Holographic subregion complexity under a thermal quench

Journal of High Energy Physics, Jul 2018

Abstract We study the evolution of holographic subregion complexity under a thermal quench in this paper. From the subregion CV proposal in the AdS/CFT correspondence, the subregion complexity in the CFT is holographically captured by the volume of the codimension-one surface enclosed by the codimension-two extremal entanglement surface and the boundary subregion. Under a thermal quench, the dual gravitational configuration is described by a Vaidya-AdS spacetime. In this case we find that the holographic subregion complexity always increases at early time, and after reaching a maximum it decreases and gets to saturation. Moreover we notice that when the size of the strip is large enough and the quench is fast enough, in AdSd+1(d ≥ 3) spacetime the evolution of the complexity is discontinuous and there is a sudden drop due to the transition of the extremal entanglement surface. We discuss the effects of the quench speed, the strip size, the black hole mass and the spacetime dimension on the evolution of the subregion complexity in detail numerically.

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Holographic subregion complexity under a thermal quench

Received: March Holographic subregion complexity under a thermal quench Bin Chen 0 1 4 5 6 7 8 Wen-Ming Li 0 1 4 7 8 Run-Qiu Yang 0 1 2 7 8 Cheng-Yong Zhang 0 1 6 7 8 Shao-Jun Zhang 0 1 3 7 8 0 No. 5 Yiheyuan Rd, Beijing 100871 , P.R. China 1 Peking University , No.5 Yiheyuan Rd, Beijing 100871 , P.R. China 2 Quantum Universe Center, Korea Institute for Advanced Study 3 Institute for Advanced Physics and Mathematics 4 Department of Physics and State Key Laboratory of Nuclear Physics and Technology 5 Collaborative Innovation Center of Quantum Matter 6 Center for High Energy Physics, Peking University 7 Zhejiang University of Technology , Hangzhou 310023 , China 8 Seoul 130-722 , Korea We study the evolution of holographic subregion complexity under a thermal quench in this paper. From the subregion CV proposal in the AdS/CFT correspondence, the subregion complexity in the CFT is holographically captured by the volume of the codimension-one surface enclosed by the codimension-two extremal entanglement surface and the boundary subregion. Under a thermal quench, the dual gravitational con guration is described by a Vaidya-AdS spacetime. In this case we nd that the holographic subregion complexity always increases at early time, and after reaching a maximum it decreases and gets to saturation. Moreover we notice that when the size of the strip is large enough and the quench is fast enough, in AdSd+1(d 3) spacetime the evolution of the complexity is discontinuous and there is a sudden drop due to the transition of the extremal entanglement surface. We discuss the e ects of the quench speed, the strip size, the black hole mass and the spacetime dimension on the evolution of the subregion complexity in detail numerically. AdS-CFT Correspondence; Gauge-gravity correspondence - The dependence of subregion complexity evolution on l The dependence of subregion complexity evolution on v0 The dependence of subregion complexity evolution on M 4 Conclusions and discussions 1 Introduction 2 General framework 2.2.1 2.2.2 2.3.1 2.3.2 3.2.1 3.2.2 3.2.3 2.1 2.2 Holographic entanglement entropy Subregion complexity 2.3 Static examples Parametrization v(z) An alternative parametrization z(v) Pure AdS Schwarzschild-AdS black hole 3 Subregion complexity in Vaidya-AdS spacetime 3.1 3.2 Evolution of holographic entanglement entropy Evolution of subregion complexity given task transferring a reference state to a target state [4{6]. However, this manipulation can not directly generalized to quantum eld theory due to the ambiguity in de ning the simple operation and the reference state in a system of in nite degrees of freedom. There have been some attempts trying to give a well-de ned complexity in quantum eld theory [7{16]. { 1 { From the AdS/CFT correspondence, there have been two di erent proposals on holographic complexity, which are referred to as the CV (Complexity=Volume) conjecture [ 1, 2, 17, 18 ] and the CA (Complexity=Action) conjecture [19, 20] respectively. The CV conjecture states that the complexity of a boundary state on a time slice is dual to the extremal volume of the corresponding codimension-one hypersurface B whose boundary is anchored at : CV = max Here Gd+1 is the gravitational constant in