97/s41/s32/s49\n/s32/s50\n/s32/s51\n/s32/s52\n/s32/s53\n/s114/s47/s103\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s46/s49/s49/s49/s48/s49/s48/s48\n/s98/s41/s32/s49\n/s32/s50\n/s32/s51\n/s32/s52\n/s32/s53\n/s69\nFigure 4. (Color online) The RDFs for the same spin projections (panel a) and\nDOS (panel b ) for a system of soft-sphere fermions at a fixed density rs= 2.1 and\ntemperature T= 60 K.\nLines: 1—ideal system; 2 — n= 0.2; 3— n= 0.6; 4— n= 1.0; 5— n= 1.4.\nSmall oscillations indicate the Monte-Carlo statistical error.\nThe results of WPIMC simulations of RDFs for fermions with the same spin\nprojection and DOS are presented in Fig. 5 at n= 0.6 and in Fig. 6 at n= 1. The\ntemperature is fixed at T= 60 K and four different values of rsare considered. With\nincreasing density the height of the peaks is rising for both hardnesses, but at n= 0.6\nthe height is about twice as high as at n= 1. This difference strongly affects the DOS\nbehavior. In general, for a low density ( rs= 2.3) the DOS at n= 0.6 is higher than the\none at n= 1. With increasing density the DOS curve goes down. Then for the highest\ndensity rs= 1.47 the DOS for n= 0.6 and n= 1 practically coincide with each other\nand are very close to the ideal DOS.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s52/s54\n/s97/s41\n/s32/s49\n/s32/s50\n/s32/s51\n/s32/s52\n/s32/s53\n/s114/s47/s103\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s46/s49/s49/s49/s48/s49/s48/s48\n/s98/s41/s32/s49\n/s32/s50\n/s32/s51\n/s32/s52\n/s32/s53\n/s69\nFigure 5. (Color online) The RDFs for the same spin projections (panel a) and Ω( E)\n(panel b ) for the system of ideal and interacting soft-sphere fermions at n= 0.6,\nT= 60 K and different densities rs.\nLines: 1—ideal system; 2— rs= 2.3; 3— rs= 2.2; 4— rs= 2.1; 5— rs= 1.47.\nSmall oscillations indicate the Monte-Carlo statistical error.Density of states of a 2Dsystem 13\n/s48 /s49 /s50 /s51/s48/s49/s50/s51\n/s97/s41/s32/s49\n/s32/s50\n/s32/s51\n/s32/s52\n/s32/s53\n/s114/s47/s103\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s46/s49/s49/s49/s48/s49/s48/s48\n/s98/s41/s32/s49\n/s32/s50\n/s32/s51\n/s32/s52\n/s32/s53\n/s69\nFigure 6. (Color online) The RDFs for the same spin projections (panel a) and\nDOS (panel b ) for a system of ideal and interacting soft-sphere fermions at n= 1,\nT= 60 K and different densities rs. Lines: 1—ideal system; 2— rs= 2.3; 3— rs= 2.2;\n4—rs= 2.1; 5— rs= 1.47. Small oscillations indicate the Monte-Carlo statistical error.\nTo calculate an RDF, W(E) and DOS Markovian chains of particle configurations\nwere generated using WPIMC (see Supplemental material). The configurations with\nnumbers 106−3×106for systems of 300, 600 and 900 particles represented by twenty\nand forty “beads” were considered as equilibrium. We used a standard basic Monte\nCarlo cell with periodic boundary conditions. The convergence and statistical error\nof the calculated functions were tested with increasing number of Monte Carlo steps,\nnumber of particles and beads at a different hardness of the pseudopotential. It turned\nout that 600 particles represented by 20 beads were enough to reach the convergence.\nWith decreasing histogram interval in the energy distribution the statistical error\nincreases due to the worsening of statistics in each interval, so the compromise between\na reasonable width of the discrete interval and the statistical error has to be achieved.\nIn our simulations the statistical error is of the order of small random oscillations of the\ndistribution functions.\n5. Discussion\nDensity of states (DOS) is known to determine the properties of matter and can be\nused to compute the thermodynamic properties of a wide variety systems of particles.\nDOSs have been considered and used many times in the literature for calculation of\nthermodynamic properties of the classical system [2–9]. This article deals with the DOS\nof quantum systems, so the Wigner formulation of quantum mechanics has been used to\nderive the new path integral representation of the quantum DOS. For the 2D quantum\nsystem of strongly correlated soft–sphere fermions we present the DOS, internal energy\ndistribution and the spin–resolved RDF obtained by the new path integral Monte Carlo\nmethod (WPIMC) [41] for the hardness of the soft–sphere potential of the order of unity\n(n= 0.2, 0.6, 1.0, 1.4). The peculiarities of the dependences of the DOS as a function\nof the hardness of the soft–sphere potential and particle density has been investigatedDensity of states of a 2Dsystem 14\nand explained. The WPIMC calculations for greater hadnesses are in progress.\nAcknowledgements\nWe thank G. S. Demyanov for comments and help in numerical aspects. We value\nstimulating discussions with Prof. M. Bonitz, T. Schoof, S. Groth and T. Dornheim\n(Kiel). The authors acknowledge the JIHT RAS Supercomputer Centre, the Joint\nSupercomputer Centre of the Russian Academy of Sciences, and the Shared Resource\nCentre Far Eastern Computing Resource IACP FEB RAS for providing computing time.\nReferences.\n[1] Harrison W A 2012 Electronic structure and the properties of solids: the physics of the chemical\nbond (Courier Corporation)\n[2] Wang F and Landau D 2001 Physical Review E 64056101\n[3] Wang F and Landau D P 2001 Physical review letters 862050\n[4] Faller R and de Pablo J J 2003 The Journal of chemical physics 1194405–4408\n[5] Vogel T, Li Y W, W¨ ust T and Landau D P 2013 Physical review letters 110210603\n[6] Liang F 2005 Journal of the American Statistical Association 1001311–1327\n[7] Moreno F, Davis S and Peralta J 2022 Computer Physics Communications 274108283\n[8] Bornn L, Jacob P E, Del Moral P and Doucet A 2013 Journal of Computational and Graphical\nStatistics 22749–773\n[9] Atchad´ e Y F and Liu J S 2010 Statistica Sinica 209–233\n[10] Martin R 2004 Cambridge Daw MS, Baskes MI (1984) Phys Rev B 296443\n[11] Seo D H, Shin H, Kang K, Kim H and Han S S 2014 The Journal of Physical Chemistry Letters\n51819–1824\n[12] Ma X, Li Z, Achenie L E and Xin H 2015 The journal of physical chemistry letters 63528–3533\n[13] Ratcliff L E, Mohr S, Huhs G, Deutsch T, Masella M and Genovese L 2017 Wiley Interdisciplinary\nReviews: Computational Molecular Science 7e1290\n[14] Galli G 1996 Current Opinion in Solid State and Materials Science 1864–874\n[15] Saad Y, Chelikowsky J R and Shontz S M 2010 SIAM review 523–54\n[16] Goedecker S 1999 Reviews of Modern Physics 711085\n[17] Vorontsov-Velyaminov P and Lyubartsev A 2003 Journal of Physics A: Mathematical and General\n36685\n[18] Lee J 1993 Physical review letters 71211\n[19] Luyten W 1971 Journal of the Royal Astronomical Society of Canada 65304\n[20] Potekhin A Y 2010 Physics-Uspekhi 531235\n[21] Feynman R P and Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York: McGraw-\nHill)\n[22] Zamalin V, Norman G and Filinov V 1977 The monte carlo method in statistical thermodynamics\n[23] Ebeling W, Fortov V and Filinov V 2017 Quantum Statistics of Dense Gases and Nonideal Plasmas\n(Berlin: Springer)\n[24] Fortov V, Filinov V, Larkin A and Ebeling W 2020 Statistical physics of Dense Gases and Nonideal\nPlasmas (Moscow: PhysMatLit)\n[25] Dornheim T, Groth S and Bonitz M 2018 Physics Reports 7441–86\n[26] Ceperley D M 1995 Reviews of Modern Physics 67279\n[27] Pollock E L and Ceperley D M 1984 Physical Review B 302555\n[28] Singer K and Smith W 1988 Molecular Physics 641215–1231\n[29] Filinov V, Syrovatka R and Levashov P 2022 Molecular Physics e2102549\n[30] Ceperley D M 1991 Journal of statistical physics 631237Density of states of a 2Dsystem 15\n[31] Ceperley D M 1992 Physical review letters 69331\n[32] Larkin A, Filinov V and Fortov V 2017 Contributions to Plasma Physics 57506–511\n[33] Larkin A, Filinov V and Fortov V 2017 Journal of Physics A: Mathematical and Theoretical 51\n035002\n[34] Wigner E 1934 Physical Review 461002\n[35] Tatarskii V I 1983 Soviet Physics Uspekhi 26311\n[36] Filinov V, Levashov P and Larkin A 2021 Journal of Physics A: Mathematical and Theoretical 55\n035001\n[37] Kubo R, Toda M and Hashitsume N 2012 Statistical physics II: nonequilibrium statistical mechanics\nvol 31 (Springer Science & Business Media)\n[38] Ses´ e L M 2020 Entropy 221338\n[39] Zamalin V and Norman G 1973 USSR Computational Mathematics and Mathematical Physics 13\n169–183\n[40] Larkin A, Filinov V and Fortov V 2016 Contributions to Plasma Physics 56187–196\n[41] Filinov V, Larkin A and Levashov P 2022 Universe 879\n[42] Filinov V, Larkin A and Levashov P 2020 Physical Review E 102033203\n[43] Demyanov G and Levashov P 2022 arXiv preprint arXiv:2205.09885\n[44] Kelbg G 1963 Ann. 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Simenel3, ‡\n1Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA\n2Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA\n3Department of Fundamental and Theoretical Physics and Department of Nuclear Physics and Accelerator Applications,\nResearch School of Physics, The Australian National University, Canberra ACT 2601, Australia\n(Dated: May 30, 2023)\nWe employ the constrained density functional theory to investigate cluster phenomena for the12C nucleus.\nThe proton and neutron densities are generated from the placement of three4He nuclei (alpha particles) geo-\nmetrically. These densities are then used in a density constrained Hartree-Fock calculation that produces an\nantisymmetrized state with the same densities through energy minimization. In the calculations no a priori\nanalytic form for the single-particle states is assumed and the full energy density functional is utilized. The geo-\nmetrical scan of the energy landscape provides the ground state of12C as an equilateral triangular configuration\nof three alphas with molecular bond like structures. The use of the nucleon localization function provides further\ninsight to these configurations. One can conclude that these configurations are a hybrid between a pure mean-\nfield and a pure alpha particle condensate. This development could facilitate DFT based fusion calculations with\na more realistic12C ground state.\nI. INTRODUCTION\nIn stellar evolution carbon plays a pivotal role through the\ncarbon burning process. The ignition of carbon burning for\nstars in the mass range M >8-10 solar masses lead to white\nNe/O dwarfs, while massive stars with masses M >25 solar\nmasses can continue burning Ne, O, and Si and end up as\nsupernovae. Similarly, Type Ia supernovae is believed to re-\nsult from an explosion of a white dwarf accreting mass from\na binary companion or a merger, inducing high enough tem-\nperatures to ignite carbon in the core leading to a supernovae\nexplosion. Superbursts are set off by the ignition of carbon in\nthe accumulated ashes of previous x-ray bursts [1–3]. Overall,\na change in the12C+12C reaction rates has a profound impact\non all these mechanisms as well as nucleosynthesis [4–11]. In\nthe stellar environment carbon is produced in a two step pro-\ncess; in the first step two4He nuclei come together to form an\nunstable8Be, which decays back to two4He nuclei with a very\nshort lifetime (10−16s). However, during the helium burning\nstage, the densities are high enough to maintain a small abun-\ndance of8Be, which renders it possible to combine with an-\nother4He to form an excited carbon nucleus through the well\nknown Hoyle state of12C. With a much lower probability a\ntriple4He combination may also lead to the same outcome.\nIn addition to its astrophysical importance, the microscopic\ndescription of the carbon nucleus, which is an essential ingre-\ndient to the reaction calculations, has proven to be a challenge.\nThis is predicated by the expectation that the structure of car-\nbon should exhibit a pronounced cluster structure. Cluster-\ning effects are widely believed to play prominent role in the\nstructure of N =Z nuclei, resulting in a molecular type phe-\nnomenon. To what degree such nuclei can be viewed as being\ncomprised of a pure alpha particle condensate is still an open\nquestion [12].\n∗umar@compsci.cas.vanderbilt.edu\n†godbey@frib.msu.edu\n‡cedric.simenel@anu.edu.auHowever, employing the standard non-relativistic density\nfunctional theory (DFT) results in a ground state of12C with-\nout any sign of clusterization [13]. This is also true for16O,\nwhile the ground states of8Be and20Ne do show some clus-\nter features [13]. Relativistic mean-field theories seem to fa-\nvor more clusterization due to deeper potentials [14]. Alter-\nnate approach of using configuration mixing with generator\ncoordinate method (GCM) calculations using the Skyrme en-\nergy density functional have also been done in Refs. [15,16].\nFurthermore, these calculations using modern energy density\nfunctionals commonly result in a spherical ground state for\nthe12C nucleus, which is experimentally known to have an\noblate deformation [17–19]. On the other hand the excited\nstates of these nuclei do seem to exhibit some cluster structure,\ne.g. the linear-chain configuration of12C, which originally\nwas thought to be the Hoyle state [20], and excited states ob-\ntained by various constraints [14,21–30]. Such formations are\nalso observed via the time-dependent Hartree-Fock studies of\nthe triple-alpha reaction [31,32] and studied in recent exper-\niments [33,34]. In these time-dependent calculations a bent-\narm intermediate configuration is observed during the decay\nof the metastable linear chain state of12C.\nThe fact that DFT calculations account for only limited\nclustering effects prompted alternate approaches to study12C\nstructure that rely more heavily on cluster wavefunctions.\nThese calculations suggest that a substantial contribution of\nalpha cluster correlations that are not accounted for in the\nmean-field description should nevertheless be present in12C\nstates. These include various ab initio calculations [35–37] as\nwell as approaches that are collectively referred to as molec-\nular dynamics that significantly extend the original Bloch-\nBrink alpha cluster model [38]. In Brink’s approach each\nquartet of nucleons where represented using harmonic oscil-\nlator wavefunctions with zero angular momentum displaced\nfrom each other by a relative coordinate. Antisymmetrization\nfollowed by normalization comprised the many-body wave-\nfunction in terms of the locations of the quartets. These quar-\ntets, interacting via an effective nucleon-nucleon interaction,\nare optimized with respect to their size and position to maparXiv:2305.17752v1 [nucl-th] 28 May 20232\nout the energy landscape showing the location of the min-\nima. This approach was extended via the resonating group\nmethod as well as generator coordinate method to better in-\ncorporate the internal structure of the clusters. The antisym-\nmetrized molecular dynamics (AMD) [39–41] approach em-\nploys a Slater determinental many-body wavefunction com-\nprised of single-particle states as Gaussian wave-packets us-\ning an advanced set of geometrical variables. Fermionic\nmolecular dynamics (FMD) [42] further extends AMD by not\nputting any restriction on the width of these Gaussians. These\ncalculations indicate a large admixture of alpha-cluster trian-\ngular states for the ground and some of the excited states con-\nfigurations of12C. The use of Gaussian basis and suitable in-\nteractions allow for very powerful extensions for the above\nmethods, such as the treatment of the center-of-mass energy,\nangular momentum projection and the use of the generator co-\nordinate method (GCM). However, many of these calculations\nassume a degeneracy between neutron and proton wave func-\ntions and do not include the full effective interaction and the\nspin-orbit force. A collection of recent reviews can be found\nin Refs. [12,43–47].\nIn this manuscript we introduce another approach for study-\ning cluster structures within the DFT framework. This is\naccomplished through the use of the density constrained\nHartree-Fock approach. Here, we start with alpha-particles as\nsolutions to the unconstrained Hartree-Fock (HF) equations.\nThese alpha particles are then geometrically arranged on the\nnumerical grid defining the total density of the system. For\neach arrangement, a mean-field solution is obtained through\nminimization of the energy by constraining the density of the\nentire system. Density constraint iterations allow for the rear-\nrangement of the single-particle states through their orthogo-\nnalization and energy minimization. This takes care of anti-\nsymmetrization as well as the overall energy dependent nor-\nmalization of the many-body wavefunction. No assumption\nabout the mathematical form of the single-particle states is\nmade and the full effective interaction, including the Coulomb\nforce, can be used. We also employ the nuclear localization\nfunction (NLF), which allows for a more precise characteri-\nzation of spatial distributions. This method blends the cluster\nbased approach with the fully microscopic approach. As we\nshall see, it has advantages and certain disadvantages.\nII. MICROSCOPIC METHODS\nIn this section we briefly outline the formalisms and meth-\nods used in our calculations. Further details can be found in\nthe cited references.\nA. Density constraint\nGiven a reference density, the density constraint proce-\ndure [48,49] allows the single-particle states, comprising the\ncombined nuclear density, to reorganize to attain their mini-\nmum energy configuration and be properly antisymmetrized\nas the many-body state is a Slater determinant of all the occu-pied single-particle wave-functions. Here, the reference den-\nsity is given by the combined density of three alpha parti-\ncles obtained from independent Hartree-Fock calculations and\nplaced in close proximity of each other. The HF minimization\nof the combined system is thus performed subject to the con-\nstraint that the local proton ( p) and neutron ( n) densities do\nnot change:\nδ*\nH−∑\nq=p,nZ\ndrλq(r)\"\n3\n∑\ni=1ρα\niq(r,Ri)# +\n=0,(1)\nwhere the λn,p(r)are Lagrange parameters at each point of\nspace constraining the neutron and proton local densities,\nρα\niq(r,Ri)is the proton/neutron densities of an alpha parti-\ncle located at position Ri, and His the effective many-body\nHamiltonian. This procedure determines a unique Slater de-\nterminant |Φ(R1,R2,R3)⟩for the combined system. The den-\nsity constraint has been extensively used in the calculation of\nion-ion interaction barriers for fusion calculations [50–52].\nB. Centre of mass correction\nA major drawback of any mean-field based microscopic\ncalculation is the uncontrolled presence of the energy associ-\nated with the center-of-mass (c.m.) motion [53]. This energy\nis particularly large for light nuclei. Most Skyrme interactions\nadopt a simple one-body correction for this energy, which may\nbe reasonable for heavy systems. This issue has been dis-\ncussed more extensively in the context of alpha clustering phe-\nnomenon for the mean-field calculations in Ref. [23], where\na constant value of 7 MeV per alpha particle was subtracted.\nHowever, the c.m. correction for a composite system may not\nbe the same as adding corrections for each alpha. Due to this\nwe cannot make any binding energy comparisons and adopted\nthe SLy4d interaction, which does not employ any center-of-\nmass correction term.\nC. The nucleon localization function (NLF)\nThe measure of localization has been originally devel-\noped in the context of a mean-field description for electronic\nsystems [54], and subsequently introduced to nuclear sys-\ntems [13,55,56]. We first realize that a fermionic mean-field\nstate is fully characterized by the one-body density-matrix\nρq(rs,r′s′). The probability of finding two nucleons with the\nsame spin at spatial locations randr′(same-spin pair proba-\nbility) for isospin qis proportional to\nPqs(r,r′) =ρq(rs,rs)ρq(r′s,r′s)−|ρq(rs,r′s)|2,(2)\nwhich vanishes for r=r′due to the Pauli exclusion principle.\nThe conditional probability for finding a nucleon at r′when\nwe know with certainty that another nucleon with the same\nspin and isospin is at ris proportional to\nRqs(r,r′) =Pqs(r,r′)\nρq(rs,rs). (3)3\nThe short-range behavior of Rqscan be obtained using tech-\nniques similar to the local density approximation [13,55]. The\nleading term in the expansion yields the localization measure\nDqsµ=τqsµ−1\n4\f\f∇ρqsµ\f\f2\nρqsµ−\f\fjqsµ\f\f2\nρqsµ. (4)\nThis measure is the most general form that is appropriate\nfor deformed nuclei and without assuming time-reversal in-\nvariance, thus also including the time-odd terms important\nin applications such as cranking or time-dependent Hartree-\nFock (TDHF). The densities and currents are given in their\nmost unrestricted form [57–59] for µ-axis denoting the spin-\nquantization axis by [55]\nρqsµ(r) =1\n2ρq(r)+1\n2σµsqµ(r), (5a)\nτqsµ(r) =1\n2τq(r)+1\n2σµTqµ(r), (5b)\njqsµ(r) =1\n2jq(r)+1\n2σµJq(r)·eµ, (5c)\nwhere σµ=2sµ=±1 and eµis the unit vector in the direc-\ntion of the µ-axis. Note that subscripts sµdenote spin along\nthe quantization axis and should not be confused by the spin-\ndensity sqµ. The dot product in Eq. (5c) is explicitly given in\nthe case of e.g.,µ=z\nJq(r)·ez=1\n2i\u0002\n(∇−∇′)sqz(r,r′)\u0003\nr=r′.\nThe explicit expressions of the local densities and currents are\ngiven in Refs. [55,57]. We note that the localization measure\nincludes the spin-density sqµ(r), the time-odd part of the ki-\nnetic density Tqµ(r), as well as the full spin-orbit tensor Jq(r),\nwhich is a pseudotensor. In this sense all of the terms in the\nSkyrme energy density functional [57] contribute to the mea-\nsure. Finally, we note that the time-odd terms contained in the\nabove definitions ( sqµ,Tqµ, and jq) are zero in static calcula-\ntions of even-even nuclei but the spin-tensor Jqis not. There-\nfore, jqsµis not zero in general.\nIt is interesting to visualize the NLF as it is also defined\nfrom the localization measure in Eq. (4). We first normalize\nthe localization measure using [55]\nDqsµ(r) =Dqsµ(r)\nτTFqsµ(r), (6)\nwhere the normalization τTF\nqsµ(r) =3\n5\u0000\n6π2\u00012/3ρ5/3\nqsµ(r)is the\nThomas-Fermi kinetic density. The NLF can then be repre-\nsented either by 1 /Dqsµor by\nCqsµ(r) =h\n1+D2\nqsµi−1\n(7)\nwhich is used here. The advantage of the latter form is that\nit scales to be in the interval [0,1], but otherwise both forms\nshow similar localization details.\nThe information content of the localization function is bet-\nter understood by considering limiting cases. The extremecase of ideal metallic bonding is realized for homogeneous\nmatter where τ=τTF\nqσ. This yields C=1\n2, a value which\nthus signals a region with a nearly homogeneous Fermi gas\nas it is typical for metal electrons, nuclear matter, or neutron\nstars. The opposite regime are space regions where exactly\none single-particle wavefunction of type qσcontributes. This\nis called localization in molecular physics. Such a situation\nyields Dqσ(r) =0, since it is not possible to find another like-\nspin state in the vicinity, and consequently C=1, the value\nwhich signals localization .\nIn the nuclear case, it is the αparticle which is perfectly\nlocalized in this sense, i.e. which has C=1 everywhere for\nall states. Well bound nuclei show usually metallic bonding\nand predominantly have C=1\n2. Light nuclei are often ex-\npected to contain pronounced α-particle sub-structures. Such\na sub-structure means that in a certain region of space only\nanαparticle is found which in turn is signaled by C=1 in\nthis region. In fact, an αsub-structure is a correlation of four\nparticles: p↑,p↓,n↑, and n↓. Thus it is signaled only if\nwe find simultaneously for all four corresponding localization\nfunctions Cqσ≈1. This localization procedure was recently\nemployed to visualize the cluster structure in N=Zlight nu-\nclei [60].\nD. Numerical details\nCalculations were done in a three-dimensional Cartesian\ngeometry with no symmetry assumptions using the code of\nRef. [61] and using the Skyrme SLy4d interaction [62], which\nhas been successful in describing various types of nuclear re-\nactions [50,63]. The three-dimensional Poisson equation for\nthe Coulomb potential is solved by using Fast-Fourier Trans-\nform techniques and the Slater approximation is used for the\nCoulomb exchange term. The static HF equations and the\ndensity constraint minimizations are implemented using the\ndamped gradient iteration method [64]. The box size used for\nall the calculations was chosen to be 24 ×24×24 fm3, with a\nmesh spacing of 1 .0 fm in all directions. These values provide\nvery accurate results due to the employment of sophisticated\ndiscretization techniques [65,66].\nIII. RESULTS\nThe placement of the three alpha particles were done as\nfollows; two alpha particles were placed on the x−axis with\na spacing denoted by d2/2 on each side of the origin. The\nthird alpha particle was placed at distance z3vertically from\nthe origin. There are numerous studies of alpha cluster mod-\nels for12C that show that this more symmetric arrangement\nleads to the minimum energy configuration [46], as antici-\npated from symmetry arguments. Moving the third alpha in\nthe y-direction would simply correspond to tilting the three-\nalpha system in the 3D space. We have scanned d2values\nranging from 1.5-6.6 fm in steps of 0.1 fm. For each value of\nd2,z3was varied from 0.7-5.0 fm in steps of 0.2 fm. When\nnecessary we have used a smaller spacing to pinpoint the de-4\nsired location more precisely. For z3<0.7 the large overlap\namong the three alphas lead to convergence problems due to\nunphysically large densities.\n1 2 3 4 51.5234566.6\n−76.5−74.3−72.1−69.9−67.8−65.6−63.4−61.2−59.1−56.9−54.7−52.5−50.4−48.2\nX+MeVd2(fm)\nz3(fm)\nFIG. 1. The 3- αenergy surface obtained from the density constraint\nprocedure as a function of the spacings d2andz3. The point marked\nbyXindicates the location of the minimum.\nIn Fig. 1 we plot the 3- αenergy surface as a function of\nthe spacings d2andz3obtained by the density constrained\nminimization procedure. The minimum energy is obtained\nford2=2.65 fm and z3=2.3 fm. This numerically ob-\ntained minimum corresponds to an equilateral triangle place-\nment of alpha particles, and is identified as the ground state.\nThese findings are in agreement with other cluster model cal-\nculations (see for example [46]). In Fig. 2(a) we plot the\nground state density as well as the localization function for\nthe ground state of12C in the x-zplane. The density has\na triangular shape and looks relatively compact with an oc-\ntupole deformation. Experimentally deduced mass radius of\n12C is 2.43 fm [67,68]. Different cluster model calculations\nyield a range of 2 .40−2.53 fm [40]. Our calculations result\nin a slightly larger radius of 2.57 fm. The quadrupole defor-\nmation for12C is experimentally deduced to be oblate with\nβ2=−0.4 [17,69]. Cluster calculations of Ref. [29] found\nβ2=−0.41 and γ=27.5◦. Our calculations find β2=−0.42\nandγ=29.7◦.\nFigure 2(b) shows the n↑localization function immediately\nafter placing the three alphas in their appropriate locations but\nbefore the start of the density constraint iterations. As we have\nmentioned above, for a single alpha particle the localization\nhas a fixed value of 1.0 throughout. Thus the mere combina-\ntion of three alphas does show a significant localization. How-\never, the dominant localization is still 1.0 suggesting a pure\nalpha makeup. The density constraint minimization modifies\n−10 −5 0 5 10−10−50510\n1.37e−120.01310.02610.03920.05220.06530.07840.09140.1040.1180.1310.1440.1570.17\n0.0\nx (fm)z (fm)\n(a)\n−10 −5 0 5 10−10−50510\n7.9e−140.0750.150.2250.30.3750.450.5250.60.6750.750.8250.90.975x (fm)z (fm)\n(b)\n0.0\n−10 −5 0 5 10−10−50510\n5.1e−170.07320.1460.220.2930.3660.4390.5130.5860.6590.7320.8060.8790.952\n0.0\nx (fm)z (fm)\n(c)FIG. 2. (a) The total density for the ground state configuration of the\nthree alpha particles plotted in the x-zplane. (b) The n↑localiza-\ntion function before the density constraint. (c) The n↑localization\nfunction of the ground state configuration after the density constraint\nminimization.\nthis localization function as shown in Fig. 2(c). The alpha sub-\nstructure of the ground state is still clearly pronounced. The5\nregions close to the value 1.0 indicate the prominent positions\nof the three alpha particles. It is clear that the alpha particles\nare connected by bond like arms. The localization function\nfor protons and spin-down components essentially show the\nsame structure.\nWe have previously shown that enforcing the Pauli ex-\nclusion principle in density-constraint HF calculations has a\nstrong impact on the spin-orbit energy [51], absorbing a large\npart of the Pauli repulsion. This is in agreement with the ob-\nservations that the spin-orbit interaction is the primary driver\nin partially dissolving the alpha clusters in the ground state of\n12C [16,70,71].\nWhat is also interesting is the evolution of the single-\nparticle parities during the density constrained minimization\nprocedure. Initially, all the alpha particles are naturally in their\ns-states. While we do not have good parity for the deformed\nstate, at the end of the minimization four of the six neutron or\nproton single-particle states acquire average negative parity\nvalues (not unity), which is appropriate for the ground state\nof12C. This has been previously observed in the dynamical\ncollapse of the metastable linear-chain state in TDHF calcu-\nlations [31]. The conclusion is that within our approach the\nground state of12C is not a pure alpha condensate but more of\na molecular type state formed by the bonding of three alphas.\n02 4 6 8 J/h\n_00.10.20.30.40.50.6PJ\nFIG. 3. Angular momentum projection of the12C ground state con-\nfiguration.\nWe have performed angular momentum projection of this\nground state configuration following the method discussed in\nRef. [15], which is shown in Fig. 3. It is interesting to see\nthat the major component is J=0, as one would expect for\nthe ground state. There is, however, a significant J=2 com-\nponent implying that, in principle, ground state observables\n(e.g., binding energy) should be evaluated from the J=0 pro-\njected state. Note that there is little contribution from J>2.\nUsing the same procedure we have also tried to identify\nthe configuration that was observed in Ref. [31], which could\nbe the candidate for the Hoyle state at DFT level. It is be-\nlieved to arise from the bending of the linear-chain three al-\npha configuration, which was seen in TDHF calculations ofthe triple-alpha reaction [31] as an intermediate state during\nthe dynamical collapse of the linear-chain state to the spheri-\ncal ground state. There, the dynamical transition of some of\nthe initial single-particle parities from an s-state to a p-state\nwas also noted. The observed metastable bent-arm configu-\n−10 −5 0 5 10−10−50510\n2.93e−120.009580.01920.02870.03830.04790.05750.06710.07660.08620.09580.1050.1150.125x (fm)z (fm)\n(a)\n0.0\n−10 −5 0 5 10−10−50510\n4.27e−170.07490.150.2250.30.3740.4490.5240.5990.6740.7490.8240.8990.974x (fm)z (fm)\n0.0(b)\nFIG. 4. (a) The total density for the bent-arm state configuration of\nthe three alpha particles plotted in the x-zplane. (b) The correspond-\ning localization function of the bent-arm state configuration.\nration occurred during this parity transition. Here, we also\nlooked at the changing parities of the single-particle states as\nwe changed the values of d2andz3. Again, these are not par-\nity projected states so this is simply a signature for changing\nsingle-particle symmetries. Dependence on parity was also\nstudied in cluster model calculations [72]. The location of this\nconfiguration is shown in Fig. 1 with a \"+\" sign. As we see\nPES is very soft in the \"d2\" direction and this point is a very\nshallow local minimum. It is interesting that the dynamical\ncollapse of the linear chain state also showed that the system\nspent some time at the bent-arm configuration but there was no6\ndiscernible minimum in the DC-TDHF potentials (see Fig. 3\nof Ref. [31]). This makes the identification of this configu-\nration more tenuous. The lowest energy configuration corre-\nsponding to this intermediate state is depicted in Fig. 4 and\ncorresponds to d2=5.0 fm and z3=2.2 fm. Similar to the\nground state case we find this bent-arm mode to be a hybrid\nconfiguration of three alphas with molecular like bonds be-\ntween the center alpha particle and the ones on each end, as\nshown in Fig. 4(a). Unlike the ground state configuration this\nconfiguration has only two main bonds and has the shape of an\nobtuse isosceles triangle. The overlap between the Slater de-\nterminant of the ground state configuration and the bent-arm\nstate is on the order of 10−3, which is small enough to con-\nsider that these are different eigenstates of the system. The\nlocalization function for the bent-arm configuration, shown in\nFig. 4(b), is very telling. We see that the two clusters that are\non each end are associated with extended C∼1 regions, indi-\ncating that they are closer to becoming pure alpha particles.\nIV . CONCLUSIONS\nWe have introduced a new framework for studying clus-\nterization in light nuclei, which is based on the constrained\ndensity functional theory. The new approach does not make\nany assumptions about the mathematical form of the single-\nparticle wavefunctions and employs the full effective interac-\ntion. Results show that the12C ground state is an equilateral\ntriangle, which has a molecular type configuration. The nu-\nclear localization function shows bond like structures being\nformed among the original alpha particles as a result of anti-\nsymmetrization and energy minimization. One can conclude\nthat these configurations are a hybrid between pure mean-fieldand a pure alpha particle condensate. From our investigation\nof the cluster energy surface it is clear that a pure alpha con-\ndensate (characterized by pure s-wave states) would only oc-\ncur if the three alphas are relatively far from each other.\nOne disadvantage of not using Gaussian type single-\nparticle states or alpha particles with custom cluster potentials\nis that we are unable to correct for the spurious center\nof mass energy. Another is that procedures like angular\nmomentum projection, generator coordinate method, etc.\nbecome numerically very challenging for the full effective\ninteraction. This makes detailed spectroscopic comparisons\nwith experiment very difficult. On the other hand one\nadvantage is that this ground state of12C may be suitable\nfor fusion barrier calculations using frozen Hartree-Fock or\ndensity constrained frozen Hartree-Fock methods [51,52],\nwhich we plan to investigate in the future. The preparation\nof the alpha clustering configuration can also be used for the\ndevelopment and quantification of new energy density func-\ntionals, particularly in sectors where the static properties are\nunder-informed by the typical data used in calibration [73,74].\nACKNOWLEDGMENTS\nOne of the authors (ASU) would like to thank the orga-\nnizers of the MCD2022 workshop, where some of ideas pre-\nsented here have been inspired. This work has been supported\nby the U.S. Department of Energy under award numbers DE-\nSC0013847 (Vanderbilt University), DE-SC0013365 (Michi-\ngan State University), DE-NA0004074 (NNSA, the Steward-\nship Science Academic Alliances program), and by the Aus-\ntralian Research Council Discovery Project (project number\nDP190100256) funding schemes.\n[1] C. A. Barnes, S. Trentalange, and S. C. Wu, Heavy-Ion Reac-\ntions in Nuclear Astrophysics, in Treatise on Heavy-Ion Sci-\nence, V ol. 6 (Plenum, New York, 1985) pp. 1–60, edited by D.\nA. Bromley.\n[2] B. M. S. Hansen and J. Liebert, Cool White Dwarfs, Annu. Rev.\nAstron. Astrophys. 41, 465 (2003).\n[3] Toonen, S., Nelemans, G., and Portegies Zwart, S., Supernova\nType Ia progenitors from merging double white dwarfs - Using\na new population synthesis model, Astron. Astrophys. 546, A70\n(2012).\n[4] G. Fruet, S. Courtin, M. Heine, D. G. Jenkins, P. Adsley,\nA. Brown, R. Canavan, W. N. Catford, E. Charon, D. Curien,\nS. Della Negra, J. 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C 106,\nL051602 (2022)." }, { "title": "2306.04023v3.Local_density_approximation_for_excited_states.pdf", "content": "Local density approximation for excited states\nTim Gould\nQueensland Micro- and Nanotechnology Centre, Griffith University, Nathan, Qld 4111, Australia∗\nStefano Pittalis\nCNR-Istituto Nanoscienze, Via Campi 213A, I-41125 Modena, Italy\nThe ground state of an homogeneous electron gas is a paradigmatic state that has been used to\nmodel and predict the electronic structure of matter at equilibrium for nearly a century. For half\na century, it has been successfully used to predict ground states of quantum systems via the local\ndensity approximation (LDA) of density functional theory (DFT); and systematic improvements\nin the form of generalized gradient approximations and evolution thereon. Here, we introduce\nthe LDA for excited states by considering a particular class of non-thermal ensemble states of the\nhomogeneous electron gas. These states find sound foundation and application in ensemble-DFT – a\ngeneralization of DFT that can deal with ground and excited states on equal footing. The ensemble-\nLDA is shown to successfully predict difficult low-lying excitations in atoms and molecules for which\napproximations based on local spin density approximation (LSDA) and time-dependent-LDA fail.\nI. INTRODUCTION\nExcitation of many-electron systems characterize novel\nstates of matter and increasingly permeate the functions\nof novel advanced technologies. In problems ranging from\nphotovoltaic devices to quantum dots to nano-particle\ncatalysts to quantum computing devices, particle-like,\ncollective, or topological excitations are exploited coher-\nently. Challenges are multidisciplinary, yet solutions can\nbe inspired – and, increasingly, predicted – by compu-\ntationally investigating quantum structures and mecha-\nnisms at the nanoscale. Density functional theory [1, 2]\n(DFT) has dominated the stage of computational elec-\ntronic structure methodologies since the 1960s, by bal-\nancing accuracy with efficiency. But DFT does not han-\ndle excited states directly, being restricted to addressing\neigenstates of lowest energy (i.e. ground states). This\nwork will show how successful DFT methods for ground\nstates can be upgraded into methods for also tackling\nexcited states.\nThe most fundamental model from which DFT gained\ninspiration, can be traced back to the seminal works\nby Thomas and Fermi [3, 4]. In 1927, they indepen-\ndently proposed a remarkable approximation for quan-\ntum physics – that the state of any many-electron sys-\ntem can be modelled by referring, via the particle den-\nsity (a local quantity), to an homogenous gas of electrons.\nDue to its poor treatment of kinetic energy contributions,\nthe resulting Thomas-Fermi approximation is not very\ngood in practice. But almost all modern modelling of\nelectronic structure employs its spiritual descendent, in\nthe form of Kohn-Sham DFT [1, 2]: 1) kinetic energy\ncontributions are treated quantum mechanically, via a\nnon-interacting auxilliary system; 2) the energy of elec-\ntrostatic interactions is treated clasically, for any given\nparticle density; 3) the HEG is only used to treat the re-\n∗t.gould@griffith.edu.aumaining quantum exchange-correlation (xc) energy con-\ntributions.\nThe homogeneous electron gas (HEG [5]) is, arguably,\nthe simplest many-electron system. It involves N→\n∞electrons interacting in response to a uniform posi-\ntive background charge of fixed density, n, and volume,\nV=N/n→ ∞ . The resulting (interacting) electronic\nstructure problem can be solved semi-analytically in its\nhigh-density and low-density limits, and to high accu-\nracy for moderate densities using quantum Monte Carlo\n(QMC) techniques. [6–8] The known paradigmatic xc be-\nhaviour of HEG may then be used to approximate the un-\nknown xc behaviour of inhomoheneous quantum systems,\nvia parametrisations. [9–11]. Crucially, it has also been\nrecognised that the LDA provides exact leading terms in\na semi-classical expansion of any quantum system, un-\nder appropriate limits; [12–14] which helps to explain the\nongoing success of the Jacob’s ladder [15] philosophy of\nsystematically improving on the LDA. [16–19]\nWhat about excited states? In the late 1980s, the time-\ndepended extension of DFT (TDDFT) was revealed to be\nan highly effective tool for simulating spectra, via a per-\nturbative (linear-response) expansion around the ground\nstate. But, despite its ongoing success, it was soon re-\nvealed [20, 21] that approximations to TDDFT could not\ndescribe important double excitations at all; and strug-\ngle to describe charge transfer excitations except by us-\ning specialized approximations. [22–24] More recently,\nsinglet-triplet inversion [25] (with great promise for pho-\ntovoltaics) has emerged as another important problem\nwhere TDDFT struggles. [26, 27]\nIn parallel with TDDFT, Kohn and collaborators put\nforward a density functional theory for stationary exci-\ntations based on mixed states (ensembles) rather than\npure states: ensemble-DFT (EDFT). [28, 29] Unlike the\nperturbation-based formalism of TDDFT, EDFT recast\nthe problem of computing excited states into an ex-\ntended “ground state”-like problem involving variational\nminima. TDDFT’s rapid success in predicting spectra,arXiv:2306.04023v3 [physics.chem-ph] 15 Jan 20242\nand challenges in constructing useful ensemble approx-\nimations, initially led to EDFT falling by the wayside.\nRecently, however, it has re-emerged as a powerful al-\nternative to TDDFT because approximations in EDFT\ncan solve precisely those excitation problems for which\nTDDFT struggles or fails. [30–42]\nMoreover, recent theoretical breakthroughs [35, 43–47]\nhave revealed aspects of the architecture of key functional\nforms in EDFT that have opened unprecedented possi-\nbilities for novel approximations for excited states. The\nchange of perspective brought about by EDFT compared\nto (TD)DFT is radical: 1) the auxiliary states of the\nKohn-Sham ensemble can acquire the form of coherent\n(finite) superposition of Slater determinants (rather than\nthe ‘disentangled’ single determinant for pure ground\nstates); 2) the ensemble Hartree energy (in contrast to\ntheclassical Hartree energy) accounts for peculiar quan-\ntum features; 3) the ensemble exchange energy does not\n(necessarily) reduce to textbook Fock-exchange expres-\nsions; 4) in addition to regular-looking state-driven cor-\nrelations, unusual density-driven correlations emerge.\nIn this work, we demonstrate that the same system of\nknowledge allows us to derive an exchange-correlation en-\nergy approximation from first principles ( ab initio ). We\nconsider the prominent example of approximations that\nare derivable from the HEG. Given nearly 100 years of\nexploration, one might expect the HEG to have given\nup all its useful secrets. Crucially this work reveals\nthat when the HEG is viewed from the perspective of\nEDFT, we can introduce a class of non-thermal ensem-\nbles from which we can derive a local approximation\nfor excited states directly . The regular LDA has pro-\nvided an highly-effective cornerstone for systematic im-\nprovements for ground states – both as the first rung of\nJacob’s ladder [15] and as a paradigmatic/semi-classical\nlimit that can constrain functional forms [16–18, 48]. The\nensemble-LDA developed in this work therefore provides\nus with a (long-sought) cornerstone for systematic im-\nprovements to approximations for excited states.\nThe remainder of this work is organized as follows:\nSection II gives an introduction to the HEG in the con-\ntext of density functional theory, and briefly introduce\nthe elements of ensemble-DFT which are exploited in the\nnovel parts of the work. Section III presents the relevant\nensemble-states of HEG, which are designed to capture\nexcited-state physics in crucial energy components of the\nHEG ensemble-states (Appendix D reports a parametri-\nsation). Section IV demonstrates the practical usefulness\nof the formal developments done by setting up and apply-\ning an ensemble-LDA to atoms and molecules. Finally,\nSection V summarizes the work, looks toward the near\nfuture, and draws conclusions.\nII. THEORETICAL BACKGROUND\nThe properties of HEGs are conventionally defined\nusing the the Wigner-Seitz radius, rs:= (3\n4πn)1/3≈0.620350 n−1/3, and spin-polarization factor, ζ=n↑−n↓\nn.\nHere, nis the density of electrons and n↑,↓are the densi-\nties of ↑,↓electrons obeying n↑+n↓=n. This combina-\ntion of terms reflects the fact that interactions between\nsame- and different-spin electrons are fundamentally dif-\nferent due to the Pauli exclusion principle, so energies\nchange not only with the total density but also the rela-\ntive contributions of majority ( ↑) and minority ( ↓) elec-\ntrons to the density.\nThis section will first motivate the standard approach\nto understanding HEGs, in the context of density func-\ntional theory. Then, it introduce ensemble density func-\ntional theory, which provides the key theoretical tool for\nthe rest of the work. Throughtout, we use atomic units so\nthat lengths (e.g. rs) are expressed in Bohr and energies\n(e.g. ϵx) are expressed in Hartree (Ha).\nA. Understanding HEGs through density\nfunctional theory\nDensity functional theory (DFT) provides an impor-\ntant tool for the analysis and parametrisation of HEGs.\nKey theorems [1, 49, 50] demonstrate that all proper-\nties of a quantum mechanical ground state are described\nby its density, n(r) (constant, nin an HEG). This is\neasily extended to spin-DFT, [51] which covers de facto\nground states like the lowest energy with a given spin-\npolarization, ζ(r) (constant ζin an HEG). DFT is typi-\ncally used synonymously with Kohn-Sham (KS) DFT, [2]\nand we shall adopt this convention throughout.\nIn Kohn-Sham DFT, the ground state energy of an N-\nelectron system in external (nuclear) potential, v(r), is\nwritten as,\nE0[n] :=Ts[n] +Z\nnvdr+EH[n] +Ex[n] +Ec[n],(1)\nwhere [ n] indicates a functional of the density, n(r), obey-\ningR\nndr=N. Useful exact energy expressions are\nknown for:\n1. The Kohn-Sham kinetic energy functional, Ts[n],\nthat includes kinetic energy effects from a non-\ninteracting system with the same density (and spin)\n– we may write Ts=P\niσ∈occR1\n2|∇ϕiσ(r)|2dr\nusing a set of occupied Kohn-Sham orbitals,\nϕiσ(r); [2]\n2. The Hartree energy functional, EH[n] =U[n], that\nincludes mean-field electrostatic interactions;\n3. The Fock exchange energy functional, Ex[n] =\n−P\nii′σ∈occU[ϕiσϕ∗\ni′σ], that includes corrections\nfor Fermionic exchange based on the same non-\ninteracting system used for Ts.\nThe unknown correlation energy functional, Ec[n], cap-\ntures classical and quantum contributions that are missed\nin the other terms.3\nTABLE I. Summary of Kohn-Sham derived properties of the HEGs considered in this work. Here, Cs= 1.10495 and Cx=\n0.458165. The cases ζ= 0 and ¯f= 2 correspond to an unpolarized gas; and ζ= 1 and ¯f= 1 are equivalent.\nType of HEG Params ts ϵx ∆ϵH\nUnpolarized gas rsCs\nr2s−Cx\nrs0\nPolarized gas rs,ζCs\nr2s(1+ζ)5/3+(1−ζ)5/3\n2−Cx\nrs(1+ζ)4/3+(1−ζ)4/3\n20\nConstant occupation factor (cof) rs,¯fCs\nr2s\u00022\n¯f\u00032/3 −Cx\nrs\u00022\n¯f\u00031/3|ϵx|(2−¯f)(¯f−1)\n¯f\nHere we introduced an electrostatic Coulomb integral,\nU[ρ] =ℜZ\nρ(r)ρ∗(r′)drdr′\n2|r−r′|(2)\nthat was adapted for complex-valued inputs to accom-\nmodate classical (here, in EHonly) and quantum (here,\ninExonly) interactions. All functionals are readily ex-\ntended to spin-polarized ground states by introducing the\nnumber, N↑≤N, of↑electrons ( N↓=N−N↑) as an\nadditional constraint, or equivalently setting ζ=N↑−N↓\nN.\nPrecise details do not matter at this point and will be\nintroduced as required.\nIn a standard HEG, the mean-field Hartree contribu-\ntion (from EH) is cancelled exactly by the positive back-\nground charge. The energy per particle, e=E/N, of an\nHEG may therefore be separated into three components,\ne(n, ζ) =ts(n, ζ) +ϵx(n, ζ) +ϵc(n, ζ), (3)\nusing eq. (1). Here, nandζare scalar constants; and\nts:=Ts/N,ϵx:=Ex/Nandϵc:=Ec/Nare energy den-\nsities per particle. The Kohn-Sham kinetic and exchange\nenergies may be obtained analytically, and are,\nts(rs, ζ) =ts(rs)(1+ζ)5/3+(1−ζ)5/3\n2:=ts(rs)fs(ζ),(4)\nϵx(rs, ζ) =ϵx(rs)(1+ζ)4/3+(1−ζ)4/3\n2:=ϵx(rs)fx(ζ),(5)\nwhere,\nts(rs) :=Ct\nr2s=3\n10\u00009π\n4\u00012/3r−2\ns= 1.10495 r−2\ns, (6)\nϵx(rs) :=−Cx\nrs=−3\n4π\u00009π\n4\u00011/3r−1\ns=−0.458165 r−1\ns,(7)\nare the kinetic and exchange energies of an unpolar-\nized HEG (in atomic units). We may alternately write,\nts(n) = 2 .87123 n2/3andϵx(n) =−0.738559 n1/3.\nThe final ingredient is the correlation energy term,\nϵc(rs, ζ) :=X\nkϵk\nc(rs)fk\nc(ζ), (8)\nwhich has known series expansions for the high- ( rs→0)\nand low-density ( rs→ ∞ ) limits, but is unknown in gen-\neral. Total energies, eQMC, of HEGs may be evaluated\nto high accuracy via quantum Monte-Carlo (QMC) sim-\nulations, which have served to supplement limiting cases\nsince pioneering work by Ceperley and Alder. [6] Then,\nϵc=eQMC−ts−ϵx, may be parametrised (e.g. [9–11])by a truncated series in the general form of (8). Models\nand parameters for ϵcare usually designed to satisfy or\napproximately satisfy limiting behaviours of HEGs, with\nsome free parameters that can be optimized to reproduce\nreference data from QMC at intermediate values.\nB. A brief introduction to ensemble DFT\nWe conclude our theory introduction by digressing\nfrom standard DFT text book material, to provide some\ntheoretical foundations for ensemble DFT (EDFT) which\naddresses a wider class of electronic structure problems\nthan are allowed by conventional DFT. [28, 50, 52–55]\nThe results from this section will then be applied to\nHEGs, to reveal some surprising results. Specifically, we\nshall focus on EDFT for excited states [28, 29].\nTo understand ensemble DFT, let us first define quan-\ntum state ensembles. A (quantum state) ensemble, ˆΓ, is\nan operator that describes a classical mixture of quantum\nstates. It may be defined using a spectral representation,\nˆΓ =X\nκwκ|κ⟩⟨κ|,0≤wκ≤1,X\nκwκ= 1,(9)\nin which an arbitrary set of orthonormal quantum states,\n|κ⟩, are assigned probabilities/weights, wκ. Operator ex-\npectation values, ¯O=⟨Ψ|ˆO|Ψ⟩, are replaced by ¯Ow=\nTr[ˆΓwˆO] =P\nκwκ⟨κ|ˆO|κ⟩which involves quantum and\nclassical averages. Ensembles are more flexible than wave\nfunctions, so can describe constrained, open and degener-\nate systems that are otherwise outside the remit of wave\nfunction mechanics or DFT. Various theorems [28, 29, 54]\nextend key results of DFT to ensembles, including impor-\ntant variational principles.\nIn excited state EDFT, the usual variational formula,\nE0= min Ψ⟨Ψ|ˆH|Ψ⟩, is replaced by the weighted average,\nEw:= inf\nˆΓwTr\u0002ˆΓwˆH\u0003\n=X\nκwκEκ, (10)\nwhere ˆΓwis an ensemble with a given set of weights\nw={w0, w1, . . .}; and Eκare eigen-energies of ˆHor-\ndered such that the lowest energies are associated with\nthe largest weights. The energies are in usual ascending\n‘excitation’ order if we define the weights to be mono-\ntonically decreasing, i.e. wκ′≤wκforEκ′≥Eκ. Note,\nwe follow the usual convention of using superscriptsw4\n(orcofelater) to identify ensemble functionals. But we\ndepart from the recent convention of using calligraphic\nletters to avoid confusion between Efor total energies of\nensembles, and ϵfor energies per particle of HEGs.\nIt is convenient to generalize eq. (1) to ensembles by\nwriting,\nEw[n] :=Tw\ns[n] +Z\nnvdr+Ew\nH[n] +Ew\nx[n] +Ew\nc[n].\n(11)\nHere, windicates the set of weights, nis the density, and\nvis the external potential. In a Kohn-Sham formalism,\nthe ensemble density is conveniently written as,\nnw(r) :=X\nkfw\nknk(r), fw\nk:=X\nκwκθκ\nk,(12)\nin terms of orbital densities, nk(r) :=|ϕk(r)|2; and av-\nerage occupation factors, fw\nk, which may be non-integer\nand involve a weighted average over the integer occupa-\ntion factors, θκ\nk∈(0,1,2) (i.e. no occupation, occupation\nin one spin, or occupation in both spins) of each KS state\nin the ensemble. The orbitals obey a spin-independent\nKS-like equation, [ ˆt+vw\ns(r)]ϕk(r) = εkϕk(r), where\nˆt≡ −1\n2∇2is the one-body kinetic energy operator.\nWith the ensemble formalism defined, we are now\nready to define the terms in eq. (11). Recent work [43,\n44, 46] has sought to rigorously define exact energy func-\ntionals for excited state ensembles, giving,\nTw\ns[n] :=X\nkfw\nkZ\n1\n2|∇ϕk|2dr, (13)\nEw\nH[n] :=X\nκκ′wmax( κ,κ′)U[ns,κκ′], (14)\nEw\nx[n] :=−X\nkk′fw\nmax( k,k′)U[ϕkϕ∗\nk′] (15)\nHere, we usedR\nϕ∗ˆtϕdr=R1\n2|∇ϕ|2dr;U[ρ] as defined\nearlier in Eq. (2); introduced ns,κκ(r) =⟨κs|ˆn(r)|κs⟩as\nthe density of Kohn-Sham state, |κs⟩; and introduced\nns,κ̸=κ′(r) =⟨κs|ˆn(r)|κ′\ns⟩as the (potentially complex-\nvalued) transition density between Kohn-Sham states\n|κs⟩and|κ′\ns⟩.\nThe remaining energy, Ew\nc:=Ew−R\nnvdr−Tw\ns−\nEw\nH−Ew\nx, is the unknown correlation energy functional.\nIt is convenient to partition,\nEw\nc[n] :=Ew,SD\nc[n] +Ew,DD\nc [n]. (16)\ninto state-driven (SD) and density-driven (DD) compo-\nnents, each with different physical origins. [44–46] The\ndensity-driven term is always zero in pure states like po-\nlarized and unpolarized HEGs.\nIt is sometimes useful to rewrite eqs (14) and (15) as,\nEw\nH/x=Z\nnw\n2,H/x(r,r′)drdr′\n2|r−r′|(17)\nkF\n kF\nkFkcofe\nF\nWavenumber, q0\nOcc., fqFIG. 1. Occupation factors as a function of wavenumber for\nan unpolarized gas (solid block), polarized gas ( ζ=1\n2, solid\nline) and cofe gas (dotted line) – all at the same electron\ndensity.\nusing the ensemble Hartree and exchange pair-densities,\nnw\n2,H(r,r′) =X\nκκ′wmax( κ,κ′)ns,κκ′(r)ns,κ′κ(r′),(18)\nnw\n2,x(r,r′) =−X\nkk′fmax( k,k′)ρk(r,r′)ρ∗\nk′(r,r′),(19)\nwhere ρk(r,r′) =ϕk(r)ϕ∗\nk(r′). It is straightforward to\nsee that using eqs (18) and (19) in (17) give the same\nenergies as (14) and (15), respectively. Details and other\nhelpful relationships for functionals will be introduced\nand used as required.\nBefore proceeding further, we make the important as-\nsumption that the results of Section II B apply to HEGs.\nThis is an assumption because all EDFT results shown\nso far are for finite systems with countable numbers of\nexcitations. By contrast, homogeneous electron gases are\ninfinite and their excitations are uncountable . The rest\nof this manuscript treats HEGs as the appropriate ther-\nmodynamic limit of finite systems whose properties are\nconsistent with the ensemble density functional theory\npresented in this section, and so obey straightforward\ngeneralizations of key equations.\nIII. CONSTANT OCCUPATION FACTOR HEGS\nWith core theory now established, let us proceed to\nexplore generalizations of HEG physics that exploit the\nadditional degrees of freedom from ensembles. Our aim is\nto develop an understanding of HEGs that spans ground-\nand excited-state physics. To that end, we will reveal\nthe properties of “constant occupation factor ensemble”\nHEGs – the meaning of the name will soon become appar-\nent. The key to generalizations is to invoke both ground\nand excited states of HEGs. As we shall show below,\nmany properties are then uniquely determined by the oc-\ncupation factors, fq, of the HEG; while others depend on\nwexplicitly, so require some extra restrictions on the na-\nture of excited states because there can be many different\nsets of weights, w, that yield a given fq.\nEqs (12), (13) and (15) reveal that the density, kinetic\nenergy and exchange energy of any ensemble system de-\npend explicitly only on the orbital occupation factors,\nfw\ni. In HEGs, we replace fw\nibyfq, i.e as a function5\nof absolute wavenumber, q. This follows from: i) the\nfact that the KS “orbitals” of an HEG are planewaves\nϕq(r)∝eiq·r; and ii) that KS minimization dictates that\nwe fill each q=|q|in full. Thus, given fqit is possible to\ndefine the density, n, as well as the kinetic and exchange\nenergies. We will therefore first discuss some HEGs from\nthe perspective of orbital occupation factors; before pro-\nceeding to refine the definition.\nThe most intuitive form of HEG is an unpolarized gas\nin the lowest energy (ground) state. In orbital (KS)\nterms, the unpolarized HEG non-interacting ground state\nis a Slater determinant of doubly occupied plane-wave\norbitals. Occupied states fill in ↑/↓pairs up to a sin-\ngle Fermi wave number, kF. Its wave-number dependent\noccupation factor and density are,\nfunpol\nq =2Θ( kF−q), k F= (3π2n)1/3, (20)\nwhere Θ( x) ={1∀x≥0; 0∀x < 0}is a Heaviside step\nfunction. The density, n, of the gas is sufficient to de-\nscribe the state.\nGround states realized by exposing the HEG to a uni-\nform external magnetic field (the corresponding vector\npotential being ignored, as in spin-DFT) have a wave-\nnumber dependent occupation factor determined by spin-\ndependent Fermi wavenumbers,\nfpol\nq=Θ(k↑\nF−q) + Θ( k↓\nF−q), k↑,↓\nF= (6π2n↑,↓)1/3.\n(21)\nThe unpolarized gas is then the special case of n↑=\nn↓=n\n2giving ζ= 0. A fully polarized gas has n↑=n,\nn↓= 0 and ζ= 1. For definiteness, we work under the\nconvention that the majority spin channel is the “up”\n(↑) channel. The density, n, and spin-polarization, ζ, are\nsufficient to describe the state.\nIn this work, we consider (non-thermal) ensembles of\nexcited states, which correspond to averaged occupation\nfactors. Specifically, we consider ensembles obeying,\nfcofe\nq=¯fΘ(¯kcofe\nF−q),¯kcofe\nF= (6π2n/¯f)1/3(22)\nThe bar on top of ¯f(and, thus, ¯kF) means that this quan-\ntity stems from an average w.r.t. an ensemble rather than\nto a pure state; and ‘cofe’ stands for ‘constant occupation\nfactor ensemble’, [56] reflecting the fact that the system\nhas the same occupation factor right up to a single (en-\nsemble) Fermi level, unlike a polarized gas. We shall\ndiscuss below that the correct interpretation associates\nfcofe\nqwith an unpolarized ensemble.\nBefore proceeding further, it is worth considering why\nwe should choose fcofe\nqto be constant or zero, rather than\nany of the infinite number of other options we could have\nchosen. The main motivation is simplicity. Firstly, we\naim to keep the number of parameters to two ( nand ¯f)\nlike the spin-polarized gas ( nandζ). We also aim to\nensure that limiting cases (unpolarized and fully polar-\nized gases) are reproduced by cofe gases – once adapted\nto inhomogeneous systems the limits respectively corre-\nspond to singlet ground states and ground and excitedstates of one-electron systems. Finally, noting that both\nlimits have the special feature that they yield constant\noccupation factors (two and one, respectively), we aim\nto retain this special feature in between the limits as a\nsensible generalization that incorporates excited states.\nThese three aims dictate the form of Eq. (22), as well\nas the kinetic and exchange energies of cofe HEGs. The\naddition of some extra restrictions (to be discussed below,\nas needed) on the excited states dictates the remaining\nproperties of cofe-HEGs. As we shall see later in Sec-\ntion IV the resulting cofe gas is effective for predicting\nground and excited states of inhomogeneous systems.\nFigure 1 illustrates the different occupation factors for\nunpolarized, polarized and cofe HEGs, all at the same\ndensity n. The polarized gas has ζ=1\n2, while the cofe\nHEG has ¯f= 1.7. The unpolarized gas has a single Fermi\nlevel with double occupations, the polarized gas has two\nFermi levels, one higher ( ↑) and one lower ( ↓) than that\nof the unpolarized gas, and is doubly occupied up to the\nlower level and then singly occupied to the higher level.\nThe cofe gas also has a single Fermi level between the\nunpolarized and ↓levels, but is only partly occupied for\nallq. The choice of ζ=1\n2and ¯f= 1.7 ensures that the\npolarized and cofe HEGs also have the same exchange\nenergy – as can be seen by evaluating eqs (5) and (26).\nOnce we accept to deal with ensembles from con-\nstrained occupation factors, we can mix with equal\nweights a polarized HEG with its time-reversed partner.\nNothing change in terms of the evaluation of the energy\ncomponents. What changes is the interpretation. Now,\nwe can find a continuum of unpolarized ensembles of cofe-\nHEGs, with energies that go from that of the regular\nunpolarized to that of the regular fully polarized HEGs.\nBut the ensembles can also accommodate ground states\nandexcited states (keeping in mind that the polarized\ngas is itself an excited state in the absence of a magnetic\nfield), in a sense that will be clarified just below.\nThe ingredients of ˆΓcofeare most easily understood by\nconsidering a finite system with four electrons:\n•The unique unpolarized state is |unpol⟩=|1222⟩,\nwhich is consistent with a Fermi level, ¯kcofe\nF=ϵ+\n2,\njust above the second orbital energy. As a singular\nstate we set wunpol = 1 and obtain f1=f2=¯f= 2.\n•The fully polarized system, |fullpol ⟩=|1↑2↑3↑4↑⟩,\nis also unique ( wfullpol = 1). It has ¯kcofe\nF=ϵ+\n4(four\norbitals allowed) and yields f1=f2=f3=f4=\n¯f= 1. The corresponding state with all ↓-electrons\nhas the same energetics (but time-reversed dynam-\nics). Ensemble averaging the ↑- and↓-spin systems\ntherefore yields a net unpolarized system with the\nsame energy terms.\n•But, if we allow three orbitals, we have three max-\nimally polarized ( N↑= 3 and N↓= 1) states:\n|cofe0⟩ ≡ |122↑3↑⟩,|cofe1⟩ ≡ |1↑223↑⟩, and|cofe2⟩ ≡\n|1↑2↑32⟩. Each state has a spin-polarization ζeff=\n3−1\n4=1\n2. The (non-interacting) Fermi level, k↑\nF,6\n012\n012\n012\nWavenumber, q012Occupation factor, fq\nFIG. 2. Like Figure 1 except showing polarized and cofe\nHEGs at a variety of ζand ¯f. Note that the polarized and\ncofe gas are, as expected, the same for ζ= 0 and ¯f= 2, or\nζ= 1 and ¯f= 1.\nfor↑electrons is always k↑\nF=ϵ+\n3. But, we can-\nnot define a level for ↓electrons due to holes in\n|cofe1⟩and|cofe2⟩. Assigning each of the three\nstates an equal weight, w0=w1=w2=1\n3,\nyields f1=f2=f3=4\n3, as desired. Thus,\n¯kcofe\nF=k↑\nF(=k↓\nF) =ϵ+\n3(after we also average over\nspin) for the whole ensemble.\nReplacing orbital indices by q, and taking the limit\nN, V→ ∞ for fixed density, n=N\nV, and ensemble Fermi\nlevel, ¯kF, yields the actual cof ensemble. It is composed\nof ground and excited states all with the same polariza-\ntion, ζeff= 2/¯f−1, (and their time reversed partners)\nwhere ¯f= 6π2n¯k−3\nFfollows from eq. (22). Sections III B\nand III D will expand a little on the specifics of states\nrequired for cofe HEGs. Here and henceforth we drop\nthe superscript from ¯kcofe\nF, and simply use ¯kF.\nIt is finally worth noting that the energy of a cofe HEG\nwith ¯f= 2 is always equal to that of an unpolarized gas\nwith ζ= 0, while the energy of a cofe HEG with ¯f= 1 is\nalways equal to that of a fully polarized gas with ζ= 1\n(keeping in mind that the ensemble averages over the\ntime-reversed state). Figure 2 shows fqfor a selection of\npolarized and cofe gases between (and at) these limits,\nall yielding the same density, n. Values of ζand ¯fare\n‘paired’ to yield the same exchange energy – we will later\nexploit this pairing in eq. (42) of Section III D.\nWe will next proceed to compute the energy compo-\nnents of the cofe HEG. Key results are summarized in\nTable I.\nA. Density, kinetic and exchange energies\nThe density, n[fq] :=R∞\n0fqq2dq\n2π2, and kinetic energy\nper particle,\nts[fq] :=1\nn[fq]Z∞\n0fqq2\n2q2dq\n2π2. (23)\nof an HEG are direct functionals of the occupation fac-\ntor distribution, fq. Prefactors deal with normaliza-tion of the orbitals and energies. The kinetic energy\nintegral follow from the fact that ϕ∗\nq(r)[−1\n2∇2ϕq(r)] =\n1\n2q2ϕ∗\nq(r)ϕq(r).\nTypically we are interested in some fixed density, n=\n3\n4πr3s, defined by its Wigner-Seitz radius, rs, which im-\nposes constraints on fq(e.g. the Fermi levels in the pre-\nvious section). Throughout we will implicitly define all\nHEGs to be at fixed Wigner-Seitz radius, rs, and vary\nother parameters under this assumption. Using the oc-\ncupation factor model for a polarized gas with fixed ζ\nandrsyields the kinetic energy given by eq. (4).\nConsider instead a cofe HEG, where fqis given by\neq. (22). We obtain, n[fq] =¯f¯k3\nF\n6π2from which we confirm\nthat ¯kF= (6π2n/¯f)1/3. The kinetic energy of a cofe\nHEG therefore has the separable expression,\ntcofe\ns(rs,¯f) =3¯kF(rs,¯f)2\n10=ts(rs)\u00142\n¯f\u00152/3\n, (24)\nusing ts(rs) from eq. (6).\nIn addition to the density and kinetic energy, the ex-\nchange energy of any HEG may also be evaluated directly\nfrom fq. Replacing sums over kandk′by integrals over\nqandq′lets us rewrite Eq. (15) as,\nϵx[fq] :=−1\nn[fq]Z∞\n0Z∞\n0fmax( q,q′)V(q, q′)q′2dq′\n2π2q2dq\n2π2,\n(25)\nwhere, V(q, q′) =R1\n−1πdx\nq2+q′2−2qq′x=π\nqq′log|q+q′|\n|q−q′|is the\nspherically averaged Coulomb potential. A little addi-\ntional work on the integral (see Appendix A for details)\nyields eq. (5) for a polarized gas; and,\nϵcofe\nx(rs,¯f) =−¯f\nnZ¯kF\n0q\nπq2dq\n2π2=ϵx(rs)\u00142\n¯f\u00151/3\n,(26)\nfor cofe HEGs, where ϵx(rs) is the unpolarized HEG ex-\npression of eq. (7).\nAlthough not necessary for computing, ϵx, we may sim-\nilarly derive an expression for the HEG exchange hole,\ndefined in eq. (19). We obtain,\nncofe\n2,x(R;rs,¯f) =Πcofe\nx(rs,¯f)N(¯kFR) (27)\nwhere,\nΠcofe\nx(rs,¯f) =−¯fZ¯kF\n0q3\n3π2q2dq\n2π2=−n2\n¯f(28)\nis the on-top pair-density of the exchange hole, and\nN(x) := 9[sin( x)−xcos(x)]2/x6is a function. We will\nuse the relationship between the exchange energy and\nexchange hole to help in deriving the properties of the\nHartree energy, in the next section.7\nB. Hartree energy\nThe ensemble Hartree energy functional is given in\neq. (14). This term is usually ignored in HEG discus-\nsions because n2,H=n2in polarized and unpolarized\ngases at arbitrary ζ, which means that ϵHexactly can-\ncels the energy of the positive background charge, ϵbg–\nthat is, ϵH[n2,H=n2] =−ϵbg. In cofe HEGs this can-\ncellation is incomplete. The singular background charge\nis guaranteed, by charge neutrality, to be cancelled in\nfull. However, the Hartree pair-density, nw\n2,H̸=n2, dif-\nfers from the background charge density, n2, and thus ϵw\nH\nincludes additional terms. The energy per particle of an\nensemble HEG is,\new[fw\nq] :=ts[fw\nq] + ∆ ϵw\nH[fw\nq] +ϵx[fw\nq] +ϵw\nc[fw\nq],(29)\nwhere superscriptswindicate an explict dependence on\nthe nature of the ensemble. The additional positive\nHartree energy contribution,\n∆ϵw\nH=ϵw\nH−ϵbg=1\nnZ\n∆nw\n2,H(R)dR\n2R, (30)\nmay be evaluated [eqs (17) and (18)] using the ensemble\nHartree pair-density deviation, ∆ nw\n2,H=nw\n2,H−n2.\nWe therefore seek closed-form expressions for ncofe\n2,Hand\n∆ϵcofe\nHfor the special case of a cofe HEGs with maxi-\nmal polarization within the ensemble, as defined earlier.\nFull details for Hartree expressions are rather involved\nso have been left to Appendix B. The rough argument\nis as follows: 1) the background charge is cancelled by\nκ=κ′terms in (14) or (18), so we need only evaluate\nκ̸=κ′terms; 2) the cof ensemble states, |κ⟩, contain ev-\nery possible combination of paired and unpaired orbitals\nup to ¯kF; 3) each of these states is weighted equally; 4)\nwe may therefore use combinatorial arguments to evalu-\nate key expressions. The final step recognises that each\nstate may be defined by a set, {q}double , of doubly occu-\npied orbitals, such that the remaining occupied orbitals\n(with |q| ≤¯kF) contain only an ↑electron. Each non-\ninteracting state is then a Slater determinant consistent\nwith the occupations, whose properties may be under-\nstood via {q}double and¯kF.\nAppendix B yields, ∆ ϵcofe\nH(rs,¯f) :=CH\nrs(2−¯f)(¯f−1)\n¯f4/3,\n[eq. (B8)] where CH= 21/3Cx. We rewrite this as,\n∆ϵcofe\nH(rs,¯f) =|ϵx(rs,¯f)|(2−¯f)(¯f−1)\n¯f, (31)\nfor use in eq. (29) and later expressions. This result fol-\nlows from the fact that,\nncofe\n2,H(R;rs,¯f) =n2+ ∆Πcofe\nH(rs,¯f)N(¯kFR), (32)\n∆Πcofe\nH(rs,¯f) =n2(2−¯f)(¯f−1)\n¯f2 =−(2−¯f)(¯f−1)\n¯fΠcofe\nx,(33)\nwhere N(x) is the same expression used in (27).C. Energies in the low-density limit\nWe cannot analytically evaluate the energy of an\nHEG at arbitrary density, n. We can, however, semi-\nanalytically evaluate it in the high density (large n, small\nrs) and low density (small n, large rs) limits. The high\ndensity limit may be obtained from a series solution\naround the Kohn-Sham solution. In the low density limit,\nthe electrons are far enough apart to undergo a process\nknown as a Wigner crystallisation. [57, 58] The resulting\n“strictly correlated electron” physics may then be under-\nstood via a classical leading order term, with quantum\ncorrections. The transition occurs at rs≈100 Bohr.\nRecent work [47] has shown that any dependence on\nensemble properties must vanish in the low-density limit\nof any finite system; so that all excited state properties\nbecome degenerate to both leading and sub-leading order .\nIt is very likely that this result also holds true in the ther-\nmodynamic limit of HEGs, as justified by the following\nintuition:\n1. As the density becomes small, the distance between\nelectrons becomes large and the particles become\neffectively classical with a quantum state defined\nby fluctuations around a classical minima;\n2. Whether the system is finite, or infinite, the fluctu-\nations may be “excited” any number of times with\nno impact on the classical leading order term of the\ninteraction energy;\n3. Furthermore, the next leading order quantum cor-\nrection from zero-point energy fluctuations around\nthe classical minima are dictated only by the den-\nsity constraint, and are therefore also independent\nof excitation structure.\nThis result has important implications for both spin-\npolarized and cofe HEGs, as both may be represented\nas ensemble of excited states – with specific properties\ngoverned by ζor¯f, respectively. It follows from the\nabove that the leading two orders of their low-density en-\nergies are independent of the excitation structure. Con-\nsequently, energies are independent of ¯fandζ. Inde-\npendence of ζhas long been theorized for spin-polarized\nHEGs. Recent QMC data [59] provides confirmation of\nthis result.\nThe leading order terms correspond to 1 /rsand 1 /r3/2\ns\nin the usual large- rsseries description of HEGs. There-\nfore, ensemble and spin effects can only contribute at\nO(1/r2\ns). The (Hartree) exchange- and correlation en-\nergy of strictly correlated electrons in the low-density\nlimit (ld) therefore obeys lim rs→∞ϵHxc(rs, ζ) =ϵld\nHxc(rs),\nwhere,\nϵld\nHxc(rs) :=−C∞\nrs+C′\n∞\nr3/2\ns+. . . , (34)\nincludes only the part of the Hartree energy that is\nnot cancelled by background charge. The best esti-8\nmates for coefficients are C∞= 0.8959≈1.95Cxand\nC′\n∞= 1.328. [60, 61]\nIn regular HEGs, the Hartree term is fully cancelled\nby background charge so can be ignored. For polarized\nHEGs we therefore obtain, lim rs→∞ϵc=ϵld\nHxc−ϵx, and,\nlim\nrs→∞ϵc(rs, ζ) :=−C∞+Cxfx(ζ)\nrs+C′\n∞\nr3/2\ns+. . . , (35)\nusing fxfrom eq. (5). By contrast, in a cofe HEG there is\na non-zero component (∆ ϵcofe\nH) in the Hartree energy. It\ntherefore follows that, lim rs→∞ϵcofe\nc=ϵld\nxc−∆ϵcofe\nH−ϵcofe\nx.\nWe finally obtain,\nlim\nrs→∞ϵcofe\nc(rs,¯f) =−C∞+Cxfcofe\nHx(¯f)\nrs+C′\n∞\nr3/2\ns+. . . ,\n(36)\nwhere,\nfcofe\nHx(¯f) =\u00142\n¯f\u00151/3(¯f−1)2+ 1\n¯f, (37)\nfollows from eqs (26) and (31). This is the appropriate\nlow-density series expansion for the correlation energy of\ncofe HEGs.\nD. State-driven correlation energies\nIn general, the correlation energy of an ensemble is\nseparable into two terms, [44, 45]\nEw\nc:=Ew,SD\nc +Ew,DD\nc , (38)\nwhere each covers different physics of the ensemble. The\n“state-driven” (SD) term is the only term present in pure\nstates, such as polarized gases. In general ensembles, it is\nlike a weighted average of conventional correlation ener-\ngies for the different states of the ensemble. The “density-\ndriven” (DD) term reflects the fact that the densities\nof the individual Kohn-Sham and interacting states that\nform the ensemble are not necessarily the same – only\nthe averaged ensemble density is the same.\nWe expect that only the SD part of the correlation en-\nergy should form part of the xc energy used in density\nfunctional approximations, so focus here on this term –\nwe explain this choice in Section IV. Our goal is there-\nfore to determine ϵSD,cofe\nc (rs,¯f) as a function of rsand\n¯f, which we will use as a basis for parameterization in\nthe next section. This involves considering the high- and\nlow-density limits of matter (and therefore cofe-HEGs),\nfor which exact results will be derived. We will also\ndiscuss how to repurpose existing data for values in be-\ntween these limits. Comprehensive analysis of both state-\nand density-driven correlation terms is reported in Ap-\npendix C. Below, we summarize key elements of the SD\ncorrelation energy analysis.The division into SD and DD terms is not unique, [44–\n46] and any explicit study of the separation into SD and\nDD terms requires accessing the properties of a variety\nof excited states of interacting HEGs. Nevertheless, dis-\ncussion near eq. (14) of Ref. 46 argues that the SD cor-\nrelation energy may be written in adiabatic connection\nand fluctuation-dissipation theorem (ACFD) form:\nϵSD\nc:=1\nnZ1\n0dλZ∞\n0−dω\nπZdrdr′\n2|r−r′|\n×\u0002\nχλ(r,r′;iω)−χ0(r,r′;iω)\u0003\n. (39)\nHere, χ0is the collective density-density response of the\nnon-interacting cofe HEG defined earlier – i.e. the en-\nsemble of part-polarized ground- and excited states that\nyield ¯f.χλis its equivalent it a scaled Coulomb inter-\naction1\nR→λ\nR. In principal, the individual states in the\ninteracting ensemble may be followed from their known\nλ= 0 values to their unknown value at arbitrary λ, al-\nthough this is not required in practice.\nThe key step toward understanding how to separate\nand parametrise terms is to use the random-phase ap-\nproximation (RPA). RPA becomes exact (to leading or-\nder) in the high-density limit. [10] More generally, RPA\nprovides an approximate solution for eq. (39), and thus\nprovides insights into the SD correlation term. Details\nare provided in Appendix C 1. Key findings are: i) that,\nϵSD,cofe\nc is approximately linear in ¯ffor high densities; ii)\nfor low densities we obtain a scaling that is similar to\nfcofe\nx(¯f). Appendix C 2 then uses the RPA results, and\nfundamental theory, to argue that,\nϵSD,cofe,hd\nc (rs,¯f) =(¯f−1)ϵc(rs,0) + (2 −¯f)ϵc(rs,1),\n(40)\nϵSD,cofe,ld\nc (rs,¯f) =ϵx(rs)\u0002C∞\nCx−fcofe\nx(¯f)\u0003\n+C′\n∞\nr3/2\ns,(41)\nare, respectively, the exact high- and low-density limits of\nϵcofe\nc. That is, lim rs→0ϵSD,cofe\nc (rs,¯f) =ϵSD,cofe,hd\nc (rs,¯f)\nand lim rs→∞ϵSD,cofe\nc (rs,¯f) =ϵSD,cofe,ld\nc (rs,¯f).\nFilling in the gaps between these limits requires quan-\ntum Monte Carlo (QMC) calculations: which, however,\nareonly available for spin-polarized ground-states of ho-\nmogenous gases. Appendix C 2 therefore shows how\nto reuse existing spin-polarized QMC data for the in-\nbetween regime, by adapting it for cofe HEGs. Specifi-\ncally, it argues that,\nϵSD,cofe,mhd\nc (rs,¯f)≡ϵQMC\nc(rs, ζ=ˆf−1\nx-map (¯f)) (42)\nis a reasonable approximation for medium-high densities\n(mhd). The key assumption behind this relationship is\nthat HEGs with same exchange energy should have a\nsimilar state-driven correlation energy. Thus, ˆfx-map (ζ)\nis a function yielding, ϵcofe\nx(rs,ˆfx-map (ζ)) =ϵx(rs, ζ) and\nϵcofe\nx(rs,¯f) =ϵx(rs,ˆf−1\nx-map (¯f)). Eq. (42) becomes exact\nin the low-density limit, but incorrect in the high-density\nlimit. More information is provided in Appendix C 2.9\n60\n40\n20\n0SD\nc [mHa]\nRef. cofe\n-200-150-100-500SD\nxc [mHa]\n1 3/2 2\nf [unitless]\n-320-300-280-260xc [mHa]\n rs=2\n100101102\nWigner-Seitz radius, rs [Bohr]100105110115120xc enhance. [%]\nFIG. 3. Correlation (top) and xc (middle) energies and xc\n(bottom) enhancement factors for HEGs as a function of rs\nand ¯f∈(2,1.85,1.50,1). Plots show the cofe (solid lines)\nparametrisation introduced here, and the adapted benchmark\nresults from Ref. 8 (circles). The inset plot shows ϵxc(cofe\nand benchmark) as a function of ¯f, for rs= 2. Line colours\nindicate the value of ¯f(see inset for values).\nAppendix D details parametrization of ϵSD,cofe\nc (rs,¯f)\nfor arbitrary densities, based on the theoretical work\nin this section. As an intermediate step, it also in-\ntroduces approximations for ˆfx-map and its inverse,\nfor use in eq. (42). Key results are visually sum-\nmarized in Figure 3, which compares the parametri-\nsation of ϵSD,cofe\nc with the (adapted) reference data\nused to fit it. The top plot shows correlation ener-\ngies, ϵSD,cofe\nc (rs,¯f). The middle plot shows deviations,\n∆ϵSD,cofe\nxc =ϵSD,cofe\nxc (rs,¯f)−ϵSD,cofe\nxc (rs,2), from unpo-\nlarized gas values. The bottom plot shows xc enhance-\nment factors, ϵSD,cofe\nxc (rs,¯f)/ϵSD,cofe\nxc (rs,2), which must\napproach one (100%) in the low-density (large rs) limit.\nIV. FROM COFE-HEG TO REAL,\nINHOMOGENEOUS SYSTEMS\nA. Adaptation to inhomogeneous systems\nThe main application of HEG work is as the founda-\ntion for approximations to inhomogeneous systems. Di-\nrect approximations based on Thomas-Fermi theory is\ngenerally not very effective for this purpose. But Kohn-\nSham density functional approximations (DFAs) based\non HEGs are wildly successful. What explains the dif-\nference? Kohn and Sham proposed to use an inhomo-geneous description of the (quantum mechanical) kinetic\nand (classical) Hartree energies, together with an HEG-\nbased approximation for the exchange and correlation\nenergies – see Eq. (1) and related discussion. It is natu-\nral to assume that ensembles will similarly benefit from\na Kohn-Sham treatment, with the goal to extend the\nsuccess of KS-based approximations for ground states to\nexcited states. Therefore, we write\nEw\neLDA = min\nn\u001a\nTw\ns[n] +Z\nn(r)v(r)dr+Ew\nH[n]\n+Z\nn(r)ϵcofe\nxc(rs(r),¯f(r))dr\u001b\n. (43)\nHere, we have replaced pure state TsandEHby their\nensemble equivalents, Tw\ns[eq. (13)] and Ew\nH[eq. (14)];\nand locally approximated the xc energy by the cofe LDA\nwith local Wigner-Seitz radius, rs(r), and local effective\noccupation factor, ¯f(r) – to be discussed below.\nIn practical terms, it is usually better to use a state-\nspecific DFA, rather than an ensemble. Eq. (43) may be\nadapted to a state-specific form [41] by defining EeLDA\nκ :=\n∂wκEw\neLDA (for any positive weight, wκ, in the ensemble).\nThen, target state |κ⟩has energy,\nEeLDA\nκ =Ts[nκ] +Z\nnκ(r)v(r)dr+EH,κ[nκ]\n+Z\nnκ(r)ϵcofe\nxc(rs,κ(r),¯fκ(r))dr, (44)\nwhere EH,κ:=∂wκEw\nHis its ensemble Hartree energy;\nand we evaluate the energy using the Wigner-Seitz radius\nand effective occupation factor from nκ.\nThe above approximation ignores density-driven (DD)\ncorrelations entirely. i.e. sets Ew,DD\nc≡0, with the as-\nsumption that they are small. More fundamentally, this\nchoice comes from the fact that the use of exact Hartree\n[Eq. (14); defined by the fluctuation dissipation theorem]\nis equivalent to a physical argument that HEGs should\nonly be used to approximate response-like properties of\nsystems [46] – consistent with work on ground state KS-\nDFT. Density-driven correlations are Hartree-like [44–46]\nso, by the same argument, should alsobe treated via the\ninhomogeneous system — not from an HEG. Therefore,\nϵcofe\nxc(rs,¯f) :=ϵcofe\nx(rs,¯f) +ϵSD,cofe\nc (rs,¯f), (45)\ninvolves only the state-driven correlation energy. From\nAppendix D, we see that the exchange energy term takes\nthe exact form,\nϵcofe\nx(rs,¯f) :=ϵx(rs)[2/¯f]1/3(46)\nwhile the state-driven correlation energy term may be\nparametrized as,\nϵSD,cofe\nc (rs,¯f) := ( ¯f−1)ϵ0\nc+ (2−¯f)ϵ1\nc\n+ (¯f−1)(2−¯f)\u0002\nM2(rs) + (3\n2−¯f)M3(rs)\u0003\n(47)10\nCore Mix Frontier1.01.11.2EnhancementLSDA\neLDA: wocc\neLDA: wdocc\nHeBe N NeMg P Ar0510152025IP [eV]\nFIG. 4. Left: Enhancement factor for different ratios of core\nand frontier occupied orbitals from LSDA (teal); and eLDA\nwith effective ¯ffrom Eq. (48) (wocc, magenta) and Eq. (49)\n(wdocc, orange). Right: Ionisation potentials (IPs) for atoms\nHe–Ar using a conventional LSDA [9] (navy), Eq. (48) (ma-\ngenta) and Eq. (49) (orange). Dashed lines indicate deviations\nfrom experimental IPs.\nforϵζ\nccomputed using eq. (D4) (parameters in Table II).\nHere, M2,3(rs) involve weighted sums (coefficients in Ta-\nble III) over functions, ϵζ\nc.\nAll expressions except ¯f(r) are thus known. The fi-\nnal step demands that we make an ansatz for ¯f(r) that\ncorrectly reproduces the properties of a cofe-HEG, but is\nalso effective for many-electron systems. A natural first\nguess is to employ a density-weighted average of occupa-\ntion factors,\n¯fwocc(r) :=X\nifw\nifw\nini(r)\nn(r)=P\ni(fw\ni)2ni(r)P\nifw\nini(r).(48)\nHere, ni=|ϕi|2is the density of orbital ϕi, and fw\niis\nits occupation factor in the ensemble. It is easily varified\nthat this ansatz is exact for any cofe HEG, so is prima\nfacie a reasonable extension to inhomogeneous systems.\nHowever, testing (to be discussed below) reveals that\nthis ansatz can yield poor results for ground states.\nThese errors come from the effective spin-enhancement\nbeing too great in regions that are partly-polarized [i.e.\nwhere 1 <¯f(r)<2]. Fortunately, we may exploit the\nfact that there are other choices of inhomogeneous ¯f(r)\nthat yield correct behaviour in HEGs, but that do not\nhamper performance in inhomogeneous ground states.\nWe therefore (see Supp. Mat. Sec. I [62] for details)\ninstead adopt a double weighted average,\n¯fdwocc(r) :=P\ni(fw\ni)1/3ni(r)P\nifw\nini(r)P\ni(fw\ni)8/3ni(r)P\nifw\nini(r),(49)\nfor calculations. The form is chosen to closely match the\nspin-enhancement of exchange in doublets and thus ex-\ntend the perfect (by construction) replication of LSDA in\none-electron (doublet) systems by an approximate repli-\ncation of LSDA in general doublet systems.\nThe left panel of Figure 4 illustrates the importance\nof choosing ¯fappropriately. It shows the exchange\nenhancement factor of a doublet system (density n=\n2nCore+nFrontier ), for different ratios of nFrontier /nCore,\nusing standard spin-polarization, and ensemble enhance-\nment with Eqs (48) and (49). It is clear that (48) over-\nC NF NHNO\nO2 O PF PHS2 S Si SO01020304050TS gap [kcal/mol]LSDA eLDA Ref.FIG. 5. Triplet–singlet gaps in atoms and diatomic systems\nfrom LSDA (navy) and eLDA (orange) calculations, compared\nto experimental reference data (black with crosses). LSDA\nand reference data from Ref. 67. TDLDA gaps are too large\nfor the figure, so have been left out.\nenhances exchange in general, relative to LSDA. By con-\ntrast, (49) matches quite closely to the spin-polarized\nenhancement of LSDA for all ratios.\nHow well does eLDA work in practice? The next sec-\ntions will address excited state energies. But, first, we\nneed to ensure that the eLDA does not make things worse\nfor ground state energies . The right panel of Figure 4\ntherefore shows the ionization potentials (IPs) of atoms\n– that is the difference in ground state energies between\nthe atom and its cation – computed with Eq. (43) using\nEq. (48) and Eq. (49). IPs provide a useful test of ¯f(r)\non ground states because the occupation factors of atoms\nand ions are always different and at least one system al-\nways involves an unpaired electron.\nThe figure reveals that Eq. (49) yields results that\nare consistently close to standard LSDA calculations,\nwhereas (48) leads to much greater deviations in some\ncases. We therefore see that using (49) yields good (rel-\native to LSDA) performance on ground states; and use\nEq. (49) for our inhomogeneous effective occupation fac-\ntor in all subsequent calculations.\nTechnical details for all atomic and molecular calcu-\nlations for ground and excited states are in Supp. Mat.\nSec. II [62]. For now it suffices to say that we carry out\nLSDA and time-dependent LDA (TDLDA) calculations\nusing standard self-consistent field (SCF) approaches im-\nplemented in psi4 [63, 64] and pyscf , [65, 66] but eval-\nuate eLDA calculations using an orbital optimized ap-\nproach with psi4 as an ‘engine’. Spin and spatial sym-\nmetries are preserved in eLDA calculations, except for\natoms which are evaluated using cylindrical spatial sym-\nmetries for consistency with standard quantum chemistry\ncodes and practice.\nB. Low-lying excitations in molecules\nWith the eLDA established and validated on ground\nstate systems, we are ready to test its predictive ability\nfor excitations. As a first test, (Figure 5) we consider the\ntwelve triplet-singlet gaps in biradicals of the TS12 [68]\ndataset. The performance of PW92 [10] energy differ-\nences (referred to as ∆SCF calculations, to differentiate11\n02468Excitation energy [eV]\nTDLDA TBETBE eLDAeLDAGlyoxal\n02468\nTDLDA TBETBE eLDAeLDABenzoquinone\n02468\nTDLDA TBETBE eLDAeLDAT etrazine\nFIG. 6. Low-lying spectra (singlets only) of glyoxal, benzo-\nquinone and tetrazine predicted using TDLDA (navy) and\neLDA (orange); compared against theoretical best estimate\n(TBE) values. [69] Connections between spectrum in approx-\nimations and TBE are shown using dotted lines, to facilitate\ncomparisons. TDLDA captures single excitations (indicated\nby single arrows on the level line) but misses the double exci-\ntations (double arrows) entirely so these connections are ex-\ncluded from the plot.\nfrom TDLDA calculations) on this dataset was explored\nin Ref. 67, using restricted, unrestricted and complex or-\nbital Kohn-Sham theory. The mean-signed errors (root\nmean squared errors) from ∆SCF calculations are −13.7\n(14.5) kcal/mol using LSDA (i.e. unrestricted Kohn-\nSham theory); and 10 .9 (11 .5) kcal/mol for restricted\ntheory. Employing complex orbitals reduces these LDA\nerrors substantially, to −1.2 (2.2) kcal/mol, albeit at the\nexpense of non-idempotent density matrices.\nUsing the eLDA formalism developed here (also a\n∆SCF method) to compute the gaps yields errors of −5.0\n(7.4) kcal/mol – respectable statistics and a major im-\nprovement on LSDA, as shown in Figure 5. Indeed, eLDA\nis closer in quality to the complex orbital performance\nthan LDA or LSDA performance, despite eLDA being\na ‘restricted’ theory that preserves idempotency (unlike\ncomplex orbitals) and avoids spin-contamination issues\n(unlike unrestricted KS).\nBy contrast, evaluating TDLDA (using VWN cor-\nrelation [9] for consistency with other results) on the\ntriplet ground states yields enormous errors of 77.2 (88.6)\nkcal/mol – with the predicted gaps being too large to in-\nclude in the figure. eLDA thus out-performs both ground\nstate (LSDA) and excited state (TDLDA) LDA-based\ncalculations, despite being constrained to yield desirable\nphysical properties of the true KS solution.\nContinuing on the theme of predicting difficult exci-\ntations, let us consider some excitations that TDLDA\ncannot predict at all: double excitations. Double excita-\ntions are singlet excited states in which the interacting\nwave function is dominated by a Slater determinant with\npaired orbitals, and in which one pair is ‘doubly pro-\nmoted’ from the dominant ground state Slater determi-nant (e.g. |ϕ2\n0ϕ2\n1ϕ2\n3⟩instead of |ϕ2\n0ϕ2\n1ϕ2\n2⟩for a six-electron\nsystem). They are impossible to predict using the adi-\nabatic approximation that is employed in all practical\nimplementations of time-dependent DFT. [20, 21]\nFigure 6 shows the low-lying singlet spectra of some\nselected molecules, computed using adiabatic time-\ndependent LDA (TDLDA) and eLDA. We choose gly-\noxal, benzoquinone and tetrazine from the QuestDB\ndataset, [69] as their low-lying spectra includes difficult-\nto-predict double excitations for which high-quality the-\noretical best estimates (TBE) results are available. They\ntherefore serve as good examples to compare the eLDA\napproach with its TDLDA counterpart.\nIt is immediately clear that, for the lowest-lying exci-\ntations involving single promotion of an electron (“single\nexcitations”, single arrows), eLDA predicts similar exci-\ntation energies to TDLDA and thus has similar perfor-\nmance – albeit with a slight tendency to underestimate\nrelative to TDLDA. However, unlike TDLDA, eLDA is\nalso able to predict excitations involving double promo-\ntion of electrons (“double excitations”, double arrows)\nwith a perfomance similar to that of single excitations.\nThus, eLDA is nearly as good as TDLDA for low-lying\nexcitations involving single promotion of an electron, but\nis also able to predict double promotions, unlike TDLDA.\nIt therefore offers a major advance on TDLDA.\nFigure 6 also provides evidence that eLDA can be\na cornerstone theory for better excited state approxi-\nmations, based on the following argument. As can be\nseen from the figure, TDLDA and eLDA yield very sim-\nilar energies for most single excitations. The similarity\nof TDLDA and eLDA energies suggests that all regular\nDFAs are likely to yield similar energies for these ex-\ncitations, whether evaluated as TDDFAs or eDFAs – a\ntheoretical justification for this argument is provided in\nSupp. Mat. Sec. III [62]. Thus, the thirty years of re-\nfinement of generalized gradient approximations (GGAs)\nand meta-GGAs (MGGAs) that has improved the quality\nof spectra predicted using TD(M)GGAs is likely to sim-\nilarly improve spectra evaluated using e(M)GGAs. But\ne(M)GGAs may alsoexploit the extra degree of freedom\nenabled by the use of cofe-gas physics and effective ¯f(r).\nIn summary, we see that TDLDA fails quite dramati-\ncally for TS12 (Figure 5 and related discussion) and can-\nnot capture double excitations (Figure. 6); in contrast\nto an excellent (TS12) or impressive (double excitations)\nperformance from eLDA on the difficult excitations. Er-\nrors in single excitation spectra (Figure. 6) from TDLDA\nand eLDA are similar. Directly, this shows that eLDA ei-\nther improves excited state predictions, or does not make\nthem worse. Indirectly, it has positive implications for\nrefinements to eLDAs, e.g. eGGAs or eMGGAs.\nV. FUTURE PROSPECTS AND CONCLUSIONS\nEnsemble density functional theory has recently ben-\nefited from a surge of fundamental understanding. This12\nhas led to rapid advancements in extending, to ex-\ncited states, the power of density functional theory for\ncomputing electronic structure of ground states. Espe-\ncially, EDFT deals seamlessly with highly “quantum”\nstates [41] (e.g. superpositions of Slater determinants\nand double excitations) of relevance to solar energy ap-\nplications and quantum technologies.\nHowever, despite an accumulation of successful appli-\ncations, EDFT currently lacks a fully consistent frame-\nwork for improving approximations: in the sense that it\nborrows density functional approximations (DFAs) which\nwere originally designed for ground states as the key\nbuilding blocks of the extended DFAs for excited states.\nThis work takes the first step toward deriving a novel\nfamily of DFAs specifically designed for excitations . It\npresents the cornerstone theory: the ‘cofe’ homogeneous\nelectron gas (HEG) and the LDA for EDFT (eLDA). The\n‘cofe’ HEG is developed using an unnoticed – thus, so\nfar, unexplored – class of non-thermal ensemble states of\nthe the HEG. Analytic expressions of the relevant (de-\nfined by two parameters, like LSDA) energy components\nare reported in Table I. Some of these components have\nno analogues in regular DFT but find home and use in\nEDFT. High- and low-density limits of the correlation\nenergy have been found analytically.\nThe eLDA is derived by dividing the DFT energy ex-\npression into terms that need to be treated using the\ninhomogeneous system, and those that are locally ap-\nproximated using a cofe gas. Parametrisations for all\nterms required by the eLDA are derived and provided.\nAn expression is also derived for the effective occupa-\ntion factor of inhomogeneous systems, and is ‘normed’\non doublet systems.\nThe novel eLDA is then tested on a suite of important\nexamples including ionization potentials, small triplet-\nsinglet gaps, and low-lying excitations. These examples\nreveal that eLDA performs similarly to LSDA and/or\ntime-dependent LDA (TDLDA) on problems where stan-\ndard theories are known to work. However, it also per-\nforms very effectively on problems where LSDA/TDLDA\nfail– yielding excellent triplet-singlet gaps and impres-\nsive double excitation energies.\neLDA therefore readily offers an effective alternative\nto standard polarized-gas based theories for both ground\nand excited state problems. But, we stress that its true\npotential lies as the cornerstone for better models and\nmethodologies. What are the next natural steps to be\nconsidered? We finish with three suggestions.\n(1) It is vital to develop a generalized gradient approx-\nimation (GGA) for cofe-HEGs, to yield an eGGA along\nthe lines of Eq. (43). The development of accurate GGAs\nin the late-1980s/early-90s greatly accelerated interest in\nDFT for ground states, by giving answers that were use-\nfully predictive. eGGAs should do the same for excited\nstates. Importantly, eGGAs would seamlessly integrate\nwith existing hybrid-EDFT successes [40–42] and remove\nreliance on combination laws that (despite working un-\nexpectedly well) are known to be incorrect for correla-tion. [41] From there, additional steps may readily be\ntaken up an excited state Jacob’s ladder, [15] to gain sys-\ntematic improvements in excited state DFT modelling.\n(2) The optimal way to model ¯f(r) remains an open\nproblem, and is entangled with (1). It would be useful\nto understand why Eq. (49) works so much better than\nEq. (48). Exploiting exact relationships, like combination\nlaws, [41] is likely to lead to improved understanding and\nadaptation of ¯f(r) in inhomogeneous systems; and thus\nimprovements to the predictive ability of eLDA and any\neDFAs built on it.\n(3) The cofe-gas is not the only excited state (ensem-\nble) HEG that we could have used. As discussed in Sec-\ntion III it is a logical and simple two-parameter model\nthat yields appropriate limits yet incorporates excited\nstate physics. But, allowing for more parameters pro-\nvides a wide scope for further generalizations. For ex-\nample, one might separate the density into core (density,\nncore) orbitals that are all double occupied, and use a\ncofe-like treatment for the remaining orbitals – yielding\na three-parameter HEG governed by n,ncore/nand ¯f\nthat includes excited states.\nPython code for studying and implementing the theory\nwork in this manuscript is provided on Github https:\n//github.com/gambort/cofHEG . Code to reproduce the\natomic and molecular tests is available on request.\nACKNOWLEDGMENTS\nTG was supported by an Australian Research Coun-\ncil (ARC) Discovery Project (DP200100033) and Future\nFellowship (FT210100663). Computing resources were\nprovided by the National Computing Merit Application\nScheme (NCMAS sp13). TG and SP would like to thank\nPaola Gori-Giorgi for interesting discussions regarding\nhomogeneous electron gases and their low density limit;\nuseful discussions with Marco Govoni on a previous ver-\nsion of the manuscript are also acknowledged.\nAppendix A: Exchange properties\nBoth ϵxand Π xinvolve integrals of form,\nX2[f] :=Z∞\n0Z∞\n0fmax( q,q′)A(q, q′)q′2dq′\n2π2q2dq\n2π2\n=2Z∞\n0Z∞\nq′fqA(q, q′)q2dq\n2π2q′2dq′\n2π2\n=2Z∞\n0Z∞\n0Θ(q−q′)fqA(q, q′)q′2dq′\n2π2q2dq\n2π2\n=Z∞\n0fq¯A(q)q2dq\n2π2(A1)13\nwhere A(q, q′) =A(q′, q) and,\n¯A(q) =2Zq\n0A(q, q′)q′2dq′\n2π2\nForϵxwe have A(q, q′) = V(q, q′) where ¯V(q) =\n2Rq\n0V(q, q′)q′2dq′\n2π2=q\nπR1\n0log|1+x|\n|1−x|xdx=q\nπ. We thus ob-\ntain eq. (25) of the main text. To compute Π xwe can\nsetA(q, q′) = 1 where ¯1(q) = 2Rq\n0q′2dq′\n2π2=q3\n3π2and so\nwe can easily compute Π xgiven fq. For cofe HEGs we\nobtain eqs (26) and (28).\nThe case of n2,x(R) is also covered by (A1), by set-\nting A(q, q′;R) =R\nei(q−q′)·Rdˆq\n4πdˆq′\n4π. However, this is\nrather painful to deal with in general. The special case\nof cofe HEGs is more easily handled by recognising that\nfmax( q,q′)=¯fΘ(¯kF−q)Θ(¯kF−q′). Then,\nncofe\n2,x(R) =−¯fZ¯kF\n0eiq·Rdq\n(2π)3Z¯kF\n0e−iq′·Rdq′\n(2π)3(A2)\n=−¯f\f\f\f\f¯k3\nF\n6π2g(¯kFR)\f\f\f\f2\n≡ −Πcofe\nxN(¯kFR) (A3)\nwhere Π x=−n2\n¯f,g(x) = 3[sin( x)−xcos(x)]/x3and\nN(x) =|g(x)|2. Thus, we obtain eq. (27).\nAppendix B: Hartree properties of cofe HEGs\nLet us consider eqs (17) and (18) for the special case\nof an HEG. First, we note that, nκκ=nfor every state\nand therefore, n2,H=n2+ ∆n2,Hwhere ∆ n2,H(r,r′) =P\nκ̸=κ′wmax( κ,κ′)nκκ′(r)nκ′κ(r′). Furthermore, the re-\nsulting pair-density can depend only on R=r−r′while\nsymmetry means it depends only on R=|r−r′|. Thus,\n∆ϵH=1\nN∆EH=1\nnZ\n∆n2,H(R)4πR2dR\n2R(B1)\nwhere we used n2to cancel the background charge,\nN=nVto cancel the integral over r, and symmetry\nto simplify the remaining integral over r′=r+R. Our\ngoal is therefore to determine ∆ n2,H(R), Note, the work-\ning in this appendix is rather involved, so we will often\ndrop superscriptswin working.\nWe are now ready to look at HEG ensembles. Con-\nsider a finite HEG of Nelectrons in a volume V, with\ndensity n=N/V. The orbitals are ϕq≈1√\nVeiq·rforq\non an appropriate reciprocal space grid. Each state, |κ⟩\nhas density ns,κκ(r) =N/V =n. The ground state is\n|0⟩=|q2\n1···q2\nN/2⟩and is unpolarized. Other states may\nbe described using |κ⟩=ˆPQκ|0⟩where ˆPpromotes Fock\norbitals in the Slater determinant and Qκ:=qa1···qap\nqi1···qip\ncontains lists of from ( i≤N/2) and to ( a > N/ 2) or-\nbitals, including spin. Cross-densities, when κ̸=κ′, are\nns,κκ′(r) =ei∆qκκ′·r/Vor zero. The former result oc-\ncurs if and only if QκandQκ′differ by a single orbitalof the same spin, giving ∆ qκκ′=q∈κ−q∈κ′. We use\n“connected” (con) to refer to any pair of states κandκ′\nthat differ only by a single orbital, and call ∆ qκκ′the\nconnection wavenumber.\nLet us now consider the case that Nelectrons are as-\nsigned to N/2≤M≤Norbitals, for a mean occupation\noff=N\nM. There are NT=\u00002M\nN\u0001\ntotal states once spin\nis accounted for, each of which is weighted by, w=1\nNT.\nEach of the NTstates, |κ⟩, has N↑,↓,κelectrons of each\nspin, giving ζκ=N↑,κ−N↓,κ\nN. State |κ⟩is connected to\nCκother states. Since only one orbital may change at a\ntime, we obtain Cκ=N↑,κ(M−N↑,κ)+N↓,κ(M−N↓,κ) =\nNM−N2\n2(1 +ζ2\nκ), where N↑,↓,κ≤M.\nOur goal is to obtain useful properties of the Hartree\npair-density. The pair-density is defined by,\n∆n2,H(R) =1\nNTX\nκ,κ′conκei∆qκκ′·R\nV2, (B2)\nwhere we used wκ=wκ′=1\nNTandr−r′:=R. The spe-\ncial case of r=r′(R=0) yields the “on-top” pair den-\nsity deviation, Π H= ∆n2H(R= 0), which is relatively\nstraightforward to evaluate using, ∆ n2,H=1\nNTV2P\nκCκ,\nwhich follows from ei∆q·R= 1 for all connected states,\nand the definition of Cκ. Using Cκfrom the above para-\ngraph yields,\n∆ΠH=1\nV2[NM +N2\n2(1 + ¯ζ2)] =n2\u00021\nf−1+¯ζ2\n2\u0003\n.(B3)\nwhere ¯ζ2=1\nNTP\nκζ2\nκis the ensemble averaged of the\nsquared spin-polarization.\nAs an initial test, consider the above analysis for the\ntwo special types of gases, unpolarized and fully polarized\ngases, which have no ensemble effects and which must\ntherefore yield ∆Π H= 0. An unpolarized gas involves\nM=N/2,¯f= 2,NT= 1 and ζκ= 0, yielding ∆Π H=\nn2(1\n2−1\n2) = 0. A fully polarized gas involves M=N,¯f=\n1,NT= 1 and ζκ= 1, yielding ∆Π H=n2(1\n1−1) = 0.\nThus, both exhibit the expected behaviour. We therefore\nsee that eq. (B3) is consistent with existing results.\nWe are now ready to generalize to constant occupation\nfactor (cof) gases, with fq=¯fΘ(¯kF−q), for 1 <¯f <2.\nAs discussed in the main text, we restricted to the special\ncase of maximally polarized states, |κ⟩, in which each\nstate has the maximum spin-polarization allowed by ¯f.\nAll these states involve N↑=MandN↓=N−Mgiving,\nζκ=2M−N\nN=2\n¯f−1 =¯ζ. Eq. (B3) then yields,\n∆ΠH=n2(2−¯f)(¯f−1)\n¯f2,ΠH=n23¯f−2\n¯f2.(B4)\nfor the on-top, R=0, pair-density.\nWe are now ready to move on from the on-top hole to\nconsider general R̸=0. We first recognise that equal\nwaiting of states is equivalent to equal weighting of con-14\nnection wavenumbers, yielding,\n∆n2,H(R) =∆Π H1\nM2X\nqX\nq′̸=qei(q−q′)·R\n=∆Π H\u0002\n|g(¯kFR)|2−1\nM\u0003\n, (B5)\nwhere g:=1\nMP\nqeiq·R. We next impose symmetry on\nthe wavenumbers, and approximate the sum by an inte-\ngral to obtain,\ng(¯kFR)≈1\nMZ(3M\n4π)1/3\n0sin(qkVR)\nqkVR4πq2dq\n=3[sin( ¯kFR)−¯kFRcos(¯kFR)]\n(¯kFR)3(B6)\nwhere kV= 2π/V1/3is the wavenumber associated with\nthe volume V; and ¯kF= (6π2M/V )1/3= (6π2n/¯f)1/3is\nthe usual Fermi wavenumber. g(x) the same expression\nfound in eq. (A3). Note, ∆ n2,H(R) integrates to zero, as\nexpected.\nFinally, eq. (B1) becomes ∆ ϵH=∆ΠH\nn[R∞\n0g(¯kFR)2\nR\n2πR2dR−¯f\n2n(9π\n2V)1/3]. In the limit V→ ∞ the second\nterm vanishes, yielding,\n∆ϵcofe\nH=¯f∆ΠH|ϵcofe\nx(rs,¯f)|=CH\nrs(2−¯f)(¯f−1)\n¯f4/3(B7)\n=|ϵcofe\nx(rs,¯f)|(2−¯f)(¯f−1)\n¯f(B8)\nwhere we used, ϵcofe\nx=−¯f\nnR∞\n0g(¯kFR)2\nR2πR2dR[which\nfollows from n2,x(R) = −¯fg(¯kFR)2] and ϵcofe\nx =\n−Cx\nrs[2/¯f]1/3derived in the main text, to obtain CH=\n21/3Cx= 0.577252. Similarly,\nn2,H(R) =n2+ ∆n2,H(R) =n2+ ∆Πcofe\nHN(¯kFR) (B9)\nwhere ∆Πcofe\nHis defined in eq. (B4); and N(x) =g(x)2=\n9[sin( x)−xcos(x)]2/x6is the unitless function defined\nnear eq. (27) or (A3).