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# Ecl17

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### Ecl17

1. 1. Thermodynamics of AdS Black Holes as Regularized Fidelity Susceptibility Aizhan Myrzakul L.N. Gumilyov Eurasian National University Eurasian International Center for Theoretical Physics Nazarbayev University ECL17: Exploring the Energetic Universe August 7, 2017 1 / 1
2. 2. Outline 1. Introduction 1.1. Entanglement Entropy 1.2. Holographic Principle 1.3. Holographic Entanglement Entropy 1.4. Fidelity Susceptibility 1.5. Holographic Complexity 2. Reissner-Nordstr¨om Anti-de Sitter black holes 3. Conclusion 4. References 2 / 1
3. 3. 1.1. Entanglement Entropy Let us consider two systems, with corresponding Hilbert spaces H1 and H2 respectively. The Hilbert space of the combined system is then H1 ⊗ H2. A state of this combined system is said to be entangled if it cannot be written in the form |Ψ⟩ = |ψ1⟩ ⊗ |ψ2⟩ (1) E.g. assume we have two particles, each of which can have one of two states. This is called a qubit. Then the state |Ψ⟩ =|↑⟩⊗ |↓⟩ (2) is not entangled. In contrast, the state |Ψ⟩ = 1 √ 2 (|↑⟩⊗ |↓⟩− |↓⟩⊗ |↑⟩) (3) is entangled. In fact, this entangled state is usually known as an EPR pair. 3 / 1
4. 4. 1.1. Entanglement Entropy The expectation value of any operator A is given by ⟨A⟩ = ∑pi ⟨ψi |A|ψi ⟩ (4) Such a state is described by an operator known as the density matrix ρ = ∑pi |ψi ⟩⟨ψi | (5) The expectation value (4) of any operator can now be written simply as ⟨A⟩ = Tr(ρA) (6) where the trace is over all states in the Hilbert space. If we know for sure that the system is described by a speciﬁc state |ψ⟩, then the density matrix is simply the projection operator ρ = |ψ⟩⟨ψ| (7) In this case, we say that we have a pure state. 4 / 1
5. 5. 1.1. Entanglement Entropy Given a classical probability distribution {pi }, the entropy is deﬁned by S = − ∑pi log pi (8) In information theory, this is called the Shannon entropy. In physics, this quantity is usually multiplied by the Boltzmann constant kB and is called the Gibbs entropy. The entropy is a measure of the uncertainty encoded in the probability distribution. For a quantum state described by a density matrix ρ, the entropy is deﬁned to be S(ρ) = −Tr(ρ log ρ) (9) This is the von Neumann entropy. If we are dealing with a reduced density matrix ρA = TrB ρ (10) then SA = −TrA(ρAlnρA) (11) is referred to as the entanglement entropy. 5 / 1
6. 6. 1.2 Holographic Principle The information hidden inside BHs is measured by the Bekenstein-Hawking black hole entropy SBH = Area(Horizon) 4GN (12) This consideration leads to the idea of entropy bound SA ≤ Area(∂A) 4GN (13) where SA-the entropy in a region A. Consequently, the d.o.f in gravity are proportional to the area instead of the volume. [t’Hooft 1993, Susskind 1994]: (d+2)-dim gravity theory ⇐⇒ (d+1)-dim non-gravity theory (QFT, CFT, etc.) [Maldacena 1997]: Gravity in AdSd+2 = CFT in Rd+1 6 / 1
