# Large $N$ theory of critical Fermi surfaces

Ilya Esterlis  Department of Physics, Harvard University, Cambridge MA 02138,
USA  Haoyu Guo  Department of Physics, Harvard University, Cambridge MA 02138,
USA  Aavishkar A. Patel  Department of Physics, University of California
Berkeley, Berkeley CA 94720, USA  Subir Sachdev  Department of Physics,
Harvard University, Cambridge MA 02138, USA

###### Abstract

We describe the large $N$ saddle point, and the structure of fluctuations
about the saddle point, of a theory containing a sharp, critical Fermi surface
in two spatial dimensions. The theory describes the onset of Ising order in a
Fermi liquid, and closely related theories apply to other cases with critical
Fermi surfaces. We employ random couplings in flavor space between the
fermions and the bosonic order parameter, but there is no spatial randomness:
consequently, the $G$-$\Sigma$ path integral of the theory is expressed in
terms of fields bilocal in spacetime. The critical exponents of the large $N$
saddle-point are the same as in the well-studied non-random RPA theory; in
particular, the entropy density vanishes in the limit of zero temperature. We
present a full numerical solution of the large $N$ saddle-point equations, and
the results agree with the critical behavior obtained analytically. Following
analyses of Sachdev-Ye-Kitaev models, we describe scaling operators which
descend from fermion bilinears around the Fermi surface. This leads to a
systematic consideration of the role of time reparameterization symmetry, and
the scaling of the Cooper pairing and $2k_{F}$ operators which can determine
associated instabilities of the critical Fermi surface. We find no violations
of scaling from time reparameterizations. We also consider the same model but
with spatially random couplings: this provides a systematic large $N$ theory
of a marginal Fermi liquid with Planckian transport.

††preprint: arXiv:2103.08615

###### Contents

  1. I Introduction
  2. II Lattice model and effective action
    1. II.1 Issues with the boson thermal mass
    2. II.2 Thermodynamics
  3. III Large $N$ critical theory
  4. IV Numerical solution of the lattice model
    1. IV.1 Lattice parameters
    2. IV.2 Results
      1. IV.2.1 Phase diagram
      2. IV.2.2 Behavior at the QCP and comparison with the patch theory
  5. V Single patch theory and time reparameterizations
    1. V.1 Time Reparameterization
    2. V.2 Sliding symmetry
    3. V.3 Feynman Diagrams
    4. V.4 Eigenvalues of the Kernel
    5. V.5 Using Sliding Symmetry
    6. V.6 Further simplification
    7. V.7 Unit eigenvalue of $K_{G}$
      1. V.7.1 $F_{I}$ even sector
      2. V.7.2 $F_{I}$ odd sector
  6. VI Diagrammatics of $G$-$\Sigma$ theory
    1. VI.1 Propagator
    2. VI.2 Vertices
    3. VI.3 $1/N$ correction to self energy
  7. VII Antipodal patch theory and fermion bilinear operators
    1. VII.1 Scaling analysis
      1. VII.1.1 $2k_{F}$ operator
      2. VII.1.2 Cooper pair operator
    2. VII.2 Pairing operator
    3. VII.3 $2k_{F}$ operator
  8. VIII Spatially Disordered Model
    1. VIII.1 Thermal Mass
    2. VIII.2 Fermion Self Energy
    3. VIII.3 Free Energy
    4. VIII.4 Conductivity
    5. VIII.5 Instabilities
  9. IX Conclusions
  10. A Amperean pairing
  11. B Large $N$ theory with spatially random couplings
    1. B.1 Thermal Mass
    2. B.2 Free Energy
      1. B.2.1 Fermion determinant
      2. B.2.2 Boson Determinant

##  I Introduction

The problem of the critical Fermi surface without quasiparticle excitations
has received extensive interest in recent years Lee (2018), given its central
role in the theory of the half-filled Landau level, quantum critical points
(QCPs) in metals, and for Fermi surfaces of fractionalized particles coupled
to emergent gauge fields in gapless spin liquids Lee (1989); Polchinski
(1994); Halperin  _et al._ (1993); Kim  _et al._ (1994); Altshuler  _et al._
(1994); Nayak and Wilczek (1994); Lee (2009); Metlitski and Sachdev (2010);
Mross  _et al._ (2010); Sur and Lee (2014); Metlitski  _et al._ (2015);
Hartnoll  _et al._ (2014); Eberlein  _et al._ (2017); Holder and Metzner
(2015a, b); Fitzpatrick  _et al._ (2014); Aguilera Damia  _et al._ (2019);
Damia  _et al._ (2020a, b); Ridgway and Hooley (2015); Patel  _et al._
(2018a); Chowdhury  _et al._ (2018); Moon and Chubukov (2010); Abanov and
Chubukov (2020); Wu  _et al._ (2020); Chubukov and Abanov (2020); Wang and
Berg (2019); Klein  _et al._ (2020); Grossman  _et al._ (2020); Chowdhury and
Berg (2020); Raether  _et al._ (2020); Patel and Sachdev (2018); Esterlis and
Schmalian (2019); Hauck  _et al._ (2020); Wang and Chubukov (2020); Aldape
_et al._ (2020); Patel and Sachdev (2017, 2019); Oganesyan  _et al._ (2001).
Much has been understood about the structure of the theory, but a fully
systematic analysis of the critical theory, its operator spectrum, and
possible low temperature instabilities to symmetry-broken or topological
states has remained elusive. Initially, it was assumed that a theory with a
large number of flavors, $N$, of fermions could provide such a systematic
theory. But, in an influential analysis, Sung-Sik Lee Lee (2009) showed that
certain higher loop corrections implied a break down of the large $N$
expansion, and that the theory remains strongly coupled even at large $N$.
Various workarounds have been proposed since then, but none are fully
satisfactory: they either involve deploying additional expansion parameters,
introduce non-analytic terms not present in the original theory, or depend
upon choosing $N$-dependent energy scales.

In this paper, we wish to apply insights gained by the study of another class
of realizations of compressible quantum matter without quasiparticle
excitations: the Sachdev-Ye-Kitaev (SYK) class of models Sachdev and Ye
(1993); Kitaev (2015); Sachdev (2015); Maldacena and Stanford (2016). These
model involve random couplings, but after an average over randomness, the
resulting theory realizes a compressible non-Fermi liquid which is amenable to
a systematic large $N$ expansion via a path integral over bilocal fields Read
_et al._ (1995); Georges  _et al._ (2001); Maldacena and Stanford (2016) often
called the $G$-$\Sigma$ theory (the low energy limit of the $G$-$\Sigma$
theory is a theory of two-dimensional quantum gravity Sachdev (2010); Kitaev
(2015); Maldacena and Stanford (2016); Maldacena  _et al._ (2016); Kitaev and
Suh (2018)). Furthermore, the model is strongly self-averaging in the non-
Fermi liquid phase i.e. for sufficiently large $N$, the properties of a single
sample are indistinguishable from those of the average, and so it is
technically far easier, and permissible, to work with the average theory.
Recent works Kim  _et al._ (2019); Klebanov  _et al._ (2020); Tikhanovskaya
_et al._ (2021a, b) have used this method to obtain operator spectra and
instabilities of non-Fermi liquids realized by SYK models. The idea of
simplification realized by an average over similar strongly-coupled theories
is also playing an important role in recent investigations in quantum gravity,
and averages over random matrices or conformal field theories yield systematic
large $N$ holographic realizations of the path integral of simple theories of
gravity Saad  _et al._ (2019); Stanford and Witten (2019); Afkhami-Jeddi  _et
al._ (2021); Maloney and Witten (2020); Pérez and Troncoso (2020); Cotler and
Jensen (2020); Engelhardt  _et al._ (2020); Datta  _et al._ (2021).

In the context of finite-dimensional systems with Fermi surfaces, Aldape et
al. Aldape  _et al._ (2020) introduced the idea of coupling the fermions to a
boson, and making the associated Yukawa-like fermion boson coupling a random
function of flavor indices, but not of space. In such a model, the sharp Fermi
surface is maintained even in the presence of randomness, and a systematic
large $N$ saddle-point with regimes of non-Fermi liquid behavior is obtained.
They used this strategy to describe non-Fermi liquid behavior across a
transition from a Fermi liquid to a fractionalized Fermi liquid (FL*). We also
note Ref. Kim  _et al._ (2021) which applies this idea to critical Dirac
fermions in 1+1 dimensions, and obtained evidence for maximal chaos in the
large $N$ limit with vanishing entropy density in the zero temperature limit.
We emphasize that in all these cases there is no spatial disorder, and the
disorder is entirely in coupling-constant space. A given sample with a
particular set of couplings will not be identical to another sample, but the
difference will vanish in the large $N$ limit. To compute the differences
between samples we have to include fluctuations of the replica off-diagonal
components of the bilocal Green’s functions Read  _et al._ (1995), but we will
not do that here. The replica off-diagonal components vanish in the $N=\infty$
saddle point theory, and it is even possible that the replica off-diagonal
fluctuations do not modify the universal critical properties at higher orders
in $1/N$, because we do not expect the critical properties to be sensitive to
the microscopic values of the couplings.

We follow these recent works Aldape  _et al._ (2020); Kim  _et al._ (2021),
and examine the non-Fermi liquid with a critical Fermi surface formed at the
QCP involving the onset of Ising order a two-dimensional Fermi liquid. We note
that the location of the critical Fermi surface in momentum space obeys an
extended Luttinger theorem Huijse and Sachdev (2011); Else  _et al._ (2020),
even in the presence of the random couplings. Our large $N$ theory of a
critical Fermi surface has the same critical behavior at $N=\infty$ as already
anticipated in the early work Lee (2018, 1989); Polchinski (1994); Halperin
_et al._ (1993). Specifically, there is no extensive entropy in the zero
temperature limit in our approach, unlike previous studies of critical Fermi
surfaces employing a large $N$ limit with random couplings Patel  _et al._
(2018a); Chowdhury  _et al._ (2018); Patel and Sachdev (2019); Chowdhury and
Berg (2020). And there is anisotropic dynamic scaling on the Fermi surface,
with the frequency $\omega\sim q_{\perp}^{3/2}$ for momenta $q_{\perp}$ normal
to the Fermi surface, and $\omega\sim q_{\parallel}^{3}$ for momenta parallel
to the Fermi surface.

We introduce our model, its averaged effective action, and the setup of the
large-$N$ expansion in Section II. This involves a path integral over a
$G$-$\Sigma$ action in which the fields are bilocal in spacetime; this should
be compared to previous studies Read  _et al._ (1995); Georges  _et al._
(2001); Maldacena and Stanford (2016) in which the fields were bilocal only in
time. The large $N$ critical theory is obtained in Section III by taking the
low energy limit on separate patches on the Fermi surface. We then present a
full numerical solution of the large $N$ equations for the complete Fermi
surface in Section IV, and find results which are in agreement with the
analytic patch analysis in Section III.

