Topological Invariant and Quantum Spin Models from Magnetic � Fluxesin Correlated Topological Insulators

F. F. Assaad, M. Bercx, and M. Hohenadler

Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany(Received 23 April 2012; revised manuscript received 5 October 2012; published 26 February 2013)

The adiabatic insertion of a � flux into a quantum spin Hall insulator gives rise to localized spin and

charge fluxon states. We demonstrate that � fluxes can be used in exact quantum Monte Carlo simulations

to identify a correlated Z2 topological insulator using the example of the Kane-Mele-Hubbard model.

In the presence of repulsive interactions, a � flux gives rise to a Kramers doublet of spin-fluxon states with

a Curie-law signature in the magnetic susceptibility. Electronic correlations also provide a bosonic mode

of magnetic excitons with tunable energy that act as exchange particles and mediate a dynamical

interaction of adjustable range and strength between spin fluxons. � fluxes can therefore be used to

build models of interacting spins. This idea is applied to a three-spin ring and to one-dimensional spin

chains. Because of the freedom to create almost arbitrary spin lattices, correlated topological insulators

with � fluxes represent a novel kind of quantum simulator, potentially useful for numerical simulations

and experiments.

DOI: 10.1103/PhysRevX.3.011015 Subject Areas: Strongly Correlated Materials, Topological Insulators

I. INTRODUCTION

A topological insulator represents a novel state of mattercharacterized by a special band structure that can result,e.g., from strong spin-orbit interaction [1,2]. In two dimen-sions, this state is called a quantum spin Hall insulator andhas deep connections with the quantum Hall effect, includ-ing the coexistence of a bulk band gap and metallic edgestates, the absence of symmetry breaking, and the possi-bility of a mathematical classification [3,4]. Importantly,because of the absence of a magnetic field, the quantumspin Hall insulator preserves time-reversal symmetry,which provides protection against interactions and disorder[5–7]. The quantum spin Hall insulator has been realized inHgTe quantum wells [8,9].

Correlated topological insulators with strong electron-electron interactions are the focus of current research [10].Intriguing concepts include electron fractionalization inthe presence of time-reversal symmetry, [11–14] spinliquids [14–16], and topological Mott insulators [17,18].Remarkably, some of the theoretical models can be studiedusing exact numerical methods. A central problem in thiscontext is the question of how to detect a topologicalstate directly from bulk properties, for example, in caseswhere the bulk-boundary correspondence breaks down.Experimentally, this issue also arises in the absence ofsharp edges in proposed cold-atom realizations as a resultof the trapping potential [19,20]. The classification interms of a Z2 Chern-Simons index relies on Bloch wave

functions and is therefore only valid for noninteractingsystems. Generalizations involve twisted boundary condi-tions [21] or Green functions [22–27] and are challengingto use in experiments or exact simulations. Indirect signa-tures such as the closing of gaps [16] or the crossing ofenergy levels [28] require, among other difficulties, experi-mental tuning of microscopic parameters.Topological insulators show a unique response to

topological defects such as dislocations [29,30] or � fluxes[12,30,31]. Upon adiabatic insertion of a � flux, Faraday’slaw, together with the quantized transverse conductivity,gives rise to midgap charge and spin-fluxon states [12,31].These states are exponentially localized around the flux[12,31]. The existence of these states is ensured, even in thepresence of interactions or disorder, by time-reversal sym-metry, and has been suggested as a bulk probe of the Z2index [12,31]. The concept of fluxons can also be general-ized to situations where spin is not conserved, such as inthe presence of Rashba coupling. In three dimensions, amagnetic flux gives rise to the wormhole effect [32].Electron-electron repulsion lifts the degeneracy of chargeand spin fluxons, but the two degenerate spin-fluxonstates constitute a localized spin with Sz ¼ �1=2 [12].Dynamical� fluxes emerge in the context of fractionalizedtopological insulators [12,13].Previous work on � fluxes in noninteracting quantum

spin Hall insulators [12,30,31] was based on square-latticemodels such as that for HgTe quantum wells [8]. Here, weconsider the half-filled Kane-Mele model on the honey-comb lattice [3] (historically the first model with a Z2topological phase), which has close connections to gra-phene [3], the integer quantum Hall effect [33], and, whenincluding interactions, to correlated Dirac fermions [15].Topological phases of interacting fermions on honeycomblattices may be realized in transition metal oxides [34],

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semiconductor structures [35], graphene [36], or coldatoms [37] (see also Ref. [10]).

Here, we use � fluxes in combination with exact quan-tum Monte Carlo simulations and show that they can beused efficiently to probe the topological invariant of corre-lated topological insulators. In particular, this method doesnot rely on an adiabatic connection to a noninteractingstate, and it may also be used for fractional states. Inaddition, we demonstrate that� fluxes permit the construc-tion of quantum spin models of almost arbitrary geometryand with tunable, dynamical interactions. The spins corre-spond to the spin fluxons created by inserting � fluxes, andthe interaction is mediated by magnetic excitons corre-sponding to collective magnetic fluctuations of the topo-logical insulator. These spin models can be studiedtheoretically with the quantum Monte Carlo method, orexperimentally. As examples, we show that a ring of threespins has a ground state with magnetization 1=2 and that aone-dimensional chain of fluxons undergoes a Mott tran-sition and is described at low energies by an XXZ model.

