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We demonstrate the inadequacy of mean-field theory by exploring the effects of initial-state correlations on the dynamics of continuous time crystals, motivating the need for 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 the cumulant expansion at the second order and demonstrate that it is sufficient to accurately capture the dynamical features overlooked by the mean field.
Mukherjee et al. (Thu,) studied this question.