Analytic Properties of the Onsager Solution of the Ising Model

Chanchal K. Majumdar

doi:10.1103/PhysRev.145.158

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Analytic Properties of the Onsager Solution of the Ising Model

Chanchal K. Majumdar

Carnegie Institute of Technology, Pittsburgh, Pennsylvania

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Phys. Rev. 145, 158 – Published 6 May, 1966

DOI: https://doi.org/10.1103/PhysRev.145.158

Abstract

The analytic properties of the Onsager solution of the two-dimensional Ising lattice in zero external magnetic field are investigated. The free energy above the critical temperature $T_{c}$ can be regarded as an analytic function in the $u$ plane, where $u = \exp (- \frac{4 J}{k_{B} T})$ , with singularities at $u = 0, \infty, and \exp (\pm \frac{4 J}{k_{B} T_{c}})$ . But its analytic continuation below the critical temperature does not describe a stable thermodynamic phase. Similar statements are true for the free energy below the critical temperature. The famous logarithmic singularity is briefly discussed. Finally, the knowledge of the analytic properties is shown, by elementary considerations, to explain the nonanalytic behavior at vanishing external field below the critical temperature—a result first enunciated by Yang and Lee for the free energy in the presence of an external magnetic field.

References (15)

C. Domb, Advan. Phys. 9, 149 (1960) ibid.9, 245 (1960) M. E. Fisher, J. Math. Phys. 4, 278 (1963) ibid.5, 944 (1964) G. Baker, Phys. Rev. 124, 768 (1961)
C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952) ibid.87, 410 (1952)
L. Onsager, Phys. Rev. 65, 117 (1944)
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, London, 1962), 4th Ed
Omitted endnote
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill Book Company, Inc., New York, 1953), Vol. 1, p. 400
Omitted endnote
[4], p. 299, problem 18. pages 288 to 291
J. Hadamard, Acta Math. 22, 191 (1898) J. Tarski, J. Math. Phys. 1, 154 (1960)
H. S. Green and C. A. Hurst, Order-Disorder Phenomena (Interscience Publishers Inc., New York, 1964), p. 228
E. C. Titchmarsh, The Theory of Functions (Oxford University Press, London, 1939), 2nd ed., p. 229
[4], p. 25
M. E. Fisher, Proc. Roy. Soc. (London) A254, 66 (1959)
M. F. Sykes, J. W. Essam, and D. S. Gaunt, J. Math. Phys. 6, 283 (1965)
[11], p. 214

[c1] C. Domb, Advan. Phys. 9, 149 (1960) ibid.9, 245 (1960) M. E. Fisher, J. Math. Phys. 4, 278 (1963) ibid.5, 944 (1964) G. Baker, Phys. Rev. 124, 768 (1961)

[c2] C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952) ibid.87, 410 (1952)

[c3] L. Onsager, Phys. Rev. 65, 117 (1944)

[c4] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, London, 1962), 4th Ed

[c5] Omitted endnote

[c6] P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill Book Company, Inc., New York, 1953), Vol. 1, p. 400

[c7] Omitted endnote

[c8] [4], p. 299, problem 18. pages 288 to 291

[c9] J. Hadamard, Acta Math. 22, 191 (1898) J. Tarski, J. Math. Phys. 1, 154 (1960)

[c10] H. S. Green and C. A. Hurst, Order-Disorder Phenomena (Interscience Publishers Inc., New York, 1964), p. 228

[c11] E. C. Titchmarsh, The Theory of Functions (Oxford University Press, London, 1939), 2nd ed., p. 229

[c12] [4], p. 25

[c13] M. E. Fisher, Proc. Roy. Soc. (London) A254, 66 (1959)

[c14] M. F. Sykes, J. W. Essam, and D. S. Gaunt, J. Math. Phys. 6, 283 (1965)

[c15] [11], p. 214

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Analytic Properties of the Onsager Solution of the Ising Model

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