Berry Curvature of interacting bosons in a honeycomb lattice

Yun Li,1 Pinaki Sengupta,2, 3 George G. Batrouni,4, 5, 6, 3 Christian Miniatura,3, 6, 7, 4 and Benoıt Gremaud3, 6, 7, 8

1Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Victoria, 3122, Australia2School of Physical and Mathematical Science, Nanyang Technological University, 21 Nayang Link, 637371 Singapore

3MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit UMI 3654, Singapore4INLN, Universite de Nice-Sophia Antipolis, CNRS; 1361 route des Lucioles, 06560 Valbonne, France

5Institut Universitaire de France, 103 boulevard Saint Michel, 75005 Paris, France.6Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117542 Singapore

7Department of Physics, National University of Singapore, 2 Science Drive 3, 117542 Singapore8Laboratoire Kastler Brossel, UPMC-Sorbonne Universites, CNRS,

ENS-PSL Research University, College de France, 4 Place Jussieu, 75005 Paris, France

We consider soft-core bosons with onsite interaction loaded in the honeycomb lattice with differentsite energies for the two sublattices. Using both a mean-field approach and quantum Monte-Carlosimulations, we show that the topology of the honeycomb lattice results in a non-vanishing Berrycurvature for the band structure of the single-particle excitations of the system. This Berry curvatureinduces an anomalous Hall effect. It is seen by studying the time evolution of a wavepacket, namelya superfluid ground state in a harmonic trap, subjected either to a constant force (Bloch oscillations)or to a sudden shift of the trap center.

I. INTRODUCTION

Topology and gauge fields, exemplified by Chern num-bers, Berry phases, Zak phase and Berry curvatures [1–3],are key concepts at the heart of many condensed-matterphenomena [4–6]. For solid-state systems, Bloch’s theo-rem introduces wave numbers k belonging to a parameterspace with the topology of a torus, the Brillouin zone,and a set of periodic wave functions depending para-metrically on k which offer natural settings for bundlesand connections [7]. Recently, ultracold atomic systemshave progressively and successfully confirmed their abil-ity to mimic or emulate some paradigmatic phenomenaof condensed-matter systems, in particular topological ef-fects. Indeed one can now load independent or interact-ing bosons and/or fermions into two-dimensional, verywell controlled, optical lattices [8–12] and one can gener-ate carefully designed synthetic magnetic fields [13–16].These advances pave the way to accurate cold atomsexperiments targeting topology-related effects such asdefects [17], color superfluidity [18], momentum-spaceBerry curvatures [19] or quantum Hall states with strongeffective magnetic fields [20–25].

Because of its remarkable low-energy electronic exci-tations, graphene has been the source of many key dis-coveries [26, 27] which sparked a vivid research flow nowreaching new territories as exemplified by ultracold atomsloaded in optical lattices [24, 28–36]. In particular, thetight-binding model on the honeycomb lattice or equiv-alently on the brick wall lattice is well known to exhibita Berry curvature in its band structure [25, 37, 38], suchthat the anomalous Hall effect can be observed [5, 39]. Inthis paper, we discuss the impact on the Berry curvatureof the interactions between bosons loaded in the honey-comb lattice with site energy imbalance, i.e. when thesite energies of the two sub-lattices are different. In thatcase, the phase diagram depicts Mott-insulating phasesat half-integer filling, a feature that is shared with the

energy imbalanced square lattice. On the other hand, aswe show below, the properties of the Bogoliubov excita-tions above the ground state are different for these twolattices: only the excitations for the honeycomb latticedepict a non-vanishing Berry curvature. This is empha-sized by looking at a dynamical situation, such as theBloch oscillations, where the anomalous Hall effect is ob-served.

