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

TL;DR

The paper develops an augmented Keldysh formalism to compute out-of-time-ordered correlators (OTOCs) in microscopic quantum models, capturing both the local exponential growth and the subsequent spread of decoherence between two initially correlated worlds.A kinetic-equation framework is derived for both diagonal (ordinary distribution) and off-diagonal (inter-world coherence) components, revealing that off-diagonal instabilities lead to a nonlinear diffusion equation with a combustion-wave-like front that propagates at a finite velocity $v_{cw}$.In electron-phonon and electron-electron systems, the instability generates a propagating coherence-decoherence front whose speed is sub-Fermi-velocity and is set by the diffusion coefficient $D_*$ and nonlinear time scale $t_*$; impurity scattering can slow the front, while impurity scattering alone is stabilizing for the off-diagonal sector.The results imply that quantum information scrambling in these systems proceeds via a laminar, front-like spread of decoherence, reconciling diffusive transport with ballistic entanglement propagation, and suggest extensions to many-body localization and related non-equilibrium phenomena.

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.

Figures (8)

• Figure 1: The traditional, $\mathcal{C}_{K}$, (a) and the augmented (b) Keldysh contours $\mathcal{C}_{aK}$. Times $t_{_{1,2}}$ label the insertion of the operators for observable or computable quantities, see text. Operators (fermionic or bosonic) are ordered according their location on the contours $\mathcal{C}_{K}$, $\mathcal{C}_{aK}$.
• Figure 2: Definition of the basic elements of the diagrammatic technique. (a) The lines (thick and thin) describe the exact and bare Green functions respectively. (b) The box describes the matrix structure of the vertices. (c) The vertices which do not conserve the number of particles (for example absorption and emission of the phonons or photons).
• Figure 3: Diagrammatic expressions for the correlators (\ref{['eq:N_rhorho']}) and (\ref{['eq:A_rhorho']}) in non-interacting problem.
• Figure 4: Vertex structure for (a) electron-electron interaction; and (b) for the electron-phonon interaction. The outgoing or ingoing vertical arrow in (b) can be any bosonic line including phonons, photons, impurity potential etc. The short dashed lines on panel (b) are the delta function in space and time.
• Figure 5: Remaining basic elements of the diagram technique for electron-phonon a) and electron-impurity b) interactions. In (a) we include the interaction constant $\lambda$into definition of the propagator to keep the vertices of Fig. \ref{['fig:Vertex-structure-for-electron-electron']} intact. Correlation function $V(r)$ describes the fluctuations due to the weak random impurities.