Entanglement Dynamics by (Non-)Unitary Local Operator Quenches in a 2D Holographic CFT

TL;DR

The work investigates entanglement dynamics after a local operator quench in a 2D holographic CFT, focusing on how the time-ordering of Euclidean (Rindler) and Lorentzian (uniform) evolutions—which do not commute—affects entanglement entropy and mutual information. Through twist-operator formalism, conformal blocks, and holographic RT/Bańados-geometries, the authors show a clear dichotomy: unitary (Lorentzian-first) evolution yields late-time logarithmic growth of EE, while non-unitary (Euclidean-first) evolution leads to late-time saturation. They map these dynamics to a gravity dual with a spacetime-dependent black-brane horizon, analyze energy-momentum densities, and demonstrate consistent CFT and gravity results across finite, semi-infinite, and two-interval configurations, including symmetry-driven mutual-information phase transitions. The paper also develops a quasiparticle picture and clarifies how regulator-dependent partition functions encode the entanglement evolution, highlighting the deep connection between local operator insertions, energy densities, and holographic geometry. Overall, it advances understanding of how time-ordering and holographic dynamics govern information scrambling and nonlocal correlations in driven CFTs.

Abstract

In this paper, we investigate the time evolution of entanglement entropy and mutual information for the spatially-infinite systems where we act with a primary operator on the vacuum state and then time-evolve it with the sequence of the Euclidean and Lorentzian time evolutions. Two-dimensional holographic conformal field theories describe the systems under consideration in this paper. The Euclidean time evolution is induced by the Rindler Hamiltonian and behaves as the regulator that tames the divergence induced by the local operator, while the Lorentzian one is induced by the uniform Hamiltonian. Under these time evolutions, we investigate the time ordering effect of the Rindler Euclidean and uniform Lorentzian time evolution operators. Consequently, we find the remarkable differences between those time evolutions are induced by whether those are unitary or non-unitary. Especially, we find that the unitary time evolution induces the late-time logarithmic growth of the entanglement entropy, while the non-unitary time evolution induces the late-time constant behavior. Furthermore, we investigate the dual gravity of the systems under consideration. Especially, we investigate the gravity duals of the systems with the insertion of the heavy primary operator and show that it is a black brane with a spacetime-dependent horizon.

Figures (11)

• Figure 1: $\frac{\langle T_{tt} \rangle_{i}}{\sin^2(a \epsilon)}$ as a function of $X$ and $t$ when the primary operator is inserted at $x= 10^3$. Left) For $i=1$. Right) For $i=2$. At the insertion point of the primary operator, two energy excitations are created and they propagate along two null directions. We set $h_{\mathcal{O}}= \bar{h}_{\mathcal{O}}=10^3$, $a=10^{-3}$ and $\epsilon=50$.
• Figure 2: $S_{A;1}$ as a function of time when the primary operator is inserted outside and on the left hand side of the subsystem. Left) The contributions of different geodesics with winding numbers $(m, \overline{m})$. Right) Comparison of the entanglement entropy on the CFT and gravity sides. Here, we set $x=10^{4}$, $x_1= 2 \times 10^4$, $x_2= 3 \times 10^4$, $a= 10^{-5}$, $c= 100$, $\epsilon = 10$ and $h_{\mathcal{O}}= 0.4 h_{0}$ where $h_{0} = \frac{c}{24}$.
• Figure 3: $S_{A;1}$ as a function of time when the primary operator is inserted inside of the subsystem. Left) The contributions of different geodesics with winding numbers $(m, \overline{m})$. Right) Comparison of the entanglement entropy on the CFT and gravity sides. Here, we set $x=2 \times 10^4$, $x_1= 10^4$, $x_2= 4 \times 10^4$, $a= 10^{-5}$, $c= 100$, $\epsilon = 10$ and $h_{\mathcal{O}}= 5 h_{0}$.
• Figure 4: The disconnected $\gamma_{12} \cup \gamma_{34}$ and connected $\gamma_{14} \cup \gamma_{23}$ Ryu-Takayanagi surfaces for the subsystem $A \cup B$. Here, $A \in [x_1, x_2]$ and $B \in [x_3, x_4]$ and they are separated by the distance $d$. The primary operator $\mathcal{O}$ is inserted at $x$ on the left hand side of the subsystem $A$.
• Figure 5: Different configurations for the calculation of the holographic mutual information: ( a) symmetric configuration where the operator is inserted between the subsystems $A$ and $B$ and at equal distances from them. ( b) asymmetric configuration where the operator is inserted between the subsystems $A$ and $B$ and closer to $A$. ( c) the operator is located on the left hand side of the subsystems $A$ and $B$. ( d) the operator is inside the subsystem $A$.