# A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph

Jintae Kim  Department of Physics, Sungkyunkwan University, Suwon 16419, Korea
Hyun-Yong Lee  Department of Applied Physics, Graduate School, Korea
University, Sejong 30019, Korea  Division of Display and Semiconductor
Physics, Korea University, Sejong 30019, Korea  Jung Hoon Han
<EMAIL_ADDRESS>Department of Physics, Sungkyunkwan University, Suwon
16419, Korea

###### Abstract

We analyze a model demonstrating the co-existence of subsystem symmetry
breaking (SSB) and symmetry-protected topological (SPT) order, or subsystem
LSPT order for short. Its mathematical origin is the existence of both a
subsystem and a local operator, both of which commute with the Hamiltonian but
anti-commute between themselves. The reason for the exponential growth of the
ground state degeneracy is attributed to the existence of subsystem
symmetries, which allows one to define both the Landau order parameter and the
SPT-like order for each independent loop.

##  I Introduction

Multiply degenerate ground states in a many-body system are usually generated
as a consequence of global symmetry breaking (GSB). A prime example of GSB is
the twofold degeneracy of the ground states in the Ising model, where all
spins are either up or all down. One of the remarkable advances in condensed
matter theory over the past decades is the identification of a new mechanism
by which multiple ground state degeneracy (GSD) is generated. One of these
routes is the topological order, as most dramatically realized in fractional
quantum Hall systems Kalmeyer and Laughlin (1987); Wen  _et al._ (1989).
Another recently discovered path to having multiple GSD is the so-called SPT
order Kitaev (2001); Chen  _et al._ (2010, 2011, 2013); Schuch  _et al._
(2011). In models with SPT order, the ground state is unique if defined on a
closed manifold but becomes multiply degenerate on an open geometry such as
the open chain. Among the tell-tale signs of SPT order is the symmetry
fractionalization of the global symmetry at the edges Pollmann  _et al._
(2010, 2012); Pollmann and Turner (2012); Else and Nayak (2014) which is also
responsible for the multiple GSD. A lot of different models exhibiting SPT
order has been examined in the past, in particular in one dimension Bondesan
and Quella (2013); Chen  _et al._ (2014); Geraedts and Motrunich (2014). Some
models demonstrate the GSB even for the closed geometry, together with the SPT
order Bondesan and Quella (2013); Geraedts and Motrunich (2014). A recent
twist to the original idea of SPT order protected by the global symmetry is
the subsystem SPT (SSPT), being protected by the symmetries in a sub-manifold
of the overall system You  _et al._ (2018).

The SPT models treated in the past considered an alternating site and link
variables arranged in a one-dimensional fashion. The case of $m$ degrees of
freedom residing on the vertices and $n$ degrees of freedom on the links was
considered in Refs. Ferreira  _et al._ , 2014, 2015; Ibieta-Jimenez  _et al._
, 2020; et al, 2020. The model exhibited a multiple GSD even for a closed
chain. When the same model was placed on a graph, the GSD increased
exponentially with the number of holes, called the first Betti number, in the
graph. In this paper, we give a clear interpretation of the results found in
Ref. et al, 2020 in terms of the notion of subsystem Landau and SPT orders.
Features that are characteristic of symmetry breaking as well as SPT can be
found in this model, and for each closed loop in the graph. We label such
order as the subsystem Landau-SPT order, or SLSPT order for short. The
observation of the co-existence of Landau and SPT orders in a given chain had
been made in the past Bondesan and Quella (2013); Geraedts and Motrunich
(2014), but not been extended to the case of a general graph. In Sec. II we
review the $V_{2}/L_{4}$ model introduced in Ref. et al, 2020, which is a
generalization of the $V_{2}/L_{2}$ model (to be defined precisely) studied by
several authors in the past Chen  _et al._ (2014); Geraedts and Motrunich
(2014); et al (2020). The ground state degeneracy this model is understood in
terms of a properly defined Landau order parameter in Sec. III. The
$V_{2}/L_{4}$ on a graph is analyzed in terms of the co-existing Landau and
SPT orders in Sec. IV. The origin of the exponential growth of GSD is
understood. We make a summary and conclude in Sec. VI.

##  II V2/L4 model

For self-consistent reading of this paper, we make a brief review of the
$V_{2}/L_{4}$ model introduced in Ref. Ferreira  _et al._ , 2015 and analyzed
in Ref. et al, 2020. In a family of Hamiltonians we will call the
$V_{m}/L_{n}$ model, there reside $m$ degrees of freedom, or levels, at the
vertices ($v$) and $n$ levels at the links ($l$). In the simplest case of a
one-dimensional chain, the vertices and links appear alternatively, as
illustrated in Fig. 1. Quantum states at the vertices and links are written
$|\alpha\rangle_{v}$ and $|\beta\rangle_{l}$, with $0\leq\alpha\leq m-1$ and
$0\leq\beta\leq n-1$, respectively.

The Hamiltonian of the $V_{m}/L_{n}$ model is a sum of mutually commuting
projectors. One type of projector called $A_{v}$ is defined with respect to a
vertex, while the other type $C_{l}$ is defined with respect to a link. Their
specific expressions depend on the choice of $(m,n)$ representing the
respective dimension of the Hilbert space at the vertices and links. It turns
out the $m=n=2$ case had been studied extensively in the past Chen  _et al._
(2014); Geraedts and Motrunich (2014). Our focus here is on the extension to
$n>2$. Striking differences from the $V_{2}/L_{2}$ situation already show up
for the $V_{2}/L_{4}$ case, where we will be devoting most of our discussion.

