\section{Introduction}

The AdS/CFT correspondence \cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}, as the most important realization of the holographic principle \cite{tHooft:1993dmi,Susskind:1994vu}, relates a gravity theory in an asymptotically anti-de Sitter (AdS) spacetime in the bulk to a conformal field theory (CFT) without gravity living on the boundary of this spacetime. It suggests non-trivial connections between different areas of physics, in particular between general relativity and quantum information theory. One of the outstanding developments in this correspondence is the seminal work of Ryu and Takayanagi \cite{Ryu:2006bv,Ryu:2006ef}, which provided a holographic dictionary for the calculation of entanglement entropy of the boundary theory. According to this proposal, the entanglement entropy of the boundary theory is equivalent to the area of a certain minimal surface in the bulk geometry. In other words, the dynamics of the bulk spacetime emerge from the quantum entanglement of the boundary theory \cite{Faulkner:2013ica}. However, the entanglement entropy may not be enough to probe the degrees of freedom in black holes interior since the volume of black holes continues growing even if spacetimes reach thermal equilibrium \cite{Susskind:2014moa}. It is believed that quantum complexity is the correct quantity which can continue to growth even after reaching thermal equilibrium, similar to the growth of black hole interior.

In the framework of quantum information theory, quantum complexity is defined by the minimal number of quantum gates needed to build a target state from a reference state \cite{watrous2009quantum,Aaronson:2016vto}.  However, the AdS/CFT correspondence provides for us two proposals to compute the complexity of states in the boundary quantum field theory of the two-sided AdS black holes. The first is the complexity=volume (CV) conjecture which assumes that the quantum complexity of the CFT on boundary is dual to the maximum volume of the Einstein-Rosen Bridge, i.e. $V$, in bulk spacetime \cite{Susskind:2014rva,Stanford:2014jda},
\begin{equation}} \def\ee{\end{equation}\label{CV} \mathcal{C}_{V}\sim \frac{V}{G\ell_{AdS}},\ee
where $\ell_{AdS}$ is the radius of curvature of the AdS spacetime and G is Newton's constant. The second is the complexity=action (CA) conjecture which assumes that the quantum complexity on boundary is associated to the gravitational action evaluated on a region of Wheeler-DeWitt (WDW) patch in the bulk spacetime \cite{Brown:2015bva,Brown:2015lvg},
\begin{equation}} \def\ee{\end{equation}\label{CA} \mathcal{C}_A \sim \frac{I_{WDW}}{\pi\hbar}.\ee
In this regard, when the WDW patch beside space/time-like boundaries includes null boundary surfaces \cite{Hayward:1993my} which can join with each other, the strategy in CA conjecture has suggested in Ref. \cite{Lehner:2016vdi}.

In general, the holographic complexity has been studied in Ref. \cite{Maldacena:2001kr}  for the eternal two-sided AdS black holes in the gravity side. From the field theory point of view, this geometry is dual to a thermofield double satate
\begin{equation}} \def\ee{\end{equation}\label{TFD} |\psi_{TFD}\rangle=\frac{1}{\sqrt{Z}}\sum_j e^{-{E_j}/({2T})} e^{-i E_{j}(t_{L}+t_{R})}|E_j\rangle_L |E_j\rangle_R\,,\ee
where $L$ and $R$ refer to the two copies of the
boundary CFTs. The entanglement between them is due to the Einstein-Rosen bridge that connects the two regions. Since the complexity is conjectured to grow with time and this property is also shared with the Einstein-Rosen bridge, it was conjectured in Refs. \cite{Maldacena:2001kr,Maldacena:2013xja} that the complexity could be identified with the volume of the maximal co-dimension one surface that ends at the boundary times $t_L$ and $t_R$.

The growth rate of complexity is a one of the noticeable outcomes in the CA conjecture that asserts the late time growth rate is proportional to $2M/\pi$, independent of the boundary curvature and the spacetime dimensions \cite{Brown:2015bva,Brown:2015lvg}. It was also suggested that this quantity has an upper bound which is proportional to the total energy of the system
\begin{equation}} \def\ee{\end{equation}\label{LB} \dot{\mathcal{C}}\leq \frac{2E}{\pi\hbar},\ee
 where known as Lloyd's bound on the rate of quantum computation \cite{Lloyd}. In the gravitational picture the mass of the black hole, $M$, is regarded as the energy, $E$. The generalization of this bound for charged and rotating black holes are given in Refs. \cite{Brown:2015lvg,Cai:2016xho}, respectively as follows
 \begin{equation}} \def\ee{\end{equation}
 \frac{dI_{WDW}}{\pi\hbar}\leq2\left[(M-\mu Q)-(M-\mu Q)_{gs}\right],\qquad
 \frac{dI_{WDW}}{\pi\hbar}\leq2\left[(M-\Omega J )-(M-\Omega J )_{gs}\right],
 \ee
where $\mu$ and $\Omega$ are the chemical potential and angular velocity of the black holes, $Q$ and $J$ are the black hole charge and angular momentum, and the subscript $gs$ denotes the ground state of the black hole. There is a large number of efforts on different aspects of both the CV and CA conjectures, such as time-evolution of complexity \cite{Brown:2017jil,Carmi:2017jqz}, subregion complexity \cite{Alishahiha:2015rta,Carmi:2016wjl,Ben-Ami:2016qex,Alishahiha:2018lfv}, UV divergencies of complexity \cite{Cai:2016xho,Chapman:2016hwi,Reynolds:2016rvl,Kim:2017lrw,Alishahiha:2019cib}, higher derivative gravities \cite{Alishahiha:2017hwg,Cano:2018aqi,Jiang:2018pfk,An:2018dbz,Ghodsi:2020qqb} , Einstein-Maxwell-dilaton gravity \cite{Swingle:2017zcd,An:2018xhv} and so on. There are also some attempts to define complexity more rigorously in quantum field theory and in a continuous way, where interestingly their results in different setups match with results from holography \cite{Chapman:2017rqy,Jefferson:2017sdb,Czech:2017ryf,Belin:2018bpg,Chapman:2018hou,Camargo:2019isp}.

The AdS/CFT correspondence has also provided us a powerful tool to study the behavior of strongly correlated materials in the condensed matter physics \cite{Hartnoll:2009sz,Herzog:2009xv,Sachdev:2011wg} which can be mapped to the classical bulk gravity using holographic methods. Especially, much attention has
been paid to the holographic description of systems with momentum relaxation. Such systems with broken translational symmetry are needed in order to give a realistic description of materials in many condensed matter systems \cite{Liu:2012tr,Ling:2013aya,Ling:2014bda,Mozaffara:2016iwm,Cremonini:2018xgj,Cremonini:2019fzz}. Since momentum is conserved in a system with translational symmetry, a constant
electric field can generate a charge current without current dissipation in the presence of
non-zero charge density. Thus, the conductivity of the system would become divergent
at zero frequency. In more realistic condensed matter materials, the momentum is not
conserved due to impurities or a lattice structure, thus leading to a finite DC conductivity.
There are various ways to achieve momentum dissipation,
such as periodic potentials, lattices and breaking diffeomorphism invariance \cite{Horowitz:2012gs,Davison:2013jba,Blake:2013bqa,Ling:2013nxa,Donos:2013eha,Andrade:2013gsa,Donos:2014yya,Davison:2015bea}.
However, there are two well-known strategies to produce momentum dissipation by inclusion matter fields that breaks translation invariance in the dual field theory. The case of scalar fields (EMA theory) that depend linearly on the horizon coordinates as given in Ref. \cite{Andrade:2013gsa}, and the case of massive gravity theories which present a broken diffeomorphism invariance in the bulk as done in Ref. \cite{deRham:2010kj}.

