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arXiv:2301.04891 · Physical review. D/Physical review. D.

Pseudoentropy for descendant operators in two-dimensional conformal field theories

Song He, Jie Yang, Yu-Xuan Zhang, Zi-Xuan Zhao

Published Jan 30, 2024• 16 citations
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Abstract

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.

DOI
10.1103/physrevd.109.025014

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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.
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.
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).
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.
Pseudo entropy extends entanglement entropy from a single quantum state to a pair of nonorthogonal states and is generally complex. Taking advantage of Cauchy-Riemann equations, Kramers-Kronig relations, and analytic continuation in state parameters, we show how and to what extent real and imaginary parts of pseudo entropy can be derived from ordinary entanglement entropy. For families with holomorphic coefficients in finite-dimensional Hilbert spaces, the reduced transition matrix equals the ordinary reduced density matrix formula evaluated at complex parameters. Then a convergent Taylor series gives the real and imaginary parts of pseudo entropy from even and odd derivatives of entanglement entropy at the real midpoint. The matrix identity also gives formulae for excess pseudo entropy and Renyi entropies, and interpolation formulae for families with polynomial coefficients. We apply these results to boundary-state quenches and thermal states in conformal field theory, and to fermionic and bosonic Gaussian states and quenches. In conformal field theory, Kramers-Kronig relations give the first moment of the imaginary part in terms of the central charge and one-point functions for boundary states.
A bstract In this paper, we explore the concept of pseudo Rényi entropy within the context of quantum field theories (QFTs). The transition matrix is constructed by applying operators situated in different regions to the vacuum state. Specifically, when the operators are positioned in the left and right Rindler wedges respectively, we discover that the logarithmic term of the pseudo Rényi entropy is necessarily real. In other cases, the result might be complex. We provide direct evaluations of specific examples within 2-dimensional conformal field theories (CFTs). Furthermore, we establish a connection between these findings and the pseudo-Hermitian condition. Our analysis reveals that the reality or complexity of the logarithmic term of pseudo Rényi entropy can be explained through this pseudo-Hermitian framework. Additionally, we investigate the divergent term of the pseudo Rényi entropy. Interestingly, we observe a universal divergent term in the second pseudo Rényi entropy within 2-dimensional CFTs. This universal term is solely dependent on the conformal dimension of the operator under consideration. For n -th pseudo Rényi entropy ( n ≥ 3), the divergent term is intricately related to the specific details of the underlying theory.
In this paper, we investigate the second and third pseudo Rényi entanglement entropies (PREE) for a locally excited state $| ψ\rangle $ and its time evolution ∣ϕ⟩=e−iHt∣ψ⟩| ϕ\rangle = e^{- i H t} | ψ\rangle∣ϕ⟩=e−iHt∣ψ⟩ in a two-dimensional conformal field theory whose field content is a free massless scalar field. We consider excited states which are constructed by applying primary operators at time t=0t=0t=0, on the vacuum state. We study the time evolution of the PREE for an entangling region in the shape of finite and semi-infinite intervals at zero temperature. It is observed that the PREE is always a complex number for t≠0t \neq 0t=0 and is a pure real number at t=0t=0t=0. Moreover, we discuss on its dependence on the location xmx_mxm​ of the center of the entangling region.