Fractional Quantum Hall Effect in Optical Lattices
Mohammad Hafezi, Anders S. Sorensen, Eugene Demler, Mikhail D. Lukin
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Fractional Quantum Hall Effect in Optical Lattices
M. Hafezi, ∗ A. S. Sørensen, E. Demler, and M. D. Lukin Physics Department, Harvard University, Cambridge, MA - 02138 QUANTOP, Danish National Research Foundation Centre of Quantum Optics,Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark.
We analyze a recently proposed method to create fractional quantum Hall (FQH) states of atomsconfined in optical lattices [A. Sørensen et al. , Phys. Rev. Lett. PACS numbers: 03.75.Lm,73.43.-f
1. INTRODUCTION
With recent advances in the field of ultra-cold atomicgases, trapped Bose-Einstein condensates (BEC’s) havebecome an important system to study many-body physicssuch as quantum phase transitions. In particular, theability to dynamically control the lattice structure andthe strength of interaction as well as the absence of im-purities in BEC’s confined in optical lattices, have led tothe recent observation of the superfluid to Mott-insulatortransition [1, 2, 3, 4, 5]. At the same time, there hasbeen a tremendous interest in studying rotating BEC’sin harmonic traps; at sufficient rotation an Abrikosov lat-tice of quantized vortices has been observed [6] and therealization of strongly correlated quantum states simi-lar to the fractional quantum Hall states has been pre-dicted to occur at higher rotation rates [7, 8, 9]. In theseproposals, the rotation can play the role of an effectivemagnetic field for the neutral atoms and in analogy withelectrons, the atoms may enter into a state describedby the Laughlin wavefunction, which was introduced todescribe the fractional quantum Hall effect. While thisapproach yields a stable ground state separated from allexcited states by an energy gap, in practice this gap israther small because of the weak interactions among theparticles in the magnetic traps typically used. In opticallattices, the interaction energies are much larger becausethe atoms are confined in a much smaller volume, andthe realization of the fractional quantum Hall effect inoptical lattices could therefore lead to a much higher en-ergy gap and be much more robust. In a recent paper[10], it was shown that it is indeed possible to realize thefractional quantum Hall effect in an optical lattice, andthat the energy gap achieved in this situation is a frac- ∗ Electronic address: [email protected] tion of the tunneling energy, which can be considerablylarger than the typical energy scales in a magnetic trap.In addition to being an interesting system in its ownright, the fractional quantum Hall effect, is also inter-esting from the point of view of topological quantumcomputation [11]. In these schemes quantum states withfractional statistics can potentially perform fault tolerantquantum computation. So far, there has been no directexperimental observation of fractional statistics althoughsome signatures has been observed in electron interfer-ometer experiments [12, 13]. Strongly correlated quan-tum gases can be a good alternative where the systemsare more controllable and impurities are not present.Therefore, realization of fractional quantum Hall statesin atomic gases can be a promising resource for topolog-ical quantum computation in the future.As noted above the FQH effect can be realized by sim-ply rotating and cooling atoms confined in a harmonictrap. In this situation, it can be shown that the Laugh-lin wavefunction exactly describes the ground state ofthe many body system [7, 14]. In an optical lattices, onthe other hand, there are a number of questions whichneed to be addressed. First of all, it is unclear to whichextent the lattices modifies the fractional quantum Hallphysics. For a single particle, the lattice modifies theenergy levels from being simple Landau levels into thefractal structure known as the Hofstadter butterfly [15].In the regime where the magnetic flux going through eachlattice α is small, one expects that this will be of minorimportance and in Ref. 10 it was argued that the frac-tional quantum Hall physics persists until α . .
3. Inthis paper, we extent and quantify predictions carriedout in Ref. 10 . Whereas Ref. 10 only considered theeffect of infinite onsite interaction, we extent the anal-ysis to finite interactions. Furthermore, where Ref. 10mainly argued that the ground state of the atoms was ina quantum Hall state by considering the overlap of theground state found by numerical diagonalization with theLaughlin wavefunction, we provide further evidence forthis claim by characterizing the topological order of thesystem by calculating Chern numbers. These calcula-tions thus characterize the order in the system even forparameter regimes where the overlap with the Laughlinwavefunction is decreased by the lattice structure.In addition to considering these fundamental featuresof the FQH states on a lattice, which are applicable re-gardless of the system being used to realize the effect, wealso study a number of questions which are of particu-lar interest to experimental efforts towards realizing theeffect with atoms in optical lattices. In particular, weshow that adding dipole interactions between the atomscan be used to increase the energy gap and stabilize theground state. Furthermore, we study Bragg spectroscopyof atoms in the lattice and show that this is a viablemethod to identify the quantum Hall states created inan experiment, and we discuss a new method to generatean effective magnetic field for the neutral atoms in thelattice.The paper is organized as follows: In Sec. 2, we studythe system with finite onsite interaction. In Sec. 3 weintroduce Chern numbers to characterize the topologicalorder of the system. The effect of the dipole-dipole inter-action has been elaborated in Sec. 4.1. Sec. 4.2 studiesthe case of ν = 1 /
2. QUANTUM HALL STATE OF BOSONS ON ALATTICE2.1. The Model
The fractional quantum Hall effect occurs for electronsconfined in a two dimensional plane under the presencesof a perpendicular strong magnetic field. If N is thenumber of electrons in the system and N φ is the num-ber of magnetic fluxes measured in units of the quantummagnetic flux Φ = h/e , then depending on the fillingfactor ν = N/N φ the ground state of the system canform highly entangled states exhibiting a rich behaviors,such as incompressibility, charge density waves, and any-onic excitations with fractional statistics. In particular,when ν = 1 /m , where m is an integer, the ground stateof the system is an incompressible quantum liquid whichis protected by an energy gap from all other states, andin the Landau gauge is well described by the Laughlinwavefunction [16]:Ψ( z , z , ..., z N ) = N Y j>k ( z j − z k ) m N Y j =1 e − y i / , (1)where the integer m should be odd in order to meet theantisymmetrization requirement for fermions. Although the fractional quantum Hall effect occurs forfermions (electrons), bosonic systems with repulsive in-teractions can exhibit similar behaviors. In particular,the Laughlin states with even m correspond to bosons.In this article, we study bosons since the experimen-tal implementation are more advanced for the ultra-coldbosonic systems. We study a system of atoms confined ina 2D lattice which can be described by the Bose-Hubbardmodel [17] with the Peierls substitution [15, 18], H = − J X x,y ˆ a † x +1 ,y ˆ a x,y e − iπαy + ˆ a † x,y +1 ˆ a x,y e iπαx + h.c. + U X x,y ˆ n x,y (ˆ n x,y − , (2)where J is the hopping energy between two neighboringsites, U is the onsite interaction energy, and 2 πα is thephase acquired by a particle going around a plaquette.This Hamiltonian is equivalent to the Hamiltonian of aU(1) gauge field (transverse magnetic field) on a squarelattice. More precisely, the non-interacting part can bewritten as − J X
