CPT theorem and classification of topological insulators and superconductors

TL;DR

This paper systematically classifies topological phases of matter, both fermionic and bosonic, using K-theory and K-matrix approaches, and clarifies the role of CPT symmetry in these classifications.

Contribution

It introduces a comprehensive classification framework for topological phases considering various symmetries and elucidates the CPT theorem's role in topological superconductor classification.

Findings

Classified topological phases using K-theory and K-matrix methods.

Linked topological superconductors through CPT symmetry.

Provided a unified view of symmetry-protected topological phases.

Abstract

We present a systematic topological classification of fermionic and bosonic topological phases protected by time-reversal, particle-hole, parity, and combination of these symmetries. We use two complementary approaches: one in terms of K-theory classification of gapped quadratic fermion theories with symmetries, and the other in terms of the K-matrix theory description of the edge theory of (2+1)-dimensional bulk theories. The first approach is specific to free fermion theories in general spatial dimensions while the second approach is limited to two spatial dimensions but incorporates effects of interactions. We also clarify the role of CPT theorem in classification of symmetry-protected topological phases, and show, in particular, topological superconductors dis- cussed before are related by CPT theorem.