The article is organized as follows. In Sec. II, we in-troduce the Kane-Mele and Kane-Mele-Hubbard models.Section III provides details about the methods. The use of� fluxes as a probe for topological states is discussed inSec. IV, whereas the construction of quantum spin modelsis the topic of Sec. V. Conclusions are given in Sec. VI, andwe provide three appendixes.

II. MODEL

The half-filled Kane-Mele model with additionalelectron-electron interactions can be studied with powerfulquantum Monte Carlo methods [16,18]. Using the spinor

notation ĉyi ¼ ðĉyi"; ĉyi#Þ, where ĉyi� is a creation operator foran electron in a Wannier state at site i with spin �, theHamiltonian reads

HKM ¼ �tXhi;ji

�ijĉyi ĉj þ i�

Xhhi;jii

�ijĉyi ð�ij � �Þĉj

þ i�Xhi;ji

�ijĉyi ðs� d̂ijÞ � ẑĉj: (1)

The notations hi; ji and hhi; jii indicate that the sites i and jare nearest neighbors and next-nearest neighbors, respec-tively, and implicitly include the Hermitian conjugateterms.

The first term describes the hopping of electrons be-tween neighboring lattice sites. The second term representsthe spin-orbit coupling which reduces the SUð2Þ spinrotation symmetry to a Uð1Þ symmetry. The third term isan additional Rashba spin-orbit coupling [38]. The addi-tional factors �ij ¼ �1 take into account any � fluxespresent, whereas the original Kane-Mele model (without� fluxes) is recovered from Eq. (1) by setting �ij ¼ 1.

The spin-orbit term corresponds to a next-nearest-neighbor hopping with a complex amplitude i� and has

been derived from the spin-orbit coupling in graphene [3].This hopping acquires a sign �1, depending on its direc-tion, the sublattice, and the electron spin. This sign isencoded in (�ij � �), where

� ij ¼dik � dkjjdik � dkjj : (2)

dik is the vector connecting sites i and k, and k is theintermediate lattice site involved in the hopping processfrom i to j. For a coordinate-independent representation,the vectors d�� are defined in three-dimensional space,

although the z component vanishes. The vector � is de-fined by � ¼ ð�x; �y; �zÞ, with the Pauli matrices ��.The last term in Eq. (1) is the Rashba spin-orbit inter-

action [3,5]. It is defined in terms of the spin vector

s ¼ �=2 and the unit vector d̂ij, which can be expressedin terms of the nearest-neighbor vectors �1, �2, �3 [39].The Rashba coupling breaks the z � �z inversionsymmetry and has to be taken into account, for example,in the presence of a substrate. Because this term includesspin-flip terms, spin is no longer conserved. The Rashbaterm has been included in the results for the noninteractingmodel (1), but cannot be included in quantum Monte Carlosimulations of the interacting model (3) due to a minus-sign problem.The model (1) can be solved exactly [3,5,40]. In the

absence of Rashba coupling, � ¼ 0, the Kane-Mele modeldescribes a Z2 quantum spin Hall insulator for any � > 0.

This state is characterized by a bulk band gap�sp ¼ 3ffiffiffi3

p�,

a spin gap �s ¼ 2�sp, and a quantized spin Hall conduc-tivity �sxy ¼ e22� . The topological state survives for Rashbainteractions �< 2

ffiffiffi3

p� (for chemical potential� ¼ 0) and

has protected, helical edge states for geometries with openboundaries [3,5,40]. We use t as the unit of energy (@ ¼ 1),take �=t ¼ 0:2, and consider periodic lattices with L� L0unit cells.To investigate the effect of electron-electron repulsion,

we consider the paradigmatic Hubbard interaction [41] andarrive at the Kane-Mele-Hubbard model [42],

HKMH ¼ HKM þHU; HU ¼ 12UXi

ðĉyi ĉi � 1Þ2: (3)

Hamiltonian (3) without Rashba coupling has been studiedintensely [16,42–48]. In particular, its symmetries permitthe application of exact quantum Monte Carlo methodswithout a sign problem [16,43,48].On a lattice with periodic boundaries, � fluxes can only

be inserted in pairs, as illustrated for the minimal numberof two fluxes in Fig. 1. The flux pair is connected by abranch cut (or string), and every hopping process crossingthe cut acquires a phase ei� ¼ �1, as encoded by �ij inEq. (1). Different choices of the branch cut for fixed fluxpositions are related by a gauge transformation.

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III. METHOD

We have used the auxiliary-field quantum MonteCarlo method [49], which was previously applied to theHubbard model on the honeycomb lattice [15], and theKane-Mele-Hubbard model [16,43,48]. The central idea ofthis stochastic method is to use a path integral representationof the interacting model (3). By means of a Hubbard-Stratonovich transformation, the Hubbard term is decoupled,leading to a problem of noninteracting fermions in anexternal, space-dependent and imaginary-time-dependentfield. The sampling is over different configurations of theseauxiliary fields in terms of local updates. For a givenconfiguration of fields, Wick’s theorem can be used tocalculate arbitrary correlation functions from the single-particle Green function. We refer to a review [50] andprevious work [15,16,48] for technical details such as thecalculation of energy gaps.

Here, we have used a projective formulation (withprojection parameter �t ¼ 40) to obtain ground-stateresults, starting from a trial wave function (the ground stateof the U ¼ 0 case) [48] and a finite-temperature formula-tion to calculate thermodynamic properties. Both variantsrely on a Trotter discretization of imaginary time (we used�� ¼ �=L ¼ �=L ¼ 0:1), but the associated systematicerror is smaller than the statistical errors. At half filling,time-reversal invariance ensures that simulations can becarried out without a minus-sign problem, even in thepresence of � fluxes.

