PHYSICAL REVIEW B 89, 165123 (2014)

Interaction-induced anomalous quantum Hall state on the honeycomb lattice

Tanja Đuric,1 Nicholas Chancellor,1 and Igor F. Herbut2,3

1London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London WC1H 0AH, UK2Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

3Max-Planck-Institut fur Physik Komplexer Systeme, Nothnitzer Str. 38, 01187 Dresden, Germany(Received 24 January 2014; published 21 April 2014)

We examine the existence of the interaction-generated quantum anomalous Hall phase on the honeycomblattice. For the spinless model at half-filling, the existence of a quantum anomalous Hall phase (Chern insulatorphase) has been predicted using mean-field methods. However, recent exact diagonalization studies for smallclusters with periodic boundary conditions have not found a clear sign of an interaction-driven Chern insulatorphase. We use the exact diagonalization method to study properties of small clusters with open boundaryconditions, and contrary to previous studies, we find clear signatures of the topological phase transition forfinite-size clusters. We also examine the applicability of the entangled-plaquette-state (correlator-product-state)ansatz to describe the ground states of the system. Within this approach the lattice is covered with plaquettesand the ground-state wave function is written in terms of the plaquette coefficients. Configurational weightscan then be optimized using a variational Monte Carlo algorithm. Using the entangled-plaquette-state ansatz westudy the ground-state properties of the system for larger system sizes and show that the results agree with theexact diagonalization results for small clusters. This confirms the validity of the entangled-plaquette-state ansatzto describe the ground states of the system and provides further confirmation of the existence of the quantumanomalous Hall phase in the thermodynamic limit, as predicted by mean-field theory calculations.

DOI: 10.1103/PhysRevB.89.165123 PACS number(s): 02.70.Ss, 71.10.Fd, 71.27.+a, 73.43.−f

I. INTRODUCTION

In recent years, the interest in topological phases of matterand topological phase transitions [1–3] has been renewed bythe discovery of topological insulators and superconductors[4–9]. For noninteracting insulators possible topological stateshave been classified [10,11] and the transitions betweendifferent topological states have been found to be characterizedby closing of the single-particle gap [1,4,5]. Much of therecent research has focused on understanding the role of in-teractions in topological insulators [12]. Of particular interestare interaction-generated topological insulators or topologicalMott insulators [13–16]. Such states appear to be due to thepresence of electronic interactions that give rise to nonlocalcomplex bond order parameters and spontaneous breaking oftime-reversal (TR) symmetry [13–15,17]. The possibility ofrealizing novel interaction-generated topologically nontrivialphases, without the requirement of strong intrinsic spin-orbitcoupling, could significantly extend the class of topologicallynontrivial materials and is thus of great importance.

Interacting spinless fermions on a honeycomb lattice areperhaps the best studied example in which an interaction-generated topologically nontrivial state, a quantum anomalousHall (QAH) state, is predicted to appear by several mean-fieldtheory calculations [13–15]. However, recent exact diagonal-ization (ED) studies of small clusters with periodic boundaryconditions [18,19] find no clear sign of a such topologicallynontrivial state. In this paper we present ED results forsmall clusters with open boundary conditions. Contrary toprevious ED studies [18,19], we find clear signatures of theinteraction-driven topological transition in the fidelity metric[20–29] and show that the transition is characterized by closingof the excitation gap. We also demonstrate the existence of theedge states in the QAH phase by calculating the density profileof a hole created out of the ground state at half-filling [25,30].

To study properties of the system for larger system sizeswe use the entangled-plaquette-state (EPS) ansatz, also calledthe correlator-product-state (CPS) ansatz [31–39], for theground-state wave function of the system and variationalMonte Carlo (VMC) [39–43]. In the EPS approach the lattice iscovered with plaquettes and the ground-state wave function iswritten in terms of the plaquette coefficients. Configurationalweights are then optimized using a VMC algorithm. Veryaccurate results have been obtained for several frustrated andunfrustrated models within the EPS approach. The approachovercomes some of the limitations of the two main numericaltechniques, density matrix renormalization group [44] andquantum Monte Carlo [45], that have been successfullyapplied to study various quantum many-body systems. Whilethe density matrix renormalization-group method gives veryaccurate results only in one dimension [46] and the quantumMonte Carlo suffers from the sign problem for Fermi systems,the EPS and VMC approach can be applied to systems of anyspatial dimensionality and is sign problem free.

