††institutetext: aInternational Centre for Theoretical Sciences (ICTS-TIFR),
Tata Institute of Fundamental Research, Shivakote, Hesaraghatta, Bengaluru
560089, India††institutetext: bDepartment of Physics,
Indian Institute of Technology Ropar, Rupnagar, Punjab 140 001, India

# More on Entanglement and Chaos near Critical Point in Strongly Coupled Gauge
Theory

Debanjan Karana and Sanjay Pantb<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We perform a holographic study of the high and low temperature behaviours of
logarithmic negativity (LN) and entanglement wedge cross section (EWCS) in a
large $N$ strongly coupled thermal field theory with critical point having a
well defined gravity dual known as 1RC black hole. The critical point is
defined via $\xi\to 2$ limit where, $\xi$ is dimensionless parameter
proportional to the charge of the 1RC black hole. We show that the logarithmic
negativity in low and high thermal limits enhances with increasing $\xi$. We
analytically compute the EWCS in low and high thermal limits and find an
agreement with the previously reported numerical results. We holographically
explore the correlation between two identical copies of thermal field theory
with critical point forming a thermofield double state (TFD) by computing the
thermo mutual information (TMI). TMI shows an increasing behaviour with
respect to the width of the boundary region. Further, we analyze the impact of
an early perturbation on the field theory by analyzing a shock wave
perturbation that grows exponentially in the dual eternal 1 RC black hole and
then estimate the degradation of TMI. However rate of such disruption of TMI
slows down as the value of critical parameter $\xi$ takes higher values.

††arxiv: 2308.00018

##  1 Introduction

In quantum information theory, entanglement entropy (EE) of a bipartite system
is synonymous to the von Neumann entropy constructed from the reduced density
matrix of one of the subsystems. Hilbert space of bipartite system made out of
two subsystems $A$ and $B$ is described as
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. The EE between subsystem A (or its
complement B) is defined as

$\mathcal{S}_{A}=-\text{Tr}(\rho_{A}\log\rho_{A}),$ (1)

where, $\rho_{A}=\text{Tr}_{B}(\rho_{AB})$ is the reduced density matrix of
$A$, obtained by taking the partial trace of the total density matrix
$\rho_{AB}$ over the degrees of freedom of $B$ Chuang . However, EE is not a
reliable measure for mixed states as it cannot differentiate between classical
and quantum correlations. One of the well-celebrated measure for mixed-state
entanglement is the mutual information (MI) that measures the total
correlations and characterizes the amount of entanglement between two
subsystems. The MI between $A$ and $B$ defined as Winter

$I(A:B)=\mathcal{S}_{A}+\mathcal{S}_{B}-\mathcal{S}_{A\cup B}$ (2)

where $\mathcal{S}_{A}$, $\mathcal{S}_{B}$, and $\mathcal{S}_{A\cup B}$ are
the von Neumann entropies of subsystem $A$, $B$ and $A\cup B$. For pure state
$\mathcal{S}_{A\cup B}=0$, and MI reduces to
$I(A:B)=\mathcal{S}_{A}=\mathcal{S}_{B}$. The positive value of MI $I(A:B)>0$
indicates the presence of entanglement between $A$ and $B$. However,
$I(A:B)=0$ implies that the subsystems may or may not be entangled and in that
case, additional entanglement measures or criteria are required. Apart from
MI, other measures such as entanglement of purification(EoP) and logarithmic
negativity (LN) are widely used to diagnose the entanglement for a mixed state
Terhal ; Peres:1996dw .

In a general context, we have the capability to transform a mixed state,
denoted as $\rho_{AB}$, residing within a Hilbert space composed of
$\mathcal{H}_{A}\otimes\mathcal{H}_{A}$, into a pure state $\ket{\psi}$ within
an expanded Hilbert space represented as
$\mathcal{H}_{A}\otimes\mathcal{H}_{A}\otimes\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}}$.
It’s important to note that there exists an infinite array of purification
pathways for $\ket{\psi}$, all satisfying the condition that
$\rho_{AB}=Tr_{A^{\prime}B^{\prime}}\ket{\psi}\bra{\psi}$. For a given
bipartite mixed state $\rho_{AB}$, we define the EoP, denoted as $E_{p}(A:B)$,
as the minimum of EE among all feasible purification

$E_{p}(A:B)=min_{\rho_{AB}=Tr_{A^{\prime}B^{\prime}}|\Psi\rangle\langle\Psi|}\\{\mathcal{S}(\rho_{AA^{\prime}})\\}$
(3)

Further, Vidal and Werner proposed a quantity termed logarithmic negativity
(LN) as a measure for the upper bound on the distillable entanglement in a
mixed state Vidal:2002zz . Unlike the MI, LN captures only the quantum
correlations and defined as,

$\mathcal{E}=\log||\rho_{AB}^{T}||$ (4)

$||\rho_{AB}^{T}||$ is the trace norm and $\rho^{T}_{AB}$ is partial transpose
of the $\rho_{AB}$ with respect to the $B$. Trace norm is directly related to
the entanglement negativity via $N=\frac{||\rho_{AB}^{T}||-1}{2}$ Vidal:2002zz
.

LN has been computed in CFT2 employing a version of the usual replica
technique involving a specific four-point function of the twist fields
Calabrese:2012nk ; Calabrese:2012ew ; Calabrese:2014yza ; Calabrese:2013mi ;
Chung:2014ed ; Wen:2015qwa ; Caste:2013nt ; Coser:2014gsa . An analytic form
of MI in $CFT_{2}$ is achieved by using the operator product expansion of
twist field Chen:2017hbk . However, field theoretic analysis for EoP hardly
exists due to the difficulties in the implementation of the minimization
procedure except for some numerical results obtained free lattice field theory
Bhattacharyya:2018sbw . The direct study of entanglement measures for strongly
coupled field theory is still an open question. Nevertheless, one can study
strongly coupled systems by exploiting the strong/weak nature of the
holographic dualities.

A concrete example of such a holographic dualities is the AdS/CFT
correspondence that suggests the information of a conformal field theory (CFT)
living on the boundary of Anti-de Sitter (AdS) space is encoded in the bulk
gravitational theory of AdS Maldacena:1997re ; Witten:1998qj . Although
general proof of the conjecture is yet to achieve, it passes numerous
consistency tests in diverse fields. The Ryu-Takayanagi formula is a crucial
example in favour of the AdS/CFT correspondence and it provides a holographic
prescription for computing the entanglement entropy in the boundary CFT, known
as holographic entanglement entropy (HEE)Ryu:2006bv ; Ryu:2006ef . It states
that the entanglement entropy of a certain region A in the CFT is given by the
area of a minimal surface (called the Ryu-Takayanagi surface) in the bulk AdS
spacetime that is homologous to the boundary region.

$\mathcal{S}_{A}=\frac{\mathcal{A}(\gamma_{A})}{4G_{N}^{d+1}}$ (5)

where $\gamma_{A}$ is a co-dimension two surface with the area
$\mathcal{A}(\gamma_{A})$ such that ${\partial}{\gamma_{A}}=\partial A$ and
$G_{N}^{d+1}$ is the $d+1$ dimensional Newton’s constant. Latter Hubeny,
Rangamani, and Takayanagi (HRT) extended this idea to the general states
including arbitrary time dependence Hubeny:2007xt . The study of entanglement
entropy in the context of AdS/CFT has provided valuable insights into quantum
phase transitions and critical phenomena in both the boundary CFT and the bulk
gravity theory Fradkin:2006mb ; Osborne:2002zz ; Baggioli:2023ynu . Quite
naturally constructing a holographic prescription of computing the
entanglement structure in a mixed state is crucial. In the context of
$AdS_{3}/CFT_{2}$, the authors of Chaturvedi:2016rcn  propose a holographic
conjecture to compute the LN of such boundary CFTs that exactly reproduced the
CFT2 results of in large central charge limit Calabrese:2014yza . See
Jain:2017aqk ; Afrasiar:2021hld  for further generalization of this proposal.

Viable holographic prescriptions for EoP are presented in Takayanagi:2017knl .
One can use the noition of purification and construct a TFD state which has
the holographic dual, a two sided eternal black hole Maldacena:2001kr . In
Shenker:2013pqa  author shows that the entanglement in a TFD state can be
destroyed via the insertion of an early time operator. The degradation of
entanglement is considered as the signature of quantum chaos. Entanglement and
quantum chaos are two distinct concepts, but they are interconnected in
various ways, especially when considering a system described by mixed density
matrix. In a chaotic system the entanglement between two causally disconnected
parts of a TFD state can be disrupted by an early perturbation which grows
exponentially in time. For a strongly coupled field theory, the shockwave
analysis and pole skipping are the mostly used holographic methods
Shenker:2013pqa ; Jahnke:2017iwi ; Blake:2018leo ; Grozdanov:2018kkt ;
Sil:2020jhr ; Mahish:2022xjz ; Chakrabortty:2022kvq

Four-dimensional, finite temperature $\mathcal{N}=4$ super Yang-Mills theory
charged under a $U(1)$ subgroup of its $SU(4)$ R-symmetry, with chemical
potential holographically corresponds to five dimensional 1RC black hole
background Gubser:1998jb ; Cvetic:1998jb . The low and high temperature limits
of HEE and HMI near the critical point are explored in 1RC black hole
backgroundEbrahim:2020qif . The author show that at and near the critical
point the leading behavior of mutual information yields a set of critical
exponents. Moreover, in Amrahi:2020jqg , a numerical investigation of the EWCS
holographically reveals that the EoP in the dual field theory at finite
temperature ($T$) and chemical potential ($\mu$) behaves as a monotonic
function of $\frac{\mu}{T}$ whereas the EoP behaves drastically different in
the presence of a critical point. The investigation of the holographic
butterfly effect is carried out within the background of a 1RC black hole. In
this context, the dynamical exponent is determined through an expansion of the
butterfly velocity in the vicinity of the critical point, as described in
Amrahi:2023xso . See Kraus:1998hv ; Cvetic:1999ne ; Cvetic:1999rb ;
Ebrahim:2018uky ; Ebrahim:2017gvk , for more holographic applications in this
background.

