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Experiments observing spin density and spin currents (responsible for, e.g.,spin-transfer torque) in spintronic devices measure only the nonequilibriumcontributions to these quantities, typically driven by injecting unpolarizedcharge current or by applying external time-dependent fields. On the otherhand, theoretical approaches to calculate them operate with both thenonequilibrium (carried by electrons around the Fermi surface) and theequilibrium (carried by the Fermi sea electrons) contributions. Thus, anunambiguous procedure should remove the equilibrium contributions, therebyrendering the nonequilibrium ones which are measurable and satisfy thegauge-invariant condition according to which expectation values of physicalquantities should not change when electric potential everywhere is shifted by aconstant amount. Using the framework of nonequilibrium Green functions, wedelineate such procedure which yields the proper gauge-invariant nonequilibriumdensity matrix in the linear-response and elastic transport regime forcurrent-carrying steady state of an open quantum system connected to twomacroscopic reservoirs. Its usage is illustrated by computing: (i) conventionalspin-transfer torque (STT) in asymmetric F/I/F magnetic tunnel junctions(MTJs); (ii) unconventional STT in asymmetric N/I/F semi-MTJs with the strongRashba spin-orbit coupling (SOC) at the I/F interface and injected currentperpendicular to that plane; and (iii) current-driven spin density within aclean ferromagnetic Rashba spin-split two-dimensional electron gas (2DEG) whichgenerates SO torque in laterally patterned N/F/I heterostructures when such2DEG is located at the N/F interface and injected charge current flows parallelto the plane.

How to construct the proper gauge-invariant density matrix in steady-state nonequilibrium: Applications to spin-transfer and spin-orbit torques