Reconciling Kubo and Keldysh Approaches to Fermi-Sea-Dependent Nonequilibrium Observables: Application to Spin Hall Current and Spin-Orbit Torque in Spintronics

TL;DR

This paper demonstrates the exact equivalence of Kubo and Keldysh formalisms in quantum transport for spintronic phenomena, resolving longstanding debates by developing numerical frameworks for their consistent application in systems with leads and disorder.

Contribution

It provides a rigorous proof of the equivalence between Kubo and Keldysh approaches for Fermi-sea-dependent observables using graphene as a testbed, with new numerical methods for their consistent evaluation.

Findings

Kubo and Keldysh formalisms are numerically equivalent for spin transport.

Proper inclusion of voltage drop is essential in Fermi-sea calculations.

New numerical frameworks enable accurate evaluation in systems with leads and disorder.

Abstract

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, etc., gives expectation values of experimentally accessible observables. For local quantities, these formulas require summing over the manifolds of {\em both} Fermi-surface and Fermi-sea states. However, debates have been raging in the literature about the vastly different physics the two formulations can apparently produce, even when applied to the same system. Here, we revisit this problem using infinite-size graphene with proximity-induced spin-orbit and magnetic exchange effects as a testbed. By splitting this system into semi-infinite leads and central active region, in the spirit of Landauer formulation of quantum transport, we prove the {\em 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 are the numerical frameworks we develop for: ({\em i}) evaluation of Kubo(-Bastin) formula for a system attached to semi-infinite leads, which ensures continuous energy spectrum and evades the need for commonly used phenomenological broadening introducing ambiguity; and ({\em ii}) proper evaluation of Fermi-sea term in the Keldysh approach, which {\em must} include the voltage drop across the central active region even if it is disorder free.