Abelian combinatorial gauge symmetry

Hongji Yu1, Dmitry Green1, 2, Andrei E. Ruckenstein1, Claudio Chamon1

1 Physics Department, Boston University, Boston, MA, 02215, USA

2 AppliedTQC.com, ResearchPULSE LLC, New York, NY 10065, USA

## Abstract

Combinatorial gauge symmetry is a principle that allows us to construct
lattice gauge theories with two key and distinguishing properties: a) only
one- and two-body interactions are needed; and b) the symmetry is exact rather
than emergent in an effective or perturbative limit. The ground state exhibits
topological order for a range of parameters. This paper is a generalization of
the construction to _any_ finite Abelian group. In addition to the general
mathematical construction, we present a physical implementation in
superconducting wire arrays, which offers a route to the experimental
realization of lattice gauge theories with static Hamiltonians.

###### Contents

  1. 1 Introduction
  2. 2 A motivating example
  3. 3 Summary of results
  4. 4 Combinatorial gauge symmetry
  5. 5 Examples of combinatorial gauge symmetric Hamiltonians
    1. 5.1 Uniform gauge transformations and reduction of number of matter spins
    2. 5.2 Tuning the spectrum of a CGS interaction
    3. 5.3 Gauge theories with more complex structure
  6. 6 Implementations
  7. 7 Summary and outlook
  8. A Lattice gauge theory conventions
  9. B Further examples of CGS coupling matrices
  10. C Breaking spurious degeneracies for $\mathbb{Z}_{2}$ CGS theories
  11. D Group of local gauge symmetries of the $A$-type operators in Haah’s code

##  1 Introduction

Physically realizing topological ordered [1] gapped quantum spin liquids in
the laboratory remains a problem at the frontier of our knowledge in both
quantum materials and quantum information sciences. Control over such states
would constitute a major step forward in building stable quantum memories,
material simulation, and ultimately scalable quantum computation. Over the
past decades, major theoretical advances have been made by formulating models
within the framework of lattice gauge theories [2], for example the well-known
$\mathbb{Z}_{2}$ toric code [3]. Recently, $\mathbb{Z}_{2}$ quantum spin
liquids have been demonstrated by preparing a state like the quantum ground
state of the static system and its elementary excitations as non-equilibrium
states in quantum simulators [4, 5]. However, as these states are prepared
dynamically, they do not exist in the absence of a an external drive or
modulation, and thus they do not possess the topological stability of the
ground states of the static Hamiltonians.

One challenge with physical realization of quantum spin liquids in the static
setting is that most well-understood theoretical models require multi-body
(e.g., 4- or 6-spin) interactions. (A notable exception is Kitaev’s honeycomb
model [6].) While it is possible to design multi-spin interactions as
effective couplings arising from physical, two-body interactions, these
constructions tend to be either perturbative (they are exact in a classical
limit) and break down in the presence of quantum dynamics [7, 8, 9, 10, 11,
12, 13, 14, 15, 16], or require capabilities beyond those achievable in real
materials or in programmable quantum devices [17, 18, 19, 20, 21].

A set of proposals utilizing the “combinatorial gauge symmetry” (CGS)
construction plots a potential alternative path to overcoming these obstacles
[22, 23]. CGS is a framework for constructing Hamiltonians out of realistic
two-body interactions where the gauge symmetry is _exact and non-perturbative_
for the full range of parameters. As such there is hope that a suitable range
of parameters can be found where the topological phase is sufficiently gapped
to be observable and stable. Earlier work has centered on special cases of
small Abelian groups ($\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$). By generalizing
the framework to all Abelian groups we further enlarge the scope of possible
implementations and parameters where one can look for such states. The
mathematical construction developed in this paper to handle the generalization
is interesting in its own right.

From a mathematical point of view, central to the construction is a
classification of two-body interactions that can be written as matrices
$W_{ij}$ that couple two quantum variables indexed by $i$ and $j$ (e.g., spins
or Josephson phases). The allowed sets of interactions $W$ must possess
several properties in order to satisfy the requirements of a lattice gauge
theory exactly. This paper is a systematic exposition of the general
requirements.

As a high level preview, the gauge symmetry arises from automorphisms of the
form $W=L^{\top}WR$ where the matrices $L$ and $R$ must be monomials in order
to preserve the commutation relations of the underlying quantum variables.
Further, because of the geometry of the lattice configuration, $R$ will be
required to be diagonal, but $L$ will not. Effectively, the problem of
classifying the allowed $W$ matrices will reduce to a study of how group
actions relate to permutations of invariant sets.

The outline of the paper is as follows. First we introduce in Sec. 2 a simple
example of a model with a combinatorial gauge symmetry corresponding to a
$\mathbb{Z}_{2}$ gauge theory on the triangular lattice to illustrate key
elements of our construction. We summarize the main results of this paper in
Sec. 3. In Sec. 4 we develop in detail the general construction for any
Abelian gauge group, as well as modifications to the models in specific cases,
and then provide illustrations for these in Sec.5. Finally, we discuss
experimental implementations in Sec. 6.

##  2 A motivating example

In this section we consider a $\mathbb{Z}_{2}$ gauge theory on a triangular
lattice with only two-body interactions. Before that, it should be made clear
what kind of gauge symmetry such a theory possesses. In an ordinary
$\mathbb{Z}_{2}$ gauge theory, e.g., the classical toric code model, the
degrees of freedom are located on the links of the lattice and the Hamiltonian
is

$H=-\lambda_{B}\sum_{p}B_{p}$ (1)

where the plaquette operator $B_{p}$ is the product of $\sigma^{z}$ operators
acting on the links bounding the plaquette $p$. In the full toric code model,
this Hamiltonian has gauge symmetries generated by star operators, that is,
products of $\sigma^{x}$ operators along a path through the dual lattice that
forms a domain wall around a subset of vertices on the lattice. At the level
of individual degrees of freedom, such symmetries act by the transformation
$\sigma^{z}\mapsto\sigma^{x}\sigma^{z}\sigma^{x}=-\sigma^{z}$. Thus, whenever
the support of a plaquette operator intersects with that of a gauge
transformation, the degrees of freedom involved will be affected by the gauge
transformation, but the plaquette operator will be invariant overall. (See
Fig. LABEL:sub@fig:triangular-pure-gauge-symmetry.) Since in the Abelian case
such a symmetry operation is always realized by a phase factor, we may
represent its effect on a plaquette term by a diagonal matrix multiplying the
vector of operators contained in this term. For example, if we collect the
$z$-components of the three spins around a star onto a vector
${\boldsymbol{\sigma}^{z}}^{\top}\equiv(\sigma_{1}^{z},\sigma_{2}^{z},\sigma_{3}^{z})^{\top}$,
then the action of the symmetry $\mathcal{R}$111In this paper we will use
different styles of letters to distinguish the abstract gauge symmetry and the
matrices that implement these symmetries by left multiplication on a vector of
gauge or matter operators. Calligraphic letters will denote the former, Roman
letters the latter. that flips the first two spins is

$\begin{pmatrix}\sigma_{1}^{z}\\\ \sigma_{2}^{z}\\\
\sigma_{3}^{z}\end{pmatrix}\xmapsto{\mathcal{R}}\begin{pmatrix}-\sigma_{1}^{z}\\\
-\sigma_{2}^{z}\\\
\sigma_{3}^{z}\end{pmatrix}=R\begin{pmatrix}\sigma_{1}^{z}\\\
\sigma_{2}^{z}\\\ \sigma_{3}^{z}\end{pmatrix}$ (2)

where $R$ is the diagonal matrix

$R=\begin{pmatrix}-\\\ &-\\\ &&+\end{pmatrix}\ .$ (3)

Under the action of $\mathcal{R}$ the product
$\sigma_{1}^{z}\sigma_{2}^{z}\sigma_{3}^{z}$ is invariant,
$(-\sigma_{1}^{z})(-\sigma_{2}^{z})\sigma_{3}^{z}=\sigma_{1}^{z}\sigma_{2}^{z}\sigma_{3}^{z}$,
because the phase factors introduced by the two spin flips multiply to one.

(a)  (b)

(c)

Figure 1: Illustration of the $\mathbb{Z}_{2}$ gauge symmetry on a triangular
lattice. (a) A gauge transformation flips certain spins, but plaquette terms
remain invariant overall, since there is always an even number of spin flips.
(b) A two-body operator satisfying the combinatorial gauge symmetry
corresponding to the gauge symmetry depicted in (a) requires four matter
degrees of freedom. (c) Under the action of the gauge symmetry in (a), the
combinatorial gauge symmetric operator in (b) stays invariant by flipping the
gauge spins just as in the pure gauge theory, while also permuting the matter
spins.

Having understood the action of the gauge symmetry on individual degrees of
freedom and on gauge invariant operators, we now construct a two-body
Hamiltonian that possesses the same gauge symmetry. To this end, in addition
to the gauge spinoperators $\sigma^{z}$ as before, we introduce matter spin
operators $\mu^{z}$ associated with each plaquette term in the pure gauge
theory. More precisely, in this example we introduce four matter spins
$\mu^{z}_{a},\,a=1,2,3,4$, which we also collect into a vector
${\boldsymbol{\mu}^{z}}^{\top}\equiv(\mu_{1}^{z},\mu_{2}^{z},\mu_{3}^{z},\mu_{4}^{z})^{\top}$.
We would like the Hamiltonian to (i) have only one- and two-body interactions
(here of the $ZZ$ type) and (ii) possess a $\mathbb{Z}_{2}$ gauge symmetry
when acting on the gauge spins. Condition (i) is consistent with an
interaction term of the form
$-J\sum_{a=1}^{4}\sum_{i=1}^{3}W_{ai}\,\mu_{a}^{z}\sigma_{i}^{z}$. Using the
vector notation, this interaction can be written compactly as
$(\boldsymbol{\mu}^{z})^{\top}W\boldsymbol{\sigma}^{z}$. As for condition
(ii), when the gauge symmetry acts on the $\sigma_{i}$’s via
$\boldsymbol{\sigma}^{z}\mapsto R\boldsymbol{\sigma}^{z}$, the interaction is
transformed into $(\boldsymbol{\mu}^{z})^{\top}WR\boldsymbol{\sigma}^{z}$,
which is equivalent to multiplying the coupling matrix $W$ on the right by
$R$. If there exists a matrix $L$ that we can multiply on the left of $W$ so
that $L^{\top}WR=W$, then $L$ is the action of the gauge symmetry on
$\boldsymbol{\mu}^{z}$, so that the transformation
$(\boldsymbol{\sigma}^{z},\boldsymbol{\mu}^{z})\mapsto(R\boldsymbol{\sigma}^{z},L\boldsymbol{\mu}^{z})$
leaves the system invariant. The $L$ and $R$ matrices arise from gauge
transformations, which ultimately come from conjugating spins operators by
some symmetry operator, so the transformation represented by $L$ should
preserve the commutation relations between the spin operators. Therefore, we
restrict our considerations only to $L$’s that permute the spins or flip
$\mu^{z}$’s. In other words, the matrix $L$ should be a monomial matrix with
entries $\pm 1$.

