Lieb-Robinson and the butterfly effect

TL;DR

The work links the Lieb-Robinson bound to the butterfly effect in strongly coupled quantum systems by introducing the butterfly velocity $v_B$ as a state-dependent, low-energy analogue of the Lieb-Robinson velocity. Using holographic methods, it derives $v_B$ for hyperscaling-violating geometries as $v_B = \left(\frac{\beta_0}{\beta}\right)^{1-1/z} \sqrt{\frac{d+z-\theta}{2(d-\theta)}}$, showing $v_B$ depends on IR data ($z,\theta$) and temperature, with $v_B$ increasing with $T$ for $z\ge1$. The analysis is complemented by free-fermion calculations that exhibit UV-sensitive, shell-like growth in the anti-commutator and a temperature-dependent effective velocity $v_{k_0} \sim T^{1-1/z}$ for low-energy excitations. TheAppendix D discussion links small commutators to bounded signaling speeds, highlighting $v_B$ as a quantitative bound on information propagation in low-energy dynamics, with prospects for experimental probing via out-of-time-ordered correlators.

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.