# Complexity Conjecture of Regular Electric Black Holes

B. Bahrami Asl1,2, S. H. Hendi1,2,3 and S. N. Sajadi1,2 email address:
<EMAIL_ADDRESS>address<EMAIL_ADDRESS>address<EMAIL_ADDRESS>1Department of Physics, School of Science,
Shiraz University, Shiraz 71454, Iran
2Biruni Observatory, School of Science, Shiraz University, Shiraz 71454, Iran
3Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada

###### Abstract

Recently, the action growth rate of a variety of four-dimensional regular
magnetic black holes in $\mathcal{F}$ frame is obtained in ElMoumni:2020nse .
Here, we study the action growth rate of a four-dimensional regular electric
black hole in $\mathcal{P}$ frame that is the Legendre transformation of
$\mathcal{F}$ frame. We also investigate the action growth rates of the
Wheeler-De Witt patch for such black hole configurations at the late time and
examine the Lloyd bound on the rate of quantum computation. We show that
although the form of the Lloyd bound formula remains unaltered, the energy
modifies due to a non-vanishing trace of the energy-momentum tensor and some
extra terms may appear in the total growth action. We also investigate the
asymptotic behavior of complexity in two conjectures for static and rotating
regular black holes.

##  I Introduction

AdS/CFT correspondence AdSCFT1 ; AdSCFT2 ; AdSCFT3  is a leading formalism
providing a consistent relation between gauge and gravitation theories and has
given us a rather deep insight on possible unification of them. Such a
correspondence captures an equivalence between the conformal field theory and
its dual theory of gravity in AdS space with a strong/weak duality point of
view. It is believed that it provides a non-perturbative formulation access to
strongly coupled regimes of quantum field theories which cannot be accessed by
the traditional perturbation approach. Indeed, AdS/CFT correspondence is one
of the important examples of entanglement between gravity and quantum
information AdSCFT4 ; AdSCFT5 ; AdSCFT6 ; AdSCFT7 . Complexity has recently
been introduced as a complement to entanglement providing additional
information on the quantum properties of black holes.

In order to calculate the holographic duality of quantum computational
complexity one may follow two proposals. The $complexity=volume$ (CV) A1 ; A2
and complexity = action (CA) B1 ; B2  conjectures. The former one states that
the complexity of the boundary state is proportional to the maximum volume of
codimension hypersurface bounded by the CFT slices and the latter conjecture
specifies that complexity on the boundary CFT is proportional to the on-shell
action of the Wheeler-DeWitt(WDW) patch in the bulk. According to these
conjectures, Lloyd showed that the growth rate of complexity for the
Schwarzschild like black holes is bounded by the total energy of the system
Lloyd2000 . Cai, et.al proved that the so-called Lloyd bound should be
modified for charges or rotating black holes Cai2016 . Although there are many
cases of black holes that confirm the mentioned upper bound L1 ; L2 ; L3 ; L4
; L5 ; L6 ; L7 , its validity is an open question since there are different
types of black hole violating the Lloyd bound V1 ; V2 ; V3 . Accordingly, the
validity of CV and CA conjectures and their possible modifications deserve
further study.

In this direction, here, we focus on the holographic complexity of nonlinearly
charged regular black holes in $\mathcal{P}$ frame. Indeed, the motivation for
studying this black hole is three folds: regularity, nonlinearity and
complexity. The first item is the regularity of the black hole with known
action, exactly. The second one is nonlinearity which has the key role to
avoid the singularity. Although such a nonlinear electrodynamics is an ad hoc
model near the black hole, it behaves like the Maxwell field far from the
horizon radius. The third one is the complexity of black holes which is
reasonable for regular black holes since the singularity may violate the
interpretation of physical quantities such as entangled entropy and
complexity.

The subject of singularity itself and its possible solution (leading to
regularity) are highly interesting from different viewpoints: mathematical and
geometrical points of view, classical gravity and its quantum standpoints,
cosmological side, effective field theory and AdS/CFT correspondence features,
string theory and supergravity aspects, etc. Indeed there are two important
singularities that their natures are not yet fully understood: the singularity
that is covered by an event horizon of black holes and the big bang
singularity. So one of the substantial questions is how one can prevent the
singularity. Although it is believed that quantum gravity can be able to
smooth out singularities, there is no consistent theory of quantum gravity
overcoming this issue despite many attempts. Thus, the question should again
be asked how we can preclude the singularity in an alternative theory.

Regarding the Einstein theory of gravity coupling minimally with an
appropriate nonlinear electrodynamics theory, various regular black hole
metrics are constructed (for an incomplete list, please take a look at
Balart2014 ; Ayon-Beato1998 ; Bronnikov2001 ; Arellano2006 ; Hassain2008 ;
Hollenstein2008 ; Balart2009 ). The exact regular solution of $f(R)$ gravity
has been studied in Hollenstein2008 . Gravitational lensing Eiroa:2010wm ,
dynamical Fernando:2012yw  and thermodynamical stability Hendi:2020knv  of
regular black hole has been studied. A new Smarr-type formula for the black
hole in nonlinear electrodynamics has been obtained in kastor09 ;
Balart:2017dzt . In this paper, we concentrate on the complexity of a simple
static and rotating regular black hole with nonlinear electrodynamics as a
source. The rest of the paper is organized as follows. In Sec. II, we briefly
review the complexity and thermodynamics of regular black hole, especially
deriving its Smarr formula. Section III is devoted to study the complexity of
large-extremal-static regular black hole in the framework of CA and CV
conjectures and comparing the results in two frameworks. In Sec. IV, we
repeated the same computation of complexity for the large-extremal rotating
regular black holes. After that, conclusions are given. In the appendix, we
also investigate two other regular cases and calculated thermodynamic
quantities, complexity and complexity of formation numerically.

##  II basic formalism

Electrically charged black hole solutions can be studied through an
alternative form of nonlinear electrodynamics obtained by the Legendre
transformation. The action of nonlinear electrodynamics in $\mathcal{P}$ frame
minimally coupled to the Einstein gravity with the York-Gibbons-Hawking
surface term and the two-dimensional joint term is given by

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi}\int
d^{4}x\sqrt{-g}\left[\frac{1}{4}(\mathcal{R}-2\Lambda)-\left(\frac{1}{2}P^{\mu\nu}F_{\mu\nu}-\mathcal{H}(\mathcal{P})\right)\right]$
(1) $\displaystyle+\frac{1}{8\pi}\int
d^{3}x\sqrt{-h}\mathcal{K}+\frac{1}{8\pi}\int d^{2}x\sqrt{-\gamma}\;a,$

where $g$ is the determinant of the metric, $\mathcal{R}$ is the Ricci scalar
and the negative cosmological constant is denoted by
$\Lambda=-\frac{3}{l^{2}}$. In addition, the anti-symmetric tensor
$P_{\mu\nu}=\mathcal{L}_{F}F_{\mu\nu}$ in which
$\mathcal{L}_{\mathcal{F}}\equiv\frac{\partial\mathcal{L}}{\partial\mathcal{F}}$,
$\mathcal{F}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $F_{\mu\nu}$ is the Faraday
tensor. Furthermore, the structure function is
$\mathcal{H(\mathcal{P})}=2\mathcal{F}\mathcal{L}_{\mathcal{F}}-\mathcal{L}$
with
$\mathcal{P}\equiv\frac{1}{4}P_{\mu\nu}P^{\mu\nu}=\mathcal{F}\mathcal{L}_{\mathcal{F}}^{2}$
since
$d\mathcal{H}=\mathcal{L}_{\mathcal{F}}^{-1}d\left(\mathcal{F}\mathcal{L}_{\mathcal{F}}^{2}\right)=\mathcal{H}_{\mathcal{P}}d\mathcal{P}$,
where
$\mathcal{H}_{\mathcal{P}}=\frac{d\mathcal{H}}{d\mathcal{P}}=\mathcal{L}_{\mathcal{F}}^{-1}$.
Moreover, $h$ and $\gamma$ are, respectively, the determinant of the induced
metrics $h_{\mu\nu}$ in three-dimensions and two-dimensional
$\gamma_{\mu\nu}$. Also, $\mathcal{K}$ is the trace of the extrinsic curvature
of the induced metric $h_{\mu\nu}$

$\mathcal{K}=n^{\mu}{}_{;\mu}=\dfrac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}n^{\mu}),$
(2)

where $n^{\mu}$ is the normal vector. The integrant $a$ is defined as

$a=\ln\left(-\frac{1}{2}N.\bar{N}\right),$

in which $N$ is the future-directed null normal to the left-moving null
surface and $\bar{N}$ denotes the future-directed null normal to the right-
moving null surface.

The first integral of Eq. (1) represents the bulk action while the second and
third ones stand for the boundary and joint parts of the WDW patch
Lehner:2016vdi . The WDW patch is used to obtain the rate of temporal change
of the action. A typical WDW patch of a black hole with two horizons is shown
in Fig. 4 which is evolved in time from $t_{0}$ to $t_{0}+\delta t$. According
to this figure, the contribution of different parts is bulk region $V_{1}$ and
$V_{2}$, and null-null surface joints $A$, $B$, $C$ and $D$ (for more details,
see Lehner:2016vdi ). Hence, we can write

$\displaystyle\partial\mathcal{A}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\pi
G}\int_{V_{1}}\sqrt{-g}~{}\mathcal{L}\;dtdrd\theta d\phi-\dfrac{1}{4\pi
G}\int_{V_{2}}\sqrt{-g}~{}\mathcal{L}\;dtdrd\theta d\phi$ (3)
$\displaystyle+\frac{1}{8\pi G}\int_{B}\sqrt{-\gamma}\ a_{B}\ d\theta
d\phi-\frac{1}{8\pi G}\int_{A}\sqrt{-\gamma}\ a_{A}\ d\theta d\phi$
$\displaystyle+\frac{1}{8\pi G}\int_{D}\sqrt{-\gamma}\ a_{D}\ d\theta
d\phi-\frac{1}{8\pi G}\int_{C}\ \sqrt{-\gamma}\ a_{C}\ d\theta d\phi\;,$

where $\mathcal{L}$ is the Lagrangian of Einstein-nonlinear electrodynamics.
In order to compute the contribution of joints, we use the following
transformation of $N$ and $\bar{N}$

$N_{\alpha}=-b_{1}\partial_{\alpha}(t-r^{\ast}),\ \ \ \ \ \
\bar{N}=b_{2}\partial_{\alpha}(t+r^{\ast}),$ (4)

in which $b_{1}$ and $b_{2}$ are two arbitrary positive constants and the
tortoise coordinate $r^{\ast}$ is defined as $r^{\ast}=\int\frac{dr}{f(r)}$.
So, the contribution of joints is calculated as

$S_{B}-S_{A}=\frac{\delta t}{4}\
\left(2rf(r)\left[\ln\left(\frac{f(r)}{b_{1}b_{2}}\right)+\frac{rf^{\prime}(r)}{2f(r)}\right]\right)^{r_{B}}_{r_{A}},$
(5) $S_{D}-S_{C}=\frac{\delta t}{4}\
\left(2rf(r)\left[\ln\left(\frac{f(r)}{b_{1}b_{2}}\right)+\frac{rf^{\prime}(r)}{2f(r)}\right]\right)_{r_{A}},$
(6)

where prime denotes derivative with respect to $r$. In Fig. 1, we assume close
cylindrical hypersurfaces consisting of a past spacelike surface $B$, a
truncated null cone $N$, a future spacelike surface $A$ and boundaries for
intersection of spacelike and null surfaces $x$ and $y$.

