Spin-glass and ferromagnetic phases of the random-bond Ising model on the Bethe lattice

Magnetic properties are numerically obtained for the random-bond Ising model of interactions + J (J > 0) and - J with probabilities p(p ⩾/ 0.5) and 1 − p, on the Bethe lattice of coordination number 3 in zero and non-zero uniform external magnetic fields. It is confirmed that the phase transitions between the ferromagnetic, the paramagnetic and the spin-glass phase occur under zero or infinitesimal uniform external field. The phase boundary between the ferromagnetic and the spin-glass phase is determined. It is shown that the susceptibility does not diverge but only has a cusp at this phase transition. The critical concentration pc at zero temperature is 0.86940 ± 0.00003. Evidence is given to show that the self-consistent solution of the effective field becomes unstable at the phase boundaries. No phase transition is found in a non-zero external field. It is concluded that the entropy and the specific heat in the Bethe approximation in the spin-glass phase are non-negative and the entropy tends to zero as the temperature approaches zero.