Yang-Lee zeros, Julia sets, and their singularity spectra

Chaotic renormalization flow in the Potts model induced by long-range competition

A proper description of spin glass remains a hard subject in theoretical physics and is considered to be closely related to the emergence of chaos in the renormalization group (RG) flow. Previous efforts concentrate on models with either complicated or nonrealistic interactions in order to achieve this chaotic behavior. Here we find that the commonly used Potts model with long-range interaction could do the job nicely in a large parameter regime as long as the competition between the ferromagnetic and antiferromagnetic interaction is maintained. With this simplicity, the appearance of chaos is observed to sensitively depend on the detailed network structure: the parity of bond number in a branch of the basic RG substituting unit; chaos only emerges for even numbers of bonds. These surprising and universal findings may shed light on the study of spin glass.

Distribution and density of the partition function zeros for the diamond-decorated Ising model

Exact renormalization map of temperature between two successive decorated lattices is given, and the distribution of the partition function zeros in the complex temperature plane is obtained for any decoration level. The rule governing the variation of the distribution pattern as the decoration level changes is given. The densities of the zeros for the first two decoration levels are calculated explicitly, and the qualitative features about the densities of higher decoration levels are given by conjecture. The Julia set associated with the renormalization map is contained in the distribution of the zeros in the limit of infinite decoration level, and the formation of the Julia set in the course of increasing the decoration level is given in terms of the variations of the zero density.

Evolution and structure formation of the distribution of partition function zeros: triangular type Ising lattices with cell decoration

The distribution of partition function zeros of the two-dimensional Ising model in the complex temperature plane is studied within the context of triangular decorated lattices and their triangle-star transformations. Exact recursion relations for the zeros are deduced for the description of the evolution of the distribution of the zeros subject to the change of decoration level. In the limit of infinite decoration level, the decorated lattices essentially possess the Sierpiński gasket or its triangle-star transformation as the inherent structure. The positions of the zeros for the infinite decorated lattices are shown to coincide with the ones for the Sierpiński gasket or its triangle-star transformation, and the distributions of zeros all appear to be a union of infinite scattered points and a Jordan curve, which is the limit of the scattered points.

Mandelbrot set in coupled logistic maps and in an electronic experiment

We suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps. The idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one-dimensional quadratic maps with a particular coupling. Hence, dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems, which relate to the Feigenbaum universality class. To substantiate the feasibility of this concept, we consider an electronic system, which exhibits dynamical phenomena intrinsic to complex analytic maps. Experimental results are presented, providing the Mandelbrot set in the parameter plane of this physical system.