Generating Lattice Non-invertible Symmetries
Weiguang Caoa,b,1 , Linhao Lic,2 , and Masahito Yamazakia,d,3

a

b

c

d

Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Department of Physics, Graduate School of Science,
University of Tokyo, Tokyo 113-0033, Japan
Department of Physics and Astronomy, Ghent University,
Krijgslaan 281, S9, B-9000 Ghent, Belgium
Trans-Scale Quantum Science Institute,
University of Tokyo, Tokyo 113-0033, Japan

Lattice non-invertible symmetries have rich fusion structures and play important roles in understanding various exotic topological phases. In this paper, we explore methods to generate newlattice non-invertible transformations/symmetries from a given non-invertible seed transformation/symmetry. The new lattice non-invertible symmetry is constructed by composing the seedtransformations on different sites or sandwiching a unitary transformation between the transformations on the same sites. In addition to known non-invertible symmetries with fusion algebras ofTambara-Yamagami ZN × ZN type, we obtain a new non-invertible symmetry in models with ZN
dipole symmetries. We name the latter the dipole Kramers-Wannier symmetry because it arises
from gauging the dipole symmetry. We further study the dipole Kramers-Wannier symmetry indepth, including its topological defect, its anomaly and its associated generalized Kennedy-Tasaki
transformation.

1

weiguang.cao@ipmu.jp
linhaoli601@163.com
3masahito.yamazaki@ipmu.jp
2

Contents
1

Introduction

1

2

Generating lattice non-invertible symmetries
2.1 The seed transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 TY(ZN × ZN ) type fusion algebra . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Dipole Kramers-Wannier transformation . . . . . . . . . . . . . . . . . . . . . .

5
6
8
9

3

Dipole Kramers-Wannier symmetry
3.1 Duality transformation: gauging on the whole spin chain . . . . . . . . . . . . .
3.2 Duality defect: half-space gauging on the spin chain . . . . . . . . . . . . . . . .
3.3 Bilinear phase map representation and symmetry-twist sectors . . . . . . . . . .

12
12
13
16

4

Anomaly of dipole Kramers-Wannier symmetry
4.1 Anomaly free condition for general N . . . . . . . . . . . . . . . . . . . . . . .
4.2 Anomaly and phase diagram for N = 3 . . . . . . . . . . . . . . . . . . . . . .

18
18
20

5

D
Generalized Kennedy-Tasaki transformation associated with ZQ
N × ZN symmetry
5.1 Construction of KT transformations . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Mapping of gapped phases under KT transformation . . . . . . . . . . . . . . .

22
23
23

6

Conclusion and discussion

27

A Kramers-Wannier duality symmetry in lattice models
A.1 Gauging ZN symmetry on the whole spin chain . . . . . . . . . . . . . . . . . .
A.2 Gauging ZN symmetry on a half of the spin chain . . . . . . . . . . . . . . . . .

28
29
30

B Topological defects of invertible symmetries in dipole Ising model

32

C Computation in bilinear phase map representation

34

1

Introduction

Lattice non-invertible symmetry: Recently, the study of non-invertible symmetries in latticemodels with tensor product Hilbert spaces has attracted much attention [1–6]. These works study
the generalized symmetries in field theory initiated by the seminal paper [7]. While an ordinary
symmetry is defined by topological operators with group-like fusion rules, there exist many examples of topological operators with more general fusion rules wherein an operator does not necessarily have an inverse. The so-called non-invertible symmetry has been extensively studied in field1

theories [8–12] and lattice models [1,2,13–16] in (1 + 1)d, and later in higher dimensions [17–23]
with potential applications in the particle phenomenology [24–26].4
One of the common methods to generate a non-invertible symmetry is to gauge a discrete
Abelian group symmetry G [8, 27]. In (1 + 1)d, gauging a 0-form discrete symmetry G leads toa gauged theory with a new 0-form symmetry Ĝ := Rep(G), where Rep(G) is the representation
category of the original group G. For a discrete Abelian group G, Rep(G) is again a discrete
Abelian group and is isomorphic to G. A well-known example is the gauging of a 0-form ZN
symmetry. Suppose the ZN symmetry is generated by an operator η with η N = 1. Let us denote
the Kramers-Wannier (KW) duality transformation from the gauging of this ZN symmetry as D.
The invertible and non-invertible operators follow the fusion algebras of the Tambara-Yamagami
type TY(ZN ) [28]:5NX
η × D = D × η = D, D × D =
ηk .
(1.1)
k=1

For N = 2, this gives the fusion rules of the Ising CFT [29, 30]. If the theory is further invariant under the gauging of the discrete Abelian group symmetry G, the non-invertible dualitytransformation will become a non-invertible symmetry.
For a lattice model with a discrete Abelian group symmetry G, we can also find a noninvertible symmetry by searching for models invariant under gauging the G symmetry. However,
non-invertible symmetries in lattice models have a richer structure than those in field theories.
Even the lattice KW symmetry D from gauging a Z2 symmetry exhibits a fusion rule mixed withthe one-site lattice translation operator T:
U × D = D × U = D,

D × D = (1 + U )T−1 ,U 2 = 1,

(1.2)

where U generates the Z2 symmetry. This lattice KW symmetry (1.2) is therefore beyond thefusion category description [5]. This observation has been generalized to the subsystem noninvertible symmetry from gauging subsystem Z2 [31] and ZN symmetry [32] in (2 + 1)d. It is
desirable to have more examples of lattice non-invertible symmetries and further explore the rich
structures in lattice non-invertible fusion algebras. Moreover, lattice non-invertible symmetries
play an important role in understanding various exotic topological phases through generalized ’t
Hooft anomalies [13, 33, 34] and a generalized Landau paradigm [16, 35–38].
In this paper, we study lattice non-invertible symmetries by composing a given non-invertible
symmetry, which we call the seed transformation, with another invertible or non-invertible symmetry. This method has been very powerful in field theories. For example, in the c = 1 compact boson, there exist non-invertible symmetries composed of a ZN -symmetry gauging and the
T-duality transformation. Another example is the non-invertible symmetry from the Adler-BellJackiw (ABJ) anomaly [24, 25], where the non-invertible rational-angle chiral rotation symmetry
4

We apologize in advance for missing references and welcome suggestions for additional relevant references.
In this paper, we will only focus on the fusion algebra and leave the discussion of other fusion data in future
exploration. Therefore we omit the specification of the bicharacter and Frobenius-Schur indicator in the descriptionof fusion category throughout this paper.
52

is constructed by stacking a (2 + 1)d fractional quantum Hall state to a non-conserved gaugeinvariant current. This method has also been studied in lattice models. The non-invertible symmetry of the cluster state is constructed by composing two KW duality operators that act separately
on even and odd sites [6,39]. Furthermore, a subsystem non-invertible symmetry is constructed by
the product of KW transformations on every line, a global Hadamard gate and the product of KWtransformations on every column in [32]. In this paper, we continue to explore this construction
in models with dipole symmetries.
Dipole symmetry: Besides non-invertible symmetry, lattice models are playgrounds for otherexotic symmetries like subsystem symmetry [40–54],6 fractal symmetry [61] and multipole symmetry [62–72]. Models with these exotic symmetries attract much interest because they admit
quasi-particles called fractons with restricted mobilities [73–83] and have UV-IR mixing in thecontinuum limit [47,84–89]. In recent years, dipole symmetry has been studied in new topologicalinsulators [90–92], exotic quantum liquids [93–96], systems with non-ergodicity [97–100], hydrodynamics with dipole conservation [101–104] and systems with anomalous diffusion [105–107].
In field theories with dipole symmetries, the charge Q and the dipole Di in the xi direction areconserved quantities:
Z
Z
Q :=

J0 ,

Di :=

space

xi J0 ,

(1.3)

space

where J0 is the time component of the conserved current. The dipole conservation constrains themobility of a single charge. If we further impose translational invariance generated by conserved
momentum Pi , the dipole symmetry mixes with the translation symmetry[Pj , Dk ] = −iδj,k Q.
One can consider the simplest example with U (1) dipole shift symmetry in (1 + 1)d [65]


I
µ0
1 2 2
2
S = dτ dx(∂τ φ) +(∂ φ) , φ ∼ φ + 2π,
2
2µ x

(1.4)

(1.5)

which is invariant under the constant and dipole shifts:φ(τ, x) → φ(τ, x) + c + cx x,

c, cx ∈ U (1).

(1.6)

To gauge the symmetry, we need to couple the system with tensor gauge fields (Aτ , Axx ) with theexotic gauge transformation
Aτ → Aτ + ∂τ α(τ, x),

Axx → Axx + ∂x2 α(τ, x),(1.7)

to cancel the contribution from shifting φ(τ, x) by an arbitrary function α(τ, x) ∈ U (1). In theliterature, the corresponding gauge theory is called the U (1) dipole gauge symmetry althoughboth charge and dipole symmetry have been gauged.
6

Subsystem symmetry is also known as gauge-like symmetry in previous literature [55–60] which is related totopological memories at finite temperature.

3

The above analysis works in parallel for a lattice model with a ZN dipole symmetry. Here, thecharge and the dipole operators are
ηQ :=

L
Y

Xi ,

L
Y
ηD :=
(Xi )i ,

i=1

N
N
ηQ
= ηD
= 1,

TηD = ηQ ηD T,

(1.8)

i=1

where Xi , Zi are ZN Pauli operators acting on site i. The discussion of boundary conditions is
subtle for a dipole symmetry, and we will for simplicity assume a periodic boundary condition
with lattice size L ≡ 0 mod N throughout this paper. The dipole symmetry acts as
ηD :Zi → ω i Zi ,ω := exp(2πi/N ),(1.9)

and the simplest symmetric interaction is Zi−1 (Zi† )2 Zi+1 .
In this paper, we illustrate the procedure of simultaneously gauging the charge and dipole
symmetries on the lattice. We find that this gauging implements a non-invertible duality transformation, which we call the dipole KW transformation. Similar to the case of field theory, we willDinterchangeably use gauging the dipole symmetry and gauging the ZQ
N × ZN symmetry.
Non-invertible dipole KW symmetry: Although there are extensive studies of non-invertible
symmetry and exotic symmetry, the two types of symmetries are often treated separately, and noninvertible symmetries in models with exotic symmetry are still rarely explored. There are a few
examples in lattice models with subsystem symmetry [31, 32] and field theories with exotic symmetry [108]. Studying non-invertible symmetries in models with exotic symmetry will deepen our
understanding of non-invertible symmetries and their fusion structures through more non-trivial
and exotic examples. For instance, the fusion of subsystem non-invertible symmetry forms a grid
operator which is a sum of non-topological invertible operators. In this paper, we further construct
and study the lattice non-invertible symmetry in (1 + 1)d spin models with dipole symmetry.
By conjugating the global Hadamard gate with the ordinary KW transformation, we obtain a
new non-invertible transformation, the dipole KW transformation, that exchanges the transverse
field and dipole interaction. Surprisingly, the fusion algebra of the dipole KW transformation
mixes with the charge conjugation symmetry instead of the lattice translation,7 further extending our understanding of the lattice non-invertible structure. In Table. 1, we list different fusion
structures that are studied in this paper.
The dipole KW transformation can be obtained from gauging the dipole symmetry. We thenstudy the topological defects by a half-gauging on the spin chain and investigate the anomaly ofthe dipole KW symmetry. In particular, the dipole Ising model with N = 3 at two self-dual points
admits an anomalous non-invertible symmetry. By gauging the ordinary Z3 symmetry, the model
is mapped to the Z3 anisotropic XZ model (or quantum torus chain [109]). The anomaly prohibits
7

In the appendix of [32], the charge conjugation also appears in the fusion rule of the subsystem non-invertible
symmetry from gauging ZN subsystem symmetry. In contrast with our discussion, the fusion rule in that reference
also mixes with the lattice translation. The fusion algebra mixed with charge conjugation also resembles the algebraof non-invertible duality symmetries found in (3 + 1)d continuum field theories [22].

