PHYSICAL REVIEW B 90, 045142 (2014)

Detection of symmetry-enriched topological phases

Ching-Yu Huang,1 Xie Chen,2 and Frank Pollmann1

1Max-Planck-Institut fur Physik komplexer Systeme, 01187 Dresden, Germany2Department of Physics, University of California, Berkeley, California 94720, USA

(Received 18 December 2013; revised manuscript received 11 July 2014; published 30 July 2014)

Topologically ordered systems in the presence of symmetries can exhibit new structures which are referredto as symmetry-enriched topological (SET) phases. We introduce simple methods to detect certain SET ordersdirectly from a complete set of topologically degenerate ground-state wave functions. In particular, we first showhow to directly determine the characteristic symmetry fractionalization of the quasiparticles from the reduceddensity matrix of the minimally entangled states. Second, we show how a simple generalization of a nonlocalorder parameter can be measured to detect SET phases. The usefulness of the proposed approach is demonstratedby examining two concrete model states which exhibit SET phases: (i) a spin-1 model on the honeycomb latticeand (ii) the resonating valence bond (RVB) state on a kagome lattice. We conclude that the spin-1 model and theRVB state are in the same SET phases.

DOI: 10.1103/PhysRevB.90.045142 PACS number(s): 64.70.Tg, 03.67.Mn, 05.30.Pr, 73.43.Cd

I. INTRODUCTION

Topologically ordered quantum systems have robust physi-cal properties, such as quasiparticle statistics and ground-statedegeneracy, which do not depend on the microscopic details ofthe Hamiltonian [1]. If the system has extra global symmetries,then the interplay between topology and symmetry can giverise to interesting “symmetry-enriched topological” (SET)phases, where the quasiparticles transform under the symmetryin a “fractional” way. The first and best understood topologicalphase—the ν = 1/3 fractional quantum Hall state—is anSET phase with charge conservation symmetry, where thequasiparticle with eiπ/3 fractional exchange statistics has e/3fractional charge [2,3]. More interestingly, it was realizedthat systems with the same topological order and the samesymmetry can be in different SET phases with differentsymmetry fractionalization on the quasiparticles. For example,in a Z2 gauge theory with spin rotation symmetry, the gaugecharges can carry half-integer spin (fractional) or integer spin(nonfractional) representations [4]. With more symmetries,more varieties of SET phases are possible and many effortshave been devoted to their classification [5–10].

An important open question is how to determine the SETorder in a model system. With the experimental prospectsto realize spin liquids in systems with various internal andlattice symmetries (e.g., herbertsmithite [11]), it is necessaryto predict theoretically which SET phase they belong to.However, this is generally hard as the SET order is intrinsicallyencoded in the global entanglement pattern of the state and nolocal order parameter can be measured to detect it. On the otherhand, several methods have been developed to determine thetopological order in the long-range entangled states [12–17],but they are insensitive to the different ways of symmetryenrichment.

In this paper, we introduce a way to detect certain SETorder by measuring a nonlocal parameter on the minimallyentangled ground states (MESs) [16] of the system on atorus. In particular, we consider those SET phases which arecharacterized by projective representation of the symmetriesin terms of the quasiparticles. The set of MESs on a torusgives us access to the quasiparticle excitations of the system

by localizing them at the ends of the cylinders when thetorus is cut into halves, as shown in Fig. 1. Now if we canmeasure the fractional symmetry representation carried by thequasiparticles, we can identify the SET order.

If the quasiparticle carries a fractional charge, such as inthe fractional quantum Hall case, we can detect it directlyby measuring charge locally near the ends of the cylinder inthe MES. A different type of symmetry fractionalization existswhere the quasiparticles carry projective representations of thesymmetry, as in the case of spin-1/2 representations on the Z2

gauge charges. We are going to focus on SET phases with thistype of fractionalization in this paper and show that they can bedetermined with the nonlocal order parameters that are relatedto the ones used to detect symmetry-protected topological(SPT) phases [18,19]. As shown in Refs. [18,19], a stringorder parameter can be designed to detect projective symmetryrepresentations on the edge of a one-dimensional (1D) gappedsystem, hence identifying the SPT order in 1D. By putting atwo-dimensional (2D) SET system onto a cylinder and pickingout the MESs, we map a 2D SET state into effectively 1D SPTstates whose order can then be detected with “nonlocal order”parameters. We demonstrate the effectiveness of this idea byapplying it to a model wave function of spin-1 bosons and alsoto the resonating valence bond state [20,21] on the kagomelattice, which has the same SET order—the Z2 topologicalorder with the Z2 charge carrying spin-1/2 representation ofthe SO(3) spin rotation symmetry.

