Rank polarization and dynamical autonomy in a projective rotor model

Abstract

We study how a higher-rank classical collective order parameter can select one dominant projective eigenline and, in a $p=2$ realization, support a dynamically autonomous low-energy director sector. In a fully connected projective-rotor model, the collective second moment is a positive semidefinite tensor $T$ of fixed trace. Conventional projective ordering drives its effective rank toward one, selecting a single dominant eigenline. Rank polarization alone, however, does not determine dynamical autonomy: extrinsic-curvature criteria depend on the chosen state-space metric, and the equilibrium tensor does not determine the transverse free-energy Hessian. A two-harmonic model makes this logical independence explicit by producing families with identical $T$ and $r_{\rm eff}$ but different transverse quadratic forms. We then construct rotationally invariant locking interactions that leave a prescribed ordered tensor exactly fixed while increasing every non-Goldstone restoring coefficient as $Θ(λ)$ within a globally controlled stable regime. For a phenomenological inertial realization with a finite positive kinetic matrix, the lowest transverse normal-mode frequency grows as $Θ(λ^{1/2})$; more generally, the low-frequency transverse susceptibility is suppressed as $O(λ^{-1})$, including in overdamped dynamics. The construction therefore proves that spectral selection and infrared transverse suppression are independent data. For $p>2$ the selected eigenline has a $(p-1)$-dimensional director manifold; the one-dimensional Goldstone sector used in the dynamical construction is specific to $p=2$. Promoting the selected internal eigenline to a physical clock requires additional soldering and integrability structure.