# Postmodern Fermi Liquids

Umang Mehta  Kadanoff Center for Theoretical Physics, University of Chicago,
Chicago, Illinois 60637, USA

(August 28, 2024)

###### Abstract

We present, in this dissertation, a pedagogical review of the formalism for
Fermi liquids developed in [1] that exploits an underlying algebro-geometric
structure described by the group of canonical transformations of a single
particle phase space. This infinite-dimensional group governs the space of
states of zero temperature Fermi liquids and thereby allows us to write down a
nonlinear, bosonized action that reproduces Landau’s kinetic theory in the
classical limit. Upon quantizing, we obtain a systematic effective field
theory as an expansion in nonlinear and higher derivative corrections
suppressed by the Fermi momentum $p_{F}$, without the need to introduce
artificial momentum scales through, e.g., decomposition of the Fermi surface
into patches. We find that Fermi liquid theory can essentially be thought of
as a non-trivial representation of the Lie group of canonical transformations,
bringing it within the fold of effective theories in many-body physics whose
structure is determined by symmetries. We survey the benefits and limitations
of this geometric formalism in the context of scaling, diagrammatic
calculations, scattering and interactions, coupling to background gauge
fields, etc. After setting up a path to extending this formalism to include
superconducting and magnetic phases, as well as applications to the problem of
non-Fermi liquids, we conclude with a discussion on possible future directions
for Fermi surface physics, and more broadly, the usefulness of diffeomorphism
groups in condensed matter physics. Unlike [1], we present a microscopic
perspective on this formalism, motivated by the closure of the algebra of
bilocal fermion bilinears and the consequences of this fact for finite density
states of interacting fermions.

To all neurodivergent people, known or unknown,
among whom I finally found a sense of community.

###### Contents

  1. I Introduction
  2. II Review and history of Fermi liquid theory
    1. II.1 “Classical Fermi liquids”: Landau’s kinetic theory
    2. II.2 “Modern Fermi liquids”: Renormalization group
    3. II.3 “Contemporary Fermi liquids”: Patch theory and traditional bosonization
      1. II.3.1 Fermionic patch theory
      2. II.3.2 Bosonization of patch fermions
  3. III Postmodern Fermi liquids: A conceptual overview
    1. III.1 The Lie algebra of fermion bilinears
    2. III.2 The space of states
    3. III.3 Schematic overview of the coadjoint orbit method
  4. IV The operator algebra and the Hamiltonian formalism
    1. IV.1 Semi-classical truncation of the Moyal algebra
    2. IV.2 Constructing the Hamiltonian formalism
      1. IV.2.1 Adjoint and coadjoint representations
      2. IV.2.2 Lie-Poisson structure and Hamiltonian
      3. IV.2.3 Equation of motion
      4. IV.2.4 An alternate route to the Hamiltonian formalism
  5. V Effective action from the coadjoint orbit method
    1. V.1 Fermi surface states and their excitations
      1. V.1.1 Coadjoint orbits and the Kirillov-Kostant-Souriau form
    2. V.2 The Wess-Zumino-Witten term and the effective action
    3. V.3 Symmetries in the postmodern formalism
      1. V.3.1 Galilean invariance
      2. V.3.2 Coupling to $U(1)$ gauge fields
      3. V.3.3 Emergent symmetries
    4. V.4 Perturbative expansion and scaling
    5. V.5 Linear and nonlinear response
      1. V.5.1 Landau damping
      2. V.5.2 Cubic response
    6. V.6 UV/IR mixing and why it is not all that bad
  6. VI A road to perturbative non-Fermi liquids
    1. VI.1 Scaling in non-Fermi liquids191919The results presented in this section are based on ongoing work, soon to appear.
  7. VII Spin and BCS extensions
    1. VII.1 Spinful Fermi surfaces
    2. VII.2 Charged fermion bilinears
  8. VIII Conclusion and Outlook
  9. A Coadjoint orbit method — mathematical details
  10. B Luttinger liquids from the coadjoint orbit method
    1. B.1 Chiral anomaly as a linear approximation

###### Acknowledgements.

It was the summer of 2008, about a month before the beginning of the school
year, and I had just got back home with my backpack full of new textbooks for
class 9. The nerd that I was (and still am), all I could think about on the
way back home was the excitement of getting to open and read the books that I
had just bought; the curious side of me was just excited to absorb all the
knowledge I could from them while the competitive side was daydreaming about
having preemptive answers to all the questions that my teachers would later
ask in class. Having already been mesmerized by science and mathematics from
the year before, my hands were drawn to the physics textbook, since it had the
best colour palette between itself, chemistry, biology, and maths. I picked up
the book, opened the cover, and, energized by that new-book-smell, flipped the
pages right past the first chapter on measurement and experimentation to the
chapters on linear motion and Newton’s laws. In no time I reached the section
on the second law of motion, and noticed a footnote that described the
inaccuracy of the linear relationship between momentum and velocity at speeds
close to the speed of light. The words ‘special theory of relativity’ were
mentioned and before I knew it, two whole years had passed with me having read
every online resource I could possibly find about special and general
relativity and non-Euclidean geometry, convinced that quantum mechanics was
not real because “Einstein didn’t believe in it”. It was in that initial spark
of interest that I knew that I wanted to pursue a career in theoretical
physics, as unorthodox as something like that would be in the culture I grew
up in. I was fortunate enough to have found abundant support for my unusual
career choice from my parents Rita and Bharat Mehta, and late grandparents
Jaya and Kantilal Mehta, for whom my education always took highest priority. I
shall forever be grateful to them for providing me the environment and
encouragement to nurture my passion for physics. My father, in particular, has
made it a point to read every single paper that I have published, even when it
makes no sense to him, and vehemently insists that I send each draft to him
for his growing collection, and I will always be glad that my work will, at
the very least, be read and appreciated by one person who I admire. I found my
first mentor in Shiraz Minwalla at the Tata Institute for Fundamental Research
(TIFR), whose wise words I will always carry with me. He instilled in me the
courage I needed to not shy away from difficult problems and even enjoy the
often long and tedious calculations that accompany them, to the point where I
now get excited at the prospect of taking on such challenges. Shiraz’s advice
was an important contributor to overcoming the many instances of impostor
syndrome that I experienced upon being thrown into the melting pot of all the
tremendously talented individuals that I encountered throughout my Ph.D. But
most importantly, it was on his suggestion that I found my advisor. I couldn’t
have asked for a better advisor than Dam Thanh Son. I switched from high
energy to condensed matter physics upon joining the University of Chicago, and
if it was not for his guidance, I would have had a much harder time with the
transition. In him I found the perfect mentor whose advising style fit with my
learning style like pieces of a jigsaw puzzle. Son’s visionary foresight is
what ultimately lead to the content in the rest of this thesis and I can only
hope to be able to replicate that in the future. I owe a lot to my unofficial
mentor, Luca V. Delacrétaz, from whom I learned various lessons ranging from
the most benign yet consequential tricks to make Mathematica compute integrals
when it is being stubborn, to the valuable philosophy behind effective field
theory. Luca is and always will be a role model to me for my career and
mentorship goals. My Ph.D. experience would not have been half as incredible
as it was if not for the extremely friendly and welcoming environment that my
office-mates cultivated. I’m grateful to Alex Bogatskiy, Harvey Hsiao, Kyle
Kawagoe, Carolyn Zhang, Yuhan Liu, Yi-Hsien Du, Ruchira Mishra, Ege Eren and
Davi Costa for all the wonderful times we had together in our little corner
office, for all the insightful discussions that helped me grow as a physicist.
I also apologize to them for likely being one of the most disruptive and
distracting office-mates that they have encountered. Everyone at the Kadanoff
Center for Theoretical Physics has been pleasantly affable and never once did
I feel like I was not welcome by the professors, postdocs and other graduate
students. My thesis committee members, Michael Levin, Jeffrey Harvey, and
Woowon Kang, were instrumental in making me think deeply about my work and
understand it from various different perspectives. The Center has only become
more social over the last six years and as much as I’m looking forward to the
next step in my career, it saddens me to have to leave behind my wonderful
colleagues and the University of Chicago. Lastly, and perhaps most
importantly, I am deeply indebted to my found family, Timothy Hoffman, Claire
Baum, and Alex Bogatskiy, (and Bowie Hoffman – Tim’s adorable little pupper)
with whom I developed a bond so strong I cannot imagine any force that can
break it. Between the Ph.D., the pandemic, and personal setbacks, the last few
years have been tumultuous and my friends stood by me with all the love and
support for which I was often too afraid to ask. Even on our various rock-
hounding vacations we couldn’t help but discuss physics and I treasure the
precious memories we made along the way. It was thanks to their support that I
persisted through the most prominent milestone of my life – the day that I
discovered that I am neurodivergent. A part of me always knew that I was
different but until then I did not have the resources or the labels that I
needed to look at it under a positive light. The online neurodivergent
community played a major role in this shift of perspective and I am eternally
grateful to have found the community and support network built by empathetic
neurodivergent strangers who likely will never truly see the scale of the
fruits of their efforts. I hope to pay it forward by continuing to advocate
for my fellow neurodivergent people. With this discovery, my life came full
circle to the realization that theoretical physics has always been a so-called
“special interest” for me – a common characteristic of the neurodivergent mind
– and will continue to hold that status for the foreseeable future. I owe my
passion for physics to my neurodivergence and therefore also a large part of
my happiness.

##  I Introduction

From metals to neutron stars, superconductors to nuclear plasmas, phases of
matter described by Fermi surfaces and their instabilities are proliferous.
The question “What are the different possible ways that interacting fermions
can behave at macroscopic scales?” is as easy to pose as it is difficult to
answer. The possibilities are endless and ever-growing and stand tall and
sturdy as a counterpoint to the traditional reductionist-constructivist
hypothesis in physics [2]. To even begin to answer this question, a broad
organizing principle is required.

One such organizing principle is obtained by counting the number of emergent
low energy degrees of freedom that govern the behaviour of such systems. The
notion of an energy gap helps categorize many-body systems into three possible
classes: gapped, gapless and ‘very gapless’.

Gapped systems do not have any propagating, low energy degrees of freedom. The
degrees of freedom here are instead topological in nature and are described by
topological quantum field theories111A new class of these that are not
described by conventional topological field theories have recently been
discovered and are collectively called ‘fracton models’ [3, 4, 5]. For a
review, see [6, 7].. Gapless systems have a finite number of propagating low
energy degrees of freedom. These often describe critical points in phase
diagrams or boundaries of topologically nontrivial gapped phases.

‘Very gapless’ systems on the other hand have infinitely many low energy
degrees of freedom. In particular, the density of states at zero energies is
finite. Systems with extended Fermi surfaces are the canonical example of such
phases, where low energy excitations can be hosted anywhere on the Fermi
surface. Within the realm of Fermi surface physics, a classification of the
possible phases of matter is still elusive, largely due to the many possible
instabilities that Fermi surfaces can have. One suitable starting point for
getting a picture of the various possibilities is to take a free Fermi gas and
turn on interactions between the fermions, allowing them to scatter off of
each other.

The interactions between fermions can then be put into one of two boxes: short
range and long range. Short range interactions are usually mediated by gapped
modes. At low energies these can effectively be thought of as point-like
interactions between fermions with corrections to this description that do not
significantly alter the physical picture. This is the realm of Fermi liquid
theory (and its instabilities), one of the pillars of modern condensed matter
physics, first developed by Landau [8] in a classic 1956 paper. Landau’s key
insight was that short range interactions in most situations do not
dramatically alter the spectrum of excitations of a free Fermi gas. The
excitations of the interacting theory are then very similar to free fermions,
and thus the notion of a quasiparticle was born. Landau’s Fermi Liquid Theory
(LFLT), the classical formalism for describing Fermi liquids, can perhaps be
called the first example of an effective theory \- a low energy description of
a system that is insouciant to microscopic details whose effects are captured
by a comparatively small number of parameters222I thank Luca V. Delacrétaz for
this succinct description of effective theories..

Despite being rather successful at describing the physics of dense,
interacting fermions, LFLT stood out among a plethora of other effective
descriptions in many body physics as one of the few theories that was not
formulated in the language of the renormalization group (RG) and was
classical333Pun intended. in nature, being described by an equation of motion
rather than an action or a Hamiltonian. Progress along these lines was made
only in 1990 in [9], which was then formalized in [10, 11] into the modern
formalism.

The effective field theory (EFT) obtained from this analysis can be simplified
at the cost of losing locality in space [12, 13, 14], so it is not a genuine
EFT in that the tower of irrelevant corrections to the scale invariant fixed
point cannot be systematically listed, for example through an expansion in
spatial and temporal derivatives. An alternate route to a local EFT for Fermi
liquids was inspired by the idea of bosonization and pioneered in [15, 16,
17]. But this approach also suffer from the same issue, in that it is unclear
how one would construct and classify irrelevant corrections to the scale
invariant fixed point. These contemporary formalisms are hence also incomplete
and in need for further refinement.

Long range interactions, on the other hand, are often mediated by gapless
degrees of freedom which cannot be ignored (i.e., integrated out) at any
energy scale, and it becomes important to keep track of the additional gapless
modes alongside the excitations of the Fermi surface. This can alter the
physics of the Fermi surface in ways that are hard to predict, since such
interactions often tend to be strong. A celebrated, now solved example of this
is the electron-phonon problem [18, 19], which accounts for the resistivity
and superconducting instability of conventional metals444For recent work on
the breakdown of the Migdal-Eliashberg theory of electron-phonon interactions,
see [20, 21]..

A more violent example of such an interaction is presented in a class of
phases dubbed non-Fermi liquids (NFL) (see, e.g., [22] and references therein
for a review). The gapless mode that couples to the Fermi surface in these
examples is usually either the critical fluctuation of an order parameter or a
gauge field in appropriate spatial dimensions. Such interactions trigger an
instability of the Fermi surface and the fate of the RG flow is one of the
biggest open problems in condensed matter physics. The list of unanswered
questions ranges from describing the phase of the end point of the RG flow
(metallic NFL or Mott insulator or unconventional superconductor) to
developing effective descriptions of the various possibilities and
understanding how they compete with one another.

Answers to these questions are crucial from an applied physics perspective
since the most common occurrence of NFL physics is in high-temperature
superconductivity [23, 24] observed in various different layered materials
such as cuprates. In many of these materials, the superconducting dome hides a
quantum critical point where the metal undergoes a magnetic phase transition,
the order parameter fluctuations of which couple to the Fermi surface and
drive the instability to a superconductor. The ultimate goal for NFL physics
would be to understand the mechanism that causes high temperature
superconductivity in order to be able to engineer materials which could
enhance this mechanism and raise the critical temperature of the
superconducting phase to larger values, possibly even to room temperature.

From a theoretical standpoint, Fermi and non-Fermi liquids provide a unique
playground to explore unconventional RG flows. Almost all tractable RG flows
in physics are between two scale invariant fixed points that have no inherent
scales. Fermi and non-Fermi liquids, however, enjoy scale invariance despite
the presence of an intrinsic scale – the Fermi momentum $p_{F}$ – and
understanding the RG flow from one to the other hence necessarily requires a
broadening of the notion of RG as well as that of a ‘scale’. Unconventional RG
flows have been gaining interest across various disciplines ranging from the
study of fractonic and exotic theories [25, 26, 27, 28, 29] to machine
learning [30, 31] and even information theory and neuroscience [32, 33], and
it is likely that Fermi surface physics can serve as a useful launchpad for
generalizing the notion of RG beyond its rigid framework and conventional
metanarrative.

A fundamental bottleneck to understanding the physics of non-Fermi liquids is
the lack of an EFT description for Fermi liquids. Since the scaling behaviour
of an NFL can differ dramatically from that of a Fermi liquid, irrelevant
corrections to any effective theory of a Fermi liquid can have important
consequences for the NFL. A classification of irrelevant corrections to Fermi
liquid theory with definite scaling properties, which is missing from the
literature so far, would thus hugely benefit the search for an effective
description for NFLs.

This is precisely the aim of the postmodern formalism developed in [1] and
expounded upon in this thesis. We find that LFLT is secretly governed by the
geometry of a rather large Lie group – that of canonical transformations of a
single-particle phase space. This constrains the structure of the effective
theory for Fermi liquids rigidly enough to be able to construct higher order
corrections to the contemporary approaches as well as classify their scaling
behaviour. The geometric structure underlying the postmodern formalism also
allows us to systematically identify and impose symmetries as well as couple
to gauge fields.

Such diffeomorphism groups are not only important for Fermi liquid theory, but
also present themselves as a useful tool across other disciplines in condensed
matter physics, such as quantum Hall states, lattices of charged monopoles or
superfluid vortices and even skyrmions in ferromagnets [34, 35], suggesting
that diffeomorphism groups have the potential to broadly understand and
constrain the properties of various many-body phases.

