Quantum Hall Effect on Odd Spheres

TL;DR

This paper solves the Landau problem for charged particles on odd-dimensional spheres, determining spectra, degeneracies, and wave functions, and explores geometric structures and connections to fuzzy complex projective spaces.

Contribution

It provides explicit solutions for Landau levels, eigenstates, and geometric interpretations on odd spheres, extending understanding of quantum Hall effects in higher dimensions.

Findings

Spectrum and degeneracies of Landau levels on odd spheres are derived.

Explicit wave functions for the lowest Landau level are constructed.

Connections between Landau levels and fuzzy ${f C}P^3$ are established.

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 ${\cal D}$-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.