# Foliated BF theories and Multipole symmetries

Hiromi Ebisu1,2***hiromi.ebisu(at)yukawa.kyoto-u.ac.jp, Masazumi
Honda2,3†††masazumi.honda(at)yukawa.kyoto-u.ac.jp and Taiichi
Nakanishi2‡‡‡taiichi.nakanishi(at)yukawa.kyoto-u.ac.jp

1Department of Physics and Astronomy, Rutgers University,
Piscataway, NJ 08854, USA
2Center for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University,
Sakyo-ku, Kyoto 606-8502, Japan
3Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS),
RIKEN, Wako 351-0198, Japan

(October 2023)

###### Abstract

Due to the recent studies of the fracton topological phases, which host
deconfined quasi-particle excitations with mobility restrictions, the concept
of symmetries have been updated. Focusing on one of such new symmetries,
multipole symmetries, including global, dipole, and quadruple symmetries, and
gauge fields associated with them, we construct a new sets of $\mathbb{Z}_{N}$
$2+1d$ foliated BF theories, where BF theories of conventional topological
phases are stacked in layers with couplings between them. By investigating
gauge invariant non-local operators, we show that our foliated BF theories
exhibit unusual ground state degeneracy depending on the system size; it
depends on the greatest common divisor between $N$ and the system size. Our
result provides a unified insight on UV lattice models of the fracton
topological phases and other unconventional ones in view of foliated field
theories.

YITP-23-129, RIKEN-iTHEMS-Report-23

###### Contents

  1. 1 Introduction
  2. 2 Warm-up: construction of the $\mathbb{Z}_{N}$ toric code
  3. 3 Dipole symmetries
    1. 3.1 Dipole and global symmetries
    2. 3.2 Ground state degeneracy
    3. 3.3 UV lattice model
  4. 4 Quadrupole
    1. 4.1 Construction of the foliated BF theory
    2. 4.2 Ground state degeneracy
    3. 4.3 UV lattice model
    4. 4.4 Multipole one-form symmetries
  5. 5 Conclusion and discussions
  6. A Derivation of (3.29)
  7. B Investigation of the lattice model by the Laplacian

##  1 Introduction

Topologically ordered phases are unconventional phases of matter beyond the
standard Landau-Ginzburg formalism and have been one of the central subjects
in condensed matter physics community [1, 2, 3, 4, 5, 6]. The prominent
feature of these phases is that they host exotic fractionalized excitations,
called anyons [7, 2, 8]. These excitations are not only theoretically
intriguing but also may find potential application in quantum information
science as exchanging these excitations can be utilized for quantum computers
[9, 10]. Theoretical framework of these phases have been well-established,
such as topological quantum field theories [11, 12, 13, 14] and the modular
tensor category [15].

While a plethora of efforts have been devoted to understand these phases from
both of theoretical and experimental viewpoints, recently new types of
topologically ordered phases have been introduced, referred to as the fracton
topological phases [16, 17, 18, 19, 20, 21]. The distinctive feature of these
phases is that a mobility constraint is imposed on quasi-particle excitations.
Because of this property, the phases exhibit the sub-extensive ground state
degeneracy (GSD), indicating that these phases are sensitive to the UV
physics. Due to this feature, conventional topological effective field theory
description of the topologically ordered phases cannot be applied to the
fracton topological phases. Fractons have attracted a lot of interests not
only in condensed matter physics, but also in the the field of high energy
physics. Indeed, fractons have been recently studied in the context of the
gravity theory [22, 23, 24, 25] and the branes [26]. Moreover, it was found
that fracton models on hyperbolic space show holographic properties [27, 28,
29]. Given the novelty of these phases, one of the challenges is to establish
a consistent framework for continuum field theories.

One of the attempts to tackle this problem is to introduce new types of
symmetries, sub-system symmetries and multipole symmetries [30, 31, 32, 33,
34, 35, 36]. Here, sub-system symmetry means that a theory is invariant under
a symmetry action which acts on a sub-manifold, such as a line or plane. The
multipole symmetry, especially, the $U(1)$ multipole symmetry, is the
generalization of the global $U(1)$ symmetry in the sense that a theory is
invariant under the global phase rotation which depends on the spatial
coordinate in polynomial form. As an example, in the case of a scalar theory
which respects the global and dipole $U(1)$ symmetries, the Lagrangian is
invariant under $\Phi\to e^{i\alpha+i\beta x}\Phi$, where $\alpha$ and $\beta$
are constants, and $x$ denotes the spatial coordinate [33]. This theory
couples to an exotic gauge theory called tensor gauge theory, which is a gauge
theory consists of an ordinary time component and 2-rank spatial-symmetric-
tensor space components [37, 38]. Investigating the fracton topological phases
in view of such new type of symmetries have been recently started [39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62].

Another strategy to construct effective field theories of the fracton
topological phases is to introduce the so-called foliated field theories [63,
64, 65, 66, 67]. Such theories are introduced in such a way that 2+1d BF
theories are stacked in layers, stemming from the fact that some of the
fracton lattice models can be decomposed into layers of the toric codes [10],
which are known to be the simplest conventional topologically ordered phases,
via unitary transformation [68, 69]. Due to this feature, we call these field
theories foliated BF theories in this paper. Important aspect of the foliated
BF theories is that the gauge transformation is modified due to the couplings
between the layers so that a constraint is imposed on the form of the gauge
invariant operators, such as the Wilson loops. This mirrors the fact that a
mobility constraint is imposed on the fractional excitations in the fracton
topological phases, which contributes to the sub-extensive GSD. To the best of
our knowledge, there are handful number of foliated BF theories that are known
to describe the fracton topological phases, such as the $X$-cube model.
Complete understanding of the foliated BF theories has yet to be elusive.

In this work, we explore new sets of 2+1d foliated BF theories based on
symmetry argument of the system. Taking one of the new type of symmetries,
multipole symmetries into account, and introducing gauge fields associated
with them, one can systematically construct new foliated topological field
theories. We further show that such theories exhibit the unusual GSD
dependence on the system size; the GSD depends on the greatest common divisor
between a quantum number $N$ characterizing the fractional charges and the
system size, which is in contrast with the previous foliated BF theories which
show the sub-extensive GSD [63, 64]. We also show that our foliated BF
theories have the UV lattice model counterparts which were studied in the
literature [70, 71, 72], sometimes referred to as the higher rank topological
phases. The way we construct the foliated topological theories is quite
general in the sense that we start with a system with multipole symmetries,
allowing us to systematically construct various kinds of foliated topological
phases. Consideration given in this work provides a unified insight on the
fracton topological phases and the higher rank topological phases [70, 71, 72,
73, 74] in view of foliated field theories.

The rest of this work is organized as follows. In Sec. 2, to see clearly how
our strategy of building up the foliated BF theories works, we discuss a
simple example of construction of a BF theory of a conventional topological
phase, starting with a theory with a global $U(1)$ symmetry. In Sec. 3, we
study a foliated BF theory with global and dipole symmetry. We also discuss
the GSD of the theory on a torus geometry and the UV lattice model
counterpart. In Sec. 4, we further explore a foliated BF theory with global,
dipole, and quadrupole symmetry. Finally, in Sec. 5, we conclude our work with
a few future directions. Technical issues are relegated into the appendices.
Throughout this paper, we study theories in $2+1$ dimensions. Also, we employ
the differential form for notational simplicity.

##  2 Warm-up: construction of the $\mathbb{Z}_{N}$ toric code

Before going into detailed discussion of the foliated BF theories, we
demonstrate a way to construct the BF theory of the $\mathbb{Z}_{N}$ toric
code [10] starting with a theory with global $U(1)$ symmetry and the
corresponding $U(1)$ gauge field for clearer illustrations. We also review the
UV stabilizer lattice model corresponding to the BF theory. While this example
may seem trivial (indeed, both of the BF theory and UV lattice model are well-
known), it lays the groundwork for more complicated cases of foliated BF
theories with multipole symmetries that we will discuss in the subsequent
sections.

We consider a theory with global $U(1)$ $0$-form symmetry and its charge $Q$.
The conserved charge is described by

$Q(V)=\int_{V}*j,$ (2.1)

where $j$ and $V$ represents the conserved 1-form current and two dimensional
spatial volume, and $*$ does the Hodge dual. This charge is global in the
sense that it commutes with the translation operation, $P_{I}\;(I=x,y)$ i.e.,

$\left[P_{I},Q\right]=0.$ (2.2)

We introduce a $1$-form $U(1)$ gauge field $a$ which couples with the current
$j$ with the coupling term being described by

$\displaystyle S_{c}=\int_{V}a\wedge*j.$ (2.3)

With the following gauge transformation ($\chi$: gauge parameter)

$a\ \to\ a+d\chi$

and the condition that the coupling term (2.3) is gauge invariant, we have the
conservation law of the current $d*j=0$.

Defining a gauge invariant flux as $f\vcentcolon=da$, we introduce the
following Lagrangian:

$\mathcal{{L}}_{TC}=\frac{N}{2\pi}b\wedge f=\frac{N}{2\pi}b\wedge da,$ (2.4)

where $b$ represents a $1$-form $U(1)$ gauge field.111 Physically $b$ is
interpreted as the gauge field for the $U(1)$ magnetic 0-form symmetry whose
conservation law is the Bianchi identity $df=0$.  The theory (2.4) is nothing
but the BF description of the $\mathbb{Z}_{N}$ toric code [10, 75]. Equation
of motion of the theory (2.4) implies that the following gauge invariant field
strength vanish

$\displaystyle B^{a}=\partial_{x}a_{y}-\partial_{y}a_{x},\quad
E^{a}_{x}=\partial_{\tau}a_{x}-\partial_{x}a_{0},\quad
E^{a}_{y}=\partial_{\tau}a_{y}-\partial_{y}a_{0},$ $\displaystyle
B^{b}=\partial_{x}b_{y}-\partial_{y}b_{x},\quad
E^{b}_{x}=\partial_{\tau}b_{x}-\partial_{x}b_{0},\quad
E^{b}_{y}=\partial_{\tau}b_{y}-\partial_{y}b_{0}.$ (2.5)

It is known that the BF theory (2.4) has non-trivial ground state degeneracy
(GSD) when we put the theory on Riemann surface with non-zero genus [10, 12,
75]. Indeed, on the torus geometry, the number of distinct non-contractible
loops of the gauge fields, $a_{x}$, and $a_{y}$ amounts to be the GSD. To see
this, we consider the following Wilson loop

$W_{0x}(y)=\exp\biggl{[}i\oint dxa_{x}(x,y)\biggr{]},$

which can be intuitively understood as the trajectory of the $\mathbb{Z}_{N}$
fractional charge going around the torus in the $x$-direction. Since the
magnetic flux vanishes, this loop does not depend on $y$. Likewise, we also
think of the following Wilson loop

$W_{0y}(x)=\exp\biggl{[}i\oint dya_{y}(x,y)\biggr{]},$

which is associated with the trajectory of the $\mathbb{Z}_{N}$ fractional
charge in the $y$-direction. Using the flux-less condition, one can verify
that it does not depend on $x$. From $W^{N}_{0x}(y)=W^{N}_{0y}(x)=1$, it
follows that there are $N^{2}$ distinct Wilson loops of the gauge field $a$,
implying that the GSD is given by $N^{2}$. The way we evaluate the GSD on the
torus by counting distinct number of the Wilson loops will be frequently used
in the case of the foliated BF theories discussed in the subsequent sections.

