Far-from-equilibrium scaling of non-abelian Goldstone modes

Abstract

We identify a broad class of nonthermal phases generated by the interplay of continuous symmetry breaking and weak nonequilibrium driving. Extending Kardar-Parisi-Zhang (KPZ) physics beyond the single $SO(2)$ chronon associated with periodically broken time translations, we construct nonequilibrium nonlinear sigma models for $SO(2)\times O(N)$ symmetry, describing non-Abelian time crystals with coexisting temporal and internal order. This symmetry structure arises naturally in driven quantum materials, active matter, and optically induced periodic states. For rotating and oscillating phases, we derive the Goldstone theories and show that chronon-$O(N)$ couplings remain finite deep in the ordered regime. One-loop renormalization group analysis reveals a KPZ-like dimensional structure: in $d=1,2$, arbitrarily weak nonequilibrium perturbations destabilize the equilibrium fixed point and generate strongly coupled nonthermal fixed points, realizing emergent equilibrium breaking. By contrast, for $d > 2$, weak perturbations are irrelevant and effective equilibrium is restored. A central result is unconventional weak dynamic scaling in the rotating phase: strongly coupled Goldstone sectors acquire distinct universal dynamical exponents despite belonging to the same order parameter. We characterize this scaling analytically and corroborate it through direct simulations in $1+1$ dimensions. In the oscillating phase, we recover and extend weak-scaling regimes known from drifting polymers. Finally, compactness and topological defects ultimately destroy long-range order but leave experimentally accessible nonthermal scaling windows. Together, these results extend KPZ universality to non-Abelian symmetry breaking.