AdSd+1 and l is some length scale associated with the bulk geometry, e.g. the anti-de Sitter (AdS) curvature scale or the radius of a The WDW patch is the bulk domain of dependence of a bulk Cauchy slice anchored at the boundary. It is the causal domain of the hypersurface de ned in the CV conjecture. Both CV and CA satisfy important requirements on the complexity such as the Lloyd's bound [21{25]. Like the entanglement entropy, it is also interesting to consider the complexity of a subregion. Instead of a pure state in the whole boundary, it is generally a mixed state produced by reducing the boundary state to a speci c subregion (donated by A). Since the mixed state is encoded in the entanglement wedge in the bulk [26, 27], the subregion complexity should involve the entanglement wedge. In [28] and [29] the CA and CV proposals have been generalized to the subregion situation respectively. For the subregion version of the CA proposal, the complexity of subregion A equals the action of the intersection of the WDW patch and the entanglement wedge [28]. As for the subregion CV proposal, the complexity equals the volume of the extremal hypersurface subregion A and corresponding Ryu-Takayanagi(RT) surface A enclosed by the boundary A [30{33]. Precisely, it can be computed by C A = V ( A ) quantity is the delity susceptibility in quantum information theory [29, 34]. The subregion CV proposal can be understood intuitively from the entanglement renormalization [ 35, 36 ] and the tensor network [37, 38]. The entanglement entropy can be estimated by the minimal number of bonds cut along a curve which is reminiscent of the entanglement curve. Then the holographic complexity can be estimated by the number of nodes in the area enclosed by the curve cutting the bonds [17]. This idea becomes more transparent from the surface/state correspondence conjecture [39] in which the complexity between two states is proportional to the number of operators enclosed by the surface corresponding to the target state and the surface corresponding to the reference state. Obviously the complexity is proportional to the volume enclosed by these two surfaces. This picture has been described in [14, 15] and also in [40]. For other works on the subregion complexity, please see [41{51]. In this paper, we would like to study the subregion complexity in a time-dependent background using the subregion CV conjecture. In particular we compute the evolution of the subregion complexity after a global thermal quench in detail. The quenched system has been viewed as an e ective model to study thermalization both in eld theory and holography [52{56]. On the gravity side, such a quench process in a conformal eld theory (CFT) is described by the process of black hole formation due to the gravitational collapse of a thin shell of null matter, which in turn can be described by a Vaidya metric. The pure state complexity in the same background has been studied analytically under the condition that the shell is pretty thin in [ 57 ]. It was found that the growth of the complexity is just the same as that for eternal black hole at the late time. This paper is organized as follows. In section2, we introduce the framework to evaluate the subregion complexity. In section3, we study holographically the evolution of the complexity after a thermal quench in detail. We summarize our results in section4. 2 General framework In this section, we introduce the general framework to study the subregion complexity in the time-dependent background corresponding to a thermal-quenched CFT. A thermal quench in a CFT can be described holographically by the collapsing of a thin shell of null dust falling from the AdS boundary to form a black hole. This process can be modeled by a Vaidya-AdS metric. The metric of the Vaidya-AdSd+1 spacetime with a planar horizon can be written in terms of the Poincare coordinate rh = m(v) 1=d: { 3 { M 2 m(v) = where v0 characterizes the thickness of the shell, or the time over which the quench occurs. Actually the quench could be taken approximately as starting at 2v0 and ending at 2v0. With this mass function, the Vaidya metric interpolates between a pure AdS in the limit v ! 1 and a Schwarzschild-AdS (SAdS) black hole with mass M in the limit v ! 1. When v0 goes to zero, the spacetime is simply the joint of a pure AdS and a SAdS at v0 = 0. The apparent horizon in the Vaidya-AdS spacetime locates at 1 " z2 ds2 = f (v; z) = 1 f (v; z)dv2 m(v)zd: 2dzdv + dx2 + X dyi2 ; with the time coordinate t on the boundary z ! 