\section{INTRODUCTION}
\label{sec:intro}

Nova events are the result of thermonuclear explosions from binary systems that occur when a white dwarf (WD) accretes H-rich material from a companion \citep[e.g.,][]{Gallagher1978}. Due to momentum conservation, the material forms an accretion disk before being accreted by the WD. The material in the disk spirals down onto the WD while its being heated up by friction. If the accretion rate onto the WD is low, the accreted material becomes degenerate, making the pressure and temperature to increase
dramatically. An explosion will ignite the material accreted on the
surface of the WD through a thermonuclear runaway event – a nova explosion. This thermonuclear runaway releases and initial energy $E_{0}\sim10^{44}$--10$^{45}$~erg,
ejecting $M_\mathrm{ej}\approx$10$^{-5}$--10$^{-4}$~M$_\odot$ at velocities as high as $v_\mathrm{ej}\gtrsim10^{3}$~km~s$^{-1}$ \citep[see, e.g.,][]{Bode2010,DellaValle2020} into the interstellar medium (ISM).

The violent ejection of material in the nova event sweeps, shocks and compresses the surrounding ISM. This interaction will create an adiabatic shock that heats up the gas inside the nova remnant creating a hot bubble similarly to that of SNe and wind-blown bubbles \citep{Gudel2008}.
The post-shock temperature of this hot bubble can be estimated as $T \propto v_\mathrm{ej}^2$ \citep{Dyson1997}. The highly supersonic ejecta should produce post-shock temperatures in excess to $10^{6}$~K, that in principle could be detected with X-ray instruments. Due to the high thermal energy of the hot bubble, it will dominate the dynamical evolution of the nova remnant until its dispersal into the ISM. During its evolution, the nova shell will experience Rayleigh-Taylor instabilities and will be disrupted \citep[see, e.g.,][]{Orlando2017}; a characteristic that is evident in narrow-band imaging of novae shells \citep[see][and references therein]{Santamaria2020}. The clumps and filaments will create ablation flows \citep[e.g.,][]{Vaytet2007}, mixing the nova shell material with that of the hot bubble, reducing its temperature to $\sim10^{6}$~K enhancing the production of soft X-ray emission \citep{Toala2018}.

There are two nova remnants with unambiguously resolved extended X-ray emission and they seem to exhibit contrasting aspects of the same event. On one hand, GK\,Per can be described as a scaled-down version of a SN remnant producing its thermal and non-thermal X-ray emission as the result of the expansion of the shock wave compressing the surrounding ISM \citep{Balman1999,Balman2005,Takei2015,Yuasa2016}. Meanwhile DQ\,Her has been recently suggested to harbour the first ever detected magnetised jet disclosing more fundamental physics acting at the level of the progenitor binary star \citep{Toala2020}.

Extended X-ray emission has been suggested to be present in other nova-like systems: the classical nova RR\,Pic \citep{Balman2004}, the cataclysmic variable (CV) DK\,Lac \citep{Takei2013} and those around the recurrent novae T\,Pyx \citep{Balman2014} and RS\,Oph \citep{Luna2009}. However, their marginal detections, in some cases unresolved, make it difficult to unveil the true spatial distribution of the X-ray emission.

\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{imagen_J210204.pdf}
\caption{Optical and X-ray colour-composite picture of IPHASX J\,210204.7$+$471015.
Red and green correspond to narrow-band images obtained with the NOT \citep[see][]{Guerrero2018}, while blue corresponds to the {\it XMM-Newton} X-ray image in the 0.7--1.2~keV energy range.
The progenitor star system is at the centre of the image.
North is up, east to the left.
}
\label{fig:image}
\end{center}
\end{figure}

In contrast, X-ray emission from the central binary system is often reported in the literature. The evolution of the X-ray properties of the binary system in a nova eruption vary significantly during the outburst and post-outburst \citep[see, e.g.,][and references therein]{Singh2020}. In the non-magnetic cases, after accretion is reestablished, the progenitor system will behave similarly to a CV \citep{Hernanz2002,Hernanz2010,Sala2017}.
Their X-ray emission is likely to be produced on post-shock regions at the base of the accreting surface on the WD \citep{Eracleous1991,Baskill2005}. In the cases in which the WD has a strong magnetic field ($B\approx$1--10~MG), as the case of the so-called intermediate polars (IPs), it quenches the inner region of the accretion disk at the magnetospheric radius of the WD. The material falls into the WD following the magnetic field lines towards the polar regions forming a shock \citep[see][]{Patterson1994} and producing X-rays \citep[see][and references therein]{Worpel2020}. In such cases the spin of the WD is not synchronised to the orbit of the binary system, and thus, some X-ray variability is seen even in quiescent state \citep[e.g.,][]{Mukai2003}.

In this paper we present an X-ray characterisation of the nova shell IPHASX J210204.7$+$471015 (hereinafter J210204; see Fig.~\ref{fig:image}) and its progenitor system. J210204 was discovered as part of the Isaac Newton Photometric H$\alpha$ Survey \citep[IPHAS;][]{Drew2005,Barentsen2014} and proposed to be a possible planetary nebula \citep{Sabin2014}, but later characterised as a nova shell \citep{Guerrero2018}.
The most obvious structure of J210204 illustrated in Figure~\ref{fig:image} is a clumpy incomplete ring-like structure mainly detected in [N\,{\sc ii}], with most of the knots exhibiting a cometary morphology.
This ring-like structure lacks emission towards the west with less dense fewer knots.
A different structure is detected in [O\,{\sc iii}], a bow-shock-like feature towards the east, which might be caused as the shock wave expands into the ISM.
The clear asymmetric morphology of J210204, reminiscent of the classical nova AT\,Cnc \citep{Shara2012}, is not common for nova shells. Their morphology represents the best piece of evidence of asymmetric explosions shaping novae shells.

We use {\it XMM-Newton} to search and characterise the X-ray emission from its central engine and to assess the presence of a hot bubble.
This paper is organised as follows.
In Section~2 we present the observations and the data analysis.
In Section~3 we present our results.
A discussion is presented in Section~4.
Finally, we summarise our results in Section~5.

\section{Observations and data preparation}

J210204 was observed with the three European Photon Imaging Cameras (EPIC) on board {\it XMM-Newton} during 2020 May 5 for a total observation time of 72~ks. The observations were obtained on the Extended Full Frame mode for the pn camera and in the Full Frame mode for the MOS 1 and MOS2 cameras.
All observations were obtained with the medium optical blocking filter.
The total exposure times on the MOS1, MOS2 and pn cameras are 67.9, 68.1 and 64.7~ks, respectively.

\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{EPIC-eps-converted-to.pdf}
\caption{
{\it XMM-Newton} EPIC (MOS1$+$MOS2$+$pn) image in the 0.3--10~keV energy range of IPHASX J\,210204.7$+$471015.
The position of the progenitor star of J210204 is shown with a red circle with an angular radius of 20~arcsec.
}
\label{fig:EPIC}
\end{center}
\end{figure}

The observation data files (ODF) were processed with the Science Analysis System ({\sc sas}) version 19.0\footnote{\url{https://www.cosmos.esa.int/web/xmm-newton/what-is-sas}}. The event files were produced using the {\it epproc} and {\it emproc} {\sc sas} tasks with the calibration files obtained on 2020 November 30. In order to excise periods of high-background we created EPIC light curves binning the data over 100~s for the 10--12~keV energy range. The background was considered high for count rates  of 0.2~counts~s$^{-1}$ for the EPIC-MOS and 0.45~counts~s$^{-1}$ for EPIC-pn.
After excising bad periods of time, the net exposures times for the MOS1, MOS2 and pn cameras resulted in 58.2, 57.9 and 40.8~ks, respectively.

The clean event files of the three EPIC cameras have been combined using the {\it merge} SAS task only for purposes of illustration. In Figure~\ref{fig:EPIC} we show this image and the position of the progenitor star of J210204.
The central source is detected in X-rays

In order to extract spectra of J210204, we defined a source circular aperture with a radius of 20~arcsec and a background aperture from a region with no contribution from point sources in the vicinity of J210204.
The resultant count rates for the central star of J210204 obtained from the MOS1, MOS2 and pn cameras are 0.70$\pm$0.15, 0.60$\pm$0.17 and 3.7$\pm$0.4~counts~ks$^{-1}$ with total detected count numbers of 40$\pm$9, 32$\pm$9 and 150$\pm$16~counts for the MOS1, MOS2 and pn cameras, respectively.
The background-subtracted EPIC-pn spectrum is presented in Figure~\ref{fig:spec}.
We do not show the MOS spectra because their low count rates result in a limited quality.

\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{spec_PN.pdf}
\caption{
Background-subtracted EPIC-pn spectrum of the central source in J210204 (black diamonds).
The solid black line represents the total model, whilst the dashed (blue) and dotted (red) lines show the contribution from the apec and power law components.
The spectrum has been extracted requesting a minimum of 10 counts per spectral bin.
The bottom panel presents the residuals of the best-fit model.
}
\label{fig:spec}
\end{center}
\end{figure}

To further search for extended X-ray emission filling the nebular shell of J210204, we also analysed the data following the Extended Source Analysis Software ({\sc esas}) tasks \citep{Snowden2004,Kuntz2008,Snowden2008}, which are currently included as part of {\sc sas}.
The {\sc esas} tasks apply more restrictive event rejection criteria, cleaning the data for the possible contamination from the astrophysical background, the soft proton background and solar wind charge-exchange reactions, which have important contributions at energies <1.5 keV.
Independent pn, MOS1 and MOS2 images were extracted, merged together, and corrected for exposure maps. This was done for three different energy ranges, namely, a soft (0.3--0.7~keV), a medium (0.7--1.2~keV) and a hard (1.2--5.0~keV) band.

No extended X-ray emission was detected in any of the three EPIC images produced with the {\sc esas} tasks, but only the presence of point sources in the vicinity of J210204.
In Figure~\ref{fig:image} we compare the medium X-ray band with optical images obtained at the Nordic Optical Telescope (NOT) taken with [O\,{\sc iii}] and [N\,{\sc ii}] narrow band filters.

\section{Analysis and results}
\label{sec:ana}

The bulk of the emission in the EPIC-pn spectrum of the central source of J210204 is in the energy range between 0.4 and 2~keV (see Fig.~\ref{fig:spec}).
The spectrum hints at the presence of three emission lines at $\sim$0.8, $\sim$1.0, and $\sim$1.8~keV.
The first two might be attributed to Ne\,{\sc ix} and Ne\,{\sc x} emission lines, respectively, whilst the latter might be due to  Si\,{\sc xiii} emission lines.
We note that a certain energetic continuum can be seen in the spectrum from 2.0 to 6.0~keV.

In order to estimate its physical properties, the EPIC-pn spectrum of the central star of J210204 was modelled using the X-Ray Spectral Fitting Package \citep[{\sc xspec};][]{Arnaud1996} version 12.9.1. A first model was attempted by accounting for a single optically-thin {\it apec} plasma emission model.
This single component model reproduced the soft part of the spectrum, but it is notably deficient for the hard X-ray emission above 2.0~keV.
Models with two-component were subsequently attempted.

Our first model included two {\it apec} plasma components with solar abundances \citep{Wilms2000} which resulted in a relatively good fit with a reduced $\chi^{2}$ of 1.22.
The soft component has a plasma temperature of $k T_1$=0.52$^{+0.24}_{-0.27}$ keV and was fitted with a hydrogen column density $N_\mathrm{H}=(8.0^{+2.7}_{-3.6})\times10^{21}$~cm$^{-2}$.
The later is consistent with the extinction measured in optical spectra \citep[see][]{Guerrero2018}.
However, {\sc xspec} had troubles fitting the second component, resulting in an undefined plasma temperature $k T_2$ between 2 and 64~keV.

A better fit to the EPIC-pn spectrum was achieved by a temperature plasma emission model plus a power law component.
This fit resulted in a similar reduced $\chi^2$ of 1.20, a column density  $N_\mathrm{H}=(8.1^{+2.5}_{-3.4})\times10^{21}$~cm$^{-2}$ and a plasma component with $kT_1=0.55^{+0.22}_{-0.27}$~keV.
The power-law component is characterised by a photon index $\Gamma=1.0^{+1.0}_{-1.4}$.
The {\sc xspec} normalization parameters\footnote{The normalization parameter is defined in {\sc xspec} as $A=10^{-14}\int n_\mathrm{e} n_\mathrm{H} \mathrm{d}V / 4 \pi d^2$, where $n_\mathrm{e}$ and $n_\mathrm{H}$ are the electron and hydrogen number densities and $d$ is the distance to the object. The integral is performed over the volume $V$.} of the two components are $A_\mathrm{apec} = 1.82\times10^{-5}$ cm$^{-5}$ and $A_\mathrm{pow}$=1.22$\times10^{-6}$ cm$^{-5}$.

The best-fit model resulted in an absorbed flux of $f_\mathrm{X}=(1.35\pm0.35)\times10^{-14}$ erg~s$^{-1}$~cm$^{-2}$ in the 0.4--6.0~keV energy range and corresponds to an intrinsic flux of $F_\mathrm{X}=(6.9\pm3.3)\times10^{-14}$ erg~s$^{-1}$~cm$^{-2}$.
Taking into account a distance of 0.6~kpc estimated by \citet{Santamaria2019}, we calculated an X-ray luminosity of
$L_\mathrm{X}=(3.0\pm1.4)\times10^{30}$~erg~s$^{-1}$.