The paper is organized as follows. In section II, weintroduce the model, summarize the basic properties ofthe energy imbalanced honeycomb lattice, in particularthe Berry curvature and we present the two approachesused in the paper: the quantum Monte-Carlo method(QMC) [40–43] and the Gutzwiller ansatz (GA) [44–46].In section III, we present our numerical results: theground state phase diagram and the excitations, empha-sizing that they depict non-vanishing Berry curvature.In section IV, we explain how to observe the anomalousHall effect resulting from the Berry curvature in the exci-tations. A summary of results and conclusions are givenin section V.

II. MODEL

A. Single-particle Hamiltonian and BerryCurvature

We start with a general bipartite lattice structure withfour nearest neighbors. Each primitive unit cell con-tains two lattice sites A and B, as depicted in Fig. 1.The whole lattice structure can be generated by repeatedtranslations of a unit cell along the Bravais primitive vec-tors

aaa1 = eeex − eeey and aaa2 = eeex + eeey (1)

where we have assumed the spacing between two sites inthe primitive unit cell is unity. The primitive reciprocal

arX

iv:1

506.

0396

4v1

[co

nd-m

at.q

uant

-gas

] 1

2 Ju

n 20

15

2

lattice vectors are

bbb1 = π(eeex − eeey) and bbb2 = π(eeex + eeey), (2)

and fulfills the relation aaai · bbbj = 2πδij .The displacements that move a A site to a neighboring

B site are parameterized by bond vectors

ccc1 = −ccc4 = eeex

ccc2 = −ccc3 = eeey(3)

a1

a2

c3

c1

c2

c4

(a) (b)

a1

a2

c3

c1

c2

c4

Sublattice A Sublattice B

y

xR

FIG. 1. (Color online) Structure of a bipartite lattice withfour nearest neighbors. Sites belonging to sublattice A arerepresented by dark (blue) points in the figure, whereas Bsites are represented by light grey (green) points. The hoppingamplitudes are all equal in (a) while in (b) the hopping termt4 along links ccc4 is different from the other three near neighborhopping terms.

Atoms trapped in the lattice can hop from one siteto the neighboring sites with amplitudes tν (ν = 1 .. 4)corresponding to the four bond vectors. In the absenceof interactions, the Hamiltonian in real space reads

H0 = −∑〈i,j〉

(tij b

†i bj + t∗ij b

†jbi

)− µA

∑i∈A

ni − µB∑j∈B

nj

(4)where tij ∈ t1, t2, t3, t4, bi is the operator annihilat-

ing one particle on lattice site i, ni = b†i bi is the atomnumber operator, µA and µB are chemical potentials forsublattice A and B respectively. When all hopping am-plitudes tν are the same, Eq. (4) describes a standardsquare lattice. If t4 = 0, the lattice becomes the brickwall lattice which is topologically equivalent to the hon-eycomb lattice. Using Fourier transform, we obtain theHamiltonian in momentum space

H0 =∑k

(b†kA b†kB

)(∆ ΓkΓ∗k −∆

)(bkAbkB

)− µN (5)

where k is the Bloch wave vector, ∆ = −(µA − µB)/2,

µ = (µA + µB)/2, N =∑

k(b†kAbkA + b†kBbkB) is the

total number of particles, and Γk = −∑4ν=1 tν e

−ik·cccν .The Hamiltonian (5) has the eigenvalues

ε±(k) = ±√

∆2 + |Γk|2 − µ (6)

and eigenfunctions [38]

|ψk+〉 =

cosβ

2

sinβ

2eiφ

, |ψk−〉 =

− sinβ

2e−iφ

cosβ

2

(7)

where φ = − arg Γk and β = arcsin( |Γk|/√

∆2 + |Γk|2 ).For ∆ = 0, the gap between the two energy bands closesat |Γk| = 0, and exhibit the so-called Dirac cone. To opena gap, one needs to imbalance the chemical potential be-tween the A and B sites. In the following, we alwaysconsider the gapped case, i.e. ∆ 6= 0.

