Hall Effect in Noncommutative Coordinates

TL;DR

This paper explores the quantum Hall effect for electrons in a plane with noncommuting coordinates, deriving spectra, currents, and phases, and discusses how noncommutativity influences observable phenomena like the Aharonov-Bohm phase.

Contribution

It provides a novel analysis of the Hall effect in noncommutative geometry, including spectrum calculation, current derivation, and path integral formulation with implications for quantum Hall physics.

Findings

Spectrum and eigenfunctions are obtained for electrons in noncommutative coordinates.

An effective magnetic field is derived from the expectation value of the electric current.

The Aharonov--Bohm phase in this setting is calculated and shown to be independent of magnetic field at strong fields.

Abstract

We consider electrons in uniform external magnetic and electric fields which move on a plane whose coordinates are noncommuting. Spectrum and eigenfunctions of the related Hamiltonian are obtained. We derive the electric current whose expectation value gives the Hall effect in terms of an effective magnetic field. We present a receipt to find the action which can be utilized in path integrals for noncommuting coordinates. In terms of this action we calculate the related Aharonov--Bohm phase and show that it also yields the same effective magnetic field. When magnetic field is strong enough this phase becomes independent of magnetic field. Measurement of it may give some hints on spatial noncommutativity. The noncommutativity parameter \theta can be tuned such that electrons moving in noncommutative coordinates are interpreted as either leading to the fractional quantum Hall effect or composite fermions in the usual coordinates.