Symmetry-resolved entanglement of 2D symmetry-protected topological states Daniel Azses1David F. Mross2and Eran Sela1 1School of Physics and Astronomy Tel Aviv University Tel Aviv 6997801 Israel

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Symmetry-resolved entanglement of 2D symmetry-protected topological states

Daniel Azses,1David F. Mross,2and Eran Sela1

1School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel

2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

Symmetry-resolved entanglement is a useful tool for characterizing symmetry-protected topolog-

ical states. In two dimensions, their entanglement spectra are described by conformal ﬁeld theories

but the symmetry resolution is largely unexplored. However, addressing this problem numerically

requires system sizes beyond the reach of exact diagonalization. Here, we develop tensor network

methods that can access much larger systems and determine universal and nonuniversal features in

their entanglement. Speciﬁcally, we construct one-dimensional matrix product operators that en-

capsulate all the entanglement data of two-dimensional symmetry-protected topological states. We

ﬁrst demonstrate our approach for the Levin-Gu model. Next, we use the cohomology formalism to

deform the phase away from the ﬁne-tuned point and track the evolution of its entanglement fea-

tures and their symmetry resolution. The entanglement spectra are always described by the same

conformal ﬁeld theory. However, the levels undergo a spectral ﬂow in accordance with an insertion

of a many-body Aharonov-Bohm ﬂux.

I. INTRODUCTION

Symmetry-protected topological states (SPTs) are

characterized by a symmetric bulk state that does not

host fractional excitations. Still, they are topological

in the sense of carrying anomalous edge states at their

boundary with a trivial state or diﬀerent SPTs. In two

dimensions, the edges are described by a one-dimensional

conformal ﬁeld theory (CFT) [1–5]. The presence of these

states is dictated by a speciﬁc structure in their entan-

glement. Yet, unlike topologically-ordered (fractional)

states, the entanglement entropy of SPTs does not con-

tain a topological term. Instead, the topological nature

of these states may be revealed by resolving the entan-

glement entropy according to symmetries or by studying

entanglement spectra (ES).

The entanglement entropy of a system with global

symmetries can be decomposed according to the asso-

ciated quantum numbers [6–9]. Speciﬁcally, it is given

by the sum of entropies for each choice of these quan-

tum numbers in one subsystem. The R´enyi moments

of the symmetry-resolved entanglement are experimen-

tally measurable [10–13] as also demonstrated for one-

dimensional SPT states [14, 15] on IBM quantum com-

puters. For such states, each symmetry sector con-

tributes equally to the total entropy [16]. This equiparti-

tion corresponds to exact degeneracies in the ES [17, 18].

These have been recognized as the source of the com-

putational power of one-dimensional SPTs [19] within

measurement-based quantum computation [20].

The ES generalizes the entanglement entropy and con-

tains additional universal information. For 2D topolog-

ical states with a chiral edge, the Li-Haldane conjec-

ture [21, 22] states that the levels of the ES correspond to

the conformal ﬁeld theory that describes a physical edge.

Extrapolating to the nonchiral case, one may expect that

the ES of SPT phases have the same universal properties

as their nonchiral edge CFTs, such as the central charge

c. For example, for SPTs stabilized by a ZNsymmetry,

the free boson CFT with c= 1 was found to describe the

edge [1, 2]. While the ES should correspond to the same

CFT as that of the physical edge, the two may diﬀer

by nonuniversal parameters such as the compactiﬁcation

radius. Other nonuniversal properties include eﬀective

ﬂuxes, which change the boundary condition of the 1D

edge.

Our motivating questions are: how does the ES de-

compose according to symmetry in 2D SPTs? How does

this decomposition ﬁt into the CFT? What are its uni-

versal properties, and how does the ES vary within a

given SPT phase? Previous work by Scaﬃdi and Ringel

exploring the emergent CFT in the ES of SPTs was lim-

ited to small system sizes [4]. It could, in principle, have

performed a symmetry resolution, which does not require

large systems. By contrast, distinguishing universal from

nonuniversal properties upon continuous variation of the

ground states, as we ﬁnd, does require large system sizes.

In order to address these issues in this work, we de-

velop an eﬃcient numerical method [23, 24] to calculate

the ES of short-range entangled states in two dimensions.

Our method is summarized in Fig. 1 and described in

Sec. II. It uses a quantum circuit representation to con-

struct gapped one-dimensional models that exhibit the

same entanglement properties as the 2D SPTs in ques-

tion. In particular, they allow us to extract the entangle-

ment spectra of large systems and their symmetry reso-

lution using tensor network methods [25–28].

