English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We investigate torque and also conventionally defined spin-Hall currents intwo-dimensional (2D) spin-orbit coupled systems of spin-1/2 particles withinthe linear response Kubo formalism. We obtain some interesting relationsbetween the conventional and torque spin-Hall conductivities for the genericeffective Hamiltonian $H_0=\epsilon_k^0+A(k)\sigma_x-B(k)\sigma_y$, where$A(k)=\eta^A_ik_i+\eta^A_{ij}k_ik_j+\eta^A_{ijl}k_ik_jk_l+...$,$B(k)=\eta^B_ik_i+\eta^B_{ij}k_ik_j+\eta^B_{ijl}k_ik_jk_l+...$, and $\eta$'sare the specific system-dependent coefficients. Specifically, we find that inthe intrinsic case the magnitude of torque spin-Hall conductivity$\sigma^{\tau_z}_{xy}(0)$ is always twice larger than the conventionalspin-Hall conductivity $\sigma^{s_z}_{xy}(0)$, and the two conductivities havethe opposite signs, i.e., $\sigma^{\tau_z}_{xy}(0)=-2\sigma^{s_z}_{xy}(0)$.This universal relation also holds in the presence of an uniform in-planemagnetic field. We also find that if the energy dispersion is rotationallyinvariant, there exists a hyper-angular momentum $I_z = (k\times\partial\theta/\partial k)_z s_z + L_z$ which is conserved. Furthermore, thehyper-angular momentum current $<{1/2}\{I_z,v_x\}>$ vanishes, and this leads toa hyper selection rule for the conventional spin-Hall current.

Torque and conventional spin-Hall currents in two-dimensional spin-orbit coupled systems: Universal relation and hyper-selection rule