On the mean-field antiferromagnetic gap for the half-filled 2D Hubbard model at zero temperature

TL;DR

This paper proves Hirsch's 1985 conjecture that the antiferromagnetic gap in the half-filled 2D Hubbard model vanishes exponentially in the weak-coupling limit, using recent mathematical results and exact integral computations.

Contribution

The paper provides a rigorous proof of the asymptotic behavior of the antiferromagnetic gap in the 2D Hubbard model at zero temperature.

Findings

Confirmed the exponential decay of the gap as U/t approaches zero

Computed an integral involving the density of states exactly

Validated the conjecture using recent mathematical results

Abstract

We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree-Fock theory. It was conjectured by Hirsch in 1985 that this gap, $\Delta$, vanishes like $\exp(-2\pi\sqrt{t/U})$ in the weak-coupling limit $U/t\downarrow 0$ ($U>0$ and $t>0$ are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.