Figure 1

(a) Magnetic insulators with magnetizations m₁ and m₂ on a superconductor. In equilibrium, this yields a spin supercurrent \({{\boldsymbol{J}}}_{{\rm{s}}}^{{\rm{eq}}} \sim {{\boldsymbol{m}}}_{1}\times {{\boldsymbol{m}}}_{2}\). A spin source injects a spin accumulation ρ_s, which exerts a torque on the spins transported by the equilibrium current, resulting in a new contribution \({{\boldsymbol{J}}}_{{\rm{s}}}^{{\rm{neq}}} \sim {{\boldsymbol{\rho }}}_{{\rm{s}}}\times ({{\boldsymbol{m}}}_{1}\times {{\boldsymbol{m}}}_{2})\). (b) If the magnets are magnetized in the x- and y-directions, an equilibrium spin-z supercurrent arises. Injection of spin-z particles does not affect its polarization. Note that a spin supercurrent is in general a rank-2 tensor, encoding both a polarization (short arrow) and transport direction (long arrow). (c) If spin-x particles are injected, however, a new spin-y supercurrent component is generated. Similarly, spin-y injection would produce a spin-x component. We model this setup as a 1D system, where the magnetic insulators connect to the superconductor at the sides; but in the diffusive limit, this should yield physically equivalent results to the setup depicted in this figure.