Abstract
Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator σ zf withquadratic fluctuations is replaced with its mean value [(σ zf )2 ≃ 〈(σ zf )2〉]. This means that the true quantum mechanical spectrum of the operator σ zf is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Abrikosov, A. A., Gor'kov, L. P., and Dzyaloshinskii, I. Ye. (1965).Quantum Field Theoretical Methods in Statistical Physics, 2nd ed., Pergamon Press, Oxford.
Brush, S. G. (1967).Reviews of Modern Physics,39, 883.
Bune, A. V., Fridkin, Y. M., Ducharme, S., Blinov, L. M., Palto, S. P., Sorokin, A. V., Yudin, S. G., and Zlatkin, A. (1998).Nature,391, 874.
Fetter, A. L., and Walecka, J. D. (1971).Quantum Theory of Many-Particle Systems, McGraw-Hill, New York.
Frank, B., and Mitran, O. (1977).Journal of Physics C: Solid State Physics,10, 2641.
Honmura, R, and Kaneyoshi, T. (1979).Journal of Physics C: Solid State Physics,12, 3979.
Jelifonov, M. P. (1971).Teoreticheskaiya i Matematicheskaiya Fizika,8, 401 [in Russian].
Kasteleijn, P. W., and Van Kranendonk, J. (1956).Physica,22, 317.
Loskutov, V. V., Mironov, G. I., and Nigmatullin, R. R. (1996).Low Temperature Physics,22, 220.
Mannari, T., and Kogeyama, A. (1968).Progress of Theoretical Physics (Supplement), 269.
Morita, T., and Horiguchi, T. (1971).Journal of Mathematical Physics,12, 981, 986.
Nigmatullin, R. R., and Toboev, V. A. (1986).Teoreticheskaiya i Matematicheskaiya Fizika,68, 88 [in Russian].
Nigmatullin, R. R., and Toboev, V. A.(1988).Teoreticheskaiya i Matematicheskaiya Fizika,74, 11 [in Russian].
Nigmatullin, R. R., and Toboev, V. A. (1989).Teoreticheskaiya i Matematicheskaiya Fizika,80, 94 [in Russian].
Oitmaa, J. (1971).Solid State Communications,9, 745.
Potashinsky, A. Z., and Pokrovsky, V. L. (1975).The Fluctuation Theory of Phase Transitions, Nauka, Moscow [in Russian].
Smart, S. (1966).Effective Field in the Theory of Magnetism, IBM Watson Research Center, Yorktown Heights, and Saunders, Philadelphia.
Stanley, H. E. (1971).Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford.
Taggart, G. B., and Fittipaldi, J. P. (1982).Physical Review B,25, 7026.
Zhang, H., and Min, B. (1981).Journal of Physics C: Solid State Physics,14, 1769.
Zubarev, D. N. (1960).Uspekhi Fizicheskikh Nauk,71, 71 [English transl.,Soviet Physics-Uspekhi,3, 320 (1960)].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nigmatullin, R.R., Khamzin, A.A. & Ghassib, H.B. One-, Two-, and Three-Dimensional Ising Model in the Static Fluctuation Approximation. International Journal of Theoretical Physics 39, 405–446 (2000). https://doi.org/10.1023/A:1003648612384
Issue Date:
DOI: https://doi.org/10.1023/A:1003648612384