Forma, Vol. 7 (No. 2), pp. 157-165, 1992
Original Paper

Hierarchical Lattice with Competing Interactions; the Case of Irrational Map in the Complex Plane

Sadao Isogami and Kyôzô Takeyama

Department of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112, Japan

(Received May 14, 1992; Accepted July 7, 1992)

Keywords: Ising system, hierarchical lattice, Yang-Lee zeros, Julia set, fractal

Abstract. The fractal structures of zeros of the partition function in the complex plane are studied for an Ising system on a diamond-type hierarchical lattice with competing ferromagnetic and antiferromagnetic interactions where the renormalization function is irrational and multi-valued. In calculating the zeros, a branch of the complex map is used which coincides with the real value transformation on the positive real axis; the argument of the variable is restricted between 0 and 2p. Varying the competition parameter, off-real-axis attractors and a "phase change" in the structure of the Julia set are found.