Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

The influence of external boundary conditions on the spherical model of a ferromagnet. I. Magnetization profiles

  • 27 Accesses

  • 10 Citations


The spherical model of a ferromagnet is investigated for various (external) boundary conditions. It is shown that, besides the well-known critical point, a second one can be produced by the boundary conditions. Although the main asymptotic of the free energy is analytic at the new critical point, theO(N1−2/d) asymptotic possesses a singularity here. A natural order parameter of the model has singularities at both critical points. The magnetization profile is studied for the whole range of the model's parameters and at different scales. It is shown that (in an appropriate regime) below the second critical temperature the magnetization profile freezes, that is, becomes temperature independent. Distributions of the single spin variables and some macroscopic observables (including normalized total spin) are studied for the whole temperature range including the critical points.

This is a preview of subscription content, log in to check access.


  1. 1.

    D. B. Abraham, Solvable model with a roughening transition for a planar Ising ferromagnet,Phys. Rev. Lett. 44:1165–1168 (1980).

  2. 2.

    D. B. Abraham, Surface structures and phase transitions—Exact results, inPhase Transitions and Critical Phenomena, Vol. 10, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1986), Chapter 1.

  3. 3.

    D. B. Abraham and P. Reed, Interface profile of the Ising ferromagnet in two dimensions,Commun. Math. Phys. 49:35–46 (1976).

  4. 4.

    D. B. Abraham and M. A. Robert, Phase separation in the spherical model,J. Phys. A: Math. Gen. 13:2229–2245 (1980).

  5. 5.

    D. B. Abraham and M. E. Issigoni, Phase separation at the surface of the Ising ferromagnet,J. Phys. A: Math. Gen. 13:L89-L91 (1980).

  6. 6.

    H. Au-Yang, Thermodynamics of an anisotropic boundary of a two-dimensional Ising model,J. Math. Phys. 14:937–946 (1973).

  7. 7.

    R. Z. Bariev, Correlation functions of the semi-infinite two-dimensional Ising model I. Local magnetization.Theor. Math. Phys. 40:623–636 (1979).

  8. 8.

    M. N. Barber and M. E. Fisher, Critical phenomena in systems of finite thickness I. The spherical model,Ann. Phys. 77:1–78 (1973).

  9. 9.

    M. N. Barber, D. Jasnow, S. Singh, and R. A. Weiner, Critical behavior of the spherical model with enhanced surface exchange,J. Phys. C: Solid State Phys. 7:3491–3504 (1974).

  10. 10.

    T. H. Berlin and M. Kac, The spherical model of a ferromagnet,Phys. Rev. 86:821–835 (1952).

  11. 11.

    R. S. Ellis and C. M. Newman, The statistics of the Curie-Weiss models,J. Stat. Phys. 19:149–161 (1978); R. S. Ellis,Entropy, Large Deviations, and Statistical Mechanics (Springer-Verlag, Berlin, 1985).

  12. 12.

    M. E. Fisher and A. E. Ferdinand, Interfacial, boundary, and size effects at critical points,Phys. Rev. Lett. 19:169 (1967).

  13. 13.

    M. E. Fisher and V. Privman, First-order transition in spherical models: Finite size scaling,Commun. Math. Phys. 103:527–548 (1986).

  14. 14.

    J. Fröhlich and C.-E. Pfister, Semi-infinite Ising model. I. Thermodynamic functions and phase diagram in absence of magnetic field,Commun. Math. Phys. 109:493–523 (1987); Semi-infinite Ising model. II. The wetting and layering transition,Commun. Math. Phys. 112:51–74 (1987).

  15. 15.

    A. Isihara, Effect of defects of spin interaction in a simple cubic lattice,Phys. Rev. 108:619–629 (1957).

  16. 16.

    J. S. Joyce, Critical properties of the spherical model, inPhase Transition and Critical Phenomena, Vol. 2, C. Domb and M. S. Green, eds. (Academic Press, New York, 1972).

  17. 17.

    H. J. F. Knops, Infinite spin dimensionality limit for nontranslationally invariant interactions,J. Math. Phys. 14:1918–1920 (1973).

  18. 18.

    J. S. Langer, A modified spherical model of a first-order phase transition,Phys. Rev. 137:A1531-A1547 (1965).

  19. 19.

    M. Lax, Relation between canonical and microcanonical ensembles,Phys. Rev. 97:1419–1420 (1955).

  20. 20.

    E. H. Lieb and C. J. Thompson, Phase transition in zero dimensions: A remark on the spherical model,J. Math. Phys. 10:1403–1406 (1969).

  21. 21.

    B. McCoy and T. T. Wu, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. IV,Phys. Rev. 162:436–475 (1967).

  22. 22.

    S. A. Molchanov and Ju. N. Sudarev, Gibbs states in the spherical model,Sov. Math. Dokl. 16:1254–1257 (1975).

  23. 23.

    A. Patrick, On phase separation in the spherical model of an ferromagnet: Quasiaverage approach,J. Stat. Phys. 72:665–701 (1993).

  24. 24.

    S. Singh, D. Jasnow, and M. N. Barber, Critical behavior of the spherical model with enhanced surface exchange: Two spherical fields,J. Phys. C: Solid State Phys. 8:3408–3414 (1975).

  25. 25.

    C. C. Yan and G. H. Wannier, Observation on the spherical model of a ferromagnet,J. Math. Phys. 6:1833–1838 (1965).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Patrick, A.E. The influence of external boundary conditions on the spherical model of a ferromagnet. I. Magnetization profiles. J Stat Phys 75, 253–295 (1994). https://doi.org/10.1007/BF02186289

Download citation

Key Words

  • Spherical model
  • magnetization profile
  • Gibbs states
  • phase transitions