We begin our discussion of the fluctuations about the large $N$ saddle point
in Section V by a consideration of the role of time reparameterization
symmetry. The special role of this symmetry, and an associated soft mode, was
first noted for the SYK model Kitaev (2015); Maldacena and Stanford (2016);
Kitaev and Suh (2018); Bagrets  _et al._ (2016, 2017); Altland  _et al._
(2019); Kruchkov  _et al._ (2020); Kobrin  _et al._ (2021) where it leads to a
violation of scaling at times of order $N$. The symmetry is also present in
the saddle-point action for the critical Fermi surface. However, we shall find
here that there is no corresponding soft mode for the critical Fermi surface,
and no associated violation of scaling. This time reparameterization analysis
is carried out in a single patch theory, which on its own realizes a ‘chiral
non-Fermi liquid’ Sur and Lee (2014). The soft mode analysis involves
examination of the eigenmodes of a ladder operator, which determine composite
operators in the particle-hole sector. We limit our consideration to momenta
orthogonal to the Fermi surface, in which case the eigenmode equations
simplify to one-dimensional integral equations. We do not find any non-trivial
operators in this sector, apart from the conserved density operator whose
correlations were studied by Kim et al. Kim  _et al._ (1994).

One operator that could have appeared in a single patch theory is the fermion
pair operator associated with Amperean pairing Lee  _et al._ (2007); Lee
(2014). This requires consideration of physics beyond the scaling limit, and
is discussed in Appendix A.

Section VI describes the structure of the $G$-$\Sigma$ theory beyond the large
$N$ saddle point. We will obtain formal expressions for the fermion self
energy at order $1/N$ in terms of the inverse of the ladder operator of the
large $N$ theory. These considerations will be quite general, and can be
applied equally to the single patch theory of Section V, the lattice theory of
Section III, or the antipodal patch theory to be considered in Section VII,
also bears a resemblance to the corresponding analysis of the SYK model in
Ref. Kitaev and Suh (2018).

Section VII turns to the examination of the scaling structure for the non-
chiral case, with a closed Fermi surface. Then, a number of singular effects
arise from antipodal pairs of patches on the Fermi surface:

  * •

Ref. Metlitski and Sachdev (2010) showed that three-loop diagrams lead to a
small correction to the fermion anomalous dimension, $\eta_{\psi}$. Our $1/N$
expansion will contain a similar correction to the value of $\eta_{\psi}$, but
it will be a systematic contribution at order $1/N$, unlike the previous
analysis which was not controlled in their large $N$ limit.

  * •

In the particle-particle sector, fermions on antipodal patches can undergo a
Cooper-pairing instability. A renormalization group analysis of this was
presented in Ref. Metlitski  _et al._ (2015) employing an expansion Mross  _et
al._ (2010) which combined large $N$ with a bare long-range interaction
controlled with small $\epsilon$. In our large $N$ expansion, the $N=\infty$
equations for the scaling dimension of the Cooper pair operator reduce to
integral equations which appeared in the $\gamma$-model of Chubukov and
collaborators Moon and Chubukov (2010); Abanov and Chubukov (2020); Wu  _et
al._ (2020); Chubukov and Abanov (2020), and also coincide with equations
studied in the SYK model Klebanov and Tarnopolsky (2017); Klebanov  _et al._
(2018). We find either a non-trivial scaling dimension for the Cooper pair
operator, or an instability to a paired ground state with no critical Fermi
surface.

  * •

In the particle-hole sector, fermions on antipodal patches yield density
fluctuations at the $2k_{F}$ wavevector. The scaling dimension of the $2k_{F}$
operator was computed in the combined large $N$/small $\epsilon$ expansion in
Ref. Mross  _et al._ (2010). Our large $N$ theory yields integral equations in
momentum and frequency for the scaling dimension of particle-hole operators on
antipodal patches, and we solve these equations numerically. These equations
have not been studied previously.

Finally, in Section VIII we consider a model in which the Yukawa coupling is
spatially random, in addition to the randomness in the flavor space. This
model also provides a systematic large $N$ theory of a sharp Fermi surface,
but for a marginal Fermi liquid. The physical properties of this model turn
out to be quite similar to a different model considered by Aldape et al.
Aldape  _et al._ (2020), including Planckian transport discussed in Section
VIII.4.

##  II Lattice model and effective action

This section will introduce a lattice model for the onset of Ising order in a
Fermi liquid on the square lattice, and described the structure of its large
$N$ saddle point.

We consider fermions $\psi_{ik}$ with a flavor index $i=1\ldots N$ and momenta
$k$ obeying periodic boundary conditions. These fermions are coupled to soft
Ising fields $\phi_{ik}$ representing Ising-nematic order corresponding to a
breaking of $C_{4}$ rotational symmetry. In imaginary time ($\tau$), the
lattice action is

$\displaystyle S=\int
d\tau\sum_{k}\sum_{i=1}^{N}\psi^{\dagger}_{ik}(\tau)\left[\partial_{\tau}-2t(\cos
k_{x}+\cos k_{y})-\mu\right]\psi_{ik}(\tau)$ $\displaystyle+\frac{1}{2}\int
d\tau\sum_{q}\sum_{i=1}^{N}\phi_{iq}(\tau)\left[-\partial_{\tau}^{2}-2J(\cos
q_{x}+\cos q_{y}-2)+m_{b}^{2}\right]\phi_{i,-q}(\tau)$ $\displaystyle+\int
d\tau\sum_{k,q}\left(\cos k_{x}-\cos
k_{y}\right)\sum_{i,j,l=1}^{N}\left[\frac{g_{ijl}}{N}\psi^{\dagger}_{i,k+q}(\tau)\psi_{jk}(\tau)\phi_{lq}(\tau)\right]\,.$
(1)

Here $t$ is the fermion hopping, $\mu$ is chemical potential, the coupling $J$
determines the dispersion of the boson, and $m_{b}$ is the bare boson mass. We
will henceforth set the nematic form factor $\cos k_{x}-\cos k_{y}$ to $1$ for
simplicity, since it does not qualitatively affect any of the physics we will
be interested in apart from at a measure zero set of special points in
momentum space with $k_{x}=\pm k_{y}$. In the absence of this form factor, the
$\phi_{ik}$ field no longer has an interpretation as a fluctuating order
parameter, as condensing $\phi_{ik}$ does not break any symmetries. However,
one may still think of $\phi_{ik}$ as a phonon field, in which case a $q=0$
instability corresponds to phase separation.

The novelty in our approach is that the Yukawa couplings $g_{ijl}$ are
independent, translationally-invariant complex Gaussian random variables with
zero mean and variance $g^{2}$, and $g_{jil}=g^{\ast}_{ijl}$. We will comment
where needed on the differences that appear upon taking real $g_{ijl}$. Upon
performing an SYK-like disorder average at large $N$, we obtain

$\displaystyle S=\int
d\tau\sum_{k}\sum_{i=1}^{N}\psi^{\dagger}_{ik}(\tau)\left[\partial_{\tau}-2t(\cos
k_{x}+\cos k_{y})-\mu\right]\psi_{ik}(\tau)$ $\displaystyle+\frac{1}{2}\int
d\tau\sum_{q}\sum_{i=1}^{N}\phi_{iq}(\tau)\left[-\partial_{\tau}^{2}-2J(\cos
q_{x}+\cos q_{y}-2)+m_{b}^{2}\right]\phi_{i,-q}(\tau)$
$\displaystyle+N\frac{g^{2}}{2}\int d\tau
d\tau^{\prime}\sum_{k,q}G(k,\tau-\tau^{\prime})G(k+q,\tau^{\prime}-\tau)D(\tau-\tau^{\prime},q)$
$\displaystyle-N\int d\tau
d\tau^{\prime}\sum_{k}\Sigma(k,\tau^{\prime}-\tau)\left[G(k,\tau-\tau^{\prime})+\frac{1}{N}\sum_{i=1}^{N}\psi_{ik}(\tau)\psi^{\dagger}_{ik}(\tau^{\prime})\right]$
$\displaystyle+\frac{N}{2}\int d\tau
d\tau^{\prime}\sum_{q}\Pi(q,\tau^{\prime}-\tau)\left[D(q,\tau-\tau^{\prime})-\frac{1}{N}\sum_{i=1}^{N}\phi_{iq}(\tau)\phi_{i,-q}(\tau^{\prime})\right].$
(2)

Here, we have introduced fermion (boson) Green’s functions $G$ ($D$) and self
energies $\Sigma$ ($\Pi$) as dynamical degrees of freedom by employing them as
Lagrange multipliers. We have already assumed the saddle point structure in
which the $G$, $\Sigma$, $D$, $\Pi$ fields are functions only of differences
in spacetime positions; but in the full path integral, these fields are
bilocal in spacetime. The action is quadratic in fermions and bosons, and
integrating them out yields the $G$-$\Sigma$-$D$-$\Pi$ action of the theory
with a prefactor of $N$.

The saddle point equations $\delta S/\delta\Sigma=\delta S/\delta G=0=\delta
S/\delta D=\delta S/\delta\Pi$, yield the familiar RPA-Dyson equations, exact
at large $N$:

$\displaystyle\Sigma(r,\tau)=g^{2}D(r,\tau)G(r,\tau),~{}~{}\Pi(r,\tau)=-g^{2}G(-r,-\tau)G(r,\tau),$
$\displaystyle G(k,i\omega_{n})=\frac{1}{i\omega_{n}+2t(\cos k_{x}+\cos
k_{y})+\mu-\Sigma(k,i\omega_{n})},$ $\displaystyle
D(q,i\Omega_{m})=\frac{1}{\Omega_{m}^{2}-2J(\cos q_{x}+\cos
q_{y}-2)+m_{b}^{2}-\Pi(q,i\Omega_{m})}.$ (3)

These equations may be solved numerically by using Fourier transforms in space
and time, and iterative updates, as will be described in Section IV. Care
needs to be taken that $D^{-1}(q,i\Omega_{m})>0$ throughout the iterative
procedure. At criticality, we must first determine $m_{b}^{2}$ by requiring
that $m_{b}^{2}-\Pi(0,0)=0$ at infinite system size and zero termperature, and
then use this value of $m_{b}^{2}$ in the finite-size and finite-temperature
problem. The number of points in the numerical analysis can be reduced
somewhat by exploiting the KMS conditions $G(\tau+\beta)=-G(\tau)$,
$D(\tau+\beta)=D(\tau)=D(-\tau)$, and the spatial $C_{4}$ symmetry
$G(x,y)=G(\pm x,\pm y)$, $D(x,y)=D(\pm x,\pm y)$.

###  II.1 Issues with the boson thermal mass

In this RPA theory, criticality is achieved when
$m_{b}^{2}-\Pi(0,0)|_{T=0}^{L=\infty}=0$, where $L$ is the system length. At
$T\neq 0$ and/or finite $L$, there is a ‘thermal mass’ for the boson given by
$M^{2}(T,L)=m_{b}^{2}-\Pi(0,0)=\Pi(0,0)|_{T=0}^{L=\infty}-\Pi(0,0)$. This
follows the finite temperature/length correction to the free fermion
compressibility, which is very small, and, for the nearest-neighbor square
lattice considered here, is negative at low $T$, which causes a first-order
transition at small $T\neq 0$ at generic fermion fillings. This is
undesirable.

