Bloch Electron in a Magnetic Field : Diagonalization of Tight-Binding Models

TL;DR

This paper establishes a connection between tight-binding models of noninteracting electrons in a magnetic field and theta functions, discovering a new symmetry that simplifies diagonalization across the parameter space.

Contribution

It introduces a novel spectrum generating symmetry that reduces the complexity of diagonalizing these models and links eigenvectors at one point to the entire parameter range.

Findings

Discovery of a new spectrum generating symmetry.

Reduction of diagonalization complexity for tight-binding models.

Connection of eigenvectors across parameter space.

Abstract

A connection of a variety of tight-binding models of noninteracting electrons on a rectangular lattice in a magnetic field with theta functions is established. A new spectrum generating symmetry is discovered which essentialy reduces the problem of diagonalization of these models. Provided that one knows one eigenvector at one point in the parameter space of the corresponding Harper equation one knows an eigenfunction of the corresponding model in the whole range of momentum singlet out by the Landau gauge.