Arresting Classical Many-Body Chaos by Kinetic Constraints

We investigate the effect of kinetic constraints on classical many-body chaos in a translationally invariant Heisenberg spin chain using a classical counterpart of the out-of-time-ordered correlator (OTOC). The strength of the constraint drives a “dynamical phase transition” separating a delocalized phase, where the classical OTOC propagates ballistically, from a localized phase, where the OTOC does not propagate at all and the entire system freezes. This is unexpected given that all spin configurations are dynamically connected to each other. We show that localization arises due to the dynamical formation of frozen islands, contiguous segments of spins immobile due to the constraints, dominating over the melting of such islands.

Figure 1

(a) Schematic of the constrained spin chain. The red spins are frozen as both their neighbors lie inside the spherical sector (gray shade) which has a polar angle $\varphi \pi$, whereas the black spins are free to evolve. (b) The classical OTOCs and decorrelators $⟨D(x,t)⟩$ as color maps in spacetime for several values of $\varphi$. Note the rapid slowing down of the light cone with $\varphi$ near ${\varphi }_c\approx 0.53$, above which the light cone is fully arrested.

Figure 2

(a) $D(x,t)$ for a randomly chosen initial condition along with the trajectory of the front $x_F^R(t)$, extracted using Eq. (6), shown in green. Inset: a small spacetime portion enlarged, as well as the definition of $\tau$. Data for $\varphi =0.53$ and $\eta =0.01$. (b) Distributions of the waiting time $P_{\tau }(\tau )$ for three different $\varphi$, each for three different maximum simulation times $t_{\max }$. While the distributions are converged with $t_{\max }$ and not tailed in the delocalized phase, they have heavy power-law tails in the localized phase that persist for longer $\tau$ for larger $t_{\max }$ and have exponents such that the mean $⟨\tau ⟩$ is divergent. (c) The numerical data suggest that $⟨\tau ⟩$ diverges as a power law with $\delta \varphi ={\varphi }_c-\varphi$ and ${\varphi }_c\approx 0.525(5)$. All statistics accumulated over $5\times {10}^5$ initial conditions.

Figure 3

Melting of an initially frozen island of length $L_F=100$ in the delocalized phase (a) or lack thereof in the localized phase (b). The plots show a color map of $A_x(t)$ with black denoting frozen spins $A_x(t)\approx 0$ and yellow denoting active with $A_x(t)\approx 1$. In each panel, the left half shows averaged data $⟨A(x,t)⟩$, whereas the right half shows it for a single initial condition. The data in (c) suggest that the mean melting time $t_M$, diverges as the critical ${\varphi }_c$ is approached from the delocalized side. The red dashed line denotes the estimated ${\varphi }_c$ from the decorrelator data.

Figure 4

(a) Collapse of $⟨D(x,t)⟩$ onto a function $\mathcal{F}[(x-x_v(t))/\xi (t)]$ for $x>0$ with $\mathcal{F}(y)=\mathrm{erfc}(y)$ denoted by the red dashed line. Inset: raw data. The plots use $\varphi =0.4$. (b) $x_v(t)$ as a function of $t$ for several values of $\varphi$ in the delocalized phase. The linear behavior indicates the presence of a well-defined butterfly velocity $v_B=x_v(t)/t$. (c) ${\xi }^2(t)$ for the same values of $\varphi$, rescaled (arbitrarily) with ${\xi }^2(300)$ for visual clarity. The linear behavior indicates the diffusive broadening of the decorrelator front.