The Berry curvature for each band of the Hamilto-nian (5) can be calculated as [38]

Ω±(k) = ±1

2

(∂ cosβ

∂kx

∂φ

∂ky− ∂ cosβ

∂ky

∂φ

∂kx

)(8)

Expanding the Bloch wave vector k over the reciprocalprimitive vectors, k = α1bbb1 + α2bbb2, we find the Berrycurvature (8) can be written as follows

Ω±(k) = ±1

2

∆

(∆2 + |Γk|2)3/2

×

[t1t3 − t2t4] sin(2πα2)− [t1t2 − t3t4] sin (2πα1)

(9)

where we have assumed real values for the hopping am-plitudes. For the square lattice (tν = t), the Berry cur-vature is always zero, reflecting the presence of both theinversion and the time-reversal symmetry in the system.However, if one of the hopping amplitudes is differentfrom the other three, for example t4 6= t1,2,3, the sym-metry under the exchange x → −x is broken. As a con-sequence, the system can exhibit a non-vanishing Berrycurvature. In this case, as the symmetry under the ex-change y → −y remains, β(k) and φ(k) are even func-tions of ky. According to Eq. (8), one can easily see thatthe Berry curvature should be an odd function of ky, i.e.Ω±(kx, ky) = −Ω±(kx,−ky).

In Fig. 2, we give an example of the Berry curvaturein the first Brillouin zone (BZ), i.e., for |αi| < 1/2, calcu-lated from (9) for the imbalanced honeycomb lattice, i.e.t1,2,3 = t, t4 = 0 and ∆ = 5t. Due to the time-reversalsymmetry, the Berry curvature has the usual propertyΩ±(−k) = −Ω±(k) such that, for each band, the Chernnumber which is the surface integral of Berry curvature,γ± =

∫BZ

dS Ω±(k), is vanishing. This emphasizes thatthe bands are topologically trivial. Furthermore, due tothe odd symmetry of the Berry curvature under the ex-change ky → −ky, it is also an even function of kx,Ω±(kx, ky) = Ω±(−kx, ky), as shown in Fig. 2: in the(α1, α2) plane, the Berry curvature is symmetric withrespect to the anti-diagonal and antisymmetric with re-spect to the diagonal.

3

FIG. 2. (Color online) Berry curvature in the first Brillouinzone for the honeycomb lattice calculated from Eq. (9). Theparameter are chosen as follows: t1, 2, 3 = t, t4 = 0, ∆ = 5t.

B. Numerical methods

When the interaction on a site is taken into account,the system can be described by the Bose-Hubbard Hamil-tonian

H = H0 +U

2

∑i

ni (ni − 1) (10)

where H0 is given by (4), and U is the on-site interactionstrength. This is the Hamiltonian we shall investigatein this study using two complementary methods that aredescribed below.

1. Quantum Monte-Carlo

The ground state of Hamiltonian (10) can be stud-ied using the stochastic series expansion (SSE) quan-tum Monte Carlo method with operator-loop updates[40–42]. The SSE is a finite-temperature QMC algo-rithm based on importance sampling of the diagonal ma-trix elements of the density matrix e−βH . There areno approximations beyond statistical errors. Using theoperator-loop cluster update, the autocorrelation timefor the system sizes we consider here is at most a fewMonte Carlo sweeps for the entire range of parameterspace explored. The simulations are carried out on finitelattices with L2 sites for L up to 32 at temperatures suffi-ciently low in order to resolve ground-state properties ofthis finite system [43]. Estimates of physical observablesin the thermodynamic limit are obtained from simulta-neous finite-size and finite-temperature extrapolation tothe L→∞, β →∞ limit. We consider here the soft-coreboson case which allows multiple occupation to occur (upto n(max) = 4 bosons per lattice site are allowed in thepresent study). Periodic boundary conditions along thex and y axis have been used.

