IPM/P-2021/51

Complexity growth in Gubser-Rocha models with momentum relaxation

H. Babaei-Aghbolagha, Davood Mahdavian Yektab, Komeil Babaei Velnic,d, H.
Mohammadzadeha

aDepartment of Physics, University of Mohaghegh Ardabili, P.O. Box 179,
Ardabil, Iran
bDepartment of Physics, Hakim Sabzevari University, P.O. Box 397, Sabzevar,
Iran
cDepartment of Physics, University of Guilan, P.O. Box 41335-1914, Rasht, Iran
dSchool of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.
Box 19395-5531, Tehran, Iran

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
The Einstein-Maxwell-Axion-Dilaton (EMAD) theories, based on the Gubser-Rocha
(GR) model, are very interesting in holographic calculations of strongly
correlated systems in the condensed matter physics. Due to the presence of
spatially dependent massless axionic scalar fields, the momentum is relaxed
and we have no translational invariance at finite charge density. It would be
of interest to study some aspects of quantum information theory for such
systems in the context of $AdS/CFT$ where EMAD theory is a holographic dual
theory. For instance, in this paper we investigate the complexity and its time
dependence for charged $AdS$ black holes of EMAD theories in diverse
dimensions via the complexity equals action (CA) conjecture. We will show that
the growth rate of the holographic complexity violates the Lloyd’s bound at
finite times. However, as shown at late times, it depends on the strength of
momentum relaxation and saturates the bound for these black holes.

##  1 Introduction

Strongly correlated systems in the condensed matter physics include some
interesting phenomena which are not easy to analyze theoretically, however in
the context of holography [1, 2], there are some practical ways which map them
to the dual weakly interacting systems [3, 4, 5, 6, 7]. For instance, linear
T-resistivity in the strange metals has a remarkable degree of universality
which does not observed in general Fermi liquids with quadratic temperature
dependence for resistivity [8, 9, 10, 11], or the divergent conductivity of
systems with momentum relaxation due to a broken translational symmetry which
occurs in more realistic condensed matter materials [12, 13, 14, 15, 16, 17].
In this way, the GR models [18] and the EMAD theories [19, 20], based on the
GR model, are of the most interesting theories which provide nice areas for
the study of electric and magnetic transport phenomena in a strongly-coupled
system. The EMAD models can also be dilatonic generalization of EMA theories
proposed in [21].

Apart from this holographic approach of solving strongly coupled field
theories in condensed matter physics, the $AdS/CFT$ correspondence [22, 23,
24, 25, 26] has been far more deep and revealing than merely providing a
classical geometrical computational tool for strongly coupled field theory
phenomena. Thinking about how field theory codes various phenomena on the
gravity side has led to the recognition of various concepts from the quantum
information. Information geometry [27], Von-Neumann [28] and Renyi entropy
[29], mutual information [30], tensor networks [31, 32], computational
complexity [33], fidelity or relative entropy [34] and quantum error
correcting codes [35] are only to name a few. Among them, entanglement entropy
has been the most fundamental thing to study as it measures the correlation
between two subsystems [36, 37, 38]. However, entanglement entropy may not be
enough to probe the degrees of freedom interior the black holes, since the
volume of black holes continues growing even if spacetimes reach thermal
equilibrium[39]. This motivates the introduction of holographic complexity in
the quantum information theory [33, 40]. As a measure, quantum complexity
describes how many simple elementary gates (unitary operators) are needed to
obtain a particular state from some chosen reference state [41, 42, 43]. (For
more definitions in quantum field theories see e.g. [44, 45, 46, 47])

In the context of $AdS/CFT$ correspondence, two holographic prescriptions have
been suggested to compute the complexity; the “complexity=volume” (CV)
conjecture [33, 40] and the “complexity= action” (CA) conjecture [48, 49]. In
the CV conjecture, the complexity of a state is proportional to the volume of
Einstein-Rosen bridge which connects two boundaries of an eternal black hole,
i.e. $\mathcal{C}_{V}=\mathcal{V}/G\,\ell$ where $G$ is the Newton’s constant
and $\ell$ is an arbitrary length scale, while in the CA picture, it is
proportional to the bulk action evaluated in a certain spacetime region known
as the Wheeler-De Witt (WDW) patch, i.e. $\mathcal{C}_{A}=I_{WDW}/\pi\hbar$.
By construction, the CA proposal is devoid of the ambiguity associated with
arbitrary length scale $\ell$ appeared in the CV picture. There are two
conceptual features here; $i)$ A gravity system for which the holographic
complexity has been studied extensively is the eternal two-sided $AdS$ black
hole [50] which is dual to a thermofield double state in the dual boundary
field theory,

$|\psi_{TFD}\rangle=\frac{1}{\sqrt{Z}}\sum_{j}e^{-{E_{j}}/({2T})}e^{-iE_{j}(t_{L}+t_{R})}|E_{j}\rangle_{L}|E_{j}\rangle_{R}\,,$
(1.1)

where $L$ and $R$ refer to the left and right entangled regions of the two
sided black hole. The entanglement is due to the Einstein-Rosen bridge that
connects the left and right boundary $CFT$s. $ii)$ For $AdS$ black holes of
mass $M$ in the CA conjecture, the late time behavior of the change rate of
complexity reaches a constant value

$\dot{\mathcal{C}}_{A}\leq{2M},$ (1.2)

where in the rest of the paper we assume that $\pi\hbar=1$. This is often
referred to the Lloyd’s bound [51], which in the case of charged and rotating
black holes is generalized respectively to [49, 52]

$\dot{\mathcal{C}}_{A}\leq 2\big{[}(M-\mu Q)-(M-\mu
Q)_{gs}\big{]},\qquad\dot{\mathcal{C}}_{A}\leq 2\big{[}(M-\Omega J)-(M-\Omega
J)_{gs}\big{]}.$ (1.3)

Here,“$gs$” denotes the ground state of the black hole. The hope that the
holographic complexity might provide useful information about the spacetime
structure behind the horizon has led to intense investigation of the CA and CV
proposals in various gravity theories, probing their structure and properties
[53]-[77].

In the light of the above discussion, it would be of interest to study the
complexity for charged $AdS$ black holes in EMAD theories which may be
examined in holographic models of quantum information theory and quantum
computation. Another motivation is to investigate that whether the
contribution of different fields in EMAD theory can affect the Lloyd’s bound
for the corresponding black holes here, or whether in general there is a
violation for this bound or not. In particular, we will compute the
holographic complexity and its time evolution for charged single-horizon
solutions of 4, 5, and $(d+1)$-dimensional EMAD theories using the CA
conjecture. We study both the early and late time limit for the growth rate of
complexity. This may not only allow us to further constrain the validity of
these holographic conjectures but also provide other examples in the growing
list of literature where the Lloyd’s bound can be explicitly violated.

It has been shown in [48] that for small Reissner-Nordstrom $AdS$ black holes
111In theoretical physics, an extremal black hole is a black hole with the
minimal possible mass that can be compatible with a given charge and angular
momentum. In other words, this is the smallest possible black hole that can
exist while rotating at a given fixed constant speed with some fixed charge.
of mass $M$ and charge $q$ in EM theory, the Lloyd’s bound at late time is
given by $\dot{\cal{C}}_{A}=2M-\mu q$, while for charged $AdS$ black holes in
a simple class of EMD theories one has $\dot{\cal{C}}_{A}=2M-\mu q-C$ where
$C$ is a constant term [78]. On the other hand, we have shown in [79] that for
neutral black branes in EMA theory, the momentum relaxation term does not
explicitly change the late time limit $\dot{\mathcal{C}}_{A}=2M$. Acually,
this fact is another main motivation for us to investigate in this paper
whether the momentum relaxation term would correct the Lloyd’s bound in EMAD
theories or not.

The outline of the paper is as follows: In section 2, we review the
holographic complexity in the CA picture. We evaluate the action on a WDW
patch that has null boundary surfaces in addition to time/space-like
boundaries. In this sense, we should consider some extra actions to remove the
ambiguities for null surfaces [80]. In section 3, we calculate the complexity
growth of charged $AdS$ black holes for EMAD holographic models in diverse
dimensions. We also provide a numerical study on $\dot{\mathcal{C}}_{A}$ to
find how the Lloyd’s bound is violated. In section 4, we summarize our results
and present a discussion about computed bounds in different theories.

##  2 Holographic Complexity via CA conjecture

Our main interest in this paper is to investigate the time dependency of
holographic complexity, however, there are two distinct counting methods of
the complexity growth rate in the CA picture. The first proposal by Brown et
al [48, 49] puts forward to calculating $\dot{\mathcal{C}}_{A}$ at late times
while in the latter proposal by Lehner et al [80], the complexity is evaluated
as well as its time evolution. In general, they are not equivalent but give
the identical very late time results [81]. The main difference between two
methods is that in the former, there is no way to violate the Lloyd’s bound,
while as we have shown the bound is violated from above in the latter.
Therefore, we use the second approach to study the holographic complexity and
its time evolution for charged $AdS$ solutions of EMAD theories. The essential
ingredient in the CA conjecture is to evaluate the action on a WDW patch [48,
49]. However, we follow the method in Ref. [54] where not only the action on
the WDW patch includes the bulk and the Gibbons-Hawking-York (GHY) boundary
terms [82, 83], but also embraces boundary segments of joint terms due to the
intersection of time-like, space-like, and null boundaries [84]. This will be
the general strategy that will be followed in the next section.