To remedy this issue, we include a fixed length constraint
$\sum_{q}\sum_{i=1}^{N}\phi_{iq}(\tau)\phi_{i,-q}(\tau)=N/\gamma$. This can
arise from the $U\rightarrow\infty$ limit on the quartic boson self-
interaction
$\sum_{r}U/(2N)(\sum_{i=1}^{N}\phi_{i,r}(\tau)\phi_{i,r}(\tau)-N/\gamma)^{2}$,
which in turn is generated by integrating out fermions in the full, non-RPA
theory. Performing a Hubbard-Stratonovich transformation and sending
$U\rightarrow\infty$ gives the action with a Lagrange multiplier
$\lambda_{r}(\tau)$,

$S_{1}=S+\frac{1}{2}\int
d\tau\sum_{r}i\lambda_{r}(\tau)\left(\sum_{i=1}^{N}\phi_{ir}(\tau)\phi_{ir}(\tau)-\frac{N}{\gamma}\right).$
(4)

At the large $N$ saddle point, $i\lambda_{r}(\tau)=m_{b}^{2}$, and we get the
model described in Sec. II, with an additional constraining equation in the
set of Dyson equations: $D(r=0,\tau=0)=1/\gamma$. We can now control the phase
diagram by tuning $\gamma$; $m_{b}^{2}$ is adjusted along with $L$ and $T$ to
keep $\gamma$ fixed. At $\gamma=\gamma_{c}$,
$m_{b}^{2}|_{T=0}^{L=\infty}-\Pi(0,0)|_{T=0}^{L=\infty}=0$. We therefore tune
$\gamma$ so that $M^{2}(T,L)$ vanishes at the lowest numerically accessible
$T$ and largest $L$. Our numerical solution, as well as analysis with the
simpilfied continuum RPA boson propagator
$D(q,i\Omega_{m})=1/(\Omega_{m}^{2}+q^{2}+|\Omega_{m}|/|q|+M^{2}(T))$, find
$M^{2}(T,\infty)\sim T\ln(1/T)$ at low $T$ at criticality, which is
parametrically smaller than the expected RPA quantum critical scaling
$M^{2}(T,\infty)\sim T^{2/3}$ Millis (1993); Hartnoll  _et al._ (2014). This
anomalous thermal mass also leads to an anomalous contribution to the fermion
self energy, from the thermal ($\Omega_{m}=0$) fluctuations of the bosons,
that doesn’t obey the expected quantum critical scaling; however, this was
shown to be in qualitative agreement with recent analysis of quantum Monte
Carlo simulation data Xu  _et al._ (2020) from the full, non-RPA theory, which
indicates that the RPA model is a reasonable description of the actual
physical problem, at least above some temperature scale.

###  II.2 Thermodynamics

The RPA grand canonical free energy, exact at the large $N$ saddle point, is
given by ($\mathcal{Z}=e^{-S_{1}}$)

$\displaystyle\mathcal{F}/N=-T\sum_{k,\omega_{n}}\ln\left[\frac{i\omega_{n}+2t(\cos
k_{x}+\cos k_{y})+\mu-\Sigma(k,i\omega_{n})}{i\omega_{n}+2t(\cos k_{x}+\cos
k_{y})+\mu}\right]-T\sum_{k}\ln\left[1+e^{(2t(\cos k_{x}+\cos
k_{y})+\mu)/T}\right]$
$\displaystyle+\frac{T}{2}\sum_{q,\Omega_{m}}\ln\left[\frac{\Omega_{m}^{2}-2J(\cos
q_{x}+\cos q_{y}-2)+m_{b}^{2}-\Pi(q,i\Omega_{m})}{\Omega_{m}^{2}-2J(\cos
q_{x}+\cos
q_{y}-2)+m_{b}^{2}}\right]+T\sum_{q}\ln\left[1-e^{-(m_{b}^{2}-2J(\cos
q_{x}+\cos q_{y}-2))^{1/2}/T}\right]$
$\displaystyle-T\sum_{k,\omega_{n}}\Sigma(k,i\omega_{n})G(k,i\omega_{n})-L^{2}\frac{m_{b}^{2}}{2\gamma}+I_{0}.$
(5)

Here expressions for free fermion and free boson free energies have been added
and subtracted to ensure numerical convergence, and $L$ is the system size.
Note that, for the boson contribution to the free energy, we are subtracting
$I_{T}=(T/2)\sum_{q,\Omega_{m}}\ln[\Omega_{m}^{2}-2J(\cos q_{x}+\cos
q_{y}-2)+m_{b}^{2}]$, which is unregulated, but only adding the (regular)
difference $I_{T}-I_{0}=T\sum_{q}\ln[1-e^{-(m_{b}^{2}-2J(\cos q_{x}+\cos
q_{y}-2))^{1/2}/T}]$, where
$I_{0}=(1/2)\int_{\Omega_{m}}\sum_{q}\ln[\Omega_{m}^{2}-2J(\cos q_{x}+\cos
q_{y}-2)+m_{b}^{2}]$. This difference represents the free energy of free boson
excitations, and vanishes as $T\rightarrow 0$. We therefore also add back the
formally infinite constant $I_{0}$ on the third line of (5), which is
physically just the ground state energy of the collection of dispersive free
boson harmonic oscillators with mass $m_{b}^{2}$:

$I_{0}=\frac{1}{2}\sum_{q}\sqrt{m_{b}^{2}-2J(\cos q_{x}+\cos q_{y}-2)}.$ (6)

The entropy is then given by
$\mathcal{S}/N=-(1/N)(\partial\mathcal{F}/\partial T)|_{\mu,\gamma}$, where
the derivative is taken numerically at fixed inverse length $\gamma$ and
chemical potential $\mu$.

##  III Large $N$ critical theory

In this section, we analyze the low-energy version of the quantum critical RPA
model around a single patch of the Fermi surface. We take $L=\infty$ to begin
with, and will be concerned only about quantum fluctuations here. We hence do
not bother about using the fixed length constraint to determine the boson
thermal mass, and we just discard the purely thermally fluctuating boson modes
of the theory by hand.

The critical singularties of the lattice $N=\infty$ theory of Section II are
described by a continuum theory which focuses on a single patch of the Fermi
surface around a chosen wavevector $k_{0}$ on the Fermi surface Metlitski and
Sachdev (2010); Polchinski (1994) – see Fig 1.

Figure 1: Patch theory for fermions in the vicinity of the points $\bm{k}_{0}$
on the Fermi surface.

We choose a point along the $x$ axis, and then the fermion dispersion near
this point is $k_{x}+k_{y}^{2}$, where we have scaled the $k_{x}$ and $k_{y}$
axes so that the co-efficients are unity. In this manner, we obtain the action
for single patch theory is ($\int_{k}\equiv\int dk_{x}dk_{y}/(2\pi)^{2}$)

$\displaystyle S=\int
d\tau\int_{k}\sum_{i=1}^{N}\psi^{\dagger}_{ik}(\tau)\left[\partial_{\tau}+k_{x}+k_{y}^{2}\right]\psi_{ik}(\tau)+\frac{1}{2}\int
d\tau\int_{q}\sum_{i=1}^{N}\phi_{iq}(\tau)\left[q_{y}^{2}\right]\phi_{i,-q}(\tau)$
$\displaystyle+N\frac{g^{2}}{2}\int d\tau
d\tau^{\prime}\int_{k,q}G(k,\tau-\tau^{\prime})G(k+q,\tau^{\prime}-\tau)D(\tau-\tau^{\prime},q)$
$\displaystyle-N\int d\tau
d\tau^{\prime}\int_{k}\Sigma(k,\tau^{\prime}-\tau)\left[G(k,\tau-\tau^{\prime})-\frac{1}{N}\sum_{i=1}^{N}\psi_{ik}(\tau)\psi^{\dagger}_{ik}(\tau^{\prime})\right]$
$\displaystyle+\frac{N}{2}\int d\tau
d\tau^{\prime}\int_{q}\Pi(q,\tau^{\prime}-\tau)\left[D(q,\tau-\tau^{\prime})-\frac{1}{N}\sum_{i=1}^{N}\phi_{iq}(\tau)\phi_{i,-q}(\tau^{\prime})\right].$
(7)

However, the $\phi_{iq}$ fields are now defined to have
$\phi_{iq}(i\Omega_{m}=0)=0$ as the thermal fluctuations are excluded.

The saddle point equations are

$\displaystyle\Sigma(r,\tau)=g^{2}D(r,\tau)G(r,\tau),~{}~{}\Pi(r,\tau)=-g^{2}G(-r,-\tau)G(r,\tau),$
$\displaystyle
G(k,i\omega_{n})=\frac{1}{i\omega_{n}-k_{x}-k_{y}^{2}-\Sigma(k,i\omega_{n})},$
$\displaystyle D(q,i\Omega_{m}\neq 0)=\frac{1}{q_{y}^{2}-\Pi(q,i\Omega_{m})}.$
(8)

We can solve the saddle point equations analytically at $T\neq 0$. First we
note that
$\mathrm{sgn}(\omega_{n}-\mathrm{Im}[\Sigma(k,i\omega_{n})])=\mathrm{sgn}(\omega_{n})$.
We then assume $\Sigma(k,i\omega_{n})=\Sigma(i\omega_{n})$ is momentum-
independent. Then

$\displaystyle\Pi(q,i\Omega_{m})=-g^{2}T\sum_{\omega_{n}}\int_{k}\frac{1}{i\omega_{n}-k_{x}-k_{y}^{2}-\Sigma(i\omega_{n})}\frac{1}{i\omega_{n}+i\Omega_{m}-k_{x}-q_{x}-(k_{y}+q_{y})^{2}-\Sigma(i\omega_{n}+i\Omega_{m})}$
$\displaystyle=-ig^{2}\frac{T}{2}\sum_{\omega_{n}}\int_{k_{y}}\frac{\mathrm{sgn}(\omega_{n}+\Omega_{m})-\mathrm{sgn}(\omega_{n})}{i\Omega_{m}-q_{x}-2k_{y}q_{y}-q_{y}^{2}+\Sigma(i\omega_{n})-\Sigma(i\omega_{n}+i\Omega_{m})}$
$\displaystyle=g^{2}\frac{T}{8|q_{y}|}\sum_{\omega_{n}}\mathrm{sgn}(\omega_{n})\left(\mathrm{sgn}(\omega_{n}+\Omega_{m})-\mathrm{sgn}(\omega_{n})\right)=-\frac{g^{2}}{8\pi}\frac{|\Omega_{m}|}{|q_{y}|}.$
(9)

