Lessons for gravity from entanglement

Transcription

1 Lessons for gravity from entanglement A thesis submitted for the degree of Doctor of Philosophy in the Faculty of Sciences Arpan Bhattacharyya Centre for High Energy Physics Indian Institute of Science Bangalore India. June 2015

2 ii

3 Declaration I hereby declare that the work presented in this thesis Lessons for gravity from entanglement is based on the research work by me under the supervision of Prof. Aninda Sinha and with my collaborators at the Centre for High Energy Physics, Indian Institute of Science, Bangalore, India. It has not been submitted elsewhere as a requirement for any degree or diploma of any other Institute or University. Proper acknowledgements and citations have been made in appropriate places while borrowing research materials from other investigations. Date : Arpan Bhattacharyya Certified by : Prof. Aninda. Sinha Centre for High Energy Physics Indian Institute of Science Bangalore India iii

4 iv

5 List of publications This thesis is based on the following publications : 1. Entanglement entropy in higher derivative holography A. Bhattacharyya, A. Kaviraj and A. Sinha. arxiv: [hep-th] JHEP 1308, 012 (2013) 2. On generalized gravitational entropy, squashed cones and holography A. Bhattacharyya, M. Sharma and A. Sinha. arxiv: [hep-th] JHEP 1401, 021 (2014) 3. Constraining gravity using entanglement in AdS/CFT S. Banerjee, A. Bhattacharyya, A Kaviraj, K. Sen and A. Sinha. arxiv: [hep-th] JHEP 1405, 029 (2014) 4. On entanglement entropy functionals in higher derivative gravity theories A. Bhattacharyya and M. Sharma. arxiv: [hep-th] JHEP 1410, 130 (2014) 5. Renormalized Entanglement Entropy for BPS Black Branes A. Bhattacharyya, S. S. Haque and A. Veliz-Osorio. arxiv: [hep-th] Phy. Rev. D 91, (2015) The following works were done during my PhD but is not included in this thesis : 1. On c-theorems in arbitrary dimensions A. Bhattacharyya, L. Y. Hung, K. Sen and A. Sinha. arxiv: [hep-th] Phy. Rev. D 86, (2012) 2. Entanglement entropy from the holographic stress tensor A. Bhattacharyya and A. Sinha. arxiv: [hep-th] Class. Quantum Grav 30, (2013) 3. Entanglement entropy from surface terms in general relativity A. Bhattacharyya and A. Sinha. arxiv: [hep-th] IJMPD 22 12, (2013) 4. Attractive holographic c-functions A. Bhattacharyya, S. S. Haque, V. Jejjala, S. Nampuri and A. Veliz-Osoio. arxiv: [hep-th] JHEP 1411, 138 (2014) v

6 vi 5. Viscosity bound for anisotropic superfluids in higher derivative gravity A. Bhattacharyya and D. Roychowdhury. arxiv: [hep-th] JHEP 1503, 063 (2015) 6. Lifshitz Hydrodynamics And New Massive Gravity A. Bhattacharyya and D. Roychowdhury. arxiv:

7 Synopsis One of the recent fundamental developments in theoretical high energy physics is the AdS/CFT correspondence [1, 2, 3, 4] which posits a relationship between Quantum Field Theories (QFT) in a given dimension and String Theory on a higher dimensional anti- de Sitter (AdS) spacetime. This has revolutionised our understanding of QFTs (more specifically conformal field theories (CFTs)) and string theory/gravity, and has far reaching consequences for explorations into a vast array of physical phenomena. Using the elegant formalism provided by this powerful duality, often called holography, one can now use fundamental physical observables in QFT to better understand the nature of quantum gravity. The theoretical tools provide a translation of calculable field theoretic observables into the language of gravity thereby leading to the construction of holographic models for several interesting QFTs. Entanglement is a fundamental physical property of all quantum systems. From models of various condensed matter systems to its application as a tool for secure and fast communication in quantum information theory [5], it serves as an intersection point between different subfields of physics [6]. From the AdS/CFT point of view quantum entanglement connects geometry with quantum information, providing a window to understand how the bulk gravity physics emerges from the holographic field theoretic viewpoint. Probing various aspects of this connection in detail will be the broad theme of this thesis. For extended, many-body systems, the most well known measure of quantum entanglement is the Entanglement Entropy (EE) which is also the best understood measure within the holographic framework. In early 2006, Ryu and Takayanagi (RT) gave a simple and elegant prescription for computing this quantity using AdS/CFT duality within Einstein gravity [7, 8]. They proposed that EE for a subsystem within an extended system (QFT), is computed by the (proper) area of a static, codimension- 2, extremal surface inside the dual AdS spacetime. The RT proposal has passed several non-trivial consistency checks, for example strong sub-additivity, area law to name a few [9]. A remarkable aspect of the proposal is the ease with which EE can now be calculated, while it is well known that obtaining EE from first principles in QFT presents several technical challenges which have so far been surmounted only in some 2d field theories using the replica method [10, 11, 12]. The most intriguing aspect of the RT proposal is its striking similarity to Bekenstein- Hawking (BH) entropy which is proportional to the area of a black hole horizon, further confirming an intimate relationship between entropy and geometry [13, 14, 15, 16]. This leads to the natural question: what is the connection between EE and BH entropy? This question has been sharpened recently by Lewkowycz and Maldacena (LM) via the concept of Generalized Gravitational Entropy which extends the QFT replica trick to a replica symmetry vii

