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.

We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and show that the characteristic length for self-averaging L small star, filled increases as exp[exp(J/Delta)] in 2D, and as exp(exp[exp(J/Delta)]) in 3D, for disorder strength Delta much less than the exchange coupling J. For system size 1<infinity for both square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and h(coer) tends to J.

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We study the hysteretic evolution of the random field Ising model at T = 0 when the magnetization M is controlled externally and the magnetic field H becomes the output variable. The dynamics is a simple modification of the single-spin-flip dynamics used in the H-driven situation and consists in flipping successively the spins with the largest local field. This allows one to perform a detailed comparison between the microscopic trajectories followed by the system with the two protocols. Simulations are performed on random graphs with connectivity z =4 ͑Bethe lattice͒ and on the three-dimensional cubic lattice. The same internal energy U͑M͒ is found with the two protocols when there is no macroscopic avalanche and it does not depend on whether the microscopic states are stable or not. On the Bethe lattice, the energy inside the macroscopic avalanche also coincides with the one that is computed analytically with the H-driven algorithm along the unstable branch of the hysteresis loop. The output field, defined here as ⌬U / ⌬M, exhibits very large fluctuations with the magnetization and is not self-averaging. The relation to the experimental situation is discussed.

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Minor hysteresis loops within the main loop are obtained analytically and exactly in the onedimensional ferromagnetic random field Ising-model at zero temperature. Numerical simulations of the model show excellent agreement with the analytical results.

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