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arXiv · 2106.07506

The noncommutative geometry of the Landau Hamiltonian: Differential aspects

Giuseppe De Nittis·Maximiliano Sandoval
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Abstract

In this work we study the differential aspects of the noncommutative geometry for the magnetic $C^*$-algebra which is a 2-cocycle deformation of the group $C^*$-algebra of $\mathbb{R}^2$. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic $C^*$-algebra based on the magnetic Dirac operator which is the square root (\`a la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic $C^*$-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

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Giuseppe De Nittis, Maximiliano Sandoval. 2021-06-14. The noncommutative geometry of the Landau Hamiltonian: Differential aspects. https://doi.org/10.1088/1751-8121%2Fac3da4

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