Topology of time-reversal invariant energy bands with adiabatic structure

TL;DR

This paper classifies the topological properties of energy bands in quantum systems with slow and fast degrees of freedom, focusing on time-reversal symmetry and phase space topology, revealing differences from traditional Bloch band classifications.

Contribution

It introduces a classification scheme for adiabatic bands in phase space using Chern numbers and Kane-Mele indices, highlighting differences from Bloch band topology.

Findings

Bands over 2D phase space are classified by Chern number.

Parity of Chern number matches band rank parity.

Even-rank bands are classified by Kane-Mele index.

Abstract

We classify the topology of bands defined by the energy states of quantum systems with scale separation between slow and fast degrees of freedom, invariant under fermionic time reversal. Classical phase space transforms differently from momentum space under time reversal, and as a consequence the topology of adiabatic bands is different from that of Bloch bands. We show that bands defined over a two-dimensional phase space are classified by the Chern number, whose parity must be equal to the parity of the band rank. Even-rank bands are equivalently classified by the Kane-Mele index, an integer equal to one half the Chern number.