PHYSICAL REVIEW B 90, 115157 (2014)

Strange correlations in spin-1 Heisenberg antiferromagnets

Keola Wierschem and Pinaki SenguptaSchool of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371

(Received 7 August 2014; revised manuscript received 10 September 2014; published 30 September 2014)

We study the behavior of the recently proposed “strange correlator” [Y.-Z. You, Z. Bi, A. Rasmussen,K. Slagle, and C. Xu, Phys. Rev. Lett. 112, 247202 (2014)] in spin-1 Heisenberg antiferromagnetic chainswith uniaxial single-ion anisotropy. Using projective quantum Monte Carlo, we are able to directly access thestrange correlator in a variety of phases, as well as to examine its critical behavior at a quantum phase transitionbetween trivial and nontrivial symmetry protected topological phases. After finding the expected long-rangebehavior in these two symmetry conserving phases, we go on to verify the topological nature of two-leg andthree-leg spin-1 Heisenberg antiferromagnetic ladders. This demonstrates the power of the strange correlator indistinguishing between trivial and nontrivial symmetry protected topological phases.

DOI: 10.1103/PhysRevB.90.115157 PACS number(s): 75.10.Jm, 75.10.Kt, 75.40.Mg, 02.70.Ss

I. INTRODUCTION

The study of many-body effects like the Haldane phaseand the fractional quantum Hall effect led to the discoveryof an entire class of quantum-mechanical ground states—topologically ordered phases—that fall outside the standardLandau framework. Unlike conventional states of matter,topological phases are not broken-symmetry ground statescharacterized by a local order parameter. Instead, they havean underlying topological structure that distinguishes themfrom disordered (topologically trivial) phases. A finite gapseparates the lowest excitations from the ground state in thebulk, but there may exist one or more gapless edge states,which is a defining characteristic of this class of phases. Thetheoretical prediction and subsequent experimental discoveryof topological (band) insulators have ushered in a period ofheightened interest in topological phases.

While many experimentally realized topological phases,such as fractional quantum Hall states and topological insula-tors, can be described within the framework of noninteractingelectrons, a natural question that arises in the study ofthese phases is the role of interactions. To address this,a minimal generalization of the free fermion topologicalphase, known as the symmetry protected topological (SPT)phase, was proposed. An SPT state is defined as the groundstate of an interacting many-body system that is comprisedof a gapped bulk state that preserves all the symmetriesof the system and a gapless nontrivial edge state that isprotected by one or more symmetries. In keeping with itsminimal character, phases with long-range topological order(defined by long-range entanglement) are excluded fromthe SPT classification. Instead, SPT states are characterizedby short-range entanglement. Significant progress has beenmade over the past few years in the understanding of SPTstates. This includes a formal mathematical classificationof these states [1,2], as well as detailed investigations ofproposed SPT phases (for both interacting fermions andbosons). The relative simplicity allows us to understandthe emergence of topological phases from the interplay ofstrong correlations, symmetry and topology in these systemsand, in turn, provides deeper insight into more complextopological states such as spin liquids and non-Fermi liquidmetals.

While several SPT states have been discovered and studiedin detail, only a handful of microscopic Hamiltonians withSPT ground states are known to date. Even more worryingly,given a Hamiltonian, there exists no well-established probe todetermine if the ground state has SPT character. Although adegenerate entanglement spectrum is often used as an indicatorof SPT order [3], this method may fail to correctly identify SPTphases protected by an off-lattice symmetry. Additionally, therelation between the low-lying entanglement spectrum and theground-state wave function has been called into question [4].Recently, a strange correlator has been proposed as a directprobe of the SPT character of a wave function and hasbeen demonstrated to identify some well known SPT phasessuccessfully [5]. In this work, we shall present details on howto evaluate the strange correlator in quantum Monte Carlo(QMC) simulations and use it to probe the topological natureof the ground state of spin-1 Heisenberg chains and ladders.

