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