Madelung–Rashba Equations Overview

• Madelung–Rashba equations are a hydrodynamic framework for quantum systems with spin–orbit coupling, integrating density, velocity, and spin evolution.
• They are derived from the Pauli equation using phase-space expansions and variational methods, capturing phenomena like drift–diffusion, quantum pressure, and geometric torques in 2D systems.
• The framework features a Lie–Poisson structure and supports accurate numerical simulation via particle-based (bohmion) schemes for applications in topological spintronics and quantum transport.

The Madelung–Rashba equations constitute a unified hydrodynamic framework for quantum systems with spin–orbit coupling (SOC), specifically incorporating the Rashba interaction. These equations extend the classical Madelung hydrodynamics of quantum mechanics to the spinorial case, yielding a coupled system for the particle density, hydrodynamic velocity, and spin vector. They arise from the Pauli equation with SOC via either phase–space kinetic expansions or variational/Hamiltonian reduction, and encode the quantum–geometric, semiclassical, and uniquely quantum correlation effects generated by spin–orbit interactions in two-dimensional electron systems and related platforms (Barletti et al., 2020, Tronci, 15 Jan 2026). Notably, the Madelung–Rashba system captures the interplay of drift–diffusion, quantum pressure, semiclassical forces, geometric torques, and quantum spin–orbit correlations stemming from the Rashba term, and forms the basis for both analytic understanding and variationally exact numerical simulation.

1. Mathematical Formulation of Madelung–Rashba Equations

The starting point is the Pauli equation with spin–orbit interaction: $$H_{\mathrm{Rashba}} = -\frac{\hbar^2}{2m^*}\Delta \,\sigma_0 - i\hbar\alpha_R\,\nabla^\perp \cdot \vec{\sigma}$$ where $\sigma_0$ is the identity, $\vec{\sigma}$ the vector of Pauli matrices, $m^*$ the effective mass, and $\alpha_R$ the Rashba coupling constant.

Applying the Madelung transform to the Pauli spinor $\Psi(\mathbf{x}, t)$,

$$\Psi(\mathbf{x}, t) = \sqrt{D(\mathbf{x}, t)}\, e^{i S(\mathbf{x}, t)/\hbar} \chi(\mathbf{x}, t) ,$$

with $D = \Psi^\dagger \Psi$ the probability density and $\chi$ the normalized local spinor, the hydrodynamic fields are:

• $D(\mathbf{x}, t)$: scalar particle (orbital) density
• $\sigma_0$0: velocity field with $\sigma_0$1 as the Berry connection and $\sigma_0$2 the SOC vector operator
• $\sigma_0$3: local spin vector

The full Madelung–Rashba hydrodynamic system for planar SOC is (Tronci, 15 Jan 2026):

$\sigma_0$4

with $\sigma_0$5 the Madelung–Bohm quantum potential, $\sigma_0$6 the semiclassical SOC torque, $\sigma_0$7 the quantum–metric torque, and $\sigma_0$8 a new Mead current operator (MCO) torque.

2. Quantum Drift–Diffusion and Kinetic Derivation

In the phase–space (Wigner function) approach (Barletti et al., 2020), the density-matrix evolution is cast into the Wigner representation and expanded via the Chapman–Enskog method with a BGK-type relaxation toward the quantum maximum entropy equilibrium. The Wigner function $\sigma_0$9 yields:

• $\vec{\sigma}$0 (scalar density)
• $\vec{\sigma}$1 (spin-density vector)

Projecting the kinetic equation onto these moments produces systems of the form

$\vec{\sigma}$2

with the Wigner-transport operator $\vec{\sigma}$3 encoding both drift and Rashba-driven terms. A crucial feature is the presence of a non-vanishing current at $\vec{\sigma}$4 in $\vec{\sigma}$5 due to SOC, an effect not present in spinless or axis-aligned spin models.

Explicitly, for the charge continuity: $\vec{\sigma}$6 with

$\vec{\sigma}$7

where $\vec{\sigma}$8 and $\vec{\sigma}$9 are Lagrange multipliers, and the Rashba SOC enters through $m^*$0 as well as direct cross-terms in the fluxes.

The corresponding spin-density equation includes leading-order spin torque $m^*$1 and spin currents with further spin–orbit and quantum geometric structure.