The rest of this dissertation is organized as follows: in section II we review
the various historic approaches to Fermi liquid theory and comment on the
benefits and drawbacks of each of them. In section III we summarize the
postmodern formalism and provide an overview that is stripped off of most
technical details for simplicity. In section IV we develop the Hamiltonian
formalism for Fermi liquids, which is then turned into an action formalism in
section V. Section V also presents how this action encodes spacetime, gauge,
and emergent symmetries, as well as how it simplifies the calculation of
correlation functions in Fermi liquids. Section VI then explores how the
postmodern formalism can be used as a stepping stone towards perturbative
NFLs. In section VII we then switch gears to present different possible
generalizations of the postmodern formalism that account for internal
symmetries, conventional superconductivity, and large momentum processes.
Finally, we conclude in section VIII with an outlook on the various potential
applications of the postmodern formalism.

##  II Review and history of Fermi liquid theory

We begin by reviewing the various approaches to describing Fermi liquids that
have been developed over the last century. This discussion is by no means
exhaustive, and we will differ to relevant references for more details.

###  II.1 “Classical Fermi liquids”: Landau’s kinetic theory

Figure 1: Hokusai’s rendition of a propagating mode in kinetic theory.

The very first description for Fermi liquids was proposed by Landau in the
form of a kinetic equation. Consider first a gas of non-interacting fermions.
Owing to Pauli’s exclusion principle, its ground state at zero temperature is
described by a occupation number function in momentum space
$f_{0}(\mathbf{p})=\Theta(\epsilon_{F}-\epsilon(\mathbf{p}))$ that takes
values 1 or 0. $\epsilon_{F}$ is the Fermi energy and $\epsilon(\mathbf{p})$
is the dispersion relation for a single fermion. The solution to the equation,

$\epsilon(\mathbf{p})=\epsilon_{F}\,,$ (1)

defines the Fermi surface at

$|\mathbf{p}|=p_{F}(\theta)\,.$ (2)

If the dispersion relation is invariant under rotations, the Fermi momentum
$p_{F}$ is a constant independent of the angles $\theta$ in momentum space.
The dynamics of this system is described by a mesoscopic555By the word
‘mesoscopic’, we mean a regime where we are concerned with physics at length
scales much larger than a characteristic length scale, here $1/p_{F}$. This
allows us to describe quantum particles in a semi-classical description using
coordinates $\mathbf{x}$ that label the mesoscopic region of size $1/p_{F}$
that the quantum particle is localized within, and momentum $\mathbf{p}$ of
the particle up to uncertainty. one-particle distribution function
$f(t,\mathbf{x},\mathbf{p})=f_{0}(\mathbf{p})+\delta
f(t,\mathbf{x},\mathbf{p})$ that obeys the collisionless Boltzmann equation:

$\partial_{t}f+\nabla_{\mathbf{p}}\epsilon(\mathbf{p})\cdot\nabla_{\mathbf{x}}f+\mathbf{F}_{\text{ext}}\cdot\nabla_{\mathbf{p}}f=0\,,$
(3)

where $\mathbf{F}_{\text{ext}}$ is the external force applied to the free
Fermi gas. The dynamics of the free Fermi gas are hence entirely captured by
the dispersion relation.

For an interacting Fermi liquid, however, the occupation number at every
momentum is not a well-defined quantum number, and we cannot characterize its
dynamics using the distribution function.

Landau’s argument to work around this issue was the following: suppose we
start with the free Fermi gas and turn on interactions adiabatically. Thanks
to Pauli exclusion principle, the available phase space for the fermions to
scatter to is significantly smaller the closer they are to the Fermi surface
initially. The low energy ($E\ll\epsilon_{F}$) part of the interacting many-
body spectrum should be continuously deformable to the spectrum of the free
theory. Since the spectrum of the free Fermi gas can be constructed from the
building block of a single fermion placed outside but close to the Fermi
surface (or a single hole inside), this building block should persist as the
interactions are adiabatically turned on and also exist in some “dressed” form
in the low energy spectrum of the interacting Fermi liquid. The remnant of
this building block in the interacting theory is what we call a quasiparticle.

In situations where this argument holds, we should have an effective single-
particle description for the dynamics of interacting Fermi liquids, analogous
to the collisionless Boltzmann equation for free fermions. In fact, Fermi
liquids are defined retroactively as fermionic phases of matter where this
argument holds. The degree of freedom describing the quasiparticle is then
also a distribution function:

$f(t,\mathbf{x},\mathbf{p})=f_{0}(\mathbf{p})+\delta
f(t,\mathbf{x},\mathbf{p})\,.$ (4)

However, since the quasiparticle only exists as part of the spectrum for
momenta close to the Fermi surface, the distribution $f$ and the fluctuation
$\delta f$ are only well defined in a narrow region $|\mathbf{p}|-p_{F}\ll
p_{F}$. All that we need in order to describe the low energy dynamics of the
interacting Fermi liquid is a dispersion relation $\epsilon_{\text{qp}}$ for
the quasiparticle. This is phenomenologically constructed as follows:

$\epsilon_{\text{qp}}(\mathbf{x},\mathbf{p})=\epsilon(\mathbf{p})+\int\frac{d^{d}p^{\prime}}{(2\pi)^{d}}F(\mathbf{p},\mathbf{p}^{\prime})\delta
f(\mathbf{x},\mathbf{p}^{\prime})\,,$ (5)

where $\epsilon(\mathbf{p})$ is the free fermion dispersion relation, and
$F(\mathbf{p},\mathbf{p}^{\prime})$ is a phenomenological function that
characterizes the interaction contribution to the energy of the quasiparticle
at $\mathbf{p}$ due to quasiparticles at $\mathbf{p}^{\prime}$. Note that the
interaction term in the quasiparticle energy is local in space, which is due
to the assumption that any interaction between the quasiparticles is short-
ranged.

At the risk of being pedantic, we emphasize again that the quasiparticle
energy, the interaction function, and the distribution are well-defined only
in a small neighbourhood of the Fermi surface. In other words the $\mathbf{p}$
derivatives of all these quantities are only well-defined at the Fermi surface
and constitute the various parameters and degrees of freedom of the effective
theory.

We can now postulate a collisionless Boltzmann equation that describes the
dynamics of the interacting Fermi liquid:

$\partial_{t}f+\nabla_{\mathbf{p}}\epsilon_{\text{qp}}[f]\cdot\nabla_{\mathbf{x}}f-\left(\nabla_{\mathbf{x}}\epsilon_{\text{qp}}[f]-\mathbf{F}_{\text{ext}}\right)\cdot\nabla_{\mathbf{p}}f=0\,.$
(6)

We will refer to this equation as Landau’s kinetic equation. One crucial
difference between the interacting Fermi liquid and the free Fermi gas is that
equation (6) is nonlinear in $\delta f$, while the collisionless Boltzmann
equation is linear. The nonlinearity comes from the dependence of the
quasiparticle energy on the distribution. This also modifies the dynamics at
the linear level, since the interaction results in internal forces
$\nabla_{\mathbf{x}}\epsilon_{\text{qp}}$ acting on the quasiparticles in
addition to any external forces.

Since the interaction function $F(\mathbf{p},\mathbf{p}^{\prime})$ is well-
defined only near the fermi surface, one often assumes that it only depends on
two points on the Fermi surface at the angles $\theta,\theta^{\prime}$, and an
angular expansion of the interaction function defines the so-called Landau
parameters,

$F(\theta,\theta^{\prime})\sim\sum_{l}F_{l}P^{(d)}_{l}(\theta,\theta^{\prime})\,,$
(7)

where $P^{(d)}_{l}(\theta,\theta^{\prime})$ form a basis of functions in $d$
dimensions that transform covariantly under the symmetries of the Fermi
surface, and $l$ is a label for the representations of those symmetries. For
example, for a spherical Fermi surface $l=0,1,2,\ldots$ is an ‘angular
momentum’ index, and the basis functions are cosines in $d=2$ and Legendre
polynomials of cosines in $d=3$.

From Landau’s kinetic equation we can calculate a plethora of physical
quantities from thermodynamic properties to correlation functions, in terms of
Landau parameters which encode the microscopic interactions. In order to
calculate correlation functions for, e.g., the particle number density and
current, we can couple the theory to background electromagnetic fields through
the Lorentz force
$\mathbf{F}_{\text{ext}}=\mathbf{E}+\mathbf{v}\times\mathbf{B}$.

One finds stability conditions for the theory as lower bounds on $F_{l}$ which
when violated, result in Pomeranchuk instabilities. For certain ranges of the
Landau parameters, Fermi liquids also exhibit a collective excitation known as
zero sound that propagates faster than the Fermi velocity
$v_{F}=\epsilon^{\prime}(p_{F})$ and is hence distinguishable from the
particle-hole continuum $\omega\leq v_{F}|\mathbf{q}|$ (figure 1). The
specific calculations that result in these various properties and more can be
found, for example, in [36, 37].

While LFLT describes many aspects of interacting Fermi liquids quite well, it
has various drawbacks. Firstly, it is unclear how such a theory would emerge
from a microscopic model. Since the kinetic equation is written down ‘by hand’
it is not even clear when one should expect a microscopic model of interacting
fermions to be described by LFLT.

Second, being an equation-of-motion based description, LFLT is in effect a
classical theory, with the only source of ‘quantumness’ being Pauli exclusion
and the Fermi-Dirac distribution that gives the ground state $f_{0}$ of the
theory. In practice this means that the theory is blind to subleading
corrections to physical quantities such as correlation functions and
thermodynamic properties.

These drawbacks would be at least partially, if not completely be remedied by
a field theoretic description - one that is amenable to the renormalization
group (RG), unlike LFLT.

###  II.2 “Modern Fermi liquids”: Renormalization group

Figure 2: Van Gogh’s visualization of scaling towards a (rectangular) Fermi
surface.

To understand the scaling behaviour of interacting Fermi liquids, we need to
pick an RG scheme. The prototypical RG scheme most commonly used in physics,
wherein we rescale length to be larger and larger, or equivalently rescale
momenta to 0, also shrinks the Fermi surface down to a point! This scheme
cannot possibly give physically relevant results since the Fermi surface is an
experimentally measurable quantity. We hence need to pick a new scaling
scheme.666It is important to note that in most commonly studied systems in
physics such as quantum or statistical field theories, the symmetries of the
system uniquely prescribe the RG scheme that can extract universal information
from it. Here, however we encounter a system where this is not immediately
obvious, so we need to look for other identifiers for the ‘correct’
prescription.

The most natural RG scheme is one where momenta are rescaled towards the Fermi
surface (figure 2). This scheme was introduced in [9, 10, 11] and is commonly
referred to as ‘Shankar-Polchinski’ RG, after the physicists who independently
formalized it.

In the spirit of effective field theory, we first identify the low energy
degrees of freedom. LFLT tells us that these are fermionic quasiparticles. We
define an operator $\psi^{\dagger}(\mathbf{p})$ that creates a quasiparticle
with momentum $\mathbf{p}$. The annihilation operator $\psi(\mathbf{p})$
creates a hole in the Fermi sea at the point $-\mathbf{p}$, so that the net
momentum of the state with a single hole is $+\mathbf{p}$777This is different
from the usual convention employed in condensed matter physics, where the
operator $c(\mathbf{p})$ creates a hole at the point $\mathbf{p}$, thereby
creating a state with momentum $-\mathbf{p}$. We use the less common
convention since in our convention, both $\psi^{\dagger}$ and $\psi$ are
Fourier transformed in the same way. This sets a uniform convention for
Fourier transforms, allowing us to Fourier transform with impunity without
having keep track of sign conventions any more than necessary.. The free
action is given by

$\int\frac{dtd^{d}p}{(2\pi)^{d}}\psi^{\dagger}(\mathbf{p})\left[i\partial_{t}-(\epsilon(\mathbf{p})-\epsilon_{F})\right]\psi(-\mathbf{p})\,.$
(8)

Each point $\mathbf{p}$ in momentum space can be written as a sum of a vector
$\mathbf{p}_{F}$ on the Fermi surface and another vector $\mathbf{k}$
orthogonal to the Fermi surface at $\mathbf{p}_{F}$:

$\mathbf{p}=\mathbf{p}_{F}+\mathbf{k}\,,\qquad
d^{d}p=d^{d-1}p_{F}\leavevmode\nobreak\ dk\,,$ (9)

where $d^{d-1}p_{F}$ is a measure for integrating over the Fermi surface. In
our RG scheme, $\mathbf{p}_{F}$ remain invariant under scaling, while
$\mathbf{k}$ get rescaled by a factor of $s\lesssim 1$ to $s\mathbf{k}$. The
dispersion can be expanded to leading order so that

$\epsilon(\mathbf{p})-\epsilon_{F}=|\mathbf{k}||\mathbf{v}_{F}(\mathbf{p}_{F})|+\mathcal{O}(k^{2})\,,$
(10)

and marginality of the free action requires

$[\partial_{t}]=[\mathbf{k}]\,,\qquad[\psi]=-\frac{1}{2}\,.$ (11)

We then write down all possible terms allowed by symmetries and analyze their
scaling behaviour, both at tree level and at loop level. The leading
nontrivial term is a quartic interaction that enables nontrivial $2\rightarrow
2$ scattering processes:

$\int_{t}\int_{\mathbf{p}_{1}\mathbf{p}_{2}\mathbf{p}_{3}\mathbf{p}_{4}}V(\mathbf{p}_{F1},\mathbf{p}_{F2},\mathbf{p}_{F3},\mathbf{p}_{F4})\psi^{\dagger}(\mathbf{p}_{1})\psi(\mathbf{p}_{2})\psi^{\dagger}(\mathbf{p}_{3})\psi(\mathbf{p}_{4})\delta(\mathbf{p}_{1}+\mathbf{p}_{2}+\mathbf{p}_{3}+\mathbf{p}_{4})\,.$
(12)

Immediately, we notice two possibilities for the scaling of the momentum
conserving delta function. If the corresponding Fermi momenta
$\mathbf{p}_{Fi}$ sum to zero, the delta function scales non-trivially under
our RG scheme, while if they do not, the delta function is (approximately)
invariant under the scale transformation.

For configurations where $\sum_{i}\mathbf{p}_{Fi}\neq 0$, we find that the
quartic term is strictly irrelevant and hence does not change the scale
invariant fixed point. For configurations with $\sum_{i}\mathbf{p}_{Fi}=0$, on
the other hand, the quartic term is marginal. All that remains is find
configurations for which the sum vanishes, and check whether loop corrections
change the scaling behaviour of the terms corresponding to the relevant
configurations.

Consider for instance $d=2$ with a circular Fermi surface. There are two
distinct classes of configurations with $\sum\mathbf{p}_{F}=0$:

$(\mathbf{p}_{F2}=-\mathbf{p}_{F1},\leavevmode\nobreak\
\mathbf{p}_{F4}=-\mathbf{p}_{F3})\,;\qquad(\mathbf{p}_{F3}=-\mathbf{p}_{F1},\leavevmode\nobreak\
\mathbf{p}_{F4}=-\mathbf{p}_{F2})\,.$ (13)

The solution with $\mathbf{p}_{F4}=-\mathbf{p}_{F1}$ is just the first
solution with the hole momenta exchanged. The first class of solutions
characterize forward scattering, i.e., incoming particles leave with nearly
the same or exchanged momenta. These correspond to particle hole pairs with a
small net momenta, such as the configuration in figure 3(a). This class of
configurations is hence often called the ‘particle-hole channel’. The form
factor
$F(\mathbf{p}_{F1},\mathbf{p}_{F3})=V(\mathbf{p}_{F1},-\mathbf{p}_{F1},\mathbf{p}_{F3},-\mathbf{p}_{F3})$
is the corresponding interaction function.

The second class of solutions has the two particles as well as the two holes
align at antipodal points on the Fermi surface respectively, with an arbitrary
angle between them, for instance in figure 3(b). This configuration
corresponds to the ‘Bardeen-Cooper-Schrieffer (BCS) channel’. The interaction
form factor
$g(\mathbf{p}_{F1},\mathbf{p}_{F2})=V(\mathbf{p}_{F1},\mathbf{p}_{F2},-\mathbf{p}_{F1},-\mathbf{p}_{F2})$
for this is independent of the forward scattering interaction, except in one
special configuration with $\mathbf{p}_{F3}=\mathbf{p}_{F2}=-\mathbf{p}_{F1}$
which imposes a constraint $F(\mathbf{p}_{F},-\mathbf{p}_{F})$ =
$g(\mathbf{p}_{F},-\mathbf{p}_{F})$. The marginal quartic terms can then be
written schematically as

$\int_{\mathbf{p}_{1}\mathbf{p}_{3}}F(\mathbf{p}_{F1},\mathbf{p}_{F3})[\psi^{\dagger}\psi\psi^{\dagger}\psi]_{\text{ph}}(\mathbf{p}_{1},\mathbf{p}_{3})+\int_{\mathbf{p}_{1}\mathbf{p}_{2}}g(\mathbf{p}_{F1},\mathbf{p}_{F2})[\psi^{\dagger}\psi\psi^{\dagger}\psi]_{\text{BCS}}(\mathbf{p}_{1},\mathbf{p}_{2})\,.$
(14)

(a) Forward scattering

(b) BCS channel

Figure 3: Scattering configurations for marginal interactions at tree level.

