Localization of information driven by stochastic resetting

This paper demonstrates that supplementing chaotic many-body dynamics with stochastic resetting induces a sharp dynamical phase transition where the Lyapunov spectrum collapses to zero, causing information to shift from ballistic scrambling to exponential localization with critical exponents $\nu=1/2$ and $z=2$.

The Gist

Imagine you have a giant, chaotic kitchen where hundreds of chefs are cooking at the same time. This is your chaotic system. In a normal chaotic kitchen, if you drop a single grain of salt (a tiny piece of information) into one pot, it doesn't just stay there. It gets whisked, stirred, and splashed around so quickly that within seconds, the flavor of that single grain is spread across the entire kitchen. You can't tell where it started anymore. In physics, we call this "scrambling." It's the reason why, if you mess up a recipe, you can't easily undo it to get back to the original ingredients.

Now, imagine a strict head chef (let's call him Mr. Reset) who walks around the kitchen. Every few minutes, on a random schedule, he blows a whistle. When he blows the whistle, every single chef instantly forgets what they were doing and goes back to their very first station, holding the exact same ingredients they started with.

This paper asks a fascinating question: What happens to the chaos if Mr. Reset blows the whistle too often?

The Two Worlds of the Kitchen

The authors discovered that there are two very different "worlds" or phases in this kitchen, depending on how often Mr. Reset blows his whistle.

1. The "Busy Kitchen" (Low Reset Rate)

If Mr. Reset blows the whistle only occasionally, the chefs still have time to scramble the salt. The chaos continues, but it's slightly slower. The information (the salt) still spreads, but it takes a bit longer to reach the other side of the kitchen. The system is still chaotic, just a little less wild.

2. The "Frozen Kitchen" (High Reset Rate)

Here is the magic part. If Mr. Reset blows the whistle too often, something dramatic happens. The chaos suddenly stops.

Imagine trying to run a race, but every time you take two steps, you are instantly teleported back to the starting line. You never get anywhere. You are stuck right where you began.

In this "Frozen Kitchen" phase:

• The Salt Doesn't Move: If you drop a grain of salt, it might wiggle a tiny bit, but it never spreads to the other pots. It stays stuck right next to where you dropped it.
• The "Butterfly Effect" Dies: In chaos theory, there's a famous idea called the "Butterfly Effect" (a butterfly flapping its wings causes a storm elsewhere). In this frozen phase, the butterfly flaps its wings, but no storm happens. The connection between the start and the end is broken.
• Information Gets "Local": The information becomes localized. It's like a ghost that haunts only one specific spot in the kitchen and refuses to leave.

The "Tipping Point" (The Phase Transition)

The paper calculates a specific "critical rate." Think of this like a speed limit.

• Below the limit: The kitchen is chaotic. Information spreads like a wildfire.
• Above the limit: The kitchen freezes. Information is trapped.

The most surprising thing the authors found is that this switch isn't gradual. It's like a light switch. One moment the chaos is there, and the next moment, it's completely gone. The "Lyapunov spectrum" (a fancy math way of measuring how chaotic the system is) suddenly collapses to zero. It's as if the entire system's ability to be unpredictable just vanishes in an instant.

Why Does This Matter?

You might wonder, "Who cares if a kitchen stops scrambling salt?"

1. Protecting Secrets: In the quantum world (the world of tiny particles and future computers), we want to protect information from getting scrambled. If we can figure out how to "reset" a system just right, we might be able to stop information from leaking out or getting destroyed. This could help build better quantum computers.
2. Understanding Measurement: The paper suggests this "frozen" state looks a lot like what happens when we measure quantum particles. When we look at a quantum system, it often stops behaving chaotically and "collapses" into a specific state. This paper shows that even in classical systems (like our kitchen), just "resetting" the system often enough can mimic this quantum behavior.
3. New States of Matter: It shows that by simply adding a random "reset" button to a system, we can create entirely new types of matter or states of physics that don't exist in nature otherwise.

