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We study scrambling, an avatar of chaos, in a weakly interacting metal in thepresence of random potential disorder. It is well known that charge and heatspread via diffusion in such an interacting disordered metal. In contrast, weshow within perturbation theory that chaos spreads in a ballistic fashion. Thesquared anticommutator of the electron field operators inherits a light-conelike growth, arising from an interplay of a growth (Lyapunov) exponent thatscales as the inelastic electron scattering rate and a diffusive piece due tothe presence of disorder. In two spatial dimensions, the Lyapunov exponent isuniversally related at weak coupling to the sheet resistivity. We are able todefine an effective temperature-dependent butterfly velocity, a speed limit forthe propagation of quantum information, that is much slower than microscopicvelocities such as the Fermi velocity and that is qualitatively similar to thatof a quantum critical system with a dynamical critical exponent $z > 1$.

Quantum butterfly effect in weakly interacting diffusive metals