In Sec. III, we apply this approach to the Levin-Gu

model [2] on an inﬁnite cylinder with circumferences as

large as L= 150. In agreement with previous studies

of much smaller systems [4], we observe the spectrum of

a CFT with central charge c= 1. Speciﬁcally the ES

can be organized in terms of primary states and their

descendants. We ﬁnd that all the descendant states have

the same subsystem symmetry quantum numbers as the

corresponding primary state. We also identify an un-

expected subtlety in the ES of the Levin-Gu model: it

distinguishes cylinders whose circumference is a multiple

of three from all others. We attribute this eﬀect to the

lattice and show that it translates into a ﬂux insertion of

arXiv:2210.12750v2 [cond-mat.str-el] 10 Mar 2023

FIG. 1. Schematics of our method: (a) We analyze a state

|Ψigiven as a quantum circuit representation as in Eq. (1).

The entanglement between subsystems Aand Bis created

by a unitary UAB acting within a ﬁnite distance from the

entanglement cut (dashed line). (b) We consider an SPT on

a cylinder consisting of regions Aand B.

the corresponding CFT.

In Sec. IV, we apply our method to explore more

generic states. We construct a continuous family of wave-

functions within one SPT phase using the framework of

cohomological classiﬁcation [29]. The ES of these states

reveal a direct relation between so-called coboundary

transformations and certain ﬂuxes aﬀecting the many-

body SPT states.

In Sec. V, we further elaborate on the gapped one-

dimensional models derived in Sections III and IV. We

demonstrate that they can be used to obtain the central

charge of the SPT edge very eﬃciently without reference

to ES. Restricting the edge Hamiltonian to terms that

act within a single subsystem results in a critical chain

that is described by the same CFT as the SPT edge.

The central charge of such one-dimensional chains can

be readily extracted from the scaling of the entanglement

entropy, which satisﬁes the Calabrese-Cardy formula [30].

II. DIMENSIONAL REDUCTION AND

TENSOR NETWORK APPROACH

Before specifying to 2D, consider a d-dimensional

space. The two subsystems Aand Bshare a (d−1)-

dimensional boundary ∂A =∂B. By virtue of their ﬁnite

depth circuit representation [31], any SPT state can be

written as

|Ψi=UAUBUAB|0i.(1)

Here |0iis a site-factorizable product state, i.e., the

ground state of a trivial gapped Hamiltonian H0that

is the sum over one-site operators. The unitaries UA

and UBact only on subsystems Aand B, respectively.

UAB acts in a (d−1)-dimensional region denoted CAB

extending a ﬁnite distance from ∂A; see Fig. 1(a). Con-

sequently, the ES is fully encoded in UAB . Indeed, the

reduced density matrix of region A is given by

ρA= TrB|ΨihΨ|=UATrBUAB |0ih0|U†

ABU†

A.(2)

Up to the unitary transformation UA, which does not

aﬀect the ES, ρAacts nontrivially only in the interface

region CAB. Consequently, the ES can be fully encoded

by a state |ψedgeithat lives only in CAB. We deﬁne this

edge state via UAB|0i=|ψedgei⊗|0iASBrCAB such that

the operator

ρedge = TrB|ψedgeihψedge|(3)

exhibits the same spectrum as ρA. The ‘edge density ma-

trix’ ρedge acts nontrivially only on a (d−1)-dimensional

region within subsystem A. It is the central object in this

paper, which we construct using tensor-network methods;

see Fig. 2(c).

The presence of an on-site symmetry generated by

S=SA⊗SB, implies that [ρA, SA] = 0. It follows that

[ρedge, U†

ASAUA] = 0. As a result, the symmetry-resolved

ES is obtained by diagonalizing ρedge simultaneously with

the edge symmetry operator

Sedge =U†

ASAUA.(4)

It is convenient to think of the pure state |ψedgeias

the ground state of a local edge Hamiltonian

Hedge =UABH0U†

AB,(5)

which has the same (gapped) spectrum as H0.Hedge

acts on the support of CAB, which contains the union

of two adjacent (d−1)-dimensional regions in Aand B.

Accordingly, Hedge can be separated as

Hedge =HA+HB+HAB.(6)

We remark that, unlike the standard “entanglement

Hamiltonian” HEdeﬁned by ρA=e−HE, the edge

Hamiltonian Hedge acts on both subsystems. However,

in Sec. V, we argue that HEand HAdescribe a CFT

with the same universal properties dictated by the bulk

SPT phase.

III. SYMMETRY-RESOLVED ES OF THE

LEVIN-GU MODEL

Next, we focus on the paradigmatic 2D Levin-Gu

model and demonstrate our method for the computation

of the ES and its symmetry resolution.

