In the rest of this thesis, we will use the simplicial cochain formalism
detailed in Appendix A. We will also work in units where the flux of field
variables, and the total angle of a circle, is quantized to unity rather than
$2\pi$.

We wish to develop a lattice model for the Chern-Simons theories described by
an action:

$S=\pi i\sum_{I,J}K_{IJ}\int a_{I}da_{J}$ (5.1)

where, as in Chapter 3, $K_{IJ}$ is a symmetric integer matrix (note that we
have adjusted the flux quantization condition relative to the standard version
we used previously). For simplicity, we assume that the theory is bosonic, so
that $K_{IJ}$ has even diagonal entries. The simplest guess for a lattice
model is to transpose (5.1) to the lattice, exchanging forms for cocycles, the
wedge product for the cup product, and $d$ for the lattice differential:

$\pi i\sum_{I,J}K_{IJ}\int
a_{I}da_{J}+\frac{1}{g}\sum_{\Box}\sum_{I}(da_{I})^{2}$ (5.2)

where the integral $``\int^{\prime\prime}$ is interpreted as evaluation
against a generator of the top cohomology of the spacetime manifold and $\Box$
labels plaquettes. This action has a similar ‘doubling’ problem as lattice
fermions at momenta $\pm\pi$, and we have introduced a Maxwell term to
suppress those points. However, this action has a more fundamental issue as
well: the field $a$ is naturally $\mathbb{R}$-valued and cannot be interpreted
as a $U(1)$ variable. In getting around this, we will discover a new Chern-
Simons action.

These technical issues with Chern-Simons lattice actions are well known, but
they have not stopped considerable work on putting the theory on the lattice.
One approach has been to attempt to construct local lattice models for Chern-
Simons theory whose many-body Hilbert space admits a tensor product
decomposition $\mathcal{V}=\bigotimes_{i}\mathcal{V}_{i}$, where
$\mathcal{V}_{i}$ is the local Hilbert space on site-$i$. The key is to find a
proper local Hamiltonian $H$ acting on $\mathcal{V}$ such that the low energy
properties of $H$ are fully described by a Chern-Simons field theory [121,
107, 37, 99, 38, 108, 64, 87, 59]. However, those lattice models are usually
not solvable. Given a lattice model, we usually do not know if it is in a
quantum-Hall topologically ordered phase, if the lattice model produce a
Chern-Simons theory at low energy, or even which Chern-Simons theory it
produces. Here we are looking for a better controlled result, where we can
derive, under a controlled approximation, the low energy effective Chern-
Simons field theory from the lattice model.

A second approach [34, 90, 13, 14] also tried to construct lattice gauge
models that produce Chern-Simons field theory at low energies. The many-body
Hilbert space $\mathcal{V}_{\text{gauge}}$ for the lattice gauge theory in
these cases is formed by gauge invariant states, which are not local, as the
$\mathcal{V}_{\text{gauge}}$ does not admit the tensor product decomposition
$\mathcal{V}_{\text{gauge}}\neq\bigotimes_{i}\mathcal{V}_{i}$. References [34,
90] proposed lattice gauge models with compact $U(1)$ gauge group. However, in
those models the gauge field in each link is not compact; rather the
compactness is enforced at global level. In contrast, the link variables in
this paper are already in compact $U(1)$ groups. References [13, 14] proposed
lattice gauge models with an entirely non-compact $U(1)$ gauge group (i.e.
$\mathbb{R}$), which is quite different from the compact $U(1)$ gauge theory
we describe here.

Our goal is to realize the most general bosonic $U(1)$ Chern-Simons theory via
a local bosonic lattice model with _compact_ degrees of freedom on each link.
In contrast to previous lattice models of Chern-Simons theory, we want our
local lattice model to be semiclassically solvable, in the sense that we can
reliably determine its low energy effective theory.

We will find local bosonic model on spacetime lattice satisfying exactly these
conditions. Under a controlled semi-classical approximation for small $g$ in
(5.1), we show that our spacetime lattice model can produce any
even-$K$-matrix CS field theory [108] of _compact_ $U(1)$’s in continuum limit
(see (5.1)).

###  5.1 Lattice Model

Now we can turn to the main lattice model. Let $\mathcal{M}^{3}$ be a three-
dimensional simplicial complex, with (possibly empty) boundary
$\mathcal{B}^{2}=\partial\mathcal{M}^{3}$. Our field variables $a$ will be
defined on the links of Since $a_{I}$ is $\mathbb{R}/\mathbb{Z}$-valued, all
physical quantities to be invariant under the following “gauge” transformation

$\displaystyle a_{I}\to a_{I}+n_{I},$ (5.3)

where $n_{I}$ are arbitrary $\mathbb{Z}$-valued 1-cochains. In this language,
the naïve action amplitude $e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}a_{I}da_{J}}$ is not locally a $U(1)$ theory
because it is not gauge invariant.

We can motivate our final action by going to to one-higher dimension, as in
Chapter 3. Instead of considering Chern-Simons theory on a three manifold
$\mathcal{M}^{3}$, we go to a four manifold $\mathcal{N}^{4}$ bounded by
$\mathcal{M}^{3}$ (i.e. $\mathcal{M}^{3}=\partial\mathcal{N}^{4}$) and
evaluate:

$i\pi\sum_{I,J}K_{IJ}\int_{\mathcal{N}^{4}}f_{I}\cup f_{J}$ (5.4)

The cup product, as well as general conventions for lattice cocycles, is
defined in Appendix A. The question is now how to define $f_{I}$. We might
expect to have $f_{I}=da_{I}$, but this cannot be correct: the field strength
is a physical quantity and so must be a periodic function of $a$. Instead, we
choose the field strength:

$f_{I}=da_{I}-\lfloor da_{I}\rceil$ (5.5)

where $\lfloor x\rceil$ denotes the nearest integer to $x$. On any plaquette
$\braket{ijk}$, we have:

$f_{I}=\frac{1}{2\pi i}\log\left[e^{2\pi ia_{I}(ij)}e^{2\pi ia_{I}(jk)}e^{2\pi
ia_{I}(ki)}\right]$ (5.6)

where the logarithm is taken with a branch cut along the negative real axis.
Hence $f$ is manifestly invariant under 5.3. The resulting action is:

$i\pi\sum_{I,J}\int_{\mathcal{N}^{4}}(da_{I}-\lfloor
da_{I}\rceil)(da_{J}-\lfloor da_{J}\rceil)$ (5.7)

Let us first assume that the field strength is weak, which implies that
$da_{I}-\lfloor da_{I}\rceil\approx 0$ and hence $d\lfloor da_{I}\rceil=0$.
Then we may rewrite the action as:

$2i\pi\sum_{I\leq J}\int_{\mathcal{N}^{4}}d\left[a_{I}(da_{J}-\lfloor
da_{J}\rceil)-\lfloor da_{I}\rceil a_{J}\right]$ (5.8)

which would lead us to the surface term action that we desired. However, we
need to add back in a term to account for cases when $d\lfloor da\rceil\neq
0$. The key here is to preserve a certain $1$-form symmetry that will be
examined in Section 5.2. This leads to another term in the action $-2\pi
i\sum_{I\leq J}\int\mathcal{M}^{3}a^{J}\cup_{1}d\lfloor da^{J}\rceil$. The
1-cup product $\underset{{\scriptscriptstyle 1}}{\smile}$ [89] is defined in
Appendix A. All told, we consider the following partition function:

$\displaystyle Z=\int[\prod da_{I}]\ e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}d\big{(}a_{I}(a_{J}-\lfloor
a_{J}\rceil)\big{)}}$ $\displaystyle e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}a_{I}(da_{J}-\lfloor da_{J}\rceil)-\lfloor
da_{I}\rceil a_{J}}$ (5.9) $\displaystyle e^{-i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}a_{J}\underset{{\scriptscriptstyle
1}}{\smile}d\lfloor
da_{I}\rceil}e^{-\int_{\mathcal{M}^{3}}\frac{|da_{I}-\lfloor
da_{I}\rceil|^{2}}{g}},$

To see that the path integral (5.1) is invariant under gauge transformation
(5.3), even when the spacetime manifold $\mathcal{M}^{3}$ has a boundary, we
first note that $e^{-i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}a_{J}\underset{{\scriptscriptstyle
1}}{\smile}d\lfloor da_{I}\rceil}$ and $da_{I}-\lfloor da_{I}\rceil$ are
invariant under (5.3). Under (5.3), the term

$e^{i2\pi\sum_{I\leq J}k_{IJ}\int_{\mathcal{M}^{3}}a_{I}(da_{J}-\lfloor
da_{J}\rceil)-\lfloor da_{I}\rceil a_{J}}$ (5.10)

changes by a factor

$e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}n^{I}da_{J}-dn^{I}a_{J}}=e^{-i2\pi\sum_{I\leq
J}k_{IJ}\int_{\partial\mathcal{M}^{3}}n^{I}a_{J}}$ (5.11)

This is exactly canceled by the change of the term

$e^{i2\pi\sum_{I\leq J}k_{IJ}\int_{\mathcal{M}^{3}}d\big{(}a_{I}(a_{J}-\lfloor
a_{J}\rceil)\big{)}}=e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\partial\mathcal{M}^{3}}\big{(}a_{I}(a_{J}-\lfloor
a_{J}\rceil)\big{)}}$ (5.12)

Hence the action amplitude of the above path integral is indeed invariant
under $\eqref{gaugeZ}$, even when $\mathcal{M}^{3}$ has a boundary.

Now we show that the bosonic lattice model (5.1) realizes a topological order
described by $U^{\kappa}(1)$ Chern-Simons topological quantum field theory in
the limit of small $g$. Taking this limit forces $da_{I}$ to be nearly
integer-valued, i.e. $da_{I}-\lfloor da_{I}\rceil\approx 0$. In turn, this
imples that $d\lfloor da_{i}\rceil=0$ since if $da_{I}=\epsilon+\lfloor
da_{I}\rceil$ where $\epsilon$ is small, then

$\displaystyle d\lfloor da_{I}\rceil=-d\epsilon+dda_{I}=-d\epsilon.$ (5.13)

As $d\lfloor da_{I}\rceil$ is quantized to be an integer, we must have
$d\lfloor da_{I}\rceil=0$. This $\mathbb{Z}$-valued 2-cocycle $\lfloor
da_{I}\rceil$ characterize the $U^{\kappa}(1)$ principle bundle on the
spacetime, since

$\displaystyle\int_{\mathcal{M}^{2}}(da_{I}-\lfloor
da_{I}\rceil)=-\int_{\mathcal{M}^{2}}\lfloor da_{I}\rceil$ (5.14)

for any closed $\mathcal{M}^{2}$. Note that
$\int_{\mathcal{M}^{2}}(da_{I}-\lfloor da_{I}\rceil)$ is the magnetic flux
through $\mathcal{M}^{2}$ which is always quantized to be an integer. In other
words, $-\int_{\mathcal{M}^{2}}\lfloor da_{I}\rceil$ is the Chern number.
Furthermore, on any local patch of spacetime, we can use the gauge
transformation (5.3) to set $\lfloor da_{I}\rceil=0$ on the patch. In this
case, the action amplitude in the path integral (5.1) becomes quadratic (i.e.
non-interacting)

$\displaystyle e^{i2\pi\sum_{I\leq J}k_{IJ}\int_{\mathcal{M}^{3}}a_{I}da_{J}}$
(5.15)

Since $da_{I}$ is close to zero, we can use a 1-form $A^{I}$ to describe the
1-cochain $a_{I}$:

$\displaystyle\int_{i}^{j}A^{I}=2\pi(a_{I})_{ij}$ (5.16)

Then the above action amplitude can be rewritten as

$\displaystyle e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}a_{I}da_{J}}\approx
e^{i\sum_{IJ}\frac{K_{IJ}}{4\pi}\int_{M^{3}}A^{I}dA^{J}}$ $\displaystyle\ \ \
\ K_{IJ}=K_{JI}\equiv\begin{cases}2k_{IJ},&\text{ if }I=J,\\\ k_{IJ},&\text{
if }I<J,\\\ \end{cases}$ (5.17)

when $A^{I}$ is nearly constant on the lattice scale. Hence the low energy
dynamics of our lattice bosonic model are described by a $U^{\kappa}(1)$
Chern-Simons field theory (5.1) at low energies.

When $f_{I}=da_{I}-\lfloor da_{I}\rceil\approx 0$, i.e. when $da_{I}$ is
nearly an integer, we also have gauge invariance under the usual gauge
symmetry:

$a_{I}\to a_{I}+d\theta_{I}$ (5.18)

for $\theta_{I}$ an ${\mathbb{R}/\mathbb{Z}}$-valued one-cochain. This is
apparent both in the formulation (5.15) and its continuum counterpart (5.1).
In order for this gauge invariance to emerge, we had to assume that spacetime
$\mathcal{M}^{3}$ had no boundary and that $d\lfloor da_{I}\rceil=0$, i.e.
that the field configurations had no monopoles. That gauge invariance should
break on the boundary is, as discussed Chapter 3, no surprise in Chern-Simons
theories. Furthermore, we should understand the monopole-free requirement in
the same way: the monopole should be thought of as trapping a defect. In
topological quantum field theories, defects can be realized as punctures in
the manifold, which therefore are boundaries. In effect, each monopole traps a
region of non-topologically ordered matter, which therefore carries a gauge-
symmetry breaking boundary.

Due to the boundaries induced by monopoles, it is not clear what the ground
state is for large $g$, and it may have a different topological order from the
one described by the $K$-matrix Chern-Simons theory. However, the higher-form
symmetries we discuss in the next section always hold.

###  5.2 Higher Symmetries and Anomalies

The most powerful aspect of the action we have constructed is its
1-symmetries. First, consider the model on a closed manifold, so that we may
ignore the surface term. Then under the shift:

$a_{I}\to a_{I}+\tilde{\beta}_{I}\hskip
28.45274pt\sum_{I}\tilde{\beta}_{I}K_{IJ}\in\mathbb{Z}$ (5.19)

where $\tilde{\beta}_{I}$ are ${\mathbb{R}/\mathbb{Z}}$-valued 1-cocycles, the
exponentiated action is invariant so long as $\sum_{I}\tilde{\beta}_{I}K_{IJ}$
are $\mathbb{Z}$-valued 1-cochains. These transformations (5.19) are the
1-symmetries of lattice model (5.1).

