Relativistic Landau models and generation of fuzzy spheres (Q2823032)

The paper provides an extensive analysis of the Dirac operator in magnetic field, referred to as the Dirac-Landau operator, on a sphere. Concise expressions for the eigenstates of the relativistic Dirac-Landau operator are found and it is shown that they are related to the eigenstates of the nonrelativistic Pauli-Schroedinger Hamiltonian through a combination of a SU(2) gauge transformation and a linear combination. By projecting to Landau level eigenstates, a family of noncommutative geometries on a sphere, referred to as the fuzzy spheres, are derived and studied. Particularly, it is observed that the radius of the zero-mode relativistic fuzzy sphere is strictly smaller than that of nonrelativistic fuzzy sphere and strictly bigger for all non-zero levels. Furthermore, effects of addition of mass to the Dirac-Landau operator on its spectrum and the fuzzy spheres are considered. It is observed that the spectrum of massive relativistic Dirac-Landau operator is symmetric with respect to the zero level, with a single exception (parity anomality). Also, it is shown that the sum of the radii of fuzzy spheres corresponding to Landau levels of opposite signs is independent of the mass. Finally, several applications of the methods to graphene are presented.