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The R\'enyi entanglement entropy (REE) of the states excited by localoperators in two-dimensional irrational conformal field theories (CFTs),especially in Liouville field theory (LFT) and $\mathcal{N}=1$ super-Liouvillefield theory (SLFT), has been investigated. In particular, the excited statesobtained by acting on the vacuum with primary operators were considered. {Westart from evaluating the second REE in a compact $c=1$ free boson field theoryat generic radius, which is an irrational CFT. Then we focus on the two specialirrational CFTs, e.g., LFT and SLFT. In these theories, the second REE of suchlocal excited states becomes divergent in early and late time limits. Forsimplicity, we study the memory effect of REE for the two classes of the localexcited states in LFT and SLFT. In order to restore the quasiparticles picture,we define the difference of REE between target and reference states, whichbelong to the same class. The variation of the difference of REE between earlyand late time limits always coincides with the log of the ratio of the fusionmatrix elements between target and reference states. Furthermore, the locallyexcited states by acting generic descendent operators on the vacuum have beenalso investigated. The variation of the difference of REE is the summation ofthe log of the ratio of the fusion matrix elements between the target andreference states, and an additional normalization factor. Since the identityoperator (or vacuum state) does not live in the Hilbert space of LFT and SLFTand no discrete terms contribute to REE in the intermediate channel, thevariation of the difference of REE between target and reference states is nolonger the log of the quantum dimension which is shown in the 1+1-dimensionalrational CFTs (RCFTs).

Conformal bootstrap to R\'enyi entropy in 2D Liouville and super-Liouville CFTs