An exact method for bosonizing the Fermi surface in arbitrary dimensions

Takamori Park1$\star$ and Leon Balents 2, 3

1 Department of Physics, University of California, Santa Barbara, CA 93106,
USA

2 Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, USA

3 Canadian Institute for Advanced Research, Toronto, Ontario, Canada

⋆<EMAIL_ADDRESS>
## Abstract

Inspired by the recent work by Delacretaz et. al. [1], we rigorously derive an
exact and simple method to bosonize a non-interacting fermionic system with a
Fermi surface starting from a microscopic Hamiltonian. In the long-wavelength
limit, we show that the derived bosonized action is exactly equivalent to the
action obtained by Delacretaz et. al. In addition, we propose diagrammatic
rules to evaluate correlation functions using our bosonized theory and
demonstrate these rules by calculating the three- and four-point density
correlation functions. We also consider a general density-density interaction
and show that the simplest approximation in our bosonic theory is identical to
RPA results.

###### Contents

  1. 1 Introduction
  2. 2 Derivation
    1. 2.1 Generating functional $Z_{0}[J]$
    2. 2.2 Legendre transform of $Z_{0}[J]$
    3. 2.3 Expressing $Z_{0}[J]$ as a functional integral over a bosonic field
    4. 2.4 Bosonized action $S[\phi]$
  3. 3 Diagrammatic calculation of correlation functions
    1. 3.1 Expanding the action and density matrix in powers of $\phi$
    2. 3.2 Diagrammatic rules
    3. 3.3 Three-point density correlation function
    4. 3.4 Four-point density correlation function
    5. 3.5 Self-energy calculations
  4. 4 Discussion
  5. A Obtaining the action derived by Delacretaz et. al. [1] from Eq. 20

##  1 Introduction

Quantum phases of matter with an extensive number of gapless modes form an
interesting class of phases in condensed matter physics that includes the
Fermi liquid phase and the broad class of strongly correlated non-Fermi
liquids. While the former is well understood, non-Fermi liquids are a
challenge to study because they exist in the strong coupling regime which
cannot be accessed using standard perturbative techniques. Instead, non-
perturbative approaches are necessary to understand how interactions destroy
the quasi-particle picture in non-Fermi liquids. In the past, bosonization of
the Fermi surface in $d>1$ attracted attention because of its potential as a
non-perturbative method to study the effects of interactions. This lead to the
proposal of several different higher-dimensional bosonization methods [2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, these approaches
involve complicated constructions (e.g. separating the Fermi surface into
patches and coarse-graining [3]).

More recently, Delacretaz et. al. proposed a new elegant method to bosonize
the Fermi surface that incorporates nonlinear effects to all orders and
appears to correctly capture Fermi liquid physics in the long-wavelength limit
[1]. Guided by the principle that their action should reproduce the Boltzmann
equation, they obtain a bosonic action through a series of inpsired
conjectures. Due to the heuristic nature of the derivation some details of
their bosonization scheme need clarifying: (a) The bosonized action is
obtained in the long-wavelength limit. How are corrections to this
incorporated into the theory? (b) Delacretaz et. al. demonstrated that tree-
level diagrams are sufficient to calculate the two- and three-point density
correlation functions. Is this generally true for all correlation functions?
What do higher-order loop diagrams contribute, and is there a way to connect
these bosonic diagrams to their corresponding fermionic diagrams? (c)
Delacretaz et. al. incorporate interaction by writing down the most general
form it can take. Naively, one would expect a density-density interaction term
to contribute the term $\int Vff$ to the action where $V$ is the potential and
$f$ is the distribution function, but is this true?

Inspired by the work by Delacretaz et. al. [1], in this work we present an
exact method for the bosonization of a system with a Fermi surface. In Section
2, we outline the derivation of our result. Our key insight is to adapt a well
known field theory result regarding the effective action that is commonly
found in field theory textbooks [17]. Applying this, we are able to exactly
express the generating functional as a restricted path integral over a bosonic
field. In the long-wavelength limit, the action that we obtained reduces to
the action obtained by Delacretaz et. al. We also briefly discuss how
interactions modify the functional integral. In Section 3, we propose
diagrammatic rules to calculate expectation values diagrammatically and
demonstrate their utility by calculating the three- and four-point density
correlation functions. Comparison with the fermion loop result is
straightforward, and we find our result is exactly correct. We also discuss
the effects of interactions and calculate approximate corrections to the two-
point density correlation function. Finally, in Section 4 we wrap up our work
with a discussion of our results. We discuss the connection between our work
and the work by Delacretaz et. al. and provide answers to the questions listed
above. In the end we also go over exciting possible future directions and
extensions of our work.

##  2 Derivation

In this section, we derive a method to bosonize a non-interacting fermion
theory with a Fermi surface from an algebraic perspective. We start with a
generating functional and first calculate its Legendre transform. Then, we
will use a well-known result found in most field theory textbooks [17] to
express the generating functional as a restricted functional integral of its
Legendre transform over a bosonic field. At the end of this section, we will
also consider how density-density interactions can be incorporated into the
bosonized theory.

As an aside, in our derivation we will deal with functions with two spatial
and one temporal arguments. Initially, we will represent these functions using
the position basis, but this is an arbitrary choice. We could also work in the
momentum basis or a mixed basis (the Wigner representation) [18, 19] via an
appropriate transformation. It is also possible to handle these functions in a
basis-free manner by noticing, for example, that a function expressed in the
position basis $J(x,x^{\prime},t)$ can be interpreted as the position-basis
matrix element of a first quantized operator, i.e. $\exists J$ such that
$\matrixelement{x}{J(t)}{x^{\prime}}=J(x,x^{\prime},t)$. As we will see later,
this compresses the notation and simplifies switching between different bases.

###  2.1 Generating functional $Z_{0}[J]$

We start with the generating functional $Z_{0}[J]$ that generates density
matrix correlation functions of a non-interacting fermionic system with a
Fermi surface at $T=0$:

$Z_{0}[J]=\expectationvalue{U_{[J]}(\infty,-\infty)}{\Omega}.$ (1)

Here, $\ket{\Omega}$ is the many-body ground state representing the Fermi sea
and $U_{[J]}$, which is a functional of the source $J$, is the time-evolution
operator defined in the usual way as

$U_{[J]}(t,t^{\prime})\equiv\begin{cases}\displaystyle\mathbb{T}\exp(-i\int_{t^{\prime}}^{t}\differential\tau
H_{0}(\tau)-i\int_{t^{\prime}}^{t}\differential\tau\int_{x,x^{\prime}}\psi^{\dagger}(x)J(x,x^{\prime},\tau)\psi(x^{\prime}))&t>t^{\prime}\\\\[10.0pt]
\displaystyle\tilde{\mathbb{T}}\exp(-i\int_{t^{\prime}}^{t}\differential\tau
H_{0}(\tau)-i\int_{t^{\prime}}^{t}\differential\tau\int_{x,x^{\prime}}\psi^{\dagger}(x)J(x,x^{\prime},\tau)\psi(x^{\prime}))&t<t^{\prime}\end{cases}.$
(2)

$H_{0}$ is the non-interacting Hamiltonian, and
$\mathbb{T},\tilde{\mathbb{T}}$ denote time ordering and anti-time ordering.
We make an important assumption that the source, $J$, satisfies

$U_{[J]}(\infty,-\infty)\ket{\Omega}\propto\ket{\Omega}.$ (3)

Then, $Z_{0}[J]$ is simply a complex phase, and if we define the “free energy”
$F_{0}[J]$ such that

$F_{0}[J]\equiv i\ln Z_{0}[J]\quad\quantity(\implies
Z_{0}[J]=e^{-iF_{0}[J]}),$ (4)

$F_{0}[J]$ must be real-valued. Using Eq. 3, the functional derivative of
$F_{0}[J]$ is

$\displaystyle f_{[J]}(x,x^{\prime},t)\equiv$ $\displaystyle\frac{\delta
F_{0}[J]}{\delta
J(x^{\prime},x,t)}=\expectationvalue{U_{[J]}(-\infty,t)\psi^{\dagger}(x^{\prime})\psi(x)U_{[J]}(t,-\infty)}{\Omega}$
(5)

which is the one-body reduced density matrix initially in the ground state
that evolves under $H_{0}$ and the source $J$.