4Duality transformation
gauging ZN : D
gauging ZN × ZN : D†o DeD† H
gauging ZQ
N × ZN : D U D

self-fusion
P
k
−1
( N
k=1 η )T
P
PN k
k
( N
k=1 ηo )(k=1 ηe )
P
PN k
k
( N
k=1 ηQ )(k=1 ηD )C

Pattern
Mixed with lattice translation T
No mixing
Mixed with charge conjugation C

Table 1: Different Fusion structures in (1 + 1)d models in this paper.
the gapped phase with a unique ground state, which is consistent with the gapless phase and thefirst order phase transition from numerical calculation [110, 111]. Based on the non-invertible
symmetry, we define the Kennedy-Tasaki (KT) transformations [112–120] associated with dipole
symmetries, as a duality relating a dipole spontaneously symmetry breaking (SSB) phase and adipole symmetry protected topological (SPT) phase. Following the construction in [3], KT transformation is a composition of the KW duality transformation and the invertible transformationthat stacks an SPT to the system. Recently, KT transformations have been applied to studying andclassifying non-trivial gapped and gapless phases [6, 32, 121–126].
The structure of this paper: In Sec. 2 we first review the construction of a non-invertible transformation with TY(ZN × ZN ) type fusion algebra and define the dipole KW transformation fromthe ordinary KW transformation as the seed transformation. In Sec. 3, we study the dipole KW
symmetry operator and defect by gauging the dipole symmetry. We also study the mapping ofsymmetry-twist sectors of the charge and dipole symmetry during the dipole KW transformation. We study the anomaly condition for the dipole KW symmetry in Sec. 4 and the generalized
Kennedy-Tasaki transformation in Sec. 5. Finally, we conclude and point out future directions in
Sec. 6.
Note added: After the completion of this manuscript, we became aware of an independent related work [127] on gauging finite modulated symmetries on spin chains, which will appear on
arXiv soon.

2

Generating lattice non-invertible symmetries

In this section, we show how to generate more lattice non-invertible symmetries in (1 + 1)d latticemodels from a given seed lattice non-invertible transformation. We study examples where theseed transformation is the KW transformation from gauging a ZN symmetry. By composing theseed transformation acting on different sites or with other invertible transformations, we either
generate a known symmetry with fusion algebras of Tambara-Yamagami type TY(ZN × ZN ), ora totally new non-invertible symmetry which appears in models with dipole symmetries.
5We work on a spin chain of L sites with the periodic boundary condition. On each site there
is a qudit |si ii ∈ Hi , si ∈ ZN . The total Hilbert space H is a tensor product of the local Hilbert
space Hi on each site i. We use |{si }i := ⊗Li=1 |si ii ∈ H for a state with qudit si at site i. The
Pauli operators acting on each site are defined as
Zi :=

N
X

si

ω |si ii hsi | ,

Xi :=

si =1

N
X

|si + 1ii hsi | ,ω := exp(2πi/N ).

(2.1)

si =1

The Pauli operators are examples of local unitary operators and satisfy the following algebra
ZiN = XiN = 1,

2.1

Zi Xi = ωXi Zi ,[Zi , Xj ] = 0

for i 6= j.

(2.2)

The seed transformation

In this paper, we use the KW transformation D of gauging the ZN symmetry as a seed transformation to generate more lattice non-invertible symmetries. The ZN symmetry is generated by
η :=

L
Y

Xi ,

η N = 1.

(2.3)

i=1

A Hamiltonian with ZN symmetry can be constructed from ZN -singlet operators {Zi† Zi+1 , Xi }.
A typical example is the ZN clock model with a transverse field
Hclock = −g −1

LX

Zi† Zi+1 − g

i=1

L
X

Xi + (h.c.),

(2.4)

i=1

where h.c. means the Hermitian conjugate. This model contains the transverse Ising model (N =2) and three states Potts model (N = 3) as special cases. In this paper, we only focus on Hermitian
8
models which automatically have the charge conjugation symmetry ZC
N generated by
X
C=|{−sj }i h{sj }| : Zi → Zi† , Xi → Xi† , i = 1, ..., L,
(2.6)
{sj }

where we sum over all possible spin configurations of {si }.
The KW transformation D acts on the operators as
D:†
Xi → Zi−1
Zi ,

†
Zi−1
Zi → Xi−1 .

(2.7)

which preserves the locality of a ZN invariant Hamiltonian. The KW transformation maps theclock model (2.4) from a symmetric gapped phase with a unique ground state (g >> 1) to a8

Throughout this paper, for a general (unitary or non-unitary) operator O, its action on other operators Oa is
OOa = Oa0 O :Oa → Oa0 .

6

(2.5)

symmetry breaking phase with N degenerate vacua (0 ≤ g << 1) and vice versa. At the criticalpoint (g = 1), the KW transformation becomes a non-invertible symmetry.
The KW transformation has several realizations in the literature [5]. In Appendix A we illustrate how to get the KW transformation D from gauging the ZN symmetry and study the topological defect of the KW duality symmetry. Such the KW transformation can be rewritten in thebilinear phase map (BPM) representation [128]
X PL
0
D=
ω i=1 (si−1 −si )si−1 |{s0i }i h{si }| ,(2.8){si },{s0i }where we sum over all possible spin configurations of {si } and {s0i }. In the BPM representation,
it is convenient to compute various operator relations. The action on the Pauli operators is




†
†
D Zj−1 Zj = Xj−1 D, DXj = Zj−1 Zj D.
(2.9)
The KW operator D commutes with the translation operator T which acts on states by a shift ofthe value of qudit to one site rightT |{si }i :=LO|si−1 ii ,(2.10)

i=1

and their fusion gives the Hermitian conjugate D†
D† = DT = TD.

(2.11)

The fusion algebra of the invertible operator η, C, the non-invertible operator D and the translation
operator T is
!
N
N
X
X
k
−1
†
CD = DC, ηD = Dη = D, D × D =η T , D ×D=ηk ,(2.12)k=1
k=1
ηC = Cη † ,Tη = ηT,

TC = CT,

C2 = 1,

TL = 1.

The detailed computation of (2.9), (2.11) and non-invertible fusion rules in (2.12) can be found in
Appendix C. Other fusion rules follow simply from the definition. Because the invertible operators
are defined in a translational invariant way, they commute with the translation operator T. D
absorbs η because of (2.9). The charge conjugation C commutes with the non-invertible KW
operator D because
X PL
X PL
0
0
CD =
ω i=1 (si−1 −si )(−si−1 ) |{s0i }i h{si }| =
ω i=1 (−(si−1 −si ))si−1 |{s0i }i h{si }| = DC.
{si },{s0i }{si },{s0i }(2.13)
Because of the appearance of lattice translation transformation, the lattice symmetry algebra involving η, D differs from the fusion algebra TY(ZN ) (1.1).
7

2.2

TY(ZN × ZN ) type fusion algebra

In this subsection, we show how to generate lattice non-invertible transformations by composing
the seed transformations acting on different sites (or different degrees of freedom in a unit cell).
We work out the example of composing the KW transformations on even and odd sites.
In this particular example, we assume that the number of sites is even (L ≡ 0 mod 2) and themodel has a ZoN × ZeN symmetry where ZoN acting on the odd sites and ZeN acting on the evensites. This symmetry is generated by
L/2
Y

ηo =

X2i−1 ,

ηe =

i=1

L/2
Y

X2i .

(2.14)

i=1†
The model with ZoN ×ZeN symmetry is constructed from the symmetry-singlet operators {Zi−1
Zi+1 , Xi }with examples of Ising type model

HIsing-type = −g

−1

L
X

L
X

†
Zi−1
Zi+1 − gi=1

Xi + (h.c.),(2.15)

i=1

and the ZN cluster model
Hcluster = −

L
X

†
Zi−1
Xi Zi+1 + (h.c.).

(2.16)

i=1

The seed transformations are the KW transformations acting on even and odd sites
Do :†X2i+1 → Z2i−1
Z2i+1 ,De :†X2i → Z2i−2
Z2i ,

†
Z2i−1
Z2i+1 → X2i−1 ,†Z2i−2Z2i → X2i−2 .

(2.17)

Since they acts on different sites, Do , De commute. We can find two more non-invertible transformations, by composing Do , De and the lattice translation T
D̃ = TDo De :†Xi → Zi−1
Zi+1 ,

D̃0 = Do De :†Xi → Zi−2
Zi ,

†
Zi−1
Zi+1 → Xi ,†Zi−2
Zi → Xi−2 ,(2.18)

with fusion algebrasD̃ × D̃ =D̃0 × D̃0 =

N
X

!
ηok

k=1
N
X

!
ηok

k=1

N
X

!
ηek

k=1
N
X

,
(2.19)

!
ηek

T−2 .

k=1

Here are some comments about the new non-invertible transformations81. For N = 2 both D̃ and D̃0 are non-invertible symmetries for the Z2 cluster state [6]. The D̃symmetry belongs to the Rep(D8 ) fusion category symmetry. The D̃0 symmetry becomesthe Rep(H8 ) fusion category symmetry in the continuum limit.9
2. Because the fusion of D̃ does not mix with lattice translation, the subalgebra of ηo , ηe , D̃
forms the fusion algebra of TY(ZN × ZN ) in the lattice level.
3. For general N , there might not exist an SPT phase that is invariant under D̃ or D̃0 symmetry.
This fact can be used as a diagnosis of the anomaly of the non-invertible symmetries.
4. It is straightforward to generalize this method and find new non-invertible transformations
and symmetries in lattice models with Abelian group G1 × G2 symmetry. The seed transformations are gauging the G1 symmetry and gauging the G2 symmetry.