II. MODEL

The Haldane phase of the spin-1 Heisenberg antiferromag-netic chain remains the earliest and most well-understood of allinteracting SPT phases. This simple model has a surprisinglyrich ground-state phase diagram. In addition to the Haldanephase, a topologically trivial quantum paramagnetic phase(the so-called large-D phase) appears when strong uniaxialeasy-plane single-ion anisotropy is introduced. For spin-1Heisenberg antiferromagnetic chains coupled into a laddergeometry, the topological nature of the ground state exhibitsan even/odd effect: trivial SPT character for even-leg laddersand nontrivial SPT character for odd-leg ladders. Our goal is todemonstrate that the strange correlator correctly identifies thevarying SPT character of these respective ground states, as wellas to examine the critical behavior of the strange correlator atthe quantum phase transition between the Haldane and large-Dphases.

To this end, we study spin-1 Heisenberg antiferromagneticchains and ladders with uniaxial single-ion anisotropy, de-scribed by the Hamiltonian

H = J∑〈ij〉

�Si · �Sj + K∑[ij ]

�Si · �Sj + D∑

i

(Sz

i

)2. (1)

1098-0121/2014/90(11)/115157(7) 115157-1 ©2014 American Physical Society

KEOLA WIERSCHEM AND PINAKI SENGUPTA PHYSICAL REVIEW B 90, 115157 (2014)

J J J

K K

(a) (b) (c)

FIG. 1. (Color online) Illustration of the three lattice geometriesconsidered in this work: (a) chain, (b) two-leg ladder, and (c) three-legladder. The spin exchange couplings J and K are shown as red andblue lines, respectively. Empty black circles represent lattice sites.

Here, 〈ij 〉 refers to neighbors along a chain, while [ij ] refersto neighbors between adjacent chains (see Fig. 1). In thefollowing, we set the spin exchange coupling J to unity,thereby defining the energy scale of our system. This leavesthe interchain coupling K and single-ion anisotropy D as ouronly Hamiltonian parameters. For the ladder geometries, wefurther set D = 0, while for the chain geometry, the parameterK becomes meaningless. We use QMC methods to studyfinite-size systems of size N and length L, where N = L,2L, and 3L for chain, two-leg ladder, and three-leg laddergeometries, respectively. Periodic boundary conditions areemployed along the length of the system.

III. METHODS

To investigate the above model in the ground-state limit, weuse a projective variant of the stochastic series expansion QMCmethod [6,7]. The main idea is as follows: instead of expandingthe density matrix as a Taylor series of Hamiltonian operations,we project out the ground state by repeated Hamiltonianoperation on a trial wave function. While the presence of atrial wave function explicitly removes the usual periodicityin the imaginary-timelike dimension of the operator string, itcan be thought of as a set of vertices of infinite weight. Thuswe can still utilize the directed loop equations of Syljuasenand Sandvik [8], minimize bounce probabilities in the loopalgorithm, and obtain efficient global updates.

Now we describe our projective QMC scheme in detail.First, let us examine the effect of m repeated Hamiltonianoperations on a trial wave function,

(H − C)m|ψ〉 = (H − C)m∑

α

cα|α〉. (2)

Here, we have expanded |ψ〉 in the basis of energy eigenstates|α〉 with coefficients given by cα = 〈α|ψ〉. The constant Cis chosen to make (H − C) negative definite. Thus, as longas c0 �= 0, the projection of |ψ〉 will be dominated by the

ground-state terms,

(H − C)m|ψ〉 =∑

α

cα(Eα − C)m|α〉. (3)

This can be made explicit by rewriting the expression as( H − CE0 − C

)m

|ψ〉 =∑

α

cα

(Eα − CE0 − C

)m

|α〉. (4)

The ground state is approached as m → ∞. In addition to thepower-based projector (H − C)m, it is also possible to use anexponential projector exp[−β(H − C)]. A detailed descriptionof the exponential projector has been given by Farhi et al. [9].

Having a valid ground-state projector, we can evaluateground-state observables as

〈O〉 = 〈ψ |(H − C)mO(H − C)m|ψ〉〈ψ |(H − C)2m|ψ〉 . (5)

These observables are calculated at the “middle” of the (open)operator string. Appropriate sampling weights can be derivedby expanding the projector (H − C)m as a summation overall possible operator strings. Each operator string consistsof a product of 2m bond operators. This is completelyanalogous to the corresponding expansion of the densitymatrix into a summation over operator strings in the standardstochastic series expansion QMC technique [6,7], with theadded simplicity that the length of our operator string remainsfixed due to our choice of a power-based projector.

Within the same formulation, we can also easily computethe overlap of the ground-state wave function with an arbitrarywave function |�〉, as required for calculating the so-calledstrange correlator [5]—defined in our case as

C(r − r ′) = 〈�|(Sxr Sx

r ′ + Syr S

y

r ′ )(H − C)2m|ψ〉〈�|(H − C)2m|ψ〉 . (6)

Here, |�〉 is a trivial symmetric product state, (H − C)2m|ψ〉projects out the ground state |�0〉, and Sx

r Sxr ′ + S

yr S

y

r ′ arestandard off-diagonal spin correlations. Thus Eq. (6) describesa correlation function at the imaginary time boundary betweenthe states |�〉 and |�0〉. This imaginary time boundary mapsonto a spatial interface between the two states via a Lorentztransformation. Now, it has been demonstrated that certainspace-time correlation functions (defined in terms of localoperators) should possess either long-range order or algebraicdecay at the interface between trivial and nontrivial SPTstates in one and two dimensions [5]. The same Lorentztransformation maps the space-time correlation at the spatialinterface to the strange correlator at the imaginary timeinterface. Thus the nature of the strange correlator providesa direct and reliable probe for the SPT nature of the groundstate of the Hamiltonian H.

Interestingly, within our projective QMC formulation, thestrange correlator is simply a correlation function measured atthe “ends” of the operator string. This can perhaps most easilybe seen by comparing Eq. (6) with an equal-time correlationfunction

�(r − r ′) = 〈ψ |(H − C)m(Sxr Sx

r ′ + Syr S

y

r ′ )(H − C)m|ψ〉〈ψ |(H − C)2m|ψ〉 .

(7)

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STRANGE CORRELATIONS IN SPIN-1 HEISENBERG . . . PHYSICAL REVIEW B 90, 115157 (2014)

This correspondence highlights the physical interpretation ofthe strange correlator as a correlation function at the temporalboundary of the time-evolved ground states |�0〉 and |�〉.

In the current work, we choose both the trivial product stateand the trial wave function to be the product state of zero spinprojection along the z axis,

|ψ〉 = |�〉 =∏

i

|0〉i . (8)

This choice conserves the on-site symmetry of the Hamiltonianin Eq. (1). Also, note that this definition has the convenientfeature that C(0) = 2. Another convenience arises when takinginto account the usual sublattice rotation of π along the z axisof the spin space that is needed to ensure negative-definiteoff-diagonal vertex weights in the bond expansion of theHamiltonian. For a projective QMC implementation, thissublattice rotation should also be applied to the trial wavefunction. Here we note that the current choice of |ψ〉 isinvariant under such a transformation. This transformationintroduces a shift of π in the momentum of spin excitations.Henceforth, we assume that the momentum vector is measuredfrom the shifted point, i.e., k → k + π .

In order to improve the statistical sampling of the strangecorrelator at the ends of the operator string, as well as normalobservables at the middle level of the operator string, weintroduce a bias in our loop updates to preferentially startthe loop at these levels of the operator string. Satisfactionof detailed balance is contingent upon equal probabilities ofstarting forward and reverse loops, which is not affected by ouradded bias. This is because loops starting at different levels ofthe operator string are not connected to each other [8].

