Eight-Dimensional Quantum Hall Effect and "Octonions"

Abstract

A generalization of the quantum Hall effect (QHE) where particles move in an eight-dimensional space under an SO(8) gauge field was constructed. A nonrelativistic Hamiltonian for particles moving on the S8 in the presence of the SO(8) gauge field above was also introduced. The particle carry SO(8) numbers and the generators ∑μν, combine nontrivally with the orbital angular momentum Lμν(0) on the equator of S8, to select some distinct special representations of the SO(9) Lab which label the particle. The two eight-dimensional quantum Hall fluids, whose dimensions vary as R20 and R 14, were also constructed.