\nAppendix C: State-driven correlation of cofe HEGs\n1. State-driven correlation energy from the\nrandom-phase approximation\nThe state-driven correlation energy [eq. (39)] involves\nthe response function at imaginary frequencies. The\nimaginary frequency density-density response of an un-\npolarized HEG is,\nχ0(q, iω;rs) :=−kF\n4π2C(q\n2kF,ω\nqkF) (C1)\n1.0 1.2 1.4 1.6 1.8 2.0\n(Average) occupation factor, f0.60.70.80.91.0c enhancement\npol. c factor\ncofe c factor\ncofe x factorFIG. 7. Correlation enhancement of spin-polarized (dash-dot\nlines) and cofe (solid lines) HEGs as a function of occupation\nfactor, f. The cofe ot exchange factor (dotted lines) is also\nshown after rescaling to yield the same vales for ¯f= 1 and\n¯f= 2.\nwhere kF= 1.9191583 /rs= (3 π2n)1/3is the Fermi\nwavenumber of an unpolarized gas. Here,\nC(Q,Γ) =1 +Γ2−Q+Q−\n4QlogQ2\n++ Γ2\nQ2\n−+ Γ2\n+ Γ\u0014\ntan−1Q−\nΓ−tan−1Q+\nΓ\u0015\n(C2)\nwhere Q±=Q±1. For brevity we shall use Q=q\n2kFand Γ =ω\nqkFto always mean unpolarized gas quantities.\nFor a polarized HEG we take half of two unpolarized\nsystems with kF↑=kF(1 + ζ)1/3:=kFh+andkF↓=\nkF(1−ζ)1/3:=kFh−. Therefore,\nχ0(q, ω;rs, ζ) =−kF\n8π2\u0002\nh+C(Q\nh+,Γ\nh+) +h−C(Q\nh−,Γ\nh−)\u0003\n(C3)\nThe cofe case of constant ¯fis easily dealt with by includ-\ning a prefactor of ¯fonχ0, and using the cofe Fermi level,\n¯kF=kF(2/¯f)1/3:=kFg. It follows from ¯f= 2/g3that,\nχcofe\n0(q, ω;rs,¯f) =−kF\n4π2g2C(Q\ng,Γ\ng). (C4)\nSetting ζ= 0 and ¯f= 2 yields h±=g= 1 and yields the\nsame response as the unpolarized gas. Similarly, setting\nζ= 1 in (C3) gives the same result as setting ¯f= 1 in\n(C4), as expected.\nFrom the response function we are able to evaluate the\nrandom-phase approximation for the correlation energy,\nvia,\nϵRPA\nc=1\n2nZ∞\n0dω\nπZ∞\n0q2dq\n2π2[χ04π\nq2+ log(1 −χ04π\nq2)].\n(C5)\nThis may be made more convenient by using n=k3\nF\n3π2,\nq= 2kFQandω=qkFΓ to write,\nϵRPA\nc=12k2\nF\nπIQΓh\nπ\nk2\nFQ2χ0i\n. (C6)15\nwhere IQΓ[f] :=R∞\n0dΓR∞\n0Q3dQ[−f+ log(1 + f)]. We\nmay also define, ¯I(P) =IQΓh\nP\nQ2C(Q,Γ)i\n.\nThus, the RPA enhancement factor for a polarized,\nrelative to an unpolarized gas at the same density, gas is,\nξRPA\nc(ζ) =IQΓh\nPh+\n2Q2C(Q\nh+,Γ\nh+) +Ph−\n2Q2C(Q\nh−,Γ\nh−)i\nIQΓh\nP\nQ2C(Q,Γ)i\n=h5\n+¯I(P\n2h+) +h5\n−¯I(P\n2h−)\n¯I(P)(C7)\nwhere h±= (1±ζ)1/3. The equivalent enhancement\nfactor of a cof ensemble HEG may be written as,\nξRPA,cofe\nc (¯f) =IQΓh\nP\ng2Q2C(Q\ng,Γ\ng)i\nIQΓh\nP\nQ2C(Q,Γ)i=g5¯I(P\ng4)\n¯I(P)(C8)\nwhere P:=1\n2πkF= 0.08293 rsandg= (2/¯f)1/3.\nThe RPA enhancement is expected to be accurate in\nthe high-density of matter kF→ ∞ . Figure 7 shows\n(state-driven) correlation energy enhancement factors,\nξRPA\nc(ζ) and ξRPA,cofe\nc (f) as a function of the (average)\noccupation factor, ¯f, using ζ=ˆf−1\nx-map (¯f) [from eq. (D8)]\nfor the effective spin-polarization. It reports ξfor high\n(n= 106), medium ( n= 1) and low ( n= 10−6) densi-\nties. We see that the state-driven correlation energy of\ncofe HEGs is: i) virtually linear in ¯f, for high densities;\nii) very similar to the (renormalized) on-top exchange\nenhancement factor, for low densities.\nThe high density ( rs→0) behaviour of ξcofe\nccan be\nshown analytically, because P→0. We may therefore\nTaylor expand the log to obtain,\nlim\nP→∞¯I(P)≈Z∞\n0Q3dQZ∞\n0dΓ1\n2\u0000P\nQ2C\u00012, (C9)\nfrom which it follows that ξcofe\nc=g5¯I(P/g4)/¯I(P) =\ng5(1\ng4)2=g−3=¯f/2 is linear in ¯f. The RPA is not\nappropriate for the low density limit, although we shall\nlater see it is qualitatively correct.\n2. State-driven correlation energies in general\nWe are now ready to use what we have learned about\ncorrelation energies from the RPA and theoretical argu-\nments to obtain general expressions for the SD correla-\ntion energies in cofe HEGs. Let us begin with the low-\ndensity limit. The main text has shown that,\nlim\nrs→∞ϵcofe\nc(rs,¯f) =ϵx(rs)\u0002C∞\nCx−fcofe\nHx(¯f)\u0003\n. (C10)\nIt can also be shown that lim rs→∞ϵDD,cofe\nc (rs,¯f)→\n−∆ϵcofe\nH(rs,¯f) – this result is a specialized case of a\nbroader relationship to be discussed in a future work.\n1.0 1.1 1.2\nx [unitless]\n0.60.81.0c [unitless]\nrs=1cofe\npol.\n1.0 1.1 1.2\nx [unitless]\nrs=2cofe\npol.\n1.0 1.1 1.2\nx [unitless]\nrs=5cofe\npol.FIG. 8. ¯ξcversus ξxusing RPA data for cofe HEG (orange,\ndashed lines) and polarized HEG (navy, solid lines). Black\ndots indicate data from Ref. 8.\nIt is thus clear that the SD enhancement factor must\ncapture the low-density scaling and cancel exchange:\nlim\nrs→∞ϵSD,cofe\nc (rs,¯f)→ϵx(rs)\u0002C∞\nCx−fcofe\nx(¯f)\u0003\n.(C11)\nSurprisingly, this is consistent with the low-density be-\nhaviour shown in Figure 7 and so reveals that the RPA\nis qualitatively correct even in the low-density limit.\nIn the high-density limit, we instead obtain,\nlim\nrs→0ϵSD,cofe\nc (rs,¯f)→(¯f−1)ϵU\nc+ (2−¯f)ϵP\nc\n=ϵU\nc+ (2−¯f)[ϵP\nc−ϵU\nc] (C12)\nwhere ϵU\nc:=ϵc(rs,0) is the correlation energy of an un-\npolarized gas and ϵP\nc:=ϵc(rs,1) is the correlation en-\nergy of a fully polarized gas and 2 −¯f= [ξRPA,cofe(¯f)−\nξRPA(0)]/[ξRPA(1)−ξRPA(0)]. This is analogous to the\nknown result that,\nlim\nrs→0ϵc(rs, ζ)→ϵU\nc+HRPA(ζ)[ϵP\nc−ϵU\nc], (C13)\nwhere HRPA(ζ) =−2[I(ζ)−1] is obtained from eq. (32)\nof Ref. 70 or, equivalently, HRPA(ζ) := [ ξRPA(ζ)−\nξRPA(0)]/[ξRPA(1)−ξRPA(0)]. We cannot say anything\nabout the DD correlation energy in this limit.\nWe thus obtain limiting behaviours for high and low-\ndensity HEGs. In typical polarized gases, one uses expan-\nsion in both limits together with QMC data, ϵQMC\nc(rs, ζ),\nto fill in the gaps for moderate and large densities. We do\nnot have QMC data for cofe gases. Thus, the final step\nin our analysis of correlation energies is to show how to\nreuse existing polarized gas QMC data for cofe HEGs.\nAs a first step, we assume that the high-density re-\nlationship [eq. (C13)] between RPA and exact results is\ntrue for moderate and large rs. That is, we expect\n¯ξc:= 1 +ξc(ζ= 1)−1\nξRPAc(ζ= 1)−1\u0002\nξRPA\nc−1\u0003\n≈ξQMC\nc (C14)\nto be approximately valid for all rs. The usefulness of this\napproximation is further supported by Figure 7, which\nshows that the RPA yields an approximately linear de-\npendence on ¯feven for low-density HEGs where the RPA\nis expected to be poor.\nThe second step is to recognise that, in low-density\ngases, we may write, ξc=C∞\nCx−ξxandξSD,cofe\nc =C∞\nCx+16\nξcofe\nxand therefore ϵSD,cofe\nc (rs→0,¯f)≈X(ϵcofe\nx(rs,¯f))\nwhere ϵc(rs→0, ζ) :=X(ϵx(rs, ζ)) – here Xis a single-\nvariable function. Figure 8 shows that a similar result\nnearly holds for moderate rsand, furthermore, that mod-\nels of both cofe and polarized gases agree rather well with\nQMC data from Spink et al [8], despite the data being\nfor polarized gases. We therefore assume that,\nϵSD,cofe\nc (rs,ˆfx-map (ζ))≈ϵQMC\nc(rs, ζ) (C15)\nfor moderate and large densities with viable QMC data,\nwhere ˆfx-map is defined such that ϵx(ζ) = ϵcofe\nx(¯f=\nˆfx-map (ζ)). This is a rather good approximation in prac-\ntice as the maximum difference between ϵSD,cofe\nc using\nRPA and ϵSD,cofe\nc using (C15) is 1 mHa for rs= 1, and\nis sub-mHa for larger rs.\nThus, equations (C12), (C15) and (C11) provide a set\nof constraints and reference values (from existing QMC\ndata) for high, moderate and low densities, respectively.\nThese three relationships are used in Appendix D to pro-\nduce the parametrisation for the state-driven correlation\nenergy of a cofe HEG.\nNote, it would be very desirable to obtain QMC or\nsimilar-quality reference data for cofe HEGs, to pro-\nvide direct inputs for parametrisations. The derivative,\ndecofe(rs,¯f)/d¯f|¯f=2, may be amenable to computation\nusing existing techniques, as it involves only low-lying\nexcited states.\nAppendix D: Parameterizations\nThe main text and previous appendices have intro-\nduced five terms that go into the cofe HEG energy as\na function of rsand ¯f. This appendix will provide a\nuseful parametrisation of the state-driven (SD) correla-\ntion energy that will allow the use of cofe HEGs in den-\nsity functional approximations. As explained in the main\ntext, the cofe-based LDA should be,\nEcofeLDA\nxc :=Z\nn(r)[ϵx(rs)fcofe\nx(¯f) +ϵSD,cofe\nc (rs,¯f)],\n(D1)\nwhere rs(r) and ¯f(r) depend on local properties of the\ninhomogeneous system.\nThe exchange term involves the closed form expression\nof eq. (26). The correlation term, ϵSD,cofe\nc (rs,¯f), needs to\nbe parametrised using:\n1. The known high-density behaviour of eq. (40);\n2. The known low-density behaviour of eq. (41).\n3. QMC data for other densities, adapted using\neq. (42);\nThe high-density limit yields, [to O(rslog(rs))]\nϵSD,cofe\nc (rs→0,¯f) :=c0(¯f) logrs−c1(¯f) (D2)TABLE II. Correlation energy parameters for selected values\nofζfrom fits to benchmark data [8] and exact constraints.\nζQMCA α β 1 β2 β3 β4\ncofe parameters\n0.00 0.031091 0.1825 7.5961 3.5879 1.2666 0.4169\n0.34 0.028833 0.2249 8.1444 3.8250 1.6479 0.5279\n0.66 0.023303 0.2946 9.8903 4.5590 2.5564 0.7525\n1.00 0.015545 0.1260 14.1229 6.2011 1.6503 0.3954\nrPW92 parameters\n0.00 0.031091 0.1825 7.5961 3.5879 1.2666 0.4169\n0.34 0.030096 0.1842 7.9233 3.7787 1.3510 0.4326\n0.66 0.026817 0.1804 9.0910 4.4326 1.5671 0.4610\n1.00 0.015546 0.1259 14.1225 6.2009 1.6496 0.3952\nwhere the parameters c0,1(¯f) are linear in ¯fand are\ntrivially related to their un- and fully-polarized counter-\nparts. [10] The low-density limit yields, [to O(1\nr2s)]\nϵSD,cofe\nc (rs→ ∞ ,¯f) :=−C∞+Cx[2/¯f]1/3\nrs+C′\n∞\nr3/2\ns.(D3)\nwhere C∞,CxandC′\n∞are universal parameters that do\nnot depend on ¯f. [47]\nPerdew and Wang [10] proposed that HEG correlation\nenergies lend themselves to a parameterization,\nF(rs;P) :=−2A(1 +αrs) log\u0014\n1 +1\n2AP4\ni=1βiri/2\ns\u0015\n(D4)\nwhere P= (A, α, β 1, β2, β3, β4) is a set of parameters that\ndepend on ζ,¯for related variables. By construction,\neq. (D4) can be made exact to leading orders for small\nand large rs. The high-density limit yields,\nAcofe=c0, βcofe\n1=e−c1/(2c0)\n2c0, βcofe\n2=2Aβ2\n1,(D5)\nwhere the coefficients are,\nc0(¯f) =0.031091 ¯f\n2, c 1(¯f) =0.00454 +0.0421 ¯f\n2.\nThe low-density limit yields,\nβcofe\n4=α\nC∞−Cxfcofex(¯f), βcofe\n3=β2\n4C′\n∞\nα,(D6)\nusing the parameters C∞≈1.95CxandC′\n∞= 1.33 [71]\nfrom Sec. III C, and fcofe\nx(¯f) = [2 /¯f]1/3from eq. (26).\nThus, only αis left undefined.\nOur goal is to find parameters, P(¯f), that can be\nused in a constant occupation factor (cof) parameteri-\nzation, ϵcofe\nc(rs,¯f∗) :=F(rs;Pcofe(¯f∗)), of the cofe HEG\nat selected values of ¯f∗; and interpolated to general ¯f.\nOur first step is to pick the values of ¯f∗. We seek to\nadapt the high-quality QMC data of Spink et al [8], who17\nTABLE III. Weighted sum parameters for M2,3(Appendix D)\nandZ2,3(Appendix E). E.g., M2=−2ϵ0\nc+ 4ϵ0.66\nc−2ϵ1\ncand\nZ3= 19.86ϵ0\nc−30.57ϵ0.34\nc+ 12.71ϵ0.66\nc−2ϵ1\nc.\nFunction ϵ0\nc ϵ0.34\nc ϵ0.66\nc ϵ1\nc\ncofe parameters\nM2 -2.00 0.00 4.00 -2.00\nM3 13.33 -22.41 11.43 -2.35\nrPW92 parameters\nZ2 -10.95 13.32 -1.47 -0.90\nZ3 19.86 -30.57 12.71 -2.00\nprovided correlation energies for, ζ∗∈(0,0.34,0.66,1),\nusing eq. (42). We therefore seek parametrizations at\n¯f∗=ˆf−1\nx-map (ζ∗), so that the right-hand side of eq. (42)\nis known.\nAs a first step, we must find ˆfx-map and its inverse.\nSetting eqs. (5) and (26) to be equal yields,\nˆfx-map (ζ)≈2−4\n3ζ2+1\n6[1.0187ζ3+ 0.9813ζ4],(D7)\nˆf−1\nx-map (¯f)≈q\n3\n4(2−¯f)\u0002\n1 +\u0000q\n4\n3−1\u0001\n(2−¯f)\u0003\n,(D8)\nwhich are exact in the polarized and unpolarized limits,\nand accurate to within 0.2% for all ζand ¯f. Eq. (D7)\ngives ¯f∗∈(2,1.85,1.50,1) for ζ∗∈(0,0.34,0.66,1),\nwhich are the ¯fvalues we use in fits. Then, for each\n¯f∗, we obtain α(¯f∗) by minimizing,\nmin\nαX\nrs∈QMC\f\fϵζQMC\nc,QMC(rs)−ϵSD,cofe\nc (rs,¯f∗)\f\f (D9)\nwhere ϵζQMC\nc,QMC(rs) is correlation energy data from Ref. 8\nandϵSD,cofe\nc (rs,¯f∗) :=F(rs, P(¯f∗)) involves the five con-\nstrained coefficients and free α(¯f∗). Optimal parameters\nfor the four values of ζ∗(called ζQMCto highlight their\norigin) are reported in Table II.\nThe next step of our parametrisation departs from\nPW92, in that we approximate the correlation energy at\narbitrary ¯fvia cubic fits (in ¯f) to the QMC data. Thus,\nϵcofe\nc(rs,¯f) := ( ¯f−1)ϵ0\nc(rs) + (2 −¯f)ϵ1\nc(rs)\n+ (¯f−1)(2−¯f)\u0002\nM2(rs) + (3\n2−¯f)M3(rs)\u0003\n,\n(D10)\nwhere ϵζ\nc(rs) :=F(rs, Pζ) is computed using eq. (D4) and\nM2andM3involve weighted sums of ϵc(rs,¯f) at selected\nvalues of ¯f. This fit becomes exact in the high-density\nlimit, as the correlation energy is linear in ¯f; and is also\nextremely accurate in the low-density limit as (2 /¯f)1/3\nfor¯f∈[1,2] may be reproduced to within 0.1% by a\ncubic fit. A cubic fit on ¯f∗∈(2,1.85,1.50,1) yields,\nM2(rs) :=2\u0002\n2ϵ0.66\nc(rs)−ϵ0\nc(rs)−ϵ1\nc(rs)\u0003\n, (D11)\nM3(rs) :=40\n357\u0002\n102ϵ0.66\nc(rs)−200ϵ0.34\nc(rs)\n+ 119 ϵ1\nc(rs)−21ϵ0\nc(rs)\u0003\n, (D12)\n60\n40\n20\n0SD\nc [mHa]\nRef.\ncofePW92\nrPW92\n-200-150-100-500SD\nxc [mHa]\n1 3/2 2\nf [unitless]\n-320-300-280-260xc [mHa]\n rs=2\n100101102\nWigner-Seitz radius, rs [Bohr]100105110115120xc enhance. [%]FIG. 9. Like Figure 3 but with the addition of polarized\nHEG results from PW92 [10] (dots) and rPW92 (dash-dot)\nforζ∈(0,0.34,0.66,1), to show differences between cofe and\npolarized gases in the high-density limit.\nwhere αis optimized on each of the four spin-\npolarizations.\nThe same strategy may also be applied to a conven-\ntional spin-polarized HEG. Thus, in addition to parame-\nters for the cofe model, Tables II and III also contains a\nset of coefficients for a “revised PW92” (rPW92) model\nthat is an analogue of the cofe model introduced here.\nDetails are provided in Appendix E. As it is based on\nsimilar principles, rPW92 is more directly comparable to\nthe cofe parametrization provided here than the original\nPW92, especially in the low-density limit.\nAppendix E: Revised PW92\nThe “revised PW92” (rPW92) parameterization is de-\nsigned as a direct replacement for the original PW92\nmodel. [10] Its main differences are: 1) the use of a cubic\nfit in ζ2, analogous to the fit to ¯fused in the main text;\n2) the use of the most up-to-date understanding of the\nlow density limit, per Sec. III C; and 3) αis found from\nthe Spink reference data. [8] Note, we fit to ζ2because\nexchange and correlation are quadratic for ζ→0, but\nlinear for ¯f→2.\nThe revised PW92 (rPW92) parameterization of cor-\nrelation energies is,\nϵrPW92\nc (rs, ζ) :=(1 −ζ2)ϵ0\nc+ζ2ϵ1\nc\n+ (1−ζ2)ζ2\u0002\nZ2(rs) +ζ2Z3(rs)\u0003\n.\n(E1)18\nwhere coefficients for Z2,3are reported in Table III. In-\nterestingly, the values we obtain for αatζ= 0 and\nζ= 1 are slightly lower than those from the original\nPW92 parametrisation, [10] most likely due to the use of\nmore modern QMC data.\nFigure 9 shows results from Figure 3 plus the LSDA\n(rPW92) parametrised along similar lines. It also in-\ncludes results from an existing LSDA (PW92 [10]). 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J´ onsson, Variational cal-\nculations of excited states via direct optimization of the\norbitals in dft, Faraday Discuss. 224, 448 (2020)." }, { "title": "2307.01489v1.Semantic_Segmentation_on_3D_Point_Clouds_with_High_Density_Variations.pdf", "content": "Semantic Segmentation on 3D Point Clouds with High Density Variations\nRyan Faulknera, Luke Haubb, Simon Ratcli ffeb, Ian Reida, Tat-Jun China\naAustralian Institute for Machine Learning - University of Adelaide, Cnr North Terrace &Frome Road, Adelaide, 5000, SA, Australia\nbMaptek, 31 Flemington St, Glenside SA, 5065, SA, Australia\nAbstract\nLiDAR scanning for surveying applications acquire measurements over wide areas and long distances, which produces\nlarge-scale 3D point clouds with significant local density variations. While existing 3D semantic segmentation models\nconduct downsampling and upsampling to build robustness against varying point densities, they are less e ffective under\nthe large local density variations characteristic of point clouds from surveying applications. To alleviate this weakness,\nwe propose a novel architecture called HDVNet that contains a nested set of encoder-decoder pathways, each handling\na specific point density range. Limiting the interconnections between the feature maps enables HDVNet to gauge the\nreliability of each feature based on the density of a point, e.g., downweighting high density features not existing in\nlow density objects. By e ffectively handling input density variations, HDVNet outperforms state-of-the-art models in\nsegmentation accuracy on real point clouds with inconsistent density, using just over half the weights.\nKeywords: Semantic segmentation, 3D point clouds, Density variation, Large scale point clouds, Multi-resolution\n1. Introduction\nLight Detection and Ranging (LiDAR) devices gen-\nerate accurate 3D measurements of their surroundings.\nWhile the generated point clouds have useful geometric\ninformation, practical application often requires seman-\ntic labels to be applied to the points. Recent progress of\ndeep models in processing 3D point clouds [1, 2, 3] has\nopened up many applications of LiDAR. In this paper,\nwe focus on semantic segmentation of LiDAR scans [4],\ni.e., assign each point a semantic label.\nMany advances in point cloud semantic segmentation\nrelate to autonomous driving, where the aim is the per-\nception of the immediate surrounds of the vehicle [5].\nTypically, automotive scans [6, 7] do not extend much\nfurther than a 100 m; indeed, the hardware limitations of\nautomotive LiDAR devices are such that scans reaching\n250 m can be considered long range [8]. The low reso-\nlution scans (approximately 105points) have a fast col-\nlection rate, making them useful for time-sensitive prob-\nlems such as obstacle avoidance. On the other hand, ter-\nrestrial LiDAR scans of surveying grade are slower, but\nof higher resolution, benefiting problems which require\nvery high precision but not real-time solutions.\nOne of the largest public datasets using a surveying-\ngrade scanner, Semantic3D [9], has high resolution\nscans of up to 108points, but only reaches physical di-\nmensions as large as 240 m horizontally, and 30 m ver-tically. In comparison, terrestrial LiDAR scans such as\nthose acquired in mining sites often have dimensions\nover a kilometre in the horizontal axes, and over 100 m\nvertically, covering a significantly larger area.\nLiDAR scans of a physically larger scale tend to suf-\nfer from high density variations; see Figs. 1 and 2. Fun-\ndamentally, fewer nearby occlusions yield more scan\npoints further from the scanner, where density is lower.\nWhile not to the same extent as surveying-grade scans,\nthe inherently lower resolution and distance limitations\nof automotive LiDAR cause it to also have density vari-\nation even in urban environments.\nState-of-the-art 3D semantic segmentation meth-\nods [4] struggle on large-scale surveying point clouds,\ndue to the higher density variation. In particular, while\nthe methods which operate directly on point clouds\n[10, 11, 12, 13, 14] extract local features in a multi-scale\nmanner through down- and up-sampling, details of how\nto best propagate and utilise features of di fferent scales\nare left to the neural network to learn. Some do com-\nbat density variation, however they only target variation\nwithin the scope of individual feature extraction steps\nand not across the entire network architecture.\nDensity variation vs class imbalance. It is vital to con-\ntrast density variation and class imbalance, both related\nfactors that influence segmentation accuracy. Classes\nPreprint submitted to ISPRS Journal of Photogrammetry and Remote Sensing July 6, 2023arXiv:2307.01489v1 [cs.CV] 4 Jul 2023(a) BEV of an open pit mine acquired for surveying. Few obstructions result in a large scan area.\n(b) BEV of an urban area acquired with terrestrial LiDAR as part of Semantic3D. Building obstructions limit coverage.\n(c) BEV of a single street acquired with automotive LiDAR (KITTI).\nFigure 1: Contrasting birds eye view (BEV) of di fferent LiDAR scan types, high to low density represented by red to blue.\n2(a) Small-Scale Terrestrial LiDAR\n0%10%20%30%40%50%\n(b) Large-Scale Terrestrial LiDAR\n0%10%20%30%40%50%\n(c) Automotive LiDAR\n0%10%20%30%40%50%\n(d) Small-Scale Terrestrial LiDAR (Processed)\n0%10%20%30%40%50%\n(e) Large-Scale Terrestrial LiDAR (Processed)\n0%10%20%30%40%50%\nDense Sparse\nFigure 2: Proportion of scan for di fferent density groups. As high\nresolution LiDAR is downsampled during preprocessing, distribution\nafter is also shown. Our preprocessing reduces the terrestrial LiDAR\nto one point every 6cm, or approximately 1100 points per metre.with fewer point samples tend to be smaller objects with\nlower point density. While this is an important chal-\nlenge to tackle, our focus in this paper is the e ffects of\ndensity variation independent of the population size of\nthe class. A single class can appear in a point cloud with\neach instance having vastly di fferent densities. A wall\nclose to the scanner for example, will have a higher den-\nsity of points than one far away; see Fig. 2 for density\ndistributions of di fferent LiDAR types.\nContributions. We highlight the importance of e ffec-\ntively accounting for local density variations in seman-\ntic segmentation on 3D point clouds, particularly those\nacquired from real-world surveying tasks. To this end:\n•We propose HDVNet (high density variation\nnetwork), a point cloud segmentation model that con-\ntains a nested set of feature extraction pipelines, each\nhandling a specific input local density; see Fig. 4. In-\nteractions between the pipelines is tightly controlled\nto exploit potential correlations between density lev-\nels. An aggregation layer applies attention scores to\nthe features accordingly, such that low density objects\nare not classified based on (potentially non-existent)\nhigh resolution features, while higher density points\nremain able to take advantage of their fine features.\n•We collected a new dataset, named HDVMine , that\nconsists of LiDAR scans from open-cut mines to\nevaluate our ideas. Our point cloud scans cover ge-\nographic areas which are kilometres in scale, making\nthem larger than existing terrestrial LiDAR datasets\n[9]. A single scan is comparable in scale to an auto-\nmotive LiDAR drive’s frames combined. In addition,\nexisting datasets comprise of “above-ground” scenes\nwhere there is a single and consistent ground plane.\nIn contrast, an open-cut mine can have multiple phys-\nical tiers, with complex structures embedded therein.\nAs we will show in Sec. 5, HDVNet yields up to 6 .7 per-\ncentage points higher accuracy in semantic segmenta-\ntion on our dataset, compared to a state-of-the-art point-\ncloud models [11] despite HDVNet using almost half as\nmany weights.\n2. Related work\nPoint clouds have useful geometric information for\neach point, but the lack of any inherent structure to the\ndata makes local context di fficult to determine. We first\nsurvey existing methods for point cloud segmentation,\nfrom those that preprocess the point cloud to alterna-\ntive representations, to those which directly take the raw\npoint cloud as input.\n32.1. Grid-based methods\nMany point cloud networks take inspiration from\nimage-processing techniques. Unlike a pixel image\nhowever, a point cloud has no inherent grid struc-\nture. For the purpose of using convolutions and sim-\nilar techniques on the point cloud, a common step is\nfirst converting from points to a grid-based represen-\ntation. These representations include two-dimensional\npixel images [15, 16], a birds eye view of the scene\n[17, 18, 19], or a three dimensional voxel grid [20, 21].\nLarge sections of empty space in the scene lead to\npoor memory scaling in grid representations. Data\nstructures such as octrees [22, 23, 24, 25, 26] avoid\nwasting memory on empty space, but information is\nstill lost where multiple points are combined into a sin-\ngle voxel. These grid structure representations have\ndemonstrated particular success for low-resolution Li-\nDAR scans where there are less fine details to be lost.\nState of the art methods for such scans range from\nmodified forms of three-dimensional voxel structures\n[27, 28] to representing the scan in two dimensions such\nas with a Range Image [29].\n2.2. Point Based Methods\nConvolutions are performed on grid structures, which\nmakes operating directly on the raw point cloud data dif-\nficult. A raw point cloud is simply a set of points, with\nno consistent ordering. PointNet [30] is a pioneering\nwork in directly processing point clouds, which demon-\nstrated the success of using network layers with Multi-\nLayer Perceptrons (MLPs). Each MLP is limited to op-\nerate only on individual points (with shared weights),\nand any operations performed on the entire point cloud\nbeing order-invariant and low-cost such as max-pooling.\nMore research rapidly followed, extending it directly\nsuch as PointNetLK [31] and PointNet ++[10], or de-\nveloping new algorithm using MLPs as a base.\nThese alternative point processing methods are de-\nsigned to better utilise the local relationship between\npoints in the scene. RandLA-Net by Hu et al. [11] does\nthis using K-Nearest Neighbours and MLPs to aggre-\ngate features for each point which represent the local\nneighbourhood. Like other MLP based methods, it is\nvery e fficient, scales well to large point clouds, and uses\nan encoder-decoder structure to get features from mul-\ntiple scales.\nAn alternative approach is to apply convolutions to\nthe raw point cloud as if it had a more grid-like struc-\nture. This requires modifying the implementation of a\nconvolution [13, 14, 32] to apply to unordered points.\nOne example of this is assigning coordinates to the con-\nvolution kernel, and using a MLP to determine howmuch each kernel weight a ffects a point based on the\npoint’s relative position to the kernel [13, 14]. This con-\ntrasts to a traditional grid-structure kernel where each\nweight fully a ffects the value in one specific pixel or\nvoxel co-ordinate and no others.\n2.3. Coarse, then fine processing\nRaw point clouds have limited features for each point\n(e.g. x,y,z,r,g,b), lacking any local context. To account\nfor this, some networks generate useful features first.\nTaeo et al. [33] have a network identify which points\nbelong to distinct objects, before then classifying each\npoint with semantic labels. Multi-pass approaches to\nfirst identify edges [34] or narrow down areas of interest\nbefore more fine processing [35, 36] are also common.\nOthers such as Varney et al. [37] and Li et al. [38] first\nextract fine features before downsampling to a sparser\npoint cloud as usual, but then go back and do so a second\ntime after the point cloud’s coarse features have been\nextracted. These methods all assume that fine features\nexist when extracting and propagating them, which does\nnot hold when scan’s density is inhomogeneous.\nAn alternative approach is to perform coarse segmen-\ntation into “superpixels” or “simple objects”, followed\nby a graph-based approach [39]. Such graph-based net-\nworks do not scale well to large and complicated scenes.\nIn addition, coarse segmentation which quickly iden-\ntifies the ground points [40] or the edges of the road\n[41], relies on assumptions such as “the lowest points\ndetected are the ground” which do not hold in contexts\nsuch as mining.\n2.4. Dealing with density variation\nConsider a point cloud of NpointsP={pi}N\ni=1. The\ndensity of the point cloud as a whole is the ratio of\npoints Nto the volume occupied by the point cloud. For\neach given point pi, we define its local density ρiusing\nthe density of its immediate neighbourhood of Knearby\npointsNi, whereNi=[Ni,1,Ni,2,...Ni,K]. Many exist-\ning methods inherently assume a homogenous density,\nsuch that the local density ρiof any point is roughly the\nsame as the average density of the entire point cloud. As\nshown previously in Fig. 2 this is not always the case,\nthe local density of points can vary greatly.\nIn both our method, and many existing pointcloud\nnetworks, the local neighbourhood of a point piis de-\ntermined using the K-Nearest neighbours, with Ka\nfixed hyperparameter, K∈Z. This creates a recep-\ntive field around each point, the Kpoints within mak-\ning up the local neighbourhood. Each time the point\ncloud is downsampled, it becomes more sparse, en-\nabling the receptive field to grow in physical size. This\n4enables early network blocks with small receptive fields\nto extract fine object features, while later blocks extract\nsparser features using larger receptive fields. These re-\nceptive fields encounter issues when density throughout\nthe point cloud is inhomogeneous.\nObjects which exist far away from the scanner or\nnear-parallel to the laser will appear in the initial scan\nwith a low point density. Early layers cannot ex-\ntract useful high-density features when the point’s local\nneighbourhood is sparse to begin with. This causes one\nof two density-variation issues, depending on whether\nthe receptive field uses a fixed number of neighbours\nK, or a fixed radius. If Kis fixed, the network layers\nare required to learn how to extract useful information\nfrom a wide variety of receptive field sizes, all using the\nsame shared weights. Alternatively, if the physical size\nof the receptive field is fixed, then the neighbourhood\nfeature will sometimes be generated from no neighbour-\ning points at all. Fig. 3 visualises this issue. In HDVNet,\nwe fix the number of neighbours K, and then take fur-\nther steps to counter the issue of inconsistent receptive\nfield size, detailed in Sec. 3.\nCentral Point Neighbouring Points Receptive Field\nFigure 3: Two variations of the receptive field when aggregating local\ninformation around a point. On the left, a fixed size field, with one\nreceptive field providing no useful information. On the right, a fixed\nnumber of K=4 neighbours, resulting in two receptive fields of vastly\ndifferent sizes being used at the same local-feature-aggregation step\nExisting methods do not address density variation\nacross the entire network like our HDVNet does, but\nthey do take steps to limit the e ffect on individual net-\nwork layers [42, 10, 12]. Alternative point cloud rep-\nresentations such as voxels tackle density by either\nweighting each voxel based on how many points it has\n[43], or implement a minimum density floor, ignoring\nsparse sections entirely [44].\nThe reliance on high density features can be ad-\ndressed by first aggregating high density points together\nto represent the scene in a more homogeneous, coarse\nmanner [45, 46]. Such approaches inevitably result in\ninformation loss as higher-density sections are down-\nsampled to achieve consistent density, although perfor-\nmance on low density objects does improve.3. High density variation network - HDVNet\nHDVNet is an architecture which processes a point\ncloud of NpointsP={pi}N\ni=1. The raw point values\npiinitially passed to the network are [ xi,yi,zi,ri,gi,bi],\nwhere x,y,zare the point’s spatial co-ordinates, and\nr,g,bare the colour values.\nThe number of points Nvaries throughout the net-\nwork as shown in Fig. 4. Each Downsampling Block\nDSremoves points, subsampling the point cloud from\none density state dto a sparser density d+1, where\nd∈{1,2,3,4,5}. The density state of the initial point\ncloud being d=1. Formally, DS dtakes Ndpoints\nas input, and returns the smaller subset of Nd+1points,\nsuch that{pj}Nd+1\nj=1=DS d({pi}Nd\ni=1). We index the point-\ncloud based on how downsampled it is, with the initial\npoint cloud being P(1)={pi}N1\ni=1, and the most down-\nsampled beingP(5)={pj}N5\nj=1such thatP(d+1)⊆P(d).\nThe number of points at each state N={Nd}5\nd=1is set\nas a hyperparemeter. We index upsampling and down-\nsampling blocks using their input pointcloud’s density\nstate d, for example the upsampling block US 5upsam-\nples the pointcloud from N5toN4points (fromP(5)to\nP(4)).\nEach point pihas a corresponding feature vector Fi,\ncontaining a total of Telements such that Fi∈RT.\nUnique to HDVNet, each feature vector Fican be sep-\narated into assigned subsections S(d)\ni⊆Fi, where the\nvector elements of each subsection are “assigned” to a\ncorresponding density state d. Each of these subfeature\nvectors contains Edelements, where Ed∈Z+. The set\nof integers{Ed}5\nd=1is defined as a hyperparameter, and\nis constant throughout the network. In Fig. 4, we visu-\nalise each feature vector subsection {S(d)\ni}Nd\ni=1as different\nshades of blue.\nEach Density Assigned Encoder Block ( DB) adds a\nnew subsection S(d)ofEdelements to each point’s cor-\nresponding feature vector. We use the number of as-\nsigned subsections ato index each feature vector F(a)\ni,\nthough the network, initialising as F(0)\niwith no assign-\nments and no elements. For example, F(3)\nihas the sub-\nsections S(1)\ni,S(2)\ni,S(3)\niand a total of E1+E2+E3ele-\nments. The total number of elements in a given F(a)\niis\ntherefore Ta, such that Ta=Pa\nd=1Ed.\nWe also index DB ablocks using the number of as-\nsigned subsections they involve. Each DB atakes as in-\nput both the existing feature vectors {F(a−1)\ni}Nd\ni=1and the\noriginal point values, with a formal definition of\n{F(a)\ni}Nd\ni=1=DB a({F(a−1)\ni}Nd\ni=1,{pi}Nd\ni=1) (1)\n5N2xk\n(1)\nDS4 US5DMLP4DMLP1Input Point Cloud\nFeature Vector Element Assignments (S(d))\nDB1\nDB4\nUS4DMLP3\nUS2\nUS3\nNetwork Blocks\nDensity Assigned Encoder Block (DB)\nDownsampling Block (DS)\nDensity Assigned Multilayer Perceptron (DMLP)Upsampling Block (US)\nClassifier BlockHDVNet\nInitial Points: No Assignment, 6 elements\nSkip Connection & Concatenation\ng4\nDB5\n DMLP5\ng3g2g1\nDS2DS1\nDB2\nDS3DB3\nDMLP2\nFinal Classifier (gfinal)DS1\nDS2\nDS3\nDS4\nN1xT1 N1x6N1xk\nN2xT2\nN3xT3\nN4xT4\nN5xT5N4x(T4+T5)N3x(T3+T4)N2x(T2+T3)N1x(T1+T2)\n(1)(1)\n(l)(1)\n(2)\n(3)\n(4)\nS(1): E1 = 12 elements\nS(2): E2 = 12 elements\nS(3):E3=40elements\nS(4): E4 = 192 elements\nS(5): E5 = 256 elements(2)\n(3)\n(4)\n(5)\n(1)Standard Network\ng1N3xk\nN4xk\nN1xkFigure 4: Proposed point cloud segmentation model HDVNet. Feature element assignments are visualised at each step of the architecture as shades\nof blue. The number of elements Edfor each subsection S(d)are those used in our experimental setup. Final features F(a)are extracted and passed\nto four classifiers gduring initial training, and a single classifier gf inal during fine tuning. Details of the classifiers are shown later in Sec. 3.5\nand Sec. 3.6. A standard U-net style network architecture for point cloud segmentation is shown in bottom left for comparison. Similar existing\nnetworks [11, 32] use this style, which does not include our density assignments, re-submission of the original point cloud, or multiple classifiers.\n6The first DB block DB 1takes only the original point\nvalues as input, with every F(0)\nibeing empty. Details on\ndensity states dare given in Sec. 3.1, while the e ffect of\nsubsection “assignments” S(d)\niare provided in Sec. 3.2\nand Sec. 3.5.\nAs shown in Fig. 4, HDVNet maps the the initial\npoint cloudP(1)to four final feature vectors {F(a)}4\na=1.\nDuring training each is passed to one of four classifiers\n{ga(F(a))}4\na=1. Each classifier maps the feature vectors\nto four sets of probability distributions {Q(d)}4\na=1. Each\nQ(d)hasNddistributions, and is created using the cor-\nresponding classifier ga, such that a=d. We define\nQ(d)={˜qi}Nd\ni=1where ˜qi=[˜qi,1,˜qi,2,..., ˜qi,k], such that\n˜qi,kis the estimated confidence that the corresponding\npoint piis of semantic class k. After a fine tuning step\nthis is simplified to a single classifier Q(f inal )={˜qi}N1\ni=1=\ngf inal({F(a)}4\na=1), used in inference and shown in Fig. 4\nas the “Final Classifier”.\n3.1. Density Groups and Density States\nOur solution to high density variation is to make point\ndensity a central part of the network architecture. To do\nso, HDVNet relies on three di fferent measures of a sin-\ngle point pi’s local density. The continuous density esti-\nmateρi, which is quantised evenly into discrete groups\nδ, and unevenly into the density states d.\nDensity is not a native measure from LiDAR, so the\nestimate comes from the the K-nearest neighbours. A\nsphere around each point is made, with the radius rbe-\ning the euclidean distance from the point to the most\ndistant of its Kneighbours. Dividing Kby the volume\ncreates a density estimate ρiin points per m3\nρi=K\n4\n3πr3(2)\nThis was chosen as a computationally e fficient way of\nestimating a point’s volume, with Nialready being cal-\nculated in order to generate a point’s local features. The\npoint clouds in Fig. 1 were coloured using ρi, so can be\nreferred to for a visualisation of the density estimate.\nWhile{ρi}Nd\ni=1is a useful estimate of every point’s den-\nsity, it is often too continuous in nature for steps in\nHDVNet which expect a more discrete input. For this\npurpose, the points are quantised into discrete grouping\nbucketsδ∈Z, withδ=0 being the first group, δ=1\nthe second, and so forth. The distributions in Fig. 2 were\ncreated using these groups δ.\nIn our experimental setup the initial grouping δ=0\nwas set to the very high density threshold of ρi>2×\n106points per m3. This value was chosen to ensure that\nvery few points fall into the first grouping, with evenhigh-density small-scale urban scans such as those in\nSemantic3D having very few points over this threshold.\nThe lower threshold tof the density grouping d=0\nis thus t0=2×106, and subsequent thresholds tδare\ncalculated to ensure that the minimum density (in m−3)\nis consistently a quarter that of the prior grouping. Each\nthreshold is thus calculated as:\ntδ=tδ−1\n4(3)\nFor better reflection of the point cloud’s density\nthroughout the network, these quantised groups δ\nare then combined into larger density states d∈\n{0,1,2,3,4,5}. Each dis the point cloud’s estimated\ndensity state at a given point in the network. There\nare more density groups δthan there are downsampling\nblocks DS, so each DSreduces the point cloud’s maxi-\nmum density by multiple groups δat a time.\nWe set which contiguous groups δmake up each den-\nsity state dby analysing the density distribution across\nall the points in the training dataset. With this average\ndistribution we estimate how many points Nwill remain\nafter downsampling to each density group δ. As the tar-\nget number of points for each density state {Nd}5\nd=1is a\nknown hyperparameter, we use the density group which\nresults in the closest number of remaining points to the\ntarget.\nDownsampling to a threshold tdtherefore results in\napproximately Ndpoints remaining, td=tδ|{ρi}≈Nd\ni=1>=\ntδ. An example of how contiguous density groups δare\ncombined into a state dis shown in Fig. 5.\nA key point to clarify is a point being within a specific\ndensity state of overall pointcloud pi∈P(d)as opposed\nto a point’s inherent density state. The initial pointcloud\nP(1)for example contains all the input points, regardless\nof how sparse they are. For when we refer exclusively to\nthe subset of points with an inherent density ρibetween\nboth that density state’s lower threshold td, and the prior\nstate’s threshold td−1we define the subset as\nI(d):={pi|td<ρ i<=td−1} (4)\nA visualisation of P(d)andI(d)is shown in Fig. 6\n3.2. Density assigned encoder block (DB)\nOur method’s key architecture modification is the as-\nsignment of feature elements to density states, as visu-\nalised in Figs. 4, 7, 8 and 13 as shades of blue for the Ed\nelements of each subsection S(d). For our feature extrac-\ntion blocks, we use our novel density assigned encoder\nblock (DB). Many of our multilayer perceptrons (MLP)\nand fully connected layers (FC) are also replaced with\n7δ\n10123 4 56 7 8 16151413121110 9\n2 34 5 dFigure 5: Each density state dis comprised of multiple groups δ\nP(d)\n1\n2\n3\n4\n5I(d)d\nFigure 6: A visualisation of P(d)andI(d)using the same hyperparam-\neters as our experiments on HDVMine. Each DSdsubsamples the\nentire pointcloud, while I(d)contains only points with a matching in-\nherent density. All points are coloured bright blue for visual clarity.\nour density aware MLP (DMLP) and density connected\nlayer (DC).\nSome of the operations performed in the density as-\nsigned encoder block’s hidden layers have higher mem-\nory requirements. As shown in Fig. 7 we reduce the\nnumber of elements in each subsection S(d)\nifrom Ed\ntoHd, with Hdalso set as a hyperparameter. We vi-\nsualise feature vector subsections with Hdelements as\nshades of maroon instead of blue. As the total num-\nber of elements normally is Ta=Pa\nd=1Ed, we define\nthe total number of elements in these hidden layers as\nUa=Pa\nd=1Hd.3.2.1. Continued input of original scene\nOne addition is the reintroduction of the original raw\npoint values after every downsampling as shown in the\nbottom left corner of Fig. 7. Each DB block requires\nnot only the input point feature vector F(a−1), but also\nthe raw point values of pi.\n{F(a)\ni}Nd\ni=1=DB a({F(a−1)\ni}Nd\ni=1,{pi}Nd\ni=1) (5)\nThis enables sparser feature extraction using original\npoint data instead of relying on unreliable propagated\nfeatures from higher densities. The features in HDVNet\nassigned to lower density states, are in this way made\nrobust to the original density of the object.\n3.2.2. Density assigned multi-layer perceptrons\nTo prevent reliance on higher density features, and\nenforce the “assigned subsections” S(d)\niof a feature vec-\ntor, element separation is applied within the encoder\nblocks. A normal MLP or FC would treat all feature el-\nements the same, so we instead use our density assigned\nMLP (DMLP) or Density Connected Layer (DC). They\nboth operate on feature vector elements without mixing\ninformation across density assigned subsections. For\nreference, we define a standard multi-layer perceptron\n(MLP) or fully connected layer (FC) as a mapping from\nFin\nifeatures to Fout\ni. The di fference being that a MLP\nalso includes layernorm (LN) and activation (A VN) lay-\ners:\nFout\ni=FC(Fin\ni) (6)\nFout\ni=MLP (Fin\ni)=AVN (LN(FC(Fin\ni))) (7)\nIn HDVNet, feature elements are each assigned either\nto the current point cloud’s density state dor a previ-\nous one ( d-1,d-2, etc). During feature extraction and\nprocessing, such as DMLPs, we allow higher density\nfeature elements to use elements assigned to lower den-\nsities as input, but not vice-versa. This rule comes from\nthe fact that an object with high density information will\nalways have low density information once it is down-\nsampled, but the same does not necessarily hold true in\nreverse. An object which is sparse to begin with will not\nhave any useful high density information to be consid-\nered.\nThe number of feature vector elements assigned to a\nspecific density is Ed. The elements of a feature vector\nF(ain)\niassigned to any of a contiguous set of subsections\nSstarttoSendis referred to as Fi,start :end. Multiple MLPs\neach viewing di fferent subsections of a point’s features\nF(ain)\niare thus combined into a Density Assigned MLP\n8Σ Concatenation Elementwise SumDensity Assigned Encoder Block 3 (DB3)\nDensity Assigned Multilayer Perceptron (DMLP)Initial Points\n DMLP\nInitial PointsΣ\nMultilayer Perceptron (MLP) Local Feature Aggregator (LFA)\nNd x (6+Ta-1)\nNd x 6Nd x Ta\nNd x Ua-1\nE1 E2 E3\nH1 H2 H3\nFeatureVector Element AssignmentsOutput Feature Vector F(a)Input Feature Vector F(a-1)\nNd x UaFigure 7: Detailed example of the density assigned encoder block DB3, wherein features assigned to three di fferent density states dare all processed.\nNdand (6 +Tl−1) are the input dimensions. The feature vector elements assigned to a=1 and a=2 are propagated from prior layers, and a=3\nassigned elements are newly added in the block. Two Local Feature Aggregation (LFA) blocks are used in addition to a skip connection. A box is\ndrawn around the series of steps which together form an example DMLP, while a second DMLP is left as a single arrow to reduce visual clutter.\n(DMLP), along with the number of element assignments\nto include in the output feature vector aout:\nDMLP (Fin\ni,aout)=MLP (Fi,1:ain,E1)\n⊕MLP (Fi,2:ain,E2)\n...\n⊕MLP (Fi,aout:ain,Eaout),(8)\nT3=10, (E1=2, E2=3, E3=5)\nT3=10, (E1=2, E2=3, E3=5) T3=10, (no density assignment)T3=(no density assignment)Density Connected Layer (DC) Fully Connected Layer (FC)\nFigure 8: A visual representation of a density connected layer ( DC3).\nFor simplicity, a feature vector with only ten elements T3=10 is\nshown. On the left is a traditional FC with no assignments. On the\nright is an example where the elements are split among three density\nassigned subsections, with a E1,E2,E3being 2,3,5 for both the input\nand output. Weights are only visualised for the first feature of each\nsubsection for clarity. A DMLP has additional layernorm and activa-\ntion layers, but follows the same density preservation rules.Where⊕is the concatenation of feature vectors, and\naout<=ain. This creates a feature extractor which is ro-\nbust to density variation, yet still extracts fine features.\nA pink square is drawn around this step in Fig. 7. As\nsparse features are generated without using fine ones,\nthey can be trusted to be robust to density variation. Our\ndensity connected layers (DC) follow the same method,\nstacking fully connected layers (FC) to preserve density\nassignments. Fig. 8 is an example of this for d=3, and\nsimilar to a DMLP, a DC can be defined as:\nDC(Fin\ni,aout)=FC(Fi,1:ain,E1)\n⊕FC(Fi,2:ain,E2)\n...\n⊕FC(Fi,aout:ain,Eaout), (9)\nOur local testing confirmed that as found in other works\n[47] low level features are enough for the majority of\nthe analysis with the benefit of features continually de-\ncreasing as they become finer. For this reason all new\nfeature vector elements added to a point cloud of density\nstate dare assigned to the new subsection S(d)\ni, maximis-\ning the number of features assigned to lower densities.\nThis is visualised in Fig. 4 and Fig. 7 by the width of\nsubsections remaining constant throughout.\n9Algorithm 1 Density Assigned MLP\nInput: Feature Vector Fin\ni, output’s number of assigned\nsubsections aout\nOutput: Feature Vector Fout\ni\nFout\ni←empty tensor\nfor each density state d,dδ tdo\nconcatenate(Ps,Pδ)\nδ←δ−1\nend while\nReturnPs\nend if\nFigure 9: Subsampling variants. Left-Right, Top-Down: Orig-\ninal, random, δ-guided random, and LiDAR-grid. Classes\nWall/Ground /Other are bright Blue /Yellow /Green for visual clarity\n103.3. LiDAR-grid subsampling\nLike existing networks, HDVNet subsamples the\npoint cloudPto smaller subsets of points. This allows\nboth for more features to be encoded into each point\nwithout running into hardware limitations, as well as\nobtaining features of lower density resolutions. In Fig. 4\nthis is represented by the downsampling step DS.\nDownsampling methods in previous models vary\nfrom random sampling [11] and farthest point sam-\npling [10, 42], to having the network itself choose which\npoints to keep [48]. Such downsampling methods do not\nretain the inherent scan ordering of terrestrial LiDAR.\nWe use a pseudo-LiDAR downsampling method similar\nto that of other works [49, 50] to preserve scan lines.\nWith the goal of making the scan more homogeneous\nin density with each downsampling step, objects with\nhigher density in the scan have more points removed,\nwhile the lowest density sections of the scan are left un-\ntouched.\nSuch terrestrial LiDAR scanners output not only the\n3D coordinates ( x,y,z) of each scan point, but often also\nthe (spherical) row and column coordinates ( r,c) of the\ncorresponding scan direction. While they can be esti-\nmated when the scanner’s co-ordinates are known (usu-\nally the point of origin for the scan), it is preferable to\nuse the original scanner’s row and column values if they\nare available for better LiDAR scan metadata. We pro-\npose that respecting the scan structure of the LiDAR\nwhile downsampling enables high-density sections of a\nscene to better resemble their low-density counterparts\nafter downsampling; see Fig. 9. When downsampled\nsections of the point cloud do not resemble naturally\nsparse objects in the LiDAR scan, the network is less\neffective at extracting coarse features.\nFor each point pi, the original metadata Mi=[ri,ci]\nis also input to the network if available. The metadata\nis used exclusively for downsampling, and not directly\nused in mapping from pito the class confidence distri-\nbution ˜qi. Instead, it enables a more accurate down-\nsampling of high-density points, removing the dispar-\nities and di fferences between di fferent density group-\nings. We define the di fference between the downsam-\npling’s target grouping δtand the point’s inherent den-\nsity grouping δias∆δ=δt−δi.\nDS lidar(ci,ri,∆δ)=ifci%2∆δ=0 and ri%2∆δ=0 1,\notherwise 0 ,\n(10)\nWe keep piifDS lidar(ci,ri,∆δ)=1, and discard it in\nthe downsampling otherwise. Downsampling based on\nthe di fference between a point’s original density andthe target density creates a more homogeneous result,\nwhile using rows and columns allows the LiDAR scan-\nline structure to be retained, as shown in the bottom row\nof Fig. 9.\nOne notable downside of this subsampling approach\nis that the number of points removed is inconsistent, as\nthe number of points Pδin each density grouping varies\nscan by scan. As a simple solution, LiDAR-grid sub-\nsampling is used for successive density groups until a\nnew target density would result less points than desired.\nThe points which would be removed when downsam-\npling to the next target density δtare then randomly re-\nmoved to achieve the desired number of points Ntas\nshown in Algorithm 2.\nWe use random subsampling as if we were to select\nthe points based on their local density it would likely\nremove a small object or section of the scan. Ran-\ndomly subsampling is both computationally e fficient\nand spreads the sampling throughout the scan. By ran-\ndomly selecting the points which would have been re-\nmoved if the LiDAR-grid subsampling was used once\nmore, we also retain scan line structure as much as pos-\nsible while avoiding subsampling of low-density areas\nof the scan.\nDS\nCentral Point Receptive Field Neighbouring Points\nFigure 10: Visualisation of how two points have had vastly di ffer-\nent receptive fields initially (left), After downsampling they are both\nneighbours of each other (right). The reliability of their dense features\nis not equal, so feature aggregation and propagation should be done\nwith care.\n3.4. Existential local feature aggregator (ELFA)\nIn HDVNet, Local features are extracted and aggre-\ngated via a nearest neighbours approach. As shown\nin Fig. 11, we have two alternative feature aggregation\nblocks. LFA is similar to that used in existing networks,\nspecifically using the LFA from RandLA-Net [11] as\na base. The point co-ordinates and features for the K\nneighbours inNiare found, and both are used to gen-\nerate a local feature vector for each point pi. The only\nmodification of note to our LFA implementation is that\nthe MLPs which involve point features are replaced with\nDMLPs, and fully connected layers (FC) are similarly\nreplaced with density connected layers (DC). The den-\nsity assignment Adof the point features is thus pre-\n11μK α Output FiInput (pi, Fi )\nNd x (3+Ua)Nd x UaLocal Feature Aggregator (LFA)\n(a)\n(a)KxNdx16 KxNdx3\nKxNdxUaKxNdx(16+Ua)\nNdx(16+Ua)\nμμ\nμ\nDensity Aware Fully Connected Layer (DFC) Multilayer Perceptron (MLP) \nPoint Co-ordinates Point Features Masked Neighbours\nK Nearest Neighbours Mean Concatenation KMask Sparse NeighboursM α α\nDensity Assigned Multilayer Perceptron (DMLP)Elementwise MultiplicationAttention Scores α\nSoftmax LayerK\nMExistential Local Feature Aggregator (ELFA)\n(a)Input (pi, Fi )\nNd x (3+Ua)KxNdx16NdxUa\nNdx2UaOutput Fi\nNd x UaNdx(16+Ua)\nNdx(16+Ua)KxNdx(16+Ua)\nKxNdx(16+Ua)KxNdx3\nKxNdxUaNdx2Ua(a)\nNFexistsNForiginalFigure 11: A standard LFA (above) and our ELFA (below). “K” represents the stacking of the point’s Kneighbouring points. HDVNet uses DMLP\nand DC’s to ensure feature assignments are upheld. As point coordinates are not density-assigned features a normal MLP is used to increase them\nfrom 3 to 16 elements. In later DMLPs involving point features and density assignments, co-ordinates are treated as d=5 (visible by all densities).\nELFA is optionally used instead to simultaneously calculate the feature based both on neighbours which “exist” and those which do not. Due to\nK-neighbours’ features multiplying memory use by K, reduced feature vector element total Uais used for LFA blocks as shown in Fig. 7\nserved. ELFA is a more modified, optional variant,\nwhich further counters density-variation.\nAs the feature elements which are assigned to higher\ndensities are calculated for sparse points, there will be\n“unreliable” or “junk” features, such as the dense fea-\ntures of the bottom right point in Fig. 10 with a larger\nreceptive field. While the network can be designed\nnot to use them at all, that removes both the ability\nof sparse points to utilise fine features of their higher-\ndensity neighbours, as well as take unreliable (due to\nvarying receptive field size) but still potentially useful\nfine features into consideration.\nIn ELFA, two neighbourhood features are created.\nAllKneighbours are used to generate NF original as\nusual, while NF exists is created from what remains af-\nter masking out points which exist at a sparser density\nthan the point cloud’s current density state d.Mask (pi,d)=ifpi∈{I(j)}d\nj=11,\notherwise 0 ,(11)\nThis masking ensures only points with the expected re-\nceptive field size contribute to NF exists. Both neighbour-\nhood features are concatenated, multiplied by an atten-\ntion score used to determine which features are most\nreliable, and then a finally passed to a DMLP. This is\nvisualised in Fig. 11. Through ELFA, the network has a\nlocal neighbourhood feature NF exists which it can learn\nwhether or not to trust. Without it, there is a higher risk\nof the network taking and using “junk” dense features\nfrom neighbouring points which have a sparser inherent\ndensity.\n3.5. Initial Training Classifiers\nAs the network architecture is assigned by density\nthroughout, we are able to utilise multiple classifiers\n12Algorithm 3 Existential Local Feature Aggregator\nInput: Local Neighbourhood Feature vector NF, In-\nherent density state of the Kneighbours IK, Inherent\ndensity state of the current point Ip\nOutput: Feature vector F\nmask←IK≤Ip\nNF exists←NF∗mask\nAttention←FC(NF)\nNF original←NF∗Attention\nNF original←DMLP (NF original,1)\nNF exists←µ(NF exists)\nF←concat (NF original,NF exists)\nAttention←FC(F)\nF←F∗Attention\nF←DMLP (F,size(F)\nReturn F\nHDVNet Feature Extractor Training Classifier ga\nNd x 64 Nd x 32Nd x 16Nd x k(a)\nClass \nProbability\nDistribution\nFully Connected Layer (FC) Softmax Layer(d)\nFigure 12: The simple classifiers used during training. g1,g2,g3and\ng4all use this architecture, using their corresponding input F(d)\ng1....g4 at the end, each predicting a class confidence\ndistribution ˜qifor each point. Each gatakes the features\nfrom a di fferent density state of the decoder as its input,\ng4usingF(4),g3the features fromF(3)and so forth.\nThe class-weighted cross-entropy loss is calculated\nfor each separate{˜qi}Nd\ni=1produced by ga, masked to in-\nclude only points with inherent densities belonging to\nthat density state or a prior one, pi∈{I(j)}d\nj=1. This spe-\ncialises each classifier for its intended density, prevent-\ningg1from being expected to classify sparse points of\nI(2),I(3),I(4)orI(5)(each density is visualised in Fig. 6).\nWe include earlier density states due to the LiDAR-\ngrid subsampling making high density objects resemble\nsparse ones, making them suitable as extra training data\nfor sparser densities.\nI(5)makes up a negligible proportion of any point\ncloud P(1), so it does not have a corresponding clas-sifier and cross-entropy loss is not calculated for it.\nAny points which belong to I(5)are treated as I(4)when\nmasking the output {˜qi}Nd\ni=1and calculating the loss.\nTo combine them together, the loss Lfor each density\nstate dis then multiplied by the square of the density\nstate number itself, so that the network can be trained\nsimultaneously for all densities.\nLtotal=12L1+22L2+......d2Ld (12)\nThe lower density weights are thus prevented from be-\ning too strongly a ffected by the higher density outputs\nwhich also use coarse features in their calculations, and\nthus a ffect the coarse features in their backpropagation.\n3.6. Fine tuning for final prediction\nWhile the loss in the section above is used during ini-\ntial training, there is a final fine-tuning step afterwards.\nSimply predicting the class label probability {˜qi}Nd\ni=1us-\ning the output from the classifier gacorresponding to the\npoint’s inherent density pi∈I(d)is sufficient. However a\nbenefit can be gained by locking the weights previously\ntrained and fine-tuning new ones which take all the the\nextracted features as input into a singular gf inalshared\nby all the points.\nAs shown in Fig. 13, the features at each density are\nfirst up-sampled to cover all the original input points,\nbefore being attention scored for each point. This at-\ntention score αiis created based on which density states\ndthe point pi“exists” in as well as it’s specific density\nestimateρi. A boolean value B(d)\niis used, with the value\nbeing true using the same “existence” definition as in\nELFA - whether the point belongs to I(d)or that of a\nprior density state (Eq. (11)).\nAs the network is initially trained for the classifiers\ng1....g4, there are no features assigned to d=5 to be\nattention scored. Therefore no boolean is made for d=\n5. At d=4 all points other than the negligible amount\nexisting in I(5)would be given a value of 1 according to\nEq. (11) so B4\niis not calculated or included either.\nαi=MLP (B1\ni,B2\ni,B3\ni,ρi) (13)\nAs the point’s density is known, and each feature is as-\nsigned to a designated density state, the network is able\nto learn which features to rely on for the final prediction\nofkclasses, and apply the attention score αiaccord-\ningly. The loss for this final step of the training is sim-\nply a class-weighted cross entropy loss using the {˜qi}N1\ni=1\noutput by gf inal.\n13US3HDVNet Feature Extractor (f)\nUS4 US2US3 US2US2\nFully Connected Layer (FC) Upsampling Block (US)\nAttention Scores Elementwise MultiplicationSoftmax Layer\nConcatenationHDVNet Final Classifier (gfinal)\nFeature Vectororigins\nElements generated from \nElements generated from \nElements generated from \nElements generated from \nElements from any density assignment\nN1 x k\nN4 x 64 N4 x 32N4 x16N3 x 64 N3 x 32N3 x 16N2 x 64 N2 x 32N2 x 16(l)(1)\n(2)\n(3)\n(4)(l)(2)\n(3)\n(4)\n{B3,B2,B1,ρi}N1i=1 i i iPoint's inherent density\n α\n αN1 x 64 N1 x 32N1 x 16\nN1 x 32N1 x 16 N1 x 64N1 x 64\nN1 x 64 N1 x 64\n(1)Figure 13: The final classifier used during inference gf inal. TheF(d)from each density state dis passed to a single classifier. Attention scoring is\nthen used to generate a reliable class probability distribution Q(1)={˜qi}N1\ni=1. Elements generated from each specific feature vector is visualised in\nshades of green to make the purpose of the density-based attention scoring clearer.\n4. Dataset - HDVMine\nWith the assistance of an industry partner, we col-\nlected 53 individual terrestrial LiDAR scans across five\ndifferent mine locations; Fig. 1a shows the point cloud\nfrom an individual scan. The scope of the individ-\nual point clouds range from 183M in one direction to\n8.4KM, with an average of 577M. Fig. 1 displays one\nof the scans from above.\nWe manually labelled the point clouds into three se-\nmantic classes: wall ,ground andother . The classes\nchosen reflect the aim to understand the overall scene\nstructure for surveying. Unlike in urban environments,\nwall andground in a mining environment vary sig-\nnificantly in smoothness and orientation. The bound-\naries between wall andground also defy simple geo-\nmetric definitions, e.g., the surfaces are not cleanly at\nright angles. Fig. 15a illustrates these challenging fea-\ntures. Class other subsumes a variety of elements such\nFigure 14: BEV of the corner of an open-cut mine in HDVMine.\n(top) Scene stitched from 7 point clouds, textured with RGB photos.\n(bottom) Ground truth semantic labels wall (blue), ground (brown),\nloose rock (grey), and manmade objects (yellow)\nas vegetation, rock piles, and man-made objects, where\nthe latter encompass less than 1% of the points; see\nFig. 15c. In total, 353 million points have been labelled.\nTab. 1 shows the population size of the classes.\n14Class Percentage in overall population\nwall 52.4\nground 31.1\nother 16.5\nTable 1: Overall proportion of each class in HDVMine dataset.\nWhile the LiDAR scans in HDVMine can be com-\nbined into contiguous scenes, in our experiments in\nSec. 5, each scan was treated as an individual input\npoint cloud. Even within a single point cloud however,\nthe local density variation is high (see Fig. 2), which in\nturn leads to significant intra-class density variation (see\nFig. 15d for wall andground examples).\n5. Experiments\nExperiments were run using three di fferent datasets,\nHDVMine (high-resolution, large-scale terrestrial Li-\nDAR), Semantic3D (high-resolution terrestrial LiDAR),\nand HelixNet (low-resolution automotive LiDAR). We\nran ablation tests with multiple variations of our archi-\ntecture:\n•HDVNet: The default network, using all methods\nas outlined in Sec. 3\n•DTC (Density aware Training Classifier): The\ntraining classifiers are modified to use DMLP and\nDC layers as the rest of HDVNet does.\n•FCO (Fine Classifer Only): Immediately train us-\ning fine classifier, instead of using the training\nclassifier from Sec. 3.5 and locking the network\nweights prior to the classifier.\n•TCO (Training Classifier Only): Inference is run\nusing the training classifiers from Sec. 3.5. Each\npoint piuses either g1,g2,g3org4according to\nwhich density state I(d)it belongs to.\n•No FA (No Feature Allocation): All DC and\nDMLP layers take features of every available den-\nsity as input. Such DC layers have no practical\ndifference to a FC layer, while each DMLP retains\nseparate layernorm (LN) and activation (A VN) for\neach small MLPs which it is constructed from.\n•No FA (small): As feature allocation reduces the\nnumber of weights used by almost half, this vari-\nant also uses less features per point throughout the\nnetwork, for an equivalent number of weights.\n•No ELFA: The Existential Local Feature Aggrega-\ntor from Sec. 3.4 is not applied\n(a)ground can be flat road or rocky bench. wall similarly varies in\nsmoothness, and the angle between their orientations is not consis-\ntent.\n(b) Classes defy simple geometric definitions. Multiple “ground\nplanes” shown blue in a side-view of a single scan with other points\nremoved.\n(c) People, Vehicles, and a pile of rock, all examples of the other\nclass.\n(d) High intra-class local density variation, even within the same in-\nstance. Wall and Ground shown losing density as distance from scan-\nner grows.\nFigure 15: Challenging features of the HDVMine dataset.\n5.1. Results on HDVMine\nAs there are multiple key di fferences between the\nimplemented architecture and RandLA-Net, additional\nablation tests were run on HDVMine. All tests were\nrun on a single 8GB Nvidia RTX 3070 for 50 Epochs\n(with each epoch being 1000 batches of batch size 4).\nTo fit the graph on the smaller GPU all networks were\ntrained with the same reduced number of features per\npoint (maxing out at 256 features per point at the end\nof the decoder). For all tests points were passed in with\nx,y,z,r,g,b, as well as the density estimate ρi.\nWhile the network takes an already-downsampled\npoint cloudP(1)as input, we upsample the labels and\ntest on the original point cloud P(0). For analysis we\n15Results on HDVMine across di fferent densities\nAll I5 I4 I3 I2 I1 I0\nProportion Of Scene 100% 0.14% 1.3% 4.4% 22.8% 14.0% 57.4%\nRandLA-Net Original 47.2 36.6 56.5 47.3 34.4 39.6 47.0\nDGCNN 5 Metre 41.8 15.0 22.7 17.6 23.1 50.3 42.6\nRandLA-Net +LN 67.8 52.1 68.2 66.8 64.4 72.0 64.1\nRandLA-Net +LN+LGS 70.8 67.6 76.3 69.6 69.5 74.7 66.1\nRandLA-Net +LN (Downsampled) 67.0 68.0 79.6 78.2 75.8 74.4 56.2\nHDVNet : DTC 69.8 54.5 74.9 70.2 70.5 74.2 64.3\nHDVNet : FCO 70.7 63.2 77.1 71.6 71.1 75.0 64.9\nHDVNet : TCO 73.5 72.5 79.4 72.8 72.5 76.6 69.9\nHDVNet : No FA 73.6 69.2 76.8 70.3 72.0 75.4 70.6\nHDVNet : No FA (small) 72.7 65.8 76.7 71.1 72.3 76.1 68.9\nHDVNet 73.6 71.8 79.1 73.2 74.6 76.2 69.3\nHDVNet : No ELFA 74.5 68.7 78.1 73.3 74.2 77.5 70.4\nTable 2: Ablation results on our high-variation dataset HDVMine. Value is MIoU for all points belonging to density state d. Best for each density\nis bolded, second best underlined. “All” is the MIoU as calculated using all the points notthe weighted average of each density’s MIoU.\nidentify the accuracy both on points with an inherent\ndensity I(1)and those with the extremely high density of\nI(0).\nIn addition, as our terrestrial LiDAR scans are too\nlarge to pass as input to a standard GPU, we used the\nsame method as RandLA-Net to break it down. Points\nwere randomly chosen from those not yet given a la-\nbel and combined with a set number of their nearest\nneighbours, passed into the network as the input point\ncloudP(1). This process was repeated until every point\nhad been processed at least once. Points processed in\nmore than one of these “spheres” had their label cho-\nsen by weighting the di fferent class distributions using\nthe point’s distance from the centre of each respective\nsphere, and then using the summed probability distribu-\ntion.\nPoints are compared at di fferent density groupings.\nI(5)is the coarsest, including all points where ρi<=0.12\npoints per m3. In comparison I(0)is the finest, in HD-\nVMine this is all the points where ρi>30,558 points\nperm3. The specific tdthresholds for d=0,1,2,3,4,5\nare (30558, 1739, 31, 1.9, 0.12, 0) respectively, based\non the known distribution of the training data.\nAs shown in Tab. 2, RandLA-Net’s use of batchnorm\nmakes it di fficult for the network to stabilise when lim-\nited GPU memory requirements require a small batch\nsize of 4. Simply swapping it for layernorm (LN)\nenabled RandLA-Net to train e ffectively. Replacing\nrandom subsampling with our Lidar-grid subsampling\n(LGS) improved results again. Even with the point’s\ndensityρidirectly passed in alongside rgb as a raw\npoint value, it was unable to learn to combat the samelevel of density variation as our HDVNet. Finally, we\nran RandLA-Net after downsampling the data heavily\nin pre-processing to obtain homogeneity in the dataset\n(if all points are sparse, there is no dense-to-sparse vari-\nation). This merely results in high accuracy on sparse\npoints coupled with poor results on high-density ones.\nUnlike HDVNet these higher results on sparse objects\ncome at too high a cost, reducing overall performance\nas fine features are completely abandoned.\nRestricting the network from using the features in\nF(a)assigned to higher densities when predicting ˜qfor a\ncoarse point was shown by “HDVNet: DTC” to reduce\nperformance. As each feature up to this final step is ex-\ntracted using only information from a specific density\nand lower, and each prediction ˜qiwith corresponding\nloss Ldis for points of a specific density, it better for\nthe network to learn to ignore an unreliable feature than\ncompletely ignore them in the final class probability cal-\nculations.\nTraining with the final classifier from the get go with\n“HDVNet: FCO” put the majority of HDVNet’s archi-\ntecture to waste. High density points make up the ma-\njority of the scene, so all else being equal their gradients\nwill overwhelm those of low-density ones making it dif-\nficult for the network to learn robust coarse features. In\ncontrast, the training classifier from Sec. 3.5 enables the\nnetwork to learn how to reliably extract coarse features.\nThe fine tuning step described in Sec. 3.6 causes a mi-\nnor improvement compared to “HDVNet: TCO” which\ndoes not use it. Applying each point’s corresponding la-\nbel provided by each of the four initial outputs remains\nsufficient if a faster training time is desired however.\n16Results on Semantic3D\nAll I4 I3 I2 I1 I0\nProportion Of Scene 100% 0.003% 0.03% 0.9% 4.9% 94.2%\nRandLA-Net 77.06 20.2 44.5 68.1 72.8 77.08\nHDVNet: Everything Implemented 67.9 31.5 36.2 62.4 67.4 67.5\nHDVNet: No ELFA 71.4 22.9 34.7 58.3 67.4 71.9\nTable 3: Results of local testing on a dataset with low density variation, Semantic3D, broken down across densities. As the point cloud is so\nhomogeneous, HDVNet’s density-aware architecture becomes a hindrance. Allowing fine object features to a ffect extraction of sparse features is\nboth reliable and beneficial when 99.1 percent of the points have the finest features seen by the network (belonging to either I(1)or the downsampled-\nin-preprocessing I(0)).\nOne point of interest in the ablation results is den-\nsity assigned feature vector subsections ( S(d)), a funda-\nmental aspect of HDVNet. As expected, removing it in\n“HDVNet: No FA” resulted in lesser results on all but\nthe (most common) highest-density category I(0). With-\nout any forced allocation of features, the network priori-\ntised the more frequent I(0)andI(1)points during train-\ning.\nIt was confirmed with “HDVNet: No FA (small)” that\nit is the explicit assignment of features to density states\ndimproving the results on sparser objects, and not a\nresult of being a simplified network with almost half the\nweights to learn. This smaller-version performed worse\nthan both the full-size “No FA” and standard HDVNet,\nas expected.\nThe existential local neighbourhood feature extrac-\ntion step (ELFA) can be considered optional, and to be\nincluded if the goal is a network which performs espe-\ncially well on sparse objects in a high density scene. Un-\nlike the other measures taken in HDVNet, the ablation\nshows that the benefit to sparse objects is outweighed\nby the cost to dense ones. Even for the high-variation\ndataset HDVMine, “HDVNet: No ELFA” performs the\nbest overall.\nUltimately HDVNet (No ELFA) achieved a MIoU 6.7\npoints above that of a RandLA-Net with minimal mod-\nifications, outperforming across all densities as well as\nagainst further simple RandLA-Net modifications.\nTests were also run using DGCNN for further com-\nparison to existing models. The standard hyperparame-\nter used by DGCNN for indoor scenes is 1.5 metre cubic\nblocks, with DGCNN taking approximately 8000 points\nfrom each block. On the HDVMine dataset, the average\nblock has 8000 points only at 5 metres, so we made this\nminor change to better accommodate the network. Even\nat 5 metres, this merely reflects the number of points\nin an \"average\" block, with many of the blocks created\nhaving less points, some substantially so. As shown in\nthe Tab. 2 models such as DGCNN which split the scene\ninto geometric sections (in this case, five metre cubes)perform poorly on high density variation data such as\nHDVMine, as they struggle to train with so many low-\npoint blocks. In inference, DGCNN shows a further de-\ncrease in performance at lower densities, as those are\nthe blocks which do not have su fficient points for the\nnetwork to e ffectively extract features. Further modifi-\ncations such as reducing the number of points expected\nfrom each block or increasing the block size further,\nwould throw away the fine features within the many 5-\nmetre blocks which do have 8000 or more points.\n5.2. Results on Semantic3D\nHDVNet was also applied to the task of Semantic3D\n[9]. The original metadata Miis not publicly available\nso angles were estimated using x,y,z, and from these\nangles rows and columns roughly approximated. Three\nof the fifteen scans typically used as part of the train-\ning set were instead put aside to use for testing. This\nwas done as the Semantic3D test dataset does not have\na public ground-truth point annotation, so detailed anal-\nysis across densities required sectioning o ffsome of the\npublicly labelled training data.\nAs shown in Fig. 2 smaller scale terrestrial LiDAR\nsuch as Semantic3D is significantly more homogeneous\nthan HDVMine. The majority of points belong to the\ndensity state I(0), which for Semantic3D is a threshold of\nρi>141,471 points per m3. Tab. 3 confirms that the im-\nproved performance seen on the HDVMine dataset does\nnot carry over to datasets with a more homogeneous\ndensity, although it continues to perform adequately. In\ncontrast to existing networks HDVNet is designed with\nthe inherent assumption of density variation in the data,\ninstead of homogeneity.\nIt should be noted that “HDVNet: Everything Im-\nplemented” performing better on “All” densities than at\nany individual one is nota calculation error but a natu-\nral result of how the MIoU is calculated. As a general\ntrend, individual classes get the highest IoU for the den-\nsity they most commonly occur, as this density state is\nalso how they commonly appeared in the training data.\n17Results on Helixnet\nAll I5 I4 I3 I2 I1\nProportion Of Scene 100% 0.34% 2.3% 12.2% 37.1% 48.0%\nRandLA-Net (LN +LGS) 49.8 14.8 24.5 37.3 52.9 56.0\nHDVNet 50.8 24.9 37.9 46.5 54.1 50.9\nHDVNet: No ELFA 53.0 23.2 34.9 46.1 55.4 54.3\nHDVNet: No ELFA, Limited FA 56.2 24.2 36.8 46.9 58.4 58.8\nTable 4: Results of local testing on automotive dataset HelixNet, broken down across densities. As the point cloud is already low resolution, there\nis no downsampling in preprocessing, resulting in no I(0)\nIn Semantic3D this is I(0)for all classes except“High\nVegetation”, which has 44% of its testing points at I(1),\ndespite that density only including 4.9% of the testing\ndataset’s points. The MIoU at I(0)averages each across\nevery class, and so is a ffected by (relatively) poorer per-\nformance of “High Vegetation”. Similarly the MIoU at\nI(1)is negatively a ffected by the IoU of classes which are\nmost populous at I(0). When calculated for “all” densi-\nties, each class IoU is a ffected primarily by the density\nwhere it has the majority of points (each of those points\nbeing either a true or false positive in the IoU calcula-\ntion). This is what results in the “All” point MIoU of\n67.9% being higher than for any of its density subsets\nI(d). The tables with the IoU of every class, at every\ndensity, for every network architecture, are not included\nin this paper for brevity.\n5.3. Results on HelixNet\nAnalysis was also performed using the automotive\nLiDAR dataset HelixNet [51]. Automotive LiDAR\ndatasets are typically much lower resolution, however\nalso have a higher variance in density than public terres-\ntrial datasets such as Semantic3D. Once again we show\nimproved performance compared to the similarly point-\nbased network RandLA-Net, with performance espe-\ncially improved on lower resolutions. Similarly ELFA\nonce more improves performance on coarse points, but\nis detrimental to the overall performance.\nHDVNet is built with the assumption that the point\ncloud still has useful features after downsampling steps.\nWe found that due to the resolution being low to begin\nwith, this assumption no longer holds. Assigning fea-\ntures to the densities I(5)andI(4)was counterproductive,\nwith a point cloud downsampled more than three times\nbecoming too sparse to still have useful features to ex-\ntract from the raw point data. Restricting the density\nassignment of features to the first three density states\nresulted in a small increase in performance.\nWhile the assumption of downsampled density states\nstill having features worth extracting is an importantweakness of our method to note, it is ultimately intended\nfor high-resolution scenes such as our HDVMine. For\nlow resolution LiDAR scans, state of the art voxel net-\nworks have demonstrated great success compared to\ndirect point cloud processing. For Helixnet, as well\nas other automotive datasets, grid-based networks sig-\nnificantly outperform our method, RandLA-Net, and\nother methods which directly process raw point clouds.\nWhether converting to a cylindrical representation, vox-\nels, pillars, etc., a low-resolution point cloud does not\nhave as much information and detail to potentially be\nlost in the conversion, reducing the need for direct point\nprocessing.\n5.4. Qualitative Results\nIn addition to the tables Tabs. 2 to 4, we have pro-\nduced qualitative results for all architectures on all\ndatasets. We visualise both the class predictions, as well\nas the point accuracy.\n18(a) Ground Truth\n (b) RandLA-Net\n (c) DGCNN\n(d) RandLA-Net +LN\n (e) RandLA-Net +LN+LGS\n (f) RandLA-Net +LN (Downsampled)\n(g) HDVNet: DTC\n (h) HDVNet: FCO\n (i) HDVNet: TCO\n(j) HDVNet: No FA\n (k) HDVNet: No FA (small)\n(l) HDVNet\n (m) HDVNet: No ELFA\nFigure 16: HDVMine qualitative results. Classes are : Wall Ground Other\n19(a) RandLA-Net\n (b) DGCNN\n (c) RandLA-Net +LN\n(d) RandLA-Net +LN+LGS\n (e) RandLA-Net +LN (Downsampled)\n (f) HDVNet: DTC\n(g) HDVNet: FCO\n (h) HDVNet: TCO\n (i) HDVNet: No FA\n(j) HDVNet: No FA (small)\n (k) HDVNet\n (l) HDVNet: No ELFA\nFigure 17: HDVMine qualitative results. Incorrect points are red, correct are green\n20(a) Ground Truth\n (b) RandLA-Net\n (c) DGCNN\n(d) RandLA-Net +LN\n (e) RandLA-Net +LN+LGS\n (f) RandLA-Net +LN (Downsampled)\n(g) HDVNet: DTC\n (h) HDVNet: FCO\n (i) HDVNet: TCO\n(j) HDVNet: No FA\n (k) HDVNet: No FA (small)\n(l) HDVNet\n (m) HDVNet: No ELFA\nFigure 18: HDVMine qualitative results. Classes are : Wall Ground Other\n21(a) Ground Truth\n (b) RandLA-Net\n(c) HDVNet\n (d) HDVNet: No ELFA\nFigure 19: Semantic3D qualitative results. Classes are : Man-made Terrain Natural Terrain High Vegetation Low Vegetation\nBuildings Hardscape Scanning Artefacts Cars\n(a) RandLA-Net\n (b) HDVNet\n (c) HDVNet: No ELFA\nFigure 20: Semantic3D qualitative results. Incorrect points are red, correct are green\n22(a) Ground Truth\n (b) RandLA-Net\n(c) HDVNet\n (d) HDVNet: No ELFA\n(e) HDVNet: Limited FA\nFigure 21: HelixNet qualitative results. Classes are :\nRoad Other Surface Building Vegetation\nTraffic Sign Static Vehicle Moving Vehicle\nPedestrian Artefact\n6. Conclusions\nIn this paper we introduced the novel network ar-\nchitecture HDVMine for direct point cloud segmenta-\ntion. We demonstrated improved performance consis-\ntent across all densities on data with high density vari-\nation, such as that from large-scale land-surveying or\nmining. The measures ingrained into the architecture\nwere each tested separately in an ablation study to con-\nfirm their individual contributions to the final results.\nWe confirmed that this performance benefit does not\ntranslate to more homogeneous terrestrial LiDAR data\nsuch as Semantic3D, and while performance in inhomo-\ngeneous low-resolution LiDAR scenes improves, grid-\n(a) RandLA-Net\n (b) HDVNet\n(c) HDVNet: No ELFA\n (d) HDVNet: Limited FA\nFigure 22: HelixNet qualitative results. Incorrect points are red, cor-\nrect are green\nbased methods remain the state of the art option for low-\nresolution LiDAR. Further research is required to de-\ntermine if the “Existential” local neighbourhood feature\nextraction step could be beneficial on data with more\nvariance than HDVMine, or if its improved performance\non sparse objects in the scene is always outweighed by\nthe detriment to the higher density objects which make\nup the majority of a scan.\n7. Acknowledgements\nThis research was carried out with support from the\ncompany Maptek, from which data was used to create\nthe dataset HDVMine, and software was used to both\nlabel and visualise point clouds.\nFunding: Ryan Faulkner was supported by an Aus-\ntralian Government Research Training Program (RTP)\nScholarship as well as a supplementary University of\nAdelaide Industry PhD (UAiPhD) Scholarship funded\nby Maptek; Tat-Jun Chin is SmartSat CRC Professorial\nChair of Sentient Satellites.\n23References\n[1] Y . 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Landrieu, Online segmentation of li-\ndar sequences: Dataset and algorithm (2022). doi:10.48550/\nARXIV.2206.08194 .\nURL https://arxiv.org/abs/2206.08194 18\n25" }, { "title": "2307.13927v1.DFR_Net__Density_Feature_Refinement_Network_for_Image_Dehazing_Utilizing_Haze_Density_Difference.pdf", "content": "JOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 1\nDFR-Net: Density Feature Refinement Network for\nImage Dehazing Utilizing Haze Density Difference\nZhongze Wang, Haitao Zhao, Lujian Yao, Jingchao Peng, Kaijie Zhao.\nAbstract —In image dehazing task, haze density is a key feature\nand affects the performance of dehazing methods. However, some\nof the existing methods lack a comparative image to measure\ndensities, and others create intermediate results but lack the\nexploitation of their density differences, which can facilitate\nperception of density. To address these deficiencies, we propose a\ndensity-aware dehazing method named Density Feature Refine-\nment Network (DFR-Net) that extracts haze density features from\ndensity differences and leverages density differences to refine\ndensity features. In DFR-Net, we first generate a proposal image\nthat has lower overall density than the hazy input, bringing in\nglobal density differences. Additionally, the dehazing residual of\nthe proposal image reflects the level of dehazing performance\nand provides local density differences that indicate localized\nhard dehazing or high density areas. Subsequently, we introduce\na Global Branch (GB) and a Local Branch (LB) to achieve\ndensity-awareness. In GB, we use Siamese networks for feature\nextraction of hazy inputs and proposal images, and we propose\na Global Density Feature Refinement (GDFR) module that can\nrefine features by pushing features with different global densities\nfurther away. In LB, we explore local density features from the\ndehazing residuals between hazy inputs and proposal images\nand introduce an Intermediate Dehazing Residual Feedforward\n(IDRF) module to update local features and pull them closer\nto clear image features. Sufficient experiments demonstrate that\nthe proposed method achieves results beyond the state-of-the-art\nmethods on various datasets.\nIndex Terms —Image Processing, Image Dehazing, Deep Learn-\ning, Density-aware.\nI. I NTRODUCTION\nHAZE is a common atmospheric phenomenon caused by\nthe accumulation of aerosol particles. It can cause severe\nquality degradation of images, which can affect subsequent\ncomputer vision tasks. Therefore, developing effective tech-\nniques for haze removal is essential to improve the quality of\nimages and ensure accurate results of downstream tasks [1–3].\nAfter decades of study, researchers [4, 5] model this atmo-\nspheric phenomenon as:\nI(x) =J(x)t(x) +A(1−t(x)) (1)\nwhere I(x)represents the hazy image; J(x)represents the\nclear image; t(x)andAstand for the transmission map (T-\nmap) and the global atmospheric light separately. And this\nmodel is commonly called the atmospheric scattering model\n(ASM).\nManuscript received XXXX 00, 0000; accepted XXXX 00, 0000. Date of\npublication XXXX 00, 0000; date of current version XXXX 00, 0000. The\nassociate editor coordinating the review of this manuscript and approving it\nfor publication was XXXX. (Corresponding authors: Haitao Zhao.)\nZhongze Wang, Haitao Zhao, Lujian Yao, Jingchao Peng, Kaijie Zhao are\nwith East China University of Science and Technology, Shanghai 200237,\nChina (e-mail: htzhao@ecust.edu.cn)\nSiamese\nStructure\nglobal feature space\n(before refinement)global feature space\n(a�er refinement)pushhaze\ndensity gap\nLocal\nFeature\nExtrac�on\nGlobal Refinement\nclear image featurehazy image feature\n—Local Refinement\nlocal feature space\n(before refinement)local feature space\n(a�er refinement)pull𝑰\n𝑷\n𝑰\n𝑷𝒓𝒆𝒔\nFig. 1. The main idea of our DFR-Net. By utilizing the density difference\nbetween the input (I) and the generated proposal image (P), global and local\ndensity features are refined in different ways: features of different global\ndensities are pushed farther apart and local density features are pulled in\ntowards clear image features.\nTo improve image quality and highlight image details cap-\ntured in hazy weather, numerous image dehazing methods have\nbeen proposed. Prior-based methods [6–10] rely on statistical\nanalysis of haze images and handcrafted priors to recover haze-\nfree images. However, these methods have limitations in their\nrobustness due to their reliance on specific assumptions, which\nmay not hold in different scenes.\nWith the success of deep neural networks in high-level\ntasks, data-driven dehazing methods [10–15] have become\nmainstream. Compared to prior-based methods, deep learning\nmethods demonstrate stronger capabilities in feature extraction\nand image restoration. However, early deep learning methods\nneglect the uneven distribution of haze, resulting in redun-\ndancy in network design and inefficiency in feature extraction\n[15]. One idea to improve these methods is to enable the\nnetwork to learn features about the haze density.\nResearch works [15–19] have proposed density-aware de-\nhazing methods. Haze density describes the distribution of\nhaze and impacts the effectiveness of a dehazing method.\nSeveral methods [16, 18, 19] estimate the T-map to obtain haze\ndensity information, which is inversely proportional to the T-\nmap [16]. However, these methods require T-map labels, which\ncan be difficult to obtain. To avoid this problem, methods\n[15, 17] directly extract density-related features. Nonetheless,\nthese methods still have shortcomings: they lack a comparator\nto measure density, and the learning process of density features\nlacks interpretability.\nAs image dehazing is an ill-posed problem, estimating aarXiv:2307.13927v1 [cs.CV] 26 Jul 2023JOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 2\nclear image directly from a hazy input using a feedforward\nnetwork is challenging [20]. Additionally, distinguishing haze\ndensities from a single hazy image is also difficult, and a\ncomparable image that helps the network learn to perceive dif-\nferent haze densities is necessary for density-awareness. While\nsome research works [15, 20, 21] have proposed methods that\ngenerate intermediate images to alleviate this problem, they\ndo not take the density differences between the intermediate\nresults and the hazy inputs into much consideration, limiting\nthe network’s ability to fully perceive haze density.\nTo address this issue, we propose a Density Feature Re-\nfinement Network (DFR-Net) that achieves density awareness\nby utilizing density difference. To fully explore haze density\nfeature, we first generate a dehazing proposal image ( P, as\nshown in Fig. 1), that can provide a comparison of density\ninformation from both global and local perspectives. Our\nproposal image is generated by a simple U-net structure and\nhas an overall lower haze density than the haze input ( I), which\nbrings global density differences. The different dehazing per-\nformance of different areas reflected in the dehazing residuals\n(res) leads to differences in local density and hints to areas\nwith haze removal challenges. Subsequently, we propose a 2-\nbranch structure consisting Global Branch (GB) and Local\nBranch (LB) based on the density difference information\nbetween PandIto comprehensively extract density features\nand refine them in different ways.\nIn detail, the scenes are the same between PandI, differing\nonly in haze density. Therefore, we propose to use Siamese\nstructures for feature extraction for Pand I, enabling the\nnetwork to understand both the different densities in one\nforward process. In addition, considering that the two features\ncontain part of the same information, to push them farther\naway and highlight the density information, we design a\nGlobal Density Feature Refinement (GDFR) module to refine\nthe features. Locally, the dehazing residual between PandI\ncontains hints of the local density information. The dehazing\nresidual represents the performance of dehazing and areas\nwith small residual values tend to be more heavy or hard\ndehazing regions. Hence, we propose to learn local density\nfeatures from the dehazing residual ( P−I) in a split and merge\n(S&M) way. To refine local features, several Intermediate\nDehazing Residual Feedforward (IDRF) modules are used,\nwhich can pull local features closer to clear image features.\nThe refinement illustration is exemplified in Fig.1. Both the\ntwo branches give a predicted haze-free image and we perform\nan adaptive fusion on them to gain the final dehazing result.\nCompared to other density-aware methods, on the one hand,\nour DFR-Net does not rely on additional T-map annotations\nand does not predict the T-map, thus reducing the manual\nworkload and the possible loss of information due to the\nT-map prediction process [22]. On the other hand, unlike\nthe way other methods extract density features, e.g., PMNet\nextracts density features from the splicing of hazy inputs and\npseudo-haze-free images via an SHA module, we analyze the\nrelationship of haze density between Pand I, and design\na network with interpretability to extract and refine density\nfeatures from the density difference. This makes our DFR-Net\ncontain density-related prior knowledge.\nhazy input (𝑰)\nProposal Image\nGenerator(PIG)\nLocal Branch\nGlobal Branchproposal image (𝑷)\nA\ndehaze residual (𝒓𝒆𝒔): cross-branch connection\nA : adaptive fusion𝑱𝑮𝑩\n𝑱𝑳𝑩final result (𝑱 )\nFig. 2. The pipeline of DFR-Net. DFR-Net first generates a proposal image\n(P) by PIG, and Pis input to the subsequent two-branch network together\nwith the hazy input ( I). Each branch predicts a pseudo-clear image and we\nperform an adaptive fusion to obtain the final result. Note that the global\ndensity features in Global Branch are fed into Local Branch by cross-branch\nconnections.\nOverall, the main contributions of our work are as follows:\n•We propose to learn and refine the haze density feature\nof a hazy image by utilizing the difference information\nbetween a generated proposal image ( P) and a hazy\ninput ( I) and an end-to-end method named DFR-Net is\ndesigned to achieve density-aware dehazing.\n•We extract haze density features both globally and locally.\nIn GB, a Global Block is introduced, which can explore\nthe image features of PandI. To highlight the features\nthat can better describe the density information, a GDFR\nmodule is proposed. In LB, local density features are\nextracted from the dehazing residual between Pand I.\nAdditionally, a IDRF module is presented to refine local\ndensity features stage by stage.\n•Sufficient experiments are conducted on our DFR-Net\nand demonstrate that it can achieve better results over\nthe existing state-of-the-art (SOTA) methods on multiple\ncommonly used datasets.\nII. R ELATED WORKS\nA. Density-aware Dehazing Methods\nIn recent years, several methods [15–18, 23–25] have at-\ntempted to improve the dehazing performance by enabling the\nnetwork to perceive haze density.\n1) Density-awareness via estimating T-map: Haze density\nis influenced by several factors and is inversely proportional to\nT-map, so some methods learn density information by estimat-\ning T-map. Lou et al. [19] predict a T-map first for nighttime\nimage dehazing. Zhang et al. [16] estimate a low-resolution T-\nmap and then jointly input the feature map and the estimated\nT-map to a Laplacian pyramid decoder to achieve a restored\nimage. Yang et al. [18] propose a semi-supervised method\nthat does not require paired data. The method estimates T-\nmap, scattering coefficient, and depth to reconstruction hazy\nimages and restores clear images. However, these methods\nrequire additional labeled data and might be inaccurate due\nto the complexity of practical scenes [12].JOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 3\nProposal Image\nGenerator(PIG)\nGlobal Block 1\nGlobal Block 7\nRestore\nBlock\nconv@k7𝑱𝑮𝑩\nGlobal Block 2\nGlobal Block 6Global\nBranch\nCC\nCSkip connection\n𝑭𝑰_𝑮𝑩𝟎𝑭𝑷𝟎\n𝑭𝑮𝟕𝑭𝑷𝟕\nGlobal Block 3\nGlobal Block 4\nGlobal Block 5C\nCC\nC\n𝑭𝑮𝟔𝑭𝑮𝟓𝑭𝑮𝟒𝑭𝑮𝟑𝑭𝑮𝟐𝑭𝑮𝟏\n: sum ·: element-wise multiply : channel-wise multiply -: subtraction\nsquare : square GAP : global average pooling : channel-wise average pooling CAP\nSiamese\nResBlock\nSiamese\nResBlock\nGDFR Share weights\n𝑭𝑰_𝑮𝑩𝒊−𝟏\n𝒄×𝒉×𝒘𝑭𝑷𝒊−𝟏\n𝒄×𝒉×𝒘\n𝑭𝑮𝒊\n𝒄×𝒉×𝒘\n𝑭𝑰_𝑮𝑩𝒊\n𝒄×𝒉×𝒘𝑭𝑷𝒊\n𝒄×𝒉×𝐰\n𝑾𝒄\n𝒄×𝟏×𝟏\n𝑭𝑰_𝑮𝑩𝒊𝑭𝑷𝒊\n—sigmoid\nGAP\nsquaresigmoid\nCAP\n𝑭𝑰𝒊\n𝒄×𝒉×𝒘1−\n·𝑭𝑮𝒊𝑾𝒔\n𝟏×𝒉×𝒘Global Block i𝐻×𝑊×𝐶𝐻\n2×𝑊\n2× 2𝐶𝐻\n4×𝑊\n4× 4𝐶𝐻\n8×𝑊\n8× 8𝐶𝐻\n4×𝑊\n4× 4𝐶𝐻\n2×𝑊\n2× 2𝐶 𝐻×𝑊×𝐶\nGDFR𝑰𝑷\nFig. 3. The illustration of GB, global block and GDFR module. Note the global density features are fed into LB by cross-branch connections.\n2) Density-awareness via extracting density features di-\nrectly: Research works [15, 17, 26] directly learn haze density\ninformation without estimating a T-map. Deng et al. [17]\ndesign a Haze-Aware Representation Distillation (HARD)\nmodule to extract global brightness and a haze-aware map.\nChen et al. [26] propose an attention mechanism based on dark\nchannel prior to describe haze concentration. However, not\nestimating the T-map would result in a lack of a comparator\nto measure density. Generating intermediate results and using\nthe information contained therein can address this issue.\nB. Dehazing Methods with Intermediate Results\nConsidering the difficulty of recovering images directly\nfrom the haze input, dehazing methods [15, 20, 21, 27] which\ngenerate intermediate results (or one result) inside the network\nto facilitate the dehazing process are proposed. Bai et al.\n[20] first generate a reference image by a deep pre-dehazer,\nand then develop a progressive feature fusion module to fuse\nthe hazy and reference features, which achieves high metrics\non several datasets. Chen et al. [21] first remove light and\nthick smoke by a Smoke Remove Network (SRN) to gain a\ncoarse output, which is concatenated with the original input\nand fed to a Pixel Compensation Network (PCN) to recover\nthe missing pixels in the thick smoke. Hong et al. [27] propose\nan Uncertainty-Driven Dehazing Network. In this method,\nintermediate results are together generated with uncertainty\nmaps for uncertainty features extraction. Ye et al. [15] alsopre-generate a pseudo-haze-free image. The hazy input and the\npseudo-haze-free image are concatenated to estimate a Density\nEncoding Matrix describing the relationship between haze\ndensity and absolute position and mixed up to the following\ndeep layers.\nDespite the above methods extracting feature from inter-\nmediate results, they do not fully consider the differences\nbetween these results and the haze inputs, especially the\ndifferences in haze density. Simple concatenation [20, 21] or\nlinear summation [15] might lead the networks to rely on the\nuncertain learning process and lose the capture of information\nabout the differences between the two images. In addition,\nthe lack of a targeted design that addresses the relationship\nbetween the intermediate results and the original input leads\nthe extracted features not fine enough and limits the dehazing\nperformance.\nOur DFR-Net improves on the aforementioned methods by\nexploring and refining density features through the utilization\nof density differences between a generated proposal image and\nthe hazy input, thereby achieving an awareness of haze density\nand superior dehazing performance.\nIII. M ETHOD\nAs illustrated in Fig. 2, DFR-Net generates a proposal\nimage using a Proposal Image Generator (PIG), which is a\nsimple U-Net, to facilitate density-awareness. Besides, DFR-\nNet consists of two primary parts: Global Branch (GB) andJOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 4\nDimensional Alignment𝑭𝑳𝑩_𝒐𝒖𝒕𝒊\n𝑭𝑳𝑩_𝒐𝒖𝒕𝟕−𝒊𝑭𝑮𝟕−𝒊S\nSC\nIDRF\nCC\nConv+LReLU C\nConv+LReLU𝑭𝑳𝒊\n(discarded)\nCSDA\nCSDA\nConv@k1𝑭𝑳𝑩_𝒊𝒏𝟖−𝒊\n𝑭𝑳𝟕−𝒊𝑭𝑰_𝑳𝑩𝒊\n𝑭𝑰_𝑳𝑩𝟕−𝒊\n𝓜𝒍𝒐𝒄𝒂𝒍𝑭𝑳𝑩_𝒐𝒖𝒕𝒊𝑭𝑮𝒊\nS\nC\nC\nConv@k1\nPU\nIDRF\nConv@k1\nPU\nC\n𝑭𝑳𝑩_𝒊𝒏𝒊+𝟏\nConv@k1\nDwconv@k3\nLeakyReLU\nConv@k1\nCAP\nsigmoid𝑭𝒊𝒏\n𝑭𝒐𝒖𝒕\nGAP\nFC\nsigmoid\n𝓜𝒍𝒐𝒄𝒂𝒍\n(a) DAFF (b) S&M (c) CSDA\nS: split C: concat\nPU : pixel-unshuffle·\n·: element-wise multiply : channel-wise multiply : sum\nProposal Image\nGenerator(PIG)\nconv@k7\nResBlockC\nResBlock\nResBlock\nDAFF\nResBlock\nRestore\nBlock\nS&M\nSkip connection𝑱𝑳𝑩Local\nBranch\nMSConv𝑭𝑰_𝑳𝑩𝟎\n𝑭𝑳𝟎—\nResBlock\nS&M\nResBlock\nS&M\nResBlock\nDAFF\nDAFF𝑭𝑮𝟏𝑭𝑮𝟐𝑭𝑮𝟑𝑭𝑮𝟒𝑭𝑮𝟓𝑭𝑮𝟔\n𝐻×𝑊× (𝐶+𝐶 𝐿)𝐻\n2×𝑊\n2× (2𝐶+𝐶 𝐿)𝐻\n4×𝑊\n4× (4𝐶+𝐶 𝐿)𝐻\n8×𝑊\n8× (8𝐶+𝐶 𝐿)𝐻\n4×𝑊\n4× (8𝐶+𝐶 𝐿)𝐻\n2×𝑊\n2× (4𝐶+𝐶 𝐿) 𝐻×𝑊× (2𝐶+𝐶 𝐿)\n𝑰\n𝑷𝒓𝒆𝒔\n: multi-scale conv layer\nMSConv\nFig. 4. The illustration of the structure of LB (top) and DAFF (a) , S&M (b), CSDA (c) modules. The locations of the IDRF usage are indicated by dashed\nlines representing plug-and-play availability.\nLocal Branch (LB). The GB and LB are responsible for\nlearning and refining global and local haze density features,\nrespectively, and generate a pseudo result each. In the end, an\nadaptive fusion is used to obtain the final dehazing result. This\nsection presents a detailed introduction of our method’s main\nideas.\nA. Proposal Image for Density-awareness\nIn DFR-Net, we first generate a proposal image ( P) that\nprovides information about the difference in haze density. P\nexhibits lower overall density than the input image ( I), and\ninconsistent dehazing performance in some local areas. Taking\nthese characteristics into account, we are motivated to extract\nand refine haze density features in the subsequent network.\nTo generate P, we employ a Proposal Image Generator\n(PIG), which is a simple U-net comprising multiple Residual\nBlocks (ResBlocks). We pre-train PIG with paired images\nand incorporate it into subsequent branches as an end-to-end\nnetwork.\nB. Global Branch\n1) Overview: The Global Branch (GB) is designed to\nextract and refine global density features using the overall\nhaze density difference. As depicted in Fig. 3 (top), GB\nconsists of a 7-stage U-Net with 7 global blocks, where\neach block has a Siamese structure with Ni\nGB(i∈[1, ...,7])\nResBlocks for feature extraction and a GDFR module forglobal density feature refinement. The inputs to GB are IandP\n∈RH×W×3, which are embedded to F0\nIGB,F0\nP∈RH×W×C\nusing a convolutional layer. Upsampling or downsampling is\nperformed between every two blocks. To obtain the predicted\ndehazing residual, the output features of the last global block\nare fed to a restore block, and the predicted dehazing residual\nis added to Ito obtain a pseudo result, ˆJGB.\nWe design the global block as a basic unit in GB, taking the\nhaze density relationship between IandPinto consideration.\nSpecifically, the i-th global block is composed of a Siamese\nstructure with Ni\nGBResBlocks for feature extraction and a\nGDFR module for global density feature refinement. For the\ni-th global block, the inputs are the outputs of the previous\nblock, Fi−1\nIGBandFi−1\nP, the outputs are Fi\nIGB,Fi\nPand the\nrefined global density feature, Fi\nG.\n2) Siamese ResBlocks for global feature extraction:\nSiamese structure shares weights and can measure similarity\n[28–30] or dissimilarity [31, 32] between samples effectively.\nSo we utilize a Siamese structure to extract image features\nfrom both Pand I, which enables the network to establish\na relationship between images with different haze densities.\nMoreover, compared to a single hazy input structure, the\nSiamese structure enables the features extracted by the network\nto better perceive the variation in haze density.\n3) GDFR for global feature refinement: To further refine\nthe extracted features, we propose the Global Density Feature\nRefinement (GDFR) module, which aims to highlight the\nfeatures that better describe global density information andJOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 5\nthus pull apart the features of images with different densities.\nFor the implementation, we compute the difference between\nthe feature maps of IandP, and then square each element in\nthe difference map. This subtractive operation filters out non-\ndensity information, and the resulting feature channels with\nlarger differences can better describe density information. We\nperform global average pooling (GAP) and sigmoid operation\non the feature differences to obtain density-related channel\nweights, denoted as Wc. We then multiply these weights with\nFi\nIGBchannel-wise:\n˜Fi\nI=Fi\nIGB⊗σ(GAP (POW ((Fi\nP−Fi\nIGB),2))) (2)\nwhere GAP (·)stands for global average pooling, σ(·)repre-\nsents sigmoid operation, ⊗stands for channel-wise multiply\nandPOW (a, b)represents a power of exponent bfor each\nelement of a.