Models with varying abundances were also attempted, in particular, models with Ne and Si abundances as free parameters.
The Ne abundances resulted in 1.2 times its solar value whilst the Si abundance converged to values $\sim$3--4 times the solar value.
However, these models did not improved considerably the previous fit with a reduced $\chi^{2}$=1.16.
Nevertheless, the estimated errors for these two elemental abundances were larger than the actual values.
For this, we will keep the model described in the paragraphs above as the best-fit to the EPIC-pn spectrum.

To assess the possible X-ray variability of the binary system of J210204, we extracted background-subtracted light curves in the 0.3--6.0, 0.4--2.0 and 1.5--6.0~keV energy ranges.
The three light curves are very similiar and, thus, we only show that of the EPIC-pn corresponding to the 0.3--6.0~keV energy range in Figure~\ref{fig:lc}.
We used the PyAstronomy Lomb-Scargle routine to produce a periodogram and calculate the most possible period of variation. This procedure suggests a period of 10.7$\pm$0.6~ks (=2.9$\pm$0.2~h) for the 0.3--6.0 and 0.4--2.0~keV light curves and a period of 11.1$\pm$0.7~ks (=3.1$\pm$0.2~hr) for the light curve in the 1.5--6.0~keV range, i.e., the periods in the different energy bands are consistent within their uncertainties.
We note that each periodogram exhibited the presence of other two secondary components. We illustrate this in Figure~\ref{fig:periodo} were we show the normalized power spectrum obtained for the light curve presented in Figure~\ref{fig:lc}.
This periodogram has three main peaks at frequencies
$\omega_1=(6.2\pm0.9)\times10^{-5}$ Hz,
$\omega_2=(9.4\pm0.4)\times10^{-5}$ Hz and
$\omega_3=(1.2\pm0.7)\times10^{-4}$ Hz.
These correspond to periods of
$P_1$=4.5$\pm$0.6 hr,
$P_2$=2.9$\pm$0.2 hr and
$P_3$=2.3$\pm$0.7 hr, respectively.
$P_2$ is the one described above while $P_1$ is consistent with that estimated from optical observations of the progenitor system of J210204 of 4.26$\pm$0.01~hr \citep[see][]{Guerrero2018}.

As shown in the previous section, no extended X-ray emission is detected with the current EPIC observations.
In order to estimate upper limits to the X-ray flux and luminosity of the possible hot bubble in J210204, we extracted a background-subtracted spectrum of a region encompassing that of the nebular emission, excising the contribution from point sources.
We estimated a 3-$\sigma$ upper limit to the EPIC-pn count rate in the 0.3--6.0~keV of 0.54 counts~ks$^{-1}$.
Using the count rate simulator PIMMS\footnote{\url{https://cxc.harvard.edu/toolkit/pimms.jsp}} version 4.10, assuming a plasma temperature for the hot bubble of $T=2\times10^{6}$~K and adopting the $N_\mathrm{H}=8\times10^{21}$~cm$^{-2}$ as obtained for the spectral fit of the progenitor system of J210204, we estimated an absorbed flux of $f_\mathrm{X,DIFF}\lesssim2.1\times10^{-16}$~erg~s$^{-1}$~cm$^{-2}$. This corresponds to an intrinsic flux of $F_\mathrm{X,DIFF}\lesssim2.5\times10^{-14}$ erg~s$^{-1}$cm$^{-2}$ and an upper limit to the X-ray luminosity of $L_\mathrm{X,DIFF}\lesssim1.1\times10^{30}$~erg~s$^{-1}$.

\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{lc.pdf}
\caption{Background-subtracted EPIC-pn light curve in the 0.3--6.0~keV energy range of the central binary system in J210204. Each bin represents 3~ks.}
\label{fig:lc}
\end{center}
\end{figure}

\section{Discussion}

Our best-fit model to the EPIC-pn spectrum of the progenitor system of J210204 suggests the presence of a thermal component with a temperature of $T_\mathrm{X}\approx(6.4\pm3.1)\times10^{6}$~K.
Interestingly, the second component is best described by a power law with a spectral index of $\Gamma=1.0$, suggesting the presence of non-thermal emission an, thus, a possible IP origin for this stellar system. We note that the number of IP systems in the Galaxy are expected to be large, contributing importantly to the Galactic X-ray properties \citep{Revnivtsev2009,Warwick2011,Warwick2014}, but not many have been confirmed in the literature\footnote{The Catalogue of IPs and IP Candidates report 24 confirmed IPs by 2014 October 29. See \url{https://asd.gsfc.nasa.gov/Koji.Mukai/iphome/catalog/alpha.html}}. Thus, the characterisation of such systems as possible IPs is crucial.

The plasma temperature of the progenitor star of J210204 is very similar to that estimated for other IPs with very similar spectra with dominant emission around 1.0~keV and emission lines above 2~keV \citep[see][and references therein]{Worpel2020}. We note that although the X-ray emission from IPs is expected to have X-ray luminosities in excees to $10^{32}$~erg~s$^{-1}$ \citep[][]{Patterson1994}, the progenitor system of J210204 has a luminosity of $10^{30}$~erg~s$^{-1}$, i.e., it is underluminous as the iconic DQ\,Her \citep[see the recently characterized cases presented in][and references therein]{Nucita2020a,Nucita2020}.

\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{periodograma2.pdf}
\caption{Normalized power spectrum of the light curve presented in Figure~\ref{fig:lc}. The three peaks marked with dotted lines correspond to frequencies of $(6.2\pm0.9)\times10^{-5}$~Hz, $9.4\pm0.4\times10^{-5}$~Hz and $(1.2\pm0.7)\times10^{-4}$~Hz which corresponds to periods of $P_1$=4.5$\pm$0.6~hr, $P_2$=2.9$\pm$0.2~hr and $P_3$=2.3$\pm$0.7~hr, respectively.}
\label{fig:periodo}
\end{center}
\end{figure}

Although the resultant light curves of the EPIC observations of the progenitor system of J210204 do not have an excellent quality, we were able to perform time series analysis. The power spectrum presented in Figure~\ref{fig:periodo} exhibits three main components.
The frequency peak $\omega_1$ implies a period of $4.5\pm0.6$~hr, which is very similar to that estimated by \citet{Guerrero2018} using optical observations. Thus, this is very likely to be the orbital period $P_\mathrm{orbit}$. If this system is indeed a IP, the dominant frequency $\omega_2$ with a period of $P_2$=2.9$\pm$0.2~hr could be attributed to the orbital spin of the WD component in the system ($P_\mathrm{spin}$). We tried to assess the origin of the third component $\omega_3$ and it seems to follow the relation
\begin{equation}
 \omega_3 =   \frac{2}{P_\mathrm{spin}} - \frac{1}{P_\mathrm{orbit}}
\end{equation}
\noindent which has been discussed thoroughly in the theoretical calculations of X-ray power spectra presented by \citet{Wynn1992}.
These authors suggested that such frequency could be used an indicator of diskless accretion mechanisms, but this seems to depend on different parameters used for the emission model.
We reckon, however, that our light curves and periodograms do not have sufficient quality to further discuss the origin of the accretion onto the progenitor system of J210204.
Indeed, broad peaks can appear in the power spectrum as the result of the combination of several components in low quality X-ray light curves \citep[e.g., see figure~8 in][]{Wang2020}.

\subsection{On the lack of extended X-ray emission}

J210204 has a complex nova shell structure, unlike the ellipsoidal shell commonly observed around classical novae \citep[see, e.g.,][]{Santamaria2020}, but it shares stark similarities to the nova shell around AT\,Cnc \citep{Shara2012}. The morphological similarities between these two novae suggest that their asymmetries are not produced by interactions with the ISM, but due to the conditions of the explosion itself.

Departures from symmetry might be attributed to the interaction of the ejecta with the companion star and material in the orbital plane of the binary system, but simulations presented in \citet{Orlando2017} show that material is mostly collimated in polar directions of the orbital plane.
The same group predicted the evolution of the X-ray luminosity for similar numerical simulations \citep{Orlando2009}.
They predicted that the total X-ray flux in their simulations achieves a maximum of $\sim10^{-9}$~erg~s$^{-1}$~cm$^{-2}$ after the fourth day of the explosion and then decays as $\sim t^{-1.2}$ mainly due to radiative cooling.
Assuming that such evolution of the X-ray flux is valid for J210204, their model predicts an X-ray flux of $(1.0-2.5)\times10^{-14}$ erg~s$^{-1}$~cm$^{-2}$ at the age of 130--170~yr suggested by \citet{Santamaria2020} for J210204, very similar to the upper limit estimated here using {\it XMM-Newton} EPIC-pn observations.
Nevertheless, we would like to remark that different parameters included in the simulations (e.g., $E_{0}$ and $M_\mathrm{ej}$) could produce a range of X-ray properties \citep[e.g.,][]{Orlando2012}.

However complex the role of the different parameters explored in the simulations discussed above, all of them produce ellipsoidal morphologies \citep[see also][]{Walder2008}, which is not the case of J210204 and AT\,Cnc.
The fact that the ejecta of these two novae appear to expand towards one side is suggestive of an explosion that originated from a specific location of the WD surface, that is, the ejecta is not only axi-symmetric, but also asymmetric.
Future simulations adopting such scenario might be explored in the future to explain the morphologies of J210204 and AT\,Cnc.

\section{Summary}

We presented the analysis of {\it XMM-Newton} observations of the nova shell
IPHASX J\,210204.7$+$471015 (J210204). Our EPIC observations detected X-ray
emission from the central binary star with properties that are very similar to the so-called IPs: a dominant plasma temperature with temperature of $T_\mathrm{X}=(6.4\pm3.1)\times10^{6}$~K with a power-law component, signature of non-thermal X-ray emission. Its luminosity, $L_\mathrm{X}=10^{30}$~erg~s$^{-1}$, is below the expected value for the X-ray emission from IPs of 10$^{32}$--10$^{34}$~erg~s$^{-1}$, but consistent with that of DQ\,Her and other underluminous IPs.

We used time series analysis of the EPIC observations and obtained three dominant frequencies to the power spectrum. The peak with period 4.5$\pm$0.6~hr seems to correspond to the orbital period of the system as observed at optical wavelengths \citep[e.g.,][]{Guerrero2018}.
The dominant component to the periodogram corresponds to a period of 2.9$\pm$0.2~hr very likely related to the spin of the WD. Future deeper X-ray observations are necessary to assess the presence of the dominat components in the power spectrum.

We used the ESAS tasks to search for extended X-ray emission within J210204 as signature of the presence of an adiabatically-shocked hot bubble without success. We estimated an upper limit of the flux and luminosity of the X-ray emission of $F_\mathrm{X,DIFF}\lesssim2.5\times10^{-14}$~erg~s$^{-1}$ and $L_\mathrm{X,DIFF}\lesssim1.1\times10^{30}$~erg~s$^{-1}$, respectively. This flux is very similar to that estimated from numerical simulations presented in the literature in which the explosion generates an ellipsoidal nova shell.
However, due to the remarkable morphology of J210204, we suggest that the explosion might have not occurred isotropically at the boundary of the WD, but it would be rather asymmetric arising from a specific region of the WD. Such claim will have to be put to test with future numerical simulations that will also serve to explain the similar morphology of AT\,Cnc.

\section*{Acknowledgements}

The authors thank the referee for a prompt report.
JAT acknowledges support from the Fundaci\'{o}n Marcos Moshinsky (M\'{e}xico).  JAT is
funded by PAPIIT project IA100720 of the Direccion General de Asuntos
del Personal Academico (DGAPA) of the Universidad Nacional
Aut\'{o}noma de M\'{e}xico (UNAM). ES and GR are thankful to Consejo Nacional de Ciencia y Tecnolog\'{i}a
(CONACyT - M\'{e}xico) for their student grants.
MAG acknowledges support of the Spanish Ministerio de Ciencia,
Innovaci\'{o}n y Universidades (MCIU) grant PGC2018-102184-B-I00 and the
State Agency for Research of the Spanish
MCIU through the “Center of Excellence Severo Ochoa” award
to the Instituto de Astrof\'{i}sica de Andaluc\'{i}a (SEV-2017-0709).
GR-L acknowledges support from CONACyT grant
263373. LS acknowledges support from PAPIIT-UNAM
grant IN101819. This work has made extensive use of
NASA’s Astrophysics Data System (ADS).

\section*{Data availability}

The data underlying this work are available in the article.
The reduced observations files will be shared on reasonable request to the corresponding author.


\section{Introduction}
\vspace{-.15cm}
Nonlinear interference (NLI) modeling in optical fiber transmission is a key tool to analyze and optimize the performance of optical communication systems. In recent years, remarkable progress has been made in this direction, and as new model's applications emerge, further work is required to meet new accuracy and computational requirements.
Constellation shaping is today a popular area of application for NLI models for two reasons: i) the performance of a given constellation significantly depends on the amount of NLI power induced during propagation; ii) NLI models can provide easy-to-compute as well as accurate cost functions for the performance optimization. However, as research focus moves towards multidimensional constellation shaping, where the different dimensions are mapped onto the degrees of freedom of the fiber channel (quadratures, polarization, wavelengths, etc),  extensions of the established NLI models for two-dimensional (2D) formats are needed.

