Scrambling with conservation laws

Quantum information scrambling [1, 2, 3] is a fundamental phenomenon in chaotic many-body systems that has been under wide discussion in recent years. On the theoretical side, this activity focused on the study of out-of-time-order correlators (OTOCs) [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. For a chaotic system with large $N$ number of degrees of freedom per unit volume, the OTOC displays an exponentially increasing deviation from its initial value which is characterized by a quantum Lyapunov exponent. This period of growth occurs after local equilibrium is achieved but before the scrambling time, when the system approaches global equilibrium. Schematically, given simple Hermitian operators $W$ and $V$, one has

$$\text{OTOC}=\langle W(t)V(0)W(t)V(0)\rangle\sim f_{0}-\frac{f_{1}}{N}e^{\lambda_{L}t}+\cdots.$$

(1)

For a large $N$ conformal field theory (CFT) holographically described by Einstein gravity, earlier works [5][6][17][18][19] calculated OTOCs through geometric methods, by studying shockwave geometries. The Lyapunov exponent in such a theory is equal to $\frac{2\pi}{\beta}$, saturating the conjectured chaos bound [20]. For times much larger than scrambling time (the late time regime), the OTOC decays to zero exponentially fast, with a different but related exponent. By contrast, it was shown in [21] that in a random circuit model with local charge conservation law, the OTOC between the charge density operator and a non-conserved operator displays a power law tail at late time. While such power law tails are expected to be generic, they have not yet been seen in holographic systems. This missing piece of physics motivated the present study. Other studies of the interplay between conservation laws, hydrodynamics, and OTOCs include [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

The bulk of this work focuses on the $U(1)$ case: we compute OTOCs between the conserved charge density and a (non-conserved) scalar operator and show that holographic systems also exhibit power law tails consistent with the random circuit result. We also argue that power law tails will be induced by energy conservation. This has not been shown before, but it is important since Hamiltonian systems generically have energy conservation but not charge conservation. In the remainder of the introduction, we review existing holographic calculations, focusing on the scattering approach. Then, in the rest of the article, we show how these calculations are modified due to $U(1)$ charge conservation and we interpret the result physically. We also discuss the extension to energy conservation. Overall, we view this work as a study of the interplay between slow modes, as in hydrodynamics, and the fast dynamics of scrambling.

The holographic calculation of OTOCs for scalar operators is discussed in many works. The approach that we follow here is based on the scattering approach discussed in [6]. In this approach, the boundary OTOC is written as an inner product of $in$ and $out$ asymptotic states.

$$\begin{split}\langle W(t_{1},x_{1})&V(t_{2},x_{2})W(t_{1},x_{1})V(t_{2},x_{2})\rangle=\langle out|in\rangle\\ &|in\rangle=W(t_{1},x_{1})V(t_{2},x_{2})|TFD\rangle\\ &|out\rangle=V^{\dagger}(t_{2},x_{2})W^{\dagger}(t_{1},x_{1})|TFD\rangle.\end{split}$$

(2)

From the bulk perspective, these $in$ and $out$ states may be written in terms of bulk wavefunctions as

$$\begin{split}|in\rangle=\int dp^{u}dx\int dp^{v}dx^{\prime}\phi_{W}(p^{v},x)\phi_{V}(p^{u},x^{\prime})|p^{u},x,p^{v},x^{\prime}\rangle_{in}\\ |out\rangle=\int dp^{u}dx\int dp^{v}dx^{\prime}\phi_{W^{\dagger}}(p^{v},x)\phi_{V^{\dagger}}(p^{u},x^{\prime})|p^{u},x,p^{v},x^{\prime}\rangle_{out}.\end{split}$$

(3)

Here $u$ and $v$ are null coordinates in the black hole geometry dual to the thermofield double state $|TFD\rangle$.

