Pole-skipping with finite-coupling corrections

Recently, it is shown that many Green’s functions are not unique at special points in complex momentum space using AdS/CFT. This phenomenon is similar to the pole-skipping in holographic chaos, and the special points are typically located at ωn = −(2πT )ni with appropriate values of complex wave number qn. We study finite-coupling corrections to special points. As examples, we consider four-derivative corrections to gravitational perturbations and four-dimensional Maxwell perturbations. While ωn is uncorrected, qn is corrected at finite coupling. Some special points disappear at particular values of higher-derivative couplings. Special point locations of the Maxwell scalar and vector modes are related to each other by the electromagnetic duality.

According to these works, many Green's functions are not unique at "special points" in complex momentum space (ω, q), where ω is frequency and q is wave number. Such a phenomenon is collectively known as "poleskipping" [13][14][15][16] and was originally discussed in the context of holographic chaos [17][18][19][20].
Main results drawn from recent works are • Various Green's functions exhibit this phenomenon. In addition to the gravitational sound mode (energy density correlators) which was originally discussed in holographic chaos, the bulk scalar field, the bulk Maxwell field (the current and charge correlators from the boundary point of view), the gravitational shear mode (momentum correlators) and the tensor mode show this behavior.
• There is a universality for ω. In all examples, special points are located at Matsubara frequencies.
Typically, they start from w := ω/(2πT ) = −i and continue w n = −in for a positive integer n 1 . For the sound mode, special points start from w −1 = +i. It is argued that the w −1 = +i special point is related to a chaotic behavior. On the other hand, the value of q n depends on the system.
The appearance of Matsubara frequencies ω n = −(2πT )ni is intriguing, but this is a strong coupling result. The appearance may be an artifact of the strong coupling limit. The purpose of this paper is to study finite-coupling corrections (higher-derivative corrections from the bulk point of view) to special points.
For the gravitational sound mode, there is a special point in the upper-half ω-plane, w −1 = +i. Higherderivative corrections to the special point have been discussed in Ref. [15].
It is argued that this special point is related to a chaotic behavior. It is conjectured that a holographic system has the maximum Lyapunov exponent λ L = 2πT [20]. It is also argued that higher-derivative corrections do not change the Lyapunov exponent. Thus, higherderivative corrections should not change the w −1 = +i special point. Ref. [15] confirms this in the context of pole-skipping. (The butterfly velocity or q −1 gets corrections.) Higher-derivative corrections to special points have been studied to some extent for the sound mode but have not been explored for the other perturbations which exhibit the pole-skipping. We study higher-derivative corrections to these "non-chaotic" special points. As examples, we consider four-derivative corrections to • pure gravity, • Einstein-Maxwell theory (in the four-dimensional neutral background).
We study the corrections to the first few special points and its implications. The main purpose is to show the universality of w n = −in. Higher-derivative corrections do not affect w n = −in but affect q n . In addition, • Special points may disappear at a particular coupling (Sec. III D).
• The four-dimensional Maxwell theory has the electromagnetic duality [21,22]. The duality has an interesting consequence to the pole-skipping: special point locations of the Maxwell scalar and vector modes are related to each other (Sec. IV C). We also comment on the relation between this property and a previous observation in Ref. [23].
• Ref. [11] introduced the notion of "anomalous points," and we make a few remarks (Sec. IV D).

II. POLE-SKIPPING
In this section, we briefly review Refs. [10][11][12]. We use the incoming Eddington-Finkelstein (EF) coordinates. For the Schwarzschild-AdS p+2 (SAdS p+2 ) black hole, the metric is given by 2 using the tortoise coordinate dr * := dr/F and v = t + r * . For simplicity, we set the AdS radius L = 1 and the horizon radius r 0 = 1. We consider the perturbations of the form As usual, we impose the incoming-wave boundary condition at the horizon.