The paper is organized as follows. In Sec. II, we review thenotion of minimally entangled states (MESs) and explain twomethods to identify SET phases from the MESs. Sections IIIand IV discuss how projective representations and nonlocalorder parameters can be used to identify the SET orderin a 2D spin-1 boson model on the hexagonal lattice andresonating valence bond state on kagome lattice, respectively.We conclude in Sec. V with a summary and short discussion.In Appendix A, we briefly review the notion of projectiverepresentation. The Hamiltonian for the spin-1 boson modelis given in Appendix B. Finally, we show the details on howto obtain the projective representation from a tensor productstate (TPS) in Appendix C and via exact diagonalization inAppendix D.

1098-0121/2014/90(4)/045142(8) 045142-1 ©2014 American Physical Society

CHING-YU HUANG, XIE CHEN, AND FRANK POLLMANN PHYSICAL REVIEW B 90, 045142 (2014)

(a) (b)

A B

Wy

-a

a xy

FIG. 1. (Color online) (a) Minimally entangled ground states ofa topological system on the torus are eigenstates of Wilson loop Wy

operators parallel to the bipartite cut. (b) Quasiparticles of type a anda are localized at the edges of the cut.

II. DETECTING SYMMETRY-ENRICHEDTOPOLOGICAL PHASES

A. Minimally entangled states

An important concept for the detection of symmetry-enriched topological (SET) phases are the minimally entangledstates (MESs) proposed in [16]. When a topologically orderedsystem is put onto a torus, the ground space has a degeneracyequal to the number of quasiparticle types N . Among allthe states in the ground subspace, one set of basis stateshas minimum bipartite entanglement when the torus is cutalong a noncontractible loop in the y direction into twocylinders, as shown in Fig. 1. These MESs are eigenstatesof the Wilson loop operators in the y direction {Wy

a } andhave a one-to-one correspondence with the quasiparticle typesa. The MES with eigenvalue 1 for all {Wy

a } corresponds tothe vacuum. The MES corresponding to a can be createdfrom the vacuum state by creating a pair of particle a andantiparticle a excitations, bringing a around a noncontractibleloop in the x direction and annihilating it with a. Therefore,when the system is cut open, the MES corresponding toa has quasiparticles a and a at each end of the cylinder.Given a model Hamiltonian with potential SET order, onecan find the MESs from a complete set of ground states on atorus by minimizing bipartite entanglement. Using techniquesintroduced in Refs. [16,22–27], the MES corresponding todifferent a can be identified by calculating the modularmatrices U and S. In the following, we follow Refs. [24,25] anduse the fact that a long cylinder is locally equivalent to a torus.In this case, the different MESs can be conveniently obtainedby changing the boundary conditions on a long cylinder. Inthis work, we use the tensor product state representation toobtain the MES (see details below). However, the MES canalso be obtained using various other numerical methods. Usingexact diagonalization (ED), one can obtain a complete set ofMESs by variationally minimizing the entanglement within theground space [26,27]. There have also been various proposalson how to obtain the MES systematically in density matrixrenormalization group (DMRG) methods [24,25,28]. Whencut into two “half cylinders,” we find a localized quasiparticlenear the edge as illustrated in Fig. 2(a) with possible projectiverepresentations.

B. Projective representation

A projective representation is like an ordinary represen-tation up to phase factors; i.e., if g,h,k are in G and fulfill

FIG. 2. (Color online) (a) Symmetry fractionalization for a local-ized quasiparticle of type a at the end of the half cylinder. (b) Thenonlocal order parameter consisting of product of onsite symmetryoperators g acting on a segment of length n on the cylinder. Thesegment is terminated by the operators XL and XR .

g · h = k, then

UgUh = eiρ(g,h)Uk. (1)

The phases ρ(g,h) are called the “factor set” of the represen-tation. In Ref. [18], a numerical approach was introduced thatallows one to directly extract the Ug from the ground-statewave functions. Furthermore, several string order parameterswere introduced which allow one to directly detect differentSPT phases, including those protected by time reversal and byinversion symmetry.

We use two methods to identify the SET phase from theMESs in this paper. We begin with the method in whichthe projective representations Ug are directly extracted fromthe Schmidt decomposition of the MES. Assume that |�a〉is the MES corresponding to quasiparticle a on an infinitecylinder and perform a Schmidt decomposition of the stateinto two Schmidt states on half cylinders,

|�a〉 =χ∑

α=1

λα

∣∣φLα,a

⟩∣∣φRα,a

⟩, (2)

where |φLα,a〉 and |φR

α,a〉 represent an orthogonal basis of the leftand right partitions, respectively. For concreteness, we assumethat the |φL

α,a〉 are defined on sites −∞ . . . 0 and |φRα,a〉 on

sites 1 . . . ∞. The λα are Schmidt values and the entanglemententropy is given by S = −∑

α λ2α log λ2

α . Note that the Schmidtstates can also be obtained by diagonalizing the reduceddensity matrix ρL (ρR) and the corresponding eigenvaluesare λ2

α . The SET phases are gapped phases with short-rangecorrelations and we assume that we have a cylinder with afinite circumference, thus the area law [29–31] guarantees thatthe values of λα decay quickly [32,33]. From now on, we areonly considering the important Schmidt states which have aSchmidt value λα > ε for a given ε > 0, so we have a finitenumber of Schmidt states.