For a one-dimensional chain the vertices and sites can be labeled as
$(v_{i},l_{i})$, with $i$ running from 1 through $L$ for a chain of length $L$
(see Fig. 1). Then we write the Hamiltonian

$\displaystyle H=-\sum\limits_{i=1}^{L}(A_{i}+C_{i}),$ (1)

where $i=(v_{i},l_{i})$ stands for a combined vertex+link unit. The vertex
operator $A_{i}$ for the $V_{2}/L_{4}$ model acts on a given vertex $v_{i}$
and its two adjacent links $(l_{i-1},l_{i})$ as

$\displaystyle A_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{n=0}^{3}(X_{l_{i-1}}x_{v_{i}}X_{l_{i}}^{3})^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{4}(1+X_{l_{i-1}}x_{v_{i}}X_{l_{i}}^{3}+X_{l_{i-1}}^{2}X_{l_{i}}^{2}+X_{l_{i-1}}^{3}x_{v_{i}}X_{l_{i}}).$

The link operator $C_{i}$ acts on the given link $l_{i}$ and its two
neighboring vertices $(v_{i},v_{i+1})$,

$\displaystyle
C_{i}=\frac{1}{2}\sum_{n=0}^{1}(z_{v_{i}}Z_{l_{i}}^{2}z_{v_{i+1}})^{n}=\frac{1}{2}(1+z_{v_{i}}Z_{l_{i}}^{2}z_{v_{i+1}}).$
(3)

Such choice of $A$ and $C$ operators is specific to the $V_{2}/L_{4}$ model.

The operator form of $A_{i}$ depends on the direction of “arrows” on the
links. The rule is to assign $X$ to a link with an “incoming” arrow to a
vertex and $X^{3}$ to a link with an “outgoing” arrow. The $A$-operator for
arbitrary arrow directions on the links is therefore given by

$\displaystyle A_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{n=0}^{3}(\prod_{l_{in}}X_{l_{in}}\prod_{l_{out}}X_{l_{out}}^{3}x_{v_{i}})^{n}.$
(4)

The products $\prod_{l_{in}}$ and $\prod_{l_{out}}$ mean, for example, that if
both arrows are “in”, then we must write
$(X_{l_{i-1}}^{3}x_{v_{i}}X_{l_{i}}^{3})^{n}$. For simplicity, we adhere to
the arrow scheme shown in Fig. 1 and the vertex operator definition in Eq.
(II).

Figure 1: Schematic figure of alternating vertices and oriented links. For a
closed chain the link $l_{L}$ attaches to the vertex $v_{1}$.

The lower-case $x_{v}$ and $z_{v}$ operators act on the vertices. Meanwhile,
upper-case $X_{l}$ and $Z_{l}$ operate on the link states. In general they
satisfy $X_{l}|g\rangle_{l}=|g+1\rangle_{l}$ (mod 4) and
$Z_{l}|g\rangle=\omega^{g}|g\rangle_{l}$ with $\omega=i$,
$g=0,~{}1,~{}2,~{}3$. Similar relations hold for the vertex operators, with
$\omega=-1$. An identity

$\displaystyle Z_{l}^{p}X_{l}^{q}=\omega^{pq}X_{l}^{q}Z_{l}^{p}$ (5)

will be used repeatedly for many of the derivations that will follow. It is
easily verified that all the projectors in the Hamiltonian are mutually
commuting and square to itself.

This model we wrote down in Eq. (1) possesses some global symmetries. For the
$V_{2}/L_{4}$ Hamiltonian, there are two global symmetry operators

$\displaystyle\phi=\prod_{i}x_{v_{i}},~{}~{}\theta=\prod_{i}Z_{l_{i}},$ (6)

which generate the $\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ symmetry of the model.
One can check that both symmetry generators commute with the Hamiltonian. A
crucial aspect of the $V_{2}/L_{4}$ model is the existence of a local
$\mathbb{Z}_{2}$ operator $X_{l_{i}}^{2}$, which commutes with the Hamiltonian
$[H,X_{l_{i}}^{2}]=0$ for any link $l_{i}$. Such local operator is absent in
the $V_{2}/L_{2}$ model. As a result, the eigenstates can be classified
according to the set of quantum numbers or “$p$-sectors”

$\displaystyle p\equiv\\{p_{1},\cdots,p_{L}\\}$ (7)

where $p_{i}=\pm 1$ stands for the eigenvalue of $X_{l_{i}}^{2}$ at the link
$l_{i}$.

Within each $p$-sector one can reduce the $V_{2}/L_{4}$ model to an effective
$V_{2}/L_{2}$ model. To accomplish this, first one organizes the link states
in terms of the eigenstates of $X_{l}$:

$\displaystyle|\overline{0}\rangle_{l}$ $\displaystyle=$
$\displaystyle(|0\rangle+|1\rangle_{l}+|2\rangle_{l}+|3\rangle_{l})/2,$
$\displaystyle|\overline{1}\rangle_{l}$ $\displaystyle=$
$\displaystyle(|0\rangle_{l}+\omega^{3}|1\rangle_{l}+\omega^{2}|2\rangle_{l}+\omega|3\rangle_{l})/2,$
$\displaystyle|\overline{2}\rangle_{l}$ $\displaystyle=$
$\displaystyle(|0\rangle_{l}-|1\rangle_{l}+|2\rangle_{l}-|3\rangle_{l})/2,$
$\displaystyle|\overline{3}\rangle_{l}$ $\displaystyle=$
$\displaystyle(|0\rangle_{l}+\omega|1\rangle_{l}+\omega^{2}|2\rangle_{l}+\omega^{3}|3\rangle_{l})/2.$
(8)

It is easily shown that

$\displaystyle X_{l}|\overline{n}\rangle_{l}$ $\displaystyle=$
$\displaystyle\omega^{n}|\overline{n}\rangle_{l},$ $\displaystyle
Z_{l}|\overline{n}\rangle_{l}$ $\displaystyle=$
$\displaystyle|\overline{n-1}\rangle_{l}.$ (9)

Both $|\overline{0}\rangle_{l_{i}}$ and $|\overline{2}\rangle_{l_{i}}$ share
the same eigenvalue $X_{l_{i}}^{2}=p_{i}=+1$. For
$|\overline{1}\rangle_{l_{i}}$ and $|\overline{3}\rangle_{l_{i}}$ the
eigenvalue is $p_{i}=-1$. Within a given $p$-sector, we have