In the next section, we proceed to analyze the strangecorrelator in the ground-state thermodynamic limit. Usingsystem sizes of length 16 � L � 96, we are able to accuratelydetermine the proper scaling limits of the strange correlatorin several symmetric phases. Analogous to the ground-statescaling of the operator string length in finite-temperaturestochastic series expansion QMC, we choose m ∝ L2 toconverge to the ground state.

IV. RESULTS

This section is organized as follows. First, we consider thebehavior of the strange correlator in the spin-1 Heisenbergantiferromagnetic chain with uniaxial single-ion anisotropy.This system has both trivial and nontrivial SPT states—thelarge-D and Haldane phases, respectively. These two phasesare separated by a continuous phase transition, which allowsfor an investigation of the critical behavior of the strangecorrelator. Subsequently, we examine the behavior of thestrange correlator in the spin-1 Heisenberg antiferromagnetictwo-leg and three-leg ladders. The ground states of spin-1ladders are expected to exhibit the following even-odd effect:an even (odd) number of legs leads to a trivial (nontrivial)SPT ground state. Hence this is a good place to test the powerof the strange correlator to distinguish trivial and nontrivialSPT states. Finally, we look at the finite-size scaling behaviorof an order parameter associated with the strange correlator.

A. Chain

Following Haldane’s seminal conjecture that the groundstate of integer spin Heisenberg antiferromagnets in one di-mension should be gapped [10,11], it was quickly realized thatthe ground state of the spin-1 Heisenberg antiferromagneticchain is closely related to the exact ground state (the so-calledAKLT state) of a Hamiltonian constructed by a sum of spin-2projectors on each bond (the AKLT model) [12]. The AKLTstate is comprised of S = 1/2 singlets on each bond. Forperiodic boundary conditions, the ground state is a gapped,valence bond solid, but for open boundary conditions, the chainhas two unpaired S = 1/2 moments at each end resulting ina fourfold degenerate ground state. In the language of SPT’s,the unpaired spins at the end form the edge states whereasthe singlets on the bonds constitute the (gapped) bulk. Thistopological character can be captured explicitly by the stringorder parameter [13], which is also finite in the Haldane phaseof the spin-1 Heisenberg antiferromagnetic chain. In otherwords, the Haldane phase is adiabatically connected to theAKLT state. In the presence of uniaxial single-ion anisotropy,there is a distinct topologically trivial quantum paramagneticphase for strong easy-plane values of the anisotropy—theso-called large-D phase. The Haldane and large-D phases areseparated by a Gaussian phase transition at the critical valueDc ≈ 0.97.

In Fig. 2, we plot the strange correlator as a function ofdistance for a 64-site chain. Results are shown for values of

0 8 16 24 32 40 48 56 64r

10-5

10-4

10-3

10-2

10-1

100

C(r

) D=0.00D=1.00D=2.00

1 10 100r

10-5

10-4

10-3

10-2

10-1

100

C(r

)

D=0.00D=1.00D=2.00

ξ=3.76

C(∞)=0.64

η=1.00

FIG. 2. (Color online) Strange correlator in the Haldane (D = 0)and large-D (D = 2) phases, as well as in the vicinity of the criticalpoint (D = 1). The long-range order and exponential decay in theHaldane and large-D phases, respectively, are readily apparent inthe log-normal plot of the upper panel, while the lower panelillustrates algebraic decay near Dc using a log-log plot. Solidlines are fits to C(r) = C(∞), C(r) ∼ r−η + (L − r)−η, and C(r) ∼e−r/ξ + e−(L−r)/ξ in the region L < 4r < 3L (black, red and bluelines, respectively). Taking J = 1, we find 2m = L2 is sufficient toconverge to the ground-state limit.

