Modular structures on trace class operators and applications to Landau levels

The energy levels, generally known as the Landau levels, which characterize the motion ofan electron in a constant magnetic field, are those of the one-dimensional harmonic oscillator,with each level being infinitely degenerate. We show in this paper how the associatedvon Neumann algebra of observables displays a modular structure in the sense of theTomita-Takesaki theory, with the algebra and its commutant referring to the two orientationsof the magnetic field. A KMS state can be built which in fact is the Gibbs state for anensemble of harmonic oscillators. Mathematically, the modular structure is shown to ariseas the natural modular structure associated to the Hilbert space of all Hilbert-Schmidtoperators.