Conformal Blocks, Entanglement Entropy & Heavy States

Transcription

1 Conformal Blocks, Entanglement Entropy & Heavy States Pinaki Banerjee The Institute of Mathematical Sciences, Chennai April 25, 2016

2 arxiv : Higher-point conformal blocks & entanglement entropy of heavy states with Shouvik Datta (ETH, Zürich) and Ritam Sinha (TIFR, Mumbai)

3 Overview Introduction Boundary Computation The Bulk Picture EE : An Application Conclusions

4 Introduction

5 Motivations AdS/CFT dominating [hep-th] for last two decades! Universal features of holography : Cardy formula, EE etc. Conformal blocks are very useful : Bootstrap, AGT, bulk locality and gravitational scattering, geodesics, Goal : To show conformal blocks with two heavy & arbitrary number of light operators factorize. Relevant in the context of EE for excited states with multiple intervals.

6 What are Conformal Blocks? Consider a p-point correlator O(z 1 )O(z 2 )O(z 3 ) O(z p ) Insert p 3 resolutions of the identity O 1 (z 1 )O 2 (z 2 ) α α O 3 (z 3 ) β ζ O p 1 (z p 1 )O p (z p ) α,β,ξ,... A typical term of this sum is called conformal block F p (z i, h i, h i ) := O 1 (z 1 )O 2 (z 2 ) α α O 3 (z 3 ) β ζ O p 1 (z p 1 )O p (z p ) These are building blocks of CFT correlators.

7 What is entanglement entropy? Density matrix of a state is defined as ρ tot = Ψ Ψ EE is Von Neumann entropy of reduced density metrix ρ A = Tr B (ρ tot ) S A = Tr A (ρ A log ρ A ). A measure of entanglement between subsystems. Vanishes for pure states.

8 What is entanglement entropy? Density matrix of a state is defined as ρ tot = Ψ Ψ EE is a geometric quantity S A = min[γ A] 4G N A measure of entanglement between subsystems. Vanishes for pure states.

9 What is entanglement entropy? Density matrix of a state is defined as ρ tot = Ψ Ψ Disjoint intervals in 1+1 dimensional systems A 1 A 2 A N 1 x 3 x 4 x 5 x 2N x 2N+1 S A = Tr A (ρ A log ρ A ). A measure of entanglement between subsystems. Vanishes for pure states.

10 Which excited states? Using the state-operator correspondence ψ = O H (0) 0 and ψ = lim z, z z2h H z 2h H 0 O H (z, z). O H (0) has very large scaling dimension. Corresponding states are heavy states.

11 Boundary Computation

12 Heavy-light correlators O 1 (z 1, z 1 )O 2 (z 2, z 2 ) O p (z p, z p ) = { hi} d { hi } F(z i, h i, h i ) F( z i, h i, h i ) We are interested in O H (z 1, z 1 ) m+1 i=2 O L (z i, z i ) O H (z m+2, z m+2 ) Our correlator of interest in terms of cross-ratios [ ] m+1 O H ( ) O L (1) O L (x i ) O H (0). We work in c limit for which i=3 F (p) (z i, h i, h i ) = exp [ c ] 6 f (p)(z i, ɛ i, ɛ i ).

13 Heavy-light correlators We shall also work in the heavy-light limit ɛ H = 6h H c O(1), = 6h L c 1 And this particular OPE channel 1 x 3 x 4 x 5 x 6 x 7 x m x m+1 ɛ p ɛ p ɛ p ɛ p 0 ɛ H ɛ Q ɛ R ɛ H

14 Monodromy method Recall a typical conformal block looks like F p (z i, h i, h i ) := O 1 (z 1 )O 2 (z 2 ) α α O 3 (z 3 ) β ζ O p 1 (z p 1 )O p (z p ) Let s insert an additional operator, ˆψ(z) Ψ(z, z i ) := O 1 (z 1 )O 2 (z 2 ) α α ˆψ(z)O 3 (z 3 ) β ζ O p 1 (z p 1 )O p (z p ) = ψ(z, z i )F (p) (z i, h i, h i ) Choose that ˆψ(z) obeys the null-state condition at level 2 [ ] 3 L 2 2(2h ψ + 1) L2 c 1 1 ψ = 0, with, h ψ = 2 9 2c

