Abstract
We discuss some aspects of the problem of the equivalence of dilute antiferromagnets and random field Ising models. We first investigate for dilute antiferromagnets the validity of the arguments of Imry and Ma. It turns out that they are applicable, but some care is required concerning the role played by the so-called internal Peierls contours. Next we consider a hierarchical version of a dilute antiferromagnetic Ising model in the presence of a uniform magnetic field and show that a renormalization group transformation maps it exactly into a hierarchical version of the random field Ising model, thus proving their equivalence as far as the critical behavior is concerned. In particular this implies that phase transition with spontaneous magnetization occurs only for dimensiond>2. Finally we show that in the absence of internal Peierls contours both models, in their hierarchical versions, exhibit phase transition already in dimensiond=2.
Similar content being viewed by others
References
A. Aharony and S. Fishman,J. Phys. C 12:L729 (1979).
J. M. G. Amaro de Matos, J. A. Baêta Segundo, and J. Fernando Perez, Fluctuations in dilute antiferromagnets: Curie-Weiss models,J. Phys. A 25: 2819–2830 (1992).
Y. Imry and S. K. Ma,Phys. Rev. Lett. 35:1399 (1975).
J. Bricmont and A. Kupiainen,Commun. Math. Phys. 116:539 (1988).
J. Bricmont and A. Kupiainen,J. Stat. Phys. 51:1021 (1988).
M. Aizenman and J. Wehr,Phys. Rev. Lett. 62:2503–2506 (1989);Commun. Math. Phys. 130:489–528 (1990).
L. F. Pontin, On the equivalence of dilute antiferromagnets and random field ferromagnetic models: A hierarchical version, Ph.D. Theses, University of São Paulo (1990) [in Portuguese].
K. Gawedzky and A. Kupiainen, Asymptotic freedom beyond perturbation theory, inLes Houches, Session XLIII, 1984, R. Stora and K. Osterwalder, eds. (North-Holland Amsterdam, 1986).
P. Collet and J.P. Eckmann, A renormalization group analysis of the hierarchical model in statistical mechanics, inLecture Notes in Physics, Vol. 74 (Springer-Verlag, Berlin, 1974).
B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pontin, L.F., Segundo, J.A.B. & Perez, J.F. Dilute antiferromagnets: Imry-Ma argument, hierarchical model, and equivalence to random field Ising models. J Stat Phys 75, 51–65 (1994). https://doi.org/10.1007/BF02186280
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02186280