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 reff 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(λ1/2); 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.
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BibTeX
@misc{wang2026rank,
title = {{Rank polarization and dynamical autonomy in a projective rotor model}},
author = {Shannon Wang and David Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15361},
eprint = {2610.15361},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15361},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rank polarization and dynamical autonomy in a projective rotor model
AU - Shannon Wang
AU - David Wang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15361
UR - https://arxiv.org/abs/2610.15361
AB - 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_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.
ER -