\nIn spatial dimension, we first multiply the channel weights\nby the squared feature difference. Next, we apply channel-wise\naverage pooling (CAP) and sigmoid operations to obtain a 2-\nD weight map that represents the spatial density difference\nWs. Higher value in Wsindicates that haze has been more\neffectively removed here. Therefore, 1−Wscan represent an\nattention map that guides the network’s focus. Here 1denots\na 2-D tensor of ones with the same shape of Ws. We then\nmultiply 1−Wsand˜Fi\nIand add the result to Fi\nIGBto obtain\nthe finally refined global density feature Fi\nG:\nFi\nG= (1−σ(CAP (POW ((Fi\nP−Fi\nIGB),2))))⊙˜Fi\nI+Fi\nIGB\n(3)\nwhere CAP (·)stands for channel-wise average pooling.\nAfter the last block, we concatenate F7\nPandF7\nGand input\nthe concatenated feature to a restore block, which is composed\nof four ResBlocks and a convolutional layer, to obtain the\ndehazing residual.\nresGB=RBGB(cat(F7\nP, F7\nG)) (4)\nwhere RBGB(·)denotes the calculation of the restore block\nin GB and cat(·)denotes the channel-wise concatenation\noperation. And the pseudo-clear image of GB can be obtained\nby:ˆJGB=I+resGB.\nC. Local Branch\n1) Overview: LB extracts local density features from the\ndehazing residual of PandI, which contains local density\ndifferences and indicates hard dehazing or high haze density\nareas. To refine these local features, we propose the IDRF\nmodule that gradually adjusts the features to match those of\nclear images.\nThe main body of LB is a 7-stage U-net, as illustrated in\nFig. 4 (top). Each stage is composed of Ni\nLB(i∈[1, ...,7])\nResBlocks. LB takes Iand res∈RH×W×3as inputs and\nembeds them to shallow features F0\nILB∈RH×W×C,F0\nL∈\nRH×W×CLusing a normal convolutional layer and a multi-\nscale convolutional layer, respectively. In the encoder stages,\nlocal and image features are extracted by a split and merge\nway, and in the decoder stages, DAFF fuses shallow image\nfeatures, deep image features, local features, and refined globalfeatures from GB. The dehazing residual can be obtained by\na restore block, and the pseudo result ˆJLBcan be obtained by\nsumming it and I.\n2) S&M for local feature extraction: The local features\nare extracted in an S&M way. Firstly, F0\nILBandF0\nLare\nmerged by concatenation and fed into stage-1, resulting in an\noutput feature with ( C+CL) channels. The output feature\nis then processed by the S&M module. As shown in Fig.\n4 (b), the input feature is split into an image feature and a\nlocal density feature. Notably, the number of channels for the\nimage feature varies from stage to stage, while the number\nof channels for the local feature is consistently fixed at CL.\nTo utilize the global density information obtained from GB,\nthe image feature is concatenated with the refined global\nfeature from GB. Afterwards, the concatenated feature and\nlocal feature are downsampled by 1×1convolutional layers\nand pixel-unshuffle, respectively. The downscaled features are\nthen concatenated and fed into the next stage. This design\nallows local features to fully interact with hazy image features,\nthus enhancing the reliability of local features.\n3) DAFF for feature fusion: Several methods introduce\nskip connection to aggregate shallow and deep features and\nsimply concatenate or add them together [33, 34], which\nmight result in a loss of information. To fully utilize the\ninformation relevant to density, we employ Density Aware\nFeature Fusion (DAFF). Specifically, given a set of features:\n{Fi\nLBout, F7−i\nLBout, F7−i\nG}(i∈[1,2,3]), which represent the\noutput feature of the i-th,(7−i)-th stage of LB and the refined\nglobal feature of the (7−i)-th block in GB, we first align them\nto a same shape and split the image features and local den-\nsity features. Then we perform convolutions and LeakyReLU\n(Conv+LReLu) on F7−i\nGand the results are concatenated with\nFi\nILBandF7−i\nILBrespectively and input to Channel-Spatial\nDensity Attention (CSDA) modules, as illustrated in Fig. 4 (c).\nMoreover, the local feature F7���i\nL is projected to a 2-D local\nattention map ( Mlocal) by Conv+LReLu and Mlocal is fed to\nCSDAs. Finally, the outputs of CSDAs are concatenated and\ncompressed to the channel number of the subsequent stage by\na1×1convolutional layer.\n4) IDRF for local feature refinement: To refine the local\ndensity feature, we further introduce IDRF module. As shown\nin Fig. 5, IDRF takes Fi\nILBas input to obtain intermediate\ndehazing residual resi\ninter by an intermediate restore block\n(IRB) composed of two ResBlocks and a convolutional layer.\nThen the resi\ninter is projected to a CL-channel feature em-\nbedding F′\nLwhich is subsequently concatenated with local\nfeatures as shown in Fig. 4 (a) and (b). With this process,\nlocal density features can be updated by the current dehazing\nresidual. To optimize this module and pull local features\nin towards clear image features, we employ a local density\nrefinement loss ( LLDR ) which will be introduced in Sec. III-D.\nSimilar to GB, LB predicts a pseudo-clear result: ˆJLB=I+\nresLB. Finally we fuse the pseudo-results of the two branches\nwith a learnable parameter α:ˆJ=α׈JGB+ (1−α)׈JLB.\nD. Loss Function\nThe overall loss fuction of our DFR-Net can be formulated\nas:L=LRec+λ1LP+λ2LRD+λ3LLDR , where LRecandJOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 6\nTABLE I\nQUANTITATIVE COMPARISON OF DFR-N ET WITH THE STATE -OF-THE-ART IMAGE DEHAZING METHODS ON DIFFERENT DATASETS (PSNR (DB)/SSIM).\nBEST RESULTS ARE BOLDED AND SECOND BEST RESULTS ARE UNDERLINED . CELLS WHERE RESULTS ARE NOT AVAILABLE ARE REPLACED BY ”-”\nMethodRESIDE-outdoor Haze4K NH-HAZE Dense-Haze\nPSNR(dB) SSIM PSNR(dB) SSIM PSNR(dB) SSIM PSNR(dB) SSIM\nNon-Density-AwareDCP [6] (TPAMI’ 10) 19.13 0.815 14.01 0.760 10.57 0.522 11.01 0.416\nDehazeNet [35] (TIP’ 16) 24.75 0.927 19.12 0.840 12.86 0.545 9.48 0.438\nAOD-Net [12] (ICCV’ 17) 24.14 0.920 17.15 0.830 15.40 0.571 12.82 0.468\nGDNet [36] (ICCV’ 19) 30.86 0.982 23.29 0.930 18.33 0.667 14.96 0.530\nMSBDN [37] (CVPR’ 20) 33.48 0.982 22.99 0.850 19.23 0.713 15.13 0.555\nFFA-Net [38] (AAAI’ 20) 33.57 0.984 26.96 0.950 19.87 0.694 12.22 0.444\nAECR-Net [14] (CVPR’ 21) - - - - 19.88 0.722 15.80 0.466\nCEEF [3] (TMM’ 22) 19.13 0.792 - - - - - -\nSGID-PFF [20] (TIP’ 22) 30.20 0.975 - - - - - -\nUDN [27] (AAAI’ 22) 34.92 0.987 - - - - - -\nQCNN-H [39] (TC’ 23) 28.74 0.964 - - - - - -\nMFINEA [40] (NN’ 23) 33.88 0.981 - - - - 18.34 0.609\nDehazeFormer-B [41] (TIP’ 23) 34.95 0.984 30.29 0.985 17.37 0.725 - -\nDensity-AwareHDDNet [16] (TC’ 22) 22.52 0.910 - - - - - -\nDeHamer [23] (CVPR’ 22) 35.18 0.986 - - 20.66 0.684 16.62 0.560\nPMNet [15] (ECCV’ 22) 34.74 0.990 33.49 0.980 20.42 0.731 16.79 0.510\nDFR-Net (ours) 35.34 0.993 34.63 0.993 21.21 0.810 18.85 0.674\nIRBProjectionConv@k3Conv@k3\nResize𝑭𝑰𝒐𝑳𝑩𝒊\nResize\n𝑱𝑱↓ 𝑰↓\n𝑰𝒓𝒆𝒔𝒊𝒏𝒕𝒆𝒓𝒊\n𝑱𝒊\n𝒊𝒏𝒕𝒆𝒓𝑭𝑳′\nIRB : intermediate restore blockℒ𝐿𝐷𝑅\nFig. 5. The illustration of IDRF module. The projected feature F′\nLwill be\nused to update local features in corresponding S&M or DAFF module.\nLPdenote L1 loss and perceptual loss [44] between the pre-\ndicted haze-free output ˆJand ground truth J,LRDandLLDR\nrepresent representation dissimilarity loss and local density\nrefinement loss, and λ1,λ2andλ3are hyper-parameters for\nloss regulation.\n1) Representation Dissimilarity Loss: In this paper, we\nintroduce a Siamese structure to learn density-related features\nfrom Iand P. To motivate the Siamese structure to learn\nmore information about the differences between the inputs,\nwe design representation dissimilarity loss:\nLRD=nX\ni=1⟨Fi\nP, Fi\nIGB⟩ (5)\nwhere ⟨a, b⟩represents the calculation of the cosine similarity\nbetween aandb. We compute cosine similarities between\nthe intermediate features of Iand Pand minimize them tolet the Siamese structure learn more representation about the\ndifference between the two inputs.\n2) Local Density Refinement Loss: To achieve the pulling\nof local features from haze images to clear images, we\nintroduce local density refinement loss.\nLLDR =1\nkkX\nj=1∥ˆJj\ninter(x)−Downj(J(x))∥1 (6)\nwhere kdenotes the number of IDRF modules which are\napplied, ˆJj\ninter(x)is the j-th intermediate predicted clear\noutput, and Downj(·)is the operation that downsamples\nground truth to the size of the corresponding intermediate\noutput.\nIV. E XPERIMENTS\nA. Datasets and Metrics\nWe conduct experiments on several datasets to train our\nmethod and test our method’s dehazing performance. The\ndatasets include: (1) RESIDE outdoor [45], which contains\n313950 synthetic outdoor hazy/clear image pairs for training\nand 500 pairs for testing; (2) Haze4K [42], which includes a\ntraining set of 3000 indoor-outdoor mixed image pairs and a\ntesting set of 1000 image pairs; (3) NH-HAZE [46], a real-\nworld dataset for the NTIRE 2020 competition, which consists\nof 55 pairs of non-homogeneous hazy images and clear images\nof real scenes (45, 5 and 5 pairs for training, validation and test\nrespectively); (4) Dense-Haze [43, 47], a real-world dataset for\nthe NTIRE 2019 competition and contains 55 pairs of dense-\nhaze images and corresponding clear images (with same data\nsplit as NH-HAZE). We evaluate the dehazing effectiveness of\nour method using two commonly used image quality metrics:\nPSNR (dB) and SSIM.JOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 7\n(g) GT (e) PMNet (f) Ours (a) hazy\n(b) DCP\n(c) AOD-Net\n(d) FFA-Net\nFig. 6. Visual comparison of various methods on Haze4K [42] dataset. Areas where our method works better are boxed out and zoomed in, or you can zoom\nin yourself to get a better view.\n(b) DCP (c) AOD-Net (f) GT (d) Dehamer (e) Ours (a) hazy\nFig. 7. Visual comparison of various methods on Dense-Haze [43] dataset. Areas where our method works better are boxed out and zoomed in, or you can\nzoom in yourself to get a better view.\nB. Implementation Details\n1) Network Configuration: In our work, we use the Basic\nBlock structure proposed in [38] as our ResBlock. {N1\nGB,\nN2\nGB, N3\nGB, N4\nGB, N5\nGB, N6\nGB, N7\nGB}and{N1\nLB, N2\nLB,\nN3\nLB, N4\nLB, N5\nLB, N6\nLB, N7\nLB}are set to {2,2,3,4,3,\n2,2}and{4,6,8,10,6,8,8}. In particular, we set the\nbasic feature dimension of GB and LB, C, to 32 and the local\ndensity dimension CLto 4. All upsampling or downsampling\noperations are implemented by 1×1convolution with pixel-\nshuffle or pixel-unshuffle. And we use IDRF module after\nevery stage in LB except the last stage.\n2) Training Settings: The PIG is pre-trained and spliced\nwith GB and LB. And the whole network is trained in an\nend-to-end fashion. We use AdamW optimizer ( β1= 0.9,\nβ2= 0.999, weight decay is 1e−4) to train the model and\niterate 600k times with the initial learning rate 1e−4reduced\nto1e−6with the cosine annealing [48]. Following [33], we\nperform progressive learning in our training process, which\nleads the network to adapt to those inputs close to the size in\npractical applications. For loss regulation, we set λ1= 0.2,\nλ2= 0.001 andλ3= 0.1. The data augmentations include\nrandom cropping, horizontal flipping, and vertical flipping.\nC. Comparison with State-of-the-art Methods\nOur comparison methods include DCP [6], DehazeNet\n[35], AOD-Net [12], GDNet [36], MSBDN [37], FFA-Net[38], AECR-Net [14], CEEF [3], SGID-PFF [20], UDN [27],\nQCNN-H [39], MFINEA [40] , DehazeFormer [41], HDDNet\n[16], DeHamer[23] and PMNet [15].\n1) Quantitative Evaluations: The quantitative results are\nshown in Table I. It demonstrates that our method achieves\nthe highest metrics on RESIDE-outdoor, Haze4K, Dense-Haze\nand NH-HAZE datasets, outperforming other SOTA methods.\nNotably, DFR-Net achieves a 1.14dB PSNR gain on the\nHaze4K dataset and significant improvements in SSIM on all\ndatasets.\nAmong the density-aware methods, our DFR-Net out-\nperforms all other methods [15–17, 23]. Compared to the\ntransmission-aware (density-related) methods DeHamer [23]\nand HDDNet [16], we directly extract density features from\nhazy images and obtain better performance. DFR-Net also\nsurpasses PMNet on metrics on multiple datasets by utilizing\ndensity difference information between Iand P, rather than\nsimply concatenating them.\n2) Qualitive Evaluations: Fig. 6 shows visual comparisons\nbetween our DFR-Net and SOTA methods on Haze4K dataset,\ndemonstrating that our DFR-Net can more effectively remove\nhaze than other methods. Specifically, AOD-Net [12] and FFA-\nNet [38] still leave haze residue in most areas, DCP [6]\nremoves some of the haze but suffers from color distorion.\nWhile PMNet [15] achieves better dehazing performance than\nprevious methods, our method is able to recover clearerJOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 8\nhazy\nPMNet\nOursVarious Haze Density\nHigh← →LowVarious Haze Density\nHigh← →Low\n(a) (b)\nPSNR / SSIM PSNR / SSIM PSNR / SSIM PSNR / SSIM\n23.86 / 0.980 25.86 / 0.981 27.29 / 0.981 27.55 / 0.986\n25.45 / 0.987 26.64 / 0.993 27.36 / 0.993 27.70 / 0.995PSNR / SSIM PSNR / SSIM PSNR / SSIM PSNR / SSIM\n31.23 / 0.989 27.86 / 0.983 30.86 / 0.989 31.92 / 0.991\n34.05 / 0.998 37.28 / 0.998 38.86 / 0.999 42.96 / 0.999\nFig. 8. Visual comparison with PMNet on Haze4K [42] dataset. Note that the densities of the input images are varied for each set of images of the same\nscene, gradually decreasing from left to right. Please zoom in for a better view.\nimages.\nIn addition, we compare the visual quality performance of\nour method with other methods on real-world dataset (Dense-\nHaze) in Fig. 7. Compared to other methods, our DFR-Net\nremoves the overall haze while achieving fine restoration in\nimage details. This is attributed to the pulling in of the image\nfeatures towards a clear image at each stage in LB, which\nreduces the occurrence of color distortion and blurring of\ndetails in our results.\nWe further compare the visual results of our DFR-Net with\nthose of PMNet [15] on the Haze4K dataset. Fig. 8 illustrates\nthe dehazing outcomes of PMNet and DFR-Net for both\nindoor and outdoor images with varying haze densities. DFR-\nNet exhibits more consistent dehazing performance across\ndifferent haze densities compared to PMNet. In particular,\nDFR-Net demonstrates superior performance in preserving\nlocal image details, as depicted in Fig. 8 (a). Conversely,\nPMNet struggles to accurately restore local details in the\ndehazed images. Additionally, DFR-Net outperforms PMNet\nin handling images with high global haze density, as shown in\nFig. 8 (b). The incorporation of both global and local density\ndifference information in DFR-Net contributes to its robust-\nness in perceiving and effectively addressing different haze\ndensities. The global density component enables the network\nto better comprehend variations in haze densities, allowing\nfor improved performance on images with diverse densities.\nOn the other hand, the local density component motivates the\nnetwork to identify and restore details in regions with high\ndensity, resulting in finer dehazed images with enhanced visual\nquality. Further details and discussions regarding these aspects\nare presented in Sec. IV-D.D. Ablation Studies\nThe main innovation of DFR-Net is the extraction and\nutilization of global and local haze density information. There-\nfore, we conduct ablation experiments and analyses in this\nsection to demonstrate the effectiveness of our utilization of\nhaze density information.\nWe conduct ablation experiments on the modules and loss\nfunctions proposed in this paper to evaluate their effectiveness\non dehazing. We first establish a base network ( ①), which\nconsists of a non-weight sharing GB (pseudo-Siamese struc-\nture), and an LB that takes the directly concatenated Iand\nPas input and aggregates features by concatenation. Only\nLRecandLPare employed to optimize the base network.\nThen we define several variants to verify the effectiveness\nof our proposed modules and loss functions on dehazing: ②\n+Siamese : Use the Siamese structure for feature extraction.\n③+LRD: Incorporate LRDinto the total loss function. ④\n+GDFR : Use GDFR to obtain refined global density features.\n⑤+DAFF : Aggregate features by DAFF. ⑥+DR : Explore\nlocal density from Dehazing Residual (DR) in LB. ⑦+IDRF :\nUse IDRF after each stage of LB, except the last one. ⑧\n+LLDR (our default setting): Include LLDR in the total loss\nfunction. Among them, variants ②-④and⑤-⑧gradually\nintroduce global and local haze density respectively. The\nablation results are presented in Table II. Subsequently, we\nwill analyze the effectiveness of introducing global and local\nhaze density separately.\n1) Ablation studies on utilizing global density difference\nand global feature refinement: We carry out experiments on\nvariants ②,③,④to verify the effectiveness of our design\nusing global density difference to extract and refine global\ndensity-related features. To ensure fairness, we adjust theJOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 9\nTABLE II\nABLATION STUDIES ON PROPOSED MODULES AND LOSS FUNCTIONS ON THE HAZE4K DATASET . NOTE THAT FLOP S AND PARAMS ARE MEASURED ON\n256×256 IMAGES .\nLabel Setting PSNR (dB) FLOPs (G) Params (M)\nw/o density difference ① base 30.52 222.46 35.76\n+ global density difference② +Siamese 31.21 241.94 35.15\n③ +LRD 31.55 241.94 35.15\n④ +GDFR 32.65 242.63 35.15\n+ local density difference⑤ +DAFF 32.89 271.54 40.52\n⑥ +DR 33.01 271.56 40.52\n⑦ +IDRF 33.52 286.60 42.11\n⑧ +LLDR (default) 34.63 286.60 42.11\n(b) base\n20.91 / 0.792(a) hazy\nPSNR / SSIM\n(d) w/ local density\n(variant�)\n21.89 / 0.850(c) w/ global density\n(variant�)\n21.22/ 0.819(e) GT\n∞/ 1\nFig. 9. Visualization of the ablation experiments on exploring and utilizing global / local haze density difference. Areas with large variance in dehazing\neffectiveness are framed out. Please zoom in for a better view.\n(a)hazyVarious Haze Density\nHigh← →Low\n(b) w/o global\ndensity\n(c) w/ global\ndensityPSNR / SSIM PSNR / SSIM PSNR / SSIM PSNR / SSIM\n32.32 / 0.978 34.42 / 0.987 35.17 / 0.988 38.15 / 0.994\n38.31 / 0.995 40.51 / 0.996 40.82 / 0.997 42.01 / 0.998\nFig. 10. Visualization of the ablation experiments on utilizing global density\ndifference and global feature refinement. Note that the haze densities of the\ninput image are varied and gradually decrease from left to right. Please zoom\nin for a better view.\nResBlock numbers of the Siamese structure to approximately\nmatch the number of parameters in the base network. Table\nII indicates that sharing parameters enables better feature\nlearning from proposal images and delivers a 0.69 dB PSNR\nimprovement. To motivate the network to learn more density-\nrelated information, LRDis employed and the inclusion of\nit leads to a 0.34 dB PSNR improvement. In addition, the\nintroduction of GDFR brings 1.10 dB PSNR performance gain\nby refining global densities features.\nWe also conducted experiments to validate the effectiveness\nof incorporating global density difference information. Fig. 10\nshowcases the dehazing results obtained without (variant ①)\n(a) hazy image (b) proposal image (c) dehazing residual\n(d) local feature from\nstage-1 of variant�\n(w/ local density)(e) local feature from\nstage-4 of variant�\n(w/ local density)(f) local feature from\nstage-6 of variant�\n(w/ local density)\n(g) local feature from\nstage-1 of variant�\n(w/o local density)(h) local feature from\nstage-4 of variant�\n(w/o local density)(i) local feature from\nstage-6 of variant�\n(w/o local density)\nFig. 11. Visualization (heatmap form) of the local density features from\nvariant ⑧and⑤. The black arrows in (d) point out the hard regions found\nby the network by utilizing the dehazing residual.\nand with (variant ④) global density information, using image\ninputs with varying densities. When global density difference\ninformation is not introduced, we observed low quantitative\nmetrics, unstable dehazing results, and inconsistent perfor-\nmance within the same scene. As depicted in Figure 10, variant\n①produces suboptimal dehazing outcomes.\nHowever, upon incorporating global density differences, the\nnetwork exhibits improved capability to perceive variations inJOURNAL OF L ATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 10\nTABLE III\nABLATION STUDIES ON THE USAGE OF IDRF. ✔INDICATES THAT THE\nOUTPUT FEATURES OF THIS STAGE WILL BE PROCESSED FOR IDRF.\nStagePSNR (dB) Params (M)\n1 2 3 4 5 6\nIDRF✔ ✔ ✔ 33.06 41.32\n✔ ✔ ✔ 32.89 41.32\n✔ ✔ ✔ ✔ ✔ ✔ 33.52 42.11\nglobal densities. This is achieved through the exploration of\nglobal density differences between the proposal image and the\ninput image using the Siamese structure, as well as the re-\nfinement of density features using the Global Density Feature\nRefinement (GDFR) module. As a result, the network becomes\nmore robust to images with different density levels, enabling\nthe generation of consistent and visually clear dehazed images.\n2) Ablation studies on utilizing local density difference and\nlocal feature refinement: We perform experiments on variants\n⑥,⑦,⑧for validation of the effectiveness of utilizing local\ndensity difference. As shown in Table II, extracting local\nfeatures from dehazing residual, IDRF and LLDR result in\n0.12, 0.41 and 1.11 dB PSNR performance gains.\nTo demonstrate the necessity of ultilizing local density\ndifference, intermediate local features of the network using\ndehazing residual to extract features (variant ⑧) or not (variant\n⑤) are visualized in Fig. 11, which shows that high density\nand hard dehazing areas are captured in the shallow stage\nof our method. And as the network deepens, the local maps\nare gradually flattened, which implies the local features are\nupdated stage by stage. However, variant ⑤fails to achieve\nthese performances.\n3) Ablation studies on the location of IDRF usage: We\ncarried out ablation experiments to determine the optimal\nplacement of the IDRF module within the network architec-\nture. We compared three cases: using IDRF only in the encoder\nstages, only in the decoder stages, and in all stages. The results,\nas shown in Table IV-D1, indicate that employing IDRF in all\nstages yields the best overall performance.\nFurthermore, we observed that incorporating IDRF in the\nencoder stage leads to improved quantitative metrics compared\nto its use in the decoder stages. This can be attributed to the\nfact that the refinement of local features should commence as\nearly as possible in the network. The primary role of IDRF is\nto update the current local features and facilitate the alignment\nof the restored image features with clear image features.\nWithout this refinement process, the network’s local features\nremain relatively unchanged, resulting in less attention being\npaid to regions with relatively high density or hard features that\nrequire careful handling at the current stage. By introducing\nIDRF at an earlier stage, the network can be guided more\neffectively to bring the image features closer to the features\nof clear images.\n4) Ablation studies on the effectiveness of DAFF: We\nfurther conduct ablation experiments on the effectiveness of\nDAFF. DAFF is designed to fully fuse the global density\nfeatures passed by cross-branch connections, the image fea-\ntures of the LB branch, and the local density features. AndTABLE IV\nABLATION STUDIES ON THE EFFECTIVENESS OF DAFF. T HE\nEXPERIMENTS ARE CONDUCTED ON THE HAZE4K DATASET .\nMethods concat [15] SK Fusion [41] DAFF\nPSNR (dB) 33.19 33.80 34.63\nwe compare the PSNR performance with other two fusion\nstrategies: concatenation [15] and SK Fusion [41]. Table IV\ndemonstrate that our DAFF achieves better results than the\nother two methods.\nAs described in Section III-C3, DAFF first combines the\nglobal features with the image features, enabling the features\nto be aware of global density information. It then introduces\nthe local features through the CSDA mechanism, facilitat-\ning pixel-level refinement of the features. In contrast, both\nconcatenation and SK Fusion methods focus primarily on\nchannel attention, without considering the attentional role of\nthe local density map on the spatial dimensions of the features.\nThe superior performance of DAFF can be attributed to its\ncomprehensive fusion strategy, which takes into account both\nglobal and local density information. This enables our method\nto capture and utilize density-related information more effec-\ntively, resulting in improved dehazing performance compared\nto concatenation and SK Fusion methods.\nV. C ONCLUSION\nIn this paper, we propose DFR-Net, a density-aware method\nfor image dehazing that utilizes haze density differences to\nextract and refine density-related features. To achieve this, we\ngenerate a proposal image and explore density representation\nfrom it and the hazy input. We use two branches to extract\nand refine global and local density features, respectively, based\non the density differences between the proposal image and\nthe hazy input. In Global Branch, features of images with\ndifferent densities are pushed away, while in Local Branch,\nhazy image features are gradually pulled closer to those of\nclear images. Our experimental results demonstrate that DFR-\nNet achieves high-performance image dehazing ability, and our\nablation studies show that our designs enable fine awareness\nand refinement of density information.\nREFERENCES\n[1] Cunyi Lin, Xianwei Rong, and Xiaoyan Yu. Msaff-net:\nMultiscale attention feature fusion networks for single image\ndehazing and beyond. IEEE transactions on multimedia , 2022.\n1\n[2] Chongyi Li, Chunle Guo, Jichang Guo, Ping Han, Huazhu\nFu, and Runmin Cong. Pdr-net: Perception-inspired single\nimage dehazing network with refinement. 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Pasteura 5, 02-093 Warszawa, Poland\n(Dated: December 7, 2023)\nWe present analytic expressions for the density of states and its consistent derivation for the two-\ndimensional Qi-Wu-Zhang (QWZ) Hamiltonian, a generic model for the Chern topological insulators\nof class A. This density of states is expressed in terms of elliptical integrals. We discuss and plot\nspecial cases of the dispersion relations and the corresponding densities of states. Spectral moments\nare also presented. The exact formulae ought to be useful in determining physical properties of\nthe non-interacting Chern insulators and within the dynamical mean-field theory for interacting\nfermions with the QWZ Hamiltonian in the non-interacting limit.\nI. INTRODUCTION\nA topological insulator (TI) is a common name for the\nnovel class of systems with non-trivial topological prop-\nerties [1, 2]. Historically, the first example of TI was a\ntwo-dimensional electron gas in a strong magnetic field\nwhere the integer quantum Hall effect was observed [3].\nAfter predicting and later discovering of other examples\nof TIs [4] the subject becomes a main stream of con-\ndensed matter physics [5], of cold atoms in optical lat-\ntices [6], of photonics [7], and even of electric engineering\n[8].\nOne of a possible path to investigate TIs is to\nstudy tight-binding models defined on particular lattices.\nAmong various interesting examples are either the Su-\nSchrieffer-Heeger model [9] and the Rice-Mele model [10]\nin one-dimension or the Haldane model [11] and the Qi-\nWu-Zhang (QWZ) model in two dimensions [12]. The\nlatter one is defined on a square lattice and the cor-\nresponding two dimensional Brillouin zone whereas the\nformer one is formulated on a hexagonal lattice.\nIn particular, the QWZ model is a well-knowntwo-\nband system in studying physics of fermions such as\nbulk and edge properties, different topological states,\nthermodynamics, transport, and many others [13]. This\nmodel is also used as a non-interacting part of the many-\nbody Hamiltonian where the two-particle interaction is\nincluded. Its further generalization to arbitrary dimen-\nsions and even to the limit of infinite dimension proved\nthat topological insulators are possible in interacting sys-\ntems as well [14].\nIn spite of such a broad interest in the QWZ model\nits density of states (DOS) is not yet determined ana-\nlytically. Although the DOS by itself is not sufficient\nto provide topological classification of a system, it is a\nbasic and very important quantity necessary to investi-\ngate thermodynamics, thermodynamic phases, response\nof the system on different probes, and many other quan-\ntities.However, its derivative, cf. Streda formula [15], canserve as a topological indicator.\nTo fill in this gap, in this article we derive analytical\nformulae of the DOS in terms ofspecial functions known\nas complete elliptic integrals, which mathematical prop-\nerties are defined and tabulated [16]. We discuss de-\ntails of the derivation and basic properties of the DOS\nwhen the relevant model parameter,a mass term control-\nling the topology, is varied.Our results are exact, ana-\nlytic and free of any numerical inaccuracy. They allow\nfor a complete and comprehensive understanding of the\nphysics(thermodynamics,dctransport,cf. STMcurrent,\nor spectral properties) of the QWZ model, which up to\nnow was mostly investigated from the point of view of\ntopological properties. Since the QWZ model plays so\nimportant role in the field, the necessity of its holistic\nunderstanding cannot be overestimated.\nThe DOS, denoted here by ρ(Ω), counts the number\nof states in a vicinity of a particular value of energy Ω,\ni.e.dN=ρ(Ω)dΩ. It can be obtained from thesingle-\nparticle Green’s function(resolvent). Analytic derivation\nof the DOS, even for two-dimensional (2D) systems, is\ntypically a challange and thus there are very few known\nanalytical results. One of the first example was obtained\nin 1953 by Hobson and Nierenberg [17]. It is an analyti-\ncal expression for the DOS of graphene with the nearest-\nneighbor hopping and represented by the complete ellip-\ntical integrals. The consequent derivation of this result\nis presented in [18]. The DOS for some others 2D lat-\nices are also obtained analytically [19]. In particular, it\ncan be done for square, triangular, honeycomb, Kagome,\nand Lieb lattices. In this paper the DOS is analytically\nderived for the QWZtwo-band Hamiltonian modeling a\nChern insulator on the square lattice. In contrast to ear-\nlier examples, our analytic DOS depends explicitly on a\nmass parameter, which allows to change the topology of\nthe sytem. In fact, we derived a whole family of DOS\nin exact, analytic forms for this two-band QWZ system\nin two dimensions. For three dimensional tight-binding\nmodels the densities of states in analytic forms were de-arXiv:2308.03681v2 [cond-mat.str-el] 6 Dec 20232\ntermined for a simple cubic lattice, a body center lattice,\nand for a face centered lattice [20–22].\nKnowledge of the DOS in analytic form is invaluable\nin further investigation of physical properties of tight-\nbinding model. Such analytic form allows for a precise\ndetermination of van Hove singularity positions and their\ntypes inside the DOS. And, as a result, it helps to achieve\nhigh accuracy when integrals involving the DOS are per-\nformed numerically.\nIn certain numerical applications analytic forms of the\nDOS improves numerical efficiency of programs. For ex-\nample, in the dynamical mean-field theory (DMFT) the\nsemi-elliptic DOS, given analytically, allows to determine\nthe Hilbert transform exactly and this simplifies the self-\nconsistency equations [23]. Also, the study of fermions in\nthe infinite dimensional limit within the DMFT used the\nanalyticGaussianformoftheDOSonhyper-cubiclattice\n[23] to investigate metal-insulator and antiferromagnetic\ntransitions, and the analytic expression of the DOS for\nface-centered lattice in the same dimension limit [24] to\ninvestigate itinerant ferromagnetism. Following the lat-\nter, the hand tailored analytic DOS with a free parame-\nter controlling its asymmetry was used within DMFT to\nstudy detailed conditions for occurring the itinerant fer-\nromagnetism inside a single band Hubbard model [25]. A\nrecent study of an extended Falicov-Kimball model also\nuseddifferentDOSprovidedinsimpleanalyticforms[26].\nSince analytic forms of the DOS are so important, some\nDOSforselectedlatticeswereincludedexplicitlyinterms\nof elliptic integrals inside the Python library devoted to\nGreen’s functions in tight-binding models [27].\nOur presentation is organized as follows: In section II\nwe define the QWZ model and discuss the dispersion re-\nlations, in Section III we introduce the DOS and present\nits analytic derivation for the QWZ Hamiltonian, Section\nIV is devoted to the discussion of the DOS in different\nparameterregimes, inSectionVweshowsomeadditional\nfeatures in the DOS, the spectral moments are discussed\nin Section VI, and we close our presentation with Sec-\ntion VII, where we offer our conclusions and outlooks. In\nAppendix A we provide mathematical definitions of the\nelliptic integrals and in the Appendix B we give selected\ndetails on calculating the spectral moments.II. QWZ MODEL HAMILTONIAN IN TWO\nDIMENSIONS\nA generic form of the two-band Hamiltonian for a 2D\nnoninteracting system in the momentum space can be\nwritten as\nˆH=X\nkˆHk=X\nkh(k)·ˆσ, (1)\nwhere k= (kx, ky)(−π/aL< kx, ky≤π/aL) is a 2D\nwave vector in the first Brillouin zone corresponding to\nthe 2D square lattice with the lattice constant aL,h(k)\nis a vector with three components being given functions\nofk, and ˆσis the vector with components represented\nby the three Pauli matrices ˆσx,ˆσy, and ˆσz.\nThe Hamiltonian (1) describes a two level system, cor-\nresponding to either the two orbitals or the spin 1/2de-\ngrees of freedom. The vector h(k)is interpreted as a Zee-\nman like magnetic field with some (perhaps non-trivial)\ndependence on the wave vector k. This model breaks the\ntime reversal symmetry and belongs to the class A in the\nten fold way classification scheme [28]. The Hamiltonian\n(1) can be easily diagonalized, giving a two band energy\nspectrum ϵ±(k) =±h(k), where h(k) =|h(k)|is the\nlength of the vector h(k).\nIn the following we consider a particular parametriza-\ntion where the length of h(k)is given by h(k)2=\nm2+ 2t2+ 2t2cos(kxaL) cos( kyaL) + 2 mt[cos(kxaL) +\ncos(kyaL)]. It corresponds to the following representa-\ntion of the vector h(k):\nhx(k) =tsin(kxaL),\nhy(k) =tsin(kyaL),\nhz(k) =m+tcos(kxaL) +tcos(kyaL),(2)\nwhere tis the hopping amplitude. In the momentum\nspace this vector h(k)has a Skyrmion configuration for\n0<|m|/t < 2[13]. In other words, the system is a\ntopological insulator with the finite Chern number ±1\nand conducting surface states at the half-filling. Hamil-\ntonian (1) can be interpreted as a tight-binding model\nof a magnetic semiconductor with the Rashba-type spin-\norbit interaction and a uniform magnetization along the\nz-axis [13]. In the following we take t= 1, which sets the\nenergy unit. We also use aL= 1for the length unit.\nForm= 0,0.5,1,2, and 2.5the corresponding eigen-\nvalues (energy bands) of the Hamiltonian (1) are plotted\nin Figs. 1-5, respectively. We see that at m= 0(Fig. 1)\nandm±2(Fig. 4) the band gap is closed at Γ,X(Y)\nandMspecialpointsinthesquareBrillouinzone, respec-\ntively, and characteristic Dirac cone(s) is (are) formed.\nForm̸= 0,±2the bands are separated by the gap as\nseen in Figs. 2, 3, and 5.\nIII. ANALYTIC DERIVATION OF DOS\nA. DOS and its symmetry\nThe band resolved DOS per lattice site for the lower\nϵ−(k) =−h(k)(upper ϵ+(k) = + h(k)) band is definedas usual\nρ±(Ω) =1\nNLX\nkδ(Ω−ϵ±(k)), (3)3\nm=0kykx\nFigure 1. Dispersion relations ϵ±(k)of the QWZ model at\nm= 0. The gap is closed at X= (±π,0)andY= (0,±π)\npoints.\nwhere δ(x)is the Dirac distribution function and NLis\nthe number of lattice sites. Due to the symmetry of the\nDirac function one can easily show that\nρ+(Ω) = ρ−(−Ω). (4)\nThetotalDOSperlatticesiteisdefinedasasumofthem,\ni.e.,\nρ(Ω) = ρ+(Ω) + ρ−(Ω). (5)\nSince the lower and upper bands do not overlap [29] the\nband resolved DOS can be directly extracted form the\ntotal DOS, i.e.,\nρ±(Ω) = θ(±Ω)ρ(Ω), (6)\nwhere θ(x)is a step function. Therefore, in the following\nwe focus on deriving a formula for the total DOS.\nB. Retarded Green’s function and total DOS\nEquivalently, the total DOS per lattice site is deter-\nmined as a trace in band indices ( ±) of the imaginary\npart of the retarded Green’s function\nρ(Ω) = −1\nπℑ[TrG(Ω)]. (7)\nThe diagonal matrix elements of the Green’s function\nG(Ω)are\nG±(Ω) =1\nNLX\nk1\nΩ−ϵ±(k) +ı0+,(8)and for a while the small imaginary part ı0+will not be\nwritten explicitly. Summation over kcan be replaced by\nthe continuous integral in the first Brillouin zoneP\nk→\nL2\n(2π)2R\nBZdkxdky, where Listhelengthofthesystemand\nL2=NLa2\nL, with aL= 1. Then the trace of the Green’s\nfunction has the form\nTrG(Ω) =Ω\n2π2Zπ\n−πZπ\n−πdkxdky\nΩ2−h(k)2.(9)\nIntegrations with respect to kxandkyare symmet-\nric. We first preform the integration of the function\n(Ω2−h(kx, ky)2)−1with respect to kx. It is convenient\nto denote\na= Ω2−(m2+ 2 + 2 mcosky),\nb=−2(m+ cos ky), (10)\nand then to use the identity (2.558.4) in [30]\nZdkx\na+bcoskx=2π√\na2−b2arctan\u0014a−b√\na2−b2tan\u0012kx\n2\u0013\u0015\n,\n(11)\nwhere a2−b2>0. For the case a2−b2≤0the result of\nthe integration (11) can be formally rewritten as\nZπ\n−πdkx\nΩ2−h2=−2πı√\nb2−a2csgn\u0014ı(b−a)√\nb2−a2\u0015\n,(12)\nwhere it is taken into account that arctan( ±tanπ/2) =\n±π/2. The complex signum function, abbreviated as\ncsgn, is equal to the sign (function sgn) of the real part\nof the argument, and sign of the imaginary part if the\nreal part is zero. We analytically continue this result by\ntaking Ω→Ω +ı0+, where we add infinitesimally small\nimaginary part as it is required in the retarded Green’s\nfunction. Then a= (Ω + ı0+)2+...gets small imaginary\npart 2ıΩ0+defining the value of csgn. Taking the limit\nof infinitisimal 0+we arrive at the following expression\nfor the trace of the Green’s function\nTrG(Ω) =2Ω\nπZπ\n0dky(sgna√\na2−b2fora2> b2,\nısgn Ω√\nb2−a2fora2≤b2,(13)\nwhereweusedthefactthattheintegrandisanevenfunc-\ntionof ky. Herethevalues aandbarethefunctionsofthe\nvariables kyandΩand the parameter mas determined\nby Eqs. (10).\nThe DOS is proportional to the imaginary part of\nTrG(Ω), therefore it is nonzero only in the region of Ω\nwhere\na2−b2≤0. (14)\nOutside the region (14) TrGis real and the DOS is equal\nto zero, i.e., ρ(Ω) = 0.\nThe next step in evaluation of Eq. (13) is to perform\nintegration over ky. It is convenient to replace y= cos ky4\nm=0.5(a)\nkykx\nm=-0.5(b)\nkykx\nFigure 2. Dispersion relations ϵ±(k)of the QWZ model at: (a) m= 0.5, and (b) m=−0.5.\nm=1(a)\nkykx\nm=-1(b)\nkykx\nFigure 3. Dispersion relations ϵ±(k)of the QWZ model at: (a) m= 1, and (b) m=−1.\nanddky=−dy(1−y2)−1/2. The boundaries of the in-\ntegration are −1≤y≤1, where changing the order of\nthe integration boundaries results an additional minus\nsign. The integration over yis performed differently de-\npending on the value of the parameter m. Therefore, in\nwhat follows we are considering three cases with |m|= 1,\n|m|>1, and|m|<1, separately.1. Case |m|= 1\nCalculations are simpler in the special case of |m|= 1.\nThe function in the denominator of the integral (13) can\nbe written explicitly as a2−b2= (Ω2−1)(Ω2−5−4my).\nIt is linear in yand changes the sign only once at the\npoint y=y0sgnm, where y0= (Ω2−5)/4. Solving\nthe condition (14) with respect to Ωand using the fact\nthat|y| ≤1we obtain that DOS is nonzero only for\n1≤ |Ω| ≤3.\nThe condition (14) considered with respect to yde-5\nm=2(a)\nkykx\nm=-2(b)\nkykx\nFigure 4. Dispersion relations ϵ±(k)of the QWZ model at: (a) m= 2, and (b) m=−2. For m= 2the gap is closed at\nM= (±π,±π)and(±π,∓π)points and for m=−2it is closed at Γ = (0 ,0)point.\nm=2.5(a)\nkykx\nm=-2.5(b)\nkykx\nFigure 5. Dispersion relations ϵ±(k)of the QWZ model at: (a) m= 2.5, and (b) m=−2.5.\ntermines the boundaries of integration in Eq. (13). For\nm= 1it is satisfied for y0≤y≤1.Then the imaginary\npart of Eq. (13) in terms of variable ygives the DOS in\nthe implicit form\nρ(Ω) =|Ω|\nπ2√\nΩ2−1Z1\ny0dyp\n(1−y2)(y−y0),(15)\nwhere the definition (7) is used. The similar expression\nwill appears for m=−1: the boundaries of integration\nare−1≤y≤ −y0and denominator in the integrandisp\n(1−y2)(−y−y0). The corresponding DOS can be\ntransformed into the result Eq. (15) by replacing the in-\ntegration variable y→ −y. So Eq. (15) is valid for both\ncases m=±1.\nThe integral over ycan be calculated in terms of the\ncomplete elliptical integral of the first kind K(x)using6\nthe identity (3.131.5) from [30]\nZu3\nu2dyp\n(u3−y)(y−u2)(y−u1)\n=2√u3−u1K\"s\n(u3−u2)\n(u3−u1)#\n, (16)\nwhere u3> u 2> u 1. In Eq. (15) these parameters are\nu1=−1,u2=y0,u3= 1and the result of integration is\nequal to√\n2Khp\n(1−y0)/2i\n. Then the DOS for |m|= 1\nis\nρ|m|=1(Ω) =\n\n√\n2\nπ2|Ω|√\nΩ2−1K\u0014q\n9−Ω2\n8\u0015\n,if1≤Ω2≤9,\n0, otherwise .\n(17)\nFor convenience, in the Appendix A we provide the defi-\nnitions of the elliptic integrals.\n2. Case |m|>1\nTo calculate the DOS for all other values of the pa-\nrameter mwe need first to analyze the function in de-\nnominator of Eq. (13). For |m| ̸= 1we can write\na2−b2= 4(m2−1)(y−y1)(y−y2). Here the left and right\nzeros of the function are denoted as y1= min(˜ y1,˜y2)and\ny2= max(˜ y1,˜y2), respectively, where\n˜y1=Ω2−1−(m+ 1)2\n2(m+ 1),\n˜y2=Ω2−1−(m−1)2\n2(m−1). (18)\nSo we have to set y1= ˜y1,y2= ˜y2if the the following\ncondition is met\nΩ2+m2−2\nm2−1>0, (19)\nand to set y1= ˜y2,y2= ˜y1, otherwise.\nLets consider the case |m|>1, then factor m2−1is\npositive. Rewriting Eq. (13) in terms of variable yand\nsubstituting it in Eq. (7), we obtain the nonzero DOS in\nthe implicit form\nρ(Ω) =|Ω|\nπ2√\nm2−1Zdyp\n(1−y2)(y1−y)(y−y2).(20)\nHere the boundaries of integration are determined by\nEq. (14), which reads y1≥y≥y2, implying |y| ≤1.\nThere are two cases, in which these conditions can be\nsatisfied by the integration variable y:\n|y1| ≤1,|y2|>1,\n|y1|>1,|y2| ≤1. (21)First one leads to the integration region y1≤y≤1, and\nthe second one to −1≤y≤y1. To perform integration\nwe use the identity (3.147.5) in [30], namely\nZu3\nu2dyp\n(u4−y)(u3−y)(y−u2)(y−u1)\n=2p\n(u4−u2)(u3−u1)K\"s\n(u3−u2)(u4−u1)\n(u4−u2)(u3−u1)#\n,\n(22)\nwhere u4> u 3> u 2> u 1. The first line of Eq. (21)\ncorresponds to u1=−1,u2=y1,u3= 1andu4=y2\nand the second line corresponds to u1=y1,u2=−1,\nu3=y2andu4= 1. The results of the integration in\nboth cases are identical\nρ(Ω) =√\n2\nπ2|Ω|√\nm2−11√y2−y1K\"s\n(1−y1)(1 + y2)\n2(y2−y1)#\n.\n(23)\nThe last step of this evaluation is to express the ob-\ntained formula in terms of mandΩ. As follows from\nEq. (19) for all Ω2>2−m2we can set y1= ˜y1,\ny2= ˜y2, where ˜y1,˜y2are given by Eq. (18). Then the\nsolution of the inequalities (21) is given by the condition\n(|m| −2)2<Ω2<(|m|+ 2)2. After making sure that\n2−m2<(|m| −2)2we substitute the same y1,y2into\nEq. (23) to obtain\nρI(Ω) =√\n2\nπ2|Ω|√\nΩ2+m2−2·\n·K\"s\n[Ω2−(m−2)2][(m+ 2)2−Ω2]\n8(Ω2+m2−2)#\n,(24)\nwhere we denote this result by ρI(Ω).\nThe final expression for the DOS for the case |m|>1\ncan be presented as\nρ|m|>1(Ω) =(\nρI(Ω),if(|m| −2)2≤Ω2≤(|m|+ 2)2,\n0,otherwise .\n(25)\nNote, that this expression reduces to the case m2= 1. In\nthiscase |m|= 1Eq.(25)coincidesexactlywithEq.(17).\n3. Case |m|<1\nWe repeat all the reasoning of the previous subsection\nfor the case |m|<1. In this case the factor |m| −1is\nnegative, and the nonzero DOS obtained from Eqs. (13)\nand (7) in terms of the variable yis of the form\nρ(Ω) =|Ω|\nπ2√\n1−m2Zdyp\n(1−y2)(y−y1)(y−y2),(26)\nwhere |y| ≤1by the definition. The boundaries of the\nintegration are determined by Eq. (14), that gives y≤y17\nory≥y2. There are three different possibilities for yto\nsatisfy these conditions,\n|y1| ≤1,|y2|>1,\n|y1|>1,|y2| ≤1,\n|y1|<1,|y2|<1. (27)\nWe are considering each of them.\nToperformintegrationinEq.(26)weusetheidentities\n(3.147.3) and (3.147.7) in [30] respectively, that for the\ncase of complete elliptical integrals have the same right\nhand side, namely\nZu2\nu1dyp\n(u4−y)(u3−y)(u2−y)(y−u1)=\n(28a)\n=Zu4\nu3dyp\n(u4−y)(y−u3)(y−u2)(y−u1)=\n(28b)\n=2p\n(u4−u2)(u3−u1)K\"s\n(u4−u3)(u2−u1)\n(u4−u2)(u3−u1)#\n,\nwhere u4> u3> u2> u1.\nThe first line of Eq. (27) gives the boundaries of inte-\ngration as −1≤y≤y1in the integral (26). The integra-\ntion can be performed by using Eq. (28a) with u1=−1,\nu2=y1,u3= 1andu4=y2. The second line of Eq. (27)\ngives the boundaries of integration as y2≤y≤1in the\nintegral (26). The integration can be performed by using\nEq. (28b) with u1=y1,u2=−1,u3=y2andu4= 1.\nThe results in these two cases are identical\nρ(Ω) =√\n2\nπ2|Ω|√\n1−m21√y2−y1K\"s\n(1 +y1)(y2−1)\n2(y2−y1)#\n.\n(29)\nThe third line of Eq. (27) splits integral (26) into the\nsum of two integrals with boundaries −1≤y≤y1and\ny2≤y≤1. The first one is performed by using Eq. (28a)\nand the second one is performed by using Eq. (28b),\nwhere we put u1=−1,u2=y1,u3=y2andu4= 1\nfor both integrals. The results of integrations are identi-\ncal and summation is reduces to multiplication by 2. The\nfinal expression reads\nρ(Ω) =2√\n2\nπ2|Ω|√\n1−m21p\n(1−y1)(1 + y2)·\n·K\"s\n(1 +y1)(1−y2)\n(1−y1)(1 + y2)#\n.(30)\nAs the last step we are expressing the formulas (29)\nand (30) in terms of mandΩ. Eq. (29) corresponds to\nthe first two lines of Eq. (27). As follows from Eq. (19)\nfor all Ω2>2−m2we can set y1= ˜y2,y2= ˜y1, where ˜y1,\n˜y2are given by Eq. (18). Note that the first two lines ofEq. (27) are equivalent to Eq. (21) and are symmetrical\nwith respect to permutation of y1andy2. So we can\nrepeat the same reasoning as for the case |m|>1. The\nsolution of this condition with respect to Ω2gives the\nsameregion (|m|−2)2<Ω2<(|m|+2)2andsubstitution\nofy1andy2into Eq. (29) results in the same DOS ρI(Ω),\ngiven by Eq. (25).\nTo evaluate Eq. (30) corresponding to the third line of\nEq.(27)wenoticethatthislineissymmetricwithrespect\nto permutation of y1andy2. Rewriting it in terms of Ω2\nwe obtain the region m2>Ω2>(|m| −2)2in which\nEq. (30) is valid. This region is split into two parts by\nEq. (19). In particular, for 2−m2>Ω2>(|m| −2)2we\nsety1= ˜y2,y2= ˜y1, and substitute them to Eq. (30) to\nobtain\nρII(Ω) =8\nπ2|Ω|\n|Ω2+m2|·\n·K\"s\n[Ω2−(m−2)2][Ω2−(m+ 2)2]\n(Ω2−m2)2#\n,(31)\nwherewedenotethisresultby ρII(Ω). Intheregion m2>\nΩ2>2−m2we set y1= ˜y1,y2= ˜y2, and substitution\nto Eq. (30) gives\nρIII(Ω) =8\nπ2|Ω|p\n[Ω2−(m−2)2][Ω2−(m+ 2)2]·\n·K\"s\n(Ω2−m2)2\n[Ω2−(m−2)2][Ω2−(m+ 2)2]#\n,(32)\nwhere we denote this result by ρIII(Ω). The final expres-\nsion for the DOS for the case |m|<1can be presented\nas\nρ|m|<1(Ω) =\n\nρI(Ω),if(|m| −2)2<Ω2≤(|m|+ 2)2,\nρII(Ω),if2−m2≤Ω2≤(|m| −2)2,\nρIII(Ω),ifm2≤Ω2<2−m2,\n0,otherwise .\n(33)\nNote, that the DOS for all values of parameter mdoes\nnot depend on the sign of mand, therefore is symmetric\nwith respect to mand−min contrast to the dispersion\nrelations.\nIV. SUMMARY OF ANALYTICAL RESULTS\nAND PLOTS OF TOTAL DOS\nInthisSectionwepresentplotsofthetotalDOSfordif-\nferent m, corresponding to the dispersion relations shown\nin Section II, and we provide consistent analyzes of them.\nThe plots are given in Figs. 6-10, that appear in the same\norder as the dispersion relations in Figs. 1-5. In these\nDOS plots the ranges of the vertical axis are different\nbut all DOS are normalized to two, corresponding to two\nbands. We see that the shapes of the DOS for the QWZ8\nmodel are much richer and with more additional features\nas compared, for example, to the hexagonal lattice with\nnearest neighbor hopping. Apart of the symmetry mand\n−m, mentioned above, the total DOS is symmetric with\nrespect to Ωand−Ω. It is clearly visible in analytic for-\nmulae of the DOS, Eqs. (17, 25, 33), where Ωis only\npresent as either Ω2or|Ω|.\nρ(Ω)\nΩ\n|m|=0\nFigure 6. Total density of states of the QWZ model at m= 0.\nThe typical shape of the DOS for 0<|m|<1is shown\nin Fig. 7. We can see that each nonvanishing part of the\nplot has three sections, in each of the section the function\nis described by ρI,ρII, orρIII, cf Eq. (33). Specifically,\nin the plot in Fig. 7 for |m|= 0.5, these sections are\nseparated by the points |Ω|=√\n1.75≈1.32and|Ω|=\n1.5. At the energies ±√\n2−m2, separating two nearby\nsections, the DOS has infinite peaks.\nAnother characteristic feature of the system with 0<\n|m|<1is the opening of the band gap of the width 2|m|.\nIt can be seen in the corresponding dispersion relations,\ncf. Fig. 2, and in the plots of DOS, where ρ(Ω) = 0in the\nrange −|m|<Ω<|m|. At the half-filling such system is\na topological insulator [12, 13].\nThe gap is closed for m= 0as seen in Fig. 6, when the\nDOS has a pseudogap at Ω = 0(DOS vanishes at a single\npoint). It corresponds to formation of the Dirac cones at\nX= (±π,0)andY= (0,±π)points in the Brillouin\nzone, that is easy to see in the plots of the dispersion\nrelations, cf. Fig. 1. At the half-filling such system is a\nsemi-metal.\nThe special case |m|= 1, given by Eq. (17), is shown\nin Fig. 8 and the corresponding dispersion relations are\nin Fig. 3. Flat parts in the dispersion relations, i.e. lines\nalong whichthe gradient of the dispersion ∇kϵ±(k) = 0,\nρ(Ω)\nΩ\n|m|=0.5Figure 7. Total density of states of the QWZ model at m=\n±0.5.\nρ(Ω)\nΩ\n|m|=1\nFigure 8. Total density of states of the QWZ model at m=\n±1.\ngive rise to the appearance of sharp peaks in the DOS.\nDespite of the fact that these flat parts are different for\nm= 1andm=−1, the shape of the DOS is the same.\nIn this |m|= 1case the system is a topological insulator\nat the half-filling [12, 13].9\nρ(Ω)\nΩ\n|m|=2\nFigure 9. Total density of states of the QWZ model at m=\n±2.\nρ(Ω)\nΩ\n|m|=2.5\nFigure 10. Total density of states of the QWZ model at m=\n±2.5.\nThe typical shape of the DOS for |m|>1(without\n|m|= 2) is shown in Fig. 10. The nonzero part of the\nDOS function is described by the single elliptic integral,\ncf. Eq. (25). It has two symmetrical infinite peaks at\nenergies ±|m|. The system has the band gap of the width2||m| −2|in the dispersion relation, shown in Fig. 5. In\nthe plot of the DOS the gap corresponds to ρ(Ω) = 0in\nthe range −||m| −2|<Ω<||m| −2|. At the half-filling\nthe system with |m|<2is a topological insulator and\nthe system with |m|>2is a trivial insulator [12, 13].\nFor|m|= 2the band gap closes and a pseudogap ap-\npears as it is shown in the plot of the DOS in Fig. 9. For-\nmation of the Dirac cones can be seen at the correspond-\ning dispersion relations in Fig. 4. For m= 2the gap\nis closed at M= (±π,±π)and(±π,∓π)points in the\nBrillouin zone and for m=−2it is closed at Γ = (0 ,0)\npoint. At the half filling such system is a semi-metal.\nHaving the exact, analytic expressions for the DOS\nin terms of the elliptic integrals we can prove rigorously\nthat the singularities, present for all mvalues, are of the\nlogarithmic divergence type, very similarly as in case of\nthe square lattice. It comes form the exact properties of\nthe elliptic integrals [16]. In numerical approaches to the\nDOS such conclusion would be hard to achieve rigorously\nsince the DOS would typically depend on an artificial\nbroadening parameter or an arbitrary truncation of an\ninfinite set of recursive equations.\nV. SUBTLE FEATURES SEEN IN TOTAL DOS\nIn this Section, we discus additional subtle features\nand general trends that can be observed for the DOS of\nthe considered QWZ model.\nA. Additional finite peaks\nAs discussed earlier, the total DOS is symmetric with\nrespect to mand−mand as well as it is symmetric with\nrespect to Ωand−Ω. It has two infinite peaks located\nat±Ω∞, where the elliptical integral K(1) =∞, i.e.,\nΩ∞=(√\n2−m2,if|m|<1,\n|m|, if|m| ≥1.(34)\nWe find, that for the values of |m|slightly larger than 1,\ntwo additional finite peaks appear at the edges of a band\ngap given by ±ΩL, for example, it is shown in Fig. 11\nfor the case of |m|= 1.15. At larger |m|these peaks\ndisappear.\nB. Widths of the band gap and the bands\nLet∆be the width of the band gap. Its value for\narbitrary m(in units with t= 1) is given by the simple\nexpression\n∆ = 2(\n|m|,if|m|<1,\n||m| −2|,if|m| ≥1.(35)10\nρ(Ω)\nΩ\n|m|=1.15ΩR ΩL\nFigure 11. Total density of states of the QWZ model at m=\n±1.15.\nEdgesofthebandgaparelocatedatenergies ±ΩL, where\nΩL= ∆/2.\nOn the other hand, the upper and lower bounds of the\nenergy spectrum (the dispersion relations ϵ±(k)) are at\n±ΩR, where ΩR=|m|+ 2. The width of each of the two\nbands ϵ±(k)is defined as W= ΩR−ΩL, since ΩR>ΩL.\nIts value is explicitly given by\nW= 2\n\n1,if|m|<1,\n|m|,if1≥ |m| ≥2,\n2,if|m|>2.(36)\nThe dependence of the gap width ∆as a function of\nmis shown in the Inset of Fig. 12. It is seen that when\n∆<2the gap of the same width is opened for the six\ndifferent values of the parameter m, at half-filling four\nof these mwill correspond to topological insulators and\nother two mwill correspond to the trivial insulators.\nTo discuss further we chose the case with the width\n��� = 1, which is possible for |m|= 0.5,1.5, and 2.5.\nThe DOS for |m|= 1.5and2.5are shown in Fig. 13\nwhereas the DOS for |m|= 0.5is in Fig. 7. Interest-\ningly, it can be seen that the DOS at the edges of the\nband gap, i.e., ρ(Ω =±ΩL), for the topological insulator\n(|m|= 0.5and1.5) is larger than the one for the trivial\ncase|m|= 2.5.Such a behavior can be explained by the\nfact that the topological phase with |m|<2is associated\nwith the overlap of the bands and the so-called band in-\nversion phenomenon. If the off-diagonal terms hx(k)and\nhy(k)were neglected the bands only overlap, resulting\nenhancement of the DOS in the overlapping regime of\nenergies. We note however, that enhancing of the DOS\ndue to this mechanism is not a sufficient condition to in-\ndicate the system as topologically non-trivial. On the\nother hand, a band inversion is a necessary condition for\nthat, at least in a broad class of tight-binding models.\nρ(Ω L)\nm/tm/t/uni0394/t\nTop Triv Top Triv\nTop TrivFigure 12. The value of the DOS at the edges of a band gap,\ni.e. at Ω = Ω L, vs. the parameter m. The Inset shows a\ndependence of the gap width vs. m. The topological and\ntrivial insulators are indicated in the figures.\nThe DOS at the edges of the band gap can be obtained\nanalytically using the properties of the elliptical integrals\nK(0) = π/2, namely\nρ(ΩL) =1\n2π(\n2|m|/√\n1−m2,if|m|<1,\n||m| −2|/(|m| −1),if|m| ≥1.(37)\nThe dependence of ρ(ΩL)as a function of mis shown in\nFig. 12. For |m|>2it is a finite smooth function. When\n|m|<2the function is singular and diverges at m=±1.\nVI. SPECTRAL MOMENTS OF TOTAL DOS\nIn this Section we present results on the spectral mo-\nmentsofthetotalDOS.Thegeneralformulaforthespec-\ntral moment of the order nreads\nMn=Z\nΩnρ(Ω)dΩ. (38)\nIntegrals of the form (38) can be calculated analytically\non the base of our analytic results from section III. Some\nof the calculations are shown in Appendix B to demon-\nstrate details of the integration technique.\nThemomentofthezeroorderrepresentsnormalization\nintegral M0= 2. All moments of the odd orders are zero\nbecause the DOS is an even function of Ω.\nThe moments of the even orders nare found as the\npolynomials in the parameter mof the same orders:\nM2= 2(m2+ 2),\nM4= 2(m4+ 8m2+ 5),\nM6= 2(m6+ 18m4+ 51m2+ 14) ,\nM8= 2(m8+ 32m6+ 210 m4+ 284 m2+ 42) .(39)11\nρ(Ω)\nΩ\n|m|=1.5 |m|=2.5\nFigure 13. Comparison of the two DOS of the QWZ model,\nwhich are characterized by the same gap. Total density of\nstates at m=±1.5(black dashed line) at half-filling corre-\nsponds to topological insulator. Total density of states at\nm=±2.5(green solid line) at half-filling corresponds to triv-\nial insulator. Enhancement of the DOS for topological case is\nseen.\nThis sequence can be continued if needed. Note, that all\nmoments have a factor of 2, which corresponds to two\nsymmetrical bands.\nVII. CONCLUSIONS AND OUTLOOKS\nIn this paper we derived the analytic formulae of the\nDOS of the QWZ Hamiltonian, a generic model forChern\ntopological insulators in two dimensions. The results\nare expressed in terms of the complete elliptic integrals.\nAnalytic expressions for the DOS are rare in general.\nOurresultsextendtheclassoftight-bindingmodelswhere\ntheexact, analytic DOS is known.\nWe discussed in details the plots of the DOS and com-\npared them with the dispersion relations for the same\nvalue of the parameter m. Some additional finite peaks\nin the DOS were identified. We provided explicit formu-\nlaeforthegapwidthandforthewidthofthebandsinthe\nQWZ model. We also found that for the same gap width\nthe topological system has larger DOS at the gap edge\nas compared with the trivial case. Apparently, in the\ntopological case a more spectral weight is redistributed\nclose to the band gap.It is due to the bands overlap and\nthe inversion band phenomenon. Finally, we obtained\nexpressions of the spectral moments of the QWZ model.\nThey are polynomials of the parameter m, controlling\nthe topology, and are of the same orders as the orders of\nthe corresponding moments.On the base of the analytic\nresults we can identify exactly the position of van Hove\nsingularities and their logarithmic type.The analytic DOS will be useful in determining ther-\nmodynamics(specific heat, compressibility, or magnetic\nsusceptibility) orlinear response(dc conductivity) of the\nQWZ Chern topological insulator. It will also sim-\nplify the dynamical mean-field theory study of the QWZ\nmodel when alocal Hubbard type of the interaction is\nadded to the Hamiltonian, cf. [31–33].A final remark,\nas mathematicians say: \"although a donut and a cup are\ntopologically equivalent, one cannot drink a coffee from a\ndonut. Theshape, i.e., thegeometry matters\". Similarly,\nall QWZ model’s physical properties, not only topolog-\nical ones, are importnt to be known in general. Wealso\nhope that by using similar methods other DOS can be\nobtained in analytic form, for example for the Haldane\nmodelof topological insulator on a hexagonal lattice.\nVIII. ACKNOWLEDGMENT\nWe thank for the financial support of the Excellence\nInitiative - Research University (IDUB) via the grant\nunder the program New Ideas - Ukraine. K.B. also\nacknowledges the support of the Deutsche Forschungs-\ngemeinschaft under the Transregional Collaborative Re-\nsearch Center TRR360.\nAppendix A: Complete elliptic integrals of the first\nand second kind\nExpressionsforcompleteellipticintegrals, whichweare\nusing, are the following\nK(q) =Zπ/2\n0dβp\n1−q2sin2β, (A1)\nE(q) =Zπ/2\n0dβq\n1−q2sin2β, (A2)\nwhere qis the modulus of the elliptic integral and q′is\nthe complementary modulus, i.e., q′2= 1−q2. The el-\nliptic integrals can also be expressed as sums or series of\nrational functions, [30]. The above definitions are tradi-\ntionally used by mathematicians. Note, that in Python\nlibraries q2is substituted as an argument of K.\nAppendix B: Calculation of spectral moments in\ndetails\nHere we show how to calculate some of the integrals\n(38). Though the spectral moment of any order can be\neasily calculated numerically, the analytical derivation is\nof special value and methodological interest.\nWe start with the simple case of |m|= 1. We express\nq2= 9−Ω2/8andq′2= (Ω2−1)/8. For all values of\nΩ2from the region where ρ(Ω)is nonzero the modulus is12\n0≤q≤1. In these terms the integral, normalizing the\nDOS, takes the form\nM0\n2=4\nπ2Z1\n0K(q)dq′. (B1)\nUsing the identity (6.141.2) in [30],\nZ1\n0K(q′)dq=π2\n4. (B2)\ngives the correct value of normalization.\nThe second moment takes the form\nM2\n2=4\nπ2Z1\n0(9−8q2)K(q)dq′. (B3)\nIt can be rewritten as M2/2 =M0/2 + 32 J/π2, where\nJ=−Z1\n0q′K(q)qdq. (B4)\nHere we used q′dq′=−qdq. The integral Jcan be\ndetermined by parts using the following substitution\ndv=K(q)qdqandu=q′. It allows us to use the in-\ndefinite integral (5.112.3) in [30], namely\nv=Z\nK(q)qdq=E(q)−q′2K(q),(B5)\nwhere E(q)is the elliptic integral of the second kind.\nIntegration by parts gives J=R1\n0E(q)dq′−J, where it\nis used that E(0) = K(0) = π/2andE(1) = 1. Using\nthe identity (6.148.2) in [30]\nZ1\n0E(q′)dq=π2\n8(B6)\nwe finally obtain J=π2/16andM2= 6.\nThe moment of the order nis expressed as a combi-\nnation of integrals of the product of an elliptic function\nand an even power of the modulus,\nMn=n/2X\ni=0CiZ1\n0q2iK(q)dq′. (B7)\nHere Ciare real numbers, compare with Eq. (B3). So the\nsame procedure with integration by parts can be used to\nobtain moment of arbitrary order.\nFor|m|>1the DOS have the infinite peaks at ±Ω∞.\nIntegral over Ωin Eq. (38) is split into two: from ΩLto\nΩ∞and from Ω∞toΩR. In these regions the comple-\nmentary modulus q′take values from 1to0, and from 0\nto1respectively. Denoting x= Ω2we can present the\nmoment of the order nin the form\nMn=√\n2\nπ2Z1\n0dq′K(q)\"\nxn/2\n2√\nx2+m2−2\u0012dx2\ndq′\u0013\n−\nxn/2\n1√x1+m2−2\u0012dx1\ndq′\u0013#\n.(B8)The substitution x=x1,2(q)is not a single-valued func-\ntion: x1,2=m2+4q′2∓4q′p\nm2−q2, wherethesign ”+”\ncorrespondstothefirstregionand ”−”tothesecondone.\nThe expression in the square brackets is transformed al-\ngebraically to a polynomial functionPn/2\ni=0Ci(m2)q2i, so\nforn= 0it is equal to 4√\n2and for n= 2it equal to\n4√\n2(m2+ 8q′2). Then moments are of the form (B7),\nwhere coefficients Ci=Ci(m2)are the functions of m2.\nThe result of calculations leads to the expressions (39).\nFor the case |m|<1two regions of integration need to\nbeconsideredseparately: |m| ≤Ω≤2−|m|and2−|m| ≤\nΩ≤2 +|m|. In each of these regions the substitution\nx=x(q)is not a single-valued, so each integral splits\ninto two ones, similar to the case with |m|>1.\nIn the first region we obtain the following: For |m| ≤\nΩ≤√\n2−m2the substitution is x1=m2−4q(q−p\n1−m2q′2)/q′2. The limits of integration x1(q′= 1) =\nm2andlimq′→0x1= 2−m2. For√\n2−m2≤Ω≤2−|m|\nthe substitution is x2=m2+ 4(1 −p\nq2+m2q′2)/q′2\nwith the limits of integration x2(q′= 1) = (2 −|m|)2and\nlimq′→0x2= 2−m2.\nIn the second region we have x3,4=m2+ 4q′2∓\n4q′p\nm2−q2valid for 2− |m| ≤ Ω≤ΩMand\nΩM≤Ω≤2 +|m|, respectively. Here ΩMis\ndefined by the condition dq/dΩ = 0 and q2=\n(Ω2−(m−2)2)((m+ 2)2−Ω2)/(Ω2+m2−2)/8. It\ncan be proven that q(ΩM) =|m|.\nThus the formula for the Mntakes the form\nMn=8\nπ2Z1\n0dq′K(q)\"\nxn/2\n2\nx2+m2\u0012dx2\ndq′\u0013\n−\nxn/2\n1p\n(x1−(m−2)2)(x1−(m+ 2)2)\u0012dx1\ndq′\u0013#\n+\n√\n2\nπ2Z|m|\n0dqK(q)\"\nxn/2\n3√x3+m2−2\u0012dx3\ndq\u0013\n−\nxn/2\n4√x4+m2−2\u0012dx4\ndq\u0013#\n.(B9)\nItshouldbenoted, thattheseintegralsaredifficulttocal-\nculate analytically. For example, for M0the substitution\nx=x(q)in Eq. (B9) results in\n(M0−1)π2\n8=Z|m|\n0dqK(q)qp\nm2−q2+\nZ1\n0dqK(q)\nq′2\"\nqp\nm2+q2(1−m2)−qp\n(1−m)2+m2q2)#\n,\n(B10)\nwhere we used the identity (6.144) in [30]\nZ1\n0K(q)1\n1 +qdq=π2\n8. (B11)13\nKnowing the fact that M0= 2, the expression (B10)\nturns into an interesting relation between integrals con-\ntaining K(q).\nIn order to determine these integrals, the integrandscan be expanded into an infinite series in q2followed by\nterm-by-term integration. But the easiest way to prove\nEq. (39) for the case |m|<1is the numerical integration.\n[1] M. Z. 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B 94, 085149\n(2016)." }, { "title": "2308.07602v1.Attraction_Domain_Analysis_for_Steady_States_of_Markovian_Open_Quantum_Systems.pdf", "content": "Attraction Domain Analysis for Steady States of Markovian\nOpen Quantum Systems⋆\nShikun Zhanga, Guofeng Zhanga,b\naDepartment of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Special\nAdministrative Region of China, China\nbShenzhen Research Institute, The Hong Kong Polytechnic University, Shenzhen 518057, China\nAbstract\nThis article concerns the attraction domain analysis for steady states in Markovian open quantum systems, which are math-\nematically described by Lindblad master equations. The central question is proposed as: given a steady state, which part of\nthe state space of density operators does it attract and which part does it not attract? We answer this question by present-\ning necessary and sufficient conditions that determine, for any steady state and initial state, whether the latter belongs to\nthe attraction domain of the former. Furthermore, it is found that the attraction domain of a steady state is the intersection\nbetween the set of density operators and an affine space which contains that steady state. Moreover, we show that steady\nstates without uniqueness in the set of density operators have attraction domains with measure zero under some translation\ninvariant and locally finite measures. Finally, an example regarding an open Heisenberg XXZ spin chain is presented. We pick\ntwo of the system’s steady states with different magnetization profiles and analyse their attraction domains.\nKey words: open quantum systems; Lindblad master equations; steady state; attraction domain.\n1 Introduction\nOpen quantum systems underpin the study of dissipative\nquantum information processing [Leghtas et al., 2015],\nwhere engineered interactions between the system and\nits environment facilitate various tasks including quan-\ntum metrology [Zhang and Gong, 2020], state stabiliza-\ntion [Ma et al., 2019,Kraus et al., 2008,Kimchi-Schwartz\net al., 2016] and autonomous error correction [F. Reiter\nand Muschik, 2017, Pan and Nguyen, 2017]. It is thus\nof both theoretical and practical significance to analyse\nthe dynamical properties of these systems.\nUnder the Markov assumption, open quantum systems\nare mathematically described by Lindblad master equa-\ntions [Breuer and Petruccione, 2001] which induce com-\npletely positive and trace preserving dynamics on the set\nof density matrices. In terms of Lindblad master equa-\ntions, it has been shown in [Schirmer and Wang, 2010]\nthat the uniqueness of a steady state is equivalent to its\n⋆This paper was not presented at any IFAC meeting. Cor-\nresponding author: Guofeng Zhang.\nEmail addresses: shikun.zhang@polyu.edu.hk (Shikun\nZhang), guofeng.zhang@polyu.edu.hk (Guofeng Zhang).global attractivity, i.e., all system trajectories with ini-\ntial states as density matrices converge to this state fol-\nlowing Lindblad evolution. This important theoretical\nresult lays the foundation for tasks related to quantum\nstate stabilization [Kraus et al., 2008,Ticozzi and Viola,\n2009,Sauer et al., 2014].\nHowever, if more than one steady states are present in\nan open quantum system, then apparently none of them\nare globally attractive. It is then natural to further in-\nvestigate what kind of initial states will be attracted to\nthem and what will not, i.e., to characterize their do-\nmain of attraction. We believe that the understanding of\nsuch locally attractive dynamics may expand our knowl-\nedge on many-body quantum systems and potentially\nbreed new mechanisms of dissipative quantum informa-\ntion processing.\nCharacterization of attraction domains has been a topic\nof intensive research in classical control theory. Since ex-\nact solutions are unattainable in many cases, Lyapunov\ntechniques are often adopted to estimate the domain\nof attraction [Tang and Daoutidis, 2019, Zarei et al.,\n2018, Bobiti and Lazar, 2018]. Such methods yield in-\nherently sufficient conditions on whether a given state\nbelongs to the attractive region and thus result in find-\nPreprint submitted to Automatica 16 August 2023arXiv:2308.07602v1 [quant-ph] 15 Aug 2023ing subsets of exact attraction domains. In this article,\nhowever, we seek to pin down the attraction domains of\nsteady states in open quantum systems in their entirety.\nTo achieve this, a necessary and sufficient condition is\npresented to verify whether a given initial density ma-\ntrix is contained in the attraction domain of a given sta-\ntionary state. Apart from pointwise verification, we also\ncharacterize the global structure of attraction domains.\nAttraction domains of steady states without uniqueness\nare clearly proper subsets of the entire state space. Then\nwe ask: how much “room” in the latter does the former\noccupy? Measure theory is a suitable tool for answering\nthis question. We prove that the attraction domains of\nnonunique steady states have measure zero under some\ntranslation invariant and locally finite measures, while\nthere exists such a measure under which the entire state\nspace has a positive finite measure. This sheds light on\nthe “almost impossibility” of stabilizing certain quan-\ntum states with Lindblad dynamics when nonlocal re-\nsources are not adequate to make them globally attrac-\ntive.\n2 Preliminaries\nLetHbe a finite dimensional Hilbert space isomorphic\ntoCN, and B(H) be the set of linear operators on H.\nFollowing Dirac’s notation in quantum mechanics, the\northonormal basis of His written as {|ϵi⟩}N\ni=1. The set\nof density operators D(H)⊂ B(H) includes all positive\nsemidefinite, trace-one operators on H, which consti-\ntutes the entire state space for finite dimensional open\nquantum systems. Let ρ∈ D(H) denote a quantum\nstate. Its evolution according to Lindblad master equa-\ntion is expressed as:\n˙ρ=−i[H, ρ] +MX\nk=1(LkρL†\nk−1\n2L†\nkLkρ−1\n2ρL†\nkLk),(1)\nwhere Hermitian operator H∈ B(H) stands for sys-\ntem Hamiltonian, and Lk∈ B(H), k= 1,2, ..., M rep-\nresent coupling operators between the system and its\nenvironment. Eq. (1) can be concisely expressed in su-\nperoperator form: ˙ ρ=L[H;L1,...,L M](ρ), where super-\noperator L[H;L1,...,L M]is a linear operator from B(H)\ntoB(H).∀A1, A2∈ B(H), their inner product is de-\nfined as ⟨A1, A2⟩≜tr(A†\n1A2). Consequently, the norm of\n∀σ∈ B(H) is defined as ∥σ∥≜p\n⟨σ, σ⟩. Let the adjoint\nof superoperators be understood with respect to such in-\nner product. Then, the superoperator L†\n[H;L1,...,L M]sat-\nisfies:\nL†\n[H;L1,...,L M]\u0000\n·\u0001\n= i[H,·] +MX\nk=1L†\nk(·)Lk−1\n2L†\nkLk(·)−1\n2(·)L†\nkLk.(2)We now give the definition of steady states.\nDefinition 1. ρss∈ D(H)is a steady state of system (1)\nifL[H;L1,...,L M](ρss) = 0 .\nA steady state ρssinduces a bipartition of D(H). Choos-\ning any ρ0∈ D(H) as the initial state, its time evolved\nstate eL[H;L1,...,LM]tρ0either converges to ρssin the long\ntime limit or it does not. This is formalised by the fol-\nlowing definition of attraction domains.\nDefinition 2. The attraction domain of steady state\nρss∈ D(H)w.r.t. system dynamics (1), denoted by\nDoA[ ρss], is defined as:\nDoA[ ρss] ={ρ∈ D(H)|lim\nt→+∞eL[H;L1,...,LM]tρ=ρss}.\n(3)\nIt is clear from this definition that the attraction domain\nof any steady state is nonempty, for it includes the steady\nstate itself at least.\nFor a steady state ρss, it has been shown in [Schirmer\nand Wang, 2010] that DoA[ ρss] =D(H) if and only if\nthere are no other steady states in D(H). That is, the\nuniqueness of ρss\u0000\ninD(H)\u0001\nis equivalent to its global\nattractivity\u0000\ninD(H)\u0001\n.\nHowever, if there is another steady state ρ′\nss∈ D(H),\nthen ρssis not globally attractive since at least ρ′\nssdoes\nnot belong to DoA[ ρss] according to Definition 1. More-\nover, any convex combination of ρssandρ′\nssis also a\nsteady state. As a matter of fact, all steady states of\nLindblad master equations in D(H) form convex sets,\nwhich may contain an uncountably infinite number of\nelements. In light of this, the exact characterization of\nattraction domains of an arbitrary steady state is a non-\ntrivial task.\nIt is worthwhile comparing Lindblad master equations\nwith Markov chains. A continuous-time homogeneous\nMarkov chain with finite state spaces must always admit\nstationary distributions. This is in analogy with the fact\nthat finite-dimensional Lindblad master equations must\nalways admit steady states. An irreducible continuous-\ntime homogeneous Markov chain must admit a unique\nstationary distribution which attracts all initial distri-\nbutions [Norris, 1997]. Irreducibility of Markov chains\nmeans that all states are accessible from one another.\nConsider a generalization to quantum systems, where\nclassical state icorresponds to quantum state |i⟩⟨i|. Ir-\nreducibility for quantum systems can be defined as: for\nany initial state |i⟩⟨i|, and any state |j⟩⟨j|, there exists a\nfinite time at which the time-evolved state has a nonzero\noverlap with |j⟩⟨j|(The overlap between two quantum\nstates ρandσmeans tr( ρσ) [Cincio et al., 2018]. In the\ncase where ρ=|i⟩⟨i|andσ=|j⟩⟨j|are pure quan-\ntum states, the overlap between ρandσis expressed as:\n2tr(ρσ) =|⟨i|j⟩|2.), which can be achieved by designing\nsuitable Hamiltonian(s) and coupling operator(s). How-\never, it is possible that the resulting Lindblad master\nequation admits more than one steady states, in which\ncase the attraction domain of each steady state can be\ndescribed by the results in our paper.\n3 Main Results\nIn this section, we answer the central question of this\narticle: if system (1) does not admit a unique steady\nstate in D(H), then which part of D(H) does each steady\nstate attract?\nMathematically, depicting the attraction domain of a\nsteady state ρssis equivalent to proposing a necessary\nand sufficient condition that is able to verify whether\nany given state in D(H) belongs to DoA[ ρss] or not. We\nthus present such a condition as one of the theoretical\nresults of this article.\nTheorem 1 Let{ωk}J\nk=1be a complete set of eigen-\noperator(s) of L†\n[H;L1,...,L M]corresponding to eigen-\nvalue(s) with zero real part(s). Let ρssbe an arbitrary\nsteady state of system (1). Then, for ρ0∈ D (H),\nρ0∈DoA[ ρss]if and only if tr(ω†\nlρ0) = tr( ω†\nlρss),\n1≤l≤J.\nPROOF. Let us denote the eigen-operator(s) of\nL†\n[H;L1,...,L M](2) with zero eigenvalue(s) by ω1, ..., ω J0\nand that with purely imaginary but nonzero eigenval-\nues, should they exist, by ωJ0+1, ..., ω J.\nSuperoperator L[H;L1,...,L M]admits no eigenvalues with\npositive real parts and no generalized eigen-operators\ncorresponding to eigenvalues with zero real parts (oth-\nerwise leading to unbounded state trajectory), and the\nsame goes for L†\n[H;L1,...,L M]. Therefore, there exist gen-\neralized eigen-operators of L†\n[H;L1,...,L M], denoted as\nωJ+1, ..., ω N2, such that ω1, ..., ω N2form a complete\nbasis of B(H).\nMeanwhile, there exist σ1, ..., σ N2∈ B (H), where\n{σk}J0\nk=1corresponds to eigen-operator(s) of L[H;L1,...,L M]\nwith eigenvalue 0, and {σk}J\nk=J0+1corresponds to\neigen-operators of L[H;L1,...,L M]with purely imaginary\nbut nonzero eigenvalues, and {σk}N2\nk=J+1corresponds\nto eigen-operators and generalized eigen-operators of\nL[H;L1,...,L M]associated with eigenvalues on the open\nleft complex plane. The operators σ1, ..., σ N2consti-\ntute a complete basis of B(H) and satisfy the following\nrelations:\ntr(ω†\njσj) =δij,1≤i, j≤N2. (4)As a result, any ρ0∈ D(H) admits the following expan-\nsion:\nρ0=N2X\nj=1tr(ω†\njρ0)σj. (5)\nIn terms of steady state ρss, since L[H;L1,...,L M](ρss) = 0,\nit must hold that\ntr(ω†\niρss) = 0 , J 0+ 1≤i≤N2. (6)\nNext, we further endow {σi}N2\ni=J+1and{ωi}N2\ni=J+1with\nthe following order:\nσλ1\n1, ..., σλ1\nn1;···;σλm\n1, ..., σλm\nnm\nω¯λ1\n1, ..., ω¯λ1\nn1;···;ω¯λm\n1, ..., ω¯λm\nnm,(7)\nwhere σλj\n1andω¯λj\n1, 1≤j≤m, are eigen-operators of\nL[H;L1,...,L M]andL†\n[H;L1,...,L M]with eigenvalues λjand\n¯λj, respectively; σλj\niandω¯λj\ni,1< i≤nj, 1≤j≤\nmare generalized eigen-operators of L[H;L1,...,L M]and\nL†\n[H;L1,...,L M]corresponding to eigenvalues λjand¯λj,\nrespectively. Also,Pm\nk=1=N2−J.\nBased on (5) and (7), the state trajectory starting from\nρ0with t≥0 is expressed as:\neL[H;L1,...,LM]tρ0\n=J0X\ni=1tr(ω†\niρ0)σi+JX\ni=J0+1eiβittr(ω†\niρ0)σi\n+mX\nk=1\u0014\n(x1,λkρ0+x2,λkρ0t+1\n2!x3,λkρ0t2+···+1\n(nk−1)!xnk,λkρ0tnk−1)σλk\n1\n+ (x2,λkρ0+x3,λkρ0t+···+1\n(nk−2)!xnk,λkρ0tnk−2)σλk\n2\n+···\n+ (xnk,λkρ0)σλknk\u0015\neλkt,\n(8)\nwhere\nL[H;L1,...,L M]σi= iβiσi, β i∈R, J 0+1≤i≤J,(9)\nand for 1 ≤j≤nk,1≤k≤m,\nxj,λkρ0= tr\u0000\n(ω¯λk\nj)†ρ0\u0001\n. (10)\nSuppose that ρ0∈DoA[ ρss]. Since Re( λk)<0, 1≤k≤\nm, it should hold that\ntr(ω†\niρss) = tr( ω†\niρ0) = 0 , J 0+ 1≤i≤J.\n3Otherwise, eL[H;L1,...,LM]tρ0does not have a limit as t\ntends to infinity. It thus follows that\nlim\nt→+∞eL[H;L1,...,LM]tρ0=J0X\ni=1tr(ω†\niρ0)σi=ρss.\nThe linear independence of σ1, ..., σ J0indicates that\ntr(ω†\niρss) = tr( ω†\niρ0) = 0 ,1≤i≤J0.\nNecessity is thus proved.\nNext, suppose that\ntr(ω†\niρss) = tr( ω†\niρ0) = 0 ,1≤i≤J.\nIt is clear from (6) and (8) that\nlim\nt→+∞eL[H;L1,...,LM]tρ0=ρss,\nwhich completes the proof of sufficiency. □\nWe then present a physical interpretation of Theorem 1.\nIn fact, based on Theorem 1, it is even possible to show\nthatρ0∈ρssif and only if there exists a linearly inde-\npendent set, denoted as {˜ωk}J\nk=1, of Hermitian opera-\ntor(s), such that tr(˜ ωkρ0) = tr(˜ ωkρss), 1≤k≤J. The\nproof is omitted for the sake of brevity. On one hand,\nthe operator(s) in {˜ωk}J\nk=1are Hermitian, and are thus\nviewed as “observables” in quantum mechanics. On the\nother hand, it is clear that each observable in {˜ωk}J\nk=1\nbelongs to the sum of eigenspace(s) of L†\n[H;L1,...,L M]cor-\nresponding to eigenvalue(s) with zero real part(s).\nWe note that the adjoint equation of (1):\n˙X=L†\n[H;L1,...,L M](X), (11)\ndescribes the evolution of observables in the Heisen-\nberg picture. Therefore, the Heisenberg evolution of ˜ ωk0,\nk0∈ {1, ..., J}, either remains constant (in this case, ˜ ωk0\nis called a “conserved quantity” in [Albert and Jiang,\n2014]) or oscillates, both displaying a non-decaying pat-\ntern in the long time limit. The Jnon-decaying ob-\nservable(s) pin down an “identification vector” for each\nρ∈ D(H), which contains the expectation value(s) of the\nobservables under state ρ. Theorem 1 and Proposition 1\nindicate that the attraction domain of a steady state ρss\nis formed by the density operator(s) equipped with the\nsame “identification vector” as that equipped by ρss.\nWe then present another result based on Theorem 1,\nwhich captures the global structure of attraction do-\nmains.Proposition 1 Consider σ1, ..., σ N2∈ B(H), where\n{σk}J\nk=1corresponds to eigen-operator(s) of L[H;L1,...,L M]\nwith eigenvalue(s) admitting zero real part(s), and\n{σk}N2\nk=J+1corresponds to eigen-operator(s) and gener-\nalized eigen-operator(s) of L[H;L1,...,L M]associated with\neigenvalues admitting negative real parts. Let ρssbe a\ndensity operator which satisfies L[H;L1,...,L M](ρss) = 0 .\nDenote the following set:\n{σ∈ B(H)|σ=ρss+N2X\nk=J+1gkσk, gk∈C, J+1≤k≤N2}\n(12)\nasAρss, which is an affine space over the subspace of\nB(H)spanned by σJ+1, ..., σ N2. Then,\nDoA[ ρss] =Aρss∩ D(H). (13)\nPROOF. We first prove that DoA[ ρss]⊂ A ρss∩ D(H).\nIt suffices to prove that ∀ρ0∈DoA[ ρss],ρ0∈ Aρss.\nLet{ηj}N2\nj=1be a complete set of eigen-operators and\ngeneralized eigen-operators of L†\n[H;L1,...,L M]satisfy-\ning tr( η†\niσj) = δij, 1≤i, j≤N2. An arbitrary\nρ0∈DoA[ ρss] can be expanded as:\nρ0=JX\nj=1tr(η†\njρ0)σj+N2X\nj=J+1tr(η†\njρ0)σj. (14)\nSince ρ0∈DoA[ ρss], according to Theorem 1, tr( η†\njρ0) =\ntr(η†\njρss), 1≤j≤J. Moreover, because\nρss=JX\nj=1tr(η†\njρss)σj,\nwe have\nρ0=ρss+N2X\nj=J+1tr(η†\njρ0)σj. (15)\nwhich implies that ρ0∈ Aρss.\nThen, it shall be proved that Aρss∩ D(H)⊂DoA[ ρss].\n∀ρ0∈ Aρss∩ D(H), we have\nρ0=ρss+N2X\nj=J+1gj\nρ0σj, gj\nρ0∈C, J+ 1≤j≤N2.(16)\nFollowing (8), it holds that\nlim\nt→+∞eL[H;L1,...,LM]t(N2X\nj=J+1gj\nρ0σj) = 0 . (17)\n4Because ρssis a steady state, we have lim t→+∞eLtρ0=\nρss, which says that ρ0∈DoA[ ρss]. □\nIf a steady state of system (1) is not unique in D(H),\nits attraction domain is a strict subset of D(H). Then,\nhow much “volume” does this attraction domain occupy\nin the set of all density operators? We show that the\nanswer is 0 under certain measures in this section. Also\npresented is an implication in the context of quantum\nstate stabilization.\nBefore presenting the main results of this section, we\nmake a few notations and definitions. Let us denote the\nset of Hermitian and trace-one operators in B(H) as\nD1(H). A subset SofD1(H) is defined as an open set\nofD1(H) if∀x∈S, there exists ϵ >0, such that any\ny∈ D 1(H) which satisfies ∥x−y∥< ϵbelongs to S. Let\nus denote the set of all open sets of D1(H) asT. Also, let\nus denote the set of steady states of system (1) as Ξ ss.\nWe are now in the position to present the following re-\nsult.\nTheorem 2 Suppose that system (1) admits non-\nunique steady states. For any measure space of the\nform (D1(H),Σ,M), with T ⊆ Σ,Mbeing translation\ninvariant on D1(H)and locally finite with respect to\n(D1(H),T), and DoA[ ρss]being measurable ∀ρss∈Ξss,\nit must hold that M(DoA[ ρss]) = 0 ,∀ρss∈Ξss.\nPROOF. We shall prove Theorem 2 by contradic-\ntion. Suppose that there exists ρ0\nss∈Ξss, such that\nM\u0000\nDoA[ ρ0\nss]\u0001\n̸= 0.\nIt is clear that D(H) is a bounded set since ∀ρ∈ D(H)\nsatisfies tr( ρ2)≤1. Also, D(H) is closed under topol-\nogyT. Based on eqs. (12) and (13), it also holds that\nDoA[ ρ0\nss] is bounded and closed under topology T.\nSince system (1) admits non-unique steady states, there\nexists ρ1\nss∈Ξsswhich is linearly independent with ρ0\nss.\nConsider the following set:\nStra≜[\np∈[0,1]\u0000\nDoA[ ρ0\nss] +p{ρ1\nss−ρ0\nss}\u0001\n.(18)\nThat is, x∈Straif and only if there exists x0∈DoA[ ρ0\nss]\nandp∈[0,1], such that\nx=x0+p(ρ1\nss−ρ0\nss). (19)\nFrom (19), it is clear that S trais bounded. We then show\nthat S trais also closed. Consider an arbitrary sequence{xn}+∞\nn=1⊂Strawith lim n→+∞xn= ˜x. Each xnin the\nsequence is decomposed as:\nxn=ρ0\nss+N2X\nr=J+1(gn\nr)σr+ ˜pn(ρ1\nss−ρ0\nss), (20)\nwhere n≥1,{gn\nr}+∞\nn=1⊂C(J+1≤r≤N2), ˜pn∈[0,1],\nand{σr}N2\nr=J+1is defined in Proposition 1. Since the\nsequence {xn}+∞\nn=1is convergent, it must be a Cauchy\nsequence. Therefore, it holds that, for k > m\nlim\nm,k→+∞N2X\nr=J+1(gk\nr−gm\nr)σr+ (˜pk−˜pm)(ρ1\nss−ρ0\nss) = 0 .\n(21)\nBecause σJ+1, ..., σ N2andρ1\nss−ρ0\nssare linearly inde-\npendent, it follows that {gn\nr}+∞\nn=1(J+ 1≤r≤N2)\nand{˜pn}+∞\nn=1are all Cauchy sequences and are therefore\nconvergent. Consequently, on one hand, the sequence\n{ρ0\nss+PN2\nr=J+1(gn\nr)σr}+∞\nn=1converges, and since each ele-\nment in the sequence belongs to the closed set DoA[ ρ0\nss],\nits limit also belongs to DoA[ ρ0\nss]. On the other hand, the\nsequence {˜pn}+∞\nn=1converges to a limit in [0 ,1], since [0 ,1]\nis a closed set. Therefore, we have shown that ˜ x∈Stra,\nwhich says that S trais closed.\nSince S tra(18) is bounded and closed, it is a compact\nset. The fact that Mis locally finite indicates that 0 ≤\nM(Stra)<+∞. Consider a infinite sequence {pn}+∞\nn=1⊂\n[0,1] with pi̸=pj,i, j≥1. Because the Mis translation\ninvariant, it holds that\nM(DoA[ ρ0\nss]) =M\u0000\nDoA[ ρ0\nss] +pn{ρ1\nss−ρ0\nss}\u0001\n,\nwhere n≥1. Based on Proposition 1, the sets DoA[ ρ0\nss]+\npn{ρ1\nss−ρ0\nss}≜Sn(n≥1) are mutually disjoint. The\ncountable additivity and monotonicity of measure M\nsay that\nM(+∞[\nn=1Sn)=+∞X\nn=1M(Sn)=+∞X\nn=1M(DoA[ ρ0\nss])≤M(Stra).\n(22)\nIfM(DoA[ ρ0\nss])̸= 0, then M(Stra) cannot be fi-\nnite, which leads to a contradiction. Therefore,\nM(DoA[ ρ0\nss]) = 0. □\nNext, we show in the following theorem that it is pos-\nsible to construct a translation invariant and locally fi-\nnite measure, under which D(H) has a finite positive\nmeasure, while attraction domains of non-unique steady\nstates have measure zero.\nTheorem 3 There exists a measure space\u0000\nD1(H),Σ0,M0\u0001\n,\nwhere T ⊆ Σ0, and M0is translation invariant on\n5D1(H)and locally finite with respect to (D1(H),T).\nWith this measure space, D(H)is measurable and\n00, the open ball\nB(x, ϵx)⊆T). Then, the mapping Festablishes a one-\nto-one correspondence between TandT1. Since mis lo-\ncally finite, T1⊆ΣL. Therefore, it is true that T ⊆ Σ0,\nand that M0is locally finite.\nIt is clear that D(H) is a convex subset of D1(H), which\nimplies that F\u0000\nD(H)\u0001\n⊆RN2−1is also convex. As a\nresult, F\u0000\nD(H)\u0001\nis Lebesgue measurable [Lang, 1986],\nand thus D(H) is measurable.\nNext, since1\nNIN∈ D(H) is positive definite, there exists\nϵ >0, such that the set:\nS(ϵ)≜{ρ|ρ=1\nNIN+N2−1X\nk=1akBk,|ak|<ϵ,1≤k≤N2−1}\n(27)\nis a subset of D(H). The set F\u0000\nS(ϵ)\u0001\nis aN2−1 dimen-\nsional hypercube in RN2−1and is thus Lebesgue mea-\nsurable. Therefore, S(ϵ) is measurable and it holds that\nM0\u0000\nS(ϵ)\u0001\n=m\u0012\nF\u0000\nS(ϵ)\u0001\u0013\n=ϵN2−1>0. (28)\nIt follows that 0 τM. On the other hand, the massive pair plasma state density ρH\nMcontributes\nto the usual matter/radiation density ρM,R. Moreover, the latter variation affects the\nformer. It implies the back-and-forth interaction between the massive pair plasma state\nand the normal matter and radiation state during the Universe’s evolution. Therefore,\nwe propose the back-and-forth interaction between the densities ρH\nMandρM,Rfollows\nthe cosmic rate equations of Boltzmann type,\n˙ρM+ 3(1 + ωM)HρM= Γ M(ρH\nM−ρM−ρR)−Γde\nMρde\nM, (2.6)\n˙ρR+ 3(1 + ωR)HρR= Γ M(ρH\nM−ρM−ρR) + Γde\nMρde\nM, (2.7)\nwhere Γde\nMandτR= (Γde\nM)−1are unstable massive pair decay rate and time. The\nterm Γde\nMρde\nMrepresents unstable massive pairs decay to light particles, such as quarks\nand leptons, gauge bosons in SM and other light sterile particles. The term 3(1 +\nωM,R)HρM,Rof the time scale [3(1 + ωM,R)H]−1represents the space-time expanding\neffect on the density ρM,R. While Γ MρH\nMis the source term and Γ M(ρM+ρR) is the\ndepletion term. The detailed balance term Γ M(ρH\nM−ρM−ρR) indicates how densities\nρH\nMandρM,Rof different time scales couple together. The ratio Γ M/H > 1 indicates the\ncoupled case, and Γ M/H < 1 indicates the decoupled case. The detailed discussions\nare in Secs. 4-5 of Ref. [22].\n2.5 Preliminary applications to inflation and reheating\nThe main aspects of the ˜ΛCDM scenario are (a) the dark energy and matter interacting\nFriedman Equations (2.1,2.2); (b) massive particle and antiparticle pairs’ production\nand oscillation; (c) holographic massive pair plasma state (2.3) and its variation rate\n(2.4); (d) cosmic rate equations (2.6) and (2.7). They form a close set of first-order\nordinary differential equations for the densities ρM, ρR, ρΛand Hubble function H. The\nsolutions are completely determined, provided initial or transition conditions are given.\nIn Ref. [21], we study the inflation when the dominant dark energy ρΛdrives\ninflation and produces massive pairs’ plasma ρH\nM, that slows down inflation ( ρΛ≫ρH\nM≈\nρM). Neglecting Eqs. (2.6,2.7) for Γ M/H < 1, we use Eq. (2.2) and ρM≈ρH\nMto obtain\nan analytical solution Hend=H∗exp−(ϵ∗Nend).Thee-folding numbers Nend≈(50 60)\nfrom the pivot inflation scale H∗to the inflation end Hend≈ΓM≈(0.42,0.35)H∗. We\nfix the mass parameter m∗byϵ∗=χ(m∗/mpl)2= (1−ns)/22. The obtained relation\nof spectral index nsand tensor-to-scalar ratio ragrees with recent CMB observations.\nWe discuss the singularity-free pre-inflation, the CMB large-scale anomaly, and dark-\nmatter density perturbations imprinting on power spectra.\nIn Ref. [22], we study the reheating when the dark energy ρΛdecreases, ρH\nMand\nρMincrease. The competition between the Hubble function H, the rates Γ Mand Γde\nM\nplay an important role in cosmic rate equations (2.6) and (2.7). First, it appears\ntheM-episode of massive pair domination when Γ M> H > Γde\nM. Then it proceeds\n2The reduced Planck mass mpl≡(8π)−1/2Mpl= 2.43×1018GeV.\n– 5 –to the R-episode of radiation domination when Γde\nM> H > ΓM. The rate equation\n(2.7) becomes a reheating equation for Γde\nM>ΓM. Unstable pairs decay to light\nparticles and the radiation energy ρRincreases, leading to reheating. We estimated\nthe mass parameter ˆ m≳20mpland obtained results agree with observations. Stable\nmassive particles remain as cold dark matter particles3. The detailed discussions are\nin Secs. 7.2-7.3 of Ref. [22].\n3˜ΛCDM solution to cosmological coincidence problem\nAt the reheating end, the radiation energy density ρRis dominant, stable cold dark\nmatter energy density ρM≪ρRand the dark energy density vanishes ρΛ≈0, namely\nρR≫ρM≫ρΛ≈0. These are the initial conditions starting the standard cosmology.\nIt then evolves to matter-dominated, dark energy-dominated epochs. We study in this\narticle the ˜ΛCDM solution after reheating, focusing on the problem of cosmological\ncoincidence between dark energy and matter.\n3.1 Dark energy interaction with matter and radiation\nTo explicitly show dark energy and matter interaction, we recast the Friedman equa-\ntions (2.1,2.2), cosmic rate equations (2.6) and (2.7) as\n˙ρΛ+ 3(1 + ωΛ)HρΛ=−2ΓM\u0000\nρH\nM−ρM−ρR\u0001\n, (3.1)\n˙ρM+ 3(1 + ωM)HρM= +Γ M\u0000\nρH\nM−ρM−ρR\u0001\n, (3.2)\n˙ρR+ 3(1 + ωR)HρR= +Γ M\u0000\nρH\nM−ρM−ρR\u0001\n, (3.3)\nwhere Γ M\u0000\nρH\nM−ρM−ρR\u0001\nrepresents the interaction between dark energy and usual\nmatter and radiation via the massive pair plasma state ρH\nM. Equations (3.2) and (3.3)\nare the cosmic rate equations (2.6) and (2.7). Here we neglect the decay term ±Γde\nMρde\nM,\nassuming unstable massive pairs ρde\nMhas decay in reheating.\nIn inflation and reheating, Γ M\u0000\nρH\nM−ρM−ρR\u0001\n>0 and ˙ ρΛ<0, dark energy\nconverts to matter and radiation energies [21,22]. After reheating, there are two cases:\n(a)ρH\nM< ρM+ρR, matter and radiation converts to dark energy ˙ ρΛ>0,\n(b)ρH\nM> ρM+ρR, dark energy converts to matter and radiation ˙ ρΛ<0.\nThese two cases are separated by ρM+ρR=ρH\nM.\n3.2 Cosmic rate equations for cosmic abundance\nWe define the cosmic abundance\nΩΛ,M,R≡ρΛ,M,R\nρtot, ρ tot≡8πG\n3H2, (3.4)\n3There, the strongly coupled case Γ M/H≫1 is assumed in the preliminary study of cold dark\nmatter abundance Ω Mevolution. We realise it should be the weakly coupled case Γ M/H < 1 after\nstudying the Ω Mevolution in this article.\n– 6 –and ΩΛ+ΩM+ΩR= 1 (2.1). The “time” variable xrelates to the scale factor a=a(t),\nx= ln( a/a 0) + ln( a0/aR) =−ln(1 + z) + ln( a0/aR). (3.5)\nThe derivative dx=Hdt anddx=−dz/(1 + z), where zis the redshift. Equation\n(2.5) becomes\nϵ= (1 + z)dH\nHdz=3\n2[(1 + ωM)ΩM+ (1 + ωR)ΩR+ (1 + ωΛ)ΩΛ]. (3.6)\nWhereas, Equations (3.1-3.3) becomes\n−(1 +z)dΩΛ\ndz+ 3(1 + ωΛ)ΩΛ=−2ΓM\nH\u0000\nΩH\nM−ΩM−ΩR\u0001\n, (3.7)\n−(1 +z)dΩM\ndz+ 3(1 + ωM)ΩM= +ΓM\nH\u0000\nΩH\nM−ΩM−ΩR\u0001\n, (3.8)\n−(1 +z)dΩR\ndz+ 3(1 + ωR)ΩR= +ΓM\nH\u0000\nΩH\nM−ΩM−ΩR\u0001\n, (3.9)\nwhere ΩH\nM= (2/3)χ( ¯m/M pl)2and the ¯ mis the mass parameter. The dark energy and\nmatter interacting rate Γ M/His characterized by the ratio\nΓM\nH=χϵ\n(4π)( ¯m/H 0)\n(H/H 0), (3.10)\nandH0is the Hubble constant at the present time a0= 0 and z= 0. We define the\ndark energy and matter exchanging amount δQ\nδQ≡ΓM\nH\u0000\nΩH\nM−ΩM−ΩR\u0001\n. (3.11)\nBoth rate Γ M/Hand amount δQare functions of redshift z.\nThese dynamical equations (3.1-3.3) are reminiscent of general modelling inter-\nacting dark energy and matter based on total mass-energy conservation, δQint∝Hρ,\nwhere ρrelates to energy density, see review [12,14]. In comparison, we find the crucial\ndifferences between the ˜ΛCDM and interacting dark energy model are: (i) the inter-\nacting term δQ(3.11) changes its sign depending on the dark energy converting to\nmatter or inverse process, while the dark energy interacting model δQint∝Hρdoes\nnot change sign in evolution; (ii) the interacting rate (3.10) is proportional to 1 /H,\nwhile δQint∝Hin dark energy interacting models. The difference (ii) shows that the\ndark energy and matter interacting rate is small in the early time of large redshift and\nbecomes large in small redshift for the later time. As will be shown, it is an important\nfeature for solving the cosmological coincidence problem.\nTo solve these dynamical equations (3.6-3.9), we adopt the initial conditions given\nby observations today ( a0= 1)\nΩ0\nΛ≈0.7,Ω0\nM≈0.3,Ω0\nR≈3×10−5, (3.12)\n– 7 –andH0. Therefore the first-order ordinary differential equations (3.6-3.9) form a close\nset, uniquely determine the solutions ΩΛ, ΩM, ΩRandHas functions of the redshift z\nfrom the today z= 0 to the past zRand to the future z→ −1.\nWe adopt (3.12) as initial conditions, because ΩR,M, Λ(zR) values are unknown.\nThe redshift zRat the reheating is given by\n(1 +zR) =a0/aR≈(g∗/2)1/3(TRH/TCMB), (3.13)\ndepending on the degeneracy g∗of relativistic particles, the reheating temperature\nTRHand CMB temperature TCMB[22,31]. The zR≫1 for TRH≫TCMB. The Hubble\nscale variation is huge after reheating, HRH∼TRH≫H0. The Universe evolves from\nradiation-, matter- and dark-energy-dominated epochs. The mass parameter can vary\nin time. Therefore, we treat the mass parameter ¯ m > H 0as an effective value for\nqualitatively studying ΩΛ, ΩM, ΩRandHevolution.\n3.3 Solution to cosmological coincidence problem\nWe present the numerical solutions in Figure 1, the left column for the future 0 >\nz >−1 and the right column for the past zR> z > 0. This solution is unique and\nindependent of where the initial condition z= 0 or z=zRis implemented. The\ndiscussions on the solution are in order:\n(i) Figures 1 (a) and (b) show ΩR,M, Λevolution in time (or inverse time): (a) from the\ntoday z= 0 (3.12) to the future ( z=−1) when Ω R→0,ΩM→0 and Ω Λ→1;\n(b) from z≈zRradiation domination Ω R≈1,ΩM≪1 and Ω Λ≈0 to the today\nz= 0 (3.12). From z=zRtoz= 0, the matter increases, radiation decreases\nand Ω R= Ω Moccurs around z∼(103−104) for the effective mass parameter\n¯m/H 0∼(5−10). The equality Ω Λ= Ω Moccurs around z≈0.2∼0.4, which\nis not sensitive to the parameter ¯ m/H 0value. Figures 1 (c) and (d) show the\nUniverse evolution ϵrate (2.5) varies from ϵ≈2 (radiation) to ϵ≈3/2 (matter),\nϵ≈0.45 (today), and then to ϵ≈0 (dark energy) domination. The quantitative\nresults mildly depend on the parameter value ¯ m/H 0. All these behaviours are\nqualitatively following the ΛCDM model.\n(ii) Figures 1 (e) and (f) show the solution of how the Hubble function in the unit of\nH0evolves from z=zRtoz= 0 then to z=−1. The approximate constancy\nH≈H0since z≈0.1 shows the Universe acceleration and dark energy Ω Λ\ndomination over matter Ω Mand radiation Ω R. Towards the future z <0,H2≈\n(8π/3)GρΛslowly varies, asymptotically approaching to constant Ω Λ≲1. The\nevolution is analogous to inflation.\n(iii) Figures 1 (g) and (h) show the dark energy and matter interacting rate Γ M/His\nsmall and exchanging amount δQ(3.11) is negative. Therefore, the matter and\nradiation energy slowly convert to dark energy from the reheating end z=zRto\ntoday z= 0, then to the future 0 > z > −1. The dark energy ΩΛ(z) increases from\nΩΛ(zR)≈0 to ΩΛ(0)≈0.7 and then to ΩΛ(−1)≈1. For values ¯ m∼(5∼10)H0\nandH0≪Mpl, we can neglect the ΩH\nM= (2/3)χ( ¯m/M pl)2≪1, and the χ¯m/H 0\nis a unique parameter for differential equations (3.7-3.9).\n– 8 –-1.0-0.8-0.6-0.4-0.20.00.00.20.40.60.81.0\n(a)zΩΛ,ΩM,ΩRΩΛΩRΩM\n0.0010.1001010001051070.00.20.40.60.81.0\n(b)zΩΛ,ΩM,ΩRΩΛΩMΩR\n-1.0 -0.8 -0.6 -0.4 -0.2 0.00.00.10.20.30.4\n(c)zϵ-rate\n0.001 0.100 10 1000 1051070.51.01.52.0\n(d)zϵ-rate\n-1.0 -0.8 -0.6 -0.4 -0.2 0.00.850.900.951.00\n(e)zH/H0\n0.001 0.100 10 1000 105107110001061091012\n(f)zH/H0\n-1.0 -0.8 -0.6 -0.4 -0.2 0.00.00000.00010.00020.00030.00040.00050.0006\n(g)zΓM/H\n0.001 0.100 10 1000 1051070.00000.00020.00040.00060.00080.0010\n(h)zΓM/H\n-1.0 -0.8 -0.6 -0.4 -0.2 0.0-0.00020-0.00015-0.00010-0.000050.00000\n(i)zδQ\n0.001 0.100 10 1000 105107-0.0006-0.0004-0.00020.0000\n(j)zδQFigure 1 . Numerically solving Eqs. (3.6-3.9) and (3.12), ¯ m/H 0= 10, we present the ˜ΛQCD\nsolutions as functions of redshift z, from today to the future (the left column) and to the\npast (the right column). See text for detailed discussions.\n– 9 –The novelty is the ˜ΛCDM results of Fig. 1 show a natural solution to the cosmic\ncoincidence problem. It requires an incredible fine-tuning on the initial value ΩΛ(zR)≈\n0 many orders of magnitude, so as to achieve ΩΛ(0)∼ΩM(0)∼ O(1) for an extreme\nlong period from the reheating era zR≫1 to the present time z= 0. We explain\nbelow how the solution works.\nThe dark energy density vanishes ( ρΛ≈0) at the reheating end zR. However, for\na long period (5 ≲z≲zR), it slowly increases ( ρΛ≳0) and closely follows up with\nradiation ρRand matter energy ρMdensities’ evolution, see Figures 1 (b) and 2 (c) for\nlarge z >5. The reason is that the dark-energy, matter, and radiation interacting rate\nΓM/H≪1 and exchanging amount δQ≪1 are very small, but not zero, see Figure 1\n(h) and (j). It is crucial for the dark energy density ρΛfollowing up the energy densities\nρRandρMsince they had been varying many orders of magnitudes in the 5 ≲z≲zR\nperiod4. To indicate the ρΛincrease following ρΛ,M, we here adopt the word “following\nup”, which has a similar sense as the word “track down” used in Ref. [2].\nWhen the redshift z≲5, the interacting rate Γ M/Hand exchanging amount\nδQ < 0 increase significantly because the Hubble function Hbecomes smaller and\nsmaller, see Figure 1 (h) and (j). The dark energy significantly increases after z≈5\nwhen the matter Ω Mdomination begins5, see Fig. 1 (b). These features are consistent\nwith the late-time interaction in the dark sector observed by data analysis [23]. As a\nresult, in a short period from z≈5 toz≈0, dark energy ΩΛincreases from ΩΛ≪1\nto the order of unit O(1). The ΩΛand ΩMcoincide ΩΛ≈ΩMatz≈0.5, and are in\nthe same order of magnitude up to z≈0. In this short period 0 .5≲z≲5, the energy\ndensities ρMandρR, and Hubble function Hvary only a few orders of magnitudes,\nsee Fig. 1 (b) and (h), in contrast with their variations in many orders of magnitudes\nsince z=zR. In other words, the recent ρMandρΛevolution are insensible to their\ninitial values at zR. Nature does not need to fine-tune the initial ratios of dark energy,\nmatter and radiation densities at the Big Bang beginning (the reheating end).\nThe dark energy, matter, and radiation interacting rate Γ M/H(3.10) and ex-\nchanging amount δQ < 0 (3.11) are small at high redshift z, and large at low redshift\nz. Due to such redshift zdependence of the interaction, ˜ΛCDM gives a dynamical\nsolution to the cosmic coincidence problem of ΛCDM in the following scenes. With-\nout any fine-tuning, the dynamical solution uniquely determines the evolution from\nΩM(0)∼ΩΛ(0)≫ΩR(zR) to ΩR(zR)≫ΩM(zR)≫ΩΛ(zR)≈0, and vice versa .\nNamely, if we would know the initial conditions ΩR,M, Λ(zR) atz≈zR(3.13), we would\nhave obtained the same dynamical solutions (Fig. 1) and the present values (3.12) by\nwithout fine-tuning. It is the main result presented in this article.\nWe have made numerical verification that the ˜ΛCDM solution from the maximal\nΩM(z≈5) to ΩM≈ΩΛ(z≈0.5) does not sensitively depend on the effective value\nof mass parameter ¯ m. It implies that the ˜ΛCDM solution should similarly function if\nwe adopt the mass parameter ¯ mweakly depending on H.\nWe do not discuss the dark energy density perturbations ( δρΛ, δpΛ) caused by its\n4It is a very long period, and that is why the ΛCDM has a fine-tuning problem.\n5This is consistent with the discussions that dark energy evolution follows first radiation then\nmatter in different ways [29,30].\n– 10 –10-30.1 10 1000 1051071090.00.20.40.60.81.0\n(a) zΩΛ,ΩM,ΩRΛCDM˜ΛCDM\n0.0010.1001010001051070.51.01.52.0\n(b)zϵ-rate˜ΛCDMΛCDM\n0.0010.0100.100110100100010-810-50.01\n(c)zΩΛ˜ΛCDMΛCDM\n0.0010.100101000105107110001061091012\n(d)zH/H0˜ΛCDMΛCDMFigure 2 . We present the ˜ΛCDM solution (Fig. 1) in comparison and contrast with the\ncanonical ΛCDM model (4.1), 4.2) and (3.12). See text for detailed discussions.\ntime-varying interaction with matter and radiation. However, we speculate that dark\nenergy undergoes transitions and becomes dominant from z≈5 to z≈0.1 should\nimpact matter density perturbation, leading to the effect on the formation of large-\nscale structures and clusters. In addition, it should induce the peculiar fluctuations of\nthe gravitational field, possibly imprinting on observations, for instance, the integrated\nSachs-Wolfe effect or galaxy positions. The reason is that the dark energy Λ results\non the gravitation field are rather different from the gravitational potential of matter.\nTo end this section, we mention that in the remote future ( z→ −1) of dominant\ndark energy, radiation and matter Ω R+ Ω Mcontinually decrease until the exchanging\namount δQ(3.11) changes sign from negative δQ < 0 to negative δQ > 0. The\ndark energy density decreases, converting to matter and radiation energy densities.\nIt shows the possibility that the Universe ends the current acceleration and starts\nrecycling again. The topic is not the scope of this article. We do not present figures\nand discussions for this situation in the remote future z→ −1.\n4 Approximated ˜ΛCDM solution for data analysis\n4.1 Comparison and contrast with ΛCDM model\nIn this section, we present the ˜ΛCDM solution in comparison and contrast with the\nΛCDM results. In the ΛCDM model, we define the cosmic abundance of radiation,\n– 11 –matter, and dark energy\nΩR= Ω0\nR(1 +z)4\nE(z)2,ΩM= Ω0\nM(1 +z)3\nE(z)2,ΩΛ= Ω0\nΛ(1 +z)0\nE(z)2, (4.1)\nand the dimensionless Hubble function E(z)≡(H/H 0)\nE(z)2= Ω0\nR(1 +z)4+ Ω0\nM(1 +z)3+ Ω0\nΛ(1 +z)0, (4.2)\nwhere E(0)2= Ω0\nR+Ω0\nM+Ω0\nΛ= 1. The evolution ϵ-rate is given by Eq. (3.6). The values\nΩ0\nR,Ω0\nM,Ω0\nΛandH0atz= 0 are the same as the initial conditions (3.12). Here we use\nthe same notations for quantities of the ˜ΛCMD and ΛCMD models. The former is the\ndark energy and matter interacting solutions to Eqs. (3.6-3.9). The latter is (4.1) and\n(4.2) for the constant dark energy density. We have implemented only one observed\ndata point (3.12) for both models.\nIn Figure 2, we compare ˜ΛCMD solutions (Fig. 1) with the ΛCMD (4.1-4.2)\nresults. The discussions are in order.\n(i) Figures 2 (a) and (b) show ΩR,M, Λand expansion rate ϵ(3.6) evolution for ΛCDM\nand˜ΛCDM. Overall they are consistent and agree with each other, particularly\nforz <10. Two ΩΛcurves overlap in (a) and the point ΩΛ= ΩMis about the\nsame. The main differences are ΩR,Mandϵ-rate are in the range z∼102−105,\nand the crossing point ΩR= ΩM. These differences mildly depend on the ˜ΛCDM\nparameter value χ¯m/H 0∼ O(10−2).