In the triangular lattice example, we can construct an interaction that
satisfies all of the above conditions with four matter spins. The coefficient
matrix is

$W=\begin{pmatrix}+&+&+\\\ +&-&-\\\ -&+&-\\\ -&-&+\end{pmatrix}\ .$ (4)

From the lattice geometry, we see that gauge transformations acting on the
gauge spins around a triangular plaquette are generated by the following two
diagonal matrices,

$\displaystyle R^{12}=\begin{pmatrix}-\\\ &-\\\ &&+\end{pmatrix}\ ,$
$\displaystyle R^{23}=\begin{pmatrix}+\\\ &-\\\ &&-\end{pmatrix}\ .$ (5)

The superscripts denote the sites where the spins are flipped. The
corresponding $L$ matrices that leave $W$ invariant under $W\mapsto
L^{\top}WR$ are

$\displaystyle L^{12}=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\end{pmatrix}\ ,$ $\displaystyle L^{23}=\begin{pmatrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{pmatrix}\ ,$ (6)

labeled by the same superscripts as the corresponding $R$ matrices. (See Fig.
LABEL:sub@fig:triangular-combinatorial-gauge-symmetry for an illustration of
the effect of $R^{12}$ and $L^{12}$ on the operators $\sigma_{i}^{z}$ and
$\mu_{a}^{z}$.) Next we take a closer look at the properties of these
matrices.

By design $W$ in (4) has the symmetry $L^{\top}WR=W$. The rows of $W$ are the
triples of $\pm 1$ that multiply to $1$. These are precisely the triples that
can be obtained from $(+,+,+)$ by flipping the signs of two entries at a time.
More formally, the rows of $W$ form the orbit of $(+,+,+)$ under the action of
the group of gauge symmetries. Orbits under a group action are examples of
invariant subsets, which means that if we flip the sign of any two columns of
$W$ via $W\mapsto WR$, i.e. act on all elements of the invariant subset by a
group element, the effect is the same as permuting the rows of $W$, i.e. the
elements of the invariant subset. Therefore, for each $R$, there will always
be a permutation matrix $L$ that “undoes” its effect by permuting the rows
back into their original positions, so that $WR=(L^{\top})^{-1}W$ is
satisfied. For example, with the pair of transformations $R^{12}$ and
$L^{12}$, we have

$WR^{12}=\begin{pmatrix}+&+&+\\\ +&-&-\\\ -&+&-\\\
-&-&+\end{pmatrix}\begin{pmatrix}-\\\ &-\\\
&&+\end{pmatrix}=\begin{pmatrix}\leavevmode\hbox to9.33pt{\vbox
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and

$L^{12}W=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\end{pmatrix}\begin{pmatrix}+&+&+\\\ +&-&-\\\ -&+&-\\\
-&-&+\end{pmatrix}=\begin{pmatrix}-&-&\leavevmode\hbox to7.78pt{\vbox
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Flipping the signs of all entries in the first two columns of $W$ has the same
effect as exchanging the first row of $W$ with the last and the second with
the third.

This interaction term needs to be supplemented with additional terms, so the
Hamiltonian of the entire system takes the form

$\displaystyle
H_{\text{CGS}}^{\mathbb{Z}_{2}}=-J\sum_{p}(\boldsymbol{\mu}_{p}^{z})^{\top}W\boldsymbol{\sigma}_{p}^{z}-h\sum_{p}\sum_{a\in
p}{\mu}_{a}^{z}-\Gamma\sum_{p}\sum_{a\in
p}{\mu}_{a}^{x}-\Gamma^{\prime}\sum_{i\in\mathcal{L}}{\sigma}_{i}^{x}\ ,$ (9)

where the subscripts $p$ on the operators $\boldsymbol{\mu}^{x,z}_{p}$ and
$\boldsymbol{\sigma}^{x,z}_{p}$ denote the plaquette that the operators are
located in, the indices $a\in p$ label the matter spins in plaquette $p$, and
the indices $i$ label the gauge spins on the set of links $\mathcal{L}$ of the
lattice. If a large transverse field $-\Gamma\sum_{p}\sum_{a\in
p}{\mu}_{a}^{x}$ is applied on the matter spins, it will effectively integrate
out the matter degrees of freedom, so that the interaction term
$-J\sum_{p}\left(\boldsymbol{\mu}_{p}^{z}\right)^{\top}W\boldsymbol{\sigma}_{p}^{z}$
will perturbatively generate a pure gauge theory. The lowest order term in
perturbation theory will be of the form $\sum_{p}\prod_{i\in
p}\sigma_{i}^{z}$. In addition, a star term can also be generated
perturbatively through the application of the transverse field on the gauge
spins $-\Gamma^{\prime}\sum_{p}\prod_{i\in p}\sigma_{i}^{x}$, thus fully
reproducing the pure gauge theory we started with. We stress that the gauge
symmetry of the system described by Eq. (9) is preserved throughout, not
merely emergent in a perturbative limit in which we justify integrating out
the matter fields, meaning that a topological phase may exist within a large
area of the parameter space of couplings $J,h,\Gamma$ and $\Gamma^{\prime}$.

For this system to be equivalent to a $\mathbb{Z}_{2}$ gauge theory whose
ground state is a spin-liquid state, it is not sufficient to only have the
gauge symmetry. We would like ground state in absence of the $\sigma^{x}$ term
to be an eigenstate of the plaquette operators $B_{p}=\prod_{i\in
p}\sigma_{i}^{z}$ with eigenvalue $1$, because this enforces a zero-flux
condition that corresponds to the confined phase in the pure gauge
theory.222In Abelian gauge theories, the charges and fluxes are dual to each
other, and can be labeled by eigenvalues of $B$ operators, or equivalently
elements of the gauge group. In this paper we will be refer to the eigenvalues
of $B$ operators as “flux”, and use the additive representation for the fluxes
in the language of modular arithmetic. See also Appendix A for a discussion of
the choosing the support of flux vs. charge variables. However, in our
motivating example on the triangular lattice, the $B_{p}=1$ state is always
degenerate with the $B_{p}=-1$ state, because a global spin flip of both
matter and gauge spins commutes with the Hamiltonian but anticommutes with
$B_{p}$. In order to split this degeneracy, we apply a field
$-h\sum_{p}\sum_{a\in p}{\mu}_{a}^{z}$ to the matter spins, with the sign of
$h$ chosen so that the $B_{p}=1$ sector is preferred. Without this term, the
ground state manifold is spanned by the flux-$0$ state
$\\{\boldsymbol{\sigma},\boldsymbol{\mu}\\}=\\{(+,+,+),(+,-,-,-)\\}$ and its
gauge symmetry partners, along with the flux-$1$ state
$\\{(-,-,-),(-,+,+,+)\\}$ and its symmetry partners. Since these two sets of
states have distinct matter spin magnetization (indeed, the CGS preserves the
magnetization of matter spins because it acts as a permutation on the matter
spins), we are able to distinguish the flux states by applying a uniform field
on the matter spins.

In the above example we outlined how to construct a combinatorial gauge theory
whose ground state has topological order. In the following sections we will
generalize these ideas.

##  3 Summary of results

The construction presented in the previous section can be repeated in full
generality for all finite Abelian gauge groups by considering cyclic gauge
groups. This is because any finite Abelian group $G$ can be decomposed into a
direct product of cyclic groups $\prod_{i}\mathbb{Z}_{k_{i}}$. Focusing on the
on the cyclic group $\mathbb{Z}_{k}$, we work with operators $\sigma^{Z}$ and
$\sigma^{X}$ acting on a $\mathbb{Z}_{k}$ clock variable, which satisfy the
relation

$\sigma^{X}\sigma^{Z}=e^{2\pi i/k}\,\sigma^{Z}\sigma^{X}\;.$ (10)

Schematically, the one- and two-body Hamiltonian for such a combinatorial
gauge theory takes the following form:

$\displaystyle
H_{\text{CGS}}=-J\sum_{p}(\boldsymbol{\mu}_{p}^{Z})^{\top}W\boldsymbol{\sigma}_{p}^{Z}-h\sum_{p}\sum_{a\in
p}{\mu}_{a}^{Z}-\Gamma\sum_{p}\sum_{a\in
p}{\mu}_{a}^{X}-\Gamma^{\prime}\sum_{i\in\mathcal{L}}{\sigma}_{i}^{X}+\mathrm{h.c.}\
,$ (11)

where we have extended the vector notation of the example of the previous
section to write the $ZZ$ term in the first line of Eq. (11), by collecting
all $q$ gauge spins and $m$ matter spins on each plaquette into vectors
${\boldsymbol{\sigma}^{Z}}^{\top}\equiv(\sigma_{1}^{Z},\sigma_{2}^{Z},\dots,\sigma_{q}^{Z})^{\top}$
and
${\boldsymbol{\mu}^{Z}}^{\top}\equiv(\mu_{1}^{Z},\mu_{2}^{Z},\dots,\mu_{m}^{Z})^{\top}$.

