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 $\ensuremath{\varepsilon}$ and particle number density $n$. 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.