0. m(v) is the mass function of the in-falling shell. In the following, we will take it to be of the form (2.1) (2.2) (2.3) We consider the subregion of an in nite strip A = x 2 L ! 1 and nite l. The pro le of the strip in a static AdS background is shown in gure 1. We are going to study the evolution of the subregion complexity of the strip holographically in the Vaidya-AdS spacetime. A As proposed in [29], the subregion complexity in a static background is proportional to the volume of a codimension-one time slice in the bulk geometry enclosed by the boundary region and the corresponding extremal codimension-two Ryu-Takayanagi (RT) surface. This proposal can be generalized to the dynamical spacetime. For a subregion A on the boundary, its holographic entanglement entropy (HEE) is captured by a codimensiontwo bulk surface with vanishing expansion of geodesics [32], i.e., the Hubeny-RangamaniTakayanagi (HRT) surface . The corresponding subregion complexity is then proportional to the volume of a codimension-one hypersurface A which takes A and A as boundaries. Note that A and hence A do not live on a constant time slice in general for a dynamical background. To get the corresponding subregion complexity, we need work out the corresponding extremal codimension-two surface A rst. 2.1 Holographic entanglement entropy Due to the symmetry of the strip, the corresponding extremal surface A in the bulk can be parametrized as HJEP07(218)34 v = v(x); z = z(x); z( l=2) = ; v( l=2) = t ; where is a cut-o . The induced metric on the extremal surface is The area is where L is the length along the spatial directions yi. Since the Lagrangian ds2 = 1 z2 Due to the symmetry of the strip, there is a turning point of the extremal surface locating at x = 0. At this point we have where z ; v are two parameters that characterize the extremal surface. The constant Taking the derivative (2.10) with respect to x and using the equation of motion for z(x), (2.11) Taking the derivative (2.10) with respect to x and using the equation of motion for v(x), we get we get 0 = 2(d 1)f (v; z)2v02 + f (v; z) 2(d 1) + 4(d 1)v0z0 (2.12) HS = 1 v = v(z): A can be solved from (2.11), (2.12) as v = v~(x), z = z~(x). Note that the surface does not live on a constant time slice for general f (v; z). Using the conserved Hamiltonian and the solution, we read the on-shell area of the extremal surface Area( A ) = 2Ld 2 Z l=2 0 z A and A. We nd that there are two equivalent ways to describe A . One way is to parametrizes A by v(z) and the other by z(v). The parametrization v(z) is more intuitive for the static backgrounds, while the parametrization z(v) is more convenient for the dynamical backgrounds. 2.2.1 Parametrization v(z) The bulk region enclosed by the extremal surface v = v~(x), z = z~(x) can be parametrized by v = v(z; x) generically. However, due to the translational symmetry of the Vaidya metric (2.1), the parametrization which characterizes the extremal surface independent of the coordinate x. Thus the extremal codimension-one hypersurface A should be be parametrized by (2.8) A (2.9) (2.10) A where x~(z) is the codimension-two extremal surface A. From the reduced Lagrangian HJEP07(218)34 dz " " This equation can be solved directly with the boundary condition determined by (v~(x); z~(x)) and A. However, there is a recipe for working out the solution A. In fact, we can get a relation v~(z~) from v~(x) and z~(x) by eliminating the parameter x on A . Then the extremal codimension-one hypersurface A can be obtained by dragging v~(z~) along the x direction. We have checked that v~(z~) is indeed the solution of (2.18). For all x on Z z 0 + 2zvzz: (2.15) (2.16) (2.17) (2.18) A = A we (2.19) (2.20) A now is (2.21) The induced metric on The volume is A is one can read the equation of motion f (v; z) 2 f (v; z) z d; This integral is more intuitive for the static background, as we will show below. However, there are situations where z is a multi-valued function of boundary time t. In this case, v~(z) and x~(z) are also multi-valued functions of z. The integral in (2.19) is then ill-de ned. In these cases, we choose another parametrization to describe the extremal codimension-one hypersurface A be parametrized by A enclosed by the extremal surface v = v~(x), z = z~(x) can also due to the translational symmetry of the Vaidya metric. The induced metric on ds2 = where x~(v) is the codimension-two extremal surface . The equation of motion gives z d; with the volume The boundary condition is determined by the codimension-two surface and A. Similar to the above subsection, the solution to eq. (2.23) can be determined by V = 2Ld 2 dv f (v; z(v)) z(v) dx~(v): (2.24) It turns out that z~(v~) is a single-valued function of v~ all the time. Thus the integral in eq. (2.24) is well de ned in the whole process of evolution. We will adopt this formula to calculate the subregion complexity for the dynamical Vaidya-AdS spacetime. De nitely, both eq. (2.19) and (2.24) give the same result when v~(z~) is singly valued. 