Nonequivalence of ensembles for long-range quantum spin systems in optical lattices
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Nonequivalence of Ensembles for Long-Range Quantum Spin Systems in OpticalLattices
Michael Kastner ∗ National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa andInstitute of Theoretical Physics, University of Stellenbosch, Stellenbosch 7600, South Africa (Dated: February 17, 2018)Motivated by the anisotropic long-range nature of the interactions between cold dipolar atomsor molecules in an optical lattice, we study the anisotropic quantum Heisenberg model with Curie-Weiss-type long-range interactions. Absence of a heat bath in optical lattice experiments suggests astudy of this model within the microcanonical ensemble. The microcanonical entropy is calculatedanalytically, and nonequivalence of microcanonical and canonical ensembles is found for a range ofanisotropy parameters. From the shape of the entropy it follows that the Curie-Weiss Heisenbergmodel is indistinguishable from the Curie-Weiss Ising model in canonical thermodynamics, althoughtheir microcanonical thermodynamics differs. Qualitatively, the observed features of nonequivalentensembles are expected to be relevant for long-range quantum spin systems realized in optical latticeexperiments.
PACS numbers: 05.30.Ch, 05.50.+q, 05.70.Fh, 67.85.Hj, 75.10.Jm
Cold dipolar gases have been in the very focus of ex-perimental and theoretical research recently [1, 2]. Inparticular, dipolar gases in optical traps have been sug-gested as laboratory realizations of lattice spin modelswhere the coupling parameters can be tuned freely, allow-ing for the realization of many Hamiltonians of interestin condensed matter physics [3].After switching off the cooling in such an experiment,total energy and number of atoms are conserved to a verygood degree. As a consequence, a statistical descriptionof such a lattice spin model should make use of the micro-canonical ensemble. For systems with short-range inter-actions, the choice of the statistical ensemble is typicallyof minor importance and could be considered a finite-size effect: differences between, say, microcanonical andcanonical expectation values are known to vanish in thethermodynamic limit of large system size, and the var-ious statistical ensembles become equivalent [4]. In thepresence of long-range interactions this is in general notthe case, and microcanonical and canonical approachescan lead to different thermodynamic properties even inthe infinite-system limit [5]. In the astrophysical con-text, nonequivalence of ensembles and the importanceof microcanonical calculations have long been known forgravitational systems [6, 7].In condensed matter physics, most systems are coupledto an environment, and therefore the canonical or grand-canonical ensembles are the ones that appropriately de-scribe the experimental situation of interest. Moreover,screening effects lead in general to interactions which areeffectively of short range, and hence equivalence of en-sembles usually can be taken for granted. As a conse-quence, calculations of thermodynamic quantities can bedone in the ensemble that is the most convenient one,which appears to be the canonical or grandcanonical, butnever the microcanonical, one. Owing to these facts, lit- tle is known about such systems in the microcanonicalensemble. Only recently, a number of toy models, con-sisting of long-range coupled classical spin variables, hasbeen studied (see [8] for a review). The study of thesestrongly simplified but analytically solvable models hasbeen very fruitful towards the aim of understanding gen-eral dynamical and thermodynamical properties of clas-sical systems with long-range interactions.Much less is known about the peculiarities of quantumspin systems with long-range interactions, and, in partic-ular, about equivalence or nonequivalence of ensembles inthis context. It is the aim of this Letter to contribute to-wards the understanding of such systems, with a focusonto the microcanonical setting as encountered in exper-iments with dipolar gases in optical lattices.To this purpose, we study the anisotropic quantumHeisenberg model with Curie-Weiss-type long-range in-teractions in the microcanonical ensemble. Such Curie-Weiss-type interactions, where each spin is interactingwith every other at equal strength, are clearly an ideal-ization of the actual interactions of dipolar atoms whichdecay like r − with the interparticle distance r . However,it is known that Curie-Weiss-type models faithfully re-produce many properties of algebraically decaying long-range interactions