This work aims to improve the understanding of classical and quantum
correlation near the critical point of the four-dimensional, finite
temperature $\mathcal{N}=4$ super Yang-Mills theory via performing holographic
computation of a few relevant quantities such as LN, EoP and TMI in the dual
five dimensional 1RC black hole background. In our analysis we find that,
adjacent configurations, at low temperature, the LN decreases as the $\xi$
parameter increases whereas at the high-temperature it increases with $\xi$
parameter. For disjoint subsystems, LN increases with $\xi$ at low
temperatures and vanishes at high temperatures. In the bipartite case, LN
increases with $\xi$ at low temperatures and decreases at high temperatures.
In all the cases, LN remain finite in critical limit $\xi\to 2$. EoP also
increases with respect to the parameter $\xi$ and remains finite in the
critical limit. We also show that TMI between two entangled subsystems forming
a TFD state increases with their individual sizes. At a fixed size of the
subsystem, TMI rises with increasing $\xi$. In order to expand our
investigation into the chaotic dynamics of strongly coupled field theories
featuring critical point, we introduce an early-time, time-dependent
perturbation. This perturbation, when realized within the holographic
framework, takes the form of an exponentially growing energy pulse, ultimately
manifesting as a shock wave. We explicitly disrupted holographic TMI with a
shockwave, and our results indicate that as $\xi$ parameter takes higher
values, the chaotic behavior of the system gets reduced.

This paper is organized as follows; in the section 2 we discussed about the
holographic dual of the strongly coupled field theory with critical points. In
section 3 we review the HEE, section 4, 5 and 6 are devoted to the HLN for two
subsystems in different configurations. In section 7 we give the analytic form
of EWCS in low and high thermal limits and in section 8 we give the detailed
computation of mutual information between two subsystems in a TFD state known
as TMI. Finally, in section 9 we summarize the results.

##  2 Background

As discussed in the introduction, we proceed with a five-dimensional geometry
which is holographic dual to a four-dimensional strongly coupled field theory
with a critical point. In the existing literature, this is usually known as
the 1RC black hole background Gubser:1998jb ; Cvetic:1998jb ; Kraus:1998hv ;
Cvetic:1999ne ; Cvetic:1999rb . Consider the following five dimensional
Einstein-Maxwell-Dilaton action

$\mathcal{S}_{\text{EMD}}=\frac{1}{16\pi G^{(5)}_{N}}\int
d^{5}x\sqrt{-g}\left[\mathcal{R}-\frac{f(\phi)}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right],$
(6)

where $A_{\mu}$ is the gauge field and $\phi$ is a scalar field. We denote the
dilaton potential as $V(\phi)$ and the coupling between the gauge field and
the dilaton is characterized by the coupling function $f(\phi)$. The functions
$f(\phi)$ and $V(\phi)$ have the following form

$f(\phi)=e^{-\sqrt{\frac{4}{3}}\phi},~{}~{}~{}V(\phi)=-\frac{1}{R^{2}}\left(8e^{\frac{\phi}{\sqrt{6}}}+4e^{-\sqrt{\frac{2}{3}}\phi}\right)$
(7)

where $R$ is the $AdS$ radius. The solution to the equations of motion of the
EDM action in equation (6) corresponds to the 1RCBH background described by

$ds^{2}=e^{2A(z)}\left(-h(z)dt^{2}+d\vec{x}^{2}_{(3)}\right)+\frac{e^{2B(z)}}{h(z)}\frac{R^{4}}{z^{4}}dz^{2}$
(8)

where,

$\begin{split}A(z)&=\ln\left(\frac{R}{z}\right)+\frac{1}{6}\ln\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right)\\\
B(z)&=-\ln\left(\frac{R}{z}\right)-\frac{1}{3}\ln\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right)\\\
h(z)&=1-\frac{M^{2}z^{4}}{R^{6}\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right)}\\\
\phi(z)&=-\sqrt{\frac{2}{3}}\ln\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right)\\\
\Phi(z)&=\frac{MQ{z_{h}}^{2}}{R^{4}\left(1+\frac{Q^{2}{z_{h}}^{2}}{R^{4}}\right)}-\frac{MQ{z}^{2}}{R^{4}\left(1+\frac{Q^{2}{z}^{2}}{R^{4}}\right)}\end{split}$
(9)

$\Phi(z)$ is the electric potential that attributes to the temporal component
of the gauge field. In this coordinate system, the boundary is situated at
$z=0$. Note that, the electric potential $\phi(z)$ is chosen in such a way
that it is regular on the boundary DeWolfe:2010he ; DeWolfe:2011ts  and
vanishes on the horizon. The parameters $M$ and $Q$ are related to the mass
and charge respectively of the black hole. One can obtain the following
expression for the blackening factor $h(z)$ using the horizon equation i.e
$h(z_{H})=0$

$h(z)=1-{\left(\frac{z}{z_{h}}\right)}^{4}\left(\frac{1+{(\frac{Qz_{h}}{R^{2}})}^{2}}{1+{(\frac{Qz}{R^{2}})}^{2}}\right)=1-{\left(\frac{z}{z_{h}}\right)}^{4}\left(\frac{1+\xi}{1+\xi(\frac{z}{z_{h}})^{2}}\right)$
(10)

where, $\xi\equiv Q^{2}z_{h}^{2}/R^{4}$. The Hawking temperature is given by,

$T=\frac{1}{2\pi
z_{h}}\left(\frac{2+{\left(\frac{Qz_{h}}{R^{2}}\right)}^{2}}{\sqrt{1+{\left(\frac{Qz_{h}}{R^{2}}\right)}^{2}}}\right)$
(11)

and the chemical potential is,

$\mu=\frac{1}{R}\lim_{z\to
0}\Phi(z)=\frac{Q}{R^{2}\sqrt{1+{\left(\frac{Qz_{h}}{R^{2}}\right)}^{2}}}$
(12)

For convenience, we rewrite the temperature, in terms of a dimensionless
quantity $\xi$ and $\hat{T}$ as

$T=\hat{T}\left(\frac{1+\frac{\xi}{2}}{\sqrt{1+\xi}}\right),~{}~{}~{}\hat{T}\equiv\frac{1}{\pi
z_{h}}$ (13)

It is shown in Ebrahim:2020qif  that 1RCBH is thermodynamic stable for
$\xi\in[0,2]$ and $\xi\to 2$ is the critical point.

##  3 Holographic Entanglement Entropy (HEE)

In this section, we provide a concise overview of the HEE calculation as
presented in Ebrahim:2020qif . We leverage the outcomes of HEE computations
conducted under various temperature conditions. Subsequently, we employ the
method outlined in KumarBasak:2020viv  to compute the HLN for the 1RCBH
background. To elaborate, we focus on a boundary subsystem characterized as a
rectangular strip denoted as $A$, with a width $l$ along the $x$ direction and
extending to a length $L$ in all the transverse directions $x^{j}$. The
coordinate $x$ is expressed in terms of the bulk coordinate $z$. This
rectangular strip can be precisely defined as follows,

$x\equiv x\in\Big{[}-\frac{l}{2},\frac{l}{2}\Big{]},\hskip
14.22636ptx^{j}\in\Big{[}-\frac{L}{2},\frac{L}{2}\Big{]},~{}~{}~{}j=2,3$ (14)

Where $L$ is exceedingly large. Determining the HEE of subsystem $A$ requires
us to calculate the smallest surface area of the co-dimension two hyper-
surface denoted as ${\gamma}_{A}$. The area functional of ${\gamma}_{A}$ is as
follows:

$\mathcal{A}({\gamma}_{A})=\int d^{3}x~{}\sqrt{\text{det}(g_{mn})}$ (15)

Where $g_{mn}$ is the induced metric of $\gamma_{A}$, the area can be written
in the following form

$\mathcal{A}=2L^{2}\int
dz~{}e^{3A(z)}\sqrt{{x^{\prime}(z)}^{2}+\frac{R^{4}}{z^{4}h(z)}e^{2(B(z)-A(z))}}$
(16)

One can find the conserved quantity corresponds to $x$ using the Lagrangian in
the area functional and obtain the following equation by imposing
$\frac{1}{x^{\prime}(z_{t})}=0$ for $z\to z_{t}$,

$x^{\prime}(z)=\frac{R^{2}}{z^{2}}\frac{e^{3A(z_{t})}e^{B(z)-A(z)}}{\sqrt{h(z)}\sqrt{e^{6A(z)}-e^{6A(z_{t})}}}$
(17)

$z_{t}$ is the turning point of the surface ($\gamma_{A}$). Using
$x^{\prime}(z)$, the area functional (16) now becomes

$\mathcal{A}=2L^{2}R^{2}\int_{0}^{z_{t}}dz~{}\frac{e^{B(z)+2A(z)}}{z^{2}\sqrt{h(z)}}\sqrt{\frac{e^{6A(z)}}{e^{6A(z)}-e^{6A(z_{t})}}}$
(18)

Finally the holographic entanglement entropy (HEE) is,

$\mathcal{S}=\frac{L^{2}R^{2}}{2G_{N}^{5}}\int_{0}^{z_{t}}dz~{}\frac{e^{B(z)+2A(z)}}{z^{2}\sqrt{h(z)}}\sqrt{\frac{e^{6A(z)}}{e^{6A(z)}-e^{6A(z_{t})}}}$
(19)

From (17) boundary parameter $l$ and the bulk parameter $z_{t}$ are related
via,

$\frac{l}{2}=\int_{0}^{z_{t}}dz~{}\frac{R^{2}}{z^{2}}\frac{e^{3A(z_{t})}e^{B(z)-A(z)}}{\sqrt{h(z)}\sqrt{e^{6A(z)}-e^{6A(z_{t})}}}$
(20)

To express the HEE in terms of boundary parameter we have to replace the
$z_{t}$ in (19) in terms of $l$. Finding a solution for the integral (20) and
expressing $z_{t}$ in relation to $l$ poses a significant challenge.
Nevertheless, in scenarios where the temperature is either low or high,
accomplishing this task becomes feasible. Equation (9) and (18) gives the
following expression