NLI power models such as the enhanced Gaussian noise (EGN) model \cite{Carena2014} introduced analitical expressions explicitly linking the properties of the transmitted constellation to the resulting NLI power after propagation. Nonetheless, such expressions have been developed only for so-called polarization-multiplexed 2D (PM-2D) formats, i.e. when a single 2D constellation is used to independently map information over 2 orthogonal polarization modes of the optical field. The resulting dual polarization four-dimensional (DP-4D) constellation is, thus, given by the Cartesian product of the component 2D constellation by itself. However, the class of PM-2D formats represents only a limited subset of all possible DP-4D constellations. A vast literature on assessing the performance of DP-4D formats that do not fall within the PM-2D class is available, (see, e.g.,~\cite{Agrell09}), and, lately, DP-4D constellations have shown improved shaping gains compared to other conventional PM-2D formats \cite{Kojima2017, Chen2019, Chen2020}. However, no NLI model has been so far available to support the design of nonlinearity-tolerant DP-4D formats.

Recently, in \cite{Liga2020}, we extended the model in \cite{Carena2014} to account for the entire DP-4D class of modulation formats. In this contribution, we present a first numerical validation of our 4D model. Moreover, we show that heuristic extensions of the EGN model to non PM-2D formats may lead to inaccuracies in the prediction of the NLI power coefficient beyond 1 dB, even for fairly regular 4D modulation formats. Our 4D model \cite{Liga2020} is instead proven to be accurate on average within 0.1 dB for all 4D modulation formats studied in this work.

\section{Analytical Formulation of the NLI coefficient for DP-4D formats}
\vspace{-.15cm}
The model we aim to validate in this work consists of an analytical formula for the computation of the vector $(\sigma^{2}_x,\sigma^{2}_y)$, where $\sigma^{2}_x$ and $\sigma^{2}_y$ represent the NLI power within the received signal bandwidth over the $x$- and $y$ polarizations, respectively. The corresponding NLI power coefficient vector $(\eta_x,\eta_y)\triangleq(\sigma^{2}_x,\sigma^{2}_y)/P^3$, where $P$ denotes the transmitted signal power. The model's expression was derived using a first-order perturbational approach under the hypothesis of single-channel transmission with quasi-rectangular pulse spectrum. A comprehensive discussion on the assumptions of the model as well as its mathematical derivation can be found in \cite{Liga2020}.

Based on \cite[eqs.~(42)-(43)]{Liga2020}, the NLI power coefficient for the $x$ polarization can be found as
\begin{align}\label{eq:4D_NLI_formula}
\begin{split}
\eta_x&=\left(\frac{8}{9}\right)^2\frac{\gamma^2}{P^3}\left[R_s^3\left(\Phi_{1}\overline{\rchi}_1+\Phi_{2}\overline{\rchi}_2+\Phi_{3} \overline{\rchi}_3\right)+R_s^2\left(\Psi_1\overline{\rchi}_4+2\Re\{\Psi_2\overline{\rchi}_5+\Psi_3\overline{\rchi}_5^*\}+\Psi_4\overline{\rchi}_6\right.\right.\\
&\left.\left.+2\Re\{\Lambda_1\overline{\rchi}_7+\Lambda_2\overline{\rchi}_7^*(f)\}+\Lambda_3\overline{\rchi}_8+2\Re\{\Lambda_4\overline{\rchi}_9+\Lambda_5\overline{\rchi}_9^*\}+\Lambda_6\overline{\rchi}_{10}\right)+R_s\Xi_1\overline{\rchi}_{11}\right],
\end{split}
\end{align}
where $\gamma$ is the fiber nonlinearity coefficient, $R_s$ is the symbol rate and the coefficients $\Phi_1$, $\Phi_2$,  $\Phi_3$, $\Psi_1$, $\Psi_2$, \dots, $\Psi_4$, $\Lambda_1$, $\Lambda_2$, \dots , $\Lambda_6$, $\Xi_1$ are functions of several different intra- and cross- polarization moments of the DP-4D transmitted modulation format, and are given in \cite[Table 8]{Liga2020}. The $\overline{\rchi}_i$ coefficients are obtained integrating over the channel bandwidth (see \cite[eq.~(42)-(43)]{Liga2020}) the frequency-dependent integrals $\rchi_i(f)$, $i=1,2,\dots,11$ in \cite[Table 8]{Liga2020}. The expression for $\eta_y$ can be found from \eqref{eq:4D_NLI_formula} by simply applying the transformation $x\rightarrow y$ and $y\rightarrow x$. As discussed in \cite[Sec.~8]{Liga2020}, \eqref{eq:4D_NLI_formula} reduces to the EGN formula for conventional PM-2D formats.

\begin{figure}[t]
\hspace{-.3cm}
\subfloat{\input{BarPlotX.tikz}}\hspace{.1cm}%
\subfloat{\input{BarPlotY.tikz}}
\vspace{-.3cm}\caption{NLI power coefficients $\eta_x$ (a), and $\eta_y$ (b), for DP-4D constellations with different cardinality $M$.}
    \label{fig:barplot}
\end{figure}

\section{Methodology}
\vspace{-.2cm}
\begin{wraptable}{r}{0.3\textwidth}\vspace{-.75cm}
\begin{footnotesize}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{c|c}
\textbf{Parameter} & \textbf{Value}  \\
\hline \hline
\multicolumn{2}{c}{\textbf{TX parameters}}\\
\hline
    Symbol rate ($R_s$) & 32 Gbaud \\
    \hline
    No. of channels & 1 \\
    \hline
    RRC rolloff & 0.01 \% \\
    \hline
    TX power ($P$) & $-20$ dBm \\
    \hline

    \multicolumn{2}{c}{\textbf{Fiber parameters}}\\

    \hline
    Attenuation coeff. ($\alpha$) & 0.2 (dB/km) \\
    \hline
    Dispersion par. ($D$) & 17 ps/nm/km \\
    \hline
    Nonlinear coeff. ($\gamma$) & 1.3 (W$\cdot \text{km})^{-1}$) \\
    \hline
    \multicolumn{2}{c}{\textbf{Link parameters}}\\
\hline
Span length & 100 km \\
\hline
No. of spans & 10 \\
\hline
 \multicolumn{2}{c}{\textbf{SSFM parameters}}\\
 \hline
Step distribution & Adaptive step \\
\hline
$\phi_{NLmax}$ & $10^{-3}$ rad \\
\hline
Sim. bandwidth & 96 GHz \\
\hline
\end{tabular}
\end{footnotesize}
\vspace{-.2cm}\caption{System parameters.}
\label{tab:sim_par}
\end{wraptable}

The numerical validation of the model in this work is performed via the estimation of the signal-to-noise ratio in split-step Fourier method (SSFM) simulations where optical nonlinearity is kept as the sole source of noise. Indeed, in such a scenario, $\mathbf{E}\approx(P_x(\text{SNR}_x\cdot P^3)^{-1}, P_y(\text{SNR}_y\cdot P^3)^{-1})$, where $P_x$, SNR$_x$, $P_y$, and SNR$_y$ are the transmitted powers and signal-to-noise ratios over $x$ and $y$ polarization, resp.
The previous formula becomes increasingly accurate as higher-order NLI terms vanish compared to the first-order one, i.e. for small values of $P$. The simulated single-channel, multi-span optical system is described in Table \ref{tab:sim_par}.
At the transmitter, 4D symbols are jointly modulated using a root-raised cosine (RRC) pulse shape. At the receiver, chromatic dispersion compensation and matched filtering followed by sampling are performed. $\text{SNR}_x$ and $\text{SNR}_y$ are then computed via a data-aided approach.
 Finally, an adaptive step size SSFM with a maximum nonlinear phase rotation $\phi_{NLmax}$ per step is used to simulate the fiber propagation.

The EGN model is also used as a reference to show the increased accuracy of \eqref{eq:4D_NLI_formula}. However, in its standard formulation, the EGN model does not provide a suitable expression to account for general DP-4D formats.
An intuitive way to extend the EGN model to general DP-4D formats is to consider the EGN expressions as separately applicable to the two 2D constellations obtained by the projection of the transmitted DP-4D format over the $x$ and $y$ polarization plane. In the following, we adopt this approach, labelled 4D-EGN model, to compute EGN-based estimates of \textbf{E}.

\section{Results}
\vspace{-.18cm}
The numerical results in this section are based on a set of DP-4D formats which combines the full list of 4D constellations in  \cite{AgrellCodesSE} (4D sphere packings), and formats used in optical communications such as the 4D 64-ary polarization ring switched (4D-64PRS) format \cite{Chen2019}, the 4D orthant-symmetric 128 (4D-OS128) format \cite{Chen2020}, and the family of 4D 2-amplitudes 8 phase-shift keying (4D-2A8PSK) formats \cite{Kojima2017}.

In Fig.~\ref{fig:barplot}, $\eta_x$ (Fig.~\ref{fig:barplot}(a)) and $\eta_y$ (Fig.~\ref{fig:barplot}(b)) are shown for some of the above mentioned constellations with different constellation cardinalities $M$, using: i) \eqref{eq:4D_NLI_formula} (blue bars); ii) the 4D-EGN model (yellow bars); iii) the SSFM (red bars). For all constellations shown, our 4D model is within 0.1 dB from the SSFM estimates for both $\eta_x$ and $\eta_y$. The 4D-EGN model leads to inaccuracies of up to 0.9 dB, for $\eta_y$ in ``l4\textunderscore128'', or 0.8 dB for $\eta_x$ in ``c4\textunderscore16''. Neither ``l4\textunderscore128'' nor ``c4\textunderscore16'' are symmetric constellations with respect to the $y=x$ plane, and this is reflected by uneven values of $\eta_x$ and $\eta_y$. However, even for symmetric constellations such as 4D-64PRS and 4D-128OS, the 4D-EGN model deviates from SSFM estimates by approx.~0.6 dB in both $\eta_x$ and $\eta_y$. As expected, for a PM-2D quadrature amplitude modulation (QAM) format such as PM-64QAM, all three estimation methods are in perfect agreement.

To further validate \eqref{eq:4D_NLI_formula}, the average deviation $\overline{\Delta\eta}\triangleq(\Delta\eta_x+\Delta\eta_y)/2$ is computed, where $\Delta\eta_x$ and $\Delta\eta_y$ are the deviations (in dB and in absolute value) for $\eta_x$ and $\eta_y$, respectively, between a given model and the SSFM estimates.
In Fig.~\ref{fig:ensamble_accuracy}, $\overline{\Delta\eta}$ is illustrated for \eqref{eq:4D_NLI_formula} and the 4D-EGN model as a function of the constellation cardinality $M$. The solid lines show the average $\overline{\Delta\eta}$ across all considered constellations, whereas dashed lines indicate the maximum and minimum deviation. The results show an average $\overline{\Delta\eta}$ within 0.1 dB for \eqref{eq:4D_NLI_formula} across all cardinalities, with maximum $\overline{\Delta\eta}$ of 0.25 dB (for $M=16$). On the contrary, the 4D-EGN model shows deviations in excess of 1 dB for low cardinality formats ($M\leq$64), with worst-case scenario $\overline{\Delta\eta}$ of 1.75 dB for $M=16$. At higher cardinalities ($M\geq$64), the 4D-EGN model accuracy improves as the average $\overline{\Delta{\eta}}$ lies between 0.3 dB and 0.5 dB.

\begin{wraptable}{r}{0.35\textwidth}\vspace{-.3cm}
\setlength{\tabcolsep}{3pt}
\begin{footnotesize}
\begin{tabular}{c|c|c}
\textbf{Const.~label} & $M$ & $(\eta_x,\eta_y)$ [dB 1/W$^2$] \\
\hline\hline
dicyclic4\textunderscore16 \cite{AgrellCodesSE} & 16 & (30.2, 30.2) \\
      4D-2A-8PSK5b \cite{Kojima2017} & 32 &  (30.3, 30.3) \\

      4D-2A-8PSK6b \cite{Kojima2017} & 64 &  (30.3, 30.3) \\
      4D-2A-8PSK7b \cite{Kojima2017} & 128 & (30.3, 30.3) \\
      w4\textunderscore256 \cite{AgrellCodesSE} & 256 & (30.7, 30.7)\\
      sphere4\textunderscore512 \cite{AgrellCodesSE} & 512 &(30.7, 30.7) \\
      120cell4\textunderscore600 \cite{AgrellCodesSE} & 600 & (30.3, 30.3)\\
      a4\textunderscore2048 \cite{AgrellCodesSE} & 2048 & (30.7, 30.8)\\
      a4\textunderscore4096 \cite{AgrellCodesSE} & 4096 & (30.8, 30.7)\\
      \hline\hline
\end{tabular}
\end{footnotesize}
\vspace{-.2cm}\caption{$\overline{\eta}$-optimal formats in Fig.~\ref{fig:opt}.}\label{tab:opt_const}
\end{wraptable}

Finally, in Fig.~\ref{fig:opt}, the minimum values of the average NLI power coefficient $\overline{\eta}\triangleq (\eta_x+\eta_y)/2$ are shown for each $M$ within the set of constellations analyzed in this work. The corresponding $\overline{\eta}$-optimal constellations are listed in Table \ref{tab:opt_const} for some values of $M$. The 4D-EGN model (yellow markers) almost always overestimates $\overline{\eta}$, with deviations up to 0.75 dB ($M=81$). Conversely, \eqref{eq:4D_NLI_formula} (blue markers) is consistently within 0.1 dB from the SSFM $\overline{\eta}$ (red markers). Among these optimal constellations, we highlight: i) ``w4\textunderscore256'' and ``a4\textunderscore4096", already studied in the context of coded modulation (see refs. in \cite{AgrellCodesSE}), which outperform PM-16QAM and PM-64QAM, respectively; ii) ``120cell600'' (see $x$ and $y$ projections in the inset in Fig.~\ref{fig:opt}), which is a constant-modulus constellation in 4D, and has the lowest $\overline{\eta}$ among all formats with $M\geq$128.

\begin{figure}[t]
\begin{floatrow}
\captionsetup{singlelinecheck=false,margin={.2cm,0cm}}
\hspace{-0.3cm}\ffigbox[\FBwidth]{\input{Gap_vs_M.tikz}}{\vspace{-.3cm}\caption{Average NLI power coeff. gap across the two polarizations ($\overline{\Delta\eta}$) as a function of the cardinality $M$.}\label{fig:ensamble_accuracy}}\hspace{-.6cm}
\captionsetup{singlelinecheck=false,margin={1cm,0cm}}
\ffigbox[\FBwidth]{\input{BestFormats_Eta_vs_M.tikz}}{\vspace{-.3cm}\caption{Minimum values of $\overline{\eta}$ vs. cardinality $M$ for \\ the 4D constellations investigated in this work.}\label{fig:opt}}
\end{floatrow}
\end{figure}

\section{Conclusions}
\vspace{-.2cm}
We tested a novel model predicting the NLI power for general DP-4D constellations. Based on this preliminary study, the assessed model shows a superior accuracy compared to a heuristic 4D extension of the EGN model. In particular, a 0.1 dB average deviation from SSFM simulations is demonstrated for the NLI power coefficient over a wide variety of regular 4D formats. Although further numerical validation is required, we foresee the model in this work as a powerful analytical tool for optimizing constellations in the DP-4D space of the optical field.