2. Mean-field

A well known mean-field method to solve the Bose-Hubbard model is the Gutzwiller ansatz [47–50], wherethe ground state wavefunction is assumed to be a tensorproduct of on-site wavefunctions:

|Ψ〉 =⊗j

|ψj〉 where |ψj〉 =

N∑n=0

fn,j |n, j〉 (11)

where |n, j〉 represents the Fock state of n atoms occu-pying the site j, N is a cutoff in the maximum numberof atoms per site, and fn,j is the probability amplitudeof having the site j occupied by n atoms.

Minimizing the mean-field energy 〈Ψ|H|Ψ〉 over theamplitudes fn,j allows us to determine the mean-fieldground state properties as functions of the different pa-rameters (U, t, µA, µB). For instance, the superfluidphase corresponds to a non-vanishing value of the orderparameter 〈Ψ|bi|Ψ〉, whereas the Mott phase correspondsto a vanishing order parameter and the |ψi〉 are pure Fockstates.

In addition, the preceding ansatz (11) can be extendedto the time domain giving us access not only to the Bo-goliubov excitations above the ground state, but also tothe full mean-field evolution of the interacting wavefunc-tion. More precisely, the time evolution is obtained bysolving the following set of equations:

idfn,j(t)

dt=∂〈Ψ(t)|H|Ψ(t)〉

∂f∗n,j. (12)

The Bogoliubov excitations are obtained by expandingthe amplitudes fn,j(t) around their ground state values

f(0)n,j , namely:

fn,j =[f(0)n,j + gn,j(t)

]e−iωjt. (13)

where ωj is the frequency of the ground state evolution.

Assuming |gn,j(t)| |f (0)n,j | and keeping only the linear

terms in the dynamical Eqs. (12), one obtains the usualBogoliubov equations:

id

dt

[g(t)g∗(t)

]= L

[g(t)g∗(t)

], (14)

where g(t) is a shorthand notation for the vector(· · · , g0,j , g1,j , · · · , gN,j , · · · )T and L has the usual Bo-goliubov structure:

L =

(A B−B∗ −A∗

). (15)

A and B are complex matrices satisfying A† = A andBT = B.

Due to the U(1) invariance, the values of f(0)n,j can be

taken real. Furthermore, as we have observed (see be-low), the mean-field ground state does not break the

4

translation invariance, such that the values f(0)n,j are the

same for all equivalent sites, namely f(0)n,j = An for all

A−sites and f(0)n,j = Bn for all B−sites. Because of trans-

lation invariance, the Bogoliubov Eqs. (14) can be di-agonalized in momentum space, taking into account thebipartite nature of the lattice:

gn,j(t) =∑k

ei(k·Aj−ωt)uk,A,n + e−i(k·Aj−ωt)v∗k,A,n(16)

gn,j(t) =∑k

ei(k·Bj−ωt)uk,B,n + e−i(k·Bj−ωt)v∗k,B,n(17)

for A sites and B sites respectively, leading to the follow-ing eigensystem

ωk

(ukvk

)= Lk

(ukvk

)(18)

where uk = (uk,A,0, · · · , uk,A,N , uk,B,0, · · · , uk,B,N ) andvk = (vk,A,0, · · · , vk,A,N , vk,B,0, · · · , vk,B,N ). Lk has thesame structure as L, which leads to the usual property: if(u, v) is an eigenstate for the energy ω, then (v∗, u∗) is aneigenstate for the energy −ω∗. Therefore one can focuson the eigenstates with positive skew-norm only, i.e. suchthat u†u−v†v = 1. The ground state is stable if all theseeigenstates have real and positive eigenenergies ωk.

III. RESULTS

A. Phase diagram

Figure 3 shows the ground state phase diagram fora honeycomb lattice (t4 = 0) obtained from ansatzEq.(11). Due to different chemical potentials betweenthe A and B sites, the insulating lobes occur not onlyat 〈n〉 = 0, 1, 2, . . . but also at 〈n〉 = 1/2, 3/2, . . .. Wecompare the mean-field result with QMC simulation inFig. 4, where we plot the density per site as a func-tion of µ = (µA + µB)/2, fixing the hopping amplitudeat t/U = 0.03, which corresponds to the white lines inFig. 3. Very good agreement is found between the mean-field approach and QMC simulations.

