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
WenMing Li 0 1 4 7 8
RunQiu Yang 0 1 2 7 8
ChengYong Zhang 0 1 6 7 8
ShaoJun 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 130722 , 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 codimensionone surface enclosed by the codimensiontwo extremal entanglement surface and the boundary subregion. Under a thermal quench, the dual gravitational con guration is described by a VaidyaAdS 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.
AdSCFT Correspondence; Gaugegravity 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
SchwarzschildAdS black hole
3
Subregion complexity in VaidyaAdS 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 wellde 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 codimensionone 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 antide 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 RyuTakayanagi(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 timedependent
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 timedependent background corresponding to a thermalquenched 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 VaidyaAdS metric. The metric of the VaidyaAdSd+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 SchwarzschildAdS (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 VaidyaAdS 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
infalling 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 VaidyaAdS spacetime.
A
As proposed in [29], the subregion complexity in a static background is proportional
to the volume of a codimensionone time slice in the bulk geometry enclosed by the
boundary region and the corresponding extremal codimensiontwo RyuTakayanagi (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
HubenyRangamaniTakayanagi (HRT) surface
. The corresponding subregion complexity is then
proportional to the volume of a codimensionone 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 codimensiontwo 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 cuto . 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 onshell 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 codimensionone hypersurface
A should be
be parametrized by
(2.8)
A
(2.9)
(2.10)
A
where x~(z) is the codimensiontwo 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 codimensionone 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 multivalued function of boundary time t. In this case,
v~(z) and x~(z) are also multivalued functions of z. The integral in (2.19) is then illde ned.
In these cases, we choose another parametrization to describe the extremal codimensionone
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 codimensiontwo extremal surface
. The equation of motion gives
z d;
with the volume
The boundary condition is determined by the codimensiontwo 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 singlevalued 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 VaidyaAdS 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 onshell 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 onshell 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
SchwarzschildAdS black hole
For the SchwarzschildAdS 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 onshell 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 onshell volume of
A
v = t + g(z);
1
f (z)
;
l
2
:
1=2
1
dz:
The equation of motion (2.18) about the codimensionone 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 codimensiontwo 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 VaidyaAdS 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 nonminimum 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 multivalued only when S^
is multivalued. The multivaluedness 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 multivalued. For the AdS4 we study here, the critical width is l = 1:6.
When z is multivalued, its evolution is subtle. The multivaluedness 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 multivaluedness 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 codimensionone
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 codimensiontwo 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 VaidyaAdS3
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 VaidyaAdS4 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 nulldust 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 codimensiontwo entanglement surface
codimensionone 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 nontrivial 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 onesided 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
onesided 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 nonzero 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.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
HJEP07(218)34
[INSPIRE].
(2013) 781 [arXiv:1306.0533] [INSPIRE].
[Addendum ibid. 64 (2016) 44 [arXiv:1403.5674] [INSPIRE].
[arXiv:1106.5875].
[6] S. Gharibian, Y. Huang, Z. Landau and S.W. Shin, Quantum Hamiltonian Complexity,
Found. Trends. Theor. Comput. Sci. 10 (2015) 159 [arXiv:1401.3916.
[7] M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Info.
[8] M.A. Nielsen et al., Quantum computation as geometry, Science 311 (2006) 1133
[9] M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quantum Info.
Comput. 6 (2006) 213 [quantph/0502070].
Comput. 8 (2008) 861 [quantph/0701004].
107 [arXiv:1707.08570] [INSPIRE].
for quantum
[10] R. Je erson and R.C. Myers, Circuit complexity in quantum eld theory, JHEP 10 (2017)
[11] S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a de nition of complexity
eld theory states, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582]
[12] R.Q. Yang, Complexity for quantum
eld theory states and applications to thermo eld
double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].
arXiv:1801.07620 [INSPIRE].
[14] P. Caputa et al., Antide Sitter space from optimization of path integrals in conformal eld
theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
[15] P. Caputa et al., Liouville action as pathintegral complexity: from continuous tensor
networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].
[16] R.Q. Yang et al., Axiomatic complexity in quantum eld theory and its applications,
[17] D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90
[21] S. Lloyd, Ultimate physical limits to computation, Nature 406 (2000) 1047.
[22] R.G. Cai et al., Action growth for AdS black holes, JHEP 09 (2016) 161
[arXiv:1606.08307] [INSPIRE].
D 95 (2017) 086017 [arXiv:1610.05090] [INSPIRE].
[23] R.Q. Yang, Strong energy condition and complexity growth bound in holography, Phys. Rev.
[24] R.G. Cai, M. Sasaki and S.J. Wang, Action growth of charged black holes with a single
horizon, Phys. Rev. D 95 (2017) 124002 [arXiv:1702.06766] [INSPIRE].
[25] Y.S. An and R.H. Peng, E ect of the dilaton on holographic complexity growth, Phys. Rev.
D 97 (2018) 066022 [arXiv:1801.03638] [INSPIRE].
[26] B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a
density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
[27] M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic
entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
[28] D. Carmi, R.C. Myers and P. Rath, Comments on holographic complexity, JHEP 03 (2017)
118 [arXiv:1612.00433] [INSPIRE].
[INSPIRE].
[29] M. Alishahiha, Holographic complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614]
[30] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,
Phys. Rev. Lett. 96 (2006) 181602 [hepth/0603001] [INSPIRE].
[31] S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006)
045 [hepth/0605073] [INSPIRE].
[32] V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement
entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
[33] X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement,
JHEP 11 (2016) 028 [arXiv:1607.07506] [INSPIRE].
115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].
[condmat/0512165] [INSPIRE].
[38] G. Evenbly and G. Vidal, Tensor network renormalization yields the multiscale entanglement
[39] M. Miyaji and T. Takayanagi, Surface/state correspondence as a generalized holography,
PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
entanglement in holography, arXiv:1701.02319 [INSPIRE].
