Hydrodynamic theory of scrambling in chaotic long-range interacting systems

TL;DR

This paper develops a hydrodynamic mean-field theory using a fractional FKPP equation to describe scrambling in chaotic quantum systems with power-law interactions, accounting for quantum fluctuations and reproducing butterfly light cone scalings.

Contribution

It introduces a renormalized fractional FKPP equation incorporating quantum fluctuations to accurately model scrambling in long-range interacting chaotic systems.

Findings

The fractional index er in the FKPP equation is renormalized from 2lpha - 1 to 2lpha - 2.

The effective theory reproduces previously observed butterfly light cone scalings.

Numerical and analytic evidence supports the cutoff and noise-modified fractional FKPP model.

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 na\"ive 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.