Noncommutative Quantum Hall Effect and Aharonov-Bohm Effect

TL;DR

This paper investigates the effects of noncommutative geometry on the quantum Hall and Aharonov-Bohm phenomena, deriving energy spectra, current expectations, and phase shifts to explore potential experimental bounds on noncommutativity parameters.

Contribution

It introduces a method to analyze noncommutative quantum systems by transforming to commuting coordinates, enabling calculation of physical observables in noncommutative space.

Findings

Derived energy spectrum and Hall conductivity in noncommutative space

Calculated phase shift for Aharonov-Bohm effect under noncommutativity

Proposed experimental bounds on noncommutativity parameters from precession measurements

Abstract

We study a system of electrons moving on a noncommutative plane in the presence of an external magnetic field which is perpendicular to this plane. For generality we assume that the coordinates and the momenta are both noncommutative. We make a transformation from the noncommutative coordinates to a set of commuting coordinates and then we write the Hamiltonian for this system. The energy spectrum and the expectation value of the current can then be calculated and the Hall conductivity can be extracted. We use the same method to calculate the phase shift for the Aharonov-Bohm effect. Precession measurements could allow strong upper limits to be imposed on the noncommutativity coordinate and momentum parameters $\Theta$ and $\Xi$.