Abstract
It is widely accepted that the free energy F(H) of the two-dimensional Ising model in the ferromagnetic phase T<T c has an essential branch cut singularity at the origin H=0. The phenomenological droplet theory predicts that near the cut drawn along the negative real axis H<0, the imaginary part of the free energy per lattice site has the form ImF[exp(±iπ)|H|]=±B|H|exp(−A/|H|) for small |H|. We verify this prediction in analytical perturbative transfer matrix calculations for the square lattice Ising model for all temperatures 0<T<T c and arbitrary anisotropy ratio J 1/J 2. We obtain an expression for the constant A which coincides exactly with the prediction of the droplet theory. For the amplitude B we obtain B=πM/18, where M is the equilibrium spontaneous magnetization. In addition we find discrete-lattice corrections to the above mentioned phenomenological formula for ImF, which oscillate in H −1.
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REFERENCES
N. S. Isakov, Commun. Math. Phys. 95:427 (1984).
D. B. Abraham and P. J. Upton, Phys. Rev. Lett. 69:225 (1992).
D. B. Abraham and P. J. Upton, Phys. Rev. Lett. 70:1567 (1993).
A. F. Andreev, Sov. Phys. JETP 18:1415 (1964).
M. E. Fisher, Physics 3:255 (1967).
H. Kastrup, Phys. Rev. Lett. 81:2280 (1998).
J. S. Langer, Ann. Phys. (N.Y.) 41:108 (1967).
N. J. Günther, D. A. Nicole, and D. J. Wallace, J. Phys. A 13:1755 (1980).
R. K. P. Zia and D. J. Wallace, Phys. Rev. B 31:1624 (1985).
M. B. Voloshin, I. Yu. Kobzarev, and L. B. Okun', Yad. Fiz. 20:1229 (1974) [Sov. J. Nucl. Phys. 20:644 (1975)].
S. Coleman, Phys. Rev. D 15:2929 (1977), Erratum: Phys. Rev. D 16:1248 (1977).
C. G. Callan and S. Coleman, Phys. Rev. D 16:1762 (1977).
P. A. Rikvold and B. M. Gorman, in Annual Review of Computational Physics I, D. Stauffer, ed. (World Scientific, Singapore, 1994).
M. J. Lowe and D. J. Wallace, J. Phys. A 13:L381 (1980).
C. K. Harris, J. Phys. A 17:L143 (1984).
C. C. A Günther, P. A. Rikvold, and M. A. Novotny, Phys. Rev. Lett. 71:3898 (1993).
C. C. A Günther, P. A. Rikvold, and M. A. Novotny, Physica A 212:194 (1994).
M. B. Voloshin, Yad. Fiz. 42:1017 (1985) [Sov. J. Nucl. Phys. 42:644 (1985)].
V. Privman and L. S. Schulman, J. Stat. Phys. 29:205 (1982).
S. B. Rutkevich, Phys. Rev. B 60:14525 (1999).
G. A. Baker, Jr. and D. Kim, J. Phys. A 13:L103 (1980).
T. D. Schultz, D. C. Mattis, and E. H. Lieb, Rev. Mod. Phys. 36:856 (1964).
M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Physica D 1:80 (1980).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics (Pergamon Press, Oxford, 1965), Vol. 3.
M. Reed and B. Symon, Methods of Mathematical Physics (Academic, New York, 1978), Vol. 4.
J. M. Ziman, Principles of the Theory of Solids (University Press, Cambridge, 1972).
R. K. P. Zia and J. E. Avron, Phys. Rev. B 25:2042 (1982).
L. Onsager, Phys. Rev. 65:117 (1944).
R. Kotecký and E. Olivieri, J. Stat. Phys. 70:1121 (1993).
J. Palmer and C. Tracy, Adv. Appl. Math. 2:329 (1981).
J. B. Kogut, Rev. Mod. Phys. 51:659 (1979).
P. A. Rikvold, private communication (2000).
A. A. Slavnov and L. D. Faddeev, Introduction to QuantumTheory of Guage Fields (Nauka, Moskow, 1988).
T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Phys. Rev. B 13:316 (1976).
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Rutkevich, S.B. Analytic Verification of the Droplet Picture in the Two-Dimensional Ising Model. Journal of Statistical Physics 104, 589–608 (2001). https://doi.org/10.1023/A:1010324620997
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DOI: https://doi.org/10.1023/A:1010324620997