To see this result, we first note that, under the transformation (5.19), the
action amplitude in (5.1) on a closed manifold changes by a factor

$e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{I}(da^{{\mathbb{R}/\mathbb{Z}}}_{J}-\lfloor
da^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil)-\lfloor
da^{{\mathbb{R}/\mathbb{Z}}}_{I}\rceil\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}-i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}\underset{{\scriptscriptstyle
1}}{\smile}d\lfloor da^{{\mathbb{R}/\mathbb{Z}}}_{I}\rceil}$ (5.20)

Because we may integrate by parts on a closed manifold and
$d\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{I}=0$, the change is of the form (see
Appendix A):

$\displaystyle e^{-i2\pi\sum_{I\leq
J}k_{IJ}\int_{\mathcal{M}^{3}}\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{I}\lfloor
da^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil+\lfloor
da^{{\mathbb{R}/\mathbb{Z}}}_{I}\rceil\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{J}+\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}\underset{{\scriptscriptstyle
1}}{\smile}d\lfloor da^{{\mathbb{R}/\mathbb{Z}}}_{I}\rceil}$
$\displaystyle=e^{-i2\pi\sum_{IJ}K_{IJ}\int_{\mathcal{M}^{3}}\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{I}\lfloor
da^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil}$ (5.21)

which remains unity for all $\lfloor da^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil$
iff $\sum_{I}\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{I}K_{IJ}$ are
$\mathbb{Z}$-valued cochains. We see that, on a fixed link $ij$, the allowed
values of $(\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{I})_{ij}$ form the
rational lattice $K^{-1}$. The 1-symmetries are given by the rational lattice
$K^{-1}$ modulo the integer lattice, which is same as the integer lattice
modulo the lattice $K$. In other words, the 1-symmetries are $Z_{k_{1}}\times
Z_{k_{2}}\times\cdots$ 1-symmetries, with $k_{i}$ being the diagonal entries
of the Smith normal form of $K$.

For example, for $U(1)$ Chern Simons theory with $\kappa=1$ and
$K_{11}=2k_{11}=k$, we have a $\mathbb{Z}_{k}$ 1-symmetry. For mutual Chern-
Simons theory (that describes a $Z_{n}$ gauge theory), with
$(K_{IJ})=\left(\begin{array}[]{cc}0&n\\\ n&0\end{array}\right)$ we have a
${Z}_{n}\times{Z}_{n}$ 1-symmetry.

Some of the above 1-symmetries are anomalous. To determine which, we need
examine which of the transformations in (5.19) changes the action amplitude
when the spacetime has a boundary. Under the transformation (5.19), the action
amplitude in (5.1) only changes by a factor defined on the boundary
$\partial\mathcal{M}^{3}$:

$e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\partial\mathcal{M}^{3}}a^{{\mathbb{R}/\mathbb{Z}}}_{I}(\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}-\lfloor\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil)+\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{I}(a^{{\mathbb{R}/\mathbb{Z}}}_{J}-\lfloor
a^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil)+\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{I}(\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}-\lfloor\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil)}...\\\
\times e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\partial\mathcal{M}^{3}}\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{J}\underset{{\scriptscriptstyle
1}}{\smile}\lfloor
da^{{\mathbb{R}/\mathbb{Z}}}_{I}\rceil-\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{I}a^{\mathbb{R}/\mathbb{Z}}_{J}}\\\
=e^{i2\pi\sum_{I\leq
J}k_{IJ}\int_{\partial\mathcal{M}^{3}}a^{{\mathbb{R}/\mathbb{Z}}}_{I}(\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}-\lfloor\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil)+\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{I}(\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}-\lfloor\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil)+\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{J}\underset{{\scriptscriptstyle
1}}{\smile}\lfloor da^{{\mathbb{R}/\mathbb{Z}}}_{I}\rceil}...\\\ \times
e^{-i2\pi\sum_{I\leq
J}k_{IJ}\int_{\partial\mathcal{M}^{3}}\tilde{\beta}^{{\mathbb{R}/\mathbb{Z}}}_{I}\lfloor
a^{{\mathbb{R}/\mathbb{Z}}}_{J}\rceil}$ (5.22)

We see that the transformations leave the action amplitude invariant if
$\sum_{I\leq J}k_{IJ}\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{J}=0$ and
$\sum_{I\leq
J}k_{IJ}\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{I}=\text{integer}$. We note
that $\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{I}$ satisfy the condition
$\sum_{IJ}K_{IJ}\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{I}\in\mathbb{Z}$, and
so the first equation implies the second. We find that the 1-symmetry
transformations in (5.19) are anomaly-free if

$\displaystyle\sum_{I\leq
J}k_{IJ}\tilde{\beta}^{\mathbb{R}/\mathbb{Z}}_{J}=0,\ \forall I$ (5.23)

For the level $k=K_{11}$ Chern-Simons theory with a single $U(1)$ gauge field,
this is simply the fact that the only $Z_{k}$ 1-symmetry is anomalous and must
break at the boundary. For the case of mutual Chern-Simons theory (ie the
$Z_{n}$ gauge theory) with $Z_{n}\times Z_{n}$ 1-symmetry, this implies that
one of the $Z_{n}$ 1-symmetries is anomalous and must break at the boundary,
while the other $Z_{n}$ 1-symmetry is anomaly-free. Note that the choice of
lattice model automatically selects which of the $Z_{n}$ 1-symmetry is
anomalous; one can select the opposite by replacing all $\sum_{I\leq J}$ with
$\sum_{I\geq J}$.

####  5.2.1 Framing Anomaly

It is well known that the Chern-Simons theory has a framing anomaly [111, 47].
After integrating out the physical degrees of freedom
$a^{\mathbb{R}/\mathbb{Z}}_{I}$ in (5.1) in the small $g$ limit, we should get
a partition function given by the 2+1D gravitational Chern-Simons term:

$\displaystyle Z(M^{3},g_{\mu\nu})\propto e^{i\frac{2\pi
c}{24}\int_{M^{3}}\Omega_{3}}$ (5.24)

where the 3-form $\Omega_{3}$ satisfies $d\Omega_{3}=p_{1}$ and $p_{1}$ is the
first Pontryagin class for the tangent bundle. Here $c$ is the chiral central
charge – the difference between the numbers of positive and negative
eigenvalues of the $K$-matrix. There is a framing anomaly when $c\neq 0$ mod
24.

This framing anomaly might prevent a local lattice realization of chiral
Chern-Simons theory with a non-zero central charge $c\neq 0$. However, our
construction shows that chiral $U(1)$ Chern-Simons theory can always be
realized on any 2+1D spacetime lattice. We believe that this is possible
because our spacetime lattice has extra structure, namely the branching
structure [30, 27, 24], which encodes the ordering of lattice vertices and
appears in the definition of the cup product. If we take different branching
structures on the same spacetime lattice, the resulting partition function $Z$
may be different. In turn, this branching structure dependence of partition
function may represent the framing anomaly.

###  5.3 Summary of the Bosonic Lattice Chern-Simons Model

We have constructed a local bosonic model which realizes any Abelian $2+1$d
topological order, represented as a lattice model of $K$-matrix Chern-Simons
theory. Moreover, this model has a local Hilbert space, and is manifestly a
function of compact rotor variables. This is a significant improvement on
previous models. Even stronger evidence for the phase produced by this model
is given by the exactly realized $1$-symmetries and their anomalies. We now
have a lattice model for any $2+1$d Abelian topological order.

Nonetheless, it is not clear how to use this achievement to get back towards a
chiral $1+1$d field theory. While these topological orders carry edge modes,
it is not clear how this could be extracted from the actions we have
constructed. In the next section, we will ‘ungauge this model,’ turning a
$2+1$d topological order into a $2+1$d SPT state. Moreover, these SPT models
will be exactly solvable, and that will point the way back towards the edge
theory.

##  Chapter 6 A $U(1)$ SPT and Discontinuous Lattice Topological Terms

In this chapter, we develop an exactly solvable model for SPT states by
‘ungauging’ the Chern-Simons model of the previous chapter. In this new,
exactly solvable SPT model, we will find examples of almost every phenomenon
discussed in Chapters 2 and 3, from chiral anomalies to the short-range
entanglement that characterizes SPTs. This will lead to a $1+1$d chiral
lattice field theory as an almost trivial corollary in the next chapter.

We begin in Section 6.1 where we make use of an ungauging process to turn the
lattice gauge theory model of topological order from Chapter 5 into a lattice
non-linear sigma model describing SPT states. This allows us to get our first
glimpse of the larger model, and by ungauging the model in the presence of a
background gauge field we will establish the first proof of the Hall
conductance of this model. In Section 6.2, we integrate this model into a two-
dimensional phase diagram describing the superfluid-Topological Mott Insulator
transition (SF-tMI). The original ‘ungauged’ action then corresponds to the
tMI fixed points, and we build an understanding of these phases as condensates
of charged vortices. We then examine the phase transitions for these models in
Section 6.5, and present a brief lattice renormalization group argument in
Section 6.6.

###  6.1 Ungauging the Chern-Simons Theory

Here we ungauge the Chern-Simons lattice action to build our SPT model, both
as-is and in the presence of a background gauge field. Let us begin with the
Chern-Simons action reduced to just a single mode:

$S=2\pi ik\int\left[a(da-\lfloor da\rceil)-\lfloor da\rceil
a-a\cup_{1}d\lfloor da\rceil+d\big{(}a(a-\lfloor a\rceil)\big{)}\right]$ (6.1)

where we have dropped the Maxwell term. Recall that, because we are working
with angular variables $a_{ij}$, and not the rotor variables $U_{ij}=e^{2\pi
ia_{ij}}$, we must impose a gauge redundancy on all physical quantities. In
order to to ensure that the degrees actually remain rotors, we require a
“rotor redundancy”, with all physical quantities being invariant under the
replacement

$a_{ij}\to a_{ij}+m_{ij}$ (6.2)

Put more simply, we require that all physical quantities must be periodic
functions of $a_{ij}$. This redundancy ensures that the action (6.1) is a
function of ${\mathbb{R}/\mathbb{Z}}\simeq U(1)$ variables, and in turn is the
source of level quantization, ensuring that $k\in\mathbb{Z}$.

The Chern-Simons action is formulated in terms of a one-cochain $a$, i.e.
variables defined on the links of the lattice. To ungauge this model we set
[106]:

$\displaystyle a_{ij}=\phi_{i}-\phi_{j},\ \ \text{or }\ \ a=d\phi$ (6.3)

where $\phi$ is zero-cochain defined on the vertices. The partition function
is now given by:

$Z=\int[\prod d\phi]e^{i2\pi k\int_{\mathcal{M}^{3}}d\phi d\lfloor
d\phi\rceil}$ (6.4)

where $\mathcal{M}^{3}$ may have boundaries, and the measure is taken to be
integration over all sites, $\int\prod
d\phi=\prod_{i}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\phi_{i}$.

This action has a similar “rotor redundancy” as the Chern-Simons action. Note
that we may shift:

$\phi_{i}\to\phi_{i}+n_{i}$ (6.5)

where $n_{i}$ is a $\mathbb{Z}$-valued zero-cochain. The term $d\lfloor
d\phi\rceil$ is invariant because it transforms as:

$d\lfloor d\phi\rceil\to d\lfloor d\phi+dn\rceil=d\lfloor
d\phi\rceil+d^{2}n=d\lfloor d\phi\rceil$ (6.6)

and so the whole action is invariant modulo $2\pi$. As in the case of the
Chern-Simons theories, this redundancy ensures that this action is a function
of the $\phi_{i}$ as ${\mathbb{R}/\mathbb{Z}}\simeq U(1)$ variables. Beyond
this rotor redundancy, our models will also possess two global symmetries. The
first is the $U(1)$ symmetry of boson number conservation, which acts as

$\phi_{i}\to\phi_{i}+\theta$ (6.7)

where $\theta$ is a constant. We also have a charge conjugation symmetry:

$\phi_{i}\to-\phi_{i},$ (6.8)

which sends $g_{i}=e^{2\pi i\phi_{i}}\to g_{i}^{*}$. Note that eq. (6.4)
enjoys all three of these symmetries. In particular, the level $k$ is
quantized as a direct result of the gauge redundancy (6.5).

Figure 6.1: _(color online). (a)_ In $2+1$ dimensions, $\lfloor d\phi\rceil$
is dual to a surface where $\phi$ has a branch cut, and $\jmath_{v}=-d\lfloor
d\phi\rceil$ is dual to the line where a branch cut ends, i.e. a vortex _(b)_.
_(c)_ Working in two dimensions for convenience, we can see on a microscopic
level that $-d\lfloor d\phi\rceil$ is the vortex number on a plaquette. The
field variables are shown as clocks, while links with nonzero $\lfloor
d\phi\rceil$ are marked in turquoise. Plaquettes with $-d\lfloor
d\phi\rceil\neq 0$ are marked in blue (vortex) and red (anti-vortex).

To understand this action, note that $\jmath_{v}\equiv\star(-d\lfloor
d\phi\rceil)$ is the vortex current. To see this, first note that $\lfloor
d\phi\rceil$ is a one-cochain which, in three dimensions, is dual to a surface
where $\phi$ has a branch cut. Hence $d\lfloor d\phi\rceil$ is dual to the
line where the branch cut surface ends (Figure 6.1a). Branch cut surfaces end
at vortex lines; a $2$d slice of a vortex line and branch cut are shown in
Figure 6.1b, where we see that the branch cut terminates in a vortex. On a
more microscopic level, consider the $2$d lattice in figure 6.1c. The $U(1)$
variables on sites are shown as clocks with zero marked as a vertical line.
The branch cut links with $\lfloor d\phi\rceil=0$ are marked in turquoise, and
the branch cut terminates in a vortex (blue) and anti-vortex (red). Hence we
see that $-d\lfloor d\phi\rceil$ is indeed the vortex number on a plaquette,
with the sign fixed so that $-d\lfloor d\phi\rceil=+1$ at a plaquette hosting
a vortex. Note that conservation of vortex number means that
$d^{\dagger}\jmath_{v}=0$, where $d^{\dagger}\equiv\star d\star$.

To gain intuition, we can also rewrite the action in terms of the group
variables $g_{i}=e^{2\pi i\phi_{i}}$. The vortex current an a plaquette $p$
becomes:

$\star\jmath_{v}=-d\lfloor d\phi\rceil=-\frac{1}{2\pi
i}d\log(g_{i}g_{j}^{*})=-\frac{1}{2\pi
i}(\log(g_{i}g_{j}^{*})+\log(g_{j}g_{k}^{*})+...)$ (6.9)

where the logarithm is taken with a branch cut along the negative real axis.
In terms of these variables, the exponentiated action becomes:

$e^{iS}=\prod_{\braket{i,j}\in\text{
links}}(g_{i}g_{j}^{*})^{\star\jmath_{v}(\Box(i,j))}$ (6.10)

where $\Box(ij)$ is the plaquette dual to the link $\braket{i,j}$ under the
cup product. This formulation is not as useful as working in terms of the
$\phi_{i}$, but we do wish to establish that this is truly a $U(1)$ theory.

