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Instabilities and phase transitions in the ising model. A review

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La Rivista del Nuovo Cimento (1971-1977)

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Notes and References

  1. For a history of the Ising model seeS. G. Brush:Rev. Mod. Phys.,39, 883 (1969).

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  2. R. Peierls:Proc. Cambridge Phil. Soc.,32, 477 (1936).

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  3. L. Onsager:Phys. Rev.,65, 117 (1944).

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  4. T. D. Lee andC. N. Yang:Phys. Rev.,87, 410 (1952).

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  5. The mean field theory, as it appears from the literature quoted below, is a theory of phase transitions due to very-long-range and weak forces. The Ising model is, in contrast, a theory based on strong and short-ranged forces. The original approach to the Van der Waals theory (also called the mean field theory) can be found in the book byS. Chapman andT. Cowling:The Mathematical Theory of Nonuniform Gases (Cambridge, 1953), p. 284. A more refined and interesting formulation is inN. G. Van Kampen:Phys. Rev.,135, A 362 (1964). A rigorous and very clear theory is inJ. L. Lebowitz andO. Penrose:Journ. Math. Phys.,7, 98 (1966). The first to understand how to rigorously formulate (and prove in particular cases) the mean field theory have beenP. Hemmer, M. Kac andG. E. Uhlenbeck in a series of papers appeared inJourn. Math. Phys. and reproduced, with introductory remarks, inE. Lieb andD. C. Mattis:Mathematical Physics in One Dimension (New York, 1966). A more phenomenological but very interesting and original theory is in the book ofR. H. Brout:Phase Transitions (New York, 1965), where the most common phase transitions are treated from the unifying point of view of the mean field theory.

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  6. The original solution of the free energy of the Ising model in 2 dimensions can be found in [3]. It was preceded by the exact location of the critical temperature byH. A. Kramers andG. H. Wannier:Phys. Rev.,60, 252 (1941). The spontaneous magnetization was found byL. Onsager:Suppl. Nuovo Cimento,6, 261 (1949), but never published; it was subsequently rediscovered byC. N. Yang:Phys. Rev.,85, 809 (1952). A modern derivation of the solution is to be found in the review article ofT. D. Schultz, D. C. Mattis andE. Lieb:Rev. Mod. Phys.,36, 856 (1964). Another interesting older review article is the paper byG. F. Newell andE. W. Montroll:Rev. Mod. Phys.,25, 353 (1953). A combinatorial solution has been found byM. Kac andM. Ward and can be found in the book byL. Landau andE. L. Lifschitz:Physique Statistique (Moscow, 1967), p. 538. This derivation is not above criticism; an up-to-date paper on the subject is due toF. A. Berezin:Russian Math. Surveys,24, 1 (1969). Another approach to the solution (the « Pfaffian method ») can be found inP. W. Kasteleyn:Physica,27, 1209 (1961).

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  7. See the review paper byM. E. Fisher:Rep. Progr. Theor. Phys.,30, 615 (1967), pag. 677–702 and appended references.

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  8. It is the case of MnCl2·4H2O seeM. E. Fisher andM. F. Sykes:Physica,28, 939 (1962); see also [7].

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  9. R. L. Dobrushin:Theory of probability and its applications,13, 197 (1968);Functional analysis and its applications,2, 292, 302 (1968);3, 22 (1968);F. Spitzer:Am. Math. Monthly,78, 142 (1971).

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  10. D. Ruelle:Statistical Mechanics (New York, 1969), p. 168, 161.

  11. R. B. Griffiths:Journ. Math. Phys.,8, 478 (1967);M. E. Fisher:Phys. Rev.,162, 475 (1967);H. S. Green andC. H. Hurst:Order Disorder Phenomena (New York, 1964).

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  12. D. C. Mattis:Theory of Magnetism (London, 1965).

  13. This term is usually omitted and in some sense its importance has only recently been recognized after the works ofR. L. Dobrushin: see ref. [9], and ofO. Lanford andD. Ruelle:Comm. Math. Phys.,13, 194 (1969). It is one of the purposes of this article to emphasize the role of this term in the theory of phase transitions.

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  14. K. Huang:Statistical Mechanics (New York, 1963).

  15. This definition is inspired byO. Lanford andD. Ruelle:Comm. Math. Phys.,13, 194 (1969), where the equivalence of the above definition with a number of other possible definitions is shown. For instance the definition in question is equivalent to the one based on the requirement that the correlation functions should be a solution of the Kirkwood-Solsburg equations. It is also equivalent to the definition of equilibrium state in terms of tangent planes (i.e. functional derivatives of a suitable functional: seeD. Ruelle:Statistical Mechanics (New York, 1969), p. 184). It should be said that these proofs of equivalence are not always explicitly derived in the quoted paper byLanford andRuelle; they are, however, an easy corollary of their results and appear, derived in detail, in the, so far, unpublished lecture notes of the lectures delivered by the author at the Courant Institute, September 1971, preprint.

