QQMUL-PH-21-01
Pole-skipping and Rarita-Schwinger fields
Nejc ˇCeplak ∗ and David Vegh † Institut de Physique Th´eorique, Universit´e Paris Saclay,CEA, CNRS, F-91191 Gif sur Yvette, France and Centre for Research in String Theory, School of Physics and AstronomyQueen Mary University of London, 327 Mile End Road, London E1 4NS, UK
In this note we analyse the equations of motion of a minimally coupled Rarita-Schwinger field nearthe horizon of an anti-de Sitter-Schwarzschild geometry. We find that at special complex values ofthe frequency and momentum there exist two independent regular solutions that are ingoing at thehorizon. These special points in Fourier space are associated with the ‘pole-skipping’ phenomenonin thermal two-point functions of operators that are holographically dual to the bulk fields. We findthat the leading pole-skipping point is located at a positive imaginary frequency with the distancefrom the origin being equal to half of the Lyapunov exponent for maximally chaotic theories.
I. INTRODUCTION
Retarded Green’s functions are one of the main objectsof interest in field theories as they encode the response ofa system in equilibrium to small perturbations. In a the-ory with a dual gravitational description [1] one can cal-culate such correlators at finite temperature by studyingthe equations of motion of bulk fields in a black hole ge-ometry which asymptotes to anti-de Sitter (AdS) space-time [2, 3]. In Lorentzian signature, one is prescribed totake the solution that is ingoing at the horizon [4] whichuniquely determines the retarded Green’s function of thedual boundary operators. Finding the precise form ofthe correlation function is generically difficult as the fullsolutions to the to the bulk equations of motion dependon the details of the background geometry.However, some features of the Green’s functions areencoded in the behaviour of the bulk fields in thenear-horizon region of the geometry. The most promi-nent example is the hydrodynamic description of holo-graphic theories, described by the low-frequency and low-momentum limit (see for example [5, 6]), where one findsthat the radial evolution of the bulk fields becomes triv-ial [7] and leads to universal results such as the ratio ofthe shear viscosity and the entropy density [8].Interestingly, the horizon region also contains informa-tion about the correlator away from the origin of Fourierspace. Namely, at certain imaginary values of the fre-quency and (generically) complex values of the momen-tum, all solutions to the equations of motion become in-going at the horizon. This phenomenon was called ‘pole-skipping’ as at such points in momentum space the asso-ciated boundary Green’s function effectively skips a pole:at these locations a zero and a pole of the correlator co-incide.Pole-skipping was initially observed in the energy den-sity correlator which is obtained by solving the linearisedEinstein’s equations in the bulk. One finds that the lead- ∗ [email protected] † [email protected] FIG. 1. Location of the leading pole-skipping point on thefrequency plane for fields of various spin. This paper focuseson the spin- case. ing pole-skipping point, that is the pole-skipping point atthe most positive imaginary frequency (see FIG. 1), is lo-cated at [9–11] ω ∗ = + iλ L , k ∗ = i λ L v B , (1)where λ L = 2 πT is the Lyapunov exponent for holo-graphic theories and v B is the butterfly velocity. Thusthis location seems to be connected to the chaotic prop-erties of holographic systems [12–14] , which is the caseeven in the presence of stringy corrections [17] (see [18–22] for further examples).Correlators associated to lower-spin fields also exhibitpole-skipping, albeit at different imaginary frequencies.Scalar and U (1) gauge fields were analysed in [23] (seealso [24–29]). In the case of a scalar field all pole-skippingpoints are located in the lower-half of the complex fre-quency plane at (imaginary) bosonic Matsubara frequen-cies ω Bn = − πiT n , where n = 1 , , . . . . The same Pole-skipping in theories which are not maximally chaotic hasbeen discussed in [15] and [16]. In such theories the relationto chaos is not straightforward and the applicability of pole-skipping as a diagnostic for chaos might be limited to only im-posing bounds on chaos. a r X i v : . [ h e p - t h ] J a n applies for the gauge field, however in the longitudinalchannel there exists an additional ‘hydrodynamic’ pole-skipping point located at ω = 0 and k = 0. Note thatsuch an infinite tower of pole-skipping points at negativeimaginary Matsubara frequencies can also be found inthe energy density correlator [23]. Similarly, for fermionfields all pole-skipping points are on the lower-half of thecomplex frequency plane [30], but they are located atfermionic Matsubara frequencies ω Fn = − πiT ( n − / n = 1 , , . . . .Note that there is a relationship between the spins ofbulk fields and the frequency values of the leading pole-skipping points as depicted in FIG. 1. The locations onthe complex ω -plane are given by ω , s = 2 πiT ( s − s denotes the spin of the field . We note thatFIG. 