Classical Decorrelator in Many-Body Systems

• Classical decorrelator is a diagnostic tool measuring the divergence of initially close trajectories in deterministic or stochastic many-body dynamics.
• It employs methods such as local Hamming distances and spin correlation metrics to trace the spread of localized perturbations across systems.
• The approach distinguishes dynamical phases—from integrable to chaotic regimes—by revealing scaling laws, Lyapunov exponents, and critical transitions.

A classical decorrelator is a spatiotemporally resolved chaos diagnostic distilled from deterministic or stochastic classical many-body dynamics. Defined as the local or global separation of a pair of trajectories initially perturbed by a microscopic difference, it quantifies the growth and spread of information, providing a classical analog to the quantum out-of-time-ordered correlator (OTOC). Classical decorrelators have emerged as a unifying tool for characterizing chaos, localization, and transitions between dynamical phases in classical spin systems, cellular automata, and related interacting models.

1. Mathematical Definitions and Protocols

The precise definition of the classical decorrelator depends on the underlying model:

• Ising-like and Cellular Automata Systems: For Boolean state variables $s_r\in\{0,1\}$, as in momentum-conserving parity check cellular automata (MCPCA), two configurations %%%%1%%%% and $s^B(0)$ are prepared differing by a localized bit-flip. Both are subsequently evolved under identical deterministic dynamics. The site-resolved decorrelator is

$$D(r,t) = \left\langle\, | s^A_r(t) - s^B_r(t) |\,\right\rangle,$$

where the average is taken over initial states and flip locations. The global Hamming distance is $H_d(t) = \sum_r D(r,t)$ (Kasim et al., 26 Sep 2025).

• Classical Spin Chains: For Heisenberg or non-reciprocal spin chains, initial configurations $\mathbf{S}_n^A(0)$, $\mathbf{S}_n^B(0)$ are identical except for a microscopic perturbation at $n=0$, $\delta\mathbf{S}_0$. The decorrelator is commonly

$$D_j(t) = 1 - \langle \mathbf{S}_j^A(t) \cdot \mathbf{S}_j^B(t) \rangle,$$

or, for small perturbations,

$$D_j(t) \simeq \tfrac{1}{2} \langle |\delta\mathbf{S}_j(t)|^2\rangle,$$

with $\delta\mathbf{S}_j(t)$ propagating via the system’s tangent-space dynamics (Bhatt et al., 15 Dec 2025).

• Stochastic and Deterministic Cellular Automata: The decorrelator is the local Hamming distance,

$$C(x,t) = \left\langle \frac{1}{2}[1 - \sigma^A(x,t)\,\sigma^B(x,t)] \right\rangle,$$

for Boolean variables $\sigma = \pm 1$, averaged over disorder realizations (Liu et al., 2021).

All protocols require preparation of trajectory pairs differing by a single-site perturbation, synchronous or identical evolution under the system’s dynamics, and extensive averaging over initial conditions.

2. Physical Interpretation: Information Spreading and Diagnosing Chaos

The decorrelator characterizes how a localized perturbation spreads and grows, quantifying the “butterfly effect” in classical many-body systems:

• Localized (Integrable) Phase: In near-integrable or localized regimes, the decorrelator remains confined. In spin-wave–dominated systems at low temperature, $D(r,t)$ exhibits ballistic Bessel propagation within a lightcone and an algebraic or exponential decay outside (Bilitewski et al., 2020, Kasim et al., 26 Sep 2025).
• Chaotic Phase: In fully chaotic regimes, the decorrelator exhibits exponential growth within a broadening ballistic front (the “light cone”), with spatial profiles saturating to nonzero values away from the origin, indicating macroscopic information spreading. This regime is associated with positive Lyapunov exponents and butterfly velocities (Bilitewski et al., 2020, Ruidas et al., 8 Jan 2026, Bhatt et al., 15 Dec 2025).
• Intermediate/Scarred Regime: At intermediate times or in weakly chaotic regimes, the decorrelator displays strong spatiotemporal heterogeneity—so-called “scars” from rare scattering events initiate secondary lightcones, producing a highly non-uniform spatial footprint (Bilitewski et al., 2020, Ruidas et al., 8 Jan 2026).
• Phase Transitions: In MCPCA, the long-time averaged decorrelator transitions from exponential decay in the localized phase ($\bar D(r) \sim \exp(-|r|/\xi)$, $\xi$ finite) to a flat profile ($\bar D(r\to\infty)>0$) in the chaotic phase, with critical scaling at the transition (Kasim et al., 26 Sep 2025).

3. Scaling, Universal Features, and Critical Regimes

Key universal features captured by the classical decorrelator include:

• Velocity-Dependent Lyapunov Exponent (VDLE): Along rays $x=vt$, the decorrelator admits the scaling $D(x=vt,t)\sim \exp[\lambda(v)t]$. The butterfly velocity $v_B$ is defined as the largest $v$ for which $\lambda(v)=0$. For $v<v_B$ the decorrelator saturates; for $v>v_B$ it decays exponentially (Liu et al., 2021, Ruidas et al., 8 Jan 2026).
• Scaling Laws at Dynamical Phase Transitions: In MCPCA, the long-distance plateau $D_\infty$ acts as an order parameter, vanishing continuously at the critical $p_c$ (ensemble parameter), $h_d \sim (p_c-p)^\beta$ with $\beta\approx 0.368$; the correlation length diverges as $\xi\sim(p-p_c)^{-\nu}$ with $\nu\approx 1$; and at $p_c$ the stationary profile decays as a power law $\bar D(r)\sim|r|^{-\eta},\,\eta\approx 0.25$ (Kasim et al., 26 Sep 2025).
• Analyticity in Cellular Automata: In chaotic Boolean automata, both $v_B$ and VDLE $\lambda(v)$ can be computed analytically from the boundary random walk of damaged regions; $\lambda(v)\sim\mu(v-v_b)^2$, establishing a universal $\beta=2$ front broadening, with scaling collapse observed for multiple parameter regimes (Liu et al., 2021).