The first term in (11) is the gauge-invariant interaction term. A gauge
transformation is represented by a pair of matrices $L_{p}$ and $R_{p}$ acting
on the vectors of operators associated with each plaquette, i.e.
$\boldsymbol{\sigma}_{p}^{Z,X}\mapsto L_{p}\boldsymbol{\sigma}_{p}^{Z,X}$ and
$\boldsymbol{\mu}_{p}^{Z,X}\mapsto R_{p}\boldsymbol{\mu}_{p}^{Z,X}$. The
diagonal $R_{p}$ matrices are determined by the behavior of the pure gauge
theory under gauge transformations, and the $L_{p}$ matrices are monomial and
constructed according to the requirement $L_{p}^{\top}WR_{p}=W$. This last
requirement ensures the gauge invariance of this interaction term. It is
possible to construct the coupling matrix $W$ for any gauge group. Moreover,
by exploiting internal symmetries of matter degrees of freedom, the number of
matter variables, or equivalently, the size of the matrix $W$ can be reduced.
A CGS Hamiltonian on a lattice with coordination number $q$ will have at least
$q$ parameters that we will be able to tune to adjust the spectrum, so that
the correct flux sector becomes the ground state.

The transverse field $-\Gamma\sum_{p}\sum_{a\in p}{\mu}_{a}^{X}$ induces
dynamics in the matter degrees of freedom. When this term is strong
($\Gamma\gg J$), the matter degrees of freedom are integrated out, and the
effective Hamiltonian to the lowest order in perturbation theory is exactly
the plaquette term in the pure gauge theory, since that will be the lowest
order term in perturbation theory that respects the gauge symmetry. Next if we
apply a weak transverse field
$-\Gamma^{\prime}\sum_{i\in\mathcal{L}}{\sigma}_{i}^{X}$, the generators of
gauge transformations, that is, the star terms in the pure gauge theory, will
be produced. Note that as long as the transverse field on the matter spins is
uniform in each plaquette, the gauge symmetry is preserved. Thus the
Hamiltonian (11) will always possess an exact gauge symmetry. This means that
even though the effective Hamiltonian only becomes exactly equal to the pure
gauge theory in the $\Gamma\to\infty$ limit, we should expect the existence of
a topological phase that can be connected adiabatically to the confined phase
of the pure gauge theory even at finite values of $\Gamma$. Indeed, in the
case of $\mathbb{Z}_{2}$ gauge theory on a square lattice, it has been shown
that such a quantum spin liquid phase exists in this limit.[24] This provides
evidence that under the $\Gamma^{\prime}\ll J\ll\Gamma$ energy hierarchy, the
topological phase is robust. Since the gap is finite, as long as symmetry
breaking perturbations are small compared to this gap, the splitting between
the topologically degenerate ground states, or indeed within any flux sector,
is exponentially small in the system size, so the topological phase should be
preserved. The presence of the term $-h\sum_{p}\sum_{a\in p}{\mu}_{a}^{Z}$ in
Eq. (11) is required to break accidental degeneracies when certain gauge-
violating transformations leave the interaction term invariant. This is
another measure to ensure the energetic separation of different flux sectors.

##  4 Combinatorial gauge symmetry

Suppose we want to construct a gauge-invariant two-body interaction whose
behavior under gauge transformations mirrors that of a pure gauge theory

$H_{\text{pure}}=-\lambda_{A}\sum_{s}A_{s}-\lambda_{B}\sum_{p}B_{p}$ (12)

where the first group of terms represents charges, i.e., generators of gauge
transformations defined on a star labeled by $s$, and the second group
enforces the constraint that the ground state is flux-free on each plaquette
labeled by $p$. To describe this behavior algebraically, we need to consider
three algebraic constructs.

First, the gauge fields can be represented by variables $\sigma$ that take
value in the gauge group $G$ and this group naturally acts on these variables.
333More rigorously, the local Hilbert space is isomorphic to the group algebra
$\mathbb{C}[G]$, spanned by the group elements as a vector space. It thus
carries an action by the gauge group via left multiplication. When an element
of the gauge group $g\in G$ acts on a gauge variable $\sigma$, we write it
schematically as $g\cdot\sigma$.

On the entire lattice, the total Hilbert space is a tensor product of all the
local gauge variables $\bigotimes_{i}\sigma_{i}$, and is acted on by the
direct product of the gauge groups $\prod_{i}G_{i}$, where the factor $G_{i}$
acts only on the variable $\sigma_{i}$. Not all elements of this direct
product are valid gauge transformations, but only the ones that are generated
by the charge operators $A_{s}$. We call this subgroup $\mathcal{G}$ the group
of gauge transformations. An operator is gauge invariant if it is invariant
under the action of this group. Nevertheless, in order to test if a local
operator $\mathcal{O}$ is gauge invariant, we need not test it against all
possible gauge transformations, but only the ones that act non-trivially on
the support $S$ of $\mathcal{O}$. This means that we are in fact interested in
the quotient group $\mathcal{G}$ under the relation $\sim$ that identifies two
gauge transformations $T=\prod_{i}g_{i}$ and
$T^{\prime}=\prod_{i}g^{\prime}_{i}$ if $g_{i}=g^{\prime}_{i}$ for all $i\in
S$, that is, their actions coincide on the support of the operator
$\mathcal{O}$. We define this group as the _local group of gauge
transformations_ $\mathcal{G}[\mathcal{O}]$ for the operator $\mathcal{O}$.
(See Fig. 2 for an illustration of the relation between the group of gauge
transformations and the local group of gauge transformations in the
$\mathbb{Z}_{2}$ gauge theory of Section 2.) For compactness, when
$\mathcal{O}$ is the flux operator $B_{p}$, we shall write $\mathcal{G}_{p}$
instead of $\mathcal{G}[B_{p}]$.

Figure 2: Two gauge transformations $T$ and $T^{\prime}$ that represent the
same element of the local group of gauge transformation (magenta) for a
plaquette operator (highlighted in black).

Now we assume that in the pure gauge theory, the flux operator $B_{p}$ is a
product of $q$ operators $\sigma_{1},\dots,\sigma_{q}$. However, since our aim
is to construct a two-body interaction term involving a set of auxiliary, or
“matter”, degrees of freedom $\mu_{1},\dots,\mu_{m}$, the resulting operator
should not be a product of all relevant operators. The most natural form is a
quadratic direct coupling between the gauge and matter variables,
$\sum_{i,a}\mu_{a}W_{ai}\sigma_{i}$. Thus we write the gauge and matter
degrees of freedom as vectors of operators $\boldsymbol{\sigma}$ and
$\boldsymbol{\mu}$, so that the action of a gauge transformation
$T\in\mathcal{G}_{p}$ on $\boldsymbol{\sigma}$ can be represented by a
diagonal matrix $R$,

$\boldsymbol{\sigma}=\begin{pmatrix}\sigma_{1}\\\ \vdots\\\
\sigma_{q}\end{pmatrix}\xmapsto{T}\begin{pmatrix}g_{1}\cdot\sigma_{1}\\\
\vdots\\\ g_{q}\cdot\sigma_{q}\end{pmatrix}=R\begin{pmatrix}\sigma_{1}\\\
\vdots\\\ \sigma_{n}\end{pmatrix}=R\boldsymbol{\sigma}$ (13)

where

$R=\begin{pmatrix}g_{1}\\\ &\ddots\\\ &&g_{q}\end{pmatrix}$ (14)

and its action on the matter variables by a matrix $L$, i.e.
$\boldsymbol{\mu}\mapsto L\boldsymbol{\mu}$. Now we define that the term
$\sum_{i,a}\mu_{a}W_{ai}\sigma_{i}=\boldsymbol{\mu}^{\top}W\boldsymbol{\sigma}$
has a _combinatorial gauge symmetry_ that reproduces the gauge symmetry of the
flux operator $B_{p}$ of the pure gauge theory
$H_{\text{pure}}=-\lambda_{B}\sum_{p}B_{p}$, if for every gauge transformation
$T$ in the local group of gauge transformations $\mathcal{G}_{p}$ of the flux
operator $B_{p}=\prod_{i\in p}\sigma_{i}$ that acts on
$\boldsymbol{\sigma}_{p}$ via the matrix $R$, we can find a matrix $L$ acting
on a vector of matter variables $\boldsymbol{\mu}$, such that
$(L\boldsymbol{\mu})^{\top}W(R\boldsymbol{\sigma})=\boldsymbol{\mu}^{\top}W\boldsymbol{\sigma}$,
or $L^{\top}WR=W$. The symmetry $L$ of the matter variables should be a
physical symmetry of the corresponding physical degrees of freedom.

The overall invariance of $\boldsymbol{\mu}^{\top}W\boldsymbol{\sigma}$ is
equivalent to the invariance of $W$ under the multiplication by $R$ on the
right and a corresponding $L$ on the left. Since an $R$ matrix multiplies $W$
on the right, it can be thought of as the same transformation applied to each
row of $W$. Thus if the rows of $W$ form an invariant set under all
$R$-transformations then multiplying $W$ on the right by a matrix $R$ has the
same effect as permuting the rows, so we can “undo” this transformation by
multiplying $W$ on the left by a permutation matrix $L^{\top}$. This gives us
the invariance

$L^{\top}WR=W$ (15)

and consequently the invariance of
$\boldsymbol{\mu}^{\top}W\boldsymbol{\sigma}$ under the transformations
$\boldsymbol{\mu}\mapsto L\boldsymbol{\mu}$ and $\boldsymbol{\sigma}\mapsto
R\boldsymbol{\sigma}$. The number of matter degrees of freedom $\mu$ is then
determined by the number of elements in this invariant set. Since invariant
sets are unions of orbits, such a coupling matrix $W$ can be constructed by
choosing one or more rows of initial entries, applying all $R$ matrix
representations of transformations in $\mathcal{G}_{p}$ to obtain the orbits,
and combining the results to form $W$.

