Conservation and persistence of spin currents and their relation to the Lieb-Schulz-Mattis twist operators

Systems with spin-orbit coupling do not conserve "bare" spin current j. A recent proposal for a conserved spin current J does not flow persistently in equilibrium. We suggest another conserved spin current J̄ that may flow persistently in equilibrium. We give two arguments for the instability of persistent current of the form 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.