[44] E. Bakhshaei, A. Mollabashi and A. Shirzad, Holographic subregion complexity for singular
surfaces, Eur. Phys. J. C 77 (2017) 665 [arXiv:1703.03469] [INSPIRE].
[45] D. Sarkar, S. Banerjee and J. Erdmenger, A holographic dual to Fisher information and its
relation with bulk entanglement, PoS(CORFU2016)092.
[46] M. Kord Zangeneh, Y.C. Ong and B. Wang, Entanglement entropy and complexity for
onedimensional holographic superconductors, Phys. Lett. B 771 (2017) 235
[arXiv:1704.00557] [INSPIRE].
[47] D. Momeni et al., Thermodynamic and holographic information dual to volume,
arXiv:1704.05785 [INSPIRE].
[48] P. Roy and T. Sarkar, Subregion holographic complexity and renormalization group ows,
Phys. Rev. D 97 (2018) 086018 [arXiv:1708.05313] [INSPIRE].
[49] D. Carmi, More on holographic volumes, entanglement and complexity, arXiv:1709.10463
[INSPIRE].
[50] R. Abt et al., Topological complexity in AdS3/CFT2, arXiv:1710.01327 [INSPIRE].
[51] L.P. Du, S.F. Wu and H.B. Zeng, Holographic complexity of the disk subregion in
(2 + 1)dimensional gapped systems, arXiv:1803.08627 [INSPIRE].
[52] P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in onedimensional systems,
J. Stat. Mech. 0504 (2005) P04010 [condmat/0503393] [INSPIRE].
[53] J. AbajoArrastia, J. Aparicio and E. Lopez, Holographic evolution of entanglement entropy,
JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].
HJEP07(218)34
holographic thermalization captured by horizon interiors and mutual information, JHEP 09
(2013) 057 [arXiv:1306.0210] [INSPIRE].
Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
theories, Phys. Rev. D 96 (2017) 126001 [arXiv:1707.03840] [INSPIRE].
of holographic complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].
theoretic complexities for time dependent thermo eld double states, JHEP 02 (2018) 082
[arXiv:1710.00600] [INSPIRE].
[arXiv:1711.02668] [INSPIRE].
01 (2016) 084 [arXiv:1509.09291] [INSPIRE].
eld theory, JHEP 06 (2018) 029 [arXiv:1804.00107] [INSPIRE].
[1] L. Susskind , Computational complexity and black hole horizons , Fortsch. Phys . 64 ( 2016 ) 24 [2] L. Susskind , Entanglement is not enough, Fortsch. Phys . 64 ( 2016 ) 49 [arXiv: 1411 .0690] [3] J. Maldacena and L. Susskind , Cool horizons for entangled black holes , Fortsch. Phys . 61 [4] J. Watrous , Quantum computational complexity, in Encyclopedia of complexity and systems science , R.A. Meyers ed., Springer Germany ( 2009 ), arXiv: 0804 . 3401 .
[5] T.J. Osborne , Hamiltonian complexity, Rept. Prog. Phys . 75 ( 2012 ) 022001 [18] L. Susskind and Y. Zhao , Switchbacks and the bridge to nowhere , arXiv: 1408 .2823 [19] A.R. Brown et al., Holographic complexity equals bulk action? , Phys. Rev. Lett . 116 ( 2016 ) [20] A.R. Brown et al., Complexity, action and black holes , Phys. Rev. D 93 ( 2016 ) 086006 [13] R. Khan , C. Krishnan and S. Sharma , Circuit complexity in fermionic eld theory , [34] M. Miyaji et al., Distance between quantum states and gaugegravity duality , Phys. Rev. Lett.
[35] G. Vidal , Entanglement renormalization, Phys. Rev. Lett . 99 ( 2007 ) 220405 [36] B. Swingle , Entanglement renormalization and holography , Phys. Rev. D 86 ( 2012 ) 065007 [37] G. Evenbly and G. Vidal. Tensor Network Renormalization , Phys. Rev. Lett . 115 ( 2015 ) [40] B. Czech , Einstein equations from varying complexity , Phys. Rev. Lett . 120 ( 2018 ) 031601 [41] O. BenAmi and D. Carmi , On volumes of subregions in holography and complexity , JHEP [42] P. Roy and T. Sarkar , Note on subregion holographic complexity, Phys. Rev. D 96 ( 2017 ) [43] S. Banerjee , J. Erdmenger and D. Sarkar , Connecting Fisher information to bulk [54] T. Albash and C.V. Johnson , Evolution of holographic entanglement entropy after thermal and electromagnetic quenches , New J. Phys . 13 ( 2011 ) 045017 [arXiv: 1008 .3027] [INSPIRE].
[55] V. Balasubramanian et al., Holographic thermalization , Phys. Rev. D 84 ( 2011 ) 026010 [arXiv: 1103 .2683] [INSPIRE].
[56] H. Liu and S.J. Suh , Entanglement growth during thermalization in holographic systems , Phys. Rev. D 89 ( 2014 ) 066012 [arXiv: 1311 .1200] [INSPIRE].
Part I , JHEP 06 ( 2018 ) 046 [arXiv: 1804 .07410] [INSPIRE].
[57] S. Chapman , H. Marrochio and R.C. Myers , Holographic complexity in Vaidya spacetimes .
[58] Y.Z. Li , S.F. Wu , Y.Q. Wang and G.H. Yang , Linear growth of entanglement entropy in [61] D. Carmi , S. Chapman , H. Marrochio , R.C. Myers and S. Sugishita , On the time dependence [62] R.Q. Yang , C. Niu , C.Y. Zhang and K.Y. Kim , Comparison of holographic and eld