Abstract

Topological phases of matter entail a prominent research theme, featuring distinct characteristics that include protected metallic edge states and possible fractionalized excitations. With the advent of symmetry protected topological (SPT) phases, many of these phenomena have effectively become accessible in the form of experimentally available band structures. Whereas SPT states protected by unitary and anti-unitary symmetries have been thoroughly understood, the inclusion of lattice symmetries provides for an active area of research. In this talk, I will present a short overview of results on defects in SPT states that directly motivate the existence of additional physics beyond the characterization based on (anti-) unitary symmetries. I will then connect these ideas to our work in which we mapped out all different gapped phases of free fermion systems in the presence of lattice symmetries. This revolves around a very simple algorithm that matches a rather involved mathematical perspective in terms of a framework called K-theory. Subsequently, I will discuss the implications of these combinatorial arguments and a plethora of related ideas that connect to this central theme of topological classification schemes, such as their impact on the description of Weyl semimetals and Wilson loop strategies to elucidate the resulting phases. Lastly I will attempt to sketch future directions, with a particular emphasis on novel SPT Floquet systems that could be accessible using tailored laser fields and invoking a natural role of interactions in the system.