Variational approaches to non-local Coulomb interactions in Mott-Hubbard systems

Abstract

From the high-temperature superconducting cuprates to novel materials such as twisted bilayer graphene, adequate modelling of two-dimensional, interacting systems is of tremendous interest. At the center of interacting electrons on a lattice stands the Hubbard model which, despite its simplicity, remains unsolved for many important cases. The three works and ideas presented in this thesis share the goal of improving computational possibilities and broaden the understanding of otherwise inaccessible parameter regimes. To that end, within the first two projects, a variational scheme has been developed which describes extended Hubbard models with both nonlocal repulsive and exchange interactions through an effective, purely local Hubbard system. Lastly, an implementation of a strong-coupling perturbation theory is presented, which has the aim of circumventing the famous, fermionic Monte Carlo sign problem. While we treat the important square lattice in all cases, the hope is that with further developments, these novel approaches may be applied to much more complex problems.