7. 7. 1.3 Holographic Entanglement Entropy Proposed by Ryu-Takayanagi (2006), the formula for holographic entanglement entropy is SA = Area(γmin) 4G (14) where A(γmin) is an area of a subsystem enclosed by the minimal surface γmin, ∂γ = ∂A, in the bulk measured in Planck scale, G - gravitational constant. 7 / 1
8. 8. 1.4 Fidelity Susceptibility The ﬁdelity sisceptibility which corresponds to the maximal volume of the bulk [Susskind 2013, Alishahiha 2015] is χF = V (γmax ) 8πRG (15) Consider N-body quantum system with Hamiltonian H0 = ∑Hi = ∑Hi ( p2 i 2m + Vi ) + ∑Vij (16) which corresponds to the system consisting of pure states |ψ(λ)⟩. If we add an external ﬁeld, e.g. magnetic ﬁeld, to this system, i.e. inﬁnitesimally transform parameter λ → λ + δλ, then the Hamiltonian for perturbed system |ψ(λ + δλ)⟩ is H ∼= H0 + λH1 (17) where |λH1| ≪ |H0|. 8 / 1
9. 9. 1.4 Fidelity Susceptibility The inner product between these two wave functions gives the ﬁdelity χF ⟨ψ(λ)|ψ(λ + δλ)⟩ = 1 − Gλλ(δλ)2 + O((δλ)3 ) (18) This metric measures the distance between two inﬁnitesimally diﬀerent states. Gλλ is also called the ﬁdelity susceptibility. When a d + 1-dimensional CFT is deformed by an exactly marginal perturbation, parametrized by λ, Gλλ is holographically estimated by Gλλ = nd · Vol(∑max ) Rd+1 (19) where nd is an O(1) constant and R is the AdS radius. The d+1 dimensional space-like surface ∑max is the time slice with the maximal volume in the AdS which ends on the time slice at the AdS boundary(ies). Fidelity measures how much |ψ(λ)⟩ is overlapped to |ψ(λ + δλ)⟩. 9 / 1
10. 10. 1.5 Holographic Complexity Following [Susskind, 2013], the formula for holographic complexity is CA = V (γmin) 8πRG (20) where R and V are the radius of the curvature and volume in the bulk, 8π was added for normalization convention. So, the holographic complexity measures the diﬃculty of connecting two diﬀerent states. 10 / 1
11. 11. 2. Reissner-Nordstr¨om Anti-de Sitter black holes The metric for RNAdS is given by ds2 = −f (r)dt2 + dr2 f (r) + r2 dΩ2 (21) with the function f being f = 1 + r2 l2 − ϵr3 + rl2 + δr4 + r2l2 (22) where ϵ and δ are deﬁned as 2Ml2 r3 + = ϵ , Q2l2 r4 + = δ . (23) Here, we also have that |Q| < l/6 and correspondingly δ < l2/(6r2 +). Let us start to compute extremal surfaces using the area functional. The area functional for a speciﬁc entangled region of the boundary in RNAdS is then given by Area = ∫ 2π 0 dϕ ∫ θ0 0 L(θ)dθ (24) where L(θ) now is L(θ) = r sin θ r2 + ( dr dθ )2 1 + r2 l2 − ϵr3 + rl2 + δr4 + r2l2 (25) 11 / 1