\n(ii) Figures 2 (c) and (d) shows ΩΛandH/H 0evolution for ˜ΛCDM and ΛCDM. The\ncrucial difference in ΩΛappears for z >10. Due to the nature of constant dark\nenergy density, the ΛCDM ΩΛ∝1/H2forz→zR≫1. It leads the fine-tuning\nproblem for achieving Ω0\nΛ∝1/H2\n0∼ O(1) from ΩΛ∝1/H2\nRH≪ O (1). Whereas,\nit is not the case for the ˜ΛCDM model, as discussed in the previous section. The\nHubble function H/H 0discrepancy between ˜ΛCDM and ΛCDM is significant\nat high red-shift ( z >1000). These comparisons and contrasts imply that the\n˜ΛCDM could relieve the H0andS8tensions between the values measured today\nand calculated in the ΛCDM model based on measurements at high red-shifts z.\nThese discussions show that (i) apart from solving the cosmic coincidence problem,\nthe˜ΛCDM’s quantities slightly deviate from the ΛCDM counterparts for z <103; (ii)\nthe˜ΛCDM represent a one-parameter ( χ¯m) extension to the ΛCDM model. These are\nbased on numerical solutions to non-linearly coupled differential equations (3.6-3.9).\nTherefore, it is not convenient in practice for quantitatively comparing the ˜ΛCDM\nwith observation data.\n4.2 Approximated ˜ΛCDM solution for phenomenological studies\nThe dynamic system formed by four differential equations (3.6-3.9) should have a fixed\npoint at low redshifts ( z≪1), where the ΛCDM realizes. From high redshifts ( z≫1),\n– 12 –the˜ΛCDM quantities approach this fixed point in scaling laws, namely, scaling factors\n(1 + z)δcorrected ΛCDM counterparts. The scaling indexes |δ| ≪ 1, because the\n˜ΛCDM approaches ΛCDM for low redshifts, as shown in Fig. 2. In Ref. [25], we expect\nthese dynamics and approximately derive analytical solutions (4.6,4.7) in the spirit of\nasymptotic safety of gravitational theories [32].\nThe view of scaling-law (1 + z)δcorrections is further supported by the smallness\nparameter χ¯m/H 0in the dark energy and matter interacting rate (3.10). For low\nredshifts, we approximately decouple Eqs. (3.1-3.3) into\n˙ρΛ+ 0HρΛ≈+δΛHρΛ, (4.3)\n˙ρM+ 3HρM≈ −δG\nRHρM, (4.4)\n˙ρR+ 4HρR≈ −δG\nMHρR. (4.5)\nThree new dimensionless parameters δG\nR,δG\nMandδΛare proportional to the primal\nparameter χ¯m/H 0. They are much smaller than the unity. Equations (4.3-4.5) yield\nthe effectively corrected densities\nρR≈ρ0\nR(1 +z)4−δR\nG, ρM≈ρ0\nM(1 +z)3−δM\nG, ρΛ≈ρ0\nΛ(1 +z)δΛ, (4.6)\nand the Hubble function\nE2(z) = Ω0\nR(1 +z)4−δR\nG+ Ω0\nM(1 +z)3−δM\nG+ Ω0\nΛ(1 +z)δΛ. (4.7)\nIn the view of Eqs. (4.6), we consider the equations of states effectively modify:\nωeff\nR≈1/3(1−δR\nG),ωeff\nM≈ −(1/3)δM\nGandωeff\nΛ≈ −1 + (1 /3)δΛ, see also Ref. [33].\nEquations (4.6,4.7) are ˜ΛCDM approximate solutions, giving scaling-law (1 + z)δcor-\nrections to ΛCDM results. The fourth independent equation (2.5) gives the constraint\nof parameters δG\nR,δG\nMandδΛ,\nδΛ≈(Ω0\nMδM\nG+ Ω0\nRδR\nG)/Ω0\nΛ, (4.8)\nand two parameters are independent. They depend actually on one primal parameter\nχm(2.3). The approximate solutions (4.6,4.7,4.8) facilitate data analysis for comparing\n˜ΛCDM with observational data.\nReferences [34,35] presents detailed numerical studies and data analysis based on\nthe approximated ˜ΛCDM solutions (4.6,4.7) and numerous data sets of observations.\nIt shows that both ΛCDM H0andS8tensions reduces to 2 σlevel with constraint\nparameters δR\nG≈ −1.5×10−2,δM\nG≈ −5.0×10−4andδΛ≈ −2.0×10−4. The negative\nparameter values support the scenario of energy conversion from radiation and matter\nto dark energy, as discussed in the previous section. The negative δΛ≲0 implies\nthat due to interactions, dark energy slightly behaves as if it was a phantom energy\nωeff\nΛ≈ −1+(1 /3)δΛ≲−1. It differs from the situation in inflation and reheating when\ndark energy converts to matter and radiation energies [22] see Fig. 3, dark energy\nbehaves as if it was a quintessence energy ωeff\nΛ>−1.\n– 13 –5 Discussions on Einstein cosmological Λterm\nWe end this article with some discussions and speculations on the gravitational (ge-\nometric) and dynamical natures of the cosmological ˜Λ term and dark-energy density\nρΛ=˜Λ/(8πG) of the Einstein theory. We discuss the dynamical solution to the cosmic\nfine-tuning problem.\n5.1 Geometric nature of ˜Λdark energy as gravitational ground state\nThe˜Λ term possibly represents [25,36–40] the non-trivial ground state (Wheeler space-\ntime foam [41, 42]) of the spacetime. The perturbative quantum gravitational field\nfluctuates upon such a ground state, and the classical gravitational field varies in such\na ground state. They are effectively described by the gravitation coupling G, the ˜Λ\nand the Ricci scalar Rterms in Einstein’s theory. Such a ground state is probably\na coherent state of the long-ranged holonomy field (see Eq. (133) of Ref. [38]). It is\na condensate state due to strongly violent quantum gravity at the Planck scale. The\nspacetime foam structure of such a ground state is most intriguing. It could be an in-\nteracting gas of gravitational instantons (wormholes), whose effective equation of state\nbehaves as pΛ=−ρΛ, see Sec. X of Ref. [43]. We are proceeding with further studies\non these aspects.\nThe ˜Λ (ξ∼1/˜Λ1/2) is the characteristic scale (correlation length) of such non-\ntrivial geometric ground state [25, 40]. It represents the intrinsic scale for effective\ngravitational field theories realized in the scaling domains of fixed points of effective\ngravitational coupling g∼GM2\nplto matter and radiation. The ˜Λ and gvarying from\none fixed point to another render its dynamic nature. It is nontrivial to demonstrate\nthese dynamical features. However, as analogies, we mention fundamental field theories\nof interactions (i) the electroweak scale v∼102GeV for electroweak field theory realized\nin the scaling domain of infrared (IR) fixed point; (ii) the scale Λ QCD∼102MeV for\nperturbative QCD field theory realized in the scaling domain of ultraviolet (UV) fixed\npoint; (iii) the low-energy hadron scale for non-perturbative QCD field theory realized\nin the scaling domain of IR fixed point.\n5.2 Asymptotically safe Einstein theory for early and present Universe\nOn the one hand, in early Universe of ˜Λ dark energy dominated inflation, H2∼\n(8πG/3)ρΛandρΛ=˜Λ/(8πG) asymptotically give ξ∼1/˜Λ1/2∼H−1. Namely, the\ncorrelation length ξis the size of the horizon. The ˜Λ1/2slowly varies from the inflation\nscale H∗to the scale Hend≈(0.42,0.35)H∗at inflation end aend. The H∗∼10−6Mpl\nis obtained from the CMB data, see Eqs. (6.5) and (6.10) of Ref. [21]. The inflation\nscale H∗is much smaller than the Planck scale. How quantum gravitation field theory\nwith the intrinsic scale ˜Λ1/2∼Mplruns to the effective Einstein theory at the scale\n˜Λ1/2∼H∗≪Mpl. How does the quantum gravity ground state evolve to the ˜Λ\nground state of effective Einstein theory? One could study it in the context of the\nasymptotically safe and effective theories of gravitation [32] and the scaling domain of\na UV unstable fixed point [25].\n– 14 –10-310-20.111010-1710-1210-710-2\n(a)x=ln(a/aend)h2,ΩRΩRh2reheatingaR\n10-310-20.111010-1710-1210-710-2\n(b)x=ln(a/aend)ΩΛ,ΩM,ΩRΩΛΩMΩRreheatingaRFigure 3 . We reproduce the ˜ΛCDM results of the Hubble function h2=H2/H2\nend(left),\nΩM, ΩRand Ω Λ(right) in reheating, see Figure 6 (a) and (b) in Ref. [22]. They are in the\nunit of ρend\nc= 3m2\nplH2\nendand the Hubble scale Hend≈(0.42,0.35)H∗at inflation end aend, see\nEq. (6.10) of Ref. [21]. The parameters ˆ m/m pl≈27.7 and χ≈1.85×10−3. It shows that in\na few numbers x= ln( a/aend), dark energy Ω Λdecreases from one to zero because of rapidly\nconverting to matter and radiation. Matter Ω Mincreases and dominates over Ω Λ. Then\nΩMdecreases because of decaying to radiation Ω R. As a result, Ω Rincreases and becomes\ndominant. At the reheating end ( aR/a0) = (1+ zR)−1, the radiation abundance is about one,\nand the dark energy and matter abundances are about zero.\nOn the other hand, in the recent Universe of ˜Λ dark energy dominated accelera-\ntion, H2\n0∼(8πG/3)ρ0\nΛasymptotically gives the scale ξ∼1/˜Λ1/2∼H−1\n0and density\nρ0\nΛ≈H2\n0/(8πG) [44]. Based on the same spirit of asymptotic safety of effective grav-\nitational theories [32], we study its realization in the scaling domain of a UV stable\nfixed point, where is the effective Einstein theory of relevant operators R/G and˜Λ/G,\nand gravitational coupling Gand cosmological ˜Λ approach their values today [25,40].\nHowever, due to the dark energy, radiation and matter interactions, as well as pair\nproduction of massive particles and antiparticles on the horizon, it is nontrivial to find\nthe scaling laws for operators R/G and˜Λ/Gby using the asymptotic safety principle.\nThe questions are how the ˜Λ dark energy varies from the inflation scale H∗to the recent\nHubble scale H0≪H∗. How the dark energy density changes from ρ∗\nΛ≈H2\n∗/(8πG)\ntoρ0\nΛ≪ρ∗\nΛin many orders of magnitudes. We use the ˜ΛCDM solutions in infla-\ntion, reheating and standard cosmology to explain the possible solution to such cosmic\nfine-tuning problem.\n5.3 Dynamical nature of ˜Λdark energy solving fine-tuning problem\nAfter the inflation end, the Universe undergoes reheating. Based on dynamical equa-\ntions (3.6-3.9), we show [22] that due to strong coupling (Γ M/H≫1) between ˜Λ dark\nenergy and matter energy densities, dark energy rapidly converts into massive matter,\nand the latter decays to radiation energy. As a result, dark energy density decreases\nfrom ρend\nΛ≈3m2\nplH2\nendtoρR\nΛ≈0, where ρR\nΛ,M,Rstand for the dark energy, matter and\nradiation densities at the reheating end aR/a0= (1 + zR)−1. We illustrate in Fig. 3\nthe dynamical reheating process from the inflation end ρend\nΛ≫ρend\nM≫ρend\nR≈0 to\n– 15 –the reheating end ρR\nR≫ρR\nM≫ρR\nΛ≈0. The radiation energy density ρR\nRbecomes\ndominant, initiating the standard cosmology. There is no fine-tuning in this process.\nThen how the standard cosmology dynamically evolves to the coincidence ρ0\nΛ∼\nρ0\nM≫ρ0\nR≈0 in the recent epoch. It is the issue addressed in this article. The initial\nvalues of the scale factor aR, Hubble constant HRHand energy densities ρR\nR,M, Λcannot\nbe completely determined. Therefore we cannot uniquely solve ordinary differential\nequations (3.6-3.9) from the reheating end zRto the present epoch z= 0. However, we\nuse the present values (3.12) to uniquely solve ordinary differential equations (3.6-3.9)\nfrom today z= 0 back to the reheating end zR≫0. As shown in Fig. 1 (b) and\nFig. 2 (c), the dynamical solutions asymptotically approach the same initial conditions\nρR\nR≫ρR\nM≫ρR\nΛ≈0 for z→zR≫1 without any fine-tuning.\nSuch qualitative matching implies a consistent dynamical solution for the cosmic\nfine-tuning problem in the following way. Converting to matter and radiation δQ≫1\n(3.11), the dark energy density decreases from the inflation scale ρ∗\nΛ≈3m2\nplH2\n∗to\ninflation end ρend\nΛ≈3m2\nplH2\nend, then to reheating end ρR\nΛ≈0. Since the standard\ncosmology starts, converted from matter and radiation δQ≲0 (3.11), the dark energy\ndensity increases from the reheating end value ρR\nΛ≈0 to the present value ρ0\nΛ≈\nH2\n0/(8πG) [44]. Such a dynamical evolution is free from fine-tuning. It can be the\nsolution to the cosmic coincidence problem. The basic reasons are that in evolution\ndark energy and matter conversion δQ(3.11) changes sign and is proportional to the\ninteracting rate Γ M/H∝χmϵ/H andm > H .\nNonetheless, we have not yet found the complete and quantitative solution to\nthe cosmic fine-tuning problem since we separately adopt the effective values of mass\nparameter m∗/H∗for inflation, ˆ m/H endfor reheating and ¯ m/H 0for standard cos-\nmology. 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Rel. 21(2018) 2 [ 1606.00180 ].\n– 19 –" }, { "title": "2310.04219v1.Control_of_the_local_photonic_density_of_states_above_magneto_optical_metamaterials.pdf", "content": "Control of the local photonic density of states above magneto-optical metamaterials\nPhilippe Ben-Abdallah1,\u0003\n1Laboratoire Charles Fabry, UMR 8501, Institut d’Optique, CNRS,\nUniversité Paris-Saclay, 2 Avenue Augustin Fresnel, 91127 Palaiseau Cedex, France\n(Dated: October 9, 2023)\nThe local density of states (LDOS) of electromagnetic field drives many basic processes associated\nto light-matter interaction such as the thermal emission of object, the spontaneous emission of\nquantum systems or the fluctuation-induced electromagnetic forces on molecules. Here, we study\nthe LDOS in the close vincinity of magneto-optical metamaterials under the action of an external\nmagnetic field and demonstrate that it can be efficiently changed over a broad or narrow spectral\nrange simply by changing the spatial orientation or the magnitude of this field. This result paves\nthe way for an active control of the photonic density of states at deep-subwavelength scale.\nTailoring the photonic density of states in the envi-\nronnement of a textured solid at subwavelength scale is\nof prime importance to control the thermal emission of\nsolids, the spontaneous emission and the decay rate (flu-\norescence) of quantum emitters placed in the neighbor-\nhood of these structures. This tuning can also be used\nto modify the force mediated by the vacuum fluctuations\non neutral objects such as atoms or molecules. Since the\npioneer work of Purcell [1] on the modification of the\nspontaneous emission of an object by changing its sur-\nrounding environnement many strategies have been pro-\nposed to sculpt the local density of states (LDOS) at a\nlength scale smaller than the wavelength of electromag-\nnetic field. Hence, nanophotonic structures such as pho-\ntonic crystals [2, 3], surface gratings [4], two dimensional\nsystems [5, 6] or complex nanostructures [7, 8] were used\nto efficiently enhance or inhibit the spontaneous emis-\nsion [9, 10] and the decay rates of molecules [11, 12]. In\nthis Letter we explore the possibility to tune the pho-\ntonic states in the close vicinity of non-reciprocal meta-\nmaterials made with magneto-optical materials using an\nexternal magnetic field and show that the LDOS can be\nefficiently controled simply by changing the spatial ori-\nentation or the magnitude of this field.\nTo start, let us consider a non reciprocal metamaterial\nmade with a network of magneto-optical nanostructures\nof permittivity \u0016\u0016\"i, deposited on an isotropic substrate\nof permittivivity \"subas sketched on Fig.1. We assume\nthis system is at equilibrium at temperature Twith its\nsurrounding environment and it is uniformly shined by\nan external magnetic field Hextin an arbitrary direction.\nWhen Hextis parallel to the normal zof the subrate, the\npermittivity tensor associated to each magneto-optical\nnanostructure takes the form [13]\n\u0016\u0016\"j(Hext) =0\n@\"aj\u0000i\"bj0\ni\"bj\"aj 0\n0 0\"cj1\nA; (1)\nwhere\"aj,\"bjand\"cjgenerally depends on the magni-\ntudeHextof magneticfield. When thespatial orientation\nofHextis changed with respect to kxby rotations in\nthe canonical basis (x;y;z)as illustrated in Fig.1, the\nT2 z \nx y Hext \n \n 𝜀 4 \n𝜀𝑠𝑢𝑏 𝜀 3 \n𝜀 2 \n𝜀 1 \nT Figure 1: Sketch of a non-reciprocal metamaterial made with\nmagneto-optical nanostructures of permittivity \u0016\u0016\"jdeposited\non an isotropic substrate of permittivity \"sub. The structure\nisassumedatequilibriumattemperature Twithitssurround-\ning environement and it is submitted to a uniform external\nmagnetic field Hextin an arbitrary angular orientation.\npermittivity tensor associated to each nanostructure is\nreadily derived from expression (1) by simple composi-\ntion with the corresponding rotation matrices.\nThe LDOS \u001a(r;!)of electromagnetic field at a given\npoint r= (x;y;z )and frequency !can be calculated\nfrom the density of energy of electromagnetic field\nu(r;!) =\u000f0+\u00160;(2)\nassociated to the average value of electric field Eand\nmagnetic field Hand using the relation\nu(r;!) =\u001a(r;!)\u0002(!;T); (3)\nbetween this density and the LDOS. Here, \u0002(T;!) =\n~!=[e~!\nkBT\u00001]denotes the mean energy of a harmonic\noscillator at temperature Tof system. When the size of\nnanostructures is much smaller than the thermal wave-\nlength\u0015th=~c=kBTand the separation distance be-\ntween these nanostructures is sufficiently large they be-\nhaves as a simple dipoles [14–16]. The electric field can\nbe related to the dipolar moments piof nanostructuresarXiv:2310.04219v1 [physics.optics] 6 Oct 20232\nx z \n2R \n𝜀𝑠𝑢𝑏 \nFigure 2: LDOS of electromagnetic field at z= 3Rin vac-\nuum above a InSb particle of radius R= 50nmplaced on\na substrate for an external magnetic field Hext(magnitude\nHext= 5T) oriented along the xandzaxis. The substrate\nis a semi-infinite germanium sample ( \"sub= 16). The system\nis at equilibrium at temperature T= 300K. The LDOS is\nnormalized by the LDOS \u001avac=!2\n\u00192c3in vacuum.\nas followed\nE(r) =!2\u00160NX\ni=1GEE(r;ri)pi; (4)\nwith\u00160the vacuum permeability and GEE(r;r0)is the\ndyadic Green tensor between the point randr0inside the\nset of N nanostructures. On the other hand each dipolar\nmoment can be decomposed into the form\npi=pfluc\ni+pind\ni (5)\nwhere the first term of rhs is its fluctuating part while\nthe second term is the part induced by all others dipolar\nand it reads\npind\ni=k2\n0\u0016\u0016\u000biX\nj6=iGEE(ri;rj)pj (6)\n\u0016\u0016\u000bibeing the nanostructure polarizability, k0=!\ncthe\nwavenumber in vacuum and \"0is the vacuum permittiv-\nity. Inserting expression (6) into relation (5) it imme-\ndiately follows the relation beween de dipolar moments\nand their fluctuating part\npi=NX\nj=1TEE;ijpfluc\nj (7)\nwith\nTEE;ij =\u000eij1\u0000(1\u0000\u000eij)k2\n0\u0016\u0016\u000biGEE(ri;rj):(8)\nAu \nHext \nx y \n(a) InSb Au \nHext \nx y InSb \n(b) Figure 3: LDOS of electromagnetic field at z= 3Rin vacuum\nabove a InSb particle of radius R= 50nmplaced in the\ncenterofanetworkmadesixgoldnanoparticlesofsameradius\nregularly arranged (a) on an ellipse with half major axe a=\n5Ralong the xaxis and the half minor axe b= 1:5R. (b)\nLDOS for an external magnetic field Hext(magnitude Hext=\n5T) along the xandyaxis. The substrate is a semi-infinite\ngermanium sample.\nIt follows that the local electric fied can be written in\nterm of fluctuating dipolar moments as [17]\nE(r) =!2\u00160NX\nj=1GEE(r;rj)pfluc\nj; (9)\nwhere the full electric-electric Green tensor takes the\nform\nGEE(r;rj) =NX\ni=1GEE(r;ri)T\u00001\nEE;ij:(10)\nFor an enemble of dipoles in free space the propagator\nreads\nGEE(r0;r00)\u0011GEE\n0(r0;r00) =\nexp(ik0r)\n4\u0019r\u0014\u0012\n1 +ik0r\u00001\nk2\n0r2\u0013\n1+3\u00003ik0r\u0000k2\n0r2\nk2\n0r2br\nbr\u0015\n;\n(11)\nwherebr\u0011r=r,r=r0\u0000r00andr=jrjand1stands for\nthe unit dyadic tensor. When the dipoles are located in3\nvacuum above a solid material the propagator reads\nGEE(r0;r00)\u0011GEE\n0(r0;r00) +GEE;sc(r0;r00);(12)\nwhere the second term of rhs describe the scattering by\nthe interface between the vacuum and material. This\nscattering term can be written as an integral with re-\nspect to the modulus \u0014of the modulus of the parallel\ncomponent \u0014= (ky;kz )of wavevector (i.e. parallel to\nthey\u0000zplane) as [18]\nGEE;sc(r0;r00) =\nZ1\n0d\u0014\n2\u0019\u0014i\n2kxexp(ikxjx+x0j)[rsSEE+rpPEE];(13)\nwherekx=p\nk2\n0\u0000\u00142is the normal component of the\nwavevector in vacuum while rsandrpare the ordinary\nFresnel coefficients of the interface associated to waves in\npolarization sandp, respectively. The scattering matrix\nSandPin (13) can be readily calculated using the Sipe\nformalism [19]. As the magnetic field is concerned, it can\nbe straighforwardly deduced from expression (9) using\nthe Faraday relation\nH=i\n!\u00160r\u0002E: (14)\nIt follows that the correlation functions of electric and\nmagnetic fields reads\n=!4\u00162\n0\n\u0002X\ni;jX\nl;k;nGEE\nlk(r;ri)GEE\u0003\nln(r;rj);(15)\n=\u0000!2\n\u0002X\ni;jX\nl;k;nGHE\nlk(r;ri)GHE\u0003\nln(r;rj);(16)\nwhere GAB\nlkdenotes the lkcomponent of Green tensor\nandpf\ni;kis thekthcomponent of ithfluctuating dipole.\nThe correlations functions of dipolar moments can be\ncalculated using the fluctuation dissipation theorem [20]\nhpf\ni;lpf\u0003\nj;ni=\u000f0\ni!(\u0016\u0016\u000bi;ln\u0000\u0016\u0016\u000b\u0003\ni;nl)\u0002(!;Ti)\u000eij\u000eln;(17)\nwhere \u0002(T;!) =~!=[e~!\nkBT\u00001]is the mean energy of a\nharmonic oscillator at temperature T,\u0016\u0016\u000biis the polariz-\nability associated to the ithdipole and \u000e\u000b\fis the usual\nKronecker symbol. As, the polarizability tensor is con-\ncerned, it can be described, by taking into account the\nradiative corrections, using the following expression [21]\n\u0016\u0016\u000bi(!) =\u0012\n\u0016\u00161\u0000ik3\n6\u0019\u0016\u0016\u000b0i\u0013\u00001\n\u0016\u0016\u000b0i; (18)\nAu \nInSb Au \nInSb \nHext Hext \nx y \nx y \n=/2 =0 (a) \n(b) \n(c) Figure 4: LDOS of electromagnetic field at the center ( z=\n0) of an hexagonal network (a) of gold-InSb nanoparticles\n(R= 50nm) regularly arranged on a circle of radius 3Rfor\ntwo different orientations of an external magnetic field (b)\nHextalong the xandzaxis. The magnitude of this field is\nHext= 5T. (c) LDOS for diifferent magnitude of magnetic\nfield oriented along the xaxis. Same substrate as in Fig.2.\nwhere \u0016\u0016\u000b0idenotes the quasistatic polarizability of the ith\nparticle and k=!=c,cbeing the speed of light in vac-\nuum. For spherical particles in vacuum, the quasistatic\npolarizability takes the simple form\n\u0016\u0016\u000b0i(!) = 4\u0019R3\u0000\u0016\u0016\"\u0000\u0016\u00161\u0001\u0000\u0016\u0016\"+ 2\u0016\u00161\u0001\u00001;(19)\nwhereRistheradiusofparticles. Ofcourseothersshapes\ncan be considered without change of the general formal-\nism introduced previously.\nTo illustrate the potential of non-reciprocal metamate-4\nrials to control the photonic states we show in Figs.3 and\n4 this LDOS in the neighborhood of composite networks\nmade with magneto-optical nanoparticles of indium anti-\nmoniure (InSb) and gold (Au) particles. In this case [22]\n\"a(Hext) =\"1\u0012\n1 +!2\nL\u0000!2\nT\n!2\nT\u0000!2\u0000i\u0000!+!2\np(!+i\r)\n![!2c\u0000(!+i\r)2]\u0013\n;\n\"b(Hext) =\"1!2\np!c\n![(!+i\r)2\u0000!2c];\n\"c=\"1\u0012\n1 +!2\nL\u0000!2\nT\n!2\nT\u0000!2\u0000i\u0000!\u0000!2\np\n!(!+i\r)\u0013\n:(20)\nFor the numerical applications we will assume that\nthe infinite-frequency dielectric constant is \"1= 15:7\n, the longitudinal optical phonon frequency !L= 3:62\u0002\n1013rad s\u00001is , the transverse optical phonon frequency\nis!T= 3:39\u00021013rad s\u00001, the plasma frequency of\nfree carriers of density !p= (ne2\nm\u0003\"0\"1)1=2with the\ndensityn= 1:36\u00021019cm\u00003. The effective mass is\nm\u0003= 7:29\u000210\u000032kg,\"0being the vacuum permit-\ntivity, \u0000 = 5:65\u00021011rad s\u00001is the phonon damping\nconstant,\r= 3:39\u00021012rad s\u00001is the free carrier damp-\ning constant, and !c=eHext=m\u0003is the cyclotron fre-\nquency with ethe electron charge. As for gold with use\nthe Drude model [23]\n\"(!) = 1\u0000!2\np\n!(!+i\r); (21)\nwith!p= 13:71\u00021015rad:s\u00001and\r= 4\u00021013s\u00001.\nWe first consider the simplest case shown in Fig.2\nwhere a single InSb nanoparticle is deposited on a ger-\nmanium (Ge) substrate ( \"sub= 16) and we calculate the\nLDOS above this particle when the external magnetic\nfield is rotated from the direction normal to the substrate\nsurfacetothedirectionparalleltoit. Inthissystemthere\nis no configurational resonance [24], so that the LDOS\nspectrum keep the same shape when the magnetic field\nis rotated. But, the situation radically changes in more\ncomplex networks. Hence, if the InSb particle is placed,\nfor instance, inside an ellipsoidal cavity (Fig.3-a) made\nwith Au particles then, by changing the orientation of\nexternal magnetic field, we see that not only the magni-\ntude of the LDOS is modified but further resonant modes\nappear in the spectrum and they can be shifted by rota-\ntion of external magnetic field[18]. These modes are the\nconfigurational or hybridization modes [25–27] which are\nrelated to the interplay between the different particles\ninside the network.\nIn the second example, plotted in Fig. 4 we show that\nthe LDOS can also be modified in a narrow spectral band\nby changing either the orientation or the magnitude of\nexternal field. The system is a simple hexagonal network\nof gold and InSb particles deposited on a Ge substrate\nas sketch in Fig. 3-a and the LDOS is calculated at the\ncenter of the network. In this case we see that the LDOScan be modulated by about two orders of magnitude at\nthe resonance frequency of the InSb particle [18] simply\nby changing the orientation of Hextbetween the normal\nto the surface and the xaxis and by about one order of\nmagnitude by changing the magnitude of Hext.\nIn conclusion, we have investigated the photonic den-\nsityofstatesnearmagnteo-opticalmetamaterialsinpres-\nence of an external magnetic field and demonstrated that\nthe LDOS can be significantly modified at subswave-\nlength scale by changing the spatial orientation or the\nmagnitude of this field. 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Molesky, A. W.\nRodriguez, NatureReviewsPhysics. 4, 8, 543-559(2022)." }, { "title": "2310.16552v1.DECWA___Density_Based_Clustering_using_Wasserstein_Distance.pdf", "content": "arXiv:2310.16552v1 [cs.LG] 25 Oct 2023DECWA : D ENSITY -BASED CLUSTERING USING WASSERSTEIN\nDISTANCE\nNabil El Malki\nIRIT\nelmalki.nabil@gmail.comRobin Cugny\nIRIT\nrobin.cugny@irit.frOlivier Teste\nIRIT\nolivier.teste@irit.fr\nFranck Ravat\nIRIT\nfranck.ravat@irit.fr\nOctober 26, 2023\nABSTRACT\nClustering is a data analysis method for extracting knowled ge by discovering groups of data called\nclusters. Among these methods, state-of-the-art density- based clustering methods have proven to\nbe effective for arbitrary-shaped clusters. Despite their encouraging results, they suffer to find low-\ndensity clusters, near clusters with similar densities, an d high-dimensional data. Our proposals are\na new characterization of clusters and a new clustering algo rithm based on spatial density and prob-\nabilistic approach. First of all, sub-clusters are built us ing spatial density represented as probability\ndensity function ( p.d.f ) of pairwise distances between points. A method is then prop osed to ag-\nglomerate similar sub-clusters by using both their density (p.d.f ) and their spatial distance. The key\nidea we propose is to use the Wasserstein metric, a powerful t ool to measure the distance between\np.d.f of sub-clusters. We show that our approach outperforms othe r state-of-the-art density-based\nclustering methods on a wide variety of datasets.\n1 Introduction\nClustering methods are popular techniques widely used to ex tract knowledge from a variety of datasets in numerous\napplications [1]. Clustering methods aim at grouping simil ar data into a subset known as cluster. Formally, the\nclustering consists in partitioning a dataset annotated X={x1,...,xn}withn=|X|intocclustersC1,...,Cc, so\nthatX=∪c\ni=1Ci[1].\nIn this paper, we only consider hard clustering [2] accordin g to which ∀i∈[1..c],∀j/ne}ationslash=i∈[1..c],Ci∩Cj=∅. We\nfocus on density-based clustering methods [3] that are able to identify clusters of arbitrary shape. Another interesti ng\nelement in these approaches is that they do not require the us er to specify the number of clusters c. Density-based\nclustering is based on the exploration of high concentratio ns (density) of points in the dataset [4]. In density-based\nclustering, a cluster in a data space is a contiguous region o f high point density separated from other such clusters\nby contiguous regions of low point density [3]. Density-bas ed clustering has difficulties to detect clusters having low\ndensities regarding high-density clusters. Low-density p oints are either considered as outliers or included in anoth er\ncluster. In the same way, near clusters of similar densities are often grouped in one cluster. Moreover, density-based\nclustering does not manage well in high-dimensional data [4 ] because the density is not evenly distributed and may\nvary severely. This paper tackles these challenging issues by defining a new density-based clustering approach.\nAmong the most known density-based algorithms are DBSCAN [5 ], OPTICS [6], HDBSCAN [7], DBCLASD [8],\nand DENCLUE [9]. Historically first, DBSCAN introduces dens ity as a minimum number of points within a given\nradius to discover clusters. Nevertheless, DBSCAN poorly m anages clusters having different densities. OPTICS\naddresses these varying density clusters by ordering point s according to a density measure. HDBSCAN improved\nthe approach by introducing a new density measure and an opti mization function aiming at finding the best clusteringAPREPRINT - OCTOBER 26, 2023\nsolution. Although these approaches solve part of the probl em of varying density clusters, they suffer from unevenly\ndistributed density and high-dimensional datasets. They s till mismanage low-density clusters by tending to consider\nthem as outliers or to merge them into a higher-density clust er.\nDBCLASD introduces a probabilistic approach of the density . DBCLASD assumes that clusters follow a uniform\nprobability law allowing it to be parameter-free [9]. Howev er, it suffers from detecting non-uniform clusters because\nof this strong assumption. As DBCLASD is also a grid-based ap proach, its density calculations are less precise when\nthe dimension increases [4].\nFinally, DENCLUE detects clusters using the probability de nsity function ( p.d.f ) of points in data space. DENCLUE\nextends approaches of clustering such as DBSCAN [5] and K-me ans [9]. Therefore it also inherits from these the\ndifficulty to detect low-density clusters when the density i s not evenly distributed.\nThe common problems of the existing density-based clusteri ng approaches are related to the difficulty of handling\nlow-density clusters, near clusters of similar densities, and high-dimensional data. The other limit concerns their\ninefficiency to properly handle nested clusters of differen t shapes and uneven distribution of densities. In this paper ,\nwe propose a new clustering approach that overcomes these li mitations. Briefly, the contributions of this article\nare as follows: (1) We propose DECWA (DEnsity-based Cluster ing using WAsserstein distance), a hybrid solution\ncombining density and probabilistic approaches. It first pr oduces sub-clusters using the p.d.f defined on pairwise\ndistances. Then, it merges sub-clusters with similar p.d.f and close in distance. (2) We propose to consider every\ncluster as a contiguous region of points where the p.d.f has its own law of probability to overcome the previously\nexplained limitations. (3) We conducted experiments on a wi de variety of datasets, DECWA outperforms state-of-the-\nart density-based algorithms in clustering quality by an av erage of 20%. Also, it works efficiently in high-dimensional\ndata comparing to the others.\n2 Contribution\nTo address the limits described earlier, we propose to consi der a cluster as a contiguous region of points where density\nfollows its own law of probability. Formally, we represent a clusterCiby a set of pairwise distances between the\npoints contained in the cluster Ci. This set is annotated Diand follows any law of probability. Diis computed via any\ndistanced:X×X→R+that has to be at least symmetric and reflexive.\nThe proposed solution consists of four consecutive steps: 1 ) The first step transforms the dataset Xinto ak-nearest\nneighbor graph representation where nodes are points and ed ges are pairwise distances, kis a hyperparameter. The\ngraph is then reduced to its Minimum Spanning Tree (MST) in or der to keep significant distances. 2) The second\nstep consists in calculating the p.d.f from the significant distances of the MST. This p.d.f is used to determines\nthe extrema, from which extraction thresholds are determin ed. 3) The third step consists of extracting sub-graphs\nfrom the MST according to extraction thresholds and identif ying corresponding sub-clusters. 4) The fourth step is\nto agglomerate sub-clusters according to spatial and proba bilistic distances. We opt for the Wasserstein distance to\nmeasure the similarity between probability distributions .\n2.1 Graph-oriented modeling of the dataset\nTo estimate the distance p.d.f , we must first calculate distances between the points of the d ataset. The computation\nstrategy of pairwise distances is based on the k-nearest neighbor method [4].\nTherefore, the first step is the construction of the k-nearest neighbor graph of the dataset. From the dataset X, an\nundirected weighted graph annotated G= (V,E)is constructed. Vis the set of vertices representing data points, and\nEis the set of edges between data points. For each point, we det erminekedges toknearest neighbor points. The\nweight of each edge is the distance between the two linked poi nts.\nTo gain efficiency and accuracy, it is possible to get rid of ma ny unnecessary distances. We favor short distances,\navoiding keeping too large distances in the k-nearest neighbors. This idea tends to obtain dense (connec ted) sub-\ngraphs. To do so, we generate a minimum spanning tree (MST) fr omG. An MST, denoted Gmin, is a connected\ngraph where the sum of the edge weights is minimal. Several al gorithms exist to generate an MST, we used Kruskal\n[10]. As kincreases, the number of connections of each node in graph Ggets bigger too. Consequently, it also\nincreases the number of possible solutions for Kruskal. As a result, the MST is more optimal. This might lead\nDECWA to better clustering results.\n2APREPRINT - OCTOBER 26, 2023\n2.2 Probability density estimation\nThe overall objective is to identify homogeneous regions wh ere linked points have similar distances. We then estimate\nthe distance p.d.f to detect groups of points having a similar spatial density.\nFor estimating the p.d.f , we use the kernel density estimation ( KDE ) [11]. Its interest lies in the fact that it makes\nno a priori hypothesis on the probability law of distances. T he density function is defined as the sum of the kernel\nfunctions Kon every distance. The commonly used kernels are Gaussian, u niform and triangular. The KDE equation\nis below :\n/hatwidefh(a) =1\n(n−1)hn−1/summationdisplay\ni=1K/parenleftBiga−ai\nh/parenrightBig\nIn our case, aandaicorrespond to the distances contained in Gmin. We have n=|X|(number of nodes in Gmin),\nandn−1is the number of distances (number of edges in Gmin).\nThe smoothing factor h∈]0;+∞[is an hyperparameter, called bandwidth. It acts as a precisi on parameter, in our case,\nit influences the number of extrema detected in the p.d.f and therefore, the number of different densities.\n2.3 Graph division\nThe overall objective is to separate different densities, h ence, the next step is to find where to cut the p.d.f . The\nsignificantly different densities are detected with the max ima of the distance p.d.f curve. In order to separate highly\nrepresented distances to less represented distances, we co nsider an extraction threshold as the mid-distance between\neach maximum and its consecutive minimum, and conversely be tween each minimum and its consecutive maximum.\nMid-distances allow capturing regions that are difficult to detect (low-density regions, or dense regions containing f ew\npoints). Another interesting consequence concerns overla pping regions that have the same densities. These regions\nare often very difficult to capture properly, whilst our mid- distance approach makes it possible to detect these cases\nbecause overlapping between regions induces a density vari ation. Once the mid-distances are identified on the p.d.f.\ncurve, we apply a process to treat successively the list of mi d-distances in descending order. We generate sub-clusters ,\nfrom the nodes of Gmin, and sets of distances of sub-clusters, from the edges of Gmin.\nFromGmin, we extract connected sub-graphs where all the nodes that ar e exclusively linked by edges greater than\nthe current mid-distance. A node linked to an edge greater th an the mid-distance and another edge less than the mid-\ndistance is not included. A sub-cluster ( Ci) is composed of points that belong to one connected sub-grap h. For each\nsub-cluster, we produce its associated set of distances ( Di) from edges between nodes of the sub-graph. A residual\ngraph is made up of edges whose distances are less than the mid -distance, and all the nodes linked by these edges.\nThis residual graph is used in the successive iterations. At the last iteration (i.e. the last mid-distance), the residu al\ngraph is also used to generate sub-clusters and associated s ets of distances. After this step, some sub-graphs are close\nto each other while having a similar p.d.f . According to our cluster characteristics, they should be m erged.\n2.4 Agglomeration of sub-clusters\nThis step aims at generating clusters from sub-cluster aggl omeration. The main difficulty of this step is to determine\nwhich sub-clusters should be merged. When the distances bet ween two sub-clusters are close, it is difficult to decide\nwhether or not to merge them. To arbitrate this decision, we u se their density, represented as the distance p.d.f .\nOnly sub-clusters with similar distance p.d.f will be merged. The agglomeration process consists in mergi ng every\nsub-clusters CiandCjthat satisfy two conditions.\nThe first condition is that CiandCjmust be spatially close enough. To ensure this, d(Ci,Cj)≤λmust be respected,\nwithλ∈R+a hyperparameter. It is necessary to determine the distance between sub-clusters ( d(Ci,Cj)) to verify\nthis condition. However, calculating every pairwise dista nce between two sub-clusters is a time-consuming operation .\nTherefore, we propose another solution based on Gmin. We consider that the value of the edge linking sub-graphs\ncorresponding to CiandCjas the distance between CiandCj. Because of the MST structure, we assume that this\ndistance is nearly the minimum distance between the points o fCiandCj.\nThe second condition relates to the similarity of the distan ce distributions DiandDj. The purpose of this condition\nis to ensure ws(Di,Dj)≤α, withα∈R+a hyperparameter, and wsthe Wasserstein distance. We have opted for\nthe Wasserstein [12] distance as a measure of similarity bet ween probability distributions because it can capture smal l\ndifferences on similar density clusters. The values of αandλallow new fusions as they increase, that tends to generate\nfewer clusters and conversely.\n3APREPRINT - OCTOBER 26, 2023\nTable 1: Experimental results\nDECWA DBCLASD DENCLUE HDBSCAN\nDataset dimensions size LD SD ARI outliers(%) ARI outliers(%) ARI outliers(%) ARI outliers(%)\ntwodiamonds 2 800 0.99 0.01 0.95 0.02 1.00 0.00 0.98 0.01\njain 2 373 1.00 0.00 0.90 0.03 0.45 0.06 0.94 0.01\ncluto-t7.10k 2 10000 0.93 0.03 0.79 0.06 0.34 0.00 0.95 0.10\ncompound 2 399 x x 0.93 0.00 0.77 0.06 0.82 0.05 0.83 0.04\npathbased 2 300 x 0.76 0.00 0.47 0.03 0.56 0.00 0.42 0.02\niris 4 150 x 0.84 0.03 0.62 0.07 0.74 0.00 0.57 0.00\ncardiotocography 35 2126 0.52 0.03 0.24 0.17 0.47 0.02 0.08 0.28\nplant 65 1600 x x 0.22 0.00 0.04 0.07 0.14 0.17 0.04 0.38\nGCM 16064 190 x x 0.48 0.03 0.18 0.18 0.24 0.06 0.27 0.27\nnew3 26833 1519 x 0.41 0.03 0.09 0.45 0.08 0.13 0.13 0.27\nkidney_uterus 10936 384 x 0.81 0.00 0.53 0.08 0.56 0.09 0.53 0.26\nAverage 0.72 0.02 0.51 0.11 0.49 0.05 0.52 0.15\nWe introduce now an iterative process for merging sub-clust ers. It consists in traversing Gminby iterating on its edges.\nFor each edge whose nodes are not in the same sub-clusters (e. g.CiandCj), we verify that d(Ci,Cj)≤λand\nws(Di,Dj)≤α. In this case, CiandCjare merged. This is operated by the union of Di, andDjand considering\nthe points of CiandCjas belonging to the same sub-cluster. The graph traversal of Gminis repeated until there are\nno longer merges. The last iteration does not result in any me rging.\n3 Experiments\nWe conducted an experimental study to show the effectivenes s and the robustness of DECWA compared to state-of-\nthe-art density-based methods. It was applied to a variety o f synthetic and real datasets, using different distances\ndepending on the data. The synthetic datasets, as well as the Decwa advantages presentation, are available on this\nlink1.\n3.1 Experiment protocol\n3.1.1 Algorithms.\nDECWA was compared to DBCLASD [8], DENCLUE [9] and HDBSCAN2[7]. For the experiments conducted in this\npaper, DECWA used Gaussian kernel in division phase and perf ormed only one graph traversal in the agglomeration\nphase. Algorithm parameter values are defined through the re petitive application of the random search approach (1000\niterations). This, in order to obtain the best score that the four algorithms can have. DBCLASD (parameter-free and\nincremental) was subject to the same approach but by randomi zing the order of the data points because it is sensitive to\nit. Clustering quality is measured by the commonly used metr ic named Adjusted Rand Index ( ARI ) [13]. In addition,\nwe also report the ratio of outliers produced by each algorit hm for each dataset.\n3.1.2 Synthetic datasets.\nThe two-dimensional diamond, jain, cluto-t7.10k, compoun d, and pathbased datasets are synthetic. They contain\nclusters of different shapes and densities. For these data, the Euclidean distance was used.\n3.1.3 Real datasets.\nThe cardiotocography dataset [14] is a set of 2126 fetal card iotocogram records, with 35 attributes providing vital\ninformation on the state of the fetus, grouped into 10 classe s. Canberra distance was applied to it. Plant dataset [15]\nconsists of 1600 leaves of 100 different classes. Each leaf i s characterized by 64 shape measurements retrieved from\nits binary image. The Euclidean distance was used on Plant. A nother known dataset, Iris [16] (150 iris flowers, 4\ndimensions, 3 classes) was used with the Manhattan distance . A collection of two very large biological datasets were\n1https://github.com/nabilEM/DECWA\n2The hierarchical structure proposed by HDBSCAN is exploite d by the Excess Of Mass (EOM) method to extract clusters, as\nused by the authors in [7].\n4APREPRINT - OCTOBER 26, 2023\ntested. The first dataset, GCM [17], is used to diagnose the ty pe of cancer (14 classes). It consists of 190 tumor\nsamples. Each one represented by the expression levels of 16 063 genes. The second dataset (kidney_uterus) [18]\nconsists of 384 tumor samples to be classified into two classe s (kidney or uterus). Each sample is the expression level\nof 10937 genes. Given that the attributes all correspond to t he same nature (gene expression) then the Bray-Curtis\ndistance was applied to it. New3 [19], a very high-dimension al dataset, is a set of 1519 documents (6 classes). Each\ndocument is represented by the term-frequency vector of siz e 26833.The cosine distance was used to measure the\nsimilarity between documents.\n3.2 Results and discussion\nThe results are reported in the table 1 (best scores are in bol d).LDcolumn stands for low-density, it means that the\ndatasets have at least one low-density cluster comparing to others.SDstands for near clusters of similar densities,\nit means that the marked datasets have at least two overlappi ng clusters with similar densities. These were detected\naccording to the intra-cluster and inter-cluster distance using ground truth.\nIn many cases, DECWA outperforms competing algorithms by a l arge margin on average 20% (e.g. jaindataset, ARI\nmargin is 1−0.45 = 0.55). DECWA has the best results in datasets with a low-density c luster (e.g. compound and\nGCM ). In this case, there is an ARI margin of 21% on average in favo r of DECWA. Though datasets with very high\ndimensions are problematic for the other algorithms, DECWA is able to give good results. Indeed, there is an ARI\nmargin of 30% on average in favor of DECWA for the last three da tasets. Near clusters of similar densities are also\ncorrectly detected by DECWA. Some datasets like iris, kidney_uterus andpathbased have overlapping clusters and\nyet DECWA separates them correctly. There is an ARI margin of 25% in favor of DECWA for datasets having this\nproblematic.\nThe outlier ratio is not relevant in case of a bad ARI score bec ause in this case, although the points are placed in\nclusters, clustering is meaningless. DECWA is the one that r eturns the least outliers on average while having a better\nARI score.\nWe conducted a statistical study, as recommended in [20], to confirm the significant difference in performance between\nthe algorithms and the robustness of DECWA comparing to the o thers. The overall concept of the study has two steps.\n1) First, a statistical test (Iman-Davenport[21]) is perfo rmed to determine if there is a significant difference betwee n\nthe algorithms at the ARI level. 2) If so, a pairwise comparis on of algorithms is performed, via a post-hoc test (Shaffer\n[22]), to identify the gain of one method over the others. Bot h tests return a p-value (in the case of the second test, it\nis returned for each comparison). The p-value allows us to de cide on the rejection of the hypothesis of equivalence\nbetween algorithms. To reject the hypothesis, the p-value m ust be lower than a significance threshold sthat we set at\n0.01. The p-value by returned the first test is 5.38e−5. It is significantly small compared to s. This means that the\nalgorithms are different in terms of ARI performance. Secon d, these differences are analyzed by the Shaffer Post-hoc\ntest. It returns a p-value for each test on a pair of algorithm s. Indeed, for the case of DECWA, the p-value is much\nlower than swhen comparing DECWA to others ( 3.6e−4with DBCLASD, 3.0e−3with DENCLUE and 1.0e−3with\nHDBSCAN), which proves that DECWA is significantly differen t from the others. For the others, the p-value is equal\nto1.0in all the tests concerning them, which statistically shows that they are equivalent. The ranking of the algorithms\naccording to the ARI was done by Friedman’s aligned rank [23] . DECWA is ranked as the best. All in all, DECWA is\nsignificantly different from the others and is the best perfo rming.\n3.3 CONCLUSION AND PERSPECTIVES\nWe proposed DECWA, a clustering algorithm based on spatial a nd probabilistic density adapted to high-dimensional\ndata. We introduced a new cluster characterization to allow efficient detection of clusters with different densities.\nExperiments were performed on various datasets and we showe d statistically that DECWA outperforms state-of-the-\nart density-based methods. Our future research integrates the application of DECWA in specific domains and on\ncomplex data as multidimensional time series.\nReferences\n[1] Anil Jain. Data clustering: 50 years beyond k-means. Pattern Recognition Letters , 31:651–666, 06 2010.\n[2] Sergios Theodoridis and Konstantinos Koutroumbas. Cha pter 14 - clustering algorithms iii: Schemes based on\nfunction optimization. In Pattern Recognition (Fourth Edition) , pages 701 – 763. Academic Press, fourth edition\nedition, 2009.\n5APREPRINT - OCTOBER 26, 2023\n[3] Hans-Peter Kriegel, Peer Kröger, Joerg Sander, and Arth ur Zimek. Density-based clustering. Wiley Interdisc.\nRew.: Data Mining and Knowledge Discovery , 1:231–240, 05 2011.\n[4] Charu C. Aggarwal and Chandan K. Reddy. Data Clustering: Algorithms and Applications . Chapman &\nHall/CRC, 1st edition, 2013.\n[5] Martin Ester, Hans-Peter Kriegel, Joerg Sander, and Xia owei Xu. 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Davis, Keh-Shin Lii, and Dimitris N. Politis . Remarks on Some Nonparametric Estimates of a Density\nFunction. In Selected Works of Murray Rosenblatt , pages 95–100. Springer New York, 2011.\n[12] Cédric Villani. Optimal transport : old and new . Springer, 2009.\n[13] L. Hubert and P. Arabie. Comparing partitions. Journal of classification , 2(1):193–218, 1985.\n[14] Diogo Ayres-de Campos, João Bernardes, Antonio Garrid o, Joaquim Marques-de Sá, and Luis Pereira-Leite.\nSisporto 2.0: A program for automated analysis of cardiotoc ograms. The Journal of Maternal-Fetal Medicine ,\n9(5):311–318, 2000.\n[15] Charles Mallah, James Cope, and James Orwell. Plant lea f classification using probabilistic integration of shape,\ntexture and margin features. Pattern Recognit. Appl. , 3842, 02 2013.\n[16] Dheeru Dua and Casey Graff. UCI machine learning reposi tory, 2017.\n[17] Sridhar Ramaswamy, Pablo Tamayo, Ryan Rifkin, Sayan Mu kherjee, Chen-Hsiang Yeang, Michael Angelo,\nChristine Ladd, Michael Reich, Eva Latulippe, Jill P. Mesir ov, Tomaso Poggio, William Gerald, Massimo Loda,\nEric S. Lander, and Todd R. Golub. Multiclass cancer diagnos is using tumor gene expression signatures. PNAS ,\n98:15149–15154, 2001.\n[18] Gregor Stiglic and Peter Kokol. Stability of ranked gen e lists in large microarray analysis studies. Journal of\nbiomedicine & biotechnology , 2010:616358, 06 2010.\n[19] Eui-Hong Han and George Karypis. Centroid-based docum ent classification: Analysis and experimental results.\nInProceedings of the 4th European Conference on Principles of Data Mining and Knowledge Discovery , page\n424–431. Springer-Verlag, 2000.\n[20] Janez Demsar. Statistical comparisons of classifiers o ver multiple data sets. Journal of Machine Learning\nResearch , 7:1–30, 01 2006.\n[21] David J. Sheskin. 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Journal of Machine Learning Research - JMLR , 9, 12 2008.\n6" }, { "title": "2310.16912v1.Transformer_based_Atmospheric_Density_Forecasting.pdf", "content": "Transformer-based Atmospheric Density Forecasting\nJulia Briden *, Peng Mun Siew†,\nMassachusetts Institute of Technology\nVictor Rodriguez-Fernandez‡,\nUniversidad Polit ´ecnica de Madrid\nRichard Linares§\nMassachusetts Institute of Technology\nABSTRACT\nAs the peak of the solar cycle approaches in 2025 and the ability of a single geomagnetic storm to significantly\nalter the orbit of Resident Space Objects (RSOs), techniques for atmospheric density forecasting are vital for space\nsituational awareness. While linear data-driven methods, such as dynamic mode decomposition with control (DMDc),\nhave been used previously for forecasting atmospheric density, deep learning-based forecasting has the ability to\ncapture nonlinearities in data. By learning multiple layer weights from historical atmospheric density data, long-term\ndependencies in the dataset are captured in the mapping between the current atmospheric density state and control input\nto the atmospheric density state at the next timestep. This work improves upon previous linear propagation methods\nfor atmospheric density forecasting, by developing a nonlinear transformer-based architecture for atmospheric density\nforecasting. Empirical NRLMSISE-00 and JB2008, as well as physics-based TIEGCM atmospheric density models\nare compared for forecasting with DMDc and with the transformer-based propagator.\n1. INTRODUCTION\nThermospheric mass density serves as the largest source of uncertainty for low Earth orbit (LEO) satellite orbit pre-\ndiction. This uncertainty is largely due to fluctuations in solar and geomagnetic activity that can occur on the order of\nhours. Solar and geomagnetic storms take place when charged particles, usually from coronal mass ejections, travel to\nEarth’s atmosphere and increase atmospheric heating and transient solar wind activity. These storms are often marked\nby high nonlinearities in the evolution of atmospheric density over time and, therefore, prove to be difficult to predict\nwith even the most advanced forecasting techniques. With the ability of a single geomagnetic storm to significantly\nalter the orbit of satellites, techniques for atmospheric density forecasting are vital for space situational awareness.\nCurrent forecasting methods use physics-based atmospheric density models, such as the Global Ionosphere-Thermosphere\nModel (GITM) and the Thermosphere-Ionosphere-Electrodynamics General Circulation Model (TIE-GCM), which\nsolve the full continuity, energy, and momentum equations required for the propagation of atmospheric density. While\nthese models are ideal for short-term forecasting, the computational cost required to run physics-based models exceeds\ncomputing capabilities for most real-time applications. Conversely, empirical models, including NRLMSISE-00 and\nJB2008, are fast to evaluate, since they are derived only from measurements, such as total mass density, tempera-\nture, and oxygen number density. Unfortunately, empirical models cannot translate their computational efficiency\nto atmospheric density prediction, since they lack the underlying dynamics required for forecasting. Recent work\nin reduced-order modeling (ROM) and dynamic mode decomposition with control (DMDc) has addressed this gap in\ncomputationally efficient atmospheric density prediction methods; ROMs use proper orthogonal decomposition (POD)\nor convolutional autoencoder-based machine learning (ML) to reduce the dimensionality of physics-based models and\ngenerate reduced-order snapshots of empirical models to enable atmospheric density propagation. To propagate a\nreduced-order atmospheric density state forward in time, DMDc models atmospheric density as a linear dynamical\n*Ph.D. Student, Department of Aeronautics and Astronautics. E-mail: jbriden@mit.edu\n†Postdoctoral Associate, Department of Aeronautics and Astronautics. E-mail: siewpm@mit.edu\n‡Associate Professor, Department of Computer Systems Engineering. E-mail: victor.rfernandez@upm.es\n§Associate Professor, Department of Aeronautics and Astronautics. E-mail: linaresr@mit.eduarXiv:2310.16912v1 [physics.ao-ph] 25 Oct 2023system with a control input, where the dynamics and input matrices are estimated from a dataset of state snapshots.\nWhile this approach captures the nominal atmospheric density dynamics relatively well, DMDc often fails when mildly\nnonlinear conditions occur [1].\nThe effectiveness of using a machine learning approach for modeling the nonlinear dynamics in atmospheric density\nhas been assessed by Turner et al. [2]. By utilizing a deep feedforward neural network (NN) for atmospheric density\nforecasting, an error reduction of over 99 %, when compared to DMDc, was achieved. While effective for short-\nterm forecasts, prediction performance was hindered by the inability of the feedforward NN to optimize with data\nfrom previous time steps. Moreover, the challenge of atmospheric density forecasting during a significant space\nweather event requires an algorithm that can capture long-term dependencies in the dataset to prevent compounding\npropagation errors. The NN must handle large inputs for the atmospheric density state and only focus on the relevant\npart of the input for accurate forecasting. With an attention component, when compared to Recurrent Neural Networks\n(RNNs), transformers train on sequential data in less time and with longer inputs by using a mechanism known as\nattention [3]. By passing all hidden states, derived from the encoded sequential atmospheric density reduced-order\nstates and space weather input, to the transformer NN, the transformer can focus its attention on only the most relevant\nhidden states. Since the occurrence of a solar or geomagnetic storm represents an anomaly in the atmospheric density\ndynamical system, the transformer’s ability to focus attention on only the relevant propagation dynamics is imperative\nfor fast and accurate atmospheric density forecasting.\nAfter reducing the dimensionality of the current atmospheric density state using a POD or ML ROM, the transformer-\nbased atmospheric density forecasting algorithm takes the current and past reduced-order states and the current space\nweather indices as an input to the transformer’s encoder network and generates a new reduced-order atmospheric\ndensity state at the next time step. Where the encoder generates a set of embeddings for the control input and the\ntime series reduced-order density states, then outputs the predicted reduced-order density states using the encoder’s\nembeddings. The training process includes dataset splits for low, medium, and high levels of space weather activity\nto improve generalization performance. The final transformer propagation model provides a mapping between time-\nseries reduced-order states and control input to the next predicted state, serving as a surrogate dynamical system.\nEmpirical NRLMSISE-00 POD ROM, JB2008 POD ROM, and JB 2008 ML ROM, as well as physics-based TIEGCM\nPOD ROM atmospheric density models are compared for forecasting with DMDc and with the transformer-based\npropagator. With almost 5,500 active satellites currently in orbit and the predicted launch of an additional 58,000 by\n2030, forecasting the nonlinear dynamics of space weather-induced changes in atmospheric density are essential for\nResident Space Object (RSO) deorbit prediction and collision avoidance.\n2. THEORY\nTo achieve a data-driven model of the global atmospheric density dynamical system, two steps are completed; first,\nthe full-order atmospheric density is reduced in space, either using proper orthogonal decomposition (POD) or a\nmachine learning-based (ML) convolutional autoencoder. Then the reduced-order snapshot is propagated forward\nusing dynamic mode decomposition with control (DMDc) or a transformer neural network. In the following section,\nPOD and ML spatial reduction theory, as well as DMDc forecasting, will be reviewed. For additional information on\nthese modeling methods, refer to [4, 5, 6, 7, 8]. Then transformer-based forecasting for time series data is covered.\n2.1 Reduced-Order Modeling\nIn the following subsection, we cover the development of the various reduced-order models used in the study.\n2.1.1 Proper Orthogonal Decomposition\nThe variation in atmospheric mass density is defined as ˜x(s,t) =x(s,t)−¯x(s). By subtracting the mean atmospheric\ndensity, ¯x(s)for a spatial grid and time index, from the true atmospheric mass density, x. Then, the reduced-order\nstate, z, can be constructed using the first r POD modes from the singular value decomposition, Ur, resulting in Eqn.\n1:\nz=U−1\nr˜x=UT\nr˜x. (1)When the full-dimensional state is desired, the first r POD modes can be multiplied by the reduced-order state while\nadding back the mean atmospheric density:\nx(s,t)≈Ur(s)z(t) +¯x(s). (2)\nFor additional information on the applications of the POD algorithm for thermospheric mass density prediction see\nMehta and Linares (2017) [6].\n2.1.2 Machine Learning-based Dimensional Reduction\nThe other dimensionality reduction technique employed in this work is the undercomplete convolutional autoencoder\nneural network [9]. Where the term undercomplete defines the encoded dimension as less than the input dimension.\nThe encoder, F, maps the input vector to a reduced-order vector:\nF:X→V. (3)\nWhile the decoder, maps a reduced-order vector to its full-dimensional form:\nG:V→X′. (4)\nTo assess the accuracy of these learned mappings, a mean squared error loss function is used for this work:\nL(X,X′) =||X−X′||2\nd. (5)\nWhere ddefines the dimension of the input vector and its reconstructed dimension. The convolutional autoencoder\nutilizes the 2D convolution for the output, g, and the input, x:\ngi,j,k,c=σ(wT\ncxi,j,k,c). (6)\nWhere gi,j,k,cis a data in the channel output and c is the filter. While training, proximal policy optimization (PPO)\nwas utilized for parameter (weights and biases) tuning. See Loffe et al. (2015) for additional information on the\nconvolutional autoencoder neural network and Briden et al. (2022) for the full neural network architecture used in this\nwork [7, 4].\nCompared to the linear POD, the ML dimensionality reduction approach has the benefit of capturing nonlinear features\nin the dataset. This benefit comes at a risk for overfitting to training data, which is mitigated by including dropout in\nthe convolutional autoencoder model used in this work.\n2.1.3 Dynamic Mode Decomposition with Control\nOnce either POD or ML are used to reduce the dimensionality of the atmospheric density state, to achieve forecasting\nfor a later time, a propagation model is formulated. In previous work, Dynamic Mode Decomposition with control\n(DMDc) was utilized to learn a linear dynamical system model from data [4]. The future state at time index k+1 is\nconstructed using Eqn. 7:\nzk+1=Azk+Buk (7)\nwhere zkis the reduced-order state at time index k, as defined in Eqn. 1 or Eqn. 3), and uis the set of space weather\nindices. The matrices, AandBare estimated using the method of least squares given the time-shifted snapshot\nmatrices for a fixed timestep, T. To convert the discrete-time dynamics into continuous-time dynamics, the relation\nfrom DeCarlo (1989) is utilized [8].\u0014\nAcBc\n0 0\u0015\n=log(\u0014\nA B\n0 I\u0015\n)/T. (8)\nAfter propagating for the desired time length, either Eqn. 2, for POD, or Eqn. 4, for the Autoencoder NN, can be used\nto project the predicted reduced-order atmospheric density to the full dimensional state.\nSimilar to the trade-offs between POD and ML for dimensionality reduction, there exists use cases for development\nof a propagation method which captures the nonlinearities in the atmospheric density dynamics. Furthermore, noise\nsensitivity serves as a prominent challenge in DMD [1]. This challenge can inhibit DMDc propagation for long\ntime periods. By constructing a transformer-based NN propagator, current challenges in robust nonlinear propagation\nduring significant space weather events can be addressed.\n2.1.4 Transformer Propagator\nTo construct the transformer forecaster, the channel-independent patch time series Transformer (PatchTST) architec-\nture was utilized [10]. By using a patching and channel-independence design, local semantic information in the time\nseries is preserved and model complexity can be reduced.\nDue to channel-independence, the transformer forecaster diverges from the DMDc linear dynamical system model;\nboth reduced-order atmospheric density states and the space weather inputs are combined into one state snapshot vec-\ntor. While this architecture enables prediction of both atmospheric density and space weather indices, the architecture\nno longer assumes atmospheric density’s dependence on the space weather input. Future work will focus on expanding\nthe Transformer propagator to include a graph neural network to learn the cross-channel relationships between space\nweather indices and atmospheric density. Although only the channel-independent transformer propagation technique\nis assessed in this work, this simplified framework serves to improve robustness to noise; channel independence miti-\ngates the possibility of projecting noise from one input channel into another input channel. A significant improvement\nfrom the noise-sensitive DMDc.\nTo propagate the state snapshot forward in time, the transformer backbone, H, maps the reduced-order state vector,\nz1:L= (z1,...,zL), to the predicted reduced-order state, ˆzL+1:L+T= (ˆzL+1,...,ˆzL+T):\nH:Z→ˆZ. (9)\nWhere Lis look-back window length to provide predictions based on and Tis the forecast window length for the\ntransformer NN to output. Rather than predicting the next density snapshot index from the previous density snapshot\nindex at a fixed rate, as DMDc does, the PatchTST architecture allows for an increase in information utilized to\ngenerate a prediction and a longer prediction window for one forward pass, resulting in potentially significant accuracy\nimprovements, as well as fast predictions for long propagation windows.\nThrough patching, each input time series is divided into patches that can overlap or not overlap. The patch length\nis denoted as Pand the stride, or non-overlapping region between two patches, is S. Then the number of patches is\nN=⌊L−P\nS⌋+2. Patching reduces memory usage and computational complexity of the attention map quadratically\nby a factor of S[10]. Since this allows the model to view longer historical data, forecasting performance can be\nsignificantly improved.\nThe encoder used in this work is a vanilla Transformer encoder that maps the observed signals to the latent represen-\ntations [10]. The patches are mapped to the Transformer latent space of dimension Dvia a trainable linear projection\nWp∈RD×P, and a learnable additive position encoding Wpos∈RD×Nis applied to monitor the temporal order of\npatches:\nz(i)\nd=Wpz(i)\np+Wpos (10)\nwhere z(i)\nd∈RD×Ndenotes the input that will be fed into the Transformer encoder. Then each head h=1,...,Hin\nmulti-head attention will transform them into query matrices Q(i)\nh= (z(i)\nd)TWQh, key matrices K(i)\nh= (z(i)\nd)TWKh, andvalue matrices V(i)\nh= (z(i)\nd)TWVh, where WQh,WKh∈RD×dkandWVh∈RD×D. After that, a scaled production is used\nfor getting attention output O(i)\nh∈RD×N:\n(O(i)\nh)T=Attention (Q(i)\nh,K(i)\nh,V(i)\nh) =Softmax \nQ(i)\nh(K(i)\nh)T\n√dk!\nV(i)\nh(11)\nThe multi-head attention block also includes BatchNorm1 layers and a feedforward network with residual connections\n[7]. Afterwards, it generates the representation denoted as z(i)∈RD×N. Finally, a flatten layer with a linear head is\nused to obtain the prediction result ˆ z(i)= (ˆz(i)\nL+1,...,ˆz(i)\nL+T)∈R1×T.\nAs in Section 2.1.2, MSE loss (Eqn. 5) is used to measure prediction accuracy. Finally, instance normalization is\nutilized in the transformer architecture to mitigate distribution shift between training and test data. For additional\ndetails on the PatchTST architecture, see Nie et al. (2023) [10, 11].\n3. METHODS\nTo assess the forecasting ability of the transformer NN propagator, test datasets consisting of high, medium, and low\nspace weather activity were constructed. POD ROMs for the JB2008, NRLMSISE-00, and TIEGCM atmospheric\ndensity models and an ML ROM for the JB2008 atmospheric density model from Briden et al. (2022), along with\ntheir associated space weather inputs, are the models used in this study [4]. The new transformer propagation method\nis compared to DMDc.\n3.1 Space Weather Inputs\nSpace weather activity in the form of space weather drivers were used for the division of training, validation, and test\ndata for the transformer propagator, as well as inputs for the DMDc propagator. Space weather indices serve as a\nproxy for levels of space weather activity. Table 1 shows the list of space weather indices used by the atmospheric\ndensity models in this work.\nTable 1: Space Weather Inputs\nIndex Description Models Used\nF10.7Solar radio noise flux at a JB2008, POD TIE-GCM ROM,\nwavelength of 10.7 cm NRLMSISE-00\nS10Activity indicator of the integratedJB200826–34 nm solar irradiance\nM10 Modified daily Mg II core-to-wing ratio JB2008\nDSTDTCTemperature change calculated fromJB2008the Disturbance Storm Time (DST) Index\nkp3-hour-range standardizedPOD TIE-GCM ROMquasi-logarithmic magnetic activity\nap3-hourly magnetic activity indexNRLMSISE-00derived from kp\nFor ROM propagation with DMDc, additional future (next-hour) and nonlinear space weather indices were used and\nare shown in Table 2. The JB2008 model uses time-lag solar flux indices, each with a different time lag. For the\nJB2008 model, future indices are defined as next-day values for each of the time-lag solar flux indices. Where F10 and\nS10 have a one day lag, M10 has a two day lag, and Y10 has a 5 day lag. These delays exist in the F10 index for the\nNRLMSISE model as well. As demonstrated in [12], the usage of future and nonlinear space weather indices improve\nthe accuracy of the DMDc prediction. These nonlinear space weather indices were used as inputs for all ROM models.\n3.2 Transformer Propagator Training\nTo develop training, validation, and test data, POD ROMs for the JB2008, NRLMSISE, and TIEGCM atmospheric\ndensity models and an ML ROM for the JB2008 atmospheric density model were split based on the level of solar\nactivity, as shown in Figure 1.Table 2: Additional Space Weather Inputs Used for the DMDc Models\nROM Model Standard input Future input Nonlinear input\nJB2008doy, hr, F10.7,F10.7,S10,S10,M10,M10,F10.7,S10,M10 DSTDTC2,\nY10,Y10,DSTDTC ,GMST ,αsun,δsun Y10,DSTDTC F10.7·DSTDTC\nTIE-GCM doy, hr, F10.7,F10.7,kp F10.7,kp kp2,kp·F10.7\nNote. doy = day of year; hr = hour in UTC; GMST = Greenwich Mean Sidereal Time. Overbars indicate the\n81-day average. Nonlinear inputs are constructed using both current inputs and future inputs.\nFig. 1: Grouped solar activity levels for the F10B solar index. Where blue is high solar activity, orange is medium\nsolar activity, and green is low solar activity.\nThe F10B solar index was used for JB2008 and the F10a solar index was used for NRLMSISE and TIEGCM solar\nactivity grouping. Both of these indices correspond to 81-day centered averages for the solar radio noise flux at a\nwavelength of 10.7 cm.\nThen each of these three space weather activity datasets were split for training, validation, and test data. The splits\nincluded 20% test data and 10% validation data. A one week forecast horizon and a 42 day look-back window were\nchosen for the transformer propagator. After each of the three datasets (high, medium, and low solar activity) were\ndivided into training, validation, and test data, a combined dataset was created by concatenating the forecasting splits\nfor all levels of solar activity. Figure 2 shows the first JB2008 POD reduced-order state for the corresponding training,\nvalidation, and test splits.\nFig. 2: Combined training, validation, and test data for the first atmospheric density POD reduced-order state.\nWhile training, validation, and test datasets are not sequential in time, this structure allows for the training process tohave representative samples from a range of solar activity levels, while having non-overlapping test samples to evaluate\nperformance for a variety of test samples. Since only one solar cycle of model data is available, this split structure is\nrequired for maximum forecasting performance. Furthermore, none of the model variables indicate the timestamp of\nthe sample. Even though this split structure does not represent a sequential use of the model, time information is not\nleaked when mixing future and past samples between training, validation and test.\nThe model architecture for the JB2008 POD transformer propagator is included in Table 3. The transformer propagator\nfor each model, including POD ROMs for the JB2008, NRLMSISE, and TIEGCM atmospheric density models and\nan ML ROM for the JB2008 atmospheric density model, have the same architecture: 1 encoder layer, 2 heads, 20-\ndimensional model, 28-dimensional fully connected netork, no attention dropout, 0.2 dropout applied to all linear\nlayers in the encoder except q, k, and v projections, a patch length of 9, and a stride length of 1.\nLayer (type) Output Shape Param # Trainable\nRevIN 1 x 34 x 168 68 True\nReplicationPad1d 1 x 34 x 1009 - -\nUnfold 1 x 9 x 1001 - -\nLinear 1 x 34 x 1001 x 20 200 True\nDropout - - -\nLinear 1 x 34 x 1001 x 20 420 True\nLinear 1 x 34 x 1001 x 20 420 True\nLinear 1 x 34 x 1001 x 20 420 True\nDropout - - -\nLinear 1 x 34 x 1001 x 20 420 True\nDropout - - -\nDropout - - -\nTranspose 1 x 20 x 1001 - -\nBatchNorm1d 1 x 20 x 1001 40 True\nTranspose 1 x 1001 x 20 - -\nLinear 1 x 1001 x 28 588 True\nGELU - - -\nDropout - - -\nLinear 1 x 1001 x 20 580 True\nDropout - - -\nTranspose 1 x 20 x 1001 - -\nBatchNorm1d 1 x 20 x 1001 40 True\nTranspose 1 x 1001 x 20 - -\nFlatten 1 x 34 x 20020 - -\nLinear 1 x 34 x 168 3363528 True\nTable 3: PatchTST Model Structure: 3,366,724 total parameters (Input shape: 1 x 34 x 1008).\nBoth the DMDc ROMs and transformer ROMs were trained on the same dataset, shown in Figure 2 in blue, which\nincludes data from low, medium, and high levels of space weather activity. Then for testing, the DMDc and transformer\npropagators were both evaluated on different levels of space weather activity to determine model performance under\nrepresentative space weather conditions.\n4. RESULTS\nTo evaluate the performance of the transformer forecasting approach, compared to the DMDc forecasting approach, the\nmean-squared-error (MSE) and mean-absolute-error (MAE) of model predictions were computed for 1 week predic-\ntion windows on the low, medium, and high solar activity test datasets. Plots were generated to show the propagation\nof MSE over 1 week periods and the DSTDTC and Kp weather inputs are plotted to show changes in space weather\nactivity. Instead of plotting F10, DSTDTC was chosen, since it proved to be a better indicator of short-term spaceweather activity. The DMDc JB2008 models use DSTDTC as an input and the DMDc TIEGCM model uses Kp as an\ninput. NRLMSISE uses ap as a input, which is derived from Kp.\nThe following plots show a comparison of DMDc and transformer-based propagation across all models for low solar\nactivity (Figure 3), medium solar activity (Figure 4), and high solar activity (Figure 5). Each hour in the forecast is\nrepresented by one data point.\nIn the low solar activity plot (Figure 3), the JB2008 Transformer ML ROM maintains the lowest MSE, remaining\nless than 2 for the entire propagation period, closely followed by the JB2008 Transformer POD ROM. While the\nJB2008 transformer models remain relatively stable over the propagation period, the NRLMSISE Transformer POD\nROM and TIEGCM Transformer POD ROM have larger spikes in MSE. On day 6, the NRLMSISE Transformer POD\nROM reaches almost 5 MSE and the TIEGCM Transformer POD ROM spikes at over 11 MSE. Additionally, a small\nspike in error occurs for the JB2008 transformer models on the same day. Upon observing DSTDTC, there is a spike\nin geomagnetic activity occurring around the same time. While the transformer models all dominated their DMDc\ncounterparts for each model, both the JB2008 DMDc ML ROM and JB2008 DMDc POD ROM maintain relatively\nlow MSE, of less than 2, for the duration of the forecast. The TIEGCM DMDc POD ROM is shown to have a long\nspike in MSE from 1 to 3 days after simulation start, jumping from less than 2 MSE to up to 9 MSE. Finally, the\nNRLMSISE DMDc POD ROM demonstrates a consistent cyclic error increase over the propagation window, reaching\nover 13 MSE on the last day.\nFig. 3: MSE for one week propagation on low space weather activity test data.\nThe medium solar activity plot (Figure 4) shows all models maintaining less than 2.5 MSE over the entire propagation\nwindow. Since this maximum error is much less than the 14 MSE maximum from the low solar activity plot, the errors\nin this plot appear to spike more. The NRLMSISE Transformer POD ROM maintains a consistent less than 0.25 MSE\nover the forecast window, with the JB2008 Transformer ML ROM maintaining a similar error range, except for the\nspike to almost 0.75 MSE before day 1. The JB2008 Transformer POD ROM and TIEGCM Transformer POD ROM\nshow much larger spikes, ranging from 0.75-1.4 MSE. For the DMDc models, the NRLMSISE DMDc POD ROM,\nsimilarly to the NRLMSISE transformer, dominates the DMDc models in MSE for much of the forecast window.\nBoth the JB2008 DMDc ML ROM and the JB2008 DMDc POD ROM grow in MSE almost monotonically, while the\nTIEGCM DMDc POD ROM fluctuates from 0 to 2.5 MSE over the entire forecast window.\nLastly, the high solar activity plot (Figure 5) illustrates differing performance between the JB2008 and NRLMSISE\nDMDc models and the TIEGCM DMDc model and transformer models. While the JB2008 DMDc ML ROM, JB2008\nDMDc POD ROM, and the NRLMSISE DMDc POD ROM increase in MSE for the propagation period, the TIEGCM\nDMDc POD ROM, JB2008 Transformer ML ROM, JB2008 Transformer POD ROM, NRLMSISE Transformer PODFig. 4: MSE for one week propagation on medium space weather activity test data.\nROM, and TIEGCM Transformer POD ROM maintain about constant MSE, of around 1.25 MSE over the propagation\nperiod. Compared to the low solar activity plot, it is apparent that the JB2008 DMDc propagated models do not\nmaintain less than 2 MSE performance when applied in high solar activity conditions. TIEGCM DMDc POD ROM,\nthe only physics-based reduced-order model in this study, is the only DMDc ROM which maintains less than 2 MSE\nat high space weather activity conditions. Interestingly, the TIEGCM DMDc POD ROM had large MSE spikes in low\nspace weather conditions. Out of the dominating transformer models, the JB2008 Transformer POD ROM, JB2008\nTransformer ML ROM, and NRLMSISE Transformer POD ROM exhibit comparable performance, of less than 1.25\nMSE for the propagation window. The TIEGCM Transformer POD ROM had an error spike of over 2.5 MSE 1 day\nafter simulation start, which occurred after a steep rise in the Kp index.\nOverall, errors often accumulate quickly over the one week time period for the DMDc propagated ROMs, while only\nspiking at specific timesteps for the transformer-based ROMs. Additionally, the transformer forecasters consistently\ndominated the DMDc forecasting method for all levels of space weather activity. With the high solar activity test\ncase showing the most significant difference between DMDc and the transformer-based approach; while the JB2008\nDMDc models were comparable to all transformer-based approaches for low space weather activity, they were no\nlonger within 2 MSE of the transformer models in high space weather conditions. In addition, the TIEGCM DMDc\nPOD ROM was shown to be the only DMDc propagated model to perform with sub-2.5 MSE for high space weather\nconditions, but similar performance was not demonstrated by the model for low space weather conditions. Finally,\nall models maintained sub-2.5 MSE in medium space weather conditions. From observing these test cases, TIEGCM\nmodel errors are shown to change with increases in the Kp index and similar variations in the other density models\noccurred in relation to changes in the DSTDTC index.\nThe results for all DMDc and Transformer-based 1 week forecasts in the test datasets, for low, medium, and high\nsolar activity, are shown in Tables 4-6. All performance metrics are computed for the reduced-order model, instead\nof projecting back into the full-order to compute each metric. This allows the true training errors to be computed,\nindependent of the low-dimensional projection errors, which are controlled by the reduced-order modeling approach\n(POD or ML).\nIn Table 4, the transformer-based models maintain the lowest MSE for 1 week forecasts across the entire low space\nweather test dataset. With the lowest error of 0.12215 MSE achieved by the JB2008 Transformer ML ROM. As in\nFigure 3, the JB2008 DMDc ML ROM and JB2008 DMDc POD ROM have comparable errors (0.38346 MSE and\n0.82024 MSE) to the transformer-based models for low space weather activity.Fig. 5: MSE for one week propagation on high space weather activity test data.\nModel Transformer MSE DMDc MSE Transformer MAE DMDc MAE\nJB2008 POD ROM 0.22178 0.82024 0.33443 0.62294\nNRLMSISE POD ROM 0.84399 7.40740 0.66705 1.88537\nTIEGCM POD ROM 0.64757 1.77646 0.54225 0.99615\nJB2008 ML ROM 0.12215 0.38346 0.25138 0.44345\nTable 4: MSE and MAE for the transformer and DMDc propagator for low solar activity.\nTable 5 also shows the lowest MSE for transformer-based models during medium space weather activity. With the\nJB2008 Tranformer ML ROM maintaining the lowest MSE of 0.33400, closely followed by the NRLMSISE Trans-\nformer POD ROM. The JB2008 DMDc POD ROM and JB2008 DMDc ML ROM now show the largest DMDc-based\nmodel errors (8.95260 MSE and 3.58280 MSE).\nModel Transformer MSE DMDc MSE Transformer MAE DMDc MAE\nJB2008 POD ROM 0.55943 8.95260 0.45291 1.98441\nNRLMSISE POD ROM 0.44961 1.40329 0.48112 0.58854\nTIEGCM POD ROM 0.65792 1.06309 0.48025 0.73588\nJB2008 ML ROM 0.33400 3.58280 0.40303 1.43284\nTable 5: MSE and MAE for the transformer and DMDc propagator for medium solar activity.\nFor high space weather activity, Table 6 shows the transformer-based forecasters dominating all empirical DMDc-\nbased models by a larger margin, by between 2.2 and 12.16 MSE. While all transformer models are very close in\npropagator error, the JB2008 Transformer POD ROM performs slightly better than all others, with about a 0.056\ndifference in MSE. While not dominating any of the transformer models, the physics-based TIEGCM DMDc POD\nROM manages to achieve the closest forecasting error to the transformer models, with about a 0.234 MSE difference.\nThe transformer-based forecaster outperforms DMDc for all levels of space weather activity for the JB2008 POD,\nNRLMSISE POD, and TIEGCM POD, and JB2008 ML ROM models. Additionally, the transformer MSE and MAE\nwere consistent between models and levels of space weather activity, varying between 0 - 0.85 and reaching a maxi-\nmum value of 0.84399 MSE for NRLMSISE POD ROM propagation during low solar activity. DMDc forecasting, on\nthe other hand, proved to be very sensitive to the input space weather indices and the initial density snapshot used forModel Transformer MSE DMDc MSE Transformer MAE DMDc MAE\nJB2008 POD ROM 0.36612 12.52918 0.34554 2.31920\nNRLMSISE POD ROM 0.42239 2.63724 0.48266 1.10854\nTIEGCM POD ROM 0.43682 0.67073 0.44843 0.60545\nJB2008 ML ROM 0.42245 6.07747 0.40448 1.74487\nTable 6: MSE and MAE for the transformer and DMDc propagator for high solar activity.\npropagation; MSE varied between 0.38346 and 12.52918, reaching a minimum for the JB2008 ML ROM propagation\nduring low solar activity and a maximum for the JB2008 POD ROM propagation during high solar activity.\n5. DISCUSSION\nFrom conducting propagation analyses and error computations for both the transformer forecaster and the DMDc fore-\ncaster for the JB2008 POD, NRLMSISE POD, TIEGCM POD, and JB2008 ML ROMs, it is apparent that transformer-\nbased forecasting provides predictions which are more robust and almost exclusively more accurate than DMDc pre-\ndictions. Likely, the transformer’s robustness can be drawn from the PatchTST architecture of enabling a look-back\nwindow. Instead of using only the previous hour of atmospheric density data, as DMDc does to make a prediction,\nthe transformer forecaster is able to use over a month of previous data (with a 42 day look-back window) to provide a\ndensity forecast. Additionally, forecasts for test data made by DMDc had very high variances. Since the matrices, A\nandB, are trained by data outside of the test dataset, the test data may be outside of the linear regime in which DMDc\ncan be applied. Leading to compounding forecasting errors in very short time periods.\nFor future work, the transformer forecaster will be augmented to include a graph neural network to learn the cross-\nchannel relationships between space weather indices and atmospheric density. This formulation will allow the space\nweather control input to affect atmospheric density as a nonlinear dynamical system, while maintaining noise robust-\nness from the channel-independent transformer architecture.\n6. CONCLUSION\nA transformer forecaster for the JB2008, NRLMSISE, and TIEGCM atmospheric density models has been formulated\nin this work and compared with DMDc forecasting methods. The transformer-based forecasting approach has been\nshown to be both accurate and robust when handling long-term dependencies in atmospheric density data, when com-\npared to the linear DMDc forecaster. Importantly, this effectiveness extends across varying levels of solar activity.\nWith the significant potential for a single geomagnetic storm to alter the orbits of Resident Space Objects (RSOs),\nour developed nonlinear transformer-based architecture for atmospheric density forecasting presents a significant ad-\nvancement over traditional linear propagation methods.\nACKNOWLEDGEMENT\nResearch was sponsored by the United States Air Force Research Laboratory and the Department of the Air Force\nArtificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000.\nThe views and conclusions contained in this document are those of the authors and should not be interpreted as repre-\nsenting the official policies, either expressed or implied, of the Department of the Air Force or the U.S. Government.\nThe U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any\ncopyright notation herein. This work was supported by a NASA Space Technology Graduate Research Opportunity\nand the National Science Foundation under award NSF-PHY-2028125. The authors would like to thank the MIT Su-\nperCloud and Lincoln Laboratory Supercomputing Center for providing HPC, database, and consultation resources\nthat have contributed to the research results reported in this paper. The authors would like to express their gratitude\nto Nicolette Clark for her substantial contributions to atmospheric density model development and analysis. Specifi-\ncally, Clark’s work on the development and uncertainty quantification of Reduced-Order Atmospheric Density Models\n(ROMs) for fast and accurate orbit propagation were essential for developing this work [5].REFERENCES\n[1] Ziyou Wu, Steven L. Brunton, and Shai Revzen. Challenges in dynamic mode decomposition. Journal of The\nRoyal Society Interface , 2021.\n[2] Turner H., Zhang M., D. Gondelach, , and R. Linares. Machine Learning Algorithms for Improved Thermospheric\nDensity Modeling , volume 12312. Springer, Cham, 2020.\n[3] Qingsong Wen, Tian Zhou, Chaoli Zhang, Weiqi Chen, Ziqing Ma, Junchi Yan, and Liang Sun. Transformers in\ntime series: A survey. arXiv preprint arXiv:2202.07125 , 2023. Accepted by 32nd International Joint Conference\non Artificial Intelligence (IJCAI 2023).\n[4] Julia Briden, Nicolette Clark, Peng Mun Siew, Richard Linares, and Tzu-Wei Fang. Impact of space weather\non space assets and satellite launches. In 23rd Advanced Maui Optical and Space Surveillance Technologies .\nAdvanced Maui Optical and Space Surveillance Technologies, 2022.\n[5] Nicolette LeAnn Clark. Reduced-Order Atmospheric Density Modeling for LEO Satellite Orbital Reentry Pre-\ndiction . PhD thesis, Massachusetts Institute of Technology, 6 2023.\n[6] Piyush M. Mehta and Richard Linares. A methodology for reduced order modeling and calibration of the upper\natmosphere. Space Weather , 15(10):1270–1287, 2017.\n[7] Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing\ninternal covariate shift. Proceedings of the 32nd International Conference on Machine Learning , PMLR 37:448-\n456, 2015.\n[8] R. A. DeCarlo. Linear systems: A state variable approach with numerical implementation . Upper Saddle River,\nNJ, USA: Prentice-Hall, Inc, 1989.\n[9] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning . MIT Press, 2016. http://www.\ndeeplearningbook.org .\n[10] Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64 words:\nLong-term forecasting with transformers. In International Conference on Learning Representations (ICLR) ,\n2023.\n[11] Ignacio Oguiza. tsai - a state-of-the-art deep learning library for time series and sequential data. Github, 2022.\n[12] David J. Gondelach and Richard Linares. Real-Time Thermospheric Density Estimation via Two-Line Element\nData Assimilation. Space Weather , 18(2):e2019SW002356, 2020." }, { "title": "2310.18675v1.Interplay_between_Chiral_Charge_Density_Wave_and_Superconductivity_in_Kagome_Superconductors__A_Self_consistent_Theoretical_Analysis.pdf", "content": "Interplay between Chiral Charge Density Wave and Superconductivity in Kagome\nSuperconductors: A Self-consistent Theoretical Analysis\nHong-Min Jiang,1,∗Min Mao,1Zhi-Yong Miao,1Shun-Li Yu,2, 3,†and Jian-Xin Li2, 3,‡\n1School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China\n2National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China\n3Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China\n(Dated: October 31, 2023)\nInspired by the recent discovery of a successive evolutions of electronically ordered states, we\npresent a self-consistent theoretical analysis that treats the interactions responsible for the chiral\ncharge order and superconductivity on an equal footing. It is revealed that the self-consistent theory\ncaptures the essential features of the successive temperature evolutions of the electronic states from\nthe high-temperature “triple- Q” 2×2 charge-density-wave state to the nematic charge-density-\nwave phase, and finally to the low-temperature superconducting state coexisting with the nematic\ncharge density wave. We provide a comprehensive explanation for the temperature evolutions of\nthe charge ordered states and discuss the consequences of the intertwining of the superconductivity\nwith the nematic charge density wave. Our findings not only account for the successive temperature\nevolutions of the ordered electronic states discovered in experiments but also provide a natural\nexplanation for the two-fold rotational symmetry observed in both the charge-density-wave and\nsuperconducting states. Moreover, the intertwining of the superconductivity with the nematic charge\ndensity wave order may also be an advisable candidate to reconcile the divergent or seemingly\ncontradictory experimental outcomes regarding the superconducting properties.\nPACS numbers: 74.20.Mn, 74.25.Ha, 74.62.En, 74.25.nj\nI. INTRODUCTION\nKagome systems, with their geometrical frustration\nand nontrivial band topology, have long served as\nparadigmatic platforms for investigating exotic quan-\ntum phases of electronic matter, including spin liquid1–9,\nvarious topological quantum phases10–17, charge density\nwave (CDW)10,18, spin density wave19, bond density\nwave20–22and superconductivity19,21–24. Of particular\ninterest is the possible phases near the van Hove filling\n(VHF), especially the superconducting (SC) state, where\nthe density-of-states (DOS) is extremely enhanced and\nthe Fermi surface (FS) exhibits perfect nesting19. These\nunique properties of the electron structure lead to the\nSC state being susceptible to competition from various\nother electronic instabilities19,21,22. Understanding the\nsuperconductivity in such a kagome material that either\navoids or even intertwines with these competing instabil-\nities remains an unsettled issue.\nThe recent discovery of superconductivity in a fam-\nily of compounds AV 3Sb5(A=K, Rb, Cs), which share\na common lattice structure with kagome net of vana-\ndium atoms, has set off a new boom of researches on the\nsuperconductivity25–73. The appealing aspects of these\ncompounds lie in that they incorporate many remark-\nable properties of the electron structure, such as VHF,\nFS nesting and nontrivial band topology25. Consistent\nwith the fairly good FS nesting and proximity to the\nvon Hove singularities, the system undergoes a “triple- Q”\n2×2 CDW transition at temperature TCDW∼78−104K,\nwith the in-plane wave vectors align with those connect-\ning the van Hove singularities25–31. While the neutron\nscattering74and muon spin spectroscopy75measurementshave ruled out the possibility of long-range magnetic or-\nder in AV 3Sb5, a significant anomalous Hall effect is still\nobserved above the onset of the SC state in this 2 ×2\nCDW phase32,35, indicating a time-reversal symmetry-\nbreaking state originating from the charge degree of free-\ndom75. So far, there are an increasing number of exper-\nimental evidences supporting that the CDW state has\na 2×2 chiral flux order26,31,33,34,76–80, i.e., the chiral\nflux phase (CFP)81–83. Furthermore, the muon spin re-\nlaxation technic observed a noticeable enhancement of\nthe internal field width, which takes place just below\nthe charge ordering temperature and persists into the\nSC state31, suggesting an intertwining of time-reversal\nsymmetry breaking charge order with superconductivity.\nNevertheless, more recent experiments revealed that\nthe high-temperature 2 ×2 CDW state does not di-\nrectly border the low temperature SC state. Instead,\nthe high-temperature 2 ×2 CDW state is separated from\nthe SC ground state by an intermediate-temperature\nregime with the two-fold ( C2) rotational symmetry of\nelectron state59,64,84,85. This electronic state with C2\nrotational symmetry is found to appear at temperature\nTnemwell below TCDW and persist into the SC state,\nas evidenced by transport47and scanning tunneling mi-\ncroscopy (STM) measurements27,64.\nApart from the exotic charge orders, the supercon-\nductivity in AV 3Sb5exhibits some unusual features as\nwell. On the one hand, the SC pairings in these com-\npounds are suggested to be of the s-wave type, supported\nby the appearance of the Hebel-Slichter coherence peak\njust below Tcin the nuclear magnetic resonance spec-\ntroscopy43and the nodeless SC gap in both the penetra-\ntion depth measurements44and the angle-resolved pho-arXiv:2310.18675v1 [cond-mat.supr-con] 28 Oct 20232\ntoemission spectroscopy (ARPES) experiment86. On the\nother hand, the indications of time-reversal symmetry\nbreaking and the C2rotational symmetry discovered in\nthe SC state27,31,47,64,86, together with the nodal SC gap\nfeature detected by some experiments28,29,42,45, hint to\nan unconventional superconductivity.\nSince the superconductivity occurs within the density\nwave ordered state, understanding the relationship be-\ntween the CDW instability and superconductivity is a\ncentral issue in the study of AV 3Sb5. Theoretical anal-\nysis has shown that a conventional fully gapped super-\nconductivity is unable to open a gap on the domains of\nthe CFP and results in the gapless edge modes in the SC\nstate87. A more direct consideration of the impact of the\nchiral 2 ×2 CDW on the SC properties has revealed that\na nodal SC gap feature shows up even if an on-site s-wave\nSC order parameter is included in the study88. However,\nthere is still limited knowledge about the rotational sym-\nmetry breaking phase that straddles the SC ground state\nand the 2 ×2 CDW state in this class of kagome met-\nals, particularly its origin, its role in the formation of the\nsuperconductivity, and its impact on the SC properties.\nIn this paper, we investigate the interplay between the\nCFP and superconductivity in a fully self-consistent the-\nory, which self-consistently treats both the chiral CDW\nand the SC pairing orders on an equal footing. The cal-\nculated results catch the essential characteristics of the\nsuccessive temperature evolutions of the electronically or-\ndered states, starting from the high-temperature 2 ×2\n“triple- Q” CFP (TCFP) to the nematic CFP (NCFP),\nand finally to the low-temperature SC state. Notably,\nthe SC state emerges in the coexistence with the NCFP,\nby which the free energy in the coexisting phase is sig-\nnificantly lowered than that in the pure SC state. The\nrotational symmetry-breaking transition of the CDW can\nbe understood from a competitive scenario, in which the\ndelicate competition between the doping deviation from\nthe VHF and the thermal broadening of the FS deter-\nmines the energetically favored state. In the coexisting\nphase of the s-wave SC pairing and the NCFP order, the\nDOS exhibits a nodal gap feature manifesting as the V-\nshaped DOS along with the residual DOS near the Fermi\nenergy. These results not only reproduce the successive\ntemperature evolutions of the ordered electronic states\nobserved in experiment, but also provide a tentative ex-\nplanation to the two-fold rotational symmetry observed\nin both the CDW and SC states. Furthermore, the in-\ntertwining of the SC pairing with the NCFP order may\nalso be an advisable candidate to reconcile the divergent\nor seemingly contradictory experimental outcomes con-\ncerning the SC properties.\nThe remainder of the paper is organized as follows. In\nSec. II, we introduce the model Hamiltonian and carry\nout analytical calculations. In Sec. III, we present nu-\nmerical calculations and discuss the results. In Sec. IV,\nwe make a conclusion.II. MODEL AND METHOD\nIt is generally considered that the scattering due to\nthe FS nesting, especially the inter-scattering between\nthree van Hove points with the nesting wave vectors\nQa= (−π,√\n3π),Qb= (−π,−√\n3π) and Qc= (2π,0)\nshown in Fig. 1(b), is closely related to the CDW in\nAV3Sb5. Meanwhile, the VHF was also proposed to be\ncrucial to the superconductivity in AV 3Sb5. A single or-\nbital tight binding model near the VHF produces the\nessential feature of the FS and the van Hove physics19.\nTherefore, to capture the main physics of the chiral CDW\nand its intertwining with the SC in AV 3Sb5, we adopt a\nminimum single orbital model. We also note that the\nsix-fold ( C6) symmetry is broken within the unit-cell of\nthe 2×2 CDW state59, without any additional reduction\nin translation symmetry. Thus, we choose the enlarged\nunit cell (EUC) with size 2 a1×2a2, as indicated by the\ndashed lines in Fig. 1(a).\nThe single orbital model can be described by the fol-\nlowing tight-binding Hamiltonian,\nH0=−tX\n⟨ij⟩αc†\niαcjα−µX\niαc†\niαciα, (1)\nwhere c†\niαcreates an electron with spin αon the site ri\nof the kagome lattice and ⟨ij⟩denotes nearest-neighbors\n(NN). tis the hopping integral between the NN sites, and\nµstands for the chemical potential. The Hamiltonian H0\ncan be written in the momentum space as,\nH0(k) =X\nkαˆΨ†\nkαˆH0\nkˆΨkα, (2)\nwith ˆΨkα= (cAkα, cBkα, cCkα)Tand\nˆH0\nk=\n−µ −2tcosk1−2tcosk2\n−2tcosk1−µ −2tcosk3\n−2tcosk2−2tcosk3−µ\n.(3)\nThe index m=A, B, C incmkσ labels the three ba-\nsis sites in the triangular primitive unit cell (PUC) as\nshown in Fig. 1(a). knis abbreviated from k·τnwith\nτ1= ˆx/2,τ2= (ˆx+√\n3ˆy)/4 and τ3=τ2−τ1denoting\nthe three NN vectors. The spectral function of H0(k) de-\nfined as A0(k, E) =−1\nπTr[Im ˆG0(k, iE→E+i0+)] with\nˆG0(k, iE) = [iEˆI−ˆH0\nk]−1. Near the VHF with 1 /6 hole\ndoping, the Hamiltonian ˆH0\nkgenerates the hexagonal FS\nand the corresponding energy band as shown in Figs. 1(b)\nand (c) respectively, which capture the essential features\nof the FS and energy band observed in the ARPES exper-\niment and the density functional theory calculations25.\nThe second part of the Hamiltonian incorporates the\norbital current order,\nHC=X\n⟨ij⟩αiWijc†\niαcjα, (4)3\nwhere Wij=−VcIm⟨χ†\nij⟩denotes the mean-field value of\nthe magnitude of the orbital current order in the CFP\nwith χij=c†\ni↑cj↑+c†\ni↓cj↓. The orbital current order could\nbe derived from the Coulomb interaction between elec-\ntrons on neighboring sites, i.e., HV=VcP\nijninjwith\nni=P\nαc†\niαciα. A straightforward algebra shows that\nninj= 2 ni−X\nαβ(c†\niαcjβ)(c†\niαcjβ)†\n= 2 ni−1\n23X\nη=0Sη\nij(Sη\nij)†, (5)\nwhere Sη\nij=P\nαβc†\niαˆση\nαβcjβ. Here, ˆ σ0is the 2 ×2 identity\nmatrix and ˆ σ1,2,3are the Pauli matrices. In the Hartree-\nFock approximation, we decouple the operator product\nSη\nij(Sη\nij)†with⟨Sη\nij⟩(Sη\nij)†+Sη\nij⟨(Sη\nij)†⟩−⟨Sη\nij⟩⟨(Sη\nij)†⟩. The\nexpectation value ⟨Sη\nij⟩defines a four-component vector\n⟨Sη\nij⟩= (⟨χij⟩,⟨Sij⟩), (6)\nwhere the mean-field amplitudes ⟨χij⟩and⟨Sij⟩=\n⟨P3\nη=1Sη\nij⟩correspond respectively to the currents in\nthe charge and spin channels. For the vanadium-based\nkagome superconductors, only the charge order is rele-\nvant. Thus we need to deal with the case that ⟨Sij⟩= 0,\nand this leads to the mean-field decoupling of the NN\nCoulomb interaction in the charge channel as\nHV,MF =−Vc\n2X\nij(⟨χ†\nij⟩χij+⟨χij⟩χ†\nij− |⟨χij⟩|2)\n+2VcX\nini. (7)\nIn this work, we focus on the CDW states with time\nreversal symmetry breaking described by the imaginary\npart of the mean-field value of χij(χ†\nij). Using the fact\nthatP\nij⟨χ†\nij⟩χij=P\nij⟨χij⟩χ†\nij, we finally arrive at the\neffective Hamiltonian in Eq. (4). In the procedure for\nobtaining Eq. (4), we also neglect the constant termP\nij|⟨χij⟩|2and absorb the term 2 VcP\niniinto the chem-\nical potential.\nThe third term accounts for the SC pairing. It reads\nHP=X\ni(∆c†\ni↑c†\ni↓+h.c.). (8)\nHere, we choose the on-site s-wave SC order parameter\n∆ =−Vs⟨ci↓ci↑⟩=Vs⟨ci↑ci↓⟩. In the calculations, we\nchoose the typical value of the effective pairing interac-\ntionVs= 1.4. Varying the pairing interaction will alter\nthe pairing amplitude, but the results presented here will\nbe qualitatively unchanged if the strength of CDW order\nchanges accordingly.\nIn the coexistence of SC and orbital current orders, the\ntotal Hamiltonian H=H0+HP+HCcan be written in\n-0.10-0.050.000.050.10E/t doping=1/6 \ndoping=9.6/60Μ\n \n \n-4-3-2-1012 \nΜΚΓ Γ \nE/t \ndoping=1/6 \ndoping=9.6/60(d)(\nc)(b)( a)C\nB\nA \nK\nyK\nxMCK\n(K')MAKQbQaQ\ncΓM\nBFIG. 1. (a) Structure of the kagome lattice, made out of three\nsublattices A(green dots), B(red dots) and C(blue dots).\nThe dashed lines in the figure denote the enlarged unit cell\nin the 2 ×2 charge density wave state. (b) Fermi surface\nproduced by the Hamiltonian H0(k) for the doping levels 1 /6\n(solid lines) and 9 .6/60 (dotted lines), respectively. (c) Tight-\nbinding dispersion along high-symmetry cuts for doping levels\n1/6 (solid curve) and 9.6/60 (dotted curve) (Note that the\nsolid and dotted curves are close to each other, so only one\nsolid curve can be discerned in a large energy scale). The\ndashed line denotes the Fermi level. (d) The enlarged view of\nthe square box in (c).\nthe momentum space within one EUC as,\nH(k) =−tX\nk,⟨˜i˜j⟩,σc†\nk˜iσck˜jσe−ik·(r˜i−r˜j)−µX\nk,˜i,σc†\nk˜iσck˜iσ\n+X\nk,⟨˜i˜j⟩,σiW˜i˜jc†\nk˜iσck˜jσe−ik·(r˜i−r˜j)\n+X\nk,˜i(∆c†\nk˜i↑c†\n−k˜i↓+h.c.), (9)\nwhere ˜i∈EUC represents the lattice site being within\none EUC, and ⟨˜i˜j⟩denotes the NN bonds with the peri-\nodic boundary condition implicitly assumed.