The Bottom Line

This paper is about a simple but powerful idea: If you keep hitting the "Reset" button on a chaotic system often enough, you can freeze the chaos.

You turn a system that is constantly changing and spreading information into a system where information gets stuck in one spot, unable to move. It's a new way to control chaos, turning a wild, unpredictable storm into a calm, frozen lake, simply by hitting the reset button at the right speed.

Technical Summary

1. Problem Statement

Extended many-body systems typically exhibit chaotic dynamics, characterized by the rapid "scrambling" of information. This scrambling renders initial conditions irretrievable and is quantified by positive Lyapunov exponents and a finite "butterfly velocity" ($v_B$) governing the spread of perturbations.

While mechanisms like integrability, strong disorder, or frequent projective measurements can suppress chaos, the specific dynamical nature of chaos suppression via stochastic resetting (returning a system to its initial state at random times) remained an open question. Previous work established that resetting drives a transition to a non-chaotic regime where the largest Lyapunov exponent ($\lambda_0$) and butterfly velocity vanish. However, the precise nature of this transition, the behavior of the full Lyapunov spectrum, and the spatiotemporal characteristics of information in the non-chaotic phase were not rigorously derived.

Core Question: What is the precise dynamical nature of the non-equilibrium, non-chaotic regime driven by stochastic resetting, and how does it alter the Lyapunov spectrum and information localization?

2. Methodology

The authors employ a combination of analytical field theory and numerical simulations:

• Theoretical Framework:

  • Model: They consider a generic scalar field $\phi(x,t)$ evolving under local, causal, chaotic dynamics, interspersed with global stochastic resetting events at rate $r$.
  • OTOC & Lyapunov Spectrum: They utilize the connection between Out-of-Time-Ordered Correlators (OTOCs) and Lyapunov spectra via Oseledets' multiplicative ergodic theorem. The OTOC $D(x, x'; t)$ measures the propagation of infinitesimal perturbations.
  • Velocity-Dependent Lyapunov Exponent (VDLE): They introduce the ansatz $D(x, x'; t) \sim \exp[\lambda(v)t]$ where $v = (x-x')/t$. The function $\lambda(v)$ characterizes the spatiotemporal spread of information.
  • Renewal Equation: To handle resetting, they derive a renewal equation relating the OTOC with resetting ($\tilde{D}$) to the OTOC without resetting ($D$). This allows them to compute the renormalized VDLE ($\tilde{\lambda}(v)$) and the Lyapunov spectrum ($\tilde{\Lambda}(k)$) analytically.
  • Saddle-Point Approximation: They use large-time asymptotic analysis (saddle-point methods) to evaluate integrals involving the exponential growth of perturbations.

• Numerical Validation:

  • Model: They simulate the Coupled Logistic Map (CLM), a discrete 1D chain of $L$ sites with diffusive coupling.
  • Parameters: Logistic parameter $a=4$ (chaotic regime) and diffusion $c=0.1$.
  • Technique: Since standard QR decomposition fails under resetting (due to trajectory averaging), they use a brute-force approach based on the renewal equation to compute the OTOC and extract the Lyapunov spectrum from its eigenvalues.

3. Key Contributions

1. Derivation of the Renormalized VDLE: The authors rigorously derive how stochastic resetting modifies the velocity-dependent Lyapunov exponent $\lambda(v)$. They show that the effect depends critically on the resetting rate $r$ relative to the unperturbed largest Lyapunov exponent $\lambda_0$.
2. Identification of a Spectral Phase Transition: They prove that the Lyapunov spectrum undergoes an abrupt collapse from a continuous distribution to a flat, zero spectrum at a critical resetting rate $r_c = \lambda_0$.
3. Characterization of Information Localization: They demonstrate that above the critical rate, information spreading is arrested in time and becomes exponentially localized in space, defining a new "localization regime."
4. Critical Exponents: They calculate the critical exponents for the divergence of the localization length ($\nu = 1/2$) and the dynamical exponent ($z=2$) near the transition.