2.3

Dipole Kramers-Wannier transformation

In this subsection, we explore yet another method to generate the lattice non-invertible transformation through sandwiching unitary operators by seed transformations acting on the same sites.
Q
This idea comes from an observation [129] that gauging the ZN symmetry η = Li=1 Xi of the
XZ model
L
L
X
X
†
−1
HXZ = −g
Xi Xi+1 − g
Zi† Zi+1 + (h.c.)(2.20)
i=1

i=1

gives the dipole Ising model with ZD
N dipole symmetry
Hdipole-Ising = −g

−1

L
X

Zi−1 (Zi† )2 Zi+1 − g

i=1

L
X

Xi + (h.c.).

(2.21)

i=1



D
C
Actually the model has a larger symmetry ZQ×Z
N o Z2 symmetry where the superscripts
N
Q, D, C means charge, dipole and charge conjugation. The generators of the charge and dipole
symmetry are
L
L
Y
Y
ηQ :=
Xi , ηD :=(Xi )i .
(2.22)i=1

i=1

For simplicity, we assume L ≡ 0 mod N to have a full ZD
N symmetry.
Starting with the seed transformation D by gauging the charge symmetry ZQ
N , we define thedipole KW transformationD̂ := TDU H D = D† U H D,
(2.23)
9

Rep(D8 ) and Rep(H8 ) have the same fusion algebra but different bicharacters. The difference in lattice transformations is a reflection of the difference of the bicharacter between the two. In this paper we only focus on the fusion
algebra and leave the study of other fusion data (such as bicharacter) in future work.

9where
U

H

:=

L
Y

UiH ,

N
X

1
UiH = √

N α,β=1i=1

Xi → Zi† ,ω −αβ |βi hα| :Z i → Xi ,(2.24)

is the product of ZN Hadamard gate UiH on every site and can be viewed as the “half-charge
conjugation” because C = (U H )2 . From the decomposition (2.23), the dipole KW transformation
first maps the dipole Ising model to the XZ model, then exchanges Xi† Xi+1 and Zi† Zi+1 , andfinally maps the model back to the dipole Ising model. The net action isD̂ :Zi−1 (Zi† )2 Zi+1 → Xi ,Xi → (Zi−1 (Zi† )2 Zi+1 )† ,(2.25)

which interchanges the dipole-interaction terms and the transverse-field terms. The above transformations are summarized in the duality web in Fig. 1.
D (or D† )Dipole Ising model with coupling g

XZ model with coupling gUHD̂D† (or D)
Dipole Ising model with coupling g −1

XZ model with coupling g −1

Figure 1: Duality web of the dipole Ising model and the XZ model.

In the next section, we will interpret the dipole KW transformation as gauging of both the
charge and dipole symmetry and study the transformation in detail. The fusion algebra of ηQ , ηD , D̂
can already be computed from the decomposition of D̂ (2.23) and the property of the seed transformation D. For example, the non-invertible fusion rule is!
NXkD̂ × D̂ = (D† U H D)(D† U H D) = D† U H
ηQ
UHD
k=1


= D† 

N
L
X
Y
k=1

where we used

!k 
Zi

N
X

C D = D† D

i=1

!
k
ηD

C=

k=1

L
Y
i=1

Zi =

L 
Y
i=1

10

Zi−1 Zi†

N
X
k=1

i

,

!
k
ηQ

N
X

(2.26)

!
k
ηD

C,

k=1

(2.27)

with the assumption L ≡ 0 mod N . The full fusion algebra of the invertible operators ηQ , ηD , C,
the non-invertible operator D̂ and the translation operator T is
! N
!
N
X
X
k
k
ηQ D̂ = ηD D̂ = D̂ηQ = D̂ηD = D̂, D̂ × D̂ =ηQηDC, D̂† = CD̂ = D̂C,
k=1†
CηQ = ηQ
C,†CηD = ηDC,

k=1†
TηD = ηQ ηD T,

TηQ = ηQ T,

C2 = 1,

TL = 1.
(2.28)

Here are a few comments about the dipole KW transformation:
1. In contrast with the ordinary KW transformation, implementing the dipole KW transformation D̂ twice does not involve a lattice translation T but a charge conjugation operation
C. The fusion subalgebra of ηQ , ηQ , D̂, C is new and is different from the fusion algebra of
TY(ZN × ZN ). The Z2 charge conjugation appears in the non-invertible fusion rule andacts non-trivially on the invertible operators ηQ , ηD . For N = 2, the charge conjugation
becomes the identity operator and the fusion subalgebra of ηQ , ηQ , D̂ becomes the fusion
algebra of Rep(D8 ).
2. For dipole Ising model at self dual point (g = 1), neither the ordinary KW transformation
D nor the global Hadamard gate U H is a symmetry, while the dipole KW transformation(2.23) is a symmetry.
Q
3. The original XZ model has another ZN symmetry generated by Li=1 Zi . In the dipole Isingmodel, this symmetry becomes the ZN dipole symmetry after the ordinary KW transformation [129]
!
!
L
L 
i
Y
Y
†
Zi D =
Zi−1 Zi
D = D ηD .
(2.29)i=1

i=1

4. SPT phases protected by dipole symmetries in (1 + 1)d have been classified in [130]. We
will study the anomaly of the dipole KW symmetry by checking the invariance of dipole
SPT phase under the dipole KW transformation for general N in Section 4.
5. By repeating the sandwiching procedure (2.23), we can in general find non-invertible transformations in models with multipole symmetries. For example, the quadruple KW transformation is
DU H D† U H D :††Xi → Zi−2 (Zi−1)3 Zi3 Zi+1,††
Zi−2 (Zi−1)3 Zi3 Zi+1
→ Xi−1 ,(2.30)

where the quadruple interaction is invariant under the quadruple symmetry operator
Y 2
ηQu :=
Xii .
(2.31)i11

3

Dipole Kramers-Wannier symmetry

In this section, we study the dipole KW symmetry in detail. We first show that the dipole KWtransformation D̂D̂ :Zi−1 (Zi† )2 Zi+1 → Xi ,Xi → (Zi−1 (Zi† )2 Zi+1 )† ,(3.1)

comes from gauging both the charge and dipole symmetry. We then study the topological interface
and defect of the dipole KW transformation by a half-gauging on the spin chain. Finally, using
the BPM representation, we study how the symmetry and twist sectors of both the charge and thedipole symmetries map into each other under this non-invertible transformation.

3.1

Duality transformation: gauging on the whole spin chain

We gauge both the charge and dipole symmetry in the following steps. First we enlarge the Hilbert
space by introducing gauge variables on the site. Then we impose the Gauss law constraints toproject to the gauge invariant sectors. Finally we rename the variables to match with the original
theory.
We illustrate the above steps in the transverse dipole Ising model
Hdipole-Ising = −g −1

LX

Zi−1 (Zi† )2 Zi+1 − g

i=1

L
X

Xi + (h.c.).

(3.2)

i=1

This Hamiltonian is invariant under the global charge and dipole transformation
Zi → ω α+βi Zi ,α, β = 0, ..., N − 1,

(3.3)

where α, β are constants. If we promote the global transformation to a gauge transformation
Zi → ω fi Zi ,(3.4)

where fi is a ZN -valued function of the position, the interaction term is not invariant
Zi−1 (Zi† )2 Zi+1 → ω fi−1 +fi+1 −2fi Zi−1 (Zi† )2 Zi+1

(3.5)

Introducing another set of Pauli operators X̃i , Z̃i on each site as gauge variables leads to thegauged Hamiltonian
Hgauged = −g

−1

L
X

X̃i Zi−1 (Zi† )2 Zi+1 − g

i=1

L
X

Xi + (h.c.),

(3.6)

i=1

which in invariant under the gauge transformation
Zi → ω fi Zi ,X̃i → ω −(fi−1 +fi+1 −2fi ) X̃i .
12

(3.7)

The gauge variables X̃i , Z̃i commutes with the original Pauli operators Xi , Zi . The gauged theoryis equivalent to coupling two isolated chains on top of each other, with an enlarged Hilbert spaceH̃ with dimension N 2L . Besides the original global symmetry (3.3), the gauged theory also has a
set of additional gauge symmetries (3.7) generated by
Gi = Xi Z̃i−1 (Z̃i† )2 Z̃i+1 ,[Gi , Hgauged ] = 0,

∀i.

(3.8)

We require that the physical state |ψi is gauge invariant
Gi |ψi = |ψi ,(3.9)

and we impose Gauss law constraints
Gi = Xi Z̃i−1 (Z̃i† )2 Z̃i+1 = 1,

Xi = (Z̃i−1 (Z̃i† )2 Z̃i+1 )† ,→∀i.

(3.10)

Afte imposing the Gauss law constraints, the enlarged Hilbert space H̃ is projected down to a new
Hilbert space with dimension 2L , which is isomorphic to the original Hilbert space H. Changing
to a new set of variablesẐi := Z̃i , X̂i := X̃i Zi−1 (Zi† )2 Zi+1 ,(3.11)
the gauged Hamiltonian becomes
−g

−1

L
X
i=1

X̂i − gL
X

(Ẑi−1 (Ẑi† )2 Ẑi+1 )† + (h.c.).

(3.12)

i=1

Finally, by renaming the variablesẐi → Zi ,X̂i → Xi ,(3.13)

we recover the original Hamiltonian with the change of coupling g → g −1 . The whole process
gives the dipole KW transformation D̂D̂ :Zi−1 (Zi† )2 Zi+1 → Xi ,Xi → (Zi−1 (Zi† )2 Zi+1 )† .

(3.14)

When the theory is self-dual with g = 1, D̂ becomes a non-invertible symmetry.

3.2

Duality defect: half-space gauging on the spin chain

In this subsection, we construct the duality defect corresponding to the dipole KW transformation
by a half-space gauging on the spin chain. The half-space gauging creates a topological interface
which further becomes a topological defect when the model is invariant under the gauging. We
also show the local unitary gates that move and fuse the duality defect. The defect, its movement
and fusion are shown in Fig. 2.

