Spin torque in a magnetic trilayer coupled to a superconductor

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A ferromagnet-normal-metal-ferromagnet trilayer does not conserve spin current and as a result, the conduction electrons can create a torque on the magnetic moments. When the trilayer is connected to a superconducting electrode, the spin torque created by a current flowing perpendicularly to the trilayer can drive the system to a configuration where the two magnetic moments are perpendicular to each other. Here we argue that in contrast to the non-equilibrium torque, the equilibrium torque (or magnetic exchange interaction) does not single out the perpendicular configuration. We calculate the equilibrium torque for a one-dimensional lattice model and find that it is sensitive to the presence of the superconductivity.