Because the Hamiltonian is quadratic in the field operators, the one-body
reduced density matrix which we henceforth simply refer to as the density
matrix, satisfies the von Neumann equation

$i\partial_{t}f_{[J]}(t)=\commutator{H_{0}+J(t)}{f_{[J]}(t)}.$ (6)

In addition, because of the condition on $J$ (Eq. 3), the density matrix
(expressed in the momentum basis) must satisfy the boundary conditions

$\lim_{t\rightarrow\pm\infty}f_{[J]}(k,k^{\prime},t)=\expectationvalue{c^{\dagger}(k^{\prime})c(k)}{\Omega}=f_{0}(k)(2\pi)^{d}\delta^{d}(k-k^{\prime}),$
(7)

where $f_{0}(k)=\theta(k_{F}-\absolutevalue*{k})$ is the zero-temperature
Fermi-Dirac distribution function. We assumed a rotationally symmetric Fermi
surface for simplicity. $f_{0}(k)$ only takes the values 0 or 1, so
$f_{0}^{2}=f_{0}$ and $f_{0}$ must be a projection operator onto states in the
Fermi sea. $f_{[J]}$ evolves unitarily from $f_{0}$, so it must also be a
projection operator but onto states in the time-evolved Fermi sea.

###  2.2 Legendre transform of $Z_{0}[J]$

Now let’s consider how to calculate the Legendre transform of $Z_{0}[J]$. As
we will see, the generating functional does not have a well defined Legendre
transform because it is generally not convex [20]. We will resolve this issue
by first restricting $Z_{0}[J]$ to a domain $\mathcal{F}_{s}$ in which it is
convex.

Let’s begin by first showing that $Z_{0}[J]$ is not convex. Consider the map
$J\mapsto f_{[J]}$. This map is not injective because given $f_{[J]}$, if we
define $\chi_{g}\equiv f_{[J]}\chi f_{[J]}+(1-f_{[J]})\chi(1-f_{[J]})\neq 0$
for an aribtrary $\chi$, then $f_{[J]}=f_{[J+\chi_{g}]}$. The proof is
straightforward. First, note that $\commutator{\chi_{g}}{f_{[J]}}=0$ using the
property that $f_{[J]}$ is a projection operator. Then, we can add this term
to the right-hand side of Eq. 6 to get

$i\partial_{t}f_{[J]}(t)=\commutator{H_{0}+J(t)+\chi_{g}(t)}{f_{[J]}(t)},$ (8)

but this is the equation of motion for $f_{[J+\chi_{g}]}$. Since, $f_{[J]}$
and $f_{[J+\chi_{g}]}$ both satisfy the same equation of motion and boundary
conditions, it must be true that $f_{[J+\chi_{g}]}=f_{[J]}$. Therefore,
$J\mapsto f_{[J]}=\frac{\delta F[J]}{\delta J}$ is not an injective map, and
$F[J]$ is not convex.

The transformation $J\rightarrow J+\chi_{g}$ defines a gauge transformation
since under this transformation, $f_{[J]}$ which is the physical degree of
freedom remains invariant. If we completely fix this gauge, then $f_{[J]}$ now
defines an injective map. Let $\mathcal{F}_{s}$ denote the space of sources
that satisfies this gauge and the condition Eq. 3, and let $\mathcal{F}_{d}$
denote the set of all density matrices that evolve unitarily in time and
satisfy the boundary conditions in Eq. 7. Then, the map $J\in\mathcal{F}_{s}$
and $f_{[J]}\in\mathcal{F}_{d}$, $J\mapsto f_{[J]}$ is bijective, and we can
calculate the Legendre transform of $F_{0}[J]$ defined on $\mathcal{F}_{s}$.

$f_{[J]}$ defines a bijective map so we can invert it. Given
$f\in\mathcal{F}_{d}$, we define $J_{[f]}\in\mathcal{F}_{s}$ such that
$f_{[J]}\evaluated{}_{J=J_{[f]}}=f$. Then, the Legendre transform of
$F_{0}[J]$ is

$\displaystyle\Gamma_{0}[f]$
$\displaystyle\equiv\quantity(F_{0}[J]-\int_{x,x^{\prime},t}\frac{\delta
F_{0}[J]}{\delta J(x^{\prime},x,t)}J(x^{\prime},x,t))\evaluated{}_{J=J_{[f]}}$
(9) $\displaystyle=F_{0}[J_{[f]}]-\Tr[J_{[f]}f]$ (10)

where $\Gamma_{0}[f]$ is defined for $f\in\mathcal{F}_{d}$. An explicit
expression for $J_{[f]}$ is obtained later in Section 2.4. The Legendre
transformation is invertible, so there is no loss of information in this
transformation. In fact, the Legendre transformation of $\Gamma_{0}[f]$ is
$F_{0}[J]$.

###  2.3 Expressing $Z_{0}[J]$ as a functional integral over a bosonic field

In this subsection, we derive an expression for the generating functional as a
functional integral over a bosonic field. This is achieved by parametrizing
the space of density matrices $\mathcal{F}_{d}$ using the Lie algebra of
unitary transformations and adapting a well-known field theory result found in
most field theory textbooks concerning the effective action [17].

As we discussed in the previous subsection, the Legendre transformation of
$\Gamma_{0}[f]$ is $F_{0}[J]$. Using the fact that the leading order term of
the saddle point approximation is equivalent to the Legendre transform, we can
write [20, 17]

$\int_{\mathcal{F}_{d}}\mathcal{D}f\exp(-i\alpha^{-1}\quantity(\Gamma_{0}[f]+\Tr[fJ]))\rightarrow\exp(-i\alpha^{-1}F_{0}[J])\qquad\textrm{as}\qquad\alpha\rightarrow
0.$ (11)

It is easy to verify that the saddle point value of the exponent on the left-
hand side is indeed $F_{0}[J]$. To evaluate the functional integral in Eq. 11,
we need to parametrize $\mathcal{F}_{d}$. One convenient choice for
parametrization is using the Lie algebra of unitary transformations. By
definition, we know that $f\in\mathcal{F}_{d}$ evolves unitarily from $f_{0}$,
so we can always write

$f(t)=e^{i\phi(t)}f_{0}e^{-i\phi(t)}$ (12)

where $e^{i\phi(t)}$ is a unitary transformation and $\phi(t)$ is an element
of the Lie algebra of unitary transformations. However, as we discussed in
Section 2.2, there is a gauge transformation that introduces redundant
descriptions of density matrices. For example, given a unitary transformation
$V(t)$ that commutes with $f_{0}$, it is clear that $e^{i\phi(t)}\rightarrow
e^{i\phi(t)}V(t)$ is a gauge transformation. We can fix the gauge by assuming
that $\phi\in\mathcal{F}_{g}$ where $\mathcal{F}_{g}$ is the space of all
unitary Lie algebra elements that satisfy the conditions

$\forall\phi\in\mathcal{F}_{g},\,f_{0}\phi
f_{0}=(1-f_{0})\phi(1-f_{0})=0\quad\textrm{ and
}\quad\lim_{t\rightarrow\pm\infty}\phi(t)=0.$ (13)

The first condition implies $\phi$ only couples momentum states inside and
outside of the Fermi sea. There are other ways to fix the gauge, but this is
the most convenient. The second condition in Eq. 13 imposes the boundary
condition for $f$ (Eq. 7).