The Levin-Gu state admits a quantum circuit form [32]

|ΨLGi=UCCZUCZ UZ|+i,(7)

where UCCZ,UCZ , and UZare, respectively, the prod-

ucts of CCZ,CZ and Zgates acting on all trian-

gles, edges, and vertices, and |+iis the ground state

of H0=−PiXi. We consider a cylinder geometry as

displayed in Fig. 1(b). Next, we use this quantum circuit

form to derive a 1D Hamiltonian that encodes the ES of

this 2D model, exemplifying the general prescription of

Sec. II.

FIG. 2. (a) The resulting 1D edge Hamiltonian corresponding

to Fig. 1(b) with the underlying triangular lattice is a zigzag

chain containing both A(even) and B(odd) sites. (b) We

construct a MPS of the ground state of Hedge living on A

and B. (c) The reduced density matrix of subsystem Ais

constructed as an MPO by contracting the odd sites (∈B) of

two copies of the MPS state.

A. Gapped 1D edge Hamiltonian

The Levin-Gu state provides an explicit example of

Eq. (1). In this case,

UA=Y

triangles∈A

Uijk

CCZ Y

links∈A

Uij

CZ Y

sites∈A

Z,(8)

and similarly for UB. The triangles and links that con-

nect the two subsystems identify the interface region CAB

as a zigzag chain with ieven ∈Aand iodd ∈B; see

Fig. 2(a). The corresponding entangling gates are

UAB =Y

Ui−1,i,i+1

CCZ Y

Ui,i+1

CZ ≡UCCZ

AB UCZ

AB .(9)

The edge Hamiltonian of Eq. (5) is then given by

Hedge =UABH0U†

AB =UCCZ

AB HclusterUCCZ

AB ,(10)

where Hcluster =UCZ

AB H0UCZ

AB =−PiZi−1XiZi+1 is the

1D cluster Hamiltonian. A more explicit form of Hedge

is readily obtained by virtue of the identity

Uijk

CCZXiUijk

CCZ =1

2Xi[I+Zj+Zk−ZjZk].(11)

We thus obtain Hedge as a sum of tensor products of

1-qubit operators,

Hedge =X

4Xi[I−Zi−2+Zi−1+Zi+1 −Zi+2

+Zi−2Zi−1+Zi−1Zi+1 +Zi+1Zi+2

−Zi−2Zi+1 −Zi−1Zi+2 −Zi−2Zi+2

+Zi−2Zi−1Zi+1 +Zi−2Zi+1Zi+2

+Zi−2Zi−1Zi+2 +Zi−1Zi+1Zi+2

−Zi−2Zi−1Zi+1Zi+2].(12)

We construct a matrix product state (MPS) of the ground

state of Hedge, denoted |ψedgeiand depicted in Fig. 2(b),

using the iTensor and Julia libraries [33]. The ground

state |ψedgeiconverges with bond dimension χ= 9 for pe-

riodic boundary conditions. Subsequently, we construct

the matrix product operator (MPO) for ρedge by con-

tracting the Bsites of the outer product |ψedgeihψedge|;

see Fig. 2(c). Finally, an excited state density ma-

trix renormalization group (DMRG) calculation on ρedge

yields the ES.

B. Symmetry resolution

The Z2symmetry operator of the Levin-Gu model is

X=QiXi. To see that the Levin-Gu state is an eigen-

state of Xand determine its eigenvalue, we use the quan-

tum circuit form [cf. Eq. (7)] along with the identities

XZiX=−Zi,(13)

XUij

CZ X=−Uij

CZ ZiZj,

XUijk

CCZX=−Uijk

CCZUij

CZ Ujk

CZ Uki

CZ ZiZjZk.

The third identity implies, in particular, that

XUCCZX= (−1)TUCCZ ,(14)

where Tis the number of triangles. The product over all

triangles includes each link and each site an even number

of times such that all Zand CZ factors cancel. Similarly,

the product over all links includes each site an even num-

ber of times, and thus

XUCZ X= (−1)LUCZ ,(15)

where Lis the total number of links. It follows that

X|ΨLGi= (−1)T+L+V|ΨLGi,(16)

where Vis the total number of vertices (i.e., sites).

For a perfect triangular lattice without a boundary,

(−1)T+L+V= 1.

According to Eq. (4), the edge symmetry operator is

given by SLG

edge =U†

AXAUA, where XA=Qi∈AXiand

UAis given by Eq. (8). Unlike the case of the full system,

the product over all triangles in subsystem Ainvolves the

links along the edge only once. Consequently, commuting

XAacross UAusing Eq. (13) produces uncanceled Zand

CZ factors. Consequently, SLG

edge acts nontrivially near

the edge and we write U†

AXAUA=SLG

edge ⊗Qi∈Ar∂A Xi.

