Conformal bootstrap to Rényi entropy in 2D Liouville and super-Liouville CFTs

Abstract

The Rényi 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 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). *

Entanglement wedge cross section from CFT: dynamics of local operator quench

“…It is very interesting to compare our result to the dynamics of the reflected entropy in other CFTs, in particular, integrable CFTs. There are many works to study entanglement entropy after a local quench in various setups [4,5,52,53,71,[86][87][88][89][90][91]. Their motivation is to characterize CFT classes by the dynamics of entanglement.…”

Correlation functions, entanglement and chaos in the $$ T\overline{T}/J\overline{T} $$-deformed CFTs

“…Moreover, the Rényi entanglement entropy has been used as a very helpful quantity to measure the properties of the vacuum state and excited states [55][56][57][58][59][60][61][62][63][64][65][66]. In the undeformed CFTs, the nth Rényi entanglement entropy S (n) A has been extensively studied in the literature [58][59][60][61][64][65][66][67][68][69][70][71][72][73][74]. In rational CFTs, it has been shown that the excess of the Rényi entanglement entropy has to be the logarithmic of quantum dimension [59] of the corresponding local operator.…”

“…As applications, we employ the formula to investigate the Rényi entanglement entropy in two-dimensional CFTs perturbatively. We just put a local primary operator following the procedure in [59][60][61][64][65][66] to obtain a locally excited state. With time evolution, the excess of Rényi entanglement entropy of deformed two-dimensional CFTs has been calculated in this paper.…”

“…In two-dimeneional quantum gravity theory, e.g. Liouville field theory, one has to redefine the proper reference state[64] to restore the causality structure and the excess of Rényi entanglement entropy will be a the log of the ratio of the fusion matrix elements between the exciting and reference states.…”

The correlation function of (1, 1) and (2, 2) supersymmetric theories with $$ T\overline{T} $$ deformation

“…Moreover, both in holography and quantum field theory, these correlation functions can be also applied to obtain various interesting quantum information quantities in the deformed field theory, e.g. the Rényi entanglement entropy of local quench in various situations [63][64][65], entanglement negativity [66], entanglement purification [67], information metric [68,69], etc.…”