Topological phases: Classification of topological insulators and superconductors of non-interacting Fermions, and beyond

TL;DR

This paper reviews the classification of topological insulators and superconductors of non-interacting Fermions, emphasizing symmetry aspects and the ten-fold way, and discusses recent results on interaction effects in three dimensions.

Contribution

It provides a comprehensive overview of classification methods for topological phases, highlighting the role of symmetries and recent advances including interactions.

Findings

Detailed analysis of the ten-fold way symmetry classification

Comparison of different classification approaches

Summary of interaction effects in three-dimensional topological phases

Abstract

After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.