3. Quantum–Geometric and Correlation Effects

Variational reduction of the Pauli action exposes two principal forms of quantum–geometric forces induced by SOC (Tronci, 15 Jan 2026):

1. Quantum Geometric Tensor (QGT) Forces: Terms proportional to $m^*$2 (with $m^*$3 the spinor density matrix) generate quantum–metric forces and torques, including diffusive spin precession and corrections to the quantum pressure.
2. Mead Current Operator (MCO) Forces: Terms of the form $m^*$4 yield strictly quantum, non-factorizing spin–orbit correlation forces and novel torques. In the hydrodynamic equations, these manifest as terms with a single derivative of the spin field, distinct from the classical and QGT contributions.

This structure leads to three evolutionary mechanisms for the spin vector:

• Semiclassical SOC torque ($m^*$5)
• Quantum geometric metric torque ($m^*$6)
• MCO quantum correlation torque (proportional to $m^*$7 and directly encoding spin–orbit correlations)

4. Structural Properties and Lie–Poisson Formulation

The full Madelung–Rashba system possesses a Lie–Poisson Hamiltonian structure (Tronci, 15 Jan 2026): $m^*$8 for momentum density $m^*$9 and spin density $\alpha_R$0.

This semidirect product structure ensures

• conservation of total energy,
• explicit identification of Casimir invariants $\alpha_R$1 for arbitrary scalar functions $\alpha_R$2,
• the presence of hydrodynamic transport, precessional, and geometric flow properties inherent to the spin–orbit coupled system.

5. Specialization to Two-Dimensional Electron Gases and Planar SOC

For planar Rashba coupling $\alpha_R$3, the hydrodynamic velocity reduces to: $\alpha_R$4 and the system further simplifies to explicit $\alpha_R$5D forms for continuity, Euler, and spin–evolution equations, including: $\alpha_R$6 with new terms arising in the spin equation from the MCO, which have no classical analogue and encode the quantum–geometric SOC correlation dynamics.

Drift–diffusion approximations of this system produce coupled equations for the particle and spin densities (including full spin vector, not just projections), suitable for modeling two-dimensional electron gases with Rashba SOC, and generalize previous treatments that were either semiclassical or only included axis-projected spin degrees of freedom (Barletti et al., 2020).

6. Numerical Implementation: Bohmion Particle Scheme

A particle-based (bohmion) discretization enables faithful and Hamiltonian-preserving simulation (Tronci, 15 Jan 2026). Key steps include:

• Mollification of density and spin densities with a regularizing kernel $\alpha_R$7, e.g. (Gaussian, width $\alpha_R$8)
• Singular ansatz: $\alpha_R$9, $\Psi(\mathbf{x}, t)$0
• Insertion into a regularized action yields finite-dimensional Hamiltonian ODEs for particle positions $\Psi(\mathbf{x}, t)$1, momenta $\Psi(\mathbf{x}, t)$2, and spins $\Psi(\mathbf{x}, t)$3
• Algorithmic steps (pseudo-code outlined in (Tronci, 15 Jan 2026)) maintain exact conservation of discrete Hamiltonian and Casimirs

This numerical framework leverages the variational foundation to robustly model spin–orbit coupled hydrodynamics, especially relevant for quantum fluids and twodimensional electron gases.

7. Physical Interpretation and Applications

The Madelung–Rashba equations provide a macroscopic, yet fully quantum, description of spin–orbit coupled dynamics in low-dimensional systems. They resolve both the transport and geometric torque effects driving phenomenology such as the spin Hall effect, persistent spin helix, and quantum geometrically-induced spin precession (Tronci, 15 Jan 2026, Barletti et al., 2020). The explicit inclusion of quantum–geometric (QGT) and MCO correlation forces distinguishes this framework from drift–diffusion and semiclassical models, allowing the study of strictly quantum correlation phenomena in SOC systems. Analysis of these equations supports theoretical and computational efforts in topological spintronics, quantum transport, and hydrodynamic electron flow in materials with strong SOC.

References:

L. Barletti, P. Holzinger, A. Jüngel, "Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interaction" (Barletti et al., 2020); G. Tronci et al., "Madelung hydrodynamics of spin-orbit coupling: action principles, currents, and correlations" (Tronci, 15 Jan 2026).