One can construct the UV Hamiltonian corresponding to the BF theory (2.4).
Focusing on the field strength of the gauge field $a$, that is, $B^{a}$,
$E^{a}_{x}$ and $E^{a}_{y}$ in (2.5), the Hamiltonian of the lattice can be
constructed in a way that the ground state does not have the electric and
magnetic charges, which are probed by the Gauss law
$G=\partial_{x}E^{a}_{x}+\partial_{y}E^{a}_{y}$ and the magnetic flux $B^{a}$
respectively. To implement this construction, we think of a 2d discrete
lattice and on each link we define a local Hilbert space spanned by

$\ket{\eta}_{(\hat{x}+1/2,\hat{y})}\quad{\rm
and}\quad\ket{\eta}_{(\hat{x},\hat{y}+1/2)},$ (2.6)

where the subscripts $(\hat{x}+1/2,\hat{y})$ and $(\hat{x},\hat{y}+1/2)$
denote the coordinate of the horizontal and vertical links, respectively while
$\eta$ labels $\mathbb{Z}_{N}$ i.e., $\eta=0,1\cdots,N-1$ $({\rm mod}\ N)$. We
also introduce the $\mathbb{Z}_{N}$ Pauli operators
$X_{(\hat{x}+1/2,\hat{y})}$ and $Z_{(\hat{x}+1/2,\hat{y})}$ acting on the
state on the horizontal link $\ket{\eta}_{(\hat{x}+1/2,\hat{y})}$ as222 In the
terminology of lattice gauge theory, $Z$ corresponds to the $\mathbb{Z}_{N}$
link variable while $X$ corresponds to the exponentiated electric field. The
state $|\eta\rangle$ corresponds to the “electric field basis” at each link.

$\displaystyle X_{(\hat{x}+1/2,\hat{y})}\ket{\eta}_{(\hat{x}+1/2,\hat{y})}$
$\displaystyle=$ $\displaystyle\ket{\eta+1}_{(\hat{x}+1/2,\hat{y})},$
$\displaystyle Z_{(\hat{x}+1/2,\hat{y})}\ket{\eta}_{(\hat{x}+1/2,\hat{y})}$
$\displaystyle=$
$\displaystyle\omega^{\eta}\ket{\eta}_{(\hat{x}+1/2,\hat{y})},$ (2.7)

with $\omega\vcentcolon=e^{2\pi i/N}$. We also define the $\mathbb{Z}_{N}$
Pauli operators $X_{(\hat{x},\hat{y}+1/2)}$ and $Z_{(\hat{x},\hat{y}+1/2)}$
acting on the state on the vertical link in a similar way. Using these Pauli
operators, we define the following two types of the operators (Fig. 1):

$\displaystyle V_{(\hat{x},\hat{y})}$ $\displaystyle=$ $\displaystyle
X^{\dagger}_{(\hat{x}-1/2,\hat{y})}X_{(\hat{x}+1/2,\hat{y})}X^{\dagger}_{(\hat{x},\hat{y}-1/2)}X_{(\hat{x},\hat{y}+1/2)},$
$\displaystyle P_{(\hat{x},\hat{y})}$ $\displaystyle=$ $\displaystyle
Z^{\dagger}_{(\hat{x},\hat{y}+1/2)}Z_{(\hat{x}+1,\hat{y}+1/2)}Z^{\dagger}_{(\hat{x}+1/2,\hat{y})}Z_{(\hat{x}+1/2,\hat{y}+1)},$
(2.8)

which correspond to the Gauss law and the magnetic flux, respectively. Then we
introduce the Hamiltonian as [10]

$H_{TC}=-\sum_{\hat{x},\hat{y}}\left[V_{(\hat{x},\hat{y})}+P_{(\hat{x},\hat{y})}\right]+({\rm
h.c.}).$ (2.9)

The ground state $\ket{\Omega}$ of this Hamiltonian is the projected state so
that it does not have electric and magnetic excitations:

$V_{(\hat{x},\hat{y})}\ket{\Omega}=P_{(\hat{x},\hat{y})}\ket{\Omega}=\ket{\Omega}\quad{\rm
for}~{}~{}^{\forall}\;\hat{x},\hat{y},$ (2.10)

and the number of the ground states on the torus agrees with the one of the BF
theory (2.4).

Figure 1: Two types of the terms defined in (2.8). The black dot represents
the $\mathbb{Z}_{N}$ Pauli operator.

In what follows, we will discuss a way to construct foliated BF theories,
starting with a theory with multipole symmetries as well as global one. The
spirit of obtaining such phases is the essentially the same as what we have
discussed here, namely, starting with conserved charges associated with
multipole symmetries, and introducing gauge fields related to them, we define
gauge invariant fluxes to introduce the BF theory. We also discuss the UV
lattice Hamiltonian in the same way as we describe the Hamiltonian (2.9) from
the BF theory. The crucial difference between the present argument and what we
will see in the following is that due to the multipole symmetries, we obtain
richer foliated BF theories.

##  3 Dipole symmetries

In this section, we turn to the case with global and dipole symmetries. To
this end, discussion presented in [76, 77] is of usefulness. We also discuss
the relation between the foliated BF theory and the UV lattice model.

###  3.1 Dipole and global symmetries

Suppose that we have a theory with conserved charges associated with global
$U(1)$ and dipole symmetries, described by $Q$, $Q_{x}$ and $Q_{y}$. These
charges are subject to the following relations333 Intuitively, the relations
can be understood by writing a charge and dipole moment as $\rho$, $x\rho$,
$y\rho$, where $\rho$ denotes the density of the $U(1)$ charge, and thinking
of shifting them by a constant in $x$\- or $y$-direction. For instance, if we
shit the dipole moment $x\rho$ by a constant in the $x$-direction, the change
of dipole moment under the shift gives $(x+\Delta x)\rho-x\rho=\Delta x\rho$,
where $\Delta x$ is constant, implying the nontrivial commutation relation
between the transnational operator and the dipole.:

$\left[P_{I},Q\right]=0,\quad\left[P_{I},Q_{J}\right]=\delta_{I,J}Q\qquad(I=x,y).$
(3.1)

where the first relation is the same as (2.2). We write the charges $Q$,
$Q_{x}$ and $Q_{y}$ via the integral expression using the conserved currents
as

$Q=\int_{V}*j,\quad Q_{I}=\int_{V}*K_{I}.$

To reproduce the relation (3.1), we demand that

$*K_{I}=*k_{I}-x_{I}*j$ (3.2)

with $k_{I}$ being a local (not necessarily conserved) current and
$(x_{1},x_{2})=(x,y)$. A straightforward calculation verifies the relation
(3.1). Corresponding to (2.3), we introduce the $U(1)$ $1$-form gauge fields
$a$ and $A^{I}$ with the coupling term defined by

$S_{dp}=\int_{V}\left(a\wedge*j+A^{I}\wedge*k_{I}\right).$ (3.3)

We need to have a proper gauge transformation in such a way that the condition
of the coupling term being gauge invariant yields the conservation law of the
currents. The following gauge transformation does the job:444 It would be
interesting to compare this with the two-group symmetry
$U(1)_{A}^{(0)}\times_{\kappa}U(1)_{B}^{(1)}$ which transforms the 1-form
gauge field $A^{(1)}$ and 2-form gauge field $B^{(2)}$ as (see e.g. [78])
$A^{(1)}\to A^{(1)}+d\lambda_{A}^{(0)},\quad B^{(2)}\to
B^{(2)}+d\Lambda_{B}^{(1)}+\frac{\kappa}{2\pi}\lambda_{A}^{(0)}dA^{(1)},$
while $\lambda_{A}^{(0)}$ and $\Lambda_{B}^{(1)}$ are the 0-form and 1-form
gauge parameters.

$a\to a+d\Lambda+\sigma_{I}dx_{I},\quad A^{I}\to A^{I}+d\sigma_{I},$ (3.4)

where $\Lambda$, $\sigma_{I}$ are the gauge parameters. Indeed, one can verify
that the gauge invariance of the coupling term $S_{dp}$ under the gauge
transformation (3.4) yields

$d*j=0,\quad d\left(*k_{I}-x_{I}*j\right)=d*K_{I}=0.$

Now we are in a good place to introduce the foliated BF theory. Defining gauge
invariant fluxes as

$f\vcentcolon=da-A^{I}\wedge dx_{I},\quad F^{I}\vcentcolon=dA^{I},$ (3.5)

we put these fluxes in the BF theory format, namely, we introduce the
following BF theory:

$\mathcal{L}_{dp}=\frac{N}{2\pi}b\wedge
f+\sum_{I=x,y}\frac{N}{2\pi}c^{I}\wedge F^{I},$ (3.6)

where $b$ and $c^{I}(I=x,y)$ represent the $U(1)$ $1$-form gauge fields. To
proceed, we also introduce the foliation fields $e^{I}$ ($I=x,y$) [63, 64].
Generally, foliation is defined to be co-dimension one sub-manifold which is
orthogonal to the $1$-form foliation field $e^{I}$. Setting the foliation
field by $e^{x}\vcentcolon=dx,e^{y}\vcentcolon=dy$, and rewriting (3.6), we
arrive at the following foliated BF theory:

$\boxed{\mathcal{L}_{dp}=\frac{N}{2\pi}a\wedge
db+\sum_{I=x,y}\frac{N}{2\pi}A^{I}\wedge dc^{I}+\frac{N}{2\pi}A^{I}\wedge
b\wedge e^{I}.}$ (3.7)

Compared with the foliated BF theories that were previously studied [63, 64],
where sub-extensive number of $2+1d$ BF theories are placed in layers, in our
theory (3.7), there are only three layers of the $2+1d$ BF theories,
corresponding to the first two terms, with coupling between the layers being
described by the last term. In the next subsection, we show that our BF theory
on a torus geometry exhibits unusual behavior of the GSD dependence on the
system size by studying the form of the non-contractible Wilson loops.

