Prog. Theor. Phys. Vol. 38 No. 6 (1967) pp. 1225-1242

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A Theory of the Second Order Phase Transitions in Spin Systems. II

— Complex Magnetic Field —

Masuo Suzuki

Department of Physics, Faculty of Science, University of Tokyo, Tokyo

(Received July 1, 1967)

Abstract:

The thermodynamic quantities of the Ising model are analyzed near T_c, using the asymptotic distribution function g(θ, t) of zeros for the partition function on a unit circle in the plane of the complex magnetic field. The distribution function is derived as g(θ, t) = t^λf(θ/t^{λ + γ}), assuming that the susceptibility behaves like χ₀⁺ ∼ t^-γ near T_c where t = T/T_c-1. It is shown that there are two types of the situation; type I, in which a critical angle θ_c(t) exists such that g(θ, t) = 0 for 0 ≦ |θ| ≦ θ_c, and type II, in which there is no critical angle. The critical indices are related to one another, using the above g(θ, t).
The relation between the magnetic equation of state M = M(h, t) and the distribution function is given as follows: g(θ, t) = (1/2π)Re M(iθ, t), where h = 2mH/kT. g(θ, t) for an exactly soluble model near T_c is given by the equation θ² ∼ t³ + a₁t²g² + a₂tg⁴ + a₃g⁶ (a_i > 0). This is a good example for the general theory.

URL : http://ptp.ipap.jp/link?PTP/38/1225/
DOI : 10.1143/PTP.38.1225

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Citing Article(s) :

Journal of the Physical Society of Japan 58 (1989) pp. 4344-4351 :
Phase Boundary of an Ising Antiferromagnet

S. Dasgupta
Progress of Theoretical Physics Vol. 38 No. 6 (1967) pp. 1243-1251 :
Note on Singularity of Specific Heat in the Second Order Phase Transition

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Progress of Theoretical Physics Vol. 39 No. 2 (1968) pp. 349-364 :
A Theory of the Second Order Phase Transitions in Spin Systems. III

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Progress of Theoretical Physics Vol. 40 No. 6 (1968) pp. 1246-1256 :
Theorems on Extended Ising Model with Applications to Dilute Ferromagnetism

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Progress of Theoretical Physics Vol. 41 No. 6 (1969) pp. 1438-1449 :
On the Distribution of Zeros for the Heisenberg Model

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Progress of Theoretical Physics Vol. 42 No. 5 (1969) pp. 1076-1085 :
Singularity of Nonlinear Response near the Critical Field. I

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Progress of Theoretical Physics Vol. 43 No. 4 (1970) pp. 882-906 :
On the Singularity of Dynamical Response and Critical Slowing Down

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Progress of Theoretical Physics Vol. 46 No. 4 (1971) pp. 1054-1070 :
Scaling with a Parameter in Spin Systems near the Critical Point. I

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Progress of Theoretical Physics Vol. 46 No. 5 (1971) pp. 1337-1359 :
Relationship among Exactly Soluble Models of Critical Phenomena. I

Masuo Suzuki
Progress of Theoretical Physics Vol. 49 No. 5 (1973) pp. 1440-1450 :
Critical Exponents for Long-Range Interactions. III

Masuo Suzuki