$\displaystyle Z_{l_{i}}^{2}|\overline{0}\rangle_{l_{i}}$ $\displaystyle=$
$\displaystyle|\overline{2}\rangle_{l_{i}},~{}(p_{i}=+1)$ $\displaystyle
Z^{2}_{l_{i}}|\overline{1}\rangle_{l_{i}}$ $\displaystyle=$
$\displaystyle|\overline{3}\rangle_{l_{i}},~{}(p_{i}=-1).$ (10)

In other words, the $Z_{l}^{2}$ operator acts effectively as a spin-1/2 Pauli
operator, $Z_{l_{i}}^{2}\rightarrow x_{l_{i}}$, in the two-dimensional
subspace of fixed $p_{i}$. One can also show that $X_{l_{i}}$ acts as the
Pauli-$z_{l_{i}}$ if $p_{i}=+1$, and as $X_{l_{i}}\equiv\omega z_{l_{i}}$ if
$p_{i}=-1$. As a result, the following replacements are allowed in a given
$p$-sector:

$\displaystyle X_{l}$ $\displaystyle\rightarrow$
$\displaystyle\omega^{(1-p_{l})/2}z_{l},$ $\displaystyle X_{l}^{2}$
$\displaystyle\rightarrow$ $\displaystyle p_{l},$ $\displaystyle X_{l}^{3}$
$\displaystyle\rightarrow$ $\displaystyle p_{l}\omega^{(1-p_{l})/2}z_{l}.$
(11)

With these considerations one can reduce the $A_{i}$ and $C_{i}$ operators in
Eqs. (II) and (3) as

$\displaystyle A_{i}$ $\displaystyle\rightarrow$ $\displaystyle
A_{i}^{(p)}=\frac{1}{4}\Bigl{[}1+p_{i-1}p_{i}$
$\displaystyle~{}~{}~{}~{}+(p_{i-1}+p_{i})\omega^{-(p_{i-1}+p_{i})/2+1}z_{l_{i-1}}x_{v_{i}}z_{l_{i}}\Bigr{]}$
$\displaystyle C_{i}$ $\displaystyle\rightarrow$ $\displaystyle
C_{i}^{(p)}=\frac{1}{2}(1+z_{v_{i}}x_{l_{i}}z_{v_{i+1}}).$ (12)

Each $p$-sector then gives rise to an effective $V_{2}/L_{2}$ Hamiltonian

$\displaystyle H^{(p)}=-\sum_{i}(A_{i}^{(p)}+C_{i}^{(p)})$ (13)

with the $A_{i}^{(p)}$ and $C_{i}^{(p)}$ terms given in Eq. (12). One can view
the original $V_{2}/L_{4}$ model as the direct sum

$\displaystyle H=\oplus_{p}H^{(p)},$ (14)

where each $H^{(p)}$ is defined in the $V_{2}/L_{2}$ subspace. From Eq. (12)
we obtain identical operators $A^{(p)}_{i}$ when all the $p_{i}$’s are
reversed, $p_{i}\rightarrow-p_{i}$, implying that each $V_{2}/L_{2}$ sector
ought to be doubly degenerate. The two degenerate subspaces are connected by
the global operation $\prod_{i}Z_{l_{i}}$ or $\prod_{i}Z_{l_{i}}^{3}$, both of
which implement $p_{i}\rightarrow-p_{i}$. In other words,

$\displaystyle H^{(-p)}$ $\displaystyle=$
$\displaystyle(\prod_{i}Z_{l_{i}})H^{(p)}(\prod_{i}Z_{l_{i}}^{3})$ (15)
$\displaystyle=$
$\displaystyle(\prod_{i}Z^{3}_{l_{i}})H^{(p)}(\prod_{i}Z_{l_{i}}).$

In particular, the ground states of the $V_{2}/L_{4}$ model comes from the
sector $p=\\{1,\cdots,1\\}$ and $p=\\{-1,\cdots,-1\\}$ where one can write
$A_{i}$ and $C_{i}$ as

$\displaystyle A_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+z_{l_{i-1}}x_{v_{i}}z_{l_{i}}),$ $\displaystyle
C_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+z_{v_{i}}x_{l_{i}}z_{v_{i+1}}).$ (16)

In fact, this $V_{2}/L_{2}$ model has been studied extensively as a model for
one-dimensional SPT Chen  _et al._ (2014); Geraedts and Motrunich (2014). The
$V_{2}/L_{4}$ model is a natural extension of the $V_{2}/L_{2}$ model. The
pure $V_{2}/L_{2}$ model, with all $p_{l}=1$ or $-1$, has the global
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry generated by

$\displaystyle\prod_{i}x_{v_{i}}~{}~{}{\rm and}~{}~{}\prod_{i}x_{l_{i}},$ (17)

but no extra local symmetry.

##  III ground states of V2/L4 model

There are two ways to go about writing down the ground states of the
$V_{2}/L_{4}$ Hamiltonian, Eq. (1). The first one is to identify the ground
states of $-\sum_{i}C_{i}$ and then act on them with the projector
$\prod_{i}A_{i}$. The other way is to first identify the ground states of
$-\sum_{i}A_{i}$, and then act on them with the projector $\prod_{i}C_{i}$. It
is not hard to see that both ways lead to states with the eigenvalues of
$A_{i}$ and $C_{i}$ all equal to +1, which by definition gives the ground
states of the projector Hamiltonian.

Given our analysis in the previous section, it seems more enlightening the
analyze the ground states in the eigenbasis of $X_{l_{i}}^{2}$, where the
operator $A_{i}$ is diagonalized easily. We obtain the two degenerate ground
states of the $V_{2}/L_{4}$ model as

$\displaystyle|G_{1}\rangle$ $\displaystyle=$ $\displaystyle
P_{C}|S_{1}\rangle,~{}~{}|G_{2}\rangle=P_{C}|S_{2}\rangle$
$\displaystyle|S_{1}\rangle$ $\displaystyle=$
$\displaystyle(\otimes_{i}|\overline{0}\rangle_{v_{i}})(\otimes_{j}|\overline{0}\rangle_{l_{j}})$
$\displaystyle|S_{2}\rangle$ $\displaystyle=$
$\displaystyle(\otimes_{i}|\overline{0}\rangle_{v_{i}})(\otimes_{j}|\overline{1}\rangle_{l_{j}}).$
(18)

