# Exact Low-Energy Solution for Critical Fermi Surfaces

Tomer Ravid111Corresponding author<EMAIL_ADDRESS>NHETC, Rutgers
University, Piscataway, NJ  Tom Banks  NHETC, Rutgers University, Piscataway,
NJ

(August 2022)

###### Abstract

We derive multidimensional bosonization directly from the electron gas in a
low-energy, low momentum regime where $\omega\gg\frac{k^{2}}{k_{F}}$, such
that the dispersion can be linearized. To reach this limit, the Fermi momentum
and the number of patches are scaled simultaneously keeping the width of each
patch finite. We apply this to obtain an exact low-energy solution of the
problem of a Fermi surface coupled to a gapless boson, free of disorder and
electron-electron scattering. Contrary to claims in the literature, we show
that the bosonized theory exactly reproduces the $\omega^{2/3}$ of electrons,
previously obtained in large-$N$ theories. We argue that correction to the
self-energy due to tangential dispersion are subdominant at sufficiently low
energies such that
$v_{F}k\gg\left(\frac{g^{4}v_{F}}{k_{F}}\right)^{1/3}\omega^{2/3}$, where $g$
is the coupling constant.

###### Contents

  1. 1 Introduction
  2. 2 Linearizing the Fermions Near the Fermi Surface
  3. 3 Exact Low-Energy Solubility of the Linearized Action
  4. 4 Exactness of Bosonization
  5. 5 Exact Boson Green’s Function
  6. 6 Exact Fermion Green’s Function
    1. 6.1 General Green’s Function
    2. 6.2 Long Comoving Distance Green’s Function
    3. 6.3 Low Energy Green’s Function
  7. 7 Discussion: Can the Dispersion Really Be Linearized?
    1. 7.1 Irrelevance of Quadratic Dispersion in Multi-Patch Theories
    2. 7.2 Relevance of $N$th Order Dispersion in Single-Patch Theories
  8. 8 Conclusion
  9. A Free Fermion Green’s Function in Bosonized Language
  10. B Behavior of the Fourier Transform of the Long Comoving Distance Green’s Function
  11. C Subdominance of Corrections Due to Quadratic Dipsersion

##  1 Introduction

A Fermi surface is a highly singular object. Electrons at each patch of the
Fermi surface lead to a separate resonance, resulting in a continuum of poles
in the Green’s function. Typically, Fermi surfaces are described by the Landau
Fermi liquid theory, which treats the phase-space density
$n_{\boldsymbol{k}}(\boldsymbol{x},t)$, and thus the Fermi surface, as a
semiclassical variable satisfying a classical kinetic equation and exhibiting
no fluctuations. However, when a Fermi surface is coupled to another highly
singular object—namely, an order parameter in the vicinity of a quantum phase
transition—fluctuations in the order parameter turn into fluctuations in the
Fermi surface, and deviations from Fermi liquid behavior are observed (see [1]
and references). This is similar to way in which mean theory becomes invalid
in the vicinity of a classical phase transition—the transition from Landau
theory to Landau-Ginzburg theory is thus analogous to the transition from
Landau Fermi liquid theory to a non-Fermi liquid theory.

The important property of the critical order parameter is that it is a
massless boson. There is another mechanism that can generate massless bosons
in real situations: a gauge field is naturally massless. However, for
conreteness we will focus on the case of an order parameter. We will consider
two spatial dimensions as this leads to the most dramatic deviations from
Fermi liquid behavior. The simplest model of a critical Fermi surface consists
of noninteracting free fermions $\psi$ coupled to an order parameter $\phi$,
viz.

$\displaystyle S\left[\psi,\psi^{\ast},\phi\right]=$ $\displaystyle\int dt\int
d^{2}p\text{
}\psi^{\ast}(\boldsymbol{p},t)\left[i\partial_{t}-\varepsilon(\boldsymbol{p})\right]\psi(\boldsymbol{p},t)$
$\displaystyle+g\int dt\text{ }d^{2}x\text{
}\phi(\boldsymbol{x},t)\left|\psi(\boldsymbol{x},t)\right|^{2}$
$\displaystyle+\int dt\text{
}d^{2}x\left[\dfrac{1}{2}\left(\partial_{\mu}\phi\right)^{2}-\dfrac{1}{2}m^{2}_{0}\phi^{2}\right]$
(1)

Here, the bare boson mass $m_{0}$ is needed in order to tune the total mass to
zero against renormalization by the interaction with the fermions. For
simplicity, here and throughout, we are ignoring the spin degrees of freedom
of the electrons, as those will have no effect on the results derived and on
the arguments leading to them. We are using units in which the speed of the
boson (which clearly differs from the speed of light in spite of the
relativistic notation) is unity to simplify the notation. The action (1)
appears to be rather simple. It ignores the effects of disorder and electron-
electron interactions. We make no claim that such effects are not important in
experiment, but merely treat (1) as a minimal model for critical Fermi
surfaces.

In spite of the apparent simplicity of the action (1), it has proven difficult
to tackle using standard techniques. It becomes strongly coupled at low
energies[2]. Thus, perturbation theory is inapplicable. Therefore, a large-$N$
theory for crticial-Fermi surfaces was introduced in [2], and the leading
order term in the $1/N$ expansion was computed. The results are interesting:
The boson becomes overdamped and rapidly decays at low energies, at a rate
scaling as $\frac{\left|\omega\right|}{\left|\boldsymbol{k}\right|}$. The
elecrons are also unstable: There remains a pole at $\omega=0$, i.e., a Fermi
surface, but the poles at nonzero energy are destroyed by the interaction with
the gapless boson, and decay at a rate proportional to $\omega^{2/3}$. The
decay of electrons agrees with experiment, at least qualitatively[1]. However,
in [3] it is shown that higher order terms in the large-$N$ expansion diverge
at low energies, so they are clearly not negligible!

In [4], a model is proposed in an attempt to improve the behavior of the
large-$N$ expansion. The model assumes that, in addition to the large number
of electron species in the standard large-$N$ theory, there is a large number
of boson species—and that the coupling between each boson and each pair of
electrons is a random number obeying some distribution. This model has a
strongly self-averaging behavior, and the $\frac{1}{N}$ corrections average to
zero, leaving one with the same results obtained by considering single-loop
diagrams in the standard theory. However, it remains unclear why the
additional structure assumed by this model should not change the physics of
the problem, and why such assumptions should yield no important deviation from
reality.

It is thus evident that a fundamental understanding of critical Fermi surfaces
requires a more direct approach, and such an approach we will attempt to
provide.

In this paper, we study the problem assuming nothing but ordinary electrons
and ordinary critical bosons, and without any kind of an approximation other
than a particular low energy, low momentum limit. We argue for what might
appear “too good to be true”—that, at least in the absence of disorder and
electron-electron interactions, critical Fermi surfaces are not merely less
complicated than previously thought, but they are in fact simple enough to be
solved exactly. We argue that the only contribution to the solution is due to
simple fermionic loops with no internal decorations—the same diagrams that
contribute in the large-$N$ theory, in spite of the absence of a large-$N$
approximation. This vast difference from previous studies owes to a nontrivial
way of taking the low-energy limit of the Fermi surface. The nontrivial low
energy limit guarantees the irrelevance of the quadratic term in the
electron’s dispersion, which is the main source of difficulty faced by
previous studies.

##  2 Linearizing the Fermions Near the Fermi Surface

The Fermi surface is a surface $k_{F}(\chi)$ satisfying

$\displaystyle\varepsilon\left(k_{F}(\chi){\hat{\boldsymbol{x}}}_{\perp}\right)=0$
(2)

where $\boldsymbol{\hat{x}}_{\perp}=\left(\cos\chi,\sin\chi\right)$ is the
normal to the Fermi surface, and $\chi$ is an angle labelling a patch of the
Fermi surface through the direction of its normal (for a spherical Fermi
surface this coincides with the polar angle). If one is interested in low-
energy degrees of freedom, then one should consider electrons in the vicinity
of the Fermi surface. Naively, this can be done by expanding
$\varepsilon(\boldsymbol{p})$ to linear order near
$\left|\boldsymbol{p}\right|=k_{F}(\chi)$. This is the approach taken, for
example, in [5]. However, this leads to a dispersion

$\displaystyle\varepsilon(\boldsymbol{p})\approxeq
v_{F}(\chi)\left(\left|\boldsymbol{p}\right|-k_{F}(\chi)\right).$ (3)

giving rise to a singular action in momentum space, and thus to an apparently
nonlocal action in position space. The singular behavior arises because the
dispersion has been linearized around a surface rather than around a point.
Linear dispersion around any specific point entails no such nonlocality. The
appropriate way to obtain a low-energy local theory of Fermi surface is thus
to divide the Fermi surface into patches, and then linearize around the Fermi
point separately for each patch.

The linearization program will result in an infinite collection of massless
fermions linearly dispersing in the direction normal to the Fermi surface. A
similar approach is presented and criticized e.g. by Sachdev in [6]. However,
Sachdev’s discussion assumes a continuum limit in which the width of each
patch is zero. As a result, each patch becomes a strictly one-dimensional
line, and (the physically important) scattering tangent to the Fermi surface
is thereby ignored, as is the curvature of the Fermi surface. The conventional
wisdom[2][3][4][6] is thus that one must expand the dispersion to quadratic
order, so that the dispersion depends on the tangential momentum, and each
patch becomes a full and rather complicated two-dimensional field theory. This
has been implemented within a multi-patch framework similar to our own in [7].
Here, we argue that the loss of tangential scattering in the naively
linearized model is not due to the linearization of the dispersion, but
instead due to the continuum limit of the patches. Instead, as one might
naively expect, the quadratic dispersion is irrelevant at low momenta compared
to the linear dispersion, but only so long as one takes a non-naive continuum
in which each patch maintains a finite width. The result is that each patch is
a “one-and-a-half-dimensional field theory”: it obeys a one-dimensional
dispersion, but its momentum lies on a two-dimensional plane. This is a
crucial point: one-dimensional massless fermions are exactly soluble (e.g. as
shown by Schwinger in the context of QED[8]), and it will be shown that the
the “one-and-a-half-dimensional” patch inherits this exact solubility.

To linearize near the Fermi momentum, note that a general momentum can be
written as a sum of the Fermi momentum of a patch and a deviation from that
Fermi momentum

$\displaystyle\boldsymbol{p}=k_{F}(\chi)\boldsymbol{\hat{x}}_{\perp}(\chi)+\boldsymbol{k}.$
(4)

One can thus linearize the dispersion around the point by taking the deviation
$\boldsymbol{k}$ to be small

$\displaystyle\varepsilon(\boldsymbol{k},\chi)$
$\displaystyle\approxeq\left.\dfrac{\partial\varepsilon}{\partial\boldsymbol{k}}\right|_{\boldsymbol{k}=0,\chi}\cdot{\boldsymbol{k}}$
$\displaystyle=v_{F}(\chi)\boldsymbol{k}\cdot\boldsymbol{{\hat{x}}}_{\perp}(\chi).$
(5)

Note that even though the resulting dispersion is only in the normal
direction, there is no underlying assumption that $\boldsymbol{k}$ is normal
to the Fermi surface, but merely that it is small in magnitude compared to the
Fermi momentum. The momentum is transferred to the fermions by the critical
boson—and provided that that the momentum carried by the boson is sufficiently
small compared to the Fermi momentum, the linear approximation holds, and the
patch behaves as a “one-and-a-half-dimensional” fermion with two dimensional
momentum but one-dimensional dispersion222For further assessment of the
validity of the lineraization of the dispersion, see Section 7. This
dispersion is highly singular and will be shown to lead to some of the
singular results obtained in large-$N$ theories of critical Fermi surfaces.

Since there is nothing special about the patch, one can split the entire Fermi
surface into a set of $N$ patches $\chi_{n}=\frac{2\pi n}{N}$, and write every
momentum as the sum of the Fermi momentum of the closest patch and its
deviation from that patch $k\lesssim\frac{\pi k_{F}}{N}$. The fermion action
in (1) can thus be linearized as

$\displaystyle S_{f}\approxeq$ $\displaystyle\sum^{N}_{n=0}\int
dt\int_{k\leq\frac{\pi k_{F}}{N}}d^{2}k\text{
}\psi^{\ast}\left(\boldsymbol{k},\chi_{n},t\right)\left[i\partial_{t}-v_{F}\left(\chi_{n}\right)k\cos\left(\chi_{n}-\theta\right)\right]\psi\left(\boldsymbol{k},\chi_{n},t\right),$
(6)

with $\theta$ the polar angle of $\boldsymbol{k}$. Unlike (3), the action (6)
is manifestly local, because the normal momentum $k_{\perp}\equiv
k\cos(\theta-\chi)$ Fourier transforms to the normal derivative
$\partial_{\perp}\equiv\cos\chi\partial_{x}+\sin\chi\partial_{y}$.

