Classification of topological insulators and superconductors with multiple order-two point group symmetries

TL;DR

This work develops a $K$-theory framework to classify topological insulators and superconductors protected by multiple $\mathbb{Z}_2$ point-group symmetries. By employing suspension isomorphisms, the authors reduce higher-dimensional classifications to zero-dimensional data, revealing a hierarchical structure controlled by a compact set of symmetry labels and flips of momentum and real-space variables. They derive a general classification formula for both real and complex AZ classes with arbitrary numbers of unitary $\mathbb{Z}_2$ symmetries and provide complete $\mathbb{Z}_2^{\times 2}$ tables, including explicit results for the zero-dimensional effective AZ classes and the corresponding periodic tables on spheres and tori. The results offer a practical framework for classifying higher-order and crystalline-protected phases in the presence of multiple $\mathbb{Z}_2$ point-group symmetries and pave the way for systematic calculations in higher symmetry settings.

Abstract

We present a method for computing the classification groups of topological insulators and superconductors in the presence of $\mathbb{Z}_2^{\times n}$ point group symmetries, for arbitrary natural numbers $n$. Each symmetry class is characterized by four possible additional symmetry types for each generator of $\mathbb{Z}_2^{\times n}$, together with bit values encoding whether pairs of generators commute or anticommute. We show that the classification is fully determined by the number of momentum- and real-space variables flipped by each generator, as well as the number of variables simultaneously flipped by any pair of generators. As a concrete illustration, we provide the complete classification table for the case of $\mathbb{Z}_2^{\times 2}$ point group symmetry.