Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory

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Citation Roberts, Daniel A., and Douglas Stanford. "Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory." Phys. Rev. Lett. 115, 131603 (September 2015). © 2015 American Physical Society

As Published http://dx.doi.org/10.1103/PhysRevLett.115.131603

Publisher American Physical Society

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Citable link http://hdl.handle.net/1721.1/98897

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 115, 131603 (2015) PHYSICAL REVIEW LETTERS 25 SEPTEMBER 2015

Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory

Daniel A. Roberts* Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

† Douglas Stanford School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA (Received 10 March 2015; revised manuscript received 14 May 2015; published 22 September 2015) We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form hWðtÞVWðtÞVi. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large c Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of ∼t − β=2π β2E E t β=2π c E ;E ð Þ log w v, where is the fast scrambling time ð Þ log and w v are the energy scales of the W;V operators.

DOI: 10.1103/PhysRevLett.115.131603 PACS numbers: 11.25.Hf, 03.67.-a, 05.45.Mt

Introduction.—In classical mechanics, the butterfly commutator, and hence a quantum butterfly effect. We effect is a vivid diagnostic of chaos: small perturbations believe that this happens for practically any [12] choice of grow rapidly to affect the entire system. In quantum W and V and that, in fact, this behavior is a basic diagnostic mechanics, studies of chaos have often focused on less of quantum chaos. direct measures, such as the statistics of level spacings in This behavior has been confirmed for theories the energy spectrum [1,2], or the local properties of energy holographically dual to Einstein gravity in recent work eigenvectors [3–5]. In this Letter, following Refs. [6–9] we [6–9,13]. There, the thermal state is represented by a black will focus on a notion of quantum chaos that is closely hole, and the V;W operators create quanta that collide near related to the classical butterfly effect. the horizon. The key effect leading to a large commutator is Consider a pair of rather general Hermitian operators V the exponential blueshift relative to later slices as the W and W in a quantum mechanical system. If the system is perturbation falls into the black hole. strongly chaotic, we expect perturbations by V to affect later The purpose of this Letter is to reproduce part of that measurements of W, for almost any choice of operators V analysis without using holography directly. We will work in and W [10]. In other words, we expect the commutator a 2D conformal field theory (CFT) where thermal expect- ½V;WðtÞ to become large. A useful diagnostic is the ation values are related to vacuum expectation values by a expectation value of the square of the commutator [11]: conformal transformation. We will also restrict attention to 2 a particular contribution to the four-point function given by −h½V;WðtÞ iβ ¼hVWðtÞWðtÞViβ þhWðtÞVVWðtÞiβ ð1Þ the large c Virasoro identity conformal block. This resums the terms corresponding to factorization on powers and − hVWðtÞVWðtÞiβ − hWðtÞVWðtÞViβ: ð2Þ derivatives of the stress tensor. The close relationship between 2 þ 1 gravity and the identity Virasoro block Here, h·iβ indicates the thermal expectation value at inverse has been demonstrated recently in Refs. [14–16]; our work temperature β. Let us assume that the operators commute at should be understood as an application of the techniques in t ¼ 0. For small times, the terms on the first line cancel the these papers to the problem studied in Refs. [6–9,13].We terms on the second line. As we move to large time, the terms will find exact agreement between the Virasoro block on the first line will each approach the order-one value calculation and the corresponding holographic calculation WW VV h iβh iβ. This can be understood by viewing the in pure 3D gravity. In particular, we will find that the VWWV WW ordering as an expectation value of in a state contribution of the identity block to hVWðtÞVWðtÞiβ given by acting with V on the thermal state. If the energy begins to decay exponentially around the fast scrambling V injected by is small, the state will relax and the expectation time [17,18] t ¼ðβ=2πÞ log c. WW value will approach the thermal value, h iβ, multiplied by A key point that will emerge in our analysis is the the norm of the state, hVViβ. following. Each of the Lorentzian correlators on the By contrast, in a suitably chaotic system, the correlation right-hand side of Eq. (1) can be obtained by analytic functions on the second line will become small for large t. continuation of the same Euclidean four-point function. From Eq. (1), it is clear that this will imply a large The sharp difference in behavior between the first line and