Further,

$\displaystyle\Sigma(k,i\omega_{n})=g^{2}T\sum_{\Omega_{m}\neq
0}\int_{q}\frac{1}{q_{y}^{2}+\frac{g^{2}}{8\pi}\frac{|\Omega_{m}|}{|q_{y}|}}\frac{1}{i\omega_{n}+i\Omega_{m}-k_{x}-q_{x}-(k_{y}+q_{y})^{2}-\Sigma(i\omega_{n}+i\Omega_{m})}$
$\displaystyle=-ig^{2}\frac{T}{2}\sum_{\Omega_{m}\neq
0}\int_{q_{y}}\frac{\mathrm{sgn}(\omega_{n}+\Omega_{m})}{q_{y}^{2}+\frac{g^{2}}{8\pi}\frac{|\Omega_{m}|}{|q_{y}|}}$
$\displaystyle=-2ig^{4/3}\pi^{1/3}\frac{T}{3\sqrt{3}}\sum_{\Omega_{m}\neq
0}\frac{\mathrm{sgn}(\omega_{n}+\Omega_{m})}{|\Omega_{m}|^{1/3}}=-i\mathrm{sgn}(\omega_{n})2^{5/3}g^{4/3}\frac{T^{2/3}}{3\sqrt{3}}H_{1/3}\left(\frac{|\omega_{n}|-\pi
T}{2\pi T}\right).$
$\displaystyle\Sigma(k,i\omega_{n},T=0)=-i\mathrm{sgn}(\omega_{n})\frac{g^{4/3}}{\pi^{2/3}\sqrt{3}}|\omega_{n}|^{2/3}.$
(10)

The function $H_{1/3}(\mathrm{x})$ is `Mathematica`’s `HarmonicNumber[x,1/3]`
and is related to generalized Reimann zeta functions Patel and Sachdev (2017).
We can therefore see that the assumptions we made about
$\Sigma(k,i\omega_{n})$ are self-consistent. The fermion self-energy vanishes
at the first Matsubara frequencies $\omega_{n}=\pm\pi T$.

##  IV Numerical solution of the lattice model

In this section we describe the numerical solution of the saddle point
equations (3) for the lattice model, along with the fixed-length constraint

$D(r=0,\tau=0)=1/\gamma.$ (11)

We solve these equations on a square lattice with periodic boundary conditions
and consider systems of linear dimensions $L=32-256$. As mentioned previously,
the equations are solved efficiently by employing fast Fourier transforms in
both space and imaginary time. Furthermore, we find convergence of the
iterative procedure is significantly enhanced by solving the equations
progressively form high to low temperature, using the solution from the
previous higher temperature as a seed for a given temperature. This procedure
is typically started at a relatively high temperature, $T\sim t$.

###  IV.1 Lattice parameters

From the bare fermion and boson dispersion in (3) we identify the following
energy scales:

$\\{t,~{}\sqrt{J},~{}g^{2}/J\\}.$ (12)

Other relevant electronic energy scales are the bandwidth $W=8t$ and the
density of states (DOS) at the Fermi energy $N_{0}\sim 1/W$. We define a
dimensionless coupling constant as

$\lambda_{0}=\frac{g^{2}}{JW}.$ (13)

The other important dimensionless parameter is $\sqrt{J}/t\sim c/v_{F}$, where
$c$ and $v_{F}$ are the boson and fermion velocities, respectively. In terms
of lattice parameters, $v_{F}\sim ta$ and $c\sim\sqrt{J}a$, where $a$ is the
lattice constant. Here we focus on the regime $c\sim v_{F}$. For all the data
presented below we fix the following parameters: $\lambda_{0}=0.125$,
$\sqrt{J}/t=2$, and $\mu=-0.5t$. For reference, this chemical potential
corresponds to a Fermi energy $\epsilon_{F}=3.5t$ and DOS $N_{0}\simeq
0.19/t$. As explained in Sec. II.1, the boson mass, $m_{b}$, is determined
self-consistently for a fixed value of $\gamma$.

###  IV.2 Results

####  IV.2.1 Phase diagram

Figure 2: Left: Phase diagram as a function of $\gamma$ in units of
$\gamma_{c}\simeq 0.15$ and temperature $T$. The color scale shows the
exponent, $x$, with which the boson mass approaches its $T=0$ value:
$M^{2}(T)-M^{2}(T=0)\sim T^{x}$. Up to logarithmic corrections, we expect
$x=1$ at the QCP and $x=2$ in the Fermi liquid regime. In the region
$T<T^{\star}$, $M^{2}(T)\sim e^{-T^{\star}/T}$ and the behavior of the system
is governed by the soft thermal fluctuations of the boson. For
$\gamma<\gamma_{c}$, the system is ordered only at $T=0$. Right: Boson mass as
a function of $T$ for $\gamma>\gamma_{c}$ (red), $\gamma=\gamma_{c}$ (black),
and $\gamma<\gamma_{c}$ (blue). The inset shows the behavior of the boson mass
at the $g=0$ QCP, where $M^{2}\sim T^{2}$. Figure 3: (a) Imaginary part of the
fermion self-energy averaged over the Fermi surface, as a function of
Mastubara frequency, $\omega_{n}$, on the Fermi liquid side ($\gamma\simeq
1.25\gamma_{c}$) for $\beta t=40-100$. Red shows the full self-energy,
$-\langle\mathrm{Im}\Sigma\rangle_{\mathrm{FS}}$, while blue and green show
the quantum, $-\langle\mathrm{Im}\Sigma_{Q}\rangle_{\mathrm{FS}}$, and
thermal, $-\langle\mathrm{Im}\Sigma_{T}\rangle_{\mathrm{FS}}$, contributions,
respectively. Different symbols correspond to different temperatures. For the
full $\Sigma$, the points for various temperatures fall essentially on the
same curve, indicating $\Sigma$ has converged to its $T=0$ value. The
components $\Sigma_{Q}$ and $\Sigma_{T}$ show a more significant temperature
dependence. We find $-\mathrm{Im}\Sigma\sim\omega_{n}$ at small frequency,
consistent with a Fermi liquid. In this regime, the contribution from
$\Sigma_{T}$ is small.
(b) Same as (a) but on the ordered side ($\gamma\simeq 0.96\gamma_{c}$) for
$\beta t=8-14$. In this regime $-\mathrm{Im}\Sigma$ is an increasing function
of $\omega_{n}$ at small frequency, where it is dominated by $\Sigma_{T}$.

To access the QCP, we first map out the phase diagram of the model as a
function of $\gamma$ and temperature $T$. The results are shown in the left
panel of Fig. 2 and are based primarily on the behavior of the renormalized
boson mass,

$M^{2}(T)=m_{b}^{2}-\Pi(0,0),$ (14)

shown in Fig. 2 (right). We find a QCP at the value $\gamma_{c}\simeq 6.7t$,
where we observe the boson mass vanishes approximately linearly, $M^{2}\sim
T$. Up to logarithmic corrections, which are hard to detect numerically, this
is consistent with earlier analytic calculations Millis (1993); Hartnoll  _et
al._ (2014). This scaling holds in the quantum-critical fan above the QCP, as
is seen from the color scale in Fig. 2. For $\gamma>\gamma_{c}$, the system is
a Fermi liquid at low temperatures, with the boson mass behaving as
$M^{2}-M^{2}(T=0)\sim T^{2}$. For $\gamma<\gamma_{c}$, the system is ordered
only for $T=0$, with the boson mass vanishing according to
$M^{2}\sim\exp(-T^{\star}/T)$ (the $T^{\star}$ line in the figure is obtained
by fitting the $M^{2}(T)$ to this functional form). The absence of a finite
temperature ordering transition is a result of the Hohenberg-Mermin-Wagner
theorem Hohenberg (1967); Mermin and Wagner (1966), as such a transition
corresponds to spontaneously breaking the $O(N)$ symmetry of the original
(disorder averaged) model. The exponentially vanishing mass as $T\to 0$ is
known from the behavior of the $O(N)$ model at large-$N$ in two dimensions. We
remark that the boson mass behaves in the same way even in the absence of the
fixed length constraint, (11). Below the temperature scale $T^{\star}$, the
system crosses over into a regime governed by soft thermal fluctuations of the
boson, and, as will be further discussed below, the fermion self-energy has a
form distinct from that of a Fermi liquid. For comparison, we also show the
behavior of the boson mass at the $g=0$ QCP of the decoupled model in the
inset of the right panel of Fig. 2, in which case $M^{2}\sim T^{2}$ (even for
$g=0$, the bosonic sector is self-interacting due to the fixed length
constraint, (11)).

To better characterize the single-fermion properties of the system across the
phase diagram, we decompose the fermion self-energy as

$\Sigma(k,i\omega_{n})=\Sigma_{T}(k,i\omega_{n})+\Sigma_{Q}(k,i\omega_{n}),$
(15)

where subscripts $T$ and $Q$ denote the “thermal” and “quantum” contributions,
respectively. The thermal contribution is defined as that from the zero
Matsubara frequency transfer term in the self-consistent equation for
$\Sigma$, while the quantum contribution comes from non-zero Matsubara
frequency transfer:

$\displaystyle\Sigma_{T}(k,i\omega_{n})$
$\displaystyle=\frac{g^{2}T}{L^{2}}\sum_{k^{\prime}}D(k-k^{\prime},\Omega=0)G(k^{\prime},i\omega_{n}),$
(16) $\displaystyle\Sigma_{Q}(k,i\omega_{n})$
$\displaystyle=\frac{g^{2}T}{L^{2}}\sum_{n^{\prime}\neq
n}\sum_{k^{\prime}}D(k-k^{\prime},i\omega_{n}-i\omega_{n^{\prime}})G(k^{\prime},i\omega_{n^{\prime}}).$
(17)

This decomposition has been used to analyze finite-$T$ corrections to quantum-
critical scaling Dell’Anna and Metzner (2006); Klein  _et al._ (2020); Xu  _et
al._ (2020); Damia  _et al._ (2020a, b) and here we find it particularly
useful in analyzing the behavior both at the QCP, $\gamma=\gamma_{c}$, and on
the ordered side, $\gamma<\gamma_{c}$.