As the main purpose of this paper, we employ the CA conjecture to study the holographic complexity and its time evolution in EMA theory with momentum relaxation, following the approach in Ref. \cite{Carmi:2017jqz}. In particular, we compute these quantities for charged and neutral AdS$_{d+1}$ black branes in this theory and investigate the Lloyd's bound for these solutions. We provide an analytical discussion for the effects of the strength of momentum relaxation on the desired quantities. The results show that the Lloyd's bound is saturated only at late time limit and there is an upper bound on the strength of momentum relaxation in each sector which provides a minimum temperature for the gravitational system to has positive energy. In addition, we examine the effect of non-linear contribution of the scalar field kinetic term  \cite{ArmendarizPicon:1999rj,Baggioli:2014roa,Cisterna:2017jmv} on the complexity growth rate in four-dimensional spacetime. This theory is known as the k-essence model of dark energy \cite{ArmendarizPicon:1999rj} in which the acceleration of the Universe (both at early and late times) can be driven by the kinetic energy instead of the potential energy of the scalar field. The evolution of holographic entanglement entropy and complexity under a thermal quench has been recently studied for EMA theory in Ref. \cite{Zhou:2019xzc}, in the context of CV conjecture.

The structure of this paper is organized as follows: in section 2, we review the EMA theory with momentum relaxation and study the time-evolution of complexity for homogeneous and isotropic charged/neutral AdS$_{d+1}$ black branes. In the context of CA conjecture, we consider WDW patches that include null sheets bounding the bulk and joint terms, and investigate how the holographic complexity approaches the late time limit on them. In section 3, we study the dyonic AdS black branes in the presence of non-linear kinetic term in the k-essence model. Beside, we consider the inclusion of a Maxwell surface term to the action that will be affected the late-time growth in the CA proposal. Finally, the section 4 is dedicated to a brief summary and concluding remarks.
\section{EMA theory with momentum relaxation}
For the sake of having momentum relaxation and finite conductivity, it is essential to construct holographic models with broken translational symmetry. In the context of AdS/CFT duality, we consider a simple holographic model in which we can study momentum relaxation \cite{Andrade:2013gsa}. In this model, the Einstein-Maxwell action in the $(d+1)$-dimensional spacetime is supplemented by $d-1$ massless scalar fields that break the translational invariance of the boundary theory. This theory is known as EMA theory in Horndeski theories of modified gravity \cite{Horndeski:1974wa,Kobayashi:2011nu}. The scalar axion fields enter the bulk action only through the kinetic term $\prt_{\mu}\psi_{I} $ and the sources are linear in the boundary, i.e. $\psi^{(0)}_{I} \propto \beta_{I i} x^i$, where $\beta$ represents the strength of momentum relaxation.

The action of this holographic model in the bulk is described by \cite{Andrade:2013gsa}
\begin{equation}} \def\ee{\end{equation}\label{act1} I_{bulk}=\frac{1}{16\pi G}\int_{M} d^{d+1}x \,\sqrt{-g}\left[R-2\Lambda-\frac14 F_{\mu\nu}F^{\mu\nu}-\frac12 \sum_{I}^{d-1} (\prt \psi_{I})^2\right],\ee
where $G$ is a $(d+1)$-dimensional gravitational constant and $\Lambda=-d(d-1)/2L^2$  is a cosmological term. The action includes the field strength $F_{\mu\nu}=\prt_{\mu}A_{\nu}-\prt_{\nu}A_{\mu}$ of a $U(1)$ gauge field $A_{\mu}$ and $d-1$ massless scalar fields $\psi_{I}$.
The model admits homogeneous and isotropic charged AdS$_{d+1}$ black brane solutions of radius $L$ with non-trivial scalar field sources. They are described by the following ansatz
\begin{equation}} \def\ee{\end{equation} \label{ansatz1} ds^2 = - f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 \delta_{i j} dx^i dx^j,  \qquad A=A_{t}(r) dt, \qquad \psi_I = \beta_{I i}\, x^i,\ee
where $i$ labels the $d-1$ spatial $x_{i}$ directions, $I$ is an internal index that labels the $d-1$ scalar fields. Substituting the ansatz (\ref{ansatz1}) in the equations of motion of action (\ref{act1}), we find that
\begin{equation}} \def\ee{\end{equation}\label{sol1} f(r)=r^2-\frac{\beta^2}{2(d-2)}-\frac{m_{0}}{r^{d-2}}+\frac{q^{2}}{r^{2(d-2)}}, \qquad A_t (r)=\frac{2q\,(d-1)}{d-2}\left(\frac{1}{r_{h}^{d-2}}-\frac{1}{r^{d-2}}\right),\ee
where
\begin{equation}} \def\ee{\end{equation} \beta^2\equiv\frac{1}{d-1}\sum_{i}^{d-1} \vec{\beta}_{i}\cdot \vec{\beta}_{i}, \qquad \vec{\beta}_{i}\cdot \vec{\beta}_{j}=\sum_{I}{\beta}_{Ii}\, {\beta}_{I j} =\beta^2 \delta_{i j} \qquad \forall \, i, j.\ee
 Note that for the AdS radius we set $L=1$ in the rest of the paper.

 The mass parameter $m_0$, which is proportional to the energy density of the brane, is computed from $f(r_h)=0$ as follows
\begin{equation}} \def\ee{\end{equation}\label{mass1} m_0 = r_h^d \left(1+ \frac{q^2}{r_h^{2(d-1)}}- \frac{1}{2(d-2)}\frac{\beta^2}{r_h^2} \right),\ee
where $r_h$ is the location of event horizon. This is related to the mass of the brane with \cite{Myers:1999psa}
\begin{equation}} \def\ee{\end{equation}\label{mass2} M=\frac{(d-1) V_{d-1}}{16\pi G}m_0.\ee
Here, $V_{d-1}$ is the dimensionless volume of the relevant spatial geometry.
The parameter $q$ is related to the charge of the brane through the Gauss's law with
\begin{equation}} \def\ee{\end{equation}\label{CH}  Q =\frac{\sqrt{2(d-1)(d-2)}\, V_{d-1}}{16\pi G }
\,q\,.\ee
The Hawking temperature and entropy of the branes are given by
\begin{equation}} \def\ee{\end{equation} \label{TS1} T= \frac{f^\prime (r_h)}{4 \pi}=\frac{1}{4\pi}\left(d\,r_+ -\frac{\beta^2}{2r_h} -\frac{(d-2) q^2}{r_h^{d-1}}\right),\qquad  S=\frac{V_{d-1}}{16\pi G} 4\pi r_h^{(d-1)}.\ee
Since the blackening factor in (\ref{sol1}) has two real roots $r_+$ and $r_-$ (where $r_+> r_-$), corresponding to the outer and inner horizons in which $f(r_{+})=f(r_{-})=0$, we can define a chemical potential in each of them as follows
\begin{equation}} \def\ee{\end{equation}\label{CP} \mu_+=\frac{\prt M}{\prt Q}\Bigg|_{V,S}=\sqrt{\frac{2(d-1)}{d-2}}\,\frac{q}{r_+^{d-2}}\,,\qquad \mu_-=\frac{\prt M}{\prt Q}\Bigg|_{V,S}=\sqrt{\frac{2(d-1)}{d-2}}\,\frac{q}{r_-^{d-2}}\,.\ee
 Various features of thermodynamics of this solution has been extensively studied in Ref. \cite{Fang:2017nse}.
\subsection{Complexity of charged black branes via CA conjecture }
We use the CA conjecture (\ref{CA}) to compute the holographic complexity for charged AdS$_{d+1}$ black branes in the EMA theory. The essential ingredient in this method is to evaluate the action on a WDW patch \cite{Brown:2015bva,Brown:2015lvg}. However, we follow the method in Ref. \cite{Carmi:2017jqz} in which not only the action in the WDW patch includes the bulk theory and the Gibbons-Hawking-York (GHY) boundary term \cite{York:1972sj,Gibbons:1976ue}, but also embraces boundary segments of joint terms due to intersection of the time-like, space-like, and null boundaries \cite{Hayward:1993my}. This will be the general strategy that would be followed in the rest of the paper. The contribution of GHY surface terms are
\begin{equation}} \def\ee{\end{equation}\label{surf} I_{bdy}=\frac{1}{8\pi G}\int_{\mathcal{B}} d^{d} x \sqrt{\gamma} K-\frac{1}{8\pi G}\int_{\mathcal{B'}} d\lambda d^{d-1}\theta \sqrt{\gamma}\,\kappa,\ee
where $K$ is the trace of the extrinsic curvature $K_{\mu\nu}=-\gamma_{\mu}^{\rho} \gamma_{\nu}^{\sigma} \nabla_{(\rho}n_{\sigma)}$ that $\gamma_{\mu\nu}$ and $n_{\mu}$ are the induced metric on the boundary and the outward pointing unit normal to the boundary, respectively. $\kappa$ is the surface term for the null segments which measures the failure of the null generators to be affinely parametrized which is assumed to vanish, i.e.,  $\kappa=0$ and does not have any contribution to CA for null segments.

