B.S., Physics, Rice University (2014)
B.A., Mathematics, Rice University (2014)
M.A.S.t., Theoretical Physics, University of Cambridge, (2015) Department of
Physics

Doctor of Philosophy in Physics

May 2022 February 11, 2022

Xiao-Gang WenCecil and Ida Green Professor of Physics

Deepto ChakrabartyProfessor of Physics, Associate Department Head

# Chiral Phases on the Lattice

Michael Austin DeMarco

While chiral quantum field theories (QFTs) describe a wide range of physical
systems, from the standard model to topological quantum matter, the
realization of chiral QFTs on a lattice has proved to be difficult due to the
Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In
this thesis, we use the connection between chiral phases of matter and chiral
quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a
huge class of exotic field theories to be simulated numerically. Our work
builds on the ‘mirror fermion’ approach to the problem of defining chiral
theories on a lattice, which defines chiral field theories as the edge modes
of chiral phases. We begin by reviewing the deep connections between chiral
phases of matter, chiral field theories, and anomalies. We then develop
numerical treatments of an $SU(2)$ chiral field theory, and provide a
semiclassically solvable definition of Abelian $2+1$ chiral topological
orders. This leads to an exactly solvable definition of chiral $U(1)$ SPT
phases with zero correlation length, which we use to extract the edge chiral
field theories exactly. These zero-correlation length models are vastly more
simple than previous approaches to defining chiral field theories on the
lattice.

### Acknowledgments

No PhD is accomplished alone. Even so, I feel that I have been particularly
fortunate to be surrounded by so many wonderful people over these past years.

I have had the most extraordinary advisor in Xiao-Gang. Writing about Sir
Isaac Newton, the economist Keynes said: “Newton was not the first of the age
of reason. He was the last of the magicians…” There is something of that magic
in Xiao-Gang. I have never met anyone like Xiao-Gang before, and I doubt I
ever will again. His mind really does pierce the mysteries of the universe to
reveal the secrets of quantum mechanics. He dreams up solutions (that even he
cannot explain how he found) which show us entirely new aspects of this
strange world. So great is his contribution that there are two historical
periods characterizing our understanding of the phases of matter: before and
after Xiao-Gang. I would like the reader to understand something even more
extraordinary about him: with his great mind and all of his achievements,
never once did he chastise me for basic questions, even at the beginning of my
PhD. Xiao-Gang was always encouraging, kind, and genius. I have been
incredibly fortunate to be his student.

I have also been lucky to have had excellent teachers and collaborators
throughout my PhD. Xiao-Gang’s other students Hamed Pakatchi and Wenjie Ji
graciously mentored me as I joined the group. Ethan Lake and Jing-Yuan Chen
taught me much of what I know about Chern-Simons Theories on the lattice, and
I collaborated with Ethan on much of the work in this thesis. Cyprian
Lewandowski and Michal Papaj were great friends and patient teachers of solid
state physics; in addition to being a great friend (and hosting a surprise
party after my defense!), Kushal Seetharam has given me a wonderful
introduction to quantum devices. Michael Pretko has been a friend and mentor,
as has Alex Dear. Josue Lopez has been a friend, lab mate, and teacher for
many years, and I was thrilled to see him marry another friend, Jessica Ruiz.
Shane Alpert has been a consistent companion through this Physics journey. I
am also grateful to my committee, Senthil Todadri and Will Detmold, for their
supervision.

I am especially grateful to the many friends and family who made this
possible. My first thanks go to my parents, who raised me, encouraged (and
later tolerated) my interest in science for all these years, always stood by
me and supported me, and graciously housed my girlfriend and I as we fled
Covid in the early days. I could never have done this without their love and
examples. Thanks also go to my mother for her extensive corrections to this
thesis. I am deeply grateful to my girlfriend, Rachel Grasfield, whose
strength I have borrowed and counsel I have relied on many times over the past
four years. Sharing the various highs and lows of graduate school with her has
been a joy in itself.

Nancy and Steve Salmon have always been there for me, and I am deeply grateful
for their love. All of my family have been wonderful throughout this journey:
Janice and Rick, Lynda and Carlos, Bob and Tracy, Bill, Katie and Nico, Jon
and Christie, Connor and Katie, Ben and Melina, Alicia, and Lauren. To my
grandfather Carlo, my grandmother Jane, my grandfather Bill, and my
grandmother Lee: I wish you could have seen this.

There are so many others. My High School teachers, Mr. Chuang and Mr. Barrows,
took a wayward teenager and led him to the light of physics. Tom and Jennifer
Pekar have been dear friends. Cathy Modica has been a deeply wise friend,
mentor, and guide to whom I will always be grateful; her advice was a light in
many confusing moments during my PhD. Sydney Miller has been essential in the
thesis process. I deeply enjoyed serving on the Federal Affairs Board with Ben
Lane, Seamus Lombardo, Jimmy McCrae, Jordan Harrod, and all my friends there;
I look forward to hearing about their work to continually strengthen our
country. To Lisa Vo and Matt Zinman, who took a plunge with me to found a
technology start-up: I will always be grateful for the faith you placed in me.
And to our advisors, Nicco Mele, Julia Lestage, and Steven Liss: thank you for
your wise teaching throughout the journey. To Rahul Srivathsa, Sameer Abraham,
Kamna Kathuria, Vinay Ramprasad, Faiyad and Sheza Ahmad, Purva Jain, Teresa
Modigel, Peeya Tak, David Chen, Alex Kendall, Joe Fisher, Francesca
Bastianello, Michael Schmid, Michael Bryden, Christi Economy, Arthur
Kouyoumidjan, Mariel Pettee, Christian Henry, and Nick Ryder: thank you for
your friendship, kindness, and antics. Jerry and Susu Meyer have been kind
friends and wise mentors; I often hear Jerry’s advice ringing in my ears and
guiding me towards the wiser path. Pastor Jon Lee guided me through one of the
toughest times of graduate school. To Dominic Valenti, my brother and friend,
and his family, Lupita, Peepaw, and others: I am so deeply grateful for your
presence in my life.

You—the family, the friends, the teachers—are what made this possible, and I
will be grateful for the rest of my life. I have absolutely loved my time here
at MIT. As much as I could not be more thrilled for what comes next, it is a
bittersweet moment. In the words of Winnie the Pooh, “How lucky am I to have
something that makes saying goodbye so hard.”

###### Contents

  1. 1 Introduction
  2. 2 Condensed Matter Quantum Field Theories
    1. 2.1 Practical Lessons in QFTs
    2. 2.2 The Chiral Fermion Problem
  3. 3 Chiral Field Theories and Quantum Matter
    1. 3.1 Chiral Field Theories, Edge Theories, and the Mirror Fermion Approach
      1. 3.1.1 Integer Quantum Hall on a Lattice
    2. 3.2 Topological Order, SPT Phases, and Local Unitary Quantum Gates
    3. 3.3 Abelian Chern-Simons Theories
    4. 3.4 Lattice Field Theories and the Group Cohomology Picture of SPTs
  4. 4 A Proposal for a Non-Perturbative Lattice Regularization of an Anomaly-Free $1+1$d $SU(2)$ Chiral Fermion Theory
    1. 4.1 The Mirror Fermion Approach and the Bulk-Boundary Correspondence
    2. 4.2 Anomaly Cancellation
    3. 4.3 Lattice Model
      1. 4.3.1 Hopping Terms
      2. 4.3.2 Higgs Field and Yukawa Coupling
    4. 4.4 Gapping Process
      1. 4.4.1 Numerical Results
      2. 4.4.2 Effective Higgs Action
    5. 4.5 Anomaly Matching and Application to a Condensed Matter System
    6. 4.6 Summary of the $SU(2)$ Model
  5. 5 A Lattice Rotor Model for $2+1$d Bosonic $U^{\kappa}(1)$ Chern-Simons Theory
    1. 5.1 Lattice Model
    2. 5.2 Higher Symmetries and Anomalies
      1. 5.2.1 Framing Anomaly
    3. 5.3 Summary of the Bosonic Lattice Chern-Simons Model
  6. 6 A $U(1)$ SPT and Discontinuous Lattice Topological Terms
    1. 6.1 Ungauging the Chern-Simons Theory
    2. 6.2 Generalization to Arbitrary $k$ and the Topological Mott Insulators
      1. 6.2.1 The Topological Mott Insulators
    3. 6.3 Model Overview
    4. 6.4 The Integral Phases
      1. 6.4.1 Ground State and SPT Order
      2. 6.4.2 Hall Conductance
      3. 6.4.3 Commuting Projector Model
    5. 6.5 Topological Transitions at Half-Integer $k$
      1. 6.5.1 Emergent Fermions
    6. 6.6 Lattice Renormalization Group
    7. 6.7 Discussion
  7. 7 A Chiral Lattice Field Theory in $1+1$d
    1. 7.1 Lattice Model
    2. 7.2 Anomalies
    3. 7.3 Continuum Theory
  8. 8 Outlook
    1. 8.1 Review
    2. 8.2 Further Lattice Field Theories
    3. 8.3 What’s Next
  9. A Cochains and Cohomology
  10. B Physical Interpretation of Level Shift Symmetry
  11. C Cech-Deligne-Beilinson Approach to Chern-Simons Theory
    1. C.1 Introduction
      1. C.1.1 Definition of a Scalar Field on the Lattice
      2. C.1.2 Definition of a Gauge Field on the Lattice
    2. C.2 Lattice Lagrangians
      1. C.2.1 Evaluation of Partition Functions
  12. D Commentary

###### List of Figures

  1. 2.1 Axial Anomaly in a Hamiltonian Model
  2. 2.2 Axial Fermion Anomaly from the Equation of Motion
  3. 2.3 Band Structure and the Nielsen-Ninomiya Theorem
  4. 3.1 Anomalous Theories as Edge Theories
  5. 3.2 Lattice Models for the Integer Quantum Hall Effect
  6. 3.3 Local Unitary Circuits and the Classification of Quantum Phases
  7. 3.4 Four Types of Quantum Phases
  8. 3.5 Gapping an SPT
  9. 3.6 Braiding Statistics for Chern-Simons Theories
  10. 4.1 The Mirror Approach
  11. 4.2 Schematic of the $SU(2)$ Model
  12. 4.3 Hopping Model for the $SU(2)$ Mirror Approach
  13. 4.4 Integrated Density of States in the $SU(2)$ Model
  14. 4.5 Strip Width and the Integrated Density of States
  15. 4.6 Mirror Theory Gap
  16. 6.1 Lattice Vortex Term
  17. 6.2 Time-reversal and Boundary Effects
  18. 6.3 Renormalization Group Flow for the $U(1)$ Topological $\theta$-term Model
  19. 6.4 Vortices and the Laughlin Thought Experiment
  20. 6.5 Commuting Hermitian Projectors
  21. 6.6 Commuting Projectors and the Hall Conductance
  22. 6.7 Emergent Fermions at the Phase Transition
  23. 6.8 Lattice Renormalization Group for the $U(1)$ $\theta$-term Model
  24. 7.1 Square Lattice
  25. C.1 Patchwise Gauge Fields
  26. C.2 Cubic Lattices, the Cup Product, and the Dual Lattice
  27. C.3 Wilson Lines in the CDB Formalism
  28. C.4 Lattice Pruning
  29. C.5 The Single Cell Lattice

###### List of Tables

  1. 3.1 Properties of Chiral Topological Orders and Chiral SPT Orders.
  2. 4.1 Anomaly-Free Chiral $SU(2)$ Representations for Fermionic Excitations.
  3. 4.2 Anomaly-Free Chiral $SU(2)\times U(1)$ Representations for Fermionic Excitations.

##  Chapter 1 Introduction

Quantum Field Theories are wild things. It has been nearly a century since
Born, Heisenberg, Jordan, and Dirac discovered Quantum Field Theories (QFTs)
[86, 97], and yet these theories remain mysterious. They are among our most
precise theories of physics, delivering predictions accurate to one part in a
trillion in some cases. However, as mathematical objects they often make no
sense, requiring careful regulation to tame their divergences. QFTs are
extremely versatile: they govern the behavior of all known quantum systems,
ranging from fireballs at trillions of Kelvin inside the Large Hadron Collider
to the cold condensates at at fractions of a billionth of a Kelvin, just one
building over from where this thesis was written. Much of theoretical physics,
from Schwinger and Feynman to Witten and Wen, is devoted to their study,
simulation, and calculation.

QFTs are so powerful and versatile because they capture the universal, long-
distance behavior of quantum systems of many particles. QFTs were discovered
in a context close to modern high-energy physics, where not only do the
energies involved create many particles but the physics itself becomes
sensitive to the immense number of virtual particles winking in and out of
existence. However, QFTs find a natural application to the theory of condensed
matter systems, where we may consider ${\sim}10^{23}$ particles interacting on
a vast lattice. From our condensed matter perspective, the ‘long-distance
behavior’ of particles on a lattice describes the characteristic properties of
a phase of matter near a gapless point, where correlation lengths grow and the
dynamics happen at long distances on the lattice scale. There the QFT can
reveal the fundamental nature of the phase: What are the low-energy
excitations? How does the system respond to a probe? QFTs capture the essence
of a system in the vicinity of a gapless point; in turn, the physics of many
particles on a lattice can be used to study and regulate quantum field
theories.

It is a testament to the wonder of physics that studying QFTs more closely
makes them more interesting and more mysterious. In the earliest days, the
precise definition of field theories was a pedagogical complication that could
be avoided with perturbative renormalization. QFTs were allowed to be
mathematically ill-defined because their predictions were so incredibly
accurate, while their formalization was left to mathematicians and
mathematical physicists. However, as our understanding of them grew, we found
that many of their most important properties—anomalies, strange conservation
laws, and beyond—were revealed when QFTs were defined precisely. This
relationship is at the core of the relationship between condensed matter and
high energy theorists that has recently ignited a golden age in the study of
topological phases and physics. In turn, a better definition of QFTs allows us
to develop a deeper understanding of the laws that govern the universe and to
use those laws to create new science and technology.

This thesis is anchored around the application of condensed matter theory to
the simulation of chiral phase and QFTs. Chiral phases and field theories,
where left and right-handed excitations behave differently, are particularly
challenging to define owing to the appearance of the ‘quantum anomalies’ that
will be defined in Chapter 2. In particular, we set out to define and regulate
chiral QFTs in the lattice, in a way which either cancels out their anomalies
or renders those anomalies explicitly calculable. We will begin with a
numerical simulation of a previously proposed approach to defining chiral
QFTs, and end with an exactly solvable approach that more efficiently creates
$U(1)$ chiral lattice field theories and reveals their subtle properties.Our
results also lead to new phases of chiral matter in condensed matter systems.

Chapters 2 and 3 form the background of this thesis. In Chapter 2, we will set
out a few of the properties of QFTs that we wish to study and demonstrate them
in the context of simple lattice models. The most important of these are
quantum anomalies. We establish a simple picture of the ‘chiral’ or ‘axial’
anomaly in $1+1$d with a lattice model, and we discuss the general relevance
of quantum anomalies to actually defining QFTs on the lattice. In Chapter 3,
we will lay out the connections between phases of matter and quantum field
theories that we defined previously. We will introduce the concepts of
‘topological’ order and ‘SPT’ order and establish the relationships between
the bulk quantum order and the anomalies of the edge theories. We will also
elaborate specific quantum field theories that we wish to simulate in the
following chapters, and comment on their classification.

Our new results are presented in Chapters 4-7. In Chapter 4, we follow a
previously proposed approach to create a new regularization of a $1+1$d
$SU(2)$ chiral field theory. This approach depends on creating a non-chiral
field theory, then giving a subset of fields a large gap so that only the
chiral theory is left at low energies, which we are able to successfully
demonstrate. Nonetheless, this approach faces technical obstacles to
generalization. In Chapter 5, we approach a related problem, the simulation of
gapped Abelian topologically ordered theories in $2+1$d. We are able to create
rotor models which can be reliably solved semiclassically, a significant
achievement for these theories. Following this, in Chapter 6, we ‘ungauge’ the
theories of Chapter 5 to create exactly solvable models of SPT ordered phases.
These exactly solvable theories are also a significant achievement, and we use
them to create commuting projector models previously thought to be impossible.
In an extraordinary turn, these SPT models immediately lead to exact chiral
field theory models as a corollary, and we explore these chiral field theories
and their anomalies in Chapter 7.

