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

Understand

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.