IV. USING � FLUXES TO PROBE CORRELATEDTOPOLOGICAL STATES

A. Thermodynamic signature of � fluxes

In the topological phase of the model (1), each � fluxgives rise to four fluxon states which are exponentiallylocalized (due to the bulk energy gap �sp) near the corre-

sponding flux-threaded hexagons [12,31] (see Fig. 1). Thestates correspond to the spin fluxons j "i, j #i (with energyE"#), forming a Kramers pair related by time reversal, andthe charge fluxons jþi, j�i (with energies Eþ, E�), relatedby particle-hole symmetry. As we show in Fig. 2, thefluxon states lie inside the bulk band gap, and for non-interacting electrons, E"# ¼ Eþ ¼ E�.

The fluxons leave a characteristic signature in the staticspin and charge susceptibilities,

s ¼ �ðhM̂2zi � hM̂zi2Þ; c ¼ �ðhN̂2i � hN̂i2Þ; (4)

which are defined in terms of the operators of total spin

in the z direction (M̂z ¼P

iĉyi �

zĉi) and of the total

charge (N̂ ¼ Piĉyi ĉi); the inverse temperature is given by� ¼ 1kBT . At low temperatures, kBT � �sp, we can restrictthe Hilbert space to fj "i; j #i; jþi; j�ig. If the spin fluxonsare independent, and for � ¼ 0, we expect a Curie laws ¼ c ¼ 12kBT per � flux, and hence s ¼ c ¼ 1kBT fortwo independent � fluxes (see Appendix A). The prefactorof the Curie law follows from the quantized spin Hallconductance in the absence of Rashba coupling [3].Similarly, a Curie law was also predicted for topologicalexcitations in polyacetylene [51].

FIG. 1. For a lattice with periodic boundaries, � fluxes can beinserted in pairs. Each flux threads a hexagon (highlighted inblue) of the honeycomb lattice, and the pair is connected by abranch cut (blue line). Hopping processes crossing the branchcut acquire a phase ei� ¼ �1.

1

10

100

1000

0.001 0.01 0.1 1 10

FIG. 3. Spin susceptibility of the Kane-Mele model(�=t ¼ 0:2) with two � fluxes at the maximal distance on an18� 18 lattice, for different Rashba couplings �. At tempera-tures kBT & 0:1t, each � flux contributes

12kBT

to the suscepti-

bility, leading to s � 1kBT . Also shown is the spin-gap energyscale 0:2�s for T ¼ 0, � ¼ 0. For �> 0, the chemical potentialis adjusted to retain a half-filled band.

FIG. 2. In a quantum spin Hall insulator, a � flux gives riseto four states (with charge q and spin Sz) localized near theflux, which lie inside the bulk energy gap between the valenceand conduction bands (labeled ‘‘VB’’ and ‘‘CB’’ in the figure,respectively) [12,31]. The states correspond to a Kramers dou-blet of spin fluxons j "i, j #i with energy E"# and a doublet ofcharge fluxons jþi, j�i with energies Eþ, E�.

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Figure 3 shows results for s as a function of tempera-ture for the Kane-Mele model with two � fluxes located atthe largest possible distance. At temperatures kBT � �s,s is dominated by bulk effects. For kBT & 0:1t, we ob-serve the expected Curie law. The latter is robust withrespect to Rashba coupling, which is crucial for possibleexperimental realizations.

B. Probing correlated topological insulators

Figure 3 establishes the existence and thermodynamicsignature of degenerate spin and charge fluxons in a quan-tum spin Hall insulator threaded by a pair of � fluxes. Wenow consider the effect of electron-electron interactions inthe framework of the Kane-Mele-Hubbard model (3). For� > 0, the phase diagram of the latter includes a correlatedquantum spin Hall insulating phase that is adiabaticallyconnected to that of the Kane-Mele model (i.e., U ¼ 0),and a Mott insulating phase with long-range antiferromag-netic order [42,48]. Figure 4(a) shows the quantum phasetransition between these two phases as a function of U=t at�=t ¼ 0:2. At the transition, the spin gap �s—as obtainedfrom finite-size scaling (see Ref. [48] for details)—closes,corresponding to the condensation of magnetic excitons[47,48]. The magnetic order is of the easy-plane type, andthe transition has 3D XY universality corresponding tothe ordering of local moments [47,48]. For U � Uc,

time-reversal symmetry is spontaneously broken, and thesingle-particle gap �sp remains open across the transition

[48] [see Fig. 4(a)].The location of the critical point can be estimated from

the scaling behavior of the real-space spin-spin correlationfunction

SxxðrÞ ¼ hSxAðrÞSxAð0Þi (5)at the largest distance r ¼ L=2. Here, we consider corre-lations between spins on the A sublattice, but results are thesame for the B sublattice. The limit limL!1SxxðL=2Þ isidentical to m2, with m being the magnetization per site.This critical value can be obtained by considering the 3DXY scaling behavior at the transition. Following Ref. [48],

we plot L2�=SxxðL=2Þ as a function of U for differentsystem sizes by using the critical exponents z ¼ 1, ¼0:6717ð1Þ, and � ¼ 0:3486ð1Þ [52]. Figure 4(b) reveals theexpected intersection of curves at the critical point andgives Uc=t ¼ 5:70ð3Þ.The well-understood magnetic transition of the model