2, since this will bethe most experimentally accessible regime.We restrict ourself to the simplest boundary conditionsfor the single particle states t s ( ~L ) ψ ( x s , y s ) = ψ ( x s , y s ),where t s ( ~L ) is the magnetic translation operator whichtranslates the single particle states ψ ( x s , y s ) around thetorus. The definition and detailed discussion of theboundary conditions will be elaborated in Section 3. Thediscussed quantities in this section, such as energy spec-trum, gap and overlap, do not depend on the boundarycondition angles (this is also verified by our numericalcalculation).In the continuum case, for the filling fraction ν = 1 / ~A = − By ˆ x ) is givenby Eq. (1) with m = 2. The generalization of the Laugh-lin wavefunction to a torus takes the form [27]Ψ( z , z , ..., z N ) = f rel ( z , z , ..., z N ) F cm ( Z ) e − P i y i / , (4)where f rel is the relative part of the wave function andis invariant under collective shifts of all z i ’s by the sameamount, and F cm ( Z ) is related to the motion of the centerof mass and is only a function of Z = P i z i . For asystem on a torus of the size ( L x × L y ), we write thewavefunction with the help of theta functions, which arethe proper oscillatory periodic functions and are definedas ϑ (cid:2) ab (cid:3) ( z | τ ) = P n e iπτ ( n + a ) +2 πi ( n + a )( z + b ) where thesum is over all integers. For the relative part we have, f rel = Y i 2, and is givenby F cm ( Z ) = ϑ (cid:20) l/ N φ − / − ( N φ − / (cid:21) (cid:18) P i z i L x | i L y L x (cid:19) (6) where l = 0 , α ≪ 1. For higher magnetic filed, the latticestructure becomes more pronounced. However, in ournumerical calculation for a moderate magnetic field α . . 4, we observe a two-fold degeneracy ground state wellseparated from the excited state by an energy gap. Wereturn to the discussion of the ground state degeneracyin Sec. 3.In the continuum limit α ≪ 1, the Laughlin wavefunc-tion is the exact ground state of the many body systemwith a short range interaction [7, 14, 29]. The reason isthat the ground state is composed entirely of states in thelowest Landau level which minimizes the magnetic partof the Hamiltonian, the first term in Eq. (2). The expec-tation value of the interacting part of the Hamiltonian,i.e. the second term in Eq. (2), for the Laughlin state iszero regardless of the strength of the interaction, since itvanishes when the particles are at the same position.To study the system with a non-vanishing α , wehave performed a direct numerical diagonalization of theHamiltonian for a small number of particles. Since we aredealing with identical particles, the states in the Hilbertspace can be labeled by specifying the number of parti-cles at each of the lattice sites. In the hard-core limit,only one particle is permitted on each lattice site, there-fore for N particles on a lattice with the number of sitesequal to ( N x = L x /a, N y = L y /a ), where a is the unitlattice side, the Hilbert space size is given by the com-bination (cid:0) NxNyN (cid:1) = N x N y ! N !( N x N y − N )! . On the other hand, incase of finite onsite interaction, the particles can be ontop of each other, so the Hilbert space is bigger and isgiven by the combination (cid:0) N + NxNy − N (cid:1) . In our simu-lations the dimension of the Hilbert can be raised upto ∼ · and the Hamiltonian is constructed in theconfiguration space by taking into account the tunnelingand interacting terms. The tunneling term is written inthe symmetric gauge, and we make sure that the phaseacquired around a plaquette is equal to 2 πα , and thatthe generalized magnetic boundary condition is satisfiedwhen the particles tunnels over the edge of the lattice[to be discussed in Sec. 3, c.f. Eq. (11)]. By diagonaliz-ing the Hamilton, we find the two-fold degenerate groundstate energy which is separated by an energy gap fromthe excited states and the corresponding wavefunction inthe configuration space. The Lauhglin wave function (4)can also be written in the configuration space by simplyevaluating the Laughlin wave function at discrete points,and therefore we can compared the overlap of these twodimensional subspaces. (a)(b) FIG. 1: (color online) (a) The overlap of the ground statewith the Laughlin wavefunction. For small α the Laughlinwavefunction is a good description of the ground state forpositive interaction strengths. The inset shows the same re-sult of small U . (b) The energy gap for N/N φ = 1 / U/J from attractive to repulsive. Fora fixed α , the behavior does not depend on the number ofatoms. The inset define the particle numbers, lattice sizes,and symbols for both parts (a) and (b). The energy gap above the ground state and the groundstate overlap with the Laughlin wavefunction for the caseof ν = 1 / α . . 2, are depicted infigure. 1. The Laughlin wavefunction remains a gooddescription of the ground state even if the strength ofthe repulsive interaction tends to zero (Fig. 1 a). Below,we discuss different limits:First, we consider U > , U ≫ Jα : If the inter-action energy scale U is much larger than the magneticone ( Jα ), all low energy states lie in a manifold, wherethe highest occupation number for each site is one, i.e.this corresponds to the hard-core limit. The ground stateis the Laughlin state and the excited states are variousmixtures of Landau states. The ground state is two-fold α U α / J G ap / J Hardcore Limit (a)(b) FIG. 2: (color online) (a) The energy gap as a function of αU and α for a fixed number of atoms (N=4). The gap is cal-culated for the parameters marked with dots and the surfaceis an extrapolation between the points. (b) Linear scaling ofthe energy gap with αU for U ≪ J, α . . 2. The results areshown for N = 2( (cid:3) ), N = 3( ∗ ), N = 4(+) and N = 5( ▽ ).The gap disappears for non-interaction system, and increaseswith increasing interaction strength ( ∝ αU ) and eventuallysaturate to the value in the hardcore limit. degenerate [28] and the gap reaches the value in for thehard-core limit at large U & J/α , as shown in Fig. 2(a). In this limit the gap only depends on the tunneling J and flux α , and the gap is a fraction of J . These resultsare consistent with the previous work in Ref. 10.Secondly, we consider | U | ≪ Jα . In this regime, themagnetic energy scale ( Jα ) is much larger than the inter-action energy scale U . For repulsive regime ( U > αU , as shown in Fig. 2 b.Thirdly, we study U = 0 where the interaction is ab-sent and the ground state becomes highly degenerate.For a single particle on a lattice, the spectrum is the fa-mous Hostadfer’s butterfly [15], while in the continuumlimit α ≪ 1, the ground state is the lowest Landau level(LLL). The single particle degeneracy of the LLL is thenumber of fluxes going through the surface, N φ . So inthe case of N bosons, the lowest energy is obtained byputting N bosons in N φ levels. Therefore, the many-body ground state’s degeneracy should be: (cid:16) N + Nφ − Nφ − (cid:17) . For example, 3 bosons and 6 fluxes gives a 336-fold de-generacy in the non-interacting ground state.If we increase the amount of phase (flux) per plaque-tte ( α ) we are no longer in the continuum limit. TheLandau level degeneracy will be replace by L L s where α = r/s is the amount of flux per plaquette and r and s are coprime [30]. Then, the many-body degeneracy willbe: (cid:16) N + L L s − L L s − (cid:17) .Fourthly, we consider U < , U ≫ Jα : when U isnegative (i.e. attractive interaction) in the limit stronginteraction regime, the ground state of the system willbecome a pillar state. In a pillar state, all bosons inthe system condensate into a single site. Therefore, thedegeneracy of the ground state is N x × N y and the groundstate manifold can be spanned as, M i √ N ! ( a † ) Ni | vac i . (7)These states will very fragile and susceptible to collapse[31].In a lattice, it is also possible to realize the fractionalquantum Hall states for attractive interaction in the limitwhen | U | ≫ Jα . Assume that the occupation number ofeach site is either zero or one. Since the attraction en-ergy U is very high and there is no channel into whichthe system can dissipate its energy, the probability for aboson to hop to a site where there is already a boson is in-finitesimally small. Therefore, the high energy attractionwill induce an effective hard-core constraint in the caseof ultra-cold system. The energy of these state shouldbe exactly equal to their hard-core ground state counter-parts, since the interaction expectation value of the inter-action energy is zero for the Laughlin state. The numeri-cal simulation shows that these two degenerate states in-deed have a good overlap with the Laughlin wavefunctionsimilar to their repulsive hard-core counterparts and alsotheir energies are equal to the hard-core ground state.These states are very similar to repulsively bound atompairs in an optical lattice which have recently been ex-perimentally observed [4].So far we have mainly considered a dilute lattice α . . 