The next natural step is to take the continuum limit of (6). However, this
limit must be taken with some care: if the width of a patch were strictly zero
(as in [6]), then arbitrarily small momentum transfer would make an electron
hop away from its patch, and by ignoring such inter-patch hopping one loses
track of all scattering tangent to the Fermi surface. Instead, the width of a
patch should be viewed as an upper cutoff on the tangential momentum (it is
the momentum an electron must acquire to leave a patch). One must take a
double scaling limit in which the the Fermi momentum is taken to infinity
simultaneously with the number of patches, so that the cutoff
$\Lambda=k_{F}\delta\chi$ is fixed. In other words,

$\displaystyle\Lambda=\dfrac{2\pi k_{F}}{N}=\text{const}.$ (7)

This limit can be visualized as follows. One approximates the Fermi surface as
a polygon with each side having a fixed width $k_{F}\delta\chi$, and takes the
limit in which number of sides approaches infinity. This is depicted for a
Fermi sphere in figure 1.

Figure 1: In the double scaling limit, one approximates the Fermi surface as a
polygon with each side having a fixed width $k_{F}\delta\chi$, and takes the
limit in which the number of sides approaches infinity.

In this limit, $\chi$ becomes a continuous variable. After rescaling $\psi$ in
such a way that $\left|\psi\right|^{2}$ represents charge density per patch
rather than charge, the action takes the form

$\displaystyle S=\int^{2\pi}_{0}d\chi\int dt\text{
}d^{2}x\psi^{\ast}(\boldsymbol{x},t,\chi)\left[i\partial_{t}-iv_{F}(\chi)\partial_{\perp}-g\phi(\boldsymbol{x},t)\right]\psi(\boldsymbol{x},t,\chi)+S_{b}\left[\phi\right],$
(8)

with $S_{b}[\phi]$ the free action of the critical boson. The free ($g=0$)
field equation derived from (8) simply displaces the fermion field at each
patch in the outgoing (thus positive) normal direction. Assuming inversion
symmetry, $v_{F}(\chi)=v_{F}(\chi+\pi)$. By analogy with the one-dimensinoal
electron gas, one can combine the incoming and outgoing modes from antipodal
patches into a single Dirac spinor, defining:

$\displaystyle\Psi\left(t,\boldsymbol{x},\chi\right)\equiv\begin{pmatrix}&\psi\left(t,\boldsymbol{x},\chi\right)\\\
&\psi\left(t,\boldsymbol{x},\chi+\pi\right)\end{pmatrix},$ (9)

so that the action can be rewritten as

$\displaystyle S$ $\displaystyle=\int^{\pi}_{0}d\chi\int dt\text{
}d^{2}x\text{
}\Psi^{\dagger}\left(t,\boldsymbol{x},\chi\right)\left[i\partial_{t}-iv_{F}\left(\chi\right)\sigma_{z}\partial_{\perp}+g\phi(\boldsymbol{x},t)\right]\Psi\left(t,\boldsymbol{x},\chi\right)+S_{b}\left[\phi\right]$
$\displaystyle=\int^{\pi}_{0}d\chi\int dt\text{ }d^{2}x\text{
}\bar{\Psi}\left(t,\boldsymbol{x},\chi\right)\left[i\gamma^{a}\partial_{a}+g\gamma^{0}\phi(\boldsymbol{x},t)\right]\Psi\left(t,\boldsymbol{x},\chi\right)+S_{b}\left[\phi\right],$
(10)

where we have introduced shorthand qausi-relativistic notation:
$\partial_{a}\equiv\left(\partial_{t},v_{F}(\chi)\partial_{\perp}\right)$,
$\gamma^{a}=\left(\sigma_{x},\sigma_{y}\right)$. The Minkowski metric raising
and lowering indices is just $\eta_{ab}=\text{diag}\left(1,-1\right)$, so the
entire dependence on the Fermi velocity (which, contrary to case of the speed
of light, is physically important and nontrivial) is contained in the
derivative vector. It has thus been established that the Fermi surface is
equivalent to an infinite collection of linearly dispersing “one-and-a-half-
dimensional” Weyl fermions, to which the critical boson couples as an electric
potential. Note that in spite of the normal dispersion, contrary to what is
argued about the naive continuum limit of [6], the curvature of the Fermi
surface is not ignored. Rather, it is directly embodied in the fact that
normals of different patches are not parallel—thus the electrons of different
patches move in different directions. We will show that this is sufficient in
order to capture the physics of effects normally attributed to the curvature
the Fermi surface: the standard electron and boson Green’s functions.

##  3 Exact Low-Energy Solubility of the Linearized Action

It is well known that the problem of one-dimensional Weyl fermions coupled to
an external field can be solved exactly, as originally shown by Schwinger [8].
Since each patch of the Fermi-surface behaves at low momenta like a “one-and-
a-half-dimensional Weyl fermion,” it is to be expected that the low-momentum
theory can be solved exactly on similar grounds. Indeed, direct path
integration establishes that this is the case. It further establishes that if
the dispersion is linear, then the only contribution to the solution comes
from simple fermionic loops—explaining why large-$N$ models give seemingly
reasonable answers.

The linearized action (18) is quadratic in fermion fields and so the fermion
functional integral can be computed exactly. According to the linearized
action (18), the expression for multi-fermion correlators can be written in
terms of the Green’s function of the fermion in an external field

$\displaystyle\left[i\gamma^{0}\gamma^{a}\partial_{a}+g\phi(t,\boldsymbol{x})\right]G\left(t,t^{\prime},\boldsymbol{x},\boldsymbol{x^{\prime}},\chi,\chi^{\prime}\right)=\delta\left(t-t^{\prime}\right)\delta^{2}\left(\boldsymbol{x}-\boldsymbol{x^{\prime}}\right)\delta(\chi-\chi^{\prime}),$
(11)

functionally averaged over the effective action

$\displaystyle S_{\text{eff}}\equiv S_{b}[\phi]+{\rm
Tr\ln}\left[i\gamma^{0}\gamma^{a}\partial_{a}+g\phi(t,\boldsymbol{x})\right],$
(12)

where the same quasi-relativistic notation from (18) has been used.

The determinant can be computed from the Green’s function via the formula

$\displaystyle\dfrac{\delta{\rm Tr\ ln}\
\left[i\gamma^{0}\gamma^{a}\partial_{a}+g\phi(t,\boldsymbol{x})\right]}{\delta\phi(t,\boldsymbol{x})}=g{\rm
Tr}\ G(x,x).$ (13)

The key to the solution of these models is the same as the one Schwinger[8]
used to solve massless quantum electrodynamics in $1+1$ dimensions. Equation
(11) states that the external field Green’s function moves right or left while
acquiring a phase depending on $\phi$. It thus has an explicit analytic
solution of the form of a free Green’s fnuction $G_{0}$ with an “Aharonov-Bohm
phase” $B$

$\displaystyle
G\left(t,t^{\prime},\boldsymbol{x},\boldsymbol{x^{\prime}},\chi,\chi^{\prime}\right)=G_{0}(t,t^{\prime},\boldsymbol{x},\boldsymbol{x^{\prime}})e^{-i\int_{0}^{\pi}d\chi\
B\left(t,t^{\prime},\boldsymbol{x},\boldsymbol{x^{\prime}},\chi\right)}\delta(\chi-\chi^{\prime}).$
(14)

Here, $G_{0}$ is the $\chi$-independent $g=0$ solution and

$\displaystyle\gamma^{0}\gamma^{a}\partial_{a}B\left(t,t^{\prime},\boldsymbol{x},\boldsymbol{x^{\prime}},\chi\right)=-g\phi(t,\boldsymbol{x}).$
(15)

In addition, we must have

$\displaystyle\int_{0}^{\chi}B(t,t,\boldsymbol{x},\boldsymbol{x},\chi)=0.$
(16)

The last condition implies that, in the expression for the logarithm of the
determinant only the term linear in the first derivative of $B$ at coinciding
points survives. The Taylor series truncates because $G_{0}$ has only a pole
at coinciding points. As a consequence, the logarithm of the determinant is
quadratic in $\phi$, which means that it is given exactly by the one loop
fermion vacuum polarization bubble! This is not a large-$N$ expansion, but an
exact property of this class of linearly dispersing critical Fermi surfaces.
The expressions for fermion propagators are all linear functionals of $\phi$,
which means that they can be evaluated exactly because the $\phi$ action is
quadratic.

Every single calculation of this paper can be performed directly from the path
integral of the resulting effective action, which is just a Hertz-Millis
action[9][10]. However, in what follows we will rewrite the theory in terms of
bosonized variables that are not strictly necessary computationally, but are
nevertheless invaluable conceptually.

##  4 Exactness of Bosonization

Eq. (14) states that the sole effect of the order parameter $\phi$ on the
electrons is to shift their phases linearly. Therefore, it is intuitively
clear that if one were to perform some kind of a “polar decomposition” of the
electron field into chiral phases and current densities, the current densities
should decouple from the order parameter while the phases would behave like an
infinite collection of bosonic fields (one per patch) linearly coupled to the
order parameter. Such a description would be a natural two-dimensional
generalization of the bosonized description of the one-dimensional Luttinger
liquid, in which the basic variable is the bosonic phase field associated with
electron-hole excitations.

Indeed, such a higher-dimension bosonized formalism was proposed by
Haldane[11] as a general framework for fluctuating Fermi surfaces. Haldane
proposed that that a Fermi surface is equivalent to an infinite collection of
linearly dispersing bosons, each corresponding to electron-hole pairs created
at a different patch—essentially filling the electron-hole continuum with an
infinite number of lines (see figure 2).

Figure 2: Haldane’s idea is essentially to fill the electron-hole continuum
(left) with an infinite collection of linearly dispersing excitations of
different slopes (right), corresponding to electron-hole pairs at different
patches.

Lawler at al. applied Haldane’s formalism to the problem of critical Fermi
surfaces[12], and showed that it reproduces the results from the random phase
approximation. In spite of these impressive results, the exactness of linear
bosonization at low energies has remained far from accepted in the literature.
Even proponents of the bosonization approach to non-Fermi liquids often assume
that the quadratic term in the electron’s dispersion is relevant and leads to
nonlinear corrections to the bosonized action, and/or that the quadratic
bosonzed action is merely a reformulation of the random phase
approximation[13][7]. By contrast, the result of Section 2 is that there is a
limit in which the electronic dispersion is fully linear, and in which each
patch behaves similarly to a one-dimensional Weyl fermion. Therefore, it
should not be surprising that in this limit, one can bosonize the Weyl fermion
of each patch similarly to the one-dimensional case, leading to an exact,
microscopic derivation of the bosonized model.

