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arXiv:2101.01313 · Physical review. B./Physical review. B

Butterfly effect and spatial structure of information spreading in a chaotic cellular automaton

Shu-Wei Liu, Josef Willsher, Thomas Bilitewski, Jinjie Li, A. Smith, Kim Christensen, Roderich Moessner, Johannes Knolle

Published Mar 17, 2021• 14 citations
arXivHTMLPDF

Abstract

Inspired by recent developments in the study of chaos in many-body systems, we construct a measure of local information spreading for a stochastic cellular automaton in the form of a spatiotemporally resolved Hamming distance. This decorrelator is a classical version of an out-of-time-order correlator studied in the context of quantum many-body systems. Focusing on the one-dimensional Kauffman cellular automaton, we extract the scaling form of our decorrelator with an associated butterfly velocity ${v}_{b}$ and a velocity-dependent Lyapunov exponent $\ensuremath{\lambda}(v)$. The existence of the latter is not a given in a discrete classical system. Second, we account for the behavior of the decorrelator in a framework based solely on the boundary of the information spreading, including an effective boundary random walk model yielding the full functional form of the decorrelator. In particular, we obtain analytic results for ${v}_{b}$ and the exponent $\ensuremath{\beta}$ in the scaling ansatz $\ensuremath{\lambda}(v)\ensuremath{\sim}\ensuremath{\mu}{(v\ensuremath{-}{v}_{b})}^{\ensuremath{\beta}}$, which is usually only obtained numerically. Finally, a full scaling collapse establishes the decorrelator as a unifying diagnostic of information spreading.

DOI
10.1103/physrevb.103.094109

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Inspired by recent developments in the study of chaos in many-body systems, we construct a measure of local information spreading for a stochastic cellular automaton in the form of a spatiotemporally resolved Hamming distance. This decorrelator is a classical version of an out-of-time-order correlator studied in the context of quantum many-body systems. Focusing on the one-dimensional Kauffman cellular automaton, we extract the scaling form of our decorrelator with an associated butterfly velocity vb{v}_{b}vb​ and a velocity-dependent Lyapunov exponent \ensuremathλ(v)\ensuremath{\lambda}(v)\ensuremathλ(v). The existence of the latter is not a given in a discrete classical system. Second, we account for the behavior of the decorrelator in a framework based solely on the boundary of the information spreading, including an effective boundary random walk model yielding the full functional form of the decorrelator. In particular, we obtain analytic results for vb{v}_{b}vb​ and the exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ in the scaling ansatz \ensuremathλ(v)\ensuremath∼\ensuremathμ(v\ensuremath−vb)\ensuremathβ\ensuremath{\lambda}(v)\ensuremath{\sim}\ensuremath{\mu}{(v\ensuremath{-}{v}_{b})}^{\ensuremath{\beta}}\ensuremathλ(v)\ensuremath∼\ensuremathμ(v\ensuremath−vb​)\ensuremathβ, which is usually only obtained numerically. Finally, a full scaling collapse establishes the decorrelator as a unifying diagnostic of information spreading.
We report a dynamical phase transition in the information spreading within a classical 2D deterministic interacting many-body system. Specifically, the transition is observed in a recently introduced momentum-conserving parity check cellular automaton (MCPCA) on the square lattice [Kasim and Prosen, Phys. Rev. Res. 7, 023230 (2025)2643-156410.1103/PhysRevResearch.7.023230]. We characterize the transition using information-theoretic quantities such as the Hamming distance and the classical decorrelator. By introducing conserved local charges of the momentum-conserving parity-check cellular automata, we show that selecting initial ensembles with specific charge values allows the system to transition from a localized information phase to a chaotic regime with ballistic information spreading. Importantly, our findings indicate that this transition is of second order, highlighting a sharp change in information spreading behavior. Furthermore, we revisit the multifractal behavior of the dynamical structure factor and show that, although present across both phases, it originates from effective local periodicities enforced by symmetry constraints.
We find that the effects of a localized perturbation in a chaotic classical many-body system-the classical Heisenberg chain at infinite temperature-spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterized by the Lyapunov exponents and the butterfly speed, respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analyzing the scaling forms, we relate the growth, spread, and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang equation.
It has been a long-standing challenge to expand the notion of exponential sensitivity to small perturbations in the initial conditions from the realm of classical chaos to that of many-body quantum systems. Recently, exponential growth in a measure known as the out-of-time-order commutator (OTOC) has been proposed as a diagnostic of chaos in the setting of many-body quantum systems. The authors examine the behavior of the OTOC along rays of different velocities for a variety of spatially local extended many-body quantum systems, both integrable and nonintegrable. They find that the velocity-dependent Lyapunov exponents are negative for velocities greater than a characteristic ``butterfly speed'', which defines the light cone for the spreading of operators. It is demonstrated that a regime with well-defined positive Lyapunov exponents inside the light-cone may only exist for classical, semiclassical, weakly interacting, or large-NNN systems, but not for fully quantum systems with strong short-range interactions and local Hilbert space dimensions of order one.
We study the chaotic dynamics in a classical many-body system of interacting spins on the kagome lattice. We characterize many-body chaos via the butterfly effect as captured by an appropriate out-of-time-ordered commutator. Due to the emergence of a spin-liquid phase, the chaotic dynamics extends all the way to zero temperature. We thus determine the full temperature dependence of two complementary aspects of the butterfly effect: the Lyapunov exponent, $\ensuremath{\mu}$, and the butterfly speed, vb{v}_{b}vb​, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant, DDD, and spin-autocorrelation time, $\ensuremath{\tau}$. We find that they all exhibit power-law behavior at low temperature, consistent with scaling of the form $D\ensuremath{\sim}{v}_{b}^{2}/\ensuremath{\mu}$ and ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\sim}T$. The vanishing of $\ensuremath{\mu}\ensuremath{\sim}{T}^{0.48}$ is parametrically slower than that of the corresponding quantum bound, $\ensuremath{\mu}\ensuremath{\sim}T$, raising interesting questions regarding the semiclassical limit of such spin systems.
Scrambling of information in a quantum many-body system, quantified by the out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos. A regime of exponential growth in the OTOC, characterized by a Lyapunov exponent, has so far mostly been observed in systems with a high-dimensional local Hilbert space and in weakly-coupled systems. Here, we propose a general criterion for the existence of a well-defined regime of exponential growth of the OTOC in spatially extended systems with local interactions. In such systems, we show that a parametrically long period of exponential growth requires the butterfly velocity to be much larger than the Lyapunov exponent times a microscopic length scale, such as the lattice spacing. As an explicit example, we study a random unitary circuit with tunable interactions. In this model, we show that in the weakly interacting limit, the above criterion is satisfied, and there is a prolonged window of exponential growth. Our results are based on numerical simulations of both Clifford and universal random circuits supported by an analytical treatment.
The speed of information propagation is finite in quantum systems with local interactions. In many such systems, local operators spread ballistically in time and can be characterized by a ``butterfly velocity", which can be measured via out-of-time-ordered correlation functions. In general, the butterfly velocity can depend asymmetrically on the direction of information propagation. In this work, we construct a family of simple 2-local Hamiltonians for understanding the asymmetric hydrodynamics of operator spreading. Our models live on a one dimensional lattice and exhibit asymmetric butterfly velocities between the left and right spatial directions. This asymmetry is transparently understood in a free (non-interacting) limit of our model Hamiltonians, where the butterfly speed can be understood in terms of quasiparticle velocities.