The joint actions are given by
\begin{equation}} \def\ee{\end{equation}\label{joint} I_{joint}=\frac{1}{8\pi G}\int_{\Sigma}  d^{d-1}x \sqrt{\sigma}\,\varrho+ \frac{1}{8\pi G}\int_{\Sigma '} d^{d-1}x \sqrt{\sigma}\,a,\ee
where the first term $\varrho$ appears when we have the intersection of two time-like or space-like boundaries while $a$ is required when one or both of the intersecting boundaries belong to null surfaces. There is also a counterterm action for the null surfaces as
\begin{equation}} \def\ee{\end{equation}\label{cterm} I_{ct}=\frac{1}{8\pi G} \int_{\mathcal{B'}} d\lambda \,d^{d-1}\theta \sqrt{\gamma}\, \Theta \log{(\ell_{c}\Theta)}\,,\ee
 which is introduced to ensure reparametrization invariance on the null boundaries. $\Theta$ is the expansion parameter denoted by
\begin{equation}} \def\ee{\end{equation}\label{amb}\Theta=\prt_{\lambda} \log{\sqrt{\gamma}},\ee
and $\ell_{c}$ is an arbitrary length scale. The precise definition of parameters, boundary metrics, and comprehensive discussions of these actions are given in Ref. \cite{Lehner:2016vdi}. In brief, the total action is defined by
 \begin{equation}} \def\ee{\end{equation}\label{tact} I_{tot}=I_{bulk}+I_{bdy}+I_{joint}+I_{ct}\,.\ee
Of course, there may be a boundary term for the Maxwell field in this action that does not change the equations of motion, but it affects the variational principle for the Maxwell field and one should change the boundary conditions consistently \cite{Goto:2018iay}. We will consider the contribution of such a term for charged geometries in the next section.

Due to the presence of null boundaries in the total action, it is more convenient to perform the calculations using the ingoing and outgoing coordinates
\begin{equation}} \def\ee{\end{equation} \label{efc} v=t+r^{*} (r)\,,\qquad u=t-r^{*} (r),\ee
where $r^{*}$ is a tortoise coordinate defined as
\begin{equation}} \def\ee{\end{equation} \label{tc} r^{*} (r) =  \int  \frac{d r}{f ( r)} \,,\ee
with asymptotic behavior
\begin{equation}} \def\ee{\end{equation} \label{tca} \lim_{r \rightarrow \infty} r_{}^{*} (r) = {r}_{\infty}^{*}.\ee

{\bf{\emph{WDW patch:}}} In order to study the evolution of complexity for action (\ref{tact}), we draw the Penrose diagram of causal structure of the charged AdS black brane described by (\ref{sol1}) in Fig.~(\ref{f1}). Following \cite{Carmi:2017jqz}, the corresponding WDW patch is denoted by the shaded region which is bounded by the light sheets sent from the two asymptotic time slices $t_L$ and $t_R$. Without lose of generality, we choose the symmetric configuration for the time slices, i.e. $t_L=t_R\equiv t/2$. In the next subsection, we evaluate the gravitational action on this patch as the boundary time increases.

The patch includes two UV cutoff surfaces near the asymptotic boundary regions at $r=r_{max}$ which are denoted by red dashed lines in Fig.~(\ref{f1}). In fact, the null boundaries of the WDW patch are started from the UV cutoff surface at $r = r_{max}$ and go through the bulk spacetime. There are two meeting points in the bulk due to intersecting the future boundaries at $r = r_{m}^1$ and the past boundaries at $r = r_{m}^2$. The time evolution of the WDW patch can be encoded in the time dependence of these points. These satisfy the following relations
\begin{equation}} \def\ee{\end{equation}\label{r1r2} \frac{t}{2}+{r}_{\infty}^{*}-r_{}^{*} (r_{m}^1)=0,\qquad \frac{t}{2}-{r}_{\infty}^{*}+r_{}^{*} (r_{m}^2)=0,\ee
which their time evolution from (\ref{tc}) is given by
\begin{equation}} \def\ee{\end{equation}\label{tr1r2} \frac{dr_m^{1}}{dt}=\frac{f(r_m^1)}{2},\qquad \frac{dr_m^2}{dt}=-\frac{f(r_m^2)}{2}.\ee
The null boundaries of the right sector of the corresponding WDW patch are
\begin{equation}} \def\ee{\end{equation}\label{BB} B_1: \frac{t}{2}=r_{}^{*} (r)-{r}_{\infty}^{*}\,,\qquad B_2: -\frac{t}{2}=r_{}^{*} (r)-{r}_{\infty}^{*}\,.\ee
These equations are important in the study of time evolution of the total action (\ref{tact}).

\begin{figure}[H]
\centering
\includegraphics[width=8cm,height=10cm]{WDWC1.jpg}
\caption{Penrose diagram of the WDW patch for charged AdS black brane in symmetric configuration ($t_L=t_R$). $r\to 0$ is the singular surface and $r\to \infty$ is the asymptotic boundary surface. The red dashed lines correspond to UV cutoff surfaces at $r=r_{max}$ and $r_m^{1}, r_m^{2}$ are the meeting points of null boundaries in the bulk.}
\label{f1}
\end{figure}

From the holographic point of view \cite{Hartnoll:2009sz}, it has been proposed in Ref. \cite{Carmi:2017jqz} that this black hole geometry with $U(1)$ symmetry is dual to a charged thermofield double state,
\begin{equation}} \def\ee{\end{equation}\label{CTFD} |\psi_{CTFD}\rangle=\frac{1}{\sqrt{Z}}\sum_{j,l} e^{-{E_j-\mu Q_l}/({2T})} e^{-i E_{j}(t_{L}+t_{R})}|E_j,-Q_l\rangle_L |E_j,Q_l\rangle_R\,,\ee
where in addition to the temperature $T$ in this neutral counterpart in (\ref{TFD}), this state have a chemical potential $\mu$ and electric charge $Q$. The density matrix of the corresponding grand canonical ensemble characterized by $T$ and $\mu$ is obtained by tracing out the states in either boundary.
\subsubsection{The growth rate of complexity}
In the following we compute the growth rate of the holographic complexity on the WDW patch associated with a charged AdS black brane - see Fig.~(\ref{f1}).  In this regard, we consider the time dependence of total action in (\ref{tact}). In the symmetric configuration of the WDW patch, we can perform the calculations only for the right side of the Penrose diagram or regions $\RN{1}$, $\RN{2}$, and $\RN{3}$ as depited in Fig.~(\ref{f1}), and then multiply the result by a factor of two.


{\bf{\emph{Bulk action:}}} The Ricciscalar tensor and kinetic term of axion fields in this background are given by
\begin{equation}} \def\ee{\end{equation}\label{rvsv} R=-d(d+1)+\frac{(d-1)\beta^2}{2r^2}\,,\qquad -\frac12 \sum_{I}^{d-1} (\prt \psi_{I})^2=-\frac{(d-1)\beta^2}{2r^2},\ee
where the contribution of $\beta^2$ term in the scalar action is canceled by its contribution from the Einstein-Hilbert action, thus from the action (\ref{act1}) we have
\begin{eqnarray}} \def\eea{\end{eqnarray} \label{bulk1} I_{bulk}&\!\!\!\!\!=\!\!\!\!\!&2\,\, \left(I_{bulk}^{\RN{1}}+I_{bulk}^{\RN{2}}+I_{bulk}^{\RN{3}}\right)\nonumber\\
&\!\!\!\!\!=\!\!\!\!\!&\frac{V_{d-1}}{8\pi G} \left[\int_{r_m^1}^{r_+}\left(\frac{t}{2}+{r}_{\infty}^{*}-r_{}^{*} (r)\right)+2\int_{r_+}^{r_{max}}\left({r}_{\infty}^{*}-r_{}^{*} (r)\right)+\int_{r_m^2}^{r_+}\left(-\frac{t}{2}+{r}_{\infty}^{*}-r_{}^{*} (r)\right) \right]\, I(r)\, dr,\nonumber\\
&\!\!\!\!\!=\!\!\!\!\!& I_{bulk}^0+I_{bulk}(t),
\eea
 where the integrand $I(r)$ is
 \begin{equation}} \def\ee{\end{equation}\label{bulkint} I(r)=r^{d-1} \left[-2 d + \frac{( d-2)\, q^2}{ 2\, r^{2(d-1) }} \right],\ee
and $I_{bulk}^0$ is the time independent part of the bulk action and only the first and third terms depend on the time through the Eqs.~(\ref{r1r2}).