Both interactions are marginal at tree level, but a one-loop calculation shows
that while forward scattering remains marginal, the BCS interaction becomes
relevant if the coupling is attractive and irrelevant if the coupling is
repulsive. Hence, attractive couplings in the BCS channel trigger a
superconducting instability that destroys the Fermi surface.

The forward scattering interaction is just the interaction function in LFLT,
but the BCS coupling is one to which LFLT is blind. The inclusion of the
pairing instability is the most important advantage of the RG approach over
LFLT, and exemplifies the power of effective field theory.

However, this approach still has its limitations. Ideally in an EFT, any
isolated term that can be written from symmetry requirements has a fixed
scaling dimension which can be calculated simply by adding the scaling
dimensions of its constituents — a principle known as power counting. But as
we saw above, understanding the scaling properties of the quartic term was a
significantly more complicated task than that, and becomes even more
complicated in higher dimensions where the number of possible configurations
with $\sum\mathbf{p}_{F}=0$ is even larger. This procedure becomes all the
more gruesome for Fermi surfaces of more complicated geometry such as those
for conduction electrons in metals.

In general, any given term in this EFT that can be written from invariance
under symmetries does not have a fixed scaling dimension and additional work
needs to be done to decompose it into a sum of terms that do. Even then one
can find constraints relating one term to another in special cases, such as
the configuration
$\mathbf{p}_{F1}=-\mathbf{p}_{F2}=-\mathbf{p}_{F3}=\mathbf{p}_{F4}$ where the
exactly marginal forward scattering coupling is identical to the marginally
relevant or irrelevant BCS coupling. These constraints need to be kept track
of by hand and do not immediately follow from any symmetry principle. Instead,
the forward scattering – BCS constraint is a consequence of hacing to
decompose a single local operator into different scattering channels that are
scaling covariant, but at the cost of an added redundancy.

Furthermore, while coupling LFLT to background gauge fields was a
straightforward task, it is much less obvious how one couples this EFT to
background gauge fields, given that the EFT lives in momentum space, where no
standard minimal coupling procedure exists.

Two remedies for the former issue have been considered, which we will
collectively refer to as the ‘contemporary’ formalism, which we review next.
Alternate functional RG schemes for Fermi surfaces which hope to capture
physics beyond Shankar-Polchinski RG have also recently been developed in [38,
39].

###  II.3 “Contemporary Fermi liquids”: Patch theory and traditional
bosonization

Figure 4: Dali’s self-portrait under a patch decomposition.

One of the key takeaways of the Shankar-Polchinski RG scheme is that, barring
BCS interactions, particle-hole pairs have a significant impact on low energy
physics only when they are sufficiently close to each other in momentum space
(compared to $p_{F}$). This suggests that one potential workaround to the
issue of interactions not having fixed scaling dimensions is the following: we
can discretize the Fermi surface to a number of patches of the same size,
labelled by a discrete index $\eta$ (figure 4), and subsequently separate
interactions into intra-patch and inter-patch scattering.

The free fermion action Fourier transformed back to coordinate space can be
written as a sum over patches,

$S=\sum_{\eta}\int d^{d-1}x_{\parallel}\int
dtdx_{\perp}\Psi_{\eta}^{\dagger}\leavevmode\nobreak\
(x_{\perp})\left(\partial_{t}+v_{F\eta}\partial_{x_{\perp}}\right)\Psi_{\eta}(x_{\perp})\,,$
(15)

where $x_{\perp}$ is a coordinate that is Fourier-conjugate to $\mathbf{k}$,
the momentum vector orthogonal to the Fermi surface, $\mathbf{x}_{\parallel}$
are coordinates conjugate to the transverse directions within a patch, and
$\Psi_{\eta}$ is the fermion on each patch defined by

$\psi(\mathbf{x})=\sum_{\eta}e^{i\mathbf{p}_{F\eta}\cdot\mathbf{x}}\Psi_{\eta}(x_{\perp})\,,$
(16)

up to normalization. This is simply a collection of chiral fermions at each
patch. Intra-patch scattering terms live within a single patch $\eta$, while
inter-patch scattering terms couple two different patches
$\eta\neq\eta^{\prime}$. If we restrict our attention to a single patch
$\eta_{0}$, the effect of the latter is simply a logarithmic renormalization
of the field strength of $\Phi_{\eta_{0}}$ as well as its dispersion relation,
so inter-patch interactions can be ignored. Intra-patch coupling can be
analyzed in the usual way under rescaling of momenta toward the Fermi surface,
transverse to the patch. Since the width of the patch is not rescaled in this
procedure, the number of patches does not change under rescaling.

####  II.3.1 Fermionic patch theory

The patch theory in the Shankar-Polchinski RG scheme has an important
drawback. Discretizing the Fermi surface makes it so that each patch is
effectively flat at low energies. To see this, consider the leading irrelevant
correction to the quadratic action, which comes from the curvature of the
Fermi surface within the patch,

$S=\int d^{d-1}x_{\parallel}\int
dtdx_{\perp}\Psi^{\dagger}\leavevmode\nobreak\
(x_{\perp})\left(\partial_{t}+v_{F}\partial_{x_{\perp}}+\frac{\kappa}{2}\nabla_{\parallel}^{2}\right)\Psi(x_{\perp})\,,$
(17)

where we have dropped the patch index $\eta_{0}$. Since
$\mathbf{x}_{\parallel}$ does not scale under the Shankar-Polchinski RG
scheme, the curvature $\kappa$ scales to zero and we lose crucial information
about the shape of the Fermi surface.

Figure 5: A single Fermi surface patch

An alternate RG scheme that is more suitable to the patch description [13,
14](see, e.g., [40] for a pedagogical description) and preserves the curvature
of the Fermi surface is one where the coordinates $\mathbf{x}_{\parallel}$
scale like $(x_{\perp})^{1/2}$. The curvature term is now scale invariant
under this scale transformation, at the expense of the width of the patch
scaling down to zero, resulting in a proliferation of the number of patches at
the scale invariant fixed point. But if we are only concerned with the low
energy properties of fermions within a single patch, we can ignore this
drawback. As far as I am aware, so systematic analysis of the consequences of
the proliferation of the number of patches exists in the literature, and in
particular it is unclear whether this blow up modifies the RG flow of a single
patch in any significant way.

One can show that intra-patch scattering from contact interactions under patch
scaling is strictly irrelevant in all dimensions, which provides some evidence
for the stability of Fermi liquids. Inter-patch couplings can at most
logarithmically renormalize the field strength of the patch fermion and the
Fermi velocity, and are often ignored. The only interactions that can modify
the RG flow are then those that are mediated by a gapless mode. Fermionic
patch theory is hence often used as an effective description for non-Fermi
liquids, since it provides an RG scheme where other interactions between patch
fermions can be safely ignored, in favour of interactions mediated by the
gapless mode which couples most strongly to patches that are tangential to its
momentum [41, 12].

Fermionic patch theory has a few more drawbacks. Firstly, in restricting the
theory to a single patch, we loose locality in position space. Secondly,
single-patch theory cannot accomodate BCS interactions either, which raises
questions about the validity of RG flows derived from it. The usual
expectation and/or hope is that the NFL fixed point obtained from patch theory
would have its own superconducting instability, which would lead it to a
superconducting fixed point with the same universal properties as the infrared
(IR) fixed point of the physical RG flow without restricting to patches.
Lastly, patch theory can only be used for understanding RG flows, but not for
calculating physical quantities such as transport properties, for which we
need to sum over all patches and be mindful about the proliferation of patches
in the IR. Furthermore, the resistance of the Shankar-Polchinski EFT to
gauging persists in fermionic patch theory as well.

Additionally, even though fermionic patch theory has attractive properties
under RG and simplifies the calculation of scaling dimensions for various
operators, the scaling behaviour of correlation functions calculated from
patch theory is still not transparent. Various cancellations among diagrams
can occur [42, 43] that alter the IR scaling form of the correlation functions
and invalidate power counting arguments. We will discuss this in more detail
in section V.5 and demonstrate how the postmodern formalism resolves this
difficulty.

####  II.3.2 Bosonization of patch fermions

Another approach that starts with the description in terms of patchwise chiral
fermions but tries to preserve locality in position space is inspired by
bosonization in 1+1d [44]. This approach was developed independently by
Haldane [15] and by Castro-Neto and Fradkin [16], and further developed by
Houghton, Kwon and Marston [17]. Since each patch fermion is a 1+1d chiral
fermion, it can be independently bosonized into a collection of chiral bosons
to give the following effective action:

$S=-p_{F}^{d-3}\sum_{\eta}\int dtd^{d}x\leavevmode\nobreak\
(\mathbf{p}_{F\eta}\cdot\nabla_{\mathbf{x}}\phi_{\eta})\left(\partial_{t}+v_{F\eta}\mathbf{p}_{F\eta}\cdot\nabla_{\mathbf{x}}\right)\phi_{\eta}\,.$
(18)

Although this formalism is local in position space, it suffers from the same
drawback as patch theory under Shankar-Polchinski scaling — it cannot
accomodate nonlinear-in-$\phi_{\eta}$ corrections from Fermi surface curvature
and the dispersion relation. This has serious consequences, since even though
the nonlinear corrections are irrelevant in Shankar-Polichinski scaling, they
contribute at leading order to various higher point correlation functions,
which traditional bosonization sans higher order corrections incorrectly
suggests would vanish. For instance, the particle number density in
traditional bosonization is linear in $\phi$, and since the action is
quadratic in $\phi$, the density $(n>2)$-point functions calculated from this
action are strictly zero, which certainly is not the case even for free
fermions.

In order to solve this issue, various authors appealed to a more algebro-
geometric picture underlying the interpretation of Fermi liquid theory as
describing the dynamics of droplets in phase space [45, 46, 47, 48, 49, 50]
similar to quantum Hall droplets on the lowest Landau level in the plane [51,
52, 53]. This approach is an early precursor to the postmodern formalism
described in this dissertation.

##  III Postmodern Fermi liquids: A conceptual overview

Figure 6: An artificial intelligence’s impression of postmodern Fermi liquid
theory.

The starting point for our theory is the observation that the operator algebra
constructed from microscopic fermions $\psi(\mathbf{x})$ has a sub-algebra
that is closed under commutators. This is the algebra of operators spanned by
(anti-Hermitian) charge 0 fermion bilinears (see section IV for details and
precise definitions),

$T(\mathbf{x},\mathbf{y})\sim i\psi^{\dagger}(\mathbf{x})\psi(\mathbf{y})\,.$
(19)

For theories whose Hamiltonian can be written entirely in terms of these
bilinears, the closure of the sub-algebra guarantees that we can restrict our
attention to the dynamics of operators in this sub-algebra in the Heisenberg
picture, or classes of states distinguished only by expectation values of such
operators in the Schrödinger picture.

What remains is to find a convenient parametrization for this large space of
operators, or equivalently, for the dual space of of states, and figure out
how to identify states with Fermi surfaces, to which the next two sections are
dedicated. While this is straightforward in principle, some assumptions and
approximations need to be made to make it useful in practice. These will be
elucidated in the following section.

Conveniently, the question of how to parametrize a Lie algebra and its dual
space has a well-established answer in mathematical literature, known as the
coadjoint orbit method [54, 55, 56]. This method was historically developed as
a procedure for finding representations of Lie groups, but can also be
interpreted as a means of setting up a dynamical system on a Lie group in the
Hamiltonian formalism, and then turning that Hamiltonian formalism into an
action. The Hamiltonian/action describe time evolution on the Lie algebra,
which in our case is the space of fermion bilinears, in the Heisenberg
picture, or equivalently on its dual space, which is the space of states, in
the Schrödinger picture888Quantization of this action then gives
representations of the Lie group under consideration..

###  III.1 The Lie algebra of fermion bilinears

Fermion bilinears $T(\mathbf{x},\mathbf{y})$ form a basis for our Lie algebra,
which we will call $\mathfrak{g}$. A general element of this algebra is a
linear combination,

$O_{F}\equiv\int d^{d}xd^{d}y\leavevmode\nobreak\
F(\mathbf{x},\mathbf{y})T(\mathbf{x},\mathbf{y})\sim i\int
d^{d}xd^{d}y\leavevmode\nobreak\
F(\mathbf{x},\mathbf{y})\psi^{\dagger}(\mathbf{x})\psi(\mathbf{y})\,,$ (20)

where $F(\mathbf{x},\mathbf{y})$ is a generic function of two variables. It
will be more convenient for us to work with the Wigner transform of the
generators:

$T(\mathbf{x},\mathbf{p})\equiv\int d^{d}y\leavevmode\nobreak\
T\left(\mathbf{x}+\frac{\mathbf{y}}{2},\mathbf{x}-\frac{\mathbf{y}}{2}\right)e^{i\mathbf{p}\cdot\mathbf{y}}\,,$
(21)

in which case, a general element of the Lie algebra,

$O_{F}\equiv\int\frac{d^{d}xd^{d}p}{(2\pi)^{d}}F(\mathbf{x},\mathbf{p})T(\mathbf{x},\mathbf{p})\,,$
(22)

is characterized instead by a function $F(\mathbf{x},\mathbf{p})$ of
coordinates and momenta instead. The function $F(\mathbf{x},\mathbf{p})$ can
be thought of as the components of the Lie algebra vector $O_{F}$, with
$\mathbf{x},\mathbf{p}$ being indices. Since we have already picked a
preferred basis for $\mathfrak{g}$, we will often refer to the the function
$F$ itself as the Lie algebra vector by a slight abuse of terminology.

Using the anti-commutation relations for the fermion creation and annihilation
operators, one can show that the commutator of two Lie algebra vectors
corresponding to functions $F(\mathbf{x},\mathbf{p})$ and
$G(\mathbf{x},\mathbf{p})$ takes the following form:

$[O_{F},O_{G}]=O_{\\{\\!\\!\\{F,G\\}\\!\\!\\}}\,,$ (23)

where the operation in the subscript of the right hand side is the Moyal
bracket of two functions,

$\\{\\!\\!\\{F,G\\}\\!\\!\\}(\mathbf{x},\mathbf{p})\equiv
2\leavevmode\nobreak\
F(\mathbf{x},\mathbf{p})\sin\left(\frac{\overleftarrow{\nabla}_{\mathbf{x}}\cdot\overrightarrow{\nabla}_{\mathbf{p}}-\overleftarrow{\nabla}_{\mathbf{p}}\cdot\overrightarrow{\nabla}_{\mathbf{x}}}{2}\right)G(\mathbf{x},\mathbf{p})\,.$
(24)

Note that up until this point, all of our formulas are exact. So far we are
working in the full quantum theory, despite the simultaneous occurrence of
both position and momentum. This is essentially achieved by a quantization
scheme that is different from but equivalent to canonical quantization, known
as Weyl quantization (or deformation quantization for more general phase
spaces).

Our Lie algebra can hence be characterized as the set of all functions of a
single-particle phase space, equipped with the Moyal bracket,

$\mathfrak{g}_{\text{Moyal}}\equiv\left(\\{F(\mathbf{x},\mathbf{p})\\};\\{\\!\\!\\{.,.\\}\\!\\!\\}\right)\,.$
(25)

We will refer to this as the Moyal algebra or the Weyl algebra999The Weyl
algebra is actually a subalgebra of the Moyal algebra, consisting of only
polynomial functions.. The associated Lie group consists of the exponents of
the bilinear operators $e^{\mathcal{O}_{F}}$. The coadjoint orbit method can
be applied directly to the Moyal algebra to yield a formal action that would
in principle exactly describe Fermi surfaces, but this action is unwieldy in
practice, owing to the fact that the Moyal bracket in equation (24) is only
defined in a power series in phase space derivatives, with convergence of the
power series having been established only for limited classes of functions
[57].

To ameliorate this issue, we can consider a truncation of the Moyal algebra to
leading order in the series expansion, which gives the Poisson bracket,

$\begin{split}\\{\\!\\!\\{F,G\\}\\!\\!\\}&=\\{F,G\\}+\mathcal{O}(\nabla_{\mathbf{x}},\nabla_{\mathbf{p}})^{3}\,,\\\
\\{F,G\\}&\equiv\nabla_{\mathbf{x}}F\cdot\nabla_{\mathbf{p}}G-\nabla_{\mathbf{p}}F\cdot\nabla_{\mathbf{x}}G\,,\end{split}$
(26)

providing an approximate, semi-classical, action-based description of Fermi
liquids via the coadjoint orbit method applied to the truncated Lie algebra of
the set of functions of a single-particle phase space, equipped with the
Poisson bracket instead of the Moyal bracket,

$\mathfrak{g}=(\\{F(\mathbf{x},\mathbf{p})\\};\\{.,.\\})\,.$ (27)

We will refer to this as the Poisson algebra. Importantly, this is the only
truncation of the Moyal algebra that preserves the Jacobi identity. We
emphasize that the Poisson algebra is not a sub-algebra of the Moyal algebra,
but rather a truncation of the Lie bracket.

