Quantum many-body operator cascade as a route to chaos

TL;DR

This paper explores quantum many-body chaos by analyzing the fractal structure of operator evolution, revealing a cascade process that leads to effective relaxation and non-locality in quantum systems.

Contribution

It introduces a spectral analysis of the operator propagator to identify fractal structures and links unitarity constraints to operator non-locality in quantum chaos.

Findings

Leading Ruelle-Pollicott eigenvectors have fractal dimensions indicating non-locality.

Unitarity constrains the decay rate of local correlations and operator fractality.

Results are validated in various quantum circuit models, including dual-unitary circuits.

Abstract

Dynamical properties of classical chaotic systems, for instance relaxation, can be understood as emerging from the time evolution of initially smooth long-wavelength densities to ever finer short-wavelength densities with fractal structure. Whether there is any analogous fractality by which one could characterize quantum many-body chaos is not known. By studying the spectral properties of the truncated operator propagator, we provide such structures. Namely, we show that the slowest-decaying operators, i.e., the leading Ruelle-Pollicott eigenvectors, have a nontrivial fractal dimension quantifying their non-locality, visible also in the divergence of their condition numbers. Furthermore, we find that unitarity imposes a constraint, i.e., an (approximate) equality, between the temporal decay rate of local correlations and this spatial operator fractal dimension. With this insight, a scenario for many-body quantum chaos becomes clear: over time, local operators evolve towards increasingly non-local ones with a quantifiable fractal structure, thereby naturally leading to effective non-unitary relaxation on the subspace of local operators - a kind of many-body Kolmogorov cascade in the space of operators. Our predictions are demonstrated in various quantum circuits: the kicked Ising model, brickwall circuits with a random 2-qubit gate, and dual-unitary circuits, where our results are exact.