Figure 1

Example of an avalanche on part of a 4-regular graph. At time $t_1$, the configuration consists of only S (stable) and F (flipped) spins. As the external field increases, a spin becomes unstable (U) at time $t_2>t_1$. This represents an immigration event for the population of U spins. At time $t_3>t_2$, the U spin flips causing the central spin, $i$, to undergo a S$\to$U transition. The central spin, $i$, had one flipped neighbor (along link 4) prior to time $t_3$, i.e., $n_i(t<t_3)=1$, and it became unstable, so that ${\zeta }_i$ is equal to the number of neighboring spins in state S, i.e., ${\zeta }_i=2$. In contrast, ${\zeta }_i$ would have been zero if the central spin had remained stable. At time $t_4$, the central spin flips and this leads to ${\xi }_i=2$ new U offspring.Reuse & Permissions

Figure 2

The rate of branching ${\mathbb{E}}^{\text{F}}\left[{\zeta }_j\right]$ [panels (a) and (d)] and $\text{Cov}[{\zeta }_j,n_j]$ [panels (b) and (e)]vs time $t$. Panels (c) and (f) show $\text{Cov}[{\zeta }_j,n_j]$ vs ${\mathbb{E}}^{\text{F}}\left[{\zeta }_j\right]$ for data presented in panels (a) and (b) and panels (d) and (e), respectively. Data corresponds to a 4-regular graph of $N={10}^9$ spins, with normally distributed disorder below $\Delta =1.65$ [panels (a)–(c)] and above $\Delta =2.5$ [panels (d)–(f)]; critical ${\Delta }_{\text{c}}\simeq 1.78215$. The external field was swept from $-\infty$ to $+\infty$ at a rate of $dH/dt=2\times {10}^{-9}$ with $H(t=0)=0$ and stochastic flips occurred at a rate of $\Gamma =1$.Reuse & Permissions

Figure 3

Time-dependent properties of avalanches. Solutions of the exact equations are plotted with lines and compared with results from numerical simulations shown by symbols. Panels (a) and (b) show the distribution of avalanche durations, ${\rho }_T$, and the integrated power spectra, $\langle S_{\text{int}}\left(\omega \right)\rangle$, respectively, for a 4-regular graph with normally distributed disorder, $\Delta ={\Delta }_{\text{c}}$ (critical disorder; solid lines and circles), $\Delta =2.0$ (dashed lines and squares), $\Delta =2.5$ (dot-dashed lines and diamonds), and $\Delta =3.0$ (double dot-dashed lines and triangles). The inset of panel (b) shows the power spectra $\langle S\left(\omega \right)\rangle$ evaluated for the same parameters as for panels (a) and (b). In both panel (a) and the inset of panel (b), $H=1.0$. The solid line in panel (a) tends to the limit ${\rho }_T\propto T^{-2}$ for $T\gg {\Gamma }^{-1}$, shown by the dotted line. In panel (b) and its inset, the dotted lines indicate $S_{\text{int}}\left(\omega \right)\propto {\omega }^{-3/2}$ and $S\left(\omega \right)\propto {\omega }^{-2}$, respectively, corresponding to the asymptotic behavior for $\omega \ll \Gamma$ near critical disorder. Panels (c) and (d) show the shape of avalanches ${\langle V\left(t\right)\rangle }_T$ vs $t/T$ for $\Delta ={\Delta }_{\text{c}}$ and $\Delta =2.5$, respectively, $H=1.0$ and $T=10$ (solid lines and circles), $T=20$ (dashed lines and squares), and $T=30$ (dot-dashed lines and diamonds). Numerical data for durations and spectra were averaged over avalanches occurring when $0.998<H<1.002$ [panel (a) and inset of panel (b)] and $0.98<H<1.02$ [panels (c) and (d)] for ${10}^3$ realizations of ${10}^6$ spins and $\Gamma =V_0=1$.Reuse & Permissions

Figure 4

Scaling function $\widehat{V}(t/T)$ in the scaling hypothesis for the pulse shape, ${\langle V\left(t\right)\rangle }_T=T^x\widehat{V}(t/T)$, with $x=1$, suggested in Refs. [1, 30]. The pulse shape was calculated for a 4-regular graph with normal disorder at criticality ($\Delta ={\Delta }_{\text{c}}$, $H=1.0$) using Eq. (42). Different curves represent different avalanche durations as marked in the figure legend, with $\Gamma =V_0=1$.Reuse & Permissions

Figure 5

Denominator of integral in Eq. (B2) vs $x$ for high disorder, $\Delta =3.0$ and $H=1.0$ (red-dashed line, subcritical regime of BP), at the critical point of the ZTRFIM characterized by ${\Delta }_{\text{c}}=1.78215$ and $H_{\text{c}}=1.0$ (black solid line, critical regime of BP), and immediately before snapping at low disorder $\Delta =1.0$ and the coercive field $H_{\text{coer}}(\Delta =1.0)=1.44754$ (blue dot-dashed line, critical regime of BP). The green double-dot dashed line corresponds to a hypothetical supercritical regime of the BP process but cannot be achieved for any values of $H$ or $\Delta$.Reuse & Permissions