2.3 Static examples Since the Vaidya metric interpolates between the pure AdS and the SAdS black hole background, let us study the HEE and the holographic subregion complexity in the pure AdS and SAdS backgrounds before we discuss the dynamical Vaidya background. 2.3.1 For the pure AdS, we have f (v; z) = 1. The equations (2.11), (2.12) have a solution Here t is the time coordinate on the boundary. Then eq. (2.10) gives where the plus sign is taken for x < 0 and the minus sign for x > 0. Integrating the above formula gives a relation between z and l. v(x) = t z(x): dz dx = z p s z2d 2 ( 2d1 2 ) = : l 2 { 7 { The on-shell area of the extremal surface reads Area( A)AdSd+1 = L d 2 L l The equation of motion (2.18) or (2.23) for pure AdS can be solved directly as Here t is the time coordinate on the AdS boundary. The on-shell volume reads 1A dz^ where we have used (2.26). Integrating it directly, we get VAdSd+1 = Ld 2 d 1 d l 1 + p Ld 2 z ( 2d1 2 ) d ( 2d1 2 ) (d 1)2 ( 2dd 2 ) ! Note that the divergent term is proportional to the volume of A. From (2.27), (2.32), it is obvious that the nite term has the same dependence of l as the nite part of the HEE. 2.3.2 Schwarzschild-AdS black hole For the Schwarzschild-AdS black hole f (v; z) = f (z) = 1 mzd. The event horizon locates at zh = m 1=d. One can show that the solution to the equations (2.11), (2.12) is v(x) = t + g(z(x)); 1 f (z(x)) : Here t is the time coordinate on the AdS boundary. The conserved Hamiltonian leads to a relation between l and z . (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) VSAdSd+1 = 2Ld 2 Z z 0 dz p 1 1 mzd z d Z z 0s z2d 2 z 1 1 1A ds: (2.38) This is the same as (2.7) in [41]. { 8 { Z z (1 mzd) 1=2 Z l=2 0 dz = dx = The on-shell area of the extremal surface turns out to be Area( A)SAdSd+1 = 2Ld 2 Z z d 1 z z2d 2 (1 + mzd) These two integrals have no explicitly analytical expression. One can verify easily that the solution is and read the on-shell volume of A v = t + g(z); 1 f (z) ; l 2 : 1=2 1 dz: The equation of motion (2.18) about the codimension-one extremal surface A becomes 0 = vz hd 4 (2 mzd)vz( 3 (1 mzd)vz) i We study the evolution of the subregion complexity after a thermal quench in this section. The thermal quench in CFT could be described holographically by the dynamical Vaidya spacetime, whose initial state corresponds to the pure AdS and the nal state corresponds to the SAdS black hole. As we have done for the static cases in the previous subsection, we need rst work out the evolution of the codimension-two extremal surface A. Evolution of holographic entanglement entropy For the Vaidya metric (2.1) with the mass function (2.2), the equations (2.11), (2.12) for We solve these two equations numerically by using the shooting method with the boundary conditions, boundary are v0(0) = z0(0) = 0; z(0) = z ; v(0) = v : Here (v ; z ) is the turning point of the extremal surface . The targets on the AdS A z(l=2) = ; v(l=2) = t where is a cuto and t is the boundary time. divergent, it is convenient to de ne subtracted HEE Once we get the solution, the HEE can be obtained from (2.13). As the HEE is S^HEE = SHEE;Vaidya SHEE;AdS; where both SHEE;Vaidya; SHEE;AdS are de ned with respect to the same boundary region. As we are discussing the strip which has a nite width but in nite length, we furthermore de ne a nite quantity from the subtracted HEE S^ = 4GN S^HEE = 2Ld 2 Area( A)Vaidya 2Ld 2 Area( A)AdSd+1 : Its evolution has two typical pro les as shown in gure 2. The rst pro le appears in the AdS3 case and also in the higher AdSd+1(d 3) cases with narrow strips. In these cases, the HEE in the Vaidya-AdS spacetime increase monotonically and reach saturation at late time. In the left panel of gure 2, we show the evolution of S^ for an interval of length l = 2 in AdS3. The other pro le appears in the higher AdSd+1(d 3) cases with wide strips. We show this pro le in the right panel of gure 2, which corresponds to