Holographic scalar and vector exchange in OTOCs and pole-skipping phenomena

We study scalar and vector exchange terms in out-of-time-order correlators (OTOCs) holographically. Applying a computational method in graviton exchange, we analyse exponential behaviors in scalar and vector exchange terms at late times. We show that their exponential behaviors in simple holographic models are related to pole-skipping points obtained from the near-horizon equations of motion of scalar and vector fields. Our results are generalizations of the relation between the graviton exchange effect in OTOCs and the pole-skipping phenomena of the dual operator, to scalar and vector fields.


Introduction
Quantum field theories (QFTs) which have the gravity duals are in a special class of QFTs, and their properties are studied from various field theoretical and holographic methods. An interesting property of quantum systems which are described by black holes is the saturation of quantum Lyapunov exponent λ L = 2π/β [1][2][3][4][5][6], where β is inverse temperature. The Lyapunov exponent λ L in four point out-of-time-ordered correlators (OTOCs) is a diagnosis of quantum chaos [5,7], and it has been proposed that λ L in quantum many-body systems with reasonable physical assumptions is bounded as λ L ≤ 2π/β [8].
As a connection between transport properties and quantum chaos, it was found that the retarded Green's function of energy density in momentum space contains information of quantum chaos in the quantum systems with the gravity duals [9]. In particular, a pole-skipping point (ω * , k * ) in the retarded Green's function of energy density would be related to the Lyapunov exponent λ L and butterfly velocity v B as follows: (1.1) Now, pole-skipping points of a Green's function G(ω, k) are defined as intersection points between lines of poles and lines of zeros in G(ω, k). This relation is called "pole-skipping phenomena" [9,10] 1 . To compute the pole-skipping points holographically, the near-horizon analysis has been established [12]. By examining special points in the equations of motion (e.o.m.) near the black hole horizon, one can obtain the pole-skipping points.
After discovering the pole-skipping phenomena of energy density, pole-skipping points of other fields such as scalar, vector, and spinor have been investigated holographically [13][14][15][16][17][18][19][20][21][22]. 2 However, the pole-skipping points of lower spin fields ( ≤ 1) do not exist in the upper half plane of complex ω-space, and quantum chaos does not seem to be related to their pole-skipping points. This observation can be interpreted in terms of the holographic computation method for the Lyapunov exponent and butterfly velocity. In holographic models, they can be computed from shock wave geometry which is made with graviton exchange (see, for example, [1,2,4,26,31,32]). Since energy momentum tensor corresponds to graviton in holography, the pole-skipping point of energy density would be related to the graviton exchange term in the OTOCs, which is relevant to quantum chaos in the holographic models. By extending this interpretation to other fields, one can expect the pole-skipping points of them to be related to "exchange terms" other than graviton exchange in the OTOCs.
In conformal field theories (CFTs), these exchange terms correspond to conformal blocks with an analytic continuation for the OTOCs. The pole-skipping points of scalar, vector, and energy density in CFTs on hyperbolic space H d with β = 2π were computed by [20,33], 3 and it was shown that they are related to late time exponential behaviors of the conformal blocks. This result matches the previous expectation and is regarded as a generalization of pole-skipping phenomena to other fields, although the pole-skipping points of scalar and vector fields are not related to maximal chaos.
From the holographic viewpoint, one can compute the pole-skipping points of CFTs on H d with β = 2π by using a d + 2 dimensional AdS-Rindler black hole geometry [20,26].
To study the generalization of pole-skipping phenomena holographically, it is useful to develop a holographic computation method of exponential behaviors in the exchange terms other than graviton exchange. In particular, computations in a planar AdS black hole is important because the pole-skipping points on flat space R d are well-studied compared to the ones on H d .
In this paper, we study the exponential behaviors of scalar and vector exchange terms in the four point OTOCs by using the holographic method. To compute the exponential behaviors, we introduce simple holographic models which have three point interactions. Our computation method is a generalization of computations for the Lyapunov exponent and butterfly velocity from the graviton exchange term. By comparing them with the nearhorizon analysis, we check that the exponential behaviors are related to the pole-skipping points in the retarded Green's function of scalar and vector fields.
The paper is organized as follows. The calculation of exponential behavior in the graviton exchange term is reviewed in Section 2. In Sections 3 and 4, we compute exponential behaviors in the scalar and vector exchange terms and compare them with the pole-skipping points derived from the near-horizon analysis. We discuss our conclusion and future work in Section 5.
We review the calculation of exponential behaviors in the graviton exchange term based on [31,32]. From this calculation, we can obtain the Lyapunov exponent and butterfly velocity in the holographic systems. In the next sections we will generalize this computation for the scalar and vector exchange terms.
For a holographic computation of OTOC , we consider the Einstein-Hilbert action and scalar fields actions: where Λ is the cosmological constant. These actions determine bulk propagators of scalars and graviton. The bulk scalar fields φ W and φ V correspond to the boundary operators W and V in the four point OTOC To compute exponential behaviors in the OTOC holographically, we assume that W is a heavy operator and treat W (t W , x W ) as a source as in [37]. Since φ W is coupled to graviton as in (2.2), the source W (t W , x W ) makes a shock wave geometry [38][39][40] in the bulk side.
As an initial metric before making the shock wave geometry, consider a black hole metric 4 where dx 2 is the squared line element of boundary space M which does not have a periodic direction. In this paper we mainly focus on M = R d and M = H d . The Hawking temperature of this black hole is T = 1/β = U (r 0 )/4π, where r 0 is the horizon radius. By using Kruskal coordinates (u, v) uv = −e U (r 0 )r * (r) , u/v = −e −U (r 0 )t , dr * = dr/U (r), (2.6) we can extend (2.5) to a two-sided black hole metric

