Scrambling and Lyapunov Exponent in Unitary Networks with Tunable Interactions

TL;DR

This paper investigates the conditions for exponential information scrambling in quantum many-body systems with local interactions, using random unitary circuits to demonstrate a prolonged scrambling regime linked to tunable interactions.

Contribution

It establishes a general criterion for exponential growth of the OTOC in spatially extended systems and demonstrates this in a tunable random circuit model.

Findings

Exponential growth regime exists when butterfly velocity exceeds Lyapunov exponent times microscopic length.

Weakly interacting limit satisfies the exponential growth criterion.

Numerical simulations confirm prolonged scrambling window in the model.

Abstract

Scrambling of information in a quantum many-body system, quantified by the out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos. A regime of exponential growth in the OTOC, characterized by a Lyapunov exponent, has so far mostly been observed in systems with a high-dimensional local Hilbert space and in weakly-coupled systems. Here, we propose a general criterion for the existence of a well-defined regime of exponential growth of the OTOC in spatially extended systems with local interactions. In such systems, we show that a parametrically long period of exponential growth requires the butterfly velocity to be much larger than the Lyapunov exponent times a microscopic length scale, such as the lattice spacing. As an explicit example, we study a random unitary circuit with tunable interactions. In this model, we show that in the weakly interacting limit the above criterion is satisfied, and there is a prolonged window of exponential growth. Our results are based on numerical simulations of both Clifford and universal random circuits supported by an analytical treatment.