4. Regime Structure: Integrable, Scarred, and Chaotic Dynamics

Across spin chains and automata, the classical decorrelator reveals a tripartite dynamical regime structure (Bilitewski et al., 2020, Ruidas et al., 8 Jan 2026):

• Short-Time Integrable Propagation: Ballistic, non-exponential decorrelator growth; propagation governed by free or weakly coupled quasiparticles (spin waves, defect wavepackets).
• Scarred Regime: Secondary lightcones seeded by inhomogeneous scatterings induce streaks (“scars”) in the spatiotemporal decorrelator, producing extreme nonuniformity and fat-tailed distributions.
• Long-Time Chaotic Regime: Decorrelator growth exponentially saturates within the lightcone, driven by an “avalanche” of overlapping scattering events, with global participation ratios decaying as $g(t)\sim1/t$—indicative of homogeneously distributed chaos.

Table of Regimes in Classical Spin Chains:

Regime	Decorrelator Behaviour	Dynamical Features
Integrable	Ballistic, algebraic	Free propagation, sharp lightcone
Scarred	Spatiotemporally sparse	Secondary lightcones, rare scattering
Chaotic	Exponential, uniform	Overlapping avalanches, saturated chaos

5. Role of Conservation Laws, Model Specifics, and Hydrodynamics

The propagation and scaling of the decorrelator reflect underlying conservation laws and nonreciprocal dynamics:

• Non-Reciprocal Spin Chains: In generalized Heisenberg chains with non-symmetric exchanges ($J_{mn}\neq J_{nm}$), the decorrelator remains ballistic despite the breakdown of simple Hamiltonian structure. In staggered or exponentially weighted variables, Hamiltonian structure and conserved quantities (magnetization-like, energy-like) are restored, yet the chaos front persists even when conservation laws degrade under further-neighbour couplings (Bhatt et al., 15 Dec 2025).
• Hydrodynamic Constraints: Hydrodynamic modes may diffuse (e.g., in the $N, E$ fields of non-reciprocal magnets), yet the decorrelator’s chaos front remains ballistic, highlighting the separation between chaotic information spreading and conserved transport (Bhatt et al., 15 Dec 2025).
• Model Classes: The decorrelator has broad applicability, from parity-conserving automata (MCPCA), nonreciprocal spin chains, to stochastic Boolean lattices (Kauffman CA), each preserving the unifying chaos diagnostic structure but differing in analytic tractability and dynamical nuances (Kasim et al., 26 Sep 2025, Bhatt et al., 15 Dec 2025, Liu et al., 2021).

6. Numerical and Analytical Results

Comprehensive numerical studies corroborate the decorrelator’s diagnostic power:

• Ballistic Propagation: Decorrelator fronts exhibit ballistic propagation with well-defined butterfly velocities, numerically extracted and analytically supported across models. In MCPCA, ballistic spreading is clear in the chaotic phase; in spin models, Bessel-wavepacket propagation is evident in integrable regimes (Kasim et al., 26 Sep 2025, Bilitewski et al., 2020, Ruidas et al., 8 Jan 2026).
• Front Broadening and Scaling Collapse: The shape and growth of the decorrelator front exhibit scaling collapse governed by universal exponents (e.g., front-broadening exponent $\alpha\approx1/3$ in nonreciprocal spin chains, $\beta=2$ in stochastic automata) (Bhatt et al., 15 Dec 2025, Liu et al., 2021).
• Phase Transition Diagnostics: Table 1 in (Kasim et al., 26 Sep 2025) summary shows order-parameter scaling ($D_\infty$, $h_d$), correlation length divergence ($\xi$), and front profiles across localized, critical, and chaotic regimes.
• Lyapunov Exponents: The largest Lyapunov exponent, and its velocity dependence, is accessible via the linearized decorrelator dynamics; in models with well-defined quasiparticles, $\lambda$ is inversely related to the spin-wave lifetime. The diffusion-chaos relation $D_s\sim v_B^2/\lambda$ connects transport and information spreading (Bilitewski et al., 2020).

7. Significance and Theoretical Implications

The classical decorrelator provides a fundamental, model-independent probe of information dynamics in classical many-body systems. It legibly distinguishes dynamical phases (localized/integrable, scarred, chaotic), realizes scaling laws traditionally associated with quantum chaos in purely classical contexts, and reveals universal behaviour across stochastic, deterministic, and nonreciprocal systems. As such, it unifies ballistic and diffusive dynamics, nonlinear chaos, and hydrodynamic constraints within a single quantitative framework. Major advances include analytic computation of scaling forms, phase transition characterization in deterministic automata, and clarification of the mechanisms by which weakly interacting quasiparticles induce many-body chaos via cascades of lightcones and avalanches of secondary perturbations (Kasim et al., 26 Sep 2025, Bilitewski et al., 2020, Ruidas et al., 8 Jan 2026, Bhatt et al., 15 Dec 2025, Liu et al., 2021).

A plausible implication is the extensibility of classical decorrelator analysis to more general classes of many-body systems, including higher-dimensional, glassy, or non-equilibrium regimes, and possible further connections between macroscopic transport and microscopic chaos measures.