Abstract
We derive a new inequality for ferromagnetic Ising spin systems and then use it to obtain information about the number of phases which can coexist in such systems. We show in particular that for even interactions only two phases (up and down magnetization) can coexist below the critical temperature at zero magnetic field (h=0) whenever the energy is a continuous function of the temperature. We also prove that the derivatives with respect toh ath=0 of the odd correlation functions (triplet,...) diverge like the susceptibility in the vicinity of the critical temperature (at least for pair interactions). Our results also apply to higher order Ising spins (not just spin 1/2).
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References
R. B. Griffiths, inPhase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds., Academic Press (1972), Vol. 1, pp. 7–109.
C. Gruber,J. stat Phys. 14:81 (1976).
D. Ruelle,Statistical Mechanics, Benjamin (1969);Thermodynamic Formalism, Addison-Wesley, to appear.
R. L. Dobrushin,Functional Anal. Appl. 2:292, 302 (1968).
O. E. Lanford and D. Ruelle,Commun. Math. Phys. 13:194 (1969); O. E. Lanford, inStatistical Mechanics and Math. Problems, A. Lenard, ed., Springer (1973).
R. B. Israel,Commun. Math. Phys. 43:59 (1975); S. A. Pirogov and Ya. G. Sinäi,Funkts. Analiz. 25 (1974); D. Ruelle, On Manifolds of Phase Coexistence, preprint (1975).
A. Messager and S. Miracle-Sole,Commun. Math. Phys. 40:187 (1975).
G. Gallavotti and S. Miracle-Sole,Phys. Rev. B 5:2555 (1972); J. Slawny,Commun. Math. Phys. 34:271 (1973);46:75 (1976); C. Gruber, A. Hinterman, and D. Merlini,Commun. Math. Phys. 40:83 (1975); C. Gruber and A. Hinterman,Physica 83A:233 (1975).
J. Ginibre,Commun. Math. Phys. 16:310 (1970).
R. B. Griffiths,J. Math. Phys. 8:478 (1967); D. G. Kelley and S. Sherman,J. Math. Phys. 9:466 (1969).
J. L. Lebowitz and A. Martin-Löf,Commun. Math. Phys. 25:276 (1972).
H. van Beyeren,Commun. Math. Phys. 40:1 (1975).
C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre,Commun. Math. Phys. 22:89 (1971).
J. L. Lebowitz,J. Stat. Phys., to appear.
J. L. Lebowitz, inMathematical Problems in Theoretical Physics, H. Araki, ed., Springer (1975).
J. Frohlich, B. Simon, and T. Spencer,Commun. Math. Phys. 50:79 (1976).
L. Onsager,Phys. Rev. 65:117 (1944).
J. L. Lebowitz,Commun. Math. Phys. 28:313 (1972).
A. Martin-Löf,Commun. Math. Phys. 25:87 (1972).
D. W. Wood and H. P. Griffiths,J. Math. Phys. 14, 1715 (1973);J. Phys. A 9:407 (1976).
R. Baxter,J. Phys. A 8:1797 (1975).
M. N. Barber,J. Phys. A (1976).
R. B. Griffiths,J. Math. Phys. 10:1559 (1969).
B. Simon and R. B. Griffiths,Commun. Math. Phys. 33:145 (1973); B. Bimon,The P(0)2 Euclidian (Quantum) Field Theory, Princeton University Press (1974).
F. Dunlop, private communication.
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Research supported in part by NSF Grant #MPS 75-20638 and USAFOR Grant #73-2430D.
John Simon Guggenheim Fellow.
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Lebowitz, J.L. Coexistence of phases in Ising ferromagnets. J Stat Phys 16, 463–476 (1977). https://doi.org/10.1007/BF01152284
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DOI: https://doi.org/10.1007/BF01152284