Novel quantum phases on graphs using abelian gauge theory

aSchool of Basic Sciences,
Indian Institute of Technology, Bhubaneswar, India bCenter for Theoretical
Physics of the Universe,
Institute for Basic Science, Daejeon, South Korea

pramod23phys<EMAIL_ADDRESS>

###### Contents

  1. 1 Introduction
  2. 2 The operators
    1. 2.1 Vertex operator or gauge transformations
    2. 2.2 Edge operators or 0-holonomy operators
    3. 2.3 The Hamiltonian
  3. 3 Ground states
    1. 3.1 GSD
    2. 3.2 Construction of ground states
    3. 3.3 Ground states ${\left|{\mbox{GS}[\alpha,\beta]}\right>}$
  4. 4 Entanglement Entropy
    1. 4.1 Bipartite EE for ${\left|{\mbox{GS}\,s}\right>}$
    2. 4.2 Bipartite EE for $\rho^{[\alpha,\beta]}$
  5. 5 Excited states
    1. 5.1 Edge excitations
    2. 5.2 Vertex excitations
    3. 5.3 Case of $m=4,n=6$
    4. 5.4 Anyon-like excitations and topological EE
    5. 5.5 Exchange statistics on graphs
  6. 6 Discussion
    1. 6.1 Summary
    2. 6.2 Outlook
  7. A Analogy to quantum field theory with $U(1)$ gauge field and a matter field
  8. B $U(1)$ gauge theory on a circle
  9. C Short review of graph homology

##  1 Introduction

Quantum phases of matter, or the ones that are beyond Landau’s classification
of spontaneous symmetry breaking with local order parameters, have gained
theoretical and experimental significance over the last few decades. Among
them, the so called topological phases of matter or the topologically ordered
phases have become increasingly important for robust methods of quantum
computation [1, 2, 3]. While topological order can manifest itself in
different ways in different dimensions, a class of solvable examples in two
dimensions are given by the quantum double models of Kitaev [4]. They are
characterized by a ground state degeneracy (GSD) that is a topological
invariant, anyonic excitations, and ground state entanglement entropy (EE)
that includes a global component depending on the superselection sectors of
the theory [5] as a subleading order term. The quantum double models are
Hamiltonian realizations of discrete gauge theories or gauge theories based on
finite groups which were well studied in the early 90’s [6]. More generally
they can also be considered for involutory Hopf algebras [7], of which the
group algebra is a special case [4, 8, 9].

Most studies consider long-ranged topologically ordered systems in two and
three dimensions. The models predominantly are located on lattices that
discretize a two- or three-dimensional differentiable manifold. In this paper
we construct exactly solvable models for quantum phases on connected graphs
which do not fall into the usual setups for physical systems, since graphs
include broader objects than discretized manifolds. However a graph is still a
topological space and can be conveniently thought of as a one-dimensional CW
complex [10]. In fact physics on graphs or networks, as it is sometimes called
in the literature, can be rather non-trivial, with early works studying the
issue of particle statistics on such spaces [11].111Graph structures also
appear in the physics and math literature under the name of quantum graphs
especially in the area of mesoscopic physics [12, 13, 14]. More importantly,
there have been studies exploring the possibilities of anyons on graphs, both
abelian and non-abelian ones [15, 16, 17, 18]. Analogous to braid groups being
the fundamental groups of the configuration space of $N$ identical particles
in $\mathbb{R}^{2}$ [19], graph braid groups play a similar role on different
types of graphs [20, 21, 22]. However, these have a fundamental difference
from the conventional braid groups, as the generators do not obey a Yang-
Baxter type relation.

It is reasonable to expect that our models on graphs share some features with
systems of the known topologically ordered phases. The goal of this paper is
to explore this possibility by analyzing models based on discrete abelian
gauge groups, which are quite similar in form to the abelian quantum double
models or the toric code. The main difference is that now we also include
‘matter’ fields on the vertices of the lattice or graph in addition to the
gauge fields on the edges/links of the lattice or graph. While the vertex
operators or the gauge transformations of the toric code are slightly modified
to act on the matter fields on the vertices as well, the plaquette operators
or the operators measuring local flux of the toric code are replaced by an
entirely new operator known as the edge operator. Before we go into the
details of the models we would like to emphasize that the models presented
here can be obtained from [23, 24, 25, 26, 27] where topological order is
discussed from the point of view of higher gauge theories [28] constructed
using 2-groups and other higher categories. The papers [23, 24] study
topological order in various dimensions using simplicial complexes and in this
context what we present here is a detailed study of the models in the simplest
such complex, namely a graph.

The rest of this paper is organized as follows. The operators on graphs,
including the Hamiltonian, are defined in section 2. The models are
parametrized by two integers, $m$ and $n$, which are the dimensions of the
local Hilbert spaces on the vertices and edges of the graph, respectively.
Following this we cover all the tell-tale signs of topological order starting
with a detailed analysis of the ground states in section 3. For general $m$
and $n$, we find the GSD to be a function of a topological invariant (the
first Betti number) and a graph invariant (the number of vertices). The latter
gives an extensive dependence on the system size.222 Graph invariants are
invariant quantities under graph isomorphisms. There are more non-trivial
graph invariants known as Tutte polynomials which also arise in statistical
physics [29]. Next we exactly compute the EE of these ground states in section
4, and find that there is a global constant term known as the topological EE,
which exists regardless of the partition of the graph. We also compute the EE
for superpositions of the ground states that are complementary to the previous
ground states, in which the topological EE turns out to vanish. Different
aspects from arguments on the minimal entropy states given in [30, 31] are
observed here. In section 5 we see that the total quantum dimension of the
system obtained from the topological EE is precisely equal to the number of
anyon-like excitations. In section 6 we summarize the result and discuss some
future directions. In appendix A, we present an analogy of the model to
quantum field theory with $U(1)$ gauge field and matter field to gain an
intuitive understanding. In appendix B, $U(1)$ gauge theory on a circle
obeying twisted boundary conditions is briefly discussed to help understand
the ground states of our models. Appendix C is devoted to some topological
aspects of graphs.

##  2 The operators

Let $G=(V,E)$ be a connected graph composed by a set of vertices $V$ and a set
of edges $E$. Each edge is endowed with an orientation and its endpoints are
attached to vertices in $V$. For each of such graphs, the adjacency matrix is
well-defined.333 The $(i,j)$-th matrix element stands for the number of edges
directed from the $i$-th vertex to the $j$-th vertex. The diagonal $(i,i)$-th
matrix element counts the number of self-loops at the $i$-th vertex.

We place finite dimensional Hilbert spaces on both the vertices and the edges
making the total Hilbert space, $\mathcal{H}=\otimes_{v\in
V}\mathcal{H}_{v}\otimes_{e\in E}\mathcal{H}_{e}$. Upon taking
$\mathcal{H}_{v}=\textrm{Span
of}\,\\{{\left|{h_{v}}\right>}|h_{v}=0,1,\cdots,m-1\\}\simeq\mathbb{C}^{m}$
for each $v$ and $\mathcal{H}_{e}=\textrm{Span
of}\,\\{{\left|{i_{e}}\right>}|i_{e}=0,1,\cdots,n-1\\}\simeq\mathbb{C}^{n}$
for each $e$, we can consider the local Hilbert spaces as carrying the
representations of the abelian groups, $\mathbb{Z}_{m}$ and $\mathbb{Z}_{n}$
respectively. In the parlance of many-body physics these are $m$ and $n$ level
systems or spins $\frac{m-1}{2}$ and $\frac{n-1}{2}$ on the vertices and edges
respectively.

Furthermore we consider a homomorphism,
$\partial:\mathbb{Z}_{n}\rightarrow\mathbb{Z}_{m}$. For any $\mathbb{Z}_{n}$
elements $a,\,b\in\\{0,1,\cdots,n-1\\}$, the homomorphism satisfies
$\partial(a+b)=\partial(a)+\partial(b)$. The $\mathbb{Z}_{n}$ degrees of
freedom on each edge are regarded as ‘gauge fields’, and the $\mathbb{Z}_{m}$
degrees of freedoms on each vertex as ‘matter fields’. The homomorphism
induces the gauge transformation property of the matter fields from that of
the gauge fields. Let $k$ be the greatest common divisor of $m$ and $n$
($\gcd(m,n)=k$). Then, we can write $m$ and $n$ as

$m=kp,\qquad n=kq,$ (2.1)

where $p$ and $q$ are coprime integers ($\gcd(p,q)=1$). The possible choices
for the homomorphism are labelled by the group $\mathbb{Z}_{k}$ and given by

$\partial^{\left[l\right]}(j)=pjl$ (2.2)

with $l\in\mathbb{Z}_{k}=\\{0,1,\cdots,k-1\\}$ labelling the homomorphisms.
These give compatible homomorphisms as $\partial^{\left[l\right]}(n)=pnl=m$,
under mod $m$ arithmetic, implying that $\partial^{\left[l\right]}$’s map the
identity of $\mathbb{Z}_{n}$ to the identity of $\mathbb{Z}_{m}$. For later
convenience, we also introduce the greatest common divisor of $k$ and $p$
which is denoted by $\xi$: $\gcd(k,p)=\xi$. Namely,

$k=\xi\tilde{k},\qquad
p=\xi\tilde{p}\qquad\textrm{with}\qquad\gcd(\tilde{k},\tilde{p})=1.$ (2.3)

Using these ingredients we define the operators that make up the Hamiltonian.
Although it is possible for a general $l$, the case $l=1$ is mainly considered
in what follows for simplicity. Then (2.2) becomes $\partial(j)=pj$.

###  2.1 Vertex operator or gauge transformations

The vertex operator, $A_{v}$, implements the gauge transformations of the
gauge group $\mathbb{Z}_{n}$, and acts nontrivially on ${\cal H}_{v}$ and
${\cal H}_{e}$ with the edges $e$ attached to $v$. Let $L_{v}$ be the set of
edges attached to the vertex $v$. $L_{v}$ is divided into a set of edges
directed inwards to $v$, $L_{v}^{+}$, and a set of edges directed outwards
from $v$, $L_{v}^{-}$: $L_{v}=L_{v}^{+}\cup L_{v}^{-}$. For the example
depicted in Fig. 1, $L_{v}^{+}=\\{e_{1},\cdots,e_{r}\\}$ and
$L_{v}^{-}=\\{e_{r+1},\cdots,e_{r+s}\\}$.

$v$$e_{1}$$e_{2}$$e_{r}$$e_{r+1}$$e_{r+2}$$e_{r+s}$ Figure 1: A vertex $v$ and
attached $r+s$ edges. The left $r$ edges ($e_{1},e_{2},\cdots,e_{r}$) are
directed to the vertex, whereas the right $s$ edges
($e_{r+1},e_{r+2},\cdots,e_{r+s}$) are outgoing from the vertex.

The operator $A_{v}$ is defined as

$A_{v}=\frac{1}{n}\sum\limits_{j=0}^{n-1}A_{v}^{(j)},\qquad A_{v}^{(j)}\equiv
x_{v}^{\partial(j)}\,{\bf X}_{L_{v}}^{pj}$ (2.4)

with

${\bf X}_{L_{v}}\equiv\left(\prod_{e\in
L_{v}^{+}}X_{e}\right)\,\left(\prod_{e\in L_{v}^{-}}X_{e}^{-1}\right).$ (2.5)

$x_{v}$ and $X_{e}$ are the shift operators on the basis of $\mathbb{C}^{m}$
and $\mathbb{C}^{n}$ respectively:

$x_{v}{\left|{h_{v}}\right>}={\left|{h_{v}+1}\right>},\qquad
X_{e}{\left|{i_{e}}\right>}={\left|{i_{e}+1}\right>},$ (2.6)

where the numbers $h_{v}$ and $i_{e}$ are evaluated in mod $m$ and mod $n$
arithmetic respectively. For the example in Fig. 1,
$A_{v}^{(j)}=x_{v}^{\partial(j)}\,\left(\prod_{a=1}^{r}X_{e_{a}}^{pj}\right)\,\left(\prod_{b=r+1}^{r+s}X_{e_{b}}^{-pj}\right)$
acts on the local Hilbert spaces as

$A_{v}^{(j)}\left({\left|{h_{v}}\right>}\prod_{a=1}^{r}{\left|{i_{e_{a}}}\right>}\prod_{b=r+1}^{r+s}{\left|{i_{e_{b}}}\right>}\right)={\left|{h_{v}+pj}\right>}\prod_{a=1}^{r}{\left|{i_{e_{a}}+pj}\right>}\prod_{b=r+1}^{r+s}{\left|{i_{e_{b}}-pj}\right>}.$
(2.7)

The vertex operator (2.4) is easily seen to be a projector
$\left(A_{v}\right)^{2}=A_{v}$, as it is a group average over
$\mathbb{Z}_{n}$. It has $n-1$ other mutually orthogonal projectors444These
properties follow from the orthogonality theorem for group characters, also
known as the Schur orthogonality relations [32]. that are labelled by the
irreducible representations (IRRs) of $\mathbb{Z}_{n}$,

$A_{v}^{\left[\alpha\right]}=\frac{1}{n}\sum\limits_{j=0}^{n-1}\chi_{\alpha,n}(j)A_{v}^{(j)},$
(2.8)

where $\alpha$ labels the IRR and $\chi_{\alpha,n}(j)$ is the character of the
element $j\in\mathbb{Z}_{n}$ in the IRR $\alpha$. Explicitly,
$\chi_{\alpha,n}(j)=\omega_{n}^{\alpha j}$ with
$\alpha\in\\{0,1,\cdots,n-1\\}$. Here and in what follows, $\omega_{d}\equiv
e^{\frac{2\pi\mathrm{i}}{d}}$ for a positive integer $d$. In this notation the
vertex operator in (2.4) corresponds to the trivial IRR ($\alpha=0$).

###  2.2 Edge operators or 0-holonomy operators

The plaquette operators in the toric code or more generally the quantum double
models [4] measure the flux of the gauge fields around a plaquette (or in
other words the smallest Wilson loop) for the discrete gauge group. We call
this the 1-holonomy operator.555We use 0-holonomy and 1-holonomy keeping in
mind that these operators can be generalized to abelian higher gauge groups as
in [23]. In the language of higher gauge theory as described in [23], matter
fields on vertices are 0-gauge fields and the gauge fields on edges are
1-gauge fields. This can be generalized to $d$-gauge fields living on
$d$-dimensional simplices of a simplicial complex. In a similar manner, we
consider the ‘0-holonomy’ operator or edge operator which acts on two adjacent
vertices and the link in-between as in Fig. 2.

$v_{1}$$v_{2}$$e$ Figure 2: A directed edge $e$ flanked by vertices $v_{1}$
and $v_{2}$.

It is defined as666 Due to the presence of the gauge field on $e$ this
operator is sometimes called as the ‘fake’ 0-holonomy in the literature [23,
24, 26, 27]. Also, when the vertices $v_{1}$ and $v_{2}$ coincide
($v_{1}=v_{2}\equiv v$) and the edge $e$ forms a self-loop, $B_{e}^{(j)}$
becomes $B_{e}^{(j)}=1_{v}Z_{e}^{qj}$ with $1_{v}$ the identity operator on
${\cal H}_{v}$.

$B_{e}=\frac{1}{k}\sum_{j=0}^{k-1}B_{e}^{(j)}\qquad\mbox{with}\qquad
B_{e}^{(j)}\equiv z_{v_{1}}^{pj}\,Z_{e}^{qj}\,z_{v_{2}}^{-pj},$ (2.9)

where $z_{v}$ and $Z_{e}$ are clock operators on the basis of $\mathbb{C}^{m}$
and $\mathbb{C}^{n}$ respectively:

$z_{v}{\left|{h_{v}}\right>}=\omega_{m}^{h_{v}}{\left|{h_{v}}\right>},\qquad
Z_{e}{\left|{i_{e}}\right>}=\omega_{n}^{i_{e}}{\left|{i_{e}}\right>}.$ (2.10)

It is easy to see that $B_{e}$ is a projector $\left(B_{e}\right)^{2}=B_{e}$
and is diagonal on the basis. It turns out that the edge operator or
0-holonomy operator (2.9) acts on the local Hilbert spaces as

$B_{e}{\left|{h_{v_{1}}}\right>}{\left|{h_{v_{2}}}\right>}{\left|{i_{e}}\right>}=\delta_{m}\left(p\left(h_{v_{2}}-h_{v_{1}}\right),\partial(i_{e})\right){\left|{h_{v_{1}}}\right>}{\left|{h_{v_{2}}}\right>}{\left|{i_{e}}\right>}$
(2.11)

with $\delta_{m}(a,b)$ being the mod $m$ Kronecker delta for integers $a$ and
$b$. Physically this can be regarded as measuring the 0-flux due to the matter
fields or the 0-gauge fields across the 1-gauge field on the edge $e$.

As with the vertex operators, we can write down the orthogonal projectors of
these edge operators by projecting to different IRRs of $\mathbb{Z}_{m}$ as

$B_{e}^{\left[\alpha\right]}=\frac{1}{k}\sum\limits_{j=0}^{k-1}\chi_{\alpha,m}(pj)\,B_{e}^{(j)}.$
(2.12)

Since $\chi_{\alpha,m}(pj)\equiv\omega_{m}^{\alpha pj}=\omega_{k}^{\alpha j}$,
independent operators are given by
$\alpha\in\mathbb{Z}_{k}=\\{0,1,\cdots,k-1\\}$ rather than $\mathbb{Z}_{m}$.

###  2.3 The Hamiltonian

For a graph $G=(E,V)$, the Hamiltonian $H$ is constructed out of the vertex
and edge operators (2.4) and (2.9) as777 For a general choice of the
homomorphism $l$, the vertex operator (2.4) with $\partial(j)$ replaced by
$\partial^{[l]}(j)$ and the edge operator (2.9) with $Z_{e}^{qj}$ changed by
$Z_{e}^{qjl}$, compose the Hamiltonian.

$H=-\sum\limits_{v\in V}A_{v}-\sum\limits_{e\in E}B_{e}.$ (2.13)

From the properties of the shift and clock operators

$x_{v}^{j_{1}}z_{v}^{j_{2}}=\omega_{m}^{-j_{1}j_{2}}z_{v}^{j_{2}}x_{v}^{j_{1}}\qquad\mbox{and}\qquad
X_{e}^{j_{1}}Z_{e}^{j_{2}}=\omega_{n}^{-j_{1}j_{2}}Z_{e}^{j_{2}}X_{e}^{j_{1}},$
(2.14)

it is easy to show that projectors $A_{v}$ and $B_{e}$ mutually commute with
each other:

$\left[A_{v},\,B_{e}\right]=\left[A_{v},\,A_{v^{\prime}}\right]=\left[B_{e},\,B_{e^{\prime}}\right]=0$
(2.15)

for any $v,\,v^{\prime}\in V$ and $e,\,e^{\prime}\in E$.888This can also be
seen from the actions on the local Hilbert spaces (2.8) and (2.11), which can
be used for generalizations to arbitrary finite non-abelian groups as well
[25].  The operators $A_{v}$ and $B_{e}$ either have no overlap or share one
edge and one vertex. In the latter case, noncommutativity of the operators on
the edge cancels with that of the operators at the vertex, which is analogous
to the abelian toric code models [4]. Thus the Hamiltonian (2.13) is a sum of
commuting projectors, and hence it is gapped and frustration-free.