We first discuss the behavior of the fermion self-energy away from the QCP. In
Fig. 3, we show the negative imaginary part of the fermion self-energy,
$\langle-\mathrm{Im}\Sigma(\omega_{n})\rangle_{\mathrm{FS}}$, where the
brackets denote averaging momentum dependence over the Fermi surface (in
general we find $\Sigma$ is weak function of the direction of $k$ along the
Fermi surface; see Fig. 5a. Fig. 3a shows the self-energy for
$\gamma>\gamma_{c}$, where we find a linear frequency dependence, consistent
with a Fermi-liquid, corresponding to fermion mass enhancement. Data are shown
for a set of temperatures in the range $\beta t=40-100$. The data points for
different temperatures fall on the same curve, indicating $\Sigma$ has
essentially converged to its $T=0$ value. In the Fermi liquid regime, the
contribution from $\Sigma_{T}$ is small. Fig. 3b shows the self-energy on the
ordered side, $\gamma<\gamma_{c}$. In this case, the low-frequency behavior is
significantly affected by $\Sigma_{T}$. On the ordered side, the exponentially
small boson mass makes it challenging to numerically access low temperatures,
and the data shown in the figure are for $\beta t=8-14$. Even here the full
self-energy is essentially temperature-independent, while the separate
contributions, $\Sigma_{Q}$ and $\Sigma_{T}$, show a stronger temperature
dependence. In contrast to the Fermi-liquid regime, on the ordered side and at
low frequency, the self-energy is essentially constant over the frequency
range $\omega_{n}\simeq(1-3)t$ and at lower frequency becomes an increasing
function of decreasing frequency. The small boson mass, satisfying $M\ll T$,
and large thermal self-energy, $\Sigma_{T}$, explain why we denote region
$T<T^{*}$ in Fig. 2 as being characterized by “strong thermal fluctuations”.
We note that the data on the ordered side of the transition are only very
slightly tuned away from the critical point, $\gamma\simeq 0.96\gamma_{c}$,
yet the behavior of the self-energy is nevertheless drastically different from
that at the QCP (to be further discussed in the next section), indicating a
rapid crossover in the behavior of the system on the ordered side. Finally, we
remark that in both regimes, $\langle-\mathrm{Im}\Sigma_{Q}(\pi
T)\rangle_{\mathrm{FS}}$ is negative; this curious fact has been explained in
Klein  _et al._ (2020).

####  IV.2.2 Behavior at the QCP and comparison with the patch theory

Figure 4: Entropy density, $S/L^{2}$, as a function of $T$ at the QCP. This is
expected to vanish as $\sim T^{2/3}$ as $T\rightarrow 0$, but our data is not
precise enough to distinguish from a linear $T$ dependence. The entropy is
obtained by numerical differentiation of the free energy density, $F/tL^{2}$,
shown in the bottom inset. The top inset is $G(k,\tau=\beta/2)$, which is
essentially the fermion spectral function averaged over an energy window of
order $T$ about the Fermi energy, for $\beta t=100$. We see the Fermi surface
remains sharply defined at the QCP, in accord with Luttinger’s theorem. Figure
5: (a) Imaginary part of fermion self-energy at the QCP, normalized by the
first Matsubara frequency $\omega_{0}=\pi T$, as a function of angle,
$\theta$, along the Fermi surface for $\beta t=100$. The variation as a
function $\theta$ is small, essentially tracking the density of states of the
non-interacting band structure. Inset shows the same quantity through the
whole first Brillouin zone, where we see it is sharply peaked at the Fermi
surface. (b) Imaginary part of fermion self-energy at the QCP, averaged over
the Fermi surface, for the range $\beta t=40-160$. For all but the lowest
frequencies, $\Sigma$ has essentially converged to its $T=0$ behavior by
$\beta t\simeq 40$. Figure 6: Decomposition of imaginary part of fermion self-
energy at the QCP, averaged over the Fermi surface, for the range of inverse
temperatures $\beta t=40-160$. (a) The quantum part, $\mathrm{Im}\Sigma_{Q}$,
shows more significant $T$-dependence than the full $\mathrm{Im}\Sigma$,
tending toward the predicted $\omega_{n}^{2/3}$ behavior as $T\to 0$. (b)
Thermal part, $\mathrm{Im}\Sigma_{T}$, displays a significant temperature
dependence, tending slowly to zero as $T\to 0$. The inset shows the thermal
part obeys the scaling behavior $-\mathrm{Im}\Sigma_{T}\sim T/\omega_{n}$
Klein  _et al._ (2020); Xu  _et al._ (2020). Figure 7: Dynamical part of boson
self-energy, $\delta\Pi$, as a function of $\Omega_{n}/ta|q|$, where $ta\sim
v_{F}$. Data are shown for representative $q$ vectors in the inverse
temperature range $\beta t=50-100$, where $\delta\Pi$ has essentially
converged to its $T=0$ limit. The small frequency behavior is linear, as
predicted from the low-energy analysis in Sec. III, albeit with a slope that
depends on the direction of $q$, as would be expected for an anisotropic Fermi
surface. Inset shows $\delta\Pi$ for $q$ vectors in the range $0<|q|a<0.8$ for
$\beta t=100$.

We now describe the behavior at the QCP, $\gamma=\gamma_{c}$. Fig. 4 shows the
entropy density, $S/L^{2}$, at the QCP, which we find vanishes as $T\to 0$.
The entropy is computed by numerical differentiation of the free energy, $F$,
which is shown in the bottom inset of Fig. 4 and computed according to (5). We
also find the Fermi surface remains sharp at the QCP, in accord with
Luttinger’s theorem. This can be seen in the top inset of Fig. 4, where we
show an imaginary-time proxy for the fermion spectral weight at zero energy:

$G(k,\tau=\beta/2)=\int_{-\infty}^{\infty}d\omega\frac{1}{2\cosh(\beta\omega/2)}A(k,\omega),$
(18)

where $A(k,\omega)$ is the fermion spectral function. This quantity is
essentially the spectral function averaged over an energy range of order $T$
about the Fermi energy and has been frequently used in numerical studies, as
it avoids the need for analytic continuation of Matsubara frequency data
Schattner  _et al._ (2016); Gerlach  _et al._ (2017); Berg  _et al._ (2019).

The behavior of the fermion self-energy at the QCP is shown in Figs. 5 and 6.
Fig. 5a shows the variation of $-\mathrm{Im}\Sigma$ along the Fermi surface.
The dependence on angle along the Fermi surface is weak, and essentially
tracks the behavior of the non-interacting DOS. The inset of the figure shows
$-\mathrm{Im}\Sigma$ across the entire Brillouin zone and we find it is peaked
on the Fermi surface. These observations are in line with the analytic
predictions of Sec. III. Fig. 5b shows the Fermi-surface average of
$-\mathrm{Im}\Sigma$. The data are shown for a set of temperatures in the
range $\beta t=40-160$. Fig. 6 shows the decomposition into $\Sigma_{Q}$ and
$\Sigma_{T}$. We find the temperature dependence of the full $\Sigma$ is weak
at all but the lowest frequencies, where the thermal contribution, shown in
Fig. 6b, is still sufficiently large to obscure the expected
$\omega_{n}^{2/3}$ scaling. The quantity $\mathrm{Im}\Sigma_{Q}$ precisely
removes this thermal contribution and, from Fig. 6a, we see that, although
$\mathrm{Im}\Sigma_{Q}$ shows a stronger temperature dependence than
$\mathrm{Im}\Sigma$, the low-frequency behavior is indeed compatible with
$\omega_{n}^{2/3}$ scaling as $T\to 0$, as in Eq. (10). In the temperature
regime shown in Fig. 6, earlier work has predicted that the the thermal self-
energy should behave as $-\mathrm{Im}\Sigma_{T}\sim T/\omega_{n}$, Klein  _et
al._ (2020); Xu  _et al._ (2020). We find this scaling is indeed well
satisfied, as seen in the inset of Fig. 6b.

Finally, Fig. 7 shows the “dynamical part” of the boson self-energy, defined
as $\delta\Pi(q,\Omega_{n})=\Pi(q,\Omega_{n})-\Pi(q,\Omega_{n}=0)$, for two
$q$ vectors and range of temperatures $\beta t=50-100$. We find the expected
linear scaling with frequency, $\delta\Pi\sim\Omega_{n}$, when
$\Omega_{n}<v_{F}|q|$ (recall $v_{F}\sim ta$). The scaling with
$\Omega_{n}/v_{F}|q|$, as in Eq. (9), is not perfectly satisfied due to the
anisotropy of the Fermi surface (in the low-energy calculations, a circular
Fermi surface is assumed). The dependence of $\delta\Pi$ on the angle of $q$
may be seen in the inset of Fig. 7, where $\delta\Pi$ is shown for a larger
set of $q$’s, in the range $0<|q|a<0.8$.

##  V Single patch theory and time reparameterizations

We now turn to a characterization of the fluctuations about the large $N$
saddle point in the spatially uniform model with a critical Fermi surface
described in Sections II and III. Here we will focus on the single patch
critical theory in (7), and will defer consideration of the special role of
antipodal patches Metlitski and Sachdev (2010) to Section VII.

In the SYK model, the flucutations are characterized by the structure of the
4-point correlators of the large $N$ saddle point Maldacena and Stanford
(2016); Kitaev and Suh (2018), and this is dominated by the time
reparameterization mode. Formally, it appears that the single patch saddle-
point of the large $N$ theory in Section III has a time reparameterization
symmetry, and so we examine it here for a corresponding soft mode. However, as
we show below, the spatial structure of the critical Fermi surface theory does
play an important role, and we find there is no special contribution from time
reparameterizations. Instead, we find that the 4-point correlators are
controlled by response functions of the conserved fermion density, which were
explored earlier in Ref. Kim  _et al._ (1994).

Integrating out the fermions from the single patch continuum action (7), leads
us to consider the following $G-\Sigma$ action

$\displaystyle\frac{S}{N}=$
$\displaystyle-\operatorname{Tr}\ln(\partial_{\tau}\delta^{(3)}-s_{f}(i\partial_{x}+\partial_{y}^{2})\delta^{(3)}+\Sigma)+\frac{1}{2}\operatorname{Tr}\ln(-s_{b}\partial_{y}^{2}\delta^{(3)}-\Pi)$
$\displaystyle~{}~{}~{}~{}+\frac{g^{2}}{2}\operatorname{Tr}(G\cdot
GD)-\operatorname{Tr}(\Sigma\cdot G)+\frac{1}{2}\operatorname{Tr}(\Pi\cdot
D).$ (19)

The $\operatorname{Tr}(\cdot)$ is defined on the indices of spacetime
$(x,y,\tau)$, similar to Gu  _et al._ (2020):

$\operatorname{Tr}(A\cdot B)\equiv\int{\rm d}^{3}x_{1}{\rm
d}^{3}x_{2}A(x_{1},x_{2})B(x_{2},x_{1}).$

We have also inserted two parameters $s_{f},s_{b}$ in front of momenta for
convenience of the analysis below.