The values of the plaquette coefficients that minimize theenergy can be found using the stochastic minimization method[39,41,42]. Due to the presence of the statistical error in thestochastic algorithm it is difficult to obtain good estimates ofthe ground-state energies for small system sizes. Having alarger number of parameters allows the optimization methodmore freedom in finding the minimum energy state and thestatistical error can be controlled by increasing the sample size.Therefore, instead of comparing the EPS ansatz results withthe ED results for small clusters, we show that the EPS resultsfor larger system sizes predict the same phases of the systemthat the ED results predict for small clusters. In particular,we calculate within the EPS approach the density profiles andthe density-functional fidelity [47] for a range of parametervalues. This then provides confirmation of the validity of the

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ĐURIC, CHANCELLOR, AND HERBUT PHYSICAL REVIEW B 89, 165123 (2014)

EPS ansatz for describing the ground states of the system andfurther confirmation of the existence of the QAH state in thethermodynamic limit, as predicted by the mean-field theorycalculations.

The paper is organized as follows. In Sec. II we introducethe extended Hubbard model for spinless fermions on ahoneycomb lattice. In Sec. III we review the mean-field phasediagram for the half-filled case. In Sec. IV we present resultsfor small clusters with open boundary conditions. The resultsfor larger system sizes obtained within the EPS and VMCapproach are presented in Sec. V. In the final section, Sec. VI,we draw our conclusions and discuss possible directions forfuture research.

II. MODEL

We consider the system of interacting spinless fermionson a honeycomb lattice. The system can be described by anextended Hubbard model,

H = −t∑〈ij〉

(c†i cj + c†j ci) + V1

∑〈ij〉

ninj + V2

∑〈〈ij〉〉

ninj ,

(1)

where ci (c†i ) are the fermion creation (annihilation) operatorsat site i, ni = c

†i ci is the number operator at site i, V1

is a repulsive nearest-neighbor (NN) interaction, and V2

is a repulsive next-to-nearest-neighbor (NNN) interaction.Here 〈ij 〉 and 〈〈ij 〉〉 denote NN and NNN sites i and j ,respectively.

The honeycomb lattice is a bipartite lattice, consisting oftwo triangular sublattices further referred to as sublattices Aand B. In the noninteracting limit (V1 = V2 = 0) the systemis in the semimetal (SM) phase with two inequivalent Fermipoints. The NN interaction V1 is the interaction between NNfermions on two different sublattices and favors a charge-density-wave phase with an order parameter ρ = (〈c†iAciA〉 −〈c†iBciB〉)/2. However, the NNN interaction V2 between NNfermions on the same sublattice introduces frustration that cancause suppression of the charge-density-wave order and theappearance of the topologically nontrivial states in the phasediagram of the system.

For the system at half-filling the predicted topologicallynontrivial state is the QAH state [13–15], characterized byspontaneously broken TR symmetry and a nonlocal bond orderparameter χij = 〈c†i cj 〉. Properties of the SM-QAH criticalpoint have been investigated in several previous studies [48,49]and the SM-QAH quantum phase transition was also found atweak interaction and in the presence of strain [50,51].

The mean-field theory calculations also predict a charge-modulated (CMs) phase [15] (charge density wave withreduced rotational symmetry) for larger values of the NNNinteraction, V2 � 2.5t , and a Kekule phase characterized byan alternating bond strength [14,15]. We review the mean-fieldphase diagram in the following section.

III. MEAN-FIELD PHASE DIAGRAM

The mean-field phase diagram for the half-filled case,obtained by Grushin et al. [15], is shown in Fig. 1. For the

FIG. 1. (Color online) Mean-field phase diagram for the half-filling case obtained in Ref. [15]. The various phases are discussedin the text. SM, semimetal phase; QAH, quantum anomalous Hallphase; CMs, charge-modulated phase; CDW, charge-density-wavephase; K, Kekule phase.

NN interaction V1 = 0 and V2 � V(1)

2c the system is in theSM phase, and with increasing V2 beyond the critical valueV

(1)2c there is a continuous transition from the SM to the QAH

phase and, further, a first-order transition to the CMs phaseat V2 = V

(2)2c . The QAH phase is found for 1.5 � V2/t � 2.5.

In the SM phase, which is connected to the noninteracting(V1 = V2 = 0) limit of the Hamiltonian, (1), there are twoinequivalent Fermi points (Dirac points) where two energybands intersect linearly. In the vicinity of these Dirac pointsthe electrons behave as relativistic two-dimensional masslessDirac fermions.

