JHEP11(2018)072 Springer November 5, 2018 November 12, 2018 September 12, 2018 : : : Accepted Received Published = 4 non-commutative super N c Published for SISSA by https://doi.org/10.1007/JHEP11(2018)072 [email protected] , , and Juan F. Pedraza b , however, both the butterfly and entanglement velocities increase θ . 3 1808.10050 Viktor Jahnke The Authors. Gauge-gravity correspondence, Black Holes in String Theory, Non- a c
Holographic theories with classical gravity duals are maximally chaotic: they , [email protected] Universidad Nacional Apartado Aut´onomade M´exico, Postal 70-543,Institute CDMX for 04510, Theoretical Physics, M´exico UniversityScience Park of 904, Amsterdam, 1098 XHE-mail: Amsterdam, Netherlands [email protected] Theory Group, Department of PhysicsThe and University Texas of Cosmology Texas Center, atDepartamento Austin, de de Austin, F´ısica Altas TX Energias, 78712, Instituto U.S.A. de Ciencias Nucleares, b c a Open Access Article funded by SCOAP Commutative Geometry ArXiv ePrint: enhance the effective light-cone for theconjectured transfer bounds of encountered quantum in information, the eludingon context previously of a local possible quantum limitation field on theory. the We comment retrieval of quantumKeywords: information imposed by non-locality. thermofield double state with definite momentum andof study the several chaos theory, related properties includingtime the and butterfly the velocity, the maximalcommutative parameter entanglement Lyapunov velocity, exponent. thewith scrambling The the latter strength of two the are non-commutativity. unaffected This by implies the that non- non-local interactions can Abstract: saturate a set of boundswhether on non-locality the can spread affect ofa such quantum prototypical bounds. information. theory In Specifically, with this weYang non-local paper Mills. consider interactions, we the We namely, question construct gravity shock dual waves of geometries that correspond to perturbations of the Willy Fischler, Chaos and entanglement spreadingnon-commutative in gauge a theory JHEP11(2018)072 34 33 8 16 9 9 28 20 14 0 r 28 12 7 17 4 – 1 – 11 11 26 2 30 15 as a function of 31 12 α 36 ) 6 0 r ( 3 K 7 2 5.1.1 Butterfly velocity 4.2.1 Entanglement velocity 3.2.1 Shocks with3.2.2 definite momentum Lyapunov exponent3.2.3 and scrambling time Butterfly velocity 5.1 Infalling particle and entanglement wedge 4.1 Mutual information4.2 in the TFD Disruption state of mutual information by shock waves 3.1 Eternal black3.2 brane geometry Shock wave geometries 2.2 Gauge invariant2.3 operators in non-commutative Out-of-time-ordered theories correlators2.4 in momentum space Entanglement entropy in non-commutative theories 1.2 Chaos and1.3 entanglement spreading Plan of the paper 2.1 Gravity dual of non-commutative SYM 1.1 Probes of quantum chaos C Divergence of 6 Conclusions and outlook A Momentum space correlator and the butterflyB velocity Shock wave parameter 5 Butterfly velocity from one-sided perturbations 4 Entanglement velocity from two-sided perturbations 3 Perturbations of the TFD state 2 Preliminaries Contents 1 Introduction JHEP11(2018)072 = V T = 1. (1.2) (1.1) i and takes . Such an WW L W h λ = , the Lyapunov ∗ i ] and the scrambling < ) non-commutative super Yang + α ) 2 2 z N ( 2 , with non-commutative parameter dx φ . The gravity dual of this theory is λ ) , ) iθ y r 4 ( 4 H ( µν 1 r r ∼ h φ iθ ] z ν − + 3 ∂ y µ 2 1 ∂ , x ] = 2 ν dx – 7 – µν ]. In the context of gauge/gravity duality, this x ], which studied a specific decoupling limit of a θ ) = 1 , x i 2 + r 81 µ ( e 2 – 44 x f , . The decoupling limit consists of scaling the string [ Moreover, dt ≡ 78 23 ) 43 4 ) -field, provided that one takes a special limit to decouple r . Notice that for future convenience, we have given the B x ( 0 . B f 2 E )( /α 2 − ds 2 2 / R 2 )-plane, i.e., [ ? φ Φ r 3 e 1 = , denotes the string coupling and , φ and large ’t Hooft coupling , x ) = 4 ) ( λ 2 / r g r 1 ( x , 2 str √ ( N , ˆ ) h , 2 4 h r 2 0 4 ds ( 3 ) r r r r R 2 h 2 ( 2 4 a a / ˆ h g 2 2 ˆ g 1 ˆ R gNα 2 ˆ g R 4 g R π = = ˆ = = = is a real and antisymmetric rank-2 tensor. The algebra of functions in a non- = 4 r 2 E 23 01 4 2Φ µν e ds C B R θ 0123 F Non-commutative theories arise naturally in string theory, as the worldvolume theory In the string frame 4 the standard relation metric above in the Einstein frame. where Hooft coupling is related to the curvature of the background and the string tension through Mills theory at large non-zero only in thetype ( IIB supergravity, with theories can be describedkind in of terms dualities was ofstack presented a of in classical D3-branes [ gravity withtension dual. non-zero to infinity, and The the closed firstlimit string example metric provided to of a zero, while this gravity keeping dual the for finite temperature SU( of D-branes with non-zero NS-NS the open and closedimplies string that sectors the [ dynamics of certain strongly-coupled, large where commutative theory can bedeformed viewed to as the an so-called algebra Moyal product, of ordinary functions with the product 2.1 Gravity dualNon-commutative of quantum field non-commutative theory SYM has been anof important great theoretical research arena interest and in a the topic pastis few that decades. space-time The basic coordinates postulatemutation do of relation non-commutativity not commute. Instead, they satisfy the following com- 2 Preliminaries JHEP11(2018)072 5 5 ), S S µ k × × (2.6) (2.7) (2.5) ( ˜ 5 O ]. ) are the is related 80 a 2.3 )[ 2.1 ) reduces to the ] 84 2.6 ]. This observation – -Schwarzschild 5 44 82 , 43 The parameter ) super Yang Mills theory. . . On the other hand, for , x θ 5 · N x · ik √ ik e 4 ) / . This parameter can be thought 1 x ? e θ ( λ ) √ O goes to zero and ( 4 correspond to the UV and IR limits, becomes large as dictated by the non- ) according to [ x / 4 4 is mapped into an energy scale in the 1 x, C H W r ( d λ W ` r 4 r ` x, C a 1 , while = Z ( ) transforming in the adjoint representation → ?W µ a ) goes over to the AdS W r ) πT x 1 + ( ) = x ( 2.3 – 8 – k = O ( -Schwarzschild solution [ O ) are parallel to the boundary and are directly H and ) = 5 r r x ( 4 t, ~x through h d → O ( θ ) → ∞ Z k ≡ ( r ˜ . µ O θ ) exhibits significant differences with respect to AdS ) = x k √ ( 4 2.3 / ˜ O 1 λ , the background ( 1 − 1, the length of the Wilson line 1, the length of the Wilson line << a H θ << θ >> r √ √ − the background ( k k r denotes the Moyal product. A few comments are in order: 1 − ? For commutativity. In this limit, the operator is dominated by the Wilson line regardless For standard operators in commutative field theory, For As usual in holography, the radial direction A simple way to understand why this background is dual to a non-commutative theory is to consider • • 5 an open string in the corresponding background, which yields the commutation relation ( where by smearing the gauge covariant operators of the gauge group over an open Wilson line of order or smaller than 2.2 Gauge invariant operatorsIn in non-commutative gauge non-commutative theories theories,the the gauge transformations non-commutativity and, of therefore, there thespace. are spacetime no However, gauge one mixes invariant can operators with in construct position gauge invariant operators in momentum space, just reflects thesuper fact that Yang Mills the atr non-commutative >>a boundary length theory scalesand, goes much in greater over particular, to than is ordinary just no means longer that asymptotically the AdS. effect of From the the non-commutativity boundary becomes perspective, pronounced this for length scales with the global SU(4) internal symmetry group, but they will playsolution, no which role in is our dual discussion. toIndeed, a it thermal can state be ofsame shown the as that standard the all SU( ones the obtained thermodynamic from the quantities AdS derived from ( parameter that will enter in every holographic computation. field theory, in such arespectively. way that The directions identified with the field theory directions. Finally, the five-sphere coordinates are associated is a function that encodes theto effects the of non-commutative the parameter non-commutativity. of as a “renormalized” non-commutative length scale at strong coupling, since this is the is the standard blackening factor, with JHEP11(2018)072 ) ]. A x ( 85 (2.8) (2.9) O is the ) given ) of the A k ), defined ρ ( r, k ( ˜ depends on O t, ~x ϕ ( θk C . One concrete θ ' 0). As expected, and the butterfly , √ ), where L W k A (0 ` λ ρ V ) has a pole precisely k ~ log t, ( A ρ C ) and . tr ( ). According to the standard . i t, ~x 2 ) − ( ) in a near-boundary expansion k r, x ( 0)] ( W = , x ˜ , O ϕ ) ) while the normalizable mode gives (0 A k x S ( − ,V ( O → ∞ ) 0 k ~ r ( t, ) as genuinely different operators at different k ϕ ( k ϕ 4 ( d ) gives the Lyapunov exponent and the butterfly – 9 – W ˜ [ k O ~ Z t, ]. −h ( ) is dual to a gauge invariant operator + C 47 0 ) = S r, k ( k ~ = ϕ t, ) contains a Wilson line whose length ( S k ( C ˜ O ) is determined from the non-normalizable mode of k ( 0 ϕ , from which we can extract the Lyapunov exponent B /v . The fact that the pole of L B ) in the sense that in the boundary theory there is a coupling of the form iλ v implies that one should not think of these operators as the “same” operator . ), for two Hermitian gauge-invariant operators 2.6 = with different momentum asone we should usually not expect do tospace. in obtain Instead, standard a one field local should operator theory.k think by of In Fourier transforming particular, to position operators are expected toexample exhibit of this a fact universal isquasinormal behavior the universal mode at dissipation analysis large time of at [ large momentum foundFinally, in the the fact that k of what operator is attached at the end. Therefore, the correlation functions of these | In the holographic context, there is a one-to-one map between gauge invariant operators ) in the boundary theory and local bulk fields 1.4 k • ~ | x ( velocity is implicit in other holographic calculations. See for instance2.4 the appendix C of [ Entanglement entropy inIn non-commutative standard theories quantum fieldcan theory be the entanglement calculated entropy by associated the to von a Neumann subsystem formula, There is nothe need spatial to coordinates. goat to In frequency the space,velocity next since section the we non-commutativity show only that acts on In order to diagnose chaos wein need to ( compute the normsuch of definition the is commutator problematic for non-commutative gaugethere theories, are because in no these gauge theories in invariant non-commutative operators theories in by position defining an space. equivalent quantity Instead, in we momentum will space, quantify i.e., chaos As usual, the source some appropriate boundary condition in the IR. 2.3 Out-of-time-ordered correlators in momentum space position space. This issuecan is solved assume by that working in theform momentum bulk ( space. field More specifically, one O dictionary, the non-normalizablecorresponds mode to of the sourceits of expectation the value. dual Thecommutative operator because, above map as explained is above, subtle there when are the no boundary gauge gauge invariant theory local is operators non- in JHEP11(2018)072 ] is 95 ¯ A (2.16) (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) for the A , . Then, one A ]. This transition can be computed in 89 volume law , A , 88 S } and its complement 0 has a coefficient which is , A ≤ A ) S ∂A 3 . . ) , x } 3 2 0 ˆ x , , x 2 ≤ 1 , . , ˆ x x ) is recovered [ . ) , 1 i A ··· ··· x γ Φ N | + + 2.11 ˆ G Φ(ˆ 1 2 4 Σ A − − = Φ → d d V A Area( is given be the classical entangling surface at a ] i – 10 – ) such that Φ 3 Φ A = . In holographic theories | 87 ∼ ∼ i , , x ˆ A Φ A A A ) such that Φ( Φ 2 86 S 3 S S {| ˆ , x Φ, with eigenvalue Φ, i.e., , x 1 = 2 can still be computed by using the standard HRRT pre- x . One possible way to define a subsystem in these theories . A , x A ) = 0 defines the boundary of the region ¯ Φ( ]. With this holographic definition, it has been shown that A A 1 3 S x ]. Very recently, the full cutoff dependence was studied in [ ( , x 89 { , 2 . Entanglement entropy in local theories follows the so-called 94 and , = 88 , x 1 A ∂A 93 A x = ]. The prescription is the following: first, one defines a region ] and has been understood as a result of the non-locality inherent of non- can then be uniquely defined as A 88 92 A – ∂γ ), where the boundary of 90 2.10 is an extremal area surface whose boundary coincides with the boundary of the , i.e., , which means that the leading UV divergence of denote the eingenvector of A A γ i Φ For holographic theories In non-commutative theories it is not always possible to precisely define the curve (or | while for large regionsfrom the a standard volume law area to law ancalculations area [ ( law behaviour has alsocommutative been theories observed [ in quantum field theory scription ( particular cut-off scale [ for small enough regions the entanglement entropy follows instead a The subsystem promotes Φ to an operator, Let was proposed in [ commutative case as where the surface Φ( The area law basicallydominated means by that contributions the coming from entanglementthe short-ranged between boundary interactions between between points close to surface) delimiting the region area law proportional to the area of the boundary of the region Σ = the bulk by the HRRT prescription [ where region reduced density matrix associated to JHEP11(2018)072 ] ]. is 14 [ 96 r and (3.7) (3.2) (3.3) (3.4) (3.5) (3.6) (3.1) B ∞ v = ≤ r , E v j dx . i dx ) r ( ij , G j + is generically larger that the dx ) = constant i 2 . H E t r v dr π dx β ( 4 ) . ) r ij ) − ( e rr . as follows, − U, V ( G 1 0 U, V c c ( = ,G ij UV dr , + tt G H r 2 tt U, V rr r G π 1 + G dt G 2 c 2 − ) π r β r r 8 = 4 ( – 11 – tt , U/V 1 = = T G ∗ dUdV ∗ ) = r ) rr − ≡ π β dr 4 β = e U, V U, V ( ( N = ,G A ) are the coordinates of the boundary theory, while A ) i dx H r UV M = 2 t, x − 2 dx ) r r ds ( ( 0 c . We assume the following near-horizon expressions for the metric H r MN = G 3. Here ( tt = , G 2 = r , 2 ds = 1 i, j Finally, we point out that the time dependence of entanglement entropy for a free In terms of these coordinates the metric reads where and then we introduce the Kruskal coordinates In the study of shock wavesnates is convenient cover to smoothly work the in two Kruskal coordinates, sides since of these the coordi- geometry. We first define the Tortoise coordinate functions The inverse Hawking temperature associated to the above metric is where the holographic radial coordinate.the horizon We at take the boundary to be located at 3 Perturbations of the3.1 TFD state Eternal blackLet brane us geometry consider a two-sided black brane geometry of the form commutativity. As explained in thetheories introduction, since this Lorentz is invariance not isdo an explicitly not issue broken and for apply. the non-commutative However, standardstrong this notions raises coupling of a regime? causality numberholds of And for questions. more general Does importantly, non-local thisquantum does information? behavior theories? the hold conjecture If in that the so, what are the implications for the transfer of coupling regime through holography. scalar field on a non-commutativeIn sphere this following paper a it quantum wascommutative quench counterpart, found was even that studied exceeding the the in entanglement speed [ velocity of light in the limit of very strong non- which found an exact match with respect to the results previously obtained in the strong JHEP11(2018)072 (3.8) (3.9) . j . dx in the past and i iθ 0 dx t ij ] = 3 T , x + 2 2 x dV , VV T MN + matter 0 2 T , one can define gauge invariant op- 0) covers the black hole (white hole) N dU 2.2 0) covers the left (right) exterior region, πG UU V < -direction. The motivation to consider such T = 0. The boundary (left or right) is located = 8 V + V > R ) changes when we add to the system a null pulse – 12 – UV 0 and = 1. We assume that the unperturbed metric is a MN 3.1 = 0 frame, the energy of this perturbation increases dUdV G 0 and t U < 1 2 UV UV T − 0 ( -direction. This pulse of energy will give rise to a shock U < 6 V = 0. We want to know how this pattern of entanglement = 2 MN 0 ( V > t R N dx V < M = 0 and moving in the 0 and , while its distance from the past horizon decreases exponentially ) is the most general stress-energy tensor which is consistent with the 0 dx i U t 0 and MN U > T U, V, x ) over a Wilson line. Despite being non-local, this type of perturbation can = . The only requirement is that the perturbation is local in time and is applied ( x k U > 1 and the singularity at ( − O MN T matter . As a results, an early enough perturbation will follow an almost null trajectory = 0 0 = 0 and moving in the T = t A possible cosmological constant term is absorbed into the definition of the stress-energy tensor. U UV 6 MN also give rise tomomentum a shock wave geometry, specifically,in a the shock asymptotic wave past. geometrythe This with non-commutativity affects is definite only perfectly two possible spatial in coordinates, non-commutative i.e. SYM [ theory, since 3.2.1 Shocks withIn definite non-commutative momentum theories isposition not space. possible However, toerators as define that explained local are in gauge section localoperator invariant in operators momentum in space. This is done by smearing a gauge covariant with very close to theat past horizon, which canwave then geometry. replaced by a null pulse of energy, located by inserting an operatorstudying in the one of evolution the ofto boundary the the theories system. creation at some ofbrane. In time a From the the perturbation gravitational point close of description,exponentially view to this with of the the corresponds boundary, which then falls into the black of energy located at a perturbation is the following.brane In geometry the is context dual of to gauge/gravityThis duality a thermofield the thermofield two-sided double double black state statetwo of has two boundaries a copies theories of very the particular at changes boundary pattern when theory. of we entanglement perturb between one the of the boundary theories far in the past. We can do that T Ricci tensor of the unperturbed geometry. 