Landau and Dirac-Landau problem on odd-dimensional spheres

2017
Coşkun, Ümit Hasan
In this thesis, solutions of the Landau and Dirac-Landau problems for charged particles on odd-dimensional spheres S(2k-1) in the background of constant SO(2k-1) gauge fields are presented. Firstly, reviews of the quantum Hall effect and in particular the Landau problem on the two-dimensional sphere S2 and and all even-dimensional spheres, S(2k), are given. Then, the key ideas in these problems are expanded and adapted to set up the Landau problem on S(2k-1). Using group theoretical methods, the energy levels of the appropriate Landau Hamiltonian together with its with degeneracies are determined. The corresponding wave functions are given in terms of the Wigner D-functions of the symmetry group SO(2k) of S(2k-1). The explicit local forms of the lowest Landau level wave functions are constructed for a particular set of SO(2k-1) gauge field background charges. We access the constant SO(2k-2) gauge field backgrounds on the equatorial S(2k-2) and obtain the differential geometric structures on the latter by forming the relevant projective modules. Finally, we examine the Dirac-Landau problem on S(2k-1) and obtain the energy spectrum, degeneracies and number of zero modes of the gauged Dirac operator on S(2k-1). 

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Citation Formats
Ü. H. Coşkun, “Landau and Dirac-Landau problem on odd-dimensional spheres,” M.S. - Master of Science, Middle East Technical University, 2017.