8 for the dual space-time [17]. This was used to prove the RT conjecture successfully by deriving the correct extremal surface equation for two derivative gravity theories. In this thesis I have studied the generalization of LM method for higher derivative gravity theories [18, 19, 20, 21, 22, 23] describing holographic duals (of QFT s with finite number of colours) and finite t Hooft coupling which takes the AdS/CFT correspondence beyond the usual supergravity limit. If one wants to use AdS/CFT to study real life systems then it is absolutely necessary to incorporate the finite coupling effect into the theory and hence the study of higher derivative effects becomes very important. In these two papers [21, 22] I have formulated a proof for the existence of the entropy functionals for certain higher derivative theories extending LM method. We have shown that the for a certain special class of higher derivative theories there exist well defined entropy functionals. To extend this proof for more general theories of gravity opens up a possibility of breaking replica symmetry in the bulk space-time [24]. For higher derivative gravity, black hole entropy for a large class of stationary black holes with bifurcate killing horizon is given by the well known Wald prescription [25, 26, 27] which relates the concept of the Noether charge with the black hole entropy. Iyer and Wald proposed a generalization for dynamical horizons. This throws up the question whether there is a relation between these EE functionals and the Noether charge, and whether we can derive them using the approach of Iyer and Wald. For a certain class of theories I have shown that there exists a relation between these two [28] but a more rigorous proof is needed. This somewhat firms up the area-entropy relation for arbitrary surfaces and proves the existence of holographic EE functionals for higher curvature theories thereby extending the applicability of Iyer-Wald formalism beyond the bifurcation surface. Apart from this, it is well known that there exist several measures of quantum entanglement, each satisfying a variety of mathematical inequalities and conditions [5]. Translating these into the language of holography constrains the dual gravity theory and will lead to general statements about the consistency of the theory. In this thesis I have discussed one such measure namely Relative entropy [29], the positivity of which has led to constraints on the underlying gravity theory [30]. Also entanglement entropy is a very useful tools for probing renormalization group (RG) flow from the holographic point of view [34, 31, 32, 35, 33]. We end with exploring the concept of renormalized entanglement entropy [36, 37] and its application in probing RG flow in the context of N = 2 gauged supergravity [38]. References [1] J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998), [arxiv:hep-th/ ]

9 [2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B428 (1998) 105, [arxiv:hep-th/ ] [3] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253, [arxiv:hep-th/ ] [4] O. Aharony, S. S. Gubser, J. Maldacena H. Ooguri and Y. Oz, Large N field theories, String theory and gravity, Phys. Rept, 323 (2000) , [arxiv:hep-th/ ]. [5] Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 23-Oct-2000 [6] J. Eisert, M. Cramer and M. B. Plenio, Area laws for the entanglement entropy - a review, Rev. Mod. Phys. 82 (2010) 277 [arxiv: [quant-ph]] and the references there in. [7] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) [hep-th/ ]. [8] T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) [arxiv: [hep-th]]. [9] S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 0608 (2006) 045 [hep-th/ ] and the references there in. [10] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424 (1994) 443 [hep-th/ ]. [11] P. Calabrese and J. L. Cardy, Entanglement entropy and conformal field theory, Journal of Physics A: Mathematical and Theoretical, Volume 42, Issue 50, article id , 36 pp. (2009). [12] P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech (2004) P06002 [hep-th/ ]. [13] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin, Quantum source of entropy for black holes, Phys. Rev. D 34, 373 [14] M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/ ]. [15] S. N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arxiv: [hep-th]] and the references there in.

10 [16] E. Bianchi and R. C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 21, [arxiv: [hep-th]]. [17] A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 1308 (2013) 090 [arxiv: [hep-th]]. [18] L. Y. Hung, R. C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher Curvature Gravity, JHEP 1104 (2011) 025 [arxiv: [hep-th]]. [19] X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP 1401 (2014) 044 [arxiv: [hep-th], arxiv: ]. [20] J. Camps, Generalized entropy and higher derivative Gravity, JHEP 1403 (2014) 070 [arxiv: [hep-th]]. [21] A. Bhattacharyya, A. Kaviraj and A. Sinha, Entanglement entropy in higher derivative holography, JHEP 1308 (2013) 012 [arxiv: [hep-th]]. [22] A. Bhattacharyya and M. Sharma, On entanglement entropy functionals in higher derivative gravity theories, JHEP 1410 (2014) 130 [arxiv: [hep-th]]. [23] R. X. Miao and W. z. Guo, Holographic Entanglement Entropy for the Most General Higher Derivative Gravity, arxiv: [hep-th]. [24] J. Camps and W. R. Kelly, Generalized gravitational entropy without replica symmetry, JHEP 1503 (2015) 061 [arxiv: [hep-th]]. [25] R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/ ]. [26] V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/ ]. [27] V. Iyer and R. M. Wald, A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/ ]. [28] A. Bhattacharyya, M. Sharma and A. Sinha, On generalized gravitational entropy, squashed cones and holography, JHEP 1401 (2014) 021 [arxiv: [hep-th]]. [29] D. D. Blanco, H. Casini, L. Y. Hung and R. C. Myers, Relative Entropy and Holography, JHEP 1308 (2013) 060 [arxiv: [hep-th]]. T. Faulkner, M. Guica, T. Hartman, R. C. Myers and M. Van Raamsdonk, Gravitation

11 from Entanglement in Holographic CFTs, JHEP 1403 (2014) 051 [arxiv: [hep-th]]. [30] S. Banerjee, A. Bhattacharyya, A. Kaviraj, K. Sen and A. Sinha, Constraining gravity using entanglement in AdS/CFT, JHEP 1405 (2014) 029 [arxiv: [hep-th]]. [31] H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/ ]. [32] H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/ ]. [33] R. C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 1101 (2011) 125 [arxiv: [hep-th]]. [34] H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) [arxiv: [hep-th]]. [35] H. Casini and M. Huerta, Positivity, entanglement entropy, and minimal surfaces, JHEP 1211 (2012) 087 [arxiv: [hep-th]]. [36] H. Liu and M. Mezei, A Refinement of entanglement entropy and the number of degrees of freedom, JHEP 1304 (2013) 162 [arxiv: [hep-th]]. [37] H. Liu and M. Mezei, Probing renormalization group flows using entanglement entropy, JHEP 1401 (2014) 098 [arxiv: [hep-th], arxiv: ]. [38] A. Bhattacharyya, S. Shajidul Haque and A. Veliz-Osorio, Renormalized Entanglement Entropy for BPS Black Branes, Phys. Rev. D 91 (2015) 4, [arxiv: [hep-th]].

