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arXiv:2602.16566

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

Norbert Mokrzański, Marcin Napiórkowski, Jacek Wojtkiewicz

Published Feb 18, 2026• 0 citations

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.

DOI: 10.48550/arxiv.2602.16566

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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.

Abstract We study dilute Bose gases in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the thermodynamic ground state energy is entirely characterized by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) [1] and Lieb–Yngvason (1998) [2], and the result for three-body interactions was proven much more recently by Nam–Ricaud–Triay (2022) [3]. The present result resolves a conjecture of Nam–Ricaud–Triay (2022) [4].

We prove the following formula for the ground state energy density of a dilute Bose gas with density

ρ

in

2

dimensions in the thermodynamic limit \begin{align*} e^{\rm{2D}}(ρ) = 4πρ^2 Y\left(1 - Y \vert \log Y \vert + \left( 2Γ+ \frac{1}{2} + \log(π) \right) Y \right) + o(ρ^2 Y^{2}). \end{align*} Here

Y= |\log(ρa^2)|^{-1}

and

a

is the scattering length of the two-body potential. This result in

2

dimensions corresponds to the famous Lee-Huang-Yang formula in

3

dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular it covers the crucial case of the hard core potential.

In the present paper we study the low density Bose gas in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the ground state energy is entirely characterised by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) and of Lieb-Yngvason (1998), and the result for three-body interactions was proven by Nam-Ricaud-Triay (2022). The present result resolves a conjecture of Nam-Ricaud-Triay (2022).

Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density

\rho

ρ in 2 dimensions in the thermodynamic limit

\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}

e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as

\rho a^2 \rightarrow 0

ρ a 2 → 0 . Here

Y= |\log (\rho a^2)|^{-1}

Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.