qualitatively, and to some extent evenquantitatively [9, 10].In this Letter the result of an exact, analytic calcula-tion of the thermodynamic limit of the microcanonicalentropy of the anisotropic quantum Heisenberg modelwith Curie-Weiss-type interactions is reported. Depend-ing on the choice of the anisotropy parameters in theHamiltonian operator, a concave entropy function isfound in some cases, and a nonconcave one in others.Correspondingly, equivalence of the microcanonical andthe canonical ensemble holds in the first case, but not inthe second.The relevance of the reported results is twofold. First,the observation of nonequivalent ensembles in long-rangequantum spin systems demonstrates that, under the ex-perimental conditions realized in cold dipolar gases inoptical traps, the choice of the statistical ensemble is ofparamount importance. Consequently, a statistical in-terpretation of the results of such experiments has togo beyond usual canonical thermodynamics. In partic-ular, differences between microcanonical and canonicalexpectation values do not diminish in importance withincreasing system size. This is in sharp contrast to mi-crocanonical computations for ideal Bose gases in traps[11], where equivalence of ensembles holds in the ther-modynamic limit. Second, the reported calculation alsoillustrates that cold dipolar gases in optical traps are ex-cellent laboratory systems in which long-range effects likethe nonequivalence of statistical ensembles or the nega-tivity of microcanonical response functions can possiblybe tested. Since these effects occur only for a certainrange of values of the anisotropy parameters, it is of par-ticular importance that coupling constants (and thereforeanisotropy parameters) in cold atom experiments can betuned with a high level of control, rendering such systemsan ideal laboratory for the study of these fundamental is-sues of thermostatistical physics. Anisotropic quantum Heisenberg model.—
The modelconsists of N spin-1 / H = ( C ) ⊗ N is the tensorproduct of N copies of the spin-1 / C ,and the Hamilton operator is given by H h = − N N X k,l =1 (cid:0) λ σ k σ l + λ σ k σ l + λ σ k σ l (cid:1) − h N X k =1 σ k . (1)The σ αk are operators on H and act like the α componentof the Pauli spin-1 / k th factor of thetensor product space H , and like identity operators on allthe other factors. The resulting commutation relation is (cid:2) σ αk , σ βl (cid:3) = 2i δ k,l ǫ αβγ σ γk , α, β ∈ { , , } , (2)where δ denotes Kronecker’s symbol and ǫ is the Levi-Civita symbol. h is the strength of an external magneticfield orientated along the 3 axis, and the constants λ , λ ,and λ determine the coupling strengths in the variousspatial directions and allow us to adjust the degree ofanisotropy. Note that it is explicitly shown in [3] thatanisotropic quantum Heisenberg models are among thesystems that can be engineered with cold polar moleculesin optical lattices.Special choices for the coupling constants in (1) yield,for example, (a) the isotropic Heisenberg model, λ = λ = λ , (b) the Ising model, λ = 0 = λ , (c) theisotropic Lipkin-Meshkov-Glick model, λ = λ and λ =0. For these special cases, the Hamiltonian (1) can be ex-pressed in terms of S and S , i.e., the square and the 3 component of a collective spin operator S = P σ k /
2. Asa consequence, an angular momentum eigenbasis simul-taneously diagonalizes H h and S and the model can besolved by elementary means.Here we consider the coupling constants λ , λ , and λ to be nonnegative, but otherwise arbitrary, and inthis case the model is known to display a transition froma ferromagnetic to a paramagnetic phase in the canonicalensemble. The exact expression for the canonical Gibbsfree energy g as a function of the inverse temperature β = 1 /T [12] and the magnetic field h is known for thismodel (and, in fact, for a larger class of systems) and canbe found, for example, in [13]. Microcanonical entropy.—
