SciPost Phys. 15, 155 (2023)

Subsystem non-invertible symmetry operators and defects
Weiguang Cao1,2 , Linhao Li2,3 , Masahito Yamazaki1,4 and Yunqin Zheng1,3
1 Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
2 Department of Physics, Graduate School of Science,
University of Tokyo, Tokyo 113-0033, Japan
3 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
4 Trans-Scale Quantum Science Institute, University of Tokyo, Tokyo 113-0033, Japan

Abstract
We explore non-invertible symmetries in two-dimensional lattice models with subsystem Z2 symmetry. We introduce a subsystem Z2 -gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wanniertransformation. The corresponding duality operators and defects are constructed bygaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion ruleswith subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both
spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss
its subtleties.
Copyright W. Cao et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 01-05-2023
Accepted 08-09-2023
Check forupdatesPublished 12-10-2023
doi:10.21468/SciPostPhys.15.4.155

Contents
1 Introduction
1.1 Subsystem non-invertible symmetry
1.2 Symmetry operators and defects
1.3 Anomalies of subsystem (non)invertible symmetry
1.4 Structure of the paper

2
2
3
4
4

2 Kramers-Wannier duality operators and defects in (1 + 1)d spin chains
2.1 Ordinary Z2 symmetry and twist operators
2.2 KW duality operators
2.3 KW duality defects
2.4 KW duality symmetry
2.5 Anomaly of KW duality symmetry

5
5
6
8
11
12

3 Subsystem Kramers-Wannier duality operators and defects in (2+1)d spin systems 13
3.1 Subsystem Z2 symmetry and twist operators
13
3.2 Subsystem KW duality operators
14
3.3 Subsystem KW duality defects
191

SciPost Phys. 15, 155 (2023)

3.4
3.5

Subsystem KW duality symmetry
Anomaly of subsystem KW duality symmetry

22
23

4 Discussion and future directions

25

A Gauging subsystem Z2 symmetry27

References30

1Introduction

1.1

Subsystem non-invertible symmetry

Symmetry is one of the most important notions in physics. In recent years, the concept ofglobal symmetry has been generalized in various directions [1]. See also the snowmass white
paper [2] for a recent overview and more references. The three most notable properties ofsymmetry operators/defects are their
1. codimensions,2. invertibility,
3. topologicalness/mobility.
We can organize generalized symmetries according to these properties (Tab. 1). An ordinary
global symmetry corresponds to codimension-one topological operators with group-like andinvertible fusion rules. The remaining symmetries differ in one or multiple properties. For
instance, higher-form symmetries relax the condition of codimension being one [1, 3–45];1
non-invertible symmetries in (1 + 1)d relax the invertibility [50–74]; subsystem symmetries in(2 + 1)d relax the topologicalness or the mobility of the symmetry generators/defects2 [75–91].3 Here and in Tab. 1 we only present a highly incomplete list of references.4
One gets further generalized symmetries by modifying more than one property simultaneously. For instance, by allowing the defect to have codimension higher than one and also
be non-invertible, we get higher categorical symmetries [101–140]; by gauging the 0-form
subsystem symmetries in (3 + 1)d, we get a fracton order with one-form subsystem symmetries [141–152].
1
Before [1], various earlier works had already hinted the existence of higher form symmetries and the dynamicalconsequences thereof [46–49].
2
Let us clarify the definition of the codimension of an operator/defect when it is not fully mobile. The (2 + 1)dsubsystem-symmetry defects are line operators, thus one would naively conclude that it is of codimension two inspacetime, and is higher-form as well. Here it is more natural to define the codimension with respect to the subspace
where the operator/defect is mobile rather than the entire spacetime. For instance, consider the subsystem Z2
symmetry where the defect is along the x direction. It is only mobile along the t direction, while is not mobile
along the y direction. So the mobile subspace is the t − x plane, and the defect with respect to the t − x plane isof codimension one.
3
There are also earlier works on subsystem symmetry from algebraic duality approach [92–96].
4
Another line of development of unconventional symmetry, motivated by commutant algebra in the study ofquantum many body scar, was explored in [97–100]. It would be interesting to see how such unconventional
symmetry fits in various entries of Tab. 1.

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SciPost Phys. 15, 155 (2023)

Table 1: Organization of global symmetries in terms of operator/defect’s codimension, invertibility and topologicalness. The goal of this paper is to explore subsystem
non-invertible symmetry, highlighted in bold font.
Symmetry

codimension

invertibility

topologicalness

ordinary symmetry
higher-form/group sym [1, 3–45]
non-invertible/categorical sym [50–74]
higher categorical sym [101–140]
subsystem sym [75–91]
higher subsystem sym [141–152]
subsystem non-invertible sym
higher subsystem non-invertible sym

=1
≥1
=1
≥1
=1
≥1
=1
≥1

✓
✓
✗
✗
✓
✓
✗
✗

✓
✓
✓
✓
✗
✗
✗
✗

The goal of this paper is to study yet another type of generalized symmetry by lifting boththe invertibility as well as the topologicalness: the corresponding symmetry is a subsystem noninvertible symmetry. We will not study a generic subsystem symmetry, and instead focus on one
typical example— the subsystem Kramers-Wannier (KW) duality symmetry associated with thegauging of a subsystem Z2 symmetry in (2 + 1)d. Here, the subsystem KW duality symmetry
has co-dimension 1 non-invertible symmetry operator and defect, which is different from theco-dimension 2 invertible subsystem Z2 symmetry operators and defects. Furthermore, thenon-invertible fusion rule will mix operators (defects) of different co-dimensions.
Our construction naturally generalizes the well-known KW duality symmetry in (1 + 1)d,which gauges an ordinary Z2 symmetry. To the authors’ knowledge, this is the first study of asymmetry where both the invertibility and topologicalness are lifted.

1.2

Symmetry operators and defects

For a given symmetry we can consider symmetry operators (along the spatial direction) andsymmetry defects (one of whose directions is along the time).5 When the symmetry is topological, the two can be deformed to each other and hence are equivalent. This naturally happens
in relativistic systems, where Lorentz symmetry ensures that time and space are on equal footing.
However, when the topologicalness is sacrificed as in the case of the subsystem symmetry,it is important to treat the symmetry operator and symmetry defect separately. The latter was
highlighted recently [87, 91] as a generator of the time-like symmetry.
In the literature, most of the discussions of non-invertible symmetries are either in continuum field theories or in the statistical models with infinite system sizes [52]. The authorsof [154] systematically discussed the properties of the KW-duality operator acting on a finitesize Hilbert space of a quantum system, and carefully discussed the effects of boundary conditions for closed spin chains. However, it is also important to study symmetry defects to fully
uncover the properties of symmetry. In particular, it is important to turn on both symmetry
operators and defects when determining the anomaly of the symmetry. Anomaly usually manifests itself as a “fractional” symmetry charge of the ground state in the presence of symmetry
defects (i.e. twisted boundary conditions) [155, 156].
5

We mainly use the terminology “operators” for maps from one Hilbert space (defined on sites) to another
Hilbert space (defined on links or plaquettes) and “defects” for interfaces between two theories. One can further
redefine the link/plaquette Hilbert spaces to be supported on sites, which we briefly discuss in Sec. 2.4 and 3.4.
This redefinition makes the operators to act within on one Hilbert space, and the defects between a single theory.
This is consistent with the recent discussion [153].

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SciPost Phys. 15, 155 (2023)

In this work, we will study both symmetry operators and defects on the lattice. Our discussion of the symmetry operators directly generalizes our previous work on (1 + 1)d KW duality
operators [154]. The discussion of symmetry defects is relatively new, and is consistent withearlier results for the (1 + 1)d KW duality defect in the Ising model on the lattice [157].
One peculiar feature we encounter is the mobility of the subsystem KW duality defect.
Because of the subsystem nature, one might naively expect that the subsystem KW duality
defect along, say, the t − x 6 direction is not mobile along the y direction. However, we find thatthe subsystem KW duality defect is mobile under translation in the entire x − y plane. Namely,
two KW duality defects defined on t − ℓ1 (x, y) and t − ℓ2 (x, y), where ℓi (x, y) specifies a pairof mutually deformable curves within the x − y plane, can be deformed to each other.7

1.3

Anomalies of subsystem (non)invertible symmetry

As with other types of symmetries, the subsystem (non)invertible symmetry may also sufferfrom anomalies. What does the anomaly of subsystem symmetry mean? We list several criteria,
not all of which are equivalent:
1. Incompatible with a gapped phase with one ground state including trivially gapped phaseand subsystem symmetry protected topological (SSPT) phases [158–160].
2. Incompatible with a gapped phase constructed from stacking lower dimensional (subsystem) topological quantum field theories (TQFTs).
3. There is a non-trivial anomaly inflow from higher dimensions [161].
In particular, point 2 is more general than point 1, because the concept of G subsystem symmetric Renormalization Group flow is believed to admit stacking layers of lower-dimensional Gsymmetric TQFTs [162,163]. Thus it is natural to define the “trivial” phase by modding out thelower dimensional symmetric gapped phases, including those whose ground state degeneracy
is higher than one. However, point 2 and point 1 are degenerate in the current case where we
only consider a Z2 subsystem symmetry in (2+1)d. This is because a non-trivial Z2 -symmetric
TQFT in (1 + 1)d must spontaneously break the Z2 symmetry. Thus in the following, we will
adopt point 1 as a criterion for the subsystem KW duality symmetry to be anomalous.
The anomaly of the KW duality symmetry in (1+1)d has been discussed in [55,56,59,106].
In this paper, we follow the approach in [106] and discuss the anomaly of the subsystem KW
duality symmetry in (2 + 1)d. The idea is to check whether an SSPT phase is invariant undera gauging of the subsystem Z2 symmetry; if not, we conclude that the subsystem KW duality
symmetry is anomalous.

1.4

Structure of the paper

The paper is organized as follows. In Sec. 2, we review the construction of ordinary KW duality
operators and defects in (1 + 1)d spin chains via gauging ordinary Z2 symmetries. We present
a comprehensive analysis of the ordinary KW duality symmetry, including the derivation of
non-invertible fusion rules, the demonstration of the mobility of duality defects, and the proofof the anomalous nature of the KW duality symmetry. In Sec. 3, we extend this formulation
to (2 + 1)d lattice models, which incorporate subsystem KW duality operators and defects.
We conduct a similar investigation of the non-invertible fusion rules, the mobility of duality
defects, and the anomaly of subsystem KW duality symmetry. Finally, in Sec. 4, we summarize
our findings and suggest potential future directions for this work.
6
We use t for both time and twist boundary condition in the main text which should not be confused. In someplaces, we also use τ for time, such as in the index of gauge fields and gauge invariant holonomy variables.
7
Note that one defect is deformable to another defect, but not to the symmetry operator.

4

SciPost Phys. 15, 155 (2023)

Kramers-Wannier duality operators and defects in (1 + 1)d spin
chains2

In this section we briefly review the non-invertible duality symmetry given by the KW transformation with respect to a non-anomalous Z2 symmetry in (1 + 1)d spin chains. The discussion
here serves as a preparation for the subsystem generalization of the non-invertible symmetryin (2 + 1)d, to be discussed in Sec. 3. Most of the materials in this section are scattered in theliterature and are not new. The goal is to gather them in one place, and also present them ina way that directly generalizes to the subsystem symmetry case.

