Crossover from three-dimensional to two-dimensional systems in the nonequilibrium zero-temperature random-field Ising model

We present extensive numerical studies of the crossover from three-dimensional to two-dimensional systems in the nonequilibrium zero-temperature random-field Ising model with metastable dynamics. Bivariate finite-size scaling hypotheses are presented for systems with sizes $L\times L\times l$ which explain the size-driven critical crossover from two dimensions ($l=\text{const}$, $L\to \infty$) to three dimensions ($l\propto L\to \infty$). A model of effective critical disorder $R_c^{\mathrm{eff}}(l,L)$ with a unique fitting parameter and no free parameters in the $R_c^{\mathrm{eff}}(l,L\to \infty )$ limit is proposed, together with expressions for the scaling of avalanche distributions bringing important implications for related experimental data analysis, especially in the case of thin three-dimensional systems.

Figure 1

(a) Integrated size distributions $D_R^{\left(\mathrm{int}\right)}\left(S\right)$ vs avalanche size $S$ obtained for the same disorder $R=2.3$ and same span $L=8192$, but various thickness $l$, shown in the legend. (b) The same as in (a), but for disorders slightly above the effective critical disorder $R_c^{\mathrm{eff}}(l,L)$ for $L\times L\times l$ systems. Inset: When shown vs $S/l^{D_f}$, the foregoing distributions partially collapse showing branching tails because the values of $R$ and $L$ are not adjusted as is required by (5).

Figure 2

Numerical estimations of the effective critical disorder $R_c^{\mathrm{eff}}(l,L)$ vs thickness $l$ and span $L$ (black balls; error bars are magnified by a factor of 5 for better visibility). The surface shows our theoretical prediction (7). Arrows show three types of characteristic directions that are used for the FSS analysis: $l/L=\text{const}$, $l=\text{const}$ (variable $L$), and both $l,L$ constant (but variable $R$).

Figure 3

Data collapsing of the integrated size distributions $D^{\left(\mathrm{int}\right)}\left(S\right)$ predicted by (a, b) (5), (c, d) (13), and (e, f) (14). (a) $l/L=1/256$. (b) $l/L=1/2$. (c) $l=1$, $R_c=0.54$. (d) $l=4,R_c=1.02$. (e) $l=8$, $L=4096$. (f) $l=32$, $L=4096$. Three-dimensional exponents are used in (a) and (b) because $l=aL$, and 2D exponents are used elsewhere.

Figure 4

(a) Duration distributions for several RFIM systems having same the disorder $R=2.00$ and thickness $l=16$, but different span $L$, given in the legend. (b) Overlapping of the distributions, which are shown in (a), is achieved by the spurious values of $L$ exponents ($-0.3$ along the horizontal axis and 0.95 along the vertical axis). These values just provide deceptive collapsing, and are neither stable (i.e., depend on the range of $L$) nor related to some true exponents.

Figure 5

$N\left(R\right)$ vs disorder $R$, where $N\left(R\right)$ is the average number per single run of 2D spanning avalanches triggered in $L\times L\times l$ systems with disorder $R$. Symbols show $N\left(R\right)$ obtained in our simulations for $l=16$ and $L=256$. The full curve shows the model function $N_{R_0,W}\left(R\right)$ of type (6) that best fits simulation data when $R_0=1.954\pm 0.001$ and $W=0.074\pm 0.002$. The position of $R_0$ is indicated by the full vertical line, and the width $W$ of the transition region is indicated by dashed lines.

Figure 6

Critical disorder $R_c\left(l\right)$ for the the $L\times L\times l$ family of lattices in the $L\to \infty$ limit, shown vs lattice thickness $l$. Inset: Symbols show the values of effective critical disorder $R_c^{\mathrm{eff}}(l,L)$ vs $L^{-1}$ for $l$ fixed to 8, 16, and 64; error bars are smaller than symbol size. For the same fixed values of $l$, solid lines show the curves $R_c^{\mathrm{th}}(l,L)$, converging to $R_c\left(l\right)$ when $1/L\to 0$.

Figure 7

Data collapsing of the integrated size distributions $D^{\left(\mathrm{int}\right)}\left(S\right)$ predicted by (5). The data are obtained in simulations of systems containing up to $\approx 1.7\times {10}^{10}$ spins. Parts (a) and (f) are already shown in the main text as parts (a) and (b) of Fig. 5, and are repeated here for the sake of completeness. (a) $l/L=1/256$. (b) $l/L=1/32$. (c) $l/L=1/16$. (d) $l/L=1/8$. (e) $l/L=1/4$. (f) $l/L=1/2$.

Figure 8

Data collapsing of the integrated duration distributions $D^{\left(\mathrm{int}\right)}\left(T\right)$ predicted by (C2). The range of $l/L$, and the values of disorder, are the same as in Fig. 7.

Figure 9

Data collapsing of the integrated energy distributions $D^{\left(\mathrm{int}\right)}\left(E\right)$ predicted by (C5). The range of $l/L$, and the values of disorder, are the same as in Figs. 7 and 8.