Minimal spin-rotor model for Barnett and Einstein--de Haas physics

Abstract: The Barnett effect is usually understood through an effective magnetic field generated by mechanical rotation, while its reciprocal Einstein--de Haas effect describes the transfer of spin angular momentum to mechanical motion. We show that this effective-field picture changes qualitatively once the mechanical degree of freedom itself is quantized. To demonstrate this, we introduce an exactly solvable minimal spin-rotor model in which a spin-$1/2$ is coupled to a quantum rotor. In a fixed angular-momentum sector, the model reproduces the conventional Barnett splitting and remains formally equivalent to a Zeeman problem. For a superposition of rotor sectors, however, the Barnett field becomes operator-valued and the resulting dynamics generates coherent spin-rotor entanglement. This is directly visible in the reduced spin purity, rotor coherence, and entanglement entropy. Our results identify a minimal quantum setting in which the Barnett effective-field picture departs from its classical form and acquires a reciprocal manifestation through spin-dependent rotor coherence.

• The paper introduces an exactly solvable spin-rotor model that clarifies the quantum-classical crossover in spin-mechanical coupling.
• It models a minimal composite system of a spin-1/2 and a compact rotor, revealing how quantum superpositions lead to entanglement and invalidate classical effective fields.
• The study quantitatively assesses quantum effects via spin purity, entanglement entropy, and rotor coherence, highlighting potential impacts on quantum sensors and spintronics.

Minimal Spin-Rotor Model for Quantum Barnett and Einstein–de Haas Effects

Overview

The paper "Minimal spin-rotor model for Barnett and Einstein--de Haas physics" (2604.23768) introduces an exactly solvable quantum mechanical model that probes the limits of the effective-field description of spin-mechanical coupling, relevant to both the Barnett and Einstein–de Haas effects. The work focuses on a minimal composite system: a single spin-$1/2$ coupled to a compact quantum rotor. By quantizing the mechanical degree of freedom and examining the crossover from classical to quantum regimes, the paper clarifies when the traditional Barnett effective-field picture breaks down, revealing entanglement and genuinely quantum reciprocal effects.

Model and Physical Context

The core of the analysis is the Hamiltonian:

${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$

where:

• $L_z$ is the quantized rotor angular momentum,
• $I$ is the moment of inertia,
• $\Delta$ is a transverse spin splitting,
• $S_x$ and $S_z$ are spin operators.

The model's simplicity isolates the essential physics of spin-rotation coupling: specifically, how angular momentum transfer between spin and mechanics is fundamentally modified when the mechanical degree of freedom is non-classical. Unlike Rabi or Dicke models where coupling is to a harmonic oscillator coordinate, here the relevant variable is the compact rotor's angular momentum, which is conserved. This distinction precludes superradiant transitions and simplifies the analysis of spin-mechanical entanglement.

The model can serve as an effective description for spin-orbit coupled systems in molecular or solid-state environments (e.g., transition metal complexes) where the mechanical environment's quantization may be relevant.

Exact Solution and Classical Limit

The spin-rotor Hamiltonian is block-diagonal in $L_z$, with each sector labeled by integer $m$:

${\cal H}_m = \frac{m^2}{2I} + \Delta S_x + m S_z.$

This is a two-level problem per ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$0, with exact eigenenergies:

${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$1

Thus, when the rotor is in a definite angular momentum eigenstate ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$2, the spin sector experiences a static effective “Barnett field” along the ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$3-direction, formally analogous to a Zeeman splitting ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$4. This is the classical Barnett regime.

Breakdown of the Classical Effective-Field Picture

The situation changes when the rotor is in a quantum superposition of different ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$5 eigenstates, e.g., ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$6. Each subspace evolves with a different effective field, and quantum correlations (entanglement) arise between the spin and rotor degrees of freedom. The time evolution of the composite system is:

${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$7

where ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$8 is the initial spin state.

Tracing over the rotor or spin degrees of freedom reveals reduced density matrices with nontrivial purity and coherence properties. If the rotor is in a superposition (e.g., ${\cal H} = \frac{L_z^2}{2I} + \Delta S_x + L_z S_z$9), spin evolution becomes sector-dependent, leading to loss of purity in the reduced spin (or rotor) subsystem. The classical picture, where the "Barnett field" is simply a constant set by angular velocity, fails: the effective field becomes operator-valued, determined by quantum correlations.

Quantitative Characterization of Quantum Effects

The entanglement between spin and rotor is quantified explicitly:

• Spin purity: $L_z$0 where $L_z$1 measures the overlap between spin states evolving under opposite angular momentum sectors.
• Entanglement entropy: $L_z$2 with $L_z$3.

At times and parameters where $L_z$4 (e.g., for balanced coupling, $L_z$5), there is maximal spin-rotor entanglement and the reduced state is maximally mixed. The reduction in spin purity and oscillatory entanglement entropy clearly distinguish the quantum regime.

On the rotor side, coherence between superposed angular momentum sectors decays due to backaction from the spin, even though total evolution remains unitary. As such, spin-induced loss of rotor coherence represents a quantum reciprocal effect analogous to Einstein–de Haas backaction, but for coherence rather than classical torque.

Theoretical and Practical Implications

The analysis demonstrates that even in a minimal, exactly solvable model, the effective-field description of spin-mechanical effects ceases to hold when the mechanical degree of freedom is quantum. The first correction to the classical Barnett effect is not a perturbative renormalization but the onset of coherent entanglement, marked by observable reductions in subsystem purity and entropy. Conversely, reciprocal effects are seen as changes in rotor coherence rather than mechanical motion.

These insights highlight the necessity of operator-based treatments for spin-mechanical coupling in quantum devices and point to nontrivial decoherence and entanglement phenomena in quantum sensors, nano-mechanical systems, and spintronics platforms where quantized mechanical motion couples to well-controlled spin states.

The framework also provides a useful platform for extensions to more complex driven, dissipative, or many-body systems, where quantum limits of angular momentum transfer may play a critical role in the dynamics of hybrid quantum devices.

Conclusion

This work establishes a rigorous quantum foundation for the Barnett and Einstein–de Haas effects, showing that in the quantum regime, the cross-coupling between spin and mechanical rotation must be treated as a source of entanglement, not simply a classical effective field. These findings have important implications for quantum technologies where mechanical and spin degrees of freedom interconvert, and they motivate further theoretical work on the quantum-classical boundary and entangled spin-mechanical dynamics in more complex settings.