Complexity Growth Following Multiple Shocks

In this paper by making use of the"Complexity=Action"proposal, we study the complexity growth after shock waves in holographic field theories. We consider both double black hole-Vaidya and AdS-Vaidya with multiple shocks geometries. We find that the Lloyd's bound is respected during the thermalization process in each of these geometries and at the late time, the complexity growth saturates to the value which is proportional to the energy of the final state. We conclude that the saturation value of complexity growth rate is independent of the initial temperature and in the case of thermal initial state, the rate of complexity is always less than the value for the vacuum initial state such that considering multiple shocks it gets more smaller. Our results indicate that by increasing the temperature of the initial state, the corresponding rate of complexity growth starts far from final saturation rate value.


I. INTRODUCTION
It is claimed that the gauge/gravity duality may shed light on better understanding of the nature of spacetime by providing a relation between entanglement and quantum gravity (see for example Ref.s [1]). In fact, with the help of this duality some certain quantum mechanical quantities can be understood classically by means of the geometry. For instance the entanglement entropy -that provides insight into quantum mechanical interpretations of the gravitational entropy-can be described holographically by minimization of an area of a codimension-two hypersurface in the bulk geometry [2]. One interesting quantum mechanical object is in fact the complexity. In principle the complexity of quantum state which originates from the field of quantum computations is defined by the number of elementary unitary operations which is required to build up a desired state from a given reference state [3].
In order to describe the complexity from the holographic point of view, there are two conjectures made by Susskind et al. The first one pointed out that the black hole interior volume at the gravity side is dual to the complexity of the boundary system at the CFT side [4].
This proposal is known as the "Complexity=Volume" or CV duality. The other one was given in Ref.s [5], in which the computational complexity of a state at time t is connected to the classical bulk on-shell action in the Wheeler-De Witt (WDW) patch. This proposal is known as the "Complexity=Action" or CA duality and is given by where C stands for the complexity of boundary state and I is the on-shell gravitational action of region inside the WDW patch. 1 .
Inspired by the Lloyd's bound [9], in Ref.s [5] it is argued that the rate of the complexity growth of a given state is bounded by the average energy of the state. Lloyd's bound states that the energy in the system puts an upper limit to the rate of computation. The conjectured Lloyd's bound is given by where E is the average energy of the state at a given boundary time t (note we take = 1).
To test this conjectured bound, several works have been done and it was shown that in some certain cases in computation of the holographic complexity, the bound is violated. For example, in the case of Schwarzschild black hole which is dual to a thermofield double state, the holographic complexity violates the bound [10]. In the time dependent Vaidya spacetime, using the CA proposal, the complexity growth has been studied in [11,12]. More recently in [13,14], Myers et al. investigated holographic complexity for eternal black hole backgrounds perturbed by shock waves. Time dependent Vaidya spacetime describes a collapsing thin shell of null matter with arbitrary energy to form a black hole. This geometry leads to quantum quenches of a system at the CFT side where this injection excites the system out of equilibrium and eventually the system evolves towards an equilibrium state [15,16].
In the shock wave geometries dual to perturbations of the thermofield double state, by making use of the CV proposal, the complexity of such states has been studied in Ref. [17].
The main goal of this paper is to further explore the Lloyd's bound when the gravity side 1 Subregion complexity was also defined in [6][7][8].
undergoes the two and many shock waves. In fact in this paper we use CA proposal and generalize the work of Ref. [11] for multiple shocks and also for the black hole initial state.
The rest of this paper is organized as follows. In section 2, we will consider the holographic complexity growth in a double black hole-Vaidya background. In this geometry there is a shock wave in the black hole background which follows a global quench of thermal state and despite the AdS-Vaidya background, the initial state is already thermal (see Ref. [18] for more details). In section 3, we will investigate the Lloyd's bound in time dependent geometry after many shocks. Finally, a brief summary will be presented in section 4. Some details of our holographic calculations can be found in Appendix.