We can lay out the construction more concretely using $\mathbb{Z}_{k}$ as the
gauge group. $\sigma$ could be represented by a clock variable
$Z=\operatorname{diag}(1,\zeta_{k},\zeta_{k}^{2},\dots,\zeta_{k}^{k-1})$ where
$\zeta_{k}=e^{2\pi i/k}$. The action of the gauge group is generated by the
$X$ operator, associated to a $k\times k$ permutation matrix

$X=\begin{pmatrix}[c|cccc]&1&&&\\\ &&1\\\ &&&\ddots\\\ &&&&1\\\ \hline\cr
1&&\end{pmatrix}\ ,$ (16)

that is, take any generator $x$ of the group $G$ and represent it by the $X$
operator, then for any group element $g$, which can be written as $x^{c}$ for
some integer $c$, its action $\sigma\mapsto g\cdot\sigma$ is represented by
$Z\mapsto(X^{\dagger})^{c}ZX^{c}$. In particular, since
$XZX^{\dagger}=\zeta_{k}Z$, such an action on $Z$ reduces to multiplication by
the phase $\zeta_{k}^{c}=e^{2\pi ic/k}$. Thus, the action of $\mathcal{G}_{p}$
that we have represented schematically as a matrix multiplication in fact is a
matrix multiplication by a diagonal $R$ matrix whose entries are $k$-th roots
of unity. For the flux term $B_{p}=\prod_{i\in p}X_{i}$, gauge invariance
requires that the entries of $R$ multiply to $1$. If $B_{p}$ is supported on
$q$ sites, the local group of gauge transformations $\mathcal{G}_{p}$ will be
a quotient of $\mathbb{Z}_{k}^{q-1}$. In many cases, $\mathcal{G}_{p}$ will be
precisely $\mathbb{Z}_{k}^{q-1}$, and when $k$ is prime, $\mathcal{G}_{p}$
will always be a direct sum of fewer than $q$ copies of $\mathbb{Z}_{k}$.
444Each charge operator will generate a subgroup of the local group of gauge
transformation isomorphic to the gauge group. The relation among them is of
the form $R_{1}^{p_{1}}R_{2}^{p_{2}}\dots R_{m}^{p_{m}}=1$, where $R_{i}$ is
the local gauge transformation generated by the $i$-th charge operator. When
$k$ is prime, this relation can be used to express $R_{m}$ in terms of the
other generators, but if not, we may only be able to eliminate a power of
$R_{m}$, resulting in a local group of gauge transformation whose direct
summands may include non-trivial quotients of $\mathbb{Z}_{k}$. This includes
the case of $\mathbb{Z}_{k}$ toric codes, but also models with much more
complicated geometry, for example the Haah’s code, which will be discussed in
the following section.

Supposing $\mathcal{G}_{p}$ is of the form $\mathbb{Z}_{k}^{q-1}$, we can
construct the coupling matrix $W$ by choosing the entries in the first row and
generating the other rows under the action of all $R\in\mathcal{G}_{p}$.
Suppose we take the first row of $W$ to be $(1,1,\dots,1)$, then the entries
of $W$ will also be $k$-th roots of unity, and the rows of $W$ consist
precisely of all $q$-tuples of $k$-th roots of unity that multiply to $1$.
There are $N=k^{q-1}$ such combinations, so the CGS model will require
$k^{q-1}$ matter degrees of freedom. Explicitly,

$W^{k,q}=\begin{pmatrix}\zeta_{k}^{a_{1,1}}&\zeta_{k}^{a_{1,2}}&\ldots&\zeta_{k}^{a_{1,q}}\vspace{.2cm}\\\
\zeta_{k}^{a_{2,1}}&\zeta_{k}^{a_{2,2}}&\ldots&\zeta_{k}^{a_{2,q}}\\\
\vdots&\vdots&\ddots&\vdots\\\
\zeta_{k}^{a_{N,1}}&\zeta_{k}^{a_{N,2}}&\ldots&\zeta_{k}^{a_{N,q}}\end{pmatrix}$
(17)

where $a_{n,j}$ are integers such that $\sum_{j=1}^{q}a_{n,j}\equiv
0\pmod{k}$, or equivalently $\prod_{j=1}^{q}\zeta_{k}^{a_{n,j}}=1$, for all
$n$. This matrix will satisfy the CGS invariance condition (15). In this case
a general gauge transformation acts on the gauge degrees of freedom via a
multiplication on the right of $W^{k,q}$ by the following diagonal matrix:

$R=\begin{pmatrix}\zeta_{k}^{c_{1}}&&&\\\ &\zeta_{k}^{c_{2}}\\\ &&\ddots\\\
&&&\zeta_{k}^{c_{q}}\end{pmatrix}\;,$ (18)

where $c_{j}$ are integers satisfying the condition
$\prod_{j=1}^{q}\zeta_{k}^{c_{j}}=1$ or $\sum_{j=1}^{q}c_{j}\equiv 0\pmod{k}$,
so that the product of each row of $W^{k,q}$ is preserved. By construction,
this right multiplication on $W$ by $R$ has the effect of permuting its rows,
so that an $N\times N$ permutation matrix $L$ exists to rearrange the rows
back in their initial order, thereby satisfying the condition $L^{\top}WR=W$.

For a specific example, consider the $k=5,q=3$ case, that is, take
$\mathbb{Z}_{5}$ as gauge group and a triangular lattice. Starting with
$(1,1,1)$ as the initial row of $W$, the remaining rows can be obtained by the
action of the local group of gauge transformations $\mathbb{Z}_{5}^{2}$,
generated by $R^{12}=\operatorname{diag}(\zeta_{5},\zeta_{5}^{-1},1)$ and
$R^{23}=\operatorname{diag}(1,\zeta_{5},\zeta_{5}^{-1})$, acting on $W$ by
right multiplication, producing a $25$-element orbit. (See Appendix B for the
full form of this matrix.) In the following discussion we will denote such a
combinatorially symmetric coupling matrix by $W^{k,q}[r_{1},\dots,r_{n}]$,
where $r_{1},\dots,r_{n}$ are representatives of distinct orbits forming the
set of its rows, so the $\mathbb{Z}_{5}$ example above is $W^{5,3}[(1,1,1)]$.
(When context makes it clear, we may also omit some or all of the arguments of
$W$.)

### Reducing the number of matter spins

In general, we do not impose anything other than permutation symmetry for the
matter variables, so in principle the $\boldsymbol{\mu}$’s in the Hamiltonian
can be any local operator. However, if the matter variables are realized as
spin-$\frac{1}{2}$ or $\mathbb{Z}_{n}$ degrees of freedom, they could possess
an internal symmetry, which generalizes $L$’s to monomial matrices. In some
cases, this allows us to reduce the number of matter variables. To be precise,
this reduction is possible whenever $\mathcal{G}_{p}$ contains a subgroup
consisting of elements of the form $R=gI$, which we may call “uniform” gauge
transformations, forming a subgroup $\mathcal{G}_{p}^{U}$ of
$\mathcal{G}_{p}$. Such uniform gauge transformations multiplies all entries
by the same group element $g$ from the right and permutes rows of the coupling
matrix differing by an overall phase factor from the left. We can factor out
all such uniform transformations and let them act as an internal symmetry of
the matter variables, or equivalently quotienting $\mathcal{G}_{p}$ by
$\mathcal{G}_{p}^{U}$. This reduces the size of $\mathcal{G}_{p}$ and
therefore the number of matter variables by a factor of
$\absolutevalue{\mathcal{G}_{p}^{U}}$. Starting from a fully constructed CGS
model with uniform gauge symmetries, we can partition rows of $W$ into sub-
orbits consisting of rows related by a uniform gauge transformation and choose
only one representative from each to constructed a reduced $W_{\text{red}}$.
For this coupling matrix, when the action of $R$ maps a row of
$W_{\text{red}}$ into rows that are not present in the original, it will
always be compensated for by a non-identity entry in the $L$ matrix. We will
see examples of this below.

### Accidental symmetries

A subtle issue can arise related to uniform gauge transformations. When the
matter variables possess internal symmetries, and when such symmetries
coincide with elements of the gauge group, they may combine into spurious
symmetries of the Hamiltonian that are not members of the local group of gauge
transformations. In the language of the coupling matrix $W$ and $L,R$
matrices, this situation is described by a pair of transformations
$(L,R)=(g^{-1}I,gI)$, when $gI$ itself is not a valid gauge transformation.
For example, this could occur when the matter variables are spin-$\frac{1}{2}$
variables, the gauge group contains an element that is represented by a phase
shift by $-1$, and the flux operator $B_{p}$ contains an odd number of
factors. The motivating example discussed in Section 2 falls into this case.
Such a spurious symmetry will collapse flux sectors that should remain
distinct, so we need to break it with an additional field
$-h\sum_{p}\sum_{a}\mu^{z}_{p,a}$ on the matter spins to remove the internal
symmetry. (See Appendix C for a demonstration that applying such a field is
sufficient to lift the spurious symmetry.)

Note that uniform spurious symmetries arise in a situation complementary to
the situation discussed above where uniform gauge transformations allow for a
reduction of the coupling matrix, so breaking the internal symmetry in the
former case will not come at a cost of increasing the number of matter
variables.

