Towards the fractional quantum Hall effect: a noncommutative geometry perspective

TL;DR

This paper surveys models of the integer and fractional quantum Hall effects using noncommutative geometry, highlighting geometric interpretations and deriving fractional conductance values aligned with experimental data.

Contribution

It introduces a geometric framework for understanding the fractional quantum Hall effect through hyperbolic geometry and orbifold Euler characteristics, extending classical models.

Findings

Derived fractional Hall conductance as orbifold Euler characteristics

Connected noncommutative geometry models with experimental data

Compared Euclidean and hyperbolic geometric models for quantum Hall effects

Abstract

In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall how one can obtain this way a single electron model of the integer quantum Hall effect. While in the case of the integer quantum Hall effect the underlying geometry is Euclidean, we then discuss a model of the fractional quantum Hall effect, which is based on hyperbolic geometry simulating the multi-electron interactions. We derive the fractional values of the Hall conductance as integer multiples of orbifold Euler characteristics. We compare the results with experimental data.