13

D̂

0

1

D̂

3

2

0

1

3

2

P

N
k
k=1 ηQ PNkk=1 ηD

0

1

2

3

D̂

D̂

D̂

(1)

(2)

(3)



C

D̂

Figure 2: (1) Half-gauging creates a topological interface/defect D̂ located at link (1, 2); (2)† 2movement operator UD̂2 = CZ2,3 (CZ2,l) (U2H )† S2,l moves the interface/defect from (1, 2) to††(2, 3); (3) Fusion operator λD̂⊗D̂ = CZ3,lCZ2,l(CZ2,l2 )2 S2,l2 fuses the defects at (1, 2) and2
1(2, 3).
(1,2,3)Half-space gauging
Consider the transverse dipole Ising model on an infinite chain and gauge the dipole symmetry
for sites i ≥ 1. The half-gauged Hamiltonian isX
X
X
Hhalf-gauged = −g −1
Zi−1 (Zi† )2 Zi+1 − g −1
X̃i Zi−1 (Zi† )2 Zi+1 − g
Xi + (h.c.). (3.15)
i≤0

i

i≥1

Imposing the Gauss law constraints that commute with the half-gauged Hamiltonian
Gi = Xi Z̃i−1 (Z̃i† )2 Z̃i+1 = 1,i ≥ 2,(3.16)

then (3.15) becomes
Hhalf-gauged = − g −1

X

Zi−1 (Zi† )2 Zi+1 − g

X

i≤0

Xi

i≤0

− g −1 X̃1 Z0 (Z1† )2 Z2 − gX1 − g −1 X̃2 Z1 (Z2† )2 Z3 − g Z̃1 (Z̃2† )2 Z̃3XX− g −1X̃i Zi−1 (Zi† )2 Zi+1 − gZ̃i−1 (Z̃i† )2 Z̃i+1 + (h.c.).
i≥3(3.17)i≥3

We can further introduce a new set of Pauli variables
Zl := Z̃1 ,Xl := X̃1 Z2 ,Ẑ2 := Z̃2 ,X̂2 := X̃2 (Z2† )2 Z3 ,Ẑi := Z̃i ,X̂i := X̃i Zi−1 (Zi† )2 Zi+1 ,14(3.18)i ≥ 3.

After renaming X̂i → Xi , Ẑi → Zi , i ≥ 2, the half-gauged Hamiltonian becomes
X
X
Hhalf-gauged = − g −1
Zi−1 (Zi† )2 Zi+1 − g
Xi
i≤0

−g

−1

i≤0

Z0 (Z1† )2 Xl − gX1 − g −1 Z1 X2 − gZl (Z2† )2 Z3− g −1X

Xi − gi≥3X(3.19)

Zi−1 (Zi† )2 Zi+1 + (h.c.).

i≥3

There is an interface located at the link (1,2) separating two theories which are exchanged byinverting the coupling g → g −1 . By inserting the duality interface, we couple the system with one
ancillary qudit |sl i with Pauli operators Xl , Zl acting on this qudit.
Under the following local unitary transformation

† 2 †


 Xl → Z2 , Zl → X2 Zl Z3 ,† 2
CZ2,3 (CZ2,l) (U2H )† S2,l :X2 → Xl (Z2† )2 ,
 X →X Z ,33 2

Z 2 → Zl ,(3.20)

the half-gauged Hamiltonian becomes
X
X
Hhalf-gauged → − g −1
Zi−1 (Zi† )2 Zi+1 − g
Xi
i≤1

−g

−1

− g −1i≤0Z1 (Z2† )2 Xl − gX2 − g −1 Z2 X3 − gZl (Z3† )2 Z4Xi≥4

Xi − gX(3.21)

Zi−1 (Zi† )2 Zi+1 + (h.c.),i≥3where the interface is moved to the link (2,3) as shown in Fig. 2. This duality interface is therefore
topological. In (3.20), CZi,j is the control-Z gate, UjH is the Hadamard gate and Si,j is the swap
gate. Their definition and transformation on local operators can be found in Table. 2.
Dipole KW duality defect
When the coupling g = 1, the duality interface becomes a duality defect. The defect Hamiltonian
with the dipole KW duality defect D̂ at link (i0 , i0 + 1) is
X(i ,i +1)
HD̂ 0 0=−(Zi−1 (Zi† )2 Zi+1 + Xi )i6=i0 ,i0 +1(3.22)† 2†
2− (Zi0 −1 (Zi0 ) Xl + Xi0 + Zl (Zi0 +1 ) Zi0 +2 + Zi0 Xi0 +1 ) + (h.c.).
This defect is topological and can be moved to link (i0 + 1, i0 + 2) by a movement operatorUD̂i0 +1 := CZi0 +1,i0 +2 (CZi†0 +1,l )2 (UiH0 +1 )† Si0 +1,l .

(3.23)

Since the defect is topological, we can insert two defects far away from each other, move them
together and fuse them. Without loss of generality, we can move the two dipole KW defects to15

links (1, 2) and (2, 3)
X(1,2),(2,3)
HD̂;D̂
=−(Zi−1 (Zi† )2 Zi+1 + Xi )i6=1,2,3− (Z0 (Z1† )2 Xl1 + X1 + Zl1 (Z2† )2 Xl2 + Z1 X2 + Zl2 (Z3† )2 Z4 + Z2 X3 ) + (h.c.),(3.24)
where Xl1 , Zl1 , Xl2 , Zl2 act on the ancillary qudit |sl1 i , |sl2 i associated with the two defects. We
need to apply the fusion operator

Xl1 → Xl1 Z2† , X → X Z2 Z† , Z → Z ,l2
2 l2 l1l2
2(1,2,3)††2
λD̂⊗D̂ = CZ3,l2 CZ2,l1 (CZ2,l2 ) S2,l2 :(3.25)
2 †

X
→
X
Z
Z
,
Z
→
Z
,

2
l
2
l
2
2
2 3



†
X3 → X3 Zl2 ,and the defect Hamiltonian after fusion isX(1,2),(2,3)(1,2)
HD̂;D̂
→ HD̂⊗D̂ = −(Zi−1 (Zi† )2 Zi+1 + Xi )(3.26)

i6=1,2− (Z0 (Z1† )2 Z2† Xl1 + X1 + Z1† (Z2† )2 Z3 Xl†2 + X2 ) + (h.c.).
This Hamiltonian has two decoupled degrees of freedom on ancillary sites l1 and l2 ; Xl1 , Xl2
become symmetry operators of the new Hamiltonian whose Hilbert space is decomposed intoa direct sum of the eigenspaces of X1 and X2 . Compared with the defect Hamiltonian withinsertion of invertible topological defects of C, ηQ , ηD in Appendix B, one finds that after picking
the eigenvalue of X1 and X2 , the fused defect Hamiltonian corresponds to different channels of
the fusion rule
! N
!
N
X
X
k
k
D̂ × D̂ =ηQηDC.
(3.27)
k=1

3.3

k=1

Bilinear phase map representation and symmetry-twist sectors

The dipole KW transformation has the following BMP representation under periodic boundary
conditions
X
PL
0
0
0
D̂ =
ω − i=1 (si+1 +si−1 −2si )si |{s0j }i h{sj }|{sj },{s0j }

=

X

ω−

PL

0
i=1 (si+1 +si−1 −2si )si|{s0j }i h{sj }| .

(3.28){sj },{s0j }where the exponent respects the dipole interaction and the minimal gauge coupling. While thefusion algebras have been studied from decomposition into the seed transformation in Sec. 2, here
we instead use the BPM representation to study the mapping of symmetry-twist sectors before
and after the dipole KW transformation. This will be useful in the analysis of the anomaly of thedipole KW symmetry in the next section.
16

Mapping between symmetry-twist sectors
Since there are charge and dipole symmetries, the states can be organized into eigenstates of ηQ
and ηD with symmetry eigenvalue ω uQ and ω uD where uQ , uD ∈ ZN . Moreover, one can also use
D
the ZQ
N and ZN symmetries to twist the boundary condition of the spins as
si+L := si + tQ + tD i,|si+L ii+L := |si + tQ + tD iii ,(3.29)

with tQ , tD ∈ ZN . The Pauli Z operators also have a boundary condition:
Zi+L = ω tQ +tD i Zi .

(3.30)

One can organize the Hilbert space into N 4 symmetry-twist sectors, labeled by (uQ , tQ , uD , tD ).
Similarly, one can also label the symmetry-twist sectors of spins after the dipole KW transformation by (buQ , btQ , ubD , btD ). We would like to find out the relation between the sectors before andafter the dipole KW transformation.
The BMP representation is modified with general boundary conditions
X
PL
0
0
0
0
0
D̂ =
ω i=1 −(si+1 +si−1 −2si )si −(tQ +tD )sL +tQ s1 |{s0j }i h{sj }|{sj },{s0j }

X

=

ω

PL

0
b b
b
i=1 −(si+1 +si−1 −2si )si −(tQ +tD )sL +tQ s1|{s0j }i h{sj }| .

(3.31){sj },{s0j }Let us first consider D̂ × ηQ acting on an arbitrary state |ψi =D̂ × ηQ |ψi = D̂XP{si } ψ{si } |{si }i.

ψ{si } |{si + 1}i

{si }

X

=

ψ{si } ω −

PL

0
b b
b
i=1 (si+1 +si−1 −2si )si −(tQ +tD )(sL +1)+tQ (s1 +1)|{s0i }i{s0i },{si }

=ω

−b
tD

(3.32)
X

ψ{si } ω

PL

0
b b
b
i=1 −(si+1 +si−1 −2si )si −(tQ +tD )sL +tQ s1|{s0i }i{s0i },{si }= ω −tD D̂ |ψi .
b

The result implies that for any eigenstate |Ψi with
ηQ |Ψi = ω uQ |Ψi ,(3.33)

ω −tD D̂ |Ψi = D̂ × ηQ |Ψi = ω uQ D̂ |Ψi ,(3.34)btD = −uQ .

(3.35)

we haveb

namely17

On the other hand, one can also consider ηbQ × D̂ acting on a arbitrary state |ψi,
X
PL
0
0
0
0
0
ηbQ × D̂ |ψi = ηbQ
ψ{si } ω i=1 −(si+1 +si−1 −2si )si −(tQ +tD )sL +tQ s1 |{s0i }i{s0i },{si }=X

ψ{si } ω

PL

0
0
0
0
0
i=1 −(si+1 +si−1 −2si )si −(tQ +tD )sL +tQ s1|{s0i + 1}i(3.36){s0i },{si }= ω tD D̂ |ψi .
This shows that the dual ZQ
N -symmetry charge sector after dipole KW transformation is identified
with the twisted sector before this transformation,ubQ = tD .

(3.37)

By similar calculations, one can determine the correspondence between the remaining sectors:btQ = uD ,ubD = −tQ .

(3.38)

In summary, we have the following map of the symmetry-twist sectors:(buQ , btQ , ubD , btD ) = (tD , uD , −tQ , −uQ ).

