The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects
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 socalled first Connes' formula) is proved.
 Publication:

SIGMA
 Pub Date:
 December 2020
 DOI:
 10.3842/SIGMA.2020.146
 arXiv:
 arXiv:2006.06785
 Bibcode:
 2020SIGMA..16..146D
 Keywords:

 Landau Hamiltonian; spectral triple; Dixmier trace; first Connes' formula;
 Mathematical Physics;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 81R60;
 58B34;
 81R15;
 81V70
 EPrint:
 Keywords: Landau Hamiltonian