The topological action suggests that this model realizes an SPT state. To see
the SPT order, we repeat the ungauging in the presence of a weak background
gauge field $\bar{a}$ and evaluate the effective action for $A$. In the
presence of background $U^{\kappa}(1)$ background gauge field $A$, the
ungauing is done via

$a=A+d\phi.$ (6.11)

Now the model is given by

$Z=e^{i2\pi k\int_{\mathcal{M}^{3}}A(dA-\lfloor dA\rceil)-\lfloor dA\rceil
A-i2\pi k\int_{\mathcal{M}^{3}}A\underset{{\scriptscriptstyle
1}}{\smile}d\lfloor dA\rceil}\\\ \int[\prod d\phi]\ e^{-i2\pi
k\int_{\mathcal{M}^{3}}d\phi\underset{{\scriptscriptstyle 1}}{\smile}d\lfloor
dA\rceil+i2\pi k\int_{\mathcal{M}^{3}}d\phi(dA-\lfloor dA\rceil)-\lfloor
dA\rceil d\phi+i2\pi k\int_{\partial\mathcal{M}^{3}}(A+d\phi)(A+d\phi-\lfloor
A+d\phi\rceil)}$ (6.12)

Note that this model has a rotor redundancy for $A$:

$A\to A+m$ (6.13)

in addition to the usual one for $\phi$. If $\mathcal{M}^{3}$ is closed, this
can be simplified to

$Z=e^{i2\pi k\int_{\mathcal{M}^{3}}A(dA-\lfloor dA\rceil)-\lfloor dA\rceil
A-i2\pi k\int_{\mathcal{M}^{3}}A\underset{{\scriptscriptstyle
1}}{\smile}d\lfloor dA\rceil}\\\ \int[\prod d\phi]e^{-i2\pi
k\int_{\mathcal{M}^{3}}d\phi\underset{{\scriptscriptstyle 1}}{\smile}d\lfloor
dA\rceil-i2\pi k\int_{\mathcal{M}^{3}}d\phi\lfloor dA\rceil+\lfloor dA\rceil
d\phi}$ (6.14)

Now we assume that the background field is “weak.” Because we are working with
${\mathbb{R}/\mathbb{Z}}$ valued fields, a weak field means that $dA$ is
nearly an integer, i.e. $|dA-\lfloor dA\rceil|<\epsilon$. Noting that this
implies that $d\lfloor dA\rceil=0$, the path integral becomes:

$\displaystyle Z=e^{i2\pi k\int_{\mathcal{M}^{3}}A(dA-\lfloor
dA\rceil)-\lfloor dA\rceil A}$ (6.15)

This is the Chern-Simons response on lattice. When $\mathcal{M}^{3}$ is
closed, the action is invariant under gauge transformations of the background
gauge field $A\to A+d\varphi$. If $\mathcal{M}^{3}$ is a disk, the redundancy
(6.13) can be used to set $\lfloor dA\rceil=0$, and the response becomes:

$\displaystyle Z=e^{i\pi\sum k\int_{\mathcal{M}^{3}}AdA}$ (6.16)

This Chern-Simons response, in terms of the level $k$, describes the Hall
conductance and is the SPT invariant for our model. We will use this action,
coupled to a background gauge field $A$ in what follows below, and will see
the Hall conductance from several different perspectives.

###  6.2 Generalization to Arbitrary $k$ and the Topological Mott Insulators

With the model (6.4) in hand, we can actually pursue a generalization to non-
integer $k$. The key to this is to note that we can replace the action by:

$S=2\pi ik\int d\phi d\lfloor d\phi\rceil=2\pi ik\int(d\phi-\lfloor
d\phi\rceil)d(d\phi-\lfloor d\phi\rceil)$ (6.17)

Now, this action is rotor redundant under (6.5) for any $k$. In this section,
we elaborate this model, adding to it a second term which reveals the
character of this action as a topological Mott Insulator.

####  6.2.1 The Topological Mott Insulators

The phase transitions that occur in lattice boson systems between Mott
insulators [71] and superfluids (SF) have been extensively studied [45, 8, 31,
80, 36, 88, 8], in part because they embody the competition between kinetic
energy and interactions which underlies much of condensed matter physics. If
the kinetic energy dominates, the bosons condense into a superfluid. On the
other hand, repulsive interactions tend to favor the bosons being localized in
real space; if they dominate the system forms a Mott Insulator.

In a seminal work [35], Fisher et. al. showed that the superfluid-Mott
insulator (SF-MI) quantum phase transition generically falls into one of two
universality classes. In the Mott insulating phase, the average density is
pinned to a fixed integer; if the the chemical potential is varied, then the
extra bosons (or holes) may condense into a SF, leading to an ‘ideal’ mean-
field transition with dynamical critical exponent $z=2$. On the other hand, if
the chemical potential is chosen so that particle-hole symmetry is maintained,
then the SF-MI transition is actually a multicritical point and lies in the XY
universality class, with $z=1$. The XY transition in particular in $2+1$d is
in the same universality class as the condensation of superfluid helium at a
finite temperature, and so has seen impressive theoretical [18, 42, 50, 48]
and experimental [62, 63] study.

This story becomes more complex in systems where time-reversal symmetry is
broken. In $2+1$d, the insulating state can develop a quantized Hall
conductance, with such states being known variously as $U(1)$ symmetry
protected topological phases [27], topological Mott Insulators (tMIs), or the
Bosonic Integer Quantum Hall Effect [66, 84, 39]. These models generated
significant excitement; recently a fermionic analogue [81] has been proposed
as the mechanism behind the Quantum Anomalous Hall state in twisted bilayer
graphene [22] (though we will focus on bosonic tMIs in this chapter). The
topological Mott insulating states are labeled by their Hall conductance,
which for the bosonic systems we consider is always an even integer multiple
of $e^{2}/h$, viz. $\sigma_{xy}=(2k)\frac{e^{2}}{h}$ for $k\in\mathbb{Z}$.

As for the SF-MI transition, we can ask about the nature of the phase
transition that occurs between a SF and a tMI. What should the the critical
exponents of this transition be? Due to the strong charge density and current
fluctuations, the answer is not immediately clear.

We will focus on the particle-hole symmetric $z=1$ multicritical point, where
we will be able to write down a lattice field theory well-suited for
describing the phase diagram at fixed integer boson density. We show that the
critical exponents of the SF-tMI $XY$ transition are exactly the same as the
regular SF-MI $XY$ transition. Moreover, we will see that the bulk dynamics of
all local excitations are identical at or even away from the critical point,
unless there is an applied background gauge field. We do this by constructing
a well defined lattice model and showing that the bulk dynamics are invariant
under a “level-shift symmetry” induced by adding a topological $\theta$-term,
that changes the Hall conductance by $2k\frac{e^{2}}{h}$, hence connecting the
dynamics of the tMIs to the $k=0$ trivial MI. As our model is well-regulated
on the lattice, we can show that this level-shift symmetry is exact, i.e.
valid at all relevant energy scales. Crucially, this means that the
extraordinary numerical, theoretical, and experimental study of the $2+1$d XY
transition is applicable to the SF-tMI transition as well.

In the presence of a background gauge field, level-shift symmetry is broken
and the topological term leads to a Chern-Simons response and quantized Hall
conductance. In particular, the topological term causes the vortices which
proliferate in the insulating phase to carry charge. We discuss how charged
vortices lead directly to the quantized Hall response. This quantized Hall
response characterizes the tMI phase and we argue that it persists in the
vicinity of the SF-tMI transition.

For two transitions related by the level-shift symmetry (i.e. for two models
differing only by the topological term), the correlations of any local
operators are identical, at and away from the transition point, even though
the two transition points have different Hall conductance. This is possible
since the level-shift symmetry, i.e. the topological term, changes the
definition of current operators.

Beyond the description of the topological Mott insulators, these models also
represent an advance on a purely theoretical front. It has been known for some
time [27, 25, 105] that topological terms for $d+1$-dimensional spacetime
lattice models are labeled by elements of the group cohomology
$H^{d+1}(G,U(1))$. The cocycles of these models provide actions that are
lattice analogs of continuum $2\pi$-quantized topological $\theta$-terms (i.e.
with $\theta=0$ mod $2\pi$), and some of our analysis parallels continuum work
[114]. It has also been known that in order to obtain a nontrivial group
cohomology class for continuous groups, one must consider discontinuous
cocycles. However, the first explicit expression of these discontinuous
cocycles was only found very recently; the $U(1)$ models discovered in [32]
and suitably generalized here are the first examples. They follow considerable
work placing quantum Hall physics on lattices [23, 91], and join similar works
aiming to describe $2+1$d systems with [96] or without [9] gappable
boundaries. They are one of several approaches demonstrating ways around [49,
32] a related no-go theorem [55, 120] preventing Hall conductance on the
lattice, which does not hold in our case due to the infinite-dimensional on-
site $U(1)$ rotor Hilbert space. These discontinuous cocycles are the key to
understanding the SF-tMI transition, and also pave the way for studying more
complicated transitions involving nonabelian Lie groups.

###  6.3 Model Overview

We will be concerned with lattice systems of bosons in $2+1d$ at fixed average
integer density. As in the traditional $XY$ model, the field variables in our
theory are $U(1)$ rotors living on the sites $i$ of a three-dimensional
Euclidean spacetime lattice. We denote the field variables by $\phi_{i}$,
which correspond to the phase of the microscopic boson operator on site $i$
(as we are working at fixed average density, we are allowed to work solely
with the phase modes $\phi_{i}$). As before, we let the $\phi_{i}$ be periodic
under shifts by unity, rather than by $2\pi$. Hence a $U(1)$ angle $\phi_{i}$
will take values in $[0,1)$, while a group element is given by $g_{i}=e^{2\pi
i\phi_{i}}$. This will avoid numerous factors of $2\pi$ below, and one may
always convert back to the usual notation by replacing
$\phi\to\tilde{\phi}/2\pi$.

The trivial (i.e. non-topological) $XY$ model Lagrangian satisfies all of
these symmetries. It is

$S=-\frac{1}{g}\sum_{\braket{i,j}}\cos(2\pi(d\phi)_{ij})$ (6.18)

Here the sum runs over all links $\braket{i,j}$ in the $2+1$d lattice, and
$(d\phi)_{ij}=\phi_{i}-\phi_{j}$. The phase of the model is controlled by $g$.
As $g\to 0$, the strong ‘kinetic’ term sets $d\phi=0$ and so
$\phi=\text{const.}$, confining all vortices and resulting in the $U(1)$
symmetry breaking superfluid phase. As $g\to\infty$, the fluctuations of
$\phi$ overpower the kinetic suppression and vortices proliferate, destroying
long-range order and leading to the Mott insulating phase.

To get the tMIs from a lattice model, we simply add our topological term:

$S_{k}=-2\pi ik\int(d\phi-\lfloor d\phi\rceil)d(d\phi-\lfloor
d\phi\rceil)=2\pi ik\int(d\phi-\lfloor d\phi\rceil)d\lfloor d\phi\rceil$
(6.19)

as $d^{2}=0$. The full model combines the kinetic energy with the theta term:

$S_{g,k}[\phi]=-\frac{1}{g}\sum_{\text{links}}\cos(2\pi d\phi)+2\pi
ik\int(d\phi-\lfloor d\phi\rceil)d\lfloor d\phi\rceil$ (6.20)

and the complete partition function is:

$Z_{k,g}=\int D\phi e^{\frac{1}{g}\sum_{\text{links}}\cos 2\pi d\phi-2\pi
ik\int(d\phi-\lfloor d\phi\rceil)d\lfloor d\phi\rceil}$ (6.21)

where the measure $\int
D\phi=\left[\prod_{i}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\phi_{i}\right]$ has
already gauge-fixed the rotor redundancy (6.5).

We thus have a two parameter phase diagram, defined in terms of a rotor-
redundant model with global $U(1)$ and particle-hole symmetries. Our task in
the remainder of this section will be to understand this phase diagram.

Let us first fix $k=0$, in which case we have reduced to the traditional $XY$
model for the SF-MI transition. This model has been extremely well studied,
and it is known that there are two phases: for small $g$, the system
spontaneously breaks $U(1)$ symmetry, setting $d\phi=0$ and resulting in a
superfluid phase. For large $g$, the system is in a disordered symmetry
preserving phase, corresponding to a Mott insulator. The transition between
the two occurs at some non-universal $g_{c}$, with $g<g_{c}$ corresponding to
the superfluid phase and $g>g_{c}$ corresponding to the Mott Insulator phase.
Physically, we can understand this in terms of spin-zero bosons on a lattice
at unit filling. When the system is a superfluid, phase rigidity
$\phi\approx\text{const.}$ develops. On the other hand, when the system is a
Mott insulator, the phase of each Boson fluctuates independently, thereby
preserving $U(1)$ symmetry and destroying any long-range order.

Adding the topological term (and thereby breaking time-reversal symmetry)
turns the one-dimensional phase diagram into a two-dimensional one. In the
superfluid phase, vortices are suppressed, and we expect that the topological
term should be unimportant, leading to just a single SF phase. On the other
hand, vortices proliferate in a disordered phase, and so we expect the
topological term to be important in the disordered phase. As we will soon see,
the tMI fixed points are given by $g\to\infty$, $k\in\mathbb{Z}$. For generic
$k$ and large $g$, we will see that the system flows to attractive fixed
points at $k\to\lfloor k\rceil$ and $g\to\infty$ which are studied in Section
6.4. For large $g$ and half-odd-integer $k$ we have the tMI-tMI transition
discussed in Section 6.5.

In terms of this vortex current $\jmath_{v}$, we may rewrite the action as:

$S_{g,k}[\phi]=-\frac{1}{g}\sum_{\text{links}}\cos(2\pi d\phi)-2\pi
ik\int(d\phi-\lfloor d\phi\rceil)\star\jmath_{v},$ (6.22)

where we see that the effect of the topological term is to couple $\phi$ to
the vortex current. Moreover, the topological term actually gives the vortices
mutual statistics. Written in terms of $\jmath_{v}$, the topological term on a
closed manifold is

$S_{k}[\phi]=-2\pi ik\int\lfloor d\phi\rceil\star\jmath_{v}=2\pi
ik\int\jmath_{v}\cup\frac{d}{\square}\jmath_{v},$ (6.23)

where $\square$ is the Laplacian. That this term gives statistical phases to
braided vortex lines can be seen by noting that it is of the same form as the
response of a Chern-Simons gauge field coupled to a background current. More
directly, current conservation $d^{\dagger}\jmath_{v}=0$ allows us to write
$\jmath_{v}=\star dL$, where the Poincare dual of $dL$ marks the vortex lines
in spacetime. In terms of $L$,

$2\pi ik\int\jmath_{v}\cup\frac{d}{\square}\jmath_{v}=2\pi ik\int L\cup dL,$
(6.24)

which indeed gives $2\pi ik$ times the linking number of the vortex
worldlines. Hence we see that the topological term changes the statistics of
vortices, with vortex lines having mutual statistics of $e^{2\pi ik}$.

Given that the topological term charges the statistics of vortices to be
$e^{2\pi ik}$, we expect that the bulk dynamics are only sensitive to the
value of $k$ mod 1. This is indeed true, and this fact is responsible for
constraining the critical exponents of the SF-tMI transition to be equal to
those of the usual SF-MI transition. To see this, note that

$S_{g,k+1}[\phi]-S_{g,k}[\phi]=2\pi i\int\Big{(}d\phi d\lfloor
d\phi\rceil-\lfloor d\phi\rceil d\lfloor d\phi\rceil\Big{)}$ (6.25)

The first term is a surface term, while the second is an integer multiple of
$2\pi i$ and may be dropped. Hence, away from a boundary, the bulk dynamics at
$k$ and $k+1$ are identical. All correlation functions of local operators, and
hence all critical exponents, are identical (see Appendix B for a physical
interpretation). We will call the bulk invariance under $k\to k+1$ the “level
shift symmetry”.