Note that the operators in the Hamiltonian are well-defined for an arbitrary
graph irrespective of it being planar or non-planar. Although we only consider
the case where the gauge group is $\mathbb{Z}_{n}$ and the matter fields
belong to $\mathbb{Z}_{m}$, the model can be extended to an arbitrary abelian
group. An analogy to quantum field theory with $U(1)$ gauge field and a matter
field is presented in appendix A.

We mention some symmetry properties of the system:

  * •

Local symmetries \- It is easy to verify that $x_{v}{\bf X}_{L_{v}}$ and
$X^{k}_{e}$ for any $v\in V$ and $e\in E$ commute with the Hamiltonian (2.13),
and generate local $\mathbb{Z}_{kpq}$ and $\mathbb{Z}_{q}$ symmetries
respectively. $z_{v}^{k}$ and $Z_{e}^{\tilde{n}}$ with

$\tilde{n}\equiv\frac{n}{\xi}=\tilde{k}q$ (2.16)

are also local operators commuting with the Hamiltonian ($\xi$ is the greatest
common divisor of $k$ and $p$ as in (2.3)).

Seemingly $x_{v}^{k}$ for any $v\in V$ is an additional local $\mathbb{Z}_{p}$
symmetry transformation, but it is equivalent to one of the above
transformations: $\left(x_{v}{\bf X}_{L_{v}}\right)^{kq}=x_{v}^{kq}$ due to
$\gcd(p,q)=1$.

  * •

Quasi-local symmetries \- Consider traversing a closed path $C$, consisting of
edges in $E$, in either the clockwise or counterclockwise direction. Then we
can write down the operator

$Z(C)\equiv\prod\limits_{e\in C}Z_{e}^{(e|C)},$ (2.17)

where $(e|C)$ is a sign factor according the orientations: $(e|C)=1$ for $e$
and $C$ parallel, and $(e|C)=-1$ for $e$ and $C$ anti-parallel. This operator
is analogous to the Wilson loop of the gauge theory, and commutes with the
Hamiltonian (2.13). The number of such independent operators is equal to the
number of independent closed paths on the graph: $|E|-|V|+1$, where $|E|$ and
$|V|$ are the numbers of edges and vertices in the graph. This is equal to the
first Betti number, a topological invariant of the graph.999Topological
aspects of the first Betti number are provided with a brief look at graph
homology theory in appendix C. For planar graphs, the first Betti number can
be interpreted as the number of one-dimensional holes.

  * •

Global symmetries \- The operator, $\prod\limits_{v\in V}x_{v}$, with support
spanning all the vertices of the graph commutes with the Hamiltonian (2.13)
and generates a global $\mathbb{Z}_{m}$ symmetry. This is deduced from the
above local symmetry as $\prod_{v\in V}x_{v}{\bf X}_{L_{v}}$ upon using the
constraint $\prod_{v\in V}{\bf X}_{L_{v}}=1$. The Hamiltonian is also
invariant under parity, which is realized on a directed graph by reversing the
orientations of all the edges, seen via taking the inverse of the shift and
clock operators on all vertices and edges.

Notice that the local transformation $X^{k}_{e}$ does not commute with the
quasi-local transformation (2.17) when $e$ is on the path $C$. Thus, any state
cannot respect both of the two symmetries. Similarly, $x_{v}{\bf X}_{L_{v}}$
does not commute with $z_{v}^{k}$ and $Z_{e}^{\tilde{n}}$.

##  3 Ground states

The Hamiltonian (2.13) is exactly solvable. Since the
$\textrm{Spec}(A_{v})=\textrm{Spec}(B_{e})=\\{0,1\\}$ for all $v$ and $e$, the
lowest energy is given by $E_{0}=-|V|-|E|$, and any ground state
${\left|{\mbox{GS}}\right>}$ satisfies
$A_{v}{\left|{\mbox{GS}}\right>}=B_{e}{\left|{\mbox{GS}}\right>}={\left|{\mbox{GS}}\right>}$
for all $v\in V$ and $e\in E$. From this it follows that the projector to the
ground state manifold is given by

$\pi_{0}=\prod\limits_{v\in V}A_{v}\prod\limits_{e\in E}B_{e}.$ (3.1)

It is clear then that the GSD is given by the trace of $\pi_{0}$ over the
total Hilbert space:

$\textrm{GSD}={\rm Tr\,}_{{\cal H}}\left(\pi_{0}\right).$ (3.2)

###  3.1 GSD

To compute the trace we observe that the non-zero contribution comes from the
term $\prod\limits_{v\in V}1_{v}\prod\limits_{e\in E}1_{e}$ in $\pi_{0}$,
where $1_{v}$ and $1_{e}$ are the identity operators on $\mathcal{H}_{v}$ and
$\mathcal{H}_{e}$ respectively. Note that ${\rm Tr\,}_{{\cal
H}}=\left(\prod_{v\in V}{\rm Tr\,}_{v}\right)\left(\prod_{e\in E}{\rm
Tr\,}_{e}\right)$ with ${\rm Tr\,}_{v}$ and ${\rm Tr\,}_{e}$ being the traces
over the Hilbert spaces ${\cal H}_{v}$ and ${\cal H}_{e}$. For the shift and
clock operators, $x_{v}^{m}=z_{v}^{m}=1_{v}$ and $x_{v}^{a}z_{v}^{b}$ is
traceless unless $a,b\in m\mathbb{Z}$. The same holds for $X_{e}$ and $Z_{e}$.
Then, from (2.4), (2.9) and (3.1), $\pi_{0}$ is written as a polynomial of
$x_{v}$, $z_{v}$, $X_{e}$ and $Z_{e}$ for all $v\in V$, $e\in E$. Among the
terms of the polynomial, only the term proportional to the identity
$\prod\limits_{v\in V}1_{v}\prod\limits_{e\in E}1_{e}$ gives non-zero
contribution under the trace operation in (3.2). Any other term contains at
least one of $x_{v}$, $z_{v}$, $X_{e}$ and $Z_{e}$ for some $v\in V$, $e\in
E$, and vanishes under the trace.

Thus computing the GSD translates into an exercise of counting the number of
$\prod\limits_{v\in V}1_{v}\prod\limits_{e\in E}1_{e}$’s in $\pi_{0}$. The
answer should be the multiplication of the three factors:

  * •

$\left(\frac{1}{n}\right)^{|V|}\left(\frac{1}{k}\right)^{|E|}$ from the
prefactors of the sums in $A_{v}$ (2.4) and $B_{e}$ (2.9)

  * •

the dimension of the total Hilbert space $m^{|V|}n^{|E|}$

  * •

the number of $\prod\limits_{v\in V}1_{v}\prod\limits_{e\in E}1_{e}$’s.

When $j=k,\,2k,\cdots,(q-1)k$ the operator $x_{v}^{pj}$ in $A_{v}^{(j)}$ (2.4)
becomes the identity, but the operators on the edges in $A_{v}^{(j)}$ remain
nontrivial. Note that $pj/k\neq 0$ (mod $q$) because of $\gcd(p,q)=1$. Also,
for $j=1,\,2,\cdots,k-1$, $Z_{e}^{qj}$ in $B_{e}^{(j)}$ (2.9) is always
nontrivial, which physically implies that there is no matter field with
neutral electric charge. These properties lead to101010 The result (3.3)
remains valid for the homomorphism with other choice of $l$, as long as
$Z_{e}^{qjl}$ in the edge operator (see footnote 7) is not trivial for any
$j\in\\{1,\,2,\,\cdots,k-1\\}$. Otherwise, matter fields with neutral electric
charge appear, and the GSD would depend on other details of the graph in
addition to $|V|$ and $|E|$.

GSD $\displaystyle=$ $\displaystyle{\rm Tr\,}_{{\cal H}}\left(\prod_{v\in
V}A_{v}\prod_{e\in E}B_{e}\right)$ (3.3) $\displaystyle=$
$\displaystyle\left(\frac{1}{n}\right)^{|V|}\left(\frac{1}{k}\right)^{|E|}\times
m^{|V|}n^{|E|}\times q$ $\displaystyle=$ $\displaystyle p^{|V|}q^{B_{1}},$

where $B_{1}=|E|-|V|+1$ is the first Betti number of the graph. For any fixed
$j\in k\mathbb{Z}_{q}=\\{0,\,k,\,2k,\,\cdots,(q-1)k\\}$, the relevant
contribution to GSD solely comes from $A_{v}^{(j)}$ and $B_{e}^{(0)}$ for all
$v\in V$ and $e\in E$, making up $\prod\limits_{v\in V}1_{v}\prod\limits_{e\in
E}1_{e}$. Note that for an edge $e$ connecting two vertices $v_{1}$ and
$v_{2}$ as in Fig 2, $A_{v_{1}}^{(j)}A_{v_{2}}^{(j)}$ acts trivially on ${\cal
H}_{e}$. The last factor $q$ on the second line counts the possible choice of
$j$ that is the number of $\prod\limits_{v\in V}1_{v}\prod\limits_{e\in
E}1_{e}$’s.

###  3.2 Construction of ground states

If we find a seed state ${\left|{s}\right>}$ satisfying
$B_{e}{\left|{s}\right>}={\left|{s}\right>}$ for any $e$, one of the ground
states is given by

${\left|{\mbox{GS}\,s}\right>}=\sqrt{{\cal N}}\,\left(\prod_{v\in
V}A_{v}\right){\left|{s}\right>},$ (3.4)

where ${\cal N}$ is a normalization constant. It is easy to see that
${\left|{s=0}\right>}\equiv\prod_{v\in V}{\left|{0_{v}}\right>}\prod_{e\in
E}{\left|{0_{e}}\right>}$ gives such a state ${\left|{s}\right>}$.

Starting with a ground state

${\left|{\mbox{GS}\,0}\right>}\equiv\sqrt{{\cal N}}\,\left(\prod_{v\in
V}A_{v}\right){\left|{s=0}\right>},$ (3.5)

we can exhaust the other ground states by acting the local operators
$x_{v}^{a}{\bf X}_{L_{v}}^{a}$ ($a\in\mathbb{Z}_{p}$) and $X_{e}^{bk}$
($b\in\mathbb{Z}_{q}$) on ${\left|{s=0}\right>}$. Note that $x_{v}{\bf
X}_{L_{v}}$ acts as a $\mathbb{Z}_{p}$-transformation on
${\left|{\mbox{GS}\,0}\right>}$ or on $A_{v}$, because of $x_{v}^{p}{\bf
X}_{L_{v}}^{p}A_{v}=A_{v}$. However, all the choices are not independent. For

$A_{v}^{(bk)}=1_{v}{\bf X}_{L_{v}}^{pbk}$ (3.6)

in (2.4), $A_{v}A_{v}^{(bk)}=A_{v}$ ($b\in\mathbb{Z}_{q}$) holds, which means
that ${\left|{s_{1}}\right>}$ and ${\left|{s_{2}}\right>}$ such that
${\left|{s_{1}}\right>}=A_{v}^{(bk)}{\left|{s_{2}}\right>}$ for any $b$ and
$v$ give the same ground state. Taking into account the constraint
$\prod_{v\in V}A_{v}^{(bk)}=1$, the number of the independent choices amounts
to

$\frac{p^{|V|}q^{|E|}}{q^{|V|-1}}=p^{|V|}q^{B_{1}}=\mbox{GSD}.$ (3.7)

Alternatively, we pick edges $\hat{e}_{L}$ ($L=1,\cdots,B_{1}$) such that the
graph $T\equiv G-\\{\hat{e}_{1},\cdots,\hat{e}_{B_{1}}\\}$ becomes a connected
tree graph, i.e., a spanning tree. Then, the independent ground states are
generated by acting $x_{v}^{a_{v}}{\bf X}_{L_{v}}^{a_{v}}$
($a_{v}\in\mathbb{Z}_{p}$, $v\in V$) and $X_{\hat{e}_{L}}^{b_{L}k}$
($b_{L}\in\mathbb{Z}_{q}$, $L=1,\cdots,B_{1}$) on ${\left|{s=0}\right>}$. The
states ${\left|{s}\right>}$ giving the independent ground states are labelled
by $\\{a_{v}\left|\right.v\in V\\}$ and
$\\{b_{L}\left|\right.L=1,\cdots,B_{1}\\}$ for a choice of $\hat{e}_{L}$’s.
The choice of $\hat{e}_{L}$’s is not unique. If one of the $\hat{e}_{L}$’s,
say $\hat{e}_{1}$, is added to the above spanning tree, a closed path
including $\hat{e}_{1}$ appears. We can choose any other edge on the closed
path instead of $\hat{e}_{1}$. For example, in a graph depicted in Fig. 3, we
can choose one among $e_{12}$, $e_{37}$, $e_{14}$, $e_{45}$, $e_{56}$ and
$e_{67}$ instead of $\hat{e}_{1}$, where $e_{ij}$ denotes the edge in $T$
connecting the vertices $v_{i}$ and $v_{j}$. The choice of the other
$\hat{e}_{L}$’s can be changed similarly. We obtain the same set of
independent ground states irrespective of the choice as we see below.

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\hat{e}_{1}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\hat{e}_{2}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\hat{e}_{3}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\hat{e}_{4}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\hat{e}_{5}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\hat{e}_{6}}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$$v_{8}$$v_{9}$$v_{10}$$v_{11}$$v_{12}$
Figure 3: A graph with a choice of $\hat{e}_{L}$’s. The red lines with arrows
represent $\hat{e}_{L}$ ($L=1,\cdots,6$), and the black lines with arrows
represent the other edges. The black lines and the vertices form a spanning
tree of the graph.

Noting that $A_{v}^{(bk)}$ acts on the vertex $v$ as identity, the
normalization is computed as111111 Similar to the computation in the previous
subsection, nonvanishing contribution arises only when the index $b$ of
$A_{v}^{(bk)}$ is the same for all $v$, which provides the last equality on
the first line of (3.8).

$\displaystyle\langle\mbox{GS}\,s{\left|{\mbox{GS}\,s}\right>}$
$\displaystyle=$ $\displaystyle{\cal N}{\left<{s}\right|}\prod_{v\in
V}A_{v}{\left|{s}\right>}={\cal
N}\left(\frac{1}{n}\right)^{|V|}{\left<{s}\right|}\left(\prod_{v\in
V}\sum_{b=0}^{q-1}A_{v}^{(bk)}\right){\left|{s}\right>}={\cal
N}\,n^{-|V|}\sum_{b=0}^{q-1}{\left<{s}\right|}\prod_{v\in
V}A_{v}^{(bk)}{\left|{s}\right>}$ (3.8) $\displaystyle=$ $\displaystyle{\cal
N}\,n^{-|V|}q,$

which determines ${\cal N}$ as

${\cal N}=\frac{n^{|V|}}{q}.$ (3.9)

#### Choice of $\hat{e}_{L}$’s

Here, we show that the same set of the independent ground states is obtained
irrespective of the choice of $\hat{e}_{L}$’s.

When we change the initial choice of one of $\hat{e}_{L}$’s, say
$\hat{e}_{L_{0}}$, we choose an edge among the edges on the closed path in the
graph $T+\\{\hat{e}_{L_{0}}\\}$, instead of $\hat{e}_{L_{0}}$. Suppose we pick
an edge $e^{\prime}$ instead of $\hat{e}_{L_{0}}$. Then the graph
$T-\\{e^{\prime}\\}$, the spanning tree $T$ after the edge $e^{\prime}$ is
removed, splits into two connected tree graphs, which are denoted by $T_{1}$
and $T_{2}$. We can see that $\prod_{v\in T_{1}}A_{v}^{(bk)}$ becomes the
product of $X_{e}^{\pm pbk}$’s with respect to the edges $e=\hat{e}_{L_{0}}$,
$e^{\prime}$ and some other $\hat{e}_{L}$’s, where $\pm$ in the power is fixed
by the orientation. $\prod_{v\in T_{2}}A_{v}^{(bk)}$ gives essentially the
same result, since $\prod_{v\in T_{2}}A_{v}^{(bk)}=\left(\prod_{v\in
V}A_{v}^{(bk)}\right)\prod_{v\in T_{1}}A_{v}^{(-bk)}=\prod_{v\in
T_{1}}A_{v}^{(-bk)}$.

Noting that

$\\{X_{e}^{bk}\left|\right.b\in\mathbb{Z}_{q}\\}=\\{X_{e}^{pbk}\left|\right.b\in\mathbb{Z}_{q}\\}$
(3.10)

due to $\gcd(p,q)=1$, we label the state ${\left|{s}\right>}$ by $a_{v}$’s and
$pb_{L}$’s:

${\left|{s}\right>}=\left(\prod_{v\in V}x_{v}^{a_{v}}{\bf
X}_{L_{v}}^{a_{v}}\right)\left(\prod_{L=1}^{B_{1}}X_{\hat{e}_{L}}^{pb_{L}k}\right){\left|{s=0}\right>}.$
(3.11)

The above result leads to

$\left(\prod_{v\in
T_{1}}A_{v}^{(-b_{L_{0}}k)}\right){\left|{s}\right>}=\left(\prod_{v\in
V}x_{v}^{a_{v}}{\bf X}_{L_{v}}^{a_{v}}\right)\left(\prod_{L\neq
L_{0}}X_{\hat{e}_{L}}^{pb^{\prime}_{L}k}\right)X_{e^{\prime}}^{-pb_{L_{0}}k}{\left|{s=0}\right>},$
(3.12)

where $b^{\prime}_{L}=b_{L}\pm b_{L_{0}}$ for the edge $\hat{e}_{L}$ appearing
in the result of $\prod_{v\in T_{1}}A_{v}^{(bk)}$, otherwise
$b^{\prime}_{L}=b_{L}$. In (3.12) $e^{\prime}$ appears with the label
$-pb_{L_{0}}$ instead of $\hat{e}_{L_{0}}$, and some other $\hat{e}_{L}$’s
remain with the label changed by $\pm pb_{L_{0}}$, compared to the initial
choice (3.11). The RHS of (3.12) describes the seed state with the choice of
$\hat{e}_{L_{0}}$ changed to $e^{\prime}$, and the LHS shows that it provides
the same ground state as (3.11) as we saw below (3.6). Thus, we can say that
the ground state is invariant under the change of $\hat{e}_{L_{0}}$
accompanied with appropriate change of the labels $\\{b_{L}\\}$. Since the
number of the labels $\\{b_{L}\\}$ does not change before and after the change
of the choice, the set of the independent ground states remains the same.