This thesis details an extended, and successful, search [15] for fruitful
lattice models of chiral QFTs. We begin by defining a chiral field theory in a
$1+1$d $SU(2)$ model and, while successful, find obstacles to future work. We
then take a meandering path through higher-dimensional phases, eventually
uncovering an SPT formalism which immediately yields a chiral field theory of
the sort we had originally sought. We hope the reader will find the journey
enlightening and useful.

##  Chapter 2 Condensed Matter Quantum Field Theories

In this chapter, we lay out some basics of quantum field theories,
establishing our notions of chirality and symmetry, for the perspective of a
reader broadly familiar with physics and calculus. The expert should refer
directly to Chapter 3.

As mentioned in the previous chapter, QFTs capture the long-distance,
universal behavior of a system of many particles. However, with
renormalization group (RG) theory, we can go further. RG leads to equivalence
classes of QFTs which have the same long-distance behavior by erasing or
‘integrating out’ short-distance degrees of freedom. Under this process, QFTs
morph into one another, until eventually only infinite-range behavior is
captured. These infinite correlation length ‘fixed-point’ QFTs are in fact
representatives of phases of matter. To study any QFT, fixed point or
otherwise, is to study a phase of matter and gain insight into the collective
behavior of a system.

Something extraordinary arises in QFTs: universality. Because there are
relatively few classes of fixed-point QFTs, there are relatively few patterns
of behavior in the physical world. This allows relationships between various
systems and their phase transitions to be derived, and is why the pattern of
cracks in metals and rocks mirror the image of lightening (eg [29]), while a
similar argument applies to the shifts of sand and snow in piles (eg [69]).
The Superfluid-Mott-Insulator transition we study in Chapter 6 is in fact in
the same universality class as the condensation of a superfluid and the
ferromagnetic ordering of two-dimensional ‘XY’ magnets, which have both been
studied extensively. This universality is a crowning triumph of modern QFTs
and is a principle behind the ordering of all matter.

Throughout the decades of QFT, significant progress has involved both the
condensed matter and high-energy communities, from the theory of symmetry
multiplets and particles to the theory of phases and Wilsonian renormalization
group (RG). We are currently in a ‘golden age’ [7] arising from just such a
collaboration, where the theory of quantum anomalies, higher symmetries, and
topological phases have been unified and the two communities are working
closer than ever, though differences of ideas and language remain. This
thesis, and the research herein, has its ideas firmly rooted in the condensed
matter side, but seeks to build tools that both the condensed matter and high-
energy communities will find useful. In particular, we set our sights on the
Chiral Fermion Problem, which has plagued simulations of both condensed matter
and high-energy theory for decades. We will demonstrate a solution to it in
Chapter 4, before developing the theory to create a far more efficient
solution in Chapter 7.

###  2.1 Practical Lessons in QFTs

A huge amount of research has gone into defining just what a QFT is: as sums
of all functions, renormalizable path integrals, quantizations of phase space,
operator-valued distributions, and more. Even more complex is the relationship
between traditional QFTs defined in continuous spacetime and lattice QFTs
defined in discrete spacetime.

We will sidestep all of this and instead take a simple-minded approach. In
this thesis, we always define QFTs as lattice QFTs. Any continuum quantities
we write down will be only useful abstractions that help us reason about the
dynamics of an underlying lattice model. This approach will render many of the
more surprising and subtle properties of QFTs extremely clear, as on the
lattice it is difficult for any complication to hide.

Let us now see this in practice. We will consider a simple $1+1$d lattice
model and use it to see some of the major features of QFTs: dynamics,
symmetry, chirality, anomalies, and the relationship between the lattice and
the continuum as well as the Hamiltonian and Lagrangian formalisms.

Consider fermions on a lattice in one spatial dimension. We label the $N$
lattice sites by $i=1,...,N$ with periodic boundary conditions, and introduce
creation and annihilation operators on each site which satisfy
$\\{\hat{c}_{i},\hat{c}_{j}^{\dagger}\\}=\delta_{ij}$,
$\\{\hat{c}_{i},\hat{c}_{j}\\}=\\{\hat{c}_{i}^{\dagger},\hat{c}_{j}^{\dagger}\\}=0$.
The Hamiltonian for our lattice model is given by:

$\hat{H}=-\frac{t}{2}\sum_{i}\left(\hat{c}_{i}^{\dagger}\hat{c}_{j}+\hat{c}_{j}^{\dagger}\hat{c}_{i}\right)-\mu\sum_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}$
(2.1)

This free theory describes particles hopping left or right, with a chemical
potential $\mu$. It can be diagonalized in momentum space by setting
$\hat{c}_{k}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{j}e^{ikj}\hat{c}_{j}^{\dagger}$,
where $k=\frac{2\pi m}{N}$, $m=1,...,N$. Doing so, we obtain the Hamiltonian:

$\hat{H}_{k}=-\sum_{k}(t\cos(k)+\mu)\hat{c}^{\dagger}_{k}\hat{c}_{k}$ (2.2)

(We will always assume unit lattice spacing). The eigenspectrum of this
Hamiltonian with $t=1,\mu=0$ is shown in Figure 2.1. At zero temperature, the
states with energy $E<\mu$ will be filled by fermions and are depicted in
blue, while states with $E>\mu$ will be unfilled. The filled states are
referred to as the ‘Fermi Sea’, while the states right at $\mu=0$ are the
‘Fermi Surface’ (It is indeed a surface in higher dimensions) and have Fermi
momenta $\pm k_{F}$.

Figure 2.1: _(Color Online)_. _(left)_ Eigenenergies of the Hamilonian (2.2)
with $\mu=0$. Filled states, below the chemical potential, are shown in blue,
while unfilled states are shown in red. _(middle)_ As we adiabatically change
the background field $A$, the energies evolve as $-\cos(k-A)$ _(right)_ After
adiabatically evolving the background gauge field from $A=0$ to
$A=\frac{2\pi}{N}$, we transfer charge from the left-moving mode to the right-
moving mode. This is the axial anomaly.

In the spirit of understanding the long distance physics, it is important to
understand what are the low-energy excitations of this system. With some
energy $\epsilon\ll t$, the only possible charge-conserving excitations
involve moving a fermion from a filled state just below the Fermi surface to
an empty state just above the Fermi surface. Crucially, each point on the
Fermi surface comes with an associated group velocity:

$v_{F}=\frac{dE}{dk}=\pm t\sin(k_{F})$ (2.3)

The disturbance we may create by exciting a particle near one of these Fermi
points will move to the left or right along our 1d spatial lattice with
velocities given by $v_{F}$.

To understand the phase of matter, we need only understand the physics near
those two points. We do so by writing down a Lagrangian that captures two
fermionic modes: one right-moving and one left moving. The correct model is:

$S=\int
dxdt\left[\psi^{\dagger}_{L}(\partial_{t}+\partial_{x})\psi_{L}+\psi^{\dagger}_{R}(\partial_{t}-\partial_{x})\psi_{R}\right]$
(2.4)

Here $\psi^{\dagger}_{L,R},\psi_{L,R}$ are anticommuting Grassman fields and
we have rescaled space so that $v_{F}=1$. Note that varying with respect to
$\psi^{\dagger}_{L,R}$ reproduces the equations of motion:

$\displaystyle(\partial_{t}+\partial_{x})\psi_{L}=0$ (2.5)
$\displaystyle(\partial_{t}-\partial_{x})\psi_{R}=0$ (2.6)

These are the wave equations in $1+1$d for an excitation moving to the left or
right, respectively.

In just a few short lines, we have written down both a Hamiltonian and
Lagrangian description of particles hopping in one dimension. Surprisingly,
quite a lot can be derived from these models, including their symmetries and a
specifically quantum phenomenon called a quantum anomaly.

The Hamiltonian models (2.1), (2.2) have an important $U(1)$ global symmetry
given by:

$\displaystyle\hat{c}_{i}\to e^{i\theta}\hat{c}_{i}\hskip
28.45274pt\hat{c}_{i}^{\dagger}\to e^{-i\theta}\hat{c}_{i}^{\dagger}$ (2.7)
$\displaystyle\hat{c}_{k}\to e^{i\theta}\hat{c}_{k}\hskip
28.45274pt\hat{c}_{k}^{\dagger}\to e^{-i\theta}\hat{c}_{k}^{\dagger}$ (2.8)

where $\theta$ is a constant. Crucially, this implies that
$\sum_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}$ commutes with $\hat{H}$,
$[\sum_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i},H]=0$, and so we have a conserved
quantity, namely conservation of charge.

We can see the same behavior in the Lagrangian formalism by sending:

$\psi_{L,R}\to e^{i\theta}\psi_{L,R}\hskip 28.45274pt\psi^{\dagger}_{L,R}\to
e^{-i\theta}\psi^{\dagger}_{L,R}$ (2.10)

which leaves (2.4) invariant. Using Noether’s theorem, we see that the charge:

$Q=\int dx(\psi^{\dagger}_{L}\psi_{L}+\psi^{\dagger}_{R}\psi_{R})$ (2.11)

is a conserved quantity.

Examining the Lagrangian (2.4), it would appear that we could rotate the phase
independently on each of the left-moving and right-moving fields. Formally, we
would implement this by, in addition to the ‘vector’ symmetry (2.10) employing
an ‘axial’ symmetry:

$\displaystyle\psi_{L}\to e^{i\theta}\psi_{L}\hskip
28.45274pt\psi^{\dagger}_{L}\to e^{-i\theta}\psi^{\dagger}_{L}$ (2.12)
$\displaystyle\psi_{R}\to e^{-i\theta}\psi_{R}\hskip
28.45274pt\psi^{\dagger}_{R}\to e^{i\theta}\psi^{\dagger}_{R}$ (2.13)

which rotates the phases on left and right-moving fields oppositely. Between
the vector and axial symmetries, we are able to independently rotate the phase
on the left and right moving modes. Instead of the the single conserved
quantity (2.11), we would obtain two:

$\displaystyle Q_{L}=\int dx~{}\psi^{\dagger}_{L}\psi_{L}$ (2.14)
$\displaystyle Q_{R}=\int dx~{}\psi^{\dagger}_{R}\psi_{R}$ (2.15)

These two charges would have the interpretation of the number of excitations
at the left and right Fermi points, respectively.

Turning back to the Hamiltonian model, we see that there is no local way to
interpret these conserved charges in terms of the
$\hat{c}_{i},\hat{c}_{i}^{\dagger}$ operators, as there is no way in the
Hamiltonian formulation to address the two Fermi points separately. This is
our first clue that something might be wrong with the axial ‘symmetry.’

More directly, we can consider coupling the model to a background gauge field,
whereby we modify the Hamiltonian to be:

$\hat{H}=-\frac{t}{2}\sum_{i}\left(\hat{c}_{i}^{\dagger}e^{iA}\hat{c}_{j}+\hat{c}_{j}^{\dagger}e^{-iA}\hat{c}_{i}\right)-\mu\sum_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}$
(2.16)

where we may take $A$ to be constant in space, but not time. In momentum
space, this Hamiltonian is:

$\hat{H}_{k}=-t\sum_{k}\cos(k-A)\hat{c}_{k}^{\dagger}\hat{c}_{k}$ (2.17)

where we have set $\mu=0$. So we see the effect of this background gauge field
is to shift the energies. Recalling that $k=\frac{2\pi m}{N}$ for $m=1,...,N$,
consider the effect of slowly changing $A$ from $A_{i}=0$ to
$A_{f}=\frac{2\pi}{N}$ as shown in Figure 2.1. The spectrum is identical for
both $A=A_{i}$ and $A=A_{f}$. However, adiabatically evolving the system would
transfer a filled state above the Fermi sea, effectively transferring charge
from the left-moving mode to the right-moving mode. Thus we see that the left
and right moving modes _do not independently conserve charge_ in the presence
of a gauge field. Only the sum of left and right moving is conserved. The
axial ‘symmetry’ is not a symmetry it all!

It is remarkable that this breaking of the axial symmetry is in fact
universal. Translating the argument above into an equation, we have seen that
the change of the axial charge is given by:

$\partial_{t}Q_{A}=\frac{2e}{2\pi}\partial_{t}A$ (2.18)

In a more general covariant theory with $e=1$, this can be rewritten as:

$\partial_{\mu}j_{A}^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu}F_{\mu\nu}$ (2.19)

where $j_{A}^{\mu}$ is the axial conserved current which is predicted to be
conserved by Noether’s theorem. This is the exact expression that can also be
derived by expanding the action for $A$ in terms of Feynman diagrams [79].
Moreover, this result is general: any lattice model that has charge-$n$ right
and left moving modes will contain a similar anomaly term, with the right-hand
side multiplied by111One factor of $n$ arises because the mode is $n$ times as
sensitive to the gauge field, and so we would lose $n$ times as many
particles. The second factor arises because each particle carries a charge of
$n$. $n^{2}$. The result is independent of the details of the model and so is
_universal_.

Figure 2.2: _(Color Online)_. A right-moving mode is created in the middle of
an open wire. If right and left moving charge are independently conserved,
then the right moving mode would continue right off the wire, which is
impossible. Hence the axial symmetry must break at a boundary.

We can also see this failure from a very physical argument. Suppose that,
instead of closed boundary conditions, we consider the theory on an open line
segment as shown in Figure 2.2. We create a right-moving excitation in the
center of the system. If the axial symmetry held, then the right-moving
excitation could only move to the right, and could not scatter into left-
moving modes. It would continue to the right, right off the end of the system,
thereby violating the axial (and vector) symmetry. This must not happen, and
so it must be true that the boundary also breaks the axial symmetry. In fact,
the symmetry breaking on a boundary should be considered as a consequence of
the breaking with a gauge field, as an electric potential could be used to
create a boundary. However, in practice the boundary formulation is often more
convenient.

Whenever a theory would seem to have a symmetry, but that symmetry is broken
in the presence of a gauge field or a boundary, we say that the theory has an
_anomalous symmetry_ or that the symmetry has a _quantum anomaly_. In this
case it is a ‘mixed anomaly’ between the global $U(1)$ and the axial $U(1)$
which breaks axial symmetry. This anomaly is our first glimpse of a ‘chiral’
anomaly. The next section will dive much deeper into these anomalies and their
relationship to a decades-old problem in lattice gauge theory.

###  2.2 The Chiral Fermion Problem

One feature we noted of the axial ‘symmetry’ is that it is not present in the
original Hamiltonian lattice model, nor is it clear how to write down a
lattice model in one dimension that would have a axial symmetry affecting the
left and right-moving modes differently. In many ways, this difficulty of the
lattice model is a saving grace: it is telling us that the theory is
anomalous. If we had some local lattice model with an axial symmetry, we could
couple it to a background gauge field and obtain a well-defined, non-anomalous
field theory. But the anomaly is universal and so this is impossible—and that
is what the difficulty in defining an axial lattice symmetry is hinting at.

The anomaly discussed in the previous section has the unique property that it
is _chiral_ , meaning that it treats left and right differently, and we now
adopt the term _chiral anomaly_ for the behavior we have seen, instead of
axial anomaly which is more common in high-energy physics. More generally, a
theory, symmetry, or field is chiral if it is not invariant under the
inversion of one spatial dimension. In general, we will be concerned with
chiral anomalies and their relations to chiral phases of matter.

In the last section, we wrote down a Lagrangian (2.4) describing left and
right moving modes. It is very tempting to attempt to take only half of that
Lagrangian, say:

$\int dxdt\psi^{\dagger}_{R}(\partial_{t}-\partial_{x})\psi_{R}$ (2.20)

as a model for a _chiral field theory_. This theory would effectively live
under a “vector $+$ axial symmetry”, with an anomaly:

$\partial_{\mu}j^{\mu}=\frac{e}{4\pi}\epsilon^{\mu\nu}F_{\mu\nu}=\frac{e}{2\pi}F_{01}$
(2.21)

This anomaly is even more severe than the axial one, as it is not ‘mixed’ but
direct: flux in the gauge field directly breaks the $U(1)$ symmetry to which
the gauge field is coupled. Just as the axial symmetry lacked a lattice
definition, there is a severe obstruction to defining this theory on the
lattice, and for the same reason: because the anomaly is universal, it must
also appear on the lattice. On the other hand, it is not immediately clear how
such an anomaly could possibly appear in a lattice model.