Note that the momentum states are only well-defined near the black hole horizon. We think of the scattering as taking place in the approximately flat near horizon region as described by Kruskal coordinates. This corresponds to the case of large time separation between $t_{1}$ and $t_{2}$, as shown in Fig 1. The wave functions are given by bulk-to-boundary propagators. So

$$\begin{split}&\phi_{W}(p^{v}_{1},x)=\int dve^{ip_{1}^{u}v}\langle\phi_{W}(u,v,x)W(t_{2},x_{2})\rangle|_{u=-\epsilon}\\ &\phi_{V}(p^{u}_{2},x^{\prime})=\int due^{ip^{v}_{2}u}\langle\phi_{V}(u,v,x)V(t_{1},x_{1})\rangle|_{v=\epsilon}\\ &\phi_{W^{\dagger}}(p^{v}_{1},x)=\int dve^{ip^{u}_{1}v}\langle\phi_{W}(u,v,x)W^{\dagger}(t_{2},x_{2})\rangle|_{u=-\epsilon}\\ &\phi_{V^{\dagger}}(p^{u}_{2},x^{\prime})=\int due^{ip^{v}_{2}u}\langle\phi_{V}(u,v,x)V^{\dagger}(t_{1},x_{1})\rangle|_{v=\epsilon}\\ \end{split}$$

(4)

These formulae have a direct interpretation as bulk scattering states sourced by boundary operators. The relevant energy scale is determined by the boundary time separation through the Mandelstam variable $s:=2p_{1}^{v}p_{2}^{u}\sim e^{\frac{2\pi}{\beta}t_{12}}$. For scrambling physics, we are interested in time scales that are larger than the relaxation time. The dominate contribution in this regime comes from summing ladder diagrams with graviton exchanges [36]. To leading order in $s$, the S-matrix approaches a pure phase, obtained from the eikonal approximation,

$$|p_{1}^{u},x,p_{2}^{v},x^{\prime}\rangle_{out}\sim e^{i\delta(s,b)}|p_{1}^{u},x,p_{2}^{v},x^{\prime}\rangle_{in}+\cdots$$

(5)

The physical interpretation of this phase factor is interaction of particle $1$ with a gravitational shockwave sourced by particle $2$ [37][38]. The shockwave metric is

$$ds^{2}=A(uv)du[dv-\delta(u)h(x)du]+B(uv)d^{d}x.$$

(6)

While passing through the shockwave located near $u\sim 0$, the scalar wave function receives a jump in the $v$ coordinate,

$$\int dp_{1}^{u}\phi(p_{1}^{u},x)e^{ip_{1}^{u}h(x)}e^{-ip_{1}^{u}v}=\phi(v-h(x),x).$$

(7)

The phase $\delta(s,b)$ is then identified with the displacement factor $h(x)=G_{N}p_{2}^{v}\frac{e^{-\mu|x|}}{|x|^{\frac{d-1}{2}}}$.

For scalar operators, the OTOC is evaluated in earlier works. In the limit $\Delta_{W}\gg\Delta_{V}\gg 1$, the heavy particle’s wavefunction is not much affected by the shockwave sourced by the light particle. So the amplitude is simply an inner product between $V$ particle wavefunctions with and without the shockwave,

$$\begin{split}\text{OTOC}&\sim\int dvdx\phi_{V}(v,x)\partial_{v}\phi_{V}(v-h(x),x)\\ &\sim\left[\frac{1}{1+\frac{G_{N}\Delta_{W}}{\Gamma}e^{\frac{2\pi}{\beta}t-\mu|x|}}\right]^{\Delta_{V}}.\end{split}$$

(8)

$\Gamma$ is a constant depending on the regularization of the correlator.

At early time, when the second term in denominator is much smaller than the first term, the OTOC is decreasing as shown in Eq. (1). Note that $e^{\frac{2\pi}{\beta}t}$ is roughly the colliding energy, and it enters through $p$ dependence of $h(x)$. The exponent is $\frac{2\pi}{\beta}$, independent of the operator. The correlator has decayed significantly when the second term in the denominator is $O(1)$. After this time, the correlator experiences an expontial decay with an operator-dependent exponent. We refer to this as the late time regime of OTOC.

The situation is changed when we consider an OTOC that involves conserved charges. For example, the R-charge in $\mathcal{N}=4$ SYM theory, and more generally the energy-momentum tensor. We will see that due to the hydrodynamical property of the conserved current, the particle sourced by these operators in the bulk spreads over a large region of space-time. As a result, the collision responsible for scrambling happens at a wide range of space-time points in the classical picture (see Fig 2). When the collision occurs near the horizon, the center of mass energy is large, but when the collision occurs further away from the horizon, the center of mass energy is smaller. This leads to a smearing of the exponential factor in Eq. (8), effectively replacing the OTOC formula with

$$\text{OTOC}\sim\int_{0}^{+\infty}ds\frac{1}{s^{\frac{d}{2}+1}}\left[\frac{1}{1+\frac{c}{N}e^{\frac{2\pi}{\beta}(t-s)-\mu|x|}}\right]^{\alpha}$$

(9)

where $c$ and $\alpha$ are some constants. One can then show that at late time, the OTOC becomes $\sim t^{-\frac{d}{2}}$.

