CPT theorem and classification of topological insulators and superconductors

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 the CPT theorem in the classification of symmetry-protected topological phases, and show, in particular, that the topological superconductors discussed before are related by the CPT theorem.

Figure 1

Two sets of topological (crystalline) insulators and superconductors related by CPT equivalence and/or T duality: (a) T symmetric TI (QSHE), CP symmetric TCI, chiral (TC) symmetric TSC with $S_z$ conservation, and P symmetric TCSC with $S_z$ conservation; (b) T symmetric TSC, P symmetric TCSC, and T and P symmetric TCI that can support gapless edge states even in the absence of charge U(1) symmetry.