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KEOLA WIERSCHEM AND PINAKI SENGUPTA PHYSICAL REVIEW B 90, 115157 (2014)

the single-ion anisotropy in the Haldane and large-D phases,as well as in the vicinity of the critical point separating thesetwo phases. By fitting the strange correlator to the appropriateasymptotic scaling forms in the region L < 4r < 3L, weconfirm the expected behavior for trivial and nontrivialSPT phases. In the nontrivial Haldane phase (D = 0), thestrange correlator approaches a constant value, C(r) ∼C(∞) = 0.64 over the fitting range indicated, in accordancewith its nontrivial SPT character. On the other hand, in thetrivial large-D phase (D = 2) the strange correlator is foundto exhibit the expected exponential decay, C(r) ∼ e−r/ξ , witha correlation length ξ = 3.76. This exponential decay is mostclearly seen as a linear regime on the log-normal plot inFig. 2 (upper panel). Near the critical point (D = 1), thestrange correlator decays algebraically as C(r) ∼ r−η withan exponent η = 1.00 just like a traditional correlationfunction does at the boundary of a continuous phase transitionas a result of the diverging correlation length. Here, thisalgebraic decay shows up as a linear regime in the log-log plotof Fig. 2 (lower panel).

B. Two-leg ladder

The two-leg spin-1 Heisenberg ladder (with D = 0) hasbeen shown to have a topologically trivial SPT groundstate [14]. This can be understood by considering the limitingcase of strong rung interactions, J � K . In this limit, theground state is well approximated as a product of rungsinglets. By definition, such a state is topologically trivial.When coupled with results from a previous QMC study thatdemonstrated a lack of any phase transition [15], this leads tothe conclusion that the ground state of the isotropic two-legspin-1 Heisenberg ladder is always topologically trivial.

Here, we calculate the strange correlator for antiferromag-netic rung interactions with J = K = 1. As seen in Fig. 3,the strange correlator at long distances quickly decays to zeroas our system size approaches the thermodynamic limit. Thisis true both for correlations within a single leg of the ladder(�x = 0) as well as for correlations between opposite legs ofthe ladder (�x = 1). Thus we see that the strange correlator

0.0 0.2 0.4 0.6 0.8 1.0r/L

0.0

0.4

0.8

1.2

1.6

2.0

C(r

)

L=16L=32L=64

0.0 0.2 0.4 0.6 0.8 1.0r/L

0.0

0.4

0.8

1.2

1.6

2.0

C(r

)

L=16L=32L=64

Δx=0 Δx=1

FIG. 3. (Color online) Strange correlator in the two-leg spin-1ladder with D = 0 and J = K = 1. We find 2m = L2 is sufficient toconverge to the ground-state limit.

0.0

0.4

0.8

1.2

1.6

2.0

C(r

)

L=16L=32L=64

0.0

0.4

0.8

1.2

1.6

2.0

C(r

)

L=16L=32L=64

0.0 0.2 0.4 0.6 0.8 1.0r/L

0.0

0.4

0.8

1.2

C(r

)

0.0 0.2 0.4 0.6 0.8 1.0r/L

0.0

0.4

0.8

1.2

C(r

)

Δx=1

Δx=0

Δx=2

Δx=0

FIG. 4. (Color online) Strange correlator in the three-leg spin-1ladder with D = 0 and J = K = 1. Correlations involving the middleleg of the ladder are shown in the left panels, while those involvingonly the outer legs are shown in the right panels. We find 2m = 3L2

is sufficient to converge to the ground-state limit.

correctly identifies the ground state of the spin-1 Heisenbergantiferromagnetic two-leg ladder as a topologically trivialstate.

C. Three-leg ladder

In contrast to the two-leg spin-1 ladder, the ground stateof the three-leg spin-1 ladder is a topologically nontrivialSPT state [16]. This can be seen in Fig. 4, where the strangecorrelator converges to a nonzero value as our system sizeapproaches the thermodynamic limit. As before, the long-distance behavior is the same for correlations within a singleleg of the ladder (�x = 0, inner or outer legs) as well asfor correlations between different legs of the ladder (�x = 1or �x = 2). Once again, we see that the strange correlatorcorrectly identifies the ground-state SPT character, this timefinding a nontrivial SPT ground state for the spin-1 Heisenbergantiferromagnetic three-leg ladder.