The difficulty in defining a chiral field theory on a lattice is the subject
of the decades-old _chiral fermion problem_ (CFP). Specifically, the CFP
refers to the difficulty to define a fermion theory in odd spatial dimensions
which satisfies:

  * •

The Hilbert space of the theory factorizes as a product of local Hilbert
spaces.

  * •

The theory can be coupled to a background gauge field.

  * •

The theory has Hamiltonian or Lagrangian formulation involving only local
terms.

  * •

The theory has symmetry $U(\theta)$ which factorizes as a product of operators
on each site $U(\theta)=\otimes_{i}U_{i}(\theta)$.

  * •

The symmetry $U$ is not inversion symmetric, i.e. it treats left and right-
handed modes differently (note that this may include a case where there are
different numbers of left and right handed modes).

In this thesis, we will solve this problem in several ways. For simplicity, we
will often relax the condition that the theory be fermionic. In the most
successful way, we will have a solvable model, but will have to allow a non-
on-site symmetry $U(\theta)$. In that case, we will in fact have a lattice
model that is local, captures the quantum anomaly, and satisfies all the other
above conditions. However, we have many chapters before we encounter that
theory.

Nielson and Ninomiya proved that that the CFP is impossible to solve without
interactions [77, 78]. The basic argument is already visible in the results of
the previous section. There we saw that the eigenvalues of the Hamiltonian
$E_{j}(k)$ are a periodic function of $k$. We assume that we fill up
eigenstates up to some Fermi energy, and we assume that $\frac{dE_{j}}{dk}\neq
0$ at the Fermi energy, so that the $E_{j}(k)$ crosses the Fermi energy
linearly (see Figure 2.3). Because the functions are periodic, we are assured
that the number of crossings with $\frac{dE_{j}}{dk}>0$ (right-movers) is
equal to the number crossings with $\frac{dE_{j}}{dk}<0$. Hence there are
equal numbers of right-and left movers. Moverover, just as the two modes we
examined share a chiral anomaly between them, in the general case modes pair
up so as to share anomalies, and this will imply that that the charges of the
right-moving modes under the non-chiral $U(1)$ symmetry will mirror exactly
the charges of the left-moving modes, and the total theory is not chiral. The
generalization to higher dimensions involves more mathematics, but the basic
idea is the same.

Figure 2.3: _(Color Online)_. _(a)_ Schematic depiction of a periodic band
structure in one spatial dimension. The chemical potential is denoted by the
dotted line. Right moving modes are highlighted in blue and left moving modes
are highlighted in red. _(b)_ Actual band structure of a random translation-
symmetric Hamiltonian with third-nearest-neighbor hopping.

The difficulty in defining a chiral theory on a lattice has been a serious
frustration for the lattice gauge theory (LGT) community because of the
critical role chiral phenomena play in the standard model. For one, all
observed neutrinos are left-handed, while all observed anti-neutrinos are
right-handed [46]. Neutrinos couple to the dynamical gauge field of the weak
sector and simulating this interaction would be useful to the lattice gauge
theory community. Beyond particle physics, simulating a chiral field theory
with its attendant anomalies would have considerable use for condensed matter
theory, which regularly sees chiral field theories in edge theories, a fact
which we will devote almost the entire next chapter to.

Considerable work to evade the Nielsen-Ninomiya result and solve the CFP has
been performed over the intervening decades. The first thing one might ask is
why can we not start with the momentum-space Hamiltonian (2.2) and restrict to
one point near the Fermi level? Doing so, one effectively creates a
discontinuous Hamiltonian in momentum space, so when it is Fourier transformed
back to real space the Hamiltonian is infinitely long ranged. Far more elegant
versions of this idea exist [75, 76, 53], and all suffer from the same non-
locality. In some cases, these models can still be coupled to a weak
background gauge field, but the non-locality violates the conditions of the
CFP that we set out above. Two other approaches soon came out of the lattice
field theory community. In the overlap formalism [72, 73, 67, 68], the chiral
theory is defined as the overlap of successive ground states. However, the
physical interpretation of this formalism can be difficult, and it is not
clear if the Hilbert space factorizes as a product of local Hilbert spaces. In
a somewhat related model, one can realize the chiral theory as fermions living
on a domain wall [54, 85] inside a higher-dimensional space. However, the
gauge field which couples to the fermions propagates in that higher-
dimensional space.

A vein of approaches to the CFP involve similar ideas to what we will explore
in the next chapter, realizing the chiral theory as the boundary of a higher-
dimensional system [101, 95, 118, 44, 33, 70, 40, 43]. This will be explored
in detail in the next chapter, in conjunction with the development of ideas
around topological order and SPT phases that we will explore in that chapter.

We have taken a short stroll through quantum field theory, with a condensed
matter lens. The great advantage of this approach, blending between continuum
arguments and lattice models, is that it can unite the best of both worlds:
the continuum arguments build intuition for the physical picture, while the
lattice model nails down technical details and clarifies subtleties. This is
particularly true in the case of anomalies, where the use of lattice models
allows us to demonstrate the $U(1)$ anomaly in just a few lines. We then
parlayed this anomaly picture to illustrate the Chiral Fermion Problem, and
the problems with chiral field theories in general. In the next section, we
will explore in detail how chiral theories can appear naturally as boundary
theories.

##  Chapter 3 Chiral Field Theories and Quantum Matter

We have discussed the relationship between field theories and anomalies in the
context of a simple lattice model. Now we will elaborate on the relationship
both of these have with quantum phases of matter, before laying out the
general properties and classification of some of those phases that we will
require in subsequent chapters.

###  3.1 Chiral Field Theories, Edge Theories, and the Mirror Fermion Approach

In the previous chapter, we discussed several chiral anomalies and found that
they indicated the failure of conservation of charge in a system. We discussed
the non-conservation of charge in a system consisting only of a right-moving
mode in one spatial dimension, and indicated that its anomaly would make the
theory difficult to define on a lattice. A theory with a single right-moving
mode in which charge is not conserved would indeed be hard to make sense of.
Particles would be appearing and vanishing whenever an electric field is
applied, and at a boundary the particles would have to continue out of the
system and into the vacuum.

However, there is a very physical way to avoid all of this and define the
theory of a chiral mode on a lattice. We can solve the seeming contradiction
of particles vanishing from the one-dimensional spacetime by understanding the
one-dimensional system as the edge of a two dimensional bulk, as shown in
Figure 3.1. We consider the edge to be gapless, and the bulk gapped. Most of
the time, any particle is confined to traveling along the edge. However, when
an electric field is applied, it can tunnel into the bulk. From the
perspective of the edge, the particle has disappeared, while from the
perspective of the bulk a particle has suddenly appeared. Accounting for both
theories, particle number is actually conserved. Mathematically, this will
mean that the edge theory and the bulk theory have equal and opposite
anomalies which cancel only when the two theories are considered together.
This approach also nullifies the anomaly which happened at the boundary of the
one-dimensional system: the one-dimensional system cannot have a boundary
because it is itself a boundary of the two-dimensional system, and boundaries
cannot have boundaries [52].

Figure 3.1: _(Color Online)_. _(a)_ In a chiral theory in one spatial
dimension, charge is not conserved, and particles (blue dotted line) may
travel along before disappearing (red X). _(b)_ We now conceive of the one-
dimensional system as the gapless edge theory of the gapped two-dimensional
bulk. Particles do not disappear, but instead travel into the bulk.

At first glance, this may seem somewhat contrived. However, exactly such a
theory describes the Quantum Hall states. In the next subsection, we describe
a detailed lattice theory for these models and extract the relevant behavior.

####  3.1.1 Integer Quantum Hall on a Lattice

The Integer Quantum Hall (IQH) effect [93] has been extremely well studied.
Here, we concoct a simple lattice model that reproduces its properties and
demonstrates the chiral edge mode.

In the IQH, we take a system with two spatial dimensions and apply a
transverse magnetic field. The IQH occurs when there are an integer number of
particles per flux quantum of the magnetic field, i.e. $\nu\in\mathbb{Z}$,
where $\nu$ is the filling fraction. To implement this on a lattice, consider
a lattice with nearest neighbor and next-nearest-neighbor hopping. Labeling
the points by $x,y\in\\{1,...,L_{x}\\}\times\\{1,...,L_{y}\\}$ We apply a
magnetic field $A_{x}=2\pi\frac{y}{2}$, $A_{y}=0$, which leads to a field
strength $F=\partial_{x}A_{y}-\partial_{y}A_{x}=-\pi$ per plaquette. With a
half flux quantum per plaquette, at half filling, the system should reproduce
the IQH state.

Figure 3.2: _(Color Online)_. _(a)_ Spatial Lattice for the Integer Quantum
Hall Effect. Black links indicate a hopping of $+1$, red a hopping of $-1$,
and green links a hopping of $+i$ in the direction of the arrow. The flux
through any plaquette is $\pi$, so at half filling this realizes a $\nu=1$ IQH
state. _(b, c)_ Fourier transforming in the $x$ direction, we plot the band
structure as a function of $k_{x}$ for _(b)_ closed boundary conditions, where
we see a gapped bulk, and _(c)_ open boundary conditions, where we see the two
gapless modes. _(d, e)_ Spacetime propagation of a particle wavefunction in an
IQH system. We consider the IQH lattice model with open boundary conditions in
the y direction and periodic boundary conditions in the $x$ direction. The two
edges are shown in purple and green, respectively. We inject a particle
wavepacket with $k\approx 0$ at time $t=0$. The amplitude for the particle to
propagate to point $(x,y,t)$ is indicated by the size of the black markers at
$(x,y,t)$, and the strength of the background gauge field is shown in the
inset.

The lattice model is shown in Figure 3.2a. Writing it out explicitly, we have:

$\hat{H}=\sum_{x}\Big{[}\hat{c}^{\dagger}_{x+1,y}\hat{c}_{x,y}+(-1)^{x}\hat{c}^{\dagger}_{x,y+1}\hat{c}_{x,y}+i\hat{c}^{\dagger}_{x+1,y+1}\hat{c}_{x,y}+h.c.\Big{]}$
(3.1)

where we have set the hopping to be unit strength. One may check that hopping
counter-clockwise around any triangle leads to a factor of $+i$, and therefore
the flux through each square plaquette is $\pi$, as desired.

To understand the edge theory, we Fourier transform in the $x$ direction. The
resulting band theory is shown in Figure 3.1b for closed boundary conditions
and Figure 3.1c for open boundary conditions. For closed boundary conditions,
we see a gapped bulk theory across the entire one-dimensional Brillouin zone.
The various bands result because we have not taken the Fourier transform in
the $y$ direction. Note that if we place the chemical potential in the range
$-1.5\leq\mu\leq 1.5$, then we fill up half the bands, leading to the desired
half filling.

For open boundary conditions, we see two modes appearing inside the bulk gap.
One is clearly right-moving ($dE/dk>0$) and the other left-moving ($dE/dk<0$).
In this case, however, the two modes are in fact spatially separated—each of
them on their own edge.

We can see the spatial separation of edge modes quite clearly by calculating
the motion of a particle added to the system within the gap. To do so, we
calculate the propagator of a spacetime lattice which uses (3.1) as its
spatial model. The results are presented in Figures 3.2d and 3.2e. There we
see a spacetime representation of the IQH system. The two $1+1$d edges are
shown in purple and green, respectively. We inject a particle with a
wavepacket with low momentum ($k\approx 0$) modes at $t=0$. The magnitude of
the amplitude for particle propagation is shown in black. In 3.2d, the
particle propagates smoothly as a slightly dispersing, right moving mode. It
remains confined to the edge and simply moves to the right. Next, we can
effect a time-varying gauge potential by rotating the boundary
conditions111This is a slightly different approach from how we treated the
Hamiltonian in the previous section but is computationally easier. Note that
in this case the wavepacket stays localized as it tunnels through the
bulk—that is an artifact of how we have chosen the gauge variation and is not
universally true. from $0$ to $2\pi$. The boundary conditions are shown in the
inset. In 3.2e, we change the boundary conditions, and this tunnels the
resulting electron through the bulk and to the other edge, where it then
becomes a left-mover. We thus have three theories each with their own
anomalies: two edges and one bulk, and taken together they must in total
conserve charge, i.e. their anomalies must cancel.

So we see that, even though the gapless modes are spatially separated (and
they may be macroscopically so), they are still connected in the same way that
the left and right moving modes of the “$\cos(k)$” model (2.16) were. One may
similarly couple in a spatially constant background gauge field and, by
varying it, transfer charge from the left-moving mode to the right-moving
mode. In this case, one also ends up transferring charge across the system.
This is the Integer Quantum Hall Effect! We have essentially reproduced
Laughlin’s argument [60], and we see that the Hall conductance is quantized
because the charge of the particles is quantized.

Thus there is a model which satisfies the rather stringent conditions that we
laid out. This model has also pointed the way to beginning to unify these
lattice field theory considerations with deep principles from condensed
matter, and we will explore those connections in the following. We return to
defining chiral field theories on the lattice in Section 4.1.

###  3.2 Topological Order, SPT Phases, and Local Unitary Quantum Gates

In the last section, we saw that the two counter-propagating gapless modes in
an IQH state may actually be spread across a sample, with one gapless mode
appearing one one edge and the other mode on the other edge. This was a
significant feat, as it allowed us to recognize that a chiral gapless mode can
indeed appear alone. However, it is not true that the modes are isolated from
one another, as they are linked by the anomaly. Exposing one edge to an
electric field actually transfers charge across the system to the other.

The source of the link between these two gapless edge modes, which may be
macroscopically separated, is the same as the source of all ‘spooky actions at
a distance’ in quantum mechanics: it is entanglement. In the quantum Hall
system, degrees of freedom in the system arbitrarily far away are entangled.
Moreover, this entanglement cannot be unwound.

We can formalize this into a working definition on how to classify quantum
phases of matter [104, 119]. Suppose we have states
$\ket{\psi},\ket{\psi^{\prime}}$ in a Hilbert space $\mathcal{H}$ which
factorizes as a tensor product over local Hilbert spaces
$\mathcal{H}=\otimes_{i}\mathcal{H}_{i}$. We wish to know under what
conditions $\ket{\psi}$ and $\ket{\psi^{\prime}}$ can be considered to belong
to the same phase of matter. Ultimately, this will cover macroscopic
observables like Hall conductance and anomalies, as well as mappings of the
low-energy Hilbert spaces of the systems. We define a local unitary operator
on a site $i$ as a unitary operator $\hat{U}_{i}$ which acts as the identity
on all the local Hilbert spaces except those in a finite radius $\xi$ of site
$i$. We define a local unitary operation as a product of local unitary
operators. We say that two states $\ket{\psi}$ and $\ket{\psi^{\prime}}$ are
in the same phase if they can be deformed into one another via a finite number
(_i.e._ independent of system size) of local unitary operations.

From the condensed matter perspective, these local unitary operations can be
roughly thought of as finite-time evolution under gapped Hamiltonians. The
locality demands the presence of a gap, since otherwise information and
entanglement could propagate at luminal velocities. Hence we are simply
considering the finite-time evolution of our system under perturbations that
are local, and gapped. This also has an appealing interpretation in terms of
quantum gates. Local unitary operations are simply (local) quantum gates, and
two states belong to the same phase if they can be deformed into one another
by a finite depth222Formally, we should say that two states belong to the same
phase if they require a depth that grows less than linearly (e.g.
logarithmically) with the system size. circuit.

The local unitary operations, or local quantum gates, allow us to partition
the set of states in a given Hilbert space. If we augment local unitary
transformations by allowing the addition of unentangled ‘ancilla’ qubits that
enlarge the physical system, we now have a proper definition of phases of
matter.