 {\bf{\emph{Boundary surface action:}}} If we choose affinely parametrization for the null normals then the null surface term vanishes ($\kappa=0$), so we only need to consider the GHY term coming from the surface at UV cutoff on the right side of WDW patch. The trace of the extrinsic curvature for ansatz (\ref{ansatz1}) is given by
 \begin{equation}} \def\ee{\end{equation} \label{EX} K=\frac{1}{2}\left(\frac{\prt_r f(r)}{\sqrt{f(r)}}+\frac{2(d-1)}{r} \sqrt{f(r)}\right). \ee
 Following \cite{Carmi:2017jqz}, we define future directed normal vectors to evaluate $K$,
 \begin{equation}} \def\ee{\end{equation} \label{NV} 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})}}.\ee
 So, we obtain the contribution of surface action in (\ref{surf}) for charged AdS solution (\ref{sol1}) as
 \begin{equation}} \def\ee{\end{equation}\label{surf2} I_{bdy}=2\,I_{bdy}^{r=r_{max}}
 =\frac{r^{d-1}\,V_{d-1}}{8\pi G}\left(\prt_r f(r) +\frac{2(d-1)}{r} f(r)\right)({r}_{\infty}^{*}-r_{}^{*} (r))\Big|_{r=r_{max}}.
 \ee
As is obvious, the cutoff term at $r = r_{max}$ is independent of time, so the boundary term has no contribution to the time evolution of complexity.
 

  {\bf{\emph{Joint action:}}} According to the WDW patch in Fig.~(\ref{f1}), there are different joint contributions at the intersection of null boundaries with the surfaces at $r=r_{max}$ and with each other at $r_{m}^1$ and $r_m^2$. It has been shown in Ref. \cite{Chapman:2016hwi} that the null joint contributions at the UV cutoff surfaces have no time dependence, so we need only to consider the last two joining points. Assume that $k_1$ and $k_2$ are the null vectors associated with two past null boundaries intersecting at $r_{m}^2$ which are defined by
  \begin{equation}} \def\ee{\end{equation}\label{kk} K_1=\xi\left(-dt+\frac{dr}{f(r)}\right), \qquad K_2=\xi\left(dt+\frac{dr}{f(r)}\right),\ee
 where $\xi$ is a normalization constant for null vectors. Those for null vectors associated with two future null boundaries (intersecting at $r_{m}^1$), i.e. $\tilde{k}_1$ and $\tilde{k}_2$, are similar. Following \cite{Lehner:2016vdi}, the joint term is defined by $a=\ln|-\frac12 k_1\cdot k_2|$ for the first set and $\tilde{a}=\ln|-\frac12 \tilde{k}_1\cdot \tilde{k}_2|$ for the second set, then the joint action (\ref{joint}) can be evaluated as
 \begin{equation}} \def\ee{\end{equation}\label{joint2} I_{joint}=-\frac{V_{d-1}}{8\pi G} \left[(r_{m}^1)^{d-1}\,\log\frac{|f(r_{m}^1)|}{\xi^2}+(r_{m}^2)^{d-1}\,\log\frac{|f(r_{m}^2)|}{\xi^2}\right],\ee
 where the time dependence of this contribution comes from Eqs. (\ref{tr1r2}).


 {\bf{\emph{Counterterm action:}}} For the sake of removing the ambiguity associated with the normalization of null vectors we need to add this boundary term to the action. Thus, we choose the affine parameter $\lambda=r/\xi$ such that the total action with the counterterm does not depend on the parametrization of null surfaces. In this parametrization, the expansion (\ref{amb}) takes the form \cite{Carmi:2017jqz}
 \begin{equation}} \def\ee{\end{equation}\label{amb2} \Theta=\frac{(d-1)\xi}{r}.\ee
 Therefore, the counterterm action in (\ref{cterm}) becomes
  \begin{equation}} \def\ee{\end{equation}\label{cterm2} I_{ct}=2\,(I_{ct}^{future}+I_{ct}^{past})=\frac{( d-1)\,V_{d-1}}{4\pi G} \left(\int_{r_m^1}^{r_{max}} r^{d-2} \log\frac{(d-1) \ell_c \xi}{r}\,dr+ \int_{r_m^2}^{r_{max}} r^{d-2} \log\frac{(d-1) \ell_c \xi}{r}\,dr\right).\ee
Again this contribution depends on the time through Eqs.~(\ref{tr1r2}).


\subsubsection{The late time behavior}
Now we can determine the rate of change of the holographic complexity by considering all of the above contributions
\begin{equation}} \def\ee{\end{equation}\label{CCAR1} \frac{d \mathcal{C}_{A}}{dt}=\frac{d}{dt} \left(\delta I_{bulk}+I_{joint}+I_{ct}\right),\ee
where $\delta I_{bulk}\equiv I_{bulk}-I_{bulk}^0$. Henceforth we will set $\pi \hbar=1$ in the CA cojecture in Eq.~(\ref{CA}) for simplicity.
Thus the growth rate of the holographic complexity yields
\begin{equation}} \def\ee{\end{equation}\label{CCAR2} \frac{d \mathcal{C}_{A}}{dt}=\frac{ (d-1)\,V_{d-1}}{16\pi G}
\left[ \frac{2 q^2}{r^{d-2}} -\, r^{d-2} f(r) \log\frac{( d-1)^2 \ell_c^2 |f(r)|}{r^2}\right]_{r_m^2}^{r_m^1}\,.\ee
At late times, the future (past) corner approaches the inner (outer) horizon, so due to the conditions $f(r_{+})=f(r_{-})=0$ the second term vanishes. This leaves the
surprising result
\begin{equation}} \def\ee{\end{equation}\label{CCAR3}\frac{d \mathcal{C}_{A}}{dt}\Bigg|_{t\to \infty}=\frac{ (d-1)\,V_{d-1}}{16\pi G} \left[\frac{2 q^2}{r^{d-2}}\right]_{r=r_+}^{r=r_-}=(M-\mu_+ Q)-(M-\mu_- Q),\ee
 where we have substituted from (\ref{CH}) and (\ref{CP}). The results in this limit are consistent with calculations in Refs. \cite{Brown:2015lvg,Cai:2016xho} for charged black holes without considering the joint and counterterm actions directly.

For the complexity growth rate of charged AdS black brane obtained in Eq.~(\ref{CCAR3}), it seems that in the $Q\to 0$ limit it vanishes, but as we know from general charged black holes, the $Q\to 0$ limit corresponds to $r_-\to0$, so we have $\mu_- Q\to 2M$ while $\mu_+ Q\to 0$. Thus in the zero charge limit we recover the case of neutral AdS branes for the Lloyd's bound, i.e. $d \mathcal{C}_A /dt=2M$.

\subsection{Complexity of neutral AdS black branes}
In order to obtain neutral black branes it is sufficient to insert $q=0$ in the ansatz (\ref{sol1}). Therefore, the mass parameter $m_0$, which is proportional to the energy density of the brane, is computed from $f(r_h)=0$ that $r_h$ is the location of event horizon,
\begin{equation}} \def\ee{\end{equation}\label{mass3} m_0=r_{h}^d\left(1- \frac{1}{2(d-2)}\frac{\beta^2}{r_h^2}\right),\ee
and this is related to the mass of the brane with
\begin{equation}} \def\ee{\end{equation}\label{mass4} M=\frac{(d-1) V_{d-1}}{16\pi G}m_0.\ee
Also the temperature and entropy of the neutral solution are given by
\begin{equation}} \def\ee{\end{equation}\label{TS2} T= \frac{f^\prime (r)}{4 \pi}\Big|_{r=r_h}=\frac{2 d r_h^2 -  \beta^2}{8 \pi r_h}\,,\qquad S=\frac{V_{d-1}}{16\pi G} 4\pi r_h^{(d-1)}.\ee

 {\bf{\emph{WDW patch:}}} The causal structure of the two-sided neutral AdS black brane with single horizon described by the Penrose diagram in Fig.~(\ref{f2}). The corresponding WDW patch is denoted by the shaded region which is bounded by the light sheets sent from the two asymptotic time slices $t_L$ and $t_R$. We choose the symmetric configuration for the time slices, i.e. $t_L=t_R\equiv t/2$. In the next subsection, we evaluate the total action (\ref{tact}) on this patch as the boundary time increases.