Continuum topological $\theta$ models have similar level shift symmetries.
From a theoretical perspective, shifting the level in this model corresponds
to adding an SPT order. In this particular case, it is remarkable that doing
so changes the action by only a surface term and therefore does not change the
bulk dynamics of the system for any local operator and hence leads to
identical critical behavior of local operators at the SF-MI and SF-tMI
critical points. More generally, this may be true not just for the SF-tMI
transition but for generic symmetry breaking transitions. If two topological
phases have the same symmetry, differ only by the addition of an SPT or
invertible topological order, and undergo symmetry-breaking phase transitions
to the same symmetry-breaking phase, similar arguments may imply that the
critical behavior of the two models is identical — but more study will be
required in this area.

Returning to the model at hand, the level-shift symmetry under $k\to k+1$ has
important implications for the action of time-reversal symmetry. Note that
time-reversal reverses the orientation of the lattice and so exchanges the
representative of the top cohomology, effectively changing the sign on the
integral in the topological term $S_{k}[\phi]$, which consequently is odd
under time reversal. This implies that if $2k\in\mathbb{Z}$, then the bulk
dynamics are time-reversal symmetric.

Figure 6.2: _(color online)_ For $2k\in\mathbb{Z}$, the action of the $W_{k}$
operators cancels out the effects of time-reversal on the boundary of an open
manifold.

While the standard time reversal symmetry does not hold on the boundary, a
modified time-reversal symmetry does. Time-reversal symmetry holds for
$2k\in\mathbb{Z}$ because $S_{g,k}$ and $S_{g,-k}$ differ only by a surface
term:

$S_{g,k}-S_{g,-k}=2\pi i(2k)\int d(\phi d\lfloor d\phi\rceil)$ (6.26)

On a closed manifold, this surface term vanishes, and the theory at $k$ and
its time-reversed conjugate at $-k$ are identical. To extend this symmetry to
open manifolds, we need to cancel the leftover effects on the boundary (See
Figure 6.2). Let us consider a spacetime manifold $\mathcal{M}^{3}$ with a
spatial boundary $\mathcal{B}^{2}$. For $2k\in\mathbb{Z}$, we define the
operators $\hat{\mathcal{E}}_{2k}=e^{-2\pi i(2k)\int_{\mathcal{B}^{2}}\phi
d\lfloor d\phi\rceil}$, and let $\hat{K}$ be the complex conjugation operator.
We define the time-reversing operators:

$\mathcal{T}_{k}=\hat{K}\hat{\mathcal{E}}_{2k}$ (6.27)

For $k=0$, this reduces to the usual time-reversal operator. For other integer
or half-odd-integer $k$, the operator $\hat{\mathcal{E}}_{k}$ corresponds to
shifting the level by $2k$. This is precisely the the boundary change of the
bulk level under shifting $k$ by an integer. For each $k$ with
$2k\in\mathbb{Z}$, the model is invariant under the $\mathcal{T}_{k}$
operator.

Figure 6.3: _(color online.)_ Because the bulk dynamics are invariant under
$k\to k+1$, we may construct a sketch of the RG flow diagram. Along the
$g=\infty$ axis are the $\mathcal{T}_{k}$ fixed points, while at $g=g_{c}$ we
have the SF-(t)MI transitions.

Together, these observations lead to the RG flow diagram shown in Figure 6.3.
The line at $g_{c}$ separates the disordered phase from the symmetry breaking
phase. As there are few vortices in the symmetry breaking phase, we expect
that the topological term will be only be relevant in the MI phase. The novel
parts of this model are the disordered fixed points with modified time-
reversal symmetry with $2k\in\mathbb{Z}$ at $g\to\infty$. Later, we will see
that the integer fixed points with $k\in\mathbb{Z}$ describe gapped SPT phases
with Hall conductance, while the half-odd-integer fixed points describe the
topological transition between tMIs at $k$ and $k+1$.

In the next section, we will study the integer $k$ phases in detail. We will
discuss in detail how to see that they describe the topological Mott
insulators, and will describe the physical mechanisms behind the SF-tMI phase
transition and the tMI Hall conductance. In Section 6.5, we will study the
half-odd-integer phases, and argue that they are the transitions between the
tMIs.

###  6.4 The Integral Phases

We now consider the fixed point phases with $g\to\infty$ and $k\in\mathbb{Z}$.
In this case, the action reduces to:

$2\pi ik\int d\phi d\lfloor d\phi\rceil=2\pi ik\int d(\phi d\lfloor
d\phi\rceil)$ (6.28)

This is the action of the exactly solvable model in [32], where it is shown
that this action creates a ground state which has a nontrivial Chern number
$2k$ and Hall response of $2k\frac{e^{2}}{h}$. In what follows, we will see
that the ground state consists of a condensate of charged vortices, and that
this condensation leads to an SPT phase with Hall conductance.

Before proceeding to that ground state physics, it will be useful to examine
the model coupled to a background gauge field. In the SF phase111Up to one-cup
products, the minimally coupled gauged action and the one derived by ungauging
the lattice Chern-Simons action differ by a term:  $4\pi
ik\int(d\phi-A)(\lfloor dA\rceil+d\lfloor d\phi-A\rceil)=2\pi
ik\int(d\phi-A)\lfloor\star j\rceil$ (6.29) which vanishes in the SF phase,
since there $\lfloor\star j\rceil=0$. In the tMI phase, we take the action
derived by ungauging. Note that the ungauged action is only rotor redundant
for $k\in\mathbb{Z}$, which means that we do not know how to couple the model
to a background gauge field in the tMI phase for non-integer $k$., we may
minimally couple the action to get:

$S_{g,k}[\phi;A]=\frac{1}{g}\sum_{\braket{i,j}}\cos
2\pi((d\phi)_{ij}-A_{ij})\\\ +2\pi ik\int(d\phi-A-\lfloor
d\phi-A\rceil)d(A+\lfloor d\phi-A\rceil)$ (6.30)

where we have used the form of the action in eq. (6.19) to create the minimal
coupling. As with $\phi$, we have also taken $A$ to have a period of unity,
rather than $2\pi$. Correspondingly we have two rotor redundancies

$\displaystyle\phi_{i}\to\phi_{i}+n_{i}$ (6.31) $\displaystyle A_{ij}\to
A_{ij}+m_{ij}$ (6.32)

with $m,n\in\mathbb{Z}$, in addition to the usual gauge symmetry:

$\displaystyle\phi_{i}\to\phi_{i}+\varphi_{i}$ $\displaystyle A_{ij}\to
A_{ij}+\varphi_{i}-\varphi_{j}$ (6.33)

and global $U(1)$ symmetry. With $k\in\mathbb{Z}$, the topological part of the
action in the superfluid phase becomes:

$S_{k}[\phi;A]=+2\pi ik\int\Big{(}(d\phi-A)dA-\lfloor d\phi-A\rceil dA\\\
+(d\phi-A)d\lfloor d\phi-A\rceil\Big{)},$ (6.34)

where the term $\int\lfloor d\phi-A\rceil d\lfloor d\phi-A\rceil$ has been
dropped on account of it being an integer. With the background field, the
vortex current becomes:

$\star\jmath_{v}=d(d\phi-A-\lfloor d\phi-A\rceil)=-(dA+d\lfloor
d\phi-A\rceil)$ (6.35)

Evaluating (6.34) on a closed manifold and substituting in the vortex density,
the topological part of the action becomes:

$S_{k}[\phi;A]=2\pi
ik\int\big{(}AdA+A\cup(\star\jmath_{v})+(\star\jmath_{v})\cup A\big{)}$ (6.36)

Thus we see a second effect of the topological term: in addition to giving the
vortices mutual statistics at non-integer $k$, at integer $k$ it gives
vortices a charge of $2k$. In turn, we will see next that it is the
condensation of these charged vortices that gives rise to the tMI phase.

####  6.4.1 Ground State and SPT Order

A closer look at the action (6.28) above reveals an apparent paradox. The
action would seem to be trivial, as the Lagrangian $d\phi d\lfloor
d\phi\rceil=d(\phi d\lfloor d\phi\rceil)$ is a total derivative, and yet we
will see [32] that the state is a stable, gapped phase with Hall conductance.
The resolution to this reflects the SPT nature of the tMI phase. In
particular, the term which the Lagrangian is a total derivative of, viz. $\phi
d\lfloor d\phi\rceil$, is itself not locally $U(1)$ symmetric. On a closed
manifold, one may rewrite the Lagrangian as a total derivative of
$-d\phi[d\phi]$, but in this case the rotor redundancy does not hold locally,
and the model thus fails to even be well-defined. There is no way to write the
Lagrangian total derivative of a term which is locally both $U(1)$ symmetric
and rotor redundant.

Thus the apparent paradox is not a paradox at all. The Lagrangian is indeed a
total derivative, but it is not a derivative of anything $U(1)$ symmetric and
rotor redundant. If we break $U(1)$ symmetry, then the model is trivial, and
the action can be canceled by an allowed total derivative term. On the other
hand, so long as $U(1)$ symmetry is preserved (along with rotor redundancy,
which is always required) then the action is nontrivial.

At the same time, the model is easily solved because it is a total derivative.
In particular, the bulk dynamics are trivial: if $\partial\mathcal{M}^{3}=0$,
then the action vanishes and the partition function simply becomes unity.
However, the ground state of the model, exposed on a spatial boundary of
spacetime, contains nontrivial physics.

We can see this behavior directly from the ground state wavefunction, by which
we mean the wavefunction created on the boundary. Specifically, we obtain the
ground state on a $2$d manifold $\mathcal{B}^{2}$ by evaluating the action
(6.28) on a spacetime $\mathcal{M}^{3}$ with (spacelike) boundary
$\mathcal{B}^{2}=\partial\mathcal{M}^{3}$. Because the action is a surface
term, we can immediately write down the $g\rightarrow\infty$ ground state:

$\ket{\psi_{k}}=\int D\phi e^{2\pi ik\int_{\mathcal{B}^{2}}\phi d\lfloor
d\phi\rceil}\ket{\\{\phi\\}}$ (6.37)

where $\ket{\\{\phi\\}}=\otimes_{i}\ket{\phi_{i}}_{i}$ and again where we
gauge-fix the measure as $\int
D\phi=\prod_{i\in\mathcal{B}^{2}}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\phi_{i}$.

To understand the ground state further, let us first examine the $k=0$ case.
In that case, the $g\rightarrow\infty$ ground state is

$\ket{\psi_{k=0}}=\int D\phi\ket{\\{\phi\\}}=\otimes_{i}\ket{0}_{i}$ (6.38)

where $\ket{0}=\int_{-\frac{1}{2}}^{\frac{1}{2}}d\phi\ket{\phi}$ is the $U(1)$
symmetric state with eigenvalue $1$. This is the ground state of a trivial
Mott insulator. Recall that $\phi_{i}$ should be thought of as the phase of a
particle at site $i$. In the Mott insulating phase, the particle wavepackets
do not overlap and their phases fluctuate independently, as in eq. (6.38) This
should be compared to deep in the symmetry breaking phase, where phase
rigidity develops and where all the $\phi_{i}$ are equal in the ground state.
For nonzero $k$, the MI wavefunction is ‘twisted’ by the operator

$\mathcal{\hat{E}}_{k}=e^{2\pi ik\int\hat{\phi}d\lfloor
d\hat{\phi}\rceil}=e^{-2\pi ik\int\hat{\phi}\hat{\rho}_{v}}$ (6.39)

so that

$\ket{\psi_{k}}=\mathcal{\hat{E}}_{k}\ket{\psi_{k=0}}$ (6.40)

Thus $\mathcal{\hat{E}}_{k}$ is the operator that provides the SPT
entanglement to twist the trivial MI into a tMI. Note that on a closed
manifold, $\hat{\mathcal{E}}_{k}$ is $U(1)$ invariant, as under
$\phi\to\phi+\theta$,

$\hat{\mathcal{E}}_{k}\to e^{2\pi
ik\theta\int\hat{\rho}_{v}}\hat{\mathcal{E}}_{k}$ (6.41)

On a closed manifold, the total vortex number must vanish, and so
$\int\hat{\rho}_{v}=0$. This is however not necessarily true on a manifold
with boundary; hence the $U(1)$ symmetry is anomalous and breaks in the
presence of a boundary.

We can also see this behavior locally. Let us split the
$\hat{\mathcal{E}}_{k}$ into operators on each plaquette $\Delta$ that
transform the $k=0$ wavefunction into a nontrivial $k$ ground state, namely:

$M_{k}[\Delta]=e^{-2\pi ik\int_{\Delta}\phi\rho_{v}}$ (6.42)

so that

$\ket{\psi_{k}}=\prod_{\Delta}M_{k}[\Delta]\ket{\psi_{k=0}}=\hat{\mathcal{E}}_{k}\ket{\psi_{k=0}}$
(6.43)

On any plaquette $\Delta$, $M_{k}[\Delta]$ is not $U(1)$ invariant. However,
when multiplied together over a closed manifold, the $M_{k}$ are indeed $U(1)$
invariant, as

$\prod_{\Delta\in\mathcal{B}^{2}}M_{k}[\Delta]=\mathcal{\hat{E}}_{k}$ (6.44)

which, as discussed above, is $U(1)$ invariant. This behavior is how SPTs are
often characterized: if we relinquish $U(1)$ symmetry, then a finite set of
local unitary transformations (the $M_{k}[\Delta]$) suffices to transform the
trivial ground state $\ket{\psi_{k=0}}$ into a general $\ket{\psi_{k}}$. On
the other hand, if we require $U(1)$ symmetry, then only the nonlocal operator
$\mathcal{E}_{k}=\prod_{\Delta}M_{k}[\Delta]$ suffices.

####  6.4.2 Hall Conductance

Figure 6.4: _(color online)._ In the Laughlin thought experiment, twisting the
boundary conditions of an open cylinder by a flux quantum transfers charge
from one edge to another. We may understand the twisting boundary conditions
as the tunneling of a vortex from one edge to another. Because the vortices
carry charge $2k$, twisting boundary conditions transfers charge $2k$ from one
edge to another and the Hall conductance is $2k$.

The topological invariant for these phases is the Hall conductance. In Section
6.4.3, the Chern number of the integer ground states will be calculated
explicitly by considering the model with twisted boundary conditions, and we
will find it to be $2k$. Physically, we can understand this result in the
context of the Laughlin thought experiment. Consider the system on an open
cylinder, as shown in Fig. 6.4. We wish to twist the boundary conditions
around the open direction by unity. Let $\ket{\psi_{k},\theta=0}$ be the
ground state at level $k$ before twisting of boundary conditions and
$\ket{\psi_{k},\theta=2\pi}$ be the ground state after an adiabatic twisting
process. The two are related by the the creation of a vortex/anti-vortex pair
at opposite edges of the sample, with the resulting branch cut providing the
$2\pi$ phase jump:

$\ket{\psi_{k},\theta=2\pi}=\hat{V}_{\Delta,\Delta^{\prime}}\ket{\psi_{k},\theta=0},$
(6.45)

where $\hat{V}_{ij}$ creates a vortex at plaquette $\Delta$ and an anti-vortex
at plaquette $\Delta^{\prime}$. We can physically imagine this as dragging a
vortex across the cylinder in order to create the branch cut. However, we have
seen that the vortices have charge $2k$, and so dragging a vortex across the
sample transfers charge from one side of the sample to the other, resulting in
a hall conductance of $2k\frac{e^{2}}{h}$.

This Hall conductance is measurable through the tMI phase, and should persist
in the vicinity of the SF-tMI phase transition. Taking into account (6.36)
near a phase transition, an effective field theory becomes:

$S_{\text{eff}}=\alpha(\rho_{SF})\int(\Pi_{T}A)^{2}+2\pi ik\int
AdA+\beta(\rho_{SF})\int(\Pi_{T}A)d(\Pi_{T}A)+...$ (6.46)

where $\Pi_{T}$ is the transverse projector. The functions
$\alpha(\rho_{SF}),\beta(\rho_{SF})$ must vanish with $\rho_{SF}\to 0$, since
they may only arise from integrating out gapless modes. In particular, as the
transition is continuous, no term may suddenly cancel the transverse
conductance due to the Chern-Simons term.

We have now examined the tMI integer phases of the model, examined the the
entanglement structure of the tMIs, understood the SF-tMI transition as the
proliferation charged vortices, and explained how this leads to a Hall
conductance in the disordered phase and near the transition. In the next
section, we create a commuting projector model for the tMI phase.

####  6.4.3 Commuting Projector Model

Given that the bulk behavior of the path integral is trivial, we expect that
we should be able to create an exactly solvable Hamiltonian model to describe
the time-evolution. Furthermore, because the path integral defines a
wavefunction on any spatial boundary independently of the bulk dynamics, we
expect that this Hamiltonian model should be a commuting projector onto a
ground state. As we shall now see, both are true.

Commuting projector models have been a central tool for understanding the new
zoo of theories. Employed most famously by Kitaev [57] to provide an exactly
solvable model for the previously proposed 2+1d $Z_{2}$ topological order [83,
98] with emergent fermions and anyons, they now describe models for a wide
class of string-net topological order [61], recently unleashed a flurry of
research on fractons [21, 94], and continue to underlie our microscopic
understanding of exotic phases.

It is quite surprising then that no commuting projector model has been
discovered for gapped phases with non-zero Hall conductance. It was commonly
believed that none could exist, and recently a no-go theorem has been proposed
[56], ruling out a large class of potential theories with a finite Hilbert
space on each site.

We can use the spacetime formalism to construct a commuting projector
Hamiltonian on a triangular lattice of the sort shown in Fig. 6.5a. To do so,
we consider the time evolution of a single site, $\phi_{4}\to\phi_{5}$, while
preserving the orientation of lattice links as shown in fig 6.5b. Evaluating
the path integral on the complex shown yields the matrix elements for the
transition $\phi_{4}\to\phi_{5}$ as a function of the surrounding $\phi_{i}$:

$M_{\phi_{4}\to\phi_{5}}(\phi_{1},...,\phi_{8})=\exp\Big{\\{}2\pi ik\Big{[}\\\
\phi_{0}\Big{(}\lfloor\phi_{5}-\phi_{2}\rceil+\lfloor\phi_{3}-\phi_{5}\rceil+\lfloor\phi_{4}-\phi_{3}\rceil+\lfloor\phi_{2}-\phi_{4}\rceil\Big{)}\\\
+\phi_{2}\Big{(}\lfloor\phi_{5}-\phi_{6}\rceil+\lfloor\phi_{2}-\phi_{5}\rceil+\lfloor\phi_{4}-\phi_{2}\rceil+\lfloor\phi_{6}-\phi_{4}\rceil\Big{)}\\\
+\phi_{3}\Big{(}\lfloor\phi_{7}-\phi_{5}\rceil+\lfloor\phi_{4}-\phi_{7}\rceil+\lfloor\phi_{3}-\phi_{4}\rceil+\lfloor\phi_{5}-\phi_{3}\rceil\Big{)}\\\
+\phi_{5}\Big{(}\lfloor\phi_{6}-\phi_{5}\rceil+\lfloor\phi_{8}-\phi_{6}\rceil+\lfloor\phi_{7}-\phi_{8}\rceil+\lfloor\phi_{5}-\phi_{7}\rceil\Big{)}\\\
+\phi_{4}\Big{(}\lfloor\phi_{7}-\phi_{4}\rceil+\lfloor\phi_{8}-\phi_{7}\rceil+\lfloor\phi_{6}-\phi_{8}\rceil+\lfloor\phi_{4}-\phi_{6}\rceil\Big{)}\Big{]}\Big{\\}}$
(6.47)

We will interpret this transition amplitude as the matrix element for an
operator $\hat{M}_{4}$ acting on site-4. However, eq. (6.47) is somewhat
daunting. Let us set consider $\hat{M}_{4}$ as an operator acting only on the
Hilbert space on site-4. If $k=0$, then
$\bra{\phi_{4}^{\prime}}\hat{M}_{4}\ket{\phi_{4}}=1$, and $\hat{M}_{4}$ is
simply the projector onto the state with zero angular momentum in each $U(1)$.
For nonzero $k$, we may rewrite the transition amplitude as
$M_{\phi_{4}\to\phi_{5}}=\exp(2\pi i(f(\phi_{5})-f(\phi_{4}))$ (note that this
implies hermiticity) where $f(\phi)$ is a function defined in the appendix
that depends on $\phi$ and takes as parameters $\phi_{1}...\phi_{8}$, but not
$\phi_{4}$ or $\phi_{5}$. This implies that, up to an overall phase, the
$\hat{M}_{i}$ act as

$\hat{M}\ket{\phi}\propto\int d\phi e^{2\pi if(\phi)}\ket{\phi}$ (6.48)

We see then that these projectors may be thought of as ‘twisting’ the zero
angular momentum state by a phase function $f(\phi)$ which depends on the
surrounding values of $\phi_{i}$. The phase itself is determined by the
cocycle of the action in (6.4). We may construct $\hat{M}_{i}$ for an entire
lattice.

Figure 6.5: _(Color Online)._ We construct a commuting projector model for the
lattice in $(a)$ by evaluating the spacetime path integral for the complex in
$(b)$ and turning the amplitude into operators. Because the path integral
contains only a surface term, we can show that $(c)$ the matrix is hermitian,
(d) the operators commute, and $(e)$ they are projectors.

The $\hat{M}_{i}$ inherit a number of remarkable properties from the fact that
the $2+1$d path integral action contains only a surface term. First, they
mutually commute: consider the three spacetime complexes in figure 6.5c, which
addresses the only nontrivial case of two adjacent $\hat{M}_{i}$. The two
colored complexes correspond to time evolving either the blue site followed by
the red or the red followed by the blue, respectively. Because the action
contains only a surface term, and the surfaces are identical, it assigns the
same amplitude to both cases. Back in the Hamiltonian picture, this implies
that $[\hat{M}_{i},\hat{M}_{j}]=0$.

For the same reason, the $\hat{M}_{i}$ are projectors. The fundamental
mechanism is illustrated in Fig. 6.5d, where we see the effect of time-
evolving twice. In the language of eq. 6.47, this is the expression
$M_{\phi_{5}\to\phi_{9}}M_{\phi_{4}\to\phi_{5}}$; in the Hamiltonian picture
this is $\hat{M}_{i}^{2}$. However, because the path integral action depends
only on the values of $\phi$ on the surface, we could equally well drop site
$5$ and its associated links; the amplitude will not change. This implies that
$\hat{M}_{i}^{2}=\hat{M}_{i}$.

The aforementioned hermiticity of the $\hat{M}_{i}$ is also due to this
reason, as reversing the orientation of the link $\braket{4,5}$ in Fig. 6.5c
changes the amplitude by complex conjugation.

There is a simple way to view $\hat{M}_{i}$. We note that the groundstate wave
function (6.37) on a spatial lattice $\mathcal{M}^{2}$ is a phase factor,
which defines the unitary operator

$\displaystyle\hat{\mathcal{E}}_{k}=e^{-2\pi
i\sum_{I,J}k\int_{\mathcal{M}^{2}}\hat{\phi}_{I}^{\mathbb{R}/\mathbb{Z}}\hat{\rho}^{v}_{J}},$
(6.49)

where $\hat{\rho}^{v}_{J}=-d\lfloor
d\hat{\phi}^{\mathbb{R}/\mathbb{Z}}_{J}\rceil$ is the vortex density operator
defined previously. Let $\hat{P}_{i}$ be the projector onto the state of zero
momenta at site $i$, with matrix elements
$\braket{\phi^{\prime}_{I}(i)}{\hat{P}_{i}}{\phi_{I}(i)}=1$. Then we find:

$\hat{M}_{i}=\hat{\mathcal{E}}_{k}\hat{P}_{i}\hat{\mathcal{E}}_{k}^{\dagger}$
(6.50)

Thus the $\hat{M}_{i}$ operators are essentially projectors onto zero angular
momentum states, twisted by the phases of the unitary operators
$\hat{\mathcal{E}}_{k}$. On a spatial lattice without boundary, the
groundstate wave function (6.37) has $U(1)$ symmetry, i.e. is invariant under
$\phi_{i}\to\phi_{i}+\theta+n_{i}$. Thus the $\hat{M}_{i}$ also have $U(1)$
symmetry. Note also that the commuting, projector, and Hermitian properties of
$\hat{M}_{i}$ immediately follow from this description. The ground state
wavefunction is given by (6.37) and can be written as
$\ket{\psi_{k}}=\hat{\mathcal{U}}_{k}\ket{\psi_{0}}$, where
$\ket{\psi_{0}}=\otimes_{i}\ket{\ell=0}_{i}$ is the product state of the zero-
angular-momentum states $\ket{\ell=0}$ on each site $i$.

Using the mutually commuting projectors $\hat{M}_{i}$, we can define a
Hamiltonian

$H=-g\sum_{i}\hat{M}_{i}$ (6.51)

to obtain a system with ground state $\ket{\psi}$ from eq. (6.37) and gap $g$.

With a model producing the ground state eq. (6.37) now in hand, we return to
the Hall conductance. We have already argued from the path integral that
coupling the system to a background gauge field leads to the expected Chern-
Simons response function, but here we appeal directly to the wavefunction of
the system on a torus to confirm the hall conductance.

Figure 6.6: _(Color Online)_. $(a)$ The commuting projectors act on a hexagon
to define a ground state. $(b)$ To calculate the Chern number of the ground
state wavefunction over the holonomy torus, we twist the boundary conditions
by $\theta_{x}n$ and $\theta_{y}m$ in the $x$ and $y$ directions,
respectively.

Consider the wavefunction on the lattice shown in Fig. 6.6. We twist the
boundary conditions by $\theta_{x}n$, $\theta_{y}m$, so that
$\phi_{x=0,y}=\phi_{x=L_{x},y}-\theta_{x}n$ and
$\phi_{x,y=0}=\phi_{x,y=L_{y}}-\theta_{y}m^{I}$, where the integral vectors
$n_{m}$ allow us to describe both direct ($n=m$) and mixed hall responses.
Denote the ground state with boundary conditions $\theta_{x},\theta_{y}$ by
$\ket{\theta_{x},\theta_{y}}$. We are interested in the phase accumulated when
we twist the boundary conditions by an integer, sending
$\theta_{x}\to\theta_{x}+\ell_{x},\theta_{y}\to\theta_{y}+\ell_{y}$ with
$\ell_{x},\ell_{y}\in\mathbb{Z}$.

Consider first the plaquettes marked in red. The contribution of these
plaquettes to the path integral is of the form:

$\exp\Big{\\{}-2\pi
ik\Big{[}(\phi_{x=L,y}+n\theta_{x}-\phi_{x=L_{x}-1,y})\lfloor\phi_{x=L_{x},y+1}-\phi_{x=L_{x},y}\rceil\\\
-\Big{(}(\phi_{x=L_{x}-1,y+1}-\phi_{x=L_{x}-1,y})\lfloor\phi_{x=L_{x},y+1}+n\theta_{x}-\phi_{x=L_{x}-1,y+1}\rceil\Big{)}\Big{]}\Big{\\}}$
(6.52)

Incrementing $\theta_{x}$ by $\ell_{x}$ changes the first term by an integer,
while the second term changes by
$(\phi_{x=L_{x}-1,y+1}-\phi_{x=L_{x}-1,y})\ell_{x}$, and so the multiplicative
change on the path integral is:

$e^{2\pi ik(\phi_{x=L_{x}-1,y+1}-\phi_{x=L_{x}-1,y})n\ell_{x}}$ (6.53)

Similarly, the plaquettes marked in blue change by:

$e^{-2\pi ik(\phi_{x+1,y=L_{y}-1}-\phi_{x,y=L_{y}-1})m\ell_{y}}$ (6.54)

On the purple plaquettes at the corner, the contribution to the path integral
is:

$\exp\Big{\\{}-2\pi
ik\Big{[}\Big{(}(\phi_{x=L_{x},y=L_{y}-1}+n\theta_{x}-\phi_{x=L_{x}-,y=L_{y}-1})\lfloor\phi_{x=L_{x},y=L_{y}}+m\theta_{y}-\phi_{x=L_{x},y=L_{y}-1}\rceil\Big{)}\\\
-\Big{(}(\phi_{x=L_{x}-1,y=L_{y}}+m\theta_{y}-\phi_{x=L_{x}-1,y=L_{y}-1})\lfloor\phi_{x=L_{x},y=L_{y}}+n\theta_{x}-\phi_{x=L_{x}-1,y=L_{y}}\rceil\Big{)}\Big{]}\Big{\\}}$
(6.55)

which will change by:

$\exp\Big{\\{}-2\pi
ik\Big{[}(\phi_{x=L_{x},y=L_{y}-1}+n\theta_{x}-\phi_{x=L_{x}-,y=L_{y}-1})m\ell_{y}\\\
-(\phi_{x=L_{x}-1,y=L_{y}}+m\theta_{y}-\phi_{x=L_{x}-1,y=L_{y}-1})n\ell_{x}\Big{]}\Big{\\}}$
(6.56)

Combining the change on the red, blue, and purple plaquettes, the overall
change to the wavefunction is:

$e^{-2\pi
ik\left[n\theta_{x}m\ell_{y}-m\theta_{y}n\ell_{x}-n\ell_{x}\int_{\gamma_{1}}d\phi+m\ell_{Y}\int_{\gamma_{2}}d\phi\right]}$
(6.57)

where $\gamma_{1},\gamma_{2}$ are the red and blue loops in Fig. 6.6,
respectively. As that $\gamma_{1},\gamma_{2}$ are closed, the sums along those
curves vanish. Hence the wavefunction transforms as:

$\ket{\theta_{x}+\ell_{x},\theta_{y}+\ell_{y}}=e^{-2\pi
ik\left[n\theta_{x}m\ell_{y}-m\theta_{y}n\ell_{x}\right]}\ket{\theta_{x},\theta_{y}}$
(6.58)

To make sense of this, apply the gauge transformation:

$\ket{\theta_{x},\theta_{y}}\to e^{-2\pi
iknm\theta_{x}\theta_{y}}\ket{\theta_{x},\theta_{y}}\ $ (6.59)

In this gauge, one may replace the phase in eq. (6.58) by:

$e^{-2\pi ik(nm+nm)\theta_{x}\ell_{y}}$ (6.60)

We should recognize this as the boundary conditions for a particle on a torus
with flux:

$2kmn$ (6.61)

We see then that our Hamiltonian system has the (mixed) Hall conductance
$n\cdot k\cdot m$, in agreement with the Chern-Simons response function
derived from the spacetime path integral. In the case of $\kappa=1$ with
$n=m=1$, this becomes a system with integer hall conductance $K=2k$ wich is an
even integer.

