From Lyapunov Growth to Weak Quantum Chaos
Abstract
Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator Cr(t), its spatial sum C̅(t) = ∑r Cr(t), and the symmetrically regularised connected dOTOC D(t) of two extensive observables. We show that Lieb--Robinson locality gives C̅(t) ≲ td. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, D(t) ≲ td, improving our earlier suboptimal t3d bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates D(t) to C̅(t) and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.
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BibTeX
@misc{grozdanov2026from,
title = {{From Lyapunov Growth to Weak Quantum Chaos}},
author = {Sašo Grozdanov and Tomaž Prosen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15880},
eprint = {2610.15880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15880},
}
RIS
TY - UNPB
TI - From Lyapunov Growth to Weak Quantum Chaos
AU - Sašo Grozdanov
AU - Tomaž Prosen
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15880
UR - https://arxiv.org/abs/2610.15880
AB - Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator C_r(t), its spatial sum C̅(t) = ∑_r C_r(t), and the symmetrically regularised connected dOTOC D(t) of two extensive observables. We show that Lieb--Robinson locality gives C̅(t) ≲ t^d. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, D(t) ≲ t^d, improving our earlier suboptimal t^3d bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates D(t) to C̅(t) and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.
ER -