The QAH phase is an example of a topological phasethat appears as a consequence of spontaneously broken TRsymmetry. The QAH state has chiral edge states and anonzero Chern number C = ±1. The state is characterizedby a complex bond order parameter χij = χ∗

ji = 〈c†i cj 〉 thatcorresponds to the complex hopping term of the Haldanemodel [1] and breaks TR symmetry.

The CMs phase is characterized by a charge modulationwithin the same sublattice. The phase is a charge densitywave with reduced rotational symmetry and it appears in theV2 � V1 regime, where it becomes energetically favorable toreduce the NNN energetic contribution ∝V2 by paying thecorresponding NN energy cost ∝V1. The CMs phase is a ratherunconventional phase. The phase is a Mott insulator since itemerges from a high NNN interaction strength V2/t . However,its band gap is determined by the hopping integral t , which isa property of band insulators.

For V1 > 0 a charge-density-wave phase and a Kekuleordered phase were also found. The charge-density-wavephase with broken inversion symmetry appears for V1 >

V2. The charge-density-wave order with charge imbalancebetween the two different sublattices reduces the amountof NN interaction energy, and with increasing V2 a Kekuleordered phase appears at V2 ∼ V1. The Kekule phase ischaracterized by an alternating bond strength, a Z3 orderparameter, and broken translational symmetry of the orig-inal honeycomb lattice that opens a gap in the energyspectrum.

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IV. EXACT DIAGONALIZATION RESULTS FOR SMALLCLUSTERS WITH OPEN BOUNDARY CONDITION

Using the ED method, we first examine signatures oftopological phase transitions for small clusters with openboundary conditions. The key signature of the topologicalphase transition is the existence of a topologically protectedlevel crossing that is robust and defines a topological phasetransition even in a finite-size system [20]. For the interaction-driven topological transition the level crossing corresponds tothe closing of the excitation gap. Accordingly, a topologicaltransition in an interacting system is marked by the closing ofthe excitation gap. The choice of the boundary condition thatcan be used to detect topological order in finite-size systemsdepends on spatial symmetries of the system.

The level crossing can also be observed in the fidelity metric[20–29]. If we denote by |ψ0(β)〉 and |ψ0(β + δβ)〉 two groundstates corresponding to slightly different values of the relevantparameter β, the fidelity between the two ground states isequivalent to the modulus of the overlap between the twostates:

F (β,β + δβ) = |〈ψ0(β + δβ)|ψ0(β)〉|. (2)

Equation (2) can be rewritten as

F = 1 − (δβ)2

2χF + . . . , (3)

where χF is the fidelity susceptibility,

χF (β) = limδβ→0

−2 ln F

δβ2= − ∂2F

∂(δβ)2. (4)

The overlap measures similarity between two states; it givesunity if two states are the same and0 if the states are orthogonal.At the point of level crossing between two orthogonal states thefidelity shows a very sharp drop that corresponds to a singularpeak in the fidelity susceptibility. The topological transitioncan then be characterized by a singular peak in the fidelitysusceptibility.

Since we are primarily interested in examining the stabilityof the QAH phase that is, according to the mean-field theorycalculations, stabilized by the NNN interaction V2, we focuson the case V1 = 0, for which the QAH phase is mostextended in (V1, V2) parameter space. In the ED calculationsthe finite-size effects can exceed the energy scale of themany-body energy gap of an incompressible ground state,so that the incompressibility cannot be recognized. For thetopologically nontrivial insulating states the choice of openboundary conditions, instead of periodic boundary conditions,changes the energy spectrum of the system by introducing edgestates within the bulk energy gap. In some cases this allowsidentification of the topological insulating states from the EDcalculations, even when such states cannot be identified fromthe ED calculations with periodic boundary conditions. Wetherefore study the properties of the system for several smallsystem sizes and with open boundary conditions.

We present the results for the cluster of 18 sites (two 3 × 3triangular sublattices) illustrated in Fig. 2, for which signaturesof the topological phase transitions predicted by the mean-fieldtheory calculations can be clearly seen. For smaller systemsizes and different aspect ratios (clusters with two 2 × 2, 2 × 3,

A(1,1)

A(2,1)

A(3,1)

A(1,2)

A(2,2)

A(3,2)

A(1,3)

A(2,3)

A(3,3)

B(1,3)B(1,1)

B(2,1)

B (3,1)

B(1,2)

B(2,2)

B(3,2) B(3,3)

B(2,3)

FIG. 2. (Color online) Illustration of the cluster of 18 sites stud-ied in Sec. IV. A and B labels correspond to two triangular sublatticesof the honeycomb lattice.

