Topological Disorder in Spin Models on Hierarchical Lattices

TL;DR

This paper introduces a new method to analyze spin models on hierarchical lattices with variable coordination numbers, revealing complex effects of disorder on phase transitions.

Contribution

It proposes a general approach for describing spin systems with variable coordination numbers and studies disorder effects on the Ising model on hierarchical lattices.

Findings

Recurrent relations for magnetization are derived.

Disorder in coordination number affects critical points.

Localization of critical points is observed.

Abstract

A general approach for the description of spin systems on hierarchial lattices with coordination number $q$ as a dynamical variable is proposed. The ferromagnetic Ising model on the Bethe lattice was studied as a simple example demonstrating our method. The annealed and partly annealed versions of disorder concerned with the lattice coordination number are invented and discussed. Recurrent relations are obtained for the evaluation of magnetization. The magnetization is calculated for the particular disorder choices, $q=2,3$ and $q=3,4$. A nontrivial localization of critical point is revealed.