Absence of Bose condensation on lattices with moat bands
AAbsence of Bose Condensation on Lattices with Moat Bands
Tigran A. Sedrakyan, Leonid I. Glazman, and Alex Kamenev William I. Fine Theoretical Physics Institute and Department of Physics,University of Minnesota, Minneapolis, Minnesota 55455, USA Department of Physics, Yale University, New Haven, Connecticut 06520, USA
We study hard-core bosons on a class of frustrated lattices with the lowest Bloch band havinga degenerate minimum along a closed contour, the moat, in the reciprocal space. We show thatat small density the ground state of the system is given by a non-condensed state, which may beviewed as a state of fermions subject to Chern-Simons gauge field. At fixed density of bosons, sucha state exhibits domains of incompressible liquids. Their fixed densities are given by fractions ofthe reciprocal-space area enclosed by the moat.
Though initially introduced for an ideal Bose gas, no-tion of Bose-Einstein condensation[1] (BEC) goes far be-yond the non-interacting case and describes, e.g., super-fluidity in such strongly correlated liquid as He[2]. Con-densation remains advantageous even at strong interac-tion, as condensed particles avoid exchange interaction,thus reducing the average potential energy. Within thispicture, elementary excitations, the quasiparticles[3], ex-hibit Bose statistics and gapless sound-like spectrum (forneutral superfluids). These predictions found countlessconfirmations in a diverse range of systems from He liq-uid to cold gases of alkali atoms[4, 5].The fundamental question is whether BEC groundstate with bosonic quasiparticle excitations is the univer-sal faith of any non-crystalline Bose substance. The goalof this paper is to present an alternative to this paradigm.To this end we discuss bosonic liquids in a family of 2Dlattices, whose band structure exhibits an energy mini-mum along a closed line – the moat , in the Brillouin zone,Fig. 1. The simplest example of the moat lattice is givenby graphene’s honeycomb lattice with nearest and next-nearest hopping[6–10]. With no interactions the groundstate is highly degenerate as bosons may condense in anystate along the moat as well as in any linear superpo-sition of such states. One may expect that interactionsremove the degeneracy and select a unique ground state.In this paper we show that at small filling factors theground state does not exhibit BEC. Instead of select-ing a single macroscopically occupied state, it involves all states along the moat and its vicinity, each one beingonly singly occupied. Such a state does not break the un-derlying U (1) symmetry, although does break the time-reversal invariance. Moreover, the elementary excitationsare not bosons, but rather fermions. At small enough fill-ing fractions, their spectrum is gapped. As a result, moatlattices provide an example of dramatic departure fromthe “BEC + Landau quasiparticles” paradigm for Boseliquids.A useful insight in the physics of the moat latticescomes from analogy with the fractional quantum Halleffect (FQHE). There too, the macroscopic degeneracyof the ground state is lifted by the interactions. It re- a b c d FIG. 1. (Color online) Lowest energy band of the honeycomblattice with t /t = 0 (a), 0 . . . M , is shown in light gray(blue). sults in incompressible (i.e. gapped) states, when theelectron density is an odd integer fraction of the lowestLandau level maximum occupation. There is a similarphenomenology associated with the lifting of degeneracyin the moat lattices. The characteristic particle densityis given by the area A M of the reciprocal space, enclosedby the moat. For a fractional filling of the form ν l = A M l + 1 + κ , l = 0 , , . . . , (1)the bosonic ground state is incompressible, here the re-ciprocal area is normalized to that of the Brillouin zone.Index κ is related to the reciprocal space Berry phaseand is given by κ = 0 if the moat encircles Γ point,Fig. 1b, and κ = 1 for moat encircling K and K (cid:48) points,Fig. 1d. For a generic lattice filling ν < /