2.1

Ordinary Z2 symmetry and twist operators

Consider a Z2 -symmetric theory on either a closed chain with L sites or an infinite chain. On
each site i there is a spin- 12 variable si ∈ {0, 1}. The Z2 operator is associated with a topologicaloperator with the Z2 fusion rule. One can either place such a topological operator along thespace or along the time direction. In the former case, the topological operator is a symmetry
generator acting on the entire Hilbert space,
Y
(1)
σix ,
U :=
i∈Z

which flips the spin on each site, i.e. si → 1 − si . The eigenvalue of U is (−1)u with u = 0, 1.
The Z2 symmetry generator satisfies two obvious properties:
1. Z2 fusion rule: U × U = 1.
2. Topological along the time direction, since it commutes with the Hamiltonian.
In the latter case, the topological operator is a symmetry defect that modifies the Hilbert space.
For a quantum field theory in the continuum with Lorentz symmetry, the space and time areon equal footing, and a Z2 operator and a Z2 defect are used interchangeably. However, it isuseful to distinguish the two, since our discussion is on the lattice, and since we would like togeneralize the discussion here to subsystem symmetries.
Let us give further details when Z2 operators are along the time direction, i.e. the Z2
defects. It is useful to first discuss Z2 defects on an infinite chain, and then on a closed chain.
Suppose the defect is localized at the origin. In the Hamiltonian formalism, such a defectcan be created by acting the original Hilbert space by a twist operator, i.e. the Z2 symmetry
generator terminated at the origin. See Fig. 1 for a schematic explanation. Concretely, thetwist operator is8
Y
U0t :=
σix .
(2)
i≤0

It creates a Z2 defect between site i = 0 and i = 1. Conjugating U0t on a Z2 symmetric Hamiltot †tnian creates a new Hamiltonian Htw,0 := (UP0 ) H U0 . For example, start with
P the Z2 symmetric
Ising model without transverse field, H = − i∈Z σzi σzi+1 , then Htw,0 = − i̸=0 σzi σzi+1 +σ0z σ1z ,and the ground state of the latter supports a Z2 domain wall. Like the Z2 symmetry generator,
the Z2 defect also satisfies the same two properties:
1. Z2 fusion rule. The fusion of Z2 defects follows from the fusion of two twist operators,
U0t × U0t = 1.
8

The superscript t either stands for “time” since the defect is along the time direction, or “twist” since it is a
twist operator creating the defect. The subscript 0 labels the endpoint of the twist operator or the locus of thedefect.

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SciPost Phys. 15, 155 (2023)

Z2 defect
U0t

Spacetime formalism on R

Hamiltonian formalism on R

Z2 defect

Ut

Ut
−L

Spacetime formalism on S 1

0

L

Hamiltonian formalism on S 1Figure 1: Z2 symmetry defects (blue) and Z2 twist operators (red) on an infinite
chain and on a ring. The black line represents the space, and the orthogonal direction
represents time. In the first row, acting a twist operator on the left half chain createsa Z2 defect along the time direction at the origin. In the second row, in order to use
the twist operator to create a symmetry defect, one needs to lift S 1 up to R and act
the twist operator on the intervals ∪k∈Z [2k L, (2k + 1)L].
2. Topological along the space direction. Given two defects at i = 0 and i = 1, generated
by U0t and U1t respectively. They give rise to the twisted Hamiltonians Htw,0 and Htw,1 .
Topologicalness of the Z2 defects means the two Hamiltonians are unitary equivalent,
i.e. there is a local unitary transformation W such that Htw,1 = W † Htw,0 W , where W
only acts on the Hilbert space around the origin. This follows if one can find W suchthat
U1t = U0t W .
(3)Indeed, W = σ1x .
We also comment on the Z2 defect on a closed chain. As explained in Fig. 1, to create a symmetry defect on a ring, we first uplift the circle to an open spin chain with periodic boundary condition |si+L 〉 = |si 〉, and act with the twist operator on the intervals∪k∈Z (2k L, (2k + 1)L]. This changes the periodic boundary condition to the twisted boundary condition |si+L 〉 = |si + 1〉. Hence on a closed chain, we can use t = 0, 1 to label the
periodic boundary condition and twisted boundary condition, corresponding to the absenceor presence of the Z2 defect, i.e. |si+L 〉 = |si + t〉.

2.2

KW duality operators

From now on, we assume that the (1 + 1)d system is invariant under the gauging of the Z2
symmetry. This means that the Z2 gauging is a symmetry of the system. Since any symmetryis equipped with a topological operator, as we discussed for the Z2 symmetry in Sec. 2.1, we
can put the operator either along the space (i.e. KW duality operator) or along the time (i.e.
KW duality defect). In this subsection, we will discuss the KW duality operator.

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SciPost Phys. 15, 155 (2023)

The KW duality operator acts on the entire Hilbert space via gauging the Z2 symmetry. We
will assume closed boundary conditions throughout this section since the change of (twisted)boundary conditions is a key feature under gauging. TheQKW duality operator was widely
known to map Pauli operators as σix → σzi σzi+1 and σzi → j≤i σix , which is well defined on
an infinitely long chain. Its action on the Hilbert space of an infinitely long chain has also
been discussed in [52]. See also [164–166] for the of Matrix-Product-State (MPS) realization.
However, on a closed chain, one has to specify how the boundary conditions transform under gauging [167–170]. In this subsection, we mainly follow the discussion in [154], where
the fusion rule, exchange of symmetry and twist sectors, and mapping of operators are all
discussed on closed chains. We only briefly summarize the results, and refer the interested
reader to [154] for further details.
Under KW, the spins {si } on the original lattice are mapped to dual spins {bsi− 1 } on the2link. We use the (bu, bt ) to label the symmetry-twist
sectors of the dual lattice. Here, (−1)ub isQLb :=b x 1 , and bt labels the boundary condition
the eigenvalue of the dual symmetry U
i=1 σ
i− 2

|b
si− 1 +L 〉 = |bsi− 1 + bt 〉.
2

2

Definition of KW duality operator: The KW transformation is realized by an operator N
acting on the Hilbert space. It is sufficient to specify how it acts on the basis states [154]N |{si }〉 =

1 X
2

L
2

PL

s j− 1 + t s L
j=1 (s j−1 +s j )b
2

(−1)

b

|{b
si+ 1 }〉 ,2{bsi+ 1 }

(4)

2

PL

1 X
N |{b
si+ 1 }〉 = L(−1)
2
2 2 {si }†s j− 1 +b
s j+ 1 )s j +tb
s1
j=1 (b222|{si }〉 ,where we use s0 = s L + t, b
s1 = bs L+ 1 + bt to identify the phases
2

2

L
L
X
X
(s j−1 + s j )bs j− 1 + bt s L =(bs j− 1 + b
s j+ 1 )s j + tb
s1 .
2

j=1

j=1

2

2

2

The exponents in (4) are reminiscent of the minimal coupling of the gauge fields. The
boundary terms in the exponents are chosen to give the correct mapping of symmetrytwist sectors. It is also straightforward to check that the duality operator is Hermitian,〈{si }|N † |{bsi+ 1 }〉 = 〈{si }|N |{bsi+ 1 }〉. By definition N commutes with the Hamiltonian (under2
2
suitable relabeling as shown below), hence N is mobile along the time direction.
Mapping between Pauli operators:
mation of Pauli operatorsb x 1 N |ψ〉 ,N σzi σzi+1 |ψ〉 = σi+ 2

KW transformation (4) induces the standard transforbz 1 σb z 1 N |ψ〉 ,N σix |ψ〉 = σ
i− 2

i+ 2

∀ |ψ〉 ∈ H .

(5)

b × N , N ×U and N † × N on a general
Non-invertible fusion rules: By acting the product of U
state |ψ〉, we find the following fusion rules:

N × U = (−1)bt N ,b × N = (−1) t N ,U

N † × N = 1 + (−1)bt U .

(6)

See [154, Sec. 2.2] for an explanation of the factors (−1) t and (−1)bt . In particular, the last
fusion rule shows that the KW duality operator N is non-invertible.

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SciPost Phys. 15, 155 (2023)

Mapping between symmetry-twist sectors: The symmetry and twist sectors before KWtransformation (u, t) and after KW transformation (bu, bt ) are related byu = bt ,

b.
t =u

(7)

Example: A canonical example with a KW duality symmetry is the Ising model with a criticaltransverse field.
L
L
X
X
z z
HIsing = −σi σi+1 −
σix .
(8)
i=1

i=1

By using (5), one obtains the dual HamiltonianÒIsing = −HLXbx 1 −
σ
i+ 2

i=1

L
X
i=1

bz 1 σbz 1 .
σ
i− 2

(9)

i+ 2

Schematically, after relabeling the spins on the sites and on the links, the two HamiltoniansÒIsing , hence the critical Ising model has a KW duality symmetry. We will
coincide, HIsing = H
make this statement more rigorous in Sec. 2.4 by performing the KW transformation on thesame lattice.

2.3

KW duality defects

In this subsection, we discuss KW duality defects in Hamiltonian formalism. Although it has
been widely discussed in the context of continuum field theories [55, 106, 107], an explicit
realization on a lattice is much less well-known (most of what we present below can be foundin [52, 106, 157]). To be self-contained, we will describe in detail the definition of the KW
duality defect via the KW duality twist operator, their fusion rule, and their mobility along thespace direction.
KW duality defect from KW duality twist operator: Inserting a KW duality defect is equivalent to performing a KW transformation on the half-space. For simplicity, we only consider 1dinfinite chains throughout this subsection. Inserting a KW duality defect at the origin amounts
to performing a KW transformation on the left half of the lattice, or equivalently applying a
KW duality twist operator N0t on the half, as shown in Fig. 2.
The KW duality twist operator terminating at i = 0 is defined as

N0t |{si }〉 =

1
2ℓ/2

X

P

(−1)

s j− 1j≤0 (s j−1 +s j )b2|{b
si− 1 }i≤0 ; {si }i>0 〉 ,2{bsi− 1 }i≤0

(10)

2

which creates the KW duality defect at i = 12 . The resulting Hilbert space is defined on halfinteger links to the left of the origin, and integer sites to the right, as shown in Fig. 2. The
overall normalization descends from that in (4) where ℓ counts the number of sites coveredby N0t . While ℓ → ∞ since N0t is infinitely long, we formally keep this normalization, which
will be useful below.9 It is also straightforward to obtain the action of (N0t )†(N0t )† |{bsi− 1 }i≤0 ; {si }i>0 〉 =
2

1 X
2

ℓ
2

P

(−1)

s j− 1j≤0 (s j−1 +s j )b2|{si }〉 .

(11){si }i≤09Another way to make sense of the normalization is to place the KW duality twist operator on a finite interval,
with both left and right boundaries.