Figure 1: Closed hypersurfaces consisting of a past spacelike surface $B$, a
truncated null cone $N$ and a future spacelike surface $A$. (a) Same direction
of normal vectors, (b) Opposite direction of normal vectors.

The contribution of surfaces $A$ and $B$ canceled each other in Fig. (1a) due
to the same direction of normal vectors. While for Fig. (1b) according to
opposite direction of normal vectors, contributions of boundaries of
intersection of surfaces, $x$ and $y$ cancel each other. Therefore depending
on what we choose for direction of normal vector, boundary terms or joint
contributions will be vanished Lehner:2016vdi .

It is notable that applying the variational principle to the action (1), one
finds the field equations as

$G_{\mu}^{\nu}+\Lambda\delta_{\mu}^{\nu}=T_{\mu}^{\nu}=\frac{1}{2}\left[\mathcal{H}_{\mathcal{P}}P_{\mu\lambda}P^{\nu\lambda}-\delta_{\mu}^{\nu}\left(2\mathcal{P}\mathcal{H}_{\mathcal{P}}-\mathcal{H}\right)\right],$
(7) $\nabla_{\nu}P^{\mu\nu}=0,$ (8)

where $G_{\mu\nu}$ is the Einstein tensor. By integrating equation (8) with
the assumption of spherical symmetry spacetime, we obtain the following static
solution

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{2}^{2},\hskip
14.22636ptf(r)=1-\frac{1}{r}\int\rho(r)r^{2}dr,$ (9)
$P_{\mu\nu}=2\delta^{t}_{[\mu}\delta^{r}_{\nu]}\frac{q}{r^{2}},\hskip
14.22636pt\mathrm{or}\hskip 14.22636pt\mathcal{P}=-\frac{q^{2}}{2r^{4}},$ (10)

where $q$ is an integration constant and $\mathcal{H}=2\rho(r)$.

In the following, for the importance of Smarr relation, we will obtain it for
the case of spherically symmetric static charged regular black holes. In order
to do that we use the Komar formula for the mass of black holes as follows

$M=-\dfrac{1}{8\pi}\oint_{\infty}\nabla^{\alpha}\xi^{\beta}dS_{\alpha\beta},$
(11)

where $\xi$ is a timelike Killing vector which satisfies Killing equation and
$dS_{\alpha\beta}$ is a two-dimensional surface element of the boundary at
infinity. Since the spacetime has two boundaries, we will write the Komar
integral for the mass as a sum of an integral over a closed null surface at
the horizon $H$ and an integral on the spacelike hypersurface $\Sigma$ which
is bounded by the horizon and infinity as

$M=-\dfrac{1}{8\pi}\oint_{H}\nabla^{\alpha}\xi^{\beta}dS_{\alpha\beta}-\dfrac{1}{8\pi}\int_{\Sigma}\nabla_{\beta}\nabla^{\alpha}\xi^{\beta}d\Sigma_{\alpha}.$
(12)

For the first term, we have

$-\dfrac{1}{8\pi}\oint_{H}\nabla^{\alpha}\xi^{\beta}dS_{\alpha\beta}=\dfrac{\kappa
A}{4\pi},$ (13)

where $A$ is the horizons surface area, $\kappa$ denotes the surface gravity
that is constant at the event horizon and satisfies
$\xi^{\alpha}{}_{;\beta}\xi_{\alpha}=\kappa\xi_{\beta}$. In order to evaluate
the second term in equation (12), we consider the Stokes theorem for an
antisymmetric tensor field $B^{\alpha\beta}$ as follows

$\oint_{S}B^{\alpha\beta}dS_{\alpha\beta}=2\int_{\Sigma}B^{\alpha\beta}{}_{;\beta}d\Sigma_{\alpha}.$
(14)

For the antisymmetric tensor $B^{\alpha\beta}=\nabla^{\alpha}\xi^{\beta}$ of
the bulk term $\Sigma$, we have

$B^{\alpha\beta}{}_{;\beta}=(\nabla^{\alpha}\xi^{\beta})_{;\beta}=-(\nabla^{\beta}\xi^{\alpha})_{;\beta}=-\square\xi^{\alpha},$
(15)

where $\square=\nabla_{\alpha}\nabla^{\alpha}$. Recalling that
$\nabla_{\rho}\nabla_{\mu}\xi_{\nu}=R_{\nu\mu\rho}{}^{\sigma}\xi_{\sigma}$, we
get

$\oint_{S}\nabla^{\alpha}\xi^{\beta}dS_{\alpha\beta}=-2\int_{\Sigma}R^{\alpha}{}_{\beta}\xi^{\beta}d\Sigma_{\alpha}.$
(16)

Now, by using the Einstein field equation
$R_{\alpha\beta}=4\pi\left(2T_{\alpha\beta}-g_{\alpha\beta}T\right)$, we
obtain

$\oint_{S}\nabla^{\alpha}\xi^{\beta}dS_{\alpha\beta}=-16\pi\int_{\Sigma}\left(T_{\beta}^{\alpha}-\dfrac{1}{2}\delta_{\beta}^{\alpha}T\right)\xi^{\beta}d\Sigma_{\alpha}.$
(17)

On the other hand, considering equation (7), the nonzero components of energy-
momentum tensor are

$T^{t}{}_{t}=T^{r}{}_{r}=-\dfrac{\mathcal{H}}{8\pi},\hskip
14.22636ptT^{\theta}{}_{\theta}=T^{\phi}{}_{\phi}=\dfrac{1}{8\pi}\left(2\mathcal{P}\mathcal{H_{P}}-\mathcal{H}\right).$
(18)

By rewriting the RHS of Eq. (17), one can obtain

$\displaystyle-16\pi\int_{\Sigma}\left(T_{\beta}^{\alpha}-\dfrac{1}{2}\delta_{\beta}^{\alpha}T\right)\xi^{\beta}d\Sigma_{\alpha}$
$\displaystyle=$
$\displaystyle-16\pi\int_{\Sigma}\left(T_{\beta}^{\alpha}-\dfrac{1}{4}\delta_{\beta}^{\alpha}T\right)\xi^{\beta}d\Sigma_{\alpha}+4\pi\int_{\Sigma}T\delta^{\alpha}{}_{\beta}\xi^{\beta}d\Sigma_{\alpha}$
$\displaystyle=$
$\displaystyle-2\int_{\Sigma}\dfrac{q}{r^{2}}Ed\Sigma_{\alpha}+4\pi\int
T\sqrt{-h}drd\theta d\phi.$ (19)

since the timelike Killing vector is $\xi^{\beta}=\delta_{t}^{\beta}$ and
$E^{2}=-2\mathcal{P}\mathcal{H_{P}}^{2}$ with $E=q/r^{2}\mathcal{H_{P}}$
identically. Finally, by inserting above results in Eq. (12), we obtain

$M=\dfrac{\kappa A}{4\pi}+q\Phi-\dfrac{1}{2}\int T\sqrt{-h}drd\theta d\phi,$
(20)

where $h$ is the trace of the induced metric of spacelike hypersurface.

##  III The complexity growth of regular electric black holes

Here, we consider a class of known regular electric black holes in AdS
spacetime and calculate its complexity growth rate. For the sake of
completeness, we will point out other regular black hole solutions in the
appendix.

Figure 2: The behavior of q in terms of $r_{+}$ for $M=l=1$. dotted line
(first case), dashed line (second case) and black line (third case).

The motivations of considering the forthcoming kind of $\mathcal{H}$ function
are its simplicity and its agreement to the correspondence which indicates
such a modified nonlinear theory should reduce to the usual Maxwell theory in
the weak-field limit. The ansatz of structure function which is inspired by
the exponential distribution is defined as Balart:2014cga

$\mathcal{H}=\mathcal{P}e^{-\left(\frac{\gamma^{\frac{3}{2}}(-\mathcal{P})^{\frac{1}{4}}}{2^{\frac{5}{4}}\chi}\right)}+\frac{6}{l^{2}},$
(21)

in which for the weak field limit ($\mathcal{P}\ll 1$), it reduces to the
Maxwell field as

$\mathcal{H}\approx\mathcal{P}+\frac{6}{l^{2}}+\mathcal{O}(\mathcal{P}^{\frac{5}{4}}).$
(22)

Considering the mentioned structure function with Eqs. (10) and (9), we can
find the following spherically symmetric exponentially black hole solution
Balart:2014cga ; Balart:2014jia ; Culetu:2014lca ; Hendi:2020knv ;
Ghosh:2014pba

$-g_{tt}=f(r)=1+\frac{r^{2}}{l^{2}}-\frac{2\chi}{r}e^{\left(-\frac{\gamma^{2}}{2\chi
r}\right)}.$ (23)

Using the series expansion of this solution for large values of $r$, it is
noticeable that its asymptotical behavior can be found by the following
expression

$-g_{tt}\approx
1+\frac{r^{2}}{l^{2}}-\frac{2\chi}{r}+\frac{\gamma^{2}}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{4}}\right),$
(24)

which is the Reissner-Nordström-AdS metric function provided $\chi$ and
$\gamma$ are associated with the mass and electric charge of the system,
respectively. In other words, the asymptotical behavior of the obtained
solution is completely matched to the Reissner-Nordström-AdS black hole.

In order to check the first law and the Smarr formula in the extended phase
space, it is convenient to consider the cosmological constant as a varying
thermodynamic quantity and interpret it as a thermodynamic pressure, as
$P=\frac{3}{8\pi l^{2}}$. In addition, its conjugate variable of the
introduced pressure is the thermodynamic volume

$V=\frac{4\pi}{3}r_{+}^{3},$ (25)

where $r_{+}$ is the event horizon radius obtained via $f(r_{+})=0$. The
temperature would be found straightforwardly through the use of surface
gravity ($\kappa$) interpretation with the following explicit relation

$T=\frac{\kappa}{2\pi}=\left.\frac{1}{4\pi}\frac{df(r)}{dr}\right|_{r=r_{+}}=\left(m-\frac{q^{2}}{2r_{+}}\right)\frac{e^{-\left(\frac{q^{2}}{2mr_{+}}\right)}}{2\pi
r_{+}^{2}}+\frac{r_{+}}{2\pi l^{2}},$ (26)

in which for the large black hole event horizon ($r_{+}\gg l$) becomes

$T\approx\dfrac{3r_{+}}{4\pi l^{2}}+\mathcal{O}\left(\dfrac{1}{r_{+}}\right).$
(27)

Since we are working in the Einstein gravity, the black hole entropy $S$
pursues the area law, yielding

$S=\frac{A}{4}=\pi r_{+}^{2}.$ (28)

In addition, the electrostatic potential $\Phi$ can be obtained at the event
horizon versus spatial infinity as the reference

$\Phi=\int_{r_{+}}^{\infty}Edr=\frac{3m}{2q}-\frac{(6mr_{+}-q^{2})e^{-\left(\frac{q^{2}}{2mr_{+}}\right)}}{4qr_{+}},$
(29)

and the asymptotic limit becomes

$\Phi\approx\dfrac{q}{r_{+}}+\mathcal{O}\left(\dfrac{1}{r_{+}^{5}}\right).$
(30)

For a black hole embedded in AdS spacetime, employing the relation between the
cosmological constant and thermodynamic pressure would result to interpret the
mass of black hole as the enthalpy. The enthalpy can be written in terms of
thermodynamic quantities as

$H=m=\frac{q^{2}}{2r_{+}\mathcal{W}\left(\frac{q^{2}l^{2}}{r_{+}^{2}(l^{2}+r_{+}^{2})}\right)},$
(31)

where $\mathcal{W}$ is the $Lambert\ W$ function. In the case of large black
hole event horizon, we have

$H\approx\dfrac{r_{+}^{3}}{2l^{2}}+\dfrac{r_{+}}{2}+\mathcal{O}\left(\dfrac{1}{r_{+}}\right).$
(32)

It is straightforward to check that these thermodynamic quantities satisfy the
first law of black hole thermodynamics in the enthalpy representation

$dH=TdS+VdP+\Phi dq,$ (33)

and the Smarr relation is given by

$\frac{H}{2}+PV-TS-\frac{q\Phi}{2}+\frac{1}{4}\int wdv=0,$ (34)

where the last term of Eq. (34) comes from the fact that the energy-momentum
tensor is not traceless, as

$\int wdv=\frac{1}{2}\int_{r_{+}}^{\infty}T^{\mu}_{\mu}\
r^{2}dr=m-\left(m+\frac{q^{2}}{2r_{+}}\right)e^{-\left(\frac{q^{2}}{2mr_{+}}\right)},$
(35)

and in the case of $r_{+}\gg l$ becomes

$\int
wdv\approx\dfrac{q^{4}l^{2}}{4r_{+}^{5}}+\mathcal{O}\left(\dfrac{1}{r_{+}^{7}}\right).$
(36)

Figure 3: The left hand side of Eq. (34) versus $r$ for $M=0.8,q=0.1,l=1$ .

