Eigenstate Thermalization Hypothesis and Free Probability

TL;DR

This work unifies the Eigenstate Thermalization Hypothesis (ETH) with Free Probability theory by showing that higher-order ETH correlations are governed by thermal free cumulants associated with non-crossing partitions. Through a diagrammatic analysis, it derives explicit expressions for thermal free cumulants $k_q^{\beta}$ in terms of on-shell ETH correlations and demonstrates consistency of ETH under operator multiplication. It also establishes bounds on large-frequency decay of on-shell correlations and provides a detailed $q=4$ calculation illustrating the free-cumulant (cumulant-moment) structure. The results suggest that Free Probability offers a powerful, generalized framework for understanding quantum thermalization beyond Gaussian random-matrix models, with implications for multi-time correlations and information scrambling.

Abstract

Quantum thermalization is well understood via the Eigenstate Thermalization Hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a `typicality' argument of invariance with respect to local rotations involving nearby energy levels. In this work, we uncover the close relation between this perspective on ETH and Free Probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the Eigenstate Thermalization Hypothesis and the structure of Free Probability, widening considerably the latter's scope and highlighting its relevance to quantum thermalization.

Figures (5)

• Figure 1: Impact of the local rotational invariance of $A_{ij}$ on the correlations between three matrix elements. The operator $A$ in the energy eigenbasis is depicted as a random matrix with a band structure. To each matrix element $A_{ij}$ is associated with a "small" $U$ (box on the diagonal) which acts as a pseudo-random unitary matrix. Matrix elements with different indices (a) are characterized by different $U$ and their average vanishes. When the indices are repeated on a loop (b) the $U$ appear in pairs and yield a finite result.
• Figure 2: ETH diagrams (a) and non-crossing partitions (b) for $q=4$. (a1-6) Loop and cactus diagrams that contribute to ETH correlators. The arrow indicates the presence of a time dependence. With $\times [n]$ we indicate that there are $n$ cyclic permutations. (a7) Non-cactus diagram. (b1-6) Non-crossing partitions for $q=4$. Each of the blocks contributes with a free cumulant $k_{n}$, where $n$ is the number of $n$ points in that partition. For completeness, we also represent the crossing partition after the dashed line.
• Figure S1: Relation between ETH diagrams (a), non-crossing partitions (b) and their dual (c) for $q=4$.
• Figure S2: ETH consistency under multiplication and non-crossing partitions of Free Probability. If the operators $A$ obeys ETH, what can we say about $B = A^2$? The lattice non-crossing partitions associated to $A^4$ in (d) can be divided in two sublattices in (c). Once summed over, this yield the non-crossing partition of the operators $B$ (a, b).
• Figure S3: Relation between dual non-crossing partitions, ETH diagrams and thermal free cumulants for $q=4$ with $k_1=0$.