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                <term xml:lang="en">Lyapunov exponent</term>
                <term xml:lang="en">superconductivity</term>
                <term xml:lang="en">metal</term>
                <term xml:lang="en">quantum information</term>
                <term xml:lang="en">inelastic scattering</term>
                <term xml:lang="en">butterfly velocity</term>
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              <p>Quenched disorder slows down the scrambling of quantum information. We formulate a kinetic theory of scrambling in a d-dimensional strongly-correlated metal in the vicinity of a superconducting phase, 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 lc = vF/ √ d where vF is 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 lc , 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. In the diffusive regime, we establish the relation v 2 /λFKPP ∼ D el where λFKPP is the Lyapunov exponent set by the inelastic scattering rate and D el is the elastic diffusion constant.</p>
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