Multiscale Structure of Eigenstate Thermalization

Imagine a giant, complex machine made of billions of tiny, interacting gears. In physics, we call this a "many-body quantum system." Usually, when we look at these machines, we expect them to eventually settle down into a calm, predictable state called "thermal equilibrium" (like a cup of coffee cooling down to room temperature).

For decades, physicists have used a rule called the Eigenstate Thermalization Hypothesis (ETH) to explain how this happens. The rule basically says: "If you look at just one specific snapshot of the machine's energy, the tiny parts inside it will already look like they are in a calm, random state."

However, this new paper by Orlov, Sharipov, and Ilievski suggests that the old rule is missing a crucial detail. They discovered that the "randomness" inside the machine depends on how wide your net is when you catch the snapshots.

Here is the breakdown of their discovery using simple analogies:

1. The Old Way: Looking Through a Narrow Keyhole

Traditionally, physicists studied these systems by looking at a very narrow slice of energy—like looking through a tiny keyhole. They would pick two snapshots of the machine that were almost identical in energy and ask, "How different are they?"

The old rule (ETH) said: "If they are close in energy, they look very similar. If they are far apart, they look completely random and unconnected."

2. The New Discovery: The Size of the Net Matters

The authors asked a new question: What happens if we don't look through a keyhole, but instead cast a wide net?

Imagine you are fishing for fish (which represent the machine's energy states).

Narrow Net (Small Fluctuations): You only catch fish that are swimming right next to each other.
Wide Net (Large Fluctuations): You cast a net that catches fish from a huge area, including fish that are far apart in the ocean.

The paper found that the "randomness" of the machine changes depending on how wide your net is.

If your net is small, the machine behaves exactly as the old rule predicted.
If your net gets wider, the machine starts behaving differently. The "connection" between the parts doesn't just fade away; it changes its mathematical shape entirely.

They call this the "Multiscale Structure." It means the machine has different "personality traits" depending on how far apart you look.

3. The "Staircase" Analogy

To prove this, the authors used a special, simplified model of a machine (an "integrable system") that is easier to solve than a chaotic one. They visualized the states of this machine as staircases made of blocks (mathematically known as Young diagrams).

The Experiment: They compared two staircases.
- Scenario A: The staircases are almost identical (a tiny difference in height).
- Scenario B: The staircases are very different (one is much taller than the other).

They calculated how likely it was for the machine to jump from one staircase to another. They found a surprising "tipping point":

Below the tipping point: The jump probability drops off slowly.
Above the tipping point: The jump probability crashes much faster, but in a specific, complex way involving logarithms (a mathematical curve that grows very slowly).

It's like driving a car: below a certain speed, the wind resistance is manageable. But once you cross a specific speed threshold, the wind resistance suddenly spikes in a way you didn't expect, changing how the car handles.

4. The "Fluctuation Scale" (The Dial)

The authors introduced a "dial" (called $\gamma$ ) that controls how wide their net is.

Dial at 0: You are looking at a tiny, precise slice (the old way).
Dial at 1: You are looking at the entire machine, including wildly different states.

They found that the statistical "rules" of the machine change abruptly when you turn this dial past a certain point (specifically, when the dial passes 0.5).

Before 0.5: The machine follows one set of rules (the standard ETH).
After 0.5: The machine follows a different set of rules where the connections between states are suppressed much more strongly.

5. The Shape of the Randomness

Finally, they looked at the "shape" of the randomness.

In the "thermal" zone (the middle of the dial), the randomness looks like a specific bell curve known as the Gumbel distribution (often used to describe extreme events, like the highest flood levels in a century).
In the "small net" zone, the randomness looks like a skewed curve (the skew-normal distribution), which is lopsided.

The Bottom Line

The paper claims that the "thermalization" of quantum systems isn't a single, fixed rule. Instead, it is a multiscale phenomenon.

Think of it like listening to a symphony:

If you listen to just one instrument (narrow net), you hear a specific melody.
If you listen to the whole orchestra (wide net), the melody changes, and the way the instruments blend together follows a different set of rules.

The authors proved that to truly understand how quantum systems settle down, you have to account for the "size of the net" you use to observe them. If you ignore this, you might miss the fact that the system behaves differently when you look at it from a broader perspective.

Technical Summary: Multiscale Structure of Eigenstate Thermalization

Problem Statement
The Eigenstate Thermalization Hypothesis (ETH) posits that isolated quantum many-body systems thermalize because individual energy eigenstates encode thermal properties. Standard ETH analysis typically focuses on microcanonical ensembles where eigenstates are confined to narrow energy windows of width $\delta E = O(1)$ (or $O(L^0)$ ). However, in physical scenarios such as quantum quenches, systems often explore ensembles with larger energy fluctuations, scaling as $\delta E = O(L^\gamma)$ with $\gamma \in [0, 1]$ .

A critical gap in current understanding is whether the statistical properties of off-diagonal matrix elements of local observables depend on this fluctuation scale $\gamma$ . Previous studies in integrable systems (e.g., Lieb-Liniger and Heisenberg models) reported "anomalous" scaling behaviors (e.g., $L \log L$ suppression) compared to the standard exponential suppression in chaotic systems. However, these studies often sampled ensembles where higher conserved charges fluctuated freely, effectively probing a specific fluctuation scale ( $\gamma = 1/2$ ) rather than a controlled parameter. The authors argue that a systematic investigation across all fluctuation scales is necessary to distinguish between macrostate-dependent properties and ensemble-dependent statistical structures.

