Holographic Entanglement Entropy and Black Holes Tadashi Takayanagi(IPMU, Tokyo) based on arXiv: JHEP 11(2011) with Tomoki Ugajin (IPMU) arXiv: with Wei Li (IPMU) Indian Strings Puri, Jan 4-11, 2011

① Introduction AdS/CFT has been a very powerful tool to understand the physics of black holes. [Strominger-Vafa 96’  AdS3/CFT2] In spite of tremendous progresses, there are still several unsolved or developing important problems on black holes in quantum gravity. The one which we would like to discuss here is about the dynamical aspects where we cannot employ the supersymmetries.

For example, consider the following problem: (1) Complete understandings on how the BH information problem is avoided in string theory ? [For static BH, there have been considerable developments: e.g. Maldacena 01’, Festuccia-Liu 07’, Hayden-Preskill 07’, Sekino-Susskind 08’, Iizuka-Polchinski 08’ …] How to measure creations and annihilations of BHs in holography ? In this talk, I would like to argue that the holographic entanglement entropy is a useful measure for this purpose.

A Quick Sketch of BH Information Problem A lot of matter ⇒ Pure state Black hole formation Thermal radiations (BH evaporation) ⇒ Mixed state ?? Contradict with QM ! Gravitational collapse

BH formation in AdS/CFT Thermalization in CFT BH formation in AdS [e.g. Chesler-Yaffe 08’, S. Bhattacharyya and S. Minwalla 09’] In particular, instantaneous excitations of CFTs are called quantum quenches. [e.g. Calabrese-Cardy 05’-10’] e.g. Mass quench Time BH AdS t m(t) m(t)=m m(t)=0: CFT

Cf. Thermalization in the probe D-brane AdS/CFT [Das-Nishioka-TT 10’] Thermalization in a probe D-brane ⇔ Horizon formation in its induced metric on the D-brane r Bulk observer Brane observer Emergent BH Probe D-brane AdS

Entropy Puzzle in AdS/CFT (a `classical analogue’ of information paradox) (i)In the CFT side, the von-Neumann entropy remains vanishing under a unitary evolutions of a pure state. (ii ) In the gravity dual, its holographic dual inevitably includes a black hole at late time and thus the entropy looks non-vanishing ! Thus, naively, (i) and (ii) contradict ! We will resolve this issue using entanglement entropy and study quantum quenches as CFT duals of BH creations and evaporations.

Another important question is (2) Entropy of Schwarzschild BH in flat spacetime ? Is there any holography in flat spacetime ? We will present one consistent outline about how the holography in flat space looks like by employing the entanglement entropy. e.g. Volume law (flat space) ⇔ Area law (AdS/CFT) highly entangled !

Contents ① Introduction ② BH Formations as Pure States in AdS/CFT ③ Entanglement Entropy as Coarse-grained Entropy ④ Holography and Entanglement in Flat Space ⑤ Conclusions

(2-1) Holographic Entanglement Entropy Divide a given quantum system into two parts A and B. Then the total Hilbert space becomes factorized We define the reduced density matrix for A by taking trace over the Hilbert space of B. A B Example: Spin Chain ② BH formations as Pure States in AdS/CFT

Now the entanglement entropy is defined by the von-Neumann entropy In QFTs, it is defined geometrically (called geometric entropy).

Holographic Entanglement Entropy The holographic entanglement entropy is given by the area of minimal surface whose boundary is given by. [Ryu-TT, 06’] (`Bekenstein-Hawking formula’ when is the horizon.)

Comments We need to employ extremal surfaces in the time-dependent spacetime. [Hubeny-Rangamani-TT, ] In the presence of a black hole horizon, the minimal surfaces typically wraps the horizon. ⇒ Reduced to the Bekenstein-Hawking entropy, consistently. Many evidences and no counter examples for 5 years, in spite of the absence of complete proof.

EE from AdS BH (i) Small A (ii) Large A

(2-2) Resolution of the Puzzle via Entanglement Entropy Our Claim: The non-vanishing entropy appears only after coase- graining. The von-Neumann entropy itself is vanishing even in the presence of black holes in AdS. First, notice that the (thermal) entropy for the total system can be found from the entanglement entropy via the formula This is indeed vanishing if we assume the pure state relation S A =S B.