The Schmidt states of |�a〉 have localized quasiparticles oftype a at the cut. Thus, on-site symmetry operations g that acton the Schmidt states transform the quasiparticles according tothe representation Ua

g (which can be either linear or projective).The Ua

g can then be directly obtained by calculating the overlapbetween the Schmidt states with their symmetry-transformedpartners,

(Ua

g

)α,β

= ⟨φR

α,a

∣∣ ( ∞∏i=1

gi

) ∣∣φRβ,a

⟩. (3)

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DETECTION OF SYMMETRY-ENRICHED TOPOLOGICAL PHASES PHYSICAL REVIEW B 90, 045142 (2014)

Equivalently, we could have chosen the left Schmidt states|φL

α,a〉. If the state |ψ0〉 is represented as the tensor product state,Eq. (3) can be efficiently evaluated by multiplying togetherall of the tensors to the right of the bond (see Appendix C).Once we have obtained the Ua

g of each symmetry operationand quasiparticle type, we can calculate the commutators andread off the factor set and hence determine in which phasethe state is. In our analysis, we assume that the edges of the1−MES are in linear representations, e.g., we do not considerthe case of having 1D SPT phases coupled to the system. Inthat case, we would have to identify the SPT properties on the1−MES and subtract the phase factor from those of the otheranyonic MESs. For non-on-site symmetries, we can obtain therepresentations in an analogous way, e.g., for time reversal, wetake the complex conjugate as discussed in [18] in detail forthe case of 1D SPT phases.

C. Nonlocal order parameter

It is desirable to have a probe for SET phases that doesnot rely on having access to the Schmidt states. For this,we construct nonlocal order parameters that are sensitiveto the type of SET order and can be directly evaluatedusing Monte Carlo methods (e.g., using the variational MonteCarlo methods applied in [16]) or potentially measured inexperiments [34]. The nonlocal order parameter we considerhere is defined as

Oan(g,XL,XR) = lim

n→∞〈�a|XL(1)

[n−1∏k=2

g(k)

]XR(n)|�a〉,

(4)

and is closely related to the string order parameter originallyintroduced by [35]. It corresponds to calculating the overlapbetween the wave function with a symmetry operation g

applied to a segment of n consecutive rings of the cylinder[as illustrated in Fig. 2(b)]. The operators XL and XR aredefined on rings terminating the segment that is transformedby g. As g is a symmetry operation, it does not change anythingin the bulk of this segment and the overlap should not vanishas n → ∞ for any cylinder with a finite circumference. It wasdemonstrated that the string order parameter of the type givenby Eq. (4) can detect SPT phases by choosing the operatorsXL and XR accordingly. The operators XL and XR are chosensuch that selection rules force the string order to be zero ina particular SPT phase (see Ref. [18] for details on how tochoose the operators). Analogously, we can use this type ofstring order to distinguish different SET phases. In practice,we can thus choose the operators XL and XR accordinglyand then evaluate Eq. (4) for each quasiparticle type to geta complete characterization of the SET phase. Note that thenonlocal order parameter [see Eq. (4)] is sensitive for a subsetof SET phases with sufficiently simple symmetries; however,more general order parameters can be constructed that workfor all symmetries (analogous to the ones used for the SPTphases in [18]).

III. SPIN-1 BOSONS ON THE HEXAGONAL LATTICE

We will now demonstrate the above by studying an exampleof an SET state that is built of spin-1 bosons. In particular, we

(a) (b)

......

...

(c)......

...

......

...

......

...

1 e

(d)

m f(e) (f)

1 2 3 4 5 6 7

0.6

0.8

1.0

1.2

1.4

0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

L

1 - MESe - MESm -MESf - MES

nn

L=2L=3L=4L=5L=6

L=2L=3L=4L=5L=6

1 - MESm - MES

e - MESf - MES

FIG. 3. (Color online) (a) The ground-state wave function isformed by an equal-weighted superposition of closed S = 1 AKLTchains on the honeycomb. The sites are either occupied by one S = 1boson or empty and the red ellipsoids represent spin-1/2 singlets.(b) An excited state with two defects which carry a spin 1/2 each.(c) The four MESs of theZ2 liquid are in a one-to-one correspondencewith the four quasiparticle types, 1,e,m, and f . (d) All MESshave a topological entanglement of γ = log 2. (e),(f) Nonlocal orderparameter shown for a string tension of t = 0.83 with identity asboundary operator [X = 1, shown in Eq. (4)] as a function of thelength of the segment n calculated for the four MESs for cylinders ofdifferent circumference L. On decays exponentially with L and wehave rescaled the quantity in the plot.