\vspace{.15cm}
 \begin{footnotesize}
\textbf{Acknowledgements:} {\setstretch{0.5} The work of G.~Liga is funded by the EuroTechPostdoc programme under the European Union’s Horizon 2020 research and innovation programme (Marie Skłodowska-Curie grant agreement No. 754462). This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 757791).\par}
\end{footnotesize}
\vspace{-.3cm}
\bibliographystyle{osajnl.bst}


\section{Introduction}

Physical properties of  Fermi gases in a large variety of different materials have been extensively studied over the past century \cite{Ashcroft}. Properties of the non-interacting Fermi gas are entirely determined by single-particle physics and the Fermi statistics \cite{Fermi}. Fermions in condensed matter physics are represented by electrons or holes that interact via the Coulomb force.
The Coulomb interaction between fermions can significantly change the properties of a Fermi gas.
For example, a one-dimensional Fermi gas forms the strongly correlated Tomonaga-Luttinger liquid at arbitrarily weak interactions \cite{Tomonaga,Luttinger,Mattis}.
In higher spatial dimensions, however, weak interactions do not spoil the properties of Fermi gases
but only slightly change the non-interacting characteristics. The gas of such weakly interacting fermions can be modeled by the gas of ``dressed'' non-interacting Landau quasiparticles. The gas of Landau quasiparticles is known as the Landau Fermi liquid (LFL) \cite{Landau}.

If the interaction is strong, the LFL breaks down and the ground state of the strongly interacting fermion system can dramatically change.
The interaction strength is measured by the dimensionless interaction parameter $r_s$:
\begin{eqnarray}
\displaystyle r_s = \frac{v_C}{E_F} \sim \frac{m e^2}{\epsilon n^{\frac{1}{D}}} ,
\label{rs}
\end{eqnarray}
where $v_C$ is the Coulomb interaction (on average), $E_F$ the Fermi energy, $n$ the fermion density, $m$ the effective mass, $e$ the elementary charge, $\epsilon$  the dielectric constant, and $D$ the spatial dimension.
Thus, in order to drive the system into the strongly interacting regime $r_s \gg 1$, we generally need a large effective mass $m$, small dielectric constant $\epsilon$ and small density $n$.
For example, the strongly interacting electron gas in near magic angle twisted bilayer graphene exhibits exotic magnetism \cite{Kotov}, charge density order \cite{Jiang}, and unconventional superconductivity \cite{Cao}, because $r_s \gg 1$ due to the low electron density and large effective mass.
The hole-doped semiconductors such as GaAs, InAs, InSb \cite{Ando}, or Ge \cite{Scappucci}
are also good candidates because of the large effective hole mass.
The hole density $n$ can be tuned to sufficiently small values by the electrostatic gates.
Taking a two-dimensional semiconductor  \cite{Ando} with $\epsilon = 10$, $m = 0.2 m_0$, where $m_0$ is the bare electron mass, and $n = 10^{11}$ cm$^{-2}$,
we get $r_s \sim 10 \gg 1$, which corresponds to the strongly interacting regime.

The Coulomb interaction between charged fermions can be divided into two physically different parts.
The first one is the classical electrostatic interaction with other electric charges via the charge density.
In quantum physics there is one more type of  interaction which comes from the quantum indistinguishability of two interacting fermions of the same type. This is the exchange interaction \cite{Heisenberg}. The exchange interaction can mix quasiparticles from different Fermi surfaces. In our study we show that, under certain conditions on the Fermi surfaces, the exchange interaction mixes the fermions into a new exotic phase.
In this new phase the fermions form a strongly interacting quantum liquid and the LFL quasiparticle picture breaks down.

In the absence of quasiparticles there is no simple visual picture to characterize quantum processes.
In order to describe quantum liquids with no quasiparticles, a quantum field description is required \cite{Fradkin}.
Excitations or quanta of the fermion fields in the LFL are long lived and they represent the Landau quasiparticles.
The Heisenberg uncertainty principle obliges all physical fields to fluctuate. For example, quantum fluctuations in the LFL result in the finite lifetime of the Landau quasiparticles \cite{Pomeranchuk}. However, quantum fluctuations in strongly interacting quantum liquids completely destroy quasiparticles \cite{Sachdev}. This means that all field excitations are strongly damped by the quantum fluctuations and cannot be considered as sharply defined long lived resonances. The single-particle methods in such quantum liquids are inadequate and, instead, the fermion correlations must be considered.

In this work we calculate the fermion Green function. The Green function is connected to some observables, e.g. to the spectral function and to linear response functions such as conductivity and spin susceptibility.
The spectral function in the new phase contains no quasiparticle poles and instead shows universal power-law scaling with frequency and momentum. The linear response functions are singular in the static limit and at the Kohn anomalies. Strongly interacting quantum liquids with these properties are called quantum critical \cite{Coleman}.

Quantum criticality in itinerant electron gases has been considered previously in application to cuprates  \cite{Daou,Keimer,Gegenwart}. Corresponding theoretical models are either based on the coupling of fermions
to a bosonic critical order parameter \cite{chubukovye,chubukov,vojta}, or on proximity to the Mott transition \cite{senthilmott},
or on the Sachdev-Ye-Kitaev (SYK) \cite{Ye,SachdevX}
scenario of  quantum criticality \cite{Parcollet,Chowdhury,Davison}.
The SYK model describes flat band fermions with long range all-to-all interactions whose matrix elements are randomly distributed. In our model,  we do not couple fermions to a critical order parameter, neither do we consider the Mott transition nor require random or even long-range interaction. The quantum criticality in our model emerges due to a resonant many-body exchange interaction between electrons that belong to different Fermi surfaces. This resonance, referred to as Fermi surface resonance (FSR), is the crucial feature of our model with far reaching consequences, both theoretically and experimentally,
as explained in great detail in the following sections.

 This work is organized as follows.
In Sec. II we describe the FSR and
provide an example of a realistic physical system which can be tuned to the FSR.
In Sec. III we construct the effective Hamiltonian which is sensitive to the FSR.
In Sec. IV we calculate the electron self-energy
corresponding to the effective Hamiltonian.
In Sec. V we investigate the strong coupling limit
which is characterized by the emergent temporal and spatial conformal symmetry.
The temporal part of the Green function is shown to obey the SYK equation, while
the spatial part obeys a generalized version of it.
In Sec. VI we fix the interaction strength
and consider the crossover between the LFL
and the quantum critical state on the $L$-$T$ diagram,
where $T$ is the temperature and $L$  the sample size. The linear response functions in the quantum critical state are calculated in Sec. VII. We find a temperature behavior of the resistance similar to the one found in strange metals. Conclusions are given in Sec. VIII. Some technical details are deferred to two appendices.

\section{Fermi surface resonance}

In our study we consider a Fermi gas with multiple non-degenerate Fermi surfaces.
This can be experimentally realized in semiconductor heterostructures \cite{Dresselhaus}.
Semiconductor heterostructures consist of  thin semiconductor layers.
The contact potential between the layers confines electrons or holes within one layer.
This leads to  quantized energy subbands corresponding to different confined modes.
Filling multiple subbands results in multiple Fermi surfaces, see Fig.~1.
Generally, the Fermi surfaces are degenerate due to spin.
The spin degeneracy can be lifted by an applied in-plane magnetic field or by spin-orbit interaction \cite{Manchon}.
The spin-orbit interaction can be precisely tuned by electric gates \cite{Zumbuhl}.
We assume that $2 N$, $N \ge 2$, of the non-degenerate Fermi surfaces can be tuned close to the FSR:
\begin{eqnarray}
\mathcal{K} = 0 ,
\label{reducedconserv}
\end{eqnarray}
where $\mathcal{K}$ is a momentum mismatch between the corresponding Fermi momenta $k_a > 0$ of the involved Fermi surfaces (we assume $\hbar=1$ throughout),
\begin{eqnarray}
\mathcal{K} = k_1 + \dots + k_{N + \sigma} - k_{N + \sigma + 1} - \dots - k_{2 N} .
\label{Delta}
\end{eqnarray}
Here, $\sigma \in \{0, 1,  \dots, N - 1\}$.
Not all Fermi surfaces are required to take part in the resonance, e.g. see Fig.~\ref{fig:ex}(a).
The Fermi surfaces are assumed to be spherically symmetric.
This is often the case in semiconductor heterostructures because the electron and hole dispersions at small densities are nearly isotropic \cite{Kriisa}.

\begin{figure}[t]
	\includegraphics[width=\columnwidth]{FSR.eps}
	\caption{
		Examples of the FSR.
		Some of the elementary resonant processes (not all of them) are shown by the arrows.
		(a)
		The simplest example of FSR
		can be realized in semiconductor heterostructures with two filled subbands, see Eq.~(\ref{example}).
		The spin degeneracy of the subbands is lifted by the applied magnetic field.
		The resonant Fermi surfaces are indicated by color.
		The gray Fermi surface is off resonance.
		(b)
		3D example of the FSR.
		Only the resonant Fermi surfaces are shown.
	}
\label{fig:ex}
\end{figure}

Equation~(\ref{reducedconserv}) can be thought of as a radial nesting of the Fermi surfaces.
Generally, nesting implies a one-dimensional character of the scattering between the nested parts of the Fermi surface.
This leads to strong enhancement of such scattering which can trigger an instability.
Celebrated examples of  instabilities driven by nesting are the charge and spin density orders \cite{Whangbo,Murayama}.
The radial nesting is also known to result in strongly interacting electron states such as fractional topological insulators with a gap~\cite{Trifunovic,Volpez}.
In our study we show that the radial nesting of the Fermi surfaces given by Eq.~(\ref{reducedconserv}) leads to a gapless quantum critical state.

In the general formulation of the problem we require all $2 N$ non-degenerate Fermi surfaces participating in the resonant $N \to N$ scattering [see Eqs.~(\ref{reducedconserv}), (\ref{Delta})] to be different.
However, we can soften this and only require
that the $N$ initial states belong to different Fermi surfaces,
and similarly for the $N$ final states; some of the initial states might then have the same Fermi surface index as some of the final states.
In the example shown in Fig.~\ref{fig:ex}(a)
the FSR corresponds to the following condition:
\begin{eqnarray}
2 k_1 + k_2 - k_3 = 0 ,
\label{example}
\end{eqnarray}
where $k_a$, again, are the Fermi momenta of corresponding Fermi surfaces.
The gray Fermi surface in Fig.~\ref{fig:ex}(a) does not participate in the resonant scattering.
This example corresponds to the soft formulation because the green Fermi surface contains initial and final states.
The integer $N$ in the soft formulation corresponds to the $N$-particle resonant scattering amplitude, e.g. $N = 2$ in Fig.~\ref{fig:ex}(a).

In order to study the new quantum critical phase experimentally, one has to satisfy the single resonant condition given by Eq.~(\ref{reducedconserv}).
In addition, one has to ensure that the Fermi gas is strongly interacting, i.e. $r_s \gg 1$.
We argue that  semiconductor heterostructures are most promising candidates for the experimental search for such new phases.
The simplest example of a semiconductor heterostructure that can host a new exotic quantum critical phase is shown in Fig.~\ref{fig:ex}(a), which represents
the Fermi surfaces of a two-dimensional electron gas with two occupied subbands.
Even though the occupation of multiple subbands is experimentally achievable \cite{Hamilton,Zheng},
the experimental research of materials with multiple Fermi surfaces is still very limited which partially explains why this new phase has never been detected before.
Each subband in Fig.~\ref{fig:ex}(a) is split by an external magnetic field.
Changing the electron density and fine tuning by the magnetic Zeeman splitting we can set the system to the FSR given by Eq.~(\ref{example}).
The given example corresponds to the $N = 2$ particle resonant scattering amplitude within the soft formulation of the FSR
because the green Fermi surface contains both initial and final states.
The FSR results in the strong mixing of three colored bands in Fig.~\ref{fig:ex}(a) which destroys quasiparticles in the vicinity of the colored Fermi surfaces.
The quasiparticles in the vicinity of the off-resonant gray Fermi surface survive.
The new phase in this example has separate Fermi-liquid and non-Fermi-liquid components.
The latter is established at the resonance given by Eq.~(\ref{example}) and can be experimentally identified from the power-law frequency and momentum dependence of the spectral function
and the singular linear response functions in the static limit and at the Kohn anomalies (see below).