The Poisson algebra has a useful physical interpretation that can be assigned
to it: it is the Lie algebra of infinitesimal canonical transformations of the
single-particle phase space. A typical element $F(\mathbf{x},\mathbf{p})$ of
the Poisson algebra generates a canonical transformation in the following way:
we can define new coordinates,

$\begin{split}\mathbf{x}^{\prime}&=\mathbf{x}-\nabla_{\mathbf{p}}F\,,\\\
\mathbf{p}^{\prime}&=\mathbf{p}+\nabla_{\mathbf{x}}F\,.\end{split}$ (28)

We can verify that the transformed coordinates
$\mathbf{x}^{\prime},\mathbf{p}^{\prime}$ are canonical pairs. This
transformation can be understood as Hamiltonian evolution for infinitesimal
time under the Hamiltonian $F(\mathbf{x},\mathbf{p})$, and we can also verify
that the commutator of two such infinitesimal transformations parametrized by
functions $F(\mathbf{x},\mathbf{p})$ and $G(\mathbf{x},\mathbf{p})$ is an
infinitesimal transformation parametrized by the Poisson bracket
$\\{F,G\\}(\mathbf{x},\mathbf{p})$. The quickest way to see this is to note
that the infitesimal transformation is generated by the phase space vector
field:

$X_{F}=\nabla_{\mathbf{x}}F\cdot\nabla_{\mathbf{p}}-\nabla_{\mathbf{p}}F\cdot\nabla_{\mathbf{x}}\,,$
(29)

and then evaluating the commutator of two vector fields $[X_{F},X_{G}]$ viewed
as differential operators acting on test functions. It is not hard to see that

$[X_{F},X_{G}]\cdot K(\mathbf{x},\mathbf{p})=X_{\\{F,G\\}}\cdot
K(\mathbf{x},\mathbf{p})\,,$ (30)

for any function $K(\mathbf{x},\mathbf{p})$.

The corresponding Lie group is naturally that of canonical transformations
under finite time. For each element $F(\mathbf{x},\mathbf{p})\in\mathfrak{g}$
of the Poisson algebra, we will define the exponent map, denoted by $\exp$
that associates with $F$ the canonical transformation $U$ obtained by time
evolving under $F$ for unit time. The set of all such $U$’s is the group of
canonical transformations that we are concerned with (known in the math
literature as the group of Hamiltonian symplectomorphisms),

$\mathcal{G}\equiv\\{U=\exp F\leavevmode\nobreak\ |\leavevmode\nobreak\
F\in\mathfrak{g}\\}\,.$ (31)

Note that the exponent map $\exp F$ from the Lie algebra to the Lie group is
different from the point-wise exponential of the function
$e^{F}=1+F+F^{2}/2+\ldots$. To avoid confusion, we will restrict ourselves to
using $\exp$ for the Lie-algebra-to-Lie-group exponent map instead of writing
it as $e^{F}$.

The truncation of the Moyal algebra to the Poisson algebra is subtle and
requires some more scrutiny. We will revisit this in section IV and clarify
the consequences of this truncation, including a discussion on which
properties this approximation succesfully captures and which ones it misses
out on.

Having understood the operator algebra of concern, we now move on to
describing the corresponding space of states that we will be interested in.

###  III.2 The space of states

In any quantum mechanical system, states are described by density matrices
$\rho$, which can be thought of as linear maps acting on operators to give the
expectation value of the operator in the chosen state,

$\rho[\mathcal{O}]\equiv\langle\mathcal{O}\rangle_{\rho}=\operatorname{Tr}(\rho\mathcal{O})\,.$
(32)

In principle, if we have access to every operator in the theory, each state is
uniquely determined by the list of expectation values of every operator in
that state. But since we are only concerned with the subalgebra of charge-
neutral fermion bilinears, we inevitably end up being unable to distinguish
all microscopic states from each other, but instead are restricted to
equivalence classes of microscopic states, where equivalence is established by
requiring identical expectation values of all fermion bilinears.

A typical representative of any such equivalence class can be described as
follows. Having chosen the basis $T(\mathbf{x},\mathbf{p})$ for the space of
fermion bilinear, we can pick a dual basis to it, which we will denote by
operators $W(\mathbf{x},\mathbf{p})$, which have the orthogonality property:

$\operatorname{Tr}\left(W(\mathbf{x}^{\prime},\mathbf{p}^{\prime})T(\mathbf{x},\mathbf{p})\right)=\delta(\mathbf{x}-\mathbf{x}^{\prime})(2\pi)^{d}\delta(\mathbf{p}-\mathbf{p}^{\prime})\,.$
(33)

A representative of the equivalence class of states can be expanded in this
dual basis with the ‘coefficients’ given by a function of
$\mathbf{x},\mathbf{p}$,

$\rho_{f}=\int\frac{d^{d}xd^{d}p}{(2\pi)^{d}}f(\mathbf{x},\mathbf{p})W(\mathbf{x},\mathbf{p})\,.$
(34)

In this state, the expectation value of a bilinear operator $O_{F}$ simplifies
to

$\operatorname{Tr}(\rho_{f}O_{F})=\int\frac{d^{d}xd^{d}p}{(2\pi)^{d}}f(\mathbf{x},\mathbf{p})F(\mathbf{x},\mathbf{p})\equiv\langle
f,F\rangle\,.$ (35)

Naturally, this set of equivalence classes is the set of linear maps from
$\mathfrak{g}_{\text{Moyal}}$ to $\mathbb{C}$, also known as the dual space of
$\mathfrak{g}_{\text{Moyal}}$, which we will denote by $\mathfrak{g}^{*}$.

$\begin{split}\mathfrak{g}^{*}&\equiv\\{f(\mathbf{x},\mathbf{p})\\}\,,\\\
f[F]&\equiv\langle
f,F\rangle\equiv\int\frac{d^{d}xd^{d}p}{(2\pi)^{d}}f(\mathbf{x},\mathbf{p})F(\mathbf{x},\mathbf{p})\,,\end{split}$
(36)

where the second line defines the action of the linear map $f$ on an element
$F$ of $\mathfrak{g}_{\text{Moyal}}$. Note that the dual space is independent
of the Lie bracket. Hence, the Moyal algebra and the Poisson algebra share the
same dual space $\mathfrak{g}^{*}$.

Ordinarily in physics, vector spaces and their dual spaces are not
distinguished between, since they are isomorphic to each other for finite
dimensional vector spaces. However, for our purposes we find it crucial to
make this pedantic distinction, since the Lie algebra and its dual space will
take different physical interpretations and consequently will be equipped with
different mathematical structures later.

That the expectation values of operators $O_{F}$ in a state $\rho_{f}$ can be
written in the form of equation (35) provides the following interpretation for
the functions $F(\mathbf{x},\mathbf{p})$ and $f(\mathbf{x},\mathbf{p})$ in the
semiclassical limit: the function $F(\mathbf{x},\mathbf{p})$ that
characterizes the linear combination of fermion bilinears will be understood
as a single-particle observable, while the function $f(\mathbf{x},\mathbf{p})$
characterizing the state is the effective single-particle phase space
distribution function (or simply the distribution for brevity) that enters the
Boltzmann equation. This connection to the Boltzmann equation will become more
precise as we develop the Hamiltonian formalism later in section IV, whose
equation of motion in the semi-classical limit is precisely the collisionless
Boltzmann equation (or Landau’s kinetic equation for interacting Fermi
liquids).

The pairing or innder product $\langle f,F\rangle$ between elements of
$\mathfrak{g}^{*}$ and $\mathfrak{g}$ is then just the average value of the
single-particle observable $F(\mathbf{x},\mathbf{p})$ in the distribution
$f(\mathbf{x},\mathbf{p})$.

###  III.3 Schematic overview of the coadjoint orbit method

Equipped with the Lie algebra $\mathfrak{g}$ consisting of single-particle
observables and its dual space $\mathfrak{g}^{*}$ consisting of distribution
functions, the coadjoint orbit method provides us an algorithm to derive an
action for our theory in broadly two steps.

First, we set up a dynamical system describing time evolution on
$\mathfrak{g}^{*}$ via a prescribed Hamiltonian. The choice of Hamiltonian
must be governed by microscopics as well as principles of effective field
theory, especially since we want to describe the theory via the truncated
Poisson algebra instead of the exact Moyal algebra. We will see that these
considerations allow us to automatically obtain Landau’s kinetic equation for
interacting Fermi liquids as the equation of motion, along with systematic
higher order corrections to Landau’s phenomenological theory.

Second, we attempt to Legendre transform the Hamiltonian into an action.
Performing this Legendre transform is a highly non-trivial task, since it
turns out that we have to restrict our state space $\mathfrak{g}^{*}$ further
in order to achieve this. This restriction, however, is natural, since the set
of all possible configurations of the distribution function
$f(\mathbf{x},\mathbf{p})$ is too large of a set to describe sharp Fermi
surfaces at zero temperature. We need only consider functions that take values
of either 0 or 1, with the boundary between the two values being the Fermi
surface. These functions must also have fixed phase space volume due to
Luttinger’s theorem. It turns out that restricting $\mathfrak{g}^{*}$ to such
states is precisely what is needed to Legendre transform the Hamiltonian to an
action. This restriction, therefore, is both physically motivated and
mathematically necessary, and we will find that Luttinger’s theorem is
automatically built into our formalism.

Consequently, the postmodern formalism for Fermi liquids essentially describes
the dynamics of a fluctuating codimension one surface in phase space whose
topology is $\mathbb{R}^{d}\times S^{d-1}$, i.e. that of a sphere at every
point $\mathbf{x}$ (figure 6).

The next two sections are devoted to the two respective steps described above,
and a survey of the necessary approximations and consequent
validity/invalidity of these steps.

##  IV The operator algebra and the Hamiltonian formalism

Before developing the Hamiltonian formalism, we first survey the algebra of
fermion bilinears more carefully. We will make a small modification to our
definition of the generators and define them instead in center of mass and
relative coordinates as

$T(\mathbf{x},\mathbf{y})\equiv\frac{i}{2}\left[\psi^{\dagger}\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\psi\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)-\psi\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)\psi^{\dagger}\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\right]\,.$
(37)

Canonical anti-commutation relations for the fermion operators
$[\psi(\mathbf{x}),\psi^{\dagger}(\mathbf{y})]_{+}=i\delta(\mathbf{x}-\mathbf{y})$
implies that this definition only differs from equation (19) by a delta
function which serves to regulate the coincidence limit $T(\mathbf{x},0)$.
Furthermore, the Hermitian conjugate takes the form,

$T^{\dagger}(\mathbf{x},\mathbf{y})=-T(\mathbf{x},-\mathbf{y})\,.$ (38)

The various Fourier transforms of this generator will be useful for later:

$\begin{split}T(\mathbf{x},\mathbf{y})&\equiv\frac{i}{2}\left[\psi^{\dagger}\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\psi\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)-\psi\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)\psi^{\dagger}\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\right]\,,\\\
T(\mathbf{q},\mathbf{p})&\equiv\frac{i}{2}\left[\psi^{\dagger}\left(\frac{\mathbf{q}}{2}+\mathbf{p}\right)\psi\left(\frac{\mathbf{q}}{2}-\mathbf{p}\right)-\psi\left(\frac{\mathbf{q}}{2}-\mathbf{p}\right)\psi^{\dagger}\left(\frac{\mathbf{q}}{2}+\mathbf{p}\right)\right]\,,\\\
T(\mathbf{x},\mathbf{p})&\equiv\int_{\mathbf{y}}T(\mathbf{x},\mathbf{y})e^{i\mathbf{p}\cdot\mathbf{y}}=\int_{\mathbf{q}}T(\mathbf{q},\mathbf{p})e^{-i\mathbf{q}\cdot\mathbf{x}}\,,\\\
T(\mathbf{q},\mathbf{y})&\equiv\int_{\mathbf{x},\mathbf{p}}T(\mathbf{x},\mathbf{p})e^{i\mathbf{q}\cdot\mathbf{x}}e^{-i\mathbf{p}\cdot\mathbf{y}}=\int_{\mathbf{x}}T(\mathbf{x},\mathbf{y})e^{i\mathbf{q}\cdot\mathbf{x}}=\int_{\mathbf{p}}T(\mathbf{q},\mathbf{p})e^{-i\mathbf{p}\cdot\mathbf{y}}\,,\end{split}$
(39)

where integrals over momenta $\mathbf{q}$ and $\mathbf{p}$ are defined with an
implicit factor of $1/(2\pi)^{d}$.

Our convention for the fermion annihilation operator $\psi(\mathbf{k})$ in
momentum space is that $\psi(\mathbf{k})$ is simply the Fourier transform of
$\psi(\mathbf{x})$. When acting on the Fermi surface it creates a state with
momentum $\mathbf{k}$. Therefore, it creates a hole at the point $-\mathbf{k}$
in the Fermi sea. This is different from the usual convention in condensed
matter physics, where the annihilation operator $c_{\mathbf{k}}$ is defined so
that it creates a hole at the point $\mathbf{k}$, thereby creating a state
with total momentum $-\mathbf{k}$.

It is worth emphasizing that in the notation we have chosen above,
$\mathbf{x}$ is the center of mass coordinate of the particle-hole pair
described by the fermion bilinear, $\mathbf{y}$ is the relative coordinate or
the separation between them. Analogously, given that $\psi(\mathbf{k})$
creates a hole with momentum $-\mathbf{k}$, the Fourier conjugate $\mathbf{q}$
to the center of mass coordinate $\mathbf{x}$ measures the momentum of the
particle-hole pair, which is the difference of the individual momenta of the
particle and hole. The Fourier conjugate $\mathbf{p}$ to the separation
$\mathbf{y}$ is the average of the individual momenta of the particle and the
hole, so the average location of the particle hole pair in momentum space
(figure 7). We shall restrict ourselves to using this notation convention
throughout this thesis, so the arguments of the generator and their specific
order should make it clear to which Fourier transform we are referring.

Figure 7: Particle-hole configuration in the parametrization of equation (39).

All of the above Fourier transforms are traceless in a fermionic Fock space.
Additionally, our definitions imply that $T(\mathbf{x},\mathbf{p})$, in
particular, is anti-Hermitian,

$T^{\dagger}(\mathbf{x},\mathbf{p})=-T(\mathbf{x},\mathbf{p})\,.$ (40)

The commutator of these generators closes, and we find

$\begin{split}[T(\mathbf{q},\mathbf{y}),T(\mathbf{q}^{\prime},\mathbf{y}^{\prime})]&=2\sin\left(\frac{\mathbf{q}^{\prime}\cdot\mathbf{y}-\mathbf{q}\cdot\mathbf{y}^{\prime}}{2}\right)T(\mathbf{q}+\mathbf{q}^{\prime},\mathbf{y}+\mathbf{y}^{\prime})\,,\\\
[T(\mathbf{x},\mathbf{p}),T(\mathbf{x}^{\prime},\mathbf{p}^{\prime})]&=2\sin\left(\frac{\nabla_{\mathbf{x}}\cdot\nabla_{\mathbf{p}^{\prime}}-\nabla_{\mathbf{x}^{\prime}}\cdot\nabla_{\mathbf{p}}}{2}\right)\left[\delta(\mathbf{x}-\mathbf{x}^{\prime})\delta(\mathbf{p}-\mathbf{p}^{\prime})T(\mathbf{x},\mathbf{p})\right]\,.\end{split}$
(41)

The coefficient functions or differential operators on the right-hand side are
the “structure constants” of the Lie algebra $\mathfrak{g}_{\text{Moyal}}$,
whose typical element is a general linear combination

$O_{F}\equiv\int_{\mathbf{x}\mathbf{p}}F(\mathbf{x},\mathbf{p})T(\mathbf{x},\mathbf{p})\,,$
(42)

where $F(\mathbf{x},\mathbf{p})$ is an arbitrary function, to be thought of a
the set of coefficients of the vector $O_{F}$ in the basis
$T(\mathbf{x},\mathbf{p})$, with $(\mathbf{x},\mathbf{p})$ playing the role of
“incidces” in this expansion. This results in the Moyal bracket for the
commutator of generic linear combinations,

$\begin{split}[O_{F},O_{G}]&=O_{\\{\\!\\!\\{F,G\\}\\!\\!\\}}\,,\\\
\\{\\!\\!\\{F,G\\}\\!\\!\\}&=2\leavevmode\nobreak\
F\sin\left(\frac{\overleftarrow{\nabla}_{\mathbf{x}}\cdot\overrightarrow{\nabla}_{\mathbf{p}}-\overleftarrow{\nabla}_{\mathbf{p}}\cdot\overrightarrow{\nabla}_{\mathbf{x}}}{2}\right)G\,.\end{split}$
(43)

Our generators also obey orthogonality relations:

$\begin{split}\operatorname{Tr}[T(\mathbf{x},\mathbf{p})T(\mathbf{x}^{\prime},\mathbf{p}^{\prime})]&=2\delta(\mathbf{x}-\mathbf{x}^{\prime})(2\pi)^{d}\delta(\mathbf{p}-\mathbf{p}^{\prime})\,,\\\
\operatorname{Tr}[T(\mathbf{q},\mathbf{y})T(\mathbf{q}^{\prime},\mathbf{y}^{\prime})]&=2(2\pi)^{d}\delta(\mathbf{q}+\mathbf{q}^{\prime})\delta(\mathbf{y}+\mathbf{y}^{\prime})\,,\end{split}$
(44)

where the trace is taken in the fermionic Fock space.

