Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves

TL;DR

This paper develops a microscopic model for the quantum butterfly effect, linking out-of-time-order correlators to combustion wave equations, revealing a propagating decoherence wave in quantum many-body systems.

Contribution

It extends the Keldysh technique to compute out-of-time-order correlators and connects their behavior to nonlinear combustion wave equations.

Findings

Out-of-time-order correlators exhibit exponential instability followed by linear propagation.

Decoherence propagation is described by a nonlinear diffusion equation with dissipation.

The propagating wave maintains a constant velocity despite diffusive dynamics.

Abstract

We extend the Keldysh technique to enable the computation of out-of-time order correlators. We show that the behavior of these correlators is described by equations that display initially an exponential instability which is followed by a linear propagation of the decoherence between two initially identically copies of the quantum many body systems with interactions. At large times the decoherence propagation (quantum butterfly effect) is described by a diffusion equation with non-linear dissipation known in the theory of combustion waves. The solution of this equation is a propagating non-linear wave moving with constant velocity despite the diffusive character of the underlying dynamics. Our general conclusions are illustrated by the detailed computations for the specific models describing the electrons interacting with bosonic degrees of freedom (phonons, two-level-systems etc.) or with each other.