Indeed, we can holographically show this as follows: [Hubeny-Rangamani-TT 07’] A Time Continuous deformation leads to S A =S B BH

Therefore, if the initial state does not include BHs, then always we have S A =S B and thus S tot =0. In such a pure state system, the total entropy is not useful to detect the BH formation. Instead, the entanglement entropy S A can be used to probe the BH formation as a coarse-grained entropy. [CFT calculation: Calabrese-Cardy 05’,..., (Finite Size effect ⇒ our work) Time evolutions of Hol EE: Hubeny-Rangamani-TT 07’ Arrastia-Aparicio-Lopez 10’, Albash-Johnson 10’ Balasubramanian-Bernamonti-de Boer-Copland-Craps- Keski-Vakkuri-Müller-Schäfer-Shigemori-Staessens 10’] t Quantum quench BH entropy S coase-grained ＝ S A (t)-S div

③ Entanglement Entropy as Coarse-grained Entropy [Ugajin-TT 10’] (3-1) Evolution of Entanglement Entropy and BH formation As an explicit example, consider the 2D free Dirac fermion on a circle. AdS/CFT: free CFT quantum gravity with a lot of quantum corrections ! Assuming that the initial wave function flows into a boundary fixed point as argued in [Calabrese-Cardy 05’], we can approximate by where is the boundary state. The constant is a regularization parameter and measures the strength of the quantum quench:.

Calculations of EE (Replica method)

The final result of entanglement entropy is given by This satisfies σ B A

cf. Finite Size System at Finite Temperature (2D free fermion c=1 on torus) [07’ Azeyanagi-Nishioka-TT] σ SASA Thermal Entropy

Time evolution of entanglement entropy Time BH Holography BH formation and evaporation in extremely quantum gravity No information paradox at all in either side ! Quantum quench Quantum quench in free CFT

(3-2) More comments First of all, we can confirm that the scalar field X (= bosonization of the Dirac fermion) is thermally excited:

Correlation functions Size of BH Radiations Exponential decay ⇒ Thermal

Semi Classical AdS BHs ? For semi classical BHs, which are much more interesting, the dual CFT gets strongly coupled as usual and thus the analysis of the time evolution is difficult. However, we can still guess what will happen especially in the AdS3 BH case. S coase-grained ＝ S A (t)-S div t Quantum quench BH entropy Universal (for any CFT) Poincare recurrence

④ Holography and Entanglement in Flat Space [Li-TT 10’] The entanglement entropy is a suitable observable in general setup of holography as in our BH example. Motivated by this, finally we would like to discuss what a holography for the flat spacetime looks like. So, simply consider Note: We may regard this as a logarithmic version of Lifshitz backgrounds:

Correlation functions We can compute the holographic n-point functions following the bulk-boundary relation just like in AdS/CFT. Assuming a scalar in R d+1 like the dilaton: Then we find that all n-point functions scale simply: ⇒ Only divergent terms appear ! Adding the (non-local) boundary counter term, all correlation functions become trivial !

Holographic Entanglement Entropy (HEE) Though this result seems at first confusing, actually it is consistent with the holographic entanglement entropy. It is easy to confirm that the HEE follows the volume law rather than the standard area law ! A B θ

This unusual volume law implies that the subsystem A gets maximally entangled with B when A is infinitesimally small. Therefore, we have the trivial correlation functions: At the same time, the volume law argues that the holographic dual of flat space is given by a highly non-local theory.

Calculations of EE in Non-local Field Theory we can calculate EE for θ=π as follows: In the end we find, Thus the volume law corresponds to the non-local action q=1/2 i.e. [Note: a bit similar to open string field theory]

Possible Relation to Schwarzschild BH Entropy This can be comparable to the Schwarzschild BH entropy: This suggests that the BH entropy may be interpreted as the entanglement entropy…

⑤ Conclusions We raised a puzzle on the entropy in the thermalization in AdS/CFT and resolved by showing the total von-Neumann entropy is always vanishing in spite that the AdS includes BHs at late time. We present a toy model of holographic dual of BH formations and evaporations using quantum quenches. We discussed a consistent picture of holography for flat space and argued that the dual theory is should be non-local and is highly entangled. Future problems: possible holography for de Sitter spaces and cosmological backgrounds