Gaussian approximation in the Ising model with long-range interaction

The infinite hierarchy of equations for the cumulants of the correlation functions of the longitudinal spin components in the Ising model with long-range interaction corresponding to a Gaussian form of the spectral density is investigated. The invariance of the cumulants with respect to permutation of the vertices is used to decouple the hierarchy. A phase transition of the second kind is described by the obtained system of equations for the cumulants of the first three orders in a three-dimensional model.