Applicability of n-vicinity method for calculation of free energy of Ising model

In a previous work, the n-vicinity method for approximate calculation of the partition function of a spin system was proposed. The equation of state was obtained in the most general form. In the present paper, we analyze the applicability of this method for the Ising model on a D-dimensional cubic lattice. The equation of state is solved for an arbitrary dimension D and the behavior of the free energy is analyzed. As expected, for large dimensions (D > 2) the system demonstrates a phase transition of the second kind. In this case, we obtain an analytical expression for the critical value of the inverse temperature. When 2 < D < 8 this expression is in a very good agreement with the results of computer simulations. In the case of small dimensions (D < 3), there is a noticeable discrepancy with the known exact results.