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SciPost Phys. 15, 155 (2023)

N0t

KW duality defect
-2
-2

-1

0

1

-1

0

1

2

1

2

2
− 52

− 32

− 12

Figure 2: KW duality defects and twist operators. The left figure is the KW duality
defect along the time direction, localized at i = 12 . Such a defect can be created,in the Hamiltonian formalism, by acting the original system with a KW duality twist
operator N0t , as shown in the right top figure. In resulting Hilbert space is represented
in the right bottom figure, where the vertical slashes represent where the spins live,
and they occupy half-integer links to the left of the origin and integer sites to the right.
The empty circles are sites to the left of the origin where no spins are supported.
Fusion rule: We first derive the fusion rule of the KW duality defects. Fusing two KW duality
defects on top of each other amounts to multiplying two KW duality twist operators at the same
position. Since the Hilbert spaces before and after N0t are different, it is most convenient tocompute the fusion (N0t )† × N0t 10(N0t )† × N0t |{si }〉 =

1
2ℓ

P

X

(−1)

s j− 1 +j≤0 (s j−1 +s j )b
2

P

′
′
s j− 1j≤0 (s j−1 +s j )b2|{si′ }i≤0 ; {si }i>0 〉{b
si− 1 }i≤0 ,{si′ }i≤02

(12)

= (1 + U0t ) |{si }〉 .
Hence the fusion of the KW duality defect with its Hermitian conjugate is equivalent to thesum of an identity defect and a Z2 defect. This is consistent with the fusion rule among two
KW duality operators in (6).11
Let us also check the fusion between the Z2 defect and the KW duality defect. Fusing theZ2 defect from the right gives

N0t × U0t |{si }〉 =

1
2ℓ/2

X

P

(−1)

s j− 1j≤0 (s j−1 +1+s j +1)b2|{bsi− 1 }i≤0 ; {si }i>0 〉2{b
si− 1 }i≤02(13)

= N0t |{si }〉 .
This is also consistent with (6). Fusing the Z2 defect from the left yieldsb t 1 × N t |{si }〉 =
U
0
−2

1

X

2ℓ/2 {bsP(−1)

s j− 1 +1)j≤0 (s j−1 +s j )(b2|{bsi− 1 }i≤0 ; {si }i>0 〉
2

}i≤0
i− 1
2

(14)

= N0t σ0z |{si }〉 .
b t 1 × N t = N t σz , where a local operator σz is located at the defectWe thus naively arrive at U
0
0 0
0
−2

locus. In the Hamiltonian formalism, note that two defects are equivalent to each other ifthey are related by a local unitary operator, see the discussions around (3) as well as [157,10

To compute the fusion rule N0t × N0t , one needs to identify the Hilbert spaces |{bsi− 1 }i≤0 ; {si }i>0 〉 and |{si }〉 by2shifting the link variables to sites.
11
Because we are working on an infinite chain, we do not see the factor (−1)bt here as opposed to (6).

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SciPost Phys. 15, 155 (2023)

Sec. VI.A].12 Hence we identify the twist operator N0t σ0z with N0t , under which we find theb t 1 × N t = N t as in (6). In summary, the fusion rules involving the KW
same fusion rule U00duality defects are−2(N0t )† × N0t = 1 + U0t ,N0t × U0t = N0t ,bt 1 × N t = N t .
U
0
0
−2

(15)

Mobility of KW duality defects: We proceed to probe the mobility of the KW duality defect,by shifting it along the space. Following the discussion around (3), we need to find a local
unitary operator W such thatN1t = N0t W .
(16)Here, N1t is defined by a shift N0t by one site,PX
s j− 11j≤1 (s j−1 +s j )bt
2 |{bN1 |{si }〉 = ℓ/2
si− 1 }i≤1 ; {si }i>1 〉 .
(−1)
2
2 {bs }

(17)

i≤1
i− 1
2

The form of W has already been found in [52, Sec. 3.3] and [157, Sec. VI.A]. Concretely,
W = CZ0,1 H1 ,(18)

where CZ0,1 is the control Z gate acting on site 0 and 1, and H1 is the Hadamard gate acting
on site 1. In terms of Pauli matrices, they are1
2

CZ0,1 = (1 + σ0z + σ1z − σ0z σ1z ) ,1

H1 = p (σ1z + σ1x ) ,2from which we can derive their action on the basis product states,
1
1 X

CZ0,1 |s0 , s1 〉 = (−1)s0 s1 |s0 , s1 〉 ,H1 |s1 〉 = p′(−1)s1 s1 |s1′ 〉 .

2 s′ =0

(19)

(20)

1

The CZ0,1 operator maps σ0x to σ0x σ1z , and leaves everything else invariant. The H1 operationexchanges σ1x ↔ σ1z . As a quick consistency check, note that Nit satisfies the fusion rule(Q
Nit )† × Nit = 1 + Uit for i = 0, 1. Combining with (16), we have U1t = W † U0t W . Since U0t isxzti≤0 σi , first conjugating by CZ0,1 appends a σ1 to U0 , and further conjugating by H1 maps
σ1z to σ1x , giving rise to U1t , as expected. With this, let us explicitly check (16),′′1 X(−1)s1 s1 +s0 s1 |{si }i≤0 ; s1′ ; {si }i>1 〉N0t W |{si }〉 = N0t p
2 s′
1
P
X
s j− 1 +(s0 +s1 )s1′
1j≤0 (s j−1 +s j )b2
= (ℓ+1)/2(−1)|{bsi− 1 }i≤0 ; s1′ ; {si }i>1 〉 .
2
2
′

(21)

{b
si− 1 }i≤0 ,s12

After relabeling the dummy variable s1′ → b
s 1 , the right-hand side in the second row is exactly2t
the same as N1 |{si }〉 in (17). This shows that the KW duality defect is mobile under translation
along the space direction.
So far, we showed that the KW duality operator is mobile under translation along the time
direction, and the KW duality defect is mobile under translation along the space direction.
To fully establish the topologicalness of the KW duality symmetry generator, we also needto discuss the mobility when we bend the time-like defect into the space-like operator. See[83,142–144] for related discussions on invertible and subsystem symmetries. We will not try
to prove the full topologicalness of the KW duality symmetry generator in the current work.
12

Note that we only identify two defects that differ by a local unitary operator. Locality here means the unitary
operator is of the same spatial dimension as the boundary of the twist operator, and in this case is zero-dimensional.
b t 1 × N t with N t because U
b t 1 is not local, although they are eventually identified
We do not a priori identify U
00
after computation.

−2

−2

10

SciPost Phys. 15, 155 (2023)

2.4

KW duality symmetry

We further redefine the spins on the links to spins on sites. Concretely,b
si+ 1 → si′ ,

(22)

2

b → u′ . Under this modification, the KW duality operator
for any i. We also relabel bt → t ′ , umaps within a single Hilbert space. In this way, the KW duality can be lifted as a symmetry of
the lattice models. Under this redefinition, (4) becomesN̄ |{si }〉 :=

PL
PL
′
′
′
′
′
1 X
1 X
j=1 (s j +s j+1 )s j +t s L+1 |{s ′ }〉 =j=1 (s j−1 +s j )s j +ts0 |{s ′ }〉 .
(−1)
(−1)
L
L
i
i
2 2 {s′ }2 2 {s′ }
i

(23)

i

Here we use a bar to distinguish it from (4). KW transformation N̄ induces the following
transformation of Pauli operatorsN̄ σzi σzi+1 |ψ〉 = σix N̄ |ψ〉 ,N̄ σix |ψ〉 = σzi−1 σzi N̄ |ψ〉 ,∀ |ψ〉 ∈ H .

(24)

and therefore leaves the Ising model with a critical transverse field invariant. One advantageof this redefinition is that it now makes sense to compute N̄ × N̄ , while before redefinition it
only makes sense to compute N × N † or N † × N . With this definition, it is straightforward to
calculate the fusion of symmetry operator N̄ :′N̄ × U = (−1) t N̄ ,′U × N̄ = (−1) t N̄ ,′N̄ × N̄ = (−1) t t (1 + (−1) t U)T ,(25)

where T is one-site translation operator
T |{si }〉 = |{si′ = si+1 }〉 .

(26)

For example, we can confirm the last identity in (25) by the following calculation

N̄ N̄ |{si }〉 =

=

=

1
2L

(−1)

′
′
j=1 (s j +s j+1 )s j +t s L+1 +

PL

′′
′′ ′
′ ′′
j=1 (s j−1 +s j )s j +t s0|{si′′ }〉{si′ },{si′′ }

L
X Y

!
′

j=1

X

L
Y

′′

δs j +s j+1 +s′′ +s′′ (−1) t (s L+1 +s0 ) |{si′′ }〉j−1{si′′ }

{si′′ }

PL

X

j

(27)
!
′

′

′′

δs j +s j+1 +s′′ +s′′ (−1) t t+t (s1 +s0 ) |{si′′ }〉
j−1

j

j=1



′
′
= (−1) t t |{si′′ = si+1 }〉 + (−1) t |{si′′ = si+1 + 1}〉 .
Under (22), the twist operator (10) is modified toN̄0t |{si }〉 =
=

1
2ℓ/2
1
2ℓ/2

X

P

′
j≤0 (s j +s j+1 )s j

P

′
′
′
j≤0 (s j−1 +s j )s j +s0 s1(−1)|{si′ }i≤0 ; {si }i>0 〉{si′ }i≤0X(−1){si′ }i≤011(28)|{si′ }i≤0 ; {si }i>0 〉 ,SciPost Phys. 15, 155 (2023)

which creates the corresponding defect at i = 0. One can directly calculate the fusion rules ofdefects. For example,

N̄0t × N̄0t |{si }〉 =

1
2ℓ

P

X

(−1)

′
j≤0 (s j +s j+1 )s j +

P

′′
′′ ′
′′
j≤0 (s j−1 +s j )s j +s0 s1|{si′′ }i≤0 ; {si }i>0 〉{si′′ }i≤0 ,{si′ }i≤0!
=Yδs j +s′′ +
j−1

Y

′′

δs j +s′′ +1 (−1)s0 s1 |{si′′ }i≤0 ; {si }i>0 〉
j−1

(29)

j≤1
′′
= (−1) |{si = si+1 }i≤0 ; {si }i>0 〉 + |{si′′ = si+1 + 1}i≤0 ; {si }i>0 〉= (σ0z + U0t )T0t |{si }〉 ,j≤1s1

where T0t is the translation twist operator at i = 0 and shifts the left half chain by one site
T0t |{si }〉 = |{si′ = si+1 }i≤0 ; {si }i>0 〉 .

(30)

We can identify σ0z T0t and T0t in the fusion rule because they are related by the local unitary
operator σ0z . In summary, we haveN̄0t × U0t = U0t × N̄0t = N̄0t ,N̄0t × N̄0t = (1 + U0t )T0t .

