Quaternion Generalization of the Laughlin State and the Three Dimensional Fractional QHE

Abstract

The 3D state of strongly correlated electrons is proposed, which in the external magnetic field $\vec B$ exhibits the fractional quantum Hall effect, with the zero temperature conductivity tensor $σ_{ij} = (e^2/h)(1/m) \sum_k ε_{ijk} B^k/\mid \vec B\mid $. The analog of Landau and Laughlin states in 3D are given using quaternion coordinates as generalization of complex coordinates. We discuss the notion of the fractional statistics in 3D introduced recently by Haldane.