In what follows we consider the general case of $D > 1$ spatial dimensions.
A $D=3$ example of the strong version of the FSR with $2 N = 4$ different Fermi surfaces
is shown in Fig.~\ref{fig:ex}(b).

\section{Effective Hamiltonian}

Now we proceed to the general case of the FSR.
Here we assume that all $2 N$ fields participating in the resonance are different, see Eq.~(\ref{reducedconserv}).
All the results that we obtain in this paper also apply to the soft version of the FSR where some initial states might have the same Fermi surface index as some final states, see Fig.~\ref{fig:ex}(a).
The FSR results in a dramatic change of the ground state because it favors resonant many-body exchange scattering.
The FSR is applied to $2 N$ different non-degenerate Fermi surfaces, so we consider an $N \to N$ scattering amplitude which is multilinear with respect to each fermion field.
As the FSR condition says nothing about initial and final states, we have to sum over all possible choices of $N$ initial and $N$ final states out of overall $2 N$ fields corresponding to the $2 N$ Fermi surfaces,
yielding the following effective Hamiltonian:
\begin{eqnarray}
\!\!\!\!\! V \! (R) = \! \sum\limits_{\{j\}} \lambda_j \Psi_{j_1}^\dagger \! (R) \dots \Psi_{j_N}^\dagger \! (R) \Psi_{j_{N + 1}} \! (R)  \dots \Psi_{j_{2 N}} \! (R) ,
\label{VN}
\end{eqnarray}
where $R = (t, {\bf r})$, ${\bf r}$ is a $D$-dimensional position vector, $t$ the time, $j$  a permutation of indices $\{1, \dots, 2 N\}$, $\Psi_a (R)$  the fermion field operator corresponding to the $a^{\mathrm{th}}$ Fermi surface, and $\lambda_j$ are the coupling constants.
We sum over all non-equivalent permutations corresponding to $C^N_{2 N} = (2 N)! / (N!)^2$ different choices of initial and final states.
For conjugate terms the corresponding $\lambda_j$ is a complex conjugate in order to ensure hermiticity of $V (R)$.
The effective Hamiltonian $V (R)$ is of exchange form as it mixes together all $2 N$ fermion fields corresponding to the $2 N$ Fermi surfaces participating in the resonance.
Similar in spirit is the effective Hamiltonian approach  widely used in condensed matter physics, in particular, in the weakly coupled wire approach where $N$-electron effective inter-wire interactions are constructed \cite{Teo,Aseev,Klino}.

In case of the soft formulation, the effective Hamiltonian has the same form as Eq.~(\ref{VN}).
The only difference is that the terms containing the square of field operators vanish due to the Fermi statistics.
This is consistent with our requirement that all $N$ initial states as well as all $N$ final states   belong to $N$ different Fermi surfaces.

The effective Hamiltonian, see Eq.~(\ref{VN}), can be constructed using  perturbation theory with respect to the two-particle Coulomb interaction, $v_C$.
The first contribution to $V (R)$ comes from the tree diagrams in the $(N-1)^{\mathrm{th}}$ order in $v_C$:
\begin{equation}
V(R) \propto \Lambda = \frac{v_C^{N - 1}}{E_F^{N - 2}} = E_F r_s^{N - 1} ,
\label{Lambda}
\end{equation}
where $\Lambda$ is the characteristic energy scale of $V (R)$, $E_F$  the Fermi energy, and $r_s$  the interaction parameter given by Eq.~(\ref{rs}).
The power of $v_C$ corresponds to the order of perturbation theory, the power of $1/E_F$ corresponds to the number of fermion propagators in the tree diagrams.
Notice that the strongly interacting regime $r_s \gg 1$ also corresponds to $\Lambda \gg E_F$.
Each of the tree diagrams can be envisioned as a sequence of $N - 1$ Coulomb exchange scattering events.
As all $2 N$ fermions are different, the momentum transfers during the exchange are all in order of the average Fermi momentum $k_F$.
We are interested in the long range correlations at relative distance $r \gg 1 / k_F$.
For such long range correlations the $N \to N$ scattering that occurs on the scale of the Fermi wavelength $\sim 1 / k_F$ is effectively local, justifying the locality of $V(R)$.
All the matrix elements that appear in the tree diagrams for $V (R)$ are hidden in the coupling constants $\lambda_j$.
Higher order diagrams for $V(R)$ only renormalize the coupling constants $\lambda_j$.
Due to symmetries of specific Hamiltonians some of the coupling constants $\lambda_j$ might be equal to zero.
However, this fact is not important for the further analysis if there are at least some non-zero $\lambda_j$.

We argue that $V(R)$ is the most important scattering amplitude close to the FSR, see Eq.~(\ref{reducedconserv}).
All other terms in the many-body scattering amplitudes are either insensitive to the FSR or contain rapidly oscillating terms on the scale of Fermi wavelength.
Here we work within the assumption that at arbitrary filling electrons or holes in semiconductors form the LFL.
This means that the interaction which is not sensitive to the FSR cannot significantly change the physics.
Interactions that contain oscillating terms can be averaged to zero on large scales $r \gg 1 / k_F$.
This allows one to include such interactions as irrelevant corrections renormalizing the LFL parameters.

The effective Hamiltonian $V(R)$ is very sensitive to the FSR condition given by Eq.~(\ref{reducedconserv}).
Further we show that
exactly at the FSR it results in the emergent long-range order
and the destruction of the quasiparticle picture
in the infrared limit.

\section{Dimensional reduction}

In this section we calculate the fermion Green function dressed by the effective interaction $V(R)$, see Eq.~(\ref{VN}).
Here we show that a strong interaction $V(R)$ completely destroys quasiparticles close to the Fermi surfaces.
The notion of Fermi surfaces is still important though because they define the sector of quantum states that are  most affected by the interaction $V(R)$.
Such Fermi surfaces without quasiparticles are known as critical Fermi surfaces \cite{Chowdhury}.

Far away from the Fermi surfaces
the resonant scattering is destroyed and
thus we expect the LFL in the ultraviolet limit
even when the FSR condition Eq.~(\ref{reducedconserv})
is satisfied.
Thus, for large $\omega_a \sim E_F$ and $\delta p_a = p_a - k_a \sim k_a$
the Green function restores the quasiparticle poles:
\begin{eqnarray}
G^{(0)}_a (\omega_a, \delta p_a) = \frac{Z_a}{\omega_a - v_a \delta p_a} ,
\label{G0}
\end{eqnarray}
where $E_F$ is the Fermi energy,
the index $a \in \{1, \dots, 2 N\}$ enumerates the Fermi surfaces,
$\omega_a$ and $p_a$ are the frequency
and the momentum,
$k_a$ is the Fermi momentum of the $a^{\rm th}$ Fermi surface,
$v_a$ is the Fermi velocity which is renormalized by irrelevant interactions, and
$1 > Z_a > 0$ is the quasiparticle residue away from the Fermi surface.
We allow the frequency $\omega_a$ to be complex.
For example,
imaginary frequencies correspond to the Matsubara formalism,
while $\omega_a \to \omega_a + i0$ yields the
retarded Green function.
In this section we calculate the Matsubara Green function at zero temperature.
Other Green functions can be obtained
via the analytical continuation
through the spectral representation \cite{lehmann}:
\begin{eqnarray}
G_a (\omega_a, \delta p_a) = \int\limits_{-\infty}^\infty  \frac{dz}{\pi}\frac{ \mathcal{A}_a (z, \delta p_a)}{\omega_a - z} ,
\label{specrep}
\end{eqnarray}
where $\mathcal{A}_a (\omega_a, \delta p_a) = - \mbox{\rm{Im}}\left[G_a (\omega_a + i0, \delta p_a)\right] > 0$ is the positively defined spectral function,
and $\mbox{\rm{Im}}$ stands for the imaginary part.

Close to the FSR, see Eqs.~(\ref{reducedconserv}) and (\ref{Delta}),
the resonant many-body exchange scattering described by the effective Hamiltonian $V(R)$, see Eq.~(\ref{VN}),
becomes important in the infrared limit.
We include it via the self-energy $\Sigma_a (i \omega_a, \delta p_a)$:
\begin{eqnarray}
G_a (i \omega_a, \delta p_a) = \frac{1}{G^{(0)}_a (i \omega_a, \delta p_a)^{-1} - \Sigma_a (i \omega_a, \delta p_a)} ,
\label{G}
\end{eqnarray}
where $G^{(0)}_a (i \omega_a, \delta p_a)$ is given by Eq.~(\ref{G0}) and contains the contributions from the irrelevant interactions,
$\omega_a$ is the fermionic Matsubara frequency.
We assume here that there is no Pomeranchuk instability \cite{pomeranchukinsta} even in the strongly interacting regime,
so the spherical symmetry of the Fermi surfaces is exact.
In this case the fermion Green function $G_a (\tau, \bm r)$ in the imaginary time-coordinate representation depends
only on the absolute value  $r=|\bm r|$.
This allows for the effective one-dimensional representation of the Green function:
\begin{eqnarray}
&& G_a(\tau, r) = \sum\limits_{\bm p_a} e^{i \bm p_a \cdot \bm r} G_a(\tau, \delta p_a) \nonumber \\
&& = \int\limits_{-k_a}^\infty \frac{d \delta p_a}{2 \pi} p_a^{D - 1} J(r p_a) G_a(\tau, \delta p_a) , \label{Gint} \\
&& \sum\limits_{\bm p_a} \equiv \int \frac{d \bm p_a}{(2 \pi)^D} = \int \frac{d \Omega_D}{(2 \pi)^{D - 1}} \frac{d \delta p_a}{2 \pi} p_a^{D - 1} , \label{pint} \\
&& J (z) = \int \frac{d \Omega_D \, e^{i z \cos \theta}}{(2 \pi)^{D - 1}} \to \frac{2 \cos \left(|z| - \frac{\pi}{2} \frac{D - 1}{2}\right)}{\left(2 \pi |z|\right)^{\frac{D - 1}{2}}} ,
\label{J}
\end{eqnarray}
where $\delta p_a = p_a - k_a$,
$k_a$ is the Fermi momentum of the $a^{\rm th}$ Fermi surface,
$d \Omega_D$ is the volume element of the $D$-dimensional solid angle.
In Eq.~(\ref{J}) we also provide the asymptotic behavior of $J(z)$ for large argument $|z|\gg 1$.
We use this to derive the long distance form of the Green function:
\begin{eqnarray}
&& G_a(\tau, r) \to \left(\frac{k_a}{2 \pi r}\right)^{\frac{D - 1}{2}}  \nonumber \\
&&\times \left[i^{-\frac{D - 1}{2}}  e^{i k_a r} \mathcal{G}_a(\tau, r) + i^{\frac{D - 1}{2}}  e^{-i k_a r} \mathcal{G}_a(\tau, -r) \right], \label{Gred}\\
&& \mathcal{G}_a(\tau, x) = \int\limits_{-\infty}^\infty \frac{d \delta p_a}{2 \pi} e^{i x \delta p_a} G_a(\tau, \delta p_a) , \label{gonedim}
\end{eqnarray}
where $r \gg 1/k_a$,
and the integral over $\delta p_a$ is extended to the interval $(-\infty, \infty)$ with negligible error.
Here, $\mathcal{G}_a(\tau, x)$ represents a one-dimensional Fourier transform, $x \in (-\infty, \infty)$.
Thus, one can consider Eqs.~(\ref{Gred})--(\ref{gonedim})
as the dimensional reduction from $D$ spatial dimensions to a single spatial dimension with coordinate $x$
which is conjugate to the momentum $\delta p_a$.

In order to calculate the effectively one-dimensional Green function $\mathcal{G}_a(\tau, x)$, see Eq.~(\ref{gonedim}),
we have to find the electron self-energy
due to the interaction Hamiltonian $V(R)$, see Eq.~(\ref{VN}).
The Feynman diagram for the exact self-energy for $N = 2$ is presented in Fig.~\ref{fig:se}(a)
and corresponds to the example shown in Fig.~\ref{fig:ex}(b).
Feynman diagrams for general $N$ can be drawn in a similar fashion.
The problem here is the renormalization of the interaction vertex, see full black square in Fig.~\ref{fig:se}(a).
In this section we omit the interaction vertex renormalization and instead consider the simpler diagram in Fig.~\ref{fig:se}(b).
Such an approximation is called the self-consistent Born approximation (SCBA).
The diagrams of the form in Fig.~\ref{fig:se}(b) are also known as ``melon'' diagrams that appear in various matrix and tensor field theories \cite{Gurau}.