(31)

The symmetry defect N̄0t is movable because when acting the local operator W ′ = CZ1,2 H1 wewill get the defect N̄1t = N̄0t W ′ acting at one site right.13

2.5

Anomaly of KW duality symmetry

We end this section by commenting on the anomaly of the KW duality symmetry. The anomaly
of non-invertible symmetries in (1 + 1)d has been discussed in [55, 106] via modular transformation, and was later more systematically explored in [56, 59]. In this subsection, we review
the approach in [106] in probing the anomaly of the KW duality symmetry, as a preparation
for the anomaly of the subsystem KW symmetry.
The anomaly of the KW duality symmetry can be seen by contradiction. Suppose it isnon-anomalous, which means that there is an invertible theory (i.e. a gapped theory with
one ground state) with KW duality symmetry. The only invertible theory is a trivial theory,and in the fixed point limit, the partition function is Ztriv = 1. Performing the KW dualitytransformation amounts to gauging the Z2 symmetry, which maps Ztriv = 1 to a Z2 symmetry
breaking phase14
X
X
R
a A
a A
ZSSB [A] =(−1) aA =(−1)Wτ Wx +Wx Wτ = δ(WτA)δ(WxA) ,
(33)
a∈Z2

Wτa ,Wxa =0,1

13

The translation defect in the fusion rule of two KW operators/defects in the transverse field Ising model wasemphasized in [153], which is termed non-invertible translation symmetry. This translation operator/defect could
not be seen if one compute N † × N when we define the Hilbert space after KW on links rather than on sites. It
is interesting to note that the same translation effect has been implicitly observed in [157, Sec. VI]. The authorsof [157] discussed the critical transverse field lsing model with two defects on the closed chain. When the two
defects fuses, e.g. at the fourth site, the corresponding Hamiltonian with L spins isXX
H Dσ ×Dσ = −σzi σzi+1 +σix + σ3z σ4x σ5x .
(32)
i̸=3,4i̸=4Since σ4x commute with this Hamiltonian, the Hibert space can be decomposed into two sectors based on the parity
xof the fourth spin. The Hamiltonian in sector labeled by σ = ±1 is just the usual lsing chain with L − 1 spins
under periodic boundary condition and twisted boundary condition. The decoupling effectively reduces the lengthof the spin chain by one, i.e. from L spins to L − 1 spins, which is equivalent to adding a defect of translation.
14
We will suppress the normalization.

12

SciPost Phys. 15, 155 (2023)

a,A
where in the second equality we assume the spacetime to be a torus, and Wτ,xis the Wilsonline of a, A along the τ, x direction respectively. Clearly, a Z2 SSB theory is not equivalent
to a trivial theory. This means that the trivial theory is not compatible with the KW duality
symmetry, i.e. the KW duality symmetry is anomalous.

3

Subsystem Kramers-Wannier duality operators and defects in(2 + 1)d spin systems

In this section, we generalize the discussion in Sec. 2 to lattice models with non-anomalous
subsystem Z2 symmetries in (2 + 1)d. We also assume that the system is invariant under thegauging of the subsystem Z2 symmetry, which generalizes the KW duality symmetry to thesubsystem KW duality symmetry. We further study various properties of its associated symmetry operators and defects, including their mobility, fusion rules, interplay with boundary
conditions, and anomalies. Unlike in (1 + 1)d where the KW duality operator and KW duality
defect share essentially the same properties, here we find that the subsystem KW duality operator and subsystem KW duality defect have very different properties. This is expected since
theories with subsystem symmetry are incompatible with Lorentz symmetry.

3.1

Subsystem Z2 symmetry and twist operators

We first review the properties of a subsystem Z2 symmetry in (2+1)d, following [89]. Consider
a non-anomalous (on-site) subsystem Z2 symmetric theory on either a square lattice with
L x × L y sites or an infinite square lattice. On each site there is a spin- 12 variable si, j ∈ {0, 1}.
When the square lattice is infinitely large, the subsystem Z2 symmetry is generated by Z2
operators on each line and column
Y
Y
y
(34)
σi,x j ′ , i, j ∈ Z ,σix′ , j , Ui =U jx =j ′ ∈Zi ′ ∈Zuxywhose eigenvalues we denote as (−1) j , (−1)ui respectively. Similar to the case of an ordinary Z2 symmetry in (1 + 1)d, the subsystem Z2 symmetry operators have Z2 self fusion
rules and they are mobile/topological along the time direction. However, they are not mobile/topological in the space direction. We should label these operators by their locations on
the lattice.
On a finite lattice (i.e. a torus), there is a constraint in the product of all symmetry generators
Ly
Lx
Y
Y
y
Ui
U jx = 1 ,
(35)
i=1

j=1

since Pauli operators on each site appear twice on the left-hand side. The total number ofsymmetry generators is 2 L x +L y −1 . The action of symmetry generators on a finite lattice isshown schematically in Fig. 3. We define the subsystem Z2 symmetry group as Zsub
2 ,
*
+
Ly
Lx
Y
Y
 2

y
y 2
y
Zsub
Ui , U jx U jx = Ui=
Ui
U jx = 1, with i = 1, ..., L x , j = 1, ..., L y . (36)
2 =
i=1

j=1

We proceed to consider subsystem Z2 defects at each site along time direction. On anytinfinite square lattice, these defects can be created by acting twist operators U0,x tj and Ui,0 on
the Hilbert space, whereYY
yt
U0,x tj =σix′ , j , Ui,0 =
σi,x j ′ .
(37)i ′ ≤0j ′ ≤013

SciPost Phys. 15, 155 (2023)

y

Ui

U jx

j

j

j

i
y

defect operator U jx

symmetry operator U jx , Ui

twist operator U0,x tj

Figure 3: Examples of subsystem Z2 symmetry operators, defect operators and twist
operators.
yt

yt

Clearly, the subsystem Z2 defects satisfy the Z2 fusion rule, U0,x tj × U0,x tj = Ui,0 × Ui,0 = 1. To seexthe mobility, note that U1,x tj and U0,x tj differ by a local unitary operator, i.e. U1,x tj = U0,x tj σ1,
j , hencext
xt
the defect created by U0, j is mobile along the x direction. On the other hand, since U0, j+1 andQ
U0,x tj differ by a string (non-local) unitary operator, i.e. U0,x tj+1 = U0,x tj i≤0 σi,x j σi,x j+1 , the defectis not mobile along the y direction. Indeed, the defects created by U0,x tj+1 and U0,x tj are regarded
yt

as inequivalent subsystem Z2 defects. The same discussion applies to Ui,0 as well. We show
some examples of symmetry operators, defect operators and twist operators diagrammaticallyin Fig. 3.
If we consider a finite square lattice instead, inserting a defect changes the boundary condition. The boundary condition is specified by [89]|si+L x , j 〉 = |si, j + t xj 〉 ,yy|si+L x , j+L y 〉 = |si, j + t x y + t xj + t i 〉 , (38)|si, j+L y 〉 = |si, j + t i 〉 ,ywhere t xj , t i , t x y ∈ {0, 1}, ∀i, j label the twisted boundary conditions along the j-th row, i-th
column and at the corner respectively. We further define boundary conditions of twist variablesyyt i+L = t i + t x y .

t xj+L = t xj + t x y ,

x

(39)

There are L x + L y + 1 twist parameters but the Hamiltonian with subsystem Z2 symmetryyyydepends only on the combinations {t x 1 := t xj + t xj+1 , t 1 := t i + t i+1 } [89]. Taking accountof the constraint

j+ 2

Ly
X
j=1

tx 1 =
j+ 2

i+ 2

Lx
X
i=1

t

y
i+ 12

= txy ,

(40)

only L x + L y − 1 twist variables are distinct.
Similar to the case of (1 + 1)d, the Hilbert space will be divided into symmetry-twist sectors labeled by eigenvalues of the symmetry generators and boundary conditions. Denote theyyyux
eigenvalues of U jx , Ui as (−1) j , (−1)ui respectively, with u xj , ui ∈ {0, 1}. The constraint on
the symmetry generators (35) leads to L x + L y − 1 independent symmetry generators, dividing the entire Hilbert space (with a fixed boundary condition) to 2 L x +L y −1 sectors. Similarly,ythe L x + L y − 1 distinct twist variables (t x 1 , t 1 ) further divide each symmetry sector into
L x +L y −1

2parts. In total, there are 4yy(u xj , ui , t x 1 , t 1 ).
j+ 2

3.2

j+ 2
L x +L y −1

i+ 2

distinguished symmetry-twist sectors labeled byi+ 2

Subsystem KW duality operators

In this subsection, we generalize the KW transformation in (1 + 1)d to the subsystem KW transformation, defined by gauging the entire subsystem Z2 symmetry in (2 + 1)d. We also assume
the theory to be invariant under the subsystem KW transformation. We will demonstrate the14

SciPost Phys. 15, 155 (2023)

construction of the codimension one surface duality operator N sub acting on the Hilbert space
implementing the subsystem KW transformation and derive its fusion rule. Additional technical details about turning on background gauge fields are included in App. A.
Dual lattice: Subsystem KW transformation maps the original square lattice with spin {si, j }to the dual lattice with spin {bsi+ 1 , j+ 1 }. Each spin b
si+ 1 , j+ 1 ∈ {0, 1} lives on the plaquette with2
2
2
2
boundary conditions|b
si+ 1 +L x , j+ 1 〉 = |b
si+ 1 , j+ 1 + bt x 1 〉 ,
2

2

2

j+ 2
y

2

|b
si+ 1 , j+ 1 +L y 〉 = |b
si+ 1 , j+ 1 + bt
2

2

2

i+ 12

2

〉,

(41)

si+ 1 , j+ 1 + bt x y + bt x 1 + bt|b
si+ 1 +L x , j+ 1 +L y 〉 = |b
2

2

2

i+ 12

j+ 2

2

y

〉,

where
bt x 1j+ 2 +L y

bt y 1= bt x 1 + bt x y ,i+ 2 +L x

j+ 2

= bt

y

i+ 12

bt x 1 , bt y 1 ,+ bt x y ,j+ 2

bt x y ∈ {0, 1} ,i+ 2

∀ i, j . (42)

We use combined boundary conditions
bt y := bt y 1 + bt y 1 .
ibt xj := bt x 1 + bt x 1 ,
j+ 2

j− 2

(43)

i+ 2

i− 2

Among them, there are only L x + L y − 1 independent variables.
Let the Pauli operator τz 1 1 , τ x 1 1 act on the dual lattice. The new subsystem Z2
symmetry is generated bybx 1 =
U
j+ 2
with the constraint

QL y
j=1

bx 1
u

i+ 2 , j+ 2

i+ 2 , j+ 2

bx 1
U

Lx
Y
i=1

QLx

j+ 2
y
b 1
u

by

τx 1 1 ,i+ 2 , j+ 2

U

i+ 12

=

Ly
Y

τx 1

j=1

i+ 2 , j+ 12

,

(44)

b y 1 = 1. Denoting the eigenvalues of the symmetry
U

i=1

i+ 2

generators as (−1) j+ 2 , (−1) i+ 2 = ±1, the Hilbert space on the dual lattice is divided intoyyb 1 ,bt xj ,bti ).
2 L x +L y −1 × 2 L x +L y −1 symmetry-twist sectors labeled by (bux 1 , u
j+ 2

i+ 2

Subsystem KW transformation: We use short-hand notations |{si, j }〉 and |{bsi+ 1 , j+ 1 }〉 for22basis states. Since the action of a symmetry generator in (34) only changes the spin along aline or a column, we use |{si, j }; {si,′ j ′ }〉 for action on j ′ -th line or |{si, j }; {si′′ , j }〉 for action oni ′ -th column. We adopt the same notation for action on the dual Hilbert space. For instance,y bx
b y 1 act as
the symmetry generators U jx , Ui , U
1,U
j+ 2