7)
At late times t W β, a geodesic between boundaries of the two-sided black hole on the t = t W slice approaches to the horizon u = 0, and an expectation value of energy Figure 1. Three point diagram as a part of the bulk tree-level graviton exchange diagram. Two straight lines represent the bulk-boundary scalar propagators, and a dotted line represents the bulk-bulk graviton propagator. The interaction region is localized around the horizon because of (2.8).
δg µν is localized on the horizon u = 0 [1,2,4,[38][39][40]: where |ψ is a dual state of the two-sided black hole with the source W (t W + iτ, x W ) [1,3], and P is related to the initial asymptotic momentum of the source. Now, we introduce Euclidean time τ for regularization. As we will see in the next sections, the t W -dependence e 2π β t W in (2.8) is related to the spin of exchange fields as e 2π β ( −1)t W , and it is consistent with the late time behavior of conformal block [3,6].
If we assume the holographic correspondence, correlation functions of QFTs can be computed from bulk scattering amplitude [41][42][43]. At late times t W β, momentum of particle around the horizon becomes exponentially large as seen in (2.8), and the bulk scattering is regarded as high energy scattering. Therefore, one can use the eikonal approximation at the late times. In the eikonal approximation with large distance limit, massless graviton ( = 2) exchange is dominant in holographic models [44][45][46].
The Lyapunov exponent and butterfly velocity are defined by the exponential behavior of sub-leading term in OTOCs which corresponds to the bulk tree-level graviton exchange diagram. To derive the exponential behavior of the tree-level diagram, let us focus on a three point diagram as shown in Fig. 1. This three point diagram corresponds to a bulk three point function at tree-level with two W and metric perturbation h µν . Since |ψ includes W , the three point function can be expressed as a classical expectation value ψ|h µν (u, v, x)|ψ . Thus, one can compute the exponential behavior by using classical analysis of the shock wave geometry which is a solution of Einstein equations.
The localized energy momentum tensor (2.8) changes the initial metric (2.7) to the shock wave geometry [38][39][40] 9) and the dynamics of h M g (x) is determined from Einstein equations with (2.8) as follows: where M is the Laplacian on M . An isotropic solution of (2.10) with M = R d is given by [2,31,32] β t W is related to ψ|h uu |ψ and the bulk tree-level graviton exchange diagram as explained above. Assuming no light higher spin fields ( > 2) for holography [47], graviton exchange is the dominant contribution to the sub-leading term in OTOCs. Since the Lyapunov exponent and butterfly velocity can be extracted from the sub-leading term, we obtain λ L and v B in the holographic model with M = R d from the exponential behavior in h R g (x)e 2π β t W as follows: As a relation between quantum chaos and energy dynamics, it was found that λ L and v B in the holographic model are related to a pole-skipping point of energy density derived from the near-horizon analysis [12]. Especially, a component of Einstein equations with ω = iλ L at the horizon has the same form as the left hand side of (2.10). In the next sections we will show that similar phenomena occur in scalar and vector exchange.
We can do the same job on the hyperbolic boundary space M = H d . We set a metric on H d as where a is a length scale of H d , and x ⊥ are transverse coordinates on R d−1 . The Laplacian on (2.13) is given by 14) and the geodesic distance d(x 1 , x 2 ) on (2.13) is defined by In this paper, we use the metric (2.13) with a = 1 as in [20,26,33]. Then the solution of (2.10) with M = H d is given by Thus, the Lyapunov exponent λ L and the butterfly velocity v B in the holographic model with M = H d are Note that the geometry of the boundary space M only affects the butterfly velocity v B not the Lyapunov exponent λ L .

