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 T2=2kB−1Jln(2+1) and TBP=kB−1Jln(3), and a line of fourth order phase transitions between TBP and ∞, where kB is the Boltzmann constant, and J is the nearest-neighbor interaction parameter.