Band topology beyond symmetry eigenvalues with applications to electronic and phononic systems

Abstract

Topology and physics share a long and eminent history. In this dissertation, we apply the tools of topology to study the physics of electrons and phonons in periodic media. The ultimate goal is to understand what topological properties are hidden in the electronic and phononic band structures, and how robust these properties are. We describe various new kinds of band topology, which emerge when comparing different ways of classifying topology. The dissertation begins by introducing some common topological classification schemes, including symmetry eigenvalue based methods, homotopy theory, cohomology theory and Wilson loops. These schemes are all applied to the same system, facilitating a systematic comparison between them. We then use the interplay between these classification schemes to describe and classify novel magnetic fragile and nodal phases in 2D electronic systems. Expanding our analysis to 3D, we uncover a new class of topological phases which we refer to as subdimensional topology. These phases exists in 3D materials which look completely trivial in symmetry eigenvalue based classification schemes. We demonstrate, however, that these phases possess intriguing topological characteristics in their bulk, surface, and spin spectra. Finally, we move on to phononic topology and discuss a novel topological invariant of phonons in 2D, which is associated with the zero wavevector band crossing of the acoustic phonons arising from the Nambu-Goldstone theorem. Our overarching aim is to showcase the diverse range of topological effects present in band structures, thereby stimulating further exploration into the complete understanding of such effects, and their physical consequences.