Eigenstate Thermalization Hypothesis correlations via non-linear Hydrodynamics

This paper utilizes non-linear hydrodynamics to predict the universal scaling of late-time free cumulants in the Eigenstate Thermalization Hypothesis, a finding that is validated by large-scale numerical simulations of non-integrable one-dimensional spin models.

The Gist

Imagine you have a giant, chaotic pot of soup. Inside, there are billions of tiny ingredients (atoms or particles) bumping into each other, swirling, and mixing. If you wait long enough, the soup becomes uniform: the temperature is the same everywhere, and the flavor is consistent. In physics, we call this thermalization.

For decades, physicists have had a rulebook for how this happens, called the Eigenstate Thermalization Hypothesis (ETH). Think of ETH as a recipe card that says, "If you look at the soup at the quantum level, the ingredients behave like random noise, but with a smooth, predictable pattern."

However, there was a missing ingredient in this recipe. ETH told us that the pattern exists, but it didn't tell us what the pattern actually looked like. It was like having a map that said, "There is a treasure here," but leaving the X marked with a question mark.

This paper, by Wang, Mishra, and colleagues, fills in that missing X. They discovered that the "shape" of this pattern is dictated by hydrodynamics—the same physics that describes how water flows, how heat spreads, and how smoke curls in the air.

Here is a breakdown of their discovery using simple analogies:

1. The "Free Cumulants" (The Secret Sauce)

To understand the soup's behavior, the authors looked at something called "free cumulants."

• The Analogy: Imagine you are trying to describe the flavor of the soup.
  • A 2-point correlation is like tasting the soup at one moment and comparing it to the next. (Is it hotter now than before?)
  • A 4-point correlation is like tasting it at four different times and seeing if the changes are just random or if they follow a specific rhythm.
  • Free Cumulants are the "pure" flavor notes. They strip away the obvious, repetitive parts to find the unique, complex interactions between the ingredients. The authors found that these "pure notes" are the key to unlocking the ETH mystery.

2. The Connection to Hydrodynamics (The Flow)

The big breakthrough is realizing that these complex quantum "flavor notes" are actually just fluid dynamics in disguise.

• The Analogy: Think of a crowd of people in a stadium.
  • If one person stands up, it's a small ripple.
  • If a wave of people stands up and sits down, that's a "hydrodynamic mode."
  • The authors found that the way these quantum particles interact over time is exactly like how a wave moves through a crowd or how heat diffuses through a metal rod.
  • The Prediction: They predicted that if you wait long enough, these complex interactions will decay (fade away) at a very specific speed, following a mathematical "power law." It's like predicting exactly how fast a drop of ink will spread in a glass of water.

3. The "Factorization" Trick (Breaking it Down)

One of the most surprising findings is that these complex, multi-layered interactions eventually break apart into simpler pieces.

• The Analogy: Imagine a complex dance routine with four dancers.
  • At the start, they are all doing a complicated, synchronized routine that is hard to predict.
  • But as time goes on, the routine simplifies. The four dancers stop doing a complex group dance and instead just pair up into two separate couples dancing independently.
  • The authors proved that in the quantum world, complex 4-way interactions eventually "factorize" (break down) into simple 2-way interactions. This means you don't need a supercomputer to predict the future of a complex system; you just need to understand how pairs of particles interact.

4. The "Finite Size" Effect (The Wall)

The paper also looked at what happens when the system isn't infinitely big (like a real computer simulation or a real lab experiment).

• The Analogy: Imagine the ink spreading in a glass of water. Eventually, the ink hits the glass walls and bounces back. The smooth spreading stops, and the whole glass just becomes a uniform color.
• In quantum systems, when the "ink" (the heat or energy) hits the edge of the system, the smooth, slow decay turns into a sudden, fast exponential drop. The authors showed that this transition happens exactly when the "wave" of particles hits the edge of the container, confirming their hydrodynamic theory.

Why Does This Matter?

Before this paper, we knew quantum systems eventually settle down to a steady state, but we didn't know the rules governing how they get there.

• The Result: The authors provided a universal rulebook. Whether you are looking at a chain of spinning atoms, a model of a magnetic material, or a complex fluid, if the system is "chaotic" (not perfectly ordered), it will follow these hydrodynamic rules.
• The Verification: They didn't just do math; they ran massive computer simulations (like running a virtual experiment millions of times) to prove that their predictions matched the data perfectly.

In a nutshell:
The authors took a mysterious, abstract rule about how quantum particles behave (ETH) and realized it's just the same old physics of flowing water and spreading heat (Hydrodynamics) wearing a fancy quantum mask. They showed that even in the chaotic quantum world, the long-term behavior is as predictable as a river flowing to the sea.

Technical Summary

1. Problem Statement

The Eigenstate Thermalization Hypothesis (ETH) posits that isolated quantum many-body systems thermalize because the matrix elements of local observables in the energy eigenbasis behave like pseudorandom variables with smooth statistical properties. While the standard ETH framework successfully describes diagonal elements and two-point off-diagonal correlations, it leaves the functional form of multi-point correlations (specifically for $n \geq 3$) undetermined.

Recent developments have linked ETH to free probability theory, identifying these multi-point correlations as "free cumulants" (denoted $\kappa_n$). However, the physical content and universal structure of these free cumulants in chaotic systems remain largely unexplored. Specifically, the paper addresses:

• What determines the shape of the smooth functions $F^{(n)}_{e_+}(\vec{\omega})$ governing ETH correlations?
• Do these correlations obey a universal hierarchy or scaling law at low frequencies (long times)?
• Can hydrodynamic principles, which govern classical fluctuations, be extended to constrain quantum ETH correlations?