2 × 4, and 3 × 4 triangular sublattices), we do not observe thelevel crossings. This is consistent with the previous findings[20,25] that the topological transitions can be detected by theED calculations only if one studies clusters with a reciprocalspace that contains Dirac points.

The energies of the ground state and first two excited statesas functions of the NNN interaction strength V2 = V2/t areshown in Fig. 3. The level crossing at V2 = V

(1)2c marks the

second-order transition from the SM to the QAH state, wherethe value of the topological index (Chern number) changes

0 0.5 1 1.5 2 2.5 3−12

−10

−8

−6

−4

−2

0

2

4

V2

E

1.6 1.65

−1.35

−1.3

−1.25

2.87 2.88 2.892.55

2.6

2.65

FIG. 3. (Color online) Ground-state and first two excited-stateenergies as functions of the NNN interaction strength V2 = V2/t .Insets: Closeup views of the level crossings at V2 = V

(1)2c and

V2 = V(2)

2c . The level crossing at V2 = V(1)

2c marks the second-ordertransition from the SM to the QAH state, where the value of thetopological index (Chern number) changes from 0 to 1. The levelcrossing at V2 = V

(2)2c corresponds to the first-order phase transition

from the QAH to the CMs phase accompanied by a change in thevalue of the Chern number from 1 to 0.

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0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V2

Δ 1.6 1.62 1.640

0.005

0.01

0.015

2.87 2.88 2.890

2

4x 10

FIG. 4. (Color online) Excitation gaps for the first three excitedstates as functions of the NNN interaction strength V2 = V2/t .Closing of the excitation gap at V2 = V

(1)2c marks the transition from

the SM to the QAH state; at V2 = V(2)

2c , the transition from the QAHto the CMs state.

from 0 to 1. The level crossing at V2 = V(2)

2c corresponds to thefirst-order phase transition from the QAH to the CMs phaseaccompanied by a change in the value of the Chern numberfrom 1 to 0. The level crossings can also be clearly seen inFig. 4, which shows the excitation gaps �(n) = En − E0 forthe first three excited states (n = 1, 2, and 3). In addition,the topological transitions are characterized by singular pointsin the fidelity susceptibility shown in Fig. 5. The fidelity andfidelity susceptibility are defined by Eqs. (2)–(4), where herethe relevant parameter β = V2.

We further examine the existence of the edge states in theV

(1)2c < V2 < V

(2)2c regime by considering the density profile of

a hole created out of the ground state [25,30]. The energy ofsingle-particle (single-hole) excitation is defined as �Ep =

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

V2

χ f

FIG. 5. (Color online) Fidelity susceptibility χF (V2,δV2) withδV2 = 0.003 as a function of the interaction strength V2 = V2/t forthe 18-site cluster shown in Fig. 2.

a b

c d

FIG. 6. (Color online) Density profile of a hole created out of theground state (�nj ) at half-filling for the 18-site cluster shown in Fig. 2and for the interaction strength (a) V2 = 0, (b) V2 = 1.5, (c) V2 = 2,and (d) V2 = 3. Here the radius of each filled circle is proportionalto the magnitude of |�nj | and open circles denote lattice sites where�nj < 0.

EN+10 − EN

0 (�Eh = EN0 − EN−1

0 ), where EN0 is the ground-

state energy of the system with N fermions. The density profileof a hole created out of the ground state is

�nj = ⟨ψN

0

∣∣nj

∣∣ψN0

⟩ − ⟨ψN−1

0

∣∣nj

∣∣ψN−10

⟩, (5)

where |ψN0 〉 is the ground-state wave function with N

fermions. In the presence of the in-gap edge states the densityprofile should be localized on the edges. The density profilefor several values of the NNN interaction strength V2/t isshown in Fig. 6. The results confirm that for the value ofV2/t = 2, when the system is in the QAH regime, the densityprofile is indeed localized on the edges, while for the values ofV2/t = 0, 1.5, and 3, corresponding to SM and CMs regimes,the density profile is delocalized. Note that for V2/t = 2 thedensity profile �nj < 0 at the sites of the central plaquetteand �nj > 0 at the sites of the plaquettes at the edges of thecluster.

We also note that for a finite cluster in the QAH regime theground state obtained from ED will be a superposition of twospontaneously generated QAH states with opposite chirality(Chern number C = ±1). More precisely, the QAH state thatwe find is a linear combination of the two Haldane states withC = +1 and C = −1, which is odd under TR. This can beseen by considering the TR symmetry of the system.

