Lieb-Robinson and the butterfly effect

TL;DR

This paper explores the relationship between the Lieb-Robinson bound and the butterfly effect in quantum systems, showing how the butterfly velocity acts as a state-dependent effective Lieb-Robinson velocity, especially in strongly-coupled theories.

Contribution

It demonstrates that the butterfly velocity can be viewed as a state-dependent Lieb-Robinson velocity and analyzes its behavior across various quantum field theories using holography.

Findings

Butterfly velocity depends on temperature and decreases towards the IR.

In strongly-coupled theories, $v_B$ differs from the traditional Lieb-Robinson velocity.

The study connects quantum chaos, information propagation, and bounds in quantum many-body systems.

Abstract

As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound---the Lieb-Robinson bound---and the butterfly effect in strongly-coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with the "butterfly" velocity $v_B$. Similarly, the Lieb-Robinson velocity places a state independent ballistic upper bound on the size of time evolved operators in non-relativistic lattice models. Here, we argue that $v_B$ is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with free particle computations to understand the role of strong coupling. We find that, depending on the way length and time scale, $v_B$ acquires a temperature dependence and decreases towards the IR. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling.