Characterization of topological states on a lattice with Chern number
Mohammad Hafezi, Anders S. Sorensen, Mikhail D. Lukin, Eugene Demler
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Characterization of topological states on a lattice with Chern number
Mohammad Hafezi , Anders S. Sørensen , Mikhail D. Lukin , Eugene Demler Physics department, Harvard university, Cambridge Massachusetts 02138 and QUANTOP, Danish National Research Foundation Centre of Quantum Optics,Niels Bohr Institute, DK-2100 Copenhagen Ø, Denmark
We study Chern numbers to characterize the ground state of strongly interacting systems on alattice. This method allows us to perform a numerical characterization of bosonic fractional quantumHall (FQH) states on a lattice where conventional overlap calculation with known continuum casesuch as Laughlin state, breaks down due to the lattice structure or dipole-dipole interaction. Thenon-vanishing Chern number indicates the existence of a topological order in the degenerate groundstate manifold.
PACS numbers: 03.75.Lm,73.43.-f
One of the most dramatic manifestations of interac-tions in many-body systems is the appearance of newquantum states of matter. Often such states can becharacterized by an order parameter and spontaneousbreaking of some global symmetries. The ‘smoking gun’evidence of such states is the appearance of Goldstonemodes of spontaneously broken symmetries. However,some types of quantum many-body phases can not becharacterized by a local order parameter. Examples canbe found in FQH systems[1], lattice gauge theories[2],and spin liquid states[3]. Such states can be charac-terized by the topological order[4] which encompassesglobal geometrical properties such as ground state de-generacies on non-trivial manifolds[5]. Topologically or-dered states often exhibit fractional excitations[6] andhave been proposed as a basis of a new approach to quan-tum computations[7]. However, in many cases, identify-ing a topologically ordered state is a challenging taskeven for theoretical analysis. Given an exact wavefunc-tion of the ground state in a finite system, how can onetell whether it describes a FQH phase of a 2D electrongas or a spin liquid phase on a lattice? One promisingdirection to identifying topological order is based on theChern number calculations[8]. The idea of this approachis to relate the topological order to the geometrical phaseof the many-body wavefunction under the change of theboundary conditions[9].Important work of Berry [10] and Simon [11] initi-ated the investigation on geometrical phase factors andsince then the field has been extensively studied in dif-ferent contexts – for a review see, for example, [12]. Inquantum Hall (QH) systems, early works on the Chernnumber analysis [13] is focused on the Hall conductanceand robustness of QH states against changes in the bandstructure[14] and the presence of disorder[5, 15]. Cur-rently, there is also considerable interest in understandingFQH states in the presence of a strong periodic potential.Such systems are important in several contexts includinganyonic spin states [16], vortex liquid states [17], and ul-tracold atoms in optical lattices[18, 19, 20, 21, 22] whichare promising candidates for an experimental realization.In this letter, we investigate a novel procedure forcalculating Chern numbers and demonstrate that this method provides insight into the topological order of theground state in regimes where other methods fails to pro-vide a definite answer for the nature of the ground statewavefunction. In particular, we study a fractional quan-tum Hall system with bosons on a lattice with a fillingfactor of ν = 1 /
2, where ν is the ratio of the number ofmagnetic flux quanta to the number of particles. In thecontinuum limit, where the flux-fraction through eachplaquette α is very small ( α ≪ α & .