2, such that ν l − < ν < ν l , the system breaks into incompressibledomains with fillings ν l − and ν l . a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Mathematically the moat bands appear, when the lat-tice Hamiltonian acquires a polynomial structure of theform ˆ H = t ˆ T + t ˆ T , ˆ T = (cid:32) G ˆ G † (cid:33) , (2)where the matrix structure is in A/B sublattice spaceand t and t are nearest and next-nearest hopping, cor-respondingly. For the case of honeycomb lattice, Fig. 2,ˆ G = G k = (cid:80) j =1 , , e i k · e j with the three lattice vectors e j connecting a site of sublattice A with three nearestneighbors of sublattice B. The Hamiltonians of the form(2) are not limited, though, to the honeycomb lattice.A generic oblique lattice with three distinct nearest andthree distinct next-nearest hopping integrals is describedby Eq. (2), if two conditions are imposed on six hop-ping constants [11] (variety of other lattices give rise toHamiltonians of the form (2)).The two energy bands of the Hamiltonian (2) are givenby E ( ∓ ) k = ∓| t || G k | + t | G k | . The lowest energy band E ( − ) k exhibits a degenerate minimum along the contour M – the moat, in the reciprocal space given by | G k | = | t | / t . For the honeycomb lattice this condition[12] issatisfied for t > | t | /
6, Fig. 1. A similar dispersionrelation appears in the context of particles with isotropicRashba spin-orbit coupling [13–19].The issue of Bose condensation for particles with sucha dispersion relation is a non-trivial one. On the non-interacting level there is no transition at any finite tem-perature. This is due to the square root, ( E − E M ) − / ,divergence of the single particle DOS near the bottomof the band. Such behavior of DOS highlights similar-ities with one-dimensional systems, where the groundstate of strongly repulsive bosons is given by the Tonks-Girardeau gas of free fermions [20–24]. Here we showthat the effective fermion picture describes the groundstate of hardcore bosons on 2D moat lattices as well. Animportant observation[17] is that the chemical potentialof fermions with the dispersion relation of Fig. 1 scales as µ F ∝ ν at small enough filling factors ν (cid:28) M , thechemical potential scales as µ B ∝ ν , due to on-site repul-sion (notice that the latter does not affect the fermionicenergy, because of the Pauli exclusion). One thus con-cludes that a small enough filling ν the fermionic state isenergetically favorable over BEC.To build a fermionic state of Bose particles one mayuse Chern-Simons flux attachments familiar in the con-text of FQHE[25–29]. This leads to composite fermions(CF) subject to a dynamic magnetic field produced bythe attached flux tubes. Following FQHE ideas, one maytreat the latter in the mean-field approximation by sub-stituting on-site density operators by their expectationvalues. In the context of FQHE this leads to a uniform t a a a e t e e FIG. 2. (Color online) The unit cell of honeycomb latticewith lattice vectors a i and e i , i = 1 , ,
3. Full (empty) citesbelong to the sublattice A ( B ). Total Chern-Simons flux is acombination of (i) πν fluxes through each of the triangles; (ii)phases exp ( − iπν ) attached to sides of the full regular triangleand exp ( iπν ) attached to the sides of the empty regular trian-gle. This arrangement of phases corresponds to the Haldanemodulation of phases with staggered φ H = − Φ / − πν/ magnetic filed, which partially compensates for the ex-ternal one. The lattice version of this procedure is some-what more subtle, however. Since the particles (and thusthe fluxes, attached to them) are confined to stay on thelattice sites, a uniform lattice filling ν does not translateinto a uniform magnetic field. As we explain below, itrather leads to a uniform magnetic flux 4 πν per unit cellsuperimposed with a staggered Haldane[30] flux arrange-ment. At small filling factors, ν (cid:28)
1, the correspondingHofstadter spectrum consists of quantized Landau lev-els, separated by cyclotron gaps. The latter protectsthe ground state from divergent fluctuation correction,rendering (local) stability of the mean-field ansatz. Thecorresponding phase diagram is schematically depicted inFig. 3.To quantify these ideas we start from the Hamilto-nian, written in terms of bosonic creation and annihila-tion operators b † r , b r , which commute at different cites,[ b ± r , b r (cid:48) ] = 0 , r (cid:54) = r (cid:48) , and fulfill the hard-core condi-tion (cid:0) b † r (cid:1) = ( b r ) = 0. For, e.g. , honeycomb latticethe Hamiltonian takes the form: H = t (cid:88) r ,j b † r b r + e j + t (cid:88) r ,j b † r b r + a j + H.c. (3)where the vectors e j and a j , j = 1 , , µ , is related to the averageon-site occupation ν through an equation of state.Motivated by the observation that the fermionic chem-ical potential is lower than that of BEC, we proceed withthe Chern-Simons transformation[25–29]. To this end we ν CF tt BEC BEC = l = l = l FIG. 3. (Color online) Phase diagram of