Holographic Entanglement Beyond Classical Gravity Xi Dong Stanford - - PowerPoint PPT Presentation

Holographic Entanglement Beyond Classical Gravity

Xi Dong

Stanford University August 2, 2013 Based on arXiv:1306.4682 with Taylor Barrella, Sean Hartnoll, and Victoria Martin See also [Faulkner (1303.7221)]

Gauge/Gravity Duality 2013, Max Planck Institute for Physics, Munich

Entanglement Entropy (EE)

A measure of quantum correlation between subsystems:

A A ρA = Tr¯

A (ρA¯ A)

SA = −TrA (ρA log ρA)

EE has found natural homes in a diverse set of areas in physics: Quantum information: robust error correction and secure communication. QFT: strong subadditivity leads to a monotonic c-function under RG flow (in certain dimensions). Condensed matter: non-local order parameter for novel phases and phenomena. Quantum gravity: likely to involve information processing in an essential way: ⋄ Connection between entanglement and spacetime? ⋄ ER=EPR? Black hole complementarity vs firewalls? Holographic entanglement entropy.

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Holographic Entanglement Entropy

A remarkably simple formula in QFTs with holographic duals:

A A

SA = (Area)min 4GN

[Ryu & Takayanagi ’06]

Satisfies strong subadditivity. [Headrick & Takayanagi ’07] Recovers known exact results for a single interval in 1+1D CFTs. E.g. at T = 0 on a line:

[Holzhey, Larsen & Wilczek ’94] [Calabrese & Cardy ’04]

S(L) = c 3 log L ǫ First derived for spherical entangling surfaces. [Casini, Huerta & Myers ’11] Proven for 2D CFTs with large c (and ∆gap). [Hartman 1303.6955] Proven for 2D CFTs with AdS3 duals. [Faulkner 1303.7221] Shown generally! (for Einstein gravity) [Lewkowycz & Maldacena 1304.4926]

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Beyond Classical Gravity

The Ryu-Takayanagi formula is valid in the limit of classical gravity (or large “c”): SA = (Area)min 4GN

• classical gravity

+ O(G 0

N)

• ne-loop bulk corrections

+ · · · One-loop terms were calculated explicitly in 2D CFTs with gravity

• duals. [Barrella, XD, Hartnoll & Martin 1306.4682]

A general prescription was recently proposed. [Faulkner, Lewkowycz &

Maldacena 1307.2892]

Why should we care about these one-loop corrections? Because sometimes they are actually the leading effect.

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Two Disjoint Intervals on a Line at Zero Temperature

L1 L2 L1 L2

In the first phase (corresponding to larger separation), the mutual information vanishes at the classical level. Recall that the mutual information is defined as I(L1 : L2) = S(L1) + S(L2) − S(L1 ∪ L2)

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One Interval on a Circle Below Hawking-Page Temperature

Thermal AdS Phase

A A

At the classical level, SA does not depend

• n T and is given by a universal formula:

SA = c 3 log R πǫ sin πLA R

• which is exact only at T = 0.

Furthermore, SA − S¯

A which measures the “pureness” of the state is

nonzero only at the one-loop order.

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One Interval on a Circle Above Hawking-Page Temperature

BTZ Black Hole Phase

A A

At the classical level, SA does not depend

• n R and is given by with another universal

formula: SA = c 3 log 1 πTǫ sinh(πTLA)

• which is exact only when the spatial circle

becomes a line (R = ∞).

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Outline

1

Constructing the Bulk

2

The Classical Level

3

One-Loop Correction

4

Conclusion Three examples

1

Two intervals on a plane

2

One interval on a torus (thermal AdS phase)

3

One interval on a torus (BTZ phase)

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Replica Trick

Introduce R´ enyi entropy: Sn = − 1 n − 1 log Tr ρn ⇒ S = lim

n→1 Sn = −Tr ρ log ρ

At integer n, R´ enyi entropy can be written in terms of partition functions: Sn = − 1 n − 1 log Zn Z n

1

Zn is the partition function on an n-sheeted branched cover (of C or T 2). For two intervals on C it is a Riemann surface

• f genus n − 1.

Goal: construct the gravity duals of these branched covers.

A B

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Goal: construct the gravity duals of these branched covers ⋄ Tremendously complicated in higher dimensions. ⋄ Must be a quotient AdS3/Γ in 3D. ⋄ Γ = “Schottky group” = discrete subgroup of isometry PSL(2, C).

λ

• torus

In Poincar´ e coordinates ds2 = dξ2 + dwd ¯ w ξ2 , L ∈ Γ acts as M¨

• bius transformations (circles to circles) at the boundary:

w → L(w) ≡ aw + b cw + d , ξ → |L′(w)|ξ , ad − bc = 1 .