1 is a compilation of several independent results.However, the analysis for spin- fields has been missingfrom the literature. In this note, by analysing the Rarita-Schwinger field in the near-horizon region of an AdS-Schwarzschild black hole, we show that indeed the rela-tion between the spin and value of the frequency of theleading pole-skipping point holds for any (half)-integerspin with s ≤ ω ps , k ps )takes the form [11, 23, 30, 33] G R ( ω ps + δω, k ps + δk ) ∝ δω − (cid:0) δωδk (cid:1) z δkδω − (cid:0) δωδk (cid:1) p δk , (2)where ( δω/δk ) p,z denote the slope of the line of p oles andline of z eros passing through the pole-skipping point .The Green’s function of all leading pole-skipping pointsdepicted in FIG. 1 has the form (2) and similarly we showthat the near-horizon analysis predicts that the corre-sponding correlator at the leading pole-skipping point ofthe Rarita-Schwinger field takes on the same form. We thank Richard Davison for bringing this order of leadingpole-skipping points to our attention. This kind of relation be-tween the frequency and spin has also appeared in the analysisof [31], where holographic correlators (and correlation functionsin theories with added higher spin currents) were studied. Sim-ilarly, in [28] (see also [32]) the exchange of vector and scalarfields in a four-point correlation function on a hyperbolic spacewas considered and the leading Regge behaviour contained thesame relation between the spin and the (imaginary) exponentialcoefficient multiplying the time coordinate. A more thorough analysis of [33] showed that there exist otherforms that a Green’s function can have near the special loca-tions. In their language, pole-skipping points at which the cor-relator takes on the form (2) are called
Type-I . Type-II pole-skipping points are associated with points that were previouslydenoted as anomalous points and are most commonly associatedwith locations at which the near-horizon analysis predicts twocoincident pole-skipping points.
Type-III pole-skipping pointsare associated with non-(half)-integer imaginary Matsubara fre-quencies and cannot be predicted from the near-horizon analysis(see also [34]).
II. GRAVITATIONAL SETUP
Let the bulk theory be described by the Einstein-Hilbert action with a negative cosmological constant S = (cid:90) d d +2 x √− g ( R − , (3)where Λ = − d ( d + 1) / L and L denotes the radius ofAdS which we set to L = 1 in all further expressions.The resulting equations of motion admit a planar blackhole solution that is asymptotically AdS and is describedby the line element ds = − r f ( r ) dv + 2 dv dr + h ( r ) dx i dx i . (4)We are using the ingoing Eddington-Finkelstein coor-dinates where r denotes the radial direction with theboundary of AdS located at r → ∞ . The usual timecoordinate can be recovered using the relation v = t + r ∗ ,where dr ∗ = dr/ ( r f ( r )), in which case one can see that { t, x i } , with i = 1 . . . d , are the coordinates of the d + 1dimensional Minkowski space at a fixed value of r .The two functions appearing in the metric are givenby f ( r ) = 1 − (cid:16) r r (cid:17) d +1 , h ( r ) = r , (5)meaning that there is a non-degenerate horizon at a fi-nite radius r = r where f ( r ) vanishes. However as weare using the ingoing Eddington-Finkelstein coordinates,such a point is a regular point of the metric. The asso-ciated (non-vanishing) Hawking temperature is given by4 πT = r f (cid:48) ( r ) = ( d + 1) r . In what follows, we keepboth f ( r ) and h ( r ) generic which allows us to identifythe source of the contributions when analysing the loca-tion of pole-skipping points. Furthermore, it allows foran easier generalisation of our results to a larger class ofbackground geometries, such as geometries with an addi-tional matter content deforming the background [35, 36],even though our derivation applies only when f ( r ) and h ( r ) take the form (5).The aim of this note is the analysis of the near-horizonbehaviour of a minimally coupled spin- field. In anAdS/CFT context, such fields were first considered in[37–41] and have later been used to study the propertiesof the charged current in the dual boundary theory (seefor example [42–45]). For a summary and discussion ofrecent results see [46].The action describing the massive Rarita-Schwingerfield Ψ M is given by S RS ∝ (cid:90) d d +2 x √− g ¯Ψ M (cid:0) Γ MNP ∇ N − m Γ MP (cid:1) Ψ P . (6) Throughout this note we use upper case Latin letters (
M, N, . . . )to denote the curved spacetime indices, whereas lower case Latinletters ( a, b, . . . ) to denote the flat space indices.
Since we are only interested in the bulk equationsof motion, we do not need the precise details ofthe overall normalisation factor or the additionalboundary terms . The anti-symmetrised products ofcurved space gamma matrices Γ MP and Γ MNP acton the spinor index of the Rarita-Schwinger field,which we suppress throughout the note. The covari-ant derivative acting on the spin- field is given by ∇ M Ψ P = ∂ M Ψ P − (cid:101) Γ NMP Ψ N + ( ω ab ) M Γ ab Ψ P , where (cid:101) Γ NMP are the Christoffel symbols and ω M is the spin con-nection one form.The equation of motion derived from (6) isΓ MNP ∇ N Ψ P − m Γ MN Ψ N = 0 , (7)however since the background metric satisfies the vacuumEinstein’s equation one can show (see for example [44])that the above equation of motion is equivalent to (cid:0) / ∇ + m (cid:1) Ψ N = 0 , (8)with additional constraintsΓ M Ψ M = 0 , ∇ M Ψ M = 0 , (9)where we defined / ∇ = Γ M ∇ M . Note that while (8) lookslike a Dirac equation, it actually couples different vectorcomponents of the field due to the term involving theChristoffel symbols in the covariant derivative, which isnot present if the covariant derivative is acting on a Diracfield. In deriving (8) we assume that the mass takes ona generic positive value in which case the constraints (9)follow naturally from the equations of motion (7). If m = d/
2, the spin- field is physically massless [47] (thenon-zero value of m is a consequence of the curvature ofspacetime) and the conditions (9) are not imposed by theequations of motion, but can be considered as a choice ofgauge. III. POLE-SKIPPING
We now show that at a specific complex value of thefrequency and momentum the equations of motion (8)and the associated constraints admit two independentsolutions that are regular at the horizon. We use thefollowing orthonormal frame E v = 1 + f ( r )2 rdv − drr , E r = 1 − f ( r )2 rdv + drr , We note that another mass term m (cid:48) g MN ¯Ψ M Ψ N can be addedto the action. This introduces a spin- degree of freedom inthe Rarita-Schwinger field (see e.g. [46]) which may change thestructure of pole-skipping. We do not consider such a term here. We use underlined indices to indicate specific flat space indices( a = v, r, . . . ) and distinguish them from particular curved spaceindices which are not underlined ( M = v, r, . . . ). E i = (cid:112) h ( r ) dx i , (10)in which case the line element (4) can be written as ds = η ab E a E b with η ab = diag( − , , , . . . , r coordinate and introduce a planewave ansatz Ψ M ( v, r, x i ) = ψ M ( r ) e − iωv + ik i x i in whichcase the equations of motion reduce to a system of cou-pled first order ordinary differential equations for ψ M ( r ).To separate these equations into tractable subsystemswe decompose the components of the field based on itseigenvalues under the action of Γ r and Γ (2) ≡ ˆ k i Γ vi ,where ˆ k i = k i /k is the normalised Euclidean vector in d -dimensional flat space and k ≡ (cid:113)(cid:80) di =1 k i its magni-tude. These two matrices commute and their eigenval-ues are equal to ±
1, hence all vector components of theRarita-Schwinger field can be decomposed as ψ M = (cid:88) α = ± (cid:88) α = ± ψ ( α ,α ) M (11)where the indices in the bracket denote the eigenvaluesunder the action of the gamma matrices asΓ r ψ ( α ,α ) M = α ψ ( α ,α ) M , Γ (2) ψ ( α ,α ) M = α ψ ( α ,α ) M , (12)with α , = ± . Each of the components in the decompo-sition contains a quarter of the total degrees of freedomof the spinor.The leading pole-skipping point for the energy-densitycorrelator is found by considering the vv -component ofthe dual bulk excitation [9–11]. Similarly, the location ofthe leading pole-skipping point for the U (1) gauge fieldis uncovered by considering the v -component of the bulkfield [23]. Expecting similar behaviour, we focus on the ψ v component of the Rarita-Schwinger field. By