Now, we want to calculate the complexity growth rate in the above background.
For this aim, we should calculate the Ricci scalar for the solution (23) as

$\mathcal{R}=-\frac{12}{l^{2}}+\frac{q^{4}e^{-\left(\frac{q^{2}}{2mr}\right)}}{2mr^{5}},$
(37)

whereas by using of Eq. (21), the Lagrangian of nonlinear electrodynamics is
given by

$\mathcal{L}=\frac{-q^{2}(4mr-q^{2})e^{-\left(\frac{q^{2}}{2mr}\right)}}{2mr^{5}}-\frac{6}{l^{2}}.$
(38)

Using Eqs. (37) and (38), we can calculate the bulk action growth as

$\frac{d\mathcal{A}_{bulk}}{dt}=\frac{1}{4}\int_{r_{-}}^{r_{+}}r^{2}\left(\mathcal{R}+\frac{6}{l^{2}}-\mathcal{L}\right)dr=m\left[e^{\left(-\frac{q^{2}}{2mr}\right)}\right]_{r_{-}}^{r_{+}},$
(39)

while the growth rate of the surface term (Eq. (1)) is given by
ElMoumni:2020nse

$\displaystyle\frac{d\mathcal{A}_{boundary}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int_{\partial\mathcal{M}}(\sqrt{-h}\mathcal{K})d\Omega_{2}=\frac{1}{2}\left[\sqrt{f(r)}\frac{\partial}{\partial
r}(r^{2}\sqrt{f(r)})\right]_{\partial\mathcal{M}}$ $\displaystyle=$
$\displaystyle\left[\frac{3r^{3}}{2l^{2}}+r-\left(\frac{3m}{2}+\frac{q^{2}}{4r}\right)e^{\left(-\frac{q^{2}}{2mr}\right)}\right]_{r_{-}}^{r_{+}},$

where we should note that in the calculation of boundary term we have used
$\sqrt{-h}=\sqrt{f}r^{2}\sin(\theta)$ and $n^{\mu}=(0,\sqrt{f},0,0)$.

Besides, the contribution of joint terms can be calculated as

$S_{D}-S_{C}=\frac{\delta t}{4}\
\left(2rf(r)\left[\ln\left(\frac{f(r)}{b_{1}b_{2}}\right)+\frac{rf^{\prime}(r)}{2f(r)}\right]\right)_{r_{C}}.$
(41)

After some manipulations, we can find

$\displaystyle S_{B}-S_{A}$ $\displaystyle=$ $\displaystyle\frac{\delta
t}{4}\left[2me^{-(\frac{q^{2}}{2mr})}-\frac{q^{2}e^{-(\frac{q^{2}}{2mr})}}{r}+\frac{2r^{3}}{l^{2}}+\left(2r-4me^{-(\frac{q^{2}}{2mr})}+\frac{2r^{3}}{l^{2}}\right)\ln\left(\frac{1-\frac{2m}{r}e^{-(\frac{q^{2}}{2mr})}+\frac{r^{2}}{l^{2}}}{b_{1}b_{2}}\right)\right]_{r_{A}},$
(42) $\displaystyle S_{D}-S_{C}$ $\displaystyle=$ $\displaystyle\frac{\delta
t}{4}\left[2me^{-(\frac{q^{2}}{2mr})}-\frac{q^{2}e^{-(\frac{q^{2}}{2mr})}}{r}+\frac{2r^{3}}{l^{2}}+\left(2r-4me^{-(\frac{q^{2}}{2mr})}+\frac{2r^{3}}{l^{2}}\right)\ln\left(\frac{1-\frac{2m}{r}e^{-(\frac{q^{2}}{2mr})}+\frac{r^{2}}{l^{2}}}{b_{1}b_{2}}\right)\right]_{r_{C}}.$
(43)

Since at the late time $r_{A}$ and $r_{C}$ approach to $r_{-}$ and $r_{+}$,
respectively, while $f(r)$ vanishes, one finds Eqs. (42) and (43) reduce to

$\displaystyle S_{B}-S_{A}$ $\displaystyle=$ $\displaystyle\frac{\delta
t}{4}\left[2me^{-\left(\frac{q^{2}}{2mr}\right)}-\frac{q^{2}e^{-\left(\frac{q^{2}}{2mr}\right)}}{r}+\frac{2r^{3}}{l^{2}}\right]_{r_{A}},$
(44) $\displaystyle S_{D}-S_{C}$ $\displaystyle=$ $\displaystyle\frac{\delta
t}{4}\left[2me^{-\left(\frac{q^{2}}{2mr}\right)}-\frac{q^{2}e^{-\left(\frac{q^{2}}{2mr}\right)}}{r}+\frac{2r^{3}}{l^{2}}\right]_{r_{C}}.$
(45)

So, the total growth rate of the action for such a black hole configuration
within WDW patch at late time approximation is simplified as

$\frac{d\mathcal{A}}{dt}=(r_{+}-r_{-})\left[1+\frac{3}{2l^{2}}(r_{+}^{2}+r_{-}^{2}+r_{+}r_{-})\right]+\left(\frac{q^{2}}{4r_{-}}+\frac{m}{2}\right)e^{\left(-\frac{q^{2}}{2mr_{-}}\right)}-\left(\frac{q^{2}}{4r_{+}}+\frac{m}{2}\right)e^{\left(-\frac{q^{2}}{2mr_{+}}\right)}.$
(46)

In order to describe Eq. (46) in terms of $r_{-}$ and $r_{+}$, we can use the
redefinitions $m$ and $q^{2}$ in terms of $r_{-}$ and $r_{+}$ as

$m=\frac{r_{-}^{2}(r_{-}^{2}+l^{2})\ln(A)A^{\left(\frac{r_{+}}{r_{+}-r_{-}}\right)}}{2(r_{+}-r_{-})l^{2}\mathcal{W}\left(\frac{\ln(A)A^{\left(\frac{r_{+}}{r_{+}-r_{-}}\right)}r_{-}^{2}(l^{2}+r_{-}^{2})}{r_{+}(r_{+}-r_{-})(l^{2}+r_{+}^{2})}\right)},\hskip
22.76228ptq^{2}=\frac{\ln(A)r_{-}^{2}(l^{2}+r_{-}^{2})r_{+}}{(r_{+}-r_{-})l^{2}e^{\frac{r_{+}\ln(A)}{r_{-}-r_{+}}}},$
(47)

where

$A=\frac{r_{+}(l^{2}+r_{+}^{2})}{r_{-}(l^{2}+r_{-}^{2})}.$ (48)

Although it is straightforward to rewrite Eq. (46) in terms of $r_{-}$ and
$r_{+}$, we ignore its explicit relation for the sake of brevity. Finally, by
using the above calculated thermodynamic quantities, it is easy to show that

$\frac{d\mathcal{A}}{dt}\leq 2\left(m-q\phi+\frac{1}{2}\int
wdv\right)_{+}-2\left(m-q\phi+\frac{1}{2}\int wdv\right)_{-},$ (49)

indicating that the action growth rate of the black hole in the WDW patch has
been bounded. For instance, for the values $m=0.8,q=0.1,l=1$ this bound is
about $1.5761$, and the differences between left and right (right mines left)
is about $0.0396$ which is clearly non-negative. It is worth mentioning that
although there are some extra terms in the right hand side of Eq. (49) due to
considering the nonlinear electrodynamics as a source, the action growth rate
is bounded.

###  III.1 Extreme case: $r_{+}\approx r_{-}$

Now, we focus on the extremal solutions. By introducing $\alpha=l/r_{+}$,
$\epsilon=1-r_{-}/r_{+}$ and in the case of extremal, large black hole,
equation (46) becomes

$\dfrac{d\mathcal{A}}{dt}\approx\left(\dfrac{9r_{+}}{4\alpha^{2}}-r_{+}\alpha^{2}+\dfrac{7r_{+}}{4}\right)\epsilon+\mathcal{O}(\epsilon^{2})+\mathcal{O}(\alpha^{4}),$
(50)

in terms of $r_{\pm}$ leads to

$\dfrac{d\mathcal{A}}{dt}\approx\dfrac{9}{4l^{2}}(r_{+}^{3}-r_{-}r_{+}^{2})+\dfrac{7}{4}(r_{+}-r_{-})+\mathcal{O}\left(\dfrac{1}{r_{+}}\right).$
(51)

By using of equations (25), (32) and ($r_{+}\approx r_{-}$) one can rewrite
above equation as follows Couch:2016exn

$\dfrac{d\mathcal{A}}{dt}\approx\dfrac{9}{2}P(V_{+}-V_{-}),\hskip
28.45274ptr_{+}\gg l,$ (52)

which shows that in the large black holes the rate of complexity is
proportional to $P\Delta V$, i.e, is controlled by thermodynamical volume.
Besides, it is notable that one can obtain such a relation in terms of entropy
for the static black holes since in the static case the thermodynamic volume
is not independent of entropy ($V=4/(3\sqrt{\pi})S^{3/2}$).

