Relaxation critical dynamics with emergent symmetry

Universal critical properties can manifest themselves not only in spatial but also in temporal directions. It has been found that critical point with emergent symmetry exhibits intriguing spatial critical properties characterized by two divergent length scales, attracting long-term investigations. However, how the temporal critical properties are affected by emergent symmetry is largely unknown. Here, we study the nonequilibrium critical dynamics in the three-dimensional ($3\mathrm{D}$) clock model, whose critical point has emergent $U(1)$ symmetry. We find that in contrast to the magnetization $M$, whose relaxation process is described by the usual dynamic exponent $z$ of the $3\mathrm{D}$ XY universality class, the angular order parameter ${\ensuremath{\phi}}_{q}$ shows a remarkable two-stage evolution characterized by different dynamic critical exponents. While in the short-time stage the relaxation dynamics is governed by $z$, in the long-time stage the dynamics is controlled by a new dynamic exponent ${z}^{\ensuremath{'}}$. Further scaling analyses confirm that ${z}^{\ensuremath{'}}$ is an indispensable dynamic critical exponent. Our results may be detected in the hexagonal $R{\mathrm{MnO}}_{3}$ ($R=\mathrm{rare}$ earth) materials experimentally.