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IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Geometric Flow Description of Minimal Surfaces andHolographic Entanglement Entropy
Ioannis MitsoulasNCSR Demokritos
based onarXiv:1910.06680 [hepth]
in collaboration with D. Katsinis and G. Pastras
INPP Annual MeetingNCSR Demokritos, November 15th 2019
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
1 IntroductionEntanglement in quantum mechanicsHolographic entanglement entropy
2 Minimal Surfaces
3 Geometric Flow EquationTwo embedding problemsThe flow equationBoundary conditions
4 Holographic Entanglement EntropyThe perturbative solutionHolographic Entanglement Entropy
5 Outlook
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Entanglement in quantum mechanicsHolographic entanglement entropy
Section 1
Introduction
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Entanglement in quantum mechanicsHolographic entanglement entropy
Consider a quantum system comprised, of two subsystems A and AC . There existstates of the total system that cannot be written as tensor product of states of thetwo subsystems. These are called entangled states.
Entangled states have the property that measurements performed in onesubsystem have effects on measurable quantities of the other subsystem.
Entanglement entropy is a measure of entanglement used in a variety of quantumsystems such as condensed matter physics or conformal field theories withholographic gravitational duals,
SEE := −Tr (𝜌A ln 𝜌A) .
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Entanglement in quantum mechanicsHolographic entanglement entropy
Through the AdS/CFT correspondence, the dynamics of strongly coupledconformal field theories are related to gravitational dynamics in spaces with AdSasymptotics.
The RyuTakayanagi conjecture relates the entanglement entropy for a region inthe boundary CFT separated from the rest degrees of freedom with an entanglingsurface to the area of a minimal codimension two surface in the bulk geometry,anchored in the same entangling surface in the boundary,
SEE =1
4GNArea
(︁Aextr
)︁.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Section 2
Minimal Surfaces
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
In the light of the RuyTakayanagi conjecture, being able to find the minimalsurface, which is anchored to a given entangling surface at the boundary, presentsa certain interest.
This task is mathematically highly nontrivial due to the nonlinear equation
K = 0,
which is obeyed by the minimal surface.
Very few minimal surfaces are explicitly known in the literature, such as thosecorresponding to spherical or striplike entangling surfaces, as well as the ellipticminimal surfaces in AdS4.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The standard approach to the problem of finding a minimal surface is to try to finda solution to the equation K = 0, for a given entangling surface.
Our approach is to study perturbatively the previous equation around theboundary for arbitrary entangling surface.
Our solution is valid only to a certain ”depth” as we move from the boundarytowards the interior of the bulk along the radial direction.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
In the context of AdSd+1/CFTd the radial direction is related to the energy scale ofthe boundary theory. The aforementioned ”depth” plays the role of a UVcutoff forthe entanglement entropy.
It is known that entanglement entropy is divergent
SEE =
{︃ad−2Λd−2 + ad−4Λd−4 + · · ·+ a0 ln Λ/R + regular terms, d even,ad−2Λd−2 + ad−4Λd−4 + · · ·+ a0 + regular terms, d odd.
The first term in this expansion is proportional to the area of the entanglingsurface, reminding of the are law for black hole entropy. The existence of the arealaw term has led to the speculation that gravity might be an entropic force.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Two embedding problemsThe flow equationBoundary conditions
Section 3
Geometric Flow Equation
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Two embedding problemsThe flow equationBoundary conditions
We focus on static asymptotically AdSd+1 spacetimes, described by the metric
ds2 = f (r) dr2 + hij(︁
r , xk)︁
dx i dx j .
We denote the holographic coordinate with r and the rest of the coordinates with x iwith i = 1 . . . d − 1. We consider two embedding problems.
The embedding of the minimal surface in the asymptotically hyperbolic space. Theminimal surface is parametrized by the coordinates {𝜌, ua} with a = 1, . . . d − 2and
r = 𝜌, x i = X i(︀𝜌, ua
)︀.
Aminimal
AdS
boun
dary
Centangling
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Two embedding problemsThe flow equationBoundary conditions
The embedding of the intersection of the minimal surface with a constant r planein this plane. The constant r plane is described by r = 𝜌, whereas the intersectionis described by
x i = x i(︀𝜌, ua
)︀.
These are functions of d − 2 variables.
Aminimal
AdS
boun
dary
Centangling
r =𝜌
plane
r =𝜌
plane
Intersectionwith Aminimal
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Two embedding problemsThe flow equationBoundary conditions
We choose a specific parametrization of the minimal surface,
𝜕x i
𝜕𝜌= a
(︀𝜌; ua
)︀ni ,
where ni is the normal unit vector to the constant r plane.
After computing the first and second fundamental forms, the equation describingthe minimal surface K = 0 assumes the form
𝜌𝜕𝜌
(︂c√det 𝛾
𝜌
)︂+
(d − 1)√det 𝛾
c𝜌= 0,
in the case of the Hd space. 𝛾 denotes the induced metric on the intersection ofthe minimal surface with the constant r plane and c denotes the quantity
c(︀𝜌; ua
)︀=
(︃a(𝜌; ua)2
f (𝜌)+ 1
)︃− 12.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Two embedding problemsThe flow equationBoundary conditions
The flow equation contains second order derivatives of the embedding functionswith respect to the holographic coordinate.
The specification of a single connected entangling surface does not determinethe minimal surface uniquely.