The article is organized as follows. For simplicity, we start with the conserved charge density of a $U(1)$ symmetry. In section 2 we give an expression for the dual photon’s wave function, to lowest order in transverse momentum and frequency. Then we analyze the inner product and equation of motion for the photon field. Using these ingredients and some additional approximations, we calculate the OTOC. In section 3 we interpret these calculations in terms of the classical picture in Fig. 2. In section 4 we discuss the case of the energy-momentum tensor. Finally, we generalize the result in section 5 to the case of generic operators with overlap with the conserved currents.

In an AdS-Schwarzschild black hole background, the metric is

$$ds^{2}=L^{2}\left[-\frac{f(R)}{R^{2}}dt^{2}+\frac{1}{R^{2}f(R)}dR^{2}+\frac{1}{R^{2}}d^{d}\vec{x}\right]$$

(10)

where $R=R_{+}$ is the horizon and boundary is at $R=0$. The inverse temperature is $\beta=\frac{4\pi R_{+}}{d+1}$ and $f(R)=1-(\frac{R}{R_{+}})^{d+1}$. It’s more convenient to use the tortoise coordinate, $r$, defined by

$$dr=-\frac{1}{f(R)}dR.$$

(11)

The domain of $r$ is $r\in(-\infty,0]$ with $-\infty$ corresponding to the horizon and $0$ corresponding to the boundary. In this coordinate, the metric components are $g_{rr}=-g_{tt}=L^{2}\frac{f(R)}{R^{2}}$. Here we use $\vec{x}$ to denote the coordinates of the transverse directions. We also consider a Maxwell field propagating in this geometry whose dynamics determines the behavior of a $U(1)$ current operator $J_{0}$ on the boundary. In this note, we will focus on the four-point correlation function with two insertions of $J_{0}$ and two insertions of a scalar operator $O$.

The Maxwell-Einstein equation is

$$\partial_{\mu}(\sqrt{-g}F^{\mu\nu})=\partial_{\mu}(\sqrt{-g}g^{\mu\rho}g^{\nu\sigma}F_{\rho\sigma})=0.$$

(12)

Consider the ansatz $A_{i}=0$ for all the transverse directions. The Fourier mode decomposition is

$$A_{\mu}(r,t,\vec{x})=\int d\omega d\vec{k}A_{\mu}(r,\omega,k)e^{-i\omega t}e^{i\vec{k}\cdot\vec{x}}$$

(13)

Using the trick in [39], we pick a transverse coordinate frame for each $\vec{k}$, such that the $x$-axis is parallel to $\vec{k}$ direction, with the other axes perpendicular to it. Then $\partial_{x}$ can be replaced by $ik$ when acting on that mode, and derivatives in other transverse direction are replaced by $0$. We also assume that the $A$ field is spherically symmetric and excited by a point source sitting at $\vec{x^{\prime}}=0$, so the momentum space $A$ only depends on norm of $\vec{k}$.

Written in component form, the equations are

$$\Bigg{\{}\begin{array}[]{ccc}\partial_{r}(\sqrt{-g}g^{rr}g^{tt}(\partial_{r}A_{t}-\partial_{t}A_{r}))+\partial_{x}(\sqrt{-g}g^{xx}g^{tt}\partial_{x}A_{t})=0\\ \partial_{t}(\sqrt{-g}g^{tt}g^{rr}(\partial_{t}A_{r}-\partial_{r}A_{t}))+\partial_{x}(\sqrt{-g}g^{xx}g^{rr}\partial_{x}A_{r})=0\\ \partial_{r}(\sqrt{-g}g^{rr}g^{xx}(-\partial_{x}A_{r}))+\partial_{t}(\sqrt{-g}g^{tt}g^{xx}(-\partial_{x}A_{t}))=0.\end{array}$$

(14)

From the second equation, one can deduce

$$\begin{split}-\omega^{2}g^{tt}A_{r}+i\omega g^{tt}\partial_{r}A_{t}-k^{2}g^{xx}A_{r}&=0\\ \implies A_{r}=\frac{i\omega g^{tt}\partial_{r}A_{t}}{\omega^{2}g^{tt}+k^{2}g^{xx}}\end{split}$$

(15)