$\mathcal{A}=2L^{2}R^{3}\int_{0}^{z_{t}}dz~{}\frac{{z_{t}}^{3}}{{z}^{6}}\sqrt{\frac{1+\xi{(\frac{z}{z_{h}})}^{2}}{1+\xi{(\frac{z_{t}}{z_{h}})}^{2}}}{\left[1-{\left(\frac{z}{z_{h}}\right)}^{4}\left(\frac{1+\xi}{1+\xi{(\frac{z}{z_{h}})}^{2}}\right)\right]}^{-\frac{1}{2}}{\left[{\left(\frac{z_{t}}{z}\right)}^{6}\left(\frac{1+\xi{(\frac{z}{z_{h}})}^{2}}{1+\xi{(\frac{z_{t}}{z_{h}})}^{2}}\right)-1\right]}^{-\frac{1}{2}}$
(21)

In similar way equation (20) can be expressed as,

$\frac{l}{2}=\int_{0}^{z_{t}}dz~{}{\left[1+\xi{\left(\frac{z}{z_{h}}\right)}^{2}\right]}^{-\frac{1}{2}}{\left[1-{\left(\frac{z}{z_{h}}\right)}^{4}\left(\frac{1+\xi}{1+\xi{(\frac{z}{z_{h}})}^{2}}\right)\right]}^{-\frac{1}{2}}{\left[{\left(\frac{z_{t}}{z}\right)}^{6}\left(\frac{1+\xi{(\frac{z}{z_{h}})}^{2}}{1+\xi{(\frac{z_{t}}{z_{h}})}^{2}}\right)-1\right]}^{-\frac{1}{2}}$
(22)

Figure 1: Turning point $z_{c}$ of RT surface with respect to width $l$.

Now it is possible to analytically solve the above two integrals by
considering several binomial and trinomial expansions. We are basically going
to employ the following series expansion formulae to write the integrands of
the above two equations

$\begin{split}&{(x+y)}^{-n}=\sum_{k=0}^{\infty}{(-1)}^{k}\frac{\Gamma(n+k)}{\Gamma(k+1)\Gamma(n)}x^{-n-k}y^{k};~{}~{}\text{given}~{}|y|<|x|\\\
&{(x+y+z)}^{-n}=\sum_{k=0}^{\infty}\sum_{j=0}^{k}\frac{\Gamma(n+k)}{\Gamma(k+1)\Gamma(n)}\frac{{(-1)}^{k}\Gamma(k+1)}{\Gamma(j+1)\Gamma(k-j+1)}x^{-n-k}y^{k-j}z^{j},~{}~{}\text{given}~{}|y+z|<|x|\end{split}$
(23)

Using equation (23) we can write the following form of the area integral

$\begin{split}\mathcal{A}=\frac{2L^{2}R^{3}}{\pi}\sum_{k=0}^{\infty}\sum_{n=0}^{k}\sum_{m=0}^{\infty}\sum_{j=0}^{\infty}\frac{{(-1)}^{k+n}\Gamma(k+\frac{1}{2})\Gamma(j+m+\frac{1}{2})}{\Gamma(n+1)\Gamma(k-n+1)\Gamma(j+1)\Gamma(m+1)}{\xi}^{k-n+m}{(1+\xi)}^{n}{\left(\frac{z_{t}}{z_{h}}\right)}^{2m}\\\
\times{\left[1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right]}^{-m-\frac{1}{2}}\int_{0}^{z_{c}}dz~{}\left[1+\xi{\left(\frac{z}{z_{h}}\right)}^{2}\right]\left[1-{\left(\frac{z}{z_{t}}\right)}^{2}\right]z^{-3}{\left(\frac{z}{z_{t}}\right)}^{6j}{\left(\frac{z}{z_{h}}\right)}^{2(k+n)}\end{split}$
(24)

The region bounded by the extremal surface exhibits divergence, primarily due
to its behavior near the boundary, which is a common expectation. Upon closer
examination, it becomes evident that when the condition $k+n+3j>1$ is met, the
final integral (and consequently, the enclosed area) remains finite.
Consequently, we must isolate and sum the terms corresponding to $(k=n=j=0)$
and $(k=1,n=j=0)$ over the variable $m$ to determine the portion of the region
containing the divergent component. By carrying out this procedure, one can
derive the subsequent outcome.

${\mathcal{A}}_{0}\equiv
L^{2}R^{2}\Bigg{\\{}\frac{1}{{\epsilon}^{2}}+\frac{3\xi}{2{z_{h}}^{2}}-\frac{1}{{z_{t}}^{2}}{\left[1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right]}^{\frac{3}{2}}\Bigg{\\}}$
(25)

In the limit as $\epsilon$ approaches zero, where $z$ takes on the value of
$\epsilon$, the cutoff surface within the bulk geometry becomes relevant as it
is intricately tied to the ultraviolet (UV) regularization of the field
theory. It becomes evident that the problematic term in equation (25) shows
behavior akin to an area law, a characteristic also shared by the associated
holographic entanglement entropy. In the context of a $d$-dimensional boundary
field theory, when the primary divergence in the UV limit, as $\epsilon$ tends
towards zero, adheres to an area law, this outcome is entirely anticipated. To
simplify calculations, we will subsequently focus on the finite component of
the area, achieved by subtracting the $1/\epsilon^{2}$ term. This can be
expressed in the following manner:

${\mathcal{A}}_{\text{finite}}=\frac{L^{2}R^{3}}{{z_{t}}^{2}}\Bigg{\\{}\frac{3\xi}{2}{\left(\frac{z_{t}}{z_{h}}\right)}^{2}-{\left[1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right]}^{\frac{3}{2}}+\frac{1+\xi}{3\xi}{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\left[{\left(1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right)}^{\frac{3}{2}}-1\right]\Bigg{\\}}\\\
+\frac{L^{2}R^{3}}{{z_{t}}^{2}}\Bigg{\\{}\sum_{k=2}^{\infty}\sum_{n=0}^{k}\sum_{m=0}^{\infty}{\Lambda}_{knm}\frac{\Gamma(m+\frac{1}{2})\Gamma(k+n-1)}{\Gamma(k+n+m+1)}{\left(\frac{z_{t}}{z_{h}}\right)}^{2(k+n+m)}\\\
\times\left[(m+1)+(k+n-1)\left(1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right)\right]\Bigg{\\}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{L^{2}R^{3}}{{z_{t}}^{2}}\Bigg{\\{}\sum_{k=0}^{\infty}\sum_{n=0}^{k}\sum_{m=0}^{\infty}\sum_{j=1}^{\infty}{\Lambda}_{knm}\frac{\Gamma(m+j+\frac{1}{2})\Gamma(k+n+3j-1)}{\Gamma(j+1)\Gamma(k+n+m+3j+1)}{\left(\frac{z_{t}}{z_{h}}\right)}^{2(k+n+m)}\\\
\times\left[(m+1)+(k+n+3j-1)\left(1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right)\right]\Bigg{\\}}$
(26)

where ${\Lambda}_{knm}$ is given by the following relation

${\Lambda}_{knm}\equiv\frac{{(-1)}^{k+n}\Gamma(k+\frac{1}{2})}{\pi\Gamma(n+1)\Gamma(k-n+1)}{\xi}^{k-n+m}{\left(1+\xi\right)}^{n}{\left[1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right]}^{-m-\frac{1}{2}}$
(27)

Hence, by incorporating the UV-divergence-dependent term into (27), we can
derive the overall surface area of the external surface associated with a
rectangular strip having a width of $l$ on the boundary. Similarly, by
following this procedure, we can determine the subsystem’s width as a function
of the turning point. Consequently, through the utilization of multinomial
expansions and the solution of the integral, as outlined in equation (22), we
can establish the ensuing relationship.

$\frac{l}{2}=z_{t}\sum_{k=0}^{\infty}\sum_{n=0}^{k}\sum_{m=0}^{\infty}\sum_{j=0}^{\infty}G_{knmj}F_{knmj}{\left(\frac{z_{t}}{z_{h}}\right)}^{2(k+n+m)}$
(28)

where the constants $G_{knmj}$ and $F_{knmj}$ are defined by the following
relations

$\begin{split}G_{knmj}&\equiv\frac{\Gamma(k+\frac{1}{2})\Gamma(j+m+\frac{1}{2})\Gamma(2+3j+k+n)}{2\pi\Gamma(n+1)\Gamma(k-n+1)\Gamma(j+1)\Gamma(3+3j+k+n+m)}\\\
F_{knmj}&\equiv{(-1)}^{k+n}{\xi}^{k-n+m}{(1+\xi)}^{n}{\left[1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right]}^{-m}\end{split}$
(29)

Note that in order to utilize the multinomial expansions for the negative
exponents, it has been verified that the subsequent relationships hold true
across the entire interval of $\xi\in[0,2]$ and for values of $z_{t}$ spanning
from the boundary to the horizon.

$\frac{\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}}{1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}}\left(1-\frac{z^{2}}{{z_{t}}^{2}}\right)<1,~{}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}~{}\xi{\left(\frac{z}{z_{h}}\right)}^{2}-(1+\xi){\left(\frac{z}{z_{h}}\right)}^{4}<1$
(30)

We now possess the analytical representation for the area of the extremal
surface associated with the subsystem of width $l$. Furthermore, we have
successfully elucidated the connection between this width and the turning
point of the RT surface, as described in equations (26) and (28). In the next
section, we are going to do this explicitly by considering low and high-
temperature limits.

