Traveling discontinuity at the quantum butterfly front

TL;DR

This paper develops a kinetic theory for quantum information scrambling near a superconducting transition, revealing maximal speed propagation, exponential growth, and shock-wave dynamics with a traveling discontinuity at the light cone boundary.

Contribution

It introduces a set of coupled PDEs describing information spreading in interacting electrons, highlighting shock-wave behavior and causal dynamics in quantum scrambling.

Findings

Scrambling propagates at the Fermi velocity.

Exponential growth occurs at early times.

Discontinuity appears at the light cone boundary.

Abstract

We formulate a kinetic theory of quantum information scrambling in the context of a paradigmatic model of interacting electrons in the vicinity of a superconducting phase transition. We carefully derive a set of coupled partial differential equations that effectively govern the dynamics of information spreading in generic dimensions. Their solutions show that scrambling propagates at the maximal speed set by the Fermi velocity. At early times, we find exponential growth at a rate set by the inelastic scattering. At late times, we find that scrambling is governed by shock-wave dynamics with traveling waves exhibiting a discontinuity at the boundary of the light cone. Notably, we find perfectly causal dynamics where the solutions do not spill outside of the light cone.