Distribution of Zeros of the Partition Function of the Ising Model

Abstract

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 z_c and 1⁄z_c when the next neighbor ferromagnetic interaction is introduced. This means that our system shows the phase transitions at finite critical fields H_c and −H_c. The locus of zeros of Fisher’s model of super-exchange antiferromagnet is also shown to cross the positive real axis at z_c and 1⁄z_c below the Néel temperature.