Figure 3.3: _(Color Online)_. _(a)_ A trivial state (blue at bottom) may be
unwound by a finite-depth local unitary circuit (orange blocks) into a product
state (blue at top). _(b)_ A topologically ordered state exhibits long-range
entanglement which cannot be unwound by a finite-depth circuit. Even after
application of the circuit, the entanglement remains (green). _(c)_ A SPT
state contains entanglement that can be unwound into a product state by a
finite-depth circuit only if we apply gates which break the symmetry (red
blocks).

Within this definition, there is one class of states that is ‘trivial,’ namely
the product states. Any state which factorizes as
$\ket{\psi}=\otimes_{i}\ket{\psi_{i}}$ with $\ket{\psi_{i}}\in\mathcal{H}_{i}$
is trivially unentangled (Figure 3.3a), and the equivalence class of states
that it generates is known as the ‘trivial phase.’

In addition to the trivial phase, there are many non-trivial phases which
cannot be deformed into the trivial phase or into each other. These are the
topologically ordered states. The basic ingredient leading to non-trivial
phases is the aforementioned long-range entanglement [57]. Any short-range
entanglement can be unwound by a finite-depth circuit. However, a long-range
pattern of entanglement cannot be unwound (Figure 3.3b) and it is this which
characterizes the nontrivial topologically ordered states. On the other hand,
these states are physically characterized by their thermal Hall conductance,
i.e. their chiral central charge, the braiding of their excitations, and their
topologically protected ground state degeneracy. Just as the Hall conductance
was connected to a $U(1)$ anomaly, the thermal Hall conductance appears in
field theory as a gravitational anomaly. Formally, each topological order is
defined by a tensor category [58], though for our purposes we will not need
the most general definition. In Section 3.3, we will review the field theory
description and classification of $2+1$ Abelian topological orders in terms of
$U(1)$ $K$-matrix Chern-Simons theory, as in Chapter 5 we will create a
lattice definition of these states.

Including a symmetry adds another layer to this classification. In doing so,
we should restrict to local unitary gates that commute with the symmetry. This
restriction leads to new phases. There will be phases that were formerly
trivial which are now non-trivial. These phases feature short range-
entanglement, but their entanglement patterns cannot be unwound by any finite-
depth circuit which respects the symmetry. However, they can be unwound by a
finite-depth circuit which breaks the symmetry (Figure 3.3c). Known variously
as Symmetry-Protected Trivial or Symmetry-Protected Topological (SPT) states,
these states have trivial ground state degeneracy, but often host nontrivial,
anomalous edges that we describe below. In Section 6, we will create an
exactly solvable lattice path integral and Hamiltonian model for a large class
of $U(1)$ SPT states.

Figure 3.4: _(Color Online)_. Phases of matter may exhibit short or long-range
entanglement as well as a symmetry that may or may not protect the state.
These two properties lead to a four-fold classification of the types of
symmetric phases of matter: trivial states, topologically ordered states,
symmetry-protected trivial states, and symmetry-enhanced topological states.

These two considerations: long-range entanglement and symmetry protection,
lead to a four-fold classification of symmetric phases shown in Figure 3.4.
There we see the trivial phase, as well as the topologically ordered phases,
the SPT phases, and the symmetry enhanced topological SET phases, which are
both topologically ordered and enjoy symmetry protection. Our focus in this
thesis will be on the topologically ordered and SPT phases333Note that this
classification is for symmetric phases. Spontaneous symmetry breaking, of the
sort we will see in the superfluid phase in Chapter 6, effectively adds a
third dimension to this classification that is not drawn in Figure 3.4..

We have already seen one of the the simplest examples of topological order:
the IQH state. There, the nontriviality of the phase is revealed not only by
the Hall conductance, which requires $U(1)$ symmetry to reveal, but by the
thermal Hall conductance and its associated gravitational anomaly. Just as it
is impossible to trivialize the bulk due to the long-range entanglement, it is
in fact impossible to gap out the edge due to the gravitational anomaly.

Figure 3.5: _(Color Online)_. _(a)_ To create an SPT system, we consider a two
layer system composed of an IQH state and its time-reversal conjugate, with
the particles in the two layers having opposite charges. _(b)_ We may gap out
the edge of the system by inserting lattice links to turn the model into an
$\text{IQH}_{+1}$ system folded over on itself, but only at the cost of
breaking the $U(1)$ symmetry defined in the text.

We have not directly seen an SPT phase, but it is easy to create a model using
the ingredients we already have. Consider the two-layer system shown in Figure
3.5a composed of the IQH state we have already examined, which we denote by
$\text{IQH}_{+1}$ and its time-reversal conjugate, which we denote by
$\text{IQH}_{-1}$. We define a $U(1)$ symmetry with the particles in the
$\text{IQH}_{+1}$ state having charge $+1$ and the particles in the $-1$ state
having charge $-1$.

If we do not consider the $U(1)$ symmetry, then the edge is trivial. As
mentioned above, both layers carry gravitational anomalies, but these cancel
because gravity couples to all particles with the same sign and the two layers
are time-reversal conjugates. The edge can be gapped out by scattering the
respective left and right-moving modes into one another. Indeed, one can
simply augment the lattice model so that it becomes the $\text{IQH}_{+1}$
state folded over on itself (Figure 3.5b). However, this explicitly breaks
$U(1)$ symmetry, as it scatters positive charges from the $\text{IQH}_{+1}$
into the $\text{IQH}_{-1}$ layer, where they become negative charges. In fact,
if we consider the $U(1)$ anomaly, we see that, because the particles have
opposite charges and the states are time-reversal conjugates of each other,
the gauge anomalies add instead of canceling. Hence the edge and the bulk are
nontrivial when we require $U(1)$ symmetry, and we have an example of an SPT
state. The model we describe in Chapter 6 has a similar edge theory to this
toy model.

| Topological Order | SPT Order
---|---|---
Entanglement | Long-Ranged | Short-Ranged but Symmetry-Protected
Bulk | Thermal Hall Conductance | Charge Hall Conductance
Boundary | Gravitational Anomaly | Gauge Anomaly
Table 3.1: Properties of Chiral Topological Orders and Chiral SPT Orders.

These two examples—the IQH state and the $U(1)$ charged SPT state composed of
$\text{IQH}_{+1}$ and $\text{IQH}_{-1}$ states—illustrate some of the
fundamental relationships that we have been exploring, which are summarized in
Table 3.1. Topologically ordered states host long-ranged entanglement in the
bulk, while the entanglement pattern for SPT order is short-ranged by
symmetry-protected. In turn, _chiral_ topologically ordered phases suffer
gravitational anomalies that correspond to bulk thermal Hall conductance,
while the _chiral_ SPT phases exhibit gauge anomalies which reflect bulk
charge Hall conductance (often referred to simply as Hall conductance)444There
is a state, the $E_{8}$ state, which hosts chiral edge modes and thermal Hall
conductance [65] but has trivial bulk statistics, and there is some discussion
as to whether $E_{8}$ is topologically ordered or an SPT phase..

In Chapter 4, we will use many of these ideas to construct a Chiral fermion
model in $1+1$d. Following that, in Chapter 5, will focus on developing a
lattice description of $K$-matrix Chern Simons theory, which encompasses both
topologically ordered and SPT states in $2+1$ dimensions. We review this
$K$-matrix theory in the following section. In Chapter 6, we will ‘ungauge’
the $K$-matrix model to yield an SPT model, and we review the classification
of SPT phases in section 3.4. In Chapter 7, we use the ungauged model of
Chapter 6 to develop a $1+1$d chiral lattice field theory that far surpasses
the model of Chapter 4.

###  3.3 Abelian Chern-Simons Theories

Much of our preliminary discussion has focused on the connection between
chiral field theories in $1+1$d and topological phases in $2+1$d. However, it
we be good to have a field theory description of those $2+1$d topological
phases. Here we review the $U(1)$ Chern-Simons theories which describe $2+1$d
Abelian topological phases [100, 2].

What the topological phases we have described have in common is a conserved
current. Starting from a phenomenological approach in terms of this conserved
current, we can build a field theoretic description of the IQH states and
$U(1)$ SPT states we have examined so far. In fact, the general $K$-matrix
formalism we will describe goes far beyond that, capturing all $2+1$d Abelian
topological orders [65].

The usual way of describing a conserved current, in terms of derivatives of
matter fields that are conserved at the level of the equation of motion, will
not be strong enough for what we want to do. We will not be concerned with the
dynamics of the theories we consider and so we do not wish to rely on an
equation of motion to conserve a current. Instead, we describe a
phenomenological conserved current in terms of a $U(1)$ gauge field $a_{\mu}$
by writing:

$j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\lambda}\partial_{\nu}a_{\lambda}\hskip
28.45274pt\text{or}\hskip 28.45274ptj=\frac{1}{2\pi}\star da$ (3.2)

where we have included a factor of $2\pi$ because while flux is quantized to
$2\pi\mathbb{Z}$ particle number should be quantized to $\mathbb{Z}$. On the
right-hand side, we have introduced the differential form notation [16] which
we will make use of liberally in this section. The conserved current is gauge
invariant under $a\to a+d\theta$ and, moreover, is automatically conserved,
as:

$\partial_{\mu}j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\lambda}\partial_{\mu}\partial_{\nu}a_{\lambda}=0\hskip
28.45274pt\text{or}\hskip 28.45274ptd^{\dagger}j=(\star
d\star)\left(\frac{1}{2\pi}\star da\right)=\frac{1}{2\pi}\star d^{2}a=0$ (3.3)

We do not need to worry about the origins of the conserved current, but
instead should take it as emergent. Much of the power of the formalism we
describe here consists in abstracting away from the microscopics of the
system.

With our current $j=\star da$ in hand, we should consider what sort of actions
can be built. One obvious choice is to give the current energy, via:

$S=-\frac{1}{g}j^{2}=-\frac{1}{(2\pi)^{2}g}(f\star f)$ (3.4)

where $f=da$. This is the usual Maxwell term familiar from electrodynamics.
Accordingly, it gives rise to dynamic suppression of current. We will
generically assume either no or a small Maxwell term to be present, but the
topological properties we are after will have to go beyond dynamics.

However, owing to the fact that we are working in $2+1$d, there is a unique
term which we can write down, namely, the Chern-Simons action:

$S=\frac{m}{4\pi}\int ada=\frac{m}{4\pi}\int
d^{3}x\epsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}$ (3.5)

At first glance, this action would appear to fail to be gauge invariant.
However, it is indeed gauge invariant under small (continuously connected to
the identity) gauge transformations on a closed manifold. Under large gauge
transformations, one can show that it is gauge invariant if $m\in\mathbb{Z}$.
(If $m$ is odd, then the theory is fermionic and will require a spin
structure.) To couple the conserved current to a background gauge field $A$,
we write $\star Aj=Ada$ and integrate by parts so that the action is always
invariant under gauge transformations of the background field. We thus add to
the term an action:

$\frac{q}{2\pi}\int adA$ (3.6)

where the coefficient has been chosen so that when $q\in\mathbb{Z}$ the action
is invariant under large gauge transformations.

Putting these together, we now have the action:

$S=\frac{m}{4\pi}\int ada+\frac{q}{2\pi}\int adA$ (3.7)

This Chern-Simons action is a gauge invariant action specified by the integers
$m$ and $q$ describing a quantum system in two spatial dimensions. $q$ should
be thought of as the charge of the quasiparticle, while $m$ is known as the
_level_ of the Chern-Simons theory.

What then are we to make of this action, and what phase should it describe? We
can match this action to real systems by understanding its Hall conductance.
Since the action is quadratic, we can work classically. Varying $a$, we see
that the equation of motion of the action is:

$mda=qdA\leftrightarrow j=\frac{q}{m}\star F$ (3.8)

where $F=dA$. Hence we see that the action ‘attaches’ quasiparticle current to
the flux of the gauge field. In the case of magnetic flux, so that $F_{xy}\neq
0$, the Chern-Simons equation of motion implies that $j^{0}\neq 0$, and so the
system localizes quasiparticles proportional to the flux. On the other hand,
suppose that couple in an electric field in the $x$ direction, so that, say
$F_{xt}\neq 0,F_{xy}=F_{yt}=0$. This equation then says that:

$j^{y}=\frac{q}{m}F_{xt}$ (3.9)

Let us set $m=1$. Then we see that applying an electric field results in
current in the $y$ direction. Noting that the electric current is related to
the quasiparticle current by $j^{e}=qj$, we see that this is the Quantum Hall
Effect, with Hall conductance $\sigma_{xy}=\frac{1}{2\pi}\frac{q^{2}}{m}$
($\hbar=e=1$). Specifically, for $m=1$, this is the basic $IQH_{+1}$ state,
while for higher $m$ this leads to the basic Laughlin series fractional
quantum Hall effects. We can obtain higher integer Hall states and more varied
fractional quantum Hall effects by including multiple quasiparticle currents
$a^{1},a^{2},...$ and allowing them to couple to both each other and the
background gauge field. The resulting action is the so-called $K$-matrix
Chern-Simons theory:

$S=\frac{1}{4\pi}\sum_{I,J}K_{IJ}\int
a^{I}da^{J}+\frac{1}{2\pi}\sum_{I}q_{I}\int a^{I}dA$ (3.10)

Here $K_{IJ}$ is a symmetric integer-valued matrix which describes the
quasiparticle currents, while $q_{I}$ is an integer-valued vector that
describes their charges.

The $K$-matrix formulation captures the essence of Abelian topological orders
in $2+1$d. To understand its properties, we begin with the Hall conductance.
Running a similar analysis as above, one can see that the Hall conductance is
given by $\sigma_{xy}=\frac{1}{2\pi}q\cdot K^{-1}\cdot q$. On the other hand,
the number of right (left) moving chiral edge modes is given by the number of
positive (negative) eigenvalues of $K$, and the thermal Hall conductance is
determined by the signature of $K$. If any of the diagonal entries is odd,
then it is a theory of fundamental fermions, and one must specify a spin
structure. On a torus, the system will have a ground state degeneracy of
$|\det K|$.

Not all theories with different $K$-matrices represent different phases. The
formulation is redundant under changing the $N\times N$ $K$-matrix by any
matrix $M\in GL(N,\mathbb{Z})$ via $K\to M^{-1}KM$; this leads to the
$K$-matrix classification of Abelian topological orders, and one can augment
this description to allow for the inclusion of internal symmetries.

One crucial aspect of these theories that will play a role in our SPT work is
the notion of braiding of excitations. Suppose we create two point-like
excitations in our two-dimensional system. If we ‘braid’ one particle around
the other, the many-body wavefunction can transit through the many-body
Hilbert space. In Abelian topological orders, we assume that when the
particles return to their original positions, the many-body wavefunction
returns to its original state (In non-Abelian topological orders there may be
a matrix acting on the excitation state manifold). While the state may remain
the same, the many-body wavefunction may pick up a $U(1)$ geometric phase
resulting from the transit of the wavefunction through the Hilbert space.

Figure 3.6: _(Color Online)_. _(a)_ We define $d^{-1}\Gamma$ as a one-form
dual to a surface bounded by $\gamma$, which is dual to the two-form $\Gamma$.
_(b)_ Calculating $\int\Gamma d^{-1}\Gamma^{\prime}$ leads to the linking
number $L(\gamma,\gamma^{\prime})$.