4(3.39)

Anomaly of dipole Kramers-Wannier symmetry

In this section, we discuss the ’t Hooft anomaly of the dipole KW symmetry by checking whether
there is a dipole SPT phase invariant under such symmetry. A theory with an anomaly-free symmetry in the low energy should be uniquely gapped. Therefore, we will find an anomaly if wecannot find a gapped phase with a unique ground state that is invariant under the dipole KWtransformation with a certain N . The anomaly imposes a nonperturbative constraint on the selfdual theories that they must have continuous or first-order phase transitions. This can help us
understand, for example, the phase diagrams of systems with Z3 dipole symmetry.

4.1

Anomaly free condition for general N

D
Suppose we have a theory with both invertible ZQ
N × ZN symmetry and the non-invertible dipole
KW symmetry. We will prove the anomaly of the dipole KW symmetry by contradiction. BecauseD
the charge and dipole symmetry ZQ
N ×ZN is anomaly free, the symmetric theory is compatible witha gapped phase with one ground state, i.e. dipole SPT phase. We exclude the trivial phase because
it is mapped to a dipole SSB phase under the dipole KW transformation and therefore does nothave a dipole KW symmetry. The dipole SPT phase has been studied [131] and classified [130],
which is given by an element of H 2 (ZN × ZN , U (1))/H 2 (ZN , U (1))2 = ZN . A simple example
is the stabilizer Hamiltonian:

HSPT-k = −

L X
N
X

[(Zi−1 Zi† )k Xi (Zi† Zi+1 )k ]m ,i=1 m=118(4.1)

where level k ∈ ZN corresponds to different classes. The SPT Hamiltonian is constructed fromthe trivial-phase Hamiltonian
L X
NX
Htriv = −(Xi )m ,(4.2)
i=1 m=1

by decorated domain wall construction
HSPT-k = TDk Htriv TD−k ,

TD =

L
Y

†
CZi−1,i CZi,i
,

(4.3)

i=1

where the transformation of control-Z gate on Pauli operators can be found in Table 2.
Since all the terms in the SPT Hamiltonian (4.1) commute with each other, the ground statesatisfies(Zi−1 Zi† )k Xi (Zi† Zi+1 )k |G.S.i = |G.S.i .
(4.4)
The SPT phases with different k are characterized by the charges of ground state |G.S.itQ ,tD with
twisted boundary conditions labelled by (tQ , tD ) [121, 132–134]. It is straightforward to calculate
the charges of symmetry operator ηQ , ηD on ground state |G.S.itQ ,tD
L
L
Y
Y
†
†
†
( Xi ) |G.S.itQ ,tD =(Zi−1
Zi2 Zi+1)k |G.S.itQ ,tD = (Z0† Z1 ZL ZL+1
)k |G.S.itQ ,tD = ω −ktD |G.S.itQ ,tD ,
i=1
L
Y

(

i=1

Xii ) |G.S.itQ ,tD =

i=1
L
Y

†
†
(Zi−1
Zi2 Zi+1)ki |G.S.itQ ,tD = (Z0† ZL )k |G.S.itQ ,tD = ω ktQ |G.S.itQ ,tD ,

i=1

(4.5)
where we used the twist boundary condition of Pauli operators in the last equality in each equation.
Therefore the ground state |G.S.itQ ,tD is in the symmetry-twist sector labelled by(uQ = −ktD , tQ , uD = ktQ , tD ).

(4.6)

If the ground states after the dipole KW transformation do not stay in the same sector, the SPT
phase is not invariant under this transformation. If for given N every k we cannot find an SPT thatis invariant under the dipole KW transformation, then the symmetry is anomalous.
Now let us check whether there is an SPT phase invariant under dipole KW transformation.
The dual Hamiltonian is
Hk0 = −

L X
N
X

†
†
[Zi−1
Zi Xik Zi Zi+1
]m .

(4.7)

i=1 m=1

The dual Hamiltonian is not necessarily an SPT but still a stabilizer model. We will classify thedual Hamiltonian with different k, N into different gapped phases in Sec. 5.2. The corresponding
ground state(s) |G.S.0 i of Hk0 satisfies††
Zi−1
Zi Xik Zi Zi+1
|G.S.0 i = |G.S.0 i .

19

(4.8)

In the dual systems, we label twist boundary conditions of dual systems using btQ , btD . Then the0
charges of ground state |G.S. ibtQ ,btD with twisted boundary conditions are
L
L
Y
Y
b
†
†
†
( Xi )k |G.S.0 ibtQ ,btD =(Zi−1
Zi2 Zi+1)−1 |G.S.0 ibtQ ,btD = (Z0† Z1 ZL ZL+1)−1 |G.S.0 ibtQ ,btD = ω tD |G.S.0 ibtQ ,btD ,
i=1
L
Y

(

i=1

Xii )k |G.S.0 ibtQ ,btD =

i=1
L
Y

†
†
(Zi−1
Zi2 Zi+1)−i |G.S.0 ibtQ ,btD = (Z0† ZL )−1 |G.S.0 ibtQ ,btD = ω −tQ |G.S.0 ibtQ ,btD ,
b

i=1

(4.9)
which shows that the ground state |G.S.0 ibtQ ,btD is in the symmetry-twist sector labelled by(buQ , btQ = −kb
uD , ubD , btD = kbuQ ).

(4.10)

Therefore, if there is an SPT phase invariant under dipole KW transformation, the ground statecharges of (4.7) and (4.1) under the same twisted boundary condition should be consistent witheach other, that is(4.6)(4.10)ubQ = −kbtD = −k 2 u
bQ ,

(4.6)

(4.10)ubD = kbtQ = −k 2 u
bD ,(4.11)

where all equations are valued modulo N . Thus k should satisfy k 2 = −1 mod N , i.e., −1
is a quadratic residue modulo N , which is the necessary anomaly-free condition for dipole KW
symmetry.10
Indeed, this condition is also a sufficient anomaly-free condition. This is because the dual
Hamiltonian (4.7) is the same as the Hamiltonian (4.1) in this case. When k 2 = −1 mod N , N
is coprime with k. Hence, for each m ∈ ZN , there exist a unique jm ∈ ZN satisfying kjm =m mod N . As a result, we have the
Hk0 = −

L X
N
X

†
†
[Zi−1
Zi Xik Zi Zi+1]jm

i=1 m=1
L X
NX(4.12)

†
†
=−
(Zi−1
Zi )jm Xim (Zi Zi+1
)jm = HSPT-k ,i=1 m=1

where in the last equation, we use the fact that jm = −k 2 jm = −km mod N .
In summary, when −1 is a quadratic residue modulo N , the dipole KW symmetry is anomalyfree and there exists a dipole SPT also protected by this non-invertible symmetry; otherwise it isan anomalous symmetry.

4.2

Anomaly and phase diagram for N = 3

When N = 3, the dipole KW symmetry is anomalous because −1 is not a quadratic residue
modulo 3. Indeed, this anomaly can help us understand the phase transition in the phase diagram
10

Similar constraints also appear in the anomaly of non-invertible duality defects in (3 + 1)d [19].

20

of the dipole Ising model:
Hdipole-Ising = −g

−1

L
X

Zi−1 (Zi† )2 Zi+1 − g

i=1

L
X

Xi + (h.c.)

i=1

L
X
p
2
−2
= g +g(cos θZi−1 (Zi† )2 Zi+1 + sin θXi ) + (h.c.),(4.13)

i=1ppwhere we defined an angle θ by cos θ = −g −1 / g 2 + g −2 , sin θ = −g/ g 2 + g −2 . The dipole
Ising model can be mapped to the XZ chain
L
X
p
†
†
2
−2
HXZ = g + g(cos θXi−1
Xi + sin θZi−1
Zi ) + (h.c.),(4.14)

i=1
D
by gauging the ordinary ZQ
a ZZ3 symmetry gener3 symmetry. The Z3 dipole symmetry becomesQ
QL
L
ated by i=1 Zi and we have a new quantum ZX
N symmetry generated by
i=1 Xi .
The phase diagram of Hamiltonian (4.14) has been determined by numerical calculation in[110, 111], under the name “quantum torus chain”. We show the phase diagram in the left part of
Fig. 3, which can be summarized as follows:

1. The phase diagram exhibits a symmetric pattern across the line of θ = 0.25π, θ = 1.25π, at
which points the model is invariant under the global Hadamard gate U H : Xi → Zi† , Zi →Xi , ∀i.
2. Around θ = 0.25π, this model hosts a large gapless phase region, θ ∈ (−0.1π, 0) ∪(0, 0.5π) ∪ (0.5π, 0.6π), with center charge c = 2. At two particular points θ = 0, 0.5π
there are first-order phase transitions with exponentially large ground state degeneracy.
3. There is a first order phase transition at θ = 1.25π, which separates the ZZ3 SSB phasewith nonzero hXi i when θ ∈ (0.6π, 1.25π) and the ZX
3 SSB phase with nonzero hZi i whenθ ∈ (1.25π, 1.9π). The first-order phase transitions at θ = 0.6π, 1.9π separate gapped SSB
phases and gapless phases.
As the KW transformation preserves the structure of the phase diagram and the center chargeof gapless region, we can obtain the phase diagram of dipole Ising model. We show the phase
diagram in the right part of Fig. 3 and summarize it here
1. The phase diagram exhibits a symmetric pattern across the line of θ = 0.25π, θ = 1.25π atwhich points the model is invariant under the dipole KW transformation D̂.
2. Around θ = 0.25π, the system is in the gapless phases with c = 2 when θ ∈ (−0.1π, 0) ∪(0, 0.5π) ∪ (0.5π, 0.6π). These phases are separated by first-order phase transitions at θ =0, 0.5π, where the Hamiltonian is dominated by the term Zi−1 (Zi† )2 Zi+1 + (h.c.) and Xi +(h.c.) respectively. therefore, there is also an exponentially large ground state degeneracy
NGS ∼ 2L .
21

3. The θ = 1.25π is a first order phase transition separating a trivial gapped phase and a dipole
SSB phase. When θ ∈ (1.25π, 1.9π), the Hamiltonian is dominated by −Xi + (h.c.) andthe system is in the trivially gapped phase. When θ ∈ (0.6π, 1.25π), the Hamiltonian isdominated by −Zi−1 (Zi† )2 Zi+1 + (h.c.). There are nine ground states when L → ∞ (with†the sequence of L ≡ 0 mod 3) with nonzero hZi−1
Zi i and hZi i and the system is in the
D
ZQ
×
Z
SSB
phase.
There
are
also
first-order
phasetransitions at θ = 0.6π, 1.9π between
33different gapped phases and gapless phases.
4. We highlight that the gapless feature of self-dual points at θ = 0.25π (g = −1) and firstorder phase transition at self-dual points 1.25π (g = 1) are consistent with the anomalous
dipole KW symmetry because the anomaly forbids the system to be uniquely gapped.

