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: ρ A = Tr ¯ A ( ρ A ¯ A ) A A S A = − Tr A ( ρ A log ρ A ) 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. Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 2 / 24
Holographic Entanglement Entropy A remarkably simple formula in QFTs with holographic duals: A S A = (Area) min A [Ryu & Takayanagi ’06] 4 G N 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] S ( L ) = c 3 log L [Calabrese & Cardy ’04] 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 AdS 3 duals. [Faulkner 1303.7221] Shown generally! (for Einstein gravity) [Lewkowycz & Maldacena 1304.4926] Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 3 / 24
Beyond Classical Gravity The Ryu-Takayanagi formula is valid in the limit of classical gravity (or large “ c ”): S A = (Area) min O ( G 0 + N ) + · · · 4 G N � �� � � �� � classical gravity one-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. Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 4 / 24
Two Disjoint Intervals on a Line at Zero Temperature L 1 L 2 L 1 L 2 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 ( L 1 : L 2 ) = S ( L 1 ) + S ( L 2 ) − S ( L 1 ∪ L 2 ) Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 5 / 24
One Interval on a Circle Below Hawking-Page Temperature Thermal AdS Phase At the classical level, S A does not depend on T and is given by a universal formula: � R � S A = c πǫ sin π L A A A 3 log R which is exact only at T = 0. Furthermore, S A − S ¯ A which measures the “pureness” of the state is nonzero only at the one-loop order. Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 6 / 24
One Interval on a Circle Above Hawking-Page Temperature BTZ Black Hole Phase At the classical level, S A does not depend on R and is given by with another universal formula: � 1 � S A = c A A 3 log π T ǫ sinh( π TL A ) which is exact only when the spatial circle becomes a line ( R = ∞ ). Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 7 / 24
Outline Constructing the Bulk 1 The Classical Level 2 One-Loop Correction 3 Conclusion 4 Three examples Two intervals on a plane 1 One interval on a torus (thermal AdS phase) 2 One interval on a torus (BTZ phase) 3 Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 8 / 24
Replica Trick Introduce R´ enyi entropy: 1 n − 1 log Tr ρ n S n = − ⇒ S = lim n → 1 S n = − Tr ρ log ρ At integer n , R´ enyi entropy can be written in terms of partition functions: 1 n − 1 log Z n S n = − Z n 1 Z n 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 of genus n − 1. Goal: construct the gravity duals of these branched covers. A B Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 9 / 24
Goal: construct the gravity duals of these branched covers ⋄ Tremendously complicated in higher dimensions. ⋄ Must be a quotient AdS 3 / Γ in 3D. ⋄ Γ = “Schottky group” = discrete subgroup of isometry PSL (2 , C ). λ torus ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� e coordinates ds 2 = d ξ 2 + dwd ¯ w In Poincar´ , ξ 2 L ∈ Γ acts as M¨ obius transformations (circles to circles) at the boundary: w �→ L ( w ) ≡ aw + b ξ �→ | L ′ ( w ) | ξ , ad − bc = 1 . cw + d , Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 10 / 24
Schottky Uniformization Every (compact) Riemann surface can be obtained as a quotient C / Γ with a suitable Schottky group Γ ⊂ PSL (2 , C ). genus 2 torus Note for genus g , Γ is freely generated by g elements L 1 , L 2 , · · · , L g . 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. Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 11 / 24
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: 4 � � � � ψ ′′ ( z ) + 1 ∆ γ i ∆ = 1 1 − 1 � ( z − z i ) 2 + ψ ( z ) = 0 z − z i n 2 2 2 i =1 The four γ i are “accessory parameters”. Take two independent solutions { ψ 1 , ψ 2 } and define w = ψ 1 ( z ) ψ 2 ( z ) ψ 1 , 2 ( z ) behaves as superposition of ( z − z i ) (1 ± 1 / n ) / 2 near z i . ⇒ w can be written in terms of in ( z − z i ) 1 / n . Single-valued! Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 12 / 24
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: 4 � � � � ψ ′′ ( z ) + 1 ∆ γ i ∆ = 1 1 − 1 � ( z − z i ) 2 + ψ ( z ) = 0 2 z − z i 2 n 2 i =1 The four γ i are “accessory parameters”. Take two independent solutions { ψ 1 , ψ 2 } and define w = ψ 1 ( z ) ψ 2 ( z ) ⋄ Solutions have monodromies: ψ 1 �→ a ψ 1 + b ψ 2 , ψ 2 �→ c ψ 1 + d ψ 2 . ⇒ Induces PSL (2 , C ) identifications! w ∼ a w + b c w + d A B Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 12 / 24
Fixing the Accessory Parameters Trivial monodromy at infinity fixes 3 of the 4 γ i : 4 4 4 4 � � � � γ i z 2 γ i z i = − 4∆ , i = − 2∆ γ i = 0 , z i . i =1 i =1 i =1 i =1 We have too many nontrivial monodromies: There are 2 g independent cycles, but Γ = � L 1 , · · · , L g � . 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. Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 13 / 24
Uniformization of the Torus 4 � � ψ ′′ ( z ) + 1 ∆ γ i � ( z − z i ) 2 + ψ ( z ) = 0 2 z − z i i =1 [Barrella, XD, Hartnoll & Martin 1306.4682] 2 ψ ′′ ( z ) + 1 � � � ∆ ℘ ( z − z i ) + γ ( − 1) i +1 ζ ( z − z i ) + δ ψ ( z ) = 0 2 i =1 Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 14 / 24
Constructing the Bulk 1 The Classical Level 2 One-Loop Correction 3 Conclusion 4 Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 15 / 24
The Classical Action �� � � � �� � d 3 x √ g d 2 x √ γ 1 R + 2 K − 1 S E = − + 2 16 π G L 2 L The on-shell action has been worked out explicitly for quotients of AdS 3 . [Krasnov ’00] It satisfies a very simple differential equation: [Faulkner 1303.7221] [Zograf & Takhtadzhyan ’88] ∂ S E = − c n 6 γ i ∂ z i Proof of Ryu-Takayagani for disjoint intervals on a plane Solves γ i 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] Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 16 / 24
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