We can actually see these particle statistics quite simply in the $K$-matrix
theory. Excitations in this theory are generated by Wilson lines
$e^{iw^{I}\oint a^{I}}$, where the integral vector $w^{I}$ describes the
composite quasiparticle character of the excitation. Including $N$ Wilson
lines with vectors $w_{n}^{N}$ along closed paths $\gamma_{n}$, the $K$-matrix
action becomes:

$\displaystyle
S=\sum_{n=1}^{N}\sum_{I}\int_{\gamma_{n}}w^{I}a^{I}+\frac{1}{4\pi}\sum_{I,J}$
$\displaystyle K_{IJ}\int a^{I}da^{J}$ (3.11)
$\displaystyle=\sum_{n=1}^{N}\sum_{I}$ $\displaystyle\int
w_{n}^{I}a^{I}\Gamma_{n}+\frac{1}{4\pi}\sum_{I,J}K_{IJ}\int a^{I}da^{J}$

where we have set the background gauge field $A$ to be zero. Here $\Gamma_{n}$
is the two-form which is Poincaré dual to $\gamma_{n}$. We assume a
contractible manifold so that we can choose one-forms that we denote by
$d^{-1}\Gamma_{n}$ such that $d(d^{-1}\Gamma_{n})=\Gamma_{n}$ since
$\gamma_{n}$ is closed and so $d\Gamma_{n}=0$. As illustrated in Figure 3.6a,
one may think of $d^{-1}\Gamma_{n}$ as being one on a surface (blue) bounded
by $\Gamma$ (black) and zero otherwise. Shifting the integration variables
$a^{I}\to a^{I}-2\pi\sum_{n}K_{IL}^{-1}w_{n}^{L}d^{-1}\Gamma_{n}$, the action
becames:

$S=\frac{1}{4\pi}\sum_{I,J}K_{IJ}\int
a^{I}da^{J}-\pi\sum_{n,n^{\prime}}(w_{n}\cdot K^{-1}\cdot
w_{n}^{\prime})\int\Gamma_{n}d^{-1}\Gamma_{n^{\prime}}$ (3.12)

We have successfully eliminated the Wilson line variable and exchanged it for
the second term, which contributes an overall phase to the path integral. We
can simplify it by noting that, as illustrated in Figure 3.6b,
$\int\Gamma_{n}d^{-1}\Gamma_{n^{\prime}}$ is the linking number
$L(\gamma_{n},\gamma_{n^{\prime}}$ between the curves
$\gamma_{n},\gamma_{n^{\prime}}$. Reducing to the case of just two curves
$\gamma_{1},\gamma_{2}$, this phase becomes

$2\pi i(w_{1}\cdot K^{-1}\cdot w_{2})L(\gamma_{1},\gamma_{2})$ (3.13)

This is the expression for the braiding statistics. Repeating a similar
analysis, one encounters the “self-statistics” for a particle turning about
itself:

$\pi i(w\cdot K^{-1}\cdot w)L(\gamma,\gamma)$ (3.14)

where $L(\gamma,\gamma)$ may be thought of as the number of times a particle
turns. (In general, care must be taken to define $L(\gamma,\gamma)$, usually
by using the framing in spacetime [113]).

These results on mutual and self statistics are a critical aspect of Chern-
Simons theory. In particular, the statistics characterize topologically
ordered states. Even more strangely, the self-statistics make clear that the
excitations themselves may be fermions ($(w\cdot K^{-1}\cdot w)$ odd), bosons
($(w\cdot K^{-1}\cdot w)$ even), or neither (general $(w\cdot K^{-1}\cdot
w)$). Note that these are all excitations described by a nominally bosonic
field $a^{I}$; instead it is the topological term which gives them statistics.
This will play a crucial role in Chapter 6, where we will encounter a
topological term which turns a bosonic lattice model into a model of emergent
fermions.

Returning to the case of a single Chern-Simons gauge field,

$\frac{m}{4\pi}\int ada$ (3.15)

there is a subtlety in this definition that we must uncover and that will play
a crucial role in Chapter 5. For any $U(1)$ connection, we may not be able to
define $a$ everywhere in spacetime. For example, we know that the flux over
any closed surface, say a sphere, is quantized to be an integer. However, if
we have a globally defined gauge field $a$, then trivially $\int_{S^{2}}da=0$,
as $S^{2}$ has no boundary. Instead, one typically defines $a$ in contractible
patches that cover spacetime, allowing the the fields on overlapping patches
to differ by a gauge transformation.

Because the Chern-Simons action is gauge invariant only after integration by
parts, if one takes a gauge field defined in patches, the action is no longer
well defined. One can add terms to the action that depend on the connections
in the various patches and the gauge transformation they differ by; this leads
to the Cech-Deligne-Beilenson formulation [3, 92]. However, we will take
another route, which instead seeks to reformulate the Chern-Simons action in
terms of the field strength $f=da$ which is gauge invariant and insensitive to
the problems of patches. We have two powers of $a$ in the Chern-Simons action
that we wish to rewrite in terms of $f$. However, $f$ is a two form, and so we
would naturally find ourselves facing a four-dimensional integral. The trick
here is to actually define the Chern-Simons term in $2+1$d as the boundary of
a $3+1$d action. Namely, let $\mathcal{N}^{4}$ be a four-manifold, with
boundary $\mathcal{M}^{3}=\partial\mathcal{N}^{4}$. Then:

$\int_{\mathcal{N}^{4}}f\wedge f=\int_{\mathcal{N}^{4}}d(a\wedge
da)=\int_{\mathcal{M}^{3}}a\wedge da$ (3.16)

So to evaluate $\int_{\mathcal{M}^{3}}ada$, we extend $a$ into a fourth
dimension and evaluate $\int_{\mathcal{N}^{4}}f\wedge f$. Because the four-
dimensional theory depends only on $f$, and not on $a$, it is insensitive to
the concerns over patches which the three-dimensional “$ada$” suffers. This
leads to the Witten’s ‘$\eta$ invariant’ formulation of Chern-Simons theory,
and we will use the $f\wedge f$ term to define a topological field theory on a
lattice in Chapter 5.

There is one remaining phenomena to address in Chern-Simons theories. We have
asserted that Chern-Simons theories may host anomalous edge theories. We have
also stated that they are gauge invariant when on a closed manifold, but we
have not discussed the case for an open manifold. In fact, the Chern-Simons
action is not gauge invariant on an open manifold, but rather suffers an
anomaly there. We can exactly match the anomaly suffered at the edge by the
Chern-Simons with an anomaly of an edge theory, and in fact this tells us that
an edge theory must appear at the boundary of the Chern-Simons theory.

One can directly match the anomaly of the edge to the boundary theory,
ensuring that they cancel. This is the field theory statement of the behavior
we encountered in Chapter 2, where charge was transferred from an edge into
the bulk and back. The resulting edge theory is:

$S_{\text{edge}}=\frac{1}{4\pi}\sum_{I,J}K_{IJ}\int
dxdt~{}\partial_{t}\phi^{I}\partial_{x}\phi^{J}+\frac{1}{2\pi}\sum_{I}q_{I}\int
dxdt~{}\phi^{I}dA$ (3.17)

This is the standard ‘bosonized’ edge theory for chiral fermion theories, the
Luttinger liquid, and more, depending on the value of $K_{IJ}$. For a bosonic
theory with $\det K=1$, one can use an $GL(2,\mathbb{Z})$ transformation to
transform any $2\times 2$ $K$-matrix into:

$K=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$ (3.18)

which we can rewrite as:

$S_{\text{edge}}=\frac{1}{2\pi}\partial_{t}\phi\partial_{x}\theta$ (3.19)

in the absence of a background field. In Chapters 6 and 7, we will discuss a
lattice field theory that has a similar continuum limit.

###  3.4 Lattice Field Theories and the Group Cohomology Picture of SPTs

In the previous section, we elaborated on the continuum field theory
description of Abelian topological and SPT phases in $2+1$ dimensions. Here we
will lay out a parallel area of progress, namely the classification of bosonic
SPT phases in all dimensions using Group Cohomology (GC) [17].

In a remarkable paper [26], Chen, et al. argued that SPT phases are in one-to-
one with the cohomology classes of maps from the group which protects the SPT
to $U(1)$. The GC result is extraordinary: it is as if one discovered the
periodic table _before_ being able to predict the physical properties of the
elements, much less actually being able to isolate any of them. In Chapter 6,
we will actually construct models for a large class of these phases and
understand their physical properties.

Our starting point for the group cohomology classification is the formalism of
lattice models on simplicial complexes [52]. Let $G$ be a group, and suppose
that $\mathcal{M}^{d+1}$ is a $d+1$ dimensional spacetime lattice with sites
labeled by $i,j,k,...$. We consider a lattice path integral in terms of a
field $g_{i}$ which assigns a $G$-valued variable to each lattice site $i$. An
action in these models is a function $\nu:G^{d+2}\to U(1)$ which assigns a
$U(1)$ phase to each $d+1$-dimensional simplex, each of which has $d+2$
lattice sites:

$\nu(g_{0},g_{1},...,g_{d+1})\equiv\nu(\\{g_{i}\\})$ (3.20)

We require that each action be symmetric under the action of $G$, namely that:

$\nu(\\{\bar{g}g_{i}\\})=\nu(\\{g_{i}\\})$ (3.21)

for any $\bar{g}\in G$.

Given any function $\nu:G^{n}\to U(1)$, we can contruct a function
$(d\nu):G^{n+1}\to U(1)$ by defining:

$(d\nu)(g_{0},...,g_{n})=\prod_{\ell=0}^{n}\nu^{(-1)^{\ell}}(\\{g_{i}\setminus
g_{\ell}\\})$ (3.22)

where $\\{g_{i}\setminus g_{\ell}\\}$ means $g_{0},...,g_{n}$ excluding
$g_{\ell}$. Because SPT states are trivial and have no topologically protected
ground state degeneracy, the action should vanish on any closed manifold. In
other words, the action should be only a surface term. In [26], Chen et al
argue that this means that the action should satisfy the cocycle condition

$d\nu=0$ (3.23)

At the same time, they argue that phases whose actions differ by an exact
cochain, i.e. $\nu,\nu^{\prime}$ such that:

$\nu^{\prime}=\nu+d\mu$ (3.24)

are equivalent. One way to understand this is to consider the ground state
wavefunction on a particular lattice. Changing the action by $d\mu$
corresponds to multiplying the ground state:

$\ket{\psi}\to\ket{\psi^{\prime}}=\prod_{S}\mu(\hat{g}_{i}\in S)\ket{\psi}$
(3.25)

where the product is over all simplices on a spatial lattice. In particular,
this is clearly a symmetric local unitary transformation as discussed
previously, and hence $\ket{\psi}$ and $\ket{\psi^{\prime}}$ belong to the
same phase.

The classifications of cocycles which satisfy eqn. (3.23) modulo symmetric
local unitary transformations (3.24) is known as group cohomology.
Specifically, the results of Chen et al argue that the classification of
bosonic SPTs protected by a symmetry $G$ in $d+1$ spatial dimensions is given
by:

$H^{d+1}(G,U(1))$ (3.26)

Moreover, the group cocycles carry a natural abelian ‘addition’ operation
which corresponds to stacking of SPT phases.

Applying the group cohomology classification to bosonic SPT phases protected
by $U(1)$ in $d$ spatial dimensions, we obtain the following classification:

$H^{d+1}(U(1),U(1))=\begin{cases}\mathbb{Z}&d=2\\\
\mathbb{Z}_{1}&d=1,3\end{cases}$ (3.27)

The nontrivial SPT classification in $2$ spatial dimensions corresponds to
states which feature a quantized Hall conductance of $2k\frac{e^{2}}{h}$.
These are the states which we will construct in Chapter 6.

Ultimately, while the group cohomology theory provides an exceptionally
powerful classification of SPT phases, it does not provide explicit field
theories for these cases. In the typical way that mathematics can be strange,
it is much easier to enumerate all possible cohomology classes than it is to
write down representatives of those classes. In turn, this is why the work of
Chapter 6 is so critical: it provides the first examples of cocycles theories
for continuous groups. These allow for a detailed physical picture as charged
vortex condensates of the $U(1)$ phases in $2$ spatial dimensions to emerge,
and the models we discuss there exhaust all the $U(1)$ bosonic SPTs in two
dimensions.

We have now examined the deep connections between anomalies and quantum phases
of matter. In the next chapter, we use these connections to develop a solution
to the Chiral Fermion Problem.

##  Chapter 4 A Proposal for a Non-Perturbative Lattice Regularization of an
Anomaly-Free $1+1$d $SU(2)$ Chiral Fermion Theory

Here we develop an early-stage numerical treatment of a novel non-perturbative
lattice regularization of a $1+1$D $SU(2)$ Chiral Gauge Theory. Our approach
follows recent proposals that exploit the connection between anomalies and
topological (or entangled) states to create a lattice regularization of any
anomaly-free chiral gauge theory, and mirrors much of the discussion of ‘mass
without mass terms’ [11, 6, 5, 12, 19, 20, 115, 117, 116]. In comparison to
other methods, our regularization enjoys (ultra) local on-site fermions and
gauge action, as well as a physically sensible on-site fermion Hilbert space.

Before proceeding further, let us further specify the problem that we mean to
solve and the conditions under which we do so. We mean to find a path integral
on a space-time lattice which produces the desired chiral fermion theory as
the low energy effective theory, subject to the following conditions:

  1. 1.

The path integral and the resulting chiral fermion theory have the same space-
time dimension.

  2. 2.

There are finitely many degrees of freedom on each lattice point.

  3. 3.

The lattice path integral is local, i.e. all terms in the Lagrangian have a
finite range (this is sometimes termed ultra-local in the lattice literature).

  4. 4.

The lattice theory has the symmetry $G$ of the chiral fermion theory. This
symmetry should act on-site, so it can be gauged [102], and is not
spontaneously broken.

  5. 5.

The emergent chiral fermion theory may or may not have Lorentz symmetry (which
would require different species of fermions to have the same velocity), but
must break parity and time-reversal symmetry. In particular, the right and
left-moving modes carry different representations of the gauge group.
(Ensuring Lorentz symmetry will be the subject of future work).

The condition of on-site symmetry discussed above merits further discussion.
The on-site condition means that the quantum operator representing any
‘internal’ symmetry factorizes as a tensor product of operators that act only
on a single site. That is, $U(g)=\otimes_{i}U_{i}(g)$, with each $U_{i}(g)$
acting only on fermions at site $i$. The conventional solution to the
Ginsparg-Wilson equation [41] is not on-site, and this is precisely why chiral
symmetry breaking is anomalous. In contrast, we are interested in models
without anomalies and with on-site symmetry.

We achieve a solution to this problem by using the mirror fermion approach
described in the next section. We first create a lattice regularization of
both the chiral theory and its mirror conjugate and then introduce
interactions induced by a Higgs field that gap out only the mirror theory. In
particular, we show that a space-time random Higgs field (which preserves
$SU(2)$ symmetry on average) can gap the fermion spectrum. We then use
topological arguments to argue that the resulting effective Higgs action can
be gapped as well.

The lattice theory can be easily gauged, and the gauged theory is guaranteed
to be $G$-gauge invariant (i.e. there is no gauge anomaly), due to the on-site
$G$-symmetry and the locality. For weak gauge coupling, the gauged lattice
path integral produces a low energy effective chiral fermion gauge theory and
the mirror sector will remain gapped, though we do not study the gauged theory
(dynamically or otherwise). Furthermore, since there are finitely many degrees
of freedom per site, the path integral is well defined for any finite space-
time volume and the chiral fermion theory is fully regulated. In this case,
the chiral fermion theory is defined non-perturbatively.

###  4.1 The Mirror Fermion Approach and the Bulk-Boundary Correspondence

Figure 4.1: _(Color Online)_. We define a chiral lattice theory as either
_(a)_ a spatial model or _(b)_ a spacetime model by considering a thin slab of
material. A chiral theory appears on one edge (purple), while the mirror of
the chiral theory appears on the other (green), and they are separated by a
bulk (pink). _(c)_ We then try to gap out the mirror sector using
interactions, leaving only the chiral theory.

Now we can take the ideas behind topological order and SPT phases and build an
approach towards defining a chiral theory on a lattice. We have seen that the
major obstacle to defining a successful chiral field theory is the anomaly,
and that one way to resolve this is to define the theory as an edge theory.
Let us formalize this slightly: we consider spatial model consisting a thin
slab of system of $L_{x}\times w$, as shown in Figure 4.1. We take periodic
boundary conditions in the $x$ direction and open boundary conditions in the
other, so that $w$ is the distance between the two edges and $L_{x}$ is the
length of the periodic edges. As before, a chiral field theory (e.g. a left-
moving mode) appears on one edge, while its mirror conjugate theory (e.g. a
right-moving mode) appears on the other edge.