By examining equation (26), we can observe the extremal surface area, which is
determined by two dimensionless parameters: $\xi$ and the ratio $z_{t}/z_{h}$.
In the subsequent two subsections, our focus will be on exploring the
holographic entanglement negativity concerning the $z_{t}/z_{h}$ ratio, a
parameter that leads to two distinct thermal limits. It’s worth noting that
$z_{h}$ is inversely related to the black hole temperature. However, for the
analysis of critical behavior, we will reserve our investigation regarding the
parameter $\xi$, which, as previously mentioned in section 2, governs the
critical behavior. Considering the ratio between the location of the extremal
surface and the horizon position, denoted as $z_{t}/z_{h}$, one can anticipate
two distinct scenarios for the area calculation in this context, specifically
when $z_{t}/z_{h}\ll 1$ and when $z_{t}/z_{h}\sim 1$. It is important to note
that the former scenario implies that the extremal surface is situated close
to the boundary at $z=0$, while the latter scenario indicates that it
approaches but does not cross the horizon. From the perspective of field
theory, we can directly translate these scenarios into two equivalent thermal
limits based on the subsystem width $l$: $\hat{T}l\ll 1$ and $\hat{T}l\gg 1$,
respectively, where $\hat{T}$ is defined in equation (13). Consequently, one
can associate the case of $z_{t}/z_{h}\ll 1$ with the low-temperature limit,
corresponding to the ground state of the CFT, and the case of
”$z_{t}/z_{h}\sim 1$ with the high-temperature limit, where the entanglement
of thermal excitations becomes significant.

Before concluding this section, we would like to explore a few key aspects of
the turning point denoted as $z_{t}$ in fig.1. These aspects are closely tied
to some well-established properties of the RT (Ryu-Takayanagi) surface. When
the parameter $l$ approaches zero, it becomes challenging to distinguish
between the turning points of the RT surfaces associated with different values
of $\xi$. However, beyond a certain threshold value of $l,"$ it becomes
possible to differentiate between the turning points corresponding to various
$\xi$ values.

It is important to note that, for a fixed $\xi$, the turning point initially
emerges from the origin and gradually increases as $l$ increases. This
behavior indicates that as the width of the boundary region expands, the RT
surfaces extend deeper into the bulk of the system. As $l$ reaches higher
values, the value of $z_{t}$ saturates, signifying that the RT surface,
associated with a boundary region of width $l$, becomes nearly parallel to the
horizon when $l$ becomes significantly large. Consequently, by examining the
plot’s characteristics, one can draw the aforementioned conclusions, which
align with our understanding of the nature of the RT surface.

##  4 Holographic Logarithmic Negativity for two adjacent subsystems

In this section, we utilize the holographic framework outlined in references
Chaturvedi:2016rcn ; Chaturvedi:2017znc ; Chaturvedi:2016rft ; Jain:2017aqk
to analyze the $1$RC black hole background and determine the holographic
entanglement negativity. This calculation involves the summation of the areas
of certain extremal surfaces, located in the bulk and associated with the
relevant subsystems. As per the conjecture, the holographic entanglement
negativity can be expressed in the following manner.

$\mathcal{E}=\frac{3}{16G_{N}^{5}}\left(\mathcal{A}_{1}+\mathcal{A}_{2}-\mathcal{A}_{12}\right)$
(31)

In this context, $\mathcal{A}_{i}$ represents the area of a co-dimension two
extremal surface that is connected to subsystem $A_{i}$ (refer to fig.2). It’s
worth noting that $\mathcal{A}_{12}$ specifically denotes the area of the
extremal surface anchored to the combined subsystem $A_{1}\cup A_{2}$. As
previously discussed in Section 3, we have already presented the formula for
calculating the extremal surface’s area related to a subsystem with a specific
width. In the following subsections, we will apply these formulas to compute
the HLN in both low and high-temperature scenarios.

###  4.1 Holographic Logarithmic Negativity for two adjacent subsystems at low
temperature

In this section, we delve into the low-temperature regime of the area
functional, along with the width parameter $l$. Additionally, we calculate the
Holographic Luttinger Number (HLN) in the low-temperature limit, considering
two neighboring subsystems with widths $l_{1}$ and $l_{2}$. To validate our
findings, we demonstrate their correspondence with those of the AdS-
Schwarzschild black hole in the $\xi\to 0$ limit, as discussed in Jain:2017xsu
.

Before proceeding with the low-temperature limit, it is essential to
acknowledge that when dealing with an infinite series, concerns about
divergence arise, depending on their growth behavior. However, in the low-
temperature regime where $z_{t}/z_{h}\ll 1$, it becomes evident that both
infinite series in equations (27) and (29) converge. Consequently, when
expanding equation (29) to the order of $z_{t}/z_{h}$, we derive the following
relationship.

$l=z_{t}\Bigg{\\{}a_{1}-\frac{a_{1}\xi}{6}{\left(\frac{z_{t}}{z_{h}}\right)}^{2}+\left[\frac{a_{2}(1+\xi)}{2}+\frac{a_{3}{\xi}^{2}}{24}\right]{\left(\frac{z_{t}}{z_{h}}\right)}^{4}+\mathcal{O}{\left(\frac{z_{t}}{z_{h}}\right)}^{6}\Bigg{\\}}$
(32)

where the constants $a_{1}$, $a_{2}$ and $a_{3}$ are,

$\begin{split}&a_{1}\equiv\sum_{j=0}^{\infty}\frac{\Gamma\left(j+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(j+1)\Gamma(2+3j)}=\frac{3\sqrt{\pi}\Gamma\left(\frac{5}{3}\right)}{\Gamma\left(\frac{1}{6}\right)}\\\
&a_{2}\equiv\sum_{j=0}^{\infty}\frac{\Gamma\left(j+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(j+1)\Gamma(4+3j)}=\frac{\sqrt{\pi}\Gamma\left(\frac{7}{3}\right)}{4~{}\Gamma\left(\frac{11}{6}\right)}\\\
&a_{3}\equiv\sum_{j=0}^{\infty}\frac{\Gamma\left(j+\frac{1}{2}\right)(4-j)}{\sqrt{\pi}\Gamma(j+1)\Gamma(2+3j)\Gamma(4+3j)}\\\
&=\frac{3}{\sqrt{\pi}}\left[\Gamma\left(\frac{5}{6}\right)\Gamma\left(\frac{5}{3}\right)-\frac{3}{5}\Gamma\left(\frac{7}{6}\right)\Gamma\left(\frac{7}{3}\right)\right]-\frac{1}{70}~{}{{}_{3}}F_{2}\left(\frac{3}{2},\frac{5}{3},\frac{7}{3};\frac{8}{3},\frac{10}{3};1\right)\end{split}$
(33)

Figure 2: Schematic diagram of the extremal surfaces, involving turning
points, corresponding to two adjacent boundary subsystems $A$ and $B$ having
widths $l_{1}$ and $l_{2}$ respectively. Here $z=0$ denotes the boundary
whereas $z=z_{h}$ denotes the horizon.

Now inverting the relation between $l$ and $z_{t}$ as,

$z_{t}=\frac{l}{a_{1}}\Bigg{\\{}1+\frac{\xi}{6a_{1}^{2}}{\left(\frac{l}{z_{h}}\right)}^{2}+\frac{1}{24a_{1}^{4}}\left[\frac{{\xi}^{2}}{6}\left(1-\frac{a_{3}}{2a_{2}}\right)-\frac{a_{2}}{a_{1}}(1+\xi)\right]{\left(\frac{l}{z_{h}}\right)}^{4}+\mathcal{O}{\left(\frac{l}{z_{h}}\right)}^{6}\Bigg{\\}}$
(34)

Similarly, one can expand the infinite series for the area functional and can
get the following equation

$\begin{split}\mathcal{A}_{\text{finite}}^{\text{low}}&=\frac{L^{2}R^{3}}{z_{t}^{2}}\left[\frac{1+\xi}{2}{\left(\frac{z_{t}}{z_{h}}\right)}^{4}-1\right]+\frac{L^{2}R^{3}}{z_{t}^{2}}\sum_{j=1}^{\infty}\frac{\Gamma\left(j+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(j+1)\Gamma(3j-1)}\\\
&\times\left[1+\frac{\xi}{3}{\left(\frac{z_{t}}{z_{h}}\right)}^{2}+\left(\frac{(-4{\xi}^{2}+9\xi+9)j-3(\xi+1)}{18j+6}\right){\left(\frac{z_{t}}{z_{h}}\right)}^{4}\right]\end{split}$
(35)

Now performing the sum and substituting the expression of the turning point we
get,

$\begin{split}\mathcal{A}_{\text{finite}}^{\text{low}}=\frac{R^{3}L^{2}}{l^{2}}\Bigg{\\{}a_{1}^{2}(w_{1}-1)+\frac{\xi}{3}{\left(\frac{l}{z_{h}}\right)}^{2}&+\frac{1}{2a_{1}^{2}}\Bigg{[}(1+\xi)\left(1-w_{3}+3w_{2}+\frac{2(w_{1}-1)a_{2}}{a_{1}}\right)\\\
&+\frac{{\xi}^{2}}{6}\left((w_{1}-1)(\frac{a_{3}}{a_{1}}-1)-8w_{2}\right)\Bigg{]}{\left(\frac{l}{z_{h}}\right)}^{4}\Bigg{\\}}\end{split}$
(36)

where the numerical constants $w_{1}$, $w_{2}$, and $w_{3}$ are,

$\begin{split}&w_{1}\equiv\frac{1}{\sqrt{\pi}}\sum_{j=1}^{\infty}\frac{\Gamma\left(j+\frac{1}{2}\right)}{\Gamma(j+1)(3j-1)}=\frac{1}{2^{2/3}}~{}{{}_{2}}F_{1}\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-1\right)\\\
&w_{2}\equiv\frac{1}{\sqrt{\pi}}\sum_{j=1}^{\infty}\frac{j\Gamma\left(j+\frac{1}{2}\right)}{\Gamma(j+1)(3j-1)(3j+1)}=\frac{1}{16}~{}{{}_{3}}F_{2}\left(\frac{2}{3},\frac{4}{3},\frac{3}{2};\frac{5}{3},\frac{7}{3};1\right)\\\
&w_{3}\equiv\frac{1}{\sqrt{\pi}}\sum_{j=1}^{\infty}\frac{\Gamma\left(j+\frac{1}{2}\right)}{\Gamma(j+1)(3j-1)(3j+1)}=\frac{3}{16}~{}{{}_{3}}F_{2}\left(\frac{2}{3},\frac{4}{3},\frac{3}{2};\frac{5}{3},\frac{7}{3};1\right)-\frac{1}{2^{1/3}}~{}{{}_{2}}F_{1}\left(\frac{4}{3},\frac{5}{3};\frac{7}{3};-1\right)\end{split}$
(37)

