Generalized Hartree-Fock Theory and the Hubbard Model

TL;DR

This paper extends Hartree-Fock theory to include quasi-free states, unifying it with BCS theory, and applies it to analyze the Hubbard model's symmetry properties and minimizers at various temperatures.

Contribution

It provides a detailed, rigorous formulation of a generalized Hartree-Fock variational principle that encompasses BCS theory and applies it to the Hubbard model.

Findings

Exact determination of minimizers for zero and nonzero temperature cases.

Analysis of broken and unbroken symmetries in the Hubbard model.

Connection between BCS states and Hartree-Fock states through particle-hole symmetry.

Abstract

The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.