3.2 Shock wave geometries In this section we study how the metric ( where the stress-energy tensor is assumed to be of the form while the region interior region. The horizonat is located at solution of Einstein’s equations The region JHEP11(2018)072 ), 0) 0), 3.10 (3.11) (3.17) (3.12) (3.13) (3.14) (3.15) (3.16) (3.10) U > U > . The pulse , E j 2 , j dx ). Note that the j ˆ x i dU d ) i dx 3.1 α dx i ˆ x ) d α , dx , ij , ) 2 + Θ can be determined from ˆ 2 T α ˆ ) to satisfy the Einstein’s U ˆ α U + Θ + d d 2 i ) , U, V and amplitude + Θ ) 3.15 ˆ ( shock i MN V ˆ U ˆ k ~ U x U, V d ( T ( ( ˆ UU V = ij + . U, V T ˆ ). ˆ ], where V α δ ˆ ( α δ i T ˆ ~x k ~ T · ˆ ˆ x ij A ˆ k V ~ 98 t, i + 2 , ) + ˆ + ( G U . i e 2 ˆ ) T α matter ) ~x ˆ MN − 97 V · i 2 U T k j α , ~ ) + i ( ˆ i x α dx ˆ − e by requiring ( δ Ud i t d ) d 2 i k N α dx π α k ~ ) β i ˆ 2 + Θ x i k ˆ Θ ) α dx t, d U in the unperturbed metric ( i i ( πG ( V ˆ U k – 13 – Θ ij δ α E e α ( i + 2 ) ˆ Θ = G i ˆ = 8 α = = ˜ + U ˆ -coordinate [ α δ ˆ + V ˆ dV V + R α ˆ V ( V ˆ V ˆ V dV ˆ V ˆ T + Θ( shock UU T dV )( dU MN 0), but only the metric in the causal future of the pulse ˆ ( ) Ud T + U, α V G − α ˆ in the U 1 2 = ˆ dU ˆ V ˆ A d U ) by ) guarantees that only the causal future of the pulse ( α ˆ + Θ U ˆ U < ˆ − T α U ˆ + Θ U + T h V = 2 h V + 2 MN + Θ U, V ( U, V → R = 2 ds ). Finally, we determine ( i ˆ V x VV U, V UV T ( T ˆ V, matter A + T ˆ U, = 2 = 2 2 ds matter T We start by replacing It turns out that the backreaction of this pulse of energy is very simple. It can be Based on this observation, we will consider the following form for the stress-energy should take the following form, respectively. The hats inevaluated these at ( expressions indicate thatequations the corresponding quantities are and in which terms the metric and the stress-energy tensor take the form while the stress-energy tensor reads For simplicity, we define the new coordinates Heaviside step function Θ( is affected by its presence. The shock wave geometry can then be written as α We will now use Einstein’s equations to determine ˜ gets modified by itssame presence. as the The unperturbed metric. metric in the causaldescribed by past, a shift on theEinstein’s other equations, as hand, we explain is below. the Given the form of the stress-energy tensor ( tensor of the shock wave, This corresponds to a pulseworld of line energy divides of definite the momentum bulkand into its two causal regions, the past causal (region future of the pulse (region JHEP11(2018)072 ) 7 L → λ 3.8 α ) and (3.23) (3.24) (3.25) (3.20) (3.21) (3.22) (3.18) (3.19) ) with as k ~ reads 3.8 t, α ( α shock T and α = 0. 2 ) . U ( δ , 2 shock UU U /β needs to satisfy the equation ), the equation for ˜ ) ∗ , πT t α ). ) and 3.1 . − t , H ( 2 ) r = 8 π ∗ ( . ) 2 t 3.14 ). Furthermore, by using ( ˜ M α 0 ii e H ij,UV (1). This case corresponds to a ho- − ) . ) t E r ∗ G − ( G + ( O H t N ) ) π ), respectively. In order to simplify the r β 3.17 BH 0 tt 2 i − 2 H , U ( t k r ( S ( e G πG ) ( ) = δ A π of metric ( π β ij,UV β 8 H 2 k ii ~ 3.16 2 − × = 0 we obtain r r log e G ) we find that ˜ G ( t, ). Using the expression for the Bekenstein- = k ( ~ 1 2 = ii π log 2 β ˜ α 2 – 14 – L = 0 in ( G π + and 3.17 ij,V β 3.21 λ 2 2 π j t = ) and ( β G . 2 k ) ∗ i in ( M = ) = E t U , we can write the leading order contribution to the ( k v ~ ∗ N 0 + 3.10 t = constant δ A k t, = G j ( ˜ − 4 α 2 k ˜ α i / . By setting ) k i M H 2 ij ij , given in ( are not affected by the non-commutative parameter, both r ∗ ( G G t 0)] ) A , BH given by ( U . With this rescaling we can recover the equations of motion ( , we will use this type of shock waves to study the disruption of S ( (0 = δ ,V 4.2 shock BH ) and matter k S ~ T β T t, ( to be diagonal, the shock wave profile is then given by → W [ ij and G −h shock T are precisely the same as for the commutative version of the SYM theory. shock ) = T ∗ k t ~ Later in section The analysis of the equations of motion simplifies when we rescale To obtain this equation we use that and t, 7 ( and two-sided mutual information. From this studythe we so-called will entanglement extract another velocity quantity of interest, and the scramblingHawking time entropy scrambling time as Note that, since where the constant ofmogeneous proportionality shock wave is geometry. ofexponent From order this profile we can readily extract the Lyapunov 3.2.2 Lyapunov exponentWe and can scrambling extract time theC chaotic properties of the boundary theory by identifying ˜ Assuming where and analyzing the terms proportional to Going back to the original coordinates notation, in the following wewe will are drop really the dealing hat with over the the coordinates symbols, but defined keeping inα in ( mind that for the unperturbed metric by setting with JHEP11(2018)072 . . ]. d H on 12 ar B (3.29) (3.30) (3.31) (3.26) (3.28) (3.27) v t >> t ) to be small and k ~ θ t, ( √ W . H . r x >> ). More specifically, it , H = k r ~ r H r = t, r φ = ( . . The dependence of r 2 2 C , ~x ), for ) . ) 4 0 11 11 H 4 H cos 4 H ) r r 11 0 11 4 ), then, this leads to H 1.4 G G r 4 4 φ r G G 4 L 11 ( a a 4 + a λ a B G + 3.22 0 tt . v 0 22 22 1 0 tt i B + G 6 + ), however, it is clear that in the limit 22 0 22 G G G -dependence disappears. 6 + 3 in the IR, and grow monotonically as /v φ G G 2 = 22 4(1 + / φ L 4(1 + 2 2 8 3 3 G iλ = k direction and = 22 -directions sin = 2 6= 2 = 3 − 11 1 + G | 1 2 x B 22 ) the G 2 k 2 11 ~ x v 2 B,x | G 2 k B,x 22 – 15 – v G v G + 2) = 0 11 11 2 1 - and = 2 k G G π/ 2) = x as a function of the non-commutative parameter from the leading pole of = 0) = 11 0 tt 1. Hence, in the limit of low momentum one can expect q for details. From ( + = π/ 2 φ = 0) = G B G ( φ = v 0 22 22 A saturates to a constant value). It is interesting to note that = ( φ 2 B 2 | B,x G G ( 1 v 2 B v k ~ φ 2 | 2 B v ( θk >> v ≡ 2 2 B,x B √ ≡ v v 1 and 2 ) has a pole precisely at ) = 1 k 2 B,x ~ φ v 2 B,x ( t, exceed the speed of light in the regime of strong non-commutativity. v 2 B,x ( 2 B v 2 α v is the angle between the 2 B,x φ v ), we get ), which also has a pole precisely at 1 and large for and k we plot ( , we expect the size of the Wilson line coupled to the operator 2.3 1 k ~ 1 2 B,x hydro -direction v θk << 1 σ For later convenience, we write the explicit formulas for the butterfly velocity along is increased (although x This seems to be consistent with the general hydrodynamic theory of quantum chaos proposed in [ √ 8 H is due to the anisotropy of the system ( Nevertheless, this result is remarkableit in represents the a context novel of violation quantumWe information of will theory, the comment since known more bounds on on this the result rate in of theIn transfer conclusions. that of paper the information. mode authors propose that the behaviour of OTOCs are controlled by a hydrodynamic chaos Both curves approach the conformalar value both As explained in theLorentz introduction, invariance is this explicitly is broken not and an the issue standard notions for of non-commutative causality theories do since not apply. and In figure Specializing these formulas to thegiven gravity by dual ( of non-commutative SYM, whose metric is the and for the butterfly velocity along In this formula φ of vanishing non-commutativity ( We refer the reader to appendix At finite for to recover an approximateMore exponential generally, behaviour we as cancan extract in be ( shown that ˜ 3.2.3 Butterfly velocity JHEP11(2018)072 . , B ∪ A,B γ A (4.1) 5.1.1 S ). The first 2.10 . The continuous H ar 5 in non-commutative on the right boundary. B v B presumably describes the B v 4 , 1 , respectively. These surfaces lie B B ∪ 2 B,x A v that stretches through the wormhole S , and so on. The above entanglement and − 3 A A B -direction, while the dashed curve represents H 1 S x ar + wormhole γ . We will confirm this intuition in section A – 16 – S C 2 -directions. The horizontal grey line corresponds to the 3 versus the dimensionless parameter ) = x 2 B v A, B ( - and I 2 2 1 x 2 B,x where the operator is inserted. In the case of non-commutative v ~x , are given by the area of the U-shaped extremal surfaces 3, while the horizontal black line corresponds to the speed of light. B / S 0 = 2 2 1 5 4 3 2 B and v 2 B v A S is the entanglement entropy of region . Butterfly velocity squared . There are two