Ordering kinetics in the random-bond $XY$ model

We present a comprehensive Monte Carlo study of domain growth in the random-bond $XY$ model with nonconserved kinetics. The presence of quenched disorder slows down domain growth in $d=2,3$. In $d=2$, we observe power-law growth with a disorder-dependent exponent on the time scales of our simulation. In $d=3$, we see the signature of an asymptotically logarithmic growth regime. The scaling functions for the real-space correlation function are seen to be independent of the disorder. However, the same does not apply for the two-time autocorrelation function, demonstrating the breakdown of superuniversality.

Figure 1

Plot of fourth-order Binder cumulant $U_4(T,L)$ vs. temperature $T$ for the $d=2$ RBXYM. The data is shown for $\epsilon =1.0$, and square lattices with $L=96,128,256$. The transition temperature $T_{\mathrm{BKT}}\left(\epsilon \right)$ is determined from the intersection of different $L$-curves. The inset shows the plot of $T_{\mathrm{BKT}}\left(\epsilon \right)$ with disorder $\epsilon$. The numerical values of $T_{\mathrm{BKT}}\left(\epsilon \right)$ are provided in Table 1.

Figure 2

Evolution snapshots of the $d=2$ RBXYM at $t={10}^6$ MCS, after a quench from $T=\infty$ to $T=0.2$ for (a) $\epsilon =0$, (b) $\epsilon =1$, and (c) $\epsilon =2$. The lattice size is ${512}^2$. In these plots, $\left\{{\theta }_i\right\}$ are marked in the interval $[{\theta }_0-0.1,{\theta }_0+0.1]$ with the following color coding: ${\theta }_0=0$ (black), ${\theta }_0=2\pi /3$ (red/gray), ${\theta }_0=-2\pi /3$ (green/light gray).

Figure 3

Vector plots for $\left\{{\theta }_i\right\}$ configurations in Fig. 2. At each lattice site $i$, we draw a vector corresponding to ${\mathit{S}}_i=(cos{\theta }_i,sin{\theta }_i)$. For a better view, we show only a ${32}^2$ corner of the ${512}^2$ lattice. Solid circles denote vortices, and solid triangles denote antivortices.

Figure 4

(a) Scaled correlation functions, $C(r,t)$ vs. $r/R\left(t\right)$, for the evolution of the $d=2$ RBXYM after a quench to $T=0.5$. We show data for $t={10}^6$ MCS, and $\epsilon =0,1,2$. (b) Scaled structure factors, $S(k,t)R{\left(t\right)}^{-2}$ vs. $kR$, for the data sets in (a). The solid curves in (a) and (b) denote the Bray-Puri-Toyoki (BPT) function in Eq. (14) for $n=2$, and its Fourier transform, respectively. The line of slope $-4$ in (b) denotes the generalized Porod law: $S(k,t)\sim k^{-(d+n)}$ for $d=n=2$.

Figure 5

Plot of the autocorrelation function, $A(t,t_w)$ vs. $R\left(t\right)/R\left(t_w\right)$, for waiting time $t_w={10}^4$ MCS, and $\epsilon =0,1,2$.

Figure 6

(a) Plot of $R\left(t\right)$ vs. $t/lnt$ on a log-log scale for the specified values of $\epsilon$. The dashed line is of slope 0.5, which indicates the growth law for the pure case: $R\left(t\right)\sim {(t/lnt)}^{1/2}$. (b) Plot of the effective exponent, $z_{\mathrm{eff}}={[d(lnR)/d(ln[t/lnt])]}^{-1}$ vs. $t$, for $\epsilon =0,0.5,1.0,1.5,2.0$. The dashed lines denote $\overline{z}\left(\epsilon \right)$.

Figure 7

Plot of the disorder-dependent growth exponent $\overline{z}\left(\epsilon \right)$ vs. $\epsilon$. The solid line is the best power-law fit: $\overline{z}=2.0+0.166{\epsilon }^{1.34}$.

Figure 8

Plot of $U_4(T,L)$ vs. $T$ for the $d=3$ RBXYM on cubic lattices of size $L=16,24,32$. We show data for $\epsilon =1.0$. In the inset, we plot $T_c\left(\epsilon \right)$ vs. $\epsilon$. The numerical values of $T_c\left(\epsilon \right)$ are provided in Table 2.

Figure 9

Evolution snapshots of the $d=3$ RBXYM for a temperature quench to $T=0.5$. The lattice size is ${128}^3$. The defects (vortex and antivortex strings) are shown at $t={10}^6$ MCS, and for $\epsilon =0,1,2$.

Figure 10

(a) Scaled correlation functions, $C(r,t)$ vs. $r/R\left(t\right)$, for the $d=3$ RBXYM after a quench to $T=1.0$. We show data sets for $t={10}^5$ MCS, and $\epsilon =0,1,2$. (b) Scaled structure factors, $S(k,t)R{\left(t\right)}^{-3}$ vs. $kR$, corresponding to the data sets in (a). The solid curves in (a) and (b) denote the BPT function [Eq. (14)] for $n=2$ and its Fourier transform, respectively. The dashed line of slope $-5$ in (b) denotes the generalized Porod law: $S(k,t)\sim k^{-(d+n)}$ for $d=3$ and $n=2$.

Figure 11

Plot of $R\left(t\right)$ vs. $t$ (on a log-log scale) for the $d=3$ RBXYM. The dashed line denotes the $t^{1/2}$-growth for the pure case.

Figure 12

(a) Plot of the effective exponent, $z_{\mathrm{eff}}$ vs. $t$. (b) Scaling collapse of $z_{\mathrm{eff}}-\overline{z}$ vs. $R/\lambda \left(\epsilon \right)$. The solid line is the best power-law fit: $z_{\mathrm{eff}}-\overline{z}\simeq 0.022{(R/\lambda )}^{1.16}$.