Machine Learning for Path Deformation and Bayesian Data Analysis in Selected Lattice Field Theories

Abstract

Our collective understanding of the laws of nature has a long history of an intricate interplay between theoretical considerations and experimental falsification. As computational power increases, simulations, at the interface between theory and experiment, have taken an increasing role in scientific discovery. In particular, first-principles calculations are indispensable for systems with non-perturbative behavior, requiring simulations to test models against experiment. One widely accepted and deployed method involves formulating the theory on a finite lattice and then applying a Monte Carlo simulation. However, with increasing interest in such simulations practical and fundamental challenges arise, such as computational demand and the numerical sign problem. In the following, I discuss selected aspects for simulations of strongly correlated systems, namely the Hubbard model and lattice quantum chromodynamics. This encompasses methods to mitigate the numerical sign problem and (Bayesian) analysis of simulation results, in particular fitting methods and the treatment of excited state contamination. <br /> The Hubbard model describes systems of strongly correlated electrons and is used often in studying chemical compounds. First principle studies of this model are almost exclusively done using Monte Carlo techniques, with the exception being very small systems where direct diagonalization methods are feasible. However, away from half filling, Monte Carlo methods struggle because of the numerical sign problem. While the sign problem is unlikely to be completely solved, methods that reduce its impact are very valuable in expanding the computable parameter space. Leveraging theoretical developments on path deformations, I demonstrate that machine learning techniques can be used to mitigate the sign problem. In particular, I train complex-valued neural networks to serve as a parameterization of a sign-optimized manifold related to Lefschetz thimbles. These methods were developed and tested on doped graphene sheets, modelled by a small number of ions with periodic boundary conditions, at fixed temporal discretization and temperature. <br /> Renewable energy is a critical aspect of modern research to reduce effects of climate change. Despite the enormous energy cost of producing solar panels, they are a valuable element in the electricity production. Organic solar cells show great promise in reducing costs and allowing for flexibility. Unfortunately, to date their efficiency falls behind their silicon-based competitors. By studying the electronic structure of certain chemical compounds that are usable for organic solar cells, further development in this area can be fostered. This motivates my work in the molecule C₂₀ H₁₂ perylene, which can be used as an acceptor material in organic solar cells. This molecule is typically not at half filling, so any simulation requires methods to mitigate the sign problem. <br /> The study of perylene shown here requires the analysis of a large data set, of the order of <em>O</em> (2000) correlators, which is only feasible with an automated analysis procedure. In this thesis I present such an automatic routine based on Bayesian analysis using the Akaike information criterion. <br /> Finally, I shift the focus to particle physics to calculate aspects of the internal structure of hadrons. Hadrons are primarily governed by the strong interaction, i.e. described by quantum chromodynamics (QCD). In this theory, the internal structure is modelled by the correlation between spatial and momentum distributions of all constituents. Many details of these distributions remain to be calculated. In this thesis, I use lattice QCD to calculate the 2^nd moment of parton distribution functions (PDFs) for the nucleon. These are the average momentum fractions carried by the considered parton of the nucleon. I analyze two ensembles at the physical pion mass to obtain the moments of unpolarized, polarized, and transversity PDF for the nucleon.