The Geometry of (non-Abelian) Landau levels

TL;DR

This paper explores the topological properties of Landau Hamiltonians, introduces a novel method for calculating topological invariants using Dixmier trace, and extends these concepts to non-Abelian magnetic field models like Jaynes-Cummings and Quaternionic models.

Contribution

It introduces a new computational tool for topological numbers in Landau levels and extends topological analysis to non-Abelian magnetic field models.

Findings

Landau Hamiltonian exhibits a generalized even time-reversal symmetry.

A new method combining Dixmier trace and harmonic oscillator resolvent for topological number calculation.

Extension of topological analysis to non-Abelian magnetic field models such as Jaynes-Cummings and Quaternionic models.

Abstract

The purpose of this paper is threefold: First of all the topological aspects of the Landau Hamiltonian are reviewed in the light (and with the jargon) of theory of topological insulators. In particular it is shown that the Landau Hamiltonian has a generalized even time-reversal symmetry (TRS). Secondly, a new tool for the computation of the topological numbers associated with each Landau level is introduced. The latter is obtained by combining the Dixmier trace and the (resolvent of the) harmonic oscillator. Finally, these results are extended to models with non-Abelian magnetic fields. Two models are investigated in details: the Jaynes-Cummings model and the "Quaternionic" model.