where h; h¯ are the conformal weights of the O operator, related to the dimension and spin by Δ ¼ h þ h¯ and J ¼ h − h¯. On the left-hand side, we have a thermal expectation value, at inverse temperature β, and on the right-hand side we have a vacuum expectation value on the z; z¯ space. It is common to work with units in which β ¼ 2π, but we prefer to keep the β dependence explicit. It will be essential in this Letter to study correlation FIG. 1 (color online). Left: The spacetime arrangement of the functions with operators at complexified times ti. In our W V and operators. Right: Their locations after the conformal convention, real t corresponds to Minkowski time and mapping, viewed in the Rindler patch on the boundary of AdS3 t x; t imaginary corresponds to Euclidean time. Notice from (gray) covered by . The union of the gray and yellow regions is z¯ z t the Poincaré patch covered by z; z¯. Eq. (3) that is the complex conjugate of only if the time is purely Euclidean. In order to make contact with standard CFT formulas for the four-point function, we will begin with a purely Euclidean arrangement of the operators. This the second line arises because the continuation defines a means a choice of z1; z¯1; …;z4; z¯4 with z¯i ¼ zi . With such a multivalued function, with different orderings correspond- configuration, the ordering of the operators is unimportant, ing to different sheets. To see the butterfly effect (via the and global conformal invariance on the z; z¯ plane implies decay of the terms on the second line), one has to move off that the four-point function can be written, the principal sheet. W z ; z¯ W z ; z¯ V z ; z¯ V z ; z¯ It is important to emphasize that although the Virasoro h ð 1 1Þ ð 2 2Þ ð 3 3Þ ð 4 4Þi identity block does reproduce the gravitational calculation 1 1 f z; z¯ ; ¼ 2h 2h ¯ ¯ ð Þ ð5Þ of the four-point function, it is not the full answer. Indeed, a w v 2hw 2hv z12 z34 z¯ z¯ single Virasoro primary with spin greater than two can easily 12 34 dominate the contribution from the identity in a certain in terms of a function f of the conformally invariant cross range of times t. In holographic terms, this sensitivity to the ratios spectrum is related to the fact that the high-energy collision z12z34 z¯12z¯34 depends on stringy corrections. Just as the gravitational z ¼ ; z¯ ¼ ;zij ≡ zi − zj: ð6Þ z13z24 z¯13z¯24 calculation is a useful model for a more accurate string- corrected analysis [19], the Virasoro identity block provides According to the general principles of CFT, we can expand f as a sum of global conformal blocks, explicitly [20,21], a model for a more complete CFT calculation. X ¯ Although it is not our purpose to prove that the four- fðz; z¯Þ¼ pðh; h¯Þzhz¯hFðh; h; 2h; zÞFðh;¯ h;¯ 2h;¯ z¯Þ; ð7Þ point-function diagnostic described above agrees with h;h¯ other definitions of quantum chaos, we will provide a F sanity check by evaluating the above four-point functions where is the Gauss hypergeometric function, the sum is SL 2 in the two-dimensional Ising model. For this system, we over the dimensions of global ð Þ primary operators, and p will see that certain out-of-time-order four-point functions the constants are related to operator product expansion p h; h¯ λ λ do not tend to zero for large t. (OPE) coefficients ð Þ¼ WWOh;h¯ VVOh;h¯ . CFT calculations.