###  3.2 Ground state degeneracy

In this subsection, we study the GSD of our foliated BF theory on a torus. In
addition to the gauge symmetry (3.4), the foliated BF theory (3.7) admits the
following gauge symmetry with respect to $b$ and $c^{I}$:

$b\to b+d\lambda,\quad c^{I}\to c^{I}+d\gamma^{I}+\lambda e^{I}.$ (3.8)

Similarly to the foliated BF theory of the X-cube model [63], the unusual
gauge symmetries (3.4) and (3.8) put constraints on the form of the gauge
invariant operators, which contributes to the unconventional GSD dependence on
the system size. To see this point more explicitly, we integrate out some of
the fields in (3.7) to transform the Lagrangian to a simpler form with fewer
number of the gauge fields.

Integrating out $b_{0}$ gives the following condition ($i,j=x,y$):

$\partial_{i}a_{j}\epsilon^{ij}-A^{I}_{i}\delta^{I}_{j}\epsilon^{ij}=0\
\leftrightarrow\ \partial_{x}a_{y}-A_{x}^{y}=\partial_{y}a_{x}-A_{y}^{x}.$
(3.9)

We also integrate out the $A^{I}_{0}$, and obtain the following condition

$\partial_{i}c^{I}_{j}\epsilon^{ij}+b_{i}\delta^{I}_{j}\epsilon^{ij}=0\
\leftrightarrow\ \begin{pmatrix}b_{x}\\\
b_{y}\end{pmatrix}=\begin{pmatrix}-\partial_{i}c^{y}_{j}\epsilon^{ij}\\\
\partial_{i}c^{x}_{j}\epsilon^{ij}\end{pmatrix}.$ (3.10)

One can eliminate the gauge fields $b$ and $A^{I}$ by substituting the
relations (3.9)(3.10) into (3.7). In doing so, we introduce gauge fields by

$A_{(ij)}\vcentcolon=\partial_{i}a_{j}-A_{i}^{j}\quad(i,j=x,y)$ (3.11)

whose gauge transformation coming from (3.4) reads

$\displaystyle A_{(ij)}\ \to\ A_{(ij)}+\partial_{i}\partial_{j}\Lambda$ (3.12)

with property that $A_{(xy)}=A_{(yx)}$ due to (3.9). After the substitution,
the Lagrangian (3.7) becomes

$\displaystyle\mathcal{L}_{dp}$ $\displaystyle=$
$\displaystyle\frac{N}{2\pi}\biggl{[}-c_{0}^{x}\left(\partial_{x}A_{(yx)}-\partial_{y}A_{(xx)}\right)-c_{0}^{y}\left(\partial_{x}A_{(yy)}-\partial_{y}A_{(xy)}\right)$
(3.13)
$\displaystyle+\tilde{c}\left(\partial_{\tau}A_{(xy)}-\partial_{x}\partial_{y}A_{0}\right)-c_{y}^{x}\left(\partial_{\tau}A_{(xx)}-\partial_{x}^{2}A_{0}\right)+c_{x}^{y}\left(\partial_{\tau}A_{(yy)}-\partial_{y}^{2}A_{0}\right)\biggr{]},$

where we have defined $A_{0}\vcentcolon=a_{0}$ and
$\tilde{c}\vcentcolon=c^{x}_{x}-c_{y}^{y}$ with the gauge transformation
$\tilde{c}\to\tilde{c}+\partial_{x}\gamma^{x}-\partial_{y}\gamma^{y}$.

Now the BF theory (3.13) consists of the gauge field
$(c^{x}_{0},c^{y}_{0},\tilde{c},c^{x}_{y},c^{y}_{x})$ and the field strength
for $(A_{0},A_{(ij)})$ and vice versa. The equations of motion of the BF
theory implies that the following gauge invariant field strength vanish:

$\displaystyle B_{x}=\partial_{x}A_{(yx)}-\partial_{y}A_{(xx)},\quad
B_{y}=\partial_{x}A_{(yy)}-\partial_{y}A_{(xy)},$ $\displaystyle
E_{(xy)}=\partial_{\tau}A_{(xy)}-\partial_{x}\partial_{y}A_{0},\quad
E_{(xx)}=\partial_{\tau}A_{(xx)}-\partial_{x}^{2}A_{0},\quad
E_{(yy)}=\partial_{\tau}A_{(yy)}-\partial_{y}^{2}A_{0},$
$\displaystyle\tilde{B}=\partial_{x}^{2}c^{x}_{y}-\partial_{y}^{2}c^{y}_{x}-\partial_{x}\partial_{y}\tilde{c},\quad
E^{x}_{y}=\partial_{\tau}c_{y}^{x}-\partial_{y}c^{x}_{0},\quad
E^{y}_{x}=\partial_{\tau}c_{x}^{y}-\partial_{x}c^{y}_{0},$
$\displaystyle\tilde{E}=\partial_{\tau}\tilde{c}-\partial_{x}c_{0}^{x}+\partial_{y}c^{y}_{0}.$
(3.14)

It is worth mentioning that the field strength $B_{x}$, $B_{y}$,
$E_{(xx)},E_{(yy)},E_{(xy)}$ has the same form as the one which was found in
the symmetric tensor gauge theory, unconventional Maxwell theory with the
2-rank spatial-symmetric-tensor space components, admitting dipole charges
[37, 38]. As we see below, our foliated BF theory is the effective field
theory description of the $\mathbb{Z}_{N}$ Higgs phase of the symmetric tensor
gauge theory.

The equation of motions ensures that there is no non-trivial local gauge
invariant operators. However, analogously to the other BF theories, the theory
has non-local gauge-invariant operators, which can be constructed from the
gauge fields either $A_{(ij)}$ or $(\tilde{c},c^{x}_{y},c^{y}_{x})$.
Especially, when placing on a torus, the theory admits the non-contractible
Wilson loops of the gauge fields which contribute to the non-trivial GSD. In
the following, we evaluate the GSD of our theory on the torus with system size
$L_{x/y}$ in the $x/y$ direction, by counting the number of distinct non-
contractible Wilson loops of the gauge fields $A_{(ij)}$.555One can evaluate
the non-contractible Wilson loops of the gauge fields, $A_{(ij)}$ and its
canonical conjugate, $(c^{y}_{x},c^{x}_{y},\tilde{c})$ to count the number of
the Heisenberg algebras to identify the GSD. Since this task is complicated as
the Wilson loops of the both of the gauge fields, $A_{(ij)}$ and
$(c^{y}_{x},c^{x}_{y},\tilde{c})$ highly depend on the spatial coordinate.
Here, we employ an alternative simpler approach to obtain the GSD by focusing
on the Wilson loops of the gauge field, $A_{(ij)}$ and count the number of
distinct configurations of the loops. To this end, we think of placing the BF
theory (3.13) on a discrete 2d lattice so that the spatial coordinate $(x,y)$
takes the integer values $(x,y)\to(\hat{x},\hat{y})$666 Here we put the hat to
emphasize that the coordinate is labeled by an integer.  in the unit of the
lattice spacing with the periodic boundary condition
$(\hat{x},\hat{y})\sim(\hat{x}+L_{x},\hat{y})\sim(\hat{x},\hat{y}+L_{y})$.
This regularization leads to a quantum theory with unusual behavior of the
GSD; instead of the sub-extensive GSD which was observed in the previous
foliated BF theories, the theory gives rise to the GSD depending on the
greatest common divisor between $N$ and the system size. By this
regularization, the integral and derivative operators are converted into the
regularized ones, e.g. $\int dx\to\sum_{\hat{x}}$, $\partial_{x}f(x,y)\to
f(\hat{x}+1,\hat{y})-f(\hat{x},\hat{y})$ (suppressing a unit of the lattice
spacing). However, without any confusion, we retain the original expressions
of the integral and derivatives.

Let us first focus on the form of the non-contractible Wilson loops of
$A_{(xx)}$ in the $x$-direction. Due to the gauge symmetry (3.11), we consider
the following two types of the closed loops:

$W_{x}(\hat{y})=\exp\biggl{[}i\oint
d\hat{x}A_{(xx)}(\hat{x},\hat{y})\biggr{]},\quad
W_{dp:x}(\hat{y})\overset{?}{=}\exp\biggl{[}i\oint
d\hat{x}\bigl{(}\hat{x}A_{(xx)}(\hat{x},\hat{y})\bigr{)}\biggr{]}.$ (3.15)

While the first loops in (3.15) describes the non-contractible loops of the
$\mathbb{Z}_{N}$ fractional charge with $W_{x}^{N}(\hat{y})=1$, which can be
also found in the other topological phases such as the toric code, a care has
to be taken in the second type of the loop. Since the integrand is
proportional to $\hat{x}$, depending on $N$ and $L_{x}$, there might be a
mismatch between initial and final value of the argument of the exponential
which is not compatible with the periodic boundary condition. To remedy this
issue, the loop has to wind around the torus multiple times to be consistent
with the periodic boundary condition. That is, the correct form of the second
loop is

$W_{dp:x}(\hat{y})=\exp\biggl{[}i\alpha_{x}\oint
d\hat{x}\bigl{(}\hat{x}A_{(xx)}(\hat{x},\hat{y})\bigr{)}\biggr{]}$ (3.16)

with an integer $\alpha_{x}$. The integer $\alpha_{x}$ is set to be the
minimal integer number so that the loop satisfies
$W_{dp:x}^{N}(\hat{y})=W_{dp:x}^{L_{x}}(\hat{y})=1$. Hence, one finds
$\alpha_{x}=\frac{N}{\gcd(N,L_{x})}$, where $\gcd$ stands for the greatest
common divisor. These two loops, $W_{x}(\hat{y})$ and $W_{dp:x}(\hat{y})$, are
characterized by quantum number $\mathbb{Z}_{N}$ and
$\mathbb{Z}_{\gcd(N,L_{x})}$, respectively. The loop (3.16) can be interpreted
as the “dipole of the Wilson loop”, i.e., the loops is formed by trajectory of
the fractional charge with its intensity increasing linearly as going around
the torus in the $x$-direction, which is reminiscent of the integration of a
dipole moment, $x\rho$. Justification of the existence of the two types of
loops is given in a close investigation of a UV lattice model, see Appendix B.
Likewise, there are two types of the non-contractible loops of $A_{(yy)}$ in
the $y$-direction, which has the form

$W_{y}(\hat{x})=\exp\biggl{[}i\oint
d\hat{y}A_{(yy)}(\hat{x},\hat{y})\biggr{]},\quad
W_{dp:y}(\hat{y})=\exp\biggl{[}i\alpha_{y}\oint
d\hat{y}\bigl{(}\hat{y}A_{(yy)}(\hat{x},\hat{y})\bigr{)}\biggr{]}$ (3.17)

with $\alpha_{y}=\frac{N}{\gcd(N,L_{y})}$, implying that these are labeled by
$\mathbb{Z}_{N}\times\mathbb{Z}_{\gcd(N,L_{y})}$.

