Partition function of the three-dimensional Ising model

I. R. Yukhnovskii

Teoretičeskaâ i matematičeskaâ fizika, Tome 36 (1978) no. 3, pp. 373-399

Résumé

The method of collective variables is used to consider the partition function of the three-dimensisnal Ising model. A rigorous transition is made from the phase space of spin variables to the phase space of collective variables. It is shown that among the set of collective variables $\{\rho_k\}^N$ there is a variable $\rho_0$ with respect to which there is a change in the form of the distribution function on the transition through the critical point. A basis distribution with respect to the collective variables describing events at the critical point is found. In the argument of the exponential, it contains the second and fourth powers of the collective variables. To within the basis distributions, the partition function of the system is integrated over layers of the phase space of the collective variables. Recursion relations are found. The critical point is determined. The method is compared with the $\varepsilon$-expansion method proposed by Wilson and Fisher. The integral form of the basis distribution is used to consider the problem of block structures of the system. Besides original results, the paper collects together the results obtained by the author, Yu. K. Rudavskii, and M. P. Kozlovskii published earlier in other journals.