Journal of Statistical Physics

, Volume 97, Issue 1–2, pp 87–144 | Cite as

On the Ising Model with Strongly Anisotropic External Field

  • F. R. Nardi
  • E. Olivieri
  • M. Zahradník


In this paper we analyze the equilibrium phase diagram of the two-dimensional ferromagnetic n.n. Ising model when the external field takes alternating signs on different rows. We show that some of the zero-temperature coexistence lines disappear at every positive sufficiently small temperature, whereas one (and only one) of them persists for sufficiently low temperature.

Ising model anisotropic field phase diagram cluster expansion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BKL]
    J. Bricmont, K. Kuroda, and J. L. Lebowitz, First order phase transitions in lattice and continuous systems: Extension of Pirogov-Sinai theory, Comm. Math. Phys. 101:501-538 (1985).Google Scholar
  2. [D]
    R. L. Dobrushin, Estimates of semi-invariants for the Ising model at low temperatures topics in statistical and theoretical physics, Amer. Math. Soc. Transl. (2) V 177:59-81 (1996).Google Scholar
  3. [DMS]
    E. I. Dinaburg, A. E. Mazel, and Ya. G. Sinai, The ANNNI model and contour models with interaction, Soviet Scientific Reviews C, Vol. 6, S. P. Novikov, ed. (Gordon and Breach, New York, 1987), pp. 113-168.Google Scholar
  4. [DO]
    R. L. Dobrushin, Existence of a phase transition in the two-dimensional and three-dimensional Ising models, Sov. Phys. Dokl. 10:111-113 (1965).Google Scholar
  5. [DS]
    E. I. Dinaburg and Ya. G. Sinai, An analysis of ANNNI model by Peierls contour method, Comm. Math. Phys. 98:119-144 (1985).Google Scholar
  6. [DS1]
    E. I. Dinaburg and Ya. G. Sinai, Contour models with interaction and their applications, Sel. Math. Sov. 7:291-315 (1988).Google Scholar
  7. [GMM]
    G. Gallavotti, A. Martin Löf, and Miracle Sole, Battelle Seattle (1971) Rendecontres, A. Lenard, ed., Lecture Notes in Physics, Vol. 20 (Springer, Berlin, 1973), pp. 162-204.Google Scholar
  8. [GR]
    R. B. Griffiths, Peierls' proof of spontaneous magnetization of a two-dimensional Ising ferrogmagnet, Phys. Rev. A 136:437-439 (1964).Google Scholar
  9. [HZ]
    P. Holický and M. Zahradník, Stratified low temperature phases of stratifies spin models. A general P. S. approach, submitted.Google Scholar
  10. [KP]
    R. Kotecký and D. Preiss, Cluster expansions for abstract polymer models, Comm. Math. Phys. 103:491-498 (1996).Google Scholar
  11. [NaO]
    F. R. Nardi and E. Olivieri, Low temperature stochastic dynamics for an Ising model with alternating field, Markov Proc. Rel. Field 2:117-166 (1996).Google Scholar
  12. [P]
    R. Peierls, On the Ising model of ferromagnetism, Proc. Cambridge Phil. Soc. 32:477-482 (1936).Google Scholar
  13. [PS]
    S. A. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems, Theor. Math. Phys. 25, 26:1185-1192, 39-49 (1975, 1976).Google Scholar
  14. [S]
    Ya. G. Sinai, Theory of phase transitions. Rigorous results (Pergamon Press, 1982).Google Scholar
  15. [Z]
    M. Zahradník, An alternate version of Pirogov-Sinai theory, Comm. Math. Phys. 93:559-581 (1984).Google Scholar
  16. [ZR]
    M. Zahradník, A short course on the Pirogov-Sinai theory, Rendiconti di Matematica 18:1-75 (1998).Google Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • F. R. Nardi
    • 1
  • E. Olivieri
    • 1
  • M. Zahradník
    • 2
  1. 1.Dipartimento di MatematicaII Università di Roma Tor VergataRomeItaly
  2. 2.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

Personalised recommendations