In the case of extremal black hole with small degenerate horizon radius, we
have

$\dfrac{d\mathcal{A}}{dt}\approx\left(\dfrac{13}{\alpha^{2}}+3\right)\dfrac{\epsilon}{4}+\mathcal{O}\left(\dfrac{1}{\alpha^{3}}\right)+\mathcal{O}(\epsilon^{2}).$
(53)

By using of CV conjecture, the rate of complexity is An:2018dbz

$\dfrac{d\mathcal{C}_{\mathcal{V}}}{dt}=\dfrac{4\pi}{Gl}\sqrt{-f(r_{min})}r_{min}^{2},$
(54)

where $r_{min}$ is the turning point of maximal surface. The late time limit
of $d\mathcal{C}_{\mathcal{V}}/dt$ becomes

$\dfrac{d\mathcal{C}_{\mathcal{V}}}{dt}=\dfrac{4\pi}{Gl}\sqrt{-f(\hat{r}_{min})}\hat{r}^{2}_{min},\hskip
28.45274ptt\longrightarrow\infty,$ (55)

where $\hat{r}_{min}$ is the extreme value point of
$\sqrt{-f(r_{min})}r_{min}^{2}$. For the present case we need to obtain the
zeros of the following equation

$l^{2}(6M\hat{r}_{min}+q^{2})\exp\left(-\dfrac{q^{2}}{2M\hat{r}_{min}}\right)-2\hat{r}^{2}_{min}(2l^{2}+3\hat{r}^{2}_{min})=0.$
(56)

in the case of small $q$, it becomes

$\hat{r}_{min}\approx r_{0min}+\varepsilon
r_{1min}=r_{0}+\dfrac{q^{2}l^{2}\varepsilon}{3Ml^{2}-12r_{0}^{3}-4r_{0}l^{2}},$
(57)

where

$r_{0}=\dfrac{(108Ml^{2}+4\sqrt{32l^{4}+729l^{4}M^{2}})^{\frac{1}{3}}}{6}-\dfrac{4l^{2}}{(108Ml^{2}+4\sqrt{32l^{4}+729l^{4}M^{2}})^{\frac{1}{3}}}.$
(58)

In the case of extremal black hole with large horizon radius, we can write

$\dfrac{\hat{r}_{min}}{r_{+}}\approx
1.5-0.5\varepsilon+\mathcal{O}(\varepsilon,\alpha^{2}),$ (59)

and correspondingly, for the late time of growth of complexity, we obtain

$\dfrac{d\mathcal{C}_{\mathcal{V}}}{dt}\approx\dfrac{40\pi}{\alpha^{2}}(r_{+}-r_{-})\approx
20P\Delta V.$ (60)

The comparison of the complexity growth from the CV and CA dualities are

$\mathcal{R}_{rate}=\dfrac{d\mathcal{C}_{\mathcal{A}}/dt}{d\mathcal{C}_{\mathcal{V}}/dt}=\dfrac{9P\Delta
V}{40P\Delta V}=0.225.$ (61)

###  III.2 Complexity of formation

The complexity of formation is the difference between complexity in the
process of forming the entangled TFD state and preparing two individual copies
of the vacuum state of the left and right boundary CFTs Chapman:2016hwi ;
Cottrell:2017ayj

$\Delta\mathcal{C}_{\mathcal{A}}=\dfrac{1}{\pi}(\mathcal{A}_{BH}-2\mathcal{A}_{AdS}).$
(62)

Figure 4: Penrose diagram for charged black holes.

For the case of charged black hole with two horizons, the complexity of
formation in CA conjecture becomes Chapman:2016hwi

$\Delta\mathcal{C}_{\mathcal{A}}=\dfrac{1}{\pi}(\Delta\mathcal{A}_{bulk}+\mathcal{A}_{joint,meet}).$
(63)

Now, we evaluate the action for the obtained black hole solutions. The
tortoise coordinate for the both horizons of the solution is

$r^{\ast}(r)=\dfrac{l^{2}r_{+}\ln\left(\dfrac{|r-r_{+}|}{r+r_{+}}\right)}{3r_{+}^{2}+l^{2}-(l^{2}+r_{+}^{2})\mathcal{W}\left(\dfrac{l^{2}q^{2}}{r_{+}^{2}(l^{2}+r_{+}^{2})}\right)}+\dfrac{l^{2}r_{-}\ln\left(\dfrac{|r-r_{-}|}{r+r_{-}}\right)}{3r_{-}^{2}+l^{2}-(l^{2}+r_{-}^{2})\mathcal{W}\left(\dfrac{l^{2}q^{2}}{r_{-}^{2}(l^{2}+r_{-}^{2})}\right)},$
(64)

The point where the ingoing null rays from the two asymptotic regions meet
inside the black hole between the two horizons ($r_{-}<r_{meet}<r_{+}$) can be
calculated numerically using Eqs. (64)-(136) which reads

$r^{\ast}(r_{meet})=0.$ (65)

Figure 5: The behavior of $r_{meet}$ in terms of $r_{+}$ (left) and $q$
(right) for $M=l=1$. red line (first case), black line (second case) and blue
line (third case).

We show the results for $r_{meet}$ in Fig. 5 for the considered solution.
According to the Fig. 5a, by increasing event horizon radius, $r_{meet}$
decreases from extremal value to $r_{-}$. Then, by increasing the electric
charge, $r_{meet}$ increases (Fig. 5b). By using of numerical curve fitting
and choosing the appropriate branches of the logarithms, one can obtain the
following relation for $r_{meet}(r_{+})$

$r_{meet}\approx 0.6825r_{+}^{2}-2.0259r_{+}+1.336,$ (66)

The above approximation functions are necessary to obtain the $A_{bulk}$ and
$A_{joint}$ in the following.
The bulk action of the black hole is

$\displaystyle A_{bulk,BH}=$ $\displaystyle 4\dfrac{4\pi}{16\pi
G}\int_{r_{meet}}^{r_{max}}\left(\mathcal{R}+\dfrac{6}{l^{2}}-\mathcal{L}\right)dr\int_{0}^{v_{\infty}-r^{\ast}(r)}dt$
$\displaystyle=$
$\displaystyle\dfrac{2q^{2}}{G}\int_{r_{meet}}^{r_{max}}dr\left(\dfrac{\exp\left(-\dfrac{q^{2}}{2Mr}\right)}{r^{4}}\right)(v_{\infty}-r^{\ast}(r))\approx\dfrac{8.27r_{+}^{2}}{Gl^{4}}-\dfrac{0.005r_{+}}{Gl^{4}}+\dfrac{0.00003r_{+}}{G\delta
l^{2}}+\mathcal{O}(1),$ (67)

where $v_{\infty}=\lim_{r\rightarrow\infty}r^{\ast}(r)$ is the constant
defining the future null boundary,
$r_{max}=\frac{l^{2}}{\delta}-\frac{\delta}{4}+\frac{M\delta^{2}}{3l^{2}}-\frac{q^{2}\delta^{3}}{8l^{4}}$
and $\delta$ is a some cutoff surface associated with a UV divergence. In
order to obtain the action of AdS spacetime, one uses

$f_{AdS}=1+\dfrac{r^{2}}{l^{2}},$ (68)

in which the tortoise coordinate for AdS spacetime is

$r_{0}^{\ast}{}(r)=l\arctan\left(\dfrac{r}{l}\right),\hskip
14.22636ptv_{0\infty}=\dfrac{\pi l}{2}.$ (69)

Thus the bulk action of AdS vacuum is

$\displaystyle A_{AdS}$ $\displaystyle=4\dfrac{4\pi}{16\pi
G}\int_{0}^{r_{0max}}dr\left(\mathcal{R}+\dfrac{6}{l^{2}}\right)\int_{0}^{v_{0\infty}-r_{0}^{\ast}{}(r)}dt=-\dfrac{6}{Gl^{2}}\int_{0}^{r_{0max}}dr(v_{0\infty}-r_{0}^{\ast}{}(r))=$
$\displaystyle\dfrac{-1}{G}\left[-l^{2}\ln\left(1+\dfrac{r_{0max}^{2}}{l^{2}}\right)+r_{0max}^{2}\left(1+\dfrac{\pi
r_{0max}}{l}-\dfrac{2r_{0max}}{l}\arctan(\dfrac{r_{0max}}{l})\right)\right],$
(70)

where $\mathcal{R}_{ads}=-12/l^{2}$ and
$r_{0max}=\frac{l^{2}}{\delta}-\frac{\delta}{4}$. Now, we should calculate the
following relation for the solution

$\Delta A_{bulk}=A_{bulk,BH}-A_{AdS}.$ (71)

The action at the meet point $r_{meet}$ is

$\mathcal{A}_{joint}=\frac{2}{G}r_{meet}^{2}\ln[f(r_{meet})],$ (72)

which in the case of large black hole becomes

$\mathcal{A}_{joint}\approx
0.0186\ln\left(\dfrac{0.0682r_{+}^{2}}{l}\right)r_{+}^{4}+\mathcal{O}(r_{+}^{3}).$

The total action is the sum of the bulk (71) and joint (72) terms.
Substituting the numerical solution for $r_{meet}$, we obtain the interesting
result plotted in Fig. 6. According to these figures, one finds the bulk
action of black hole and complexity of formation are linear functions of
entropy. For the large black hole, we have

$\Delta\mathcal{C}_{\mathcal{A}}\approx\dfrac{8.27C_{T}S}{l^{2}}+\mathcal{O}(S^{1/2}),$
(73)

where $C_{T}=3l^{2}/\pi^{3}G$ is the boundary central charge.

Figure 6: Plot of $\Delta\mathcal{C}_{\mathcal{V}}$ (up) and
$\Delta\mathcal{C}_{\mathcal{A}}$ (down) in terms of $r_{+}$ for static
exponential solution..

Here, we want to obtain the complexity of formation by using of CV conjecture.
To do that, we consider the maximal volume functional for the $t=0$ timeslice
(the straight line connecting the two boundaries through the bifurcation
surface in the Penrose diagram) as follows Chapman:2016hwi ; Balushi:2020wkt

$\mathcal{V}=8\pi\int_{r_{+}}^{r_{max}}dr\dfrac{r^{2}}{\sqrt{f}}=4\pi
lr_{+}^{2}+2\pi l^{3}+\dfrac{\pi l\delta^{2}}{4}+8\pi
l^{3}\ln\left(\dfrac{l}{\sqrt{\delta
r_{+}}}\right)+\mathcal{O}\left(\dfrac{l^{5}}{r_{+}^{5}}\right),$ (74)

where the corresponding contribution from two copies of the vacuum AdS
background is

$\mathcal{V}_{AdS}=8\pi\int_{0}^{r_{max}}dr\dfrac{r^{2}}{\sqrt{f_{AdS}}}=4\pi
l\left[r_{max}\sqrt{l^{2}+r_{max}^{2}}-l^{2}\ln\left(\dfrac{r_{max}+\sqrt{l^{2}+r_{max}^{2}}}{l}\right)\right]=\dfrac{4\pi
l^{5}}{\delta^{2}}-4\pi l^{3}\ln\left(\dfrac{2l}{\delta}\right)-\dfrac{\pi
l\delta^{2}}{4}.$ (75)

The complexity of formation is obtained by subtracting from (74) the volume of
the AdS vacuum

$\Delta\mathcal{C}_{\mathcal{V}}=\dfrac{\mathcal{V}-2\mathcal{V}_{AdS}}{G_{N}R}=\dfrac{8\pi}{G_{N}R}\left[\int_{r_{+}}^{r_{max}}\dfrac{r^{2}dr}{\sqrt{f(r)}}-\int_{0}^{r_{max}}\dfrac{r^{2}dr}{\sqrt{f_{AdS}}}\right]=\dfrac{41.3C_{T}S}{l^{3}}+\mathcal{O}(S^{1/2}).$
(76)

To understand the behaviour of complexity of formation, we obtain the plot of
$\Delta\mathcal{C}_{\mathcal{V}}$ as a function of $r_{+}$ in Fig. 6.

It is interesting, of course, to compare the results of CA and CV duality. The
ratio of the complexity of formation for large black holes $r_{+}\gg l$ is

$\mathcal{R}_{form}=\dfrac{\Delta\mathcal{C}_{\mathcal{A}}}{\Delta\mathcal{C}_{\mathcal{V}}}\approx
0.2,$ (77)

and by using (61) and (77) one gets

$\mathcal{R}_{rate}-\mathcal{R}_{form}=0.02.$ (78)

As can be seen, the differences between two ratios is small. So, the two
ratios agree very well and this comparison suggests that the two holographic
approaches to complexity are consistent.