D. Finite-size scaling

In order to make a more quantitative statement on the scal-ing properties of the strange correlator in the thermodynamiclimit, we define a finite-size order parameter �L = 1

N

∑�r C(�r)

based on the corresponding static structure factor [see Eq. (A1)in Appendix]. In Fig. 5, we show the system size dependence of�L for various SPT phases. As expected, within the nontrivialSPT phases (chain and three-leg ladder with D = 0), we find�L approaches a constant value exponentially with system sizeas �L = �∞ + (a − be−L/ξ )/L [Eq. (A6) in Appendix]. Fromthe fit, we extract �∞ = 0.64 and ξ = 9.58 for the Haldanephase of the spin-1 chain, in agreement with our previous fitof C(r) for a single system size. For the three-leg ladder,the corresponding estimates are �∞ = 0.29 and ξ = 29.3.The finite value of the order parameter in the thermodynamiclimit confirms the nontrivial SPT character of the ground state.For the trivial SPT phases (chain with D = 2, two-leg ladderwith D = 0), �L instead scales exponentially to zero withsystem size. Fitting the simulation data to the same form of

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STRANGE CORRELATIONS IN SPIN-1 HEISENBERG . . . PHYSICAL REVIEW B 90, 115157 (2014)

0.00 0.02 0.04 0.06 0.081/L

0.0

0.2

0.4

0.6

0.8

1.0

ΨL

D=0.00D=1.00D=2.00

0.00 0.02 0.04 0.06 0.081/L

0.0

0.2

0.4

0.6

0.8

1.0

ΨL

2-Leg Ladder3-Leg Ladder

ξ≈4.15

ξ≈9.58, Ψ∞≈0.64

η≈0.84

ξ≈13.1

ξ≈29.3, Ψ∞≈0.29

FIG. 5. (Color online) Finite-size scaling of the order parameterfor the strange correlator in chain (left panel) and ladder (right panel)geometries. Lines are obtained as fits to the data using the finite-sizescaling forms derived in the Appendix. Parameters are the same as inearlier figures.

the finite-size order parameter, Eq. (A6), we obtain �∞ = 0and ξ = 4.15 for the chain and �∞ = 0 and ξ = 13.1 forthe two-leg ladder. The correlation length of the chain isreasonably close to the value obtained from a direct fit toC(r). In the last scenario, we probe the behavior of �L nearthe critical point (chain with D = 1) and find that it followsa scaling form �L = a/L + b/Lη [Eq. (A11) in Appendix].The extracted exponent η = 0.84 is substantially less thanthe value estimated directly from C(r), but this is not asurprise considering the quasi-long-range nature of correlationfunctions at the critical point. The behavior of the finite sizeorder parameter in the different parameter regimes confirmsthat the strange correlator follows the usual critical behavioracross the phase boundary between trivial and nontrivialSPT phases.

The critical behavior at the quantum phase transitionbetween the Haldane and large-D phases is captured bya conformal field theory that maps onto a free Gaussianmodel [17]. The critical exponents for this Gaussian transitioncan be expressed in terms of a single parameter K (theLuttinger parameter, not to be confused with the interchaincoupling defined earlier). For the equal-time Green’s functionG(r) = 〈Sx

0 Sxr + S

y

0 Syr 〉, the anomalous dimensionality should

be given by ηG = 1/2K (we introduce a subscript to distin-guish the critical exponents of the Green’s function and thestrange correlator), while the critical exponent governing thecorrelation length is expected to be ν = 1/(2 − K). Similarrelations can be derived through the well-known bosonizationtechnique.

