Dynamics of Ising models with damping

We show for the Ising model that it is possible to construct a discrete time stochastic model analogous to the Langevin equation that incorporates an arbitrary amount of damping. It is shown to give the correct equilibrium statistics and is then used to investigate nonequilibrium phenomena, in particular, magnetic avalanches. The value of damping can greatly alter the shape of hysteresis loops, and for small damping and high disorder, the morphology of large avalanches can be drastically affected. Small damping also alters the size distribution of avalanches at criticality.

Figure 1

(Color online) Plot of results obtained for the two dimensional Ising model on a ${128}^2$ lattice for two different values of the damping parameter. This is a plot of the average magnetization per spin $m$ vs $T$. The $\times$’s are for no dissipation, $\alpha =1$, which is a purely microcanonical simulation. The $+$’s are for $\alpha =0.5$. The dashed curve is the exact solution to this model in the thermodynamic limit.

Figure 2

(Color online) The major branch of the descending hysteresis loop for ${64}^2$ systems using different values of the damping parameter and the spin coupling. Strong damping, $\alpha =0.5$ (dashed-dotted line), is shown in the left most curve as judged from the top of the plot for coupling $j=0.3$, which starts decreasing from $M=1$ at $h=-0.2$, and does not have large abrupt changes. All the other curves are for weak damping, $\alpha =0.99$. In this case but also for $j=0.3$, we see that although $M$ starts to decrease at the same location as for strong damping, it abruptly drops as the field is lowered. As the coupling $j$ is decreased, smooth curves are eventually seen again. Going from left to right, as judged from the top, $j=0.3$, 0.25, 0.2, and 0.15.

Figure 3

(a) Snapshot of the spin configuration for a ${256}^2$ system with $j=0.35$, $\alpha =0.9$, during a system-size avalanche at the field $h=-0.400007$. (b) A grayscale plot of the auxiliary variables at the same time.

Figure 4

Snapshots of the spin configurations for a ${256}^2$ system with $j=0.25$, $\alpha =0.99$, during an avalanche at the field $h=-7\times {10}^{-5}$. (a) The beginning of the avalanche. (b) When the avalanche is of order of half the system size. (c) The final configuration of the avalanche.

Figure 5

(Color online) The major branch of the descending hysteresis loops in two ${32}^3$ systems with $j=0.19$ for two different values of the dissipation: upper curve: $\alpha =0.5$ and lower curve: $\alpha =0.99$.

Figure 6

(Color online) (a) Magnetization versus field for the Ising model with damping described in the text. The system size is ${32}^3$ and the two lines represent two runs close to criticality: one with a coupling of $j=0.165$ and the other of 0.167. (b) The plot for relaxational dynamics (large damping) with couplings of 0.21 and 0.212.

Figure 7

(Color online) The avalanche size distribution, which is measured for the entire hysteresis loop for $\alpha =0.99$ $(+)$ and $\alpha =0.9$ $(\times )$. The $x$ axis is the number of avalanches normalized by its mean size. The $y$ axis is the normalized distribution of sizes. The less negative sloped straight line is a fit of the $\alpha =0.99$ curve and has a slope of $-1.4$. The more strongly sloped one has a slope of $-2$.