Using extensive simulations, we study the dynamic critical phenomena of the ${\mathbb{Z}}_q$ model in an oscillating field. We find Berezinskii-Kosterlitz-Thouless transitions for $q=6$ in two dimensions and a first-order transition for $q=3$ in three dimensions, hence the equilibrium-nonequilibrium correspondence is conditionally validated. Surprisingly, the correspondence is apparently violated at $q=5$, 6, and 7 in three dimensions, for which the equilibrium and nonequilibrium scenarios feature a single transition and two transitions, respectively. In the nonequilibrium case, the discrete $q$-fold order is washed away by the first dynamic transition at the period $P_{\mathrm{z}}$ of the oscillating field, where an O(2) symmetry-breaking Nambu-Goldstone phase starts to emerge, persisting up to a higher period $P_{\mathrm{o}}$. The dynamic transition at $P_{\mathrm{o}}$ falls into the $\mathrm{O}\left(2\right)$ critical universality class. We relate the highly nontrivial dynamic behaviors to the two-length criticality arising from dangerously irrelevant fields and find that the dynamics decode the extremely subtle equilibrium criticality. Our findings significantly expand the current understanding of emergent phenomena and offer a promising methodology for elucidating critical phenomena that are notoriously challenging.

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• Continuous phase transition
• Dynamic critical phenomena
• Monte Carlo methods