Ground state energy of the dilute Bose-Hubbard gas on Bravais lattices

TL;DR

This paper proves that the ground state energy of dilute Bose-Hubbard gases on Bravais lattices follows a universal formula similar to continuous systems, depending only on the scattering length, regardless of lattice details.

Contribution

It establishes the leading-order ground state energy asymptotics for dilute Bose-Hubbard gases on Bravais lattices, extending Dyson and Lieb-Yngvason results to lattice systems.

Findings

Ground state energy density scales as 4πaρ² with corrections

Leading order depends only on scattering length, not lattice specifics

Results are valid in the dilute limit as density approaches zero

Abstract

We study interacting bosons on a three-dimensional Bravais lattice with positive hopping amplitudes and on-site repulsive interactions. We prove that, in the dilute limit $\rho\to 0$, the ground state energy density satisfies $$e_0(\rho) = 4\pi a \rho^2 \big(1+O(\rho^{1/6})\big),$$ where $a$ is the lattice scattering length defined through the corresponding two-body problem. This establishes the analogue of the Dyson and Lieb-Yngvason theorems for the Bose-Hubbard gas. Our result shows that the leading-order energy is universal: although the lattice geometry affects the microscopic dispersion relation, it enters the leading order asymptotics only through the scattering length. In particular, it is independent of other features of the underlying Bravais lattice.