### Product gauge groups

The framework for CGS is applicable to general Abelian gauge groups, but it is
sufficient to consider the more restricted case of cyclic groups, because any
finite Abelian group can be decomposed into a direct product of cyclic groups.
A CGS Hamiltonian whose gauge group is a direct product can be built up from
CGS Hamiltonians with the factor groups as the gauge group. When the gauge
group $G$ is a direct product $G_{1}\times G_{2}$, the group of gauge
transformations and the local groups of gauge transformations are also direct
products. This means that if
$H_{1}=-J\sum_{p}(\boldsymbol{\mu}_{p}^{1})^{\top}W^{1}\boldsymbol{\sigma}_{p}^{1}$
has combinatorial gauge symmetry with respect to a $G_{1}$ gauge theory
defined on a lattice and so does
$H_{2}=-J\sum_{p}(\boldsymbol{\mu}_{p}^{2})^{\top}W^{2}\boldsymbol{\sigma}_{p}^{2}$
with respect to a $G_{2}$ gauge theory on the same lattice, we can take the
tensor product of the two systems, and write

$H=H_{1}\otimes\operatorname{id}_{2}+\operatorname{id}_{1}\otimes
H_{2}=-J\sum_{p}\left[(\boldsymbol{\mu}_{p}^{1}\otimes\operatorname{id}_{2})^{\top}W^{1}(\boldsymbol{\sigma}_{p}^{1}\otimes\operatorname{id}_{2})+(\operatorname{id}_{1}\otimes\boldsymbol{\mu}_{p}^{2})^{\top}W^{2}(\operatorname{id}_{1}\otimes\boldsymbol{\sigma}_{p}^{2})\right]\
,$ (19)

where $\operatorname{id}_{1,2}$ are identity operators acting on the two
tensor sectors. This Hamiltonian will then be a $G_{1}\times G_{2}$
combinatorial gauge theory.

Note that even in cases where the gauge group is already cyclic, the tensor
product construct could reduce the number of matter spins. For example, a
$\mathbb{Z}_{6}$ gauge theory on the triangular lattice can be constructed
with $36$ matter spins, and the number can be reduced to $12$ if the matter
degrees of freedom have a $\mathbb{Z}_{3}$ internal symmetry. If the gauge
group $\mathbb{Z}_{6}$ is treated as $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ so
that the gauge theory is constructed through the tensor product construction,
only $13$ matter spins are needed, with $4$ matter spins for the
$\mathbb{Z}_{2}$ sector (see Section 2) and $9$ for the $\mathbb{Z}_{3}$
sector (see Section 5.2). The coupling matrix for the $\mathbb{Z}_{3}$ sector
can be further reduced (also see Section 5.2), so that in total only $7$
matter degrees of freedom is required, of which $3$ need to be
$\mathbb{Z}_{3}$ variables.

### Ground state flux and tuning the spectrum of the CGS coupling term

We note that there is no fundamental constraint on the initial row(s) of the
coupling matrix $W$, so each of these entries can be regarded as a tunable
parameter, which gives us a lot of freedom to adjust the spectrum of the CGS
coupling term. This is fortuitous, because while the combinatorial gauge
symmetry construction guarantees that any model we generate has an exact gauge
symmetry, this does not mean that the ground state of our two-body model also
corresponds to the ground state of the gauge theory we are constructing, as we
saw in the case of a $\mathbb{Z}_{2}$ gauge theory on a triangular lattice. To
find the energy of a particular flux sector, we can fix a set of value of the
$\sigma^{Z}$ operators that satisfy the flux constraint, then (supposing that
the matter variables are spin-$\frac{1}{2}$’s) choose the signs of the
$\mu^{z}$’s, so that the energy is minimized,

$E_{0}=-\sum_{a}\absolutevalue{2\operatorname{Re}\left(\sum_{i}W_{ai}\;\sigma^{Z}_{i}\right)}\
.$ (20)

Due to the combinatorial gauge symmetry, this expression only depends on the
flux $\prod_{i\in p}\sigma_{i}^{Z}$ and the initial coupling values that
generate $W$. Given any specific combinatorial gauge symmetry, these initial
couplings can be tuned to find a coupling matrix that has the zero flux sector
as the lowest energy one. This argument also shows that when the matter
degrees of freedom have internal symmetries, transforming them under such
symmetries does not change the spectrum of the CGS Hamiltonian. In other
words, if the matter degrees of freedom are spin-$\frac{1}{2}$’s, for example,
flipping the sign of entire rows of the coupling matrix $W$ will not affect
the spectrum.

Take the coupling matrix (17) as an example, if a more general set of coupling
constants $(b_{1},b_{2},\dots,b_{q})$ are taken as the starting point of the
construction, the resulting coupling matrix will become $W^{k,q}$ with column
$j$ rescaled by $b_{j}$, and the ground state energy will now become

$E_{0}\mapsto
E_{0}^{b}=-\sum_{a}\absolutevalue{2\operatorname{Re}\left(\sum_{i}W_{ai}^{k,q}\;b_{i}\;\sigma_{i}^{Z}\right)}\;.$
(21)

In the $W^{5,3}[(1,1,1)]$ example mentioned earlier, the ground state contains
both flux-$0$ and flux-$1$ and sectors. If instead $r=(-1,1,1)$ or
$r=(0.5,1,1)$ is the first row, which corresponds to rescaling the first
column of the coupling matrix (40) shown in the appendix by $-1$ or $0.5$,
respectively, the ground state will be a flux-$0$ state.

##  5 Examples of combinatorial gauge symmetric Hamiltonians

In examples to follow, we will examine a few CGS Hamiltonians. Through these
examples we shall further illustrate both the flexibility of these models and
the necessary conditions for realizing a topological state.

###  5.1 Uniform gauge transformations and reduction of number of matter spins

Consider a $\mathbb{Z}_{2}$ gauge theory on a square lattice, e.g. the toric
code. Using the initial row $(-1,1,1,1)$, we will get the $8\times 4$
interaction matrix

$W^{2,4}=\begin{pmatrix}-&+&+&+\\\ +&-&-&-\\\ +&-&+&+\\\ -&+&-&-\\\ +&+&-&+\\\
-&-&+&-\\\ +&+&+&-\\\ -&-&-&+\end{pmatrix}\ .$ (22)

On the other hand, the $4\times 4$ Hadamard matrix

$W^{\prime}=\begin{pmatrix}-&+&+&+\\\ +&-&+&+\\\ +&+&-&+\\\
+&+&+&-\end{pmatrix}$ (23)

with the same initial row was used to construct a $\mathbb{Z}_{2}$ gauge
theory [22]. Both matrices possess combinatorial gauge symmetries
corresponding to a $\mathbb{Z}_{2}$ gauge theory on a square lattice, but (22)
has twice as many rows as (23). We will show that the two constructions are
equivalent, and $W^{4,2}$ can be reduced to $W^{\prime}$ procedurally. (Thus
we may refer to the latter as $W^{4,2}_{\text{red}}$.)

That the rows of $W^{4,2}_{\text{red}}$ form an incomplete orbit under gauge
transformations might suggest that $W^{4,2}_{\text{red}}$ cannot have a
combinatorial gauge symmetry. Indeed, multiplying $W^{4,2}_{\text{red}}$ on
the right by $R^{12}=\operatorname{diag}\\{(-,-,+,+)\\}$ we get

$W^{4,2}_{\text{red}}R=\begin{pmatrix}+&-&+&+\\\ -&+&+&+\\\ -&-&-&+\\\
-&-&+&-\end{pmatrix}$ (24)

which not only permutes the rows of $W^{4,2}_{\text{red}}$ but also flips the
signs of two of them. However, if we relax the restriction that the $L$
matrices be permutations, but also allow for $-1$ in addition to $1$ in its
non-zero entries, i.e., by letting $L$ be monomial matrices, we can recover
the combinatorial gauge symmetry. Physically, this corresponds to not only
permuting, but also flipping the matter spins. This operation commutes with a
transverse field, so the $\mathbb{Z}_{2}$ gauge symmetry is preserved in the
full Hamiltonian. What makes this manipulation possible is that while
$W^{4,2}_{\text{red}}$ does not include every element of the orbit under the
action of the gauge transformations, it does include all such entries up to
overall signs, so that the action of an $R$ matrix permutes the rows of
$W^{4,2}_{\text{red}}$ up to signs, which can be compensated for by the signs
of the entries in the $L$ matrix. In this particular case, the corresponding
$L$ matrix is

$L^{12}_{\text{red}}=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{pmatrix}\ .$ (25)

The full and reduced versions of the coupling matrices, $W^{4,2}$ and
$W^{4,2}_{\text{red}}$, are mapped to each other as follows. To reconstruct
$W^{4,2}$, append to each row $r_{a}\in W^{4,2}_{\text{red}}$ a row $-r_{a}$.
Conversely, observe that $W^{4,2}$ consists of four pairs of rows that differ
by an overall sign, so by choosing one representative from each pair, we form
$W^{4,2}_{\text{red}}$. This procedure also produces a correspondence between
the full and reduced versions of $L$ matrices. A reduced $L$ matrix is
augmented into the full version by replacing each entry by a $2\times 2$
block. Zeros and ones become zero and identity matrices, while $-1$ becomes a
$2\times 2$ permutation matrix that swaps the two rows that differ by an
overall sign. For example, from $L^{12}_{\text{red}}$ in (25), we get

$L^{12}=\begin{pmatrix}[c|c|c|c]\text{\LARGE 0}&\begin{matrix}1&0\\\
0&1\end{matrix}&\text{\LARGE 0}&\text{\LARGE 0}\\\
\hline\cr\begin{matrix}1&0\\\ 0&1\end{matrix}&\text{\LARGE 0}&\text{\LARGE
0}&\text{\LARGE 0}\\\ \hline\cr\text{\LARGE 0}&\text{\LARGE 0}&\text{\LARGE
0}&\begin{matrix}0&1\\\ 1&0\end{matrix}\\\ \hline\cr\text{\LARGE
0}&\text{\LARGE 0}&\begin{matrix}0&1\\\ 1&0\end{matrix}&\text{\LARGE
0}\end{pmatrix}\;.$ (26)

Compare the transformation of the full $W$ matrix under the action of
$R^{12}$,

$WR^{12}=\begin{pmatrix}+&-&+&+\\\ -&+&-&-\\\ -&+&+&+\\\ +&-&-&-\\\ -&-&-&+\\\
+&+&+&-\\\ -&-&+&-\\\ +&+&-&+\end{pmatrix}\ .$ (27)

This is a permutation of the rows $1$ and $3$, $2$ and $4$, $5$ and $8$, and
$6$ and $7$, which precisely represented by $L^{12}$. To reduce from a full
$L$ matrix, we first sort the rows of the coupling matrix so that rows related
by a uniform gauge transformation are adjacent to each other, so that the full
$L$ matrix will take a block form. Using a permutation representation of
$\mathbb{Z}_{2}$, each block in the full $L$ matrix is mapped to $+1$ or $-1$.