(3) provides a test case for the use of � fluxes to probea correlated quantum spin Hall state, as well as to trackthe interaction-driven transition to a topologically trivialstate. We solve the interacting model with two � fluxes byusing exact quantum Monte Carlo simulations. Spin flux-ons can be detected by calculating the lattice-site-resolved,dynamical spin-structure factor at T ¼ 0, defined as

Sði; !Þ ¼ �Xn

jhnjĉyi �zĉij0ij2�ðEn � E0 �!Þ: (6)

Here, HKMHjni ¼ Enjni, and j0i denotes the ground state.Sði; !Þ corresponds to the spectrum of spin excitations atlattice site i. A real-space map of the spin-fluxon states j "i,j #i is obtained by integrating Sði; !Þ up to an energy scale�=t ¼ 0:2, well within the charge gap �c � 2�sp, givingS�ðiÞ ¼

R�0 d!Sði; !Þ. For U=t ¼ 4, corresponding to the

quantum spin Hall phase [see Fig. 4(a)], we see in Fig. 5(a)very sharply defined spin fluxons localized at the two flux-threaded hexagons. The value of S�ðiÞ is about 3 orders ofmagnitude smaller at lattice sites that are further awayfrom a flux so that the spin fluxons can easily be detectednumerically. In Fig. 5(b), we show results for the magneticinsulating phase at U=t ¼ 6. As expected for this topologi-cally trivial state, no well-defined spin fluxons exist.The dependence of S�ðiÞ on U=t across the magnetic

quantum phase transition is shown in Fig. 5(c). A clearsignal is found deep in the topological-insulator phase,whereas a strong drop is observed on approaching thecritical point at Uc=t ¼ 5:70ð3Þ. Hence, the spin-fluxonsignal can be used in quantum Monte Carlo simulationsto distinguish topological and nontopological phases.As for the noninteracting case (Fig. 3), the spin fluxons

created by the fluxes give rise to a characteristic Curie lawin the spin susceptibility. Figure 6(a) shows quantumMonte Carlo results for the spin and charge susceptibilities

0.0

0.5

1.0

1.5

2.0

3 4 5 6 7 8

(a)

Topologicalinsulator

AntiferromagneticMott insulator

0.0

0.2

0.4

0.6

0.8

1.0

1.2

5.3 5.4 5.5 5.6 5.7 5.8 5.9 6

(b)

FIG. 4. (a) Spin gap �sðq ¼ 0Þ and single-particle gap�spðq ¼ KÞ in the thermodynamic limit as a function of theHubbard repulsion U, at T ¼ 0 (�=t ¼ 0:2, � ¼ 0). (b) Scalingof SxxðL=2Þ using the critical exponents of the 3D XY model,z ¼ 1, ¼ 0:6717ð1Þ, and � ¼ 0:3486ð1Þ [52]. The intersectiongives the critical point Uc=t ¼ 5:70ð3Þ. The lattice size is L� L.Error bars are smaller than the symbols.

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defined in Eq. (4) in the topological phase (U=t ¼ 4).We again consider two fluxes at the maximal separation.At low temperatures, kBT � �s, s is well described bys ¼ 2kBT , or 1kBT per � flux. The additional factor of 2compared to the case U ¼ 0 comes from the splitting ofspin and charge states, which only leaves the Kramers

doublet fj "i; j #ig at low energies (see Appendix A). TheCurie law holds down to the lowest temperatures consid-ered in Fig. 6(a). Finally, the charge susceptibility isstrongly suppressed at low temperatures and reveals theabsence of low-energy charge fluxons as a result of theHubbard repulsion.

C. Interaction between spin fluxons

So far, we have exploited the thermodynamic andspectral signatures of independent spin-fluxon excitations(i.e., free spins). On periodic lattices, spin fluxons can onlybe created in pairs, and it is therefore interesting to con-sider their mutual interactions. Such interactions will playa key role in Sec. V, where we consider quantum spin Hallinsulators with multiple � fluxes to create and simulatesystems of interacting spins.Interaction effects due to a coupling between two spin

fluxons in a lattice with one pair of� fluxes become visiblefor larger U=t, i.e., closer to the magnetic transition.Figures 6(b) and 6(c) show a deviation from the Curielaw below a temperature scale determined by the interac-tion between spin fluxons. In the model (3), this interactionis mediated by the exchange of collective spin excitations(magnetic excitons), which are the lowest-lying excitationsof the correlated topological insulator, and they evolve intothe gapless Goldstone mode of the magnetic state. Sincemagnetic order is of the easy-plane type, the dominantcontribution of the resulting interaction is expected tohave the general form

Sint ¼ �g2Xr�r0

ZZ �0d�d�0½Sþr ð�ÞDðr� r0; �� �0Þ

� S�r0 ð�0Þ þ H:c:�; (7)where S�r ð�Þ are spin-flip operators acting on a spin fluxonat position r at time �, Dðr� r0; �� �0Þ is the Fourier

(a) Topological insulator

(b) Antiferromagnet

0.00

0.01

0.02

0.03

0.04

0.05

3 4 5 6 7 8

(c)Topologicalinsulator

AntiferromagneticMott insulator

FIG. 5. Integrated dynamical spin-structure factor S�ðiÞ atT ¼ 0 on a 9� 9 lattice. (a) Localized spin fluxons created inthe topological-insulator phase at U=t ¼ 4. (b) Absence ofspin fluxons in the magnetic phase at U=t ¼ 6. (c) Maximumof S�ðiÞ, as a function of U=t. Here �=t ¼ 0:2, � ¼ 0, and�=t ¼ 0:2.