Let us illustrate the above in the graph in Fig. 3. We consider the case
$\hat{e}_{L_{0}}=\hat{e}_{5}$, and take $e^{\prime}=e_{48}$ in the closed path
on the graph $T+\hat{e}_{5}$. Then, $T-\\{e_{48}\\}$ splits into the two
connected tree graphs: $T_{1}$ composed by the vertices $v_{1},\cdots,v_{7}$
and the black edges connecting them, and $T_{2}$ composed by
$v_{8},\cdots,v_{12}$ and the black edges connecting them. We have

$\prod_{v\in
T_{1}}A_{v}^{(bk)}=\left(X_{\hat{e}_{5}}X_{e_{48}}X_{\hat{e}_{4}}^{-1}X_{\hat{e}_{6}}^{-1}\right)^{pbk}.$
(3.13)

Acting $\prod_{v\in T_{1}}A_{v}^{(-b_{5}k)}$ on the initial choice (3.11) with
$B_{1}=6$ leads to

$\left(\prod_{v\in
T_{1}}A_{v}^{(-b_{5}k)}\right){\left|{s}\right>}=\left(\prod_{v\in
V}x_{v}^{a_{v}}{\bf
X}_{L_{v}}^{a_{v}}\right)\left(\prod_{L=1}^{3}X_{\hat{e}_{L}}^{pb_{L}k}\right)X_{\hat{e}_{4}}^{p(b_{4}+b_{5})k}X_{\hat{e}_{6}}^{p(b_{6}+b_{5})k}X_{e_{48}}^{-pb_{5}k}{\left|{s=0}\right>}.$
(3.14)

Thus, the ground state (3.4) remains the same under the change of
$\hat{e}_{5}$ to $e_{48}$ together with labels changed as $b_{5}\to-b_{5}$,
$b_{4}\to b_{4}+b_{5}$ and $b_{6}\to b_{6}+b_{5}$ (mod $q$).

Since this procedure can be repeated for changes of the other $\hat{e}_{L}$’s,
we can say that the same set of the independent ground states is obtained
irrespective of the choice of $\hat{e}_{L}$’s.

The obtained ground states ${\left|{\mbox{GS}\,s}\right>}$ in (3.4) and (3.11)
with $s=0,1,\cdots,(\mbox{GSD})-1$ are eigenstates of the local operator
$z_{v}^{k}$ (${}^{\forall}v\in V$) and the operator of quasi-local symmetry
$Z(C)$ in (2.17) for any closed path $C$. Let $C_{L}$ denote a closed path
appearing when $\hat{e}_{L}$ is added to the spanning tree $T$.
${\left|{\mbox{GS}\,s}\right>}$ is distinguished by the eigenvalues of
$z_{v}^{k}$’s and $Z(C_{L})$’s which measure $a_{v}$’s and $b_{L}$’s
respectively:

$\displaystyle
z_{v}^{k}\,{\left|{\mbox{GS}\,s}\right>}=\omega_{p}^{a_{v}}\,{\left|{\mbox{GS}\,s}\right>},$
(3.15) $\displaystyle
Z(C_{L})\,{\left|{\mbox{GS}\,s}\right>}=\omega_{q}^{pb_{L}\,(\hat{e}_{L}|C_{L})}\,{\left|{\mbox{GS}\,s}\right>}.$
(3.16)

In order to give a physical interpretation of (3.16), let us pick an embedding
space of the graph in which a simply connected domain bounded by $C_{L}$ can
be defined.121212 Graphs consist of vertices and edges as mentioned in the
beginning of section 2. Since the domain bounded by $C_{L}$ lies outside the
graph, we need to mention the embedding space in order to consider magnetic
flux penetrating the domain. This is analogous to the global magnetic fluxes
penetrating the hole of the torus in the toric code. Incidentally they also
distinguish the ground states in the toric code just as how the local magnetic
fluxes distinguish the ground states in our models.  From the analogy to the
field theory in appendix A, $a_{v}$ represents some degrees of freedom of the
matter field $\phi$ at the point $v$ on the ground state, while
$pb_{L}(\hat{e}_{L}|C_{L})$ is interpreted as magnetic flux penetrating the
inside of $C_{L}$ since the Wilson loop measures magnetic flux penetrating the
domain surrounded by the loop. This is valid even if $C_{L}$ is a
topologically nontrivial cycle under the setting of the embedding space. Note
that for any $L^{\prime}(\neq L)$, $\hat{e}_{L^{\prime}}$ does not belong to
$C_{L}$, which is seen from the above definition of $C_{L}$. $U(1)$ gauge
field on a circle has nontrivial topological structure as briefly summarized
in appendix B. In particular, the nontrivial topological configuration of the
gauge field generates magnetic flux as seen in (B.8), which is analogous to
the twist by $X_{\hat{e}_{L}}^{pb_{L}k}$ providing the $\mathbb{Z}_{q}$
magnetic flux $pb_{L}$.

On the other hand, an individual ground state is not invariant under the local
transformations $x_{v}{\bf X}_{L_{v}}$ and $X_{e}^{k}$, but mapped to another
individual ground state.

###  3.3 Ground states ${\left|{\mbox{GS}[\alpha,\beta]}\right>}$

Next, we construct eigenstates with respect to the local transformations
$x_{v}{\bf X}_{L_{v}}$ and $X_{e}^{k}$ by taking appropriate linear
combinations of the ground states ${\left|{\mbox{GS}\,s}\right>}$
($s=0,1,\cdots,(\mbox{GSD})-1$). Let us introduce operators

$\displaystyle P_{v}^{[\alpha_{v}]}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{p}\sum_{a=0}^{p-1}\omega_{p}^{a\alpha_{v}}x_{v}^{a}{\bf
X}_{L_{v}}^{a}\qquad(\alpha_{v}\in\mathbb{Z}_{p}),$ (3.17) $\displaystyle
P_{e}^{[\beta_{e}]}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{q}\sum_{b=0}^{q-1}\omega_{q}^{b\beta_{e}}X_{e}^{bk}\qquad(\beta_{e}\in\mathbb{Z}_{q}).$
(3.18)

Note that

$P_{v}^{[\alpha_{v}]}=A_{v}P_{v}^{[\alpha_{v}]}=\frac{1}{np}\sum_{j=0}^{np-1}\omega_{p}^{j\alpha_{v}}x_{v}^{j}{\bf
X}_{L_{v}}^{j}$ (3.19)

holds on the ground states. As the RHSs of (3.18) and (3.19) show, they are
projection operators on the ground states.

Acting the operator

${\cal P}={\cal P}^{[\alpha,\beta]}\equiv\left(\prod_{v\in
V}P_{v}^{[\alpha_{v}]}\right)\left(\prod_{L=1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right).$
(3.20)

on ${\left|{\mbox{GS}\,0}\right>}$ generates a desirable linear combination of
all the ground states with the coefficients being phases:

${\left|{\mbox{GS}[\alpha,\beta]}\right>}={\cal
P}{\left|{\mbox{GS}\,0}\right>}.$ (3.21)

Compared with the ground states ${\left|{\mbox{GS}\,s}\right>}$ in (3.4), the
labels $a_{v}$’s and $b_{L}$’s are converted to $\alpha_{v}$’s and
$\beta_{L}$’s by the discrete Fourier transformations.

Associated to the ground state (3.21), we can regard the graph as an electric
circuit in which the ‘current’ $\beta_{L}$ flows on the line $\hat{e}_{L}$ to
the direction of its orientation. Then, the currents on the other lines which
are not $\hat{e}_{L}$’s are determined by the ‘current conservation’ at the
vertices. The current conservation follows from the relation
$A_{v}^{(bk)}{\left|{\mbox{GS}\,s}\right>}={\left|{\mbox{GS}\,s}\right>}$,
namely

$\prod_{e\in L_{v}^{+}}X_{e}^{pbk}=\prod_{e\in
L_{v}^{-}}X_{e}^{pbk}\qquad\mbox{on}\quad{\left|{\mbox{GS}\,s}\right>}$ (3.22)

for any $v$ and $b\in\mathbb{Z}_{q}$. We can see that

$\displaystyle x_{v}{\bf
X}_{L_{v}}{\left|{\mbox{GS}[\alpha,\beta]}\right>}=\omega_{p}^{-\alpha_{v}}{\left|{\mbox{GS}[\alpha,\beta]}\right>},$
(3.23) $\displaystyle
X_{e}^{k}\,{\left|{\mbox{GS}[\alpha,\beta]}\right>}=\begin{cases}\omega_{q}^{-\beta_{L}}{\left|{\mbox{GS}[\alpha,\beta]}\right>}&(e=\hat{e}_{L})\\\
\omega_{q}^{-\bar{\beta}_{e}}{\left|{\mbox{GS}[\alpha,\beta]}\right>}&(e\notin\\{\hat{e}_{1},\cdots,\hat{e}_{B_{1}}\\}),\end{cases}$
(3.24)

where $\bar{\beta}_{e}$ represents the current on the edge $e$, a linear
combination of $\beta_{L}$’s determined by the current conservation. Note that
(3.18) can also be written as

$P_{e}^{[\beta_{e}]}=\frac{1}{q}\sum_{b=0}^{q-1}\omega_{q}^{pb\beta_{e}}X_{e}^{pbk}\qquad(\beta_{e}\in\mathbb{Z}_{q})$
(3.25)

since $p$ and $q$ are coprime. As an example, for the graph in Fig. 3,
$\bar{\beta}_{e}$’s are determined as
$\bar{\beta}_{e_{12}}=\bar{\beta}_{e_{14}}=\bar{\beta}_{e_{37}}=\beta_{1}$,
$\bar{\beta}_{e_{67}}=-\beta_{1}+\beta_{6}$,
$\bar{\beta}_{e_{56}}=-\beta_{1}-\beta_{5}+\beta_{6}$,
$\bar{\beta}_{e_{45}}=-\beta_{1}+\beta_{4}-\beta_{5}+\beta_{6}$,
$\bar{\beta}_{e_{48}}=\beta_{4}-\beta_{5}+\beta_{6}$, and so on.

By the discrete Fourier transformations of (3.11) and (3.12), we can see that
when the choice of $\hat{e}_{L_{0}}$ is changed to $e^{\prime}$ as discussed
in the previous subsection, the initial ground state
${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ remains the same form with the
current $\beta_{L_{0}}$ on $\hat{e}_{L_{0}}$ replaced to
$\bar{\beta}_{e^{\prime}}$ on $e^{\prime}$. From a set of the ground states
for any one choice of $\hat{e}_{L}$’s, the ground states for the other choices
are derived.

Thus ${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ is invariant under the local
$\mathbb{Z}_{p}$ and $\mathbb{Z}_{q}$ transformations (up to phase factors) as
in (3.23) and (3.24). As $A_{v}$ can be regarded as the Gauss law operator,
$x_{v}^{p}{\bf X}_{L_{v}}^{p}$ corresponds to an operator of ‘small gauge
transformations’, i.e., topologically trivial gauge transformations connected
to the identity. Then, the local $\mathbb{Z}_{p}$ and $\mathbb{Z}_{q}$
transformations, which are not generated by $x_{v}^{p}{\bf X}_{L_{v}}^{p}$,
can be interpreted as ‘large gauge transformations’, topologically nontrivial
gauge transformations not connected to the identity.
${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ is similar to the $\theta$ vacuum
in gauge theory when the vacuum has nontrivial topological structure, as seen
in (B.6).

On the other hand, $z_{v}^{k}$ and $Z(C_{L})$ act on (3.21) as

$\displaystyle
z_{v}^{k}\,{\left|{\mbox{GS}[\alpha,\beta]}\right>}={\left|{\mbox{GS}[\tilde{\alpha},\beta]}\right>},$
(3.26) $\displaystyle
Z(C_{L})\,{\left|{\mbox{GS}[\alpha,\beta]}\right>}={\left|{\mbox{GS}[\alpha,\tilde{\beta}]}\right>},$
(3.27)

where

$\tilde{\alpha}_{v^{\prime}}\equiv\begin{cases}\alpha_{v^{\prime}}&(v^{\prime}\neq
v)\\\
\alpha_{v}+1&(v^{\prime}=v)\end{cases}\qquad\mbox{and}\qquad\tilde{\beta}_{L^{\prime}}\equiv\begin{cases}\beta_{L^{\prime}}&(L^{\prime}\neq
L)\\\ \beta_{L}+(\hat{e}_{L}|C_{L})&(L^{\prime}=L).\end{cases}$ (3.28)

From the analogy to the field theory in appendix A, $\alpha_{v}$ represents
some degrees of freedom of the momentum $\pi$ of the matter field at the point
$v$ on the ground state, while $\beta_{L}$ is interpreted as electric flux or
current flowing along $\hat{e}_{L}$ since the Wilson loop creates the unit
electric flux along the loop.131313 We can say that the two ground states
${\left|{\mbox{GS}\,s}\right>}$ and ${\left|{\mbox{GS}[\alpha,\beta]}\right>}$
are complementary in the sense of quantum theory. Namely, they are analogous
to the position eigenstate ${\left|{x}\right>}$ and the momentum eigenstate
${\left|{p}\right>}$ with the commutation relation of the corresponding
operators $[\hat{x},\hat{p}]=\mathrm{i}$. The relation
${\left|{p}\right>}=\int dx\,e^{\mathrm{i}px}{\left|{x}\right>}$ is similar to
the ground states related by the discrete Fourier transformations. The
operator $e^{\mathrm{i}a\hat{x}}$, which is analogous to $z_{v}^{k}$ or
$Z(C_{L})$, measures the position when it acts on the position eigenstate,
whereas it creates a shift of the momentum when it acts on the momentum
eigenstate.

#### Norm of the ground state ${\left|{\mbox{GS}[\alpha,\beta]}\right>}$

Since the operator (3.20) satisfies ${\cal P}={\cal P}^{\dagger}$, ${\cal
P}^{2}={\cal P}$ and commutes with $A_{v}$ for any $v$,

$\displaystyle\langle\mbox{GS}[\alpha,\beta]{\left|{\mbox{GS}[\alpha,\beta]}\right>}={\cal
N}{\left<{s=0}\right|}\left(\prod_{v\in V}A_{v}\right){\cal
P}{\left|{s=0}\right>}$ (3.29) $\displaystyle=$ $\displaystyle{\cal
N}\left(\prod_{e\in E}{\left<{0_{e}}\right|}\right)\left\\{\prod_{v\in
V}{\left<{0_{v}}\right|}A_{v}P_{v}^{[\alpha_{v}]}{\left|{0_{v}}\right>}\right\\}\left(\prod_{L=1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\left(\prod_{e\in
E}{\left|{0_{e}}\right>}\right).$

Then,

${\left<{0_{v}}\right|}A_{v}P_{v}^{[\alpha_{v}]}{\left|{0_{v}}\right>}={\left<{0_{v}}\right|}A_{v}\left(P_{v}^{[\alpha_{v}]}\right)^{\dagger}{\left|{0_{v}}\right>}=\frac{1}{np}\sum_{j=0}^{n-1}\sum_{a=0}^{p-1}\omega_{p}^{-a\alpha_{v}}{\bf
X}_{L_{v}}^{pj-a}\delta_{m}(pj,a),$ (3.30)

where ${\left<{0_{v}}\right|}x_{v}^{pj-a}{\left|{0_{v}}\right>}$ gives
$\delta_{m}(pj,a)$. The nonzero contribution comes from $a=0$ and $j=ku$
($u=0,1,\cdots,q-1$), which leads to

${\left<{0_{v}}\right|}A_{v}P_{v}^{[\alpha_{v}]}{\left|{0_{v}}\right>}=\frac{1}{np}\sum_{j=0}^{q-1}{\bf
X}_{L_{v}}^{kpj},$ (3.31)

and then

$\langle\mbox{GS}[\alpha,\beta]{\left|{\mbox{GS}[\alpha,\beta]}\right>}={\cal
N}\left(\frac{1}{np}\right)^{|V|}\left(\prod_{e\in
E}{\left<{0_{e}}\right|}\right)\left(\prod_{v\in V}\sum_{j=0}^{q-1}{\bf
X}_{L_{v}}^{kpj}\right)\left(\prod_{L=1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\left(\prod_{e\in
E}{\left|{0_{e}}\right>}\right).$ (3.32)

In evaluating this, we combine the results

${\left<{0_{e}}\right|}X_{e}^{kpj}X_{e}^{-kpj^{\prime}}{\left|{0_{e}}\right>}=\delta_{q}(j,j^{\prime})$
(3.33)

and141414(3.34) is derived as follows. Using (3.25), the LHS becomes
$\frac{1}{q}\sum_{b=0}^{q-1}\omega_{q}^{bp\beta_{L}}{\left<{0_{\hat{e}_{L}}}\right|}X_{\hat{e}_{L}}^{kp(j-j^{\prime}+b)}{\left|{0_{\hat{e}_{L}}}\right>}=\frac{1}{q}\sum_{b=0}^{q-1}\omega_{q}^{bp\beta_{L}}\delta_{n}(kp(j-j^{\prime}+b),0).$
The mod $n$ Kronecker delta is nonzero only when $b=j^{\prime}-j$ mod $q$,
which gives the RHS of (3.34).

${\left<{0_{\hat{e}_{L}}}\right|}X_{\hat{e}_{L}}^{kpj}X_{\hat{e}_{L}}^{-kpj^{\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}{\left|{0_{\hat{e}_{L}}}\right>}=\frac{1}{q}\,\omega_{q}^{\beta_{L}\,p(j^{\prime}-j)}.$
(3.34)

$v$$v^{\prime}$$j$$j^{\prime}$$e$ Figure 4: The indices $j$ and $j^{\prime}$
in the figure represents the operators ${\bf X}_{L_{v}}^{kpj}$ and ${\bf
X}_{L_{v^{\prime}}}^{kpj^{\prime}}$ sitting at the vertices $v$ and
$v^{\prime}$, respectively. The edge $e$ connects the vertices. These
operators yield the operators in the LHS of eq.(3.33). The result of (3.33)
shows that only the case $j=j^{\prime}$ (mod $q$) is relevant.

Let us first consider (3.33) for $e\in
E-\\{\hat{e}_{1},\cdots,\hat{e}_{B_{1}}\\}$. As seen in Fig. 4, The indices
$j$ and $j^{\prime}$ are associated to the vertices $v$ and $v^{\prime}$ at
both ends of the edge $e$. For giving nonzero contribution, $j$ and
$j^{\prime}$ should be equal (mod $q$). From the fact that the graph
$T=G-\\{\hat{e}_{1},\cdots,\hat{e}_{B_{1}}\\}$ is a connected tree graph, all
the vertices are connected by the mod $q$ Kronecker delta (3.33), which makes
all the indices $j$ equal. Then $\omega_{q}$-factor in (3.34) becomes 1.
Finally we have

$\langle\mbox{GS}[\alpha,\beta]{\left|{\mbox{GS}[\alpha,\beta]}\right>}={\cal
N}\left(\frac{1}{np}\right)^{|V|}\left(\sum_{j=0}^{q-1}1\right)\frac{1}{q^{B_{1}}}=\frac{1}{\textrm{GSD}},$
(3.35)

where we used (3.9).