The vertex eigenstates
$|\overline{0}\rangle_{v_{i}}=(|0\rangle_{v_{i}}+|1\rangle_{v_{i}})/\sqrt{2}$
and
$|\overline{1}\rangle_{v_{i}}=(|0\rangle_{v_{i}}-|1\rangle_{v_{i}})/\sqrt{2}$
diagonalize the $x_{v}$ operator. The ground states of the $V_{2}/L_{4}$ model
are obtained as the projection by $P_{C}=\prod_{i}C_{i}$ on the two “seed
states” $|S_{1}\rangle$ and $|S_{2}\rangle$. The (unique) ground state of the
$V_{2}/L_{2}$ model is obtained from $|G_{1}\rangle$ above, by rewriting
$C_{i}$ as in Eq. (16). The two ground states of the $V_{2}/L_{4}$ model share
the properties

$\displaystyle(\prod_{i}Z_{l_{i}})|G_{1}\rangle$ $\displaystyle=$
$\displaystyle|G_{2}\rangle,$ $\displaystyle X_{l_{i}}^{2}|G_{1}\rangle$
$\displaystyle=$ $\displaystyle+|G_{1}\rangle$ $\displaystyle
X_{l_{i}}^{2}|G_{2}\rangle$ $\displaystyle=$ $\displaystyle-|G_{2}\rangle.$
(19)

We also show how to write down the ground states in the $Z$-basis, where the
$C_{i}$’s are diagonalized first and $P_{A}=\prod_{i}A_{i}$ acts as a
projector. The two ground states are

$\displaystyle|G^{\prime}_{1}\rangle$ $\displaystyle=$ $\displaystyle
P_{A}\left[(\otimes_{i}\ket{0}_{v_{i}})(\otimes_{j}\ket{0}_{l_{j}})\right],$
$\displaystyle|G^{\prime}_{2}\rangle$ $\displaystyle=$ $\displaystyle
P_{A}\left[(\otimes_{i}\ket{0}_{v_{i}})(\otimes_{j\neq
j^{\prime}}\ket{0}_{l_{j}})\otimes\ket{2}_{l_{j^{\prime}}}\right]$ (20)

where $j^{\prime}$ is arbitrary. These ground states share the properties

$\displaystyle X_{l_{i}}^{2}|G^{\prime}_{1}\rangle$ $\displaystyle=$
$\displaystyle|G^{\prime}_{2}\rangle,$
$\displaystyle(\prod_{i}Z_{l_{i}})|G^{\prime}_{1}\rangle$ $\displaystyle=$
$\displaystyle+|G^{\prime}_{1}\rangle,$
$\displaystyle(\prod_{i}Z_{l_{i}})|G^{\prime}_{2}\rangle$ $\displaystyle=$
$\displaystyle-|G^{\prime}_{2}\rangle.$ (21)

Comparing Eqs. (19) and (21), one concludes

$\displaystyle|G_{1}\rangle$ $\displaystyle=$
$\displaystyle(|G^{\prime}_{1}\rangle+|G^{\prime}_{2}\rangle)/\sqrt{2}$
$\displaystyle|G_{2}\rangle$ $\displaystyle=$
$\displaystyle(|G^{\prime}_{1}\rangle-|G^{\prime}_{2}\rangle)/\sqrt{2}.$ (22)

In what follows, we provide a geometrical interpretation of the ground state
$|G_{1}\rangle$. Due to $z_{v}^{2}=1$, the product of neighboring
$z_{v_{i}}Z_{l_{i}}^{2}z_{v_{i+1}}$ is simply
$Q_{12}=z_{v_{1}}\left(\prod_{i_{\mathcal{S}}}Z^{2}_{l_{i}}\right)z_{v_{2}}$
where $i_{\mathcal{S}}$ stands for the links between the left-most vertex
$v_{1}$ and the right-most one $v_{2}$. Its action on the seed state
$|S_{1}\rangle$ permutes the link states along $\mathcal{S}$, i.e.,
$|\overline{0}\rangle_{l}\rightarrow|\overline{2}\rangle_{l}$ and flips the
vertex states at edges
$|\overline{0}\rangle_{v_{1/2}}\rightarrow|\overline{1}\rangle_{v_{1/2}}$, or
graphically illustrated as

$\displaystyle\includegraphics[width=65.04034pt]{string.pdf},$

where blue dot stands for each edge vertex while the red string for all links
and vertices along $\mathcal{S}$. Consequently, the expansion of the projector
$P_{C}$ leads to the super position of all possible string configurations on
the circle:

$\displaystyle\includegraphics[width=186.45341pt]{geometric_gs.pdf}.$ (23)

Due to $|G_{2}\rangle=\prod_{i}Z_{l_{i}}|G_{2}\rangle$, the geometrical
interpretation of $|G_{2}\rangle$ is given in the same manner. Only difference
is that all link states are raised by one, i.e.,
$|\overline{n}\rangle_{l}\rightarrow|\overline{n+1}\rangle_{l}$.

In an open chain of the same model, some edge states appear as a consequence
of SPT, as thoroughly analyzed in Ref. et al, 2020.

##  IV V2/L4 Model on a graph

###  IV.1 GSD on a Graph

The discussion of the ground states of the $V_{2}/L_{4}$ model both in the
closed and the open chain thus far might suggest that we are merely dealing
with what seems to be two copies of the well-known $V_{2}/L_{2}$ model.
Interestingly, the real point of departure between the two families of models
occurs when these models are put on a graphet al (2020). The closed circle is
a simplest example of a graph with the first Betti number $B_{1}=1$. Now, one
can imagine putting the model on a more intricate graph such as shown in Fig.
2(b), which has $B_{1}=2$. Intuitively, the first Betti number measures the
number of cycles or independent loops in a given graph.

Figure 2: Examples of closed and connected planar graphs with various Betti
numbers (a) $B_{1}=1$, (b) $B_{1}=2$, and (c), (d) $B_{1}=3$. Dots are the
vertices and the arrows represent the links of the graph. Dashed lines are the
independent closed loops where we can perform a subsystem $\mathbb{Z}_{4}$
symmetry transformation. Because of the orientation, the loop operators
correspond to the right semi-circular loop are different for (c) and (d).