Note that in the limit where $\xi\to 0$, we get $z_{h}=1/\pi T$, and the
subleading terms become $2^{\text{nd}}$ and $4^{\text{th}}$ order in $Tl$. To
express this relation in a more simplified way, we define

$\begin{split}&c\equiv a_{1}^{2}(w_{1}-1)\\\
&f(\xi)\equiv(1+\xi)\frac{\left(1-w_{3}+3w_{2}+\frac{2(w_{1}-1)a_{2}}{a_{1}}\right)}{a_{1}^{2}}+\frac{{\xi}^{2}}{6}\frac{\left((w_{1}-1)(\frac{a_{3}}{a_{1}}-1)-8w_{2}\right)}{a_{1}^{2}}\end{split}$
(38)

Therefore, using the above definitions we finally get the area functional of a
boundary subsystem which was thought of a rectangular strip with width $l$

$\mathcal{A}_{\text{finite}}^{\text{low}}=R^{3}{\left(\frac{L}{l}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}l\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}l\right)}^{4}\Bigg{\\}}$
(39)

In the context of two adjoining subsystems, we designate these subsystems as
$A_{1}$ and $A_{2}$, each defined by the width of their respective rectangular
strips, denoted as $l_{1}$ and $l_{2}$. Utilizing equation (31), we derive the
subsequent expression for the HLN within the low-temperature regime for the
scenario involving two adjacent subsystems.

$\begin{split}\mathcal{E}_{low}=\frac{3R^{3}}{16G_{N}^{5}}\Bigg{[}c\Bigg{\\{}{\left(\frac{L}{l_{1}}\right)}^{2}+{\left(\frac{L}{l_{2}}\right)}^{2}-{\left(\frac{L}{l_{1}+l_{2}}\right)}^{2}\Bigg{\\}}+\frac{\xi}{3}L^{2}{\pi}^{2}{\hat{T}}^{2}-f(\xi)\left(\pi
L^{2}{\hat{T}}^{4}\right)l_{1}l_{2}\Bigg{]}\end{split}$ (40)

This serves as a reminder that the HLN expression above is derived by
considering the finite portion of the extremal areas of the adjacent
subsystems. Consequently, the UV-divergence component is not evident in this
expression. Nevertheless, if we work with the complete area expression, the UV
divergence term will also appear in the negativity expression. Now one can
compare this HLN expression with the one in Jain:2017xsu  for the AdSd+1
Schwarzschild black hole. Please note that this comparison is valid in the
limit of $Q\to 0$, which can be achieved by setting $\xi\to 0$. In the
provided expression, the first three terms within the curly braces are
inversely proportional to the squares of the lengths corresponding to the
relevant boundary regions. The third term, which depends on $f(\xi)$, accounts
for the product of the widths of two subregions and includes the $\hat{T}^{4}$
dependence. The second term will naturally disappear in the critical limit of
$\xi\to 0$, rendering the HLN finite in the $\xi\to 2$ limit.

###  4.2 Holographic Logarithmic Negativity for two adjacent subsystems at
high temperature

As mentioned in the previous section, there is always a concern regarding the
divergence of infinite series. Fortunately, various methods for summation or
regularization are available to address the issue of divergence in a given
series. We observe that in the high-temperature limit where $z_{t}$ approaches
$z_{h}$, the infinite series in equation (26) does not exhibit convergence.
Nevertheless, it is possible to regularize the series by reordering its terms
in a manner that allows us to recover the component proportional to $l$. In
the limit where $z_{t}$ tends toward $z_{h}$, the expression for the area of
the RT-surface takes the following form.

$\mathcal{A}_{\text{finite}}^{\text{high}}=R^{3}{\left(\frac{L}{z_{h}}\right)}^{2}\Bigg{\\{}\sqrt{1+\xi}\left(\frac{l}{z_{h}}\right)+(S_{1}+S_{2}+S_{3})\Bigg{\\}}$
(41)

where $S_{1}$, $S_{2}$ and $S_{3}$ dependent on $\xi$ and given by

$\begin{split}&S_{1}\equiv\frac{3\xi}{2}-\frac{1}{3}-\frac{11}{5\xi}-\frac{244}{105{\xi}^{2}}-\frac{32}{35{\xi}^{3}}-\frac{16}{35{\xi}^{4}}+\sqrt{\xi+1}\left(-\frac{64\xi}{105}-\frac{124}{105}+\frac{26}{21\xi}+\frac{214}{105{\xi}^{2}}+\frac{24}{35{\xi}^{3}}+\frac{16}{35{\xi}^{4}}\right)\\\
&S_{2}\equiv\sum_{k=2}^{\infty}\sum_{n=0}^{k}\sum_{m=0}^{\infty}\frac{\Gamma\left(k+\frac{1}{2}\right)\Gamma\left(m+\frac{1}{2}\right)\Gamma(k+n+2){\left(-1\right)}^{k+n}{\xi}^{k-n+m}{\left(1+\xi\right)}^{n-m-\frac{1}{2}}}{\pi\Gamma(n+1)\Gamma(k-n+1)\Gamma(k+n+m+3)}\\\
&\times\Bigg{\\{}\frac{m+1}{k+n-1}\left[1+\frac{m+1}{k+n}\left(2+\frac{m}{k+n+1}\right)\right]+\frac{(1+\xi)(m+1)}{k+n}\left(2+\frac{m}{k+n+1}\right)\Bigg{\\}}\end{split}$
(42)
$\begin{split}&S_{3}\equiv\sum_{k=2}^{\infty}\sum_{n=0}^{k}\sum_{m=0}^{\infty}\sum_{j=1}^{\infty}\frac{\Gamma\left(k+\frac{1}{2}\right)\Gamma\left(j+m+\frac{1}{2}\right)\Gamma(k+n+3j+2)}{\pi\Gamma(n+1)\Gamma(j+1)\Gamma(k-n+1)\Gamma(k+n+m+3j+3)}\\\
&\times{(-1)}^{k+n}{\xi}^{k-n+m}{(1+\xi)}^{n-m-\frac{1}{2}}\times\Bigg{\\{}\frac{m+1}{k+n+3j-1}\left[1+\frac{m+1}{k+n+3j}\left(2+\frac{m}{k+n+3j+1}\right)\right]\\\
&+\frac{(1+\xi)(m+1)}{k+n+3j}\left(2+\frac{m}{k+n+3j+1}\right)\Bigg{\\}}\end{split}$
(43)

Finally, in terms of temperature $\hat{T}$, we can write

$\mathcal{A}_{\text{finite}}^{\text{high}}=R^{3}{\left(\frac{L}{l}\right)}^{2}\Bigg{\\{}\sqrt{1+\xi}{\left(\pi\hat{T}l\right)}^{3}+S_{4}{\left(\pi\hat{T}l\right)}^{2}\Bigg{\\}}~{}~{}~{}\text{where}~{}~{}S_{4}=S_{1}+S_{2}+S_{3}$
(44)

Therefore, using the formula for the HLN as given by equation (31) we find the
HEN at a high-temperature regime for two adjacent subsystems $A_{1}$ and
$A_{2}$

$\mathcal{E}_{high}=\frac{3R^{3}}{16G_{N}^{5}}\Bigg{\\{}S_{4}L^{2}{\left(\pi\hat{T}\right)}^{2}\Bigg{\\}}$
(45)

Similarly, as previously noted, the negativity expression above is derived by
considering the finite part of the extremal areas of the adjacent subsystems.
Consequently, the UV-divergence component does not manifest in this
expression. To establish a comparison, one can now contrast this entanglement
negativity expression with the result obtained in Jain:2017xsu  for the AdSd+1
Schwarzschild black hole in the limit as $\xi\to 0$. To observe an exact match
in the high-temperature scenario, it is necessary to expand the exponential
terms in Jain:2017xsu  to linear order in $l$, resulting in a dependence
solely on $T^{d-2}$.

Before concluding this section, it’s worth noting that in equation (44), the
initial term dependent on temperature scales with the volume of the
rectangular strip, represented as $L^{2}l$, while the subsequent term is
related to the area. Consequently, the first term characterizes thermal
entropy, while the second term represents the entanglement entropy between the
strip region and its complement. However, in the case of negativity, the
volume-dependent component is absent. This indicates that at high
temperatures, entanglement entropy and thermal entropy become equivalent and
exhibit a temperature dependence of $\hat{T}^{2}$, as deduced from the area
calculation in Ebrahim:2020qif .

