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

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) = 2k(B)(-1)J In(root 2 + 1) and T(BP) = k(B)(-1)J In(3), and a line of fourth order phase transitions between T(BP) and infinity, where k(B) is the Boltzmann constant, and J is the nearest-neighbor interaction parameter. (C) 2008 Elsevier B.V. All rights reserved.