Using this parametrization, $f=f[\phi]$ is now a functional of $\phi$ and the
functional integral in Eq. 11 in the limit of small $\alpha$ is

$I[J,\alpha]\equiv\int_{\mathcal{F}_{g}}\mathcal{D}\phi\exp(-i\alpha^{-1}\quantity(\Gamma_{0}[f[\phi]]+\Tr[f[\phi]J])).$
(14)

The Jacobian $\mathcal{J}[\phi]=\det[\frac{\delta f[\phi]}{\delta\phi}]$ that
comes from the change of variables is a subleading contribution to the action
as $\alpha\rightarrow 0$, so it can be ignored. Technically, the unitary group
is compact, but extending the integral over $\phi$ to all values only changes
the functional integral up to an overall factor. Therefore, $I[J,\alpha]$ is a
Gaussian integral with perturbations, and it can be evaluated using Feynman
diagrams.

Let’s consider what type of diagrams we need to consider as $\alpha\rightarrow
0$. It is known that $\alpha$ is a loop expansion parameter. Each vertex
contributes a factor of $\alpha^{-1}$ and each propagator contributes a factor
of $\alpha$, so Feynman diagrams with $L$ loops are proportional to
$\alpha^{L-1}$. Since $I[J,\alpha]\rightarrow\exp(-i\alpha^{-1}F_{0}[J])$ as
$\alpha\rightarrow 0$, $-iF_{0}[J]$ must correspond to connected diagrams with
the weight $\alpha^{-1}$ which are tree diagrams (L=0). Therefore,

$\displaystyle F_{0}[J]=$ $\displaystyle i\underset{\textrm{connected
tree}}{\int_{\mathcal{F}_{g}}}\mathcal{D}\phi
e^{-i(\Gamma_{0}[f[\phi]]-\Gamma_{0}[f_{0}])-i\Tr[(f[\phi]-f_{0})J]}+\Tr[f_{0}J]$
(15)

up to an additive constant, and the generating functional itself is the sum of
all tree diagrams (that don’t need to be connected),

$Z_{0}[J]=\underset{\textrm{tree}}{\int_{\mathcal{F}_{g}}}\mathcal{D}\phi
e^{-i\Gamma_{0}[f[\phi]]-i\Tr[f[\phi]J]}.$ (16)

We refer to these functional integrals that are limited to certain diagrams as
restricted functional integrals. As we have already mentioned, this is an
adaptation of a well-known field theory result on effective actions found in
most field theory textbooks [17].

###  2.4 Bosonized action $S[\phi]$

Here, we define the bosonized action $S[\phi]$ and find an explicit expression
for it. We also consider how interactions modify the action.

The definition for the action is given by

$S[\phi]=-\Gamma_{0}[f[\phi]]+\Gamma_{0}[f_{0}].$ (17)

In order to find an explicit expression for $\Gamma_{0}[f[\phi]]$, we first
identify the correspondence

$U_{[J_{[f]}]}(t,-\infty)=e^{i\phi(t)}.$ (18)

There are other ways to relate $U$ and $\phi$ because of the gauge structure,
but this is most convenient. Under this correspondence, it is clear that
$Z_{0}[J_{[f]}]=\expectationvalue{U_{[J_{[f]}]}(\infty,-\infty)}{\Omega}=1$
and $F_{0}[J_{[f]}]=0$ because of the second condition in Eq. 13. Therefore,
the first term in Eq. 10 vanishes. Next, taking the time derivative of Eq. 18
gives us $i\partial_{t}e^{i\phi(t)}=(H_{0}+J_{f[t]}(t))e^{i\phi(t)}$, so

$J_{[f]}=i(\partial_{t}e^{i\phi(t)})e^{-i\phi(t)}-H_{0}.$ (19)

Therefore, the action must be

$S[\phi]=\Tr[J_{[f]}f[\phi]]+\Tr[H_{0}f_{0}]=\Tr[f_{0}e^{-i\phi}(i\partial_{t}-H_{0})e^{i\phi}]+\Tr[f_{0}H_{0}].$
(20)

The free energy and generating functional is given by

$\displaystyle F_{0}[J]=$ $\displaystyle i\underset{\textrm{connected
tree}}{\int_{\mathcal{F}_{g}}}\mathcal{D}\phi
e^{iS[\phi]-i\Tr[(f[\phi]-f_{0})J]}+\Tr[f_{0}J]$ (21) $\displaystyle
Z_{0}[J]=$
$\displaystyle\underset{\textrm{tree}}{\int_{\mathcal{F}_{g}}}\mathcal{D}\phi
e^{iS[\phi]-i\Tr[f[\phi]J]}.$ (22)

Up until now, we have exclusively considered non-interacting systems, but
incorporating interactions is straightforward. Consider a general two-body
interaction

$H_{\textrm{int}}=\frac{1}{2}\int_{x,y}V(x-y)\psi^{\dagger}(x)\psi^{\dagger}(y)\psi(y)\psi(x).$
(23)

The generating functional of the interacting system $Z[J]$ can be expressed
using its non-interacting counter part according to

$Z[J]=\exp(\frac{i}{2}\int_{x,y,t}V(x-y)\frac{\delta}{\delta
J(x,x,t)}\frac{\delta}{\delta J(y,y,t)})Z_{0}[J].$ (24)

This adds an additional term to the action $S[\phi]\rightarrow
S[\phi]+S_{\textrm{int}}[\phi]$ where

$\displaystyle S_{\textrm{int}}[\phi]=$
$\displaystyle-\frac{1}{2}\int_{x,y,t}V(x-y)f[\phi](x,x,t)f[\phi](y,y,t)$ (25)
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{q,k,k^{\prime},t}\tilde{V}_{q}f[\phi](k+q,k,t)f[\phi](k^{\prime}-q,k^{\prime},t).$
(26)

The action that we obtained (Eq. 20) looks similar to the action derived by
Delacretaz et. al. [1]. In Section 4 and Appendix A we discuss the link
between the two actions in detail.