12

13 Acknowledgements First and foremost, I would like to convey my sincere thanks to my advisor, Aninda Sinha for his generous support as well as outstanding guidance during the entire tenure of my doctoral research. Being his first PhD student was always a stimulating experience. I thank him for guiding me throughout my doctoral work and helping me to complete my PhD in just 3 years. Apart from learning a great deal of physics from him, he has helped me a lot to improve my soft skills. I am also thankful to Menika Sharma, Ling-Yan (Janet) Hung, Shajid Haque, Alvaro Veliz Osorio, Vishnu Jejjala, Suresh Nampuri and Dibakar Roychowdhury for useful collaborations and numerous productive discussions which has added a lot towards my understanding of the subject itself. I would also like to thank Rajesh Gopakumar, Jose Edelstein, Axel Kleinschmidt, Johanna Erdmenger, Tadashi Takayanagi, Heng-Yu Chen, Janet Hung, Shamik Banerjee, Vishnu Jejjala for inviting me to give seminars. I am also thankful to them for numerous stimulating discussions in various occasions. I am also thankful to Ashoke Sen, Rob Myers and Joan Camps for valuable discussions. I would also like to thank all the professors of the Department of Physics and the Centre for High Energy Physics for providing beautiful courses. In particular, I would like to thank Prof. Justin David for providing a beautiful course on QFT. I thank the Chairman of CHEP, B Ananthanarayan for striving to maintain a vibrant and simulating atmosphere in the department. I am indeed grateful to my Integrated PhD batchmates and colleagues at the Centre for High Energy Physics for creating friendly and competitive atmosphere. During my stay at IISc, I found various departmental activities like the weekly math-phys meets, journal club sessions, seminars and colloquia etc. as quite stimulating and in particular playing a very crucial role in developing the scientific mind. Finally, I would specially like to thank Apratim and Shouvik for helping me enormously with all the diagrams and the Latex. I thank the Indian Institute of Science for their generous financial support for attending numerous conferences and visiting other research institutes in India and abroad during my tenure. Finally, I m very thankful to my parents for giving me constant support and never giving up hope on me. xiii Arpan Bhattacharyya Bangalore, June 2015.

14 xiv

15 xv To my parents.

16 xvi

17 xvii Somewhere, something incredible is waiting to be known. Carl Sagan

18 xviii

19 Contents 1 Introduction Introduction Quantum Entanglement Entanglement entropy and Holography Holographic entanglement entropy functionals: A derivation Introduction Entropy functional for general theories of gravity Test of the entropy functional for R 2 theory Minimal surface condition from the entropy functional Minimal surface condition from the Lewkowycz-Maldacena method The stress-energy tensor from the brane interpretation Quasi-topological gravity The entropy functional Universal terms Minimal surface condition Discussion Entanglement entropy from generalized entropy Introduction Generalized entropy and Fefferman-Graham expansion Four derivative theory New Massive Gravity Quasi-Topological Gravity α 3 IIB supergravity Comment about singularities in the metric Discussion Connection between entanglement entropy and Wald entropy 81 xix

20 xx CONTENTS 4.1 Introduction Wald Entropy Four derivative theory Cylinder Sphere Quasi-Topological gravity α 3 IIB supergravity Connection with Ryu-Takayanagi Comments on the connection with the Iyer-Wald prescription Universality in Renyi entropy Discussion Constraining gravity using entanglement entropy Introduction Smoothness of entangling surface Discussion Relative entropy Introduction Relative entropy considerations Relative entropy in Gauss-Bonnet holography Linear order calculations Quadratic corrections Constant T µν Shockwave background Correction from additional operators Relative entropy for an anisotropic plasma Discussion Coding holographic RG flow using entanglement entropy Introduction Renormalized Entanglement Entropy BPS black objects in AdS

21 CONTENTS xxi 7.4 REE for BPS black branes Discussion Conclusions 145

22 xxii

23 1 Introduction 1.1 Introduction The concept of entanglement is a very old one and dates back to early 1930 s when quantum mechanics was born. One of the measures of quantum entanglement is the entanglement entropy. Although it has played a crucial role in understanding some aspects of quantum mechanics, its application remains rather limited mainly due to its non local behaviour until late 90 s. In early 1970 s Bekenstein proposed that the black hole entropy (S BH ) follows an area law [1]. S BH = A 4G N (1.1) where A is the area of the horizon and G N is the Newton constant. This formula is quite counter-intuitive as entropy is usually an extrinsic quantity, depends on the volume of the system. Later Stephen Hawking showed that a black hole emits radiation with a well defined temperature [2], thereby establishing the concept of black hole entropy. It was observed that the calculation of the entropy associated with the radiation emitted from the black hole is plagued by the presence of ultraviolet divergences. These divergences can be associated with the particles close to the horizon [3] and one has to regulate them to get a finite answer for the entropy. Then in early 1980 s Bombelli, Koul, Lee and Sorkin in their seminal work [4] showed that one can possibly understand that black hole entropy using the concept of entanglement. To the observers outside the black hole horizon there is no information about the spacetime inside the horizon. They considered scalar fields in the black hole background and traced out the spacetime inside the horizon, thereby defining a reduced density matrix for the system. Using this, they computed the von-neumann entropy and it was shown that the entropy follows an area law. Later it was generalized by Srednicki [5] for massless scalar fields in flat spacetime. He also showed that if one divides the spacetime in two parts, then the entropy associate with the reduce density matrix for one of these two parts is proportional to the area of the boundary between these two halves. Later this concept of entanglement entropy (EE) was made concrete by Callan, Holzhey, Larsen and Wilczek [6] and separately by Susskind and Uglum [7]. From their work it is evident that the EE exhibits a universal 1