The space of all charge-0 bosonic operators hence forms an infinite
dimensional Lie algebra, known as the Moyal algebra. We will restrict
ourselves to a class of microscopic Hamiltonians that can be expanded in a
polynomial expansion in the generators of this algebra,

$\begin{split}H_{\text{micro}}&=\int_{\mathbf{p}}\epsilon(\mathbf{p})\psi^{\dagger}(\mathbf{p})\psi(-\mathbf{p})\\\
&+\int_{\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3},\mathbf{p}_{4}}V(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3},\mathbf{p}_{4})\psi^{\dagger}(\mathbf{p}_{1})\psi(\mathbf{p}_{2})\psi^{\dagger}(\mathbf{p}_{3})\psi(\mathbf{p}_{4})\delta(\mathbf{p}_{1}+\mathbf{p}_{2}+\mathbf{p}_{3}+\mathbf{p}_{4})\\\
&+\mathcal{O}(\psi^{\dagger}\psi)^{3}\,,\end{split}$ (45)

where $\epsilon(\mathbf{p})$ is the free particle dispersion,
$V(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3},\mathbf{p}_{4})$ characterizes
$2\rightarrow 2$ scattering processes, and so on for higher order terms.

###  IV.1 Semi-classical truncation of the Moyal algebra

While the discussion so far has been exact, in practice, using the Moyal
algebra can be extremely tedious since the star product and the Moyal bracket
are defined as series expansions. A remedy for this is provided by the Poisson
truncation discussed in section III.1,

$\begin{split}\\{\\!\\!\\{F,G\\}\\!\\!\\}&=\\{F,G\\}+\mathcal{O}(\nabla_{\mathbf{x}},\nabla_{\mathbf{p}})^{3}\,,\\\
\\{F,G\\}&\equiv\nabla_{\mathbf{x}}F\cdot\nabla_{\mathbf{p}}G-\nabla_{\mathbf{p}}F\cdot\nabla_{\mathbf{x}}G\,.\end{split}$
(46)

The Poisson bracket is, in fact, the only truncation of the Moyal bracket that
satisfies the Jacobi identity. This truncation, however, comes at a cost, and
limits the validity of the theory to regimes where the Poisson bracket is a
good approximation to the Moyal bracket. This is only true when

$\nabla_{\mathbf{x}}\cdot\nabla_{\mathbf{p}}\ll 1\,,$ (47)

which can be rephrased in three other ways by Fourier transforming
$\mathbf{x}$ and/or $\mathbf{p}$:

$\nabla_{\mathbf{x}}\cdot\nabla_{\mathbf{p}}\ll
1\,,\quad\Leftrightarrow\quad\mathbf{q}\cdot\mathbf{y}\ll
1\,,\quad\Leftrightarrow\quad\nabla_{\mathbf{x}}\cdot\mathbf{y}\ll
1\,,\quad\Leftrightarrow\quad\mathbf{q}\cdot\nabla_{\mathbf{p}}\ll 1\,.$ (48)

Recall that $\mathbf{x}$ corresponds to the center of mass coordinate of a
particle-hole pair, $\mathbf{y}$ is the separation, $\mathbf{q}$ measures the
net momentum of the particle-hole excitation, and $\mathbf{p}$ is the average
of the momenta of the particle and the hole. With these in mind, equation (48)
implies that the Poisson truncation of the Moyal algebra of fermion bilinears
is applicable in situations where we have a separation of scales, with
$(\mathbf{x},\mathbf{q})$ characterizing the long distance or infrared (IR)
scale, and $(\mathbf{y},\mathbf{p})$ characterizing the short distance or
ultraviolet (UV) scale.

In position space, this means that we are restricting ourselves to probing
physics at length-scales much larger than the typical separation of a
particle-hole pair. In momentum space, a typical particle-hole excitation over
a Fermi surface has $|\mathbf{p}|\sim p_{F}$, and the Poisson truncation is
valid for pairs whose net momentum is much smaller than that, i.e.,

$|\mathbf{q}|\ll p_{F}\,.$ (49)

The corrections to the Poisson truncation can then be thought of as a
derivative expansion with higher derivatives terms being suppressed owing to
the fact that

$\nabla_{\mathbf{x}}\cdot\nabla_{\mathbf{p}}\sim\frac{|\nabla_{\mathbf{x}}|}{p_{F}}\ll
1\,.$ (50)

With this analysis in mind, let us try to understand what consequences the
Poisson truncation has for interactions between the fermions. We will consider
the quartic term in the microscopic Hamiltonian, which can be written in the
following way:

$\begin{split}H_{\text{micro}}^{\text{int}}&=\int_{\mathbf{q},\mathbf{p};\mathbf{q}^{\prime},\mathbf{p}^{\prime}}V(\mathbf{q},\mathbf{p};\mathbf{q}^{\prime},\mathbf{p}^{\prime})\psi^{\dagger}\left(\frac{\mathbf{q}}{2}+\mathbf{p}\right)\psi\left(\frac{\mathbf{q}}{2}-\mathbf{p}\right)\psi^{\dagger}\left(\frac{\mathbf{q}^{\prime}}{2}+\mathbf{p}^{\prime}\right)\psi\left(\frac{\mathbf{q}^{\prime}}{2}-\mathbf{p}^{\prime}\right)\delta(\mathbf{q}+\mathbf{q}^{\prime})\\\
&\simeq\int_{\mathbf{q},\mathbf{p};\mathbf{q}^{\prime},\mathbf{p}^{\prime}}V(\mathbf{q},\mathbf{p};\mathbf{q}^{\prime},\mathbf{p}^{\prime})T(\mathbf{q},\mathbf{p})T(\mathbf{q}^{\prime},\mathbf{p}^{\prime})\delta(\mathbf{q}+\mathbf{q}^{\prime})\,,\end{split}$
(51)

where the symbol $\simeq$ means that we have ignored the quadratic terms
generated upon replacing $\psi^{\dagger}(\mathbf{k}_{1})\psi(\mathbf{k}_{2})$
with its antisymmetrized version.

The above Hamiltonian characterizes $2\rightarrow 2$ scattering processes. In
general, the momenta
$(\mathbf{q},\mathbf{p};\mathbf{q}^{\prime},\mathbf{p}^{\prime})$ could take
any values allowing for generic scattering configurations on the Fermi
surface. However, the semi-classical limit captures those configurations with
$|\mathbf{p}|\sim|\mathbf{p}^{\prime}|\sim p_{F}$, and
$\mathbf{q},\mathbf{q}^{\prime}\ll p_{F}$. This corresponds to particle-hole
pairs close to the Fermi surface with small net momentum, such as the
configuration in figure 3(a). Higher derivative corrections to the
semiclassical limit then systematically account for particle-hole pairs with a
larger separation in momentum space.

###  IV.2 Constructing the Hamiltonian formalism

To recap the discussion in section III, we find a Lie algebra in the operator
algebra, whose generators are fermion bilinears $T(\mathbf{x},\mathbf{p})$,
whose structure constants can be read off from the commutation relations,

$\begin{split}[T(\mathbf{q},\mathbf{y}),T(\mathbf{q}^{\prime},\mathbf{y}^{\prime})]&=2\sin\left(\frac{\mathbf{q}^{\prime}\cdot\mathbf{y}-\mathbf{q}\cdot\mathbf{y}^{\prime}}{2}\right)T(\mathbf{q}+\mathbf{q}^{\prime},\mathbf{y}+\mathbf{y}^{\prime})\\\
&=\left(\mathbf{q}^{\prime}\cdot\mathbf{y}-\mathbf{q}\cdot\mathbf{y}^{\prime}\right)T(\mathbf{q}+\mathbf{q}^{\prime},\mathbf{y}+\mathbf{y}^{\prime})+\mathcal{O}(\mathbf{q},\mathbf{y})^{3}\,.\end{split}$
(52)

The pair $(\mathbf{q},\mathbf{y})$ or its Fourier conjugate
$(\mathbf{x},\mathbf{p})$ can be thought of as a Lie algebra index. Generic
elements of the Lie algebra are linear combinations of the generators,

$O_{F}=\int_{\mathbf{x},\mathbf{p}}F(\mathbf{x},\mathbf{p})T(\mathbf{x},\mathbf{p})\,,$
(53)

characterized by functions $F(\mathbf{x},\mathbf{p})$. The commutator of two
such functions specifies the Lie bracket,

$[O_{F},O_{G}]=O_{\\{\\!\\!\\{F,G\\}\\!\\!\\}}=O_{\\{F,G\\}}+\mathcal{O}(\nabla_{\mathbf{x}},\nabla_{\mathbf{p}})^{3}\,,$
(54)

and we can succinctly define the (truncated) Lie algebra as the set of
functions of $\mathbf{x}$ and $\mathbf{p}$ equipped with the Poisson bracket:

$\begin{split}\mathfrak{g}&\equiv\\{F(\mathbf{x},\mathbf{p})\\}\,,\\\
\\{F,G\\}&=\nabla_{\mathbf{x}}F\cdot\nabla_{\mathbf{p}}G-\nabla_{\mathbf{p}}F\cdot\nabla_{\mathbf{x}}G\,.\end{split}$
(55)

The corresponding Lie group consists of the set of exponentials $e^{O_{F}}$ of
the operators $O_{F}$, and in the semi-classical limit takes on the
interpretation of canonical transformations $U$ of the single-particle phase
space $\mathbb{R}^{2d}$ generated by the function $F$ viewed as a Hamiltonian.

$\mathcal{G}\equiv\\{U=\exp F\leavevmode\nobreak\ |\leavevmode\nobreak\
F\in\mathfrak{g}\\}\,.$ (56)

We also saw in section III.2 that the space of states was given by the dual
space $\mathfrak{g}^{*}$, whose elements are also functions
$f(\mathbf{x},\mathbf{p})$ which are interpreted as quasiprobability
distribution functions, which act on elements of the Lie algebra to give the
average value of a single-particle observable $F(\mathbf{x},\mathbf{p})$ in
the state $f(\mathbf{x},\mathbf{p})$.

$\begin{split}\mathfrak{g}^{*}&\equiv\\{f(\mathbf{x},\mathbf{p})\\}\,,\\\
\langle
f,F\rangle&\equiv\int_{\mathbf{x},\mathbf{p}}F(\mathbf{x},\mathbf{p})f(\mathbf{x},\mathbf{p})\,.\end{split}$
(57)

$\mathfrak{g}^{*}$ is the effective phase space for Fermi liquids and we need
to define a Hamiltonian and a Poisson structure on this to get an equation of
motion. In order to do so, let us first define the action of the Lie group and
Lie algebra on the Lie algebra and its dual space.

####  IV.2.1 Adjoint and coadjoint representations

The Lie bracket furnishes a natural action of the Lie algebra on itself, known
as the Lie algebra adjoint action:

$\begin{split}\text{ad}_{F}\leavevmode\nobreak\ &:\leavevmode\nobreak\
\mathfrak{g}\rightarrow\mathfrak{g}\,,\\\
\text{ad}_{F}G&\equiv\\{F,G\\}\,.\end{split}$ (58)

This can be exponentiated to obtain an action of the Lie group on the Lie
algebra, called the Lie group adjoint action:

$\begin{split}\text{Ad}_{U}\leavevmode\nobreak\ &:\leavevmode\nobreak\
\mathfrak{g}\rightarrow\mathfrak{g}\,,\\\ \text{Ad}_{U=\exp F}G\equiv
UGU^{-1}\equiv
e^{\text{ad}_{F}}G&=G+\\{F,G\\}+\frac{1}{2!}\\{F,\\{F,G\\}\\}+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ .\end{split}$ (59)

We will often use $UGU^{-1}$ as alternate notation for the adjoint action to
make it clear that intuition from quantum mechanics (and matrix Lie groups)
applies more or less straightforwardly to our case as well.

The action of the Lie group and Lie algebra on the Lie algebra are called the
adjoint representation.

From the above, we can also define the action of the Lie algebra and Lie group
on the dual space $\mathfrak{g}^{*}$, known as the coadjoint actions:

$\begin{split}\text{ad}^{*}_{F},\leavevmode\nobreak\
\text{Ad}^{*}_{U}\leavevmode\nobreak\ &:\leavevmode\nobreak\
\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}\,,\\\
\text{ad}^{*}_{F}f&\equiv\\{F,f\\}\,,\\\ \text{Ad}^{*}_{U=\exp F}f\equiv
UfU^{-1}\equiv
e^{\text{ad}^{*}_{F}}f&=f+\\{F,f\\}+\frac{1}{2!}\\{F,\\{F,f\\}\\}+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ .\end{split}$ (60)

Together these define the coadjoint representation.

####  IV.2.2 Lie-Poisson structure and Hamiltonian

Next, we need a Poisson structure for functionals of $\mathfrak{g}^{*}$. This
requires a bilinear map that takes in two functionals $\mathscr{F}[f]$ and
$\mathscr{G}[f]$, and spits out a third functional $\mathscr{H}[f]$ in a way
consistent with the product rule as well as with the Jacobi identity. Such a
structure is provided by the Lie-Poisson bracket, defined as follows:

$\\{\mathscr{F},\mathscr{G}\\}_{\text{LP}}[f]\equiv\left\langle
f,\left\\{\delta\mathscr{F}|_{f},\delta\mathscr{G}|_{f}\right\\}_{\text{Poisson}}\right\rangle\,.$
(61)

The above formula is dense, so let us unpack it in a few sentences.
$\mathfrak{g}^{*}$ is a vector space. A typical point in this vector space is
the function $f(\mathbf{x},\mathbf{p})$. Being a vector space, the tangent
space $T_{f}\mathfrak{g}^{*}$ to $\mathfrak{g}^{*}$ at the point $f$ is
isomorphic to $\mathfrak{g}^{*}$. Therefore any tangent vector at a point in
$\mathfrak{g}^{*}$ can be equivalently thought of as an element of
$\mathfrak{g}^{*}$. Analogously, the cotangent space
$T^{*}_{f}\mathfrak{g}^{*}$ to $\mathfrak{g}^{*}$ at the point $f$ is
isomorphic to the space $\mathfrak{g}^{**}\cong\mathfrak{g}$ that is dual to
$\mathfrak{g}^{*}$, which is just the Lie algebra. So cotangent vectors at a
point are elements of $\mathfrak{g}$.

The variation $\delta\equiv\delta/\delta f$ of a functional $\mathscr{F}$ is
an exterior derivative of a function of $\mathfrak{g}^{*}$. Therefore
$\delta\mathscr{F}$ is a cotangent field on $\mathfrak{g}^{*}$. Its value
$\delta\mathscr{F}|_{f}$ at the point $f$, being a cotangent vector, is an
element of the Lie algebra. The same holds for $\delta\mathscr{G}|_{f}$. Since
these are both elements of the Lie algebra, i.e., functions of
$(\mathbf{x},\mathbf{p})$, we can take their Lie bracket, which in our case is
the Poisson bracket. The resulting function, when paired with $f$ using our
inner product, gives us the value of the Lie-Poisson bracket functional
$\\{\mathscr{F},\mathscr{G}\\}_{\text{LP}}$ evaluated at the point $f$.

That the Lie-Poisson bracket obeys the product rule and Jacobi identity
follows from the fact that the Poisson bracket itself obeys both.

All that remains is to construct a Hamiltonian functional $H[f]$. Instead of
deriving this from the microscopic Hamiltonian in equation (45), we will use
effective field theory to write down a Hamiltonian in a systematic expansion.
We will assume translation and rotational invariance in the continuum limit,
even though the requirement of rotational invariance can be relaxed further to
account for materials with more complicated electronic Fermi surfaces.