2, where the difference between a lattice and the con-tinuum is very limited. We shall now begin to inves-tigate what happens for larger values of α , where theeffect of the lattice plays a significant role. Fig. 3, showsthe ground state overlap with the Laughlin wavefunc-tion as a function of the strength of magnetic flux α andthe strength of the onsite interaction U . As α increasesthe Laughlin overlap is no longer an exact descriptionof the system since the lattice behavior of the system ismore pronounced comparing to the continuum case. Thisbehavior doesn’t depend significantly on the number of particles for the limited number of particles that we haveinvestigated N ≤ 5. We have, however, not made anymodification to the Laughlin wave function to take intoaccount the underlying lattice, and from the calculationspresented so far, it is unclear whether the decreasing wavefunction overlap represents a change in the nature of theground state, or whether it is just caused by a modifica-tion to the Laughlin wave function due the difference be-tween the continuum and the lattice. To investigate this,in the next Section, we provide a better characterizationof the ground state in terms of Chern numbers, whichshows that the same topological order is still present inthe system for higher values of α .As a summary, we observe that the Laughlin wavefunc-tion is a good description for the case of dilute lattice( α ≪ (a)(b) FIG. 3: (color online) Ground state overlap with the Laughlinwavefunction. (a) and (b) are for 3 and 4 atoms on a lattice,respectively. α is varied by changing the size of the lattice(the size in the two orthogonal directions differ at most byunity). The Laughlin state ceases to be a good description ofthe system as the lattice nature of the system becomes moreapparent α & . 25. The overlap is only calculated at the posi-tions shown with dots and the color coding is an extrapolationbetween the points. becomes smaller for weaker values of interaction and inthe perturbative regime U ≪ J is proportional to αU for α . . 3. CHERN NUMBER AND TOPOLOGICALINVARIANCE3.1. Chern number as a probe of topological order In the theory of quantum Hall effect, it is well un-derstood that the conductance quantization, is due tothe existence of certain topological invariants, so calledChern numbers. The topological invariance was first in-troduced by Avron et al. [32] in the context of Thouless et al. (TKNdN)’s original theory [33] about quantiza-tion of the conductance. TKNdN in their seminal work,showed that the Hall conductance calculated from theKubo formula can be expressed into an integral over themagnetic Brillouin zone, which shows the quantizationexplicitly. The original paper of TKNdN deals with thesingle-particle problem and Bloch waves which can not begeneralized to topological invariance. The generalizationto many-body systems has been done by Niu et al. [34]and also Tao et al. [35], by manipulating the phases de-scribing the closed boundary conditions on a torus (i.e.twist angles), both for the integer and the fractional Hallsystems. These twist angles come from natural general-ization of the closed boundary condition.To clarify the origin of these phases, we start witha single particle picture. A single particle with charge( q ) on a torus of the size ( L x , L y ) in the presence of amagnetic field B perpendicular to the torus surface, isdescribed by the Hamiltonian H s = 12 m "(cid:18) − i ~ ∂∂x − qA x (cid:19) + (cid:18) − i ~ ∂∂y − qA y (cid:19) , (8)where ~A is the corresponding vector potential ( ∂A y ∂x − ∂A x ∂y = B ). This Hamiltonian is invariant under the mag-netic translation, t s ( a ) = e i a · k s / ~ (9)where a is a vector in the plane, and k s is the pseudo-momentum, defined by k sx = − i ~ ∂∂x − qA x − qByk sy = − i ~ ∂∂y − qA y + qBx (10)The generalized boundary condition on a torus is givenby the single-particle translation t s ( L x ˆ x ) ψ ( x s , y s ) = e iθ ψ ( x s , y s ) t s ( L y ˆ y ) ψ ( x s , y s ) = e iθ ψ ( x s , y s ) (11) where θ and θ are twist angles of the boundary. The ori-gin of these phases can be understood by noting that theperiodic boundary conditions corresponds to the torus inFig. 4(a). The magnetic flux through the surface arisesfrom the field perpendicular to the surface of the torus.However, in addition to this flux, there may also be fluxesdue to a magnetic field residing inside the torus or pass-ing through the torus hole, and it is these extra fluxeswhich give rise to the phases. The extra free angles areall the same for all particles and all states in the Hilbertspace, and their time derivative can be related to thevoltage drops across the Hall device in two dimensions.The eigenstates of the Hamiltonian, including theground state will be a function of these boundary anglesΨ ( α ) ( θ , θ ). By defining some integral form of this eigen-state, one can introduce quantities, that do not dependon the details of the system, but reveal general topolog-ical features of the eigenstates.First we discuss the simplest situation, where theground state is non-degenerate, and later we shall gener-alize this to our situation with a degenerate ground state.The Chern number is in the context of quantum Hall sys-tems related to a measurable physical quantity, the Hallconductance. The boundary averaged Hall conductancefor the (non-degenerate) α th many-body eigenstate ofthe Hamiltonian is [34, 35]: σ αH = C ( α ) e /h , where theChern number C ( α ) is given by C ( α ) = 12 π Z π dθ Z π dθ ( ∂ A ( α )2 − ∂ A ( α )1 ) , (12)where A ( α ) j ( θ , θ ) is defined as a vector field based on theeigenstate Ψ ( α ) ( θ , θ ) on the boundary torus S × S by A ( α ) j ( θ , θ ) . = i h Ψ ( α ) | ∂∂θ j | Ψ ( α ) i . (13)It should be noted that the wave function Ψ ( α ) ( θ , θ )is defined up to a phase factor on the boundary-phasespace. Therefore, Ψ ( α ) ( θ , θ ) and e if ( θ ,θ ) Ψ ( α ) ( θ , θ )are physically equivalent for any smooth function f ( θ , θ ). Under this phase change, A ( α ) j ( θ , θ ) trans-forms like a gauge: A ( α ) j ( θ , θ ) → A ( α ) j ( θ , θ ) − ∂ j f ( θ , θ ) (14)Ψ ( α ) ( θ , θ ) → e if ( θ ,θ ) Ψ ( α ) ( θ , θ ) . Hence, the Chern number integral is conserved un-der this gauge transformation and it encapsulates generalproperties of the system. Chern numbers has been usedextensively in the quantum Hall literature, for character-ization of the localized and extended states (Ref. 36 andrefs. therein). In this paper, we use the Chern number asan indicator of order in the system. Moreover, it enablesus to characterize the ground state in different regimes,especially where the calculation of the overlap with theLaughlin wave function fails to give a conclusive answer. (a)(b) FIG. 4: (a) Twist angles of the toroidal boundary condition.In addition to the flux going through the surface there mayalso be a flux inside the torus or going through the hole in themiddle. When encircling these fluxes the wave function ac-quire an extra phase represented by the boundary conditionsin Eq. (11) (b) Redefining the vector potential around the sin-gularities: A j is not well-defined everywhere on the torus ofthe boundary condition. Therefore, another vector field A ′ j with different definition should be introduced around each sin-gularity ( θ n , θ n ) of A j . A j and A ′ j are related to each otherby a gauge transformation χ and the Chern number dependsonly on the loop integrals of χ around those singularities re-gions, c.f., Eq.