The Non-Commutative Geometry of the Quantum Hall Effect

TL;DR

This paper presents a mathematical framework based on Non-Commutative Geometry to explain the quantization of Hall conductivity and the occurrence of plateaux in the Integer Quantum Hall Effect.

Contribution

It introduces a novel non-commutative geometric approach to analyze the quantum Hall effect, proving quantization and plateau formation within this framework.

Findings

Hall conductivity is quantized within the framework

Plateaux occur when Fermi energy varies in localized states

Mathematical proof of quantization using non-commutative geometry

Abstract

We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using Non-Commutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.