Then one can use the first equation to find a second order differential equation for $A_{t}$,

$$\partial_{r}(\sqrt{-g}g^{rr}g^{tt}\frac{k^{2}g^{xx}\partial_{r}A_{t}}{\omega^{2}g^{tt}+k^{2}g^{xx}})-k^{2}\sqrt{-g}g^{xx}g^{tt}A_{t}=0.$$

(16)

This equation has two independent solutions. As $r\rightarrow\infty$, they behave like $e^{\pm i\omega r}$, corresponding to out-going and in-falling boundary condition at the horizon. We focus on one of them, since the other one is obtained by complex conjugation. The solution can be found explicitly in the small frequency and small momentum limit by setting $\omega\rightarrow\lambda\omega$, $k\rightarrow\lambda k$, and taking $\lambda\ll 1$. Up to first order in $\lambda$, we have

$$A_{t}(r,\omega,k)=e^{-i\omega r}C(\omega,k)[1+i\omega\int_{-\infty}^{r}dr(1-(\frac{R}{R_{+}})^{d-2})+\frac{ik^{2}}{\omega}\int_{-\infty}^{r}dr(\frac{R}{R_{+}})^{d-2}f+O(\lambda^{2})].$$

(17)

The normalization constant $C$ is fixed by requiring $\lim_{r\rightarrow 0}A_{t}(r,\omega,k)\rightarrow 1$. Then we find that

$$A_{t}(r,\omega,k)=e^{-i\omega r}\left[\frac{1+i\omega H(r)+i\frac{k^{2}}{\omega}R_{+}\frac{1-(\frac{R}{R_{+}})^{d-1}}{d-1}}{1+i\omega H(0)+i\frac{k^{2}}{\omega}R_{+}\frac{1}{d-1}}+O(\lambda^{2})\right]$$

(18)

The factor $\frac{1}{d-1}R_{+}$ is identified as the diffusion constant $D$. $H(r)$ is the indeterminate integral of the second term in Eq. (18). If we ignore the $\omega H(0)$ term in the denominator, then this function has a pole at $\omega=-iDk^{2}$. With this $\omega\sim k^{2}$ scaling, the $\omega H(0)$ is indeed subleading compared to the $ik^{2}D/\omega$ term at small $\omega$ and $k$. Moreover, we can anticipate that $\omega^{2}$ is of order $k^{4}$ after using the residue theorem. Hence, to the leading order in small $k$ and $\omega$, we can neglect $\omega^{2}$ and higher order terms. Another approximation is to set $(\frac{R}{R_{+}})\sim 1$, corresponding to the near horizon region. Since the scrambling time is large, the relevant physics indeed happens within this region. In summary, we have

$$A_{t}(r,\omega,k)\sim e^{-i\omega r}\frac{\omega}{\omega+iDk^{2}}.$$

(19)

The real space form is

$$A_{t}(r,t,\vec{x})\sim\partial_{t}\left[\frac{1}{(t-t^{\prime}+r)^{\frac{d}{2}}}e^{-\frac{|\vec{x}-\vec{x}^{\prime}|^{2}}{4D(t-t^{\prime}+r)}}\theta(t-t^{\prime}+r)\right]$$

(20)

where $t^{\prime}$ and $\vec{x}^{\prime}$ label the position of the boundary source. We expect this expression to hold near the horizon when the transverse separation from the source is large. Now we use Eq. (15) to obtain the other components,

$$\begin{split}A_{r}&\sim\frac{i}{\omega}\partial_{r}A_{t}=A_{t},\\ F_{tr}=\partial_{t}A_{r}-\partial_{r}A_{t}&=\frac{-k^{2}g^{xx}\partial_{r}A_{t}}{\omega^{2}g^{tt}+k^{2}g^{xx}}\sim\frac{ik^{2}}{\omega}g^{xx}g_{tt}A_{t}.\end{split}$$

(21)

Since $g_{tt}\rightarrow 0$ near horizon, $F_{tr}$ is also very small there. (Note that the physical electric field $E_{r}=\sqrt{g^{tt}}\sqrt{g^{rr}}F_{tr}$ is still finite).

