Distribution of Zeros of the Partition Function in the First and Second Order Phase Transitions : Extension of the Lee- Yang Theorem

概要

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The distribution of zeros of the partition function for one-dimensional lattice system of generalized Ising spins_ with magnitude unity is studied, where it is assumed that the component spins interact with the nearest neighbours through a ferromagnetic coupling and each spin state has a certain degeneracy. The spin states are denoted by a variable σ which can take either of three values, plus and minus unities and zero with the respective degeneracies, D(1), D(0) and D(-1), where it is assumed D(1)=D(-1). Zeros of the partition function Z(ζ) of the system as exposed to an external magnetic field are studied on the ζ-plane, where ζ=exp(H) indicates the exponential function of the external field divided by temperature H and it is regarded as a complex number. The distribution of zeros is classified into three typical patterns, corresponding to the respective three regions on the plane of temperature(T) versus degeneracy ratio (ρ= D(0)/D(1)): Zeros never lie on the unit circle in the region of larger ρ, all on the unit circle in the region of smaller ρ and only a part on the unit circle in the remaining region. On the basis of this result, it is expected in the case of two- or three-dimensional system that phase transitions of first and second orders occur in the cases of ρ larger and smaller than a critical value ρ_c, respectively.
理論物理学刊行会の論文
1981-05-25