Our Hamiltonian will take the form of a double expansion, one in
nonlinearities in $f(\mathbf{x},\mathbf{p})$, and the other in spatial
derivatives. The latter will be justified by the semi-classical limit (50),
since derivatives must be suppressed by the Fermi momentum. To justify the
former, we must organize our Hamiltonian in a polynomial expansion in
fluctuations around the ground state,

$f_{0}(\mathbf{p})=\Theta(p_{F}-|\mathbf{p}|)\,.$ (62)

Defining fluctuations around this reference state as

$\delta f(\mathbf{x},\mathbf{p})\equiv
f(\mathbf{x},\mathbf{p})-f_{0}(\mathbf{p})\,,$ (63)

we can write the most general effective Hamiltonian as follows

$\begin{split}H[f]&=\int_{\mathbf{x}\mathbf{p}}\epsilon(\mathbf{p})f(\mathbf{x},\mathbf{p})\\\
&+\frac{1}{2}\int_{\mathbf{x}\mathbf{p}\mathbf{p}^{\prime}}F^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})\delta
f(\mathbf{x},\mathbf{p})\delta
f(\mathbf{x},\mathbf{p}^{\prime})+\mathbf{F}^{(2,1)}(\mathbf{p},\mathbf{p}^{\prime})\cdot\left(\frac{\nabla_{\mathbf{x}}}{p_{F}}\delta
f(\mathbf{x},\mathbf{p})\right)\delta
f(\mathbf{x},\mathbf{p}^{\prime})+\ldots\\\
&+\frac{1}{3}\int_{\mathbf{x}\mathbf{p}\mathbf{p}^{\prime}\mathbf{p}^{\prime\prime}}F^{(3,0)}(\mathbf{p},\mathbf{p}^{\prime},\mathbf{p}^{\prime\prime})\delta
f(\mathbf{x},\mathbf{p})\delta f(\mathbf{x},\mathbf{p}^{\prime})\delta
f(\mathbf{x},\mathbf{p}^{\prime\prime})+\ldots\\\ &+\leavevmode\nobreak\
\ldots\,\leavevmode\nobreak\ \leavevmode\nobreak\ .\end{split}$ (64)

In the above, $\epsilon(\mathbf{p})$ is the free fermion dispersion relation
and the various coefficient functions $F^{(m,n)}$ parametrize interactions. In
our notation, the $m$-index of $F^{(m,n)}$ labels the nonlinearity of the
interaction, while the $n$-index labels the number of $\mathbf{x}$-derivatives
in that coupling. Of course, there can be multiple independent terms or order
$(m,n)$ in which case additional indices are required to distinguish their
coefficient functions. The various couplings $(\epsilon,F^{(m,n)})$ are
functional analogues of Wilson coefficients in an effective field theory, and
we will often refer to them as Wilson coefficients by a slight abuse of
terminology, or Wilson coefficient functions if we want to be precise.

####  IV.2.3 Equation of motion

Armed with the Lie-Poisson structure (61) as well as the Hamiltonian (64), we
can write down Hamilton’s equation of motion for our system on
$\mathfrak{g}^{*}$,

$\partial_{t}f=\\{f,H\\}_{\text{LP}}[f]\,.$ (65)

The Lie-Poisson bracket can be evaluated from its definition in terms of the
Poisson bracket, by using the fact that $\delta
f(\mathbf{x},\mathbf{p})/\delta
f(\mathbf{x}^{\prime},\mathbf{p}^{\prime})=\delta(\mathbf{x}-\mathbf{x}^{\prime})\delta(\mathbf{p}-\mathbf{p}^{\prime})$
and integrating by parts, to obtain

$\partial_{t}f(t,\mathbf{x},\mathbf{p})+\left\\{f(t,\mathbf{x},\mathbf{p}),\frac{\delta
H}{\delta f(t,\mathbf{x},\mathbf{p})}\right\\}_{\text{Poisson}}=0\,.$ (66)

The variation of the Hamiltonian can be calculated straightforwardly, and
defines the quasiparticle dispersion relation,

$\epsilon_{\text{qp}}[f]\equiv\frac{\delta H}{\delta
f}=\epsilon(\mathbf{p})+\int_{\mathbf{p}^{\prime}}F^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})\delta
f(t,\mathbf{x},\mathbf{p}^{\prime})+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (67)

in terms of which the equation of motion turns into Landau’s kinetic equation
(6):

$\partial_{t}f+\nabla_{\mathbf{p}}\epsilon_{\text{qp}}[f]\cdot\nabla_{\mathbf{x}}f-\nabla_{\mathbf{x}}\epsilon_{\text{qp}}[f]\cdot\nabla_{\mathbf{p}}f=0\,.$
(68)

We see that $F^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})$ is simply Landau’s
interaction function, but we also find an infinite series of higher order
corrections to the quasiparticle energy.

The study of the algebra of fermion bilinears, paired with EFT philosophy,
hence provides a a formalism that captures LFLT as well as higher derivative
corrections to LFLT in a systematic expansion.

Note that the formalism and equation of motion itself applies generally to any
state $f(\mathbf{x},\mathbf{p})$, irrespective of whether it describes the
excitations of a Fermi surface at zero temperature. The only place that the
Fermi surface has entered in this discussion so far is in justifying the
series expansion of the Hamiltonian (64). For other systems, a different
choice of Hamiltonian should suffice, as long as time evolution in such a
system can be described by canonical transformations.

####  IV.2.4 An alternate route to the Hamiltonian formalism

An alternate way to arrive at the Hamiltonian formalism described in this
section, without relying on the algebra of fermion bilinears, is the
following:

Landau’s kinetic equation is simply a non-linear modification of the
collisionless Boltzmann equation. Time evolution as determined by the
collisionless Boltzmann equation not only preserves volume in the single-
particle phase space, as shown by Liouville’s theorem, but also preserves the
symplectic form (or equivalently Poisson brackets) in the single-particle
phase space. This implies that any solution $f(t,\mathbf{x},\mathbf{p})$ to
the collisionless Boltzmann equation can be described as the action of a one-
parameter family of canonical transformations, parametrized by $t$, acting on
the initial state $f(t=0,\mathbf{x},\mathbf{p})$.

The dynamical system described by the collisionless Boltzmann equation is
hence equivalent to a dynamical system on the Lie group of canonical
transformations, since the solutions to the equations of motion are simply
curves on the group manifold. The method described in the above section is a
well-established method to formulate dynamical systems on Lie groups [54, 58],
and hence automatically applies to our case [59]. This formalism requires a
prescribed Hamiltonian to describe time evolution, and the most natural one is
the double expansion (64). As we have already seen, this immediately gives us
LFLT at the equation of motion.

In [1], this was the perspective that was primarily presented in the main
body, with the connection to fermion bilinears being relegated to the
appendices. In this section, we have instead surveyed in detail the more
microscopic approach to constructing the Hamiltonian, with the aim to clarify
the connection to microscopics as well as expound upon what approximations and
assumptions are required at the microscopic level in order to obtain this
effective description. While we have largely appealed to EFT philosophy in
order to construct the effective Hamiltonian (64), it remains to see whether
it is possible to derive the effective Hamiltonian for certain classes of
microscopic Hamiltonians such as the ones in equation (45), using the
properties of the fermion bilinear algebra.

##  V Effective action from the coadjoint orbit method

The second step towards obtaining an action description for Fermi liquids is
to Legendre transform the Hamiltonian. Let us briefly described how this is
usually achieved for a Hamiltonian system on a general phase space manifold
$\Gamma$, equipped with some choice of Poisson brackets. Defining
$\partial_{I}$ as a derivative on the phase space manifold, the Poisson
bracket of two functions $F$ and $G$ on $\Gamma$ can always be locally written
in the following way:

$\\{F,G\\}=\Pi^{IJ}\partial_{I}F\partial_{J}G\,,$ (69)

where $\Pi^{IJ}$ is an anti-symmetric rank 2 tensor on $\Gamma$, known as the
Poisson bi-vector. To switch from a Hamiltonian formalism to an action
formalism, we invert the Poisson bivector to obtain a closed, anti-symmetric,
non-degenerate symplectic form:

$\omega=\Pi^{-1}\,,\qquad\omega_{IJ}\Pi^{JK}=\delta_{I}^{K}$ (70)

The symplectic form allows us to write down a ‘$p\dot{q}$’ term in the
following way: introduce an extra dimension $s\in[0,1]$ in addition to time
$t$ so that $s=1$ corresponds to physical time, and use boundary conditions in
$s$ so that all degrees of freedom vanish at $s=0$. Let $\phi^{I}$ be
coordinates on phase space, i.e., the phase space degree of freedom. The
$p\dot{q}$ term is then given by

$\int dt\int_{0}^{1}ds\leavevmode\nobreak\
\omega(\partial_{t}\phi,\partial_{s}\phi)=\int
dt\int_{0}^{1}ds\leavevmode\nobreak\
\omega_{IJ}\partial_{t}\phi^{I}\partial_{s}\phi^{J}\,,$ (71)

with an additional spatial integral involved if $\phi^{I}$ are fields in
space101010The symplectic form is closed ($d\omega=0$) by definition, or as a
consequence of the Jacobi identity for the Poisson bracket. This implies that
the $\mathbf{p}\dot{q}$ term is independent of the choice of “bulk”
extension.. The Legendre transform of the Hamiltonian $H[\phi]$ is then

$S=\int dt\int_{0}^{1}ds\leavevmode\nobreak\
\omega(\partial_{t}\phi,\partial_{s}\phi)-\int dtH[\phi]\,.$ (72)

This entire construction relies on the ability to invert the Poisson bi-
vector. However, this invertibility is, in general, not guaranteed by the
definition of the Poisson bracket, and when it is not, we cannot find an
action that gives the same equation of motion without changing the phase space
either by finding a description in terms of different degrees of freedom or by
eliminating redundant ones. This is the case for the Hamiltonian formalism
described in section IV, so the Legendre transformation is not as
straightforward as we could have hoped for. Before describing the remedy for
this barrier, let us first revisit the microscopic description of the space of
states from section III.2.

###  V.1 Fermi surface states and their excitations

To recap the discussion in section III.2, the space of states
$\mathfrak{g}^{*}$ is given by the vector space dual to the algebra of fermion
bilinears. These are equivalence classes of density matrices that cannot be
distinguished by the expectation values of fermion bilinears. A typical
representative of such an equivalence class is characterized by the
distribution function $f(\mathbf{x},\mathbf{p})$ in the following way:

$\rho_{f}=\int_{\mathbf{x},\mathbf{p}}f(\mathbf{x},\mathbf{p})W(\mathbf{x},\mathbf{p})\,,$
(73)

where $W(\mathbf{x},\mathbf{p})$ is the basis dual to
$T(\mathbf{x},\mathbf{p})$, defined by

$\operatorname{Tr}[W(\mathbf{x},\mathbf{p})T(\mathbf{x}^{\prime},\mathbf{p}^{\prime})]=\delta(\mathbf{x}-\mathbf{x}^{\prime})(2\pi)^{d}\delta(\mathbf{p}-\mathbf{p}^{\prime})\,.$
(74)

The expectation value of a general operator
$O_{F}=\int_{\mathbf{x}\mathbf{p}}F(\mathbf{x},\mathbf{p})T(\mathbf{x},\mathbf{p})$
in the state $\rho_{f}$ can be written as

$\langle
O_{F}\rangle_{\rho_{f}}=\operatorname{Tr}[\rho_{f}O_{F}]=\int_{\mathbf{x},\mathbf{p}}F(\mathbf{x},\mathbf{p})f(\mathbf{x},\mathbf{p})=\langle
f,F\rangle\,,$ (75)

and the distribution function $f_{\rho}(\mathbf{x},\mathbf{p})$ that
represents any given state $\rho$ itself can be obtained from the state as

$f(\mathbf{x},\mathbf{p})=\left\langle
T(\mathbf{x},\mathbf{p})\right\rangle_{\rho}\,.$ (76)

Of course, this is generically true for any (pure or mixed) state, not just
states with a Fermi surface. The distribution functions corresponding to these
correspond to a subset of $\mathfrak{g}^{*}$.

Consider, for instance, a spherical Fermi surface with Fermi momentum $p_{F}$.
The state that describes is a pure state obtained by filling every momentum
within the spherical Fermi surface with a fermion,

$\ket{\text{FS}}=\prod_{|\mathbf{k}|\leq
p_{F}}\psi^{\dagger}(\mathbf{k})\ket{0}\,,$ (77)

where $\ket{0}$ is the vacuum. It is straightforward to show using fermion
anticommutation relations that

$f_{0}(\mathbf{p})=\braket{\text{FS}}{T(\mathbf{x},\mathbf{p})}{\text{FS}}=\frac{1}{2}\text{sign}(p_{F}-|\mathbf{p}|)\,.$
(78)

For later convenience, let us define instead the distribution function of a
state as

$f(\mathbf{x},\mathbf{p})=\langle
T(\mathbf{x},\mathbf{p})\rangle_{\rho}+\frac{1}{2}\,,$ (79)

so that

$f_{0}(\mathbf{p})=\Theta(p_{F}-|\mathbf{p}|)\,,$ (80)

is the occupation number function for a spherical Fermi surface111111This
shift is equivalent to saying that the distribution is defined by the
expectation value of the Wigner transform of
$\psi^{\dagger}(\mathbf{x}_{1})\psi(\mathbf{x}_{2})$, instead of it anti-
Hermitian part.. This shift also ensures that the integral used to define the
pairing $\langle f,F\rangle$ converges for states with a sharp Fermi surface,
since the domain of integration is effectively bounded in momentum space.

Excitations on top of the Fermi surface take the form of particle-hole pairs,
which are created by the action of fermion bilinears on $\ket{\text{FS}}$. A
state with a single particle hole excitation is then given by

$\ket{\mathbf{k}_{1};\mathbf{k}_{2}}\equiv\psi^{\dagger}(\mathbf{k}_{1})\psi(-\mathbf{k}_{2})\ket{\text{FS}}\,.$
(81)

Fermion anticommutation relations ensure that this state is different from
$\ket{\text{FS}}$ only if $\mathbf{k}_{1}\notin\text{FS}$ and
$\mathbf{k}_{2}\in\text{FS}$. Antisymmetrizing over the particle and the hole
to regulate the coincidence singularity
$\mathbf{k}_{1}\rightarrow\mathbf{k}_{2}$, and Wigner transforming allows us
to write such states in an alternate basis:

$\ket{\mathbf{x};\mathbf{p}}\equiv T(\mathbf{x},\mathbf{p})\ket{\text{FS}}\,.$
(82)

In the semi-classical limit, where $|\nabla_{\mathbf{x}}|\ll\mathbf{p}\sim
p_{F}$, the state $\ket{\mathbf{x};\mathbf{p}}$ is interpreted as a particle
hole pair created at the point $\mathbf{p}$ on the Fermi surface, locally in a
mesoscopic region of size $1/p_{F}$ at the position labelled by the spatial
coordinate $\mathbf{x}$. The momentum $\mathbf{p}$ has no relation to the net
momentum $\mathbf{q}$ of the particle-hole pair, and only labels on which
‘patch’ of the Fermi surface the particle-hole pair lives.

Another equivalent basis that will be more convenient for us is that of
coherent states defined as

$\ket{F(\mathbf{x},\mathbf{p})}\equiv
e^{\int_{\mathbf{x}\mathbf{p}}F(\mathbf{x},\mathbf{p})T(\mathbf{x},\mathbf{p})}\ket{\text{FS}}\,,$
(83)

whose distribution function is given by the following:

$f_{F}(\mathbf{x},\mathbf{p})=f_{0}(\mathbf{p})+\\{\\!\\!\\{F,f_{0}\\}\\!\\!\\}+\frac{1}{2!}\\{\\!\\!\\{F,\\{\\!\\!\\{F,f_{0}\\}\\!\\!\\}\\}\\!\\!\\}+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (84)

This is just the coadjoint action of $F(\mathbf{x},\mathbf{p})$ on
$f_{0}(\mathbf{p})$ in the Moyal algebra! The set of unitary operators
$U_{F}=e^{\int FT}$ form the corresponding group and we find that particle-
hole coherent states of a Fermi surface is obtained by the action of all
possible group transformations on the spherical Fermi surface. This applies to
the parametrization of the states in terms of their distribution functions as
well, in that the distribution function for a particle-hole coherent state is
obtained by acting on the spherical Fermi surface distribution with a group
transformation.

In the semi-classical limit, the Moyal brackets are replaced by Poisson
brackets and the semi-classical distribution function for a coherent state is
given by

$f_{F}=\text{Ad}^{*}_{\exp
F}f_{0}=f_{0}+\\{F,f_{0}\\}+\frac{1}{2!}\\{F,\\{F,f_{0}\\}\\}+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (85)

which is interpreted as the action of the canonical transformation $U=\exp F$
on the spherical Fermi surface state. An intuitive picture for this is the
following: take all the points within the Fermi surface. The canonical
transformation $U$ maps each one of these to a new point. Being a smooth
coordinate transformation, this preserves the proximity of points and
transforms the initial spherical swarm of points into a new shape that is
topologically equivalent to a filled sphere (see figure 8). The precise shape
of boundary of this region can be parametrized by a function
$p_{F}(\mathbf{x},\theta)$, where $\theta$ are angular coordinates in momentum
space. We then have

$f_{F}(\mathbf{x},\mathbf{p})=\Theta(p_{F}(\mathbf{x},\theta)-|\mathbf{p}|)\,,$
(86)

which is entirely characterized by a shape in phase space. The space of states
for particle-hole excitations is then just the space of closed surfaces in
phase space [16].

This space of states is described mathematically by what is called a coadjoint
orbit, which we define below.