The saddle point equations are (we focus at zero temperature and ignore the
zero-mode subtraction for bosons)

$\displaystyle\Sigma(r,\tau)$ $\displaystyle=$ $\displaystyle
g^{2}G(r,\tau)D(r,\tau)$ (20) $\displaystyle\Pi(r,\tau)$ $\displaystyle=$
$\displaystyle-g^{2}G(-r,-\tau)G(r,\tau)$ (21) $\displaystyle
G(k,i\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{i\omega_{n}-s_{f}(k_{x}+k_{y}^{2})-\Sigma(k,i\omega_{n})}$
(22) $\displaystyle D(q,i\Omega_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{q_{y}^{2}-\Pi(q,i\Omega_{n})}$ (23)

We recall the solutions of these equations at zero temperature obtained in
Section III:

$\displaystyle\Sigma(k,i\omega)$ $\displaystyle=$ $\displaystyle-i{\rm
sgn\,}(\omega)2^{5/3}g^{4/3}\frac{T^{2/3}}{3\sqrt{3}}H_{1/3}\left(\frac{|\omega|-\pi
T}{2\pi T}\right)\quad(T\to 0)\,,$ (24) $\displaystyle\Pi(q,i\Omega)$
$\displaystyle=$ $\displaystyle-\frac{g^{2}}{8\pi}\frac{|\Omega|}{|q_{y}|}\,.$
(25)

We describe the structure of fluctuation around the saddle point. We introduce
a collective notation for Green’s functions ${\mathcal{G}}=(D,G)$ and self
energies $\Xi=(\Pi,\Sigma)$, and let $\Lambda=\mathrm{diag}(-1/2,1)$ acting on
the two component space of $(D,G)$ or $(\Pi,\Sigma)$. Following derivations in
Gu  _et al._ (2020); Guo  _et al._ (2020), we can expand the $G-\Sigma$ action
around the saddle point to quadratic order as

$\delta
S=\frac{1}{2}\begin{pmatrix}\delta\Xi^{T}&\delta{\mathcal{G}}^{T}\end{pmatrix}\Lambda\begin{pmatrix}W_{\Sigma}&-1\\\
-1&W_{G}\end{pmatrix}\begin{pmatrix}\delta\Xi\\\
\delta{\mathcal{G}}\end{pmatrix}\,,$ (26)

where $T$ means transpose on spacetime indices. Here $W_{\Sigma}$ and $W_{G}$
are defined as

$W_{\Sigma}=\frac{\delta{\mathcal{G}}_{*}[\Xi]}{\delta\Xi},\qquad
W_{G}=\frac{\delta\Xi_{*}[{\mathcal{G}}]}{\delta{\mathcal{G}}}\,,$ (27)

where ${\mathcal{G}}_{*}[\Xi]$ is the saddle point expression of
${\mathcal{G}}$ viewed as a functional of $\Xi$, and similarly for
$\Xi_{*}[{\mathcal{G}}]$.

We can further integrate out $\delta\Xi$ in (26), to obtain

$\delta S=\frac{1}{2}\delta{\mathcal{G}}^{T}\Lambda
W_{\Sigma}^{-1}(\underbrace{W_{\Sigma}W_{G}}_{K_{G}}-1)\delta{\mathcal{G}}\,,$
(28)

where we have defined the kernel $K_{G}=W_{\Sigma}W_{G}$. Therefore, soft
fluctuations are related to unit eigenvalue of $K_{G}$.

###  V.1 Time Reparameterization

We now note the time reparameterization symmetry of (V). Consider the
following reparameterization

$\tau=f(\sigma),\quad{\rm d}x=(f^{\prime}(\sigma))^{1/z}{\rm
d}\tilde{x},\quad{\rm d}y=(f^{\prime}(\sigma))^{1/(2z)}{\rm d}\tilde{y}.$ (29)

Then ignoring the irrelevant $\partial_{\tau}$ term, the action is invariant
under the change of variables

$\displaystyle
G(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})=\frac{1}{(f^{\prime}(\sigma)f^{\prime}(\sigma^{\prime}))^{a}}\tilde{G}(\tilde{x},\tilde{x}^{\prime},\tilde{y},\tilde{y}^{\prime},\sigma,\sigma^{\prime})$
(30)
$\displaystyle\Sigma(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})=\frac{1}{(f^{\prime}(\sigma)f^{\prime}(\sigma^{\prime}))^{1+3/(2z)-a}}\tilde{\Sigma}(\tilde{x},\tilde{x}^{\prime},\tilde{y},\tilde{y}^{\prime},\sigma,\sigma^{\prime})$
(31) $\displaystyle
D(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})=\frac{1}{(f^{\prime}(\sigma)f^{\prime}(\sigma^{\prime}))^{1+3/(2z)-2a}}\tilde{D}(\tilde{x},\tilde{x}^{\prime},\tilde{y},\tilde{y}^{\prime},\sigma,\sigma^{\prime})$
(32)
$\displaystyle\Pi(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})=\frac{1}{(f^{\prime}(\sigma)f^{\prime}(\sigma^{\prime}))^{2a}}\tilde{\Pi}(\tilde{x},\tilde{x}^{\prime},\tilde{y},\tilde{y}^{\prime},\sigma,\sigma^{\prime})$
(33) $\displaystyle
s_{f}(i\partial_{x}+\partial_{y}^{2})\delta(x,x^{\prime})\delta(y,y^{\prime})\delta(\tau,\tau^{\prime})=$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{s}_{f}\frac{1}{(f^{\prime}(\sigma)f^{\prime}(\sigma^{\prime}))^{1/2+5/(4z)}}(i\partial_{\tilde{x}}+\partial_{\tilde{y}}^{2})\delta(\tilde{x},\tilde{x}^{\prime})\delta(\tilde{y},\tilde{y}^{\prime})\delta(\sigma,\sigma^{\prime})$
(34) $\displaystyle
s_{b}\partial_{y}^{2}\delta(x,x^{\prime})\delta(y,y^{\prime})\delta(\tau,\tau^{\prime})=\tilde{s}_{b}\frac{1}{(f^{\prime}(\sigma)f^{\prime}(\sigma^{\prime}))^{1/2+5/(4z)}}\partial_{\tilde{y}}^{2}\delta(\tilde{x},\tilde{x}^{\prime})\delta(\tilde{y},\tilde{y}^{\prime})\delta(\sigma,\sigma^{\prime})$
(35)

The consistency with saddle point equation yields $a=2/3,z=3/2$, and
$s_{b},s_{f}$ are marginal couplings.

Let us consider the consequence of reparameterization symmetry on the kernel
$K_{G}$. In the low-energy conformal limit (ignoring $\partial_{\tau}$ term),
the saddle point equations can be written as

$\begin{split}G\cdot(s_{f}(i\partial_{x}+\partial_{y}^{2})\delta^{(3)}-\Sigma)&=1,\\\
D\cdot(-s_{b}\partial_{y}^{2}\delta^{(3)}-\Pi)&=1.\end{split}$ (36)

Due to the reparameterization symmetry, it’s valid to consider an
infinitesimal reparameterization
$\delta_{\epsilon}:f(\tau)=\tau+\epsilon(\tau)$ on both sides. Because the
Schwinger-Dyson equation holds, we can rewrite
$\delta_{\epsilon}\Sigma,\delta_{\epsilon}\Pi$ in terms of
$\delta_{\epsilon}G,\delta_{\epsilon}D$, and therefore we obtain

$(1-K_{G})\delta_{\epsilon}{\mathcal{G}}=W_{\Sigma}\delta_{\epsilon}k.$ (37)

This is a two component equation for
$(\delta_{\epsilon}D,\delta_{\epsilon}G)$. On the RHS, $\delta_{\epsilon}k$ is
the reparameterization of momentum term

$\delta_{\epsilon}k=(s_{f}\delta_{\epsilon}(i\partial_{x}+\partial_{y}^{2})\delta^{(3)},-s_{b}\delta_{\epsilon}\partial_{y}^{2}\delta^{(3)}).$
(38)

In the original SYK model, the RHS is absent and the reparameterization mode
is an eigenvector of $K_{G}$ with eigenvalue one Maldacena and Stanford
(2016). This eigenvalue one is responsible for the $\beta J$ enhancement in
four-point functions. In the current model, the presence of
$\delta_{\epsilon}k$ term will destroy the dominance of the unit eigenvalue
mode in the action for fluctuations, and the reparameterization fluctuation
will not have $\beta J$ enhancements. Therefore, the low-energy theory will
contain not only the reparameterization but also other fluctuations.

We can also repeat the above discussion for the $U(1)$ gauge symmetry

$\begin{split}\delta_{\lambda}G(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})&=i(\lambda(\tau)-\lambda(\tau^{\prime}))G(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})\,,\\\
\delta_{\lambda}\Sigma(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})&=i(\lambda(\tau)-\lambda(\tau^{\prime}))\Sigma(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})\,.\end{split}$
(39)

This symmetry is emergent at low-energy given that the $\partial_{\tau}$ term
in the action is irrelevant. Running the above argument for this $U(1)$
symmetry, we obtain an eigenvector of $K_{G}$ with unit eigenvalue:

$(1-K_{G})\delta_{\lambda}{\mathcal{G}}=0\,,$ (40)

and there is no momentum term on the RHS because the symmetry is uniform in
space. In what follows, we will demonstrate that
$\delta_{\lambda}{\mathcal{G}}$ is the only unit eigenvector of $K_{G}$ that
obeys sliding symmetry.

###  V.2 Sliding symmetry

An important symmetry of the patched Fermi surface problem is the sliding
symmetry

$\begin{split}\phi(x,y)&\to\phi(x,y+\theta x),\\\ \psi(x,y)&\to
e^{-i[(\theta/2)y+(\theta^{2}/4)x]}\psi(x,y+\theta x)\,.\end{split}$ (41)

In Fourier space, the representation is simpler

$\begin{split}\phi(q_{x},q_{y})&\to\phi(q_{x}-\theta q_{y},q_{y}),\\\
\psi(k_{x},k_{y})&\to\psi(k_{x}-\theta
k_{y}-\frac{\theta^{2}}{4},k_{y}+\frac{\theta}{2}).\end{split}$ (42)

We can see there are two different representations of the sliding group. One
is the boson class $[0]$, and another is the fermion class $[1]$. Given two
momenta $k_{1},k_{2}$ that transforms under representation $[n_{1}]$ and
$[n_{2}]$ respectively, we can fuse them into a new representation simply by
momentum addition

$k_{3}=\alpha k_{1}+\beta k_{2},$

which transforms under representation $[n_{3}]$ and $n_{3}=\alpha n_{1}+\beta
n_{2}$. Here $\alpha,\beta$ are rational numbers such that $n_{3}$ is an
integer. In our problem, we will only encounter class $[0]$ and $[1]$.

Each representation is associated with some invariants. For example $\bm{q}$
of $[0]$ has invariant $q_{y}$ and $\bm{k}$ of $[1]$ has invariant
$k_{x}+k_{y}^{2}$.

The eigenfunctions of the kernel $K_{G}$ are bosonic or fermionic two point
functions $B(k,p)$ and $F(k,p)$ (see (48)), where $k$ is the relative momentum
and $p$ is the CoM momentum. For $B(k,p)$, both $(k,p)$ are class $[0]$. For
$F(k,p)$, $k$ is class $[1]$ and $p$ is class $[0]$. Therefore $B$ is in class
$[0]\otimes[0]$ and $F$ is in class $[0]\otimes[1]$. They are in tensor
product representations.

