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arXiv:2509.22368 · Physical review. E

Phase transition from localization to chaos in a classical many-body system

Yusuf Kasim, Pavel Orlov, Tomaž Prosen

Published Jun 8, 2026• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/6y6k-35l9

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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.
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Information theoretic measures have helped to sharpen our understanding of many-body quantum states. As perhaps the most well-known example, the entanglement entropy (or more generally, the bipartite mutual information) has become a powerful tool for characterizing the dynamical growth of quantum correlations. By contrast, although computable, the bipartite mutual information (MI) is almost never explored in classical many particle systems; this owes in part to the fact that computing the MI requires keeping track of the evolution of the full probability distribution, a feat which is rarely done (or thought to be needed) in classical many-body simulations. Here, we utilize the MI to analyze the spreading of information in 1D elementary cellular automata (CA). Broadly speaking, we find that the behavior of the MI in these dynamical systems exhibits a few different types of scaling that roughly correspond to known CA universality classes. Of particular note is that we observe a set of automata for which the MI converges parametrically slowly to its thermodynamic value. We develop a microscopic understanding of this behavior by analyzing a two-species model of annihilating particles moving in opposite directions. Our work suggests the possibility that information theoretic tools such as the MI might enable a more fine-grained characterization of classical many-body states and dynamics.
Local measurements in quantum systems are projective operations which act to counteract the spread of quantum entanglement. Recent work has shown that local, random measurements applied to a generic volume-law entanglement generating many-body system are able to force a transition into an area-law phase. This work shows that projective operations can also force a similar classical phase transition; we show that local projections in a chaotic system can freeze information dynamics. In rough analogy with measurement-induced phase transitions, this is characterized by an absence of information spreading instead of entanglement entropy. We leverage a damage-spreading model of the classical transition to predict the butterfly velocity of the system both near to and away from the transition point. We map out the full phase diagram and show that the critical point is shifted by local projections, but remains in the directed percolation universality class. We discuss the implication for other classical chaotic many-body systems and the relation to synchronization transitions.
The fundamental question of how information spreads in closed quantum many-body systems is often addressed through the lens of the bipartite entanglement entropy, a quantity that describes correlations in a comprehensive (nonlocal) way. Among the most striking features of the entanglement entropy are its unbounded linear growth in the thermodynamic limit, its asymptotic extensivity in finite-size systems, and the possibility of measurement-induced phase transitions, all of which have no obvious classical counterpart. Here, we show how these key qualitative features emerge naturally also in classical information spreading, as long as one treats the classical many-body problem on par with the quantum one, that is, by explicitly accounting for the exponentially large classical probability distribution. Our analysis is supported by extensive numerics on prototypical cellular automata and Hamiltonian systems, for which we focus on the classical mutual information and also introduce a ``classical entanglement entropy.'' Our study sheds light on the nature of information spreading in classical and quantum systems, and opens avenues for quantum-inspired classical approaches across physics, information theory, and statistics.
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.