##  5 Holographic Logarithmic Negativity for two disjoint subsystems

In this section, we will determine the HLN for two distinct subsystems within
the background of a $1$RC black hole. Similar to our analysis of neighboring
subsystems, we will establish a connection between our findings and the
general results outlined in KumarBasak:2020viv  for a $(d+1)$-dimensional AdS
Schwarzschild black hole. Specifically, we focus on two non-overlapping
intervals, denoted as $A_{1}$ and $A_{2}$, each with widths $l_{1}$ and
$l_{2}$ respectively, as illustrated in fig.3. These intervals collectively
constitute the mixed-state subsystem $A$, with a gap separating them
corresponding to a subsystem $A_{m}\subset B$ of width $l_{m}$, where
$B=A^{c}$ represents the remainder of the system. To clarify, we define the
three intervals as follows

$A_{1}~{}:~{}~{}x^{1}\equiv
x\in\left[-\frac{l_{1}}{2},\frac{l_{1}}{2}\right],\hskip
22.76219ptx^{(j)}\in\left[-\frac{L}{2},\frac{L}{2}\right]~{}~{}\text{where}~{}j=2,3$
(46) $A_{2}~{}:~{}~{}x^{2}\equiv
x\in\left[-\frac{l_{2}}{2},\frac{l_{2}}{2}\right],\hskip
22.76219ptx^{(j)}\in\left[-\frac{L}{2},\frac{L}{2}\right]~{}~{}\text{where}~{}j=2,3$
(47) $A_{m}~{}:~{}~{}x^{m}\equiv
x\in\left[-\frac{l_{m}}{2},\frac{l_{m}}{2}\right],\hskip
14.22636ptx^{(j)}\in\left[-\frac{L}{2},\frac{L}{2}\right]~{}~{}\text{where}~{}j=2,3$
(48)

where $L$ for the transverse coordinates are taken to be very large
$L\to\infty$.
Now following the conjecture as given in KumarBasak:2020viv ; Malvimat:2018txq
one can write the entanglement negativity corresponding to the disjoint
intervals as

$\mathcal{E}=\frac{3}{16G_{N}^{5}}\Big{(}\mathcal{A}_{A_{1}\cup
A_{m}}+\mathcal{A}_{A_{m}\cup A_{2}}-\mathcal{A}_{A_{1}\cup A_{m}\cup
A_{2}}-\mathcal{A}_{A_{m}}\Big{)}$ (49)

where $\mathcal{A}_{A_{1}\cup A_{m}}$ and $\mathcal{A}_{A_{m}\cup A_{2}}$ are
the areas of the extremal surfaces anchored with respect to the region
$A_{1}\cup A_{m}$ and $A_{2}\cup A_{m}$ respectively. $\mathcal{A}_{A_{1}\cup
A_{m}\cup A_{2}}$ is the area of the extremal surface anchored with respect to
the region $A_{1}\cup A_{m}\cup A_{2}$ and $A_{m}$ follows a similar meaning.
Note that all the three surfaces corresponding to the intervals $A_{1}$,
$A_{2}$ and $A_{m}$ have the turning points labeled as $z_{t_{1}}$,
$z_{t_{2}}$ and $z_{t_{m}}$. By utilizing the surface area of the RT surfaces
in their respective regions, we can calculate the HLN. Furthermore, we will
examine the low and high-temperature limits separately in the following two
subsections.

###  5.1 Holographic Logarithmic Negativity for two disjoint subsystems at low
temperature

In our prior analysis, we derived the expression for the extremal surface area
corresponding to a region of width $l$ in the low-temperature limit.
Consequently, we shall reformulate the area expression as follows

$\mathcal{A}_{\text{finite}}^{\text{low}}=R^{3}{\left(\frac{L}{l}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}l\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}l\right)}^{4}\Bigg{\\}}$
(50)

To calculate the HEN, we must utilize this relation to formulate the
expressions for the areas of the extremal surfaces associated with all the
intervals specified in equation (49). By performing this procedure, we derive
the following relationships

$\begin{split}&\mathcal{A}_{A_{1}\cup A_{2}\cup
A_{m}}=R^{3}{\left(\frac{L}{l_{1}+l_{2}+l_{m}}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}(l_{1}+l_{2}+l_{m})\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}(l_{1}+l_{2}+l_{m})\right)}^{4}\Bigg{\\}}\\\
&\mathcal{A}_{A_{1}\cup
A_{m}}=R^{3}{\left(\frac{L}{l_{1}+l_{m}}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}(l_{1}+l_{m})\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}(l_{1}+l_{m})\right)}^{4}\Bigg{\\}}\\\
&\mathcal{A}_{A_{2}\cup
A_{m}}=R^{3}{\left(\frac{L}{l_{2}+l_{m}}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}(l_{2}+l_{m})\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}(l_{2}+l_{m})\right)}^{4}\Bigg{\\}}\\\
&\mathcal{A}_{A_{m}}=R^{3}{\left(\frac{L}{l_{m}}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}l_{m}\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}l_{m}\right)}^{4}\Bigg{\\}}\end{split}$
(51)

Using the above equation in (49) one would obtain the HLN at low temperatures
for two disjoint subsystems

$\begin{split}\mathcal{E}_{low}=\frac{3R^{3}}{16G_{N}^{5}}\Bigg{[}c\Bigg{\\{}{\left(\frac{L}{l_{1}+l_{m}}\right)}^{2}+{\left(\frac{L}{l_{2}+l_{m}}\right)}^{2}-{\left(\frac{L}{l_{1}+l_{2}+l_{m}}\right)}^{2}-{\left(\frac{L}{l_{m}}\right)}^{2}\Bigg{\\}}\\\
+\frac{1}{2}f(\xi)\left(\pi
L^{2}\hat{T}^{4}\right)\Bigg{\\{}{\left(l_{1}+l_{m}\right)}^{2}+{\left(l_{2}+l_{m}\right)}^{2}-{\left(l_{1}+l_{2}+l_{m}\right)}^{2}-l_{m}^{2}\Bigg{\\}}\Bigg{]}\end{split}$
(52)

Figure 3: Schematic diagram of the extremal surfaces at low effective
temperature, involving the turning points, corresponding to the subregions
$A_{1}$ and $A_{2}$ separated by an interval $A_{m}$.

Note that we are dealing exclusively with the finite portion of the area. This
is the reason why, in previous instances, the HLN (Hartman-Maldacena-Niarchos)
did not incorporate the UV divergence term, denoted as $L^{2}/\epsilon^{2}$.
However, in the scenario of disjoint intervals, a closer examination reveals
that even if we consider the entire area expression, including the divergent
portion, the HLN remains independent of the cutoff. This stands in contrast to
the situation encountered in the case of mixed-state configurations involving
adjacent intervals. The first term on the right-hand side of the equation
above originates from the contribution of the AdS5 vacuum and remains
unaffected by temperature changes. The remaining terms represent finite-
temperature corrections to the HLN at low temperatures, which closely resemble
the conditions observed in the mixed-state scenario of adjacent intervals.

A similar outcome has been previously documented in Malvimat:2018txq .
Additionally, one can naturally anticipate that as the limit
$l_{m}\to\epsilon$ is approached, the entanglement negativity (HLN) for
separate subsystems will replicate the outcome expressed in equation (40) for
adjacent subsystems. When taking the limit $l_{m}\to\epsilon$ in equation
(52), it becomes possible to recreate both the first component (which is
independent of temperature) and the third component (dependent on
$\hat{T}^{4}l_{1}l_{2}$). Furthermore, a term reliant on the cutoff emerges,
expressed as $\frac{2}{d-2}{\left(\frac{L}{\epsilon}\right)}^{d-2}$.
Intriguingly, this term would have been present in the HLN expression at low
temperatures if the cutoff-dependent part within the region of the RT surfaces
for multiple subregions had been considered. Hence, we can deduce that as
$l_{m}$ approaches $\epsilon$, the entanglement negativity for separate
subsystems converges to that of adjacent subsystems.

###  5.2 Holographic Logarithmic Negativity for two disjoint subsystems at
high temperature

In our previous analysis, we obtained the expression of the area of the
extremal surface corresponding to a region with a width of $l$ at a high-
temperature limit. Therefore we rewrite the expression of the area as follows

$\mathcal{A}_{\text{finite}}^{\text{high}}=R^{3}{\left(\frac{L}{l}\right)}^{2}\Bigg{\\{}\sqrt{1+\xi}{\left(\pi\hat{T}l\right)}^{3}+S_{4}{\left(\pi\hat{T}l\right)}^{2}\Bigg{\\}}$
(53)

Now to compute the HLN we employ this relation to write down the expressions
of the area of the extremal surfaces of all the intervals required in equation
(49). By doing so we obtain the following relations

$\begin{split}&\mathcal{A}_{A_{1}\cup A_{2}\cup
A_{m}}=R^{3}{\left(\frac{L}{l_{1}+l_{2}+l_{m}}\right)}^{2}\Bigg{\\{}\sqrt{1+\xi}{\left(\pi\hat{T}(l_{1}+l_{2}+l_{m})\right)}^{3}+S_{4}{\left(\pi\hat{T}(l_{1}+l_{2}+l_{m})\right)}^{2}\Bigg{\\}}\\\
&\mathcal{A}_{A_{1}\cup
A_{m}}=R^{3}{\left(\frac{L}{l_{1}+l_{m}}\right)}^{2}\Bigg{\\{}\sqrt{1+\xi}{\left(\pi\hat{T}(l_{1}+l_{m})\right)}^{3}+S_{4}{\left(\pi\hat{T}(l_{1}+l_{m})\right)}^{2}\Bigg{\\}}\\\
&\mathcal{A}_{A_{2}\cup
A_{m}}=R^{3}{\left(\frac{L}{l_{2}+l_{m}}\right)}^{2}\Bigg{\\{}\sqrt{1+\xi}{\left(\pi\hat{T}(l_{2}+l_{m})\right)}^{3}+S_{4}{\left(\pi\hat{T}(l_{2}+l_{m})\right)}^{2}\Bigg{\\}}\\\
&\mathcal{A}_{A_{m}}=R^{3}{\left(\frac{L}{l_{m}}\right)}^{2}\Bigg{\\{}\sqrt{1+\xi}{\left(\pi\hat{T}l_{m}\right)}^{3}+S_{4}{\left(\pi\hat{T}l_{m}\right)}^{2}\Bigg{\\}}\end{split}$
(54)

Using the above equation in (49) we obtain the expression for HLN at high
temperature,

$\begin{split}\mathcal{E}_{high}=\frac{3R^{3}L^{2}}{16G_{N}^{5}}\Bigg{\\{}\sqrt{1+\xi}{(\pi
T)}^{3}(l_{1}+l_{m})+S_{4}{(\pi T)}^{2}+\sqrt{1+\xi}{(\pi
T)}^{3}(l_{2}+l_{m})+S_{4}{(\pi T)}^{2}\\\ -\sqrt{1+\xi}{(\pi
T)}^{3}(l_{1}+l_{2}+l_{m})-S_{4}{(\pi T)}^{2}-\sqrt{1+\xi}{(\pi
T)}^{3}l_{m}-S_{4}{(\pi T)}^{2}\Bigg{\\}}\end{split}$ (55)

By simplifying the expression above, we readily demonstrate that the HLN
evaluates to zero. Consequently, when dealing with two separate subsystems
under high-temperature conditions, the HLN becomes null. This outcome aligns
with expectations since entanglement negativity exclusively quantifies quantum
correlations, whereas high temperatures primarily entail thermal entropy. To
validate this outcome, it would be beneficial to examine the HLN expression
for disjoint subsystems at elevated temperatures as provided in the reference
Malvimat:2018txq . This reference offers the following expression for the same
within a generic background of a $(d+1)$-dimensional AdS Schwarzschild black
hole.