In this patch, $r_m$ is the point in which the past light sheets from the left and right boundaries intersect before hitting the past singularity at some critical time $t_c$ in the symmetric configuration
\begin{equation}} \def\ee{\end{equation}\label{critt} \frac{t_c}{2}={r}_{\infty}^{*}-r_{}^{*} (0).\ee
It also contains a cutoff surface near the future singularity at $r=\epsilon$ and two surfaces near the asymptotic boundary regions at $r=r_{max}$. These surfaces are specified by the dashed red lines in Fig.~(\ref{f2}). Also the boundaries $B_1$ and $B_2$ are given by Eqs.~(\ref{BB}).
\begin{figure}[H]
\centering
\includegraphics[width=8cm,height=6cm]{WDWN.jpg}
\caption{Penrose diagram of the WDW patch of neutral AdS black brane in symmetric configuration ($t_L=t_R$).}
\label{f2}
\end{figure}
\subsubsection{The growth rate of complexity}
In the following we compute the growth rate of the holographic complexity on the WDW patch associated with a two-sided AdS black brane for times $t>t_c$ - see Fig.~(\ref{f2}).  In this regard, we consider the time dependence of total action in (\ref{tact}). In the symmetric configuration of the WDW patch, we can perform the calculations only for the right side of the Penrose diagram or regions $\RN{1}$, $\RN{2}$, and $\RN{3}$, and then multiply the result by a factor of two.


{\bf{\emph{Bulk action:}}} The contribution of bulk comes from the action (\ref{act1}) by omitting the Maxwell term. Therefore, according the relations in (\ref{rvsv}), for $t>t_c$ we have
\begin{eqnarray}} \def\eea{\end{eqnarray} \label{bulk2} I_{bulk}&\!\!\!\!\!=\!\!\!\!\!&2\,\, \left(I_{bulk}^{\RN{1}}+I_{bulk}^{\RN{2}}+I_{bulk}^{\RN{3}}\right)\nonumber\\
&\!\!\!\!\!=\!\!\!\!\!&-\frac{d\, V_{d-1}}{4\pi G}\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]\, r^{d-1} dr,\nonumber\\
&\!\!\!\!\!=\!\!\!\!\!&I_{bulk}^{0}-\frac{d\, V_{d-1}}{4\pi G} \int_{\epsilon}^{r_m}\left(\frac{t}{2}+{r}_{\infty}^{*}-r_{}^{*} (r)\right) \,r^{d-1} dr,
\eea
 where as is clear, only the second term depends on the time.


 {\bf{\emph{Boundary surface action:}}} If we choose affinely parametrization for the null normals then the null surface term vanishes ($\kappa=0$), so we only need to consider the GHY term coming from the regulator surface at the future singularity and the surface at the UV cutoff.
 So, we obtain the contribution of surface action in (\ref{surf}) for neutral AdS solution as
 \begin{eqnarray}} \def\eea{\end{eqnarray}\label{surf3} I_{bdy}&\!\!\!\!\!=\!\!\!\!\!&2\,(I_{bdy}^{r=\epsilon}+I_{bdy}^{r=r_{max}})\\
 &\!\!\!\!\!=\!\!\!\!\!&-\frac{d\,V_{d-1}}{8\pi G}\!\!\left[(2{r}^d-r_{h}^d)- \frac{2(d-1)r^{d-2}-dr_h^{d-2}}{2d(d-2)}\beta^2\right]\!\!\left((t/2+{r}_{\infty}^{*}-r_{}^{*} (r))\Big|_{r=\epsilon}-({r}_{\infty}^{*}-r_{}^{*} (r))\Big|_{r=r_{max}}\right).\nonumber
 \eea
As seen, the cutoff term at $r = r_{max}$ is independent of time, so we can rewrite (\ref{surf3}) as
 \begin{equation}} \def\ee{\end{equation}\label{surf4}  I_{bdy}= I_{bdy}^0-\frac{d\,V_{d-1}}{8\pi G}\!\!\left[(2{r}^d-r_{h}^d)- \frac{2(d-1)r^{d-2}-dr_h^{d-2}}{2d(d-2)}\beta^2\right] (t/2+{r}_{\infty}^{*}-r_{}^{*} (r))\Big|_{r=\epsilon},\ee
where $I_{bdy}^0$ is independent of time and has no contribution to the growth rate.
 

  {\bf{\emph{Joint action:}}} According to the WDW patch in Fig.~(\ref{f2}), there are different joint contributions at the intersection of null boundaries with the surfaces at $r=\epsilon$ and $r=r_{max}$, and with each other at $r_{m}$. However, the joints at singular and cutoff surfaces are independent of time. Assuming that $k_1$ and $k_2$ are given by the relations in (\ref{kk}), then the joint action (\ref{joint}) can be evaluated as
 \begin{equation}} \def\ee{\end{equation}\label{joint3} I_{joint}=-\frac{V_{d-1}}{8\pi G} \left[r^{d-1}\,\log\frac{|f(r)|}{\xi^2}\right]_{r=r_{m}}.\ee
 The time evolution of this contribution is through the implicit time dependence of $r_m$ with
 \begin{equation}} \def\ee{\end{equation}\label{rmeq} \frac{t}{2}-{r}_{\infty}^{*}+r_{}^{*} (r_m)=0.\ee


 {\bf{\emph{Counterterm action:}}} Using the parameter expansion introduced in (\ref{amb2}), the counterterm action becomes
 \begin{eqnarray}} \def\eea{\end{eqnarray}\label{cterm3} I_{ct}&\!\!\!\!\!=\!\!\!\!\!&2\,(I_{ct}^{future}+I_{ct}^{past}) \\
 &\!\!\!\!\!=\!\!\!\!\!& \frac{(d-1) V_{d-1}}{4\pi G}\left (\int_{{\epsilon}}^{r_{max}}+\int_{r_m}^{r_{max}}\right) r^{d-2} \log\frac{(d-1) \ell_{c}\xi}{r} dr.
 \eea
 It is clear that the cutoff bounds have no time dependence, thus in the limit ${\epsilon}\rightarrow 0$, the counterterm action depends on the time trough the Eq.~(\ref{rmeq}).
\subsubsection{The late time behavior}
In the case of neutral branes we should also consider the contribution of boundary term, that is, the growth rate for $t>t_c$ is calculated from
\begin{equation}} \def\ee{\end{equation}\label{CAR1} \frac{d \mathcal{C}_{A}}{dt}=\frac{d}{dt} \left(\delta I_{bulk}+\delta I_{bdy}+I_{joint}+I_{ct}\right),\ee
where $\delta I_{bulk}\equiv I_{bulk}-I_{bulk}^0$ and $\delta I_{bdy}\equiv I_{bdy}-I_{bdy}^0$. From Eqs.~(\ref{bulk2}) and (\ref{surf4}) for the bulk and boundary actions,
\begin{eqnarray}} \def\eea{\end{eqnarray} \label{joint4} \frac{dI_{joint}}{dt} &=&\frac{V_{d-1}}{8\pi G} \Bigg[r_m^d + \frac{1}{2} (d-2) r_h^d + \frac{1}{2} ( d-1) (r_m^d -  r_h^d) \log\frac{\bigl|f(r_m)\bigr|}{\xi^2} \\
	&&\hspace{1cm}+\left( \frac{( d-1) (  r_h^{ d-2}- r_m^{ d-2} ) }{4 ( d-2) }\log\frac{\bigl|f(r_m)\bigr|}{\xi^2}- \frac{1}{4} r_h^{ d-2}  \right)  \beta^2 \Bigg], \nonumber\eea
 for the joint term, and the following for counterterm action
 \begin{equation}} \def\ee{\end{equation}\label{cterm4} \frac{dI_{ct}}{dt}=\frac{ ( d-1)\,V_{d-1}}{16\pi G} \left[ (r_m^d -  r_h^d) - \frac{( r_m^{ d-2} - r_h^{ d-2})  \beta^2}{2 ( d-2)}\right] \log\frac{( d-1) \ell_c \xi}{r_m}\,,\ee
we obtain the growth rate of the complexity in the CA conjecture as
\begin{equation}} \def\ee{\end{equation} \label{CAR2} \frac{d \mathcal{C}_{A}}{dt}=\frac{(d-1)\,V_{d-1}}{8\pi G}\left[ r_h^d -  \frac{r_h^{d-2}  \beta^2}{2 ( d-2)}+ \tfrac{1}{2} r_m^{ d-2} f(r_m) \log\frac{( d-1)^2 \ell_c^2 |f(r_m)|}{r_m^2}\right],\ee
or equivalently, from Eq.~(\ref{mass4}) it can be recast in the form
\begin{equation}} \def\ee{\end{equation}\label{CAR3}
\frac{d \mathcal{C}_{A}}{dt}=2M+\frac{V_{d-1}}{16\pi G}\left[( d-1) r_m^{ d-2} f(r_m) \log\frac{( d-1)^2 \ell_c^2 |f(r_m)|}{r_m^2}\right],
\ee
where the time dependence of the meeting point $r_m$ is obtained from Eq.~(\ref{rmeq}) and definition of tortoise coordinate in Eq.~(\ref{tc}) as
\begin{equation}} \def\ee{\end{equation} \label{rmeq2} \frac{dr_m}{dt}=-\frac{f(r_m)}{2}.\ee

One can observed that though the result (\ref{CAR3}) does not satisfy the Lloyd's bound on the rate of quantum computation \cite{Lloyd} for $t>t_c$, but at late time limit in which $r_m$ approaches $r_h$ and $f(r_m)\rightarrow 0$, the contribution of second term vaniches and this yields
\begin{equation}} \def\ee{\end{equation}\label{lbs} \frac{d\mathcal{C}_{A}}{dt}\Bigg|_{t\to \infty}=  2 M,\ee
which is consistent with the expected rate of growth at late time in Refs. \cite{Brown:2015bva,Brown:2015lvg} even in the presence of momentum relaxation. Though the joint and counterterm contributions of both the charged and neutral solutions are sensitive to the ambiguities of null boundaries through the normalization constant $\xi$, but according to (\ref{CCAR2}) and (\ref{CAR2}) the result for total action is independent of it.


More specifically, the relation (\ref{mass3}) yields a bound on the value of $\beta$ that the larger the value of $\beta$ leads to negative value for the mass which has no physical interpretation. For the neutral branes studied in this section, this maximum value is obtained from $\beta^2_{max}\!=\!2(d-2)r_h^2$. We investigate that for $\beta\leq \beta_{max}$, the equation $f(r)=0$ always has a real positive root (single horizon $r_h$) even when $m_0=0$. Further, from the definition of temperature in (\ref{TS2}) this value then yields a finite temperature $T={r_h}/{2\pi}$ in this case. In other words, the vacuum metric ($M=0$) has the form of an AdS black hole that in the context of AdS/CFT, it can be interpreted in terms of an entangled state of two copies of the
CFT on a hyperbolic hyperplane \cite{Casini:2011kv}. The entangled state of two copies of the CFT on a hyperbolic plane appearing above can then be understood as a conformaly transformed description of the global vacuum state which entangles the CFT degrees of freedom on the interior with those on the exterior of the sphere.

This result provides strong motivations to investigate the vanishing of complexity growth rate at some finite temperature other than zero. For $\beta=\beta_{max}$, actually this is a minimum temperature, i.e. $T_{min}={r_h}/{2\pi}$. Therefore, from the equation  (\ref{lbs}) we expect that the variation of the complexity becomes zero in this temperature only  in the Lloyd limit. There is a similar discussion on the temperature for resistivity and conductivity of field theories which are dual to EMA-Dilaton theories in Ref. \cite{Gouteraux:2014hca}. 

We also note that regardless of the unphysical values obtained for the mass, in the case of $T<T_{min}$ or $\beta>\beta_{max}$, we observed that equation $f(r)=0$ has two real positive roots and the geometry have a causal structure similar to that encountered for charged AdS black holes. We have plotted the behavior of blackening factor $f(r)$ for different values of $\beta$ in Fig.~(\ref{f3}), for instance in four-dimensional spacetime. The case that happens here is denoted by the solid red curve in figure. Studying the complexity growth rate for this charged-like geometry similar to what was done in the previous subsection, we find that $d{\mathcal{C}_A/dt}$ vanishes at late times. It has been shown in Ref. \cite{Chapman:2016hwi} that a similar thing happens in the case of AdS black holes with hyperbolic geometry. For temperatures below $T=1/(2\pi L)$, the small hyperbolic black holes (i.e. $r_h < L$) have a causal structure similar to that of charged AdS black holes and the rate of growth of complexity at late times vanishes. In fact, though we consider the AdS branes with planar geometry ($k=0$) in (\ref{sol1}), for $\beta=\beta_{max}$ the neutral brane metric behaves as well as a hyperbolic geometry ($k=-1$).

\begin{figure}[H]
\centering
\includegraphics[width=8cm,height=6cm]{horizon.pdf}
\caption{The structure of blackening factor in terms of different values of momentum relaxation for $d=3$ and $r_h=1$. The solid red line corresponds to $\beta>\beta_{max}$. }
\label{f3}
\end{figure}
In order to analyze in detail the behavior of time evolution of complexity growth rate, we investigate the time derivative of the action given in Eq.~(\ref{CAR3}) as a function of time in four-dimensional spacetime. The results are depicted for some typical locations of the event horizon, $r_h=1,1.5,2.5$, in Fig.~(\ref{f4}a) and for some values of momentum relaxation constant, $\beta=0,1,1.2$ , in Fig.~(\ref{f4}b). For the sake of the numerical precision we first solve the equation of $r_m$ in (\ref{rmeq2}) numerically and then plot the diagrams. We can see from figures that in the initial times after $t_c$, the growth rate of action violates the Lloyd's bound for all values of $r_h$ and $\beta$ while at large times $t>>t_c$ it saturates this, i.e. $2M$. Although the general behavior of the plots is the same, the curves behave differently in each panel. For instance, in the left panel larger black holes saturate the bound sooner than smaller ones while all violate the bound with the same strength (the curves have the same peak) in the initial times. In the right panel, the larger value of $\beta$ corresponds to the stronger violation of bound and it saturates the bound in some later times. The other point that can be inferred from
Figs.~(\ref{f4}a) and (\ref{f4}b) is that the rate of complexity saturates the Lloyd's bound from above at late times. It seems to be consistent with the positive sign of the second term in Eq.~(\ref{CAR3}).

\begin{figure}[H]
\centering
\subfigure[]
{\includegraphics[width=.48\textwidth]{RCrhn.pdf}}
\subfigure[]
{\includegraphics[width=.48\textwidth]{RCbetan.pdf}}
	    \caption{The rate of complexity growth in CA conjecture versus time $t$ for different values of (a) event horizon, and (b) momentum relaxation.}
\label{f4}
\end{figure}
\section{The growth rate of complexity in the k-essence sector }
The previous studies can be generalized to the case in which the kinetic term for the scalar fields can have non-linear contributions. Such a case can be implemented by the
so-called k-essence models \cite{ArmendarizPicon:1999rj}, in which the mentioned kinetic term are generalized to be a function $P( \psi,(\prt\psi)^2)$. A simple case contained on this setup is the one in which the scalar fields apart from the standard kinetic term possess a kinetic non-linear contribution given by higher powers of the kinetic term. In this subsection, we are going to study the growth rate of complexity for dyonic AdS black branes in this holographic model from the CA proposal. However, for later convenience we consider the four-dimensional bulk spacetime.