(16). Before explaining the method for calculating the Chernnumber, we clarify some issues related to the degeneracyof the ground state. In some systems, the ground statecan be degenerate, this can be intrinsic or it can be as aresult of the topology of the system. If the ground state isdegenerate, we should generalize the simple integral formof Eq. (12) to take into account the redundancy insidethe ground state manifold. For example, in the case ofa two-fold degenerate ground state, an extra gauge free-dom related to the relative phase between two groundstates, and this freedom should be fixed. In other words,as we change the twist angles, we can not keep track ofthe evolution of both states, since one can not distinguishthem from each other. Therefore, to uniquely determinethe Chern number of the ground state(s), we should re-solve this gauge invariance, which is treated in Section3.2 and 3.3.It important to note that the degeneracy in the non-interacting regime is fundamentally different from the de-generacy in the interacting case. In the non-interactinglimit, the degeneracy can be lifted by a local perturba-tion e.g. impurity, while in the hardcore case, the de-generacy remains in the thermodynamic limit [37]. Thelatter degeneracy in the ground state is a consequenceof the global non-trivial properties of the manifold onwhich the particles move rather than related to a sym-metry breaking which happens in conventional modelse.g. Ising model. The topological degeneracy is not aconsequence of breaking of any symmetry only in theground state, instead it is the entire Hilbert space whichis split into disconnected pieces not related by any sym-metry transformation. With a general argument, Wen[38] showed that if the degeneracy of a chiral spin sys-tem moving on a simple torus is k , then it should be k g on a torus with g handles (Riemann surface with genus g ), therefore the topological detergency is an intrinsicfeature of the system. In particular, in the context ofquantum Hall effect, this multicomponent feature of theground state on a torus has a physical significance, whilethe single component ground state on a sphere bound-ary condition gives zero conductance, the torus geome-try with multicomponent ground state results in a correctconductance measured in the experiment, since the torusboundary condition is more relevant to the experiment.Changing twists angles of the boundary will rotate thesecomponents into each other and gives an overall non-zeroconductance [34].As studied in a recent work by M. Oshikawa and T.Senthil [39], as a universal feature, it has been shownthat in presence of a gap, there is a direct connectionbetween the fractionalization and the topological order.More precisely, once a system has some quasiparticleswith fractional statistics, a topological degeneracy shouldoccur, which indicates the presence of a topological or-der. Therefore, the amount of the degeneracy is relatedto the statistics of the fractionalized quasiparticles e.g.in the case of ν = 1 / 2, the two-fold degeneracy is relatedto 1/2 anyonic statistics of the corresponding quasiparti-cles. Chern number has been also studies for spin 1 / et al. [41] studiesChern number for disordered Fermi system using a non-commutative geometry.To resolve the extra gauge freedom related to the twodegenerate ground states, we consider two possibilities:I. lifting the degeneracy by adding some impurities, II.fixing the relative phase between the two states in theground state. Below, we explore both cases. In the firstcase, we introduce some fixed impurities to lift the de-generacy in the ground state for all values of the twistangles. This is an artifact of the finite size of the systemwhich we take advantage of. In the presence of pertur-bation, we show that the system has a topological orderin spite of poor overlap with the Laughlin state. In thesecond approach, we use a scheme recently proposed byHatsugai [42, 43] which is a generalized form for degen-erate manifolds. In this section, we introduce some perturbation in thefinite system in form of local potentials (similar to localimpurities in electronic systems) to split the degeneracyof the ground state and resolve the corresponding gaugeinvariance, which allows us to compute the Chern num-ber. Furthermore, the fact that we can still uniquely de-termine the Chern number in the presence of impurities,shows that the system retains its topological order, evenwhen the impurities distort its ground state wavefunctionaway from the Laughlin wavefunction.In the context of the quantum Hall effect, the conven-tional numerical calculation of various physical quantitiessuch as the energy spectrum, screening charge densityprofile, wave functions overlaps, and the density-densitycorrelation functions, can not be used for understandingthe transport properties of electrons in the presence ofimpurities (although useful for studying of isolated im-purities [44, 45]). Recently, D. N. Sheng et al. [36] cal-culated the Chern number as an unambiguous way todistinguish insulating and current carrying states in the ν = 1 / ν = 1 / 3) for a finite num-ber of electrons. The energy splitting between the lowestthree states then decreased with increasing number ofparticles, which indicates the recovery of the degeneracyin the thermodynamic limit. Moreover, the mobility gapcan be determined by looking at the energy at which theChern number drops towards zero. This energy gap iscomparable to the energy gap obtained from the experi-ment and it is not necessarily equal to the spectrum gapwhich separates the degenerate ground state from the ex-cited states This shows the significance of Chern numbercalculations for understanding these systems. In a finite system, the coupling to a single-body inter-action, e.g. impurities, can lift the degeneracy and onecan uniquely determine the Chern number for the indi-vidual states by direct integration of Eq. (12). On onehand, the impurity should be strong enough to split theenergy in the ground state (in this case E , E , where E j denotes the energy of the j th energy level) for all valuesof the twist angles. On the other hand, these impuri-ties should be weak enough so that the energy splittingin the ground state remains smaller than the thermody-namic gap (in this case E − E ≪ E − E ).To calculate the Chern number of individual level, asmentioned in the pervious section, we have to fix thephase of the wavefunction. The method that we explorein this section can be considered as a simplified versionof the general method developed by Hatsugai [43] whichwe will explore in the next section. Following Kohmoto’sprocedure [46], we assume that the ground state Ψ( θ , θ )may be expanded for all twist angles on a s -dimensionalHilbert discrete space Ψ( θ , θ ) = ( c , c , ..., c s ). If A j ( θ , θ ) in Eq. (12) is a periodic function on the torusof the boundary condition, then by application of Stoke’stheorem, the Chern number will be always zeros. Thenon-triviality (non-zero conductance in the case of quan-tum Hall system) occurs because of the zeros of the wavefunction, where the phase is not well-defined. Therefore, A ( θ , θ ) is not well defined everywhere and its flux inte-gral