Considering a $B_{1}=2$ graph such as shown in Fig. 2(b), one can write the
same $A_{i}$ and $C_{i}$ projectors as in the $B_{1}=1$ graph, namely a
circle, except at the two vertices $v_{i}$ and $v_{j}$ where two lines become
connected at the vertex. At these vertices, there are three (not two) links
which are connected to a single vertex. The definition of the vertex operators
which are projectors and commute with other vertex operators must generalize
accordingly,

$\displaystyle A_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1\\!+\\!X_{l_{i-1}}X_{l_{i}}^{3}X_{l_{i^{\prime}}}^{3}x_{v_{i}}\\!$
$\displaystyle+\\!X_{l_{i-1}}^{2}X_{l_{i}}^{2}X_{l_{i^{\prime}}}^{2}\\!+\\!X_{l_{i-1}}^{3}X_{l_{i}}X_{l_{i^{\prime}}}x_{v_{i}})$
$\displaystyle A_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1\\!+\\!X_{l_{j-1}}X_{l_{j}}^{3}X_{l_{i^{\prime\prime}}}x_{v_{j}}\\!$
(24)
$\displaystyle+\\!X_{l_{j-1}}^{2}X_{l_{j}}^{2}X_{l_{i^{\prime\prime}}}^{2}\\!+\\!X_{l_{j-1}}^{3}X_{l_{j}}X_{l_{i^{\prime\prime}}}^{3}x_{v_{j}})$

where $v_{i}$ ($v_{j}$) has one (two) arrowhead towards it and two (one)
arrowheads away from it. In fact, a completely general definition of the
vertex operator for arbitrary graph is possible as

$\displaystyle A_{i}=\frac{1}{4}\sum_{j=0}^{3}\left[(\prod_{q\in
in}X_{l_{q,i}}\prod_{q^{\prime}\in
out}X_{l_{q^{\prime},i}}^{3})x_{v_{i}}\right]^{j}.$ (25)

There is a factor $X$ ($X^{3}$) for the links whose arrows come into (out of)
the vertex. The link operator $C_{i}$ is the same as in Eq. (3) regardless of
the graph type. Despite the much complex forms, $A_{i}$ and $C_{j}$ always
remain mutually commuting projectors. The $V_{2}/L_{4}$ Hamiltonian on an
arbitrary graph generalizes accordingly,

$\displaystyle H=-\sum_{i=1}^{N_{v}}A_{i}-\sum_{j=1}^{N_{l}}C_{j},$ (26)

spanning all the vertices and the links in the graph and using Eq. (25) for
the vertex operator. The number of vertices ($N_{v}$) and of links ($N_{l}$)
are no longer equal for a general graph but are rather related by the first
Betti number,

$\displaystyle N_{l}-N_{v}+1=B_{1}.$ (27)

###  IV.2 Subsystem Symmetries

We can discuss how to explicitly construct the multitude of degenerate ground
states on a graph. For example, the $V_{2}/L_{4}$ model defined on a graph
shown in Fig. 2(b) allows two loop operators that commute with the
Hamiltonian. We call them $\theta_{1}$ and $\theta_{2}$, and they consist of
the product of $Z_{l}$’s along the left semi-circular loop and the large
circular loop, respectively. These two loops are drawn as dashed lines in Fig.
2(b). Note that the definition of these loop operators are obtained by simply
“following the arrows” drawn on the graph. On a $B_{1}=2$ graph we have two
such independent loops. Accordingly one can write down four independent ground
states, which are
$|G_{1}\rangle=P_{C}\left[(\otimes_{i}|\overline{0}\rangle_{v_{i}})(\otimes_{j}|\overline{0}\rangle_{l_{j}})\right]$,
and

$\displaystyle|G_{2}\rangle$ $\displaystyle=$
$\displaystyle\theta_{1}|G_{1}\rangle,$ $\displaystyle|G_{3}\rangle$
$\displaystyle=$ $\displaystyle\theta_{2}|G_{1}\rangle,$
$\displaystyle|G_{4}\rangle$ $\displaystyle=$
$\displaystyle\theta_{1}\theta_{2}|G_{1}\rangle.$ (28)

At first sight the construction of degenerate ground states on a graph bears
resemblance to the way that topologically distinct states are generated on a
finite-genus space. The GSD formula ${\rm GSD}=2^{B_{1}}$ has resemblance to
the formula for the topological ${\rm GSD}=2^{2g}$, which applies to two-
dimensional topological models such as the toric code, where $g$ (also known
as the second Betti number) is the genus of two-dimensional surface.

Unlike the topologically ordered states which do not have local order
parameters, the four ground states derived above can be distinguished by their
“order parameter” $X_{l}^{2}$. With $|G_{1}\rangle$, the expectation value is
$\langle G_{1}|X_{l}^{2}|G_{1}\rangle=+1$ on every link of the graph. For
$|G_{2}\rangle$, the links along the loop where $\theta_{1}$ acts have
$\langle X_{l}^{2}\rangle=-1$. For $|G_{3}\rangle$, it is the links along the
outer perimeter where the order parameters are reversed. Finally in the fourth
ground state $|G_{4}\rangle$ it is the other inner loop where the links have
$\langle X_{l}^{2}\rangle=-1$. The four order parameter patterns are depicted
in Fig. 3. There is a clear parallel to the usual classification of states by
order parameters, but one must carefully note that its nature is not entirely
global. Perhaps a better termnology is the subsystem symmetry breaking (SSB,
not to be confused with the spontaneous symmetry breaking) and distinguish it
from the global symmetry breaking (GSB) of most many-body models.