usingthe constraints (9), we find that only ψ r couples to ψ v sowe focus on that subsystem of equations. This subsys-tem then separates into two decoupled sets of differentialequation – one containing components whose Γ r and Γ (2) eigenvalues are equal, and one with components whoseeigenvalues under the action of the aforementioned ma-trices are opposite. But the two decoupled systems of dif-ferential equations are related by k → − k . We can thusfocus only on one and obtain the results for the other by areversal of the momentum k . For concreteness, we choosethe subsystem dealing with the components whose eigen-values under Γ r and Γ (2) are equal. These equations aregiven in full detail in the appendix (see (A5)), togetherwith some additional information about their derivation.In our analysis the detailed form of the equations ofmotion is not needed as we are merely interested intheir near-horizon expansion. Because in the ingoingEddington-Finkelstein coordinates the horizon is a regu-lar point, we can expand the field components in a seriesas ψ ( α ,α ) M ( r ) = ∞ (cid:88) l =0 ψ ( α ,α ) M,l ( r − r ) l , (13)where ψ ( α ,α ) M,l are constant coefficients. Similarly, theequations of motion themselves can be expanded in aseries at the horizon after which solving the differentialequations translates into solving a system of algebraicequations at each order of the series expansion.At a generic point in Fourier space these algebraicequations halve the number of free parameters in theRarita-Schwinger field which corresponds to choosing thesolution to the equations of motion that is ingoing at thehorizon. For example, by evaluating the equations of mo-tion directly at the horizon, one finds, amongst others,the following equation (see (A7)) (cid:34) − mr − iω + 2 ikr (cid:112) h ( r ) − r f (cid:48) ( r ) (cid:35) Γ v ψ ( − , − ) v, + (cid:34) mr − iω − ikr (cid:112) h ( r ) − r f (cid:48) ( r ) (cid:35) ψ (+ , +) v, = 0 , (14)which for generic ω and k relates ψ ( − , − ) v, to ψ (+ , +) v, . Sim-ilarly, equations of motion at higher order in the near-horizon expansion relate all other coefficients (includingthose involving ψ r components) to the ones appearing in(14) allowing us to perturbatively construct a solutionwith half of the total number of degrees of freedom.Equation (14) is trivially satisfied if both coefficientsin the square brackets vanish, which is the case if thefrequency and momentum are precisely ω ≡ ω = ir f (cid:48) ( r )4 = iπT , (15a) k ≡ k = − im (cid:112) h ( r ) = − md + 1 iπT . (15b)Then both ψ (+ , +) v, and ψ ( − , − ) v, remain unconstrained andby using other algebraic equations from the near horizonexpansion of the equations of motion we can constructtwo linearly independent solutions that are regular at thehorizon .The plots in FIG. 2 show the location of poles of theboundary Green’s function in the complex ω -plane atdifferent values of the momentum, calculated using theLeaver method [48]. We see that at the special point(15b) there is a pole located exactly at (15a) – this is thepole predicted by the near-horizon analysis.Finally, recall that there exist an additional pole-skipping point at the same frequency (15a) but at the If we set ω = iπT but leave k generic, the complete near-horizonexpansions, such as (13), contain additional logarithmic termslog( r − r ) similar to the terms appearing in the scalar [23] orfermion [30] field expansions. However, one finds that by impos-ing the equations of motion, all coefficients multiplying termswith a logarithmic divergence at the horizon have to vanish, un-less the momentum is finely tuned to the pole-skipping value(15b). ■■■■ - - - - - - - Re ω π T I m ω π T ■■■■ ■■ - - - - - - - Re ω π T I m ω π T FIG. 2. Numerical analysis of the poles of the retardedGreen’s function in AdS for a Rarita-Schwinger field withmass m = 3 .