When taking a thermodynamic limit, we scale the lengths $L_{x}$ and, in a
spacetime lattice, $L_{t}$. However, we keep $w$ fixed. This ensures that the
system is truly $1+1$d in the thermodynamic limit.

Now it would seem that we have successfully defined a lattice chiral theory.
However, while it is true that we have a chiral theory on one edge, this model
also contains the mirror conjugate of the chiral theory on the opposite edge,
and the total low-energy theory remains non-chiral. Moreover, the chiral
theory and its mirror are still coupled, as in the presence of a background
gauge field charge may tunnel from one side to another. The key then is to
choose an anomaly-free chiral theory, so that the edges decouple.

However, even if the edges are decoupled, the low-energy sector still contains
the chiral theory and its mirror conjugate and so is non-chiral. This is
exactly where the results of the previous chapter are crucial: it has been
conjectured [103], and in some cases proven [95], that being free of anomalies
is equivalent to having trivial bulk ordering. In turn, this implies that the
mirror edge can be gapped. Hence we can introduce interactions to the mirror
edge to gap it out, and all that should remain is the chiral theory. This is
the mirror fermion approach [101, 118, 44, 70, 40, 43] we mentioned
previously. It is related to the Eichten-Preskill approach [33], which seeks
to use composite fermions to gap out a mirror edge, but the sufficient
conditions we propose are much more strict (it it has been shown [101] that
the Eichten-Preskill conditions cannot be strict enough, as they propose to
gap out an anomalous theory). This chapter will use this approach to define an
anomaly-free $SU(2)$ gauge theory on the lattice.

###  4.2 Anomaly Cancellation

We require that our edge theory be free of all anomalies. As discussed in
Chapter 3, this implies that the bulk theory is trivially ordered. Confirming
that a theory is free of all anomalies is not generally an easy task. The
cancellation of all Adler-Bell-Jackiw (ABJ) [10, 1] type anomalies can be
ensured using the usual anomaly cancellation conditions [79, 95], which
examine the Lie Algebra of the gauge group $G$ to provide powerful constraints
(see Section 4.5). However, anomalies beyond those detectable from the Lie
Algebra can still occur (e.g. [112]) which result from the non-trivial
homotopic structure of the gauge group. For our $1+1d$ system defined on the
edge of a $2+1d$ bulk, we ought to naïvely require that $\pi_{n}(G)=0$, $n\leq
3$.

In practice, our sufficient condition is slightly more forgiving. Consider a
$d+1$ dimensional theory with a chiral theory on one edge, the mirror
conjugate theory on the other, and an otherwise gapped bulk. In order to gap
out the mirror edge, we couple the fermion fields there to a Higgs field
$\phi$ transforming in the fundamental representation of $G$. Suppose that, in
the symmetry-breaking ($\phi=\text{const.}$) case, the Higgs field gaps out
the mirror edge and breaks the symmetry group $G$ symmetry down to
$G_{\text{grnd}}$. If $\pi_{n}(G/G_{\text{grnd}})=0$ for $n\leq d+2$, then the
connection with entangled states conjectures that we should be able to restore
the symmetry by demanding that $\phi$ be in a disordered phase. In this paper,
we provide numerical support for this conjecture for a specific $1+1$d theory.

We take the Lie Group $G=SU(2)$. To ensure ABJ anomaly cancellation, we
consider the $U(1)<SU(2)$ $S^{z}$ subgroup of $SU(2)$ and ensure that it
cancels. This requires:

$\sum_{L}\frac{4}{3}s_{i}(s_{i}+1)(2s_{i}+1)=\sum_{R}\frac{4}{3}s_{i}(s_{i}+1)(2s_{i}+1)$
(4.1)

We also require the cancellation of all gravitational anomalies, which is
equivalent to:

$\sum_{L}(2s_{i}+1)=\sum_{R}(2s_{i}+1)$ (4.2)

We do not consider any other symmetries for now.

The simplest chiral $SU(2)$ representation that satisfies the anomaly
cancellation conditions is
$1_{R}\oplus(0_{R})^{5}\oplus(\nicefrac{{1}}{{2}}_{L})^{4}$, where subscripts
indicate a collection of left or right-movers. Topologically, $SU(2)\simeq
S^{3}$, and so while $\pi_{1}(SU(2))=\pi_{2}(SU(2))=0$,
$\pi_{3}(SU(2))=\mathbb{Z}$. Fortunately, this simply reflects the possibility
of a Wess-Zumino-Witten (WZW) [109] term, and corresponds to the perturbative
ABJ anomalies which are absent in our model by design. Hence we can replace
the requirement that $\pi_{d+2}(G/G_{\text{grnd}})=0$ by the weaker condition
$\pi_{d+2}(G/G_{\text{grnd}})=\mathbb{Z}$, given that the theory is free of
all ABJ anomalies. (In this paper, $d=1$, $G=SU(2)$, and $G_{\text{grnd}}=1$.)
Although there is no $SU(2)$ symmetric mass term, we will show that the
$SU(2)$ chiral fermion theory can be fully gapped by strong interactions
without breaking the $SU(2)$ symmetry.

Note that this sufficient condition is far from necessary. A theory with
topological defects in the Higgs field could form a gas of defects and
(possibly) still gap out the mirror sector. But this restrictive, sufficient
condition keeps the theory simple, and it is more than enough for an
interesting system. In fact this condition can regularize far more complicated
and topical theories: a similar proposal uses the same condition to suggest a
regularization for an $SO(10)$ gauge theory in $3+1$ dimensions [101].

###  4.3 Lattice Model

We consider a lattice path integral in imaginary time, which has a form

$\displaystyle Z$
$\displaystyle=\int[\prod_{i}\phi_{i}]\text{e}^{-S_{\phi}(\phi)}\int\prod_{i}[\text{d}\psi_{ia}\text{d}\psi_{ia}^{\dagger}]\text{e}^{-\psi_{ia}^{\dagger}M^{1+1D}_{ia,jb}(\phi)\psi_{jb}}$
$\displaystyle=\int
D[\phi(x)]\text{e}^{-S_{\phi}(\phi)}\text{Det}[M^{1+1D}(\phi)]$ (4.3)

where $\psi_{ia}$ is the fermion field, $\phi_{i}$ is the Higgs field on the
site-$i$ of 1+1D space-time lattice, and index $a$ labels different components
of the fermion field. Here $\phi_{i}\in SU(2)$ and $\psi_{a}$ is a
representation of $SU(2)$. Also, the $\phi$-action $S_{\phi}(\phi)$ and the
$\phi$ dependent matrix $M^{1+1D}_{ia,jb}(\phi)$ have a $SU(2)$ symmetry. We
claim that if we choose $S_{\phi}(\phi)$ and $M^{1+1D}_{ia,jb}(\phi)$
properly, the above model is described by the following $SU(2)$ chiral fermion
effective theory

$\displaystyle{\cal
L}=\Psi_{R\alpha}^{\dagger}(\partial_{\tau}-v^{R}_{\alpha}\text{i}\partial_{x})\Psi_{R\alpha}+\Psi_{L\alpha}^{\dagger}(\partial_{\tau}+v^{L}_{\beta}\text{i}\partial_{x})\Psi_{L\alpha}$
(4.4)

at low energies, where $\alpha=1,\cdots,8$. $\Psi_{R\alpha}$ and
$\Psi_{L\alpha}$ transform as two different $SU(2)$ reducible representations:
$\Psi_{R\alpha}$ is formed by one $SU(2)$ triplet and five $SU(2)$ singlets,
while $\Psi_{L\alpha}$ is formed by four $SU(2)$ doublets. The velocity
parameters in eq. 4.4 are all positive, but may different in magnitude. Our
model breaks parity, which is already prohibited by the Nielsen-Ninomiya
theorem, while promoting this to full Lorentz invariance will be the subject
of a future work.

Our lattice model consists of a hopping part and a Yukawa coupling to a
(quenched) Higgs field. We label the 1+1D space-time lattice site by a pair
$i=(i_{\tau},i_{x})$, where $i_{x}=1,\cdots,L_{x}$ and
$i_{\tau}=1,\cdots,L_{\tau}$. We label the fermion species by three indices
$a=(o,i_{w},\alpha)$, where $o=R,L$, $i_{w}=1,\cdots,L_{w}$ and
$\alpha=1,\cdots,8$. We choose the fermion lattice action to have a form

$\displaystyle\ \ \ \ \psi_{ia}^{\dagger}M^{1+1D}_{ia,jb}(\phi)\psi_{jb}$
$\displaystyle=\psi_{R,i_{\tau}i_{x}i_{w},\alpha}^{\dagger}M^{1+2D,R}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}\psi_{R,j_{\tau}j_{x}j_{w},\alpha}$
$\displaystyle+\psi_{L,i_{\tau}i_{x}i_{w},\alpha}^{\dagger}M^{1+2D,L}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}\psi_{L,j_{\tau}j_{x}j_{w},\alpha}$
$\displaystyle+[\psi_{R,i_{\tau}i_{x}i_{w},\alpha}^{\dagger}M^{1+2D,RL}_{i_{\tau}i_{x}i_{w};\alpha\beta}(\phi)\psi_{L,i_{\tau}i_{x}i_{w},\beta}+h.c.].$
(4.5)

Here $\psi_{R,\alpha}$ is formed by one $SU(2)$ triplet and five $SU(2)$
singlets. $\psi_{L,\alpha}$ is formed by four $SU(2)$ doublets. This fermion
action can be viewed as a fermion action on 2+1D space-time, where the
$w$-direction has a finite thickness $L_{w}$ measured in lattice spacing.

Viewing the above as a 2+1D system and following the mirror fermion approach,
[33, 70, 40, 43] we choose
$M^{1+2D,R}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}$
($M^{1+2D,L}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}$) such that the fermions
$\psi_{R,\alpha}$ are gapped in the 2+1D bulk, and have 8 massless right-
moving (left-moving) modes on the $i_{w}=1$ boundary and 8 massless left-
moving (right-moving) modes on the $i_{w}=L_{w}$ boundary. Note that
$M^{1+2D,R}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}$ and
$M^{1+2D,L}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}$ are independent of the
$SU(2)$ Scalar field $\phi$. Their detailed expression will be given in
Section 4.3. Using condensed matter terminology,
$M^{1+2D,R}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}$ and
$M^{1+2D,L}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}$ both describe a single
filled Landau level but with opposite magnetic fields.

When viewed as a 2+1D system, the $SU(2)$ scalar field $\phi_{i_{\tau},i_{x}}$
only lives on the $i_{w}=L_{w}$ boundary and only couples to the fermions on
the $i_{w}=L_{w}$ boundary. We choose the coupling form in
$M^{1+2D,RL}_{i_{\tau}i_{x}i_{w};\alpha\beta}(\phi)$ such that a constant
$SU(2)$ scalar field $\phi$ can give all the right-moving and left-moving
chiral fermions on the $i_{w}=L_{w}$ boundary a finite mass $M_{\phi}$ and
then consider configurations where $\phi$ varies smoothly over the $1+1$d
spacetime.

####  4.3.1 Hopping Terms

Figure 4.2: _(Color Online)_. _(a)_ Layers of fermion hoping model with Chern
number $+1$ and $-1$, which give rise to right-hand and left-hand chiral
fermions on the 1+1D boundary. _(b)_ General schematic of our model. Each
black line represents a layer of our hopping model on a $2+1$D dimensional
lattice. We stack $8$ Chern number $+1$ layers (blue) with $8$ Chern number
$-1$ layers (red). Next, we organize the fermions on those layers into $SU(2)$
representations. Each representation is illustrated by a white square with the
spin label except the trivial spin-0 representations which we omit. This
results in a $1_{R}\oplus(0_{R})^{5}\oplus(\nicefrac{{1}}{{2}}_{L})^{4}$
theory on the left edge and its mirror $(L\leftrightarrow R)$ theory on the
right edge. We couple the fermions on the right edge to a Higgs field that
gaps out the right edge, leaving only the chiral theory on the left edge.

Here we present the details of our lattice model. Our model does not directly
come from a discretization of the Weyl or Dirac Lagrangian; instead, we create
a spacetime lattice description of states with nonzero Chern number, like
those discussed in Chapter 2.

We first create a hopping model with two edges. One edge will be described by
a gapless chiral theory, with the other edge described by the mirror conjugate
gapless theory and the bulk otherwise gapped. For $1+1$D chiral theories, the
left- and right-handed excitations simply become spinless left- and right-
moving complex fermions. In our model, the 8 left- and 8 right-moving fermions
carry the following $SU(2)$ representations
$1_{R}\oplus(0_{R})^{5}\oplus(\nicefrac{{1}}{{2}}_{L})^{4}$, which, as
discussed in the previous Section, are the simplest which satisfy the ABJ
anomaly cancellation conditions.

Our approach begins by creating a fermion hopping model on a 2+1D space-time
lattice which contains 16 layers. Each layer has a finite gap in the 2+1D bulk
and one chiral complex fermion mode on the 1+1D boundary. 8 layers have right-
handed chiral fermions on the edge, and the other 8 layers have left-handed
chiral fermions. We choose the 2+1D space-time lattice to be a thin slab of
size $L_{t}\times L_{x}\times L_{w}$. We will fix $L_{w}$ while taking
$L_{t}=L_{x}\equiv L\to\infty$, so the system is effectively $1+1$D. One
surface of the slab is the normal sector and the other surface is the mirror
sector (see Fig. 4.2). We will add interactions between the fermions in the
mirror sector (i.e. only on one surface). The interactions are induced by an
$SU(2)$ Higgs field in the fundamental representation, which breaks the
$U(8)\times U(8)$ symmetry of the non-interacting system down to $SU(2)$. The
8 right-moving and 8 left-moving chiral fermions form the following $SU(2)$
representations: $1_{R}\oplus(0_{R})^{5}\oplus(\nicefrac{{1}}{{2}}_{L})^{4}$
in the mirror sector (and
$1_{L}\oplus(0_{L})^{5}\oplus(\nicefrac{{1}}{{2}}_{R})^{4}$ in the normal
sector, see Fig. 4.2).

Figure 4.3: _(Color Online)_. _(a)_ : Chern number $+1$ hopping model. Fermion
sites are shown as spheres, with hopping terms as links. A yellow link
indicates a hopping of $+1$, a red link a hopping of $-1$, and a green link a
hopping of $+i$ in the direction of the arrow and $-i$ in the opposite
direction. Hopping around any plaquette generates a phase of $\pi$, hence with
$1$ fermion per site this is a Chern number $1$ (Integer Quantum Hall) state.
_(b)_ : Spacetime lattice with $L_{w}=2$. Each spatial component is just a
slice of a IQH state shown in _(a)_ , while the purple links represent a
Hopping of $t$ in _only_ the direction of the double arrows. In addition, each
site is given the onsite chemical potential $-t\psi^{\dagger}_{i}\psi_{i}$; we
later set $t=3$. The Hopping matrix corresponding to this model is our
spacetime Lagrangian. _(c)_ Dispersion relation for the spatial lattice with
$L_{w}=2$. _(d)_ Complex Eigenvalues of the spacetime hopping model.

We split our hopping model into spatial and temporal hopping, i.e. splitting
the imaginary time Lagrangian into ‘$\partial_{\tau}$’ and Hamiltonian terms
as $\mathcal{L}=\partial_{\tau}-H$. The spatial (Hamiltonian) terms, detailed
below, are motivated by Chern number states discussed previously, whereas the
temporal hopping is provided by a doubling-free hopping term. Our approach
breaks discrete rotational (Lorentz) symmetry, which only reappears at low
energies. While all fermions will still have linear velocities, it is possible
that, after the mirror edge is gapped, fermions on the chiral edge may have
differing velocities.