are going to construct a simple SET state, namely, an “Affleck,Kennedy, Lieb and Tasaki (AKLT) string model state,” andshow how we can then extract all characteristic properties. Thestate we consider here is defined on a honeycomb lattice whereeach site is either unoccupied or contains one spin-1 boson. Theground-state wave function is an equal-weighted superpositionof loop coverings on the honeycomb lattice, where along theloops the spin-1 bosons form AKLT chains [36] and away fromthe loops the sites are unoccupied, as shown in Fig. 3(a). InAppendix B, we describe a local Hamiltonian which has thestate as its ground state. Neglecting the internal structure ofthe loops, this state is exactly the ground state of the toric codemodel [37]—which is the fixed point of a Z2 topologically

045142-3

CHING-YU HUANG, XIE CHEN, AND FRANK POLLMANN PHYSICAL REVIEW B 90, 045142 (2014)

ordered phase [38,39]. The topological entanglement entropy(TEE), usually denoted by γ , is the constant term in theentanglement entropy S = cL − γ , where L is the length of theboundary of the region [12,13]. The characteristic topologicalentanglement entropy of the Z2 phase is γ = log 2. The Z2

phase has a fourfold ground-state degeneracy on a torus,which corresponds to four different types of quasiparticleexcitations. These are the electric particles e [showing up asthe ends of open strings in Fig. 3(b)], the magnetic particles m

(corresponding to defects on plaquettes), fermions f (boundpairs of e and m), and the identity particle 1. The fourquasiparticle types are related to the ±1 eigenspaces of {Wy

a }.In the AKLT string model, the sites have integer spin S = 0

or S = 1. However, the e particles appear at ends of openAKLT loops and therefore carry fractionalized spins S = 1/2,as shown in Fig. 3(b). Let us now assume the presence of asymmetry, e.g., SO(3), which is a discrete subgroup such asZ2 × Z2 or time-reversal symmetry. We find that the on-siterepresentations are always linear, while the half-integer spincarried by the e particle has a projective representation. Them particle carries integer spin and their bound state f hashalf-integer spin. Numerically, the AKLT string state can berepresented exactly by a tensor product state (TPS) [40], whichsimplifies the calculations considerably.

A. TPS representation of the AKLT string model

Let us first recall the toric code model [37] for which theground state is the fixed point of a Z2 topologically orderedphase. The model is defined on the honeycomb lattice whereeach link is occupied by an Ising spin. The Hamiltonian isgiven by the spins around each vertex v, −∑

v

∏i∈v Zi , and

plaquette p of the lattice, −∑p

∏i∈p Xi . Here, X and Z

are Pauli operators. If we define that the state |↑〉 of spincorresponds to a string on a link, the ground-state wavefunction is an equal-weighted superposition of all closed-loopconfigurations |�Z2〉 = ∑ |D〉 on the lattice.

Now we start from a state |Ds〉, which is a tensor product ofsinglets |↑↓〉 − |↓↑〉 between two connected sites in the loopcovering |D〉. The state |Ds〉 can be regarded as a coveringof the honeycomb lattice with 1D S = 1 AKLT chains. Eachsite is occupied by either two spin-1/2 (that have a total spinS = 1) or a vacuum state (no string crosses this site). Theequal weighted superposition of all AKLT loop coverings,|�〉 = ∑ |Ds〉, on the honeycomb lattice forms the “AKLTstring” model state. This state has Z2 topological order and isa simple example of a SET phase.

To obtain a TPS representation of the AKLT string model,we can place the entangled state |ω〉 = 1√

2(|01〉 − |10〉) + |22〉

along all edges of the lattice. Each physical site has three virtualparticles on the honeycomb lattice, and each of the virtualparticles can be regarded as spin 1/2 ⊕ 0. The virtual indices“0,1” provide the spin-1/2 degrees of freedom which holdssinglet along the edge. The third index “2” that belongs to thespin-0 subspace is used to indicate there is no singlet along theedge. At each physical site, if two of the virtual particles stayin spin-1/2 subspace and the other one stays in spin 0, thesethree virtual particles would be mapped to a physical spin-1boson (S = 1,0, − 1). If all of the virtual particles stay in spin0, it forms a physical vacuum state, ∅. The following projector

realizes the spin-1 boson on the honeycomb lattice:

P = t |1〉(〈002| + 〈020| + 〈200|)+ t |0〉(〈012| + 〈120| + 〈201| + 〈021| + 〈210| + 〈102|)+ t | − 1〉(〈112| + 〈121| + 〈211|)+ |∅〉〈222|, (5)

where t is a string tension. Each projector P on site with t = 1represents an equal-weighted superposition of all AKLT loopcoverings. By contracting the virtual particles with projectorsP , we can obtain the TPS representations directly.

