The geometric structure of the Landau bands

TL;DR

This paper provides a semiclassical analysis of the geometric structure of Landau bands in a two-dimensional system with a periodic electric field, revealing localized and extended states within the bands.

Contribution

It introduces a novel semiclassical method to analyze Landau bands without extra assumptions, using iterative averaging and Reeb graphs to describe their geometric structure.

Findings

Landau bands contain localized states on the wings and extended states near the middle.

Different Landau bands exhibit distinct geometric structures.

The approach accurately separates variables and characterizes the spectrum of Harper-like operators.

Abstract

We have proposed a semiclassical explanation of the geometric structure of the spectrum for the two-dimensional Landau Hamiltonian with a two-periodic electric field without any additional assumptions on the potential. Applying an iterative averaging procedure we approximately, with any degree of accuracy, separate variables and describe a given Landau band as the spectrum of a Harper-like operator. The quantized Reeb graph for such an operator is used to obtain the following structure of the Landau band: localized states on the band wings and extended states near the middle of the band. Our approach also shows that different Landau bands have different geometric structure.