Analytical Expressions for Ising Models on High Dimensional Lattices

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions [Formula: see text]. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice [Formula: see text]. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with the results of computer simulations. For [Formula: see text] , our theoretical estimates match the numerical results both qualitatively and quantitatively. For [Formula: see text] , our method is sufficiently accurate for the calculation of the critical temperatures; however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice ([Formula: see text]), this contradicts the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice ([Formula: see text]), the character of the singularity is under current discussion. For the dimensions [Formula: see text] the m-vicinity method is not applicable.