We must also understand how this model evades the no-go theorem. The infinite
dimensional on-site Hilbert space of the rotors, combined with an action that
is not a continuous function of the field variables, is what allows this model
to exist. The discontinuous action is critical for commuting projectors; if it
were continuous, then one could truncate the Hilbert space to low-angular
momentum modes and render the on-site Hilbert space finite while retaining the
full $U^{\kappa}(1)$ symmetry and commuting-projector property, hence running
afoul of the no-go theorem [56]. Conversely, the no-go theorem assumes that
the ground state wavefunction is a finite Laurent polynomial in
$e^{i\theta_{x}},e^{i\theta_{y}}$, an assumption which is violated in our
model (See the details of the Chern number calculation in the Appendix for an
example). This commuting projector model represents a fixed-point theory for
$U^{\kappa}(1)$ SPT phases with nonzero Hall conductance; it may be that any
such fixed-point theory requires an infinite on-site Hilbert space.

###  6.5 Topological Transitions at Half-Integer $k$

We have now established that the phases at integral $k$ are gapped SPT states.
We also understand the $XY$ transition caused by increasing $g$ along $k=0$
and, by level-shift symmetry, for all integer $k$. In this section, we examine
a different phase transition, that obtained by changing $k$ along the line
$g=\infty$. We argue that that this must be a topological transition, and map
it to a model of superconducting fermions. Whether the resulting order
parameter is complex or real, and thus the superconductor breaks time-reversal
symmetry or is gapless, corresponds to whether the transition is first-order
or second order. We are, however, unable to discern which of the two
occurs222The nature of the transition may depend on lattice details..

Taking $g\to\infty$ and setting $k=m-\frac{1}{2}$, with $m\in\mathbb{Z}$, the
action (6.20) becomes:

$S_{g\to\infty,k=m-\frac{1}{2}}[\phi]=\pi
i(2m-1)\int_{\mathcal{M}^{3}}(d\phi-\lfloor d\phi\rceil)d\lfloor d\phi\rceil$
(6.62)

This action is invariant under the time-reversing operator
$\mathcal{T}_{m+\frac{1}{2}}$ defined previously. If
$\partial\mathcal{M}=\emptyset$, then the action reduces to:

$S_{g\to\infty,k=m-\frac{1}{2}}[\phi]=-\pi
i(2m-1)\int_{\mathcal{M}^{3}}\lfloor d\phi\rceil d\lfloor d\phi\rceil.$ (6.63)

As a first measure, note that we may rewrite the bulk action as the boundary
of a time-reversal symmetric term in one higher dimension. For a four-manifold
$\mathcal{N}^{4}$ such that $\partial\mathcal{N}^{4}=\mathcal{M}^{3}$,

$-\pi i(2m-1)\int_{\mathcal{M}^{3}}\lfloor d\phi\rceil d\lfloor
d\phi\rceil=-\pi i(2m-1)\int_{\mathcal{N}^{4}}d\lfloor d\phi\rceil d\lfloor
d\phi\rceil$ (6.64)

Note that this action, after exponentiation, is invariant under the usual
time-reversal. For the our purposes, the critical fact is that time-reversal
forces $m$ to be an integer in the four dimensional theory, and hence the four
dimensional theory is a $U(1)\times\mathcal{T}$ SPT. Our three-dimensional
theory, as the boundary of an SPT, must break the $U(1)$ or $\mathcal{T}$
symmetry, be gapless, or develop topological order.

We do not believe that these theories develop topological order. On the one
hand, the integer $k$ phases are stable $U(1)$ SPTs; if the half-odd-integer
phases were stable topological orders, then there would be to be another class
of critical $k$ points, somewhere between the integer and half-odd-integer $k$
(recall Figure 6.3). On the other hand, in Section 6.6, we show a numerical RG
calculation that suggests that $k$ flows to the nearest integer, with the
points where $k$ is a half-odd-integer separating different phases. We find no
evidence of a stable phase at half-odd-integer $k$, nor do we find any fixed
points besides the integer and half-odd-integer $k$.

Hence we are left with the options of the half-odd-integer phases breaking
$U(1)\times\mathcal{T}$ symmetry (in which case the transition between the
integer $k$ $U(1)$ SPTs would be first order) or gaplessness (in which case
the transition is second order). We now argue that when $k$ is a half-odd-
integer, the model admits a description in terms of emergent fermions, which
order in a way dependent on whether or not the transition is continuous.

####  6.5.1 Emergent Fermions

Let us work with the bulk action (6.63). Recall that that the vortex current
is given by $\jmath_{v}=-\star d\lfloor d\phi\rceil$. Using this, we can
rewrite this action as

$-\pi i(2m-1)\int_{\mathcal{M}^{3}}\jmath_{v}\frac{d}{\Box}\jmath_{v}$ (6.65)

Recall that in $2+1$d, the $1$-cochain $\lfloor d\phi\rceil$ is dual to a two
dimensional surface which describes the branch cut emanating from a vortex.
The $2$-cochain $d\lfloor d\phi\rceil$ is dual to a one-dimensional vortex
line. The topological $\lfloor d\phi\rceil d\lfloor d\phi\rceil$ term counts
the intersections of the vortex lines with the branch cut walls, i.e. twice
the linking number (See figures 6.7a, 6.7b).

Figure 6.7: _(color online)._ $(a)$ The topological term at half integer $k$
assigns a factor of $(-1)$ for each intersection (red $x$) of a vortex line
(black) with a branch cut wall (blue). $(b)$ Disconnected lines must have two
such intersections and so may not have anyonic statistics. $(c)$ Rotor
redundancy $\phi\to\phi+n$ locally moves the branch cut walls, and always
create intersections in pairs in the bulk. $(d)$ in the presence of a boundary
(red), moving a domain wall can create a single intersection, and rotor
redundancy is preserved by a charge at the end of the vortex line (magenta
star) implemented by the $d\phi$ term. A single vortex line describing the
exchange of two particles has a single intersection, as does the
(topologically equivalent) case of $(d)$ a $2\pi$ rotation of a single
particle. Hence the vortices are fermions.

In the bulk, this action (6.63) is invariant under the rotor redundancy
because sending $\phi\to\phi+n$ corresponds to a moving a branch cut wall or
creating/annihilating them in pairs. As shown in figure 6.7, in the bulk, a
local move of a branch cut wall must generate intersections in pairs, hence
the action is invariant.

We can further understand the character of these vortices by examining their
braiding. First, consider two disconnected, contractible lines. As shown in
figure 6.7b, these two lines must intersect in two places. In fact $\lfloor
d\phi\rceil d\lfloor d\phi\rceil$ is twice their linking number; because the
action assigns a factor of $(-1)$ to each intersection, these two lines have
no braiding phase, and the vortices do not have anyonic statistics.

However, a more complex situation arises when we consider just a single vortex
line. In figure 6.7e, we see a line representing the creation of two vortex-
antivortex pairs, the exchange of two vortices, and the annihilation of the
swapped pairs. Figure 6.7f is topologically equivalent, and shows the case of
a $2\pi$ rotation of a vortex. In either case, a single intersection is
created, resulting in a factor of $(-1)$. Hence the vortices are fermions.

Now that we have identified the vortices as fermions, it may seem that the
task is complete. The content of the action (6.63) is exhausted, and a simple
guess for the emergent behavior of the system may be completely free fermions.
However, this misses a crucial factor. There is a Jacobian we neglected
passing from the rotor field $\phi$ to the branch cut and vortex fields
$\lfloor d\phi\rceil$ and $\jmath=-\star d\lfloor d\phi\rceil$. This Jacobian
leaves a statistical imprint on the theory which will generate interactions
for the fermions.

First, recall that we have defined the path integral measure as

$\int D\phi=\left[\prod_{i}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\phi_{i}\right]$
(6.66)

As each $\phi_{i}\in[-\frac{1}{2},\frac{1}{2})$, the branch cut field $\lfloor
d\phi\rceil_{ij}=\lfloor\phi_{i}-\phi_{j}\rceil$ may take values in
$\\{-1,0,1\\}$. However, $\lfloor d\phi\rceil$ is not distributed uniformly
over these three values. If we let $\phi_{i}$ and $\phi_{j}$ be chosen
randomly, the probability $p$ that $\lfloor d\phi\rceil$ takes on a given
value is

$\lfloor d\phi\rceil=\begin{cases}-1&p=\nicefrac{{1}}{{8}}\\\
~{}0&p=\nicefrac{{3}}{{4}}\\\ ~{}1&p=\nicefrac{{1}}{{8}}\end{cases}$ (6.67)

This immediately leads to a probability for the observation of a vortex. On a
given plaquette, the vortex number is given by $\rho_{v}=-d\lfloor
d\phi\rceil$, and on a triangular lattice must take values in
$\\{-3,-2,...,3\\}$. One can extrapolate the probabilities from those for
$\lfloor d\phi\rceil$ to get:

$\rho_{v}=-d\lfloor d\phi\rceil=\begin{cases}-3&p=\nicefrac{{1}}{{512}}\\\
-2&p=\nicefrac{{18}}{{512}}\\\ -1&p=\nicefrac{{111}}{{512}}\\\
~{}0&p=\nicefrac{{252}}{{512}}\\\ ~{}1&p=\nicefrac{{111}}{{512}}\\\
~{}2&p=\nicefrac{{18}}{{512}}\\\ ~{}3&p=\nicefrac{{1}}{{512}}\\\ \end{cases}$
(6.68)

We see that when the $\phi$ are completely random, although having no vortex
is the most likely single option, it is still less than $50\%$ probability.
More than half of the time there is a vortex on a plaquette.

The details of the induced interactions are lattice dependent. However, the
generic behavior can be described simply as follows. For random $\phi$,
approximately one quarter of the lattice plaquettes will host a vortex, while
another quarter will host an anti-vortex. Vortices and anti-vortices on
adjacent sites have a weak attractive interaction and may annihilate, while
both vortices and anti-vortices may hop to adjacent plaquettes with real
amplitude.

Recalling that the vortices and anti-vortices are statistically fermions for
half-odd-integer $k$, we may convert this rough theory into an effective
fermion theory. We consider a honeycomb lattice (the dual of the triangular
lattice) with real hopping coefficients. We denote vortices as spin-up
fermions, anti-vortices as spin down, and impose a strong on-site repulsive
interaction to prevent overlap. In this context, conservation of vortex
current density corresponds to $S^{z}$ symmetry, while vortex-antivortex
annihilation results in superconducting pairing terms. A Hamiltonian for this
model is accordingly

$H=\sum_{\braket{i,j}}\Big{(}t\sum_{\sigma}(c^{\dagger}_{i,\sigma}c_{j\sigma}+c^{\dagger}_{j,\sigma}c_{i,\sigma})+\Delta(c^{\dagger}_{i\uparrow}c^{\dagger}{j\downarrow}+c_{j\downarrow}c_{i\uparrow})-gc^{\dagger}_{i,\uparrow}c_{i,\uparrow}c^{\dagger}_{j,\downarrow}c_{j,\downarrow}\Big{)}\\\
-\mu\sum_{i}\sum_{\sigma}c^{\dagger}_{i,\sigma}c_{i,\sigma}+U\sum_{i}c^{\dagger}_{i,\uparrow}c_{i,\uparrow}c^{\dagger}_{i,\downarrow}c_{i,\downarrow}$
(6.69)

Here the $t$ term represents the fermion hopping; $\Delta$ the vortex-
antivortex annihilation; $g>0$ the nearest neighbor attractive interactions;
$U\to\infty$ the strong on-site repulsion; and $\mu$ the chemical potential.
The parameters should be tuned to roughly reproduce the fermion density,
hopping, pairing, and attractive interactions from the statistical analysis.

Within this framework, we can again see the question of symmetry breaking or
gaplessness (or, equivalently, of a first-order or second-order transition) in
the topological transition between $k$ and $k+1$ phases. The question is of
the form of the pairing wavefunction
$\psi_{ij}=\braket{c^{\dagger}_{i,\uparrow}c^{\dagger}_{j,\downarrow}}$. The
strong on-site repulsion will suppress s-wave pairing, and we are left with
p-wave or d-wave pairing. These must either break time-reversal symmetry or be
gapless; whichever the dynamics favor will determine the nature of the
topological transition. However, determining which case occurs (which may
depend on the lattice), would have to be determined in a future work.

###  6.6 Lattice Renormalization Group

Figure 6.8: _(Color Online)_. _(a)_ We integrate out $\phi_{0}$ in the simplex
shown to perform a simple Lattice RG calculation. _(b), (c)_ : Phase of the
quantity $\Phi_{c}$ calculated in eq. (6.75) (blue) and $\Phi_{k}$ from eq.
(6.73) (orange) for _(b)_ $k=.8$ and _(c)_ $k=1.2$. _(d), (e)_ : Magnitude of
$\Phi_{c}$ for _(d)_ $k=.8$ and _(e)_ $k=1.2$.

