Yi, Jinmin2026-06-012026-06-012026-06-012026-05-14https://hdl.handle.net/10012/23491The 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.enNATURAL SCIENCES::Physics::Condensed matter physicstopological semimetalquantum anomalydisordered systemssymmetry protected topological phasesDoctoral Thesis