This document outlines a presentation on thermodynamics of anti-de Sitter black holes as regularized fidelity susceptibility. It introduces concepts like entanglement entropy, the holographic principle, and holographic entanglement entropy. It then discusses fidelity susceptibility and holographic complexity. The document derives formulas for these quantities for Reissner-Nordström anti-de Sitter black holes. Specifically, it derives an expression for the difference in entanglement entropy between the pure background and metric deformation. It also computes the holographic complexity and fidelity susceptibility volumes for RN black holes through integrals involving the metric function.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. THANK YOU FOR ATTENTION!!!
29 / 1