candidates for this extremal surface. The first one is the surface A , while the second one is a surface B S B γ ∪ In order to compute the two-sided mutual information we consider a strip-like region Finally, it is instructive to discuss the physical meaning of ∪ A on the left boundary of the geometry and an identical region A whose boundary coincide withoutside the the boundary event of horizon, inis the given left by and the right areaof regions, of respectively. the The extremal last surfaceγ term, whose boundary coincides with the boundary where entropies can be computedtwo holographically using terms, the HRRT prescription ( disruption of the mutualin information holographic in theories. the second case characterizes the butterflyA effect The mutual information is defined as 4 Entanglement velocity from two-sidedIn perturbations this sectiongeometry we compute and the in two-sided the mutual presence information of both a in shock the wave. unperturbed As explained in the introduction, the theories. In the commutative case,operator the around butterfly a velocity point describes thetheories, spatial the growth operator of an isgrowth of smeared this over operator a around the Wilson curve line. Here Figure 1 curve represents the butterflythe velocity butterfly velocity along along the conformal the result JHEP11(2018)072 . . right boundary 2 γ ∪ 1 ) = 0, γ → −∞ has less = 0 t A, B ) decreases ( I A, B ), which implies B wormhole ( wormhole γ B γ γ I are small enough, γ B )( → wormhole b ( → and . Eventually, the mutual 0 ) + Area( A , then we have A for a schematic illustration. A γ γ 2 . In this case the appropriate t > t wormhole Area( → ∞ γ 1 2 < are the angles of the five-sphere. The γ γ L 0, therefore, it effectively increases the size ) i θ B < etboundary left γ ). On the other hand, if 0 ∪ t B γ A = 0 slice of (a) the unperturbed two-sided black γ 2, with (in the right side of the geometry). The red curves t – 17 – 0. B L/ ), where i γ right boundary > ) ≤ connecting the two sides of the geometry. The extremal 3 , r, θ ) + Area( , 2 3 2 γ A A, B x γ , x ( 2 I ≤ and , x ) 2 1 r γ ( L/ the region dubbed as the “commutative strip” are the set of points , x ]. B − has less area than the surface ) = Area( γ ) 88 B B a = 0. In both cases the blue curves represent the U-shaped extremal surfaces → γ γ = (0 and t , then we have that Area( ∪ A Horizon Horizon ∪ ` γ m B defined in the text is given by the union of these two surfaces, A A γ γ X γ ≤ ∪ 1 A x γ . Schematic representation of the wormhole 2 1 ≤ γ γ γ Before proceeding further, let us explain the general expectations. In the unperturbed (in the left side of the geometry) and etboundary left A defined in reference [ Commutative strip: with 0 embedding is is sensitive to arbitrarily small perturbations sent in the asymptotic past. 4.1 Mutual information in theWe will TFD discuss state two cases, the “commutative strip” and the “non-commutative strip”, as the amount of mutualinformation information must drop at to a zeroAs given as explained time we in move the slice the introduction,left-right shock the pattern wave positive farther mutual of into information entanglement the characterizes(and of the past eventually vanishes) special the in a TFD shock state, wave geometry and shows that the this pattern fact of that entanglement area than a positive mutual information geometry, the mutual information mustand be become zero positive if for the regions large regions. The presence of the shock wave should decrease connecting the two boundariesIf of the the surface geometry. Seebecause figure Area( brane geometry and (b)assume the that two-sided the black shock braneof wave geometry the is in wormhole sent the at at presenceγ some of time a shockrepresent wave. extremal We surfaces surface Figure 2 JHEP11(2018)072 (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.14) (4.10) (4.13) (4.11) (4.12) wormhole γ x . By solving , . 2 → ∞ / 6 0 1 x /r 2 . ) 6 m , r r f 1 ( 6 r 0 − √ , x /r 1 , f 6 m 11 dr r at which r p 2 √ 1 G / ∞ H , 1 m f − are , r r + r . The above functional does not dr 5 Z 3 1 m √ 5 2 B r S rr 8 . = ) ∞ S H p . r r 2 dr R G r = ( ) Z f N 2 0 r r 2 ∞ 1 and 8 m x 0 L ( √ G 2 , r 0 5 / x R ) Z − x 5 θθ A + , 1 2 0 Ω r dr 8 + ; 6 G L x 6 m 11 f 0 22 r r 5 3 R 1 4 = N f ∞ m , respectively. The entanglement entropy of metric on G r 22 1 2 4 G r r ` r ) G Z L x, x × + f , 2 5 = ( 8 – 18 – 1 4 = q 3 at the point Ω L r θθ rr dr G R ab 33 1 ) = 4Ω p 2 g 3 = g G G x N = drr dr = L 0 5 dx wormhole 2 = = = G ) L 0 det Z Z γ 2 4 A ∂ i ∂x x 8 8 θ γ N p i rr 22 dx is computed from the area of the extremal surface θ R R = g σ g wormhole G g 2 2 is the volume of a unit 8 γ 4 B p Z ) = 2Ω d L L Area( 5 = 0 and ∪ 5 5 5 A Area( A γ 1 Z = S x = Area( we get B = Ω = Ω = Ω ∪ 0 B ) = A x S Area( , and Ω A S ) we can write the on-shell area of the surface as 3 γ = dx A ) for 2 4.10 S Area( dx 4.9 R , and so there is a conserved quantity associated to translations in x = 2 is then given by L B ∪ where the factor of 4disconnected comes surfaces, from at the factA that we have two sides in the geometry and two The entanglement entropy connecting the two sides of the geometry so, the entanglement entropies of the subregions Using equation ( where in the last equalitythe we computed equation ( where depend on The area functional to be extremized is given by components of the induced metric are the following: JHEP11(2018)072 , 3 (4.26) (4.21) (4.22) (4.15) (4.16) (4.18) (4.19) (4.20) (4.23) (4.24) (4.25) (4.17) x , . By solving m , r 2 , / 1 # → ∞ 2 0 f ) r ”. This means that x r √ a ( 0 , x 1 dr 22 − ∞ H r G 6 m Z at which + , /r , − 1 . In this case the appropriate 2 6 m , rr / r r 6 3 . The above functional does not m 1 5 G 5 r S p /r . . = S 2 f = 2 6 m 2 ) r → ∞ . Note, however, that both quantities ) / r r ` √ 1 2 1 r 5 θθ 0 ( ( 0 L 2 0 − x G − x r dr 1 x 2 1 0 + 6 / , 6 m + r x 1 22 22 ∞ r m ) p 3 r r hf G metric on G r 1 . We can plot the mutual information as a Z f ; 4 ) versus hf 0 2 r 1 r m ) for the commutative strip is shown in figure / 4 × + , , √ = 0 (black curve). 2, with r hf 1 11 – 19 – r 1 at the point 4 q a = 2 22 θθ rr 11 x, x r L/ dr , A, B ( p ) and the components of the induced metric are G G G G i A, B ( 3 = L dr ( dr G ∞ I ab = m 0 ≤ 0 I r = = = = g 3 x 2 L Z 0 3 drr dr i , , r, θ ∂ ∂x x " θ 1 dx 3 i Z rr 11 33 det Z Z x θ 1 8 g g g = g , x is the volume of a unit 8 8 = by writing the later quantity as a function of R p ) the “non-commutative strip” is given by the set of points dx p ≤ N 2 R R 5 r σ ` 2 2 ( 8 L G dx 2 Z 5 d L L we get , x 5 5 5 Ω 1 Z L/ 0 Z x − , x = , and Ω = Ω = Ω = Ω ) = ` 3 ) = and A = (0 ) for dx γ 1 A, B ` m ( are independent of the non-commutative parameter “ I dx 4.25 X ≤ ` R , and so there is a conserved quantity associated to translations in 2 Area( x x = 2 ≤ L ) and A, B ( where in the last equalitythe we computed equation ( where depend on The area functional to be extremized is given by embedding is Schwarzschild geometry. The plot of and corresponds to the curve labeled by Non-commutative strip: with 0 and then making a parametric plotI of the results for the commutative strip are the same as for a strip in a 5-dimensional AdS- which is a functionfunction of of the the strip’s turning width point From the above expressions we can compute the mutual information, JHEP11(2018)072 ], 88 and ) = r ) and ( (4.31) (4.27) (4.29) (4.30) (4.32) (4.28) h crit wormhole ` γ A, B , = ( I m ` r , . # 6 hf r /r √ . , 6 m ) as a function of the , r 1 1 6 dr hf r − , − /r ∞ A, B √ H 1 ( r 6 m 6 m I Z r hf r p dr 1 /r √ 6 − 1 − ∞ are H r hf r . Both quantities 6 1 r ` dr Z p √ B /r p 8 ∞ H 6 m r hf dr R we plot r Z hf N r 2 1 √ and ∞ 8 3 m L − √ G 2 r ”, because they have factors of reduces the critical length 5 R A r 1 dr Z a 2 Ω ) versus dr a 8 L p ∞ m 5 R , respectively. The entanglement entropy of r = N ∞ . We can plot the mutual information as a m ` 2 r Z ) G hf m r A, B Z L r ( – 20 – 2 5 = 8 √ I Ω = 2 2 R ) = 4Ω 2 x dr N = L dr 0 ∞ 5 m = 1). In figure wormhole G ) x r γ 4 A Z h γ N " Z is computed from the area of the extremal surface wormhole G 8 by writing the later quantity as a function of B = γ 4 ) = 2Ω R = 0 and ∪ Area( ` N 2 A A Area( 2 dx γ L G S = x 5 = Area( Z B Ω ∪ B = A S Area( ) we can write the on-shell area of the surface as ` S ) = = 4.26 A S A, B for several values of the non-commutative parameter at a fixed temperature. ( I ` = 0 (or equivalently . Also, note that we can recover the expressions for the commutative strip 1 a − ) 4 r is then given by 4 a B ∪ depend on the non-commutative parameter “ the commutative case. 4.2 Disruption ofLet mutual us information now by study how shockwave the waves two-sided geometry. mutual information In changes in the the following, presence we of a will shock specialize to the case of a homogeneous shock strip’s width As expected fromincreasing the in results the of non-commutativehence mutual parameter lowers information the threshold for for the thenon-commutativity phase introduces transition one-sided of more mutual black correlations information. between brane This two implies [ sub-systems that as compared to and then making` a parametric plot(1 of + by setting which is a functionfunction of of the the strip’s turning width point From the above expressions we can compute the mutual information, where the factor of 4disconnected comes surfaces, from at the factA that we have two sides in the geometry and two The entanglement entropy connecting the two sides of the geometry so, the entanglement entropies of the subregions Using equation ( JHEP11(2018)072 , - α α for ` (4.33) (4.34) (4.35) = 0 (black has smaller a . The mutual . /β ) 0 ) has various α ( πt α 2 ( B e wormhole B ∪ γ ∪ reg A × A S S 0. That means that the − , > 1.5 , ) = 0) so that the mutual information α = 0) ( 2 (red curve) . In all cases we have α . = 0) = constant ; B α crit ( ∪ α α = 1 A B ; ∪ S a ) as a function of the strip’s width A, B A ( ` > ` − I S N (5) A, B ` B ( − /G I S 1.0 ) 3 ) = α R + α ( ( 2 A B L – 21 – B S ∪ ∪ A A S S ) = generically depends on the shock wave parameter − α B ; ) = B ∪ α 2 (purple curve) and S A ( . 