As previously stated, the intutive idea is to decompose the eletcron field
into a phase and a magntiude. This idea is essentially correct, and would lead
to the desired result; yet it is hard to make it rigorous, since the electron
field is a Grassmann coordinate, and because of the chiral anomaly. Instead of
decomposing the electron field into a magntiude and a phase (i.e., shifting
the phases to zero), consider an arbitrary local shift of the phase. This
amounts to defining a new field $\Gamma$ as

$\displaystyle\Gamma\left(\boldsymbol{x},t,\chi\right)\equiv\exp\left[-i\sigma_{z}\overline{\zeta}\left(\boldsymbol{x},t,\chi\right)-i\overline{\eta}\left(\boldsymbol{x},t,\chi\right)\right]\Psi\left(\boldsymbol{x},t,\chi\right).$
(17)

for some phase functions $\overline{\zeta}$ and $\overline{\eta}$. Plugging
this into the classical action gives

$\displaystyle S$ $\displaystyle\rightarrow\int^{\pi}_{0}d\chi\int dt\text{
}d^{2}x\text{
}\bar{\Gamma}\left(t,\boldsymbol{x},\chi\right)\left[i\gamma^{a}\left(\partial_{a}+i\epsilon_{ab}\partial^{b}\overline{\zeta}+i\partial_{a}\overline{\eta}\right)+g\gamma^{0}\phi(\boldsymbol{x},t)\right]\Gamma\left(t,\boldsymbol{x},\chi\right)+S_{b}\left[\phi\right],$
(18)

Since the entire effect of the critical field on the electron field is encoded
in the phase of (14), it is natural to suspect that the critical boson can be
decoupled from the electrons by appropriately choosing the phase shifts. This
is indeed the case, because by the Helmholtz decomposition any two-dimensional
vector field can be written as the sum of “irrotational flow”
$\partial_{a}\overline{\eta}$ and “incompressible flow”
$\epsilon_{ab}\partial^{b}\overline{\zeta}$. Hence, by choosing
$\overline{\eta}$ and $\overline{\zeta}$ such that

$\displaystyle\epsilon_{ab}\partial^{b}\overline{\zeta}\left(\boldsymbol{x},t,\chi\right)+\partial_{a}\overline{\eta}\left(\boldsymbol{x},t,\chi\right)=-g\phi\left(\boldsymbol{x},t\right)\delta^{0}_{a},$
(19)

the interaction term disappears. It thus appears that by integrating the
fermions out, one recovers the free boson action, with no effect from the
fermions! However, there is an additional effect of the fermions for which the
classical action does not account: by the chiral anomaly, the measure of the
path integral changes under the transformation (17), and the transformation of
the measure will give rise to an additional term in the action. Computing the
transformation of the measure is similar to the standard Fujikawa method for
1+1D fermions in an external electromagnetic potential. The main difference is
that the effective potentials, which depend on $\phi$ and on
$\overline{\zeta}$, change under the chiral transformation. Hence, one should
break the transformation into infinitesimal blocks and compute the change of
the measure under each one of them. This is done in the one dimensional case
in [14], where the full details of the derivation can be found. Here, we will
just outline the steps.

Let $\alpha$ be a variable parameterizing the “stage” of the transformation.
Similarly to [14], we will choose $\alpha$ such that $\alpha=0$ labels the
initial stage, which is just the original action (18), in which the effective
potential is $g\phi\delta^{a}_{0}$, and $\alpha=0$ labels the final state, in
which the fermions have been fully decoupled and thus the effective potential
vanishes. The effective potential at any intermediate state is

$A^{a}(\alpha)=\left(1-\alpha\right)\epsilon^{ab}\partial_{b}\overline{\zeta},$
(20)

where (19) has been used, with the $\overline{\eta}$-term discarded as a gauge
transformation which will have no effect on the result. The total measure of
the fermions will be a product of the $1+1$-dimensional measures of all
patches $\chi$ and all positions in the direction tangent to the Fermi surface
$x_{\parallel}\equiv x\sin\chi-y\cos\chi$. One can thus apply the standard
1+1-dimensional Fujikawa formula[15] to the each patch and each tangential
position, and at each infinitesimal stage of the transformation

$\displaystyle
d\overline{\Gamma}\left(x_{\parallel},\chi\right)d\Gamma\left(x_{\parallel},\chi\right)=$
$\displaystyle\exp\left[i\int\dfrac{dtdx_{\perp}}{2\pi
v_{F}(\chi)}\int^{1}_{0}d\alpha\overline{\zeta}\left(x_{\perp},x_{\parallel},t,\chi\right)\epsilon^{ab}\partial_{a}A_{b}(\alpha)\right]$
$\displaystyle\times
d\overline{\Psi}\left(x_{\parallel},\chi\right)d{\Psi}\left(x_{\parallel},\chi\right)$
$\displaystyle=$ $\displaystyle\exp\left[\dfrac{i}{4\pi}\int\dfrac{dt\text{
}dx_{\perp}}{v_{F}(\chi)}\tilde{\zeta}\left(t,x_{\perp},x_{\parallel},\chi\right)\partial_{a}\partial^{a}\tilde{\zeta}\left(t,x_{\perp},x_{\parallel},\chi\right)\right]$
$\displaystyle\times
d\overline{\Psi}\left(x_{\parallel},\chi\right)d{\Psi}\left(x_{\parallel},\chi\right).$
(21)

The total transformation of the measure
$\mathcal{D}\Gamma\mathcal{D}\overline{\Gamma}$ will then be a product over
patches and tangential positions. Because of subtleties of the double scaling
limit, computing the product requires lattice regularization, so $\chi$ is
replaced with some discrete $\chi_{n}$ and $x_{\parallel}$ with
$x_{\parallel,m}$. The resulting transformation is

$\displaystyle\mathcal{D}\overline{\Gamma}\mathcal{D}\Gamma$
$\displaystyle=\prod_{n,m}d\overline{\Gamma}\left(x_{\parallel,m},\chi_{n}\right)d\Gamma\left(x_{\parallel_{m}},\chi_{n}\right)$
$\displaystyle=\mathcal{D}\overline{\Psi}\mathcal{D}\Psi\exp\left[\dfrac{i}{4\pi}\sum_{n,m}\int\dfrac{dt\text{
}dx_{\perp}}{v_{F}(\chi)}\tilde{\zeta}\left(t,x_{\perp},x_{\parallel,m},\chi_{n}\right)\partial_{a}\partial^{a}\tilde{\zeta}\left(t,x_{\perp},x_{\parallel,m},\chi_{n}\right)\right].$
(22)

The sum looks worryingly divergent in the continuum limit, owing to the
absence of an infinitesimal measure. However, observe that in the double
scaling limit the Fermi momentum is also “divergent” compared to any momentum
$k$, and yet it is a parameter of the theory. It turns out that the divergence
of the sum is precisely of the same order as the divergence of the Fermi
momentum. The double scaling limit necessitates that the discretization of the
tangential position is inversely related to the upper momentum cutoff—which,
in turn, is proportional to the width of the patch (i.e., the discretization
of $\chi$) via eq. (7):

$\delta x_{\parallel}=\dfrac{2\pi}{\Lambda}=\dfrac{2\pi}{k_{F}\delta\chi},$
(23)

in other words, the missing integration measure $\delta\chi\delta
x_{\parallel}$ is simply given by the Fermi wavelength:

$\delta x_{\parallel}\delta\chi=\dfrac{2\pi}{k_{F}},$ (24)

and plugging this into (4) gives

$\displaystyle\mathcal{D}\overline{\Gamma}\mathcal{D}\Gamma$
$\displaystyle=\mathcal{D}\overline{\Psi}\mathcal{D}\Psi\exp\left[i\dfrac{1}{8\pi^{2}}\sum_{n,m}k_{F}\left(\chi_{n}\right)\delta\chi\delta
x_{\perp}\int\dfrac{dt\text{
}dx_{\perp}}{v_{F}\left(\chi_{n}\right)}\tilde{\zeta}\left(t,x_{\perp},x_{\parallel,m},\chi_{n}\right)\partial_{a}\partial^{a}\tilde{\zeta}\left(t,x_{\perp},x_{\parallel,m},\chi_{n}\right)\right]$
$\displaystyle\overset{\text{continuum}}{\rightarrow}\mathcal{D}\overline{\Psi}\mathcal{D}\Psi\exp\left[i\int^{\pi}_{0}d\chi\dfrac{k_{F}(\chi)}{8\pi^{2}}\int\dfrac{dt\text{
}d^{2}x}{v_{F}(\chi)}\tilde{\zeta}\left(t,\boldsymbol{x},\chi\right)\partial_{a}\partial^{a}\tilde{\zeta}\left(t,\boldsymbol{x},\chi_{n}\right)\right]$
(25)

$\tilde{\zeta}$ is a nonlocal functional of $\phi$, defined by (19)—or,
equivalently, by its curl

$\partial_{a}\partial^{a}\overline{\zeta}=-gv_{F}(\chi)\partial_{\perp}\phi.$
(26)

This can be used to simplify the factor of
$\partial_{a}\partial^{a}\overline{\zeta}$ in the transformation of the
measure, so the resulting path integral is

$Z=\int\mathcal{D}\phi\mathcal{D}\overline{\Gamma}\mathcal{D}{\Gamma}\exp\left[ig\int^{\pi}_{0}d\chi\dfrac{k_{F}(\chi)}{8\pi^{2}}\int{dt\text{
}d^{2}x}\tilde{\zeta}\left(t,\boldsymbol{x},\chi\right)\partial_{\perp}\phi\left(t,\boldsymbol{x}\right)+iS_{b}\left[\phi\right]+iS_{f}\left[\Gamma,\overline{\Gamma}\right]\right],$
(27)

with $S_{f}$ the free fermion action of $\Gamma$, completely decoupled from
$\phi$. The direct dependence on $\overline{\zeta}$ of the term that
originated from the anomaly means that this action still depends on $\phi$
nonlocally. Of course, this dependence is identical to the one in the Hertz-
Millis action obtained by the method of Section 3. This nonlocality of the
effective action can be traced back to the fact that $\overline{\zeta}$ is not
an integration variable, but a classical field constrained to solve the
classical field equation (26). While the classical field equation is local,
its solution depends nonlocally on $\phi$. However, the classical field
equation minimizes a quadratic action, and it is well-known that integrating
out a field with a quadratic equation (e.g. the electromagnetic field in QED)
is equivalent to simply plugging a classical solution into the path integral.
The implication is that one can rewrite the nonlocal term as a local term with
an additional integration variable $\zeta\left(\boldsymbol{x},t,\chi\right)$:

$\displaystyle\exp\left[ig\int^{\pi}_{0}d\chi\dfrac{k_{F}(\chi)}{8\pi^{2}}\int{dt\text{
}d^{2}x}\tilde{\zeta}\left(t,\boldsymbol{x},\chi\right)\partial_{\perp}\phi\right]=\int\mathcal{D}\zeta\exp$
$\displaystyle\left[i\int^{\pi}_{0}d\chi\dfrac{k_{F}(\chi)}{16\pi^{2}v_{F}(\chi)}\int{dt\text{
}d^{2}x}\left(\partial_{a}\zeta\partial^{a}\zeta\right.\right.$
$\displaystyle\left.+2gv_{F}(\chi){\zeta}\left(t,\boldsymbol{x},\chi\right)\partial_{\perp}\phi\left(t,\boldsymbol{x}\right)\right)\bigg{]}.$
(28)

All in all, integrating out the decoupled fermions $\Gamma$, one finds that a
path integral of an infinite collection of bosonic phase fields, coupled to
the critical order parameter via a quadratic action

$\displaystyle
S[\zeta,\phi]=\dfrac{1}{16\pi^{2}}\int^{\pi}_{0}d\chi\dfrac{k_{F}(\chi)}{v_{F}(\chi)}\int
dt\text{ }d^{2}x$
$\displaystyle\left[\partial_{a}\zeta\left(t,\boldsymbol{x},\chi\right)\partial^{a}\zeta\left(t,\boldsymbol{x},\chi\right)\right.$
$\displaystyle\left.+2gv_{F}(\chi)\phi(t,\boldsymbol{x})\partial_{\perp}\zeta\left(t,\boldsymbol{x},\chi\right)\right]+S_{b}[\phi].$
(29)

This is nothing but a multi-dimensional bosonized action for the critical
Fermi surface—derived directly and exactly as a low energy limit of the
microscopic action (1).

##  5 Exact Boson Green’s Function

The bosonized action (4) is quadratic, and so the correlation functions can be
easily computed. Because of the quadraticity of the action, one might assume
that it leads to no nontrivial correlators. However, as shown in [12], this
action reproduces the non-Fermi liquid Green’s functions computed within the
large-$N$ approximation.

The nontriviality of the results derived within an innocuous-looking quadratic
action owes to the abundance of poles of the Fermi surface, as alluded in the
introduction. This abundance of poles implies, in particular, that an incoming
boson will always resonate with some patch of the Fermi-surface, regardless of
its specific energy and momentum (but provided that they are within the
electron-hole continuum)—thus, there is always some on-shell state to which
the boson can decay. This is clearly an effect of the curvature of the Fermi
surface, for if all patches moved in the same direction there would be no
multitude of resonances.

The above is the source of Landau damping of the boson, and is seen clearly in
the explicit calculation of the boson’s Green’s function. The basic
calculation from the bosonized theory was laid out in [12] and is
recapitulated in [13], but we would like to stress an interesting result
regarding the analytic behavior of the boson self-energy that was not
discussed in these references, so we will briefly outline the calculation.