This is due to the fact that the entangling surface might be part of a more complexdisconnected entangling surface.
A1
A2A3
C1
C2C3
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The perturbative solutionHolographic Entanglement Entropy
Section 4
Holographic Entanglement Entropy
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The perturbative solutionHolographic Entanglement Entropy
We assume an expansion for the embedding functions of the minimal surfacearound 𝜌 = 0 of the form
x i(︀𝜌; ua
)︀=
∞∑︁m=0
x i(m)(︀ua)︀𝜌m.
There are similar expansions for the quantities c and√𝛾 that appear in the flow
equation.
c =∞∑︁
m=0
c(m)𝜌m and
√︀det 𝛾 =
√det𝒢𝜌d−2
∞∑︁m=0
𝛾(m)𝜌m.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The perturbative solutionHolographic Entanglement Entropy
It turns out that the embedding functions up to order O(︀𝜌d)︀
are determined by theDirichlet boundary data solely, i.e. the local characteristics of the entanglingsurface.
Up to this order only even terms appear.
For orders higher than O(︀𝜌d)︀
 odd terms appear, when d is odd, 𝜌2n log 𝜌  terms appear, when d is even,
which depend on the nonlocal characteristics of the entangling surface.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The perturbative solutionHolographic Entanglement Entropy
The first terms of the previous expansions are shown here.
x i(2) = −𝒦
2 (d − 2)𝒩 i ,
c(2) = −𝒦2
2 (d − 2)2, 𝛾(2) = −
𝒦2
2 (d − 2),
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The perturbative solutionHolographic Entanglement Entropy
x i(4) =𝒦
8 (d − 2) (d − 4)
(︃−
2𝒦2
(d − 2)2+𝒦ab𝒦ab +
�𝒦𝒦
)︃𝒩 i −
𝒢cb𝒦𝜕b𝒦8 (d − 2)2
𝜕c𝒳 i ,
𝛾(4) =(d − 3)𝒦2
4 (d − 2)2 (d − 4)
(︃(d − 3)2 + 1
2 (d − 2) (d − 3)𝒦2 − 𝒦ab𝒦ab −
�𝒦𝒦
)︃
and
c(4) =
⎧⎨⎩𝒦2
2(d−2)2(d−4)
(︁3d−4
4(d−2)2 𝒦2 −𝒦ab𝒦ab − �𝒦𝒦
)︁, d ≥ 5,
−𝒦4
8 +𝒦�𝒦
2 −9x i(3)x
i(3)
2 , d = 3.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The perturbative solutionHolographic Entanglement Entropy
The area of the minimal surface is given by
A (Λ) =∫︁ 𝜌max
1/Λd𝜌∫︁
dd−2u√det Γ =
∫︁ 𝜌max1/Λ
d𝜌∫︁
dd−2u
√︀f (𝜌) det 𝛾
c.
We expand it in a similar fashion.
A (Λ) = a0 ln Λ +d−2∑︁n=1
anΛn + nondivergent terms.
The logarithmic term appears when d is even.
It turns out that all divergent terms (including the universal logarithmic ones) aredetermined by the Dirichlet data solely.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
The perturbative solutionHolographic Entanglement Entropy
For any d ≥ 3, the most divergent term is indeed the ”area law” term
ad−2 =1
d − 2
∫︁dd−2u
√det𝒢 =
1d − 2
𝒜.
For d ≥ 4 there is at least one more divergent term
ad−4 =
{︃− d−3
2(d−2)2(d−4)
∫︀dd−2u
√det𝒢𝒦2, d ≥ 4,
− 18∫︀
d2u√det𝒢𝒦2, d = 4.
At d = 4, this term is the universal logarithmic term.For d ≥ 6
ad−6 =
⎧⎨⎩d−5
4(d−2)2(d−4)(d−6)
∫︀dd−2u
√det𝒢
[︁d2−5d+82(d−2)2 𝒦
4 −𝒦2𝒦ab𝒦ab −𝒦�𝒦]︁, d ≥ 6,
1128
∫︀d4u
√det𝒢
[︁7
16𝒦4 −𝒦2𝒦ab𝒦ab −𝒦�𝒦
]︁, d = 6.
At d = 6 this is a universal logarithmic term.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
Section 5
Outlook
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
We present a systematic method to determine all divergent terms in theexpansion of the holographic entanglement entropy, for arbitrary entanglingsurface.
The method can be applied to entangling surfaces possessing nonsmooth points.
It would be interesting to apply the method for general higher derivative gravitytheories, in which case the RyuTakayanagi functional gets modified.
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy

IntroductionMinimal Surfaces
Geometric Flow EquationHolographic Entanglement Entropy
Outlook
This research is supported by the General Secretariat for Research and Technology(GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) in theframework of the “First Postdoctoral researchers support”, which funds the program“Holographic APPlications of quantum ENtanglement (HAPPEN)”, based in NSCR“Demokritos”, under grant agreement No 2595.
Thank you for your attention!
Ioannis Mitsoulas Geometric Flow Description of Minimal Surfaces and Holographic Entanglement Entropy
IntroductionEntanglement in quantum mechanicsHolographic entanglement entropy
Minimal SurfacesGeometric Flow EquationTwo embedding problemsThe flow equationBoundary conditions
Holographic Entanglement EntropyThe perturbative solutionHolographic Entanglement Entropy
Outlook