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Equivalence of ensembles, condensation and glassy dynamics in the Bose-Hubbard Hamiltonian

François Huveneers, Theil, Elias

Published Jan 1, 2019• 5 citations
OpenAlex

Abstract

We study mathematically the equilibrium properties of the Bose-Hubbard Hamiltonian in the limit of a vanishing hopping amplitude. This system conserves the energy and the number of particles. We establish the equivalence between the microcanonical and the grand-canonical ensembles for all allowed values of the density of particles ρ and density of energy ε. Moreover, given ρ, we show that the system undergoes a transition as ε increases, from a usual positive temperature state to the infinite temperature state where a macroscopic excess of energy condensates on a single site. Analogous results have been obtained by S. Chatterjee (2017) for a closely related model. We introduce here a different method to tackle this problem, hoping that it reflects more directly the basic understanding stemming from statistical mechanics. We discuss also how, and in which sense, the condensation of energy leads to a glassy dynamics.

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Journal of Statistical Mechanics Theory and Experiment•Feb 1, 2020•4 citations
We study mathematically the equilibrium properties of the Bose-Hubbard Hamiltonian in the limit of a vanishing hopping amplitude. This system conserves the energy and the number of particles. We establish the equivalence between the microcanonical and the grand-canonical ensembles for all allowed values of the density of particles ρ and density of energy ε. Moreover, given ρ, we show that the system undergoes a transition as ε increases, from a usual positive temperature state to the infinite temperature state where a macroscopic excess of energy condensates on a single site. Analogous results have been obtained by S. Chatterjee (2017) for a closely related model. We introduce here a different method to tackle this problem, hoping that it reflects more directly the basic understanding stemming from statistical mechanics. We discuss also how, and in which sense, the condensation of energy leads to a glassy dynamics.
We investigate the thermodynamic properties of a bosonic Josephson junction in the full quantum approach and, in particular, we concentrate in studying the thermal averages of one- and two-body properties below and above the transition from delocalized to self-trapped regimes. This temperature dependence is determined by using the fact that at equilibrium the microcanonical and canonical ensembles should be equivalent. To establish the robustness of the equilibrium state, we first study a one body property and show numerically that any arbitrary state localized in energy, when evolved, reaches a stationary or equilibrium state. Comparison among averages of one- and two-body properties in the microcanonical and canonical ensembles reveals discrepances, thus leading to non-equivalence among these ensembles. Such averages differences can be attributed to the fact that the Hilbert space of the system scales as its size N, and consequently, the entropy does not scale as N. We further find as a natural consequence of studying the finite bosonic Josephson junction in the two-mode Bose Hubbard context, that positive and negative temperatures are obtained. This result can be generalized for any finite optical lattice.
We study the equilibrium properties of the repulsive quantum Bose-Hubbard model at high temperatures in arbitrary dimensions, with and without disorder. In its microcanonical setting the model conserves energy and particle number. The microcanonical dynamics is characterized by a pair of two densities: energy density $\ensuremath{\varepsilon}$ and particle number density nnn. The macrocanonical Gibbs distribution also depends on two parameters: the inverse non-negative temperature $\ensuremath{\beta}$ and the chemical potential $\ensuremath{\mu}$. We prove the existence of non-Gibbs states, that is, pairs $(\ensuremath{\varepsilon},n)$ which cannot be mapped onto $(\ensuremath{\beta},\ensuremath{\mu})$. The separation line in the density control parameter space between Gibbs and non-Gibbs states $\ensuremath{\varepsilon}\ensuremath{\sim}{n}^{2}$ corresponds to infinite temperature $\ensuremath{\beta}=0$. The non-Gibbs phase cannot be cured into a Gibbs one within the standard Gibbs formalism using negative temperatures.
We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.
In this paper we study the relation between long cycles and Bose–Einstein condensation in the infinite-range Bose–Hubbard model. We obtain an expression for the cycle density involving the partition function for a Bose–Hubbard Hamiltonian with a single-site correction. Inspired by the approximating Hamiltonian method we conjecture a simplified expression for the short cycle density as a ratio of single-site partition functions. In the absence of condensation we prove that this simplification is exact and use it to show that in this case the long cycle density vanishes. In the presence of condensation we can justify this simplification when a gauge-symmetry breaking term is introduced in the Hamiltonian. Assuming our conjecture is correct, we compare numerically the long cycle density with the condensate and find that although they coexist, in general, they are not equal.
ABSTRACT This paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of statesN(E) ∝Enfor high quantum numbers, a condition for condensation beingn> 0. These results are obtained on the basis of the following assumptions: (i) For large volumesV(a) all energy levels behave likeV−θ, and (b) there exists a finite integerMsuch that it is justifiable to put for thejth energy levelEj=c V−θand to use the continuous spectrum approximation, wheneverj ≥ Mc θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.
Abstract We investigate a system of interacting bosons with random intersite tunnelling amplitudes. We describe these by introducing Gaussian-distributed hopping integrals into the standard Bose–Hubbard model. This system has been recently shown to exhibit a quantum phase transition to a glassy state. The latter is characterized by a quenched disorder of boson wave-function phases. In this aspect, the system resembles quantum spin-glass systems that attracted much attention. By exploiting this analogy, we employ the well-established methodology originated by Sherrington and Kirkpatrick, which bases on the replica trick and the Trotter–Suzuki expansion. This treatment transforms the original quantum problem into an effective classical one with an additional time-like dimension. Here, we focus on autocorrelation functions of canonical variables of the effective system in the time-like domain. Deep in the disordered phase, we find a highly dynamical nature of correlations in agreement with the expected short memory of the system. This behaviour weakens while approaching and passing the phase boundary, where in the glassy phase asymptotically non-vanishing correlations are encountered. Thus, the state features infinite memory, which is consistent with the quenched nature of glassy disorder with random but frozen boson phases.