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arXiv:math-ph/0611034 · Journal of Mathematical Physics

Ground state energy of the low density Hubbard model: An upper bound

Alessandro Giuliani

Published Feb 1, 2007• 13 citations
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Abstract

We derive an upper bound on the ground state energy of the three-dimensional (3D) repulsive Hubbard model on the cubic lattice agreeing in the low density limit with the known asymptotic expression of the ground state energy of the dilute Fermi gas in the continuum. As a corollary, we prove an old conjecture on the low density behavior of the 3D Hubbard model, i.e., that the total spin of the ground state vanishes as the density goes to zero.

DOI
10.1063/1.2437650

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We compute an upper bound on the ground state energy of a dilute two-dimensional Fermi gas with repulsive short-range interactions. Our bound can be viewed as the two-dimensional analogue of a formula derived by Huang and Yang in the three-dimensional case. It captures the first three terms in an asymptotic expansion for small ϱa2\varrho a^2ϱa2, where ϱ\varrhoϱ denotes the density and aaa the scattering length of the interaction potential.
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→0ρ\to 0ρ→0, the ground state energy density satisfies e0(ρ)=4πaρ2(1+O(ρ1/6)),e_0(ρ) = 4πa ρ^2 \big(1+O(ρ^{1/6})\big),e0​(ρ)=4πaρ2(1+O(ρ1/6)), where aaa 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.