0.5 S B A, B + ∪ ( = 1 A reg I A a S S ) = α ; remain outside the horizon, while the shock wave only affects quantities 0.1 0.0 0.4 0.3 0.2 0.5 do not. This can be easily understood, since the corresponding extremal B ) B A, ( I γ A, B B ( S I 8 (blue curve), . and A and . Mutual Information (in units of = 0 γ = 1. A a H S r In the following, we will consider cases where As expected on general grounds, the entanglement entropy is positive in theextremal unperturbed surface geometry, stretching i.e., between the two sides of the geometry independent divergences. In practiceglement entropy we find it convenient to define a regularized entan- and rewrite the mutual information as where we have indicatedwhile that surfaces that probe the black hole interior. fixed wave, in which the shock waveinformation parameter in has the a form shock wave geometry will be denoted as Figure 3 non-commutative SYM theory. Thecurve), curves correspond from the right to the left to JHEP11(2018)072 ) = 4.42 (4.46) (4.44) (4.40) (4.41) (4.42) (4.43) (4.45) (4.36) (4.37) (4.38) (4.39) m . When increases X B α γ , , we can expect and a 2 / 1 A γ 2 , ˙ r = 0. By solving ( , f , rr . r ) 2 0 2 6 G r . / r 1 ( + + fr f f 2 2 2 4 tt 4 − r r − − 2 ˙ G E , r r . fr + p 2 5 at which ˙ 3 0 2 2 dr 4 . E S r / 0 r + 6 − ) 1 + 5 θθ r t r p − f ; 2 G fr p ˙ r 2 − E = dr f , 22 − 2 r, 2 4 p ( , r ˙ 2 E G Z 3 r ˙ r 3 fr L 8 22 rr dr Z R metric on dx -translations, there is an associated conserved f dt r dt G + G 1 + 2 t 2 3 Z = × f r L + = 2 Z Z 8 – 22 – 5 at the point ˙ − r 8 8 θθ tt R dr f 33 E dtdx N 2 R R 2 q g G G 2 2 L r G Z Z 5 L L 4 = = = = 5 5 5 ) = 2Ω = i = 2Ω θ tt i 22 2 θ dt g ˙ g − L r = g ˙ = 2Ω = 2Ω r Z B ˙ r ) = 2Ω L ∪ wormhole ∂ ∂ the appropriate embedding in this case is A along the extremal surface as γ ) = S t r = ( t E wormhole Area( γ ). The components of the induced metric are i , θ is given by ) Area( t ( B ∪ , r A 3 S , x 2 we obtain 0, the wormhole becomes longer and the area of the extremal surface probing the , x r 0 t, the disruption of mutual informationHowever, to we be will the proceed sameentropy as with for the the analysis commutative SYM for theory. illustrative purposes. The entanglement and the time coordinate Since these expressions do not depend on the non-comutative parameter Using the above result we can write the on-shell area as where in the last equalityfor we computed ˙ Since the above functional is invariant under quantity, and the functional to be extremized is ( α > interior also increases, resultingthe in mutual a information decrease of eventuallysided correlations. the drops mutual to information. zero, As signalingCommutative the total disruption strip: of two- area than the two extremal surfaces lying outside the black brane JHEP11(2018)072 . 0 . r # f (4.51) (4.52) (4.47) (4.49) (4.50) 2 r + 2 4 , r − # r 2 f 2 E r can be written as p + B can then be written 2 ∪ 4 dr have the same area and r reg A − B H have the same area, we . , r r ∪ S 0 2 r A ! III ! Z E (4.48) S . The shock wave is absent 6 6 ) III H p H r fr r fr + 2 and , ( 2 2 dr ) ≤ − B and 0 − ! H r II ∪ E 0 1 r E ( 1 0 f r 3 r A 2 II 2 surface (blue, dashed curve, defined by Z S K r r r 1 + 1 + )+ − p 0 p + 2 ) r p ( 0 − 2 r f − − ( 2 K f 1 r 2 1 B )+ r ∪ 0 , + 2 α r A ( + 2 f 4 f f S – 23 – 1 . The segments 1 2 r 2 1 2 4 1 2 − III K r r r r r − 0 e r 2 r III dr 2 dr E dr II can also be written as a function of the turning point E 0 H . Since the regions r ∞ r p H 0 α and = 0) = ¯ r r r p ) = 2 . The entanglement entropy = 0), and its effects become stronger as one decreases 4 Z Z Z 0 dr α H at which the constant- 0 r II r ( E r π π π ( 0 I ∞ β β β , H R B 4 2 4 r α r dr I ∪ Z A = = = ∞ H " +2 S r 1 2 3 8 H Z ∞ r − R K K K " R N 2 ) 8 is controlled by the turning point α L G = R ( 5 B N 2 for details). The final result reads B Ω ∪ L ∪ G B III A 5 A ∪ S Ω S ) = (or, equivalently, , as shown in figure II 0 ∪ H r = I ( r R III ) = B 0 . Extremal surface (horizontal, red) in the shock wave geometry. We divide the left half = ∪ r ( A 0 r and B ) intersects the extremal surface. S 0 ∪ r reg A (see appendix II S = , 0 where Finally, the shock wave parameter r where the extra factorshock of wave 2 on accounts forwhen the two sides ofIn terms the of geometry. this The parameter, effect the regularized of entanglement the entropy can write more explicitly as they are separated by ther point It is convenient to divideI the region of integration of the above integral into three regions, Figure 4 of the surface into three parts, JHEP11(2018)072 , ) = 6 for ), as 4.60 0 (4.58) (4.59) (4.60) (4.61) (4.63) (4.64) (4.62) (4.53) (4.54) (4.55) (4.56) (4.57) C m r and ( X 5 α , 2 / 1 2 = 0 in the results ˙ r , ), even in the limit a (see appendix rr α f ; 2 G 4 = 0. By solving ( , r / H . . 1 r r ) + ) , 3 + 0 0 f h f A, B tt r 2 r 2 6 ( h ( / ( = r r 4 G I 1 h f 2 c 2 − . This means that the region + r − r 2 r c − 4 / 2 r . h E 2 , 5 θθ ˙ 4 r E s 5 at which ˙ fr 3 2 0 dr G − = = 0 (black curves). S 0 r r r p 2 h f 1 + 0 r + 2 2 / a − r 6 / 1 22 E . 1 f q r G 2 ) = h f , − t p 2 2 − 2 2 ; 4 / r ˙ ˙ / 2 ) increases monotonically as one decreases E dr r r 1 11 ˙ 2 r 1 0 2 fr / rr r r, h / G 3 1 Z Z ( ( 1 metric on r -translations, there is an associated conserved 3 G − + 1 + 8 t h α L dr = × , , f R diverge at + dx fh – 24 – at the point = 2 2 dt dt ˙ Z 1 3 − r 0 11 22 θθ tt dr L r t E α 8 5 f G G G G Z Z q 2 R 8 8 Z ∝ r N 2 = = = = dt dx = R R L G ) = i 2 2 5 0 θ 4 ) for the commutative strip are shown in figures Z = is then given by tt i L L 11 33 r θ ) = 2Ω dt g 5 5 5 ( g g 2 − L 2Ω g the appropriate embedding in this case is t B ˙ r ˙ ∪ r Z = ˙ A r L S ∂ and = 2Ω = 2Ω ∂ A, B, α B along the extremal surface as ( ) = ∪ ) = 2Ω t 3 wormhole I r A = ( γ K t S E ) and ). The components of the induced metric are Area( wormhole i α ( γ , θ B ) ) = 0) and it diverges at some critical radius t ∪ ( H reg A r S ( , r Area( 3 α . The results for the commutative strip can be found by setting , x and, in particular, the singularity cannot be probed by 0 we obtain c , 1 r (with → ∞ 0 t, x 0 The entanglement entropy Using the above result we can write the on-shell area as and the time coordinate where in the last equalityfor we computed ˙ Since the above functional is invariant under quantity, and the functional to be extremized is well as for respectively, and correspond to the curves labeledNon-commutative by strip: ( r details). Indeed, both r < r t for the non-commutative strip (which will be presented below). The plots for As expected, the shock wave parameter JHEP11(2018)072 . ) 0 0 r α II r ( . , B I (with # (4.68) (4.69) (4.70) (4.66) (4.67) (4.71) (4.65) ∪ reg 0 f A r 2 S . r # + f 2 4 r − 2 ! + r hr f can be written as 2 4 2 . , E − r B ) 2 4 2 ∪ . p , r / hr H p r reg A 2 1 2 r 2 / ( / S ! E ! 1 ! 