Time Evolution of Multi-Party Entanglement Signals

TL;DR

Time-evolution of multiparty entanglement signals is explored in chaotic quantum many-body systems using the membrane theory of entanglement dynamics, with a focus on signals like $R_3$, $I_3$, and $Q_4$ and their higher-party generalizations. The authors show that scrambling generates rich, signal-dependent multipartite entanglement, including dynamical phase transitions evidenced by discontinuities in $S_R$-based constructions, non-monotonic growth, and regimes where entanglement becomes entirely multipartite in certain regions. The membrane framework, extended to time-dependent reflected entropy and, in 2d, a generalised membrane, provides a universal tool to compute these dynamics in the late-time, large-region limit across holographic and chaotic systems. The results underscore the necessity of multiple, carefully chosen signals to capture the full multipartite entanglement structure and reveal notable differences in dynamics between higher-dimensional holographic theories and 2d CFTs, where extra symmetry modifies propagation and saturation patterns.

Abstract

We study the real-time dynamics of multi-party entanglement signals in chaotic quantum many-body systems including but not necessarily restricted to holographic conformal field theories. We find that scrambling dynamics generates multiparty entanglement with rich structure including: (a) qualitatively different dynamical behaviours for different signals, likely reflecting different dynamics for different kinds of entanglement patterns, (b) discontinuities indicating dynamical phase transitions in the entanglement structure, (c) transient and non-monotonic multiparty entanglement, and (d) periods during which the extensive entanglement of some regions is entirely multipartite. Our main technical tool is the membrane theory of entanglement dynamics.

Figures (24)

• Figure 1: Cartoon of a minimal membrane (orange) extending in spacetime. This membrane computes entanglement entropy between the subregion $A$ and its complement $\overline{A}$ at time $t$.
• Figure 2: Time evolution of entanglement entropy of a single bounded interval in an unbounded system as described by the effective membrane theory. The entanglement entropy first grows linearly and then saturates, and is described in these two regimes by a vertical (blue) and cone (red) membranes, respectively.
• Figure 3: Membrane configurations computing $S(AB)$ when $B$ is semi-infinite, at $0< t<\frac{\ell_c}{2}$ (left) and $t>\frac{\ell_c}{2}$ (right). The $AB$ entanglement wedge is shaded. At $t<\frac{\ell_c}{2}$, the EW changes from disconnected to connected.
• Figure 4: Minimal cross-section (thick) bisecting the $AB$ entanglement wedge (shaded region) at intermediate (left) and late (right) time. These entanglement wedge cross-sections compute the reflected entropy $S_R$. $t_0$ is the vertical distance between the top tip of the $v_B$ cone and the $t=T$ slice.
• Figure 5: An example of different $v_B$ cone configurations for computing the saturation value of $S(AB)$. The solid lines correspond to the disconnected configurations, whereas the dashed lines represent the connected configurations. In the large subregion limit, the disconnected configurations always dominate.