12. 12. 2. Reissner-Nordstr¨om Anti-de Sitter black holes The Euler-Lagrange equation corresponding to the above L(θ) is d2r dθ2 = 1 −2r2 sin (θ) l2ϵr3 + + 2r sin (θ) l2δr4 + + 2r5 sin (θ) l2 + 2r3 sin (θ) l4 × × [4 sin (θ) r8 + 8 sin (θ) l2 r6 − 8r5 sin (θ) ϵr3 + − 2r5 cos (θ) ( dr dθ ) l2 + 4r4 sin (θ) l4 + + 8r4 sin (θ) ( dr dθ )2 l2 + 8r4 sin (θ) δr4 + − 2r3 cos (θ) ( dr dθ ) l4 − − 8r3 sin (θ) l2 ϵh3 + 2r2 cos (θ) ( dr dθ ) l2 ϵr3 + + 4r2 sin (θ) ϵ2 r6 ++ + 6r2 sin (θ) ( dr dθ )2 l4 + 8r2 sin (θ) l2 δr4 + − 5r sin (θ) ( dr dθ )2 l2 ϵr3 +− − 2r (θ) cos (θ) ( dr dθ ) l2 δr4 + − 8r sin (θ) ϵr7 +δ − 2r cos (θ) ( dr dθ )3 l4 + + 4 sin (θ) δ2 r8 + + 4 sin (θ) ( dr dθ )2 l2 δr4 +]. (26) 12 / 1
13. 13. 2. Reissner-Nordstr¨om Anti-de Sitter black holes We can solve this equation by expanding in series of θ. We set here the AdS radius l = 1 r (θ) = ρ + 1 2 ( −ρϵr3 + + δr4 + + ρ4 + ρ2 ) θ2 ρ + 1 96ρ2 [( 9ρϵ2 r6 + − 9ϵh7 δ − 45ρ4 ϵ r3 +− − 29ρ2 ϵr3 + + 36ρ3 δr4 + + 20ρδr4 + + 36ρ7 + 56ρ5 + 20ρ3 ) θ4 ] + O ( θ6 ) . (27) then again expanding it in δ up to second order, we ﬁnd that the ﬁnite part of the entanglement entropy. This part is the diﬀerence between the pure background and the AdS deformation of the metric. Doing that, we ﬁnd ∆S = − 1440ρ 4G × [θ2 0(675 θ4 0r3 +ϵ ρ4 + 540 θ2 0r3 +ϵ ρ2 + 375 θ4 0r7 +ϵ δ + 495 θ4 0r3 +ϵ ρ2 −450 θ4 0ρ7 − 272 θ4 0ρ3 − 720 θ4 0ρ5 − 540 ρ5 θ2 0 − 480 ρ3 θ2 0 − 420 θ4 0ρ δ r4 + −600 θ4 0ρ3 δ r4 + − 720 ρ3 − 540 ρ θ2 0δ r4 +)]−1 . (28) 13 / 1
14. 14. 2. Reissner-Nordstr¨om Anti-de Sitter black holes The complete expressions for L(θ) and the integral related with the area are L = 1 240 θ ( 240l4ρ4 + 320l4ρ4θ2 + 450ρ8θ4 + 360θ2ρ6l2 + 272l4ρ4θ4 + 720θ4l2ρ6 ) l4ρ2 + 1 240 θ ( −360θ2ρ3l2r3 + − 675ρ5θ4r3 + − 495θ4ρ3l2r3 + ) ϵ l4ρ2 + ( 1 240 θ ( 420r4 +θ4ρ2l2 + 360r4 +θ2ρ2l2 + 600θ4ρ4r4 + ) l4ρ2 − 25 16 θ5r7 +ϵ l4ρ ) δ. (29) By integrating we ﬁnd ∫ θ0 0 Ldθ = − 1 1440 θ2 0 ( 675θ4 0ρ4r3 + + 540ρ2θ2 0r3 + + 375θ4 0r7 +δ + 495θ4 0ρ2r3 + ) ϵ ρ + 1 1440 θ2 0 ( 450θ4 0ρ7 + 272θ4 0ρ3l4 + 720θ4 0ρ5 + 540ρ5θ2 0 + 480ρ3θ2 0 + 420θ4 0ρδr4 + ρ + 1 1440 θ2 0 ( +600θ4 0ρ3δr4 + + 720ρ3 + 540ρθ2 0δr4 + ) ρ . (30 14 / 1
15. 15. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Let us know compute the holographic complexity and ﬁdelity susceptibility dual volumes for a RN black hole. These quantities can be written as follows Vc = 2π ∫ θ0 0 sin θdθ ∫ r(θ) r+ r2dr √ f , (31) VFid = 2π ∫ 2π 0 sin θdθ ∫ r∞ r+ r2dr √ f . (32) Now, by expanding the integrand r2/ √ f in Taylor series up to linear terms in ϵ and θ, we obtain r2 √ 1 + r2 l2 − ϵr3 + rl2 + δr4 + r2l2 = − 1 4 lr3 + ( −2r2ϵl2 − 2r4ϵ + 2r+δrl2 + 2r+δr3 + 3r4 +δϵ ) r (l2 + r2)5/2 .(33) 15 / 1