24 INTRODUCTION behaviour, it goes as a logarithm of correlation length in dimensions after suitably regulating the ultraviolet divergences. But still its application remain rather limited as in general it is very hard to compute this quantity for generic field theories. But in early 2000, Calabrese and Cardy used the Replica Trick formulated by Callan, Holzey, Larson and Wilczek successfully to compute the EE for many cases in the context of 1+1 dimensional conformal field theory (CFT) [8], thereby increasing its physical importance. After that EE has been calculated extensively not only for 1+1 dimensional CFT but also for various other simple quantum field theories both analytically and numerically. Also in recent times it has been successfully computed numerically using the lattice technique for quantum many body systems [9]. Although the techniques employed for computing EE is very hard and yet to be developed fully, still in recent times, we have lots of data regarding quantum entanglement and EE coming from both analytical and numerical approaches [10, 11]. Recently EE has found many applications in various branches of physics like, quantum information, Figure 1.1: Entanglement and its diverese applications condensed matter system, statistical physics and in AdS/ CFT Prof. Robert. C. Myers in the Conference Entangle- in physics. (Picture courtesy- From the talk given by [9, 10, 11, 12, 13, 14]. It serves as an ment from gravity, ICTS, 2014, Bangalore, India) intersecting point between various subfields of physics. In recent times it has played a crucial role in understanding the nature of holography (AdS/ CFT correspondence). AdS/ CFT correspondence, commonly known as Holography is one of the most important dualities in physics. It postulates that gravity emerges from a certain class of field theories. It is still not known rigorously how holography works from first principles. EE has the merit to shed light on this problem as it connects quantum information of the system with geometry. There exists an interesting connection between geometry and EE and our goal in this thesis will be to explore some aspects of this connection. Recently apart from EE,

25 CHAPTER 1. INTRODUCTION 3 many different tools of quantum entanglement like, entanglement negativity, differential entropy, quantum error coding, relative entropy, cmera (continuous multiscale entanglement renormalization ansatz) etc, have been used in an attempt to build geometry from the field theory data, is some sense trying to prove the holographic principle [15, 16, 17, 18, 19, 20]. But still EE plays the central character in this program and we will use this to learn many lessons of gravity in the holographic set up Quantum Entanglement Let us first consider a quantum mechanical system consists of two spin 1 particles. We 2 denote the total Hilbert space by H. Now H A and H B denote the Hilbert spaces of the two individual particles. Also, H = H A H B. Now consider the following state belonging to H, ψ 1 >= 1 2 ( + >) A ( + >) B. (1.2) This state is not an entangled state as there is no correlation between the two particles in this state. On the other hand, if we consider the following state ψ 2 >= 1 2 ( A B > B A >), (1.3) then one cannot factorize this state in terms of the individual particle states. So this state is an example of entangled state. In other words, entanglement is a property of a quantum mechanical system that tells us, that one cannot describe the underlying pair of particles belonging to this particular state independently. All the physical properties of the two individual particles are correlated with each other. This argument can be extended for any number of particles and interesting things shows up when one consider many body systems due to the non local nature of the entanglement. One of the measures of entanglement is the entanglement entropy (EE). Let us first see how we can define this. First step is to define a density matrix for the full system. If ψ > is the wavefunction characterizing the total system then the density matrix can be defined in the following way, ρ = ψ >< ψ. (1.4) Next step is to define the reduced density matrix. Suppose we want to compute the EE for the sub-system A. Then we first trace out the degrees of freedom corresponding the system B. ρ A = T r B ρ =< b ρ b >, (1.5)

26 INTRODUCTION where ρ A is the reduced density matrix for the subsystem A. Now we define EE by defining the von-neumann entropy which is, S EE (A) = T r A ρ A ln ρ A. (1.6) Here we have completed the trace over the subsystem A. Now this is a good time to cook up some examples and elucidate the process of computing EE. First consider the state as mentioned in Eq. (1.2). We trace out B and the corresponding reduced density matrix is, ( ) ρ A = (1.7) Now to compute the EE as defined in Eq. (1.6) one first diagonalize this and find the eigenvalues. In terms of the eigenvalues Eq. (1.6) becomes 2 S EE = λ i ln λ i. (1.8) i=1 This in turn gives, S EE = 0. (1.9) So the entropy is zero and hence the state is not entangled. Now consider the state as shown in Eq. (1.3). Corresponding reduced density matrix is, ( ) ρ A = (1.10) The entropy is, S EE = ln 2. (1.11) It is an entangled state, in fact it is a maximally entangled state. So whenever S EE is nonzero the state is entangled. Now let us consider a more complicated system, a system of two coupled oscillators. This type of systems are considered in [5] and we will review their calculation here to demonstrate the increasing difficulty of computing this quantity when one consider quantum many body systems. Let us start by writing the hamiltonian that describes the two coupled oscillators. H = 1 [p 21 + p 22 + k 1 (x 21 + x 22) ] + k 2 (x 1 x 2 ) 2, (1.12) 2 1 and 2 respectively denote the two oscillators. Then one defines the canonical coordinate x A = x 1 + x 2 2, x B = x 1 + x 2 2. (1.13)

