Subexponential decay of local correlations from diffusion-limited dephasing

The paper argues that in one-dimensional chaotic quantum systems with conservation laws, local correlations decay subexponentially (as stretched exponentials or slower) due to the coherent persistence of inert "void" regions, a phenomenon that standard hydrodynamics fails to capture and which vanishes under extrinsic dephasing.

The Gist

Imagine a chaotic quantum system (like a complex, jiggling collection of particles) as a crowded, noisy dance floor. Usually, if you try to keep a delicate secret (a "quantum superposition") in one spot, the noise of the crowd destroys it very quickly. In physics terms, we say the secret "dephases" or decays exponentially fast, like a battery running out.

However, this paper argues that in certain one-dimensional systems (think of the dance floor as a single, long hallway), there is a special trick that keeps secrets alive much longer than expected. Instead of fading away quickly, they fade away extremely slowly, following a "stretched" pattern.

Here is the simple breakdown of how and why this happens, using analogies from the paper:

1. The "Void" (The Empty Room)

The key to this slow decay is the existence of "voids."
Imagine the crowded dance floor. Occasionally, purely by chance, a large section of the hallway becomes completely empty. The paper calls these "voids."

• Why they matter: If you place your delicate secret (a quantum particle) inside this empty room, it is safe. The noisy crowd outside cannot reach it yet.
• The Catch: These empty rooms are rare. The bigger the room, the rarer it is.

2. The "Melting" Iceberg (Diffusion)

The empty room doesn't stay empty forever. The crowd from the edges slowly "melts" into the void, filling it up. This process is called diffusion.

• The Analogy: Think of the void as a block of ice in a warm room. The heat (the crowd) slowly melts the ice from the outside in.
• The Result: As long as the void is big enough, your secret particle stays safe inside. The secret only starts to fade once the void is filled in enough for the crowd to reach the particle.

3. The Race Against Time

The paper calculates a race between two things:

1. How rare is the void? (Bigger voids are harder to find).
2. How fast does the void fill up? (Diffusion takes time).

The authors found that the "sweet spot" is a specific size of void that lasts just long enough to protect the particle for a surprisingly long time. Because the filling-up process is slow (diffusion-limited), the decay of the secret is subexponential.

• Normal Decay: Like a lightbulb burning out quickly (Exponential).
• This Paper's Decay: Like a slow leak in a boat that takes ages to sink (Stretched Exponential).

4. Two Different Speeds

The paper identifies two scenarios for how fast the void fills up, depending on the type of system:

• Scenario A: The Random Circuit (The "Random Walk")
  • Imagine the crowd moving randomly. The void fills up at a standard diffusion rate.
  • Result: The secret decays as $e^{-\sqrt{t}}$. (Think of this as a "square root" slowdown).
• Scenario B: The Ordered System (The "Ballistic" Walk)
  • Imagine the crowd moves in a more organized, wave-like pattern. The void fills up faster, but the math changes slightly.
  • Result: The secret decays as $e^{-t^{2/3}}$. (This is even slower than the square root case).

5. The "Noise" Test (Why it's Quantum)

To prove this isn't just a weird classical trick, the authors added "extrinsic noise" (like a loudspeaker blaring static over the dance floor).

• The Result: As soon as they added this external noise, the slow, stretched decay vanished, and the secrets died quickly again.
• The Lesson: This slow decay relies entirely on quantum coherence (the delicate, wave-like nature of the particles). If you break that coherence with outside noise, the "void" protection fails.

Summary

In chaotic quantum systems with conservation laws (like a rule that total "spin" must stay the same), local secrets don't die quickly. Instead, they hide in rare, temporary "empty rooms" (voids) within the system. These rooms slowly fill up from the edges, acting as a shield. Because it takes a long time for the crowd to fill these rooms, the secrets survive for a very long time, decaying in a slow, stretched-out way that is unique to quantum mechanics.

What the paper does NOT claim:

• It does not claim this can be used to build better batteries or medical devices.
• It does not claim this happens in all dimensions (it focuses on 1D).
• It does not claim this works if there is no conservation law (like a rule keeping the total number of particles constant).

Technical Summary

Problem Statement
Chaotic quantum systems at finite energy density are generally expected to thermalize, acting as their own heat baths that rapidly dephase local quantum superpositions. In such systems, the long-time dynamics of local operators are typically governed by classical fluctuating hydrodynamics, where correlations decay algebraically (hydrodynamic long-time tails) or exponentially if the operator has zero overlap with hydrodynamic modes (e.g., charge-raising operators in charge-conserving systems). The standard expectation is that operators orthogonal to hydrodynamic modes should relax exponentially, with a timescale set by microscopic physics. This paper challenges that expectation, arguing that in one-dimensional chaotic systems with conservation laws, the dephasing of such local correlations is actually subexponential.

Methodology
The authors employ a combination of analytical arguments based on hydrodynamic scaling and rare-region physics, alongside numerical simulations using Tensor Network methods (specifically Time-Evolving Block Decimation, TEBD) and random circuit techniques.

