Geometric Flow Description of Minimal Surfaces and ... · Geometric Flow Equation Holographic...

IntroductionMinimal Surfaces

Geometric Flow EquationHolographic Entanglement Entropy

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Geometric Flow Description of Minimal Surfaces andHolographic Entanglement Entropy

Ioannis MitsoulasNCSR Demokritos

based onarXiv:1910.06680 [hep-th]

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



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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




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Entanglement in quantum mechanicsHolographic entanglement entropy

Section 1

Introduction




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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) .




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Through the AdS/CFT correspondence, the dynamics of strongly coupledconformal field theories are related to gravitational dynamics in spaces with AdSasymptotics.

The Ryu-Takayanagi 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 co-dimension two surface in the bulk geometry,anchored in the same entangling surface in the boundary,

SEE =1

4GNArea

(︁Aextr

)︁.




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Section 2

Minimal Surfaces




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In the light of the Ruy-Takayanagi 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 non-trivial due to the non-linear 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 strip-like entangling surfaces, as well as the ellipticminimal surfaces in AdS4.




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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.




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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 UV-cutoff forthe entanglement entropy.

It is known that entanglement entropy is divergent

SEE =

{︃ad−2Λ

d−2 + ad−4Λd−4 + · · ·+ a0 ln Λ/R + regular terms, d even,

ad−2Λd−2 + ad−4Λ

d−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.




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Two embedding problemsThe flow equationBoundary conditions

Section 3

Geometric Flow Equation




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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 i

with 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




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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




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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

.




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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




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The perturbative solutionHolographic Entanglement Entropy

Section 4

Holographic Entanglement Entropy




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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.




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It turns out that the embedding functions up to order O(︀𝜌d)︀ are determined by the

Dirichlet 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 non-local characteristics of the entangling surface.




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The first terms of the previous expansions are shown here.

x i(2) = −

𝒦2 (d − 2)

𝒩 i ,

c(2) = −𝒦2

2 (d − 2)2 , 𝛾(2) = −𝒦2

2 (d − 2),




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x i(4) =

𝒦8 (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)2𝒦2 −𝒦ab𝒦ab − �𝒦

𝒦

)︁, d ≥ 5,

−𝒦4

8 + 𝒦�𝒦2 −

9x i(3)x

i(3)

2 , d = 3.




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The area of the minimal surface is given by

A (Λ) =

∫︁ 𝜌max

1/Λd𝜌∫︁

dd−2u√det Γ =

∫︁ 𝜌max

1/Λd𝜌∫︁

dd−2u

√︀f (𝜌) det 𝛾

c.

We expand it in a similar fashion.

A (Λ) = a0 ln Λ +

d−2∑︁n=1

anΛn + non-divergent 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.




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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

𝒦4 −𝒦2𝒦ab𝒦ab −𝒦�𝒦]︁, d ≥ 6,

1128

∫︀d4u

√det𝒢

[︁7

16𝒦4 −𝒦2𝒦ab𝒦ab −𝒦�𝒦

]︁, d = 6.

At d = 6 this is a universal logarithmic term.




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Section 5

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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 non-smooth points.

It would be interesting to apply the method for general higher derivative gravitytheories, in which case the Ryu-Takayanagi functional gets modified.




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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 Post-doctoral 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!


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