In this section we calculate the effects of integrating out a rotor on a given
site, which reveals the flow of $k$ to integer fixed points. Taking
$g\to\infty$ recall that the action in this limit is:

$S=2\pi ik\int d\phi d\lfloor d\phi\rceil.$ (6.70)

To evaluate this, we place it on a tetrahedral lattice shown in Figure 6.8a
and perform a course-graning. Consider integrating out point $0$ in Figure
6.8a. We wish to calculate:

$\int d\phi_{0}\exp\Bigg{\\{}2\pi
ik\Bigg{[}\left(\phi_{1}-\phi_{0}-\lfloor\phi_{1}-\phi_{0}\rceil\right)\Big{[}e-\left(\lfloor\phi_{2}-\phi_{1}\rceil+\lfloor\phi_{3}-\phi_{2}\rceil-\lfloor\phi_{3}-\phi_{1}\rceil\right)\\\
+\left(\lfloor\phi_{2}-\phi_{1}\rceil+\lfloor\phi_{4}-\phi_{2}\rceil-\lfloor\phi_{4}-\phi_{1}\rceil\right)-\left(\lfloor\phi_{3}-\phi_{1}\rceil+\lfloor\phi_{4}-\phi_{3}\rceil-\lfloor\phi_{4}-\phi_{1}\rceil\right)\Big{]}\\\
+\left(\phi_{2}-\phi_{0}-\lfloor\phi_{2}-\phi_{0}\rceil\right)\left[\lfloor\phi_{3}-\phi_{2}\rceil+\lfloor\phi_{4}-\phi_{5}\rceil-\lfloor\phi_{4}-\phi_{2}\rceil\right]\Bigg{]}\Bigg{\\}}$
(6.71)

which can be rewritten as:

$\int d\phi_{0}\exp\Bigg{\\{}-2\pi
ik\Bigg{[}(\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{0}\rceil-\lfloor\phi_{0}-\phi_{1}\rceil)\left(\lfloor\phi_{3}-\phi_{2}\rceil+\lfloor\phi_{4}-\phi_{3}\rceil-\phi_{4-\phi_{2}}\right)\Bigg{]}\Bigg{\\}}$
(6.72)

Or, setting
$\rho_{234}=-\lfloor\phi_{3}-\phi_{2}\rceil+\lfloor\phi_{4}-\phi_{3}\rceil-\lfloor\phi_{4}-\phi_{2}\rceil$,

$\displaystyle\underbrace{e^{-2\pi
ik(\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{1}\rceil)\rho_{v}}}_{\Phi_{k}}\underbrace{\int
d\phi_{0}e^{2\pi
ik(\lfloor\phi_{2}-\phi_{0}\rceil+\lfloor\phi_{0}-\phi_{1}\rceil-\lfloor\phi_{2}-\phi_{1}\rceil)\rho_{234}}}_{\Phi_{c}}$
(6.73)

Now the amplitude has decomposed into the expected phase $\Phi_{k}$, times a
correction term $\Phi_{c}$. Turning our attention to the integral in
$\Phi_{c}$, we may use $U(1)$ symmetry to set $\phi_{2}=0$. Then The integral
becomes:

$\Phi_{c}=\int d\phi_{0}e^{2\pi
ik(\lfloor\phi_{0}-\phi_{1}\rceil+\lfloor\phi_{1}\rceil-\lfloor\phi_{0}\rceil)}=1-|\phi_{1}-\lfloor\phi_{1}\rceil|+|\phi_{1}-\lfloor\phi_{1}\rceil|e^{2\pi
ik\rho_{234}\text{sgn}(\phi_{1}-\lfloor\phi_{1}\rceil)}$ (6.74)

Restoring $U(1)$ symmetry, this is:

$\Phi_{c}=1-|\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{1}\rceil|+|\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{1}\rceil|e^{-2\pi
ik\rho_{234}\text{sgn}(\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{1}\rceil)}$
(6.75)

To understand this expression, fix $\rho_{234}=1$. The phase of both
$\Phi_{k}$ and $\Phi_{c}$ functions of $\phi_{2}-\phi_{1}$ are plotted in
figures 6.8b and 6.8c for $k=.8$ and $k=1.2$, respectively. For $k=.8$, the
phase of $\Phi_{k}$ and $\Phi_{c}$ vary jointly, and $\Phi_{c}$ is pushing the
effective $k$, back towards $k=1$. For $k=1.2$, they vary oppositely, and
$\Phi_{c}$ is pushing the effective $k$ down, again back towards $k=1$.

This trend continues across the spectrum. For $k$ neither integer nor half
integer, $\Phi_{c}$ acts to push the effective $k$ back towards the nearest
integer. For $k\in\mathbb{Z}$, $\Phi_{c}=1$, and there is no correction,
reflecting the fact that the integral theories are time-reversal-invariant,
fixed point theories. For $k\in\mathbb{Z}+\frac{1}{2}$ $\Phi_{c}$ is real,
reflecting the time-reversal invariance of the half-odd-integer theories.

A further question arises because $|\Phi_{c}|<1$ when $k\notin\mathbb{Z}$ and
$|\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{1}\rceil|>0$. Figures 6.8d and 6.8e
show $|\Phi_{c}|$ for $k=.8$ and $k=.5$, respectively. For $k=.8$, the damping
effect is somewhat small, while for $k=.5$ it is extreme, and $|\Phi_{c}|=0$
for $|\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{1}\rceil|=\pm.5$. For general
$k$ and small $|\phi_{2}-\phi_{1}-\lfloor\phi_{2}-\phi_{1}\rceil|$, this
implies that there should be a logarithmic term in the action. Note that this
only applies when $\rho_{234}\neq 0$; when $\rho_{234}=0$, $\Phi_{c}=1$ in all
cases.

###  6.7 Discussion

In this Chapter, we have used a new discontinuous cocycle model to shed light
on the transition between a superfluid and a topological Mott Insulator.
Without a background gauge field, the correlation functions of all local
operators are identical at and away from the SF-tMI and SF-MI transitions,
while introducing a background gauge field leads to different Hall responses
for MI and tMI. This immediately implies that all critical exponents of
correlation functions of $e^{i2\pi\phi_{i}}$ and its conjugate momentum must
be the same in the SF-MI and SF-tMI phases, which allows the considerable
numerical results of the XY model to be applied to the SF-tMI transition.

We may also ask what other topological phases may be described by a similar
framework. An obvious $U(1)\times\mathcal{T}$ phase in $1+1$d is one coming
from the action:

$S=i\pi k\int_{\mathcal{M}^{2}}d\lfloor d\phi\rceil$ (6.76)

On the other hand, $d+1$ dimensional, with even $d>0$, generalizations of our
model are given by:

$S=2\pi ik\int_{\mathcal{M}^{d+1}}(d\phi-\lfloor d\phi\rceil)(d\lfloor
d\phi\rceil)^{\frac{d}{2}}$ (6.77)

where the exponent is taken using the cup product. If, as in $2+1$ dimensions,
these are SPT states for integral $k$, they would exhaust all the $U(1)$ SPTs
predicted by group cohomology [28]. Many more possibilities arise with
multiple $U(1)$ fields $\phi^{I}$. One could also consider extending this
formalism to SPTs with nonabelian symmetry. Could the Haldane phase be
described by a similar exactly solvable discontinuous path integral? In every
case, the existence of a model of the sort we have described here would carry
the same implication of level-shift symmetry: in the absence of a gauge field,
all correlations of local operators would be the same in the symmetry-breaking
to topological transition as in a similar symmetry-breaking to trivial state
transition. The SPT order would then by diagnosed by the response to an
applied gauge field.

Even just the model we have described has further applications. Because the
edge theory may be treated alone, this proposes a solution to the $U(1)$
“chiral fermion problem:” The edge theory must describe pairs of counter-
propagating modes with differing charges, so that the total chiral central
charge of the edge theory vanishes, while the modes carry a chiral
representation of $U(1)$. This is the subject of the next chapter.

##  Chapter 7 A Chiral Lattice Field Theory in $1+1$d

We can now create the promised simple chiral lattice field theory. All the
ingredients for this theory are actually in the last chapter, where we
developed an exactly solvable lattice model for a $U(1)$ SPT. Here we extract
the bosonic chiral edge theory and examine its properties. This $1+1$d chiral
boson edge theory also paves the way for simulation of more complicated chiral
QFTs, including in higher dimensions and for nonabelian symmetries. The
simplest extension of this theory is a chiral fermion theory that may be
obtained by introducing a spin structure, which solves the Chiral Fermion
Problem.

All of the properties we need in the SPT theory from the previous chapter
arise because that theory is a fixed-point theory. In turn, the key to writing
a fixed-point theory on the lattice was first discovered in the group
cohomology formalism [26] but not implemented until the works described in
this thesis: in order to capture topological actions, we must allow for
actions which are not continuous functions of the field variables. Once we
have a fixed-point topological action in $2+1$d, we have a gapped bulk and a
gapless $1+1$d edge which decouple exactly, since the penetration of the
gapless mode into the bulk must be zero at the fixed point. This is what
enables us to write down the $1+1d$ model and study its properties.

In retrospect, that we should require physical quantities that are not
continuous functions of the field variables in fixed-point theories follows
from a simple argument. Consider, as we will shortly, a lattice QFT consisting
of $U(1)$ variables on the sites of a lattice. The space of field
configurations is $U(1)^{\text{n}_{\text{sites}}}$. Consider a function $\rho$
of the field variables which indicates the vortex number on each plaquette. In
a fixed-point theory, the output of $\rho$ should be an integer for each
plaquette, i.e. an element of $\mathbb{Z}^{\text{n}_{\text{plaquettes}}}$. As
a function from a connected manifold, $U(1)^{\text{n}_{\text{sites}}}$, to a
discrete space, $\mathbb{Z}^{\text{n}_{\text{plaquettes}}}$, $\rho$ must be
either discontinuous or constant, and constant would be useless. Hence when
describing vortices in a fixed point theory, we should allow for discontinuous
physical quantities; this holds for topological defects in many theories.

###  7.1 Lattice Model

Figure 7.1: The model we study works on any lattice with a simplicial
structure, but we will write down explicit expressions on this square lattice.

The starting point for our $1+1$d chiral lattice model is actually the $2+1$d
action from the last section, with $k\in\mathbb{Z}$ and $g\to\infty$. In this
Chapter, we will be focused on its $1+1$d boundary, so we change notation and
let $\mathcal{N}^{3}$ be the $2+1$d lattice, with
$\mathcal{M}^{2}=\partial\mathcal{N}^{3}$ its $1+1$d boundary. Hence the
action is:

$S_{k}[\phi]=-2\pi ik\int_{\mathcal{N}^{3}}(d\phi-\lfloor d\phi\rceil)\cup
d(d\phi-\lfloor d\phi\rceil)\\\ =2\pi ik\int_{\mathcal{N}^{3}}d\phi\cup
d\lfloor d\phi\rceil=-2\pi ik\int_{\mathcal{N}^{3}}d\phi\cup\rho_{v}$ (7.1)

where used the fact that $d^{2}=0$ and that the action appears as $e^{iS}$ to
simplify the action and introduced $\rho_{v}\equiv-d\lfloor d\phi\rceil$.
Recall that we have a gauge redundancy which ensures that the theory is
genuinely a function of $U(1)$ rotors:

$\phi\to\phi+n$ (7.2)

The full path integral is:

$Z=\int D\phi e^{iS_{k}[\phi]}\hskip 28.45274pt\int
D\phi=\prod_{i}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\phi_{i}$ (7.3)

where the integral measure is gauge-fixed under (7.2).

The most important aspect of the action (7.1) for our present purposes is that
it is a surface term which vanishes on a closed manifold. Hence we may obtain
a theory solely on $\mathcal{M}^{2}=\partial\mathcal{N}^{3}$:

$S_{k}=2\pi ik\int_{\mathcal{M}^{2}}\phi d\lfloor d\phi\rceil=-2\pi
ik\int\phi\rho_{v}$ (7.4)

This is the model which we wish to present and, together with its gauged
version we will see later, is the chiral lattice field theory. It is very
simple. In indices on a square lattice, the action is:

$S_{k}[\phi]=-2\pi
ik\sum_{i\in\mathcal{M}^{2}}\phi_{i}\Big{(}\lfloor\phi_{i}-\phi_{i+x}\rceil+\lfloor\phi_{i+x}-\phi_{i+x+y}\rceil\\\
-\lfloor\phi_{i+y}-\phi_{i+x+y}\rceil-\lfloor\phi_{i}-\phi_{i+y}\rceil\Big{)}$
(7.5)

where $i$ sums over the sites of the lattice. As before, under the redundancy
(7.2), the action is invariant, since $\rho_{v}$ is invariant and $n\rho_{v}$
is integer-valued and so does not affect the exponential.

###  7.2 Anomalies

This action has the unusual, anomalous $U(1)$ symmetry discussed in the
previous chapter. Under $\phi\to\phi+\theta$, with $\theta$ a global constant,
the action transforms as:

$S_{k}[\phi]\to S_{k}[\phi]-2\pi ik\theta\int_{\mathcal{M}^{2}}\rho_{v}$ (7.6)

If $\mathcal{M}^{2}$ is closed, then the total vortex number is zero, as
$\int_{\mathcal{M}^{2}}\rho_{v}=-\int_{\mathcal{M}^{2}}d\lfloor d\phi\rceil=0$
by summation by parts, and so the theory is $U(1)$ symmetric. If
$\mathcal{M}^{2}$ has a boundary $\mathcal{B}^{1}=\partial\mathcal{M}^{2}$,
then the action is not $U(1)$ invariant. The breaking of $U(1)$ symmetry in
the presence of a boundary is one characterization of the $U(1)$ anomaly (we
saw this behavior in the context of the ground state wavefunction in the
previous chapter). It should be considered as a consequence of the familiar
expression of the anomaly in a background gauge field (which we investigate
next), as a background gauge field could impose an electric potential that
would create a boundary. The resolution, of course, is to recall that the
$1+1$d action (7.4) should be considered as the boundary of the $2+1$d system
(7.1). Hence
$\mathcal{B}^{1}=\partial\mathcal{M}^{2}=\partial^{2}\mathcal{N}^{3}=\emptyset$,
as boundaries do not have boundaries.

We can see the anomaly in a different light by coupling to a background gauge
field. The resulting $2+1$d action was given in Chapter 6 and is:

$S_{k}[\phi;A]=-2\pi ik\int_{\mathcal{N}^{3}}\Big{\\{}(d\phi-A)(dA-\lfloor
dA\rceil)\\\ -\lfloor dA\rceil(d\phi-A)-d\Big{[}(d\phi-A)(d\phi-A-\lfloor
d\phi-A\rceil)\Big{]}\Big{\\}}$ (7.7)

As before, we quantize $A$ to have cycles in unity and this action has three
gauge redundancies. Two are of the same “rotor redundancy” form:

$\displaystyle\phi_{i}\to\phi_{i}+n_{i}$ (7.8) $\displaystyle A_{ij}\to
A_{ij}+m_{ij}$ (7.9)

for $n_{i},m_{ij}\in\mathbb{Z}$. The third is the typical gauge invariance:

$\phi\to\phi+\theta\hskip 28.45274ptA\to A+d\theta$ (7.10)

for $\theta$ an $\mathbb{R}/\mathbb{Z}$-valued field. We also assume $A$ to be
weak, in the sense that $dA-\lfloor dA\rceil\approx 0$. This implies that
$d\lfloor dA\rceil=0$, i.e. that field configurations are free of monopoles.

Now we separate (7.7) into boundary terms and bulk terms. We can rewrite the
action as:

$2\pi ik\int_{\mathcal{N}^{3}}\Big{\\{}-A(dA-\lfloor dA\rceil)+\lfloor
dA\rceil{A}-d(A(A-\lfloor A\rceil))\Big{\\}}\\\ +2\pi
ik\int_{\mathcal{M}^{2}}\Big{\\{}\phi(dA-\lfloor dA\rceil)-\lfloor
dA\rceil\phi\\\ -(d\phi-A)(d\phi-A-\lfloor d\phi-A\rceil)+A(A-\lfloor
A\rceil)\Big{\\}}$ (7.11)

We have now split the action into ‘bulk terms’ consisting only of $A$ and
‘boundary terms’ which contain all the $\phi$. Each of them are separately
invariant under $\phi\to\phi+n$, and we have added and subtracted terms to
ensure that they are invariant under $A\to A+m$. They are not separately
invariant under the gauge symmetry (7.10).