27 CHAPTER 1. INTRODUCTION 5 In terms of these coordinates the ground state wave function can be written as ψ 0 (x 1, x 2 ) = ( ω Aω B π 2 ) 1/4 e (ω Ax2 A +ω B x2 B )/2, (1.14) where ω A = k 1/2 1 and ω B = (k 1 + 2k 2 ) 1/2 are the two frequencies corresponding to the two normal modes. Now suppose we integrate out the oscillator 2. The reduced density matrix is defined as, This gives ρ(x 1, x 1) = dx 2 ψ 0 (x 1, x 2 )ψ 0 (x 1, x 2 ). (1.15) ρ(x 1, x 1) = ( δ π )1/2 e α(x2 1 +x 2 1 )/2+βx 1x 1, (1.16) where β = 1 (ω A ω B ) 2 4 (ω A +ω B and δ = α β = 2ω Aω B. Then we have to just compute the S ) (ω A +ω B ) EE as defined in Eq. (1.6). To do that we have to find the eigenvalues of this reduced density matrix. In this case we are fortunate, as one can easily solve this problem and eigenvalues are given as, Then λ n = ( 1 β )( β ) n. (1.17) α + (α 2 β 2 ) 1/2 α + (α 2 β 2 ) 1/2 S EE = n After performing this sum, which is somewhat tedious, we get λ n ln λ n. (1.18) S EE ( k ( 1 β ) β ( ) = ln 1 k 2 α + (α 2 β 2 ) 1/2 α β + (α 2 β 2 ) ln β ). 1/2 α + (α 2 β 2 ) 1/2 Ultimately S EE is just a function of the ratio of k 1 and k 2. (1.19) Now the stage is prepared for us to generalize this concept for field theory. In the field theory the problem becomes much more difficult and subtle. One key issue is to factorize the Hilbert space. One way is to discretize the system over a lattice. 1 However one can still use the von-neumann formula as defined in Eq. (1.6), but one has to deal with the ultraviolet divergences that are present in the field theory. As shown in the Fig. (1.2), we can consider a particular region in the field theory denoted by A. To compute the EE for this region we trace out the remaining portion. The system is discretized over the full space. The S EE (A) for the subsystem A is roughly proportional to the number of links cut by the boundary of the region A. So it is telling us that, indeed EE 1 Several ambiguities might enter in the calculation because of the discretization, specially for the gauge theory. But still one can extract a meaningful answer for EE in the field theory.

28 INTRODUCTION Figure 1.2: System is discretized on a lattice and H = i H i is proportional to the area of the boundary dividing the two regions. If the total system is in the pure state, then one can show S EE (A) = S EE (B), (1.20) where B denotes the remaining portion of the spacetime. From this one can intuitively guess that S EE is proportional to the number of degrees of freedom that live at the boundary between the two regions. Although there is no formal proof of area law, it has been checked for many instances. For almost all the cases when one considers a ground state of a local hamiltonian, one indeed gets the area law. It is more or less robust, although the violation of it has been observed for the excited states and also for non-local hamiltonians [21]. Let us close this section by briefly sketching an argument for the area law. We will follow [5] and consider scalar field theory. To show this let us go back to the oscillator case as almost all the field theory can be described effectively using the coupled oscillators model. To start with let us write down the Hamiltonian, H = 1 d 3 x[π 2 + φ(x) 2 ]. (1.21) 2 Here φ(x) denotes the scalar field and π is the canonical momentum. After this we express this Hamiltonian in terms of the partial wave expansion of the scalar field, φ lm = x dωz lm φ(x). (1.22) Z lm are the spherical harmonics. The above relation stems from the fact that we can expand the scalar field in the basis of spherical harmonics. A similar relation can be written for the

29 CHAPTER 1. INTRODUCTION 7 conjugate momentum. We impose the canonical quantization relation, [φ lm (x), π l m (x )] = iδ ll δ mm δ(x x ). (1.23) In terms of this partial wave, H = l,m H lm. (1.24) Now, H lm = ( dx πlm(x) 2 + x 2 [ x ( φ lm x )]2 + l(l + 1) ) φ x 2 lm (x). (1.25) Then we discretize the system. We put it on a lattice with a lattice spacing 1. The M plays M the role of the uv cutoff. The boundary condition imposed on φ lm (x) is such that it vanishes when x L where L is the length of the box in which the system is placed. Also L = (N + 1) 1 M, (1.26) where N is a integer and this relation shows that the system is discretized. So the Hamiltonian becomes H lm = M 2 N j=1 [ π 2 lm + (j )2 ( φlm,j j φ lm,j+1 j + 1 ) 2 + l(l + 1) j 2 φ 2 lm,j ]. (1.27) Now this looks exactly the same as the N coupled oscillators hamiltonian. We can proceed as before extending the result of 2 coupled oscillators. We trace over the first n number of sites to obtain the EE. Finally we get S EE (n, N) = l a l (n)[ ln a l (n) + 1], (1.28) where a l (n) = n(n+1)(2n+1)2 64l 2 (l+1) 2 + O( 1 l 6 ). At this level we are only interested in the leading result in l, as that will give after summing all values of l s, the area like term. We perform the sum over l numerically. We define a radius R midway between the outermost point which was traced over and the innermost point which was not as, Then it can be shown that the leading term of the EE is, R = (n ) 1 M. (1.29) S EE = 0.30M 2 R 2. (1.30) From this it is clearly evident, that the EE corresponding to the ground state wavefunction of a local hamiltonian of the scalar field satisfies the area law. A more intuitive way to understand the area law [22] is to consider a particular entangling region A as shown in Fig. (1.3). For simplicity let us stick to the massless scalar field model

30 INTRODUCTION Figure 1.3: Modes straddling the boundary A is responsible for S EE in dimensions. We expand the scalar field in terms of its modes. These modes are quasi-localized and each has an momentum k. Now we know, k = 2π λ, (1.31) where λ is the usual wavelength. The total number of modes inside the region A is given by, N = kmax k min dn = kmax k min V d 3 k (2π) 3, (1.32) where V is the volume of A. Now k min = 2π and k 2R min = 2π. As all the wavelengths are ɛ localized inside A, maximum wavelength can atmost be equivalent to 2R and the minimum wavelength is 0. But then k min will be divergent, hence we have to put a uv cut-off ɛ. From this the necessity of a cut-off becomes quite clear. Now we count the fraction of the modes which resides at the boundary of A, responsible for the EE. N s kmax k min αãdn, (1.33) V where N s denotes the number of modes straddling the boundary and it is a fraction of the total number of modes living inside A. α denotes the thickness of the boundary (α << R) and à denotes its area. Now only the mode localized near the boundary is responsible for S EE, so we can approximate α by 2π k. Also d 3 k = 4πk 2 dk and αd 3 k = 8π 2 kdk. We next perform the integration and the entropy is proportional to N s upto some phase-space factors. We get, S EE 2πà ɛ 2. (1.34) So EE is proportional to the area, hence proportional to number of degrees of freedom residing at the boundary between the two region. It can be also shown in the same way that for dimensions it is proportional to the circumference and in it goes as logarithm.

