Third and Fourth Order Phase Transitions: Exact Solution for the Ising Model on the Cayley Tree

TL;DR

This paper provides an exact analytical solution showing that the Ising model on the Cayley tree exhibits both third and fourth order phase transitions across specific temperature ranges, revealing complex critical behavior.

Contribution

It offers the first exact derivation of the phase transition lines and orders for the Ising model on the Cayley tree, clarifying the nature of these transitions.

Findings

Identifies a line of third order phase transitions between specific temperatures.

Establishes a line of fourth order phase transitions at higher temperatures.

Provides exact analytical expressions for transition points.

Abstract

An exact analytical derivation is presented, showing that the Ising model on the Cayley tree exhibits a line of third order phase transition points, between temperatures $T_2=2k_B^{-1}J\ln ({\sqrt 2}+1) $ and $T_{BP}=k_B^{-1}J\ln (3)$, and a line of fourth order phase transitions between $T_{BP}$ and $\infty$, where $k_B$ is the Boltzmann constant, and $J$ is the nearest-neighbor interaction parameter.