##  IV Generalization to rotating solution:

The metric of rotating charged regular black hole in the Boyer-Lindquist
coordinates is obtained as

$dS^{2}=-\dfrac{\Delta_{r}}{\Sigma}\left(dt-\dfrac{a\sin^{2}(\theta)}{\Xi}d\phi\right)^{2}+\dfrac{\Sigma}{\Delta_{r}}dr^{2}+\dfrac{\Sigma}{\Delta_{\theta}}d\theta^{2}+\dfrac{\Delta_{\theta}\sin^{2}(\theta)}{\Sigma}\left(adt-\dfrac{r^{2}+a^{2}}{\Xi}d\phi\right)^{2},$
(79)

where

$\displaystyle\Delta_{r}$
$\displaystyle=(r^{2}+a^{2})(1+\dfrac{r^{2}}{l^{2}})-2f,\;\;\;\;\Sigma=r^{2}+a^{2}\cos^{2}(\theta),\;\;\;\;\Xi=1-\dfrac{a^{2}}{l^{2}},$
$\displaystyle\Delta_{\theta}$
$\displaystyle=1-\dfrac{a^{2}}{l^{2}}\cos^{2}(\theta),\;\;\;\;f(r)=Mr\exp\left(-\dfrac{q^{2}}{2Mr}\right).$
(80)

The outer and inner horizons are determined by the equation
$\Delta(r_{\pm})=0$, respectively. The required thermodynamical quantities to
check the first law are Hendi:2020knv

$\displaystyle P=$ $\displaystyle\dfrac{3}{8\pi l^{2}},\hskip
14.22636ptV=\dfrac{r_{+}A}{3}+\dfrac{4\pi J^{2}}{3\mathcal{M}},\hskip
14.22636ptA=\dfrac{4\pi(r_{+}^{2}+a^{2})}{\Xi},\hskip
14.22636ptJ=\dfrac{Ma}{\Xi^{2}},\hskip
14.22636pt\mathcal{M}=\dfrac{M}{\Xi^{2}},\hskip 14.22636ptQ=\dfrac{q}{\Xi},$
$\displaystyle T=$
$\displaystyle-\dfrac{(2Mr_{+}+q^{2})e^{-\left(\dfrac{q^{2}}{2Mr_{+}}\right)}}{4\pi(r_{+}^{2}+a^{2})r_{+}}+\dfrac{r_{+}(r_{+}^{2}+2r_{+}^{2}+a^{2})}{2\pi
l^{2}(r_{+}^{2}+a^{2})},\hskip
14.22636pt\Phi=\dfrac{qr_{+}}{r_{+}^{2}+a^{2}},\hskip
14.22636pt\Omega=\dfrac{a}{r_{+}^{2}+a^{2}}\left(1+\dfrac{r_{+}^{2}}{l^{2}}\right).$
(81)

where

$M=\dfrac{q^{2}}{2r_{+}W\left(\dfrac{l^{2}q^{2}}{(r_{+}^{2}+a^{2})(r_{+}^{2}+l^{2})}\right)}.$
(82)

So, the first law of thermodynamic is satisfied as Hendi:2020knv

$d\mathcal{M}=TdS+PdV+\Phi dQ+\Omega dJ,$ (83)

Integrating the on-shell Einstein-Hilbert bulk action, directly, in the case
of $\frac{q}{M}\ll 1$ and $\frac{a}{l}\ll 1$ leads to

$\displaystyle\frac{d\mathcal{A}_{bulk}}{dt}$
$\displaystyle=\frac{1}{8}\int_{r_{-}}^{r_{+}}\int_{0}^{\pi}\sqrt{-g}\left(\mathcal{R}+\frac{6}{l^{2}}-\mathcal{L}\right)drd\theta=$
$\displaystyle-\left[\dfrac{24Mr^{7}-24Ml^{2}q^{2}r^{3}+9l^{2}q^{4}r^{2}+24Ma^{2}r^{5}+24Ma^{2}q^{2}l^{2}r-11a^{2}l^{2}q^{4}}{48\Xi
ml^{2}r^{4}}\right]_{r_{-}}^{r_{+}},$ (84)

since $\sqrt{-g}=\dfrac{\sin\theta}{\Xi}\Sigma$ and

$\mathcal{L}=\dfrac{2q^{2}e^{-\dfrac{q^{2}}{2Mr}}((Mr+q^{2})a^{4}\cos^{4}\theta+(-6Mr+q^{2})a^{2}r^{2}\cos^{2}\theta+Mr^{5})}{Mr(r^{2}+a^{2}\cos^{2}\theta)^{4}}.$
(85)

The growth rate of the surface term in the case of $\frac{q}{M}\ll 1$ and
$\frac{a}{l}\ll 1$ (Eq. (1)) is

$\dfrac{d\mathcal{A}_{boundary}}{dt}=\frac{1}{8\pi}\int_{\partial\mathcal{M}}(\sqrt{-h}\mathcal{K})d\Omega_{2}=\left[\dfrac{\Delta_{r}^{{}^{\prime}}}{4\Xi}\right]_{r_{-}}^{r_{+}}=\left[\dfrac{-8M^{2}r^{2}l^{2}+8Mr^{3}(l^{2}+a^{2}+2r^{2})+q^{4}l^{2}}{16Ml^{2}\Xi
r^{2}}\right]_{r_{-}}^{r_{+}},$ (86)

since the determinant of the induced metric on the null hypersurface $r_{\pm}$
is $h=-\dfrac{\sin^{2}\theta\Sigma\Delta_{r}}{\Xi^{2}}$ and the trace of
extrinsic curvature can be calculated from Eq. (2) and the normal vector is
$n_{\mu}=(0,\sqrt{\frac{\Delta_{r}}{\Sigma}},0,0)$. Combining the bulk action
and boundary term, we can obtain the growth rate of total action as follows

$\dfrac{d\mathcal{A}}{dt}=\left[\dfrac{24Mq^{2}l^{2}r^{3}-6q^{4}l^{2}r^{2}-24Ma^{2}q^{2}l^{2}r+11a^{2}q^{4}l^{2}-24M^{2}l^{2}r^{4}+24Ml^{2}r^{5}+24Mr^{7}}{48\Xi
Ml^{2}r^{4}}\right]_{r_{-}}^{r_{+}},$ (87)

where

$q^{2}=\dfrac{r_{+}(a^{2}l^{2}+a^{2}r_{-}^{2}+r_{-}^{2}l^{2}+r_{-}^{4})\ln\left(\dfrac{r_{-}(l^{2}+r_{+}^{2})(a^{2}+r_{+}^{2})}{r_{+}(a^{2}+r_{-}^{2})(l^{2}+r_{-}^{2})}\right)}{(r_{+}-r_{-})l^{2}\left(\dfrac{r_{-}(l^{2}+r_{+}^{2})(a^{2}+r_{+}^{2})}{r_{+}(a^{2}+r_{-}^{2})(l^{2}+r_{-}^{2})}\right)^{\frac{r_{+}}{r_{-}-r_{+}}}}.$

In the case of $\alpha\ll 1$, $\beta\ll 1$ and $\epsilon\ll 1$, one finds

$\displaystyle q^{2}/r_{+}^{2}$ $\displaystyle=$
$\displaystyle\left(\dfrac{60.26}{\alpha^{2}}-100.43+\left(-\dfrac{100.43}{\alpha^{2}}+140.6\right)\beta^{2}\right)+$
(88)
$\displaystyle\left(-\dfrac{120.51}{\alpha^{2}}+100.43+\dfrac{100.43\beta^{2}}{\alpha^{2}}\right)\epsilon+\mathcal{O}(\epsilon^{2})+\mathcal{O}(\alpha^{2})+\mathcal{O}(\beta^{4}),$

while for $\alpha\gg 1$, $\beta\ll 1$ and $\epsilon\rightarrow 0$, we have

$\displaystyle q^{2}/r_{+}^{2}$ $\displaystyle=$ $\displaystyle
2.72+\dfrac{13.59}{\alpha^{2}}-\left(8.15+\dfrac{29.9}{\alpha^{2}}\right)\beta^{2}-$
(89)
$\displaystyle\left(2.72+\dfrac{27.2}{\alpha^{2}}-\dfrac{29.9\beta^{2}}{\alpha^{2}}\right)\epsilon+\mathcal{O}(\epsilon)+\mathcal{O}(\beta^{4})+\mathcal{O}(1/\alpha^{4}).$

In addition, in the case of $\alpha\ll 1$, $\beta\ll 1$ and
$\epsilon\rightarrow 0$, we obtain

$\dfrac{d\mathcal{A}}{dt}\approx\dfrac{22.3r_{+}^{3}}{l^{2}}-\dfrac{9.32r_{-}r_{+}^{2}}{l^{2}}+\mathcal{O}(r_{+}),$
(90)

and for $\alpha\ll 1$, $\beta\ll 1$ and $\epsilon\rightarrow 1$, one achieves

$\dfrac{d\mathcal{A}}{dt}\approx\dfrac{0.21r_{+}\alpha^{4}\beta^{6}}{\epsilon^{12}}+\mathcal{O}(\epsilon^{-11}),$
(91)

since

$\displaystyle V/r_{+}^{3}\approx$ $\displaystyle\dfrac{0.8\times
10^{-9}\alpha^{6}\beta^{6}}{\epsilon^{12}},\hskip
28.45274pt\epsilon\rightarrow 1,$ $\displaystyle V/r_{+}^{3}\approx$
$\displaystyle 84.13+560.9\alpha^{6},\hskip 28.45274pt\epsilon\rightarrow 0.$
(92)

As can be seen in both cases $\epsilon\rightarrow 1,0$, the rate of complexity
proportional to the volume of black holes instead of entropy of black holes.

###  IV.1 Complexity of formation

In this section complexity of formation for rotating black holes in the limit
$\frac{a}{l}\ll 1$ and $\frac{q}{M}\ll 1$ and in the conjectures Complexity-
Action and Complexity-Volume is calculated.

To do so, we calculated Lagrangian of the bulk and vacuum AdS in the limit
$\frac{a}{l}\ll 1$ and $\frac{q}{M}\ll 1$, which in the second-order
approximation are written the following form

$L_{bulk}=\frac{-4r^{3}l^{2}-4r^{3}a^{2}-4a^{2}l^{2}r+\frac{4q^{2}l^{2}(l^{2}+a^{2})}{r}-\frac{4a^{2}q^{2}l^{4}}{r^{3}}}{8l^{4}},\;\;\;\;\;\;L_{AdS}=-\frac{2r^{3}}{l^{2}}.$

For obtaining the complexity of formation we should compute the following
relation:

$\pi\Delta
C_{A}=A_{bulk}-A_{Ads}+A_{joints}=\int\left(\int\sqrt{-g}\pounds_{bulk}(r_{\infty}^{\ast}-r^{\ast})dr-\int\sqrt{-g}\pounds_{AdS}(r_{\infty}^{\ast}-r^{\ast})dr\right)d\theta+A_{joints},$

where

$\pounds_{bulk}=R+\frac{6}{l^{2}}-L,\;\;\;\;\pounds_{AdS}=R+\frac{6}{l^{2}},\;\;\;\;r^{\ast}=\int\sqrt{\frac{g_{rr}}{g_{tt}}}dr.$

Since $\lim_{r\rightarrow\infty}r^{\ast}=0\ $and$\ r^{\ast}$ in the limit
$\frac{q}{M}\ll 1$ and $\frac{a}{l}\ll 1$ can be calculated analytically, but
the integral of the first term with respect to $r$ can not be calculated
analytically thus it is actually more convenient to use integration by parts
to eliminate the appearance of $r^{\ast}$ inside this expression, because
after that we must integrating with respect to $\theta$ therefore the first
integral must be calculated analytically,

$A_{bulk}=\int\int\sqrt{-g}\pounds_{bulk}(-r^{\ast})drd\theta=\int\left((-r^{\ast})\int\sqrt{-g}\pounds_{bulk}dr+\int\left(\sqrt{-\frac{g_{rr}}{g_{tt}}}\int\sqrt{-g}\pounds_{bulk}dr\right)dr\right)d\theta.$

$A_{Ads}$ is obtained in the same way. After some calculations complexity of
formation for slowly rotating black holes in the limit $\frac{q}{M}\ll 1$
within Complexity-Action conjecture is obtained which is shown in Fig. 7.