0.6π 0.5π

0.6π 0.5π
0.25π

ZZ3 SSB

Two component
Luttinger liquid
Dipole SSB

0Gapless region
with c = 2

−0.1π

0
−0.1π

ZQ
3 SSB

Trivially gapped

1.25π

1.25π

Figure 3: The phase diagram of XZ model (4.14) (left) and the dipole Ising model (right)
where dashed lines are first order phases transition and solid lines are continuous phase transitions. The red dots θ = 0.25π, 1.25π (right) are critical points where the dipole Ising modelhas the anomalous non-invertible symmetry.

5

Generalized Kennedy-Tasaki transformation associated withDZQ
N × ZN symmetry

In this section, we will discuss generalized Kennedy-Tasaki (KT) transformations associated withDZQ
N × ZN symmetry [120], which relate dipole SSB phases and dipole SPT phases.

22

5.1

Construction of KT transformations

In Sec 4.1, we defined a TD transformation for systems with dipole symmetries
TD :=

L
Y

†
CZi−1,i CZi,i
,

(5.1)

i=1

which generates the dipole SPT phases from the trivial phase:TDk :Htriv → HSPT-k .

(5.2)

A single TD transformation will increase the SPT level by one11
TD :HSPT-k → HSPT-(k+1) .

(5.3)

TD is an invertible transformation and TD−k maps a dipole SPT with level k to a trivial phase (withlevel 0). Because we can map a trivial phase to the dipole SSB phase
HSSB = −

L X
N
X

†
†
(Zi−1
Zi2 Zi+1
)m ,

(5.4)

i=1 m=1

via the dipole KW transformation D̂, and further the SSB phase is invariant under the TD transformation, we can define a duality transformation, the generalized KT transformation, between the
SSB phase and the SPT phase with each k
KTk := TDk D̂TD−k :HSPT-k → HSSB ,HSSB → HSPT-k .

The duality web is shown in Fig. 4.
This KT transformation is non-invertible with the fusion rule
! N
!
N
X
X
m
m
KTk × KTk =
ηQ
ηD
C.
m=1

(5.5)

(5.6)

m=1

D
Thus when acting KTk twice, the Hamiltonian with ZQ
N × ZN and charge conjugation symmetry
is invariant, which implies KTk is indeed a duality transformation for such Hamiltonian, e.g. theSPT Hamiltonian (4.1).12

5.2

Mapping of gapped phases under KT transformation

In the previous subsection, we define the KT duality transformation KTk for each dipole SPT withlevel k. Now, we would like to study the general transformation of KTp on SPT with a different
11Here we use the notation that the trivial phase is the SPT phase with level 0.
Indeed, if we apply charge conjugation to the SPT phase and SSB phase, one can show the features of thesephases, e.g., ground state charge under twisted boundaries and ground state degeneracy, are kept invariant.
12

23

KTk

TDk

SSB

TDk

D̂

Trivial

TD−k

SPTk

Figure 4: Transformations between the trivial phase, the SSB phase and the SPT phase withlevel k. KT transformation TDk D̂TD−k is a duality map between the SSB phase and SPT phasewith level k.

level p + k with k 6= 0. We can calculate the transformation in sequence because of the definition
KTp = TDp D̂TD−p . In the first step, TD−p will map the SPT with level p + k to the SPT with level k.
The second step is to determine which gapped phase the SPT will become after the dipole KWtransformation, which depends on whether k and N are coprime. Then in the third step, TD−p willfurther stack some SPTs to the model. Suppose after the KT transformation KTp on HSPT-(p+k) we
get a dual Hamiltonian. We summarize the results here and leave the derivation later. The duality
web is shown in Fig. 5.
1. When gcd(k, N ) > 1, the dual Hamiltonian is a partial SSB phase with unbroken ZQ
N/gcd(k,N ) ×
D
ZN/gcd(k,N ) symmetry and stacked with a SPT phase of unbroken symmetry with leveln + p × gcd(k, N ), where nk = −gcd(k, N ) mod N .
2. When gcd(k, N ) = 1, the dual Hamiltonian is a SPT phase with level n0 + p, where n0 k =−1 mod N .
Mapping of SPT phases under dipole KW transformation
This part provides the technique details about the duality web of KT transformation on SPT withat a general level.
Start from the second step and suppose after the dipole KW transformation on HSPT-k we
get the dual Hamiltonian Hk0 . It is sufficient to study the ground-state degeneracy of the dual
Hamiltonian with the periodic boundary condition. Recall the analysis in Sec. 4.1 and (4.10),
where the ground states of the dual Hamiltonian are in symmetry-twist sector labeled by (buQ , btQ =b−kbuD , ubD , tD = kb
uQ ). In the periodic boundary condition, we havebtQ = −kb
uD = 0,btD = kb
uQ = 0, mod N.

(5.7)

Therefore, ubD and ubQ must be multiples of N/gcd(k, N ) and the ground state degeneracy is
2
gcd(k, N ) .
24

SPTp+n1
n1
KT p+

SSB

KTp

KT

p+k

1

KTp+k2

SPTp+k1

SPTp+k2

KTp

G1 SSB ⊗ G2 SPTp+n2

Figure 5: Duality web between various gapped phases. Unless specified, the SSB phaseDbreaks all G = ZQ
N × ZN symmetry and the SPT phase is protected G symmetry. k1 , n1
are coprime with N and n1 k1 = −1 mod N . k2 is not coprime with N and k2 n2 =QDD−gcd(k2 , N ) mod N . G1 = ZQ
N/gcd(k2 ,N ) × ZN/gcd(k2 ,N ) and G2 = Zgcd(k2 ,N ) × Zgcd(k2 ,N ) .

When gcd(k, N ) > 1: The dual Hamiltonian has degenerate ground states and the remaining
gcd(k,N )gcd(k,N )D
unbroken subgroup ZQ
and ηD, as its groundN/gcd(k,N ) × ZN/gcd(k,N ) is generated by ηQ
state charge under PBC is trivial. The dual Hamiltonian is therefore in an SSB phase stacked withD
a ZQ
N/gcd(k,N ) × ZN/gcd(k,N ) SPT phase. Then we need to determine the level of the SPT phase.
Because of the broken symmetry, the twist sector label btQ , btD takes value in gcd(k, N )ZN/gcd(k,N ) .
It is convenient to use the reduced twist variables(btQ , buD ) = (btQ /gcd(k, N ), b
tD /gcd(k, N )) ∈ Z2N/gcd(k,N )(5.8)

for twist sectors of the unbroken symmetry. Moreover, we can also define the symmetry sector of
unbroken symmetry (buQ , buD ) = (buQ , ubD ) mod N/gcd(k, N ) where (buQ , buD ) ∈ Z2N/gcd(k,N ) . ThisD
is because any state |ψi with ZQ
uQ , ubD ) satisfies
N × ZN charge (bgcd(k,N )ηQ|ψi = ω gcd(k,N )buQ |ψi = (ω 0 )ubQ |ψi = (ω 0 )buQ |ψi,

gcd(k,N )|ψi = ω gcd(k,N )buD |ψi = (ω 0 )ubD |ψi = (ω 0 )buD |ψi,

ηD

(5.9)

gcd(k,N )where ηQ(D) is the generator of the unbroken symmetry and ω 0 = ω gcd(k,N ) . To further determine which class the stacked SPT belongs to, we need to check the symmetry-twist sector of theDground states in terms of the unbroken symmetry ZQ
N/gcd(k,N ) × ZN/gcd(k,N ) to follow (4.6), i.e.
(buQ = −nbtD , btQ , b
uD = nbtQ ) mod (N/gcd(k, N )),(5.10)

if the level is n. On the other hand, the symmetry-twist sector after the dipole KW transformation
should satisfy (4.10) and we havebtQ = −kbuD ,btD = kb
uQ mod N,
25(5.11)

andbtQ = −(k/gcd(k, N ))buD ,btD = (k/gcd(k, N ))buQ mod (N/gcd(k, N )),(5.12)

in terms of the reduced variables. Since N/gcd(k, N ) is coprime with k/gcd(k, N ), we can finda unique integer n ∈ ZN/gcd(k,N ) satisfying n(k/gcd(k, N )) = −1 mod (N/gcd(k, N )). Then we
have
nbtQ = −n(k/gcd(k, N ))buD = buD ,nbtD = n(k/gcd(k, N ))buQ = −buQ mod (N/gcd(k, N )).
(5.13)
Q
D
Compared with (5.10), the stacked ZN/gcd(k,N ) × ZN/gcd(k,N ) SPT has level n, wheren(k/gcd(k, N )) = −1 mod (N/gcd(k, N ))→nk = −gcd(k, N ) mod N.

(5.14)

D
In the third step, TDp transformation stacks a ZQ
N × ZN SPT phase with level p. To detect which
Q
D
ZN/gcd(k,N ) × ZN/gcd(k,N ) class such an SPT belongs to, we consider a symmetry-twist sector ofD
4
unbroken ZQ
N/gcd(k,N ) × ZN/gcd(k,N ) symmetry labeled by (uQ , tQ , uD , tD ) ∈ ZN/gcd(k,N ) , which has
D
the relation with symmetry-twist sectors of ZQ
N × ZN symmetry:(uQ , uD ) = (uQ , uD ) mod N/gcd(k, N ),(tQ , tD ) = (gcd(k, N )tQ , gcd(k, N )tD ).

(5.15)

D
Recall the ground state of ZQ
N × ZN SPT with level p is in the symmetry-twist sector labelled by(uQ = −p tD , tQ , uD = p tQ , tD ).

(5.16)The ground state then also satisfies(uQ = −p gcd(k, N )tD , uD = p gcd(k, N )tQ ) mod N/gcd(k, N ),(5.17)

D
which implies such SPT belongs to ZQ
N/gcd(k,N ) × ZN/gcd(k,N ) SPT phase with level p × gcd(k, N ).
Thus we can conclude that the KT-dual system is in a partial SSB phase with unbroken
Q
ZN/gcd(k,N ) × ZD
N/gcd(k,N ) symmetry and stacked with an SPT phase of unbroken symmetry withlevel n + p × gcd(k, N ).

When gcd(k, N ) = 1: Because k is coprime with N , the dual Hamiltonian is in an SPT phase.
The symmetry-twist sector after the dipole KW transformation should satisfy (4.10)btQ = −kbuD ,btD = kb
uQ mod N.

(5.18)

Because k is coprime with N , we can find an unique integer n ∈ ZN such that nk = −1 mod N .
therefore,nb
tQ = −nkb
uD = ubD , nbtD = nkb
uQ = −buQ mod N.
(5.19)
D
Compared with (4.6), the dual system is in the ZQ
N × ZN SPT phase with level n. Then in the thirdpstep, TD maps the SPT with level n to an SPT with level n + p.

