Phase ordering kinetics of the long-range Ising model

We use an efficient method that eases the daunting task of simulating dynamics in spin systems with long-range interaction. Our Monte Carlo simulations of the long-range Ising model for the nonequilibrium phase ordering dynamics in two spatial dimensions perform significantly faster than the standard Metropolis approach and considerably more efficiently than the kinetic Monte Carlo method. Importantly, this enables us to establish agreement with the theoretical prediction for the time dependence of domain growth, in contrast to previous numerical studies. This method can easily be generalized to applications in other systems.

Figure 1

Evolution snapshots at different times, demonstrating the ferromagnetic ordering in LRIM with $L=1024$ for different $\sigma$. Only the up spins ($+$) are marked.

Figure 2

(a) Correlation functions $C(r,t)$ at different times for $\sigma =0.6$. (b) Demonstration of the scaling of $C(r,t)$ as a function of $r/\ell \left(t\right)$ for the same times as in (a). (c) Scaling plots for the structure factor $S(k,t)$. The solid line there corresponds to the Porod tail behavior of $S(k,t)\sim k^{-3}$.

Figure 3

(a) Scaled correlation function $C(r,t)$ at $t=100$ MCSs for different $\sigma$ as mentioned. (b) Same as (a) but for the scaled structure factor $S(k,t)$. The solid line again corresponds to the Porod tail.

Figure 4

(a) Time dependence of the length scale $\ell \left(t\right)$ for $\sigma =0.6$ for three different $L$ as indicated. The solid and the dashed lines correspond to $t^{1/(1+\sigma )}$ and $t^{1/2}$, respectively. The inset illustrates the finite-size scaling using the same data with $t_0=3$ and ${\ell }_0=5$. The solid line shows the expected $y^{-1/(1+\sigma )}$ behavior. (b) Time dependence of $\ell \left(t\right)$ for different $\sigma$ as indicated with $L=2048$. The solid lines are the respective predictions in (2). The inset here shows the length-scale data for $\sigma =0.6$ with $L=1024$ when using different cutoff distances $r_c$ in Eq. (1). The cutoff $r_c=8$ is close to the value of $r_c\approx 8.3$ used in Ref. [12]. The lines have the same meaning as in the main frame of (a).