1. Entanglement Entropy from AdS/CFT Tadashi Takayanagi (Kyoto Univ.) Based on hep-th/0603001, 0605073, 0608213, 0611035, arXiv:0704.3719, 0705.0016, 0710.2956, 0712.1850 with S. Ryu (KITP), V. Hubeny, M. Rangamani (Durham) M. Headrick (Stanford) T. Azeyanagi, T. Hirata, T. Nishioka (Kyoto) A. Karch and E. Thompson (Washington) Miami 2007 Celebrating ten years of AdS/CFT

2. AdS/CFT has been successfully studied for ten years as the best example of holography. Holography is expected to relate quantum gravity to various quantum mechanical systems. To understand holography from general viewpoints, we need to find a universal quantity as a physical observable. In this talk, we would like to point out that the quantity called entanglement entropy is such a candidate. ① Introduction

3. What is the entanglement entropy (EE) ? A measure how much a given quantum state is quantum mechanically entangled (or complicated). (A) It is universal in that it is well defined in any quantum mechanical systems. (B) In QFT, it has the properties of both entropy and correlation functions.

4. ?? ? Entangled Not Entangled The simplest example: two electrons (two qubits)

5. Definition of entanglement entropy Divide a given quantum system into two parts A and B. Then the total Hilbert space becomes factorized A B Example: Spin Chain

6. Now the entanglement entropy is defined by the von-Neumann entropy We define the reduced density matrix for A by taking trace over the Hilbert space of B.

7. In this talk we consider the entanglement entropy in quantum field theories on (d+1) dim. spacetime Then, we divide into A and B by specifying the boundary. A B

8. An analogy with black hole entropy As we have seen, the entanglement entropy is defined by smearing out the Hilbert space for the submanifold B. EE ~ `Lost Information’ hidden in B This origin of entropy looks similar to the black hole entropy. The boundary region ~ the event horizon.

9. Area law divergences Its leading term is proportional to the area of the (d-1) dim. boundary [Bombelli-Koul-Lee-Sorkin 86’, Srednicki 93’] where is a UV cutoff (i.e. lattice spacing). Very similar to the Bekenstein-Hawking formula of black hole entropy

10. ② Holographic Entanglement Entropy We assume the AdS/CFT in the supergravity limit. But we can generalize this to other setups of holography. Example. Poincare coordinate

11. Holographic Computation [06’ Ryu-TT] (1)Divide the space N is into A and B. (2) Extend their boundary to the entire AdS space. This defines a d dimensional surface. (3)Pick up a minimal area surface and call this. (4)The E.E. is given by naively applying the Bekenstein-Hawking formula as if were an event horizon.

13. Comments (i)We can compare AdS and CFT results exactly in AdS3/CFT2 case and find perfect agreements. (ii) The holographic formula can be derived from the bulk-boundary relation of GKPW. [Fursaev 06’] (iii) It is straightforward to derive the area law divergence of EE, induced from the warp factor of AdS spaces.

14. (iv) In the presence of the horizon, the EE for the total system coincides with the BH entropy. [ This is consistent with earlier pioneering works: Maldacena 01’ and Hawking-Maldacena-Strominger 00’ ; See also Emparan 06’] (v) The EE generally obeys the inequality called strong subadditivity. Our holographic formula clearly satisfies this. [Headrick-TT 07’] (vi)In the time-dependent b.g., we need a covariant description. It turns that the minimal surface is replaced with the extremal surface in any Lorentzian spacetime. [Hubeny-Rangamani-TT 07’]

15. (3-1) EE from AdS3/CFT2 Consider AdS3 in the global coordinate In this case, the minimal surface is a geodesic line which starts at and ends at ( ). ③ Various Examples Boundary

16. The length of, which is denoted by, is found as Thus we obtain the prediction of the entanglement entropy where we have employed the celebrated relation [Brown-Henneaux 86’]

17. Furthermore, the UV cutoff a is related to via In this way we reproduced the known formula [94’ Holzhey-Larsen-Wilczek, 04’ Calabrese-Cardy ] (In 2D CFT, we can analytically compute EE in various setups owing to the conformal map technique. But this is not so in higher dimensions.) [04’ Calabrese-Cardy ]

18. Finite temperature case We assume the length of the total system is infinite. In this case, the dual gravity background is the BTZ black hole and the geodesic distance is given by This again reproduces the known formula at finite T.

19. Geometric Interpretation (i) Small A (ii) Large A

20. Let us compute the holographic EE dual to a CFT on R 1,d. We concentrate on the following two examples. (a) Infinite Strip (b) Circular disk (3-2) Higher Dimensional Case A A

21. Entanglement Entropy for (a) Infinite Strip from AdS

22. Entanglement Entropy for (b) Circular Disk from AdS Conformal Anomaly (~central charge) A universal quantity which characterizes odd dimensional CFT

23. (3-3) Entanglement Entropy in 4D CFT from AdS5 Consider the basic example of type IIB string on, which is dual to 4D N=4 SU(N) super Yang-Mills theory. In this case, we obtain the prediction from supergravity (dual to the strongly coupled Yang-Mills) We would like to compare this with free Yang-Mills result.

24. Comparison with free field theory result On the other hand, the AdS results numerically reads The order one deviation is expected since the AdS result corresponds to the strongly coupled Yang-Mills.. [cf. 4/3 problem in thermal entropy, Gubser-Klebanov-Peet 96’]

25. (3-4) EE in confining gauge theories In many confining examples (AdS soliton, KS solution), we find a phase transition when we change the width l of A. [Nishioka-TT 06’, Klebanov-Kutasov-Murugan 07’] S A -S div l Confinement / deconfinement transition z l Minimal Surface Disconnected Surfaces l

26. Twist parameter Entropy (finite part) Free Yang-Mills (sum of KK modes) AdS side (Strongly coupled YM) Supersymmetric Point Comparison with Free Yang-Mills on Twisted Circle AB

27. ⑥ Conclusions We have found the holographic area formula of entanglement entropy via AdS/CFT duality. [Other our recent topics] If we apply our argument to the AdS2/CFT1, we find EE of two CQMs = Wald entropy formula. [Azeyanagi-Nishioka-TT, 07’] In the defect or interface CFTs, we can extract the boundary entropy (g-function) from EE. It is holographically computable in Janus b.g. [Azeyanagi-Karch-Thompson-TT, 07’] Can we reconstruct the spacetime metric from the information of EE in the dual CFT ??

28. Boundary entropy obtained from 2D Interface CFT and Janus Free interface CFT 3D Janus Deformation Boundary entropy （ =log g)