By trial and error, we find the following invariants of the above tensor
product representations up to quadratic order:

$\begin{split}&[0]\otimes[0]:k_{y},~{}p_{y},~{}p_{x}k_{y}-p_{y}k_{x}\\\
&[0]\otimes[1]:p_{y},~{}k_{x}+k_{y}^{2},~{}p_{x}+2p_{y}k_{y}\end{split}$ (43)

###  V.3 Feynman Diagrams

In this part we give a diagrammatic prescription to compute $K_{G}$. We have
the following diagrammatic representations for $\delta_{\epsilon}G$ and
$\delta_{\epsilon}D$

$\begin{pmatrix}\delta_{\epsilon}D(3;4)\\\
\delta_{\epsilon}G(3;4)\end{pmatrix}=\begin{pmatrix}\leavevmode\hbox
to45.48pt{\vbox to35.18pt{\pgfpicture\makeatletter\hbox{\hskip
30.08301pt\lower-17.58856pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.0pt}{-12.0pt}\pgfsys@moveto{-15.0pt}{-12.0pt}\pgfsys@lineto{-15.0pt}{12.0pt}\pgfsys@lineto{15.0pt}{12.0pt}\pgfsys@lineto{15.0pt}{-12.0pt}\pgfsys@closepath\pgfsys@moveto{15.0pt}{12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-20.0pt}{12.0pt}\pgfsys@lineto{-15.0pt}{12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-20.0pt}{-12.0pt}\pgfsys@lineto{-15.0pt}{-12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.75pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.75pt}{9.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.63722pt}{-2.86945pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\delta_{\epsilon}D$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\ \leavevmode\hbox
to45.48pt{\vbox to35.18pt{\pgfpicture\makeatletter\hbox{\hskip
30.08301pt\lower-17.58856pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.0pt}{-12.0pt}\pgfsys@moveto{-15.0pt}{-12.0pt}\pgfsys@lineto{-15.0pt}{12.0pt}\pgfsys@lineto{15.0pt}{12.0pt}\pgfsys@lineto{15.0pt}{-12.0pt}\pgfsys@closepath\pgfsys@moveto{15.0pt}{12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-20.0pt}{12.0pt}\pgfsys@lineto{-15.0pt}{12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-20.0pt}{-12.0pt}\pgfsys@lineto{-15.0pt}{-12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.75pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.75pt}{9.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.29pt}{-2.86945pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\delta_{\epsilon}G$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{pmatrix},$ (44)

where numbers are short-hands for spacetime coordinates.

Using the saddle point equations, we can also write down the Feynman diagrams
for $W_{\Sigma}$ and $W_{G}$:

$W_{\Sigma}(1,2;3,4)=\begin{pmatrix}\leavevmode\hbox to63.17pt{\vbox
to35.18pt{\pgfpicture\makeatletter\hbox{\hskip
10.08301pt\lower-17.58856pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{40.0pt}{12.0pt}\pgfsys@lineto{0.0pt}{12.0pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-12.0pt}\pgfsys@lineto{40.0pt}{-12.0pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.75pt}{9.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.75pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.25pt}{9.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.25pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&0\\\
0&\leavevmode\hbox to63.17pt{\vbox
to35.18pt{\pgfpicture\makeatletter\hbox{\hskip
10.08301pt\lower-17.58856pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{40.0pt}{12.0pt}\pgfsys@lineto{0.0pt}{12.0pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-12.0pt}\pgfsys@lineto{40.0pt}{-12.0pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.75pt}{9.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.75pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.25pt}{9.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.25pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{pmatrix},$ (45)
$W_{G}(1,2;3,4)=\begin{pmatrix}0&-g^{2}\left(\leavevmode\hbox to39.17pt{\vbox
to35.18pt{\pgfpicture\makeatletter\hbox{\hskip
10.08301pt\lower-17.58856pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{20.0pt}{12.0pt}\pgfsys@lineto{0.0pt}{12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-12.0pt}\pgfsys@lineto{20.0pt}{-12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{20.0pt}{12.0pt}\pgfsys@lineto{20.0pt}{-12.0pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{-12.0pt}\pgfsys@lineto{20.0pt}{-12.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{\pgfsys@beginscope\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.75pt}{9.74445pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.75pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.25pt}{9.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.25pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{pmatrix}\,,$ (46)

where a black arrowed line denotes fermion propagator, a dashed arrowed line
denotes boson propagator, and an unarrowed dashed line denotes spacetime
$\delta$-function. The first entry is boson and the second entry is fermion.
Recalling $\Lambda=\mathrm{diag}(-1/2,1)$, we see that $\Lambda W_{\Sigma}$
and $\Lambda W_{G}$ are explicitly symmetric as required by quadratic
expansion. Therefore we can obtain the diagram for $K_{G}=W_{\Sigma}W_{G}$ as

$K_{G}(1,2;3,4)=\begin{pmatrix}0&-g^{2}\left(\leavevmode\hbox to39.17pt{\vbox
to35.18pt{\pgfpicture\makeatletter\hbox{\hskip
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.25pt}{-14.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\end{pmatrix}$ (47)

Examination of these diagrams shows that summing the series $(1-K_{G})^{-1}$
is equivalent to summing ladder diagrams of fermions with the so-called ‘Maki-
Thomson’ and ‘Aslamazov-Larkin’ corrections to all orders; the first order
terms of this type were examined by Kim et al. Kim  _et al._ (1994).

###  V.4 Eigenvalues of the Kernel

Let us investigate the eigenvalues of the kernel (47). Unit eigenvalues will
correspond to composite scaling operators Gross and Rosenhaus (2017); Klebanov
and Tarnopolsky (2017); Klebanov  _et al._ (2018); Tikhanovskaya  _et al._
(2021a, b) appearing in the operator product expansion of a particle and a
hole in the single patch theory, as they lead to singularities in ladder
expansion $(1-K_{G})^{-1}$.

Assume the eigenvectors have the following ansatz

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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=V(\omega,\bm{k},\Omega,\bm{p})=\begin{pmatrix}B(k,p)\\\
F(k,p)\end{pmatrix},$ (48)

where $k=(\omega,\bm{k})$ is the relative momentum (frequency), and
$p=(\Omega,\bm{p})$ is the conserved center of mass momentum (frequency).

We now calculate how $K_{G}$ acts on the above ansatz, and obtain

$\begin{pmatrix}\tilde{B}\\\ \tilde{F}\end{pmatrix}=K_{G}\begin{pmatrix}B\\\
F\end{pmatrix},$ (49)

where

$\begin{split}&\tilde{B}(k_{1},p)=-g^{2}D(k_{1}+p/2)D(k_{1}-p/2)\int\frac{{\rm
d}^{3}k_{2}}{(2\pi)^{3}}\left[G(k_{2}-k_{1})F(k_{2},p)+G(k_{1}-k_{2})F(-k_{2},p)\right],\end{split}$
(50) $\begin{split}\tilde{F}(k_{1},p)&=g^{2}G(k_{1}+p/2)G(k_{1}-p/2)\\\
&\times\int\frac{{\rm
d}^{3}k_{2}}{(2\pi)^{3}}\left[\frac{1}{2}G(k_{1}-k_{2})\left(B(k_{2},p)+B(-k_{2},p)\right)+D(k_{1}-k_{2})F(k_{2},p)\right]\end{split}$
(51)

###  V.5 Using Sliding Symmetry

We can use sliding symmetries to simplify the kernel $K_{G}$. The generic
sliding symmetric eigenfunctions $B,F$ depend on the invariants discussed in
(43) (we assumed that higher order invariants can be factorized into lower
order ones):

$\displaystyle B(k_{2},p)$ $\displaystyle=$ $\displaystyle
B(\omega_{2},\Omega,k_{2y},p_{y},k_{2y}p_{x}-p_{y}k_{2x}),$ (52)
$\displaystyle F(k_{2},p)$ $\displaystyle=$ $\displaystyle
F(\omega_{2},\Omega,k_{2x}+k_{2y}^{2},p_{y},p_{x}+2p_{y}k_{2y}).$ (53)

One can verify that the kernel $K_{G}$ actually preserves the above sliding
symmetric ansatz.

###  V.6 Further simplification

Because the CoM momentum $p$ is conserved by $K_{G}$, we are free to specify
its value. Similarly for the frequency $\Omega$. We will limit ourselves here
to the case $p_{y}=0$ because then, as shown below, the integral equations can
be simplified to one over frequency alone. So we will be restricting attention
to longitudinal density fluctuations of the fermions on the Fermi surface, in
the terminology of Kim et al. Kim  _et al._ (1994). The case with $p_{y}\neq
0$ corresponds to the transverse ‘diamagnetic’ sector Kim  _et al._ (1994),
which we do not analyze below.

The kernel further simplifies if we set $p_{y}=0$, which simplifies one of the
arguments in the ansatz:

$\displaystyle B(k_{2},p)$ $\displaystyle=$ $\displaystyle
B(\omega_{2},k_{2y},\Omega,p_{x})$ (54) $\displaystyle F(k_{2},p)$
$\displaystyle=$ $\displaystyle F(\omega_{2},k_{2x}+k_{2y}^{2},\Omega,p_{x})$
(55)

Therefore, in (50), for the first term we do $k_{2x}\to k_{2x}-k_{2y}^{2}$
followed by $k_{2y}\to k_{2y}+k_{1y}$, and for the second term we do
$k_{2x}\to k_{2x}+k_{2y}^{2}$, and then integrate over $k_{2y}$:

$\begin{split}&\tilde{B}(k_{1},p)=-g^{2}D(k_{1}+p/2)D(k_{1}-p/2)\int\frac{{\rm
d}^{3}k_{2}}{(2\pi)^{3}}\\\
&\times\Big{[}G(\omega_{2}-\omega_{1},k_{2x}-(k_{1x}+k_{1y}^{2})-2k_{1y}k_{2y})F(\omega_{2},k_{2x},p)\\\
&+G(\omega_{1}-\omega_{2},(k_{1x}+k_{1y}^{2})-k_{2x}-2k_{1y}k_{2y})F(-\omega_{2},-k_{2x},p)\Big{]}\\\
&=-g^{2}D(k_{1}+p/2)D(k_{1}-p/2)\int\frac{{\rm d}\omega_{2}{\rm
d}k_{2x}}{(2\pi)^{2}}\\\ &\times\frac{i}{4|k_{1y}|}{\rm
sgn\,}(\omega_{1}-\omega_{2})\left[F(\omega_{2},k_{2x},p)-F(-\omega_{2},k_{2x},p)\right],\end{split}$
(56)

and in the second term of last line we also flipped $-k_{2x}\to k_{2x}$.