We calculate the SCBA self-energy $\Sigma_1 (\tau, \delta p_1)$, see Fig.~\ref{fig:se}(b),
in the imaginary time-coordinate representation:
\begin{eqnarray}
&& \Sigma_1 (\tau, r) = (-1)^{N - 1} \sum\limits_{\{\eta\}} |\lambda_\eta|^2 \prod\limits_{a = 2}^{2 N} G_{a} (\eta_a \tau, r) , \label{sigmageneral}
\end{eqnarray}
where $\lambda_\eta$ are the bare interaction coupling constants, see Eq.~(\ref{VN}).
We consider the self-energy corresponding to the electrons near the 1$^{\mathrm{st}}$ Fermi surface,
the result is similar for other Fermi surface indices.
Each $\eta = (\eta_2, \dots, \eta_{2 N})$, $\eta_b = \pm 1$, corresponds to one of the choices to draw arrows on the Feynman diagram,
one of such choices is shown in Fig.~\ref{fig:se}(b).
The charge conservation imposes the following constraint:
\begin{eqnarray}
\sum\limits_{b = 2}^{2 N} \eta_b = 1 .
\label{charge}
\end{eqnarray}
This gives overall
$C^N_{2 N - 1} = (2 N - 1)! / (N! (N - 1)!)$
terms in the sum over $\eta$.
The long-distance asymptotics of the self-energy
takes the form of Eq.~(\ref{Gred}):
\begin{eqnarray}
&& \Sigma_1(\tau, r) \to \left(\frac{k_1}{2 \pi r}\right)^{\frac{D - 1}{2}}  \nonumber \\
&&\times \left[i^{-\frac{D - 1}{2}}  e^{i k_1 r} \mathcal{S}_1(\tau, r) + i^{\frac{D - 1}{2}}  e^{-i k_1 r} \mathcal{S}_1(\tau, -r) \right], \label{Sigred}\\
&& \mathcal{S}_1(\tau, x) = \int\limits_{-\infty}^\infty \frac{d \delta p_1}{2 \pi} e^{i x \delta p_1} \Sigma_1(\tau, \delta p_1) , \label{sigonedim}
\end{eqnarray}
where $\mathcal{S}_1(\tau, x)$ is a one-dimensional self-energy.
In order to find $\mathcal{S}_1(\tau, x)$, we
substitute the asymptotic expansion Eq.~(\ref{Gred}) of the Green functions
in Eq.~(\ref{sigmageneral})
and compare with Eq.~(\ref{Sigred}):
\begin{eqnarray}
&& \!\!\!\!\!\!\!\! \mathcal{S}_1 (\tau, x) =  (-1)^{N - 1} c_1  \sum\limits_{\{\eta\}}  |\lambda_\eta|^2  \prod\limits_{a = 2}^{2 N}  \mathcal{G}_{a} (\eta_a \tau, s_a x) \mathcal{D} (x) ,
\label{sigmaxt} \\
&& \!\!\!\!\!\!\!\! \mathcal{D} (x) = \frac{e^{- i \mathcal{K} x}}{|x|^{2\nu}}
i^{\sigma (D - 1) {\rm sgn} (x)}, \,\,\, \nu = \frac{1}{2}(N - 1) (D - 1) ,
\label{Dapp}
\end{eqnarray}
where $s_a = -1$ ($ s_a = +1$) for $a \in \{2, \dots, N + \sigma\}$ ($a \in \{N + \sigma + 1, \dots , 2N \}$),
$|\mathcal{K}| \ll k_a$ and $\sigma$ are defined in Eq.~(\ref{reducedconserv})
and $c_1$ is the following constant:
\begin{eqnarray}
&& \!\!\! c_1 = \left(\frac{k_F}{k_1}\right)^{D - 1} \left(\frac{k_F}{2 \pi}\right)^{2 \nu} , \,
\left(k_F\right)^{2 N} \equiv k_1 \cdot \dots \cdot k_{2 N} ,
\label{c1}
\end{eqnarray}
where $k_F$ is the average Fermi momentum.
In Eq.~(\ref{sigmaxt}) we only retained slowly varying terms because we are interested
in the long distance correlations.
Note that $1 / \mathcal{K}$ is the only
length scale that survived the dimensional reduction.
The fluctuations at $|x| \gg 1 / |\mathcal{K}|$ are not important due to the oscillating exponential in Eq.~(\ref{Dapp}),
and thus $1/|\mathcal{K}|$ defines the finite range of the interaction.
At $\mathcal{K} = 0$ there are no internal length scales that enter Eq.~(\ref{sigmaxt})
which results in the emergent long range order.

\begin{figure}[t]
	\includegraphics[width=\columnwidth]{SE.eps}
	\caption{
		Electron self-energy.
		(a) Feynman diagram for the exact self-energy for the case $N = 2$.
		Solid lines correspond to the exact Green functions, see Eq.~(\ref{G}).
		The Fermi surface index $a \in \{1,2,3,4\}$ is indicated by color and corresponds to the example given in Fig.~\ref{fig:ex}(b).
		The black (white) square corresponds to the exact (bare) interaction vertex.
		There are two more contributions to the self-energy that differ by the arrow directions.
		The Feynman diagrams for general $N$ can be drawn similarly.
		(b) The SCBA.
		The interaction vertex correction is neglected which is shown by two bare vertices (white squares).
	}
	\label{fig:se}
\end{figure}

\section{Strong coupling limit}

In this section we consider
the strong coupling regime,
when
the Green function close to the Fermi surfaces, see Eq.~(\ref{G}),
is entirely defined by its self-energy:
\begin{eqnarray}
G_a (i \omega_a, \delta p_a) \approx - \frac{1}{\Sigma_a (i \omega_a, \delta p_a)} .
\label{green}
\end{eqnarray}
This approximation breaks down far away from the Fermi surfaces
where the many-body scattering is off-resonant.
In other words, the LFL is restored in the
ultraviolet limit far away from the Fermi surface.
Only the quantum states close to the Fermi surface
are strongly affected by the interaction.
These quantum states represent the infrared sector of the problem.

In the $(\tau, x)$ representation Eq.~(\ref{green}) takes the integral form:
\begin{eqnarray}
\!\!\!\!\!\!\!\! \delta (\tau) \delta (x) = - \int d\tau' dx' \, \mathcal{G}_a (\tau - \tau', x - x') \mathcal{S}_a (\tau', x') ,
\label{strongcoup}
\end{eqnarray}
where $\mathcal{G}_a(\tau, x)$ and $\mathcal{S}_a(\tau, x)$ are defined in Eqs.~(\ref{gonedim}) and (\ref{sigonedim}), respectively.
Substituting Eq.~(\ref{sigmaxt}) into Eq.~(\ref{strongcoup}) results in the integral Dyson equation for the Green function
calculated within the SCBA and in the strong coupling limit:
\begin{eqnarray}
&& \delta (\tau) \delta (x) = (-1)^N c_1 \int d\tau' dx' \, \mathcal{G}_1 (\tau - \tau', x - x')   \nonumber \\
&&\times \sum\limits_{\{\eta\}} |\lambda_\eta|^2 \prod\limits_{a = 2}^{2 N} \mathcal{G}_{a} (\eta_a \tau', s_a x') \mathcal{D} (x') .
\label{gequation}
\end{eqnarray}

At the FSR the effective interaction $\mathcal{D}(x)$ is quasi-long-range, i.e. it looks the same at all scales,
see Eq.~(\ref{Dapp}) at $\mathcal{K} = 0$.
In contrast to Eqs.~(\ref{G}) and (\ref{sigmageneral}) for the total Green function and the
self-energy,
which contain the information about the physical scales
such as the Fermi momenta $k_a$
and corresponding band splittings,
Eq.~(\ref{gequation})
is free from any physical energy or length scale.
This observation suggests that the Green functions in Eq.~(\ref{gequation})
are also universal scaling functions.
As all $2 N$ Green functions occur in the product in Eq.~(\ref{gequation}),
they are equivalent, and, thus,
we expect the same critical exponents for all of them regardless of the index $a$.
A simple dimensional analysis of Eq.~(\ref{gequation}) yields:
\begin{eqnarray}
\mathcal{G}_a (\tau, x) \propto \frac{1}{|\tau|^{2 h}} \frac{1}{|x|^{2 l}}, \,\,\,\,\, h = \frac{1}{2 N}, \, l = \frac{1 - \nu}{2 N} ,
\label{scalingapp}
\end{eqnarray}
where $\nu$ is given by Eq.~(\ref{Dapp}).
The time and coordinate scalings of the Green function
 are different due to the effective interaction $\mathcal{D}(x)$.
This naturally suggests the separation of temporal and spatial dynamics:
\begin{eqnarray}
\mathcal{G}_a (\tau, x) = C_a g (\tau) \gamma (x) ,
\label{separation}
\end{eqnarray}
where $C_a$ is some constant that might be different for different $a$.
This ansatz is different from the truly one-dimensional case $D = 1$ in which the temporal and spatial scalings are the same and the separation argument does not apply for $D=1$.
Instead, left and right linear combinations of time and coordinate must be used for $D=1$.
Thus, our current analysis of Eq.~(\ref{gequation}) is only applicable for $D > 1$ where the effective interaction $\mathcal{D}(x)$
breaks the equivalence between time and coordinate.
The Dyson Eq.~(\ref{gequation}) then separates into a time and a coordinate equation:
\begin{eqnarray}
&& \!\!\!\!\!\!\!\!\!\! \delta (\tau) = (-1)^{N} \int d\tau' \, g (\tau - \tau') g(-\tau')^{N - 1} g(\tau')^N ,
\label{g}\\
&& \!\!\!\!\!\!\!\!\!\! \delta (x) \! = \! \!\! \int  \!\! dx' \gamma (x - x') \gamma(-x')^{N + \sigma - 1} \gamma(x')^{N - \sigma} \mathcal{D} (x') . \label{gamma}
\end{eqnarray}
As only the product of $g(\tau)$ and $\gamma(x)$
is important, see Eq.~(\ref{separation}),
we have an additional symmetry under the transformation $g(\tau) \to C g(\tau)$ and $\gamma(x) \to \gamma(x) / C$ for any complex $C\neq 0$.
The coefficients $C_a$ satisfy the following algebraic equation:
\begin{eqnarray}
c_1 \lambda^2 \prod\limits_{a = 1}^{2 N} C_a = 1 , \,\,\,\,\, \lambda = \sqrt{\sum\limits_{\{\eta\}}|\lambda_\eta|^2 } .
\label{Ca}
\end{eqnarray}

Equation~(\ref{Ca}) will be different
if we consider the Dyson equation Eq.~(\ref{strongcoup}) for some other Fermi surface index $a \ne 1$.
This is due to the non-universal coefficient $c_1 \propto k_1^{1 - D}$, see Eq.~(\ref{c1}),
which is in general replaced by $c_a \propto k_a^{1 - D}$.
At the same time the product of all coefficients $C_a$ in Eq.~(\ref{Ca})
is independent of the Fermi surface index.
This
apparent conflict
arises due to inadequacy of the long distance
expansion, Eq.~(\ref{Gred}), at small
distances $r \lesssim 1/k_F$,
$k_F$ is an average Fermi momentum.
Indeed, the right hand side of Eq.~(\ref{gequation}) has to diverge at $x = 0$
which corresponds to the ultraviolet divergence
of the propagators $\mathcal{G}_a(\tau, x)$ at $x = 0$, see Eq.~(\ref{scalingapp}).
On the other hand, such a divergence at small $r \lesssim 1/k_F$  is regularized (and thus absent) in the exact Green function $G_a(\tau, r)$.
In other words, the long-distance infrared limit
that we take here does not allow us to define the exact
numerical prefactor in Eq.~(\ref{gequation})
as it depends on the ultraviolet regularization.
At the same time,
the scale-independent limit of the Dyson Eq.~(\ref{gequation})
at $\mathcal{K} = 0$ implies that $c_1$ has to be replaced by a universal coefficient $c$
which we estimate via  dimensional analysis:
\begin{eqnarray}
&& c_1 \to c \sim \left(\frac{k_F}{2 \pi}\right)^{2 \nu} . \label{c}
\end{eqnarray}

We can choose all coefficients $C_a$ positive, i.e. $C_a > 0$.
Non-trivial phases can come from solutions for $g(\tau)$ and $\gamma(x)$ which we consider later.
All coefficients $C_a > 0$ are combined in the single product in Eq.~(\ref{Ca}) with $c_1$ replaced by $c$, see Eq.~(\ref{c}), so it is not possible to determine them separately without connecting the infrared interaction-dominated limit
with the ultraviolet limit far away from the Fermi surfaces which is given by the LFL.
From Eq.~(\ref{Ca}) we conclude that the coefficients $C_a$ scale with the interaction strength as $C_a \propto \lambda^{-1/N}$.
This corresponds to the following self-energy scaling:
\begin{eqnarray}
&& \Sigma_a \propto \lambda^{1/N} . \label{siglam}
\end{eqnarray}
This scaling justifies the strongly interacting limit $\lambda \to \infty$ at which the self-energy correction is dominant over the single-particle spectral part.
Apart from the considered interaction scaling, the coefficients $C_a$ play no role in the infrared physics that we study here,
so we can concentrate on the universal functions $g(\tau)$ and $\gamma(x)$.

