Variational Monte Carlo and Markov Chains for Computational Physics

Abstract

In this chapter the basic concepts of Markov-processes and Monte Carlo methods, as well as a detailed description of the variational Monte Carlo technique will be presented. Particular emphasis will be devoted to the apparent mistery of Monte Carlo techniques that allows us to sample a correlated many electron wave function defined in an Hilbert space that is exponentially large with the number N of electrons, in an affordable computational time, namely scaling with a modest power of N. This is a direct consequence of two key properties that are not common to all Monte Carlo techniques: (i) the possibility to define a Markov process and appropriate stochastic variables with a finite correlation time and variance, respectively; (ii) both these quantities should increase at most polynomially with N. In principle, the above properties should be proven a priori, because their numerical validations could be very difficult in practice. It will be shown that this is the case for the simplest variational Monte Carlo technique for quite generic wave functions and model Hamiltonians.