We turn to introducing non-contractible loops of the gauge field $A_{(xy)}$ in
either $x$\- or $y$-direction. To this end, we consider the following gauge
invariant loops

$W_{xy:x}(\hat{y})=\exp\biggl{[}i\oint
d\hat{x}A_{(xy)}(\hat{x},\hat{y})\biggr{]},\quad
W_{xy:y}(\hat{x})=\exp\biggl{[}i\oint
d\hat{y}A_{(xy)}(\hat{x},\hat{y})\biggr{]}.$ (3.18)

Naively, there are $N^{2}$ distinct loops, however, a sets of constraints
reduce this number, as we discuss below.

After having defined gauge invariant non-contractible loops
(3.15)(3.17)(3.18), we need to check the coordinate dependence of the loops to
count the distinct number of the loops. As we will see, there is a several
constraints imposed on the loops of the gauge field $A_{(xy)}$. We first check
the $\hat{y}$ dependence of the loops, $W_{x}(\hat{y})$, $W_{dp:x}(\hat{y})$
(3.15). From the equation of motion of the BF theory $B_{y}=0$, where $B_{y}$
is given in (3.14), and performing integral along the $y$-direction, we have

$\oint d\hat{y}\left(\partial_{x}A_{(yx)}-\partial_{y}A_{(xx)}\right)=0\
\leftrightarrow\ \partial_{y}\oint d\hat{x}A_{(xx)}=0,$ (3.19)

implying that the loop $W_{x}(\hat{y})$ (3.15) does not depend on $y$.
Furthermore, multiplying $\hat{x}$ on the equation of motion $B_{x}=0$ and
perform integral in the $x$-direction, one finds

$\partial_{y}\oint d\hat{x}\hat{x}A_{(xx)}=-\oint d\hat{x}A_{(xy)},$ (3.20)

from which we have

$W_{dp:x}(\hat{y})=W_{dp:x}(\hat{y}-1)\biggl{(}W_{xy:x}^{\alpha_{x}}(\hat{y})\biggr{)}^{\dagger}.$
(3.21)

The iterative use of (3.21) gives

$W_{dp:x}(\hat{y})=W_{dp:x}(\hat{y}_{0})\biggl{(}W_{xy:x}^{\alpha_{x}(\hat{y}-\hat{y}_{0})}(\hat{y}_{0})\biggr{)}^{\dagger},$
(3.22)

implying that in order to deform the loop $W_{dp:x}(\hat{y})$ from $\hat{y}$
to $\hat{y}_{0}$, we need to multiply additional loops. Due to the periodic
boundary condition, $W_{dp:x}(\hat{y})=W_{dp:x}(\hat{y}+L_{y})$, we must have

$W_{xy:x}^{\alpha_{x}L_{y}}(\hat{y})=1.$ (3.23)

Therefore, there are $N\times\gcd(N,L_{x})$ distinct loops (3.15) with the
condition of (3.23).

We turn to checking the $\hat{x}$ dependence of the loops, $W_{y}(\hat{x})$,
$W_{dp:y}(\hat{x})$ (3.17). The similar line of thoughts leads to that
$W_{y}(\hat{x})$ does not depend on $\hat{x}$, and that

$W_{dp:y}(\hat{y})=W_{dp:y}(\hat{x}_{0})\biggl{(}W_{xy:y}^{\alpha_{y}(\hat{x}-\hat{x}_{0})}(\hat{x}_{0})\biggr{)}^{\dagger},$
(3.24)

from which one finds

$W_{xy:y}^{\alpha_{y}L_{x}}(\hat{x})=1.$ (3.25)

Hence, there are $N\times\gcd(N,L_{y})$ distinct loops (3.17) with the
condition of (3.25).

We move onto the investigating the coordinate dependence of the loops
$W_{xy:x}(\hat{y})$ and $W_{xy:y}(\hat{x})$ in (3.18). From the equation of
motion $B_{x}=0$, and performing integral along the $x$-direction, we have

$\oint d\hat{x}(\partial_{x}A_{(yy)}-\partial_{y}A_{(xy)})=0\ \leftrightarrow\
\partial_{y}\oint d\hat{x}A_{(xy)}=0,$ (3.26)

from which it can be shown that the loop $W_{xy:x}(\hat{y})$ can be deformed
so that it can go up or down in the $y$-direction, namely,
$W_{xy:x}(\hat{y})=W_{xy:x}(\hat{y}-1)=\cdots=W_{xy:x}(\hat{y}_{0})$ where
$\hat{y}_{0}$ denotes an arbitrary $y$-coordinate (i.e., the loop
$W_{xy:x}(\hat{y})$ does not depend on $\hat{y}$). Likewise, one can show that
the loop $W_{xy:y}(\hat{x})$ can go to left or right, i.e,
$W_{xy:y}(\hat{x})=W_{xy:y}(\hat{x}-1)=\cdots=W_{xy:y}(\hat{x}_{0})$ with
$\hat{x}_{0}$ being an arbitrary $x$-coordinate. Using this property, it
follows that

$W_{xy:x}^{L_{y}}(\hat{y})=W_{xy:y}^{L_{x}}(\hat{x}).$ (3.27)

With the conditions (3.23), (3.25) and (3.27), one can count the distinct
number of the Wilson loops involving $A_{(xy)}$ (3.18). Let us consider a set
of loops (3.18) with the following form

$W_{xy:x}^{k_{x}}(\hat{y})W_{xy:y}^{k_{y}}(\hat{x})$ (3.28)

with $(k_{x},k_{y})$ being integers. We need to count the distinct combination
of $(k_{x},k_{y})$. Relegating the details to the Appendix A, we find that
there are $N\times\gcd(N,L_{x},L_{y})$ distinct loops of $A_{(xy)}$.

To recap the argument, we have counted the distinct number of the non-
contractible loops of the gauge fields $A_{(ij)}$. For the loops involving
$A_{(xx)}$ and $A_{(yy)}$, there are
$(N\times\gcd(N,L_{x}))\times(N\times\gcd(N,L_{y}))$ distinct loops whereas
there are $N\times\gcd(N,L_{x},L_{y})$ distinct loops of $A_{(xy)}$. The total
number of the distinct loops amounts to be the GSD. Hence, we arrive at

$\displaystyle\boxed{GSD=N^{3}\times\gcd(N,L_{x})\times\gcd(N,L_{y})\times\gcd(N,L_{x},L_{y}).}$
(3.29)

Compared with previous foliated BF theories [63, 64, 65], where coupling
between the layers of the toric codes yields the sub-extensive GSD, in our
foliated BF theory, the coupling between the three layers of the toric codes
gives rise to unusual GSD dependence on the system size.

###  3.3 UV lattice model

(a)

(b)

Figure 2: Two types of the terms that constitute Hamiltonian (3.33). The red
(blue) square represents the Pauli operator acting on the local Hilbert space
$\ket{a}_{(\hat{x},\hat{y})}$ ($\ket{b}_{(\hat{x},\hat{y})}$), whereas small
square with black line denotes the Pauli operator that act on
$\ket{c}_{(\hat{x}+1/2,\hat{y}+1/2)}$. A purple square indicates the composite
of the Pauli operators which act on both of $\ket{a}_{(\hat{x},\hat{y})}$ and
$\ket{b}_{(\hat{x},\hat{y})}$.

Analogously to the fact that the BF theory description of the $\mathbb{Z}_{N}$
toric code can be mapped to the stabilizer model in the UV lattice, one can
implement the similar mapping from the BF theory to the lattice model.
Focusing on gauge invariant field strength, $B_{x}$, $B_{y}$, $E_{(ij)}$
(3.14), we introduce the UV lattice Hamiltonian in such a way that the ground
state does not admit the $\mathbb{Z}_{N}$ electric and magnetic charge
excitations.

To this end, we envisage a 2d square lattice, and introduce two types of local
Hilbert space of the $\mathbb{Z}_{N}$ spin, denoted by
$\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}$ with
$a,b=0,1,\cdots,N-1\;\mod N$ as well as the $\mathbb{Z}_{N}$ Pauli operators
$X_{i,(\hat{x},\hat{y})},Z_{i,(\hat{x},\hat{y})}(i=1,2)$ that act on the
Hilbert space with

$\displaystyle
Z_{1,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\omega^{a}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})},\quad
Z_{2,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\omega^{b}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}$
$\displaystyle
X_{1,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\ket{a+1}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})},\quad
X_{2,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\ket{a}_{(\hat{x},\hat{y})}\ket{b+1}_{(\hat{x},\hat{y})}.$
(3.30)

At each node of the dual lattice, labeled by the coordinate
($\hat{x}+1/2,\hat{y}+1/2$), we also introduce a local Hilbert space
represented by $\ket{c}_{(\hat{x}+1/2,\hat{y}+1/2)}$ with
$c=0,\cdots,N-1\>\mod N$ and the $\mathbb{Z}_{N}$ Pauli operators,
$X_{0,(\hat{x}+1/2,\hat{y}+1/2)}$, $Z_{0,(\hat{x}+1/2,\hat{y}+1/2)}$ which act
on the state as

$\displaystyle
Z_{0,(\hat{x}+1/2,\hat{y}+1/2)}\ket{c}_{(\hat{x}+1/2,\hat{y}+1/2)}$
$\displaystyle=$ $\displaystyle\omega^{c}\ket{c}_{(\hat{x}+1/2,\hat{y}+1/2)},$
$\displaystyle
X_{0,(\hat{x}+1/2,\hat{y}+1/2)}\ket{c}_{(\hat{x}+1/2,\hat{y}+1/2)}$
$\displaystyle=$ $\displaystyle\ket{c+1}_{(\hat{x}+1/2,\hat{y}+1/2)}.$ (3.31)

With these preparations, we define the following mutually commuting operators:

$\displaystyle V_{(\hat{x},\hat{y})}$ $\displaystyle\vcentcolon=$
$\displaystyle
X_{1,(\hat{x}+1,\hat{y})}X_{1,(\hat{x}-1,\hat{y})}(X_{1,(\hat{x},\hat{y})}^{\dagger})^{2}X_{2,(\hat{x},\hat{y}+1)}X_{2,(\hat{x},\hat{y}-1)}(X_{2,(\hat{x},\hat{y})}^{\dagger})^{2}$
$\displaystyle
X_{0,(\hat{x}+1/2,\hat{y}+1/2)}X^{\dagger}_{0,(\hat{x}-1/2,\hat{y}+1/2)}X_{0,(\hat{x}-1/2,\hat{y}-1/2)}X^{\dagger}_{0,(\hat{x}-1/2,\hat{y}-1/2)},$
$\displaystyle P_{(\hat{x},\hat{y}+1/2)}$ $\displaystyle\vcentcolon=$
$\displaystyle
Z_{1,(\hat{x},\hat{y}+1)}^{\dagger}Z_{1,(\hat{x},\hat{y})}Z_{0,(\hat{x}+1/2,\hat{y}+1/2)}Z^{\dagger}_{0,(\hat{x}-1/2,\hat{y}+1/2)},$
$\displaystyle Q_{(\hat{x}+1/2,\hat{y})}$ $\displaystyle\vcentcolon=$
$\displaystyle
Z_{2,(\hat{x}+1,\hat{y})}Z^{\dagger}_{2,(\hat{x},\hat{y})}Z_{0,(\hat{x}+1/2,\hat{y}+1/2)}Z^{\dagger}_{0,(\hat{x}+1/2,\hat{y}-1/2)},$
(3.32)

which are portrayed in Fig. 2(a) and 2(b). The operator
$V_{(\hat{x},\hat{y})}$ corresponds to the Gauss law term involving the
eclectic fields, $E_{(ij)}$,
$G=\partial_{x}^{2}E_{(xx)}+\partial_{y}^{2}E_{(yy)}+\partial_{x}\partial_{y}E_{(xy)}$,
whereas the operator $P_{(\hat{x},\hat{y}+1/2)}$ and
$Q_{(\hat{x}+1/2,\hat{y})}$ is associated with the magnetic flux $B_{x}$ and
$B_{y}$, respectively [37]. The Hamiltonian is defined in such a way that the
ground state does not contain the electric and magnetic excitations, namely,

$H_{dp}\vcentcolon=-\sum_{\hat{x},\hat{y}}\left[V_{(\hat{x},\hat{y})}+P_{(\hat{x},\hat{y}+1/2)}+Q_{(\hat{x}+1/2,\hat{y})}\right]+({\rm
h.c.})~{}.$ (3.33)

It is interesting to note that in [70, 74], the same Hamiltonian was obtained
by Higgsing the tensor gauge theory [37], and that the same GSD as (3.29) was
derived by counting superselection sectors (i.e., the number of anyons).

##  4 Quadrupole

After having seen the example of the BF theories with dipoles symmetries, in
this section, we extend the previous argument to the case with quadrupole
symmetry.

###  4.1 Construction of the foliated BF theory

We envisage a theory with quadrupole $U(1)$ symmetry in addition to the global
charge and dipole ones whose charges are denoted as $Q$, $Q_{x}$, $Q_{y}$,
$Q_{xy}$. These charges are subject to the following commutation relation with
the transnational operators777The last commutation relation can be intuitively
understood by writing the dipoles and quadruple moment as $x\rho$, $y\rho$,
$xy\rho$ and thinking of translating the quadrupole $xy\rho$ by a constant.
For instance, if we translate $xy\rho$ in the $x$-direction, the change of the
quadruple under such a translation gives $(\Delta x)y\rho$ ($\Delta x$:
constant), which is noting but the $y$-dipole. :

$[P_{I},Q]=0,\quad[P_{I},Q_{J}]=\delta_{I,J}Q,\quad[P_{I},Q_{xy}]=Q_{\bar{I}},$
(4.1)

where

$\bar{I}=\begin{cases}x&{\rm for}\ I=y\\\ y&{\rm for}\ I=x\end{cases}.$

We also write the charges via integral expression of $1$-form currents as

$Q=\int_{V}*j,\quad Q_{I}=\int_{V}*K_{I},\quad Q_{xy}=\int_{V}*\ell.$ (4.2)

If we set

$*K_{I}=*k_{I}-x_{I}*j,\quad*\ell\vcentcolon=*J+xy*j-x_{I}*k_{I},$ (4.3)

where $k_{I}$ and $J$ represent non-conserved local current, then a simple
calculation shows that the form (4.2) jointly with (4.3) yields the
commutation relations (4.1). We introduce $1$-form $U(1)$ gauge fields as $a$,
$A^{I}$, and $a^{\prime}$ that are coupled with the currents with coupling
term

$\displaystyle S_{qp}=\int_{V}a\wedge*j+A^{I}\wedge*k_{I}+a^{\prime}\wedge*J.$
(4.4)

With the following gauge transformation, ($\Lambda$, $\sigma_{I}$,
$\Lambda^{\prime}$ :gauge parameters)

$a\to a+d\Lambda+\sigma_{I}dx_{I},\quad A^{I}\to
A^{I}+d\sigma_{I}+\Lambda^{\prime}dx_{\bar{I}},\quad a^{\prime}\to
a^{\prime}+d\Lambda^{\prime},$ (4.5)

jointly with the condition $S_{qp}$ is invariant under the gauge
transformation, one can verify the three kinds of currents are conserved,
i.e., $d*j=d*K_{I}=d*\ell=0$.

We further introduce gauge invariant fluxes by

$f\vcentcolon=da-A^{I}\wedge dx^{I},\quad
F^{I}\vcentcolon=dA^{I}-a^{\prime}\wedge dx^{\bar{I}},\quad
f^{\prime}\vcentcolon=da^{\prime}.$ (4.6)

Using these fluxes, we defined a BF theory as

$\mathcal{L}_{qp}=\frac{N}{2\pi}\biggl{[}b\wedge f+\sum_{I=x,y}c^{I}\wedge
F^{I}+D\wedge f^{\prime}\biggl{]}.$ (4.7)

Writing the foliated field as $e^{I}=dx^{I}$, (4.7) is transformed into the
following foliated BF theory:

$\boxed{\mathcal{L}_{qp}=\frac{N}{2\pi}a\wedge
db+\frac{N}{2\pi}a^{\prime}\wedge dD+\sum_{I=x,y}\frac{N}{2\pi}A^{I}\wedge
dc^{I}+\frac{N}{2\pi}A^{I}\wedge b\wedge e^{I}+\frac{N}{2\pi}a^{\prime}\wedge
c^{I}\wedge e^{\bar{I}}}.$ (4.8)

Compared with the previous foliated BF theory (3.7), we now have four layers
of the toric codes, corresponding to the first three terms in (4.8), with the
coupling terms between the layers being given by the last two terms. In the
next subsection, we show that the foliated BF theory exhibits unusual GSD
dependence on the system size on torus geometry by studying the non-
contractible Wilson loops.

###  4.2 Ground state degeneracy

In addition to (4.5), the theory respects the following gauge symmetry:

$D\to D+d\lambda+\xi^{I}e^{\bar{I}},\quad c^{I}\to c^{I}-d\xi^{I}.$ (4.9)

Analogously to the previous foliated BF theory (3.7), the theory admits
unusual gauge symmetries due to the presence of the couplings between the
layers. This puts constraints on the form of the gauge invariant operators,
contributing to the lattice dependence of the GSD. To see this point, we
simplify the Lagrangian (4.8) by integrating out some of the fields.

By integrating out $b_{0}$, and $a^{\prime}_{0}$ we have

$\partial_{i}a_{j}\epsilon^{ij}-A^{I}_{i}\delta^{I}_{j}\epsilon^{ij}=0\
\leftrightarrow\ \partial_{x}a_{y}-A_{x}^{y}=\partial_{y}a_{x}-A_{y}^{x},$
(4.10)

and

$\partial_{i}D_{j}\epsilon^{ij}+\sum_{I\neq
J}c_{i}^{I}\delta^{J}_{j}\epsilon^{ij}=0\ \leftrightarrow\
c^{x}_{x}-c^{y}_{y}=-\partial_{i}D_{j}\epsilon^{ij}.$ (4.11)

Furthermore, integrating out the gauge field $A^{I}_{0}$ and $c^{I}_{0}$ gives
rise to the following two conditions:

$\partial_{i}c^{I}_{j}\epsilon^{ij}+b_{i}\delta^{I}_{j}\epsilon^{ij}=0,\quad\partial_{i}A^{I}_{j}\epsilon^{ij}+a^{\prime}_{i}\delta^{\bar{I}}_{j}\epsilon^{ij}=0,$
(4.12)

which can be rewritten as

$\begin{pmatrix}b_{x}\\\
b_{y}\end{pmatrix}=\begin{pmatrix}-\partial_{i}c^{y}_{j}\epsilon^{ij}\\\
\partial_{i}c^{x}_{j}\epsilon^{ij}\end{pmatrix},\quad\begin{pmatrix}a^{\prime}_{x}\\\
a^{\prime}_{y}\end{pmatrix}=\begin{pmatrix}\partial_{i}A^{x}_{j}\epsilon^{ij}\\\
-\partial_{i}A^{y}_{j}\epsilon^{ij}\end{pmatrix}.$ (4.13)

To proceed, we introduce gauge fields as

$A_{(xx)}\vcentcolon=A_{x}^{x}-\partial_{x}a_{x},\quad
A_{(yy)}\vcentcolon=A_{y}^{y}-\partial_{y}a_{y},$

and eliminate the gauge fields $a^{\prime}$ and $b$ by use of the relations
(4.10)-(4.13), the Lagrangian is rewritten as

$\boxed{\mathcal{L}_{qp}=\frac{N}{2\pi}\biggl{[}A_{0}\left(\partial_{x}^{2}B_{(yy)}-\partial_{y}^{2}B_{(xx)}\right)-A_{(xx)}\left(\partial_{\tau}B_{(yy)}-\partial_{y}^{2}B_{0}\right)+A_{(yy)}\left(\partial_{\tau}B_{(xx)}-\partial_{x}^{2}B_{0}\right)\biggr{]},}$
(4.14)

where

$A_{0}\vcentcolon=a_{0},\quad B_{0}\vcentcolon=D_{0},\quad
B_{(xx)}\vcentcolon=\partial_{x}D_{x}+c^{y}_{x},\quad
B_{(yy)}\vcentcolon=\partial_{y}D_{y}+c^{x}_{y}.$ (4.15)

Recalling (4.5)(4.9), the gauge transformation of $B_{0},B_{(xx)},B_{(yy)}$
and that of $A_{0}$, $A_{(xx)}$, $A_{(yy)}$ reads (we relabel the gauge
parameters)

$\displaystyle B_{0}\to B_{0}+\partial_{\tau}\phi,\quad B_{(xx)}\to
B_{(xx)}+\partial_{x}^{2}\phi,\quad B_{(yy)}\to
B_{(yy)}+\partial_{y}^{2}\phi,$ $\displaystyle A_{0}\to
A_{0}+\partial_{\tau}\phi^{\prime},\quad A_{(xx)}\to
A_{(xx)}+\partial_{x}^{2}\phi^{\prime},\quad A_{(yy)}\to
A_{(yy)}+\partial_{y}^{2}\phi^{\prime}.$ (4.16)