For the first term of (51), we directly integrate over $k_{2x}$, and for the
second term, we shift $k_{2x}\to k_{2x}-k_{2y}^{2}$ and then integrate over
$k_{2y}$:

$\begin{split}\tilde{F}(k_{1},p)&=g^{2}G(k_{1}+p/2)G(k_{1}-p/2)\int\frac{{\rm
d}\omega_{2}}{2\pi}\\\ &\times\Big{[}\int\frac{{\rm
d}k_{2y}}{2\pi}\frac{-i}{4}{\rm
sgn\,}(\omega_{1}-\omega_{2})(B(\omega_{2},k_{2y})+B(-\omega_{2},k_{2y}))\\\
&+\int\frac{{\rm
d}k_{2x}}{2\pi}\frac{4\pi^{1/3}}{3\sqrt{3}g^{2/3}|\omega_{1}-\omega_{2}|^{1/3}}F(\omega_{2},k_{2x},p)\Big{]}.\end{split}$
(57)

Plugging $p=(\Omega,p_{x},0)$ into (56) and (57), we see that the sliding
symmetry is preserved. Furthermore, the action of $K_{G}$ is highly degenerate
because it only cares about the integration of $F,B$ over all spatial momenta.
We can therefore integrate out all spatial momenta to get a functional only in
frequency space. Let

$\displaystyle B_{I}(\omega)$ $\displaystyle=$ $\displaystyle\int\frac{{\rm
d}k_{y}}{2\pi}B(\omega,k_{y}),$ (58) $\displaystyle F_{I}(\omega)$
$\displaystyle=$ $\displaystyle\int\frac{{\rm d}k_{x}}{2\pi}F(\omega,k_{x}).$
(59)

For simplicity we have suppressed $(\Omega,p_{x})$ dependence in the
arguments.

The projected action of $K_{G}$ is

$\begin{split}\tilde{B}_{I}(\omega_{1})&=-\frac{8i\pi^{4/3}}{3\sqrt{3}g^{2/3}|\omega_{1}^{2}-\Omega^{2}/4|^{1/3}\left(|\omega_{1}-\Omega/2|^{2/3}+|\omega_{1}+\Omega/2|^{2/3}+|\omega_{1}^{2}-\Omega^{2}/4|^{1/3}\right)}\\\
&\times\int\frac{{\rm d}\omega_{2}}{2\pi}{\rm
sgn\,}(\omega_{1}-\omega_{2})\left[F_{I}(\omega_{2})-F_{I}(-\omega_{2})\right],\end{split}$
(60)

and

$\begin{split}\tilde{F}_{I}(\omega_{1})&=\frac{ig^{2}}{2}\frac{{\rm
sgn\,}(\omega_{1}+\Omega/2)-{\rm sgn\,}(\omega_{1}-\Omega/2)}{i\Omega-
p_{x}-\left[\Sigma(\omega_{1}+\Omega/2)-\Sigma(\omega_{1}-\Omega/2)\right]}\int\frac{{\rm
d}\omega_{2}}{2\pi}\\\ &\times\Big{[}\frac{-i}{4}{\rm
sgn\,}(\omega_{1}-\omega_{2})(B_{I}(\omega_{2})+B_{I}(-\omega_{2}))+\frac{4\pi^{1/3}}{3\sqrt{3}g^{2/3}|\omega_{1}-\omega_{2}|^{1/3}}F_{I}(\omega_{2})\Big{]}.\end{split}$
(61)

In the conformal limit $|\Omega|,|p_{x}|\ll g$, we can ignore the $i\Omega-
p_{x}$ term in (61), and after rescaling $b=g^{2/3}B$, we can rewrite the
above equations to be independent of $g$:

$\begin{split}\tilde{b}_{I}(\omega_{1})&=-\frac{8i\pi^{4/3}}{3\sqrt{3}|\omega_{1}^{2}-\Omega^{2}/4|^{1/3}\left(|\omega_{1}-\Omega/2|^{2/3}+|\omega_{1}+\Omega/2|^{2/3}+|\omega_{1}^{2}-\Omega^{2}/4|^{1/3}\right)}\\\
&\times\int\frac{{\rm d}\omega_{2}}{2\pi}{\rm
sgn\,}(\omega_{1}-\omega_{2})\left[F_{I}(\omega_{2})-F_{I}(-\omega_{2})\right],\end{split}$
(62) $\begin{split}\tilde{F}_{I}(\omega_{1})&=-\frac{i}{2}\frac{{\rm
sgn\,}(\omega_{1}+\Omega/2)-{\rm
sgn\,}(\omega_{1}-\Omega/2)}{\left[\bar{\Sigma}(\omega_{1}+\Omega/2)-\bar{\Sigma}(\omega_{1}-\Omega/2)\right]}\int\frac{{\rm
d}\omega_{2}}{2\pi}\\\ &\times\Big{[}\frac{-i}{4}{\rm
sgn\,}(\omega_{1}-\omega_{2})(b_{I}(\omega_{2})+b_{I}(-\omega_{2}))+\frac{4\pi^{1/3}}{3\sqrt{3}|\omega_{1}-\omega_{2}|^{1/3}}F_{I}(\omega_{2})\Big{]}\,,\end{split}$
(63)

where $\bar{\Sigma}=g^{-4/3}\Sigma$.

###  V.7 Unit eigenvalue of $K_{G}$

The action of $K_{G}$ in (62),(63) can be classified into two sectors. The
first one is $b_{I}=0$ and $F_{I}(\omega)$ even. The second sector is
$b_{I}(\omega)$ even and $F_{I}(\omega)$ odd.

####  V.7.1 $F_{I}$ even sector

In this case the conformal $K_{G}$ reduces to (63) with $b_{I}=0$. By
numerical diagonalization, we found only one mode with unit eigenvector. It is
generated by the U(1) gauge symmetry

$\begin{split}\delta_{\lambda}G(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})&=i(\lambda(\tau)-\lambda(\tau^{\prime}))G(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})\,,\\\
\delta_{\lambda}\Sigma(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})&=i(\lambda(\tau)-\lambda(\tau^{\prime}))\Sigma(x,x^{\prime},y,y^{\prime},\tau,\tau^{\prime})\,.\end{split}$
(64)

There is no action on $D,\Pi$ or the kinetic term. The Fourier transform of
$\delta_{\lambda}G$ is

$\delta_{\lambda}G(\Omega,\omega,\bm{k})=\lambda_{\Omega}\left[iG(\omega-\Omega/2,\bm{k})-iG(\omega+\Omega/2,\bm{k})\right]\,,$
(65)

where $\Omega$ is the CoM frequency, $\omega$ is the relative frequency and
$\bm{k}$ is the relative momentum. $\lambda_{\Omega}=\int{\rm d}\tau
e^{i\Omega\tau}\lambda(\tau)$.

Integrating out the spatial momentum, we get

$F_{I}(\Omega,\omega)=\int\frac{{\rm
d}k_{x}}{2\pi}\delta_{\lambda}G(\Omega,\omega,k_{x})=\frac{1}{2}\left({\rm
sgn\,}(\omega-\Omega/2)-{\rm sgn\,}(\omega+\Omega/2)\right),$ (66)

and $B_{I}(\Omega,\omega)=0$. We can analytically verify that this is an exact
eigenvector of the conformal $K_{G}$ with unit eigenvalue. If we retain the
$i\Omega-p_{x}$ term, the correction is of order $\Omega^{1/3}g^{-4/3}$.

####  V.7.2 $F_{I}$ odd sector

To numerically diagonalize the kernel, we first substitute (62) into (63) to
eliminate $b_{I}$:

$\tilde{F}_{I}(\omega)=\frac{2}{3}\frac{\theta(\Omega^{2}/4-\omega^{2})}{|\omega-\Omega/2|^{2/3}+|\omega+\Omega/2|^{2/3}}\int_{-\Omega/2}^{\Omega/2}{\rm
d}\omega^{\prime}\left[\frac{1}{|\omega-\omega^{\prime}|^{1/3}}+h(\omega,\omega^{\prime})\right]F_{I}(\omega^{\prime})\,,$
(67)

where

$h(\omega,\omega^{\prime})=\int_{-\infty}^{\infty}{\rm d}\nu\frac{{\rm
sgn\,}(\nu-\omega){\rm
sgn\,}(\nu-\omega^{\prime})}{|\nu^{2}-\Omega^{2}/4|^{1/3}\left(|\nu-\Omega/2|^{2/3}+|\nu+\Omega/2|^{2/3}+|\nu^{2}-\Omega^{2}/4|^{1/3}\right)}\,.$
(68)

Here we have used the explicit form of $\Sigma(\omega)$.

In (67), $\Omega$ is the only external scale so we can safely set $\Omega=1$.
By numerically diagonalizing $K_{G}$, we found no eigenvalue close to 1.

##  VI Diagrammatics of $G$-$\Sigma$ theory

In this section we discuss the diagrammatics of the $G$-$\Sigma$ theory, with
the goal of developing a systematic $1/N$ expansion. As the large $N$ limit is
expressed as the saddle point of a $G$-$\Sigma$ action, and the self energy
does not have a prefactor of $1/N$ in the Dyson equation, the difficulties
described in Ref. Lee (2009) do not arise here. The structure of the expansion
for the bilocal fields is dictated by the form of the $G$-$\Sigma$ action, and
the bilocal field propagator resums an infinite number of terms from the
previous approach Lee (2009).

Using notations in Sec. V, we can expand the $G$-$\Sigma$ action around the
saddle point as

$S=NS_{0}+NS_{2}[\delta{\mathcal{G}},\delta\Xi]+NS_{3}[\delta{\mathcal{G}},\delta\Xi]+NS_{4}[\delta{\mathcal{G}},\delta\Xi]+\dots$
(69)

Here $S_{n}$ means the term which contains $n$-th power of
$\delta{\mathcal{G}}$ and $\delta\Xi$. In particular, $S_{2}$ is given by
(26). For convenience of power counting, we decide to make propagators of
$\delta{\mathcal{G}}$ and $\delta\Xi$ independent of $N$ and push $N$ power
counting into vertices. This can be done by rescaling
$(\delta{\mathcal{G}},\delta\Xi)\to N^{-1/2}(\delta{\mathcal{G}},\delta\Xi)$,
and therefore $NS_{n}[\delta{\mathcal{G}},\delta\Xi]\to
N^{1-n/2}S_{n}[\delta{\mathcal{G}},\delta\Xi]$.

###  VI.1 Propagator

Using (26) and (28), we can write down the propagator of $\delta{\mathcal{G}}$
as

$G_{{\mathcal{G}}{\mathcal{G}}}(1,2;3,4)\equiv\braket{\delta{\mathcal{G}}(1,2)\delta{\mathcal{G}}(4,3)}=\left[\frac{1}{K_{G}-1}W_{\Sigma}\Lambda^{-1}\right](1,2;3,4)\equiv\leavevmode\hbox
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Here the numbers in the argument denote space time indices. For example $1$
means $(x_{1},y_{1},\tau_{1})$. On the RHS we also defined its diagrammatic
representation.

Similarly we can derive the propagator for $\delta\Xi$ to be

$G_{\Xi\Xi}(1,2;3,4)\equiv\braket{\delta\Xi(1,2)\delta\Xi(4,3)}=\left[W_{G}\frac{1}{K_{G}-1}\Lambda^{-1}\right](1,2;3,4)\equiv\leavevmode\hbox
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