For spinless fermions the (antiunitary) TR symmetryoperator T = σ1K , where σ1 denotes the Pauli matrix andK complex conjugation. The TR symmetry operator T is anexact symmetry operator of Hamiltonian (1), and therefore itcommutes with the Hamiltonian. In the basis of the two statesthat are related by TR symmetry, |ψ+〉 and |ψ−〉 = T |ψ+〉,the Hamiltonian then has to be proportional to the unit matrixand to σ1. However, although the system has TR symmetry theantiunitary TR symmetry operator squares to 1 (T 2 = +1) and,as such, does not imply Kramers’ degeneracy. Accordingly,there is no twofold degeneracy of the QAH phase in thefinite-size system we are studying.

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INTERACTION-INDUCED ANOMALOUS QUANTUM HALL . . . PHYSICAL REVIEW B 89, 165123 (2014)

The symmetry associated with the two states that cross isTR symmetry. The QAH state has a different quantum numberfor the TR symmetry operator than the SM and CMs phases.The QAH ground state that we find is the linear combination ofthe two TR symmetry-breaking states which is odd under TR,whereas the SM and CMs phases are even under TR. Since theHamiltonian cannot have matrix elements between states thatare odd and states that are even under TR the level crossingsare allowed.

In summary, our ED results show clear signs of aninteraction-driven Chern insulator (QAH) phase, contrary torecent ED studies of small clusters with periodic boundaryconditions [18,19]. This confirms the importance of theboundary condition and the symmetry of the cluster to detecttopological order in finite-size systems. In the followingsection we study the ground-state properties of the systemfor larger system sizes to further confirm the existence of theQAH phase in the thermodynamic limit.

V. THE EPS ANSATZ AND VARIATIONALMONTE CARLO ALGORITHM

Studying larger system sizes with the ED method is notpossible due to the rapid increase in the size of the Hilbert spacewith an increase in the size of the system. However, recentlythere has been much success using tensor network methods tonumerically simulate a variety of strongly correlated models.In this section we examine the applicability of the EPSansatz [31–39] to describe the ground states of the system.Within this approach the lattice is covered with plaquettesand the ground-state wave function is written in terms of theplaquette coefficients. Configurational weights can then beoptimized using a VMC algorithm. This then allows study ofthe ground-state properties of the system for larger systemsizes and further confirmation of the existence of the QAHphase in the thermodynamic limit.

For a lattice with l sites an arbitrary quantum many-bodywave function can be written as

|ψ〉 =∑

n1,n2,...,nl

Wn1,n2,...,nl|n1,n2, . . . ,nl〉

=∑

n

Wn|n〉, (6)

where n denotes the vector of occupancies (ni ∈ {0,1} in afermion system) and Wn is the amplitude or weight of a givenconfiguration n. In the EPS description, also called the CPSdescription, the weight W (n) is expressed as a product ofthe plaquette coefficients over the lattice. A correlator is anoperator that is diagonal in the lattice basis and can be chosento act on an arbitrary number of sites. Such a general correlatorfor a plaquette p with lp sites can be written as

cp =∑np

cnp

p Pnp, (7)

where np = {np1,np2, . . . ,nplp } is the occupancy vector of thesites of the plaquette p and Pnp

is the projection operator

Pnp= |np〉〈np|. (8)

The EPS (CPS) is then obtained by applying the product ofcorrelators to a reference wave function |〉,

|ψ〉EPS = C|〉 =∏p

cp|〉. (9)

For a fermionic system the Jastrow-Slater CPS fermion wavefunction is obtained by applying a set of correlators (Jastrowfactors) to a Slater determinant reference of orbitals [33],

|〉 = det |φ1φ2 . . . φk|, (10)

and the wave function amplitudes Wn in Eq. (6) are, in thiscase, given by

Wn1,n2,...,nl=

∏p

cnp

p × det |φ1(r1)φ2(r2) . . . φl(rk)|, (11)

where r1,r2, . . . ,rk label the positions of k occupied sites inthe occupancy vector |n1,n2, . . . ,nl〉.

Here the Slater determinant, (10), orbitals can be obtainedby solving the noninteracting limit of Hamiltonian (1) withV1 = V2 = 0. By applying the product of correlators to theSlater determinant reference wave function the correlationsbetween sites are introduced into the wave-function EPSansatz and are explicitly encoded in correlator building blocks.The wave function of the original system, expressed as theproduct of the wave functions for sub-blocks (plaquettes),gives reasonable estimates of the ground-state energy andshort-range correlations.

