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.
Understanding the ground state energy of interacting quantum many-body systems is a central problem in mathematical physics. Although a complete description is generally out of reach, rigorous results can be obtained in suitable asymptotic regimes. One particularly tractable regime is the dilute limit, where the particle density ρ is sufficiently small compared to the interaction scale. In this setting, Bose gases exhibit a remarkable universality: to leading order, the ground state energy depends only on a single effective parameter, the two-body scattering length a.
For three-dimensional continuum Bose gases with repulsive interactions, this is seen in the so-called Lee-Huang-Yang-Wu [24,41] formula e 0 (ρ) = 4πaρ 2 1 + 128 15 √ π (ρa 3 ) 1/2 + 8( 4π 3 -√ 3)ρa 3 ln(ρa 3 ) + . . . , which captures in the dilute regime ρa 3 → 0 the correct ground state energy per unit volume, up to corrections of order aρ 2 (ρa 3 ), which are expected to no longer be universal in a. The leading-order term was rigorously established by Dyson [10] as an upper bound and, over 40 years later, by Lieb and Yngvason [26] as a lower bound (see also [43]). An upper bound matching the second order term was established by Yau and Yin [42] (see also [11,4,3]), while a lower bound finishing the proof of the Lee-Huang-Yang conjecture was established by Fournais and Solovej in [15] (see also [16]). In [21,22], a new, simpler proof that also establishes the free energy expansion in the positive temperature case was given (see also [38,44,2]). Finally, recently, Brooks, Oldenbrug, Saint Aubin and Schlein [7] established for the first time an upper bound that includes the third order term (so-called Wu term). A natural question is whether this universality persists in discrete settings. Bose gases realized in optical lattices are described by lattice Hamiltonians, most prominently by the Bose-Hubbard model [17,14], which has become a standard effective model for interacting bosons and has been extensively studied in both theory and experiment. Such systems arise on a variety of lattice geometries [34] beyond the simple cubic case, which motivates the consideration of general Bravais lattices. In this setting the single-particle dispersion and the associated low-energy kinematics depend strongly on the geometry and hopping amplitudes of the underlying lattice. In contrast to the continuum, both the interaction and the lattice structure influence the two-body problem, and it is not a priori clear whether the leading-order energy retains a universal form independent of the microscopic details.
In this work we show that such universality indeed survives on lattices as far as the leading order term of the energy is concerned. We consider interacting bosons on an arbitrary three-dimensional Bravais lattice with positive hopping amplitudes and on-site repulsive interactions and prove that, in the dilute limit e 0 (ρ) = 4πaρ 2 (1 + O(ρ 1/6 ))
where a denotes the lattice scattering length (cf. Appendix C). Thus all microscopic information -both the interaction strength and the lattice geometry -is absorbed into this effective parameter, and the leading-order energy is independent of other details of the underlying lattice. This provides a discrete analogue of Dyson and Lieb-Yngvason theorems for the Bose-Hubbard gas.
It is worth emphasizing that this universality is specific to the leading-order term. While in the continuum the second and (expectedly) third order terms also exhibit a universal structure depending only on the scattering length, in the lattice setting one expects higher-order terms to depend explicitly on the single-particle dispersion and hence on the geometry of the underlying lattice. In particular, beyond order aρ 2 the energy is not determined solely by the scattering length. Our result therefore identifies the precise regime in which lattice effects are completely absorbed into this effective parameter. We expect that the same leading-order universality holds for more general short-range lattice potentials.
Related universality results have been obtained for fermionic systems, both in the continuum [25,13,18,19,8] and on the cubic lattice (with nearest neighbor hopping) [20,39], where the leading-order energy is again determined by an appropriate scattering parameter. The available lattice proofs for fermions employ techniques that do not readily transfer to bosons. In particular, arguments based on Dyson-type lemmas have no direct counterparts. Therefore, in order to prove our main result, we rely on techniques that have been developed more recently in the context of continuous bosonic systems. For the lower bound we use a localization method coupled with Bogoliubov theory [5,32,9]. To make it work we need to develop estimates for the eigenvalues of Neumann Laplacians on general Bravais lattices. In fact, these bounds lead to the relative error of order O(ρ 1/6 ) in the lower bound. The upper bound is an adaptation of the a
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