The Gaussian transition has been well studied [17–21], withrecent results from the density matrix renormalization groupobtaining Dc = 0.96845(8) and K = 1.321(1) [21]. To testour method, in the top panels of Fig. 6, we plot the equal-timeGreen’s function at half the system size, G(L/2), multipliedby LηG to obtain a dimensionless parameter. As can be seen inthe left panel, this dimensionless parameter is independent ofthe system size at the critical point Dc, while the right panel

0.68

0.72

0.76

0.80

G(L

/2) L

η G

L=16L=32L=48L=64L=80L=96 0.68

0.72

0.76

0.80

G(L

/2) L

η G

0.85 0.90 0.95 1.00 1.05 1.10D

1.0

1.5

2.0

2.5

3.0

C(L

/2) L

η S

-3 -2 -1 0 1 2 3(D-Dc) L

1/ν

1.0

1.5

2.0

2.5

3.0

C(L

/2) L

η S

1/ν=0.679ηG=0.3785, ηS=2ηG

Dc=0.96845

FIG. 6. (Color online) Finite-size scaling of the order parameteracross the Gaussian critical point of the Heisenberg AFM chain withsingle-ion anisotropy for the normal Green’s function (top panels) andthe strange correlator (bottom panels). Lines are shown as guides tothe eye. We find 2m = 4L2 is sufficient to converge to the ground-statelimit.

demonstrates finite-size data collapse. Interestingly, when weplot the strange correlator using an exponent ηS = 2ηG, wesee a similar curve crossing and finite-size collapse (lowerpanels, Fig. 6). This gives a value ηS = 0.757 that is lowerthan our estimate from fitting �L in Fig. 5, yet reasonablyclose considering the distance of D = 1 from Dc. It remainsto be seen why this relation works so well here, and whether ornot it applies generally at the boundary between two distinctSPT phases.

V. DISCUSSION

In one dimension, a nonlocal order parameter can bedefined to distinguish between trivial and nontrivial SPTphases [22]. For the Haldane phase of the spin-1 Heisenbergantiferromagnetic chain, this is none other than the originalstring order parameter [13]. Similar nonlocal order parameterscan also be defined on two-leg [15] and three-leg [16] ladders.However, note that such nonlocal order parameters can eithermeasure hidden symmetry conservation or hidden symmetrybreaking, depending upon their construction [22]. Thus thefour-body string order parameter of Todo et al. [15] is in factevidence for a trivial SPT state in the two-leg ladder.

While carefully constructed string order parameters mayallow for the characterization of SPT phases in one dimension,they are only valid in the presence of dihedral symmetry (D2

or Z2 × Z2) [22]. Additionally, they do not extrapolate easilyto higher dimensions. The strange correlator is thus a powerfultool for the generic investigation of SPT properties regardlessof on-site symmetry, and extends easily to higher dimensions.As demonstrated by You et al. [5], the strange correlator caneasily distinguish trivial and nontrivial SPT states in one andtwo dimensions. In three dimensions, a long-range or quasi-long-range strange correlator still implies nontrivial SPT order;however, due to the possibility of topologically ordered edge

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KEOLA WIERSCHEM AND PINAKI SENGUPTA PHYSICAL REVIEW B 90, 115157 (2014)

states, it becomes possible for a nontrivial SPT state to have ashort-range strange correlator [5].

Further insight into the strange correlator may be obtainedby taking into consideration the edge modes in nontrivialSPT states. Taking the Haldane ground state as a concreteexample, and recalling its relation to the AKLT state, we cansee that there should be free spin-1/2 degrees of freedomat the spatial boundary between the Haldane phase and thetrivial large-D phase. These free states will evolve in time assteady states, meaning they possess long-range correlations intime. After a Lorentz transformation, temporal correlationsat the spatial boundary between the Haldane and large-Dphases transform into real-space correlations at the temporalboundary between these nontrivial and trivial SPT states. Thusa long-range strange correlator can be viewed as evidence forfree edge modes in one dimension.