can be non-zero. To uniquely determine the Chernnumber, we assume that the wave function and the vec-tor field are not defined for certain points ( θ n , θ n ) in S n regions on the torus of the boundary condition. For sim-plicity, we first discuss this procedure, in the case of anon-degenerate ground state. For calculating the inte-gral, we should acquire another gauge convention for thewave function inside these S n regions, e.g. in a discretesystem, we may require an arbitrary element of the wavefunction to be always real, and thereby we can define anew vector field A ′ ( α ) j ( θ , θ ), which is well defined insidethese regions. These two vector fields differ from eachother by a gauge transformation (Fig. 4): A ( α ) j ( θ , θ ) − A ′ ( α ) j ( θ , θ ) = ∂ j χ ( θ , θ ) , (15)and the Chern number reduces to the winding numberof the gauge transformation χ ( θ , θ ) over small loopsencircling ( θ n , θ n ), i.e. ∂S n , C ( α ) = X n π I ∂S n −→∇ χ · d −→ θ . (16)The one-dimensional gauge Eq. (15) should be re-solved by making two conventions. For example, in oneconvention the first element and in the other the sec-ond element of the wavefunction in the Hilbert spaceshould be real i.e. transforming the ground state Ψinto Ψ Φ = P Φ = ΨΨ † Φ where Φ = (1 , , ..., † isa s -dimensional vector and P is a projection into theground state and similarly with the other reference vec-tor Φ ′ = (0 , , ..., † . Since the gauge that relates twovector fields is the same as the one that relates the cor-responding wavefunctions (similar to Eq.(15)), we canuniquely determine the gauge transformation function χ by evaluating Ω( θ , θ ) = e iχ = Φ † P Φ ′ . Therefore, theChern number will be equal to the number of windingsof χ around regions where Λ φ = Φ † P Φ = | c | is zero.Counting the vorticities has a vigorous computational ad-vantage over the conventional method of direct integra- tion of Eq. (12). In the direct integration, we need tochoose a large number of mesh points for the boundaryangles, because of the discrete approximation of deriva-tives in Eq. (12), and this makes the calculation computa-tionally heavy. We note that for the system on a lattice,we should exactly diagonalize the Hamiltonian which isa sparse matrix as opposed to the continuum case wherethe Hamiltonian is a dense matrix residing on a smallerprojected Hilbert space (lowest Landau level). FIG. 5: (color online) Chern number associated to low-lying energy states, in the presences of impurities. Due to the impuritypotential (a), two-fold degenerate ground state splits and the wavefunction overlap with the Laughlin state drops to 52% and65%, for the first and the second energy state, respectively. The results are for 3 atoms on a 6x6 lattice ( α = 0 . 17) in thehard-core limit. (b) Ω( θ , θ ) for the first level has no vorticity. However, for the second level, as shown in (c), Ω( θ , θ ) hasvorticity equal to one associated with regions where either Λ φ or Λ ′ φ vanishes. For removing the degeneracy, in our numerical sim-ulations, we add a small amount of impurity which ismodeled as delta function potentials located at randomsites with a random strength, of the order of the tunnel-ing energy J . This is described by a Hamiltonian of theform H = P i U imi ˆ n i , where i numerates the lattice site,ˆ n i is the atom number operator and U imi is the strengthof the impurity at site i .We choose reference states Φ and Φ ′ to be eigenvectors of the numerically diagonalized Hamiltonian at two dif-ferent twist angles. In Fig. 5, vorticities of e iχ associatedwith the first and the second energy level is depicted. Itis easy to see that the Chern number associated to thetwo ground states is one. Number of vortices may varyfor the first and second ground states, but their sum isalways equal to one. The hard-core limit ( U ≫ J ) isvery similar to the case of fractional quantum Hall ef-fect, which in the context of Hall systems, means a share0of 1 / e /h unit) for each ground state [35]. Whenthe onsite interaction strength is small ( U < J ), the ther-modynamic gap becomes comparable to the ground stateenergy splitting E − E ∼ E − E , the Chern numbercan not be uniquely determined, and the system doesn’thave topological order. On the other hand, in the limitof strong interaction ( U ≫ J ), the total Chern numberassociated to the ground states is equal to one, regardlessof the impurity configuration. Moreover, in the hard-corelimit, although the ground state is not described by theLaughlin wavefunction, since it is distorted by the impu-rity ( in our model it can be as low as 50%), the Chernnumber is unique and robust. This is an indication thatthe topological order is not related to the symmetries ofthe Hamiltonian and it is robust against arbitrary localperturbations [37]. These results indicate the existenceof a topological order in the system and robustness of theground states against local perturbations. The method developed in the previous section has thegraphical vortex representation for the Chern numberwhich makes it computationally advantageous comparedto the direct integration of Eq. (12). It can not, how-ever, be applied directly to a degenerate ground state,and therefore we had to introduce an impurity potentialwhich lifted the degeneracy. On the other hand, a signif-icant amount of impurity in the system may distort theenergy spectrum, so that the underlying physical proper-ties of the lattice and fluxes could be confounded by theartifacts due to the impurities, especially for large α . Toaddress this issues, in this section, we explore a gener-alized method of the previous section based on Refs. 42and 43, which works for a degenerate ground state.By generalizing the Chern number formalism for a de-generate ground state manifold, instead of having a sin-gle vector field A ( α ) j ( θ , θ ), a tensor field A ( α,β ) j ( θ , θ )should be defined, where α, β = 1 , , ..., q for a q -folddegenerate ground state A ( α,β ) j ( θ , θ ) . = i h Ψ ( α ) | ∂∂θ j | Ψ ( β ) i (17)Similar to the non-degenerate case, when A ( α,β ) j is notdefined, a new gauge convention should be acquired forthe regions with singularities. This gives rise to a tensorgauge transformation χ ( α,β ) ( θ , θ ) on the border of theseregions A ( α,β ) j ( θ , θ ) − A ′ ( α,β ) j ( θ , θ ) = ∂ j χ ( α,β ) ( θ , θ ) . (18)Following Hatsugai’s proposal[43] for fixing the groundstate manifold gauge, we take two reference multiplets Φand Φ ′ which are two arbitrary s × q matrices; q is theground state degeneracy (equal to 2 in our case). In our (a) π 2 π0π2 π00.51 (b) π 2 π0 π2 π00.51 (c) FIG. 6: (color online) (a) shows the argument of Ω( θ , θ )as arrows for fixed Φ and Φ ′ . (b) and (c): surface plots ofdetΛ Φ and detΛ ′ Φ (blue is lower than red). θ and θ changesfrom zero to 2 π . These plots have been produced for 3 atomswith N φ = 6 ( α = 0 . 24) in the hard-core limit on a 5x5 lattice.The total vorticity corresponding to each of the reference wavefunctions (Φ or Φ ′ ) indicates a Chern number equal to one. FIG. 7: (color online) Ω( θ , θ ) for fixed Φ and Φ ′ . θ and θ changes form zero to 2 π . This plot has been produced for4 atoms with Nφ = 8 in the hard-core limit on a 5x5 lattice( α = 0 . ′ ) indicates a Chern number equal to one. numerical simulation, we choose the multiplets to be twosets of ground state at two different twist angles far fromeach other, e.g. (0 , 0) and ( π, π ). We define an overlapmatrix as Λ φ = Φ † P Φ where P = ΨΨ † is again theprojection into the ground state multiplet, and considerthe regions where detΛ Φ or detΛ Φ ′ vanishes (similar tozeros of the wave function in the non-degenerate case).Hence, the Chern number for q degenerate states, will beequal to the total winding number of Tr χ ( α,β ) for smallneighborhoods, S n , in which detΛ Φ vanishes C (1 , , ..., q ) = X n π I ∂S n −→∇ Tr χ ( α,β ) · d −→ θ (19)which is the same as the number of vortices of Ω(Φ , Φ ′ ) =det(Φ † P Φ ′ ). It should be noted that the zeros of det Λ Φ and detΛ ′ Φ should not coincide in order to uniquely de-termine the total vorticity. In Fig. 6, we have plotted Ω,detΛ Φ , and detΛ Φ ′ , found by numerical diagonalizationof the Hamiltonian for a mesh (30 × 30) of winding an-gles θ and θ . In this figure, the Chern number can bedetermined be counting the number of vortices and it isreadily seen that the winding number is equal to one forthe corresponding zeros of detΛ Φ (or detΛ Φ ′ ).We have calculated the Chern number for fixed ν = 1 / α ’s by the approach described above. Theresult is shown in Table I. For low α ≪ 1, we know fromSec. 