Non-equilibrium dynamics of the disordered Power of Two model

This study investigates the non-equilibrium dynamics of the disordered Power-of-Two model, revealing that while strong disorder induces localization and unique non-monotonic scrambling signatures, the system ultimately remains ergodic in the thermodynamic limit for any finite disorder strength.

The Gist

Imagine a crowded dance floor where everyone is paired up with a specific partner to dance with. In most dance floors, you only dance with the people standing right next to you. But in the Power-of-Two (PWR2) model studied in this paper, the rules are weird and magical: you only dance with partners who are standing at specific distances away from you—exactly 1 step, 2 steps, 4 steps, 8 steps, and so on.

This creates a unique, "sparse" network of connections. In a perfect world (no chaos, no noise), this dance floor is a super-fast information highway. If you whisper a secret to one person, it spreads to the entire room almost instantly. Physicists call this "fast scrambling." It's like a rumor that travels so fast it reaches everyone before you can even finish your sentence.

The Experiment: Introducing the "Noise"

The researchers asked: What happens if we mess up the dance floor? They introduced disorder, which is like having random, loud music playing in different corners, or having some dancers who are too distracted to follow the rhythm. In physics, this is called "random magnetic fields."

They wanted to see if this noise would stop the information from spreading, causing the dancers to get "stuck" in their own little groups. This phenomenon is called Many-Body Localization (MBL). If MBL happens, the system stops mixing, remembers its starting state forever, and never reaches a "thermal" (equilibrium) state.

What They Found: The "Ghost" Connections

Here is where the story gets fascinating:

1. The Noise Does Slow Things Down: When the disorder was strong, the information did spread much slower. The dancers stayed closer to their starting spots, and the system remembered its initial state longer.
2. The "Ghost" Effect: In normal systems, if you have noise, information spreads in a smooth wave (like a ripple in a pond). But in this Power-of-Two model, the information spread was bumpy and weird. Because the dancers were connected to people far away (the "power of two" distances), the information didn't just move forward; it jumped around in a non-monotonic pattern. It was as if the rumor jumped to the back of the room, then back to the front, skipping people in between. This is a unique signature of this specific model.

The Big Surprise: The System Never Actually Stops

The researchers dug deeper to see if the system would eventually get "stuck" (localized) if they made the noise strong enough. They looked at the math of the energy levels and how entangled the dancers were.

The Verdict:
For any fixed amount of noise, if you make the dance floor big enough (add more and more dancers), the system never gets stuck.

Think of it like this:

• If you have a small room (small system), the noise can easily stop the rumor from spreading. The system looks "localized."
• But as you keep adding more dancers (increasing the system size), the "Power-of-Two" connections become so powerful and far-reaching that the noise can't stop the information from eventually reaching everyone.

The "critical noise level" needed to stop the system keeps getting higher and higher as the room gets bigger. In the limit of an infinitely large room (the thermodynamic limit), you would need infinite noise to stop the scrambling.

The Takeaway

This paper tells us that the Power-of-Two model is incredibly robust. Even if you throw a lot of chaos at it, its unique, long-distance connections allow it to keep mixing and scrambling information. It refuses to get "frozen" or localized, no matter how big the system gets.

In simple terms:
Imagine a game of "Telephone" where players can only pass notes to people at specific distances (1, 2, 4, 8 seats away). Even if you try to disrupt the game with loud noise, the unique way the players are connected ensures the message eventually reaches everyone, provided the game is played with enough people. The system is too clever to be stopped by disorder.

Technical Summary

Here is a detailed technical summary of the paper "Non-equilibrium dynamics of the disordered Power of Two model" by Kunal Singh and Sayan Choudhury.

1. Problem Statement

The paper investigates the interplay between long-range interactions and disorder in isolated quantum many-body systems. Specifically, it focuses on the Power-of-Two (PWR2) model, a system characterized by sparse long-range couplings between spin-1/2 particles separated by distances $d = 2^n$ (where $n$ is an integer).

While the clean (disorder-free) PWR2 model is known to be a "fast scrambler" exhibiting rapid thermalization, the authors aim to determine how on-site disorder affects its non-equilibrium dynamics. The central question is whether the PWR2 model can host a Many-Body Localized (MBL) phase in the thermodynamic limit, or if its unique interaction geometry prevents localization even under strong disorder.