U jx′ |{si, j }〉 = |{si, j }; {1 − si, j ′ }〉 ,i+ 2

b x 1 |{bsi+ 1 , j+ 1 }〉 = |{bsi+ 1 , j+ 1 }; {1 − bsi+ 1 , j ′ + 1 }〉 .
U
′
j +2

2

2

2

2

2

2

(45)

Compared with (1 + 1)d, we define the action of subsystem KW transformation N sub on theentire Hilbert space by attaching a phase to the dual basis state and summing over all thedual basis states. The exponent of the phase contains the term reminiscent of the minimal
coupling of a subsystem gauge field on the plaquette as well as the boundary term that gives
the following mapping between symmetry-twist sectors
bx 1 = t x 1 ,
u
j+ 2

j+ 2

b
u

y

i+ 21

=t

y
i+ 21

,

as shown in Fig. 4.
15

bt xj = u xj ,bt y = u y ,
i
i

(46)

SciPost Phys. 15, 155 (2023)

y

bt y
i−1

t

y

b
u

ui−1

i+ 12
y
i+ 21

j+1

bx 1 t x 1
u
j+ 2

j

j+ 2

u xj−1 bt xj−1

j−1
i−1

i+1

i

Figure 4: Mapping of symmetry-twist sectors.
Concretely, the subsystem KW duality operator is defined as

N sub |{si, j }〉 =1

X

2(L x +L y )/2 {bs 1(−1)Cbulk +Cbdy |{bsi+ 1 , j+ 1 }〉 ,2}
i+ 2 , j+ 1
2

2

(47)

where in the exponent Cbulk is the bulk minimal coupling between the original spins on sites
and dual spins on the plaquette, and Cbdy is the boundary coupling ensuring the correct
symmetry-twist sector transformation (46),
Ly
Lx X
X
(si−1, j−1 + si, j−1 + si−1, j + si, j )bsi− 1 , j− 1 ,Cbulk :=
2

i=1 j=1

Cbdy :=

LyX
j=1

bt x 1 (s L x , j + s L x , j−1 ) +
j− 2

Lx
X

2

(48)
bt

i=1

y

i− 12

(si,L y + si−1,L y ) + bt x y s L x ,L y ,where the spin with index 0 equal to the one with L x /L y shifted by proper boundary condition,ye.g. s0,0 = s L x ,L y + t x y + t Lx + t L . As in (1 + 1)d, we adopt the same phase for the definitionxyÒ. The subsystemof the subsystem KW transformation N sub acting on the dual Hilbert space H
KW transformation N sub is Hermitian.

Mapping between Pauli operators:

From (47) and (48), the Pauli operators transform as

bx 1
N sub σzi, j σzi+1, j σzi, j+1 σzi+1, j+1 |ψ〉 = σi+ 2 , j+ 12

bz 1
N sub σi,x j |ψ〉 = σi− 2 , j− 21bz 1
σi− 2 , j+ 21

bz 1
σi+ 2 , j− 12

N sub |ψ〉 ,bz 1
σi+ 2 , j+ 12∀ |ψ〉 ∈ H ,N sub |ψ〉 ,∀ |ψ〉 ∈ H ,(49)which is a natural generalization of the ordinary KW transformation in (1 + 1)d.
Fusion rules: Fusion rules can be derived straightforwardly by acting the P
symmetry operay bx
ysubb
tors U jx , Ui , U
,
U
and
the
dualityoperatorNonageneral
state
|ψ〉=
1
1
{si, j }ψ{si, j } |{si, j }〉,
j+ 2

i+ 2

where ψ{si, j } = 〈{si, j }|ψ〉 is the wavefunction coefficient. For example, acting N sub × U jx′ on|ψ〉XN sub × U jx′ |ψ〉 =ψ{si, j };{1−si, j′ } N sub |{si, j }〉 ,(50)
{si, j },{b
si }

16

SciPost Phys. 15, 155 (2023)

where we use redefinition of spin si, j ′ → 1 − si, j ′ in the j ′ -th line. Then we perform the subsystem KW transformation and redefine spins to send ψ{si, j };{1−si, j′ } back to ψ{si, j } . Under redefinition, the changes in the bulk minimal coupling terms cancel because spins with index j ′always appear in pairs. The only contribution is from the boundary term
bt x′ 1 (s L x , j ′ + s L x , j ′ −1 ) + bt x′ 1 (s L x , j ′ +1 + s L x , j ′ )
j +2

j −2

(51)

→ bt x′ 1 (1 − s L x , j ′ + s L x , j ′ −1 ) + bt x′ 1 (s L x , j ′ +1 + 1 − s L x , j ′ ) ,
j −2
j +2
bt x

′

bx
1 + t j′ + 1
2

leading to an extra factor (−1) j − 2

bt x= (−1) j′ . Therefore, the fusion rule isbt x

N sub × U jx′ = (−1) j′ N sub .

(52)

By a similar argument, one can work out fusions with other symmetry generators. Here we
only list the results:
tx

t

y

y

b x 1 × N sub = (−1) j′ + 12 N sub , N sub × U y′ = (−1)bti′ N sub , U
b y 1 × N sub = (−1) i′ + 12 N sub . (53)
U
′
′
i
j +2

i +2

For self fusion of the duality operator, by acting (N sub )† × N sub on a general state |ψ〉 we
get two copies of the phase
Ly
Lx X
X
′
′
′
′
si− 1 , j− 1
(si−1, j−1 + si, j−1 + si−1, j + si, j + si−1,
j−1 + si, j−1 + si−1, j + si, j )b2

i=1 j=1

+

Ly
X
j=1
bx y

+t

Lx
X

bt x 1 (s L x , j + s L x , j−1 + s′L , j + s′L , j−1 ) +
j− 2

x

x

i=1

2

′
′
bt y 1 (si,L y + si−1,L y + si,L)+ si−1,L
i− 2

(54)

y

y

(s L x ,L y + s′L ,L ) .
xy

We then sum over states in the dual Hilbert space to get a product of delta function constraints
on states in the original Hilbert space H′′′′
si−1, j−1 + si, j−1 + si−1, j + si, j + si−1,
j−1 + si, j−1 + si−1, j + si, j = 0 ,∀ i, j .

(55)

The general solutions of the constraints areyysi,′ j = si, j + mi + m xj ,mi , m xj ∈ {0, 1} ,y∀ i, j ,(56)

ywhere mi , m xj having value 1 means one insertion of Ui , U jx respectively. We should then sum
over all distinct solutions,15
¬
¶
y
y
y
y
M = (mi , m xj ) mi , m xj ∈ {0, 1}, (mi , m xj ) ≃ (mi + 1, m xj + 1) ,∀i, j .

(57)

The final fusion rule is
(N

sub †

) ×N

sub

=

X

PL y

(−1)

bt x m x +
j
j=1 j

y
(mi ,m xj )∈M

PL x

bt y m y
i
i=1 i

Lx
Y
i=1

15

y

y m
Ui i

Ly
Y


U jx

m x
j

,

(58)

j=1

The global constraint can be seen as follows. Each configuration of m’s specifies a profile of operators U’s.
Since the product of all U operators is a trivial operator as shown in (35), and the product of all U corresponds toyy
all m’s being 1, we should identify (mi , m xj ) ≃ (mi + 1, m xj + 1) for all i, j.

17

SciPost Phys. 15, 155 (2023)

N sub

t
y

=

x

Grid{bt y ,bt x }

N sub

i

j

Figure 5: Fusion between two subsystem KW duality operators gives rise to a grid
operator, where the grid is along the space direction.
generalizing the ordinary Ising fusion rule. The right-hand side is a sum of the generators of
Zsub
2 , reminiscent of the condensation defect that appears in the fusion rule of duality defectsyin (3 + 1)d gauge theories [106, 107, 110, 114, 116, 118]. As the operators U jx and Ui haverestricted mobility, and they form grids, we denote the right-hand side as the grid operator,

Gridy{bti ,bt xj }

PL y

X

=

(−1)

bt x m x +
j
j=1 j

PL x

bt y m y
i=1 i
i

y

(mi ,m xj )∈M

Lx
Y

y

y m
Ui i

Ly
Y


i=1

U jx

m x
j

(59)

j=1

Ly

Lx


Y
1 Y
bt x
by y
=
1 + (−1)ti Ui
1 + (−1) j U jx ,
2 i=1
j=1

(60)

(N sub )† × N sub = Grid{bt y ,bt x } .

(61)

so thatij

We also present this fusion rule in Fig. 5.
To the authors’ knowledge, the subsystem KW duality operator N sub is the first explicit
example of a non-invertible operator in models with subsystem symmetry, which generates a
non-invertible subsystem KW duality symmetry.
Example: A canonical example with subsystem KW duality is the plaquette Ising model withthe critical transverse field in (2 + 1)d:
HPIsing = −

Ly
Lx X
X

σzi, j σzi+1, j σzi, j+1 σzi+1, j+1 −

Ly
Lx X
X

σi,x j .

(62)

i=1 j=1i=1 j=1

This model can describe a square array of superconductor grains with frustrating geometricphases and the states with eigenvalue σz = ±1 are associated with p ± i p order of each grain[171, 172]. The first term of Hamiltonian represents the phase acquired by a Cooper pair in
the process of encircling a plaquette, while the second term arises from tunneling between thep ± i p order parameters. Applying the subsystem KW transformation, one obtains the dual
HamiltonianÒPIsing = −
H

Ly
Lx X
X
i=1 j=1

bx 1 1 −σi+ 2 , j+ 2

Ly
Lx X
X

bz 1
σ

i=1 j=1

i− 2 , j− 12bz 1
σi+ 2 , j− 12bz 1
σi− 2 , j+ 12bz 1
σi+ 2 , j+ 12

.

(63)

After relabeling the spins on the sites and on the plaquettes, the two Hamiltonians coincide,ÒPIsing , hence the critical Plaquette Ising model has a subsystem KW duality symmetry.
HPIsing = H
We will also formulate this symmetry by performing the subsystem KW transformation on thesame lattice, as shown in Sec. 3.4.
18

SciPost Phys. 15, 155 (2023)

Moreover, the plaquette Ising model (62), as a (2 + 1)d quantum spin model, has a corresponding 3d anisotropic classical model on the cube lattice [172, 173]:X
X(64)s⃗j s⃗j+⃗ez ,−β E = −Ks⃗j s⃗j+⃗e x s⃗j+⃗e y s⃗j+⃗e x +⃗e y − Jz⃗j⃗jwhere s = ±1 is put on each vertex. The first term is between four spins of all plaquettes in thex y planes and the second term is between two spins over all bonds in the z direction. In thisclassical model, the subsystem KW transformation represents a duality between the partition
function of high temperature and low temperature. Numerical calculations indicate that thephase transition at the self-dual point is first order [174].

