Entanglement Entropy for Descendent Local Operators in 2D CFTs

TL;DR

The paper extends the study of entanglement measures in 2D CFTs from primary to descendant local operators. Using the replica trick, it shows that for several descendant classes (notably L^(-) barL^(-)O and ∂O, L^(-)O), the Rényi and entanglement entropies reproduce the universal value log d_O, the quantum dimension of the primary, while more general descendants introduce additional, nonuniversal corrections due to holomorphic–antiholomorphic mixing. In BCFTs, the maximal entropy remains unchanged, with boundary effects affecting only the time evolution. The authors also develop a perturbative CFT framework to handle deformations, finding additive contributions to RE from local excitations and global deformations, expanding the toolkit for entanglement studies of descendant states in rational and deformed 2D CFTs.

Abstract

We mainly study the Rényi entropy and entanglement entropy of the states locally excited by the descendent operators in two dimensional conformal field theories (CFTs). In rational CFTs, we prove that the increase of entanglement entropy and Rényi entropy for a class of descendent operators, which are generated by $\cal{L}^{(-)}\bar{\cal{L}}^{(-)}$ onto the primary operator, always coincide with the logarithmic of quantum dimension of the corresponding primary operator. That means the Rényi entropy and entanglement entropy for these descendent operators are the same as the ones of their corresponding primary operator. For 2D rational CFTs with a boundary, we confirm that the Rényi entropy always coincides with the logarithmic of quantum dimension of the primary operator during some periods of the evolution. Furthermore, we consider more general descendent operators generated by $\sum_{} d_{\{n_i\}\{n_j\}}(\prod_{i} L_{-n_i}\prod_{j}{\bar L}_{-n_j})$ on the primary operator. For these operators, the entanglement entropy and Rényi entropy get additional corrections, as the mixing of holomorphic and anti-holomorphic Virasoro generators enhance the entanglement. Finally, we employ perturbative CFT techniques to evaluate the Rényi entropy of the excited operators in deformed CFT. The Rényi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.

Figures (2)

• Figure 1: This figure is to show our setup in two dimensional plane $\omega=x+i t$ . The system will be triggered at $x=-l$ and there are left- and right-moving quasi-particles at $t=0$.
• Figure 2: The fusion transformations to obtain $\Delta S^{(n)}_A$.