The gauge invariant inner product between two gauge field configurations $A_{1}$ and $A_{2}$ is

$$(A_{1},A_{2})=\int\sqrt{h}n^{\mu}({F_{1}}^{*}_{\mu\nu}{A_{2}}^{\nu}-{F_{2}}_{\mu\nu}{A_{1}^{*}}^{\nu})$$

(22)

Where the integration is over a Cauchy surface. Following [6], we choose to integrate on constant $u\sim 0$ slice. Then this inner product becomes

$$\int dvd\vec{x}\left(\frac{R}{L}\right)^{-d}g^{uv}({F_{1}}_{vu}^{*}{A_{2}}_{v}-{F_{2}}_{vu}{A_{1}^{*}}_{v}),$$

(23)

with $u$ and $v$ related to $t$ and $r$ by

$$\begin{split}u=-e^{\frac{2\pi}{\beta}(r-t)},\\ v=e^{\frac{2\pi}{\beta}(t+r)},\\ (\frac{\beta}{2\pi})^{2}g^{uv}=2uvg^{tt}.\end{split}$$

(24)

The various components of gauge field are related to the old ones by

$$\begin{split}A_{v}=&\frac{\partial t}{\partial v}A_{t}+\frac{\partial r}{\partial v}A_{r},\\ =&\frac{\beta}{2\pi}\frac{1}{2v}(A_{t}+A_{r}),\\ F_{uv}=&\frac{\partial t}{\partial u}\frac{\partial r}{\partial v}F_{tr}+\frac{\partial r}{\partial u}\frac{\partial t}{\partial v}F_{rt},\\ =&-(\frac{\beta}{2\pi})^{2}\frac{1}{2uv}F_{tr}.\end{split}$$

(25)

Following Eq. (19) and Eq. (21),

$$\begin{split}A_{v}=\frac{\beta}{2\pi}\frac{1}{v}A_{t}=&\frac{1}{v}v\partial_{v}\left(\int d\omega d\vec{k}\frac{i}{\omega+iDk^{2}}v^{-i\frac{\beta\omega}{2\pi}}e^{i\vec{k}\cdot\vec{x}}\right)\\ \coloneqq&\partial_{v}\phi(v,\vec{x}),\\ g_{xx}g^{tt}F_{tr}=&\int d\omega d\vec{k}\frac{ik^{2}}{\omega+iDk^{2}}v^{-i\frac{\beta\omega}{2\pi}}e^{i\vec{k}\cdot\vec{x}}=-\vec{\nabla}^{2}\phi(v,\vec{x}).\end{split}$$

(26)

Plug this into the inner product expression Eq. (23), we get

$$\begin{split}(A_{1},A_{2})=&\int dvd^{d}\vec{x}\left(\frac{R}{L}\right)^{2-d}[-\nabla^{2}\phi_{1}^{*}(v,\vec{x})\partial_{v}\phi_{2}(v,\vec{x})+\nabla^{2}\phi_{2}(v,\vec{x})\partial_{v}\phi_{1}^{*}(v,\vec{x})]\\ =&\int dvd^{d}\vec{x}\left(\frac{R}{L}\right)^{2-d}[\vec{\nabla}\phi_{1}^{*}(v,\vec{x})\cdot\partial_{v}\vec{\nabla}\phi_{2}(v,\vec{x})-\vec{\nabla}\phi_{2}(v,\vec{x})\cdot\partial_{v}\vec{\nabla}\phi_{1}^{*}(v,\vec{x})]\end{split}$$

(27)

where $\phi(v,\vec{x})$ is the diffusion kernel given in Eq. (26). We will see that the inner product written in this way is very convenient when discussing the shockwave.

The original definition of OTOC is complicated by a UV divergence arising from coincident operator insertions. To avoid this, one can consider a regularized version obtained by inserting operators at different values of imaginary time,

$$\mathcal{C}_{reg}=Tr[\rho^{1-\epsilon}V(t_{2},\vec{x}_{2})W(t_{1},\vec{x}_{1})\rho^{\epsilon}V(t_{2},\vec{x}_{2})W(t_{1},\vec{x}_{1})]$$

(28)

To simplify the discussion, we choose a symmetric regularization. In the following, we will consider the correlator

$$\mathcal{C_{J_{0}O}}=Tr[\rho^{\frac{1}{2}}J_{0}(t_{2},\vec{x}_{2})O(t_{1},\vec{x}_{1})\rho^{\frac{1}{2}}J_{0}(t_{2},\vec{x}_{2})O(t_{1},\vec{x}_{1})].$$

(29)