1 we plot the shock wave 6 B 6 can then be written more h 8 H h r ∪ p r 5 r . The shock wave is absent 1 2 1 A have the same area, we can 8 B dr H / − − − a S ∪ 1 r H , A h r f 0 ) fh − fh ≤ 2 r S 0 III 2 + 9 2 r ) r Z dr 0 1 ( 1 − 4 H 0 − 4 H r 3 H r r + E r E r ( 0 4 K 4 r 4 and + 2 a B a . In figure Z )+ − ∪ c 0 f 1 + 2 1 + r r A 2 II r ( hr r +2 S 2 p 2 p = 9 + 2 K E + − − 0 )+ 4 p p r 1 0 1 2 − r for several values of the non-commutative pa- / ( 2 , 1 – 25 – 1 0 r hr = 0) = gets repealed from the singularity as we increase h f r K f 2 f 1 2 1 e 2 1 2 α 0 E + 3 + r r ( ) increases monotonically as one decreases r r can be written as a function of the turning point 0 B p 4 H r dr dr dr 2 dr ∪ r α ( 3 / 4 H 0 A ) = 2 1 α ∞ diverge at H ∞ r r = 0), and its effects become stronger as one decreases a H 0 0 S r h ¯ r r r r 0 Z E − = Z Z Z ( t − . Since the regions dr " α π π π ) . The entanglement entropy α β β β 8 4 2 4 4 4 ∝ ∞ α H H 0 / r R ( r r H 1 N 2 ) Z = = = R B r 0 L " ∪ G 10 r 1 2 3 5 A ( 8 +2 t Ω K K K is controlled by the turning point S = R H N 2 ∞ c r B = r L R G ∪ 5 and ) = A we show the behavior of the regularized entanglement entropy for details). The final result reads 0 = Ω 3 S r 6 ( (or, equivalently, B versus the ‘turning point’ K B III H ) = ∪ α r ∪ 0 reg A r II = S ( , as shown in figure ∪ I 0 B R r ∪ ) = 0) and it diverges at some critical radius III A H S r ( The shock wave parameter parameter rameter. In general, wethe observe strength of that the non-commutativity, meaninginterior. that the This extremal might surface be probeetry. a less In of consequence the figure of the fuzzy nature of the non-commutative geom- α Indeed both where Finally, the shock wave parameter (see appendix where the extra factorshock of wave 2 on accounts forwhen the two sides ofIn terms the of geometry. this The parameter, effect the regularized of entanglement the entropy and write explicitly as Again, we divide the region of integration of the above integral into three regions, JHEP11(2018)072 = α ) as a 6 8 (blue . A, B , and this = 1. = 0 ( = 1 (purple α I H a a r . In the next 4 a for non-commutative )( α b ( H 2 r 1.00 log 8 (blue curve), . 0 = 0 (black curves), = 0 divided by a 0.95 0 a r = 1. -2 ) grows linearly with log H and (b) mutual information α r ( 1.0 0.8 0.6 0.4 0.2 0.0 B H B 0) ; B A, ( /I ) α ; B A, ( 0.90 I ∪ ∪ at which the system was perturbed ( reg A /r A S 0 0 S t r 6 – 26 – 0.85 = 0 (black curve), 5 (red curves). In all cases we have fixed . a = 1 4 a versus the ‘turning point’ α 0.80 ) α a 2 ). As we can see from these plots, the disruption of the mutual α log ; 0.75 0 A, B 50 0 ( 100 250 200 150 I ) . Both in (a) and (b) the curves correspond to 0 α 2 (purple curves) and r . ( = 2 (red curve). In all cases we have fixed α -2 a = 1 0.2 0.0 0.6 0.4 1.0 0.8 . (a) Regularized entanglement entropy . Shock wave parameter a B ∪ A ,B A, ( /I S 0) ; reg entanglement velocity. 4.2.1 Entanglement velocity Before saturation, the entanglementimplies entropy that it grows linearly with the time in a shocktual wave information geometry and howinformation this occurs results faster as insection we the we increase disruption will the of quantify non-commutative the this parameter two-sided statement mu- more clearly by the calculation of the so-called Figure 6 function of log curves), Figure 5 SYM theory. The curvescurve) and correspond to JHEP11(2018)072 , 7 (4.77) (4.72) (4.73) (4.74) (4.75) (4.76) (4.78) , ). Finally, com- ) ). In the vicinity exceeds the speed , the above result B 12 H . 2 r /β c with the shock wave ∪ 0 r 4.71 12 entanglement velocity E,x B πt A a v 2 ( ( ∪ ≈ e reg ∂ A 0 O r S × + , 4 / for ! 3 , , given by ( 8 H ) ) increases monotonically as we ) 4 r c c ) 0 2 23 c / c 8 r r = 0 in all the formulas below. r 3 r ) α , . ( r ( a √ √ ( c 3 ( a ) 5 f r α = f c h ) ( log = constant c − r 0 − 108 h ( r , r ( π α f s β p p 3 4 H + h . We observe that 3 c 3 c 3 H N (5) − r H r ) 4 ) r r 3 = 0) = c c / G 3 3 s r r ar 4 R H 4 a ( ( r 3 c ( 3 R H (5) N Σ 23 4 h f 2 r 2 r = – 27 – G a A L √ − . The rate of change of = th E,x 3 th 0 s v s t = s 2 3 c + r = = B 4 E,x 8 2 ∪ 0 1 / v B 3 reg A R √ = 0 in this formula we obtain the standard entanglement dt 3 ∪ 0 2 N E,x reg A ) we can then extract the entanglement velocity for the a dS L v dt G = 5 as a function of dS we obtain, ) 2 c ) 1.11 2Ω c r H ( r E,x ( ∼ = f ar increases as we increase the non-comutative parameter! In figure v h − B 2 ∪ is the five-dimensional Newton constant. Using the formula for the reg A s E,x and grows linearly with 5 3 S c 3 H is the area of the 4 hyperplanes defining Σ = v grows exponentially with time, and diverges at a critical radius N 1 R r r B 5 2 G H ∪ α Ω L r = E,x A v 2 S ). From this linear behaviour we can define the so-called = ≤ 0 = 4 t 0 E,x π r β v 2 (5) N Σ e G A × , one can show that As shown in the previous section, the function c r which shows that we plot both which also applies forexpanding the in commutative powers strip of in non-commutative SYM. More generally, One can check that byvelocity setting for a strip in ordinary SYM theory, where paring with thenon-commutative formula strip, ( which we denote as thermal entropy density, we can rewrite the above equation as implies that time is where of Since the shift which is a quantity that characterizesfollowing, the we spread will of specialize entanglement incommutative to chaotic strip the system. can case In be of the obtained the simply non-commutative by strip. setting Thedecrease results for the const JHEP11(2018)072 ] 14 in a V . The blue curve H generically, for any ar i 1.5 B,x v 1 ≤ E,x i v that does not rely on the shock 2 E,x v (1) and grows without bound in B E,x v v is based on entanglement wedge sub- ∼ O 1.0 ]. Here we extend their results for the B H H v 14 ar ar – 28 – of order versus the dimensionless parameter H 0.5 E ar v ]. Finally, we note that 96 0.0 0.6 0.5 1.1 1.0 0.9 0.8 0.7 1.2 ], and it was first proposed in [ 99 E . The horizontal grey line represents the speed of light. v 4 / 3 3 . Entanglement velocity / 2 √ = E 5.1 Infalling particle andThe derivation entanglement goes wedge as follows.black Consider brane the geometry. application of This a operatoreventually localized falls creates bulk into a the operator black one hole particle and state thermalizes. in As the the bulk particle falls theory into which the black hole, In this section wewave present results. an alternative This derivation alternative of region way of duality [ computing kind of anisotropic metrics that we consider in this paper. value of the non-commutativeholds parameter. for our This non-commutative implies(possibly setup, that non-local) and the quantum suggests system. conjecture that made it in might [ indeed be5 true for any Butterfly velocity from one-sided perturbations of light already atthe some limit of value strong of non-commutativity.results This obtained behavior for is the infollowing qualitative entanglement a agreement entropy quantum with for quench the a [ free scalar field on the fuzzy sphere Figure 7 represent the entanglement velocityrepresent for the entanglement a velocity non-commutative forv a strip, commutative while strip, the which is horizontal equal black to line the conformal result JHEP11(2018)072 ] is , . (5.6) (5.2) (5.3) (5.4) (5.5) (5.1) 14 # V 2 . This ) ρ A i ∂ . . j )( ) H H r dx r i ( and the horizon , in which terms ii − ) dx G H r ∞ r # ( 1 2 )( 0 tt ρ H = ) + G r 2 of the boundary theory H ( r 2 r 0 ij ( A π 0 ii β π G , β 2 2 G j ) )), the area functional can be H ) + ) i ) dx ) r i H H x H H ( ( r r r r ii ( ( ( dx also lies very close to the horizon, , ρ − ij 0 tt G 0 ij ij is the holographic radial coordinate i ) V r G G G G . r H , x contains the particle created by t r . 2 = + ( π β = ( ρ 1 0 2 A 0 tt 2 ) + c c ij 2 − H = (0 G ρ e dr r r ( 0 2 rr m π ρ ij ,G G X G – 29 – π H β r 2 " = 4 + ) = 1 ], which implies that the information of the particle t c 2 β − ( + ρ r 2 dt 102 tt – 1 + dρ = G " + − x rr 100 1 2 = − dt d 2 2 ,G d ρ ) ds 2 . In the following, we compute the butterfly velocity by requiring H Z r ) A H π 1. We assume the boundary is located at − β r , the inverse Hawking temperature reads 2 ( 1 r − ( c ij 0 c G , such that this operator is delocalized in a very large region − V = and = det 0 tt 2 , . . . , d c ) are the boundary theory coordinates, . Considering the embedding q G i ds . We now consider a fixed time slice of the geometry, at a long time after the A H = 1 t, x r = i, j We assume the following near-horizon expressions Let us assume a generic black hole metric of the form In this context, the butterfly velocity can be calculated using the entanglement wedge Area = r gets scrambled with an increasing number of degrees of freedom and, as a result, the wedge of written as The infalling particle getsapproaches blue the shifted horizon as exponentially it falls into the black hole and, atNow late times, we it proceed to calculate the position of the RT surface defining the entanglement It is convenient tothe go above to metric Rindler becomes coordinates, In terms of limit simplifies the analysisentanglement for wedge linearise, two because reasons. theblack corresponding First, hole RT horizon. the surface equations lies Second,having of very a the close motion simple particle to description defining created the in the by terms of Rindler coordinates. where ( and at application of subregion duality. According to thiscan duality be a certain completely subregion entanglement described wedge of by athat subregion the entanglement in wedge the of a bulk certain geometry, region which is called the operator effectively grows in space.holographic This UV/IR connection is [ consistent withgets the delocalized standard