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 T ⁽⁴⁾_c = 2k ⁻¹_B J/ln{1+2/[(z−1)^3/4−1]}, confirming a result obtained by Müller-Hartmann and Zittartz [Phys. Rev. Lett. 33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andk _B 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.