4. The poles are calculated using the methodspresented in [48] adapted to the spin-3 / ω -plane nearthe value of the momentum where pole-skipping is observed.The momentum is varied linearly as k = k + (cid:15)e iπ , where (cid:15) ∈ R . We notice that the red curve passes through ω as (cid:15) = 0 which we denote with a red square. In blue we depicttwo additional poles which pass through ω = − πi as (cid:15) → k = k e i(cid:15) .Pole-skipping occurs at the filled squares. opposite momentum to (15b) that is obtained from thenear-horizon analysis of the components, whose Γ r andΓ (2) eigenvalues are opposite.As is known from previous results [10, 11, 23, 28, 30,33], a location in Fourier space where there exist multi-ple independent ingoing solutions at the horizon corre-sponds to a point where a pole and a zero of the bound-ary retarded Green’s function coincide. As such, (15) isthe first location at which we observe pole-skipping forthe Rarita-Schwinger field. Most notably, this point islocated at a positive imaginary frequency. Up to now,the only other example of a pole-skipping point beingfound on the upper complex ω half-plane is the leadingpole-skipping location for the energy-density correlator,in which case the precise location in Fourier space hasbeen conjectured to be related to the chaotic propertiesof the theory, as seen in (1).For the Rarita-Schwinger field, the modulus (15a) isexactly half the value of the maximal Lyapunov exponentbound of [14]. In our opinion this suggests that there islittle relation to chaos, despite the seeming exponentialgrowth of the solution, if the values (15) are inserted intothe plane-wave ansatz. This is further substantiated bythe fact that the value of the momentum at which pole-skipping occurs depends on the mass of the field and thusthe location is sensitive both to the background and theprobe.Finally, recall that the equations of motion near thehorizon also predict the form of the Green’s function nearthe pole-skipping point. To that end evaluate (14) at anearby point in Fourier space, which we denote by ω = ω + (cid:15)δω and k = k + (cid:15)δk , where (cid:15) > k fixed. The analysis of [30] appliesto this case and we refer the reader to that reference fora detailed calculation. The final result of [30] applieshere as well: working at linear order in (cid:15) , one finds thatnear (15) the retarded Green’s function takes the pole-skipping form (2). IV. DISCUSSION
It has recently been shown that quantum chaos in fieldtheories with holographic duals manifests itself in thethermal energy density two-point functions. These cor-relators exhibit the ‘pole-skipping’ phenomenon at spe-cial (imaginary) values of the frequency and momentum.At such points poles and zeroes collide and the Green’sfunction becomes ill-defined. It was found [9–11] that thepole-skipping frequency lies at ω = +2 πiT . The modu-lus | ω | = 2 πT is thought to be related to the maximalLyapunov exponent [14].For bulk fields with spins other than two, the corre-sponding boundary two-point functions also show pole-skipping at certain Matsubara frequencies. Scalar fieldsfirst exhibit pole-skipping at ω = − πiT , while for con-served currents this occurs at ω = 0 [23]. Fermionicfields show pole-skipping at ω = − πiT , i.e. at the firstnegative fermionic Matsubara frequency [30]. The cal-culations rely on an analysis of the near-horizon regionof the bulk geometry and give non-trivial constraints onGreen’s functions at frequencies ω ∼ T .Although the relevant pole-skipping momenta do notseem to be universal, there is a definite structure in pole-skipping frequencies; see FIG. 1. Namely, there is a rela-tionship between the frequency of the first pole-skipping point and the spin of the bulk field.In this paper we have dealt with the spin- case whichhas been missing from previous analyses. In partic-ular, we