🌀 nonlinear sciences

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

This paper reports a second-order dynamical phase transition from localized to ballistic information spreading in a classical 2D momentum-conserving parity check cellular automaton, driven by conserved local charges and characterized by information-theoretic metrics and multifractal behavior arising from symmetry-enforced periodicities.

Original authors: Yusuf Kasim, Pavel Orlov, Tomaž Prosen

Published 2026-08-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yusuf Kasim, Pavel Orlov, Tomaž Prosen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Information Shuffle: From Frozen Puddles to Wild Storms

Imagine you are watching a massive, invisible dance floor filled with thousands of tiny dancers. In the world of physics, this dance floor is a "many-body system"—a collection of countless particles interacting with each other. For a long time, scientists have been obsessed with a specific question: How does order turn into chaos? In small groups, we know the rules; the dancers might follow a predictable rhythm. But when you add millions of dancers, things get messy. Sometimes, a tiny nudge to one dancer stays local, like a ripple in a pond. Other times, that same nudge explodes, sending shockwaves across the entire floor until everyone is dancing wildly out of sync. This shift from a calm, predictable state to a chaotic, spreading one is the "regularity-to-chaos transition," a fundamental mystery in nature that explains everything from why metals conduct electricity to how black holes might behave.

To study this, physicists often use simplified models called "cellular automata." Think of these as a giant grid of light switches (or bits) that can be either ON or OFF. Every second, a set of strict rules tells each switch how to flip based on its neighbors. It's like a digital game of "Life," but with specific, unbreakable laws. One of the most exciting things to track in these games is "information spreading." If you flip one switch at the start, does that change stay stuck in one corner, or does it race across the board, changing everything it touches? This is the digital version of the "butterfly effect," where a small flap of a wing can eventually cause a storm. Understanding when and why this happens helps us grasp the very limits of predictability in our universe.

The Paper's Discovery: A Frozen Grid vs. A Chaotic Storm

In this study, the authors Yusuf Kasim, Pavel Orlov, and Tomaž Prosen introduce a new, highly organized version of this digital dance floor called the "Momentum-Conserving Parity Check Cellular Automaton" (MCPCA). Imagine a grid where every switch is connected to its neighbors in a very specific way, and there's a hidden rule: the total "momentum" of the system must always stay the same, like a perfectly balanced seesaw. Because of this rule, the system develops a special kind of "memory" called a "loop charge." You can think of this as a secret code written in loops around the grid. If you draw a closed loop around a group of switches, the code inside that loop never changes, no matter how long the dance goes on.

The researchers asked a simple but profound question: What happens if we start the dance with different amounts of this secret code? They found that the answer depends entirely on how many "empty" loops (where the code is zero) versus "full" loops (where the code is at its maximum) are present at the start.

The Two Phases of the Dance

Through massive computer simulations, the team discovered that the system can exist in two completely different states, separated by a sharp tipping point:

The Localized Phase (The Frozen Puddle): When the starting conditions are "full" of the secret code (specifically, when the probability of a certain bit being '1' is high, around 0.25 or more), the system gets stuck. If you flip one switch to start a ripple, that ripple tries to spread but hits invisible walls. It bounces around a tiny area and dies out. The information never leaves its neighborhood. In the paper's language, the "Hamming distance" (a measure of how different the system becomes from its original state) stays tiny, no matter how big the grid is. It's like throwing a pebble into a frozen lake; the water doesn't ripple.
The Delocalized Phase (The Wild Storm): When the starting conditions are "empty" of the secret code (when the probability is low, below 0.25), the rules loosen up. Now, if you flip one switch, the ripple doesn't stop. It races across the entire grid, eventually changing the state of almost every switch. The information spreads ballistically, like a wildfire in a dry forest. In this phase, the Hamming distance grows huge, filling up the whole system.

The Tipping Point

The most exciting part of their finding is that the switch between these two states isn't gradual; it's a second-order phase transition. This means there is a critical point (at a probability of about 0.25) where the behavior changes abruptly.

