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Random Currents and Continuity of Ising Model’s Spontaneous Magnetization

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Abstract

The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on \({\mathbb{Z}^d}\) in d = 3 dimensions. The analysis also applies to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. The proof employs in an essential way an extension of the Ising model’s random current representation to the model’s infinite volume limit. Using it, we relate the continuity of the magnetization to the vanishing of the free boundary condition Gibbs state’s long range order parameter. For reflection positive models the resulting criterion for continuity may be established through the infrared bound for all but the borderline lower dimensional cases. The exclusion applies to the one dimensional model with 1/r 2 interaction for which the spontaneous magnetization is known to be discontinuous at T c .

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References

  1. Aizenman M., Barsky D.J., Fernández R.: The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47(3–4), 343–374 (1987)

    Article  ADS  Google Scholar 

  2. Aizenman M., Chayes J.T., Chayes L., Newman C.M.: Discontinuity of the magnetization in one-dimensional \({1/|x-y|^2}\) Ising and Potts models. J. Stat. Phys. 50(1–2), 1–40 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Aizenman M., Fernández R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44(3–4), 393–454 (1986)

    Article  ADS  MATH  Google Scholar 

  4. Aizenman M., Graham R.: On the renormalized coupling constant and the susceptibility in \({\varphi^{4}_{4}}\) field theory and the Ising model in four dimensions. Nucl. Phys. B 225(2), 261–288 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  5. Aizenman M.: Geometric analysis of \({\varphi^4}\) fields and Ising models. Commun. Math. Phys. 86(1), 1–48 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Baxter R.J.: Potts model at the critical temperature. J. Phys. C Solid State Phys. 6(23), L445 (1973)

    Article  ADS  Google Scholar 

  7. Biskup M., Chayes L., Crawford N.: Mean-field driven first-order phase transitions in systems with long-range interactions. J. Stat. Phys. 122(6), 1139–1193 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Biskup M.: Reflection positivity and phase transitions in lattice spin models. In: Methods of Contemporary Mathematical Statistical Physics, pp. 1–86. Springer, New York (2009)

    Google Scholar 

  9. Burton R.M., Keane M.: Density and uniqueness in percolation. Commun. Math. Phys. 121(3), 501–505 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Bodineau T.: Translation invariant Gibbs states for the Ising model. Probab. Theory Relat. Fields 135(2), 153–168 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, L.-C., Sakai, A.: Critical two-point functions for long-range statistical-mechanical models in high dimensions. To appear in Ann. Probab. (arXiv:1204.1180)

  12. Duminil-Copin, H.: Parafermionic observables and their applications to planar statistical physics models, Ensaios Matematicos, vol. 25. Brazilian Mathematical Society (2013)

  13. Duminil-Copin H., Hongler C., Nolin P.: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Commun. Pure. Appl. Math. 64(9), 1165–1198 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Duminil-Copin, H., Sidoravicius, V., Tassion, V.: Continuity of the phase transition for planar Potts models with \({1\leq q\leq 4}\) . Preprint, 50 pages (2013)

  15. Dobrushin R.L.: Prescribing a system of random variables by the help of conditional distributions. Probab. Theory Appl. 15, 469–497 (1970)

    Article  Google Scholar 

  16. Dyson F.J.: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12(2), 91–107 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  17. Fröhlich J., Israel R., Lieb E.H., Simon B.: Phase transitions and reflection positivity. I. General theory and long range lattice models. Commun. Math. Phys. 62(1), 1–34 (1978)

    Article  ADS  Google Scholar 

  18. Fisher M.E.: Critical temperatures of anisotropic Ising lattices, II general upper bounds. Phys. Rev. 162, 480 (1967)

    Article  ADS  Google Scholar 

  19. Fortuin C.M., Kasteleyn P.W., Ginibre J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22(2), 89–103 (1971)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Fröhlich J., Simon B., Spencer T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50(1), 79–95 (1976)

    Article  ADS  Google Scholar 

  21. Georgii H.O.: Gibbs measures and phase transitions, 2nd edn. In: de Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter & Co, Berlin (2011)

    Book  Google Scholar 

  22. Griffiths R.B., Hurst C.A., Sherman S.: Concavity of magnetization of an ising ferromagnet in a positive magnetic field. J. Math. Phys. 11, 790 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  23. Glimm J., Jaffe A.: \({\phi^4_2}\) Quantum field model in the single-phase region: differentiability of the mass and bounds on critical exponents. Phys. Rev. D 10, 536 (1974)