A. Power series expansion
As a typical example of special points, consider the field equation of the form (2.3a) The horizon r = 1 is a regular singularity, and P and Q are expanded as in the EF coordinates. One typically has P −1 = 1 − iw and Q −1 = Q −1 (w, q 2 ), where w := ω/(2πT ) and q := q/(2πT ). The field equation takes this form, e.g., for 2 We use upper-case Latin indices M, N, . . . for the (p + 2)dimensional bulk spacetime coordinates and use Greek indices µ, ν, . . . for the (p + 1)-dimensional boundary coordinates. The boundary coordinates are written as x µ = (t, x i ) = (t, x) = (t, x, y, z · · · ).
• the bulk scalar field, • the bulk Maxwell field (scalar mode and vector mode), • the gravitational perturbations (tensor mode and shear mode).
We mainly focus on these perturbations, where field equations take the form (2.3).
The solution can be written as a power series: Substituting this into the field equation, one obtains the indicial equation at the lowest order: The coefficient φ n is obtained by a recursion relation. The λ 1 (λ 2 )-mode represents the incoming (outgoing) mode, and we choose the incoming mode λ = λ 1 = 0 hereafter. In the incoming EF coordinates, the incoming wave is a Taylor series. The λ 2 -mode is not a Taylor series for a generic w.
The situation changes when iw is a nonnegative integer. Then, the λ 2 -mode is also a Taylor series naively. But λ 1 and λ 2 differ by an integer. In such a case, the smaller root fails to produce the independent solution since the recursion relation breaks down at some φ n . Instead, the second solution would contain a ln(r − 1) term and is not regular at r = 1.
However, this log term disappears for special values of q. Therefore, one has two regular solutions at iw n = n with appropriate q n . Such a point is called a "special point" or a "pole-skipping point." In order to obtain (w n , q n ) systematically, write the rest of the field equation in a matrix form [11]: (2.7) Here, In particular, M n,n+1 = n(n − 1 + P −1 ) = n(n − iw). The matrix M (n) is obtained by keeping the first n rows and n columns of M . The special points at iw n = n are obtained from det M (n) (w n , q n ) = 0 . (2.9) For example, consider the first row: Normally, this equation determines φ 1 from φ 0 . However, when M 12 = M 11 = 0, both φ 0 and φ 1 are free parameters. The former condition gives M 12 = P −1 = 1 − iw = 0. The latter condition is M 11 = Q −1 = 0. Since Q −1 contains q 2 , there are 2 solutions of q and 2 special points. The horizon r = 1 is a regular singularity, but the horizon becomes a regular point at (w 1 , q 1 ) because P −1 = Q −1 = 0. Ref. [12] uses this criterion to find (w 1 , q 1 ). Also, λ 2 = 1 at w 1 , so the extra regular solution is the "outgoing" solution we did not select previously.
Similarly, when M 23 = det M (2) = 0, φ 0 and φ 2 become free parameters. The former condition gives M 23 = 2(2 − iw) = 0. The latter condition is a degree-4 polynomial in q since M ij contains q 2 . Thus, there are 4 solutions of q and 4 special points. One gets As is clear from this analysis, the appearance of Matsubara frequencies iw n = n comes from λ 2 −λ 1 = iw, and this is the consequence of the field equation of the form (2.3). We pay attention to this point when we examine field equations with higher-derivative corrections.

B. Nonuniqueness
At a special point, the bulk solution is not unique in the sense that it is parametrized by φ n /φ 0 . This is also written by the "slope dependence" δq/δw. Consider the φ n -equation and expand near the special point w = w n + δw and q = q n + δq. The field equation becomes The solution depends on φ n /φ 0 and this is written in terms of δq/δw how one approaches the special point. As a result of nonuniqueness of the bulk solution, the boundary Green's function is not unique. Generically, one would write a Green's function as (2.14) Near the special point, the Green's function takes the form G R = δω(∂ ω b) n + δq(∂ q b) n + · · · δω(∂ ω a) n + δq(∂ q a) n + · · · (2.15a) = (∂ ω b) n + δq δω (∂ q b) n + · · · (∂ ω a) n + δq δω (∂ q a) n + · · · , (2.15b) and the Green's function at the special point is not uniquely determined. Rather, it depends on the slope δq/δω 3 .

C. Tensor decomposition
We consider Maxwell and gravitational perturbations of the form e −iωv+iqx . We consider these perturbations in neutral backgrounds, so they do not couple to each other. The perturbations are decomposed under the transformation of boundary spatial coordinate x i . The Maxwell perturbations A M are decomposed as For the gravitational perturbations, the tensor mode h yz is gauge-invariant by itself. For the shear mode, the gauge-invariant variables are (2.17b) 3 We should point out that the slope dependence may not take the form δq/δω. In this paper, we consider the slope dependence in a broader sense. It is a little subtle how one writes the slope dependence or how one approaches special points. This is related to "anomalous points" in Ref. [11]. See Sec. IV D for more details.