It follows from what has been said that the state with n sitescan be expressed as a (translationally invariant) TPS with bonddimension χ = 3 as follows:

|�〉 =∑

s1,s2,...,sn

tTr(T [s1]T [s2] · · · T [sn])|s1,s2, . . . ,sn〉, (6)

where the si ∈ {1,0, − 1,∅} correspond to the three S =1 states and the vacuum state, respectively. The nonzeroelements of the tensors are

T[∅]

222 = 1,

T[0]

201 = T[0]

021 = T[0]

012 = −T[0]

210 = −T[0]

120 = −T[0]

102 = t√2,

T[−1]

112 = T[−1]

121 = T[−1]

211 = −T[1]

002 = −T[1]

020 = −T[1]

200 = −t,

(7)

where t is a string tension. If t = 1, this state is an equal-weighted superposition of all “AKLT loop” coverings.

B. Numerical results

Starting from this set of tensors, we can find the TPS rep-resentation for all of the MESs (for details, see Appendix C).Note that the MESs can be identified by calculating themodular matrices (such as U , S, and T matrices) directly fromthe TPS representation [22,23]. As described in Refs. [41,42],we can obtain the reduced density matrix for a TPS on thecylinder of circumference L for each MES. In particular,the Schmidt states |φL

α,a〉 of each MES can be obtained bydiagonalizing the reduced density matrix on the cylinder. Fromthe entanglement entropy, which scales with L as cL − γ , wecan then directly obtain the constant γ . We obtain for all MESsa γ which converges to log 2 as L increases [see Fig. 3(d)]. Thisis the expected result for a Z2 topologically ordered phase.

By inserting the symmetry operators, such as the time-reversal operator or the π rotation operators [Rx = exp(iπSx)and Rz = exp(iπSz)] into the transfer matrix, we directlyobtain the overlap given by Eq. (3) yielding the desired Ua

g .From the Ua

g , we can then calculate the commutators whichcharacterize the SET phase:

1 e m f

UaT R

(Ua

T R

)∗1.0 −1.0 1.0 −1.0

UaRx U

aRz

(Ua

Rx

)†(Ua

Rz

)†1.0 −1.0 1.0 −1.0

We find that the commutators of Uag reveal nontrivial phase

factors for the time- reversal symmetry andZ2 × Z2 symmetry

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DETECTION OF SYMMETRY-ENRICHED TOPOLOGICAL PHASES PHYSICAL REVIEW B 90, 045142 (2014)

in the e− and f −MES. This is the fingerprint of the specificSET phase. We also demonstrate how to detect SET phasesby using the nonlocal order parameter given by Eq. (4). Thenonlocal order parameter with X = 1 and g = Rz shown inFigs. 3(e) and 3(f) reveals the projective representations ofthe e and f quasiparticles. The selection rule implies thatthe nonlocal order parameter On(Rz,1,1) vanishes in e− andf −MES, as they have nontrivial phase factors under Rz.

IV. RESONATING VALENCE BOND STATE ON THEKAGOME LATTICE

We will now show that the resonating valence bond (RVB)state on the kagome lattice [shown in Fig. 4(a)] is in the sameSET phase as the spin-1 model above. The RVB state onthe kagome lattice is a good approximation of the groundstate for the S = 1/2 Heisenberg [43] and represents a statewith Z2 topological order [44]. In the RVB state, a singletcan fractionalize into two spinons which carry a spin 1/2[see Fig. 4(b)]. Thus the quasiparticles again carry projectiverepresentations.

Again we can make use of the TPS description of the stateas follows: The tensor representation of the RVB state also canbe obtained by using the χ = 3 entangled state. In Ref. [42],a three-particle entangled state, |ε〉 = ∑2

i,j,k=0 εijk|ijk〉 +|222〉, was used and placed on a triangle of kagome lattice.The ε is the Levi-Civita symbol with ε012 = ε120 = ε201 = 1and ε021 = ε210 = ε102 = −1. Applying the projector

P =|↑〉(〈02| + 〈02|) + |↓〉(〈12| + 〈21|) (8)

to each vertex yields the desired RVB state.

(a) (b)

(c) (d)

2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

2 3 4 5 6 7 8-1.0

-0.5

0.0

0.5

1.0

1.5

LL

1 -MESe -MESm-MESf -MES

1 -MESe -MESm-MESf -MES

FIG. 4. (Color online) (a) The RVB state on the kagome lattice isgiven by an equal-weighted superposition of nearest-neighbor singletcoverings. The red ellipsoids represent spin-1/2 singlets. (b) A pairof spinon excitations carrying a spin 1/2 each. (c) All MESs have atopological entanglement of γ = log 2. (d) Nonlocal order parameterfor each MES.

