Unconventional criticality in $O(D)$-invariant loop-constrained Landau theory

We study an unconventional phase transition in ferroelectrics where the polarization field is constrained to be divergence-free, allowing only loop-like configurations. This local constraint fundamentally alters the critical behavior, driving the system beyond the Landau-Ginzburg-Wilson paradigm. A renormalization group analysis shows that the polarization acquires an unusually large anomalous dimension, $\eta\approx 0.239$ in three dimensions, far exceeding the typical values in $O(3)$-invariant systems. We attribute this effect to a naturally induced gauge symmetry originating from the zero divergence constraint. Such gauge-field behavior is reminiscent of fractionalized phases, revealing a fundamental connection between constrained ferroelectrics and emergent gauge phenomena in correlated matter.