3.3

Subsystem KW duality defects

We switch to discussing the defects associated with the subsystem KW duality symmetry byplacing one direction of the duality operator along the time. These defects are 2d surfaces
with one direction along the time, and another direction along the space. For simplicity, we
work on the infinite 2d square lattice. After defining the subsystem KW duality defect via
the subsystem KW duality twist operator, we discuss the fusion rules as well as the mobility
under translation along the space direction. For simpler conventions, we will adopt the same
notation for the twist operator and the corresponding duality defect.
Subsystem KW duality defect from subsystem KW duality twist operator: On the infinite
2d lattice, consider a subsystem KW duality defect localized at a line S that divides the space
into two halves S+ and S− . Insertion of such a defect is realized by a twist operator NSsub thatacts on the half-space S− and terminates at S. The twist operator NSsub is defined asX
half
1
NSsub |{si, j }〉 = (ℓ )/2(65)(−1)CS |{bsi− 1 , j− 1 }S− ; {si, j }S+ 〉 ,22
2 S− {bs 1 1 }i− 2 , j− 2 S−

where
CShalf :=X(si−1, j−1 + si, j−1 + si−1, j + si, j )bsi− 1 , j− 1 .
2

S−

(66)

2

In both equations, the summation is over the half-space S− , and ℓS− is a formal parametercounting the number of sites covered by S− . The action of the Hermitian conjugate of thetwist operator is defined asXhalf1
si− 1 , j− 1 }S− ; {si, j }S+ 〉 = (ℓ )/2(−1)CS |{si, j }〉 .
(67)
(NSsub )† |{b22S2 − {s }i, j S−In Fig. 6, we give several typical examples of subsystem KW duality defects.
Fusion rule: We can derive the fusion rules of the subsystem KW duality defects by acting
the corresponding twist operators on the same region. Here we list fusion rules of the twist
operators shown in Fig. 6,
X Y
m x
y Y
j
y m
(N sub1 )† × N sub1 =Ui i
U0,x tj
,
i= 2

i= 2

y

mi ,m xj =0,1 i≤0XY

mi ,m xj =0,1

i

X

Y

(N sub1 )† × N sub1 =
j= 2

j= 2

(N⋆sub )† × N⋆sub =

j

y

yt

U0,i



y
mi

Y

U jx

m x
j

,

(68)

j≤0
y

y  miUiymi ,m xj =0,1 i≤0



y

 Y
y t m1

U0,1

j>0

19

U0,x tjm x Y jj≤0U1,x tjm xj,

SciPost Phys. 15, 155 (2023)

S

S−

S+

S

S−

S+
2

S+

1
S
0
S−
− 23

− 12

1
2

1

− 23

2

(a) Defect localized at j ≥ 12 .

(b) Defect localized at j ≥ 12 .

− 12

1
2

3
2

2

3

−1

(c) Defect localized at ⋆ : i = 12 , j ≥ 12 ,j = 12 , 12 ≤ i ≤ 23 , and i = 23 , j ≤ 12 .

Figure 6: Three examples of subsystem KW duality defects. An action of a twist
operator on the half-space S− amounts to an insertion of a duality defect localized atline S.
where twist operators N sub1 , N sub1 and N⋆sub correspond to defects located at i = 12 , j = 12 and
i= 2

j= 2

⋆ respectively.
Let us interpret the first fusion rule in (68). On the right-hand side, each term in theysum is a product of symmetry operators Ui and twist operators U0,x tj . Note that appendingya symmetry operator, e.g. Ui , to a twist operator does not change the twisted Hamiltonian(since the symmetry operator commutes with the Hamiltonian). Hence the twist operator on
the right-hand side of (68) should be defined up to symmetry operators. Hence (68) simplifies
to
X Y
m x
j
,
U0,x tj(N sub1 )† × N sub1 =
i= 2

i= 2

m xj =0,1(N sub1 )† × N sub1 =
j= 2

j= 2

y
mi =0,1(N⋆sub )† × N⋆sub =

j

X Y

y


y t mi

U0,i

,

i

(69)

y

m x
m x Y  Yjjy t m1U1,x tj.
U0,x tj
U0,1
y
j≤0
j>0
m1 ,m xj =0,1
XThe right-hand side is again a grid defect, but the grid is only along the time direction. See
Fig. 7 for the first fusion rule in (69). The fusion rules of subsystem KW duality defects in(69) are different from those of the subsystem KW duality operators in (58). That there aredifferences between space-like operators and time-like defects is a typical feature of subsystemsymmetries, which are inherently non-relativistic.
Next, let us consider the fusion rules of the Z2 defects and the subsystem KW duality
defects. To fuse them, the Z2 defect should be inserted in the surface where the duality defectresides. In other words, the corresponding Z2 twist operator should terminate on S whensubsystem KW twist operator NSsub is considered. For simplicity, consider the fusion of N sub1 ,i= 2

b x t1
U0,x tj and U

1 . The fusion rules are− 2 , j− 2

N sub1 × U0,x tj = N sub1 ,
i= 2

i= 2

b x t1 1 × N sub1 = N sub1 σzUσz ∼ N sub1 .
− 2 , j− 2i= 2i= 2 0, j−1 0, j
i= 2

(70)

Here we identify two defects related by a local unitary operator in the same equivalent class.
Therefore, we find the same fusion rules as (52) and (53).
20

SciPost Phys. 15, 155 (2023)

t
=

y
x
(N sub1 )†i= 2
N sub1

P

i= 2m xj =0,1

Q 
jU0,x tjm x
jFigure 7: Fusion between two subsystem KW duality defects gives rise to a grid defect,
where the grid is along the time direction.
Mobility of subsystem KW duality defects: We proceed to examine the mobility of thesubsystem KW duality defects. Intuitively, since the subsystem symmetry is not mobile under ageneric translation along the space, so does the subsystem KW duality defect. However, below
we will find that this is not true. We will demonstrate the mobility by studying the example inFig. 8, where the deformation happens only around the origin.
We find a local unitary operator W relating the two duality twist operators= NSsub W ,
NSsub
′

(71)

W = CZ(0,0),(1,1) CZ(1,0),(1,1) CZ(0,1),(1,1) H(1,1) .

(72)

where
We adopt the same convention as in the previous section, i.e. CZ(i, j),(i ′ , j ′ ) is the control Z gate
acting on sites (i, j) and (i ′ , j ′ ), and H(1,1) is the Hadamard gate acting on site (1, 1). (71) can
S

S−

− 32

− 21

1
2

S−′

S+

3
2

2

3

S+′

S′

2

2

1

1

0

0

−1

− 32

(a) Defect localized at S : i = 12 , j ≥ 12 ,j = 21 , 12 ≤ i ≤ 23 and i = 32 , j ≤ 12

− 12

1
2

3
2

2

3

−1

(b) Defect localized at S ′ : i = 21 , j ≥ 23 ,j = 12 , 12 ≤ i ≤ 23 and i = 32 , j ≤ 32

Figure 8: Duality defects located on S and S ′ . We show that one can be moved tothe other by a local unitary operator.

21

SciPost Phys. 15, 155 (2023)

be checked directly′
1 X′
NSsub W |{si, j }〉 = NSsub p(−1)(s0,0 +s0,1 +s1,0 +s1,1 )s1,1 |{si, j }S− ; s1,1; {si, j }S+′ 〉
2 s′
1,1
X
half
′
1
′
= (ℓ +1)/2(−1)CS +(s0,0 +s0,1 +s1,0 +s1,1 )s1,1 |{b; {si, j }S+′ 〉si− 1 , j− 1 }S− ; s1,1
2
2
2 S−
{b
s 1 1 } ,s′i− 2 , j− 2 S−

′
s1,1
→b
s1,1
2 2

1

=

2

half

X

(ℓS ′ )/2−

1,1|{si, j }〉 .
(−1)CS′ |{bsi− 1 , j− 1 }S−′ ; {si, j }S+′ 〉 = NSsub′2{bsi− 1 , j− 1 }S ′
2

2

2

−

(73)
This shows that the two defects NSsuband NSsub are equivalent, hence the defect is mobile′
under a spatial deformation. As commented at the end of Sec. 2.3, this does not prove that thesubsystem KW generator is topological under arbitrary spacetime deformation. We will not try
to prove this in this work.

3.4

Subsystem KW duality symmetry

We briefly comment on the subsystem KW transformation with a single Hilbert space. Similar
to the (1 + 1)d transformation, we will define a new transformation N̄ sub by redefining theplaquette spins to sites,b
si+ 1 , j+ 1 → si,′ j .
(74)
2

2

Concretely, (47) becomesN̄ sub |{si, j }〉 =1

X

2(L x +L y )/2 {s′ }(−1)Cbulk +Cbdy |{si,′ j }〉 ,

(75)

i, j

where the bulk and boundary terms are
Ly
Lx X
X
Cbulk :=
(si, j + si, j+1 + si+1, j + si+1, j+1 )si,′ j ,i=1 j=1

Cbdy :=LyX

t ′x
j s L x +1, j +

j=1

Lx
X

(76)
′y
t i si,L y +1 + t ′x y s L x +1,L y +1 .

i=1N̄ sub is a symmetry of the critical plaquette Ising model because it exchanges the plaquette-σz
terms and the transverse-σ x termN̄ sub σzi, j σzi+1, j σzi, j+1 σzi+1, j+1 |ψ〉 = σi,x j N̄ sub |ψ〉 ,N̄

sub

σi,x j |ψ〉 = σzi, j σzi−1, j σzi, j−1 σzi−1, j−1 N̄ sub |ψ〉 ,∀ |ψ〉 ∈ H ,∀ |ψ〉 ∈ H .

(77)

In the untwisted sector, the nontrivial fusion rules is

N̄

sub

× N̄

sub

=

Lx
Y

y
1 + Ui

i=1

Ly
Y



1 + U jx Tx y ,

(78)

j=1

where Tx y implements the lattice translation in the diagonal direction
Tx y |{si, j }〉 = |{si,′ j = si+1, j+1 }〉 .
22

(79)

SciPost Phys. 15, 155 (2023)

In a similar way, the symmetry defects are created by acting the twist operators

N̄Ssub |{si, j }〉 =1
2

X(ℓS− )/2′
S− (si, j +si+1, j +si, j+1 +si+1, j+1 )si, j

P

(−1)

|{si,′ j }S− ; {si, j }S+ 〉 ,{si,′ j }S−(80)defined on the half space. The fusion rules and mobility of these defects follows in the similar
discussion as Sec. 3.3.