#### Pure-state density matrix

Next, when ${\cal P}$ in (3.20) acts on other ${\left|{\mbox{GS}\,s}\right>}$
with $s\neq 0$, it also gives ${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ up to
some phase factors:

${\cal
P}{\left|{\mbox{GS}\,s}\right>}=e^{\mathrm{i}\theta_{s}}{\left|{\mbox{GS}[\alpha,\beta]}\right>},$
(3.36)

which leads to

${\cal P}{\left|{\mbox{GS}\,s}\right>}{\left<{\mbox{GS}\,s}\right|}{\cal
P}^{\dagger}={\left|{\mbox{GS}[\alpha,\beta]}\right>}{\left<{\mbox{GS}[\alpha,\beta]}\right|}$
(3.37)

for any $s$.

Finally, we find that the desired pure-state density matrix is given by the
${\cal P}$ transformation to the projector (3.1), $\pi_{0}=\prod_{v\in
V}A_{v}\prod_{e\in
E}B_{e}=\sum_{s=0}^{(\textrm{GSD})-1}{\left|{\mbox{GS}\,s}\right>}{\left<{\mbox{GS}\,s}\right|}$:

$\rho^{[\alpha,\beta]}\equiv{\cal P}\pi_{0}{\cal
P}^{\dagger}=\sum_{s=0}^{(\textrm{GSD})-1}{\cal
P}{\left|{\mbox{GS}\,s}\right>}{\left<{\mbox{GS}\,s}\right|}{\cal
P}^{\dagger}=(\textrm{GSD})\times{\left|{\mbox{GS}[\alpha,\beta]}\right>}{\left<{\mbox{GS}[\alpha,\beta]}\right|}.$
(3.38)

(3.35) means that $\rho^{[\alpha,\beta]}$ has the correct normalization ${\rm
Tr\,}\rho^{[\alpha,\beta]}=1$. From the properties of ${\cal P}$,
$\rho^{[\alpha,\beta]}$ is simplified as

$\rho^{[\alpha,\beta]}=\left(\prod_{v\in V}A_{v}\prod_{e\in
E}B_{e}\right){\cal P}.$ (3.39)

##  4 Entanglement Entropy

To better understand the ground states of the Hamiltonian (2.13), we compute
their EE. As the system is gapped, we expect that the leading order term is
proportional to the ‘area’ of the boundary of a bipartition of the system (the
area law), which is proven in gapped one-dimensional systems [34]. Further
interesting features are expected in a constant sub-leading term, called
topological EE [5, 35, 36, 37, 38], which is a speculated signal for a
topologically ordered state. This is a global term to the EE that is present
regardless of the partition of the system.

In this section, we exactly compute the EE (and thus the topological EE) both
for the individual ground states ${\left|{\mbox{GS}\,s}\right>}$ and their
linear combinations ${\left|{\mbox{GS}[\alpha,\beta]}\right>}$, with respect
to a bipartite separation of the system. We first split the total system given
by the graph $G$ into the three parts:

$G=G_{1}+G_{2}+E_{12},$ (4.1)

where each of $G_{1}$ and $G_{2}$ is a connected subgraph, and $E_{12}$ is a
set of edges connecting $G_{1}$ (at the vertices $\nu_{1},\cdots,\nu_{r}$) and
$G_{2}$ (at the vertices $\bar{\nu}_{1},\cdots,\bar{\nu}_{r^{\prime}}$). An
example of the division (4.1) is depicted in Fig. 5.

$\nu_{1}$$\nu_{2}$$\nu_{3}$$\nu_{r}$$\bar{\nu}_{1}$$\bar{\nu}_{2}$$\bar{\nu}_{3}$$\bar{\nu}_{r^{\prime}}$$G_{1}$$G_{2}$
Figure 5: An example of the division (4.1). The orange (light blue) circle and
its interior represent the region where the connected subgraph $G_{1}$
($G_{2}$) is located. Edges and vertices in the interior are suppressed. The
black lines with arrows are edges belonging to $E_{12}$, which connect $G_{1}$
at the vertices $\nu_{1},\nu_{2},\cdots,\nu_{r}$ and $G_{2}$ at the vertices
$\bar{\nu}_{1},\bar{\nu}_{2},\cdots,\bar{\nu}_{r^{\prime}}$.

$N_{v}$ and $N_{e}$ denote the numbers of vertices and edges of $G_{1}$, and
$M_{v}$ and $M_{e}$ denote those of $G_{2}$. $E_{12}$ consists of $f$ edges.
Then, the first Betti numbers of $G_{1}$ and $G_{2}$ are given by

$B_{1}^{\prime}=N_{e}-N_{v}+1\qquad\mbox{and}\qquad
B_{1}^{\prime\prime}=M_{e}-M_{v}+1,$ (4.2)

respectively. Then the first Betti number of $G$ is

$B_{1}=|E|-|V|+1=(N_{e}+M_{e}+f)-(N_{v}+M_{v})+1=B_{1}^{\prime}+B_{1}^{\prime\prime}+f-1.$
(4.3)

For computing the bipartite EE, we take a subsystem $A$ as $G_{1}$ and trace
out the degrees of freedom of the rest $B=G_{2}+E_{12}$. For each
$t=1,\cdots,r$, we divide a set of the edges attaching to the vertex
$\nu_{t}$, $L_{t}(\equiv L_{\nu_{t}})$,151515In this section, we often write
the subscripts $\nu_{t}$ ($t=1,\cdots,r$) and $\bar{\nu}_{t}$
$(t=1,\cdots,r^{\prime}$) as $t$ and $\bar{t}$ for notational simplicity. Then
$\bar{t}$ runs over $\bar{1},\cdots,\bar{r^{\prime}}$.  into a set of those
belonging to $G_{1}$, $L^{\prime}_{t}(=L_{t}\cap G_{1})$, and a set of those
belonging to $E_{12}$, $\tilde{L}_{t}(=L_{t}\cap E_{12})$:

$L_{t}=L^{\prime}_{t}+\tilde{L}_{t}\qquad(t=1,\cdots,r).$ (4.4)

Likewise, for $\bar{t}=\bar{1},\cdots,\bar{r^{\prime}}$, $L_{\bar{t}}(\equiv
L_{\bar{\nu}_{t}})$ is divided into $L^{\prime}_{\bar{t}}(=L_{\bar{t}}\cap
G_{2})$ and $\tilde{L}_{\bar{t}}(=L_{\bar{t}}\cap E_{12})$:

$L_{\bar{t}}=L^{\prime}_{\bar{t}}+\tilde{L}_{\bar{t}}\quad(\bar{t}=\bar{1},\cdots,\bar{r^{\prime}}).$
(4.5)

Correspondingly, ${\bf X}_{L_{t}}$ and ${\bf X}_{L_{\bar{t}}}$ are factored as

${\bf X}_{L_{t}}={\bf X}_{L^{\prime}_{t}}{\bf
X}_{\tilde{L}_{t}}\quad(t=1,\cdots,r)\qquad\mbox{and}\qquad{\bf
X}_{L_{\bar{t}}}={\bf X}_{L^{\prime}_{\bar{t}}}{\bf
X}_{\tilde{L}_{\bar{t}}}\qquad(\bar{t}=\bar{1},\cdots,\bar{r^{\prime}}).$
(4.6)

Then,
$E_{12}=\\{\tilde{L}_{1},\cdots,\tilde{L}_{r}\\}=\\{\tilde{L}_{\bar{1}},\cdots,\tilde{L}_{\bar{r^{\prime}}}\\}$.
For any edge connecting the vertices $\nu_{t}\in G_{1}$ and
$\bar{\nu}_{t^{\prime}}\in G_{2}$, when the edge is incoming to $\nu_{t}$, it
is outgoing from $\bar{\nu}_{t^{\prime}}$, and vice versa. Thus,

$\prod_{t=1}^{r}{\bf
X}_{\tilde{L}_{t}}=\prod_{\bar{t}=\bar{1}}^{\bar{r^{\prime}}}{\bf
X}_{\tilde{L}_{\bar{t}}}^{-1}$ (4.7)

holds.

###  4.1 Bipartite EE for ${\left|{\mbox{GS}\,s}\right>}$

We start with a pure state described by the density matrix

$\rho_{s}={\left|{\mbox{GS}\,s}\right>}{\left<{\mbox{GS}\,s}\right|}\qquad\mbox{with}\qquad
s=0,1,\cdots,(\mbox{GSD})-1,$ (4.8)

and compute the reduced density matrix by tracing out the degrees of freedom
of $B=G_{2}+E_{12}$:

$\rho_{s,\,A}={\rm Tr\,}_{B}\,\rho_{s}=\frac{n^{N_{v}+M_{v}}}{q}\,{\rm
Tr\,}_{B}\left[\left(\prod_{v\in
V}A_{v}\right){\left|{s}\right>}{\left<{s}\right|}\left(\prod_{v\in
V}A_{v}\right)\right],$ (4.9)

where (3.4) and (3.9) are used. On the RHS of

$\prod_{v\in V}A_{v}=\left(\prod_{v\in
G_{1}-\\{\nu_{1},\cdots,\nu_{r}\\}}A_{v}\right)\left(\prod_{t=1}^{r}A_{t}\right)\left(\prod_{v\in
G_{2}}A_{v}\right),$ (4.10)

the first factor is irrelevant to the trace, whereas the last factor is fully
traced out and the trace cyclicity can be applied. ${\left|{s}\right>}$ in
(3.11) can be expressed as a product state

${\left|{s}\right>}=\prod_{v\in V}{\left|{h_{v}}\right>}\prod_{e\in
E}{\left|{i_{e}}\right>}\qquad\mbox{for some
}h_{v}\in\mathbb{Z}_{m},\,i_{e}\in\mathbb{Z}_{n}$ (4.11)

which is similarly decomposed. Then (4.9) reads

$\rho_{s,\,A}=\frac{n^{N_{v}+M_{v}}}{q}\,{\cal A}_{1}\,{\rm
Tr\,}_{B}\left[{\cal A}_{\rm bdy}\left(\prod_{v\in
G_{2}}A_{v}{\left|{h_{v}}\right>}{\left<{h_{v}}\right|}\right)\left(\prod_{e\in
B}{\left|{i_{e}}\right>}{\left<{i_{e}}\right|}\right){\cal A}_{\rm
bdy}^{\dagger}\right]{\cal A}_{1}^{\dagger}$ (4.12)

with

$\displaystyle{\cal A}_{1}\equiv\left(\prod_{v\in
G_{1}-\\{\nu_{1},\cdots,\nu_{r}\\}}A_{v}\right)\left(\prod_{e\in
G_{1}-\\{L^{\prime}_{1},\cdots,L^{\prime}_{r}\\}}{\left|{i_{e}}\right>}\right),$
(4.13) $\displaystyle{\cal A}_{\rm
bdy}\equiv\prod_{t=1}^{r}A_{t}{\left|{h_{t}}\right>}{\left|{i_{L^{\prime}_{t}}}\right>},\qquad{\left|{i_{L^{\prime}_{t}}}\right>}\equiv\prod_{e\in
L^{\prime}_{t}}{\left|{i_{e}}\right>}.$ (4.14)

Note that ${\cal A}_{1}$ and ${\cal A}_{\rm bdy}$ act nontrivially on the
Hilbert space on $e\in\\{L^{\prime}_{1},\cdots,L^{\prime}_{r}\\}\subset G_{1}$
and that on $e\in\\{\tilde{L}_{1},\cdots,\tilde{L}_{r}\\}=E_{12}$,
respectively.

#### Computation of ${\rm Tr\,}_{G_{2}}$

In the computation of ${\rm Tr\,}_{B}={\rm Tr\,}_{E_{12}}{\rm Tr\,}_{G_{2}}$,
let us first compute ${\rm Tr\,}_{v\in G_{2}}$ and then ${\rm Tr\,}_{e\in
G_{2}}$.

The relevant part of the former is only the second factor in ${\rm
Tr\,}_{B}[\cdots]$ in (4.12):

${\rm Tr\,}_{v\in G_{2}}\left[\prod_{v\in
G_{2}}A_{v}{\left|{h_{v}}\right>}{\left<{h_{v}}\right|}\right]=\prod_{v\in
G_{2}}{\left<{h_{v}}\right|}A_{v}{\left|{h_{v}}\right>}=\left(\frac{1}{n}\right)^{M_{v}}\prod_{v\in
G_{2}}\sum_{b=0}^{q-1}{\bf X}_{L_{v}}^{pbk}.$ (4.15)

As is seen in (3.6), only $j=bk$ ($b\in\mathbb{Z}_{q}$) terms in (2.4) give
nonvanishing contribution. Next, ${\rm Tr\,}_{e\in G_{2}}$ is computed as

$\displaystyle{\rm Tr\,}_{e\in G_{2}}\left[\left(\prod_{v\in
G_{2}}\sum_{b=0}^{q-1}{\bf X}_{L_{v}}^{pbk}\right)\left(\prod_{e\in
G_{2}}{\left|{i_{e}}\right>}{\left<{i_{e}}\right|}\right)\right]=\left(\prod_{e\in
G_{2}}{\left<{i_{e}}\right|}\right)\left(\prod_{v\in
G_{2}}\sum_{b=0}^{q-1}{\bf X}_{L_{v}}^{pbk}\right)\left(\prod_{e\in
G_{2}}{\left|{i_{e}}\right>}\right)$
$\displaystyle=\sum_{b=0}^{q-1}\,\prod_{\bar{t}=\bar{1}}^{\bar{r^{\prime}}}{\bf
X}_{\tilde{L}_{\bar{t}}}^{pbk}=\sum_{b=0}^{q-1}\,\prod_{t=1}^{r}{\bf
X}_{\tilde{L}_{t}}^{pbk},$ (4.16)

where (4.7) and the change of the summation index $b\to q-b$ were used at the
last equality.

Now we find

$\rho_{s,\,A}=\frac{n^{N_{v}}}{q}\,{\cal A}_{1}\,{\rm
Tr\,}_{E_{12}}\left[{\cal A}_{\rm
bdy}\left(\sum_{b=0}^{q-1}\,\prod_{t=1}^{r}{\bf
X}_{\tilde{L}_{t}}^{pbk}\right)\left(\prod_{e\in
E_{12}}{\left|{i_{e}}\right>}{\left<{i_{e}}\right|}\right){\cal A}_{\rm
bdy}^{\dagger}\right]{\cal A}_{1}^{\dagger}.$ (4.17)

#### Computation of ${\rm Tr\,}_{E_{12}}$

From (4.4), $A_{t}(\equiv A_{\nu_{t}})$ can be expressed as
$A_{t}=\frac{1}{n}\sum_{j=0}^{n-1}x^{pj}_{t}{\bf X}_{L^{\prime}_{t}}^{pj}{\bf
X}_{\tilde{L}_{t}}^{pj}$. ${\rm Tr\,}_{E_{12}}[\cdots]$ in (4.17) becomes

$\displaystyle(\mbox{${\rm Tr\,}_{E_{12}}[\cdots]$ in
(\ref{rhosA3})})=\sum_{b=0}^{q-1}\prod_{t=1}^{r}\left\\{\left(\frac{1}{n}\right)^{2}\sum_{j,j^{\prime}=0}^{n-1}x_{t}^{pj}{\left|{h_{t}}\right>}\,{\bf
X}_{L^{\prime}_{t}}^{pj}{\left|{i_{L^{\prime}_{t}}}\right>}\right.$
$\displaystyle\left.\times{\rm Tr\,}_{\tilde{L}_{t}}\left[{\bf
X}_{\tilde{L}_{t}}^{pj+pbk}\left(\prod_{e\in\tilde{L}_{t}}{\left|{i_{e}}\right>}{\left<{i_{e}}\right|}\right){\bf
X}_{\tilde{L}_{t}}^{-pj^{\prime}}\right]{\left<{i_{L^{\prime}_{t}}}\right|}{\bf
X}_{L^{\prime}_{t}}^{-pj^{\prime}}\,{\left<{h_{t}}\right|}x_{t}^{-pj^{\prime}}\right\\},$
(4.18)

in which ${\rm Tr\,}_{\tilde{L}_{t}}[\cdots]$ gives
$\delta_{n}\left(p(-j^{\prime}+j+bk),0\right)$. From (2.3), the Kronecker
delta means that $\tilde{p}(-j^{\prime}+j+bk)=0$ mod $\tilde{k}q$. Note that
$\gcd(p,q)=\gcd(\xi\tilde{p},q)=1$ is equivalent to $\gcd(\xi,q)=1$ and
$\gcd(\tilde{p},q)=1$. Combining this and (2.3), we find
$\gcd(\tilde{p},\tilde{k}q)=1$. Thus, $j^{\prime}$ giving nonzero contribution
is $j^{\prime}=j+bk$ mod $\tilde{k}q$, i.e.,

$j^{\prime}=j+bk-\tilde{k}qu\qquad(u\in\mathbb{Z}_{\xi}),$ (4.19)

which leads to

$(\mbox{${\rm Tr\,}_{E_{12}}[\cdots]$ in
(\ref{rhosA3})})=\left(\sum_{b=0}^{q-1}\,\prod_{t=1}^{r}{\bf
X}_{L^{\prime}_{t}}^{-pbk}\right)\prod_{t=1}^{r}\left\\{\frac{\xi}{n^{2}}\left(\sum_{j=0}^{n-1}P_{t,\,j}P_{L^{\prime}_{t},\,j}\right)Q_{t}\right\\}.$
(4.20)

Here $P_{t,\,j}$, $P_{L^{\prime}_{t},\,j}$ and $Q_{t}$ are projection
operators defined by

$P_{t,\,j}\equiv
x_{t}^{pj}{\left|{h_{t}}\right>}{\left<{h_{t}}\right|}x_{t}^{-pj},\qquad
P_{L^{\prime}_{t},\,j}\equiv{\bf
X}_{L^{\prime}_{t}}^{pj}{\left|{i_{L^{\prime}_{t}}}\right>}{\left<{i_{L^{\prime}_{t}}}\right|}{\bf
X}_{L^{\prime}_{t}}^{-pj},\qquad
Q_{t}\equiv\frac{1}{\xi}\sum_{u=0}^{\xi-1}x_{t}^{p\tilde{k}qu}.$ (4.21)

In deriving (4.20), $pbk=mb$ and $p\tilde{k}qu=\tilde{p}kqu=\tilde{p}nu$ were
used.