##  3 Diagrammatic calculation of correlation functions

In this section, we propose a set of diagrammatic rules that simplifies the
task of calculating correlation functions from the bosonic theory. We
demonstrate their use by calculating the three- and four-point density
correlation functions. Since the bosonic theory is exact, we can directly
compare results with the equivalent fermionic theory. We find that the
diagrams obtained from the bosonic theory are partitions of the corresponding
diagrams obtained from the fermionic theory. In other words, the process of
bosonization separates each fermion diagrams into several bosonic diagrams. In
the long wavelength limit, fermion 1-loop diagrams exhibit partial
cancellation and extracting the correct leading order behavior can be tricky
[21, 22]. However, we find that the diagrams calculated from the bosonic
theory are well-behaved, and calculating the correct long wavelength behavior
becomes straightforward.

###  3.1 Expanding the action and density matrix in powers of $\phi$

We first begin by expanding the action in powers of $\phi$ to find an
expression for the quadratic term and the higher-order

$S[\phi]=\sum_{n=1}^{\infty}\frac{(-i)^{n}}{n!}\Tr[f_{0}\quantity(-i\textrm{ad}_{\phi}^{n-1}\partial_{t}\phi-\textrm{ad}_{\phi}^{n}H_{0})],$
(27)

where $\textrm{ad}_{\phi}(\cdot)\equiv\commutator{\phi}{\cdot}$ is the adjoint
action of the Lie algebra. The action is most conveniently expressed in the
momentum basis since $H_{0}$ and $f_{0}$ are diagonal in this basis. The
symmetrized action in the momentum basis is

$\displaystyle S[\phi]=\sum_{\begin{subarray}{c}n=2\\\
n:\text{even}\end{subarray}}^{\infty}S_{(n)}[\phi],$ (28)

where

$\displaystyle
S_{(n)}[\phi]\equiv\frac{i^{n}2^{n-2}}{n!}\int_{\begin{subarray}{c}p_{1},\cdots,p_{n}\\\
\omega_{1},\cdots,\omega_{n}\end{subarray}}$
$\displaystyle\quantity[\frac{-\omega_{1}+\omega_{2}-\cdots+\omega_{n}}{n}-2\frac{-\varepsilon_{p_{1}}+\varepsilon_{p_{2}}-\cdots+\varepsilon_{p_{n}}}{n}]F^{(n)}(p_{1},\cdots,p_{n})$
$\displaystyle\qquad\times\phi(p_{1},p_{2},\omega_{1})\cdots\phi(p_{n},p_{1},\omega_{n})2\pi\delta(\omega_{1}+\cdots+\omega_{n}).$
(29)

$\varepsilon_{p}=\expectationvalue{H_{0}}{p}$ is the dispersion which for
simplicity we have assumed is $\varepsilon_{p}=\frac{p^{2}}{2m}-E_{F}$. The
factor $F^{(n)}$ is a function that only takes the values $-1,0,1$ and is
defined as

$\displaystyle F^{(n)}(p_{1},\cdots,p_{n})$ $\displaystyle\equiv$
$\displaystyle\begin{cases}f_{0}(p_{1})(1-f_{0}(p_{2}))f_{0}(p_{3})\cdots
f_{0}(p_{n})-(1-f_{0}(p_{0}))f_{0}(p_{1})(1-f_{0}(p_{2}))\cdots(1-f_{0}(p_{n})),&n\text{:
even}\\\ (1-f_{0}(p_{1}))f_{0}(p_{2})(1-f_{0}(p_{3}))\cdots
f_{0}(p_{n})-f_{0}(p_{1})(1-f_{0}(p_{2}))f_{0}(p_{3})\cdots(1-f_{0}(p_{n})),&n\text{:
odd}\end{cases}.$ (30)

Adjacent momenta, $p_{i},p_{i+1}$, in the argument cannot both be in or out of
the Fermi sea. This is a consequence of the first condition in Eq. 13. For
example, if $f_{0}(p_{i})=f_{0}(p_{i+1})$, then
$F^{(n)}(p_{1},\cdots,p_{n})=0$. In addition, as a result of this condition,
odd $n$ terms in the action must vanish because they are proportional to
$f_{0}(p_{1})(1-f_{0}(p_{1}))$ which vanishes because $f_{0}$ is a projection
operator.

The quadratic contribution to the action is

$S_{0}[\phi]\equiv
S_{(2)}[\phi]=\int\frac{1}{2}\quantity[\omega-\varepsilon_{k}+\varepsilon_{k}^{\prime}]\absolutevalue{\phi(k,k^{\prime},\omega)}^{2}F^{(2)}(k,k^{\prime}),$
(31)

so the propagator evaluated with respect to the quadratic action is

$\displaystyle\expectationvalue{\phi(k,k^{\prime},\omega)\phi(p^{\prime},p,\omega^{\prime})}_{0}=$
$\displaystyle
D_{0}(k,k^{\prime},\omega)(2\pi)^{2d+1}\delta^{d}(p-k)\delta^{d}(p^{\prime}-k^{\prime})\delta(\omega+\omega^{\prime}).$
(32)

where

$D_{0}(k,k^{\prime},\omega)=D_{0}(k^{\prime},k,-\omega)\equiv-i\frac{f_{0}(k)-f_{0}(k^{\prime})}{\omega-\varepsilon_{k}+\varepsilon_{k^{\prime}}}$
(33)

Note, $F^{(2)}(k,k^{\prime})=f_{0}(k^{\prime})-f_{0}(k)$. The quadratic action
in Ref. [1] can be obtained from Eq. 31 by taking the limit $q\equiv
k-k^{\prime}\rightarrow 0$ and fixing the center of momentum to the Fermi
surface $\absolutevalue{\frac{k+k^{\prime}}{2}}=k_{F}$. Because of the non-
linear relationship between $\phi$ and $f$, expanding in $\phi$ leads to an
infinite number of terms. However, only connected tree diagrams are relevant,
so the calculation of a correlation function only involves finite number of
terms.

The expansion of the density matrix in the momentum basis can also be
expressed using $F^{(n)}$.

$\displaystyle f(k,k^{\prime},t)=$
$\displaystyle\sum_{n=0}^{\infty}\underbrace{\frac{i^{n}2^{n-1}}{n!}\int_{k_{1},\cdots,k_{n-1}}\phi(k,k_{1},t)\phi(k_{1},k_{2},t)\cdots\phi(k_{n-1},k^{\prime},t)F^{(n+1)}(k,k_{1},\cdots,k_{n-1},k^{\prime})}_{\textstyle\equiv
f_{(n)}(k,k^{\prime},t)}.$ (34)

###  3.2 Diagrammatic rules

Correlation functions can be calculated using the bosonized theory without the
use of diagrams, but they often involve lengthy and tedious algebra resulting
from integrating over multiple delta functions. This is because the field
$\phi$ has two spatial indices rather than one which doubles the number of
spatial delta functions. This can be avoided by using Feynman diagrams. Below,
we establish Feynman diagram rules for the bosonized theory:

  * •

Points represent momenta.

  * •

The field $\phi(k,k^{\prime},\omega)$ is represented by a line with an arrow
that starts at $k^{\prime}$ and ends at $k$. A dashed line with an arrow is
used to keep track of the frequency and momentum exchange. (Fig. 1(a))

  * •

The propagator $D_{0}(k,k^{\prime},\omega)$ is represented by a loop
connecting the momenta $k$ and $k^{\prime}$. (Fig. 1(b))

$k$
---
$k^{\prime}$
---
$k-k^{\prime},\omega$
---
$\phi(k,k^{\prime},\omega)$
---
(a) $k^{\prime}$
---
$k$
---
$k-k^{\prime},\omega$
---
$k^{\prime}-k,-\omega$
---
$D_{0}(k,k^{\prime},\omega)$
---
(b)

Figure 1: Feynman diagram elements for (a) the field
$\phi(k,k^{\prime},\omega)$ and (b) the propagator
$D_{0}(k,k^{\prime},\omega)$.

