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

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-$1\mathrm{/}2$ 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.