2 that the ground state is the Laughlin state and weexpect to get a Chern number equal to one. For higher α , the lattice structure becomes more apparent and theoverlap with the Laughlin state decreases. However, inour calculation, the ground state remains two-fold degen-erate and it turns out that the ground state Chern num-ber tends to remain equal to one before reaching somecritical α c ≃ . 4. Hence, also in this regime we expect tohave similar topological order and fractional statistics of Atoms Lattice α Chern/state Overlap3 6x6 .17 1/2 0.994 6x6 .22 1/2 0.983 5x5 .24 1/2 0.983 4x5 .3 1/2 0.914 5x5 .32 1/2 0.783 4x4 .375 1/2 0.29TABLE I: Chern Number for different configurations in thehard-core limit for fixed filling factor ν = 1 / 2. The Laughlinstate overlap is shown in the last column. although it deviatesfrom the Laughlin state. Although the ground state deviatesfrom the Laughlin state, the Chern number remains equal toone half per state before reaching some critical α c ≃ . the excitations above these states on the lattice.For the arguments above to be applicable, it is essen-tial that we can uniquely identify a two-fold degenerateground state which is well separated from all higher ly-ing states. For higher flux densities, α > α c , the two-foldground state degeneracy is no longer valid everywhereon the torus of the boundary condition. In this regime,the issue of degeneracy is more subtle, and the finite sizeeffect becomes significant. The translational symmetryargument [28], which was used in the Section 2, is notapplicable on a lattice and as pointed out by Kol et al .[47] the degeneracy of the ground state may vary period-ically with the system size. Some of the gaps which ap-pear in the calculation may be due to the finite size andvanish in the thermodynamic limit, whereas others mayrepresent real energy gaps which are still present in thethermodynamic limit. To investigate this, we study theground state degeneracy as a function of boundary an-gles ( θ , θ ) which are not physical observable and there-fore the degeneracy in thermodynamic limit should notdepend on their value. In particular, Fig. 8 shows theenergy levels of five particles at α = 0 . θ = θ = 0), while they touch each otherat ( θ = θ = π ). We have observed similar behavior fordifferent number of particles and lattice sizes e.g. 3 and4 atoms at α = 0 . 5. In this case, the system seems to nothave a two-fold degeneracy. Therefore, the ground stateenters a different regime which is a subject for furtherinvestigation.For having the topological order, it is not necessaryto be in the hard-core limit. Even at finite interactionstrength U ∼ Jα , we have observed the same topologicalorder with the help of the Chern number calculation. If U gets further smaller, the energy gap above the groundstate diminishes (as seen in Sec. 2) and the topologicalorder disappears.We conclude that the Chern number can be unambigu-ously calculated for the ground state of the system in aregime where Laughlin’s description is not appropriate2 π 2 π0π2 π−12−11.5−11 θ θ E / J (a) E / J (b) FIG. 8: (color online) Low-lying energy levels as a function oftwist angles. For high α the degeneracy of the ground state isa function of twist angles. The shown results are for 5 atomson a 5x5 lattice i.e. α = 0 . θ = π for sevenlowest energy levels. The first and the second energy levelsget close to each other at θ = θ = π . for the lattice. The non-zero Chern number of a two-folddegenerate ground state, in this case equal to one halfper state, is a direct indication of the topological orderof the system. 4. EXTENSION OF THE MODEL In Sections 2 and 3 above, we have investigated theconditions under which the fractional quantum Hall ef-fect may be realized for particles on a lattice. The mo-tivation for this study is the possibility to generate thequantum Hall effect with ultra cold atoms in an opti-cal lattice but the results of these sections are applicableregardless of the method used to attain this situation. In this and the following sections, we investigate somequestions which are of particular relevance to ultra coldatoms in an optical lattice. First, we introduce a longrange, e.g., dipole-dipole, interaction which turn out toincrease the energy gap and thereby stabilizes the quan-tum Hall states. We then turn to the case of ν = 1 / In an experimental realization of the quantum Hall ef-fect on a lattice, it is desirable to have as large an en-ergy gap as possible in order to be insensitive to exter-nal perturbations. So far, we have studied effect of theshort range interaction, and we have shown that the gapincreases with increasing interaction strength, but thevalue of the gap saturates when the short range interac-tion becomes comparable to the tunneling energy J .In this section, we explore the possibility of increasingthe gap by adding a long-range repulsive dipole-dipoleinteraction to the system. Previously, such dipole-dipoleinteraction has also been studied in Ref.onlinecitecooper2005 as a method to achieve Read-Rezayistates [48] of rapidly rotating cold trapped atoms for ν = 3 / H d − d = U dipole X ≤ i 0, but ex-perimentally time varying fields may be introduced whicheffectively change the sign of the interaction[50]. Forcompleteness, we shall therefore both investigate posi-tive and negative U dipole , but the repulsive interactioncorresponding to static dipoles will be the most desirablesituation since it stabilizes the quantum Hall states.Experimentally the dipole-dipole interaction will nat-urally be present in the recently realized Bose-Einsteincondensation of Chromium[51] which has a large mag-netic moment. However, for a lattice realization, polarmolecules which have strong permanent electric dipolemoments is a more promising candidate. For typical po-lar molecules with the electric moment ℘ ∼ a ∼ . µ m, U dipole can be up to3 (a)(b) FIG. 9: (color online) (a) The overlap of the ground statewith the Laughlin wavefunction (dashed lines) and four lowlying energies of the system (solid lines) versus the dipole-dipole interaction for four atoms on a 6x6 lattice. (b) Gapenhancement for a fixed repulsive dipole-dipole interactionstrength U dipole = 5 J versus α . The results are shown for N = 2( (cid:3) ), N = 3( ∗ ), N = 4(+) few kHz, an order of magnitude greater than the typicaltunneling J/ π ~ which can be few hundreds of Hz [1].To study the effect of the dipole-dipole interaction, weagain numerically diagonalize the Hamiltonian for a fewhardcore bosons ( U ≫ J ), in the dilute regime α . U dipole ≫ J ), theground state wave function deviates from the Laughlinwavefunction, but the topological order remains the sameas for the system without dipole interaction. We veri-fied this by calculating the Chern number as explainedin Sec. 3 for different values of the dipole-dipole interac-tion strength U dipole and observed that the total Chernnumber of the two-fold degenerate ground state is equalto one. Moreover, as it is shown in Fig. 9 (b) addingsuch an interaction can increase the gap: the lower curvecorresponds to the hard-core limit discussed in the pre-vious work [10] and the upper curve corresponds to