###  3.3 Three-point density correlation function

In this section, we calculate the three-point density correlation function
using diagrams of the bosonized theory. For simplicity, we ignore the effects
of interaction. The correlation function is defined as

$C_{(3)}(1,2,3)=\expectationvalue{\rho(1)\rho(2)\rho(3)}_{c},$ (35)

where $i=(q_{i},\omega_{i})$ and $\rho(i)=\int_{k}f(k+q_{i},k,\omega_{i})$ for
$i=1,2,3$ . We define the $n$-th order term of the expansion of $\rho$ in
powers of $\phi$ as
$\rho_{(n)}(i)\equiv\int_{k}f_{(n)}(k+q_{i},k,\omega_{i})$. As we showed
above, only the connected tree diagrams need to be considered. Schematically,
there is only one such term as shown in Fig. 2(a), so

$C_{(3)}(1,2,3)=\expectationvalue{\rho_{(2)}(1)\rho_{(1)}(2)\rho_{(1)}(3)}_{0,c}+\expectationvalue{\rho_{(1)}(1)\rho_{(2)}(2)\rho_{(1)}(3)}_{0,c}+\expectationvalue{\rho_{(1)}(1)\rho_{(1)}(2)\rho_{(2)}(3)}_{0,c}.$
(36)

We evaluate the first term diagramatically. The other two terms are obtained
by permuting the indices of the arguments. The first term generates two
diagrams that are related by exchange of
$(q_{2},\omega_{2})\leftrightarrow(q_{3},\omega_{3})$ as shown in Fig. 3. The
diagrams evaluate to

$\displaystyle\expectationvalue{\rho_{(2)}(1)\rho_{(1)}(2)\rho_{(1)}(3)}_{0,c}=$
$\displaystyle-\int_{k}\bigg{(}\frac{F^{(3)}(k+q_{1},k+q_{1}+q_{2},k)}{(\omega_{2}-\varepsilon_{k+q_{1}+q_{2}}+\varepsilon_{k+q_{1}})(\omega_{3}-\varepsilon_{k}+\varepsilon_{k+q_{1}+q_{2}})}$
$\displaystyle\qquad+\frac{F^{(3)}(k+q_{1},k+q_{1}+q_{3},k)}{(\omega_{2}-\varepsilon_{k}+\varepsilon_{k+q_{1}+q_{3}})(\omega_{3}-\varepsilon_{k+q_{1}+q_{3}}+\varepsilon_{k+q_{1}})}\bigg{)}$
$\displaystyle\qquad\times(2\pi)^{d+1}\delta^{d}(q_{1}+q_{2}+q_{3})\delta(\omega_{1}+\omega_{2}+\omega_{3}).$
(37)

The first and second terms correspond to Fig. 3(a) and Fig. 3(b) respectively.
By permuting the external indices and reorganizing the terms we find

$\displaystyle C_{(3)}(1,2,3)=$
$\displaystyle-\int_{k}\bigg{(}\frac{F^{(3)}(k+q_{1},k+q_{1}+q_{2},k)}{(\omega_{3}-\varepsilon_{k}+\varepsilon_{k+q_{1}+q_{2}})(\omega_{2}-\varepsilon_{k+q_{1}+q_{2}}+\varepsilon_{k+q_{1}})}$
$\displaystyle\qquad+\frac{F^{(3)}(k+q_{2},k+q_{2}+q_{3},k)}{(\omega_{1}-\varepsilon_{k}+\varepsilon_{k+q_{2}+q_{3}})(\omega_{3}-\varepsilon_{k+q_{2}+q_{3}}+\varepsilon_{k+q_{2}})}$
$\displaystyle\qquad+\frac{F^{(3)}(k+q_{3},k+q_{1}+q_{3},k)}{(\omega_{2}-\varepsilon_{k}+\varepsilon_{k+q_{1}+q_{3}})(\omega_{1}-\varepsilon_{k+q_{1}+q_{3}}+\varepsilon_{k+q_{3}})}+(2\leftrightarrow
3)\bigg{)}$ (38)

where we omitted the delta function factors that conserve momentum and
frequency. The first three terms correspond to the fermion loop diagram that
describe the momentum scattering process, $k\rightarrow k+q_{1}\rightarrow
k+q_{1}+q_{2}\rightarrow k$ which we refer to as the $(123)$ scattering
process. The $(2\leftrightarrow 3)$ term corresponds to the $(132)$ scattering
process which is the reversed process. A comparison with the fermion loop
results shows that Eq. 38 is exactly correct.

$\rho_{(1)}$
---
$\rho_{(2)}$
---
$\rho_{(1)}$
---
(a) $\rho_{(1)}$
---
$\rho_{(1)}$
---
$\rho_{(1)}$
---
$\rho_{(3)}$
---
(b) $\rho_{(2)}$
---
$\rho_{(2)}$
---
$\rho_{(1)}$
---
$\rho_{(1)}$
---
(c) $\rho_{(1)}$
---
$\rho_{(1)}$
---
$\rho_{(1)}$
---
$\rho_{(1)}$
---
$S_{(4)}$
---
(d)

Figure 2: Schematic tree-level diagrams for the three- and four-point density
correlation functions. (a) corresponds to the three-point correlation function
and (b–e) correspond to the four-point correlation function.

$q_{1},\omega_{1}$
---
$q_{3},\omega_{3}$
---
$q_{2},\omega_{2}$
---
$k$
---
$k+q_{1}$
---
$k+q_{1}+q_{2}$
---
(a) $q_{1},\omega_{1}$
---
$q_{2},\omega_{2}$
---
$q_{3},\omega_{3}$
---
$k$
---
$k+q_{1}$
---
$k+q_{1}+q_{3}$
---
(b)

Figure 3: Feynman diagrams for
$\expectationvalue{\rho_{(2)}(1)\rho_{(1)}(2)\rho_{(1)}(3)}_{0,c}$.

Notice, that in the low-momentum limit, $F^{(3)}$ in the numerator of each
term vanishes if $k$ is not near the Fermi surface. Moreover, if we take the
limit $\absolutevalue{q_{i}}\rightarrow 0$, the $F^{(3)}$ factors becomes a
delta function $\delta(\absolutevalue{k}-k_{F})$ and the integral over $k$ is
reduced to an angular integral. The leading and sub-leading behavior can be
straightforwardly extracted from this.

