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Critical slowing down (CSD) has been a trademark of critical dynamics forequilibrium phase transitions of a many-body system, where the relaxation timefor the system to reach thermal equilibrium or quantum ground state divergeswith system size. The time crystal phase transition has attracted muchattention in recent years for it provides a scenario of phase transition ofquantum dynamics, unlike conventional equilibrium phase transitions. Here, westudy critical dynamics near the Floquet time crystal phase transition. Itscritical behavior is described by introducing a space-time coarse grainedcorrelation function, whose relaxation time diverges at the critical pointrevealing the CSD. This is demonstrated by investigating the Floquet dynamicsof one-dimensional disordered spin chain. Through finite-size scaling analysis,we show the relaxation time has a universal sub-exponential scaling near thecritical point, in sharp contrast to the standard power-law behavior for CSD inequilibrium phase transitions. This prediction can be readily tested in presentquantum simulation experiments.

The Sub-Exponential Critical Slowing Down at Floquet Time Crystal Phase Transition