The bulk actions for holographic EMAD models are denoted by $I_{bulk}$ and
essentially are different in diverse dimensions. However, the contribution of
GHY surface terms are the same for all of them as follows

$I_{surf}=\frac{1}{8\pi G}\int_{\mathcal{B}}d^{d}x\sqrt{h}K-\frac{1}{8\pi
G}\int_{\mathcal{B^{\prime}}}d\lambda\,d^{d-1}\theta\sqrt{\gamma}\,\kappa,$
(2.1)

where $K$ is the extrinsic curvature of the time-like or space-like boundaries
($\mathcal{B}$) and $\kappa$ is extrinsic curvature of the null segments
($\mathcal{B}^{\prime}$). Here, $h_{\mu\nu}$ and $\gamma_{\mu\nu}$ are the
induced metrics on $\mathcal{B}$ and $\mathcal{B}^{\prime}$, respectively, so
that $K_{\mu\nu}=-h_{\mu}^{\rho}h_{\nu}^{\sigma}\nabla_{(\rho}n_{\sigma)}$ and
$\kappa=-N_{\mu}k^{\nu}\nabla_{\nu}k^{\mu}$ where $n_{\mu}$ and $k_{\mu}$ are
outward pointing unit normal vectors on them. $\lambda$ parameterizes the null
generator of the null boundary such that for affinely parametrization
$\kappa=0$ [80].

The joint actions are given by

$I_{jnt}=\frac{1}{8\pi
G}\int_{\Sigma}d^{d-1}x\sqrt{\sigma}\,\eta+\frac{1}{8\pi
G}\int_{\Sigma^{\prime}}d^{d-1}x\sqrt{\sigma}\,a,$ (2.2)

where $\eta$ corresponds to the intersection of time-like or space-like
boundaries while $a$ is related to intersecting of null surfaces. As shown in
Fig. (1), the WDW patch have intersections of null surfaces with time-like or
space-like boundaries at $r={\epsilon}$ and $r=r_{max}$ but since they have no
time dependence [60], it is not necessary to consider them here. Thus, only
the contribution of null meeting point $r_{m}$ remains for joint action.

There is also a counterterm action for the null surfaces as

$I_{ct}=\frac{1}{8\pi
G}\int_{\mathcal{B^{\prime}}}d\lambda\,d^{d-1}\theta\sqrt{\gamma}\,\Theta\log{(\ell_{c}\Theta)}\,,$
(2.3)

which is introduced to ensure reparametrization invariance on the null
boundaries and does not change the variational principle. Indeed, the
parameter expansion $\Theta$ is the relative rate of change of the cross-
sectional area of a bundle of null generators which is denoted by

$\Theta=\partial_{\lambda}\log{\sqrt{\gamma}},$ (2.4)

and $\ell_{c}$ is an arbitrary length scale (for more discussions see
e.g.[80]). Therefore, we should investigate the following total action on the
WDW patch

${I}_{WDW}=I_{bulk}+I_{surf}+I_{jnt}+I_{ct}\,.$ (2.5)

In Fig. (1) we have plotted the Penrose diagram of a two-sided $AdS$ black
hole with single horizon $r_{h}$. The grey shaded region is the WDW patch
bounded by the light sheets sent from the two asymptotic symmetric time slices
$t_{L}$ and $t_{R}$ such that $t_{L}=t_{R}\equiv t/2$. The WDW patch includes
two equivalent sectors where the future and past null boundaries of the right
sector are given respectively by

${t_{L}}=r^{*}(r)-{r}_{\infty}^{*}\,,\qquad-{t_{R}}=r^{*}(r)-{r}_{\infty}^{*}\,.$
(2.6)

In general, $r^{*}$ is a tortoise coordinate defined as
$dr^{*}(r)={dr}/{f(r)}$ upon which the new coordinates $v=t+r^{*}(r)$ and
$u=t-r^{*}(r)$ are defined and
$\lim_{r\rightarrow\infty}r^{*}(r)={r}_{\infty}^{*}$. Also $r_{m}$ is the
meeting point of null boundaries before reaching the past singularity at
critical time $t_{c}=2({r}_{\infty}^{*}-r^{*}(0))$.

Figure 1: Penrose diagram of the WDW patch for $AdS$ black holes. $r_{0}$ is
the physical singularity and $r={\varepsilon}$ and $r=r_{max}$ are the IR and
UV cut off surfaces, respectively.

##  3 Holographic Complexity for EMAD theories

In this section we consider the holographic complexity and its time evolution
for three families of holographic EMD theory based on the GR model in the
presence of momentum relaxation term constructed from axionic scalar fields.
The general bulk action of EMAD theories in arbitrary dimensions is given by
[19]

$I={1\over 16\pi
G_{d+1}}{\int}d^{d+1}x\sqrt{-g}\left[R-\frac{1}{2}(\partial_{\mu}\phi)^{2}+V(\phi)-\frac{1}{4}Z(\phi)F_{\mu\nu}^{2}-\frac{1}{2}\sum_{I=1}^{d-1}(\partial_{\mu}\psi_{I})^{2}\right],$
(3.1)

where $G_{d+1}$, $\phi(r)$, and
$F_{\mu\nu}\\!=\\!\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ are the
${d+1}$-dimensional Newton’s constant, dilaton field and field strength of
$U(1)$ gauge field $A_{\mu}$, respectively. Also, $V(\phi)$ is a dilatonic
scalar potential and $Z(\phi)$ is a coupling function between the Maxwell and
dilaton fields. The axionic scalar fields enter the bulk action only through
the kinetic term $\partial_{\mu}\psi_{I}$ and the sources are linear in the
boundary, i.e. $\psi^{(0)}_{I}\propto\beta_{Ii}x^{i}$ [21], where $\beta$
represents the strength of the momentum relaxation for which

$\beta^{2}\equiv\frac{1}{2}\sum_{i=1}^{d-1}\vec{\beta}_{i}\cdot\vec{\beta}_{i},\qquad\vec{\beta}_{i}\cdot\vec{\beta}_{j}=\sum_{I=1}^{d-1}{\beta}_{Ii}\,{\beta}_{Ij}=\beta^{2}\delta_{ij}\qquad\forall\,i,j=1,2,...,d-1.$
(3.2)

The field equations of the action (3.1) are given by

$\displaystyle
R_{\mu\nu}-\frac{V(\phi)}{d-1}g_{\mu\nu}-\frac{1}{2}\partial_{\mu}\phi\,\partial_{\nu}\phi-\frac{1}{2}Z(\phi){F_{\mu}}^{\rho}F_{\nu\rho}-\frac{1}{4(d-1)}g_{\mu\nu}Z(\phi)F^{2}-\frac{1}{2}\sum_{I=1}^{d-1}{\partial_{\mu}\psi_{I}\,\partial_{\nu}\psi_{I}}=0,$
$\displaystyle\nabla_{\mu}\left(Z(\phi)F^{\mu\nu}\right)=0,$ (3.3)
$\displaystyle\Box\psi_{I}=0,\qquad I=1,..,d-1,$
$\displaystyle\Box\phi+\frac{dV(\phi)}{d\phi}-\frac{1}{4}\frac{dZ(\phi)}{d\phi}F^{2}=0.$

In general, the analytical charged solutions of these equations are described
by the following ansatz

$ds^{2}=e^{2A(r)}\left(-h(r)\,dt^{2}+d\vec{x}^{2}\right)+\frac{e^{2B(r)}}{h(r)}dr^{2},$
(3.4)

where $d\vec{x}^{2}$ is the line element of $(d-1)$-dimensional spatial flat
space 222To describe the line element of the black holes, in this paper, we
use the convention in Ref. [85] in which the holographic renormaliation of EMD
theories have been considered.. In this regard, we evaluate the total action
(2.5) on the WDW patch shown in Fig. (1) for each theory. We also determine
the Lloyd’s bound on the rate of complexity at late times.

Since we are going to analyze the holographic complexity of charged black
holes in asymptotically $AdS$ space, it is expected that one should consider
the causal structure of Reissner-Nordstrom $AdS$ black hole given in figure 14
of [60] in which the WDW patch have two intersecting points correspond to
either future or past null boundaries where at the late time coincide with the
inner and outer horizons respectively. But, as will be shown for charged black
holes in this paper, the inner Cauchy horizon is replaced by a curvature
singularity at $r=0$, therefore, the causal structure is described by the one
for $AdS$-Schwarzschild black hole as depicted in Fig. (1).

###  3.1 $AdS_{4}$ black holes

In four dimensions the EMAD theory is constructed from two parts: an EMD
theory obtained from dimensional reduction of $AdS_{4}\times S^{7}$ in $M$\-
theory to triple intersection of $M5$-branes that has the three-equal-charge
black hole [86] and a kinetic term from two massless axionic scalar fields
$\psi_{I}$ with $I=1,2$ [19]. This theory is described by the following bulk
action

${I}_{bulk}\\!=\\!{1\over 16\pi G_{4}}\int d^{4}x\sqrt{-g}\left[R-{1\over
4}e^{\phi}F_{\mu\nu}^{2}-{3\over 2}(\partial_{\mu}\phi)^{2}+{6\over
L^{2}}\cosh\phi-\tfrac{1}{2}\sum_{I=1}^{2}{(\partial_{\mu}\psi_{I})}^{2}\right]\\!=\\!{1\over
16\pi G_{4}}\int d^{4}x\sqrt{-g}\mathcal{L}_{1},\,$ (3.5)

where $\mathcal{L}_{1}$ is the Lagragian density introduced for later
convenience. For $\phi(r)=0$, the model reduces to the action in Ref. [21, 87]
for which we have extensively considered its holographic complexity in Ref.
[79]. Holographic calculations of this model have been studied in Refs. [88,
89, 90, 91]. Substituting the ansatz (3.4) in the equations of motion, we
obtain

$\displaystyle
A(r)=-B(r)=\log\frac{r}{L}+\frac{3}{4}\log\left(1+\frac{Q}{r}\right),\quad\phi(r)=\frac{1}{2}\log\left(1+\frac{Q}{r}\right),$
$\displaystyle
h(r)=1-\frac{L^{4}\beta^{2}}{2(Q+r)^{2}}-\frac{(Q+r_{h})^{3}}{(Q+r)^{3}}\bigl{(}1-\frac{L^{4}\beta^{2}}{2(Q+r_{h})^{2}}\bigr{)},\quad\psi_{I}=\beta_{Ii}x^{i},$
(3.6) $\displaystyle A_{\mu}dx^{\mu}=A_{t}(r)dt,\quad
A_{t}(r)=\frac{r-r_{h}}{Q+r}\sqrt{\frac{3Q(Q+r_{h})}{L^{2}}\bigl{(}1-\frac{L^{4}\beta^{2}}{2(Q+r_{h})^{2}}\bigr{)}},$

where $Q$ represents a charge parameter and $L$ is a length scale that
henceforth we set $L=1$. We can recast the solution (3.4) in the form of a
general $AdS$ black hole as

$ds^{2}=-f(r)\,dt^{2}+{U(r)}d\vec{x}^{2}+\frac{1}{f(r)}dr^{2},$ (3.7)

in which

$f(r)=\frac{r^{1/2}(r-r_{h})\left(3Q^{2}+r_{h}^{2}+r_{h}r+r^{2}+3Q(r_{h}+r)\right)}{(Q+r)^{3/2}}-\frac{r^{1/2}(r-r_{h})\beta^{2}}{2(Q+r)^{3/2}},\quad
U(r)=r^{1/2}(Q+r)^{3/2}.$ (3.8)

This expression of $f(r)$ explicitly shows that the solution has two
singularities at $r=r_{h}$ and $r=0$ where the first corresponds to the event
horizon and the latter is the spacetime singularity. Substituting the
functions (3.8) and parameters (3.1) in to the Lagrangian (3.5), we have

$\sqrt{-g}{\mathcal{L}}_{1}=-\frac{3\bigl{(}Q^{4}+4r^{4}+Q^{3}(-3r_{h}+10r)-3Q^{2}(r_{h}^{2}-6r^{2})-Q(r_{h}^{3}-14r^{3})\bigr{)}}{2(Q+r)^{2}}-\frac{3Q(Q+r_{h})\beta^{2}}{4(Q+r)^{2}}.$
(3.9)

As alluded in the introduction, it is costum to write the late time behavior
of $\dot{\mathcal{C}}_{A}$ in terms of the physical parameters of black holes,
thus we should calculate them for each solution. The mass of the black hole is
given by

$M={V_{2}\over 16\pi
G_{4}}2\omega,\qquad\omega=(Q+r_{h})^{3}\left(1-\frac{\beta^{2}}{2(Q+r_{h})^{2}}\right),$
(3.10)

where $\omega$ is a mass parameter and $V_{2}$ is the dimensionless volume of
2-dimensional spatial geometry $\vec{x}$. In the zero dissipation
$\beta\rightarrow 0$, the mass parameter reduces to $\omega=Q^{3}$ for
extremal limit $r_{h}=0$ in Ref. [18]. Also, in the limit $Q\rightarrow 0$,
one achieves the mass of neutral black hole given by Eq. (2.36) in Ref. [79].
The chemical potential and black hole charge are obtained from $A_{t}$ in
(3.1) as

$\mu=\sqrt{3Q(Q+r_{h})\left(1-\frac{\beta^{2}}{2(Q+r_{h})^{2}}\right)},\qquad
q=\frac{V_{2}}{16\pi
G_{4}}\sqrt{3Q(Q+r_{h})^{3}\left(1-\frac{\beta^{2}}{2(Q+r_{h})^{2}}\right)},$
(3.11)

where the chemical potential can be read off from the asymptotic value of the
electric potential $A_{t}(r\rightarrow\infty)$ and $q$ is computed from the
Gauss’s law. The temperature and entropy of the black hole are obtained from

$T=\frac{f(r)^{\prime}}{4\pi}\Big{|}_{r=r_{h}}=\frac{r_{h}^{1/2}\bigl{(}6(Q+r_{h})^{2}-\beta^{2}\bigr{)}}{8\pi(Q+r_{h})^{3/2}},\quad
S=\frac{A_{h}}{4G_{4}}\Big{|}_{r=r_{h}}=\frac{V_{2}}{4G_{4}}r_{h}^{1/2}(Q+r_{h})^{3/2},$
(3.12)

where prime is the derivative with respect to $r$ and $A_{h}=V_{2}U(r_{h})$ is
the area of the event horizon, such that

$TS=\frac{V_{2}}{8\pi
G_{4}}\frac{r_{h}}{4}\left(6(Q+r_{h})^{2}-\beta^{2}\right).$ (3.13)

####  3.1.1 The growth rate of complexity

We will now proceed to compute the time evolution of the holographic
complexity via the CA proposal. Due to the left-right symmetry of the WDW
patch for planar $AdS$ black hole in Fig. (1) we only work on the right side
for times $t>t_{c}$ and then multiplied the final result by a factor of 2 for
symmetric left side. The contribution of the bulk action (3.5) is given by

$\displaystyle I_{bulk}$ $\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle
2\,\,\left(I_{bulk}^{\textup{I}}+I_{bulk}^{\textup{II}}+I_{bulk}^{\textup{III}}\right)$
(3.14) $\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle\frac{V_{2}}{8\pi
G_{4}}\left[\int_{\epsilon}^{r_{h}}\left(\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)+2\int_{r_{h}}^{r_{max}}\left({r}_{\infty}^{*}-r^{*}(r)\right)+\int_{r_{m}}^{r_{h}}\left(-\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)\right]\,\sqrt{-g}\,\mathcal{L}_{1}\,dr,$
$\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$ $\displaystyle
I_{bulk}^{0}+\frac{V_{2}}{8\pi
G_{4}}\int_{\epsilon}^{r_{m}}\left(\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)\,\sqrt{-g}\,\mathcal{L}_{1}\,dr,$

where $I_{bulk}^{0}$ is the time independent part, therefore we have

$\frac{dI_{bulk}}{dt}=\frac{V_{2}}{16\pi
G_{4}}\left[-3Q^{2}r-\frac{9}{2}Qr^{2}-2r^{3}-\frac{3Q(Q+r_{h})^{3}}{2(Q+r)}+\frac{3Q(Q+r_{h})\beta^{2}}{4(Q+r)}\right]_{\epsilon}^{r_{m}}.$
(3.15)

According to Fig. (1), the boundary action includes two distinguished parts: a
time-like surface at the IR cut off and a space-like surface at the UV cut off
point. The extrinsic curvature for the ansatz (3.7) is as fallows

$K=\frac{1}{2}\left(\frac{\partial_{r}f(r)}{\sqrt{f(r)}}+2\sqrt{f(r)}\,\partial_{r}{\ln\left(U(r)\right)}\right),$
(3.16)

where the normal vectors corresponding to these surfaces are given by

$r={\epsilon}:\quad{\bf
t}=t_{\mu}dx^{\mu}=-\frac{dr}{\sqrt{-f(\epsilon)}},\qquad\qquad
r=r_{max}:\quad{\bf s}=s_{\mu}dx^{\mu}=\frac{dr}{\sqrt{f(r_{max})}}.$ (3.17)

Again by supposing a factor of 2 for the left sector of patch we have

$I_{surf}=2\,(I_{surf}^{r=\epsilon}+I_{surf}^{r=r_{max}})=I_{surf}^{0}+\frac{V_{2}}{4\pi
G_{4}}\sqrt{h}K\,\left(\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)\Big{|}_{r=\epsilon}.$
(3.18)

Here $I_{surf}^{0}$ includes the contribution of the UV cutoff surfaces whish
is independent of time, thus the time evolution of the GHY surface term is