Let’s study the long-wavelength behavior of the correlation function. As we
mentioned above, in the original fermionic theory, partial cancellation
between fermion loop diagrams occur when $\omega_{i}\rightarrow
0,q_{i}\rightarrow 0$, and the long-wavelength behavior of the density
correlation function is determined by the sub-leading term [21, 22]. This
cancellation can be tricky to see in the original theory, but we demonstrate
it is straightforward to see in our bosonic theory. Consider the term Eq. 37.
The numerator of the first term expands to

$F^{(3)}(k+q_{1},k+q_{1}+q_{2},k)=f_{0}(k)-f_{0}(k+q_{2})+\quantity(\begin{subarray}{c}\textrm{terms
symmetric under }\\\ \textrm{permutation of
}\end{subarray}\quantity{q_{1},q_{1}+q_{2},q_{1}+q_{2}+q_{3}}).$ (39)

The symmetric term vanishes if we consider all other diagrams, so we only need
to consider the first two terms. The leading order behavior of the first term
in Eq. 37 is

$\int_{k}\frac{q_{2}\cdot\hat{k}\delta(\absolutevalue{k}-k_{F})}{\quantity(\omega_{2}-q_{2}\cdot\nabla_{k}\varepsilon_{k})\quantity(\omega_{3}+(q_{1}+q_{2})\cdot\nabla_{k}\varepsilon_{k})}.$
(40)

Clearly this diverges as $q_{i},\omega_{i}\rightarrow 0$. The first and second
terms in Eq. 37 are related by a change in sign of the external momenta and
frequency, $q_{i},\omega_{i}\rightarrow-q_{i},-\omega_{i}$. Therefore, the
leading order behavior of the two terms cancel and the long-wavelength
behavior turns out to not divergent. Instead the sub-leading term of the two
terms in Eq. 37 determines the scaling.

###  3.4 Four-point density correlation function

Let’s now consider the four-point density correlation function defined as

$C_{(4)}(1,2,3,4)=\expectationvalue{\rho(1)\rho(2)\rho(3)\rho(4)}_{c}.$ (41)

The three possible connected tree diagrams are schematically identified in
Figs. 2(b)–2(d). They correspond to the terms below.

$\displaystyle C_{(4)}(1,2,3,4)=$
$\displaystyle\quantity(\expectationvalue{\rho_{(3)}(1)\rho_{(1)}(2)\rho_{(1)}(3)\rho_{(1)}(4)}_{0,c}+\textrm{(other
permutations)})$
$\displaystyle+\quantity(\expectationvalue{\rho_{(2)}(1)\rho_{(2)}(2)\rho_{(1)}(3)\rho_{(1)}(4)}_{0,c}+\textrm{(other
permutations)})$
$\displaystyle+\expectationvalue{\rho_{(1)}(1)\rho_{(1)}(2)\rho_{(1)}(3)\rho_{(1)}(4)iS_{(4)}}_{0,c}$
(42)

To simplify the comparison with the corresponding fermion loop diagram, we
organize the diagrams by the scattering process they describe. There are $3!$
distinct scattering processes. For simplicity, we only consider diagrams that
describe the fermion scattering process, $k\rightarrow k+q_{1}$ $\rightarrow
k+q_{1}+q_{2}$ $\rightarrow k+q_{1}+q_{2}+q_{3}$ $\rightarrow k$ which we
refer to as the $(1234)$ scattering process as we did in the previous
subsection. The other diagrams can be obtained by permuting the external
momenta and frequency. It’s straightforward to iterate through the diagrams
that contribute to the $(1234)$ scattering process.

The first term in Eq. 42 only leads to the diagram Fig. 4(a). The second term
leads to three diagrams shown in Figs. 4(b)–4(c), and the last term is given
by Fig. 4(d).

$q_{i},\omega_{i}$
---
$q_{i+1},\omega_{i+1}$
---
$q_{i+3},\omega_{i+3}$
---
$q_{i+2},\omega_{i+2}$
---
$k+p_{i}$
---
$k+p_{i+1}$
---
$k+p_{i+2}$
---
$k+p_{i+3}$
---
(a)

$q_{i},\omega_{i}$
---
$q_{i+1},\omega_{i+1}$
---
$q_{i+2},\omega_{i+2}$
---
$q_{i+3},\omega_{i+3}$
---
$k+p_{i}$
---
$k+p_{i+1}$
---
$k+p_{i+2}$
---
$k+p_{i+3}$
---
(b) $q_{i+1},\omega_{i+1}$
---
$q_{i+2},\omega_{i+2}$
---
$q_{i+3},\omega_{i+3}$
---
$q_{i},\omega_{i}$
---
$k+p_{i+3}$
---
$k+p_{i+2}$
---
$k+p_{i+1}$
---
$k+p_{i}$
---
(c)

$q_{i},\omega_{i}$
---
$q_{i+1},\omega_{i+1}$
---
$q_{i+2},\omega_{i+2}$
---
$q_{i+3},\omega_{i+3}$
---
$k+p_{i}$
---
$k+p_{i+1}$
---
$k+p_{i+2}$
---
$k+p_{i+3}$
---
(d) $q_{1},\omega_{1}$
---
$q_{2},\omega_{2}$
---
$q_{4},\omega_{4}$
---
$q_{3},\omega_{3}$
---
$k+p_{1}$
---
$k+p_{2}$
---
$k$
---
$k+p_{3}$
---
(e)

Figure 4: Feynman diagrams of $C_{(4)}$ describing the $(1234)$ scattering
process. (a) corresponds to terms of the form
$\expectationvalue{\rho_{(3)}\rho_{(1)}\rho_{(1)}\rho_{(1)}}_{0,c}$, (b–d)
corresponds to terms of the form
$\expectationvalue{\rho_{(2)}\rho_{(2)}\rho_{(1)}\rho_{(1)}}_{0,c}$, and (e)
corresponds to
$\expectationvalue{\rho_{(1)}\rho_{(1)}\rho_{(1)}\rho_{(1)}iS_{(4)}}_{0,c}$

Let

$g_{ij}(k)=g_{ji}(k)\equiv\frac{f_{0}(k+p_{i})-f_{0}(k+p_{j})}{\Omega_{i}-\Omega_{j}-\varepsilon(k+p_{i})+\varepsilon(k+p_{j})},$
(43)

where $p_{i}\equiv\sum_{j=1}^{i}q_{j}$ and
$\Omega_{i}\equiv\sum_{j=1}^{i}\omega_{i}$ for $i=1,\cdots,4$. The indices are
defined mod $4$ for convenience. All the diagrams can be expressed as an
integral over products of these functions. Below we list the contribution from
each diagram.

$\displaystyle\textrm{(\lx@cref{creftype~refnum}{fig:c4a})}=$
$\displaystyle-i\frac{2}{3}\sum_{i=1}^{4}\int_{k}g_{i,i+1}(k)g_{i+1,i+2}(k)g_{i+2,i+3}(k)$
(44) $\displaystyle\textrm{(\lx@cref{creftype~refnum}{fig:c4b})}=$
$\displaystyle-i\sum_{i=1}^{2}\int_{k}g_{i+3,i+2}(k)g_{i+2,i}(k)g_{i,i+1}(k)$
(45) $\displaystyle\textrm{(\lx@cref{creftype~refnum}{fig:c4c})}=$
$\displaystyle-i\sum_{i=1}^{2}\int_{k}g_{i,i+1}(k)g_{i+1,i+3}(k)g_{i+3,i+2}(k)$
(46) $\displaystyle\textrm{(\lx@cref{creftype~refnum}{fig:c4d})}=$
$\displaystyle i\sum_{i=1}^{4}\int_{k}g_{i+1,i}(k)g_{i+2,i}(k)g_{i+3,i}(k)$
(47) $\displaystyle\textrm{(\lx@cref{creftype~refnum}{fig:c4e})}=$
$\displaystyle\frac{i}{6}\sum_{i=1}^{4}\int_{k}g_{i,i+1}(k)g_{i+1,i+2}(k)g_{i+2,i+3}(k)$
(48)

