Streda-like formula in spin Hall effect

TL;DR

This paper derives a generalized Streda formula for spin transport in spin-orbit coupled systems, accounting for non-conservation of spin and explaining conditions for quantized spin Hall conductivity.

Contribution

It extends the Streda formula to include spin transport, highlighting the effects of spin non-conservation on spin Hall conductance quantization.

Findings

Extra contribution to spin Hall conductance when spin is not conserved

Quantized spin Hall conductivity is exact only when spin z-component is conserved

The generalized formula explains deviations from quantization in realistic systems

Abstract

A generalized Streda formula is derived for the spin transport in spin-orbit coupled systems. As compared with the original Streda formula for charge transport, there is an extra contribution of the spin Hall conductance whenever the spin is not conserved. For recently studied systems with quantum spin Hall effect in which the z-component spin is conserved, this extra contribution vanishes and the quantized value of spin Hall conductivity can be reproduced in the present approach. However, as spin is not conserved in general, this extra contribution can not be neglected, and the quantization is not exact.