The basic process that contributes to the boson’s Green function is the
following: the boson can emit an electron-hole pair at each patch, then
reabsorb it. This process gives an energy and momentum-depedent number, and
can happen any number of times, resulting in a geometric series for the
boson’s Green’s function (see figure 3). The number is just the self energy,
which is proportional to the electron-hole field’s (i.e., $\zeta$’s) Green’s
function and to the squared (momentum-dependent) coupling, averaged over all
patches:

$\displaystyle\Pi(\omega,\boldsymbol{k})=\dfrac{g^{2}}{4\pi^{2}}\int^{\pi}_{0}d\chi\dfrac{k_{F}(\chi)}{v_{F}(\chi)}\dfrac{v_{F}^{2}k^{2}\cos^{2}(\theta-\chi)}{\omega^{2}-v^{2}_{F}(\chi){k^{2}}\cos^{2}\left(\chi-\theta\right)+i\epsilon}.$
(30)

Figure 3: A boson (wavy line) can emit a density wave (dashed line) and re-
absorb it any number of times. The amplitude for each such event is some
number, and the sum of all such amplitudes is a geometric series, and the
amplitude of each emission and reabsorption process if the self-energy.

where $\theta$ is the polar angle of momentum,
$\boldsymbol{k}=\left(k\cos\theta,k\sin\theta\right)$. This is most easily
calculated in the case of a Fermi sphere, in which $k_{F}$ and $v_{F}$ are
independent of $\chi$. In this case, there is a symmetry under simultaneous
rotation of $\theta$ and $\chi$, and the result could be computed without loss
of generality by choosing coordinates in which $\theta=0$.

When $\omega<v_{F}k$, the boson’s phase velocity is smaller than the Fermi
velocity, and there is always some resonant patch satisfying $\omega=\pm
v_{F}k\cos\chi^{\star}$, leading to a pole in the denominator of (30). We will
call this case the “subsonic regime.” The result would be then a sum of the
principal value and a contribution from the pole

$\displaystyle\Pi(\omega,\boldsymbol{k})=$
$\displaystyle\dfrac{k_{F}g^{2}}{4\pi^{2}v_{F}}\mathcal{P}\int^{\pi}_{0}d\chi\dfrac{\cos^{2}\chi}{\dfrac{{\omega}^{2}}{{v_{F}^{2}k}^{2}}-\cos^{2}\chi}$
$\displaystyle+i\dfrac{k_{F}g^{2}}{\pi
v_{F}}\int^{\pi}_{0}d\chi\cos^{2}\chi\delta\left(\dfrac{{\omega}^{2}}{{v_{F}^{2}k}^{2}}-\cos^{2}\chi\right).$
(31)

The principal value in the subsonic regime is found to simply be a constant by
partial fraction expansion:

$\dfrac{k_{F}g^{2}}{4\pi^{2}v_{F}}\mathcal{P}\int^{\pi}_{0}d\chi\dfrac{\cos^{2}\chi}{\dfrac{{\omega}^{2}}{{v_{F}^{2}k}^{2}}-\cos^{2}\chi}=-\dfrac{k_{F}g^{2}}{4\pi
v_{F}}.$ (32)

The result is thus solely a (negative) contribution to the squared mass of the
boson:

$m^{2}=m_{0}^{2}-\dfrac{k_{F}g^{2}}{4\pi v_{F}}.$ (33)

which can be cancelled by renormalizing the boson’s bare mass

$m_{0}^{2}=\dfrac{k_{F}g^{2}}{4\pi v_{F}}.$ (34)

The lesson is that the critical point should be defined in a way that takes
the interaction with the Fermi surface into account. Ignoring the mass
renormalization, one is thus left with the contribution from the pole, leading
to

$\Pi\left(\omega,\boldsymbol{k}\right)=-i\dfrac{k_{F}g^{2}}{4\pi
v_{F}}\left({\dfrac{v_{F}^{2}k^{2}}{\omega^{2}}-1}\right)^{-\frac{1}{2}},$
(35)

so the interesting contribution comes from a single resonant patch, which
always exists in the subsonic regime—as advertized. This is a standard result
also found in e.g. [12] and [13]. It is plotted in figure 4. Note that the
result only depends on the ratio of $\omega$ and $k$—the only energy scale is
the momentum. The low-energy and low-momentum limit thus do not commute—the
low energy limit is the high momentum limit. Of course, the entire validity of
the result depends on the momentum’s being much lower than $k_{F}$, but both
the “low momentum” and “high momentum” limits as used in this context are
consistent with this requirement. In the low energy limit, the result
coincides with the famous
$\frac{\left|\omega\right|}{\left|\boldsymbol{k}\right|}$ behavior of the
boson’s self energy, obtained by large-$N$ methods in [2][4].

Figure 4: In the case of a Fermi sphere, the boson’s self energy starts out
linear in $\frac{\left|\omega\right|}{\left|\boldsymbol{k}\right|}$. It has a
pole at $\omega=v_{F}k$, below which it is purely imaginary and above which it
is purely real. It then asymptotes to a finite “mass” for small momenta/high
energies.

The formal result (35) also turns out to analytically continue to the
supersonic regime, $\omega>v_{F}k$—except that since (35) turns from purely
imaginary to purely real, its interpretation changes from pure decay into a
pure correction to the dispersion. This dramatic change is to be expected
because there are no electron-hole pairs to which the boson can decay in the
supersonic regime—moving faster than the speed of the Fermi surface, it
behaves like a shock wave that cannot be scattered by the medium. In the “low
momentum” limit (or “high energy” limit, which can actually involve
arbitrarily small $\omega$!), the correction to the dispersion just adds a
mass, which is nothing but the “Schwinger mass” calculated exactly in [7] for
the critical Fermi surface in the same low-momentum limit using the chiral
anomaly. Again, the analogy with Schwinger’s exact solution of 1+1-dimensional
QED is striking[8].

When the boson moves at exactly the Fermi velocity, $\omega=v_{F}k$, there is
a pole in the self-energy. This pole is analogous to the ${\left(\omega-
v_{F}k\right)^{-\alpha}}$ pole found in the one-dimensional Luttinger liquid,
and can be viewed as a resonance between the boson and the Fermi surface’s
density waves. However, the behavior of the pole becomes more surprising in
the non-isotropic case, and this is the novel result that was mentioned above.

In the non-isotropic case, the principal value is no longer a mere constant,
but the imaginary part of the self-energy can still be easily calculated:

$\displaystyle\Im\Pi(\omega,\boldsymbol{k})$
$\displaystyle=-\dfrac{g^{2}}{\pi}\int^{\pi}_{0}d\chi\dfrac{k_{F}(\chi)}{v_{F}(\chi)}\delta\left(1-v_{F}(\chi)^{2}\frac{k^{2}}{\omega^{2}}\cos^{2}\left(\theta-\chi\right)\right)$
$\displaystyle=-\dfrac{g^{2}}{4\pi}\dfrac{k_{F}\left(\chi^{\star}\right)}{v_{F}\left(\chi^{\star}\right)}{\left|\sqrt{v_{F}^{2}\left(\chi^{\star}\right)\dfrac{k^{2}}{\omega^{2}}-1}-\left|\dfrac{{v^{\prime}_{F}}\left(\chi^{\star}\right)}{v_{F}\left(\chi^{\star}\right)}\right|\right|}^{-1},$
(36)

where $\chi^{\star}$ is now defined implicitly as the solution of the
resonance equation

$v_{F}\left(\chi^{\star}\right)\cos\left(\theta-\chi^{\star}\right)=\pm\dfrac{\left|\omega\right|}{\left|\boldsymbol{k}\right|}.$
(37)

In general, there is no pole when the boson moves at the Fermi velocity of the
patch, $\omega=v_{F}\left(\chi^{\star}\right)k$. Nevertheless, for any given
momentum, (5) does diverge at some subsonic frequency, i.e., lower than
$v_{F}\left(\chi^{\star}\right)k$. Observe that the denominator vanishes if

$\sqrt{1-\dfrac{\omega^{2}}{v_{F}^{2}\left(\chi^{\star}\right)k^{2}}}+\dfrac{{v^{\prime}_{F}}\left(\chi^{\star}\right)}{v_{F}\left(\chi^{\star}\right)}\dfrac{\left|\omega\right|}{v_{F}\left(\chi^{\star}\right)\left|\boldsymbol{k}\right|}=0.$
(38)

Note that the resonance condition (37) has a solution for any (subsonic)
combination of $\omega$, $\left|\boldsymbol{k}\right|$ and $\chi^{\star}$,
provided that one sets $\theta$ appropriately—thus, one can regard
$\chi^{\star}$ as some fixed, given constant. For given $\chi^{\star}$,
equation (38) describes the intersection between a quarter circle and a
straight line:

$\displaystyle\sqrt{1-x^{2}}+\dfrac{{v^{\prime}_{F}}}{v_{F}}x=0$
$\displaystyle\text{where }\text{ }0\leq x\leq 1,$ (39)

which has a unique solution:

$x=\dfrac{1}{\sqrt{1+\frac{v^{\prime 2}_{F}}{v_{F}^{2}}}}$ (40)

(see figure 5). This solution depends on the direction of the incoming boson’s
momentum through $\chi^{\star}$, so rather than a single pole, one finds
different poles for bosons moving in different directions.

Figure 5: Any positive line passing through the origin intersects with a
quarter circle centered around the origin. Hence, the self-energy (5) has a
pole for any $\chi^{\star}$.

While some pole has survived the angular dependence, this pole is not quite
the pole obtained for a Fermi sphere. In the spherically symmetric case, the
self-energy diverges as

$\Pi(\epsilon)\sim\dfrac{k_{F}g^{2}}{v_{F}}{\dfrac{1}{\sqrt{2\epsilon}}}$ (41)

where $\epsilon=\frac{\omega-v_{F}k}{v_{F}k}$. In equation (5), by contrast,
$\sqrt{\frac{v_{F}^{2}k^{2}}{\omega^{2}}-1}$ is analytic at the pole, and so
the self-energy diverges linearly:

$\Pi(\epsilon)\sim-\dfrac{ik_{F}g^{2}\left|v^{\prime}_{F}\right|}{\left(1+\frac{v^{\prime
2}_{F}}{v_{F}^{2}}\right)^{\frac{3}{2}}v_{F}^{2}}{\dfrac{1}{\left|\epsilon\right|}}$
(42)

where $\epsilon=\frac{\omega-\omega_{\text{pole}}}{\omega_{\text{pole}}}$.

The pole in the self-energy has thus been shifted to subsonic frequencies.
However, $\omega=v_{F}\left(\chi^{\star}\right)k$ remains the threshold above
which no on-shell density waves can be excited (again, treating $\chi^{\star}$
as a fixed constant by choosing $\theta$ appropriately)—above it, the self-
energy is purely real. Therefore, rather than diverging, the self-energy jumps
at $\omega=v_{F}k$—its imaginary part jumps from some finite value down to
zero. To our knowledge, these results regarding the pole and the jump in the
self-energy are new.

In a future publication, we will provide a simple explanation of the full
functional dependence of the boson’s self energy based on fundamental physical
principles.

##  6 Exact Fermion Green’s Function

###  6.1 General Green’s Function

We now turn to the evaluation fermion Green’s function. Lawler et al.
calculated it for equal times and equal positions [12], assuming a quadrupolar
Landau interaction between fermions at different patches (we are assuming no
such coupling). However, the full Green’s function has an interesting scale
invariance missing from the equal-time and equal-position results, and is
necessary in order to reproduce the standard $\omega^{2/3}$ self-energy, so we
will discuss it at length.

In light of the derivation in Section 4 in which $\zeta$ arises as a phase
field, and by analogy with the one-dimensional Luttinger liquid, one can
identify the electron’s field with an exponential of the chiral mode of the
phase. This can be derived more rigorously by inserting powers of the electron
field $\psi$ to the path integral, and repeating the derivation of Section 4
(i.e., shifting the chiral phase of the Dirac spinor $\Psi$ to cancel the
interaction term and integrating the fermion field out). Note also that if the
phase is a generator of shifts of the charge, then its exponential would shift
the charge by one unit.

