Abstract
We study the anisotropic quantum mechanical ferromagnetic Heisenberg model. By anisotropic we mean that thex andy exchange constants are equal but smaller than thez exchange constant. We show that for any amount of anisotropy there is long range order in two or more dimensions at low enough temperature. We also develop a convergent low temperature expansion and use it to prove exponential decay of the truncated correlation functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Battle, G., Federbush, P.: A phase cell cluster expansion for Euclidean field theories. Ann. Phys.142, 95–139 (1982)
Battle, G.: A new combinatoric estimate for cluster expansions. Commun. Math. Phys.94, 133–139 (1984)
Brydges, D.: A short course on cluster expansions. In: Proceeding of 1984 Les Houches Summer School. Osterwalder, K. (ed.) (to be published)
Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Commun. Math. Phys.85, 517–528 (1982)
Dyson, F., Lieb, E., Simon, B.: Phase transitions in quantum spin systems with isotropic and non-isotropic interactions. J. Stat. Phys.18, 335–383 (1978)
Fröhlich, J., Lieb, E.: Phase transitions in anisotropic lattice spin systems. Commun. Math. Phys.60, 233–267 (1978)
Gallavotti, G., Martin-Löf, A., Miracle-Sole, S.: Some problems connected with the description of coexisting phases at low temperatures in the Ising model. Lecture Notes in Physics, Vol. 20. Berlin, Heidelberg, New York: Springer 1971
Ginibre, J.: Existence of phase transitions for quantum lattice systems. Commun. Math. Phys.14, 205–234 (1969)
Kirkwood, J.: Phase transitions in the Ising model with transverse field. J. Stat. Phys.37, 407–417 (1984)
Malyshev, V.: Phase transitions in classical Heisenberg ferromagnets with arbitrary parameter of anisotropy. Commun. Math. Phys.40, 75–82 (1975)
Mermin, N., Wagner, H.: Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett.17, 1133–1136 (1966)
Robinson, D.: A new proof of the existence of phase transitions in the anisotropic Heisenberg model. Commun. Math. Phys.14, 195–204 (1969)
Seiler, E.: Gauge theories as a problem of constructive quantum field theory and statistical mechanics. Lecture Notes in Physics, Vol. 159. Berlin, Heidelberg, New York: Springer 1982
Thomas, L., Yin, Z.: Low temperature expansions for the Gibbs states of quantum Ising lattice systems. J. Math. Phys.25, 3128–3134 (1984)
Thomas, L., Yin, Z.: Low temperature expansions for the Gibbs states of weakly interacting quantum Ising lattice systems. Commun. Math. Phys.91, 405–417 (1983)
Bricmont, J., Lebowitz, J.L., Pfister, C.E.: Some inequalities for anisotropic rotators (preprint)
Bricmont, J., Fontaine, J.-R.: Infrared bounds and the Peierls argument in two dimensions. Commun. Math. Phys.87, 417–427 (1982)
Speer, E.R.: Failure of reflection positivity in the quantum Heisenberg ferromagnet. To appear in Lett. Math. Phys.
Author information
Authors and Affiliations
Additional information
Communicated by J. Fröhlich
Research partially supported by U.S. National Science Foundation under Grant PHY8116101 AO 3
Rights and permissions
About this article
Cite this article
Kennedy, T. Long range order in the anisotropic quantum ferromagnetic Heisenberg model. Commun.Math. Phys. 100, 447–462 (1985). https://doi.org/10.1007/BF01206139
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01206139