VAIDYA BACKGROUND
In this section we study the holographic complexity growth after a global quench of a thermal state. Basically, in this setup the thermalization starts from non-zero temperature and after this sharp perturbation, the system thermalizes and reaches equilibrium. The holographic dual of this process is provided by an injection of a thin shell of null matter (shock wave) in the black hole background and the corresponding geometry can be well described by a double black hole-Vaidya metric. In other words, in this geometry one deals with a black hole evolving from the initial state with the horizon r h 2 to the final state with the horizon r h 1 . Such an injection causes the temperature of the underlying system increases from its initial value T i to T f . In this section we study the evolution of holographic complexity for double black hole-Vaidya background; this geometry is given by where and R is a typical length scale that we set it to one. The Penrose diagram of this spacetime is shown in Fig. (1).
In order to compute the complexity for the boundary state at time t according to the CA conjecture, we need to find the on-shell action in the corresponding WDW patch; following [11,12] the corresponding patch can be separated into two parts v > 0 and v < 0. Thus, in what follows we want to find I v>0 and I v<0 that are the gravitational action for shaded regions v > 0 and v < 0, respectively and the total on-shell action becomes since the on-shell action of a null dust vanishes. and are labeled by P 1 and P 2 . The null boundary between points B and P 1 which is denoted by r 1 (v, t) satisfies the following integral equation Inserting the explicit form of the f 1 results in The coordinates of the points A, B, and P 1 are It is noted that the coordinate for B is obtained by expanding Eq. (7) around r P ∼ δ that leads to The bulk contribution to the gravitational action is given by the Einstein-Hilbert term.
However, the Ricci scalar is constant in this region and the bulk contribution to I (v>0) becomes proportional to the volume of the corresponding region. Thus, we have where we have defined as the volume of d-dimensional boundary system parametrized by x i , i = 1, · · · , d. Also G N is the Newton's constant and R and Λ are respectively the Ricci scalar and cosmological constant given by Now we should compute the boundary (surface) contributions to I (v>0) . The surface gravity of the null boundaries would be vanished by using the affine parametrization for the null directions, so these null boundaries have no contribution in computing the on-shell action. On the other hand, for the timelike boundary there is no time dependency in the corresponding surface term. Since our ultimate goal is to compute the time derivative of the complexity, we ignore the calculation of the boundary term for the timelike surface.
Therefore, we have to consider the spacelike surface at r = r ∞ . The contribution of this boundary is given by Gibbons-Hawking term. To compute this term, we first consider the spacelike surface at r = r ∞ r h 1 and then take the limit r ∞ → ∞. The future-directed normal vector to the r = r ∞ surface is Thus, the corresponding Gibbons-Hawking term is given by where γ ij ≡ g ij +n i n j is the inverse induced metric on the r = r ∞ surface. The negative sign in Eq. (16) is due to the fact that the shaded region is to the past of the r = r ∞ boundary.
Solving the integrals in Eq. (16) yields Finally, we consider the contributions of the joint points where a null boundary intersects with another boundary. These points are located at A, B and P 1 and their contributions to the on-shell action are given by [19] I joint where √ γ ind = r −d is the determinant of the induced metric of the codimension-two corners, the s i stands for the normal vector of the cut-off boundary (r = δ) which is given by and k in and k out are the null generators which are It is straightforward to show that the joint terms resulted from A and B are in fact timeindependent, thus it is only needed to compute the joint term coming from the P 1 . Therefore, one obtains Thus, the corresponding gravitational action for v > 0 region of Fig. (1) is given by the sum of the contributions from the bulk, surfaces and joint terms which is given by noting that we have dropped the time-independent terms.

B. Calculation of action for v < 0 region
In this region there are three boundaries. One of them is the spacelike boundary at r = r ∞ , the other is the in-falling null shell at v = 0 and the third one is the null boundary that connects points P 1 and P 2 denoted by r 2 (v, t) which is satisfying the following integral Similar to the v > 0 region, the bulk contribution to I (v<0) is given by the Einstein-Hilbert term which is indeed proportional to the volume of the shaded region for v < 0 in Fig. (1).

It is given by
which leads to where we have used R − 2Λ = −2(d + 1).
We now consider the boundary contributions to I (v<0) . As long as the surface gravity of the null generators vanishes, they do not contribute to our calculations. On the other hand, the contribution of the spacelike surface located at r = r ∞ would be time-independent so that we do not take its effect into account in our calculation. Therefore, we only have to focus on the null counter terms. The contribution from the counter terms at P 1 is given by 2 where in this case the null generator k i in is given by Eq. (20) and also one has Therefore, one obtains Putting the results together, the time dependent parts of the gravitational action for the present region becomes (30)

C. On-shell action and complexity growth
The total gravitational action of the WDW patch for the double black hole-Vaidya geometry is given by The important fact is that since we have used the same free parameters in writing the null vectors for both v > 0 and v < 0 regions, the total action is independent of α and β. 3 2 The corner P 2 has no contribution because the volume density of the codimension-two surface falls off as r −d . 3 In general, different free parameters can be used, but by adding proper counter terms, it can be shown that the final result will not change.
The time derivative of the above expression gives us where E is the energy of the equilibrium state in the boundary which is given in terms of the radius of the last horizon as follows and also the corresponding temperature is T = d+1 4π r h 1 . In fact, the expression (32) gives us the bound on the rate of complexity. It is worth mentioning that since r P 1 never crosses the event horizon r h 1 (i.e. r h 1 > r P 1 ), and also r h 2 > r h 1 , it can be concluded that the second and third terms on the right side of Eq. (32) are always negative. Consequently, one can which is in agreement with the Lloyd's bound. On the other hand, at the late time t r h 1 which means r P 1 → r h 1 , the rate of complexity growth eventually saturates and its saturation value becomes 2E π .