It has a representation as an inner product of $in$ and $out$ states

$$\begin{split}|in\rangle=O^{L}(t_{1},\vec{x}_{1})J_{0}^{R}(t_{2},\vec{x}_{2})|S-AdS\rangle\\ |out\rangle=J_{0}^{L}(t_{2},\vec{x}_{2})O^{R}(t_{1},\vec{x_{1}})|S-AdS\rangle\end{split}$$

(30)

where $|S-AdS\rangle$ denotes the Schwarzschild-AdS black hole thermal double state,

$$|S-AdS\rangle=\sum_{n}e^{-\frac{\beta}{2}E_{n}}|E_{n}\rangle_{L}|E_{n}\rangle_{R}.$$

(31)

The superscript $L$ and $R$ on each operator means that the excitation is created on the left or right side, respectively. For example, $J^{R}_{0}(t_{2})$ creates a photon on the right. We denote the corresponding gauge field by $A_{\mu}^{R}$. In general it is a linear superposition of in-falling and out-going solutions, depending on the state we are constructing. Following [40][41], we choose the coefficients such that the field has positive Kruskal frequency for in-falling mode and negative Kruskal frequency for out-going mode. For this purpose, we use the following combination

$$A_{\mu}^{R}(r,\omega,k)=(1+n(\omega)){A_{\mu}^{R}}_{\tiny{\text{in-falling}}}(r,\omega,k)-n(\omega){A_{\mu}^{R}}_{\text{in-falling}}^{*}(r,\omega,k),$$

(32)

$n(\omega)$ is the Boltzmann factor.

For the in-falling part, we use the ansatz:

$$\begin{split}{A^{R}_{v}}_{\text{in-falling}}&=\partial_{v}\phi^{R}_{\text{in-falling}}(v,\vec{x}),\\ \phi^{R}_{\text{in-falling}}(v,\vec{x})&=\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(ve^{-\frac{2\pi}{\beta}t_{2}}-1)^{\frac{\beta Dk^{2}}{2\pi}}}\theta(v-e^{\frac{2\pi}{\beta}t_{2}})e^{i\vec{k}\cdot(\vec{x}-\vec{x_{2}})}.\end{split}$$

(33)

This is just rewritten from Eq. (26), except that we have inserted an extra $1$ in the denominator. This change doesn’t modify the long time behavior of the wave function, but it does provide some convenience in the analysis. On the other hand, the out-going part in $A_{v}$ is proportional to $u$. To evaluate the OTOC, we choose to calculate the inner product on the surface $u\sim 0$. Hence, the out-going mode’s contribution can be neglected. The thermal factor $(1+n(\omega))$ is proportional to $\frac{1}{2\sin(\frac{\beta Dk^{2}}{2})}$, evaluated at the diffusive pole¹¹1The actual pole contains a small real part that is higher order in $k$ which keeps the integrand finite.. It suggests the following ansatz for $A_{v}^{R}$

$$\begin{split}A_{v}^{R}&=\partial_{v}\phi^{R}(v,\vec{x}),\\ \phi^{R}(v,\vec{x})&=\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{2\sin(\frac{\beta Dk^{2}}{2})}\frac{1}{(1-ve^{-\frac{2\pi}{\beta}t_{2}})^{\frac{\beta Dk^{2}}{2\pi}}}e^{i\vec{k}\cdot(\vec{x}-\vec{x_{2}})}.\end{split}$$

(34)

A nice property of the above expression is that the real space form of $\phi^{R}$ and $\phi^{R}_{in-falling}$ are related by analytical continuation, if we treat it as a complete solution of equation of motion. To see this, note that

$$\frac{1}{(1-e^{\frac{2\pi}{\beta}(t-t_{2}+r+i0^{+})})^{\frac{\beta Dk^{2}}{2\pi}}}-\frac{1}{(1-e^{\frac{2\pi}{\beta}(t-t_{2}+r-i0^{+})})^{\frac{\beta Dk^{2}}{2\pi}}}=2i\sin(\frac{\beta Dk^{2}}{2})\frac{1}{(ve^{-\frac{2\pi}{\beta}t_{2}}-1)^{\frac{\beta Dk^{2}}{2\pi}}}\theta(v-e^{\frac{2\pi}{\beta}t_{2}})$$

(35)