FIG. 6. (a) Spin (s) and charge (c) susceptibilities of the Kane-Mele-Hubbard model (�=t ¼ 0:2, � ¼ 0) at U=t ¼ 4. We considerL� L lattices with one pair of � fluxes placed at the maximal distance. At low temperatures, the spin susceptibility reveals a Curie laws ¼ 2kBT , whereas the charge susceptibility is suppressed by the charge gap. (b), (c) Spin susceptibility as a function of temperature fordifferent values of U=t (�=t ¼ 0:2, � ¼ 0). (a)–(c) show that with increasing U=t, the range of the interaction between spin fluxonsincreases, leading to deviations from the Curie law s ¼ 2kBT at low temperatures. (d) For U >Uc ¼ 5:70ð3Þt, s reflects the presenceof long-range magnetic order in the bulk. Error bars are smaller than the symbol size. The arrows indicate the energy scale associatedwith the spin gap.

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transform of the exciton propagator Dðq; i�mÞ(q: momen-tum; �m ¼ 2n�=�: bosonic Matsubara frequency), and gis a coupling constant. At long wavelengths, the dispersionrelation of the collective spin mode can be written as

!ðqÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2jq�Qj2 þ�2sp , where v is the spin velocity,�s is the spin gap, and Q is the magnetic-orderingwave vector. The minimal exciton energy is given by!ðq ¼ QÞ ¼ �s. Fourier transformation of the propagator(see Appendix C) gives, in the limit of low energies andlong wavelengths,

Dðr; �Þ expðiQ � rÞ expð��s�Þ exp��jrj

2�s2v2�

�: (8)

The first term determines the sign of the interaction. Thedecay at large imaginary time � is governed by the spin gap�s. The fast decay as a function of jrj underlies the clearCurie law seen, e.g., in Fig. 6. The interaction rangeand strength can be tuned via the spin gap and hence[cf. Fig. 4(a)] by varying U=t.

From Eq. (8), we expect the interaction range to in-crease with increasing U=t due to the decrease of �s[see Fig. 4(a)]. Indeed, Figs. 6(b) and 6(c) reveal anenhanced effect of the spin-fluxon separation at low tem-peratures with increasing U=t. In particular, for U=t ¼ 5:5(close to the magnetic transition), Fig. 6(c) shows a Curielaw corresponding to two free spin fluxons, only for thelargest system sizes (L ¼ 18). As U ! Uc, the interactionrange diverges, and free spin fluxons can no longer exist.For U >Uc, time-reversal invariance is broken, and �fluxes do not create spin fluxons. Instead, the spin suscep-tibility in Fig. 6(d) is that of an antiferromagnet; the finitevalue of s=L

2 at T ¼ 0 reflects the density of spin-waveexcitations.

To illustrate the dependence of the interaction strengthon �s, we consider two fluxes at a fixed, small distance, asillustrated in the inset of Fig. 7. We show the spin suscep-tibility for different values of U=t in Fig. 7. For U=t ¼ 3, aCurie law s � 2kBT may be inferred at temperatures kBT �0:1t. Increasing U=t, the interaction between the spinfluxons becomes too large to observe free spin fluxonsbelow the temperature range set by the bulk spin gap �s.The downturn of s occurs at higher and higher tempera-tures with increasing U=t, and it reflects a tunable,correlation-induced energy scale for the interactionbetween spin fluxons that is absent in Fig. 3.

V. �-FLUX QUANTUM SPIN MODELS

The possibility of inserting � fluxes to create localizedspin fluxons with a tunable interaction mediated by mag-netic excitons provides a toolbox to engineer interactingspin models in correlated topological insulators. The com-putational effort for quantum Monte Carlo simulationsdoes not depend on the number of � fluxes, and the lattercan be arranged in almost arbitrary geometries on thehoneycomb lattice.

A. Three-spin system

As a first extension of the two-spin cases considered sofar, we consider four spin fluxons emerging from two pairsof � fluxes. The fluxes are arranged so that three spinfluxons form a ring, and the fourth spin fluxon is locatedat the largest distance from the center of the ring. For largeenough lattices, the separated spin fluxon will not couple tothe other three, and the physical problem is similar toexperiments on coupled quantum dots [53] or flux qubits[54] in the context of quantum computation. The three spinfluxons experience a transverse interaction of the form (7)

1

10

100

1000

0.01 0.1 1

FIG. 7. Spin susceptibility as a function of temperature for two� fluxes arranged as shown in the inset (�=t ¼ 0:2, � ¼ 0, 9� 9lattice). With increasing U=t, the strength of the interactionbetween spin fluxons increases, as revealed by the shift of thetemperature below which deviations from a 2kBT

Curie law occur.

Statistical errors are smaller than the symbol size. Inset: S�ðiÞfor U=t ¼ 4, using the same color coding as in Fig. 5(a).

1

10

100

1000

0.01 0.1 1

FIG. 8. Spin susceptibility as a function of temperature for four� fluxes arranged as shown in the inset (�=t ¼ 0:2, � ¼ 0,U=t ¼ 4) on L� L lattices. The data reveal a Curie law 4kBT atintermediate temperatures and 2kBT

at low temperatures.

Statistical errors are smaller than the symbol size. Inset: S�ðiÞfor L ¼ 15, using the same color coding as in Fig. 5(a).

