Quantum Hall Conductivity in a Landau Type Model with a Realistic Geometry II

TL;DR

This paper explores the geometrical and quantum mechanical properties of a finite-size quantum Hall system with various boundary conditions, revealing how the Hall conductivity quantization depends on the ratio of charge carriers to magnetic flux.

Contribution

It introduces a mathematical framework to analyze the geometrical structures influencing Hall conductivity quantization in finite systems with different boundary conditions.

Findings

Quantization depends on the ratio NB/2π, being integral or fractional.

Wave functions are constructed on free group graphs or punctured Riemann surfaces depending on NB/2π.

Results are discussed from a phenomenological perspective.

Abstract

We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractionnal quantization of the Hall conductivity depending on the value of $NB/2\pi$ ($N$ is the number of charge carriers and $B$ is the magnetic field). When $NB/2\pi$ is irrationnal, we show that monovalued wave functions can be constructed only on the graph of a free group with two generators. When $NB/2\pi$ is rationnal, the relevant space becomes a puncturated Riemann surface. We finally discuss our results from a phenomenological viewpoint.