The estimates improve with an increase in plaquette sizeand also, in general, a greater overlap between the plaquettesgives a more accurate description of the ground state. Uponincreasing the number of sites covered by correlators (largerplaquette size) the EPS becomes an exact family of states.However, in many cases even correlators with a small numberof parameters can describe the qualitative behavior of largesystems. Here we choose eight-site plaquettes illustrated inFig. 7 and consider as an example a cluster of 128 sites (two8 × 8 triangular sublattices) of the same shape as the clusterin Fig. 7 and with open boundary conditions.

Expectation values of the energy and other operators can beobtained by using Monte Carlo importance sampling [39–42].In the VMC the energy E is written as

E = 〈ψ |H |ψ〉 =∑

n,n′ W ∗(n′)〈n′|H |n〉W (n)∑n |W (n)|2 , (12)

FIG. 7. (Color online) Illustration of the eight-site plaquette cor-relators used in the calculation of the ground-state properties of thesystem for larger system sizes. Here the underlying triangular latticehas a two-site basis, where the basis sites are labeled A and B.

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ĐURIC, CHANCELLOR, AND HERBUT PHYSICAL REVIEW B 89, 165123 (2014)

where we assume that the wave function |ψ〉 is not normalized.The energy can then be further rewritten as

E =∑

n

P (n)E(n), (13)

where E(n) is the local energy

E(n) =∑

n′

W ∗(n′)W ∗(n)

〈n′|H |n〉, (14)

and the probability P (n) is given by

P (n) = |W (n)|2∑n |W (n)|2 . (15)

The expectation value of any operator O can be written in thesame form by replacing the Hamiltonian H with the operatorO in the previous expressions. According to the variationalprinciple minimization of expression (13) with respect to theweights gives an upper bound of the ground-state energy. Theprobability of a given configuration P (n) is never explicitlycalculated from Eq. (15). Instead, the Metropolis algorithm[43] is used to sample the probability distribution and toefficiently compute the overall energy as an average of thesampled local energies.

The plaquette coefficients that minimize the energy can befound by using the stochastic minimization method [39–42],which requires only the first derivatives of the energy withrespect to the plaquette coefficients given by

∂E

∂cnp

p

= 2∑

n

{P (n)�

np

p

[E(n) −

∑n′

P (n′)E(n′)]}

, (16)

where the wave function |ψ〉 in Eq. (12) is approximated bythe EPS wave function, (9), and

�np

p = 1

W (n)

∂W (n)

∂cnp

p

= bp

cnp

p

, (17)

with W (n) given by Eq. (11). Here bp denotes the number oftimes the plaquette coefficient c

np

p appears in the product, (11),for the amplitude W (n) for configuration n. If each correlatoris used only once, bp = 1.

To calculate the ground-state energy and the ground-stateexpectation value of any operator O, we follow the variationalalgorithm used in several previous studies of various stronglycorrelated models [39,41,42]. For a given set of plaquettecoefficients the energy, (13), and its first derivative, (16),can be efficiently computed using the Metropolis algorithm.In our calculation the total number of fermions N is fixed.We start from a randomly chosen initial configuration |n〉 =|n1,n2, . . . ,nl〉, with

∑li=1 ni = N , and then generate via

the Metropolis algorithm a large set of new configurationsby exchanging the occupancy numbers ni and nj at twoneighboring sites i and j . Starting from configuration |n〉,the acceptance probability of a new configuration |n′〉 isgiven by

PA = min

[ |W (n′)|2|W (n)|2 ,1

]. (18)

The overall energy can then be efficiently computed as theaverage of the local energies sampled by a Markov chain in

the Metropolis algorithm. The first derivative can be calculatedequivalently from the same sample.

It is also important to note that there is no need to computethe wave-function normalization at any point in the calculation.The most time-consuming step in the calculation is the evalu-ation of the fraction W (n′)/W (n). However, the configurationweights W (n) are given by a simple product of numbers, and tocalculate the fraction W (n′)/W (n) it is necessary to calculateonly products of the plaquette coefficients that representsites {i}, where the occupancy numbers n{i} have changedto n′

{i} = n{i}. Also, if the Hamiltonian is local, there are onlya few nonzero matrix elements 〈n|H |n′〉 in expression (14) forthe local energy, and only a few fraction terms W (n′)/W (n)need to be calculated for each contribution to the energy.

The values of the plaquette coefficients that minimize theenergy can be found using the stochastic minimization method[39,41,42]. The steps of the variational algorithm to calculatethe ground-state energy are as follows: (i) start from therandomly chosen complex values for the plaquette coefficients,(ii) evaluate the energy and its gradient vector, (iii) update allthe plaquette coefficients c

np

p according to

cnp

p → cnp

p − rδ(k) · sign

(∂E

∂cnp

p

)∗, (19)

and (iv) iterate from (ii) until convergence of the energy isreached. Here r is a random number between 0 and 1, and δ(k)is the step size for a given iteration k.

The energy and its derivative are estimated from 3 ×F (k) × l sampled values in each iteration k, where l is thenumber of lattice sites and F (k) is called the number ofsweeps per sample. In a given sweep each lattice site is visitedsequentially and a move n → n′ to a new configuration isproposed by exchanging the occupancy numbers ni and nj

of two neighboring sites i and j (3 NN for each lattice site).Also, to achieve convergence and reach the optimal energyvalue it is important to carefully tune the gradient step δ(k).For each iteration k, the number of sweeps F is increasedlinearly, F = F0k, and the procedure of evaluating the energyand updating the coefficients is repeated G = G0k times. Thestep size is gradually reduced per iteration. Here we use ageometric form, δ = δ0Q

k , with Q = 0.9.The number of sweeps per iteration is increased because

the derivatives become smaller as the energy minimum isapproached and require more sampling in order not to bedominated by noise. An increasing G effectively correspondsto a slower cooling rate. Here we take F0 = 10, G0 = 10, andQ = 0.9. The initial minimization routine is performed withδ0 = 0.05 for 50 iterations. The resulting plaquette coefficientsare then used as a starting point for a new run of 50 iterationswith δ0 = 0.005. After the minimization is complete theexpectation values are calculated by repeating the procedurefor a single iteration with zero step size and large F and G toobtain more accurate estimates of the expectation values. Theresults for the ground-state energy as a function of the NNNinteraction strength V2/t (V1 = 0) for a 128-site cluster withopen boundary conditions are shown in Fig. 8.

To confirm the topological phase transitions foundfor the smaller system sizes we further calculate thedensity-functional fidelity and its susceptibility [47] for the

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FIG. 8. Ground-state energy at half-filling as a function of theNNN interaction strength V2/t (V1 = 0) for a 128-site cluster of thesame shape as the cluster in Fig. 7 and with open boundary conditions,obtained using the EPS ansatz with eight-site plaquettes (Fig. 7) andstochastic minimization method. Statistical errors are smaller thanthe symbol size.

128-site cluster. The density-functional fidelity measures thesimilarity between the density distributions of two groundstates in parameter space. As explained in Sec. IV thetopological transition can be characterized by a singular peakin the fidelity susceptibility, (4). However, according to theHohenberg-Kohn theorems [52] the ground-state properties ofa quantum many-body system are uniquely determined by thedensity distribution nr that minimizes the ground-state energyfunctional E0[nr]. Therefore the most relevant informationabout the ground state of the system is captured by the densitydistribution and any change in the structure of the wavefunction corresponds to a change in the density distribution.Accordingly, the topological phase transition can be found bycalculating the similarity between density distributions andis characterized by a singular peak in the density-functionalfidelity susceptibility. Moreover, the density-functional fidelitycan be easily measured in experiments and provides a strategyto study quantum critical phenomena both theoretically andexperimentally.

The density distribution can be obtained as

nr = 〈�0(β)|nr|�0(β)〉, (20)

where r = (x,y) denotes sites on the honeycomb lattice, nr =c†rcr is the number operator at site r, and β is the relevant

parameter in the Hamiltonian (here β = V2/t). The density-functional fidelity for two ground states at β and β ′ is givenby [47]

Fn(β,β ′) = tr√

n(β)n(β ′), (21)

where

n =∑

r

nr|r〉〈r|. (22)

Equation (21) can be rewritten as

Fn(β,β + δβ) = 1 − (δβ)2

2χFn

+ . . . , (23)

FIG. 9. Density-functional fidelity susceptibility χFn(V2,δV2),

with δV2 = 0.05, as a function of the interaction strength V2 = V2/t

(V1 = 0) for a 128-site cluster of the same shape as the clusterin Fig. 7 and with open boundary conditions. Statistical errors aresmaller than the symbol size. The SM-to-QAH phase transition (a)and QAH-to-CMs phase transition (b) are characterized by a singularpeak (discontinuity) in χFn

(V2,δV2).

where δβ = β − β ′ and the density-functional fidelity suscep-tibility χFn

has the form

χFn=

∑r

1

4nr

(∂nr

∂β

)2

. (24)

Density-functional fidelity is a functional of nr and ∂nr/∂β,which both maximize the density-functional fidelity suscepti-bility, (24), at the critical point (for example, if the density ina certain region vanishes rapidly).