\nBased on the Bogoliubov transformation, we obtain the\nfollowing Bogoliubov-de Gennes equations in the EUC,\nX\nk˜j\u0012H˜i˜j,σ∆˜i˜j\n∆∗\n˜i˜j−H∗\n˜i˜j,¯σ\u0013\nexp[ik·(r˜j−r˜i)] \nuk\nn,˜j,σ\nvk\nn,˜j,¯σ!\n=Ek\nn \nuk\nn,˜i,σ\nvk\nn,˜i,¯σ!\n,(10)\nwhere H˜i˜j,σ= (−t+iW˜i˜j)δ˜i+τ˜j,˜j−µδ˜i,˜jwith τ˜jdenoting\nthe four NN vectors and ∆ ˜i˜j= ∆δ˜i,˜j.uk\nn,˜i,σandvk\nn,˜i,¯σare4\nthe Bogoliubov quasiparticle amplitudes on the ˜i-th site\nwith momentum kand eigenvalue Ek\nn. The amplitudes\nof the SC pairing and the orbital current order, as well as\nthe electron densities, are obtained through the following\nself-consistent equations,\n∆ =Vs\n2X\nk,nuk\nn,˜i,σvk∗\nn,˜i,¯σtanh(Ek\nn\n2kBT)\nW˜i˜j=Vc\n2Im{X\nk,n(uk\nn,˜i,σuk∗\nn,˜j,σ+vk\nn,˜i,¯σvk∗\nn,˜j,¯σ)\n×exp[−ik·(r˜j−r˜i)] tanh(Ek\nn\n2kBT)}\nn˜i=X\nk,n{|uk\nn,˜i,↑|2f(Ek\nn) +|vk\nn,˜i,↓|2[1−f(Ek\nn)]}.(11)\nDue to the fairly good FS nesting, the proximity to\nthe VHF, and the presence of multiple electronical or-\nders, the self-consistent calculations may yield several\nsolutions with local energy minima at the same tempera-\nture and doping. In cases where multiple solutions arise\nfrom the self-consistent calculations at the same temper-\nature but with different sets of initially random input\nparameters, we compare their free energy defined as\nF=−2kBTX\nk,n,Ek\nn>0ln[2 cosh(Ek\nn\n2kBT)] +N|∆|2\nVs\n+X\nk,⟨˜i˜j⟩|W˜i˜j|2\n2Vc, (12)\nso as to find the most favorable state in energy.\nThen, the single-particle Green’s functions\nG˜i˜j(k, iω) = −Rβ\n0dτexpiωτ⟨Tτck˜i(iτ)c†\nk˜j(0)⟩can be\nexpressed as\nG˜i˜j(k, iω) =X\nn uk\nn,˜i,↑uk∗\nn,˜j,↑\niω−Ekn+vk\nn,˜i,↓vk∗\nn,˜j,↓\niω+Ekn!\n.(13)\nThe spectral function A(k, E) and the DOS ρ(E) can\nbe derived from the analytic continuation of the Green’s\nfunction as,\nA(k, E) =−1\nNPπX\n˜iImG˜i˜i(k, iE→E+i0+),(14)\nand\nρ(E) =1\nNkX\nkA(k, E), (15)\nwhere NPandNkare the number of PUCs in the EUC\nand the number of k-points in the Brillouin zone, respec-\ntively.\n10.09.99.89.79.69.59.49.30.000.020.040.060.08\n60×I\nV Nematic CFP+SCIII Nematic CFPII Triple-Q CFPI Normal state \n TemperatureD\noping\nNematic CFPT riple-Q CFP(a)(\nb)( c)FIG. 2. (a) Phase diagram as a function of doping and tem-\nperature. The orbital current configurations for the “triple-\nQ” CFP (b), and the nematic CFP (c), respectively. The\narrows in (b) with their respective colors indicated by the la-\nbels of “High”, “Medium” and “Low” signal the magnitudes\nof the bond current orders. The arrows in (c) with their re-\nspective colors indicated by the labels of “High” and “Low”\nsignal the magnitudes of the bond current orders. In (c), the\nlarge (green dots) and small (red and blue dots) sizes of lattice\nsite also signal respectively the “High” and “Low” values of\nthe on-site SC pairing amplitude in the coexisting phase(see\ntext and Table II for reference).\nIII. RESULTS AND DISCUSSION\nA. Phase Diagram\nIn the following analysis, the chemical potential µis\nadjusted to achieve the desired filling. Right at the VHF,\nthe FS possesses a hexagonal shape, with the saddle\npoints MA/B/C located exactly on the FS. This unique\nFS possesses a perfect nesting property and facilitates\nthe inter-scatterings between three van Hove singulari-\nties connected by the nesting vectors Qa= (−π,√\n3π),\nQb= (−π,−√\n3π) and Qc= (2π,0), which has been con-\nsidered as the primary factors promoting the so-called\n“triple- Q” 2×2 CDW in AV 3Sb525–31. Away from\nthe VHF, the FS becomes more rounded, and the nest-\ning is weakened, particularly around the saddle points\n[MA/B/C in Fig. 1(b)]. We focus on the situation where\nthe hole doping is deceased from 1 /6, such that the saddle\npoints move slightly below the Fermi level as displayed\nin Figs. 1(c) and (d), being consistent with the density\nfunctional theory calculations27,89–91.\nAs a function of doping and temperature, we find a5\nrich phase diagram of the model at and near the VHF,\nwhich is summarized in Fig. 2(a) for the typical val-\nues of Vs= 1.4 and Vc= 1.2. Right at the VHF for\n1/6 doping, where the DOS at the Fermi level is max-\nimally enhanced and the nesting features of the FS are\nstrongest, the system prefers the TCFP [Fig. 2(b)] be-\nfore entering the SC state. Once the system deviates\nfrom the VHF, the NCFP order [Fig. 2(c)] develops in\nbetween the TCFP and the low-temperature SC state.\nIn this case, when decreasing the temperature, the sys-\ntem starts from the high-temperature normal state and\npasses through the NCFP state, and finally transits into\nthe SC state. When the filling further departs from the\nvan Hove point, the region of the NCFP expands grad-\nually towards higher temperatures with the concomitant\nshrinking of the TCFP region, and eventually the TCFP\nis completely displaced by the NCFP state. Interestingly,\nthe SC state always coexists with the NCFP state in the\nlow temperature region of the phase diagram, with its\nfree energy being significantly lower than those of the\npure states. Although the variations of VsandVcmay\naffect the phase boundaries, the essential feature of the\nphase diagram, namely the consecutive evolvement of dif-\nferent ordered states with temperature, remains qualita-\ntively unchanged.\nIt is remarkable that the successive temperature evo-\nlutions from the TCFP phase to the NCFP state, occur-\nring at a doping level slightly deviating from the van Hove\npoint in a self-consistent manner, exhibits the same trend\nas the experimental observations59,64,84,85. Particularly,\nthe ground state characterized by the coexistence of the\nNCFP and SC orders may be related to the C2symme-\ntry and the time-reversal-symmetry breaking observed in\nthe SC state27,47,64.\nSince the effect of doping on the phase diagram is\nclosely related to the VHF, we will focus on the physics\nassociated with the three van Hove points, in addition\nto the nesting properties of the FS. At each van Hove\npoint, the electronic states come exclusively from one of\nthe three distinct sublattices19. As a result, the scat-\ntering between low-energy electronic states connected by\neach nesting wave vector occurs solely between two sub-\nlattices. This unique property creates the necessary con-\nditions for the transition from the TCFP to the NCFP\nthrough doping. At VHF, the electronic states at the\nthree van Hove points are mutually coupled by the CDW\norders with three wave vectors Qa,QbandQcin the\nTCFP in an end-to-end manner [Fig. 1(b)]. This is to\nsay, the CDW orders with three wave vectors are mutu-\nally coupled in pairs with the strongest coupling strength\nat the van Hove points. Thus, as depicted in the phase\ndiagram [Fig. 2(a)], a stable charge order pattern that\nsimultaneously satisfies the three wave vectors can be\nfound at the VHF. However, when the system deviates\nfrom the VHF by reducing the hole doping, the chem-\nical potential µis elevated, and accordingly the saddle\npoints move below the Fermi level, as demonstrated in\nFigs. 1(c) and (d). The deviation of the FS from the sad-\n10.09.99.89.79.69.5-0.0040.0000.0040.008\n60× \n ΔFD\noping FTCFP-FNCFP\n0.000 .020 .040 .06-21.22-21.20-21.18-21.16 \nNCFP normal state \nNCFP+SC \n Free energyT\nemperature pure SC(b)( a)FIG. 3. (a) Doping dependence of the free energy difference\nper site between the “triple- Q” CFP and nematic CFP. In\nobtaining the results in (a), the CDW states are each inten-\ntionally kept in their respective forms during the calculations.\n(b) The evolutions of the free energy per site from the normal\nstate (dashed line) to pure SC state (dashed ling with square\nsymbol), and from the nematic CFP (dotted line) to the co-\nexisting phase with nematic CFP and SC state (dotted line\nwith round symbol) at doping level 9 .8/60.\ndle points weakens the mutual couplings between pairs of\nthe three wave vectors and, correspondingly, the TCFP.\nIn this situation, there is a significant decrease in the en-\nergy difference between the TCFP, which satisfies three\nordered wave vectors, and the NCFP, which has only one\nordered wave vector. Moreover, the NCFP will deform\nthe FS, suppressing the other two NCFP with different\nordered wave vectors while further enhancing itself [refer\nto Fig. 2(c) and Table I]. Consequently, within a cer-\ntain doping range, the NCFP becomes more stable than\nthe TCFP. For better clarity, we present the evolutions\nof the free energy deference between the TCFP and the\nNCFP with doping at a specific temperature T= 0.04 in\nFig. 3(a). It shows that the TCFP has a lower free en-\nergy than that of the NCFP, when the doping level has\nnot much deviations from the VHF. Nevertheless, as the\nsystem deviates appreciably from the VHF, the NCFP\nacquires the lower free energy. As a result, a spontaneous\nrotational symmetry-breaking transition occurs from the\nTCFP to the NCFP at the doping level defined by the\nzero point of the free energy difference.\nThe temperature effects on the CDW states are also\nclosely related to the van Hove physics. Although the\nvan Hove points shift below the Fermi level for dop-\ning levels deviating from the VHF, the thermal broad-\nening effect becomes prominent at relatively high tem-\nperatures, thereby increasing the effectiveness of the van\nHove points. This enhances the mutual coupling among\nthree CDW orders associated with different wave vectors\nQa,QbandQc. As a result, for doping levels that devi-\nate from the VHF, the TCFP is stabilized by the thermal\nbroadening effect at relatively high temperatures, while\nthe NCFP becomes more favorable due to the reduction\nof thermal broadening effect at low temperatures.\nPreviously, the transition to the CDW with C2sym-\nmetry was proposed to arise from interlayer interactions\nbetween adjacent kagome planes with already existing6\nC6-symmetry charge orders in each single layer89,92. As\na secondary outcome of the C6-symmetry charge orders\nin this interlayer coupling scenario, the nematicity typ-\nically occurs at a lower temperature Tnemwell below\nTCDW . However, as shown in Fig. 2(a), our theory\nshows a regime of less than 9 .4/60 hole doping where the\nNCFP directly straddles the normal and SC states, de-\nspite in the high doping level regime of the phase diagram\nthe appearance of nematic CDW at low temperatures is\nwell below TCDW . Interestingly, a recent experiment has\nindeed observed an immediate development of nematic-\nity and possible time-reversal symmetry breaking in the\nCDW state of CsV 3Sb593, providing further support on\nour theory.\nAs the temperature continues to decrease, several in-\ngredients promote the development of the coexisting\nphase of the NCFP and SC state. First of all, as depicted\nin Figs. 3(b) and A1(a), the NCFP exhibits a lower en-\nergy compared to the normal and TCFP states before\nthe SC transition, making it energetically favorable as\nthe parent state for the formation of the SC order. Sec-\nondly, the well-preserved portions of the FS, especially\nthose portions near the saddle points [the MApoints in\nFig 4(b)], within the NCFP provide sufficient electronic\nstates for the formation of the SC pairing. Thirdly, apart\nfrom the gaped portions of the FS, the one-wave-vector\nscattering with the rotational symmetry breaking in the\nNCFP state induces a slight deformation of the FS [refer\nto Figs. 4(b) and A1(b)], which brings the remaining FS\ncloser to the van Hove points compared to the normal\nstate. As a result, the coexisting phase of NCFP and SC\norders possesses the significantly lower free energy than\nthat for the pure SC state [see Fig. 3(b)].\nB. Characteristics of electronic states\nNext, we will investigate in detail on the electronic\nstructures in different regime of the phase diagram,\nnamely the TCFP state, the NCFP state, and the coex-\nisting state of the NCFP and SC orders. To demonstrate\nour results, we will focus on the typical cases with a dop-\ning level of 9 .8/60 at temperatures T= 0.05,T= 0.03\nandT= 1×10−5, corresponding to the TCFP state, the\nNCFP state and the coexisting phase of the NCFP and\nSC orders, respectively.\nFor the TCFP, the bond current orders obtained from\nself-consistent calculations can be classified into three\nlevels of magnitude: “High”, “Medium” and “Low”, as\ndisplayed in Fig. 2(b) and listed in Table I. These or-\nders give rise to a special pattern known as the “Star\nof David” state. Fig. 4(a) presents the distribution of\nspectral weight A(k, E) atE= 0, which is unfolded in\nthe primitive Brillouin zone. Consistent with the previ-\nous non-self-consistent results88, the zero-energy spectral\nweight distribution clearly reveals partially gapped Fermi\nsegments and preserves the C6symmetry. Correspond-\ningly, the DOS shown in Fig. 5(a) exhibits a “pseudogap-\nK\nyK\nx03060MA\nK\nyK\nx03060MA\n-4-3-2-1012Γ\nΜ BΚ \n E/tΓ\n-4-3-2-1012Γ\nΜ Κ \n E/tΓ\n(\nd)( c)(b)( a)M\nAMAFIG. 4. Zero-energy spectral weight distribution A(k, E) un-\nfolded in the primitive Brillouin zone for hole doping 9 .8/60 at\nT= 0.05 (a), and at T= 0.03 (b), respectively. The former\nsituates in the “triple- Q” CFP region, while the latter lies\nin the region of the nematic CFP state. The unfolded dis-\npersions along high-symmetry cuts in the primitive Brillouin\nzone are shown correspondingly in (c) for the “triple- Q” CFP\nstate, and in (d) for the nematic CFP state, respectively.\nlike” feature, with non-zero minima occurring at E= 0.\nIn contrast, the DOS for the normal state, represented\nby the black dashed curve in the same figure, displays\nthe typical van Hove peak near E= 0.\nOn the other hand, as the temperature decreases into\nthe NCFP regime, such as T= 0.03, the magnitude\ndistribution of the three inequivalent bond current or-\nders transforms into that of two inequivalent orders.\nThe “Low” intensity is associated with two inequivalent\nbonds, which are approximately one order of magnitude\nsmaller than the other one labeled as “High” intensity. In\nthis case, only the strong current order between sublat-\ntices BandCcorresponds to the charge order scattering\nbetween MBandMCpoints with a momentum transfer\nQa. As a result, the depletion of the zero-energy spec-\ntral weight only occurs in regions connected by one of the\nthree wave vectors, such as Qain Fig. 4(b), manifesting\nthe characteristics of the C2symmetry. The DOS for the\nNCFP depicted by the black solid curve in Fig. 5(b) still\nexhibits two peaks resembling gap edges, but a significant\nDOS shows up within the peak edges, accompanied by a\nresidue van Hove peak near the zero bias, resulting from\nthe large portion of the unspoilt Fermi segments and the\npreserved van Hove points MA.\nLet’s further analyze the characteristics of the band\nstructures in the two CDW states. In the TCFP, the\nbands along different high-symmetry cuts Γ →K/K′→\nMA/B/C →Γ remain the same due to the preservation of\ntheC6symmetry, while in the NCFP, a gap opens near7\nHigh Medium Low\n“triple- Q” CFP 0.043805 0.035128 0.026666\nnematic CFP 0.071768 0.00841\nTABLE I. Magnitude of the bond current order |Wij|in the\nself-consistent calculations at filling level 9 .8/60 with T= 0.05\nfor the “triple- Q” CFP and T= 0.03 for the nematic CFP.\nHigh Low\nnematic CFP 0.072113 0.008144\nSC 0.050711 0.025567\nTABLE II. Magnitudes of the bond current order |Wij|and\nthe SC pairing |∆|in the self-consistent calculations at filling\nlevel 9 .8/60 with T= 1×10−5for the coexisting phase of the\nnematic CFP and SC orders.\ntheMB/Cpoints but not near the MApoint, as shown\nin Figs. 4(d), A2(a) and A2(b). In addition, near the\nsaddle point, the band is triply split in the TCFP, while it\nis only double split near the MB/Cpoints in the NCFP,\nas illustrated in Figs. 4(c), (d) and Fig. A2(a). This\nbehavior can be understood through the “patch model”,\nwhich provides an approximate description of the low-\nenergy scatterings between the saddle points94–96. In the\nTCFP state, the patch model involving the “triple- Q”\nscatterings reads,\nHTCFP (M) =\nεMA iλAB iλAC\n−iλAB εMBiλBC\n−iλAC−iλBCεMC\n.(16)\nNevertheless, the charge order scattering in the NCFP in-\nvolves only one wave vector, such as Qathat corresponds\nto the bond current configuration in Fig. 2(c). As a re-\nsult, the patch model in the NCFP state is reduced to\nHNCFP (M) = \nεMBiλBC\n−iλBCεMC!\n. (17)\nHere, εMA(εMB,εMC) stands for the energy at the sad-\ndle point MA(MB,MC) that originates from the sublat-\nticeA(B,C), and λAB(λAC,λBC) represents the scat-\ntering strength of the bond current order between MA\nandMB(MAandMC,MBandMC). Near the VHF,\nwhere εMA=εMB=εMC≈0, one can immediately find\nthat the Hamiltonian HTCFP (M) has three eigenvalues\nE0(M) = 0 and E±(M) =±p\nλ2\nAB+λ2\nAC+λ2\nBC. The\nHamiltonian HNCFP (M), on the other hand, has two\neigenvalues E±(M) =±|λBC|. It is worth pointing out\nthat the unique change of the electronic structures from\nthe TCFP to the NCFP can serve as an indirect evidence\nto identify the electronic nematicity in AV 3Sb5.\nThen, we turn to the coexisting phase of the NCFP\nand SC orders. On the one hand, as indicated in Ta-\nble II, the strength of the bond current orders changes\n-0.4-0.20.00.20.40246810 \n DOSE\n/t TCFP \nNormal state\n10.09.99.89.79.69.59.40.0280.0300.0320.0340.036T\nemperature \nTnemAmplitudeD\noping W \nΔ\n60×0\n.020.040.060.08\n-0.4-0.20.00.20.40246810 \n DOSE\n/t NCFP \nNormal state\n-0.4-0.20 .00 .20 .403691215 \n DOSE\n/t NCFP+SC \nPure SC(d)( c)(b)( a)FIG. 5. Energy dependence of the DOSs for the “triple- Q”\nCFP (a), the nematic CFP (b), and the coexisting phase of\nthe nematic CFP and SC orders (c), respectively. (d) The\ndoping dependence of the averaged SC pairing amplitude ¯∆,\nthe averaged magnitude of the bond current orders ¯W, and\nthe transition temperature of the nematic CFP Tnem.\nlittle upon entering the coexisting phase. On the other\nhand, as displayed in Fig. 2(c), the distribution of SC\npairing amplitudes depends not only on the strength but\nalso on the direction of the surrounding bond current\norders. Specifically, the “High” value of the SC pairing\namplitude appears at the sublattice site [the sublattice A\nin Fig. 2(c)] where the surrounding bond current orders\nare weak and the bonds connected to the same sublat-\ntice sites carry either the same inflow current directions\nor the same outflow current directions. On the contrary,\nthe “Low” value of the SC pairing amplitude appears on\nthe sublattice sites where a pair of bonds connected to\nthe same sublattice sites have respective inflow and out-\nflow current directions. The uneven distribution of the\nSC pairing amplitude can be understood from the low\nenergy spectral distribution shown in Fig. 4(b). In the\ncoexisting phase of the NCFP and SC orders, since the\ndepletion of the low energy spectral weight only occurs\nat portions between MBandMC, the remained spec-\ntral weights, including the perfect van Hove points MA\ncaused by the deformation of the FS, mainly come from\nthe sublattice A. Consequently, the SC pairing ampli-\ntude on sublattice Ais significantly larger than those on\nthe sublattices BandC.\nDue to the uneven distributions of the SC pairing am-\nplitude and the scattering of the CDW order, a V-shaped\nDOS can be observed in Fig. 5(c), accompanied by multi-\nple sets of coherent peaks and residual zero-energy DOS,\nconstituting a characteristic of a nodal multi-gap SC pair-\ning state as having been observed in the STM experi-\nments28,29,42. For comparison, the dotted curve in the8\nsame figure portrays a typical U-shaped full gap struc-\nture for the DOS in the state with pure on-site s-wave\nSC pairing.\nConsidering the nature of the mean-field approxima-\ntion, it should be noted that the transition tempera-\nture between different phases in the calculated results\nis just qualitative rather than quantitative. Neverthe-\nless, the anti-correlation trend between the SC pairing\namplitude and the transition temperature of the nematic\nphase at different doping levels can be clearly observed\nin Fig. 5(d), as supported by the STM experiments85.\nIV. CONCLUSION\nIn conclusion, we have investigated the origin of the\nchiral CDW and its interplay with superconductivity in a\nfully self-consistent theory considering the orbital current\norder and the on-site SC pairing, which determines both\nthe CDW and the SC orders self-consistently. It was re-\nvealed that the self-consistent theory captures the salient\nfeature for the successive temperature evolutions of the\nordered electronic states from the high-temperature 2 ×2\nTCFP to the NCFP, and to the low-temperature s-wave\nSC state in a coexisting manner with the NCFP order.\nThe rotational symmetry breaking transition of the CDW\ncould be understood from a scenario in which the com-\npetition between the deviation from the VHF and the\nthermal broadening of the FS determines which state is\nit in. The intertwining of the s-wave SC pairing with the\nNCFP order produced a nodal gap feature manifesting\nas the V-shaped DOS along with the residual DOS near\nthe Fermi energy. The self-consistent theory not only\nproduced the successive temperature evolutions of the\nelectronically ordered states observed in experiment, but\nmight also offer a heuristic explanation to the two-fold\nrotational symmetry of electron state detected in both\nthe CDW and the SC states. Moreover, the intertwin-\ning of the SC pairing with the NCFP order, which was\nfound to be a ground state in the self-consistent theory\nat the low temperature regime, might also be a promis-\ning alternative for mediating the divergent or seemingly\ncontradictory experimental outcomes regarding the SC\nproperties. Overall, our study sheds light on the intri-\ncate relationship between the chiral CDW and supercon-\nductivity, providing valuable insights into the underlying\nmechanisms and experimental observations.\nV. ACKNOWLEDGEMENT\nThis work was supported by National Key Projects\nfor Research and Development of China (Grant No.\n2021YFA1400400), and the National Natural Science\nFoundation of China (Grants No. 12074175, No.\n12374137 and No. 92165205).Appendix: Temperature evolution of free energy,\ndetails of spectrum and band structures along\nhigh-symmetry cuts\nIn Fig. A1(a), we show the temperature evolution\nof the free energy in the TCFP and in the NCFP. In\nFig. A1(b), we display the momentum cut of the spectral\nweight along the MA→Γ→MAdirection [see Fig. 4(b)\nin the main text] at a doping level deviating from the\nVHF for the normal state and for the NCFP.\n0.020.030.040.050.060.07-21.22-21.21-21.20-21.19 \n Free energyT\nemperature TCFP \nNCFP\n0204060\n0204060 \nΓΜ ΑΜΑ \n Γ\nΜ ΑΜΑ \n IntensityM\nomentum cut normal state \nNCFP(b)( a)\nFIG. A1. (a) Temperature evolution of the free energy per\nsite for the doping level 9 .8/60 in the “triple- Q” CFP (solid\nline) and in the nematic CFP (dotted line). In obtaining the\nresults in (a), the CDW states are intentionally kept in their\nrespective forms during the calculations. (b) The momentum\ncut of the spectral weight along the MA→Γ→MAdirection\nat a doping level deviating from the VHF for the normal state\n(solid curve) and for the nematic CFP (dotted curve). The\npeaks of the spectral weight intensity denote the position of\nthe Fermi surface [refer to Fig. 4(b) in the main text]. The\nsame figure of (b) is replotted in the inset by breaking the x-\naxis in order to have a better view of the Fermi surface shift.\nIn Fig. A2, we present the unfolded dispersions of the\nspectral weight along different high-symmetry cuts for\nthe NCFP. Owing to the C2symmetry of the NCFP,\nthe energy bands exhibit different features along different\nhigh symmetry cut. Specifically, an energy gap opens\nnear the MB/C points but not near the MApoint, as\npresented respectively in Figs. A2(a) and (b).\n-4-3-2-1012Γ\nΜ CΚ' \n E/tΓ\n-4-3-2-1012Γ\nΜ ΑΚ \n E/tΓ\n(b)( a)\nFIG. A2. The unfolded dispersions along high-symmetry cuts\nΓ-K′-MC-Γ (a), and Γ- K-MA-Γ (b), respectively.9\n∗monsoonjhm@sina.com\n†slyu@nju.edu.cn\n‡jxli@nju.edu.cn\n1Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, Phys. Rev.\nLett. 98, 117205 (2007).\n2H. C. Jiang, Z. Y. Weng, and D. N. 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B 104,\n045122 (2021)." }, { "title": "2312.08067v1.Homogenization_of_2D_materials_in_the_Thomas_Fermi_von_Weizsacker_theory.pdf", "content": "Homogenization of 2D materials in the Thomas-Fermi-von\nWeizs¨ acker theory\nSaad Benjelloun∗,3, Salma Lahbabi†1,2, and Abdelqoddous Moussa‡,1\n1College of Computing, Universit´ e Mohamed 6 Polytechnique, Benguerir,\nMorocco\n2EMAMI, LRI, ENSEM, Hassan II University of Casablanca, Morocco\n3Makhbar Mathematical Sciences Research Institute, Casablanca, Morocco\nDecember 14, 2023\nAbstract\nWe study the homogenization of the Thomas-Fermi-von Weizs¨ acker (TFW) model for\n2D materials introduced in [3]. It consists in considering 2D-periodic nuclear densities with\nperiods going to zero. We study the behavior of the corresponding ground state electronic\ndensities and ground state energies. The main result is that these three dimensional\nproblems converge to a limit model that is one dimensional, similar to the one proposed\nin [9]. We also illustrate this convergence with numerical simulations and estimate the\nconverging rate for the ground state electronic densities and the ground state energies.\nKeywords— Thomas-Fermi-von Weizs¨ acker model, Homogenization, 2D materials, Periodic\nmodels, Crystals\nContents\n1 Introduction and main results 3\n1.1 Thomas-Fermi-von Weizs¨ acker model for finite systems . . . . . . . . . . . . . . . . . . 4\n1.2 Thomas-Fermi-von Weizs¨ acker model for 2D crystals . . . . . . . . . . . . . . . . . . . 4\n1.3 Homogenization of 2D materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2 Hartree interaction 8\n∗saad.benjelloun@makhbar.ma\n†s.lahbabi@ensem.ac.ma\n‡abdelqoddous.moussa@um6p.ma\n1arXiv:2312.08067v1 [math-ph] 13 Dec 20233 Homogenization procedure: proof of Theorem 1.5 10\n4 Numerical illustration 15\n21 Introduction and main results\nIn recent years, 2D materials became an active research field [8] due to their promising physical,\nelectrical, chemical, and optical properties that their 3D materials do not have [16, 7, 10]. Electronic\nstructure simulations are highly useful in the discovery of these properties and their tuning for the\npotential applications [17, 11, 15]. Thus the need of mathematical models and simulation algorithms\ntailored for 2D materials [5, 14, 18].\nDensity Functional Theory is one of the most widely used simulation tool in electronic structure\ncalculations. It consists in describing the electrons by their density ρand the energy of the system\nby a functional of ρ. A famous model in this class is the (orbital free) Thomas-Fermi-Von Weizs¨ acker\n(TFW) model [12, 19]. From a mathematical point of view, an important result in the study of crystals\nis the thermodynamic limit problem. It consists in proving that when a finite cluster converges to\nsome periodic perfect crystal, the corresponding ground state electronic density and ground state\nenergy per unit volume converge to the periodic equivalent. This program has been carried out for\nthe Thomas-Fermi-von Weizs¨ acker model for three dimensional (3D) crystals in [6] and for one and\ntwo dimensional (1D and 2D) crystals in [3].\nIn this paper, we investigate the homogenization of the TFW model for 2D crystals. Our goal is to\nfind a homogeneous material equivalent to a 2D crystal in the limit when the lattice parameter goes to\nzero. This corresponds to putting more and more (normalized) nuclei in the unit cell, or equivalently\nlooking at the crystal from further and further away. It turns out that the homogenized material can\nbe described by a 1D model, in the same spirit as in [9], which allows to reduce computational time\nand resources required to simulate a 2D material, at least in a zero order approximation. Our proof\ncan be generalized if we substitute theR\nρ5/3term in the kinetic energy byR\nρpfor some3\n2< p. Note\nthat the strict convexity of the energy functional gives the uniqueness of the ground state, which plays\nan important role in the proof.\nUp to our knowledge, the closest work to ours is [4], where the authors use the 2D TFW model [3]\nto derive macroscopic features of a crystal from the microscopic structure in the presence of an external\nelectric field. The crystal is modeled in the band R2×[−1,1] and micro-macro limit is taken when\nthe ration between the atomic spacing and the size of the crystal goes to zero.\nThe article is organized as follows. We start by recalling the Thomas-Fermi-von Weizs¨ acker model\nfor finite systems in Section 1.1, and for 2D crystals in Section 1.2. In Section 1.3, we define the\nhomogenization process, along with the limit problem, and state our main result, whose proof is de-\ntailed in Section 3. Intermediate results about the 2D Coulomb interaction are presented in Section 2.\nFinally, numerical illustrations are gathered in Section 4.\nAcknowledgments\nThe research leading to these results has received funding from OCP grant AS70 “Towards phosphorene\nbased materials and devices”. Prof. S. Lahbabi thanks the CEREMADE for hosting her during the\nfinal writing of this article.\n31.1 Thomas-Fermi-von Weizs¨ acker model for finite systems\nWe present in this section the TFW model for finite systems. Let m∈L1(R3) be a finite nuclear\ncharge density. The state of the electrons is described by a non negative electronic density ρ∈L1(R3).\nThe energy functional is given by\nEm(ρ) =Z\nR3|∇√ρ|2+Z\nR3ρ5/3+1\n2D(ρ−m, ρ−m), (1)\nwhere the first two terms represent the kinetic energy of the electrons and D(f, g) is the Coulomb\ninteraction between charge densities fandgin the Coulomb space C=n\nf∈ S′(R3),bf\n|·|∈L2(R3)o\n.\nIt is defined by\nD(f, g) =Z\nR3Z\nR3f(x)g(y)\n|x−y|dxdy= 4πZ\nR3bf(k)bg(k)\n|k|2dk,\nwhere ˆf(k) denotes the k−th Fourier coefficient for f. The ground state is given by the following\nminimization problem\nIm= inf\u001a\nEm(ρ), ρ⩾0,√ρ∈H1(R3),Z\nR3ρ=Z\nR3m\u001b\n. (2)\nIt is well known that problem (2) has a unique minimizer ρ(see for instance [2]). u=√ρis the unique\nsolution of the corresponding Euler-Lagrange equation\n(\n−∆u+5\n3u7/3+uΦ = λu,\n−∆Φ = 4 π(u2−m), (3)\nwhere λ∈R,and Φ = ( u2−m)∗1\n|·|is the mean-field potential.\n1.2 Thomas-Fermi-von Weizs¨ acker model for 2D crystals\nWe present in this section the TFW model for 2D crystals introduced in [3]. 2D crystals are charac-\nterized by a nuclear density mthat has the periodicity of a 2D lattice R=a1Z+a2Z, where ( a1, a2)\nare two linearly independent vectors in R2(see Figure 1), namely\nm(x1+k1, x2+k2, x3) =m(x1, x2, x3),∀x∈R3,∀(k1, k2)∈ R.\nFrom now on, we denote by Qthe unit cell of R ⊂R2and by Γ = Q×Rthe unit cell of Rseen as a\nlattice in R3. For x∈R3, we denote by x= (x1, x2)∈R2so that x= (x, x3).\nBy means of a thermodynamic limit procedure, it has been shown in [3] that 2D crystals can be\ndescribed by a model similar to (1)-(2) posed on the unit cell Γ. The main difference is that the 3D\nGreen function1\n|x|is replaced by the 2D periodic Green function Gsolution of\n−∆G= 4πX\nk∈R×{ 0}δk.\n4t t t t t tt t t t t tt t t t t tt t t t t tt t t t t t\nr r\nr rQ\n\u0003\u0003\u0003\n\u0003\u0003\u0003\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\u0001 \u000bx1-x26x3\u0003\u0003\u0003\n\u0003\u0003\u0003\n\u0003\u0003\u0003\n\u0003\u0003\u0003\nΓ\nFigure 1: Example of a 2D lattice and its unit cell Γ.\nAn explicit formula of Gis given by [3, equation 11]\nG(x) =−2π\n|Q||x3|+X\nk∈R \n1\n|x−(k,0)|−1\n|Q|Z\nQdy\f\fx−\u0000\ny+k,0\u0001\f\f!\n. (4)\nIt can be seen as the sum over the lattice of the Coulomb potential created by a point charge placed at\nthe lattice sites, screened by a uniform background of negative unit charge. A Fourier decomposition\nofGcan be found in [9]. The 2D crystals energy functional then reads\nEm\nper(ρ) =Z\nΓ|∇√ρ|2+Z\nΓρ5/3+1\n2DG(m−ρ, m−ρ), (5)\nwhere the Hartree interaction DGis given by\nDG(f, g) :=Z\nΓZ\nΓf(x)g(y)G(x−y) dxdy.\nRemark 1.1. In [3], there is an Coulomb correction term in the energy which comes from the Dirichlet\nboundary condition at infinity considered in the thermodynamic limit procedure, so that the energy is\ngEmper(ρ) =Em\nper(ρ) +1\n|Q|Z\nΓZ\nΓf(x)g(y)\n|x−y|.\nIn our study, we omit this correction term as it does not affect the problem from a mathematical point\nof view.\nThe ground state is given by the minimization problem\nIm\nper= inf\u001a\nEm\nper(ρ), ρ⩾0,√ρ∈Xper,Z\nΓρ=Z\nΓm\u001b\n(6)\nwhere\nXper=n\nv∈H1\nper(Γ),(1 +|x3|)1/2v∈L2\nper(Γ)o\n.\nThis problem has been studied in [3] along with some basic properties of the solution, that we sum-\n5marize in the following Theorem.\nTheorem 1.2 ([3, Theorem 3.2]) .Letm̸= 0 be a smooth non-negative R-periodic function with\ncompact support with respect to x3. Then, the minimization problem (6)has a unique minimizer ρ.\nu=√ρis the unique solution of the corresponding Euler-Lagrange system\n\n\n−∆u+5\n3u7/3+uΦ =λu,\n−∆Φ = 4 π(u2−m),(7)\nwhere λ∈R. In addition, u∈L∞(R3)and|u(x)|⩽C\n1 +|x3|3/2, for|x3|>1,C >0being a constant\nindependent of the density m.\nRemark 1.3. The fact that, in the inequality |u(x)|⩽C\n1 +|x3|3/2, for|x3|>1, the constant is\nindependent of the density m, comes from a supersolution method applied to the equation (7)(see [3,\nTheorem 3.1-Theorem 2.3]).\n1.3 Homogenization of 2D materials\nIn the framework of the Thomas-Fermi (TF) model and the reduce Hartree Fock model (rHF), the\nrecent work [9] studies reduced models for 2D homogeneous materials. The idea is that if the material\nis homogeneous, the 3D model is equivalent to a 1D model; simpler and less costly to simulate. In the\npresent work, we are interested in the homogenization procedure, which models looking at a crystal\nmacroscopically, from further and further away. Namely, we put more and more nuclei in the unit\ncell, with the right charge normalization, and we ask the following questions:\n•What is the nuclear density at the limit?\n•Does the ground state electronic density, potential and energy converge?\n•What is the model describing the limit electronic structure?\nLetmbe a nuclear density satisfying the conditions of Theorem 1.2 and consider the following\nsequence of nuclear densities which consists in putting Nsmall nuclei in the unit cell\nmN(x1, x2, x3) :=m(Nx1, Nx 2, x3),∀x∈Γ. (8)\nWe note that mNis1\nNR-periodic, which describes a more homogeneous material than the initial one.\nThe limit when N→+∞describes a homogeneous material (see Figure 2). We will show that this\nmodel ”converges”, in some sense that we will precise later, to the following limit model. For a 1D\nnuclear density µ, we introduce the energy functional\nEµ\n1(ρ) =Z\nR|∇√ρ|2+Z\nRρ5/3+1\n2D1(ρ−µ, ρ−µ) (9)\nwhere the 1D Hartree interaction D1is defined for functions decaying fast enough by\nD1(f, g) =−2πZ\nR\u0012Z\nR|s−t|f(s)g(t) dt\u0013\nds\n6Figure 2: The homogenization process for the nuclear density mN, illustrated at N= 1,4 and\n16 from left to right.\n(more details about the 1D Hartree interaction can be read in [9]). The ground state of this model is\ngiven by the minimization problem\nIµ\n1= min {Eµ(ρ), ρ⩾0,√ρ∈X1,Z\nRρ=Z\nRµ}, (10)\nwhere\nX1=n\nv∈H1(R),(1 +|t|)1/2v∈L2(R)o\n.\nThe following Theorem, which is a direct consequence of Theorem 1.2 and the definitions of Hartree\ninteraction terms DGandD1, is similar to [9, Theorem 2.2, Theorem 2.8], which treat Thomas-Fermi\nand reduced Hartree-Fock models.\nTheorem 1.4. Letµ∈ D(R). Problem (10) has a unique minimizer. u=√ρis the unique solution\nof the corresponding Euler Lagrange equations\n\n\n−u′′+5\n3u7/3+uΦ =λu,\n−Φ′′= 4π(u2−µ),\nwhere λ∈R, and there exists C >0, independent of µ, such that for any |t|⩾1,|u|⩽C\n1+|t|3/2.\nOur main contribution is the following Theorem.\nTheorem 1.5. Letmbe a nuclear density satisfying the hypothesis of Theorem 1.2 and for N∈N\\{0},\nletmNbe defined as in (8). We denote by IN=ImNand by ρNthe corresponding ground state given\nby Theorem 1.2. Let m0(t) =1\n|Q|R\nQm(x, t) dxand denote by I0=Im0\n1and by ρ0the corresponding\nground state given by Theorem 1.4. The following holds\ni.lim\nN→∞IN=I0,\nii.ρNconverges to ρ0inL1(Γ), inLp\nloc(Γ)∀1⩽p⩽3and almost everywhere,\niii.√ρNconverges√ρ0weakly in H1\nper(Γ).\nOur result is a zero order approximation of a finite Nsituation. In homogenization theory of partial\ndifferential equations with periodically oscillating coefficients, the two scale convergence technique is\nusually used to find higher order terms in the approximation [1]. Applying a similar approach to our\nproblem is a perspective of this work.\n72 Hartree interaction\nWe recall in this section some properties of the Green function G, the Hartree interaction DGand\nprove a uniqueness result on the mean-field potential Φ. In the subsequent proposition, we introduce\na convenient decomposition for the kernel G, that proves to be useful throughout the paper.\nProposition 2.1. We have\nG(x) =−2π\n|Q||x3|+1\n|x|+ψ(x),\nwhere ψ∈L∞(Γ).\nProposition 2.1 is partially proved in [3]. We detail the proof here for consistency.\nProof. Using (4), we have\nψ(x) =−1\n|Q|Z\nQdy\f\fx−(y,0)\f\f+X\nk∈(R2)∗ \n1\n|x−(k,0)|−1\n|Q|Z\nQdy\f\fx−(y+k,0)\f\f!\n.\nLet us show that ψ∈L∞(Γ). The function x7→1\n|Q|Z\nQdy\f\fx−(y,0)\f\fis continuous on Γ and goes to\nzero as |x3| → ∞ . Moreover, the sum defining ψconverges normally on a neighborhood of 0 R3[3,\nProposition 3.2]. Therefore it is continuous on that neighborhood, which implies that ψis bounded\nonQ×[−ε, ε], for some ε >0. From [3, Equation 3.12], there exists C >0 such that\nX\nk∈(R2)∗ \n1\n|x−(k,0)|−1\n|Q|Z\nQdy\f\fx−(y+k,0)\f\f!\n⩽C\n|x3|α,∀α <1.\nIt follows that the x7→P\nk∈(R2)∗ \n1\n|x−(k,0)|−1\n|Q|R\nQdy\f\fx−(y+k,0)\f\f!\nis bounded on x∈Q×(R\\\n[−ε, ε]), thus ψis also bounded on the same interval.\nAs a consequence of the above proposition, we prove useful properties of the potential Φ. Let us\nintroduce some notations. For some domain Ω ⊂Rdand 1 ⩽p⩽∞, we denote by\nLp\nunif(Ω) =\u001a\nf∈Lp\nloc(Ω),∀r >0,sup\nx+Br⊂Ω∥f∥Lp(B(x,r))<∞\u001b\n,\nwhere B(x, r) is the ball of radius rcentered at x. For f∈Lp(Γ) and g∈Lq(Γ), the convolution\n(f∗Γg)(x) =Z\nΓf(x−y)g(y) dy\nis well defined in Lr(Γ), where1\np+1\nq= 1 +1\nr.\nProposition 2.2. Let\nYper=\u001a\nf∈L1(Γ)∩L5/3(Γ),Z\nΓf= 0,|x|f∈L1(Γ)\u001b\n.\n8The map\nYper→L∞(Γ)\nf7→G∗Γf\nis well defined and continuous. Moreover, for f∈Yper,G∗Γfis the unique solution, up to an additive\nconstant, of the Poisson equation(\nΦ∈L1\nunif(R3)\n−∆Φ = 4 πf.(11)\nProof. From Proposition 2.1, we have that G(x) =1\n|x|−2π\n|Q||x3|+ψ(x), with ψ∈L∞(Γ). We are\nthus going to bound the three functions\nf∗Γ|x3|, f∗Γ1\n|x|andf∗Γψ\ninL∞(Γ). First, since ψ∈L∞(Γ), we have for any x∈Γ\n\f\f\f\fZ\nΓf(x−y)ψ(y) dy\f\f\f\f⩽∥ψ∥L∞(Γ)∥f∥L1(x−Γ).\nThus\n∥f∗Γψ∥L∞(Γ)⩽∥ψ∥L∞(Γ)∥f∥L1(Γ).\nWe move to f∗Γ1\n|x|. We have\n\f\f\f\ff∗Γ1\n|x|(x)\f\f\f\f⩽Z\nΓ|f(x−y)|\n|y|dy⩽Z\nΓ|f(x−y)|dy+Z\nΓ|f(x−y)|\n|y|1|y|<1dy.\nFor the second term, we use H¨ older inequality with f∈L5/3(Γ) and1\n|y|1|y|<1∈L5/2(Γ). We conclude\nthat\r\r\r1\n|x|∗Γf\r\r\r\nL∞⩽∥f∥L1(Γ)+∥f∥L5/3(Γ)\r\r\r\r1\n|y|1|y|<1\r\r\r\r\nL5/2(Γ).\nRegarding the last term, we use the neutrality assumption to write\nZ\nΓf(x−y)|y3|dy=Z\nΓf(x−y)(|y3| − |x3|) dy.\nThus, using the triangle inequality and the R-periodicity of x7→x3f(x), we obtain\n\f\f\f\fZ\nΓf(x−y)|y3|dy\f\f\f\f⩽Z\nΓ|f(x−y)||x3−y3|dy=Z\nΓ|f(y)||y3|dy.\nFinally, we need to prove that Φ −G∗Γfis constant for any solution Φ of (11). Notice first that\nh= Φ−G∗Γfis a harmonic function over R3. By the mean value Theorem for a harmonic functions,\nwe have that\n(Φ−h) (x) =Z\nB(x,1)(Φ−h) (y) dy⩽sup\nx∈ΓZ\nB(x,1)(Φ−h) (y) dy⩽∥Φ−h∥L1\nunif.\nTherefore, Φ −his harmonic and bounded in Γ .By Louisville’s Theorem, we conclude that Φ −his\n9constant.\nCorollary 2.3. Letf∈Yper, then ∇(G∗Γf)∈\u0000\nL2(Γ)\u00013and there exists C >0, such that\nDG(f, f) =∥∇(G∗Γf)∥2\nL2(Γ)⩽C∥f∥L1(Γ)\u0010\n∥f∥L1(Γ)+∥f∥L5/3(Γ)+∥|x3|f∥L1(Γ)\u0011\n.\nProof. We have\nDG(f, f) =Z\nΓ(G∗Γf)×f=Z\nΓ(G∗Γf)×∆(G∗Γf) =Z\nΓ|∇(G∗Γf)|2.\nBesides, by the previous proposition G∗Γf∈L∞(Γ) and\n∥G∗Γf∥L∞(Γ)⩽(∥ψ∥L∞(Γ)+ 1)∥f∥L1(Γ)+∥f∥L5/3(Γ)\r\r\r\r1\n|y|1|y|<1\r\r\r\r\nL5/2(Γ)+∥|x|f∥L1(Γ)⩽C∥f∥Yper,\nwhich proves the inequality stated in the corollary.\n3 Homogenization procedure: proof of Theorem 1.5\nThis section is devoted to the proof of Theorem 1.5. The strategy of the proof is as follows. We start by\nproving the convergence of the nuclear densities ( mN). Indeed, when Nincreases, the nuclear density\nmNbecomes more homogeneous and the sequence ( mN) converges to the 2D homogeneous density\nm0(x3) =1\n|Q|R\nQm(x) dx(see Lemma 3.3). The electronic densities ρNare uniformly bounded with\nrespect to N. We can thus extract convergent subsequences (see Proposition 3.5). Both convergences\ngive the upper bound I⩽lim inf IN. To prove the lower bound I0⩾lim sup IN, we use ρ0as a test\nfunction. The proof is divided into four steps.\nStep 1: Properties of the sequence (mN)We state in this section two properties of the\nsequence ( mN).\nLemma 3.1. Forp⩾1andf∈L1\nloc(R)such that x7→f(x3)mp(x)∈L1(Γ), we have\nZ\nQmp\nN(x, x3) dx=Z\nQmp(x, x3) dx,\nand Z\nΓf(x3)mp\nN(x) dx=Z\nΓf(x3)mp(x) dx.\nProof. We have\nZ\nQmp\nN(x1, x2, x3) dx1dx2=Z\nQmp(Nx1, Nx 2, x3) dx1dx2=1\nN2Z\nNQmp(y1, y2, x3) dy1dy2.\nAsmisQ-periodic then,Z\nNQmp(y1, y2, x3) dy1dy2=N2Z\nQmp(y1, y2, x3) dy1dy2. Hence, the first\nclaim is proved. The second claim easily follows.\n10Remark 3.2. By the second point of Lemma 3.1, we have thatR\nΓmN=R\nΓm=R\nΓm0, so that any\nadmissible state for INis also an admissible state for I0, and vice versa.\nLemma 3.3. The sequence (mN)converges to m0weakly in Lp(Γ)∀1⩽p <+∞.\nProof. From the Q-periodicity of m, we have for any x3∈R(see for instance [13])\nmN(·,·, x3)⇀ m 0(·,·, x3) in Lp(Q)∀1⩽p <∞. (12)\nForφ∈Lq(Q) and ψ∈Lq(R), with1\nq+1\np= 1, we have fN(x3) :=Z\nQmN(x)φ(x)ψ(x3) dx→f(x3) :=\nm0(x3)ψ(x3)Z\nQφ(x) dxa.e. and\n|fN(x3)|⩽∥φ∥Lq(Q)|ψ(x3)|\u0012Z\nQ|mN(x, x3)|pdx\u00131/p\n=∥φ∥Lq(Q)|ψ(x3)|\u0012Z\nQ|m(x, x3)|pdx\u00131/p\n.\n(13)\nThus, by dominated convergence Theorem,\nZ\nRfN(x3) dx3=Z\nΓmN(x)φ(x)ψ(x3) dx→Z\nRf(x3) dx3=Z\nΓm0(x3)φ(x)ψ(x3) dx.\nBy the density of Lq(Q)⊗Lq(R) inLq(Γ), the proof is complete.\nStep 2: Convergence of electronic densities The following proposition gives uniform bounds\non quantities of interest with respect to N.\nProposition 3.4. There exist various constants Cindependent of Nsuch that the following bounds\nhold.\nI.IN⩽C,\nII.∥√ρN∥H1(Γ)⩽C,\nIII.∥ρN∥Lp(Γ)⩽Cfor all 1⩽p⩽3,\nIV.DG(ρN−mN, ρN−mN)⩽C,\nProof. AsρNis the minimizer of INand since mis an admissible test function for EmNper, we have\nIN=EmNper(ρN)⩽EmNper(m) =Z\nΓ\f\f∇√m\f\f2+Z\nΓm5/3+1\n2DG(m−mN, m−mN).\nBy corollary 2.3 applied to f=m−mNand Lemma 3.1, we have\nDG(m−mN, m−mN)⩽C∥m−mN∥L1(Γ)\u0010\n∥m−mN∥L1(Γ)+∥m−mN∥L5/3(Γ)\n+∥|x3|(m−mN)∥L1(Γ)\u0011\n⩽C∥m∥L1(Γ)\u0010\n∥m∥L1(Γ)+∥m∥L5/3(Γ)+∥|x3|m∥L1(Γ)\u0011\n.\nPoints II-IV are easily deduces from point point I.\n11Proposition 3.5. There exists a non negative function ρ0∈L1(Γ)∩L5/3(Γ)such that, up to a\nsubsequence,\n•(ρN)converges to ρ0strongly in L1(Γ)andLp\nloc(Γ)for all 1⩽p⩽3, weakly in Lp(Γ)and almost\neverywhere on R3,\n•(∇√ρN)converges to ∇√ρ0weakly in (L2(Γ))3\n•ρ0−m0∈Yper\n•The sequence ∇ΦN=∇(G∗Γ(ρN−mN))converges to ∇Φ0=∇(G∗Γ(ρ0−m0))weakly in\n(L2(Γ))3.\nAs a consequence\nEm0per(ρ0)⩽lim inf IN.\nProof. The sequence (√ρN)Nis uniformly bounded with respect to NinH1(Γ) by Proposition 3.4.\nThen, up to a subsequence, it converges to some non negative function u0weakly in H1(Γ), strongly\ninLp\nloc(Γ) for all 2 ⩽p⩽6 and almost everywhere on R3. Besides, by Theorem 1.2, there exists C⩾0\nsuch that for any Nand any x∈R3such that |x3|⩾1, it holds\n|uN(x)|⩽C\n1 +|x3|3/2.\nBy the almost everywhere convergence, u0satisfies the same estimate, and we have for ρ0:=u2\n0the\nfollowing estimate\n|x3|ρ0(x)⩽ρ0(x)1|x3|⩽1+C\n1 +|x3|3∈L1(Γ).\nThus ρ0∈L1(Γ)∩L5/3(Γ) and |x3|ρ0∈L1(Γ).\nWe show now that ρNconverges to ρ0strongly in L1(Γ). Let ε >0 and Rlarge enough such that\nfor any M, N ∈N\\ {0}\nZ\n|x3|>R|ρN−ρM|⩽1\nRZ\n|x3|>R|x3||ρN−ρM|⩽C\nR⩽ε\n2.\nBy the strong convergence of ρNinL1\nloc(Γ), for M, N large enough, we have\nZ\n|x3|0. We take ε= 10−6in our\nsimulations.\nWe adopt a periodic setting, with the unit cell Γ = Q×[−L\n2,L\n2], with Q=\u0002\n−1\n2,1\n2\u00032. We use a\nFourier series decomposition approach to solve the linearized equation (21). The solution is expected\nto be R−periodic with respect to x1andx2and decaying at infinity with respect to |x3|. Taking a\nlarge value for L, it is reasonable to impose period conditions in the third direction as well. Therefore,\nwe consider the Fourier basis\nfk(x) =1√\nLei2π\u0010\nx·k+k3x3\nL\u0011\n.\nWe illustrate the convergence results in Theorem 1.5 using the series of nuclei densities mNdefined\nas\nmN(x1, x2, x3) =µ(N·x1)m0(x3) =π\n2|cos(Nπx 1)| ×5 exp\u0012\n−x2\n3\n8\u0013\n.\nWe note thatR\nQµ(N·x1) dx1dx2=π\n2R\nQ|cos(N ·πx1)|= 1. We take L= 2π, which proves effective\nas the function m0(x3) = 5 exp\u0010\n−x2\n3\n8\u0011\nrapidly decreases in the x3direction. The same behaviour\nis assumed for the solutions uN, and is validated with 1D simulations for ( u0, ρ0), the 1D solution\ncorresponding to m0.\nThe different functions are sampled on a grid of dimensions (200 ×N,4,300). We keep the dis-\ncretization constant along the yandz-axis as the homogenization is only applied in the x3dimension.\nAs we will only compute the zero Fourier mode along x2, four points are largely enough. We compute\nthe positive Fourier modes up to indices K= (K1, K2, K3) = (4 ×N,0,6). On the x3-axis, we use the\nfirst 7 Fourier modes, found to be sufficient for accurate reconstruction of mNandm0along that axis.\nSimulations in 1D prove that the solution ( u0(x3), ρ0(x3)), is also accurately represented by these 7\nmodes. We assume that this is also the case in 3D for all values of N. The choice of (4 N+ 1) Fourier\nmodes in the x1direction, is motivated by the need for increased Fourier modes as Ngrows, to capture\nthe changing characteristics and new harmonics for mNand ’eventually’ for the solution ( ρN, uN).\nIn fact, we know that ( ρN, uN) will rather converge to a constant along the axis x1by Theorem 1.5,\nhowever we enforce a hard check for this by computing also large modes for ρNanduN.\nDifferent Lp-norms of the error eN=ρN−ρ0are presented in Figure 3 for values of N= 1, . . . , 5.\nWe, indeed, observe the convergence stated in Theorem 1.5. The same figure shows some convergence\nrates estimates. We conjecture that we have a theoretical convergence rate of 1 /Nrwith10\n3< r < 4\nfor the different norms of enand the gradient L2norm∥∇un− ∇u0∥2. The discrepancies in Figure 3\natN= 4,5 are probably due to numerical errors.\nFigure 4 illustrates the convergence of the ground state energy IN=EmNper(ρN) to the 1D energy\nI0=Em0\n1(ρ0), proved in Theorem 1.5. The estimated convergence rate for the energy is of the\norder≈1\nN3/2.\nRemark 4.1. It is noteworthy to mention that, for N⩾6, and using higher order Fourier modes,\n16Figure 3: Convergence analysis for ennorms (left) and convergence rates estimation for en\nnorms and the gradient L2norm∥∇un− ∇u0∥2(right).\nFigure 4: Convergence analysis for the energies IN(left) and convergence rate estimation\n(right).\nthe presence of numerical errors hinders a clear observation of further convergence for eN.\nReferences\n[1]G. Allaire ,Homogenization and two-scale convergence , SIAM Journal on Mathematical Anal-\nysis, 23 (1992), pp. 1482–1518.\n[2]R. Benguria, H. Brezis, and E. Lieb ,The Thomas-Fermi-von weizs¨ acker theory of atoms\nand molecules , Communications in Mathematical Physics, 79 (1981), pp. 167–180.\n[3]X. Blanc and C. Le Bris ,Thomas-fermi type theories for polymers and thin films , Advances\nin Differential Equations, 5 (2000), pp. 977–1032.\n[4]X. Blanc and R. Monneau ,Screening of an applied electric field inside a metallic layer de-\nscribed by the Thomas-Fermi-von Weizs¨ acker model , Advances in Differential Equations, 7 (2002),\npp. 847 – 876.\n[5]A. Carvalho, P. E. Trevisanutto, S. Taioli, and A. H. C. Neto ,Computational methods\nfor 2d materials modelling , Reports on Progress in Physics, 84 (2021), p. 106501.\n17[6]I. Catto, C. Le Bris, and P. L. Lions ,Mathematical theory of thermodynamic limits :\nThomas- Fermi type models , Oxford University Press, 1998.\n[7]V. Chaudhary, P. Neugebauer, O. Mounkachi, S. Lahbabi, and A. E. Fatimy ,Phos-\nphorene—an emerging two-dimensional material: recent advances in synthesis, functionalization,\nand applications , 2D Materials, 9 (2022), p. 032001.\n[8]A. Geim and I. Grigorieva ,Van der Waals heterostructures , Nature, 499 (2013), pp. 419–425.\n[9]D. Gontier, S. Lahbabi, and A. Maichine ,Density functional theory for two-dimensional\nhomogeneous materials , Communications in Mathematical Physics, 388 (2021), pp. 1475–1505.\n[10]D. Gupta, V. Chauhan, and R. Kumar ,A comprehensive review on synthesis and applications\nof molybdenum disulfide MoS 2material: Past and recent developments , Inorganic Chemistry\nCommunications, 121 (2020), p. 108200.\n[11]F. Hussain, M. Imran, and H. Ullah ,Density Functional Theory (DFT) Study of Novel 2D\nand 3D Materials , Springer Singapore, Singapore, 2017, pp. 269–284.\n[12]E. H. Lieb and B. Simon ,The Thomas-Fermi theory of atoms, molecules and solids , Advances\nin Mathematics, 23 (1977), pp. 22–116.\n[13]D. Lukkassen and P. Wall ,On weak convergence of locally periodic functions , Journal of\nNonlinear Mathematical Physics, 9 (2002), pp. 42–57.\n[14]A. Patra, S. Jana, P. Samal, F. Tran, L. Kalantari, J. Doumont, and P. Blaha ,Effi-\ncient band structure calculation of two-dimensional materials from semilocal density functionals ,\nThe Journal of Physical Chemistry C, 125 (2021), p. 11206–11215.\n[15]K. Ren, M. Sun, Y. Luo, S. Wang, J. Yu, and W. Tang ,First-principle study of electronic\nand optical properties of two-dimensional materials-based heterostructures based on transition\nmetal dichalcogenides and boron phosphide , Applied Surface Science, 476 (2019), pp. 70–75.\n[16]J. Saha and A. Dutta ,A review of graphene: Material synthesis from biomass sources , Waste\nand Biomass Valorization, 13 (2022), p. 1385–1429.\n[17]Q. Tang, Z. Zhou, and Z. Chen ,Innovation and discovery of graphene-like materials via\ndensity-functional theory computations , WIREs Comput Mol Sci, 5 (2015), pp. 360–379.\n[18]S. Tawfik, O. Isayev, C. Stampfl, J. Shapter, D. Winkler, and M. J. Ford ,Efficient\nprediction of structural and electronic properties of hybrid 2d materials using complementary DFT\nand machine learning approaches , ChemRxiv, (2018).\n[19]C. F. v. Weizs ¨acker ,Zur theorie der kernmassen , Zeitschrift f¨ ur Physik, 96 (1935), pp. 431–\n458.\n18" }, { "title": "2401.14809v1.Ground_state_energy_of_Bogoliubov_energy_functional_in_the_high_density_limit.pdf", "content": "arXiv:2401.14809v1 [math-ph] 26 Jan 2024GROUND STATE ENERGY OF BOGOLIUBOV ENERGY FUNCTIONAL IN\nTHE HIGH DENSITY LIMIT\nNORBERT MOKRZAŃSKI AND BARTOSZ PAŁUBA\nAbstract. We consider the Bogoliubov energy functional proposed by Na piórkowski, Reuvers and\nSolovej and analize it in the high density regime. We derive a two term asymptotic expansion of\nthe ground state energy.\n1.Introduction\nIn 1947 Nikolay Bogoliubov published the work \"On the theory of superfluidity\" [3], which\nturned out to be a fundamental result in the theory of interac ting Bose gases. The main goal of\nthat paper was a microscopic derivation of Landau’s criteri on of superfluidity which is quantum\neffect observed in liquid helium4He. Bogoliubov, using a heuristic argument, proposed that t he\nHamiltonian of the system can be replaced by an effective simp ler one. There where two main\nassumptions made by Bogolibov. The first one was that there is Bose-Einstein Condensation in\nthe system. The second one was that the strength of interacti on between two particles outside\nthe condensate is negligible. Hence, to obtain an effective H amiltonian, terms involving more than\ntwo creation or annihilation operators of non-zero modes ca n be dropped. This led to an effective\nquadratic Hamiltonian which could be diagonalized explici tly, leading to an excitation spectrum\nthat allowed to explain the superfluidity phenomenon using t he Landau argument [13].\nBogoliubov’s approach, although it was lacking mathematic al rigor, appeared to be very useful\nfor describing bosonic interacting systems. When Bose-Ein stein condensates had begun to be\nobtained in experiments involving cold atoms [1, 7], this to pic became particularly interesting in\nboth theoretical and mathematical physics communities. Si nce then significant progress regarding\nunderstanding of this theory has been made (see [15, 20] for r ecent reviews).\nIn this paper we want to revisit the approach proposed by Napi órkowski, Reuvers and Solovej\nin [16, 17, 18, 9] (and before that by Critchley and Solomon in [6]). The authors analyze the\nthermodynamic properties of the interacting Bose gas syste m using variational perspective based on\nBogoliubov theory. They introduce the free energy (density ) functional Fobtained by evaluating\nthe free energy expectation value on quasi-free states and p assing to the thermodynamic limit (see\nnext section for more detailed discussion). In the first pape r [16] the authors prove the existence of\nthe minimizers under certain assumptions on the interactio n potential V(in [19] it has been shown\nthat some of those assumptions can be weakened). They also pr ove that this model predicts the\nphase transition in the system. In the second paper [17] auth ors consider the dilute limit and obtain\nresults on the critical temperature and on the expansion of t he free energy in terms of temperature\nand total density ρof the system. In [9] the authors considered the dilute two-d imensional model\nand obtained a ground state energy expansion. The main reaso n why the authors analyzed this\nvariational model is that the many-body problem (i.e. start ing from the many-body Schrödinger\nequation) in the thermodynamics limit is very difficult and th e results are rather limited (see\n[11, 2, 10] for most recent ones).\nThe goal of this paper is to investigate the opposite, high de nsity limit of the Bogoliubov\nfunctional in the case of zero temperature. From the many-bo dy perspective this problem was for\nthe first time considered by Lieb in [14]. In that paper he argu ed how the two leading terms of the\nground state energy should look like. The lower bound is in fa ct a straightforward consequence of\nthe Onsager lemma (see [8, Lemma 7.2]). Lieb also claims the u pper bound, but some details are\n12 N. MOKRZAŃSKI AND B. PAŁUBA\nmissing there (we plan to fill the gaps in a future work). Later the high density asymptotics have\nbeen derived within an effective model introduce by Lieb and t hen analyzed by Carlen, Jauslin and\nLieb [4, 5, 12]. Our analysis is motivated by the question of o ne of the authors above who asked\nwhether the Bogoliubov functional predicts the high densit y free energy expansion in agreement\nwith the works above. We show that this is indeed the case. Our second result shows that imposing\nsome extra conditions on the regularity of the potential lea ds to obtaining better, more precise\nestimations of the error term.\nThe paper is structured as follows: in the next section we wil l sketch the derivation of the\nfunctional and formulate the main result. The following sec tion is devoted to the proof of the\nmain theorem. In the last section we obtain additional error estimates.\nAcknowledgments. We want to thank Prof. Marcin Napiórkowski for many helpful d iscussions\nand support. The work of NM was supported by the Polish-Germa n NCN-DFG grant Beethoven\nClassic 3 (project no. 2018/31/G/ST1/01166).\n2.The functional and the main result\nWe will start with a brief sketch of the derivation of the Bogo liubov energy functional. We refer\nto the appendix of [16] for full details of the derivation, al so for the non-zero temperature case\nconcerning the entropy.\nThe Hamiltonian of the interacting Bose gas in the box of size L(in the momentum represen-\ntation) is given by:\nH=/summationdisplay\np∈2π\nLZd/parenleftbig\np2−µ/parenrightbig\na∗\npap+1\n2L3/summationdisplay\np,q,k∈2π\nLZd/hatwideV(k)a∗\np+ka∗\nq−kaka−k, (1)\nwherepis a momentum of the particles, µis a chemical potential of the gas, /hatwideV(k)is a Fourier\ncoefficient of the (periodized) potential function Vandap,a∗\npare respectively annihilation and\ncreation operators of a particle with momentum p. Inspired by the fact that the ground states of\nquadratic Hamiltonians are quasi-free states, we will rest rict our attention to such states only. By\ndefinition these are the states that satisfy Wick rule and the refore are fully determined by density\nfunction γand function of pairing α:\nγ(p) =/an}bracketle{ta∗\npap/an}bracketri}ht, α(p) =/an}bracketle{tapa−p/an}bracketri}ht=/an}bracketle{ta∗\npa∗\n−p/an}bracketri}ht.\nIn order to include the possibility of Bose-Einstein conden sation in the system, we use Weyl\ntransformation to introduce the density of the condensate ρ0. This operation transforms the\nannihilation operators as:\nap→ap+δ0,p/radicalbig\nL3ρ0,\nanalogously for the creation operators.\nHaving the transformed Hamiltonian, we consider its expect ation value in the quasi-free, trans-\nlation invariant state. Dividing this expectation value by the volume of the system L3and formally\npassing to the thermodynamic limit ( N→ ∞, L→ ∞,ρ:=N\nL3=const) we conclude that the\nenergy (density) may be described by the functional Fgiven by the formula:\nF(γ,α,ρ 0) = (2π)−3ˆ\nR3p2γ(p)dp+1\n2/hatwideV(0)ρ2\n+ρ0(2π)−3ˆ\nR3/hatwideV(p)[γ(p)+α(p)]dp\n+1\n2(2π)−6¨\nR3×R3/hatwideV(p−q)[α(p)α(q)+γ(p)γ(q)]dpdq(2)GROUND STATE ENERGY OF BOGOLIUBOV ENERGY FUNCTIONAL IN THE H IGH DENSITY LIMIT 3\nHereρis the total density of the system, defined as:\nρ=ρ0+(2π)−3ˆ\nR3γ(p)dp=:ρ0+ργ, (3)\nwhereργwas introduced to denote the density of the particles outsid e the condensate. For our\npurpose, the convention of the Fourier transform and its inv erse is:\n/hatwideV(p) =ˆ\nR3V(x)e−ipxdx, V(x) = (2π)−3ˆ\nR3/hatwideV(p)eipxdp.\nThe domain of the functional is as follows:\nD=/braceleftbig\n(γ,α,ρ 0):γ∈L1/parenleftbig/parenleftbig\n1+p2/parenrightbig\ndp/parenrightbig\n, γ(p)≥0, α(p)2≤γ(p)2+γ(p), ρ0≥0/bracerightbig\n. (4)\nThe inequalities in the above definition correspond to the ph ysical assumptions that the number\nof particles cannot be negative and the associated generali zed two-particle reduced density matrix\nhas to be positive defined.\nThe goal of this paper is to study asymptotic behavior of the c anonical ground state energy, i.e.\nthe quantity:\nE(ρ) = inf\nDρF(γ,α,ρ 0), (5)\nwhereDρis the subset of the domain with total density (3) fixed to be eq ualρ. Note that this\nis not a standard meaning of a canonical formulation of the pr oblem as it is only the expectation\nvalue of the number of particles that is fixed. We are interest ed in deriving the leading order\nterms of the asymptotic expansion of the energy in the high de nsity regime ρ→ ∞ and give an\nestimation of the error. Our result can be considered as the c ounterpart to the result of [17] where\nthe authors investigated the behavior of F(γ, α, ρ 0)in the dilute limit ρ1/3a≪1, whereais a\nscattering length of the interaction potential V.\nThe existence of both canonical and grand canonical (i.e. wh en the total density ρof the system\nis not fixed) minimizers for this functional was proven in [16 ] under certain assumptions on the\npotential V. In [19] it was shown that those assumptions in the grand cano nical version can be\nreduced to:\nV≥0, V∈L1(R3), V/ne}ationslash≡0,/hatwideV≥0,/hatwideV∈L1(R3). (6)\nIn this paper we will work under the same assumptions on the po tentialV. After the proof of\nthe first theorem we will discuss the possibilities of weaken ing the assumptions even further (cf.\nRemark 3.3 at the end of next section). We would like to stress out that our result is completely\nindependent of the existence of the minimizers, yet we impos e assumptions (6) as they happen to\nbe well suited for the considered problem.\nBefore stating the main result, to avoid misinterpretation , we will recall the notation used to\ndenote asymptotic errors: we say that some function h(x)is of order o(f(x))x→∞iflimx→∞f(x)\nh(x)= 0.\nSimilarly h(x)is of order O(f(x))iff(x)\nh(x)stays bounded for x→ ∞.\nNow we can state the theorems.\nTheorem 2.1. Assume the potential Vsatisfies (6). The ground state energy E(ρ)defined as in\n(5)in the limit of high densities ρ→ ∞ can be expanded as:\nE(ρ) =1\n2/hatwideV(0)ρ2−1\n2V(0)ρ+o(ρ)ρ→∞. (7)\nThe leading terms in the above expansion correspond to the on es proposed in [14] (where the\nclaim was phrased in terms of energy per particle). To prove t his theorem we will establish suitable\nlower and upper bound of the ground state energy E(ρ)– this will be done in the next section.\nBefore that, we will formulate the next result concerning th e error order estimation.\nTheorem 2.2. LetVbe a function satisfying assumptions (6).4 N. MOKRZAŃSKI AND B. PAŁUBA\ni) Assume furthermore that /hatwideVdecays polynomially at infinity, that is, for sufficiently lar ge\n|p|, as:\n/hatwideV(p)≤C\n|p|k, (8)\nfor some constants C >0,k >3. Then there exist a θ∈/parenleftbig2\n3,1/parenrightbig\n(dependent on k)\nsuch that:\nE(ρ) =1\n2/hatwideV(0)ρ2−1\n2V(0)ρ+O(ρθ)ρ→∞. (9)\nThe exact value of θcan be chosen as:\nθ=2k\n3k−3.\nii) Assume that instead of (8)function /hatwideVdecays exponentially, i.e. satisfies inequality\n/hatwideV(p)≤Ce−c|p|\nfor some constants C >0andc >0. Then\nE(ρ) =1\n2/hatwideV(0)ρ2−1\n2V(0)ρ+O/parenleftbig\nρ2/3(lnρ)6/parenrightbig\nρ→∞. (10)\nAs we will see in the following sections, the proof of the Theo rem 2.2 will follow by obtaining a\nstronger estimation on one of the terms in the functional. Le t us proceed to the proof of the first\ntheorem.\n3.Proof of Theorem 2.1\nAs mentioned before, the proof is based on derivation of prop er lower and upper bound of E(ρ).\n3.1.The lower bound. We will start with the derivation of the lower bound as it is ra ther\nstraightforward. Note that the only term possibly yielding negative values in the functional (2) is:\nρ0(2π)−3ˆ\nR3[α(p)+γ(p)]dp. (11)\nIndeed, every term aside the quadratic one in αis positive due to the assumptions ˆV≥0and\nγ≥0. To prove positivity of this particular quadratic term we wi ll show that\n(2π)−6¨\nR3×R3/hatwideV(p−q)α(p)α(q)dpdq=ˆ\nR3V(x)|ˇα(x)|dx, (12)\nwhich is positive as V≥0. Asαis not necessarily a L2function, we cannot use Plancherel\nTheorem directly. We need to proceed more carefully. First n ote that α∈L1(R3)+L2(R3)due\nto the decomposition α=α1+α2withα1=αχγ>1∈L1andα2=αχγ≤1∈L2. It follows that\nthe inverse Fourier transform ˇαis well defined. Now we can rewrite α(p)α(q)appearing inside the\nintegral on the left hand side of (12) as\nα(p)α(q) =α1(p)α1(q)+α1(p)α2(q)+α2(p)α1(q)+α2(p)α2(q).\nWith this expansion we can express the last three terms as L2inner products of αiand the\nconvolutions /hatwideV∗αj(i,j= 1,2). This is possible as the convolutions /hatwideV∗α2and/hatwideV∗α1areL2\nfunctions due to the facts that /hatwideV∈L1,/hatwideV∈L2(by interpolation with the L∞norm) and Young\ninequality. Now, for those terms, Plancherel Theorem is app licable. As for the first term, we can\nexpress it by integral of Vand inverse transform of αby using the definition of Fourier transformGROUND STATE ENERGY OF BOGOLIUBOV ENERGY FUNCTIONAL IN THE H IGH DENSITY LIMIT 5\nforVand changing the order of integration by Fubini theorem (we u se the assumptions on V). In\nthe end we obtain\n(2π)−6¨\nR3×R3/hatwideV(p−q)α(p)α(q)dpdq\n=ˆ\nR3V(x)|ˇα1(x)|dx+ˆ\nR3V(x)ˇα1(x)ˇα2(x)dx\n+ˆ\nR3V(x)ˇα2(x)ˇα1(x)dx+ˆ\nR3V(x)|ˇα2(x)|dx\n=ˆ\nR3V(x)|ˇα(x)|dx,\nwhich is equality (12).\nNeglecting all terms aside (11) and the one concerning the to tal density squared leads to the\nestimate\nF(γ,α,ρ 0)≥1\n2/hatwideV(0)ρ2+ρ0(2π)−3ˆ\nR3/hatwideV(p)[α(p)+γ(p)]dp.\nBy the inequality α2≤γ2+γwe can deduce that\nα(p)+γ(p)≥ −1\n2. (13)\nUsing this fact we can easily find lower bound for the integral´/hatwideV(γ+α):\nρ0(2π)−3ˆ\nR3/hatwideV(p)[γ(p)+α(p)]dp≥ −1\n2ρ0(2π)−3ˆ\nR3/hatwideV(p)dp\n=−1\n2V(0)ρ0\n≥ −1\n2V(0)ρ,\nwhere the last inequality follows from the assumption V≥0(actually just the fact V(0)≥0, see\nRemark 3.3 at the end of this section). We deduce that for any s tate(γ,α,ρ 0)∈ Dρwe have\nF(γ,α,ρ 0)≥1\n2/hatwideV(0)ρ2−1\n2V(0)ρ. (14)\nTaking infimum over the domain Dρleads to obtaining the lower bound required for proving (7).\nRemark 3.1.Note that obtained estimation does not include any error ter m. This in particular\nmeans that the correction term in (7) is positive.\n3.2.The upper bound. We proceed to the derivation of the upper bound. Note that obt aining\njust the leading ρ2term is easy, it suffices to consider the trial state (0,0,ρ). The functional\nevaluated at it is equal to:\nF(0,0,ρ0=ρ) =1\n2/hatwideV(0)ρ2.\nTo obtain the second term −1\n2V(0)ρwe will construct a family of trial states that yield this res ult\nwith an error of order o(ρ).\nBefore doing that, we shall give some intuition behind the co nstruction. The main idea is\nto asympotically saturate the inequality (13) for p’s on sufficiently large set. As proven in [16,\nCorollary 2.1] and [19, Corollary 2], the minimizing triple (αmin,γmin,ρmin\n0)of the functional F\ncorresponds to the quasi-free pure state, which means that i t satisfies α2(p) =γ(p)(γ(p)+1) .\nThis suggests considering the states γandα(up to a multiplicative constant) of the form:\nγ=λχA, α=−√\nλ2+λχA,6 N. MOKRZAŃSKI AND B. PAŁUBA\nwhereχAis a characteristic function of a suitably chosen set Aof finite measure, in order to\npreserve the integrability of γ. Asymptotically for λ→ ∞ the expression γ+αbehaves as:\nγ(p)+α(p) =/parenleftBig\nλ−√\nλ2+λ/parenrightBig\nχA(p) =−1\n1+/radicalBig\n1+1\nλχA(p) =/parenleftbigg\n−1\n2+O/parenleftbigg1\nλ/parenrightbigg/parenrightbigg\nχA(p).(15)\nIn order to successfully use the above bound, the set Aneeds to cover sufficiently large space. In\nthis case the integral of /hatwideVon the complement of this set can be treated as a sufficiently sm all\nerror term. Simultaneously, we need to remember about remai ning terms of the functional. Their\nimpact should be negligible with respect to total density ρ. To summarize, the requirements for\nthe trial states γandαare as follows:\n•both functions may have arbitrarily large value on their sup port,\n•supports of those functions can cover arbitrarily large spa ce,\n•certain integrals in (2) have to be small with respect to tota l density ρ.\nLet us proceed to the construction. For L >0define set PL⊂R3as\nPL=/bracketleftbigg\n−L\n2,L\n2/bracketrightbigg3\nand letχPLbe its characteristic function. Let γandαbe defined accordingly to the introduction\nof this section, that is as:\nγ(p) = (2π)3λχPL(p), α(p) =−(2π)3√\nλ2+λχPL(p).\nThe relation between parameters λ,Land total density ρis to be specified later, for this moment\nwe will only assume that ρis sufficiently large so that ργ≤ρ. As the density is fixed, we also have\nρ0=ρ−ργ.\nNow, we will estimate each term in energy functional (2) eval uated at the above trial state. First\nwe will consider the kinetic term:\n(2π)−3ˆ\nR3p2γ(p)dp=λˆ\nPLp2dp=λˆL/2\n−L/2dpxˆL/2\n−L/2dpyˆL/2\n−L/2dpz/parenleftbig\np2\nx+p2\ny+p2\nz/parenrightbig\n=\n= 3λˆL/2\n−L/2dpxˆL/2\n−L/2dpyˆL/2\n−L/2p2\nzdpz= 3λL2/parenleftbiggp3\nz\n3/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleL/2\n−L/2=1\n4λL5.(16)\nNext we will focus on the term, ρ0(2π)−3´/hatwideV(γ+α). Using (15) we obtain:\nρ0(2π)−3ˆ\nR3/hatwideV(p)(γn(p)+αn(p))dp= (ρ−ργ)ˆ\nPL/hatwideV(p)/parenleftbigg\n−1\n2+O(1/λ)/parenrightbigg\ndp\n=−1\n2ρˆ\nPL/hatwideV(p)dp+1\n2ργˆ\nPL/hatwideV(p)dp+(ρ−ργ)O(1/λ)ˆ\nPL/hatwideV(p)dp.(17)\nThe first term is the only negative one, we can decompose it fur ther:\n−1\n2ρˆ\nPL/hatwideV(p)dp=−1\n2ρ/parenleftbiggˆ\nR3/hatwideV(p)dp−ˆ\nR3\\PL/hatwideV(p)dp/parenrightbigg\n=−1\n2ρV(0)+1\n2ρˆ\nR3\\PL/hatwideV(p)dp.\nThe first expression in the above expansion is the term that we need in order to prove (7). We will\nnow show that the second term in the above expansion and the re st of terms in (17) are negligible\nup to correct order. We have:\n1\n2ρˆ\nR3\\PL/hatwideV(p)dp=ρ·o(1)L→∞, (18)GROUND STATE ENERGY OF BOGOLIUBOV ENERGY FUNCTIONAL IN THE H IGH DENSITY LIMIT 7\nwhere we used the facts /hatwideV∈L1(R3),/uniontext\nLPL=R3and Lebesgue Dominated Convergence Theorem.\nFor the remaining terms in (17) we get:\n1\n2ργˆ\nPL/hatwideV(p)dp≤1\n2ργˆ\nR3/hatwideV(p)dp=1\n2V(0)ργ (19)\nand\n(ρ−ργ)O(1/λ)ˆ\nPL/hatwideV(p)dp≤ρ·O(1/λ)λ→∞·ˆ\nR3/hatwideV(p)dp=ρ·O(1/λ)λ→∞. (20)\nUsing (18), (19) and (20) to estimate (17) we get:\nρ0(2π)−3ˆ\nR3/hatwideV(p)(γn(p)+αn(p))dp≤ −1\n2V(0)ρ+ρ·o(1)L→∞+1\n2V(0)ργ+ρ·O(1/λ)λ→∞.(21)\nIt remains to bound the convolution term of F. As we consider high density limit, we can\nassumeλ >1and therefore γ≥1on its support PL. On this set we obtain pointwise estimate:\n|α| ≤/radicalbig\nγ(γ+1) =γ/radicalbig\n1+1/γ≤√\n2γ.\nForp’s outside PLthe above inequality is also satisfied in a trivial way. Now we can make the\nestimation:\n1\n2(2π)−6¨\nR3×R3/hatwideV(p−q)[γ(p)γ(q)+α(p)α(q)]dpdq≤\n≤3\n2(2π)−6¨\nR3×R3/hatwideV(p−q)γ(p)γ(q)dpdq\n=3\n2(2π)−6ˆ\nR3/parenleftBig\n/hatwideV∗γ/parenrightBig\n(p)γ(p)dp≤3\n2(2π)−6/bardbl/hatwideV∗γ/bardbl∞ˆ\nR3γ(p)dp\n=3\n2(2π)−3/bardbl/hatwideV∗γ/bardbl∞ργ≤3\n2(2π)−3/bardbl/hatwideV/bardbl∞/bardblγ/bardbl1ργ=3\n2/bardbl/hatwideV/bardbl∞ρ2\nγ,(22)\nwhere in the last line we used Young inequality for convoluti on.\nWe now combine all of the established estimates (16), (21) an d (22). Estimating ργ≤ρ2\nγ=λ2L6\nand denoting constants independent of the parameters in the error term by C1andC2we obtain\na bound:\nF(γ,α,ρ 0)≤1\n2/hatwideV(0)ρ2−1\n2V(0)ρ\n+C1λL5+ρ·/parenleftbig\no(1)L→∞+O(1/λ)λ→∞/parenrightbig\n+C2λ2L6.(23)\nWe will specify λandLto get desired result. We will look for relation of the form:\nλ=ρr, L=ρs,\nfor appropriately chosen powers r >0ands >0. With this selection the error term (as ρ→ ∞)\nis of order:\nC1ρr+5s+ρ/parenleftbig\no(1)+O(ρ−r)/parenrightbig\n+C2ρ2r+6s=o(ρ)+O(ρ2r+6s). (24)\nNow, choose any parameters randssuch that 2r+6s <1. Then the error term is of order o(ρ)\nand therefore:\nF(γ,α,ρ 0=ρ−ργ)≤1\n2/hatwideV(0)ρ2−1\n2V(0)ρ+o(ρ)ρ→∞.\nCombining it with lower bound (14) we have finished the proof o f Theorem 2.1.\nWe will end this section with some discussion concerning the obtained result.\nRemark 3.2.We shall briefly discuss the structure of the trial states rea lizing the optimal upper\nbound of the ground state energy. First note that the kinetic energy does not explicitly contribute\nto the energy expansion (7) as both terms are related to the in teraction potential V. This obser-\nvation is with agreement with trial states used in the proof, where the kinetic energy contribution\nwas a part of the error term and it becomes negligible in the hi gh density limit.8 N. MOKRZAŃSKI AND B. PAŁUBA\nAnother observation is that for the trial states the ratio ρ0/ρof the condensate density and total\ndensity of the system tends to one in this limit. It follows th at, in this limit, it is energetically\nfavorable for particles to almost fully occupy the zero mome ntum state.\nRemark 3.3.The essential estimates in the above proof were obtained usi ng only the assumption\n/hatwideV≥0(and its consequence V(0)≥0) without considering the sign of V. The assumption\nV≥0was the technical one, necessary in (12) to bound the quadrat ic term involving α. As the\nquadratic term involving both γandαin the construction of the functional arises from evaluatin g\nthe expectation value of /hatwideVin a quasi-free state, it is expected that only the assumptio n/hatwideV≥0\nshould be sufficient. Similar observation on the potential wa s also made in [14]. This suggest that\nthe estimates concerning the lower bound of F(in particular (12)) might be refined by obtaining\nan estimate of the entire convolution term of the functional .\nRemark 3.4.Presented method is also applicable for dimensions other th and= 3. Repeating the\nargument for any dimension d, the analogue of (24) is an error of order:\no(ρ)+O(ρm(r,s)),\nwhere\nm(r,s) = max{r+(d+2)s,2r+2ds}.\nChoosing r,ssuch that m(r,s)<1yields the result. Note that the maximum of the above two\nexpressions in nontrivial only if d= 1.\n4.Proof of Theorem 2.2 and further discussion\nIn this section we shall prove that adding some extra assumpt ions on the potential V(or rather\nits Fourier transform /hatwideV) leads to obtaining lower order error term in (7). Note that t he worst\nerror estimate in the proof of Theorem 2.1 comes from (18) whe n estimating the integral of /hatwideVon\nthe complement of the set PL. This justifies the need of assumption on the rate of decay of /hatwideVat\ninfinity.\nProof of Theorem 2.2. The proof is divided into two steps, each one concerning part icular assump-\ntion on/hatwideV.\nStep 1\nWe repeat every step in the proof of the previous theorem. Wit h the extra assumption on /hatwideVwe\ncan make further estimation in (18) and write:\n1\n2ρˆ\nR3\\PL/hatwideV(p)dp≤1\n2ρˆ\nR3\\PLC\n|p|kdp≤˜Cρˆ∞\n2Lr2\nrkdr=ρ·O/parenleftbigg1\nLk−3/parenrightbigg\nL→∞. (25)\nThe analogue of the upper bound (23) becomes as follows:\nF(γ,α,ρ 0)≤1\n2/hatwideV(0)ρ2−1\n2V(0)ρ\n+C1λL5+ρ·/parenleftbig\nO/parenleftbig\n1/Lk−3/parenrightbig\nL→∞+O(1/λ)λ→∞/parenrightbig\n+C2λ2L6.\nOnce again we will look for relation of the parameters of the f orm:\nλ=ρr, L=ρs\nfor appropriately chosen powers r >0ands >0. The error term is then of order:\nC1ρr+5s+ρ/parenleftBig\nO/parenleftbig\nρ−(k−3)s/parenrightbig\n+O(ρ−r)/parenrightBig\n+C2ρ2r+6s=O(ρm(r,s)),\nwhere\nm(r,s) = max{2r+6s,1−(k−3)s,1−r}.\nTo get the error of desired order, parameters randsneed to be chosen in a way that m(r,s)≤θ.GROUND STATE ENERGY OF BOGOLIUBOV ENERGY FUNCTIONAL IN THE H IGH DENSITY LIMIT 9\nFirst, we will derive the necessary conditions for the exist ence of such θ. Assume there exist\nr >0ands >0satisfying m(r,s)≤θ. Then, combining inequalities 1−r≤θand2r+6s≤θ\nresults in\nθ >2\n3, (26)\nso this is the lowest order of error possible to obtain by prop osed method (note that this inequality\nis strict). Next, as r≥1−θ, we can write:\nr= 1−θ+δ,\nfor some δ≥0and plug it to the first inequality, that is:\n2r+6s= 2−2θ+2δ+6s≤θ⇐⇒s≤3θ−2−2δ\n6.\nUsing the fact that (k−3)s≥1−θ, the following inequality must be fulfilled:\n1−θ\nk−3≤3θ−2−2δ\n6⇐⇒δ≤3θ(k−1)−2k\n2(k−3).\nAsδ≥0we need to have:\n3θ(k−1)−2k≥0⇐⇒θ≥2k\n3k−3.\nNote that the first obtained estimate (26) is the limiting cas e (fork→ ∞) of the above inequality.\nReversing the order of this reasoning we observe that derive d condition for θis also the sufficient\none for the existence of randsyieldingm(r,s)≤θ. Thus the expansion (9) stated in the theorem\nis valid for θ=2k\n3k−3.\nStep 2\nWhen/hatwideVdecays exponentially, we can do analogous computation as in (25) and obtain\n1\n2ρˆ\nR3\\PL/hatwideV(p)dp≤1\n2ρˆ\nR3\\PLCe−c|p|dp≤C′ρˆ∞\n2Lr2e−crdr≤C′′ρL2e−cL=ρ·O/parenleftbig\nL2e−cL/parenrightbig\nL→∞\nand therefore\nF(γ,α,ρ 0)≤1\n2/hatwideV(0)ρ2−1\n2V(0)ρ\n+C1λL5+ρ·/parenleftbig\nO/parenleftbig\nL2e−cL/parenrightbig\nL→∞+O(1/λ)λ→∞/parenrightbig\n+C2λ2L6.\nThis time we will choose parameters λandLas\nλ=ρ2/3, L=1\n3clnρ,\nwherecis the same constant as in the exponent in asymptotic term. Wi th this selection we can\nexpress the error term as:\n˜C1ρ1/3(lnρ)2+ρ/parenleftbigg\nO/parenleftbigg(lnρ)2\nρ1/3/parenrightbigg\n+O/parenleftbigg1\nρ1/3/parenrightbigg/parenrightbigg\n+˜C2ρ2/3(lnρ)6=O/parenleftbig\nρ2/3(lnρ)6/parenrightbig\n.\n/square\nWe shall end with some general remarks.\nRemark 4.1.Similarly like in the Remark 3.4 it is possible to generalize this result to dimensions\nother than d= 3. In the polynomial decay case the assumption (8) needs to hol d fork > d .\nFollowing the same reasoning as above, we deduce that θ, for which the result is valid, can be\nchosen as:\nθ=2k\n3k−d.\nThe limiting value ( k→ ∞) is the same as before, equal to2\n3.10 N. MOKRZAŃSKI AND B. PAŁUBA\nIn the case of exponential decay, the asymptotic error will b e of order O/parenleftbig\nρ2/3(lnρ)2d/parenrightbig\nas the\nlogarithm term is related to the squared volume of the set PL, which is equal to L2d.\nRemark 4.2.We would like to note that considered assumptions on the poly nomial or exponential\nrate of decay are very general as they hold for many reasonabl e potentials (e.g. for Vbeing a\nSchwartz function). Presented method, however, can be easi ly generalized to rates of decay of /hatwideV\nof other form.\nAlso note that, by using similar argument preceding inequal ity (26), regardless of the rate of\ndecay it is not possible by this method to obtain error of orde r lower than ρ2/3. As the considered\ntrial state appears to be well suited for this problem, this s uggest that the explicit form of the third\nleading order term in the expansion of the ground state energ y might be much more dependent on\nthe properties of the interaction potential V.\nReferences\n[1]M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A . Cornell ,Observation of\nBose-Einstein Condensation in a Dilute Atomic Vapor , Science 269 (1995), pp. 198–201\n[2]G. Basti, S. Cenatiempo, A. Giuliani, A. Olgiati, G. Pasqualet ti and B. Schlein ,Upper bound for\nthe ground state energy of a dilute Bose gas of hard spheres , arXiv:2212.04431 [math-ph]\n[3]N. N. Bogoliubov ,On the theory of superfluidity , J. Phys. (USSR), 11 (1947), p. 23.\n[4]E. Carlen, I. Jauslin and E.H. Lieb ,Analysis of a simple equation for the ground state energy of t he Bose\ngas, Pure and Applied Analysis 2(3), 659-684 (2019)\n[5]E. Carlen, M. Holzmann, I. Jauslin and E.H. Lieb ,A simplified approach to the repulsive Bose gas from\nlow to high densities and its numerical accuracy , Phys. Rev. A. 103, 053309 (2020)\n[6]R.H. Critchley and A. Solomon ,A Variational Approach to Superfluidity , J. Stat. Phys., 14, 381–393\n(1976)\n[7]K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durf ee, D. M. Kurn, and W.\nKetterle ,Bose-Einstein Condensation in a Gas of Sodium Atoms , Phys. Rev. Lett. 75 (1995), pp. 3969–3973\n[8]J. Dereziński and M. Napiórkowski ,Excitation spectrum of interacting bosons in the mean-field i nfinite-\nvolume limit , Ann. Henri Poincaré, 15 (2014), pp. 2409–2439.\n[9]S. Fournais, M. Napiórkowski, R. Reuvers and J.P. Solovej ,Ground state energy of a dilute two-\ndimensional Bose gas from the Bogoliubov free energy functi onal, J. Math. Phys. 60, 071903 (2019)\n[10]S. Fournais and J.P. Solovej ,The energy of dilute Bose gases , Ann. Math. 192(3), 893-976 (2020)\n[11]F. Haberberger, C. Hainzl, P.T. Nam, R. Seiringer and A. Tria y,The free energy of dilute Bose\ngases at low temperatures , arXiv:2304.02405 [math-ph]\n[12]I. Jauslin ,Review of a Simplified Approach to study the Bose gas at all den sities, The Physics and Mathematics\nof Elliott Lieb, The 90th anniversary Volume 1, pages 609-63 5 (2022)\n[13]L. Landau ,Theory of the Superfluidity of Helium II , Phys. Rev., 60, 356–358 (1941)\n[14]E. H. Lieb ,Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas , Phys. Rev., 130 (1963),\np. 2518–2528.\n[15]M. Napiórkowski ,Dynamics of interacting bosons: a compact review , in: Density Functionals for Many-\nParticle Systems - Mathematical Theory and Physical Applic ations of Effective Equations, World Scientific (2023)\n[16]M. Napiórkowski, R. Reuvers and J.P. Solovej ,Bogoliubov free energy functional I. Existence of min-\nimizers and phase diagram , Arch. Ration. Mech. Anal. 229 (3) (2018), 1037-1090.\n[17]M. Napiórkowski, R. Reuvers and J.P. Solovej ,Bogoliubov free energy functional II. The dilute limit ,\nCommun. Math. Phys. 360 (1) (2018), 347-403.\n[18]M. Napiórkowski, R. Reuvers and J.P. Solovej ,Calculation of the Critical Temperature of a Dilute\nBose Gas in the Bogoliubov Approximation , EPL 121 (1) (2018), 10007\n[19]J. Oldenburg ,On Ground States Of The Bogoliubov Energy Functional: A Dire ct Proof , Journal of Mathe-\nmatical Physics 62, 071902 (2021)\n[20]B. Schlein ,Bose gases in the Gross-Pitaevskii limit: a survey of some rigo rous results , arXiv:2203.10855\n[math-ph]GROUND STATE ENERGY OF BOGOLIUBOV ENERGY FUNCTIONAL IN THE H IGH DENSITY LIMIT 11\nDepartment of Mathematical Methods in Physics, Faculty of Phy sics, University of Warsaw,\nPasteura 5, 02-093 Warszawa, Poland\nEmail address :norbert.mokrzanski@fuw.edu.pl\nInstitute of Geophysics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa,\nPoland\nEmail address :bartosz.paluba@fuw.edu.pl" }, { "title": "2402.00432v1.Density_fluctuations_for_Squeezed_Number_State_and_Coherent_Squeezed_Number_State_in_Flat_FRW_Universe.pdf", "content": "Density fluctuations for Squeezed Number State\nand Coherent Squeezed Number State in Flat\nFRW Universe\nDhwani Gangal1, Sudhava Yadav1, K.K. Venkataratnam1*\n1Department of Physics, Malaviya National Institute of Technology\nJaipur, Jaipur, Jaipur, 302017, India.\n*Corresponding author(s). E-mail(s): kvkamma.phy@mnit.ac.in;\nAbstract\nWe study the density fluctuations for Coherent Squeezed Number State (CSNS)\nand Squeezed Number States (SNS) formalism in Semiclassical theory of grav-\nity in flat FRW universe. We used Number state evolution of oscillatory phase\nof inflaton for coherent squeezed number state and squeezed number states for-\nmalisms. We analyzed that density fluctuations for SNS depends upon squeezing\nparameter and number state while for CSNS density fluctuations depends upon\nsqueezing parameter, number state and coherent state parameter. These param-\neters plays an important role for quantum consideration of SNS and CSNS. The\nresults of the analysis shows that increase in density fluctuations for both SNS\nand CSNS, demonstrate quantum behavior of SCEE as well as production of\nvarious kind of particles in these states.\nKeywords: Squeezing Parameter,Energy Momentum Tensor, Squeezed Number State\n(SNS), Coherent Squeezed Number State (CSNS), Density Fluctuations,Semiclaasical\ngravity\n1 Introduction\nThe theory of inflation postulates that the prompt expansion of Universe started just\nafter Big Bang [1, 2]. After the inflationary era there must be numerous particles those\nare responsible for re-thermalization of Universe [3]. The transition between inflation\nand further evolutionary phase of universe is reheating [4–7]. This transition era is well\nparameterized using various reheating parameters [8–10] and plays a crucial role in\n1arXiv:2402.00432v1 [gr-qc] 1 Feb 2024understanding standard matter production. The quantum properties and fluctuations\nrelated to inflaton provides an important information of early universe and gravity.\nFurther, that it is related to classical gravity in Friedmann-Robertson-Walker (FRW)\nUniverse [11, 12]. According to cosmological principle Universe is homogeneous and\nisotropic. Cosmological, description of universe is based on Friedmann equations with\nscalar field(s). Under the classical gravity, Friedmann equations assume that those are\nvalid even at initial stage of universe too. Though, quantum properties and quantum\nfluctuations of matter fields are anticipated to play a significant role. It is worth\nmentioning here that quantum effects into classical gravity can be better understood\nby Semi-Classical Theory of Gravity (SCTG). Further, the SCTG is based on fact\nthat gravitational field depends upon quantized matter field having curved space-\ntime. That has drawn special attention from theoretical physicists [13–16] . Though\nquantum gravity effects can be consider negligible at this stage of universe. To describe\nthe universe both gravity and matter fields are to be preserved quantum mechanically,\nbut at present no consistent quantum theory is available to describe gravity. Hence,\nproper picture of early universe with a suitable cosmological model can be constructed\nin terms of the semi-classical Friedmann equations, where gravity can be preserved\nas classical with quantized matter field(s). The elementary idea of the Friedmann-\nLemaˆ ıtre-Robertson-Walker metric is based on particular solution of Einstein’s field\nequations of general relativity [17–20].\nConcept of squeezed states was firstly introduced by Kennard as an analysis of\nGaussian wave packet of harmonic oscillator [21]. The quantum effect of inflaton\nis based on semi-classical gravity in quantum optics considerations [22, 23]. So the\nquantum mechanical number state representation of scalar and complex scalar fields\ncoupled to semi-classical FRW universe [24–26], language of quantum optics is an\nimportant to understand the quantum optical effects in cosmology [27]. The num-\nber state representation was further analyzed to identify the expanding universe and\nanisotropic universe [28–31]. Initial and final states of particle creation in expanding\nuniverse were also described using number state [32–36]. Consider a suitable, quantum\nmechanical states of the universe such as Squeezed Vacuum State (SVS), Squeezed\nNumber State (SNS), and Coherent Squeezed Number States (CSNS) are used to\nunderstand the multifaceted nature of flat FRW Universe [37–50].\nQuantum mechanical considerations of particle production due to Coherent Squeezed\nNumber State (CSNS) and Squeezed Number States (SNS) in the oscillatory region of\nFRW universe under the light of semi-classical gravity [30, 31]. In this paper, we study\ndensity fluctuations for Squeezed Number State (SNS) and Coherent Squeezed Num-\nber States (CSNS) using quantum mechanical squeezed operator and displacement\noperator using number states.\n2 Energy-Momentum Tensor in Semiclassical\nGravity\nVarious models of modern cosmology are constructed based upon classical gravity of\nEinstein field equations with scalar field on the FRW metric. Friedmann equations\nare further enumerated quantum mechanically for background metric. There is no\n2significant description of quantum theory of gravity is available, so contemporary\ncosmological models, for background metric is considered as classical, while the matter\nfield as quantum mechanical. Such type of theoretical models are called as semiclassical\ntheory of gravity. In this circumstances, the description of the gravitational field using\nsemi-classical theory of gravity, can be written as (here onwards we use the unit system\nc=ℏ=1 and G=1\nm2p)\nEµν=8π\nm2p⟨Tµν⟩. (1)\nWhere Eµνis the Einstein tensor and ⟨Tµν⟩is the expectation value of energy-\nmomentum tensor in a suitable quantum state. The quantum state satisfy the time\ndependent Schrodinger equation and it can be written as\ni∂\n∂tψ=∧\nHψ, (2)\nwhere∧\nHis Hamiltonian operator. Equation for Friedmann-Robertson-Walker (FRW)\nspace-time having generalized coordinates ( r1,r2,r3,r4) can be written as\nds2=−dr2\n4+G2(t)\u0000\ndr2\n1+dr2\n2+dr2\n3\u0001\n, (3)\nwhere G(t) is the scale factor.\nLagrangian density Lfor massive inflaton for flat FRW universe is given as\nL=−1\n2p\n(−g)\u0000\ngµν∂µΦ∂νΦ +m2Φ2\u0001\n. (4)\nKlein-Gordon (K-G) equation for massive inflaton Φ can be derived from equations\n(3) and (4) as\n¨Φ + 3˙G(t)\nG(t)˙Φ +m2Φ = 0 , (5)\nwhere Hubble parameter H=˙G(t)\nG(t)and Φ is the Homogeneous scalar field for gravity to\nbe consider, ˙Φ and ¨Φ are the first and second order derivatives with respect to time.\nΠ is the momentum conjugate to∧\nΦ is represented as\nΠ =∂L\n∂˙Φ. (6)\nConsider the canonical quantization, the Hamiltonian for the inflaton can be\nwritten as\nH=1\n2G3(t)Π2+1\n2G3(t)m2∧\nΦ2\n, (7)\nwhile temporal component of the energy-momentum tensor can be calculated as\nT00=G3(t)\u00121\n2˙Φ2+1\n2m2∧\nΦ2\u0013\n, (8)\n33 Formulation of Squeezed Number State (SNS)\nSingle mode squeezed coherent state is defined as\n|Υ, ζ⟩=∧\nW(ρ,Ψ)D(Υ)|0⟩, (9)\nwhere D(Υ) is displacement operator and is defined as\nD(Υ) = exp(Υ∧e†\n−Υ∗∧e), (10)\nand the squeezing operator∧\nW(ρ,Ψ) is given by\n∧\nW(ρ,Ψ) = expρ\n2\u0012\nexp(−iΨ)∧e2\n−exp(iΨ)∧e†2\u0013\n, (11)\nwhere ρcan take values between 0 ≤ρ≤ ∞ is squeezing parameter to regulates order\nof squeezing, Ψ can take values between -Π ≤Ψ≤Π is squeezing angle that governs\ndistribution of conjugate variables. Squeezing operator have following property\n∧\nW†∧e∧\nW=∧ecoshρ−∧e†\nexp(iΨ)sinh ρ, (12)\n∧\nW†∧e†∧\nW=∧e†\ncoshρ−∧eexp(−iΨ)sinh ρ. (13)\nOperation of squeezing parameter on number state gives as Squeezed Number State\nand it is defined as\n|ζ, n⟩=∧\nW(ρ,Ψ)|n⟩. (14)\nThe operation of annihilation operator∧eand creation operator∧e†\non number state\nis given as\n∧e|n⟩=√n|n−1⟩, (15)\n∧e†\n|n⟩=√\nn+ 1|n+ 1⟩. (16)\nEigenstates for the Hamiltonian can be written as\n∧e†∧\n(t)e(t)|n,Φ,t⟩=n|n,Φ,t⟩. (17)\n4 Formulation of Coherent Squeezed Number States\n(CSNS)\nCoherent Squeezed Number States (CSNS) is an important state in quantum optics.\nSingle mode coherent state is descirbed as\n4|Υ⟩=D(Υ)|0⟩. (18)\nThe action of∧eon the coherent state is given as\n∧e|Υ⟩= Υ|Υ⟩. (19)\nAnnihilation operator∧eand creation operator∧e†\ncombined with single mode\ndisplacement operator D(Υ) which is given by Eq. (10) and satisfy the following\nproperties\nD†∧e†\nD=∧e†\n+ Υ∗, (20)\nD†∧eD=∧e+ Υ. (21)\nOperation of single mode Displacement operator D(Υ) and squeezing parameter\noperator∧\nW(ρ,Ψ) on Number State gives the coherent squeezed number state (CSNS)\nand it is defined as\n|Υ, ζ, n⟩=D(Υ)∧\nW(ρ,Ψ)|n⟩. (22)\n5 Oscillatory phase of inflaton and Semiclassical\nEinstein euquations for SNS and CSNS\nConsider an oscillatory phase of massive inflaton minimally coupled to a flat FRW\nmetric in a nonclassical states. Among nonclassical states, firstly we consider it in\nsqueezed Number State (SNS) and after that we consider it in coherent squeezed\nNumber State (CSNS) in a flat FRW metric minimally coupled to massive inflaton. The\ntemporal component of classical Einstein equation under classical gravity is written as\n ˙G(t)\nG(t)!2\n=8π\n3m2pT00\nG3(t), (23)\nwhere, inflaton energy density T00is given by Eq. (8). For the suitable quantum\nstate with normal ordered expectation value of Hamiltonian\u001c\n:∧\nHm:\u001d\n, Friedmann\nequation in the semiclassical theory of gravity can be written as\n ˙G(t)\nG(t)!2\n=8π\n3m2p1\nG3(t)⟨:∧\nHm:⟩. (24)\nWe know that the annihilation operator (∧e) and creation operator (∧e†\n) obey the\ncommutation relation as\n\u0014\n∧e,∧e†\u0015\n= 1. (25)\n5Further, annihilation and creation operators can be computed as\n∧e(t) = Φ∗(t)∧\nΠ− G3(t)˙Φ∗(t)∧\nΦ,\n∧e†\n(t) = Φ( t)∧\nΠ− G3(t)˙Φ(t)∧\nΦ. (26)\nUsing Eqs.(17, 25-26),∧\nΦ and Π holds the relations as\n∧\nΦ =1\ni\u0012\nΦ∗∧e†\n−Φ∧e\u0013\n, (27)\n∧\nΦ2\n=\u0012\n2∧e†∧e+ 1\u0013\nΦ∗Φ−\u0012\nΦ∗∧e†\u00132\n−\u0010\nΦ∧e\u00112\n, (28)\n∧\nΠ = iG3(t)\u0012\n˙Φ∧e−˙Φ∗∧e†\u0013\n, (29)\n∧\nΠ2\n=G3(t)\"\u0012\n2∧e†∧e+ 1\u0013\n˙Φ∗˙Φ−\u0012\n˙Φ∗∧e†\u00132\n−\u0010\n˙Φ∧e\u00112#\n. (30)\nHamiltonian in semiclassical Friedmann equation computed using Eqs. (17, 24-26)\nfor the number state as\n ˙G(t)\nG(t)!2\n=8π\n3m2p\u0014\u0012\nn+1\n2\u0013\u0010\n˙Φ∗(t)˙Φ(t) +m2Φ∗(t)Φ(t)\u0011\u0015\n, (31)\nwhere Φ( t) and Φ∗(t) satisfy the equation (5) and the Wronskian condition given as\nG3(t)\u0014\n˙Φ∗(t)Φ(t)−Φ∗(t)˙Φ(t)\u0015\n=i. (32)\nSemiclassical Einstein equation for SNS and CSNS has an important consequences\nof above formulation. To determine the Semiclassical Einstein equation for SNS, by\nusing Eq. (11-16, 30) and after applying the SNS properties and computation we get\nnormal ordered expection value of\u001c\n:∧\nΠ2\n:\u001d\nfor Squeezed Number State is\n\u001c\n:∧\nΠ2\n:\u001d\nSNS=2G6(t)\u0014\u0012\nnCosh2ρ+ (n+1\n2)Sinh2ρ+1\n2\u0013\n˙Φ∗˙Φ\n−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\n˙Φ∗2−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\n˙Φ2\u0015\n. (33)\nusing Eqs. (11-16, 28) and after computation, we get the normal ordered expecta-\ntion value of\u001c\n:∧\nΦ2\n:\u001d\nfor Squeezed Number State (SNS) is\n\u001c\n:∧\nΦ2\n:\u001d\nSNS=2\u0014\u0012\nnCosh2ρ+ (n+1\n2)Sinh2ρ+1\n2\u0013\nΦ∗Φ\n6−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\nΦ∗2−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\nΦ2\u0015\n. (34)\nusing Eqs.(7, 33-34), we compute the normal ordered expectation value of the\nHamiltonian in Squeezed Number State (SNS) as\n\u001c\n:∧\nH:\u001d\nSNS=G3(t)\u0014\u0012\nnCosh2ρ+ (n+1\n2)Sinh2ρ+1\n2\u0013\u0012\n˙Φ∗˙Φ +m2Φ∗Φ\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\u0012\n˙Φ∗2+m2Φ∗2\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\u0012\n˙Φ2+m2Φ2\u0013\u0015\n. (35)\nusing Eqs.(35) in Eqs.(24), the semiclassical Einstein equation for Squeezed\nNumber State (SNS) is\n ˙G(t)\nG(t)!2\nSNS=8π\n3m2p\u0014\u0012\nnCosh2ρ+ (n+1\n2)Sinh2ρ+1\n2\u0013\u0012\n˙Φ∗˙Φ +m2Φ∗Φ\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\u0012\n˙Φ∗2+m2Φ∗2\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ\u0013\u0012\n˙Φ2+m2Φ2\u0013\u0015\n. (36)\nNow, we determine the Semiclassical Einstein Equation for Coherent Squeezed\nNunber State (CSNS), by using Eq. (11-22, 30) and after applying the CSNS properties\nand computation we get normal ordered expection value of\u001c\n:∧\nΠ2\n:\u001d\nfor Coherent\nSqueezed Number State (CSNS) is\n⟨:∧\nΠ2\n:⟩CSNS =2G6(t)\u0014\u0012\nncosh2ρ+ (n+1\n2) sinh2ρ+1\n2+Υ∗Υ\u0013\n˙Φ∗˙Φ\n−\u0012\n(n+1\n2) sinh ρcoshρ−Υ∗2\n2\u0013\n˙Φ∗2\n−\u0012\n(n+1\n2) sinh ρcoshρ−Υ2\n2\u0013\n˙Φ2\u0015\n. (37)\nusing Eqs. (11-22, 28) after computation, we get normal ordered expection value\nof\u001c\n:∧\nΦ2\n:\u001d\nfor Coherent Squeezed Number State (CSNS) is\n7\u001c\n:∧\nΦ2\n:\u001d\nCSNS =2\u0014\u0012\nnCosh2ρ+ (n+1\n2)Sinh2ρ+1\n2+Υ∗Υ\u0013\nΦ∗Φ\n−\u0012\n(n+1\n2)Sinh ρCoshρ−Υ∗2\n2\u0013\nΦ∗2\n−\u0012\n(n+1\n2)Sinh ρCoshρ−Υ2\n2\u0013\nΦ2\u0015\n. (38)\nusing Eqs.(7, 37-38), we get the normal ordered expectation value of Hamiltanian\nin Coherent Squeezed Number State (CSNS) as\n\u001c\n:∧\nH:\u001d\nCSNS=G3(t)\u0014\u0012\nnCosh2ρ+ (n+1\n2)Sinh2ρ+1\n2+Υ∗Υ\u0013\u0012\n˙Φ∗˙Φ +m2Φ∗Φ\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ−Υ∗2\n2\u0013\u0012\n˙Φ∗2+m2Φ∗2\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ−Υ2\n2\u0013\u0012\n˙Φ2+m2Φ2\u0013\u0015\n. (39)\nusing Eqs.(39) in Eqs.(24), the semiclassical Einstein equation for Coherent\nSqueezed Number State (CSNS) is\n ˙G(t)\nG(t)!2\nCSNS=8π\n3m2p\u0014\u0012\nnCosh2ρ+ (n+1\n2)Sinh2ρ+1\n2+Υ∗Υ\u0013\u0012\n˙Φ∗˙Φ +m2Φ∗Φ\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ−Υ∗2\n2\u0013\u0012\n˙Φ∗2+m2Φ∗2\u0013\n−\u0012\n(n+1\n2)Sinh ρCoshρ−Υ2\n2\u0013\u0012\n˙Φ2+m2Φ2\u0013\u0015\n. (40)\n6 Density fluctuations for Oscillatory phase of\ninflaton for Squeezed Number State and Coherent\nSqueezed Number State\nThe semiclassical theory of FRW universe predicts fluctuations for energy momen-\ntum tensor. The energy momentum tensor of the inflaton for quantum states related\nto particle production in FRW universe can be studied with the help of semiclassi-\ncal Einstein equation. Semiclassical Einstein equation for flat FRW universe can be\ndetermine using following relation\n△=⟨:T2\nµν:⟩ − ⟨:Tµν:⟩2. (41)\n8Here\n:T2\nµν:⟩and\n:Tµν:⟩2represents normal ordered expectation value of square\nof energy momentum tensor and square of the normal ordered expectation value of\nenergy momentum tensor respectively. Symbol : : used to demonstrate that normal\nordered value of physical quantities are to be computed with respect to normal order-\ning of scalar field. In this study we have determined density fluctuations for Squeezed\nNumber State and Coherent Squeezed Number State using single mode temporal\ncomponent of the energy momentum tensor.\n6.1 Density fluctuations for Squeezed Number State (SNS)\nDensity fluctuations for Squeezed Number State is calculated using Eq. (41),normal\nordered expectation value of square of energy momentum tensor\n:T2\n00:\u000b\nfor Squeezed\nNumber State is given as\n\n:T2\n00:\u000b\nSNS=1\n4G6(t)m4\u001c\n:∧\nΦ4\n:\u001d\nSNS+m2\n4\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nSNS\n+m2\n4\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nSNS+1\n4G6(t)\u001c\n:∧\nΠ4\n:\u001d\nSNS. (42)\nUsing Eqs. (12-13, 25-30) the values of\u001c\n:∧\nΠ4\n:\u001d\nSNScan be computed as\n\u001c\n:∧\nΠ4\n:\u001d\nSNS=G6(t)\n4m2t4\u0014\n3 + Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2+ 48n+ 24\u0001\n+ Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n+ Cosh ρSinhρ(24n+ 12) + Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Sinh2ρ(12n+ 12) + Cosh4ρ(6n2−6n) + Cosh2ρ(12n)\u0015\n. (43)\nUsing Eqs. (12-13, 25-30) the values of\u001c\n:∧\nΦ4\n:\u001d\nSNScan be computed as\n\u001c\n:∧\nΦ4\n:\u001d\nSNS=1\n4m2G6(t)\u0014\n3 + Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2\n+ 48n+ 24) + Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n+ Cosh ρSinhρ(24n+ 12) + Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Sinh2ρ(12n+ 12) + Cosh4ρ(6n2−6n) + Cosh2ρ(12n)\u0015\n. (44)\n9Using Eqs. (12-13, 25-30) the values of\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nSNScan be computed as\n\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nSNS=1\n4m2t2\u0014\n3 + Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2\u0001\n+ 48n+ 24) + Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n+ Cosh ρSinhρ(24n+ 12) + Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Sinh2ρ(12n+ 12) + Cosh4ρ(6n2−6n)\n+ Cosh2ρ(12n)\u0015\n. (45)\nUsing Eqs. (12-13, 25-30) the values of\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nSNScan be computed as\n\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nSNS=1\n4m2t2\u0014\n3 + Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2\u0001\n+ 48n+ 24) + Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n+ Cosh ρSinhρ(24n+ 12) + Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Sinh2ρ(12n+ 12) + Cosh4ρ(6n2−6n)\n+ Cosh2ρ(12n)\u0015\n. (46)\nSubstituting the values of\u001c\n:∧\nΦ4\n:\u001d\nSNS,\u001c\n:∧\nΠ4\n:\u001d\nSNS,\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nSNS,\n\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nSNSin Eq. (42), the normal ordered expection value of square of energy\nmomentum tensor can be calculated as\n\n:T2\n00:\u000b\nSNS=\u00121\n16m2t4+1\n8t2+m2\n16\u0013\u0014\n3 + Cosh3ρSinhρ(24n2\n+ Cosh ρSinh3ρ\u0000\n24n2+ 48n+ 24\u0001\n+ Cosh2ρSinh2ρ(36n2\n+36n+ 12) + Cosh ρSinhρ(24n+ 12) + Sinh4ρ(6n2\n+18n+ 12) + Sinh2ρ(12n+ 12) + Cosh4ρ(6n2−6n)\n+ Cosh2ρ(12n)\u0015\n. (47)\nNow, normal ordered expectation value of temporal component of energy momen-\ntum tensor is\n10⟨:T00:⟩SNS=1\n2G3(t)m2\u001c\n:∧\nΦ2\n:\u001d\nSNS+1\n2G3(t)\u001c\n:∧\nΠ2\n:\u001d\nSNS. (48)\nUsing Eqs. (12-13, 25-30) the values of\u001c\n:∧\nΠ2\n:\u001d\nSNScan be computed as\n\u001c\n:∧\nΠ2\n:\u001d\nSNS=G3(t)\n2mt2\u0014\nCosh2ρ(2n) + Sinh2ρ(2n+ 2) + 1 + Cosh ρSinhρ(4n+ 2)\u0015\n.(49)\nUsing Eqs. (12-13, 25-30) the values of\u001c\n:∧\nΦ2\n:\u001d\nSNScan be computed as\n\u001c\n:∧\nΦ2\n:\u001d\nSNS=1\n2mG3(t)\u0014\nCosh2ρ(2n) + Sinh2ρ(2n+ 2) + 1\n+ Cosh ρSinhρ(4n+ 2)\u0015\n, (50)\nusing Eqs. (49) and (50) in Eq.(48), normal ordered value of temporal component of\nenergy momentum tensor will be\n⟨:T00:⟩SNS=\u0014m\n4+1\n4mt2\u0015\u0014\nCosh2ρ(2n) + Sinh2ρ(2n+ 2) + 1\n+ Cosh ρSinhρ(4n+ 2)\u0015\n. (51)\nSquare of normal ordered value of temporal component of energy momentum tensor\ncan be computed as\n⟨:T00:⟩2\nSNS=\u00121\n16m2t4+1\n8t2+m2\n16\u0013\u0014\n1 + Cosh3ρSinhρ\u0000\n16n2+ 8n\u0001\n+ Cosh ρSinh3ρ\u0000\n16n2+ 24n+ 8\u0001\n+ Cosh2ρSinh2ρ\u0000\n24n2\n+24n+ 4) + Cosh ρSinhρ(8n+ 4) + Sinh4ρ\u0000\n4n2\n+ 8n+ 4) + Sinh2ρ(4n+ 4) + Cosh4ρ\u0000\n4n2\u0001\n+ Cosh2ρ(4n)\u0015\n. (52)\nSubstituting the values of\n:T2\n00:\u000b\nSNSand⟨:T00:⟩2SNSin Eq. (41), density\nfluctuations for Squeezed Number State (SNS) is\n11△SNS=\u00121\n16m2t4+1\n8t2+m2\n16\u0013\u0014\n2 + Cosh3ρSinhρ\u0000\n8n2−8n\u0001\nCoshρSinh3ρ\u0000\n8n2+ 24n+ 16\u0001\n+ Cosh2ρSinh2ρ\u0000\n12n2\n12n+ 8) + Cosh ρSinhρ(16n+ 8) + Sinh4ρ\u0000\n2n2+ 10n+ 8\u0001\n+ Sinh2ρ(8n+ 8) + Cosh4ρ(2n2−6n) + Cosh2ρ(8n)\u0015\n. (53)\nEq. (53) shows the △SNSdensity fluctuations for squeezed number state depend on\nsqueezing parameter ( ρ) as well as number state (n), but it is more strongly depend\non number state rather than the squeezing parameter ( ρ). Numerical values of density\nfluctuations △SNSfor various values of squeezed number state can be calculated using\nEq. (53), for n=1, 2, 3, 4 is as shown in table 1, 2 , 3, while assuming m= t=1. Table 1\nand 2 shows, for ρranging between 0 .001 to 0.09 the variation in density fluctuations\nis very small for all number state taken into consideration, but table 3 shows that,\nforρranging 0.1 to 0.09 shows large variation in △SNS. As we take n=0 in Eq. (53),\nit reduces into density fluctuations for squeezed vacuum state [44–46], showing the\nvalidity of approximation used in SCEE. Fig. 1 shows variation of density fluctuations\n△SNS, for various values of squeezed number state with squeezing parameter ρ. It can\nbe deduced from Fig. 1 that with increasing value of ρas well as n, density fluctuations\n△SNSincreases. In Fig. 2, 3-D plot between ρ, n and △SNSshows variation of density\nfluctuations △SNSwith various number state and squeezing parameter ρ\nTable 1 Numerical values of density\nfluctuations △SNSfor various squeezed number\nstate for squeezing parameter ρ <<< 1\nρ n=1 n=2 n=3 n=4\n0.001 2.50601 4.51403 6.52605 8.54208\n0.002 2.51206 4.52812 6.55221 8.58432\n0.003 2.51813 4.54227 6.57847 8.62672\n0.004 2.52423 4.55648 6.60484 8.66929\n0.005 2.53035 4.57075 6.63131 8.71201\n0.006 2.53651 4.58509 6.65789 8.75490\n0.007 2.54269 4.59948 6.68457 8.79796\n0.008 2.54890 4.61394 6.71136 8.84118\n0.009 2.55515 4.62846 6.73826 8.88456\n6.2 Density fluctuations for Coherent Squeezed Number\nStates (CSNS)\nDensity fluctuations for coherent squeezed number state is being calculated using Eq.