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Schottky Uniformization

Every (compact) Riemann surface can be obtained as a quotient C/Γ with a suitable Schottky group Γ ⊂ PSL(2, C).

torus genus 2

Note for genus g, Γ is freely generated by g elements L1, L2, · · · , Lg. Strategy: find Γ, extend it to the bulk, and obtain the gravity dual. There can be more than one Schottky group Γ that generates the same Riemann surface. (E.g. the torus.) They give different bulk solutions (saddles) for the same boundary. Strategy: find all Γ thus giving all bulk solutions for a given

• boundary. Choose the dominant solution.

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Finding the Schottky Group

Need to construct the w coordinate which is acted on by Γ. Consider two intervals on a plane (with z coordinate). z is not single-valued on the branched cover. Consider the differential equation: ψ′′(z) + 1 2

4

• i=1
• ∆

(z − zi)2 + γi z − zi

• ψ(z) = 0

∆ = 1 2

• 1 − 1

n2

• The four γi are “accessory parameters”.

Take two independent solutions {ψ1, ψ2} and define w = ψ1(z) ψ2(z) ψ1,2(z) behaves as superposition of (z − zi)(1±1/n)/2 near zi. ⇒ w can be written in terms of in (z − zi)1/n. Single-valued!

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Finding the Schottky Group

Need to construct the w coordinate which is acted on by Γ. Consider two intervals on a plane (with z coordinate). z is not single-valued on the branched cover. Consider the differential equation: ψ′′(z) + 1 2

4

• i=1
• ∆

(z − zi)2 + γi z − zi

• ψ(z) = 0

∆ = 1 2

• 1 − 1

n2

• The four γi are “accessory parameters”.

Take two independent solutions {ψ1, ψ2} and define w = ψ1(z) ψ2(z) ⋄ Solutions have monodromies: ψ1 → aψ1 + bψ2, ψ2 → cψ1 + dψ2. ⇒ Induces PSL(2, C) identifications! w ∼ a w + b c w + d

A B

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Fixing the Accessory Parameters

Trivial monodromy at infinity fixes 3 of the 4 γi:

4

• i=1

γi = 0 ,

4

• i=1

γizi = −4∆ ,

4

• i=1

γiz2

i = −2∆ 4

• i=1

zi . We have too many nontrivial monodromies: There are 2g independent cycles, but Γ = L1, · · · , Lg.

A B

Require trivial monodromy on either the A or B cycle. ⇒ This fixes the remaining accessory parameter! This can be done either numerically, or analytically in certain regimes. We have obtained two Schottky groups that generate the branched cover.

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Uniformization of the Torus

ψ′′(z) + 1 2

4

• i=1
• ∆

(z − zi)2 + γi z − zi

• ψ(z) = 0

ψ′′(z) + 1 2

2

• i=1
• ∆ ℘(z − zi) + γ(−1)i+1ζ(z − zi) + δ
• ψ(z) = 0

[Barrella, XD, Hartnoll & Martin 1306.4682]

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1

Constructing the Bulk

2

The Classical Level

3

One-Loop Correction

4

Conclusion

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The Classical Action

SE = − 1 16πG

• d3x √g
• R + 2

L2

• + 2
• d2x √γ
• K − 1

L

• The on-shell action has been worked out explicitly for quotients of AdS3.

[Krasnov ’00]

It satisfies a very simple differential equation:

[Faulkner 1303.7221] [Zograf & Takhtadzhyan ’88]

∂SE ∂zi = −c n 6 γi Proof of Ryu-Takayagani for disjoint intervals on a plane Solves γi to linear order in n − 1 and integrates the above equation.

[Faulkner 1303.7221]

Proof of Ryu-Takayagani for one interval on a torus Use the uniformization equation for branched covers of the torus.

[Barrella, XD, Hartnoll & Martin 1306.4682]

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1

Constructing the Bulk

2

The Classical Level

3

One-Loop Correction

4

Conclusion

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One-Loop Bulk Correction

Given by the functional determinant of the operator describing quadratic fluctuations of all the bulk fields. For AdS3/Γ there is an elegant expression. [Giombi, Maloney & Yin

(0804.1773); Yin (0710.2129)]

For metric fluctuations: log Z|one-loop = −

• γ∈P

∞

• m=2

log |1 − qm

γ |

⋄ P is a set of representatives of the primitive conjugacy classes of Γ. ⋄ qγ is defined by writing the two eigenvalues of γ ∈ Γ ⊂ PSL(2, C) as q±1/2

γ

with |qγ| < 1. ⋄ Similar expressions exist for other bulk fields. ⋄ This one-loop correction is perturbatively exact in pure 3D gravity.

