A version of the Ising model is developed in which the spin variables can be treated accurately in the continuum approximation. The perturbation series, both above and below the critical temperature
Tc, is examined, and it is shown that there is a shift of
Tc from its mean-field value proportional to ${q}^{\ensuremath{-}2}\mathrm{ln}q$, as well as the well-known shift proportional to ${q}^{\ensuremath{-}1}$; here
q is the number of mutually interacting particles. It is shown, using renormalization theory, that there is a perturbation series in ${q}^{\ensuremath{-}1}{|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}\frac{1}{2}}$ for which all terms are finite in the limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$, if the shift of
Tc is put in correctly. For the two-dimensional model, the shift is shown to be proportional to ${q}^{\ensuremath{-}1}\mathrm{ln}q$. Conditions are derived for a finite system to display critical behavior characteristic of three, two, one, or zero dimensions. It is shown how similar results can be obtained for a model similar to the Heisenberg model and for the standard Ising and Heisenberg models with interactions extending over many neighbors. A comparison is made between previously calculated numerical results for
Tc and the asymptotic forms derived here.
We consider Ising models with long-range ferromagnetic pair interactions decaying as for . We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For we find which can be compared with recent results of Luijten and Blöte who found , and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of decreases significantly. In addition to we obtain approximate values for the critical exponents , and using the coherent anomaly method. For we obtain , , and - all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of .
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.
The authors present the results of the coherent anomaly method when applied to Ising models in one dimension with long-range interactions. This class of systems acts as an interesting and challenging test for the method because the critical exponents as well as the critical temperature values of which are given by the method, depend on the rate of fall-off of the interactions. Thus one can see how accurately the method gives results which correctly reflect this dependency. The results obtained from this method are compared with results obtained by a variety of other methods.
Critical behavior of Ising ferromagnetic systems with the competition between short- and long-range interactions near the point of a second-order phase transition has been investigated. The effects of long-range interaction lead to the need to calculate the influence of next-nearest spins, which lie inside the sphere of a specific radius. The study was carried out for a power law of decreaseing the exchange integral with distance for a three-dimensional Ising model with a cubic lattice. A comparison with the results of a field-theoretical approach and data of real experiments has been carried out. It has been shown that the critical temperature grows linearly with an increase in the parameters of long-range interaction. Critical exponents have been calculated at different parameters of long-range interaction. It is shown that at small dimensions of a system it is the short-range interaction that exerts the dominant effect on the critical behavior of the system. With an increase in the linear dimension of the system, long-range forces begin to dominate.