31 CHAPTER 1. INTRODUCTION 9 Lastly, S EE satisfies one more important property, namely strong subadditivity. For a bipartite system it tells us that, S EE (A) + S EE (B) S EE (A B) + S EE (A B). (1.35) This result can be extended for any arbitrary number of subsystems. This inequality provides a non trivial constraints on S EE and any consistent holographic proposal should pass this test Entanglement entropy and Holography In this section we will review various facts about holographic entanglement entropy. The main goal of this thesis is to understand the connection between EE and geometry, thereby learning important lessons about underlying gravity theory in the holographic set up. By holography we will mean AdS/ CFT correspondence. Two main character of this play is Anti-de Sitter space (AdS) and conformal field theory (CFT). So before proceeding further let us briefly comment on the structure of the conformal group and AdS spacetime [23]. Conformal group and structure of AdS Conformal isometries keep the metric invariant upto a scale transformation. The conformal transformations form a group by themselves. Poincaré group comes as a subgroup under the broad structure of the conformal group. The conformal transformation preserves the angle between the two curves. These transformations consist of the following four kinds of transformations. Translation x µ = x µ + a µ. Lorentz x µ = R µ ν x ν, where infinitesimal matrix R µν is antisymmetric. Dilatation x µ = cx µ. Special Conformal Transformation(SCT) x µ = xµ c µ x c.x+c 2 x 2. For SCT the conformal factor is (1 2 c.x+c 2 x 2 ) 2. SCT is nothing but a translation preceded and followed by an inversion. The corresponding generators for the infinitesimal transformations are listed below. For a generic field Translation(P µ ) i µ. Rotation (J µν ) i(x µ ν x ν µ ) + S µν.

32 INTRODUCTION Dilatation(D) i(d + (x. )). SCT(K µ ) i((2x µ x ν 2g µν x 2 ) ν + 2d.x µ ) + 2x ν S µν, where S µν is an anti-symmetric spin matrix for a given field and satisfies the Lorentz algebra. d is a real number that depends on the nature of the fields that are present in the underlying theory. 2 These generators satisfy the following commutation relations among themselves. [D, D] = 0, [P a, P b ] = 0, [D, P a ] = ip a, [J ab, P c ] = i(g ac P b g bc P a ), [J ab, J cd ] = i(g ad J bc + g bc J ad g ac J bd g bd J ac ), (1.36) [J ab, D] = 0, [D, K a ] = ik a, [ik a, K b ] = 0, [K a, P b ] = 2i(g ab D J ab ). For example, in dimensional flat spacetime we have the following 10 conformal generators, J 1 = a = ip a, J 2 = x b a x a b = ij ab, J 3 = (x a a ) = id, J 4 = (2x a (x d d ) (x d x d ) a ) = ik a. (1.37) a, b runs from 1 to 3. These generators satisfy the usual conformal commutation rules. Now we will see what are the corresponding isometry generators of AdS 4. We first write the AdS 4 metric in poincare coordinates. This is the coordinate system we will often use throughout this thesis. ds 2 = L2 (dz 2 + dx 2 + dy 2 + dt 2 ) z 2. (1.38) t denotes the Euclidean time. Then we do the following substitution It gives 2 e.g for Fermion d = 3 2 and for Boson d = 1. r = L z. (1.39) ds 2 = L2 r 2 dr2 + r2 L 2 (dt2 + dx 2 + dy 2 ). (1.40)

33 CHAPTER 1. INTRODUCTION 11 Next we list all the 10 generators. J 1 = t, J 2 = x, J 3 = y, J 4 = x t t x, J 5 = y x x y, J 6 = t y y t, J 7 = r r t t x x y y, (1.41) J 8 = rt r 1 2 t2 t tx x ty y, J 9 = rx r tx t 1 2 x2 x xy y, J 10 = ry r ty t xy x 1 2 y2 y. We make suitable identifications and t it, such that the generators satisfy the usual SO(3, 2) algebra [J ab, J cd ] = i[g ad J bc + g bc J ad g ac J bd g bd J ac ], where a, b, c, d {0, 1, 2, 3, 4}, So basically they satisfy the same algebra as the CFT generators in one lower dimensions. AdS/CFT Now we describe what exactly this correspondence is. There are many dualities that exist in the physics [24]. Among them AdS/CFT connects a strongly coupled field theory with a weakly coupled gravity in Anti-de Sitter (AdS) space time [25]. It is a strong weak duality. It has been observed that there exists an equivalence between a strongly coupled N = 4 supersymmetric SU(N) Yang-Mills (SYM) theory and Type IIB string theory on AdS 5 S 5 in the large N limit. Now consider a stack of N D3-branes. Open strings describe the excitations of the D3-branes and the low energy dynamics is governed by N = 4 SYM gauge theory. For this theory one can define a t-hooft coupling λ = gy 2 M N = g sn. We can do a perturbation theory when λ << 1 (also g s << 1). On the other hand we have closed string excitations in the vacuum. This gives rise to the gravity multiplate in 10 dimensions, low energy description of which is effectively given by Type IIB supergravity. One can construct a metric solution for this theory for which the near horizon geometry looks like, ds 2 = α r 2 [ 4πgs N ( dt2 + dx dx dx 2 3) + 4πg s N dr r + 4πg 2 s NdΩ 2 5] (1.42) We have assumed that α 0 so that we can neglect stringy effects and work in the supergravity regime. Identifying L 2 = α 4πg s N, where L is the AdS radius we can see that