The spatial component of our lattice is provided by the Chern number $\pm 1$
states shown in Figure 4.3a, and leads to an $L_{w}L_{x}\times L_{w}L_{x}$
matrix. We can write this matrix explicitly as:

$\footnotesize H_{L,(i_{x},i_{w},i_{\tau}),(j_{x},j_{w},j_{\tau})}=\\\
\delta_{i_{\tau},j_{\tau}}\left((-1)^{w}(i\delta_{i_{x},j_{x}+1}\delta_{i_{w},j_{w}}+\delta_{i_{x},j_{x}+1}\delta_{i_{w},j_{w+1}})\right.\\\
\left.+\delta_{i_{x},j_{x}}\delta_{i_{w},j_{w}+1}\right)+\text{H.C.}$ (4.6)

However, it is easier to understand either using Figure 4.3a. The essential
feature is that the hopping clockwise around any triangular half-plaquette
yields a phase of $\frac{\pi}{2}$, so that at the lattice model describes
particles hopping in an applied magnetic field. The physics of particles
hopping on a lattice with varying fluxes is rich; here we only study particles
with $\pi$ flux per plaquette. At half filling (ie filling only the lower
band), there is one flux quantum per fermion. The model we have chosen leads
to a lattice version of Landau levels: semiclassically, particles will travel
in closed cyclotron orbits, leading to level quantization. In the bulk (i.e.
with periodic boundary conditions in both the $x$ and $w$ directions) the
model we have chosen has two bands, at energies $\pm 1$, with the chemical
potential set to zero.

In condensed matter physics, our lattice model describes a state known as a
Chern number $+1$ state, due to the quantized curvature of the eigenstates of
the lower (occupied) band. Crucially, states with nonzero Chern number cannot
be smoothly deformed without closing the band gap to a state with zero Chern
number, such as the vacuum. If we take open boundary conditions in the $w$
direction, there must be gapless chiral edge modes localized at the edges of
the lattice. In this case, there is is a right mover at $w=1$, which we call
the chiral edge, at a left mover at $w=L_{w}$, which we call the mirror edge.
In this way, we have a condensed matter-inspired realization of the ‘Domain
Wall’ idea; instead of a standard lattice Dirac operator with a region with
positive mass term situated next to a region with negative mass term, we have
a Chern number $+1$ state next to the (zero Chern number) vacuum.

The essential features of this model can also be understood in momentum space.
Taking the Fourier transform in the $x$-direction, with open boundary
conditions in the $w$ direction, the Hamiltonian becomes an $L_{w}\times
L_{w}$ matrix of the form:

$H_{R}={\footnotesize\left(\begin{array}[]{ccccc}2\sin
k_{x}&1-e^{ik_{x}}&0&0&...\\\ 1-e^{-ik_{x}}&-2\sin k_{x}&1+e^{ik_{x}}&0&...\\\
0&1+e^{-ik_{x}}&2\sin k_{x}&1-e^{-ik_{x}}&...\\\ 0&0&1-e^{-ik_{x}}&-2\sin
k_{x}&\ddots\\\ \vdots&\vdots&\vdots&\ddots&\ddots\end{array}\right)}$

Near $k_{x}=0$, this becomes:

$H_{R}\approx{\footnotesize\left(\begin{array}[]{cccccc}2k_{x}&0&0&0&0&...\\\
0&-2k_{x}&2&0&0&...\\\ 0&2&2k_{x}&0&0&...\\\ 0&0&0&-2k_{x}&2&...\\\
0&0&0&2&2k_{x}&\ddots\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots\end{array}\right)}$

This Hamiltonian has two low energy modes: a right mover at $w=1$ (the chiral
edge), which appears above, and a left mover other at $w=L_{w}$ (the mirror
edge), which would appear in the lower-right corner above (at $k_{x}=0$ for
even $L_{w}$). Otherwise, one can check that the system has a gap of
approximately $2$ for all $k_{x}$. If we were to implement periodic boundary
conditions in the $w$ direction, we would add a term $1\pm e^{ik_{x}}$ in the
upper right corner and $1\pm e^{-ik_{x}}$ in the lower left, and again one can
check that this leads to a completely gapped Hamiltonian. We also define a
left moving Hamiltonian $H_{L}=H_{R}^{*}$ in real space or
$H_{L}(k_{x})=H_{R}(-k_{x})$ in momentum space, which has a left mover at
$w=1$ and a right mover at $w=L_{w}$.

The above Hamiltonian, with open boundary conditions in the $w$ direction,
provides the spatial hopping $H$ in our Lagrangian “$\partial_{\tau}-H$”. Now
we need to add the “$\partial_{\tau}$” term. In lattice gauge theory, there
are many standard ways to construct fermion hopping models, but many of them
have fermion doubling in time direction. We use the trick in Ref. [74] to
prevent time-direction fermion doubling, which introduces a hopping in
frequency space of $3(1-e^{-i\omega})$. The total (one-flavor) hopping model
is then given by

$M^{1+2d}_{L,(i_{x},i_{w},i_{\tau}),(j_{x},j_{w},j_{\tau})}=3(\delta_{i_{\tau},j_{\tau}+1}-\delta_{i_{\tau},j_{\tau}})\delta_{i_{x},j_{x}}\delta_{i_{w},j_{w}}-H_{L,(i_{x},i_{w},i_{\tau}),(j_{x},j_{w},j_{\tau})}$
(4.7)

and similarly for $M^{1+1d}_{R}$. For example, for $L_{w}=2$, the Lagrangian
Hopping matrix in frequency momentum space becomes:

$M_{L}={\footnotesize\delta_{\alpha,\beta}\left(\begin{array}[]{cc}2\sin
k_{x}+3(1-e^{-i\omega})&1-e^{ik_{x}}\\\ 1-e^{-ik_{x}}&-2\sin
k_{x}+3(1-e^{-i\omega})\end{array}\right)}$ (4.8)

where $M_{L}$ acts diagonally on the flavor indices $\alpha,\beta$. The right-
handed version acts as $M_{R}(k_{x},\omega)=M_{L}(-k_{x},\omega)$. These
define the matrices $M^{1+2D,R}$ and $M^{1+2D,L}$ shown in eq. (4.3).

####  4.3.2 Higgs Field and Yukawa Coupling

The mass term that gaps out all the mirror fermions can be generated by an
$SU(2)$-Higgs field $\phi$ in the fundamental representation that breaks
$G=SU(2)$ down to $G_{\text{grnd}}=1$. Topologically,
$G/G_{\text{grnd}}=SU(2)\simeq S^{3}$, and so while
$\pi_{1}(SU(2))=\pi_{2}(SU(2))=0$, $\pi_{3}(SU(2))=\mathbb{Z}$. Fortunately,
this simply reflects the possibility of a Wess-Zumino-Witten (WZW) [109] term,
and corresponds to the perturbative ABJ anomalies which are absent in our
model by design.

Let us use a 8-component $\psi_{R}$ to describe the 8-layers of fermions with
Chern number 1, and $\psi_{L}$ for the 8-layers with Chern number $-1$. The
Higgs coupling on the mirror surface is given by

$\mathcal{L}_{\text{int}}=\psi_{R,i_{\tau}i_{x}i_{w},\alpha}^{\dagger}M^{1+2D,RL}_{i_{\tau}i_{x}i_{w};\alpha\beta}\psi_{L,i_{\tau}i_{x}i_{w},\beta}$
(4.9)

where

$M^{1+2D,RL}_{i_{\tau}i_{x}i_{w};\alpha\beta}=g\delta_{i_{w},L_{w}}\delta_{i,j}\Theta_{L}[\phi(i_{\tau},i_{x})]\Theta_{R}[\phi(i_{\tau},i_{x})]^{\dagger}$
(4.10)

where $\delta_{i_{w},L_{w}}$ constrains the action to the mirror surface. We
have assumed the Higgs field $\phi$ to have a unit ampltude $|\phi|=1$. In
this case, the Higgs field can be viewed as an element in the $SU(2)$ group.
$\Theta_{R}[\phi]$ and $\Theta_{L}[\phi]$ (as well as
$\Theta_{L}[\phi]\Theta_{R}^{\dagger}[\phi]$ and
$\Theta_{R}[\phi]\Theta_{L}^{\dagger}[\phi]$) are representation matrices of
$SU(2)$ that make the above Higgs coupling $SU(2)$ invariant. The explicit
$8\times 8$ matrices $\Theta_{R}[\phi]$ and $\Theta_{L}[\phi]$ are listed
below:

$\displaystyle\Theta_{R}[\phi]$ $\displaystyle=\Theta_{1}[\phi]\oplus
I_{5\times 5},$ $\displaystyle\Theta_{L}[\phi]$
$\displaystyle=\Theta_{\nicefrac{{1}}{{2}}}[\phi]\oplus\Theta_{\nicefrac{{1}}{{2}}}[\phi]\oplus\Theta_{\nicefrac{{1}}{{2}}}[\phi]\oplus\Theta_{\nicefrac{{1}}{{2}}}[\phi].$
(4.11)

where $\Theta_{s}(\phi)$ is the spin-$s$ representation (with dimension
$2s+1$). Because $\psi_{R}$ contains three fermions in the spin-$1$
representation and five singlets, $\Theta_{R}^{\dagger}\psi_{R}$ is invariant
under $SU(2)$, and similarly for $\Theta_{L}^{\dagger}\psi_{L}$. Hence the
coupling Lagrangian, composed of
$\psi_{L}^{\dagger}\Theta_{L}[\phi]\Theta_{R}^{\dagger}[\phi]\psi_{R}$ and its
conjugate, is as well.

With the above Higgs coupling on the mirror surface, the total Lagrangian is
given by eq. (4.3), reproduced below:

$\displaystyle\ \ \ \ \psi_{ia}^{\dagger}M^{1+1D}_{ia,jb}(\phi)\psi_{jb}$
$\displaystyle=\psi_{R,i_{\tau}i_{x}i_{w},\alpha}^{\dagger}M^{1+2D,R}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}\psi_{R,j_{\tau}j_{x}j_{w},\alpha}$
$\displaystyle+\psi_{L,i_{\tau}i_{x}i_{w},\alpha}^{\dagger}M^{1+2D,L}_{i_{\tau}i_{x}i_{w};j_{\tau}j_{x}j_{w}}\psi_{L,j_{\tau}j_{x}j_{w},\alpha}$
$\displaystyle+[\psi_{R,i_{\tau}i_{x}i_{w},\alpha}^{\dagger}M^{1+2D,RL}_{i_{\tau}i_{x}i_{w};\alpha\beta}(\phi)\psi_{L,i_{\tau}i_{x}i_{w},\beta}+h.c.].$

Schematically, this is:

$\psi^{\dagger}\not{D}\Psi=\Psi^{\dagger}\left(\begin{array}[]{cc}M_{R}&g\delta_{w,L_{w}}\Theta_{R}^{\dagger}\Theta_{L}\\\
g\delta_{w,L_{w}}\Theta_{L}^{\dagger}\Theta_{R}&M_{L}\end{array}\right)\Psi$
(4.12)

where $\Psi^{\dagger}=(\psi_{R}^{\dagger},\psi_{L}^{\dagger})$. On the mirror
surface and in the continuum limit, we have

$\Psi^{\dagger}\not{D}\Psi=\Psi^{\dagger}\left(\begin{array}[]{cc}\partial_{t}-\text{i}\partial_{x}&g\Theta_{R}^{\dagger}\Theta_{L}\\\
g\Theta_{L}^{\dagger}\Theta_{R}&\partial_{t}+\text{i}\partial_{x}\end{array}\right)\Psi$
(4.13)

which has the familiar field theory form. On the normal surface, we have

$\Psi^{\dagger}\not{D}\Psi=\Psi^{\dagger}\left(\begin{array}[]{cc}\partial_{t}+\text{i}\partial_{x}&0\\\
0&\partial_{t}-\text{i}\partial_{x}\end{array}\right)\Psi$ (4.14)

which is the gapless chiral theory we seek. Our task now is to ensure that the
Higgs interaction gaps out the mirror theory (4.13) while preserving (4.14)
Denoting the full content of eq. (4.3) as $\Psi^{\dagger}\not{D}\Psi$, the
full partition function of our system is now:

$\displaystyle Z=\int D\phi e^{-S_{\text{H}}[\phi]}\int
D\Psi^{\dagger}D\Psi\exp[-\Psi^{\dagger}\not{D}\Psi]$ (4.15)
$\displaystyle=\int D\phi e^{-S_{\text{H}}[\phi]}\det(\not{D})$

where $S_{H}[\phi]=-U[\phi]$ is the action for the Higgs and we have neglected
a proportionality constant.

We choose the Higgs action $S_{H}[\phi]$ such that the Higgs field is in a
disordered $SU(2)$ symmetric state with correlation length $\xi>1$ (with unit
lattice spacing). If we took $\xi\to\infty$, i.e. if the Higgs field $\phi$
were a constant (which breaks $SU(2)$ symmetry), the mirror sector would be
fully gapped and all eigenvectors of $\not{D}$ with eigenvalues $\lambda$ with
$|\lambda|<\Delta$ would be localized near the normal surface. The central
result of this chapter is that the mirror edge is still gapped for a finite
correlation length, and the normal sector is hardly affected by the gapping
process.

Performing the full integral of $\phi$ disordered Higgs phase is intractable.
Instead, we study the above path integral in two steps. In the next Section,
we consider a typical random space-time Higgs fields with a finite correlation
length $\xi$ which preserves $SU(2)$ symmetry on average. We will show that
the mirror sector can be fully gapped by these random Higgs fields when
$\xi\gtrsim 7$ for $g=1$ and system sizes up to $L=80$. Then we can safely
integrate out the mirror fermions and obtain a non-linear $\sigma$-model for
the Higgs field. Lastly, we argue that the non-linear $\sigma$-model can be in
the $SU(2)$ symmetric disordered phase with a gap, the only low energy
excitations coming from the massless chiral fermions in the normal sector
which is unaffected by the gapping process in the mirror sector.

The choice of dynamical Higgs field is of central importance. If at any point
$\phi$ fluctuates too rapidly, a low-energy fermion mode may be localized
there. An ideal configuration would have $|\nabla\phi(x)|=\text{const.}>0$. In
the lattice model, we first choose a random $\phi(x)$ and then smooth it,
taking care to apply the most smoothing in regions of largest $|\nabla\phi|$.
This nonlinear smoothing process leads to a $\phi$ with nearly constant but
nonzero $|\nabla\phi|$. In a continuum sense, this can be achieved by a an
action of the form $S(\phi)=-\int
d^{2}x(-|\nabla\phi|^{2}+|\nabla\phi|^{4}+...)$. Since we cannot yet perform
the integral over Higgs configurations, we do not specify a lattice action at
this point.

###  4.4 Gapping Process

We first show that introducing a Higgs field opens a spectral gap in the
fermion determinant. Next, we confirm that this gap remains in a thermodynamic
limit and check that the gapped mirror edge is indeed decoupled from the
chiral edge. Finally, we address the resulting non-linear sigma model for the
Higgs field obtained by integrating out the mirror fermions.

####  4.4.1 Numerical Results

Figure 4.4: _(Color Online)_. Integrated Density of States (IDOS) for various
choices of the coupling strength $g$ and the correlation length of the Higgs
field $\xi$ with $L=80$ and $L_{w}=2$. The horizontal axis is the eigenvalue
magnitude, while the vertical axis is the eigenvalue number (in order of
increasing magnitude). We first find the low-magnitude eigenvalues of the
matrix $M^{1+1D}$ in eqn. (4.3), and order them
$|\lambda_{n}|\leq|\lambda_{n+1}|$. Each plot shows $n$ vs. $|\lambda_{n}|$.
States localized on the normal surface (the weight of the eigenvector on the
normal surface is larger than $0.8$) are denoted by green circles, others are
denoted by blue squares. A black dashed line indicates the magnitude of the
lowest mirror-surface eigenvalue. In the upper panels, we use the same
configuration of the Higgs field with $\xi\approx 10.4$ and different Higgs
couplings $g$. For $g=0$, the IDOS is two copies of our chiral edge theory. As
the interaction strength increases, states not localized to the normal surface
are gapped out until at $g=1$ only the normal sector remains (below
$|\lambda|\lesssim.45$). In the lower panels, we fix $g=1$ and smooth the
Higgs field, increasing the correlation length from $\xi\approx.6$ until
$\xi\approx 11.4$. For $\xi\approx 3.8$, there are many low-lying mirror
states. As we increase $\xi$, the mirror gap increases until at $\xi\approx
11.4$ no mirror-edge states remain (below $|\lambda|\lesssim.45$). Normal
sector: In the lower-right panel, we calculate the IDOS for the normal sector
using (4.14). This IDOS matches that of our model with Higgs coupling $g=1$
and $\xi>10$. Thus the low lying modes of our model are exactly described by
the normal sector only, and the mirror sector is fully gapped. The slight
$\lambda_{n}$ renormalization can be mitigated by increasing $L_{w}$.