over intuition a from larger the that, region at as late it times, falls the deeper rate into of the growth bulk. of The this proposal region of is [ controlled by the butterfly velocity. V JHEP11(2018)072 , a β I (5.7) (5.8) (5.9) (5.11) (5.12) (5.13) (5.14) (5.15) (5.10) exceeds ), in which H ρ r ( direction. This ii . Requiring the . i i G 2 . x R / ) p ) H ], when / H 3) i r r ( x ( 14 − ii t , 0 kk d = π G β i G 2 by solving the equation ) σ p = ( along the ) H ≥ r H a = ( . i r min A , ( ρ ) , R kk ) 0 tt σi ) H ) i σ G r H G R a ( x r ) | ( ( 0 ii q H σ p ρ ) ii ρ . r . G MR 2 R 2 H ( with ) ( | G σ r 0 tt a H ( t , M M I r i p G ii ( MR = in terms of ii G -direction is calculated as -coordinates. The approximate solution for a − M B,x i ) ) = e σ ρ µ | G v i x p + 1) A 2 2 σ x 2 | a ≈ ( − a ≥ – 30 – | = 2 ] ρ i i σ M 2 i Γ( | π ( β R ∂ 14 ∂ min 2 ⇒ a M in the ∂σ ) ρ I ) H min H r ρ A a r ( is the size of the region along the = ( 1 ii = i i ii 2 M + 1) ) G 2 t R to be contained in the entanglement wedge implies β G a ( is different along the different directions and so is the size of − M B,x ρ 2 v p V to be diagonal. The equations of motion that follows from the Γ( B v ≤ π ij β is 2 -coordinates by the equation min G σ σ ρ min = R ρ i is interpreted as the radius of closest approach to the horizon and ) = i B,x σ ). min v ( ρ ρ 3.29 . Let us say that A )–( is the size of the region σ R 3.28 This formula is intions complete ( agreement with the ones obtained from shock wave calcula- or, equivalently where the butterfly velocity For an anisotropic system, the region is related to itsinfalling size particle in created by where this equation at large 5.1.1 Butterfly velocity is a modified Besselthe function surface of exits the the second nearpossible kind. horizon to region determine As and the explained reaches size the in of boundary [ the very quickly. region It is then The solution of the above equation is [ In this formula, In order to solve thisterms equation the equation we define of motion the becomes new coordinates above functional are where where we have assumed JHEP11(2018)072 1 ∼ ] and B,x 3 . In v ~x , x ). The 2 x → ∞ 2.6 -direction. 1 x W are shown in . Let us now to depend on ` directly at the θk B θT v V ' √ 4 ]. However, in non- plane, i.e. [ / W 1 ` 3 = 4 non-commutative x 103 πλ -direction, and the latter N − 1 , according to ( = is the angle between 2 x x H W φ - directions. We observe that, 3 a r . This causes x 33 , roughly as G k , where t = - and ) 2 φ x 22 ( G B to be very large, such that v 6= 1 k 11 as a function of – 31 – or G are exactly the same as the corresponding values B ) over a Wilson line θ v x ∗ ( t O and the corresponding one particle state, in our non- -direction is and V ~x , there is a natural set of gauge invariant operators that L λ 2.2 scales with the momentum . Since neither the temperature nor the entropy are affected by S W , while the scrambling time scales logarithmically with the entropy ` log goes as before, except that now the entanglement surface will not π π/β β B 2 v = = 2 ∗ L t . In particular, the butterfly velocity will be the same as before and will λ 1 x , where the first one is the butterfly velocity along the 2 . Since the non-commutativity is introduced along the -direction. In more general cases, when the Wilson line is not very large, we expect 1 B,x 1 ) to describe the expansion of the operator in a region around the Wilson line that v In contrast, the butterfly velocity is largely affected by the non-commutative parameter As explained in section Before closing this section we would like to offer some intuition about the bound- x φ ( -axis. As the system evolves in time, the information will get delocalized in a cylindrical , the gravity dual is hence anisotropic, 1 , specially in the UV. The results for B figure iθ the direction of the perturbation.and For simplicity, we onlyone computed corresponds the to the components butterfly velocity along the chaos bound, of the system, the non-commutativity, both in ordinary SYM theory. θ In this paper we haveSYM studied theory. shock From the waves shock in wavethis the profiles, system, we gravity extracted dual namely, several the to chaos-related butterflynent. properties of velocity, As the expected scrambling on time, general and grounds, the we Lyapunov find expo- that the Lyapunov exponent saturates the the v defines it. 6 Conclusions and outlook contained inside the entanglementThe wedge, which derivation will of alsodepend display a on cylindrical symmetry. describe how fast the‘cylinder’ whose information radius about along the the smeared operator gets delocalized inside a imagine having a veryThis large can Wilson line be along achievedthis one by approximation of taking the the either information directions,x about say the the perturbation isregion initially around localized along this the axis. In the bulk, the information about this perturbation will be commutative gauge theories there are noto local explain operators how in the position above space, prescription so it works is in necessary thecan present be case. defined in non-commutativeordinary gauge theories, gauge which covariant can operators besize obtained of by this smearing Wilson the line ary picture of thecommutative bulk setup. operator Without loss ofboundary, generality, and we follow can the imagine evolutionIn inserting of ordinary the AdS/CFT created this particlespace, would which as can mean it in falls that turn into we be the are interpreted black turning as brane. a on local an quench, operator see localized e.g. [ in JHEP11(2018)072 - 9 2 x due B v increases ], namely 2 14 B,x v directly in a non- B v ]. 96 by studying the disruption ]. . Right before the transition, E 104 ] or in the gravity dual of the v [ 45 grows linearly, with a slope given θk → ∞ B ∼ 0 ∪ t ⊥ A ` S ) for various values of the non-commutative A, B ( saturates a constant value, while – 32 – I 1 -direction), the results are the same as for an AdS B,x and 1 v x become larger than the speed of light. For large values B ∪ = 4 super Yang Mills [ 2 A ]. It will also interesting to compute S B,x N 46 v so the standard notions of causality do not apply. Nevertheless, θ and 1 B,x in general quantum systems. Indeed, we find that this is valid in our v we show the behavior of the entanglement velocity as a function of the i exceeds the speed of light in the limit of strong non-locality. This behavior 7 2 B,x v E,x v ≤ 1, both i ∼ we show the results for E,x H . In figure 6 v -direction) the entanglement velocity increases with the non-commutative parameter. E We also confirmed the expectation based on the conjecture proposed in [ Finally, we also computed the entanglement velocity The fact that the butterfly velocity exceeds the speed of light in the regime of strong 3 a r Recall that in non-commutative theories, the information of any degree of freedom moving with a large v x 9 momentum is highly delocalized in the transverse directions Acknowledgments It is a pleasure toraad thank Schalm Jan for de useful Boer, discussions Jose and Edelstein, comments on Philippe the Sabella-Garnier manuscript. and Koen- The research of WF test further this conjectureof in the other dipole non-local deformationso-called theories, of little string for theory example,commutative [ field in theory the (using perturbation gravity theory)results and dual compare obtained with in the strong this coupling paper. free scalar field on the fuzzy sphere following athat quantum quench [ setup for any value of theindeed non-commutative parameter be suggesting that true the conjecture for might any (possibly non-local) quantum system. It would be interesting to (strip with finite widthblack along brane, the while foror a “non-commutative” stripEventually, (strip with finiteis width in along qualitative the agreement with the results obtained for the entanglement entropy for a parameter. In general, wethe find shock that wave, the and mutualthe eventually information entanglement vanishes entropy is as of reduced one theby in let two the sub-systems presence of non-commutative parameter. We considered two geometries. For a “commutative strip” cost” or ainteresting decrease to on understand the this phenomenon amount better. of “usefulof information” the at two-sided fixedfigure mutual time. information It in would the be presence of homogeneous shock waves. In this result is remarkable ina the novel violation context of of the quantum knownthough, information bounds that theory, on in since the this it rate limitso of represents the transfer the information of is information. implementation highly We ofexponentially delocalized comment, longer due a time to local than theto UV/IR protocol the the mixing, commutative to non-commutativity case. is retrieve necessarily As compensated the a by information result, an increase an might in increase require the on “computational an of the