Bloch waves and Non-commutative Tori of Magnetic Translations

TL;DR

This paper reviews the Landau problem, exploring how magnetic translations generate non-commutative tori in quantum systems with rational magnetic flux, and analyzes the structure of Bloch wavefunctions within this framework.

Contribution

It demonstrates that Bloch wavefunctions form modules over non-commutative tori and elucidates the bi-module structure linking Morita equivalent tori in the context of magnetic translations.

Findings

Bloch wavefunctions form finite-dimensional modules over non-commutative tori.

Magnetic translations generate non-commutative tori at rational flux densities.

The bi-module structure connects Morita equivalent non-commutative tori.

Abstract

We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A K\"ahler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction the magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant which is the non-commutative torus of magnetic translation in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori.