Abstract
We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that in a sufficiently large external field the Lee-Yang theorem implies uniqueness of the equilibrium state.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bricmont, J.: Correlation inequalities for two-component fields. Ann. Soc. Sc. Brux.90, 245–252 (1976)
Dobrushin, R. L., Shlosman, S. B.: Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics. Commun. math. Phys.42, 31–40 (1975)
Dunlop, F.: Correlation inequalities for multicomponent rotators. Commun. math. Phys.49, 247–256 (1976)
Dunlop, F., Newman, C. M.: Multicomponent field theories and classical rotators. Commun. math. Phys.44, 223–235 (1974)
Fortuin, C. M., Ginibre, J., Kastelyn, P. W.: Correlation inequalities on some partially ordered sets. Commun. math. Phys.22, 89–103 (1971)
Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. math. Phys.50, 79–85 (1976)
Ginibre, J.: General formulation of Griffiths' inequalities. Commun. math. Phys.16, 310–328 (1970)
Griffiths, R. B.: Phase transitions. In: Statistical mechanics and quantum field theory (Les Houches), pp. 241–279. New York, London, Paris: Gordon and Breach 1970
Israël, R.: Thesis, Princeton University
Israël, R. B.: High-temperature analyticity in classical lattice systems. Commun. math. Phys.50, 245–257 (1976)
Kunz, H., Pfister, Ch.Ed.: First order phase transitions in the plane rotator ferromagnetic model in two dimensions. Commun. math. Phys.46, 245–251 (1976)
Kunz, H., Pfister, Ch. Ed., Vuillermot, P. A.: Inequalities for some classical spin vector models. Bielefeld preprint
Lanford, O. E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. math. Phys.13, 194–215 (1969)
Lebowitz, J. L.: Coexistence of phases in Ising ferromagnets. Preprint
Lebowitz, J. L.: GHS and other inequalities. Commun. math. Phys.35, 87–92 (1974)
Lebowitz, J. L., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. math. Phys.25, 276–282 (1976)
Mermin, N. D.: Absence of ordering in certain classical systems. J. Math. Phys.6, 1061–1064 (1967)
Monroë, J. L.: Correlation inequalities for two-dimensional vector spin system. J. Math. Phys.16, 1809–1812 (1975)
Ruelle, D.: On the use of “small external fields” in the problem of symmetry breakdown in statistical mechanics. Ann. Phys.69, 364–374 (1972)
Ruelle, D.: Statistical mechanics. New York: Benjamin 1969
Holsztynski, W., Slawny, J.: Phase transitions in ferromagnetic spin systems at low temperatures. Preprint
Sylvester, G. S.: Thesis, Harvard University
Suzuki, M., Fisher, M.: Zeros of the partition function for the Heisenberg, Ferroelectric and general Ising models. J. Math. Phys.12, 235–246 (1971)
Messager, A., Miracle, S., Pfister, Ch. E.: to be published
Author information
Authors and Affiliations
Additional information
Communicated by E. Lieb
Rights and permissions
About this article
Cite this article
Bricmont, J., Fontaine, J.R. & Landau, L.J. On the uniqueness of the equilibrium state for plane rotators. Commun.Math. Phys. 56, 281–296 (1977). https://doi.org/10.1007/BF01614213
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01614213