Spin Hall conductivity in insulators with non-conserved spin

TL;DR

This paper derives a formula for spin Hall conductivity in 2D insulators with non-conserved spin, highlighting how spin non-conservation affects quantized spin Hall responses in topological insulators.

Contribution

It introduces a new formula incorporating a correction term for non-conserved spin, extending the understanding of spin Hall effects in topological insulators.

Findings

Correction term scales quadratically with spin-breaking amplitude

Spin Hall conductivity deviates from quantized value when spin is not conserved

Formula applies to models like BHZ and Kane-Mele for topological insulators

Abstract

We study the linear response of a spin current to a small electric field in a two-dimensional crystalline insulator with non-conserved spin. We adopt the spin current operator proposed in [J. Shi et al., Phys. Rev. Lett. 96, 076604 (2006)], which satisfies a continuity equation and fits the Onsager relations. We use the time-independent perturbation theory to present a formula for the spin Hall conductivity, which consists of a "Chern-like" term, reminiscent of the Kubo formula obtained for the quantum Hall systems, and a correction term that accounts for the non-conservation of spin. We illustrate our findings on the Bernevig-Hughes-Zhang model and the Kane-Mele model for time-reversal symmetric topological insulators and show that the correction term scales quadratically with the amplitude of the spin-conservation-breaking terms. In both models, the spin Hall conductivity deviates from the quantized value when spin is not conserved.