n-vicinities method for three dimensional Ising Model

The n -vicinities method for approximate calculations of the partition function of a spin system was proposed previously. The equation of state was obtained in the most general form. In the present publication these results are adapted to the Ising model on the D - dimensional cubic lattice. The state equation is solved for an arbitrary dimension D and the behavior of the free energy is analyzed. For large values of D (D > 2) the obtained results are in good agreement with the ones obtained by means of computer simulations. For small values of D, there are noticeable discrepancies with the exact results.


Introduction
In the papers [1]- [3] we develop the n -vicinities method for approximate calculation of the partition function. The method can be described as follows. Let 1 N  be the dimension of the problem, and 0 N  sR be the initial configuration whose n -vicinity n  is the set of all the configurations that differ from 0 s by the values of n coordinates ( 0,1,..., nN  ). The distribution of the energies of the states from n  fits the Gaussian density reasonable well [3]. The mean energy n E and the variance 2 n  can be accurately expressed in terms of the parameters of the connection matrix [1], [2]. Then in the asymptotic limit summing over the states from n  in the expression for the partition function can be replaced by integration of exp( ) E   over the Gaussian measure, and summing over the all nvicinities by integration between 0 and 1 over the parameter / x n N  . As a result the partition function takes the form of the integral , which is calculated with the aid of the saddle-point method. The function of two variables ( , ) F x E depends on the following parameters: the inverse temperature  , the external magnetic field (if it is present), the trace of the connection matrix squared and some other numerical parameters. That is ( , ) ( , , , ...) . In the most general form the expression for ( , ) F x E was obtained in [2], [3]. In the present publication we adapt this expression for the Ising model on the D -dimension hypercube, derive the equation of state, plot and examine the graphs of the free energy. Because of the size of the publication here is only the summary of our calculations. The details will be presented in [4].

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(1 ) In [1]- [3] it was shown that the expression for the partition function can be reduced to the form: Solving the first equation for  and substituting the result into the second one, we obtain the minimization problem for the function of one variable The restriction means that among all the minimums we examine only those for which 0 We use the other normalization and more convenient notations: Then the final form of the problem is as follows: for each value of the "renormalized" inverse temperature b we have to find the global minimum of the function ( When b exceeds 1, the right-hand boundary of the interval where we are looking for the global minimum becomes depending on b.
The free energy is equal to . In the general case,  is the effective coordination number that describes the interaction of the spin with the nearest vicinity.

Analysis of the state equation when H=0
Let us set 0 H  . The case of the nonzero magnetic field will be described in [4]. We seek the global minimum of the function () fx (1) with 0 h  . For that we have to solve the state equation It is evident that (see figure 1b). In this case, to calculate the free energy one has to substitute 1 () xb in equation (1). When c bb  the phase transition of the second kind takes place in the system. A further increase of the parameter b is accompanied by the deepening of the minimum and steady shifting of the minimum point towards 0. When b becomes larger than 1, the right boundary of the interval becomes the variable that is equal to b x (see equation (2)). When b  the interval (0, ) b x contracts to the origin of coordinates and the global minimum 1 () xb tends to zero. As it has to be, when the temperature tends to zero, the spin system tends to its ground state  numbers are shown in the upper row of the table 1. In the lower row we present the values of c b that are the results of computer simulations (see [5]- [7]). We see a good agreement between our theoretical estimates and the results of computer simulations.
Thus, equation (5) provides us with a reasonable estimate of the critical temperature for the Ising model on the D-dimension hypercube when 8 / 3 D  . We would like to remind that the parameter  has not to be even integer-valued numbers (see the note in the end of the previous section).
During  The essential fault of our approach is the presence of the jump of the magnetization in the 2D Ising model. This error is the result of distortions due to the approximation of the true distribution of energies by the Gaussian density. Without getting into details, let us note that analyzing the equation of state we see how a small distortion of the curve's shape allows to eliminate the unnecessary jump of the magnetization. We hope that this can be done by an improvement of our model. For example, a more accurate approximation can be used in place of the Gaussian density. This is the subject of our further analysis. Even in the present form when we ignore the jumps of the magnetization and are interested in the behavior of the free energy only, the results obtained with the aid of our approach are reasonable enough.
In figure 4 for the 2D Ising model we show the dependence of the inverse critical temperature c  on the ratio of the interaction constants along the two axes: 1 J  and K varies from 0.2 to 1. We see that our results are in sufficiently good agreement with the exact solution.  The work was supported by the financial support of RBRF grants №15-07-04861 and 16-01-00626.