Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux
FFractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux
L. Hormozi , G. M¨oller , and S. H. Simon Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, Maryland 20899, USA TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom
We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquetteis close to a rational fraction, the low-energy physics is mapped to a multispecies continuum model: bosonsin the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin . We findthat the interaction potential between the bosons involves terms that do not conserve pseudospin, correspondingto umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that inexperimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these termsare crucial for determining the nature of allowed ground states. In particular, we show numerically that certainpaired wave functions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certainother wave functions can be destabilized when umklapp processes become strong.
Recent advances in the field of topological phases and theirpotential application in implementing an intrinsically fault-tolerant quantum computer [1, 2] have revitalized interestin fractional quantum Hall (FQH) states as the most promi-nent examples of topologically ordered phases of matter [3].Even though it has only been observed in (fermionic) solid-statesystems, the FQH effect can also exist for bosons [4–7]. Promising candidates are systems of interacting ultra-cold atoms where the necessary magnetic fields are simu-lated by rapid rotation [8, 9] or by laser-induced syntheticgauge fields [10]. At low temperatures when the filling frac-tion ν (the ratio of the particle density n to the magnetic fluxdensity n φ ) is sufficiently small, one can expect to observebosonic versions of the FQH effect [5]. For example, the ex-act ground state of bosons with contact interaction at fillingfraction ν = 1 / is the Laughlin state [4, 11], while at ν = 1 the ground state is in the same topological phase as the non-Abelian Moore-Read Pfaffian state [5, 12].A major advantage of optical and atomic systems over con-ventional solid-state systems is the possibility of creating andcontrolling quasiparticle excitations more naturally and withhigher precision (e.g., by shining focused laser beams on theatomic gas) [11]. A number of proposals suggest that the FQHregime for cold atoms can be most easily achieved using op-tical lattices [13–17]. The question naturally occurs whetherthere is new physics that may arise for a system of interactingbosons in the FQH regime due to the effects of an underlyinglattice. It has been shown that in the limit when the flux den-sity n φ , or equivalently the number of flux quanta per latticeplaquette, is small, one can ignore the existence of the latticeand treat the system in the continuum limit [15, 18]. When n φ is large, however, the presence of the lattice can potentiallylead to new correlated states of matter that are absent in thecontinuum [19–21]. This is the limit we will focus on.The starting point for our analysis of the many-bodyphysics in this problem is the observation that when n φ isclose to a rational fraction, the lowest energy bands in the Hof-stadter butterfly, a fractal structure realizing the single-particleenergy spectrum of particles hopping on a lattice in a magneticfield [22], are reminiscent of Landau levels in the continuum. This resemblance can be formalized by mapping the single-particle states of the system to a continuum model when theflux density is near simple rational fractions [19].The main result of this Letter is the following. In agree-ment with [19], we find that for flux per plaquette close toa rational fraction, n φ = p/q + (cid:15) with p, q small integers,and (cid:15) sufficiently small, one can map the system to an effec-tive continuum model with Landau levels and an added degreeof freedom for the particles, a sub-band index or pseudospin,which can take q possible values. However, in addition tothe density-density interactions between bosons of differentpseudospin found in [19], we find anomalous “pairing” inter-actions that do not conserve the number of particles of eachpseudospin species. We find that these pairing terms, whichbecome increasingly strong as (cid:15) is increased, are crucial in