Some rigorous results concerning the uniform metallic ground states of single-band Hamiltonians in arbitrary dimensions

TL;DR

This paper reviews fundamental results about the Fermi surface of uniform metallic ground states in single-band Hamiltonians, emphasizing the limitations of non-self-consistent calculations and the importance of the exact Hartree-Fock framework.

Contribution

It clarifies the relationship between the exact Fermi surface and Hartree-Fock theory, and critiques previous second-order interaction calculations for their deficiencies.

Findings

Exact Fermi surface is a subset of Hartree-Fock Fermi surface.

Non-self-consistent self-energy calculations fail to capture the true Fermi surface.

Number-density distribution in the Hubbard model is not non-interacting v-representable.

Abstract

We reproduce and review some of the main results of three of our earlier papers, utilizing in doing so a considerably more transparent formalism than originally utilized. The most fundamental result to which we pay especial attention in this paper, is that the exact Fermi surface (FS) of the uniform metallic ground state (GS) of any single-band Hamiltonian, describing fermions, is a subset of the FS within the framework of the exact Hartree-Fock theory. We also review some of the physical implications of the latter result. Our considerations reveal that the interacting FS of a uniform metallic GS cannot be calculated exactly to order \nu (\nu \ge 2) in the coupling constant \lambda of the interaction potential in terms of the self-energy calculated to order \nu in a non-self-consistent fashion. We show this to be interlinked with the failure of the Luttinger-Ward identity, and thus of the Luttinger theorem, for a self-energy that is not appropriately related to the single-particle Green function from which the FS is deduced. We further show that the same mechanism that embodies the Luttinger theorem within the framework of the exact theory, accounts for a non-trivial dependence of the exact self-energy on \lambda that cannot be captured within a non-self-consistent framework. We thus establish that the extant calculations that purportedly prove deformation of the interacting FS of the metallic GS of the single-band Hubbard Hamiltonian with respect to its Hartree-Fock counterpart at the second order in the on-site interaction energy U, are fundamentally deficient. In an appendix we show that the number-density distribution function, to be distinguished from the site-occupation distribution function, corresponding to the GS of the Hubbard Hamiltonian is not non-interacting v-representable, a fact established earlier numerically. [Abridged Abstract]