Non-Gibbs states on a Bose-Hubbard 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 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 can not be mapped onto (β,μ). The separation line in the density control parameter space between Gibbs and non-Gibbs states ε∼n² corresponds to infinite temperature β=0. The non-Gibbs phase can not be cured into a Gibbs one within the standard Gibbs formalism using negative temperatures.