Constrained Dynamics and Directed Percolation

In a recent work [A. Deger et al., Phys. Rev. Lett. 129, 160601 (2022).] we have shown that kinetic constraints can completely arrest many-body chaos in the dynamics of a classical, deterministic, translationally invariant spin system with the strength of the constraint driving a dynamical phase transition. Using extensive numerical simulations and scaling analyses we demonstrate here that this constraint-induced phase transition lies in the directed percolation universality class in both one and two spatial dimensions.

Figure 1

Instances of active clusters for constrained spin dynamics in the active phase (left), at criticality (middle), and in the frozen phase (right) in the space-time graph in $1+1\mathrm{D}$ (top) and $2+1\mathrm{D}$ (bottom). In all cases we start from an all-sites-active configuration. In the active phase, there is always a finite density of active sites at arbitrarily long times whereas in the frozen phase the system goes into an absorbing state where all spins are frozen.

Figure 2

Critical scaling in the $1+1\mathrm{D}$ case. Left: The density of active sites $\rho$ as a function of $t$ for different values of ${\theta }_c$. At the critical point, $\rho$ decays with $t$ following a power law with an exponent $\alpha =0.159$ consistent with DP in $1+1\mathrm{D}$ (black dashed line). Middle: Scaling $\rho$ in the left panel with $t^{\alpha }$ and plotting it against $t|{\theta }_c-{\theta }_{c,\mathrm{crit}}{|}^{{\nu }_t}$ with ${\theta }_{c,\mathrm{crit}}\approx 0.5234\pi$ shows perfect scaling collapse for the DP universality exponents of $\alpha =0.159$ and ${\nu }_t=1.734$. Right: Finite-size scaling at the critical point by plotting $\rho t^{\alpha }$ against $t{L}^{-z}$ with the DP universality exponent, $z=1.581$, again shows excellent collapse. The inset shows the unscaled data. Results for $g=0.4$, $h=0.1$, and $T=2\pi$.

Figure 3

Critical scaling in the $2+1\mathrm{D}$ case. Analogous figure to Fig. 2 but for $d=2$. In the left panel, the dashed black line corresponds to $t^{-\alpha }$ with $\alpha =0.451$. The middle panel shows the collapse of $\rho t^{\alpha }$ onto a universal function of $t|{\theta }_c-{\theta }_{c,\mathrm{crit}}{|}^{{\nu }_t}$ with ${\theta }_{c,\mathrm{crit}}\approx 0.7084\pi$ and ${\nu }_t=1.295$, consistent with DP universality. Right panel shows finite-size scaling collapse at ${\theta }_{c,\mathrm{crit}}$ of $\rho t^{\alpha }$ as function of $t{L}^{-z}$ with $z=1.76$, the DP universality value. Results for $g=0.4$, $h=0.1$, and $T=2\pi$.