Systems with spin-orbit coupling do not conserve "bare" spin current $\bf{ j}$. A recent proposal for a conserved spin current $\bf{ J}$ [J. Shi { \it et.al} Phys. Rev. Lett. { \bf 96}, 076604 (2006)] does not flow persistently in equilibrium. We suggest another conserved spin current $\bar{ \bf{ J}}$ that may flow persistently in equilibrium. We give two arguments for the instability of persistent current of the form $\bf{ J}$: one based on the equations of motions and another based on a variational construction using Lieb-Schulz-Mattis twist operators. In the absence of spin-orbit coupling, the three forms of spin current coincide.
Categories: cond-mat.mes-hall