The outgoing mode of the phase-field is obtained in the path integral
formalism by projecting out the incoming mode:

$\zeta_{+}(\boldsymbol{k},\omega,\chi)=\left(1-\dfrac{\omega}{v_{F}k_{\perp}}\right)\zeta(\boldsymbol{k},\omega,\chi).$
(43)

In terms of it, the fermion field is given by

$\psi\left(\boldsymbol{x},t,\chi\right)=\exp\left[i\zeta_{+}\left(\boldsymbol{x},t,\chi\right)\right],$
(44)

Using the rules of Gaussian integration, the fermion propagator is thus given
by

$\displaystyle G_{+}(t,x)$
$\displaystyle=\left<\exp\left[i\zeta_{+}\left(\boldsymbol{x},t,\chi\right)\right]\exp\left[-i\zeta_{+}\left(\boldsymbol{0},0,\chi\right)\right]\right>$
$\displaystyle=\exp\left[\left<\zeta_{+}\left(\boldsymbol{x},t,\chi\right)\zeta_{+}\left(\boldsymbol{0},0,\chi\right)\right>\right].$
(45)

The electron-hole pair propagator is given by the sum of the free propagator
and an interacting correction related to the self energy. The reason is that
an electron-hole pair can either emit or not emit a boson, and the
contribution to the propagator in the latter case is nothing but a free
propgagtor. Hence

$\left<\zeta_{+}\left(\boldsymbol{x},t,\chi\right)\zeta_{+}\left(\boldsymbol{0},0,\chi\right)\right>\equiv\Delta_{+}(\boldsymbol{x},t)=\Delta^{(0)}_{+}(\boldsymbol{x},t)+\Delta^{\prime}_{+}(\boldsymbol{x},t),$
(46)

so the Fermion propagator (6.1) is a product

$G_{+}(\boldsymbol{x},t)=G^{(0)}_{+}(\boldsymbol{x},t)G^{\prime}_{+}(\boldsymbol{x},t).$
(47)

Unsurprisingly, $G^{(0)}_{+}(\boldsymbol{x},t)$ is essentially the propagator
of a free, 1+1D massless fermion. However, some subtleties arise in its
derivation due to the highly singular nature of tangential (non-)dispersion.
Since this is a rather subtle discussion, and since the free fermion problem
is more standard and less interesting than the critical Fermi surface, we
leave the derivation to Appendix A. Here, we just state the final answer:

$\displaystyle
G^{(0)}_{+}\left(\boldsymbol{x},t\right)=\delta\left(x_{\parallel}\right)\dfrac{1}{x_{\perp}-v_{F}t+i\epsilon\text{sign}\left(x_{\perp}\right)}.$
(48)

Figure 6: The interacting correction to the density wave propagator is as
follows: A density wave can scatter into a boson, which can do whatever it can
do, but in the end has to scatter back into a density wave.

The interacting correction to the electron-hole propagator
$\Delta^{\prime}\left(\boldsymbol{x},t\right)$ is just the amplitude that the
electron-hole pair scatter into a dressed boson (i.e., one that emit an
electron-hole pair any number of times) and that the boson then scatters again
into an electron-hole pair, as shown in figure 6. This amplitude is
proportional the product of two chiral density wave propagators and the full
boson propagator

$\displaystyle\Delta^{\prime}_{+}(\omega,\boldsymbol{k})$
$\displaystyle=\dfrac{g^{2}k_{F}^{2}}{16\pi^{2}}k_{\perp}^{2}\Delta^{(0)}_{+}(\omega,\boldsymbol{k})D(\omega,\boldsymbol{k})\Delta^{(0)}_{+}(\omega,\boldsymbol{k})$
$\displaystyle\approxeq-\dfrac{4\pi^{2}g^{2}}{\left(\omega-
v_{F}k_{\perp}\right)^{2}}\dfrac{1}{k^{2}_{\parallel}+i\dfrac{g^{2}k_{F}}{4\pi
v_{F}}\dfrac{\left|\omega\right|}{\left|k_{\parallel}\right|}}.$ (49)

Here, we have taken a low energy limit: we have thus neglected the
$\omega^{2}$ term in the boson’s propagator, which is subdominant at low
energies compared to the
$\frac{\left|\omega\right|}{\left|\boldsymbol{k}\right|}$ term. We have also
approximated $\left|\boldsymbol{k}\right|$ as $\left|k_{\parallel}\right|$,
which is exact at low energy (where tangential scattering dominates, since it
costs zero energy).

Equation (6.1) is a product of a term with no $k_{\parallel}$-dependence and a
term with no $k_{\perp}$-dependence. Transforming into position space is thus
rather simple: the 2D spatial Fourier transform factorizes into a product of
two 1D Fourier transforms. The normal Fourier transform in particular is
straightfowrad, and it is similar to the Fourier transform of a free
particle’s Green’s function in classical mechanics

$\int^{\infty}_{-\infty}\dfrac{dk_{\perp}}{2\pi}\dfrac{e^{-ik_{\perp}x_{\perp}}}{\left(\omega-
v_{F}k_{\perp}\right)^{2}}=-\dfrac{1}{2v_{F}^{2}}\left|x_{\perp}\right|\exp\left(\dfrac{i}{v_{F}}\omega
x_{\perp}\right),$ (50)

so the only dependence on $\omega$ is an operator that shifts time by
$x_{\perp}/v_{F}$—which is to be expected since the electrons move outward at
the Fermi velocity.

The tangential Fourier transform looks slightly more threatening. However,
because of the $\delta\left(x_{\parallel}\right)$ from the free propagator,
the only contribution to the fermion propagator will be from
$x_{\parallel}=0$. The $x_{\parallel}=0$ Fourier transform can be computed by
dimensional analysis, owing to the scale invariance of the boson propagator:

$\displaystyle\int^{\infty}_{-\infty}\dfrac{dk_{\parallel}}{2\pi}\dfrac{1}{k_{\parallel}^{2}+i\dfrac{g^{2}k_{F}}{4\pi
v_{F}}\frac{\left|\omega\right|}{\left|k_{\parallel}\right|}}$
$\displaystyle=\left(\dfrac{g^{2}k_{F}}{4\pi
v_{F}}\right)^{-1/3}\left|\omega\right|^{-1/3}\int^{\infty}_{-\infty}\dfrac{ds}{2\pi}\dfrac{1}{s^{2}+\frac{i}{\left|s\right|}}$
$\displaystyle=\dfrac{2}{3\sqrt{3}}\left(\dfrac{g^{2}k_{F}}{\pi
v_{F}}\right)^{-1/3}e^{-i\pi/6}\left|\omega\right|^{-1/3},$ (51)

and so the only dependence on $\omega$ is a power law which will Fourier
transform into another power law. This power law would diverge at small
comoving distance $x_{\perp}-v_{F}t$. Consequently, the pole of (48) at
$x_{\perp}=v_{F}t$ is wiped out—yet it is precisely the residue from this pole
that determines the free off-shell propagator. However, the divergence of the
Fourier transform owes to the contribution from indefinitely high frequencies,
and by setting an appropriate high frequency cutoff $\Gamma$ no divergence
arises. With the cutoff, one obtains

$\displaystyle\Delta^{\prime}_{+}\left(x_{\perp},x_{\parallel}=0,t\right)$
$\displaystyle=-\beta
e^{-i\pi/6}\left|x_{\perp}\right|\int^{\Gamma}_{-\Gamma}\dfrac{d\omega}{2\pi}\left|\omega\right|^{-1/3}\exp\left[\dfrac{i}{v_{F}}\omega\left(x_{\perp}-v_{F}t\right)\right]$
$\displaystyle=-\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}H\left(\Gamma\left|x_{\perp}-v_{F}t\right|\right),$
(52)

where

$\beta=\Gamma^{2/3}\dfrac{16\pi^{2}}{3\sqrt{3}}\left(\dfrac{\pi
g^{4}}{k_{F}v_{F}^{4}}\right)^{1/3},$ (53)

and

$\displaystyle H(z)$ $\displaystyle=\dfrac{1}{\pi}\int^{1}_{0}ds\text{
}s^{-1/3}\cos\left(zs\right)$
$\displaystyle=\dfrac{3}{2\pi}\prescript{}{1}{F}_{2}\left(\dfrac{1}{3};\dfrac{1}{2},\dfrac{4}{3};-\dfrac{1}{4}z^{2}\right),$
(54)

with $F$ the generalized hypergeometric function. The important properties of
the complicated-looking $H$ function are its two extreme limits: At comoving
distances much smaller than the cutoff, it is just a finite constant
insensitive to the distance

$H(0)=\dfrac{1}{\pi}\int^{1}_{0}ds\text{ }s^{-1/3}=\dfrac{3}{2\pi}$ (55)

which will govern the residue from the $x_{\perp}=v_{F}t$ pole. At comoving
distances much larger than the cutoff, the result is a power law insensitive
to the cutoff

$\displaystyle H(z)$
$\displaystyle\overset{z\rightarrow\infty}{\sim}\dfrac{1}{\pi}\int^{\infty}_{0}ds\text{
}s^{-1/3}\cos\left(zs\right)$
$\displaystyle=\dfrac{\Gamma\left(2/3\right)}{2\pi}\left|z\right|^{-2/3}$ (56)

which will govern the behavior of the propagator far from the pole.

All in all, one finds

$\displaystyle G_{+}(\boldsymbol{x},t)=$ $\displaystyle
G^{(0)}_{+}(\boldsymbol{x},t)\exp\left(\Delta^{\prime}_{+}\left(x_{\perp},x_{\parallel}=0,t\right)\right)$
$\displaystyle=$
$\displaystyle\dfrac{\delta\left(x_{\parallel}\right)}{x_{\perp}-v_{F}t+i\epsilon\text{sign}\left(x_{\perp}\right)}\exp\left[-\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}H\left(\Gamma\left|x_{\perp}-v_{F}t\right|\right)\right]$
$\displaystyle=$
$\displaystyle-i\pi\text{sign}\left(x_{\perp}\right)\delta\left(x_{\parallel}\right)\delta\left(x_{\perp}-v_{F}t\right)\exp\left(-\frac{3}{2\pi}\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}\right)$
$\displaystyle+\mathcal{P}\dfrac{\delta\left(x_{\parallel}\right)}{x_{\perp}-v_{F}t}\exp\left[-\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}H\left(\Gamma\left|x_{\perp}-v_{F}t\right|\right)\right].$
(57)

Equation (6.1) is the exact fermion Green’s function. While it is valid under
the assumption that the fermion emits bosons with $\omega\ll v_{F}k$, the
propagator’s Fourier transform is clearly no longer a function of
$\frac{\omega}{v_{F}k}$ alone, and so $v_{F}k$ is no longer the only energy
scale—there will be a variety of different behaviors at different energies.
The celebrated $\omega^{2/3}$ self-energy is obtained in the special limit of
low energies, in which the Green’s function simplifies considerably.

###  6.2 Long Comoving Distance Green’s Function

The low energy limit is related, but not equivalent, to the asymptotic
behavior of the real-space propagator far from its pole,
$x-v_{F}t\rightarrow\infty$. The difference between the two limits is that the
delta-singularity contributes to all energy scales indiscriminately.
Nevertheless, the limit of large $x-v_{F}t$ already illustrates many of the
essential properties of the less obvious low energy limit, and is interesting
in its own right. Hence, we will start by discussing it.

For large $x-v_{F}t$, one may ignore the cutoff and the pole, and the Green’s
function can be written as

$\displaystyle
G_{+}(\boldsymbol{x},t)\approxeq\dfrac{\delta\left(x_{\parallel}\right)}{x_{\perp}-v_{F}t}\exp\left(-\gamma
e^{-i\pi/6}\dfrac{\left|x_{\perp}\right|}{\left|x_{\perp}-v_{F}t\right|^{2/3}}\right),$
(58)

where we have defined

$\gamma=\dfrac{4\Gamma(2/3)}{3\sqrt{3}}\left(\dfrac{\pi^{4}g^{4}}{k_{F}v_{F}^{2}}\right)^{1/3}.$
(59)

The asymptotic Green’s function (58) is consistent with the equal-time result
obtained by Lawler et al. in [12], which is of the form
$\frac{1}{|x|}\exp\left(-C\left|x\right|^{1/3}\right)$ but not with the equal-
position result, which is
$\frac{1}{\left|t\right|}\exp\left(-D\left|t\right|^{2/3}\right)$—instead,
their equal position result behaves like our result after a long time and at a
finite but smaller distance, $x\ll v_{F}t$. This may be a result of the fact
they explicitly impose a cutoff on the normal momentum in their calculation of
the Fourier integrals, breaking the scale invariance of the result. In this
case, our result should be valid for positions exceeding the cutoff, but not
at strictly equal positions.

