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 one can determine the covariance matrix corresponding to the fermionic Gaussian state that optimally approximates the quantum state of fermions. The methods apply to relatively large systems, because their complexity only scales quadratically with the number of lattice sites. Moreover, they are specially suited to describe inhomogeneous 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 10×10 lattice with and without external confinement.
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