Since it can be seen that the property

$\left(P_{t,\,j}P_{L^{\prime}_{t},\,j}\right)\left(P_{t,\,j^{\prime}}P_{L^{\prime}_{t},\,j^{\prime}}\right)=\delta_{j,j^{\prime}}\left(P_{t,\,j}P_{L^{\prime}_{t},\,j}\right)$
(4.22)

holds, we introduce more projection operators as

$P_{t}\equiv\sum_{j=0}^{n-1}P_{t,\,j}P_{L^{\prime}_{t},\,j},\qquad
Q^{\prime}\equiv\frac{1}{q}\sum_{b=0}^{q-1}\,\prod_{t=1}^{r}{\bf
X}_{L^{\prime}_{t}}^{-pbk},$ (4.23)

and obtain

$(\mbox{${\rm Tr\,}_{E_{12}}[\cdots]$ in
(\ref{rhosA3})})=\left(\frac{\xi}{n^{2}}\right)^{r}q\,Q^{\prime}\left(\prod_{t=1}^{r}P_{t}Q_{t}\right).$
(4.24)

Plugging (4.24) to (4.17), we end up with

$\rho_{s,\,A}=n^{N_{v}-2r}\xi^{r}\,{\cal
A}_{1}\,Q^{\prime}\left(\prod_{t=1}^{r}P_{t}Q_{t}\right){\cal
A}_{1}^{\dagger}.$ (4.25)

Notice that the projection operators $Q^{\prime}$, $P_{t}$ and $Q_{t}$
mutually commute. $P_{t}x_{t}^{p\tilde{k}qu}=x_{t}^{p\tilde{k}qu}P_{t}$ and
$P_{t}{\bf X}_{L^{\prime}_{t}}^{-pbk}={\bf X}_{L^{\prime}_{t}}^{-pbk}P_{t}$
are verified by shifting $j$ in the sum in the definition of $P_{t}$ (4.23) as
$j\to j+\tilde{k}qu$ and $j\to j-bk$, respectively. This leads to
$P_{t}Q_{t}=Q_{t}P_{t}$ and $P_{t}Q^{\prime}=Q^{\prime}P_{t}$.

#### Bipartite EE

A similar computation of (4.15) and (4.16) gives

${\cal A}_{1}^{\dagger}{\cal
A}_{1}=n^{-N_{v}+r}\sum_{b=0}^{q-1}\prod_{t=1}^{r}{\bf
X}_{L^{\prime}_{t}}^{-pbk}=n^{-N_{v}+r}q\,Q^{\prime}.$ (4.26)

Then we find

$\rho_{s,\,A}^{2}=\tilde{n}^{-r}q\,\rho_{s,\,A}$ (4.27)

with (2.16). It means ${\rm
Spec}\,\rho_{s,\,A}=\left\\{\tilde{n}^{-r}q,0\right\\}$. As it should be from
${\rm Tr\,}\rho_{s}=1$, we can directly check ${\rm Tr\,}\rho_{s,\,A}=1$ from
the expression (4.25). Thus, it is seen that $\rho_{s,\,A}$ has the eigenvalue
$\tilde{n}^{-r}q$ with multiplicity $\tilde{n}^{r}q^{-1}$.

Finally, the bipartite EE is obtained as

$\displaystyle S_{s,\,A}$ $\displaystyle=$ $\displaystyle-{\rm
Tr\,}\left[\rho_{s,\,A}\log_{2}\rho_{s,\,A}\right]=-\left\\{\left(\tilde{n}^{-r}q\right)\log_{2}\left(\tilde{n}^{-r}q\right)\right\\}\times\tilde{n}^{r}q^{-1}$
(4.28) $\displaystyle=$
$\displaystyle\left(\log_{2}\tilde{n}\right)r-\log_{2}q.$

The result (4.28) is independent of $a_{v}$’s and $b_{L}$’s which labels the
state ${\left|{s}\right>}$. Namely, all the individual ground state
${\left|{\mbox{GS}\,s}\right>}$ gives the same EE. As $r$ grows the linear
term of $r$ dominates, which shows that contribution around the boundary
between the subsystems becomes dominant in the EE. Namely the EE obeys the
area law. On the other hand, the constant term $-\log_{2}q$ characterizes a
global feature of the entanglement of the ground state, which is called the
topological EE [5].161616 In [5, 38], prescriptions are presented to obtain
the constant contribution eliminating the short range effects, in which it is
not necessary to identify the linear term to be subtracted from the whole
expression.  The topological EE is denoted by $-\gamma$. Here we have

$\gamma=\log_{2}q$ (4.29)

and the total quantum dimension is $D=q$. For $q=1$ ($m$ is an integer
multiple of $n$: $m=pn$), the topological EE vanishes.

For $r\geq 2$, (4.28) vanishes only when $\tilde{k}=q=1$ (i.e., $m$ is an
integer multiple of $n^{2}$: $m=\tilde{p}n^{2}$). Then, the vertex operator
$A_{v}$ reduces to a strictly local operator (nontrivially acting only on the
vertex $v$):
$A_{v}=\frac{1}{\xi}\sum_{j=0}^{\xi-1}x_{v}^{\xi\tilde{p}j}1_{L_{v}}$. Since
the $A_{v}$ does not generate entanglement, the ground state (3.4) becomes a
product state.

The ground states ${\left|{\mbox{GS}\,s}\right>}$ have definite magnetic flux
for each independent closed path on the graph. This corresponds to the basis
state which maximizes the negative of the topological EE, $\gamma$, according
to [30, 31]. However, in the next subsection we will see that it does not
always mean minimizing the whole EE, which is the case in the toric code [30,
31].

###  4.2 Bipartite EE for $\rho^{[\alpha,\beta]}$

In the computation of the EE for the density matrix (3.39), let us consider
the case that among the $\hat{e}_{L}$’s, the first $B_{1}^{\prime}$,
$\\{\hat{e}_{1},\cdots,\hat{e}_{B_{1}^{\prime}}\\}$, are in $G_{1}$, the next
$B_{1}^{\prime\prime}$,
$\\{\hat{e}_{B_{1}^{\prime}+1},\cdots,\hat{e}_{B_{1}^{\prime}+B_{1}^{\prime\prime}}\\}$,
in $G_{2}$, and the rest,
{$\hat{e}_{B_{1}^{\prime}+B_{1}^{\prime\prime}+1},\cdots,\hat{e}_{B_{1}}\\}$,
in $E_{12}$. From (4.3), we see that $(f-1)$ of the $f$ edges of $E_{12}$ are
$\hat{e}_{L}$’s. Notice that any choice of $\hat{e}_{L}$’s can be reduced to
the case as discussed in section 3.3.

The reduced density matrix reads

$\displaystyle\rho_{A}^{[\alpha,\beta]}$ $\displaystyle=$ $\displaystyle{\rm
Tr\,}_{B}\,\rho^{[\alpha,\beta]}={\rm
Tr\,}_{B}\left[\left(\prod_{v}A_{v}P_{v}^{[\alpha_{v}]}\right)\left(\prod_{e}B_{e}\right)\left(\prod_{L=1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\right]$
(4.30) $\displaystyle=$ $\displaystyle\left(\prod_{v\in
G_{1}-\\{\nu_{1},\cdots,\nu_{r}\\}}A_{v}P_{v}^{[\alpha_{v}]}\right)\left(\prod_{e\in
G_{1}}B_{e}\right)\left(\prod_{L=1}^{B_{1}^{\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)$
$\displaystyle\times{\rm
Tr\,}_{B}\left[\left(\prod_{t=1}^{r}A_{t}P_{t}^{[\alpha_{t}]}\right)\left(\prod_{v\in
G_{2}}A_{v}P_{v}^{[\alpha_{v}]}\right)\left(\prod_{e\in
B}B_{e}\right)\left(\prod_{L=B_{1}^{\prime}+1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\right].$

Here it is easy to see that for $\left(\prod_{e\in B}B_{e}\right)$ only
$\frac{1}{k}B_{e}^{(0)}$ in $B_{e}$ (2.9) gives nonzero contribution. So we
may replace $\left(\prod_{e\in B}B_{e}\right)$ with the factor $k^{-M_{e}-f}$.
The last line of (4.30) becomes

$\displaystyle(\textrm{last line of (\ref{rhoabA})})$ $\displaystyle=$
$\displaystyle k^{-M_{e}-f}\,{\rm
Tr\,}_{E_{12}}\left\\{\left(\prod_{t=1}^{r}A_{t}P_{t}^{[\alpha_{t}]}\right)\left(\prod_{L=B_{1}^{\prime}+B_{1}^{\prime\prime}+1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\right.$
(4.31) $\displaystyle\left.\times{\rm Tr\,}_{G_{2}}\left[\left(\prod_{v\in
G_{2}}A_{v}P_{v}^{[\alpha_{v}]}\right)\left(\prod_{L=B_{1}^{\prime}+1}^{B_{1}^{\prime}+B_{1}^{\prime\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\right]\right\\}.$

#### Computation of ${\rm Tr\,}_{G_{2}}$

For the computation of ${\rm Tr\,}_{G_{2}}$ in (4.31), we first evaluate ${\rm
Tr\,}_{v\in G_{2}}$ and then ${\rm Tr\,}_{e\in G_{2}}$.

In a similar manner to (3.31),

${\rm
Tr\,}_{v}\left(A_{v}P_{v}^{[\alpha_{v}]}\right)=\frac{1}{q}\sum_{j=0}^{q-1}{\bf
X}_{L_{v}}^{kpj},$ (4.32)

which leads to

$(\mbox{${\rm Tr\,}_{G_{2}}[\cdots]$ in (\ref{TrB})})=q^{-M_{v}}\,{\rm
Tr\,}_{e\in G_{2}}\left[\prod_{v\in G_{2}}\left(\sum_{j=0}^{q-1}{\bf
X}_{L_{v}}^{kpj}\right)\cdot\left(\prod_{L=B_{1}^{\prime}+1}^{B_{1}^{\prime}+B_{1}^{\prime\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\right].$
(4.33)

Similar to (3.33) and (3.34),

$\displaystyle{\rm
Tr\,}_{e}\left(X_{e}^{kpj}X_{e}^{-kpj^{\prime}}\right)=n\delta_{q}(j,j^{\prime}),$
(4.34) $\displaystyle{\rm
Tr\,}_{\hat{e}_{L}}\left(X_{\hat{e}_{L}}^{kpj}X_{\hat{e}_{L}}^{-kpj^{\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)=\frac{n}{q}\,\omega_{q}^{\beta_{L}\,p(j^{\prime}-j)}.$
(4.35)

Since the graph
$G_{2}-\\{\hat{e}_{B_{1}^{\prime}+1},\cdots,\hat{e}_{B_{1}^{\prime}+B_{1}^{\prime\prime}}\\}$
becomes a connected tree graph, all the $j$-indices become the same due to the
mod $q$ Kronecker delta (4.34) from each edge, and the $\omega_{q}$-factors
from (4.35) all become 1. Then we find

$(\mbox{${\rm Tr\,}_{G_{2}}[\cdots]$ in
(\ref{TrB})})=n^{M_{e}}q^{-M_{v}-B_{1}^{\prime\prime}}\,\sum_{j=0}^{q-1}\,\prod_{t=1}^{r^{\prime}}{\bf
X}_{\tilde{L}_{\bar{t}}}^{kpj},$ (4.36)

where (4.5) and (4.6) are used.

Plugging (4.36) to (4.31) we have

$(\textrm{last line of (\ref{rhoabA})})=k^{-f}q^{-1}\,{\rm
Tr\,}_{E_{12}}\left\\{\left(\prod_{t=1}^{r}A_{t}P_{t}^{[\alpha_{t}]}\right)\left(\prod_{L=B_{1}^{\prime}+B_{1}^{\prime\prime}+1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\sum_{j=0}^{q-1}\,\prod_{t=1}^{r}{\bf
X}_{\tilde{L}_{t}}^{-kpj}\right\\},$ (4.37)

after (4.2) and (4.7) are used.

#### Computation of ${\rm Tr\,}_{E_{12}}$

Using (3.19) and (4.6), we express (4.37) as

$\displaystyle(\textrm{last line of (\ref{rhoabA})})$ $\displaystyle=$
$\displaystyle
k^{-f}q^{-1}(np)^{-r}\sum_{j^{\prime}=0}^{q-1}\sum_{j_{1},\cdots,j_{r}=0}^{np-1}\left(\prod_{t=1}^{r}\omega_{p}^{j_{t}\alpha_{t}}x_{t}^{j_{t}}\,{\bf
X}_{L^{\prime}_{t}}^{j_{t}}\right)$ (4.38) $\displaystyle\times{\rm
Tr\,}_{E_{12}}\left\\{\left(\prod_{t=1}^{r}{\bf
X}_{\tilde{L}_{t}}^{j_{t}-kpj^{\prime}}\right)\left(\prod_{L=B_{1}^{\prime}+B_{1}^{\prime\prime}+1}^{B_{1}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)\right\\}.$

Note that $j_{t}$ is associated to the vertex $\nu_{t}$ and $j^{\prime}$ is
associated to the vertices $\bar{\nu}_{1},\cdots,\bar{\nu}_{r^{\prime}}$.

Computation of the trace on each edge goes as

${\rm
Tr\,}_{e}\left(X_{e}^{-j_{t}}X_{e}^{kpj^{\prime}}\right)=n\delta_{n}(j_{t},kpj^{\prime}),$
(4.39)

from which $j_{t}$ giving nonzero contribution is

$j_{t}=kpj^{\prime}+kqu\qquad(\mbox{mod $np$})$ (4.40)

with $j^{\prime}=0,1,\cdots,q-1$ and $u=0,1,\cdots,p-1$. Also,

${\rm
Tr\,}_{\hat{e}_{L}}\left(X_{\hat{e}_{L}}^{-j_{t}}X_{\hat{e}_{L}}^{kpj^{\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)=\frac{1}{q}\sum_{b=0}^{q-1}\omega_{q}^{b\beta_{L}}\,{\rm
Tr\,}_{\hat{e}_{L}}X_{\hat{e}_{L}}^{-j_{t}+kpj^{\prime}+bk}=\frac{n}{q}\sum_{b=0}^{q-1}\omega_{q}^{b\beta_{L}}\,\delta_{n}(j_{t},k(b+pj^{\prime})).$
(4.41)

Here, $j_{t}$ giving nonzero contribution is

$j_{t}=k\tilde{j}_{t}\qquad(\tilde{j}_{t}=0,1,\cdots,pq-1),$ (4.42)

and then we find

${\rm
Tr\,}_{\hat{e}_{L}}\left(X_{\hat{e}_{L}}^{-j_{t}}X_{\hat{e}_{L}}^{kpj^{\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)=\frac{n}{q}\,\omega_{q}^{\beta_{L}\,(\tilde{j}_{t}-pj^{\prime})}.$
(4.43)

As mentioned above, $(f-1)$ of the $f$ edges in $E_{12}$ are $\hat{e}_{L}$’s.
We assume that the only one edge in $E_{12}$ which is not $\hat{e}_{L}$
attaches to the vertex $\nu_{1}$. It does not lose generality, since this
situation can be always realized by appropriately renaming the vertices
$\nu_{1},\cdots,\nu_{r}$.

Plugging (4.39)-(4.43) to (4.38) leads to

$\displaystyle(\textrm{last line of (\ref{rhoabA})})$ $\displaystyle=$
$\displaystyle(np)^{-r}\sum_{j^{\prime}=0}^{q-1}\sum_{u=0}^{p-1}\omega_{p}^{kq\alpha_{1}u}x_{1}^{kqu}\,{\bf
X}_{L^{\prime}_{1}}^{kpj^{\prime}}$ (4.44)
$\displaystyle\times\sum_{\tilde{j}_{2},\cdots,\tilde{j}_{r}=0}^{pq-1}\left(\prod_{t=2}^{r}\omega_{p}^{k\alpha_{t}\tilde{j}_{t}}\omega_{q}^{\beta_{t}^{\prime}\,(\tilde{j}_{t}-pj^{\prime})}\,x_{t}^{k\tilde{j}_{t}}\,{\bf
X}_{L^{\prime}_{t}}^{k\tilde{j}_{t}}\right),$

where $\beta_{t}^{\prime}$ is the sum of the currents $\beta_{L}$’s flowing
from the vertex $\nu_{t}$ (with the index $j_{t}$) to the vertices
$\bar{\nu}_{1},\cdots,\bar{\nu}_{r^{\prime}}$ (with $j^{\prime}$). For
example, in case that the vertex $\nu_{t}$ is attached to three
$\hat{e}_{L}$’s in $E_{12}$, say $\hat{e}_{L_{1}}$, $\hat{e}_{L_{2}}$ and
$\hat{e}_{L_{3}}$, as in Fig, 6, we have
$\beta_{t}^{\prime}=\beta_{L_{1}}+\beta_{L_{2}}+\beta_{L_{3}}$.

$\nu_{t}$$\bar{\nu}_{1},\cdots,\bar{\nu}_{r^{\prime}}$$j_{t}$$j^{\prime}$$\hat{e}_{L_{1}}$$\hat{e}_{L_{2}}$$\hat{e}_{L_{3}}$
Figure 6: The index $j_{t}$ is associated to the vertex $\nu_{t}$, which is
attached to the three edges $\hat{e}_{L_{1}}$, $\hat{e}_{L_{2}}$ and
$\hat{e}_{L_{3}}$ in $E_{12}$. The other ends of the edges are either of the
vertices $\bar{\nu}_{1},\cdots,\bar{\nu}_{r^{\prime}}$ which are depicted by a
single dot, because they are endowed with the common index $j^{\prime}$ as a
result of the computation of ${\rm Tr\,}_{G_{2}}$. In this case,
$\beta_{t}^{\prime}$ in (4.44) is given by
$\beta_{t}^{\prime}=\beta_{L_{1}}+\beta_{L_{2}}+\beta_{L_{3}}$.

We now define projection operators as

$\displaystyle
Q_{1,p}\equiv\frac{1}{p}\sum_{u=0}^{p-1}\omega_{p}^{kq\alpha_{1}u}x_{1}^{kqu},\qquad
Q_{1,q}\equiv\frac{1}{q}\sum_{j^{\prime}=0}^{q-1}\omega_{q}^{-pj^{\prime}(\sum_{t=2}^{r}\beta_{t}^{\prime})}\,{\bf
X}_{L_{1}^{\prime}}^{kpj^{\prime}},$
$\displaystyle\tilde{Q}_{t}\equiv\frac{1}{pq}\sum_{\tilde{j}=0}^{pq-1}\omega_{p}^{k\alpha_{t}\tilde{j}}\omega_{q}^{\beta_{t}^{\prime}\tilde{j}}\,x_{t}^{k\tilde{j}}\,{\bf
X}_{L^{\prime}_{t}}^{k\tilde{j}}$ (4.45)

for $t=2,\cdots,r$, and express (4.44) as

$(\textrm{last line of
(\ref{rhoabA})})=k^{-r}Q_{1,p}Q_{1,q}\prod_{t=2}^{r}\tilde{Q}_{t}.$ (4.46)

Finally, (4.30) becomes

$\rho_{A}^{[\alpha,\beta]}=k^{-r}\left(\prod_{v\in
G_{1}-\\{\nu_{1},\cdots,\nu_{r}\\}}A_{v}P_{v}^{[\alpha_{v}]}\right)\left(\prod_{e\in
G_{1}}B_{e}\right)\left(\prod_{L=1}^{B_{1}^{\prime}}P_{\hat{e}_{L}}^{[\beta_{L}]}\right)Q_{1,p}Q_{1,q}\prod_{t=2}^{r}\tilde{Q}_{t}.$
(4.47)

#### Result of EE

It is easy to see that

$\textrm{Spec}\left(\rho_{A}^{[\alpha,\beta]}\right)=\\{k^{-r},0\\},$ (4.48)

because the RHS of (4.47) is the product of commuting projectors except the
factor $k^{-r}$. It can be directly checked that ${\rm
Tr\,}\rho_{A}^{[\alpha,\beta]}=1$ holds as it should be from ${\rm
Tr\,}\rho^{[\alpha,\beta]}=1$. This shows that the reduced density matrix has
the eigenvalue $k^{-r}$ with the multiplicity $k^{r}$.