However, at positions exceeding the cutoff and at arbitrary times, the result
reveals a remarkable nature of the Green’s function not apparent from the
equal-position and at equal-time results. It is similar to the diffusive
Green’s function

$G_{\text{diffusion}}(x,t)=\dfrac{1}{\sqrt{4\pi
Dt}}\exp\left(-\dfrac{x^{2}}{4Dt}\right),$ (60)

with the two obvious differences being that in (58) the exponents of $x$ and
$t$ are anomalous, and that the null coordinate $x_{\perp}-v_{F}t$ plays the
role of time. Just like its diffusive counterpart, (58) has scale
invariance—an emergent low-energy scale invariance not apparent in the
microscopic action (18). This scale invariance mixes space and time:

$\displaystyle x_{\perp}$ $\displaystyle\rightarrow\alpha x$ $\displaystyle
x_{\perp}-v_{F}t$
$\displaystyle\rightarrow\alpha^{3/2}\left(x_{\perp}-v_{F}t\right)$
$\displaystyle G_{+}(\boldsymbol{x},t)$
$\displaystyle\rightarrow\alpha^{-3/2}G_{+}(\boldsymbol{x},t).$ (61)

One can easily convert this to momentum space, leading to a scaling of the
frequency $\omega$ and the null momentum $\omega-v_{F}k$:

$\displaystyle\omega-v_{F}k_{\perp}$
$\displaystyle\rightarrow\alpha^{-1}\left(\omega-v_{F}k_{\perp}\right)$
$\displaystyle\omega$ $\displaystyle\rightarrow\alpha^{-3/2}\omega$
$\displaystyle G_{+}(\boldsymbol{k},\omega)$ $\displaystyle\rightarrow\alpha
G_{+}(\boldsymbol{k},\omega).$ (62)

The general form of $G_{+}(\boldsymbol{k},\omega)$ consistent with this
scaling, with parity and with time reversal, is

$G_{+}(\boldsymbol{k},\omega)=\dfrac{A\left(\frac{\text{sign}(\omega)\omega^{2/3}}{\omega-
v_{F}k_{\perp}}\right)}{\omega-
v_{F}k_{\perp}+B\left(\frac{\text{sign}(\omega)\omega^{2/3}}{\omega-
v_{F}k_{\perp}}\right)\text{sign}(\omega)\omega^{2/3}}.$ (63)

The two extreme limits of this expression, are, similarly to the bosonic case,
low energies and high null momenta $\omega-
v_{F}k_{\perp}\gg\left(\frac{g^{4}v_{F}}{k_{F}}\right)^{1/3}\omega^{2/3}$, and
high energies and low momenta $\omega-
v_{F}k_{\perp}\ll\left(\frac{g^{4}v_{F}}{k_{F}}\right)^{1/3}\omega^{2/3}$. If
the coefficients $A$ and $B$ were not singular at zero energy, then the
leading order low energy propagator would be the standard result:

$G_{+}(\boldsymbol{k},\omega)=\dfrac{A_{0}}{\omega-
v_{F}k_{\perp}+B_{0}\text{sign}(\omega)\omega^{2/3}}.$ (64)

However, because of the antisymmetry of (6.3) in $x_{\perp}-v_{F}t$, the zero
frequency Fourier transform actually vanishes:

$G_{+}(\omega=0,\boldsymbol{x})=\int
d^{2}x\dfrac{d\left(x_{\perp}-v_{F}t\right)}{v_{F}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}G_{+}(x_{\perp}-v_{F}t,\boldsymbol{x})=0,$
(65)

and in Appendix B it is shown that at low energies the Fourier transform
scales as $\frac{\omega^{3/2}}{\left(\omega-v_{F}k_{\perp}\right)^{2}}$. This
apparent discrepancy with the standard result arises because the long
(comoving-)distance limit is not equivalent to the low energy limit—the low
energy limit depends crucially on the residue from the $x=v_{F}t$ pole. Such
is the reason why it was necessary to explicitly impose the high-energy cutoff
in Subsection 6.1.

###  6.3 Low Energy Green’s Function

It turns out that the standard $\omega^{2/3}$ result is recovered in the
strongest possible low-energy limit, in which the null momentum
$k_{\perp}-\omega/v_{F}$ is much greater not only than the momentum scale
constructed from the energy $\omega$, but than the one constructed from the
high-energy cutoff $\Gamma$:

$\omega-
v_{F}k_{\perp}\gg\left(\dfrac{g^{4}v_{F}}{k_{F}}\right)^{1/3}\Gamma^{2/3}.$
(66)

While it may appear odd that the high-energy cutoff is to be treated as a
small parameter, observe that it is suppressed by a factor of $k_{F}^{-1/3}$,
and $k_{F}$ is taken to infinity in the double scaling limit. Hence, (66) does
not require that the cutoff $\Gamma$ be small or even finite, but merely that
$k_{F}$ be sufficiently large compared to $\frac{\Gamma^{2}}{\left(\omega-
v_{F}k_{\perp}\right)^{3}}$. Of course, this is perfectly consistent with the
largeness of the cutoff compared to the energy, $\Gamma\gg\omega$—in fact, it
will be shown that the undetermined cutoff $\Gamma$ drops out of the leading
order result.

Recall that the full propagator is

$\displaystyle G_{+}(\boldsymbol{x},t)=$
$\displaystyle-i\pi\text{sign}\left(x_{\perp}\right)\delta\left(x_{\parallel}\right)\delta\left(x_{\perp}-v_{F}t\right)\exp\left(-\frac{3}{2\pi}\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}\right)$
$\displaystyle+\mathcal{P}\dfrac{\delta\left(x_{\parallel}\right)}{x_{\perp}-v_{F}t}\exp\left[-\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}H\left(\Gamma\left|x_{\perp}-v_{F}t\right|\right)\right].$
(67)

In the high-momentum, low energy limit (66), the dominant contribution to the
residue from the pole will be from the jump of the sign function, so one can
approximate

$\exp\left(-\frac{3}{2\pi}\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}\right)\approxeq 1.$ (68)

Corrections will be small since the exponent decays very slowly ($\beta\ll
v_{F}k_{\perp}-\omega$), and it can be readily shown that they scale as
$\frac{\Gamma^{4/3}}{\left(\omega-v_{F}k_{\perp}\right)^{2}}$.

Furthermore, since $\Gamma\gg\omega$, the dominant contribution to the
principal value will be from comoving distances $x-v_{F}t$ far exceeding the
inverse energy cutoff, so one can approximate333Note that it would be going
too far to approximate the exponent as unity, as was done in the case of the
residue from the pole. Doing so would erase all dependence of the principal
value on $x_{\perp}$, thereby turning it into a delta-function of the null
momentum—too crude an approximation if one wishes to understand the energy and
momentum-dependence of the propagator off-shell.

$\exp\left[-\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}H\left(\Gamma\left|x_{\perp}-v_{F}t\right|\right)\right]\approxeq\exp\left(-\gamma
e^{-i\pi/6}\dfrac{\left|x_{\perp}\right|}{\left|x_{\perp}-v_{F}t\right|^{2/3}}\right).$
(69)

Corrections will be from the small interval $-v_{F}\Gamma^{-1}\lesssim
x-v_{F}t\lesssim v_{F}\Gamma^{-1}$, and their effect would likewise be small,
as can be easily assessed using the antisymmetry of the principal value term

$\displaystyle\int^{v_{F}\Gamma^{-1}}_{-v_{F}\Gamma^{-1}}du\text{
}e^{i\frac{\omega}{v_{F}}u}\dfrac{\exp\left[-\beta
e^{-i\pi/6}{\left|x_{\perp}\right|}H\left(\Gamma\left|u\right|\right)\right]}{u}$
$\displaystyle\sim\int^{v_{F}\Gamma^{-1}}_{-v_{F}\Gamma^{-1}}du\dfrac{\sin\left[\frac{\omega}{v_{F}}u\right]}{u}$
$\displaystyle=\int^{\Gamma/v_{F}}_{-\Gamma/v_{F}}du\dfrac{\omega}{v_{F}}+\mathcal{O}\left(\dfrac{\omega^{2}}{\Gamma^{2}}\right)$
$\displaystyle\sim\dfrac{\omega}{\Gamma},$ (70)

where $u\equiv x-v_{F}t$.

The total low-energy propagator is thus identical to the long-distance
propagator (6.3) up to a simple residue from the pole

$\displaystyle G_{+}(\boldsymbol{x},t)\approxeq$
$\displaystyle-i\pi\delta\left(x_{\perp}-v_{F}t\right)\delta\left(x_{\parallel}\right)\text{sign}\left(x_{\perp}\right)$
$\displaystyle+\dfrac{\delta\left(x_{\parallel}\right)}{x_{\perp}-v_{F}t}\exp\left(-\gamma
e^{-i\pi/6}\dfrac{\left|x_{\perp}\right|}{\left|x_{\perp}-v_{F}t\right|^{2/3}}\right).$
(71)

As claimed before and shown in Appendix B, the long-distance propagator
Fourier transforms to

$\dfrac{\delta\left(x_{\parallel}\right)}{x_{\perp}-v_{F}t}\exp\left(-\gamma
e^{-i\pi/6}\dfrac{\left|x_{\perp}\right|}{\left|x_{\perp}-v_{F}t\right|^{2/3}}\right)\rightarrow-B\text{sign}(\omega)\dfrac{\omega^{2/3}}{\left(\omega-
v_{F}k_{\perp}\right)^{2}}+\mathcal{O}\left({\dfrac{\omega^{2}}{k^{4}}}\right),$
(72)

where

$B\equiv\dfrac{4}{\sqrt{3}}e^{-i\pi/3}\left(\dfrac{\pi^{4}g^{4}}{k_{F}v_{F}^{4}}\right).$
(73)

The Fourier transform of the residue from the pole can also be easily
computed, and gives

$-i\pi\delta\left(x_{\perp}-v_{F}t\right)\delta\left(x_{\parallel}\right)\text{sign}\left(x_{\perp}\right)\rightarrow\dfrac{1}{\omega-
v_{F}k_{\perp}}.$ (74)

Combining, one finds

$\displaystyle G_{+}\left(\omega,\boldsymbol{k}\right)$
$\displaystyle=\dfrac{1}{\omega-
v_{F}k_{\perp}}-B\text{sign}(\omega)\dfrac{\omega^{2/3}}{\left(\omega-
v_{F}k_{\perp}\right)^{2}}+\mathcal{O}\left({\dfrac{\omega^{2}}{k^{4}}}\right)$
$\displaystyle=\dfrac{1}{\omega-
v_{F}k_{\perp}+B\text{sign}(\omega)\omega^{2/3}}+\mathcal{O}\left({\dfrac{\omega^{2}}{k^{4}}}\right),$
(75)

which is the standard and celebrated result obtained in the large-$N$
theory[2].

All the important of the large-$N$ theory have thus been reproduced without
any large-$N$ approximation, and without any tangential term in the electron
dispersion. In what follows, we will assess whether ignoring the tangential
term has truly been justified.

##  7 Discussion: Can the Dispersion Really Be Linearized?

###  7.1 Irrelevance of Quadratic Dispersion in Multi-Patch Theories

Presumably, our assertion most prone to the reader’s suspicion is that the the
tangential dispersion is irrelevant at low momenta, as all the rest follows
from it. In particular, one might think that since $k_{\perp}$ and
$k_{\parallel}$ are independent variables, it is conceivable that the
quadratic order of the latter is comparable to the linear order of the former.
In fact, at low energies, it seems like it must be the case, because the
normal momentum will be essentially zero, so all the scattering will be in the
tangential direction.

