Zero-temperature hysteresis in the random-field Ising model on a Bethe lattice

We consider the single-spin-flip dynamics of the random-field Ising model on a Bethe lattice at zero temperature in the presence of a uniform external field. We determine the average magnetization as the external field is varied from − ∞ to + ∞ by setting up the self-consistent field equations, which we show are exact in this case. The qualitative behavior of magnetization as a function of the external field unexpectedly depends on the coordination number z of the Bethe lattice. For z = 3, with a gaussian distribution of the quenched random fields, we find no jump in magnetization for any non-zero strength of disorder. For z ≥ 4, for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field, which disappears for a larger variance of the quenched field. We determine exactly the critical point separating smooth hysteresis curves from those with a jump. We have checked our results by Monte Carlo simulations of the model on 3- and 4- coordinated random graphs, which for large system sizes give the same results as on the Bethe lattice, but avoid surface effects altogether. PACS nos: 75.60, 05.50, 75.40G 1 I.