Similarly, on the left side boundary, the operator $J_{0}^{L}$ sources a wave function $A_{v}^{L}$. By symmetry, $A_{v}^{L}$ is related to $A_{v}^{R}$ via the transformation $(u,v)\rightarrow(-u,-v)$. Correspondingly, we define $\phi^{L}(v,\vec{x})=\phi^{R}(-v,\vec{x})$. Then $A_{v}^{L}$ can be written as

$$\begin{split}A_{v}^{L}&=\partial_{v}\phi^{L}(v,\vec{x}),\\ \phi^{L}(v,\vec{x})&=\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{2\sin(\frac{\beta Dk^{2}}{2})}\frac{1}{(1+ve^{-\frac{2\pi}{\beta}t_{2}})^{\frac{\beta Dk^{2}}{2\pi}}}e^{i\vec{k}\cdot(\vec{x}-\vec{x_{2}})}.\end{split}$$

(36)

The scalar operator $O^{R}$ and $O^{L}$ create scalar mode in the bulk. For simplicity, we assume that the scalar operator $O$ has a large conformal dimension, corresponding to a particle in bulk with a large mass. As a consequence, we can treat the scalar particle semi-classically. As it moves deep into the bulk, the scalar mode carries the shockwave along with it, which modifies the photon’s wave function as we will analyze in the next section. In contrast, we will neglect the back-reaction from the photon on this scalar mode.

Since we are interested in the large time limit of the OTOC, we will take $t_{1}\gg\beta$ and $t_{2}\ll-\beta$. As a result, the geodesic of the scalar particle is approximated by $u=\epsilon\sim 0$. The shockwave geometry is described by a metric that contains a singularity near the horizon,

$$ds^{2}=2g_{uv}du[dv-\delta(u)h(\vec{x})du]+g_{xx}d\Omega_{d},$$

(37)

where $h(\vec{x})=\frac{\Delta_{O}}{N}e^{\frac{2\pi}{\beta}t_{1}-\mu|\vec{x}-\vec{x}_{1}|}$ and $\mu=\sqrt{\frac{2d}{d+1}}\frac{2\pi}{\beta}$.

The Maxwell-Einstein equation in this background is

$$.\begin{split}\partial_{v}(\sqrt{\gamma}F_{uv}g^{uv})+\sum_{i=1}^{d}\partial_{i}(\sqrt{\gamma}g^{ii}F_{iv})=0,\\ \partial_{u}(\sqrt{\gamma}F_{vi}g^{ii})+\partial_{v}(\sqrt{\gamma}F_{ui}g^{ii})+\partial_{v}(\sqrt{-g}g^{vv}F_{vi}g^{ii})=0,\end{split}$$

(38)

where the $i$’s run from $1$ to $d$ and label the transverse directions. The effect of the shockwave is encoded in the $g^{vv}$ component. $\gamma$ is the metric determinant in the transverse directions, so $-g=g_{uv}^{2}\gamma$.

As before, we simplify the equation by approximating $\frac{R}{R_{+}}\sim 1$ such that we can set $\gamma=(\frac{L}{R})^{2d}$ and $g^{ii}=(\frac{L}{R})^{2}$. Then using $g^{vv}=-(g^{uv})^{2}g_{uu}=2g^{uv}h(\vec{x})\delta(u)$, the second equation becomes

$$-\partial_{u}\partial_{i}A_{v}-\partial_{v}\partial_{i}A_{u}-2\partial_{v}(\partial_{i}A_{v}h(\vec{x})\delta(u))=0.$$

(39)

In the region $u<\epsilon$ and $u>\epsilon$, the first two terms are finite. At $u=\epsilon$, we are searching for a solution where $A_{v}$ jumps and $A_{u}$ may contain a delta function $\delta(u)$. On the other hand, from the first equation of Eq. (38), $F_{uv}$ must be finite. Therefore the delta functions in $\partial_{u}A_{v}$ and $\partial_{v}A_{u}$ must cancel each other at $u=\epsilon$. Applying this condition, Eq. (39) can be written as

$$-2\partial_{u}\partial_{i}A_{v}-2\partial_{v}(\partial_{i}A_{v}h(\vec{x})\delta(u))=0.$$

(40)

Integrating over $u$ we obtain

$$\partial_{i}A_{v}(v,\vec{x})|_{u=\epsilon+0^{+}}=\partial_{i}A_{v}(v-h(\vec{x}),\vec{x})|_{u=\epsilon-0^{+}}.$$

(41)