B. Excitations

The phase diagram in Fig. 3 itself is not a signatureof the Berry curvature induced by the topology of thehoneycomb lattice. Indeed, the Hubbard model on thesquare lattice (i.e. with a non-vanishing hopping param-eter t4) with the same imbalance µA − µB exhibits asimilar phase diagram, including the half-integer fillingMott phases. The reason is that the ground state corre-sponds, roughly speaking, to k = 0, precisely where theBerry curvature Ω(k) vanishes. Therefore, the impactof the Berry curvature has to be found in the excitationproperties.

0

0.05

0.10

0.51

1.5

1

2

µ / Ut / U

⟨ n ⟩

(a)

FIG. 3. (Color online) Mean-field phase diagram of the Bose-Hubbard Hamiltonian Eq. (10) on the honeycomb lattice.(a) Density per site; (b) the superfluid order parameter. Theparameters used in the calculation are t1, 2, 3 = t, t4 = 0,∆/U = 0.15, N = 10. The white lines correspond to t/U =0.03.

As explained above, within the mean-field approach,the Bogoliubov excitations are obtained from the diag-onalization of Lk, leading to a band structure for theBogoliubov spectrum. For instance, in Fig. 5, we showthe lowest excitation band for the honeycomb lattice fortwo different phases, namely the Mott phase (µ/U = 0)and the superfluid phase (µ/U = 0.15). The rest of theparameters are the same as in Fig. 4. As expected, theexcitations are gapped in the Mott phase, whereas, inthe superfluid phase, they are gapless and exhibit a lin-ear behavior at small momenta.

In general the Bogoliubov bands are isolated allowingthe numerical computation [51] of the Berry curvatureassociated with each band, as shown in Fig. 6, for the twoexcitation bands of Fig. 5. One clearly sees that in bothsituations, the Berry curvature is non-vanishing, beingmaximum (in absolute value) along the anti-diagonal, i.e.the ky axis, like in the non-interacting case. In addition,the time reversal symmetry and the y → −y symmetry,

5

0 0.5 1 1.50

0.5

1

1.5

2

2.5

µ / U

⟨ n

⟩

Mean−field

SSE QMC

FIG. 4. (Color online) Comparison of QMC and mean-fieldresults for the density per site as a function of total chemicalpotential µ = (µA + µB)/2. Parameters are t1, 2, 3 = t, t4 =0, ∆/U = 0.15, as in Fig. 3, and the hopping amplitude isfixed to t/U = 0.03. The blue solid curve is obtained fromthe mean-field approach (corresponding to the white line inFig. 3), the red dots are obtained from QMC simulations. Thelattice size for QMC simulations is 16× 16, with a maximumatom number per lattice site n(max) = 4.

like in the non-interacting case, imply that the Berrycurvature is odd under both transformations k→ −k andky → −ky and that it is even under the transformationkx → −kx. In both Fig. 6 and Fig. 5, these propertiesare clearly seen: (i) antisymmetric under inversion andwith respect to the diagonal; (ii) symmetric with respectto the anti-diagonal.

Similarly, one could define a pseudo Berry curvaturefor the excitations obtained from QMC computations:because of the bipartite nature of the honeycomb lattice,the equal time Green function in k-space is a 2×2 matrix:

G(k) =

(GAA(k) GAB(k)GBA(k) GBB(k)

), (19)

where GLL′(k) = 〈b†kLbkL′〉. Since G(k) is hermitian,one can write G(k) = g0(k)11 + g(k) · σ, where σ =(σx, σy, σz) are the Pauli matrices. Thereby, one can de-fine the spherical angles βg(k) and φg(k), correspondingto the direction of g(k), or, equivalently, obtained fromthe diagonalization of G(k), similarly to diagonalizingH0, see Eq.(5) and Eq.(7). From these two angles, onecan define a pseudo Berry curvature:

Ωg(k) =1

2

(∂ cosβg∂kx

∂φg∂ky− ∂ cosβg

∂ky

∂φg∂kx

). (20)

This curvature, being computed from the equal-timeGreen function, is not directly related to a genuine Berryphase that could be measured from the excitations of thesystem. For that, one would need to define a curvaturefrom the real frequency Green function G(k, ω). That en-tails computing the imaginary time Green function and

FIG. 5. (Color online) Bogoliubov excitation spectrum ω(k).Parameters are the same as in Fig. 4. The top panel cor-responds to the Mott Phase µ/U = 0 and the bottom oneto the superfluid phase µ/U = 0.15. As expected, the excita-tions are gapped in the Mott phase, whereas, in the superfluidphase, they are gapless and exhibit a linear behavior at smallmomenta.

subsequent analytic continuation to the real axis, whichis well known to be a delicate task. Therefore, in orderto emphasize the impact of the lattice topology on theexcitations, we shall focus on the pseudo Berry curvaturedefined in Eq. (20). The results are shown in figure 7 cor-responding to a ground state in the n = 1/2 Mott phase.For comparison, we also show the pseudo Berry curvatureobtained in the same ground state but for the square lat-tice. Only the curvature for the honeycomb lattice is non-vanishing and is largest (in absolute value) around thesame position as for the non-interacting case, i.e. aroundthe location of the conical intersections. In addition, thispseudo Berry curvature has the same symmetry proper-ties: Ωg(k) = −Ωg(−k), (time reversal symmetry), oddunder ky → −ky and even under kx → −kx.

In the superfluid phase, the 2× 2 matrix is dominatedby its diagonal elements and the resulting pseudo Berrycurvature is smaller than the QMC accuracy.

6

FIG. 6. (Color online) Berry curvature of the Bogoliubovexcitations displayed in Fig. 5. One can clearly observeΩ(k) = −Ω(−k) (time reversal symmetry), the antisymmetrywith respect to the diagonal (ky → −ky) and the symmetrywith respect to the anti-diagonal (kx → −kx).

-1

-0.5

0

0.5

1

kx/π

-1-0.5

0 0.5

1

ky/π

-0.100

-0.050

0.000

0.050

0.100

Ωg

-1

-0.5

0

0.5

1

kx/π

-1-0.5

0 0.5

1

ky/π

-0.002

-0.001

0.000

0.001

0.002

Ωg

FIG. 7. (Color online) Pseudo Berry curvature, Eq.(20), ob-tained with the QMC equal-time Green functions for the hon-eycomb and square lattices. The upper panel correspondsto the n = 1/2 Mott phase for the honeycomb lattice, i.e.t1, 2, 3 = t, t4 = 0, ∆/U = 0.15, µ/U = 0 and t/U = 0.03.The lower panel corresponds to the n = 1/2 Mott phase forthe square lattice, i.e. t1, 2, 3, 4 = t. For the square lattice, oneobtains only a random pattern with an amplitude two ordersof magnitude lower than the honeycomb lattice results. Thisemphasizes that the interacting system must really depict anon-vanishing Berry curvature in the excitation spectrum. Inaddition, one can see that the pseudo curvature has the sameproperties as the non-interacting system: it attains the largest(absolute) values around the location of the conical intersec-tions and fulfills Ωg(k) = −Ωg(−k), due to the time reversalsymmetry. It is also odd under ky → −ky and even underkx → −kx.