\n(41), where normal ordered expectation value of square of energy momentum tensor\n:T2\n00:\u000b\nCSNS for squeezed number state is given as\n12Table 2 Numerical values of density\nfluctuations △SNSfor various squeezed number\nstate for squeezing parameter ρ << 1\nρ n=1 n=2 n=3 n=4\n0.01 2.56142 4.64304 6.76527 8.92811\n0.02 2.62573 4.79231 7.04137 9.37291\n0.03 2.69305 4.94804 7.32873 9.83511\n0.04 2.76347 5.11049 7.62782 10.31550\n0.05 2.83711 5.27992 7.93912 10.81470\n0.06 2.91410 5.45659 8.26312 11.33370\n0.07 2.99453 5.64079 8.60034 11.87320\n0.08 3.07856 5.83282 8.95133 12.43410\n0.09 3.16630 6.03296 9.31664 13.01730\nTable 3 Numerical values of density fluctuations\n△SNSfor various squeezed number state for squeezing\nparameter ρ <1\nρ n=1 n=2 n=3 n=4\n0.1 3.25790 6.24150 9.69680 13.62380\n0.2 4.42100 8.87210 14.46600 21.20270\n0.3 6.17420 12.81450 21.58080 32.47310\n0.4 8.79540 18.70150 32.19470 49.27500\n0.5 12.70040 27.47850 48.02890 74.35150\n0.6 18.50970 40.55610 71.65060 111.79300\n0.7 27.14900 60.03830 106.89000 167.70500\n0.8 39.99840 89.06340 159.46100 251.19200\n0.9 59.11520 132.31200 237.88900 375.84600\n\n:T2\n00:\u000b\nCSNS=1\n4G6(t)m4\u001c\n:∧\nΦ4\n:\u001d\nCSNS\n+m2\n4\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nCSNS+m2\n4\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nCSNS\n+1\n4G6(t)\u001c\n:∧\nΠ4\n:\u001d\nCSNS. (54)\nUsing Eqs. (12-13, 20-30) the values of\u001c\n:∧\nΠ4\n:\u001d\nCSNS can be computed as\n\u001c\n:∧\nΠ4\n:\u001d\nCSNS=G6(t)\n4m2t4\u0014\n3 + Υ∗4+ Υ4+ 6Υ∗2Υ2+ 12Υ∗Υ−6Υ∗2−6Υ2\n−4Υ∗3Υ−4Υ∗Υ3+ Cosh4ρ(6n2−6n)\n+ Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n13Fig. 1 Variation of density fluctuations △SNSfor various values of squeezing parameter ρ\nFig. 2 3-D plot showing variation of density fluctuations △SNSwith the values of n and squeezing\nparameter ρ\n+ Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2+ 48n+ 24\u0001\n+ Cosh ρSinhρ(24n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Sinh2ρ(12n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Cosh2ρ(12n)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\u0015\n. (55)\n14Using Eqs. (12-13, 20-30) the values of\u001c\n:∧\nΦ4\n:\u001d\nCSNScan be computed as\n\u001c\n:∧\nΦ4\n:\u001d\nCSNS=1\n4m2G6(t)\u0014\n3 + Υ∗4+ Υ4+ 6Υ∗2Υ2+ 12Υ∗Υ\n−6Υ∗2−6Υ2−4Υ∗3Υ−4Υ∗Υ3+ Cosh4ρ\u0000\n6n2−6n\u0001\n+ Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n+ Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2+ 48n+ 24\u0001\n+ Cosh ρSinhρ(24n+ 12){1 + 2Υ∗Υ}\n−Υ∗2−Υ2+ Sinh2ρ(12n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Cosh2ρ(12n)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\u0015\n. (56)\nUsing Eqs. (12-13, 20-30) the values of\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nCSNScan be calculated as\n\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nCSNS=1\n4m2t2\u0014\n3 + Υ∗4+ Υ4+ 6Υ∗2Υ2+ 12Υ∗Υ−6Υ∗2\n−6Υ2−4Υ∗3Υ−4Υ∗Υ3+ Cosh4ρ(6n2−6n)\n+ Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n+ Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2+ 48n+ 24\u0001\n+ Cosh ρSinhρ(24n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Sinh2ρ(12n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Cosh2ρ(12n)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\u0015\n. (57)\nUsing Eqs. (12-13, 20-30) the values of\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nCSNS can be calculated as\n\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nCSNS=1\n4m2t2\u0014\n3 + Υ∗4+ Υ4+ 6Υ∗2Υ2+ 12Υ∗Υ\n−6Υ∗2−6Υ2−4Υ∗3Υ−4Υ∗Υ3+ Cosh4ρ(6n2−6n)\n+ Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Cosh2ρSinh2ρ\u0000\n36n236n+ 12\u0001\n+ Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2+ 48n+ 24\u0001\n+ Cosh ρSinhρ(24n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Sinh2ρ(12n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n15+ Cosh2ρ(12n)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\u0015\n. (58)\nSubstituting the values of\u001c\n:∧\nΦ4\n:\u001d\nCSNS,\u001c\n:∧\nΠ4\n:\u001d\nCSNS,\u001c\n:∧\nΦ2∧\nΠ2\n:\u001d\nCSNS,\n\u001c\n:∧\nΠ2∧\nΦ2\n:\u001d\nCSNSin Eq. (54), The Normal ordered value of square of energy momentum\ntensor T00can be calculated as\n\n:T2\n00:\u000b\nCSNS=\u00121\n16m2t4+1\n8t2+m2\n16\u0013\u0014\n3 + Υ∗4+ Υ4\n+ 6Υ∗2Υ2+ 12Υ∗Υ−6Υ∗2−6Υ2−4Υ∗3Υ−4Υ∗Υ3\n+ Cosh4ρ(6n2−6n) + Sinh4ρ\u0000\n6n2+ 18n+ 12\u0001\n+ Cosh2ρSinh2ρ\u0000\n36n2+ 36n+ 12\u0001\n+ Cosh3ρSinhρ\u0000\n24n2\u0001\n+ Cosh ρSinh3ρ\u0000\n24n2+ 48n+ 24\u0001\n+ Cosh ρSinhρ(24n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Sinh2ρ(12n+ 12)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\n+ Cosh2ρ(12n)\b\n1 + 2Υ∗Υ−Υ∗2−Υ2\t\u0015\n. (59)\nNow, normal ordered value of temporal component of energy momentum tensor\n⟨:T00:⟩CSNS=1\n2G3(t)m2\u001c\n:∧\nΦ2\n:\u001d\nCSNS +1\n2G3(t)\u001c\n:∧\nΠ2\n:\u001d\nCSNS. (60)\nUsing Eqs. (12-13, 20-30) the values of\u001c\n:∧\nΠ2\n:\u001d\nCSNS can be calculated as\n\u001c\n:∧\nΠ2\n:\u001d\nCSNS =G3(t)\n2mt2[Cosh2ρ(2n) + Sinh2ρ(2n+ 2)\n+ Cosh ρSinhρ(4n+ 2) + 1 + Υ∗Υ−Υ∗2−Υ2], (61)\nUsing Eqs. (12-13, 20-30) the values of\u001c\n:∧\nΦ2\n:\u001d\nCSNS can be calculated as\n\u001c\n:∧\nΦ2\n:\u001d\nCSNS=1\n2mG3(t[Cosh2ρ(2n) + Sinh2ρ(2n+ 2)\n+ Cosh ρSinhρ(4n+ 2) + 1 + Υ∗Υ−Υ∗2−Υ2]. (62)\nUsing Eqs. (61) and (62) in Eqs. (60), normal ordered average value of temporal\ncomponent of energy momentum tensor can be computed as\n16⟨:T00:⟩CSNS=\u0014m\n4+1\n4mt2\u0015\n[Cosh2ρ(2n) + Sinh2ρ(2n+ 2)\n+ Cosh ρSinhρ(4n+ 2) + 1 + Υ∗Υ−Υ∗2−Υ2]. (63)\nSquare of of normal ordered value of temporal component of energy momentum\ntensor can be calculated as\n⟨:T00:⟩2\nCSNS =\u00121\n16m2t4+1\n8t2+m2\n16\u0013\u0014\n1 + Υ∗4+ Υ4+ 3Υ∗2Υ2\n+ 2Υ∗Υ−2Υ∗2−2Υ2−2Υ∗3Υ−2Υ∗Υ3\n+ Cosh4ρ\u0000\n4n2\u0001\n+ Sinh4ρ\u0000\n4n2+ 8n+ 4\u0001\n+ Cosh2ρSinh2ρ\u0000\n24n2+ 24n+ 4\u0001\n+ Cosh3ρSinhρ(16n2\n+ 8n) + Cosh ρSinh3ρ\u0000\n16n2+ 24n+ 8\u0001\n+ Cosh ρSinhρ(8n+ 4)\b\n1 + Υ∗Υ−Υ∗2−Υ2\t\n+ Sinh2ρ(4n+ 4)\b\n1 + Υ∗Υ−Υ∗2−Υ2\t\n+ Cosh2ρ(4n)\b\n1 + Υ∗Υ−Υ∗2−Υ2\t\u0015\n. (64)\nSubstituting the values of\n:T2\n00:\u000b\nCSNS and⟨:T00:⟩2CSNS in Eq. (41) then the\ndensity fluctuations for coherent squeezed number state is\n△CSNS =\u00121\n16m2t4+1\n8t2+m2\n16\u0013\u0014\n2 + 3Υ∗2Υ2+ 10Υ∗Υ−4Υ∗2\n−4Υ2−2Υ∗3Υ−2Υ∗Υ3+ Cosh4ρ(2n2−6n)\n+ Sinh4ρ\u0000\n2n2+ 10n+ 8\u0001\n+ Cosh2ρSinh2ρ\u0000\n12n2+ 12n+ 8\u0001\n+ Cosh3ρSinhρ\u0000\n8n2−8n\u0001\n+ Cosh ρSinh3ρ\u0000\n8n2+ 24n+ 16\u0001\n+ Cosh ρSinhρ(16n+ 8)\b\n1−Υ∗2−Υ2\t\n+ Cosh ρSinhρ(40n\n+ 20){Υ∗Υ}+ Sinh2ρ(8n+ 8)\b\n1−Υ∗2−Υ2\t\n+ Sinh2ρ(20n+ 20){Υ∗Υ}+ Cosh2ρ(8n)\b\n1−Υ∗2−Υ2\t\n+ Cosh2ρ(20n){Υ∗Υ}\u0015\n. (65)\nEq. (65) shows the △CSNS (density fluctuations for Coherent Squeezed Number\nState (CSNS)) that depend on squeezing parameter ( ρ), Coherent state parameter\n(Υ) as well as number state (n) but its dependency on Coherent state parameter and\nnumber state of consideration is more prominent than the squeezing parameter ρ.\nNumerical values of density fluctuations △CSNS for various values ofcoherent squeezed\n17number state calculated using Eq. (65) for n =1 to 4 is as shown in Table 4, 5, 6.\nFor simplification we considered, Υ*=Υ=1, and for squeezing parameter ( ρ) ranging\nbetween 0 .001 to 0.09 the variation in density fluctuations is very small for all number\nstate taken into consideration in Table 4, 5 but in Table 6 it is shown that for ρranging\n0.1 to 0.09 shows large variation in △CSNS. If we consider coherent state parameter\nΥ*=Υ=0, Eq. (65) reduces to Eq. (53) that is for density fluctuations for squeezed\nnumber state. For substituting both coherent state parameter Υ*=Υ=0 and number\nstate n=0 in Eq. (65), it produces density fluctuations for squeezed vacuum state [44–\n46], showing the validity of approximation used in SCEE. Fig. 3 shows variation of\ndensity fluctuations △CSNS, for various values of coherent squeezed number state with\nsqueezing parameter ρ. It can be deduced from Fig. 3 that with increasing value of ρ\nas well as n, density fluctuations △CSNS increases. 3-D plot between ρ, n and △CSNS,\nshows variation of density fluctuations △CSNS with various values of coherent squeezed\nnumber state and squeezing parameter ρ(Fig. 4).\nTable 4 Numerical values of density fluctuations\n△CSNS for various coherent squeezed number state\nfor squeezing parameter ρ <<< 1\nρ n=1 n=2 n=3 n=4\n0.001 2.75902 5.76903 9.78306 14.80110\n0.002 2.76806 5.78813 9.81624 14.85240\n0.003 2.77714 5.80730 9.84953 14.90380\n0.004 2.78624 5.82653 9.88295 14.95550\n0.005 2.79538 5.84583 9.91648 15.00730\n0.006 2.80454 5.86520 9.95014 15.05940\n0.007 2.81374 5.88463 9.98392 15.11160\n0.008 2.82297 5.90413 10.01780 15.16400\n0.009 2.83223 5.92370 10.05180 15.21660\nTable 5 Numerical values of density fluctuations\n△CSNS for various coherent squeezed number\nstate for squeezing parameter ρ << 1\nρ n=1 n=2 n=3 n=4\n0.01 2.84152 5.94334 10.08600 15.26940\n0.02 2.93615 6.14353 10.43420 15.80820\n0.03 3.03400 6.35083 10.79520 16.36700\n0.04 3.13520 6.56551 11.16930 16.94670\n0.05 3.23986 6.78784 11.55720 17.54800\n0.06 3.34812 7.01811 11.95940 18.17190\n0.07 3.46011 7.25664 12.37630 18.81910\n0.08 3.57596 7.50372 12.80860 19.49060\n0.09 3.69582 7.75970 13.25690 20.18740\n18Table 6 Numerical values of density fluctuations\n△CSNS for various coherent squeezed number state for\nsqueezing parameter ρ <1\nρ n=1 n=2 n=3 n=4\n0.1 3.81984 8.02489 13.72180 20.91050\n0.2 5.32605 11.26900 19.43740 29.83140\n0.3 7.46335 15.92570 27.70820 42.81080\n0.4 10.51790 22.64950 39.73410 61.77180\n0.5 14.91100 32.40740 57.29290 89.56740\n0.6 21.26490 46.63140 83.02110 130.43400\n0.7 30.49980 67.44430 120.83300 190.66700\n0.8 43.97830 97.99640 176.54700 279.63000\n0.9 63.72180 142.96800 258.81200 411.2550\nFig. 3 Variation of density fluctuations △CSNS for various values of squeezing parameter ρ\n7 Results and Discussion\nWe analyzed quantum effect of Squeezed Number State and Coherent Squeezed Num-\nber State on Friedmann-Robertson-Walker universe in the framework of semiclassical\ntheory of gravity. Non-classical state of gravity plays a significant role in number state\nrepresentation and particle production. As there was no consistent quantum theory of\ngravity is available, so semiclassical methods at appropriate situations, where back-\nground metric has classical consideration with matter field as quantum effects [32, 33].\nHere, we consider number state evolution for oscillatory phase inflaton for Coherent\nSqueezed Number State and Squeezed Number State formalisms were analyzed. We\nanalyzed quantum consideration for coherent state depends on number state and\ncoherent state parameter while squeezed states are concomitant on number state\n19Fig. 4 3-D plot showing variation of density fluctuations △CSNS with number state and squeezing\nparameter ρ\nand parameter of squeezing. Both these formulations obtained universe at squeezed\nvacuum state [44] shows the validity of the semiclassical Einstein equation and energy\ndensity of the inflaton. Oscillatory phase of inflaton for Coherent Squeezed Num-\nber State and Squeezed Number State has similar power-law expansion of Einstein\nequation [48–50]. In our study, the results revealed that for both Squeezed number\nstates and Coherent squeezed number states and density fluctuations are inversely\nproportional to various powers to t.\nThe results deduced that inflaton is minimally coupled to flat FRW universe can\nbe represented by SCEE. Quantum effects and density fluctuations of inflaton are\nanalyzed in context of number state representation for Squeezed Number State (SNS)\nand Coherent Squeezed Number State (CSNS), further generalized to demonstrate\nthe validity of approximations in terms of energy density. 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URL https://linkinghub.elsevier.com/\nretrieve/pii/S0921452607002256.\n25" }, { "title": "2402.07161v1.Extended__N__centered_ensemble_density_functional_theory_of_double_electronic_excitations.pdf", "content": "Extended N-centered ensemble density functional theory of double electronic\nexcitations\nFilip Cernatic1and Emmanuel Fromager1\n1Laboratoire de Chimie Quantique, Institut de Chimie, CNRS/Universit´ e\nde Strasbourg, 4 rue Blaise Pascal, 67000 Strasbourg, France\nA recent work [arXiv:2401.04685] has merged N-centered ensembles of neutral and charged elec-\ntronic ground states with ensembles of neutral ground and excited states, thus providing a general\nand in-principle exact (so-called extended N-centered) ensemble density functional theory of neu-\ntral and charged electronic excitations. This formalism made it possible to revisit the concept of\ndensity-functional derivative discontinuity, in the particular case of single excitations from the high-\nest occupied Kohn–Sham (KS) molecular orbital, without invoking the usual “asymptotic behavior\nof the density” argument. In this work, we address a broader class of excitations, with a particular\nfocus on double excitations. An exact implementation of the theory is presented for the two-electron\nHubbard dimer model. A thorough comparison of the true physical ground- and excited-state elec-\ntronic structures with that of the fictitious ensemble density-functional KS system is also presented.\nDepending on the choice of the density-functional ensemble as well as the asymmetry of the dimer\nand the correlation strength, an inversion of states can be observed. In some other cases, the strong\nmixture of KS states within the true physical system makes the assignment “single excitation” or\n“double excitation” irrelevant.\nI. INTRODUCTION\nIn the mean-field (or noninteracting) description of\nelectronic structures, such as Hartree–Fock (HF) the-\nory [1, 2] and Kohn–Sham density-functional theory (KS-\nDFT) [3], a double excitation refers to the promotion of\ntwo electrons from occupied orbitals in a reference con-\nfiguration (usually a ground-state Slater determinant)\ninto two virtual orbitals, resulting in a new configura-\ntion, “doubly-excited” relative to the reference. In prac-\ntice, this simple picture of distributing electrons among\norbitals in a single configuration is often used as a start-\ning point for describing neutral excitation processes ( i.e.,\nprocesses involving two states with the same number of\nelectrons) in interacting many-electron systems. In the\nlatter, doubly-excited configurations alone no longer re-\nflect the full details of the electronic structure of ex-\ncited states, which are in general described by config-\nuration expansions with single and multiple (double and\nhigher) excitations from the reference [4]. Contributions\nfrom the doubles are absolutely essential in many appli-\ncations, such as the study of excited states in conjugated\nmolecules [5–8], singlet fission [9, 10], and autoionizing\nresonances [11], to cite a few examples.\nOne of the standard and computationally affordable\nmethods for computing neutral excitations and excited-\nstate properties in molecules and extended systems is the\nlinear-response time-dependent DFT (TD-DFT) [12–15].\nIn linear response TD-DFT, single excitations are ex-\nplicitly encoded in the KS density-density response func-\ntion, from which any true interacting excitation energy\n(not only single excitation ones) can in principle be re-\ntrieved viathe frequency-dependent Hartree-exchange-\ncorrelation (Hxc) kernel, which relates to the functional\nderivative of the time-dependent density-functional Hxc\npotential. However, the development of practically appli-\ncable and accurate Hxc kernels is far from trivial [16–18],and, in the most commonly used adiabatic approxima-\ntion, the frequency-independent ground-state Hxc kernel\nis employed. As a result, double and higher excitations\nare completely absent from the computed spectra (the\nreader is referred to Refs. [11, 14, 15, 19] for more com-\nprehensive discussions on this matter).\nAlternatively, a time-independent and variational ap-\nproach to excited states that has recently gained an\nincreasing interest is the theory of many-electron en-\nsembles [20–45]. Ensemble DFT, which extends reg-\nular ground-state DFT to ensembles of ground and\n(neutral) excited states, was originally introduced by\nTheophilou [21, 46] for equi-ensembles and then further\ngeneralized by Gross, Oliveira and Kohn [23, 47, 48],\nhence the name TGOK-DFT [49]. Unlike linear response\nTD-DFT, TGOK-DFT can describe explicitly any (sin-\ngle or multiple) excitation process, in principle exactly,\nwith essentially the same computational cost as a regu-\nlar ground-state DFT calculation. A single calculation is\nin principle sufficient to retrieve the energy levels of all\nthe states that belong to the ensemble [50]. Providing\na proper description of the true physical ensemble en-\nergy, through an appropriate ensemble weight-dependent\nHxc density functional is, however, a very challenging\ntask [25, 33, 35, 37–39, 51–53].\nAs shown recently by the authors and co-workers [49],\nthe weight dependence of the ensemble Hxc density\nfunctional can be explicitly connected to the density-\nfunctional exactification of the one-electron KS picture\n(excitation energy-wise), through the formulation of ex-\nact Koopmans’ theorems for specific ionization processes.\nIndeed, by combining the ionization of the ground state\nwith that of the neutrally-excited state of interest, as\noriginally proposed by Levy [54], it becomes possible to\nexactify the KS orbital energies in the evaluation of neu-\ntral excitation energies. This can be achieved within the\nso-called extended N-centered (e Nc) ensemble density-arXiv:2402.07161v1 [cond-mat.str-el] 11 Feb 20242\nfunctional formalism [49], where, by construction, the\nensemble density still integrates to the integer number\nNof electrons in the reference ground state, like in\nTGOK-DFT, despite the incorporation of charged ex-\ncited states into the ensemble. This trick allows for an\nexactification of Koopmans’ theorem without invoking\nthe asymptotic behavior of the density away from the sys-\ntem under study, unlike in more conventional approaches\nto density-functional ensembles of ground and excited\nstates [37, 54]. An immediate consequence of such an\nexactification is the appearance of a density-functional\nderivative discontinuity in the Hxc potential following the\ninclusion of a given excitation into the ensemble. Even\nthough e Nc ensemble DFT is a very general approach,\nonly single excitations from the highest occupied molec-\nular orbital (HOMO) have been discussed in detail in\nRef. 49. In the present work, we extend the discussion\nto any type of single or double excitation process, with a\nparticular focus on the derivative discontinuity that the\nlatter induces and the connection between the ensemble\ndensity-functional KS electronic structure and that of the\ntrue physical system.\nThe paper is organized as follows. After a brief review\nof eNc ensemble DFT in Sec. II A, we present in Sec. II B\na general density-functional exactification of Koopmans’\ntheorem and its application to the evaluation of any sin-\ngle or double neutral excitation energy. The degree of\nexcitation in the ensemble density-functional KS system\nand its connection to the physical process is also dis-\ncussed (in Sec. II C). A more explicit derivation of the\ntheory for an e Nc ensemble with two neutral excited\nstates, in addition to the cationic ground state, is pre-\nsented in Sec. III. Its exact implementation within the\nHubbard dimer model is finally discussed (in Secs.IV A\nand IV B), and the results obtained for various ensemble\nweight values, correlation, and asymmetry regimes are\nanalyzed in Sec. IV C. Conclusions are given in Sec. V.\nII. THEORY\nA. Brief review of extended N-centered ensemble\nDFT\nWhile a regular N-centered ensemble consists of a ref-\nerence N-electron ground state complemented by the\ncationic [( N−1)-electron] and anionic [( N+ 1)-electron]\nground states, to which (possibly different) ensemble\nweights are assigned [55, 56], an e Nc ensemble incor-\nporates neutral excitation processes [49]. In Ref. [49],\nthese processes have been considered explicitly for the\nN-electron system only but in fact, as it will become\nclear and useful in the following, excited states of the\n(N±p)-electron system, where p= 1,2, . . ., can be triv-\nially incorporated into the ensemble too, thus making the\nformalism very general. Mathematically, an e Nc ensem-\nble, that we simply refer to as ensemble from now on, isdescribed by the following density matrix operator,\nˆΓξeNc=\n1−X\nν̸=0Nν\nNξν\n|Ψ0⟩⟨Ψ0|+X\nν̸=0ξν|Ψν⟩⟨Ψν|\n(1a)\nshorthand=\nnotationX\nνξν|Ψν⟩⟨Ψν|. (1b)\nΨ0≡ΨN\n0denotes the (normalized) reference ground-\nstate wavefunction of N0=Nelectrons with Hamilto-\nnian ˆH=ˆT+ˆWee+ˆVext, where ˆT≡ −1\n2PN\ni=1∇2\nriis the\nkinetic energy operator, ˆWee≡PN\n1≤i0 of interest:\nΩN\nν≡Eν−E0= (Eκ−E0)−(Eκ−Eν) (24a)\n=\u0010\nEξ[0→κ]\nκ − Eξ[0→κ]\n0\u0011\n−\u0010\nEξ[ν→κ]\nκ − Eξ[ν→κ]\nν\u0011\n(24b)\n=X\nk(nκ,k−n0,k)εξ[0→κ]\nk−X\nk(nκ,k−nν,k)εξ[ν→κ]\nk.\n(24c)\nInterestingly, in the above mathematical construction,\nthe Hxc potentials associated with each ionization pro-\ncess reproduce the same ensemble density nξ(r). Con-\nsequently, they differ by a constant which, according to\nEqs. (22) and (23), simply corresponds to a weight deriva-\ntive of the Hxc ensemble density functional:\nZdr\nN\u0010\nvξ[ν→κ]\nHxc (r)−vξ[0→κ]\nHxc (r)\u0011\nnξ(r)ν>0=∂Eξ\nHxc[n]\n∂ξν\f\f\f\f\f\nn=nξ.\n(25)\nEq. (25) generalizes previous work [37, 54] to any type\nof neutral excitation, without invoking the asymptotic\nbehavior of the ensemble density away from the system\nof interest (see Refs. 65 and 39 for a detailed comparison\nof the two approaches for charged excitations).\nIfκcorresponds, in the noninteracting KS world (see\nSecs. II C and IV C 2 for further discussion on this point),\nto an ionized state with a hole in the KS orbital i\n(1≤i≤N) while ν >0 corresponds to a single i→aex-\ncitation ( a > N ), then the corresponding exact physical\nexcitation energy simply reads, according to Eq. (24c),\nΩN\nν=εξ[ν→κ]\na −εξ[0→κ]\ni . (26)\nOn the other hand, if ν >0 now corresponds (still in\nthe KS world) to a double ( i, j)→(a, b) excitation (1 ≤\nj≤Nandb > N ), and κis still the singly-ionized state\nwith a hole in orbital i, then the exact physical excitation\nenergy expression becomes, according to Eq. (24c),\nΩN\nν=−εξ[0→κ]\ni −εξ[ν→κ]\nj +εξ[ν→κ]\na +εξ[ν→κ]\nb(27a)\n=εξ[ν→κ]\na −εξ[0→κ]\ni +\u0010\nεξ[ν→κ]\nb−εξ[ν→κ]\nj\u0011\n,(27b)\nor, equivalently,\nΩN\nν=εξ[ν→κ]\na −εξ[0→κ]\ni +\u0010\nεξ[0→κ]\nb−εξ[0→κ]\nj\u0011\n,(28)\nbecause the Hxc potentials vξ[ν→κ]\nHxc (r) and vξ[0→κ]\nHxc (r) only\ndiffer by a constant expressed in Eq. (25). Eqs. (26) and\n(28) provide an exactification of the KS orbital energies\nin the evaluation of single- and double-electron excitation\nenergies, respectively. They generalize Eq. (54) of Ref. 49\nwhich is only applicable to single excitations from the\nHOMO.5\nC. What are we supposed to learn from the KS\nensemble about physical excitation processes?\nAs already mentioned in the introduction, the descrip-\ntion of double electronic excitations ( i.e., the modelling\nof two-hole/two-particle states) in the context of lin-\near response TD-DFT is very challenging [15, 18]. In-\ndeed, in the latter regime, only single excitations ( i.e.,\none-hole/one-particle states) are treated explicitly . Dou-\nble electron excitation energies can in principle be re-\ntrieved by using a proper frequency-dependent Hxc ker-\nnel [15, 18]. The situation is quite different in the con-\ntext of ensemble DFT, since multiple electronic excita-\ntions can be explicitly incorporated into the KS ensem-\nble. What is far from clear, however, is how informative\nthe KS ground and excited states are about the true in-\nteracting eigenstates. Let us first comment on a com-\nmon misunderstanding of the statement “ensemble DFT\ncan describe double excitations”. Obviously, the latter\ndoes not mean that the true physical excitation process\n(to which double excitations may contribute) matches\nthe one occuring in the ensemble density-functional KS\nsystem. It simply means that two-hole/two-particle ex-\ncitation processes can be treated explicitly within the\nensemble KS orbital space. Despite the loss of informa-\ntion about the true interacting states, which is a common\nfeature of density-functional theories, ensemble DFT still\nprovides an in-principle exact description of single and\nmultiple excitations, ensemble density-wise. Indeed, for\na given number Mof lowest N-electron states ( M= 3 in\nthe following) and given ensemble weight values, the non-\ninteracting KS ensemble, which contains the same num-\nberMof lowest N-electron KS eigenstates (Slater deter-\nminants or configuration state functions) as the physical\none, is expected to reproduce the true interacting ensem-\nble density. It is a priori its only connection with the\ntrue physical ensemble but it is sufficient to determine,\nin principle exactly, the energy levels of all the states\nthat belong to that ensemble, according to Eqs. (11b)\nand (18c) [see also Refs. 50 and 49]. The identification of\nexcitations is clear in the noninteracting KS picture. For\nexample, in the Hubbard dimer model (see Sec. IV), the\nfirst excited state is singly-excited and the second one\nis doubly-excited. However, true interacting electronic\nstructures are much more complex. They can be mix-\ntures of ground, singly-excited, and doubly-excited KS\nstates, for example. In some specific asymmetry and cor-\nrelation regimes, a reordering of the eigenstates may also\noccur when switching from the noninteracting ensemble\nKS picture to the interacting one. These different scenar-\nios are illustrated and further discussed in Sec. IV C 2.III. EXPLICIT FORMULATION INVOLVING\nTHE GROUND CATIONIC STATE AND TWO\nNEUTRAL EXCITED STATES\nWe consider in this section the particular case (studied\nlater in the Hubbard dimer model) of an e Nc ensemble\nconsisting of the reference N-electron ground state, the\ntwo lowest N-electron excited states (with weights ξ1and\nξ2, respectively), and the ( N−1)-electron ground state\n(with weight ξ−):\nˆΓξ=\u0012\n1−(N−1)\nNξ−−ξ1−ξ2\u0013\f\fΨN\n0\u000b\nΨN\n0\f\f\n+ξ−\f\fΨN−1\n0\u000b\nΨN−1\n0\f\f+ξ1\f\fΨN\n1\u000b\nΨN\n1\f\f+ξ2\f\fΨN\n2\u000b\nΨN\n2\f\f,\n(29)\nwhere the collection of independent weights reduces to\nξ≡(ξ−, ξ1, ξ2). (30)\nNote that, in order to allow for a variational evaluation\nof the corresponding ensemble energy,\nEξ=\u0012\n1−(N−1)ξ−\nN−ξ1−ξ2\u0013\nEN\n0+ξ−EN−1\n0\n+ξ1EN\n1+ξ2EN\n2,(31)\nwhich is necessary to set up an ensemble DFT, the fol-\nlowing inequalities should be fulfilled:\nξ−≥0 (32)\nand [23]\nξ0= 1−(N−1)ξ−\nN−ξ1−ξ2≥ξ1≥ξ2≥0.(33)\nConsequently, we have\n2ξ2≤1−(N−1)ξ−\nN−ξ1≤1−(N−1)ξ−\nN−ξ2,(34)\nthus leading to the following allowed range of ensemble\nweight values,\n0≤ξ2≤1\n3\u0012\n1−(N−1)ξ−\nN\u0013\n(35)\nand\nξ2≤ξ1≤1\n2\u0012\n1−(N−1)ξ−\nN−ξ2\u0013\n. (36)\nTurning to the general construction in Eq. (22) of the\n(unique) Hxc potential that satisfies Koopmans’ theorem\nexactly, for a given ionization process that will be indexed\nwithI= 0,1,2 in the following, we obtain from Eq. (23)\nthe more explicit expressions\nDξ[0]\nHxc[n] =Eξ\nHxc[n]\nN−\u0012\n1 +ξ−\nN\u0013∂Eξ\nHxc[n]\n∂ξ−−ξ1\nN∂Eξ\nHxc[n]\n∂ξ1\n−ξ2\nN∂Eξ\nHxc[n]\n∂ξ2,\n(37)6\nDξ[1]\nHxc[n] =Eξ\nHxc[n]\nN−\u0012\n1 +ξ−\nN\u0013∂Eξ\nHxc[n]\n∂ξ−\n+\u0012\n1−ξ1\nN\u0013∂Eξ\nHxc[n]\n∂ξ1−ξ2\nN∂Eξ\nHxc[n]\n∂ξ2,(38)\nand\nDξ[2]\nHxc[n] =Eξ\nHxc[n]\nN−\u0012\n1 +ξ−\nN\u0013∂Eξ\nHxc[n]\n∂ξ−−ξ1\nN∂Eξ\nHxc[n]\n∂ξ1\n+\u0012\n1−ξ2\nN\u0013∂Eξ\nHxc[n]\n∂ξ2,\n(39)\nfor the ionization of the ground state ( I= 0), the ioniza-\ntion of the first excited state ( I= 1), and the ionization\nof the second excited state ( I= 2), respectively. An ex-\nact implementation of the three Hxc potentials from the\nabove ensemble density-functional quantities is presented\nin the next section within the Hubbard dimer model, as\na proof of concept.\nIV. EXACT IMPLEMENTATION FOR THE\nTWO-ELECTRON HUBBARD DIMER\nA. Introduction to the model\nThe Hubbard dimer is a simple but nontrivial two-\nsite lattice model that can be used, for example, for de-\nscribing diatomic molecules [66]. As it can be solved\nexactly [67], it is often used as a toy system for test-\ning new ideas in connection with the many-body prob-\nlem [28, 39, 50, 51, 66–74]. The basic idea of the model is\nto simplify the (second-quantized) ab initio Hamiltonian\nas follows,\nˆH→ˆH=ˆT+ˆU+ˆVext, (40)\nwhere the analogue for the kinetic energy operator ˆT\n(the so-called hopping operator), the on-site electron re-\npulsion operator ˆU, and the local (external) potential op-\nerator ˆVextread\nˆT=−tX\nσ=↑,↓(ˆc†\n0σˆc1σ+ ˆc†\n1σˆc0σ), (41a)\nˆU=U1X\ni=0ˆni↑ˆni↓, (41b)\nˆVext=∆vext\n2(ˆn1−ˆn0), (41c)\nrespectively. The index i∈ {0,1}labels the two atomic\nsites, ˆ niσ= ˆc†\niσˆciσis the spin-site occupation operator,\nand ˆni=P\nσ=↑,↓ˆniσplays the role of the density opera-\ntor (on site i). The asymmetry of the model is controlled\nby the difference ∆ vextin external potential between sites\n1 and 0, while electron correlation effects can be tunedthrough the ratio U/t. In this context, the electron den-\nsity is the collection of site occupations {ni=⟨ˆni⟩}i=0,1.\nIn the following, the central number of electrons will be\nfixed to N=n0+n1= 2, so that the density reduces to\na single number nthat we choose to be the occupation of\nsite 0, i.e.,n:=n0. Note that, in the symmetric dimer\n(which would correspond to the hydrogen molecule in a\nminimal basis, for example), we have n= 1. The asym-\nmetric dimer can be used, on the other hand, as a model\nfor heteronuclear diatomic molecules such as LiF [66], for\nexample.\nWe consider in the following the e Nc ensemble de-\nscribed in Sec. III, where the two neutral (singlet) excited\nstates are, in the noninteracting KS picture, singly and\ndoubly excited, respectively. The hopping parameter is\nset to t= 1/2 throughout the paper.\nB. Computation of exact ensemble\ndensity-functional energies and potentials\nThe implementation of e Nc ensemble DFT for a tri-\nensemble ( i.e., in the particular case where ξ2= 0) has\nbeen extensively discussed in Ref. 49. As shown in Ap-\npendix A, the more general 4-state ensemble case studied\nin the present work can be recast into an effective tri-\nensemble problem, simply because the three two-electron\nground- and excited-state (singlet) energies sum up to\n2U[50]. This simplification, which applies to the Hub-\nbard dimer only and is not general, leads to the following\nexpression for the interacting ensemble density functional\nintroduced in Eq. (10),\nFξ(n) = 2 Uξ2+ (1−3ξ2)Fζ(ν), (42)\nwhere ζ= (ζ−, ζ1) is an effective tri-ensemble weights\ncollection defined as follows,\nζ−≡ζ−(ξ) =ξ−\n1−3ξ2, (43a)\nζ1≡ζ1(ξ) =ξ1−ξ2\n1−3ξ2, (43b)\nand\nν≡ν(n,ξ) =n−3ξ2\n1−3ξ2(44)\nis an effective tri-ensemble density. From Eq. (42), taken\natU= 0, which gives\nTξ\ns(n) = (1 −3ξ2)Tζ\ns(ν), (45)\nand the following expression for the tri-ensemble nonin-\nteracting kinetic energy functional [49],\nTζ\ns(ν) =−2tp\n(1−ζ1)2−(1−ν)2, (46)\nwe can express exactly and analytically the 4-state en-\nsemble density-functional noninteracting kinetic energy\nas follows,\nTξ\ns(n) =−2tp\n(1−ξ1−2ξ2)2−(1−n)2. (47)7\nNote that Tξ\ns(n) does not depend on ξ−[49]. More-\nover, according to Eqs. (42) and (45), the 4-state en-\nsemble density-functional Hxc energy can be evaluated\nfrom the tri-ensemble one (which can be computed ex-\nactly through a Lieb maximization [49]) as follows,\nEξ\nHxc(n) =Fξ(n)−Tξ\ns(n) (48a)\n= 2Uξ2+ (1−3ξ2)Eζ\nHxc(ν). (48b)\nTurning to the ensemble density-functional potentials,\nthe difference in KS potential between sites 1 and 0,\n∆vξ\nKS(n), is the maximizer [24] for U= 0 of the e Nc\nensemble Lieb functional introduced in Appendix A (see\nEq. (A1)), thus leading to\n∆vξ\nKS(n) =∂Tξ\ns(n)\n∂n, (49)\nor, equivalently (see Eq. (47)),\n∆vξ\nKS(n) =2t(n−1)p\n(1−ξ1−2ξ2)2−(1−n)2. (50)\nOn the other hand, the ensemble Hxc potential difference\ncan be evaluated as follows (see Eq. (17)),\n∆vξ\nHxc= ∆vξ\nKS(nξ)−∆vext, (51)\nwhere nξis the true physical ensemble density. The\nlatter can be determined, for given ensemble weights ξ\nand external potential difference ∆ vextvalues, from the\nHellmann–Feynman theorem [24]:\n1−nξ=∂Eξ(∆v)\n∂∆v\f\f\f\f\n∆v=∆vext. (52)\nNote that ∆ vextis the maximizing potential of the inter-\nacting ensemble Lieb functional in Eq. (A1), when eval-\nuated for n=nξ. Consequently,\n∆vext=∂Fξ(n)\n∂n\f\f\f\f\nn=nξ, (53)\nand, according to Eqs. (49) and (51),\n∆vξ\nHxc= ∆vξ\nHxc(nξ), (54)\nwhere the Hxc ensemble density-functional potential dif-\nference formally equals\n∆vξ\nHxc(n) =∂Tξ\ns(n)\n∂n−∂Fξ(n)\n∂n=−∂Eξ\nHxc(n)\n∂n.(55)\nNote that the minus sign on the right-hand side of the\nabove equation originates from the arbitrary choice we\nmade to compute potential differences between sites 1\nand 0 while referring to the occupation of site 0 as the\ndensity n[24, 75].\nWe can now construct from ∆ vξ\nHxcthe ensemble Hxc\npotential, which is in principle defined up to a constantthat we denote −µξ\nHxc. Its value on site i(i= 0,1) reads\n(see Eq. (41c))\nvξ\nHxc,i= (−1)i−1∆vξ\nHxc(nξ)\n2−µξ\nHxc. (56)\nFor a given ionization process I(see Sec. III), the con-\nstant−µξ\nHxcis uniquely defined from the constraint of\nEq. (22), which becomes in the Hubbard dimer model\n(we recall that N= 2),\nZdr\nNvξ[I]\nHxc(r)nξ(r)\n→1\n21X\ni=0vξ[I]\nHxc,inξ\ni=\u0000\n1−nξ\u0001\n∆vξ\nHxc(nξ)\n2−µξ[I]\nHxc\n!=Dξ[I]\nHxc(nξ),(57)\nthus ensuring the exactification of Koopmans’ theorem\nfor that specific ionization. We finally conclude from\nEq. (56) that the value of the corresponding Hxc po-\ntential on site 1 equals\nvξ[I]\nHxc:=vξ[I]\nHxc,1=nξ∆vξ\nHxc(nξ)\n2+Dξ[I]\nHxc(nξ). (58)\nC. Results and discussion\n1. Derivative discontinuities induced by double excitations\nIn a recent work [49], we investigated the derivative\ndiscontinuity that the Hxc potential exhibits when the\nfirst singlet excited state is incorporated into the ensem-\nble under study. For that purpose, we compared two\nscenarios which are reproduced in Fig. 1 in the mod-\nerately correlated U/t= 2 regime, for both symmetric\n(∆vext= 0) and asymmetric (∆ vext= 1) dimers. In the\nfirst scenario, where 0 < ξ−≤2 and ξ1=ξ2= 0, the Hxc\npotential is uniquely defined from the ionization of the\ntwo-electron ground state, previously labelled as I= 0\n(see Eqs. (22) and (37)). In the second scenario, where\nξ−→0+, 0< ξ1<1/2, and ξ2= 0, the Hxc potential\nis defined from the ionization of the first excited state\n(I= 1), according to Eqs. (22) and (38). In this work,\nwe focus on the modification of the Hxc potential when\nthe ensemble density-functional KS system undergoes a\ndouble excitation (note that its connection with the exci-\ntation process that the true interacting system undergoes\nwill be discussed in detail in the next section). For that\npurpose, we introduce a third scenario that differs from\nthe second one only by the infinitesimal incorporation of\nthe second excited state into the ensemble, i.e.,ξ−→0+,\n0< ξ1<1/2, and ξ2→0+, so that the Hxc potential can\nnow be uniquely defined from the ionization of the latter\nstate ( I= 2), according to Eq. (39). As shown in Fig. 1,\nthe Hxc potential does exhibit a derivative discontinu-\nity when switching from the first to the second excited8\n0.0 0.5 1.0 1.5 2.0\nensemble weight – ξ1(blue, violet) or ξ−(red)0.20.40.60.81.0Hxc potential on site 1DD 2\nfor∆vext= 1\nDD 1\nfor∆vext= 1t= 0.5\nU= 1\nvξ[I=0 ]\nHxc,∆vext= 1\nvξ[I=1 ]\nHxc,∆vext= 1\nvξ[I=2 ]\nHxc,∆vext= 1vξ[I=0 ]\nHxc,∆vext= 0\nvξ[I=1 ]\nHxc,∆vext= 0\nvξ[I=2 ]\nHxc,∆vext= 0\nFIG. 1. Exact Hxc potential on site 1 (see Eqs. (51) and (58)) plotted as a function of an ensemble weight ( ξ−orξ1, depending\non the considered ionization process) for symmetric (dotted lines) and asymmetric (∆ vext= 1, solid lines) Hubbard dimers\nwith U/t= 2. For the ionization of the ground state ( I= 0, red curves), ξ−varies in the range 0 < ξ−≤2 while ξ1=ξ2= 0\n(see Eqs. (35) and (36)). As for the ionization of the first excited state ( I= 1, blue curves), ξ−→0+,ξ2= 0, and ξ1varies in\nthe range 0 < ξ1<1/2. The Hxc potential associated with the ionization of the second excited state ( I= 2, violet curves) is\nalso plotted as a function of ξ1(in the range 0 < ξ1<1/2) for ξ−→0+andξ2→0+. The vertical cyan blue and magenta\narrows show the derivative discontinuities that the Hxc potential exhibits when crossing regular two-electron ground-state DFT\n(where all the weights equal zero) from the ionized ground state toward the first and second excited states, respectively.\nstate, as expected from Eq. (25). We also note that, in\nthe asymmetric case (∆ vext= 1), the two Hxc poten-\ntials differ substantially in their variation with respect\nto the ensemble weight ξ1, especially when approaching\ntheξ1= 1/2 limit. This can be rationalized as follows.\nAccording to the final Hxc potential expression (on site\n1) given in Eq. (58), and Eq. (50), the deviation in Hxc\npotential between the first and second excited states can\nbe expressed exactly as follows,\nv(0+,ξ1,0+)[2]\nHxc −v(0+,ξ1,0)[1]\nHxc\n=\n0<ξ1≤1/2h\nDξ[2]\nHxc(nξ)−Dξ[1]\nHxc(nξ)i\nξ=(0,ξ1,0),(59)or, equivalently (see Eqs. (38), and (39)),\nv(0+,ξ1,0+)[2]\nHxc −v(0+,ξ1,0)[1]\nHxc\n=\"\n∂Eξ\nHxc(n)\n∂ξ2\f\f\f\f\f\nn=nξ−∂Eξ\nHxc(n)\n∂ξ1\f\f\f\f\f\nn=nξ#\nξ=(0,ξ1,0),(60)\nwhere, according to the reduction in ensemble size dis-\ncussed in appendix A (see also Eqs. (43), (44), and (48b)),\n∂Eξ\nHxc(n)\n∂ξ2\f\f\f\f\f\nξ=(0,ξ1,0)= 2U\n−\"\n3Eξ\nHxc(n)−∂ζ1(ξ)\n∂ξ2∂Eξ\nHxc(n)\n∂ξ1\n−∂ν(n,ξ)\n∂ξ2∂Eξ\nHxc(n)\n∂n#\nξ=(0,ξ1,0),(61)9\nthus leading to (see Eq. (55))\n\"\n∂Eξ\nHxc(n)\n∂ξ2−∂Eξ\nHxc(n)\n∂ξ1#\nξ=(0,ξ1,0)= 2U\n−\"\n3Eξ\nHxc(n) + (2 −3ξ1)∂Eξ\nHxc(n)\n∂ξ1\n+ 3(n−1)∆vξ\nHxc(n)#\nξ=(0,ξ1,0).(62)\nAs readily seen from Eqs. (60) and (62), the difference in\nHxc potentials consists of three density-functional contri-\nbutions to which 2 Uis added. One of them, which reads\nmore explicitly as follows,\n−h\n3(n−1)∆vξ\nHxc(n)\f\f\f\nn=nξi\nξ=(0,ξ1,0)(63a)\n=−h\n3(n−1)∆vξ\nKS(n)\f\f\f\nn=nξi\nξ=(0,ξ1,0)\n+ 3(nξ1−1)∆vext (63b)\n=−6t(nξ1−1)2\np\n(1−ξ1)2−(1−nξ1)2+ 3(nξ1−1)∆vext,(63c)\nwhere we used the shorthand notation nξ1:=nξ=(0,ξ1,0),\nrelates to the ensemble KS potential (see Eqs. (50) and\n(63b)). In the asymmetric U= 2t= ∆vext= 1 regime\ndepicted in Fig. 1, the ensemble density nξ1varies\nweakly with ξ1in the range 1 .4≤nξ1<1.5 [24, 49].\nThis explains why the Hxc potential for the second\nexcited state ( I= 2) decreases sharply with ξ1when\napproaching the limit ξ1= 1/2 (see the denominator in\nthe first term of Eq. (63c)).\nLet us finally note that, when the dimer is symmetric\n(i.e., ∆vext= 0), the ensemble density equals nξ= 1\nand [49]\nEξ\nHxc(n)\f\f\f\nn=1ξ=(0,ξ1,0)=U(1 +ξ1)\n2\n+ (1−ξ1)\u0012\n2t−1\n2p\nU2+ 16t2\u0013\n,\n(64)\nso that (see Eq. (48b))\nEξ\nHxc(n)\f\f\f\nn=1ξ=(0,ξ1,ξ2)=U(1 +ξ1)\n2\n+ (1−ξ1−2ξ2)\u0012\n2t−1\n2p\nU2+ 16t2\u0013\n.\n(65)\nThus we conclude that, like the Hxc potential defined\nfrom the ionization of the first excited state [49], the one\ndeduced from the ionization of the second excited state\nis weight-independent and it deviates from the latter asfollows, according to Eq. (60),\nv(0+,ξ1,0+)[2]\nHxc −v(0+,ξ1,0)[1]\nHxc =\n∆vext=0−2t−U\n2+1\n2p\nU2+ 16t2,\n(66)\nwhich is in perfect agreement with Fig. 1.\n2. Analysis of the physical eigenstates in the ensemble\ndensity-functional KS representation\nThe infinitesimal ξ−→0+incorporation of the ionized\nground state, which was essential for describing deriva-\ntive discontinuities in the previous section, is of no use\nin the following discussion since we are interested in the\nphysical and KS states, which are invariant under any\nuniform shift in potential. Therefore, we can simply set\nξ−= 0 and study the regular TGOK ensemble consist-\ning of the two-electron ground state and the two lowest\n(singlet) neutral excited states (with weights ξ1andξ2,\nrespectively). The (weight-independent) physical eigen-\nstates can be decomposed as follows in the lattice ( i.e.,\nlocalized) representation,\n|Ψν⟩=2X\nK=0⟨ΞK|Ψν⟩|ΞK⟩, ν= 0,1,2, (67)\nwhere [75]\n|Ξ0⟩= ˆc†\n0↑ˆc†\n0↓|vac⟩, (68a)\n|Ξ1⟩= ˆc†\n1↑ˆc†\n1↓|vac⟩, (68b)\n|Ξ2⟩=1√\n2\u0010\nˆc†\n0↑ˆc†\n1↓−ˆc†\n0↓ˆc†\n1↑\u0011\n|vac⟩. (68c)\nWe are in fact interested in the representation of the\neigenstates in the ( a priori weight-dependent, according\nto Eq. (13)) ensemble density-functional KS basis, i.e.,\n|Ψν⟩=2X\nµ=0⟨Φξ\nµ|Ψν⟩\f\fΦξ\nµ\u000b\n, ν= 0,1,2. (69)\nThe derivation of both representations is discussed in de-\ntail in Appendix B.\nLet us first consider the symmetric dimer (∆ vext= 0).\nSince, in this case, the ensemble density equals 1 [24],\nthe KS potential difference equals zero. Consequently,\nthe KS states are weight-independent and equivalent to\nthe solutions of the regular H¨ uckel (or tight binding)\nproblem for the hydrogen molecule in a minimal basis.\nThe configuration weights of the interacting eigenstates\nin the KS representation are plotted in Fig. 2 as func-\ntions of U(we recall that t= 1/2 throughout this work).\nFor analysis purposes, the configuration weights obtained\nin the lattice representation ( i.e., in the basis of the 1 s\natomic orbitals if we pursue the analogy with the hydro-\ngen molecule) are also plotted in the left panels of Fig. 3.\nFor symmetry reasons, the first (singlet) excited state is10\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight∆vext= 0, ξ1= 0, ξ2= 0\n|/angbracketleftΦξ\n0|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ0/angbracketright|2\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ1/angbracketright|2\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ2/angbracketright|2\nFIG. 2. Configuration weights in the ensemble density-functional KS basis\b\nΦξ\nµ\t\nof the true interacting eigenfunctions {Ψν}\nplotted as functions of the interaction strength Uin the symmetric Hubbard dimer (∆ vext= 0). The KS basis is weight-\nindependent in this case. Top panel: ground-state expansion ( ν= 0). Middle panel: first excited state ( ν= 1). Bottom panel:\nsecond excited state ( ν= 2). See text for further details.\nU-independent (it equals1√\n2(|Ξ0⟩ − |Ξ1⟩) and its energy\nisU) and, therefore, it matches the singly-excited KS\nstate. On the other hand, as Uincreases, both ground\nand second excited states (which belong to the same spa-\ntial symmetry) become mixtures of ground and doubly-\nexcited KS states, as expected. Referring to the second\nexcited state as “doubly-excited” is relevant in this case\nbut we should remember that the ground-state KS config-\nuration contributes significantly and, ultimately, equally,\nwhen the symmetric dimer becomes strictly correlated\n(i.e., when the hydrogen molecule dissociates).\nThe impact of asymmetry on the interacting ground-and excited-state configuration expansions within the\n(now weight-dependent) ensemble density-functional KS\nrepresentation is investigated in the moderately corre-\nlated U/t= 2 regime in Fig. 4. The stronger U/t= 6\ncorrelation regime is investigated in Fig. 5. We focus\nhere on equi-ensembles [45], which are commonly used\nin wavefunction theory calculations. Let us first consider\nthe bi-ensemble density-functional case, i.e.,ξ2= 0 and\nξ1=ξ0= 1/2 (see the left panels of both Figures). As\nsoon as we slightly deviate from the symmetric case ( i.e.,\nfor ∆ vext>0), the second excited state (which does\nnot belong to the bi-ensemble) rapidly reduces to the11\n0 1 2 3 4\nU0.000.250.500.751.00Configuration weight∆vext= 0\n0 1 2 3 4\nU0.000.250.500.751.00Configuration weight∆vext= 1\n|/angbracketleftΞ0|Ψ0/angbracketright|2\n|/angbracketleftΞ1|Ψ0/angbracketright|2\n|/angbracketleftΞ2|Ψ0/angbracketright|2\n0 1 2 3 4\nU0.000.250.500.751.00Configuration weight\n0 1 2 3 4\nU0.000.250.500.751.00Configuration weight|/angbracketleftΞ0|Ψ1/angbracketright|2\n|/angbracketleftΞ1|Ψ1/angbracketright|2\n|/angbracketleftΞ2|Ψ1/angbracketright|2\n0 1 2 3 4\nU0.000.250.500.751.00Configuration weight\n0 1 2 3 4\nU0.000.250.500.751.00Configuration weight|/angbracketleftΞ0|Ψ2/angbracketright|2\n|/angbracketleftΞ1|Ψ2/angbracketright|2\n|/angbracketleftΞ2|Ψ2/angbracketright|2\nFIG. 3. Configuration weights in the (local) site-based representation {ΞK}of the true interacting eigenfunctions {Ψν}plotted\nas functions of the interaction strength Uin the symmetric (∆ vext= 0, left panels) and asymmetric (∆ vext= 1, right panels)\nHubbard dimers. Top panels: ground-state expansion ( ν= 0). Middle panels: first excited state ( ν= 1). Bottom panels:\nsecond excited state ( ν= 2). See text for further details.\ndoubly-excited (bi-ensemble density-functional) KS de-\nterminant, as ∆ vextincreases (see the bottom left panel\nof Fig. 4 and the bottom panels of Fig. 5). On the other\nhand, for U= 1, both ground and first excited states are\nmixtures of ground and singly-excited KS states, in the\nrange 0 <∆vext≤3. Referring to the first excited state\nas singly-excited is relevant in this case but we should\nof course remember that, because of electron correlation,\nthe ground-state KS configuration may contribute signif-\nicantly. Actually, in the vicinity of ∆ vext=t= 1/2,\nwe notice that the latter contributes even more than the\nfirst excited KS one (see the top and middle left panelsof Fig. 4). In the stronger U= 3 correlation regime, this\nfeature is even more pronounced when 0 .1<∆vext< U\n(see the top and middle panels of Fig. 5). For complete-\nness, we plot in the left panels of Fig. 6 the configu-\nration weights as functions of Ufor the fixed ∆ vext= 1\nasymmetric potential value. As readily seen from the top\nand middle panels, as we approach the strictly correlated\nU/t→+∞limit, the physical interacting eigenstates\nbecome pure KS states with a major difference though:\nThe first excited state turns out to be the ground KS\nstate, and vice versa . The reason is the following. In\nthis regime, the ground- and first excited-state densities12\nare close to 1 (because the ground-state wavefunction is\nessentially that of the strongly correlated and symmetric\ndimer) and 2, respectively, as deduced from the top and\nmiddle right panels of Fig. 3 (see also Eqs. (67) and (68)).\nConsequently, the equi-bi-ensemble density is close to 1.5,\nwhich means that the equi-bi-ensemble KS potential dif-\nference is approaching + ∞(see Eq. (50)). As a result,\nin the ground KS state, the two electrons are essentially\nlocalized on site 0, which corresponds to the firstinteract-\ning excited state. On the other hand, in the first excited\nKS state, the density equals 1 on both sites, exactly like\nin the interacting ground state. We note finally that, in\nthe strongly asymmetric ∆ vext>> U regimes depicted\nin the left panels of Figs. 4 and 5, physical and KS states\nbecome essentially identical as ∆ vextapproaches + ∞.\nIndeed, in this regime, the equi-bi-ensemble density is\nstill close to 1.5, as deduced from the top and middle\npanels of Fig. 7. Therefore, the KS states are unchanged\nbut the interacting ground state now consists of two elec-\ntrons localized on site 0 while the first excited state has\na density equal to 1 on both sites, exactly like in the KS\nworld.\nLet us now turn to the equi-tri-ensemble density-\nfunctional case. As clearly illustrated in Figs. 4 and 5,\nmoving from a bi- to a tri-ensemble completely changes\nthe ensemble density-functional KS basis and, therefore,\nthe representation of the physical eigenstates (which are\nunchanged) in the latter basis. For the fixed U= 1\ninteraction strength value, the doubly-excited KS state\ncontributes to both ground and first excited interacting\nstates for a broader range of ∆ vextvalues. In the lat-\nter asymmetric regime, we also notice that, unlike in the\nequi-bi-ensemble case, the ground KS state gives a rel-\natively good description of the true ground state (see\nthe top panels of Fig. 4), while both first and second ex-\ncited states are mixtures of singly- and doubly-excited KS\nstates (see the middle and bottom right panels of Fig. 4).\nThe overall change in ensemble density-functional KS\nrepresentation of the true eigenstates, when moving from\na bi- to a tri-equi-ensemble, can be rationalized as follows.\nAs pointed out in Sec. IV C 1, when U= 2t= ∆vext= 1,\nfor example, the equi-bi-ensemble density is relatively\nclose to 1.5 (both ground- and first excited-state densities\nare close to the latter value [24]), which means that the\nequi-bi-ensemble KS potential (for which ξ1= 1/2 and\nξ2= 0 in Eq. (50)) is very attractive on site 0. Therefore,\nin this case, the KS ground state essentially consists of\ntwo electrons localized on site 0, which does not reflect at\nall the true ground-state electronic structure (see the top\nleft panel of Fig. 7). On the other hand, when ξ1= 1/3\nandξ2= (1/3)−η, where η→0+, so that we can ap-\nproach the equi-tri-ensemble case, the density equals\nntri=\u00121\n3+η\u0013\nnΨ0+1\n3nΨ1+\u00121\n3−η\u0013\nnΨ2(70a)\n≈1 + 3 η(nΨ0−1), (70b)\nwhere we used the fact that nΨ2= 3−nΨ0−nΨ1[50] and,\nin the considered regime, nΨ0≈nΨ1. [24] Consequently,the KS potential difference can be simplified as follows\n(see Eq. (50)),\n∆vtri\nKS≈6tη(nΨ0−1)p\n4η2−(ntri−1)2=6t(nΨ0−1)q\n4−9 (nΨ0−1)2.\n(71)\nAs readily seen from the above equation, unlike in the\nequi-bi-ensemble case, the KS potential does not become\nsingular when nΨ0is approaching 1.5, which is the case\nin the considered regime. Therefore, the KS potential\nis now much less attractive on site 0 and the electrons\nare more delocalized in the KS ground state, like in the\ninteracting ground state.\nNote that, in this moderately correlated case, each\nKS excited state still gives a qualitatively correct de-\nscription of each physical excited state. In the stronger\n1 = ∆ vext<< U correlation regime, where the equi-bi-\nensemble density is even closer to 1.5 (since nΨ0≈1 and\nnΨ1≈2 [24], thus leading to nΨ2≈0), the equi-tri-\nensemble density reduces to ntri≈1 +η(see Eq. (70a))\nand\n∆vtri\nKS≈2tηp\n4η2−(ntri−1)2=2t√\n3, (72)\nwhich is again finite, unlike the equi-bi-ensemble KS po-\ntential difference which tends to + ∞. This explains the\ndrastic change in representation of the interacting eigen-\nstates when moving from the bi- to the tri-ensemble case\n(see the left and right panels of Fig. 6). For example, the\ntrue ground state is described, for large Uvalues, through\nan equal mixing of ground and doubly-excited KS states,\nlike in the strongly correlated symmetric dimer. On\nthe other hand, both first and second excited states are\ncombinations of ground (25%), singly-excited (50%), and\ndoubly-excited (25%) KS states. In this strongly corre-\nlated regime, the one-particle picture of electronic excita-\ntions completely breaks down, as expected, thus making\nlabels such as ”single excitation” or ”double excitation”\nirrelevant for the true physical excitation processes.\nV. CONCLUSIONS AND OUTLOOK\nAs a complement to a recent work [49], where the e Nc\nensemble density functional theory of electronic excita-\ntions has been introduced, we extended in the present\npaper the theory to the description of single-electron ex-\ncitations from anyoccupied orbital in the KS ground\nstate and, most importantly, to the challenging dou-\nble excitations. The exactification of Koopmans’ theo-\nrem for single-electron ionization processes and the re-\nlated concept of density-functional Hxc derivative dis-\ncontinuity, whose mathematical construction fully relies\non the weight-dependent ensemble density-functional Hxc\nenergy, still play a central role. The theory has been\nimplemented within the two-electron Hubbard dimer13\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weightU= 1, ξ1= 0.5, ξ2= 0\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weightU= 1, ξ1= 1/3, ξ2= 1/3\n|/angbracketleftΦξ\n0|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ0/angbracketright|2\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weight\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ1/angbracketright|2\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weight\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ2/angbracketright|2\nFIG. 4. Same as Fig. 2 but the configuration weights are now plotted for U= 1 (and t= 1/2) as functions of ∆ vextin the\nequi-bi- (left panels) and equi-tri-ensemble (right panels) density-functional KS representations ( ξ−= 0 in both cases). See\ntext for further details.\nmodel. Nontrivial modifications of the exact Hxc poten-\ntial upon neutral excitation processes, including the ex-\npected derivative discontinuities, have been highlighted\nand rationalized. Finally, in order to clarify the state-\nment “ensemble DFT can describe double excitations”,\nand also discuss what labels like “single excitation” or\n“double excitation” actually mean in the context of en-\nsemble DFT, we have analyzed the representation of\nthe three lowest two-electron (singlet) eigenstates of the\nHubbard dimer in both equi-bi- and equi-tri-ensemble\ndensity-functional KS bases. Even though the true in-\nteracting and KS ensembles share the same density, they\ncan be drastically different. In some regimes, the statescan be similar but their ordering in energy is different.\nIn some other regimes, the physical states are mixtures\nof ground and excited KS states. This analysis also re-\nveals that the KS representation of the physical eigen-\nstates can be very sensitive to the choice of ensemble,\nthrough its dependence on the ensemble density. While\nthe present work focused on the exact theory, the next\nchallenging task consists in developing density-functional\napproximations in this context. Combining a generalized\nKS formulation of the theory with perturbative ensem-\nble DFT [37] would, for example, be an interesting path\nto follow. We may also learn from the time-dependent\nlinear response of density-functional ensembles. Indeed,14\n0 2 4 6\n∆vext0.000.250.500.751.00Configuration weightU= 3, ξ1= 0.5, ξ2= 0\n0.0 0.1 0.2 0.3\n∆vext0.000.250.500.751.00Configuration weightU= 3, ξ1= 0.5, ξ2= 0\n|/angbracketleftΦξ\n0|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ0/angbracketright|2\n0 2 4 6\n∆vext0.000.250.500.751.00Configuration weight\n0.0 0.1 0.2 0.3\n∆vext0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ1/angbracketright|2\n0 2 4 6\n∆vext0.000.250.500.751.00Configuration weight\n0.0 0.1 0.2 0.3\n∆vext0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ2/angbracketright|2\nFIG. 5. Same as Fig. 4 for U= 3 (and t= 1/2) in the equi-bi-ensemble density-functional KS representation only. Right\npanels show details of the left panels in the range 0 ≤∆vext≤0.3.\nlike the static formulation of ensemble DFT, the latter\nresponse is expected to give us access to the excitation\nenergies. Work is currently in progress in these different\ndirections.\nACKNOWLEDGEMENTS\nThe authors thank ANR (CoLab project, grant no.:\nANR-19-CE07-0024-02) for funding as well as P.-F. Loos\nand B. Senjean for fruitful discussions.Appendix A: Computation of exact ensemble\ndensity functionals and reduction to a tri-ensemble\nThroughout Sec. IV, we employ the density functional\nFξ(n) introduced in Eq. (10) for an e Nc ensemble con-\nsisting of the ground, first and second excited (singlet)\ntwo-electron states, and the cationic ground one-electron\nstate. It is characterized by the collection of weights\nξ= (ξ−, ξ1, ξ2). The functional Fξ(n) is the analog for\neNc ensembles of the Levy–Lieb functional [64, 76], that\nwe assume to be equivalent to a Lieb functional [76] for\ndensities under study, like in Ref. [49]. Consequently, it\ncan be evaluated through a Legendre–Fenchel transform,15\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight∆vext= 1, ξ1= 0.5, ξ2= 0\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight∆vext= 1, ξ1= 1/3, ξ2= 1/3\n|/angbracketleftΦξ\n0|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ0/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ0/angbracketright|2\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ1/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ1/angbracketright|2\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight\n0 2 4 6 8 10\nU0.000.250.500.751.00Configuration weight|/angbracketleftΦξ\n0|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n1|Ψ2/angbracketright|2\n|/angbracketleftΦξ\n2|Ψ2/angbracketright|2\nFIG. 6. Same as Fig. 2 for the asymmetric (∆ vext= 1) dimer in the equi-bi- (left panels) and equi-tri-ensemble (right panels)\ndensity-functional KS representations ( ξ−= 0 in both cases). See text for further details.\nas follows,\nFξ(n) = sup\n∆v\u001a\u0012\n1−ξ−\n2−ξ1−ξ2\u0013\nEN\n0(∆v)\n+ξ−EN−1\n0(∆v) +ξ1EN\n1(∆v)\n+ξ2EN\n2(∆v) + ∆ v(n−1)\u001b\n.(A1)\nSince the (singlet) two-electron energies in the Hub-\nbard dimer sum up to 2 U[50], we can afford\nthe reduction of the above four-state ensemble to\nan effective tri-ensemble (consisting of the ground,\nfirst excited, and cationic states) by substituting\nEN\n2(∆v) = 2 U−EN\n1(∆v)−EN\n0(∆v) into the aboveequation. This reduction is completely analogous to\nthe TGOK tri-to-bi-ensemble reduction in Ref. 50 (see\nEqs. (A2)-(A4) therein), leading to a similar expression\nforFξ(n), which reads as follows,\nFξ(n) = 2 Uξ2+ (1−3ξ2)Fζ(ν), (A2)\nwhere ζ= (ζ−, ζ1) is the collection of weights for\nthe tri-ensemble, with the two effective weights equal\ntoζ−=ξ−/(1−3ξ2) and ζ1= (ξ1−ξ2)/(1−3ξ2).\nThe effective tri-ensemble density ν reads\nν= (n−3ξ2)/(1−3ξ2). The tri-ensemble Lieb func-\ntional Fζ(ν), which has been extensively used in Ref. 49,16\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weightU= 1\n0 1 2 3 4 5\n∆vext0.000.250.500.751.00Configuration weightU= 3\n|/angbracketleftΞ0|Ψ0/angbracketright|2\n|/angbracketleftΞ1|Ψ0/angbracketright|2\n|/angbracketleftΞ2|Ψ0/angbracketright|2\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weight\n0 1 2 3 4 5\n∆vext0.000.250.500.751.00Configuration weight|/angbracketleftΞ0|Ψ1/angbracketright|2\n|/angbracketleftΞ1|Ψ1/angbracketright|2\n|/angbracketleftΞ2|Ψ1/angbracketright|2\n0 1 2 3 4\n∆vext0.000.250.500.751.00Configuration weight\n0 1 2 3 4 5\n∆vext0.000.250.500.751.00Configuration weight|/angbracketleftΞ0|Ψ2/angbracketright|2\n|/angbracketleftΞ1|Ψ2/angbracketright|2\n|/angbracketleftΞ2|Ψ2/angbracketright|2\nFIG. 7. Same as Fig. 3 but the (localized) site-based configuration weights are now plotted for U= 1 (left panels) and U= 3\n(right panels) as functions of ∆ vextwith t= 1/2.\ncan be evaluated as follows,\nFζ(ν) = sup\n∆v\u001a\u0012\n1−ζ−\n2−ζ1\u0013\nEN\n0(∆v) +ζ−EN−1\n0(∆v)\n+ζ1EN\n1(∆v) + ∆ v(ν−1)\u001b\n.\n(A3)Appendix B: Computation of wavefunction\nexpansion coefficients in the lattice and ensemble\nKS representations\nIn the Hubbard dimer, the singlet subspace of the two-\nelectron Hilbert space comprises three configurations. In\nthe lattice site basis, they are expressed as follows,\n|Ξ0⟩= ˆc†\n0↑ˆc†\n0↓|vac⟩, (B1a)\n|Ξ1⟩= ˆc†\n1↑ˆc†\n1↓|vac⟩, (B1b)\n|Ξ2⟩=1√\n2\u0010\nˆc†\n0↑ˆc†\n1↓−ˆc†\n0↓ˆc†\n1↑\u0011\n|vac⟩. (B1c)\nAny singlet two-electron eigenstate can be expanded17\nin the basis of above configurations as follows,\n|Ψν⟩=2X\nK=0CKν|ΞK⟩, ν= 0,1,2, (B2)\nwhere Ψ ν≡ΨN\nν(N= 2 here), and {CKν=⟨ΞK|Ψν⟩}\nare the expansion coefficients, which can be obtained\nby inserting Eq. (B2) into the Schr¨ odinger equation\nˆH|Ψν⟩=Eν|Ψν⟩for the ground, first and second excited\nstates ( ν= 0,1,2, respectively), and projecting into the\nsite many-body basis ⟨ΞL|,\nX\nKCKν⟨ΞL|ˆH|ΞK⟩=EνCLν. (B3)\nUsing Eq. (41) to evaluate the Hamiltonian matrix ele-\nments in the site basis (see also Ref. 75), ⟨ΞL|ˆH|ΞK⟩,\n⟨Ξ0|ˆH|Ξ0⟩=U−∆vext,\n⟨Ξ1|ˆH|Ξ1⟩=U+ ∆vext,\n⟨Ξ2|ˆH|Ξ2⟩= 0,\n⟨Ξ0|ˆH|Ξ1⟩= 0,\n⟨Ξ0|ˆH|Ξ2⟩=⟨Ξ1|ˆH|Ξ2⟩=−√\n2t,(B4)\nwe obtain from Eq. (B3) a system of three linear equa-\ntions for the coefficients CKν,\n(U−∆vext−Eν)C0ν−√\n2tC2ν= 0\n(U+ ∆vext−Eν)C1ν−√\n2tC2ν= 0\n−√\n2t(C0ν+C1ν)−EνC2ν= 0.(B5)\nAssuming nondegeneracy, we fix C0νand express C1νand\nC2νas follows,\nC1ν=U−∆vext−Eν\nU+ ∆vext−EνC0ν, C 2ν=U−∆vext−Eν√\n2tC0ν.\n(B6)\nThen, C0νis determined by normalizing the squared\nnorm of the coefficients to unity,\n1 =2X\nK=0|CKν|2\n=|C0ν|2\"\n1 +\u0012U−∆vext−Eν\nU+ ∆vext−Eν\u00132\n+(U−∆vext−Eν)2\n2t2#\n=|C0ν|22t2\u0002\n(Eν−U−∆vext)2+ (Eν−U+ ∆vext)2\u0003\n2t2(U+ ∆vext+Eν)2\n+|C0ν|2\u0002\n(Eν−U)2−∆v2\next\u00032\n2t2(U+ ∆vext+Eν)2.\n(B7)\nUsing the fact that\n(Eν−U−∆vext)2+(Eν−U+∆vext)2= 2\u0002\n(Eν−U)2+ ∆v2\next\u0003\n,\n(B8)and, by introducing the following notation,\nGν=\u0002\n(Eν−U)2−∆v2\next\u00032+ 4t2\u0002\n(Eν−U)2+ ∆v2\next\u0003\n,\n(B9)\nit follows that\n|C0ν|2=2t2(U+ ∆vext−Eν)\nGν. (B10)\nFor the coefficient C0ν, one may choose the positive\nsquare root of the above expression, which gives\nC0ν=√\n2t|U+ ∆vext−Eν|√Gν\n= sgn ( U+ ∆vext−Eν)√\n2t(U+ ∆vext−Eν)√Gν.\n(B11)\nThen, C1νandC2νcan be expressed as follows,\nC1ν=U−∆vext−Eν\nU+ ∆vext−Eν√\n2t|U+ ∆vext−Eν|√Gν\n= sgn ( U+ ∆vext−Eν)√\n2t(U−∆vext−Eν)√Gν,\n(B12)\nC2ν=U−∆vext−Eν√\n2t√\n2t|U+ ∆vext−Eν|√Gν\n= sgn ( U+ ∆vext−Eν)(Eν−U)2−∆v2\next√Gν.(B13)\nThe above expressions for CKνare completely general.\nFor instance, in a weight-dependent ensemble KS system\n(we consider the particular case where ξ−= 0, like in\nSec. IV C 2), for which the ground, singly- and doubly-\nexcited KS wavefunctions ( ν= 0,1,2,respectively) are\nexpressed in the site basis as follows,\n\f\fΦξ\nν\u000b\n=2X\nK=0Dξ\nKν|ΞK⟩, ν= 0,1,2, (B14)\nthe expansion coefficientsn\nDξ\nKν=⟨ΞK|Φξ\nν⟩o\nare ob-\ntained from Eqs. 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We introduce a special family of distributional α-densities and give a trans-\nversality criterion stating when their product is defined, closely rela ted to H¨ ormander’s\ncriterion for general distributions. Moreover, we show that for t he subspace of distri-\nbutional half-densities in this family the distribution product natura lly yields a pairing\nthat extends the usual one on smooth half-densities.\n1.Introduction\nThe notion of state in quantum mechanics is generally defined as a unit vector, or ray,\nin a Hilbert space [9], and the probability of transition from one state ψ1to another\nstateψ2is given using the Hilbert space inner product as |/angbracketleftψ1, ψ2/angbracketright|2. When these states\nare realized as the solutions of the Schr¨ odinger equation a number of technical problems\narise with the Hilbert space formulation of the states. In particular , since its solutions\n(or initial conditions) can have singularities, they are distributions r ather than honest\nfunctions in a L2Hilbert space [9]. The necessity to include certain distributions as sta tes\nmakes the axiomatic requirement of the state space to be a Hilbert s pace difficult to fulfill\nin practice. This has led to considering weaker definitions than that o f a Hilbert space as\nnatural state spaces for quantum mechanics. For example, part ial inner product spaces\n[1, 2] are vector spaces where the inner product (and hence the t ransition probability) is\nonly partially defined. This issue is closely related to the product of dis tributions, which\nis well-known not to be possible in general [4, 8]. H¨ ormander gave a cr iterion involving\nproperties of the distributions’ wave-front sets governing when their product is possible\n[8]. Extending theproduct fromfunctions todistributions hasalsoim portant implications\nin quantum field theory and renormalization theory; see [5, 6] for ex ample.\nIn this note, we take a geometric approach to the notionof state r ather than ananalytic\none, based on the concept of α-densities as described in [3, 7]. For instance, the space of\nhalfdensities ona manifold Xis anatural candidate foraquantum statespace asit comes\nequipped with an intrinsic Hilbert space product (upon taking suitable completions) and\nlocally it can be modeled as the space of square-integrable functions . To include singular\n2020Mathematics Subject Classification. 46C50, 46F10, 53C99, 81S10.\n1GEOMETRIC STATES IN MEMORY OF STEVE ZELDITCH 2\nstates, we still need to consider the dual space of distributional h alf densities though on\nwhich the natural intrinsic inner product is unfortunately not alway s defined.\nThe main contribution of this work is to give an explicit geometric descr iption of a\nspecial family of distributional α-densities, which we call geometric states , for which the\nproduct is well-defined provided that their spatial components (wh ich we call cores) inter-\nsect transversally (which is a special case of H¨ ormander’s criterio n [8]). When specialized\ntoα= 1/2, this product allows us to extend the intrinsic inner-product of ha lf densities\nto the whole space of geometric states, assuming transversal int ersection of their cores.\n2.Geometric States\nBefore giving a definition of a geometric state, let us start by recallin g the notion of α\n-densities; see [3, 7] for more details. Consider a n-dimensional vector space VoverC. We\ndenote byV∗its dual, that is, the vector space of linear maps V→C, and byF(V) its set\nof frames, that is, the set of ordered bases e= (e1,...,e n). For each α∈C, the space of\nα-densities on Vis the space of mappings η:F(V)→Csuch thatη(Ae) =|detA|αη(e),\nwhereAis a non-singular linear map and |detA|is the absolute value of its determinant.\nWe denote by |V|αthe space of α-densities on V, which is a complex vector space of\ndimension 1.\nNow consider a smooth manifold X. The space of α-densities on X, which we denote\nby|Ω|α(X), is the space of sections of the line-bundle |TX|α→Xwhose fiber at xis\n|TxX|α. More generally, given a vector bundle E→X, we can form the line bundle\n|E|α→Xwhose fiber at xis the 1-dimensional vector space over Cofα-densities |Ex|α.\nWeconsiderthespaceofsmoothsectionsΓ( X,|E|α)asaC∞(X)-module. Wewillconsider\nthe tensor product of such modules over C∞(X). There are a few canonical isomorphisms\nof importance to us that we now enumerate for convenience. First of all|E∗|α→X\nnaturally identifies with |E|−α→X. Second, given an exact sequence 0 →A→B→\nC→0 of vector bundles over X, we have the canonical isomorphism |B|α≃ |A|α⊗|C|α,\nwhere we use the shorthand notation |E|αto denote the vector bundle |E|α→X. At last,\nwe have that |E|α⊗|E|β≃ |E|α+βwhich specializes to |Ω|α(X)⊗|Ω|β(X)≃ |Ω|α+β(X)\nwhenEis the tangent bundle to X.\nWe can now define our space of geometric states:\nDefinition 1. We define the space Hα\nX,C, whereCis a smooth submanifold of a smooth\nmanifoldXandα∈C, as the subspace of the following density bundle sections\n|Ω|α(C)⊗Γ(|N∗C|1−α).\nThe first factor of the tensor product above is the space of smoo thα-densities on C,\nwhile the second factor is the space of the smooth sections of the ( 1−α)-density bundle\nassociated to the conormal bundle N∗C→CtoC. We call an element in either of this\nspaces interchangeably a geometric state or ageometric distributional α-density .\nThis family of spaces above has interesting well-known extreme case s:\nExample 2. WhenC=X,the spaces Hα\nX,Ccoincide with the usual smooth α-densities\non the full space X. In particular, when α= 0, this space identifies with the space\nEXof the smooth functions on X, which, in turn, can be regarded as a subspace of\nthe distributions on X(with test “functions” taken in the smooth compactly supported\n1-densities on X).GEOMETRIC STATES IN MEMORY OF STEVE ZELDITCH 3\nExample 3. Whenα= 1,we have the identification of Hα\nX,Cwith the space of smooth 1-\ndensities (or measures) supported on the submanifold C. This space can also be regarded\nas a subspace of the distributions on Xsupported on C(now with test functions the\ncompactly supported smooth functions on X).\nExample 4. WhenCis reduced to a single point x∈X, thenHα\nX,Cidentifies with the\n(α−1)-densities |TxX|α−1at that point, which further identifies with Rwhenα= 1, the\ninverse volume elements 1 /|TxX|whenα= 0 (which will see are related with the delta\ndistributions), and the volume elements |TxX|whenα= 2.\nA last example comes from the following Proposition\nProposition 5. LetCbe a smooth submanifold of X. Consider the conormal bundle\nN∗CtoCinX. Then the restriction of a half-density in |Ω|1\n2(N∗C)to the zero section\nofN∗C→Ccan be identified with an element of |Ω|1\n2(C)⊗Γ(|N∗C|1\n2), that is, with an\nelement of H1\n2\nX,C.\nProof.This comes directly from the fact that the tangent space to N∗Crestricted to\nthe zero section can be identified with the bundle TC⊕N∗Cand the usual canonical\nisomorphism for α-density bundles. /square\nThe proposition above motivates the following example:\nExample 6. Half-densitiesonlagrangiansubmanifolds, suchastheconormalbu ndleN∗C\nto a submanifold C, are typically related to some form of “geometric quantum states”\n(see [3] for example). As shown in Proposition 5, a half-density on N∗Cinduces a element\nofH1\n2\nX,C. This space can be regarded as a subspace of the distributional ha lf-densities\nonXthanks to Proposition 8, which makes it clear that the pairing of an ele ment of\nH1\n2\nX,Cwith a compactly supported half density in |Ω0|1\n2(X) yields a compactly supported\n1-density on Cthat can be integrated over Cinto a complex number. This yields the\nnatural pairing:\n/angbracketleft·,·/angbracketright:H1\n2\nX,C×|Ω0|1\n2(X)−→C.\nThe examples above suggest the name of “geometric states” for t he spaces Hα\nX,C, since\nthe 1-density example is related to classical state spaces, while the half-density example is\nrelated to the quantum state spaces, both of which are given here in term of geometrical\ndata.\n3.Geometric states as distributions\nIn this section, we show that elements in the geometric state space H1−α\nX,Cnaturally\nidentify with distributional (1 −α)-densities, that is, elements in the dual to the space of\nthe compactly supported smooth α-densities on X. In the following, we will denote the\nspaces of compactly supported smooth α-densities on Xby\nDα\nX:=|Ω0|α(X).\nNote that D0\nXidentifies with the space DXof compactly supported functions (test func-\ntions) onX.\nLemma 7. LetCbe a smooth submanifold of a smooth manifold X. The restriction to\nCof theα-densities on Xyields the following canonical map\n(3.1) |Ω|α(X)−→ |Ω|α(C)⊗Γ(|NC|α)GEOMETRIC STATES IN MEMORY OF STEVE ZELDITCH 4\nProof.The proof follows immediately from the exact sequence 0 →TC→TX→NC→\n0 and the canonical isomorphism |A⊕B|α≃ |A|α⊗|B|α. /square\nProposition 8. There is a natural bilinear pairing\n/angbracketleft·,·/angbracketright:H1−α\nX,C×Dα\nX−→C,\nturningH1−α\nX,Cinto a subset of the distributional (1−α)-densities (Dα\nX)′onX.\nProof.Thanks to Lemma 7, the restriction of an element φ∈ Dα\nXto the submanifold C\ncan be identified with a compactly supported section\nφ|C∈ |Ω0|α(C)⊗Γ0(|NC|α).\nNow taking the tensor product of this restriction with a section\nθ∈ |Ω|1−α(C)⊗Γ(|N∗C|α)\n(i.e. with an element θ∈ H1−α\nX,C), we obtain a compactly supported 1-density\nθ⊗φ|C∈ |Ω0|1(C)\nHence, we can integrate this tensor product, yielding the following n atural pairing:\n/angbracketleftθ, φ/angbracketright:=/integraldisplay\nCθ⊗φ|C.\n/square\n4.Product of geometric distributions\nOn theα-density test spaces Dα\nX, that is, the spaces of compactly supported smooth\nα-densities on X, we have the natural product of α-densities\nDα\nX×Dβ\nX−→ Dα+β\nX\ngiven by the tensor product of the density bundle sections. (When α=β= 0, it\ndegenerates to the product of smooth compactly supported fun ctions.) The question\nof whether and when this product can be extended to the distribut ional spaces ( DX)′\nhas wide-reaching implications for both partial differential equation s and quantum-field\ntheory. In general, it is not possible, and H¨ ormander [8] has given a criterion on pairs of\ndistributions which is sufficient to multiply them together in a meaningfu l way. Roughly,\nthe H¨ ormander criterion states that the product of distribution s makes sense when their\nwave front sets do not interact.\nHere we are considering the spaces of distributional α-densities (not only the case\nα= 0), and we specialize to the case of the distributional α-density spaces introduced in\nthe previous sections. The main result is that the product of two ge ometric states in Hα\nX,C\nand inHβ\nX,Dis defined when CandDintersect transversally along a smooth submanifold.\nLet’s start to see that it is true on a particular sub-family of the geo metric states\nConsider the degenerate case when C=D=X. Here, we obviously have a transverse\nintersection. Now, we have that Hα\nX,Xcoincides with the smooth α-densities on Xand\nHβ\nX,Dcoincides with the smooth β-densities on X. In this case, we already know that the\ntensor product gives a well defined product\nHα\nX,X×Hβ\nX,X→ Hα+β\nX,X\nthanks to the canonical isomorphism |Ω|α(X)⊗ |Ω|β(X)≃ |Ω|α+β(X).The following\ntheorem states that this product can be also extended to geomet ric states, provided thatGEOMETRIC STATES IN MEMORY OF STEVE ZELDITCH 5\ntheir supports intersect transversally. Moreover, in the particu lar case when α=β= 1/2,\nthis product can be used to define a pairing, since the product betw een two geometric\ndistributional half-densities specializes to a 1-density on the inters ection, which can then\nbe integrated, resulting into a pairing between geometric half-dens ities with transverse\nintersection.\nTheorem 9. LetCandDbe two smooth submanifolds of a manifold Xwith smooth\ntransverse intersection C∩D. Then, we have the canonical mapping\nHα\nX,C⊗Hβ\nX,D→ Hα+β\nX,C∩D\ndefining a product on the space of geometric distributional d ensities.\nProof.The proof rests on two simple properties. The first one is always valid , and it was\nalready implicitly used in the proof of Lemma 7. We state it explicitly now a t the level\nof the density bundles:\n(4.1) |TX|U|δ≃ |TU|δ⊗|NU|δ,\nfor any smooth submanifold Uof smooth manifold X. The second property is valid only\nwhen the intersection C∩Dof the two smooth submanifolds CandDofXis smooth\nand transverse; in this case, we have that\n(4.2) |N(C∩D)|δ≃ |NC|δ⊗|NC|δ.\nNow consider θ1∈ Hα\nX,Candθ2∈ Hβ\nX,D. The tensor product θ1⊗θ2of the restriction to\nthe intersection C∩Dis a section of the following tensor products of density bundles\n(4.3) |TC|α⊗|TD|β⊗|N∗C|1−α⊗|N∗D|1−β.\nUsing (4.1), we can decompose the first two factors as\n|TC|α≃ |TX|α⊗|N∗C|αand|TD|β≃ |TX|β⊗|N∗D|β.\nNow substituting the new expression of these factors into (4.3), g rouping the terms to-\ngether, and using the usual canonical isomorphisms of density bun dles, we obtain that\nthe product space (4.3) identifies with\n|TX|α+β⊗|N∗C|1⊗|N∗D|1.\nSince we are assuming that CandDare transverse, we can then use the identification\n(4.2) for the two last factors above yielding\n|TX|α+β⊗|N∗(C∩D)|1.\nNow, using again the decomposition (4.1), we obtain that (4.3) can fu rther be identified\nwith\n|T(C∩D)|α+β⊗|N∗(C∩D)|1−(α+β).\nThus, the tensor product θ1⊗θ2of the restrictions coincide with a section of this last\nbundle, that is, with an element of Hα+β\nX,C∩D. /squareGEOMETRIC STATES IN MEMORY OF STEVE ZELDITCH 6\n5.Pairing of geometric distributions\nThe space of all compactly supported smooth α-densities does not form an inner-\nproduct space (unless α=1\n2) but there is always a pairing\n/angbracketleft·,·/angbracketright:Dα\nX× D1−α\nX−→C,\nwhere the pairing between a (compactly supported) α-density and a (compactly sup-\nported) (1 −α)-density comes from integrating the 1-density obtained by taking their\nproduct. The global structure on DX=⊕αDα\nX, on which the inner product is only par-\ntially defined, is reminiscent of that of a Partial Inner-Product Space introduced in\n[1] (see also [2]).\nThe main theorem in the previous section tells us that this structure can be extended\nto the space of geometric distributions. Namely, if we restrict the g eometric state product\nto pairs of geometric subspaces with density degree in the same rela tion as above and\nwith cores intersecting transversally, we obtain the following specia lization:\nHα\nX,C⊗ H1−α\nX,D→ H1\nX,C∩D≃ |Ω|1(C∩D).\nNow, integrating this last 1-density, we obtain a pairing that genera lizes the Partial Inner-\nProduct space structure to the space of geometric distributiona lα-densities. Let us sum-\nmarize this in a proposition:\nProposition 10. Consider the geometric distribution spaces Hα\nX,CandHβ\nX,D. If we\nhave further that α+β= 1and thatCandDintersect transversally along a compact\nsubmanifold, there is a natural bilinear pairing between th ese spaces:\n/angbracketleft·,·/angbracketright:Hα\nX,C× H1−α\nX,D−→C.\nIn particular, these pairings endow the geometric distribu tional half-densities ( α= 1/2)\nwith a partially defined inner-product, extending the natur al inner-product on the space of\nsmooth compactly supported half-densities.\nIf we regard the smooth half-density space D1/2\nXonXas modeling the states of a\nquantum system, then the collection of geometric distribution spac es\nH1/2\nX:=⊕CH1\n2\nX,C\nextends this model. The partially-defined inner product should be th en thought as giving\nthe probability transition from one geometric state to another, un der the compatibility\ncondition that the cores of these states are intersecting transv ersally. This implies that\na geometric state spatially supported on Ccan transition to a geometric state spatially\nsupported on Donly when these states have some amount of spatial overlap.\nReferences\n[1] Antoine, J.-P., and Grossmann, A., Partial Inner Product Space s. I General Properties, Journal of\nFunctional Analysis, Vol. 23, (4), pp. 369-389, 1976.\n[2] Antoine, J.-P., and Trapani, C., Partial Inner Product Spaces: T heory and Applications. Lecture\nNotes in Mathematics 1986, Springer-Verlag, 2009.\n[3] Bates, S., and Weinstein, A., Lectures on the Geometry of Quant ization. Berkeley Mathematics\nLecture Notes 8, American Mathematical Society,1997.\n[4] Brouder, C., Dang V. N., and H´ elein F., Continuity of the fundamen tal operations on distributions\nhaving a specified wave front set (with a counter example by Semyon Alesker). Studia Mathematica,\nVol. 232, pp. 201-226, (3) 2016.GEOMETRIC STATES IN MEMORY OF STEVE ZELDITCH 7\n[5] Dang, N. V., Renormalization of quantum field theory on curved sp ace- times, a causal approach.\nPh.D. thesis, Paris Diderot University, 2013.\n[6] Dang, N. V., The extension of distributions on manifolds, a microloc al approach. Annales de l’institut\nHenri Poincar´ e, Vol. 17, pp. 819-859, (4) 2016.\n[7] Guillemin, V., and Sternberg, S., Geometric Asymptotics. Mathema tical Surveys and Monographs\n14, American Mathematical Society, 1977.\n[8] H¨ ormander, L., The Analysis of Linear Partial Differential Opera tors IV. Springer-Verlag, 2009.\n[9] Isham, C. J., Lectures On Quantum Theory: Mathematical And S tructural Foundations. Allied\nPublishers, 2001.\nGoogle, Mountain View, CA 94043, USA\nEmail address :dherin@google.com\nDepartment of Mathematics, University of California, Berk eley, CA 94720, USA, and\nDepartment of Mathematics, Stanford University, Stanford , CA 94305, USA\nEmail address :alanw@math.berkeley.edu" }, { "title": "2402.17943v1.Sequential_transport_maps_using_SoS_density_estimation_and__α__divergences.pdf", "content": "Sequential transport maps using SoS density\nestimation and α-divergences\nBenjamin Zanger∗, Tiangang Cui†, Martin Schreiber‡, Olivier Zahm§\nFebruary 29, 2024\nAbstract\nTransport-based density estimation methods are receiving growing interest be-\ncause of their ability to efficiently generate samples from the approximated density.\nWe further invertigate the sequential transport maps framework proposed from [10,\n11], which builds on a sequence of composed Knothe-Rosenblatt (KR) maps. Each\nof those maps are built by first estimating an intermediate density of moderate\ncomplexity, and then by computing the exact KR map from a reference density to\nthe precomputed approximate density. In our work, we explore the use of Sum-\nof-Squares (SoS) densities and α-divergences for approximating the intermediate\ndensities. Combining SoS densities with α-divergence interestingly yields convex\noptimization problems which can be efficiently solved using semidefinite program-\nming. The main advantage of α-divergences is to enable working with unnormalized\ndensities, which provides benefits both numerically and theoretically. In particu-\nlar, we provide two new convergence analyses of the sequential transport maps:\none based on a triangle-like inequality and the second on information geometric\nproperties of α-divergences for unnormalizied densities. The choice of intermediate\ndensities is also crucial for the efficiency of the method. While tempered (or an-\nnealed) densities are the state-of-the-art, we introduce diffusion-based intermediate\ndensities which permits to approximate densities known from samples only. Such\nintermediate densities are well-established in machine learning for generative mod-\neling. Finally we propose and try different low-dimensional maps (or lazy maps) for\ndealing with high-dimensional problems and numerically demonstrate our methods\non several benchmarks, including Bayesian inference problems and unsupervised\nlearning task.\n1 Introduction\nDensity estimation is a fundamental problem in data sciences. Transport-based methods\nare receiving growing interest because of their ability to sample easily from the approx-\n∗Universit´ e Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France (ben-\njamin.zanger@inria.fr)\n†School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia (tian-\ngang.cui@sydney.edu.au)\n‡Universit´ e Grenoble Alpes, LJK, 38000 Grenoble, France\nTechnical University of Munich, Germany (martin.schreiber@univ-grenoble-alpes.fr)\n§Universit´ e Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France\n(olivier.zahm@inria.fr)\n1arXiv:2402.17943v1 [stat.ML] 27 Feb 2024imated density [26, 18, 31, 34]. These methods aim at building a deterministic diffeo-\nmorphism T, also called a transport map, which pushes forward an arbitrary reference\nprobability density ρrefto a given target probability density πto be approximated. This\npushforward density, denoted by T♯ρref, is the density of the random vector T(Ξ), where\nΞ∼ρref. Variational methods consist in solving a problem of the form\nmin\nT ∈MD(π||T♯ρref), (1)\nwhere the statistical divergence D( ·∥·) and the set of diffeomorphisms Mare typically\nchosen so that problem (1) is a tractable problem. Typically, the Kullback-Leibler (KL)\ndivergence is employed for D( ·∥·) when only samples from πare available (learning from\ndata), while the reverse KL divergence is utilized when πcan be evaluated up to a\nnormalizing constant (learning from unnormalized density). Minimizing the reverse KL\ndivergence, however, fails to capture multi-modal densities π, a well known problem for\nsuch zero-forcing divergence, see e.g. [5, 29, 16]. Regarding the set M, the first corner-\nstone for handling high-dimensional diffeomorphic maps is the monotone triangular map,\nwhere each i-th map component depends on the first ivariables only and is monotone in\nthei-th variable [3, 19, 14]. As to increase the approximation power, the second corner-\nstone is to compose multiple monotone triangular map. This is the basic idea of many\nmap parametrization used in the Normalizing flows literature [30, 16, 34] to cite just a\nfew. Let us emphasis that, in principle, there exists infinitely many maps Twhich sat-\nisfyT♯ρref=π. This is contrarily to the Monge optimal transport problem which seeks\nthe map Tthat minimizes some transport-related lossR\n(x− T(x))2dρref(x) under the\nconstraint T♯ρref=π, see [43, 32]. Instead, the goal of (1) is to build a tractable map\nT ∈ M such that T♯ρref≈π. In other terms, problem (1) is not concerned with the\npaths by which Ttransports ρreftoπ; it solely focuses on the resulting distance between\nT♯ρrefandπ.\nAn emerging strategy for solving (1) is to first approximate πwith an approximate\ndensity eπand then to compute a map Twhich exactly pushes forward ρreftoeπ. Among\nthe infinitely many maps Twhich satisfy T♯ρref=eπ, the Knothe-Rosenblatt (KR) map\nis rather simple to evaluate since it requires only computing the conditional marginals\nofeπ. In the seminal work [15], eπis built in the tensor-train format, an approximation\nclass which permits to efficiently compute the KR map, see also [8, 10, 9]. Polynomial\napproximation methods have also been used recently in [11, 44] for building transport\nmaps. In all these papers, the statistical divergence D( ·∥·) is chosen to be the Hellinger\ndistance, which is the L2distance between square root densities. Conveniently, this dis-\ntance permits building approximations eπtoπusing standard L2function approximation\ntechniques, like polynomial interpolation, least-squares or tensor methods. In general,\nhowever, the variational problem min eπD(π||eπ) is difficult to solve when πis multimodal\nor when it concentrates on a low-dimensional manifold. The solution proposed in [8]\nconsists in introducing an arbitrary sequence of bridging densities\nπ(1), π(2), . . . , π(L)=π, (2)\nwith increasing complexity. Similar to Sequential Monte Carlo Samplers [12], this se-\nquence allows for breaking down the challenging approximation problem (1) into a se-\nquence of intermediate problems of more manageable complexity. A classical choice of\nbridging densities are tempered densities π(ℓ)(x)∝π(x)βℓwith parameterss 0 ≤β0≤\n. . .≤βL= 1 [8, 11, 10], although other option are possible depending on the application\n2[9, 13]. In practice, the strategy consists in building Ltransport maps Q1, . . . ,QLone\nafter the other by solving the following variational problems sequentially\nmin\nQℓ∈MD(π(ℓ)||(Tℓ−1◦ Qℓ)♯ρref),where Tℓ−1=Q1◦. . .◦ Qℓ−1. (3)\nThese problems are equivalent to estimating the pullback density ( Tℓ−1)♯π(ℓ)with an\nintermediate approximation ρ(ℓ)= (Qℓ)♯ρref. Again, this can be done by first building\nthe approximate density ρ(ℓ)and then by extracting the KR map Qℓwhich pushes forward\nρrefto the precomputed ρ(ℓ). An illustration of such an sequential approximation using\nL= 3 steps is depicted in Figure 1.\nρref\n(T1)♯ρref\n (T2)♯ρref\n (T3)♯ρref\nπ Q1 Q2 Q3\nFigure 1: Visualization of the approximation of a bimodal distribution π(right) using\nL= 3 intermediate tempered densities estimated using SoS (4) and a guassian reference\ndistribution ρref.\nIn this paper, we pursue the above methodology by extending it to Sum-of-Squares\n(SoS) density estimation using α-divergences. In details, we construct the each interme-\ndiate density ρ(ℓ)under the form\nρ(ℓ)(x)∝\u0000\nΦ(x)⊤AℓΦ(x)\u0001\nρref(x), (4)\nwhere Aℓ⪰0 is a positive semidefinite matrix to be determined, and Φ is a vectorization\nof orthonormal basis function. This approximation class (4) is a generalization of the\npolynomial squared method used in [11, 44] where ρ(ℓ)(x)∝(v⊤\nℓΦ(x))2ρref(x). SoS den-\nsities also permit to efficiently compute the KR map Qℓsuch that ( Qℓ)♯ρref=ρ(ℓ), which\nis a crucial property for the proposed methodology. SoS functions have also been used in\n[19, 37] for parametrizing the map Tdirectly in order to ensure the monotonicity of each\nmap components. Contrarily to the previous works which, depending on the application,\nutilize either the Hellinger distance, the KL divergence or the reverse KL divergence, we\nhere employ the class of α-divergences D α(·∥·) with parameter α∈R. This class of diver-\ngence encompasses all the previously mentioned divergences and, consequently, permits\nto unify the convergence analysis for problem (3). Using the notion of α-geodesic, we\nalso propose a novel convergence analysis for (3) which relies on the geometry induced by\nDα(·∥·). Most importantly, the use of α-divergence for learning SoS densities (4) results in\naconvex optimization problems for (3) which can be efficiently solved using off-the-shelf\ntoolboxes. This unified framework we propose enables estimation of the density πnot\nonly from data (when only samples from πare available) but also from point-evaluations\nof the unnormalized density π.\nFinally, we explore different ways to define the sequence of bridging densities (2).\nWhen learning from data, we rather consider the following bridging densities defined by\nconvoluting πwith a diffusion kernel κt(·,·), that is\nπ(ℓ)(x) =Z\nκtℓ(x,y)π(y)dy. (5)\n3with time parameter ∞=t0≥t1≥. . .≥tL= 0. The main reason for employing (5) is\nthat samples from π(ℓ)are readily obtained by simulating an Ornstein-Uhlenbeck process.\nThis idea is at the root of diffusion models [39, 40].\nWe introduce α-divergences and SoS functions in Section 2 and 3. We then explain\ndiffusion bridging densities and provide convergence analysis of sequential transport maps\nin Section 4. To address the curse of dimensionality, in Section 5 we give two methods\nto scale to higher dimensions. Finally, in Section 6, numerical examples demonstrate the\nfeasibility of the proposed methods.\n2 Variational density estimation using α-divergence\nWe propose to use α-divergences [5] for the variational density estimation. Using the\ndefinition of α-divergences from [45], for a given α∈R, the α-divergence between two\nunnormalized densities fandg(meaning integrable and positive functions) reads\nDα(f||g) =Z\nϕα\u0012f(x)\ng(x)\u0013\ng(x)dx,with ϕα(t) =\n\ntα−1\nα(α−1)−t−1\nα−1α /∈ {0,1}\ntlog(t)−t+ 1 α= 1\n−log(t) +t−1α= 0.(6)\nNotice that for all α∈R, the function ϕα:R≥0→R≥0is positive, convex ϕ′′\nα(t)>0, and\nis minimial at t= 1, meaning ϕ′\nα(1) = 0. Denoting by πf(x)∝f(x) and πg(x)∝g(x)\nthe probability densities obtained by normalizing respectively fandg, the choices α∈\n{0,1/2,1,2}yield respectively to\nD1(πf||πg) = D KL(πf||πg), D0(πf||πg) = D KL(πg||πf), (7)\nD1/2(πf||πg) = 4 D Hell(πf||πg)2, D2(πf||πg) =1\n2χ2(πf||πg), (8)\nwhere D KL(πf||πg) =R\nlog(πf\nπg)dπfis the Kullback-Leibler divergence, D Hell(πf||πg) =\n(1\n2R\n(√πf−√πg)2dx)1/2the Hellinger distance and χ2(πf||πg) =R\n(πf\nπg−1)2dπgthe chi-\nsquare divergence.\nOne practical advantage of using α-divergences with unnormalized density functions is\nthat there is no need to enforce the approximate density to integrate to one while learning\nit. Additionally, one avoid the requirement of knowing the normalizing constant of the\ntarget density, which is convenient for Bayesian inverse problems where the posterior\ndensity is known up to a multiplicative constant. For instance, it is shown in [16] that a\ndirect discretization of the KL divergence (for probability densities) might not converge\nwell: in Appendix B, we show that the discretization of D 1(f||g) (the extension of the KL\ndivergence for unnormalized measures) does not suffer from this instability and also refer\nto [28, Appendix A] in which the same issue is shown for discretization of f-divergences.\nDenoting by Zf=R\nf(x)dxandZg=R\ng(x)dxthe normalizing constant of fandg, we\nshow in Appendix A.3 that\nDα(f||g) =Zα\nf\nZα−1\ngDα(πf||πg) +Zgϕα\u0012Zf\nZg\u0013\n, (9)\nholds for any unnormalized densities f, g. This relation suggests that for a density f,\nminimizing g7→Dα(f||g) permits to control both D α(πf||πg) and ϕα(Zf\nZg) so that, after\nnormalizing g, the probability density πgyields a controlled approximation to πf.\n4According to [27], minimizing g7→Dα(f||g) with α≤0 forces g(x)≪1 in regions\nwhere f(x)≪1, thus avoiding false positives (zero forcing property). Reciprocally,\nminimizing g7→Dα(f||g) with α≫1 forces g(x)≫1 in regions where f(x)≫1,\nwhich avoids false negatives (zero avoiding property). This is coherent with the duality\nproperty\nDα(f||g) = D 1−α(g||f), (10)\nwhich holds for any α∈R, see e.g. [1]. Another fundamental property of α-divergences\nis the stability under transportation by a diffeomorphism T, meaning\nDα(f||T♯g) = D α\u0000\nT♯f||g\u0001\n. (11)\nAs detailed later in Section 4, this property permits us to reformulate problem (3) as an\napproximation of T♯\nℓ−1π(ℓ), the pullback of the ℓ-th bridging density via the map Tℓ−1.\nFor the sake of completeness, the derivation of Equation (11) is given in Appendix A.2.\nIn practice, we distinguish the cases where either the target probability density πcan\nbe evaluated up to a normalizing constant, or where the target probability density is\nonly known via samples from π. The first case typically corresponds to Bayesian inverse\nproblems, where the goal is to sample from the posterior density\nπ(x)∝f(x) :=L(x, y)π0(x). (12)\nHere, π0denotes the prior and L(x, y) the likelihood of a given data set yconditioned on\nx. A related application is rare event estimation where L(x, y) is the indicator function\n(of level y) of the event which we need to compute the probability of. One can use a\nMonte-Carlo estimation of D α(f||g) of the form of\nbDα(f||g) =1\nNNX\ni=1ϕα\u0012f(X(i))\ng(X(i))\u0013g(X(i))\nρref(X(i)), (13)\nwhere X(1), . . . ,X(N)∼ρrefare independent samples drawn from a given reference mea-\nsureρref. We discuss later in Section 3.3 the choice of ρref.\nProperty 1 (Convexity of α-divergence) .The function t7→tϕα(u/t) is convex for any\nα∈Randu∈R≥0. As a consequence, the functions g7→Dα(f||g) and g7→bDα(f||g)\nare also convex.\nIn the case where only samples from πare known, setting α= 1 yields the KL\ndivergence D KL(π||πg) =R\nlog(π)dπ−R\nlog(πg)dπ. A Monte-Carlo estimate of the last\nterm yields\nbJKL(πg) =−1\nNNX\ni=1log\u0000\nπg(X(i))\u0001\n, (14)\nwhere here X(1), . . . ,X(N)are samples from π. Note that πg7→bJKL(πg) is also convex\nover the set of probability measures πg.\n53 Sum-of-Squares densities\nLet{ϕ1, . . . , ϕ m}be a family of mfunctions in L2\nρref(X), where X ⊂Rdis the support of\na reference density ρref. Given a symmetric positive semidefinite matrix A⪰0 inRm×m,\nwe introduce the Sum-of-Squares (SoS) functions as\ngA(x) =\u0000\nΦ(x)⊤AΦ(x)\u0001\nρref(x), (15)\nwhere Φ( x) = ( ϕ1(x), . . . , ϕ m(x)). By construction, gA:X →R+is positive and inte-\ngrable. Indeed, denoting by A=Pm\ni=1λiuiu⊤\nithe eigenvalue decomposition of Awith\nλi≥0 and ui∈Rm, the function x7→Φ(x)⊤AΦ(x) =Pm\ni=1λi(u⊤\niΦ(x))2is the sum of\nmsquared functions, which explains the terminology SoS. Note that gAdoes not neces-\nsarily integrate to one. We call SoS functions which integrate to one SoS densities and\nintroduce them in Section 3.3. Since the parametrization A7→gAis linear, the function\nA7→bDα(f||gA) as in (13) remains convex, as first mentioned in [25]. The resulting convex\nproblem min A⪰0bDα(f||gA) can be efficiently solved by using semidefinite programming\n(SDP). Toolboxes to solve such problems computationally are for example JuMP.jl [24]\nandCVX [17].\nNext in Section 3.1 we describe how to construct Φ using orthonormal basis function.\nIn Section 3.2 we show how to efficiently perform integration of SoS densities. Finally\nin Section 3.3 we show how to compute the Knothe-Rosenblatt map of a SoS probability\ndensity.\nRemark 3.1.While A⪰0 is sufficient to ensure the positivity of gA, this condition can be\nrelaxed depending on Xand Φ. For instance, for polynomial basis Φ and a semi-algebraic\nsetX, the condition A⪰0 can be replaced with a weaker moment-sequence condition ,\nfor example see [22, 33]. This is, however, not considered in the present paper.\n3.1 Tensor product basis\nAssuming X=X1× ··· × X dis a product space and ρref(x) =ρref1(x1). . . ρ refd(xd) a\nproduct measure, we can construct Φ : X →Rmby tensorizing univariate orthonormal\nfunctions as follow. Let\b\nϕk\ni\t∞\ni=1be a set of orthonormal functions on L2\nρrefk(Xk), 1≤k≤d\nand, given a multi-index α∈Nd, consider the following tensorization\nϕα(x) =ϕ1\nα1(x1). . . ϕd\nαd(xd).\nThen{ϕα}α∈Ndforms an orthonormal basis of L2\nρref(X). In our work, we use multivari-\nate polynomials to define ϕαalthough, in principle, any set of orthonormal functions on\nL2\nµk(Xk) can be used (trigonometric functions, wavelets,...). In the following, we give\nexamples of orthogonal bases, including Legendre polynomial on [ −1,1], Hermite poly-\nnomials on R, and transformed Legendre polynomials on generic domains.\nExample 1 (Orthogonal polynomials) .A property of any orthogonal polynomials basis\n{Pn}n≥0inL2\nρrefis that they can be computed efficiently via a recurrence relation, given\nby\nPn(x) = (anx+bn)Pn−1(x) +cnPn−2(x), (16)\nwhere the series an, bn, and cndepend on ρref[21]. For the uniform density ρref=\nU([−1,1]) on the bounded domain [ −1,1], we obtain the Legendre polynomials with\nan=2n+ 1\nn+ 1bn= 0 cn=n\nn+ 1. (17)\n6The normalized Legendre polynomials Lnare obtained by\nLn(x) =r\nn+1\n2Pn(x). (18)\nFor the Gaussian measure ρref=N(0,1), on the unbounded domain R, we obtain the\nprobabilist’s Hermite polynomials Hn(x) with the recurrence relation\nan= 1 bn= 0 cn=−n. (19)\nUsing univariate Hermite polynomials for Φ, it is assured that lim x→∞fA(x)ω(x) = 0 and\nthat the integralR∞\n−∞fA(x)ω(x)dx <∞forω(x) = exp( −x2/2). However, for numerical\nreasons, we prefer to use transformed Legendre polynomials when working in Rd, which\nwe explain in the next example.\nExample 2 (Transformed Legendre polynomials [38]) .Another way to obtain an orthog-\nonal univariate basis of L2\nρref(X) is to consider Legendre polynomials LnonL2\nµ([−1,1])\nwith µ=U([−1,1]) and the map Rsuch that R♯µ=ρref. From the normalized Legendre\npolynomials {Li}i≥0we can create functions {ϕi}i≥0given by\nϕi(x) =Li(R−1(x)) (20)\nwhich are orthonormal in L2\nρref(X),\nZ\nϕi(x)ϕj(x)ρref(x)dx=Z\nϕi(R(x′))ϕj(R(x′))R♯ρref(x′)dx′(21)\n=Z\nLi(x′)Lj(x′)µ(x′)dx′=δi,j. (22)\nWe show two transformations from [11] and [38] to transform Legendre polynomials to\northogonal functions on Rin the following table.\nR(x) |∇R(x)| R−1(x)|∇R−1(x)|\nlogarithmic tanh( x)1−tanh( x)2 1\n2log\u00001+x\n1−x\u00011\n1−x2\nalgebraicx√\n1+x21\n(1+x2)3/2x√\n1−x21\n(1−x2)3/2\nIn practice, we work with a subset of indices K ⊂Ndwith finite cardinality |K|=m.\nIn our implementation we use\nK=(\nα∈Nd:dX\ni=1αi≤p)\n, m =\u0012p+d\nd\u0013\n, (23)\nwhich yields the set of polynomials of total degree bounded by p. In order to vectorize\n{ϕα(x)}α∈Kin a vector Φ( x)∈Rm, we consider a bijective map\nσ:K → { 1, . . . , m },\nwhich lists the elements of Kin the lexicographical order. Other sets are possible, e.g.\ndownward closed sets which allow for adaptivity, see [7]. We then define the feature map\nΦ :Rd→Rmsuch that Φ i(x) =ϕσ−1(i)(x).\n73.2 Integration of SoS functions\nLet us rewrite the SoS functions from Eq. (15) to the form gA(x) = trace\u0000\nAΦ(x)Φ(x)⊤ρref(x)\u0001\n.\nThis allows to apply linear operators Lacting on functions of x, such as integration and\ndifferentiation, to Φ( x)Φ(x)⊤ρref(x) as follow (also see [25])\nLgA(x) = trace ( AW(x)),where W(x) =L(ΦΦ⊤ρref)(x).\nAs shown in the following proposition, a SoS function remains SoS after integrating over\none variable.\nProposition 1 (Integration over one variable) .LetLbe the integration operator over\nthe variable xℓwith ℓ∈ {1, . . . , d }defined by\nLg(x) =Z\nXℓg(x−ℓ, xℓ)dxℓ.\nLetgAbe a SoS function as in (15) with ρref(x) =Qd\ni=1ρrefi(xi) and (Φ( x))σ(α)=Qd\ni=1ϕi\nαi(xi), where {ϕℓ\n1, ϕℓ\n2, . . .}are orthonormal functions in L2\nρrefℓ(Xℓ). Then we have\nLgA(x) =\u0000\nΦ−ℓ(x−ℓ)⊤A−ℓΦ−ℓ(x−ℓ)\u0001\nρref(x−ℓ), (24)\nwhere A−ℓand Φ −ℓ(x−ℓ) are respectively the PSD matrix and the feature map defined as\nfollow. Denote by K−ℓ={α−ℓ:α∈ K} the multi-index obtained by removing the ℓ-th\ncomponents of the elements of K, and by σ−ℓ:K−ℓ→ {1, . . . ,|K−ℓ|}the lexicographical\norder for K−ℓ. The feature map Φ −ℓis defined by (Φ −ℓ(xℓ))σ−ℓ(α−ℓ)=Q\ni̸=ℓϕi\nαi(xi). The\nPSD matrix A−ℓis defined by\nA−ℓ=P(A⊙M)P⊤, (25)\nwhere ⊙denotes the Hadamard product and P∈R|K−ℓ|×mandM∈Rm×mare given by\nMσ(α)σ(β)=δαℓ,βℓand Pσ−ℓ(α−ℓ)σ(β)=Y\nk̸=ℓδαk,βk, (26)\nfor all α, β∈ K.\nProof. See Appendix A.1\nThe next corollary, given without proof, shows that Proposition 1 permits to integrate\nover severall variables.\nCorollary 1 (Integration over severall variables) .Proposition 1 can be applied iteratively\nto integrate over an arbitrary set of variables indexed by ℓ⊂ {1, . . . d}, meaning LgA(x) =R\ngA(x−ℓ,xℓ)dρrefℓ, where xℓ= (xi)i∈ℓ. In particular, the integration over all the variables\nLgA=R\ngA(x)dρref(x) is given by\nZ\nXΦ(x)⊤AΦ(x)dρref(x) = trace ( A). (27)\nRemark 3.2.For the fully tensorized set K=K1×. . .× K dwithKk={1, . . . , r k}the\nmatrices MandPas in (26) are\nP⊤=Ir1⊗ ··· ⊗ Irℓ−1⊗1rℓ⊗Irℓ+1⊗ ··· ⊗ Ird,\nM=1r1×r1⊗ ··· ⊗ 1rℓ−1×rℓ−1⊗Irℓ⊗1rℓ+1×rℓ+1⊗ ··· ⊗ 1rℓ×rℓ,\nwhere 1rℓ∈Rrℓand1rℓ∈Rrℓ×rℓare respectively the vector and the matrix full with ones\nandIridentity matrices of size r.\n8Remark 3.3 (Non orthonormal basis) .If the basis {ϕℓ\n1, ϕℓ\n2, . . .}is not orthonormal, then\nMfrom Proposition 1 is replaced with\nMσ(i)σ(j)=Z\nXlϕil(xl)ϕjl(xl)dρrefl(xl).