Emergence of cooperative dynamics in fully packed classical dimers

The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional “strings” of high mobility, which govern both local and long-wavelength dynamical properties. We introduce a coarse-grained description of the strings, based on the Edwards-Wilkinson model, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We discuss the implications of our results for the dynamics of constrained systems more generally.

Figure 1

Left: A dimer configuration with a single string, relative to a fully staggered configuration with maximal flux ${\mathrm{\Phi }}_x=\frac{1}{2}{L}^2$ along the horizontal direction. A single string spanning the system once reduces the flux ${\mathrm{\Phi }}_x$ by $-L$, the smallest possible amount, irrespective of its path. Flippable plaquettes, which appear when the dimers are shifted, are marked with stars; the fully staggered configuration has none. Middle and right: Configurations with ${\varphi }_x=\frac{1}{4}$ evolved for time $t$. Persistent plaquettes are white, while those that have flipped are blue; strings are yellow. The dynamics is spatially heterogeneous: Even at relatively long times, extended regions are unvisited by strings and hence persistent.

Figure 2

Normalized height correlations ${\mathcal{G}}_{\zeta }(\mathit{q},t)=G_{\zeta }(\mathit{q},t)/G_{\zeta }(\mathit{q},0)$ at wave vector $\mathit{q}=\{\frac{\pi }{16},\frac{\pi }{4}\}$ and flux $\mathbf{\varphi }=\{{\varphi }_x,0\}$. Symbols show MC results (error bars are smaller than symbols), while the thick black line is the theoretical prediction for the large-flux limit (i.e., close to the maximum ${\varphi }_x=1$). The short- and long-time limits, Eq. (15), are shown with dot-dashed and dashed lines, respectively. The system size is $L=256$; besides restricting ${\varphi }_x$ to discrete values, finite-size effects are minimal. Inset: Same data with double-logarithmic vertical scale. On this plot, a stretched exponential $e^{-{(t/\tau )}^{\beta }}$ appears as a straight line with slope $-\beta$. A dot-dashed straight line with a slope of 1 is shown for comparison; the data deviate from this slope, indicating stretching, even for zero flux.

Figure 3

MC results for persistence $p\left(t\right)$ at flux ${\varphi }_x$ labeled as in Fig. 2, except where indicated. (Error bars are smaller than symbols.) System size $L=4096$ is required to reach flux ${\varphi }_x=2047/2048$ (see Sec. 2a); for the other flux values, $L=256$. As expected based on the string picture, an initial exponential decay (straight line with a slope of 1) is followed by a stretched exponential (slope $<1$), lasting until $\mathrm{\Lambda }t\sim {(1-{\varphi }_x)}^{-1/\beta }$. The dashed line on the right, with a slope of $1/4$, is the function $p\left(t\right)\propto e^{-c{t}^{1/4}}$, in agreement with Eq. (16) for the limit ${\varphi }_x\to 1$.

Figure 4

Four types of segments that can be combined to form a string. The first two are steps at which the vertical position of the string changes by $\pm 1$, while the last two are the two segments of a horizontal step. (This step is split so that all four segments involve the same horizontal displacement.)

Figure 5

Mean flippability in equilibrium $\langle f\rangle$ as a function of deviation from maximum flux, $\theta =1-{\varphi }_x$. The solid green line shows Monte Carlo results for system size $L=256$ (error bars are smaller than symbols), while the dashed black line shows the analytical prediction from Eq. (A16).

Figure 6

Mean-square transverse displacement of a string $y(x,t)$ in equilibrium, given by the equal-time correlation $〈{[y(x,t)-y(x^{\prime },t)]}^2〉$, as a function of displacement along the string, $|x-x^{\prime }|$. The solid green line shows Monte Carlo results using a dimer configuration containing a single string in a system of size $L=1024$. (Error bars are smaller than symbols.) The dashed black line shows the analytical result of Eq. (B7), with $D=1/\sqrt{2}$ determined using Eq. (A19).

Figure 7

Growth of mean-square transverse displacement of a string, $〈{[y(x,t)-y(x,0)]}^2〉$, at a fixed position $x$ as a function of time $t$. The solid green line shows Monte Carlo results using a dimer configuration containing a single string in a system of size $L=1024$. (Error bars are smaller than symbols.) The dashed black line shows the analytical result of Eq. (B9), using values of $D$ and $\mathrm{\Lambda }$ fixed using Eqs. (A19) and (A16).

Figure 8

Normalized height correlations ${\mathcal{G}}_{\zeta }(\mathit{q},t)=G_{\zeta }(\mathit{q},t)/G_{\zeta }(\mathit{q},0)$ at flux $\mathbf{\varphi }=\{{\varphi }_x,0\}$ and wave vectors (a) $\mathit{q}=\{\frac{\pi }{8},\frac{\pi }{8}\}$, (b) $\mathit{q}=\{\frac{\pi }{4},\frac{\pi }{4}\}$, and (c) $\mathit{q}=\{\frac{\pi }{4},\frac{\pi }{16}\}$. In each case, the symbols show MC results with system size $L=256$ (error bars are smaller than symbols), while the thick black line shows the theoretical prediction for the large-flux limit (i.e., close to the maximum ${\varphi }_x=1$). The short- and long-time limits are shown with dot-dashed and dashed lines, respectively. Insets: Same data with double-logarithmic vertical scale; stretched exponentials appear as straight lines. The saturation at long times is an artifact resulting from the statistical uncertainty.

Figure 9

Inverse persistence time ${\tau }_{\text{p}}^{-1}$ as a function of deviation from maximum flux, $\theta =1-{\varphi }_x$. The symbols show Monte Carlo results for system sizes $L=64$ (red), 128 (blue), and 256 (green). (Error bars are smaller than symbols.) The dashed black line shows the analytical prediction ${\tau }_{\text{p}}\sim {\theta }^{-4}$, which applies for $0.05\lesssim \theta \lesssim 0.2$ (close, but not too close, to maximum flux). For larger $\theta$, the density of strings is sufficiently high that their interactions become important, and the picture of independent strings breaks down. When $\theta \lesssim L^{-1}$, finite-size effects become important.