have shown from a bulk perspective that aRarita-Schwinger field in an AdS-Schwarzschild back-ground exhibits pole-skipping precisely at the expectedfirst fermionic Matsubara frequency on the upper-halfplane. Since the analysis only concerned the near-horizonregion, we expect that the results can be extended tomore general spacetimes with regular horizons. Withthis result, we complete the hierarchy of pole-skippinglocations for various fields of different spin.It is important to note that the results for individ-ual fields were calculated independently and in no wayrelied on supersymmetry. Furthermore, the above men-tioned locations are merely the first pole-skipping pointsand in some sense are the simplest ones - the locationsin momentum space depend only on the values of thebackground metric at the horizon and/or the nearbyregion through first derivatives, whereas higher orderpole-skipping points generically depend on higher orderderivatives as well [23]. Thus it is enticing to conjecturethat the hierarchy is the property of the near-horizonregion of spacetime itself.It would be interesting to investigate the origin of thishierarchy (for a CFT analysis, see [34, 49, 50]). Note thatthe pole-skipping points are located at (imaginary) Mat-subara frequencies which means that the static bulk hasa Euclidean counterpart (obtained by Wick-rotation).Since the geometry smoothly caps off, a shift along theEuclidean time circle translates into a rotation at thetip (which is the Euclidean analog of the event horizon).Fourier modes are therefore connected to spin, provid-ing an explanation for the observed hierarchy of pole-skipping points.In the above analysis we have merely shown the ex-istence of a single pole-skipping point, but previous re-sults on other fields would suggest that there exists anentire tower of higher-order pole-skipping points at neg-ative imaginary frequencies, which we have not focusedon. Furthermore, the above analysis can be expanded toinclude matter fields or allow for an additional mass termin the action of the Rarita-Schwinger field. It would beinteresting to see if there exist a configuration at whichthe pole-skipping points vanish.Finally, the Rarita-Schwinger fields in backgroundswith simple horizons have been used in calculations offermionic currents [42–45]. It would be interesting tostudy the pole-skipping points in that context and seethe consequences on the spectral function of the dualfermionic currents in the boundary theory. ACKNOWLEDGEMENTS
We are grateful to Richard Davison for helpful discus-sions. N ˇC is supported by the ERC Grant 787320 - QBHStructure. DV is supported by the STFC Ernest Ruther-ford grant ST/P004334/1.
Appendix A: Equations of motion
In this appendix we spell out some of the details thathave been omitted in the main part of this note. Wekeep the same notational conventions as in the main text.In addition, throughout the note we use the followingconvention for the flat space Clifford algebra (cid:8) Γ a , Γ b (cid:9) = 2 η ab , (A1)where η ab = diag( − , +1 , . . . , +1), which most impor-tantly implies that Γ v squares to − h ( r ) and f ( r ) un-evaluated to keep track of the origin of individual terms inthe final expressions. However, in deriving (8) from (7),one assumes that the Ricci tensor and the Ricci scalarsatisfy R MN = − ( d + 1) g MN , R = − ( d + 1)( d + 2) , (A2)which is the case if the two aforementioned functions takethe form (5), as for example for a black brane solution.So while the results presented in this note hold only forbackgrounds that satisfy (A2), we believe that the gener-alisation of our results to more complicated backgroundsshould be straightforward and that the findings wouldresemble those of this note.Recall that due to the presence of the Christoffel termsin the covariant derivative, different vector componentsof the