Third and fourth order phase transitions: Exact solution for the Ising model on the Cayley tree

Abstract

In this work we present the first exact solution of a system of interacting particles with phase transitions of order higher than two. The presented analytical derivation shows that the Ising model on the Cayley tree exhibits a line of third order phase transition points, between temperatures T 2 = 2 k B − 1 J ln ( 2 + 1 ) and T B P = k B − 1 J ln ( 3 ) , and a line of fourth order phase transitions between T B P and ∞ , where k B is the Boltzmann constant, and J is the nearest-neighbor interaction parameter.