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We present numerical methods to solve the Generalized Hartree-Fock theory forfermionic systems in lattices, both in thermal equilibrium and out ofequilibrium. Specifically, we show how to determine the covariance matrixcorresponding to the Fermionic Gaussian state that optimally approximates thequantum state of the fermions. The methods apply to relatively large systems,since their complexity only scales quadratically with the number of latticesites. Moreover, they are specially suited to describe inhomogenous systems, asthose typically found in recent experiments with atoms in optical lattices, atleast in the weak interaction regime. As a benchmark, we have applied them tothe two-dimensional Hubbard model on a 10x10 lattice with and without anexternal confinement.

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