Quantum Hall Effect on Odd Spheres

Abstract

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 $\mathcal{D}$-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 $\pm 1$ line bundles over $\mathbb{C}{P}^3$ at level $K$, respectively.

14+1 pages

References in corpus (20)

• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• A Four Dimensional Generalization of the Quantum Hall Effect
• Quantum Hall Effect in Higher Dimensions
• The Eight Dimensional Quantum Hall Effect and the Octonions
• The effective action for edge states in higher dimensional quantum Hall systems
• How Bob Laughlin Tamed the Giant Graviton from Taub-NUT space
• Higher-Dimensional Quantum Hall Effect in String Theory
• Edge States for quantum Hall droplets in higher dimensions and a generalized WZW model
• The Spectrum of the Dirac Operator on Coset Spaces with Homogeneous Gauge Fields
• Quantum Hall effect on $S^3$, edge states and fuzzy $S^3/{\mathbf{Z}}_2$
• Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
• Dimensional Hierarchy in Quantum Hall Effects on Fuzzy Spheres
• High-Dimensional Topological Insulators with Quaternionic Analytic Landau Levels
• Topological insulators with SU(2) Landau levels
• Isotropic Landau levels of Dirac fermions in high dimensions
• Chiral Topological Insulator on Nambu 3-Algebraic Geometry
• Dirac Operators on Coset Spaces
• A Fuzzy Three Sphere and Fuzzy Tori
• Quantum Hall Effect on the Flag Manifold F_2
• Quantum Hall Effect on the Grassmannians ${\mathbf{Gr}}_2({\mathbb{C}}^N)$

Cited by in corpus (5)

• Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy
• $SO(4)$ Landau Models and Matrix Geometry
• $SO(5)$ Landau Model and 4D Quantum Hall Effect in The $SO(4)$ Monopole Background
• Extended Dynamical Symmetries of Landau Levels in Higher Dimensions
• A Unified Construction of Skyrme-type Non-linear sigma Models via The Higher Dimensional Landau Models