non-commutative parameter, indefinitely. non-locality is not surprising.commutative Indeed, parameter Lorentz invariance is explicitly broken by the non- for JHEP11(2018)072 , the 0 (or (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) (A.4) , ]). In 22 85 2 G → φ = 2 k , ) can indeed 33 cos 2 k ~ G φ φ 0 and the shock t, ( 2 2 α → cos sin at finite momentum, W φ 33 ` α 2 G plane, i.e. sin + , 3 2 33 x φ φ G 2 2 − , , + . This is a consequence of the 2 2 2 sin sin φ 2 x φ φ . φ φ 22 2 2 2 sin cos G /β M ) φ , φ φ sin sin ∗ . Interestingly, at the pole, the modulus + 1 t + φ − 22 1 φ t 2 2 cos sin sin i ( iM 2 G k π k k k = 0. By writing the momentum components 2 . ii ) ) ) sin = + e 2 H H H cos G 22 r r r k 22 φ ( ( ( – 33 – 11 M G 2 G 11 22 33 G + ) = still depends on 6= + G G G cos 2 k i ~ p φ k 11 B t, p p p 11 µ v ii 2 ( G ) gives the Lyapunov exponent and the butterfly velocity G ˜ α G = = = k ~ cos 2 1 2 3 ) = t, ( k k k µ φ 11 ( C 2 L G B λ v p . If we assume isotropy in the − 2 µ φ = , 2 3 ). Note that ) -direction, i.e. k ) = 2 1 2 and is the Lyapunov exponent and x + , the butterfly velocity is completely anisotropic and depends on the two 3.27 2 L φ 2 2 φ, φ λ 33 φ, φ k ( ( G π/β ). We start from the solution of the shock wave profile ˜ 2 B B + v v 2 1 6= k − = 2 ), in which the size of the Wilson line is vanishingly small 3.27 θ 22 L = = G √ λ 2 gives us the ratio of the Lyapunov exponent and the butterfly velocity 6= k The fact that the pole of >> k ~ | 11 ~x is implicit in otherour holographic setup, calculations we (see can forbe confirm instance identified that the with appendix the| the C quantity of appearing butterfly [ in velocity. the We pole do of so ˜ by looking at the limit above formula simplifies to which leads to ( anisotropy in the is the butterfly velocity alongG an arbitrary direction.spherical Note angles that, in the most general case, where the position of the poleof can be specified as The above function has ain pole spherical at coordinates In this appendix we studyformula ( shock wave geometries and present a detailed derivation of the PHY-1620610. VJ is supported(CONACyT) under by grant Mexico’s CB-2014/238734. National JFPzation Council is for of supported Scientific Science by Research and the (NWO) Technology Netherlands under Organi- the VENI grantA 680-47-456/1486. Momentum space correlator and the butterfly velocity is based upon work supported by the National Science Foundation under Grant Number JHEP11(2018)072 B ∪ A (A.8) (A.9) S (A.12) (A.13) (A.14) (A.15) (A.10) (A.11) , | ~σ | M 10 − e | ) it is indeed the ) k ~σ ∗ ~ defines a constant- | t t, 0 − ( t r ( α π β , 2 . e ) 2 2 µ 33 | 2 G ~x . φ, φ | ( 22 ~x π , 2 · β π B 0 2 G k ~ 4 v i r ) M e 11 2 π ) . . , β ∗ | 2 G + | t 2 ≡ 2 ~x φ,φ ~σ | ( | 2 i . The parameter | φ − √ φ ) t ≡ ~σ B k M ( | 4 H v − ii π ii r = β e 2 − sin cos . In this appendix determine the relation ( ) G ∗ 0 G e φ , ~σ t 0 ii φ φ H = · r 2 2 3 r − ) G i~q 2 t ) i ( k ~ , ) e 3 cos sin sin π x M M 0 tt ) π H α π ( | | | d 4 β ∗ + – 34 – r 2 t 2 G (2 2 ~x ~x ~x + ( ) appearing at the pole of ˜ q e | | | − µ 2 ii 2 t ~q · | ( Z G = = = , the above integral can be written as i π i~σ ~q β i 2 | ) = e as a function of X 2 φ, φ q 1 2 3 e 3 ( ) q ii x x x ) = s 3 3 π B α t, ~x d ) π G ~q (2 v ( π β 3 β 2 π t, ~x 2 α √ d R ( (2 α = → = ) we obtain | Z i ~σ 2 k | 33 A.9 M ) are defined such that M G 2 22 or by the ‘turning point’ G ) in ( φ, φ α 11 G A.11 p = 0 in all the formulas below. ) = a t, ~σ ( By symmetry considerations we know that the extremal surface whose area gives α In the last equality we use that 10 surface inside the black horizon which intersect the extremal surface exactly at the point setting divides the bulk into two halves,r as shown in figure In the case of homogeneousby shocks, the the parameter strength ofbetween the these shock wave two can parameters. becommutative either In strip. measured the The following case we of will a specialize commutative to strip the can be case recovered of from the our non- results by which is the wellconfirms known that shock that wave the profilebutterfly quantity for velocity. the case of localized perturbations.B This Shock wave parameter Substituting ( where the angles ( With the above definitions we can write where all the metricpression, functions we are write evaluated at the horizon. To further simplify this ex- position space as the Fourier transform of ˜ By changing variables as wave becomes approximately local. In this limit we can write the shock wave profile in JHEP11(2018)072 . ) ) 2 r 0). ( , t III ,V 1 (B.9) (B.4) (B.5) (B.6) (B.7) (B.8) (B.1) (B.2) (B.3) 2 U U and as , . ) = ( 2). In what ) = ( V II . The third ! ! 0 , r 1 and obtain an . I , α/ U, V U, V 0 and 2 2 1 2 , − + 1 = r -coordinate along . ) U U . 0 f f U , r V 1 1 r ) ( = − − ) = (0 0 1 from the boundary to !# f r 0 h h !# 2 ( 1 r 6 6 0 1 1 U f r r !# r − 0 can be expressed as 2 2 2 U, V 1 = − 0 1 1 − − r r f α dr 0 − f E E 1 1 in terms of . − ∞ surface at , . f dr to − r 1 ) h 2 r h f 0 r 1 + 1 + 6 Z H ¯ − r 0) to the point ( V 6 1 r r r , ( 1) to the horizon ( f Z r h − 2 3 − p p 1 1 2 2 6 1 − h − r = 1 r K U − = , 6 = 2 E and r r E ∗ 1 )+ − ∗ 2 f f dr = 0. On the other hand, the time 0 ). 2 1 1 2 2 r E − 0 ( ∗ r r r ) = ( 1 + 2 E , r 1 + r ¯ ,U r dr , r ) = (1 ) K 1 from Z t ) 4.70 p dr dr 1 + p t U + )+ π + 1 + U, V U ∗ 0 β ) to the horizon at ( p ∗ )–( Z Z 4 r r 2 U, V r ( ( f p ( f – 35 – 1 π π π 1 2 β 1 β β π 2 f 2 ,V β 4 4 r K 2 2 r f 2 4.68 e e 1 e 2 r exp U r du = − dr 2 ) = ) = Z 1 H t t U r = ∞ du 0 H ) = 2 r r ∆ ) = ( 0 0 r Z Z = H ,V ) = r r − + ∆ ) ( r 2 π π t Z ( β β ∗ ∗ ,V V α 4 4 t U, V − r r ) π ∗ t " " β r 4 ( − (∆ (∆ ∗ " π β r 2 ( π π β β e π 4 4 β 2 = exp = exp = e = = 2 2 1 1 can be calculated considering the variation of 2 1 = exp can be written as ). = U 2 2 U 0 U U 1 2 2 r 2 2 U V ( U we consider the variation of U V V α α 4 2 ’s are given by equations ( III U i can then be computed by considering the variation in the = 0. We split the left part of the surface into three segments ∆ log ∆ log K α r is a point behind the horizon at which r In the left exterior region, the Kruskal coordinates are defined as where the Finally, after some simplifications we find that the parameter The coordinate The shift the segment To compute The coordinate the horizon Using the above equations we can express variation in the coordinates defined as where ¯ along the extremal surface can be written as expression for while, inside the black hole and in the right side of the geometry, these coordinates are The first segment goes fromThe the second boundary ( segment goes fromwhere the the horizon extremal ( surfacesegment connects intersects the with point the ( follows constant- we compute the unknown quantities at which ˙ JHEP11(2018)072 ] (C.3) (C.4) (C.1) (C.2) Int. J. , 2 ) 0 (1998) 253 . r ) diverges. Ac- 0 2 − r ]. ( ) r 3 0 ( hep-th/9711200 r K O . − + SPIRE . ! r ( ) 2 IN 6 0 ) 0 r r 0 r ][ 1 r = − r − − at which 0 1 (1998) 231] [ r fh ( r c 6 2 6 2 0 ( r r r 1 r − 1 1 O = . − E ) and approaches the standard value − r = such that 0 fh fh 6 0 0 ) + c r r = 0 1 + 6 Gauge theory correlators from noncritical 4.71 0 Adv. Theor. Math. Phys. r 1 r r c − , − 1 r p ) hep-th/9802109 → − − = s 0 r [ − r 0 r ( ( 0 r fh − 1 0 h 6 r 1 ) 6 – 36 – r 0 r = 6 0 + 1 r r 1 f r ) one finds ( 6 0 1 − 2 1 ) − 0 r f r 0 − 6 1 6 r ) 6 fh r C.1 fh (1998) 105 ( r − 0 1 r dr 1 ) 1 r f 1 0 − ( limit of superconformal field theories and supergravity H 2 0 − Adv. Theor. Math. Phys. − r r 0 r h [ ( ), which permits any use, distribution and reproduction in r ) ]. fh N fh h Z 0 fh dr ) ) r B 428 0 ( π 0 H β r r 4 f 0 r 0. ( r ( f SPIRE − Z 3 = ) is given by → IN π 3 0 = 1 + = = β K a 4 r can be obtained by considering the integrand of the above equation ][ The large (1999) 1113 K ( CC-BY 4.0 6 c 3 r r . Notice that in this limit 1 Phys. Lett. 38 K This article is distributed under the terms of the Creative Commons 0 ≈ − , − r Anti-de Sitter space and holography 3 ), fh K ]. → 2 − r 4.70 E in the limit 4 SPIRE / H 1 IN hep-th/9802150 r [ string theory [ Theor. Phys. 3 S.S. Gubser, I.R. Klebanov and A.M. Polyakov, E. Witten, J.M. Maldacena, → [2] [3] [1] c any medium, provided the original author(s) and source areReferences credited. The solution to this equation isr given by equation ( Open Access. Attribution License ( Indeed, the above expression diverges when Using the above result in equation ( The critical radius in the limit In this appendix wecording determine to the ( critical radius C Divergence of JHEP11(2018)072 , ] , (2017) ] Phys. , 05 (2014) 067 ]. (2016) 106 JHEP (2016) 070 , talk given at 03 , (2016) 086014 08 05 ]. SPIRE arXiv:1312.3296 ]. IN [ D 94 JHEP ][ arXiv:0808.2096 arXiv:1705.01728 ]. (2017) 064 , [ , SPIRE JHEP JHEP (2017) 155131 , SPIRE IN , 05 ]. IN ][ ]. 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