  We demonstrate the inadequacy of mean-field theory by exploring the effects
of initial state correlations on the dynamics of continuous time crystals,
necessitating higher-order cumulant expansions. We exemplify this using cat
states for which the mean field fails to predict a phase transition but the
second order cumulant expansion theory captures it. Motivated by the symmetries
of the system, we choose a truncation of cumulant theory at the second-order
and demonstrate that it is sufficient to accurately capture the dynamical
features overlooked by the mean-field.
