Efficient Monte Carlo Sampling of Lattice Field Theories

Abstract

Monte Carlo Sampling of Quantum Field Theories suffers from many inefficiencies. These inefficiencies, among other things, make determination of the QCD phase diagram and calculation of the correlation functions numerically difficult. As a step towards eventually overcoming these issues, the problem of infinite variance due to the exceptional configurations in fermionic Lattice Field Theories and probability distributions of the two-point functions in bosonic Lattice Field Theories are investigated. In the context of four-fermion interactions, a family of discrete Hubbard-Stratonovich sampling schemes are developed to avoid exceptional configurations. It is then shown that, while this sampling schemes work in principle, the estimations of uncertainties are unreliable. To overcome this limitation, a reweighting method is developed and shown to be efficient and reliable for the models investigated. As a study of the probability distributions of the correlators in a simple model, the probability distributions of the two-point functions are exactly calculated for interacting 𝑂(𝑁) models in the disordered phase. It is shown that, by utilizing the probability distribution of the two-point function, improved estimators of the mean can be constructed. Taken together, these techniques show that the statistical properties of Monte Carlo sampling in simple LQFTs can be exploited to improve calculations of physical quantities and set up the groundwork for future applications to phenomenologically relevant QFTs such as QCD.