Derivation of the Bose-Hubbard model in the multiscale limit

TL;DR

This paper rigorously derives the Bose-Hubbard model from a one-dimensional nonlinear Schrödinger equation with a periodic potential, demonstrating phase transition behavior in Bose-Einstein condensates.

Contribution

It provides a mathematical proof connecting the nonlinear Schrödinger equation to the Bose-Hubbard model in the multiscale limit, including phase transition analysis.

Findings

Stationary solutions of the Bose-Hubbard model approximate those of the NLSE in the semiclassical limit.

Large nonlinearity leads to localized solutions on a single lattice site.

Phase transition from superfluid to Mott-insulator is rigorously established.

Abstract

In this paper we consider a one-dimensional non-linear Schroedinger equation (NLSE) with a periodic potential. In the semiclassical limit we prove that the stationary solutions of the Bose-Hubbard equation approximate the stationary solutions of the (NLSE). In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential; as a result the phase transition from superfluid to Mott-insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is rigorously proved.