Expansion of the Many-body Quantum Gibbs State of the Bose-Hubbard Model on a Finite Graph

TL;DR

This paper develops a detailed expansion of the many-body quantum Gibbs state for the Bose-Hubbard model on finite graphs, connecting it to a nonlinear Schrödinger equation and providing explicit recursive formulas for the coefficients.

Contribution

It introduces an expansion of the Gibbs state in inverse temperature, with computable coefficients, extending previous convergence results to higher orders.

Findings

Explicit recursive formulas for expansion coefficients

First two coefficients computed explicitly

Extension of convergence results to higher-order expansions

Abstract

We consider the many-body quantum Gibbs state for the Bose-Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schr\"odinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.