Generalized TKNN-equations

TL;DR

This paper derives generalized TKNN-equations using bundle representations of the noncommutative torus, relating Chern numbers to Hall conductances in magnetic Bloch electron systems, including for irrational deformation parameters.

Contribution

It introduces a novel derivation of generalized TKNN-equations via bundle representations, extending their applicability to irrational deformation parameters.

Findings

Derived generalized TKNN-equations for rational deformation parameters.

Extended equations to irrational deformation parameters.

Connected Chern numbers with Hall conductances in magnetic systems.

Abstract

We derive generalized TKNN-equations via bundle representations of the noncommutative torus with rational deformation parameter, the bundle coming from spectral projections in the torus algebra. These equations relate Chern numbers of dual bundles which we interpret as Hall conductances for Dirac-like Hamiltonians describing magnetic Bloch electrons in a strong magnetic field. We also present their generalizations for irrational values of the deformation parameter.