A quantum hydrodynamical description for scrambling and many-body chaos

TL;DR

This paper introduces a quantum hydrodynamics framework to unify the description of chaos and operator spreading in many-body quantum systems, capturing exponential growth and butterfly spreading of OTOCs.

Contribution

It formulates a novel quantum hydrodynamics theory valid at finite 5, explaining chaos phenomena through a shift symmetry and pole-skipping in correlation functions.

Findings

Hydrodynamic cloud buildup describes operator scrambling.

Shift symmetry explains exponential growth of OTOCs.

Pole-skipping indicates hydrodynamic origin of chaos.

Abstract

Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit calculations in specific models. In this paper we propose a unified description of the exponential growth and ballistic butterfly spreading of OTOCs across different systems using a newly formulated "quantum hydrodynamics," which is valid at finite $\hbar$ and to all orders in derivatives. The scrambling of a generic few-body operator in a chaotic system is described as building up a "hydrodynamic cloud," and the exponential growth of the cloud arises from a shift symmetry of the hydrodynamic action. The shift symmetry also shields correlation functions of the energy density and flux, and time ordered correlation functions of generic operators from exponential growth, while leads to chaotic behavior in OTOCs. The theory also predicts an interesting phenomenon of the skipping of a pole at special values of complex frequency and momentum in two-point functions of energy density and flux. This pole-skipping phenomenon may be considered as a "smoking gun" for the hydrodynamic origin of the chaotic mode. We also discuss the possibility that such a hydrodynamic description could be a hallmark of maximally chaotic systems.