[Paper Review] A numerical exact solution of the Bose-Hubbard model

N. Elstner, H. Monien|ArXiv.org|May 25, 1999

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper presents a numerically exact solution of the one- and two-dimensional Bose-Hubbard model using a strong-coupling expansion, enabling precise calculation of the single-particle and single-hole excitation spectra and the structure factor in the Mott insulating phase. The method confirms reentrance behavior in one dimension near the critical point and accurately maps the zero-temperature phase diagram, including the critical endpoints of the Mott lobes.

ABSTRACT

In this paper we report results from a systematic strong-coupling expansion of the Bose-Hubbard model in one and two spatial dimensions. We obtain numerically exact results for the structure factor and the spectrum of single particle and single hole excitations in the Mott insulator. This enables the determination of the zero-temperature phase diagram and the location of the critical endpoints of the Mott lobes. In one dimension we confirm the occurrence of reentrance behavior from the compressible to the insulating phase in a region close to the critical point.

Motivation & Objective

• To develop a numerically exact method for solving the Bose-Hubbard model in one and two dimensions.
• To determine the zero-temperature phase diagram of the Bose-Hubbard model, particularly the Mott insulating lobes.
• To locate the critical endpoints of the Mott lobes with high precision.
• To investigate the nature of the superfluid-to-Mott insulator transition, including reentrance behavior in one dimension.
• To compute the structure factor and excitation spectra (single-particle and single-hole) in the Mott insulating phase.

Proposed method

• A systematic strong-coupling expansion is employed to solve the Bose-Hubbard Hamiltonian in one and two spatial dimensions.
• The method enables numerically exact calculations of the ground state and low-energy excitations in the Mott insulating phase.
• The structure factor and the spectra of single-particle and single-hole excitations are computed using this expansion.
• The approach allows for the determination of critical points in the phase diagram by analyzing the excitation gap closure.
• The method is applied to both one- and two-dimensional lattices to compare dimensional effects.
• The numerical results are validated by comparison with known analytical limits and by consistency checks in the strong-coupling regime.

Experimental results

Research questions

• RQ1 What is the precise location of the critical endpoints of the Mott lobes in one and two dimensions?
• RQ2 Does reentrance from the compressible to the insulating phase occur near the critical point in one-dimensional systems?
• RQ3 How do the single-particle and single-hole excitation spectra behave in the Mott insulating phase?
• RQ4 What is the structure factor of the Mott insulator, and how does it relate to the phase transition?
• RQ5 How accurate is the strong-coupling expansion in capturing the zero-temperature phase diagram of the Bose-Hubbard model?

Key findings

• The study confirms the presence of reentrance behavior in the one-dimensional Bose-Hubbard model near the critical point, indicating a non-monotonic transition path.
• The critical endpoints of the Mott lobes are determined with high numerical precision using the excitation spectrum and structure factor.
• The single-particle and single-hole excitation spectra are computed numerically exactly in the Mott insulating phase for both one and two dimensions.
• The structure factor is calculated and found to be consistent with the expected behavior near the Mott transition.
• The method successfully reproduces known analytical limits in the strong-coupling regime, validating its accuracy.
• The zero-temperature phase diagram is mapped with improved resolution, particularly in the vicinity of the critical points.