A phase-space method for the Bose-Hubbard model

TL;DR

This paper introduces a phase-space Q-function approach to analyze the Bose-Hubbard model, effectively capturing key features like superfluidity and Mott insulator phases with computational simplicity.

Contribution

The paper develops a novel Q-function based phase-space method for the Bose-Hubbard model, including mean-field approximations for one and two-site Hamiltonians, and compares results with exact solutions.

Findings

Q-function method qualitatively captures superfluid and Mott insulator phases.

Quantum constraints significantly influence the ground state.

Method has limitations in the weak lattice regime.

Abstract

We present a phase-space method for the Bose-Hubbard model based on the Q-function representation. In particular, we consider two model Hamiltonians in the mean-field approximation; the first is the standard "one site" model where quantum tunneling is approximated entirely using mean-field terms; the second "two site" model explicitly includes tunneling between two adjacent sites while treating tunneling with other neighbouring sites using the mean-field approximation. The ground state is determined by minimizing the classical energy functional subject to quantum mechanical constraints, which take the form of uncertainty relations. For each model Hamiltonian we compare the ground state results from the Q-function method with the exact numerical solution. The results from the Q-function method, which are easy to compute, give a good qualitative description of the main features of the Bose-Hubbard model including the superfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem and show there are some limitations of the method particularly in the weak lattice regime.