1. Analytical Framework: The core argument relies on the existence of "voids"—rare regions in equilibrium states that resemble zero-entropy charge sectors (e.g., the particle vacuum or ground state). In these voids, the dynamics of a local excitation (such as a magnon created by a charge-raising operator) is coherent and decoupled from the thermal bath until the void is filled by diffusive transport from the edges.

   • Random Circuits: For $U(1)$-symmetric random unitary circuits, the authors construct a lower bound on the correlation function by considering the probability of finding a void of size $\ell$ and the diffusion time required to fill it. They utilize a replica statistical mechanics model to compute circuit-averaged moments of correlation functions.
   • Translationally Invariant Systems: For Floquet and Hamiltonian systems, the authors argue that while transport inside a void may be ballistic, the filling of the void is limited by the diffusive transport of particles from the high-density edges. They derive a scaling solution for the density profile $n(x,t)$ near a void, leading to a specific optimization of the void size $\ell$ that dominates the late-time decay.

2. Numerical Simulations:

   • Random Circuits: Simulations of the transfer matrix evolution for the second moment of the correlation function $Z(x,t) = \mathbb{E}_U |\langle \sigma^+_x(t) \sigma^-_0 \rangle|^2$.
   • Floquet Systems: TEBD simulations of chaotic, translationally invariant Floquet models with conserved magnetization to measure the decay of non-hydrodynamic autocorrelators $C(t) = \langle \sigma^+_0(t) \sigma^-_0 \rangle$.
   • Noise Sensitivity: Simulations introducing extrinsic dephasing noise to test the stability of the observed decay mechanisms.
   • Hydrodynamic Verification: Direct simulation of nonlinear diffusion equations and stochastic gas models to verify the density-dependent diffusivity $D(n) \sim 1/n$ and the resulting scaling collapse.

Key Contributions and Results

1. Subexponential Decay Bound: The paper demonstrates that in 1D chaotic systems with a conserved charge and zero-entropy charge sectors, all local correlation functions decay subexponentially.

   • Random Circuits: For systems with a finite diffusion constant inside voids (e.g., random charge-conserving circuits), local correlations decay as a stretched exponential $\exp(-O(\sqrt{t}))$.
   • Translationally Invariant Systems: For systems where the diffusion constant diverges at low density (e.g., $D(n) \sim 1/n$ in Hamiltonian or Floquet systems), the decay is slower, scaling as $\exp(-O(t^{2/3}))$. This exponent arises from optimizing the trade-off between the rarity of large voids (probability $\sim e^{-\ell}$) and the time required for the void to fill (diffusion time $\sim \ell^2$ or ballistic filling limited by edge diffusion).

2. Mechanism of "Diffusion-Limited Dephasing": The authors identify the physical mechanism as the persistence of coherent dynamics within rare void regions. A local superposition inserted into a void does not dephase until the void is filled by hydrodynamic processes. This mechanism is distinct from standard hydrodynamic tails and applies specifically to operators orthogonal to hydrodynamic modes.

3. Quantum Coherence Requirement: The subexponential decay is shown to be a quantum coherent effect. Numerical simulations with extrinsic dephasing noise (which preserves conservation laws but destroys local coherence) result in a breakdown of the stretched exponential behavior, reverting to standard exponential decay. This suggests that maintaining coherence over large scales is essential for this phenomenon.

4. Numerical Verification:

   • In random circuits, the data confirms the $\exp(-\sqrt{t})$ scaling and the sensitivity to dephasing noise.
   • In translationally invariant Floquet models, the data supports the $\exp(-t^{2/3})$ scaling, with fitted exponents $\alpha \approx 0.60 - 0.65$, consistent with the theoretical bound.
   • The paper verifies that the decay rate of non-hydrodynamic correlations in low-density thermal states is linearly proportional to the magnon density, supporting the collision-based dephasing model used in the derivation.

Significance and Claims
The paper claims that the expectation of exponential relaxation for non-hydrodynamic operators in charge-conserving systems is incorrect. Instead, the presence of conservation laws and zero-entropy sectors leads to a universal, slow, subexponential relaxation governed by the diffusion-limited filling of rare voids.

• Theoretical Implication: This result lies beyond standard fluctuating hydrodynamics, which typically predicts algebraic tails for hydrodynamic modes and exponential decay for non-hydrodynamic ones. The authors argue that the "void" mechanism is a generic feature of 1D chaotic systems with conservation laws.
• Computational Challenge: The results suggest a fundamental limit for numerical algorithms that simulate quantum dynamics by introducing noise and extrapolating to the weak-noise limit (e.g., dissipation-assisted operator evolution). Since stretched exponential relaxation relies on maintaining coherence over large scales, such methods may fail to capture the correct late-time behavior even for local correlation functions.
• Experimental Witness: The authors propose that observing this specific subexponential decay could serve as an experimentally accessible witness of quantum coherence in present-day quantum devices, distinguishing true quantum dynamics from noisy classical approximations.

The paper remains modest regarding higher dimensions, noting that while a naive application of their arguments suggests different scaling exponents, a thorough investigation is required, particularly in $d=2$ where the contributions might become exponentially decaying. Similarly, the reconciliation of these results with Ruelle-Pollicott resonances is identified as an open question for future work.