de-termining the possible ground states of the system for realisticvalues of the parameters of the problem.As a detailed example, we consider the most (experimen-tally) realistic case n φ = 1 / (cid:15) and study several effec-tive filling fractions ˜ ν = n/(cid:15) . We find a new FQH state at ˜ ν = 1 , which does not exist without the pairing interactionsbut becomes stabilized by the increase in (cid:15) and the concomi-tant increase in these interactions. This new state is related totwo copies of the non-Abelian Moore-Read Pfaffian state [12];however, it is a topologically distinct phase of matter. Incontrast, we find that the pairing terms destabilize the statespredicted at fillings ˜ ν = 2 / previously discussed by [19], ˜ ν = 4 / [20], and ˜ ν = 2 [21]. We present detailed numericalevidence for our conclusions and argue that experiments aremost likely to be in a regime where these pairing terms areimportant.We consider bosons with onsite repulsive interaction, hop-ping on a two-dimensional square lattice, subject to a uni-form perpendicular effective magnetic field. This system isdescribed by a modified Bose-Hubbard Hamiltonian [23], H = − J (cid:88)
1) + 2 cos(2 πn φ x − k ) φ ( x ) = ( E/J ) φ ( x ) and k is the momentum in the y -direction. Note that x and y are both integers.Consider the case of n φ = (cid:15) (cid:28) , where a continuumapproximation of the discrete Harper’s equation can be usedfor the low-energy eigenstates. In this limit, it is conve-nient to use a Wannier basis localized near minima of thecosine potential. These Wannier functions can be approxi-mated by harmonic oscillator (Landau level) solutions withoscillator length (magnetic length) l = 1 / √ π(cid:15) centeredat x k = k/ (2 π(cid:15) ) , i.e., φ k ( x ) ∼ exp( − π(cid:15) ( x − x k ) ) . Thebandwidth of the lowest band arises from tunneling betweenadjacent minima of the potential and for small (cid:15) it scales as ∼ e − C/(cid:15) where the constant C ≈ . can be obtained bythe WKB approximation [24]. Note that for small (cid:15) , the band-width is much smaller than the band gap ∆ = 4 Jπ(cid:15) , makingthis limit of the Hofstadter problem an example of an (almost)flat Chern band [25].Now let us consider flux densities close to a rational frac-tion. For simplicity, we focus on n φ = 1 / (cid:15) , forwhich Harper’s equation becomes φ ( x + 1) + φ ( x −
1) +2( − x cos(2 π(cid:15)x − k ) φ ( x ) = ( E/J ) φ ( x ) . This form sug-gests a Wannier solution analogous to the above case, but witha two-site form factor to account for the rapidly oscillatingfactor ( − x . We propose the ansatz solution, φ ks ( x ) ∼ (1 + A ( − x + s ) e − π(cid:15) ( x − x ks ) , (2)where we have defined x ks = ( k − sπ ) / (2 π(cid:15) ) and s = 0 , isa sub-band index. Thus for each momentum k , there are twopossible wave functions, which are spatially separated (due tothe shift in the center of the oscillator) and also have their mainweight on either the even or odd sites of the lattice. Choosingthe value A = √ − π(cid:15) ( √ − / solves Harper’s equationto order O ( (cid:15) ) and higher order terms can be added to A tosatisfy the equation to still higher order.If we interpret the sub-band index s as a quantum numberrepresenting a new degree of freedom, the low-energy bandsin the lattice at n φ = 1 / (cid:15) are equivalent to the energybands of a two-species system at n φ = (cid:15) , which is the con-tinuum (Landau level) limit. Thus at n φ = 1 / (cid:15) we canunderstand (cid:15) as the effective flux density giving rise to an ef-fective filling fraction defined as ˜ ν = n/(cid:15) . Similarly, for gen-eral n φ = p/q + (cid:15) (with p and q coprime), q solutions can be found and the system can be treated as a q -species model witheffective flux density (cid:15) [19, 20]. With increasing q , the band-width increases as ∼ e − C/ ( q (cid:15) ) whereas the band gap, whileremaining proportional to (cid:15) , decreases with increasing q .In order to be in the FQH regime, it is necessary that theinteraction energy be larger than the bandwidth (so that the in-teraction dominates over the kinetic energy). In addition, wewould like the interaction to be smaller than the band gap sothat all of the physics occurs within the lowest Landau band;however this requirement may not be crucial [26]. Finally,the temperature must be less