16. 16. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Hence, the volume corrsponding to the holographic complexity of the RN black hole becomes Vc = − ∫ θ0 0 sin θdθ ( 1 2 r3 +ϵr2 (1 + r2)3/2 + 1 2 r3 +ϵ (1 + r2)3/2 + 1 2 δr4 +r √ 1 + r2 + 1 4 ϵr7 +δ (1 + r2)3/2 + 3 4 ϵr7 +δ √ 1 + r2 − 3 4 ϵr7 +δ ln [ 2 + 2 √ 1 + r2 r ] ) r(θ) r+ . (34) After computing this integral and expanding up to sixth order in θ and by taking asymp- totic expansion in ρ we ﬁnd the following compacted expression Vc = − 1 48 ( −6 θ2 0ϵ br2 + − 6 θ2 0ϵ b ) θ2 0r3 +π ρ b3 − 1 48b3 × [ − 3 θ2 0r6 +δ bϵ ln ( 1 + b r+ ) + 2 θ2 0r4 +δ + 4 θ2 0r3 +δ b − 48 r4 +δ ϵ − 24 r4 +δ +2 θ2 0ϵ r2 + + 4 θ2 0r+δ b + 24 r+δ b − 3 θ2 0r4 +δ bϵ ln ( 1 + b r+ ) + 2 θ2 0r2 +δ + 4 θ2 0r4 +δ ϵ −24 ϵ + 2 θ2 0ϵ − 24 ϵ r2 + + 24 r3 +δ b − 36 r6 +δ ϵ + 36 r4 +δ ϵ b ln ( 1 + b r+ ) − 24 r2 +δ +36 r6 +δ ϵ b ln ( 1 + b r+ ) + 3 θ2 0r6 +δ ϵ ] θ2 0r3 +π , (35) where a = √ 1 + ρ2 and b = √ 1 + r2 +. 16 / 1
17. 17. 2. Reissner-Nordstr¨om Anti-de Sitter black holes The mass, temperature and complexity pressure of the RN black hole are deﬁned as follows M = r+ 2 ( 1 + r2 + l2 + Q2 r2 + ) (36) T = 1 4π ( 3r4 + + r2 + − Q2 r3 + ) (37) P = − ∂M ∂VFid = − ∂M ∂r+ ∂VFid ∂r+ (38) The explicit expression for the complexity pressure is very long for the space-time stud- ied. Now, we need to express the complexity pressure and volume in terms of the temperature. In order to do that, we need to solve (??) for r+. Thus, we need to solve the following equation r4 + − 4πT 3 r3 + + r2 + 3 − Q2 3 = 0 (39) The roots of this equation are given by x1,2 = − ˜b 4˜a − S ± 1 2 √ −4S2 − 2p + q S x3,4 = − ˜b 4˜a + S ± 1 2 √ −4S2 − 2p + q S (40) 17 / 1
18. 18. 2. Reissner-Nordstr¨om Anti-de Sitter black holes where p, q and S are deﬁned by p = 8˜a˜c − 3˜b2 8˜a2 , q = ˜b3 − 4˜a˜b + 8˜a2 ˜d 8˜a3 , S = 1 6 √ −6 + 3G + 3 ∆0 G , G = 3 √ 1 2 ∆1 + 1 2 √ ∆2 1 − 4∆3 0 , ∆0 = ˜c2 − 3˜b ˜d + 12˜a˜e , ∆1 = 2˜c3 − 9˜b˜c ˜d + 27˜b2 ˜e + 27˜a ˜d2 − 72˜a˜c ˜e ,(41) and ∆ determined as ∆2 1 − 4∆3 0 = −27∆ (42) is a determinant of the fourth order polynomial. If ∆ > 0, then all four roots of the equation are either real or complex. 18 / 1
19. 19. 2. Reissner-Nordstr¨om Anti-de Sitter black holes From (39), we have that ˜a = 1, ˜b = −(4πT)/3, ˜c = 1/3, ˜d = 0 and ˜e = −Q2/3. Therefore, for our case the roots are given by r1,2 + = 1 3 π T − 1 6 √ −6 + 3 3 √ k + 3 1 9 − 4 Q2 3 √ k ± 1 6 −3 3 √ k − 3 1 9 − 4 Q2 3 √ k + 12 π2T2 + 54 ( − 8 27 π3T3 + 2 3 π T ) √ −6 + 3 3 √ k + 3 1/9−4 Q2 3√ k (43) r3,4 + = 1 3 π T − 1 6 √ −6 + 3 3 √ k + 3 1 9 − 4 Q2 3 √ k ± 1 6 −3 3 √ k − 3 1 9 − 4 Q2 3 √ k + 12 π2T2 + 54 ( − 8 27 π3T3 + 2 3 π T ) √ −6 + 3 3 √ k + 3 1/9−4 Q2 3√ k (44) where k = 1 27 − 8 π2 T2 Q2 + 4 Q2 + 1 2 √( 2 27 − 16 π2T2Q2 + 8 Q2 )2 − 4 ( 1 9 − 4 Q2 )3 . (45) 19 / 1
20. 20. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Since k must be real, the inequality ( 2 27 − 16 π2 T2 Q2 + 8 Q2 )2 − 4 ( 1 9 − 4 Q2 )3 ≥ 0 (46) must hold. Equivalently, the above inequality can be expressed as T ≤ 1 4Qπ − ( 2 ( 1 9 − 4 Q2 )3/2 − 2 27 − 8 Q2 ) . (47) Now, by taking series in Q up to order 6 in the above equation, we ﬁnd T ≤ 1 2 √ 3 π − 3 4 √ 3Q2 π − 81 16 √ 3Q4 π + O ( Q6 ) , (48) or T ≤ 0.27567 − 0.41350 Q2 − 2.7912 Q4 + O ( Q6 ) . (49) From this expression, we can note that the temperature will be maximum when Q = 0. 20 / 1