The action of non-linear EMA theory was studied in Refs. \cite{Baggioli:2014roa,Cisterna:2017jmv} and is given by
\begin{equation}} \def\ee{\end{equation}\label{nlact} I_{bulk}=\frac{1}{16\pi G}\int_{M} d^{4}x \,\sqrt{-g}\left[R-2\Lambda-\frac14 F_{\mu\nu}F^{\mu\nu}- \sum_{I=1}^{2} (\chi_I+ \gamma \chi_I^k)\right], \ee
where $\chi_I=\frac12 \prt_{\mu} \psi_{I} \prt^{\mu}\psi_{I}$ and $\gamma$ is the coupling of non-linear axionic term. The AdS black brane solution of this model is described by the ansatz (\ref{ansatz1}). Therefore, the blackening factor and axionic scalar fields are
\begin{equation}} \def\ee{\end{equation}\label{sol2} f(r) = r^2 - \frac{\beta^2}{2}  + \frac{Q_e^2+Q_m^2}{4r^2} - \frac{2 m_0}{r} +\gamma \,\frac{ \beta^{2 k}}{2^{k} (2 k-3) r^{2(k-1)} },\quad
 \psi_1=\beta x_1,\,\,\psi_2=\beta x_2\,.\ee
The Maxwell equations is easily solved by
\begin{equation}} \def\ee{\end{equation}\label{GF}A=-\frac{Q_e}{r}\,dt+\frac{Q_m}{2}\left(x_1 dx_2-x_2 dx_1\right).\ee
where $Q_e$ and $Q_m$ are the electric and magnetic monopole charges.
The Hawking temperature, entropy, and mass of the brane are given by
\begin{equation}} \def\ee{\end{equation}\label{mc}  T\!=\!\! \frac{f^\prime (r_+)}{4 \pi}\!=\!\frac{1}{4\pi}\!\left(3r_+  -\frac{\beta^2}{2r_+}-\frac{Q_e^2+Q_m^2}{4r_+^3}-\gamma \frac{\beta^{2k}}{2^k  r_+^{2k-1}}\right)\!\!,\quad S\!=\!\frac{V_2}{16\pi G} 4\pi r_+^{2},\quad M\!=\!\frac{4V_2}{16\pi G}m_0,\ee
where again the mass parameter is obtained from condition $f(r_+)=0$ as
\begin{equation}} \def\ee{\end{equation} \label{mass5} m_0 = \frac{Q_e^2+Q_m^2}{8 r_+} + \frac{1}{2} r_+^3 -  \frac{1}{4} r_+ \beta^2 -\gamma \frac{\beta^{2 k} }{2^{k+1}  (2 k-3)  r_+^{ 2 k-3} },\ee
and $r_+$ is the location of event (outer) horizon. The extended thermodynamics of the above solution has been studied in Refs. \cite{Cisterna:2017jmv,Hu:2018prt}. We note also that in particular, for $k=2$ the metric (\ref{sol2}) behaves as a double horizons black hole, i.e. $r_+$ and $r_-$, so one can use the WDW patch in Fig.~(\ref{f1}) to compute the evolution of holographic complexity of state which is dual to this geometry.
\subsection{The action growth rate}
The total time derivative of the holographic complexity for dyonic charged AdS solutions in this model is calculated from (\ref{CCAR1}). However, as mentioned in the previous section, we can also consider the contribution of a boundary term for the Maxwell field in the total action as \cite{Goto:2018iay}
\begin{equation}} \def\ee{\end{equation}\label{mq} I_{\mu Q}=\eta \int_{\prt \mathcal{M}} d\Sigma_{\mu} F^{\mu\nu} A_{\nu},\ee
which does not change the equations of motion. In general, employing a Dirichlet boundary condition results in a well-posed variational principle, but due to the boundary term (\ref{mq}) we instead need to impose a Neumann boundary condition for $\eta=1$, or a mixed boundary condition for general $\eta$. A comprehensive discussion about this boundary action is given in Ref. \cite{Goto:2018iay}. On the other hand, using the Stokes's theorem and the Maxwell equations we can convert the boundary term (\ref{mq}) to the bulk Maxwell action as
\begin{equation}} \def\ee{\end{equation}\label{mq1} I_{\mu Q}=\frac{\eta}{2} \int_{\mathcal{M}} d^4 x \sqrt{-g} F_{\mu\nu} F^{\mu\nu} ,\ee
so the boundary action (\ref{mq}) contributes in the complexity growth rate through this bulk term.

Using similar discussion for the three bulk regions on the WDW in Fig.~(\ref{f1}), the complexity of the bulk action is written as follows
\begin{equation}} \def\ee{\end{equation} \label{bulk3} I_{\text{bulk}}+I_{\mu Q}=\frac{V_2}{16\pi G} \int_{WDW}\!\!\! dr\,dt \, r^{2} \Bigg[ -6 - \gamma \frac{\beta^4 }{2 r^4} -  ( 2 \eta-1) \frac{Q_e^2 -  Q_m^2}{2 r^4}\Bigg],\ee
where in four dimensions the volume of boundary surface becomes $V_2$, then for the time evolution of the bulk term we have
\begin{equation}} \def\ee{\end{equation}\label{NCAR1} \frac{d }{dt }(\delta I_{\text{Bulk}}+I_{\mu Q})=  \frac{V_2}{8 \pi G } \Bigg[  r^3- \gamma\frac{ \beta^4 }{4 r} - ( 2 \eta-1 )  \frac{Q_e^2 -  Q_m^2}{4 r}  \Bigg]{\Bigg|}_{r_m^2}^{r_m^1}\,.\ee
For the joints of null boundaries in $r_{m}^1$ and $r_{m}^2$ we have
\begin{equation}} \def\ee{\end{equation}\label{NCAR2}  \frac{dI_{joint}}{dt}=- \frac{V_2}{8\pi G} \left[r^3-\frac{Q_e^2+Q_m^2}{4r}-\gamma \frac{\beta^4}{4\,r}+r f(r) \log\frac{|f(r)|}{\xi^2}\right]{\Bigg|}_{r_m^2}^{r_m^1},\ee
and also the contribution of counterterm action becomes
\begin{equation}} \def\ee{\end{equation}\label{NCAR3} \frac{dI_{ct}}{dt}= \frac{V_2}{8\pi G} \left[4 m_0 -  \frac{Q_e^2 + Q_m^2}{2 r} -  \frac{ \beta^4 \gamma}{2 r} + \beta^2 r - 2 r^3  \right]\log\frac{2 \ell_c \xi}{r}  {\Bigg|}_{r_m^2}^{r_m^1}.\ee

As asserted in the previous section, the contribution of GHY boundary term for the cutoff surfaces on WDW patch in Fig.~(\ref{f1}) is time independent, so we ignore it.
Combining the above results we can determine the final expression for the complexity growth rate as
\begin{equation}} \def\ee{\end{equation}\label{NCAR4} \frac{d \mathcal{C}_{A}}{dt}=\frac{V_2}{ 8\pi G }
\left[\frac{  (1 -  \eta)\,Q_e^2 +  \eta\, Q_m^2}{2r}- r f(r) \log\frac{4 \ell_c^2 |f(r)|}{r^2}\right]_{r_m^2}^{r_m^1}.\ee
It is obvious that the normalization constant $\xi$ and the coupling of non-linear axionic action $\gamma$ do not affect the final result explicitly. Also, one can check that in the limit $\gamma\to 0$, Eqs.~(\ref{NCAR1})- (\ref{NCAR3}) give the results in the previous subsection for electrically ($Q_m=0$) charged AdS$_{d+1}$ black branes when $d=3$. According to Eq.~(\ref{NCAR1}), for $\eta=1/2$ the boundary and bulk Maxwell terms cancel each other and do not affect the gravitational action, in spite of their geometric contribution in the background (\ref{sol2}) to the action.