The boundary integral is the proper gauged action for the edge mode, i.e. our
$1+1$d chiral boson model. Specifically, it is:

$S=2\pi ik\int_{\mathcal{M}^{2}}\Big{\\{}\phi(dA-\lfloor dA\rceil)-\lfloor
dA\rceil\phi\\\ -(d\phi-A)(d\phi-A-\lfloor d\phi-A\rceil)+A(A-\lfloor
A\rceil)\Big{\\}}$ (7.12)

Or, writing it on a square lattice,

$S=2\pi
ik\sum_{i}\phi_{i}\Big{(}A_{i,i+x}+A_{i+x,i+x+y}-A_{i+x,i+x+y}-A_{i+y,i+x+y}\\\
-\lfloor A_{i,i+x}+A_{i+x,i+x+y}-A_{i+x,i+x+y}-A_{i+y,i+x+y}\rceil\Big{)}\\\
-\lfloor
A_{i,i+x}+A_{i+x,i+x+y}-A_{i+x,i+x+y}-A_{i+y,i+x+y}\rceil\phi_{i+x+y}\\\
-(\phi_{i}-\phi_{i+x}-A_{i,i+x})(\phi_{i+x}-\phi_{i+x+y}-A_{i+x,i+x+y}-\lfloor\phi_{i+x}-\phi_{i+x+y}-A_{i+x,i+x+y}\rceil)\\\
+(\phi_{i}-\phi_{i+y}-A_{i,i+xy})(\phi_{i+y}-\phi_{i+x+y}-A_{i+y,i+x+y}-\lfloor\phi_{i+y}-\phi_{i+x+y}-A_{i+y,i+x+y}\rceil)\\\
+A_{i,i+x}(A_{i+x,i+x+y}-\lfloor
A_{i+x,i+x+y}\rceil)-A_{i,i+y}(A_{i+y,i+x+y}-\lfloor A_{i+y,i+x+y}\rceil)$
(7.13)

Together with the ungauged $(A=0)$ model (7.4), eq. (7.12) describes the
chiral lattice field theory. One can check that the action is invariant under
both of the symmetries (7.8) and (7.9). However, it is not invariant under
(7.10), i.e. $\phi\to\phi+\theta$, $A\to A+d\theta$. The anomaly structure is
in general complicated, but takes a simple form when we set $d\theta=0$. In
that case, the action changes by a term:

$-2\pi(2k)i\theta\int_{M^{2}}\lfloor dA\rceil$ (7.14)

Now, $\int_{\mathcal{M}^{2}}(dA-\lfloor
dA\rceil)=-\int_{\mathcal{M}^{2}}\lfloor dA\rceil$ is the total flux of the
gauge field over $\mathcal{M}^{2}$, and so this is precisely
$2\pi(2k)\int\theta F$, i.e. the anomaly required by the Hall conductance. In
the context of the Laughlin thought experiment, we twist boundary conditions
by $2\pi$, which sets $-\int\lfloor dA\rceil=1$. In that case, eq. (7.14)
tells us that charge $2k$ has appeared in the edge theory, i.e. has been added
to the edge theory from the bulk. We should recognize this as the expected
Adler-Bell-Jackiw anomaly, i.e. as a lattice, discrete generalization of $4\pi
ik\int\theta dA$. As is usual, this failure of gauge invariance is cancelled
by an equal and opposite term from the bulk terms in eq. (7.11).

We have now examined the $1+1$d lattice model in detail and seen the $U(1)$
anomaly through both symmetry breaking in the presence of a boundary and
through direct coupling to a background gauge field. Now we write down a
continuum model for the edge theory and explain how it creates a chiral
representation of $U(1)$ and a nonzero quantized Hall conductance.

###  7.3 Continuum Theory

The ungauged action (7.4) describes a bosonic field $\phi$ coupled to its
vortices. To develop a continuum description, we describe the vortices using a
hydrodynamic approach:

$\rho_{v}\sim\partial_{x}\partial_{t}\theta(x,t)$ (7.15)

We consider an action which ensures that that vortices shift $\phi$ by $2\pi$
i.e. that $[\phi(x),\partial_{x}\theta]=2\pi i$:

$S\sim 2\pi i\int\phi\partial_{x}\partial_{t}\theta$ (7.16)

where $\phi$ has charge $1$ and, as discussed in the previous chapter, the
vortex field $\theta$ has charge $2k$. We define the composite fields
$\phi_{R}=\frac{1}{2}(\phi+\theta)$, $\phi_{L}=\frac{1}{2}(\phi-\theta)$ to
get:

$2\pi
i\int(\phi_{R}\partial_{x}\partial_{t}\phi_{R}-\phi_{L}\partial_{x}\partial_{t}\phi_{L})$
(7.17)

Thus the chiral model consists of a right-moving mode $\phi_{R}$ and a left
moving mode $\phi_{L}$. As there are equal numbers of left and right moving
modes, there is no gravitational anomaly or, equivalently, thermal Hall
conductance.

On the other hand, there is a $U(1)$ anomaly which arises because $\phi_{L}$
and $\phi_{R}$ have differing charges. Denote the change of a field $\varphi$
by $C[\varphi]$, so that $C[\phi]=1$. We have seen that vortices have charge
$2k$ and so $C[\theta]=2k$. Hence the anomaly has coefficient:

$C[\phi_{R}]-C[\phi_{L}]=C[\theta]=2k$ (7.18)

This is consistent with the Hall conductance of the bulk system, which is
$2k\frac{e^{2}}{h}$ [32].

We have written down a boson theory with chiral $U(1)$ symmetry in $1+1$d by
extracting the edge theory from an exactly solvable $2+1$d chiral model. We
then demonstrated the $U(1)$ anomaly by both inspection of the ungauged theory
and by explicitly coupling in a background gauge field and calculating the
variation of the edge action. Finally, we wrote down a continuum theory for
our model and showed that it carries chiral $U(1)$ charge as expected. They
key to our model, and indeed the main lesson from this thesis, is that
discontinuous actions are essential when considering exactly solvable fixed-
point theories on lattices, and that with these we can build exactly solvable
theories that we previously believed to be impossible.

##  Chapter 8 Outlook

This thesis has been a deep exploration of chiral phases, and their attendant
chiral field theories, on the lattice. The key to our work was the
relationship between a renormalization-group class of Quantum Field Theories
and phases of matter. In essence, we defined our field theories as the low-
energy effective theories (on the edge or in the bulk) of phases of matter,
and then used techniques from condensed matter physics to understand the field
theories. Here we review our work, lay out immediate next steps, and speculate
on the sources of future progress in the field.

###  8.1 Review

We began with a numerical realization of the ‘Mirror Fermion’ approach. There
we went beyond previous Abelian models to create an $SU(2)$ chiral gauge
theory. We saw the gap opened, and numerically demonstrated the preservation
of symmetry, but were faced with significant technical challenges: we could
not sufficiently address how to perform the integral over Higgs
configurations, nor could we reduce the model in such a way that the Higgs
field was unnecessary. We won the battle of opening a gap in the fermion
spectrum, but the Higgs problem means that this victory was a Pyrrhic one.

Following the difficulties in defining $1+1$d chiral edge modes, we changed
our approach in two key ways: we pivoted to the bulk theory in $2+1$
dimensions, and also restricted to Abelian Chern-Simons theories. This led to
a significant success: we were able to define a lattice model for any Abelian
topological order, the low-energy behavior of which we could reliably
determine semi-classically. Moreover, this theory was one of local rotor
degrees of freedom, a feature which no one had been able to reliably
incorporate into Chern-Simons theories. The added appearance of exact
$1$-symmetries then provided extremely strong evidence for the validity of the
model.

The next step was the mathematically obvious one. Given a model of topological
order in terms of a gauge field, we ungauged it to get an SPT model. However,
the utility of the model went far beyond that. On the heels of a number of
results arguing that no commuting projector model could host Hall conductance,
our model provided exactly that. Moreover, we later connected it to the theory
of the $XY$ transition between a superfluid and a topological Mott insulator,
and understood both the transition and the Hall conductance as a condensation
of charged vortices.

Of course, the ungauged model had another critical application. Because it is
a zero-correlation-length, fixed-point model, the edge theory could be exactly
decoupled from the bulk. This gave us an exact, $1+1$d $U(1)$ chiral lattice
field theory. Moreover, we could directly compute its anomaly and confirm that
it had the expected behavior. This is remarkable: we retreated from gapless
chiral edge theories, went to the bulk theory and, after several intermediate
steps, found a perfect chiral edge theory handed back to us. Moreover, that
formulation paves the way for far more complicated chiral field theories,
which we elaborate on in the next section.

###  8.2 Further Lattice Field Theories

The discontinuous cocycle models we have discovered are an extraordinary tool
with which we can create new exactly solvable models of SPT phases and their
chiral gapless edge theories. An obvious $U(1)\times\mathcal{T}$ phase in
$1+1$d is one coming from the action:

$S=i\pi k\int_{\mathcal{M}^{2}}d\lfloor d\phi\rceil$ (8.1)

On the other hand, $d+1$ dimensional, with even $d>0$, generalizations of our
model are given by:

$S=2\pi ik\int_{\mathcal{M}^{d+1}}(d\phi-\lfloor d\phi\rceil)(d\lfloor
d\phi\rceil)^{\frac{d}{2}}$ (8.2)

where the exponent is taken using the cup product. If, as in $2+1$ dimensions,
these are SPT states for integral $k$, they would exhaust all the $U(1)$ SPTs
predicted by group cohomology [28]. This leads to a $4+1$d SPT action:

$S=2\pi ik\int_{\mathcal{M}^{4+1}}d\phi d\lfloor d\phi\rceil d\lfloor
d\phi\rceil$ (8.3)

which would host a $3+1$d $U(1)$ chiral edge theory on the boundary
$\mathcal{B}^{3+1}=\partial\mathcal{M}^{4+1}$:

$S=2\pi ik\int_{\mathcal{B}^{3+1}}\phi d\lfloor d\phi\rceil d\lfloor
d\phi\rceil$ (8.4)

This theory thus defines chiral fermions with $U(1)$ symmetry propagating in
spacetime just like ours. Many more possibilities arise with multiple $U(1)$
fields $\phi^{I}$, and it is probably that this formalism would capture all of
them. We are also preparing a discontinuous model which captures all the time-
reversal SPTs.

One could also consider extending this formalism to SPTs with non-Abelian
symmetry. The most obviously relevant theories are the $SU(N)$ SPTs in
$D=3+4n$ spacetime dimensions.

In the particular case of $SU(2)$ SPTs, we can use the fact that $SU(2)\simeq
S^{3}$ to establish how these theories should be constructed. Let $g_{i}$ be
the zero-cochain lattice variable. In that case, any tetrahedron with elements
$g_{i}$ of $S^{3}$ defined at its corners defines a tetrahedron on the surface
of $S^{3}$. Let the three-cochain $\Omega[g]$ be the volume of that
tetrahedron in $S^{3}$ (this is easier to thing about in one lower dimension,
i.e. with elements of $S^{2}$ on the corner of triangles, and $\Omega$ a solid
angle). At the same time, any four-simplex (i.e. one higher dimension than a
tetrahedron) with elements of $S^{3}$ at its corners defines a wrapping number
from its surface to $S^{3}$. Let $W[g]$ be the four-cochain which is that
wrapping number. Now, these $SU(2)$ theories would have the form:

$S\propto ik\int_{\mathcal{M}^{3+4n}}\Omega\cup W^{n}$ (8.5)

where $k$ is the level and $n$ is adjusted for the appropriate dimension.
Moreover, one can see that $\Omega$ must be exact up to an integral part, and
$W$ is always exact, so we once again have a surface term formalism and
commuting projector model. All of this follows as an immediate generalization
of our work here; the difficulty lies in actually writing down formulae for
$\Omega$ and $W$.

While the discontinuous cochain formalism describes SPT phases and their
chiral edges which suffer gauge anomalies, we are also interested in
topological orders and their gravitational anomalies. The formalism we
described in Chapter 5 provides a description of all Abelian topological
orders which is semi-classically solvable. However, we would like to develop
an exactly solvable formalism. Doing so is more difficult, and probably cannot
be described quite the same sense as the SPT phases: we have seen that the
surface term formalism leads to a commuting projector model. However, it is
believed that commuting projector models cannot have thermal Hall conductance,
which corresponds to gravitational anomalies, because thermal Hall conductance
would lead to energy flow through the system. The change in energy at a site
$i$ is given by:

$\dot{E}_{i}=\partial_{t}\braket{\hat{H}_{i}}=-i\left[\hat{H},\hat{H}_{i}\right]=-i\sum_{j}\left[\hat{H}_{j},\hat{H}_{i}\right]=0$
(8.6)

Here $\hat{H}_{i}$ are the commuting projectors and
$\hat{H}=\sum_{i}\hat{H}_{i}$ is the Hamiltonian. Hence commuting projector
models cannot host gravitational anomalies and so we do not expect one for
chiral topological orders. Of course, that has not stopped us from some trials
towards exactly solved models of topologically ordered phases, and we describe
one incomplete approach in Appendix C.

###  8.3 What’s Next

In Chapter 3, we briefly mentioned that the meeting of condensed matter
physics and high-energy theory had triggered a golden age. This continues
apace, and has been the source of enormous progress through the past decade.
There are new materials, field theories, and phases of matter being discovered
almost daily.

A new player has also arrived: the exploding fields of quantum computing and
quantum information (QC/QI). New hardware, materials, and algorithms have
unleashed an entire new field of science and an entire new industry. Progress
in quantum algorithms is proceeding extremely fast, and the first
demonstrations of quantum supremacy have been achieved in the past few years.
Huge teams of quantum engineers are being hired, while a quantum computing
startup recently received a $\$2$B valuation. The future is extraordinarily
bright.

I will be working in quantum computing after graduation, so I am biased, but I
do genuinely believe that what is next in physics will come from the
intersection of QC/QI and other fields. Indeed, the entire group cohomology
formalism arose when a QI student under Ike Chuang began working with Xiao-
Gang. Now I make the reverse journey, as I begin a fellowship working with Ike
and I look forward to what’s next.

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##  Appendix A Cochains and Cohomology

Let us first set some notation. We consider a three-dimensional simplicial
complex $M$, which we take to contain $0$-simplices (vertices), $1$-simplices
(links), $2$-simplices (faces), and $3$-simplices (faces). In this thesis, we
consider our spacetime lattice to be a simplicial complex with matter fields
living on the vertices and gauge fields living on the links. Moreover, we
assume that these fields take values in an Abelian group.

The matter fields, denoted $g_{i}$, form a map from the $0$-simplices of $M$
(or more formally, form a map from the space of $0$-chains of $M$) to the
target space. This map is called a $0$-cochain, as it defines a linear map
from the free abelian group on the $0$-simplices of $M$ to the target space.
Similarly, a gauge field $a_{ij}$ living on the links of the lattice defines a
$1$-cochain. One can continue this, with $n$-cochains defining maps from the
$n$-simplices to the target space.

To see the explicit action of an $n$-cochain, let us label simplices by their
vertices, so that an $n$-simplex is given by $[v_{0},...,v_{n}]$. Then an
$n$-cochain $a$ assigns a target space element $a([v_{0},...,v_{n}])$ to any
$n$-simplex. Furthermore, this map is multilinear over a formal sum of
$n$-simplices with coefficients in $\mathbb{Z}$. For example, let