34 INTRODUCTION metric defined in Eq. (1.42) describes AdS 5 S 5 geometry. Also string length l s = α and this description is valid when, ( L l s ) 4 = 4πgs N >> 1. (1.43) This means that classical gravity description is valid when the AdS length scale is much bigger than the string length and one can use this supergravity language when the t-hooft coupling becomes very large. Also we know g 2 Y M N = 4πg sn. All these things point to the fact that we have a classical gravity description when L >> l s in the bulk and it is equivalent to a strongly coupled gauge theory at the boundary. Although this conjecture has not been proved yet, it passes several important checks, for eg, matching of the spectrum of chiral operators, correlation functions, supersymmetric indices etc. We obtain a precise dictionary between field theory correlators and correlators of fields living inside the AdS space time. One example is that, currents in the conformal field theory (CFT) side correspond to a gauge field living inside the bulk spacetime. One can easily see that the isometry group of AdS 5 is SO(4, 2) and isometry group for S 5 is SO(6). On the other hand the gauge theory remains invariant under the action of SO(4, 2) conformal group and also possess an SO(6) R-symmetry. So we have obtained a geometric realization of the field theory degrees of freedom. Based on this, one can study systems described by strongly coupled field theories by using equivalent classical gravity description. Holography is being used to study hydrodynamic transports of quark gluon plasma, phase transitions in condensed matter systems etc [26]. 3 Holographic entanglement entropy Although there exist several evidences supporting holographic principle, but it is still not clear how gravity emerges from field theory. To understand this several tools have been employed, EE is one of them. In AdS/CFT set up we will investigate EE and will see that it will provide us with a nice geometrical problem. We will see that we can extract important information about the underlying geometry, hence the gravity theory using this quantity. Now as the AdS/CFT is a two way street, let us start by reviewing some basics about EE in the CFT side of the story. 3 An analogous duality has been observed between the near horizon geometry of AdS 3 S 3 M and that of the low energy description on the branes in D1-D5 system in terms of 1+1 dimensional CFT. Also in recent times holographic principles are being applied for other spacetimes, for eg, Lifshitz, de-sitter etc [27].

35 CHAPTER 1. INTRODUCTION 13 Entanglement entropy in CFT Most general structure of EE for a CFT in even spacetime dimensions is, S EE = c 1 R d 2 ɛ d c 2 ln ( R) + (1.44) ɛ First term is the area term. R denotes the radius of the entangling surface i.e the surface for which the EE is computed. ɛ is the uv-cutoff. In even dimensions one gets a logarithmic term known as universal term in EE. The coefficient proportional to this term is cut-off independent and carries the information of the central charges of the underlying CFT. So this term is also known as universal term and we will be interested in computing this term throughout this thesis for various theories of gravity. Also we will be considering time independent scenarios and choose a particular time slice t = 0. Let us take an example. In d = 4 dimensions, c 2 takes the following form [28], c 2 = A d 2 xr + C d 2 x(w abcd h ac h bd Ks K sabk sab ). (1.45) We have chosen t = 0 slice and rest of the 3 dimensional space is divided into two halves. So the boundary between the entangling region and the rest of the space is a two dimensional space. We will call it a codimension-2 surface. The integration defined in Eq. (1.45) is essentially over this boundary. c 2 depends on the geometrical property of this codimension-2 surface. R is the Ricci scalar, W abcd is the Weyl tensor and K sab is the extrinsic curvature of this surface. h ac is the projection operator to the surface from the 4 dimensional spacetime. h ab = η ab n s an bs. (1.46) s denotes the two transverse directions and a, b are the surface indices. The extrinsic curvature can be defined as, K sab = e α ae β b hµ αh ν β µ n sν. (1.47) It has also an index (s) corresponding to the two transverse directions and two normals are defined for that. K s is the trace of K sab. h µ α is the bulk to surface projection operator and e α a is the tangent vector 4. A and C are related to the two anomaly coefficient that are present for the 4 dimensional CFT. A = A 16π, C = C 2 16π. (1.48) 2 A is known as Euler anomaly and C is know as Weyl anomaly. They show up in the non vanishing part of trace of the stress energy tensor. < T i i >= C 16π 2 W 2 4 e α a = Xα X a, where µ and a denote respectively bulk and surface indices. A 16π 2 E 4 (1.49)

36 INTRODUCTION where, E 4 = R ijkl R ijkl 4R ij R ij + R 2 is the 4 dimensional Euler tensor and W 2 is the square of 4 dimensional Weyl tensor. i, j, k, l denote the 4 dimensional indices. In general to compute EE form field theory one uses Replica Trick [6, 8]. For that, first step is to define Renyi-entropy (S n ), S n = 1 n 1 ln tr Aρ n A. (1.50) Then one has to evaluate this quantity on a Replica space. First we have to write down the reduced density matrix in path integral formalism [6, 8]. For simplicity consider a dimensional space. We will closely follow the notations and conventions of [13]. We choose a t E = 0 slice and consider an interval (A) as our entangling region as shown in the figure. We denote all the dynamical fields that are present collectively as φ(t E, x) where t E is the Euclidean time. We also impose the following boundary condition on the fields living inside A. Figure 1.4: Path integral formulation of density matrix ( Aspects of holographic entanglement entropy, S. Ryu and T. Takayanagi, arxiv:-:hep-th/ ) φ 0 (t E = 0+, x) = φ + (x). (1.51) and for other fields living outside A we demand that, φ 0 (x) = φ (x). (1.52) The ground state wave function can be written as, Ψ(φ 0 (x)) = Complex conjugate of this is defined as, Ψ(φ 0 (x)) = φ0 (t E =0+,x)=φ + (x) t E = te = φ 0 (t E =0,x)=φ (x) Dφe S(φ(x)). (1.53) Dφe S(φ(x)). (1.54) From this one can construct the density matrix easily, ρ Aφ+ φ = 1 te = Dφe S(φ(x)) Π x A δ(φ(t E + = 0, x φ + (x))δ(φ(t E = 0, x φ (x)). Z 1 t E = (1.55)

