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

Here we apply the n-vicinity method of approximate calculation of the partition function to an Ising Model with the nearest neighbor interaction on D-dimensional hypercube lattice. We solve the equation of state for an arbitrary dimension D and analyze the behavior of the free energy. As expected, for large dimensions (D≥3) the system demonstrates a phase transition of the second kind. In this case, we obtain a simple analytical expression for the critical value of the inverse temperature. When 3≤D≤7 this expression is in a very good agreement with the results of computer simulations. In the case of small dimensions (D=1,2), there is a noticeable discrepancy with the known exact results.