3.5

Anomaly of subsystem KW duality symmetry

We conclude this section by generalizing the approach in Sec. 2.5 to show that the noninvertible subsystem KW duality symmetry is anomalous.
We proceed to prove the anomaly by contradiction. Assume that the subsystem Z2 symmetry is anomaly free, hence it is compatible with a gapped phase with one ground state, i.e.
subsystem symmetry protected topological phase. Such a phase has been classified in [159],
which is given by H 2 (Z2 × Z2 , U(1))/(H 2 (Z2 , U(1)))3 = Z2 .16 Hence there is one trivial phaseand one non-trivial phase. The trivial phase has partition function Ztriv = 1, and the non-trivial
phase has the partition function [161]Z
R

Dφ x y (−1) φ

ZSSPT [Aτ , A x y ] =xy(∂τ A x y −∂ x ∂ y Aτ )+Aτ A x y

,

(81)

where φ x y ∈ 0, 1 is an auxiliary field, integrating which constraints the flatness conditionof the background field Aτ , A x y of the subsystem Z2 gauge field. See Appendix A or [89] forfurther discussions on the relation between the subsystem Z2 symmetry operators (34) and thesubsystem Z2 symmetry background fields on the square lattice. The Lagrangian is invariant
under gauge transformation φ x y → φ x y − α, Aτ → Aτ + ∂τ α, A x y → A x y + ∂ x ∂ y α.
Let us check whether Ztriv and ZSSPT are separately invariant under the subsystem KWtransformation, i.e. the gauging of the subsystem Z2 symmetry. For Ztriv , gauging subsystem
Z2 symmetry gives a spontaneous subsystem symmetry broken (SSSB) phase, whose partition
function is17
Z
R
ZSSSB ==

D a (−1) aτ A x y +a x y Aτ

X

PL x

(−1)

i=1



A
Wτ,a y;i W y;i
+W A

Wa 1
τ, y;i+ 1
y;i+ 2
2



+

PL y
j=1



a
A
A
Wτ,x;
j Wx; j +Wa
1 Wx; j+ 1
τ,x; j+ 2
2

(82)



,

conf. of awhere we regulate the integration of the dynamical gauge field on a finite cubic lattice with
L x × L y sites and suppressed the overall normalization. The sum runs over all configurationsof gauge field a, which can be converted to the summation over holonomy variables. However,there are only 2(L x + L y − 1) independent holonomy variables out of 2(L x + L y ) variables
Wτ,a y;i , W ay;i+ 12aa, Wτ,x;j, Wx; j+ 21

16

∈ {0, 1} .

(83)

The classification is only for strong SSPT phases, while there are also weak SSPT phases. The weak SSPT
yphases only made use of a subset of symmetry generators, say {U jx } but not {Ui }, hence the background fields
are different. In [161], the weak SSPT with subsystem Z2 symmetry was discussed, whose field theory is A t ∂ x A y .
Below, we would like to discuss the anomaly of the entire subsystem Z2 symmetry whose background fields are
A t , A x y , thus we will only discuss the SSPT whose field theory is in terms of A t , A x y . It turns out that the only SSPT
with such a background field is the strong SSPT.
17
Again we will suppress the normalization factor.

23

SciPost Phys. 15, 155 (2023)

To pick the independent variables, recall the constraint on the space holonomy variables
Lx
X

Wa

1 +

Ly
X

y;i+ 2

i=1

Wa

1 =

Lx
X

x; j+ 2

j=1

A
W y;i
+

Ly
X

i=1

A
Wx;
j = 0,(84)

j=1

which also shows that the partition function is invariant under a global unit shift of the timeholonomy variables. With the above redundancy, we can choose the following gauge fixing
conditionWτ,a y;1 = W A= 0,τ, y; 23

Wa 3 =
y; 2
A
W y;1
=

Lx
X

Wa 1 +
y;i+ 2

i=2
Lx
X

A
W y;i
+

i=2

Ly
X

Wa

x; j+ 12

j=1

Ly
X

,

(85)

A
Wx;
j.

j=1

Then we can evaluate the partition function (82)ZSSSB =



PL x

X

(−1)

i=2

A
Wτ,a y;i W y;i
+W A

a
1W



+

y;i+ 12

τ, y;i+ 2



PL y
j=1

a
A
A
Wτ,x;
j Wx; j +Wa1W

τ,x; j+ 21
x; j+ 2

conf. of a

=

Lx
Y
i=2

A
)
δ(W A
)δ(W y;iτ, y;i+ 21

(86)

Ly

Y
δ WA
j=1

 

A
δ Wx;
1
j .

τ,x; j+ 2

Indeed, the ground state degeneracy is extensive and spontaneously breaks the subsystem Z2
symmetry. This means that the trivial phase is not invariant under gauging.
For ZSSPT , gauging subsystem Z2 symmetry yields a partially spontaneous subsystem symmetry broken (PSSSB) phase, whose partition function isZ
ZRZPSSSB ==Dφ x y (−1) φ

Da

Z

xy

(∂τ a x y −∂ x ∂ y aτ )+aτ a x y +aτ A x y +a x y Aτ(87)R

D a (−1)aτ a x y +aτ A x y +a x y Aτ,

where we first integrated out the auxiliary field φ x y enforcing the flatness condition∂τ a x y − ∂ x ∂ y aτ = 0. PSSSB means the ground state degeneracy is not extensive and only
part of the subsystem symmetry is broken.
To see this, let us integrate out aτ , a x y in (87). Naively, one would attempt to simply
integrate out aτ which enforces a x y = A x y . Such a naive integration, however, is obstructed
due to subtle global constraints originating from the flatness condition for the gauge field.
Below, we evaluate the integration carefully by first converting the integration of gauge fields
in terms of summation over holonomy variables. Concretely,
ZPSSSB =

PL x

X

(−1)

i=1

Wa

1
y;i+ 2aa(Wτ;y,i +Wτ, y;i+1 )+

PL y
j=1

a
a
1 (Wτ,x; j +Wτ,x; j+1 )x; j+ 2Wa

{conf. of a}

(88)

PL x

× (−1)

Aa
Wτ,a y;i W y;i+W A
1 W y;i+ 1
i=1τ, y;i+ 2
2

24



+

PL y



a
A
A
a
Wτ,x;
j Wx; j +Wτ,x; j+ 1 Wx; j+ 1
j=1
2
2



.

SciPost Phys. 15, 155 (2023)

Using the gauge fixing condition (85) and summing over the spatial holonomy variables of the
dynamical gauge field W a 1 , W a 1 for i = 2, ..., L x and j = 1, ..., L y , we will gety;i+ 2x; j+ 2PL x

X

ZPSSSB =Wa

δ̃A,a (−1)

AWτ,a y;i W y;i
+

PL y
j=1

a
A
Wτ,x;
j Wx; j,(89)

1 =0,1a
1 ,Wτ, y;i+ 2

i=2

τ,x; j+ 2

where the delta constraint δ̃A,a includesWA= Wτ,a y;2 + Wτ,a y;i + Wτ,a y;i+1 ,WA= Wτ,a y;2 + Wτ,a y;L ,τ, y;i+ 21

τ, y;L x + 12WAτ,x; j+ 12

i = 2, ..., L x − 1 ,
(90)

x

a
a
= Wτ,a y;2 + Wτ,x;
j + Wτ,x; j+1 ,j = 1, ..., L y .

The first two lines are constraints on the y direction and the last line is on the x direction.
Further summing over the delta constraints on y and x directions separately leads to
Lx
X
i=2
Ly

X
j=1

WA

τ, y;i+ 21= L x Wτ,a y;2 ,
(91)

WA

τ,x; j+ 12= L y Wτ,a y;2 .

This means that the background field A obey global constraints, depending on the parity of L x
and L y :PLx
PL y
1. (L x , L y ) = (even, even): two independent constraints i=2
WA
WA
1 =
1 = 0,j=1
τ, y;i+ 2

leading to four-fold ground state degeneracy.
2. (L x , L y ) = (even, odd): one independent constraint

PLx

WA

= 0, leading to two-WA= 0, leading to two-

WA

+

i=2

fold ground state degeneracy.
3. (L x , L y ) = (odd, even): one independent constraint

PL y
j=1

fold ground state degeneracy.
4. (L x , L y ) = (odd, odd): one independent constraint
leading to two-fold ground state degeneracy.

PLx
i=2

τ,x; j+ 2

τ, y;i+ 21

τ,x; j+ 21

τ, y;i+ 21

PL y
j=1

WA

τ,x; j+ 21= 0,

For all cases, the theory after subsystem KW transformation spontaneously breaks part of thesymmetry, and therefore ZSSPT is not invariant under gauging subsystem Z2 symmetry.18
Because both Ztriv and ZSSPT are not compatible with subsystem KW duality symmetry, we
can conclude that this symmetry is anomalous. We remark that this anomaly protects thephase transition of the plaquette Ising model with the critical transverse field in (62).

4Discussion and future directions

In this paper, we gave the first example of a subsystem non-invertible symmetry — the subsystem KW duality symmetry — in lattice models in (2 + 1)d, thereby filling in one missing
18
The feature where the gauged strong
P SSPT has 4 or 2 GSD depending on the parity of (L x , L y ) can also beseen for the lattice model HSSPT = − i, j σi,x j σzi, j+1 σzi+1, j σzi+1, j+1 σzi, j−1 σzi−1, j σzi−1, j−1 . We thank Trithep Devakul forpointing out this exactly solvable model.

25

SciPost Phys. 15, 155 (2023)

Table 2: Comparison between the KW duality symmetry in (1+1)d and the subsystem
KW duality symmetry in (2 + 1)d.

dimension
operationfusion rules of operators
fusion rules of defects
invertible defect mobility
duality defect mobility
anomaly of duality symmetry

duality symmetry(1 + 1)dgauging Z2(6)(15)
mobilemobileanomalous

subsystem duality symmetry(2 + 1)dgauging subsystem Z2(52), (53), (61)(69)
restricted mobile
mobile along spatial translation
anomalouscorner of generalized symmetries in Tab. 1. The discussion of subsystem duality symmetry in(2 + 1)d is a direct generalization of the ordinary duality symmetry in (1 + 1)d. We listed thecomparison in Tab. 2. Here we comment on several future directions.
To have a generic understanding of subsystem non-invertible symmetry, one can employ
similar analysis to models with general abelian subsystem symmetry [83, 175] or theories
with global dipole symmetry [87, 91]. Another interesting direction is to find the last missing
corner in Tab. 1, a higher subsystem non-invertible symmetry, by studying duality operatorsand defects from gauging higher-form subsystem symmetry [85, 90, 142–144, 146, 148, 176–
178].
As a duality transformation, the subsystem KW transformation provides a new method
to study SSPT phases. For example, we can perform a parallel analysis for (2 + 1)d manybody systems with Z2 × Z2 subsystem symmetries where subsystem KW transformation N sub
amounts to gauging the Z2 × Z2 subsystem symmetries. Moreover, there exists a unitary decorated domain wall (DW) transformation UDW relating Z2 × Z2 strong SSPT phase and triviallygapped phase19 in (2 + 1)d [79, 179]. Similar to the cases in (1 + 1)d [154, 180, 181], one candefine the subsystem Kennedy-Tasaki (KT) transformation

NKT = N sub UDW N sub ,(92)

which relates the SSPT to SSSB phase, while leaving the trivial gapped phase invariant. See
Fig. 9 for a summary. The KT duality transformation thus offers a hidden symmetry-breaking
interpretation for strong SSPT phase [182–184]. It is interesting to explore the explicit expression of this duality transformation and its applications to gapped and gapless SSPTs [181,185–188].
In [89], we discussed the subsystem Jordan-Wigner (JW) transformation. It is known[169, 189, 190] that in (1 + 1)d, the JW transformation, the KW transformation, and stacking
a fermionic SPT, i.e. the Arf invariant, satisfy the following relation(Stack fermionic SPT) · JW = JW · KW .

