Hydrodynamic theory of scrambling in chaotic long-range interacting systems

Abstract

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 the 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 $\frac{1}{r^{\alpha }}$, 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 $\frac{1}{N}$ 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 that we previously found for $N=1$ and generic finite $N$. In order to reproduce these scalings, the fractional index $\mu$ in the FKPP equation needs to be shifted from the naïve value of $\mu =2\alpha -1$ to a renormalized value $\mu =2\alpha -2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.

17+1 pages, 14 figures

References in corpus (2)

• Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves
• The acceleration of evolutionary spread by long-range dispersal

Cited by in corpus (5)

• Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions
• Reaction-diffusive dynamics of number-conserving dissipative quantum state preparation
• Instability of the engineered dark state in two-band fermions under number-conserving dissipative dynamics
• Kinetics of information scrambling in correlated electrons: disorder-driven transition from shock-wave to FKPP dynamics
• Mechanisms of scrambling and unscrambling of quantum information in the ground state in spin chains: domain-walls, spin-flips and scattering phase shifts