Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1711.00624 · Physical review. D/Physical review. D.

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

Song He

Published Jan 7, 2019• 30 citations
arXivHTMLPDF

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).

DOI
10.1103/physrevd.99.026005

Related papers

Entanglement entropy for descendent local operators in 2D CFTs

Bin Chen, Wu-zhong Guo, Song He +1 more

We mainly study the Rényi entropy and entanglement entropy of the states locally excited by the descendent operators in two dimensional conformal field theories (CFTs). In rational CFTs, we prove that the increase of entanglement entropy and Rényi entropy for a class of descendent operators, which are generated by $$ {\mathrm{\mathcal{L}}}^{\left(-\right)}{\overline{\mathrm{\mathcal{L}}}}^{\left(-\right)} $$ onto the primary operator, always coincide with the logarithmic of quantum dimension of the corresponding primary operator. That means the Rényi entropy and entanglement entropy for these descendent operators are the same as the ones of their corresponding primary operator. For 2D rational CFTs with a boundary, we confirm that the Rényi entropy always coincides with the logarithmic of quantum dimension of the primary operator during some periods of the evolution. Furthermore, we consider more general descendent operators generated by $$ {\displaystyle \sum {d}_{\left\{{n}_i\right\}\left\{{n}_j\right\}}\left({\displaystyle {\prod}_i{L}_{-}{{}_n}_{{}_i}{\displaystyle {\prod}_j{{\overline{L}}_{-n}}_{{}_j}}}\right)} $$ on the primary operator. For these operators, the entanglement entropy and Rényi entropy get additional corrections, as the mixing of holomorphic and anti-holomorphic Virasoro generators enhance the entanglement. Finally, we employ perturbative CFT techniques to evaluate the Rényi entropy of the excited operators in deformed CFT. The Rényi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.

arXiv:1507.01157•Journal of High Energy Physics•Oct 1, 2015•59 citations

Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs

Wu-zhong Guo, Song He

We study Rényi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly, we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The Rényi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the Rényi entropy of locally excited states in 2D CFT with a boundary. We show that the time evolution of Rényi entropy is affected by the boundary, but does not depend on the boundary condition. Moreover, we show that the maximal value of Rényi entropy always coincides with the log of quantum dimension of the primary operator. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards it.

arXiv:1501.00757•Journal of High Energy Physics•Apr 1, 2015•52 citations

Pseudoentropy for descendant operators in two-dimensional conformal field theories

Song He, Jie Yang, Yu-Xuan Zhang +1 more

We study the late-time behaviors of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories. To construct the transition matrix, we utilize two nonorthogonal locally excited states that are created by the application of different descendant operators to vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudoentropy and pseudo-Rényi entropy corresponds to the logarithm of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(Rényi) entropy compared to that for (Rényi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(Rényi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(Rényi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.

arXiv:2301.04891•Physical review. D/Physical review. D.•Jan 30, 2024•16 citations

On the real-time evolution of pseudo-entropy in 2d CFTs

Wu-zhong Guo, Song He, Yu-Xuan Zhang

Abstract In this work, we study the real-time evolution of pseudo-(Rényi) entropy, a generalization of entanglement entropy, in two-dimensional conformal field theories (CFTs). We focus on states obtained by acting primary operators located at different space points or their linear combinations on the vacuum. We show the similarities and differences between the pseudo-(Rényi) entropy and entanglement entropy. For excitation by a single primary operator, we analyze the behaviors of the 2nd pseudo-Rényi entropy in various limits and find some symmetries associated with the subsystem and the positions of the inserted operators. For excitation by linear combinations, the late time limit of the nth pseudo-Rényi entropy shows a simple form related to the coefficients of the combinations and Rényi entropy of the operators, which can be derived by using the Schmidt decomposition. Further, we find two kinds of particular spatial configurations of insertion operators in one of which the pseudo-(Rényi) entropy remains real throughout the time evolution.

arXiv:2206.11818•Journal of High Energy Physics•Sep 13, 2022•22 citations

Renyi entropy for local quenches in 2D CFT from numerical conformal blocks

Yuya Kusuki, Tadashi Takayanagi

We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of log t term. Our analysis covers the entire parameter regions with respect to the replica number n and the conformal dimension h O of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov’s recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge c, drastically changes when the dimensions of external primary states reach the value c/32. In particular, when h O ≥ c/32 and n ≥ 2, we find a new universal formula $$ \varDelta {S}_A^{(n)}\simeq \frac{nc}{24\left(n-1\right)} $$ log t. Our numerical results also confirm existing analytical results using the HHLL approximation.

arXiv:1711.09913•Journal of High Energy Physics•Jan 1, 2018•41 citations

Entanglement entropy, OTOC and bootstrap in 2D CFTs from Regge and light cone limits of multi-point conformal block

Yuya Kusuki, Masamichi Miyaji

Abstract We explore the structures of light cone and Regge limit singularities of n-point Virasoro conformal blocks in c > 1 two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full 1/c corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time. We first show that, the n-th (n > 2) Renyi entropy after a local quench in our CFT grows logarithmically at late time, for any c and any conformal dimensions of excited primary. In particular, we find that this behavior is independent of c, contrary to the expectation that the finite c correction fixes the late time Renyi entropy to be constant. We also show that the constant part of the late time Renyi entropy is given by a monodromy matrix. We also investigate OTOCs by using the monodromy matrix. We first rewrite the monodromy matrix in terms of fusion matrix explicitly. By this expression, we find that the OTOC decays exponentially in time, and the decay rates are divided into three patterns, depending on the dimensions of external operators. We note that our result is valid for any c > 1 and any external operator dimensions. Our monodromy matrix approach can be generalized to the Liouville theory and we show that the Liouville OTOC approaches constant in the late time regime. We emphasize that, there is a number of other applications of the fusion and the monodromy matrix approaches, such as solving the conformal bootstrap equation. Therefore, it is tempting to believe that the fusion and monodromy matrix approaches provide a key to understanding the AdS/CFT correspondence.

