Z3and Z33symmetry protected topological paramagnets Hrant TopchyanaVasilii IugovbcMkhitar MirumyanaShahane Khachatryana

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Z3and (×Z3)3symmetry protected topological

paramagnets

Hrant Topchyan,aVasilii Iugov,b,c Mkhitar Mirumyan,aShahane Khachatryan,a

Tigran Hakobyan,a,d Tigran Sedrakyane,1

aAlihkanyan National Laboratory, Yerevan Physics Institute, Alikhanian Br. 2, 0036, Yerevan,

Armenia

bSimons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY, 11794-3636,

USA

cC. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794-

3636, USA

dYerevan State University, Alex Manoogian 1, 0025, Yerevan, Armenia

eDepartment of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA

E-mail: hranttopchyan1@gmail.com,vasyayugov@gmail.com,

mkhitar@gmail.com,shah@yerphi.am,tigran.hakobyan@ysu.am,

tsedrakyan@umass.edu

Abstract: We identify two-dimensional three-state Potts paramagnets with gapless edge

modes on a triangular lattice protected by (×Z3)3≡Z3×Z3×Z3symmetry and smaller

Z3symmetry. We derive microscopic models for the gapless edge, uncover their symmetries

and analyze the conformal properties. We study the properties of the gapless edge by

employing the numerical density-matrix renormalization group (DMRG) simulation and

exact diagonalization. We discuss the corresponding conformal ﬁeld theory, its central

charge, and the scaling dimension of the corresponding primary ﬁeld. We argue, that

the low energy limit of our edge modes deﬁned by the SUk(3)/SUk(2) coset conformal

ﬁeld theory with the level k= 2. The discussed two-dimensional models realize a variety of

symmetry-protected topological phases, opening a window for studies of the unconventional

quantum criticalities between them.

1Corresponding author.

arXiv:2210.01187v4 [cond-mat.str-el] 29 Dec 2023

Contents

1 Introduction 2

2 A Three-state Potts model 4

3 Classiﬁcation of SPT states via cohomologies 5

3.1 Introduction to cohomologies of ﬁnite abelian groups and application to Z3

and Z3×Z3×Z3groups 5

3.2 Third cohomology and cocycles of Z3group 7

3.3 Third cohomology and cocycles for Z3×Z3×Z3group 8

3.4 SPT ground state and a parent Hamiltonian 10

4Z3×Z3×Z3SPT paramagnet 11

5 Symmetry protected edge states 12

5.1 Edge Hamiltonian 15

5.2 Dual edge transformation with the Z3gauge ﬁeld 17

5.3 Alternative description of the edge Hamiltonian 18

6 Symmetries of the boundary model and its conformal properties 19

6.1 ’t Hooft anomaly 19

6.2 Winding symmetry and laterality 19

6.2.1 Associated U(1) symmetry and the current algebra 21

6.3 Gapless excitations and their conformal dimension 23

6.4 Entanglement entropy 25

7 Discussion and perspectives of the low energy theory of the edge Hamil-

tonian of the Z3×Z3×Z3SPT paramagnet 26

7.1 Naïve search for matching conformal properties 26

7.2 Conjectured CFT: the coset SU(3)2/SU(2)2model 28

7.3 Further perspectives for the phase transitions in the edge Hamiltonian 28

7.4 On the Z3SPT paramagnet 29

7.5 Concluding remarks 30

A Permutation group S331

B Nontriviality of 3-cocycles 31

– 1 –

1 Introduction

Over the past decade, symmetry-protected topological (SPT) phases [1–7] have generated

a lot of research interest [8–33]. They are essentially diﬀerent from Landau deﬁnition of

phases via local order parameters and carry topological characterization.

It is well-established that symmetry-broken ordered states are all characterized by

group theory describing the order parameter manifold. Such states are short-range en-

tangled. The SPT states are also short-range entangled[34], in contrast to topologically

ordered states [35–48] that are long-range entangled. Thus, short-range entangled phases

are generally symmetry broken (described by the Landau paradigm), SPT (outside of the

Landau paradigm), or support simultaneous coexistence of symmetry-breaking and SPT

order. Importantly, SPT orders support the symmetry-protected gapless boundary exci-

tations. These states, often with non-standard statistics, are substantial for the basics

of topological quantum computation. The topological systems near criticality are gener-

ally remarkable for their universal ﬁnite-size scaling behavior [49,50] and sensitivity to

symmetry-breaking perturbations [51,52]. They may possess both paramagnetic [9] and

spin-ordered phases, with the latter being closely related to Neel orders [53–55].