Scalar exchange
In this section we analyze the exponential behavior of scalar exchange term in the four point OTOC by using a simple holographic model. Our computation is a generalization of the method reviewed in Section 2. Specifically, we investigate the exponential behaviors on planar and hyperbolic black holes. We show that the exponential behavior is the same as the one for the leading pole-skipping point derived from the near-horizon analysis of scalar field.

Exponential behavior with scalar exchange
On the background geometry (2.7), we consider actions of scalar fields where S W and S V are given by (2.2) We want to determine t W -dependence of N ϕ (t W ) based on [3]. By integrating (3.5) over v = 0, we obtain Let us first start with the denominator. The norm ψ|ψ is expressed as a Klein-Gordon inner product [3,4] where K(t W , x W ; u, v, x) is a bulk-to-boundary propagator of φ W which is determined from (2.2) on (2.7). Since the black hole metric (2.5) does not depend on t, we assume that the propagator K(t W , x W ; u, v, x) has a time translation symmetry. This assumption means that K(t W , x W ; u, v, x) is a function with respect to t W − t, and we can express β t derived from (2.6). With the assumption and transformation u = e 2π β t W u, we obtain and therefore ψ|ψ does not depend on t W . Next, for the numerator, we estimate By contracting φ W with the boundary operator W in |ψ , we obtain By using (3.6), (3.9), and (3.11), we determine the t W -dependence of N ϕ (t W ) Unlike the exponential boost e + 2π β t W in graviton exchange, the exponential behavior e − 2π β t W in scalar exchange decays at late times t W β. This is the reason why scalar exchange is excluded in computations of the Lyapunov exponent with the eikonal approximation.
With (3.2) and (3.5), an e.o.m. of ϕ is To solve it, we use an ansatz (3.14) Without loss of generality, we set f ϕ (0) = 1. After we put this ansatz into (3.13), the e.o.m. becomes Using uδ (u) = −δ(u) and uδ(u) = 0 [2], we obtain a relation on the horizon u = 0 where B (uv) := ∂ uv B(uv). Note that (3.16) does not depend on derivatives of f ϕ (uv). Furthermore, using expressions on the horizon u = 0 (r = r 0 ): we can express (3.16) as follows: This equation determines the exponential behavior in h M ϕ (x). As an explicit example, an isotropic solution of (3.19) is the SO(d − 1, 1) invariant geodesic distance between x and x W in H d (2.15).

