Bott periodicity for the topological classification of gapped states of matter with reflection symmetry

TL;DR

This paper reveals Bott periodicity in the classification of topological insulators and superconductors with reflection symmetry, using scattering theory and dimensional reduction, extending understanding of topological phases across dimensions.

Contribution

It introduces a scattering-matrix based dimensional reduction scheme that uncovers Bott periodicity in topological classifications with reflection symmetry, including the second descendant  phase.

Findings

Classification exhibits two period-two and four period-eight cycles.

Derived classification in arbitrary dimensions from one-dimensional case.

Confirmed consistency with recent comprehensive classification by Shiozaki and Sato.

Abstract

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-two and four period-eight cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme the classification in arbitrary dimensions $d \ge 1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)]. The use of a scattering-matrix inspired method allows us to address the second descendant $\bZ_2$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.