Prabodh Shukla

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 behaviour 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 , 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 three- and four-coordinated random graphs, which for large system sizes give the same results as on the Bethe lattice, but avoid surface effects altogether.

Hysteresis in the random-field Ising model and bootstrap percolation.

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<<L<L small star, filled, the coercive field h(coer) varies as 2J-DeltalnlnL for the square lattice, and as 2J-DeltalnlnlnL on the cubic lattice. Its limiting value is 0 for L-->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.

Distribution of Avalanche Sizes in the Hysteretic Response of the Random-Field Ising Model on a Bethe Lattice at Zero Temperature

We consider the zero-temperature single-spin-flip dynamics of the random-field Ising model on a Bethe lattice in the presence of an external field h. We derive the exact self-consistent equations to determine the distribution Prob(s) of avalanche sizes s as the external field increases from −∞ to ∞. We solve these equations explicitly for a rectangular distribution of the random fields for a linear chain and the Bethe lattice of coordination number z=3, and show that in these cases, Prob(s) decreases exponentially with s for large s for all h on the hysteresis loop. We find that for z≥4 and for small disorder, the magnetization shows a first-order discontinuity for several continuous and unimodal distributions of the random fields. The avalanche distribution Prob(s) varies as s−3/2 for large s near the discontinuity.

Exact solution of return hysteresis loops in a one-dimensional random-field ising model at zero temperature

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

Exact expressions for minor hysteresis loops in the random field Ising model on a Bethe lattice at zero temperature.

We obtain exact expressions for the minor hysteresis loops in the ferromagnetic random field Ising model on a Bethe lattice at zero temperature in the case when the driving field is cycled infinitely slowly.

Critical hysteresis in random-field XY and Heisenberg models.

We study zero-temperature hysteresis in random-field XY and Heisenberg models in the zero-frequency limit of a cyclic driving field. We consider three distributions of the random field and present exact solutions in the mean-field limit. The results show a strong effect of the form of disorder on critical hysteresis as well as the shape of hysteresis loops. A discrepancy with an earlier study based on the renormalization group is resolved.

Persistence in one-dimensional Ising models with parallel dynamics

We study persistence in one-dimensional ferromagnetic and antiferromagnetic nearest-neighbor Ising models with parallel dynamics. The probability P(t) that a given spin has not flipped up to time t, when the system evolves from an initial random configuration, decays as P(t) approximately 1/t(straight theta(p)) with straight theta(p) approximately 0.75 numerically. A mapping to the dynamics of two decoupled A+A-->0 models yields straight theta(p)=3/4 exactly. A finite size scaling analysis clarifies the nature of dynamical scaling in the distribution of persistent sites obtained under this dynamics.

Hysteresis in random-field Ising model on a Bethe lattice with a mixed coordination number

We study zero-temperature hysteresis in the random-field Ising model on a Bethe lattice where a fraction c of the sites have coordination number z = 4 while the remaining fraction have z = 3. Numerical simulations as well as probabilistic methods are used to show the existence of critical hysteresis for all values of . This extends earlier results for c = 0 and c = 1 to the entire range , and provides new insight in non-equilibrium critical phenomena. Our analysis shows that a spanning avalanche can occur on a lattice even in the absence of a spanning cluster of z = 4 sites.

Analysis of wasp-waisted hysteresis loops in magnetic rocks.

The random-field Ising model of hysteresis is generalized to dilute magnets and is solved on a Bethe lattice. Exact expressions for the major and minor hysteresis loops are obtained. In the strongly dilute limit the model provides a simple and useful understanding of the shapes of hysteresis loops in magnetic rock samples.

Magnetization-driven random-field Ising model at T=0

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=4Bethe lattice and on the three-dimensional cubic lattice. The same internal energy UM 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.