####  V.1.1 Coadjoint orbits and the Kirillov-Kostant-Souriau form

Figure 8: Fermi surface states from canonical transformations

As we saw above, the space of states relevant for zero temperature Fermi
surface physics is not all of $\mathfrak{g}^{*}$, but a subset of it
consisting of functions that take values $1$ or $0$ separated by a closed
surface. This restriction is formally achieved by picking a reference state,
$f_{0}(\mathbf{p})$ in our case, and acting on it via all possible canonical
transformations. Canonical transformations act on $\mathfrak{g}^{*}$ via the
coadjoint action, so the set generated from this procedure is known as the
coadjoint orbit of $f_{0}$:

$\mathcal{O}_{f_{0}}\equiv\\{f=\text{Ad}^{*}_{U}f_{0}\in\mathfrak{g}^{*}\leavevmode\nobreak\
|\leavevmode\nobreak\ U\in\mathcal{G}\\}\,.$ (87)

Two different canonical transformations acting on the same reference state can
indeed generate the same element of the coadjoint orbit, owing to the fact
that there is a nontrivial subgroup that leaves $f_{0}$ invariant, called the
stabilizer subgroup of $f_{0}$, which we will denote by $\mathcal{H}$.

$\begin{split}\mathcal{H}&\equiv\\{V\in\mathcal{G}\leavevmode\nobreak\
|\leavevmode\nobreak\ \text{Ad}^{*}_{V}f_{0}=f_{0}\\}\\\
&=\\{V=\exp\alpha\leavevmode\nobreak\ |\leavevmode\nobreak\
\alpha\in\mathfrak{g},\leavevmode\nobreak\
\text{ad}^{*}_{\alpha}f_{0}=0\\}\,.\end{split}$ (88)

So the canonical transformations $U$ and $UV$ create the same state from
$f_{0}$, since

$\text{Ad}^{*}_{UV}f_{0}=UVf_{0}(U_{V})^{-1}=U(Vf_{0}V^{-1})U^{-1}=\text{Ad}^{*}_{U}f_{0}\,.$
(89)

Each state $f$ in the coadjoint orbit is hence represented by a left coset
$U\mathcal{H}$, and the coadjoint orbit is then the left coset space,

$\mathcal{O}_{f_{0}}\cong\mathcal{G}/\mathcal{H}\,.$ (90)

Since every element of the coadjoint orbit is related to every other by
canonical transformations, we find an important result for time evolution
under any Hamiltonian $H[f]$. Infinitesimal time evolution occurs by the
action of the infinitesimal canonical transformation $\delta
H|_{f}\in\mathfrak{g}$, while finite time evolution occurs by exponentiating
the sequence of infinitesimal canonical transformations, which itself is a
canonical transformation. Therefore, time evolution takes an initial state to
another state in the same coadjoint orbit as the initial state.

The coadjoint orbit $\mathcal{O}_{f_{0}}$ is hence preserved by time
evolution, and can hence be thought of as a reduced phase space for Fermi
liquids. The Hamiltonian and Lie-Poisson structure can both be restricted to
the coadjoint orbit with complete consistency, and the entire Hamiltonian
formalism can be defined solely for $\mathcal{O}_{f_{0}}$ instead of all of
$\mathfrak{g}^{*}$.

Unlike the Lie-Poisson structure for $\mathfrak{g}^{*}$, however, the Lie-
Poisson structure restricted to $\mathcal{O}_{f_{0}}$ is invertible, and
permits the definition of a closed, non-degenerate symplectic form, known as
the Kirillov-Kostant-Souriau (KKS) form. Being a 2-form, it is defined by its
action on a pair of vectors tangent to the coadjoint orbit at any given point.

Consider the point $f\in\mathcal{O}_{f_{0}}$. Since the coadjoint orbit is a
submanifold of $\mathfrak{g}^{*}$, the tangent space
$T_{f}\mathcal{O}_{f_{0}}$ to $\mathcal{O}_{f_{0}}$ at the point $f$ is a
subspace of the tangent space $T_{f}\mathfrak{g}^{*}$ to $\mathfrak{g}^{*}$.
Tangent vectors of $\mathfrak{g}^{*}$ can be thought of as elements of
$\mathfrak{g}^{*}$, so defining the KKS form amounts to defining its action
$\omega_{\text{KKS}}(g,k)$ on any two arbitrary functions
$g,k\in\mathfrak{g}^{*}$ which are tangent to $\mathcal{O}_{f_{0}}$.

It can be shown that the tangents $g$ and $k$ at the point $f$ can be obtained
from the coadjoint action of two Lie algebra elements $G,K\in\mathfrak{g}$ on
$f$ (see, for instance, [58]), i.e.,

$\text{ad}^{*}_{G}f=g,\qquad\text{ad}^{*}_{K}f=k\,.$ (91)

$G$ and $K$ are not uniquely determined by $g$ and $k$ respectively, but
rather representatives of equivalence classes of Lie algebra elements. The KKS
form is then defined in terms of $G$ and $K$ as follows:

$\omega_{\text{KKS}}(g,k)\equiv\left\langle
f,\\{G,K\\}_{\text{Poisson}}\right\rangle\,.$ (92)

The pairing of the Poisson bracket with $f$ makes it clear that any other
choice of representative of the equivalence classes of $G$ and $K$
respectively gives the same answer, using the fact that if $G$ and
$G^{\prime}$ are two elements of the same equivalence class, then
$\text{ad}^{*}_{G-G^{\prime}}f=0$. To show that the KKS form is closed, note
that the differential $d\omega_{\text{KKS}}$ acts on three instead of two
tangents, and it is not difficult to show that

$d\omega_{\text{KKS}}(g,k,l)=\left\langle
f,\\{\\{G,K\\},L\\}\right\rangle+\text{cyclic permutations}\,,$ (93)

where $L\in\mathfrak{g}$ is such that
$\text{ad}^{*}_{L}f=l\in\mathfrak{g}^{*}$. The right hand side then vanishes
due to the Jacobi identity.

Armed with the Kirillov form, we can formally write down an action for Fermi
liquids in terms of the field $f\in\mathcal{O}_{f_{0}}$, which looks like

$\begin{split}S_{\text{FL}}[f]&=S_{\text{WZW}}[f]-\int dt\leavevmode\nobreak\
H[f]\,,\\\ S_{\text{WZW}}[f]&=\int dt\int_{0}^{1}ds\leavevmode\nobreak\
\omega_{\text{KKS}}\left(\partial_{t}f,\partial_{s}f\right)\,,\end{split}$
(94)

where $S_{\text{WZW}}$ is the Wess-Zumino-Witten (WZW) term, $H[f]$ is the
Hamiltonian in equation (64), and $f$ obeys the following boundary conditions
on the $(t,s)$-strip:

$f(t,s=1)=f(t)\,,\qquad f(t,s=0)=0\,.$ (95)

###  V.2 The Wess-Zumino-Witten term and the effective action

The action (94), while exact (in the semi-classical limit corrected by the
derivative expansion) is written in a rather formal way that cannot really be
used for calculations. In order to make it more useful, we need to find a
convenient parametrization of the coadjoint orbit. The simplest one is
obtained directly from the definition of the orbit, i.e., by acting on the
reference state $f_{0}$ by all possible canonical transformations, generated
by the field $-\phi(\mathbf{x},\mathbf{p})\in\mathfrak{g}$. In this
parametrization, the field $\phi(\mathbf{x},\mathbf{p})$ is our degree of
freedom. The minus sign is conventional and chosen for later convenience.

Elements $f(\mathbf{x},\mathbf{p})$ of the coadjoint orbit can be parametrized
as follows:

$\begin{split}f_{\phi}(\mathbf{x},\mathbf{p})=\text{Ad}^{*}_{\exp(-\phi)}f_{0}&=f_{0}+\\{\phi,f_{0}\\}+\frac{1}{2!}\\{\phi,\\{\phi,f_{0}\\}\\}+\ldots\\\
&=\Theta(p_{F}-|\mathbf{p}|)+(\mathbf{n}_{\theta}\cdot\nabla_{\mathbf{x}}\phi)\delta(|\mathbf{p}|-p_{F})+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ ,\end{split}$ (96)

where $\mathbf{n}_{\theta}$ is the unit normal to the spherical Fermi surface
at the angular coordinates $\theta$ in momentum space.

The stabilizer $\mathcal{H}$ of $f_{0}$ can be described by its Lie subalgebra
$\mathfrak{h}$ which corresponds to functions
$\alpha(\mathbf{x},\mathbf{p})\in\mathfrak{g}$ that obey the following
condition:

$\begin{split}\text{ad}^{*}_{\alpha}f_{0}=\\{\alpha,f_{0}\\}=0\,,\\\
\implies(\mathbf{n}_{\theta}\cdot\nabla_{\mathbf{x}}\alpha)|_{|\mathbf{p}|=p_{F}}=0\,.\end{split}$
(97)

Consequently, the canonical transformation $\exp\alpha$ leaves $f_{0}$
invariant,

$\text{Ad}^{*}_{\exp\alpha}f_{0}=e^{\text{ad}^{*}_{\alpha}}f_{0}=f_{0}\,.$
(98)

The equivalence $U\simeq UV$ then leads to an equivalence relation for $\phi$,

$\begin{split}\exp(-\phi)&\simeq\exp(-\phi)\exp(\alpha)\,,\\\
\implies\phi&\simeq\phi-\alpha+\frac{1}{2}\\{\phi,\alpha\\}+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ ,\end{split}$ (99)

which allows us to “gauge fix” $\phi$ to be independent of the radial momentum
coordinate,

$\phi=\phi(\mathbf{x},\theta)\,,$ (100)

where $\theta$ are angular coordinates in momentum space. A suitable choice of
$\alpha$ that achieves this, for example, at leading order in the
transformation (99), is

$\alpha_{\phi}(\mathbf{x},\mathbf{p})=\phi(\mathbf{x},\mathbf{p})-\phi(\mathbf{x},\theta)|_{|\mathbf{p}|=p_{F}}\,.$
(101)

It is easy to check that $\\{\alpha_{\phi},f_{0}\\}=0$. While we use a
$|\mathbf{p}|$-independent parametrization of our degree of freedom for
convenience, any other choice is equally valid and will result in the same
physical quantities, with the various choices being related by field
redefinitions.

What remains is to write down the WZW term in terms of this field to obtain an
action description for Fermi liquids. The definition of the KKS form requires
that we find functions $G$ and $K$ such that

$\text{ad}^{*}_{G}f_{\phi}=\partial_{t}f\,,\qquad\text{ad}^{*}_{K}f_{\phi}=\partial_{s}f\,.$
(102)

Using the fact that $f_{\phi}=\text{Ad}^{*}_{U}f_{0}=Uf_{0}U^{-1}$ where
$U=\exp(-\phi)$, we can show that the required functions are121212The simplest
way to do this is to pretend that $U,f_{0},f$ are all matrices, replace all
Poisson brackets with matrix commutators, simplify the expressions and finally
replace all commutators back with Poisson brackets.

$G=\partial_{t}UU^{-1}\,,\qquad K=\partial_{s}UU^{-1}\,,$ (103)

so that the KKS form evaluates to

$\begin{split}\omega_{\text{KKS}}(\partial_{t}f,\partial_{s}f)&=\left\langle
f,\\{\partial_{t}UU^{-1},\partial_{s}UU^{-1}\\}\right\rangle\\\ &=\left\langle
f_{0},\\{U^{-1}\partial_{t}U,U^{-1}\partial_{s}U\\}\right\rangle\,,\end{split}$
(104)

with boundary conditions $\phi(t,s=1)=\phi(t)$ and $\phi(t,s=0)=0$. The above
expression can be simplified to a sum of total $s$\- and $t$-derivatives,
which allows us the write the WZW term as

$S_{\text{WZW}}=\int dt\leavevmode\nobreak\ \langle
f_{0},U^{-1}\partial_{t}U\rangle\,.$ (105)

This is a subtle point, since it suggests that the KKS form is necessarily
exact, which is not true generally for a Lie group, especially for a coadjoint
orbit with non-trivial topology. Since the group of canonical transformations
is a diffeomorphism group, the topology of its coadjoint orbits is unknown,
and it is unclear whether the KKS form on the coadjoint orbit
$\mathcal{O}_{f_{0}}$ is exact or not.

The expression (104), on the other hand, is exact, owing to the fact that we
are describing a generic point $f$ on the coadjoint orbit as a canonical
transformation $U$ acting on the reference state $f_{0}$. Furthermore, we are
restricting ourselves to canonical transformations that are connected to the
identity by expressing $U$ as the exponent of a Lie algebra element $-\phi$.
This parametrization of the coadjoint orbit is hence incomplete, and only
captures the largest possible patch of the coadjoint orbit around $f_{0}$,
missing out on information about disconnected components of the orbit as well
as the global topology of the component containing $f_{0}$.

This choice of parametrization suffices, however, to describe a perturbative
expansion around the reference state $f_{0}$, since all states accessible to
such a perturbative expansion necessarily live in a patch around $f_{0}$,
making the choice of the reference state somewhat crucial for this method to
work. To account for nonperturbative properties of Fermi liquids, a different
parametrization of the coadjoint orbit is required, which we leave to future
work.

Finally, we obtain a perturbative action that describes Fermi liquids,

$S_{\text{FL}}=\int dt\leavevmode\nobreak\ \langle
f_{0},U^{-1}\partial_{t}U\rangle-\int dt\leavevmode\nobreak\
H[f_{\phi}=Uf_{0}U^{-1}]\,,$ (106)

with $U=\exp(-\phi)$. The action can be expanded order by order in $\phi$, and
we will find that higher order terms are suppressed by powers of $p_{F}$,
which takes on the role of the UV cutoff of the theory.

Of course, since this action is just the Legendre transformation of the
Hamiltonian (64), the equation of motion is guaranteed to be equation (68).
But this can also be verified directly by varying the action under

$U\rightarrow U^{\prime}=\exp\delta\phi\cdot U\,,$ (107)

with $\delta\phi(t,\mathbf{x},\mathbf{p})\in\mathfrak{g}$. To linear order in
$\delta\phi$, we have

$\delta[U^{-1}\partial_{t}U]=U^{-1}(\partial_{t}\delta\phi)U\,,\qquad\delta
H[Uf_{0}U^{-1}]=\left\langle
f_{\phi},\\{\epsilon_{\text{qp}}[f_{\phi}],\delta\phi\\}_{\text{Poisson}}\right\rangle\,,$
(108)

where $\epsilon_{\text{qp}}[f]=\delta H/\delta f$ is the quasiparticle energy.
This gives us the following result for the variation of the action:

$\delta S=-\int
dt\left\langle\partial_{t}f_{\phi}+\left\\{f_{\phi},\epsilon_{\text{qp}}[f_{\phi}]\right\\}_{\text{Poisson}},\delta\phi\right\rangle\,,$
(109)

from which we can read off the equation of motion,

$\partial_{t}f_{\phi}+\left\\{f_{\phi},\epsilon_{\text{qp}}[f_{\phi}]\right\\}=0\,,$
(110)

which is, as expected, identical to equation (68).

###  V.3 Symmetries in the postmodern formalism

This geometric perspective for Fermi liquids, in part, powerful because of how
it encodes symmetries through the algebra of canonical transformations. We
will categorize the symmetries we want to introduce into the formalism into
three different groups: spacetime, gauge and internal symmetries. The last of
these three requires an extension of the algebra of canonical transformations
and will hence be dealt with later in section VII.1.

Let us first discuss some key aspects of how symmetries act in the postmodern
formalism, and focus in particular on the unintuitive consequences of the fact
that the algebra of canonical transformations is in fact a diffeomorphism
algebra as opposed to a global symmetry algebra.

Recall that the coadjoint orbit
$\mathcal{O}_{f_{0}}\cong\mathcal{G}/\mathcal{H}$ is the left coset space of
the group of canonical transformations. Therefore every state
$f\in\mathcal{O}_{f_{0}}$ is identical to an equivalence class of canonical
transformations under the equivalence relation,

$U\simeq UV\,,\quad V\in\mathcal{H}\,.$ (111)

The explicit map from $\mathcal{G}/\mathcal{H}$ to $\mathcal{O}_{f_{0}}$ is
given by131313The discussion below equation (104) of the subtlety of not being
able to capture every state in the coadjoint orbit does not apply here since
we are not requiring $U\in G$ to be the exponent of any Lie algebra element.

