Gaussian approximation in the ising model with long-range interaction

Gaussian Approximation for Ising Model. Variational Approach

“…The considered approximation gives a bit overestimated magnitude for the temperature of a phase transition (t C ≈ 0, 81) (we shall recall that an Onsager solution gives t C ≈ 0, 57). In this manner, the suggested approximation will also describe qualitatively the two-dimensional system, while in a two-tails approximation the solution for an order parameter is missing [8].…”

“…In this manner, we come to a known system of equations (2.12) and (2.13) [4][5][6][7][8], which will describe the behaviour of an order parameter in the two-tails approximation, whereS determines a variance of Gaussian distribution of fluctuations of an order parameter. The formula (2.11) determines a free energy of a system in this approximation and was obtained in [4] by a method of diagram techniques.…”

“…But one of the shortcomings of the mentioned approximations is that at the self-consistent method including the mean field and fluctuations, the ruptures of the order parameter temperature dependence arise which formally can be considered as the first order phase transition. A number of methods with the purpose of removing these shortcomings is known too [6][7][8]. In [6] the Brillouin zone (for the primitive cubic lattice) is modified for this purpose, in [7] a certain number of two-tails is limited.…”

“…In [6] the Brillouin zone (for the primitive cubic lattice) is modified for this purpose, in [7] a certain number of two-tails is limited. A more consequent method is considered in [8], where the necessity is pointed out for taking into consideration the additional series of diagrams which have the same order of quantity as the two-tails approximation diagrams. For this purpose the author of [8] considered the system of equations for spin cumulants and suggested the splitting way of the 4-th order spin cumulant for getting the closed system of equations for the spin cumulants up to the 3-rd order.…”

“…A more consequent method is considered in [8], where the necessity is pointed out for taking into consideration the additional series of diagrams which have the same order of quantity as the two-tails approximation diagrams. For this purpose the author of [8] considered the system of equations for spin cumulants and suggested the splitting way of the 4-th order spin cumulant for getting the closed system of equations for the spin cumulants up to the 3-rd order. Though the author showed that in the vicinity of the critical point the method led to the second order phase transition, the dependence of the order parameter in the wide temperature range is not given which seems to be connected with the numerical analysis complicity of the obtained system of equations.…”