Topological spin current induced by noncommuting coordinates: An application to the spin-Hall effect

Abstract

A method for computing the spin-Hall conductivity for a two-dimensional electron gas in the presence of the spin-orbit interaction is presented. The spin current is computed using the many body wave function which is degenerate at zero momentum. The degeneracy at $\stackrel{P\vec}{K}=0$ gives rise to noncommuting Cartesian coordinates. The noncommuting Cartesian coordinates are a result of an effective Aharonov-Bohm vortex at $\stackrel{P\vec}{K}=0$. An explicit calculation which avoids the use of a diverging spin-density current in the Rashba model is presented. The conductivity is determined by the linear response theory which has two parts: a static spin-Hall conductivity which is determined by the noncommuting coordinates and has the value $\frac{\ensuremath{\mid}e\ensuremath{\mid}}{4\ensuremath{\pi}}$ and a time-dependent conductivity which renormalizes the static conductivity. The value of this renormalization depends on the spin-orbit polarization energy and Zeeman energy. As a result the spin-Hall conductivity varies between $\frac{\ensuremath{\mid}e\ensuremath{\mid}}{4\ensuremath{\pi}}$ and $\frac{\ensuremath{\mid}e\ensuremath{\mid}}{8\ensuremath{\pi}}$. In the absence of a Zeeman field we find that the long time behavior is given by the renormalized conductivity $\frac{\ensuremath{\mid}e\ensuremath{\mid}}{8\ensuremath{\pi}}$. For relatively small magnetic fields, the Zeeman field allows one to probe continuously the spin-Hall conductivity from the static unrenormalized value $\frac{\ensuremath{\mid}e\ensuremath{\mid}}{4\ensuremath{\pi}}$ to the fully renormalized value $\frac{\ensuremath{\mid}e\ensuremath{\mid}}{8\ensuremath{\pi}}$. When the Zeeman energy exceeds the Fermi energy, only one Fermi-Dirac band is occupied and as a result the static Hall conductivity is half the static spin-Hall conductivity. We compute the uniform magnetization without the Zeeman field and show that the spin current is covariantly conserved and satisfies effectively the continuity equation. The effect of a time-reversal scattering potential due to a single impurity restores the commutation of the coordinates. As a result the spin-Hall current vanishes.