统一Kubo和Keldysh方法用于费米海依赖的非平衡可观测量：应用于自旋霍尔电流和自旋轨道扭矩在自旋电子学中
Reconciling Kubo and Keldysh Approaches to Fermi-Sea-Dependent Nonequilibrium Observables: Application to Spin Hall Current and Spin-Orbit Torque in Spintronics
AI总结：
本文通过无限大石墨烯模型，证明Kubo和Keldysh方法在计算自旋霍尔电流和自旋轨道扭矩时的数值等价性，解决了两种方法在非平衡可观测量中的分歧。
摘要：
在固体中研究自旋依赖现象的量子输运研究通常使用Kubo或Keldysh公式来计算稳态线性响应 regime 中的非平衡密度算子的迹。与感兴趣的算子（如自旋密度、自旋电流密度等）的迹给出实验可测量可观测量的期望值。对于局部量，这些公式要求对费米面和费米海状态的曼尼弗进行求和。然而，文献中关于两种公式似乎能产生截然不同的物理现象的争论一直持续。在这里，我们使用无限大小的石墨烯作为测试平台，该平台具有近邻诱导的自旋轨道和磁交换效应。通过将此系统分为半无限的导线和中央活性区域，在Landauer量子输运公式的精神下，我们证明了Kubo和Keldysh方法的数值精确等价性，通过计算清洁和无序极限下的自旋霍尔电流密度和自旋轨道扭矩。两种方法的统一的关键在于我们为：（i）评估连接半无限导线的系统的Kubo（-Bastin）公式，这确保了连续的能量谱并避免了通常使用的现象学展宽引入的不确定性；以及（ii）正确评估Keldysh方法中的费米海项，该项必须包括中央活性区域的电压降，即使该区域是无序的。
英文摘要：
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.
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