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

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 and whose contribution was overlooked in the past. The distinction between different 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 current shift in the spin Hall effect and the correlation-induced quantum torques. Finally, we illustrate the framework via the Madelung--Rashba equations for planar SOC configurations and propose a particle-based scheme for numerical implementation.