arXiv:1905.02191•Journal of High Energy Physics•Aug 1, 2019•58 citations
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 N=1\mathcal{N}=1N=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=1c=1c=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+11+11+1-dimensional rational CFTs (RCFTs).
We mainly study the Rényi entropy and entanglement entropy of the states locally excited by the descendent operators in two dimensional conformal field theories (CFTs). In rational CFTs, we prove that the increase of entanglement entropy and Rényi entropy for a class of descendent operators, which are generated by L(−)L‾(−) {\mathrm{\mathcal{L}}}^{\left(-\right)}{\overline{\mathrm{\mathcal{L}}}}^{\left(-\right)} L(−)L(−) onto the primary operator, always coincide with the logarithmic of quantum dimension of the corresponding primary operator. That means the Rényi entropy and entanglement entropy for these descendent operators are the same as the ones of their corresponding primary operator. For 2D rational CFTs with a boundary, we confirm that the Rényi entropy always coincides with the logarithmic of quantum dimension of the primary operator during some periods of the evolution. Furthermore, we consider more general descendent operators generated by ∑d{ni}{nj}(∏iL−ni∏jL‾−nj) {\displaystyle \sum {d}_{\left\{{n}_i\right\}\left\{{n}_j\right\}}\left({\displaystyle {\prod}_i{L}_{-}{{}_n}_{{}_i}{\displaystyle {\prod}_j{{\overline{L}}_{-n}}_{{}_j}}}\right)} ∑d{ni​}{nj​}​(∏i​L−​n​i​​∏j​L−n​j​​) on the primary operator. For these operators, the entanglement entropy and Rényi entropy get additional corrections, as the mixing of holomorphic and anti-holomorphic Virasoro generators enhance the entanglement. Finally, we employ perturbative CFT techniques to evaluate the Rényi entropy of the excited operators in deformed CFT. The Rényi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.
We study Rényi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly, we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The Rényi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the Rényi entropy of locally excited states in 2D CFT with a boundary. We show that the time evolution of Rényi entropy is affected by the boundary, but does not depend on the boundary condition. Moreover, we show that the maximal value of Rényi entropy always coincides with the log of quantum dimension of the primary operator. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards it.
We study the late-time behaviors of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories. To construct the transition matrix, we utilize two nonorthogonal locally excited states that are created by the application of different descendant operators to vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudoentropy and pseudo-Rényi entropy corresponds to the logarithm of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(Rényi) entropy compared to that for (Rényi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(Rényi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(Rényi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.
Abstract In this work, we study the real-time evolution of pseudo-(Rényi) entropy, a generalization of entanglement entropy, in two-dimensional conformal field theories (CFTs). We focus on states obtained by acting primary operators located at different space points or their linear combinations on the vacuum. We show the similarities and differences between the pseudo-(Rényi) entropy and entanglement entropy. For excitation by a single primary operator, we analyze the behaviors of the 2nd pseudo-Rényi entropy in various limits and find some symmetries associated with the subsystem and the positions of the inserted operators. For excitation by linear combinations, the late time limit of the nth pseudo-Rényi entropy shows a simple form related to the coefficients of the combinations and Rényi entropy of the operators, which can be derived by using the Schmidt decomposition. Further, we find two kinds of particular spatial configurations of insertion operators in one of which the pseudo-(Rényi) entropy remains real throughout the time evolution.
We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of log t term. Our analysis covers the entire parameter regions with respect to the replica number n and the conformal dimension h O of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov’s recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge c, drastically changes when the dimensions of external primary states reach the value c/32. In particular, when h O ≥ c/32 and n ≥ 2, we find a new universal formula ΔSA(n)≃nc24(n−1) \varDelta {S}_A^{(n)}\simeq \frac{nc}{24\left(n-1\right)} ΔSA(n)​≃24(n−1)nc​ log t. Our numerical results also confirm existing analytical results using the HHLL approximation.
Abstract We explore the structures of light cone and Regge limit singularities of n-point Virasoro conformal blocks in c > 1 two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full 1/c corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time. We first show that, the n-th (n > 2) Renyi entropy after a local quench in our CFT grows logarithmically at late time, for any c and any conformal dimensions of excited primary. In particular, we find that this behavior is independent of c, contrary to the expectation that the finite c correction fixes the late time Renyi entropy to be constant. We also show that the constant part of the late time Renyi entropy is given by a monodromy matrix. We also investigate OTOCs by using the monodromy matrix. We first rewrite the monodromy matrix in terms of fusion matrix explicitly. By this expression, we find that the OTOC decays exponentially in time, and the decay rates are divided into three patterns, depending on the dimensions of external operators. We note that our result is valid for any c > 1 and any external operator dimensions. Our monodromy matrix approach can be generalized to the Liouville theory and we show that the Liouville OTOC approaches constant in the late time regime. We emphasize that, there is a number of other applications of the fusion and the monodromy matrix approaches, such as solving the conformal bootstrap equation. Therefore, it is tempting to believe that the fusion and monodromy matrix approaches provide a key to understanding the AdS/CFT correspondence.