the strip of width l = 5 in AdS4. Di erent from the rst pro le, this pro le shows that though the { 9 { (3.2) (3.3) (3.4) (3.5) (3.6) 0 4 3 1 0 = 1 and v0 = 0:01 here. The transition point in the right panel locates HEE increases rst as well, it exhibits a swallow tail before reaching the saturation. This phenomenon was rst discovered in [54]. The swallow tail implies that there are multiple solutions to the di erential equationns at a given boundary time. We should choose the one which gives the surface of the minimum area. The solutions which correspond to the surfaces of non-minimum area are marked in grey in the right panel of gure 2. In any case, the HEE is always increasing continuously before reaching the saturation. For more details on the evolution of the HEE after a thermal quench, please refer to [54{56, 58]. In gure 3 we show the corresponding evolution of z . It is multi-valued only when S^ is multi-valued. The multi-valuedness depends on the spacetime dimension and the strip width. In AdS3, z is always singly valued no matter how large l is. However, in the spacetime with dimension d 4, z is singly valued only when l is small. When l is large enough, z becomes multi-valued. For the AdS4 we study here, the critical width is l = 1:6. When z is multi-valued, its evolution is subtle. The multi-valuedness means that there are multiple extremal surfaces at a given time. The requirement [32] that the HRT surface should be of the minimal area leads to the transition at some point. In the right panel of gure 3, the evolution of z follows the line in orange, which has discontinuity. The transition point is at t = 3:3248. The details of the corresponding evolution of the extremal surface A are shown in gure 4 and gure 5. In gure 4, A evolves smoothly from the initial state to the nal -0.5 0.5 1.0 0.0 x HJEP07(218)34 A = (z~(x); v~(x)) for AdS3 and l = 2. We x M = 1 and v0 = 0:01 here. The left panel shows the evolution in (x; v; z). The right panel shows their projection on to the (x; z) plane. The extremal surface evolves from left to right in the left panel and from up to down in the right panel. z 4 AdS4,l=5 3 z2 1 0 -2 -1 1 2 0 x state. In gure 5, the evolution of A has a gap marked in gray before it reaches the nal state. These gray surfaces correspond to the swallow tail in gure 2. They are not the smallest area surfaces at the given boundary times. More precisely, the multi-valuedness not only depends on the spacetime dimension and the size of the strip, but also depends on the parameter v0. In the above discussion, we x M = 1 and v0 = 0:01. As we will show later, the swallow tail would disappear if we choose a large enough v0, which corresponds to a slow quench. 3.2 Evolution of subregion complexity Once we get the HRT surface A = (v~(x); z~(x)), we can determine the codimension-one extremal surface A by dragging the points on A along the x direction, as we have stressed in the subsection 2.2. The evolution of the HEE, the volume of normalized subtracted volume A has the pro le shown in gure 6. Similar to A which can be obtained by (2.19) is divergent, thus we de ne a C^ = 8 RGC 2Ld 2 A = VV aidya VAdS 2Ld 2 (3.7) subregion complexity enclosed by the codimension-two extremal surface A which characterizes the A and A. We take AdS4; l = 5; M = 1 and v0 = 0:01 here. 0.10 AdS3,l=2 0.08 l 0.06 ` C 0.04 0.02 0.00 t where R is the AdS radius which has been set to 1, and the volumes are de ned with respect to the same boundary region. It is nite and can be used to characterize the evolution of the subregion complexity. saturation in the late time. As shown in gure 7, the evolution of the subregion complexity has a common feature: it increases at the early stage and reaches a maximum, then it decreases and gets to Another important feature of the subregion complexity under a global quench is that it may evolves discontinuously, as shown by the orange line in the right panel of gure 7. This is due to the transition of the HRT surface shown in gure 3. As a result, the subregion complexity exhibits a sudden drop in the evolution. The gray dashed part in gure 7 corresponds to the swallow tail in gure 2. In other words, even though the HEE always evolves continuously, the subregion complexity does not. 3.2.1 The dependence of subregion complexity evolution on l The evolutions of the holographic entanglement entropy and the subregion complexity for di erent l are displayed in gure 8. As shown in the lower left panel for the Vaidya-AdS3 spacetime, the