Rarita-Schwinger field are coupled. We can makethis explicit by putting such terms on the right-hand side,resulting in (cid:0) /D + m (cid:1) Ψ M = Γ N (cid:101) Γ PMN Ψ P , (A3)where D M = ∂ M + ( ω ab ) M Γ ab denotes the covariantderivative acting on a spinor field. The left-hand side of equation (A3) with the choice of vielbein (10) has beenworked out in [30]. One can then use the gamma-tracelesscondition (9) to arrive at two coupled equations contain-ing only the components Ψ v and Ψ r (cid:0) /D + m (cid:1) Ψ v = ∂ r ( r f ( r ))2 r (cid:26)(cid:2) Γ v + Γ r (cid:3) Ψ v + r (cid:2) (1 + f ( r ))Γ v − (1 − f ( r )) Γ r (cid:3) Ψ r (cid:27) , (A4a) (cid:0) /D + m (cid:1) Ψ r = − ∂ r ( r f ( r ))2 r (cid:2) Γ v + Γ r (cid:3) Ψ r − ∂ r h ( r )2 rh ( r ) × (cid:26)(cid:2) Γ v + Γ r (cid:3) Ψ v + r (cid:20) (1 + f ( r ))Γ r − (1 − f ( r ))Γ v (cid:21) Ψ r (cid:27) . (A4b)Since the components of the metric depend only on thecoordinate r , one can write the field in the form of aplane wave Ψ M ( r ) = ψ M ( r ) e − iωv + ik i x i , and insert thisansatz into the above equations. After decomposing thespinors in terms of their behaviour under the action ofΓ r and Γ (2) ≡ ˆ k i Γ vi matrices, as described in (11) and(12), one obtains eight coupled first order ordinary differ-ential equations for the different components of ψ v and ψ r . The system of equations can be split into two de-coupled subsystems of 4 equations, with one describingthe components whose Γ r and Γ (2) eigenvalues are equaland the other containing the components with oppositeeigenvalues under the action of the two matrices. Thetwo subsystems of equations are related by k → − k , soit is sufficient to analyse only one of the subsystems andinfer the results of the other by simply reversing the mo-mentum.Hence in what follows we focus at the system of equa-tions that involve ψ ( ± , ± ) v and ψ ( ± , ± ) r which are given,after some algebra, by equations S = (cid:34) r f ( r ) ∂ r − iω − rf ( r ) + r f (cid:48) ( r )4 + d r f ( r ) h (cid:48) ( r )4 h ( r ) + mr (1 + f ( r ))2 − ikr (1 − f ( r ))2 (cid:112) h ( r ) (cid:35) ψ (+ , +) v + Γ v (cid:34) − iω − rf ( r ) + r f (cid:48) ( r )4 − mr (1 − f ( r ))2 + ikr (1 + f ( r ))2 (cid:112) h ( r ) (cid:35) ψ ( − , − ) v − r f ( r )2 ∂ r ( r f ( r ))Γ v ψ ( − , − ) r , (A5a) S = Γ v (cid:34) r f ( r ) ∂ r − iω − rf ( r ) + r f (cid:48) ( r )4 + d r f ( r ) h (cid:48) ( r )4 h ( r ) − mr (1 + f ( r ))2 + ikr (1 − f ( r ))2 (cid:112) h ( r ) (cid:35) ψ ( − , − ) v + (cid:34) − iω − rf ( r ) + r f (cid:48) ( r )4 + mr (1 − f ( r ))2 − ikr (1 + f ( r ))2 (cid:112) h ( r ) (cid:35) ψ (+ , +) v − r f ( r )2 ∂ r ( r f ( r )) ψ (+ , +) r , (A5b) S = (cid:34) r f ( r ) ∂ r − iω + 3(2 rf ( r ) + r f (cid:48) ( r ))4 + ( d + 2) r f ( r ) h (cid:48) ( r )4 h ( r ) + mr (1 + f ( r ))2 − ikr (1 − f ( r ))2 (cid:112) h ( r ) (cid:35) ψ (+ , +) r + Γ v (cid:34) − iω + 4 rf ( r ) + 3 r f (cid:48) ( r )4 − mr (1 − f ( r ))2 + ikr (1 + f ( r ))2 (cid:112) h ( r ) (cid:35) ψ ( − , − ) r + h (cid:48) ( r )2 h ( r ) (cid:16) ψ (+ , +) v + Γ v ψ ( − , − ) v (cid:17) , (A5c) S = Γ v (cid:34) r f ( r ) ∂ r − iω + 3(2 rf ( r ) + r f (cid:48) ( r ))4 + ( d + 2) r f ( r ) h (cid:48) ( r )4 h ( r ) − mr (1 + f ( r ))2 + ikr (1 − f ( r ))2 (cid:112) h ( r ) (cid:35) ψ ( − , − ) r + (cid:34) − iω + 4 rf ( r ) + 3 r f (cid:48) ( r )4 + mr (1 − f ( r ))2 − ikr (1 + f ( r ))2 (cid:112) h ( r ) (cid:35) ψ (+ , +) r + h (cid:48) ( r )2 h ( r ) (cid:16) ψ (+ , +) v + Γ v ψ ( − , − ) v (cid:17) . (A5d)For later convenience we have labelled them as S i , with i = 1 , , ,
