Localized shocks

TL;DR

This paper investigates how products of localized operators evolve over time in chaotic systems, revealing a linear growth in spatial extent and establishing a geometric correspondence between tensor networks and shock waves in gravity duals.

Contribution

It introduces a generalized geometric correspondence between tensor network representations and localized shock waves in gravity duals, extending previous homogeneous case results.

Findings

Single precursor fills a region growing linearly with time

Products of precursors can be represented as tensor networks

Correspondence established between tensor networks and Einstein-Rosen bridges with shock waves

Abstract

We study products of precursors of spatially local operators, $W_{x_{n}}(t_{n}) ... W_{x_1}(t_1)$, where $W_x(t) = e^{-iHt} W_x e^{iHt}$. Using chaotic spin-chain numerics and gauge/gravity duality, we show that a single precursor fills a spatial region that grows linearly in $t$. In a lattice system, products of such operators can be represented using tensor networks. In gauge/gravity duality, they are related to Einstein-Rosen bridges supported by localized shock waves. We find a geometrical correspondence between these two descriptions, generalizing earlier work in the spatially homogeneous case.