The topological insulators of free fermions, which feature an SPT phase protected

by U(1) and time-reversal symmetry, are well-studied in the literature. There are two

distinct types of such time-reversal invariant band insulators: topological insulators and

conventional insulators [56–59]. The two families of insulators are distinguished by the fact

that topological insulators have protected gapless boundary modes, while trivial insulators

do not. This is because time reversal and charge conservation symmetry play a crucial role

in this physics: the boundary modes will gain a gap if any of these two symmetries are

broken explicitly or spontaneously. Then the distinction between topological insulators and

conventional insulators disappears.

A deeper understanding of SPT phases followed the work of Xiao-Gang Wen and collab-

orators, where a classiﬁcation of SPT states was formulated based on cohomology classes of

discrete groups [1,3–5,10,11]. Full classiﬁcation of SPT phases in 1D system was presented

in [3,4,60,61] This classiﬁcation has become a powerful mathematical tool for characteriz-

ing and distinguishing between diﬀerent SPT states, and predicting their properties, such as

the number of protected edge modes and the nature of the ground state degeneracy. Along

these lines of research, an interesting and important result was reported in Ref. [9], where

the authors show how an ordinary paramagnetic Ising model with Z2symmetry can be

modiﬁed to produce a gapped SPT state with gapless Z2symmetry-protected edge states.

According to the standard deﬁnition, gapped SPT states have several characteristic

properties. The system should have an internal symmetry Gand a ground state with no

spontaneous symmetry breaking. The SPT state should be distinct from the "trivial" state

(a kind of product spin/boson state) and can not be continuously (in some parameters)

connected with a "trivial" state without closing the gap in the bulk. The last property

is that the two states can be connected continuously without closing the gap, but one or

more symmetries of the Hamiltonian should be broken. As a general rule, an SPT system

has massless edge states, which at some momentum have zero energy. Furthermore, the

– 2 –

presence of the gapless edge (or its absence) can, in principle, distinguish the SPT state

from the trivial ground state. Therefore, some characteristic classes topologically protect

these edge states. Refs. [8,10,15] report classiﬁcation of SPT states in ddimensional

spin models according to cohomology groups Hd+1(G, U(1)) of the site symmetry group G

of the models with coeﬃcients in wave functions phase factor U(1). This bulk-boundary

link and appearance of massless edge states is a hallmark of t’Hooft anomaly [62], linked to

well known Lieb-Schultz-Mattis (LSM) theorem, which is an anomaly between translational

invariance and global symmetry G of the system [20,21]. In Ref. [21], it was shown how

the presence of this anomaly helps to formulate the low energy limit of lattice theories.

Namely, if the anomaly symmetry of the lattice model coincides with the one in quantum

ﬁeld theory, then both low energy limits may also coincide.

One approach to understanding SPT phases is through the use of group cohomology,

that allows both for the characterization of topological phases in terms of the underlying

symmetry group and their construction[10]. Speciﬁcally, Ref. [10] examines how the concept

of a group extension can be used to construct SPT models with a variety of diﬀerent

symmetries, including discrete and continuous symmetries. Here one has to derive an

explicit group cocycle and then construct a lattice SPT model following the procedure

outlined in that work. The group cohomology classiﬁcation also oﬀers physical insites

into the implications of these models, including their relevance to topological insulators,

superconductors, and other exotic quantum materials.

In this respect, Ref. [9] also reports a nice procedure formulated to generate massless

edge modes in the paramagnetic phase of two dimensional Z2Ising model in bulk. One

can make any unitary transformation of the Hamiltonian and operators and generate a

new theory in bulk. Since the transformation is unitary, the spectrum in the bulk of the

model will remain the same. However, the excitations may gain nontrivial statistics upon

gauging[22–25] . Within this approach, to ﬁnd the Hamiltonian of the boundary modes,

one ﬁxes the operators of unitary transformation on the system boundary by ﬁxing external

spins and summing up those spins for a new Hamiltonian to have the same symmetries as

the parent one. Due to this summation, the new Hamiltonian is not unitary equivalent to

the initial one but diﬀers only by the appearance of new edge states, which can become

gapless. Applying this type of transformation to the paramagnetic Z2Ising model, Levin

and Gu have identiﬁed the edge states with the one-dimensional (1D) Hamiltonian of the

gapless XX model (conformal theory with central charge c= 1). Hence, this transformed

2D model with a gapless edge forms an SPT paramagnet.