    Article  ADS  Google Scholar 

  24. Griffiths R.B.: Correlations in Ising ferromagnets. I. J. Math. Phys. 8, 478 (1967)

    Article  ADS  Google Scholar 

  25. Griffiths R.B.: Correlations in Ising ferromagnets. II. external magnetic fields. J. Math. Phys. 8, 484 (1967)

    Article  ADS  Google Scholar 

  26. Grimmett G.R.: The random-cluster model, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math. Sciences], vol. 333. Springer, Berlin (2006)

    Google Scholar 

  27. Heydenreich M., van der Hofstad R., Sakai A.: Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Stat. Phys. 132(6), 1001–1049 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Kaufman B.: Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev. 76(8), 1232 (1949)

    Article  ADS  MATH  Google Scholar 

  29. Kaufman B., Onsager L.: Crystal statistics. III. Short-range order in a binary Ising lattice. Phys. Rev. 76(8), 1244 (1949)

    Article  ADS  MATH  Google Scholar 

  30. Kotecký R., Shlosman S.B.: First-order phase transitions in large entropy lattice models. Commun. Math. Phys. 83(4), 493–515 (1982)

    Article  ADS  Google Scholar 

  31. Lebowitz J.L.: More inequalities for Ising ferromagnets. Phys. Rev. B 5, 2538–2540 (1972)

    Article  ADS  Google Scholar 

  32. Lebowitz J.: Coexistence of phases in Ising ferromagnets. J. Stat. Phys. 16(6), 463–476 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  33. Lenz W.: Beitrag zum verständnis der magnetischen eigenschaften in festen körpern. Phys. Zeitschr. 21, 613–615 (1920)

    Google Scholar 

  34. Lebowitz J.L., Martin-Löf A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. Math. Phys. 25, 276–282 (1972)

    Article  ADS  Google Scholar 

  35. Laanait L., Messager A., Ruiz J.: Phases coexistence and surface tensions for the Potts model. Commun. Math. Phys. 105(4), 527–545 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  36. Lee T.D., Yang C.N.: Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87(2), 410–419 (1952)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Messager A., Miracle-Sole S.: Correlation functions and boundary conditions in the Ising ferromagnet. J. Stat. Phys. 17(4), 245–262 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  38. Onsager L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65(3–4), 117 (1944)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Peierls R.: On Ising’s model of ferromagnetism. Math. Proc. Camb. Philos. Soc. 32, 477–481 (1936)

    Article  ADS  MATH  Google Scholar 

  40. Sakai A.: Lace expansion for the Ising model. Commun. Math. Phys. 272(2), 283–344 (2007)

    Article  ADS  MATH  Google Scholar 

  41. Schrader R.: New correlation inequalities for the Ising model and P(ϕ) theories. Phys. Rev. B (3) 15(5), 2798–2803 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  42. Slade, G.: The lace expansion and its applications. Lecture Notes in Mathematics, vol. 1879, Springer, Berlin (2006). (Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, Edited and with a foreword by Jean Picard.)

  43. Smirnova-Nagnibeda T., Pak I.: On non-uniqueness of percolation in nonamenable cayley graphs. Comptes Rendus Acad. Sci. (Paris) Sér I 330, 1–6 (2000)

    MathSciNet  Google Scholar 

  44. Thouless D.J.: Long-range order in one-dimensional Ising systems. Phys. Rev. 187, 732–733 (1969)

    Article  ADS  Google Scholar 

  45. Werner, W.: Percolation et modèle d’Ising, Cours Spécialisés [Specialized Courses], vol. 16. Société Mathématique de France, Paris (2009)

  46. Yang C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85(5), 808–816 (1952)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Departments of Physics and Mathematics, Princeton University, Princeton, NJ, USA

    Michael Aizenman

  2. Département de Mathématiques, Université de Genève, Geneva, Switzerland

    Hugo Duminil-Copin

  3. IMPA, Rio de Janeiro, Brazil

    Vladas Sidoravicius

Authors
  1. Michael Aizenman
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  2. Hugo Duminil-Copin
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  3. Vladas Sidoravicius
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Correspondence to Michael Aizenman.

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Communicated by H. Spohn

To Ruty (1987–2014)

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Aizenman, M., Duminil-Copin, H. & Sidoravicius, V. Random Currents and Continuity of Ising Model’s Spontaneous Magnetization. Commun. Math. Phys. 334, 719–742 (2015). https://doi.org/10.1007/s00220-014-2093-y

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