The bipartite EE is found as

$S_{A}^{[\alpha,\beta]}=-{\rm
Tr\,}\left(\rho_{A}^{[\alpha,\beta]}\log_{2}\rho_{A}^{[\alpha,\beta]}\right)=-\left(k^{-r}\log_{2}k^{-r}\right)\times
k^{r}=(\log_{2}k)\,r.$ (4.49)

This is proportional to $r$ (the ‘area’ of the boundary), which exhibits the
area law. The result is independent of the choice of $\alpha_{v}$’s or
$\beta_{L}$’s. There is no constant term, namely the topological EE vanishes.

Comparing to the result for the individual ground state
${\left|{\mbox{GS}\,s}\right>}$ (4.28), we can see that when $\xi=1$ (i.e.,
$k$ and $p$ are coprime),

$S_{A}^{[\alpha,\beta]}\leq S_{s,A}$ (4.50)

always holds. (4.50) is equivalent to $q^{r-1}\geq 1$, which is valid for any
positive integers $q$ and $r$. When $\xi\neq 1$, (4.50) is equivalent to

$q\leq\left(\frac{q}{\xi}\right)^{r},$ (4.51)

which holds when $q>\xi$ for $r$ large. On the other hand, $S_{s,\,A}$ is
smaller than $S_{A}^{[\alpha,\beta]}$ when $\xi>q$ for any $r$.

Note that in the case $S_{A}^{[\alpha,\beta]}<S_{s,A}$ the basis state
maximizing the minus of the topological EE does not minimize the EE due to the
contribution from the leading term proportional to $r$. In [30], since the
leading term of the EE is common among the bases of the degenerate ground
states, the basis states which maximize the negative of the topological EE are
called the minimum entropy states. However this does not always hold here,
because the basis change affects the leading term as well as the constant term
of the EE. For $q>\xi$ this can be seen as a distinguishing feature of our
model from the toric code.

##  5 Excited states

In this section we obtain the first and second excited states of the model.
There are anyon-like excitations among them, and their relevance to the
obtained topological EE is discussed. Further we discuss the statistics of the
obtained anyons for exchange processes that are peculiar to graphs.

Excited states of the models governed by the Hamiltonian in (2.13) appear when
at least one of the edge or vertex operators ($B_{e}$ or $A_{v}$) assumes the
zero-eigenvalue. Recall that each of these operators has the eigenvalues 0 and
1, since they are projection operators. In particular when some edge operators
(vertex operators) take the zero-eigenvalues we will denote them as edge
excitations (vertex excitations). When both edge and vertex operators take
zero-eigenvalues we end up with an example of a combined excitation.
Henceforth we use the phrase, ‘the edge or vertex operators are excited’, when
they assume the zero-eigenvalues.

In what follows we will show that every edge operator can be excited
independently, or in other words all the edge excitations are isolated. On the
other hand, some of the vertex excitations are isolated and the remaining can
only be excited in pairs, that is they are deconfined and there is no energy
cost in moving them around. This can be contrasted with the situation in the
abelian quantum double models where all the excitations are deconfined.

In sections 5.1 and 5.2, we discuss excitations on the ground states
${\left|{\mbox{GS}\,s}\right>}$. Excitations on the ground states
${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ are similarly constructed.

###  5.1 Edge excitations

The edge operator (2.9) having the zero-eigenvalue implies that one of its
orthogonal complements, $B_{e}^{[\alpha]}$ in (2.12) with $\alpha\neq 0$ mod
$k$, has the eigenvalue 1. To check if a single edge operator on $e^{\prime}$
is excited we follow the computations of the GSD in section 3.1 to evaluate

${\rm
Tr\,}_{\mathcal{H}}\left(B_{e^{\prime}}^{\left[\alpha\right]}\prod\limits_{v\in
V}A_{v}\prod\limits_{e\in
E-\\{e^{\prime}\\}}B_{e}\right)=p^{|V|}q^{B_{1}}=\mbox{GSD}\neq 0,$ (5.1)

which implies the existence of isolated edge excitations for all values of
$\alpha\in\\{1,2,\cdots,k-1\\}$. This exhausts the elementary edge excitations
of the theory meaning that a pair of edge excitations has to be composed of
two isolated edge excitations.

#### Excited states $X_{e}^{\beta}{\left|{\mbox{GS}\,s}\right>}$:

To obtain these isolated edge excitations, let us first pick the state
$X_{e}^{\beta}{\left|{\mbox{GS}\,s}\right>}$ with
$\beta\in\\{1,2,\cdots,n-1\\}$. The ground state
${\left|{\mbox{GS}\,s}\right>}$ is given by (3.4) and (3.11). Since
$X_{e}^{k}$ is a local symmetry mapping the ground state to some other ground
state, the above state for any $\beta$ reduces to the form
$X_{e}^{\beta}{\left|{\mbox{GS}\,s^{\prime}}\right>}$ with
$\beta\in\\{1,2,\cdots,k-1\\}$. Hence we may consider the case
$\beta\in\\{1,2,\cdots,k-1\\}$ without loss of generality.

From (2.14) we can see that

$B_{e}^{(j)}X_{e}^{\beta}=\omega_{k}^{\beta j}X_{e}^{\beta}B_{e}^{(j)}$ (5.2)

and thus

$B_{e}X_{e}^{\beta}=X_{e}^{\beta}B_{e}^{[\beta]}.$ (5.3)

In addition, since ${\left|{s}\right>}$ is an eigenstate of $B_{e}$ with the
eigenvalue 1,

$B_{e}^{[\alpha]}{\left|{s}\right>}=0\quad\mbox{for $\alpha\neq 0$ mod $k$}.$
(5.4)

(5.3) and (5.4) lead to

$B_{e}X_{e}^{\beta}{\left|{\mbox{GS}\,s}\right>}=0\quad\mbox{for
$\beta\in\\{1,2,\cdots,k-1\\}$},$ (5.5)

which implies that $X_{e}^{\beta}{\left|{\mbox{GS}\,s}\right>}$ are first
excited states with the energy $E_{0}+1$. Here, $E_{0}=-|V|-|E|$ is the ground
state energy.

We can also apply the operators $x_{v}^{\beta}$
($\beta\in\\{1,2,\cdots,k-1\\}$) on the vertices to excite all the edge
operators corresponding to the edges attached to the vertex $v$. As discussed
in section 2.3, any power of $x_{v}$ reduces to the above $x_{v}^{\beta}$ up
to the multiplications of the local symmetry operator $\left(x_{v}{\bf
X}_{L_{v}}\right)^{kq}=x_{v}^{kq}$. However, $x_{v}^{\beta}$ is not an
independent excitation but a collection of the isolated excitations on the
edges attached to the vertex $v$. This follows from

$x_{v}^{\beta}=\left(x_{v}^{\beta}{\bf X}_{L_{v}}^{\beta}\right){\bf
X}_{L_{v}}^{-\beta},$ (5.6)

where the factor in the parentheses generates a local symmetry. Thus,
$x_{v}^{\beta}{\left|{\mbox{GS}\,s}\right>}={\bf
X}_{L_{v}}^{-\beta}{\left|{\mbox{GS}\,s^{\prime}}\right>}$ with
${\left|{s^{\prime}}\right>}=\left(x_{v}^{\beta}{\bf
X}_{L_{v}}^{\beta}\right){\left|{s}\right>}$.

###  5.2 Vertex excitations

The eigenvalue 0 for the vertex operator in (2.4) corresponds to the
eigenvalue 1 for one of the orthogonal vertex operators, $A_{v}^{[\alpha]}$ in
(2.8) with $\alpha\neq 0$ mod $n$. First we look at the possibility for a
single vertex operator to be excited, or an isolated vertex excitation, at the
vertex $v^{\prime}$ by computing

${\rm
Tr\,}_{\mathcal{H}}\left(A_{v^{\prime}}^{\left[\alpha\right]}\prod\limits_{v\in
V-\\{v^{\prime}\\}}A_{v}\prod\limits_{e\in
E}B_{e}\right)=\frac{(\mbox{GSD})}{q}\sum_{b=0}^{q-1}\omega_{q}^{\alpha b},$
(5.7)

which does not vanish only when $\alpha=0$ mod $q$. It implies that $k-1$
isolated vertex excitations exist corresponding to
$\alpha\in\\{q,2q,\cdots,(k-1)q\\}$. As we will see later, the remaining
possibilities $\alpha\neq 0$ mod $q$ contribute to deconfined excitations.

#### Excited states $z_{v}^{\beta}{\left|{\mbox{GS}\,s}\right>}$:

As in the previous subsection, to create the isolated vertex excitations let
us pick the state $z_{v}^{\beta}{\left|{\mbox{GS},s}\right>}$ with
$\beta\in\\{1,\cdots,m-1\\}$. Since $z_{v}^{k}$ generates a local symmetry,
the cases $\beta\in\\{1,\cdots,k-1\\}$ are candidates for the independent
excitations. The relation

$z_{v}^{\beta}A_{v}^{(j)}=\omega_{m}^{\beta pj}A_{v}^{(j)}z_{v}^{\beta}$ (5.8)

together with $\omega_{m}^{\beta pj}=\omega_{k}^{\beta j}=\omega_{n}^{\beta
qj}$ leads to

$z_{v}^{\beta}A_{v}=A_{v}^{[\beta q]}z_{v}^{\beta}.$ (5.9)

Then, we have

$z_{v}^{\beta}{\left|{\mbox{GS}\,s}\right>}=\omega_{m}^{\beta
a_{v}}\sqrt{{\cal N}}\,A_{v}^{[\beta q]}\left(\prod_{v^{\prime}\in
V-\\{v\\}}A_{v^{\prime}}\right){\left|{s}\right>}.$ (5.10)

Note that $A_{v}^{[\beta q]}$ is orthogonal to $A_{v}$ only when $\beta\neq 0$
mod $k$. Thus, $z_{v}^{\beta}{\left|{\mbox{GS}\,s}\right>}$ with
$\beta\in\\{1,2,\cdots,k-1\\}$ are independent isolated vertex excitations
with the energy $E_{0}+1$ (first excited states).

#### Excited states $Z_{e}^{\gamma}{\left|{\mbox{GS}\,s}\right>}$:

Next we turn our attention to the deconfined vertex excitations that occur in
pairs. These are similar to the abelian quantum double models, and hence we
first pick the states $Z_{e}^{\gamma}{\left|{\mbox{GS}\,s}\right>}$ for
$\gamma\in\\{1,2,\cdots,n-1\\}$. Since $Z_{e}^{\tilde{n}}=Z_{e}^{\tilde{k}q}$
generates a local symmetry as mentioned around (2.16), we may consider the
cases $\gamma\in\\{1,2,\cdots,\tilde{k}q-1\\}$. Furthermore,

$Z^{qj}_{e}=z_{v_{1}}^{-pj}\left[z_{v_{1}}^{pj}Z_{e}^{qj}z_{v_{2}}^{-pj}\right]z_{v_{2}}^{pj}=z_{v_{1}}^{-pj}z_{v_{2}}^{pj}B_{e}^{(j)}\quad\mbox{for
$j\in\\{1,\cdots,k-1\\}$}$ (5.11)

implies that whenever $\gamma$ is a multiple of $q$,
$Z_{e}^{\gamma}{\left|{\mbox{GS}\,s}\right>}$ reduces to a composition of two
isolated vertex excitations on $v_{1}$ and $v_{2}$, since $B_{e}^{(j)}$ can be
written as a linear combination of the edge operator $B_{e}$ and its
orthogonal complements $B_{e}^{[\alpha]}$ in (2.12). Thus, we find candidates
for independent deconfined vertex excitations as those for
$\gamma\in\\{1,2,\cdots,q-1\\}$.

From (2.14) we obtain for $e\in L_{v}^{\pm}$

$Z_{e}^{\gamma}A^{(j)}_{v}=\omega_{n}^{\pm\gamma
pj}A_{v}^{(j)}Z_{e}^{\gamma}\qquad(j\in\mathbb{Z}_{n}),$ (5.12)

and thus

$Z_{e}^{\gamma}A_{v}=A_{v}^{[\pm\gamma p]}Z_{e}^{\gamma}.$ (5.13)

Here, $\gcd(p,q)=\gcd(\xi\tilde{p},q)=1$ provides $\gcd(\tilde{p},q)=1$. This
and $\gcd(\tilde{k},\tilde{p})=1$ give $\gcd(\tilde{k}q,\tilde{p})=1$, which
leads to $\omega_{n}^{\gamma p}=\omega_{\tilde{k}q}^{\gamma\tilde{p}}\neq 1$
and thus $A_{v}^{[\pm\gamma p]}$ are orthogonal to $A_{v}$ for any $\gamma$ in
the above range. For an edge $e$ belonging to $L_{v_{1}}^{-}$ and
$L_{v_{2}}^{+}$ as in Fig. 2, we explicitly see

$Z_{e}^{\gamma}{\left|{\mbox{GS}\,s}\right>}=\omega_{q}^{p\gamma\sum_{L=1}^{B_{1}}\delta_{e,\hat{e}_{L}}b_{L}}\sqrt{{\cal
N}}\,A_{v_{1}}^{[-\gamma p]}A_{v_{2}}^{[\gamma p]}\left(\prod_{v^{\prime}\in
V-\\{v_{1},v_{2}\\}}A_{v^{\prime}}\right){\left|{s}\right>},$ (5.14)

which indicates that $Z_{e}^{\gamma}{\left|{\mbox{GS}\,s}\right>}$ with
$\gamma\in\\{1,2,\cdots,q-1\\}$ are second excited states with the energy
$E_{0}+2$. The excitations are paired and occur at the both ends of the edge
$e$, namely the vertices $v_{1}$ and $v_{2}$. Now it is clear that among the
possibilities of $\alpha\neq 0$ mod $q$ mentioned below (5.7), $q-1$ of them
are independent and corresponds to the excitations (5.14).

$v_{1}$$v_{2}$$v_{3}$$e_{1}$$e_{2}$ Figure 7: Three vertices $v_{1}$, $v_{2}$
and $v_{3}$ are connected by two edges $e_{1}$ and $e_{2}$ such that $e_{1}$
is directed from $v_{1}$ to $v_{3}$, and $e_{2}$ is directed from $v_{3}$ to
$v_{2}$.

Likewise, for two edges ($e_{1}$ and $e_{2}$) and three vertices ($v_{1}$,
$v_{2}$ and $v_{3}$), where $e_{1}\in L_{v_{1}}^{-},L_{v_{3}}^{+}$ and
$e_{2}\in L_{v_{3}}^{-},L_{v_{2}}^{+}$ as in Fig. 7, consecutive two
excitations read

$\displaystyle
Z_{e_{1}}^{\gamma_{1}}Z_{e_{2}}^{\gamma_{2}}{\left|{\mbox{GS}\,s}\right>}$
$\displaystyle=$
$\displaystyle\omega_{q}^{p\sum_{a=1}^{2}\gamma_{a}\sum_{L=1}^{B_{1}}\delta_{e_{a},\hat{e}_{L}}b_{L}}\sqrt{{\cal
N}}$ (5.15) $\displaystyle\times
A_{v_{1}}^{[-\gamma_{1}p]}A_{v_{3}}^{[(\gamma_{1}-\gamma_{2})p]}A_{v_{2}}^{[\gamma_{2}p]}\left(\prod_{v^{\prime}\in
V-\\{v_{1},v_{2},v_{3}\\}}A_{v^{\prime}}\right){\left|{s}\right>}$

with $\gamma_{1},\gamma_{2}\in\\{1,2,\cdots,q-1\\}$. Note that the relation

$Z_{e_{1}}^{\gamma_{1}}A_{v_{3}}^{[-\gamma_{2}p]}=A_{v_{3}}^{[(\gamma_{1}-\gamma_{2})p]}Z_{e_{1}}^{\gamma_{1}}$
(5.16)

holds. When $\gamma_{1}=\gamma_{2}$, the excitation at $v_{3}$ disappears and
(5.15) become second excited states. Based on this observation, we can claim a
general statement. Let $P$ be an arbitrary path directed from the vertex
$v_{1}$ to the vertex $v_{2}$ on the graph. For the Wilson line operator along
$P$,171717$(e|P)$ is a sign factor defined similarly to $(e|C)$ at (2.17).

$Z(P)\equiv\prod_{e\in P}Z_{e}^{(e|P)},$ (5.17)

the states $Z(P)^{\gamma}{\left|{\mbox{GS}\,s}\right>}$ are at the second
excited level with the energy $E_{0}+2$ for $\gamma\in\\{1,2,\cdots,q-1\\}$.
The excitations occur at the endpoints of $P$, $v_{1}$ and $v_{2}$.

###  5.3 Case of $m=4,n=6$

For illustrative purposes of the peculiarities of the excited states, we
concretely present the case of $m=4$ and $n=6$, i.e., $k=p=2$ and $q=3$. In
this case there are six mutually orthogonal vertex operators,
$A_{v}^{[\alpha]}$, $\alpha\in\\{0,1,\cdots,5\\}$, and two mutually orthogonal
edge operators, $B_{e}^{[\alpha]}$, $\alpha\in\\{0,1\\}$. These operators are
given in (2.8) and (2.12) respectively and for clarity we write their full
expressions here.