25, the Laughlin wavefunction ceases to be agood description of the system, indicated by a decreas-ing overlap between the ground state and the Laughlinwavefunction. From this study, it is unclear whether thisrepresented a change in the nature of the ground state,or just that the lattice structure distorts the state. Here,we use the Chern number calculation to provide an un-ambiguous characterization of the ground state even out-side the regime where there is a significant overlap withthe Laughlin wavefunction. In particular, we show thatthe Chern number and hence the topological order of thesystem remains undisturbed until α . . q on a torus T ( L x × L y ) in the presence of a magneticfield B perpendicular to the surface. The correspondingHamiltonian is invariant under the magnetic translationof single particle s , t s ( a ) = e i a · k s / ~ where a is a vector onthe torus, and k s is the pseudo-momentum of particle s ,defined by k x ( y ) = − i ~ ∂∂x ( y ) − qA x ( y ) ∓ qBy ( x ) in x and(y) direction, respectively, and ~A is the correspondingvector potential. The generalized boundary condition isgiven by the translation, t s ( L i ˆ i ) ψ ( x s , y s ) = e iθ i ψ ( x s , y s ),where ( i = 1 ,
2) refer to two directions (x,y) on the torus T and the θ i ’s are twist angles of the boundary (Fig.1a).The magnetic phase through each plaquette (2 πα ) arisesfrom the field perpendicular to the surface of the torus.The Chern number for non-degenerate state α is definedby, FIG. 1: (a) Twist angles of the boundary condition as the re-sult of two magnetic fluxes threaded the torus (b) Redefiningthe vector potential around the singularities: A j is not well-defined everywhere on the torus of the boundary condition.Therefore, another vector field A ′ j with different definitionshould be introduced around each singularity ( θ n , θ n ) of A j .However, A j and A ′ j are related to each other with a gaugetransformation χ and the Chern number depends only on theloop integrals of χ around those singularity regions. C ( α ) = 12 π Z π dθ Z π dθ ( ∂ A ( α )2 − ∂ A ( α )1 ) (1)where A ( α ) j ( θ , θ ) is defined as a vector field basedon the eigenstate Ψ ( α ) ( θ , θ ) on T by A ( α ) j ( θ , θ ) . = i h Ψ ( α ) | ∂∂θ j | Ψ ( α ) i .In the context of QH systems, the time derivative oftwist angles could be considered as voltage drops acrossthe Hall device in two dimensions and the boundary av-eraged Hall conductance of the any state is related to theChern number of that state [9]: σ αH = C ( α ) e /h .The non-trivial behavior (non-zero conductance in thecase of quantum Hall system) occurs because of singulari-ties of the vector field. If for a given non-degenerate state the corresponding vector field is not defined for certainangles ( θ n , θ n ) in S n regions (Fig.1b), then we should in-troduce a new well-defined vector field A ′ ( α ) j ( θ , θ ), in-side those regions. These two vector fields differ fromeach other by a gauge transformation, A ( α ) j ( θ , θ ) −A ′ ( α ) j ( θ , θ ) = ∂ j χ ( θ , θ ) and the Chern number re-duces to the winding number of the gauge transformation χ ( θ , θ ) over small loops encircling ( θ n , θ n ), i.e. ∂S n [13], C ( α ) = X n π I ∂S n −→∇ χ · d −→ θ . (2)For the case of degenerate ground state a general-ization of the above argument can be made, where in-stead of having a single vector field A ( α ) j ( θ , θ ), a ten-sor field A ( α,β ) j ( θ , θ ) should be defined, α, β = 1 ...q for a q -fold degenerate ground state: A ( α,β ) j ( θ , θ ) . = i h Ψ ( α ) | ∂∂θ j | Ψ ( β ) i .Therefore, when A ( α,β ) j is not defined, similar to thenon-degenerate case, a new gauge convention shouldbe acquired for those regions with singularities. Thisgives rise to a tensor gauge transformation on the bor-der of these regions, ∂ j χ ( α,β ) ( θ , θ ) = A ( α,β ) j ( θ , θ ) −A ′ ( α,β ) j ( θ , θ ) and consequently the Chern number isgiven by the trace of the tensor χ , C (1 , , ..., q ) = X n π I ∂S n −→∇ Tr χ ( α,β ) · d −→ θ . (3)We focus on a system of bosons on a square latticedescribed by the Hamiltonian [19]: 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 − , (4)where J is the hopping energy between two neighbor-ing sites, U is the on-site interaction energy, and 2 πα isthe phase acquired by a particle going around a plaque-tte. We concentrate on the hardcore limit ( U ≫ J ) and ν = 1 / ν is the ratio of the number of particles tothe total number of flux in the system. The experimen-tal proposal for realizing such a Hamiltonian for atomicgases confined in an optical lattice has already been in-vestigated [18, 19]. The ground state of the system forvery dilute lattice α . . α increases thelattice structure becomes more apparent and the overlapwith Laughlin wavefunction breaks down. However, bynumerical calculation, we show that Chern number char-acterizes system better and remains the same, i.e. 1/2 FIG. 2: Low-lying energy levels as a function of twist angles.For finite α the ground state energy oscillates as a functionof twist angles and for high α & . α = 0 .
