Fixed Point Properties of the Ising Ferromagnet on the Hanoi Networks

TL;DR

This paper investigates the fixed-point behavior of the Ising ferromagnet on Hanoi networks using exact renormalization group analysis, revealing non-universal critical phenomena and tunable phase transitions influenced by network structure.

Contribution

It provides an exact renormalization group analysis of the Ising model on Hanoi networks, highlighting how small structural changes lead to diverse critical behaviors and phase transition types.

Findings

Non-universal fixed points depend on temperature and initial couplings.

Diverse behaviors emerge from small structural modifications.

Identifies tunable transitions between power-law and essential singularities.

Abstract

The Ising model with ferromagnetic couplings on the Hanoi networks is analyzed with an exact renormalization group. In particular, the fixed-points are determined and the renormalization-group flow for certain initial conditions is analyzed. Hanoi networks combine a one-dimensional lattice structure with a hierarchy of small-world bonds to create a mix of geometric and mean-field properties. Generically, the small-world bonds result in non-universal behavior, i.e. fixed points and scaling exponents that depend on temperature and the initial choice of coupling strengths. It is shown that a diversity of different behaviors can be observed with seemingly small changes in the structure of the networks. Defining interpolating families of such networks, we find tunable transitions between regimes with power-law and certain essential singularities in the critical scaling of the correlation length, similar to the so-called inverted Berezinskii-Kosterlitz-Thouless transition previously observed only in scale-free or dense networks.