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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.
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A semiclassical constrained Hamiltonian system which was established to study dynamical systems of matrix valued non-Abelian gauge fields is employed to formulate spin Hall effect in noncommuting coordinates at the first order in the constant noncommutativity parameter theta . The method is first illustrated by studying the Hall effect on the noncommutative plane in a gauge independent fashion. Then, the Drude model type and the Hall effect type formulations of spin Hall effect are considered in noncommuting coordinates and theta deformed spin Hall conductivities which they provide are acquired. It is shown that by adjusting theta different formulations of spin Hall conductivity are accomplished. Hence, the noncommutative theory can be envisaged as an effective theory which unifies different approaches to similar physical phenomena.
In order to investigate whether space coordinates are intrinsically noncommutative, we make use of the Hall effect on the two-dimensional plane. We calculate the Hall conductivity in such a way that the noncommutative U(1) gauge invariance is manifest. We find that the noncommutativity parameter theta does not appear in the Hall conductivity itself, but the particle number density of electron depends on theta. We point out that the peak of particle number density differs from that of the charge density.
We discuss the quantum Hall effect on a single-layer graphene in the framework of noncommutative (NC) phase space. We find it induces a shift in the Hall resistivity. Furthermore, comparison with experimental data reveals an upper bound on the magnitude of the momentum NC parameter $eta$ in about $sqrt{eta}leq 2.5 , mathrm{eV}/c$.
When phase space coordinates are noncommutative, especially including arbitrarily noncommutative momenta, the Hall effect is reinvestigated. A minimally gauge-invariant coupling of electromagnetic field is introduced by making use of Faddeev-Jackiw formulation for unconstrained and constrained systems. We find that the parameter of noncommutative momenta makes an important contribution to the Hall conductivity.
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 $Xi$.
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