arXiv2026
We study the time evolution of the entanglement structure of holographic conformal field theories after a local quench. Using the mutual information between two spatial intervals as a probe, we find that $1+1$-dimensional conformal field theories exhibit a rich pattern of dynamical phase transitions. In the large-central-charge limit, mutual information develops sharp non-analyticities at critical times, providing a concrete entanglement-based realization of dynamical quantum phase transitions. We find that the dynamics organize into six distinct phases of mutual information, each controlled by the dominance of a different conformal block, or equivalently, a different holographic geodesic configuration. This phase structure goes beyond the standard quasi-particle picture, explaining non-analytic features that are not captured by simple light-cone propagation from the quench points. We further identify a dynamical $D_4$ symmetry acting on the interval endpoints that controls the presence or absence of mutual information. The onset of mutual information is governed by the breaking of this symmetry to a $\mathbb{Z}_2 \times \mathbb{Z}_2$ subgroup, suggesting a symmetry-based characterization of non-equilibrium entanglement dynamics analogous to the role of symmetry in equilibrium critical phenomena. Finally, numerical studies of critical spin chains indicate that finite-$c$ effects smooth out the sharp large-$c$ transitions between different mutual-information phases, while the transitions between phases with and without mutual information appear to remain non-analytic. These results offer a unifying perspective on real-time entanglement dynamics and their critical features in conformal many-body systems.