subregion complexity increases at the early stage and then decreases and HJEP07(218)34 lower panels show the corresponding subregion complexity density C^=l (thick lines) for di erent l. We x M = 1; v0 = 0:01 here. l=2 AdS4 1 1 l=1 l=2 l=1 l=2 2 t l=3 2 t l=3 l=4 l=4 l=5 3 3 4 4 ` S 0.4 Ai(i = 1; 2; 3; 4; 5) are the entanglement surfaces corresponding to the boundary times t = 0:01; 1:4660; 3:3248; 3:3248 and 3:9373, respectively. A3 and A4 have the same area at boundary time t = 3:3248, which corresponds to the transition point. The red dashed line is the apparent horizon. We x M = 1; v0 = 0:01 here. maintains to be a constant value at late time. The situation in the Vaidya-AdS4 spacetime is similar except that when the size l is large enough, there is a sudden drop of the subregion complexity in the evolution, as shown in the lower right panel. This corresponds exactly to the kink in the evolution of HEE shown in the upper right panel. We plot the transition point in gure 9. The entanglement surface evolves from left to right. Its pro le experiences a transition at time t = 3:3248. The corresponding surfaces area. But the volumes they enclosed are di erent. This leads to a sudden drop of the A3 and A4 have the same subregion complexity. l=12 l=16 1.0 AdS3 ` lC0.6 Remarkably, we nd that the growth rate of the complexity density for di erent l is almost the same at the early stage. This is very similar to the evolution of the entanglement entropy for di erent l. It has been argued that for the geometry of strip, the area of the boundary of the subregion A does not change, so the initial propagation of excitation from the subregion A to outside which contributes to the entanglement is not a ected by the strip width [54]. Since in the early time the complexity density growth is mainly caused by the local excitations, which is independent of l, the same rate of increasing for di erent l could be expected. On the other hand, the nonlocal excitations have important contributions to the subregion complexity at later time such that the evolutions present di erent behaviors. For the cases that l is large enough, we nd that the complexity density grows for a long time before it drops down. The evolutions of the subregion complexities for di erent l in AdS3 are shown in the right panel of gure 10. The complexity presents two increasing stages: it increases faster in the early time, then it increases at a slower rate. At the second stage, it evolves almost linearly, the larger l is, the longer it stands, with the slope being proportional to the mass parameter, Besides, we also notice that, the maximum value of the complexity density in the evolution is proportional to the size l The proportional factor is a function of spacetime dimension d and the mass parameter M . Due to the limitation of our numerical method, the more detailed analysis on the evolution of the complexity for di erent l in AdS4 is absent here. Nevertheless, from the right panel of gure 8, we see that the linear growth in the second stage persists, and the larger the size, the longer the complexity increases. In fact, the linear growth of the complexity has been found in many di erent nonholographic systems [ 1, 60 ] and also appears in the CV and CA conjectures at late time limit [61, 62]. It is also reminiscent of the time evolution of the entanglement entropy from black hole interiors [59]. In our model, if we set l ! 1, we may expect that the behavior of the complexity will turn to the behavior for whole boundary region and so the complexity C^=l / M t: C^max=l / l: l=16 l=20 (3.8) (3.9) -2 -2 0.6 AdS4,l=5 0 0 v0=0.9 v0=0.01 2 t 2 t v0=0.9 v0=0.01 4 4 6 6 C^=l on v0. We take v0 = 0:01; 0:3; 0:6; 0:9 and complexity evolution disappears when v0 > 0:57. x M = 1 here. The sudden drop in the subregion would increase linearly at late time as well. Since the brutal numerical method is not able to study the cases of an extremal large l, one may turn to the analytical way adopted in [ 56 ] to study the linear growth of the subregion complexity. Actually, the recent studies of the complexity following a global quench based on the CA and CV conjectures show that the late time behavior of the complexity for the whole boundary region is linear [ 57, 63 ]. 