$\displaystyle\frac{{dI_{surf}}}{dt}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{V_{2}}{8\pi
G_{4}}\Big{[}-\frac{3\bigl{(}Q^{3}(3r_{h}-5r)+3Q^{2}(r_{h}^{2}+2r_{h}r-5r^{2})+Q(r_{h}^{3}+6r_{h}^{2}r-13r^{3})+2r(r_{h}^{3}-2r^{3})\bigr{)}}{4(Q+r)}$
(3.19)
$\displaystyle\quad\qquad-\frac{(8r^{2}+5Qr-6r_{h}r-3Qr_{h})\beta^{2}}{8(Q+r)}\Big{]}_{r=\epsilon}.$

It has been shown in Ref. [60] that the null joint contributions at the UV
cutoff surfaces have no time dependence, so we need only to consider the joint
at $r_{m}$. Assume that $k_{1}$ and $k_{2}$, the null vectors associated with
two past null boundaries intersecting at $r_{m}$, are defined by

$k_{1}=\xi\left(-dt+\frac{dr}{f(r)}\right),\qquad
k_{2}=\xi\left(dt+\frac{dr}{f(r)}\right),$ (3.20)

where $\xi$ is a normalization constant for null vectors. Following [80], the
joint term $a\\!=\\!\ln|-\frac{k_{1}\cdot k_{2}}{2}|$ in the action (2.2) can
be evaluated as

$I_{jnt}=\frac{V_{2}}{8\pi
G_{4}}\Bigg{[}-\log\bigl{(}\frac{f(r)}{\xi^{2}}\bigr{)}U(r)\Bigg{]}_{r=r_{m}}.$
(3.21)

Due to the implicit time dependence of $r_{m}$ with equation
$\frac{t}{2}-{r}_{\infty}^{*}+r^{*}(r_{m})=0$, or

$\frac{dr_{m}}{dt}=-\frac{f(r_{m})}{2},$ (3.22)

and ${dI_{jnt}}/{dt}=-\frac{f(r_{m})}{2}\,{dI_{jnt}}/{dr_{m}}$, the time
evolution of this contribution is

$\frac{dI_{jnt}}{dt}=\frac{V_{2}}{16\pi
G_{4}}\Bigg{[}U(r)\partial_{r}{f(r)}+f(r)\partial_{r}{U(r)}\log\bigl{(}\frac{f(r)}{\xi^{2}}\bigr{)}\Bigg{]}_{r=r_{m}},$
(3.23)

after substituting from (3.8) we obtain

$\displaystyle\frac{dI_{jnt}}{dt}$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{V_{2}}{32\pi
G_{4}}\Bigg{[}\frac{Q^{3}(Q\\!+\\!7r)\\!+\\!Q{r}^{2}(15Q\\!+\\!13{r})\\!+\\!4{r}^{4}\\!-\\!(Q\\!-\\!2{r})(Q\\!+\\!r_{h})^{3}}{(Q\\!+\\!{r})}\\!-\\!\frac{\left(2rr_{h}\\!+\\!Q(3r\\!-\\!r_{h})\right)\beta^{2}}{2(Q\\!+\\!r)}$
(3.24) $\displaystyle\\!\\!\\!\\!+\\!\\!\\!\\!$
$\displaystyle\log\bigl{(}\frac{f({r})}{\xi^{2}}\bigr{)}(Q+4{r})({r}-r_{h})\left(\frac{3Q^{2}+{r}^{2}+{r}r_{h}+r_{h}^{2}+3Q({r}+r_{h})}{(Q+{r})}-\frac{\beta^{2}}{2(Q+{r})}\right)\Bigg{]}_{r=r_{m}}.$

In the late time limit where ${r_{m}}$ reaches the horizon at $r_{h}$, we have

$\frac{dI_{jnt}}{dt}{\Bigg{|}}_{t>>t_{c}}=\frac{V_{2}}{8\pi
G_{4}}\Bigg{[}\frac{3}{2}r_{h}(Q+r_{h})^{2}-\frac{1}{4}r_{h}\beta^{2}\Bigg{]}=TS,$
(3.25)

where as expected in the extremal limit becomes zero. The counterterm action
(2.3) for parametrization of the past and future null boundaries is

$I_{ct}=2\,(I_{ct}^{future}+I_{ct}^{past})=\frac{V_{2}}{4\pi
G_{4}}\left(\int_{{\epsilon}}^{r_{max}}+\int_{r_{m}}^{r_{max}}\right)\partial_{r}{U(r)}\log\left(\frac{\ell_{c}\xi\partial_{r}U(r)}{U(r)}\right)dr,$
(3.26)

here, we have used the affine parameter $\lambda=r/\xi$, where $\xi$ is a
constant, so the total action with the counterterm does not depend on the
parametrization of the null surfaces [54] and the expansion (2.4) takes the
form

$\Theta=\frac{\xi\partial_{r}U(r)}{U(r)}.$ (3.27)

Then from (3.22), the time evolution of the counterterm action becomes

$\frac{dI_{ct}}{dt}=\frac{V_{2}}{8\pi
G_{4}}\Bigg{[}\frac{(r-r_{h})(Q+4r)\bigl{(}6Q^{2}+2r_{h}^{2}-\beta^{2}+2r_{h}r+2r^{2}+6Q(r_{h}+r)\bigr{)}}{8(Q+r)}\log{\frac{l_{c}\xi(Q+4r)}{2r(Q+r)}}\Bigg{]}_{r={r_{m}}}.$
(3.28)

Although the gowth rate of the joint and counterterm actions depend on the
parametrization constant $\xi$ for null surfaces, but it is found that the
rate of complexity from the total action (2.5) is independent of it. In other
words, the contribution of the joint action for $\xi$ is eliminated by the one
for counterterm action.

####  3.1.2 Late time behavior

To understand how $\dot{\cal{C}}_{A}$ violates the Lloyd’s bound, we first
investigate its late time behavior. Using the quantities in (3.10) and (3.11),
we obtain

$\dot{\cal{C}}_{A}\Big{|}_{LT}\\!=\\!\frac{dI_{WDW}}{dt}{\Bigg{|}}_{t>>t_{c}}\\!=\\!\frac{V_{2}}{8\pi
G_{4}}\left[\frac{9}{2}Qr_{h}^{2}+2r_{h}^{3}+r_{h}(3Q^{2}-\beta^{2})\right]=2M-\mu
q-\frac{V_{2}}{16\pi G_{4}}\left[Q^{3}-\frac{1}{2}Q\beta^{2}\right].$ (3.29)

It is obvious that in the extremal limit ($r_{h}=0$), the late time behavior
vanishes. Also, the result shows that not only the contribution of dilaton
theory changes the late time limit of charged black holes in (1.3), see e.g.
Ref. [78], but also the presence of momentum dissipation makes a difference.
The ratios $\dot{\cal{C}}_{A}/\big{(}\dot{\cal{C}}_{A}\big{)}_{LT}$ have
plotted for different values of $Q$ and $\beta$ in Fig. (2) where “$LT$”
stands for the late time behavior given in equation (3.29). As is obvious,
$\dot{\cal{C}}_{A}$ violates the Lloyd’s bound for this kind of black hole
from above.

Figure 2: The growth rate of complexity relative to its late time limit for
different values of charge and momentum relaxation.

###  3.2  $AdS_{5}$ black holes

The extension of GR model [18] in five dimensions to EMAD theories with
momentum relaxation has been proposed in Ref. [19]. The EMD part comes from
10D type IIB string theory with geometry $AdS_{5}\times S^{5}$ as the near
horizon limit of D3-branes [92]. With a similar trend, the translational
invariance is broken by adding a kinetic term of axionic scalar fields
$\psi_{I}$’s. The holographic charge transport and linear resistivity for
charged $AdS_{5}$ black holes in this model have been reviewed in Refs. [19,
90]. The bulk action of 5D EMAD theory is

$I_{bulk}\\!=\\!{1\over 16\pi
G_{5}}\\!\int\\!d^{5}x\sqrt{-g}\left[R\\!-\\!{1\over
4}e^{4\phi}F_{\mu\nu}^{2}\\!-\\!12(\partial_{\mu}\phi)^{2}\\!+\\!{1\over
L^{2}}(8e^{2\phi}\\!+\\!4e^{-4\phi})\\!-\\!\frac{1}{2}\sum_{I=1}^{3}{(\partial_{\mu}\psi_{I})}^{2}\right]\\!=\\!{1\over
16\pi G_{5}}\\!\int\\!d^{5}x\sqrt{-g}\mathcal{L}_{2},\,$ (3.30)

here the field content of this model is analogous to 4D case discussed in
section 3.1. Analytic charged solution in this theory is given by the general
ansatz (3.4) but with three-dimensional spatial space $d\vec{x}^{2}$, such
that

