Dartmouth Events

Abstract: Topological phase transitions have opened doors to new phases of matter and physical phenomena, many of which have been already observed experimentally. A hallmark of topological phases is the existence of localized boundary excitations, that are robust against disorder satisfying certain symmetry constraints. I will present a generalization of Bloch's theorem applicable to systems where translation symmetry is only mildly broken by arbitrary non-periodic boundary conditions. This provides an exact structural characterization of all energy eigenstates, by mapping the diagonalization problem to one of solving a system of polynomial equations. I will highlight applications to models of relevance to topological matter, with a focus on s-wave topological superconductors, for which an unconventional 4 pi-periodic Josephson effect without a fermionic parity switch is unveiled. I will also discuss rigorous connections between the topology of the bulk and the boundary excitations in one-dimensional topological phases, and how they are instrumental in understanding "stability" of the boundary excitations. For example, I will show how the Majorana modes at the edges of the s-wave topological superconducting wire previously discussed are stabilized by multiple symmetries, a scenario outside the purview of the standard classification of topological phases.