Madelung hydrodynamics of spin-orbit coupling: action principles, currents, and correlations

TL;DR

This paper develops a Hamiltonian-based quantum hydrodynamics framework for spin-orbit coupling, revealing distinct force mechanisms, spin transport phenomena, and proposing a new approach to define spin currents in spintronics.

Contribution

It introduces a novel variational and Hamiltonian formulation of spin-orbit coupled quantum hydrodynamics, distinguishing different quantum force mechanisms and addressing open questions in spin current definitions.

Findings

Identifies SOC-induced quantum forces acting on orbital trajectories.

Reveals two mechanisms for quantum spin-orbit correlations.

Proposes a new definition of transport spin current addressing spintronics questions.

Abstract

We exploit the variational and Hamiltonian structures of quantum hydrodynamics with spin to unfold the correlation and torque mechanisms accompanying spin-orbit coupling (SOC) in electronic motion. Using Hamilton's action principle for the Pauli equation, we isolate SOC-induced quantum forces that act on the orbital Madelung--Bohm trajectories and complement the usual force terms known to appear in quantum hydrodynamics with spin. While the latter spin-hydrodynamic forces relate to the quantum geometric tensor (QGT), SOC-induced orbital forces originate from a particular current operator that contributes prominently to the spin current. This distinction between force terms reveals two fundamentally different mechanisms generating quantum spin-orbit correlations. Leveraging the Hamiltonian structure of the hydrodynamic system, we also elucidate spin transport features such as the correlation-induced quantum torques and the current shift in the spin Hall effect. This Hall shift leads to a new definition of the transport spin current thereby addressing an open question in spintronics. Finally, we illustrate the framework via the Madelung--Rashba equations for planar SOC configurations and propose a particle-based scheme for numerical implementation.