Reconciling Kubo and Keldysh approaches to Fermi-sea-dependent nonequilibrium observables:Application to spin Hall current and spin-orbit torque in spintronics

Quantum transport studies of spin-dependent phenomena in solids commonly employ the Kubo or Keldysh formulas for the nonequilibrium density operator in the steady-state linear-response regime. Its trace with operators of interest, such as the spin density, spin current density, and so on, gives expectation values of experimentally accessible observables. As is well known, for local observables, these formulas require summing over the manifolds of both Fermi-surface and Fermi-sea states. However, the significantly different results yielded by the two formulations when applied to the same system have ignited vigorous debates. Here, we revisit this problem using an infinite-size graphene system with proximity-induced spin-orbit and magnetic exchange effects as a test bed. By considering such system as being composed of central active region in between two semi-infinite leads, in the spirit of the Landauer setup for quantum transport, we prove the numerically exact equivalence of the Kubo and Keldysh approaches via the computation of spin Hall current density and spin-orbit torque in both clean and disordered limits. The key to reconciling the two approaches is the numerical frameworks we put forward for (i) evaluation of the Kubo(-Bastin) formula for a system attached to semi-infinite leads, which ensures a continuous energy spectrum and evades the need for commonly used phenomenological broadening otherwise responsible for ambiguities, and (ii) proper evaluation of the Fermi-sea term in the Keldysh approach, which must include the voltage drop across the central active region even if it is disorder free.