Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups

TL;DR

This paper applies the Atiyah-Hirzebruch spectral sequence to band theory, providing a comprehensive framework to classify topological invariants across all 230 space groups, revealing numerous torsion invariants.

Contribution

It introduces a formalism connecting AHSS with band topology, enabling complete classification of topological invariants for all space groups without additional symmetries.

Findings

Complete list of topological invariants for 230 space groups

Identification of torsion invariants in symmorphic space groups

Integration of AHSS with band theory concepts

Abstract

We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant K-theory in the context of band theory. Various notions in the band theory such as irreducible representations at high-symmetric points, the compatibility relation, topological gapless and singular points naturally fits into the AHSS. As an application of the AHSS, we get the complete list of topological invariants for 230 space groups without time-reversal or particle-hole invariance. We find that a lot of torsion topological invariants appear even for symmorphic space groups.