hard-core bosons ona honeycomb lattice. CF and BEC are composite fermion andBose condensate states respectively. Also shown incompress-ible states with fractionally quantized filling fractions ν l . write the bosonic operators as b ( † ) r = c ( † ) r e ± i (cid:80) r (cid:48)(cid:54) = r arg[ r − ˜r ] n ˜r , (4)where the summation runs over all sites of the lattice.Since the bosonic operators on different sites commute,the newly defined operators c r and c † r obey fermioniccommutation relations. Also notice that the number op-erator is given by n r = c † r c r . Upon transformation (4)hopping terms of the Hamiltonian (3) acquire phase fac-tors e i (cid:80) ˜r φ ˜r , r , r (cid:48) n ˜r , where φ ˜r , r , r (cid:48) is an angle at which thelink (cid:104) r , r (cid:48) (cid:105) is seen from the lattice site ˜r . In terms of thefermionic operators the Hamiltonian (3) reads as H = t (cid:88) r ,j c † r c r + e j e i (cid:80) ˜r φ ˜r , r , r + ej n ˜r + t (cid:88) r ,j c † r c r + a j e i (cid:80) ˜r φ ˜r , r , r + a j n ˜r + H.c. (5)Notice that the hard-core condition is taken care of bythe Pauli principle and thus fermions may be consideredas non-interacting. Using the expression for φ ˜r , r , r (cid:48) , onecan directly check that (cid:80) ˜r φ ˜r , r , r + e i n ˜r − (cid:80) ˜r φ ˜r , r , r + e j n ˜r = (cid:80) ˜r φ ˜r , r , r + e i − e j n ˜r , for any two vectors e i/j , i, j = 1 , , a l = e i − e j , whilethe left hand side represents the phase of two consecutivenearest-neighbor (NN) hops along vectors e i and − e j .As a result, the Hamiltonian (5) retains the algebraicstructure of Eq. (2), where operator ˆ T describes fermionsin NN graphene lattice subject to CS fluxes.To analyze the consequences of these phase factors weadopt the mean-field ansatz[25, 27], n ˜r ≈ (cid:104) n ˜r (cid:105) ≡ ν . Thissubstitutes fluctuating CS phases with an external mag-netic filed, carrying flux Φ = 4 πν per unit cell (two cites, (cid:81) tE FIG. 4. Hofstadter energy spectrum vs. filling fraction ν ∈ [0 , / t = t / i.e. moat M is around the Γpoint. Notice that the bottom of the Hofstadter spectrum isflat, which is a consequence of the fact that all Landau levelsexhibit minima at the same energy E = − t / t . each with the occupation ν and 2 π flux per particle).While NN hoping operator ˆ T is sensitive only to this totalflux, the next-NN operator ˆ T implies that the magneticfiled exhibits Haldane modulation[30] within the unit cell.Indeed the phase factor, corresponding to a link (cid:104) rr (cid:48) (cid:105) is ϕ rr (cid:48) = (cid:80) ˜r (cid:54) = r , r (cid:48) φ ˜r , rr (cid:48) ν + (arg[ r − r (cid:48) ] − arg[ r (cid:48) − r ]) ν .For a counterclockwise travel along any elemental (i.e.not encircling any lattice points) triangle, the first termhere accumulates the net phase πν . The second termbrings phase − πν for small 120 ◦ triangles and phase3 πν for large equilateral triangle, Fig. 2. As a result,the entire flux Φ is concentrated into a half of the unitcell – the large empty triangle. This corresponds toHaldane modulation[30] with the staggering parameter φ H = − Φ / superimposed with the uniform flux Φ. No-tice, that only such configuration of fluxes results in thealgebraic Hamiltonian (2), while, e.g., a constant mag-netic field does not admit representation (2).This algebraic structure (2) greatly simplifies spectralproblem by reducing it to diagonalization of the NN oper-ator ˆ T . As mentioned above, the latter is sensitive only tothe total flux Φ, but not to the staggered component φ H .At small filling factors (i.e. magnetic fields) its spectrummay be analyzed in the semiclassical approximation[31].Accordingly, the eigenvalues of ˆ T , denoted as G l (Φ),where l = 0 , , . . . , can be found by: (i) considering theconstant energy contours | G k | = const = G of the bareoperator in the reciprocal k -space, and (ii) identifying G l (Φ) with energy G of contours having normalized re-ciprocal area A l = (cid:18) l + 12 − κ (cid:19) Φ2 π , (6)where 2 πκ is the Berry phase[32, 33]. Finally, the spec-trum of the Hamiltonian (2), which describes the latticesubject to the uniform magnetic flux Φ and Haldane mod-ulation φ H = − Φ /
