Emergent Nambu-Goldstone phase of Z_{q} model from dynamics

Using extensive simulations, we study the dynamic critical phenomena of the 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_{z} of the oscillating field, where an O(2) symmetry-breaking Nambu-Goldstone phase starts to emerge, persisting up to a higher period P_{o}. The dynamic transition at P_{o} falls into the O(2) 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.