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Chaotic Many-Body Quantum Dynamics, Spectral Correlations, and Energy Diffusion

J. T. Chalker and Dominik Hahn

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Phys. Rev. X 16, 031076 – Published 23 September, 2026

DOI: https://doi.org/10.1103/bjj5-49pg

Abstract

We study chaotic many-body quantum dynamics in a minimal model with spatial structure and local interactions. It has a time-independent Hamiltonian, in contrast to quantum circuits and Brownian models, and is simple at the single-site level, in contrast to Sachdev-Ye-Kitaev chains. It is analytically tractable for large local Hilbert space dimension and weak intersite coupling. In this limit, we show that energy dynamics is described by a classical master equation and is diffusive. We also show that the spectral form factor can be expressed exactly in terms of the solution to this master equation. For a two-site system, we obtain closed-form expressions for both the two-point correlator of energy density and the spectral form factor, in essentially perfect agreement with numerical simulations. For an L-site system we show at late times how a linear ramp emerges in the spectral form factor, as universally expected from level repulsion in chaotic quantum systems. Conversely, at earlier times we identify two distinct mechanisms for an increase of the spectral form factor above its ramp value. One of these is associated with energy diffusion and is effective until the Thouless time, which varies as L2. The other involves contributions like those that would appear if the system were composed of many uncoupled subsystems: They generate a large enhancement of the spectral form factor and are suppressed on a timescale varying as (lnL)2. Besides being exact for the limit considered, we believe our approach provides the natural approximation even for small local Hilbert space dimension and strong intersite coupling. We present a numerical study of a spin-half chain, finding an early-time enhancement of the spectral form factor which is qualitatively similar to that in our solvable model.

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The behavior of generic many-body quantum systems (such as a chain of coupled spins) is hard to treat theoretically, because exact solutions are not usually possible and numerical simulations are limited to small system sizes. In this paper we introduce a model that can be solved analytically by averaging over an ensemble of similar systems, and that is expected to be representative more generally of systems like spin chains. We study two aspects of this model: the dynamics of energy density, and the statistical properties of the energy levels of the Hamiltonian. We demonstrate that energy spreads diffusively, which is expected but hard to establish in general. We also show that the energy-level correlations are intimately linked to the energy dynamics, and combine features specific to spatially extended systems with other features that are universal for all quantum chaotic systems. We expect that the model and approach we have introduced will be the starting point for future studies of quantum chaos in spatially extended many-body systems.

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Phys. Rev. X 16, 031076– Published 23 September, 2026
  • Received 19 December 2025
  • Revised 19 June 2026
  • Accepted 25 August 2026
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