Discontinuity in the distribution of field increments between avalanches in non-abelian random field Blume-Emery-Griffiths model with no passing violation

Abstract

We study the zero-temperature quasi-statically driven dynamics of the random field Blume--Emery--Griffiths model (RFBEGM) as a minimal framework to investigate the consequences of violating the no-passing property in driven disordered systems. While the random field Ising model obeys no-passing and displays abelian relaxation dynamics, we show that this property is generically violated in the RFBEGM. By systematically exploring the full parameter space of the fully connected model, we identify the regimes in which no-passing is broken and demonstrate that, when this violation is combined with frustration induced by a repulsive biquadratic coupling, it leaves a clear dynamical signature. Specifically, the distribution of the minimal field increment required to trigger successive avalanches develops a discontinuity that is absent both in no-passing dynamics and in unfrustrated no-passing-violating regimes. We provide analytical arguments that locate the onset of this discontinuity, in excellent agreement with numerical simulations. Our results establish this discontinuity as a robust diagnostic of frustration-induced blocking in non-abelian avalanche dynamics within a mean-field setting, without making claims about new universality classes.

Figures (14)

• Figure 1: Distribution of the minimal field increment $\delta H_{\min}$ required to destabilize the system from a metastable state. (a) No-passing (NPP) regime with $K = -1.0$, $\Delta = -1.5$, where the distribution is continuous and nearly flat for small $\delta H_{\min}$. (b) No-passing-violating (NPV) regime with repulsive biquadratic coupling ($K = -1.5$, $\Delta = -0.5$), where the distribution develops a clear discontinuity. Insets show scaling collapses of the form $P(\delta H_{\min}) = N P(N \delta H_{\min})$. For the frustrated NPV case, the jump occurs at $N \delta H_{\min} = |K| - 1 = 0.5$, highlighting a qualitative dynamical difference between NPP and frustrated NPV regimes.
• Figure 2: The symbol $\textcolor{ForestGreen}{\blacktriangleright}$ is used for regions with NPP, $\textcolor{red}{\CIRCLE}$ denotes regions with NPV with $R_c \neq 0$ and $\textcolor{red}{\blacksquare}$ denotes the region with NPV and $R_c=0$. A $\bigcirc$ around a symbol implies that the fraction of zero spins, $N_0 \neq 0$.The $K=\Delta$ line separates the $N_0=0$ regions from $N_0 \neq 0$. The symbols represent the actual points in the $K-\Delta$ plane where we have explicitly verified the dynamics in simulations.
• Figure 3: In the first row $m-H$ is plotted for a typical realization of disorder. In each case forward plot is obtained by starting with large negative $H$ and backward plot by starting with large positive $H$. For $R<R_c$, the system shows first order hysteresis that vanishes at $R_c$. The second column shows evolution of the fraction of $0$ spins ($N_0$) in each case for the forward plot. Specifically, (a) and (d) are for NPV with frustration ($K=-1.5; \Delta =-0.5$) with $R_c \approx 0.7$; (b) and (e) are for NPP ($K=-1.0; \Delta =-1.5$) with $R_c \approx 0.8$; and (c) and (f) are for NPV without frustration ($K=1.5; \Delta =2.5$) with $R_c \approx 1.0$
• Figure 4: Distribution of the field increment $\delta H$ required to destabilize the system from a metastable state at fixed $H$ at a random site, $P_H(\delta H)$ for Panel a) : NPP with $K=-1.0; \Delta =-1.5$ and Panel b) : NPV with $K=-1.5; \Delta =-0.5$
• Figure 5: Plot of the probability of avalanche of size $1$ ( $p_A(1,R)$) as a function of $R$ for NPV ($K= -1.5,\Delta= -0.5$) and NPP ($K=-1.0,\Delta=-1.5$).