We observe here that Eq.~(\ref{g}) is the Dyson equation for the generalized SYK model,
referred to as $q$-SYK model~\cite{Ye,SachdevX}, with $q = 2 N$ in our case.
The SYK model describes $(0+1)$-dimensional strongly correlated fermions with all-to-all interactions whose matrix elements are randomly distributed.
Quite remarkably,
the temporal dynamics for our $(D + 1)$-dimensional system without having any randomness in our model and for short-range interactions given by the Hamiltonian $V(R)$, see Eq.~(\ref{VN}),
maps exactly onto the $q$-SYK model with $q = 2 N$.
The SYK model is the central example
of the AdS/CFT correspondence \cite{maldacena}
and plays an important role in understanding
the nature of strange metals \cite{Parcollet,Chowdhury,Davison}.
In contrast to Eq.~(\ref{g}),
Eq.~(\ref{gamma})
contains the effective interaction
$\mathcal{D}(x)$
which can be interpreted as a propagator of an emergent complex boson.
This emergent boson is merely a result of the dimensional reduction.
Its propagator $\mathcal{D}(x)$
contains the momentum mismatch $\mathcal{K}$ (which is zero at the FSR) and the geometric factors coming from the $s$-wave expansion of the fermion propagators, see Eqs.~(\ref{Gred}) and (\ref{Dapp}).

The Matsubara Green function $G_a (\tau, r)$, see Eq.~(\ref{Gred}), has to be
real-valued which is evident from the spectral representation, see Eq.~(\ref{specrep}).
This is equivalent to the following constraint:
\begin{eqnarray}
&& \mathcal{G}_a^* (\tau, r) = \mathcal{G}_a (\tau, -r) ,
\end{eqnarray}
where $\mathcal{G}_a (\tau, r)$ is defined in Eq.~(\ref{gonedim}).
Together with the separation of variables, see Eq.~(\ref{separation}),
this results in the following conditions on the universal functions $g(\tau)$ and $\gamma(x)$:
\begin{eqnarray}
&& g^*(\tau) = g(\tau), \, \gamma^*(-x) = \gamma(x) . \label{sym}
\end{eqnarray}
In other words,
$g(\tau)$ and $\gamma(\delta p)$
have to be real-valued functions.
Following Refs.~\cite{SachdevX,Parcollet},
we find the universal functions $g(\tau)$ and $\gamma(x)$:
\begin{eqnarray}
&& g(\tau) \propto \frac{{\rm sgn}(\tau)}{|\tau|^{2h}} , \label{gphsym} \\
&& \gamma(x) \propto \frac{1}{|x|^{2 l}} ,  \label{gamx}
\end{eqnarray}
where we used the symmetry of Eq.~(\ref{sym}).

Equation~(\ref{g})
can be easily extended to the case of finite temperature $T$
due to the emergent conformal symmetry, for details see Refs.~\cite{SachdevX,Parcollet}:
\begin{eqnarray}
&& g_T (\tau) \propto {\rm sgn}(\tau) \left|\frac{\pi T}{\sin \left(\pi T \tau\right)}\right|^{2 h} , \label{gT}
\end{eqnarray}
where $\tau \in (-1/T, 1/T)$ is the imaginary time, with the Boltzmann constant set to $k_B=1$.
Using Eqs.~(\ref{Gred}), (\ref{separation}), (\ref{gphsym}), and (\ref{gamx}),
we find explicitly the long time and long distance asymptotics
of the finite temperature Matsubara Green function:
\begin{eqnarray}
&& \!\!\!\!\! G_a (\tau, r) \propto {\rm sgn}(\tau) \left|\frac{\pi T}{\sin \left(\pi T \tau\right)}\right|^{2 h} \frac{\cos \left(k_a r - \frac{\pi}{2}\frac{D - 1}{2}\right)}{r^{(D + 1) h}} , \label{greenasympt}
\end{eqnarray}
where $h = 1/(2 N)$ is the temporal scaling dimension.
The corresponding Fourier transform $G_a (i\omega_a, \delta p_a)$ becomes:
\begin{eqnarray}
&& \!\!\!\!\!\!\!\!\!\! G_a (i\omega_a, \delta p_a) \!=\! - \frac{i}{T}\! \left|\frac{T}{\Lambda}\right|^{2 h} \left|\frac{k_a}{\delta p_a}\right|^{1 - 2 l} \!\!\! \frac{\displaystyle \Gamma \left(h + \frac{\omega_a}{2 \pi T}\right)}{\displaystyle \Gamma \left(1 - h + \frac{\omega_a}{2 \pi T}\right)} , \label{matw}
\end{eqnarray}
where $\omega_a$ is the discrete Matsubara frequency, $k_a$  the Fermi momentum corresponding to the $a^{\rm th}$ Fermi surface.
Here we restored the physical dimensions
using Eqs.~(\ref{Lambda}), (\ref{siglam}).
The unknown numerical factors are hidden in the
phenomenological ultraviolet scale $\Lambda$
that defines the interaction strength, see Eq.~(\ref{Lambda}).

Analytically continuing the Matsubara Green function
Eq.~(\ref{matw}) to real frequencies,
we find the retarded Green function:
\begin{eqnarray}
&& \!\!\!\!\!\!\!\!\!\! G_a^{R} (\omega_a, \delta p_a) \! = \! - \frac{i}{T} \! \left|\frac{T}{\Lambda}\right|^{2 h} \! \left|\frac{k_a}{\delta p_a}\right|^{1 - 2 l} \!\!\! \frac{\displaystyle \Gamma \left(h - \frac{i\omega_a}{2 \pi T}\right)}{\displaystyle \Gamma \left(1 - h - \frac{i\omega_a}{2 \pi T}\right)} ,
\label{gret}
\end{eqnarray}
where $\omega_a$ is a real frequency.
The advanced Green function
is complex conjugate to the retarded one.
In the limit of large frequency
$\omega \gg T$
we restore the power-law frequency scaling
$G_a(\omega_a, \delta p_a) \propto |\omega_a|^{2 h - 1}$, where
$2 h - 1 < 0$ signaling the singularity at $\omega \to 0$.
At finite temperature $T$
this singularity is cut
as $G_a (\omega_a = 0, \delta p_a) \propto T^{2 h - 1}$, see Eqs.~(\ref{matw}) and (\ref{gret}).

As we know the exact retarded Green function,
we can calculate the fermion spectral density:
\begin{eqnarray}
&& \mathcal{A}_a (\omega_a, \delta p_a) = - \frac{1}{\pi} \mbox{\rm{Im}}\left[G_a^R(\omega_a, \delta p_a)\right] = \frac{\sin (\pi h)}{\pi^2 T}  \nonumber \\
&&\times \left|\frac{T}{\Lambda}\right|^{2 h} \left|\frac{k_a}{\delta p_a}\right|^{1 - 2 l} \cosh \left(\frac{\omega_a}{2 T}\right) \left|\Gamma\left(h + \frac{i \omega_a}{2 \pi T}\right)\right|^2 . \label{den}
\end{eqnarray}
The temperature and frequency dependence of
the spectral density correspond to
the $q$-SYK, $q = 2N$, model, see Ref. \cite{Parcollet}.
The electron spectral function, see Eq.~(\ref{den}),
does not contain any sharply defined
quasiparticle peaks.
Instead, there is the power law
scaling in the momentum $\delta p_a$ and frequency $\omega_a \gg T$.
At low frequency $\omega_a \ll T$, the spectral density
is a power-law function of temperature.
Thus, measuring the momentum, frequency, and temperature scaling of the spectral
density close to the Fermi surface
allows one to identify experimentally the scaling dimensions $h$ and $l$
that determine the quantum critical point.

\section{Exactness of SCBA and stability of the quantum critical point}

In order to calculate the electron self-energy,
we used the SCBA,
i.e. we neglected the interaction vertex corrections,
see Fig.~\ref{fig:se}.
Here we aim to show that the SCBA
is exact in the strong coupling limit.
For this we investigate the symmetries of the Dyson Eqs.~(\ref{g}) and (\ref{gamma}).
It turns out that both equations are invariant under  time and coordinate reparametrizations which constitutes the
two-dimensional conformal symmetry.
This means that if $g(\tau, \tau')$ [$\gamma(x, x')$]
are  solutions of the Dyson Eq.~(\ref{g}) [Eq.~(\ref{gamma})],
then the following functions $\tilde g (t_1, t_2)$ [$\tilde \gamma (\xi_1, \xi_2)$]
are also solutions of the corresponding Dyson equations:
\begin{eqnarray}
&& \tau = f_1 (t), \, x = f_2(\xi) , \\
&& \tilde g (t_1, t_2) = \left|f_1'(t_1) f_1'(t_2) \right|^h g(f_1(t_1), f_1(t_2)) , \\
&& \tilde \gamma(\xi_1, \xi_2) = \left|f_2'(\xi_1) f_2'(\xi_2)\right|^{l} \gamma(f_2 (\xi_1), f_2(\xi_2)) ,
\end{eqnarray}
where $f_{1,2}(z)$ is a real-valued monotonic function, with
$f'_{1,2}(z)$ being its derivative with respect to $z$.
In contrast to the SYK Dyson Eq.~(\ref{g}),
the spatial Dyson Eq.~(\ref{gamma})
contains the function $\mathcal{D} (x)$
which transforms as a propagator of an emergent
boson field with the scaling dimension $\nu$:
\begin{eqnarray}
&& \tilde \mathcal{D}(\xi_1, \xi_2) = \left|f_2'(\xi_1) f_2'(\xi_2)\right|^\nu \mathcal{D}(f_2(\xi_1), f_2(\xi_2)) .
\end{eqnarray}
The conformal symmetry of both temporal and spatial Dyson Eqs.~(\ref{g}) and (\ref{gamma})
is signaling an interaction fixed point in the renormalization group sense.
In other words,
our initial ansatz that the interaction vertex
is not renormalized in the infrared limit
turned out to be correct in the strong coupling limit
due to the emergent conformal symmetry.

So far, we have solved exactly
the problem of a strongly interacting electron gas that is
tuned to the FSR.
However, we neglected the single-particle spectral part
compared to the self-energy, see Eq.~(\ref{green}).
The strongly interacting limit is justified
if $|\Sigma_a (i \omega_a, \delta p_a)| \gg \mbox{\rm{max}} \{\omega_a, v_a \delta p_a\}$, see Eq.~(\ref{G}),
where $v_a$ is the Fermi velocity
at the $a^{\rm th}$ Fermi surface.
Here we consider the case of finite temperature $T$ and finite size $L$ of the system.
In this case the infrared and long distance limits correspond to $\omega_a \sim T$ and $\delta p_a \sim 1/L$.
Using Eqs.~(\ref{green}) and (\ref{matw}),
we can estimate the self-energy at the smallest frequencies and momenta:
\begin{eqnarray}
&& \Sigma_a (i \omega_a \sim i T, \delta p_a \sim 1/L) \propto \Lambda^{2 h} T^{1 - 2 h} (k_a L)^{2 l - 1} .
\end{eqnarray}
Therefore,
the strong coupling limit corresponds to
the following constraint:
\begin{eqnarray}
&& \Lambda^{2 h} T^{1 - 2 h} (k_F L)^{2 l - 1} \gg \mbox{\rm{max}} \left\{T, \frac{v_F}{L} \right\} , \label{cond}
\end{eqnarray}
where we introduced the average Fermi momentum $k_F$
and average Fermi velocity $v_F$.
This condition can be represented in
simpler form:
\begin{eqnarray}
&& \frac{r_s^{N - 1}}{(k_F L)^{\frac{1 - 2 l}{2 h}}} \gg \frac{T}{E_F} \gg  \frac{1}{r_s} \left(k_F L\right)^{- \frac{2 l}{1 - 2 h}}, \label{Trange}
\end{eqnarray}
where we used Eq.~(\ref{Lambda})
to express everything in terms of $r_s \gg 1$.
Here it is important that
$1 - 2 l > 0$ for any $N \ge 2$, $D > 1$,
see Eq.~(\ref{scalingapp}).
This results in an upper bound $L^*$ for the
system size $L$:
\begin{eqnarray}
&& L \ll L^* = \frac{1}{k_F} r_s^{\frac{1 - 2 h}{1 - 2h - 2 l}} . \label{Lstar}
\end{eqnarray}
Thus,
the quantum critical point corresponding to $\mathcal{K} = 0$
is stable
only for finite size $L \ll L^*$
and in the temperature range given by Eq.~(\ref{Trange}),
see Fig.~\ref{fig:LT}.

\begin{figure}[t]
	\includegraphics[width=\columnwidth]{QC.eps}
	\caption{
		The $L$-$T$ phase diagram illustrating the crossover
		between the LFL (blue) and quantum critical (QC) state (red).
		The dashed lines correspond to upper and
		lower bounds in Eq.~(\ref{Trange}).
		The upper bound for the wire length
		$L^*$ corresponds to Eq.~(\ref{Lstar}).
	}
	\label{fig:LT}
\end{figure}

In the dimensional reduction we rely on the rotational symmetry of the Fermi surfaces.
However, main conclusions of our paper will still be true in case of small asymmetry
of the Fermi surfaces.
Let the Fermi momentum of the $a^{\rm th}$ Fermi surface range from $k_a - \Delta k_a$ to $k_a + \Delta k_a$, where
$\Delta k_a \ll k_a$.
Then in addition to Eq.~(\ref{Trange})
the temperature must also satisfy
$T \gg v_a \Delta k_a$. Here, $v_a$ is the Fermi velocity of the $a^{\rm th}$ Fermi surface,
so that the Fermi surface asymmetry is smoothened out by thermal effects.
Comparing with Eq.~(\ref{Trange}) this gives
an upper bound for $\Delta k_a$:
\begin{eqnarray}
&& \frac{v_a \Delta k_a}{E_F} \ll \frac{r_s^{N - 1}}{(k_F L)^{\frac{1 - 2 l}{2 h}}} .
\end{eqnarray}