Figure 7: Complexity of formation in the CA conjecture versus
$\frac{r_{-}}{r_{+}}$ for rotating black hole.

The behavior of $\frac{r_{meet}}{r_{+}}$ ($r_{meet}$) versus $r_{-}/r_{+}$
($r_{+}$) is plotted in Fig. 8. We find that $r_{meet}$ is approximately a
linear function of event horizon and the best fitting curve relation is
$r_{meet}=-2.5r_{+}+2.6$.

Figure 8:  The behavior of $\frac{r_{meet}}{r_{+}}$ versus
$\frac{r_{-}}{r_{+}}$ (up) and $r_{meet}$ versus $r_{+}$ (down).

Complexity of formation in CV conjecture

For calculating the complexity in CV conjecture, we must find the surface with
maximal volume. For this purpose we should write the line element in the
tortoise coordinate

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
d\upsilon^{2}\left(-\frac{\Delta_{r}}{\Sigma}+\frac{a^{2}\Delta_{\theta}\sin^{2}(\theta)}{\Sigma}\right)+dr^{2}\left(-\frac{\Delta_{r}}{\Sigma}\frac{g_{rr}}{g_{tt}}+\frac{a^{2}\Delta_{\theta}\sin^{2}(\theta)}{\Sigma}\frac{g_{rr}}{g_{tt}}+\frac{\Sigma}{\Delta_{r}}\right)$
$\displaystyle+2drd\upsilon\sqrt{-\frac{g_{rr}}{g_{tt}}}\left(\frac{\Delta_{r}}{\Sigma}-\frac{a^{2}\Delta_{\theta}\sin^{2}(\theta)}{\Sigma}\right)+\frac{\Sigma}{\Delta_{\theta}}d\theta^{2}+d\phi^{2}\left(-\frac{\Delta_{r}}{\Sigma}\frac{a^{2}\sin^{4}(\theta)}{\Xi^{2}}+\frac{(a^{2}+r^{2})^{2}\Delta_{\theta}\sin^{2}(\theta)}{\Xi^{2}\Sigma}\right)$
$\displaystyle-d\phi
d\upsilon\left(\frac{a\sin^{2}(\theta)}{\Sigma\Xi}\right)\left(2\Delta_{r}-(a^{2}+r^{2})\Delta_{\theta}\right)+drd\phi\sqrt{-\frac{g_{rr}}{g_{tt}}}\left(-2\frac{\Delta_{r}}{\Sigma}+\frac{a(a^{2}+r^{2})\Delta_{\theta}\sin^{2}(\theta)}{\Xi\Sigma}\right),$

where

$\upsilon=t+r^{\ast},\;\;\;\;r^{\ast}=\int\sqrt{-\frac{g_{rr}}{g_{tt}}}dr=\int\sqrt{\frac{\Sigma^{2}}{\Delta_{r}^{2}-a^{2}\Delta_{r}\Delta_{\theta}\sin^{2}(\theta)}}dr.$

To find the maximal surface we describe the surface with the parametric
relations $r=r(\lambda)$ and $\upsilon=\upsilon(\lambda)$ with some parameter
$\lambda$, then the volume of the surface becomes in the following form

$V=\int\sqrt{\sigma}d^{2}x=4\pi\int\int drd\theta
r^{2}\left(\sqrt{\left[-\frac{\Delta_{r}}{\Sigma}+\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Sigma}\right]\dot{\upsilon}^{2}+2\dot{r}\dot{\upsilon}\sqrt{1-\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Delta_{r}}}}\right),$
(93)

where dots indicate derivatives with respect to $\lambda$ and the factor 2 is
originated from this fact that the surface is composed from two equivalent
parts. We are free to choose $\lambda$ to keep the radial volume element fixed
as follow

$r^{2}\sqrt{\left[-\frac{\Delta_{r}}{\Sigma}+\frac{\Delta_{\theta}a^{2}\sin^{2}}{\Sigma}\right]\dot{\upsilon}^{2}+2\dot{r}\dot{\upsilon}\sqrt{1-\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Delta_{r}}}}=1,$
(94)

this Lagrangian is independent of $\upsilon$ and hence there is a conserved
quantity given by

$E=-\frac{\partial\pounds}{\partial\upsilon}=\left(\frac{\Delta_{r}}{\Sigma}-\frac{\Delta_{\theta}a^{2}\sin(\theta)^{2}}{\Sigma}\right)\dot{\upsilon}-\sqrt{1-\frac{\Delta_{\theta}a^{2}\sin(\theta)^{2}}{\Delta_{r}}}\dot{r}.$
(95)

By substituting $\dot{\upsilon}$ from (94) into the (95), one can obtain

$E^{2}=r^{4}\left(\frac{\Delta_{r}}{\Sigma}-\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Sigma}\right)+r^{8}\dot{r}^{2}\sqrt{1-\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Delta_{r}}},$

with substituting above equations into the relation (93) for maximal surface
volume one can find that

$V=4\pi\int_{r_{\min}}^{r_{\max}}\int_{0}^{\pi}\frac{dr}{\dot{r}}=2\Omega\int_{r_{\min}}^{r_{\max}}\int_{0}^{\pi}\frac{r^{4}\sqrt[4]{1-\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Delta_{r}}}d\theta
dr}{\sqrt{r^{4}(\frac{\Delta_{r}}{\Sigma}-\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Sigma})+E^{2}}}.$

Here we wish to take $r_{\max}$ to be infinity, but this will yield a
divergent result in general. A finite result can be obtained by performing a
carefully matched subtraction of the AdS vacuum. Here $r_{min}$ is the turning
point of the surface which determined by the condition $\dot{r}=0$

$E^{2}+r_{\min}^{4}\left(-\frac{\Delta_{r}}{\Sigma}+\frac{\Delta_{\theta}\sin^{2}(\theta)}{\Sigma}\right)_{r_{\min}}=0.$

A simple calculation shows that $r_{min}$ will be on or inside the (outer)
horizon, and so we have that, $g_{tt}(r_{\min})<0$ ,
$\upsilon(\lambda_{\min})>0$ $\Longrightarrow$ E$<0$ and we recall that
$g_{tt}(r_{\min})<0$ in the region between the inner and event horizon. We are
interested in maximal slice which $\tau=0$ in this case $r_{\min}=r_{+}$ which
gives $E=0$. Thus, the complexity of formation becomes

$\displaystyle\Delta
C_{V}=\frac{V_{bulk}-2V_{ads}}{G_{N}L}=\frac{4\pi}{G_{N}L}\left(\int_{r_{+}}^{r_{\max}}\int_{0}^{\pi}\frac{r^{2}\sqrt{\frac{\Sigma}{\Delta_{r}}}}{(1-\frac{\Delta_{\theta}a^{2}\sin^{2}(\theta)}{\Delta_{r}})^{\frac{1}{4}}}d\theta
dr-\int_{0}^{r_{0\max}}\int_{0}^{\pi}2r^{2}\sqrt{\frac{\Sigma}{r^{2}(1+\frac{r^{2}}{l^{2}})-2Mr}}d\theta
dr\right),$

integrating with respect to $\theta$ gives us

$\displaystyle\Delta C_{Vbulk}$ $\displaystyle=$
$\displaystyle\frac{8l\pi^{2}}{G_{N}L}\int_{r_{+}}^{r_{\max}}\frac{dr}{\left[r^{3}-(-r^{2}+2m)l^{2}\right]^{\frac{5}{2}}}\left[\left(\frac{r^{7}}{2}-r^{6}m+\left(-\frac{q^{2}}{8}-\frac{3}{32}a^{2}\right)r^{5}+\frac{5a^{2}q^{2}r^{3}}{32}\right)l^{2}+\frac{r^{9}}{4}-\frac{a^{2}r^{7}}{16}+\right.$
$\displaystyle\left.\left(\frac{1}{4}r^{5}-mr^{4}+\left(-\frac{q^{2}}{8}+m^{2}-\frac{a^{2}}{32}\right)r^{3}+\left(\frac{q^{2}}{4}m-\frac{a^{2}m}{16}\right)r^{2}+\frac{a^{2}}{4}\left(m^{2}+\frac{7q^{2}}{16}\right)r+\frac{a^{2}q^{2}m}{16}\right)l^{4}\right]-\frac{8l\pi^{2}}{G_{N}L}\times$
$\displaystyle\int_{0}^{r_{0\max}}dr\frac{l\sqrt{r}}{\sqrt{(r-2m)l^{2}+r^{3}}}.$

As the next step, we have to integrate the above relation with respect to $r$.
In this case the integral cannot be calculated analytically, and therefore, we
should calculate it numerically which the results are shown in Fig. 9.

Figure 9: Complexity of formation in the CV conjecture versus
$\frac{r_{-}}{r_{+}}$.

According to Figs. 7 and 9, it is obvious that complexity of formation in two
mentioned conjectures has same behavior and it is bounded without divergency.
In the limit $\frac{r_{-}}{r_{+}}\rightarrow 0$ the value of the complexity of
formation (calculated via CV conjecture) is reduced and tends to $4\times
10^{21}$ while at the large value of $\frac{r_{-}}{r_{+}}\sim 0.25$, it’s
value approximately tends to a constant. Since increasing the electric charge
leads decreasing (increasing) inner horizon (event horizon), it is easy to
find that $\frac{r_{-}}{r_{+}}$ is an increasing function of electric charge,
$q$, and therefore, the complexity of formation in both conjectures increases
as one increases the electric charge.

##  V Conclusion

In this paper, we studied the complexity of regular static and rotating black
holes in the presence of nonlinear electrodynamics . For the case of the
static black hole, by looking at the action growth rates of the Wheeler-De
Witt patch, we showed the Lloyd bound is satisfied and its form remains
unaltered . Then, we have obtained the rate of complexity and complexity of
formation by using both CA and CV conjectures. From the comparison of the
result, we concluded that the two approaches to the complexity are consistent.
In the case of slowly rotating one and with assumption $\frac{q}{M}\ll 1$, we
obtained the rate of complexity proportional to the volume of the black hole
(in the limit $\alpha\ll 1$ and $\beta\ll 1$). So, one can conclude that the
rate of complexity for both static and rotating black holes is proportional to
thermodynamical volume instead of entropy.

The complexity of formation is numerically obtained by using both conjectures.
Its behavior is reflected in Figs. 7 and 9 which shows that for both
conjectures, the complexity of formation has the same behavior and is bounded.

In the appendix, we investigated two other regular cases and obtained the rate
of complexity and complexity of formation by using of both conjectures for
them. According to Figs. 11 and 12, it is obvious that for each case, the
complexity of formation obtained from two approaches are relatively consistent
and on the other hand behavior of complexity of formation for these two
regular cases have interesting similarities with each other.

Acknowledgements
The authors thank Shiraz University Research Council.

##  VI Appendix

For the sake of completeness, here, we investigate complexity conjecture of
two more examples of static regular black holes in $\mathcal{P}$.