A. Higher-derivative corrections
Higher-derivative corrections have been widely discussed in AdS/CFT. See, e.g., Refs. [22][23][24][25][26][27][28][29]. AdS/CFT has two couplings, 't Hooft coupling λ and the number of colors N c . The leading Einstein gravity results are the large-N c limit, i.e., λ → ∞, N c → ∞. The 1/λcorrections correspond to higher-derivative corrections or α ′ -corrections. The 1/N c -corrections correspond to string loop corrections or quantum gravity corrections. We focus on the former corrections since the latter is difficult to evaluate in general and little is known.
From string theory point of view, the bulk action is an effective action expanded in the number of derivatives. Schematically, where L i denotes i-derivative terms. L 2 is the leading order Lagrangian: for pure gravity, We focus on the first nontrivial corrections with four derivatives. In general, one should include all possible independent terms 4 . For pure gravity, where α i ∼ α ′ /L 2 ≪ 1 (we set L = 1 below). The values depend on the theory one considers, but we assume that such a theory exists. For example, heterotic string theory does contain such terms. Also, for pure gravity, these are the only possible corrections at O(α ′ ). But in the presence of matter fields such as the Maxwell field and a scalar field, one should include all possible four-derivative terms (see next section for the Einstein-Maxwell theory.)

B. Field redefinitions
For higher-derivative corrections, it is important to take field redefinitions into account. Many coefficients α i are actually ambiguous due to field redefinitions [30]. In the absence of an off-shell formalism, the effective action is derived from the string theory S-matrix (see, e.g., [31]), but the S-matrix does not change under field redefinitions.
As a simple example, consider a pure scalar theory Assume that the scalar has a shift symmetry φ → φ + c so that it appears only as ∇φ in the action. There are 3 four-derivative terms: But under the field redefinition, the leading term changes as so β i change as Note that the field redefinition changes O(α ′2 ) terms as well.
The effective action has ambiguities at higher order, but this does not affect on-shell physics. The field redefinition for example changes the metric but does not change dimensionless physical quantities. Then, what one should do is to eliminate ambiguous terms as many as possible.
This of course simplifies analysis. But, more importantly, one should check whether any nontrivial term is left. If there were none, one would not obtain nontrivial results. Also, we consider linear perturbations in this paper. In such a case, some further terms may be dropped because they are higher order in perturbations.
For the pure scalar theory, there are 3 four-derivative terms and 2 field redefinition parameters. This leaves one term β 3 , but it involves 4 powers of perturbations (in the φ = 0 background), so no nontrivial term is left. Consequently, higher-derivative corrections are trivial for the pure scalar theory. Similarly, the pure Maxwell theory in a neutral background has no nontrivial correction (see Sec. IV). For nontrivial corrections, we analyze pure gravity and the Einstein-Maxwell theory.
Finally, as mentioned above, the field redefinition changes higher order terms in α ′ as well, so the equivalence under the field redefinition holds only at O(α ′ ) perturbatively. For nonperturbative results in α ′ , the equivalence holds only if one takes into account higher order terms in α ′ .
For pure gravity, under the field redefinition with the rescaled cosmological constant (Appendix A): So, one can set α 1 = α 2 = 0. Another choice is the Gauss-Bonnet combination: . This combination is particularly useful because the field equation is at most second order in derivatives. We consider this Gauss-Bonnet correction below.

C. Pole-skipping
For Gauss-Bonnet gravity, the field equation is given by The black hole background of Gauss-Bonnet gravity is obtained in Ref. [32]. In the incoming EF coordinates, The constant N GB is chosen so that the boundary metric takes the form ds 2 = r 2 (−dv 2 + d x 2 p ). The Hawking temperature is given by where we restored the horizon radius r 0 . The other thermodynamic quantities are These quantities can be obtained from the Euclidean computation [25]. Alternatively, one can use the Wald formula and the first law dε = T ds. The entropy obeys the area law for planar Gauss-Bonnet black holes even in the presence of higher-derivative corrections.
We consider the p = 3 tensor perturbation of the form (See Appendix D for p > 3 Gauss-Bonnet gravity.) When λ GB = 0, the tensor mode equation takes the form of a minimally-coupled massless scalar field. With the Gauss-Bonnet term, the tensor mode equation is rather lengthy, so we do not present it explicitly. But recall that special points iw n = n come from λ 2 − λ 1 = iw. In the EF coordinates, the roots λ are obtained from the near-horizon limit of the φ ′′ and φ ′ terms of the field equation. In this limit, where w is normalized by α ′ -corrected temperature. The field equation takes the same form as Eq. (2.3). Thus, the indicial equation gives (λ 1 , λ 2 ) = (0, iw), and the higher-derivative correction does not affect iw n = n. Following Sec. II A, the first few special points are obtained from where q N := N GB q. One then obtains We used the field redefinition to consider the Gauss-Bonnet combination. But as long as the results are expressed at O(α ′ ) perturbatively, special point locations do not change under the field redefinition. See Appendix A for the details. So, consider the generic curvature-squared theories (3.3). For example, iw 1 = 1 special point is given by We confirmed the result explicitly by analyzing special points for Eq. (3.3).
For the shear mode, the field equations can be written as first-order differential equations of gauge-invariant variables. Schematically,  .3), so the higher-derivative correction does not affect iw n = n. The first few special points of the shear mode are In the λ GB → 0 limit, these results coincide with known results.
For the sound mode, the pole-skipping analysis is a little intricate because the field equation does not always take the same form as Eq. (2.3). In this sense, the sound mode is not our main concern, but for completeness and for its importance, we discuss it in Appendix B.

D. Disappearance of special points
So far we discussed higher-derivative corrections in a perturbative framework and presented results to O(λ GB ). Field redefinitions do not affect the results. But in this subsection, we go beyond the perturbative analysis and consider some particular values of λ GB .
Not all values of the coupling are allowed though. The consistency of the dual theory prevents the coupling from becoming very large. As is clear from the metric, λ GB ≤ 1/4, but there is a more stringent constraint from the causality of the dual theory [29]: (3.23) The upper bound reduces to 1/4 in the p → ∞ limit. For p = 3, −7/36 ≤ λ GB ≤ 9/100. One should keep in mind that we truncate the action at O(α ′ ) here. When one considers particular values of λ GB , one can no longer ignore the other higher-derivative terms at O(α ′2 ) and higher. Also, the equivalence under field redefinitions no longer holds. As mentioned in Sec. III B, the field redefinition of the truncated action in general produces the other higher-derivative terms. Thus, statements as rigorous as the perturbative analysis are not possible here. One should regard the truncated action as a toy model. However, going beyond the perturbative analysis, one has a qualitatively new feature. Some special points "disappear" at a particular λ GB .
The first special point is determined by When iw 1 = 1 and M 11 = 0, both φ 0 and φ 1 become free parameters, and one has two regular solutions. The condition M 11 = 0 is satisfied by choosing an appropriate q 2 . However, at finite coupling, we have one more parameter λ GB . One can fine-tune λ GB so that the q 2coefficient of M 11 vanishes. Then, M 11 = 0 and φ 0 must vanish. As a result, there is a unique regular solution. In fact, M 11 = Q −1 = 0, so the horizon remains a regular singularity: another solution should not be regular, and one expects a ln(r − 1) solution.
For the tensor mode, M 11 = det M (1) is given in Eq. (3.18a), and the q 2 -coefficient vanishes at The special point w 1 disappears there. This lies inside the bound (3.23). Since we use the truncated action, the precise value of λ × is likely to change by the other higher-derivative corrections.
For the shear mode, the special point w 1 does not disappear inside the bound (Appendix C). Actually, for theories considered in this paper, only the tensor mode with p = 3, 4 has a disappearance point inside bounds. The disappearance is a new interesting phenomenon, and it can occur in principle. But combined with such bounds, the disappearance does not seem to occur frequently. Special points may be protected from disappearance.
The disappearance is particularly interesting if it occurs in the sound mode of gravitational perturbations because its special point is related to a chaotic behavior. The sound mode has the first special point at w −1 = +i which reflects the maximum Lyapunov exponent λ L = 2πT .
The higher-derivative correction to the w −1 special point has been discussed in Ref. [15] for the p = 3 Gauss-Bonnet gravity (see Appendix B for p ≥ 3 Gauss-Bonnet gravity). The special point is corrected as (3.26) The result is valid nonperturbatively in λ GB . Since N GB = 0, the disappearance of the special point does not occur in the sound mode of Gauss-Bonnet gravity. However, it would be interesting to examine whether the disappearance of the sound mode special point never occurs or not even if one uses the other higher-derivative corrections. Also, if it occurs, it would be interesting to study its implication to chaos. The out-of-time-ordered correlators (OTOC) are often used to study quantum many-body chaos, and it would be interesting to look at OTOCs at the disappearance point λ × .