The ﬁrst factor only acts on ∂A, which contains all even

sites. It is given by

SLG

edge =Y

i=even

XiY

i=even

Ui,i+2

CZ Zi,(17)

up to an additional overall factor (−1)TA+LA+VA= 1

accounting for the total number of triangles, links, and

vertices in subsystem A. As we can see, in addition to the

on-site factor Qi=even Xi, there is a non-on-site factor [1].

The latter is the manifestation of the nontrivial SPT. In

fact, it is this factor, which is classiﬁed [1] by the third

cohomology group H3(Z2, U(1)) = Z2. One can rewrite

the edge symmetry operator as

SLG

edge =Y

j=even

XjY

j=even

eiπ

4(ZjZj+2−1),(18)

which shows explicitly that the on-site and non-on-site

factors commute.

Having constructed the edge symmetry Eq. (18), we

conﬁrm that the eigenstates |ψiiof ρedge that we obtained

from DMRG satisfy SLG

edge|ψii=si|ψii, where si=±1 is

the symmetry eigenvalue.

1. Entanglement Hamiltonian

The above algorithm yields the eigenvalues λiof the re-

duced density matrix ρedge, from which we obtain a list

of quasienergies ξi=−log λi, being the eigenvalues of

the entanglement Hamiltonian deﬁned by ρedge =e−HE.

According to the Li-Haldane conjecture [21], in topologi-

cal systems, the latter displays the spectrum of a physical

edge. In the present SPT, this spectrum is known to be

a nonchiral free boson CFT [1].

We match the list of quasienergies {ξ0, ξ1, . . . }in in-

creasing order to the form

ξi−ξ0=v

L∆i,(19)

where vis a free parameter corresponding to the velocity

of the CFT, Lis the circumference of the cylinder, and

∆iare “scaling dimensions” of the CFT, as given below.

C. Entanglement spectrum for Ldivisible by 3

As discussed in detail in Appendix A, we can see that

the numerically obtained eigenvalues of the entanglement

Hamiltonian approximate the free boson spectrum

∆(`, m, R) = `2

R2+R2m2

4+ integers (`, m ∈Z),(20)

with compactiﬁcation radius R=√2. As reviewed in

Appendix D, this spectrum can be viewed as an inﬁnite

FIG. 3. Entanglement spectrum of the Levin-Gu model on

a cylinder with circumference L= 12 and its symmetry res-

olution. The low-lying energy levels and their degeneracies

match Eq. (20), with the second term “integers” being given

by Pi>0i(ni+ ¯ni) with ni,¯ni∈Nbeing the i’th integer

(i≥1) excitation on top of the primary ﬁeld denoted (`, m).

The right panel displays the ES levels and their degeneracy

according to Eq. (20). The symmetry eigenvalue si=±1 is

denoted by blue and red, respectively.

set of primary states |`, mi. Each of these generates an

inﬁnite tower of descendant states denoted in Eq. (20) by

“integers”.

The low-lying levels of this spectrum can already be

seen to match this pattern in a short system of L= 12,

as seen in Fig. 3. Results for longer systems, shown in

Appendix A, conﬁrm this pattern for higher energy levels.

As required, each eigenvector of ρedge has a well-

deﬁned subsystem symmetry. We ﬁnd that the corre-

sponding quantum number is given by si= (−1)`+m, as

predicted from a ﬁeld theory analysis [1, 3]. In particular,

all states generated by a given primary ﬁeld inherit its

symmetry properties. In Fig. 3, the symmetry eigenval-

ues si=±1 are indicated in blue and red, respectively.

D. Entanglement spectrum for Lnondivisible by 3

Our numerical results for lengths Lthat are not mul-

tiples of 3 follow a reproducible sequence diﬀerent from

Eq. (20). Instead, they approximate the pattern shown

in Table I (cf. Appendix A). This sequence is captured

by the modiﬁed free boson spectrum

∆(`, m) = (`−φ)2−φ2

2+m2

2+ integers,(21)

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Symmetry-resolvedentanglementof2Dsymmetry-protectedtopologicalstatesDanielAzses,1DavidF.Mross,2andEranSela11SchoolofPhysicsandAstronomy,TelAvivUniversity,TelAviv6997801,Israel2DepartmentofCondensedMatterPhysics,WeizmannInstituteofScience,Rehovot7610001,IsraelSymmetry-resolvedentanglementisausefultoo...

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Symmetry-resolved entanglement of 2D symmetry-protected topological states Daniel Azses1David F. Mross2and Eran Sela1 1School of Physics and Astronomy Tel Aviv University Tel Aviv 6997801 Israel

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