Topological meaning of Z$_2$ numbers in time reversal invariant systems

TL;DR

This paper reveals that the Z₂ topological invariant in time reversal invariant insulators is fundamentally connected to the global anomaly, and it simplifies the second Chern number to a Z₂ classification through the relative phase of Kramers doublets.

Contribution

It establishes a deep theoretical link between the Z₂ invariant and global anomalies, clarifying the topological significance of the Z₂ classification.

Findings

Z₂ invariant relates to global anomaly

Relative phase reduces Z to Z₂

Second Chern number underpins Z₂ classification

Abstract

We show that the Z$_2$ invariant, which classifies the topological properties of time reversal invariant insulators, has deep relationship with the global anomaly. Although the second Chern number is the basic topological invariant characterizing time reversal systems, we show that the relative phase between the Kramers doublet reduces the topological quantum number Z to Z$_2$.