Entanglement Entropy for Descendent Local Operators in 2D CFTs

We mainly study the R\'enyi 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 R\'enyi 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 R\'enyi 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 R\'enyi 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 R\'enyi 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 R\'enyi entropy of the excited operators in deformed CFT. The R\'enyi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.