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and behave as an effective spin with Sz ¼ �1=2 atlow temperatures (see Appendix B). The spin susceptibilityfor U=t ¼ 4 shown in Fig. 8 reveals that, at low tempera-tures, the two independent spins indeed give rise to theexpected Curie law s ¼ 2kBT . At higher temperatureskBT � 0:1t, we find s ¼ 4kBT , corresponding to four in-dependent spin fluxons. In the regime where s ¼ 2kBT , thesign of the interaction between the spin fluxons determinesthe ground-state degeneracy of the three-spin cluster. A netferromagnetic interaction results in a spin-1=2 doublet,whereas an antiferromagnetic coupling gives rise to a four-fold degenerate, chiral ground state (see Appendix B). Inprinciple, the sign of the exchange coupling can be deter-mined from entropy measurements. Since Q ¼ 0 for themodel (3), Eq. (8) suggests that the interaction isferromagnetic.

B. Simulation of one-dimensional fluxon chains

Whereas the study of systems with a small number ofspins is relevant for applications such as quantum comput-ing, many questions in condensed matter physics are re-lated to periodic spin lattices. In this section, we thereforeconsider one-dimensional chains of � fluxes in the honey-comb lattice with periodic boundary conditions.

We begin with the noninteracting Kane-Mele modelwith a periodic flux chain. The fluxon excitations arevisible in Fig. 9, which shows the integrated local densityof states, A�ðiÞ ¼

R�0 d!Aði; !Þ; the single-particle spec-

tral function Aði; !Þ is defined as usual in terms of thesingle-particle Green function, Aði; !Þ ¼ �ImGði; !Þ.Whereas the fluxons are well localized in the directionnormal to the chain, the overlap of neighboring fluxons inthe chain gives rise to a tight-binding band inside thetopological band gap, which can be seen in the spectrumshown in Fig. 10. The specific form of the band structurecan be attributed to the fact that the smallest unit cell forthe fluxon chain contains two flux-threaded hexagons

(and is four hexagons wide) (see Fig. 9). Exploitingthe fact that the four possible fluxon states per hexagon,fj "i; j #i; jþi; j�ig, can formally be written in terms of thefermion Fock states fj "i; j #i; j0i; j "#ig, and assumingnearest-neighbor hopping, a suitable Hamiltonian isgiven by

H ¼ �~tXi�

ð�yi�c i� � c yi��iþa�� þ H:c:Þ; (9)

where �, c refer to the two flux-threaded hexagonsin the unit cell, and i numbers the unit cells. The resultingband dispersion ðkÞ ¼ �2~t sinð2kaÞ matches the low-energy bands in the spectrum (Fig. 10). The form of theeffective low-energy Hamiltonian, and especially thegapless nature of the spectrum, stems from the fact thatthe unit cell is a gauge choice; a translation by halfa lattice vector, a�=2, corresponds to a gauge transfor-mation. This symmetry allows the intercell and intracellhopping integrals to differ only by a phase ei�. Imposingtime-reversal symmetry pins the phase factor to � ¼ 0and � ¼ �, thus leading to the dispersion relations�2~t cos½ðkþ �=a�Þa�=2�. The choice � ¼ � producesthe above-mentioned dispersion relation, and the choice� ¼ 0 merely corresponds to translating the reciprocallattice by half a reciprocal lattice unit vector.In contrast to the helical edge states of a quantum spin

Hall insulator, each of the two fluxon bands is spin degen-erate. As a result, and because the system is half filled, weexpect a Mott transition of charge fluxons for any nonzeroelectron-electron repulsion. Figure 11 shows the spin andcharge susceptibilities of the Kane-Mele-Hubbard modelon L� 12 lattices with L=2 � fluxes and U=t ¼ 4. TheHubbard U causes an exponential suppression of thecharge susceptibility at low temperatures [see Fig. 11(b)and inset], whereas low-energy spin-fluxon excitations

FIG. 9. Integrated local density of states A�ðiÞ (� ¼ 0:2t; seetext) at T ¼ 0 for the Kane-Mele model (�=t ¼ 0:2, � ¼ 0) witha periodic chain of � fluxes. We show a part of the 72� 12lattice used and the size of the magnetic unit cell containing two� fluxes. The latter has width a� ¼ 4a, where a 1 corre-sponds to the norm of the lattice vectors of the underlyinghoneycomb lattice.

FIG. 10. Spectrum of eigenvalues of the Kane-Mele modelwith a periodic chain of � fluxes (cf. Fig. 9). Here, �=t ¼ 0:2,� ¼ 0, and the honeycomb lattice has dimensions 72� 12.Points correspond to eigenvalues and lines to the band structure

ðkÞ ¼ �2~t sinð2kaÞ, with ~t � 0:126t and a 1.

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remain [Fig. 11(a)]. Hence, similar to the one-dimensionalHubbard model, the fluxon chain undergoes a Motttransition to a state with a nonzero charge gap but gaplessspin excitations.

