Abstract
The R\'enyi entanglement entropy (REE) of the states excited by local operators in two-dimensional irrational conformal field theories (CFTs), especially in Liouville field theory (LFT) and $\mathcal{N}=1$ super-Liouville field theory (SLFT), has been investigated. In particular, the excited states obtained by acting on the vacuum with primary operators were considered. {We start from evaluating the second REE in a compact $c=1$ free boson field theory at generic radius, which is an irrational CFT. Then we focus on the two special irrational CFTs, e.g., LFT and SLFT. In these theories, the second REE of such local excited states becomes divergent in early and late time limits. For simplicity, we study the memory effect of REE for the two classes of the local excited states in LFT and SLFT. In order to restore the quasiparticles picture, we define the difference of REE between target and reference states, which belong to the same class. The variation of the difference of REE between early and late time limits always coincides with the log of the ratio of the fusion matrix elements between target and reference states. Furthermore, the locally excited states by acting generic descendent operators on the vacuum have been also investigated. The variation of the difference of REE is the summation of the log of the ratio of the fusion matrix elements between the target and reference states, and an additional normalization factor. Since the identity operator (or vacuum state) does not live in the Hilbert space of LFT and SLFT and no discrete terms contribute to REE in the intermediate channel, the variation of the difference of REE between target and reference states is no longer the log of the quantum dimension which is shown in the 1+1-dimensional rational CFTs (RCFTs).
Highlights
One can define some observables to detect the property of the vacuum or excited states in a local quantum field theory
The entanglement entropy and Renyi entropy, both of them are defined as a function of the reduced density matrix ρA, which can be obtained by tracing out the degrees of freedom of the complement of A in the original density matrix ρ
This study mainly focuses on the variation of Renyi entropy SðAnÞ between excited states and a reference state, where the excited states are acquired by acting primary or descendent fields on the vacuum in irrational conformal field theories (CFTs)
Summary
One can define some observables to detect the property of the vacuum or excited states in a local quantum field theory. To understand whether or not these connections are AdS/CFT-like, we would like to work out the large central charge properties of local excited states by primary fields in LFT or SLFT; because EE and REE can be probed on the field theory side and the holographic side, both of them will be good objects with which to test the properties of these connections.
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