3.2.2 The dependence of subregion complexity evolution on v0 In this subsection, we study the e ect of the parameter v0 on the evolution of the subregion complexity. The parameter v0 characterizes the thickness of the null-dust shell in the gravity, its inverse could be taken as the speed of the quench. The numerical results are shown in gure 11. All the processes evolve from the pure AdS background to an identical SAdS black hole background. It is obvious that the thinner the shell is, the sooner the quench happens, and the earlier the system reaches equilibrium. The thicker the shell is, the earlier the system starts to evolve, but the maximum complexity the system can reach is smaller. Thus the subregion complexity is closely related to the change rate of a state. Especially, the sudden drop in the complexity evolution disappears when v0 is large enough. For the AdS4 case, the critical point is v0 = 0:57. Namely, if the quench happens slowly enough, the subregion complexity evolves continuously. 3.2.3 The dependence of subregion complexity evolution on M Now we study the e ect of the mass parameter M on the evolution. We x the shell thickness v0 = 0:01 here. The numerical results are shown in gure 12. The system evolves HJEP07(218)34 1 1 2 t 2 t M=1 M=0.5 M=0.25 M=0.05 3 3 M=0.5 M=0.25 M=0.05 4 4 0.6 AdS4,l=5 ` S0.15 x v0 = 0:01 here. Note that there is still a sudden drop of complexity in the evolution when M = 0:05 in the right lower panel. C^max=l is also proportional to l. Thus we have from a pure AdS background to the SAdS black holes with di erent mass M . The maximum complexity C^max the system can reach in the evolution depends on M . For AdS3; l = 2, we get C^max=l / 0:12M . For AdS4; l = 5, we get C^max=l / 0:62M . As we discussed above, C^max=l f (d)M l (3.10) where the coe cient f (d) is a function of spacetime dimension. Unlike the parameter v0, the increases of M can not change the qualitative behavior of the evolution, as shown in the lower right panel. Moreover, the larger the M is, the sooner the subregion complexity reaches the constant value, as shown more obviously in the right lower panel. If we zoom in the nal stage of the evolution shown in the left lower panel in gure 8 and gure 12, we nd that the di erence of the complexity between the initial state and the nal state C^f decreases with M and l. C^f is more involved in the right lower panels in gure 8 and gure 12. In this subsection, we study the dependence of the nal subregion complexity on M and l in detail. For AdS3 in the left upper panel of gure 13, we see that the complexity of the nal state is always smaller than the initial state. The complexity density decreases with M linearly for di erent l and has almost the same rate 0:004M . The situation is more complicated for AdS4 shown in the right upper panel of gure 13. The complexity density decreases with almost the same rate for di erent l at the beginning. Then it begins to increases with M . These coincide with the behaviors we have found in gure 8 and gure 12. AdS3 0.2 0.4 upper left panel is for AdS3 and l = 1; 2; 3; 4; 5. The upper right panel is for AdS4 and l = 1; 2; 3; 4; 5. We also compare the dependence of complexity density on the spacetime dimension. From the left lower panel, we see that the complexity always decreases with M when the strip size l is not big enough. However, when the strip size is large, the complexity density would decreases with M rst and then increases almost linearly when M is large enough in AdSd+1 with d 3. 4 Conclusions and discussions In this paper, we analyzed the evolution of the subregion complexity under a global quench by using numerical method. We considered the situation where the boundary subregion A is an in nite strip on a time slice of the AdS boundary. We followed the subregion CV proposal, which states that the subregion complexity is proportional to the volume of a A enclosed by A and the codimension-two entanglement surface codimension-one surface A corresponding to A. We found the following qualitative picture: the subregion complexity increases at early time after a quench, and after reaching the maximum it decreases surprisingly to a constant value at late time. This non-trivial feature is also observed in [64] where the local quench is used to study the subregion CV proposal. The decrease of complexity is also observed in some space like singular bulk gravitational background [65, 66]. It was argued that the decrease of complexity has something to with the entanglement structure. However, as pointed out in [2], entanglement is not enough to explain the complexity change. There should be other mechanism for this phenomenon. The evolution of complexity following a quench in free