###  VI.1 Second Case:

Here, we consider another structure function inspired by the log-logistic
distribution as the second class of solution which is given by Balart:2014cga
; Dymnikova:2004zc

$\mathcal{H}=\frac{\mathcal{P}}{(1+\gamma_{0}\sqrt{-\mathcal{P}})^{2}}+\frac{6}{l^{2}},$
(96)

where $\gamma_{0}=\frac{r_{0}^{2}}{\sqrt{2\gamma^{2}}}$, and $r_{0}$ and
$\gamma$ are two parameters. In the weak field limit ($\mathcal{P}\ll 1$), Eq.
(96) reduces to

$\mathcal{H}\approx\mathcal{P}+\frac{6}{l^{2}}+\mathcal{O}(\mathcal{P}^{\frac{3}{2}}).$
(97)

Taking such a structure function into account with $\mathcal{H}=2\rho(r)$ and
Eqs. (10) and (9), we can, directly, achieve the following $arctan$ form of
the solution Dymnikova:2004zc ; Sajadi:2017glu

$f(r)=1-\frac{4\chi}{\pi
r}\left[arctan(x)-\frac{x}{1+x^{2}}\right]+\frac{r^{2}}{l^{2}},\;\;\;r_{0}=\frac{\pi\gamma^{2}}{8\chi},\;\;\;x=\frac{r}{r_{0}},$
(98)

with the following asymptotical behavior for large $r$

$-g_{tt}\approx
1+\frac{r^{2}}{l^{2}}-\frac{2\chi}{r}+\frac{\gamma^{2}}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{4}}\right).$
(99)

As a result, the obtained solution behaves like Reissner-Nordström-AdS,
asymptotically, when we adjust $\chi$ and $\gamma$ as the mass and electric
charge of the system, respectively.

Such as before, we can obtain the following thermodynamic quantities for the
recent $arctan$ solution

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{3}{8\pi l^{2}},$ (100)
$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{4\pi r_{+}^{3}}{3},$
(101) $\displaystyle S$ $\displaystyle=$ $\displaystyle\pi r_{+}^{2},$ (102)
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{m\left((1+x_{+}^{2})^{2}\arctan(x_{+})-x_{+}-3x_{+}^{3}\right)}{\pi^{2}r_{+}^{2}(1+x_{+}^{2})^{2}}+\frac{r_{+}}{2\pi
l^{2}},$ (103) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\int_{r_{+}}^{\infty}Edr=\frac{3m}{\pi
q}\left(\frac{\pi}{2}-\arctan(x_{+})+\frac{(1+\frac{5}{3}x_{+}^{2})x_{+}}{(1+x_{+}^{2})^{2}}\right),$
(104)

where $x_{+}=x|_{r=r_{+}}$. Having the mentioned thermodynamic quantities, one
can directly examine the first law of black hole thermodynamics in the
enthalpy representation

$dH=TdS+VdP+\Phi dq,$ (105)

where

$H=m=\frac{\pi
r_{+}(1+\frac{r_{+}^{2}}{l^{2}})}{4\left(\arctan(x_{+})-\frac{x_{+}}{1+x_{+}^{2}}\right)}.$
(106)

Moreover, the Smarr relation is given by

$\frac{H}{2}+PV-TS-\frac{q\Phi}{2}+\frac{1}{4}\int wdv=0,$ (107)

where

$\int wdv=\frac{1}{2}\int_{r_{+}}^{\infty}T_{\mu}^{\mu}\
r^{2}dr=m-\frac{2mx_{+}(x_{+}^{2}-1)}{\pi(1+x_{+}^{2})^{2}}-\frac{2m}{\pi}\arctan(x_{+}).$
(108)

Now, we calculate joint terms according to joint contribution which is
described in the first class. After simplification, we can write

$\displaystyle{S_{B^{\prime}B}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\delta t\
\left[{\frac{{4m}}{{\pi{r^{2}}}}\left({\arctan(x)-\frac{{x}}{1+x^{2}}}\right)-\frac{{4m}}{{\pi
r}}\left({\frac{{2x{{}^{2}}}}{{{{{r_{0}}(1+x^{2})}^{2}}}}}\right)+\frac{{2r}}{{{l^{2}}}}}\right]_{r_{B}},$
$\displaystyle{S_{C^{\prime}C}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\delta
t\left[{\frac{{4m}}{{\pi{r^{2}}}}\left({\arctan(x)-\frac{{x}}{1+x^{2}}}\right)-\frac{{4m}}{{\pi
r}}\left({\frac{{2x{{}^{2}}}}{{{{{r_{0}}(1+x^{2})}^{2}}}}}\right)+\frac{{2r}}{{{l^{2}}}}}\right]_{r_{C}}.$
(109)

Considering the metric function obtained here, it is easy to show that the
Ricci scalar is calculated as

$\mathcal{R}=\frac{4q^{2}}{r_{0}^{4}(1+x^{2})^{3}}-\frac{12}{l^{2}},$ (110)

and also, the Lagrangian of nonlinear electrodynamics can be calculated at the
event horizon with the following explicit form

$\mathcal{L}=\frac{16m(1-x_{+}^{2})}{\pi
r_{0}^{3}(1+x_{+}^{2})^{3}}-\frac{6}{l^{2}}.$ (111)

Consequently, the growth rate of bulk action and surface term are,
respectively,

$\frac{d\mathcal{A}_{bulk}}{dt}=\left[\frac{2m}{\pi}\left(\arctan(x)-{\frac{{x}}{1+x^{2}}}\right)\right]_{r_{-}}^{r_{+}},$
(112)

and

$\frac{d\mathcal{A}_{boundray}}{dt}=\left[r-\frac{3m}{\pi}\arctan(x)+\frac{mx(3+x^{2})}{\pi(1+x^{2})^{2}}+\frac{3r^{3}}{2l^{2}}\right]_{r_{-}}^{r_{+}}.$
(113)

Finally, we find that the total growth rate of the action for such black hole
configuration within WDW patch at late time approximation can be collected as

$\frac{d\mathcal{A}}{dt}=\left[r-\frac{m}{\pi}\arctan(x)-\frac{mx(1-x^{2})}{\pi(1+x^{2})^{2}}+\frac{3r^{3}}{2l^{2}}\right]_{r_{-}}^{r_{+}}.$
(114)

Thus, by using the above calculated thermodynamic quantities, it is easy to
show that the Lloyd bound is satisfied. As an example, for the values
$m=0.8,q=0.1,l=1$ this bound is about $1.6872$, and the differences between
left and right (right mines left of Eq. (49)) is about $0.0666$.

###  VI.2 Third Case:

As the third case, we consider the following structure function which is
inspired by the Fermi-Dirac distribution functionAyonBeato:1999rg ;
Balart:2014cga

$\mathcal{H}=\frac{\mathcal{P}}{\cosh^{2}\left(\frac{\gamma^{\frac{3}{2}}(-\mathcal{P})^{\frac{1}{4}}}{2^{\frac{5}{4}}\chi}\right)}+\frac{6}{l^{2}},$
(115)

with the following weak field limit ($\mathcal{P}\ll 1$)

$\mathcal{H}\approx\mathcal{P}+\frac{6}{l^{2}}+\mathcal{O}(\mathcal{P})^{\frac{3}{2}}.$
(116)

Considering the spherically symmetric spacetime with previous relations for
$\mathcal{H}$, the third class of solution is

$f(r)=1-\frac{2\chi}{r}\left[1-\tanh\left(\frac{\gamma^{2}}{2\chi
r}\right)\right]+\frac{r^{2}}{l^{2}}.$ (117)

According to the series expansion of the metric function for large $r$, one
can find this solution behaves like the Reissner-Nordström-AdS,
asymptotically, as

$-g_{tt}\approx
1+\frac{r^{2}}{l^{2}}-\frac{2\chi}{r}+\frac{\gamma^{2}}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{4}}\right),$
(118)

whereas $\chi$ and $\gamma$ will be associated with the mass and electric
charge of the system, respectively.

Using the previous approach in the extended phase space, the following
thermodynamical quantities can be obtained, directly,

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{3}{8\pi l^{2}},$ (119)
$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{4\pi r_{+}^{3}}{3},$
(120) $\displaystyle S$ $\displaystyle=$ $\displaystyle\pi r_{+}^{2},$ (121)
$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{m}{2\pi
r_{+}^{2}}\left[1-\tanh\left(\frac{q^{2}}{2mr_{+}}\right)\right]-\frac{q^{2}}{4\pi
r_{+}^{3}}\left[1-\tanh^{2}\left(\frac{q^{2}}{2mr_{+}}\right)\right]+\frac{r_{+}}{2\pi
l^{2}},$ (122) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\int_{r_{+}}^{\infty}Edr=\frac{3m}{2q}-\frac{3mr_{+}\left(1+e^{\frac{q^{2}}{mr_{+}}}\right)-q^{2}e^{\frac{q^{2}}{mr_{+}}}}{qr_{+}\left(1+e^{\frac{q^{2}}{mr_{+}}}\right)^{2}}.$
(123)

It is evident that these thermodynamic quantities satisfy the first law of
black hole thermodynamics as

$dH=TdS+VdP+\Phi dq,$ (124)

where in the enthalpy representation, we have

$H=m.$ (125)

In addition, the related Smarr formula is given by

$\frac{H}{2}+PV-TS-\frac{q\Phi}{2}+\frac{1}{4}\int wdv=0,$ (126)

where

$\int
wdv=\frac{1}{2}\int_{r_{+}}^{\infty}T_{\mu}^{\mu}r^{2}dr=m-\frac{2mr_{+}\left(1+e^{\frac{q^{2}}{mr_{+}}}\right)+2q^{2}e^{\frac{q^{2}}{mr_{+}}}}{r_{+}\left(1+e^{\frac{q^{2}}{mr_{+}}}\right)^{2}}.$
(127)

According to the previous section, joint contribution can be calculated as
below

$S_{B^{\prime}B}=\frac{1}{4}\delta
t\left({\frac{{2m}}{{r^{2}}}}\left[{{1-\tanh}\left({\frac{{q^{2}}}{{2mr}}}\right)}\right]{-\frac{{q^{2}}}{{r^{3}}}}\left[{{1-{{{\tanh}^{2}\left({\frac{{q^{2}}}{{2mr}}}\right)}}}}\right]{+\frac{{2r}}{{l^{2}}}}\right)_{r_{B}},$
(128) ${S_{C^{\prime}C}}=\frac{1}{4}\delta
t\left({\frac{{2m}}{{r^{2}}}}\left[{{1-\tanh}\left({\frac{{q^{2}}}{{2mr}}}\right)}\right]{-\frac{{q^{2}}}{{r^{3}}}}\left[{{1-{{{\tanh}^{2}\left({\frac{{q^{2}}}{{2mr}}}\right)}}}}\right]{+\frac{{2r}}{{l^{2}}}}\right)_{r_{C}}.$
(129)

Moreover, the Ricci scalar for this solution is simplified as

$\mathcal{R}=\frac{q^{4}\sinh\left(\frac{q^{2}}{2mr}\right)}{mr^{5}\cosh^{3}\left(\frac{q^{2}}{2mr}\right)}-\frac{12}{l^{2}},$
(130)

and the nonlinear Lagrangian is as follows

$\mathcal{L}=\frac{q^{2}\left(q^{2}\tanh\left(\frac{q^{2}}{2mr}\right)-2mr\right)}{mr^{5}\cosh^{2}\left(\frac{q^{2}}{2mr}\right)}-\frac{6}{l^{2}}.$
(131)

It is notable that the growth rate of the bulk action and its related surface
term can be written as

$\displaystyle\frac{d\mathcal{A}_{bulk}}{dt}$ $\displaystyle=$
$\displaystyle\left[\frac{2m}{1+e^{\left(\frac{q^{2}}{mr}\right)}}\right]_{r_{-}}^{r_{+}},$
(132) $\displaystyle\frac{d\mathcal{A}_{boundray}}{dt}$ $\displaystyle=$
$\displaystyle\left[r-(\frac{3m}{2}+\frac{q^{2}}{4r})+\frac{3m}{2}\tanh\left(\frac{q^{2}}{2mr}\right)+\frac{q^{2}}{4r}\tanh^{2}\left(\frac{q^{2}}{2mr}\right)+\frac{3r^{3}}{2l^{2}}\right]_{r_{-}}^{r_{+}}.$
(133)

As a result, the total growth rate of action for such a black hole
configuration within WDW patch at late time approximation is

$\frac{d\mathcal{A}}{dt}=\left(r+\frac{3r^{3}}{2l^{2}}-\frac{m}{2}\left[1-\tanh\left(\frac{q^{2}}{2mr}\right)\right]-\frac{q^{2}}{4r}\left[1-\tanh^{2}\left(\frac{q^{2}}{2mr}\right)\right]\right)_{r_{-}}^{r_{+}}.$
(134)

For instance, for the values $m=0.8,q=0.1,l=1$ this bound is about $8.5783$,
and the differences between left and right (right mines left of Eq. (49)) is
$7.0399$.