15 Monodromy method The differential operator representation gives an ODE d 2 ψ(z) dz 2 + T (z)ψ(z) = 0, with, T (z) = p [ ɛ i (z z i ) 2 + c ] i z z i i=1 Here, ɛ i = 6h i /c and c i are the accessory parameters c i = f (p)(z i, ɛ i, ɛ i ) z i satisfying c i z j = c j z i Solve for the c i, by using the monodromy properties of the solution ψ(z) around the singularities of T (z).

16 Monodromy method c i can be obtained by studying monodromy properties of ψ(z), around contours containing the operator insertions. Monodromy around a contour γ k = info about the resultant operator which arises upon fusing the operators within γ k ( ) e +πiλ 0 M(γ k ) = 0 e πiλ, Λ = 1 4 ɛ p Perturbation theory in (heavy-light limit) ψ(z) = ψ (0) (z) + ψ (1) (z) + ψ (2) (z) +, T (z) = T (0) (z) + T (1) (z) + T (2) (z) +, c i (z) = c (0) i (z) + c (1) i (z) + c (2) i (z) +, for i = 3, 4,..., m + 1.

17 Monodromy method Choice of monodromy contour = Choice of OPE channel We choose the contours such that each of them contains a pair of light operators within. This is equivalent to looking at the OPE channel in which light operators fuse in pairs. This choice is geared towards entanglement entropy calculations. 1 x 3 x 4 x 5 γ 1 γ 2 = 1 x 3 x 4 x 5 ɛ p ɛ p 0 ɛ H ɛ H

18 Monodromy method Choice of monodromy contour = Choice of OPE channel We choose the contours such that each of them contains a pair of light operators within. This is equivalent to looking at the OPE channel in which light operators fuse in pairs. For 5-pt function (H-L-L-L-H) 1 ɛ p = x 3 x 4 γ 1 γ2 1 x 3 x 4 0 ɛ H ɛ H

19 Monodromy method The monodromy conditions for all the contours form a coupled system of equations for the accessory parameters. Performing the exercise for 5- and 6-point blocks provides sufficient intuition to guess the solutions. For light operators located at x p and x q living with in a contour, the corresponding accessory parameters are c p = (x α q (α 1) + x α p (α + 1)) + (x p x q ) α/2 α ɛ p x p (x α p x α, q ) c q = (x α p (α 1) + x α q (α + 1)) + (x q x p ) α/2 α ɛ p x q (x α q x α. p )

20 Monodromy method The accessory parameters can now be used to obtain the conformal block c i = f (p)(z i, ɛ i, ɛ i ) z i F (p) (z i, h i, h [ i ) = exp c ] 6 f (p)(z i, ɛ i, ɛ i ) Even-point conformal blocks The (m + 2)-point block factorizes into a product of m/2 4-point conformal blocks [ F (m+2) ({x i };, ɛ H ; ɛ p ) = exp c ] 6 f (4)(x p, x q ;, ɛ H ; ɛ p ) = Ω i {(p,q)} Ω i {(p,q)} F (4) (x p, x q ;, ɛ H ; ɛ p ). Ω i : Indicates the OPE channels / monodromy contours.

21 Monodromy method Odd-point conformal blocks The (m + 2)-point block factorizes into a product of (m 1)/2 4-point conformal blocks and a 3-point function [ F (m+2) ({x i };, ɛ H ; ɛ a ) = (1) c ] 6 f (4)(x p, x q ;, ɛ H ; ɛ a ) = (1) Ω B i {(p,q)} exp Ω B i {(p,q)} F (4) (x p, x q ;, ɛ H ; ɛ a ). where, f (4) (x i, x j ;, ɛ H ; ɛ p) = ( (1 α) log x i x j + 2 log xα i xα j α ) + 2 ɛ p log 4α xα/2 j x α/2 j + x α/2 i x α/2 i.