\section{Linear response functions}

In this section
we consider the linear response functions
that can also be used for experimental
identification \cite{Stano}
of the predicted quantum critical state.
In case of non-degenerate Fermi surfaces
the linear response functions are represented by the
following charge susceptibilities
that we calculate within the Matsubara technique:
\begin{eqnarray}
\chi_{ab} (\tau, r) = - G_a (\tau, r) G_b (-\tau, r),
\label{chi}
\end{eqnarray}
where $a$ and $b$ are the Fermi surface indices,
$r$ is an absolute value of a $D$-dimensional
coordinate vector,
$\tau$ is imaginary time.
The vertex correction
is neglected in Eq.~(\ref{chi}).
In Appendix A we show that the vertex correction
to a linear response function
does not affect the scaling properties.
Using the long time and long distance asymptotics of the finite temperature Matsubara Green function
which is given by Eq.~(\ref{greenasympt}),
we find the asymptotics of the linear response functions:\\
\begin{eqnarray}
&& \chi_{ab} (\tau, r) \propto \left|\frac{\pi T}{\sin (\pi T \tau)}\right|^{4 h}  \nonumber \\
&&\hspace{40pt} \times \frac{\cos\left(k_+ r - \pi \frac{D - 1}{2}\right) + \cos\left(k_- r\right)}{r^{2 (D + 1) h }} , \label{chias}
\end{eqnarray}
where $k_\pm = k_a \pm k_b$.
The temporal part coincides with the SYK susceptibility,
see Ref.~\cite{Parcollet}.
The spatial part is dominated by the Kohn anomalies \cite{Kohn}.
In case $a = b$ the Matsubara susceptibility $\chi_{aa}(\tau, r)$ has a non-oscillatory quasi-long-range component.
The Fourier transform of Eq.~(\ref{chias}) yields $\chi_{ab} (i\omega, q)$:
\begin{eqnarray}
\chi_{ab} (i\omega, q) \propto \chi_M (i\omega) \chi_{ab} (q) , \label{chiqw}
\end{eqnarray}
where
\begin{eqnarray}
 \chi_M (i\omega) = \frac{\Gamma(1 - 4 h)}{\Lambda} \left|\frac{T}{\Lambda}\right|^{4 h - 1} \frac{\displaystyle \Gamma\left(2 h + \frac{ \omega}{2 \pi T}\right)}{\displaystyle \Gamma\left(1 - 2 h + \frac{\omega}{2 \pi T}\right)}  \label{chim}
\end{eqnarray}
and
\begin{eqnarray}
&& \chi_{ab} (q \approx |k_\pm|) = \frac{(k_a k_b)^{D - (D + 1) h} \Gamma\left(\frac{D + 1}{2} (1 - 4 h)\right)}{\displaystyle q^{\frac{D - 1}{2}} |q - |k_\pm||^{\frac{D + 1}{2} (1 - 4 h)}}  \nonumber \\
&& \times \cos\left(\frac{\pi}{4} \left[(D + 1) (1 - 4 h) \pm {\rm sgn} (q - |k_\pm|) (D - 1)\right]\right).\,\,\,\,\,\,\,\,\,\, \label{chiq}
\end{eqnarray}
Here, the index $M$ in $\chi_M (i\omega)$ stands for Matsubara susceptibility,
and $\omega$ is the bosonic Matsubara frequency.
We also restored the physical dimensions
using Eq.~(\ref{matw}),
even though the overall numerical factor is suppressed.
Equation~(\ref{chiq}) is also valid in the case
$a = b$ with $k_- = 0$ where $\chi_{aa}(i\omega, q)$ is divergent for $q \to 0$.
Analytically continuing $\chi_M(i\omega)$
to real frequencies we find the retarded linear response function:
\begin{eqnarray}
&& \chi_R (\omega) = \frac{\Gamma(1 - 4 h)}{\Lambda} \left|\frac{T}{\Lambda}\right|^{4 h - 1} \frac{\displaystyle \Gamma\left(2 h - \frac{i \omega}{2 \pi T}\right)}{\displaystyle \Gamma\left(1 - 2 h - \frac{i \omega}{2 \pi T}\right)} ,  \,\,\,\,\, \label{chiret}
\end{eqnarray}
where $\omega$ is a real frequency.
The imaginary part of the retarded linear response,
describing the dissipation due to interaction effects, becomes:
\begin{eqnarray}
&& {\rm Im} \left(\chi_{ab}^R (\omega, q)\right) \propto \chi_{ab}(q) {\rm Im} \left(\chi_R (\omega)\right) , \\
&& {\rm Im} \left(\chi_R (\omega)\right) = \cos(2 \pi h) \frac{\Gamma(1 - 4 h)}{\pi \Lambda} \left|\frac{T}{\Lambda}\right|^{4 h - 1}  \nonumber\\
&& \times \sinh \left(\frac{\omega}{2 T}\right) \left|\Gamma\left(2 h + \frac{i \omega}{2 \pi T}\right)\right|^2 , \label{imchi}
\end{eqnarray}
where $\chi_{ab} (q)$ is given by Eq.~(\ref{chiq}).

For $N = 2$ we have $h = 1/4$,
which results in a divergence in Eq.~(\ref{chiqw}).
Regularizing this divergence,
we find the retarded susceptibility
for $h = 1/4$:
\begin{eqnarray}
&& \!\!\!\!\! \chi_R (\omega) = \frac{1}{\Lambda} \left[\ln \left(\frac{\Lambda}{2 \pi T}\right) - \psi\left(\frac{1}{2} - \frac{i \omega}{2 \pi T}\right)\right], \, h = \frac{1}{4} , \label{quarter} \\
&& \!\!\!\!\! {\rm Im} \left(\chi_R (\omega)\right) = \frac{\pi}{2 \Lambda} \tanh\left(\frac{\omega}{2 T}\right) .
\end{eqnarray}
Here $\psi(z) = \Gamma'(z)/\Gamma(z)$ is the digamma function.
The real part of $\chi_R (\omega)$ is
logarithmically divergent at $\omega \gg T$.
These equations correspond to the standard SYK susceptibility, see Ref.~\cite{Parcollet}.
For $N = 2$ the Kohn anomalies are
logarithmically divergent at $q \approx |k_\pm| = |k_a \pm k_b|$:
\begin{eqnarray}
&& \!\!\!\!\! \!\!\!\chi_{ab} (q \approx |k_\pm|) \propto (k_a k_b)^{\frac{D}{2}} \left|\frac{k_a k_b}{q^2}\right|^{\frac{D - 1}{4}} \ln\left|\frac{k_a k_b}{(q -|k_\pm|)^2}\right| . \label{chiq2}
\end{eqnarray}
From Eq.~(\ref{chiq2}) we see that
the susceptibility at $a = b$ and $q \to 0$
diverges as a power law
with additional logarithmic divergence $\chi_{aa} (q \to 0) \to \ln(k_a/q) / q^{(D - 1)/2}$.

The divergent Kohn anomalies given by Eq.~(\ref{chiq})
and the one-dimensional character of the radially nested scattering resonance may lead to spatially inhomogeneous orders.
For example, coupling to intrinsic phonons
can result in a charge density order due to the Peierls instability \cite{Peierls}.
Another example comes from the Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange interaction between  magnetic impurities which is mediated by itinerant fermions \cite{Ruderman}.
Magnetic helical order can be established if the spin susceptibility of the itinerant fermions has a divergent Kohn anomaly \cite{Braunecker,Scheller}.

Next, we  derive the optical conductivity
which takes the following form
in the Matsubara representation \cite{Parcollet,Georges}:
\begin{eqnarray}
&& \sigma_{\alpha \beta} (i \omega, q) \nonumber \\
&& = \frac{1}{\omega} \sum\limits_{a, \bm p} T \sum\limits_{\varepsilon} v_a^\alpha v_a^\beta G_a (i \varepsilon, \bm p) G_a (i (\varepsilon + \omega), \bm p + \bm q) , \label{sig}
\end{eqnarray}
where we sum over the fermionic Matsubara frequencies $\varepsilon$
and $v_a^\alpha$ is the $\alpha$ component of the
current vertex.
In Appendix A we argue that
the current vertex renormalization
does not affect the scaling in the conformal limit, i.e. we
can use Eq.~(\ref{sig})
with current vertices represented by the corresponding Fermi velocity $\bm v_a$.
Due to the separation of variables in the Green function,
the $\omega$-dependent part of the Matsubara optical conductivity
is simply given by $\chi_M (i\omega) / \omega$,
for $\chi_M(i \omega)$ see Eq.~(\ref{chim}),
while the $q$-dependent part is given by a power law
because $G_a (i \omega, \delta p_a) \propto 1 / |\delta p_a|^{1 - 2l}$, see Eq.~(\ref{matw}).
Analytically continuing to  real frequencies,
we find the retarded optical conductivity:
\begin{eqnarray}
&& \sigma_{\alpha \beta} (\omega, q) = \delta_{\alpha \beta} \sigma(\omega, q) , \,\,\,\, \sigma(\omega, q) \propto  \frac{\chi_R(\omega)}{i \omega}  \frac{1}{q^{1 - 4 l}} , \label{conduct}
\end{eqnarray}
where $\delta_{\alpha\beta}$ appeared
after the angular average over $v_a^\alpha v_a^\beta$,
$\omega$ is a real frequency,
and $l$ is the spatial scaling dimension, see Eq.~(\ref{scalingapp}).
We see from Eq.~(\ref{conduct}) that the dissipative real part of the optical
conductivity is proportional
to the imaginary part of the retarded
susceptibility:
\begin{eqnarray}
&& {\rm Re} (\sigma (\omega, q)) \propto \frac{{\rm Im} \left(\chi_R(\omega)\right)}{\omega} \frac{1}{q^{1 - 4 l}} .
\end{eqnarray}

In the limit $q \to 0$ the optical conductivity diverges due to the
translational invariance in our system.
The translational symmetry can be broken by
either spatially varying electric field that results in a finite $q$
or by the finite size of the system (with open boundary conditions) which
results in the infrared cut-off at $q \sim 1/L$,
$L$ is the system size.
In this work we do not consider the effects of
momentum non-conserving interactions like
disorder or Umklapp
scattering, i.e. we assume $L \ll \ell$,
where $\ell$ is the mean free path.
In such a clean limit $L \ll \ell$ the dc resistivity $\rho_{dc}(T, L)$ exhibits anomalous scaling with temperature and system size:
\begin{eqnarray}
&& \rho_{dc}(T, L) = \frac{1}{{\rm Re} (\sigma (\omega = 0, q = 1/L))} \propto \frac{T^{2 - 4 h}}{ L^{1 - 4 l}}. \label{rhodc}
\end{eqnarray}
As $1 - 4 l > 0$, the resistivity $\rho_{dc}(T, L)$ tends to zero for $L \to \infty$,
as it should in absence of  momentum non-conserving interactions.
Thus, measuring the temperature and sample size dependence for $L \ll \ell$ of the static resistivity
gives both the temporal $h$ and spatial $l$
conformal dimensions.

The case of $N = 2$ is particularly interesting.
In this case, the dc resistivity is linear in temperature $\rho_{dc} \propto T$,
see Eq.~(\ref{rhodc}).
This is the characteristic feature of strange metals which has been observed experimentally in cuprates \cite{Daou,Keimer} and heavy fermion metals \cite{Gegenwart}.
Current theories of strange metals \cite{Parcollet,Davison,Chowdhury} are based on the SYK model \cite{Ye} that requires long-range interaction and random distribution of the interaction matrix elements.
In our model the effective interaction is short range, see Eq.~(\ref{VN}).
Moreover, there is no randomness involved in the problem.
Therefore, the new physical mechanism of the quantum criticality based on the resonant many-body exchange scattering that we propose in our study might play an important role in understanding the nature of strange metals.\\

\section{Conclusions}

In our study we theoretically discovered a novel physical state of strongly interacting fermions which can be realized in materials with multiple Fermi surfaces that are subject to a special resonant condition, FSR, given by Eq.~(\ref{reducedconserv}).
This phase can be experimentally identified by the spectral function showing no Landau quasiparticles close to the Fermi surface, by the anomalous power-law temperature and size dependence of the dc resistivity,
and by the divergent susceptibilities in the static limit and at the Kohn anomalies.
We believe that the new exotic phase that we predict in this work can be found, for instance, in semiconductor heterostructures because of the large interaction parameter $r_s \gg 1$.
Moreover, the high quality and tunability of  semiconductor devices makes it possible to tune the system to the FSR, see Eq.~(\ref{reducedconserv}), which is required for establishing  the new quantum critical state. \\

\section{ACKNOWLEDGEMENTS}
We thank Oleg P. Sushkov
for inspiring and stimulating discussions.
We acknowledge support by the Georg H. Endress foundation,
the Swiss National Science Foundation, and NCCR QSIT. This project received funding from the European Union's Horizon 2020 research
and innovation program (ERC Starting Grant, grant
agreement No 757725).