2. Methodology

The authors employ a combination of analytical field theory and large-scale numerical simulations.

A. Analytical Framework: Non-linear Hydrodynamics

• Hydrodynamic Effective Field Theory (EFT): The authors utilize the EFT of fluctuating hydrodynamics to describe the late-time dynamics of conserved densities (e.g., energy density $\hat{h}$) in non-integrable systems.
• Classical vs. Free Cumulants: They distinguish between "classical cumulants" (standard connected correlation functions) and "free cumulants" (ETH correlations).
  • Classical cumulants are suppressed by powers of time at late times due to the hierarchy of non-linear hydrodynamic responses.
  • Using the combinatorial relationship between classical and free cumulants (involving non-crossing partitions), they derive the scaling behavior of free cumulants.
• Scaling Prediction: By expanding local operators into hydrodynamic fields (e.g., $\hat{A} \sim \alpha_1 \hat{h} + \alpha_2 \partial_x \hat{h} + \dots$), they predict that time-ordered free cumulants $\kappa_n$ factorize into products of two-point functions at sufficiently long times.
  • Even-order cumulants factorize completely into products of two-point functions.
  • Odd-order cumulants factorize into products of two-point functions and a single three-point function.

B. Numerical Verification

• Models: The predictions are tested on three non-integrable 1D spin models:
  1. Spin-1 Mixed-Field Ising Model (Main focus).
  2. Spin-1/2 Mixed-Field Ising Model.
  3. Floquet XXZ Model.
• Observables: Energy density ($\hat{h}_i$), local magnetization ($\hat{s}^z_i$), and energy current ($\hat{j}_i$).
• Methods:
  • Exact Diagonalization (ED): Used for small system sizes ($L \leq 19$).
  • Dynamical Quantum Typicality (DQT): Used for larger systems ($L$ up to 32) to access the thermodynamic limit and long-time dynamics. DQT allows the calculation of thermal expectation values using a single random pure state, significantly reducing computational cost.
• Analysis: The authors compute free cumulants $\kappa_n(t)$ and classical cumulants $r_n(t)$ (defined as the connected part after subtracting factorized terms) to verify power-law decays.

3. Key Contributions

1. Universal Hydrodynamic Description of ETH: The paper establishes that the smooth functions in the ETH framework are not arbitrary but are constrained by non-linear hydrodynamics at low frequencies.
2. Hierarchy of Free Cumulants: It demonstrates that ETH free cumulants obey a strict hierarchy where higher-order correlations decay faster than lower-order ones, governed by the scaling dimensions of the underlying hydrodynamic fields.
3. Factorization at Late Times: A central theoretical result is that at late times, even-order free cumulants asymptotically factorize into products of two-point functions, and odd-order ones factorize into two-point functions times a three-point function. This implies that the "new" information in higher-order ETH correlations is limited to the three-point function.
4. Bridging Free Probability and Hydrodynamics: The work provides a concrete physical mechanism (hydrodynamic tails) for the emergence of free probability structures in chaotic quantum dynamics.

4. Key Results

• Power-Law Scaling: Numerical simulations confirm the predicted power-law decays for free cumulants in the thermodynamic limit.
  • For energy density ($\hat{h}$) in 1D ($d=1$):
    • $\kappa_2(t) \sim t^{-1/2}$
    • $\kappa_3(t) \sim t^{-1}$
    • $\kappa_4(t) \sim (\kappa_2)^2 + t^{-3/2}$ (The connected part $r_4(t) \sim t^{-3/2}$).
  • For energy current ($\hat{j}$), which has a different scaling dimension, the exponents shift (e.g., $\kappa_2 \sim t^{-3/2}$), matching the hydrodynamic prediction based on the operator's symmetry.
• Finite-Size Effects: At very late times, the power-law decay crosses over to an exponential decay $\sim e^{-\Gamma t}$. The decay rate $\Gamma$ scales as $L^{-2}$, consistent with the global relaxation time (Thouless time) of a diffusive system.
• Temperature Dependence: The hydrodynamic scaling holds for both infinite temperature ($\beta=0$) and finite temperatures ($\beta > 0$).
• Agreement: There is excellent quantitative agreement between the hydrodynamic predictions and numerical data for various local observables and system sizes, validating the universality of the approach.

5. Significance

• Completing the ETH Framework: This work fills a critical gap in the ETH paradigm by determining the specific functional form of multi-point correlations, moving beyond the assumption of "smoothness" to a predictive "hydrodynamic" form.
• New Diagnostic for Chaos: The scaling of free cumulants provides a new, robust diagnostic for quantum chaos and thermalization, distinct from standard measures like level statistics or OTOCs.
• Experimental Relevance: Since hydrodynamic tails are universal, these predictions offer testable signatures for future experiments in cold atoms and quantum simulators, particularly in measuring higher-order correlation functions.
• Theoretical Unification: It unifies the description of thermalization in quantum systems (ETH) with the classical description of transport (Hydrodynamics), showing that the statistical properties of quantum eigenstates are deeply rooted in the macroscopic transport laws of the system.

In conclusion, the paper demonstrates that non-linear hydrodynamics dictates the structure of ETH correlations, providing a universal, predictive framework for the statistics of matrix elements in chaotic quantum systems.