Communications in Mathematical Physics

, Volume 364, Issue 2, pp 655–682 | Cite as

High-Fugacity Expansion, Lee–Yang Zeros, and Order–Disorder Transitions in Hard-Core Lattice Systems

  • Ian Jauslin
  • Joel L. Lebowitz


We establish existence of order–disorder phase transitions for a class of “non-sliding” hard-core lattice particle systems on a lattice in two or more dimensions. All particles have the same shape and can be made to cover the lattice perfectly in a finite number of ways. We also show that the pressure and correlation functions have a convergent expansion in powers of the inverse of the fugacity. This implies that the Lee–Yang zeros lie in an annulus with finite positive radii.


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We are grateful to Giovanni Gallavotti and Roman Kotecký for enlightening discussions. The work of J.L.L. was supported by AFOSR Grant FA9550-16-1-0037. The work of I.J. was supported by The Giorgio and Elena Petronio Fellowship Fund and The Giorgio and Elena Petronio Fellowship Fund II.


  1. 1.
    Alder B.J., Wainwright T.E.: Phase transition for a hard sphere system. J. Chem. Phys. 27(5), 1208–1209 (1957)ADSCrossRefGoogle Scholar
  2. 2.
    Baxter R.J.: Hard hexagons: exact solution. J. Phys. A Math. Gen. 13(3), L61–L70 (1980)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, Cambridge (1982)zbMATHGoogle Scholar
  4. 4.
    Bernard, E.P., Krauth, W.: Two-step melting in two dimensions: First-order liquid-hexatic transition. Physical Review Letters 107(15), 155704 (2011)Google Scholar
  5. 5.
    Bovier A., Zahradník, M.: A simple inductive approach to the problem of convergence of cluster expansions of polymer models. J. Stat. Phys. 100(3/4): 765–778 (2000)Google Scholar
  6. 6.
    Brascamp H.J., Kunz H.: Analyticity properties of the ising model in the antiferromagnetic phase. Commun. Math. Phys. 32(2), 93–106 (1973)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dobrushin R.L.: The problem of uniqueness of a gibbsian random field and the problem of phase transitions. Funct. Anal. Appl. 2(4), 302–312 (1969)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eisenberg E., Baram A.: A first-order phase transition and a super-cooled fluid in a two-dimensional lattice gas model. Europhys. Lett. (EPL) 71(6), 900–905 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    Gallavotti G., Bonetto F., Gentile G.: Aspects of the Ergodic, Qualitative and Statistical Properties of Motion. Springer, Berlin (2004)CrossRefGoogle Scholar
  10. 10.
    Gallavotti G., Miracle-Sole S., Robinson D.: Analyticity properties of a lattice gas. Phys. Lett. A 25(7), 493–494 (1967)ADSCrossRefGoogle Scholar
  11. 11.
    Gaunt D.S.: Hard-sphere lattice gases. II. Plane-triangular and three-dimensional lattices. J. Chem. Phys. 46(8), 3237–3259 (1967)ADSCrossRefGoogle Scholar
  12. 12.
    Gaunt D.S., Fisher M.E.: Hard-sphere lattice gases. i. Plane-square lattice. J. Chem. Phys. 43(8), 2840–2863 (1965)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghosh A., Dhar D.: On the orientational ordering of long rods on a lattice. Europhys. Lett. (EPL) 78(2), 20003 (2007)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Groeneveld J.: Two theorems on classical many-particle systems. Phys. Lett. 3(1), 50–51 (1962)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Heilmann O.J., Praestgaard E.: Phase transition in a lattice gas with third nearest neighbour exclusion on a square lattice. J. Phys. A Math. Nucl. Gen. 7(15), 1913–1917 (1974)ADSCrossRefGoogle Scholar
  16. 16.
    Isobe M., Krauth W.: Hard-sphere melting and crystallization with event-chain monte carlo. J. Chem. Phys. 143(8), 084509 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    Jauslin I., Lebowitz J.L.: Crystalline ordering and large fugacity expansion for hard-core lattice particles. J. Phys. Chem. B 122(13), 3266–3271 (2017)CrossRefGoogle Scholar
  18. 18.
    Joyce G.S.: On the hard-hexagon model and the theory of modular functions. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 325((1588), 643–702 (1988)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kotecký, R., Preiss, D.: An inductive approach to the pirogov-sinai theory. In: Proceedings of the 11th Winter School on Abstract Analysis, Rendiconti del Circolo Metematico di Palermo, Serie II, supplemento, vol. 3, pp. 161–164 (1984)Google Scholar
  20. 20.
    Kotecký R., Preiss D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103(3), 491–498 (1986)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Lebowitz J.L., Ruelle D., Speer E.R.: Location of the lee-yang zeros and absence of phase transitions in some ising spin systems. J. Math. Phys. 53(9), 095211 (2012)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lee T.D., Yang C.N.: Statistical theory of equations of state and phase transitions. II. lattice gas and ising model. Phys. Rev. 87(3), 410–419 (1952)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Mayer J.E.: The statistical mechanics of condensing systems. i.. J. Chem. Phys. 5(1), 67–73 (1937)ADSCrossRefGoogle Scholar
  24. 24.
    McCoy, B.M.: Advanced statistical mechanics, vol.146 of International Series of Monographs on Physics. Oxford University Press, Oxford (2010)Google Scholar
  25. 25.
    Peierls R., Born M.: On Ising’s model of ferromagnetism. Math. Proc. Camb. Philos. Soc. 32((03), 477 (1936)ADSCrossRefGoogle Scholar
  26. 26.
    Penrose O.: Convergence of fugacity expansions for fluids and lattice gases. J. Math. Phys. 4(10), 1312–1320 (1963)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Pirogov S.A., Sinai Y.G.: Phase diagrams of classical lattice systems. Theor. Math. Phys. 25(3), 1185–1192 (1975)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pusey P.N., van Megen W.: Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. Nature 320(6060), 340–342 (1986)ADSCrossRefGoogle Scholar
  29. 29.
    Richthammer T.: Translation-invariance of two-dimensional gibbsian point processes. Commun. Math. Phys. 274(1), 81–122 (2007)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Ruelle D.: Correlation functions of classical gases. Ann. Phys. 25(1), 109–120 (1963)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Ruelle D.: Statistical Mechanics: Rigorous Results. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  32. 32.
    Strandburg K.J.: Two-dimensional melting. Rev. Mod. Phys. 60(1), 161–207 (1988)ADSCrossRefGoogle Scholar
  33. 33.
    Ursell H.D.: The evaluation of Gibbs’ phase-integral for imperfect gases. Math. Proc. Camb. Philos. Soc. 23(06), 685 (1927)ADSCrossRefGoogle Scholar
  34. 34.
    Wood W.W., Jacobson J.D.: Preliminary results from a recalculation of the monte carlo equation of state of hard spheres. J. Chem. Phys. 27(5), 1207–1208 (1957)ADSCrossRefGoogle Scholar
  35. 35.
    Yang C.N., Lee T.D.: Statistical theory of equations of state and phase transitions. i. Theory of condensation. Phys. Rev. 87(3), 404–409 (1952)ADSMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Departments of Mathematics and PhysicsRutgers UniversityPiscatawayUSA
  3. 3.Simons Center for Systems BiologyInstitute for Advanced StudyPrincetonUSA

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