Hall Conductivities for Confined System in Noncommutative Plane

TL;DR

This paper explores the spin Hall effect in a noncommutative plane using generalized quantum mechanics, deriving conductivities as functions of the noncommutative parameter and connecting to known results in the commutative limit.

Contribution

It introduces a novel approach to analyze the spin Hall effect on a noncommutative plane, linking algebraic structures to physical conductivities and reproducing known wavefunctions.

Findings

Charge and spin Hall conductivities depend on the noncommutative parameter .

Standard results are recovered when  is set to zero.

The approach reproduces Laughlin wavefunctions in the commutative limit.

Abstract

We propose an approach based on the generalized quantum mechanics to deal with the basic features of the spin Hall effect. We begin by considering two decoupled harmonic oscillators on the noncommutative plane and determine the solutions of the energy spectrum. We realize two algebras in terms of the quadratic observables and show their importance in filling the shells with fermions. Under some transformation we show that our system is submitted to an effective Lorentz force similar to that acting on one particle in an external magnetic field. From equation of motions, we end up with the charge and spin Hall conductivities as function of the noncommutative parameter \theta. By switching off \theta we recover standard results developed on the subject and in the limit \theta --> 0 we show that our approach can reproduce the Laughin wavefunctions.