III. COMPLEXITY GROWTH AFTER SHOCK WAVES
In this section we are interested in the rate of complexity in a strongly coupled CFT after a sudden change. At the gravity side, despite the previous section, the initial geometry is supposed to be AdS. In the case of one shock, the complexity growth has been studied in [11]. It was shown that the rate of complexity growth saturates the bound soon after the system reaches local equilibrium. The result is as follows with E being the average energy of the state at time t. In this case, one gets d dt C ≤ 2E π and the saturation takes place at late time when r P 1 → r h 1 . In what follows, we extend this consideration for two and more shock waves. (left) and d = 3 (right). In both of the plots r h 1 is set to one. It is clear that by increasing the temperature of the initial state (i.e., for smaller r h 2 ), the rate of complexity growth initiates far from final saturation value.

A. Two shocks
Let us consider black hole-Vaidya with two in-falling shells. At the field theory side, this means the system deals with energy injection twice in the conformal field theory. After first injection the system thermalizes and after the second one, the system gets more energy and hence becomes more complicated. Here, we want to study the rate of complexity due to two quenches.
The metric is still given by (3) where f (v, r) is now defined as follows (we set the AdS radius to one) In this case we have three regions: two black holes and one AdS; we are going to calculate the on-shell action in each region. The mathematical details are almost the same as previous section, so we only write the on-shell action in each region.
• Region one, v > 0: The corresponding on-shell action for region one is already computed in Eq. (23) and is given by • Region two, t 1 < v < 0: In this region there is an additional bulk contribution which is To obtain the contribution due to the joint terms we note that there are two corners P 1 and P 2 which have almost the same contribution with different signs. 4 Thus, one can show • Region three, v < t 1 : It is noted that the Poincare horizon does not contribute to the on-shell action in this region, therefore, it is straightforward to check that the following expression is indeed the on-shell action in this region We already have all ingredients to write the total gravitational action for this case which becomes The time derivative of the above expression leads to where the energy of the final black hole is denoted by E which is the value that the rate of complexity obtained above, finally saturates to at late time i.e. r P 1 → r h 1 . As long as r P 1 ≤ r P 2 ≤ r h 1 ≤ r h 2 , one can check that in this case the log terms are always negative which means the Lloyd's bound is also satisfied.
In the following we will generalize the discussion to n collapsing null shells.

B. Multiple shocks
In this subsection we would like to study the complexity evolution after many collapsing null mass shells. Clearly, we examine the rate of change of the complexity after n shocks.
In this geometry, f (v, r) in the metric (3) takes the following form In this case we are dealing with n + 1 patches, hence the total action is given by where one can write and the on-shell action for the region i (1 < i ≤ n) is given by The collapsing null shells take place at boundary times t 0 , · · · t n−1 . The shaded region denotes the WDW patch corresponding to boundary time t. The intersections of the past null boundary of the WDW patch and the collapsing shells are denoted by P 1 , P 2 , · · · , P n , noting that r h j , j = 1, · · · , n stand for the radius of the horizon in each case. and finally, for the AdS part one has To summarize, by making use of the CA conjecture, the following expression is achieved to study the bound on the complexity growth in the case of the n shocks: where in our notation f n+1 (r) = 1. Similar to the case of two shocks, E is the energy of the final state and in deriving Eq. (48), we make use of the following relation: for j = 1 , f j (r P j ) f j (r P j−1 ) It is easy to check that in this case one also has d dt C ≤ 2E π and the late time saturation value becomes 2E π .
We would like to thank Mohsen Alishahiha for his very kind and generous support and also for his edifying contribution to the content. We would like to acknowledge M. Reza Mohammadi for his useful comments. We also thank A. Akhavan, A. Naseh, F. Omidi M.
Vahidinia and Mostafa Tanhayi for some related discussions. This work has been supported in parts by Islamic Azad University Central Tehran Branch.
APPENDIX: SOME USEFUL FORMULA In this appendix some details of calculation are given. Some details of the time derivative for v > 0 region is presented below: . (52) It is noted that we have dr 1 (v,t) dt = − dr 1 (v,t) dv . For the null boundary in region i (1 < i ≤ n), one can write Taking the time derivative of Eq. (53) results in and setting v = t i−1 in the above expression leads to Moreover, it can be readily checked that