Along a similar line of reasoning, the loop operator consisting of the product
of $Z_{l}$’s and $Z_{l}^{3}$’s along the independent loops as dictated by the
flow of arrows commute with the Hamiltonian and performs the subsystem
$\mathbb{Z}_{4}$ symmetry transformation. In the case of Fig. 2(c), all such
loop operators consist of the product of $Z_{l}$’s only. On the other hand,
the loop operator corresponding to the right semi-circuplar loop in Fig. 2(d)
is $\prod_{i}Z_{l_{i}}\prod_{i^{\prime}}Z_{l_{i^{\prime}}}^{3}$ where $i$’s
are the links on the arc and $i^{\prime}$’s are the links on the inner
segment. The number of independent loops equals the Betti number $B_{1}$, and
the degenerate ground states are generated by applying the subsystem loop
operators and their products on one particular ground state $|G\rangle$. There
are exactly $2^{B_{1}}-1$ different products of subsystem loop operators
available, for a total of $2^{B_{1}}$ ground states on a graph with the Betti
number $B_{1}$.

Figure 3: Four kinds of configurations of $p$’s that reduce the $V_{2}/L_{4}$
model to the $V_{2}/L_{2}$ model when $B_{1}=2$. The $\pm$ signs on the links
represent the eigenvalues of $X_{l}^{2}$, or equivalently, the $p_{l}$’s.

The $V_{2}/L_{4}$ Hamiltonian on a graph still has the exact symmetry
$[H,X_{l}^{2}]=0$ for all the links $l$. As a result, $X_{l}^{2}$ operators in
the graph model can be replaced by their respective quantum numbers $p_{l}$.
In particular, the vertex operators at the junction given in Eq. (24) become,
in a given $p$-sector,

$\displaystyle A_{i}^{(p)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\bigl{[}1\\!+p_{i-1}p_{i}p_{i^{\prime}}$
$\displaystyle+$
$\displaystyle(p_{i}p_{i^{\prime}}\\!+\\!p_{i-1})\omega^{(3-p_{i-1}-p_{i}-p_{i^{\prime}})/2}z_{l_{i-1}}z_{l_{i}}z_{l_{i^{\prime}}}x_{v_{i}}\bigr{]}$
$\displaystyle A_{j}^{(p)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\bigl{[}1\\!+p_{j-1}p_{j}p_{i^{\prime}}$
$\displaystyle+$
$\displaystyle(p_{j-1}p_{i^{\prime}}\\!+\\!p_{j})\omega^{(3-p_{j-1}-p_{j}-p_{j^{\prime}})/2}z_{l_{j-1}}z_{l_{j}}z_{l_{i^{\prime}}}x_{v_{i}}\bigr{]}.$

One can show, by explicit calculation, that $A_{i}^{(p)}$ becomes
$\frac{1}{2}\bigl{(}1+z_{l_{i-1}}z_{l_{i}}z_{l_{i^{\prime}}}x_{v_{i}}\bigr{)}$
for $\\{p_{i-1},p_{i},p_{i^{\prime}}\\}$ $=\\{1,1,1\\}$, $\\{-1,1,-1\\}$,
$\\{-1,-1,1\\}$. In a similar manner, we have
$A_{j}^{(p)}=\frac{1}{2}\bigl{(}1+z_{l_{j-1}}z_{l_{j}}z_{l_{i^{\prime}}}x_{v_{j}}\bigr{)}$
for $\\{p_{j-1},p_{j},p_{i^{\prime}}\\}=$ $\\{1,1,1\\}$, $\\{-1,-1,1\\}$,
$\\{1,-1,-1\\}$. The $V_{2}/L_{2}$ model on a graph comes from having the
following choice of vertex operators:

$\displaystyle A_{i}^{(p)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}1+z_{l_{i-1}}z_{l_{i}}z_{l_{i^{\prime}}}x_{v_{i}}\bigr{)}$
$\displaystyle A_{j}^{(p)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}1+z_{l_{j-1}}z_{l_{j}}z_{l_{i^{\prime}}}x_{v_{j}}\bigr{)}$
(30)

at the two junctions $i$ and $j$ shown in Fig. 2(b). For all other vertices
and all the links one has the usual definition of the vertex and link
operators given in Eq. (16). After some enumeration, one finds the four
configurations of $p$’s shown in Fig. 3 can reduce the $V_{2}/L_{4}$ model to
the $V_{2}/L_{2}$ model on the graph with $B_{1}=2$. This argument again shows
why there is fourfold degeneracy of the ground states on the $B_{1}=2$ graph.

It is worth examining the general character of the pure $V_{2}/L_{2}$ model on
a graph. It can be shown that GSD of the $V_{2}/L_{2}$ model remains at GSD=1
regardless of the Betti number of the graph. The global
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry of the $V_{2}/L_{2}$ model on a
simply connected graph with $B_{1}=1$ is partially lost due to the vertex
terms at the junction, Eq. (30). One can prove quite easily that although
$\prod_{i}x_{v_{i}}$ remains a symmetry, $\prod_{i}x_{l_{i}}$ no longer
commutes with the vertex operators at the junction and hence fails to be a
symmetry operator. The global symmetry of the $V_{2}/L_{2}$ model is lowered
from $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ to $\mathbb{Z}_{2}$ on a multiply
connected graph. On further observation, however, we realize that a partial
product of link operators $\prod^{\prime}_{i}x_{l_{i}}$ for the links forming
a closed loop does commute with the junction terms in Eq. (30) and restore the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry for that loop. Contrasted with
the usual SPT, this is a realization of the subsystem SPT, or SSPT You  _et
al._ (2018). By implication, when we cut open any segment of the graph, the
emerging edge behavior and symmetry fractionalization will be exactly those of
the open chain case already analyzed. Since the ground states of the pure
$V_{2}/L_{2}$ model are ground states of the $V_{2}/L_{4}$ model, even in
$V_{2}/L_{4}$ model there is SSPT.

##  V Excitations

The concept of excitation arises naturally in frustration-free models to which
our $V_{2}/L_{4}$ model belongs. The ground state(s) has all of the
eigenvalues of $A_{i}$ and $C_{i}$ equal to +1, and excited states should have
one of these equal to zero instead et al (2020). Depending on whether the link
or the vertex operator eigenvalues change from 1 to 0, one can make a
distinction between link excitations and vertex excitations. Another useful
way to classify excitations is in terms of the changes in the $p$-eigenvalues,
$p=\\{p_{1},\cdots,p_{L}\\}$, of the chain. Changes in any of the eigenvalues
in the $p$-set leads to different sectors of the block, $H=\oplus_{p}H^{(p)}$,
where $H$ is the original $V_{2}/L_{4}$ Hamiltonian and each $H^{(p)}$
represents some realization of the $V_{2}/L_{2}$ Hamiltonian. We will examine
the nature of excitations from both perspectives. Eigenvalues of $A_{i}$ and
$C_{i}$ will be denoted $a_{i}$ and $c_{i}$ from now on.

