Abstract
The spherical model of a ferromagnet is investigated in the framework of the generalized quasiaverage approach where an external field positive in one half of a square lattice and negative in the other half is used. It is shown that in addition to the well-known critical point, a second one can be produced by the field. Although the main asymptotic of the free energy is analytic at this point, the next-to-leading asymptotic possesses a singularity here, as well as at the point where the free energy per site is nonanalytic. An order parameter of the model also has singularities at both critical points. The magnetization profile is studied at different scales. It is shown that (in an appropriate regime), below the new critical temperature the magnetization profile freezes, that is, becomes temperature independent.
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References
D. B. Abraham and P. Reed, Interface profile of the Ising ferromagnet in two dimensions,Commun. Math. Phys. 49:35–46 (1976).
D. B. Abraham and M. A. Robert, Phase separation in the spherical model,J. Phys. A: Math. Gen. 12:L129–L132 (1979).
D. B. Abraham and M. A. Robert, Phase separation in the spherical model,J. Phys. A: Math. Gen. 13:2229–2245 (1980).
N. Angelescu and V. A. Zagrebnov, A generalized quasiaverage approach to the description of the limit states of then-vector Curie-Weiss ferromagnet,J. Stat. Phys. 41:323–334 (1985).
M. N. Barber, Finite-size scaling, inPhase Transition and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1983), Chapter 2.
M. N. Barber and M. E. Fisher, Critical phenomena in systems of finite thickness I. The spherical model,Ann. Phys. 77:1–78 (1973).
M. van den Berg, J. T. Lewis, and J. V. Pulè, A general theory of Bose-Einstein condensation,Helv. Phys. Acta 59:1271–1288 (1986).
T. H. Berlin and M. Kac, The spherical model of a ferromagnet,Phys. Rev. 86:821–835 (1952).
N. N. Bogoliubov,Lectures on Quantum Statistics, Vol. 2, Quasi-Averages (Gordon and Breach, New York, 1970).
J. G. Brankov and D. M. Danchev, On the limit Gibbs states of the spherical model.J. Phys. A: Math. Gen. 20:4901–4913 (1987).
J. G. Brankov, V. A. Zagrebnov, and N. S. Tonchev, Description of the limit Gibbs states for the Curie-Weiss-Ising model,Theor. Math. Phys. 66:72–80 (1986).
R. L. Dobrushin, Gibbs states describing coexistence of phases for a three-dimensional Ising model,Theory Prob. Appl. 17:582–600 (1972).
G. Gallavotti, The phase separation in the two-dimensional Ising model,Commun. Math. Phys. 27:103–136 (1972).
J. Gough, and J. V. Pulè, The spherical model of Bose-Einstein condensation,Helv. Phys. Acta. 66:17–52 (1993).
J. D. Gunton and M. J. Buckingham, Condensation of the ideal Bose gas as a cooperative transition,Phys. Rev. 166:152–158 (1968).
A. E. Ferdinand and M. E. Fisher, Bounded and inhomogeneous Ising models. I. Specific-heat anomaly of a finite lattice,Phys. Rev. 185:832–846 (1969).
H. O. Georgii,Gibbs Measures and Phase Transitions (de Gruyter, Berlin, 1988).
Y. Imry and S.-K. Ma, Random-field instability of the ordered state of continuous symmetry,Phys. Rev. Lett. 35:1399–1401 (1975).
H. J. F. Knops, Infinite spin dimensionality limit for nontranslationally invariant interactions,J. Math. Phys. 14:1918–1920 (1973).
V. A. Malyshev and R. A. Minlos,Gibbs Random Fields. The Method of Cluster Expansions (Reidel, Dordrecht, 1991).
S. A. Molchanov and Ju. N. Sudarev, Gibbs states in the spherical model,Sov. Math. Dokl. 16:1254–1257 (1975).
L. A. Pastur, Disordered spherical model,J. Stat. Phys. 27:119–151 (1982).
H. Van Beijeren, Interface sharpness in the Ising system,Commun. Math. Phys. 40:1–6 (1975).
C. C. Yan and G. H. Wannier, Observation on the spherical model of a ferromagnet,J. Math. Phys. 6:1833–1838 (1965).
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Patrick, A.E. On phase separation in the spherical model of a ferromagnet: Quasiaverage approach. J Stat Phys 72, 665–701 (1993). https://doi.org/10.1007/BF01048028
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DOI: https://doi.org/10.1007/BF01048028