Hydrodynamic theory of scrambling in chaotic long-range interacting systems

The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a mean-field theory of out-of-time-ordered commutators in locally interacting quantum chaotic systems at high energy density. In systems with power-law interactions, the corresponding fractional-derivative FKPP equation provides an analogous mean-field theory. However, the fractional FKPP description is potentially subject to strong quantum fluctuation effects, so it is not clear a priori if it provides a suitable effective description for generic chaotic systems with power-law interactions. Here we study this problem using a model of coupled quantum dots with interactions decaying as $1/{r}^{\ensuremath{\alpha}}$, where each dot hosts $N$ degrees of freedom. The large-$N$ limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by $1/N$ corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for $N=1$ and generic finite $N$. In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.