Critically Slow Operator Dynamics in Constrained Many-Body Systems

The far-from-equilibrium dynamics of generic interacting quantum systems is characterized by a handful of universal guiding principles, among them the ballistic spreading of initially local operators. Here, we show that in certain constrained many-body systems the structure of conservation laws can cause a drastic modification of this universal behavior. As an example, we study operator growth characterized by out-of-time-order correlations (OTOCs) in a dipole-conserving fracton chain. We identify a critical point with sub-ballistically moving OTOC front, that separates a ballistic from a dynamically frozen phase. This critical point is tied to an underlying localization transition and we use its associated scaling properties to derive an effective description of the moving operator front via a biased random walk with long waiting times. We support our arguments numerically using classically simulable automaton circuits.

Figure 1

Dynamical phase diagram of operator spreading. Our dipole conserving system exhibits a localization transition at a critical density $n_c$. In the limit of long timescales, operators spread ballistically on the ergodic side $n<n_c$, and are frozen on the nonergodic side $n>n_c$ of the transition. At the critical density, the operator front spreads sub-ballistically as $\sim t^{1/2z}$ with $z=4$. The lower panels show the associated OTOC; Eq. (2). The sub-ballistic spread is visible also away from the critical density at finite times within a critical fan, bounded by a crossover time that diverges as $|n-n_c{|}^{-2z}$ upon approaching the transition.

Figure 2

Time evolution. (a) The time evolution is given by a layered circuit structure consisting of random unitary automaton gates of range four. (b) The gates map local strings of occupation numbers to other such strings under the constraints of charge and dipole conservation.

Figure 3

Origin of waiting times for operator spreading. (a) The single-shot OTOC, shown for a randomly chosen initial state at half-filling $n=1.0$, experiences waiting times that slow its spread. (b) A generic state chosen at an average density $n<n_c$ locally features regions of size $\ell$ whose density exceeds the critical value $n_c$. An operator inserted into such a localized region cannot escape from within—a property inherent to the fractonic constraint of dipole-conservation. Instead, the operator is subject to a waiting time $\tau \sim {\ell }^z={\ell }^4$ set by the time required for transport to melt the frozen region from the outside.

Figure 4

OTOC front and waiting time distributions. (a) Average $\overline{⟨x_r(t)⟩}$ of the position of the OTOC front as defined in Fig. 3. For $n<n_c$, an upward bending towards an expected ballistic growth can be observed on the simulated timescales for densities deep in the ergodic phase. At the critical density $n_c=1.5$, the dynamics of the front shows the sub-ballistic power law $\overline{⟨x_r(t)⟩}\sim t^{1/8}$, and the spread remains bounded $⟨x_r(t)⟩<\text{const}$ in the localized phase. The densities shown are $n=1.0–1.5$ in steps of 0.1 (light to dark red), as well as $n=1.7$. (b) Probability distributions $p_n(\tau )$ of waiting times in the automaton circuit dynamics evaluated at densities below and at $n_c$ [same values as in (a)]. The distribution $p_{n_c}(\tau )$ at the critical point follows an algebraic decay $\sim {\tau }^{-9/8}$. For values $n<n_c$ in the ergodic phase, $p_n(\tau )$ bends downwards to a faster-than-algebraic decay. (c) This faster decay is of the form of a stretched exponential, as confirmed by inspecting $g_n(\tau )=p_n(\tau )/p_{n_c}(\tau )\sim \exp \{-(\tau /{\tau }_c{)}^{1/4}\}$. Plotting this quantity as a function of $\tau |n-n_c{|}^8$ leads to a scaling collapse at late times, consistent with ${\tau }_c\sim |n-n_c{|}^{-2z}=|n-n_c{|}^{-8}$. The densities shown are $n=1.2–1.4$ in steps of 0.05.