Quantum Hall Effect in Higher Dimensions

TL;DR

This paper extends the study of the quantum Hall effect to higher-dimensional complex projective spaces, solving the Landau problem and analyzing quantum Hall states with detailed examples on ${f C}P^2$ and ${f C}P^3$.

Contribution

It provides explicit solutions for the Landau problem on ${f C}P^k$, including wavefunctions and state properties, expanding understanding of quantum Hall effects in higher dimensions.

Findings

Finite spatial density achievable with finite internal states on ${f C}P^k$

Wavefunctions for quantum Hall states on ${f C}P^2$ are derived

Incompressibility of the quantum Hall states is demonstrated

Abstract

Following recent work on the quantum Hall effect on $S^4$, we solve the Landau problem on the complex projective spaces ${\bf C}P^k$ 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 ${\bf C}P^2$ 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 ${\bf C}P^3$ is related to the case of $S^4$.