Unusual scaling for two-dimensional avalanches: Curing the faceting and scaling in the lower critical dimension

The nonequilibrium random-field Ising model is well studied, yet there are outstanding questions. In two dimensions, power-law scaling approaches fail and the critical disorder is difficult to pin down. Additionally, the presence of faceting on the square lattice creates avalanches that are lattice dependent at small scales. We propose two methods which we find solve these issues. First, we perform large-scale simulations on a Voronoi lattice to mitigate the effects of faceting. Second, the invariant arguments of the universal scaling functions necessary to perform scaling collapses can be directly determined using our recent normal form theory of the renormalization group. This method has proven useful in cleanly capturing the complex behavior which occurs in both the lower and upper critical dimensions of systems and here captures the two-dimensional nonequilibrium random-field Ising model behavior well. The obtained scaling collapses span a range of a factor of 10 in the disorder and a factor of ${10}^4$ in avalanche cutoff. They are consistent with a critical disorder at zero and with a lower critical dimension for the model equal to 2.

Figure 1

A plot of the avalanches at different values of the disorder (a) $r=0.5$, (b) $r=1.0$, (c) $r=5.0$, (d) $r=50.0$. Different colors show different avalanches.

Figure 2

Segment of the Voronoi lattice. Spins are at circled sites, randomly scattered in the two-dimensional plane. The spins interact with neighbors across bonds denoted by black lines. The Voronoi lattice in green determines the neighbors of each spin.

Figure 3

Scaling collapse of the area-weighted avalanche size distribution $A\left(s\right|w)$ for $w$ ranging from 0.8 to 8.0. There is a slight bulge at $s/\mathrm{\Sigma }\left(w\right)\sim {10}^{-2}$ for small $w$. The shape of this curve $\mathcal{A}$ is universal; it should be reproduced in experiments and other simulations in the same universality class.

Figure 4

Scaling collapse of the change in magnetization of the sample with respect to the field $\frac{dM}{dh}\left(h\right|w)$ for values of $w$ ranging from 0.8 to 8.0. Again, the shape of this curve is universal.

Figure 5

Comparison of the best fit of $\mathrm{\Sigma }\left(w\right)$ and $\eta \left(w\right)$ derived with different functional forms of $\frac{dw}{dl}$. We have $w=(r-r_c)/s_s$ such that $\mathrm{\Sigma }\left(r\right)=\mathrm{\Sigma }\left(w\right)$ and $\eta \left(r\right)=\eta \left(w\right)$. “NF” corresponds to $\mathrm{\Sigma }$ and $\eta$ derived from the transcritical normal form, “Power Law” the hyperbolic (power-law) form, and “Pitchfork” the pitchfork form.

Figure 6

Comparison of $1/log\mathrm{\Sigma }\left(w\right)$ for the best fit of $\mathrm{\Sigma }\left(w\right)$ derived with different functional forms of $\frac{dw}{dl}$. We have $w=(r-r_c)/s_s$ such that $\mathrm{\Sigma }\left(r\right)=\mathrm{\Sigma }\left(w\right)$. “NF” corresponds to $\mathrm{\Sigma }$ derived from the transcritical normal form, “${\mathrm{NF}}_{\mathrm{alt}}$” an alternative normal form (Appendix pp1-s3) constraining $r_c=0$, “Power Law” the hyperbolic (power-law) form, and “Pitchfork” the pitchfork form.

Figure 7

Fit comparisons ${\mathrm{\Sigma }}_{\mathrm{th}}\left(w\right)$ for the transcritical form.

Figure 8

Fit comparisons ${\eta }_{\mathrm{th}}\left(w\right)$ for the transcritical form.

Figure 9

Fit comparisons ${\mathrm{\Sigma }}_{\mathrm{alt}}\left(w\right)$ for the alternative transcritical form.

Figure 10

Fit comparisons ${\eta }_{\mathrm{alt}}\left(w\right)$ for the alternative transcritical form.

Figure 11

Density fluctuations of Voronoi cells give effective random fields. Our model puts spins on a randomly chosen lattice of sites (circled) and assigns bonds to their Voronoi neighbors (black lines connecting sites). The energy per unit length of a domain wall between up and down spins will be proportional to the square root of the density of points. Here the red curve denotes a boundary between artificially created low- and high-density regions. An invading avalanche of up spins entering from the left will likely get pinned as it approaches the red curved boundary, just as if there were random fields pointing downward along that curve.