7

IV. ANOMALOUS HALL EFFECT

Although the preceding analysis has shown that theimpact of the Berry curvature can be found in the prop-erties of the excitations, one can still raise the questionwhether it can be observed. It is well known that in thissituation, the system is expected to exhibit an anoma-lous Hall effect (AHE) – a Hall effect without applyingan external magnetic field – in the non-interacting case.In this section, we shall demonstrate that one can stillobserve the AHE in the presence of the interaction. How-ever, the AHE can be observed only if the initial stateis a wavepacket well-localized in real space. Indeed, ifone starts with a pure Bloch state, the effect of a con-stant force only leads to Bloch oscillations, i.e. a periodicvariation of the quasi-momentum along the direction ofthe force, and the Berry curvature simply results in a

modification of the phase accumulated along the path.

A simple way to prepare a wavepacket is to start fromthe ground state in a harmonic trap, as depicted in Fig. 8:the atoms are well localized in space, still, the wavepacketis also well localized in the Brillouin zone around k = 0.The harmonic trap is such that the chemical potential atthe center corresponds to the superfluid phase describedin the previous section, i.e. µ/U = 0.15 and t/U = 0.03.The site energy of the A-sublattice is lower than that ofthe B-sublattice, corresponding to ∆/U = −0.15, result-ing, in the present case, in a Mott-like state for the A-sitewith a flat density at unit filling, see Fig.8, top-left. Onthe contrary, the filling of the B-sublattice is low, suchthat the system is in a superfluid phase, with a smooth,gaussian like, density, see Fig.8, top-right. In both cases,the density in the k-space is peaked at k = 0.

-50

0

50X -50

0

50

Y

0.0

0.2

0.4

0.6

0.8

1.0

nA

-50

0

50X -50

0

50

Y

0.0

0.1

0.2

0.3

0.4

0.5

0.6

nB

-0.4-0.2

0 0.2

0.4

α1-0.4

-0.2

0

0.2

0.4

α2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

nA(k)

-0.4-0.2

0 0.2

0.4

α1-0.4

-0.2

0

0.2

0.4

α2

0.0

0.5

1.0

1.5

nB(k)

FIG. 8. (Color online) Initial wavepacket in a harmonic trap. The atomic density in the A-sublattice (resp. B-sublattice) isdisplayed in the top left (resp. top right) plot. The harmonic trap is such that the chemical potential at the center correspondsto the superfluid phase described in the previous section. Since the chemical potential µA is larger than µB , i.e. ∆/U = −0.15,the A-sites are in a Mott-like state with unit filling. On the contrary, the filling of the B-sublattice is low, such that the systemis in a superfluid phase. Densities in the k-space for both the sublattices are displayed in the bottom row and are found to bepeaked around k = 0.

A. Bloch oscillations

To simulate Bloch oscillations, starting from thewavepacket, one removes abruptly the harmonic trap andadiabatically increases the linear potential along the y

axis at time τ = 0 mimicking the effect of an electric fieldE along the y axis. The evolution of the wavepacket isobtained by solving the time-dependent equations for themean-field amplitudes Eq.(12). In k-space, this leads to asmooth evolution of the wavepacket (with some broaden-

8

ing) along ky, see Fig. 9. This is emphasized in Fig. 10,where we have plotted the average value of ky (withinthe B-sublattice) as as function of time. After the adi-abatic transition, the linear increase of ky with time isa clear signature of the Bloch oscillation. The AHE isdemonstrated by the bottom plot in Fig. 10, where weshow the displacement from the center of the trap of thewavepacket along the x-axis, i.e. perpendicular to the ap-plied force. One can clearly see the impact of the Berrycurvature, in particular when comparing with a similarevolution but for the square lattice, for which there isno displacement. In addition, changing the sign of thelinear potential i.e. the direction of the applied force, re-sults in changing the direction of the Bloch oscillationsk(τ) → −k(τ), i.e. changing the direction of the groupvelocity of the wavepacket. Together with the fact thatthe Berry curvature is odd with respect to the changek→ −k, this results in a Hall displacement in the samedirection along the x-axis, as depicted in Fig. 10: thetwo curves coincide with each other. Note that we havechecked that this effect is independent of the exact lo-cation of the center of the trap. Finally, we have alsoverified that if the force is applied along the x-axis, noAHE is observed since the Berry curvature vanishes forky = 0.