2. Methodology

The authors employ a combination of dynamical simulations and spectral analysis to characterize the system:

• Model Definition: The Hamiltonian is given by:
  $$H_{PWR2} = \sum_{i \neq j} J_{ij} (S^x_i S^x_j + S^y_i S^y_j) + \sum_i h_i S^z_i$$
  where couplings $J_{ij} = J$ only if $|i-j| = 2^n$, and $h_i$ are random magnetic fields uniformly distributed in $[-h, h]$.
• Dynamical Probes:
  • Single-Magnon Dynamics: Tracking the time evolution of a localized spin excitation ($\langle n_j \rangle$) to observe information spreading.
  • Out-of-Time-Ordered Correlators (OTOCs): Calculating $C_{ij}(t) = \langle [S^z_i(t), S^z_j(0)]^\dagger [S^z_i(t), S^z_j(0)] \rangle$ to measure information scrambling and chaos.
  • Survival Probability ($L(t)$): Measuring the retention of the initial state memory after a quantum quench.
  • Entanglement Entropy ($S(t)$): Computing the half-chain entanglement entropy to distinguish between thermal (volume law) and localized (area law/logarithmic growth) phases.
• Spectral and Eigenstate Analysis:
  • Spectral Statistics: Analyzing the ratio of adjacent energy gaps ($\langle r \rangle$) to distinguish between Wigner-Dyson (chaotic) and Poisson (localized) statistics.
  • Eigenstate Entanglement: Examining the average entanglement of mid-spectrum eigenstates.
  • Inverse Participation Ratio (IPR): Calculating $P = \sum |\langle \mu_j | \psi_n \rangle|^4$ to quantify the delocalization of eigenstates in the computational basis.

3. Key Contributions and Results

A. Suppression of Scrambling and Emergence of Non-Monotonicity

• Disorder-Free Limit: The system exhibits fast scrambling with a scrambling time scaling as $t^* \propto \log(N)$, consistent with its role as a fast scrambler.
• Disordered Regime:
  • Strong disorder suppresses the growth of OTOCs and entanglement entropy, leading to the retention of initial state memory (non-ergodic behavior).
  • Unique Spatial Profile: Unlike conventional long-range models where OTOCs decay monotonically with distance, the PWR2 model exhibits a non-monotonic spatial dependence in the OTOC profile ($C_{1j}$) under strong disorder. This arises from the intrinsic non-locality of the $2^n$ couplings, creating a distinct scrambling signature.

B. Finite-Size vs. Thermodynamic Limit Behavior

The paper presents a critical finding regarding the existence of the MBL phase:

• Finite Systems: For any fixed system size $L$, there exists a critical disorder strength $h_c$ where the system transitions from a chaotic (Wigner-Dyson) phase to a localized (Poisson) phase. Indicators such as $\langle r \rangle$, entanglement entropy, and IPR all show signatures of localization when $h > h_c$.
• Thermodynamic Limit ($L \to \infty$):
  • The critical disorder strength $h_c$ diverges with system size ($h_c \to \infty$ as $L \to \infty$).
  • At a fixed disorder strength $h$, as the system size increases:
    • The eigenstate-averaged entanglement entropy increases.
    • The inverse participation ratio decreases (indicating enhanced delocalization).
  • The peak of the entanglement variance ($\delta S$), which typically marks the transition point, shifts to higher disorder values as $L$ grows.

C. Conclusion on Localization

The authors conclude that while strong disorder induces localization in finite-sized PWR2 chains, the system does not host a Many-Body Localized phase in the thermodynamic limit for any finite disorder strength. The system remains ergodic (chaotic) in the thermodynamic limit, suggesting that the sparse long-range interactions of the PWR2 model are robust enough to overcome disorder.

4. Significance

• Robustness of Chaos: The work demonstrates that specific sparse long-range interaction geometries (like the Power-of-Two model) can sustain quantum chaos and thermalization even in the presence of disorder, challenging the universality of MBL in long-range systems.
• Distinct Scrambling Signatures: The discovery of non-monotonic OTOC profiles provides a new diagnostic tool for identifying systems with non-local, tree-like interaction graphs, distinguishing them from standard power-law or all-to-all interacting models.
• Experimental Relevance: Given that the PWR2 model has been realized in cold-atom experiments, these findings offer theoretical guidance for experimentalists exploring the boundaries between localization and thermalization in programmable quantum simulators.
• Future Directions: The paper suggests future investigations into spin squeezing, quantum Fisher information, and the effects of periodic driving (Floquet dynamics) or dissipation in these disordered sparse-coupled systems.