Spin Hall effect on a noncommutative space

Abstract

We study the spin-orbital interaction and the spin Hall effect of an electron moving on a noncommutative space under the influence of a vector potential $\mathrm{AP\vec}$. On a noncommutative space, we find that the commutator between the vector potential $\mathrm{AP\vec}$ and the electric potential ${V}_{1}(\mathrm{rP\vec})$ of the lattice induces a new term, which can be treated as an effective electric field, and the spin Hall conductivity obtains some correction. On a noncommutative space, the spin current and spin Hall conductivity have distinct values in different directions, and depend explicitly on the noncommutative parameter. Once this spin Hall conductivity in different directions can be measured experimentally with a high level of accuracy, the data can then be used to impose bounds on the value of the space noncommutativity parameter. We have also defined a new parameter, $\ensuremath{\varsigma}=\ensuremath{\rho}\ensuremath{\theta}$ ($\ensuremath{\rho}$ is the electron concentration, $\ensuremath{\theta}$ is the noncommutativity parameter), which can be measured experimentally. Our approach is based on the Foldy-Wouthuysen transformation, which gives a general Hamiltonian of a nonrelativistic electron moving on a noncommutative space.