The factor
$(2\pi)^{d+1}\delta^{d}(q_{1}+q_{2}+q_{3}+q_{4})\delta(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4})$
was omitted. Notice, that (C4a) and (C4e) are identical up to an overall
factor. Adding all these contributions together gives us

$\displaystyle C_{(4)}(1,2,3,4)=$
$\displaystyle\bigg{(}-\frac{i}{2}\sum_{i=1}^{4}\int_{k}g_{i+3,i+2}(k)g_{i+2,i+1}(k)g_{i+1,i}(k)+i\sum_{i=1}^{4}\int_{k}g_{i+1,i}(k)g_{i+2,i}(k)g_{i+3,i}(k)$
$\displaystyle-i\sum_{i=1}^{2}\int_{k}g_{i+3,i+2}(k)g_{i+2,i}(k)g_{i,i+1}(k)-i\sum_{i=1}^{2}\int_{k}g_{i+2,i+3}(k)g_{i+3,i+1}(k)g_{i+1,i}(k)\bigg{)}$
$\displaystyle+(\textrm{permutations of
}\\{(\omega_{1},q_{1}),(\omega_{2},q_{2}),(\omega_{3},q_{3}),(\omega_{4},q_{4})\\})$
(49)

where we omitted the overall delta function factors that conserve momentum and
frequency. The first four terms correspond to the $(1234)$ scattering process.
With some algebra it is possible to show it is exactly equivalent to the
corresponding fermion loop diagram. As we discussed above for $C_{(3)}$, in
the low momentum regime each factor of $g_{ij}(k)$ is non-zero only if
$\absolutevalue{k}\sim k_{F}$, and if we take
$\absolutevalue{q_{i}}\rightarrow 0$, we get a delta function
$\delta(\absolutevalue{k}-k_{F})$.

###  3.5 Self-energy calculations

Let us now consider the effects of interaction. As we discussed above,
interaction is given by Eq. 26. Let us define
$S_{\textrm{int}}^{(n,n^{\prime})}$ as

$S_{\textrm{int}}^{(n,n^{\prime})}=S_{\textrm{int}}^{(n^{\prime},n)}=-\frac{1}{2}\int_{q,k,k^{\prime},t}\tilde{V}_{q}f_{(n)}(k+q,k,t)f_{(m)}(k^{\prime}-q,k^{\prime},t)$
(50)

so that
$S_{\textrm{int}}=\sum_{n,n^{\prime}=1}^{\infty}S_{\textrm{int}}^{(n,n^{\prime})}$.
Let’s now think about what kind of diagrams can be written down if we include
these interaction terms. The interaction term is introduced by taking a
second-order functional derivative with respect to $J$, so diagrammatically,
interactions can be thought of as a diagram element that takes two external
legs of tree diagrams and stitches them together. Here, when we refer to
external legs we refer to the dashed lines in the full diagrams or the points
in the schematic diagrams. The external legs can be from the same tree diagram
or from different tree diagrams.

For simplicity, let’s consider the two-point density correlation function. The
corrections to the correlation function up to second order in interaction are
shown in Fig. 5. In general, the diagrams are constructed using two-, three-,
and four-point density correlation functions of the bare theory. Clearly,
higher order corrections will contain higher order correlation functions. A
simple approximation we can consider is to only include diagrams that involve
two-point density correlation functions. It is then straightforward to show
that the two-point density correlation function is

$\displaystyle C_{(2),\textrm{int}}(1,2)$
$\displaystyle=\int_{k}D_{0}(k+q_{1},k,\omega_{1})\sum_{n=0}^{\infty}\quantity(-i\tilde{V}_{q}\int_{k}D_{0}(k+q_{1},k,\omega_{1}))^{n}$
(51)
$\displaystyle=\frac{\int_{k}D_{0}(k+q_{1},k,\omega_{1})}{1+i\tilde{V}_{q}\int_{k}D_{0}(k+q_{1},k,\omega_{1})}$
(52)

where the momentum and frequency delta functions are omitted. This is simply
the RPA approximation.

(a) (b)

(c) (d)

Figure 5: Schematic representations of the corrections to the two-point
density correlation function up to second order in interaction. The circles
represent the bare two-, three-, and four-point density correlation functions
depending on the number of legs they have.

##  4 Discussion

In this work, we derived an exact method to express a fermionic theory with a
Fermi surface as a restricted functional integral over a bosonic field.
Starting from a generating functional with a source coupled to density, we
calculated the Legendre transform of the free energy. We then showed that the
integral of the Legendre transformed free energy over the Lie algebra of the
unitary group when restricted to tree level diagrams reproduces the generating
functional. In order to demonstrate the utility of our formalism, we evaluated
the three- and four-point density correlation functions using Feynman diagram
rules that we proposed. The results from our bosonized theory matched the
fermion loop diagram results exactly.

This work was inspired by the work by Delacretaz et. al. in Ref. [1]. Using
heuristic arguments, they derived an elegant bosonization scheme for a system
with a Fermi surface that incorporates nonlinear effects. It turns out that
the long-wavelength limit of the bosonized action that we derived (Eq. 20) is
equivalent to the bosonic action in Ref. [1]. We can see this correspondence
by expressing our bosonized action in the Wigner representation. In this
representation, operator products are replaced by the non-commutative moyal
product. The action derived by Delacretaz et. al. can then be obtained by
expanding the moyal product to first order in spatial derivatives. Then, the
Lie algebra of the unitary group becomes the Poisson algebra of the canonical
transformations and the two actions become equivalent. Therefore, our work
serves as a rigorous derivation of the results presented in Ref. [1]. In
Appendix A we go over this argument in more detail.

In addition to providing a proof, our work also clarifies some ambiguities in
Ref. [1] that we listed earlier in Section 1. First of all, the bosonized
theory in Ref. [1] is valid in the long-wavelength limit, so a question one
can ask is how can corrections to this be obtained? As we discussed above, the
long-wavelength limit is a result of using the Poisson algebra so corrections
can be obtained by expanding to the Lie algebra of the unitary group. Our
bosonized action is exact and considers fluctuations of the Fermi surface
generated by the unitary group, so our action can be be used to obtain
corrections. Secondly, Ref. [1] doesn’t explain why correlation functions can
be obtained just from tree-level diagrams. We explained in Section 2 that this
is due to taking the saddle point approximation to obtain the Legendre
transform. Lastly, the link between a four-fermion interaction term and the
interaction term in the bosonic action was not explicitly stated in Ref. [1].
We also clarified this at the end of Section 2.

The bosonization formalism in this paper is an adaptation of an old field
theory result concerning the effective action [17]. One might then ask what is
new in this work. The result on the effective action is usually applied to
bosonic systems with a source coupled linearly to the field, but in this work
we started with a fermionic theory and coupled the source to a fermion
bilinear. The Legendre transform which produces a bosonic action from an
fermionic action is inherently non-trivial. In addition, the change of
variables from the density matrices to the Lie algebra of the unitary group is
a non-linear transformation that partitions the fermion diagrams in a non-
trivial manner.