Just below 0.25, the information spreads everywhere.
Just above 0.25, the information gets trapped.
Right at 0.25, the system is in a strange, critical state where the "correlation length" (how far the ripple reaches) becomes infinite, and the decay of the ripple follows a specific mathematical power law, dropping off as $|r|^{-0.25}$ .

The authors measured this carefully, finding that the "magnetization" of the information spreading (how much of the grid gets changed) grows like $(p_c - p)^{0.37}$ as you approach the critical point from the chaotic side. This suggests a very specific, sharp type of transition, similar to how a magnet suddenly loses its magnetism when heated past a certain temperature.

What It Is NOT

The paper is very careful to rule out some ideas that might seem obvious. For instance, the researchers also looked at something called the "multifractal dynamical structure factor." Previously, this complex, jagged pattern in the system's energy spectrum was thought to be a sign of the system's special nature. However, the authors found that this pattern exists in both the frozen phase and the stormy phase. It doesn't change when the system switches from localized to chaotic. Therefore, they conclude that this multifractal pattern is not a sign of the phase transition itself. Instead, it comes from the fact that even in the chaotic phase, small pockets of the system still have their own little, repeating rhythms (local periodicities) that create the pattern. So, while the "fingerprint" of the system looks complex everywhere, it doesn't tell you which phase the system is in.

How Sure Are They?

The authors are confident in their results, but they are honest about the limits. Their findings are based on numerical simulations on grids of various sizes (up to $256 \times 256$ ). They didn't prove this mathematically with a rigorous theorem (which is incredibly hard for these systems), but their data shows a very clear, consistent pattern. They estimate the critical point is $p_c \approx 0.25$ and the critical exponents are $\beta \approx 0.368$ and $\nu \approx 1$ . They also noted that this transition only happens on a square grid; if you change the shape of the grid to a honeycomb, the transition disappears, and the system stays chaotic.

In short, this paper shows that by tweaking the initial "secret code" of a simple, rule-bound digital universe, you can force it to either freeze in place or explode into chaos. It's a new, clean example of how order and chaos can coexist in the same set of rules, waiting for the right starting conditions to flip the switch.

Technical Summary: Phase Transition from Localization to Chaos in Classical Many-Body Systems

Problem Statement
The transition from regularity to chaos in interacting many-body systems remains a fundamental challenge in statistical mechanics, particularly when coupling strengths push systems beyond the reach of perturbation theory. While the Kolmogorov–Arnold–Moser (KAM) theory describes this transition in few-particle systems, rigorous results for many-body systems are scarce. A primary open question is identifying minimal, exactly solvable models that exhibit a transition from regular (localized) to chaotic (ergodic) behavior. This work addresses this by investigating the Momentum-Conserving Parity Check Cellular Automaton (MCPCA) on a square lattice, a reversible deterministic system recently shown to exhibit multifractal dynamical structure factors. The authors aim to determine if this system undergoes a dynamical phase transition in information spreading and to characterize the nature of such a transition.

Methodology
The study utilizes the MCPCA defined on a bipartite square lattice where edges carry binary dynamical variables ( $s_e \in \mathbb{Z}_2$ ). The dynamics are governed by local bijective maps that conserve parity and momentum. A crucial feature of the model is the existence of 1-form symmetries, manifested as conserved "loop charges" ( $M_\gamma$ ) associated with closed loops on the lattice. These charges constrain the dynamics, effectively fragmenting the phase space.

To probe the dynamical behavior, the authors:

Control Initial Ensembles: Instead of sampling from a maximal entropy state ( $p=0.5$ ), they generate initial states using a Bernoulli process with probability $p$ for a bit to be 0. This allows them to tune the density of plaquette charges ( $n_q$ ), specifically the density of plaquettes with zero charge ( $n_0$ ) versus non-zero charges ( $n_{\pm 1}, n_{\pm 2}$ ).
Quantify Information Spreading: They employ two primary observables:
- Hamming Distance ( $H_d$ ): The number of differing bits between two trajectories starting from nearly identical initial conditions (differing by one bit). This serves as a classical analogue to the Lyapunov exponent.
- Classical Decorrelator ( $D(r, t)$ ): A spatially resolved measure of the difference between trajectories, defined as the ensemble average of $|s^A_r(t) - s^B_r(t)|$ .
Analyze Scaling: They examine the long-time saturation values of these observables and their dependence on system size ( $N$ ) and the sampling parameter ( $p$ ) to identify critical points and exponents.
Investigate Multifractality: They revisit the dynamical structure factor (power spectrum of density-density correlations) to determine if its multifractal nature acts as an order parameter for the transition.

