Phase diagrams of lattice models on Cayley tree and chandelier network: a review

TL;DR

This review comprehensively surveys phase diagrams of Ising and Potts models on Cayley trees and chandelier networks, analyzing phase transitions, dynamical behaviors, and providing computational tools for these lattice models.

Contribution

It systematically compiles known results, presents recursive equations, and analyzes phase transitions and dynamical behaviors on various Cayley-tree-like lattices.

Findings

Phase diagrams of Ising and Potts models are thoroughly analyzed.

Transition mechanisms between different phases are characterized by Lyapunov exponents.

Dynamical behaviors on chandelier networks are explored and compared.

Abstract

The main purpose of this review paper is to give systematically all the known results on phase diagrams corresponding to lattice models (Ising and Potts) on Cayley tree (or Bethe lattice) and chandelier networks. A detailed survey of various modelling applications of lattice models is reported. By using Vannimenus's approach, the recursive equations of Ising and Potts models associated to a given Hamiltonian on the Cayley tree are presented and analyzed. The corresponding phase diagrams with programming codes in different programming languages are plotted. To detect the phase transitions in the modulated phase, we investigate in detail the actual variation of the wave-vector $q$ with temperature and the Lyapunov exponent associated with the trajectory of our current recursive system. We determine the transition between commensurate ($C$) and incommensurate ($I$) phases by means of the Lyapunov exponents, wave-vector, and strange attractor for a comprehensive comparison. We survey the dynamical behavior of the Ising model on the chandelier network. We examine the phase diagrams of the Ising model corresponding to a given Hamiltonian on a new type of "Cayley-tree-like lattice", such as triangular, rectangular, pentagonal chandelier networks (lattices). Moreover, several open problems are discussed.