Topological phases and families of free fermions

Abstract

The first part of this thesis will involve reviewing some preliminaries on topological phases and families, and their modern classification schemes. For the first part, we then study phase diagrams of charge-conserving systems in the class `A' of the Altland-Zirnbauer classification. We introduce topological textures permeating the phase diagram and discuss the global properties, including edge mode loci, of the same. We then show using field theoretic techniques that these properties are robust to perturbations. Further, using the suspension construction recipe owing to a conjectured classification, we build higher dimensional generalities of the well-known topologically non-trivial families of the Rice-Mele and Berry's model. We aim to connect this to the DDKS invariant introduced in prior work to characterise higher-dimensional topological families. For the second part, we review some work and efforts on describing a subset of gapless topological phases: topological metals. We aim to introduce a semi-metal with novel edge mode structuring arising from a torsion invariant from the non-orientable nature of the Brillouin zone.