In the Mott phase of the fluxon chain, the low-energyphysics is expected to be described by spin fluctuationsand hence by an effective spin model with spins corre-sponding to Kramers doublets of localized spin fluxons.Because the interaction range depends exponentially onthe spin gap, we expect nearest-neighbor interactionsJxy, Jzz between spin fluxons to dominate, except for theclose vicinity of the magnetic transition. As argued be-fore, the magnetic exciton is of predominantly easy-planetype, and we therefore expect anisotropic interactions,jJxyj � jJzzj. The minimal model for the spin chain isthe one-dimensional XXZ Hamiltonian,

H ¼ JzzXi

Szi Sziþ1 þ Jxy

Xi

ðSþi S�iþ1 þ S�i Sþiþ1Þ: (10)

Using the ALPS 1.3 implementation [55], we can simulatethis model in the stochastic-series-expansion representa-tion to calculate the spin susceptibility as a function oftemperature. There is one free parameter, Jxy=Jzz, which

is varied to obtain the best fit to the low-temperaturesusceptibility (at high temperatures, bulk states of thetopological insulator begin to play a role) of the Kane-Mele-Hubbard model. For example, considering sixspins, a rather good match between the spin-fluxon dataand the XXZ model is obtained for Jzz=jJxyj ¼ �0:1(the sign of Jxy is irrelevant) (see Fig. 12). Importantly,taking the same parameters, and simulating ten spins withboth spin fluxons and the XXZ model, equally goodagreement is found in Fig. 12. These results demonstratethat the spin fluxons form a one-dimensional spin systemwith well-defined interactions and that a quantum spinHall insulator with � fluxes can indeed be used as aquantum simulator.

VI. CONCLUSIONS

In this work, we have presented quantum Monte Carloresults for a correlated quantum spin Hall insulator withtopological defects in the form of � fluxes. Such fluxesrepresent a universal probe for the topological index thatcan be used in the presence of electronic correlations anddoes not rely on spin conservation or an adiabatic connec-tion to a noninteracting topological insulator. Our resultsdemonstrate that � fluxes can be combined with exactnumerical simulations and lead to clear signatures ofnontrivial topological properties in spectral and thermody-namic properties. As a concrete example, we have studiedthe magnetic quantum phase transition of the Kane-Mele-Hubbard model at which time-reversal symmetry is spon-taneously broken. In principle, � fluxes can also be used inconnection with fractional quantum spin Hall states.More generally, � fluxes in correlated topological

insulators allow one to construct and simulate quantumspin models and hence lead to a novel class of quantum

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

(a)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

(b)

0 5 10 15 20 25

(no fluxes)

FIG. 11. (a) Spin and (b) charge susceptibility of theKane-Mele-Hubbard model (�=t ¼ 0:2, � ¼ 0) at U=t ¼ 4.We consider L� 12 lattices with L=2 � fluxes arranged in aperiodic one-dimensional chain. The inset in (b) shows thecharge susceptibility as a function of inverse temperature on alogarithmic scale. The key in (a) applies to all panels.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06 0.08 0.1

XXZ model-flux model-flux model

FIG. 12. Spin susceptibility as a function of temperature.Symbols correspond to quantum Monte Carlo results for theKane-Mele-Hubbard model (�=t ¼ 0:2, � ¼ 0, U=t ¼ 4) onL� 12 lattices with L=2 � fluxes arranged in a periodic one-dimensional chain. Lines are quantum Monte Carlo results forthe one-dimensional XXZ model with Jzz=jJxyj ¼ �0:1 and L=2lattice sites (spins).

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simulators. This finding is not restricted to the Kane-Mele-Hubbard model considered here. In particular, magnetismdriven by electronic correlations—the origin of theinteraction between spin fluxons—is a common phenome-non. The physics described here relies on the coexistenceof magnetic correlations and time-reversal symmetry andcannot be captured by static mean-field descriptions. Thespin models share the topological protection of their hostagainst, for example, disorder. In general, they are charac-terized by a dynamical, time-dependent interaction remi-niscent of spin-boson problems. The detailed form and signof the interaction, whose strength and range can be tunedvia the electronic interactions, depend on the electronicHamiltonian and the lattice geometry of the underlyingtopological insulator. Because of the spin-orbit interaction,the spin symmetry is Uð1Þ, and—similar to cold-atomrealizations of the quantum Ising model [56]—the spin-spin interaction is generically anisotropic. We have pro-vided explicit evidence for the feasibility of our idea interms of simulations of two and four spins, as well as ofone-dimensional spin chains. Additional Rashba termslead to spin models with a discrete Z2 Ising symmetry.Although spin-fluxon states are still well defined [12,31], itis a priori not clear which operators have to be measured inthe numerical simulations. Finally, the concept of fluxonsoriginating from � fluxes carries over to three-dimensionaltopological insulators [12,32].

An open question of central importance is whether theuse of � fluxes will enable us to study quantum spinsystems that are currently not accessible to numericalmethods, for example, due to a sign problem in the pres-ence of frustrated interactions. Whereas we have providedevidence for the possibility to simulate arrays and chains ofquantum spins, and to tune the interaction strength andrange, entropy measurements are required to determine thesign of the interactions. However, the latter are extremelydemanding to carry out using discrete-time quantumMonte Carlo methods. A systematic effort to study spinfluxon chain and ladder geometries is currently in progress.

Our idea may potentially also be used in experiments.A strongly correlated topological insulator on the honey-comb lattice may be realized with Na2IrO3 [34] or withmolecular graphene [57]. It has been suggested that �fluxes can be created in a quantum spin Hall insulator bymeans of an adjacent superconductor and a magnetic field[31]. This idea can be generalized to arrays of � fluxesusing Abrikosov lattices. Alternatively, � fluxes may berealized using SQUIDs. A potential problem is that thediameter of the � fluxes will not be of the order of thelattice constant. Other exciting recent proposals that arerelevant for the realization of our idea include artificialsemiconductor honeycomb structures [35], cold atoms inoptical lattices [19], and cold atoms on chips [58]. In solid-state setups, � fluxes can also be created by dislocations[29,30] or wedge disclinations [59].