The topological phase transitions from the SM to the QAHphase and from the QAH phase to the CMs phase for a128-site cluster, characterized by singular points in the density-functional fidelity susceptibility, are clearly shown in Fig. 9.Also, a charge modulation within the same sublattice in theCMs phase can be clearly seen in the density distribution asshown in Fig. 11, while the charge modulation is absent in theQAH phase (Fig. 10). Our results indicate that for the 128-sitecluster and at V1 = 0 the QAH phase is found for 1.6 � V2/t �2.6, which is very close to the mean-field theory predictionsfor the system in the thermodynamic limit and to the ED resultobtained for a much smaller system size (18-site cluster).

We also note that in the thermodynamic limit the two lowestenergy states, which are odd and even under TR, becomeexactly degenerate in the QAH regime. Consequently, thereare two degenerate QAH ground states with opposite chirality(Chern number C = ±1) in the thermodynamic limit. This

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FIG. 10. (Color online) Ground-state density distribution for a128-site cluster with open boundary conditions at V2/t = 1.8 (V1 =0), when the system is in the QAH phase. Here the radius of eachcircle is proportional to the magnitude of the density.

also leads to the disappearance of the cusp in the energy as afunction of V2/t at the SM-QAH phase transition and confirmsthat the SM-QAH transition is a continuous phase transitionas predicted by the mean-field theory calculations.

In summary, our results obtained using the EPS ansatzfor the ground-state wave function and VMC indicate theexistence of the QAH state for larger system sizes and providefurther confirmation of the presence of the QAH phase inthe thermodynamic limit, as predicted by several mean-fieldtheory calculations [13–15]. This is in contrast to the previousresults [18,19] obtained using the ED method for small clusterswith periodic boundary conditions and cluster perturbationtheory [18].

VI. CONCLUSIONS

We have studied the system of interacting spinless fermionson a honeycomb lattice. Using the ED method for smallsystem sizes, and the EPS ansatz and VMC method forlarger system sizes, we find evidence of the existence of theinteraction-generated quantum Hall state previously predictedby several mean-field theory calculations [13–15]. In par-ticular, we have presented results for an 18-site cluster anda 128-site cluster with open boundary conditions. We findclear signs of the predicted topological phase transitions (fromthe SM to the interaction-generated quantum Hall state andfrom the quantum Hall state to the charge-modulated state)in the fidelity metric and also demonstrate the appearanceof the sublattice charge modulation in the CMs phase bycalculating the density profile for the 128 sites cluster. Thequantum Hall state was not found in the previous ED studies ofsmall clusters with periodic boundary conditions [18,19]. Ourresults thus indicate the importance of the boundary conditions

FIG. 11. (Color online) Ground-state density distribution for a128-site cluster with open boundary conditions at V2/t = 2.8 (V1 =0), when the system is in the CMs phase. Here the radius of eachcircle is proportional to the magnitude of the density.

and the symmetry of the cluster for detecting topological orderin finite-size systems. The results also confirm the validity ofthe Jastrow-Slater EPS (CPS) ansatz to describe the groundstates of the system.

Further work is necessary for more direct comparison ofthe EPS ansatz results with the ED results for small clusters.As mentioned before, a disadvantage of the VMC stochasticalgorithm is the presence of a statistical error, which can becontrolled by increasing the system size. To obtain results thatcan be directly compared with the ED results for small clustersit is therefore necessary to use a nonstochastic algorithm [33]to evaluate the energy and to optimize the amplitudes of theEPS wave function. Also, further insights could be obtainedby considering the results using different types of correlators(plaquettes of different shapes and sizes). Additional directionsof future research are to study the ground states of the systemat nonzero NN interaction strength V1 and possible topologicalphases of the system away from half-filling that are predictedby mean-field theory calculations [15,53]. We hope that theresults presented in this paper will motivate further studies ofthe extended Hubbard model for fermions on a honeycomblattice and other models for which interaction-generatedtopologically nontrivial phases have been predicted.

ACKNOWLEDGMENTS

We thank Fakher F. Assaad, Derek K. K. Lee, and ChrisHooley for very helpful suggestions and discussions. We alsothank Andrew G. Green for very useful suggestions and forcareful reading of the manuscript. This work was funded bythe EPSRC under Grant No. EP/K02163X/1. N.C. was fundedby Lockheed Martin Corporation at the time this work wascarried out. I.F.H. was supported by the NSERC of Canada.

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