Pole-skipping points of scalar field
As an alternative method to obtain the Lyapunov exponent λ L and butterfly velocity v B , we seek for the pole-skipping points. Pole-skipping points are the points in momentum space which makes the Green's function non-unique: 0/0. One of the method to diagnose the pole-skipping points is the near-horizon analysis. This analysis detect the non-uniqueness of the Green's function by the enhancement of the number of free parameters at the horizon r = r 0 . In this section, we review the near-horizon analysis of minimally coupled scalar field [14] on the general boundary space M . The near-horizon analysis in the planar space (M = R d ) and the hyperbolic space (M = H d ) are well studied, see for instance, [13,15,20].
To perform the near-horizon analysis of the minimally coupled scalar field ϕ, we only consider the action (3.2). Its e.o.m. is where is the Laplacian with respect to the general metric. Using the incoming Eddington-Finkelstein coordinates v EF = t+r * with the tortoise coordinate defined in (2.6), the metric in our purpose is ds 2 = −U (r)dv 2 EF + 2dv EF dr + V (r)dx 2 . (3.23) Using this metric (3.23) and the scalar field perturbation of the form ϕ(v EF , r, x) ∼ φ(r, x)e −iωv EF , the e.o.m. becomes where M is the Laplacian of the general boundary space M , prime is the derivative with respect to r, and we omitted the arguments of scalar field φ(r, x) and U (r), V (r). The e.o.m.(3.24) has a regular singular point at the horizon because U ∼ (r − r 0 ) and V ∼ (r − r 0 ) 0 . The general solutions of a second order differential equation with regular singular points are well known, and we seek for the conditions where the solution φ contains two independent regular solutions. In this case the solution is determined by two parameters so the holographic Green's function becomes a function of the ratio of two parameters, yielding non-unique Green's function.
As a first step, we examine the Frobenius series solution near the horizon r = r 0 , Argument φ 0 in φ(r; φ 0 ) denotes the free coefficient of the series which determines all the other coefficients for a given α. After we insert (3.25) into (3.24), we can find the so-called indicial equation at the lowest power of (r − r 0 ) which determines the α. Solving this indicial equation gives two possible α where iω = iωβ 2π = 2iω U (r 0 ) . Then, the forms of two independent solutions of (3.24) depend on the difference between the two roots of indicial equation : α 2 −α 1 = iω. The solutions of (3.24) are classified as three cases : iω is i) non-integer ii) zero iii) non-zero integer.
There are two Frobenius series solutions for α 1 = 0 and α 2 = iω. As the second solution φ (2) has non-integer exponents (r − r 0 ) iω+n , φ (2) is not regular. Thus, the regularity condition picks up only one solution φ (1) . As the solution can be uniquely determined by the single free coefficient φ (1) 0 , 5 so does the holographic Green's function. 5 Indeed, this parameter can be set to be 1, because the equation is linear.
ii) iω is zero As first solution φ (1) contains log term, it is not regular. By the regularity condition, the second solution φ (2) is only allowed. Thus, the holographic Green's function can be uniquely determined.
iii) iω is non-zero integer 0 . Thus, in this case, the holographic Green's function is not uniquely defined and such points are called pole-skipping points.
The leading (smallest) pole-skipping point is iω = 1 (ω = −i2π/β) and which can be obtained by plugging the form of (3.27) into the e.o.m (3.24). Here, k i is encoded in M as explained in footnote 6. In summary, the pole-sipping conditions are These conditions coincide with the coefficient of the exponential behavior of scalar field e − 2π β t W (3.12) (massive scalar field in near-horizon analysis behaves like ϕ ∼ e −iωv EF = e − 2π β v EF ) and the condition for the spatial part h M ϕ (x) (3.19) obtained by the scalar exchange in the previous subsection.
Alternative way to obtain the leading pole-skipping point is as follows. By plugging (3.25) to the e.o.m. at the near-horizon limit, we can get the expression at the lowest order as One can observe that φ 0 and φ 1 cannot be determined when ω = −i2π/β and M − m 2 ϕ V (r 0 ) − iωdV (r 0 )/2 = 0. These conditions give the leading pole-skipping points of the massive scalar field.
We leave some comments on resemblance and difference between the two analysis methods in this section.
• Both analysis methods depend strongly on the metric at the black hole horizon.
• The procedure to determine the late time behavior e − 2π β t W in Subsection 3.1 depends on the three point interaction (3.3) as in (3.11). On the other hand, in Subsection 3.2, the late time behavior can be determined from the e.o.m. of ϕ only.
• In the four point OTOC with M = R d , space propagation is expressed in terms of isotropic propagation e ik|x| as in (3.21). On the other hand, in the near-horizon analysis, we often use Fourier expansion with e ik i x i instead of e ik|x| . See, for instance, [12].

Vector exchange
Here we study exponential behaviors of vector exchange terms by considering complex scalar fields interacted with a vector field. As well as the scalar case, we show that they are related to the leading pole-skipping points in the near-horizon analysis of vector field.