\n3.3 SoS densities and Knothe-Rosenblatt map\nConsider the normalized SoS function, which we refer to as SoS density, defined by\nπA(x) =Φ(x)⊤AΦ(x)\ntrace( A)ρref(x). (28)\nBy (27), πAintegrates to one and is a well defined probability density function. Next,\nwe show how to compute the Knothe-Rosenblatt (KR) map QAwhich is is the unique\ntriangular map (up to variable ordering) such that ( QA)♯ρref=πA, see [36]. Let us\nfactorize πAas the product of its conditional marginals\nπA(x) =πA(x1)πA(x2|x1). . . π A(xn|x1, . . . , x n−1), (29)\nwhere πA(xi|x1, . . . , x i−1) is the conditional marginal density given by\nπA(xi|x1, . . . , x i−1) =πA(x1, . . . , x i)\nπA(x1, . . . , x i−1), (30)\nπA(x1, . . . , x i) =Z\nπA(x1, . . . , x i, x′\ni+1, . . . , x′\nd)dx′\ni+1. . .dx′\nd. (31)\nWe also denote by Π A(xi|x1, . . . , x i−1) the cumulative distribution function (CDF) of the\ni-th conditional of πA, given by\nΠA(xi|x1, . . . , x i−1) =Zxi\n−∞πA(x′\ni|x1, . . . , x i−1)dx′\ni. (32)\nLetQA: [0,1]d→ X be the triangular map defined by recursion as follow\nQA(ξ1, . . . , ξ d) =\nx1\nx2\n...\nxd\n=\nΠ−1\nA(ξ1)\nΠ−1\nA(ξ2|x1)\n...\nΠ−1\nA(ξd|x1, . . . , x d−1)\n, (33)\nwhere Π−1\nA(·|x1, . . . , x i−1) is the reciprocal function of the monotone function Π A(·|x1, . . . , x i−1).\nBy construction, the map QApushes forward the uniform measure µ=U([0,1]d) toπA,\nmeaning ( QA)♯µ=πA. LetQ0be the KR map of ρrefsuch that ( Q0)♯µ=ρref. Finally,\nwe let QA:X 7→ X be the KR map of πAsuch that\n(QA)♯ρref=πA,where QA(x) =QA◦Q−1\n0(x). (34)\nHence, we can define the set Mof the variational problem in Eq. (3) as KR maps\nparametrized with A, given by\nMSoS={QA:X → X as in (34), where A⪰0}. (35)\nThe procedure of constructing QA(and thus QA) is summarized in Figure 2. By Proposi-\ntion 1, all the marginals are readily computable and, by using an orthonormal polynomial\nbasis, all the antiderivatives are computable in closed form.\n9πA(x≤n)\nπA(xn|x≤n−1) =πA(x≤n)\nπA(x≤n−1)\nΠA−1(ξn|x≤n−1)/integraltextξndxn\nπA(x≤n−1)\nπA(xn−1|x≤n−2) =πA(x≤n−1)\nπA(x≤n−2)\nΠA−1(ξn−1|x≤n−2)/integraltextξn−1dxn−1\nπA(x≤n−2)\nπA(xn−2|x≤n−3) =πA(x≤n−2)\nπA(x≤n−3)\nΠA−1(ξn−2|x≤n−3)/integraltextξn−2dxn−2\nπA(x≤n−3)\nπA(xn−3|x≤n−4) =πA(x≤n−3)\nπA(x≤n−4)\nΠA−1(ξn−3|x≤n−4)/integraltextξn−3dxn−3. . .\n/integraltextdxn−3\n/integraltextdxn−2\n/integraltextdxn−1\n/integraltextdxnFigure 2: Visualization of the construction of QAfrom a given density πA. First, one\ncreates the needed marginalization of πAand the conditionals πA(xi|x1, . . . , x n−1) =\nπ(xi|x≤i−1). Then, the CDFs are computed by calculating the antiderivatives, which\ngives access to the RT. The inverse RT is constructed by inversion of the CDFs according\nto formula (33).\nRemark 3.4 (Identity map) .Because ρrefis a product measure, Q−1\n0(x) is a diagonal map\nwith ( Q−1\n0(x))i=Rxi\n−∞ρrefi(x′\ni)dx′\nibeing the CDF of ρrefi. In addition, if the basis Φ\ncontains the constant function, meaning u⊤\n0Φ(x) = 1 for some u0∈Rm, then Q0=QA0\nwith A0=u0u⊤\n0. Under this condition, the set Mcontains the identify map QA0◦Q−1\n0=\nid.\nExample 3 (Defining ρrefon indefinite domains) .Assume supp( π) =Rd. The mappings\nfrom Example 2 can be modified (by an affine change in coordinates from [ −1,1] to [0 ,1])\nto satisfy ( Q0)♯µ=ρref.\nRemark 3.5 (Mapped basis functions on µ).Assume a basis of functions orthonormal\ninL2\nµ([0,1]) which are mapped to X, in the manner of Example 2. Note that, then,\nQ0=R. The map QAcan be decomposed to QA=R ◦eQAwhere eQA: [0,1]d→[0,1]d.\nFurthermore,\nQA=R ◦eQA◦ R−1. (36)\nApplying such maps sequentially allows to save computation by using R−1◦ R=id.\n4 Sequential Transport Map with α-divergence\nWe build sequential transport maps by using a sequence of bridging densities {π(ℓ)}L\nℓ=1so\nthat the sequential maps Tℓ=Q1◦ ··· ◦ Q ℓpush ρreftoeπ(ℓ)≈π(ℓ), with ( Tℓ)♯ρref=eπ(ℓ).\nTℓis built recursively as given in Eq. (3).\nThe classical choice of bridging densities for smooth densities are tempered densities,\nπ(ℓ)=πβℓwith 0 < β 1≤ ··· ≤ βL= 1. For other less widely used bridging methods, we\n10refer to the literature of sequential Monte Carlo methods [13]. In Section 4.1 we explore\nthe use of bridging densities from diffusion processes when learning from data.\nAfterwards, we present two convergence analysis for sequential transport using α-\ndivergences. A previous analysis given in [10] holds solely for the Hellinger distance,\na distance for which the triangle inequality holds. Both of our convergence analyses\nhold for all α-divergences, for which, in general, the triangle inequality does not hold.\nThe determined error bounds by both analyses depend on the distances between two\nconsecutive bridging densities. Let us define a bound for the maximal distance.\nDefinition 1 (maximal α-divergence between consecutive bridging densities) .We define\nthe maximum α-divergence between two consecutive bridging densities by η,\nDα\u0000\nπ(ℓ)||π(ℓ−1)\u0001\n≤η(L)∀ℓ∈[L]. (37)\nFurthermore, both analyses require that every sequential estimation actually improves\nupon the previous. We assume this holds.\nAssumption 1. We assume there exists ω <1 such that\nDα\u0000\nπ(ℓ)||eπ(ℓ)\u0001\n≤ωDα\u0000\nπ(ℓ)||eπ(ℓ−1)\u0001\n∀ℓ∈ {1, . . . , L }. (38)\nRemark 4.1 (ωfor unnormalized densities) .The bound ˜ ωsatisfying\nsup\n˜ω(\n˜ω: Dα\u0010\nf(ℓ)||˜f(ℓ)\u0011\n−Z˜fℓϕα \nZf(ℓ)\nZ˜fℓ!\n≤˜ωDα\u0010\nf(ℓ)||˜f(ℓ−1)\u0011)\n(39)\nbounds ωby\nω \nZ˜f(ℓ+1)\nZ˜f(ℓ)!α−1\n≤˜ω. (40)\nThe first convergence analysis, presented in Section 4.2, uses a quasi triangular in-\nequality for α-divergences, while the second convergence analysis, given in Section 4.3, is\nmore general by utilizing the information geometric properties of α-divergences.\n4.1 Bridging densities by diffusion process\nWe propose a new type of bridging density via the solution of a diffusion process, which\nare the root of diffusion models in machine learning [39, 40]. In particular, we consider the\nOrnstein-Uhlenbeck process. For an intial distribution πof the process, the distribution\nat time t≥0 is\nπt(x) =Z\nκt(x, y)π(y)dy, (41)\nwhere\nκt(x, y) =1\n(2π(1−e−2t))d/2exp\u0012\n−∥x−e−ty∥2\n2(1−e−2t)\u0013\n. (42)\n11(a) double gaussian\n (b) box function\nFigure 3: Ornstein-Uhlenbeck process at different time steps for X(0) being distributed\naccording to a double gaussian (a) and a box function (b).\nAt time t= 0,πtequals π, while for t→ ∞ , it approaches to a Gaussian. We then define\nour bridging densities along the time inverse path of the process, so that\nπ(ℓ)=πtℓ (43)\nwith t1> t2> . . . > t L= 0. Hence, this provides us with a path from the reference to\nthe final distribution. Figure 3 visualizes the Ornstein-Uhlenbeck distribution at different\ntimes tfor two different initial distributions. Evaluating this integral is challenging,\nespecially in cases in which the evaluation of πis computationally expensive. However,\nthis can be avoided when working with samples xfrom π,x∼π, since there exist an\nSDE in particle form describing the process given by\ndX(t) =∇Xlogρ(X(t))dt+ dW (44)\nwhere ρis a Gaussian. This exact method to create a diffusion path is known in the\nmachine learning community as Langevian flow orLangevin diffusion [35].\nBased on this idea, we propose an algorithm for learning the distribution of a given\nset of samples. Given X={xi}N\ni=1, with xi∼π, we split the set into a training Xtrainand\nvalidation set Xval, so that Xtrain∪Xval=X. We estimate a sequence of densities from\ntime evolved samples from Xtrainof the time reverse Langevin diffusion with time steps\napproaching exponentially to 0 chosen by the function F(in our case, depending on a\ngiven parameter βand the data dimension d). We do this until we begin to overfit, which\nwe determine by crossvalidation using the negative log-likelihood of Xval. The algorithm\nis summarized in Algorithm 1.\nWe propose a specific time step function Fgiven by\nF(tℓ−1;β, d) :=−1\n2ln\u0012\n1−1−exp (−2tℓ−1)\nβ2/d\u0013\n, (45)\nbased on heuristics and the idea that the ∥πt∥∞is bounded and exponentially decaying\nwith tand some heuristics, which we explain in more details in Appendix C. More gener-\nally, any function Fcan be used which approaches to 0, optimally, limiting ωβ,ω(1 +ϵ),\norη.\n12Algorithm 1: Sequential transport from data with diffusion process and stop-\nping criterion.\nInput: Xtrain,Xval,β,N\n1R(T) =−P\nx∈XvallogT♯ρref(x);\n2ℓ= 0;\n3tℓ=∞;\n4Tℓ= id;\n5repeat\n6 ℓ←ℓ+ 1;\n7 tℓ=F(tℓ−1;β, d);\n8 Xℓ={x(ℓ)\ni}, where x(ℓ)\ni∈Xtrainis one solution of Eq. (44) for time tℓand\nX(0) = xi;\n9eπ(ℓ)(x) = arg min˜π∈SoS−P\nx∈Xtrainlogeπ(ℓ)(x);\n10 Tℓ=Tℓ−1◦\u0000\nRosenblatt transport of eπ(ℓ)(x)\u0001\n;\n11until R(Tℓ)≤R(Tℓ−1)ORℓ > N ;\nOutput: Tℓ\n4.2 Convergence analysis using density ratio\nIn the following, we establish a property of the α-divergences which we can use instead\nof a triangular inequality in our first convergence analysis.\nProposition 2 (Triangular-like inequalities for probabilities) .Letp, ν, q be three propob-\nability densities. Furthermore, we assume there exists γx\nα>1 for x∈ {p, q}so that\nsup\u0010x\nν\u0011α\n≤γx\nα. (46)\nThen, for α /∈ {0,1}, we have\nDα(p||q)≤γq\n1−αDα(p||ν) + D α(ν||q) +\f\f\f\fγq\n1−α\nα−1\f\f\f\f(47)\nDα(p||q)≤Dα(p||ν) +γp\nαDα(ν||q) +\f\f\f\fγp\nα\nα\f\f\f\f. (48)\nFor the KL-divergence ( α= 1) we have\nDKL(p||q)≤DKL(p||ν) +γp\n1DKL(ν||q) (49)\nand the reverse KL divergence α= 0 by duality.\nProof. See AppendixA.4.\nOur analysis only uses the second inequality, hence, we assume a finite bound for γ\nwhich holds for all sequential approximations.\nAssumption 2. We assume\nβ(L) := sup\nℓ≤Lsup\nx∈X\u0012π(ℓ+1)(x)\nπ(ℓ)(x)\u0013α\n<+∞. (50)\n13Based on these definitions and assumptions, we give our first convergence bound in\nthe following proposition.\nProposition 3. Assume Assumptions 1 and 2 hold. Furthermore, let D α(π(0)||eπ(0)) = 0,\nfor example by π(0)=ρref. Ifωβ < 1, then the α-divergence of the estimation of π(L)by\n˜π(L), with α̸= 0, converges with\nDα(π(L)||eπ(L))≤ωβ\n1−ωβ\u0014\nη(L) +β\nα\u0015\n. (51)\nProof. See Appendix A.5.\nRemark 4.2 (Comparison with bound for Hellinger distance in [10]) .In [10], for the same\npreconditioning procedure an error bound for the Hellinger distance of\nDHell\u0000\nπ(L)||eπ(L)\u0001\n≤1\n2√ω\n1−√ωp\nη(L) (52)\nis given, where ωandηare the bounds for α= 0.5, the squared Hellinger distance\n(see Eq. (7)). For the derivation see A.7. The bound we derived here translates for the\nHellinger distance to\nDHell\u0000\nπ(L)||eπ(L)\u0001\n≤1\n2s\nωβ\n1−ωβp\nη(L) + 2β (53)\nRemark 4.3 (Bound for KL divergence) .For the KL divergence, ηcan be bounded by\nη≤log(β) and it holds that\nDKL\u0000\nπ(L)||eπ(L)\u0001\n≤ωβ\n1−ωβ(logβ+β). (54)\n4.3 Convergence analysis using α-geodesic\nInstead of the triangular like property of α-divergence used in the previous analysis, we\nuse the geometric properties of α-divergences in our second convergence analysis. While\nwe summarize the geometric properties needed for our analysis, for a detailed introduction\ninto the geometry of α-divergences, we refer to [1].\nα-divergences give Rn\n+a dually flat structure. In general, this is not the case for the\nspace of probability distributions, except for the KL and reverse KL-divergence, see [1,\np. 74]. Hence, in the following, we always work in Rn\n+. Let us state the definition of the\n(dual) α-geodesic.\nDefinition 2 (αandα∗-geodesic [1]) .Theα-geodesic connecting two unnormalized den-\nsities pandqis given by\nµt(x) =(\n{(1−t)p(x)1−α+tq(x)1−α}1\n1−αforα̸= 1\np(x)1−tq(x)tforα= 1\nFurthermore, the (1 −α)-geodesic is the α∗-geodesic (dual α-geodesic).\nIn our analysis, we make use of the projection theorem and generalized Pythagorean\ntheorem, which we state in the following.\n14π(ℓ)π(ℓ+1)\nf(ℓ)\nproj\n˜π(ℓ)\nSα-geodesic\nα-projectionα∗-projectionFigure 4: Visualization of the α-geodesic going through π(ℓ)andπ(ℓ+1)as well as the\napproximation submanifold Swith the approximation eπ(ℓ)being the α-projection of π(ℓ)\nontoS. The α∗-projection back on the α-geodesic is used in order to use the generalized\nPythagorean theorem between eπ(ℓ),f(ℓ)\nproj, and π(ℓ+1).\nTheorem 4.4 (generalized Pythagorean theorem [1]) .Given three measures, f, g, h , so\nthat the α-geodesic between fandgand the α∗-geodesic between gandhis orthogonal.\nThan,\nDα(f||h) = D α(f||g) + D α(g||h). (55)\nTheorem 4.5 (Projection theorem [1]) .Consider a measure gand a submanifold S.\nmin\n˜g∈MDα(g||˜g) (56)\nis an α-projection onto M. The minimizer is unique if Mis an α∗-flat submanifold\n(sometimes referred to as α∗-convex set).\nLet us define an α-geodesic µtwhich connects two consecutive bridging densities π(ℓ)\nandπ(ℓ+1), as shown in Figure 4. An approximation of π(ℓ)byeπ(ℓ)is found by minimizing\nthe variational density estimation problem, which in the geometric interpretation is a\nprojection from π(ℓ)onto the approximation manifold S, following an α-geodesic. From\nan information geometric view, this has an unique solution, if Sis an α∗-flat submanifold\nofRn\n+, which means, that all geodesics connecting two points in Sare contained in S. In\nour work, this is not necessarily the case and not further necessary for this convergence\nanalysis.\nNext, we project the approximation eπ(ℓ)back onto a point f(ℓ)\nprojon the α-geodesic\nbetween π(ℓ)andπ(ℓ+1). This projection is defined by\nf(ℓ)\nproj= arg min\nµt∈α−geodesicDα\u0000\nµt||eπ(ℓ)\u0001\n(57)\nand follows the α∗-geodesic between these points. Furthermore, f(ℓ)\nprojis unique, since, by\ndefinition, the α∗-geodesic is α∗-flat. For an visualization of these projections, see Fig. 4.\nThis allows us to use the generalized Pythagorean theorem between eπ(ℓ),f(ℓ)\nproj, and\nπ(ℓ+1). In the following, we make an assumption about the distance between f(ℓ)\nprojand\nπ(ℓ).\nAssumption 3. We assume that there exists an ϵ≥0 so that\nDα\u0010\nπ(ℓ+1)||f(ℓ)\nproj\u0011\n≤(1 +ϵ) Dα\u0000\nπ(ℓ+1)||π(ℓ)\u0001\n∀l∈[L]. (58)\n15Based on these two assumptions, we can establish a convergence bound for α-divergences.\nProposition 4 (Self-reinforced error bound for α-divergences) .Let assumptions 1 and 3\nbe true. Furthermore, let D α(π(0)||eπ(0)) = 0, for example by π(0)=ρref. Ifω <1, than\nthe estimation of π(L)is bounded with\nDα\u0000\nπ(L)||eπ(L)\u0001\n≤ω(1 +ϵ)\n1−ωη(L). (59)\nProof. See Appendix A.6.\nRemark 4.6 (Hellinger distance) .The Hellinger distance is symmetric. Since the α∗-\ngeodesic is (by definition) an α∗-flat manifold, the α-projection to fℓ\nprojis unique. Fol-\nlowing, that for the Hellinger distance the α-geodesic and α∗-geodesic are equivalent,\nπ(ℓ)=f(ℓ)\nprojifeπ(ℓ)is unique (that is, if Sis an α-flat submanifold). This is for example\nthe case for squared functions, so that a probability density π∝p2which are used in [10,\n44, 11], but not for SoS functions. For squared function ϵ= 0 and the bound we derive is\nDHell\u0000\nπ(L)||eπ(L)\u0001\n≤1\n2rω\n1−ωp\nη(L). (60)\nNotice that this error bound is strictly better compared to the bound in [10] given in\nEq. (52), which we show Appendix A.7.\n5 Density estimation in high dimensions\nThe SoS densities presented do not scale well in high dimension due to the exponential\nincrease of |K|with respect to d, see (23). In the following, we summarize two methods\nwhich can be used to scale to higher dimensions.\n5.1 Subspace projection: the lazy maps\nOne strategy is to iteratively select a subspace of Xin which to perform density esti-\nmation. In the following, let X=Rd. A low dimensional map on a linear subspace\nU∈Rd×r, where r≪dis the reduced dimension, writes\nQℓ(x) =UℓeQℓ(U⊤\nℓx) +\u0000\n1−UℓU⊤\nℓ\u0001\nx (61)\nwhere eQ:Rr→Rris a low-dimensional map acting on Rrwith r≪d. By iteratively\nworking on different subspaces, density estimation can be performed in high dimensions.\nThis strategy, in the case of working with evaluations of the unnormalized densities,\nwas used in [4] by choosing the subspace by minimizing an upper bound of the KL-\ndivergence. For working with samples out of a distribution, a more sophisticated version\nwas proposed in [2], where score ratio matching is used to find a subspace.\nIn our work, we use randomly chosen subspaces and let an exploration of other meth-\nods open for future work.\n165.2 Knowing the conditional independence structure\nConsider a given conditional independence structure by an undirected graph G(V, E).\nFor an introduction to such graphical models, we refer to [23]. In this graph, Vare\nthe vertices, describing random variables, and Eedges, describing dependencies between\nvariables. The conditional independence in this graph is given as follows; Given two\nvertices which are not adjacent a, b∈G, so that {a, b}/∈E, than aandbare conditional\nindependent given all other variables, written as\na⊥ ⊥b|V/{a, b} ⇔ π(a, b|V/{a, b}) =π(a|V/{a, b})π(b|V/{a, b}). (62)\nThe properties of such a model allows to factorize a probability density πaccording to\nG. Given a factorization of Gby the set of cliques C, there exist non-negative functions\n(also called couplings) ϕc(x) for c∈Cwhich only act on a subset of x,xc. Hence, πhas\nthe form\nπ(x) =Y\nc∈Cϕc(x) (63)\nwhere ϕcare not unique. It is known that this property can be used to reduce the\ncomplexity of density estimation and that a sequential composition of lower dimensional\nmaps can be learned to estimate π, as explored in detail in [41].\nWe provide a conceptual example in Fig. 5 of how to do this, while we refer to [41]\nfor the full explanation. The graphical model we consider consists of 4 variables, where\nthe density can be learned using two sequential maps. We define a fist map by\nx1x2\nx3\nx4ϕ1 ϕ2\nϕ3ϕ4π\nx1x2\nx3\nx4(ϕ2·ϕ3)◦ T1 (ϕ2·ϕ3)◦ T1\nx1x2\nx3\nx4 ρref\nT♯\n1 T♯\n2\nFigure 5: Example of learning a graphical model with 4 variables using 2 sequential maps.\nA first map only acts on x1, x2, and x4and removes any coupling to x1while introducing\na new coupling between x2andx4. In a second map, the remaining couplings are removed\nto end up with the uncorrelated variables distributed according to ρref(ρrefis a product\nmeasure).\nT♯\n1(ϕ1·ϕ4) (x)∝ρref(x), (64)\nwhere for our example we assume ρref(x) =U(x), so that π(x)ρref(x)∝π(x). It is\nrequired that the map T1in its second and fourth variable do not depend on the first, for\nexample\nT1(x) =\nT1,1(x1, x2, x4)\nT1,2(x2, x4)\nx3\nT1,4(x4)\n, (65)\n17so that all other variables in T♯\n1πare independent of x1(depicted in the middle Figure\nof Fig. 5). Consecutively, a second map satisfying\nT♯\n2(ϕ2·ϕ3) (T1(x))∝ρref(x) (66)\nis enough to learn π,\n(T2◦ T1)♯π(x)∝ T♯\n2(ϕ2·ϕ3) (T1(x))∝ρref. (67)\nHowever, this is based on the assumption that the maps are exact. Let us consider\nx1x2\nx3\nx4ϕ1 ϕ2\nϕ3ϕ4π\nx1x2\nx3\nx4(ϕ2·ϕ3)◦ T1 (ϕ2·ϕ3)◦ T1\nx1x2\nx3\nx4 ρref\nT♯\n1 T♯\n2\nFigure 6: Fig. 5 with assuming an imperfect map T1which creates an error with a\ndependency shown in dotted red. Consecutively, a general T2spreads this error to more\nvariables, shown in dotted blue.\nimperfections, where T1creates and error ϵ1which depends on the variables it acts on,\nT♯\n1(ϕ1·ϕ4) (x)∝ϵ1(x1, x2, x4). Further applying T2, even if this is an exact map, spreads\nthe error ϵ1, so that T♯\n2ϵ1depend on all variables in the network. This is depicted in\nFig. 6. Hence, when considering imperfect maps, the estimation error does not follow the\nconditional independence structure of the original model.\n6 Numerical Examples\nWe demonstrate the method presented in this paper with numerical examples, imple-\nmented in Julia. Our implementations can be found at https://github.com/benjione/\nSequentialMeasureTransport.jl . For the optimization, we use the JuMP.jl library [24]\ntogether with Hypatia.jl , as an interior point solver [6]. For more details regarding the\nimplementation, see Appendix D.\n6.1 Multimodal density from data by diffusion process\nWe demonstrate the diffusion process given in Section 4.1 by estimating a two modal\ndensity, given by\nπX(x) with x∼\n\nN\u0012\u0010\n2 2\u0011⊤\n,diag\u0010\n0.1 0.5\u0011\u0013\nwith probability 0 .5\nN\u0012\u0010\n−2−2\u0011⊤\n,diag\u0010\n0.5 0.1\u0011\u0013\nelse.\nThe density is learned from 1000 independent samples. For the diffusion process, we\nchoose 20 bridging densities with timesteps between t1≈1.0≤...≤t20= 0.0. The SoS\n18(a)t1≈1.0\n (b)t2≈0.71\n(c)t6≈0.28\n (d)t20= 0\nFigure 7: Visualization of the reverse diffusion process at 4 different time steps, where\nthe samples evolved after the given times are visualized as green dots and the estimated\ndensity as a contour.\nfunctions we employ for estimating the densities use Legendre polynomials up to order\n4. In Fig. 7, we provide a little cartoon of diffused samples and its density estimation at\ndifferent timesteps in the process. In the first timestep, the distribution is simple enough\nthat our model is able to capture it well. Following the reverse process, the previous\nestimations are used as preconditioners in order to learn the more and more complicated\ndensities in the reverse diffusion process.\n6.2 Density from datasets\nWe estimate the density of UCI datasets and follow approaches from [3] and [42]. We\nremove all discrete coefficients and coefficients with Pearson correlation higher than 0 .98.\nWe then randomly split the data into a training and test dataset, where 90 % are for\ntraining and 10 % for testing and evaluate the negative log likelihood function of the test\ndataset to evaluate how well the density fits the data. We do this a total of 10 times\nwith different splits in train and test data for a better comparison. This setup is identical\nwith the one in [3] and we compare the results of the negative log likelihood directly.\nThese results are for the adaptive transport map (ATM) algorithm [3] and a multivariate\nGaussian estimation.\nATM estimates densities using approximations of the type g(f(x)), where g:X →R+\nandfbeing a linear function space (polynomials), with an adaptive amount of coefficients\n19using downward closed sets. Finally, a KR map is build from the approximation.\nFor our method, we use Algorithm 1 with β= 2.0 and apply density estimations\nto randomly picked 8-dimensional subspaces, in order to reduce computation time. For\neach of this subspaces, we choose polynomials of up to order 3 so that A∈R70×70. The\nTable 1: Comparison of log likelihood function for different UCI datasets and density\nestimation methods. The best results are highlighted in bold.\nDataset (d, N) SoS Gaussian SoS [s] ATM # layers SoS\nWhite wine (11,4898) 11.1±0.213.2±0.51613±250 11.0±0.2 16.4±3.0\nRed wine (11,1599) 10.0±0.513.2±0.3353±64 9.8±0.4 12.5±2.8\nParkinsons (15,5875) 4.4±0.410.8±0.44022±333 2.8±0.4 29.0±2.7\nBoston (10,506) −4.1±1.611.3±0.5646±81.53.1±0.6 28.5±4.7\nresults are depicted in Table 1. For the red and white wine dataset SoS and ATM per-\nform similarly, while for Parkinsons ATM performs significantly better and for Boston\nSoS performs significantly better. We explain the worse result on Parkinsons with using\nrandom subspaces and the higher dimension of this dataset. From the lower dimensional\nBoston dataset we conclude that our methodology works well in general. We want to\nmention that Algorithm 1 can also be employed with the adaptive functions from ATM\nand we believe that this would outperform the SoS, since these scale better to higher\ndimensions. Our results should only be considered as a proof of concept for the applica-\nbility of SoS and diffusion bridging densities, while further work to mitigate the curse of\ndimensionality is needed.\n6.3 Susceptible-Infected-Removed (SIR) model\nWe consider an example from epidemiology and follow the setting in [11]. Having access\nto the SIR model, a model describing the spread of a disease, see [20], we want to calibrate\nparameters of the model given observations on the amount of infected persons at different\npoints in time. The SIR model is given by\n˙Sk=−βkIkSk+1\n2X\nj∈Ik(Sj−Sk)\n˙Ik=βkSkIk−γkIk+1\n2X\nj∈Ik(Ij−Ik)\n˙Rk=γkIk+1\n2X\nj∈Ik(Rj−Rk)\nwhere k∈ K describes the amount of compartments in the model. In the following, we\nwill only consider |K|= 1. The parameters to be determined are γkandβk. The ODE\nis simulated for t∈[0,5] with 6 equidistant observations yk,jinIkperturbed with noise\nϵk,j∼ N(0,1), so that yk,j=Ik(tj)+ϵk,j. We pose the calibration problem in an Bayesian\nsetting, where the prior belief on the parameters is uniform in [0 ,2] for all parameters\nand the likelihood function is gaussian and given by\nL(x|y)∝exp \n−PK,6\nk=1,j=1(Ik(5j\n6;x)−yk,j)2\n2!\n20We approximate the posterior using 4 self-reinforced layers with downward closed ten-\nsorized polynomials in Φ of maximum degree 6. We choose bridging densities of type\nπ(l)=πpost(x)βlwith β1= 1/8,β2= 1/4,β3= 1/2 and β4= 1 and use 1000 evaluations\nof the unnormalized posterior for each layer. A comparison between the contours of the\n(a) Variational posterior ( α= 2)\n (b) Unnormalized posterior\nFigure 8: Comparison of the variational density (a) and the unnormalized posterior\ndistribution (b) for the parameter space around the center of mass, for γ∈[0.85,1.1] and\nβ∈[0.08,0.15].\nvariational posterior and unnormalized posterior around the center of mass is depicted in\nFig. 8. Visually, the density estimation fits the true density well. To quantify our results,\nwe consider the estimated density for the task of self-normalized importance sampling\nand calculate the effective sampling size (ESS). To do this, we run the density estimation\n10 times with drawing new noise ϵk,jin every iteration and calculate the ESS for density\nestimation using α-divergences between 0 .4 and 3 .0. The results are shown in Figure 9.\nAs expected, for higher αa better ESS is achieved. This is because the ESS can be\nFigure 9: ESS for α-divergence between α= 0.4 and α= 3.0 for 10 runs on the SIR\nmodel with different realizations of noise. The ESS is given as effective samples for 1000\nsamples.\nwritten as ESS = 1 /(1 + Var eπ[Zf]) and minimizing the χ2-divergence is equivalent to\nminimizing Var eπ[Zf] (e.g. see [27]). Since higher α-divergences are upper bounds on the\nχ2-divergence, it is expected that minimizing over them performs similarly. Furthermore,\nbecause of the zero avoiding property for α≥1, which is a requirement for successful\n21importance sampling in a lot of cases, higher αfor estimations of densities for importance\nsampling is desirable.\n6.4 Gaussian graphical model\nWe give a demonstration of estimating graphical models using a sequence of maps. For\nsimplicity, we use a Gaussian graphical model, in which the distributions remains Gaus-\nsian. We use the conditional independence structure from Fig. 5 where the total dis-\ntribution, π∝ϕ1ϕ2ϕ3ϕ4is a normal distribution with expectation and covariance given\nby\nµ=\n0\n0\n0\n0\nand Σ =\n0.26 0 .2 0 .16−0.12\n0.2 0 .24 0 .19−0.14\n0.16 0 .19 0 .24−0.18\n−0.12−0.14−0.18 0 .29\n. (68)\nWe perform tempering, by first tempering on ϕ1·ϕ4using 4 layers. Afterwards, we learn\nthe density ϕ2·ϕ3, also using tempering with 4 layers. To demonstrate our result, we\ncompare samples of the marginal distribution π(x1, x3) with the density function of the\ntrue marginal, given in Fig. 10.\n(a) True density\n (b) Samples from estimated density.\nFigure 10: Comparison of marginal density π(x1, x3) in (a) with samples from the esti-\nmated density (b).\n7 Conclusion and Outlook\nWe introduced SoS functions for density approximation utilizable with a wide range of\nstatistical divergences. We embedded this tool into the framework of measure trans-\nport [26] and used it in the context of sequential transport, as in [10]. The SoS functions\nwe proposed are a generalization of [11] and [44] with an extension to general divergences.\nTo the best of our knowledge, our work is the first in measure transport to propose a\nlinear function space in Rn\n+to be utilizable with any convex statistical divergence.\nWe provided two convergence analysis for greedy density estimation, a first with in-\ntuitive assumptions and a second one which allowed us to improve the bound for the\n22Hellinger distance in [10]. We demonstrated the applicability of our method with numer-\nical examples. In order to make our proposed method better applicable, future work is\nnecessary, from which in our opinion the most important ones are:\n•Sampling\nWhile optimal sampling is well established for least squares in linear approximation\nspaces, we do not know how to do this for SoS functions.\n•Approximation rates\nSince SoS functions generalize squared polynomials, results from [44] are applicable,\nbut a work on more general approximation rates for SoS functions is missing.\n•Bridging densities\nThe bridging densities obtained by tempering work well for continuous distributions\nand the diffusion when working with samples out of π. However, we believe that\nbridging densities for discontinuous distributions and with guarantees of conver-\ngence are missing.\n•High dimensionality\nOur method does not scale well with high dimensions. In our numerical example,\nin Sec. 6, we used random subspaces to deal with the higher dimensions. We think\nthat our method can be transformed into a tensor version to make it scale better\nwith high dimensions.\nA Proofs\nA.1 Proof of Proposition 1\nThis proof is inspired by [11] where the marginalization is done for squared polynomials\nonly. We can write\nLgA(x) =Z\nΦ(x−ℓ, x′\nℓ)⊤AΦ(x−ℓ, x′\nℓ)dµℓ(x′\nℓ)\n=X\nα,β∈KAσ(α)σ(β) Y\ni̸=ℓϕi\nαi(xi)! Y\ni̸=ℓϕi\nβi(xi)!Z\nϕαl(x′\nl)ϕβl(x′\nl)dµℓ(x′\nl)\n=X\nα,β∈KˆΦσ(β)Aσ(α)σ(β)Mσ(α)σ(β)ˆΦσ(α)\n=ˆΦ(x−ℓ)⊤(A⊙M)ˆΦ(x−ℓ)\nwhere\nMσ(α)σ(β)=Z\nϕαl(x′\nl)ϕβl(x′\nl)dµℓ(x′\nl),\nandˆΦσ(α)(x−ℓ) =Q\ni̸=ℓϕi\nαi(xi). Note that the vector ˆΦ(x−ℓ) has size m=|K|and con-\ntains duplicated entries of the vector Φ −ℓ(x−ℓ) defined by ˆΦσ−ℓ(α−ℓ)(x−ℓ) =Q\ni̸=ℓϕi\nαi(xi).\nBy definition (26) of the matrix P∈R|K−ℓ|×m, we have ˆΦσ(α)(x−ℓ) =P⊤Φ−ℓ(x−ℓ). This\nshows M−ℓ=P(A⊙M)P⊤and concludes the proof.\n23A.2 Proof of pushforward and pullback in α-divergence\nWe have\nDα(T♯p||q) =Z\nϕα\u0012p◦T−1· ∇xT−1\nq\u0013\ndq (69)\n=Z\nϕα\u0012p◦T−1◦T· ∇xT−1◦T\nq◦T\u0013\ndT♯q (70)\n=Z\nϕα\u0012p\nq◦T· ∇xT\u0013\ndT♯q (71)\n= D α\u0000\np||T♯q\u0001\n(72)\nTrivially, this does not only hold for probability distributions but also measures.\nA.3 Proof of relation between measures and probabilities in α-\ndivergence\nDα(f||g) =Z\nϕα\u0012f\ng\u0013\ndg (73)\n=ZZα\nf\nZα\ng\u0010\nπf\nπg\u0011α\n−\u0010\nZf\nZg\u0011α\n−1 + 1\nα(α−1)Zgdπg+ZZf\nZgπf\nπg−Zf\nZg−1 + 1\nα−1Zgdπg(74)\n=Zα\nf\nZα−1\ngDα(πf||πg) +Zg\nZα\nf\nZαg−1\nα(α−1)+Zf\nZg−1\nα−1\n (75)\n=Zα\nf\nZα−1\ngDα(πf||πg) +Zgϕα\u0012Zf\nZg\u0013\n. (76)\nA.4 Proof of Proposition 2\nFor the inequalities:\nWe work with probability densities.\nLet\n˜ϕα(t) =tα−1\nα(α−1)(77)\nFor probabilities, we have\nDα(p||q) =Z\nϕα\u0012p\nq\u0013\ndq=Z\n˜ϕα\u0012p\nq\u0013\ndq. (78)\nNote that ϕα(t)≥0 for t >0 while this is not the case for ˜ϕα. We assume there exists a\nγx\nα>0\nsup\u0010x\nν\u0011α\n≤γx\nα (79)\n24forx∈ {p, q}. For α /∈ {0,1}\nϕα\u0012p\nq\u0013\nν>0=\u0000p\nν\u0001α\u0010\nν\nq\u0011α\n−1\nα(α−1)−p\nq−1\nα−1(80)\n=\u0012ν\nq\u0013α\u0000p\nν\u0001α−1\nα(α−1)+\u0010\nν\nq\u0011α\n−1\nα(α−1)±\u0012ν\nq\u0013αp\nν−1\nα−1−p\nq−1\nα−1(81)\n=\u0012ν\nq\u0013α\nϕα\u0010p\nν\u0011\n+˜ϕα\u0012ν\nq\u0013\n+\u0012ν\nq\u0013αp\nν−1\nα−1−p\nq−1\nα−1. (82)\nGoing to the D αwe get\nDα(p||q) =Z\u0012ν\nq\u0013α\nϕα\u0010p\nν\u0011\n+˜ϕα\u0012ν\nq\u0013\n+\u0012ν\nq\u0013αp\nν−1\nα−1−p\nq−1\nα−1dq (83)\n≤sup\u0012ν\nq\u0013α−1\nDα(p||ν) + D α(ν||q) +Z\u0012ν\nq\u0013α−1p\nν−1\nα−1dν (84)\nForα >1:\n≤sup\u0012ν\nq\u0013α−1\nDα(p||ν) + D α(ν||q) +sup\u0010\nν\nq\u0011α−1\n−inf\u0010\nν\nq\u0011α−1\nα−1(85)\n≤γq\n1−αDα(p||ν) + D α(ν||q) +γq\n1−α−inf\u0010\nν\nq\u0011α−1\nα−1(86)\n≤γq\n1−αDα(p||ν) + D α(ν||q) +γq\n1−α\nα−1(87)\nForα <1:\n≤sup\u0012ν\nq\u0013α−1\nDα(p||ν) + D α(ν||q) +inf\u0010\nν\nq\u0011α−1\n−sup\u0010\nν\nq\u0011α−1\nα−1(88)\n≤γq\n1−αDα(p||ν) + D α(ν||q) +\f\f\f\f\f\f\fγq\nα−1−inf\u0010\nν\nq\u0011α−1\nα−1\f\f\f\f\f\f\f(89)\n≤γq\n1−αDα(p||ν) + D α(ν||q) +\f\f\f\fγq\n1−α\nα−1\f\f\f\f(90)\nBy duality and α >0 (hence, 1 −α <1)\nDα(p||q) = D 1−α(q||p) (91)\n≤γp\nαD1−α(q||ν) + D 1−α(ν||p) +γp\nα\nα(92)\n= D α(p||ν) +γp\nαDα(ν||q) +γp\nα\nα(93)\nand duality with α <0 (hence, 1 −α >1)\nDα(p||q) = D 1−α(q||p) (94)\n≤γp\nαD1−α(q||ν) + D 1−α(ν||p)−γp\nα\nα(95)\n= D α(p||ν) +γp\nαDα(ν||q) +\f\f\f\fγp\nα\nα\f\f\f\f(96)\n25Next, for α= 1.\nϕα\u0012p\nq\u0013\n=p\nqlog\u0010p\nν\u0011\n+p\nνν\nqlog\u0012ν\nq\u0013\n−p\nq+ 1 (97)\n=p\nqlog\u0010p\nν\u0011\n+p\nνϕα\u0012ν\nq\u0013\n+ν\nq−p\nq(98)\nInserting for D KL,\nDKL(p||q)≤DKL(p||ν) + supp\nνDKL(ν||q) + 1−1 (99)\n≤DKL(p||ν) +γp\nαDKL(ν||q) (100)\nWe do α= 0 by duality\nD0(p||q) = D KL(q||p) (101)\n≤DKL(q||ν) +γq\n1DKL(ν||p) (102)\n=γq\n1−αD0(p||ν) + D 0(ν||q). (103)\nA.5 Proof of Proposition 3\nLet us first work with normalized densities. We have to assume that α≥1. Note that\nβ≥γπℓ\nαin Proposition 2.\nDα\u0000\nπ(L)||eπ(L)\u0001Ass. 1\n≤ωDα\u0000\nπ(L)||eπ(L−1)\u0001\n(104)\nProp. 2\n≤ω\u0014\nDα\u0000\nπ(L)||π(L−1)\u0001\n+βDα\u0000\nπ(L−1)||eπ(L−1)\u0001\n+β\nα\u0015\n. (105)\nBy recursion and D α\u0000\nπ(0)||˜π(0)\u0001\n= 0, we get\nDα\u0000\nπ(L)||eπ(L)\u0001\n≤ωLX\ni=1(ωβ)L−i\u0014\nDα\u0000\nπ(i)||π(i−1)\u0001\n+β\nα\u0015\n(106)\nDef. 1\n≤ωLX\ni=1(ωβ)L−i\u0014\nη(L) +β\nα\u0015\n(107)\n≤ω\u0014\nη(L) +β\nα\u0015\n+ωL−1X\ni=1(ωβ)i\u0014\nη(L) +β\nα\u0015\n(108)\nωβ<1\n≤ω\u0014\nη(L) +β\nα\u0015\n+ω2β\n1−ωβ\u0014\nη(L) +β\nα\u0015\n(109)\n≤ωβ\n1−ωβ\u0014\nη(L) +β\nα\u0015\n. (110)\n26A.6 Proof of Proposition 4\nLet ˜π(0)=π(0), e.g. by setting π(0)=ρrefor with a known diagonal map R♯\n0π(0)=µ, so\nthat ˜π(0)can be build exactly.\nDα\u0000\nπ(L)||˜π(L)\u0001Ass. 1\n≤ωDα\u0000\nπ(L)||˜π(L−1)\u0001\nPythag.=ωh\nDα\u0010\nπ(L)||f(L)\nproj\u0011\n+ D α\u0010\nf(L)\nproj||˜π(L−1)\u0011i\nAss. 3\n≤ω(1 +ϵ) Dα\u0000\nπ(L)||π(L−1)\u0001\n+ωDα\u0010\nf(L)\nproj||˜π(L−1)\u0011\nEq.(57)\n≤ω(1 +ϵ) Dα\u0000\nπ(L)||π(L−1)\u0001\n+ωDα\u0000\nπ(L−1)||˜π(L−1)\u0001\nBy recursion,\nDα\u0000\nπ(L)||˜π(L)\u0001\n≤ω(1 +ϵ)LX\nℓ=1ωL−ℓDα\u0000\nπ(l)||π(l−1)\u0001\n+ωLDα\u0000\nπ(0)||˜π(0)\u0001\n=ω(1 +ϵ)LX\nl=1ωL−ℓDα\u0000\nπ(l)||π(l−1)\u0001\nDef. 1\n≤ω(1 +ϵ)LX\nℓ=1ωL−ℓη(L)\n≤ω(1 +ϵ)η(L) +ω(1 +ϵ)L−1X\nℓ=1ωℓη(L)\n=ω(1 +ϵ)L−1X\nℓ=0ωℓη(L)\n≤ω(1 +ϵ)\n1−ωη(L)\nA.7 Proof of Remark 4.2 and Remark 4.6\nFirst, we translate the variables used in [10] to the ones used in this paper. We call the\nvariables from [10] ωHandηH, defined by\nDHell\u0000\nπ(ℓ)||eπ(ℓ)\u0001\n≤ωHDHell\u0000\nπ(ℓ)||eπ(ℓ−1)\u0001\n(111)\nand\nDHell\u0000\nπ(ℓ)||π(ℓ−1)\u0001\n≤ηH∀ℓ∈[L]. (112)\nWe translate, using α= 0.5,\nDα\u0000\nπ(ℓ)||eπ(ℓ)\u0001\n≤ωDα\u0000\nπ(ℓ)||eπ(ℓ−1)\u0001\n(113)\n4 DHell\u0000\nπ(ℓ)||eπ(ℓ)\u00012≤ω4 DHell\u0000\nπ(ℓ)||eπ(ℓ−1)\u00012(114)\nDHell\u0000\nπ(ℓ)||eπ(ℓ)\u00012≤√ωDHell\u0000\nπ(ℓ)||eπ(ℓ−1)\u0001\n, (115)\n27hence, ωH=√ω(equal sign by doing the reverse and ωH≤√ω≤ωH), and\nDα\u0000\nπ(ℓ)||π(ℓ−1)\u0001\n≤η (116)\n4 DHell\u0000\nπ(ℓ)||π(ℓ−1)\u00012≤η (117)\nDHell\u0000\nπ(ℓ)||π(ℓ−1)\u0001\n≤1\n2√η, (118)\nhence, ηH=1\n2√η(same as before to get the equality). Therefore, the bound given in [10]\nwrites\nDHell\u0000\nπ(L)||eπ(L)\u0001\n≤ωH\n1−ωHηH=1\n2√ω\n1−√ω√η. (119)\nLet us show that the bound derived by the geometric properties is strictly better\ncompared to the bound in [10]. Note that for 0 < ω < 1,ω≤√ω.\n1\n2rω\n1−ω√η≤1\n2√ω\n1−√ω√η (120)\n√ω√1−ω≤√ω\n1−√ω(121)\n1−√ω≤√\n1−ω (122)\n1−2√ω+ω≤1−ω (123)\n1−2√ω+ω≤1−2ω+ω= 1−ω. (124)\nEquality holds for ω→0,1.\nB Comparison of the KL divergence for positive mea-\nsures and probability distributions\nIn [16], the paper is motivated in the appendix by the fact that a naive discretization\nof the KL divergence does not converge to the true density and is unstable. In a naive\ndiscretization, the minimization problem\nmin\n˜πDKL(π||˜π)\nis discretized by\nmin\n˜πDKL(π||˜π) = min\n˜π−Z\nlog(˜π)dπ\n≈min\n˜π−X\nilog(˜π(xi))π(xi)\np(xi)xi∼p\nwith a distribution pused for importance sampling. The authors consider a gradient\ndescent approach, in which the gradient\nd˜π\ndt=−∇ ˜πDKL(π||˜π)\n28of the discretized KL divergence is followed. Hence, we can calculate this evaluated at a\nsingle discretization point xi,\nd˜π\ndt|x=xi=π\n˜π|x=xi.\nA solution to this is (see [16])\n˜π(xi) =p\n2tπ(xi).\nHence, it is necessary to satisfyR\nd˜π= 1 in order to converge to π(see [16]). We show\nthat for the extension of the KL divergence, gradient descent convergences ´for any single\npoint xitoπ(xi). Note that\n∇˜πDKL(π||˜π) =−Zπ\n˜πdx−Z\ndx\n∇˜πDKL(π||˜π)|x=xi=π(xi)\n˜π(xi)−1.\nWhile before, for π,˜π >0 the gradient never reached 0, here, the minima is reached for\nπ(xi) = ˜π(xi).\nC A Heuristic for choosing the timesteps in the in-\nverse diffusion process\nIn the following, we try to find optimal time steps in the reversed Ornstein-Uhlenbeck\nprocess so that βfrom Assumption 2 is small. First, note that we can bound the ∥•∥∞\nofπtin the Ornstein-Uhlenbeck process.\nProposition 5 (Bound on ∥•∥∞for Ornstein-Uhlenbeck process) .Letπ(0)=ρrefbe the\nstationary solution of the Ornstein-Uhlenbeck process and πa normalized distribution.\nWe have that\n∥πt(x)∥∞≤1\n(2π(1−e−2t))d/2\nwhere πtas in Eq. (41).\nProof. Using Young’s convolution inequality\n∥π ⋆ κ t∥∞≤ ∥π∥1∥κt∥∞. (125)\nBecause we assume probability densities, ∥π∥1=∥κt∥1= 1.\nWe can calculate ∥κt(y)∥∞, since it is a Gaussian pdf:\n∥κt∥∞=1\n(2π(1−e−2t))d/2.\nNote that in general this bound is not tight. Nevertheless, it gives a decay of the\nmaximum value of the distribution in the diffusion process and we use it as an heuristic.\nWe need to work with an assumption that for small tthe bound is less tight compared\nto larger t.\n29Assumption 4. Define a constant Ct≤1, so that\n∥πt(x)∥∞=Ct1\n(2π(1−e−2t))d/2. (126)\nWe make the assumption that Ct1≤Ct2fort1≤t2.\nWe are not able to bound β, but instead to bound a variable ˜β=∥π(l+1)∥∞\n∥π(l)∥∞which is\na lower bound of βand assume that β≈˜β.\nProposition 6 (˜βis a lower bound of β).˜β=∥π(ℓ+1)∥∞\n∥π(ℓ)∥∞∀ℓ∈[L] is a lower bound of β.\nProof.\nβ= sup\nx∈Xπ(ℓ+1)\nπ(ℓ)≥sup\nx∈Xπ(ℓ+1)\n∥π(ℓ)∥∞=\r\rπ(ℓ+1)\r\r\n∞\n∥π(ℓ)∥∞=˜β.\nWith these, we are able to choose a timestep tℓ+1given tℓso that ˜βis smaller than\nsome given value.\nCorollary 2 (Bound on ˜βby diffusion process) .Given tℓand˜β, assuming Assumption 4\nholds, a subsequent time step tℓ+1so that∥π(ℓ+1)∥∞\n∥π(ℓ)∥∞≤˜βis satisfied for\ntℓ+1≥ −1\n2ln\u0012\n1−1−exp (−2tℓ)\n˜β2/d\u0013\n.\nProof. Given ˜βandtℓ, we want to find tℓ+1. Note that tℓ+1≤tℓand therefore, from\nAssumption 4, Ctℓ+1≤Ctℓ. This gives\n˜β=\r\rπ(ℓ+1)\r\r\n∞\n∥π(ℓ)∥∞=Ctℓ+1[2π(1−exp(−2tℓ))]d/2\nCtℓ[2π(1−exp(−2tℓ+1))]d/2Ass. 4\n≤[2π(1−exp(−2tℓ))]d/2\n[2π(1−exp(−2tℓ+1))]d/2\n˜β(1−e−2tℓ+1)d/2≥(1−e−2tℓ)d/2\n1−e−2tℓ+1≥1\n˜β2/d(1−e−2tℓ)\ne−2tℓ+1≤1−1−e−2tℓ\n˜β2/d\ntℓ+1≥ −1\n2ln(1−1−e−2tℓ\n˜β2/d).\n30D Implementation details\nWe implement our method in Julia. To optimize the SoS functions, we use the JuMP.jl\nlibrary [24], an interface for implementing SDP and other optimization problems. As a\nsolver, we use Hypatia.jl [6], an interior point solver.\nNext, we explain in more details how the minimization of α-divergences is performed\nusing SDP. First, we discretize α-divergences using evaluations at xi∼µ, so that\nmin\ngDα(f||g) = min\ngZ\nϕα(f\ng)dg (127)\n= min\ngZfαg1−α−g\nα(α−1)−f−g\nα−1dµ (128)\n= min\ngZfαg1−α\nα(α−1)+g\nαdµ (129)\n= min\ng1\nNNX\ni=1f(xi)αg(xi)1−α\nα(α−1)+Zg\nαdµ with xi∼µ. (130)\ngitself is an SoS-function, where A⪰0. This is handeled in SDP by using a PSD cone for\nA. To integrateR\ng/αdµ, we use thatR\ngdµ= trace( A). The main difficulty remaining is\nto map g(xi)1−αto an SDP problem. To do so, we use an auxiliary variable ti, so that ti\nupper or lower bounds f(xi)αg(xi)1−αand minimize over this variable at the same time.\nFor this, we use the power cone for α̸= 0,1, given by\nKp=\b\n(x, y, z )∈R3:xpy1−p≥ |z|, x≥0, y≥0\t\n. (131)\nFirst, consider α <0. We use another variable ti, so that ti≥f(xi)αg(xi)1−αand write\nti≥f(xi)αg(xi)1−α(132)\ntif(xi)−α≥g(xi)1−α(133)\nt1\n1−α\ni\u0000\nf(xi)−1\u0001α\n1−α≥g(xi). (134)\nSince f(xi)−1can be very high, for numerical reasons it is better to use\nt1\n1−α\ni≥f(xi)α\n1−αg(xi) (135)\nand map this to the powercone with ( ti,1, f(xi)α\n1−αq(xi))∈K1/(1−α).\nNext, we consider 0 < α < 1. Note, that α(α−1) is negative in this region, hence,\nwe use a variable ti, so that ti≤f(xi)αg(xi)1−α. This directly maps to the powercone,\nby using ( f(xi), g(xi), ti)∈Kα.\nLast, we consider α >1 and again we seek for ti≥f(xi)αg(xi)1−αm but we can not\nuse the same formulation as for α < 0, since in the powercone p∈[0,1]. Hence, we\nformulate\nti≥f(xi)αg(xi)1−α(136)\ntig(xi)α−1≥f(xi)α(137)\nt1\nα\nig(xi)1−1\nα≥f(xi). (138)\nThis maps to the powercone as ( ti, g(xi), f(xi))∈K1/α.\n31Forα= 0,1, the powercone does not work. Instead, the exponential cone could be\nused. However, JuMP has the possibility of using the relative entropy cone , which directly\nworks with the Hypatia.jl solver. This cone is given by\nKn=(\n(u, v, w )∈R1+2n|u≥nX\ni=1wilog\u0012wi\nvi\u0013)\n. (139)\nBoth the KL divergence, as well as the reverse KL divergence, can be implemented using\nthis cone.\nReferences\n[1] Shun-ichi Amari. Information Geometry and Its Applications . en. Vol. 194. Applied\nMathematical Sciences. 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In: Neural Processing Letters 2.6 (Dec. 1995), pp. 28–31. issn: 1370-4621,\n1573-773X. doi:10.1007/BF02309013 .url:http://link.springer.com/10.\n1007/BF02309013 (visited on 12/07/2023).\n35" }, { "title": "2403.01099v1.Diffusive_Decay_of_Collective_Quantum_Excitations_in_Electron_Gas.pdf", "content": "arXiv:2403.01099v1 [cond-mat.mes-hall] 2 Mar 2024Diffusive Decay of Collective Quantum Excitations in Electr on Gas\nM. Akbari-Moghanjoughi\nFaculty of Sciences, Department of Physics,\nAzarbaijan Shahid Madani University,\n51745-406 Tabriz, Iran\nmassoud2002@yahoo.com\n(Dated: March 5, 2024)\nAbstract\nIn this work the multistream quasiparticle model of collect ive electron excitations is used to study\nthe energy-density distribution of collective quantum exc itations in an interacting electron gas with\narbitrary degree of degeneracy. Generalized relations for the probability current and energy density\ndistributions is obtained which reveals a new interesting q uantum phenomenon of diffusive decay of\npurequasiparticle states at microscopic level. The effects i s studied for various cases of free quasiparti-\ncles, quasiparticle in an infinite square-well potential an d half-space collective excitations. It is shown\nthat plasmon excitations have the intrinsic tendency to dec ay into equilibrium state with uniform\nenergy density spacial distribution. It is found that plasm on levels of quasipaticle in a square-well\npotential are unstable decaying into equilibrium state due to the fundamental property of collective\nexcitations. Thedecay rates of pureplasmonstates aredete rminedanalytically. Moreover, fordamped\nquasiparticle excitations the non-vanishing probability current divergence leads to imaginary energy\ndensity resulting in damping instability of energy density dynamic. The pronounced energy density\nvalley close to half-space boundaryat low level excitation s predicts attractive forceclose to thesurface.\nCurrent research can have implications with applications i n plasmonics and related fields. Current\nanalysis can be readily generalized to include external pot ential and magnetic field effects.\n1I. THEORETICAL BACKGROUND\nCollective excitations play critical role in natural phenomena. They p roduce large-scale\npatterns such as solitons, dispersive and shock waves and even ch aotic behavior in most cases\n[1]. The plasma state definition strictly relies on the collective nature o f oscillation of charged\nspecies [2]. Large class of instabilities arise due to delicate interplay of collective interactions\namong different species in plasmas via the dispersion and dissipation. H owever, as compared\nto other states of matter plasma occupies a large span of density- temperature regime making\nalmost 99 percent of the observable universe to consist of plasmas . In the two extreme density\nlimits, on the other hand, the definition of plasmas (due to collective n ature of act) becomes\nobscure. At very low concentration the number of particles inside t he Debye sphere become\ninsufficient for collective effects to manifest. In such a case only sing le particle aspects of\nphenomena become visible. At the other extreme, when the number density of charged species\nincreases unboundedly for agiven temperature, the plasmonblack out effect comes into playand\nprohibits collective effects. Therefore, collective quantum plasma, which arise due to definition\nwhenthedeBroglie’swavelengthcomparestotheinter-particlespa cing, requiresamoredelicate\ndefinition in terms of plasma parameters. Such a characteristic defi nition requires a brand new\nequation of state (EoS) which has been discussed elsewhere [3, 4].\nQuantum mechanics, by its virtue, has been originally designed to stu dy the single-particle\neffects[5,6]. However, itsbuilt-instatisticalpredictionabilityleadst oanomaloussingle-particle\naspects which due to Copenhagen indeterministic-type interpreta tions has baffled scientists for\nmany years and even led to rejection by pioneers in the field including E instein with the famous\nquote: ”God does not play dice”. The apparent inconsistencies in ph ysical interpretations of\nquantumtheoryhadconsequently ledtoappearanceofdeterminis tictheoriessuchasdeBroglie-\nBohm pilot-wave and Madelung’s fluid-like formalism. Due to inherent ab ility in statistical\nprediction beside simplicity, which has brought quantum mechanics en ormous success in many\nfields of modern physics, the theory has been effectively used to inc orporate many body effects\nby original works of Wigner and Moyal [7–10]. The development of man y body formulation\nin collective quantum theory of plasmas is indebted to many pioneering works [11–27]. There\nhas been many successful recent developments in the fields of qua ntum kinetic [28, 29], density\nfunctional [31] and quantum hydrodynamic theories [30] which have been explored through\nrecent decade [32–57]. However, due to intrinsic complexity of many body quantum effects,\ndetailed understanding of quantum effects has faced fundamenta l difficulties [58–60].\n2Plasmonics is a newly emerging field of applied science relying on the unde rstanding of\ncollective quantum effects. Plasmonic research due to appealing tec hnological applications has\nattracted growing interest in past few years. Plasmonic devices ca n find applications in optical\nemitters [61], plasmon focusing [62], nanoscale waveguiding [63] and op tical antennas [64],\nnanoscale swiches [65] and plasmonic lasers [66]. Moreover, collective plasma excitations may\nbecome a more efficient way in solar energy conversion leading toward s improved photovoltaic\nandcatalyticdesignsbycollectiveenergytransport[67,68]. Incur rent researchweexploresome\nnew aspects of quasiparticle model of collective electron excitation s concerning diffusion of pure\ncollective electron states which can have important implications to th efields of nanoplasmonics.\nThe phase space evolution of quasiparticle excitations in electron ga s has been recently studied\nusing this model [69]. We briefly discuss the model and the plasmon disp ersion properties in\nSec. II. Generalized probability current definition for damped collec tive excitations is given in\nSec. III. The energy density of interacting electron gas is derived in Sec. IV. Application of the\nmodel to free quasiparticle, quasiparticle in a square-well potentia l and damped quasiparticle\nare respectively presented in Secs. V, VI and VII and conclusions a re drawn in Sec. VIII.\nII. QUASIPARTICLE MODEL AND DISPERSION RELATION\nIn this section we briefly introduce the quasiparticle model of collect ive excitations in elec-\ntron gas using multistream model and review the energy band struc ture of damped plasmon\nexcitations. Despite conventional many body theories in the multist ream model electrons are\nlocalized in momentum space couples to each other via the Poisson equ ation. By discarding the\nprobability concept of finding electrons, one avoids the redundant mean field approximations in\norder to solve the many body problem. We picture each electron as a quantum stream governed\nby universal Hamiltonian which encompasses all essential ingredient s operating on individual\nparticles in the system. The act of this hamiltonian is through the follo wing single-particle\nSchr¨ odinger equation\ni¯h∂Nj(r,t)\n∂t=HNj(r,t), (1)\nwherejdenotes the stream number and the generalized Hamiltonian is H=−(¯h2/2m)∆−(κ·\n∇)−eφ(r) +µwhich, respectively from the left, includes the kinetic, damping, elec trostatic\npotential and chemical potential operators. Here, κis parameter to introduce the damping\neffect in a phenomenological manner and contains the missing dimensio nal parameters in this\nterm. Although, (1) is called the single-particle, the electrons are r elated through acting of\n3the universal Hamiltonian operator which is itself under the influence of each particle via field\nequation such as Poisson’s relation and appropriate equation of sta tes (EoS) which relate given\npotentials in the Hamiltonian to the local density of electrons. There fore, the wavefunction\nNjdescribes not only the dynamics of given j-th electron in the system, but also its relation\nto others. The quantum electron streams are coupled via the Poiss on’s relation (assuming a\nknown EoS) as follows\n∆φ(r)+2(κ·∇)φ(r) = 4πe/bracketleftBiggN/summationdisplay\nj=1Nj(r,t)N∗\nj(r,t)−n0/bracketrightBigg\n, (2)\nin whichn0denotes the number density of the static neutralizing background charge. The\nlocal electron number density uses the standard definition in terms of wavefunction, n=\nN/summationtext\nj=1Nj(r,t)N∗\nj(r,t). We now define the multistream wavefunction as N(r,t) =N/summationtext\nj=1Nj(r,t)\nwhich is the analogous counterpart of many body wavefunction in st andard treatments. How-\never, in the multistream model the summation form replaces the pro ducts of single particle\nwavefunctions which causes main difficulties in many body treatments . Using this new def-\ninition one arrives at the following coupled system of differential equa tion for quasiparticle\nexcitations\ni¯h∂N(r,t)\n∂t=−¯h2\n2m∆N(r,t)−2(κ·∇)N(r,t)−eφ(r)N(r,t)+µN(r,t),(3a)\n∆φ(r)+2(κ·∇)φ(r) = 4πe/bracketleftBigg\nN(r,t)N∗(r,t)−N/summationdisplay\nk/negationslash=jNk(r,t)N∗\nj(r,t)−n0/bracketrightBigg\n.(3b)\nIn the limit of large electron number, ( N≫1), the second term in rhs of (3) vanishes due\nto phase mixing effect and one arrive at the following simplified system [3 ]\ni¯h∂N(r,t)\n∂t=−¯h2\n2m∆N(r,t)−2(κ·∇)N(r,t)−eφ(r)N(r,t)+µN(r,t),(4a)\n∆φ(r)+2(κ·∇)φ(r) = 4πe[N(r,t)N∗(r,t)−n0]. (4b)\nNote that the summation in (3) which includes the mixed states can be come important in few-\nbody quantum systems and other electron interaction effects. In current analysis however we\nignore the electron exchange and correlation effect which stand in t he next order of importance.\nThe quasiparticle model of the electron gas in Eq. (4) leads to the line arized stationary\nsolution under conditions ψ0= 1,φ0= 0,µ0=µ0and separation of variable. The normalized\n4-3-2-1012301234\nkkp(a)E\n\u00002Ep\n\u0001PlasmonEnergyBand,\u0002=0,0.4\n151617181920212205101520253035\nLogn0cm\n-3(b)lp\n\u0000nm\n\u0001Plasmonlength\n15161718192021220.00.51.01.52.02.53.0\nLogn0cm\n-3(c)Ep\n\u0000eV\n\u0001PlasmonEnergy\n1012141618202201234\nLogn0cm\n-3(d)vp/c(10-3)PlasmonSpeed\nFIG. 1. (a) Energy dispersion of undamped (thin solid curve) , damped (thick curve) and free electron\n(dashed curve) excitations. (b) Variation of plasmon lengt h versus free electron number density (c)\nVariation of plasmon energy versus free electron number den sity (d) Variation of plasmon speed versus\nfree electron number density.\nsystem follows\ni¯hdϕ(t)\ndt=ǫϕ(t), (5a)\n∆Ψ(r)+(κ·∇)Ψ(r)+Φ(r)+2EΨ(r) = 0, (5b)\n∆Φ(r)+(κ·∇)Φ(r)−Ψ(r) = 0. (5c)\nin which N(r,t) =ψ(r)ϕ(t) with Ψ( r) =ψ(r)/√n0and Φ(r) =eφ(r)eiθ/2Epare normalized\n5quantities with Ep=/radicalbig\n4πe2n0/mbeing the plasmon energy. Note that upon the linearization\nthephaseofwavefunction Ψislost, hence, theelectrostaticpote ntialinthissystemisconsidered\nimaginary in order to compensate this effect. Here, the normalized e nergy parameter E=\n(ǫ−µ0)/2Epdenotes the scaled total kinetic energy of the collective excitation s withǫ=N/summationtext\nj=1ǫj\nbeing the energy eigenvalue of collective excitations with ǫjbeing the energy of j-th electron in\nthe system. The normalization scheme leads to the space and time be ing also normalized with\nrespect to the plasmon length lp= 1/kpwithkp=/radicalbig\n2mEp/¯hbeing the plasmon wavenumber\nand the inverse plasmon frequency, ωp=Ep/¯h. Being in the plasmon unit scale requires all\nspeeds to be normalized with respect to plasmon speed vp= ¯hkp/mand temperatures with\nthat of the plasmon Tp=Ep/kB. The time dependent of stationary quasiparticle excitations is\nreadily found to be ϕ(t) = exp(−iωt) withω=ǫ/¯h.\nFourier analysis of the system (5) leads to the generalized quasipar ticle energy band of the\nformE= (k2+κ2)/2 + 1/2(k2+κ2). In the absence of damping the well-known dispersion\nof collective excitation is retained which can also be written in a more us eful form of E=\nk2\n1/2 +k2\n2/2 wherek1=/radicalbig\nE−√\nE2−1 andk2=/radicalbig\nE−√\nE2+1 denote the two distinct\nscale lengths known as the generalized de-Broglies wavenumbers ch aracterizing wave-like and\nparticle-like oscillations in the system and satisfy the complementarit y relationk1k2= 1.\nFigure 1 depicts the generalized quasiparticle energy dispersion rela tion and the related\nplasmon parameter variations. Figure 1(a) shows the free electro n band as dashed curve, the\nfreequasiparticle bandwiththinsolidcurve anddampedquasiparticle bandasthickcurve. The\npresence of collective excitation energy gap of ∆ E= 2Epis apparent for quasiparticle bands.\nBoth undamped and damped quasiparticle dispersions approach tha t of the free electron in the\nhigh wavenumber limit. The novel aspect of quasiparticle model is clea rly reflected in this plot\nwhich captures both single particle and collective electron excitation s in a single frame. Figure\n1(b) shows the variations of plasmon scaling length in quasiparticle th eory in nanometer unit.\nIt varies from few tenth of a nanometer in typical metals up to tens of nanometers for electron\ngas in doped semiconductors. Figure 1(c) shows the variations of p lasmon energy as the scaling\nunit of energy in this analysis and it varies from few electron volts in ty pical metals to relatively\nsmall values in semiconductors. Finally, Fig. 1(d) shows the plasmon s peed variations with\nelectron number density. It shows typical orders of 108cm/s in typical metals as expected.\n6III. PROBABILITY CURRENT DENSITY OF DAMPED QUASIPARTICLES\nStarting from the time-dependent damped 3D Schr¨ odinger-Poiss on equation, we have [71]\n2i∂N(r,t)\n∂t=−∆N(r,t)−2(κ·∇)N(r,t)−Φ(r)N(r,t)+µN(r,t), (6a)\n∆Φ(r)+2(κ·∇)Φ(r)−N(r,t)N∗(r,t) = 0, (6b)\nwhere the factor 2 again appears in (6) because the time is normalize d to 1/(2ωp). The conti-\nnuity equation gives\n∂n\n∂t=∂N(r,t)N∗(r,t)\n∂t=N(r,t)∂N∗(r,t)\n∂t+N∗(r,t)∂N(r,t)\n∂t, (7)\nwhere, the algebraic manipulation of time-dependent Schr¨ odinger -Poisson equation leads to\n2iN∗∂N\n∂t=−N∗∆N −N∗(κ·∇)N −N∗ΦN+N∗µN (8a)\n2iN∂N∗\n∂t=−N∆N∗−N(κ·∇)N∗−NΦN∗+NµN∗. (8b)\nCombining the relations (7) and (8), we find\n−∂n\n∂t=i\n2(N∆N∗−N∗∆N)+i[N(κ·∇)N∗−N∗(κ·∇)N]−i\n2NN∗[Φ−Φ∗].(9)\nFrom the Poisson’s relation, we obtain\nNN∗(Φ−Φ∗) = (Φ∆Φ∗−Φ∗∆Φ)+2i[Φ(κ·∇)Φ∗−Φ∗(κ·∇)Φ], (10)\nNow considering the generalized continuity equation ∂n/∂t+∇ ·Jn=Sn, where, JnandSn\ndenote the probability current density and the corresponding sou rce term, we find that\nJn(r,t) =i\n2[N(r,t)∇N∗(r,t)−N∗(r,t)∇N(r,t)]−i\n2[Φ(r)∇Φ∗(r)−Φ∗(r)∇Φ(r)].(11)\nand\nSn(r,t) =−i[N(κ·∇)N∗−N∗(κ·∇)N]+i[Φ(κ·∇)Φ∗−Φ∗(∇·κ)Φ].(12)\nNote that the source term vanishes for the undamped excitations .\nIV. GENERALIZED ENERGY DENSITY OF COLLECTIVE EXCITATIONS\nIn order to obtain the energy density, we use the standard definit ion for kinetic energy\nexpected value of /an}b∇acketle{tǫ/an}b∇acket∇i}ht=/integraltext\nN∗HNdτin which His the Hamiltonian of the interacting electron\nsystem with the damping effect and dτis the volume element. The Hamiltonian is given as\nH=−∆−2(κ·∇)−Φ(r)+µ. (13)\n7For quasiparticle excitations with scaled energy E=ǫ−µwe have\n/an}b∇acketle{tE/an}b∇acket∇i}ht=−/integraldisplay\nN∗(r,t)∆N(r,t)dτ−2/integraldisplay\nN∗(r,t)(κ·∇)N(r,t)dτ−/integraldisplay\nN∗(r,t)Φ(r)N(r,t)dτ.\n(14)\nIt follows that the energy density of the system is given by\nρE=−N∗(r,t)∆N(r,t)−2N∗(r,t)(∇·κ)N(r,t)−N∗(r,t)Φ(r)N(r,t).(15)\nFrom the damped Poisson’s relation we have\nΦ∆Φ∗+2Φ(κ·∇)Φ∗−N∗ΦN= 0, (16)\nwhich consequently leads to\nρE=−N∗∆N −2N∗(κ·∇)N −Φ∆Φ∗+2Φ(κ·∇)Φ∗. (17)\nNowusingtheidentities N∗∆N=∇·(N∗∇N)−∇N∗·∇NandΦ∆Φ∗=∇·(Φ∇Φ∗)−∇Φ∗·∇Φ\nwe arrive at the following expression for the energy density\nρE=∇N∗·∇N+∇Φ∗·∇Φ−∇·(N∗∇N)−∇·(Φ∇Φ∗)−2N∗(κ·∇)N −2Φ(κ·∇)Φ∗.(18)\nThe first term in (18) is the normal kinetic energy density [70]. Howev er, the second term\nrepresents the electrostatic potential energy density of the sy stem since E=−∇Φ represents\nthe normalized local electric field in the electron gas. The remaining te rms in (18) are cast in\nterms of the ℜ(real) and ℑ(imaginary) components given as follows\nℜ[N∗(κ·∇)N] =1\n2[N∗(κ·∇)N+N(κ·∇)N∗], (19a)\nℑ[N∗(κ·∇)N] =i\n2[N(κ·∇)N∗−N∗(κ·∇)N], (19b)\nℜ[Φ(κ·∇)Φ∗] =1\n2[Φ(κ·∇)Φ∗+Φ∗(κ·∇)Φ], (19c)\nℑ[Φ(κ·∇)Φ∗] =i\n2[Φ∗(κ·∇)Φ−Φ(κ·∇)Φ∗]. (19d)\nThis results in a compact form\nρE=∇N∗·∇N+∇Φ∗·∇Φ+∇·(Jρ−iJn)−(Sρ−iSn). (20)\nwhereJρ=−1\n2∇ρis the quantum diffusion current density with ρ=NN∗+ ΦΦ∗being the\ngeneralizeddensity(energy-density) ofthesystem. Thecontinu ityequationfordiffusioncurrent\n8is∂ρ/∂t+∇·Jρ=Sρwith the diffusion source term given as Sρ= (κ·∇)ρ. The final form of\nthe energy density is given as\nρE= (∇N∗·∇N+∇Φ∗·∇Φ)−∂ρ\n∂t+i∂n\n∂t, (21)\nwheren=NN∗is the normal density. Note that the appearance of imaginary term in the\ngeneralizedenergydensityrelation(21)representstheexistenc e ofdampingorgrowingquantum\ninstability. Equation (21) can be regarded as the generalized form o f the equation given in [70].\nV. ONE-DIMENSIONAL FREE QUASIPARTICLE EXCITATIONS\nWenowapplythefindingsoftheproceedingsectiontothefreequas iparticleexcitationsinthe\narbitrary degenerate electron gas. The one-dimensional solution of typeN= Ψ(x)exp(−iωt)\n(ω=ǫ/¯h) to (6) with κ= 0 and initial values Ψ′\n0= Φ′\n0= 0 reads [71]\n\nΦ(x)\nΨ(x)\n=Aρ\n2α\nΨ0+k2\n2Φ0−(Ψ0+k2\n1Φ0)\n−(Φ0+k2\n1Ψ0) Φ0+k2\n2Ψ0\n\nexp(ik1x)\nexp(ik2x)\n, (22)\nwhereAρis the normalization factor to ensure the consistency with the equilib rium state,\nk1=√\nE−α,k2=√\nE+αwithα=√\nE2−1 admitting a complementarity-like relation\nbetween wave-like and particle-like length scales ( k1k2= 1). The initial values are set Ψ 0= 1\nand Φ 0= 0 throughout the analysis, as is expected for positions far from p erturbations. The\nfree quasiparticle theory is useful in modeling of linear collective excit ations in a dense electron\nbeam or other unbounded electron fluid. The excitations which may b e caused by external\nstimuli such as electromagnetic interactions. In this case the part icle, potential and generalized\ndensities are given respectively, as\nρψ=A2\nρ\n4α2/braceleftbig\nk4\n1+k4\n2−2cos[(k1−k2)x]/bracerightbig\n, (23a)\nρφ=A2\nρ\nα2sin/bracketleftbigg1\n2(k1−k2)x/bracketrightbigg2\n, (23b)\nρ=ρψ+ρφ=A2\nρ\n4α2/braceleftbig\n2+k4\n1+k4\n2−4cos[(k1−k2)x]/bracerightbig\n, (23c)\nwhere the normalization factor Aρ(E) = 2α//radicalbig\n2+k4\n1+k4\n2equals the invariant quantity of the\nsystem,ρψ−ρφ= 1−1/E2, for given quasiparticle orbital energy, E. This must be compared\nto the case of free electron wavefunction which is nonnormalizable.\n9051015202530-1.0-0.50.00.51.0\nx\u0003lp\n\u0004\n(a)Re(Ψ),Re(Φ)E=1.5\n051015202530-1.0-0.50.00.51.0\nxlp\n(b)Im(Ψ),Im(Φ)E=1.5\n0510152025300.00.51.01.52.0\nxlp\n(c)ρψ,ρϕ,ρParticle-FieldDensities,E =1.5\n0510152025300.00.51.01.52.0\nxlp\n(d)ρGeneralizedDensity,E =1.1,1.5,3\nFIG. 2. (a) Real part of free quasiparticle wavefunction (th in curve) and electrostatic energy (thick\ncurve) profiles. (b) Imaginary part of free quasiparticle wa vefunction (thin curve) and electrostatic\nenergy (thick curve) profiles. (c) The generalized density d ue to particle (thin solid curve), field\n(dashed curve) and total (thick curve). (d) The total genera lized density for different quasiparticle\nenergy orbital. Increase in thickness in (d) reflects the inc rease in varied parameter in this plot.\nFigure 2 depicts normalized wavefucntion (thin curves) and electro static energy (thick\ncurves) profiles of free quasiparticle excitations along with genera lized density variations. Fig-\nure 2(a) and 2(b) shows the real and imaginary parts at the quasip article orbital E= 1.5\nindicating the dual-tone nature of oscillations. The generalized dens ityρ= Ψ∗Ψ+Φ∗Φ (thick\ncurve), with ρψ= Ψ∗Ψ (thin curve) and ρφ= Φ∗Φ (dashed curve) profiles is depicted in\n10051015202530012345\nxlp\n(a)Eψ,Eϕ,ΕρEnergyDensity,E =1.5\n05101520253001234\nxlp\n(b)ΕρEnergyDensity,E =1.1,1.2,1.3\n02468100.00.20.40.60.81.01.21.4\nE2Ep\n(c)ΕlEnergyperUnitLength,L =30\n0510152025300.00.20.40.60.81.01.21.4\nllp\n(d)ΕlEnergyperUnitLength,E =1.5\nFIG. 3. (a) Energy density of free quasiparticle excitation s due to kinetic (thin solid curve), potential\n(dashedcurve) andtotal (thick curve). (b)Total energyden sityfordifferentorbital energies withcurve\nthickness increasing with increase in the energy. (c) The ex pected energy per unit length variation\nwith quasiparticle orbital energy. (d) The expected energy per unit length varied with length.\nFig. 2(c). Note that the quantity Ψ∗Ψ−Φ∗Φ is space invariant in the system. The general-\nized density oscillates around the equilibrium value of ρ0= Ψ2\n0+ Φ2\n0. Figure 2(d) shows the\ngeneralized density profiles at different quasiparticle orbital. The am plitude of oscillations is\nincreased/decreased as its wavelength is increased/decreased w hich is a charcteristic behavior\nof the quasiparticle dispersion, shown in Fig. 1(a).\nThe energy densities corresponding to kinetic, potential and gene ralized densities are respec-\n11tively given as\nEψ=dΨ∗\ndxdΨ\ndx=A2\nρ\n4α2/braceleftbig\nk6\n1+k6\n2−2cos[(k1−k2)x]/bracerightbig\n, (24a)\nEφ=dΦ∗\ndxdΦ\ndx=A2\nρ\n4α2/braceleftbig\nk2\n1+k2\n2−2cos[(k1−k2)x]/bracerightbig\n, (24b)\nEρ=Eψ+Eφ=A2\nρ\n2α2/braceleftbig\nk6\n1+k6\n2−2cos[(k1−k2)x]/bracerightbig\n. (24c)\nNote thatEρ= 2Eψ.\nFigure 3 depicts the variations in energy density of free quasipartic le excitations in the\nelectron gas at given energy orbital. Figure 3(a) reveals that the o scillation wavelength in\nkinetic (Eψ) and potential ( Eφ) energy densities is the same, so that, the oscillate in-phase\ntogether. It is because the kinetic energy comes from the particle aspect of the gas where as\nthe potential energy from the charge which are indeed attached t o each other in the electron\ngas. Figure 3(b) shows that increase in orbital energy leads to incr ease the level but decrease\nin wavelength of the total energy density ( Eρ=Eψ+Eφ). The expected quasiparticle energy\nper unit length is shown in terms of orbital energy in Fig. 3(c) for L= 30 in plasmon length\nunit. It is shown that this quantity increase with orbital energy, mo notonically. On the other\nhand, Fig. 3(d) shows that for the orbital energy E= 1.5 (in twice plasmon energy unit) the\nexpected energy per unit length decreases with increase in length L.\nOn the other hand, the diffusion current densities can be written as\nJρψ=Jρφ=A2\nρ\n4α2(k2−k1)sin[(k1−k2)x], (25)\nand the total diffusion current reads Jρ= 2Jρψ. Moreover, the diffusion current flux, given that\nthe diffusion source is zero, reads\n−dJρ\ndx=−A2\nρ\n2α2(k1−k2)2cos[(k1−k2)x], (26)\nwhich indicates the local density flow in or out of the point.\nThe space variations in diffusion current densities are depicted in Fig. 4. Figure 4(a)\nreveals that the current densities at orbital E= 1.5, contributed from kinetic ( Jρψ) and po-\ntential (Jρφ) energies, amount to the same value in all space. It is remarked fro m Fig. 4(b)\nthat magnitude/wavelength of the diffusion current densities incre ase/decrease with the orbital\nquasiparticle energy. The probability current density variations ar e shown in Figs. 4(c) and\n4(d). It is shown in Fig. 4(c) that contributions from kinetic ( Jnψ) and potential ( Jnφ) vary\nout of phase and amounts to a constant value of Jn=Jnψ+Jnφ= (k5\n1+k5\n2−k1−k2)/4E2\n12051015202530-0.4-0.20.00.20.4\nxlp\n(a)Jρψ,Jρϕ,JρDiffusionCurrentDensities,E =1.5\n051015202530-0.4-0.20.00.20.4\nxlp\n(b)JρDiffusionCurrentDensity,E =1.1,1.2,1.3\n051015202530-0.50.00.51.01.5\nxlp\n(c)Jnψ,Jnϕ,JnProbabilityCurrentDensity,E =1.5\n05101520253002468\nE2Ep\n(d)JnProbabilityCurrentDensity\nFIG.4. (a) Diffusioncurrentdensitiesforfreequasiparticl eexcitations duetokinetic(thinsolidcurve),\npotential (dashedcurve) andtotal (thick curve). (b) Total diffusioncurrentdensity for different orbital\nenergies with curve thickness increasing with increase in t he energy. (c) probability current densities\nfor free quasiparticle excitations due to kinetic (thin sol id curve), potential (dashed curve) and total\n(thick curve). (d) Variation of total probability current w ith quasiparticle orbital energy.\nfor the given orbital energy E. Variation of probability current density with orbital energy is\ndepicted in Fig. 4(d) showing a monotonic increase in this quantity.\n13FIG. 5. (a) Variation of diffusion current density (dashed cur ve) and generalized density (solid curve).\n(b)Variation ofdiffusiondensityflux(dashedcurve)andgene ralizeddensity(solidcurve). (c)Diffusion\nof excited free quasiparticle state from orbital with energ yE= 1.5. (d) Diffusion of excited free\nquasiparticle state from orbital with energy E= 2.\nThe probability current densities are given as\nJnψ=A2\nρ\n4α2/braceleftbig\nk5\n1+k5\n2−(k1+k2)cos[(k1−k2)x]/bracerightbig\n, (27a)\nJnφ=A2\nρ\n4α2(k1+k2){cos[(k1−k2)x]−1}, (27b)\nand the total probability current reads Jn=A2\nρ(k5\n1+k5\n2−k1−k2)/4α2which is constant. Also,\nthe probability flux ∂n/∂tis zero for all given quasiparticle energy orbital. The energy per unit\n14length of the 1D electron gas at given orbital Eis\nEl=1\nll/integraldisplay\n0Eρdx=2/l\n2+k4\n1+k4\n2/braceleftbigg\nk6\n1+k6\n2−2sin[(k1−k2)l]\n(k1−k2)l/bracerightbigg\n. (28)\nThe time evolution the generalized density in the system is given by the diffusion-like equation\n∂ρ(x,t)/∂t= (1/2)∂2ρ(x,t)/∂x2with the initial condition ρ(x,0) given by Eq. (38)c. The time\ndependent solution then reads\nρ(x,t) = 1−4e−1\n2(k1−k2)2tcos[(k1−k2)x]. (29)\nNote thatρ(x,∞) = 1, as is dictated by the normalization value. The characteristic qu antum\ndiffusionortheperturbationdampingtimeis η= 2/(k1−k2)2= 1/(E−1)inverselyproportional\nto the quasiparticle orbital energy. This is to say that collective exc itations with higher energy\ndiffuse faster towards the equilibrium state. Notealso that thedam ping timediverges for E= 1\nwhich corresponds to quantum beating or ground-state quasipar ticle orbital where k1=k2.\nFigure 5 shows the density flux due to quantum diffusion in the electro n gas. Figure 5(a)\nshows the diffusion current density (dashed curve) and generalize d density (solid curve) in\nthe same plot. The diffusion current varies sinusoidally according to t he generalized den-\nsity distribution in the electron gas. The governing continuity equat ion for this variations is\n∂ρ(x,t)/∂t+∂Jρ(x,t)/∂x= 0. The variation in diffusion density flux ∂ρ(x,t)/∂twithρis de-\npicted in Fig. 5(b). It is clearly remarked that the flux is inward/outw ard (positive/negative)\non low/high density regions trying to retain the uniformity of the gen eralized density distribu-\ntion in the electron gas. This effect is simulated in a 3D plot in Figs. 5(c) a nd 5(d) for different\nvalues of orbital energy. They clearly depict the quantum diffusion e ffect on collective excited\nstates in the system. The later effect predicts a unstable charact er of all excited quasiparticle\nstates except the ground state level for which the decay time bec omes infinite. It is therefore\nessential for a long lived collective excitations to reside as close to th e quantum beating point\nfor which the exciting photon energy is ¯ hω≃2Ep.\nVI. COLLECTIVE ELECTRON EXCITATIONS IN A 1D BOX\nWe now turn into the solution of undamped quasiparticle excitations in one-dimensional\ninfinite square-well of length L. The solution, with previously taken initial values, is given by\n1501020304050-2-1012\nxlp\n(a)Ψ(x)Wavefunction,n =1,L=50\n01020304050-2-1012\nxlp\n(b)Ψ(x)WavefunctionDensity,n =2,L=50\n01020304050-2-1012\nxlp\n(c)Φ(x)ElectrostaticPotential,n =1,L=50\n01020304050-2-1012\nxlp\n(d)Φ(x)ElectrostaticPotential,n =2,L=50\nFIG. 6. (a) Wavefunction of quasiparticle in a box at ground s tate. (b) Wavefunction of quasiparticle\nin a box at first excited state. (c) Electrostatic energy of qu asiparticle in a box at ground state. (d)\nElectrostatic energy of quasiparticle in a box at first excit ed state.\n[72]\n\nΦ(x)\nΨ(x)\n=Aρ\n2α\nk2\n2−k2\n1\n−1 1\n\ncos(k1x)\ncos(k2x)\n, (30)\nwhereAis a normalization factor satisfying the relation ρ(x,t)|t→∞= 1 given as\nAρ(L,n) =2√\n2α/radicalbig\n2+k4\n1+k4\n2. (31)\n16The quasiparticle orbital in this case are quantized with the energy e igenvalues given by En=\n(L2+2n2π2)/L2.\nThe generalized density profiles for L= 40 and n= 1,2 is depicted in Fig. 7. The\ncontribution from particle probability ( ρψ) is shown in Fig. 7(a). This contribution clearly\nvanished at the wall locations. The contribution from charge proba bility (ρφ) is depicted in\nFig. 7(b) showing a sinilar profile as in Fig. 7(a), but, with non-vanishin g values at the walls.\nThe total generalized density ρ=ρψ+ρφis depicted for n= 1,2 in Figs. 7(c) and 7(d). These\nplots clearly indicate the characteristic features of quantum inter ference effect due to single\nelectron and collective oscillations in the electron gas.\nFigure 6 depicts the normalized wavefunction and corresponding ele ctrostatic energy distri-\nbutions for a quasiparticle state in an infinite square well of length L= 50 in the quantized\nlevelsn= 1,2. It shows the characteristic wavefucntion showing variation bot h due to single\nelectron as well as collective oscillations in the box. note that in this ca se, as compared to\nthe elementary problem of particle in a box, the fine structure dens ity oscillation is due to the\nsingle particle excitations. Also note that while the wavefucntion van ishes at the potential wall\nthe electrostatic energy remains finite at this locations due to finite probability of charge close\nto the confining walls.\nThe generalized density of quasiparticles in this case reads\nρ(L,n) =A2\n4α2/bracketleftbig/parenleftbig\n1+k4\n2/parenrightbig\ncos2(k1x)−4cos(k1x)cos(k2x)+/parenleftbig\n1+k4\n1/parenrightbig\ncos2(k2x)/bracketrightbig\n.(32)\nNote that the generalized density does not vanish at the boundary of potential well, since, it\nincludes the potential squared values by the definition.\nOn the other hand, the energy density distributions due to particle and field presence are\ngiven as\nEψ=dΨ∗\ndxdΨ\ndx=A2\nρ\n4α2[k1sin(k1x)−k2sin(k2x)]2, (33a)\nEφ=dΦ∗\ndxdΦ\ndx=A2\nρ\n4α2[k2sin(k1x)−k1sin(k2x)]2, (33b)\nEρ=Eψ+Eφ=A2\nρ\n4α2/braceleftbig\n[k1sin(k1x)−k2sin(k2x)]2+[k2sin(k1x)−k1sin(k2x)]2/bracerightbig\n.(33c)\nAlso the total energy inside the potential well is given as the orbital energy, width of the\n170 10 20 30 400.00.51.01.52.0\nxlp\n(a)ρψParticleDensity,n=1,L=40\n0 10 20 30 400.00.51.01.52.0\nxlp\n(b)ρϕFieldDensity,n=1,L=40\n0 10 20 30 4001234\nxlp\n(c)ρGeneralizedDensity,n=1,L=40\n0 10 20 30 4001234\nxlp\n(d)ρGeneralizedDensity,n=2,L=40\nFIG. 7. (a) Generalized density due to particle of quasipart icle in a box at ground state. (b) General-\nized density due to charge of quasiparticle in a box at first ex cited state. (c) Total generalized density\nof quasiparticle in a box at ground state. (d) Total generali zed density of quasiparticle in a box at\nfirst excited state.\nconfining box and the quantum number, as\nEl=L/integraldisplay\n0Eρdx=A2\nρ\n16α2/braceleftbig\nk1/parenleftbig\n1+k4\n2/parenrightbig\n[2k1L−sin(2k1L)]+k2/parenleftbig\n1+k4\n1/parenrightbig\n[2k2L−sin(2k2L)] (34a)\n+8[k1sin(2k2L)−k2sin(2k1L)]\nk2\n1−k2\n2. (34b)\nFigure 8(a) shows the energy density distribution of quasiparticle in a box. While the\n18contributions from kinetic (thin solid curve) and potential (dashed curve) energies amount\napproximately to the same level, there are variances when approac hing the walls. Close to the\nwalls there is an out of phase oscillations between these two compone nts. The total energy\ndensity (thick curve) shows quantized distribution of the energy in side the box which is due to\nthe known interference effect. The total energy density in the fir st excited quasiparticle state\nis depicted in Fig. 8(b). The normalized energy per unit length of box is shown for different\norbitalenergyinFig. 8(c). It showsalmostlinear increaseforgiven quantumnumber asthebox\nlength increases which is due to increase in the electron number dens ity. The diffusion current\ndensities are depicted in Fig. 8(d) showing quantized state in this qua ntity and vanishing at\nthe wall positions. The maximum diffusion current density appears to be in the middle of the\nbox.\nMoreover, while the probability current density of stationary stat es is zero the diffusion\ncurrent densities do not vanish in the box and are given as\nJρψ=A2\nρ\n4α2[cos(k1x)−cos(k2x)][k1sin(k1x)−k2sin(k2x)], (35a)\nJρφ=A2\nρ\n4α2/bracketleftbig\nk2\n1cos(k2x)−k2\n2cos(k1x)/bracketrightbig\n[k2sin(k1x)−k1sin(k2x)], (35b)\nJρ=A2\nρ\n8α2/bracketleftbig\nk1/parenleftbig\n1+k4\n2/parenrightbig\nsin(2k1x)+k2/parenleftbig\n1+k4\n1/parenrightbig\nsin(2k2x) (35c)\n−4k1cos(k2x)sin(k1x)−4k2cos(k1x)sin(k2x)]. (35d)\nThe temporal diffusion of the excited states is given by ∂ρ(x,t)/∂t= (1/2)∂2ρ(x,t)/∂x2in\nthe absence of the source term. The solution to this equation read s\nρ(x,t) =A2\nρ\n8α2/braceleftBig\n2+k4\n1+k4\n2+e−2k2\n1t/parenleftbig\n1+k4\n2/parenrightbig\ncos(2k1x) +e−2k2\n2t/parenleftbig\n1+k4\n1/parenrightbig\ncos(2k2x) (36a)\n−4e−1\n2(k1−k2)2tcos[(k1−k2)x]−4e−1\n2(k1+k2)2tcos[(k1+k2)x]/bracerightBig\n, (36b)\nfor the initial condition ρ(x,0) given by Eqs. (31). Note that the diffusion in this case is\nnot monotonic but the excited states diffuse to equilibrium value ρ(x,∞) = 1 similar to the\ncase of a free quasiparticle excitation. Note also that, first the ra pid density oscillations due\nto single-electron excitations diffuse and then after relatively longe r time the equilibrium is\nreached.\nThe diffusion flux along with the corresponding generalized density pr ofile is shown in Figs.\n9(a) and 9(b) for given values of box and quantum number. They re veal maximum outward\ndiffusion at density hills and minimum diffusion at density valleys, as expec ted. The diffusion\n19profiles for different times is shown in Figs. 9(c) and 9(d). It is beaut ifully shown in Fig. 9(c)\nthatat thegroundstatequasiparticle level first theexcitations d ueto singleelectron excitations\ndecay and then the collective density profile diffuses later. The same feature also takes place\nfor the first excited state, as shown in Fig. 9(d). The temporal dy namic of pure state diffusion\nis shown in a clear density plot view in Fig. 10.\nVII. ENERGY DENSITY OF DAMPED QUASIPARTICLES\nFinally, wewouldliketoapplythecollectivequantumdiffusiontheorytot heone-dimensional\ndamped quasiparticle excitations. This case can have quite general application in surface plas-\nmon excitations and the spill-out electron at the half-space electro n gas. The non-transient\nsolution in this case with the boundary values Ψ′\n0= Φ′\n0= 0 is [71]\n\nΦ(x)\nΨ(x)\n=e−κx\n2α\nΨ0+k2\n2Φ0−(Ψ0+k2\n1Φ0)\n−(Φ0+k2\n1Ψ0) Φ0+k2\n2Ψ0\n\n(1+iκ/β1)eiβ1x\n(1+iκ/β2)eiβ2x\n,(37)\nwhereβ1=/radicalbig\nk2\n1−κ2andβ2=/radicalbig\nk2\n2−κ2. We have taken Ψ 0= 1 and Φ 0= 0 for all\nsimulations.\nThe wavefunction (thin curves) and corresponding electrostatic (thick curves) energy dis-\ntribution in half space damped quasiparticle excitations are depicted in Figs. 11(a) and 11(b)\nshowing the characteristic decaying dual tone oscillations. The gen eralized density profiles ρψ\n(solid thin curve) ρφ(dashed) and total ρ(thick curve) are shown in Figs. 11(c). Periodic\nvariations in density profile is apparent due to the interference effe ct. Such density variations\nis related to the spill out electron density at plasmonic material surf ace.\nThe generalized density components are given as\nρψ=e−2κx\n4α2β2\n1β2\n2/braceleftbig\nk4\n1β2\n2/parenleftbig\nβ2\n1+κ2/parenrightbig\n+k4\n2β2\n1/parenleftbig\nβ2\n2+κ2/parenrightbig\n(38a)\n+2β1β2/braceleftbig\nκ(β2−β1)sin[(β1−β2)x]−/parenleftbig\nβ1β2+κ2/parenrightbig\ncos[(β1−β2)x]/bracerightbig/bracerightbig\n(38b)\nρφ=e−2κx\n4α2β2\n1β2\n2/braceleftbig\n2β2\n1β2\n2+κ2/parenleftbig\nβ2\n1+β2\n2/parenrightbig\n(38c)\n−2β1β2/braceleftbig/parenleftbig\nβ1β2+κ2/parenrightbig\ncos[(β1−β2)x]+κ(β1−β2)sin[(β1−β2)x]/bracerightbig/bracerightbig\n(38d)\nρ=e−2κx\n4α2β2\n1β2\n2/braceleftbig\nβ2\n2κ2/parenleftbig\nk4\n1+1/parenrightbig\n+β2\n1β2\n2/parenleftbig\n2+k4\n1+k4\n2/parenrightbig\n+κ2β2\n1β2\n2/parenleftbig\nk4\n2+1/parenrightbig\n(38e)\n+4β1β2/braceleftbig\nκ(β2−β1)sin[(β1−β2)x]−/parenleftbig\nβ1β2+κ2/parenrightbig\ncos[(β1−β2)x]/bracerightbig/bracerightbig\n.(38f)\n20The energy densities for damped quasiparticle excitation are given a s\nEψ=e−2κx\n4α2β2\n1β2\n2/bracketleftbig\nk2\n1β2(β1−iκ)2e−iβ1x−k2\n2β1(β2−iκ)2e−iβ2x/bracketrightbig\n(39a)\n×/bracketleftbig\nk2\n1β2(β1+iκ)2eiβ1x−k2\n2β1(β2+iκ)2eiβ2x/bracketrightbig\n, (39b)\nEφ=e−2κx\n4α2β2\n1β2\n2/bracketleftbig\nβ2(β1−iκ)2e−iβ1x−β1(β2−iκ)2e−iβ2x/bracketrightbig\n(39c)\n×/bracketleftbig\nβ2(β1+iκ)2eiβ1x−β1(β2+iκ)2eiβ2x/bracketrightbig\n. (39d)\nThe long expression for the total energy density Eρ=Eψ+Eφis avoided here. Note that the\ntotal energy density has imaginary component due to damping. This gives rise to the imaginary\nenergy values which is the characteristic of damping/growing instab ility of the excitation. The\nimaginary part of the energy density reads\nEρi=κe−2κx\n2α2β1β2/bracketleftbig\nβ1β2/parenleftbig\nβ1+β2−k4\n1β1−k4\n2β2/parenrightbig\n+κ2/parenleftbig\nβ1+β2−k4\n1β2−k4\n2β1/parenrightbig/bracketrightbig\n.(40)\nThis leads to the total imaginary energy component as\nEi=−β1(k4\n2−1)(β2\n2+κ2)+β2(k4\n1−1)(β2\n1+κ2)\n4α2β1β2, (41)\nwhich indicates a negative constant value characteristics of an exp onential decay of the form\nexp(−Eit) with the rate given by (41). The existence of damping in this case or iginates from\nthe fact that Jndoes not vanishes in this case. The components of diffusion current densities\nare given below\nVariations in the energy density distribution of damped quasiparticle excitations is shown\nin Fig. 12. The energy density of damped excitations is imaginary due t o the non-vanishing\nnature of probability current in this case, e.g. see Eq. (20). The to tal energy density (thick\ncurve) shows a pronounced valley close to the boundary ( x= 0). Figure 12(b) shows the total\nenergy density for different orbital energy revealing the fact tha t with decrease of the energy\nthe energy density valley gets deeper. This is a manifestation of att ractive force of the half-\nspace electron gas very close to the surface which is quite similar to t he well known Casimir\neffect. The imaginary part of the energy density is shown to be nega tive increasing outward\nthe boundary by Fig. 12(c). The real (solid curve) and imaginary (d ashed) parts of the total\nenergyEtat the electron spill out region as varied with the damping parameter is shown in\nFig. 12(d). While the real part decreases with the decrease in the d amping parameter, which\n21is also related to the collective quantum electron tunneling [73], the ima ginary part is constant\nnegative which indicates a damping effect of the energy-density osc illations with the rate Ei.\nJρψ=e−2κx\n4α2β2\n1β2\n2/braceleftbig\nκk4\n1β2\n2/parenleftbig\nβ2\n1+κ2/parenrightbig\n+κk4\n2β2\n1/parenleftbig\nβ2\n2+κ2/parenrightbig\n(42a)\n+β1β2/braceleftbig\nκ/parenleftbig\nβ2\n1−4β1β2+β2\n2−2κ2/parenrightbig\ncos[(β1−β2)x]−(β1−β2)/parenleftbig\nβ1β2+3κ2/parenrightbig\nsin[(β1−β2)x]/bracerightbig/bracerightbig\n(42b)\nJρφ=e−2κx\n4α2β2\n1β2\n2/braceleftbig\n2κβ2\n1β2\n2+κ3/parenleftbig\nβ2\n1+β2\n2/parenrightbig\n(42c)\n+κβ1β2/parenleftbig\nβ2\n1−4β1β2+β2\n2−2κ2/parenrightbig\ncos[(β1−β2)x]−β1β2(β1−β2)/parenleftbig\nβ1β2+3κ2/parenrightbig\nsin[(β1−β2)x]/bracerightbig\n(42d)\nJρ=e−2κx\n4α2β2\n1β2\n2/braceleftbig\nβ2\n2κ3/parenleftbig\n1+k4\n1/parenrightbig\n+κβ2\n1β2\n2/bracketleftbig/parenleftbig\n2+k4\n1+k4\n2/parenrightbig\n+κ2k4\n2/bracketrightbig\n(42e)\n2β1β2/braceleftbig\nκ/parenleftbig\nβ2\n1−4β1β2+β2\n2−2κ2/parenrightbig\ncos[(β1−β2)x]−(β1−β2)/parenleftbig\nβ1β2+3κ2/parenrightbig\nsin[(β1−β2)x]/bracerightbig/bracerightbig\n.\n(42f)\nThe corresponding sources are Sρψ=−2κJρψ,Sρφ=−2κJρφandSρ=−2κJρ. The probability\ncurrent densities are given as follows\nJnψ=e−2κx(β1+β2)\n4α2β1β2/braceleftbig/parenleftbig\nβ1β2+κ2/parenrightbig\n{cos[x(β1−β2)]−1}+κ(β1−β2)sin[(β1−β2)x]/bracerightbig\n,\n(43a)\nJnφ=e−2κx\n4α2β1β2/braceleftbig\nβ1β2/parenleftbig\nβ1−k4\n1β1+β2−k4\n2β2/parenrightbig\n+κ2/parenleftbig\nβ1−k4\n2β1+β2−k4\n1β2/parenrightbig/bracerightbig\n, (43b)\nJn=e−2κx\n4α2β1β2{(β1+β2)κsin[(β1−β2)x/2]} (43c)\n×/braceleftbigg\nκ(β2−β1)cos/bracketleftbigg(β1−β2)\n2x/bracketrightbigg\n+/parenleftbig\nβ1β2+κ2/parenrightbig\nsin/bracketleftbigg(β1−β2)\n2x/bracketrightbigg/bracerightbigg\n. (43d)\nThe corresponding sources are Snψ=−2κJnψ,Snφ=−2κJnφandSn=−2κJn. The temporal\ndiffusion of the excited states is given by ∂ρ(x,t)/∂t= (1/2)∂2ρ(x,t)/∂x2+κ∂ρ/∂xin the\npresence of the source term Sρ. The solution to this equation can not be obtained analytically\nand the numerical analysis of this problem is left for future investiga tion.\nVariations of diffusion and probability current densities due to kinetic (thin solid curve),\npotential (dashed curve) and total (thick curve) is shown in Figs. 13(a) and 13(b). While\nthe kinetic and potential diffusion current components are closely v arying their probability\ncounterparts which appear to contribute to imaginary energy den sity appear out of phase. The\n22corresponding sourcesfordiffusionandprobabilitycurrent aresh ownrespectivent inFigs. 13(c)\nand 13(d).\nVIII. CONCLUDING REMARKS\nIn this paper we explored new features of collective quantum excita tions in the framework of\nmultistream quasiparticle excitations. Wededuced ageneralized pro bability current formula for\ndamped collective quantum excitations and studied the energy dens ity of arbitrary degenerate\nelectron gas. 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Plasmas 29, 082112 (2022); doi.org/10.1063/5.0102151\n2705101520253001234\nxlp\n(a)Εψ,Εϕ,ΕρEnergyDensity,n =1,L=30\n05101520253001234\nxlp\n(b)ΕρEnergyDensity,n =2,L=30\n051015202530051015202530\nLlp\n(c)ΕtTotalEnergy,n =1,2,3\n051015202530-2-1012\nxlp\n(d)Jψ,Jϕ,JρDiffusionCurrentDensity,n =1,L=30\nFIG. 8. (a) Energy density of quasiparticle in a box kinetic ( thin solid curve), potential (dashed\ncurve) energies and total (thick curve). (b) Total energy de nsity of quasiparticle in a box at first\nexcited state. (c) Variation of normalized expected energy inside the box for quantum number values\nn= 1,2,3 with resoect to the box length. Increase in curve thickness indicates the increase in varied\nparameter above the panel. (d) Variation of diffusion current density of quasiparticle in a box due to\nkinetic (thin solid curve), potential (dashed curve) energ ies and total diffusion current density (thick\ncurve).\n280 5 10 15 20-4-202\n4\nxlp\n(a)ρ,ρtDiffusionFlux,L=20,n=1\n0 5 10 15 20-4-2024\nxlp\n(b)ρ,ρtDiffusionFlux,L =20,n=2\n05101520253001234\nxlp\n(c)ρDiffusion,L =30,n=1,t=0,0.5,1,10\n05101520253001234\nxlp\n(d)ρDiffusion,L =30,n=2,t=0,0.5,1,10\nFIG. 9. (a) Generalized density (solid curve) and diffusion de nsity flux (dashed curve) for quasipar-\nticle in a box at ground state level. (b) Generalized density (solid curve) and diffusion density flux\n(dashed curve) for quasiparticle in a box at first excited sta te. (c) Variation of generalized density for\nquasiparticle excitation in a box at ground state level with time. (d) Variation of generalized density\nfor quasiparticle excitation in a box at first excited state w ith time. The increase in curve thickness\nindicates the increase in time in plots (c) and (d).\n29FIG. 10. (a) The density plot of quasiparticle in a box genera lized density at ground state level at\ninitial state. (b) Diffusion of the initial state at t= 0.5. (c) Diffusion of the initial state at t= 1. (d)\nDiffusion of the initial state at t= 10. The time in the unit of inverse of twice the plasmon frequ ency.\n30051015202530-1.0-0.50.00.51.0\nxlp\n(a)Re(Ψ),Re(Φ)E=1.5,κ=0.1\n051015202530-1.0-0.50.00.51.0\nxlp\n(b)Im(Ψ),Im(Φ)E=1.5,0.1\n0510152025300.00.51.01.52.0\nxlp\n(c)ρψ,ρϕ,ρParticle-FieldDensities,E =1.5,κ=0.1\n05101520253001234\nxlp\n(d)ρGeneralizedDensity,E =1.1,1.5,2, κ=0.1\nFIG. 11. (a) Real part of damped quasiparticle wavefunction (thin curve) and corresponding elec-\ntrostatic energy (thick curve) distribution. (b) Imaginar y part of damped quasiparticle wavefunction\n(thin curve) and electrostatic energy (thick curve) profile s. (c) The generalized density due to particle\n(thin solid curve), field (dashed curve) and total (thick cur ve). (d) The total generalized density\nfor different quasiparticle energy orbital. Increase in thic kness in (d) reflects the increase in varied\nparameter in this plot.\n3105101520253001234\nxlp\n(a)Eψ,Eϕ,Re(Eρ)EnergyDensity,E =1.1\n05101520253001234\nxlp\n(b)Re(Ερ)EnergyDensity,E =1.1,1.2,1.3, κ=0.1\n051015202530-0.5-0.4-0.3-0.2-0.10.0\nxlp\n(c)Im(Ερ)EnergyDensity,E =1.1,2,3, κ=0.1\n0.00.10.20.30.40.50.6-5051015202530\nκkp\n(d)Re(E),Im(E)Energy,E =1.1\nFIG. 12. (a) Energy density of damped quasiparticle excitat ions due to kinetic (thin solid curve),\npotential (dashed curve) and total (thick curve). (b) Total real energy density for different orbital\nenergiesofdampedquasiparticleexcitations withcurveth icknessincreasingwithincreaseintheenergy.\n(c) Total imaginary energy density for different orbital ener gies of damped quasiparticle excitations\nwith curve thickness increasing with increase in the energy . (d) The real and imaginary expected\nenergy of damped quasiparticle excitation with respected t o the damping parameter.\n32051015202530-0.4-0.20.00.20.4\nxlp\n(a)Jψ,Jϕ,JρDiffusionCurrentDensities,E =1.1,κ=0.1\n051015202530-2-10123\nxlp\n(b)Jψ,Jϕ,JρProbabilityCurrentDensity,E =1.1,κ=0.1\n051015202530-0.10-0.050.000.050.10\nxlp\n(c)Sρψ,Sρϕ,SρDiffusionS\u0005 \u0006 \u0007\b \t \nE=1.1,κ=0.1\n051015202530-0.6-0.4-0.20.00.20.4\nxlp\n(d)Snψ,Snϕ,Sn\nC \u0006 \u0007\u0007 \t \u000b \f S\u0005 \u0006 \u0007\b \t \nE=1.1,κ=0.1\nFIG. 13. (a) Variation of diffusion current density of damped q uasiparticle due to kinetic (thin\nsolid curve), potential (dashed curve) energies and total d iffusion current density (thick curve). (b)\nVariation of probability current density of damped quasipa rticle due to kinetic (thin solid curve),\npotential (dashed curve) energies and total diffusion curren t density (thick curve). (c) Variation of\ndiffusion source of damped quasiparticle due to kinetic (thin solid curve), potential (dashed curve)\nenergies andtotal diffusioncurrentdensity(thick curve). ( d)Variation ofprobability sourceofdamped\nquasiparticle due to kinetic (thin solid curve), potential (dashed curve) energies and total diffusion\ncurrent density (thick curve).\n33" }, { "title": "2403.01502v1.Oscillating_charged_Andreev_Bound_States_and_Their_Appearance_in_UTe__2_.pdf", "content": "arXiv:2403.01502v1 [cond-mat.supr-con] 3 Mar 2024Oscillating-charged Andreev Bound States and Their Appear ance in UTe 2\nSatoshi Ando,1,∗Shingo Kobayashi,2,∗Andreas P. Schnyder,3Yasuhiro Asano,4and Satoshi Ikegaya1,5,∗\n1Department of Applied Physics, Nagoya University, Nagoya 4 64-8603, Japan\n2RIKEN Center for Emergent Matter Science, Wako, Saitama, 35 1-0198, Japan\n3Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany\n4Department of Applied Physics, Hokkaido University, Sappo ro 060-8628, Japan\n5Institute for Advanced Research, Nagoya University, Nagoy a 464-8601, Japan\n(Dated: March 5, 2024)\nIn a superconductor with a sublattice degree of freedom, we fi nd unconventional Andreev bound\nstates whose charge density oscillates in sign between the t wo sublattices. The appearance of these\noscillating-charged Andreevboundstates is characterize d bya Zak phase, rather than aconventional\ntopological invariant. In contrast to conventional Andree v bound states, for oscillating-charged\nAndreev bound states the proportionality between the elect ron-like spectral function, the local\ndensity of states and the tunneling conductance is broken. W e examine the possible appearance of\nthese novel Andreev bound states in UTe 2and locally noncentrosymmetric superconductors.\nIntroduction .—The characterization of electronic\nstates in terms of topology has been a hot research\ntopic [1–6] in condensed matter physics. By the\nbulk-boundary correspondence, a nontrivial topolog-\nical number in the bulk implies the appearance of\ngapless states at the surface. One example is the\nthree-dimensional winding number that describes the\ntopology and surface states of the B phase of superfluid\n3He [7, 8]. Another example is the Zak phase (or Berry\nphase) [9, 10] which describes the quantum mechanical\nproperties of the electronic states in insulators and\n(semi-)metals. For instance, a quantized Zak phase of π\nindicates the topological nontriviality of one dimensional\ninversion symmetric insulators [10]. In the absence\nof inversion and mirror symmetry, however, the Zak\nphase is not quantized. Nevertheless it is a measurable\nquantity as it is related to the bulk polarization, as\ndescribed by the modern theory of polarization [11–16].\nAt the surface, the Zak phase leads to in-gap surface\nstates and charge accumulation, whereby the surface\ncharge is defined only up to an integer multiple of\n2π[12, 17–19] (see also Refs. [20, 21] for alternative\ncharacterizations of surface topology).\nSo far, the Zak phase has been mostly used in insu-\nlators and semimetals to describe their polarization and\ntopological properties. The Zak phase can also be used\nto characterize superconductors [4], whose particle-hole\nsymmetry quantizes it to zero or π. However, unquan-\ntized Zak phases have, to the best of our knowledge, not\nbeen discussed in the context of superconductors.\nIn this Letter, we show that the topology of super-\nconductors with a sublattice degree of freedom can be\ncharacterized by an unqunatized Zak phase, which by\nthe bulk-boundary correspondence leads to a novel type\nof Andreev bound states (ABSs) at the surface. Such\nsystems are described by Bogoliubov-de Gennes (BdG)\nHamiltonians that can be separated into two subsec-\ntors. While particle-hole symmetry is satisfied for the\nfull Hamiltonian, each subsector breaks it, thereby al-lowing the definition of a unquantized Zak phase for each\nsubsector. By the bulk-boundary correspondence this\nunquantized Zak phase leads to unusual surface ABS\nwhose key properties are: (i) The ABS energy is in gen-\neral non-zero, (ii) the charge density of the ABSs oscil-\nlates in sign between the two sublattices [see Figs. 1(a)-\n(c)], and (iii) the proportionality between the electron-\nlike charge density ρe, the local density of states ρ, and\nthe tunneling conductance Gis broken [see Figs. 1(d)-\n(f)]. Due to the first two properties we call these ABS\noscillating-chargedAndreev bound states(OCABS). The\nthird property is particularly remarkable, since the pro-\nportionality between ρe,ρ, andGhas been up to now\naccepted as a general feature of superconductors, as de-\nscribed by quasiclassical theory [22–24]. Finally we dis-\ncuss the possible appearance of the OCABS in the super-\nconductor UTe 2[25], which has attracted much attention\nrecently due to a field-induced reentrant spin-triplet su-\nperconductivity [26–32].\nMinimal model .—First, we introduce a generic model\nwith sublattice degree of freedom that exhibits OCABS\nwith unquantized Zak phase. We use the Rice-Mele\nmodel as the starting point [41], since it is the mini-\nmal model of an insulator harboring in-gap states char-\nacterized by the Zak phase. Our basic scheme is to\nreplace the inter-sublattice hopping terms in the Rice-\nMele Hamiltonian with the inter-sublattice pair poten-\ntial. Specifically, we consider a sublattice superconduc-\ntor illustrated in Fig. 1(a). There are two sites within\nthe primitive unit cells, i.e., the a- andb-sites located at\nxandx+1\n2a0, respectively, forming two distinct sublat-\ntices. In what follows, we set the lattice constante a0to\none,suchthat xtakesonintegervalues. Forsimplicitywe\nconsider first the extreme case, where finite hopping only\noccurs between sites of the same sublattice [single lines\nin Fig. 1(a)], while pairing happens only between differ-\nent sublattices [double lines in Fig. 1(a)]. This extreme\nlimit may be unrealistic, but it is useful to bring out the\nkey properties of the OCABSs. A straightforward ex-2\n(a)\nFIG. 1. (a) Schematicimage ofthesublattice superconducto r.\nThe single (double) line describes the coupling through t(∆).\n(b) and (c) Charge density of OCABS, i.e., n˜x,+defined by\nEq. (7), as a function of the distance from the edge. We\nchooseµ= 0.1tand ∆ = 0 .01t. In (b) and (c), the results for\n1≤˜x≤10 +1\n2and that for 1 ≤˜x≤500 +1\n2are plotted,\nrespectively. (d) ρe(1,E) and (e) ρ(1,E) as a function of the\nenergy. (f) G(eV) as a function of the bias voltage.\ntension of our minimal model describes a locally noncen-\ntrosymmetric superconductor [33–35] with an inter-layer\npairing [36, 37], which is proposed to describe the super-\nconductor CeRh 2As2[38, 39]. In the Supplemental Ma-\nterial (SM) [40], we show the explicit BdG Hamiltonian\nfor the locally noncentrosymmetric superconductor and\ndemonstrate the appearance of the OCABSs. Further-\nmore, we will show later that the low-energy excitations\nof superconducting UTe 2[25] are effectively described by\nour minimal model. Let us consider the BdG Hamilto-\nnian of our minimal model in momentum space:\nH=1\n2/summationdisplay\nk[Ψ†\nk,+,Ψ†\nk,−]/bracketleftbigghk,+0\n0hk,−/bracketrightbigg/bracketleftbiggΨk,+\nΨk,−/bracketrightbigg\n,\nhk,s=/bracketleftBigg\ntcos(k)−µ∆e−isk\n2cos(k/2)\n∆eisk\n2cos(k/2)−tcos(k)+µ/bracketrightBigg\n,\nΨk,+= [ψk,a,ψ†\n−k,b]T,Ψk,−= [ψk,b,ψ†\n−k,a]T(1)\nwheres= +1 (−1),ψk,αis the annihilation operator of\nan electron at sublattice αwith momentum k,tdenotes\nthehoppingintegral, µrepresentsthechemicalpotential,\nFIG. 2. Zak phase as a function of the chemical potential.\nand ∆ is the pair potential. We remark that the block\ncomponent hk,shas a form similar to the Hamiltonian\nof the Rice-Mele model [41], where the inter-sublattice\nhopping terms and the on-site potentials are replaced by\nthe inter-sublattice pair potential and the normal-state\nkinetic term, respectively. The particle-hole symmetry\noperator in this basis is given by\n˜C=/bracketleftbigg\n0C\nC0/bracketrightbigg\n, C=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\nK, (2)\nwhereKrepresents the complex conjugation operator\nand˜C2=−1, due to the absence of spin-degrees of free-\ndom. Importantly, the particle-hole symmetry is broken\nwithin each block component, Chk,sC−1=−h−k,−s∝ne}ationslash=\n−h−k,s, while the operation of ˜Cadditionally inter-\nchanges the two blocks. Thus, the Zak phase defined\nfor each block,\nγs=i/integraldisplayπ\n−πdk∝an}bracketle{tuk,s|∂k|uk,s∝an}bracketri}ht, (3)\nis not quantized and can take on any value between\n−πandπ. Here |uk,s∝an}bracketri}htdescribes the occupied eigen-\nstateofhk,sandsatisfiesthe periodicboundarycondition\n|uk,s∝an}bracketri}ht=|uk+2π,s∝an}bracketri}ht∗. In Fig. 2, we show γsas a function\nofµ, where ∆ = 0 .001t. We indeed obtain γs∝ne}ationslash= 0,π\nfor|µ|< t, whereasγs= 0 for|µ|> t, due to the ab-\nsence of Fermi points. Here, particle-hole symmetry im-\nposesγ++γ−= 0 mod 2 πsince the full Hamiltonian\nis topologically trivial. The fractional part of γs, i.e.,\nγs/2πmod 1, characterizes the increase of the probabil-\nity density at the surface [17–19]. The BdG Hamiltonian\nin Eq. (24) has the complete block-diagonal form. Even\nin the presence of small block-off diagonal components,\nwe can still define the unquantized Zak phase in terms\nof a projection, i.e., a projection into the block diagonal\n∗In this paper, we only focus on the intercellular part of the Z ak\nphase [17].3\nspace, before computing the Zak phase, as is the case\nwith the spin Chern number [42–44].\nTo see the relation between γsand ABSs, we solve the\nBdG equation in real space:\nTsϕx−1,s+T†\nsϕx+1,s+Kϕx,s=Esϕx,s,(4)\nwith\nTs=/bracketleftBigg\nt\n2(1+s)∆\n4(1−s)∆\n4−t\n2/bracketrightBigg\n, K=/bracketleftbigg\n−µ∆\n2∆\n2µ/bracketrightbigg\n,\nϕx,+= [ux,+,vx+1\n2,+]T, ϕx,−= [ux+1\n2,−,vx,−]T,\nux+1\n2,+=vx,+=ux,−=vx+1\n2,−= 0,(5)\nwhereux,s(vx,s) denotes the electron (hole) component\nin the sublattice a, andux+1\n2,s(vx+1\n2,s) represents the\nelectron (hole) component in the sublattice b. We con-\nsider the semi-infinite system for x >0 and apply an\nopen-boundary condition, ϕ0,s= 0. We indeed find the\nsolutions of the ABSs for |µ|0 irrespective of q. Thus, the\nlow-energy excitation is effectively described by the 2 ×2\nHamiltonian,\nˆheff\nq,s=/bracketleftbiggǫq−Vq−s∆αq,−s\n−s∆α∗\nq,−s−ǫq+Vq/bracketrightbigg\n.(19)\nInterestingly, although we start from the momentum-\nindependent pair potential, the resultant superconduct-\ning gap in the band basis ends up with the momentum-\ndependent inter-sublattice pair potential:\ns∆αq,−s∝e−sikx\n2e−siky\n2cos(kx/2)cos(ky/2),(20)\nwhere the inter-sublattice pair potential is attributed to\nthe inter-rung pairing and the inter-sublattice inter-rung\nhopping terms. For a fixed kx(y), the effective Hamilto-\nnian in Eq. (19) is equivalent to Eq. (24). Therefore, we\nexpect the appearance of the OCABSs associated with\nthe Zak phase.\nTo verify this expectation, we calculate the Zak phase5\n(b)\nFIG. 3. (a) Zak phase as function of kx. (b)-(c) Electron part\nof angle-resolved spectral function at kz=πas a function of\nkxandtheenergy E. (d)-(e)Electronpartofspectralfunction\nas a function of the energy E, where the contributions from\ndifferent kxandkzare integrated. In (b)and (d), we show the\nresults for the outermost site, i.e., ˜ y= 1, and in (c) and (e),\nwe plot the results for the second-outermost site ˜ y= 1+1\n2.\nfor the effective Hamiltonian,\nγs,kx=i/integraldisplayπ\n−πdky∝an}bracketle{tuq,s|∂ky|uq,s∝an}bracketri}ht, (21)\nwhich characterizesthe probability density accumulation\nat the surface parallel to the xdirection, where |uq,s∝an}bracketri}htde-\nscribes the occupied state of ˇheff\nq,s. As shown in Fig. 3(a),\nthe Zak phase becomes finite for |kx|<0.5π. In Fig. 3(b)\nand 3(c), we show the electron part of angle-resolved\nspectral function ρe(˜y,kx,kz=π,E) as a function of kx\nand energy E, where ˜y=yory+1\n2measures the distant\nfrom the surface with ybeing an integer number. The\nGreen’s function is calculated using the recursiveGreen’sfunction techniques [45, 46]. In Fig. 3(b) and 3(c), the\nresult for the outermost site ρe(1,kx,π,E) and that for\nthe second-outermost site ρe(1+1\n2,kx,π,E) are plotted,\nrespectively. The dotted white lines denote the bulk dis-\npersion, obtained by diagonalizing Hqin Eq. (18). We\nfind the ABSs for |kx|<0.5πwhere the Zak phase is fi-\nnite. Moreover, ρe(˜y,kx,π,E) forthe outermost(second-\noutermost) site has the significant enhancement only for\nE <0 (E >0), which means that the resultant ABSs\nexhibit the oscillating-charge nature. In Figs. 3(d) and\n3(e), we show the electron part of the spectral function\nat ˜y= 1 and ˜y= 1+1\n2as a function of E, respectively.\nThecontributionsfromdifferent kxandkzareintegrated,\ni.e.,ρe(˜y,E) =/summationtext\nkx,kzρe(˜y,kx,kz,E). Forρe(˜y,E) at\nthe outermost (second-outermost) site, we find the sig-\nnificant hump only for E <0 (E >0). Although the\nlow-energy effective Hamiltonian in Eq. (19) is valid only\natkz=π, we confirm that the signature of the OCABSs\nremainssignificantevenwhenthe contributionsfromvar-\niouskzare integrated.\nSummary .—In summary, we have demonstrated the\nappearance of the unconventional ABSs characterized\nby the Zak phase, i.e., OCABSs. The OCABSs cause\nthe clear breakdown in the proportionality among ρe,\nρ, andG. We also describe the possible emergence of\nOCABSs in the explicit model of UTe 2and locally non-\ncentrosymmetric superconductors. Proposing conclusive\nexperimentsforobservingthe oscillating-chargenatureof\nthe OCABSs would be a desirable future task. Finding\nfurther material candidates hosting the OCABSs is also\nan important future work; superconducting ladder sys-\ntems [53–60] and superconducting layered systems [61–\n68] are promising candidates. 