A. Higher-derivative corrections and field redefinitions
In this section, we consider the four-dimensional Einstein-Maxwell theory: In the absence of sources, the four-dimensional bulk Maxwell theory is (Hodge) self-dual. From the boundary point of view, the duality is interpreted as "particlevortex" duality [21]. As we see below, the self-duality has an interesting implication to the pole-skipping. Thus, we consider a neutral black hole background. Then, the Maxwell perturbations decouple from gravitational perturbations. The background is the SAdS 4 black hole: The Hawking temperature is given by 2πT = 3/2. Again we consider all possible four-derivative terms. In the Einstein-Maxwell theory, there are 6 new independent terms [22,27,33]: where F 4 := F AB F BC F CD F DA . First, we reduce the number of terms by field redefinitions. There are 6 new terms in the action and 3 new field redefinition parameters (Appendix A). This leaves 3 terms in the action: one can choose α 5 , α 6 , and α 7 . Second, the Maxwell field has no background. α 5 and α 6 terms involve 4 powers of perturbations, so they do not contribute to linear perturbation problems. For the pure gravity part, the fourdimensional Gauss-Bonnet term is a total derivative, so one can ignore them.
Therefore, we end up with the Maxwell theory with only one nontrivial correction (in the SAdS 4 background): Instead, one often uses where C ABCD is the Weyl tensor: This does not affect perturbative analysis because these two corrections are related by field redefinitions. Ref. [22] introduces this higher-derivative correction to break the self-duality of the Maxwell theory. We consider how the correction affects special points of the Maxwell theory. It is convenient to write the action in a general form: where The field equation is given by The field equation is at most second order in derivatives for Maxwell perturbations. We first consider the vector mode A y e −iωv+iqx . The special points iw n = n come from λ 2 − λ 1 = iw. In the EF coordinates, the roots λ are obtained from A ′′ y and A ′ y terms of the field equation, so it is enough to focus on this part of the field equation. The vector mode equation can be written as 10a) Near the horizon r = 1, F (r) ∼ 4πT (r − 1), and G(r) ∼ 1 + 4γ which is nonvanishing from Eq. (4.14) below. So, the field equation is approximately given by The field equation takes the same form as Eq. (2.3). Thus, the correction γ does not affect iw n = n. Following Sec. II A, the first few special points of the Maxwell vector mode are The scalar mode can be analyzed in a manner similar to the shear mode in Sec. III C. The first few special points are In the γ → 0 limit, these results coincide with known results. Note that • When γ = 0, the vector and scalar modes have special points at the same locations.
• To O(γ), the scalar mode special points are obtained from the vector mode ones by γ → −γ.
Just like pure gravity, one may consider particular values of γ. The dual theory respects causality [22] if |γ| ≤ 1 12 . (4.14) For the Maxwell vector and scalar modes, the first special points w 1 do not disappear inside the bound (Appendix C).

C. Electromagnetic duality
The special point locations of the Maxwell vector and scalar modes are related to each other. This is understood from the duality of the four-dimensional bulk Maxwell theory.
Thus, the conductivity is constant and is frequencyindependent: The self-duality has an interesting consequence to special point locations. Suppose that K T has a special point (ω n , q n ) and is not unique there. In order to retain Eq. (4.17), K L is not unique there as well. For example, Ref. [12] obtained the Green's functions at (ω 1 , q 1 ) which satisfy Eq. (4.17) 5 .
When γ = 0, the self-duality is lost. But one can still construct a dual theory and the correlators still satisfy some relations [22]. Add the following term in the action (4.6) and perform the functional integrals over F MN andÂ M .
Here, ǫ 0123 = √ −g. The duality comes from the functional integrations in two different orders. Performing the integral overÂ M gives the Bianchi identity ǫ ABCD ∂ B F CD = 0, which implies F MN = ∂ M A N − ∂ N A M . What remains is the standard Maxwell theory with functional integral over A M .
Instead, if one integrates out F MN first, the resulting action is given bŷ whereĝ 4 := 1/g 4 , andF MN := ∂ MÂN − ∂ NÂM . Also, The correlators of the original theory and the dual theory satisfy The"self-energies" Π used in Ref. [12] is related to K as Π = √ k 2 K. While q 1 = 0, note that discussion here is different from the last paragraph one. In fact, K T (w 1 , q 1 ) = K L (w 1 , q 1 ). We define the Green's function at the special point by the limit δw, δq → 0. What is really meaningful here is the q = 0 expressions.
For the standard Maxwell theory,X CD AB = I CD AB , so the theory is self-dual. When γ is small, one can show that so the dual transformation maps γ → −γ to O(γ). Then, from Eq. (4.22), K T andK L have a special point at the same location (ω n , q n (γ)). Because the dual transformation maps γ → −γ, K L has a special point at (ω n , q n (−γ)). When q = 0, Eq. (4.22) implies that the conductivities of the dual theory pair are the inverse of each other: .
So, the poles and zeros of σ are interchanged in the dual theory. Ref. [23] studies these poles and zeros since they are equally important. In the limit γ → 0, the poles and the zeros approach each other in the complex ω-plane. They "annihilate" at Matsubara frequencies since σ must be constant [23]. In retrospect, what they observed is a precursor of the pole-skipping: they study the overlaps of poles and zeros. They do not see nonuniqueness however because they take q = 0 first.

