Strong-coupling expansion for the momentum distribution of the Bose Hubbard model with benchmarking against exact numerical results

TL;DR

This paper develops a strong-coupling expansion method for the Bose Hubbard model's momentum distribution, achieving high accuracy and benchmarking against numerical simulations across different dimensions.

Contribution

It introduces a comprehensive strong-coupling expansion formalism that includes inhomogeneous effects and provides highly accurate analytical expressions for the momentum distribution.

Findings

Accurate momentum distribution results for the Mott phase in 1D, 2D, and 3D.

Phenomenological expressions for Mott phase lobes surpass existing approximations.

Benchmarking shows the method's high accuracy against QMC and DMRG simulations.

Abstract

A strong-coupling expansion for the Green's functions, self-energies and correlation functions of the Bose Hubbard model is developed. We illustrate the general formalism, which includes all possible inhomogeneous effects in the formalism, such as disorder, or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third-order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator to superfluid transition along with a generalization of the RPA-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions; the accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling theory results are benchmarked against numerically exact QMC simulations in two and three dimensions and against DMRG calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.