Classification of crystalline topological insulators through K-theory

TL;DR

This paper develops a detailed method using K-theory and spectral sequences to classify topological phases of crystalline insulators, providing explicit calculations and insights into their topological invariants.

Contribution

It introduces a systematic approach to compute K-theory groups for crystalline topological insulators using spectral sequences and symmetry decompositions, extending previous results.

Findings

Explicit computation of K-theory groups for various crystal symmetries

Identification of topological invariants via spectral sequence analysis

Discussion of limitations and open problems in the classification method

Abstract

Topological phases for free fermions in systems with crystal symmetry are classified by the topology of the valence band viewed as a vector bundle over the Brillouin zone. Additional symmetries, such as crystal symmetries which act non-trivially on the Brillouin zone, or time-reversal symmetry, endow the vector bundle with extra structure. These vector bundles are classified by a suitable version of K-theory. While relatively easy to define, these K-theory groups are notoriously hard to compute in explicit examples. In this paper we describe in detail how one can compute these K-theory groups starting with a decomposition of the Brillouin zone in terms of simple submanifolds on which the symmetries act nicely. The main mathematical tool is the Atiyah-Hirzebruch spectral sequence associated to such a decomposition, which will not only yield the explicit result for several crystal symmetries, but also sheds light on the origin of the topological invariants. This extends results that have appeared in the literature so far. We also describe examples in which this approach fails to directly yield a conclusive answer, and discuss various open problems and directions for future research.