Topological properties of quantum periodic Hamiltonians

TL;DR

This paper calculates the topological Chern indices of periodic quantum Hamiltonians using a semi-classical approach, linking quantum topological invariants to classical trajectories on phase space.

Contribution

It introduces a semi-classical method to compute Chern indices for Harper-like Hamiltonians, relating them to classical homotopy of quasi-modes.

Findings

Chern index equals the homotopy of quasi-mode paths on phase space.

Chern indices can be expressed through classical trajectory properties.

The method applies to generic spectral bands of periodic quantum systems.

Abstract

We consider periodic quantum Hamiltonians on the torus phase space (Harper-like Hamiltonians). We calculate the topological Chern index which characterizes each spectral band in the generic case. This calculation is made by a semi-classical approach with use of quasi-modes. As a result, the Chern index is equal to the homotopy of the path of these quasi-modes on phase space as the Floquet parameter (\theta) of the band is varied. It is quite interesting that the Chern indices, defined as topological quantum numbers, can be expressed from simple properties of the classical trajectories.