The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

TL;DR

This paper constructs a noncommutative geometric framework for the quantum Hall effect, focusing on the spectral triple of magnetic operators and analyzing its metric properties, including a key formula from Bellissard's theory.

Contribution

It introduces a spectral triple for magnetic operators in the quantum Hall effect and proves a fundamental metric formula within this noncommutative setting.

Findings

Established a spectral triple for magnetic operators

Proved the first Connes' formula in this context

Analyzed the metric properties of the spectral triple

Abstract

This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.