Critical fluctuations of time-dependent magnetization in a random-field Ising model

Cooperative behaviors near the disorder-induced critical point in a random-field Ising model are numerically investigated by analyzing time-dependent magnetization in ordering processes from a special initial condition. We find that the intensity of fluctuations of time-dependent magnetization, $\chi \left(t\right)$, attains a maximum value at a time $t=\tau$ in a normal phase and that $\chi \left(\tau \right)$ and $\tau$ exhibit divergences near the disorder-induced critical point. Furthermore, spin configurations around the time $\tau$ are characterized by a length scale, which also exhibits a divergence near the critical point. We estimate the critical exponents that characterize these power-law divergences by using a finite-size scaling method.

Figure 1

Ordering process of magnetization, $m\left(t\right)$. $L=40$ and $R=2.16$.

Figure 2

$m\left(t_{\infty }\right)$ as a function of $h$ for systems with three different sizes.

Figure 3

Phase diagram. The circles represent frozen states. $L=40$.

Figure 4

$\chi \left(t\right)$ for three values of $h$. $R=2.16$.

Figure 5

Finite-size scaling for ${\chi }_m\left(h,L\right)$ and $\tau \left(h,L\right)$ (inset). The three graphs for systems with different sizes are collapsed into a single curve. The statistical error bars are within the size of symbols. $h_{\text{c}}=1.44$ and $R=2.16$.

Figure 6

$S\left(\mathit{k}\right)/S\left(\mathbf{0}\right)$ as a function of $k\xi \left(h,L\right)$ Inset: Finite-size scaling for $\xi \left(h,L\right)$. Filled symbols represent those of the system for $L=64$ and the others for $L=32$. $R=2.16$.

Figure 7

Schematic phase diagram in the $(R,H,T)$ space.