Bloch waves and Non-commutative Tori of Magnetic Translations

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 Kähler 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.

23 pages

References in corpus (4)

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• Hall viscosity and geometric response in the Chern-Simons matrix model of the Laughlin states
• Functional equations for quantum theta functions
• Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory