Approximate method of free energy calculation for spin system with arbitrary connection matrix

The proposed method of the free energy calculation is based on the approximation of the energy distribution in the microcanonical ensemble by the Gaussian distribution. We hope that our approach will be effective for the systems with long-range interaction, where large coordination number q ensures the correctness of the central limit theorem application. However, the method provides good results also for systems with short-range interaction when the number q is not so large.

coincides perfectly with the density of the Gaussian distribution (see figure 1). Outside of this interval the densities differ, however for our purposes these distinctions on the tails are not pivotal. Then the summation over the class n  can be replaced by integration over the Gaussian measure, and the expression for the partition function takes the form where the minimal and maximal values of the energy ( ) E s are taken as the inferior and superior limits of integration. This technique is essential for our approach. Now the calculation of the partition function reduces to the calculation of the integral with the aid of the saddle-point method.

The Main Equation
In the asymptotic limit   N the expressions for n E and 2 n  take the form: Using the Stirling asymptotic and replacing the summation over n with the integration over the variable N n x /  , we obtain: , and then  (3), we finally obtain: The free energy per site is equal to

The Ising Model on a Hypercube
Our approach is theoretically based for 1  q . Nevertheless, rather good results can also be obtained for comparatively small values of q .
For D -dimensional Ising model in each row of the matrix T there are only 2 q D  nonzero elements, and they all are equal to J . Here the ground state is the configuration 0 (1,1,. It is easy to solve the equation d / d 0 f x  and to examine the properties of the solution (see [7]). At , and the difference between this value and the estimate 0.149 obtained in [5] is even less. In general, when q is large enough, from Eq.(6) we obtain 1 / c q   . This is in agreement with the supposition that for large dimensionalities D the Ising model shows indices typical for the mean field model [5], [6]. Now let us show the results for 16 / 3 q  . For 2D-Ising model ( 4 q  ) there are some notable distinctions in the scenario described above. (Because of the length of this paper it is impossible to present them in this paper.) Here no analytic expression for the critical value c  can be obtained; is a solution of a transcendental equation. This value differs from the known exact solution 0.4407 [3]. For 1D-Ising model ( 2 q  ) our approach does not work at all: the state of the   is the ground state again, and Eq.(5) is still true. In particular, it works for Ising model on the Bete lattice [3], when each spin interacts with q nearest neighbors. Equation (6) is in qualitative accord with the exact expressions obtained for the Bete lattice. Both results coincide, when q   .
When for any n we have 0    . This simplifies all the calculations, and we easily obtain the classical Bragg-Williams equation [3]. For all other matrices the dispersion 2 0 x   , and our approach allows one to obtain more reasonable estimates than the mean field model. The work is supported by the Russian Basic Research Foundation (grants 12-07-00295 and 13-01-00504).