###  VI.3 Complexity of formation

Similar to the complexity of formation of the first case we would like to
study the complexity of formation of second and third cases. In this way,
$r^{\ast}(r)$ and $r_{meet}$ of the second and the third solutions can be
obtained as follows:

$\displaystyle r^{\ast}(r)=$ $\displaystyle\dfrac{\pi
l^{2}r_{+}^{2}\ln\left(\dfrac{|r-r_{+}|}{r+r_{+}}\right)}{4ml^{2}\arctan(\dfrac{r_{+}}{r_{0}})+\dfrac{2\pi
r_{+}^{5}}{r_{+}^{2}+r_{0}^{2}}+\dfrac{2r_{0}r_{+}(\pi
r_{0}^{3}r_{+}^{2}-6ml^{2}r_{+}^{2}-4ml^{2}r_{0}^{2})}{(r_{+}^{2}+r_{0}^{2})^{2}}}+$
$\displaystyle\dfrac{\pi
l^{2}r_{-}^{2}\ln\left(\dfrac{|r-r_{-}|}{r+r_{-}}\right)}{4ml^{2}\arctan(\dfrac{r_{-}}{r_{0}})+\dfrac{2\pi
r_{-}^{5}}{r_{-}^{2}+r_{0}^{2}}+\dfrac{2r_{0}r_{+}(\pi
r_{0}^{3}r_{-}^{2}-6ml^{2}r_{-}^{2}-4ml^{2}r_{0}^{2})}{(r_{-}^{2}+r_{0}^{2})^{2}}}\hskip
76.82234ptsecond\;case$ (135) $\displaystyle r^{\ast}(r)=$
$\displaystyle\dfrac{l^{2}r_{+}^{3}\ln\left(\dfrac{|r-r_{+}|}{r+r_{+}}\right)}{2mr_{+}l^{2}\left(1-\tanh\left(\dfrac{q^{2}}{2mr_{+}}\right)\right)-q^{2}l^{2}\left(1-\tanh^{2}\left(\dfrac{q^{2}}{2mr_{+}}\right)\right)+2r_{+}^{4}}+$
$\displaystyle\dfrac{l^{2}r_{-}^{3}\ln\left(\dfrac{|r-r_{-}|}{r+r_{-}}\right)}{2mr_{-}l^{2}\left(1-\tanh\left(\dfrac{q^{2}}{2mr_{-}}\right)\right)-q^{2}l^{2}\left(1-\tanh^{2}\left(\dfrac{q^{2}}{2mr_{-}}\right)\right)+2r_{-}^{4}}\hskip
71.13188ptthird\;case$ (136)

and

$\displaystyle r_{meet}$ $\displaystyle\approx
0.068r_{+}^{2}-1.23r_{+}+1.1433,\hskip 42.67912ptsecond\;\;case$ (137)
$\displaystyle r_{meet}$
$\displaystyle\approx-0.35387r_{+}^{2}-0.64296r_{+}+0.9821.\hskip
14.22636ptthird\;\;case$ (138)

Accordingly, the bulk action of the black holes are

$\displaystyle A_{bulk,BH}=$ $\displaystyle\dfrac{16Mr_{0}}{\pi
G}\int_{r_{meet}}^{r_{max}}dr\left(\dfrac{1}{(r^{2}+r_{0}^{2})^{2}}\right)(v_{\infty}-r^{\ast}(r))\hskip
42.67912ptsecond\;case$ (139) $\displaystyle A_{bulk,BH}=$
$\displaystyle\dfrac{2q^{2}}{G}\int_{r_{meet}}^{r_{max}}dr\left(\dfrac{1-\tanh^{2}\left(\dfrac{q^{2}}{2Mr}\right)}{r^{4}}\right)(v_{\infty
5}-r^{\ast}(r))\hskip 14.22636ptthird\;case$ (140)

here $r_{max}$ is the same as the first case. The relation for $A_{joint}$ for
large black holes are written in the following form:

$\displaystyle\mathcal{A}_{joint}\approx$ $\displaystyle
0.0185\ln\left(\dfrac{0.068r_{+}^{2}}{l}\right)r_{+}^{4}+\mathcal{O}(r_{+}^{3}).\;\;\;second\;case$
(141) $\displaystyle\mathcal{A}_{joint}\approx$ $\displaystyle
0.1252\ln\left(\dfrac{-0.35387r_{+}^{2}}{l}\right)r_{+}^{4}+\mathcal{O}(r_{+}^{3}).\;\;\;third\;case$
(142)

Figure 10: Plot of $A_{joint}$ for second and third cases which are shown in
black and dotted line respectively.

The figure of $A_{joint}$ for second and third cases are plotted in Fig. 10.
According to Fig. 10, it is obvious that for small black hole the contribution
of the joints to the action is large for large black holes, this contribution
becomes smaller and goes to zero.

Finally according to above relations and the results related to the $AdS$
spacetime (68)-(III.2), $\Delta C_{A}$ from (63) can be obtained. The plot of
$\Delta C_{A}$ for second and third cases are shown in Fig. 11.

Figure 11 describes that complexity of formation for second and third cases
increases for large black holes. Finally $\Delta C_{V}$ according to Eq. (76)
for these two cases can be obtained.

Figure 12 shows that $\Delta C_{V}$ for both cases has the same behavior as
with $\Delta C_{A}$, increasing with radius of the black hole. To summarize
the behavior of $\Delta C_{A}$ and $\Delta C_{V}$ for three cases versus
$r_{+}$ are plotted in Fig. 13.

Figure 11: Plots of $\Delta C_{A}$ for second and third cases are shown in the
right and left panels respectively. Figure 12: Plots of $\Delta C_{V}$ for
second and third cases are shown in the right and left panels respectively.
Figure 13: Plot of Complexity of formation for the different geometries. For
first, second and third cases respectively are shown from up to down, and left
panels describe $\Delta C_{A}$ and right ones indicate $\Delta C_{V}$.

## References

  * (1) H. El Moumni and K. Masmar, Nucl. Phys. B 950, 114837 (2020).
  * (2) G. ’t Hooft, Conf. Proc. C 930308, 284 (1993).
  * (3) L. Susskind, J. Math. Phys. 36, 6377 (1995).
  * (4) J. M. Maldacena, Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)].
  * (5) S. Ryu and T. Takayanagi, JHEP 08, 045 (2006).
  * (6) V. E. Hubeny, M. Rangamani and T. Takayanagi, JHEP 07, 062 (2007).
  * (7) H. Casini, M. Huerta, and R. C. Myers, JHEP 05, 036 (2011).
  * (8) A. Lewkowycz and J. Maldacena, JHEP 08, 090 (2013).
  * (9) L. Susskind, Fortsch. Phys. 64, 24 (2016).
  * (10) D. Stanford and L. Susskind, Phys. Rev. D 90, 126007 (2014).
  * (11) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Phys. Rev. Lett. 116, 191301 (2016).
  * (12) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Phys. Rev. D 93, 086006 (2016).
  * (13) S. Lloyd, Nature 406, 6799 (2000).
  * (14) R. G. Cai, S. M. Ruan, S. J. Wang, R. Q. Yang and R. H. Peng, JHEP 09, 161 (2016).
  * (15) W. J. Pan and Y. C. Huang, Phys. Rev. D 95, 126013 (2017).
  * (16) P. A. Cano, R. A. Hennigar and H. Marrochio, Phys. Rev. Lett. 121, 121602 (2018).
  * (17) A. Ovgun and K. Jusufi, [arXiv:1801.09615].
  * (18) K. Meng, Eur. Phys. J. C 79, 984 (2019).
  * (19) X. H. Feng and H. S. Liu, Eur. Phys. J. C 79, 40 (2019).
  * (20) J. Jiang, Eur. Phys. J. C 79, 130 (2019).
  * (21) J. Jiang, Phys. Rev. D 98, 086018 (2018).
  * (22) E. Yaraie, H. Ghaffarnejad and M. Farsam, Eur. Phys. J. C 78, 967 (2018).
  * (23) K. Nagasaki, Phys. Rev. D 98, 126014 (2018).
  * (24) Z. Y. Fan and H. Z. Liang, Phys. Rev. D 100, 086016 (2019).
  * (25) L. Balart and E. C. Vagenas, Phys. Rev. D 90, 124045 (2014).
  * (26) E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80, 5056 (1998).
  * (27) K. A. Bronnikov, Phys. Rev. D 63, 044005 (2001).
  * (28) A. V. B. Arellano and F. S. N. Lobo, Class. Quantum Grav. 23, 5811 (2006).
  * (29) M. Hassain, Class. Quantum Grav. 25, 246 (2008).
  * (30) L. Hollenstein and F. S. N. Lobo, Phys. Rev. D 78, 124007 (2008).
  * (31) L. Balart, Mod. Phys. Lett. A 24, 2777 (2009).
  * (32) E. F. Eiroa and C. M. Sendra, Class. Quant. Grav. 28, 085008 (2011).
  * (33) S. Fernando and J. Correa, Phys. Rev. D 86, 064039 (2012).
  * (34) S. H. Hendi, S. N. Sajadi and M. Khademi, [arXiv:2006.11575].
  * (35) D. Kastor, S. Ray and J. Traschen, Class. Quant. Grav. 27, 235014 (2010).
  * (36) L. Balart and S. Fernando, Mod. Phys. Lett. A 32, 1750219 (2017).
  * (37) L. Lehner, R. C. Myers, E. Poisson and R. D. Sorkin, Phys. Rev. D 94, 084046 (2016).
  * (38) L. Balart and E. C. Vagenas, Phys. Rev. D 90, 124045 (2014).
  * (39) L. Balart and E. C. Vagenas, Phys. Lett. B 730, 14 (2014).
  * (40) H. Culetu, Int. J. Theor. Phys. 54, 2855 (2015).
  * (41) S. G. Ghosh, Eur. Phys. J. C 75, 532 (2015).
  * (42) J. Couch, W. Fischler and P. H. Nguyen, JHEP 03, 119 (2017).
  * (43) Y. S. An, R. G. Cai and Y. Peng, Phys. Rev. D 98, 106013 (2018).
  * (44) S. Chapman, H. Marrochio and R. C. Myers, JHEP 01, 062 (2017).
  * (45) W. Cottrell and M. Montero, JHEP 02, 039 (2018).
  * (46) A. A. Balushi, R. A. Hennigar, H. K. Kunduri and R. B. Mann, [arXiv:2008.09138].
  * (47) I. Dymnikova, Class. Quant. Grav. 21, 4417 (2004).
  * (48) S. N. Sajadi and N. Riazi, Gen. Rel. Grav. 49, 45 (2017).
  * (49) E. Ayon-Beato and A. Garcia, Phys. Lett. B 464, 25 (1999).