Gaussian Fermionic Matrix Product States for generalized Hartree-Fock in quasi-1D systems

In many approximate approaches to fermionic quantum many-body systems, such as Hartree-Fock and density functional theory, solving a system of non-interacting fermions coupled to some effective potential is the computational bottleneck. In this work, we demonstrate that this crucial computational step can be accelerated using recently developed methods for Gaussian fermionic matrix product states (GFMPS). As an example, we studied the generalized Hartree-Fock method, which unifies Hartree-Fock and self-consistent BCS theory, applied to Hubbard models with an inhomogeneous potential. We demonstrate that for quasi-one-dimensional systems with local interactions, our approach scales approximately linear in the length of the system while yielding a similar accuracy to standard approaches that scale cubically in the system size.