Higher-order susceptibilities of the regular and the random Ising model on the Cayley tree. I

Abstract

Explicit expressions for the fourth-order susceptibility χ(4), the fourth derivative of thebulk free energy with respect to the external field, are given for the regular and the random-bond Ising model on the Cayley tree in the thermodynamic limit, at zero external field. The fourth-order susceptibility for the regular system diverges at temperature Tc(4)= 2kB−1J/ln{1+2/[(z−1)3/4−1]}, confirming a result obtained by Muller-Hartmann and Zittartz [Phys. Rev. Lett.33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andkB is the Boltzmann constant. The temperatures at which χ(4) and the ordinary susceptibility χ(2) diverge are given also for the random-bond and the random-site Ising model and for diluted Ising models.