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

Song He

TL;DR

This work analyzes the memory of the second and higher Rényi entanglement entropies for locally excited states in 1+1D irrational CFTs, focusing on Liouville and super-Liouville theories. By applying the replica trick and conformal bootstrap, it shows that, unlike rational CFTs, the late-time REE memory is controlled by ratios of fusion-matrix elements rather than a quantum dimension, due to the continuous spectrum and absence of the vacuum in the Hilbert space. The authors define a finite memory observable ΔS_EE^{(n)} by comparing target and reference states within the same class, and derive explicit early- and late-time behaviors for primary and descendant excitations in LFT and SLFT, with the late-time memory governed by F^L_{Q/2,Q/2} type fusion data. They extend the analysis to generic nth REE and to descendants, highlighting the key difference between irrational CFTs and RCFTs, and discuss potential holographic interpretations and future directions, including OTOCs and large-central-charge behavior.

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 $\mathcal{N}=1$ 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).

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

TL;DR

This work analyzes the memory of the second and higher Rényi entanglement entropies for locally excited states in 1+1D irrational CFTs, focusing on Liouville and super-Liouville theories. By applying the replica trick and conformal bootstrap, it shows that, unlike rational CFTs, the late-time REE memory is controlled by ratios of fusion-matrix elements rather than a quantum dimension, due to the continuous spectrum and absence of the vacuum in the Hilbert space. The authors define a finite memory observable ΔS_EE^{(n)} by comparing target and reference states within the same class, and derive explicit early- and late-time behaviors for primary and descendant excitations in LFT and SLFT, with the late-time memory governed by F^L_{Q/2,Q/2} type fusion data. They extend the analysis to generic nth REE and to descendants, highlighting the key difference between irrational CFTs and RCFTs, and discuss potential holographic interpretations and future directions, including OTOCs and large-central-charge behavior.

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 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 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).

Paper Structure

This paper contains 20 sections, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Second Rényi Entanglement Entropy
  3. Setup in 2D CFT
  4. Convention
  5. The Variation of second Rényi Entanglement Entropy
  6. The Second REE in $c=1$ free boson field theory
  7. The Second REE in Liouville Field Theory
  8. The second REE in Super Liouville field theory
  9. The $n$th REE in LFT and SLFT
  10. The $n$th REE for generic descendent states
  11. Discussions and conclusions
  12. Appendix
  13. The Notations of LFT
  14. The Notations of SLFT
  15. The function $\Gamma_b(x)$
  16. ...and 5 more sections

Figures (1)

  • Figure 1: This figure is to show our basic setup in two dimensional plane $w = x + it$. We consider the subsystem A $0 <x<L$ with $L\to \infty$. We just put the local operators at $x = -l, t=0$. The local operator will trigger left- and right-moving quasi-particles with time evolution.