$\displaystyle A(r)=\log{r\over L}+{1\over 3}\log\left(1+{Q\over
r^{2}}\right),\quad B(r)=-\log{r\over L}-{2\over 3}\log\left(1+{Q\over
r^{2}}\right),\quad\psi_{I}=\beta\delta_{Ii}x^{i},$ $\displaystyle
h(r)=1-\frac{\beta^{2}}{4(Q+r^{2})}-\frac{(Q+r_{h}^{2})^{2}}{(Q+r^{2})^{2}}\bigl{(}1-\frac{\beta^{2}}{4(Q+r_{h}^{2})}\bigr{)},\quad\phi(r)={1\over
6}\log\left(1+{Q\over r^{2}}\right),$ (3.31) $\displaystyle
A_{t}(r)=\sqrt{2Q\bigl{(}1-\frac{\beta^{2}}{4(Q+r_{h}^{2})}\bigr{)}}(1-\frac{(Q+r_{h}^{2})}{(Q+r^{2})}).$

Again we set $L=1$ without lose of generality. This is an asymptotically $AdS$
black hole which can be rewritten with a warped factor $w(r)$ as

$ds^{2}=e^{w(r)}\left(-f(r)\,dt^{2}+{U(r)}d\vec{x}^{2}+\frac{1}{f(r)}\,{d}r^{2}\right),$
(3.32)

where

$w(r)=-{1\over 3}\log\left(1+{Q\over r^{2}}\right),\quad
f(r)=\frac{(r^{2}-r_{h}^{2})(2Q+r_{h}^{2}+r^{2})}{Q+r^{2}}-\frac{(r^{2}-r_{h}^{2})\beta^{2}}{4(Q+r^{2})},\quad{U(r)}=(Q+r^{2}).$

The mass of the black hole is given by

$M={V_{3}\over 16\pi
G_{5}}\,3\omega,\qquad\omega=(Q+r_{h}^{2})^{2}\left(1-\frac{\beta^{2}}{4(Q+r_{h}^{2})}\right),$
(3.33)

where $V_{3}$ is the volume of the 3-dimensional flat space. In the limit
$\beta=0$, it gives the mass parameter in Ref. [18] while in the limit $Q=0$,
yields the mass of neutral black branes we obtained in [79] for $d=4$. The
chemical potential and charge of the black hole are computed from the gauge
potential in (3.2) as

$\mu=\sqrt{2Q\bigl{(}1-\frac{\beta^{2}}{4(Q+r_{h}^{2})}\bigr{)}},\qquad
q={V_{3}\over 8\pi
G_{5}}(Q+r_{h}^{2})\sqrt{2Q\bigl{(}1-\frac{\beta^{2}}{4(Q+r_{h}^{2})}\bigr{)}}.$
(3.34)

On the other hand, the temperature and entropy of the black hole are

$T=\frac{1}{8\pi}r_{h}(8-\frac{\beta^{2}}{Q+r_{h}^{2}}),\qquad
S=\frac{V_{3}}{4G_{5}}r_{h}(Q+r_{h}^{2})$ (3.35)

so that one has

$TS=\frac{V_{3}}{8\pi
G_{5}}\frac{r_{h}^{2}}{4}\bigl{(}8(Q+r_{h}^{2})-\beta^{2}\bigr{)}.$ (3.36)

####  3.2.1 The growth rate of complexity

Following the prescription in section 2, we obtain the growth rate of
holographic complexity for charged $AdS_{5}$ black holes of 5D EMAD theory in
the CA picture. The WDW patch in this case is also described by Fig. (1) and
we need to compute the total action (2.5) on it. In this regard, the
contribution of the bulk action on three different regions I, II, and III for
times $t>t_{c}$ is given by

$\displaystyle I_{bulk}$ $\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle
2\,\,\left(I_{bulk}^{\textup{I}}+I_{bulk}^{\textup{II}}+I_{bulk}^{\textup{III}}\right)$
(3.37) $\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$
$\displaystyle\frac{V_{3}}{8\pi
G_{5}}\left[\int_{\epsilon}^{r_{h}}\left(\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)+2\int_{r_{h}}^{r_{max}}\left({r}_{\infty}^{*}-r^{*}(r)\right)+\int_{r_{m}}^{r_{h}}\left(-\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)\right]\,\sqrt{-g}\,\mathcal{L}_{2}\,dr,$
$\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!$ $\displaystyle
I_{bulk}^{0}+\frac{V_{3}}{8\pi
G_{5}}\int_{\epsilon}^{r_{m}}\left(\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)\,\sqrt{-g}\,\mathcal{L}_{2}\,dr,$

where $I_{bulk}^{0}$ is the independency of the time and $\mathcal{L}_{2}$ is
the Lagrangian density of (3.30) which on the background (3.2) we have

$\sqrt{-g}{\mathcal{L}}_{2}=-\frac{8r\bigl{(}Q^{3}+Q^{2}(7r^{2}-2r_{h}^{2})+Q(8r^{4}-r_{h}^{4})+3r^{6}\bigr{)}}{3(Q+r^{2})^{2}}-\frac{2rQ(Q+r_{h}^{2})\beta^{2}}{3(Q+r^{2})^{2}}.$
(3.38)

Thus the time derivative of (3.37) yields

$\frac{dI_{bulk}}{dt}=\frac{V_{3}}{8\pi
G_{5}}\left[-\tfrac{4}{3}Qr^{2}-r^{4}-\frac{2Q(Q+r_{h}^{2})^{2}}{3(Q+r^{2})}+\frac{Q(Q+r_{h}^{2})\beta^{2}}{6(Q+r^{2})}\right]_{\epsilon}^{r_{m}}.$
(3.39)

Note that the time evolution of this result comes from the differential
equation of $r_{m}$ for the past null junction given in (3.22).

In order to find the extrinsic curvature for geometry (3.32), we define the
normal vector $n_{\mu}=(0,n_{r},0,0,0)$ where $n_{r}=\sqrt{\frac{f(r)}{W(r)}}$
and for simplicity we choose $W(r)=e^{w(r)}$. Thus, from the definition in
section 2 one can achieve the following expression

$K=\frac{4f(r)\,U(r)\,W(r)^{\prime}+W(r)\bigl{(}3f(r)\,U(r)^{\prime}+f(r)^{\prime}\,U(r)\bigr{)}}{2f(r)^{1/2}\,U(r)\,W(r)^{3/2}},$
(3.40)

where prime is the derivative with respect to $r$. For times $t>t_{c}$ the
boundary action (2.1) becomes

$I_{surf}=I_{surf}^{0}+\frac{V_{3}}{4\pi
G_{5}}\sqrt{h}K\,\left(\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)\Big{|}_{r=\epsilon}.$
(3.41)

Therefore its time derivative gives

$\frac{dI_{surf}}{dt}=\frac{V_{3}}{8\pi
G_{5}}\sqrt{h}{K{\Bigg{|}}_{r=\epsilon}},$ (3.42)

in which

$\displaystyle\sqrt{h}K$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{2\bigl{(}Q^{2}(7r^{2}-4r_{h}^{2})+2Q(7r^{4}-r_{h}^{4}-3r_{h}^{2}r^{2})+3r^{2}(2r^{4}-r_{h}^{4})\bigr{)}}{3(Q+r^{2})}$
(3.43) $\displaystyle\\!\\!\\!-\\!\\!\\!$
$\displaystyle\frac{\bigl{(}6r_{h}^{2}r^{2}-9r^{4}+Q(4r_{h}^{2}-7r^{2})\bigr{)}\beta^{2}}{12(Q+r^{2})}.$

For the null joint contribution of intersecting the two past null boundaries
at $r=r_{m}$ with outward-directed null vectors defined in (3.20) we have

$I_{jnt}=-\frac{V_{3}}{8\pi
G_{5}}\,\left[U(r)^{3/2}\,W(r)^{3/2}\log\Bigl{(}\frac{\bigl{|}W(r)f(r)\bigr{|}}{\xi^{2}}\Bigr{)}\right]_{r=r_{m}},$
(3.44)

where the implicit time dependence of this term is through the Eq. (3.22). For
the background functions in (3.32), the joint term can then be evaluated as

$\displaystyle\frac{dI_{jnt}}{dt}$ $\displaystyle=$
$\displaystyle\frac{V_{3}}{8\pi
G_{5}}\Bigg{[}\frac{(Q+3r^{2})(r^{2}-r_{h}^{2})(2Q+r^{2}+r_{h}^{2})}{3r^{1/3}(Q+r^{2})^{4/3}}\,\log\Bigl{(}\frac{\bigl{|}W(r)f(r)\bigr{|}}{\xi^{2}}\Bigr{)}$
(3.45) $\displaystyle+$
$\displaystyle\frac{8Q^{2}r^{2}+7Qr^{4}+3r^{6}-(Q-3r^{2})(2Q+r_{h}^{2})r_{h}^{2}}{3r^{1/3}(Q+r^{2})^{4/3}}$
$\displaystyle-$
$\displaystyle\left(\frac{(Q+3r^{2})(r^{2}-r_{h}^{2})}{12r^{1/3}(Q+r^{2})^{4/3}}\,\log\Bigl{(}\frac{\bigl{|}W(r)f(r)\bigr{|}}{\xi^{2}}\Bigr{)}+\frac{Q(r^{2}-r_{h}^{2})+3r^{2}(Q+r_{h}^{2})}{12r^{1/3}(Q+r^{2})^{4/3}}\right)\beta^{2}\Bigg{]}_{r=r_{m}},$

in which

$\log\Bigl{(}\frac{\bigl{|}W(r)f(r)\bigr{|}}{\xi^{2}}\Bigr{)}=\log\bigl{(}\frac{r^{2/3}(r^{2}-r_{h}^{2})(2Q+r^{2}+r_{h}^{2}-{\beta^{2}}/4)}{\xi^{2}(Q+r^{2})^{4/3}}\bigr{)}.$
(3.46)

In the late time limit one can find that the growth rate of the joint action
is proportional to the product of the temperature and entropy

$\frac{dI_{jnt}}{dt}{\Bigg{|}}_{t>>t_{c}}=\frac{V_{3}}{8\pi
G_{5}}\Bigg{[}2r_{h}^{2}(Q^{2}+r_{h}^{2})-\tfrac{1}{4}r_{h}^{2}\beta^{2}\Bigg{]}=T\,S,$
(3.47)

where in the extremal limit vanishes.

