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  • Open Access

Free Cumulants in Full Eigenstate Thermalization from Boundary Scrambling

Felix Fritzsch1,*, Gabriel O. Alves1,*,†, Michael A. Rampp1, and Pieter W. Claeys1,2

  • *These authors contributed equally to this work.
  • †Contact author: alves.go.co@gmail
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PRX Quantum 7, 043003 – Published 5 October, 2026

DOI: https://doi.org/10.1103/6n9v-b3b7

Abstract

Out-of-time-order correlation functions (OTOCs) and their higher-order generalizations present important probes of quantum information dynamics and scrambling. We introduce a solvable many-body quantum model, which we term boundary scrambling, for which the full dynamics of higher-order OTOCs is analytically tractable. Following recent extensions of the eigenstate thermalization hypothesis, these dynamics are encoded in free cumulants describing higher-order correlations between matrix elements. Using this correspondence, we obtain analytical predictions for time- and frequency-resolved free cumulants and show these to be stable away from the solvable point. The solvability is enabled by the identification of a higher-order Markovian influence matrix, capturing the effect of the full system on a local subsystem. These results provide insight into the emergence of random-matrix behavior from structured Floquet dynamics and show how techniques from free probability can be applied in the construction of exactly solvable many-body models.

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Popular Summary

A puzzling question for physicists is how an isolated quantum system locally relaxes to equilibrium. The eigenstate thermalization hypothesis (ETH) is currently one of the main tools available to understand the emergence of statistical physics in this setting. Motivated by random matrix theory, ETH has recently been extended to more refined predictions for this thermalization process in such systems, as probed by higher-order correlators. However, it remains an open question how structured quantum dynamics leads to emergent random-matrix behavior. In this work, we introduce a solvable quantum circuit model which can be reduced to an effective bath picture via tensor network techniques that enable exact calculations. This construction leads to new analytical predictions for such higher-order correlators, which establish emergent random-matrix behavior in a highly structured quantum model. Our work paves the way for further studies at the interface between thermalization and solvable models of quantum many-body dynamics.

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PRX Quantum 7, 043003– Published 5 October, 2026
  • Received 8 October 2025
  • Revised 30 June 2026
  • Accepted 2 September 2026
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