###  5.2 Tuning the spectrum of a CGS interaction

Next we shall analyze the case that has been studied in [23], namely a
$\mathbb{Z}_{3}$ gauge theory on a triangular lattice (see Appendix A for the
change in underlying lattice) in the context of the general method of building
a CGS model as well as techniques for reducing the number of matter degrees of
freedom and adjusting the spectrum.

A plaquette in a triangular lattice has coordination number $3$ and we are
working with the gauge group $\mathbb{Z}_{3}$, so starting from the initial
row $\left(1,1,1\right)$, we construct a $W$ matrix

$W^{3,3}[(1,1,1)]=\begin{pmatrix}1&1&1\\\ \omega&\omega&\omega\\\
\overline{\omega}&\overline{\omega}&\overline{\omega}\\\
1&\omega&\overline{\omega}\\\ \omega&\overline{\omega}&1\\\
\overline{\omega}&1&\omega\\\ 1&\overline{\omega}&\omega\\\
\omega&1&\overline{\omega}\\\ \overline{\omega}&\omega&1\end{pmatrix}$ (28)

where $\omega=e^{2\pi i/3}$. Note that the rows can be partitioned into three
sets, and within each set the rows differ by an overall phase factor.
Following the reduction procedure illustrated by the previous example, we pick
one representative from each set to construct a reduced coupling matrix,

$W^{3,3}_{\text{red}}[(1,1,1)]=\begin{pmatrix}1&1&1\\\
1&\omega&\overline{\omega}\\\ 1&\overline{\omega}&\omega\end{pmatrix}$ (29)

and take the matter degrees of freedom to be $\mathbb{Z}_{3}$ clock variables.

However the ground state of this CGS Hamiltonian does not lie in the
$\mathbb{Z}_{3}$ flux $0$ sector, instead it is in the flux $1$ and flux $2$
sector, which can be verified from equation (20). To fix this issue, we can
adjust the initial row of the $W$ matrix, for example to $(-1,1,1)$. An
alternative solution is presented in the appendix of [23], that is, a $W$
matrix that combines two orbits under the gauge transformations,

$W^{3,3}_{\text{red}}[(1,1,\omega),(1,1,\overline{\omega})]=\begin{pmatrix}1&1&\omega\\\
1&\omega&1\\\ 1&\overline{\omega}&\overline{\omega}\\\
1&1&\overline{\omega}\\\ 1&\overline{\omega}&\omega\\\
1&\omega&\omega\end{pmatrix}\ .$ (30)

In Sec. 4 we noted that as long as the rows of $W$ forms an invariant set
under the action of the gauge transformations, it possesses a combinatorial
gauge symmetry. The simplest invariant sets are orbits of a single row, but
they can also combine multiple distinct orbits. This is not obviously
advantageous, because the size of such a $W$ matrix is not minimal. However,
the above $W$ for the $\mathbb{Z}_{3}$ CGS turns out to have the correct flux
state as the ground state, while the simple $W$ matrix formed out of the orbit
of $(1,1,1)$ does not. Thus we have another method of adjusting the spectrum
of CGS Hamiltonians.

###  5.3 Gauge theories with more complex structure

Figure 3: Illustration of the two types of operators in the Hamiltonian of the
Haah’s code.

To illustrate the power of the CGS construction, we now construct a model with
the symmetry of Haah’s code. Recall that Haah’s code is defined on three-
dimensional cubic lattice, where there are two spins on each site. The
Hamiltonian is the sum of two types of generators of a stabilizer code, each
supported on each cubic cell, $c$, of the lattice,

$H_{\text{HC}}=-\sum_{c}A_{c}-\sum_{c}B_{c}\ .$

Each operator $A_{c}$ and $B_{c}$ is a product of $\sigma^{z}$’s and
$\sigma^{x}$’s, respectively, acting on a subset of the eight spins on the
cubic cell, as illustrated in Fig. 3. Reformulated as a gauge theory, we may
treat the $A$’s as charge operators and the $B$’s a generators of the gauge
symmetry. Thus to construct a CGS model with the same type of symmetry, we
need to compute the group of local gauge transformations on an $A$-type
operator generated by conjugating it by $B$-type operators. This is computed
in Appendix D. Despite the complex arrangement of sites, the local group of
gauge symmetries is very simple, $\mathbb{Z}_{2}^{7}$. This means that we
could emulate an $A$-type operator by a CGS model of the $(2,8)$-type.

##  6 Implementations

(a)

(b)

Figure 4: Superconducting wire arrays that implement combinatorial gauge
symmetries for (a) a $\mathbb{Z}_{2}$ gauge theory on a square lattice [25]
and (b) a $\mathbb{Z}_{3}$ gauge theory on a honeycomb lattice [23]. (a) The
array is a grid of “waffles” in which horizontal wires extending across
neighboring waffles (in black) represent the gauge degrees of freedom, and the
vertical wires (in gray) are the matter degrees of freedom. Within each
waffle, the wires are coupled by Josephson junctions with no phase shift that
corresponding to $+1$ in the coupling matrix (black cross on white square) and
junctions with a $\pi$ phase shift that correspond to a $-1$ in the coupling
matrix (white cross on black square). (b) In the $\mathbb{Z}_{3}$ case, the
grid of waffles is constructed by the same principle, but within each waffle,
the wires are coupled via Josephson junctions, and the phase shift required by
the CGS Hamiltonian is realized by a uniform magnetic field that generates a
flux of $\Phi=(n+\frac{1}{3})\Phi_{0}$ within each elementary plaquette of the
waffle, where $\Phi_{0}=h/2e$ is the flux quantum.

Proposals have been made for realizing a $\mathbb{Z}_{2}$ CGS model on a
square lattice with a superconducting circuit [25], and a classical
$\mathbb{Z}_{2}$ spin liquid with the same combinatorial gauge symmetry was
built and observed [26]. These implementations suggest generalizations.
Following the approach in [25], superconducting wires can be arranged into an
array of “waffles”, or grids of intersecting wires, as illustrated in Fig.
LABEL:sub@fig:wire-array-toric. Generally, such wire arrays are described by
the Hamiltonian (see also [23])

$H=H_{J}+H_{C}\ ,$

with the Josephson coupling

$H_{J}=-E_{J}\sum_{p}\left[\sum_{a,i\in
p}W_{ai}e^{i(\theta_{i}-\phi_{a})}+\mathrm{h.c.}\right]$

and a quadratic capacitance term $H_{C}$ that depends on the conjugate charges
to the phase variables $\theta_{i},\phi_{a}$. When the coupling coefficients
$W_{ai}$ are those of a CGS model, the Josephson energy possesses a
combinatorial gauge symmetry, where the action of gauge groups corresponds to
shifts in the location of the minima of Josephson energies. (One can ensure
that $H_{C}$ does not break the symmetry by requiring permutation invariance
of the capacitance matrix.) In the regime where $H_{J}$ is dominant, the
minima of the Josephson energy correspond to the minima of the CGS model.

(a)

(b)  (c)

Figure 5: Flux-based superconducting wire array realizations. The general
configuration of gauge and matter wires in a waffle and the phase shift
through a Josephson junction induced by a magnetic field is depicted in (a).
Correspondences between CGS coupling matrices and flux configurations are
shown in (b) for the $\mathbb{Z}_{2}$ CGS model on a triangular lattice and a
waffle with a uniform flux $\Phi_{2}=\frac{1}{2}\Phi_{0}$ except for one
plaquette, and (c) for the $\mathbb{Z}_{3}$ CGS theory with the coupling
matrix $W^{3,3}_{\text{red}}[(1,1,\omega),(1,1,\overline{\omega})]$ and a
waffle with a uniform flux $\Phi_{3}=\frac{1}{3}\Phi_{0}$ except for one
plaquette and one junction (represented by a dotted box) that has a
$\frac{2\pi}{3}$ phase shift.

To get the phase factors $W_{ai}$ in the Josephson Hamiltonian, two strategies
are available. The junctions can be built as Josephson junctions or
$\pi$-Josephson junctions [27, 28, 29, 30] when the $W$ matrix consists only
of $+1$ and $-1$ entries, or by a more general approach given in [23], shown
in Fig. LABEL:sub@fig:wire-array-z3, where the $W$ matrix can be realized by
threading a uniform flux of $\frac{1}{3}$ flux quantum in every elementary
plaquette of the waffle. The latter strategy can in fact be applied to other
CGS Hamiltonians.

Generically, the Hamiltonian of a waffle of superconducting wires is the sum
of the Josephson energies at the intersections of the wires, whose minima will
acquire phase shifts in the presence of a magnetic field. Let $\theta_{i}$ be
the superconducting phase of the wire representing the $i$-th gauge variable,
and let $\phi_{a}$ be the phase of the $a$-th matter wire, both measured at
the edges of the waffle (the left edge for $\phi_{a}$ and the top edge for
$\theta_{i}$). The Josephson junction energy at the intersection of the
$\theta_{i}$ and $\phi_{a}$ wires is equal to

$H_{ai}=-E_{J}\cos\left(\theta_{i}-\phi_{a}+A_{ai}\right)\ ,$ (31)

where

$A_{ai}=\frac{2\pi}{\Phi_{0}}\;\left(\int_{(x_{i},y_{1})}^{(x_{i},y_{a})}\mathbf{A}\cdot\differential{\mathbf{l}}-\int_{(x_{1},y_{a})}^{(x_{i},y_{a})}\mathbf{A}\cdot\differential{\mathbf{l}}\right)$
(32)

is the phase accumulated by integrating the vector potential $\mathbf{A}$,
which originates from magnetic fluxes, along the path from the top of gauge
wire $i$ to the left of matter wire $a$ (See Fig. LABEL:sub@fig:waffle-phase-
shift-general), and $\Phi_{0}$ is the flux quantum.