$\displaystyle A_{v}^{[0]}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[1_{v}1_{L_{v}}+x^{2}_{v}{\bf
X}^{2}_{L_{v}}+1_{v}{\bf X}^{4}_{L_{v}}+x^{2}_{v}1_{L_{v}}+1_{v}{\bf
X}^{2}_{L_{v}}+x^{2}_{v}{\bf X}^{4}_{L_{v}}\right],$ $\displaystyle
A_{v}^{[1]}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[1_{v}1_{L_{v}}+\omega_{6}x^{2}_{v}{\bf
X}^{2}_{L_{v}}+\omega_{6}^{2}1_{v}{\bf
X}^{4}_{L_{v}}+\omega_{6}^{3}x^{2}_{v}1_{L_{v}}+\omega_{6}^{4}{\bf
X}^{2}_{L_{v}}+\omega_{6}^{5}x^{2}_{v}{\bf X}^{4}_{L_{v}}\right],$
$\displaystyle A_{v}^{[2]}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[1_{v}1_{L_{v}}+\omega_{3}x^{2}_{v}{\bf
X}^{2}_{L_{v}}+\omega_{3}^{2}1_{v}{\bf
X}^{4}_{L_{v}}+x^{2}_{v}1_{L_{v}}+\omega_{3}{\bf
X}^{2}_{L_{v}}+\omega_{3}^{2}x^{2}_{v}{\bf X}^{4}_{L_{v}}\right],$
$\displaystyle A_{v}^{[3]}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[1_{v}1_{L_{v}}-x^{2}_{v}{\bf
X}^{2}_{L_{v}}+1_{v}{\bf X}^{4}_{L_{v}}-x^{2}_{v}1_{L_{v}}+1_{v}{\bf
X}^{2}_{L_{v}}-x^{2}_{v}{\bf X}^{4}_{L_{v}}\right],$ $\displaystyle
A_{v}^{[4]}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[1_{v}1_{L_{v}}+\omega_{3}^{2}x^{2}_{v}{\bf
X}^{2}_{L_{v}}+\omega_{3}1_{v}{\bf
X}^{4}_{L_{v}}+x^{2}_{v}1_{L_{v}}+\omega_{3}^{2}{\bf
X}^{2}_{L_{v}}+\omega_{3}x^{2}_{v}{\bf X}^{4}_{L_{v}}\right],$ $\displaystyle
A_{v}^{[5]}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[1_{v}1_{L_{v}}+\omega_{6}^{5}x^{2}_{v}{\bf
X}^{2}_{L_{v}}+\omega_{6}^{4}1_{v}{\bf
X}^{4}_{L_{v}}+\omega_{6}^{3}x^{2}_{v}1_{L_{v}}+\omega_{6}^{2}{\bf
X}^{2}_{L_{v}}+\omega_{6}x^{2}_{v}{\bf X}^{4}_{L_{v}}\right],$

and

$\displaystyle B_{e}^{[0]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1_{v_{1}}1_{e}1_{v_{2}}+z_{v_{1}}^{2}Z_{e}^{3}z_{v_{2}}^{2}\right],$
$\displaystyle B_{e}^{[1]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1_{v_{1}}1_{e}1_{v_{2}}-z_{v_{1}}^{2}Z_{e}^{3}z_{v_{2}}^{2}\right].$

Here, $\omega_{6}=e^{\frac{2\pi\mathrm{i}}{6}}$,
$\omega_{3}=e^{\frac{2\pi\mathrm{i}}{3}}$, $x_{v}^{4}=z_{v}^{4}=1_{v}$ and
$X_{e}^{6}=Z_{e}^{6}=1_{e}$. While the ground states are the +1 eigenstates of
$A_{v}^{[0]}$ and $B_{e}^{[0]}$, the excited states are +1 eigenstates of the
remaining orthogonal operators, as discussed in sections 5.1 and 5.2. We will
exhaust them using the results of sections 5.1 and 5.2.

The first excited states are given by $X_{e}{\left|{\mbox{GS}\,s}\right>}$ and
$z_{v}{\left|{\mbox{GS}\,s}\right>}$ for any $e\in E$ and $v\in V$, which are
isolated (immobile) and appear when $k\neq 1$ in general. They are the
isolated edge and vertex excitations and are easily seen as the +1 eigenstates
of the operators, $B_{e}^{[1]}$ and $A_{v}^{[3]}$, respectively. The second
excited states are given by $Z(P){\left|{\mbox{GS}\,s}\right>}$ and
$Z(P)^{2}{\left|{\mbox{GS}\,s}\right>}$ with $Z(P)$ being (5.17) for any open
path on the graph $G$, which are deconfined (mobile) and appear when $q\neq 1$
in general.

The operators $Z(P)$ and $Z(P)^{2}$ create vertex excitations at the endpoints
of the path $P$, which are the +1 eigenstates of the vertex operators,
$A_{v}^{[2]}$ and $A_{v}^{[4]}$. Finally appending the operators $z_{v}$ at
the end points of the path operators, $Z(P)$ and $Z(P)^{2}$ we create the +1
eigenstates of the operators, $A_{v}^{[1]}$ and $A_{v}^{[5]}$. Interestingly,
these combined excitations are also at the second excited level.181818 We did
not considered such combined excitations in section 5.2 because the analysis
seems to be complicated for general $m$ and $n$.

Note that there are no deconfined excited states created by $X_{e}$. This
realizes the exchange phases of ‘anyons’ which are not exactly the same as
what appear in the toric code or the two-dimensional quantum double models of
Kitaev (see the following subsections).

###  5.4 Anyon-like excitations and topological EE

In this section we discuss anyon-like excitations and their relevance to the
topological EE computed in sections 4.1 and 4.2.

Let us first recall the toric code model [4]. The toric code model is defined
on the square lattice191919The toric code models are well-defined on any
triangulation of the two dimensional space, and the square lattice is usually
chosen for simplicity. with the Pauli spin operators, $\bar{X}_{e}$ and
$\bar{Z}_{e}$, acting on each link.202020 We put the bar to the operators of
the toric code model in order to distinguish the operators in our model. The
Hamiltonian consists of two kinds of interaction terms – the star term $A_{v}$
consisting of $\bar{X}_{e}$’s and the plaquette term $B_{p}$ consisting of
$\bar{Z}_{e}$’s. The former energetically imposes the Gauss law constraints,
and the latter gives a standard gauge kinetic term on the lattice. It is
analogous to the $\mathbb{Z}_{2}$ lattice gauge theory. There are two kinds of
deconfined excitations. One is ‘electric excitations’ which are constructed by
Wilson line operators of $\bar{Z}_{e}$ acting on the ground states. Associated
to a path on the (original) lattice, the corresponding Wilson line operator is
defined by the product of $\bar{Z}_{e}$’s along the path. The excitations
occur at the endpoints of the path, which can be interpreted as electric
charges. The other is ‘magnetic excitations’ constructed by acting ’t Hooft
line operators on the ground states. Associated to a path on the dual lattice,
the corresponding ’t Hooft line operator is defined by the product of
$\bar{X}_{e}$’s on the edges $e$ intersecting with the path. The excitations
appear at the endpoints of the path, which can be interpreted as magnetic
fluxes. When an electric charge moves around a magnetic flux (and vice versa),
an anyon phase appears due to the Aharonov-Bohm effect. See Fig. 8 for an
example of ‘t Hooft and Wilson line operators.

Figure 8: An example of ’t Hooft and Wilson line operators in the toric code
model on the square lattice. For the original lattice drawn in the gray lines,
the dual lattice is drawn in the light blue lines. The blue line represents a
path on the dual lattice, and its associated ’t Hooft line operator is given
by the product of $\bar{X}_{e}$’s on the red edges. The blue crosses represent
the ends of the ’t Hooft line, at which magnetic excitations occur. The black
line is a path on the original lattice. The product of $\bar{Z}_{e}$’s along
the path gives the associated Wilson line. Electric excitations appear at the
endpoints of the path (the black dots).

Clearly the electric excitations correspond to the Wilson line operators
$Z(P)^{\gamma}$ ($\gamma=1,2,\cdots,q-1$) acting on the ground states in our
case. However, there seems to be no counterpart to the magnetic excitations in
excitations discussed in sections 5.1 and 5.2. We see that the
$\mathbb{Z}_{q}$ magnetic fluxes $pb_{L}$’s on the ground states
${\left|{\mbox{GS}\,s}\right>}$ play an analogous role to the magnetic
excitations, except the point that the magnetic fluxes do not cost any energy,
or they are condensed into the ground state. In our case, since the plaquette
terms of $Z_{e}$’s are absent in the Hamiltonian (2.13), the ground states can
accommodate the zero-energy magnetic fluxes.212121Interestingly, adding the
plaquette terms of $Z_{e}^{q}$’s rather than $Z_{e}$’s to the Hamiltonian does
not alter the ground states.  From (3.16), after one of the endpoints of the
Wilson line circulates along the closed path $C_{L}$, it acquires the
Aharonov-Bohm phase222222 A similar phenomenon is observed in topological flux
phases in the string net models [39]. Although the string net models normally
allow ground states with zero flux, the topological flux phases are realized
by modifying the Hamiltonians so that nonzero flux states are energetically
favored [40].

$\omega_{q}^{p\gamma b_{L}\,(\hat{e}_{L}|C_{L})}.$ (5.18)

Since a general closed path on the graph is a linear combination of $C_{L}$’s
with the coefficients $\pm 1$, the phase appearing after moving along the
general path is given by the product of the phases for each $C_{L}$. In the
phase (5.18), $\gamma$ and $b_{L}$ are $\mathbb{Z}_{q}$-valued (including the
trivial case), which leads to the total quantum dimension $D=\sqrt{q^{2}}=q$.
This accounts for the topological EE term obtained in section 4.1.

On the other hand, the ground states
${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ have the $\mathbb{Z}_{q}$ electric
flux $\beta_{L}(\hat{e}_{L}|C_{L})$ along $C_{L}$ as shown in section 3.3.
$\beta_{L}$ corresponds to $\gamma$ in the above. Thus the same process
acquiring the anyon phase (5.18) occurs by inserting the local operator
$X_{\hat{e}_{L}}^{-pbk\,(\hat{e}_{L}|C_{L})}$ ($b\in\mathbb{Z}_{q}$). As is
seen from (3.24),

$X_{\hat{e}_{L}}^{-pb_{L}k\,(\hat{e}_{L}|C_{L})}{\left|{\mbox{GS}[\alpha,\beta]}\right>}=\omega_{q}^{pb_{L}\beta_{L}(\hat{e}_{L}|C_{L})}{\left|{\mbox{GS}[\alpha,\beta]}\right>}.$
(5.19)

This operator is local and does not contribute to the topological EE, which
explains the reason why the topological EE vanishes for
${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ in the result (4.49).

###  5.5 Exchange statistics on graphs

Before closing this section, we mention about connections to the analysis in
[16, 17, 18] to identify exchange statistics that are peculiar to graphs.
There, quantum particles sit on vertices of graphs, and it is investigated
which phases can appear for various patterns of the exchange of the particles.
The setting is different from the case of the abelian toric code models, in
which an anyon phase appears only when an electric excitation at a vertex of
the (original) lattice moves around a magnetic one at a vertex of the dual
lattice and vice versa. When an electric excitation moves around another
electric one, no phase appears.

Similarly, in our case, movable excitations at vertices are only of the
electric type, and their exchange as in Fig. 9 does not provide a nontrivial
phase. However, in case that the ground state ${\left|{\mbox{GS}\,s}\right>}$
has magnetic flux penetrating the region enclosed by the loop $C:v_{1}\to
v_{2}\to v_{3}\to v_{4}\to v_{1}$, the Aharonov-Bohm phase appears as already
mentioned.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{0}$$e$ Figure 9: Two Wilson lines on a graph.
The thick lines denote the Wilson lines. The orientations are suppressed. When
the excitation at the vertex $v_{1}$ move around the excitation at $v_{0}$
along the loop $C:v_{1}\to v_{2}\to v_{3}\to v_{4}\to v_{1}$ (as the blue
arrow indicates), there appears no anyon phase because they are both electric
excitations. Interestingly, if there is an isolated excitation on some edge
$e$ on the loop $C$, the electric excitation moves along $C$ on
$X_{e}^{\beta}{\left|{\mbox{GS}\,s}\right>}$ ($\beta=\\{1,\cdots,k-1\\}$)
provides a phase.

Interestingly, even when the ground state does not have the magnetic flux, and
instead an isolated excitation $X_{e}^{\beta}$ ($\beta\in\\{1,\cdots,k-1\\}$)
exists on some edge $e$ on the loop, a nontrivial phase appears as

$Z(C)^{\gamma}X_{e}^{\beta}{\left|{\mbox{GS}\,s}\right>}=\omega_{n}^{\beta\gamma\,(e|C)}X_{e}^{\beta}Z(C)^{\gamma}{\left|{\mbox{GS}\,s}\right>}=\omega_{n}^{\beta\gamma\,(e|C)}X_{e}^{\beta}{\left|{\mbox{GS}\,s}\right>},$
(5.20)

where $\gamma\in\\{1,\cdots,q-1\\}$, and the last equality comes from (3.16)
with $b_{L}=0$ (no magnetic flux). This can be thought of as a generalization
of the Aharnov-Bohm effect in the presence of the 0-holonomy operator or the
edge operator.

In addition, the exchange on a T- (or Y-) junction is considered in [16, 17,
18], where two particles at the vertices $(v_{3},v_{2})$ move as
$(v_{3},v_{2})\to(v_{3},v_{4})\to(v_{2},v_{4})\to(v_{1},v_{4})\to(v_{1},v_{2})\to(v_{1},v_{3})\to(v_{2},v_{3})$
in Figs. 10 and 11. In our case no phase appears in the process as seen below.

$v_{3}$$v_{2}$$v_{1}$$v_{4}$$e_{1}$$e_{2}$$e_{3}$ Figure 10: A graph of a T-
(or Y-) junction.  Figure 11: The exchange process of two particles (the red
and blue dots) in a T-(Y-) junction of the graph. For simplicity the
orientations and other vertices are suppressed.

Let us consider two deconfined excitations of strings $Z(P_{1})^{\gamma_{1}}$
and $Z(P_{2})^{\gamma_{2}}$ ($\gamma_{1},\gamma_{2}\in\\{1,\cdots,q-1\\}$) on
any fixed state ${\left|{\psi}\right>}$. Suppose the T-(or Y-) junction is a
part of the graph and the red and blue dots in Fig. 11 are endpoints of
$Z(P_{1})^{\gamma_{1}}$ and $Z(P_{2})^{\gamma_{2}}$, respectively. When
$e_{3}\in P_{1}$,232323We can similarly show for other configurations of the
path. the initial state is written as

${\left|{\psi_{i}}\right>}=Z_{e_{3}}^{\gamma_{1}}\cdots{\left|{\psi}\right>},$
(5.21)

where $(\cdots)$ expresses the strings outside the T-(Y-) junction and does
not change in the process (see Fig. 12).

$v_{3}$$v_{2}$$v_{1}$$v_{4}$$e_{1}$$e_{2}$$e_{3}$$P_{1}$$P_{2}$ Figure 12: An
initial configuration of the strings $Z(P_{1})^{\gamma_{1}}$ (red line) and
$Z(P_{2})^{\gamma_{2}}$ (blue line).

We trace the path of each of the strings in the exchange process. The endpoint
of the string $Z(P_{1})^{\gamma_{1}}$ (the red dot) moves as $v_{2}\to
v_{4}\to v_{2}\to v_{3}$ which gives the additional contribution to the string

$Z_{e_{3}}^{-\gamma_{1}}Z_{e_{2}}^{-\gamma_{1}}Z_{e_{2}}^{\gamma_{1}}=Z_{e_{3}}^{-\gamma_{1}},$
(5.22)

whereas the endpoint of the string $Z(P_{2})^{\gamma_{2}}$ (the blue dot)
moves as $v_{3}\to v_{2}\to v_{1}\to v_{2}$ which yields

$Z_{e_{1}}^{\gamma_{2}}Z_{e_{1}}^{-\gamma_{2}}Z_{e_{3}}^{\gamma_{2}}=Z_{e_{3}}^{\gamma_{2}}.$
(5.23)

Thus, $Z_{e_{3}}^{-\gamma_{1}+\gamma_{2}}$ is obtained in total. Including
this, the final state becomes

${\left|{\psi_{f}}\right>}=Z_{e_{3}}^{-\gamma_{1}+\gamma_{2}}{\left|{\psi_{i}}\right>}=Z_{e_{3}}^{\gamma_{2}}\cdots{\left|{\psi}\right>}.$
(5.24)

(5.24) is the same as (5.21) after $\gamma_{2}$ is replaced by $\gamma_{1}$,
which realizes the exchange without a phase. Note that this holds for any
state ${\left|{\psi}\right>}$242424As a remark we would like to add that it is
possible to obtain exchange phases in this process if we move the deconfined
vertex excitations with $Z_{e}^{q-\gamma}$ instead of $Z_{e}^{-\gamma}$. This
is allowed as $\gamma\in\\{1,2,\cdots,q-1\\}$. However this exchange process
is not adiabatic as $Z_{e}^{q}$ (See (5.11)) creates extra vertex excitations
changing the energy of the state during the exchange process. The phase, upon
exchanging the deconfined vertex excitations in the background of three
isolated edge excitations created by $X_{e_{1}}^{\beta_{1}}$,
$X_{e_{2}}^{\beta_{2}}$ and $X_{e_{3}}^{\beta_{3}}$ is,
$\omega_{n}^{q\beta_{1}+q\beta_{2}+2q\beta_{3}}$..

##  6 Discussion

###  6.1 Summary

In this paper we have initiated a detailed study of abelian gauge theories on
graphs that host quantum phases not classified as phases of spontaneous
symmetry breaking with local order parameters. In general the models realize
such quantum phases with a mixture of topological and extensive
aspects.252525According to Theorem IV.6 in [23], the GSD is given by
$\mbox{GSD}=\prod_{n}\left|H^{n}(C,H_{n}(G)\right|=\underbrace{\left|\mbox{Hom}(H_{0}(C),H_{0}(G)\right|}_{p^{|V|}}\underbrace{\left|\mbox{Hom}(H_{1}(C),H_{1}(G)\right|}_{p^{|E|-|V|+1}}$
since the Ext and Tor functors are null. This implies that the extensive part
of the GSD, $p^{|V|}$, can also be regarded as a topological contribution from
the mathematical perspective. However, we should remark that each of the
ground states contributing to the extensive part is fragile under local
perturbations, as mentioned in the last paragraph of this section, defying the
physicist’s definition of topological phases.  They possess features that are
reminiscent of the two-dimensional quantum double models of Kitaev [4], with
some subtle differences in their properties, notably in the GSD and in the
nature of the anyonic excitations.

In some cases occurring for specific families of $m$ and $n$ values, our
result reads:

  * •

When $p=1$, $n$ is an integer multiple of $m$: $n=mq$. We obtain the purely
topological case in which the GSD reduces to $q^{B_{1}}$. The EE for the
ground states, ${\left|{\mbox{GS}\,s}\right>}$, includes the constant term
(topological EE) as $-\gamma=-\log_{2}q$, accounted for by the anyons in the
model.

  * •

When $q=1$, $m$ is an integer multiple of $n$: $m=np$. The GSD is extensive as
seen by the expression $p^{|V|}$. For the ground states
${\left|{\mbox{GS}\,s}\right>}$, the topological EE vanishes.

  * •

When $m=n$, there is a unique ground state obeying the area law for the EE. It
is also interesting to note that the Hamiltonian for $m=n=2$ is unitarily
equivalent to the cluster state Hamiltonians representing a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry protected topological (SPT)
phase [41], when the graph forms either an open or closed chain.

  * •

When $k=1$, $m=p$ and $n=q$ are co-prime to each other. We do not have valid
homomorphisms, $\partial^{\left[l\right]}$ between $\mathbb{Z}_{n}$ and
$\mathbb{Z}_{m}$ except for the trivial homomorphism. In this case, the edge
operators become trivial ($B_{e}=1$), and the vertex operators reduce to
$A_{v}=\frac{1}{q}\sum_{j=0}^{q-1}1_{v}{\bf X}_{L_{v}}^{pj}$ which impose the
Gauss law constraints of the pure gauge theory. Although only the operators
$X_{e}$ appear in the Hamiltionian, the result for the GSD, $p^{|V|}q^{B_{1}}$
is still valid, and the EE for the ground states
${\left|{\mbox{GS}\,s}\right>}$ includes the global constant term
$-\gamma=-\log_{2}q$. All the excitations are deconfined and given by the
Wilson line operators, which constitute anyon-like excitations in the
interplay with the magnetic fluxes in the ground states.