It is straightforward to provide a numerical study on the rate of change of complexity for the dyonic black hole given by Eq.~(\ref{sol2}). In this respect, we have plotted the full time-dependence of the growth rate action in Eq.~(\ref{NCAR4}) for different values of $\gamma$ and $Q_e/Q_{m}$ in Fig.~(\ref{f5}), and of $\eta$ in Fig.~(\ref{f6}). We see from Fig.~(\ref{f5}a) that irrespect to the behavior of $dI_{WDW}/dt$ at the initial times, by increasing the coupling of non-linear axionic term the action growth rate decreases while in contrast, by increasing the ratio between electric and magnetic charges, it increases which is illustrated in Fig.~(\ref{f5}b). This opposite behavior relative to these constant parameters is expectable due to the relations (\ref{sol2}) and (\ref{NCAR4}). In other words, for dominant electric solution, i.e. $\eta<1/2$, the sign of charges ratio and $\gamma$ term are different. Of course, we have checked that this opposition also occurs for dominant magnetic solution ($\eta>1/2$). 
\begin{figure}[H]
\centering
\subfigure[$Q_e/Q_m=2$]
{\includegraphics[width=.48\textwidth]{nonlinear.pdf}}
\subfigure[$\gamma=4$]
{\includegraphics[width=.48\textwidth]{charges.pdf}}
	    \caption{The growth rate of complexity for different values of (a) the coupling of non-linear action, and (b) charges ratio with $\beta=0.2, \eta=0.2, \ell_c=1$.}
\label{f5}
\end{figure}

As observed in Fig.~(\ref{f6}), the time evolution of complexity in CA proposal for different values of the coupling of boundary Maxwell action is similar to its behavior for $\gamma$. That is, the rate of growing the action decreases by increasing the value of $\eta$.   

\begin{figure}[H]
\centering
\includegraphics[width=8cm,height=5cm]{dyonic.pdf}
\caption{The growth rate of complexity for different values of $\eta$ with $Q_e/Q_m=2, \beta=0.2, \gamma=5, \ell_c=1$.}
\label{f6}
\end{figure}
\subsection{The late time behavior}
 Since in the late time limit the meeting points reach to the horizons of the geometry on the patch, the final result for the growth rate can be read from (\ref{NCAR4}) as
\begin{equation}} \def\ee{\end{equation}\label{NCAR5} \frac{d\mathcal{C}_{A}}{dt}\Bigg|_{t>>t_c}=\frac{V_2}{16\pi G}\Big[\frac{  (1 -  \eta)\,Q_e^2 +  \eta\, Q_m^2}{r}\Big]_{r_+}^{r_-}.\ee
The result shows that in the absence of Maxwell boundary term, i.e. $\eta=0$, the late time behavior is similar to the case of charged black branes in (\ref{CCAR3}) even in the presence of magnetic charge. Also for $\eta=1$, the late-time growth rate is only proportional to the magnetic charge, i.e. it vanishes for electrically charged black branes.



\section{Conclusions and outlook}
In this paper, we extended the study of holographic complexity via AdS black branes with momentum relaxation in $(d+1)$-dimensional EMA theory by using the CA conjecture. The momentum relaxation introduced by linear massless axion fields in the bulk breaks the translational symmetry of the dual field theory and gives finite conductivity. In this regard, to retain homogeneity of the bulk theory the axion fields have been assumed to be a linear function of the boundary spatial coordinates with the equal constant coefficients determining the strength of momentum relaxation. Particularly, we have investigated the effects of this parameter on the holographic complexity and its time evolution.

We computed the complexity for both the charged and neutral branes as homogeneous and isotropic solutions of the bulk theory. Following the approach in \cite{Carmi:2017jqz} to evaluate the complexity on the WDW patch in each sector, we considered the general GHY surface terms and contribution of joints and counterterm of the corresponding null boundaries. The late time behavior of the growth rate of complexity was studied for two solutions and we found that it is always finite and well defined, and satisfies the Lloyd's bound as given in Eqs.~(\ref{CCAR3}) and (\ref{lbs}). It has been found in Ref. \cite{Brown:2015bva} that the rate of change of action at late times for neutral AdS black holes is $d I_{WDW}/dt=2M$ independent of the size of black hole and spacetime dimension. We have also observed in this paper that neutral AdS black brane confirms this statement even with momentum relaxation.

We also obtained an upper bound for the strength of momentum relaxation $\beta_{max}$, in the case of neutral branes, such that for $\beta>\beta_{max}$ the brane mass becomes negative which has no physical meaning. We studied the time evolution of complexity growth rate numerically in Figs.~(\ref{f3}) for different values of $r_h$ and $\beta$. The plots showed that in spite of different rates, the complexity growth rates for different sizes of the brane and different strengths of momentum relaxation saturate the Lloyd's bound from above. An outstanding outcome corresponding to $\beta_{max}$ is that there is a minimum temperature given by $T_{min}={r_h}/{2\pi}$ for which the growth rate of complexity vanishes in spite of the fact that it should be vanishes at zero temperature for neutral AdS branes.


In addition, we have studied a holographic model including the non-linear contribution of axionic kinetic term while preserve the homogenity and isotropy of the solutions. We assumed a particular branch of solutions that has been described by a dyonic charged black brane with momentum relaxation and then calculated the rate of complexity in this model. We have also considered a boundary action for the Maxwell field in this model. The Maxwell boundary term contributed as a bulk action in the change of complexity such that for $\eta=1/2$ the bulk actions had no contribution in the complexity rate.
The results showed that even though the coupling of non-linear term ($\gamma$) affects the contribution of different actions in the total complexity, it does not change the growth rate at late times as denoted in Eq.~(\ref{NCAR5}). 

In the absence of Maxwell surface term ($\eta=0$), the growth rate vanished at late time for purely magnetic charged branes, while for non-zero electric charges it gave the known expression for general charged branes as in (\ref{CCAR3}). In contrast, for $\eta=1$ the behavior is reversed, that is the rate of growth is non-vanishing for pure magnetically charged branes and vanishes for electrically ones. The numerical investigation for the full time dependence of the complexity growth rate in CA proposal illustrated in Figs.~(\ref{f5}) and (\ref{f6}) have shown that $dI_{WDW}/dt$ increases by increasing the charges ratio while reduces by increasing $\gamma$ and $\eta$. 

It would be of interest to consider the effects of momentum relaxation on the growth rate of complexity in the case of charged dilatonic backgrounds. In general, the action (\ref{act1}) in the presence of dilaton and axion fields with momentum relaxation recasts as follows \cite{Gouteraux:2014hca}
 \begin{equation}} \def\ee{\end{equation}\label{dil}  I=\frac{1}{16\pi G}\int_{M} d^{d+1}x \,\sqrt{-g}\left[R-\frac12 (\prt \phi)^2+V(\phi)-\frac14 Z(\phi) F_{\mu\nu}F^{\mu\nu}-\frac12 \sum_{j}^{d-1} (\prt \psi_{j})^2\right].\ee
The related discussions about the growth of holographic complexity for dilatonic metrics without axion fields have been done in Ref. \cite{Swingle:2017zcd}. For instance, it has been shown in Ref. \cite{Cai:2017sjv} that the total rate of holographic complexity at late times is given by
\begin{equation}} \def\ee{\end{equation}\label{CRD} \frac{d \mathcal{C}_{A}}{dt}\Bigg|_{t\to \infty}=2M-\mu Q-D, \qquad \qquad D\equiv \frac{e^{2\phi}Q^2}{2M}.\ee
We proposed that in the presence of dilaton field a similar term like $D$ would added to the Lloyd's bound of complexity. One can also study the complexity growth rate of a non-relativistic but isotropic boundary theory \cite{Ling:2016ien} which
is dual to a bulk geometry with momentum relaxation for Lifshitz and hyperscaling violating metrics, as done in Ref. \cite{Alishahiha:2018tep} for EMD theory without momentum relaxation. Another proposal in the context of CA conjecture for these holographic models is to investigate the complexity growth rate of AdS black branes at a finite cutoff. The concept of this geometric cutoff at $r = r_c$ comes from the $T\bar{T}$ deformation of a conformal field theory in AdS/CFT dictionary \cite{Zamolodchikov:2004ce,McGough:2016lol}, such that the coupling of this operator removes the asymptotic region of the AdS spacetime. In this regard, there have been made some efforts in Refs. \cite{Akhavan:2018wla,Alishahiha:2019lng,Hashemi:2019xeq}.  
\section*{Acknowledgment}
The authors would like to thank M. Alishahiha, A. Ghodsi, M. R. Mohammadi Mozaffar and G. Jafari for valuable comments and discussions. HBA and HM would like to thank Z. Ebadi, M. Nattagh Najafi and M. Maleki for useful discussions.
