Quantum Hall Effect in Higher Dimensions

Abstract

Following recent work on the quantum Hall effect on $S^4$, we solve the Landau problem on the complex projective spaces $\mathbf{C}{P}^k$ and discuss quantum Hall states for such spaces. Unlike the case of $S^4$, a finite spatial density can be obtained with a finite number of internal states for each particle. We treat the case of $\mathbf{C}{P}^2$ in some detail considering both Abelian and nonabelian background fields. The wavefunctions are obtained and incompressibility of the Hall states is shown. The case of $\mathbf{C}{P}^3$ is related to the case of $S^4$.

15 pages, LaTeX

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