Yang-Lee zeros of the Ising model on the closed Cayley tree

Abstract

The Yang-Lee zeros of the partition function of the ferro-, antiferro- and of the partially antiferromagnetic anisotropic Ising models defined on the closed symmetric Cayley tree are studied. The applicability of the Yang-Lee theorem to the antiferromagnetic systems is shown to be a consequence of the invariance of the unit circle under the Bethe-Peierls map. The relationship as well as the distinction between the set of zeros and the Julia set is established. The fractal dimension of the Julia set is shown to be equal to one in the low temperature phase and to be a decreasing function of the temperature in the paramagnetic phase of the three systems.