than the energy gap of the FQHstate, which is typically set by the interaction energy (althoughit could also be set by the band gap if that is smaller). Statescompeting with quantum Hall liquids include Bose-Einsteincondensates: these describe the physics at n φ = 1 / , for ex-ample [27, 28].Given these restrictions, and given that experimentally ob-taining low temperatures will always be a challenge, it is clearthat the FQH effect is most likely to be observed in the regimeof intermediate (cid:15) where, the band gap is not too small and thebandwidth is not too large. Indeed, it is perhaps optimal towork in a regime where bandwidth and band gap are compa-rable. One can simply look at the Hofstadter spectrum to seewhere these inequalities are best satisfied [22]. The most ex-perimentally favorable case occurs for n φ = (cid:15) (cid:28) . Here, (cid:15) might be as large as . before the bandwidth is on the order ofthe band gap, and the band gap may be as large as J . This par-ticular case has been studied extensively previously [15, 18].The case of n φ = 1 / (cid:15) , which we focus on here, is alsofairly favorable for the observation of FQH effect. The param-eter (cid:15) can be as large as . before the band gap is on the orderof the bandwidth, and the band gap may be as large as about . J . While n φ = 1 / (cid:15) is still experimentally plausible,the cases of n φ = p/q + (cid:15) with q > have extremely tinyband gaps and, hence, seem less accessible. We note that de-spite the fact that these inequalities of energy scales are harderto satisfy for n φ = 1 / (cid:15) than for n φ = (cid:15) , the former hasricher physics associated with the new quantum number, thesub-band index introduced above.We now turn to consider the effect of the interaction term inthe Hamiltonian Eq. (1). Using any basis of (single-particle)states ψ a ( x, y ) with corresponding creation and annihilationoperators ˆ ψ † a and ˆ ψ a , the interaction may be decomposed as, ˆ U = (cid:88) abcd U abcd ˆ ψ † a ˆ ψ † b ˆ ψ c ˆ ψ d , (3)where U abcd = U (cid:88) x,y ψ ∗ a ( x, y ) ψ ∗ b ( x, y ) ψ c ( x, y ) ψ d ( x, y ) . (4)For n φ = (cid:15) (cid:28) , we use the basis for the lowestband, i.e., ψ k ( x, y ) = φ k ( x ) e iky with φ k ( x ) the Gaus-sian form as described above (properly normalized). In thislimit we may convert the sums into integrals, then, pro-jected to the lowest energy band, we obtain U k k k k = ε E [ U ε / ] N=8N=10N=12 ε E [ U ε / ] ν =2/3, N=10 ν =4/3, N=12 ν =2, N=14 ν=1 : gap others: gap ~~ ~~ ε | < Ψ e x | Ψ t r i a l > | N=8N=12 ν=1: ~ overlap (a) (b) (c) FIG. 1: (a) The FQH gap E at effective filling fraction ˜ ν = 1 asa function of (cid:15) , where the flux density is n φ = 1 / (cid:15) . Data areshown for N = 8 , bosons. Increasing (cid:15) increases the strength ofthe pairing terms of the Hamiltonian and stabilizes this state. (b) TheFQH gap for ˜ ν = 2/3 ( N = 10 ), 4/3 ( N = 12 ), and 2 ( N = 14 ) asa function of (cid:15) . Increasing (cid:15) decreases the FQH gap and destabilizesthe corresponding states. (c) Overlap between the exact ground stateof the system at effective filling fraction ˜ ν = 1 and the trial wavefunction Eq. (6) vs. (cid:15) . The overlap for (cid:15) > . exceeds
95 % . √ (cid:15) U e − (cid:80) i 95 % for N = 12 particles, which is an excellentindicator of the validity of our proposed wave function. Notethat, although outside the regime of validity for our model, at (cid:15) (cid:39) . , Eq. (6) is nearly an exact ground state of the two-body interaction, Eq. (5), to an accuracy of about − . Wealso find that the inverse /z dependence of the paired wavefunction in Eq. (6) is optimal, as introducing variational pa-rameters to change its shape [33] does not increase the overlapsignificantly.We have also studied the quasihole spectrum of Eq. (6) inthe presence of additional flux. For the model H − Hamil- tonian, the quasihole spectrum is precisely that of two decou-pled Moore-Read layers — the quasiholes of each layer cor-responding to the so-called half-quantum vortices of He-A.However, for our Hamiltonian of interest, Eq. (5), the umk-lapp pairing terms lock the direction of the d -vector, thus re-quiring that the quasiholes pair between layers, confining thehalf quantum vortices and leaving the system with effectivelyAbelian excitations. To establish with clarity