Universality classes and crossover behaviors in non-Abelian directed sandpiles

We study universality classes and crossover behaviors in non-Abelian directed sandpile models in terms of the metastable pattern analysis. The non-Abelian property induces spatially correlated metastable patterns, characterized by the algebraic decay of the grain density along the propagation direction of an avalanche. Crossover scaling behaviors are observed in the grain density due to the interplay between the toppling randomness and the parity of the threshold value. In the presence of such crossovers, we show that the broadness of the grain distribution plays a crucial role in resolving the ambiguity of the universality class. Finally, we claim that the metastable pattern analysis is important as much as the conventional analysis of avalanche dynamics.

Figure 1

A two-dimensional $\frac{\pi }{4}$-rotated lattice for DSMs.

Figure 2

Toppling rules of non-Abelian deterministic DSMs on a two-dimensional tilted lattice [20]: (a) aND, (b) bND, and (c) cND. The gray colored grains and arrow represent the state before toppling and the black colored ones represent the state after toppling.

Figure 3

(Color online) Typical metastable grain (a) patterns and (b) scars of six DSMs (from left to right, AD, AS, NS, aND, bND, and cND) at $z_c=2$: occupied sites and scars are shown as black dots and the typical shapes of dissipative avalanches consisting of toppled sites as blue (shaded) areas on a lattice with $L=150$ (horizontally) and $T=250$ (vertically).

Figure 4

(Color online) Numerical results of the aND for various $z_c$ values: (a) grain occupation densities ${\rho }_{\theta }\left(t\right)$ versus $t$ on a lattice with $T=2^{14}$ for $z_c=3$, 5, 7, and 9 ($T=2^{13}$ otherwise) and $L=2T$. (b) Finite-size scaling of $P\left(t,T\right)$ with the MF value $\left({\tau }_t=2\right)$ for $z_c=2$ (lines) and $z_c=4$ (open symbols) and with the NS value $\left({\tau }_t=7/4\right)$ at $z_c=3$ (filled symbols). Here, $T=L=2^{10}$, $2^{11}$, $2^{12}$, and $2^{13}$ from right to left and $D_t=1$ for all cases. (c) Crossover scaling of grain occupation densities for odd $z_c$ values by ${\rho }_{\theta }\left(t\right)z_c^2$ as a function of $t/z_c^2$, which are the same data as (a). (d) Rescaled grain distributions $Q_r\left(z,t\right)$ at two different layers, $t=16$ (open symbols) and 4096 (filled symbols) for $z_c=50$ (△) and 51 (◇) on a lattice with $T=2^{13}$ and $L=2T$.

Figure 5

(Color online) (a) Grain occupation densities ${\rho }_{\theta }\left(t\right)$ versus $t$ in the pND at a fixed $z_c=3$ as $p$ increases from 0.5 (the aND as the thin line at the top) to 1 (the bND as the thick line at the bottom). We used a lattice with $T=2^{13}$ and $L=2T$. (b) Crossover scaling of grain occupation densities at $z_c=3$ by ${\rho }_{\theta }\left(t\right)/{\left(1-p\right)}^{1/2}$ as a function of $t{\left(1-p\right)}^{1/2}$.

Figure 6

(Color online) Numerical results of the bND for various $z_c$ values: (a) grain occupation densities ${\rho }_{\theta }\left(t\right)$ versus $t$ on a lattice with $T=2^{14}$ and $L=2T$ ($T=2^{13}$ for $z_c=50$ and 51), where lines for $z_c=2$, 4, and 50 are from top to bottom. (b) Finite-size scaling of $P\left(t,T\right)$ with MF values for $z_c=2$ (lines), $z_c=3$ (filled symbols), and $z_c=4$ (open symbols). Here, $T=L=2^{10}$, $2^{11}$, $2^{12}$, and $2^{13}$ from right to left. (c) Rescaled grain distributions $Q_r\left(z,t\right)$ at two different layers, $t=16$ (open symbols) and 4096 (filled symbols) for $z_c=50$ (△) and 51 (◇) on a lattice with $T=2^{13}$ and $L=2T$.

Figure 7

(Color online) (a) Grain occupation densities ${\rho }_{\theta }\left(t\right)$ versus $t$ in the NS for various $z_c$ values (lines for $z_c=2$, 20, 50, and 100 from top to bottom at the right). We used a lattice with $T=2^{13}$ and $L=2T$. (b) In particular to the cases for $z_c=25$ and 100, rescaled grain distributions, $Q_r\left(z,t\right)$, are measured at two different layers, $t=16$ (open symbols) and 4096 (filled symbols).