Figure 4: Schematic diagram of the extremal surfaces at high effective
temperature, involving the turning points, corresponding to the subregions
$A_{1}$ and $A_{2}$ separated by an interval $A_{m}$.

$\begin{split}\mathcal{E}=\frac{3}{16G_{N}^{5}}{\left(\frac{4\pi}{d}\right)}^{d-1}\frac{C_{1}}{4\pi}\sqrt{2d(d-1)}L^{d-2}T^{d-2}\Bigg{\\{}-e^{-\sqrt{\frac{d-1}{2d}}4\pi
T(l_{1}+l_{m})}-e^{-\sqrt{\frac{d-1}{2d}}4\pi T(l_{2}+l_{m})}\\\
+e^{-\sqrt{\frac{d-1}{2d}}4\pi
T(l_{1}+l_{2}+l_{m})}+e^{-\sqrt{\frac{d-1}{2d}}4\pi
Tl_{m}}\Bigg{\\}}\end{split}$ (56)

If one expands the exponential terms on the right-hand side of the above
equation, the following expression can be obtained

$\begin{split}\mathcal{E}=\frac{3}{16G_{N}^{5}}{\left(\frac{4\pi}{d}\right)}^{d-1}\frac{C_{1}}{4\pi}\sqrt{2d(d-1)}L^{d-2}T^{d-2}\Bigg{\\{}-1+\sqrt{\frac{d-1}{2d}}4\pi
T(l_{1}+l_{m})-1\\\ +\sqrt{\frac{d-1}{2d}}4\pi
T(l_{2}+l_{m})+1-\sqrt{\frac{d-1}{2d}}4\pi
T(l_{1}+l_{2}+l_{m})+1-\sqrt{\frac{d-1}{2d}}4\pi Tl_{m}\Bigg{\\}}\end{split}$
(57)

From the equation presented above, it becomes clear that the HLN expression
evaluates to zero, mirroring the result we derived in this subsection.
Justification for expanding the exponential terms to linear order in $l$ can
be found in the behavior of the extremal area at high temperatures, as it
exclusively encompasses linear order dependence in $l$. Consequently, we can
confidently affirm that our high-temperature entanglement negativity result
for disjoint subsystems aligns with the findings in Malvimat:2018txq .

##  6 Holographic Logarithmic Negativity for bipartite systems

In this section, we will calculate the HLN for a bipartite configuration.
Similar to the preceding sections, we will determine the entanglement
negativity for both low and high-temperature regimes. To validate our
findings, we will establish their consistency by comparing them to previously
obtained results for the general $(d+1)$-dimensional AdS-Schwarzschild black
hole, as documented in Chaturvedi:2016rft .

We provide a brief overview of the setup for the bipartite system. To gain a
clear understanding of this setup, it is essential to begin by partitioning
the boundary CFT into two subsystems, denoted as $A$ and its complement
$A^{c}$. Furthermore, we will consider two additional subsystems, namely
$B_{1}$ and $B_{2}$, which are situated adjacent to $A$ and positioned on
either side of it in such a way that we have the union $B=B_{1}\cup B_{2}$. As
in the preceding sections, we will use the notation $A_{\gamma}$ to represent
the area of co-dimension $2$ static minimal surfaces within the bulk geometry,
anchored on these subsystems. Consequently, in a general context, the HLN
(Holographic entanglement negativity) for the bipartite system formed by the
union of $A$ and $A^{c}$ is expressed as follows

$\mathcal{E}=\lim_{B\to
A^{c}}\biggl{[}\frac{3}{16G_{N}^{(d+1)}}\Big{(}2\mathcal{A}_{A}+\mathcal{A}_{B_{1}}+\mathcal{A}_{B_{2}}-\mathcal{A}_{A\cup
B_{1}}-\mathcal{A}_{A\cup B_{2}}\Big{)}\biggr{]}$ (58)

In the above equation, $G^{(d+1)}_{N}$ represents the Newton constant in a
$(d+1)$-dimensional context. It’s important to note that we can interpret the
bipartite limit, denoted as $(B\to A^{c})$, by extending the subsystems
$B_{1}$ and $B_{2}$ to the extent that $B$ effectively becomes equal to the
complement of $A$, denoted as $A^{c}$. To provide precise definitions for the
subsystems in question namely, $A$, $B_{1}$, and $B_{2}$ we will describe them
in the context of the $4$-dimensional boundary CFT as follows:

$\begin{split}&A:~{}~{}x^{1}\equiv
x\in\left[-\frac{l}{2},\frac{l}{2}\right],\hskip
27.0301ptx^{(j)}\in\left[-\frac{L_{2}}{2},\frac{L_{2}}{2}\right]~{}~{}\text{where}~{}j=2,3\\\
&B_{1}:~{}~{}x^{1}\equiv x\in\left[-L,-\frac{l}{2}\right],\hskip
14.22636ptx^{(j)}\in\left[-\frac{L_{2}}{2},\frac{L_{2}}{2}\right]~{}~{}\text{where}~{}j=2,3\\\
&B_{2}:~{}~{}x^{1}\equiv x\in\left[\frac{l}{2},L\right],\hskip
32.72066ptx^{(j)}\in\left[-\frac{L_{2}}{2},\frac{L_{2}}{2}\right]~{}~{}\text{where}~{}j=2,3\end{split}$
(59)

The bipartite limit, denoted as $B\to A^{c}$, can be applied when considering
the limit as $L$ approaches infinity within the intervals $B_{1}$ and $B_{2}$.
It’s important to note that the choice of subsystem $A$ has been symmetrically
made along the partitioning direction, resulting in equal extremal areas,
namely, $\mathcal{A}_{B_{1}}=\mathcal{A}_{B_{2}}$ and $\mathcal{A}_{A\cup
B_{1}}=\mathcal{A}_{A\cup B_{2}}$. This identification is particularly
elegant, as it simplifies the expression presented in equation (58) for the
HEN to the following form

$\mathcal{E}=\lim_{B\to
A^{c}}\frac{3}{8G_{N}^{5}}\Big{(}\mathcal{A}_{A}+\mathcal{A}_{B_{1}}-\mathcal{A}_{A\cup
B_{1}}\Big{)}$ (60)

To facilitate future reference, note that we designate the turning points
associated with the RT surfaces anchored on regions $B_{1}$, $A$, and $A\cup
B_{1}$ as $z_{t_{1}}$, $z_{t_{2}}$, and $z_{t_{3}}$. Consequently, we will
utilize this equation in the following sections to calculate the HEN under
varying temperature conditions.

###  6.1 Holographic Logarithmic Negativity for bipartite systems at low
temperature

In this section, we compute the HLN for the bipartite state in the low-
temperature regime. This regime corresponds to the temperature limit
$\hat{T}l\ll 1$, which in the bulk corresponds to the case where the horizon
is at a large distance from the turning point $z_{t_{2}}$ of the extremal
surface anchored on the subsystem $A$.

Figure 5: Schematic diagram of the extremal surfaces corresponding to the
bipartite subsystem at low effective temperature.

At the low-temperature limit, we are already aware of the perturbative
solution of the infinite series of $\frac{l}{2}$ as discussed in section 4. By
doing so, one could obtain the relation between the turning point of the RT
surface anchored of the subsystem $A$ and the width of the subsystem as

$z_{t_{2}}=\frac{l}{a_{1}}\Bigg{\\{}1+\frac{\xi}{6a_{1}^{2}}{\left(\frac{l}{z_{h}}\right)}^{2}+\frac{1}{24a_{1}^{4}}\left[\frac{{\xi}^{2}}{6}\left(1-\frac{a_{3}}{2a_{2}}\right)-\frac{a_{2}}{a_{1}}(1+\xi)\right]{\left(\frac{l}{z_{h}}\right)}^{4}+\mathcal{O}{\left(\frac{l}{z_{h}}\right)}^{6}\Bigg{\\}}$
(61)

Using the above relation we obtained the area of the extremal surface
corresponding to the subsystem $A$ at a low-temperature regime as follows
111Likewise the previous cases, note that the equation below does not contain
the UV cut-off dependent part as we have considered the finite part of the
area of $\mathcal{A}$ only.

$\mathcal{A}_{A}=R^{3}{\left(\frac{L}{l}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}l\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}l\right)}^{4}\Bigg{\\}}$
(62)

The subsystems $B_{1}$ and $A\cup B_{1}$ in the boundary with lengths
$(L-l/2)$ and $(L+l/2)$ along the $x^{1}$ direction are very large in the
limit $B\to A^{c}$ which corresponds to the limit $L\to\infty$. Therefore, the
extremal surfaces described by the areas $\mathcal{A}_{B_{1}}$ and
$\mathcal{A}_{A\cup B_{1}}$ will extend deep into the bulk approaching the
black hole horizon even at low temperatures i.e., $(z_{t_{1}}\sim z_{h})$ and
$(z_{t_{3}}\sim z_{h})$. Hence, for computing the expressions for the areas
$\mathcal{A}_{B_{1}}$ and $\mathcal{A}_{A\cup B_{1}}$ we employ the method
developed in Fischler:2012ca  for the case when the RT surfaces approach the
black hole horizon. Following the procedure, we can write the turning points
of the extremal surfaces $\mathcal{A}_{B_{1}}$ and $\mathcal{A}_{A\cup B_{1}}$
as follows for a $(d+1)$-dimensional AdS Schwarzchild black hole 222Although,
we are writing $d+1$ dimensional result but we will use $d=4$ in final
expressions.