Key Contributions and Results

Discovery of a Dynamical Phase Transition: The authors identify a sharp transition between two distinct regimes of information spreading controlled by the density of conserved loop charges:
- Delocalized (Chaotic) Phase ( $p < p_c$ ): When the density of zero-charge plaquettes ( $n_0$ ) dominates, local perturbations spread ballistically across the system. The Hamming distance density ( $h_d = \lim_{N\to\infty} \bar{H}_d / 2N^2$ ) remains finite, and the decorrelator reaches a non-zero plateau at large distances.
- Localized Phase ( $p > p_c$ ): When the density of non-zero charge plaquettes ( $n_{\pm 1}, n_{\pm 2}$ ) dominates, information spreading is confined. The Hamming distance density vanishes in the thermodynamic limit ( $h_d = 0$ ), and the decorrelator decays exponentially with distance ( $D(r) \sim e^{-|r|/\xi}$ ).
Characterization of the Transition:
- The transition is identified as second-order.
- The critical sampling probability is estimated at $p_c \approx 0.25$ .
- Critical Exponents:
  - The order parameter (Hamming distance density) scales as $h_d \sim (p_c - p)^\beta$ with $\beta \approx 0.368$ for $p < p_c$ .
  - The correlation length $\xi$ diverges as $\xi \sim (p - p_c)^{-\nu}$ with $\nu \approx 1$ as the system approaches the critical point from the localized side.
  - At the critical point, the decorrelator follows a power-law decay $\bar{D}(r) \sim |r|^{-\eta}$ with $\eta \approx 0.25$ .
Decoupling of Multifractality and Localization:
- The authors demonstrate that the multifractal dynamical structure factor, previously observed in MCPCA, persists across both the localized and delocalized phases.
- They attribute this multifractality to the emergence of effective local periodicities enforced by the 1-form symmetry constraints. Regions of the system with specific loop charges evolve with small, approximate periods, creating a superposition of frequencies that results in a multifractal spectrum.
- Crucially, the paper concludes that multifractality is not an indicator of the information-spreading phase transition; it is a separate phenomenon arising from the same underlying symmetries.
Lattice Dependence: The phase transition is observed specifically on the square lattice. The authors note that for MCPCA on a honeycomb lattice, no such transition occurs, and the system remains delocalized.

Significance and Claims
The paper claims to present a minimal, exactly solvable classical many-body model that exhibits a well-defined second-order phase transition from localized to chaotic information spreading. By utilizing conserved local charges (1-form symmetries) to tune the initial ensemble, the authors provide a mechanism to control the ergodicity of the system.

The work highlights that:

Symmetry-Induced Constraints: Conservation laws (specifically 1-form symmetries) can induce dynamical localization even in deterministic, reversible systems without disorder.
Distinct Phenomena: The multifractal behavior of the spectral response and the localization-delocalization transition are physically distinct consequences of the model's symmetries. One does not imply the other.
Analytical Potential: The existence of such a transition in a simple cellular automaton suggests a promising avenue for analytical treatment (e.g., via renormalization group techniques) to understand the universality class of regularity-to-chaos transitions in many-body systems.

The authors remain modest regarding the universality of these findings, noting that the transition is specific to the square lattice geometry and that the transport properties of the topological charges remain an open area for future investigation. They also identify the extension of these results to quantum circuits constructed from similar symmetry-conserving gates as a natural next step.

The Great Information Shuffle: From Frozen Puddles to Wild Storms

The Paper's Discovery: A Frozen Grid vs. A Chaotic Storm

Technical Summary: Phase Transition from Localization to Chaos in Classical Many-Body Systems

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