The theory (4.14) has the same form as the BF theory description of the
$\mathbb{Z}_{N}$ toric code (2.4) with difference being that the spatial
derivatives are replaced with the second order. The equation of motions in the
Lagrangian imply that the following gauge invariant field strength vanish:

$\displaystyle F_{A}=\partial_{x}^{2}A_{(yy)}-\partial_{y}^{2}A_{(yy)},\quad
E_{xA}=\partial_{\tau}A_{(xx)}-\partial_{x}^{2}A_{0},\quad
E_{yA}=\partial_{\tau}A_{(yy)}-\partial_{y}^{2}A_{0},$ $\displaystyle
F_{B}=\partial_{x}^{2}B_{(yy)}-\partial_{y}^{2}B_{(yy)},\quad
E_{xB}=\partial_{\tau}B_{(xx)}-\partial_{x}^{2}B_{0},\quad
E_{yB}=\partial_{\tau}B_{(yy)}-\partial_{y}^{2}B_{0}.$ (4.17)

As in the case with the other BF theories, the equations of motions ensure
that there is no non-trivial local gauge invariant operators, yet, the theory
admits non-local gauge-invariant operators, contributing to the GSD. With the
simplified Lagrangian (4.14), we evaluate the GSD on the torus geometry by
counting the distinct number of the non-contractible Wilson loops of the gauge
fields $A_{(xx)}$ and $A_{(yy)}$. Below, similarly to the previous section, we
think of the theory (4.14) on a discrete 2d lattice with periodic boundary
condition and evaluate the GSD.888Again, we put the theory on the discrete 2d
lattice with coordinate ($\hat{x},\hat{y}$), and due to the periodic boundary
condition,
$(\hat{x}+L_{x},\hat{y})\sim(\hat{x},\hat{y}+L_{y})\sim(\hat{x},\hat{y})$.

As for the non-contractible loops of the gauge field, $A_{(xx)}$, referring to
the gauge transformation (4.16), and argument around (3.16), we have two types
of such loops in the form of

$W_{x}(\hat{y})=\exp\biggl{[}i\oint
d\hat{x}A_{(xx)}(\hat{x},\hat{y})\biggr{]},\quad
W_{dp:x}(\hat{y})=\exp\biggl{[}i\alpha_{x}\oint
d\hat{x}\bigl{(}\hat{x}A_{(xx)}(\hat{x},\hat{y})\bigr{)}\biggr{]}$ (4.18)

with $\alpha_{x}=\frac{N}{\gcd(N,L_{x})}$. Justification of the existence of
these loops is given in Appendix. B, where we discuss thoroughly the Wilson
loops in the UV lattice model. In order to count the number of distinct such
loops, we need to check whether these loops depend on $\hat{y}$. Let us first
focus on the loop $W_{x}(\hat{y})$. From the equation of motion,
$F_{A}=\partial_{x}^{2}A_{(yy)}-\partial_{y}^{2}A_{(yy)}=0$, and performing
integral along the $x$-direction, we have

$\partial_{y}^{2}\oint d\hat{x}A_{(xx)}(\hat{x},\hat{y})=0\ \leftrightarrow\
\partial_{y}\oint d\hat{x}\partial_{y}A_{(xx)}(\hat{x},\hat{y})=0,$ (4.19)

from which it follows that $\oint
d\hat{x}\partial_{y}A_{(xx)}(\hat{x},\hat{y})$ is independent of $\hat{y}$.
For an arbitrary $\hat{y}_{0}$, we set

$\displaystyle\oint d\hat{x}\partial_{y}A_{(xx)}(\hat{x},\hat{y})=\oint
d\hat{x}\partial_{y}A_{(xx)}(\hat{x},\hat{y}_{0}),$ (4.20)

which can be rewritten as

$\displaystyle\oint d\hat{x}A_{(xx)}(\hat{x},\hat{y}+1)-\oint
d\hat{x}A_{(xx)}(\hat{x},\hat{y})=\oint
d\hat{x}\partial_{y}A_{(xx)}(\hat{x},\hat{y}_{0}).$ (4.21)

Iterative use of (4.21) gives

$\oint d\hat{x}A_{(xx)}(\hat{x},\hat{y})=\oint
d\hat{x}A_{(xx)}(\hat{x},\hat{y}_{0})+(\hat{y}-\hat{y}_{0})\oint
d\hat{x}\partial_{y}A_{(xx)}(\hat{x},\hat{y}_{0}),$ (4.22)

from which we have

$W_{x}(\hat{y})=W_{x}(\hat{y}_{0})(\tilde{W}_{x}(\hat{y}_{0}))^{(\hat{y}_{0}-\hat{y})},\;\;\tilde{W}_{x}(\hat{y}_{0})\vcentcolon=\exp\biggl{[}i\oint
d\hat{x}A_{(xx)}(\hat{x},\hat{y}_{0}+1)-A_{(xx)}(\hat{x},\hat{y}_{0})\biggr{]},$
(4.23)

indicating that in order to deform the loop $W_{x}(\hat{y})$ from $\hat{y}$ to
$\hat{y}_{0}$, we need to multiply other loops. The relation (4.23) is
corroborated by a close investigation of the UV lattice model presented in
Appendix. B.

The distinct loops of $W_{x}(\hat{y})$ is labeled by two quantum numbers,
corresponding to two loops, $W_{x}(\hat{y}_{0})$ and
$\tilde{W}_{x}(\hat{y}_{0})$. The loop $W_{x}(\hat{y}_{0})$ is labeled by
$\mathbb{Z}_{N}$ as we have $W_{x}^{N}(\hat{y}_{0})=1$. Due to the periodic
boundary condition, $W_{x}(\hat{y}+L_{y})=W_{x}(\hat{y})$ and (4.23), the loop
$\tilde{W}_{x}(\hat{y}_{0})$ is subject to
$\tilde{W}_{x}^{L_{y}}(\hat{y}_{0})=\tilde{W}_{x}^{N}(\hat{y}_{0})=1$, from
which it follows that the loop $\tilde{W}_{x}(\hat{y}_{0})$ is labeled by
$\mathbb{Z}_{\gcd(N,L_{y})}$. In total, the distinct number of the loop
$W_{x}(\hat{y})$ is given by $N\times\gcd(N,L_{y})$.

The similar consideration shows that there are
$\gcd(N,L_{x})\times\gcd(N,L_{x},L_{y})$ distinct loops of
$W_{dp:x}(\hat{y})$. Overall, there are
$N\times\gcd(N,L_{x})\times\gcd(N,L_{x})\times\gcd(N,L_{x},L_{y})$ distinct
non-contractible loops of the gauge field $A_{(xx)}$ in the $x$-direction. One
can analogously count the number of the distinct non-contractible loops of
$A_{(yy)}$ in the $y$-direction, arriving at the same number. Therefore, the
GSD, which is equivalent to the distinct number of loops of the gauge fields
$A_{(xx)}$ and $A_{(yy)}$, is given by

$\boxed{GSD=[N\times\gcd(N,L_{x})\times\gcd(N,L_{x})\times\gcd(N,L_{x},L_{y})]^{2}.}$
(4.24)

Analogously to the previous case, the foliated BF theory show the GSD
dependence on the UV lattice, involving the greatest common divisor between
$N$ and the system size, which is in sharp contrast with the sub-extensive GSD
dependence found in the preexisting foliated BF theories.

###  4.3 UV lattice model

Similarly to the previous section, we can map from the BF theory to the UV
lattice stabilizer model. To start, we introduce a 2d square lattice where we
place two types of $N$-qubit state ($\mathbb{Z}_{N}$ clock states) on each
vertex. We represent basis of the two types of the clock states as
$\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}$ with
$a,b\in\mathbb{Z}_{N}$, and the $\mathbb{Z}_{N}$ Pauli operators acting on the
state as

Figure 3: Two types of the terms defined in (4.26). The red and blue squares
distinguish between the Pauli operators that act on the state $\ket{a}$ and
those on $\ket{b}$ respectively, whereas the pink square represents the
composite of the Pauli operators that act on both of $\ket{a}$ and $\ket{b}$.

$\displaystyle
Z_{1,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\omega^{a}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})},\quad
Z_{2,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\omega^{b}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})},$
$\displaystyle
X_{1,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\ket{a+1}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})},\quad
X_{2,(\hat{x},\hat{y})}\ket{a}_{(\hat{x},\hat{y})}\ket{b}_{(\hat{x},\hat{y})}=\ket{a}_{(\hat{x},\hat{y})}\ket{b+1}_{(\hat{x},\hat{y})}.$
(4.25)

With this preparation, we introduce the following terms by (see Fig. 3)

$\displaystyle V_{(\hat{x},\hat{y})}$ $\displaystyle\vcentcolon=$
$\displaystyle
X_{1,(\hat{x}+1,\hat{y})}X_{1,(\hat{x}-1,\hat{y})}(X_{1,(\hat{x},\hat{y})}^{\dagger})^{2}X_{2,(\hat{x},\hat{y}+1)}X_{2,(\hat{x},\hat{y}-1)}(X_{2,(\hat{x},\hat{y})}^{\dagger})^{2},$
$\displaystyle P_{(\hat{x},\hat{y})}$ $\displaystyle\vcentcolon=$
$\displaystyle
Z_{1,(\hat{x},\hat{y}+1)}^{\dagger}Z_{1,(\hat{x},\hat{y}-1)}^{\dagger}Z^{2}_{1,(\hat{x},\hat{y})}Z_{2,(\hat{x}+1,\hat{y})}Z_{2,(\hat{x}-1,\hat{y})}(Z_{2,(\hat{x},\hat{y})}^{\dagger})^{2},$
(4.26)

Note that $V_{(\hat{x},\hat{y})}$ and $P_{(\hat{x},\hat{y})}$ is what
corresponds to the Gauss law and flux operator in the BF theory (4.8), where
it reads as $G=\partial_{x}^{2}E_{xA}+\partial_{y}^{2}E_{yA}$,
$F_{A}=\partial_{x}^{2}A_{(yy)}-\partial_{y}^{2}A_{(yy)}$, respectively. It is
straightforward to check that these operators mutually commute. Hamiltonian is
defined by

$H_{qp}=-\sum_{\hat{x},\hat{y}}\left[V_{(\hat{x},\hat{y})}+P_{(\hat{x},\hat{y})}\right]+({\rm
h.c.}).$ (4.27)

This model is exactly solvable as individual term in the Hamiltonian commutes
with one another. Further, the model exhibits unusual behaviour of the ground
state degeneracy on torus geometry, depending on the system size [72].
Differing the details to the Appendix B, we derive the GSD of the lattice
model, arriving the same value as (4.24).

