Variational Monte Carlo for the Hubbard Model

In these lecture notes, we introduce the variational Monte Carlo method for the Hubbard model. First, we briefly introduce the single-orbital Hubbard model and its multi-orbital extension, focusing only on electronic models. Then, we introduce the general scheme under which variational Monte Carlo works and we present the variational wave functions that can properly describe superconductive and magnetic states in single-orbital and multi-orbital Hubbard models. We also detail how to perform Monte Carlo calculations for the given wave functions. We also describe how to optimize a variational wave function with the stochastic reconfiguration method. Finally, we present a few results for correlation functions that can be computed in variational Monte Carlo, to determine, for instance, the presence of charge or spin order and superconductivity.