22 Caveats This factorization is true only... 1 at large central charge. 2 in the heavy-light limit 3 for this specific choice of OPE channels 4 ɛ p

23 Bulk Picture

24 The dual geometry The heavy excited state is dual to the conical defect geometry ds 2 = ( dt α2 2 cos 2 + 1α ) ρ 2 dρ2 + sin 2 ρ dφ 2, with α = 1 24h H /c. The light operators are dual to bulk scalars of masses of O(c) and can be approximated by worldlines. w ɛ p w i w j

25 The dual geometry The heavy excited state is dual to the conical defect geometry ds 2 = ( dt α2 2 cos 2 + 1α ) ρ 2 dρ2 + sin 2 ρ dφ 2, with α = 1 24h H /c. The conformal blocks can be reproduced by considering lengths of suitable worldline configurations in the bulk. w ɛ p w i w j

26 4-point block from bulk O H ( )O L (x i )O L (x j )O H (0) The worldline action : S = l L + ɛ p l p The R-T lengths ( l L (w ij ) = 2 log sin αw ij 2 ( l p (w ij ) = log tan αw ij 4 ) + 2 log ). ( ) Λ, 2

27 4-point block from bulk O H ( )O L (x i )O L (x j )O H (0) The worldline action : S = l L + ɛ p l p The (w i dependent) contribution to the correlator ( αw G(w i, w j ) = e c 6 S(w i,w j ) = e h L l L (w ij ) h p l p(w ij tan ij ) hp ) 4 = ( αw sin ij ) 2hL. 2

28 4-point block from bulk O H ( )O L (x i )O L (x j )O H (0) The worldline action : S = l L + ɛ p l p From cylinder to plane : x i = e iw i and x j = e iw j F (4) (x i, x j ) = x h L i x h L j G(w i, w j ) wi,j = i log x i,j (Matches!)

29 Higher point block from bulk Even-point conformal blocks w 7 w 6 1 x 3 x 4 x 5 x 6 x 7 ɛ p ɛ p ɛ p ɛ p 0 ɛ p ɛ p w 5 0 ɛ H ɛ H w 3 w 4

30 Higher point block from bulk Odd-point conformal blocks w 6 1 x 3 x 4 x 5 x 6 ɛ p ɛ p 0 ɛ p ɛ p w 5 0 ɛ H ɛ H w 3 w 4

31 EE : An Application

32 EE for excited states A 1 A 2 A N 1 x 3 x 4 x 5 x 2N x 2N+1 EE from Rényi entropy : S (n) A = 1 1 n log tr A (ρ A ) n ; n 1 Effectively need to compute ( for n 1) G n (x i, x i ) = Ψ σ(1) σ(x 3 )σ(x 4 ) σ(x 5 )σ(x 6 ) σ(x 7 )... σ(x 2N ) σ(x 2N+1 ) Ψ = 0 Ψ( ) σ(1) σ(x 3 ) 2N i=4,6, Dimensions of the twist and anti-twist operators h σ = h σ = c ( n 1 ) 24 n σ(x i ) σ(x i+1 ) Ψ(0) 0

33 EE for excited states In the limit n 1 σ, σ : Light operators Ψ : Heavy operator S A = lim S (n) n 1 A = c 3 min i { Ω i {(p,q)} } log (xα p x α q ). α(x p x q ) α 1 2 with, α = 1 24h H /c

34 Conclusions & Outlook

35 Summary Higher point conformal blocks are tractable in the heavy-light limit. These conformal blocks can be reproduced precisely from the dual gravity picture. This is applied to find entanglement entropy of disjoint intervals in heavy states. This conformal block can be rewritten in terms of geodesic lengths (bulk locality?)

36 Outlook Applications 1 Tripartite information 2 Mutual information in local quenches 3 Scrambling, chaos,... Extensions 1 Higher spin holography 2 One-loop corrections 3 Higher dimensions,...

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