The crucial point is that since the momentum and the energy of an electron-
hole pair are transferred to it by the incoming boson, the negligibly of the
tangential dispersion is nothing but a condition on the incoming boson’s
state. To explicitly assess whether the condition is reasonable, consider for
simplicity an isotropic dispersion

$\varepsilon\left(\boldsymbol{p}\right)=\varepsilon\left(\left|\boldsymbol{p}\right|\right).$
(76)

Now, using

$\left|\boldsymbol{p}\right|=\sqrt{\left(k_{\perp}+k_{F}\right)^{2}+k_{\parallel}^{2}},$
(77)

one can expand the dispersion while keeping the leading tangential term to
obtain

$\varepsilon(\boldsymbol{k})=v_{F}k_{\perp}+\dfrac{v_{F}}{2k_{F}}k_{\parallel}^{2}+\mathcal{O}\left(k^{3}\right).$
(78)

At low enough energies, one can assume that the entire momentum is tangential,
$k_{\parallel}^{2}\approxeq k^{2}$. The ratio between the tangential term and
the normal term is then the dimensionless parameter

$\eta\equiv\dfrac{k^{2}}{k_{F}k_{\perp}}=\dfrac{1}{\dfrac{k_{F}}{k}\dfrac{\omega}{v_{F}k}-1}.$
(79)

It is seen that $\eta$ is small insofar as

$\dfrac{\omega}{v_{F}k}\gg\dfrac{k}{k_{F}}.$ (80)

This definitely holds when the phase velocity of the boson is at least of the
order of the Fermi velocity, in which case $\frac{\omega}{v_{F}k}$ is not a
small parameter. But it also holds for infinitesimal $\frac{\omega}{v_{F}k}$,
provided only that the other infinitesimal parameter $\frac{k}{k_{F}}$ is even
smaller. This defines a consistent IR limit in which $k\rightarrow 0$ and
$\omega\rightarrow 0$ in such a way that there is the hierarchy of
dimensionless parameters shown in table 1. Differently put, $\omega$ should be
bounded between a linear function of $k$ and a quadratic function of $k$.

1
---
$\downarrow$
$\dfrac{\omega}{v_{F}k}$
$\downarrow$
$\dfrac{k}{k_{F}}$

Table 1: Hierarchy of dimensionless parameters in the limit under which: (1)
normal dispersion dominates; (2) the subsonic low-energy limit of Section 5
holds.

Observe that the statement that normal dispersion dominates over tangential
dispersion does not assume the normal momentum dominates over the tangential
momentum (which is clearly false at low energy). The validity of the limit
depends not on the smallness of $k_{\parallel}$ relative to $k_{\perp}$, but
instead on the smallness of both relative to the Fermi momentum $k_{F}$.

This establishes that the linearized calculation of the boson self-energy is
valid at low energies, as it does not couple different frequencies and
momenta, so one can freely choose $\omega$ and $\boldsymbol{k}$ consistent
with the condition (80). Yet the calculation of the electron self-energy in
Section 6 involved a sum over all energies and all momenta. The validity of
the linearized limit thus depends on whether the dominant contribution to the
sums satisfies it. In [16], it is pointed out that the peak of the boson
propagator is at momentum of the order of
$\left(\frac{g^{2}k_{F}}{v_{F}}\omega\right)^{1/3}$ (see (6.1)), so that at
low energies the typical tangential energy exceeds the typical normal energy,
suggesting that the former dominates.

However, this argument ignores the spread around those typical values—if the
spread is much greater than the peak, then the precise location of the peak is
washed out. For the free fermion propagator, this is certainly the case: it is
just a scale-free power law around $\omega=v_{F}k_{\perp}$, and shifting
$\omega$ down would merely shift the pole in $k_{\perp}$. For the boson
propgator, there is a momentum scale
$\left(\frac{g^{2}k_{F}}{v_{F}}\omega\right)^{1/3}$ characterizing the
sharpness of the peak, and the propagator (viewed as a distribution function)
can be normalized. Shifting $\omega$ down would not merely shift the location
of the peak, but also make it increasingly sharper. This suggests that in
spite of the smaller “typical value” the tangential dispersion, due to its
equally small spread the answer is dominated by regions of phase space where
the normal dispersion dominates, and eq. (80) is satisfied.

The counter-argument might seem suspicious in light of the long, power-law
tail of the boson propagator at large momenta. It turns out to work
nonetheless—the point is that at low energies, the quadratic dispersion $\beta
k^{2}_{\parallel}$ would vary infinitely slowly compared to
$D(\omega,k_{\parallel})$ around the latter’s sharp peak, so the former can
just be approximated as zero. This is confirmed by a calculation. Chubukov et
al. argue [16] that the quadratic dispersion is relevant for the evaluation of
the equal-time and equal position propagators that had been computed by Lawler
et al.[12] However, the low energy limit is defined by the condition $\omega-
v_{F}k_{\perp}\gg\left(\frac{g^{4}v_{F}}{k_{F}}\right)^{1/3}\omega^{2/3}$—or
(aside from the $x_{\perp}=v_{F}t$ pole),
$x_{\perp}\ll\left(\frac{g^{4}v_{F}}{k_{F}}\right)^{1/3}\left(x_{\perp}-v_{F}t\right)^{2/3}$,
while the double scaling limit requires $x_{\perp}\gg k_{F}$. The former is
clearly not satisfied for the equal time propagator, while the latter is
clearly not satisfied for the equal position propagator. Instead, it is the
long-comoving distance and finite $x_{\perp}$ propagator (as well as the
$x_{\perp}=v_{F}t$ pole) that governs the low-energy behavior of
$G(\omega,\boldsymbol{k})$, and leads to the $\omega^{2/3}$ self-energy.

In Appendix C, we add quadratic dispersion in heuristic way suggested by [16],
but focus on the long comoving distance limit rather than the equal time or
equal position limits. We show that corrections to the fermion propagator are
subodminant.

###  7.2 Relevance of $N$th Order Dispersion in Single-Patch Theories

At first glance, the above might seem to contradict the finding by many
authors[4] [17] [18] that under the renormalization group, the quadratic,
tangential term in the dispersion is relevant. The reason why there is no
contradiction is that these authors were considering a different class of
models than our own, in which only a single patch of the Fermi surface, and
possibly its antipode, are taken into account. Within such models, it is
hardly surprising that the quadratic term in the dispersion is relevant—after
all, without it one would be unable to derive the correct Landau
damping444Incidentally, the Landau damping is the only step in the one-loop
calculation that depends on the quadratic dispersion—the one loop electron
self-energy is independent of the dispersion so long as one can derive the
correct Landau damping.. Indeed, the quadratic dispersion is the last trace of
the curvature of the Fermi surface in single patch theories, and discarding it
amounts to ignoring the curvature.

However, in linearized multi-patch theories, the curvature is taken into
account through the fact that different patches move in different directions,
and the correct Landau damping is reproduced. To compare single-patch and
multi-patch theories, one should express them in the same language. This can
be done by rewriting the entire multi-patch action (18) relative to a
specific, fixed patch of $\boldsymbol{p}=\left(k_{F},0\right)$—where recall
that $\boldsymbol{p}$ is the absolute momentum, to be distinguished from the
relative momentum $\boldsymbol{k}$. To this end, define the momentum relative
to a specific patch

$\boldsymbol{q}=\left(p_{x}-k_{F},p_{y}\right).$ (81)

In the continuum limit, the multi-patch model should converge to the
linearized dispersion (again, assuming spherical symmetry for simplicity)

$\displaystyle\varepsilon(\boldsymbol{p})=v_{F}\left(\left|\boldsymbol{p}\right|-k_{F}\right)$
(82)

In terms of the single-patch relative momentum $\boldsymbol{q}$, one can
rewrite this as

$\varepsilon(\boldsymbol{q})=v_{F}\sqrt{\left(q_{x}+k_{F}\right)^{2}+q_{y}^{2}}-v_{F}k_{F}.$
(83)

Note that in spite the formal similarity to (77), (83) has an entirely
different interpretation: $q_{x}$ is and $q_{y}$ are the $x$ and $y$
components of momentum relative to a specific Fermi point with momentum in the
$x$-direction, while $k_{\perp}$ and $k_{\parallel}$ are the components of the
momentum relative to the nearest patch, locally normal and tangential to the
Fermi surface, respectively. The implication is that contrary to (77), at low
energies, there is no reason to suppose that $q_{x}$ and $q_{y}$ are small—a
patch different from the “reference patch” of the theory would have
$\boldsymbol{q}$ of the order of $k_{F}$. Hence, to recover the full multi-
patch theory one should keep track of all orders of $q_{x}$ and $q_{y}$:

$\varepsilon(\boldsymbol{q})=v_{F}\left[q_{x}+\dfrac{q_{y}^{2}}{2k_{F}}-\dfrac{q_{y}^{2}q_{x}}{2k_{F}^{2}}+...\right].$
(84)

While $\boldsymbol{q}$ need not be small, one might nevertheless hope that
similarly to standard relativistic field theory, under the renormalization
group the expansion truncates, and that high order terms are irrelevant at low
energies. Indeed, when keeping only the quadratic term, the one loop results
for the boson and fermion Green’s functions precisely agree with those
obtained in the multi-patch theory[2]. However, higher loop corrections
diverge at low energies in the quadratic single-patch theory[3], whereas in
the multi-patch theory it has been shown that they vanish. We are thus led to
conjecture that while in multi-patch theories the quadratic dispersion is
irrelevant, in single-patch theories one must go beyond quadratic dispersion
to correctly reproduce the higher loop results. Far from ignoring the
curvature of the Fermi surface, the linearized multi-patch theory takes
curvature into account more precisely than quadratic multi-patch theories! The
distinction between multi-patch theories and single-patch theories is
summarized in table 2.

|  Single-patch theories |  Multi-patch theories
---|---|---
Linear dispersion |  Relevant but insufficient at one loop |  Relevant and sufficient at one loop; higher loops vanish
Quadratic dispersion |  Relevant and sufficient at one loop; higher loops diverge |  Irrelevant in the double-scaling limit and under the conditions of Section 7.1
Higher than quadratic dispersion |  Relevant at higher loops |  Irrelevant
Table 2: Relevance of various powers in the dispersion in single-patch
theories and in multi-patch theories.

##  8 Conclusion

We have shown that there is a well-defined low-energy and low-momentum limit
under which the simplest possible critical Fermi surface behaves simply, and
can be solved exactly. The appeal of our approach is in that it is seemingly
fully microscopic, and assumes no structure other than ordinary electrons and
an ordinary bosonic order parameter. Moreover, the absence of any inessential
structure makes it particularly easy to interpret the mechanisms yielding some
otherwise counter-intuitive results. In this paper we have established that
this approach reproduces the essential results for critical Fermi surfaces
obtained in the large-$N$ theory.

Of course, there remains the question of whether our approach assumes enough
structure to capture all the physics of critical Fermi surfaces—or whether it
neglects important effects. One important assumption that we have made is that
the electrons are free save for the interaction with the critical boson. This
does not seem too unreasonable given that the interaction with the boson is
long-ranged and singular, while normal interactions are short-ranged and thus
potentially subdominant. Yet any interaction between electrons would result in
a coupling between patches, thus challenging the picture that the boson only
couples to a single patch. Therefore, it remains an open question whether the
coupling between different patches is relevant. Lawler et al. [12] already
considered the effect of a quadrupolar Landau interaction, but they obtained
results consistent with our own. We have been trying to consider a different
kind of an interaction between patches, and our initial results suggest that
it may lead to new qualitative effects. This is a subject for a future
publication.

In addition to the assumption that electron-hole pairs at different patches
are decoupled, our approach assumes that the electron and the hole always
reside at the patch. This is certainly a valid assumption when the electrons
are only coupled to the critical boson—provided that the critical boson
carries too small a momentum to excite an electron and a hole at two distant
patches. Yet, a simple white noise disorder acts on all momenta
indiscriminately, and thus excites electron-hole pairs at all momenta. The
effect of disorder, or of some other momentum relaxation mechanism, is
essential for understanding the celebrated linear-$T$ resistivity (for without
scattering, electrons accelerate in the presence of a constant electric
field). It it for future research to study whether a “gentler” kind of
disorder which acts on each patch separately and decays at high momenta is
sufficient to understand linear-$T$ resistivity—or whether the results of the
model where electrons cannot hop between different patches can somehow be
extended to the presence of disorder—or whether the model should be
generalized to the case in which electron-hole pairs at two different patches
can be excited (this can be done, e.g., by considering a bosonized field that
depends on two patches $\zeta\left(\chi,\chi^{\prime}\right)$).

We make no claim that the effects of disorder and inter-patch-coupling are not
essential in realistic experiments. Rather, we regard the simplest model as a
ground on which to build in the future—as a zeroth order approximation to be
perturbed. In our view, it is necessary to understand the principles behind
individual effects before one can understand the interplay of effects—and what
is the complex case but an interplay of individually simple effects?

