On the real-time evolution of pseudo-entropy in 2d CFTs

TL;DR

The work extends the concept of entanglement to pseudo-entropy in 2d CFTs and develops a real-time framework for its evolution under local excitations. Using the replica method, it derives the excess and real-time behavior of the second pseudo-Rényi entropy and general n, uncovering universal saturation to log d_O in RCFTs and characteristic large-c growth patterns. It reveals symmetry properties of the evolution, identifies special spatial configurations where the pseudo-entropy remains real, and shows that linear combinations of operators retain memory of insertion data in the late-time limit. The results illuminate how pseudo-entropy encodes correlation structures and quasiparticle dynamics, with implications for holography, information scrambling, and potential higher-dimensional generalizations.

Abstract

In this work, we study the real-time evolution of pseudo-(Rényi) entropy, a generalization of entanglement entropy, in two-dimensional conformal field theories (CFTs). We focus on states obtained by acting primary operators located at different space points or their linear combinations on the vacuum. We show the similarities and differences between the pseudo-(Rényi) entropy and entanglement entropy. For excitation by a single primary operator, we analyze the behaviors of the 2nd pseudo-Rényi entropy in various limits and find some symmetries associated with the subsystem and the positions of the insertion operators. For excitation by linear combinations, the late time limit of the $n$th pseudo-Rényi entropy shows a simple form related to the coefficients of the combinations and Rényi entropy of the operators, which can be derived by using the Schmidt decomposition. Further, we find two kinds of particular spatial configurations of insertion operators in one of which the pseudo-(Rényi) entropy remains real throughout the time evolution.

Figures (7)

• Figure 1: The 2-sheeted space $\Sigma_2$. The dashed box represents the subsystem $A$.
• Figure 2: The real-time evolution of $\text{Re}[\Delta S_A^{(2)}]$ in $c=1$ free scalar, where the insertion operator is chosen to be $\frac{1}{\sqrt{2}}(e^{i\phi}+e^{-i\phi})$ and the regulator $\epsilon=10^{-5}$. (a): $A=[0,\infty)$, $l=2$; (b): $A=[0,20],~l=2$; (c): $A=[0,10]$, $x_m=-5$; (d): $A=[0,20]$, $x_m=10$.
• Figure 3: (a) and (c): The time evolution of $\text{Re}[\Delta S_A^{(2)}]$ under the $\phi_{(2,1)}$-excitation in minimal models. The regulator is chosen to be $\epsilon=10^{-5}$. We have $A=[0,20]$, $l=2$, $x_m=-5$ for (a), and $A=[0,\infty)$, $x_m=0$, $l=10$ for (c). The dashed lines correspond to $\log d_{(2,1)}$ for different $p/p'$; (b): The time evolution of $\text{e}^{S^{(2)}_{A;vac}}\cdot\text{Tr}[(\mathcal{T}_A^{1|2})^2]$ in the case of $A=[0,\infty)$, where $S^{(2)}_{A;vac}$ is the 2nd Rényi entropy of $A$ when the total system is in the vacuum state. The parameters are selected as $\epsilon=10^{-5}$, $x_m=0$, $l=10$; (d): The time evolution of $\Delta S_A^{(2)}$ in the case of $A=[0,L]$. The parameters are selected as $L=20$, $\epsilon=10^{-5}$, $x_m=10$, $l=2$. $\Delta S^{(2)}_A(\eta,\bar{\eta})=\Delta S^{(2)}_A(1/2,1/2)$ at the dashed lines.
• Figure 4: The real-time evolution of $\text{Re}[\Delta S_A^{(2)}]$ or $\Delta S_A^{(2)}$ in $\text{SU}(N)_k$ WZW models with $g^\alpha_\beta$-excitation. The regulator is chosen to be $\epsilon=10^{-5}$. (a), (b), (c): $A=[0,\infty)$, $l=10$, $x_m=0$. The dashed lines correspond to $\log d_g$ for different $N$ and $k$; (d): $A=[0,20]$, $x_m=10$, $l=2$. $\Delta S^{(2)}_A(\eta,\bar{\eta})=\Delta S^{(2)}_A(1/2,1/2)$ at the dashed lines.
• Figure 5: (a) and (c): The late time limits of $\Delta S_A^{(2)}$ with respect to the mixing factor $q$ in $\sigma$+$\mathbb{I}$-excitation. The regulator is chosen to be $\epsilon=10^{-5}$; (b): The full-time evolution of $\text{Re}[\Delta S_A^{(2)}]$ in $\sigma$+$\mathbb{I}$-excitation. Parameters are selected as $q=0.5$, $x=\tilde{x}=-5$, $\epsilon=10^{-5}$. The dashed lines are the theoretical limits derived from Eq.\ref{['resultsfinal']} for the corresponding parameters; (d): The full-time evolution of $\Delta S_A^{(2)}$ in $\sigma$+$\mathbb{I}$-excitation. Parameters are selected as $x=-\tilde{x}=\pm5$, $\epsilon=10^{-5}$. The dashed lines are the theoretical limits derived from Eq.\ref{['resultsfinal']} for the corresponding parameters.