In our work, we briefly discussed the effects of interaction, and we obtained
the RPA approximation by only considering the simplest series of diagrams that
contribute to the two-point density correlation function. In general, because
the bosonization scheme partitions fermion diagrams into several bosonic
diagrams this increases the variety of diagrams that we need to consider in
the bosonized theory. In future works, it would be interesting to consider all
of these diagrams and see how they can be organized compared to the original
fermionic theory.

Another direction to consider in the future is to apply the bosonization
method we outlined in this work to other systems where the relevant degrees of
freedom are different. In principle, our approach can be simply generalized to
different cases by coupling the source $J$ in the generating functional to
different fermion bilinears. For example, $J$ can couple to
$\psi(x,t)\psi(x^{\prime},t)$ to describe a BCS theory or to
$\psi_{\alpha}^{\dagger}(x,t)\frac{\overrightarrow{\sigma}_{\alpha\beta}}{2}\psi_{\beta}(x^{\prime},t)$
where $\alpha,\beta=\uparrow,\downarrow$ to describe spin fluctuations. One
can also consider the case where the source is coupled to two fermion
operators at different times,
$\psi^{\dagger}(x,t)\psi(x^{\prime},t^{\prime})$. Han et. al. considered this
case in their proposal for a bosonization procedure for non-Fermi liquids in
Ref. [23].

## Acknowledgements

The authors thank Yi-Hsien Du, Léo Mangeolle, and Lucile Savary for insightful
discussions. T.P. was supported by a quantum Foundry fellowship through the
National Science Foundation through Enabling Quantum Leap: Convergent
Accelerated Discovery Foundries for Quantum Materials Science, Engineering and
Information (Q-AMASE-i) award number DMR-1906325, supplemented by the NSF CMMT
program under Grant No. DMR-2116515, which also supported L.B. We also
acknowledge support from the Simons Collaboration on Ultra-Quantum Matter,
which is a grant from the Simons Foundation (651440).

##  Appendix A Obtaining the action derived by Delacretaz et. al. [1] from Eq.
20

Here, we show how to explicitly obtain the action derived by Delacretez et.
al.in Ref. [1]. For convenience we replicate the action in Eq. 20 here.

$S[\phi]\equiv-\Gamma[f[\phi]]+\Gamma[f_{0}]=\Tr[f_{0}e^{-i\phi}(i\partial_{t}-H_{0})e^{i\phi}]+\Tr[f_{0}H_{0}].$
(53)

To obtain the action in Ref. [1], we need to express the equation above in the
Wigner representation. The Wigner representation is a mixed basis
representation. Given a single-particle operator $A$, its Wigner
representation is defined as [18, 19]

$A\rightarrow A(x,p)=\int_{y}e^{-iy\cdot
p}\matrixelement{x+\frac{y}{2}}{A}{x-\frac{y}{2}}.$ (54)

The product of operators can be represented as the Moyal product
($\star$-product) of their Wigner representations.

$AB\rightarrow A(x,p)\star B(x,p)\equiv
A(x,p)e^{\frac{i}{2}\quantity(\overleftarrow{\nabla}_{x}\cdot\overrightarrow{\nabla}_{p}-\overleftarrow{\nabla}_{p}\cdot\overrightarrow{\nabla}_{x})}B(x,p)$
(55)

The arrows of the derivatives indicate the directions in which they act.

The trace of an operator is its integral over phase space.

$\tr[A]=\int_{x,p}A(x,p)$ (56)

The trace of the product of two operators is the phase-space integral of the
products of their Wigner representation.

$\tr[AB]=\int_{x,p}A(x,p)B(x,p)$ (57)

The commutator of two operators becomes

$\commutator{A}{B}\rightarrow\\{\\{A(x,p),B(x,p)\\}\\}=A(x,p)\star
B(x,p)-B(x,p)\star A(x,p).$ (58)

The commutator in the Wigner representation defines a Lie algebra which we
refer to as the $\star$ algebra. If $A(x,p)$ and $B(x,p)$ vary slowly in
space, we can expand the Moyal product in powers of the gradient. The lowest
order term of the commutator turns out to be the Poisson bracket.

$\\{\\{A(x,p),B(x,p)\\}\\}\approx
A(x,p)i\quantity(\overleftarrow{\nabla}_{x}\cdot\overrightarrow{\nabla}_{p}-\overleftarrow{\nabla}_{p}\cdot\overrightarrow{\nabla}_{x})B(x,p)=i\poissonbracket{A(x,p)}{B(x,p)}_{\textrm{p.b.}}$
(59)

Hence, the Poisson algebra is obtained from the $\star$ algebra in the long-
wavelength limit.

Now that we’ve reviewed the necessary facts on the Wigner representation
needed to proceed, let’s consider the Hamiltonian term of the action. As we
discussed in Section 3, it can be expanded using the adjoint action of $\phi$.

$e^{-i\phi}H_{0}e^{i\phi}=\sum_{n=0}\frac{(-i)^{n}}{n!}\textrm{ad}^{n}_{\phi}{H_{0}}.$
(60)

In the Wigner representation this becomes

$(e^{-i\phi}H_{0}e^{i\phi})(x,p)=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\quantity(\textrm{ad}^{(\star)}_{\phi(x,p)})^{n}{H_{0}(p)}$
(61)

where
$\textrm{ad}^{(\star)}_{\phi(x,p)}(\cdot)\equiv\\{\\{\phi(x,p),\cdot\\}\\}$ is
the adjoint action of the $\star$ algebra. Assuming $\phi(x,p)$ varies slowly
in space, this becomes

$(e^{-i\phi}H_{0}e^{i\phi})(x,p)\rightarrow\sum_{n=0}^{\infty}\frac{1}{n!}\quantity(\textrm{ad}^{(P)}_{\phi(x,p)})^{n}{H_{0}(p)}$
(62)

where
$\textrm{ad}^{(P)}_{\phi(x,p)}(\cdot)\equiv\poissonbracket{\phi(x,p)}{\cdot}_{\textrm{p.b.}}$.
Following the notation $U=e^{-\phi}$ used in Ref. [1], we get

$(e^{-i\phi}H_{0}e^{i\phi})(x,p)\rightarrow U^{-1}H_{0}U.$ (63)

where the right-hand side is to be interpreted as the adjoint action of the
Lie group that represents the canonical transformations. The same argument
follows for the term with the time-derivative, so

$(e^{-i\phi}(i\partial_{t}-H_{0})e^{i\phi})(x,p)\rightarrow
U^{-1}(\partial_{t}-H_{0})U.$ (64)

Using the notation in Ref. [1], that the integral over the phase-space can be
expressed as an inner-product and using Eq. 57, the action becomes

$S[\phi]=\Tr[f_{0}e^{-i\phi}(i\partial_{t}-H_{0})e^{i\phi}]\rightarrow\int\differential
t\left<f_{0},U^{-1}(\partial_{t}-H_{0})U\right>$ (65)

Note, this is not an equality since we assumed that $\phi(x,p)$ varies slowly
in space. This assumption is equivalent to the fact that in the momentum
representation, the field $\phi(k,k^{\prime},t)$ is non-negligible only when
$\absolutevalue{k-k^{\prime}}\ll k_{F}$. When calculating the correlation
functions, it can also be understood as the small momentum limit,
$\absolutevalue{q_{i}}\rightarrow 0$.

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