Edge states can also explain the even/odd effect in spin-1ladders with time-reversal symmetry. For even-legged ladders,the spin-1/2 edge states along each chain become coupled intoan overall integer spin state, and it is possible to form a singlet,thereby removing the edge degrees of freedom. However, forodd-leg ladders the total edge spin will be half an odd integer,and by Kramers theorem must be at least doubly degenerate.In this case, the edge state can only be removed by breakingthe symmetry or undergoing a bulk phase transition.

VI. CONCLUSION

In conclusion, we have implemented a projective QMCmethod that is able to calculate the strange correlator ina wide variety of phases. Using this method to study thespin-1 Heisenberg antiferromagnetic chain, we have verifiedthe topological nature of this prototypical SPT system. Addinga uniaxial single-ion anisotropy, we find evidence of criticalbehavior in the strange correlator at the quantum phasetransition between the Haldane and large-D phases. Thusthe strange correlator can be used as an order parameter forphase transitions between trivial and nontrivial SPT states. Wehave also calculated the strange correlations in two-leg andthree-leg ladders to verify their relative trivial and nontrivialSPT phases. Although the topological characterization ofthese phases was known from past work, the QMC methodsimplemented here are easily extended to higher dimensions.Further, the strange correlator should continue to maintain atleast quasi-long-range behavior in two dimensions [5]. Thispaves the way for applications to systems in two dimensions,where string order becomes ill defined [23].

ACKNOWLEDGMENTS

We thank Anders Sandvik and Yi-Zhuang You for helpfuldiscussions. We also thank Cenke Xu for useful comments.This research used resources of the National Energy ResearchScientific Computing Center, a DOE Office of Science UserFacility supported by the Office of Science of the US Depart-ment of Energy under Contract No. DE-AC02-05CH11231.This work was supported in part by the Ministry of Education,Singapore under Grant No. MOE2011-T2-1-108.

APPENDIX: FINITE-SIZE SCALING FORMS

Here we derive finite-size scaling forms for the orderparameter, which can be defined as the structure factor persite:

�L = S(0)

N= 1

N

∑�r

C(�r). (A1)

This expression can be converted into an integral in thecontinuum limit that is obtained as L → ∞:

S(0) =∫ L

0C(r)dr. (A2)

Assuming that the correlation function approaches the asymp-totic form C(r) ∼ e−r/ξ for r > R, we replace the integral from0 to R by a “core charge” C0. This yields a simple expressionfor the structure factor,

S(0) = C0 + A

∫ L−R

R

[e−r/ξ + e−(L−r)/ξ ]dr, (A3)

where we take into account the periodicity of C(r) for finitesimulation cells with periodic boundary conditions. This gives

S(0) = C0 + 2Aξ [e−R/ξ − e−(L−R)/ξ ], (A4)

which ultimately is the same as

S(0) = a − be−L/ξ , (A5)

i.e., we cannot uniquely determine C0 and R. This leads to afinite-size scaling form:

�L = �∞ + (a − be−L/ξ )/L, (A6)

where �∞ �= 0 allows for an exponential decay to a nonzerovalue, as is the case for ordered phases.

At a critical point, the expected asymptotic scaling formbecomes C(r) ∼ r−η, so instead we find

S(0) = C0 + A

∫ L−R

R

[r−η + (L − r)−η]dr, (A7)

which after integration becomes

S(0) = C0 + 2A

1 − η[(L − R)1−η − R1−η]. (A8)

Again, we cannot uniquely determine C0 and R, which leavesthe general scaling form

S(0) = a + b(L − R)1−η. (A9)

For small R/L, this can be replaced by the simpler form

S(0) = a + bL1−η. (A10)

Thus we have used the following finite size scaling form forthe order parameter near the critical point:

�L = a/L + b/Lη. (A11)

The above finite-size scaling forms can also be derived forhigher dimensions d > 1 by using the relation �L = S(0)L−d .In this case, the correlation function at a critical point isdefined as C(r) ∼ r2−(d+z+η), where z is the dynamic criticalexponent for ground-state phase transitions. For the Gaussianphase transition separating the Haldane and large-D phases,we expect z = 1 [20].

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