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Kuroki, arXiv:2309.09462\n(2023).8\nSupplemental Material for “Oscillating-charged Andreev B ound States and Their\nAppearance in UTe 2”\nSatoshi Ando1, Shingo Kobayashi2, Andreas P. Schnyder3, Yasuhiro Asano4, and Satoshi Ikegaya1,5\n1Department of Applied Physics, Nagoya University, Nagoya 4 64-8603, Japan\n2RIKEN Center for Emergent Matter Science, Wako, Saitama, 35 1-0198, Japan\n3Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany\n4Department of Applied Physics, Hokkaido University, Sappo ro 060-8628, Japan\n5Institute for Advanced Research, Nagoya University, Nagoy a 464-8601, Japan\nOSCILLATING-CHARGED ANDREEV BOUND STATES\nIN LOCALLY NONCENTROSYMMETRIC SUPERCONDUCTORS\nIn this section, we show the appearance of the oscillating-charged Andreev bound state (OCABS) in a locally\nnoncentrosymmetricsuperconductor. The Bogoliubov–deGenne s(BdG) Hamiltonian ofalocallynoncentrosymmetric\nsuperconductor with an inter-layer pairing, which is proposed in Ref s. [36, 37], is given by\nH=1\n2/summationdisplay\nk,σ[ψ†\nkσ,ψT\n−k¯σ]Hk/bracketleftbiggψkσ\nψ∗\n−k¯σ/bracketrightbigg\n+Hsoc,\nψ†\nkσ= [ψ†\nk,σ,a,1,ψ†\nk,σ,a,2,ψ†\nk,σ,b,1,ψ†\nk,σ,b,2],\nHk=/bracketleftbigg˜h(k)˜h∆(k)\n˜h†\n∆(k)−˜h∗(−k)/bracketrightbigg\n,\n˜h(k) =/bracketleftbiggˆKkˆT†\nkˆTkˆKk/bracketrightbigg\n,\nˆKk=/bracketleftbigg\nh00(k) 0\n0h00(k)/bracketrightbigg\n,ˆTk=eikx\n2eiky\n2eikz\n2/bracketleftbigg\n0h10(k)−ih20(k)\nh10(k)+ih20(k) 0/bracketrightbigg\n,\nh00(k) = 2tp[cos(kx)+t2cos(ky)]−µ,\nh10(k) = 4(tu+td)cos(kx/2)cos(ky/2)cos(kz/2),\nh20(k) =−4(tu−td)cos(kx/2)cos(ky/2)sin(kz/2),\n˜h∆(k) =/bracketleftbigg0ˆ∆(k)\nˆ∆†(k) 0/bracketrightbigg\n,ˆ∆ =eikx\n2eiky\n2eikz\n2/bracketleftbigg0d13(k)\nd13(k) 0/bracketrightbigg\n,\nd13(k) = ∆cos(kx/2)cos(ky/2)sin(kz/2),\nHsoc=1\n2/summationdisplay\nk[ψ†\nk↑,ψ†\nk↓,ψT\n−k↑,ψT\n−k↓]Λk\nψk↑\nψk↓\nψ∗\n−k↑\nψ∗\n−k↓\n,\nΛk=/bracketleftbiggλk0\n0−λ∗\n−k/bracketrightbigg\n, λk=/bracketleftbigg˜λ33(k)˜λ31(k)−i˜λ32(k)\n˜λ31(k)+i˜λ32(k)−˜λ33(k)/bracketrightbigg\n,\n˜λ3ν(k) =/bracketleftbiggˆλ3ν(k) 0\n0ˆλ3ν(k)/bracketrightbigg\n,ˆλ3ν(k) =/bracketleftbiggh3ν(k) 0\n0−h3ν(k)/bracketrightbigg\n,\nh31(k) =−αsin(ky), h32(k) =αsin(kx), h33(k) =λsin(kx)sin(ky)sin(kz)[cos(kx)−cos(ky)],(22)\nwhereψk,σ,α,lis the annihilation operator of an electron with momentum kwith spinσat thel-th layer of the\nsublatticeα. The index ¯ σmeans the opposite spin of σ. Each hoping term is schematically illustrated in Fig. 4. The\nstrength of the Rashba (Ising-type) spin-orbit coupling are desc ribed byα(λ). Note that Eq. (22) is modified to take\ninto account the sublattice degrees of freedom explicitly. We assum e the inter-layer-even spin-triplet odd-parity pair\npotential [36, 37]. When we neglect inter-layer hopping terms and th e spin-orbit coupling terms, i.e.,\nh10(k) =h20(k) =h31(k) =h32(k) =h33(k) = 0, (23)9\nFIG. 4. Schematic image of the hopping terms.\nthe BdG Hamiltonian is rewritten as\nH=1\n2/summationdisplay\nk,σ,l[Ψ†\nk,σ,l,+,Ψ†\nk,σ,l,−]/bracketleftbigghk,+0\n0hk,−/bracketrightbigg/bracketleftbiggΨk,σ,l,+\nΨk,σ,l,−/bracketrightbigg\n,\nhk,s=/bracketleftBigg\ntpcos(kx)−µ(ky) ∆ s(ky,kz)e−iskx\n2cos(kx/2)\n∆∗\ns(ky,kz)eiskx\n2cos(kx/2)−tpcos(kx)+µ(ky)/bracketrightBigg\n,\nµ(ky) =µ−tpcos(ky),∆s(ky,kz) = ∆e−isky\n2e−iskz\n2cos(ky/2)sin(kz/2),\nΨk,σ,l,+= [ψk,σ,a,l,ψ†\n−k,σ,b,¯l]T,Ψk,σ,l,−= [ψk,σ,b,l,ψ†\n−k,σ,a,¯l]T,(24)\nwheres= +1 (−1) and¯lindicates the opposite layer of l. With fixed kyandkz,hk,scoincides with the BdG\nHamiltonian of the minimal model in Eq. (1) of the main text. Therefor e, we can expect the appearance of the\nOCABSs in the locally noncentrosymmetric superconductor with the inter-layer pairing.\nTo demonstrate the appearance of the OCABSs, we consider the e lectron part of the spectral function, which is\ndefined by\nρe(˜x,E) =/summationdisplay\nky,kzρe(˜x,ky,kz,E), (25)\nρe(˜x,ky,kz,E) =−1\nπIm[gky,kz(˜x,˜x,E)], (26)\nwhere ˜x=xorx+1\n2measures the distant from the surface with xbeing integer numbers, and gky,kz(˜x,˜x,E) is the\nretarded Green’s function obtained using the recursive Green’s fu nction techniques [45, 46]. In Figs. 5(a) and 5(b),\nwe showρe(˜x,E) at the outermost site (˜ x= 1) and the second-outermost site (˜ x= 1+1\n2) as a function of the energy,\nrespectively. We assume the dominant inter-layer hopping case, i.e., tu,td> α. Specifically, we choose ( µ, tp, tu,\ntd, α, λ,∆)=(0,1,0.2,0.2,0.05,0,0.001). Here we ignore the Ising-type spin-orbit coupling due to the in ter-layer\nlong-ranged hopping. In Figs. 6(a) and 6(b), we show ρe(˜x,E) with the dominant Rashba spin-orbit coupling case,\ni.e.,tu,td< α. Specifically, we choose ( µ, tp, tu, td, α, λ,∆)=(0,1,0.05,−0.05,0.2,0,0.001). As shown in Fig. 5\nand Fig. 6, in both cases, ρe(˜x,E) at the outermost (second-outermost) has the significant peak only forE <0\n(E >0). Thus, we confirm the appearance of the OCABSs in the locally non centrosymmetric superconductor with\nthe inter-layer pairing.10\n(a) (b)\nFIG. 5. Electron part of the spectral function with the domin ant inter-layer hopping case (i.e., tu,td> α) as a function of the\nenergyE. In (a), we show the result for the outermost site, i.e., ˜ x= 1, and in (b), we plot the result for the second-outermost\nsite ˜x= 1+1\n2.\n(a) (b)\nFIG. 6. Electron part of the spectral function with the domin ant Rashba spin-orbit coupling case (i.e., tu,td< α) as a function\nof the energy E. In (a), we show the result for the outermost site, i.e., ˜ x= 1, and in (b), we plot the result for the second-\noutermost site ˜ x= 1+1\n2.11\nENERGY EIGENVALUE AND WAVE FUNCTION\nOF THE OSCILLATING-CHARGED ANDREEV BOUND STATE\nIn this section, we calculate the energy eigenvalue and the wave fun ction of the OCABS in the minimal model. We\nconsider the BdG equation, which is equivalent to Eq. (4) in the main te xt:\nTsϕx−1,s+T†\nsϕx+1,s+Kϕx,s=Esϕx,s, (27)\nwith\nTs=/bracketleftBigg\nt\n2(1+s)∆\n4(1−s)∆\n4−t\n2/bracketrightBigg\n, K=/bracketleftbigg−µ∆\n2∆\n2µ/bracketrightbigg\n,\nϕx,+= [ux,+,vx+1\n2,+]T, ϕx,−= [ux+1\n2,−,vx,−]T,\nux+1\n2,+=vx,+=ux,−=vx+1\n2,−= 0,(28)\nwhereux,s(vx,s) denotes the electron(hole) componentin the a-sublattice, and ux+1\n2,s(vx+1\n2,s) representsthe electron\n(hole) component in the b-sublattice. We consider the semi-infinite system for x >0 and apply an open-boundary\ncondition,ϕ0,s= 0.\nWe first focus on the solution belonging to s= +. Substituting\nϕx,+=/bracketleftbiggux,+\nvx+1\n2,+/bracketrightbigg\n=/bracketleftbigguk\nvk/bracketrightbigg\neikx, (29)\ninto Eq. (27), we obtain:\n/bracketleftbigg\ntcosk−µ e−ik/2∆cos(k/2)\neik/2∆cos(k/2)−tcosk+µ/bracketrightbigg/bracketleftbigg\nuk\nvk/bracketrightbigg\n=E/bracketleftbigg\nuk\nvk/bracketrightbigg\n. (30)\nFrom Eq. (30), we find\nE=±/radicalbig\n(tcos(k)−µ)2+∆2cos2(k/2), (31)\nand\n/bracketleftbigguk\nvk/bracketrightbigg\n∝/bracketleftbiggE+(tcos(k)−µ)\neik/2∆cos(k/2)/bracketrightbigg\n∝/bracketleftbigg\ne−ik/2∆cos(k/2)\nE−(tcos(k)−µ)/bracketrightbigg\n. (32)\nTo investigate the Andreev bound states, we assume\nk=q+iκ (33)\nwhereqandκare real numbers. By substituting Eq. (33) into Eq. (31), we obta in\nE2=t2(cos2(q)cosh2(κ)−sin2(q)sinh2(κ))−2tµcos(q)cosh(κ)+µ2+∆21+cos(q)cosh(κ)\n2\n+i/parenleftbigg\n−2t2cos(q)cosh(κ)sin(q)sinh(κ)+2tµsin(q)sinh(κ)−∆2sin(q)sinh(κ)\n2/parenrightbigg\n. (34)\nFrom the imaginary part of Eq. (34), we obtain\ncos(q)cosh(κ) =4tµ−∆2\n4t2. (35)\nFrom the real part of Eq. (34) and Eq. (35), we obtain\nsin2(q)sinh2(κ) =8tµ∆2−∆4+8t2∆2−16t2E2\n16t4. (36)12\nHere we assume ∆ /t≪1 and approximate Eq. (35) and Eq. (36) as\ncos(q)cosh(κ) =µ\nt, (37)\nsin2(q)sinh2(κ) =tµ∆2+t2∆2−2t2E2\n2t4, (38)\nrespectively. In general, the decay length of the Andreev bound s tate increases (i.e., κdecreases) by decreasing ∆.\nThus, in the limit of ∆ /t≪1, we can expect κ≪1. On the basis of this expectation, we approximate Eq. (37) and\nEq. (40) as\ncos(q) =µ\nt, (39)\nκ2sin2(q) =tµ∆2+t2∆2−2t2E2\n2t4, (40)\nrespectively. The expectation of κ≪1 is justified later. From Eq. (39), we obtain\nq=±qF, qF= arccos/parenleftBigµ\nt/parenrightBig\n, (41)\nwhere−1<µ/t< 1 is satisfied. From Eq. (40) and Eq. (41), we obtain\nκ=±κE, κE=/radicalBigg\ntµ∆2+t2∆2−2t2E2\n2t2(t2−µ2). (42)\nAs a result, the wave function of the Andreev bound state is repre sented by\nϕx,+=a/bracketleftbiggE+(tcos(k+)−µ)\neik+/2∆cos(k+/2)/bracketrightbigg\neik+x+b/bracketleftbigg\neik−/2∆cos(k−/2)\nE−(tcos(k−)−µ)/bracketrightbigg\ne−ik−x, (43)\nk±=qF±iκE (44)\nwhereaandbare numerical coefficients. The wave function of ϕx,+satisfies\nlim\nx→∞ϕx,+= 0. (45)\nFrom the conditions of ∆ /t≪1 andκ≪1, we can approximate\nϕx,+≈a/bracketleftbiggE−itsin(qF)κE\neiqF∆cos(qF/2)/bracketrightbigg\neik+x+b/bracketleftbigg\neiqF/2∆cos(qF/2)\nE−itsin(qF)κE/bracketrightbigg\ne−ik−x. (46)\nFrom the boundary condition of ϕx=0,+= 0, we obtain\n/bracketleftbiggE−itsin(qF)κEeiqF/2∆cos(qF/2)\neiqF/2∆cos(qF/2)E−itsin(qF)κE/bracketrightbigg/bracketleftbigga\nb/bracketrightbigg\n= 0, (47)\nwhich leads\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleE−itsin(qF)κEeiqF/2∆cos(qF/2)\neiqF/2∆cos(qF/2)E−itsin(qF)κE/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0. (48)\nBy solving Eq. (48), we obtain the energy eigenvalue of the Andreev bound state as\nE=E+=−∆t+µ\n2t. (49)\nBy substituting Eq. (49) into Eq. (42), we find\nκE=κ0=∆\n2t, (50)13\nwhich satisfies κ0≪1 for ∆/t≪1. By substituting Eq. (49) into Eq. (47), we obtain\n/bracketleftbigg\n1−1\n−1 1/bracketrightbigg/bracketleftbigg\na\nb/bracketrightbigg\n= 0, (51)\nwhich leads a=b. As a result, we find\nϕx,+=a/parenleftbigg/bracketleftbigg1\n−1/bracketrightbigg\neik+x+/bracketleftbigg−1\n1/bracketrightbigg\ne−ik−x/parenrightbigg\n= 2ia/bracketleftbigg1\n−1/bracketrightbigg\nsin(qFx)e−κ0x. (52)\nFrom the normalization condition,\n/summationdisplay\nx|ϕx,+|2= 1, (53)\nwe obtain\nϕx,+=/bracketleftbigg\n1\n−1/bracketrightbigg\nφx, φx=√2κ0sin(qFx)e−κ0x. (54)\nThe energy eigenvalue in Eq. (49) and the wave function in Eq. (54) a re equivalent to those in Eq. (6) in the main\ntext. In a similar manner, the energy eigenvalue and the wave funct ion of the Andreev bound states belonging to\ns=−are obtained as\nE−= ∆t+µ\n2t, ϕx,−=/bracketleftbigg\n1\n1/bracketrightbigg\nφx, (55)\nrespectively.\nGREEN’S FUNCTION AT E≈E±\nIn this section, we calculate the retarded Green’s function at E≈E±, which is given by Eq. (12) in the main text.\nAt first, we consider the field operator\nΨx,s= Ψc\nx,s+Ψa\nx,s, (56)\nΨx,+=/bracketleftBigg\nψx\nψ†\nx+1\n2/bracketrightBigg\n,Ψx,−=/bracketleftbiggψx+1\n2\nψ†\nx/bracketrightbigg\n, (57)\nwhereψ˜x(ψ†\n˜x)with ˜x=xorx+1\n2representsthe annihilation(creation)operatorofthe electrona t ˜x. Thecontribution\nfrom the continuum states to the field operator is given by\nΨc\nx,s=/summationdisplay\nν/parenleftBig\nϕx,s,νγs,ν+ ¯ϕx,s,νγ†\n−s,ν/parenrightBig\n, (58)\n(59)\nwhereϕx,s,νsatisfies\nTsϕx−1,s,ν+T†\nsϕx+1,s,ν+Kϕx,s,ν=Es,νϕx,s,ν, (60)\nwithEs,ν>0 and\nlim\nx→∞ψx,s,ν∝ne}ationslash= 0. (61)\nThe wave function of ¯ ϕx,s,νis given by\n¯ϕx,s,ν=Cϕx,−s,ν, C=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\nK (62)14\nwithKbeing the complex conjugation operator, where ¯ ϕx,s,νsatisfies\nTs¯ϕx−1,s,ν+T†\ns¯ϕx+1,s,ν+K¯ϕx,s,ν=−E−s,ν¯ϕx,s,ν. (63)\nThe annihilation (creation) operator of the Bogoliubov quasiparticle is given by γs,ν(γ†\ns,ν). The contribution from\nthe Andreev bound state to the filed operator is given by\nΨa\nx,s=/bracketleftbigg1\n−s/bracketrightbigg\nφxγs,a (64)\nwhereγs,adenotes the annihilation operator of the Andreev bound state. Th e retarded Green’s function is defined\nby\nG+(x,t;x′,t′) =/bracketleftbiggg(x,t;x′,t′)f(x,t;x′+1\n2,t′)\nf(x+1\n2,t;x′,t′)g(x+1\n2,t;x′+1\n2,t′)/bracketrightbigg\n=−iΘ(t−t′)/bracketleftBigg\n{ψx(t),ψ†\nx′(t′)} {ψx(t),ψx+1\n2(t′)}\n{ψ†\nx+1\n2(t),ψ†\nx(t′)} {ψ†\nx+1\n2(t),ψx+1\n2(t′)}/bracketrightBigg\n, (65)\nG−(x,t;x′,t′) =/bracketleftbiggg(x+1\n2,t;x′+1\n2,t′)f(x+1\n2,t;x′,t′)\nf(x,t;x′+1\n2,t′)g(x,t;x′,t′)/bracketrightbigg\n=−iΘ(t−t′)/bracketleftBigg\n{ψx+1\n2(t),ψ†\nx′+1\n2(t′)} {ψx+1\n2(t),ψx(t′)}\n{ψ†\nx(t),ψ†\nx+1\n2(t′)} {ψ†\nx(t),ψx(t′)}/bracketrightBigg\n, (66)\nwhere\ng(x,t;x′+1\n2,t′) =g(x+1\n2,t;x′,t′) =g(x,t;x′+1\n2,t′) =g(x+1\n2,t;x′,t′) = 0, (67)\nf(x,t;x′,t′) =f(x+1\n2,t;x′+1\n2,t′) =f(x,t;x′,t′) =f(x+1\n2,t;x′+1\n2,t′) = 0. (68)\nIn the spectral representation, the retarded Green’s function is given by\nG+(x,x′,E) =/bracketleftbiggg(x,x′,E)f(x,x′+1\n2,E)\nf(x+1\n2,x′,E)g(x+1\n2,x′+1\n2,E)/bracketrightbigg\n=Gc\n+(x,x′,E)+Ga\n+(x,x′,E), (69)\nG−(x,x′,E) =/bracketleftbiggg(x+1\n2,x′+1\n2,E)f(x+1\n2,x′,E)\nf(x,x′+1\n2,E)g(x,x′,E)/bracketrightbigg\n=Gc\n−(x,x′,E)+Ga\n−(x,x′,E), (70)\nwhere\nGc\ns(x,x′,E) =/summationdisplay\nν/parenleftbigg\nϕx,s,ν1\nE−Es,ν+iδϕ†\nx′,s,ν+ ¯ϕx,s,ν1\nE+E−s,ν+iδ¯ϕ†\nx′,s,ν/parenrightbigg\n, (71)\nGa\ns(x,x′,E) =φxφx′\nE−Es+iδ/bracketleftbigg1−s\n−s1/bracketrightbigg\n. (72)\nWhen we focus on E≈Es, the contribution from the continuum states to the Green’s funct ion, i.e.,Gc\ns(x,x′,E) is\nnegligible. Therefore, we obtain\nGs(x,x′,E)≈Ga\ns(x,x′,E), (73)\nforE≈Es. As a result, we find\ng(x,x,E) =g(x+1\n2,x+1\n2,E) =|φx|2\nE+Ec+iδ,\ng(x+1\n2,x+1\n2,E) =g(x,x,E) =|φx|2\nE−Ec+iδ,(74)\nwithEc=|Es|, which is equivalent to Eq. (12) in the main text.15\nDIFFERENTIAL CONDUCTANCE\nIN THE PRESENCE OF ANDREEV BOUND STATES\nIn this section, we derive the differential conductance given by Eq. (14) in the main text. We start with a tight-\nbinding BdG Hamiltonian describing a normal-metal–superconductor j unction,\nH=HS+HN+HT, (75)\nHS=/summationdisplay\nx,x′>0[c†\nx,cx]/bracketleftbigg\nh(x,x′) ∆(x,x′)\n−∆∗(x,x′)−h∗(x,x′)/bracketrightbigg/bracketleftbiggcx′\nc†\nx′/bracketrightbigg\n, (76)\nHN=−t/summationdisplay\nx<0(a†\nx+1ax+h.c.), (77)\nHT=−t′(c†\nx=1ax=0+h.c.), (78)\nwherecx(ax) represents the annihilation operator of the electron in the super conducting (normal) segment. For the\nsuperconducting segment (i.e., x >0), the kinetic energy in the normal state is described by h(x,x′), and the pair\npotential is given by ∆( x,x′). The hopping integral in the normal segment ( x≤0) is given by t. The hopping integral\nat the interface is represented by t′. The Bogoliubov transformation is denoted by\n/bracketleftbiggcx\nc†\nx/bracketrightbigg\n=/summationdisplay\nν/parenleftbigg/bracketleftbigg\nuν(x)\nvν(x)/bracketrightbigg\nγν+/bracketleftbigg\nv∗\nν(x)\nu∗\nν(x)/bracketrightbigg\nγ†\nν/parenrightbigg\n, (79)\nwhere\n/summationdisplay\nx′>0/bracketleftbiggh(x,x′) ∆(x,x′)\n−∆∗(x,x′)−h∗(x,x′)/bracketrightbigg/bracketleftbigguν(x′)\nvν(x′)/bracketrightbigg\n=Eν/bracketleftbigguν(x)\nvν(x)/bracketrightbigg\n,(x>0) (80)\nandγν(γ†\nν) represents the annihilation (creation) operator of a Bogoliubov q uasiparticle having the energy Eν. In\nwhat follows, we assume\n00). As a result, we confirm the appearance of the OCABSs even with the spin-orbit coupling potentials.\nBand basis Hamiltonian at kz=π\nIn this section, we calculate the band-basis Hamiltonian at kz=π, which is given by Eq. (18) in the main text.\nHere, we ignore the spin-orbit coupling Hsoc. Atkz=π, the Hamiltonian Hkin Eq. (96) is rewritten as\nHq=/bracketleftbigg˜hq˜h∆\n˜h†\n∆−˜h∗\n−q/bracketrightbigg\n, (104)\n˜hq=\nǫqmg0−χ∗\nqfz,q\nmgǫqχ∗\nqfz,q0\n0χqfz,qǫqmg\n−χqfz,q0mgǫq\n, χq=eikx\n2eiky\n2, (105)19\nwhereq= (kx,ky,kz=π). By using a unitary operator,\nUA=U4U3U2U1, (106)\nU1=/bracketleftbigg˜U10\n0˜U1/bracketrightbigg\n,˜U1=1√\n2\n1 1 0 0\n−1 1 0 0\n0 0 1 1\n0 0−1 1\n, (107)\nU2=/bracketleftbigg˜U20\n0˜U2/bracketrightbigg\n,˜U2=\n1 0 0 0\n0 0 0 1\n0 1 0 0\n0 0 1 0\n, (108)\nU3=\nˆ1 0 0 0\n0 0 0ˆ1\n0ˆ1 0 0\n0 0ˆ1 0\n,ˆ1 =/bracketleftbigg\n1 0\n0 1/bracketrightbigg\n, (109)\nU4=/bracketleftbigg˜U40\n0˜U4/bracketrightbigg\n,˜U4=\n1 0 0 0\n0 1 0 0\n0 0 0 1\n0 0 1 0\n, (110)\nwe deform the Hamiltonian Hqas\nH′\nq=UAHqU†\nA=/bracketleftbiggˇH′\nq,+0\n0ˇH′\nq,−/bracketrightbigg\n, (111)\nˇH′\nq,s=\nǫq+smg−sχ∗\nqfz,q0 s∆\n−sχqfz,qǫq−smg−s∆ 0\n0 −s∆−ǫq−smg−sχqfz,q\ns∆ 0 −sχ∗\nqfz,q−ǫq+smg\n. (112)\nFurthermore, by using a unitary operator,\nU5=/bracketleftbiggˇU50\n0ˇU5/bracketrightbigg\n,ˇU5=/bracketleftbiggˆU50\n0ˆU5/bracketrightbigg\n, (113)\nˆU5=1/radicalbig\n2Vq(Vq+mg)/bracketleftbiggVq+mg−χ∗\nqfz,q\n−χqfz,q−(Vq+mg)/bracketrightbigg\n, (114)\nˆU5=1/radicalbig\n2Vq(Vq+mg)/bracketleftbiggVq+mgχqfz,q\nχ∗\nqfz,q−(Vq+mg)/bracketrightbigg\n, (115)\nVq=/radicalBig\nm2g+f2z,q, (116)\nwe eventually obtain the band-basis Hamiltonian,\nHband\nq=U5H′\nqU†\n5=/bracketleftbiggˇHq,+0\n0ˇHq,−/bracketrightbigg\n(117)\nˇHq,s=\nǫq+sVq0s∆αq,−−s∆βq\n0ǫq−sVqs∆βqs∆αq,+\ns∆αq,+s∆βq−ǫq−sVq0\n−s∆βqs∆αq,−0−ǫq+sVq\n, (118)\nαq,±=e±ikx/2e±iky/2fz,q\nVq, βq=mg\nVq, (119)\nwhich is equivalent to Eq. (18) in the main text.20\nElectron part of spectral function with other pairing symme tries\nIn this section, we show the electron part of spectral function wit h other possible pairing symmetries discussed in\nRef. [25]. Specifically, we consider the inter-rung-odd spin-triplet s-wave pair potential belonging to B2uandB3u\nsymmetries. The BdG Hamiltonian is given by\nH=1\n2/summationdisplay\nk,σ[ψ†\nkσ,ψT\n−kσ]H2u(3u)\nk/bracketleftbiggψkσ\nψ∗\n−kσ/bracketrightbigg\n+Hsoc,\nH2u(3u)\nk=\n˜hk˜h2u(3u)\n∆,σ/braceleftBig\n˜h2u(3u)\n∆,σ/bracerightBig†\n−˜h∗\n−k\n,\n˜h2u\n∆,σ=−sσ/bracketleftbiggˆ∆ 0\n0ˆ∆/bracketrightbigg\n,˜h3u\n∆,σ=i/bracketleftbiggˆ∆ 0\n0ˆ∆/bracketrightbigg\n,(120)\nwheres↑(↓)= 1 (−1). In what follows, we focus on the Andreev bound states at the s urface parallel to the xdirection.\nIn Figs. 9(a) and 9(b), we show ρe(˜y,E) withB2upairing symmetry at the outermost site (˜ y= 1) and the second-\noutermost site (˜ y= 1+1\n2) as a function of the energy, respectively. In Figs. 10(a) and 10( b), we show ρe(˜y,E) with\nB3upairing symmetry at ˜ y= 1 and ˜y= 1+1\n2as a function of E, respectively. In both cases, we find that ρe(˜y,E)\nat the outermost (second-outermost) has the significant hump o nly forE <0 (E >0). As a result, we confirm the\nappearance of the OCABSs even with B2uandB3usymmetries.\n(a) (b)\nFIG. 8. Electron part of the spectral function in the presenc e of the spin-orbit coupling potentials as a function of the e nergy\nE. The pair potential is belonging to Aupairing symmetry. In (a), we show the result for the outermos t site, i.e., ˜ y= 1, and\nin (b), we plot the result for the second-outermost site ˜ y= 1+1\n2.21\n(a) (b)\nFIG. 9. Electron part of the spectral function with B2upairing symmetry as a function of the energy E. In (a), we show the\nresult for the outermost site, i.e., ˜ y= 1, and in (b), we plot the result for the second-outermost si te ˜y= 1+1\n2.\n(a) (b)\nFIG. 10. Electron part of the spectral function with B3upairing symmetry as a function of the energy E. In (a), we show the\nresult for the outermost site, i.e., ˜ y= 1, and in (b), we plot the result for the second-outermost si te ˜y= 1+1\n2." }, { "title": "2403.01949v1.Spatially_dispersing_in_gap_states_induced_by_Andreev_tunneling_through_single_electronic_state.pdf", "content": "1 \nSpatially dispersing in-gap states induced by Andreev tunneling \nthrough single electronic state \n \nRuixia Zhong1,*, Zhongzheng Yang1,*, Qi Wang1,*, Fanbang Zheng1, Wenhui Li1,2, Juefei Wu1, \nChenhaoping Wen1, Xi Chen2, Yanpeng Qi1,3,4,†, Shichao Yan1,3,† \n \n1 School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China \n2State Key Laboratory of Low -Dimensional Quantum Physics, Department of Physics, \nTsinghua University, Beijing 10084, China \n3ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, \nChina \n4Shanghai Key Laboratory of High -resolution Electron Microscopy, ShanghaiTech University, \nShanghai 201210, China \n \n*These authors contributed equally to this work \n†Email : yanshch@shanghaitech.edu.cn; qiyp@shanghaitech.edu.cn \nAbstract \nBy using low -temperature scanning tunneling microscopy and spectroscopy (STM/STS), we \nobserve superconducting in-gap states induced by Andreev tunneling through single impurity state \nin a low -carrier -density superconductor (NaAlSi). The energy -symmetric in-gap states appear when \nthe impurity state is located within the superconducting gap. Superconducting i n-gap states can \ncross the Fermi level, and show X-shaped spatial dispersion . We interpret the in-gap states as a \nconsequence of the Andreev tunneling through the impurity state, which involves the formation or \nbreakup of a Cooper pair . Due to the low carrier density in NaAlSi, the in -gap state is tunable by \ncontrolling the STM tip -sample distance. Under strong external magnetic fields, the impurity state \nshows Zeeman splitting when it is located near the Fermi level. Our findings not only demonstrate \nthe Andreev tunneling involv ing single electronic state , but also provide new insights for \nunderstanding the spatially dispersing in-gap states in low -carrier -density superconductors. \nMain text \nSuperconducting in-gap state s may arise when the superconducting order parameter varies in \nspace [1]. For instance, magnetic impurities in s-wave superconductors can lead to Yu -Shi-Rusinov \n(YSR) state [2-8], and weak links between two superconductors can generate Andreev states [9]. \nDue to the protection of the superconducting gap, superconducting in-gap states also provide a \npromising platform for applications in quantum compu ting [10-12]. 2 Tunneling experiments can provide a wealth of information on superconducting in -gap states , \nsuch as the energy position , orbital structure , and quasiparticle relaxation through the in -gap states \n[6,13,14] . Andreev tunneling , which involv es the creation or annihilation of a Cooper pair , is a key \nprocess when tunneling into or out of a superconductor [15,16] . For instance, tunneling through \nYSR state can be considered as carried by the Andreev pr ocess [14,17] . However, o bserving the \nAndreev tunneling through a pure electronic state within the superconducting gap is challenging . \nThis is because superconductivity often emerges in high carrier density systems where the single \nelectronic state could be strongly screened. The screening effect is reduced in low carrier density \nsuperconductors [18,19] , which may make it possible to observe the Andreev tunneling involving \nsingle electronic state. \nHere we report a low-temperature scanning tunneling microscopy and spectroscopy (STM/STS) \nstudy on a low -carrier -density superconductor ( NaAlSi ). We detect the energy -symmetric \nsuperconducting in -gap states induced by Andreev tunneling through a single impurity state. We \nfind that the superconducting in -gap states in NaAlSi show strong spatial dispersion , and investigate \nthe impurity state behavior under external magnetic fields. We also discover that due to the low \ncarrier density in NaAlSi, this kind of superconducting in -gap states are highly tunable . \nNaAlSi is an intrinsically self -doped low carrier density superconductor with superconducting \ntransition temperature ( Tc) at ∼7 K [20-26]. NaAlSi single crystals were grown by Ga flux method \n(see Supplemental Material for more details) . STM/STS experiments were conducted with a \nUnisoku low -temperature and high -magnetic -field STM (USM1600). The tungsten tips were used \nin the STM measurements, and they were flashed by electron -beam bombardment for several \nminutes before use. STS was measured using a standard lock -in detection technique with a \nfrequency of 914 Hz. \nAs shown in Fig. 1(a), NaAlSi has the same structure as the “111” type ion -pnictide \nsuperconductors [20,21] . Cleaving NaAlSi single crystals result s in a Na-terminated surface with \nsquare lattice . Figure 1(c) is the typical constant -current STM topography taken on the surface of \nNaAlSi . As marked by the red and yellow arrow s, we can see that there are typically two kinds of \natomic defects [Fig. 1 (c) and Supplement al Material Fig. S 1]. One defect appears as a dark hole, \nwhich is likely the Na vacancy. T he other defect is cross -shaped . In the zoom -in STM topography \nwith cross -shaped defects [Fig. 1d ], we can still observe the square lattice of the Na atoms , which \nindicates the cross -shaped defects are likely from the underneath AlSi layer. \nWe first characterize superconductivity in NaAlSi by measuring the temperature and magnetic \nfield dependent differential conductance (dI/dV) spectra . As shown in Fig. 1 (e), there is full -gap \nsuperconducting spectrum at ~ 130 mK, and it is gradually suppressed as increasing the temperature \n[Fig. 1 (e)]. In order t o quanti fy the size of the superconducting gap, we fit the temperature -\ndependent dI/dV spectra with an isotropic s -wave gap function [see Supplement al Material Fig. S2], \nand the temperature dependence of the superconducting gap (Δ) shows a BCS -gap behavior [Fig. \n1f] [27,28] . The extracted Tc for NaAlSi is about ~7 K , which is consistent with the Tc obtained in \nthe electrical transport measurement [Fig. 1 (b)]. The low -temperature superconducting gap is about \n1 meV, which yields a ratio of 2 Δ/kBTc ∼3.32. In addition to the temperature -dependent dI/dV \nspectra, we also carry out dI/dV measurements with the magnetic field s perpendicular to the NaAlSi \nsurface. As increasing the strength of the magnetic fields, the superconducting gap feature gradually 3 disappears [Fig. 1(g)]. Fitting the magnetic -field-dependent superconducting gap with Δ(H) = \nΔ(0)[1 − (H/Hc2)2]1/2 result s in an upper critical field about 0. 8 ± 0.1 T [Fig. 1(h)]. \nAlthough we can measure the full-gap superconducting spectrum on the NaAlSi surface [Fig. \n1(e)], at some locations we also observe the superconducting in -gap states. As shown in Fig s. 2(b) \nand 2 (c), due to the presence of the superconducting in -gap states, there are ring -shaped features in \nthe d I/dV maps taken with the bias voltages within the superconducting gap [0 mV for Fig. 2 (b), \n−0.625 mV for Fig. 2 (c)]. In this manuscript, we use “ring” to label the ring-shaped feature in the \ndI/dV maps for simplicity. We find that the atomic defects as shown in the STM topography are not \ndirectly related with superconducting in -gap states (see Supplemental Material Fig. S3). In \ncomparison with the concentration of the defects on the surface of NaAlSi, t he ring s shown in the \ndI/dV maps are sparse, and there are roughly 15 rings in a 1 20 by 120 nm2 area (see Supplemental \nMaterial Fig. S4). \nTo understand the origin of the superconducting in -gap states, we perform detailed dI/dV \nmeasurements at the locations of the rings . Figure 2 (d) shows the d I/dV linecut profile taken across \nthe ring in Fig. 2 (b), and Figure 2 (f) are the typ ical d I/dV spectra marked by the coloured arrows in \nFig. 2 (d). We can see that there are three kinds of areas : (1) Inside the ring, besides the \nsuperconducting gap, there is an electronic state that is located above the superconducting coherence \npeak [spectrum 1 in Fig. 2 (f)]. We name this electronic state as “impurity state”. As moving towards \nthe ring, th is impurity state gradually shifts towards the Fermi level [spectrum 2 in Fig. 2 (f)]; (2) \nNear the ring , there are sharp energy -symmetric resonances inside the superconducting gap that \nextend over ~5 nm in real space [spectra 3 -5 in Fig. 2 (f)]. Importantly, in this area, the energies of \nthe superconducting in -gap states show strong spatial variation . The sub -gap states shift towards the \nFermi level , then t hey cross at the Fermi level and split again . This results in the X -shaped \nsuperconducting in -gap states [Fig. 2(d)]; (3) Outside the ring, the full -gap superconducting \nspectrum recovers [spectrum 6 in Fig. 2 (f)]. \nTo reveal the influence of superconductivity to the in-gap states , we apply external magnetic \nfields above the critical magnetic field to suppress superconductivity in NaAlSi. Figure 2 (e) is the \ndI/dV linecut profile taken with 1.5 T magnetic field along the same arrow as for Fig. 2 (d). Figure \n2(g) shows the d I/dV spectra in Fig. 2 e taken at the same positions as the corresponding spectra in \nFig. 2 (f). We can see that as long as superconductivity is suppressed by the external magnetic field s, \nthe sharp energy -symmetric superconducting in -gap states disappear, and there is a broader impurity \nstate near the Fermi level [spectra 3 -5 in Fig. 2 (g)]. This indicates that the superconducting in -gap \nstates emerge where the impurity state is located within the superconducting gap . The width of the \nin-gap states is significantly reduced than that of the impurity state [Fig. 2d ]. This could be due to \nthe protection of the superconducting gap, the lifetime of the impurity state located within the \nsuperconducting gap is greatly enhanced [29]. \nFor different rings show n in the d I/dV maps in Figs. 2 (b) and 2 (c), their relative energy position \nbetween the impur ity state and the Fermi level could be different. Figures 2(h) and 2(i) are the d I/dV \nlinecut profile s across another ring shown in Fig. 2(c) with 0 T and 1.5 T magnetic fields , \nrespectively . We can see that even inside the ring, the electronic state is still slightly below the Fermi \nlevel [Fig. 2 (i)]. The energy -symmetric superconducting in -gap state appear around the center of \nthe ring where the impurity state is located within the superconducting gap [Fig. 2h]. The different 4 energy position s of the impurity state could be due to the different location s of the impurities in \nNaAlS i. In the d I/dV spectrum taken with larger energy range, in addition to the impurity state \nlocated around the Fermi level, we can also observe the other discrete energy states ( see \nSupplemental Material Fig. S5). The energy level spacing for the discreate energy states is about 15 \nmeV. \nThe X -shaped spatially dispersing superconducting in -gap states have also been observed in \nthe intrinsic impurities in FeTe 0.55Se0.45 and in the Pb/Co/Si(111) sandwiched structures [30,31] . In \nFeTe 0.55Se0.45, they have been interpreted as magnetic impurity induced Y SR states [30], and they \nare suggested to be topological in the Pb/Co/Si(111) system [31]. However, t hese two works focus \non the energy scales within the superconducting gap, and no clear impurity state is reported in these \ntwo systems. Now we discuss the mechanism for observing the energy -symmetric superconducting \nin-gap states in NaAlSi. We divide it into two scenarios: one scenario is that the impurity state ( Vp) \nis slightly above the Fermi level [0 < Vp < Δ, Figs. 3(a) and 3(b)]; and the other is the impurity state \nis slightly below the Fermi level [0 < Vp < Δ, Figs. 3(c) and 3(d)]. For the first scenario, when we \napply positive sample bias voltage ( Vs > 0), electrons tunnel from the STM tip to NaAlSi. The \nelectrons can tunnel into the impurity state when Vs equals Vp [Fig. 3(a)], and we could observe a \npeak feature in the d I/dV spectrum. With negative sample bias voltage ( Vs < 0), electrons tunnel \nfrom NaAlSi to the STM tip. When the absolute value of Vs equals Vp, the Cooper pair breakup \noccurs in NaAlSi, one electron tunnels inelastically into the STM tip and excites the other electron \ninto the impurity state [Fig. 3(b)] [32]. Due to this Andreev tunneling process with the breakup of a \nCooper pair , the impurity state located above the Fermi level can also contribute to the tunneling \ncurrent even with negative bias voltage [Fig. 3(b)]. \nFor the second scenario [Figs. 3(c) and 3(d)], the impurity state is located slightly below the \nFermi level (−Δ < Vp < 0), and it is singly occupied. When we apply negative sample bias voltage \n(Vs < 0), electrons tunnel from NaAlSi to STM tip. The electron in the electronic state tunnel into \nSTM tip when Vs equals Vp [Fig. 3(d)], and we measure a peak feature at Vs in the d I/dV spectrum. \nWith positive bias voltage ( Vs > 0), electrons tunnel from STM tip to NaAlSi. When Vs equals the \nabsolute value of Vp, the electron from STM tip tunnels inelastically into NaAlSi and excites the \nelectron in the impurity state to form a Cooper pair in NaAlSi [Fig. 3(c)] [32]. With this Andreev \ntunneling process involving the formation of a Cooper pair , we can also observe a peak feature in \nthe positive bias voltage side of the d I/dV spectrum. \nHaving understood the origin of the particle -hole symmetr ic superconducting in -gap states in \nNaAlSi, the next question is why the in-gap states exhibit spatial d ispersion ? First, our dI/dV spectra \ndata clearly indicate that the in-gap states are induced by the impurity state, and the energy positions \nof the sub -gap states are also determined by the relative energy position between the impurity state \nand the Fermi level . Second, due to the different materials, NaAlSi could have significant different \nwork function as the STM tip, which induce s a voltage drop between NaAlSi and STM tip [30]. \nThird, because of the low carrier density in NaAlSi, there would be non -zero screening length that \nresult s in the penetration of the electric field from STM into NaAlSi. Based on the above three \nfactors , we propose that the STM tip can behave as a gate electrode that tunes the energy levels of \nthe impurity [30,33,34] . When the moving the STM tip laterally towards the impurity location, the \nlocal electric field effect from the STM tip gets stronger. It reach es the maximum when the STM 5 tip is right above the impurity. This can nicely explain the spatial dependence of the impurity state \nshowing in the d I/dV linecut profiles in Fig. 2 . \nThis gating effect of the STM tip can also be confirmed by tuning the impurity state with tip -\nsample distance. We control the tip -sample distance by using the STM -based constant -current \nfeedback, and increase the tunneling current setpoint while keeping the bias voltage constant. Figure \n4(a) shows three d I/dV linecut profiles measured with different tip -sample distances along the arrow \nin the inset of Fig. 4(a), and the tip -sample distance is measured relative to the setpoint: Vs = −10 \nmV, I = 150 pA. When the STM tip is brought closer to the NaAlSi surface, the local electric field \nbecomes stronger and the impurity electronic state is shifted to higher energy. In order to see the \ninfluence of the tip -sample distance to the superconducting in -gap state s, we measure the d I/dV \nlinecut profiles taken near the ring and with smaller energy scale [Fig. 4(b)]. The tip -sample distance \nis measured relative to the setpoint: Vs = −3 mV, I = 150 pA. In this case, as reducing the tip-sample \ndistance , the locations showing the X -shaped in-gap states and the crossing point at zero bias voltage \nappear at larger distance from the center of the ring [Fig. 4(b)]. \nFinally, we investigate the response of the impurity state to strong external magnetic fields that \nare significantly larger than the critical magnetic field in NaAlSi. Figure 4 (d) is a d I/dV map that \nshows another ring feature. Figures 4 (e-g) are the evolution of the d I/dV spectra under external \nmagnetic fields taken at the locations marked in Fig. 4 (d). Similar as the other rings shown in Fig. \n2, as moving towards the center of the ring, the energy position of the impurity state shifts away \nfrom the Fermi level [Figs. 4 (e-g)]. When the impurity electronic state is located at the Fermi level, \nthere is clear energy splitting under strong external magnetic field [Fig. 4 (e)]. By fitting the peak \npositions under 9 T magnetic fields, the extracted energy splitting is about 1.5 m eV, which indicates \nthat the g-factor for this impurity state is about 2.87. When the impurity state is shifted away from \nthe Fermi level, the magnetic -field-induced energy splitting effect is reduced [Figs. 4 (f) and 4 (g)]. \nIn conclusion, by using high -resolution STM/STS , we observe superconducting in -gap states \nwith strong spatial dispersion in NaAlSi . The energy -symmetric in -gap resonances are induced by \nAndreev tunneling through the local ized impurity state. Due to the low carrier density in NaAlSi, \nthe superconducting in -gap states can be tuned by the local gating effect from the STM tip , which \nalso explain s their spatial disper sion. The mechanism for the spatially dispersing superconducting \nin-gap states revealed here advances the understanding the inter play between localized impurity \nstate and superconductivity with low carrier density , which may open a new route for developing \nnano electronic devices based on low -carrier -density superconductors . \nAcknowledgements \nS.Y. acknowledges the financial support from the National Key R&D program of China \n(2020YFA0309602, 2022YFA1402703) and the start -up funding from ShanghaiTech University. \nY.Q. acknowledges the financial support from the National Key R&D program of China \n(2023YFA1607400, 2018YFA0704300) and the National Natural Science Foundation of China \n(52272265). \n \n 6 References \n[1] E. Prada et al. , Nature Reviews Physics 2, 575 (2020). \n[2] L. 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Ho, Science 280, 1732 (1998). \n[33] A. P. Wijnheijmer, J. K. Garleff, K. Teichmann, M. Wenderoth, S. Loth, and P. M. Koenraad, \nPhysical Review B 84, 125310 (2011). \n[34] I. Battisti, V. Fedoseev, K. M. Bastiaans, A. de la Torre, R. S. Perry, F. Baumberger, and M. \nP. Allan, Physical Review B 95, 235141 (2017). \n \nFigur e 1 \n \nFig. 1 (a) Schematic showing the crystal structure of NaAlSi. (b) Temperature -dependent electrical \nresistance of NaAlSi , which shows a sharp superconducting transition at ~7 K. (c) Constant -current \nSTM topography taken on the NaAlSi surface ( Vs = −100 mV, I = 100 pA). (d) Zoom -in view of \nthe NaAlSi surface ( Vs = 500 mV, I = 200 pA). (e, f) Temperature dependence s of the d I/dV spectra \n(e) and the size of the superconducting gap ( f). The solid line in (f) is a BCS fitting for the \nsuperconducting gap. (g, h) Magnetic -field dependence s of the d I/dV spectra ( g) and the size of the \nsuperconducting gap (h). The solid line in (h) is a calculated curve by using the Ginzburg -Landau \n(GL) theory. \n \n8 Figure 2 \n \nFig. 2 (a) Constant -current STM topography taken on the NaAlSi surface ( Vs = −50 mV, I = 100 \npA). (b, c) dI/dV maps on the area shown in (a) with 0 mV and −0.625 mV bias voltage s, respectively . \n(d, e) dI/dV linecut profiles taken along the white arrow in (b) with 0 T ( d) and 1.5 T (e) magnetic \nfields. Setup condition: Vs = −10 mV, I = 300 pA . (f, g) The typical d I/dV spectra marked by the \ncolored arrows in (d) and (e), respectively. (h, i) dI/dV linecut profiles taken along the white arrow \nin (c) with 0 T ( h) and 1.5 T ( i) magnetic fields. Setup condition: Vs = −10 mV, I = 300 pA. The \ndata shown in this figure are taken at 130 mK. \n \n9 Figure 3 \n \nFig. 3 (a) and (b) The electron tunneling model when the impurity state (IS) in NaAlSi is located \nslightly above the Fermi level (0 < Vp < Δ). Vp denotes the energy position of the impurity state. (c) \nand (d) The electron tunneling model when the impurity state (IS) in NaAlSi is located slightly \nbelow the Fermi level (−Δ < Vp < 0). For (a) and (c), a positive bias voltage ( Vs > 0) is applied to \nNaAlSi, the electron tunnels from STM tip to NaAlSi. For (b) and (d), a negative bias voltage ( Vs < \n0) is applied to NaAlSi, the electron tunnels from NaAlSi to STM tip. \n \n10 Figure 4 \n \nFig. 4 (a) dI/dV linecut profiles taken along the arrow shown in the inset of (a) with different tip -\nsample distance s (Δz) at 30 mK. The tip -sample distance is measured relative to the setpoint Vs = \n−10 mV, I = 150 pA. (b) dI/dV linecut profiles taken along the orange arrow in Fig. 2 (b) with \ndifferent tip -sample distance s (Δz) at 130 mK. The tip -sample distance is measured relative to the \nsetpoint Vs = −3 mV, I = 150 pA. “ r” denotes the distance from the cross point at zero bias voltage \nto the starting point for the d I/dV linecut. (c) and (d) Constant -current STM topography ( c) and \ndI/dV map taken with 0 mV bias voltage (d). (e-g) Magnetic -field dependent d I/dV spectra on the \npositions marked in (c) and (d). Setup condition: Vs = 10 mV, I = 1.2 nA. The black lines in (e) and \n(f) are the fitting results with two Voigt profiles shown by the black dashed lines. The dI/dV spectra \nin (e-g) are vertically offset for clarity and taken at 130 mK. \n" }, { "title": "2403.04604v1.Foundation_for_the_ΔSCF_Approach_in_Density_Functional_Theory.pdf", "content": "Foundation for the ∆SCF Approach in Density Functional Theory\nWeitao Yang\nDepartment of Chemistry and Department of Physics,\nDuke University, Durham, North Carolina 27708∗\nPaul W. Ayers\nDepartment of Chemistry and Chemical Biology,\nMcMaster University, Hamilton, Ontario L8S 4M1†\n(Dated: March 8, 2024)\nWe extend ground-state density-functional theory to excited states and provide the theoretical\nformulation for the widely used ∆ SCF method for calculating excited-state energies and densities.\nAs the electron density alone is insufficient to characterize excited states, we formulate excited-state\ntheory using the defining variables of a noninteracting reference system, namely (1) the excita-\ntion quantum number nsand the potential ws(r) (excited-state potential-functional theory, nPFT),\n(2) the noninteracting wavefunction Φ (Φ-functional theory, ΦFT), or (3) the noninteracting one-\nelectron reduced density matrix γs(r,r′) (density-matrix-functional theory, γsFT). We show the\nequivalence of these three sets of variables and their corresponding energy functionals. Importantly,\nthe ground and excited-state exchange-correlation energy use the same universal functional, regard-\nless of whether ( ns, ws(r)), Φ, or γs(r,r′) is selected as the fundamental descriptor of the system. We\nderive the excited-state (generalized) Kohn-Sham equations. The minimum of all three functionals\nis the ground-state energy and, for ground states, they are all equivalent to the Hohenberg-Kohn-\nSham method. The other stationary points of the functionals provide the excited-state energies and\nelectron densities, establishing the foundation for the ∆ SCF method.\nThe mathematical framework of density-functional\ntheory (DFT), in its original formulations, is based on\nthe minimum-energy variational principle and is thus re-\nstricted to ground states.[1–8] However, the importance\nof photochemical and nonadiabatic dynamics provides\nimpetus for a treatment of excited states with DFT.\nEarly successes allowed ground-state DFT to be extended\nto the lowest-energy excited state of a given symmetry.[9–\n14], but this requires symmetry-specific functionals and\nonly provides access to a tiny fraction of excited states.\nThe theoretical formulation of ground-state DFT can-\nnot be directly extended to excited states because there\nis no excited-state extension of the Hohenberg-Kohn\ntheorem.[15–18] Specifically, it is possible for the mth\nexcited state density of one system to be the nthex-\ncited state density of a different system. Therefore one\nneeds more than just the excited-state electron den-\nsity to fully specify the state of the system. This ad-\nditional information can come from: (1) A real num-\nber, or pair of integers, specifying the excitation level,\n[12, 19–22] (2) The ground-state density and the excita-\ntion level,[19, 22–28] (3) ensemble weights,[29–42] or (4)\na matrix of densities.[43–46]\nThere are also approaches based on ground-state re-\nsponse properties. For example, the dynamic linear re-\nsponse of the ground state diverges when the frequency\nis matched to an excitation energy. This motivates time-\ndependent DFT (TDDFT),[47–50] as well as closely re-\nlated approaches based on the random phase approxima-\n∗weitao.yang@duke.edu\n†ayers@mcmaster.cation (including particle-hole and particle-particle/hole-\nhole flavors), equations of motion, and propagator\ntheory.[51–56] Establishing the exactness of such ap-\nproaches is significantly harder than the analogous Levy-\nLieb formulations of ground-state DFT.[57–59]\nAn attractive practical approach is the ∆SCF method,\nwhich was first proposed in the context of what we\nnow call DFT by Slater.[60, 61] In ∆SCF, one con-\nstructs the excited-state density by choosing a non-\naufbau population of the Kohn-Sham (KS) orbitals and\nconverges the associated self-consistent field (SCF) cal-\nculation; the energy is evaluated using the ground-state\ndensity functional even though one is targeting an ex-\ncited state.[62, 63] Results are often excellent, compara-\nble to the accuracy of TDDFT for well-behaved excited\nstates but superior for excitations with charge-transfer or\nmultiple-excitation character.[64–73] This begs the ques-\ntion: why can one apply a density functional that was\ndeveloped for ground states to excited states?\nTo formulate the ∆SCF approach as a density-\nfunctional method, one needs to define an energy func-\ntional that is stationary for excited excited states.[20]\nFor the restricted set of excited-state electron densities\nthat are not the ground-state density for any system, the\nLevy constrained search functional is stationary.[74] To\nextend this result to arbitrary excited states, G¨ orling re-\nplaced the minimization of the Levy constrained search\nfunctional with a stationary principle and labeled the\nstationary states with a real number ν.[20], leading to\na procedure where the functionals depend on the νthat\nspecifies the system and state of interest. Indeed, the\ntheoretical foundation of the ∆SCF method as it is used\nin practice has not been established.[75, 76]arXiv:2403.04604v1 [physics.chem-ph] 7 Mar 20242\nHere, we show that the ∆SCF method is exact and re-\nveal the universal ground-and-excited-state energy func-\ntional. Instead of the density, our approach uses the po-\ntential as the basic variable and uses the fact that ∆SCF\ntheory only requires that excited-state electron densities,\nwhich are inherently interacting v-representable, also be\nnoninteracting v-representable. We use the fundamen-\ntal excited-state potential functionals to build practical\n∆SCF methods based on the stationary-state wavefunc-\ntions and density matrices of the noninteracting system.\nLet Ψλ\nn,wandρλ\nn,w(r) denote the wavefunction and\ndensity of the nth eigenstate of a N-electron Hamilto-\nnian with a local potential,\nˆHλ\nw=ˆT+λVee+NX\ni=1w(ri). (1)\nExtending the concept of ground-state v-representability,\nthe wavefunctions, density matrices, and densities of\neigenstates from Hamiltonians of this form are said to\nbe excited-state v-representable. Extending the ground-\nstate potential functional theory for a physical system\nwith external potential v(r),[77] define the excited-state\npotential functional:\nEλ\nv[n, w] =⟨Ψλ\nn,w|ˆHλ\nv|Ψλ\nn,w⟩ (2)\n=⟨Ψλ\nn,w|ˆHλ\nw|Ψλ\nn,w⟩+Z\ndr(v(r)−w(r))ρλ\nn,w(r).\nTheorem 1. Eλ\nv[n, w]is stationary with respect to varia-\ntions in the trial potential, w(r), ifw(r)and the physical\npotential, v(r), differ by at most a constant.\nProof. Consider a variation δw(r). For a nondegenerate\neigenstate, δ\nΨλ\nn,w\f\fˆHλ\nw\f\fΨλ\nn,w\u000b\n=R\ndrδw(r)ρλ\nn,w(r) ,\nδEλ\nv[n, w] =δ\nΨλ\nn,w\f\fˆHλ\nw\f\fΨλ\nn,w\u000b\n−Z\ndrδw(r)ρλ\nn,w(r)\n+ZZ\ndr′dr(v(r′)−w(r′))δρλ\nn,w(r′)\nδw(r)δw(r)\n=ZZ\ndr′dr(v(r)−w(r))δρλ\nn,w(r)\nδw(r′)δw(r′) (3)\nδEλ\nN,v[n, w(r)] = 0 when v(r)−w(r) is a constant.\nNote. See the supplemental material for the theorem’s\nconverse and its extension to degenerate states. This re-\nsult, and all subsequent analysis, is easily extended to\nspin-resolved (unrestricted) calculations: just replace the\nspatial variable (r)with the spin variable (x) = (r, σ).\nTheorem 1 allows us to shift variables from the many-\nelectron wavefunction to the potential: the nthexcited\nstate can be determined by making the energy functional\nE1\nv[n, w], stationary with respect to the potential w. The\npotential and nthen suffice to determine the excited-\nstate wavefunction, Ψλ\nn,v, (by solving the Schr¨ odinger\nequation).Conversely, because Hamiltonians, ˆHλ\nw, with differ-\nent potentials have different eigenfunctions and different\neigenfunctions of the same Hamiltonian are not only dif-\nferent, but orthogonal,\nLemma. There is a mapping from the space of excited-\nstate v-representable wavefunctions to their correspond-\ning potentials and excitation levels.\nThis lemma is proved in the supplementary material.\nWe now address practical KS-DFT calculations. The\nground-state Kohn-Sham assumption is that for the\nground state of any physical system, there exists a\nnoninteracting system with the same ground-state den-\nsity, ρ1\n0,w(r) = ρ0\n0,ws(r). (I.e., every ground-state v-\nrepresentable density is assumed to be ground-state\nnoninteracting v-representable.) The Hohenberg-Kohn\ntheorem then guarantees that the ground-state den-\nsity uniquely determines the (non)interacting potential,\nws(r), and the number of electrons, ergo the noninter-\nacting Hamiltonian\nˆHs=ˆT+NX\niws(ri) =ˆH0\nws, (4)\nand its eigenfunctions, which can be chosen to be Slater\ndeterminants, Φ.\nWe first recognize that ∆SCF calculations rely upon\ntheexcited-state Kohn-Sham assumption , namely that\nfor any bound state of any physical system, there ex-\nists a bound state of a noninteracting system with the\nsame electron density, ρ1\nn,w(r) =ρ0\nns,ws(r). (I.e., each\nexcited-state v-representable density is also excited-state\nnoninteracting v-representable.) For the same electron\ndensity, there may be multiple noninteracting systems\nwith different excitation quantum numbers nsand local\npotentials ws(r). Starting from any of these noninteract-\ning systems, the many-electron wavefunction Ψ n,wcan\nbe approached from the noninteracting Φ nswsthrough\nan adiabatic connecting Hamiltonian, ˆHλ\nwλ(cf. Eq. (1)),\nwhere wλ(r) is any parameterized path of local potentials\nthat satisfies the boundary conditions: w0(r) =ws(r)\nandw1(r) =w(r).[78–80] (For example, one may choose\nwλ(r) = (1 −λ)ws(r) +λw(r).[81]) The adiabatic con-\nnection maps the excitation quantum number nand the\nlocal potential w(r), or equivalently, Ψ, of a bound state\nof an interacting system to the corresponding quantities\nof a computationally convenient noninteracting system,\nnsandws(r), or equivalently, Φ nsws.\nOur excited state KS assumption is parallel to the KS\nassumption for the ground states, with one critical differ-\nence: we do not assume the one-to-one mapping between\nexcited-state densities and local potentials. It is not even\nnecessary to choose the noninteracting reference so that\nits excitation number, ns, matches that of the physical\nsystem: to construct the adiabatic connection we only\nneed assume that there exists some eigenstate of some\nnoninteracting reference system with the same density\nas the targeted eigenstate of the physical system.3\nWe now use the excited-state KS assumption to con-\nstruct the energy functionals and variational principles\nfor ∆SCF calculations. Using the excited state KS map-\nping from {ns, ws(r)}to{n, w(r)}, we can reexpress the\nexcited-state potential functional, Eq. (2), in terms of\nthe noninteracting reference system,\nEv[ns, ws] =E1\nv[n, w] =D\nΨ1\nn,w\f\f\fˆH1\nv\f\f\fΨ1\nn,wE\n.(5)\nAs a consequence of the Lemma, an excited-state nonin-\nteracting v-representable wavefunction, Φ, uniquely de-\ntermines nsandws; we can thus define a functional of\nthe KS wavefunction,\nEv[Φ] = Ev[ns, ws]. (6)\nSimilarly, as shown in the supplementary material, an\nexcited-state one-electron density matrix, γs(r,r′), de-\ntermines nsandws. We thus define a (noninteracting)\ndensity-matrix functional,\nEv[γs(r,r′)] =Ev[ns, ws]. (7)\nWe call these three equivalent approaches the excited-\nstate potential-functional theory ( nPFT), the Φ-\nfunctional theory (ΦFT), and the γs-functional theory\n(γsFT). These functionals, together with the follow-\ning variational principle, provide the foundation for the\n∆SCF method.\nTheorem 2. Ev[ns, ws]is stationary with respect to\nvariations in the trial potential ws(r), when w(r), the\nmap of ws(r)to the potential at full interaction strength,\nequals the physical potential v(r)(up to a constant).\nProof. Consider a variation δws(r). From Eqs. (2) and\n(5),\nδEv\nδws(r)=Z\ndr′\u0014δ⟨Ψn,w|Hw|Ψn,w⟩\nδw(r′)−ρn,w(r)\n+Z\ndr′′(v(r′′)−w(r′′))δρn,w(r′′)\nδw(r′)\u0015δw(r′)\nδws(r)\n=Z\ndr′′(v(r′′)−w(r′′))δρn,w(r′′)\nδws(r)(8)\nifv(r)−w(r) is a constant, δEv[ns, ws(r)] = 0 .\nNote. See the supplemental material for the theorem’s\nconverse and its extension to degenerate states.\nThe variational principle for Ev[ns, ws] with respect to\nwsimplies corresponding variational principles for Ev[Φ]\nandEv[γs]. E.g., from the definition, (6),\nEv[Φ] =⟨Φ|ˆT|Φ⟩+J[ρ] +Exc[Φ] +Z\ndrv(r)ρ(r)\nwhere we use ρ(r) =ρn,w(r) =γs(r,r) =⟨Φ|ˆρ(r)|Φ⟩,\ndenote the classical Coulomb energy as J[ρ], and define\nthe exchange-correlation energy functional as\nExc[Φ] =⟨Ψ1\nn,w|ˆT+Vee|Ψ1\nn,w⟩ − ⟨Φ|ˆT|Φ⟩ −J[ρ] (9)Subject to the ground-state KS assumption, there ex-\nists a ground-state wavefunction for a system with full\nelectron interaction, Ψ1\n0,w, and these expressions for the\nexchange-correlation energy are equivalent to the tradi-\ntional KS definition. Moreover, as the ground-state den-\nsityρ(r) uniquely determines Φ and γs, we can use the\ndensity as the basic variable and define the exchange-\ncorrelation energy (and, if we wish, even the noninter-\nacting kinetic energy) as implicit functionals of ρ(r).\nSimilarly, subject to the excited-state KS assumption,\nthere exists an excited-state wavefunction for a system\nwith full electron interaction, Ψ1\nn,w, and Exccan be ex-\npressed as a functional of the noninteracting wavefunc-\ntion, Φ, or the noninteracting density matrix, γs. This\nleads to Eqs. (9), which are applicable to both ground\nand excited states. The basic variable is ( n, ws),γsor Φ;\nthe electron density ρ(r) no longer suffices. Thus, while\nground-state KS theory is a DFT, excited-state KS the-\nory is a nPFT, ΦFT, or ρsFT. Thus, as shown in Table I,\nexcited-state KS theory subsumes ground-state KS-DFT.\nWe now derive the working equations for the stationary\npoints. For convenience, we will work directly with the\nset of Norbitals, {ϕi(r)}, that are occupied in the nth\ns\nexcited state of the noninteracting Hamiltonian (cf. Eq.\n(4)), each of which is an eigenfunction of the one-electron\nHamiltonian with the trial potential ws(r):\n\u0000\n−1\n2∇2+ws(r)\u0001\nϕi(r) =εiϕi(r). (10)\nWe can then express Theorem 2’s stationarity condition\nin terms of the occupied orbitals or, equivalently, Φ or\nγs. E.g.,\n0 =δEv[γs]\nδws(r′)=X\niZ\ndrδEv[γs]\nδϕ∗\ni(r)δϕ∗\ni(r)\nδws(r′)+c.c.\n=X\niZ\ndrδϕ∗\ni(r)\nδws(r′)\u0014\u0012\n−1\n2∇2+v(r) +vJ\u0013\nϕi(r)\n+δExc[γs(r,r′)]\nδϕ∗\ni(r′)\u0015\n+c.c. (11)\nHere vJ(r) denotes the classical Coulomb potential. This\nstationary condition is the excited-state generalization of\nthe optimized effective potential (OEP) minimum energy\ncondition for the ground states.[82, 83]\nWhen Exc[γs] isapproximated by an explicit functional\nof the electron density, Exc[ρ(r)], as in the local density\nfunctional approximation or a generalized gradient ap-\nproximation, we have\nδExc[γs]\nδϕ∗\ni(r)=δExc[ρ]\nδϕ∗\ni(r)=δExc[ρ]\nδρ(r)ϕi(r) =vxc(r)ϕi(r),\nwhere vxc(r) =δExc[ρ]\nδρ(r)is the (local) exchange-correlation\npotential. The OEP stationary condition (11) is then\nsatisfied when the trial potential is equal to the Kohn-\nSham potential constructed from the density composed\nby the orbitals occupied in the nth\nsexcited state,\nws(r) =veff(r) =v(r) +vJ(r) +vxc(r). (12)4\nTABLE I. Ground- and excited-state energy functionals and\ntheir variables. The new functionals are in the last row.\nTheoretical Formulations\nTheory States DFT nPFT ΦFT γsFT\nHK ground Ev[ρ(r)]E1\nv[0, w]\nKS ground Ev[ρ(r)]Ev[0, ws]Ev[Φ]Ev[γs(r,r′)]\nall N/A Ev[ns, ws]Ev[Φ]Ev[γs(r,r′)]\nThe ∆SCF approach is to reach the stationary condition\nby solving, self-consistently, the excited-state KS equa-\ntions obtained by inserting Eq. (12) into Eq. (10).\nWhen Exc[γs] is given as an explicit functional of\nγs(r,r′), as in Hartree-Fock and hybrid DFT,\nδExc[Φ]\nδϕ∗\ni(r)=δExc[γs]\nδϕ∗\ni(r)=Z\ndr′vxc(r,r′)ϕi(r′) (13)\nwhere the nonlocal exchange-correlation operator is de-\nfined as vxc(r,r′) =δExc[γs]\nδγs(r′,r). The noninteracting poten-\ntial can be determined by substituting (13) into the OEP\nequation (11). Alternatively, a different stationary point\ncan be obtained using a generalized Kohn-Sham (GKS)\napproach where one allows the noninteracting reference\nsystem described by the one-electron equations (10) to\nhave a nonlocal effective potential,[84]\nws(r,r′) =veff(r,r′) = (v(r) +vJ(r))δ(r−r′)+vxc(r,r′).\n(14)\nThis GKS approach was already developed in the orig-\ninal work of Kohn and Sham and was called Hartree-\nFock-Kohn-Sham method[2, 85]. GKS calculations are\nmuch easier computationally than OEP calculations, but\nthe difference in total energy and electron density, for\nground states, is usually small. However, there is a ma-\njor difference in the values of the (virtual) orbital energies\nand their interpretation.[86, 87]\nTheorem 2 establishes the foundation for ∆SCF calcu-\nlations in KS, OEP and GKS calculations. As expressed\nin the Figure, while γsor Φ are used as explicit vari-\nables in ground-state KS theory, they are inferred from\nthe noninteracting Hamiltonian and its potential ws(r),\nwhich are determined uniquely (up to a trivial constant)\nby the ground-state density. Thus ground-state KS the-\nory is still a bona fide DFT. Because excited-state den-\nsities do not uniquely determine their associated poten-\ntials, excited-state KS theory cannot be based only on the\ndensity, but it can be based on the noninteracting poten-\ntial. The excited-state electron density, as the functional\nderivative of the eigenenergy with respect to the poten-\ntial, still plays a key role.[88] Specifically, at stationary\npoints of our new functionals the density of the nonin-\nteracting system is equal to the density of corresponding\nphysical system, for both ground and excited states.\nTo construct the exchange-correlation energy func-\ntional, one uses the adiabatic connection. It is simplest\nand most conventional to choose a constant-density adia-\nbatic connection, where nλandwλ(r) are chosen so that\nFIG. 1. Variables and Their Relationship for Describing Ex-\ncited States.\nthe specified excited state density, ⟨Ψλ\nnλ,wλ|ˆρ(r)|Ψλ\nnλ,wλ⟩,\nof the adiabatic connection Hamiltonian (cf. Eq. (1)) re-\nmains constant.[20] By tracing this pathway, ( ns, ws) =\u0000\nn0, w0\u0001\n(equivalently Φ and γs) determine the proper-\nties of the relevant excited state of the interacting system,\u0000\nn1, w1\u0001\n, and the exchange-correlation functional can be\nexplicitly written,\nExc[γs] =Z1\n0dλ⟨Ψλ\nnλ,wλ|ˆVee|Ψλ\nnλ,wλ⟩ −J[ρ] (15)\nAs a functional of γsorΦ, Eq. (15) is the same for\nground and excited states. This justifies the using approx-\nimations to Excthat were originally designed for ground\nstates for excited states in the ∆SCF method. We are\ncertainly not the first to note the utility of Φ- or γs-\nfunctionals for treating excited states, as Φ or γsare al-\nways the operational variables in ∆SCF calculations.[19–\n21, 41, 60–63, 89] Our work defines the corresponding\nenergy functionals and formulates their variational prin-\nciples, based on the potential-functional formulation.\nFor ground states, the original Hohenberg-Kohn-Sham\ntheorems were restricted to v-representable densities,[1,\n2] motivating the development of functionals defined\non the broader set of N-representable densities.[3–7]\nHowever, because there is no excited-state Hohenberg-\nKohn theorem, N-representable DFT frameworks can-\nnot be directly extended to excited states.[6, 19–26] Our\nground-state potential functional theory preserves the\nv-representable framework of the original Hohenberg-\nKohn-Sham DFT[77] and has now been extended to ex-\ncited states, establishing the rigor of the ∆SCF method.\nSummarizing, we present three universal ground-and-\nexcited-state functionals Ev[n, ws] = Ev[Φ] = Ev[γs].\nThe first functional generalizes the ground-state PFT\nformulation.[77] Similarly, Ev[Φ] and Ev[γs] extend the\nground-state (generalized) Kohn-Sham approach. 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Franzke1and Christof Holzer2\n1)Otto Schott Institute of Materials Research, Friedrich Schiller University Jena, Löbdergraben 32, 07743 Jena,\nGermany\n2)Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Straße 1, 76131 Karlsruhe,\nGermany\n(*Email for correspondence: christof.holzer@kit.edu)\n(*Email for correspondence: yannick.franzke@uni-jena.de)\n(Dated: 22 March 2024)\nSpin–orbit coupling induces a current density in the ground state, which consequently requires a generalization for\nmeta -generalized gradient approximations. That is, the exchange-correlation energy has to be constructed as an ex-\nplicit functional of the current density and a generalized kinetic energy density has to be formed to satisfy theoretical\nconstraints. Herein, we generalize our previously presented formalism of spin–orbit current density functional theory\n[Holzer et al. , J. Chem. Phys. 157, 204102 (2022)] to non-magnetic and magnetic periodic systems of arbitrary di-\nmension. Besides the ground-state exchange-correlation potential, analytical derivatives such as geometry gradients\nand stress tensors are implemented. The importance of the current density is assessed for band gaps, lattice constants,\nmagnetic transitions, and Rashba splittings. For the latter, the impact of the current density may be larger than the\ndeviation between different density functional approximations.\nI. INTRODUCTION\nIn the constantly evolving field of density functional theory\n(DFT), especially the construction of meta -generalized gradi-\nent approximations (meta-GGAs) has received great attention\nover the last two decades.1–4Modern meta-GGAs use the iso-\norbital constraint and the von-Weizäcker inequality to iden-\ntify one-electron regions and, thus cancelling self-interaction\nerrors in single electron regions.3,5According to benchmark\ncalculations,4,6–21meta-GGAs outperform the preceding gen-\neralized gradient approximations (GGAs) at roughly the same\ncomputational cost. However, external magnetic fields22–25\nor spin–orbit coupling26–28necessitate further generalizations\nfor meta-GGAs to meet theoretical constrains, as the current\ndensity alters the curvature of the Fermi hole in its second-\norder Taylor expansion.29–38Density functional approxima-\ntions (DFAs) constructed by taking into account this change\nin curvature are termed “CDFT” functionals.22,39CDFT based\napproximations are for example necessary for the iso-orbital\nindicator to remain bounded between 0 and 1.35,36In external\nmagnetic fields, this correction is even necessary to guarantee\ngauge-invariance.22,25,36\nFurthermore, certain current-carrying ground states also\nheavily depend on the correct introduction of the current\ndensity, with a failure to account for them leading to\nlarge deviation for meta-GGAs.36,40We recently presented\na current-dependent formulation of density functional the-\nory for spin–orbit coupling in the molecular regime.36Here,\ninclusion of the current density leads to notable changes\nfor both closed-shell Kramers-restricted (KR) and open-shell\nKramers-unrestricted (KU) calculations.\nTo account for the change in Fermi hole curvature, the ki-\nnetic energy density τis generalized with the current density\n⃗j. In a two-component (2c) non-collinear formalism,41–51thisresults in the generalized current density\n˜τ↑,↓=τ↑,↓−|⃗j↑,↓|2\n2n↑,↓(1)\nbased on the spin-up and down quantities.36These are formed\nwith the particle and spin-magnetization contributions, e.g.\nthe spin-up and down electron density n↑,↓follows as\nn↑,↓(⃗r) =1\n2[n(⃗r)±|⃗m(⃗r)|] =1\n2[n(⃗r)±s(⃗r)] (2)\nwith the particle density n, the spin-magnetization vector ⃗m,\nand the spin density s. Therefore, the exchange-correlation\n(XC) energy of a “pure” functional2explicitly depends on the\ncurrent density\nEXC=Z\nfXCh\nn↑,↓(⃗r),γ↑↑,↑↓,↓↓(⃗r),τ↑,↓(⃗r),⃗j↑,↓(⃗r)i\nd3r\n=Z\ngXC\u0002\nn↑,↓(⃗r),γ↑↑,↑↓,↓↓(⃗r),˜τ↑,↓(⃗r)\u0003\nd3r(3)\nwhere fXCdescribes the density functional approximation\nandγ↑↓(⃗r) =1\n4\u0010\n⃗∇n↑(⃗r)\u0011\n·\u0010\n⃗∇n↓(⃗r)\u0011\n, hence γ↑↓=γ↓↑. For a\nKramers-restricted system, the spin-magnetization vector and\nthe particle current density vanish. However, the spin-current\ndensity is generally non-zero and thus it is still necessary to\nform the generalized kinetic energy density.\nIn this work, we will extend our previous CDFT\nformulation36to non-magnetic and magnetic periodic sys-\ntems. We assess the importance of the current density for\nband gaps, cell structures, magnetic transitions, and Rashba\nsplittings with common meta-GGAs. Together with previous\nstudies on the impact of spin–orbit-induced current densities\nfor meta-GGAs36,38,52–54this will further help to set guide-\nlines and recommendations for the application of CDFT to\nthe different properties and functionals for discrete and peri-\nodic systems.arXiv:2403.14420v1 [physics.chem-ph] 21 Mar 2024CDFT for Spin–Orbit Coupling: Extension to Periodic Systems 2\nII. THEORY\nThe one-particle density matrix associated with the two-\ncomponent spinor functions ⃗ψi⃗kat a given ⃗kpoint reads\nn(⃗r,⃗r′) =1\nVFBZn\n∑\ni=1Zε⃗k\ni<εF\nFBZ⃗ψi⃗k(⃗r)\u0010\n⃗ψi⃗k(⃗r′)\u0011†\nd3k\n=\u0012\nnαα(⃗r,⃗r′)nαβ(⃗r,⃗r′)\nnβα(⃗r,⃗r′)nββ(⃗r,⃗r′)\u0013 (4)\nwhere VFBZis the volume of the first Brillouin zone (FBZ),\nεithe energy eigenvalue, and εFthe Fermi level. The spinor\nfunctions are expanded with Bloch functions, φµ, based on\natomic orbital (AO) basis functions, χµ, as\n⃗ψi⃗k(⃗r) = \nψα,⃗k\ni(⃗r)\nψβ,⃗k\ni(⃗r)!\n=∑\nµ\ncα,⃗k\nµi\ncβ,⃗k\nµi\nφ⃗k\nµ(⃗r) (5)\nφ⃗k\nµ(⃗r) =1√NUC∑\n⃗Lei⃗k·⃗Lχ⃗L\nµ(⃗r). (6)\nNUCdenotes the number of electrons in the unit cell (UC) and\n⃗Lthe lattice vector. Thus, all density variables are available\nfrom the AO density matrix in position space given by\nDσσ′,⃗L⃗L′\nµν =1\nVFBZ∑\niZε⃗k\ni<εF\nFBZei⃗k·[⃗L−⃗L′]\u0010\ncσ,⃗k\nµic∗σ′,⃗k\nνi\u0011\nd3k(7)\nwith the expansion coefficients cµiandσ,σ′∈ {α,β}. The\ncomplete two-component AO density matrix reads\nD⃗L⃗L′=\u0012\nDααDαβ\nDβαDββ\u0013⃗L⃗L′\nwith\u0010\nD⃗L⃗L′\u0011†\n=D⃗L′⃗L. (8)\nIn the spirit of Bulik et al. ,47the real symmetric (RS), real an-\ntisymmetric (RA), imaginary symmetric (IS), and imaginary\nantisymmetric (OA) linear combinations\nh\nDσσ′\nRS, RAi⃗L⃗L′\n=1\n2h\nRe\u0010\nDσσ′±Dσ′σ\u0011i⃗L⃗L′\n(9)\nh\nDσσ′\nIA, ISi⃗L⃗L′\n=1\n2h\nIm\u0010\nDσσ′±Dσ′σ\u0011i⃗L⃗L′\n(10)\nare formed. Of course, the same-spin antisymmetric contri-\nbutions are zero. The electron density and its derivatives are\navailable from the symmetric linear combinations\nn(⃗r) =∑\nµν∑\n⃗L⃗L′h\nDαα\nRS+Dββ\nRSi⃗L⃗L′\nµνχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r)(11)\nmx(⃗r) =∑\nµν∑\n⃗L⃗L′2h\nDαβ\nRSi⃗L⃗L′\nµνχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r) (12)\nmy(⃗r) =∑\nµν∑\n⃗L⃗L′2h\nDαβ\nISi⃗L⃗L′\nµνχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r) (13)\nmz(⃗r) =∑\nµν∑\n⃗L⃗L′h\nDαα\nRS−Dββ\nRSi⃗L⃗L′\nµνχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r).(14)The particle current density ⃗jpand the spin-current densities\n⃗ju, with u∈ {x,y,z}, are obtained from the antisymmetric lin-\near combinations\n⃗jp(⃗r) =−i\n2∑\nµν∑\n⃗L⃗L′h\nDαα\nIA+Dββ\nIAi⃗L⃗L′\nµνξ⃗L⃗L′\nµν (15)\n⃗jx(⃗r) =−i\n2∑\nµν∑\n⃗L⃗L′2h\nDαβ\nIAi⃗L⃗L′\nµνξ⃗L⃗L′\nµν (16)\n⃗jy(⃗r) =−i\n2∑\nµν∑\n⃗L⃗L′2h\nDαβ\nRAi⃗L⃗L′\nµνξ⃗L⃗L′\nµν (17)\n⃗jz(⃗r) =−i\n2∑\nµν∑\n⃗L⃗L′h\nDαα\nIA−Dββ\nIAi⃗L⃗L′\nµνξ⃗L⃗L′\nµν (18)\nwith\nξ⃗L⃗L′\nµν=nh\n⃗∇χ⃗L\nµ(⃗r)i\nχ⃗L′\nν(⃗r)−χ⃗L\nµ(⃗r)h\n⃗∇χ⃗L′\nν(⃗r)io\n. (19)\nFollowing our molecular ansatz,36the scalar part of the XC\npotential is obtained as\nVXC,⃗L⃗L′\nµν,0=1\n2Z\u0014∂gXC\n∂n↑+∂gXC\n∂n↓\u0015\nχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r)d3r\n+1\n2Z\"\n|⃗j↑|2\n2n2\n↑∂gXC\n∂n↓+|⃗j↓|2\n2n2\n↓∂gXC\n∂˜τ↓#\nχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r)d3r\n−1\n2Z\u0014\n2∂gXC\n∂γ↑↑⃗∇n↑+2∂gXC\n∂γ↓↓⃗∇n↓+∂gXC\n∂γ↑↓(⃗∇n↑+⃗∇n↓)\u0015\nhn\n⃗∇χ⃗L\nµ(⃗r)o\nχ⃗L′\nν(⃗r)+χ⃗L\nµ(⃗r)n\n⃗∇χ⃗L′\nν(⃗r)oi\nd3r\n+Z1\n2\u0014∂gXC\n∂˜τ↑+∂gXC\n∂˜τ↓\u0015h\n⃗∇χ⃗L\nµ(⃗r)i\n·h\n⃗∇χ⃗L′\nν(⃗r)i\nd3r\n+Zi\n2\"⃗j↑\nn↑∂gXC\n∂˜τ↑+⃗j↓\nn↓∂gXC\n∂˜τ↓#\nξ⃗L⃗L′\nµνd3r\n(20)\nand the spin-magnetization part with u∈ {x,y,z}reads\nVXC,⃗L⃗L′\nµν,u=mu\n2sZ\u0014∂gXC\n∂n↑−∂gXC\n∂n↓\u0015\nχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r)d3r\n+mu\n2sZ\"\n|⃗j↑|2\n2n2\n↑∂gXC\n∂n↓−|⃗j↓|2\n2n2\n↓∂gXC\n∂˜τ↓#\nχ⃗L\nµ(⃗r)χ⃗L′\nν(⃗r)d3r\n−mu\n2sZ\u0014\n2∂gXC\n∂γ↑↑⃗∇n↑−2∂gXC\n∂γ↓↓⃗∇n↓−∂gXC\n∂γ↑↓(⃗∇n↑−⃗∇n↓)\u0015\nhn\n⃗∇χ⃗L\nµ(⃗r)o\nχ⃗L′\nν(⃗r)+χ⃗L\nµ(⃗r)n\n⃗∇χ⃗L′\nν(⃗r)oi\nd3r\n+Zmu\n2s\u0014∂gXC\n∂˜τ↑−∂gXC\n∂˜τ↓\u0015h\n⃗∇χ⃗L\nµ(⃗r)i\n·h\n⃗∇χ⃗L′\nν(⃗r)i\nd3r\n+iZmu\n2s\"⃗j↑\nn↑∂gXC\n∂˜τ↑−⃗j↓\nn↓∂gXC\n∂˜τ↓#\nξ⃗L⃗L′\nµνd3r.\n(21)CDFT for Spin–Orbit Coupling: Extension to Periodic Systems 3\nFIG. 1. (a) Dependence of the energy on the cell parameter in units of Hartree per atom for one-component (1c) restricted Kohn–Sham (RKS)\nand 1c unrestriced Kohn–Sham (UKS) at the TPSS/dhf-SVP-2c level. (b) Dependence of the energy on the cell parameter in units of Hartree\nper atom for 2c KR and 2c KU with the spin aligned along x(Sx). (c) Magnetic moment in units of Bohr’s magneton µBper atom for the\nspin contribution of 1c UKS, 2c KU SxDFT and CDFT approach. Periodicity is along the xdirection for one-dimensional systems.55The\nopen-shell solutions are energetically favored compared to the respective closed-shell solutions and the Sxalignment is preferred over Sy,z.\nResults without the current-independent TPSS functional are taken from Ref. 56. Detailed results are listed in the Supplementary Material.\nThe spin blocks of the Kohn–Sham equations are formed\nby combining the scalar and spin-magnetization contributions\nwith the respective Pauli matrices, c.f. Refs. 36 and 56. After\ntransformation to the kspace, the Kohn–Sham equations can\nbe solved as usual.\nFor non-magnetic or closed-shell system, ⃗mand⃗jpvanish\nso that a Kramers-restricted framework can be applied and\ntime-reversal symmetry57may be exploited. However, the\nspin current densities are still non-zero and hence the spin-up\nand spin-down quantities ⃗j↑,↓contribute to the XC potential\nthrough the diamagnetic or quadratic terms.36\nThe CDFT approach outlined herein is implemented in the\nriper module56,58–64of TURBOMOLE.65–67The numerical\nintegration of the exchange-correlation potential is carried\nwith the algorithm of Ref. 59. Note that the current density\nalso leads to antisymmetric contributions. Integration weights\nare constructed according to Stratmann et al.68Geometry gra-\ndients and stress tensors are implemented based on previous\nwork, as it essentially involves further derivatives of the basis\nfunctions.56,61,63We note that all integrals and the exchange\npotential for the Kohn–Sham equations are evaluated in the\nposition space, which allows to exploit sparsity. The increase\nin the computational cost by calculating the current density\ncontributions on a grid is modest—especially compared to\nthe inclusion of the current density through Hartree–Fock ex-\nchange with hybrid functions.\nIII. COMPUTATIONAL METHODS\nFirst, we study the impact of the current density on the mag-\nnetic transition of one-dimensional linear Pt chains.69–73TwoPt atoms are placed in the unit cell with the cell parameter\ndranging from 4.0 Å to 6.0 Å. Calculations are performed\nat the TPSS/dhf-SVP-2c74,75level employing Dirac–Fock ef-\nfective core potentials (DF-ECPs),76replacing 60 electrons\n(ECP-60). A Gaussian smearing of 0.01 Hartree77is used and\nak-mesh with 32 points is applied. Integration grids, con-\nvergence settings, etc. are given in the Supplementary Ma-\nterial. We note in passing that the Karlsruhe dhf-type basis\nsets were optimized for discrete systems,75however, the cor-\nresponding pob-type basis sets for periodic calculations78–83\nare not yet available with tailored extensions for spin–orbit\ntwo-component calculations.84Therefore and for consistency\nwith previous studies, we apply the dhf-type basis sets.\nSecond, the band gaps and the Rashba splitting of\nthe transition-metal dichalcogenide monolayers MoCh 2and\nWCh 2(Ch = S, Se, Te) in the hexagonal (2H) phase are stud-\nied with the M06-L,85r2SCAN,86,87TASK,88TPSS,74Tao–\nMo,5and PKZB89functionals. The GGA PBE90serves as\nreference. The dhf-TZVP-2c basis set75is applied with DF-\nECPs for Mo (ECP-28), W (ECP-60), Se (ECP-10), Te (ECP-\n28).76,91,92Structures are taken from Ref. 93. A k-mesh of\n33×××33 points is applied.\nThird, the CDFT framework is applied to complement\nour previous meta-GGA study on silver halides.56Here,\nwe consider the TPSS,74revTPSS,94,95Tao–Mo,5PKZB,89\nr2SCAN,86,87and M06-L.85We use the dhf-SVP basis set75\nand DF-ECPs are applied for Ag (ECP-28) and I (ECP-\n28).96,97Ak-mesh of 7 ×7×7 is employed.CDFT for Spin–Orbit Coupling: Extension to Periodic Systems 4\nIV. RESULTS AND DISCUSSION\nA. Linear Pt Chain\nThe one-dimensional linear Pt chain is a common reference\nsystem for a transition to a magnetic system. The closed-shell\nconfiguration constitutes the electronic ground state with a\nsmall cell parameter, whereas the magnetic open-shell solu-\ntion becomes lower in energy with increasing cell size.69–73\nThis is also confirmed at the scalar and spin–orbit TPSS\nlevel in Fig. 1. For small cell parameters from d=4.0 to\nd=4.9 Å, the open-shell initial guess converges to a closed-\nshell non-magnetic solution in the self-consistent field (SCF)\nprocedure. The most favorable total energy is found for\nd=4.7 Å in agreement with previous studies based on GGA\nfunctionals.56,72Further, CDFT and DFT lead to a very simi-\nlar potential energy surface or a similar behavior of the relative\nenergy with respect to the cell parameter d.\nAt the scalar level, the transition to a magnetic material oc-\ncurs between d≈5.2 Å and d≈5.3 Å. Inclusion of spin–orbit\ncoupling shifts this transition to a smaller dparameter of about\n5.0 Å. Here, the current-dependent variant of TPSS (cTPSS)\nleads to a lower total energy for both closed-shell and open-\nshell solutions, i.e. a more negative energy, and a larger mag-\nnetic moment. The impact of the current density is generally\nlarger for the magnetic solution than for the closed-shell state,\nsee also the Supplementary Material for detailed results. For\nthe closed-shell solution, the difference in the total energy by\ninclusion of the current density is too small to be visible in\npanel (b) of Fig. 1. This finding is qualitatively in line with\nour previous studies on molecular systems.36\nInclusion of the current density also consistently leads to a\nlarger magnetic moment. For instance, TPSS predicts a mag-\nnetic moment of 1.04 µ B/atom at d=6.0 Å, whereas a value\nof 1.13 µ B/atom is found with cTPSS. Generally, an increase\nin the range of 5–8 % is observed after the transition to a mag-\nnetic solution.\nB. Transition-Metal Dichalcogenide Monolayers\nTransition-metal dichalcogenide monolayers have many in-\nteresting physical properties such as the quantum spin Hall,98\nnon-linear anomalous Hall,99and Rashba effects.100In the\nH2 phase, time-reversal symmetry holds for the non-magnetic\nsystems but space-inversion symmetry is lost. Thus, spin–\norbit coupling lifts the degeneracy of the valence band at\nthe K point in the Brillouin zone. For the Mo and W sys-\ntems, this Rashba splitting is very pronounced and values be-\ntween 0.1 and 0.5 eV are obtained with relativistic all-electron\nmethods.93,101\nBand gaps and Rashba splittings at the K point obtained\nwith DFT and CDFT approaches are listed in Tab. I. One\nthe one hand, the impact on the band gaps is rather small,\nwith TASK and r2SCAN showing the largest changes but not\nexceeding 0.1 eV . Here, the deviation between the different\nDFAs is far larger. One the other hand, the Rashba splitting\nis very sensitive to the inclusion of the current density. ForTABLE I. Band gaps and Rashba splittings of the valence band at\nthe K point for transition-metal dichalcogenide monolayers at the 1c\nDFT, 2c DFT, and 2c CDFT level with the dhf-TZVP-2c basis set\nand DF-ECPs for all atoms. All values in eV . Results for MoS 2and\nWS 2are listed in the Supplementary Material.\nBand Gap Rashba Splitting\nSystem DFA 1c DFT 2c DFT 2c CDFT 2c DFT 2c CDFT\nMoSe 2PBE 1 .554 1 .453 – 0 .166 –\nPKZB 1 .558 1 .455 1 .455 0 .166 0 .169\nTao–Mo 1 .561 1 .459 1 .458 0 .166 0 .170\nTPSS 1 .563 1 .459 1 .458 0 .167 0 .174\nM06-L 1 .563 1 .460 1 .455 0 .170 0 .181\nr2SCAN 1 .657 1 .554 1 .544 0 .166 0 .189\nTASK 1 .735 1 .629 1 .607 0 .172 0 .218\nMoTe 2PBE 1 .157 1 .039 – 0 .185 –\nPKZB 1 .160 1 .042 1 .042 0 .186 0 .189\nTao–Mo 1 .168 1 .051 1 .049 0 .185 0 .188\nTPSS 1 .170 1 .051 1 .048 0 .185 0 .195\nM06-L 1 .182 1 .060 1 .053 0 .192 0 .209\nr2SCAN 1 .235 1 .116 1 .101 0 .186 0 .218\nTASK 1 .287 1 .161 1 .126 0 .197 0 .269\nWSe 2PBE 1 .627 1 .324 – 0 .409 –\nPKZB 1 .652 1 .377 1 .374 0 .407 0 .410\nTao–Mo 1 .667 1 .371 1 .367 0 .403 0 .411\nTPSS 1 .660 1 .379 1 .371 0 .409 0 .420\nM06-L 1 .668 1 .413 1 .405 0 .422 0 .440\nr2SCAN 1 .774 1 .482 1 .452 0 .419 0 .457\nTASK 1 .867 1 .604 1 .571 0 .440 0 .514\nWTe 2PBE 1 .219 0 .933 – 0 .423 –\nPKZB 1 .227 0 .953 0 .951 0 .423 0 .427\nTao–Mo 1 .241 0 .963 0 .957 0 .419 0 .426\nTPSS 1 .237 0 .960 0 .954 0 .422 0 .435\nM06-L 1 .253 0 .968 0 .957 0 .431 0 .456\nr2SCAN 1 .325 1 .039 1 .005 0 .436 0 .483\nTASK 1 .388 1 .096 1 .050 0 .465 0 .566\ninstance, the results change from 0.436 eV to 0.483 eV and\n0.465 eV to 0.566 eV for WTe 2with r2SCAN and TASK, re-\nspectively.\nThe most pronounced current-density effects are found for\nTASK throughout all systems, which is in line with molecular\nstudies.36,53,102Here, the current density increase the Rashba\nsplitting by 25 % on average. r2SCAN ranks second in this re-\ngard with 13 % followed by M06-L with 6 %. For PKZB, Tao–\nMo and TPSS, changes of only 1–3 % are observed. Among\nthe different monolayers, MoTe 2leads to the largest impact\nof the current density for all DFAs, with a relative deviation\nof 36 % between DFT and CDFT for TASK. The impact of\nthe current density for the DFAs can be rationalized by the\nenhancement factor.102,103\nC. Silver Halide Crystals\nThe band gaps and optimized lattice constant of AgI with\nvarious meta-GGAs are listed in Tab. II. Results for AgCl and\nAgBr are presented in the Supplementary Material. Overall,\nthe current density is of minor importance for the band gaps.CDFT for Spin–Orbit Coupling: Extension to Periodic Systems 5\nTABLE II. Optimized lattice constant a(in Å, rocksalt structure)\nof three-dimensional AgI and band gaps (in eV) at high symmetry\npoints of the first Brillouin zone with various density functional ap-\nproximations and the dhf-SVP basis sets. A “c” indicates the current-\ndependent variant of the DFA. Results for DFAs without inclusion of\nthe current density are taken from Ref. 56, except for M06-L. Calcu-\nlations are performed without dispersion correction (no D3) and with\nthe D3 correction using Becke–Johnson damping (D3-BJ).\nDFA Dispersion a L–L Γ–Γ X–X L–X\nTPSS no D3 6 .153 3 .249 2 .047 2 .937 0 .583\ncTPSS no D3 6 .150 3 .244 2 .052 2 .940 0 .579\nrevTPSS no D3 6 .116 3 .149 2 .132 2 .998 0 .537\ncrevTPSS no D3 6 .098 3 .122 2 .161 3 .014 0 .512\nTao–Mo no D3 6 .071 3 .099 2 .343 3 .108 0 .616\ncTao–Mo no D3 6 .069 3 .086 2 .339 3 .109 0 .619\nPKZB no D3 6 .200 3 .409 2 .153 2 .987 0 .826\ncPKZB no D3 6 .210 3 .422 2 .138 2 .979 0 .839\nr2SCAN no D3 6 .159 3 .776 2 .501 3 .237 0 .911\ncr2SCAN no D3 6 .156 3 .772 2 .504 3 .240 0 .907\nM06-L no D3 6 .325 3 .511 1 .891 2 .976 1 .116\ncM06-L no D3 6 .327 3 .512 1 .889 2 .974 1 .117\nTPSS D3-BJ 5 .982 2 .988 2 .352 3 .098 0 .343\ncTPSS D3-BJ 5 .980 2 .985 2 .354 3 .099 0 .341\nrevTPSS D3-BJ 5 .949 2 .883 2 .443 3 .156 0 .291\ncrevTPSS D3-BJ 5 .952 2 .889 2 .435 3 .152 0 .296\nTao–Mo D3-BJ 6 .068 3 .095 2 .347 3 .110 0 .613\ncTao–Mo D3-BJ 6 .062 3 .087 2 .358 3 .116 0 .604\nr2SCAN D3-BJ 6 .156 3 .773 2 .506 3 .240 0 .907\ncr2SCAN D3-BJ 6 .155 3 .771 2 .506 3 .241 0 .905\nChanges are in the order of meV . These results are in qualita-\ntive agreement with the two-dimensional MoTe 2monolayer,\nwhich consists of atoms from the same row of the periodic\ntable of elements.\nLikewise the current density does not lead to major changes\nfor the cell structure and the lattice constant. The small impact\non the lattice constant can be rationalized by the comparably\nsmall impact of spin–orbit coupling on the cell structure.56D3\ndispersion correction with Becke–Johnson damping8,104–107\n(D3-BJ) leads to much larger changes than the application of\nCDFT. Therefore, other computational parameters than the in-\nclusion of the current density for meta-GGAs are more impor-\ntant for the cell structures of the silver halide crystals.\nV. CONCLUSION\nWe have extended our previous formulation of spin–orbit\ncurrent density functional theory to periodic systems of arbi-\ntrary dimension. The impact of the current density was as-\nsessed for various properties of non-magnetic and magnetic\nsystems. Here, the band gaps and lattice constants are not no-\ntably affected. In contrast, the Rashba splitting which is only\ndue to spin–orbit coupling is substantially affected. The in-\nclusion of the current density for a given functional may lead\nto larger changes than the deviation of the results among dif-\nferent functionals.With the present work, CDFT is now applicable to a wide\nrange of chemical and physical properties of discrete and pe-\nriodic systems, including analytic first-order property calcula-\ntions. We generally recommend to include the current density\nfor r2SCAN and the Minnesota functionals, if available in the\nused electronic structure code. For TASK, inclusion of the\ncurrent density is clearly mandatory as it leads to substantial\nchanges of the results.\nSUPPLEMENTARY MATERIAL\nSupporting Information is available with all computational\ndetails and data.\nACKNOWLEDGMENTS\nWe thank Fabian Pauly (Augsburg) and Marek Sierka\n(Jena) for helpful comments. Y .J.F. gratefully acknowledges\nsupport via the Walter–Benjamin programme funded by the\nDeutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) — 518707327. C.H. gratefully acknowledges\nfunding by the V olkswagen Foundation.\nAUTHOR DECLARATIONS\nConflict of Interest\nThe authors have no conflicts to disclose.\nAuthor Contributions\nYannick J. Franzke : Conceptualization (equal); Data cu-\nration (lead); Formal analysis (lead); Investigation (equal);\nMethodology (lead); Software (lead); Validation (equal); Vi-\nsualization (lead); Writing – original draft (lead); Writing –\nreview & editing (equal).\nChristof Holzer : Conceptualization (equal); Data cura-\ntion (supporting); Formal analysis (supporting); Investigation\n(equal); Methodology (supporting); Software (supporting);\nValidation (equal); Writing – original draft (supporting); Writ-\ning – review & editing (equal).\nDATA AVAILABILITY STATEMENT\nThe data that support the findings of this study are available\nwithin the article and its supplementary material.\nREFERENCES\n1K. Burke, J. Chem. Phys. 136, 150901 (2012).\n2A. D. Becke, J. Chem. Phys. 140, 18A301 (2014).CDFT for Spin–Orbit Coupling: Extension to Periodic Systems 6\n3J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, 036402\n(2015).\n4N. Mardirossian and M. Head-Gordon, Mol. 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Phys. 154, 061101 (2021).Supplementary Material for “Current density functional framework for spin–orbit\ncoupling: Extension to periodic systems”\nYannick J. Franzke1and Christof Holzer2\n1)Otto Schott Institute of Materials Research, Friedrich Schiller University Jena,\nLöbdergraben 32, 07743 Jena, Germany\n2)Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT),\nWolfgang-Gaede-Straße 1, 76131 Karlsruhe, Germany\n(*Email for correspondence: christof.holzer@kit.edu)\n(*Email for correspondence: yannick.franzke@uni-jena.de)\n(Dated: 21 March 2024)\n1CONTENTS\nI. Linear One-Dimensional Pt Chain 3\nA. Computational Details and Detailed Results 3\nII. Two-Dimensional Transition-Metal Dichalcogenide Monolayers 5\nA. Computational Details 5\nB. Results 6\nIII. Three-Dimensional Silver Halide Crystals 10\nA. Computational Details 10\nB. Results 11\nReferences 14\n2I. LINEAR ONE-DIMENSIONAL PT CHAIN\nA. Computational Details and Detailed Results\nTwo Pt atoms are placed in a unit cell with a cell parameter d. Relativistic effects are introduced\nthrough small-core Dirac–Fock effective core potentials (ECP-60).1The dhf-SVP-2c orbital and\nauxiliary basis sets are applied2and large integration grids3,4(grid size 4) are employed for the\nnumerical integration of the exchange-correlation potential5with the TPSS functional.6Note that\nthe riper module uses the resolution of the identity approximation (RI- J) in combination with the\ncontinuous fast multipole method (CFMM).7,8Default settings are employed for RI- J, CFMM,\nand the transformation to the orthogonal basis set for the diagonalization of the Kohn–Sham or\nFock matrix.8,9A Gaussian smearing of 0 .01 Hartree is applied10and a k-mesh of 32 points is\nused. The self-consistent field (SCF) procedure is converged with a threshold of 10−8Hartree.\nKramers-restriced (KR) and Kramers-unrestricted (KU) calculations are started from a molecular\nextended Hückel guess with a closed-shell configuration and an open-shell configuration using\nfour unpaired electrons, respectively. For the KU calculations, the initial wavefunction is chosen\nto be an eigenfunction of the spin operator ˆSx. Periodicity is also assumed along the xdirection.9\nStructures in TURBOMOLE format9,11–15are available from Ref. 16. Results are listed in Tab. I.\nHere, the CDFT energies are always lower than the DFT energy, i.e. a more negative energy is\nfound.\n3TABLE I. Total SCF energies in Hartree obtained with the 2c KR DFT, 2c KU DFT, 2c KR CDFT, and 2c KU CDFT approach, and energies for the limit\nof a vanishing Gaussian smearing (limit). The expectation values of spin Sxare listed for the open-shell calculations. The TPSS functional6is employed\nwith the dhf-SVP-2c basis set2and a Dirac–Fock ECP for Pt.12c DFT results are taken from Ref. 16.\nd KR DFT limit KR DFT KR CDFT limit KR CDFT KU DFT limit KU DFT KU DFT ⟨Sx⟩KU CDFT limit KU CDFT KU CDFT ⟨Sx⟩\n4.0−239 .0074 −239 .0065 −239 .0077 −239 .0067 −239 .0074 −239 .0065 3 .62×10−5−239 .0076 −239 .0067 2 .64×10−8\n4.1−239 .0592 −239 .0580 −239 .0594 −239 .0583 −239 .0592 −239 .0580 2 .35×10−5−239 .0594 −239 .0583 −1.00×10−14\n4.2−239 .0975 −239 .0961 −239 .0977 −239 .0963 −239 .0975 −239 .0961 4 .72×10−5−239 .0977 −239 .0963 1 .00×10−14\n4.3−239 .1251 −239 .1234 −239 .1254 −239 .1237 −239 .1251 −239 .1234 3 .41×10−4−239 .1253 −239 .1236 2 .08×10−4\n4.4−239 .1443 −239 .1423 −239 .1446 −239 .1425 −239 .1443 −239 .1423 4 .77×10−4−239 .1446 −239 .1425 7 .00×10−14\n4.5−239 .1568 −239 .1544 −239 .1571 −239 .1547 −239 .1568 −239 .1544 5 .10×10−4−239 .1570 −239 .1546 1 .69×10−3\n4.6−239 .1641 −239 .1613 −239 .1644 −239 .1616 −239 .1641 −239 .1613 5 .05×10−4−239 .1643 −239 .1615 3 .40×10−3\n4.7−239 .1672 −239 .1640 −239 .1675 −239 .1643 −239 .1672 −239 .1640 3 .94×10−4−239 .1675 −239 .1642 5 .12×10−3\n4.8−239 .1672 −239 .1636 −239 .1675 −239 .1639 −239 .1672 −239 .1636 3 .97×10−3−239 .1675 −239 .1638 3 .72×10−2\n4.9−239 .1648 −239 .1607 −239 .1651 −239 .1610 −239 .1650 −239 .1611 2 .54×10−1−239 .1656 −239 .1618 3 .81×10−1\n5.0−239 .1606 −239 .1560 −239 .1609 −239 .1564 −239 .1614 −239 .1575 4 .91×10−1−239 .1621 −239 .1583 5 .62×10−1\n5.1−239 .1550 −239 .1501 −239 .1554 −239 .1504 −239 .1565 −239 .1525 6 .27×10−1−239 .1573 −239 .1534 6 .78×10−1\n5.2−239 .1486 −239 .1432 −239 .1489 −239 .1436 −239 .1506 −239 .1464 7 .13×10−1−239 .1515 −239 .1474 7 .59×10−1\n5.3−239 .1415 −239 .1357 −239 .1418 −239 .1361 −239 .1439 −239 .1395 7 .76×10−1−239 .1449 −239 .1406 8 .18×10−1\n5.4−239 .1340 −239 .1278 −239 .1343 −239 .1282 −239 .1367 −239 .1320 8 .30×10−1−239 .1379 −239 .1332 8 .76×10−1\n5.5−239 .1262 −239 .1197 −239 .1266 −239 .1201 −239 .1293 −239 .1241 8 .84×10−1−239 .1306 −239 .1256 9 .39×10−1\n5.6−239 .1183 −239 .1115 −239 .1187 −239 .1118 −239 .1217 −239 .1162 9 .27×10−1−239 .1232 −239 .1179 9 .98×10−1\n5.7−239 .1104 −239 .1032 −239 .1108 −239 .1036 −239 .1141 −239 .1083 9 .69×10−1−239 .1159 −239 .1104 1 .05\n5.8−239 .1025 −239 .0950 −239 .1029 −239 .0954 −239 .1067 −239 .1006 1 .00 −239 .1087 −239 .1030 1 .09\n5.9−239 .0948 −239 .0869 −239 .0952 −239 .0873 −239 .0994 −239 .0931 1 .03 −239 .1017 −239 .0958 1 .11\n6.0−239 .0873 −239 .0791 −239 .0877 −239 .0795 −239 .0924 −239 .0859 1 .04 −239 .0948 −239 .0887 1 .13\n4II. TWO-DIMENSIONAL TRANSITION-METAL DICHALCOGENIDE\nMONOLAYERS\nA. Computational Details\nBand gaps and Rashba splittings are calculated for the transition-metal dichalcogenide mono-\nlayers MoS 2, MoSe 2, MoTe 2, WS 2, WSe 2, and WTe 2with the PBE,17M06-L,18r2SCAN,19,20\nTASK,21TPSS,6Tao–Mo,22and PKZB23density functional approximations (DFAs). Note that\nwe use Libxc24–26for M06-L, r2SCAN, TASK, Tao–Mo, and PKZB. The dhf-TZVP-2c orbital\nand auxiliary basis sets are applied.2Originally, the dhf-type basis sets only employ ECPs for el-\nements beyond Kr. That is, Dirac–Fock ECPs are employed for Mo (ECP-28), W (ECP-60), and\nTe (ECP-28).1,27,28However, this leads to rather large deviations from the relativistic all-electron\ncalculations at the PBE level for MoSe 2and WSe 2.29,30See Tabs. II and III. Therefore, we applied\na small-core Dirac–Fock ECP for Se (ECP-10) as well.27,31The corresponding results are shown\nin Tabs. IV and V.\nLarge integration grids3,4(grid size 3) are employed for the numerical integration of the\nexchange-correlation potential.5Again, default settings are applied for RI- J, CFMM, and the\ntransformation to the orthogonal basis set for the diagonalization of the Kohn–Sham or Fock\nmatrix.8,9The SCF procedure is converged with a threshold of 10−7Hartree and a k-mesh of\n33×33 points is used. Structures are taken from Ref. 29.\n5B. Results\nTABLE II. Band gaps and Rashba splittings of the valence band at the K point for transition-metal dichalco-\ngenide monolayers at the 1c DFT, 2c DFT, and 2c CDFT level with the dhf-TZVP-2c basis set and DF-ECPs\nfor Mo and Te. All values in eV .\nBand Gap Rashba Splitting\nSystem DFA 1c DFT 2c DFT 2c CDFT 2c DFT 2c CDFT\nMoS 2 PBE 1 .827 1 .757 1 .755 0 .102 0 .102\nM06-L 1 .814 1 .772 1 .772 0 .107 0 .112\nr2SCAN 1 .940 1 .884 1 .879 0 .103 0 .116\nTASK 2 .035 1 .979 1 .968 0 .105 0 .130\nTPSS 1 .829 1 .775 1 .774 0 .102 0 .106\nTao–Mo 1 .828 1 .774 1 .773 0 .101 0 .104\nPKZB 1 .829 1 .775 1 .774 0 .101 0 .103\nMoSe 2PBE 1 .560 1 .504 1 .504 0 .102 0 .102\nM06-L 1 .567 1 .509 1 .508 0 .107 0 .113\nr2SCAN 1 .658 1 .602 1 .596 0 .104 0 .119\nTASK 1 .738 1 .681 1 .669 0 .107 0 .135\nTPSS 1 .566 1 .510 1 .508 0 .103 0 .107\nTao–Mo 1 .564 1 .508 1 .508 0 .102 0 .105\nPKZB 1 .562 1 .506 1 .506 0 .102 0 .104\nMoTe 2PBE 1 .157 1 .039 1 .039 0 .185 0 .185\nM06-L 1 .182 1 .060 1 .053 0 .192 0 .209\nr2SCAN 1 .235 1 .116 1 .101 0 .186 0 .218\nTASK 1 .287 1 .161 1 .126 0 .197 0 .269\nTPSS 1 .170 1 .051 1 .048 0 .185 0 .195\nTao–Mo 1 .168 1 .051 1 .049 0 .185 0 .188\nPKZB 1 .160 1 .042 1 .042 0 .186 0 .189\n6TABLE III. Band gaps and Rashba splittings of the valence band at the K point for transition-metal dichalco-\ngenide monolayers at the 1c DFT, 2c DFT, and 2c CDFT level with the dhf-TZVP-2c basis set and DF-ECPs\nfor W and Te. All values in eV .\nBand Gap Rashba Splitting\nSystem DFA 1c DFT 2c DFT 2c CDFT 2c DFT 2c CDFT\nWS 2 PBE 1 .999 1 .697 1 .697 0 .343 0 .343\nM06-L 2 .024 1 .803 1 .798 0 .357 0 .369\nr2SCAN 2 .156 1 .868 1 .845 0 .351 0 .379\nTASK 2 .261 2 .037 2 .006 0 .364 0 .416\nTPSS 2 .031 1 .758 1 .752 0 .343 0 .351\nTao–Mo 2 .040 1 .754 1 .749 0 .337 0 .343\nPKZB 2 .023 1 .752 1 .750 0 .339 0 .342\nWSe 2 PBE 1 .654 1 .385 1 .385 0 .354 0 .354\nM06-L 1 .676 1 .447 1 .440 0 .366 0 .381\nr2SCAN 1 .782 1 .534 1 .507 0 .363 0 .395\nTASK 1 .872 1 .637 1 .610 0 .378 0 .440\nTPSS 1 .672 1 .436 1 .429 0 .353 0 .362\nTao–Mo 1 .680 1 .433 1 .427 0 .347 0 .354\nPKZB 1 .664 1 .435 1 .433 0 .350 0 .353\nWTe 2 PBE 1 .219 0 .933 0 .933 0 .423 0 .423\nM06-L 1 .253 0 .968 0 .957 0 .431 0 .456\nr2SCAN 1 .325 1 .039 1 .005 0 .436 0 .483\nTASK 1 .388 1 .096 1 .050 0 .465 0 .566\nTPSS 1 .237 0 .960 0 .954 0 .422 0 .435\nTao–Mo 1 .241 0 .963 0 .957 0 .419 0 .426\nPKZB 1 .227 0 .953 0 .951 0 .423 0 .427\n7TABLE IV . Band gaps and Rashba splittings of the valence band at the K point for transition-metal dichalco-\ngenide monolayers at the 1c DFT, 2c DFT, and 2c CDFT level with the dhf-TZVP-2c basis set and DF-ECPs\nfor Mo, Se, and Te. All values in eV .\nBand Gap Rashba Splitting\nSystem DFA 1c DFT 2c DFT 2c CDFT 2c DFT 2c CDFT\nMoS 2PBE 1 .827 1 .757 1 .755 0 .102 0 .102\nM06-L 1 .814 1 .772 1 .772 0 .107 0 .112\nr2SCAN 1 .940 1 .884 1 .879 0 .103 0 .116\nTASK 2 .035 1 .979 1 .968 0 .105 0 .130\nTPSS 1 .829 1 .775 1 .774 0 .102 0 .106\nTao–Mo 1 .828 1 .774 1 .773 0 .101 0 .104\nPKZB 1 .829 1 .775 1 .774 0 .101 0 .103\nMoSe 2PBE 1 .554 1 .453 – 0 .166 –\nPKZB 1 .558 1 .455 1 .455 0 .166 0 .169\nTao–Mo 1 .561 1 .459 1 .458 0 .166 0 .170\nTPSS 1 .563 1 .459 1 .458 0 .167 0 .174\nM06-L 1 .563 1 .460 1 .455 0 .170 0 .181\nr2SCAN 1 .657 1 .554 1 .544 0 .166 0 .189\nTASK 1 .735 1 .629 1 .607 0 .172 0 .218\nMoTe 2PBE 1 .157 1 .039 – 0 .185 –\nPKZB 1 .160 1 .042 1 .042 0 .186 0 .189\nTao–Mo 1 .168 1 .051 1 .049 0 .185 0 .188\nTPSS 1 .170 1 .051 1 .048 0 .185 0 .195\nM06-L 1 .182 1 .060 1 .053 0 .192 0 .209\nr2SCAN 1 .235 1 .116 1 .101 0 .186 0 .218\nTASK 1 .287 1 .161 1 .126 0 .197 0 .269\n8TABLE V . Band gaps and Rashba splittings of the valence band at the K point for transition-metal dichalco-\ngenide monolayers at the 1c DFT, 2c DFT, and 2c CDFT level with the dhf-TZVP-2c basis set and DF-ECPs\nfor W, Se, and Te. All values in eV .\nBand Gap Rashba Splitting\nSystem DFA 1c DFT 2c DFT 2c CDFT 2c DFT 2c CDFT\nWS 2 PBE 1 .999 1 .697 1 .697 0 .343 0 .343\nM06-L 2 .024 1 .803 1 .798 0 .357 0 .369\nr2SCAN 2 .156 1 .868 1 .845 0 .351 0 .379\nTASK 2 .261 2 .037 2 .006 0 .364 0 .416\nTPSS 2 .031 1 .758 1 .752 0 .343 0 .351\nTao–Mo 2 .040 1 .754 1 .749 0 .337 0 .343\nPKZB 2 .023 1 .752 1 .750 0 .339 0 .342\nWSe 2PBE 1 .627 1 .324 – 0 .409 –\nPKZB 1 .652 1 .377 1 .374 0 .407 0 .410\nTao–Mo 1 .667 1 .371 1 .367 0 .403 0 .411\nTPSS 1 .660 1 .379 1 .371 0 .409 0 .420\nM06-L 1 .668 1 .413 1 .405 0 .422 0 .440\nr2SCAN 1 .774 1 .482 1 .452 0 .419 0 .457\nTASK 1 .867 1 .604 1 .571 0 .440 0 .514\nWTe 2PBE 1 .219 0 .933 – 0 .423 –\nPKZB 1 .227 0 .953 0 .951 0 .423 0 .427\nTao–Mo 1 .241 0 .963 0 .957 0 .419 0 .426\nTPSS 1 .237 0 .960 0 .954 0 .422 0 .435\nM06-L 1 .253 0 .968 0 .957 0 .431 0 .456\nr2SCAN 1 .325 1 .039 1 .005 0 .436 0 .483\nTASK 1 .388 1 .096 1 .050 0 .465 0 .566\n9III. THREE-DIMENSIONAL SILVER HALIDE CRYSTALS\nA. Computational Details\nSilver halide systems are commonly studied with relativistic methods32–34and we have previ-\nously calculated the band gaps and cell structure with various meta-GGAs in DFT framework.16\nThus, we complement these results with the CDFT treatment. Computational settings are chosen\naccordingly. That is, the dhf-SVP orbital and auxiliary basis sets2are applied with small-core\nDirac–Fock ECPs (ECP-28) for Ag and I.35,36The TPSS,6revTPSS,37,38Tao–Mo,22PKZB,23\nr2SCAN,19,20and M06-L18density functional approximations are used with large integration grids\n(grid size 4).3,4We use Libxc24–26for revTPSS, Tao–Mo, PKZB and M06-L. The D3 dispersion\ncorrection is applied if available for the given DFA.39–43Default settings are chosen for RI- J,\nCFMM, and the transformation to the orthogonal basis set for the diagonalization of the Kohn–\nSham or Fock matrix.8,9An SCF convergence threshold of 10−7Hartree is applied and a k-mesh\nof 7×7×7 point is employed. Cell structures are optimized with weight derivatives and an energy\nthreshold of 10−6Hartree.44Results for AgCl, AgBr, and AgI are listed in Tabs. VI, VII, and VIII,\nrespectively. Inclusion of the current density in the exchange-correlation potential is indicated by\na “c” for the functional acronym. Initial cell structures were taken from Ref. 16.\n10B. Results\nTABLE VI. Optimized lattice constant a(in Å, rocksalt structure) of three-dimensional AgCl and band gaps\n(in eV) at high symmetry points of the first Brillouin zone with various density functional approximations\nand the dhf-SVP basis sets.2A Dirac–Fock ECP is applied for Ag (ECP-28).35A “c” indicates the current-\ndependent variant of the DFA. Results for DFAs without inclusion of the current density are taken from\nRef. 16, except for M06-L. Calculations are performed without dispersion correction (no D3) and with the\nD3 correction using Becke–Johnson damping (D3-BJ).39,40\nDFA Dispersion a L–L Γ–Γ X–X L– Γ\nTPSS no D3 5 .586 4 .553 3 .042 4 .135 0 .966\ncTPSS no D3 5 .578 4 .539 3 .056 4 .138 0 .964\nrevTPSS no D3 5 .561 4 .381 3 .023 4 .126 0 .890\ncrevTPSS no D3 5 .542 4 .350 3 .059 4 .135 0 .885\nTao–Mo no D3 5 .541 4 .242 3 .099 4 .238 0 .971\ncTao-Mo no D3 5 .518 4 .212 3 .150 4 .251 0 .971\nPKZB no D3 5 .636 4 .593 3 .099 4 .259 1 .176\ncPKZB no D3 5 .644 4 .604 3 .086 4 .256 1 .178\nr2SCAN no D3 5 .576 5 .017 3 .545 4 .564 1 .413\ncr2SCAN no D3 5 .564 5 .001 3 .570 4 .569 1 .413\nM06-L no D3 5 .636 4 .947 3 .357 4 .654 1 .464\ncM06-L no D3 5 .640 4 .951 3 .349 4 .649 1 .464\nTPSS D3-BJ 5 .498 4 .400 3 .213 4 .175 0 .943\ncTPSS D3-BJ 5 .461 4 .334 3 .293 4 .193 0 .937\nrevTPSS D3-BJ 5 .476 4 .233 3 .204 4 .169 0 .873\ncrevTPSS D3-BJ 5 .445 4 .179 3 .274 4 .184 0 .870\nTao–Mo D3-BJ 5 .511 4 .203 3 .166 4 .255 0 .971\ncTao-Mo D3-BJ 5 .499 4 .186 3 .194 4 .262 0 .972\nr2SCAN D3-BJ 5 .517 4 .943 3 .677 4 .592 1 .416\ncr2SCAN D3-BJ 5 .514 4 .939 3 .683 4 .592 1 .416\n11TABLE VII. Optimized lattice constant a(in Å, rocksalt structure) of three-dimensional AgBr and band gaps\n(in eV) at high symmetry points of the first Brillouin zone with various density functional approximations\nand the dhf-SVP basis sets.2A Dirac–Fock ECP is applied for Ag (ECP-28).35A “c” indicates the current-\ndependent variant of the DFA. Results for DFAs without inclusion of the current density are taken from\nRef. 16, except for M06-L. Calculations are performed without dispersion correction (no D3) and with the\nD3 correction using Becke–Johnson damping (D3-BJ).39,40\nDFA Dispersion a L–L Γ–Γ X–X L– Γ\nTPSS no D3 5 .812 3 .988 2 .899 3 .677 0 .950\ncTPSS no D3 5 .802 3 .971 2 .918 3 .679 0 .949\nrevTPSS no D3 5 .784 3 .871 2 .963 3 .667 0 .936\ncrevTPSS no D3 5 .764 3 .839 2 .999 3 .672 0 .936\nTao–Mo no D3 5 .738 3 .781 3 .140 3 .792 1 .079\ncTao-Mo no D3 5 .736 3 .779 3 .143 3 .792 1 .079\nPKZB no D3 5 .868 4 .128 2 .985 3 .812 1 .189\ncPKZB no D3 5 .875 4 .138 2 .973 3 .812 1 .190\nr2SCAN no D3 5 .811 4 .501 3 .361 4 .094 1 .416\ncr2SCAN no D3 5 .800 4 .486 3 .384 4 .095 1 .418\nM06-L no D3 5 .909 4 .440 3 .134 4 .204 1 .452\ncM06-L no D3 5 .916 4 .447 3 .120 4 .199 1 .450\nTPSS D3-BJ 5 .708 3 .815 3 .092 3 .708 0 .951\ncTPSS D3-BJ 5 .673 3 .754 3 .163 3 .719 0 .956\nrevTPSS D3-BJ 5 .690 3 .715 3 .143 3 .695 0 .942\ncrevTPSS D3-BJ 5 .655 3 .653 3 .218 3 .707 0 .948\nTao–Mo D3-BJ 5 .743 3 .789 3 .128 3 .790 1 .077\ncTao–Mo D3-BJ 5 .731 3 .771 3 .155 3 .794 1 .080\nr2SCAN D3-BJ 5 .774 4 .448 3 .444 4 .104 1 .423\ncr2SCAN D3-BJ 5 .758 4 .429 3 .473 4 .107 1 .426\n12TABLE VIII. Optimized lattice constant a(in Å, rocksalt structure) of three-dimensional AgI and band gaps\n(in eV) at high symmetry points of the first Brillouin zone with various density functional approximations\nand the dhf-SVP basis sets.2Dirac–Fock ECPs are applied for Ag (ECP-28) and I (ECP-28).35,36A “c” in-\ndicates the current-dependent variant of the DFA. Results for DFAs without inclusion of the current density\nare taken from Ref. 16, except for M06-L. Calculations are performed without dispersion correction (no\nD3) and with the D3 correction using Becke–Johnson damping (D3-BJ).39,40\nDFA Dispersion a L–L Γ–Γ X–X L–X\nTPSS no D3 6 .153 3 .249 2 .047 2 .937 0 .583\ncTPSS no D3 6 .150 3 .244 2 .052 2 .940 0 .579\nrevTPSS no D3 6 .116 3 .149 2 .132 2 .998 0 .537\ncrevTPSS no D3 6 .098 3 .122 2 .161 3 .014 0 .512\nTao–Mo no D3 6 .071 3 .099 2 .343 3 .108 0 .616\ncTao–Mo no D3 6 .069 3 .086 2 .339 3 .109 0 .619\nPKZB no D3 6 .200 3 .409 2 .153 2 .987 0 .826\ncPKZB no D3 6 .210 3 .422 2 .138 2 .979 0 .839\nr2SCAN no D3 6 .159 3 .776 2 .501 3 .237 0 .911\ncr2SCAN no D3 6 .156 3 .772 2 .504 3 .240 0 .907\nM06-L no D3 6 .325 3 .511 1 .891 2 .976 1 .116\ncM06-L no D3 6 .327 3 .512 1 .889 2 .974 1 .117\nTPSS D3-BJ 5 .982 2 .988 2 .352 3 .098 0 .343\ncTPSS D3-BJ 5 .980 2 .985 2 .354 3 .099 0 .341\nrevTPSS D3-BJ 5 .949 2 .883 2 .443 3 .156 0 .291\ncrevTPSS D3-BJ 5 .952 2 .889 2 .435 3 .152 0 .296\nTao–Mo D3-BJ 6 .068 3 .095 2 .347 3 .110 0 .613\ncTao–Mo D3-BJ 6 .062 3 .087 2 .358 3 .116 0 .604\nr2SCAN D3-BJ 6 .156 3 .773 2 .506 3 .240 0 .907\ncr2SCAN D3-BJ 6 .155 3 .771 2 .506 3 .241 0 .905\n13REFERENCES\n1D. 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