The Measure of a Phase

Abstract

One of the main advances in condensed matter physics over the last few decades has been the remarkable realization that free-Fermion descriptions are both surprisingly rich in physics and good descriptions of actual experimentally realizable systems. While foundational work has already classified such phases in simple settings, this work pushes these classifications into new territory by formalizing bulk-boundary correspondence in the language of Green's functions and developing novel tools such as the projected Green's function formalism. The results in this dissertation consist of three distinct applications of these ideas. First, in the context of translation-invariant, Hermitian systems, projected Green's functions reduce the study of topological phases to the study of polynomial zeros in the complex plane. Second, in the context of non-Hermitian systems, this formalism is extended to uncover new classes of topological phases and explain the underlying mechanism behind the non-hermitian skin effect. Third, by applying this toolkit to 1D quasi-periodic systems through a series of a rational approximates, we directly connect quasi-periodic topology to the famous Aubry-André metal-insulator transition and show that the topological invariants remain well-defined across the phase transition (addressing the Dry Ten Martini Problem).