Scrambling and quantum chaos indicators from long-time properties of operator distributions

TL;DR

This paper investigates quantum scrambling and chaos by analyzing long-time properties of operator distributions across various models, offering a refined perspective beyond traditional OTOC measures.

Contribution

It introduces a novel approach to characterize operator scrambling through the structure of operator expansion coefficients over time, linking it to quantum chaos indicators.

Findings

Long-time operator distribution features are consistent across different models.

Operator distribution properties can serve as proxies for quantum chaos onset.

Analyzes the experimental cost of probing operator distributions via OTOCs.

Abstract

Scrambling is a key concept in the analysis of nonequilibrium properties of quantum many-body systems. Most studies focus on its characterization via out-of-time-ordered correlation functions (OTOCs), particularly through the early-time decay of the OTOC. However, scrambling is a complex process which involves operator spreading and operator entanglement, and a full characterization requires one to access more refined information on the operator dynamics at several timescales. In this work we analyze operator scrambling by expanding the target operator in a complete basis and studying the structure of the expansion coefficients treated as a coarse-grained probability distribution in the space of operators. We study different features of this distribution, such as its mean, variance, and participation ratio, for the Ising model with longitudinal and transverse fields, kicked collective spin models, and random circuit models. We show that the long-time properties of the operator distribution display common features across these cases, and discuss how these properties can be used as a proxy for the onset of quantum chaos. Finally, we discuss the connection with OTOCs and analyze the cost of probing the operator distribution experimentally using these correlation functions.