Quantum Spin-Hall Effect in a Quantum Wire

In this paper, we show that spatially displaced helical states that transport electrons with opposite spins in opposite directions are formed in a harmonic quantum wire due to the spin-orbit (SO) coupling induced by lateral confinement. These states give rise to a transverse spin imbalance in response to the longitudinal charge current as well as a transverse charge imbalance in response to the longitudinal spin current. We analyze the consequential direct and inverse spin-Hall effects in terms of the spin-charge reciprocity relations in a unified manner by virtue of spin conservation. Through a numerical calculation based on the nonequilibrium Green's function formalism, we demonstrate that a generalized spin-Hall resistance that identically describes charge-to-spin and spin-to-charge conversions exhibits a series of quantum plateaus when the SO coupling is sufficiently strong.