The Non-Dissipative Spin-Hall Conductivity and The Identification of the Conserved Current

The two dimensional Rashba Hamiltonian is investigated using the momentum representation.One finds that the SU(2) transformation which diagonalizes the Hamiltonian gives rise to non commuting Cartesian coordinates for K=0 and zero otherwise.This result corresponds to the Aharonov -Bohm phase in the momentum space. The Spin -Hall conductivity is given by $\frac{e}{4\pi}$ which disagree with the result $\frac{e}{8\pi}$ given in the literature.Using Stokes theorem we find that the Spin -Hall current is carried equally by the up and down electrons on the Fermi surface. We identify the Magnetic current and find that for an electric field with a finite Fourier component in the momentum space a non zero Spin -Hall current is obtained.For the electric field which is constant in space, the orbital magnetic current cancels the spin current. In order to measure the Spin-Hall conductance we propose to apply a magnetic field gradient $\Delta H_{2}$ in the $i=2$ direction and to measure a Charge- Hall current $\frac{e}{h}(\frac{g}{2})\mu_{B}\Delta H_{2}$ in the $i=1$ direction.