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

TL;DR

This work proves Hirsch’s conjecture for the mean-field antiferromagnetic gap in the half-filled 2D Hubbard model at zero temperature by deriving a sharp asymptotic formula from the Hartree–Fock gap equation expressed through the 2D tight-binding density of states $N_0(\epsilon)$. The authors obtain a precise leading behavior $\Delta(U)=32 e^{-\frac{2\pi}{\sqrt{U}}}$ with controlled corrections, after showing the auxiliary constant $b_1=0$ via exact evaluation of a regularized integral and an explicit computation of $a_0=\frac{(\ln 2)^2}{\pi^2}-\frac{1}{24}$. The strategy splits the problem into two propositions: a detailed asymptotic analysis of the gap (prop.\ 1) and a rigorous computation of $a_0$ (prop.\ a0prop), connecting the result to related mean-field quantities such as $T_N(U)$ and the gap ratio, and revealing links to BCS-type equations. The work advances analytic control over Hartree–Fock theory for Hubbard-like models and provides exact constants that clarify the low-$U$ behavior and its physical implications, including the Néel temperature and the gap-to-temperature ratio.

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, $Δ$, vanishes like $\exp(-2π\sqrt{t/U})$ 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.

Theorems & Definitions (5)

• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Lemma 3.1
• proof : Proof of Lemma \ref{['lemma:key']}