In thermodynamics, the en-ergy e is the variable conjugate to the inverse tempera-ture β , and the magnetization m is conjugate to − βh . Soin the same way that g ( β, h ) represents the fundamentalquantity of the quantum Heisenberg model in the canon-ical ensemble, the microcanonical entropy s ( e, m ) servesas a starting point for a microcanonical description inthe thermodynamic limit. However, for a pair of vari-ables ( e, m ) corresponding to the pair of noncommutingoperators ( H , M = 2 S ), it is not even well establishedhow to define a quantum microcanonical entropy, sym-bolically given by s N ( e, m ) = 1 N ln Tr [ δ ( N e − H ) δ ( N m − M )] . (3)Note that the symbolic expressions make little mathe-matical sense and require some physically reasonable reg-ularization. Extending a suggestion by Truong [14] tointeracting systems, the definition s N ( e, m ) = 1 N ln X ¯ e, ¯ m Tr[ P H (¯ e ) P M ( ¯ m )] δ ∆ (¯ e − e ) δ ∆ ( ¯ m − m )(4)seems to be physically reasonable, but difficult to applyin practice. Here, ¯ e and ¯ m denote eigenvalues of the op-erators H /N and M/N , respectively. δ ∆ is the charac-teristic function of the interval [ − ∆ , δ ∆ ( x ) = 1 if x ∈ [ − ∆ , P H (¯ e ), P M ( ¯ m ) denotethe eigenprojections of the operators H and M belong-ing to the eigenvalues ¯ e and ¯ m , respectively.We here report results for s ( e, m ) = lim N →∞ s N ( e, m )obtained by using a different regularization. The ana-lytic calculation of s uses, among others, some ingredi-ents from a related canonical calculation by Tindemansand Capel [15]. Details will be reported elsewhere, butthe main steps of the calculation can be sketched as fol-lows. (i) The deltas in (3) are replaced by their Fourierintegral representations. (ii) The Lie-Trotter formulais applied to separate the resulting exponential of theHamiltonian into exponentials of the type exp { c α ( S α ) } , α ∈ { , , } , with constants c α and collective spin com-ponents S α . (iii) These exponentials are transformedinto exponentials exp { ˜ c α S α } by applying the Hubbard-Stratonovich trick. The trade-off for steps (ii) and (iii) isa 3 n +2-dimensional integral, to be considered in the limit n → ∞ . The advantage, however, is that the Hilbertspace trace of exp { ˜ c α S α } factorizes into traces over thesingle-spin Hilbert spaces C , which can be easily per-formed. (iv) The resulting high-dimensional complex in-tegral can be solved in the thermodynamic limit N → ∞ ,for example, by the method of steepest descent.The final result for the microcanonical entropy of theCurie-Weiss anisotropic quantum Heisenberg model inthe thermodynamic limit is s ( e, m ) = ln 2 −
12 [1 − f ( e, m )] ln[1 − f ( e, m )] −
12 [1 + f ( e, m )] ln[1 + f ( e, m )] (5)with f ( e, m ) = s m (cid:18) − λ λ ⊥ (cid:19) − eλ ⊥ , (6)and λ ⊥ = max { λ , λ } [16], where s ( e, m ) is defined onthe subset of R for which0 < m ( λ ⊥ − λ ) − e < λ ⊥ and 2 e < − m λ . (7)The result is remarkably simple, in the sense that anexplicit expression for s ( e, m ) can be given. This is incontrast to the canonical ensemble, where g ( β, h ) is givenimplicitly as the solution of a maximization [13]. Plotsof the domains and graphs of s ( e, m ) are shown in Fig. 1for a number of coupling strengths λ ⊥ , λ . Nonequivalence of ensembles.—
On a thermodynamiclevel, equivalence or nonequivalence of the microcanon-ical and the canonical ensembles is related to the con-cavity or nonconcavity of the microcanonical entropy [5].By inspection of rows three to seven in Fig. 1 [or by sim-ple analysis of the results in (5)–(7)], the entropy s for λ ⊥ > λ is seen to be a concave function on a domainwhich is a convex set. For λ ⊥ < λ , the domain is not aconvex set and therefore the entropy is neither convex norconcave. In the latter case, microcanonical and canoni-cal ensembles are not equivalent, in the sense that it isimpossible to obtain the microcanonical entropy s ( e, m )from the canonical Gibbs free energy g ( β, h ), althoughthe converse is always possible by means of a Legendre-Fenchel transform.The physical interpretation of ensemble equivalence isthat every thermodynamic equilibrium state of the sys-tem that can be probed by fixing certain values for e and m can also be probed by fixing the correspondingvalues of the inverse temperature β ( e, m ) and the mag-netic field h ( e, m ). In the situation λ ⊥ < λ wherenonequivalence holds, this is not the case: only equilib-rium states corresponding to values of ( e, m ) for which s coincides with its concave envelope can be probed byfixing ( β, h ); macrostates corresponding to other values FIG. 1. (Color online) Domains (left) and graphs (right) ofthe microcanonical entropy s ( e, m ) of the anisotropic quan-tum Heisenberg model for some combinations of the couplings λ ⊥ , λ . From top to bottom: ( λ ⊥ , λ ) = (1 / , / , , , / , / , / , e and the ordinate is the magnetization m , and the entropy is defined on the shaded area. of ( e, m ), however, are not accessible as thermodynamicequilibrium states when controlling temperature and fieldin the canonical ensemble. In this sense, microcanonicalthermodynamics can be considered not only as differ-ent from its canonical counterpart, but also as richer,allowing to probe equilibrium states of matter which areotherwise inaccessible. The realization of a long-rangequantum spin system by means of a cold dipolar gas inan optical lattice offers the unique and exciting possibil-ity to study such states in a fully controlled laboratorysetting [17]. Thermodynamic equivalence of models.—
