Applications of Semiclassical Theory in Statistical and Quantum Mechanics

University of Minnesota Ph.D. dissertation. June 2018. Major: Physics. Advisor: Alex Kamenev. 1 computer file (PDF); v, 96 pages.Since the initiation of quantum theory in the early 20th century, semiclassical methods have been a perenniel source of insight into a diverse range of phenomena. In spite of this history, however, there remain interesting and insightful applications of semiclassical theory to physics. This thesis advances this programme in several directions. First, we consider the statistical mechanics of multivalent 1D Coulomb gases and demonstrate how the semiclassical WKB method may be used to expose its thermodynamic properties. In doing so, we develop ideas from algebraic topology and complex Riemann surfaces. Moving to quantum theory proper, these tools are applied fruitfully to the phenomenon of spin tunneling oscillations in magnetic molecules with large instrinsic spin. Moving away from the WKB approximation, these ideas from complex analysis also proved crucial in exposing universal finite-size scaling effects in 1D lattice systems such as the Su-Schrieffer-Heeger model of polyacetylene and the Kitaev chain. Finally, we end by considering the the weak noise theory of the KPZ equation and thereby discover a novel phase transition in its large deviation statistics