Non-Gibbs states on a Bose-Hubbard lattice

Abstract

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 ε and particle number density n. The macrocanonical Gibbs distribution also depends on two parameters: the inverse nonnegative temperature β and the chemical potential µ. We prove the existence of non-Gibbs states, that is, pairs (ε, n) which cannot be mapped onto (β, µ). The separation line in the density control parameter space between Gibbs and non-Gibbs states ε ∼ n 2 corresponds to infinite temperature β = 0. The non-Gibbs phase cannot be cured into a Gibbs one within the standard Gibbs formalism using negative temperatures.

Quench dynamics in disordered two-dimensional Gross-Pitaevskii Lattices

“…All microcanonical states y(x) > y ∞ (x) can not be described by a Gibbs distribution with a positive temperature, and negative temperature assumptions lead to a divergence of the partition function (technically this happens only on infinite systems; we will assume here that our considered system sizes are large enough for this statement to apply). Recently these results were generalized to Gross-Pitaevskii lattices with any lattice dimension and disorder, and even to corresponding quantum many-body interacting Bose-Hubbard lattices [9]. While the zero-temperature line y 0 (x) renormalizes in the presence of a disorder potential, the infinite temperature line y ∞ (x) = x 2 is invariant under the addition of disorder.…”

“…Therefore the DGP model can be expected not to possess a many-body localization-delocalization transition. However, the classical DGP model, as well as its quantum BH counterpart, exhibit a non-Gibbs phase, which is characterized by at least partial nonergodic properties and absence of full thermalization [8,9]. An intriguing question is therefore whether these non-Gibbs phases have an impact on the outcome of the quench dynamics.…”

Statistical Mechanics of Systems with Negative Temperature

“…More recently, the statistical mechanics of the disordered DNLS Hamiltonian has been analyzed making use of the grand-canonical formalism [178]: the authors conclude that for weak disorder the phase diagram looks like the one of the non-disordered model, while correctly pointing out that their results apply to the microcanonical case, whenever the equivalence between ensembles could be established. In a more recent paper the thermodynamics of the DNLS model and of its quantum counterpart, the Bose-Hubbard model, has been analyzed by the canonical ensemble [179]. The main claim of these authors is that the Gibbs canonical ensemble is, conceptually, the most convenient one to study this problem and conclude that the high-energy phase is characterized by the presence of non-Gibbs states, that cannot be converted into standard Gibbs states by introducing negative temperatures.…”

Density resolved wave packet spreading in disordered Gross-Pitaevskii lattices

“…The entire non-Gibbs density range h > a 2 is not captured by a positive temperature, while negative temperature assumptions lead to a divergence of partition functions, and microcanonical dynamics show strong deviations from expected ergodic behavior including self-trapping [17,19]. The Gibbs-nonGibbs separation line in the density space was recently shown to persist for entire classes of generalized GP lattice equations as well as their Bose-Hubbard quantum counterparts, for any lattice dimension, and in the presence of disorder [20]. Remarkably the addition of disorder for (1) leaves the infinite temperature line h = a 2 invariant [16].…”