21. 21. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Figure 1 : T as a function of Q . This graph is conﬁned with the condition |Q| < l 6 . 21 / 1
22. 22. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Let us now ﬁnd the horizon of RNAdS black hole. To do that, we need to rewrite (??) using (??), which gives us f (r) = 1 + r2 l2 − 2M r + Q2 r2 (50) Now, the horizon condition f (r) = 0 can be written as ξ4 + ξ2 − 2M l ξ + ( Q l )2 = 0 (51) where ξ = r/l. Here our aim is to ﬁnd the largest root of the above equation which will correspond to the outer horizon of the RN charged black hole. The largest root is given by ξ3 = 1 6 ( −6 + 3k + 3 1+12Q2 k )1/2 + 1 6 √ −12 − 3m − 3(1+12Q2) m − 108M√ −6+3m+3 1+12Q2 m (52) where m = ( 1 + 54 M2 + 6 √ 3 M2 + 81 M4 − Q2 − 12 Q4 − 48 Q6 )1/3 [Momeni, Myrza- kul and et.al. 2016]. 22 / 1
23. 23. 2. Reissner-Nordstr¨om Anti-de Sitter black holes a) b) c) Figure 2 : Location of the roots ξ1, ξ2 ξ3 for the RN black hole as a function of Q, M. 23 / 1
24. 24. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Figure 3 : Figure showing a P-V diagram for the thermodynamic volume V = 4 3 πr3 + and pressure P = 3 8πl2 for RNSAdS black holes. 24 / 1
25. 25. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Figure 4 : Figure showing a P-V diagrams for ﬁdelity versus pressure. For various temperatures of RNSAdS black holes, indicating that ﬁdelity does, indeed, represent the thermodynamic volume. 25 / 1
26. 26. 2. Reissner-Nordstr¨om Anti-de Sitter black holes Figure 5 : A P-V diagram between holographic complexity and pressure, showing a totally diﬀerent behaviour than the thermodynamic P-V diagram. 26 / 1
27. 27. 3 Conclusion Future problems 1. HEE for non-AdS spacetimes? 2. What is an analogue of the Einstein eq. for HEE? 3. A new formulation of QG in terms of Quantum Entanglement 27 / 1
28. 28. 4 References 1. t’ Hooft G, DIMENSIONAL REDUCTION in QUANTUM GRAVITY. [arxiv: 9310026]. 2. Susskind L, The World as a Hologram, [arxiv: 9409089] 3. Maldacena J, he Large N limit of superconformal ﬁeld theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)]. 4. Ryu S. and Takayanagi T, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006). 5. Susskind L, Computational Complexity and Black Hole Horizons, [hep-th/1402.5674],[hep- th/1403.5695]. 6. Alishahiha M, Holographic Complexity, Phys. Rev. D 92, 126009 (2015). 7. Momeni D, Myrzakul A, Myrzakulov R, Faizal M, Alsaleh S, Alasfar L Thermody- namics of AdS Black Holes as Regularized Fidelity Susceptibility. [arXiv:1704.05785]. 28 / 1
29. 29. THANK YOU FOR ATTENTION!!! 29 / 1