Pseudo-entropy for descendant operators in two-dimensional conformal field theories

TL;DR

The paper investigates the late-time behavior of pseudo-(Rényi) entropy for descendant-state excitations in 2D RCFTs. Using the replica method and conformal mapping, it shows that when two descendants arise from the same primary via a single Virasoro generator, the late-time pseudo-Rényi entropy excess equals the primary’s quantum-dimension log, mirroring entanglement entropy. For more general descendants, especially linear combinations that mix holomorphic and antiholomorphic sectors, an additional entanglement contribution appears, captured by an effective transition matrix in a finite-dimensional space. This framework reveals how fusion data and holomorphic-antiholomorphic mixing shape pseudo-entanglement, and it points to extensions to broader CFTs such as Liouville or non-diagonal theories.

Abstract

We study the late-time behaviors of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories (RCFTs). To construct the transition matrix, we utilize two non-orthogonal locally excited states that are created by the application of different descendant operators to the vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudo-entropy and pseudo-Rényi entropy corresponds to the logarithmic of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(Rényi) entropy compared to that for (Rényi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(Rényi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(Rényi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.

Figures (2)

• Figure 1: $k-1$ fusion transformations to obtain $\Delta S_A^{(k)}$
• Figure 2: The late-time excess of the 2nd Rényi entropy (in blue) or the 2nd pseudo-Rényi entropy (in orange) of the linear combination operator $(C_1\partial+(1-C_1)\bar{\partial})\mathcal{\varepsilon}$, where $\varepsilon$ is the energy density operator in the critical Ising model. we have $d_{\varepsilon}=1$. The hollow circles represent the numerical data obtained by using the known four-point function of $\varepsilon$, while the solid lines represent the theoretical result obtained by using Eq.\ref{['kthpee2']}(or \ref{['kthpee3']}) (corresponding to the blue line) and Eq.\ref{['kthpee1']} (corresponding to the orange line). It should be noted that when the linear combination operator is the equally-weighted sum of $L_{-1}\varepsilon$ and $\bar{L}_{-1}\varepsilon$, i.e., $C_1=1/2$, the late-time excess of the Rényi entropy ($x_1=x_2$) is $\log2$, while the late-time excess of the pseudo-Rényi entropy ($x_1\neq x_2$) is $\log\frac{18}{17}\approx0.057$.