The boundary counterterm (2.3) requires evaluating the expansion scalar (2.4)
in the null boundaries of the WDW patch, thus with the parametrization
$\lambda=r/\xi$ we obtain

$\Theta=\frac{3\xi}{2}\left(\frac{U(r)^{\prime}}{U(r)}+\frac{W(r)^{\prime}}{W(r)}\right).$
(3.48)

Similar to the action (3.26) the evaluation for $\epsilon$ vanishes and the UV
regulator surface at $r_{max}$ only contributes a fixed constant, therefore
the time dependence comes only from the term evaluated at the meeting point
$r_{m}$. Now, by virtue of dynamical equation (3.22) we can determine time
evolution of the counterterm action as

$\frac{dI_{ct}}{dt}=\frac{V_{3}}{16\pi
G_{5}}\Bigg{[}\frac{(r^{2}-r_{h}^{2})(Q+3r^{2})(2Q+r_{h}^{2}+r^{2}-\beta^{2}/4)}{(Q+r^{2})}\,\log\left(\frac{l_{c}\xi(Q+3r^{2})}{r(Q+r^{2})}\right)\Bigg{]}_{r=r_{m}}.$
(3.49)

####  3.2.2 Late time behavior

The rate of the complexity at very late times for the total action (2.5) is
given by

$\frac{dI_{WDW}}{dt}{\Bigg{|}}_{t>>t_{c}}=\frac{V_{3}}{8\pi
G_{5}}\Bigg{[}4Qr_{h}^{2}+3r_{h}^{4}-\tfrac{3}{4}r_{h}^{2}\beta^{2}\Bigg{]},$
(3.50)

where by using the quantities in Eqs. (3.33) and (3.34) we can rewrite it as

$\dot{\cal{C}}_{A}\Big{|}_{LT}=\frac{dI_{WDW}}{dt}{\Bigg{|}}_{t>>t_{c}}=2M-\mu
q-\frac{V_{3}}{16\pi G_{5}}\left[2Q^{2}-\frac{1}{2}Q\beta^{2}\right].$ (3.51)

As is obvious, the momentum relaxation and dilatonic field terms have explicit
contribution in the Lloyd’s bound due to the charge parameter $Q$. The ratios
$\dot{\cal{C}}_{A}/\big{(}\dot{\cal{C}}_{A}\big{)}_{LT}$ have plotted for
different values of $Q$ and $\beta$ in Fig. (3). As seen in figure,
$\dot{\cal{C}}_{A}$ violates the Lloyd’s bound for this kind of black hole
from above.

Figure 3: The growth rate of complexity relative to its late time limit for
different values of the charge and momentum relaxation.

###  3.3 Warped $AdS_{d+1}$ black holes

Since the 4D and 5D EMAD theories used to studied had analytic solutions, it
would be of interest to investigate a generalized theory in arbitrary
dimensions. This model has been proposed in Ref. [19] which still allows
analytic charged solutions. The holographic characteristics of strange metals
for this model have been studied in Refs. [19, 90].

Let us consider a general $(d+1)$-dimensional EMAD theory given by the action
(3.1) with the following Lagrangian density

$\mathcal{L}_{3}=R-\frac{1}{2}(\partial_{\mu}\phi)^{2}+V(\phi)-\frac{1}{4}Z(\phi)F_{\mu\nu}^{2}-\frac{1}{2}\sum_{i=1}^{d-1}{(\partial_{\mu}\psi_{I})}^{2},$
(3.52)

where the functions $Z(\phi)$ and $V(\phi)$ are assumed to have the following
specific forms [19]

$Z(\phi)=e^{-(d-2)\delta\phi},\qquad
V(\phi)=V_{1}e^{\frac{(d-2)(d-1)\delta^{2}-2}{2(d-1)\delta}\phi}+V_{2}e^{-\frac{2}{(d-1)\delta}\phi}+V_{3}e^{(d-2)\delta\phi}.$
(3.53)

Here $V_{1}$, $V_{2}$, and $V_{3}$ are three constant parameters given by

$\displaystyle V_{1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{8(d-2)(d-1)^{3}\delta^{2}}{\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}^{2}},\qquad
V_{2}=\frac{(d-2)^{2}(d-1)^{2}\bigl{(}d(d-1)\delta^{2}-2\bigr{)}\delta^{2}}{\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}^{2}},$
$\displaystyle V_{3}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{4d(d-1)-2(d-1)^{2}(d-2)^{2}\delta^{2}}{\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}^{2}},$
(3.54)

where $\delta$ is a free parameter. It should be noticed that for $d=3$ by
choosing $\delta=\sqrt{\frac{1}{3}}$ and redefinition
$\phi\rightarrow-\sqrt{3}\phi$ the bulk action (3.52) reduced to 4D EMAD
theory given in (3.5), while for $d=4$, $\delta=\sqrt{\frac{1}{6}}$ and
redefinition $\phi\rightarrow-\sqrt{24}\phi$, it yields the 5D theory
described by (3.30).

The Eqs. (3) admit an analytical charged black hole solution defined by the
ansatz (3.4) with the following quantities

$\displaystyle
A=\frac{1}{2}\log\Bigl{(}r^{2-\frac{4}{(d-1)(d-2)\delta^{2}+2}}(Q+r^{d-2})^{\frac{4}{(d-2)\left((d-1)(d-2)\delta^{2}+2\right)}}\Bigr{)},\quad\psi_{I}=\beta\delta_{Ii}x^{i},$
$\displaystyle
B=\frac{1}{2}\log\bigl{(}r^{\frac{4(d-2)}{(d-1)(d-2)\delta^{2}+2}-2}(Q+r^{d-2})^{-\frac{4}{(d-1)(d-2)\delta^{2}+2}}\bigr{)},\quad\phi(r)=-\frac{2(d-1)\delta}{(d-1)(d-2)\delta^{2}+2}\log\left(1+\frac{Q}{r^{d-2}}\right),$
$\displaystyle
h(r)=\bigl{(}1-\frac{\beta^{2}}{2(d-2)(Q+r^{d-2})^{\frac{2}{d-2}}}\bigr{)}-\frac{(Q+r_{h}^{d-2})^{\frac{d}{d-2}}}{(Q+r^{d-2})^{\frac{d}{d-2}}}\bigl{(}1-\frac{\beta^{2}}{2(d-2)(Q+r_{h}^{d-2})^{\frac{2}{d-2}}}\bigr{)},$
(3.55) $\displaystyle
A_{t}(r)=\sqrt{\frac{dQ}{d-2}}(Q+r_{h}^{d-2})^{-\frac{d-4}{2(d-2)}}\sqrt{1-\frac{\beta^{2}}{2(d-2)(Q+r_{h}^{d-2})^{\frac{2}{d-2}}}}(1-\frac{Q+r_{h}^{d-2}}{Q+r^{d-2}}).$

Here, $r_{h}$ is the location of the event horizon and we have set here $L=1$.
In analogy to the previous sections, we can recast this geometry as an
asymptotically $AdS_{d+1}$ black hole

$ds^{2}=e^{w(r)}\left(-f(r)\,dt^{2}+{U(r)}\,d\Sigma_{d-1}^{2}+\frac{1}{f(r)}\,\,\mathrm{d}r^{2}\right),$
(3.56)

where $W(r)=e^{w(r)}$ and the corresponding functions are obtained as

$\displaystyle
f(r)=g(r)^{-\frac{2(d-1)}{(d-2)\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}}}\mathcal{F}(r),\quad
U(r)=g(r)^{\frac{d-1}{2(d-2)\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}}},$
$\displaystyle
W(r)=g(r)^{-\frac{2(d-3)}{(d-2)\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}}},\quad
g(r)=1+\frac{Q}{r^{d-2}},$ (3.57)
$\displaystyle\mathcal{F}(r)=r^{2}\Bigl{(}-\frac{r_{h}^{d}}{r^{d}}g(r_{h})^{\frac{4(d-1)}{(d-2)\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}}}+g(r)^{\frac{4(d-1)}{(d-2)\bigl{(}(d-1)(d-2)\delta^{2}+2\bigr{)}}}\Bigr{)}-\frac{\beta^{2}}{2(d-2)}(1-\frac{r_{h}^{d-2}}{r^{d-2}}).$

