Potts models on hierarchical lattices and Renormalization Group dynamics

TL;DR

This paper demonstrates that the renormalization group generator for Potts models on hierarchical lattices can be represented as a rational map, enabling analysis of phase transition zeros via complex dynamics.

Contribution

It introduces a novel representation of the renormalization group as a rational map on complex projective spaces, extending analysis to models with external fields and multiple-spin interactions.

Findings

Lee-Yang and Fisher zeros are in the unstable set of the renormalization map for certain lattices.

The rational map framework allows studying phase transition zeros through complex dynamics.

The approach generalizes to models with external magnetic fields and multi-spin interactions.

Abstract

We prove that the generator of the renormalization group of Potts models on hierarchical lattices can be represented by a rational map acting on a finite-dimensional product of complex projective spaces. In this framework we can also consider models with an applied external magnetic field and multiple-spin interactions. We use recent results regarding iteration of rational maps in several complex variables to show that, for some class of hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of the renormalization map.