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  16. This solution of the Ising model in a zero field can be found in the paper byG. F. Newell andE. W. Montroll:Rev. Mod. Phys.,25, 353 (1953).

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  17. This expansion can be used as a starting point for the combinatorial solution mentioned in [6]. SeeL. Landau andE. L. Lifshitz:Physique Statistique (Moscow, 1967).

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  18. Of course we do not really attach a deep physical meaning to the difference between these two approaches. Clearly they should become equivalent if one pretended to extract all the possible information from them. What is really important is that the first questions raised by both approaches are very interesting and relevant from a physical point of view. One of the goals of the analytic theory of the phase transitions is to understand the nature of the singularity at the critical point and along the break of the isotherms. A lot of interest has been devoted to this point and a number of enlightening phenomenological results are available. However the number of rigorous results on the matter is rather limited. An idea of the type of problems that are of interest can be gotten by reading the papers ofP. W. Kasteleyn: inFundamental Problems in Statistical Mechanics, II, edited byE. G. D. Cohen (Amsterdam, 1968), or the more detailed paper byM. E. Fisher:Rep. Progr. Theor. Phys.,30, 615 (1967), and the paper byM. E. Fisher:Physics, Physica, Fizika,3, 255 (1967).

  19. This geometric picture of the spin configurations can be traced back at least as far as Peierls’ paper, ref. [2], and has been used, together with formula (4.11), to derive (6.8) (« Kramer’s and Wannier’s duality relation ») and (6.9) byH. A. Kramers andG. H. Wannier: ref. [6]. A recent interesting generalization of the duality concept has been given byF. J. Wegner:Journ. Math. Phys.,12, 2259 (1971), where some very interesting applications can be found as well as references to earlier works. The duality relation between (+)-or (—)-boundary conditions and open boundary conditions (which is used here) has been developed in a conversation withBenettin, Jona-Lasinio andStella. The reader can find other interesting applications of the duality relation in their paper to appear inLett. Nuovo Cimento (June 1972).

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  20. The above proof is due toR. B. Griffiths and, independently, toR. L. Dobushin and is inspired byR. Peierls:Proc. Cambridge Phil. Soc.,32, 477 (1936).

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  21. M. E. Fisher:Physics, Physica, Fizika,3, 255 (1967).

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  22. This theorem is due toR. A. Minlos andJ. G. Sinai:Trans. Moscow Math. Soc.,19, 121 (1968);Math. USSR Sbornik,2, 335 (1967).

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  23. F. A. Berezin andJ. G. Sinai:Trans. Moscow Math. Soc.,19, 219 (1967).

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  24. Here the symbolϱβ(θ)(dθ/2π) has not to be taken too seriously; it really denotes a measure on the circle and this measure is not necessarily dθ-continuous. Also the « convergence » statement really means the existence of a measure such that (9.2) holds for all realz. The original proof of this theorem is due toT. D. Lee andC. N. Yang:Phys. Rev.,87, 410 (1952). A much stronger and general theorem leading, in particular, to Lee-Yang’s theorem is inD. Ruelle:Phys. Rev. Lett.,26, 303 (1971). Ruelle’s theorem has been the last of a series of improvements and generalizations of Lee-Yang’s theorem; see references in Ruelle’s paper.

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  25. D. Ruelle:The use of small external fields etc., preprint (1971), to appear inJourn. Math. Phys.

  26. A. Martin-Löf andJ. L. Lebowitz:Comm. Math. Phys.,25, 276 (1972).

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  27. K. B. Griffiths:Journ. Math. Phys.,8, 478 (1967);M. E. Fisher:Lectures in Physics, Vol.7 C (Boulder Colo., 1965).

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  28. The definition below is due toD. Ruelle:Statistical Mechanics (New York, 1969), p. 161.

  29. This is an unpublished result ofR. B. Griffiths. His proof is reported inG. Gallavotti, A. Martin-Löf andS. Miracle-Sole: to appear in theLecture Notes of the 1971 Battelle-Seattle Summer Rencontres in Mathematics and Physics, edited byA. Lenard (Berlin).

  30. R. A. Minlos andJ. G. Sinai:Trans. Moscow Math. Soc.,19, 121 (1968);Math. USSR Sbornik,2, 335 (1967).

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  31. G. Gallavotti andS. Miracle-Sole:Phys. Rev. B,5, 2555 (1972); see alsoA. Martin-Löf:Comm. Math. Phys.,24, 253 (1972).