In this paper we will consider the case with $\delta=\sqrt{\frac{2}{d(d-1)}}$
which corresponds to the GR model in higher dimensions and the IR geometry is
conformal to $AdS_{2}\times R^{d-1}$ [19]. It is noticed that $\delta=0$
corresponds to a Reissner-Nordstrom $AdS$ black hole but in general one can
consider an arbitrary value of
$\delta\in\left[0,\sqrt{\frac{2}{d(d-1)}}\right]$. Further discussion for
$\delta=\frac{1}{\sqrt{3}}$ could be found in Ref. [90]. Due to this fact, the
mass of the black hole is given by

$M={V_{d-1}\over 16\pi
G_{d+1}}(d-1)\,\omega,\quad\omega=(Q+r_{h}^{d-2})^{\frac{d}{d-2}}\bigl{(}1-\frac{\beta^{2}}{2(d-2)(Q+r_{h}^{d-2})^{\frac{2}{d-2}}}\bigr{)},$
(3.58)

where $V_{d-1}$ is the volume of ($d-1$)-dimensional flat space with the line
element $d\Sigma_{d-1}^{2}$. The chemical potential and charge of the black
hole can be found from $A_{t}(r)$ in (3.3) as

$\displaystyle q$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle{V_{d-1}\over 16\pi
G_{d+1}}\sqrt{d(d-2)Q(Q+r_{h}^{d-2})^{\frac{d}{(d-2)}}\left(1-\frac{\beta^{2}}{2(d-2)(Q+r_{h}^{d-2})^{\frac{2}{d-2}}}\right)}\,,$
$\displaystyle\mu$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\sqrt{\frac{dQ}{d-2}\,(Q+r_{h}^{d-2})^{-\frac{d-4}{(d-2)}}\left(1-\frac{\beta^{2}}{2(d-2)(Q+r_{h}^{d-2})^{\frac{2}{d-2}}}\right)}\,.$
(3.59)

The temprature and entropy of the black hole at the event horizon are given by

$T=\frac{r_{h}^{d/2-1}}{4\pi}\biggl{(}d(Q+r_{h}^{d-2})^{-\frac{d-4}{2(d-2)}}-\frac{1}{2}(Q+r_{h}^{d-2})^{-\frac{d}{2(d-2)}}\beta^{2}\biggr{)},\quad
S=\frac{V_{d-1}}{4G_{d+1}}r_{h}^{d/2-1}(Q+r_{h}^{d-2})^{\frac{d}{2(d-2)}},$
(3.60)

such that

$TS=\frac{V_{d-1}}{16\pi
G_{d+1}}\bigg{[}dr_{h}^{d-2}(Q+r_{h}^{d-2})^{\frac{2}{d-2}}-\tfrac{1}{2}r_{h}^{d-2}\beta^{2}\bigg{]}.$
(3.61)

####  3.3.1 The growth rate of complexity

The time evolution of holographic complexity for $AdS$ black holes of
generalized EMAD theory on the WDW patch shown in Fig. (1) is determined by
the rate of action (2.5) via CA proposal. Therefore, for the bulk action (3.1)
on the right sector of WDW patch denoted by regions I, II and III we have

$\displaystyle
I_{bulk}(t>t_{c})=2\left(I_{bulk}^{\textup{I}}+I_{bulk}^{\textup{II}}+I_{bulk}^{\textup{III}}\right)=I_{bulk}^{0}+\frac{V_{d-1}}{8\pi
G_{d+1}}\int_{\epsilon}^{r_{m}}\left(\frac{t}{2}+{r}_{\infty}^{*}-r^{*}(r)\right)\,\sqrt{-g}\,\mathcal{L}_{3}\,dr,$
(3.62)

where similar to the previous subsections, $I_{bulk}^{0}$ is independent of
the time and $\mathcal{L}_{3}$ is the Lagrangian density in Eq. (3.52)
evaluated on the background (3.3). Thus, the growth rate of the bulk
contribution becomes

$\frac{dI_{bulk}}{dt}=\frac{V_{d-1}}{8\pi
G_{d+1}}\Bigg{[}-\frac{\bigl{(}\frac{(d-2)Q}{2(d-1)}+r^{d-2}\bigr{)}}{(Q+r^{d-2})^{-\frac{2}{d-2}}}+\frac{dQ(Q+r_{h}^{d-2})^{\frac{d}{d-2}}}{2(d-1)(Q+r^{d-2})}+\frac{dQ(Q+r_{h}^{d-2})\beta^{2}}{4(d-1)(d-2)(Q+r^{d-2})}\Bigg{]}_{\epsilon}^{r_{m}}.$
(3.63)

The extrinsic curvature for the boundary GHY surface term calculated for the
normal vector with $n_{r}=\sqrt{\frac{f(r)}{W(r)}}$ is obtained from (3.56) as

$\sqrt{h}K=\frac{1}{2}\,U(r)^{\frac{(d-1)}{2}}W(r)^{\frac{(d-1)}{2}}f(r)^{\prime}+\frac{1}{2}\,f(r)\,U(r)^{\frac{(d-1)}{2}}\,W(r)^{\frac{(d-3)}{2}}\left((d-1)W(r)U(r)^{\prime}+d\,U(r)W(r)^{\prime}\right),$
(3.64)

which from (3.3) it yields

$\displaystyle\sqrt{h}K$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\,d(Q+r^{d-2})^{\frac{d}{d-2}}+\frac{d^{2}Q}{2(d-1)(Q+r^{d-2})^{-\frac{2}{d-2}}}+\frac{d\bigl{(}\frac{(d-2)Q}{d-1}+r^{d-2}\bigr{)}(Q+r_{h}^{d-2})^{\frac{d}{d-2}}}{2(Q+r^{d-2})}$
(3.65)
$\displaystyle+\frac{Q\bigl{(}(2d-1)r^{d-2}-dr_{h}^{d-2}\bigr{)}\beta^{2}}{4(d-1)(Q+r^{d-2})}+\frac{r^{d-2}\bigl{(}2(d-1)r^{d-2}-dr_{h}^{d-2}\bigr{)}\beta^{2}}{4(d-2)(Q+r^{d-2})}.$

As noticed in Eq. (3.18) the boundary contribution coming from the time-like
surface at the UV cutoff regulator $r=r_{max}$ yields a fixed constant, i.e.,
they do not contribute to the time derivative of the action, and with
affinely-parametrized null normals ($\kappa=0$), the null surface term
vanishes. Thus, we only need to consider the boundary surface at the future
singularity which is given by

$\frac{dI_{surf}}{dt}=\frac{V_{d-1}}{8\pi
G_{d+1}}\sqrt{h}K{\Bigg{|}}_{r=\epsilon}.$ (3.66)

The final result for the joint action at the meeting point $r=r_{m}$, shown in
Fig. (1), is given by

$I_{jnt}=-\frac{V_{d-1}}{8\pi
G_{d+1}}\left[U(r)^{\frac{d-1}{2}}W(r)^{\frac{d-1}{2}}\log\bigl{(}\frac{f(r)W(r)}{\xi^{2}}\bigr{)}\right]_{r=r_{m}},$
(3.67)

where $\xi$ is the normalization constant appearing in the null normals
(3.20), i.e., $k\cdot\partial_{t}\big{|}_{r\rightarrow\infty}=\pm\xi$.
According to the time dependence of $r_{m}$ through Eq. (3.22) the evolution
of this action has the following form

$\displaystyle\frac{dI_{jnt}}{dt}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{V_{d-1}}{8\pi
G_{d+1}}\Bigg{[}\frac{(Q+r^{d-2})^{\frac{2}{d-2}}\bigl{(}(d-2)Q+2(d-1)r^{d-2}\bigr{)}\bigl{(}2+(d-1)\log(\frac{\mathcal{F}(r)}{\xi^{2}})\bigr{)}}{4(d-1)}$
$\displaystyle\\!\\!\\!-\\!\\!\\!$
$\displaystyle\frac{(d-2)\bigl{(}Q-(d-1)r^{d-2}\bigr{)}(Q+r_{h}^{d-2})^{\frac{d}{d-2}}}{2(d-1)(Q+r^{d-2})}+\frac{\bigl{(}(1+\frac{Q}{r^{d-2}})-d(1+\frac{Q}{r_{h}^{d-2}})\bigr{)}(r\,r_{h})^{d-2}\beta^{2}}{4(d-1)(Q+r^{d-2})}$
$\displaystyle\\!\\!\\!-\\!\\!\\!$
$\displaystyle\frac{\bigl{(}(d-2)Q+2(d-1)r^{d-2}\bigr{)}(Q+r_{h}^{d-2})^{\frac{d}{d-2}}\log(\frac{\mathcal{F}(r)}{\xi^{2}})}{4(Q+r^{d-2})}$
$\displaystyle\\!\\!\\!-\\!\\!\\!$
$\displaystyle\frac{\bigl{(}(d-2)Q+2(d-1)r^{d-2}\bigr{)}(r^{d-2}-r_{h}^{d-2})\beta^{2}\log(\frac{\mathcal{F}(r)}{\xi^{2}})}{8(d-2)(Q+r^{d-2})}\Bigg{]}_{r=r_{m}},$

in which

$\mathcal{F}(r)=r^{\frac{d-2}{d-1}}(Q+r^{d-2})^{-\frac{d}{d-1}}\left((Q+r^{d-2})^{\frac{d}{d-2}}-(Q+r_{h}^{d-2})^{\frac{d}{d-2}}-\frac{\beta^{2}}{2(d-2)}\Bigl{(}r^{d-2}-r_{h}^{d-2}\Bigr{)}\right).$
(3.69)