that this is thecorrect physics we have been able to predict the entire low-energy quasihole spectrum of the He-A model, using a gen-eralization of the approach introduced in [36], which preciselymatches the low-lying spectrum of the microscopic Hamil-tonian, Eq. (5), for every case we could numerically access.These results will be presented elsewhere.Signatures for our proposed state can be derived from arange of experimental probes for the detection of quantumHall states in cold gases, such as measurements of ground-state incompressibility [37], noise correlations [20, 38], andpossibly a direct measurement of quasihole statistics [11].The methods described here can be generalized to flux den-sity n φ = p/q + (cid:15) although, as discussed above, FQH effectwith larger q is likely to be harder to realize in experiments. Inthis case there would be a sub-band index s = 0 , . . . ( q − and the umklapp terms of the interaction would allow noncon-servation of this sub-band index via s + s − s − s = 0mod q , which could lead to new pairing terms and possiblynew physics.To summarize, we have shown that anomalous pairing(umklapp) interaction terms are crucial to the physics of FQHeffect for interacting bosons on a lattice at flux density n φ =1 / (cid:15) . We find that the pairing terms greatly modify theground state at various effective filling fractions ˜ ν = n/(cid:15) .At ˜ ν = 1 , we demonstrate that these terms stabilize a newpaired FQH state, which is effectively two coupled copies ofthe Moore-Read Pfaffian state. At ˜ ν = 2 / , / and , we findthat the incompressible states are destabilized by the pairingterms.Discussions with S. Adam, E. Ardonne, N. R. Cooper,M. Hafezi, L. Mathey, R. Palmer, M. Peterson, and espe-cially E. Tiesinga are gratefully acknowledged. The authorsacknowledge the hospitality of Nordita and the Aspen Cen-ter for Physics and support from NIST/NRC (L.H.), Trin-ity Hall Cambridge, the Newton Trust, and the LeverhulmeTrust under Grant ECF-2011-565 (G.M.) and EPSRC GrantEP/I032487/1 (S.H.S.). [1] A. Y. Kitaev, Ann. Phys. , 22003 (2003).[2] C. Nayak et al. , Rev. Mod. Phys. ,1083 (2008).[3] X. -G. Wen and Q. Niu, Phys. Rev. B , 9377 (1990).[4] N.R. Cooper and N.K. Wilkin, Phys. Rev. B , R16279 (1999);N.K. Wilkin and J. Gunn, Phys. Rev. Lett. , 6 (2000).[5] N.R. Cooper et al. , Phys. Rev. Lett. , 120405 (2001).[6] N.R. Cooper, Advances in Physics , 539 (2008). [7] A.G. Morris and D.L. Feder, Phys. Rev. Lett. , 240401(2007).[8] K.W. Madison et al. , Phys. Rev. Lett. , 806 (2000).[9] J.R. Abo-Shaeer et al. , Science , 476 (2001).[10] I.B. Spielman Phys. Rev. A , 063613 (2009); Y.-J. Lin et al. ,Nature , 628 (2009).[11] B. Paredes et al. , Phys. Rev. Lett. , 010402 (2001).[12] G. Moore and N. Read, Nucl. Phys. B360 362 (1991).[13] D. Jaksch and P. Zoller, N. J. Phys. , 56 (2003).[14] E.J. Mueller, Phys. Rev. A , 041603(R) (2004).[15] A.S. Sorensen et al. , Phys. Rev. Lett. , 086803 (2005).[16] F. Gerbier and J. Dalibard, N. J. Phys. , 033007 (2010).[17] N.R. Cooper, Phys. Rev. Lett. et al. , Phys. Rev. A , 023613 (2007).[19] R.N. Palmer and D. Jaksch, Phys. Rev. Lett. , 180407 (2006).[20] R.N. Palmer et al. , Phys. Rev. A , 013609 (2008).[21] G. M¨oller and N.R. Cooper, Phys. Rev. Lett. , 105303(2009).[22] D.R. Hofstadter, Phys. Rev. B , 2239 (1976).[23] D. Jaksch et al. , Phys. Rev. Lett. , 3108 (1998).[24] G.I. Watson, J. Phys. A24 et al. , Phys. Rev. Lett. , 236802 (2011); Kai Sun etal. , ibid et al. , ibid , 063625 (2010).[28] S. Powell et al. , Phys. Rev. A, , 013612 (2011).[29] B. Chakraborty, P. Pietil¨ainen, The Quantum Hall Effects: Frac-tional and Integral , 2 nd ed., Springer, 1995.[30] F.D.M. Haldane, Phys. Rev. Lett. , 605 (1983).[31] F.D.M. Haldane and E.H. Rezayi, Phys. Rev. Lett. , 237(1985).[32] N. Read and D. Green, Phys. Rev. B , 10267 (2000); D. A.Ivanov, Phys. Rev. Lett. , 268 (2001).[33] G. M¨oller and S. H. Simon, Phys. Rev. B , 075319 (2008).[34] FQH states have an additional quantum number σ , known asshift, such that N φ = ˜ ν − N − σ in finite spherical systems.Our states have (˜ ν, σ ) = (2 / , , (1 , , (4 / , , and (2 , ,respectively.[35] E. Ardonne, and K. Schoutens, Phys. Rev. Lett. , 5096(1999), E. Ardonne et al. , Nucl. Phys. B 607 , 16864 (1996).[37] N.R. Cooper et al. , Phys. Rev. A , 063622 (2005).[38] E. Altman et al. , Phys. Rev. A70