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

TL;DR

The paper analyzes a dilute Bose gas on a 3D Bravais lattice with repulsive on-site interactions and positive hopping, proving the leading ground state energy density is universal: $e_0(\rho)=4\pi a\rho^2\bigl(1+O(\rho^{1/6})\bigr)$ where $a$ is the lattice scattering length. The authors adapt Dyson–Lieb–Yngvason type techniques to the lattice: an upper bound via a grand-canonical Bogoliubov trial state and a lower bound via localization into sub-lattices with Neumann spectral estimates, together with a lattice Fourier analysis for the scattering length. The leading term depends only on $a$; lattice geometry and dispersion influence higher-order corrections. This establishes a discrete analogue of continuum universality results for Bose gases in optical lattices and clarifies the regime where lattice effects are completely absorbed into the scattering length $a$.

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 $ρ\to 0$, the ground state energy density satisfies $$e_0(ρ) = 4πa ρ^2 \big(1+O(ρ^{1/6})\big),$$ 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.

Theorems & Definitions (26)

• Theorem 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• Lemma 3.4
• proof