$f_{U}\equiv Uf_{0}U^{-1}\,.$ (112)

Now the group of canonical transformations $G$ can itself act on the coset in
one of two different ways, called the left and right actions, respectively
given by the transformations

$U\xrightarrow{\text{left}}WU\,,\qquad U\xrightarrow{\text{right}}UW\,,\qquad
W\in\mathcal{G}\,.$ (113)

Both of these induce transformations on the coadjoint orbit as follows:

$f_{U}\xrightarrow{\text{left}}Wf_{U}W^{-1}\,,\qquad
f_{U}\xrightarrow{\text{right}}UWf_{0}W^{-1}U^{-1}\,,$ (114)

but only the left action can be naturally and directly written as a
transformation of $\mathcal{G}$ on the coadjoint orbit, independent of the
choice of reference state $f_{0}$. Therefore symmetries must act on the coset
space via the left action. The right action instead is reserved for
transformations by elements $V$ of the stabilizer $\mathcal{H}$, resulting in
a coset redundancy that is a gauge symmetry of our theory (not to be confused
with the gauge symmetry when we couple to background $U(1)$ gauge fields
later). The WZW term is invariant under a larger gauge symmetry of all
canonical transformations under the right action, since these simply pick out
a different reference state to parametrize the coadjoint orbit, but the
Hamiltonian breaks this $\mathcal{G}$ gauge symmetry down to a $\mathcal{H}$
gauge symmetry by uniquely picking $f_{0}$ as the ground state.

Note also that the WZW term is invariant under the left action of every
canonical transformation that does not depend on time, since

$(WU)^{-1}\partial_{t}(WU)=U^{-1}\partial_{t}U\,,$ (115)

but the Hamiltonian is not. The rule of thumb for imposing symmetries on this
theory will be the following:

  * •

Identify the subalgebra of canonical transformations that generates the
symmetry

  * •

If the symmetry being considered is a spacetime symmetry, impose invariance of
the action under the transformation $U\rightarrow WU$

  * •

If the symmetry in consideration is a gauge symmetry, turn on background
fields that make the state $f$ invariant under the transformation $WfW^{-1}$.

The last point is unusual and not how we typically gauge a theory, and will be
discussed in more detail later. But before imposing any symmetry on our
theory, let us describe a global symmetry that does not act on the state $f$,
but is instead a consequence of our choice of parametrization of the coadjoint
orbit. Recall that we chose to define the canonical transformation $U$ that
generates $f$ as the exponent of a Lie algebra element,

$U=\exp(-\phi)\,,\qquad\phi(\mathbf{x},\mathbf{p})\in\mathfrak{g}\,.$ (116)

Elements of the Lie algebra have a symmetry built into them, which corresponds
to constant shifts141414The more mathematically inclined reader might worry
that in order for the pairing $\langle f,F\rangle$ between $\mathfrak{g}^{*}$
and $\mathfrak{g}$ to be well-defined, suitable boundary conditions need to be
imposed on functions which a constant shift would violate. However, this shift
symmetry can be interpreted as a transformation of the boundary conditions to
make the pairing well-defined.:

$\phi(\mathbf{x},\mathbf{p})\rightarrow\phi(\mathbf{x},\mathbf{p})+c\,.$ (117)

These shifts preserve the action of the canonical transformation on any state,
since $f_{\phi}$ only depends on $\phi$ through its derivatives. While such
shifts leave $f$ invariant, they will not leave the WZW term invariant if $c$
is promoted to a function of time, and it is not difficult to show that

$\delta S_{\text{WZW}}=\int dt\leavevmode\nobreak\ \langle
f,\partial_{t}c(t)\rangle=-\int dt\leavevmode\nobreak\
\langle\partial_{t}f,c(t)\rangle\,.$ (118)

Noether’s theorem then tells us that we must then have

$\partial_{t}\int_{\mathbf{x},\mathbf{p}}f(\mathbf{x},\mathbf{p})=0\,,$ (119)

i.e., the total particle number,

$N=\int_{\mathbf{x},\mathbf{p}}f(\mathbf{x},\mathbf{p})\,,$ (120)

is conserved.

####  V.3.1 Galilean invariance

As an example of a spacetime symmetry, let us demonstrate how invariance under
Galilean boosts constrains our action. The first step is to identify the
subalgebra of canonical transformations that generates Galilean boosts. A
typical elements of this algebra is given by the time-dependent function,

$B_{v}=\mathbf{v}\cdot(\mathbf{p}t-m\mathbf{x})\,,$ (121)

with $W=\exp B_{v}$ being the corresponding canonical transformation. Under
this transformation, we have

$f(\mathbf{x},\mathbf{p})\rightarrow(\text{Ad}^{*}_{W}f)(\mathbf{x},\mathbf{p})=f(\mathbf{x}-\mathbf{v}t,\mathbf{p}-m\mathbf{v})\,,$
(122)

as can be obtained by observing that the expansion of the coadjoint action
takes the form of a Taylor series and then resumming the Taylor series.

Let us first evaluate the constraint on the free fermion action obtained from
Galilean invariance. The action can be written as follows:

$S_{\text{free fermion}}=\int dt\left\langle
f_{0},U^{-1}\partial_{t}U\right\rangle-\int dt\left\langle
f,\epsilon\right\rangle\,.$ (123)

The WZW term transforms to

$\left\langle f_{0},U^{-1}W^{-1}\partial_{t}(WU)\right\rangle=\left\langle
f_{0},U^{-1}\partial_{t}U\right\rangle+\left\langle
f,W^{-1}\partial_{t}W\right\rangle\,,$ (124)

while the Hamiltonian term becomes

$\left\langle WfW^{-1},\epsilon\right\rangle=\left\langle f,W^{-1}\epsilon
W\right\rangle\,,$ (125)

so the change in the action is given by the following

$\delta S=\int dt\left\langle
f,W^{-1}(\partial_{t}-\epsilon)W-\epsilon\right\rangle\,.$ (126)

Invariance under boosts then requires that

$W^{-1}\partial_{t}W=W^{-1}\epsilon W-\epsilon\,,$ (127)

where $W^{-1}\epsilon
W=\text{Ad}^{*}_{W^{-1}}\epsilon=\epsilon(\mathbf{p}+m\mathbf{v})$ owing to
the fact that $W^{-1}=\exp(-B_{v})=\exp B_{-v}$. The left hand side can now be
expanded using the following formula,

$W^{-1}\partial_{t}W=\partial_{t}B_{v}+\frac{1}{2!}\\{\partial_{t}B_{v},B_{v}\\}+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (128)

and compared order by order in $\mathbf{v}$ with the Taylor expansion of the
right hand side to obtain the following:

$\mathbf{p}=m\nabla_{\mathbf{p}}\epsilon$ (129)

which tells us that the dispersion relation must be quadratic:

$\epsilon(\mathbf{p})=\frac{p^{2}}{2m}+\text{constant}\,.$ (130)

This exactly what is expected for a free fermion with Galilean invariance.
Next, we derive the effective mass of Landau quasiparticles by imposing
Galilean invariance on the interacting theory truncated to quadratic order in
the fluctuation $\delta f=f-f_{0}$:

$H[f]=\int_{\mathbf{x}\mathbf{p}}\epsilon(\mathbf{p})f(\mathbf{x},\mathbf{p})+\frac{1}{2}\int_{\mathbf{x}\mathbf{p}\mathbf{p}^{\prime}}F^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})\delta
f(\mathbf{x},\mathbf{p})\delta
f(\mathbf{x},\mathbf{p}^{\prime})+\mathcal{O}(\delta
f^{3},\nabla_{\mathbf{x}})\,.$ (131)

We have already seen that the transformation of the WZW term under a Galilean
boost is cancelled by the transformation of a linear-in-$f$ Hamiltonian term
with the dispersion $\epsilon=p^{2}/2m$. Therefore, invariance of the
interacting theory can be achieved by demanding invariance of the shifted
Hamiltonian:

$\tilde{H}[f]=\int_{\mathbf{x}\mathbf{p}}\left(\epsilon(\mathbf{p})-\frac{p^{2}}{2m}\right)f(\mathbf{x},\mathbf{p})+\frac{1}{2}\int_{\mathbf{x}\mathbf{p}\mathbf{p}^{\prime}}F^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})\delta
f(\mathbf{x},\mathbf{p})\delta
f(\mathbf{x},\mathbf{p}^{\prime})+\mathcal{O}(\delta
f^{3},\nabla_{\mathbf{x}})\,.$ (132)

To obtain constraints from boost invariance, it suffices to consider
infinitesimal transformations,

$f\rightarrow
f+\\{B_{v},f\\}+\mathcal{O}(v^{2})=f-\mathbf{v}\cdot(t\nabla_{\mathbf{x}}+m\nabla_{\mathbf{p}})f+\mathcal{O}(v^{2})\,,$
(133)

under which the fluctuation transforms as

$\delta f\rightarrow-\leavevmode\nobreak\
m\mathbf{v}\cdot\nabla_{\mathbf{p}}f_{0}+\delta
f-\mathbf{v}\cdot(t\nabla_{\mathbf{x}}+m\nabla_{\mathbf{p}})\delta f\,.$ (134)

Note that the transformation of the fluctuation $\delta f$ is inhomogeneous in
$\delta f$. In particular, it can reduce the degree of a monomial by up to 1.
This results in constraints that mix the various Wilson coefficient functions,
so that $F^{(m,n)}$ will be constrained by $F^{(m-1,n)}$.

The transformation of the shifted Hamiltonian under a boost is given by

$\begin{split}\tilde{H}\rightarrow\tilde{H}&-m\mathbf{v}\cdot\int_{\mathbf{x}\mathbf{p}}\left(\epsilon-\frac{p^{2}}{2m}\right)\nabla_{\mathbf{p}}f_{0}\\\
&+m\mathbf{v}\cdot\int_{\mathbf{x}\mathbf{p}}\left(\nabla_{\mathbf{p}}\epsilon-\frac{\mathbf{p}}{m}-\int_{\mathbf{p}^{\prime}}F^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})\nabla_{\mathbf{p}^{\prime}}f_{0}(\mathbf{p}^{\prime})\right)\delta
f(\mathbf{x},\mathbf{p})\\\ &+\mathcal{O}(\delta
f^{2},\nabla_{\mathbf{x}})\,.\end{split}$ (135)

Rotational invariance kills the term in the first line, while the second line
gives us a non-trivial constraint,

$\nabla_{\mathbf{p}}\epsilon-\frac{\mathbf{p}}{m}=\int_{\mathbf{p}^{\prime}}F^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})\nabla_{\mathbf{p}^{\prime}}f_{0}(\mathbf{p}^{\prime})\,,\qquad||\mathbf{p}|-p_{F}|\ll
p_{F}\,.$ (136)

The requirement of $\mathbf{p}$ being sufficiently close to $p_{F}$ comes from
the fact that $\delta f$ must be localized near the Fermi surface for a
perturbative expansion in $\delta f$ to be valid. It suffices to set
$\mathbf{p}$ to a point $p_{F}\mathbf{n}_{\theta}$ on the Fermi surface and
write $\nabla_{\mathbf{p}}\epsilon|_{p_{F}}=p_{F}\mathbf{n}_{\theta}/m^{*}$,
where $m^{*}=p_{F}/v_{F}$ is the effective mass of the quasiparticle.
Furthermore, the $\nabla_{\mathbf{p}^{\prime}}f_{0}$ term in the integral sets
$\mathbf{p}^{\prime}$ to be on the Fermi surface as well, and we can expand
the Landau interaction function in angular channels using rotational
covariance. For example, in $d=2$, we write

$F^{(2,0)}(p_{F}\mathbf{n}_{\theta},p_{F}\mathbf{n}_{\theta}^{\prime})=\frac{8\pi^{2}v_{F}}{p_{F}^{2}}\sum_{l\geq
0}F_{l}\cos l(\theta-\theta^{\prime})\,,$ (137)

to simplify the boost invariance constraint to

$p_{F}\left(\frac{1}{m^{*}}-\frac{1}{m}\right)\mathbf{n}_{\theta}=-v_{F}F_{1}\mathbf{n}_{\theta}\,.$
(138)

Solving for the effective mass in terms of the Galilean boost parameter $m$
and the first Landau parameter $F_{1}$, we find the known result:

$m^{*}=m(1+F_{1})\,.$ (139)

####  V.3.2 Coupling to $U(1)$ gauge fields

As mentioned briefly before, the procedure for coupling our theory to
background gauge fields is very different from the usual procedure of gauging
a global symmetry. A systematic procedure for coupling Fermi liquids to a
gauge field has been difficult to achieve in the past owing to the fact that
effective theories live in momentum space, and here we present a new approach
that provides a solution.

The key observation is that the set of gauge transformations, characterized by
functions $\lambda(t,\mathbf{x})$, forms a subalgebra of infinitesimal
canonical transformations. All such functions Poisson-commute with each other,
since they do not depend on $\mathbf{p}$, so this subalgebra is abelian. It is
not difficult to show that under the canonical transformation $W=\exp\lambda$,
we have

$(\text{Ad}^{*}_{W}f)(\mathbf{x},\mathbf{p})=f(\mathbf{x},\mathbf{p}+\nabla_{\mathbf{x}}\lambda)\,.$
(140)

These then act on the coset representative $U=\exp(-\phi)$ as

$U\rightarrow
WU\,,\qquad\phi\rightarrow\phi-\lambda+\frac{1}{2}\\{\lambda,\phi\\}+\ldots\,\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (141)

The above transformation makes it clear why the usual procedure of gauging the
global $U(1)$ symmetry (117) by promoting the transformation to depend on
space and time is ambiguous when applied to the current theory, since simply
promoting the transformation parameter to a function misses out on the
nonlinear corrections in the Baker-Campbell-Haussdorff formula. The minimal
coupling procedure then is blind to nonlinear couplings to the gauge field as
well as contact terms required to ensure gauge invariance.

Naturally, the Fermi liquid action is not invariant under these
transformations, so we need to turn on background gauge fields
$A_{\mu}(t,\mathbf{x})$ that transform under the gauge transformation as

$A_{\mu}(t,\mathbf{x})\rightarrow
W^{-1}(A_{\mu}-\partial_{\mu})W=A_{\mu}(t,\mathbf{x})-\partial_{\mu}\lambda(t,\mathbf{x})\,,$
(142)

where $\mu=(t,\mathbf{x})$ is a spacetime index. The WZW term and the
Hamiltonian can be made invariant separately under gauge transformations. Let
us start with the WZW term, whose transformation is given by

$\begin{split}U^{-1}\partial_{t}U&\rightarrow
U^{-1}\partial_{t}U+U^{-1}(W^{-1}\partial_{t}W)U\,,\\\
\implies\delta_{\lambda}S_{\text{WZW}}&=\int dt\left\langle
f_{0},U^{-1}(\partial_{t}\lambda)U\right\rangle\,.\end{split}$ (143)

Evidently, making this invariant amounts to modifying it to the following:

$S_{\text{WZW}}[\phi,A_{0}]=\int dt\left\langle
f_{0},U^{-1}(\partial_{t}-A_{0})U\right\rangle\,,$ (144)

which is now invariant under the simultaneous transformation

$U\rightarrow WU\,,\qquad A_{0}\rightarrow W^{-1}(A_{0}-\partial_{t})W\,.$
(145)

Next, to make the Hamiltonian invariant, it suffices to ensure the invariance
of $f$ under gauge transformations by coupling it to the background gauge
fields. One can see that the appropriate modification is

$f_{A}(t,\mathbf{x},\mathbf{p})\equiv
f(t,\mathbf{x},\mathbf{p}+\mathbf{A}(t,\mathbf{x}))\,,$ (146)

where $\mathbf{A}$ is the spatial part of the gauge field. Since $\mathbf{x}$
does not transform at all under the gauge transformation, the transformation
of $\mathbf{p}$ is cancelled by the gauge transformation of $\mathbf{x}$.
While $f_{A}$ is now gauge invariant, its spatial derivatives are not, since

$(\nabla_{\mathbf{x}}f)(\mathbf{x},\mathbf{p})\rightarrow(\nabla_{\mathbf{x}}f)(\mathbf{x},\mathbf{p}+\nabla_{\mathbf{x}}\lambda)+\\{\nabla_{\mathbf{x}}\lambda,f\\}(\mathbf{x},\mathbf{p}+\nabla_{\mathbf{x}}\lambda)\,.$
(147)

But this is straightforwardly remedied by replacing partial derivatives by
covariant derivatives:

$D_{\mathbf{x}}f\equiv\nabla_{\mathbf{x}}f-\\{\mathbf{A},f\\}\,.$ (148)

While $f$ transforms covariantly under canonical transformations, the
fluctuation $\delta f=f-f_{0}$ does not, so it is convenient to re-expand the
Hamiltonian in $f$ instead of $\delta f$, with modified Wilson coefficient
functions $\tilde{F}^{(m,n)}$ that can be related straightforwardly to the
original ones in equation (64). The modified gauge-invariant Hamiltonian is
then

$\begin{split}H_{\text{gauged}}[f,\mathbf{A}]=H[f_{A}]&=\int_{\mathbf{x}\mathbf{p}}\epsilon(\mathbf{p})f(\mathbf{x},\mathbf{p}+\mathbf{A})\\\
&+\frac{1}{2}\int_{\mathbf{x}\mathbf{p}\mathbf{p}^{\prime}}\tilde{F}^{(2,0)}(\mathbf{p},\mathbf{p}^{\prime})f(\mathbf{x},\mathbf{p}+\mathbf{A})f(\mathbf{x},\mathbf{p}^{\prime}+\mathbf{A})\\\