Once we obtain the RVB phase represented by a TPS, thecalculations of the nonlocal order parameters can be performedanalogously. We first obtain the TEE of each of the fourMESs that tend to log 2 for large L, as shown in Fig. 4(c)—asexpected for a Z2 phase. The projective representations Ua

g arethe same as the result in the previous example. The nonlocalorder parameter vanishes in e− and f −MES and tends to aconstant in 1− and m−MES, as shown in Fig. 4(d). Fromthe TEE and nonlocal parameter, we can conclude that theRVB state is in the same phase as the AKLT string model.That is, the two states can be transformed into each otherusing symmetric local unitary transformation. In the senseof the quantum state renormalization group (QSRG) [45]or multiscale entanglement renormalization (MERA) [46]scheme, this implies that both states flow to the same fixedpoint.

V. CONCLUSIONS

In this paper, we have introduced two simple methodsto detect certain SET phases which are characterized byprojective representations of the quasiparticles. The firstapproach is achieved by the Schmidt states of minimallyentangle states (MESs) on a cylinder. We can measure thesymmetry representations of quasiparticle type a at the endof a cylinder. Thus it is a very convenient method if weuse exact diagonalization (see Appendix D), density matrixrenormalization group [47], or tensor product state based [41]techniques. The second approach is to do a segment ofmeasurements on the real spins. The selection rules obtaina SET characterization. This way is more physical and can beused by other methods, e.g., quantum Monte Carlo methods,or potentially measured experimentally. We demonstrated theusefulness of the approaches by considering first an AKLTstring model and the RVB state on the kagome lattice. Finally,we find that the AKLT string model and the RVB state are inthe same SET phases.

ACKNOWLEDGMENTS

X.C. is supported by the Miller Institute for Basic Researchin Science at the University of California, Berkeley. We aregrateful for discussions with Ashvin Vishwanath, Ari Turner,Mike Hermele, Mike Zaletel, Norbert Schuch, and SiddhardhMorampudi.

APPENDIX A: PROJECTIVE REPRESENTATION

Matrices u(g) form a projective representation of symmetrygroup G if

u(g1)u(g2) = ω(g1,g2)u(g1g2), g1,g2 ∈ G. (A1)

Here, ω(g1,g2)’s are U(1) phase factors, which is called thefactor system of the projective representation. The factorsystem satisfies

ω(g2,g3)ω(g1,g2g3) = ω(g1,g2)ω(g1g2,g3), (A2)

for all g1,g2,g3 ∈ G. If ω(g1,g2) = 1, this reduces to the usuallinear representation of G.

045142-5

CHING-YU HUANG, XIE CHEN, AND FRANK POLLMANN PHYSICAL REVIEW B 90, 045142 (2014)

FIG. 5. Vertex configuration allowed by the hv term. The emptycircle represents the vacuum sector while the solid circle representsthe spin-1 boson sector.

A different choice of prefactor for the representationmatrices u′(g) = β(g)u(g) will lead to a different factor systemω′(g1,g2):

ω′(g1,g2) = β(g1g2)

β(g1)β(g2)ω(g1,g2). (A3)

We regard u′(g) and u(g) that differ only by a prefactor asequivalent projective representations and the correspondingfactor systems ω′(g1,g2) and ω(g1,g2) as belonging to thesame class ω.

Suppose that we have one projective representation u1(g)with factor system ω1(g1,g2) of class ω1 and anotheru2(g) with factor system ω2(g1,g2) of class ω2; obviouslyu1(g) ⊗ u2(g) is a projective presentation with factor groupω1(g1,g2)ω2(g1,g2). The corresponding class ω can be writtenas a sum ω1 + ω2. Under such an addition rule, the equivalenceclasses of factor systems form an Abelian group, which iscalled the second cohomology group of G and denoted asH2[G,U(1)]. The identity element 1 ∈ H2[G,U(1)] is the classthat corresponds to the linear representation of the group.