Note that this shift in $v$ doesn’t commute with derivatives in the transverse directions, so this simple shift rule only applies to $\vec{\nabla}A$ instead of $A$ itself. The first equation in Eq. (38) gives the same relation between $F_{uv}$ and $A$ as in Eq. (21). Comparing with Eq. (26), we find that this simple shift rule also applies to $\vec{\nabla}\phi(v,\vec{x})$. Hence,

$$\vec{\nabla}\phi(v,\vec{x})|_{u=\epsilon+0^{+}}=\vec{\nabla}\phi(v-h(\vec{x}),\vec{x})|_{u=\epsilon-0^{+}},$$

(42)

which is the reason to write the inner product in the form of Eq. (27). In summary, after scattering with the shockwave, the wave function sourced by $J_{0}^{R}$ is modified according to the rule Eq. (42). If we choose to evaluate the inner product, Eq. (27), just after the scattering, along $u=\epsilon+0^{+}$, then $\phi_{1}$ and $\phi_{2}$ (in Eq. (27)) should be

$$\begin{split}\vec{\nabla}\phi_{1}(v,\vec{x})&=\vec{\nabla}\phi^{L}(v,\vec{x}),\\ \vec{\nabla}\phi_{2}(v,\vec{x})&=\vec{\nabla}\phi^{R}(v-h(\vec{x}-\vec{x_{1}}),\vec{x}).\end{split}$$

(43)

In this section, we evaluate the inner product in Eq. (27). It turns out that the relevant integral in is easier to do in momentum space. Define

$$\phi(p,\vec{x})=\int dve^{ipv}\phi(v,\vec{x}),$$

(44)

so that

$$\begin{split}\vec{\nabla}{\phi_{1}}(p,\vec{x})=&\int\frac{d^{d}\vec{k}}{(2\pi)^{d}}\frac{-i\vec{k}}{2\sin(\frac{\beta Dk^{2}}{2})}\frac{p^{\frac{\beta Dk^{2}}{2\pi}-1}e^{Dk^{2}t_{2}}}{\Gamma(\frac{\beta}{2\pi}Dk^{2})}e^{ipe^{\frac{2\pi}{\beta}t_{2}}+iph(\vec{x}-\vec{x}_{1})}e^{i\vec{k}\cdot(\vec{x}-\vec{x_{2}})}\theta(p),\\ \vec{\nabla}{\phi_{2}}(p,\vec{x})=&\int\frac{d^{d}\vec{k}}{(2\pi)^{d}}\frac{-i\vec{k}}{2\sin(\frac{\beta Dk^{2}}{2})}\frac{p^{\frac{\beta Dk^{2}}{2\pi}-1}e^{Dk^{2}t_{2}}}{\Gamma(\frac{\beta}{2\pi}Dk^{2})}e^{-ipe^{\frac{2\pi}{\beta}t_{2}}}e^{i\vec{k}\cdot(\vec{x}-\vec{x_{2}})}\theta(p).\end{split}$$

(45)

Note that if we only keep the leading order terms in $k^{2}$, then the term $\sin(\frac{\beta Dk^{2}}{2})$ will cancel with $\Gamma(\frac{\beta}{2\pi}Dk^{2})$. As long as we are considering large transverse coordinate separation, this approximation should be qualitatively correct. Plugging these into Eq. (27) and approximating $\frac{R}{R_{+}}\sim 1$, we obtain

$$\begin{split}&(A_{1},A_{2})=\\ &\frac{1}{\pi}\int dp\int d^{d}\vec{x}\int\frac{d^{d}k}{(2\pi)^{d}}\int\frac{d^{d}k^{\prime}}{(2\pi)^{d}}(\vec{k}\cdot\vec{k^{\prime}})pp^{Dk^{2}-1}e^{Dk^{2}t_{2}}p^{Dk^{\prime 2}-1}e^{Dk^{\prime 2}t_{2}}e^{-2ipe^{t_{2}}}e^{-iph(\vec{x}-\vec{x_{1}})}e^{i(\vec{k^{\prime}}-\vec{k})\cdot(\vec{x}-\vec{x_{2}})}.\end{split}$$

(46)

To save space, we have suppressed the factor $\frac{\beta}{2\pi}$ setting the units of time. After some changes of variable and approximations shown in Appendix B, we obtain the final expression for the OTOC as

$$\begin{split}(A_{1},A_{2})\sim\frac{1}{\left[\ln{\left(2+\frac{\Delta_{O}}{N}e^{\frac{2\pi}{\beta}t_{12}-\mu|\vec{x}_{12}|}\right)}\right]^{\frac{d}{2}}}.\end{split}$$

(47)