4, so that the equations of motion can be sum-marized in a compact way as S i = 0.We are interested in the near-horizon expansion ofthese equations. As we are working with the ingoingEddington-Finkelstein coordinates in which the horizonis a regular point, we assume that all functions and fieldscan be expanded in a series around the horizon at r = r .The expansion of (5) is trivial and we use the field ex-pansion (13). Then (A5) also get expanded around thehorizon S i = ∞ (cid:88) l =0 S ( l ) i ( r − r ) l = 0 , ⇒ S ( l ) i = 0 , i = 1 , , , , (A6)in which case the equations of motion become an infi-nite set of algebraic equations that we can solve order byorder.Our main interest lies in evaluating the equations ofmotion directly at the horizon, or in other words, thezeroth order equations S (0) i = 0. While the four equa-tions in (A5) are independent in general, one finds thatdirectly at the horizon there are only two independentequations as S (0)1 = S (0)2 and S (0)3 = S (0)4 , with S (0)1 = (cid:34) − mr − iω + 2 ikr (cid:112) h ( r ) − r f (cid:48) ( r ) (cid:35) Γ v ψ ( − , − ) v, + (cid:34) mr − iω − ikr (cid:112) h ( r ) − r f (cid:48) ( r ) (cid:35) ψ (+ , +) v, = 0 , (A7a) S (0)3 = (cid:34) − mr − iω + 2 ikr (cid:112) h ( r ) + 3 r f (cid:48) ( r ) (cid:35) Γ v ψ ( − , − ) r, + (cid:34) mr − iω − ikr (cid:112) h ( r ) + 3 r f (cid:48) ( r ) (cid:35) ψ (+ , +) r, + 2 h (cid:48) ( r ) h ( r ) (cid:16) ψ (+ , +) v, + Γ v ψ ( − , − ) v, (cid:17) = 0 . (A7b)In fact (A7a) is presented in the main text as (14).In order to find the locations of pole-skipping points,we need to look for the values of ω and k at which (A7)and their higher-order analogues do not impose enough constraints on the solutions of the equations of motionto uniquely determine the retarded Green’s function ofthe boundary theory. The determination of the lead-ing pole-skipping point is described in the main text(see the discussion around equation (15)), but in factthere also exists an infinite tower of pole-skipping pointsat negative imaginary fermionic Matsubara frequencies ω Fn = − πiT ( n − /
2) where n = 1 , , . . . .The procedure of identifying such locations mimics theanalysis presented for the fermion case [30], but withsome additional caveats, and wont be discussed in detailhere. We just report some partial results which explainthe additional poles found in the numerical analysis thatwe present in FIG. 2. In order to find the pole-skippingpoints at ω F = − iπT , one needs to analyse the equationsat linear order in the expansion (A6) and the locationsare obtained by looking for points at which a linear com-bination of ψ (+ , +) v, and ψ ( − , − ) v, represents an additionalfree parameter of the solutions to the equations of mo-tion. We find that for a generic value of the mass m there are three pole-skipping points at this value of thefrequency.Pole-skipping points at ω F = − iπT are perhapsmore interesting as they arise due to both ψ ( α ,α ) v, and ψ ( α ,α ) r, . In fact, one finds that two special points coin-cide at ω = − i r f (cid:48) ( r ) = − iπT , (A8a) k = − im (cid:112) h ( r ) = − iπd + 1 mT , (A8b)and are thus responsible for the additional two poles thatcan be seen in FIG. 2. A simple way to confirm that thisis indeed the case is to insert these values into (A7) andobserve that the prefactors in the square brackets (A7b)vanish, hence leaving ψ (+ , +) r, and ψ ( − , − ) r, unconstrained.Furthermore, one finds that at these values of the fre-quency and momentum (A7a) and (A7b) give equivalentconstraints. We have not investigated the significanceof this “double” pole-skipping point (see the analysis of[33]), but we attribute this occurrence to the interplaybetween the ψ r and ψ v components in the equations ofmotion. [1] J. M. Maldacena, “The Large N limit of superconformalfield theories and supergravity,” Int. J. Theor. Phys. (1999) 1113–1133, arXiv:hep-th/9711200 [hep-th] .[Adv. Theor. Math. Phys.2,231(1998)].[2] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov,“Gauge theory correlators from noncritical stringtheory,” Phys. Lett.
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