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arXiv:2108.09210 · Journal of High Energy Physics

Symmetry-resolved entanglement for excited states and two entangling intervals in AdS3/CFT2

Konstantin Weisenberger, Suting Zhao, Christian Northe, René Meyer

Published Dec 16, 2021• 59 citations
arXivHTMLPDF

Abstract

Abstract We test the proposal of [1] for the holographic computation of the charged moments and the resulting symmetry-resolved entanglement entropy in different excited states, as well as for two entangling intervals. Our holographic computations are performed in U(1) Chern-Simons-Einstein-Hilbert gravity, and are confirmed by independent results in a conformal field theory at large central charge. In particular, we consider two classes of excited states, corresponding to charged and uncharged conical defects in AdS3. In the conformal field theory, these states are generated by the insertion of charged and uncharged heavy operators. We employ the monodromy method to calculate the ensuing four-point function between the heavy operators and the twist fields. For the two-interval case, we derive our results on the AdS and the conformal field theory side, respectively, from the generating function method of [1], as well as the vertex operator algebra. In all cases considered, we find equipartition of entanglement between the different charge sectors. We also clarify an aspect of conformal field theories with a large central charge and $$ \hat{\mathfrak{u}}{(1)}_k $$ u ̂ 1 k Kac-Moody symmetry used in our calculations, namely the factorization of the Hilbert space into a gravitational Virasoro sector with large central charge, and a $$ \hat{\mathfrak{u}}{(1)}_k $$ u ̂ 1 k Kac-Moody sector.

DOI
10.1007/jhep12(2021)104

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Abstract We test the proposal of [1] for the holographic computation of the charged moments and the resulting symmetry-resolved entanglement entropy in different excited states, as well as for two entangling intervals. Our holographic computations are performed in U(1) Chern-Simons-Einstein-Hilbert gravity, and are confirmed by independent results in a conformal field theory at large central charge. In particular, we consider two classes of excited states, corresponding to charged and uncharged conical defects in AdS3. In the conformal field theory, these states are generated by the insertion of charged and uncharged heavy operators. We employ the monodromy method to calculate the ensuing four-point function between the heavy operators and the twist fields. For the two-interval case, we derive our results on the AdS and the conformal field theory side, respectively, from the generating function method of [1], as well as the vertex operator algebra. In all cases considered, we find equipartition of entanglement between the different charge sectors. We also clarify an aspect of conformal field theories with a large central charge and u^(1)k\hat{\mathfrak{u}}{(1)}_ku^(1)k​ u ̂ 1 k Kac-Moody symmetry used in our calculations, namely the factorization of the Hilbert space into a gravitational Virasoro sector with large central charge, and a u^(1)k\hat{\mathfrak{u}}{(1)}_ku^(1)k​ u ̂ 1 k Kac-Moody sector.
Abstract We consider symmetry-resolved entanglement entropy in AdS3/CFT2coupled to U(1) Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS3, namely the Ryu-Takayanagi geodesic minimally coupled to the U(1) Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged U(1) vertex operators inserted at the endpoints of the entangling interval. Furthermore, we devise a new method to calculate the symmetry resolved entanglement entropy by relating the generating function for the charged moments to the amount of charge in the entangling subregion. We calculate the subregion charge from the U(1) Chern-Simons gauge field sourced by the bulk Wilson line. We use our method to derive the symmetry-resolved entanglement entropy for Poincaré patch and global AdS3, as well as for the conical defect geometries. In all three cases, the symmetry resolved entanglement entropy is determined by the length of the Ryu-Takayanagi geodesic and the Chern-Simons levelk, and fulfills equipartition of entanglement. The asymptotic symmetry algebra of the bulk theory is of u^(1)k \hat{\mathfrak{u}}{(1)}_k u^(1)k​ û1k Kac-Moody type. Employing the u^(1)k \hat{\mathfrak{u}}{(1)}_k u^(1)k​ û1k Kac-Moody symmetry, we confirm our holographic results by a calculation in the dual conformal field theory.
We consider the entanglement entropy in 2d conformal field theory in a class of excited states produced by the insertion of a heavy local operator. These include both high-energy eigenstates of the Hamiltonian and time-dependent local quenches. We compute the universal contribution from the stress tensor to the single interval Renyi entropies and entanglement entropy, and conjecture that this dominates the answer in theories with a large central charge and a sparse spectrum of low-dimension operators. The resulting entanglement entropies agree precisely with holographic calculations in three-dimensional gravity. High-energy eigenstates are dual to microstates of the BTZ black hole, so the corresponding holographic calculation is a geodesic length in the black hole geometry; agreement between these two answers demonstrates that these individual microstates of holographic CFTs effectively thermalize at the level of the single-interval entanglement entropy. For local quenches, the dual geometry is a highly boosted black hole or conical defect. On the CFT side, the rise in entanglement entropy after a quench is directly related to the monodromy of a Virasoro conformal block.
Higher spin entanglement and W N WN {\mathcal{W}}_{\mathrm{N}} WN​ conformal blocks
Two-dimensional conformal field theories with extended W \mathcal{W} W -symmetry algebras have dual descriptions in terms of weakly coupled higher spin gravity in AdS3 at large central charge. Observables that can be computed and compared in the two descriptions include Rényi and entanglement entropies, and correlation functions of local operators. We develop techniques for computing these, in a manner that sheds light on when and why one can expect agreement between such quantities on each side of the duality. We set up the computation of excited state Rényi entropies in the bulk in terms of Chern-Simons connections, and show how this directly parallels the CFT computation of correlation functions. More generally, we consider the vacuum conformal block for general operators with Δ ∼ c. When two of the operators obey Δc≪1 \frac{\varDelta }{c}\ll 1 cΔ​≪1 , we show by explicit computation that the vacuum conformal block is computed by a bulk Wilson line probing an asymptotically AdS3 background with higher spin fields excited, the latter emerging as the effective bulk description of the excited state produced by the heavy operators. Among other things, this puts a previous proposal for computing higher spin entanglement entropy via Wilson lines on firmer footing, and clarifies its relation to CFT. We also study the corresponding computation in Toda theory and find that this provides yet another independent way to arrive at the same result.
Abstract We report a throughout analysis of the entanglement entropies related to different symmetry sectors in the low-lying primary excited states of a conformal field theory (CFT) with an internal U (1) symmetry. Our findings extend recent results for the ground state. We derive a general expression for the charged moments, i.e. the generalised cumulant generating function, which can be written in terms of correlation functions of the operator that define the state through the CFT operator-state correspondence. We provide explicit analytic computations for the compact boson CFT (aka Luttinger liquid) for the vertex and derivative excitations. The Fourier transform of the charged moments gives the desired symmetry resolved entropies. At the leading order, they satisfy entanglement equipartition, as in the ground state, but we find, within CFT, subleading terms that break it. Our analytical findings are checked against free fermions calculations on a lattice, finding excellent agreement. As a byproduct, we have exact results for the full counting statistics of the U (1) charge in the considered excited states.
A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N, R) × SL(N, R) Chern-Simons gauge theories in the (2 + 1)-dimensional bulk space-time, and CFTs with WN symmetry algebras on the (1 + 1)-dimensional boundary. The topological character of the bulk theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the WN CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS3 gravity, corresponding to SL(2, R) × SL(2, R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the bulk theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.
This is an extended version of our short report hep-th/0603001, where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.