Abstract
We study numerically the magnetic susceptibility of the hierarchical model with Ising spins (σ=±1) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly using recursive methods which exploit the symmetries of the model. Lattices with up to 218 sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form (1-β/β 0)g for thewhole temperature range considered. This approximate law implies a simple approximate formula for the coefficients of the high-temperature expansion, and, more importantly, approximate relations among the coefficients themselves. We found that some of these approximate relations hold with errors less then 2%. On the other hand,g differs significantly from the critical exponent γ calculated with the epsilon expansion, even when the fit is restricted to intervals closer toβ c. We discuss this discrepancy in the context the renormalization group analysis of the hierarchical model.
Similar content being viewed by others
References
K. Wilson,Phys. Rev. B 4:3185 (1971); K. Wilson and J. Kogut,Phys. Rep. 12:75 (1974).
F. Dyson,Commun. Math. Phys. 12:91 (1969); G. Baker,Phys. Rev. B 5:2622 (1972).
P. Bleher and Y. Sinai,Commun. Math. Phys. 45:247 (1975); P. Collet and J. P. Eckmann,Commun. Math. Phys. 55:67 (1977).
P. Collet, J.-P. Eckmann, and B. Hirsbrunner,Phys. Lett. 71B:385 (1977).
P. Collet and J.-P. Eckmann,Lecture Notes in Physics, No. 74 (Springer-Verlag, Berlin, 1978).
K. Gawedzki and A. Kupiainen, InLes Houches 1985, K. Osterwalder and R. Stora, eds.
Y. Meurice, InProceedings of International Europhysics Conference 1993, to appear; University of Iowa preprint 93-12, hep-th 9307128 (1993).
Y. Meurice,Mod. Phys. Lett. 7:3331 (1992);J. Math. Phys. 35:769 (1994).
A. Polyakov,Phys. Lett. 82B:247 (1979);Phys. Lett. 103B:211 (1981).
J. L. Lucio and Y. Meurice,Mod. Phys. Lett. 6:1199 (1991); Y. Meurice,Phys. Lett. 265:377 (1991).
G. Parisi,Statistical Field Theory (Addison-Wesley, New York, 1988), and references therein.
C. Domb and M. S. Green,Phase Transition and Critical Phenomena, Vol. 3 (Academic Press, New York, 1982).
M. Levy, J. C. Le Guillou, and J. Zinn-Justin, eds.,Phase Transitions, Cargese 1980 (Plenum Press, New York, 1982).
Z. Roskies,Phys. Rev. B 24:5305 (1981); J. Adler, M. Moshe, and V. Privman,Phys. Rev. B 26:3958 (1982); G. Nickel and M. Dixon,Phys. Rev. B 26:3965 (1982).
Y. Meurice and G. Ordaz, work in progress.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Meurice, Y., Ordaz, G. & Rodgers, V.G.J. A numerical study of the hierarchical Ising model: High-temperature versus epsilon expansion. J Stat Phys 77, 607–626 (1994). https://doi.org/10.1007/BF02179452
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179452