As shown before, in the late time limit $r_{m}$ coincides with the event
horizon $r_{h}$, then from Eq. (3.61) the joint term (3.3.1) leads to

$\frac{dI_{jnt}}{dt}{\Bigg{|}}_{t>>t_{c}}=\frac{V_{d-1}}{16\pi
G_{d+1}}\bigg{[}dr_{h}^{d-2}(Q+r_{h}^{d-2})^{\frac{2}{d-2}}-\frac{1}{2}r_{h}^{d-2}\beta^{2}\bigg{]}=TS.$
(3.70)

By fixing the parametrization of null generators as $\lambda=r/\xi$, the
boundary counterterm action for the null boundaries is defined by the
following expansion parameter

$\Theta=\frac{(d-1)\xi}{2}\left(\frac{U(r)^{\prime}}{U(r)}+\frac{W(r)^{\prime}}{W(r)}\right)=\frac{(d-2)Q+2(d-1)r^{d-2}}{2r(Q+r^{d-2})}\xi,$
(3.71)

such that the time evolution of counterterm action becomes

$\displaystyle\frac{dI_{ct}}{dt}$ $\displaystyle=$
$\displaystyle\frac{V_{d-1}}{16\pi
G_{d+1}}\Bigg{[}\frac{(d-2)Q+2(d-1)r^{d-2}}{(Q+r^{d-2})}\biggl{(}\frac{\bigl{(}(Q+r^{d-2})^{\frac{d}{d-2}}-(Q+r_{h}^{d-2})^{\frac{d}{d-2}}\bigr{)}}{2}$
(3.72)
$\displaystyle\qquad\qquad-\frac{(r^{d-2}-r_{h}^{d-2})}{4(d-2)}\beta^{2}\biggr{)}\times\log\biggl{(}\frac{(d-2)Q+2(d-1)r^{d-2}}{2r(Q+r^{d-2})}l_{c}\xi\biggr{)}\Bigg{]}_{r=r_{m}}.$

####  3.3.2 Late time behavior

Now we are ready to calculate the rate of holographic complexity at large
times for the total action $I_{WDW}$ by sum over the above results according
to Eq. (2.5) as

$\displaystyle\dot{\cal{C}}_{A}\Big{|}_{LT}$ $\displaystyle=$
$\displaystyle\frac{dI_{WDW}}{dt}{\Bigg{|}}_{t>>t_{c}}$ $\displaystyle=$
$\displaystyle\frac{V_{d-1}}{16\pi
G_{d+1}}\Bigg{[}-(d-2)Q^{\frac{d}{d-2}}+(Q+r_{h}^{d-2})^{\frac{2}{d-2}}\left((d-2)Q+2(d-1)r_{h}^{d-2}\right)-\frac{(d-1)r_{h}^{d-2}\beta^{2}}{(d-2)}\Bigg{]}.$

Substituting from (3.58) and (3.3) in Eq. (3.3.2) we obtain

$\dot{\cal{C}}_{A}\Big{|}_{LT}=2M-\mu q-\frac{V_{d-1}}{16\pi
G_{d+1}}\Bigg{[}(d-2)Q^{\frac{d}{d-2}}-\frac{1}{2}Q\beta^{2}\Bigg{]}.$ (3.74)

In the absence of momentum relaxation term this result reduces to the
following Lloyd’s bound for the $(d+1)$-dimensional charged $AdS$ black holes
in the generalized GR model

$\dot{\cal{C}}_{A}\Big{|}_{LT}=2M_{0}-\mu_{0}q_{0}-D,\qquad
D=\frac{V_{d-1}}{16\pi G_{d+1}}(d-2)Q^{\frac{d}{d-2}},$ (3.75)

where the quantities $M_{0}$, $\mu_{0}$ and $q_{0}$ are given by the relations
(3.58) and (3.3) when $\beta=0$. It should be also noted that both of Eqs.
(3.74) and (3.75) vanish in the extremal limit $r_{h}=0$.

##  4 Conclusions and outlook

One of the main implications in the context of $AdS/CFT$ duality, which
relates the physics of the black holes in gravity side to the information
theory in dual field theory, is the notion of quantum complexity. In other
words, holographic methods are powerful tools to study the properties of
strongly correlated systems by mapping them to the dual weakly interacting
systems. In this regard, the GR models are of great importance. In this paper,
we investigated the holographic complexity for the charged $AdS$ black holes
in these models when the translational invariance is broken in diverse
dimensions. Following the prescription in [54] with the inclusion of joint and
counterterm actions for null boundaries [80], we studied the growth rate of
complexity on the WDW patch shown in Fig. (1) for a modification of GR model
to holographic theory containing axion fields which yields the momentum
relaxation (EMAD theories).

We have shown that though the time evolution of the joint and counterterm for
null surfaces depend separately on the normalization condition $\xi$, which is
needed to remove the reparametrization ambiguity of null boundaries, the sum
of the two terms in $\dot{\cal{C}}_{A}=dI_{WDW}/dt$ is independent of $\xi$.
We examined the standard calculations for the three most used models, i.e., 4,
5, and $(d+1)$-dimensional EMAD theories with single-horizon charged $AdS$
black holes. According to the curves in Figs. (2) and (3), the growth rate of
holographic complexity is finite for early times in each case although it
violates the Lloyd’s bound. However, at late times it saturates the
corresponding bound which is not $2M$. In particular, the results for GR model
and EMAD theories in arbitrary dimensions are given by Eqs. (3.75) and (3.74),
respectively.

To have a comparison of our results with other literature, we have summarized
the late time behavior of $\dot{\cal{C}}_{A}$ for neutral and charged $AdS$
black holes in different theories in Tab. (1). It is shown that for neutral
black holes of mass $M$, the Lloyd’s bound is equal $2M$ even by adding the
momentum relaxation term or higher order curvature corrections. For charged
solutions with single horizon, we have the contribution of chemical potential
term too. When there are non-linear expressions of dilatonic or gauge fields,
we have extra term in the bound due to these corrections 333In the table we
have set $V_{d-1}=16\pi G$ and the constants are
$C=b_{0}^{1/2}Q^{3/2}\frac{\Gamma\left(\frac{1}{4}\right)\Gamma\left(\frac{5}{4}\right)}{3\Gamma\left(\frac{1}{2}\right)}$,
$D=\frac{Q^{2}e^{2\phi_{0}}}{2M}$ and
$C_{0}=\frac{g^{2}r_{\star}^{3}(\beta\gamma+4\kappa)}{10\pi}$.. Finally, we
have shown that in the presence of momentum relaxation term in EMAD theory we
have a correction of $Q$ and $\beta$ which are given in the last two rows in
Tab. (1).

Table 1: Late time limit of $\dot{\cal{C}}_{A}$ for single horizon black holes in different theories. Model |  $\big{(}\dot{\cal{C}}_{A}\big{)}_{LT}$
---|---
Neutral $AdS$ BHs of Einstein theory [49, 54] |  $2M$
Neutral $AdS$ BHs of EMA theory [79] |  $2M$
Neutral $AdS$ BHs of higher curvature theories [66, 93, 70] |  $2M$
Anisotropic BHs of Mateos and Trancanelli theory [94] |  $2M$
Extremal RN-$AdS$ BHs of EM theory [48] |  $2M-\mu\,q$
Charged $AdS$ BHs of Born-Infeld theory [78] |  $2M-\mu\,q-C$
Charged $AdS$ BHs of EMD theory [78][95] |  $2M-\mu\,q-D$
Charged $AdS$ BHs of Horndeski theory [96] |  $2M-\mu\,q-C_{0}$
Charged $AdS$ BHs of generalized GR theory |  $2M-\mu\,q-(d-2)Q^{\frac{d}{d-2}}$
Charged $AdS$ BHs of EMAD theory |  $2M-\mu\,q-(d-2)Q^{\frac{d}{d-2}}+\frac{1}{2}Q\beta^{2}$

It would be also of interest to study the holographic complexity for other
generalization of the holographic axion models in [21, 87] by considering a
special gauge-axion higher derivative term [97, 98] of the form

$S=\frac{1}{16\pi G_{4}}\int
d^{4}x\sqrt{-g}\left[R-2\Lambda-V(X)-\frac{1}{4}\left(1+\mathcal{K}Tr[X]\right)F_{\mu\nu}F^{\mu\nu}\right],$
(4.1)

where $X^{I}=\beta\delta_{i}^{I}x^{i}$ are the axionic scalar fields, or other
holographic axion models in [99, 100].

## Acknowledgment

The authors would like to thank M. R. Mohammadi Mozaffar for his valuable
comments and discussion.

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