The link excitation is attained by rewriting one of the operators $C_{i}$ in
the projector $P_{C}$ by its orthogonal complement et al (2020)

$\displaystyle
C_{i}^{\perp}=\frac{1}{2}(1-z_{v_{i}}Z_{l_{i}}^{2}z_{v_{i+1}}),~{}~{}C_{i}^{\perp}C_{i}=0.$
(31)

We can define the projector

$\displaystyle P_{C}(j)=(\prod_{i<j}C_{i})C_{j}^{\perp}(\prod_{i>j}C_{i}),$
(32)

and use such projector to create a link-excited state

$\displaystyle|l_{j}\rangle=P_{C}(j)|S\rangle$ (33)

with $|S\rangle$ being one of the two seed states. Since
$C_{j}|l_{j}\rangle=0$, one concludes $c_{j}=0$, qualifying it as a link
excitation. Note that $X_{l_{j}}C_{j}=C_{j}^{\perp}X_{l_{j}}$, and therefore

$\displaystyle X_{l_{j}}|G\rangle$ $\displaystyle=$ $\displaystyle
X_{l_{j}}P_{C}|S\rangle=P_{C}(j)X_{l_{j}}|S\rangle$ (34) $\displaystyle=$
$\displaystyle\pm P_{C}(j)|S\rangle=\pm|l_{j}\rangle,$

where the $\pm$ sign comes from having $|S\rangle=|S_{1}\rangle$ or
$|S\rangle=|S_{2}\rangle$. In other words, link excitations are created by
applying $X_{l_{i}}$ on a given link to the ground states. By a similar
consideration, we learn that $x_{v_{j}}$ acting on the ground states generate
a link-pair excitation:

$\displaystyle x_{v_{j}}|G\rangle=|l_{j-1},l_{j}\rangle.$ (35)

The proof comes from the observation,

$\displaystyle
x_{v_{j}}P_{C}=(\prod_{i<j-1}C_{i})C_{j-1}^{\perp}C_{j}^{\perp}(\prod_{i>j+1}C_{i}).$
(36)

A string of $x$-excitations acting on the ground states gives

$\displaystyle x_{v_{2}}\cdots x_{v_{k}}\ket{G}$ $\displaystyle=$
$\displaystyle\ket{l_{1},l_{k}}$ (37)

and when the string forms a closed loop, the state gets back to the original
ground state, e.g. $\prod_{i\in loop}x_{v_{i}}\ket{G}=\ket{G}$.

Both types of link-excited states $X_{l_{j}}|G\rangle$ and
$x_{v_{j}}|G\rangle$ share the same $p$-eigenvalues as the ground state. This
is seen by the fact that the operator that defines the $p$-eigenvalue,
$X_{l}^{2}$, commutes with with $X_{l}$ and $x_{v}$ which create the link
excitations. We conclude that the link excitations $X_{l_{j}}|G\rangle$ and
$x_{v_{j}}|G\rangle$ occurs in the same $p$-sector as the ground state
(intra-$p$ excitation). In this regard, the link excitations we write down
here are the same as what the $V_{2}/L_{2}$ model would give.

The $p$-altering or inter-$p$ excitations are created by acting with
$Z_{l_{j}}$ directly on the seed states. Since $Z_{l_{i}}$ commutes with the
projector $P_{C}$, the $p$-altering excited state is given by

$\displaystyle P_{C}(Z_{l_{j}}|S\rangle)=Z_{l_{j}}|G\rangle.$ (38)

One can easily check that $X_{l_{j}}^{2}$ anti-commutes with $Z_{l_{j}}$,
therefore $p_{l_{j}}$ changes from +1 to -1. This state, interestingly,
continues to remain an eigenstate of $C_{i}$ with $c_{i}=+1$ everywhere.
Contrary to naive expectation, $Z_{l_{j}}|G\rangle$ is not a link excitation.
When one applies $A_{i}$ to this state, some nontrivial changes are
discovered:

$\displaystyle A_{j}Z_{l_{j}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}Z_{l_{j}}(1+\omega^{3}X_{l_{j-1}}x_{v_{j}}X_{l_{j}}^{3}$
$\displaystyle+\omega^{2}X_{l_{j-1}}^{2}X_{l_{j}}^{2}+\omega
X_{l_{j-1}}^{3}x_{v_{j}}X_{l_{j}})$ $\displaystyle A_{j+1}Z_{l_{j}}$
$\displaystyle=$ $\displaystyle\frac{1}{4}Z_{l_{j}}(1+\omega
X_{l_{j}}x_{v_{j+1}}X_{l_{j+1}}^{3}$ (39)
$\displaystyle+\omega^{2}X_{l_{j}}^{2}X_{l_{j+1}}^{2}+\omega^{3}X_{l_{j}}^{3}x_{v_{j+1}}X_{l_{j+1}}).$

The appearance of new terms on the right side of the equations necessitates
that we define some new vertex projectors as follows:

$\displaystyle A_{i}(n)$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1+\omega^{n}X_{l_{i-1}}x_{v_{i}}X_{l_{i}}^{3}$ (40)
$\displaystyle~{}~{}+\omega^{2n}X_{l_{i-1}}^{2}X_{l_{i}}^{2}+\omega^{3n}X_{l_{i-1}}^{3}x_{v_{i}}X_{l_{i}}).$

The vertex projector used to define the Hamiltonian is recovered for $n=0$,
and the relations derived in Eq. (39) are now succinctly summed up:

$\displaystyle A_{j}Z_{l_{j}}$ $\displaystyle=$ $\displaystyle
Z_{l_{j}}A_{j}(3)$ $\displaystyle A_{j+1}Z_{l_{j}}$ $\displaystyle=$
$\displaystyle Z_{l_{j}}A_{j+1}(1).$ (41)

