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The geometry of (non-abelian) Landau levels. (English) Zbl 1440.81074

Summary: The purpose of this paper is threefold: First of all the topological aspects of the Landau Hamiltonian are reviewed in the light of (and with the jargon of) the theory of topological insulators. In particular it is shown that the Landau Hamiltonian has a generalized even time-reversal symmetry (TRS). Secondly, a new tool for the computation of the topological numbers associated with each Landau level is introduced by combining the Dixmier trace and the (resolvent of the) harmonic oscillator. Finally, these results are extended to models with non-abelian magnetic fields. Two models are investigated in details: the Jaynes-Cummings model and the “Quaternionic” model.

MSC:

81V70 Many-body theory; quantum Hall effect
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
81T13 Yang-Mills and other gauge theories in quantum field theory
82D37 Statistical mechanics of semiconductors
46L60 Applications of selfadjoint operator algebras to physics
58J42 Noncommutative global analysis, noncommutative residues
57R22 Topology of vector bundles and fiber bundles
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
55N25 Homology with local coefficients, equivariant cohomology
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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