A gauge transformation $\mathbf{A}\mapsto\mathbf{A}+\gradient\varphi$ shifts
the value of $A_{ai}$, up to a factor of ${2\pi}/{\Phi_{0}}$, by

$\varphi(x_{i},y_{a})-\varphi(x_{i},y_{1})-\varphi(x_{i},y_{a})+\varphi(x_{1},y_{a})=-\varphi(x_{i},y_{1})+\varphi(x_{1},y_{a})\
.$ (33)

Since these two terms on the right-hand side of (33) only depend on the
position where $\theta_{i}$ and $\phi_{a}$ are measured, they can be absorbed
into $\theta_{i}$ and $\phi_{a}$, respectively. Thus fixing a gauge amounts to
choosing the reference point for the phases in the matter and gauge wires. We
choose the gauge such that

$\int_{(x_{1},y_{1})}^{(x_{1},y_{a})}\mathbf{A}\cdot\differential{\mathbf{l}}=\int_{(x_{1},y_{1})}^{(x_{i},y_{1})}\mathbf{A}\cdot\differential{\mathbf{l}}=0\;,$
(34)

in which case $A_{ai}$ is tied to the total magnetic flux $\Phi_{ai}$ within
the area enclosed by the left and top edges of the waffle and gauge wire $i$
and matter wire $a$:

$A_{ai}=2\pi\frac{\Phi_{ai}}{\Phi_{0}}\;.$ (35)

These shifts in the Josephson energy minima can be used to implement the
coupling matrix $W$ of a CGS model. To obtain the most general $W$ matrix, all
junctions are regular Josephson junctions with the exception of those
involving the top wire of the waffle. These latter junctions need to be
tunable, so that the Josephson phase is that of $W_{1i}$. This can be achieved
using the asymmetric DC SQUID design in [25]. Then an appropriate magnetic
field should be applied, so that at the junction between the $\theta_{i}$ and
$\phi_{a}$ wires a phase shift $\Phi_{ai}$ is induced, satisfying

$W_{ai}=e^{i2\pi\frac{\Phi_{ai}}{\Phi_{0}}}\;W_{1i}\;.$ (36)

This relation gives us a one-to-one correspondence between $W$ matrices and
flux configurations in waffles, because excluding the first column and first
row, an $n\times m$ coupling matrix contains $(n-1)\times(m-1)$ entries, and
there are precisely $(n-1)\times(m-1)$ elementary plaquettes in a waffle with
$m$ gauge wires and $n$ matter wires.

For example, the $\mathbb{Z}_{2}$ triangular CGS coupling matrix (4)
corresponds to the following flux configuration.

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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7.69968pt\hfil&\hfil\hskip 7.69968pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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7.69968pt\hfil\cr\vskip 0.0pt\cr\hfil\hskip
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.69968pt\hfil&\hfil\hskip 6.033pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}}
{{}\pgfsys@rect{-5.83301pt}{-3.33301pt}{11.66602pt}{13.11044pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.033pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ ,$

where $\Phi_{0}=h/2e$ is the flux quantum. Note that a coupling matrix with
its rows reordered or its rows transformed according to an internal symmetry
of the matter degrees of freedom will have the same combinatorial gauge
symmetry and spectrum, but in general it will correspond to a different flux
configuration in the waffle realization. For example, if we swap the second
and third rows of (4) and flip the sign of the second and last row of the
resulting matrix, the coupling matrix will correspond to the following flux
arrangement (as illustrated in LABEL:sub@fig:z2-honeycomb-flux),

${\Phi^{2,3}}^{\prime}=\frac{1}{2}\Phi_{0}\leavevmode\hbox to37.46pt{\vbox
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7.69968pt\hfil&\hfil\hskip 7.69968pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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7.69968pt\hfil\cr\vskip 0.0pt\cr\hfil\hskip
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6.033pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$

Similarly, while the waffle corresponding to (30) is threaded by
$\frac{1}{3}\Phi_{0}$ and $\frac{2}{3}\Phi_{0}$ fluxes, switching the fourth
and fifth rows allows us to implement it as (see also Fig
LABEL:sub@fig:z3-two-orbit-flux).

$\Phi^{3,3}=\frac{1}{3}\Phi_{0}\leavevmode\hbox to37.46pt{\vbox
to74.22pt{\pgfpicture\makeatletter\hbox{\hskip 18.73236pt\lower-3.33301pt\hbox
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Given the total Josephson energy of a waffle

$H^{\text{waf}}=-E_{J}\sum_{p}\left[\sum_{i,a\in
p}W_{ai}\;e^{i(\theta_{i}-\phi_{a})}+\mathrm{h.c.}\right]$ (37)

that corresponds to a CGS model with $W$ as its coupling matrix, the minimum
is achieved at

$E^{\text{waf}}_{0}=-2E_{J}\sum_{a}\absolutevalue{\sum_{i}W_{ai}\;e^{i\theta_{i}}}$
(38)

by a phase configuration $(\theta_{1},\dots,\theta_{q})$ on the gauge wires
that corresponds to the ground state of the CGS Hamiltonian. (Compare to
(20).) And as argued in [23], the phases $(\phi_{1},\dots,\phi_{N})$ on the
matter wires are tethered to the $\theta_{i}$’s via

$e^{i\phi_{a}}=\frac{\sum_{i}W_{ai}\;e^{i\theta_{i}}}{\absolutevalue{\sum_{i}W_{ai}\;e^{i\theta_{i}}}}\
,$ (39)

in the same way that the signs of the matter spins in the ground state are
fixed by the configuration of the gauge spins. This shows that the ground
state degeneracy of the waffle system corresponds exactly to that of the CGS
Hamiltonian, so when the Josephson energy is dominant, the superconducting
wire array will exhibit a topological phase described by the CGS Hamiltonian.

##  7 Summary and outlook

We have presented a general method for constructing a class of one-and two-
body-interaction Hamiltonians with an _exact_ gauge symmetry for any given
Abelian group. The work systematically encompasses earlier sporadic examples
of combinatorial gauge symmetry. We illustrated the principles and the
construction for a variety of groups and lattice geometries.

The core physical concept was to build a lattice with matter and gauge degrees
of freedom, which can be spins or superconducting wires. The core mathematical
concept was to classify the allowed interactions between the matter and gauge
degrees of freedom. This problem reduced to ensuring consistency between the
interaction matrix and action of the gauge group on the underlying quantum
variables.

The Hamiltonians with the exact gauge symmetry should contain gapped
topological phases for a wide range of parameters. We are hopeful that the
construction here is sufficiently general that one or more of these examples
can lead to realizations of topologically ordered quantum spin liquids in
engineered or programmable quantum devices. At the very least, the
mathematical construction here may offer a direction for further theoretical
explorations.

## Acknowledgments

We thank Garry Goldstein for insightful discussions as well as for valuable
input. The work of H.Y. and C.C. is supported by the DOE Grant DE-
FG02-06ER46316 (work on designing topological phases of matter and foundations
of combinatorial gauge theory) and by the NSF Grant DMR-1906325 (work on
interfaced topological states in superconducting wires).

##  Appendix A Lattice gauge theory conventions

In this section we make a comment on the lattice gauge theory convention
adopted in this paper and contrast it with the conventions in previous work on
combinatorial gauge theory [22, 23]. Briefly, in previous examples of
$\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$ combinatorial gauge theories, the flux
variables of the gauge theory are located on vertices, so the flux operators
are supported on stars, of a square lattice and a honeycomb lattice,
respectively. Under the conventions of this paper, these constructions would
be for a square plaquette-based $\mathbb{Z}_{2}$ flux operator, and a
triangular plaquette-based $\mathbb{Z}_{3}$ flux operator.

Figure 6: A $\mathbb{Z}_{2}$ gauge theory on a triangular lattice where the
flux operators are supported on plaquettes (blue lattice) can be reinterpreted
as a theory on the honeycomb lattice with flux operators supported on the
vertices (yellow lattice).

In 2D Abelian lattice gauge theories, the flux and charge variables are dual
to each other. Under a transformation that maps each face of graph to a vertex
of its dual graph and vice versa, the flux variables can be interpreted as the
charge variable of a theory defined on the dual graph. This means that, in
terms of physical properties, it makes no difference if the flux variables are
supported on the plaquettes of a graph or on the vertices. The more popular
convention is to put flux variables on plaquettes, which is also the
convention adopted in this paper. See Fig. 6 for an illustration of the
$\mathbb{Z}_{2}$ gauge theory in Section 2. Note that the position of the
degrees of freedom and the Hamiltonian do not depend on the underlying lattice
chosen.

We also note that the convention of situating flux variables on vertices of
the lattice is the source of the terminology “matter” degrees of freedom.
These matter variables are directly coupled to the gauge degrees of freedom,
and are analogously supported on the vertices of the lattice. While strictly
speaking individual “matter” variables don’t carry representations of the
gauge group, their Hilbert spaces may be rearranged into a direct sum of
subspaces that do. Regardless of the true nature of these matter variables, in
this paper we are interested in the confined phase of the gauge theory, which
is most straightforwardly achieved when the matter variables are integrated
out, so we need not dwell on their role in the lattice gauge theory.

##  Appendix B Further examples of CGS coupling matrices

Here we show two further examples of CGS coupling matrices.