D. Comments on "anomalous points"
Ref. [11] introduced the notion of "anomalous points," and we make some remarks. At a special point, a Green's function is not unique, but at an anomalous point, the Green's function does not take the "pole-skipping form," namely it is not written as δq/δw.
When γ = 0, q 1 = 0 (and q 2,1 = 0). This is an example of anomalous points 6 . But, first of all, the Green's function is not unique at (w 1 , q 1 ). Ref. [12] explicitly shows that the Green's function depends on δ(q 2 )/δw. However, one would write the Green's function in terms of δq/δw and assume a finite δq/δw. Then, 25) and the slope dependence vanishes. Namely, whether a special point is anomalous or not is merely the matter of how one approaches the special point. Moreover, we saw that q 1 = 0 in the presence of the higher-derivative correction. While q 1 is an anomalous point in the large-N c limit, it is no longer true at finite coupling. At anomalous points, a Green's function is not written as δq/δw but is not uniquely determined.
In our opinion, it is not really necessary to distinguish anomalous points from the other special points.
If one uses expressions of Sec. II A, anomalous points satisfy both Eq. (2.9) and ∂ q det M (n) (w n , q n ) = 0 .
(4. 26) In such a case, the first term of Eq. (2.13) vanishes, so the solution does not depend on δq/δw. But one could equally expand the equation in terms of q 2 and may replace the first term by ∂ q 2 det M (n) (w n , q n ) δ(q 2 ), which may not vanish. For example, for the Maxwell vector mode, det M (1) = M 11 ∝ q 2 .

V. MORE ON THE UNIVERSALITY
Many Green's functions are not unique at Matsubara frequencies, and we have shown that this is valid even at finite coupling, but our analysis is far from complete. If one focuses on the universality of iw n = n, one can consult previous works on higher-derivative corrections.
One often uses the Schwarzschild coordinates, so note the relation between the EF coordinates and the Schwarzschild coordinates. In the EF coordinates, we consider the perturbation e −iωv φ ∼ e −iωt (r − 1) −iw/2 φ and incoming: φ ∼ 1 , On the other hand, in the Schwarzschild coordinates, one sets e −iωt φ and incoming: φ ∼ (r − 1) −iw/2 , Either way, a special point arises when λ 2 − λ 1 = iw is a nonnegative integer, where w is normalized by α ′corrected temperature. Thus, in the Schwarzschild coordinates, special points iw n = n eventually come from the well-known results There is a large literature of higher-derivative corrections, and we list only a few. One can see iw n = n from the following works but cannot see how q n is corrected: • Ref. [26] considers the N = 4 SYM which has O(α ′3 ) corrections and analyze the p = 3 shear and sound modes.
• Ref. [25] considers the p = 3 Gauss-Bonnet gravity and analyze the tensor, shear, and sound modes, and our result of the universality is implicitly known from this work.
• Ref. [29] considers Gauss-Bonnet gravity in arbitrary dimensions and analyze the tensor, shear, and sound modes. This reference provides the master equations for these modes. While the near-horizon behavior is not explicitly stated, one can show λ 2 − λ 1 = iw from their formulae and can carry out the pole-skipping analysis. In Appendix D, we list a first few special points.
• Ref. [22] considers the p = 2 Einstein-Maxwell theory in a neutral black hole background and analyze the Maxwell vector perturbation, and our result of the universality is implicitly known from this work.
• Refs. [27,28] consider the p = 3 Einstein-Maxwell theory in a charged black hole background and analyze the tensor mode.
In the Schwarzschild coordinates, if the solution with exponent −iw/2 exists, the time-reversal symmetry of gravity guarantees the existence of the solution with exponent +iw/2. What is nontrivial is that the difference is an integer. It is useful to write the field equation in the form of Schrödinger equation. Use the tortoise coordinate r * and define a new field φ =: G(r)ϕ. By choosing G(r) appropriately, the field equation becomes Incidentally, one often uses this form to derive the bound on couplings such as Eqs. (3.23) and (4.14). The effective potential V (r) typically behaves as V ∼ (r − 1) near the horizon. Then, the near-horizon solution is where 4πT r * ∼ ln(r − 1). Thus, the near-horizon behavior (5.1) follows from the following assumptions: 1. The background is static.
2. There exists a master field φ and its field equation takes the form (5.2).