37 CHAPTER 1. INTRODUCTION 15 We have taken care of the fact, that the boundary conditions imposed are obeyed by inserting the delta functions. Then to evaluate the n th Renyi-entropy we have to find the product of the n copies of this density matrix glued with each other by making suitable identifications which we do in the next step. T r A ρ n A = ρ Aφ1+ φ 1 ρ Aφn+ φ n. (1.56) Also we have imposed φ i = φ i+1,+ for all i = 1, n. So this quantity gives rise to the Replica space which is similar to an n sheeted Riemann surface and has a discrete Z(n) symmetry coming from permuting the n copies of the replica. So finally we have to compute the path integral on this replicated manifold which is denoted by Z n. T r A ρ n = (Z 1 ) n Dφe S(φ) = Z n (Z 1 ). (1.57) n replicaspace This is an important formula for computing EE from field theory using the replica trick. We will see in the later sections, what is its implication in the context of holography. Replica method has been employed successfully for computing EE for dimensional CFT s [8], but for higher dimensions it is hard to apply this method [29]. Entanglement entropy in Holography Now let us turn our attention to the holography, main hero of our story. Importance of EE in this context is profound. We will see that it geometrize the problem in the bulk space time. Before going into the details let us take an example, which will show the connection between EE and geometry. We start by drawing an analogy with the quantum mechanics. We consider CFTs on two spatially disconnected regions [30]. Next we consider a wavefunction for this system of the form, Ψ >= Ψ A > Ψ B > (1.58) where A and B denote the individual wavefunctions of the two CFTs. So it is evident that the state is not an entangled state as it is written as a direct product of two states. This kind of state in holography corresponds to a disconnected geometry. Now let us consider two disconnected CFTs placed on S d. E i is the energy corresponding to the ith eigenstate. Now let us consider the following state, ψ >= i e βe i 2 Ei E i >. (1.59) This state is not a direct product state. So it is an entangled state. From holography we know that this corresponds to a thermofield double state and the dual geometry is an eternal

38 INTRODUCTION Figure 1.5: On the left hand side, we have shown a thermofield double state and its holographic dual eternal black hole is shown on the right hand side. (Picture courtesy- Building up spacetime with quantum entanglement by Mark Van Raamsdonk, arxiv: ) black hole [31]. So this shows that quantum superposition of two states of two classically disconnected CFTs corresponds to a classically connected geometry. This analogy makes things more interesting as one can possibly hope to understand geometry using EE. In the context of AdS/CFT one can ask, whether it is possible to associate a concept of entropy for any arbitrary region of field theory sitting at the boundary of AdS? If so, then the next obvious thing is to ask which portion of the bulk spacetime capture that information? The answer comes in the form of Ryu-Takayanagi (RT) proposal for Einstein gravity [32]. The proposal is very simple, it says that one can attach a notion of entanglement entropy for any arbitrary region at the boundary of AdS at a constant time slice and the corresponding entropy is given by the area of some special codimension 2 surface (γ A ) inside the bulk spacetime. So consider a d dimensional bulk space time. Then the EE associated with a region A at the boundary is given by, S EE (A) = 2π lp d 2 Area(γ A ). (1.60) l p is the planck length. To remind ourselves, this proposal is made for Einstein gravity. S = 1 d d+1 x[r 2Λ]. (1.61) 2l d 1 p The AdS metric is a solution of the action mentioned in Eq. (1.61) with a negative cosmological constant Λ = d(d 1) 2L 2.

39 CHAPTER 1. INTRODUCTION 17 Now let us demystify this proposal and see how it works. Let us for simplicity consider a AdS 5 metric and a spherical region (A) at the boundary as shown in the Fig. (1.6). Figure 1.6: holographic prescription for EE: γ A is the extremal surface ds 2 = L 2 z 2 (dz2 +dt 2 +dr 2 +r 2 dθ 2 +r 2 sin(θ) 2 dφ 2 ). (1.62) z is the radial coordinate of the AdS space. We consider a constant time slice and set the Euclidean time t = 0. Now to evaluate EE associated with the region A we will employ RT method which tells us to find a particular surface (γ A ) extending inside the bulk spacetime which minimizes the area enclosed. Area of that particular surface will give us the EE and γ A is known as the minimal surface. To elucidate this further, we put t = 0 and r = f(z) in the (1.62) to obtain an induce metric for the minimal surface. Then we evaluate the area for this. S EE = 8π2 L3 dz f(z)2 1 + f (z) 2. (1.63) l 3 p z 3 We minimize S EE defined in Eq. (1.63), thereby obtaining an equation for f(z). Solving that equation with the following boundary condition, f(z = 0) = f (z = 0) = 0, (1.64) where the prime denotes the derivative with respect to z, we get 5 f(z) = R 2 z 2. (1.65) R is the radius of the spherical region (A) at the boundary. We plug this into Eq. (1.63) and expand the resulting expression around z=0 which is divergent. We introduce a uv-cutoff ɛ and finally we get, [ S EE = 4π2 L3 R 2 ] l 3 p ɛ ln(r 2 ɛ ). (1.66) The leading term follows the area law and also we get an universal term. One can consider more general surfaces and still get the expected results for the universal term (1.45). RT proposal passes several consistency checks, it produces correct universal 5 For any generic entangling surface, the extremization condition in the context of Einstein gravity can be written as K s = 0 where, K s is the trace of the extrinsic curvature of this codimension-2 surface.