Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 146, 50 pages      arXiv:2006.06785      https://doi.org/10.3842/SIGMA.2020.146
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

Giuseppe De Nittis a and Maximiliano Sandoval b
a) Facultad de Matemáticas & Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile
b) Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

Received June 12, 2020, in final form December 22, 2020; Published online December 28, 2020

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.

Key words: Landau Hamiltonian; spectral triple; Dixmier trace; first Connes' formula.

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