Winding number order in the Haldane model with interactions
WWinding number order in the Haldane model withinteractions
E. Alba , J. K. Pachos , J. J. Garc´ıa-Ripoll Instituto de F´ısica Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006,Spain School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UKE-mail: [email protected]
Abstract.
We study the Haldane model with nearest-neighbor interactions.This model is physically motivated by the associated ultracold atomsimplementation. We show that the topological phase of the interacting modelcan be characterized by a physically observable winding number. The robustnessof this number extends well beyond the topological insulator phase towardsattractive and repulsive interactions that are comparable to the kinetic energyscale of the model. We identify and characterize the relevant phases of the model.
Keywords : Topological phases, topological insulators, ultracold atoms, DMRG.
1. Introduction
The integer quantum Hall effect initially [1] and the more extended set of topologicalinsulators more recently [2], represent a family of many-body systems with exotictransport properties. These properties originate from a robust and hidden topologythat appears in the bands of the system [3]. More precisely, if such two-dimensionalnon-interacting models are embedded in a torus and diagonalized in momentum space,they give rise to a collection of bands, ε n ( k ), and eigenstates, ψ n ( k ), which arecharacterized by the Chern numbers ν n = i π (cid:90) d k (cid:2) ∂ k x (cid:104) ψ n | ∂ k y ψ n (cid:105) − ∂ k y (cid:104) ψ n | ∂ k x ψ n (cid:105) (cid:3) ∈ Z . (1)When the Fermi energy lays in the gap between two bands and the total Chern numberof the occupied bands is nonzero, the resulting state is a topological insulator. Its keyphysical property is that it is non-conducting in the bulk, but it has robust conductingstates that live on the boundary of the lattice. These edge states are well known fortheir robustness under perturbations: the nonlocal nature of the topological phase thatsupports them, allows edge states to survive under extreme conditions of distortion,impurities and other external perturbations. This resilience of topological edge statesis highly favorable when considering their realization and detection in the laboratory.An important open question in topological insulators is what happens to theirtopological properties in the presence of interactions. One special situation is when weforce the energy bands of the insulator to be extremely narrow, having ε n ( k ) almostindependent of k , and introduce a strong repulsion or attraction. A paradigmatic caseis the fractional quantum Hall effect (FQHE) [4], where states with fractional filling a r X i v : . [ c ond - m a t . s t r- e l ] M a y inding number order in the Haldane model with interactions winding number . This is a mathematical object ν = 14 π (cid:90) d k S ( k ) · [ ∇ S ( k ) × ∇ S ( k )] , (2)which can be constructed when the unit cell of the lattice is represented as a pseudospinand the eigenstates of the model are represented by unique Bloch vectors, S ( k ) := (cid:104) σ (cid:105) = tr( σ | ψ ( k ) (cid:105) (cid:104) ψ ( k ) | ) . (3)For the non-interacting case the winding number is identical to the Chern number[15]. It has been shown that ν is an observable that can be directly measured inultracold atoms experiments [12]. This approach has been generalized to topologicalsuperconductors [16] and to composite problems (i.e. unit cell dimensionality 2 n ),where it still signals the existence of topological phase [15]. In this respect, we regardthe winding number as a global order parameter in its own right that can be measuredexperimentally. a) b) Figure 1. (a) Honeycomb lattice model depicting the nearest neighbor, t , next-nearest-neighbor hoppings, t a,b , and the interaction U . (b) Lattice unit cell andindices, together with the orientation of the phase in the nearest neighbor hopping. inding number order in the Haldane model with interactions at all times detectable through the winding number (2).For very repulsive interactions a charge density wave order (CDW) develops, as alreadyindicated by exact diagonalizations in finite lattices with a fixed number of particlesper site [6, 7]. For very attractive interactions the lattice fills up, as we work atzero chemical potential regime. This phenomenology can be observed with a verystraightforward mean-field theory that works with the vector field (3) as an orderparameter. This theory shows both the survival of the winding number as well asthe phase transitions into the topological order and into the CDW. These mean-fieldpredictions are confirmed by Matrix-Product-States (MPS, also known as DMRG)ground state computations with up to 50 lattice sites, a size that doubles what iscurrently achieved through exact diagionalizations. Finally, the destruction of thetopological insulator that takes place for large attractive interactions is modeled usingan extension of the mean-field theory by Poletti et al. [17], which shows the appearanceof a superconducting phase in such regimes.The structure of this paper is as follows. In section 2 we introduce the form ofthe Haldane model used here. Section 3 introduces a mean-field theory that workswith the spin field (3) as order parameter and thus gives at all times a proper valueof the winding number. Section 4 introduces Matrix-Product-States, the variationalansatz that we use to get numerical estimates of the ground state properties in finite-size problems. Since both our mean field and our DMRG fail for very attractiveinteractions, section 5 introduces a BCS ansatz that we can use to describe that regime,generalizing the ideas in [17]. Section 6 introduces the main results, describing thephase transitions that take place in this model and how they are characterized throughthe winding number, the mean field and DMRG, or the BCS theory. Finally, section7 discusses further implications of this work and possible outlooks.
