Phase ordering in disordered and inhomogeneous systems

We study numerically the coarsening dynamics of the Ising model on a regular lattice with random bonds and on deterministic fractal substrates. We propose a unifying interpretation of the phase-ordering processes based on two classes of dynamical behaviors characterized by different growth laws of the ordered domain size, namely logarithmic or power law, respectively. It is conjectured that the interplay between these dynamical classes is regulated by the same topological feature that governs the presence or the absence of a finite-temperature phase transition.

Figure 1

Parameter space of the model. The blue lines on the left, right, and bottom of the figure correspond to a pure system. Red and green heavy dots are the strong-disorder ($S$) and percolative ($W$) fixed points with $\epsilon =J_0$. Dot-dashed red and continuous green heavy lines are the line of fixed points originating from $S$ and $W$ upon changing $\epsilon$ (see Secs. 3b and 3c).

Figure 2

Panel (a) (left) $L\left(t\right)$ is plotted against $t$ in a double-log plot for a quench of the ARBIM to $T=\epsilon$, with $\epsilon =J_0$ and different values of $d$ specified in the key. The black-dashed line is the pure-case power law $L\left(t\right)\propto t^{1/2}$ and the green-dashed line is the fit $L\left(t\right)\sim t^{0.22}$ at $d=d_c$. In the upper inset, some of the curves of the main picture are plotted against $lnt$ on a double-log plot. The orange-dashed and blue-dashed lines are the fits $y=0.03{(lnt)}^{2.5}$ and $y=0.04{(lnt)}^{2.1}$, respectively. In the inset, the effective exponent $1/{\zeta }_{\text{eff}}$ is plotted against $d$. Panel (b) (right) $L\left(t\right)$ is plotted against $t$ in a double-log plot for a quench of the ARBIM to different final temperatures (specified in the key, curves lower upon decreasing temperature), with $\epsilon =J_0$ and $d=0.5$. In the inset, the exponent $1/\zeta (\epsilon =J_0,T,d=d_c)$, obtained from a fit of the curves from $t={10}^2$ onward, is plotted against $T$.

Figure 3

$L\left(t\right)$ is plotted against $t$ in a double-log plot for a quench of the ARBIM to $T=\epsilon$, with $\epsilon =3/4J_0$ and different values of $d$ specified in the key. Panel (a) (left) shows the cases with $0\le d\le d_c$, the right one those with $d_c\le d\le 1$. The inset is a zoom on the curves in the last two decades. The black-dashed line is the pure-case power law $L\left(t\right)\propto t^{1/2}$. The green-dashed line is the power law $L\left(t\right)\propto t^{0.2}$. In the upper inset, some of the curves of the main picture are plotted against $lnt$ on a double-log plot. The orange-dashed line is the fit $y=0.03{(lnt)}^{2.4}$. Panel (b) (right) shows the cases with $d_c\le d\le 1$. In the inset, the behavior of the effective exponent $1/{\zeta }_{\text{eff}}$ is shown for the whole range of $0\le d\le 1$.

Figure 4

Upper panels: recursive construction of the SG and of the SC. Lower panel: the SG (in black) embedded in a triangular lattice (dashed-red line). Bonds marked with a blue X are the cutting bonds.

Figure 5

Sketch of the movement of an interface on the SG.

Figure 6

$L\left(t\right)$ for the GSGIM with $\epsilon /J_0={10}^{-1}$ [panel (a), left] and $\epsilon /J_0=0.9$ [panel (b), right].

Figure 7

$L\left(t\right)$ for the GSCIM with $\epsilon /J_0={10}^{-1}$ [panel (a), left] and $\epsilon /J_0=0.9$ [panel (b), right].