Operator Spreading, Duality, and the Noisy Long-Range FKPP Equation

TL;DR

This paper explores the growth of operator size in quantum chaos, linking discrete stochastic models and a noisy long-range FKPP equation, and demonstrates their equivalence through mathematical duality and numerical analysis.

Contribution

It introduces a duality between discrete and continuum models of operator spreading, clarifies the behavior of the noisy long-range FKPP equation, and provides numerical evidence of their equivalence.

Findings

Superlinear butterfly light cones with stretched exponential or power-law scaling.

Numerical agreement between discrete and continuum models in light cone scaling.

Demonstration of the duality via an intermediate model with no finite size effects.

Abstract

Operator spreading provides a new characterization of quantum chaos beyond the semi-classical limit. There are two complementary views of how the characteristic size of an operator, also known as the butterfly light cone, grows under chaotic quantum time evolution: A discrete stochastic population dynamics or a stochastic reaction-diffusion equation in the continuum. When the interaction decays as a power function of distance, the discrete population dynamics model features superlinear butterfly light cones with stretched exponential or power-law scaling. Its continuum counterpart, a noisy long-range Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation, remains less understood. We use a mathematical duality to demonstrate their equivalence through an intermediate model, which replaces the hard local population limit by an equilibrium population. Through an algorithm with no finite size effect, we demonstrate numerically remarkable agreements in their light cone scalings.