6, is found in terms of G l (Φ) as E l (Φ) = − t G l (Φ) + t (cid:2) G l (Φ) (cid:3) . (7)Since we have attached exactly one flux quantum perfermion, all states at the lowest Landau level (LLL) areoccupied. As a result, the many-body ground state en-ergy follows LLL. The peculiarity of the moat dispersionis that LLL is not necessarily l = 0 one, but a level with l ≈ A M /ν , see inset in Fig. 5. Indeed, Landau levels(7) are non-monotonic functions of flux. They reach theminimum at G = t / t , i.e. exactly at the very bot-tom of the moat. Recalling that Φ = 4 πν , one obtainsthe set of the filling factors ν l , Eq. (1), where LLL (andthus the ground state energy) reaches its minima. Asan illustration, consider the moat closely encircling K and K (cid:48) points, Fig. 1d. In this case G k ≈ | k | / κ = 1 / G l (Φ) = (cid:112) √ l ,leading to E l (Φ) = − t (cid:112) √ l + t √ l . The non-monotonic dependence on Φ is evident.To go beyond the semiclassical approximation we con-sider the Hofstadter problem on the lattice, includingHaldane modulation. For a rational flux Φ = 4 πp/q ( p and q are positive integers) diagonalization of the opera-tor ˆ T reduces to Harper equation, which can be analyzednumerically. For such fluxes the spectrum splits onto q non-overlapping subbands, labeled by m = 1 , , . . . q .The corresponding spectrum E m, k (Φ), Fig. 4, acquiresthe form of the Hofstadter butterfly[34]. Notice theflatness of the lower edge of the spectrum, which re-flects the divergent DOS at this energy. Fig. 5 ampli-fies the lowest part of the Hofstadter spectrum. Non-monotonic Landau levels, Eq. (7), are clearly visible atsmall filling fractions. The ground state energy per par-ticle E GS ( ν ) = qNp (cid:80) pm =1 (cid:80) N/q k E m, k (4 πp/q ), where N is number of lattice sites, is shown In Fig. 5. For smallfilling fractions it closely follows the semiclassical LLL(7), exhibiting the minima at the fractionally quantizedfilling fractions ν l , Eq. (1). Due to Maxwell phase separa-tion rule, this leads to the macroscopic chemical potentialof the staircase shape with the jumps at the fractionallyquantized filling fractions (1), see Fig. 5. The flat regionsof the staircase imply phase separation into domains withfillings ν l and ν l +1 .As seen in Fig. 5, for ν (cid:28) ν = 1 /
2, Fig. 4, (indeed, fluxper cell is Φ = 2 π and may be gauged away), suggestingthat the mean-field ansatz may be inapplicable (at leastnot in the form adopted above).To conclude, we considered the nature of the ground tE − − ν FIG. 5. (Color online) Bottom part of Fig. 4. Thick (red) linerepresents the ground state energy per particle, E GS ( ν ), ob-tained numerically from the Hofstadter energy spectrum. Ar-rows show fractionally quantized filling fractions (1). Dashedline is the macroscopic chemical potential exhibiting jumps atthe fractionally quantized filling fractions. Dotted line is thechemical potential of the Bose condensed state. Inset: Semi-classical Landau levels as functions of the filling fraction. state and low-energy excitations of repulsive bosons onlattices with moat bands. The optical lattices with ap-propriate characteristics have been reported [7–10] veryrecently, opening a way for experiments on cold bosonicatoms with moat dispersion. We have shown that atsmall filling factors the expected ground state is not BEC,but is rather a filled LLL of composite fermions. The ex-citations are gapped and have fermionic statistics. Thismanifests itself in discontinues jumps of the chemical po-tential at fractional filling fractions, Eq. (1).We are grateful to V. Galitski, D. Huse and O. Starykhfor useful discussions. This work was supported by DOEcontract DE-FG02-08ER46482. [1] A. Einstein, Sitzungsberichte der Preussischen Akademieder Wissenschaften , 3 (1925).[2] P. Nozieres and D. Pines, Theory of quantum liquids ,Westview Press (1999).[3] L. D. Landau, J. Phys. USSR , 71-90 (1941); Phys. Rev. , 356-358 (1941).[4] A. J. Leggett, Rev. Mod. Phys. , 307 (2001).[5] L. P. Pitaevskii and S. Stringari, Bose-Einstein Conden-sation , Clarendon Press, Oxford, 2003.[6] C. N. Varney, K. Sun, V. Galitski, and M. Rigol, New J.of Phys. , 115028 (2012).[7] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T.Esslinger, Nature , 302 (2012).[8] K. K. Gomes, Warren Mar, Wonhee Ko, F. Guinea, andH. C. Manoharan, Nature , 306 (2012).[9] J. Simon and M. Greiner, Nature , 282 (2012). [10] T. Uehlinger, G. Jotzu, M. Messer, D. Greif, W. Hofstet-ter, U. Bissbort, and T. Esslinger, Phys. Rev. Lett. ,185307 (2013).[11] For three distinct t , t (cid:48) , t (cid:48)(cid:48) and t , t (cid:48) , t (cid:48)(cid:48) the Hamiltonianhas the structure of (2), if t t = t (cid:48) t (cid:48) = t (cid:48)(cid:48) t (cid:48)(cid:48) .[12] The antiferromagnetic sign t >
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