Distribution of Zeros of the Partition Function of the Ising Model

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To study the relations between Yang-Lee’s theory of the phase transitions and the distribution of the zeros of the partition functions in the antiferromagnetic case, the partition functions of finite (4×4 and 4×6) Ising model with the nearest and the next nearest neighbor interactions are calculated. The distributions of the zeros in the fugacity z plane of the partition function for various combinations of the values of J⁄kT−J′⁄kT or tanh (J⁄2kT)−tanh (J′⁄2kT) are investigated. The patterns of the distribution of the zeros show several types corresponding to ferromagnetic state, antiferromagnetic state and superantiferromagnetic state. In the antiferromagnetic ordered state the locus of the zeros is found to be nearly two concentric circles and to cross the positive real axis at two points zc and 1⁄zc when the next neighbor ferromagnetic interaction is introduced. This means that our system shows the phase transitions at finite critical fields Hc and −Hc. The locus of zeros of Fisher’s model of super-exchange antiferromagnet is also shown to cross the positive real axis at zc and 1⁄zc below the Néel temperature.

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