###  6.2 Outlook

We present some directions for further study:

  * •

The toric code model can be extended to a lattice discretizing surfaces with
boundary [42, 43]. It is worth considering a similar extension in the models
on graphs presented here. In general, it seems nontrivial to divide a graph
into bulk and boundary parts. Tree graphs and finite regular lattices are
examples in which such a division is possible. In the tree graph, vertices
with valency 1 are identified as boundaries. In the finite square lattice,
vertices with valency 2 or 3 and edges connecting them form the boundary. It
is interesting to find some other class of graphs such that the division is
possible. If the models are defined on these graphs with appropriate
modifications to the Hamiltonian at the boundary, we expect to see the
appearance of edge states similar to what happens in the SPT case. This will
lead to an extra degeneracy in the number of ground states in addition to the
topological and the extensive degeneracy already present. The tree graph is a
special case where we do not expect any topological degeneracy as $B_{1}=0$ in
this case. The interesting thing to note is that these edge states may not
result from a fractionalization of a global symmetry as it happens in the SPT
case [44].

  * •

Generalizations of these models to finite non-abelian groups are possible
along the lines presented in [25]. These are however much harder to analyze.
It is also natural to see if these models can be generalized to other algebras
much like the quantum double models of Kitaev. With the machinery developed in
[25] this might be possible as well.

  * •

Locally the vertex and plaquette operators of the quantum double models
satisfy the relations of a quasitriangular Hopf algebra [7]. In fact
Drinfeld’s quantum double construction is tailored to construct such algebras
that have the $R$-matrix satisfying the Yang-Baxters equation encoded in them.
This gives rise to the anyon excitations which are the IRR’s of this algebra
[6]. It would be very interesting to study the way in which this algebra is
modified for the operators presented here. Naturally we may expect them to
have some generator that can realize the relations of the graph braid groups.

  * •

The result obtained here is valid for the same model defined on lattices in
arbitrary dimensions, because graphs includes lattices in any dimensions. It
is expected that these results apply to a broad class of lattice models.

  * •

We have seen that the GSD of these models are a mixture of a topological
invariant (first Betti number) and an extensive quantity depending on the
lattice size. Moreover apart from the anyonic excitations that account for the
topological EE, there are other immobile excitations that cannot be moved
around the graph. These features are very similar to those present in the
fracton phases of matter as discussed in [45, 46, 47]. We hope to explore more
of this connection.

  * •

These systems can also be viewed as stabiliser codes and it is natural to look
for an application in quantum computation. However for the case when $q\neq 1$
there are weight 1 symmetries implying that single qubit operations move us
within the logical space making them undetectable. Thus direct application of
our models to quantum error correction seems to be restrictive to the case
$p=1$. For $p\neq 1$, the $p^{|V|}$ ground states
${\left|{\mbox{GS}[\alpha,\beta]}\right>}$ with fixed $\beta_{L}$’s are
flipped to one another by local perturbations as seen in (3.26). From (3.27),
the $p=1$ case seems interesting especially when the lengths of all the closed
paths $C_{L}$ grow as the size of the graph $G$, $|V|$ or $|E|$, increases.
Then, the ground states ${\left|{\mbox{GS}[-,\beta]}\right>}$ (there is no
$\alpha$ parameter when $p=1$) are expected to be robust against local
perturbations and useful for error correcting purposes. We present such an
example in Fig. 13. The size of the graph, $|E|$ or $|V|$, grows as $N$
increases. Then, both the size of the holes and the number of holes also
scale. The logarithm of the GSD is subextensive:
$\log_{2}(\mbox{GSD})=O(N^{2})=O\left(|V|^{2/3}\right)=\left(|E|^{2/3}\right)$.
From the quantum computation perspective,
${\left|{\mbox{GS}[-,\beta]}\right>}$’s are logical qudits. Starting from any
state of ${\left|{\mbox{GS}[-,\beta]}\right>}$’s, we can exhaust all the
logical qudits by successively acting $Z(C_{L})$’s as seen in (3.27). The
logical operations grow with the system size just as in the case of surface
codes. The example seems to realize robustness under local perturbations and
the growth of the number of logical qudits as $N$ increases.

$N$$N$ Figure 13: A graph whose shape is a coarse net. The orientations of the
edges are suppressed. The size of each hole of the net is $N\times N$, and the
number of the holes is $B_{1}=N^{2}$. $|E|=2N^{3}+2N^{2}$ and
$|V|=2N^{3}+N^{2}+1$. The case $N=4$ is depicted.

  * •

As a final remark we note that it is possible to go beyond the chain
complex/higher gauge formulation presented in [23, 24] with the following
example :

$A_{v}=\frac{1}{m}\sum\limits_{j=0}^{m-1}~{}x_{v}^{qj}{\bf
X}_{L_{v}}^{qj},\qquad
B_{e}=\frac{1}{k}\sum\limits_{j=0}^{k-1}z_{v_{1}}^{pj}Z_{e}^{qj}z_{v_{2}}^{-pj}.$
(6.1)

This model can be thought of as the one where the 0-gauge fields on the
vertices act on the 1-gauge fields on the edges,

$\tilde{\partial}:\mathbb{Z}_{m}\rightarrow\mathbb{Z}_{n};~{}~{}~{}\tilde{\partial}(j)=qj,$
(6.2)

which is a valid homomorphism as $\tilde{\partial}(m)=qm=pn\equiv
0~{}\textrm{mod}~{}n$. It is opposite to the direction of the homomorphism
$\partial$. This model has a ground state degeneracy given by

$\mbox{GSD}=q^{|E|}\eta,$ (6.3)

with $\eta=\gcd(q,m)$. We hope to analyze these directions in the future.

### Acknowledgements

FS is supported by the Institute for Basic Science in Korea (IBS-R018-D1).

##  Appendix A Analogy to quantum field theory with $U(1)$ gauge field and a
matter field

To gain an intuitive understanding of our model, we consider an analogy to
quantum field theory with $U(1)$ gauge field $\vec{A}$ and a complex scalar
field $\phi$ for charged matter. $\vec{A}$ means space components of the gauge
field, and the time component $A_{0}$ is supposed to be fixed at $A_{0}=0$.

The second term in the Hamiltonian (2.13), $-\sum_{e\in E}B_{e}$, can be
regarded as the kinetic term of the matter field $|\vec{D}\phi|^{2}$, where
$\vec{D}$ denotes the covariant derivative associated to the gauge field.
Then, we can interpret $Z_{e}$ and $z_{v}$ as (the exponential of) the gauge
field $\vec{A}$ and the matter field $\phi$, respectively. (2.17) corresponds
to the Wilson loop, and ${\left|{h_{v}}\right>}\in{\cal H}_{v}$
(${\left|{i_{e}}\right>}\in{\cal H}_{e}$) correspond to the eigenstate of the
matter field $\phi$ (the gauge field $\vec{A}$).

As mentioned in section 2.1, the first term in (2.13), $-\sum_{v\in V}A_{v}$,
imposes the gauge invariance energetically. Namely, the gauge symmetry is
preserved on the eigenstates of $A_{v}$ with the eigenvalue 1 for all $v\in
V$. $A_{v}$ is analogous to an operator imposing the Gauss law constraint
${\rm div}\,\vec{E}-\rho_{\rm matter}=0$ ($\vec{E}$ and $\rho_{\rm matter}$
stand for the electric field and the matter charge density, respectively). We
can interpret $X_{e}$ and $x_{v}$ as the electric field (the conjugate
momentum of the gauge field) and the conjugate momentum $\pi$ of the matter
field.

The Hamiltonian (2.13) does not possess the plaquette terms of $Z_{e}$ which
corresponds to the gauge kinetic terms $\vec{E}^{2}+\vec{B}^{2}$ with
$\vec{B}\equiv{\rm rot}\,\vec{A}$ in the gauge field theory.

Note that the Wilson loop
$\exp\left(\mathrm{i}\int_{C}d\vec{x}\cdot\vec{A}\right)$ along a loop $C$ in
the space creates the unit electric flux along the loop, which is seen from
its canonical commutation relation with $\vec{E}$:
$[A_{i}(t,\,\vec{x}),\,E_{j}(t,\vec{x^{\prime}})]=\mathrm{i}\delta_{ij}\delta(\vec{x}-\vec{x^{\prime}})$.

##  Appendix B $U(1)$ gauge theory on a circle

In this appendix we briefly review $U(1)$ gauge theory defined on a circle and
the property of its vacuum, in order to help understanding the ground states
in sections 3.2 and 3.3.

For a mathematically well-defined treatment, we also impose the periodic
boundary conditions in the time direction $x^{0}=t\in[0,T]$ as well as the
space direction $x^{1}=x\in[0,L]$. After final results are obtained, we can
send $T$ to infinity. In gauge theory it is sufficient that gauge fields are
periodic modulo gauge transformations. In general, gauge fields
$A_{\mu}(t,\,x)$ ($\mu=0,1$) satisfy

$A_{\mu}(T,\,x)=A_{\mu}(0,\,x)+\mathrm{i}h(x)\partial_{\mu}h(x)^{-1},\qquad
A_{\mu}(t,\,L)=A_{\mu}(t,\,0)+\mathrm{i}g(t)\partial_{\mu}g(t)^{-1},$ (B.1)

where $h(x),\,g(t)\in U(1)$ are transition functions at $t=T$ and $x=L$,
respectively. In order to obtain topologically nontrivial configurations
(nontrivial $U(1)$ bundles over the 2-torus), we take

$h(x)=\exp\left(\frac{2\pi\mathrm{i}}{L}n_{x}x\right),\qquad
g(t)=\exp\left(\frac{2\pi\mathrm{i}}{T}n_{t}t\right)\qquad(n_{x},\,n_{t}\in\mathbb{Z})$
(B.2)

as an example. Those with nontrivial $n_{x}$ and $n_{t}$ cannot be obtained by
continuous deformations from the identity: $h(x)=1$ and $g(t)=1$. As discussed
in [48], we can undo one of the twists, say $g(t)$, using a gauge
transformation $\Omega(t,\,x)$ such that $\Omega(t,\,0)=1$ and
$\Omega(t,\,L)=g(t)$. For example, we take

$\Omega(t,\,x)=\exp\left(\frac{2\pi\mathrm{i}}{TL}n_{t}tx\right),$ (B.3)

and then obtain

$A_{\mu}(T,\,x)=A_{\mu}(0,\,x)+\delta_{\mu,1}\frac{2\pi\nu}{L},\qquad
A_{\mu}(t,\,L)=A_{\mu}(t,\,0)$ (B.4)

with $\nu=n_{x}-n_{t}$.

In the boundary conditions (B.4), we can take the $A_{0}=0$ gauge. Then,
topologically nontrivial gauge transformations labelled by an integer
$\nu\in\mathbb{Z}$ are given by
$h_{\nu}(x)=\exp\left(\frac{2\pi\mathrm{i}}{L}\nu x\right)$ times
topologically trivial gauge transformations. The topologically trivial gauge
transformations are connected to the identity by continuous deformations. The
configuration space of the gauge field $A_{1}$ is also divided into the
sectors. Namely, configurations satisfying

$A_{1}(T,\,x)=A_{1}(0,\,x)+\frac{2\pi\nu}{L}$ (B.5)

belong to the sector $\nu$. Correspondingly the Hilbert space is classified by
the topological number $\nu$. Given an initial state (at $t=0$) with the
topological number $\nu_{0}$, time evolution of the system under the condition
(B.5) leads to a final state (at $t=T$) with the topological number
$\nu+\nu_{0}$. The vacuum with the topological number $\nu$,
${\left|{\Omega_{\nu}}\right>}$ ($\nu\in\mathbb{Z}$), is changed to the one
with different $\nu$ by topologically nontrivial gauge transformations.
However, the $\theta$-vacuum defined by

${\left|{\theta}\right>}\equiv\sum_{\nu\in\mathbb{Z}}e^{\mathrm{i}\nu\theta}{\left|{\Omega_{\nu}}\right>}$
(B.6)

becomes an eigenstate for any gauge transformation.

From (B.5) it can be seen that $\nu$ is equal to the first Chern number:

$c_{1}\equiv\frac{1}{2\pi}\int_{0}^{T}dt\int_{0}^{L}dx\,F_{01}=\frac{1}{2\pi}\int_{0}^{T}dt\int_{0}^{L}dx\,\partial_{0}A_{1}=\frac{1}{2\pi}\int_{0}^{L}dx\,\left[A_{1}(T,\,x)-A_{1}(0,\,x)\right]=\nu.$
(B.7)

This formula indicates that the vacuum in the nontrivial topological sector
${\left|{\Omega_{\nu}}\right>}$ has a nontrivial background field strength
$F_{01}$. Suppose we can take a simply connected domain surrounded by the
circle $[0,L]$. Magnetic flux penetrating the domain can be expressed as
$\Phi(t)=\int^{L}_{0}dx\,A_{1}(t,\,x)$. Then (B.7) immediately gives

$\Phi(T)-\Phi(0)=2\pi\nu,$ (B.8)

which means that the twist (B.5) provides the magnetic flux $2\pi\nu$.

For a system on a graph as we are discussing in the text, we can consider a
subsystem on each closed path $C_{L}$ analogously to the $U(1)$ theory on a
circle here, at least regarding the topological structure of the ground
states.

##  Appendix C Short review of graph homology

Algebraic topology helps distinguish topological spaces systematically. The
fundamental group and higher homotopy groups classify topological spaces by
characterizing the holes of different dimensions in these spaces but they
quickly become hard to interpret as we increase the dimension of the
topological space. A commutative alternative to homotopy is given by homology
theory which we are concerned with.

If $X$ is a topological space we can construct a sequence of groups,
$H_{n}(X)$ for $n=0,1,2,\cdots$, termed as the homology groups. These are
commutative and their rank measures the number of $n$ dimensional holes in
$X$. We will illustrate these groups with the simplest example of $X$, a
graph. Consider the graph shown in Fig. 14.

$x$$y$$z$$a$$b$$c$$d$ Figure 14: A directed graph, $X$ with three vertices,
$\\{x,y,z\\}$ and four edges, $\\{a,b,c,d\\}$.

This graph is made up of three vertices $x,~{}y,~{}z$ and four edges
$a,~{}b,~{}c,~{}d$. The edges are directed as shown in Fig. 14. The vertices
are also called 0-simplices and the edges, 1-simplices. Together the graph $X$
is a simplicial complex. Naturally higher dimensional surfaces correspond to
higher simplices but here we restrict ourselves to 0- and 1-dimensional
simplices as we are interested in graphs.

The set of vertices is denoted by $C_{0}$ and is the free abelian group
generated by the vertices $x,y,z$. A general element of $C_{0}$ is $\alpha
x+\beta y+\gamma z$ with the coefficients $\alpha,\beta,\gamma$ being numbers
in some field which we take to be the integers, $\mathbb{Z}$. Likewise, the
set of edges is denoted by $C_{1}$ and is the free abelian group generated by
$a,b,c,d$. In the literature the elements of $C_{0}$ and $C_{1}$ are called
zero- and one-dimensional chains, respectively.

We now consider a group homomorphism,

$C_{1}\overset{\partial_{1}}{\longrightarrow}C_{0},$

which is called the boundary map. As the name implies it maps the edge in
$C_{1}$ to its boundary in $C_{0}$. For the case of the graph $X$ in Fig. 14
we obtain

$\partial_{1}(a)=y-x,\qquad\partial_{1}(b)=z-y,\qquad\partial_{1}(c)=x-z,\qquad\partial_{1}(d)=x-z.$
(C.1)

Clearly the 0-chains $y-x$, $z-y$, etc are the boundaries of the 1-chains or
edges. We can now think of special 1-chains called cycles whose boundary is
null. For the graph $X$ in Fig. 14 we obtain three cycles, $a+b+c$, $a+b+d$
and $c-d$, each of whose boundaries evaluate to 0. A crucial property of the
boundary map,

$\partial^{2}=0,$ (C.2)

can be verified by evaluating $\partial_{1}^{2}$ in $X$.

Consider the short exact sequence

$0\overset{\partial_{2}}{\longrightarrow}C_{1}\overset{\partial_{1}}{\longrightarrow}C_{0}\overset{\partial_{0}}{\longrightarrow}0.$
(C.3)

The homology groups $H_{n}$ are defined as

$H_{n}(X)=Z_{n}/B_{n},$ (C.4)

where $Z_{n}$ are the group of cycles and $B_{n}$ are the group of boundaries.
More precisely $Z_{n}={\rm Ker}(\partial_{n})$ and $B_{n}={\rm
Im}(\partial_{n+1})$. The quotient $Z_{n}/B_{n}$ collects $n$-chain cycles
that are not boundaries of $n+1$-chains. Thus it is the group generated by the
independent $n$-dimensional cycles or holes.

Thus for the graph $X$ in Fig. 14 we can compute $H_{1}={\rm
Ker}(\partial_{1})/{\rm Im}(\partial_{2})$. ${\rm
Ker}(\partial_{1})=\mathbb{Z}\oplus\mathbb{Z}$ is generated by $a+b+c$ and
$a+b+d$ (two of the obtained three cycles are linearly independent), and ${\rm
Im}(\partial_{2})=0$ as there are no 2-chains for the graph $X$. Thus
$H_{1}(X)=\mathbb{Z}\oplus\mathbb{Z}$ essentially enumerates the number of
independent one-dimensional cycles in $X$. The rank of $H_{1}(X)$ is known as
the first Betti number, $B_{1}(X)$, which is equal to 2 for the graph $X$.

For a general graph with $|E|$ edges and $|V|$ vertices
$H_{1}=\left(\oplus\mathbb{Z}\right)^{|E|-|V|+1}$ and hence $B_{1}=|E|-|V|+1$.
From a familiar result in graph theory we identify $|E|-|V|+1$ to be the
number of independent cycles of the graph under consideration.

We can also compute $H_{0}(X)={\rm Ker}(\partial_{0})/{\rm Im}(\partial_{1})$
for the graph $X$ in Fig. 14. Now ${\rm
Ker}(\partial_{0})=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ is generated by
$x,y,z$ and ${\rm
Im}(\partial_{1})=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ is generated by
$y-x,z-y,x-z$. To take the quotient we equate each element in
$Im(\partial_{1})$ to 0 which implies $H_{0}(X)=\mathbb{Z}$. From this example
we can convince ourselves that all vertices in a connected component of a
general graph will be identified. Thus $H_{0}$ for a general graph measures
the number of connected components of the graph and denotes the zeroth Betti
number. Clearly for the graph $X$ in Fig. 14, $B_{0}(X)=1$.

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