Conservation of spin current: Model including self-consistent spin-spin interaction

Spin current defined conventionally, namely, the spin density multiplied by the group velocity, is not a conserved quantity due to possible spin rotations caused by spin-orbit interaction. However, in a model with spin-spin interactions, rotation of a spin causes a dynamic response of surrounding spins that opposes the rotation. Such a many-body effect restores the spin-current conservation. Here we prove that the nonconservation problem of the spin current can be resolved if a self-consistent spin-spin interaction is included in the analysis. We further derive a spin-conductance formula which partitions the spin current into different leads of a multilead conductor.