Ground state energy of the low density Hubbard model

TL;DR

This paper establishes a precise leading order correction to the ground state energy of the low density Hubbard model, extending continuum results to lattice systems, by deriving bounds that confirm the correction term involving scattering length and particle densities.

Contribution

It provides the first rigorous derivation of the low density correction to the Hubbard model's ground state energy, matching known continuum results and extending them to lattice systems.

Findings

The correction term is $8 ext{ extbackslash pi} a ho_u ho_d$ in the low density limit.

The derived bounds confirm the correction's leading order behavior.

Extension of continuum model results to lattice Hubbard models.

Abstract

We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani, our result proves that in the low density limit, the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by $8\pi a \rho_u \rho_d$, where $\rho_{u(d)}$ denotes the density of the spin-up (down) particles, and $a$ is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.