Matrix models for eigenstate thermalization

TL;DR

This work develops an ETH matrix-model framework that couples energy-level statistics to generalized, non-Gaussian ETH correlations via a two-matrix ensemble, enabling a unified description of quantum chaos and gravity. By introducing higher-order couplings $G^{(n)}$ and a carefully tuned potential $V(H,\mathcal{O})$, the authors reproduce thermal mean-field theory and extend to JT gravity with scalar matter, matching disk correlators and cylinder amplitudes through a double-scaling limit. They show that non-Gaussian ETH corrections are essential for crossing symmetry and higher-point functions, and they implement this through regulators (Selberg and $q$-deformed) and a constraint-squared potential, achieving control over disk data and conjecturally higher-genus amplitudes. The framework suggests a principled, topological interpretation of ETH anomalies via the $e^{S_0}$ expansion and offers a path to embedding quantum-chaotic dynamics into gravitational path integrals, potentially extending to higher dimensions. Overall, the ETH matrix model bridges random-matrix theory, ETH, and low-dimensional gravity, providing concrete prescriptions to encode non-Gaussian correlations and to reproduce JT+matter physics across topologies.

Abstract

We develop a class of matrix models which implement and formalize the `eigenstate thermalization hypothesis' (ETH) and point out that in general these models must contain non-Gaussian corrections, already in order to correctly capture thermal mean-field theory, or to capture non-trivial OTOCs as well as their higher-order generalizations. We develop the framework of these `ETH matrix models', and put it in the context of recent studies in statistical physics incorporating higher statistical moments into the ETH ansatz. We then use the `ETH matrix model' in order to develop a matrix-integral description of JT gravity coupled to a single scalar field in the bulk. This particular example takes the form of a double-scaled ETH matrix model with non-Gaussian couplings matching disk correlators and the density of states of the gravitational theory. Having defined the model from the disk data, we present evidence that the model correctly captures the JT+matter theory with multiple boundaries, and conjecturally at higher genus. This is a shorter companion paper to the work [1], serving both as a guide to the much more extensive material presented there, as well as developing its underpinning in statistical physics.

Figures (1)

• Figure 1: Topology of the ETH matrix model. The left panel shows a planar diagram contributing to the two-point function of operators \ref{['eq.ETHMatrixModelTwoPoint']} which, un-normalized, gives an answer of order $e^{S_0}$. Any non-planar diagram, exemplified by the diagram on the right, contributes at lower-order in the $e^{S_0}$ expansion, the example above contributing at ${\cal O}\left( e^{-S_0} \right)$. The order of an arbitrary diagram is determined by its Euler characteristic $e^{\chi S_0 }$ when viewing its double-line representation as a triangulation of a 2D Riemann surface. This well-known feature of matrix models is the reason why non-Gaussian contributions in the matrix model action are needed for the ETH matrix model in order to reproduce basic facts even about thermal mean-field theory.