Holographic Entanglement Beyond Classical Gravity

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 ρ A = Tr Ā (ρ AĀ ) 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 A S A = (Area) min 4G N [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] S(L) = c 3 log L ɛ [Calabrese & Cardy 04] 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 4G N) 0 + }{{ 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 A A At the classical level, S A does not depend on T and is given by a universal formula: S A = c ( R 3 log πɛ sin πl ) A R which is exact only at T = 0. Furthermore, S A S Ā 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 A A At the classical level, S A does not depend on R and is given by with another universal formula: S A = c 3 log [ 1 πt ɛ sinh(πtl 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 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) Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 8 / 24

Replica Trick Introduce Rényi entropy: S n = 1 n 1 log Tr ρn S = lim n 1 S n = Tr ρ log ρ At integer n, Rényi entropy can be written in terms of partition functions: S n = 1 n 1 log Z n Z1 n 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 0000000 1111111 000000000000000 111111111111111 000000000000000 111111111111111 0000000000000 1111111111111 In Poincaré coordinates ds 2 = dξ2 + dwd w ξ 2, L Γ acts as Möbius transformations (circles to circles) at the boundary: w L(w) aw + b cw + d, ξ L (w) ξ, ad bc = 1. 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). torus genus 2 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: ψ (z) + 1 2 4 ( i=1 (z z i ) 2 + γ i z z i ) ψ(z) = 0 = 1 (1 1n ) 2 2 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: ψ (z) + 1 2 4 ( i=1 (z z i ) 2 + γ i z z i ) ψ(z) = 0 = 1 (1 1n ) 2 2 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 aψ 1 + bψ 2, ψ 2 cψ 1 + dψ 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 γ i = 0, 4 γ i z i = 4, 4 γ i zi 2 = 2 4 z i. i=1 i=1 i=1 i=1 We have too many nontrivial monodromies: There are 2g 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 ψ (z) + 1 2 4 ( i=1 (z z i ) 2 + γ i z z i ) ψ(z) = 0 [Barrella, XD, Hartnoll & Martin 1306.4682] ψ (z) + 1 2 2 i=1 ( ) (z z i ) + γ( 1) i+1 ζ(z z i ) + δ ψ(z) = 0 Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 14 / 24

1 Constructing the Bulk 2 The Classical Level 3 One-Loop Correction 4 Conclusion Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 15 / 24

The Classical Action S E = 1 16πG [ d 3 x g (R + 2L ) 2 + 2 d 2 x γ ( K 1 )] 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] S E z i = c n 6 γ i Proof of Ryu-Takayagani for disjoint intervals on a plane [Zograf & Takhtadzhyan 88] 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

1 Constructing the Bulk 2 The Classical Level 3 One-Loop Correction 4 Conclusion Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 17 / 24

One-Loop Bulk Correction Given by the functional determinant of the operator describing quadratic fluctuations of all the bulk fields. For AdS 3 /Γ there is an elegant expression. [Giombi, Maloney & Yin (0804.1773); Yin (0710.2129)] For metric fluctuations: log Z one-loop = γ P log 1 qγ m m=2 P is a set of representatives of the primitive conjugacy classes of Γ. q γ is defined by writing the two eigenvalues of γ Γ PSL(2, C) as q ±1/2 γ with q γ < 1. Similar expressions exist for other bulk fields. This one-loop correction is perturbatively exact in pure 3D gravity. [Maloney & Witten (0712.0155)] Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 18 / 24

Strategy 1 Find the Schottky group Γ corresponding to the n-sheeted covers. M1 M2 L1 = M2 M1-1 L 2 = M 2 L 1 M 2 2 Generate P = {primitive conjugacy classes} = {non-repeated words up to conjugation}. E.g. P = {L 1, L 2, L 1 1, L 1 2, L 1L 2 L 2 L 1, L 1 L 1 2, } 3 Compute eigenvalues of these words & sum over their contributions. log Z one-loop = γ P log 1 qγ m m=2 4 Analytically continue the (summed) one-loop result to n 1. Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 19 / 24

Numerical Results for Two Disjoint Intervals I n 1 10 6 6 5 n 2 4 n 3 3 n 4 2 1 I n 1 10 6 250 200 150 100 50 n 1 n 2 n 3 n 4 0 0.0 0.2 0.4 0.6 0.8 x 0 0.0 0.2 0.4 0.6 0.8 1.0 Solid = numerics, dashed (dotted) = 4th (5th) order analytic expansion in small x (to be explained on the next slide). x Mutual Rényi information between two intervals [z 1, z 2 ], [z 3, z 4 ] I n (L 1 : L 2 ) = S n (L 1 ) + S n (L 2 ) S n (L 1 L 2 ) only depends on the cross ratio x (z 2 z 1 )(z 4 z 3 ) (z 3 z 1 )(z 4 z 3 ). Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 20 / 24

Analytic Expansion in Small Cross Ratio 1 Find the generators L i of Γ: 1 y y 1 x = 4y (y + 1) 2 2 Form non-repeated words. For small y, we (1) find γ i 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 L i. 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: {L k+m L k+m 1 L m+1 } = {L 1, L 2,, L 2 L 1, L 3 L 2,, L 3 L 2 L 1, } Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 21 / 24

Nice feature 2: at integer n the sum can be done explicitly in terms of rational functions of n: S n one-loop = n n 1 [ csc 8 n 1 256n 8 x 4 + (n2 1) csc 8 + csc 10 ] 128n 10 x 5 + O(x 6 ) k=1 = (n + 1)(n2 + 11)(3n 4 + 10n 2 + 227) 3628800n 7 x 4 + O(x 5 ) ( ) πk where csc csc n (4) Analytically continue the one-loop result to n 1: ( x 4 S one-loop = 630 + 2x 5 693 + 15x 6 4004 + x 7 234 + 167x 8 ) 36936 + O(x 9 ) Exactly agrees with known results at leading order: ( x ) 2h π Γ(2h + 1) S = N 4 4 Γ ( ) 2h + 3 + [Calabrese, Cardy & Tonni 11] 2 Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 22 / 24

One-Loop Corrections in the Torus Case A Nice feature: only single-letter words {L i, L 1 i } contribute to the leading order in the low / high T limit. Thermal AdS: A S n one-loop = 1 n 1 S one-loop = [ 8πL R cot S A S Ā = 8π cot ( πla R [ 2 sin 4 ( ) πl R n 3 sin 4 ( πl nr ) 2n ( ) ] πl + 8 R ] ) e 4π TR + O ( e 4π TR + O ( e 4π TR + O ( e 6π TR ) e 6π TR e 6π TR Agrees (morally) with a free field calculation in [Herzog & Spillane 1209.6368]. ) ) A BTZ: A [ ] S n one-loop = 1 2 sinh 4 (πtl) n 1 n 3 sinh 4 ( ) 2n e 4πTR + O ( e 6πTR) πtl n S one-loop = [ 8πTL coth(πtl) + 8] e 4πTR + O ( e 6πTR) Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 23 / 24

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? Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 24 / 24

Back up slides Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 1 / 2

I n 0 c 0.8 0.6 0.4 n 1 n 1.1 n 2 n 3 I n 0 c 0.003 0.002 n 1 n 1.1 n 2 n 3 0.2 n 10 0.001 n 10 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x 0.000 0.0 0.1 0.2 0.3 0.4 Xi Dong (Stanford University) Holographic Entanglement Beyond Classical Gravity 2 / 2