Optimal Gaussian approximation in the Ising model

The effect of spin fluctuations on the magnetic phase transition in the Ising model is studied. The calculation of basic characteristics is reduced to the integration over configurations of a stochastic (fluctuating) field. To evaluate the integrals, the optimal Gaussian approximation of the fluctuating field is constructed. An explicit expression for the system of nonlinear equations that defines the parameters of the optimal Gaussian approximation at each value of temperature is obtained. It is shown that, for weak interaction of spins, the temperature of the phase transition becomes smaller than that in the mean-field theory, but the phase transition remains second order. With an increase of interaction, the solution becomes nonunique at high temperatures and a jump first-order phase transition is observed.