Abstract
We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method ofinfrared bounds and the chessboard method of estimating contour probabilities in Peierls arguments. We illustrate the ideas by applying them to models with long range interactions in one and two dimensions. Additional applications are discussed in a second paper.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brascamp, H., Lieb, E.H.: Some inequalities for Gaussian measures and the long range order of the one dimensional plasma. In: Functional integration and its applications (ed. A. M. Arthurs), pp. 1–14. Oxford: Clarendon Press 1975
Donoghue, W.F., Jr.: Monotone matrix functions and analytic continuation. Berlin-Heidelberg-New York: Springer 1974
Dyson, F.: Existence of a phase transition in a one-dimensional ising ferromagnet. Commun. math. Phys.12, 91 (1969). Non-existence of spontaneous magnetization in a one-dimensional ising ferromagnet. Commun. math. Phys.12, 212 (1969)
Dyson, F., Lieb, E.H., Simon, B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys.18, 335–383 (1978). See also: Phase transitions in the quantum Heisenberg model. Phys. Rev. Letters37, 120–123 (1976)
Fröhlich, J.: Phase transitions, goldstone bosons and topological superselection rules. Acta Phys. Austriaca Suppl.XV, 133–269 (1976).
Fröhlich, J.: The pure phases (harmonic functions) of generalized processes. Or: mathematical physics of phase transitions and symmetry breaking. Invited talk at Jan. 1977 A.M.S., St. Louis meeting. Bull. Am. Math. Soc. (in press)
Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase transitions and reflection positivity. II. Short range lattice models. J. Stat. Phys. (to be submitted)
Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase transitions and reflection positivity. III. Continuous models. Commun. math. Phys. (to be submitted)
Fröhlich, J., Lieb, E.H.: Phase transitions in anisotropic lattice spin systems. Commun. math. Phys.60, 233–267 (1978)
Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions, and continuous symmetry breaking. Commun. math. Phys.50, 79 (1976)
Gallavotti, G., Miracle-Sole, S.: Equilibrium states of the Ising model in the two-phase region. Phys. Rev.5, 2555–2559 (1972)
Gel'fand, I.M., Vilinkin, N.Ya.: Generalized functions, Vol. 4. New York: Academic Press 1964
Ginibre, J.: General formulation of Griffiths' inequality. Commun. math. Phys.16, 310–328 (1970)
Glimm, J., Jaffe, A., Spencer, T.: Phase transitions for φ 42 quantum fields. Commun. math. Phys.45, 203 (1975)
Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear non-selfadjoint operators. Providence, RI: American Mathematical Society 1969
Griffiths, R.: Phase transitions. In: Statistical mechanics and quantum field theory, Les Houches, 1970, pp. 241–280. New York: Gordon and Breach 1971
Hegerfeldt, G.C.: Correlation inequalities for Ising ferromagnets with symmetries. Commun. math. Phys.57, 259–266 (1977)
Hegerfeldt, G.C., Nappi, C.: Mixing properties in lattice systems. Commun. math. Phys.53, 1–7 (1977)
Heilmann, O.J., Lieb, E.H.: Lattice models for liquid crystals (in preparation)
Holsztynski, W., Slawny, J.: Peierls condition and number of ground states. Commun. math. Phys.61, 177–190 (1978)
Horn, A.: On the singular values of a product of completely continuous operators. Proc. Nat. Acad. Sci. USA36, 374–375 (1950)
Israel, R.: Convexity and the theory of lattice gases. Princeton, NJ: Princeton University Press 1978
Israel, R.: Phase transitions in one-dimensional lattice systems. Proc. 1977 IUPAP Meeting, Haifa
Jolley, C.B.W.: Summation of series. New York: Dover 1961
Klein, A.: A characterization of Osterwalder-Schrader path spaces by the associated semigroup. Bull. Am. Math. Soc.82, 762–764 (1976)
Kunz, H., Pfister, C.E.: First order phase transition in the plane rotor ferromagnetic model in two dimensions. Commun. math. Phys.46, 245 (1976)
Lieb, E.H.: New proofs of long range order. Proceedings of the International Conference on the Mathematical Problems in Theoretical Physics, Rome, 1977. Lecture notes in physics. Berlin-Heidelberg-New York: Springer (in press)
Elliott, R.J.: Phenomenological discussion of magnetic ordering in the rare-earth metals. Phys. Rev.124, 346–353 (1961)
Mermin, N.D.: Absence of ordering in certain classical systems. J. Math. Phys.8, 1061–1064 (1967)
Osterwalder, K., Schrader, R.: Axioms for Euclidean green's functions. Commun. math. Phys.31, 83 (1973)
Osterwalder, K., Seiler, E.: Gauge field theories on the lattice. Ann. Phys.110, 440–471 (1978)
Kim, D., Thompson, C.J.: A lattice model with an infinite number of phase transitions. J. Phys. A. Math. Gen.9, 2097–2103 (1976)
Pirogov, S.A., Sinai, Ya. G.: Phase transitions of the first kind for small perturbations of the Ising model. Funct. Anal. Pril.8, 25–30 (1974). [Engl. translation: Funct. Anal. Appl.8, 21–25 (1974)]
Pirogov, S.A., Sinai, Ya. G.: Phase diagrams of classical lattice systems. Teor. Mat. Fiz.25, 358–369 (1975). [Engl. translation: Theor. Math. Phys.25, 1185–1192 (1975)]
Pirogov, S.A., Sinai, Ya. G.: Phase diagrams of classical lattice systems. Continuation. Theor. Mat. Fiz.26, 61–76 (1976). [Engl. translation: Theor. Math. Phys.26, 39–49 (1971)]
Redner, S., Stanley, H.E.: The R-S model for magnetic systems with competing interactions: series expansions and some rigorous results. J. Phys. C. Solid State Phys.10, 4765–4784 (1977)
Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. II: Fourier analysis, self-adjointness. New York: Academic Press 1975
Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. IV: Analysis of operators. New York: Academic Press 1978
Ruelle, D.: Statistical mechanics. New York: Benjamin 1969
Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc.44, 522–536 (1938)
Schrader, R.: New Correlation inequalities for the Ising model andP(φ) theories. Phys. Rev. B15, 2798–2803 (1977)
Simon, B.: TheP(φ)2 Euclidean (quantum) field theory. Princeton, NJ: Princeton University Press 1974
Simon, B.: New rigorous existence theorems for phase transitions in model systems. Proc. 1977 IUPAP Meeting, Haifa
Author information
Authors and Affiliations
Additional information
Communicated by J. Glimm
Research partially supported by US National Science Foundation under Grant MPS-75-11864
Research partially supported by Canadian National Research Council under Grant A4015
Research partially supported by US National Science Foundation under Grant MCS-75-21684-A01
Rights and permissions
About this article
Cite this article
Fröhlich, J., Israel, R., Lieb, E.H. et al. Phase transitions and reflection positivity. I. General theory and long range lattice models. Commun.Math. Phys. 62, 1–34 (1978). https://doi.org/10.1007/BF01940327
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01940327