Methodology
To address this, the authors propose a general framework for probing the multiscale structure of matrix elements by introducing a tunable "fluctuation scale" parameter $\gamma$ .

Distance Metric: Instead of relying solely on energy separation, the authors define a distance between eigenstates based on the $L^1$ -norm of the differences in their conserved charge densities. This allows for a unified treatment of both chaotic and integrable systems.
Model Selection: The authors utilize a specific integrable quantum field theory: a massless bosonic field theory (specifically the $\beta=1$ $β = 1$ case of the quantum Benjamin-Ono hierarchy). This model is chosen because:
- Its spectrum and eigenstates can be mapped to integer partitions (Young diagrams).
- Matrix elements of vertex operators (a class of local observables) admit an explicit, factorizable closed-form expression involving the "arms" and "legs" of the Young diagrams.
- It permits efficient numerical sampling of eigenstates from ensembles with arbitrary fluctuation scales up to system sizes $L \sim 10^6$ , overcoming the limitations of exact diagonalization in chaotic systems.
Ensemble Construction: The authors employ "modified Vershik ensembles." Starting from the standard Vershik ensemble (canonical Gibbs state on Young diagrams with Gaussian fluctuations of scale $O(L^{1/2})$ ), they rescale the fluctuation magnitude to $d = O(L^\gamma)$ . This allows for the systematic sampling of eigenstate pairs separated by a distance $d$ corresponding to a specific $\gamma$ .
Analytical and Numerical Analysis:
- Analytical: They derive exact asymptotic formulas for the suppression rate of matrix elements for a specific class of states called "staircase diagrams" (partitions $\lambda = (k, k-1, \dots, 1)$ ).
- Numerical: They compute the distribution of the pseudo-random variable $\kappa_{mn} = -\frac{1}{L} \log |A_{mn}|^2$ (representing the suppression rate) and its width $\delta \kappa$ across the modified Vershik ensembles for various $\gamma$ .

Key Contributions and Results

Multiscale Dependence of Scaling Exponents:
The study reveals that the algebraic scaling exponents governing the mean suppression rate ( $p(\gamma)$ ) and the distribution width ( $q(\gamma)$ ) are non-analytic functions of the fluctuation scale $\gamma$ .
- Suppression Rate ( $p(\gamma)$ ):
  - For $\gamma < 1/2$ : $p(\gamma) = 0$ . The suppression is exponential ( $|A_{mn}|^2 \sim e^{-O(L)}$ ), consistent with standard ETH expectations for states within the same macrostate.
  - For $1/2 \le \gamma < 1$ : $p(\gamma) = 2\gamma - 1$ . The suppression becomes parametrically stronger, scaling as $|A_{mn}|^2 \sim e^{-O(L^{2\gamma} \log L)}$ .
  - For $\gamma = 1$ (distinct macrostates): $p(1) = 1$ , leading to $|A_{mn}|^2 \sim e^{-O(L^2)}$ .
- Distribution Width ( $q(\gamma)$ ):
  - For $\gamma < 1/3$ : $q(\gamma) = -\gamma$ .
  - For $\gamma \ge 1/3$ : $q(\gamma) = 2\gamma - 1$ .
    These results demonstrate that the statistical properties of matrix elements are not solely determined by the macrostate (charge densities) but depend explicitly on the ensemble's fluctuation scale.
Resolution of Anomalous Scaling:
The authors explain the previously observed "anomalous" $L \log L$ suppression in integrable systems (Refs. [53, 54]). They show that these studies effectively sampled at the thermal fluctuation scale $\gamma = 1/2$ . In this regime, the derived scaling $p(1/2) = 0$ (with a logarithmic correction arising from the specific functional form) matches the observed behavior. The "anomaly" is thus identified as a consequence of the specific fluctuation scale inherent in thermal ensembles of integrable systems, rather than a fundamental breakdown of ETH.
Universal Distribution Functions:
The probability distribution of the normalized matrix elements changes depending on the fluctuation scale:
- For $\gamma > 1/3$ (including the thermal scale $\gamma=1/2$ ): The distribution follows the Gumbel distribution, consistent with previous findings in other integrable models.
- For $\gamma < 1/3$ : The distribution deviates from Gumbel and is well-approximated by a skew-normal distribution.
Analytical Verification:
The analytical results derived for staircase diagrams perfectly reproduce the scaling laws found numerically in the broader modified Vershik ensembles, confirming the robustness of the scaling variable $u = \frac{d^2}{L} \log(L/d)$ .

Significance
The paper claims to establish that the statistical properties of matrix elements in equilibrium macrostates possess an underlying multiscale structure. The primary significance lies in demonstrating that the choice of the averaging ensemble (specifically the scale of charge fluctuations) fundamentally alters the statistical behavior of off-diagonal matrix elements.

The authors emphasize that while the diagonal ETH (relating diagonal elements to macrostate parameters) remains valid, the off-diagonal ETH ansatz requires a more nuanced formulation that accounts for the fluctuation scale $\gamma$ . This work provides a unified explanation for the scaling behaviors observed in both chaotic and integrable systems, clarifying that "anomalous" behaviors in integrable models are often artifacts of probing specific fluctuation scales ( $\gamma \approx 1/2$ ) rather than intrinsic failures of thermalization. The findings suggest that a complete understanding of thermalization dynamics, particularly in quantum quench protocols where fluctuation scales vary, must incorporate this multiscale dependence.