26

6Conclusion and discussion

In this paper, we constructed the dipole KW transformation by composing the seed transformation
and by gauging the charge and dipole symmetry. Then we study this new non-invertible symmetry
through fusion algebras (2.28), topological defects (in Fig. 2) and the anomaly (and its constraints
in the phase diagram in Fig. 3). As an application, we constructed generalized KT transformations(in Fig. 4) that connect various gapped phases with dipole symmetry in a duality web (in Fig. 5).
We studied the fusion algebra (2.28) of the dipole KW symmetry and other invertible symmetry operators. The fusion rule of non-invertible dipole KW symmetry mixes with the charge
conjugation symmetry and therefore the whole fusion algebra is different from the TY(ZN × ZN )fusion algebra. It is interesting to further determine the fusion category data, in particular the Fsymbols which are related to the associativity of the fusion algebra and the anomaly of the fusioncategory symmetry.
We have shown that due to the anomaly of the non-invertible symmetry, the two self-dual
points in the phase diagram of dipole Ising model with N = 3 cannot be uniquely gapped. Thisis consistent with numerical calculations in the anisotropic XZ model (which is the ordinary KW
dual of the dipole Ising model), where one self-dual point is a first order phase transition and theother one is in a large gapless region with center charge c = 2. It will be interesting to investigate
numerically whether the gapless phase will become unstable with symmetric perturbations when
we choose N such that the non-invertible symmetry is not anomalous.
It is also interesting to extend the composing construction. Consider a theory A with a global
symmetry generated by an operator G and a non-invertible duality transformation K between theories A and B. There is a non-invertible operator K† GK that commutes with the Hamiltonianof the theory B. Although whether or not K† GK is an interesting non-invertible symmetry depends on specific models, this might give a systematic construction and perhaps classification ofcertain types of lattice non-invertible symmetries. Here are examples in the literature with similar constructions. In (2 + 1)d, the subsystem non-invertible symmetry has been constructed [32]by composing the KW transformation acting on lines and columns. Recently in [135], the authors constructed a non-invertible symmetry in (2 + 1)d with cosine function fusion rule using the
sandwich construction.
In the end, we sketch another application inspired by [126]. In this paper, the authors studied(0)(1)
the non-invertible symmetry in (2 + 1)d lattice model with Z2 × Z2 symmetry. An example isthe lattice Z2 gauge theory with the Ising matter
XY
XY
X
X
X
H=−
σlx −
σlz −
σlz −
Xv −
Zv Zv0 ,
(6.1)
v

l3v

f

l∈f

l

v

hv,v 0 ix(z)where σl denotes a gauge spin on the link, Xv , Zv a matter spin on the site, and hv, v 0 i denotethe link of nearest sites v, v 0 . The non-invertible symmetry is
1
D = SC :
2

Xv ←→

Y

σlx ,

l3v

27

z
Zv Zv0 ←→ σhv,v
0i,

(6.2)

where S is the non-invertible swap operator and C is the condensation operator whose precise
definition can be found in [126]. We will take (6.2) as the seed transformation and construct a
new non-invertible symmetryYY
DU H D : Xv ←→Zv Zv0 =Zv 0 ,(6.3)
l=hv,v 0 i3v

l=hv,v 0 i3v

Qwhere U H = l UlH is the product of Hadamard gate on every link. This non-invertible symmetry can be found in lattice models with subsystem Z2 symmetry acting on the diagonal lines.
Especially, the (2 + 1)d cluster modelXY
H2d cluster = −
Xv
Zv0 ,
(6.4)
v

l=hv,v 0 i3v

is protected by this non-invertible symmetry. We leave the detailed investigation of this noninvertible symmetry (6.3) for future work.

Acknowledgements
We would like to thank Yunqin Zheng for initial collaboration and for inspiring suggestions anddiscussions. We thank Shuheng Shao for useful comments after reading the draft. W.C. thanks
Shinsei Ryu, Ramanjit Sohal, Sahand Seifnashri, Kantaro Ohmori, Shuheng Shao for useful discussions. W.C. is 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 (20H05860, 23K17689, 23K25865), and by JST, Japan (PRESTO
Grant No. JPMJPR225A, Moonshot R&D Grant No. JPMJMS2061). He would also like to thankthe Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial
support during the completion of this work. The authors of this paper were ordered alphabetically.

A

Kramers-Wannier duality symmetry in lattice models

In this Appendix, we review KW transformation obtained by gauging a ZN symmetry on a 1d spin
chain. The main idea has been discussed e.g. in [5]. For reference, the definitions of ZN unitary28

Quantum Gate

Nontrivial transformation

CZi,jXi → Xi Zj , Xj → Xj Zi

UiH

Xi → Zi† , Zi → Xi

Si,jXi → Xj , Xj → Xi , Zi → Zj , Zj → Zi

Table 2: Nontrivial actions of the ZN quantum gates.
gates areN

1 X −αβ α βCZi,j =
ωZi Zj ,
N α,β=1NX

1UiH = √

N α,β=1

ω −αβ |βi hα| ,

(A.1)

N

Si,j =

1 X αβ α βω (Xi Zi )(Xj−α Zj−β ),N α,β=1

and their actions are summarized in table 2.

A.1

Gauging ZN symmetry on the whole spin chain

For simplicity, consider ZN clock model at critical point on a chain with L sites
H=−

L
X
†
(Zi−1
Zi + Xi ) + (h.c.).

(A.2)

i=1

Q
Since the ZN symmetry η = Li=1 Xi is onsite, and thus non-anomalous, we can gauge thissymmetry by coupling the gauge variables X̃i+1/2 , Z̃i+1/2 on the link. The gauged Hamiltonian is
Hgauged = −

L
X

†
(Zi−1
X̃i−1/2 Zi + Xi ) + (h.c.).

(A.3)

i=1

which has a enlarged Hilbert space with dimension N 2L and gauge symmetries generated by†
Gi = Z̃i−1/2 Xi Z̃i+1/2,[Gi , Hgauged ] = 0,

∀i.

(A.4)

We need to project to gauge-invariant sector by imposing Gauss law constraints†
Gi = Z̃i−1/2 Xi Z̃i+1/2
= 1,

→
29

†
Xi = Z̃i−1/2Z̃i+1/2 ,

∀i.

(A.5)

The gauged Hamiltonian has a new 0-form ZN symmetry which is easily seen by introducing a
new set of Pauli operatorsẐi−1/2 = Z̃i−1/2 ,†X̂i−1/2 = Zi−1
X̃i−1/2 Zi .

(A.6)

The gauged Hamiltonian becomesL
X†
Hgauged = −(X̂i−1/2 + Ẑi−1/2Ẑi+1/2 ) + (h.c.).

(A.7)

i=1

We recover the original Hamiltonian by renaming the variablesẐi−1/2 → Zi−1 ,X̂i−1/2 → Xi−1 .

(A.8)

The whole process gives the famous KW transformation on the ZN -symmetric operator†
Zi−1
Zi → Xi−1 ,†Xi → Zi−1
Zi .

(A.9)

The renaming is an isomorphism between two Hilbert spaces, both of which we denote by H.
Notice that the identification (A.9) involves a “half-translation” on the lattice, i.e., the fusion ruleof KW duality operator will involve the one-site translation operator T.

A.2

Gauging ZN symmetry on a half of the spin chain

Suppose we only gauge the ZN symmetry on the half of the closed chain. Without loss of generality, we couple the gauge variables X̃i , Z̃i on J links from (L, 1) to (J − 1, J) with J < L. The
half-gauged Hamiltonian is
Hhalf-gauged = −

J
X

†
(Zi−1
X̃i−1/2 Zi + Xi ) −

i=1

L−1
X

†
(Zi−1
Zi + Xi ) + (h.c.).

(A.10)

i=J+1

The dimension of the enlarged Hilbert space is N L+J . However, there are only J − 1 generatorsof gauge symmetry†
Gi := Z̃i−1/2
Xi† Z̃i+1/2 , i = 1, ..., J − 1,
(A.11)
that commute with the half-gauged Hamiltonian Hhalf-gauged . After imposing the J − 1 Gauss law
constraints Gi = 1, i = 1, ..., J − 1 and projecting to the gauge invariant sector, the resulting
Hilbert space has dimension N J+1 , with an extra degree of freedom compared with the original
Hilbert space. We can then introduce a new set of Pauli variables X̂i , Ẑi for the gauged half lineẐ1/2 := Z̃1/2 ,X̂1/2 := X̃1/2 Z1 ,Ẑi−1/2 := Z̃i−1/2 ,†X̂i−1/2 := Zi−1
X̃i−1/2 Zi−1 ,ẐJ−1/2 := Z̃J−1/2 ,†X̂J−1/2 := ZJ−1
X̃1/2 .

30

i = 2, ..., J − 2(A.12)

The boundary terms are defined so that they have standard commutation relation with the operators
at site J and L. The gauged Hamiltonian then becomes†
Hhalf-gauged = − (ZL† X̂1/2 + Ẑ1/2
Ẑ3/2 ) −− (X̂J−1/2 ZJ + XJ ) −J−1
X(X̂i−1/2 + Ẑi−1/2 Ẑi+1/2 )

i=2
L−1
X

(A.13)†(Zi−1
Zi + Xi ) + (h.c.).

i=J+1

After renaming X̂i−1/2 → Xi , Ẑi−1/2 → Zi for i = 1, ..., J −2 and X̂J−1/2 → X(J−1,J) , ẐJ−1/2 →Z(J−1,J) , which is the extra degree of freedom after imposing the Gauss law, the gauged Hamiltonian becomes
Hhalf-gauged = − (ZL† X1 + Z1† Z2 ) −J−2X(Xi + Zi† Zi+1 )

i=2†− (XJ−1 + ZJ−1 Z(J−1,J) + X(J−1,J) ZJ + XJ )

−

L−1
X

(A.14)†(Zi−1
Zi + Xi ) + (h.c.).

i=J+1†
By conjugating the Hamiltonian with the unitary operator CZ(J−1,J),J, and reshuffling the terms,
we get a simpler form

Hhalf-gauged = − (ZL† X1 ) −

J−2
X

†
(Zi−1
Zi + Xi )i=2††− (ZJ−1 Z(J−1,J) + X(J−1,J) + Z(J−1,J)
XJ )

−

L−1
X

(A.15)†(Zi−1
Zi + Xi ) + (h.c.).