Exponential behavior with vector exchange
On the black hole background metric (2.7), we consider actions of complex scalar fields and a vector field The bulk complex scalar fields φ W and φ V are dual to the boundary operators W and V in the four point OTOC The bulk interactions (4.5) and (4.6) as three point interaction 7 mean that W and V interact with a boundary vector operator which is dual to A µ .
As (3.5) in the previous section, we consider a localized expectation value at late times t W β for vector exchange and determine t W -dependence of N A µ (t W ). With (4.5), for A u , we obtain where we use the time translation symmetry (3.8). Since (4.8) and ψ|ψ do not depend on t W , we conclude that N A u (t W ) for vector exchange does not depend on t W at late times: Similarly, one can estimate t W -dependence of the other components This difference of the t W -dependence between components seems to be related to different values of ω at the leading pole-skipping points in different channels. With (4.4), (4.7) and (4.10), an e.o.m. of A ν is where we ignore O(e − 2π β t W ) terms. To solve the above equation, we use an ansatz where we set f A (0) = 1. Using uδ (u) = −δ(u) and uδ(u) = 0, it turns out that the e.o.m. of vector field has only δ(u) dependent terms. On the horizon u = 0, the equation with ν = u becomes 8 This relation determines the spatial exponential behaviour in h M A (x). When M = R d , an isotropic solution of (4.13) is at large distance d(x, x W ) 1.

Pole-skipping points of vector field
The action of bulk vector field (4.4) yields the e.o.m.
With the Eddington-Finkelstein coordinate (3.23), the e.o.m. of ν = v EF , r components with the Lorenz condition decouple from the other components ν = v EF , r. Such sector is called diffusive or longitudinal channel and is relevant to the leading pole-skipping points for the vector field [13][14][15]. Thus, we only consider the longitudinal channel in this section. By using an ansatz A µ (v EF , r, x) ∼ A µ (r, x)e −iωv EF , the e.o.m. with ν = v EF , r are where i is the index of the coordinates x, and prime is the derivative with respect to r. We omitted the arguments of vector fields A µ (r, x) and U (r), V (r) for compact expressions.
Using the Lorenz condition For most cases, if A 0 v EF and A 0 r are fixed, the higher order fields A 1 v EF , A 1 r can be determined by (4.21) and (4.22). However, in case of ω = 0 and M − m 2 A V (r 0 ) = 0, A 1 v EF cannot be determined by A 0 v EF from (4.22). It is the enhancement of the number of free parameters. This condition gives the leading pole-skipping points of the vector field and coincide with the exponential behavior of the vector exchange: independence of t W at late times and (4.13) in the previous subsection.

Summary and discussion
We have studied exponential behaviors of scalar and vector exchange terms in four point OTOCs by investigating simple holographic models. We have shown that the exponential behaviors are related to special points in the near-horizon e.o.m. for scalar and vector fields. Let us summarize the results of our calculations.

Scalar field
The exponential behavior in scalar exchange at late times t W β is h M ϕ (x)e − 2π β t W , where h M ϕ (x) is determined by (3.19). In the near-horizon analysis, the leading pole-skipping points are determined by the non-uniqueness conditions of the holographic Green's function in (3.30), and they coincide with e − 2π β t W and (3.19).
Vector field The exponential behavior in vector exchange at the late times does not depend on t W as h M A (x), where h M A (x) is determined by (4.13). This time-independence and (4.13) can be extracted from the near-horizon analysis by imposing that (4.22) is trivial at ω = 0.
Originally, the pole-skipping phenomena in the near-horizon analysis are relations between the exponential behavior in graviton exchange in OTOC and the near-horizon Einstein's equations [12]. Our results imply a generalization of the pole-skipping phenomena for arbitrary bosonic fields. Note that the pole-skipping phenomena of graviton or energy momentum tensor are related to maximal chaos, however the pole-skipping phenomena of other fields are not related to maximal chaos as one can see from the t W -dependence of exponential behaviors.
We comment on some future directions of this work. One future direction is to study more complicated holographic actions which are proposed from the viewpoint of AdS/CMT. For example, the dilaton coupling was considered in [12]. Another future direction is to change the decomposition of vector field in the near-horizon analysis for comparison with the exponential behavior. It may be useful to decompose the vector field in the near-horizon analysis by Kruskal coordinates. It would also be interesting to compute the exponential behavior with exchange of the other channel in vector field although it is a sub-leading term in vector exchange at late times.