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  32. This theorem is due toR. A. Minlos andJ. G. Sinai:Trans. Moscow Math. Soc.,19, 121 (1968);Math. USSR Sbornik,2, 335 (1967). ActuallyMinlos andSinai prove a more difficult theorem under slightly different conditions. The adaptation to the deduction of the results given here can be found inG. Gallavotti andA. Martin-Löf:Comm. Math. Phys.,25, 87 (1972); or, better, inG. Gallavotti, A. Martin-Löf andS. Miracle-Sole: to appear in theLecture Notes of the 1971 Battelle-Seattle Summer Rencontres in Mathematics and Physics, edited byA. Lenard (Berlin).

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  33. G. Gallavotti andA. Martin-Löf:Comm. Math. Phys.,25, 87 (1972).

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  34. For a comparison of various old definitions of surface tension, new ones and a proof of their equivalence seeD. Abraham, G. Gallavotti andA. Martin-Löf:Surface tension in the two-dimensional Ising model, preprint (1972).

  35. The results of this Section are due toG. Gallavotti andH. Van Beyeren for the two-dimensional case and toR. L. Dobrushin for the 3-dimensional case. SeeG. Gallavotti andH. Van Beyren: preprint (1972);G. Gallavotti:Comm. Math. Phys., to appear;R. L. Dobrushin: to appear inTeor. Mat. Fiz.

  36. R. J. Burford andM. E. Fisher:Phys. Rev.,156, 583 (1967).

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  37. The reader may consult the book ofD. Ruelle:Statistical Mechanics (New York, 1969), p. 125;R. A. Minlos andJ. G. Sinai:Trans. Moscow Math. Soc.,19, 237 (1967);F. A. Berezin andJ. G. Sinai:Trans. Moscow Math. Soc.,19, 219 (1967);R. L. Dobrushin:Functional analysis and its applications,3, 22 (1968); and the review article byJ. Ginibre: inColloques du CNRS (Gif-sur-Yvette, 1970).

  38. D. Ruelle:Statistical Mechanics (New York, 1969), p. 112.

  39. R. L. Minlos andJ. G. Sinai:Teor. Mat. Fiz,2, No. 2 (1970);W. J. Camp andM. E. Fisher:Phys. Rev. Lett.,26, 73, 565 (1971).

  40. D. Abraham:Studies Appl. Math.,50, 71 (1971).

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  41. O. Lanford andD. Ruelle:Comm. Math. Phys.,13, 194 (1969).

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  42. Metastability should be a dynamical phenomenon as suggested, for instance, byO. Lanford andD. Ruelle:Comm. Math. Phys.,13, 194 (1969). An interesting rigorous treatment of the metastability phenomenon in the case of very weak and very-long-ranged forces can be found inJ. L. Lebowitz andD. Penrose: to appear.

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  43. See the review paper byJ. Ginibre: inColloques du CNRS (Gif-sur-Yvette, 1970). See alsoJ. L. Lebowitz andG. Gallavotti:Journ. Math. Phys.,12, 1129 (1971).

  44. D. Ruelle:Phys. Rev. Lett.,27, 1041 (1971).

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  45. Some interesting results are to be expected in these cases. See, for instance,D. Mermin:Phys. Rev. Lett.,26, 168 (1971).

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  46. D. Abraham:Studies Appl. Math.,50, 71 (1971).

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  47. R. L. Minlos andJ. G. Sinai:Teor. Mat. Fiz,2, No. 2 (1970);W. J. Camp andM. E. Fisher:Phys. Rev. Lett.,26, 73, 565 (1971).

  48. W. J. Camp andM. E. Fisher:Phys. Rev. Lett.,26, 73, 565 (1971).

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General References

  1. D. Ruelle:Statistical Mechanics (New York, 1969).

  2. R. A. Minlos:Lectures in Statistical Physics, inRussian Math. Surveys,23, 137 (1968).

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  3. The interested reader will be helped by the very recent and complete review of rigorous results (many of which refer to the Ising model and its generalizations) in the article by:R. B. Griffiths:Rigorous results and theorems, to appear inPhase Transitions and Critical Points, edited byC. Domb andM. S. Green (New York,).

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  1. Istituto di Fisica dell’Università, Roma

    G. Gallavotti

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Lavoro finanziato dall’Istituto di Fisica dell’Università, Roma, ed eseguito nell’ambito del CNR-GNAFA.

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Gallavotti, G. Instabilities and phase transitions in the ising model. A review. Riv. Nuovo Cim. 2, 133–169 (1972). https://doi.org/10.1007/BF02747778

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