Finite-time scaling with two characteristic time scales: Driven critical dynamics with emergent symmetry

Critical points with emergent symmetry exhibit intriguing scaling properties induced by two divergent length scales, attracting extensive investigations recently. We study the driven critical dynamics in a three-dimensional q-state clock model, in which the ordered phase breaks the Zq discrete symmetry, while an emergent U (1) symmetry appears at the critical point. By increasing the temperature at a finite velocity v to traverse the critical point from the ordered phase, we uncover rich dynamic scaling properties beyond the celebrated Kibble-Zurek mechanism. Our findings reveal the existence of two finite-time scaling (FTS) regions, characterized by two driving-induced time scales ζd v^-z/r and ζd' v^-z/r', respectively. Here z is the dynamic exponent, r is the usual critical exponent of v, and r' represents an additional critical exponent of v associated with the dangerously irrelevant scaling variable. While the square of the order parameter M² obeys the usual FTS form, the angular order parameter ϕq shows remarkably distinct scaling behaviors controlled by both FTS regions. For small v, ϕq is dominated by the time scale ζd, whereas for large v, ϕq is governed by the second time scale ζd'. We verify the universality of these scaling properties in models with both isotropic and anisotropic couplings. Our theoretical insights provide a promising foundation for further experimental investigations in the hexagonal RMnO₃ (R=rare earth) materials.