2. The model
We work with a variant of the Haldane model that introduces flux as a phase on thenearest neighbor hopping of a honeycomb lattice, as explained in Ref. [12]. The modelis extended to consider also nearest-neighbor interactions that take place between thedifferent sublattices of the Haldane model. Overall, we can write the Hamiltonian inthe form H = H + H U , with the Haldane model describing the hopping of particlesin the honeycomb lattice. Introducing pairs of indices, s = ( i, j ), running over thelattice, as shown in Fig. 1, we can write H = − (cid:88) v ∈{ v , v , v } (cid:88) s ( t v b † s + v a s + H . c . ) − (cid:88) s (cid:15) ( a † s a s − b † s b s ) (4) − (cid:88) v ∈{ v , v , v } (cid:88) s ( t a a † s + v a s + t b b † s + v b s )and a nearest-neighbor interaction H U = (cid:88) v ∈{ v , v , v } (cid:88) s U n a, s n b, s + v (5)Here, v i ∈ { (0 , , (1 , , (0 , , (1 , − } represent displacements between lattice cells; t v = t exp( iφ v ), and the intralattice hoppings, t a,b , as well as the nearest-neighborinteraction U and a possible lattice imbalance, (cid:15) , that will be made zero in this work. inding number order in the Haldane model with interactions t and t a,b , implements a Dirac-typeHamiltonian with two distinct singularities in the Brillouin zone. The t term givesrise to the kinetic terms of the effective model, while the t a , t b terms implement amomentum dependent mass. The combination of both terms can be rewritten as apseudospin model H = − (cid:90) BZ d k ε ( k ) S ( k ) · [Ψ † ( k ) σ Ψ( k )] , (6)where ± ε ( k ) are the energies of the two bands at the given quasimomentum k . Athalf filling, that is one atom per unit cell, the ground state is obtained by placing oneparticle in the lowest band, which is a single particle state with an orientation of thepseudospin Ψ( k ) † = ( a † k , b † k ) dictated by S .As explained in Ref. [12], this model could be implemented using two separateoptical lattices that store atoms in two different internal states, one for sublattice Aand a different state for sublattice B. The nearest neighbor hopping would then beimplemented by laser assisted tunneling between internal states, so that the hopping t ij would carry the phase difference between internal states at their respective positions, t xy ∝ exp[ k · ( x + y )]. In our setup we have chosen one fixed direction for the phase, k , shown in figure 1b, so that the complex hoppings can be written as t v = e i Φ = t ∗ v . (7)This approach has the characteristic that the overlap between neighboring sites, whichmakes it possible to have nearest-neighbor assisted tunneling, also allows for a strongnearest-neighbor interaction (c.f. Ref. [14]).
