Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bloch waves and Non-commutative Tori of Magnetic Translations

Tekin Dereli, Todor Popov

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.

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.

Paper Structure

This paper contains 11 sections, 5 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Landau levels and their degeneracy
  3. Quantization and Kähler polarization.
  4. Magnetic Translations
  5. Quantum Tori
  6. Magnetic Translation Eigenfunctions
  7. Modular Invariance and Coherent States
  8. Metaplectic Representations and Squeezed States
  9. Matrix algebras and Morita equivalence
  10. Conclusion and Outlook
  11. Acknowledgement

Key Result

Proposition 6.1

Let the parameter $\kappa$ be a rational number $\kappa = \frac{N}{M}$ with $M$ and $N$ coprimes, $gcd(M,N)=1$. Then the degree of degeneracy of each Landau level is $MN$.

Theorems & Definitions (8)

  • Definition 4.1
  • Proposition 6.1
  • Definition 6.2
  • Proposition 6.3
  • Lemma 7.1
  • Lemma 9.1
  • Definition 9.2
  • Proposition 9.3