Super slowing down in the bond-diluted Ising model

Abstract

In models in statistical physics, the dynamics often slows down tremendously near the critical point. Usually, the correlation time $\tau$ at the critical point increases with system size $L$ in power-law fashion: $\tau \sim L^z$, which defines the critical dynamical exponent $z$. We show that this also holds for the 2D bond-diluted Ising model in the regime $p>p_c$, where $p$ is the parameter denoting the bond concentration, but with a dynamical critical exponent $z(p)$ which shows a strong $p$-dependence. Moreover, we show numerically that $z(p)$, as obtained from the autocorrelation of the total magnetisation, diverges when the percolation threshold $p_c=1/2$ is approached: $z(p)-z(1) \sim (p-p_c)^{-2}$. We refer to this observed extremely fast increase of the correlation time with size as {\it super slowing down}. Independent measurement data from the mean-square deviation of the total magnetisation, which exhibits anomalous diffusion at the critical point, supports this result.