###  4.4 Multipole one-form symmetries

(a)

(b)

(c)

(d)

(e)

Figure 4: (a) Two type of the terms $V_{(\hat{x},\hat{y})}$ and
$P_{(\hat{x},\hat{y})}$ constituting Hamiltonian (4.27) with $N=2$. (b)-(d)
Four closed loops of the operators $Z_{1}$’s and $Z_{1}$’s that commute with
the Hamiltonian. The red and blue squares distinguish between the Pauli
operators that act on the state $\ket{a}$ and those on $\ket{b}$.

It is known that in the $2+1d$ toric code, there are $1$-form symmetries [79],
which are prototype examples of the generalized global symmetries [80]. In
condensed matter physics language, the $1$-form symmetries in the toric code
are nothing but the closed loops of the fractionalized charges on the lattice.
Since our UV lattice model (4.27) has the simple form, one can explicitly see
that there are such $1$-form symmetries, corresponding to closed loops. In
addition, such symmetries follow the similar relation of the multipole
symmetries that we started with (4.1), which is not seen in the regular toric
code.

To this end, we focus on the case with $N=2$ while other $N$ cases can be
argued in a similar way. In this case, two type of the terms
$V_{\hat{x},\hat{y}}$ and $P_{(\hat{x},\hat{y})}$ constituting the Hamiltonian
(4.27) becomes (Fig. 4(a))

$\displaystyle V_{(\hat{x},\hat{y})}$ $\displaystyle=$ $\displaystyle
X_{1,(\hat{x}+1,\hat{y})}X_{1,(\hat{x}-1,\hat{y})}X_{2,(\hat{x},\hat{y}+1)}X_{2,(\hat{x},\hat{y}-1)},$
$\displaystyle P_{(\hat{x},\hat{y})}$ $\displaystyle=$ $\displaystyle
Z_{1,(\hat{x},\hat{y}+1)}Z_{1,(\hat{x},\hat{y}-1)}Z_{2,(\hat{x}+1,\hat{y})}Z_{2,(\hat{x}-1,\hat{y})}.$
(4.28)

There are four types of closed loops of the operators $Z_{1}$’s and $Z_{2}$’s
that commute with the Hamiltonian, as portrayed in Fig. 4(b)-4(e).999 The
other four types of closed loops of the operators $X_{1}$’s and $X_{2}$’s can
be similarly discussed.  The first loop (Fig. 4(b)), denoted by
$\tilde{Q}_{xy}$, consists of two horizontal lines of $Z_{1}$’s and two
vertical lines of $Z_{2}$’s, where each $Z_{1}$ ($Z_{2}$) is separated by a
single site in the horizontal (vertical) direction. The second loop (Fig.
4(c)), labeled by $\tilde{Q}_{x}$, is formed in such a way that there are four
horizontal lines of $Z_{1}$’s, each of which is separated by a single site,
and there are two vertical lines of $Z_{2}$’s. The role of $Z_{1}$’s and
$Z_{1}$’s is switched in the third loop, $\tilde{Q}_{y}$ (Fig. 4(d)). The
fourth loop, $\tilde{Q}$ consists of four horizontal lines of $Z_{1}$’s and
four vertical lines of $Z_{2}$’s. We consider translating these operators by
one lattice site in either $x$\- or $y$-direction, and see how these loops
change. Denoting a transnational operator in the $I$-direction by one lattice
site by $T_{I}$, we investigate the change of the translation by taking the
ratio between a loop after the translation and the one before the translation.
We have

$\frac{T^{-1}_{x}\tilde{Q}_{xy}T_{x}}{\tilde{Q}_{xy}}=\tilde{Q}_{y},\quad\frac{T^{-1}_{y}\tilde{Q}_{xy}T_{y}}{\tilde{Q}_{xy}}=\tilde{Q}_{x},\quad\frac{T^{-1}_{x}\tilde{Q}_{x}T_{x}}{\tilde{Q}_{x}}=\tilde{Q},\quad\frac{T^{-1}_{y}\tilde{Q}_{y}T_{y}}{\tilde{Q}_{y}}=\tilde{Q}.$
(4.29)

This relation is inherited from our original consideration of a theory with
multipole symmetries (4.1) in the sense that if we translate a loop, it gives
rise to another loop, similarly to the fact that translating a quadruple gives
a dipole (or translating a dipole gives a charge) in (4.1). It would be
interesting to study these $1$-form symmetries from the perspective higher-
form gauging and anomaly. Also, since these $1$-form symmetries are
qualitatively different from those found in the regular toric code due to
(4.29), it could be intriguing to address whether the lattice model (4.27) has
different features in view of quantum error corrections [9]. We will leave
these issues for future works.

##  5 Conclusion and discussions

Symmetry has been one of the fundamental laws of physics, and is a guiding
principle, allowing us to analyze various physical properties. In this work,
we have demonstrated that symmetry plays a pivotal role to construct new field
theories in the context of the fracton topological phases. We have
demonstrated a way to construct the $2+1d$ foliated BF theories based on the
argument of the multipole symmetries. By introducing gauge fields associated
with these symmetries, one can systematically construct new foliated BF
theories. These theories exhibit unusual GSD dependence on the system size,
involving the greatest common divisor between $N$ and the length of the
lattice. We have also shown that these foliated BF theories are effective
field theory description of unconventional topological phases, referred to as
the higher rank topological phases. Our consideration provides a unified
insight on various unconventional topological phases such as fracton and
higher rank topological phases in terms of the foliated topological field
theories.

There are several future research directions regarding the present work that
we would like to pursue for future studies. In this paper, we focus on the
construction of the $2+1d$ foliated BF theories, which can be generalized in a
several ways. For instance, one could study the construction of the $3+1$d or
even higher dimensional foliated BF theories, with various multipole
symmetries higher than dipole or quadrupole, such as ocutopoles. Also, one
could think of higher-form symmetries analogue of the multipole symmetries in
the same way as we did in this paper and see what the resulting foliated BF
theories are. Studying a boundary theory and topological defects of our
foliated BF theories to see how multipole degrees of freedom are incorporated
could be an interesting direction.

It would be also interesting to study whether our model can be useful in the
context of quantum computing. The UV stabilizer lattice models that we have
discussed here are distinct from the conventional toric code due to the
sensitivity to the local geometry. Also, as seen from Sec. 4.4, the model
admits $1$-form symmetries which have the similar commutation relation with
the transnational operators as the ones of the multipole symmetries that we
started with. In the case of the toric code, it is known that the capability
of the quantum error correction is characterized by the classical Ising
universal class [9]. Since the UV lattice model that we discuss respects the
multipole symmetries, one naively would expect that robustness of the model
against errors is qualitatively different (See [81] for relevant discussion.).
It would be intriguing to investigate whether the UV lattice models discussed
in the present paper show different universality classes that characterize the
capability of the error correction.

## Acknowledgement

We thank Biswarup Ash, Bo Han, Rohit Kalloor, Ken Shiozaki, Masataka Watanabe
for discussions. H. E. is supported by KAKENHI-PROJECT-23H01097. M. H. is
supported by MEXT Q-LEAP, JSPS Grant-in-Aid for Transformative Research Areas
(A) “Extreme Universe” JP21H05190 [D01] and JSPS KAKENHI Grant Number
22H01222. M. H. and T. N. are supported by JST PRESTO Grant Number JPMJPR2117.
T. N is supported by JST, the establishment of university fellowships towards
the creation of science technology innovation, Grant Number JPMJFS2123.

##  Appendix A Derivation of (3.29)

In this appendix, we provide an argument to derive (3.29). The composite of
loops (3.28) is subject to the conditions

$(k_{x},k_{y})\sim(k_{x}+N,k_{y})\sim(k_{x}+\alpha_{y}L_{y},k_{y})$ (A.1)

and

$(k_{x},k_{y})\sim(k_{x},k_{y}+N)\sim(k_{x},k_{y}+\alpha_{x}L_{y})$ (A.2)

with $\alpha_{x/y}=\frac{N}{\gcd(N,L_{x/y})}$. Also, $(k_{x},k_{y})$ has to
satisfy

$\displaystyle(k_{x},k_{y})\sim(k_{x}+L_{x},k_{y}-L_{y}).$ (A.3)

From (A.1) and (A.3), there are $\gcd(N,\alpha_{y}L_{x},L_{x})$ distinct
number of $k_{x}$ and for these distinct values of $k_{x}$, there are
$\gcd(N,\alpha_{x}L_{y})$ distinct number of $k_{y}$. In total, there are
$\gcd(N,\alpha_{y}L_{x},L_{x})\times\gcd(N,\alpha_{x}L_{y})$ distinct number
of the composite loops (3.28). Since
$\gcd(N,\alpha_{y}L_{x},L_{x})=\gcd(N,L_{x})$, it follows that

$\displaystyle\gcd(N,\alpha_{y}L_{x},L_{x})\times\gcd(N,\alpha_{x}L_{y})$
$\displaystyle=$
$\displaystyle\gcd(N,L_{x})\times\gcd\biggl{(}N,\frac{N}{\gcd(N,L_{x})}L_{y}\biggr{)}$
(A.4) $\displaystyle=$ $\displaystyle
N\gcd(\gcd(L_{x},N),L_{y})=N\gcd(N,L_{x},L_{y}).$

Hence, there are $N\gcd(N,L_{x},L_{y})$ distinct loops of the gauge field
$A_{(xy)}$. Taking into the other non-contractible loops, we arrive at that
the GSD, which is the number of the distinct non-contractible loops of the
gauge fields $A_{(ij)}$ is given by (3.29).

##  Appendix B Investigation of the lattice model by the Laplacian

In this appendix, we analyze the UV lattice model (4.27) obtained from the
foliated BF theory, especially investigate the Wilson loops that contributes
to the non-trivial GSD on a torus geometry from a different perspective. As we
discussed in the main text, the model contains the second order spatial
derivatives, involving the nearest neighboring states. Due to this property,
one can make use of the Laplacian, which is the graph theoretical analogue of
the second order derivatives [82], allowing us to systematically investigate
the lattice model, such as the Wilson loops. To see how, we first recall the
Hamiltonian has the following form,

$H_{qp}=-\sum_{\hat{x},\hat{y}}\left[V_{(\hat{x},\hat{y})}+P_{(\hat{x},\hat{y})}\right]+({\rm
h.c.}),$ (B.1)

and that the ground state $\ket{\Omega}$ satisfies
$V_{(\hat{x},\hat{y})}\ket{\Omega}=P_{(\hat{x},\hat{y})}\ket{\Omega}=\ket{\Omega}$,
i.e., the ground state does not admit an electric and magnetic excitation. To
discuss the Wilson loops, we also investigate the excitations. When acting an
operator $Z_{1,(\hat{x},\hat{y})}$ on the ground state, it violates the
condition $V_{(\hat{x},\hat{y})}=1$, that is,

$V_{(\hat{x},\hat{y})}(Z_{1,(\hat{x},\hat{y})}\ket{\Omega})=\omega^{-2}(Z_{1,(\hat{x},\hat{y})}\ket{\Omega}),\quad
V_{(\hat{x}\pm
1,\hat{y})}(Z_{1,(\hat{x},\hat{y})}\ket{\Omega})=\omega(Z_{1,(\hat{x},\hat{y})}\ket{\Omega})$
(B.2)

yielding electric charges. This implies that by acting an operator
$Z_{1,(\hat{x},\hat{y})}$, an electric charge is induced at the coordinate
$(\hat{x}\pm 1,\hat{y})$ and two conjugate of the electric charges are
obtained at $(\hat{x},\hat{y})$. Denoting the $\mathbb{Z}_{N}$ eclectic charge
as $e_{(\hat{x},\hat{y})}$ (and its conjugate as
$\overline{e}_{(\hat{x},\hat{y})}$), (B.2) can be described by the fusion
rule, reading

$I\ \to\
e_{(\hat{x}-1,\hat{y})}\overline{e}^{2}_{(\hat{x},\hat{y})}e_{(\hat{x}+1,\hat{y})}.$
(B.3)

The fusion rule of the magnetic charges can be similarly discussed.