—Conventions and review: In this Continuation to the second sheet: In order to apply the Letter, we will study thermal four-point correlation func- above formulas to the correlators (1) and (2), we need to tions of W and V of the form in Eqs. (1) and (2). Eventually, understand how to obtain them as analytic continuations of these operators will be arranged in the timelike configu- the Euclidean four-point function. That this is possible ration shown in Fig. 1, where V is at the origin and W is at follows from the fact that all Wightman functions are position t>x>0. However, we will obtain these corre- analytic continuations of each other [22]. The procedure lation functions by starting with the Euclidean correlator involves three steps. First, one starts with the Euclidean and analytically continuing. In 2D CFT, we can map function, assigning small and different imaginary times thermal expectation values to vacuum expectation values tj ¼ iϵj to each of the operators. Second, with the imagi- through the conformal transformation nary times held fixed, one increases the real times of the operators to the desired Lorentzian values. Finally, one zðx; tÞ¼eð2π=βÞðxþtÞ; z¯ðx; tÞ¼eð2π=βÞðx−tÞ: ð3Þ smears the operators in real time and then takes the Here, x; t are the original coordinates on the spatially imaginary times fϵig to zero [24]. The result will be a infinite thermal system and z; z¯ are coordinates on the Lorentzian correlator ordered such that the leftmost oper- vacuum system. Explicitly, ator corresponds to the smallest value of ϵ, the second 2πz h 2πz¯ h¯ operator corresponds to the second smallest, and so on. hOðx; tÞiβ ¼ hOðz; z¯Þi; ð4Þ This elaborate procedure is necessary because Eq. (5) is a β β multivalued function of the independent complex variables 131603-2 week ending PRL 115, 131603 (2015) PHYSICAL REVIEW LETTERS 25 SEPTEMBER 2015 X ¯ fðz; z¯Þ ≈ p~ ðh; h¯Þz1−hz¯h; ð14Þ h;h¯ where p~ has been defined to absorb the transformation coefficient. On this sheet, as z; z¯ become small, global z primaries with large spin become important. As a function FIG. 2 (color online). The paths taken by the cross ratio during x; t the continuations corresponding to (from left to right) hWVWVi, of , individual terms in this sum grow like h−h¯−1 t − h h¯−1 x hWWVVi, and hWVVWi. Only in the first case does the path pass eð Þ e ð þ Þ . For sufficiently large t,thissumdiverges, around the branch point at z ¼ 1. and it must be defined by analytic continuation. In other words, we must do the sum over h; h¯ before we continue the fzi; z¯ig. The interesting multivaluedness comes from cross ratios. In a CFT dual to string theory in AdS3, we expect fðz; z¯Þ. By crossing symmetry, this function is single valued this divergence even at a fixed order in the large c expansion, on the Euclidean section z¯ ¼ z, but it is multivalued as a because of the sum over higher spin bulk exchanges [19]. function of independent z and z¯, with branch cuts extending Virasoro identity block: The primary focus of this Letter from one to infinity. Different orderings of the W;V is reproducing the Einstein gravity calculation of the corre- operators correspond to different sheets of this function. lation function. This calculation was done by studying free To determine the correct sheet, we must assign iϵ’s as above, propagation on a shock wave background, which implicitly and follow the path of the cross ratios, watching to see if they sums an infinite tower of ladder exchange diagrams. In the pass around the branch loci at z ¼ 1 and z¯ ¼ 1. CFT, these diagrams are related to terms involving powers To carry this out directly, we write and derivatives of the stress tensor in the OPE representation