ACKNOWLEDGMENTS

We thank N. Cooper, T. L. Hughes, L. Molenkamp,J. Moore, J. Oostinga, X.-L. Qi, C. Xu, and S. C. Zhangfor insightful conversations, and acknowledge supportfrom DFG Grants No. FOR1162 and No. As 120/4-3, aswell as generous computer time at the LRZMunich and theJSC. We made use of ALPS 1.3 [55].

APPENDIX A: SPIN SUSCEPTIBILITYFOR TWO � FLUXES

A single � flux in a topological insulator gives rise tofour states, j "i, j #i, jþi, j�i. In the absence of correla-tions, these states are degenerate. At low temperatures, thespin susceptibility, defined in Eq. (4), can be calculatedusing the Hilbert space formed by only these states.Defining an effective Hamiltonian H� ¼

PcE

c jc ihc jwith c 2 fþ;�; "; #g and Eþ ¼ E� ¼ E" ¼ E# ¼ E"#, weobtain

s¼�ðhM̂2zi�hM̂zi2Þ¼ 1kBTP

c hc jM̂2ze��H� jc iPc hc je��H� jc i

¼ 12kBT

:

(A1)

For U � kBT, the spin fluxons j "i, j #i are the only low-energy excitations, and s can be calculated by restrictingc to f"; #g. Since E" ¼ E# ¼ E"# due to time-reversal sym-metry, we get

s ¼ 1kBT : (A2)

For the case of two independent � fluxes, the aboveresults imply s ¼ 1kBT at U ¼ 0 and s ¼ 2kBT for U > 0.These results agree with the numerical results shown inFig. 3 for U ¼ 0 and in Figs. 6 and 7 for U > 0.Our derivation is only valid in the absence of the Rashba

spin-orbit coupling �. However, the numerical results inFig. 3 show that the low-temperature Curie law in s is thesame also for � � 0.

APPENDIX B: SPIN SUSCEPTIBILITYAND GROUND-STATE DEGENERACY

FOR FOUR � FLUXES

The results for the Kane-Mele-Hubbard model with four� fluxes shown in Fig. 8 reveal a 2kBT Curie law at low

temperatures and a 4kBTCurie law at higher temperatures.

This finding can be understood as corresponding to eithertwo or four noninteracting spins. The latter case corre-sponds to the spatially separate spin fluxon and an effectivespin-1=2Kramers doublet (formed by the three nearby spinfluxons) in the regime where s � 2kBT and to four non-interacting spin fluxons in the regime where s � 4kBT .The cluster formed by the three nearby spin fluxons has

the possible configurations

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jMzj ¼ 32: fj """i; j ###ig;

jMzj ¼ 12: fj "##i; j #"#i; j ##"i; j #""i; j "#"i; j ""#ig;

(B1)

whereMz denotes the total spin in the z direction. Since theexciton-mediated interaction in the Kane-Mele-Hubbardmodel has the form given in Eq. (7), and hence promotesspin flips, the ground state can be expected to have jMzj ¼1=2. The above-mentioned effective spin-1=2 doublet thencorresponds to the two possible values Mz ¼ �1=2.

The degeneracy of the ground state depends on the sign ofthe interaction. ConsideringMz¼1=2, we have the allowedstates j #""i, j "#"i, and j""#i. The spin-flip terms that con-nect these states are of the form JðSþiþ1S�i þSþi S�iþ1Þ, withperiodic boundary conditions. An equivalent representa-tion is given by the Hamiltonian

H ¼ JXj

ðjjþ 1ihjj þ jjihjþ 1jÞ; (B2)

which describes the hopping of a particle (the spin-down)on a three-site ring, with j1i ¼ j #""i, etc. The eigenstatesare obtained by Fourier transformation and have the form

jki ¼ 1ffiffiffi3

p X3j¼1

eikjjji; k ¼ 0;� 2�3

: (B3)

The eigenvalues are given by

EðkÞ ¼ 2J cosk: (B4)For J < 0, the ground state has k ¼ 0 and energy equal to2J. For J > 0, the ground state is chiral, with k ¼ �2�=3and energy equal to �J. Taking into account the sectorMz ¼ �1=2, we find a total ground-state degeneracy oftwo in the ferromagnetic case (J < 0) and four in theantiferromagnetic case (J > 0).

APPENDIX C: FOURIER TRANSFORM OF THEEXCITON PROPAGATOR

The exciton propagator in Eq. (7) takes the form

Dðq; �Þ ¼ e�!ðqÞ

e�!ðqÞ � 1�e��!ðqÞ

e��!ðqÞ � 1 ; (C1)

with the exciton energy

!ðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2jq�Qj2 þ �2s

q: (C2)

In the low-temperature limit � ! 1, the propagator be-comes Dðq; �Þ ¼ e��!ðqÞ. Setting q0 ¼ q�Q, the Fouriertransform is given by

Dðr; �Þ ¼ eiQ�rZ

d2q0eiq0�re�!ðq0þQÞ�: (C3)

Assuming �s � v, we can expand to obtain

!ðq0 þQÞ � �s�1þ v

2jq0j22�2s

�: (C4)

Taking the continuum limit, the Fourier transform involvesthe product of two Gaussian integrals, and the result isgiven by Eq. (8).

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