thesystem including the dipole-dipole interaction. This en- hancement varies linearly with the flux density α in adilute lattice and doesn’t depend on the number of par-ticles and consequently, it is expected to behave similarlyin the thermodynamic limit.One of the impediment of the experimental realizationof Quantum Hall state is the smallness of the gap whichcan be improved by adding dipole-dipole interaction. Inthis section, we showed that this improvement is possibleand moreover, by Chern number evaluation, we verifiedthat adding dipole interaction doesn’t change the topo-logical behavior of the ground state manifold. ν = 1 / So far we have concentrated on the case of ν = 1 / 2. Inthis section, we briefly investigate the case of ν = 1 / q , where ν = 1 /q . Following Haldane’sargument[28], due to the center of mass motion, the de-generacy of the ground state is expected to be q -fold ona torus. Similar to the case of ν = 1 / 2, the Laughlinwavefunction should be a suitable description for any q provided that the magnetic field is weak so that we areclose to the continuum limit, i.e. α ≪ 1. Also the Chernnumber is expected to be equal to one for the q-fold de-generate ground state, which in the context of quantumHall effect means a share of 1 /q of the conduction quan-tum e /h for each state in the q-fold degenerate groundstate manifold.We have done both overlap and the Chern number cal-culation to check those premises. In the case of ν = 1 / α . . 1. The averagewave function overlap of four lowest energy eigenstateswith the Laughlin wavefunction is depicted in figure 10,where we have used a generalization of the Laughlinwavefunction for periodic boundary conditions similar toEq. (4) [27].We observe that the Laughlin wavefunction is a reliabledescription of the system with ν = 1 / α . . 1) compared to ν = 1 / α . . 3. Contrary to ν =1 / 2, where the gap is a fraction of the tunneling energy J , the gap for ν = 1 / m = 4 state is thus equal to the m = 2 state.It thus costs a negligible energy to compress the ν = 1 / ν = 1 / ν = 1 / FIG. 10: (color online) The overlap of the first four low-lying energy states with the Laughlin wavefunction for thecase of ν = 1 / U dipole = 5 J ). The Laughlin state is only a good descrip-tion for a more dilute lattice α . . ν = 1 / α . . see a non vanishing gap.Even though that with short range interactions, thegap is very small in our numerical calculations, it is stillsufficiently large that it allows us to unambiguously de-termine the Chern number for the ground state mani-fold as described in Sec. 3.3. As expected the calculationshows that the Chern number is equal to one correspond-ing to a four-fold degenerate ground state consistent withthe generalization of the fermionic case in the fractionalquantum Hall theory [34, 35]. In Fig. 10, the overlapof the first four lowest energy state with the Laughlinwavefunction is depicted. In the absence of the dipole in-teraction, the ground state overlap is significant only for α . . 1, however, by adding a moderate dipole interac-tion ( U dipole = 5 J ), the overlap becomes more significantfor a larger interval of the flux density, i.e. α . . 25. Thisis due to the fact that state with lower density becomemore favorable in the presence of a long-range repulsiveinteraction.We observed that adding a dipole interaction wouldlead to an improvement of the gap for ν = 1 / α in the case of ν = 1 / 4. There-fore this long-range interaction can be used as an tool forstabilizing the ground state and make the realization ofthese quantum states, experimentally more feasible. 5. DETECTION OF THE QUANTUM HALLSTATE In an experimental realization of the quantum Hallstates, it is essential to have an experimental probe whichcan verify that the desired states were produced. In mostexperiments with cold trapped atoms, the state of thesystem is probed by releasing the atoms from the trap and imaging the momentum distribution. In Ref. 10, itwas shown that this technique provide some indicationof the dynamics in the system. This measurement tech-nique, however, only provides limited information, sinceit only measures the single particle density matrix, andprovides no information about the correlations betweenthe particles. In Refs. 52 and 53 a more advanced mea-surement techniques were proposed, where the particlecorrelation is obtained by looking at the correlations inthe expansion images. In this section, we study Braggscattering as an alternative measurement strategy whichreveals the excitation spectrum of the quantum system.In Ref. 54 bosonic quantum Hall system responses to aperturbative potential is studied. We focus on Braggscattering where two momentum states of the same in-ternal ground state are connected by a stimulated two-photon process [55]. By setting up two laser beams withfrequencies ω and ω and wave vectors ~k and ~k in theplane of the original lattice, a running optical superlat-tice will be produced with frequency ω − ω and wavevector ~k − ~k . (Both frequencies ω and ω should beclose to an internal electronic dipole transition in theatoms). The beams should be weak and sufficiently de-tuned so that direct photon transitions are negligible, i.e. E , E , γ ≪ ω − ω , ω − ω , where ω is the frequencyof the transition, γ is the corresponding spontaneous de-cay rate and E , E are the Rabi frequencies related to thelaser-atom coupling. In this perturbative regime, the in-elastic scattering of photons will be suppressed; the atomstarts from the internal ground state, absorbs one pho-ton from e.g. beam 1 by going to a virtual excited stateand then emits another photon into beam 2 by return-ing to its internal ground state. After this process, thesystem has acquired an energy equal to ~ ω = ~ ( ω − ω )and a momentum kick equal to ~q = ~k − ~k . There-fore, the overall effect of the recoil process is a movingAC Stark shift as a perturbing moving potential, andthe effective Hamiltonian represents the exchange of thetwo-photon recoil momentum and the energy differenceto the system and is proportional to the local density i.e. H ∝ ρ ( r ) e − i ( ω − ω ) t + i ( ~k − ~k ) .~r + c.c. .This process can be used to probe density fluctuationsof the system and thus to measure directly the dynamicstructure factor S ( q, ω ) and static structure factor S ( q ).This kind of spectroscopy has been studied for a BEC ina trap by Blakie et al. [56] and Zambelli et al. [57] andhas been realized experimentally in Refs. 58, 59, 60, 61,62, 63.Also, Bragg spectroscopy in an optical lattice is dis-cussed in Ref. 64 in a mean-field approach and also inRef. 65 as a probe of the Mott-insulator excitation spec-trum. On the other hand, in the context of quantumHall effect, the static structure factor has been studiedfor probing the magnetoroton excitations [66] and chargedensity waves [67]. The dynamic and static structurefactors are given respectively as5 (a)(b) FIG. 11: (color online) (a) Structure factor and (b) energyspectrum for a 11x11 lattice with 3 atoms on a torus. Pointsshows the momentums allowed by the boundary conditions.The dotted line in (b) shows for comparison the low energyspectrum of a free particle which equals J ( qa ) . S ( ~q, ω ) = X n, |h n | ρ † ( ~q ) | i| δ ( ω − E n + E ) , (21) S ( ~q ) = X n, |h n | ρ † ( ~q ) | i| = X n, |h n | X ~r i e i~q · ~r i | i| , (22)where the density fluctuation operator is defined as ρ † ( ~q ) = P m,n A ~q ( m, n ) c † m c n and the coefficient are de-fined as Fourier transforms of the Wannier functions: A ~q ( m, n ) = R d ~r e i~q · ~r φ ∗ ( ~r − ~r m ) φ ( ~r − ~r n ), where theWannier function φ ( ~r − ~r n ) is the wave function of anatom localized on a site centered at ~r n . Below, we focuson deep optical lattices, where A ~q ( m, n ) = e i~q · ~r m δ m,n .In the structure factor, there is a sum over the ex-cited states | n i and ground states | i and the self-termis thus excluded. The ground state on a torus is two-fold degenerate and therefore in our numerics, we addthe contribution of both.Since we are working on a discrete lattice, there willbe a cut-off in the allowed momentum given by the lat-tice spacing q max = π/a , where a is the distance between lattice sites. Fig. 11(a) shows the structure factor forthe case ν = 1 / α calculated from our nu-merical diagonalizations. In the data presented here, wehave chosen −→ q = q ˆ x and but the result should be simi-lar in other directions in the lattice plane. We see that S ( q ) is modulated at a momentum corresponding to themagnetic length. For the parameters that we have in-vestigated, the general features of the structure factor isindependent of the size of the system.We obtain the excitation spectrum shown in Fig. 11(b) similar to Ref. 66 by the Feynman-Bijl approxima-tion. In the continuum limit ( α ≪ ∝ ρ k | i . Therefore, the variational es-timate for the excitation energy is ω ( q ) ≃ ~ q / mS ( q ).At zero momentum and at the momentum correspondingto the magnetic length, there are gaps, and we also ob-serve a deviation from the free particle spectrum similarto the magneoroton case as a reminiscent of the Wignercrystal. It should be noted that the deviation does notdepend on the size or the number of particles in the sys-tem. As clearly seen in the Fig. 11 the energy spectrumand structure factor deviate from those of free particles,therefore, it could be used as an experimental probe ofthe system.The structure factor and excitation spectrum implysome general features that are very different from thatof the Mott-insulator and superfluid states, and can beused a powerful experimental indication of the quantumHall states. 6. GENERATING MAGNETIC HAMILTONIANFOR NEUTRAL ATOMS ON A LATTICE Recently, there has been several proposals for produc-ing artificial magnetic field for neutral atoms in opti-cal lattices [10, 21, 22], however, the implementation ofeach of them is still experimentally demanding. Recently,there has been an experimental demonstration of a rotat-ing optical lattice [23] which is equivalent to an effectivemagnetic field (see below). The technique used in thisexperiment, however, generates a lattice with a large lat-tice spacing, because it uses laser beam which are not ex-actly counter propagating. This longer spacing reducesthe energy scale in the lattice and thereby also reducesquantities such as energy gaps. Here, we shall now in-troduce an alternative method for generating a rotatingoptical lattice, which does not result in an increased lat-tice spacing. This method consists of rotating the opticallattice by manipulating laser beams.In a frame of reference rotating with angular velocity ω around the z − axis , the Hamiltonian for a particle ofmass m in an (planar) harmonic trap of natural frequency ω is6 FIG. 12: Proposal for realizing a rotating optical lattice. FourAOMs (black boxes) changes the direction of the lattice beam,which are subsequently focussed in the middle of the setup byfour lenses (grey). Simultaneously varying the four diffractionangles in the AOMs will generate a rotating optical lattice. H = p m + 12 mω ( x + y ) − ω ˆ z.r × p = ( p − mω ˆ z × r ) m + 12 m ( ω − ω )( x + y )(23)At resonance ω = ω the form is equivalent to the Hamil-tonian of a particle of charge q experiencing an effectivemagnetic field B = ▽× ( mω ˆ z × r/q ) = (2 mω/q )ˆ z . There-fore, by simply rotating the optical lattice, we can mimicthe magnetic field for neutral atoms.To rotate the optical lattice, we propose to set up fouracousto-optic modulators (AOM) and four focusing com-posite lenses as shown in the Fig. 12. By sweeping theacoustic wave frequency, the beams can be focused andmake a rotating optical lattice.In an AOM, for the first order diffracted light we havesin θ B = λ , where Λ is the wavelength of sound inthe medium, λ is the light wavelength and θ B is halfof the angle between a diffracted beam and the non-diffracted beam (Fig. 12). By increasing the frequency ofthe acoustic wave, the diffraction angle increases. How-ever, the beam should be focused by a large aperture lensso that it always passes the region where we want to makethe optical lattice. By focusing a similar but counter-propagating beam, we can make a rotating standing wave(Fig. 12). By repeating the same configuration in thetransverse direction, we can make a rotating optical lat-tice. In particular, if the AOM is far from the compositelenses, D ≫ d , then x/D = λ/ Λ and x/d = tan θ where − π/ ≤ ( θ = ωt ) ≤ π/ 4, where the parameters are de-fined in Fig. 12. If we consider a square lattice withdimensions N x = N y = N , the number of magnetic fluxgiven by rotation is N φ = BA Φ = π N ωω r (24)where ω r = ~ k / M is the atomic recoil frequency.On the other hand, the upper limit for the magneticfield comes from the single Bloch band approximationwhich we made in writing the Hamiltonian for the opti-cal lattice. In order for the particles to remain in the firstband, the traveling lattice beams should move them adi-abatically. From Ref. 68, the adiabaticity condition for amoving lattice with an acceleration η equal to ω N λ/ mηλ ≪ ~ ω p ω r , where ω p is the frequencydifference between the first and the second band in thelattice. This puts a limit on how large the lattice canbecome N ≪ ω p ω ω r .Hence, for ν = 1 / N N ∼ and a typical recoil frequency ω r = (2 π )4 kHz, onecan enter the regime of fractional quantum Hall effect byrotating the lattice at ω ∼ (2 π )650 Hz. If a deep opticallattice is used e.g. ω p ∼ ω r , the adiabaticity conditionis easily satisfied for a lattice of size N ∼ θ = π/ 7. CONCLUSIONS An extended study of the realization of the fractionalquantum Hall states in optical lattices has been pre-sented. We showed that a Hamiltonian similar to that ofa magnetic field for charged particles can be constructedfor neutral ultra cold atoms and molecules confined inan optical lattice. By adding an onsite interaction forthe case of ν = 1 / 2, an energy gap develops betweenthe ground state and the first excited state, which in-creases linearly as αU and saturates to its value in thehardcore limit U ≫ J . We learned that the Laughlinwavefunction is a reliable description of the system forlow flux densities α . . 25. However, for higher α ’s, thelattice structure becomes more pronounced and a bet-ter description of the system can be carried out by in-vestigating the Chern number associated to the groundstate manifold. The Chern number indicates that the sys-tem has topological order up to some critical flux density α c ≃ . 4, where the properties of the ground state man-ifold starts to change. We have also studied ν = 1 / ν = 1 / 2, the Laughlin wavefunctiononly describes the ground state for lower values of the flux α . . 1. We showed that a dipole-dipole interaction canenhance the gap and stabilize the system, and therefore7make the ground state more experimentally realizable.Bragg spectroscopy has been studied as a potential ex-perimental tool to diagnose theses incompressible states.Characterization of the ground state by evaluating theChern number, developed in Sec. 3.3, can be general-ized to other interesting many-body systems where theconventional overlap calculation fails to work. In partic-ular, this method can be applied to ground states withnon-Abelian statistics which are appealing candidates forfault-tolerant quantum computation. Acknowledgments We thank K. Yang, M. Greiner, S. Girvin and J. I.Cirac for fruitful discussions. This work was partiallysupported by the NSF Career award, Packard Founda-tion, AFSOR and the Danish Natural Science ResearchCouncil. [1] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, andI. Bloch, Nature , 39 (2002).[2] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. 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