Topological Spin Current

We show that the $SU(2)$ transformation which diagonalizes the two dimensional spin-orbit hamiltonian has a singularity in the momentum space at $\vec{K} = 0$ which gives rise to non-commuting cartesian coordinates. When an external electric field is applied, the non-commuting cartesian coordinates induce a Hall current. The presence of a random potential in an infinite system causes the single particle occupation at $\vec{K} = 0$ and the Hall current to vanishes. For a finite system, the spin-Hall conductance is quantized in units of $\frac{e g \mu_{B}}{2 h}$, and the charge-Hall conductivity increases with the strength of the Zeeman magnetic field. We propose an experiment to test our theory.