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A Strategy for Proving the Strong Eigenstate Thermalization Hypothesis : Chaotic Systems and Holography

Taishi Kawamoto

TL;DR

This work provides sufficient conditions for the strong ETH in highly chaotic theories by linking long-time real-time thermal correlators to probabilistic bounds that decay doubly exponentially with entropy. The key mechanism relies on clustering of Wightman functions and quantum mixing of two-point functions, together enabling sub-Gaussian tail bounds and, in turn, double-exponential suppression of diagonal and off-diagonal ETH violations. The authors illustrate the approach with bottom-up holographic toy models, including large-$N$ generalized free fields and controlled $1/N$ corrections, and discuss how wormhole contributions and back-reaction may affect late-time behavior. They also connect these results to the standard ETH ansatz and discuss energy-window considerations, offering a framework to understand thermalization in chaotic quantum many-body systems and their gravity duals. The findings advance a quantitative pathway to proving strong ETH in holographic contexts and suggest robust thermalization mechanisms in theories with gravity duals.

Abstract

The strong eigenstate thermalization hypothesis (ETH) provides a sufficient condition for thermalization and equilibration. Although it is expected to be hold in a wide class of highly chaotic theories, there are only a few analytic examples demonstrating the strong ETH in special cases, often through methods related to integrability. In this paper, we explore sufficient conditions for the strong ETH that may apply to a broad range of chaotic theories. These conditions are expressed as inequalities involving the long-time averages of real-time thermal correlators. Specifically, as an illustration, we consider simple toy examples which satisfy these conditions under certain technical assumptions. This toy models have same properties as holographic theories at least in the perturbation in large $N$. We give a few comments for more realistic holographic models.

A Strategy for Proving the Strong Eigenstate Thermalization Hypothesis : Chaotic Systems and Holography

TL;DR

This work provides sufficient conditions for the strong ETH in highly chaotic theories by linking long-time real-time thermal correlators to probabilistic bounds that decay doubly exponentially with entropy. The key mechanism relies on clustering of Wightman functions and quantum mixing of two-point functions, together enabling sub-Gaussian tail bounds and, in turn, double-exponential suppression of diagonal and off-diagonal ETH violations. The authors illustrate the approach with bottom-up holographic toy models, including large- generalized free fields and controlled corrections, and discuss how wormhole contributions and back-reaction may affect late-time behavior. They also connect these results to the standard ETH ansatz and discuss energy-window considerations, offering a framework to understand thermalization in chaotic quantum many-body systems and their gravity duals. The findings advance a quantitative pathway to proving strong ETH in holographic contexts and suggest robust thermalization mechanisms in theories with gravity duals.

Abstract

The strong eigenstate thermalization hypothesis (ETH) provides a sufficient condition for thermalization and equilibration. Although it is expected to be hold in a wide class of highly chaotic theories, there are only a few analytic examples demonstrating the strong ETH in special cases, often through methods related to integrability. In this paper, we explore sufficient conditions for the strong ETH that may apply to a broad range of chaotic theories. These conditions are expressed as inequalities involving the long-time averages of real-time thermal correlators. Specifically, as an illustration, we consider simple toy examples which satisfy these conditions under certain technical assumptions. This toy models have same properties as holographic theories at least in the perturbation in large . We give a few comments for more realistic holographic models.

Paper Structure

This paper contains 30 sections, 6 theorems, 193 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Short Review of Strong and Weak ETH
  3. Initial State Independent Upper Bound: Strong ETH
  4. Initial State Dependent Upper Bound
  5. Sufficient Condition for the Double Exponential Scaling
  6. Concentration Inequality and Sub-Gaussian Properties
  7. Sub-Gaussian Random Variables: A Review
  8. Quadratic Generalization of sub-Gaussian variables
  9. Almost Sub-Gaussianity of $\Delta_{\mathrm{diag,ETH}}^{\epsilon}\qty[\mathcal{O}]$ and $\Delta_{\mathrm{off,ETH}}^{\epsilon}\qty[\mathcal{O}]$
  10. Ensemble Equivalence
  11. Off Diagonal Elements
  12. Examples: Bottom-Up Holographic Theories
  13. Holographic Theories and Large $N$ Generalized Free Field Theories
  14. Clustering Property
  15. Quantum Mixing
  16. ...and 15 more sections

Key Result

Theorem 1

For any $\epsilon>0$, for initial state $\ket{\psi_0}=\sum_{\ket{E_n}\in\mathcal{H}_{E,\Delta E}} c_n \ket{E_n}$ and bounded operator $\mathcal{O} \in \mathcal{B}(\mathcal{H})$, it holds that where $\lVert \mathcal{O} \rVert_{\rm{op}}$ denotes the operator norm $\lVert \mathcal{O} \rVert_{\rm{op}} := \sup_{\ket{\psi}\in\mathcal{H}}\sqrt{\frac{\bra{\psi}\mathcal{O}^\dag \mathcal{O} \ket{\psi}}

Figures (13)

  • Figure 1: Sketch of the Witten diagram for the four-point function. We denote $K$ as the bulk-to-boundary propagator, and $G$ denotes the boundary correlators. $\Gamma_4$ represents the sum of the 1PI diagrams for the four-point vertex.
  • Figure 2: Sketch of the 1PI diagram $\Gamma_4$ in the perturbation series of $g_4$.
  • Figure 3: The picture for the energy injection changes the shape of the black horizon. The yellow region corresponds to the shock wave like fluid by the huge number of the operators. We compute the two point function of the probe operators far above the yellow region.
  • Figure 4: Sketch of the wormhole contribution to the for point function. The red line corresponds to the worldline whose contribution to the correlation functions are not decaying.
  • Figure 5: The plots of LHS and RHS with $M=10,y=0.1,m_{\max}=10$.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2: Equilibration
  • Definition 3: Thermalization
  • Definition 4: Strong ETH
  • Theorem 1
  • Theorem 2
  • Definition 5
  • Proposition 3
  • Conjecture 4
  • Theorem 5
  • ...and 4 more