Entanglement Tsunami in (1+1)-Dimensions

TL;DR

The paper investigates how entanglement entropy evolves after a global quench in (1+1)-D CFTs at large $c$ with sparse spectra, contrasting the free quasiparticle picture against holographic calculations. It shows that while a simple quasiparticle model can capture single-interval growth, it fails for disjoint intervals, where holography yields a non-decreasing entropy. The authors adopt the entanglement tsunami as a practical rule that reproduces the holographic time dependence for one or two intervals and provides a meaningful upper bound for many intervals. They connect these results to the large-$c$ vacuum and post-quench correlators via the replica trick and BCFT techniques, highlighting how identity-block dominance yields the holographic RT/HRT results. The work also outlines extensions to higher dimensions and discusses the potential role of interacting quasiparticles as a future direction.

Abstract

We study the time dependence of the entanglement entropy of disjoint intervals following a global quantum quench in (1+1)-dimensional CFTs at large-$c$ with a sparse spectrum. The result agrees with a holographic calculation but differs from the free field theory answer. In particular, a simple model of free quasiparticle propagation is not adequate for CFTs with a holographic dual. We elaborate on the entanglement tsunami proposal of Liu and Suh and show how it can be used to reproduce the holographic answer.

Figures (6)

• Figure 1: The entropy production as a function of time for a region consisting of two disjoint intervals of length $L$, separated by a distance $R>L$. The quasiparticle model (left) shows decreasing behavior between $2t=R$ and $2t=L+R$. The holographic calculation (right) is monotonically increasing before saturation at $2t=L$, after which the entropy remains constant.
• Figure 2: An EPR pair produced at the points marked as green at the bottom of the figure. When the constituent particles are at the positions marked as red at the intermediate time, they contribute to the entanglement entropy. At the later time when the particles are at the positions marked as blue, they do not contribute to the entanglement entropy. This process leads to a decrease in the entanglement entropy in the quasiparticle picture.
• Figure 3: Penrose diagram of the time dependent geometry following the quench. The red vertical line on the right is the AdS boundary ($z=0$ in Poincare patch). The green diagonal line is the infalling shell, and the blue diagonal line is the horizon. The dashed curve is a late time extremal surface, which asymptotes to the critical surface, indicated by the solid curve. The linear growth of entanglement entropy comes from the portion of the extremal surface lying along the critical surface behind the horizon.
• Figure 4: Here we display the extremal surfaces for two intervals. The first candidate HRT surface is the union of the two smaller arcs (marked in red and labeled $\mathcal{A}_1$ and $\mathcal{A}_2$). The second candidate is the union of the two larger arcs (marked in green and labeled $\mathcal{A}_3$ and $\mathcal{A}_4$).
• Figure 5: The quench for two intervals of length $L$ separated by a distance $R$ when $L >R$. On the left, we show the entanglement tsunami wavefront as a function of time (jagged black line.) The region $A$ is marked as red. The intervals between the disconnected components of $A$ are marked as blue. On the right we show the entanglement entropy as a function of time.