The $A_{i}(n)$ projectors commute with $C_{j}$ and
$A_{i}(n)A_{i}(n^{\prime})=0$ unless $n=n^{\prime}$. Furthermore,
$A_{i}(n)|S\rangle=0$ unless $n=0$. Due to these properties, we can show

$\displaystyle A_{j}Z_{l_{j}}|G\rangle=A_{j+1}Z_{l_{j}}|G\rangle=0.$ (42)

In other words, a pair of vertex excitations with $a_{j}=a_{j+1}=0$ has been
created. We conclude that different $p$-sectors of the $V_{2}/L_{4}$ model are
connected through the creation of vertex-pair excitations. A
$p_{l_{j}}=1\rightarrow-1$ implies that a pair of adjacent vertices have been
excited, $a_{j}=a_{j+1}=0$. We may sum up the situation as

$\displaystyle Z_{l_{j}}|G\rangle$ $\displaystyle=$
$\displaystyle|\omega^{3}_{v_{j}},\omega_{v_{j+1}}\rangle,$ $\displaystyle
Z^{3}_{l_{j}}|G\rangle$ $\displaystyle=$
$\displaystyle|\omega_{v_{j}},\omega^{3}_{v_{j+1}}\rangle.$ (43)

The symbol on the right $|\omega^{n}_{v}\rangle$ means that the seed state has
been acted on by $A_{v}(n)$ before the projector $P_{C}$ is applied. The
vertex excitations carrying $\omega^{3}$ and $\omega$ as quantum numbers must
exist as a pair, and serve to connect different $p$-sectors.

On the other hand, exciting the vertex state by $z_{v}$ or a link state by
$Z_{l}^{2}$ does not change the $p$-eigenvalues, and leads to the following
excitations:

$\displaystyle z_{v_{j}}|G\rangle$ $\displaystyle=$
$\displaystyle|\omega^{2}_{v_{j}}\rangle,$ $\displaystyle
Z_{l_{j}}^{2}|G\rangle$ $\displaystyle=$
$\displaystyle|\omega^{2}_{v_{j}},\omega^{2}_{v_{j+1}}\rangle=z_{v_{j}}z_{v_{j+1}}|G\rangle.$
(44)

The vertex excitation with the quantum number $\omega^{2}=-1$ can exist in
isolation. The link projector $C_{i}$ has the useful identity
$C_{j}=z_{v_{j}}z_{v_{j+1}}Z_{l_{j}}^{2}C_{j}$, which one can verify directly
from its definition. As a result, $P_{C}$ acting on some seed state
$|S\rangle$ leads to the same consequence as when acting on a different seed
state $z_{v_{j}}z_{v_{j+1}}Z_{l_{j}}^{2}|S\rangle$. This leads to the identity
mentioned in the second equation above.

The $p$-changing vertex-pair excitation created by $Z_{l_{j}}|G\rangle$ can be
generalized by considering a string of $Z$-excitations given by the product

$\displaystyle Z_{l_{1}}\cdots
Z_{l_{k}}|G\rangle=|\omega_{v_{1}}^{3},\omega_{v_{k+1}}\rangle.$ (45)

By the time the product $\prod_{i}Z_{l_{i}}$ forms a closed loop, one reaches
the other ground state, e.g. $(\prod_{i\in
loop}Z_{l_{i}})|G_{1}\rangle=|G_{2}\rangle$. To sum up, there are intra-$p$
excitations in the form of link excitations [Eq. (34)] and vertex excitations
[Eq. (44)], and inter-$p$ excitations in the form of vertex-pair excitations
[Eq. (43)]. This gives the complete classification of the elementary
excitations in the $V_{2}/L_{4}$ model.

##  VI Discussion

In this paper we have analyzed the properties of the $V_{2}/L_{4}$ model on a
general graph et al (2020). The Hilbert space of the model is block-
diagonalized by a set of local quantum numbers $\\{p_{l_{i}}=\pm 1\\}$, and we
have shown that the $V_{2}/L_{4}$ model maps to a general $V_{2}/L_{2}$ model
in each $p$-sector.

The GSD of our model grows exponentially with the Betti number $B_{1}$
characterizing the number of cycles in a graph. In fact one can easily show
that the exponential growth of GSD with the Betti number is not unique to the
$V_{2}/L_{4}$ model. Even a simple Potts model on a graph can be defined in a
way that exhibits the same GSD behavior. For that, one considers the same kind
of graph as before and place $n$ degrees of freedom at each link labeled by
the variable $z_{l}=0,\cdots,n-1$, and none on the vertices. We then have the
Potts interaction $-\delta_{z_{l},z_{l^{\prime}}}$ for the neighboring links
$(l,l^{\prime})$, except when more than two lines meet at a vertex. In that
case we have the three-link interaction
$-\delta_{z_{l_{1}},z_{l_{2}},z_{l_{3}}}$ among the three links
$(l_{1},l_{2},l_{3})$ joined at a vertex. For more than three links, one
simply takes the delta function of all the links
$-\delta_{z_{l_{1}},z_{l_{2}},\cdots}$ with an overall minus sign. It is
easily verified that the ground states have the $z_{l}$’s distributed in
exactly the same pattern as those of $X_{l}^{2}$’s (or $p_{l}$’s) shown in
Fig. 3 when $n=2$. For general graphs and $n$ degrees of freedom, GSD equals
$n^{B_{1}}$, but no feature of SPT or symmetry fractionalization would be
present in such models.

To sum up, the subsystem symmetry and its breaking observed in the
$V_{2}/L_{4}$ model is not related to the SPT nature of the phase but rather
co-exist with it. The idea of subsystem symmetries is applicable for both the
Landau order parameter characterized by $X_{l}^{2}$, and the SPT order.

###### Acknowledgements.

H. J. H. was supported by the Quantum Computing Development Program (No.
2019M3E4A1080227). H.-Y.L. was supported by a Korea University Grant and
National Research Foundation of Korea (NRF-2020R1I1A3074769). We appreciate
enlightening discussion with Munjip Park.

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