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

Abstract

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.

TheT= 0 random-field Ising model on a Bethe lattice with large coordination number: hysteresis and metastable states

“…[8] and the expansion of these equations to O(1/z) is worked out in Appendix B. Let us recall again that the main feature is the existence of an outof-equilibrium phase transition for z ≥ 4 in the thermodynamic limit.…”

“…Therefore, these reentrant branches, which correspond to the "unstable" root of the self-consistent equation of Ref. [8], bear no relation to the quenched complexity and do not seem to have a physical meaning. As H increases from 0 and m * 0 (H) decreases along the reentrant part of the meanfield curve, A (0) decreases and the minimum value of the complexity Σ Q (g → ±∞, H) also decreases.…”

“…The hysteresis loop obtained from the equations of Ref. [8] is shown in Fig. 3 (for clarity, the mean-field magnetization curve m 0 (H) is only shown in Fig.…”

“…In this Appendix, we compute the hysteresis loop at the order 1/z, starting from the equations derived by Dhar et al [8]. The approach to the mean-field behavior has been studied numerically in Ref.…”

“…According to Ref. [8], the magnetization m(H) along the ascending branch of the hysteresis loop is given by…”