3. Mean field theory
Our approach towards doing a mean-field study of the Haldane model builds on ourknowledge of the non-interacting solution: in this case, the ground state wavefunctionis exactly parameterized by the pseudospin orientation S ( k ) of the fermion thatpopulates the k quasimomentum in the Brillouin zone. Consequently, we may writethe ground state wavefunction at half filling as | GS (cid:105) = ⊗ k | S ( k ) (cid:105) , (8)where the tensor product is taken over the Hilbert spaces of each quasimomentummode.In the interacting case we expect the nearest-neighbor repulsion to have amoderate effect on the fermion distribution, more or less preserving the Fermi seaand the topological order associated to the vector field S . In other words, we willestimate variationally the ground state properties of the interacting model using thewavefunction (8) and the orientations of the pseudospins as variational parameters.This mean-field approach is quite unique in that we do not work with twovariational parameters describing the unit cell in position space, but rather employsa quasi-continuous normalized vector field S ( k ) defined on a lattice of N equispacedpoints in the Brillouin zone. Overall we have to work with 2 N degrees of freedomrepresenting the orientation of the pseudospins in such sampling. It is important toremark that this method becomes exact in the U = 0 limit. inding number order in the Haldane model with interactions U → + ∞ , whereatoms localize in either of the sublattices. This variational ansatz, however, doesnot capture other orders, such as a superconducting phase that should arise for largenegative U ∼ − | t | , which have to be analyzed using a different ansatz. We rewrite the Haldane model in momentum space, including the interaction. Forthat we introduce the Fourier transformed operators a k = 1 N / (cid:88) s e − i k s a s , (9)and similarly for the b operators. Here k is a set of N discrete momentum operatorsthat span the first Brillouin zone (BZ), and r is the lattice position.The interaction term is written in position space as H U = U (cid:88) v m (cid:88) s b † s + v m b s + v m a † s a s , (10)with three displacements v , , that connect one site to its neighboring cells. TheFourier transform of this Hamiltonian becomes H = UN (cid:88) k , , , b † k b k a † k a k δ ( k − k + k − k ) f ( k − k ) . (11)The central exponential, summed over r has lead to a δ ( · ) that enforces theconservation of momentum. Finally, we have the weight function f ( q ) = (cid:88) v m e i q · v m . (12)At this point we notice that H U can be decomposed into terms that connect twomomenta and terms that connect four different momenta. Since we are going to usea variational wavefunction of the form (8), the latter terms do not contribute. OurHamiltonian thus reads H = − (cid:88) k ε ( k ) S ( k ) · σ + UN (cid:88) p (cid:54) = q (cid:2) b † p b p a † q a q f (0) + b † p b q a † q a p f ( p − q ) (cid:3) . (13)Within our variational subspace of one particle per momentum orbital, we have therelations b † q b q = 12 (1 + σ z q ) , a † q a q = 12 (1 − σ z q ) , b † q a q = σ +q . (14)This allows us to rewrite, up to global constants, H = − (cid:88) k ε ( k ) S ( k ) · σ − U N (cid:32)(cid:88) p σ z p (cid:33) + UN (cid:88) p (cid:54) = q σ + p σ − q f ( p − q ) . (15)The second term induces charge density wave order, forcing all atoms to be on onesublattice or the other, while the third term scatters particles to different momenta.Both terms counteract the effect of the magnetic field ε ( p ) S ( p ) that tries to enforcetopological order. inding number order in the Haldane model with interactions Figure 2.
Matrix Product State structure running through the 2D honeycomblattice. Dashed lines represent the underlying honeycomb lattice, while solid linesrepresent the MPS bond dimensions.
4. Matrix Product States ground state
One alternative to confirm the mean-field predictions would be to do some exactdiagonalization to compute the ground state wavefunction, evaluating the observablesthat build up the winding number. However, this becomes unfeasible for two combinedreasons.The first one is that we need minimum size of the problem to detect the windingnumber. If we perform a simulation with L × L unit cells in position space, usingperiodic boundary conditions, that would amount to sampling L × L points in theBrillouin zone. A quick numerical calculation revealed that this sampling is insufficientto capture the winding number accurately if L ≤
3. This means that the smallestsimulation that could conceivable reproduce the non-interacting result would be a4 × × × | Ψ[ A ] (cid:105) = (cid:88) n ...n N tr( A n A n · · · A n N ) | n , n . . . n N (cid:105) , (16)where n m ∈ { , } is the occupation number of the m -th lattice site. Using a rather inding number order in the Haldane model with interactions A = argmin A (cid:104) Ψ | H | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) . (17) After computing the ground state we still have to recover the winding number asmeasured in a time-of-flight experiment [12]. In such experiments, the expansionof the atoms in the lattice makes them adopt a Fourier transform of their originalwavefunctions [18], a † i → (cid:90) d p ˜ w ( p ) e i pr i ˜ ψ a ( p ) † , (18) b † i → (cid:90) d p ˜ w ( p ) e i pr i ˜ ψ b ( p ) † , (19)where the quasimomentum is mapped to the position of the expanding