Magnetization-driven random-field Ising model atT=0

Abstract

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.

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

“…[16]) and the mean-field behavior described by Eq. (42) is recovered when z → ∞ [16]. From a mathematical point of view the transition for ∆ < ∆ c (z) is due to a saddle-node bifurcation [8]: the self-consistent equation (B.2) (that corresponds to an increasing applied field) has three real roots in the range H 1 (∆) < H < H 2 (∆) and two of the solutions coalesce and become complex at the branching fields H 1 and H 2 (see Fig.…”

“…Although the precise behavior of the complexity in the vicinity of the "knees" is still unknown (and is certainly different on the Bethe and on euclidian lattices), it is now clear that the discontinuity in the hysteresis loop associated with a macroscopic avalanche is due to the existence of a gap in the magnetization of the metastable states for a range of applied field. Consequently, by controlling the magnetization instead of the magnetic field [9], one should observe a reentrant loop, as indeed observed in some magnetic materials [1] and other disordered systems [21,22]. Finally, one may wonder whether the most numerous, hence probable, states in the middle of the loop are accessible dynamically; it is, however, dubious that this can be achieved by performing a deep quench from T = ∞ to T = 0 at a fixed field [23].…”

“…The approach to the mean-field behavior has been studied numerically in Ref. [16], but, as far as we know, the explicit calculation has only been performed in the limit z → ∞. According to Ref.…”

Hysteresis in a system driven by either generalized force or displacement variables: Martensitic phase transition in single-crystallineCu−Zn−Al

“…13 However, there is no reason to expect that out of equilibrium hysteresis loops obtained when controlling the force will be similar to those obtained when the control parameter is the generalized displacement. Such a situation has been very recently theoretically studied 14,15 by making use of the random field Ising model ͑RFIM͒, 16 which is a prototype model for athermal hysteresis. Results predict significant differences in the hysteresis loops obtained in the force-driven case ͑external field H͒ to those obtained by driving the generalized displacement ͑magnetization M͒.…”

Diluted three-dimensional random field Ising model at zero temperature with metastable dynamics

“…6,7 Moreover, many properties of the critical point have also been studied analytically on Bethe lattices. [8][9][10][11][12] Experimental evidence for the occurrence of such a critical point has been found in different magnetic systems. 13,14 Another interesting result of the RFIM with metastable dynamics is that it reproduces the experimental observation that m͑H͒ trajectories of such athermal systems are discontinuous on small scales.…”