APPENDIX B: HAMILTONIAN FOR THEAKLT STRING MODEL

In this section, we describe a Hamiltonian which has thespin-1 boson wave function—an equal-weighted superpositionof AKLT loops—as its ground state. The Hamiltonian containsa vertex term and a plaquette term,

H =∑

v

hv +∑

p

hp. (B1)

Each hv acts on four vertices on the hexagonal lattice, with onevertex in the center and three neighboring vertices around it.First hv projects onto the following subspace: if the centervertex is in the vacuum sector, then all three neighboringvertices are in the vacuum sector; if the center vertex is in theone boson sector, then two of the neighboring vertices are alsoin the one boson sector while the other one is in the vacuumsector, as shown in Fig. 5. Moreover, the vertex term containsa coupling term �Si · �Sj + 1

3 (�Si · �Sj )2 between the spin 1 at thecenter and each of the two spin 1’s at the neighboring sites.The low-energy space of all the vertex terms is then composedof loop configurations of AKLT chains. The plaquette term hp

then allows the AKLT loop configurations to fluctuate fromone to another. More specifically, if a plaquette does not havea loop initially, the plaquette operator would attach an AKLTloop to it, as shown in Fig. 6. If a loop overlaps with part of theplaquette, then the plaquette term would flip the loop to be on

FIG. 6. (Color online) Plaquette operator allows Haldane loopsto fluctuate. Small spins on the side indicate edge spins of the Haldanechain segment.

the other side of the plaquette. Note that a segment of an AKLTchain has four low-energy states associated with the two edgespin 1/2’s. When mapping the segment from one side of theplaquette to another side, we need to match the state of theedge spin.

APPENDIX C: DETAILS ON HOW TO OBTAIN THEPROJECTIVE REPRESENTATIONS OF THE ANYONS IN

THE Z2 MODEL FROM A TPS

In this section, we show the details of how to obtainthe projective representations Ua

g of the anyons from TPSrepresentations. The main procedure is now as follows: wefirst obtain TPS representations of the anyons from a completeset of the ground state. We then map a 2D TPS to an effective1D matrix product state (MPS) representation A, and findthe canonical form of the matrix A. Finally, we can obtain theprojective representations Ua

g of the anyons from the symmetrytransformations.

Let us consider a wave function |�〉 of an N1 × N2 spinlattice in a cylindrical geometry. It can be expressed in termsof a TPS as

|ψ〉 =∑

s1,1,s1,2,...,sN1 ,N2

tTr(T s1,1T s1,2 · · · T sN1 ,N2 ) (C1)

× |s1,1,s1,2, . . . ,sN1,N2〉,

where the physical spin si,j = 1, . . . ,ds , and T si,j are rank-fivetensors for a square lattice. In the AKLT string model weconsidered, the honeycomb lattice also can be mapped onto aneffective square lattice [see Fig. 7(a)].

Our proposed method is implemented as follows.(1) Find the TPS representations of MESs:We reiterate how the TPS representation of the minimally

entangled states corresponding to the anyons [42] can beobtained. In Z2 topologically ordered phases, the TPS rep-resentations of four anyons can be obtained directly fromthe complete set of ground states. Suppose that we have theground states of a Z2 topologically ordered phase in TPScorresponding to Eq. (C1). The MESs of quasiparticles 1 ande, which are related to the ±1 eigenspaces of W

ya , can be

obtained from the TPS for the ground state with even andodd parity number of the boundaries of a cylinder. The paritynumber is defined by counting the number of singlets that crossa vertical line of the cylinder. These two MESs can be writtenas simple TPS representations.

We also can insert a string operator Z ⊗ Z · · · ⊗ Z [as theblue line shown in Fig. 3(c)] to TPSs for 1− and e−MES tocreate magnetic fluxes. The representation of the operator Z is

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DETECTION OF SYMMETRY-ENRICHED TOPOLOGICAL PHASES PHYSICAL REVIEW B 90, 045142 (2014)

FIG. 7. (Color online) (a) Tensors are denoted as boxes with onephysical index s and four virtual indices α,β,γ,δ. (b) The tensorsmeet all of the spins as a ring and form a new tensor A. (c) Theeffective MPS can be obtained by replacing the tensors around a ringwith a new tensor A and contracting the virtual indexes along thehorizontal direction.

given by

Z =

⎛⎜⎝

1 0 0

0 1 0

0 0 −1

⎞⎟⎠ . (C2)

Again, we can obtain TPS representations for m− and f −MESwith a string operator.

(2) Map 2D TPS to 1D MPS:We consider a MES corresponding to quasiparticle a in the

TPS given by Eq. (C1) (here the index of the anyon is omitted).Following Refs. [41,42], we block all spins that are in the firstk columns to form a ring, and define a new tensor,

Ak,Sk

αk,γk=

∑β1,k ,...,δN1 ,k

T s1,k T s2,k · · · T sN1 ,k , (C3)

as shown in Fig. 7(b). Here, Sk denotes the combination ofall physical indices s1,k,s2,k, . . . ,sN1,k , and αk and γk denotethe combination of all inner indices α1,k,α2,k, . . . ,αN1,k andγ1,k,γ2,k, . . . ,γN1,k , respectively. In terms of those tensors A,the MES can be expressed as

|ψ〉 =∑

S1S2···SN2

Tr(A1,S1A2,S2 · · · AN2,SN2 ) (C4)

× |S1S2 · · · SN2〉,where Si is a physical index of a new tensor which includesall physical indices around a ring. A 2D TPS thus can bemapped to an effective 1D MPS with large physical and innerdimensions, as shown in Fig. 7(c).