G(1) is constant.
We are unaware of any general theorem, but not all systems satisfy these assumptions. As an example, consider the Maxwell vector mode with γ = −1/4. The effective potential is given in Eqs. (5.11)-(5.13) of Ref. [22]. When γ = −1/4, V ∼ (constant) which violates the above assumption. In the EF coordinates, the field equation does not take the same form as Eq. (2.3). One can see this from Eq. (4.10). However, γ = −1/4 is outside the bound (4.14). Thus, a generic bulk system does not satisfy the universality. One may need to impose some additional inputs such as the causality of the dual theory. rameters (a 1 , a 2 , a 3 , a 4 , b 2 ), so there are 4 invariant couplings. Two are α 3 and α 7 , and the other two are One would expect that physical results depend only on these combinations of couplings. The field redefinitions also affect two-derivative terms as In order to keep the canonical normalization of L 2 , choose rescaling parameters as Then, the rescaled cosmological constant becomes One expects that dimensionless physical quantities do not change at O(α ′ ) under field redefinitions. But it is not entirely obvious that special point locations (w n , q n ) do not depend on "schemes." For our theories, this can be checked in a few ways.
First, one can explicitly check this. For the p = 3 tensor mode, we explicitly carry out analysis for generic curvature-squared theories (3.3). For the p = 2 Einstein-Maxwell theory, we explicitly carry out analysis both for Eq. (4.3) and for Eq. (4.4).
Second, at a special point, a field φ is not unique. A field redefinition φ =φ + δφ subtracts δφ perturbatively in O(α ′ ) from the field. Under such a perturbative change, the nonuniqueness should remain.
Third, for our theories, Thus, the field redefinition (A1) is just an overall scaling at O(α ′ ) 7 . The scaling can be compensated by an isotropic scaling of x M and L, and the metric returns to the original one. Since the scaling is involved, ω and T can scale in general, but dimensionless quantities such as ω/T do not change.
For the root λ 1 , M n,n+1 = n(n − iw), so the potential pole-skipping points are iw n = n. But in this case, w and q are related, so it is not always possible to satisfy pole-skipping conditions. One can check that the poleskipping condition is satisfied only for n = 1.
For the root λ 2 , M n,n+1 = n(n + iw), so the potential pole-skipping points are iw n = −n. But Taylor series solutions are possible only for n = 1. Also, the poleskipping condition is satisfied for n = 1. The w −1 special point appears in this way. Imposing Eq. (B1) on these two cases, one obtains Appendix C: Pole-skipping conditions • The p = 3 shear mode: The O(q 2 ) term of Eq. (C1a) vanishes at λ × = 1/4, and det M (1) = −5/2. Actually, one can show that the q 2 -dependence completely disappears from the field equations at λ × = 1/4, but this is outside the bound (3.23), so we do not consider this case further.

Appendix D: Gauss-Bonnet gravity in arbitrary dimensions
Ref. [29] derived the master equations for the tensor, shear, and sound modes of Gauss-Bonnet gravity in arbitrary dimensions. The master equations for the tensor and shear modes take the same form as Eq. (2.3), so the higher-derivative correction does not affect iw n = n.
Special points are obtained from det M (n) (w n , q n ) = 0. For the tensor mode, det M (1) (w 1 ) = M 11 (w 1 ) = (p + 1) 2 C 1 C 2 , where q N := N GB q. The expression of det M (2) is rather lengthy, so we do not present it explicitly. To O(λ GB ),