32 (4atoms on a 5x5 lattice) while (b) shows the five lowest energylevels for α = 0 . θ = π and θ is varied from zero to 2 π . for each state in the ground state manifold, for systemswith higher flux density α . . α ≪
1, i.e. themagnetic length is much larger than the lattice spacing,and hence the lattice Hamiltonian approaches the con-tinuum limit. According to Haldane [23], the magnetictranslational symmetry of the center of mass results ina two-fold degeneracy of the ground state for ν = 1 / U ≪ J ), the degeneracy can belifted by a local perturbation e.g. an impurity, while inthe hardcore case, the degeneracy remains in the thermo-dynamic limit [5]. The latter degeneracy is a consequenceof the global non-trivial properties of the manifold onwhich the particles move rather than symmetries of theHamiltonian (e.g. the Ising model)[4]. Recently, it wasshown [6] that in presence of a gap, there is a direct con-nection between the fractionalization and the topologicaldegeneracy. In particular, the amount of the degeneracyis related to the statistics of the fractionalized quasipar- ticles e.g. in the case of ν = 1 /
2, the two-fold degeneracyis related to 1/2 anyonic statistics of the correspondingquasiparticles.The ground state degeneracy prevents the direct inte-gration of Eq. (1) since wavefunctions would mix togetherwhen twist angles vary. Therefore, one has to use Eq. (3)and also resolve the extra gauge related to the groundstate. We can consider two possibilities: fixing the rel-ative phase between the two states in the ground state,or lifting the degeneracy by adding some impurities. Inthe latter case, we can show that the system has a topo-logical order in spite of poor overlap with the Laughlinstate [19]. On the other hand, a significant amount ofimpurity in the system may distort the energy spectrum,so that the underlying physical properties of the latticeand fluxes could be confounded by the artifacts due tothe impurities, especially for large α . Therefore, in thisletter we focus on the degenerate case.We start with the simple case of a non-degenerateground state on a discrete s -dimensional Hilbert space,Ψ( θ , θ ) = ( c , c , ..., c s ). The one-dimensional gaugecan be resolved by making two conventions: in oneconvention the first element and in the other the sec-ond element of the wavefunction in the Hilbert spaceshould be real i.e. we transform 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 , , ..., † . Hence, we can uniquely determinethe gauge χ which relates the two corresponding vectorfields: e iχ = Φ † P Φ ′ . Therefore, the Chern number willbe equal to the number of vorticities of χ around regionswhere Λ φ = Φ † P Φ = | c | is zero.For fixing the q -dimensional ground state manifoldgauge, we take two reference multiplets Φ and Φ ′ whichare two s × q matrices ( q = in our case). We definean overlap matrix as Λ φ = Φ † P Φ, and consider the re-gions where det Λ Φ or det Λ ′ Φ vanishes (similar to zerosof the wave function in the non-degenerate case). Hence,the Chern number for q degenerate states will be equal tothe total winding number of Tr χ ( α,β ) for small neighbor-hoods S n , in which detΛ Φ vanishes. It should be notedthat the zeros of detΛ Φ and detΛ ′ Φ should not coincidein order to uniquely determine the total vorticity. In ournumerical calculation, we choose multiplets Φ and Φ ′ tobe two sets of two degenerate ground states at two dif-ferent twist angles far apart e.g. (0 ,
0) and ( π, π ). InFig. 3, we have plotted Ω = det(Φ † P Φ ′ ), detΛ Φ , anddetΛ ′ Φ , found by numerical diagonalization of the Hamil-tonian over a grid (30 ×
30) of twist angles θ and θ .The Chern number can be determined by counting thenumber of vortices and it is readily seen that the wind-ing number is equal to one for the corresponding zeros ofdetΛ ′ Φ and detΛ Φ .We have calculated the Chern number for fixed ν = 1 / α ’s by the method described above. The re-sult is shown in Tab.I. For α ≪
1, we know from previ-ous calculation [18] that the ground state is the Laughlin π 2 π0 π2 π (b) π 2 π0π2 π (c)
