Partition function of the Potts model on self-similar lattices as a dynamical system and multiple transitions

Abstract

We present an analytic study of the Potts model partition function on two different types of self-similar lattices of triangular shape with non integer Hausdorff dimension. Both types of lattices analyzed here are interesting examples of non-trivial thermodynamics in less than two dimensions. First, the Sierpinski gasket is considered. It is shown that, by introducing suitable geometric coefficients, it is possible to reduce the computation of the partition function to a dynamical system, whose variables are directly connected to (the arising of) frustration on macroscopic scales, and to determine the possible phases of the system. The same method is then used to analyse the Hanoi graph. Again, dynamical system theory provides a very elegant way to determine the phase diagram of the system. Then, exploiting the analysis of the basins of attractions of the corresponding dynamical systems, we construct various examples of self-similar lattices with more than one critical temperature. These multiple critical temperatures correspond to crossing phases with different degrees of frustration.