Saddle-point scrambling without thermalisation

TL;DR

This paper investigates how to differentiate between chaos-induced and saddle point-driven growth in out-of-time-order correlators (OTOCs) in quantum systems, using the Dicke model and driven Bose-Hubbard dimer, highlighting long-term behaviour and entanglement entropy as key markers.

Contribution

The study introduces methods to distinguish chaos from saddle point effects in OTOC growth by analyzing long-term dynamics and entanglement entropy in specific quantum models.

Findings

Chaotic systems show steady long-term OTOC growth, saddle points cause oscillations.

Entanglement entropy in chaotic systems follows a Page curve.

Long-term behaviour and entropy provide markers beyond initial OTOC growth.

Abstract

Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalisation in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analysing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behaviour. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behaviour in quantum systems, beyond the initial exponential growth in OTOCs.