Pseudo R\'enyi Entanglement Entropies For an Excited State and Its Time Evolution in a 2D CFT

In this paper, we investigate the second and third pseudo R\'enyi entanglement entropies (PREE) for a locally excited state $| \psi \rangle$ and its time evolution $| \phi \rangle = e^{- i H t} | \psi \rangle$ in a two-dimensional conformal field theory whose field content is a free massless scalar field. We consider excited states which are constructed by applying primary operators at time $t=0$, on the vacuum state. We study the time evolution of the PREE for an entangling region in the shape of finite and semi-infinite intervals at zero temperature. It is observed that the PREE is always a complex number for $t \neq 0$ and is a pure real number at $t=0$. Moreover, we discuss on its dependence on the location $x_m$ of the center of the entangling region.