Green’s Function Approach to the Bose-Hubbard Model

Matthias Ohliger,Axel Pelster

doi:10.4236/wjcmp.2013.32020

Abstract

We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green’s function approach allows for calculating the excitation spectrum both for zero and finite temperature and for determining the effective masses of particles and holes.

Highlights

Ultracold bosonic gases trapped in the periodic potential of optical lattices represent tunable model systems for studying the physics of quantum phase transitions [1,2,3,4]
We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures
Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by high-precision Monte-Carlo simulations of the Bose-Hubbard model

Summary

Introduction

Ultracold bosonic gases trapped in the periodic potential of optical lattices represent tunable model systems for studying the physics of quantum phase transitions [1,2,3,4]. In the opposite case, where the on-site interaction dominates over the hopping term, the ground state is a Mott insulator (MI) where each boson is trapped in one of the respective potential minima This characteristic quantum phase transition of the Bose-Hubbard model has been studied extensively both with analytical [5,6,7,8,9,10,11] and numerical [12,13,14] methods for zero temperature, while less literature exists on the finitetemperature properties of this transition [15,16,17]. These effects could be taken into account by applying the local density approximation where the external potential is taken into account in form of a spatially dependent chemical potential

Green’s Function Approach

Phase Boundary

Excitation Spectrum

Conclusion and Outlook