Expansion of the Many-body quantum Gibbs state of the Bose–Hubbard model on a finite graph
Zied Ammari
Université de Rennes, Rennes, FranceShahnaz Farhat
Constructor University Bremen, Bremen, GermanySören Petrat
Constructor University Bremen, Bremen, Germany

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 Schrödinger 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.
Cite this article
Zied Ammari, Shahnaz Farhat, Sören Petrat, Expansion of the Many-body quantum Gibbs state of the Bose–Hubbard model on a finite graph. Doc. Math. 30 (2025), no. 2, pp. 475–496
DOI 10.4171/DM/1001