To see what the process of gapping out the mirror theory looks like, let us
first fix the system size $L_{x}=L_{t}=L=80$ and Higgs correlation length
$\xi$. Figure 4.4 (upper panels) show the integrated density of states (the
number of eigenvalues whose absolute values are less then $|\lambda|$) as we
turn on the interaction from $g=0$ to $g=1$. At $g=0$, there are 32 gapless
modes—16 from the chiral theory and 16 from the mirror conjugate. As we turn
on the interaction, the fluctuating Higgs field smoothly gaps out the mirror
theory modes, leaving only the chiral theory at low energies. Using eq. (4.7),
we can calculate the integrated density of states (IDOS) for the normal sector
alone, which we present in the lower-right panel of Figure 4.4. Comparing the
normal sector IDOS to the lattice results, we see that the low-lying modes of
the lattice models are indeed described by the unaffected normal sector alone.

We can determine if a given eigenstate $\ket{v}$ is localized on the normal
edge by examining the expectation value of the $w$ position operator with
respect to $\ket{v}$. If a state were indeed perfectly localized at $w=1$,
then we would have $\braket{w}_{v}=1$. For our purposes, we say that a state
is localized on the normal edge if $\braket{w}_{v}<1.2$. While the precise
value of this cutoff is arbitrary, for larger $L_{w}$, this is a very strong
condition, as it implies that the wavefunction of $\ket{v}$ vanishes except
for very near the $w=1$ edge and implies that $\ket{v}$ has essentially no
probability near the $w=L_{w}$ edge.

Next, we fix $g=1$ and instead vary the Higgs field correlation length $\xi$
in Figure 4.4 (lower panels). For $\xi\lesssim 4$, the mirror edge modes
appear at small magnitude, though their momentum structure is wiped out by the
rapidly fluctuating Higgs field. As $\xi$ is increased, the mirror theory
modes again are driven to large magnitude, leaving only the chiral theory at
small eigenvalues.

Figure 4.5: _(Color Online)_. _(a-b)_ : Integrated Density of States (IDOS)
for two values of the width $L_{w}$. The horizontal axis is the eigenvalue
magnitude, while the vertical axis is the eigenvalue number (in order of
increasing magnitude). We fix $L_{x}=L_{t}=60$, choose a single Higgs
configuration $\phi$ with correlation length $\xi\approx 8.1$, and examine the
low-energy IDOS as we vary $L_{w}$. States localized on the normal edge (i.e.
with $\braket{w}<1.2$) are denoted by green circles, others are denoted by
blue squares. Vertical dotted lines indicate the magnitude of the free (no
Higgs) eigenvalues. For $L_{w}=2$ _(a)_ , the gapless modes remain but the
momentum structure is substantially affected by the gapping of the mirror
edge. For $L_{w}=4$, the low-lying states match with their free (Higgs-less)
eigenvalues very well. No mass, and minimal momentum renormalization, is
transmitted to the normal sector by the gapping of the mirror sector. The
computation can be extended to larger values of $L_{w}$ at high computational
cost. _(c)_ IDOS for the normal sector, calculated artificially using the
momentum-space Lagrangian. The $L_{w}=4$ lattice IDOS matches this artificial
normal sector IDOS, though higher-magnitude modes are shifted upwards by the
lowest-lying gapped mirror modes.

The mirror edge is effectively decoupled from the chiral edge. To see this, we
fix $L_{x}=L_{t}=60$, choose a single Higgs configuration $\phi$ with
correlation length $\xi\approx 8.1$, and examine the low-energy IDOS as we
vary $L_{w}$. In Figure 4.5(a-b), we plot the IDOS for $L_{w}=2,4$. States
localized on the normal edge are marked in green while all others are marked
in blue. Further, we have plotted the $g=0$ (no Higgs coupling) eigenvalues as
vertical dotted lines. For $L_{w}=2$, we see that while the mirror sector is
indeed gapped out, the low-lying excitation energies are smeared away from
their $g=0$ values. For $L_{w}=4$, the normal sector excitation energies are
almost exactly their $g=0$ eigenvalues, decoupling from the gapped mirror
sector with no residual mass. One can compute this for higher values of
$L_{w}$ (though at significant computational cost) and find that the low-lying
eigenvalues almost exactly match their $g=0$ values. Additionally, we can
calculate ‘by hand’ what the integrated density of states for the normal
sector alone and finite lattice size would look like. We plot this in Figure
4.5c. Comparing the lattice results (a-b) to the ‘by-hand’ calculation (c), we
see that at low magnitude, the lattice results reproduce the expected ‘by-
hand’ result very well for larger $L_{w}$, thus demonstrating that the normal
sector decouples from the mirror sector with no residual mass. Surprisingly,
we find that even just $L_{w}=2$ is enough to capture the gapping process and
allows us to reach large system sizes with modest computational resources. In
Figures 4.4 and 4.5 we took $L_{w}=2$, as we will in the rest of this paper.

Figure 4.6: Mirror theory gap as a function of correlation length $\xi$ for
various choices of $L$. We couple the fermions on the mirror surface to a
Higgs field fluctuating with correlation length $\xi$ and look for the
smallest magnitude eigenvalue for states not localized on the chiral theory
edge. $\xi=L$ would correspond to the usual symmetry-breaking Higgs mechanism.
We choose a finite $\xi$, independent of $L$. For $\xi\gtrsim 8$, the mirror
edge is gapped with $\Delta\approx.35$, and this gap is independent of $L$ for
$L\gtrsim 40$, indicating thermodynamic behavior. Note that, in the
$\xi=L\to\infty$ ‘symmetry-breaking’ limit, $\Delta=g=1$.

The Higgs interaction in the mirror sector could leak into the normal sector
and cause residual couplings in the normal sector. The only relevant couplings
in the normal sector would be mass terms, but by design the $SU(2)$ chiral
fermions in the normal sector have no $SU(2)$ symmetric mass terms. Hence
residual couplings from the mirror sector can generate only exponentially
small $SU(2)$ symmetric interactions on the chiral theory edge. Since there is
no $SU(2)$ symmetric mass term, the effect of the exponentially small $SU(2)$
symmetric interactions is weak and the gapless chiral theory is effectively
decoupled from the gapped mirror theory.

Now we turn to the thermodynamic behavior, fixing $g=1$ and examining the gap
given to the mirror modes. Figure 4.6 shows how this mirror-mode gap scales
with the system size $L$ and the correlation length $\xi$. At $\xi\approx 8$,
for system sizes $L\gtrsim 40$, we see that the gap is roughly constant at
$\Delta\approx.35$, and largely independent of system size. This indicates
that we are indeed seeing thermodynamic-limit behavior and that we have
successfully gapped out only the mirror theory.

####  4.4.2 Effective Higgs Action

The spectral gap of $\not{D}$ in the mirror sector allows us to safely
integrate out the mirror fermions and obtain a 1+1D non-linear $\sigma$-model
of the Higgs field $\phi$ on the mirror surface. Since $\not{D}$ is not
Hermitian, this determinant may be complex and its phase can fluctuate. The
complex phase could give rise to the topological terms in the non-linear
$\sigma$-model which would make the Higgs effective theory gapless. However,
the target space of the $SU(2)$ Higgs non-linear $\sigma$-model is
$SU(2)=S^{3}$ which has no topological defects in 1+1D space-time. The only
potential topological term in the 1+1D $S^{3}$ non-linear $\sigma$-model is
the WZW term from $\pi_{3}(S^{3})=\mathbb{Z}$. This topological term would
correspond to the ABJ $SU(2)$ anomaly. Such a topological term does not appear
since our model is free of ABJ anomalies by design. It is believed that a non-
linear $\sigma$-model with no topological defects or topological term can be a
gapped disordered phase that does not break any symmetry. This is the phase we
believe our model to be in, though further numerical study of nonlinear
$\sigma$-models is needed to conclusively confirm this.

We have demonstrated that coupling the mirror edge to a disordered Higgs field
introduces a gap in the fermion spectrum on the mirror edge, and have argued
that the resulting non-linear $\sigma$-model for the Higgs field is gapped.
Because the chiral theory is effectively decoupled from the mirror edge, all
that remains at low energy is our low energy, chiral, $SU(2)$ theory.

###  4.5 Anomaly Matching and Application to a Condensed Matter System

The model we considered in this Chapter needs a significant modification to be
considered as a true condensed matter system, as it violates the spin
statistics theorem (SST). In particular, the SST states that fermions have
half-integer spin, while bosons have integer spin. However, we have considered
a fermion model with one spin-$1$ mode. Here we propose several other models
that escapes this issue. However, these new models result in larger on-site
matrices which introduce further technical challenges. We also include $U(1)$
symmetry, and ensure that, as holds in condensed matter systems, all fermions
have odd change.

Representation | Left Anomaly | Left Chiral Central Charge
---|---|---
$\nicefrac{{1}}{{2}}_{L}\oplus\nicefrac{{9}}{{2}}_{L}^{2}\oplus\nicefrac{{3}}{{2}}_{R}\oplus\nicefrac{{5}}{{2}}_{R}\oplus\nicefrac{{11}}{{2}}_{R}$ | $662$ | $22$
$\nicefrac{{1}}{{2}}_{L}\oplus\nicefrac{{17}}{{2}}_{L}^{2}\oplus\nicefrac{{7}}{{2}}_{R}^{2}\oplus\nicefrac{{21}}{{2}}_{R}$ | $3878$ | $38$
$\nicefrac{{1}}{{2}}_{L}\oplus\nicefrac{{3}}{{2}}_{L}\oplus\nicefrac{{13}}{{2}}_{L}\oplus\nicefrac{{17}}{{2}}_{L}\oplus\nicefrac{{5}}{{2}}_{R}^{3}\oplus\nicefrac{{19}}{{2}}_{R}$ | $2870$ | $38$
Table 4.1: A Few Anomaly-Free Chiral $SU(2)$ Representations for Fermionic
Excitations. We ensure that the Adler-Bell-Jackiw anomalies cancel for every
$U(1)$ subgroup of $SU(2)$. We assume that all excitations are fermionic, and
spin-statistics requires that the fermionic excitations have odd spin.
$(\nicefrac{{1}}{{2}},3)_{L}\oplus(\nicefrac{{5}}{{2}},3)_{L}\oplus(\nicefrac{{9}}{{2}},3)_{L}\oplus(\nicefrac{{9}}{{2}},11)_{L}(\oplus\nicefrac{{3}}{{2}},7)_{R}\oplus(\nicefrac{{5}}{{2}},7)_{R}^{2}\oplus(\nicefrac{{11}}{{2}},7)_{R}\hskip
56.9055pt$
---
Left $SU(2)$ Anomaly | Left $U(1)$ Anomaly | Left Chiral Central Charge
$732$ | $1372$ | $28$
$(\nicefrac{{1}}{{2}},3)_{L}\oplus(\nicefrac{{9}}{{2}},3)_{L}^{2}\oplus(\nicefrac{{9}}{{2}},11)_{L}\oplus(\nicefrac{{3}}{{2}},3)_{R}\oplus(\nicefrac{{5}}{{2}},7)_{R}\oplus(\nicefrac{{9}}{{2}},7)_{R}\oplus(\nicefrac{{11}}{{2}},7)_{R}$
Left $SU(2)$ Anomaly | Left $U(1)$ Anomaly | Left Chiral Central Charge
992 | 1408 | 32
Table 4.2: A Few Anomaly-Free Chiral $SU(2)\times U(1)$ Representations for
Fermionic Excitations. We ensure cancellation of the Adler-Bell-Jackiw
anomalies independently for the $U(1)$ and $SU(2)$ symmetries. In addition to
spin-statistics requiring that fermionic excitations have odd spin, in
condensed matter systems all fermions will have odd charge.

Recall that in order for the $U(1)\leqslant SU(2)$ ABJ anomalies to cancel, we
require that:

$\sum_{L}\frac{4}{3}s_{i}(s_{i}+1)(2s_{i}+1)=\sum_{R}\frac{4}{3}s_{i}(s_{i}+1)(2s_{i}+1)$
(4.16)

We also require the cancellation of all gravitational anomalies, which is
equivalent to:

$\sum_{L}(2s_{i}+1)=\sum_{R}(2s_{i}+1)$ (4.17)

Combining these, and ensuring that we only use half-integer $SU(2)$
representations, we can search for chiral representations, several of which
are shown in in Table 4.1. If we also include $U(1)$ symmetry, with charges
$q_{i}$, then we have a further anomaly cancellation condition:

$\sum_{L}(2s_{i}+1)q_{i}^{2}=\sum_{R}(2s_{i}+1)q_{i}^{2}$ (4.18)

Including this condition, we can search for chiral $SU(2)\times U(1)$
theories, two of which are given in Table 4.2.

Finally, let us note the mathematical perspective on these spin-statistics
restrictions. In any fermionic system, there is an additional fermionic parity
symmetry $\mathbb{Z}_{2}^{f}$. A system with symmetry group $G$ would naively
have the actual symmetry group $g\times\mathbb{Z}_{2}^{f}$. However, in
reality $\mathbb{Z}_{2}^{f}$ is realized as a subgroup of $G$. In the case of
$SU(2)$, $\mathbb{Z}_{2}^{f}$ is given by the subgroup $\\{I,-I\\}$, while for
$SU(2)\times U(1)$, the subgroup appears as $\\{(I,1),(-I,1)=(I,-1)\\}$.

###  4.6 Summary of the $SU(2)$ Model

The method that we have demonstrated in this chapter leads to a lattice
regularization for chiral QFTs. Both fermions and the gauge symmetry $G$ are
defined on on a space-time lattice with no extra dimensions, and the lattice
model is entirely local. Our approach should work for any $d+1$D chiral
fermion theory that satisfies: (1) there exist mass terms that make all the
chiral fermions massive and break the $G$ symmetry down to $G_{\text{grnd}}$;
and (2) $\pi_{n}(G/G_{\text{grnd}})=0$ for $n\leq d+2$.

While this work presents progress towards a solution to the long standing
chiral fermion problem, three important considerations remain. First, although
all low-energy modes in our model are linear, the velocities of fermions in
different $SU(2)$ representations may be different; in the future we will try
to include a ‘shift’ symmetry that fixes the velocities to be equal. Second,
the integral over Higgs considerations can be performed numerically in an
approximate way. This is not a trivial task, as the system suffers from a sign
problem. Finally, coupling to a weak gauge field (with $U=\exp[iA_{ij}]$ near
unity) should be studied as well.

These remaining challenges are considerable. Because the problem requires
interactions, it will be almost always be difficult to solve. This rather
severe technical challenge both limits the future utility of the approach, and
obscures the proof of its concept. In the next Chapter, we will move up a
dimension and consider a lattice regularization of Chern-Simons theories,
which describe the $2+1$d bulk that a chiral theory might live on. The Chern-
Simons theory will lead to an ungauged SPT in that we study in Chapter 6,
which, as an almost trivial consequence, will yield a much more tractable
chiral edge theory in Chapter 7.

##  Chapter 5 A Lattice Rotor Model for $2+1$d Bosonic $U^{\kappa}(1)$ Chern-
Simons Theory

In the last chapter, we saw a lattice model for a fermionic chiral field
theory. That model faced significant challenges in its implementation due to
strong interactions. In this chapter, we attack a related problem in one
higher dimension: the description of Abelian topological order through
$U^{\kappa}(1)$ Chern-Simons theory. We will develop a lattice model which
describes these theories at low energy. This will be a theory of $U(1)$
lattice rotor variables, and is to our knowledge the first of its kind. The
semiclassical approximations we use to extract this low-energy behavior are
more reliable and under better control than the Higgs integration of the
previous chapter; in turn, this theory itself will lead us back to the chiral
edge in the following chapter.