Generalized Hartree-Fock theory for the dispersion relations of interacting fermionic lattice systems

Abstract

We study the variational solution of generic interacting fermionic lattice systems using fermionic Gaussian states and show that the process of gaussification, leading to a nonlinear closed equation of motion for the covariance matrix, is locally optimal in time by relating it to the time-dependent variational principle. By linearizing our nonlinear equation of motion around the ground-state fixed point we describe a method to study low-lying excited states leading to a variational method to determine the dispersion relations of generic interacting fermionic lattice systems. This procedure is applied to study the attractive and repulsive Hubbard model on a two-dimensional lattice, as well as the stability of the Hofstadter butterfly structure in the presence of interactions.