0 0 ð2π=βÞðt þiϵ1Þ −ð2π=βÞðt þiϵ1Þ of the four-point function. In a two-dimensional CFT, all z1 ¼ e ; z¯1 ¼ e ; ð8Þ 0 0 such terms can be treated simultaneously using the Virasoro ð2π=βÞðt þiϵ2Þ −ð2π=βÞðt þiϵ2Þ z2 ¼ e ; z¯2 ¼ e ; ð9Þ conformal block of the identity operator, which itself is an ð2π=βÞðxþiϵ3Þ ð2π=βÞðx−iϵ3Þ z3 ¼ e ; z¯3 ¼ e ; ð10Þ infinite sum of SLð2Þ conformal blocks. Including only these

ð2π=βÞðxþiϵ4Þ ð2π=βÞðx−iϵ4Þ terms in the OPE amounts to replacing z4 ¼ e ; z¯4 ¼ e ð11Þ fðz; z¯Þ → FðzÞF¯ ðz¯Þ; ð15Þ as a function of the continuation parameter t0. When t0 ¼ 0, we have a purely Euclidean correlator, on the principal where F is the Virasoro conformal block with dimension sheet of the function fðz; z¯Þ. When t0 ¼ t>x,wehavean zero in the intermediate channel. This substitution is appro- arrangement of operators as shown in Fig. 1. priate for a large N CFT with a sparse spectrum of single- The cross ratios z; z¯ are determined by these coordinates trace higher spin operators [29]. as in Eq. (6). Their paths, as a function of t0, depend on the The function F is not known explicitly, but there are ordering of operators through the associated iϵ prescription. several methods for approximating it [14–16]. We will use a Representative paths for the three cases of interest are formula from Ref. [15], which is valid at large c, with hw=c shown in Fig. 2. The variable z¯ never passes around the fixed andsmall andhv fixed and large. Here, the formula reads branch point at one, and the z variable does so only in the −6h =c 2h zð1 − zÞ w v case corresponding to WVWV [25]. FðzÞ ≈ : ð16Þ 1 − 1 − z 1−12hw=c In the final configuration with t0 ¼ t, the cross ratios are ð Þ small. For t ≫ x,wehave This function has a branch point at z ¼ 1, as expected. z 1 z ð2π=βÞðx−tÞ −ð2π=βÞðxþtÞ Following the contour around ¼ and taking small, we z ≈−e ϵ ϵ34; z¯ ≈−e ϵ ϵ34; ð12Þ 12 12 find where we introduced the abbreviation 1 2hv F z ≈ : ð2π=βÞiϵi ð2π=βÞiϵj ð Þ 24πih ð17Þ ϵij i e − e : w ¼ ð Þ ð13Þ 1 − cz For the orderings WWVV and WVVW, no branch cuts are The trajectory of z¯ does not circle the branch point at z¯ ¼ 1,so crossed, so the limit of small cross ratios can be taken on for small z¯, we simply have F¯ ðz¯Þ ≈ 1, the contribution of the the principal sheet of Eq. (7). The contribution from identity operator itself. Substituting Eq. (17) in Eq. (15) and the identity operator dominates, verifying our statement then in Eq. (5),wefind in the Introduction that both hWðtÞVVWðtÞiβ and hWðt þ iϵ1ÞVðiϵ3ÞWðt þ iϵ2ÞVðiϵ4Þiβ hWðtÞWðtÞVViβ approach hWWihVViβ for large t. WVWV z hWðiϵ1ÞWðiϵ2ÞiβhVðiϵ3ÞVðiϵ4Þiβ For , passes around the branch point at one. The hypergeometric function Fða; b; c; zÞ has known monodromy 1 2hv ≈ ; z 1 24πih 2π=β t−t −x ð18Þ around ¼ , returning to a multiple of itself, plus a multiple 1 w eð Þð Þ þ ϵ ϵ of the other linearly independent solution to the hypergeo- 12 34 z1−cF 1 a − c; 1 b − c; 2 − c; z t metric equation, ð þ þ Þ.For where we define the fast scrambling time [17,18] with the small z; z¯, we then have convention 131603-3 week ending PRL 115, 131603 (2015) PHYSICAL REVIEW LETTERS 25 SEPTEMBER 2015

