Ising spin glass in the Bethe approximation at zero temperature

We study the properties of the Ising spin glass in the Bethe approximation at zero temperature for the case of a ±J distribution of bonds. We show that there is an infinite number of solutions for the probability distribution function of the effective field. Each one is a sum of 2N+1 delta functions located at Jn/N, n=0,±1, ±2,…,±N. In the limit N→∞ we obtain a solution with a continous part in addition to delta functions located at 0 and ±J. For each case we calculate the spin glass order parameter, the energy and entropy.