Let us leaveaside for a moment the question of experimental realiza-tion and discuss a different kind of equivalence specific tothe anisotropic quantum Heisenberg model. It had beenobserved already in the 1970s that the isotropic Heisen-berg model and the Ising model are thermodynamicallyequivalent in the sense that their canonical free energiescoincide [18]. One can verify by Legendre-Fenchel trans-forming the entropy in (5) that the same is in fact true forall coupling strengths satisfying λ ⊥ λ . Geometrically,this thermodynamic equivalence corresponds to the factthat the entropies s ( e, m ) for those couplings share thesame concave hull (which is equal to the entropy of theisotropic Heisenberg model plotted in row three of Fig. 1).Identical concave hulls of entropies imply, however, iden-tical canonical free energies, and hence thermodynamicequivalence of the two models follows in the canonicalensemble. Remarkably, however, thermodynamic equiv-alence does not hold in the microcanonical ensemble, asis obvious from the different shapes of entropies in rowsone to three of Fig. 1. Discussion.—
The microcanonical entropies (3) and(4) discussed in this Letter describe the physical situ-ation of fixed energy e and magnetization m . In a coldatom experiment, energy is conserved to a very good de-gree due to the absence of a heat bath. For apolar gaseswhere s -wave scattering is dominant, the total magneti-zation is also fixed, and the resulting short-range inter-acting microcanonical spin systems have been discussedin [19]. For dipolar gases where long-range interactionsare present and nonequivalent ensembles can occur, themagnetization is not conserved in general (unless an ex-perimentalist comes up with an ingenious trick the authoris not aware of). More easily, nonequivalence of ensem-bles could be observed in long-range quantum spin sys-tems undergoing a temperature-driven first-order transi-tion. In this case, nonequivalence is signalled by a non-concave microcanonical entropy s ( e ), corresponding toconservation of energy e , but fluctuating magnetization.Although the anisotropic quantum Heisenberg modeldiscussed in this Letter is among the systems which canbe engineered with cold polar molecules in optical lattices[3], this model is chosen here not for its particular fea-tures, but to illustrate general, and possibly even generic,properties of long-range interacting quantum spin sys- tems: nonconcave entropies, nonequivalence of statisti-cal ensembles, or other phenomena like negative micro-canonical response functions must be expected to showup under the experimental conditions realized in exper-iments with cold dipolar gases in optical lattices in gen-eral. Finally, note that dipolar atoms or molecules arenot the only possible realization of long-range quantumspin systems in optical lattices: Following a suggestion byO’Dell et al. [20], long-range interactions decaying withthe interparticle distance r like r − can be engineered byshining appropriately tuned laser light onto atoms, evenin the absence of a permanent dipole moment. Summary.—
A calculation of the microcanonical en-tropy of the anisotropic Curie-Weiss quantum Heisenbergmodel was reported. The results illustrate peculiaritiesof long-range quantum spin systems, like nonconcave en-tropies and nonequivalence of statistical ensembles. Themicrocanonical setting models the conditions relevant forexperiments with dipolar gases in optical lattices. Theresults point out the importance of nonstandard thermo-dynamics beyond the canonical ensemble for such exper-iments on the one hand, and on the other hand suggestthe use of optical lattice experiments for the study offundamental issues of thermostatistics. ∗ [email protected][1] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, andT. Pfau, Phys. Rev. Lett., , 160401 (2005).[2] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Peer,B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne,D. S. Jin, and J. Ye, Science, , 231 (2008).[3] A. Micheli, G. K. Brennen, and P. Zoller, Nature Phys., , 341 (2006).[4] D. Ruelle, Statistical Mechanics: Rigorous Results (Ben-jamin, Reading, 1969).[5] H. Touchette, R. S. Ellis, and B. Turkington, Physica A, , 138 (2004).[6] D. Lynden-Bell and R. Wood, Mon. Not. R. Astron. Soc., , 495 (1968).[7] W. Thirring,
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