eld theory is studied recently [67]. It was found that whether the complexity grow or decrease depending on the quench parameters. To compare with the holographic result, the evolution of complexity following a quench in conformal eld theory is required. Another important feature in the subregion complexity under a global quench we found here is that when the size of the strip is large enough and the quench is fast enough, in AdSd+1 spacetime with d 3 the evolution of the complexity is discontinuous and there is a sudden drop due to the transition of the HRT surface. Moreover, at the early time of the evolution, the growth rates of the subregion complexity densities for the strips of di erent sizes are almost the same. This implies that the complexity growth is related to the local operators excitations. On the other hand, for a large enough strip, the subregion complexity grows linearly with time. If we set the strip size l ! 1, we may expect that the late time behavior of subregion complexity is linearly increasing. However, the large l ! 1 limit should be considered carefully, due to the presence of the holographic entanglement plateau [68{70]. In this limit, the HRT surface could be the union of the black hole horizon and the HRT surface for the complementary region. One has to take into account of this possibility in discussing the large l limit. Actually, the complexity we considered here for strip with limit l ! 1 should be reduced to the CV proposal for one-sided black hole. This case has been studied in [ 57 ] where it was found that the late time limit of the growth rate of the holographic complexity for the one-sided black hole is precisely the same as that found for an eternal black hole. Thus the complexity for strip with in nite width will not decrease and there will not be a plateau at late time. In asymptotic AdS3 black hole case, our results show that the complexity and the corresponding entanglement entropy for subregion will both keep a constant approximately if the evolutional time t & l=2. This can be understood from the thermalization of local states. ref. [71] has shown that, for a given quench in 2D CFT, the density matrix of subsystem will be exponentially close to a thermal density matrix if the time is lager than l=2. Its correction to thermal state will be suppressed by e 4 min(t l=2)= . Here is the inverse temperature and min is the smallest dimension among those operators which have a non-zero expectation value in the initial state. Thus, we can expect that the complexity and entanglement entropy will suddenly go to their values in corresponding thermal state when the time t is larger than l=2. This kind of behavior has been shown clearly in our gure 10. The similar behaviours can also be observed in higher dimensional cases, however, the critical time is not l=2 but depends on the dimension. This sudden saturation is one characteristic phenomenon in subregion complexity. One can easy see that the critical time of saturation will approach to in nity if the size of subregion l approaches to in nity. We also analyzed the dependence of the subregion complexity on various parameters, including the quench speed, the strip size, the black hole mass and the spacetime dimension. For slow quenches or small strip, the sudden drop in the subregion complexity evolution disappear such that the complexity evolves continuously. The mass parameter does no change the qualitative behavior in the evolution when other parameters are xed. Our study can be extended in several directions. Besides the large size limit we mentioned above, it would be interesting to consider the evolution of the subregion complexity under a charge quench or in higher derivative gravity. It would be certainly interesting to study the subregion complexity by using the CA proposal in order to understand the holographic complexity better. Acknowledgments We thank Davood Momeni for correspondence. B. Chen and W.-M. Li are supported in part by NSFC Grant No. 11275010, No. 11325522, No. 11335012 and No. 11735001. C.-Y. Zhang is supported by National Postdoctoral Program for Innovative Talents BX201600005. S.-J. Zhang is supported in part by National Natural Science Foundation of China (No.11605155). Open Access. 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Bin Chen, Wen-Ming Li, Run-Qiu Yang, Cheng-Yong Zhang, Shao-Jun Zhang. Holographic subregion complexity under a thermal quench, Journal of High Energy Physics, 2018, 34, DOI: 10.1007/JHEP07(2018)034