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

## TL;DR

This paper rigorously analyzes the equilibrium properties of the Bose-Hubbard model at low hopping, demonstrating ensemble equivalence, phase transition to energy condensation, and connections to glassy dynamics.

## Contribution

It introduces a new method to prove ensemble equivalence and characterizes the energy condensation transition in the Bose-Hubbard Hamiltonian.

## Key findings

- Ensemble equivalence holds for all particle densities.
- A phase transition occurs from positive to infinite temperature states.
- Energy condensation on a single site leads to glassy dynamics.

## 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 $\rho$ and density of energy $\varepsilon$. Moreover, given $\rho$, we show that the system undergoes a transition as $\varepsilon$ 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.