-0.4-0.2

0 0.2

0.4

α1-0.4

-0.2

0

0.2

0.4

α2

0.0

0.5

1.0

nB(k)

-0.4-0.2

0 0.2

0.4

α1-0.4

-0.2

0

0.2

0.4

α2

0.0

0.3

0.6

nB(k)

FIG. 9. (Color online) Evolution of the wavepacket in momen-tum space. The upper panel displays the wavepacket after atime τ = 25/U , the lower one after τ = 50/U . The centerof the wavepacket moves at a constant velocity along the kyaxis, i.e., α1 + α2 = 0. The broadening of the wavepacket isweak.

FIG. 10. (Color online) Bloch oscillations and AHE for awavepacket. The top plot shows the average value of ky(within the B-sublattice) as a function of time. After theadiabatic transition, the linear increase of ky with time is aclear signature of the Bloch oscillation. The two curves, i.e,the continuous (black) line and the dashed (red) line withthe circles, correspond to opposite directions of the effectiveelectric field. The AHE is demonstrated by the bottom plot,where one plots the displacement from the center of the trapof the wavepacket along the x-axis, i.e. perpendicular to theapplied force: the continuous (black) line with the squares andthe dashed (red) line with the circles correspond to oppositedirections of the effective electric field. For comparison, thecorresponding displacement on a square lattice is also shownby the continuous (green) line with the diamonds.

B. Shift of the trap center

The amplitude of the AHE discussed above is rathersmall. One way to get a larger (measurable) effect con-sists of shifting abruptly the center of the trap at τ = 0along the y-axis, such that the wavepacket experiences anet force in that direction. Figure 11 shows the evolutionof the expectation value of the center of the wavepacket,i.e., 〈r〉, after a shift of the trap center by 20 lattice spac-ings in the y direction. As one can see, not only does thewavepacket evolve along the shifted center, but, at thesame time, it moves along the transverse direction, i.e.the x-axis. Similarly to the Bloch oscillations, one hasthe following results: (i) nothing happens on the squarelattice; (ii) nothing happens when the shift is along the x-axis where the Berry curvature always vanishes; (iii) thedisplacement along x is in the same direction whether the

9

trap center is shifted towards the +y or the −y direction.

FIG. 11. (Color online) Evolution of the average center ofthe wavepacket, i.e., 〈r〉, after a shift of the trap center by20 lattice spacings in the y direction (continuous (black) linewith the squares. The wavepacket clearly exhibits motionalong the transverse direction, i.e. the x-axis. Similarly toBloch oscillations, one has the following results: (i) nothinghappens on the square lattice, see the (green) line with thediamonds; (ii) nothing happens when the shift is along the x-axis where the Berry curvature always vanishes, see the (blue)line with the triangles; (iii) the displacement along x is in thesame direction whether the trap center is shifted towards the+y or the −y direction, see the dashed (red) line with thecircles.

V. CONCLUSION

In summary, we have shown that for bosons in thehoneycomb lattice with energy imbalance, the Berry cur-vature originally seen in the band structure, i.e. for thenon-interacting system, is also present in the excitationsabove the ground state of the interacting system, in boththe Mott-insulating phase and the superfluid phase. Inaddition, we have shown that one consequence of theBerry phase, the anomalous Hall effect, could be observedin dynamical experiments, such as Bloch oscillations.

The Centre for Quantum Technologies is a ResearchCentre of Excellence funded by the Ministry of Educa-tion and National Research Foundation of Singapore. PSacknowledges financial support from the Ministry of Edu-cation, Singapore via grant MOE2014-T2-2-112. LY ac-knowledges support from the ARC Discovery Projects(Grant Nos DE150101636 and DP140103231).

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