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Different from usual critical point characterized by a single length scale,critical point with emergent symmetry exhibits intriguing critical propertiescharacterized by two relevant length scales, attracting long-terminvestigations from both theoretical and experimental aspects. A naturalquestion is how the critical dynamics is affected by the presence of tworelevant length scales. Here we study the relaxation critical dynamics in thethree-dimensional ($3$D) clock model, whose critical point has emergent $U(1)$symmetry. We find that in contrast to the magnatization $M$, whose relaxationprocess is described by the usual dynamic exponent $z$ of the $3$D $XY$universality class, the angular order parameter $\phi_q$ shows a two-stageevolution characterized by different dynamic critical exponents. While in theshort-time stage the relaxation dynamics is governed by $z$, in the long-timestage the dynamics is controlled by a new dynamic exponent $z'$. We also showthe off-critical-point effects in the critical relaxation. Our results may beexperimentally detected in the hexagonal RMnO$_3$ (R$=$rare earth) materials.

Relaxation Critical Dynamics with Emergent Symmetry