Quantum Hall Effect on Odd Spheres
Autor: | Ü. H. Coşkun, Goksu Can Toga, Seckin Kurkcuoglu |
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Jazyk: | angličtina |
Rok vydání: | 2016 |
Předmět: |
Physics
High Energy Physics - Theory 010308 nuclear & particles physics Winding number Hilbert space FOS: Physical sciences Landau quantization Quantum Hall effect Dirac operator 01 natural sciences symbols.namesake High Energy Physics - Theory (hep-th) Quantum mechanics Irreducible representation 0103 physical sciences symbols Gauge theory 010306 general physics Hamiltonian (quantum mechanics) Mathematical physics |
Popis: | We solve the Landau problem for charged particles on odd-dimensional spheres $S^{2k-1}$ in the background of constant SO(2k-1) gauge fields carrying the irreducible representation $\left ( \frac{I}{2}, \frac{I}{2}, \cdots, \frac{I}{2} \right)$. We determine the spectrum of the Hamiltonian, the degeneracy of the Landau levels and give the eigenstates in terms of the Wigner ${\cal D}$-functions, and for odd values of $I$ the explicit local form of the wave functions in the lowest Landau level (LLL). Spectrum of the Dirac operator on $S^{2k-1}$ in the same gauge field background together with its degeneracies is also determined and in particular the number of zero modes is found. We show how the essential differential geometric structure of the Landau problem on the equatorial $S^{2k-2}$ is captured by constructing the relevant projective modules. For the Landau problem on $S^5$, we demonstrate an exact correspondence between the union of Hilbert spaces of LLL's with $I$ ranging from $0$ to $I_{max} = 2K$ or $I_{max} = 2K+1$ to the Hilbert spaces of the fuzzy ${\mathbb C}P^3$ or that of winding number $\pm1$ line bundles over ${\mathbb C}P^3$ at level $K$, respectively. 14+1 pages |
Databáze: | OpenAIRE |
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