i=J+1

From the above expression, the gauged Hamiltonian is also given by the original Hamiltonian byinserting the KW defect D at the link (L, 1) and the Hermitian conjugate D† at the link (J − 1, J).
Now we can study the movement and fusion of the non-invertible topological defects. The
defect Hamiltonian with insertion of a single KW defect D on the link (J − 1, J) is(J−1,J)
HD
=−

J−1
X

†
†
(Zi−1
Zi + Xi ) − ZJ−1
XJ −

i=1

L−1
X

†
(Zi−1
Zi + Xi ) + (h.c.),(A.16)

i=J+1†
and we can use the unitary operator UDJ = CZ(J,J+1)
HJ† , implementing the transformation

XJ → ZJ ,ZJ → XJ† ZJ+1

XJ+1 → ZJ† XJ+1 ,31

ZJ+1 → ZJ+1 ,(A.17)

to move this defect to the link (J, J + 1).
To fuse the defects considering the insertion of two KW defects on the adjacent links (J −1, J)
and (J, J + 1)
(J−1,J),(J,J+1)
HD;D
=−

J−1
L−1
X
X
†
†
†
†
(Zi−1 Zi + Xi ) − (ZJ−1 XJ + ZJ XJ+1 ) −(Zi−1
Zi + Xi ) + (h.c.).
i=1

i=J+1

(A.18)
The fusion operator λJD⊗D = (UDJ )† will map this Hamiltonian to(J−1,J+1)HD⊗D(J−1,J),(J,J+1)=λJD⊗D HD;D
=−

J−1
X

(λJD⊗D )†

†
†
(Zi−1
Zi + Xi ) − (ZJ† ZJ−1
ZJ+1 + XJ+1 ) −

i=1

L−1
X

†
(Zi−1
Zi + Xi ) + (h.c.).

i=J+1(A.19)In the fused defect Hamiltonian, the degrees of freedom on site J is decoupled and is equivalent
to an insertion of the T−1 defect [5], while ZJ becomes a symmetry of the new Hamiltonian.
The Hilbert space is a direct sum of the eigenspaces of ZJ . When ZJ takes an eigenvalue ω k ,the new Hamiltonian is equivalent to the original Hamiltonian with a defect η k inserted between
J − 1, J + 1. Therefore the topological defect D follows the same fusion rule as its operator
counterpart:
!
N
X
D×D=
η k T−1 .
(A.20)
k=1

B

Topological defects of invertible symmetries in dipole Isingmodel

In this section, we review the topological defects of invertible symmetries ηQ , ηD , C in the dipole
Ising modelXX
H = −g −1
Zi−1 (Zi† )2 Zi+1 − g
Xi + (h.c.).
(B.1)
i

i

Much of the discussion can be generalized to other models with dipole symmetry. We study thedefects through the defect Hamiltonian which is obtained by conjugating the original Hamiltonian
with a twist operator, the symmetry transformation on a half of the infinite chain.
ηQ defect
The charge symmetry ZQ
N is generated by
ηQ :=

Y

Xi ,

i

32

N
ηQ
= 1,

(B.2)

while the ZN twist operator acting on i ≤ 0 isηQ,0 =

Y

Xi .

(B.3)

i≤0

The defect Hamiltonian with a ηQ defect at the link (0,1) is
(0,1)

HQ

†
= ηQ,0 HηQ,0
X
X= − g −1
Zi−1 (Zi† )2 Zi+1 − g
Xi

(B.4)

i

i6=0,1− g −1 ωZ−1 (Z0† )2 Z1 − g −1 ω −1 Z0 (Z1† )2 Z2 + (h.c.),where we label the defect Hamiltonian with the type and location of defect. We also show the
movement operators, which are local unitary operators and move the defect by one site. For
example, the movement operator for the ηQ defect is
UQ1 := X1 :

(0,1)

HQ

(1,2)

→ HQ

(0,1)

= Uη1 HQ(UQ1 )† .

(B.5)

Since the movement operators are local unitary operators, the movement does not cost energy andthe defect is topological. One can also put several defects and fuse them. The fusion of topologicaldefects shares the same rule of the fusion of corresponding operators.
ηD defect
The dipole symmetry ZD
N is generated by
ηD =

Y

Xii ,

N
ηD
= 1.

(B.6)

(Xi )i ,(B.7)

i

The twist operator acting on sites i < i0 is
ηD,i0 −1 =Y
i<i0with the movement operator (Xi0 )i0 . The defect Hamiltonian with a ηD defect inserted at the link(i0 − 1, i0 ) isX
X
(i −1,i )HD0 0 = − g −1
Zi−1 (Zi† )2 Zi+1 − gXiii6=i0 −1,i0
(B.8)
†
† 2
−1 −i02−1 i0 −1
− g ω Zi0 −2 (Zi0 −1 ) Zi0 − g ω
Zi0 −1 (Zi0 ) Zi0 +1 + (h.c.).
Charge conjugation defectThe charge conjugation symmetry ZC
2 is generated byYC=Ci , Ci : Zi → Zi† ,i

33

Xi → Xi† ,(B.9)

The twist operator is the truncation of C on half of the chain and the movement operator is Ci . The
defect Hamiltonian with a charge conjugation defect inserted at link (0, 1) is
X
X
(0,1)
HC = − g −1
Zi−1 (Zi† )2 Zi+1 − gXiii6=0,1(B.10)
† 2 †
† 2
−1
−1 †
− g Z−1 (Z0 ) Z1 − g Z0 (Z1 ) Z2 + (h.c.).

C

Computation in bilinear phase map representation

In this appendix, we compute all the operator relations and fusions rigorously in the BMP representation. For reference, we list the action of Pauli operators on bra and ket states
Zi |si i =Xi |si i =

N
X
α=1
N
X

ω α |αi hα|si i = ω si |si i ,hsi | Zi =|α + 1i hα|si i = |si + 1i ,NX

ω α hsi |αi hα| = ω si hsi | ,

α=1
N
X

hsi | Xi =

α=1

(C.1)
hsi |α + 1i hα| = hsi − 1| .

α=1

The bra and ket states are eigenstates of the Zi , while Xi shifts the ket state by +1 and the bra
state by −1. Starting from a state |{si }i = ⊗Li=1 |si ii , we define the state after a lattice translationas |{si−1 }i := T |{si }i := ⊗Li=1 |si−1 ii , where all values of qudit on each site has been move to
one site right. The lattice translation operator is therefore defined asXT=|{si−1 }i h{si }|
(C.2)
{si }

and its action on Pauli operators is
TZi T−1 = Zi+1 ,TXi T−1 = Xi+1 .

(C.3)

KW transformation
The KW transformation is given by
X PL
X PL 0 00D=ω i=1 (si−1 −si )si−1 |{s0i }i h{si }| =
ω i=1 (si −si−1 )si |{s0i }i h{si }| .
{si },{s0i }{si },{s0i }

34

(C.4)

The transformation of Pauli operators is




X PL
0
†
†
D Zj−1Zj =ω i=1 (si−1 −si )si−1 |{s0i }i h{si }| Zj−1
Zj{si },{s0i }

X

=

ω

PL

0
i=1 (si−1 −si )si−1|{s0i }i h{si }| ω −(sj−1 −sj ){si },{s0i }

X

=

ω

P

0
0
i6=j−1 (si−1 −si )si−1 +(sj−1 −sj )(sj−1 −1)ωP0i (si−1 −si )si−1|{s0i }i h{si }|(C.5){si },{s0i }X=|{s0i6=j−1 ; s0j−1 + 1}i h{si }|{si },{s0i }= Xj−1 D,
where we rename s0j−1 − 1 by ŝ0j−1 after the third equality and drop the hat in the fourth equality.
The notation |{s0i6=j−1 ; s0j−1 + 1}i differs from |{s0i }i only by the shift of qudit on the j − 1 site.
Similarly,
X PL 0 0
DXj =ω i=1 (si −si−1 )si |{s0i }i h{si }| Xj{si },{s0i }

=

X

ω

PL

0
0
i=1 (si −si−1 )si|{s0i }i h{si6=j ; sj − 1}|{si },{s0i }

=

X

(C.6)
ω

P

0
0
0
i6=j (si −si−1 )si +(sj −sj−1 )(sj +1)|{s0i }i h{si }|{si },{s0i }†= Zj−1
Zj D.

The fusion of translation operator with the KW operator is
X PL 0 0
TD =ω i=1 (si −si−1 )si T |{s0i }i h{si }|{si },{s0i }

=

X

ω

PL

|{s0i−1 }i h{si }|PL|{ŝi }i h{si }|00i=1 (si −si−1 )si{si },{s0i }

=

X

ω

i=1 (ŝi+1 −ŝi )si(C.7){si },{ŝi }

=

X

ω

PL

i=1 −(ŝi −ŝi+1 )si|{ŝi }i h{si }| ,0i=1 −(si−1 −si )si−1|{si }i h{s0i }| .

{si },{ŝi }and the conjugate of D is
D† =

X

ω

PL

{si },{s0i }

35

(C.8)

Therefore, by renaming the dummy variables we show that TD = D† . Similarly,
X PL 0 0DT =ω i=1 (si −si−1 )si |{s0i }i h{si }| T{si },{s0i }

=

X

ω

PL

|{s0i }i h{si+1 }|PL|{s0i }i h{ŝi }|00
i=1 (si −si−1 )si{si },{s0i }

=

X

ω

0
0
i=1 (si −si−1 )ŝi−1(C.9){s0i },{ŝi }

=

X

ω

PL

0
0
i=1 −(si−1 −si )ŝi−1|{s0i }i h{ŝi }| = D† .

{si },{ŝi }Therefore we confirmed that the translation operator commutes with the KW operator and theirfusion is the Hermitian conjugate of the KW operator:D† = TD = DT.

(C.10)The non-invertible fusion rule is given by
X
PL
PL
0
0
0
D×D=
ω i=1 (si −si−1 )si ω i=1 (mi−1 −mi )mi−1 |{s0i }i h{si }|{m0i }i h{mi }|={si },{s0i },{mi },{m0i }
X
PL

ω

0
0
i=1 (si −si−1 +mi −mi+1 )si|{s0i }i h{mi }|{si },{s0i },{mi }

=

L
X Y

(C.11)

δs0i −s0i−1 +mi −mi+1 ,0 |{s0i }i h{mi }| .

{s0i },{mi } i=1

The solution for the constraints s0i − s0i−1 + mi − mi+1 = 0, i = 1, .., L is
s0i = mi+1 + k,

k = 1, ..., N, i = 1, ..., L.

(C.12)

Therefore, the fusion rule is
D×D=

N
XX

|{mi+1 + k}i h{mi }|{mi } k=1

=

N
X

!
X

ηk

=

(C.13)

{mi }

k=1
N
X

|{mi+1 }i h{mi }|

!
ηk

T−1 .

k=1

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