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arXiv:1805.08782 · SciPost Physics

Bulk entanglement entropy in perturbative excited states

Alexandre Belin, Nabil Iqbal, Sagar F. Lokhande

Published Sep 20, 2018• 46 citations
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Abstract

We compute the bulk entanglement entropy across the Ryu-Takayanagi surface for a one-particle state in a scalar field theory in AdS _3 3 . We work directly within the bulk Hilbert space and include the spatial spread of the scalar wavefunction. We give closed form expressions in the limit of small interval sizes and compare the result to a CFT computation of entanglement entropy in an excited primary state at large c c . Including the contribution from the backreacted minimal area, we find agreement between the CFT result and the FLM and JLMS formulas for quantum corrections to holographic entanglement entropy. This provides a non-trivial check in a state where the answer is not dictated by symmetry. Along the way, we provide closed-form expressions for the scalar field Bogoliubov coefficients that relate the global and Rindler slicings of AdS _3 3 .

DOI
10.21468/scipostphys.5.3.024

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We compute the bulk entanglement entropy across the Ryu-Takayanagi surface for a one-particle state in a scalar field theory in AdS _3 3 . We work directly within the bulk Hilbert space and include the spatial spread of the scalar wavefunction. We give closed form expressions in the limit of small interval sizes and compare the result to a CFT computation of entanglement entropy in an excited primary state at large c c . Including the contribution from the backreacted minimal area, we find agreement between the CFT result and the FLM and JLMS formulas for quantum corrections to holographic entanglement entropy. This provides a non-trivial check in a state where the answer is not dictated by symmetry. Along the way, we provide closed-form expressions for the scalar field Bogoliubov coefficients that relate the global and Rindler slicings of AdS _3 3 .
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A bstract We consider single-particle excitations of the massive Chern-Simons field of mass M in AdS 3 and evaluate their contribution at the first sub-leading order in G N to the entanglement entropy across the Ryu-Takayanagi surface. Quantizing the Chern-Simons field in AdS 3 , we evaluate the corrections to the holographic entanglement entropy using the Faulkner-Lewkowycz-Maldacena formula. The massive Chern-Simons field also obeys the equations of motion of a massive vector in AdS 3 . The lowest-energy single-particle excitation of this field is dual to the primary operator of conformal dimension M + 1 with spin one in the dual CFT; all other single-particle excitations are dual to its global descendants. We compare the entanglement entropy result from the FLM formula to the single-interval entanglement entropy in large-charge holographic CFT obtained using the replica trick for the primary and its tower of holomorphic descendants. The two results agree precisely in the leading and sub-leading terms of the short interval expansion. We evaluate the contribution of the edge mode to the vacuum-subtracted entanglement and show that it vanishes, which is crucial for the FLM formula to agree with the CFT result. On taking the massless limit, the result coincides with the contribution of a U (1) current to the single interval entanglement entropy. This is surprising since an earlier calculation in the literature reproduced the CFT result entirely from the edge U (1) degrees of freedom on the RT surface.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
Abstract Quantum corrections to the entanglement entropy of matter fields interacting with dynamical gravity have proven to be very important in the study of the black hole information problem. We consider a one-particle excited state of a massive scalar field infalling in a pure AdS3 geometry and compute these corrections for bulk subregions anchored on the AdS boundary. In the dual CFT2, the state is given by the insertion of a local primary operator and its evolution thereafter. We calculate the area and bulk entanglement entropy corrections at order O(N0), \mathcal{O}\left({N}^0\right), O(N0), O N 0 , both in AdS and its CFT dual. The two calculations match, thus providing a non-trivial check of the FLM formula in a dynamical setting. Further, we observe that the bulk entanglement entropy follows a Page curve. We explain the precise sense in which our setup can be interpreted as a simple model of black hole evaporation and comment on the implications for the information problem.
Quantum corrections to the entanglement entropy of matter fields interacting with dynamical gravity have proven to be very important in the study of the black hole information problem. We consider a one-particle excited state of a massive scalar field infalling in a pure AdS3geometry and compute these corrections for bulk subregions anchored on the AdS boundary. In the dual CFT2, the state is given by the insertion of a local primary operator and its evolution thereafter. We calculate the area and bulk entanglement entropy corrections at order O(N0), \mathcal{O}\left({N}^0\right), O(N0), ON0, both in AdS and its CFT dual. The two calculations match, thus providing a non-trivial check of the FLM formula in a dynamical setting. Further, we observe that the bulk entanglement entropy follows a Page curve. We explain the precise sense in which our setup can be interpreted as a simple model of black hole evaporation and comment on the implications for the information problem.
A bstract Using recent developments in expressing one-loop partition functions in Euclidean AdS 2 space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in AdS 2 (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global AdS 2 boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the CFT 1 (CQM), each living near one of the two boundaries of global AdS 2 , in a state at global time τ → – ∞. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean AdS 2 disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the AdS 2 / CFT 1 version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in AdS 2 . Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian AdS 2 and show that it precisely matches the boundary entanglement entropy.