Date of Award
Doctor of Philosophy (PhD)
In this work we utilize principles from optimization and machine learning to solve variational problems in solid-state physics. First we extend the quantum variational method to include an additional objective function term that ensures ground state solutions will have localized character, obtaining the so called Wannier functions. Here the additional term relies on dictionary learning to extract localized Wannier features from a dataset of known Wannier functions. Our approach displays a systematically controllable energy-localization trade-off, has an objective function that allows for the use of fast numerical solvers, and is capable of being used in highly-efficient self-consistent algorithms. Next we improve upon this approach by replacing the manual tasks required of dictionary learning with a three-variable optimization problem so that we can learn Wannier features ”on the fly”. We also demonstrate how to use the many-body quantum variational method to guide the training of deep neural networks (DNN)s that approximate the many-body ground state wave function. These DNNs are also used to instruct the basis sampling of variational Monte Carlo (VMC). We apply space group symmetry to both VMC and DNN training which reduces the size of DNN training data by a factor of 50 while achieving 99.98% energy accuracy for a two-dimensional SU(n) spin chain model.
Magnetta, Bradley, "Machine Learning Based Variational Approaches for Problems in Solid-State Physics" (2021). Yale Graduate School of Arts and Sciences Dissertations. 375.