# Weak Quantum Chaos

## TL;DR

This paper introduces the concept of weak quantum chaos, characterized by polynomial growth of OTOC density in non-integrable quantum systems, and demonstrates its properties through numerical and analytical methods.

## Contribution

It proposes the OTOC density as a more effective measure of quantum chaos in bounded local operators and establishes its polynomial growth in non-integrable systems.

## Key findings

- OTOC density grows at most polynomially in time in non-integrable systems
- In integrable cases, OTOC density saturates to a plateau
- Numerical and analytical results support the weak quantum chaos concept

## Abstract

Out-of-time-ordered correlation functions (OTOC's) are presently being extensively debated as quantifiers of dynamical chaos in interacting quantum many-body systems. We argue that in quantum spin and fermionic systems, where all local operators are bounded, an OTOC of local observables is bounded as well and thus its exponential growth is merely transient. As a better measure of quantum chaos in such systems, we propose, and study, the density of the OTOC of extensive sums of local observables, which can exhibit indefinite growth in the thermodynamic limit. We demonstrate this for the kicked quantum Ising model by using large-scale numerical results and an analytic solution in the integrable regime. In a generic case, we observe the growth of the OTOC density to be linear in time. We prove that this density in general, locally interacting, non-integrable quantum spin and fermionic dynamical systems exhibits growth that is at most polynomial in time---a phenomenon, which we term weak quantum chaos. In the special case of the model being integrable and the observables under consideration quadratic, the OTOC density saturates to a plateau.