The Bose-Hubbard ground state: extended Bogoliubov and variational methods compared with time-evolving block decimation

TL;DR

This paper compares various methods for determining the ground state of a one-dimensional Bose-Hubbard model, highlighting the limitations of approximate theories and demonstrating the effectiveness of the time-evolving block decimation as a reference.

Contribution

The study applies time-evolving block decimation to periodic systems and benchmarks it against approximate methods for the Bose-Hubbard model.

Findings

HFB methods do not improve upon Bogoliubov theory in strong interactions.

HFB theories are accurate only in the weakly-interacting superfluid regime.

The variational Bijl-Dingle-Jastrow method captures the superfluid-Mott insulator transition qualitatively.

Abstract

We determine the ground-state properties of a gas of interacting bosonic atoms in a one-dimensional optical lattice. The system is modelled by the Bose-Hubbard Hamiltonian. We show how to apply the time-evolving block decimation method to systems with periodic boundary conditions, and employ it as a reference to find the ground state of the Bose-Hubbard model. Results are compared with recently proposed approximate methods, such as Hartree-Fock-Bogoliubov (HFB) theories generalised for strong interactions and the variational Bijl-Dingle-Jastrow method. We find that all HFB methods do not bring any improvement to the Bogoliubov theory and therefore provide correct results only in the weakly-interacting limit, where the system is deeply in the superfluid regime. On the other hand, the variational Bijl-Dingle-Jastrow method is applicable for much stronger interactions, but is essentially limited to the superfluid regime as it reproduces the superfluid-Mott insulator transition only qualitatively.