[Maloney & Witten (0712.0155)]

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Strategy

1

Find the Schottky group Γ corresponding to the n-sheeted covers.

M1 M2 L1 = M2 M1 L = M L M

2

• 1

2 1 2

2

Generate P = {primitive conjugacy classes} = {non-repeated words up to conjugation}. E.g. P = {L1, L2, L−1

1 , L−1 2 , L1L2 ∼ L2L1, L1L−1 2 , · · · }

3

Compute eigenvalues of these words & sum over their contributions. log Z|one-loop = −

• γ∈P

∞

• m=2

log |1 − qm

γ |

4

Analytically continue the (summed) one-loop result to n → 1.

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Numerical Results for Two Disjoint Intervals

n 2 n 3 n 4 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7

x In

1106

n 2 n 3 n 4 n 1 0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 200 250 300

x In

1106

Solid = numerics, dashed (dotted) = 4th (5th) order analytic expansion in small x (to be explained on the next slide). Mutual R´ enyi information between two intervals [z1, z2], [z3, z4] In(L1 : L2) = Sn(L1) + Sn(L2) − Sn(L1 ∪ L2)

• nly depends on the cross ratio x ≡ (z2 − z1)(z4 − z3)

(z3 − z1)(z4 − z3).

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Analytic Expansion in Small Cross Ratio

1

Find the generators Li of Γ:

1 1 y y

x = 4y (y + 1)2

For small y, we

(1) find γi by imposing trivial monodromy; (2) solve the differential equation for ψ(z) in two regimes: |z| ≪ 1 and |z| ≫ y; (3) match the solutions and construct Li.

2

Form non-repeated words.

3

Compute eigenvalues & sum over their contributions.

Nice feature 1: only finitely many words contribute to each order in y. At leading order, only “consecutively decreasing words” (and their inverses) contribute:

{Lk+mLk+m−1 · · · Lm+1} = {L1, L2, · · · , L2L1, L3L2, · · · , L3L2L1, · · · }

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Nice feature 2: at integer n the sum can be done explicitly in terms of rational functions of n: Sn|one-loop = − n n − 1

n−1

• k=1

csc8 256n8 x4 + (n2 − 1) csc8 + csc10 128n10 x5 + O(x6)

• = (n + 1)(n2 + 11)(3n4 + 10n2 + 227)

3628800n7 x4 + O(x5) where csc ≡ csc πk n

• (4) Analytically continue the one-loop result to n → 1:

S|one-loop = − x4 630 + 2x5 693 + 15x6 4004 + x7 234 + 167x8 36936 + O(x9)

• Exactly agrees with known results at leading order:

S = −N x 4 2h √π 4 Γ(2h + 1) Γ

• 2h + 3

2

+ · · ·

[Calabrese, Cardy & Tonni ’11]

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One-Loop Corrections in the Torus Case

Nice feature: only “single-letter” words {Li, L−1

i

} contribute to the leading order in the low / high T limit.

Thermal AdS:

A A

Sn|one-loop = − 1 n − 1

• 2 sin4 πL

R

• n3 sin4 πL

nR

− 2n

• e− 4π

TR + O

• e− 6π

TR

• S|one-loop =
• −8πL

R cot πL R

• + 8
• e− 4π

TR + O

• e− 6π

TR

• ⇒

SA − S¯

A = −8π cot

πLA R

• e− 4π

TR + O

• e− 6π

TR

• Agrees (morally) with a free field calculation in [Herzog & Spillane 1209.6368].

BTZ:

A A

Sn|one-loop = − 1 n − 1

• 2 sinh4(πTL)

n3 sinh4 πTL

n

− 2n

• e−4πTR + O
• e−6πTR

S|one-loop = [−8πTL coth(πTL) + 8] e−4πTR + O

• e−6πTR

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Conclusion

We calculated the corrections to the Ryu-Takayanagi formula by computing the one-loop determinant in the bulk geometry dual to the n-sheeted cover and analytically continuing to n → 1. One-loop corrections are perturbatively exact for pure gravity in 3D. We focused on two intervals on a plane and one interval on a torus in various limits (small x, low / high T). Our calculations agree with and go beyond known results. Questions and Future Directions Is there an exact formula for the one-loop correction to the entanglement entropy that is valid for all cross ratio? How does our calculations relate to the minimal surface? Can [Faulkner, Lewkowycz & Maldacena 1307.2892] reproduce our results in a simpler way?

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Back up slides

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n 1 n 1.1 n 2 n 3 n 10 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

x In

0c

n 1 n 1.1 n 2 n 3 n 10 0.0 0.1 0.2 0.3 0.4 0.5 0.000 0.001 0.002 0.003 0.004

x In

0c Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 2 / 2