β 0 −jx0−xj t c: hðx Þ¼2πGNPe : ðA4Þ ¼ 2π log ð19Þ To relate this geometry to the state jψi, we need to fix P. Equation (18) is the main result of our Letter. The correlation In other words, we need to evaluate function (18) begins to decrease at a time t∼ðβ=2πÞlogc=hw. R 0 0 This formula agrees precisely with the bulk analysis hψj dx du Tuujψi ; ðA5Þ (reviewed in Appendix A) in the above scaling. It is hψjψi also interesting to consider the scaling hv;hw fixed, hw ≫ where we take the integral to run over the slice v ¼ 0.We hv ≫ 1 with c → ∞. The bulk analysis suggests that will assume that W is dual to a single-particle operator in Eq. (18) is also correct in this scaling, but even without the bulk, so that the state jψi can be described by a Klein- the Virasoro block analysis, the SLð2Þ block of the stress Gordon wave function K. This wave function is a bulk-to tensor [which gives a contribution ∝ hwhv=ðczÞ on the boundary propagator from the location (x; t) of the W second sheet] is enough to show that, for general dimen- operator. It is given in terms of the regularized geodesic sions, the time until the identity Virasoro block is affected is distance d from the boundary point, as K ∝ ðcosh dÞ−2hw . of order t − ðβ=2πÞ log hwhv, where the second term is At v 0, we find order one in this scaling. This gives a field-theoretic ¼ explanation for the origin of the fast scrambling time. N Kðt; x; u0;x0Þ¼ : ðA6Þ etu0 x − x0 2hw We are grateful to Ethan Dyer, Tom Hartman, Alexei ½ þ coshð Þ Kitaev, and Steve Shenker for helpful discussions. D. A. R. The norm hψjψi is a Klein-Gordon inner product is supported by the Fannie and John Hertz Foundation and Z is very thankful for the hospitality of the Stanford Institute hψjψi¼2i dx0du0Kðt þ iτ;x; u0;x0Þ for Theoretical Physics during the completion of this 0 0 work. D. A. R. also acknowledges the U.S. Department × ∂u0 Kðt þ iτ;x; u ;xÞ: ðA7Þ of Energy under cooperative research agreement Contract The u0 integral can be done using contour integration, and No. DE-SC00012567. D. S. is supported by NSF Grant the x0 integral can be done in terms of Γ functions: No. PHY_1314311/Dirac. 4π3=2 Γ 4h ψ ψ N 2 ½ w : h j i¼ 1 ðA8Þ 2 τ 4hw Γ 2h Γ 2h APPENDIX A: BULK CALCULATIONS ð sin Þ ½ w ½ w þ 2 Here, we will compute the correlation function (18) using For the numerator, the stress tensor for the Klein- T ∂ φ∂ φ the gravitational shock wave methods of Refs. [6,8].If Gordon field is given by the expression uu ¼ u u . hw ≫ hv ≫ 1, we can calculate the correlation function by Contracting bulk operators with boundary operators using W K,wehave treating the operator as creating a shock wave and Z Z calculating the two-point function of the V operator on that ψ dx0du0T ψ 2 dx0du0∂ K t iτ;x; u0;x0 background. The analysis breaks into two parts: (i) finding the h j uuj i¼ u0 ð w þ Þ geometry of the shock sourced by W and (ii) computing the ∂ K t iτ;x; u0;x0 ; correlation function of the V operators in that background. × u0 ð w þ Þ ðA9Þ The metric of a localized shock wave [30] in (2 þ 1)- where the factor of 2 comes from the two different ways of dimensional AdS-Rindler space is [6,8,26,31,32] doing the contractions. The integrals can be done the same 4 1 − uv 2 way as before: ds2 − dudv ð Þ dx2 Z ¼ 1 uv 2 þ 1 uv 2 ð þ Þ ð þ Þ 0 0 hψj dx du Tuujψi þ 4δðuÞhðxÞdu2; ðA1Þ 3=2 t 1 8π e w Γ½4hwΓ½2hw þ where h will be defined below. We will consider a shock ¼ N 2 2 : ðA10Þ 4hwþ1 3 sourced by a stress tensor, ð2 sin τÞ Γ½2hw Taking the ratio at large hw, we find 0 0 0 0 0 Tuuðu ;v;xÞ¼Pδðu Þδðx − xÞ; ðA2Þ t 2hwe w appropriate for a particle sourced by the W operator, P : ¼ τ ðA11Þ traveling along the u ¼ 0 horizon at transverse position sin x. The metric (A1) can be understood as two halves of AdS- The second step, following Ref. [6], is to compute the Rindler, glued together at u ¼ 0 with a shift two-sided correlation function of the V operators in this δvðxÞ¼hðxÞðA3Þ shock background. We will do this using the geodesic approximation in the v direction. Plugging into Einstein’s equations, we ψ V V ψ ∝ e−md; determine h as h j L Rj i ðA12Þ 131603-4 week ending