Generalized Hartree-Fock Theory for Interacting Fermions in Lattices: Numerical Methods

TL;DR

This paper introduces efficient numerical methods for solving the Generalized Hartree-Fock theory in lattice fermionic systems, capable of handling large, inhomogeneous systems in and out of equilibrium, with applications demonstrated on the 2D Hubbard model.

Contribution

The paper develops scalable numerical algorithms for the Generalized Hartree-Fock theory that are effective for large, inhomogeneous lattice fermion systems, especially in weak interaction regimes.

Findings

Methods successfully applied to 10x10 Hubbard model.

Algorithms scale quadratically with system size.

Effective for inhomogeneous and out-of-equilibrium systems.

Abstract

We present numerical methods to solve the Generalized Hartree-Fock theory for fermionic systems in lattices, both in thermal equilibrium and out of equilibrium. Specifically, we show how to determine the covariance matrix corresponding to the Fermionic Gaussian state that optimally approximates the quantum state of the fermions. The methods apply to relatively large systems, since their complexity only scales quadratically with the number of lattice sites. Moreover, they are specially suited to describe inhomogenous systems, as those typically found in recent experiments with atoms in optical lattices, at least in the weak interaction regime. As a benchmark, we have applied them to the two-dimensional Hubbard model on a 10x10 lattice with and without an external confinement.