Uniform Density Theorem for the Hubbard Model

TL;DR

This paper proves a uniform density property for the Hubbard model and related bipartite lattice models at half-filling, revealing specific behaviors of the single-particle density matrix and correlations.

Contribution

It establishes a general uniform density theorem for a class of bipartite lattice models, including the Hubbard and Falicov-Kimball models, at half-filling.

Findings

Single-particle density matrix is constant on the diagonal.

Density matrix vanishes off-diagonal and between different sublattices.

No higher-order correlations between sites of the same sublattice.

Abstract

A general class of hopping models on a finite bipartite lattice is considered, including the Hubbard model and the Falicov-Kimball model. For the half-filled band, the single-particle density matrix $\uprho (x,y)$ in the ground state and in the canonical and grand canonical ensembles is shown to be constant on the diagonal $x=y$, and to vanish if $x \not=y$ and if $x$ and $y$ are on the same sublattice. For free electron hopping models, it is shown in addition that there are no correlations between sites of the same sublattice in any higher order density matrix. Physical implications are discussed.