The variational-cumulant-expansion (VCE) method is adopted to calculate the Heisenberg model. The free energy of the model is expanded to the third order, and the analytical formulas for the critical temperature
Tc are given for each order. The first order of VCE corresponds to the mean-field theory, the second order gives the same result for
Tc as the 1/z expansion, and the third order gives a more complicated correction. As a comparison, the results of the two-dimensional (2D) Ising model for VCE to the seventh order are shown. We carry out a trial to predict the critical temperature of the infinite order of the VCE, and get a very accurate result for the 2D Ising model. A prediction of
Tc for the Heisenberg model is also made.
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.