Ferromagnetism of the Hubbard Model at Strong Coupling in the Hartree-Fock Approximation

TL;DR

This paper rigorously demonstrates that Hartree-Fock theory predicts saturated ferromagnetism in the Hubbard model at strong coupling and low density, revealing a fundamental flaw in the approximation.

Contribution

It proves that Hartree-Fock approximation incorrectly predicts ferromagnetism in the Hubbard model at low density and strong coupling, contrary to known results.

Findings

Hartree-Fock predicts saturated ferromagnetism at low density and strong coupling.

This prediction contradicts the known absence of magnetization at low density.

The lowest energy state corresponds to maximum total spin at fixed particle number.

Abstract

As a contribution to the study of Hartree-Fock theory we prove rigorously that the Hartree-Fock approximation to the ground state of the d-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential mu) is small and the coupling constant U is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any U, and thus shows that HF theory is wrong in this case. As in the usual Hartree-Fock theory we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, {S}_z = sum_x n_{x,\uparrow} - n_{x,\downarrow}, and we find that the choice 2{S}_z = N = particle number gives the lowest energy at fixed 0 < mu < 4d.