Topological phases protected by point group symmetry

TL;DR

This paper develops a comprehensive framework to classify and understand topological phases protected by crystalline point group symmetries, including bosonic and fermionic states, using lower-dimensional topological building blocks.

Contribution

It introduces a general classification scheme for point group SPT phases, applicable to various symmetries and dimensions, and illustrates it with multiple examples including electronic TCIs.

Findings

Classified bosonic and fermionic pgSPT phases in 3D

Identified a $Z_8 \times Z_2$ classification for electronic TCIs

Proposed a framework linking lower-dimensional topological states to higher-dimensional phases

Abstract

We consider symmetry protected topological (SPT) phases with crystalline point group symmetry, dubbed point group SPT (pgSPT) phases. We show that such phases can be understood in terms of lower-dimensional topological phases with on-site symmetry, and can be constructed as stacks and arrays of these lower-dimensional states. This provides the basis for a general framework to classify and characterize bosonic and fermionic pgSPT phases, that can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimension. We develop and illustrate this framework by means of a few examples, focusing on three-dimensional states. We classify bosonic pgSPT phases and fermionic topological crystalline superconductors with $Z_2^P$ (reflection) symmetry, electronic topological crystalline insulators (TCIs) with ${\rm U}(1) \times {Z}_2^P$ symmetry, and bosonic pgSPT phases with $C_{2v}$ symmetry, which is generated by two perpendicular mirror reflections. We also study surface properties, with a focus on gapped, topologically ordered surface states. For electronic TCIs we find a $Z_8 \times Z_2$ classification, where the $Z_8$ corresponds to known states obtained from non-interacting electrons, and the $Z_2$ corresponds to a "strongly correlated" TCI that requires strong interactions in the bulk. Our approach may also point the way toward a general theory of symmetry enriched topological (SET) phases with crystalline point group symmetry.