Generalized Hartree-Fock theory and the Hubbard model

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 non-zero 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