Abstract
Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's ‘effective dimension’. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component Ising (and more genrally Griffiths=Simon class) spin variables. In particular, it is established that even in dimensions d<4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as J x=1/|x|d+δ with 0<δ≤d/2, then the exponents \(\hat \beta \), δ, γ and Δ4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with δ>-2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
FisherM. E., MaS., and NickelB. G., Phys. Rev. Lett. 29, 917–920 (1972).
PitzerK. S., ConceicaoM., deLimaP., and SchriberD. R., J. Phys. Chem. 89, 1854–1855 (1985).
FröhlichJ., SimonB., and SpencerT., Commun. Math. Phys. 50, 79–85 (1976).
GriffithsR. B., J. Math. Phys. 10, 1559–1656 (1969). Simon, B. and Griffiths, R. B., Commun. Math. Phys. 33, 145–164 (1973).
LeeT. D. and YangC. N., Phys. Rev. 87, 410–419 (1952).
AizenmanM. and FernándezR., J. Stat. Phys. 44, 393–454 (1986).
AizenmanM., Phys. Rev. Lett. 54, 839–842 (1985).
AizenmanM., BarskyD. J., and FernándezR., J. Stat. Phys. 47, 343–374 (1987).
GriffithsR. B., J. Math. Phys. 8, 478–484 (1967).
SimonB., Commun. Math. Phys. 77, 111–126 (1980).
AizenmanM., in A.Jaffe, D.Sasz, and D.Petz (eds.), Statistical Mechanics and Dynamical Systems (Proceedings Kosheg 1984), Birkhauser, Boston, 1985.
GlimmJ. and JaffeA., Phys. Rev. D10, 536 (1974).
AizenmanM. and GrahamR., Nucl. Phys. B225 [FS9], 261–288 (1983).
Fröhlich, J. and Sokal, A. D., The random walk representation of classical spin systems and correlation inequalities. III. Nonzero magnetic field, in preparation.
SokalA. D., Phys. Lett. 71A, 451–453 (1979).
SokalA. D., Ann. Inst. Henri Poincaré A37, 317 (1982).
AizenmanM., Phys. Rev. Lett. 47, 1 (1981); Commun. Math. Phys. 86, 1–48 (1982).
Fröhlich, Nucl. Phys. B20 [FS4], 281 (1982).
FröhlichJ., IsraelR., LiebE. H., and SimonB., Commun. Math. Phys. 62, 1–34 (1978).
FortuinC., KasteleynP., and GinibreJ., Commun. Math. Phys. 22, 89 (1971).
GriffithsR. B., J. Math. Phys. 8, 484–489 (1967).
GriffithsR. B., HurstC. A., and ShermanS., J. Math. Phys. 11, 790–795 (1970).
AizenmanM. and BarskyD. J., Commun. Math. Phys. 108, 489–526 (1987).
Newman, C. M., Appendix to contribution in Proceedings of the SIAM workshop on multiphase flow, G. Papanicolau (ed.), to appear.
LebowitzJ. L., Commun. Math. Phys. 35, 87 (1974).
McBryanO. and RosenJ., Commun. Math. Phys. 51, 97 (1967).
AizenmanM. and NewmanC. M., J. Stat. Phys. 36, 107–143 (1984).
ChayesJ. T. and ChayesL., Phys. Rev. Lett. 56, 1619–1622 (1986).
AizenmanM. and NewmanC. M., Commun. Math. Phys. 107, 611–644 (1986). Aizenman, M., Chayes, J. T., Chayes, L., and Newman, C. M., J. Stat. Phys. (1988); Imbrie, J. Z. and Newman, M., submitted to Commun. Math. Phys.
Author information
Authors and Affiliations
Additional information
Also in the Physics Department. Research supported in part by the National Science Foundation Grant PHY 86-05164.
Rights and permissions
About this article
Cite this article
Aizenman, M., Fernández, R. Critical exponents for long-range interactions. Lett Math Phys 16, 39–49 (1988). https://doi.org/10.1007/BF00398169
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00398169