## Acknowledgements

We wish to thank Hart Goldman, Subir Sachdev and Piers Coleman for helpful and
intriguing comments.

##  Appendix A Free Fermion Green’s Function in Bosonized Language

As previous calculation, the calculation of the free fermion Green’s function
in the bosonized language requires that $\chi$ and the tangential position
$x_{\parallel}$ be discretized. The free discretized action of a fermion is
given by

$\displaystyle S_{0}[\zeta]=\dfrac{1}{8\pi}\int dt\text{
}dx_{\perp}\sum_{n,m}\dfrac{1}{v_{F}(\chi_{n})}$
$\displaystyle\left[\partial_{a}\zeta\left(t,x_{\perp},x_{\parallel,m},\chi_{n}\right)\partial^{a}\zeta\left(t,x_{\perp},x_{\parallel,m},\chi_{n}\right)\right],$
(85)

where a factor of $\frac{k_{F}}{2\pi}$ has not disappeared—rather, recall that
the way in which it arose in the double scaling limit is through its relation
to the the integration measure $\delta
x_{\parallel}\delta\chi=\frac{2\pi}{k_{F}}$, and if not for the continuum
limit it would not have appeared in the first place. In this case, the chiral
electron-hole propagator is just a chiral Green’s function of the $1+1$D wave
equation, satisfying

$\dfrac{1}{2\pi}\partial_{\perp}\left(\partial_{t}-v_{F}\left(\chi_{n}\right)\partial_{\perp}\right)\Delta^{(0)}_{n,n,m,m^{\prime}}\left(t,t^{\prime},x_{\perp},x^{\prime}_{\perp}\right)=\delta\left(t-t^{\prime}\right)\left(x_{\perp}-x_{\perp}^{\prime}\right)\delta_{m,m^{\prime}},$
(86)

whose general solution is defined up to a shift by a function of
$x_{\parallel}-x^{\prime}_{\parallel}$, of which the equation is independent

$\Delta^{(0)}_{n,n,m,0}\left(t,0,x_{\perp},0\right)=\log\left[x_{\perp}-v_{F}t+i\epsilon\text{sign}\left(x_{\perp}\right)\right]+C\left(x_{\parallel,m}\right).$
(87)

This shift ambiguity can be traced back to the chiral symmetry of the
classical theory, under which $\zeta$ is shifted by an arbitrary function of
$x_{\parallel}$. This results in a free electron Green’s function defined up
to a factor of $x_{\parallel}$

$\displaystyle G^{(0)}_{+}\left(\boldsymbol{x},t\right)$
$\displaystyle=\exp\left(\Delta^{(0)}_{+}\left(\boldsymbol{x},t\right)\right)$
$\displaystyle=f\left(x_{\parallel}\right)\dfrac{1}{x_{\perp}-v_{F}t+i\epsilon\text{sign}\left(x_{\perp}\right)}.$
(88)

The correct continuum limit is clearly obtained by setting

$\displaystyle G^{(0)}_{+}\left(\boldsymbol{x},t\right)$
$\displaystyle=\delta\left(x_{\parallel}\right)\dfrac{1}{x_{\perp}-v_{F}t+i\epsilon\text{sign}\left(x_{\perp}\right)},$

and, indeed, it clear from the action that variables at different
$x_{\parallel}$ are not correlated.

##  Appendix B Behavior of the Fourier Transform of the Long Comoving Distance
Green’s Function

Let $u\equiv x_{\perp}-v_{F}t$. The Fourier transform of the long comoving
distance Green’s function, (6.3), is given by

$\displaystyle
G_{+}(\omega,\boldsymbol{k})=\int^{\infty}_{-\infty}du\int^{\infty}_{-\infty}dx_{\perp}\dfrac{e^{i\left(\frac{\omega}{v_{F}}-k_{\perp}\right)x_{\perp}-i\frac{\omega}{v_{F}}u}}{u}\exp\left(-\gamma
e^{-i\pi/6}\dfrac{\left|x_{\perp}\right|}{\left|u\right|^{2/3}}\right)$ (89)

The $x_{\perp}$ integral can be easily performed, giving

$G_{+}(\omega,\boldsymbol{k})=\int^{\infty}_{-\infty}du\text{
}{e^{-i\frac{\omega}{v_{F}}u}}\dfrac{2\gamma
e^{-i\pi/6}}{u^{1/3}\left[\left(\omega-
v_{F}k_{\perp}\right)^{2}u^{4/3}+\gamma^{2}e^{-i\pi/3}\right]}.$ (90)

One can then expand (90) in powers of $\omega$, but for constant $\omega-
v_{F}k_{\perp}$ (it is natural to treat $\omega$ and $\omega-v_{F}k_{\perp}$,
rather than $\omega$ and $k$, as independent variables since they do not
intermix under the scaling transformation (6.2)). Clearly, the zeroth order
term vanishes by antisymmetry in $u$. The first order term can be easily shown
to diverge, consistent with an $\omega^{\alpha}$ behavior with some exponent
$1>\alpha>0$, which by the scaling symmetry (6.2) should be given by
$\frac{2}{3}$. Indeed, this can be confirmed by an explicit calculation. The
small-$\omega$ propagator can be found by taking the limit of large $u$ in the
integrand:

$\displaystyle G_{+}(\omega,\boldsymbol{k})$
$\displaystyle\approxeq\int^{\infty}_{-\infty}du\text{
}{e^{-i\frac{\omega}{v_{F}}u}}\dfrac{2\gamma e^{-i\pi/6}}{\left(\omega-
v_{F}k_{\perp}\right)^{2}u^{5/3}}$
$\displaystyle=-B\dfrac{\text{sign}(\omega)\omega^{2/3}}{\left(\omega-
v_{F}k_{\perp}\right)^{2}},$ (91)

where

$B=\dfrac{4}{\sqrt{3}}e^{-i\pi/3}\left(\dfrac{\pi^{4}g^{4}}{k_{F}v_{F}^{4}}\right).$
(92)

##  Appendix C Subdominance of Corrections Due to Quadratic Dipsersion

For simplicity, we focus on $x_{\parallel}=0$, as is done in [16], though the
basic argument can be generalized to $x_{\parallel}\neq 0$. The long comoving-
distance fermion propagator can then be written as

$\displaystyle
G_{+}(x_{\perp},x_{\parallel}=0,t)\propto\dfrac{1}{x_{\perp}-v_{F}t}\exp\left[-Q\left(t,x_{\perp}\right)\right],$
(93)

for some function $Q$. In the linearized theory, we have obtained

$Q\left(t,x_{\perp}\right)=-\gamma
e^{-i\pi/6}\dfrac{\left|x_{\perp}\right|}{\left|x_{\perp}-v_{F}t\right|^{2/3}}.$
(94)

Chubukov et al.[16] suggest heuristically accounting for the effects of
quadratic dispersion simply by adding it to the free electron propagators in
(6.1), i.e., by setting

$Q(\omega,k_{\perp})=-\int^{\infty}_{-\infty}\dfrac{dk_{\parallel}}{2\pi}\dfrac{4\pi^{2}g^{2}}{\left(\omega-
v_{F}k_{\perp}-\beta
k_{\parallel}^{2}\right)^{2}}\dfrac{1}{k^{2}_{\parallel}+i\dfrac{g^{2}k_{F}}{4\pi
v_{F}}\dfrac{\left|\omega\right|}{\left|k_{\parallel}\right|}},$ (95)

with $\beta$ some coefficient which will be left unspecified so as to study
the effects of neglecting it (physically, it scales as $k_{F}^{-1}$). The
Fourier transform of $Q$ with respect to $k_{\perp}$ is the same as in (50)
except with the frequency shifted by $-\beta k_{\parallel}^{2}$, giving

$\displaystyle Q(\omega,x_{\perp})$
$\displaystyle=-\dfrac{2\pi^{2}g^{2}}{v_{F}^{2}}\int^{\infty}_{-\infty}\dfrac{dk_{\parallel}}{2\pi}\dfrac{1}{k^{2}_{\parallel}+i\dfrac{g^{2}k_{F}}{4\pi
v_{F}}\dfrac{\left|\omega\right|}{\left|k_{\parallel}\right|}}\left|x_{\perp}\right|\exp\left[\dfrac{i}{v_{F}}\left(\omega-\beta
k_{\parallel}^{2}\right)x_{\perp}\right]$
$\displaystyle=-\dfrac{2\pi^{2}g^{2}}{v_{F}^{2}}\left(\dfrac{g^{2}k_{F}}{4\pi
v_{F}}\omega\right)^{-1/3}\left|x_{\perp}\right|F\left(i\dfrac{\beta}{v_{F}}\left(\frac{g^{2}k_{F}}{4\pi
v_{F}}\omega\right)^{2/3}x_{\perp}\right)\exp\left(\dfrac{i}{v_{F}}\omega
x_{\perp}\right),$ (96)

where

$F\left(z\right)\equiv\int^{\infty}_{-\infty}\dfrac{ds}{2\pi}\dfrac{1}{s^{2}+\frac{i}{\left|s\right|}}\exp\left(-s^{2}z\right).$
(97)

Since $F$ depends on $\omega^{2/3}$ and $\beta$ solely through their product,
it is immediately clear that the small frequency limit is equivalent to the
small $\beta$ limit555It might seem inconsistent to neglect $\omega^{2/3}$ in
the argument of $F$ while keeping the factor of $\omega x_{\perp}$ in the
exponent. However, the sole effect of the latter is to replace time with
comoving distance. It is certainly “legal” to neglect this latter factor,
which amounts to replacing $\omega-v_{F}k_{\perp}$ with $-v_{F}k_{\perp}$ and
will have no effect on the $\omega^{2/3}$ self-energy.:

$Q(\omega,x_{\perp})\approxeq-\dfrac{2\pi^{2}g^{2}}{v_{F}^{2}}\left(\dfrac{g^{2}k_{F}}{4\pi
v_{F}}\omega\right)^{-1/3}\left|x_{\perp}\right|F\left(0\right),$ (98)

i.e., the low-energy contribution to $Q$ is exactly that obtained within the
linearized theory.

The low-energy contributions to $Q$ will govern the long-time behavior of the
propagator, and thus the low-energy behavior of the propagator. It thus
follows that the low-energy propagator is unaffected by the quadratic
dispersion. This can be confirmed by computing $F(z)$ exactly666Note that it
is necessary to compute the integral first and only then expand in $\omega$
rather than the other way around. The reason is that since the expansion of
$F(z)$ involves fractional powers of $z$, its derivatives diverge at $z=0$.,
which can be done on a computer. The result is

$\displaystyle F(z)=$
$\displaystyle-\dfrac{1}{\sqrt{\pi}}\prescript{}{1}{F}_{3}\left(1;\dfrac{1}{2},\dfrac{5}{6},\dfrac{7}{6};-\dfrac{1}{27}z^{3}\right)$
$\displaystyle+\dfrac{1}{6}e^{-z/2}\left[e^{3z/2}+\cos\left(\dfrac{\sqrt{3}z}{2}\right)+\sin\left(\dfrac{\sqrt{3}z}{2}\right)\right],$
(99)

where $\prescript{}{p}{F}_{q}$ is (again) the generalized hypergeometric
function. $Q\left(\omega,x_{\perp}\right)$ can then be expanded for small
frequencies, and since each term is just a function $\omega$ it can be easily
transformed to the time domain. The leading order correction due to the
quadratic dispersion is of the form

$Q^{(1)}\left(t,x_{\perp}\right)\propto\beta^{1/2}\text{sign}\left(x_{\perp}\right)\delta\left(x_{\perp}-v_{F}t\right)\left(x_{\perp}\right)^{3/2},$
(100)

which is just a delta function in the comoving distance. For $n>1$, the $n$th
order correction is of the form

$Q^{(n)}\left(t,x_{\perp}\right)\propto\beta^{n/2}\text{sign}\left(x_{\perp}\right)\left(x_{\perp}-v_{F}t\right)^{-\frac{1}{3}(2+n)}\left(x_{\perp}\right)^{1+n/2}.$
(101)

It is thus clear that all the corrections due to quadratic dispersion decay
faster at long times than the result of the linearized theory. They will thus
correspond to corrections to the self energy of higher orders in
$\frac{\omega^{2/3}}{\omega-v_{F}k_{\perp}}$.

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