Disorder Effects in Topological Phases of Matter

Abstract

The thesis is divided into two parts, both focusing on the disorder effects in topological phases of matter.

The first part explores the properties of Weyl semimetals with quenched disorder. A fundamental fact of condensed matter physics is that sufficient disorder typically drives a Fermi liquid metal into an Anderson insulator: a compressible but non-conducting phase. Recently, topological semimetals have emerged as another way a metallic phase can be realized. We show that, unlike ordinary metals, at least some topological semimetals are immune to localization and become a diffusive metal with a nonzero density of states at arbitrarily weak disorder. We present several physical arguments, based on diagrammatic perturbation theory and Keldysh field theory, as well as an exact mapping onto a two-dimensional array of coupled replicated Hubbard chains, to back up this claim.

The second part focuses on the disorder effects in one-dimensional spin chains. We define a new notion of order and disorder parameters for Ising-symmetric spin chains with quenched disorder, and establish a rigorous trade-off theorem between them. We show that in such a disordered ensemble, the system must have one and only one of the following: a nonzero $O(1)$ order parameter or a nonzero $O(1)$ disorder parameter with even parity under the Ising symmetry. We also present a rigorous treatment of the rare region effects in the disordered Ising chain, and show that the rare regions do not destroy the trade-off theorems. This theorem also provides a foundation for string order parameters in disordered average symmetry-protected topological (SPT) phases.