Localized shocks

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.

23 pages plus appendices, 12 figures. v2: minor error in Appendix B corrected. v3: figure added to the introduction comparing the butterfly effect cone with the standard light cone

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