atoms, p = m r /t . In this formula r i represents the center of the i -th lattice cell in the givensublattice, s ∈ { a, b } . The weight ˜ w is the Fourier transform of the wavefunction of anatom trapped in one lattice site, and for deep enough lattices it can be taken constantover a large domain. Finally, N is an overall normalization.The previous expression implies that if we measure the spin texture of theexpanding atoms v ( p ) = (cid:104) Ψ † ( p ) σ Ψ( p ) (cid:105) , (20)with ˜Ψ † = ( ˜ ψ † a , ˜ ψ † b ), this texture will be related to the spin texture of the originalbands in the lattice v ( p ) ∝ | ˜ w ( p ) | (cid:88) m , n (cid:104) Ψ † m σ Ψ n (cid:105) e i p ( r m − r n ) . (21)In particular, when p = k coincides with a quasimomentum in the Brillouin zone, v ( k ) ∝ S ( k ), up to factors that drop out when we normalize v to recover S .Consistently with the previous reasoning, we have worked with the MPSwavefunction computing the vector field v k = 1 M (cid:88) m , n (cid:104) Ψ † m σ Ψ n (cid:105) e i k ( r m − r n ) , (22)where k ∈ π/M × [ − M/ , M/ ⊗ , is a finite sampling of the Brillouin zone. Usingthis sampling, we then compute the winding number using an accurate formula forthe solid angle spanned by every three neighboring pseudospins, S = v / | v | , on themomentum space lattice [19], ν = (cid:88) (cid:104) k,p,q (cid:105) Ω( S k , S p , S q ) , (23)with the solid angle approximationtan (cid:20) Ω( a , b , c )2 (cid:21) = a · ( b × c ) | a || b || c | + ( a · b ) | c | + ( a · c ) | b | + ( b · c ) | a | , (24)that is valid for small Ω. inding number order in the Haldane model with interactions There are several important technical remarks regarding the MPS simulations. Thefirst one regards the use of conserved quantities to restrict the simulation, imposingfor instance, a total number of particles in the lattice. We have chosen not to do this,looking for the ground state of the full Hamiltonian, which is equivalent to minimizingthe free energy at zero chemical potential. This means that in these simulationsan increase or decrease of the filling is possible, though, as we will see below, it isirrelevant for the topological order as detected by the winding number –a signature ofthe robustness of this quantity, as shown already in [12].The second remark regards the size and type of lattice. MPS simulations weredone for lattices with 4 ×
4, 4 × × bond dimension .This dimension is the size of the matrices in (16) and it relates to the maximum amountof entanglement that is available in a bipartition of the state. The simulations thatwe show here are done with bond dimension χ = 100. This restriction is basedon the need to scan in detail the whole parameter space and the use of long-rangeinteractions induced by the 2D-to-1D mapping. However, in this particular study,for the observables that we computed, including density correlations and the fullwinding number, we have verified that convergence starts already at very low bonddimensions, such as 30 for the 5 × χ → χ × L , where L is the latticediameter), our MPS representation, which is easier to operate numerically, does notneed to be too complex to reproduce many of the relevant physical properties.
5. BCS Ansatz
We can further analyze our system, in the case of attractive (
U <
0) interactions, bymeans of a using a BCS ansatz. In order to do so, and to facilitate of comparison, wewill follow the procedure used in Ref. [17], which deals with a similar tight-bindingHamiltonian. Up to trivial factors, the Hamiltonian presented there corresponds tothe Φ = 0 case in Eq. (7). The BCS ansatz is based on a Wick expansion of theinteraction term (overwrite term here) in momentum space which givesˆ H int = U (cid:88) k (cid:16) ∆ (cid:63) k ˆ b − k ˆ a k + ∆ k ˆ a † k ˆ b †− k − ∆ k (cid:104) ˆ a † k ˆ b †− k (cid:105) (cid:17) , (25)where ∆ k = − /N (cid:80) k (cid:48) f ( k − k (cid:48) ) (cid:104) ˆ b − k (cid:48) ˆ a k (cid:48) (cid:105) is computed through a self-consistentcalculation. The order parameters which characterize the phase are nearest-neighbortwo particle pairing correlators δ = (cid:16) (cid:104) ˆ a s ˆ b s + v (cid:105) , (cid:104) ˆ a s ˆ b s + v (cid:105) , (cid:104) ˆ a s ˆ b s + v (cid:105) (cid:17) , (26) inding number order in the Haldane model with interactions Figure 3. (a-b) Winding number as a function of the imparted phase Φ andthe nearest neighbor interaction, U , using t = 1 , t a = − t b = 0 .
1. We plot theoutcome of the mean-field calculation (a) and of a MPS simulation (b) with 5 × |(cid:104) σ z (cid:105)| . (d) MPS expected value of the phase separation, (cid:104) n a, x n b, x (cid:104) (solid), and of the average number of particles per site, (cid:104) n a,b (cid:105) (dash-dot), as a function of the interaction, U/t , for various fluxes Φ /π = 0 (line), 0 . . where v m are again the directions spanned by the three nearest neighbors. FollowingRef. [17], we rewrite the order parameter as δ / | δ | = (cid:80) m w m u m , where the u m are abasis for the S cyclical permutation group: u = (1 , , / √ u = (2 , − , − / √ u = (0 , , − / √
3. A non-zero value of any w m coefficient (that is, of | δ | )signals a superfluid phase, while the particular coefficient renders information aboutthe symmetry of that phase. As we will see below, the results that we obtain in theattractive regime are consistent with those of Poletti et al [17].