(93)

In (2 + 1)d systems with subsystem Z2 symmetries, besides the subsystem JW transformation,
subsystem KW transformation, and stacking a subsystem fermionic SPT (i.e. the subsystem
Arf invariant in [89, Sec. 4]), we can also stack a bosonic subsystem Z2 strong and weakSPT [158, 161, 191, 192], see also the discussion in Sec. 3.5. Hence these four operators
generate a more complicated diagram. It would be interesting to explore the relation between
them in the future.
Finally, it has been realized recently [164,193] that gaugings and measurements can be implemented in the quantum circuits to prepare interesting topological states, and has potential
19

Here we consider the product state, which excludes the weak SSPT phase.

26

SciPost Phys. 15, 155 (2023)

KT

Trivial

KW

DW

SSSB

DW

KT

SSPT

KW

Figure 9: Three gapped phases with Z2 × Z2 subsystem symmetry and the dualities
between them. Note that the KW duality transformation in the figure applies to theentire subsystem Z2 × Z2 symmetry.
application in fault-tolerant quantum computation. It is also widely appreciated [194–199]
that quantum codes with subsystem symmetries behave better in protecting quantum information from errors. It would be interesting to investigate whether the subsystem KW duality
operators can be implemented in the quantum circuits and to search for possible applicationsin quantum computation.

Acknowledgements
W.C. thanks Kantaro Ohmori for his wonderful lectures on non-invertible symmetry in the
Kavli Winter Asian School in IBS, Korea and his insights during the discussion. We also thank
Trithep Devakul, Jie Wang, Satoshi Yamaguchi, Shi Chen, and Han Yan for helpful discussions,and Masaki Oshikawa for the collaboration on a related work [154] and discussions.
The authors of this paper were ordered alphabetically.
Funding information This work is partially supported by World Premier International Research Center Initiative (WPI) Initiative, MEXT, Japan at Kavli IPMU, the University of Tokyo.
W.C. and L.L. are supported by the Global Science Graduate Course (GSGC) program of the
University of Tokyo. W.C. also acknowledges support from JSPS KAKENHI grant numbers
JP19H05810, JP22J21553 and JP22KJ1072. M.Y. is also supported in part by the JSPS Grantin-Aid for Scientific Research (19K03820, 19H00689, 20H05860, 23H01168), and by JST,Japan (PRESTO Grant No. JPMJPR225A, Moonshot R&D Grant No. JPMJMS2061).

A

Gauging subsystem Z2 symmetry

The subsystem KW transformation gauges a subsystem Z2 symmetry. We will review the technical details [89] in this appendix.

27

SciPost Phys. 15, 155 (2023)(i, j, k + 1)Bτi, j,k+ 12

τ

y
x

B

(i + 1, j + 1, k)xyi+ 12 , j+ 21 ,k(i, j, k)(i + 1, j, k)

Figure 10: Background gauge field for subsystem Z2 symmetry on lattice.
Turing on background gauge fields: To couple the theory to a background gauge field,
consider a spacetime cubic lattice with T sites along the time direction and L x × L y sites alongxy
the space direction. The background gauge fields B τ∈ {0, 1} are defined on
1,B1
1i, j,k+ 2

i+ 2 , j+ 2 ,kx; j+ 2y;i+ 12the time link and the spatial plaquette respectively, see Fig.10.
B
B
B
The gauge invariant holonomy variables Wτ;i,1,Wj, W
ming gauge fields over particular spacetime circles.
B
Wτ;i,
j =

T
X

Bτ

WB

x; j+ 21WBy;i+ 21

=

B

i=1
Ly

=

X
j=1

B
B
= Wτ,x,j + Wτ, y,i ,i, j,k+ 12

k=1
Lx
X

B

∈ {0, 1} are defined by sum-xy
i+ 12 , j+ 21 ,kxyi+ 12 , j+ 21 ,k= tx 1 ,

(A.1)

j+ 2

=t

y
i+ 21

.

The number of holonomy variables grows with system size. Because the L x × L y variables
B
B
B
Wτ,i,
j are highly reducible, instead we use Wτ,x; j , Wτ, y,i ∈ {0, 1}, detecting the insertion
WB

y WB

of symmetry operator (U jx ) τ,x; j and (Ui ) τ, y;i . The constraint on the symmetry generator
QLx
y QL yU
U x = 1 induces the redundancy
i=1 i
j=1 j
B
B
Wτ,x;
j → Wτ,x; j + 1 ,Wτ,B y;i → Wτ,B y;i + 1 ,(A.2)

which is obvious from (A.1). Along the spatial cycle, the holonomy variables W B

x; j+ 12detect the boundary conditions with the following constraint
Lx
X
i=1

WB 1 =
y;i+ 2

Ly
X

WB

j=1

x; j+ 12= txy .

,WB

y;i+ 12(A.3)

In summary, there are L x + L y − 1 independent holonomy variables along the time and space
directions separately. The partition function with background fields is
!

Ly
Lx
Y
Y
B
B
y
W
W
B
B
B
B


Z(Wτ,x;(U jx ) t,x; j(Ui ) t, y;i e−β H .
1,W1 ) := Tr
j , Wτ, y;i , W
x; j+ 2y;i+ 2WB

B
1 ,W y;i+ 1x; j+ 2
2

28

j=1

i=1

(A.4)

SciPost Phys. 15, 155 (2023)

Gauging subsystem Z2 symmetry: Consider a theory X with a non-anomalous subsystem
Z2 symmetry on the spacetime 3-torus T 3 . After gauging, we obtain a theory X /Zsub
liv2
ing on the dual lattice with a new subsystem Z2 symmetry. According to the new subsystem Z2 symmetry, the Hilbert space is divided into symmetry and twist sectors labeled byyyb 1 },bt := {bt xj ,bti }. Equivalently, we can couple the new theory with a new backb
u := {bux 1 , u
j+ 2

i+ 2

ground subsystem Z2 gauge field.
The gauging procedure contains two steps: first attach to the partition function a phase
whose exponent is the cup product of the gauge fields in X and X /Zsub
2 and then sum over all
distinct gauge field configurations in the theory X to promote its gauge field into a dynamical
field. In terms of gauging ordinary Z2 symmetry in (1 + 1)d, the background for the quantum
symmetry couples to the dynamical gauge field via the standard couplingZbτ B x + b x Bτ = WτB Wxb + WxB Wτb .

(A.5)(A.5) can be derived using the flatness of gauge fields B, b. We show it on lattice for later convenience of the generalization in subsystem symmetry. After discretization on the spacetime
lattice with L x × T sites, the flatness means that the gauge field B is closedBτ

i, j+ 21+ Bτi+1, j+ 12+ Bx 1 + Bx 1
i+ 2 , j= 0,

i+ 2 , j+1(A.6)

equivalent to the statement that the gauge field B is exactBτB x 1 = Bi, j + Bi+1, j ,i, j+ 12i+ 2 , j= Bi, j + Bi, j+1 ,(A.7)

where Bi, j ∈ {0, 1} is the potential field of the gauge field. By definition, the holonomy variables measure the twist boundary of the potential field
WτB =

T
X

Bτ

i, j+ 21

j=1

WxB =

Lx
X

(A.8)
B x 1 = B1, j + B L x +1, j .
i+ 2 , ji=1(A.5) then follows
ZBτ b x + B x bτ =

Lx X
T
X
i=1 j=1== Bi,1 + Bi,T +1 ,Bτi, j+ 21bx

i, j+ 12

+ B x 1 bτ 1

Lx X
T
X
(Bi, j + Bi, j+1 )b x

i, j+ 12i=1 j=1=

i+ 2 , j i+ 2 , j

Lx X
T
X

Bi, j (b xi, j− 2i=1 j=1

+

Lx
X
i=1

x
1 + b

+ (Bi, j + Bi+1, j )bτ 1

i+ 2 , j(A.9)τ+ bτ 1 )1
1 + b
i− 2 , ji, j+ 2b x 1 (Bi,1 + Bi,L x +1 ) +i,T + 2TX

i+ 2 , jbτ

j=1

L x + 12 , j(B1, j + B T +1, j )= WτB Wxb + WxB Wτb .
Similarly, we can regularize the coupling to subsystem symmetry gauge fields. Consider
the spacetime cub lattice with L x × L y × T sites. The flatness means that the gauge field B isclosedBτ

i, j,k+ 21+ Bτ

i+1, j,k+ 12+ Bτ

i, j+1,k+ 12+ Bτ

i+1, j+1,k+ 12

29

+B

xy
i+ 12 , j+ 12 ,k+B

xyi+ 21 , j+ 12 ,k+1

= 0,

(A.10)

SciPost Phys. 15, 155 (2023)

equivalent to the exactness
Bτ= Bi, j,k + Bi, j,k+1 ,xy= Bi, j,k + Bi+1, j,k + Bi, j+1,k + Bi+1, j+1,k ,i, j,k+ 12

B

i+ 12 , j+ 12 ,k(A.11)

where Bi, j,k ∈ {0, 1} is the potential field for the subsystem gauge field. The holonomy variables
measure the twist boundary of the potential field
B
B
B
Wτ,x,
j + Wτ, y,i = Wτ,i, j =WB=

x, j+ 21

WB

T
X
k=1
Lx
X

Bτ

i, j,k+ 12

B

i=1
Ly

1 =

y,i+ 2

X
j=1

B= Bi, j,1 + Bi, j,T +1 ,xy
i+ 12 , j+ 21 ,kxyi+ 12 , j+ 21 ,k= B1, j,k + B L x +1, j,k + B1, j+1,k + B L x +1, j+1,k ,(A.12)= Bi,1,k + Bi,L y +1,k + Bi+1,1,k + Bi+1,L y +1,k .

The standard regularization of the coupling to subsystem gauge field is
Z

Ly T
Lx X
X
X

Bτ b x y + B x y bτ =B

i=1 j=1 k=1

xyxybτ 1 1 + B τb.
i, j,k+ 12 i, j,k+ 21
i+ 12 , j+ 12 ,k i+ 2 , j+ 2 ,k

(A.13)

After a lengthy but straightforward derivation, we obtainZbτ B x y + b x y Bτ =

Lx
X
(W b

τ, y;i+ 2

i=1bB
1 + Wτ, y;i W y;i ) +B
1Wy;i+ 2

Ly
X
(W b
j=1

τ,x; j+ 12

WBx; j+ 12b
B
+ Wτ,x;j Wx; j ) .

(A.14)
Following the above procedure, the partition function after gauging is
B
B
B
ZX /Zsub (Wτ,x;j , Wτ, y;i , W=,WB 1)
y;i+ 2Xx; j+ 212

1
2

2(L x +L y −1)Wbτ,x; j+ 12,W bτ, y;i+ 1
2

ZX (W bτ,x; j+ 12b,Wx;b j ,W y;i
=0,1

PL x

× (−1)

i=1

(W b

,W b

τ, y;i+ 12BBb
1 W y;i+ 1 +Wτ, y;i W y;i )+τ, y;i+ 22b), Wx;b j , W y;i

PL y
j=1

(W b

B
B
b
1 Wx; j+ 1 +Wτ,x; j Wx; j )τ,x; j+ 2
2.
(A.15)

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