PRL 115, 131603 (2015) PHYSICAL REVIEW LETTERS 25 SEPTEMBER 2015 where d is the regularized geodesic distance and the mass m [5] M. Srednicki, Chaos and quantum thermalization, Phys. is approximately 2hv, the conformal weight of V. Rev. E 50, 888 (1994). Following Ref. [6], we find [6] S. H. Shenker and D. Stanford, Black holes and the butterfly effect, J. High Energy Phys. 03 (2014) 067. d ¼ 2 log 2r∞ þ log½1 þ hð0Þ: ðA13Þ [7] S. H. Shenker and D. Stanford, Multiple shocks, J. High After subtracting the divergent distance in the unper- Energy Phys. 12 (2014) 046. [8] D. A. Roberts, D. Stanford, and L. Susskind, Localized turbed thermofield double state, d 2 log 2r∞, we plug TFD ¼ shocks, J. High Energy Phys. 03 (2015) 051. the distance into Eq. (A12), finding [9] A. Kitaev, Hidden correlations in the hawking radiation and ψ V V ψ 1 2hv thermal noise, in Talk given at the Fundamental Physics h j L Rj i : ¼ 4πG h t−x ðA14Þ Prize Symposium (2014). hψjψihVLVRi 1 N w e þ sin τ [10] We take V;W to be approximately local operators, smeared This agrees with Eq. (18) after (i) using GN ¼ 3=2c to over a thermal scale, and with one-point functions subtracted. express the gravitational constant in terms of the central [11] A. Larkin and Y. Ovchinnikov, Quasiclassical method in the 28 charge, (ii) plugging in ϵ1 ¼ −τ, ϵ2 ¼ τ, ϵ3 ¼ 0, ϵ4 ¼ β=2, theory of superconductivity, J. Exp. Theor. Phys. , 1200 and (iii) using β ¼ 2π. (1969). [12] Here, “practically any” should include all local operators with both h; h¯ nonzero. APPENDIX B: ISING MODEL [13] S. 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B599, 459 across the branch cut for the two correlators that do have a (2001). second sheet and taking z small, we find [22] Theorem 3.6 of Ref. [23]. [23] R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, hσσσσiβ hσϵσϵiβ hϵϵϵϵiβ 0; −1; 1: and All That (W. A. Benjamin, New York, 1964). 2 ¼ ¼ 2 ¼ ðB4Þ hσσiβ hσσiβhϵϵiβ hϵϵiβ [24] In fact, we will omit this final step in this Letter. However, we will omit it consistently on both sides of the bulk and Only hσσσσi vanishes at large t. boundary calculations that we are comparing. [25] A very similar continuation was discussed for high-energy scattering kinematics in Refs. [26–28]. [26] L. Cornalba, M. S. Costa, J. Penedones, and R. Schiappa, *[email protected] Eikonal approximation in AdS=CFT: From shock waves † [email protected] to four-point functions, J. High Energy Phys. 08 (2007) 019. [1] M. V. Berry and M. Tabor, Level clustering in the regular [27] L. Cornalba, M. S. Costa, J. Penedones, and R. Schiappa, spectrum, Proc. R. 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with spin greater than two and dimension parametrically since the geometry is locally pure AdS3 away from the independent of c will dominate over the Virasoro identity source. t ’ block near time . This means that the universal contribution [31] T. Dray and G. t Hooft, The gravitational shock wave of a to the correlator identified in this Letter applies only to massless particle, Nucl. Phys. B253, 173 (1985). CFTs with a very sparse spectrum of single-trace operators. [32] G. T. Horowitz and N. Itzhaki, Black holes, shock waves, This matches the sensitivity of high-energy scattering in and causality in the AdS=CFT correspondence, J. High gravity to massive higher spin fields, which grow more Energy Phys. 02 (1999) 010. rapidly with energy. In a consistent bulk theory of higher spin [33] A. Belavin, A. M. Polyakov, and A. Zamolodchikov, fields, such as string theory, a Regge-type resummation of Infinite conformal symmetry in two-dimensional quantum an infinite number of stringy operators is necessary. For field theory, Nucl. Phys. B241, 333 (1984). further discussion of this point in the present context, see [34] M. P. Mattis, Correlations in two-dimensional critical Ref. [19]. theories, Nucl. Phys. B285, 671 (1987). [30] Although we refer to these geometries as shock waves, the [35] P. H. Ginsparg, Applied conformal field theory, terminology is somewhat misleading in 2 þ 1 dimensions, arXiv:hep-th/9108028.

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