For a $\mathbb{Z}_{5}$ gauge theory on a triangular lattice, the following
coupling matrix is constructed from the initial row $(1,1,1)$,

$W^{5,3}[(1,1,1)]=\begin{pmatrix}1&1&1\\\ 1&\zeta&\zeta^{4}\\\
1&\zeta^{2}&\zeta^{3}\\\ 1&\zeta^{3}&\zeta^{2}\\\ 1&\zeta^{4}&\zeta\\\
\zeta&\zeta^{4}&1\\\ \zeta&1&\zeta^{4}\\\ \zeta&\zeta&\zeta^{3}\\\
\zeta&\zeta^{2}&\zeta^{2}\\\ \zeta&\zeta^{3}&\zeta\\\ \zeta^{2}&\zeta^{3}&1\\\
\zeta^{2}&\zeta^{4}&\zeta^{4}\\\ \zeta^{2}&1&\zeta^{3}\\\
\zeta^{2}&\zeta&\zeta^{2}\\\ \zeta^{2}&\zeta^{2}&\zeta\\\
\zeta^{3}&\zeta^{2}&1\\\ \zeta^{3}&\zeta^{3}&\zeta^{4}\\\
\zeta^{3}&\zeta^{4}&\zeta^{3}\\\ \zeta^{3}&1&\zeta^{2}\\\
\zeta^{3}&\zeta&\zeta\\\ \zeta^{4}&\zeta&1\\\ \zeta^{4}&\zeta^{2}&\zeta^{4}\\\
\zeta^{4}&\zeta^{3}&\zeta^{3}\\\ \zeta^{4}&\zeta^{4}&\zeta^{2}\\\
\zeta^{4}&1&\zeta\end{pmatrix}\ .$ (40)

Here $\zeta=e^{2\pi i/5}$ is one of the fifth roots of unity. The group of
gauge transformations are generated by

$\displaystyle R^{12}$
$\displaystyle=\operatorname{diag}\\{\zeta,\zeta^{4},1\\}\ ,$ $\displaystyle
R^{23}$ $\displaystyle=\operatorname{diag}\\{1,\zeta,\zeta^{4}\\}\ ,$

and their corresponding $L$ matrices,

$L^{12}=\operatorname{perm}\\{(1,6,11,16,21)(2,7,12,17,22)\\\
(3,8,13,18,23)(4,9,14,19,24),(5,10,15,20,25)\\}$

and

$L^{12}=\operatorname{perm}\\{(1,2,3,4,5)(6,7,8,9,10)\\\
(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)\\}$

where $\operatorname{perm}\\{\sigma\\}$ denotes the permutation matrix
corresponding to the permutation $\sigma$.

For a $\mathbb{Z}_{2}$ gauge theory on a triangular lattice, using the initial
row $(1,1,1,1,1,1)$ we can construct a $2^{6-1}\times 6=32\times 6$ CGS
coupling. However, since $\gcd(2,6)=2$, we can reduce the number of rows of
this matrix by a factor of $2$, to the following $16\times 6$ matrix,

$W^{2,6}_{\text{red}}[(1,1,1,1,1,1)]=\begin{pmatrix}+&+&+&+&+&+\\\
+&+&+&+&-&-\\\ +&+&+&-&-&+\\\ +&+&+&-&+&-\\\ +&+&-&-&+&+\\\ +&+&-&-&-&-\\\
+&+&-&+&-&+\\\ +&+&-&+&+&-\\\ +&-&-&+&+&+\\\ +&-&-&+&-&-\\\ +&-&-&-&-&+\\\
+&-&-&-&+&-\\\ +&-&+&-&+&+\\\ +&-&+&-&-&-\\\ +&-&+&+&-&+\\\
+&-&+&+&+&-\end{pmatrix}\ .$ (41)

The generators of the gauge transformation can be taken to be

$\displaystyle R^{12}$ $\displaystyle=\operatorname{diag}\\{-,-,+,+,+,+\\}\ ,$
$\displaystyle R^{23}$ $\displaystyle=\operatorname{diag}\\{+,-,-,+,+,+\\}\ ,$
$\displaystyle R^{34}$ $\displaystyle=\operatorname{diag}\\{+,+,-,-,+,+\\}\ ,$
$\displaystyle R^{45}$ $\displaystyle=\operatorname{diag}\\{+,+,+,-,-,+\\}\ ,$
$\displaystyle R^{56}$ $\displaystyle=\operatorname{diag}\\{+,+,+,+,-,-\\}\ ,$

and

$\displaystyle L^{12}$
$\displaystyle=-\operatorname{perm}\\{(1,6)(2,5)(3,8)(4,7)(9,14)(10,13)(11,16)(12,15)\\}\
,$ $\displaystyle L^{23}$
$\displaystyle=\operatorname{perm}\\{(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)\\}\
,$ $\displaystyle L^{34}$
$\displaystyle=\operatorname{perm}\\{(1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)\\}\
,$ $\displaystyle L^{45}$
$\displaystyle=\operatorname{perm}\\{(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)\\}\
,$ $\displaystyle L^{56}$
$\displaystyle=\operatorname{perm}\\{(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)\\}\
.$

This example corresponds to the color code [31] on a honeycomb lattice.

##  Appendix C Breaking spurious degeneracies for $\mathbb{Z}_{2}$ CGS
theories

For a $\mathbb{Z}_{2}$ CGS theory on a lattice with an odd coordination number
$q$, the basic construction outlined in Sec. 4 gives us a $2^{q-1}\times q$
$W$ matrix. The rows of $W$ consist of $q$-element vectors of $\pm 1$ that
contain an even number of $-1$’s. To find the ground state of this CGS
Hamiltonian, we need only minimize the energy under the constraint $\sigma=1$,
since all other flux $0$ states are gauge symmetry partners of this state, and
the flux $1$ states can be obtained by a global spin flip. In this case, the
Hamiltonian reduces to $-\sum_{a}\mu_{a}\sum_{i}W_{ai}$. This is minimized
when each $\mu_{a}$ takes the sign that makes $\mu_{a}\sum_{i}W_{ai}$
negative, which is in turn determined by the sign of $\sum_{i}W_{ai}$. Among
these, there are $\binom{q}{2j}$ rows with $2j$ negative signs. When $2j<q/2$,
the sum of all entries in the row is positive, otherwise it is negative. Thus
in the ground state, the total $\mu$-magnetization is

$\sum_{j=0}^{\lfloor(q-1)/4\rfloor}\binom{q}{2j}-\sum_{j=\lceil(q-1)/4\rceil}^{(q-1)/2}\binom{q}{2j}\
.$

Note that the first sum is over the even binomial coefficients up to
$(q-1)/2$, and the latter sum is over the remaining even binomial
coefficients. Since $\binom{q}{2j}=\binom{q}{q-2j}$ and $q$ is odd, the latter
sum is equivalent to a sum over the odd binomial coefficients up to but not
including the $(q-1)/2$-th one. This means that we can rewrite the total
$\mu$-magnetization as

$\sum_{j=0}^{(q-1)/2}(-1)^{j}\binom{q}{j}\ .$

A simple inductive argument shows that this is equal to

$(-1)^{(q-1)/2}\binom{q-1}{(q-1)/2}\ .$

This shows that the $\mu$-magnetization in the CGS Hamiltonian for $k=2$ and
$q$ odd is always non-zero, thus guaranteeing that a uniform field on the
matter spins will serve to split the ground state degeneracy between flux $0$
and flux $1$ states.

Figure 7: Notations used for writing down the generators of the local group of
gauge symmetries. The sites that are acted on non-trivially by type-$A$
operators are numbered from $1$ through $8$ as in the diagram. The type-$B$
operators that generate non-trivial transformations of one particular
$A$-operator are labeled by their coordinates relative to it, in the
coordinate system illustrated on the right.

##  Appendix D Group of local gauge symmetries of the $A$-type operators in
Haah’s code

Table 1: Generators of the group of local gauge symmetries Generator | Decomposition
---|---
$P_{12}$ |  | $B_{(0,-1,-1)}$ |
$P_{23}$ |  | $B_{(0,-1,-1)}\cdot B_{(-1,0,-1)}$ |
$P_{34}$ |  | $B_{(-1,0,0)}$ |
$P_{45}$ |  | $B_{(-1,-1,1)}$ |
$P_{56}$ |  | $B_{(0,-1,1)}$ |
$P_{67}$ |  | $B_{(1,0,1)}$ |
$P_{78}$ |  | $B_{(0,1,1)}$ |

To describe the local group of gauge symmetries $\mathcal{G}_{A}$ of a
type-$A$ operator in the Haah’s code Hamiltonian, it is convenient to label
sites that it acts on as in Fig. 7. The gauge symmetry generated by a type-$B$
operator can then be written as the a multiplication by a matrix $R$,

$\begin{pmatrix}Z_{1}\\\ \vdots\\\
Z_{8}\end{pmatrix}\xmapsto{B[-]B^{-1}}R\begin{pmatrix}Z_{1}\\\ \vdots\\\
Z_{8}\end{pmatrix}\ .$

Since $B$ is a product of Pauli $x$-operators, the matrices $R$ are diagonal
matrices with $\pm 1$ along the diagonal. Thus for $A$ to be invariant under
this transformation, there must be an even number of $-1$ along the diagonal.
In other words, the local group of gauge symmetries must be a subgroup of the
group consisting of operations flipping an even number of the spins in
$(Z_{1},\dots,Z_{8})$, which we shall denote by $\tilde{\mathcal{G}}$. A
generating set of $\tilde{\mathcal{G}}$ can be taken to be
$\\{P_{i,i+1}\\}_{i=1,\dots,7}$, where $P_{i,i+1}$ can be represented by the
diagonal matrix

$R_{i,i+1}=\operatorname{diag}\\{+1,\dots,+1,\overset{i\text{-th}}{-1},\overset{(i+1)\text{-th}}{-1},+1,\dots,+1\\}\
.$

This also shows that $\tilde{\mathcal{G}}\cong\mathbb{Z}_{2}^{7}$. Now by
showing that every element in this generating set can be expressed in terms of
symmetry operations generated by type-$B$ operators, we can prove that the
local group of symmetries $\mathcal{G}_{A}$ is exactly equal to
$\tilde{\mathcal{G}}$. The generators of $\tilde{\mathcal{G}}$ and their
decompositions in terms of $B$-operators are tabulated in Table 1. Note that
there are $14$ type-$B$ operators that generate non-trivial symmetries of $A$,
so this computation also shows that they are not all independent of each
other.

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