6. Results
Figure 3 summarizes the main results from the previous numerical methods. In figures3a-b we plot the winding number of the mean-field or MPS simulation of the groundstate. Both figures show a strong qualitative agreement, exhibiting a phase transitionfrom a trivial phase around zero flux, Φ = 0, into a topological phase at larger fluxeswhere the Haldane phase is obtained for a wide range of interactions.Along the vertical axis we find that the topological phase described by the windingnumber disappears for repulsive interactions around U ∼ t . As shown in figure 3c, this inding number order in the Haldane model with interactions Figure 4.
Results from the BCS ansatz. In figures (b-d) we plot the absolutevalue of the order parameters, w k for k = 1 , ,
3, from the BCS ansatz (26),while figure (a) shows the the norm | δ | . The solid line delimits where thesuperconducting order parameter appears. Note that this happens well belowthe line U = − transition is due to a spontaneous symmetry breaking of the σ z expectation values,signaling the transition into a charge density wave where particles are polarized intoone sublattice or the other.To confirm the phase separation or CDW order, we have studied a similar orderparameter in the MPS simulation. In particular, we have computed the expectationvalue (cid:104) n a s n b s (cid:105) , which measures the coexistence of atoms in neighboring sites in thehoneycomb lattice. For large U this value decreases continuously down to zero,confirming the separation of species in the lattice. Interestingly, from the point ofview of the CDW order, this looks like a cross-over, but from the point of view ofthe winding number this looks like sharp phase transition into a disordered regime.Whether this is an artifact of poor resolution on the numerical side (for instancebecause the norm of v becomes so small that our computation of the winding numberis inaccurate), is something that our MPS simulations cannot resolve.We have also studied what happens for negative or attractive interactions. When U <
0, the ground state configuration at zero chemical potential has a filling fractionlarger than 1 /
2, that is more than one particle per unit cell. Despite this enlargedfilling, the ordered phase still persists until U = −
1, as evidenced by the windingnumber [Figure 3a-b]. At this critical interaction the lattice becomes perfectlyfilled [Figure 3d], forming a Mott insulator with two particles per site. This trivialconfiguration cannot be reproduced with the mean-field, because that wavefunctionassumed half-filling. In order to study this region of strong attractions we have to usethe BCS ansatz developed above (5). Figures 4a-d show the outcome of that ansatz, inding number order in the Haldane model with interactions U = 0 Hamiltonian.
7. Discussion
Summing up, our study reveals that the topological insulator phase survives for a widerange of interaction and that this phase is faithfully detected by the winding numberoperator. Given that earlier exact diagionalizations computed the Chern numberin selected regimes of parameters [6, 7], this leads us to conclude that the windingnumber can also reproduce the Chern number in interacting systems with moderatecorrelations.The ideas put forward in this work have very straightforward extensions to othermodels, including other topological insulators and composite systems [15], and othertypes of interactions, such as on-site Hubbard terms or long-range interactions. In allthese systems it would be interesting to see whether the winding number may act asa precursor of other strongly correlated phases.One may also wonder about the applicability of our results to the recentbeautiful experiment demonstrating the Haldane model in an optical lattice [10]. Thisexperiment uses laser assisted tunneling to implement t a,b , relying on the originallattice to supply t . Compared with our original proposal [12], it has the problem thatthe small overlap between neighboring sites would lead to a small value of U . In thiscase it would be more advantageous to implement other types of interactions, suchas relying on two-level atoms and implementing on-site repulsion or attraction. Suchmodels fall out of the scope of this work, but could be studied using a generalization ofour MPS and mean-field methods above, combined with the study of partial windingnumbers [15] and how they relate to the global topological order.Finally, we would like to emphasize the mean-field ansatz 8, which could be ofinterest to other contexts and models. Such momentum-representation mean-fieldtheory differs from other mean-field theories that have been developed in positionspace [21], and connect to long-range interaction classical spin models that have beenlong analyzed in the literature. The connection between these models, the symmetriesof the underlying topological insulator and how these are broken by the interactionterms could be a profitable avenue to understand the nature of the phase transitionsthat have been found in this study. Acknowledgments
The authors acknowledge useful discussions with Zlatko Papic. This work has beensupported by Spanish MINECO Project FIS2012-33022 and CAM Research NetworkQUITEMAD+.
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