Noncommutative Description of Quantum Spin Hall Effect

TL;DR

This paper introduces a noncommutative quantum mechanics framework to analyze the intrinsic spin Hall effect, revealing a magnetic-field-free Hall conductivity dependent on noncommutativity parameters.

Contribution

It presents a novel approach using noncommutative geometry to model the spin Hall effect and connects different existing interpretations through a unified framework.

Findings

Hall conductivity exists without external magnetic field

Conductivity depends on noncommutativity parameter 

Different approaches to spin Hall effect are recoverable

Abstract

We propose an approach based on a generalized quantum mechanics to deal with the basic features of the intrinsic spin Hall effect. This can be done by considering two decoupled harmonic oscillators on the noncommutative plane and evaluating the spin Hall conductivity. Focusing on the high frequency regime, we obtain a diagonalized Hamiltonian. After getting the corresponding spectrum, we show that there is a Hall conductivity without an external magnetic field, which is noncommutativity parameter \theta-dependent. This allows us to make contact with the spin Hall effect and also give different interpretations. Fixing \theta, one can recover three different approaches dealing with the phenomenon.