Neural network variational Monte Carlo for quantum phase transitions and multi-component systems

Computational solutions of the time-independent Schrodinger equation are now a commodity research tool in the natural sciences. However, advancing our understanding of novel states of matter requires the development of methods with greater accuracy, efficiency, or applicability to systems where current approaches fall short. Recently, deep learning has emerged as a revolutionary paradigm in the computational sciences, driving state-of-the-art developments in computer vision and natural language processing. Now, a race is afoot to extend this revolution to quantum chemistry: neural network representations of many-body electronic wavefunctions, combined with the variational Monte Carlo (VMC) method, have been shown to produce highly accurate solutions to the time-independent Schrodinger equation. Still a nascent field, neural network VMC (NNVMC) promises an ideal tradeoff between accuracy and asymptotic efficiency. This thesis explores novel advantages of NNVMC, enabled by the lack of dependence on a set of basis functions. We study two applications of the method: totally ab initio treatment of quantum phase transitions in solid-state systems, exemplified by the Wigner transition in the homogeneous electron gas; and the accurate description of multicomponent systems, exemplified by positronic bound states of ordinary molecules, which are challenging to describe within electronic basis sets. We demonstrate that the accuracy of our method rivals or exceeds, with minimal inductive bias, existing state-of-the-art methods. Finally, we unify these themes to study nuclear quantum effects without resorting to the Born-Oppenheimer approximation. Our proof-of-principle suggests that neural network trial wavefunctions offer a route to an unambiguous determination of the phases of solid hydrogen at high pressure. More generally, we believe that NNVMC offers a universal method, and expect that neural-network-based methods will become a powerful tool in the study of systems where the fundamental physics is not yet fully understood.