Critical behavior of the Ising model on a hierarchical lattice with aperiodic interactions

TL;DR

This paper investigates how aperiodic interactions affect the critical behavior of the Ising model on hierarchical lattices, revealing conditions under which fluctuations alter phase transition properties.

Contribution

It derives exact renormalization-group relations for the Ising model with aperiodic interactions and establishes a criterion for the relevance of geometric fluctuations.

Findings

Small fluctuations do not change critical behavior.

Large fluctuations can prevent reaching the uniform fixed point.

A criterion for the relevance of geometric fluctuations is provided.

Abstract

We write exact renormalization-group recursion relations for nearest-neighbor ferromagnetic Ising models on Migdal-Kadanoff hierarchical lattices with a distribution of aperiodic exchange interactions according to a class of substitutional sequences. For small geometric fluctuations, the critical behavior is unchanged with respect to the uniform case. For large fluctuations, as in the case of the Rudin-Shapiro sequence, the uniform fixed point in the parameter space cannot be reached from any physical initial conditions. We derive a criterion to check the relevance of the geometric fluctuations.