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The variational-cumulant expansion (VCE) method is used to calculate the quantum XY model, for which the trial action does not commute with the action , with arbitrary magnitude of spin on various one-, two- and three-dimensional lattices. The free energy of the model is expanded to the third order and the analytical formulae of critical temperature are given for each order. To testify this method in a wider range, the Heisenberg model on different lattices is also calculated, which has more comparable results. Weighted fitting method of statistical mathematics is adopted to predict of infinite order.