In this paper, we construct an SPT phase on a two-dimensional triangular lattice

based on the higher spin system, namely the three-state Potts model in its paramagnetic

phase. We discuss the construction of the model in detail and formulate the SPT lattice

paramagnets with Z3symmetry. We start from the three-state Potts model on a triangular

lattice in the paramagnetic phase, whose spectrum is gapped. The paramagnetic Potts

model has larger, Z3×Z3×Z3symmetry, where each Z3is deﬁned on three diﬀerent

triangular lattice sites. The triangular lattice contains three triangular sub-lattices of larger

size, denoting with index i= 1,2,3. Then, following the approach of Ref. [9], we reformulate

the Z3Potts model using symmetry-protected unitary transformation, which has nontrivial

– 3 –

cocycle property in the Z3×Z3×Z3and Z3groups. According to Refs. [8,10,15], the

variety of SPT states in 2D models is deﬁned by elements of the third cohomology group of

the symmetry group of the model. In our case it leads to H3(Z3×Z3×Z3, U(1)) = (×Z3)7

[25,27] and H3(Z3, U(1)) = Z3. We show how Z3×Z3×Z3and Z3symmetric unitary

transformations preserve the spectrum in bulk and lead to the appearance of gapless edge

states manifesting the presence of the SPT phase. Models with Z3and Z3×Z3×Z3

symmetry were discussed also in [18] and [20,25] respectively. Finally, we identify the

Hamiltonian operators describing the edge states in Z3×Z3×Z3symmetric case and

investigate their symmetry and conformal properties.

Importantly, we found that our edge Hamiltonian has hidden U(1) anomalous symme-

try, which is responsible for the current algebra of the corresponding low-energy CFT and

shrinks the possible theories considerably, helping us to identify the low-energy CFT can-

didate as the coset SU(3)2/SU(2)2model. We have used the technique developed in [76]

for U(1) subgroup in XXZ Heisenberg chain to detect corresponding Kac-Moody algebra

and its central extension in our model.

2 A Three-state Potts model

The Hamiltonian of the three-state Potts model can be derived from its action formulation

following the standard prescription [63]

HP=γX

r∈2D,µ εSz

rε−Sz

r+µa+ε−Sz

rεSz

r+µ−X

r∈2D

(X+

r+X−

r),(2.1)

X+

r= (X−

r)†=





0 1 0

0 0 1

1 0 0





, Sz

r=





1 0 0

0 0 0

0 0 −1





,(2.2)

where ε=e2πi/3is a basic element of the group Z3,µa, a = 1,2,3are three basic lattice

vectors of the triangular lattice, and γis the interaction constant. The operators X±

rand

ε±Sz

rbelong to the cyclic spin-1 representation of quantum group SUq(2) with q= 3 and

they obey the algebra

X±

rεSz

r=ε∓1εSz

rX±

r,(X+

r)3= (X−

r)3= 1.(2.3)

Note that the above matrices are related to the parafermion generators[64].

We consider the three-state Potts model on a triangular lattice, R, with boundary, ∂R,

in the paramagnetic phase when γ= 0. Then the Hamiltonian is simple and divided into

two non-interacting parts on the boundary and in bulk: HP=Hbulk +Hedge

Hbulk =−X

r∈R

(X+

r+X−

r), Hedge =−X

r∈∂R

(X+

r+X−

r).(2.4)

The Hamiltonian (2.4) has a much larger symmetry in its paramagnetic phase. The sites

of the triangular lattice can be divided into three groups. Each site of the basic triangles

– 4 –

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Z3and(×Z3)3symmetryprotectedtopologicalparamagnetsHrantTopchyan,aVasiliiIugov,b,cMkhitarMirumyan,aShahaneKhachatryan,aTigranHakobyan,a,dTigranSedrakyane,1aAlihkanyanNationalLaboratory,YerevanPhysicsInstitute,AlikhanianBr.2,0036,Yerevan,ArmeniabSimonsCenterforGeometryandPhysics,StonyBrookUniversity,S...

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