4. Key Results

A. The Velocity-Dependent Lyapunov Exponent (VDLE)

The renormalized VDLE, $\tilde{\lambda}(v)$, exhibits two distinct phases:

• Chaotic Phase ($r < r_c = \lambda_0$):
  The spectrum is simply shifted down by the resetting rate:
  $$\tilde{\lambda}(v) = \lambda(v) - r$$
  The system remains chaotic ($\tilde{\lambda}_0 > 0$), but the butterfly velocity $\tilde{v}_B$ vanishes as $\tilde{v}_B \sim (r_c - r)^{1/2}$ as $r \to r_c$.
• Localized Phase ($r > r_c$):
  The system becomes non-chaotic. The VDLE develops a non-analytic cusp at $v=0$:
  $$\tilde{\lambda}(v) = \begin{cases} \lambda'(v^*)|v| & \text{for } |v| < v^* \\ \lambda(v) - r & \text{for } |v| \geq v^* \end{cases}$$
  Here, $v^*$ is a velocity determined by the condition $\lambda'(v^*) = (\lambda(v^*) - r)/v^*$. Crucially, $\tilde{\lambda}(v) \leq 0$ for all $v$, meaning no exponential growth of perturbations occurs. The maximum is pinned at $\tilde{\lambda}(0) = 0$.

B. The Lyapunov Spectrum Collapse

The Lyapunov spectrum $\tilde{\Lambda}(k)$ (the set of all Lyapunov exponents) undergoes a sharp phase transition:

• For $r < r_c$: $\tilde{\Lambda}(k) = \Lambda(k) - r$. The spectrum shifts uniformly but retains its shape (a square-root singularity at the upper edge).
• For $r \geq r_c$: $\tilde{\Lambda}(k) = 0$ for all $k$.
  The entire spectrum collapses to zero. This indicates a transition from a regime with a broad distribution of growth rates to one where all modes are maximally degenerate and non-growing.

C. Spatiotemporal Localization

In the localized regime ($r > r_c$), the OTOC does not grow ballistically but saturates to a steady-state spatial profile:
$$\lim_{t \to \infty} \tilde{D}(x, x'; t) \sim \exp\left(-\frac{|x - x'|}{\xi_r}\right)$$

• Localization Length ($\xi_r$): The characteristic length scale diverges as the critical point is approached from above:
  $$\xi_r \sim \frac{1}{\sqrt{r - r_c}} \implies \nu = 1/2$$
• Timescale: The approach to this steady state is governed by $\tau_r = 1/(r - r_c)$.
• Dynamical Exponent: The scaling $\xi_r \sim \tau_r^{1/z}$ yields $z = 2$.

5. Significance and Implications

• Mechanism of Chaos Suppression: The paper provides a rigorous analytical mechanism showing how stochastic resetting acts as a "non-equilibrium drive" that fundamentally alters the Lyapunov spectrum, leading to a complete arrest of information scrambling.
• Connection to Measurement-Induced Phase Transitions (MIPTs): The authors draw a compelling parallel between this classical resetting transition and quantum MIPTs. Both involve a transition from a volume-law (scrambling) to an area-law (localized) phase.
  • Unlike previous classical attempts to model MIPTs (which often failed to reproduce the $z=1$ exponent associated with emergent spacetime symmetry), this resetting model naturally generates a $z=2$ dynamical exponent.
  • The "projection" of the system to its initial state mimics the Born rule in quantum measurements, suggesting resetting could be a powerful classical proxy for studying quantum measurement dynamics.
• Universality: The results are derived for generic chaotic dynamics and validated on the CLM, suggesting the transition is a universal feature of extended chaotic systems subject to global resetting.

In summary, the paper establishes that stochastic resetting drives a sharp dynamical phase transition where information localization emerges, characterized by a collapse of the Lyapunov spectrum to zero and a divergence of the localization length with critical exponents $\nu=1/2$ and $z=2$.