Using this property, one can study the GSD of the model on a torus geometry by
counting the distinct number of Wilson loops. To this end, the Laplacian comes
into play. We consider the model (B.1) on a 2d lattice with periodic boundary
condition and the system size being $L_{x}\times L_{y}$. In this lattice, we
think of a closed loop of the electric charge in the horizontal direction at
$\hat{y}$, described by
$\prod_{\hat{x}=1}^{L_{x}}Z_{1,(\hat{x},\hat{y})}^{a_{\hat{x}}}$ with
$a_{\hat{x}}\in\mathbb{Z}_{N}$. From (B.3), the electric charges induced by
acting this operator on the ground state are described by the fusion rule

$I\to\prod_{\hat{x}=1}^{L_{x}}\otimes
e^{r_{\hat{x}}}_{(\hat{x},\hat{y})}\quad(r_{\hat{x}}\in\mathbb{Z}_{N})$ (B.4)

with

$\mathbf{r}=-L_{{L_{x}\times L_{x}}}\mathbf{a}.$ (B.5)

Here, $\mathbf{a}\vcentcolon=(a_{1},\cdots,a_{L_{x}})^{T}$,
$\mathbf{r}\vcentcolon=(r_{1},\cdots,r_{L_{x}})^{T}$, and $L_{x}\times L_{x}$
matrix, $L_{{L_{x}\times L_{x}}}$, is the Laplacian, the graph theoretical
analogue of the second order spatial derivative, which has the following form

$L_{L_{x}\times L_{x}}=\begin{pmatrix}2&-1&&&-1\\\ -1&2&-1&&\\\
&-1&2&\ddots&\\\ &&\ddots&\ddots&-1\\\ -1&&&-1&2\end{pmatrix}.$ (B.6)

Since the closed non-contractible loop
$\prod_{\hat{x}=1}^{L_{x}}Z_{1,(\hat{x},\hat{y})}^{a_{\hat{x}}}$ has to
commute with the Hamiltonian, the fusion rules (B.4) and (B.5) have to be
trivial, i.e., $\mathbf{r}=\mathbf{0}\mod N$. Thus, the closed loops of the
electric charges are characterized by the kernel of the Laplacian.

By evaluating the Laplacian (B.6), the solution of $L_{L_{x}\times
L_{x}}\mathbf{a}=\mathbf{0}$ is given by [72]

$\mathbf{a}=\alpha_{0}(1,1,\cdots,1)^{T}+\alpha_{1}\times\frac{N}{\gcd(N,L_{x})}(1,2,3,\cdots,L_{x})^{T}.$
(B.7)

(a)

(b)

Figure 5: (a) Examples of the two types of the loops given in (B.8) in the
case of $N=L_{x}=6$. The red dots denote the operators
$Z_{1,(\hat{x},\hat{y}})$. The integer numbers correspond to the power
entering in the form of the loops (B.8). The periodic boundary condition is
imposed so that the left and right edges are identified. (b) Configuration
corresponding to the second term of (B.14), which is the composite of the
loops with opposite charge located adjacent in the $y$-direction.

Here, $\alpha_{0}\in\mathbb{Z}_{N}$ and
$\alpha_{1}\in\mathbb{Z}_{\gcd(N,L_{x})}$. Hence, there are two kinds of the
closed loops in the horizontal direction
($\alpha_{x}=\frac{N}{\gcd(N,L_{x})}$):

$W_{x}(\hat{y})=\prod_{\hat{x}=1}^{L_{x}}Z_{1,(\hat{x},\hat{y})},\quad
W_{dp:x}(\hat{y})=\prod_{\hat{x}=1}^{L_{x}}(Z^{\hat{x}}_{1,(\hat{x},\hat{y})})^{\alpha_{x}}.$
(B.8)

The first loop is interpreted as the conventional Wilson loop (Fig. 5(a)
left), which is formed by the trajectory of the $\mathbb{Z}_{N}$ electric
charge, traveling around the torus in the $x$-direction. This corresponds to
the first term of (4.18) in the BF theory. The second loop can be interpreted
as the “dipole of the Wilson loop” (Fig. 5(a) right) in the sense that the
loops is formed by the $\mathbb{Z}_{N}$ electric charge going around the torus
with its intensity increasing linearly. This corresponds to the second term in
(4.18). The first loop, $W_{x}(\hat{y})$ is labeled by $\mathbb{Z}_{N}$
whereas the second one $W_{dp:x}(\hat{y})$ is by $\mathbb{Z}_{\gcd(N,L_{x})}$.
By evaluating the fusion rules, one can similarly discuss the configuration of
the (part of) closed loops in the lattice model (3.33) obtained from the
foliated BF theory with dipole symmetry (3.7). For instance, the form of the
loops given in (3.16) and (3.15) can be similarly derived.

Having identified loops of the electric charges in the $x$-direction (B.8),
one need to check whether these loops are $\hat{y}$ dependent or not to count
the distinct number of configuration of the loops. Let us focus on the loop
$W_{x}(\hat{y})$. By multiplying the following operator

$M(\hat{y})=\prod_{\hat{x}=1}^{L_{x}}P_{(\hat{x},\hat{y})},$ (B.9)

where $P_{(\hat{x},\hat{y})}$ is given in (4.26), the loop $W_{x}(\hat{y})$ is
deformed as

$M(\hat{y})W_{x}(\hat{y})=W_{x}(\hat{y}-1)W_{x}^{\dagger}(\hat{y})W_{x}(\hat{y}+1).$
(B.10)

We need to count the distinct configuration of the loop $W_{x}(\hat{y})$ up to
the deformation (B.10). To this end, we think of deforming the composite of
the loops, $\prod_{\hat{y}=1}^{L_{y}}W^{b_{\hat{y}}}_{x}(\hat{y})$ with
$b_{\hat{y}}\in\mathbb{Z}_{N}$. Multiplying it with the following operator

$\prod_{\hat{y}=1}^{L_{y}}M^{c_{\hat{y}}}(\hat{y})\quad(c_{\hat{y}}\in\mathbb{Z}_{N})$
(B.11)

and referring to (B.10), it follows that the composite of the loops after
multiplying (B.11), becomes

$\prod_{\hat{y}=1}^{L_{y}}W^{\tilde{b}_{\hat{y}}}(\hat{y})\quad(\tilde{b}_{\hat{y}}\in\mathbb{Z}_{N})$
(B.12)

with

$\tilde{\mathbf{b}}=\mathbf{b}-L_{L_{y}\times L_{y}}\mathbf{c}.$ (B.13)

Here, $\mathbf{b}=(b_{1},\cdots,b_{L_{y}})^{T}$ with other vectors
$\tilde{\mathbf{b}},\mathbf{c}$ being similarly defined and the $L_{y}\times
L_{y}$ matrix, $L_{L_{y}\times L_{y}}$ denotes the Laplacian, which has the
same form as (B.6), where the matrix size being replaced with $L_{y}\times
L_{y}$. It follows that the composite of loops,
$\prod_{\hat{y}=1}^{L_{y}}W^{b_{\hat{y}}}_{x}(\hat{y})$, labeled by the vector
$\mathbf{b}$ and the ones
$\prod_{\hat{y}=1}^{L_{y}}W^{\tilde{b}_{\hat{y}}}_{x}(\hat{y})$ labeled by
$\tilde{\mathbf{b}}$ which are related via (B.13) with
${}^{\exists}\mathbf{c}$ are identified. Therefore, the number of distinct
configurations of the loop $W_{x}(\hat{y})$ found to be
$\mathbf{s}\vcentcolon=\mathbb{Z}_{N}^{L_{y}}/\text{Im}(L_{L_{y}\times
L_{y}})$, which is the cokernel of the Laplacian. Note that this corresponds
to the discussion around Eq. (4.19) in the main text, where we count the
distinct number of the loops $W(\hat{y})$ subject to the condition (4.19).
Namely, in (4.19), we discuss the distinct configuration of the loops up to
the operation of the second order derivative $\partial_{y}^{2}$, which
corresponds to the present consideration where we need to find distinct
configurations of the loops up to the Laplacian, $L_{L_{y}\times L_{y}}$.

The cokernel is given by [72]

$\displaystyle\mathbf{s}=\beta_{0}(0,\cdots,0,1)^{T}+\beta_{1}(0\cdots,0,1,-1)^{T}$
(B.14)

with $\beta_{0}\in\mathbb{Z}_{N}$, $\beta_{1}\in\mathbb{Z}_{\gcd(N,L_{y})}$.
Eq. (B.14) indicates that the distinct configurations of the loops are
characterized by a single loop and the composite of loops (Fig. 5(b)) which
have the opposite charge, located adjacent to each other in the $y$-direction.
These two types of loops are labeled by $\mathbb{Z}_{N}$ and
$\mathbb{Z}_{\gcd(N,L_{y})}$. This corresponds to the discussion around
(4.23).

Overall, we have counted the distinct number of loops $W_{x}(\hat{y})$ which
is given by $N\times\gcd(N,L_{y})$. Analogous line of thoughts leads to that
there are $\gcd(N,L_{x})\times\gcd(N,L_{x},L_{y})$ distinct configurations of
the loop $W_{dp:x}(\hat{y})$. Thus, in total, there are
$N\times\gcd(N,L_{y})\times\gcd(N,L_{x})\times\gcd(N,L_{x},L_{y})$ different
configurations of the loops of the electric charge. So far, have considered
closed loops of the electric charges. Regarding the closed loops of the
magnetic charges, the similar argument follows as the electric charges, thus
they are labeled by the same quantum numbers. Taking it into account, we
finally arrive at that the GSD is the same value as (4.24).

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