(3) Determine the canonical form:Now, we have an effective 1D MPS and need to determine

the canonical form [48] of the same state. The procedure iscovered in detail. First, we form a positive double tensor Eby merging two layers of tensor A and A∗ with the physical

FIG. 8. (Color online) (a) Diagrammatic representation of theright-hand side vector. (b) Condition of a MPS to be a canonical form.The transfer matrix has the identity as eigenvectors with eigenvalue 1.(c) Operators inserted into a TPS to find the projective representation.

indices contracted, namely,

Eαα′,ββ ′ =∑

S

A∗S

α′,β ′AS

α,β. (C5)

Next, we find the VR that is the dominant right eigenvectorof double tensor E with eigenvalue η [see Fig. 8(a)]. Weprepare an initial vector Vi and apply the double tensorsN times. By using the power method, in the limit N →∞, (Eαα′,ββ ′)N (Vi)ββ ′ will converge to the dominant right

eigenvector of E. Here, we normalize VR such that VRV†R =

1 and ignore a constant phase factor which results from theright end. Then, we decompose the matrix (VR)αα′ , which isHermitian and non-negative, as VR = W

√λ√

λW † = XX†.Finally, we arrange the tensors A and X into a new tensor,

A′Sα,β

=∑γ ,δ

X−1α,γ AS

γ ,δXδ,β . (C6)

The tensor A′ is thus in the canonical form that is defined bythe right eigenvector of E, as shown in Fig. 8(b).

(4) Determine projective representations:To obtain the projective representations of symmetries, we

need to insert symmetry operators R to the measured state.The generalized transfer matrix G is given by

Gαα′,ββ ′ =∑SS ′

A∗S

α′,β ′RSS ′

AS ′α,β

, (C7)

where the operator RSS ′are applied to the tensors. As R is

a symmetry operator, the generalized transfer matrix G has alargest eigenvalue η of modulo 1, as shown in Fig. 8(c). Againwe apply the generalized transfer matrix many times to aninitial state, and the dominant state is related to a projectiverepresentations Ua

g .

APPENDIX D: EXTRACTING THE PROJECTIVEREPRESENTATIONS FROM EXACT DIAGONALIZATION

In this section, we will show how to obtain the projectiverepresentation via exact diagonalization. For simplicity, we

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CHING-YU HUANG, XIE CHEN, AND FRANK POLLMANN PHYSICAL REVIEW B 90, 045142 (2014)

demonstrate this here only for SPT phases. First of all, weconsider open boundary conditions for a Hamiltonian with2N + 1 sites, and obtain the ground-state wave function |�〉by using exact diagonalization. Then, we want to determinethe MPS representation of the middle site from |�〉. However,the degenerate ground states are likely to lead to ambiguities.

In particular, a 1D SPT phase has generical degeneraciesdue to the edge modes at the ends of the chain. For example,the S = 1 antiferromagnetic chain in the Haldane phase hasspin-1/2 edge states, which lead to the degeneracy of theground state. We can add a small symmetry-breaking fieldat the boundaries to split the edge state degeneracy and obtaina unique ground state |�〉.

The procedure is covered in detail as follows. We do aSchmidt decomposition of the bipartite splitting: {1 . . . N −1|N . . . 2N + 1} of ground state |�〉, so that

|�〉 =χ∑

i=1

λi

∣∣α[1...N−1]i

⟩L

∣∣α[N...2N+1]i

⟩R, (D1)

where |α[1...N−1]i 〉L and |α[N...2N+1]

i 〉R form an orthogonal basis.Again we give a Schmidt decomposition of |�〉 according to[1 . . . N] : [N + 1 . . . 2N + 1],

|�〉 =χ∑

i=1

λi

∣∣β[1...N]i

⟩L

∣∣β[N+1...2N+1]i

⟩R. (D2)

After the above decompositions, the ground state can beexpressed as

|�〉 =χ∑

i=1

AsN

αi ,βi

∣∣α[1...N−1]i

⟩L|sN 〉∣∣β[N+1...2N+1]

i

⟩R, (D3)

where sN = 1, . . . ,ds , where ds denotes the physical dimen-sion and χ is the number of the Schmidt values. We thenarrive at the standard representation of the MPS. To have atrue translationally invariant MPS, we have to fix the phaseambiguity of the virtual indices. For inversion symmetricsystems, this can be done by expressing, for example, the leftSchmidt states as the inverted right ones. It is easy to obtainthe projective representation of the matrix As

α,β by followingthe procedures of Appendix C.

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