Kinetics of information scrambling in correlated electrons: disorder-driven transition from shock-wave to FKPP dynamics

Quenched disorder slows down the scrambling of quantum information. Using a bottom-up approach, we formulate a kinetic theory of scrambling in a correlated metal near a superconducting transition, following the scrambling dynamics as the impurity scattering rate is increased. Within this framework, we rigorously show that the butterfly velocity $v$ is bounded by the light cone velocity $v_{\mathrm{l}\mathrm{c}}$ set by the Fermi velocity. We analytically identify a disorder-driven dynamical transition occurring at small but finite disorder strength between a spreading of information characterized at late times by a discontinuous shock wave propagating at the maximum velocity $v_{\mathrm{l}\mathrm{c}}$, and a smooth traveling wave belonging to the Fisher or Kolmogorov-Petrovsky-Piskunov (FKPP) class and propagating at a slower, if not considerably slower, velocity $v$. In the diffusive regime, we establish the relation $v^2\mathrm{/}{\lambda }_{\mathrm{F}\mathrm{K}\mathrm{P}\mathrm{P}}\sim D_{\mathrm{e}\mathrm{l}}$ where ${\lambda }_{\mathrm{F}\mathrm{K}\mathrm{P}\mathrm{P}}$ is the Lyapunov exponent set by the inelastic scattering rate and $D_{\mathrm{e}\mathrm{l}}$ is the elastic diffusion constant.