FIG. 3: (a) Ω( θ , θ ) for fixed Φ and Φ ′ . θ and θ changesform zero to 2 π . This plot has been produced for 4 atomsin the hard-core limit on a 5x5 lattice ( α = 0 . Φ and detΛ ′ Φ (blue is lower thanred). θ and θ changes form zero to 2 π . The total vorticitycorresponding to each of trial function (Φ or Φ ′ ) indicates aChern number equal to one for the two dimensional groundstate manifold. state and we expect to obtain a Chern number equal to1/2 for each state i.e. total Chern number equal to one.For higher α , the lattice structure becomes more appar-ent and the overlap with the Laughlin state decreasesrapidly. However, in our calculation, the ground stateremains two-fold degenerate and the associated Chern Atoms Lattice α gap /J Chern/state
Overlap3 6x6 .17 0.24 1/2 0.994 6x6 .22 0.24 1/2 0.983 5x5 .24 0.23 1/2 0.983 4x5 .3 0.18 1/2 0.914 5x5 .32 0.15 1/2 0.783 4x4 .375 0.03 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 ≃ . number remains equal to one before reaching some criti-cal α c ≃ .
4. Hence, we expect to have similar topolog-ical order and fractional statistics of the excitations onthe lattice in this regime.For higher flux densities, α > α c , the two-fold groundstate degeneracy is no longer valid everywhere on thetorus of the boundary condition. In this regime, the is-sue of degeneracy is more subtle, and finite size effectbecomes significant. The translational symmetry argu-ment [23] is no longer valid and the degeneracy of theground state varies periodically with the system size [25].Some gaps might be due to the finite size and vanish inthe thermodynamic limit. To investigate this, we studythe ground state degeneracy as a function of boundaryangles ( θ , θ ) which are not physical observable. There-fore, the degeneracy in thermodynamic limit should notdepend on the their value. In particular, Fig. 2b showsthe energy levels of five particles at α = 0 . θ = θ = 0), while they touch each other at( θ = θ = π ). We have observed similar behavior fordifferent number of particles and lattice sizes e.g. 3 and4 atoms at α = 0 .
5. Therefore, the ground state enters adifferent regime which is a subject for further investiga-tion. Existence of the topological order does not requirea very strong interaction i.e. hard-core limit. Even atfinite interaction strength U ∼ Jα , we have observed thesame topological order with the help of the Chern num-ber calculation. If U gets further smaller, the energy gapabove the ground state diminishes [19] and the topologi-cal order disappears.One of the impediment of the experimental realizationof Quantum Hall state is the smallness of the gap whichcan be improved in presence of the dipole-dipole inter-action [19]. The dipole interaction can be representedas extra term P ij U d n i n j / | r i − r j | in the HamiltonianEq. (4), where n i is the number of particles at location r i in the units of lattice spacing and U d is the strengthof the interaction. The magnetic dipole-dipole interac-tion has been achieved in Bose-Einstein condensation ofChromium [26], however, for a lattice realization, polarmolecules with strong permanent electric dipole momentsare more promising candidates, where the dipole interac-tion can be an order of magnitude greater than the tun-neling energy. In the presence of such strong long-rangeinteraction, the ground state deviates from the conven-tional FQH state even in the continuum case (i.e. foreven α < .
2, the overlap with the Laughlin wavefunc-tion decreases by increasing the strength of the dipoleinteraction). However, by evaluating Chern number, weare able to identify the topological order of the system,that turns out to be intact i.e. Chern number equal toone for the two-fold degenerate ground state.In conclusion, we have investigated a method to un-ambiguously calculate the Chern number for the ground state of a system. For the FQHE system on a lattice thatwe have investigated, the Laughlin wavefunction ceasesto be a good description of the ground state for highfluxes α & .
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