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

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, $$\Delta $$ Δ , vanishes like $$\exp (-2\pi \sqrt{t/U})$$ exp ( - 2 π t / U ) in the weak-coupling limit $$U/t\downarrow 0$$ U / t ↓ 0 ( $$U>0$$ U > 0 and $$t>0$$ 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.