$\begin{split}z_{t_{1}}=z_{h}(1+\epsilon_{1})=z_{h}\Bigg{[}(1+k_{2}e^{-\sqrt{\frac{d(d-1)}{2}}z_{h}\left(L-\frac{l}{2}\right)}\Bigg{]}\\\
z_{t_{3}}=z_{h}(1+\epsilon_{1})=z_{h}\Bigg{[}(1+k_{2}e^{-\sqrt{\frac{d(d-1)}{2}}z_{h}\left(L+\frac{l}{2}\right)}\Bigg{]}\end{split}$
(63)

where $k_{2}$ has the following form

$k_{2}=\frac{1}{d}e^{\sqrt{\frac{d(d-1)}{2}}c_{1}}$ (64)
$c_{1}=\frac{2\sqrt{\pi}\Gamma\left(\frac{d}{2(d-1)}\right)}{\Gamma\left(\frac{1}{d-1}\right)}+\sum_{n=1}^{\infty}\Bigg{\\{}\frac{2}{(1+nd)}\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma(n+1)}\frac{\Gamma\left(\frac{d(n+1)}{2(d-1)}\right)}{\Gamma\left(\frac{nd+1}{2(d-1)}\right)}-\frac{\sqrt{2}}{\sqrt{d(d-1)}n}\Bigg{\\}}$
(65)

We can now find out the area of the extremal surface for the subsystems
$\mathcal{A}_{B_{1}}$ and $\mathcal{A}_{A\cup B_{1}}$ by substituting (63) in
(26). We then take the sum as an expansion of $\epsilon_{1}$ and
$\epsilon_{3}$ respectively and consider the terms in linear order of them.
Therefore we obtain the following expressions

$\begin{split}{\mathcal{A}}_{B_{1}}=\frac{L^{2}R^{3}}{z_{h}^{2}}\Big{\\{}\alpha(\xi)+\gamma(\xi)+\mu(\xi)\Big{\\}}+\frac{L^{2}R^{3}}{z_{h}}\left(L-\frac{l}{2}\right)\Big{\\{}\beta(\xi)+\delta(\xi)+\nu(\xi)\Big{\\}}\\\
\mathcal{A}_{A\cup
B_{1}}=\frac{L^{2}R^{3}}{z_{h}^{2}}\Big{\\{}\alpha(\xi)+\gamma(\xi)+\mu(\xi)\Big{\\}}+\frac{L^{2}R^{3}}{z_{h}}\left(L+\frac{l}{2}\right)\Big{\\{}\beta(\xi)+\delta(\xi)+\nu(\xi)\Big{\\}}\end{split}$
(66)

where all the $\xi$-dependent functions have been defined in Appendix-A.
Finally using equation (66) in (60) we can get the HLN for the bipartite
system at the low-temperature limit

$\mathcal{E}_{low}=\frac{3}{8G_{N}^{5}}\Bigg{[}R^{3}{\left(\frac{L}{l}\right)}^{2}\Bigg{\\{}c+\frac{\xi}{3}{\left(\pi\hat{T}l\right)}^{2}+\frac{1}{2}f(\xi){\left(\pi\hat{T}l\right)}^{4}\Bigg{\\}}-R^{3}L^{2}l\hat{T}g(\xi)\Bigg{]}$
(67)

where the function $g(\xi)$ can be written as
$g(\xi)=\pi(\beta(\xi)+\delta(\xi)+\nu(\xi))$.

Note that in the equation presented earlier, the last term is directly
proportional to $L^{2}l$, representing the three-dimensional volume of
subsystem A. With this correspondence, one can infer that the final term is
proportional to the thermal entropy associated with subsystem A. To further
scrutinize our findings, we can examine the behavior as $\xi$ approaches zero.
It is expected that in the limit where $Q$ tends to zero (which can also be
achieved by setting $\xi$ to zero), the aforementioned result will coincide
with the AdS Schwarzschild black hole result found in Chaturvedi:2016rft . It
is worth emphasizing that the initial term enclosed within the curly brackets,
with the appropriate scaling factor, precisely reproduces the entanglement
entropy of subsystem A at low temperatures.

Now, as the limit $\xi\to 0$ is taken, we can examine the behavior of the
function $g(\xi)$, revealing that $g(\xi)$ scales as $1/\xi$. Consequently, in
terms of temperature, we can express this as $g(\xi)\propto T^{2}$. By
combining the aforementioned arguments, it becomes apparent that the final
term, which is proportional to the volume $V=L^{2}l$, exhibits an explicit
temperature dependence of order $T^{3}$. Hence, one can interpret this last
term in the preceding equation as the thermal entropy (in $(d+1)$-dimensional
AdS Schwarzschild geometry, thermal entropy is proportional to $VT^{d-1}$) of
the system A. Consequently, we can now reformulate equation (67) in the
following manner

$\mathcal{E}_{low}=\frac{3}{2}\Big{\\{}S_{A}-\mathcal{C}S_{A}^{\text{Th}}\Big{\\}},~{}~{}\text{where}~{}\mathcal{C}~{}\text{is
a constant}$ (68)

Surprisingly, the equation presented above reveals that the HLN effectively
quantifies distillable entanglement by eliminating the thermal contribution in
low-temperature conditions. This characteristic stands as a universal trait of
entanglement negativity within finite-temperature mixed states.

###  6.2 Holographic Logarithmic Negativity for Bipartite Systems at High
Temperature

In the high-temperature regime, the turning point $z_{t_{2}}$ of the extremal
surface, representing the area $\mathcal{A}A$, converges close to the black
hole horizon. This convergence is characterized by the condition $z{t_{2}}\sim
z_{h}$, as illustrated in fig. 6. Consequently, we can employ the same
methodology to calculate the area of the extremal surface corresponding to
subsystem $A$ as we did for $\mathcal{A}{B_{1}}$ and $\mathcal{A}{A\cup
B_{1}}$ in the preceding section.

It’s worth noting that, as previously explained, these surfaces consistently
explore the vicinity of the black hole horizon, both at low and high
temperatures. This behavior is a consequence of the limit $B\to A^{c}$, or
equivalently, $L\to\infty$. Therefore, we can utilize equation (66) to compute
the Holographic Entanglement Negativity (HEN) in the high-temperature regime.
We can write the following expression of the turning point corresponding to
the extremal surface of the subsystem $A$

Figure 6: Schematic diagram of the extremal surfaces corresponding to the
bipartite subsystem at high effective temperature.

$z_{t_{2}}=z_{h}(1+\epsilon_{1})=z_{h}\Bigg{[}(1+k_{2}e^{-\sqrt{\frac{d(d-1)}{2}}z_{h}l}\Bigg{]}$
(69)

where $k_{2}$ is,

$k_{2}=\frac{1}{d}e^{\sqrt{\frac{d(d-1)}{2}}c_{1}}$ (70)
$c_{1}=\frac{2\sqrt{\pi}\Gamma\left(\frac{d}{2(d-1)}\right)}{\Gamma\left(\frac{1}{d-1}\right)}+\sum_{n=1}^{\infty}\Bigg{\\{}\frac{2}{(1+nd)}\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma(n+1)}\frac{\Gamma\left(\frac{d(n+1)}{2(d-1)}\right)}{\Gamma\left(\frac{nd+1}{2(d-1)}\right)}-\frac{\sqrt{2}}{\sqrt{d(d-1)}n}\Bigg{\\}}$
(71)

Using the above equations we can write,

${\mathcal{A}}_{A}=\frac{L^{2}R^{3}}{z_{h}^{2}}\Big{\\{}\alpha(\xi)+\gamma(\xi)+\mu(\xi)\Big{\\}}+\frac{L^{2}R^{3}}{z_{h}}l\Big{\\{}\beta(\xi)+\delta(\xi)+\nu(\xi)\Big{\\}}$
(72)

Ultimately, by incorporating equations (72) and (66) into (60), we arrive at
the following outcome for the HLN in the bipartite scenario under high-
temperature limit

$\mathcal{E}_{high}=\frac{3}{8G_{N}^{5}}\Bigg{[}\frac{L^{2}R^{3}}{z_{h}^{2}}\Big{\\{}\alpha(\xi)+\gamma(\xi)+\mu(\xi)\Big{\\}}+\frac{L^{2}R^{3}}{z_{h}}l\Big{\\{}\beta(\xi)+\delta(\xi)+\nu(\xi)\Big{\\}}-L^{2}R^{3}l\hat{T}g(\xi)\Bigg{]}$
(73)

Note that, as previously demonstrated for the low-temperature scenario, we can
similarly reformulate the equation above for the high-temperature regime. This
can be achieved by applying the same analysis in the limit as $\xi$ approaches
zero, resulting in a more concise expression.

$\mathcal{E}_{high}=\frac{3}{2}\Big{\\{}S_{A}-\mathcal{C}S_{A}^{\text{Th}}\Big{\\}},~{}~{}\text{where}~{}\mathcal{C}~{}\text{is
a constant}$ (74)

Much like in the case of low temperatures, in the high-temperature regime, the
HLN also facilitates the extraction of distillable quantum entanglement. This
extraction process involves eliminating the thermal contribution, a universal
characteristic of the entanglement negativity observed in finite-temperature
mixed states of a holographic CFT.

##  7 Entanglement Wedge Cross Section (EWCS)

In this section, we compute the analytic form of the EWCS and perform limiting
analysis for low and high-temperature regimes. To delineate the concept of the
entanglement wedge, we take into account two subsystems, labeled as $A$ and
$B$, situated on the boundary. The Ryu-Takayanagi (RT) surface, represented as
$\gamma_{AB}$, characterizes the region encompassing $A\cup B$. Subsequently,
the entanglement wedge is defined as the volume corresponding to the boundary
$A\cup B\cup\gamma_{AB}$. The Entanglement Wedge Cross Section (EWCS) is
established through the extremal area surface $\Gamma_{W}$, which bifurcates
the regions $A$ and $B$, as illustrated in fig. 7. In a numerical analysis
conducted in Amrahi:2020jqg , the EWCS has been computed.

Nevertheless, the analytic expression for the EWCS in the background of 1RCBH
has not yet been documented. In this context, we establish the boundary
subsystems $A$ and $B$, each with a length of $l$ and separated by a distance
$D$.

$\begin{split}&A:~{}~{}x^{1}\equiv
x\in\left[-l-\frac{D}{2},-\frac{D}{2}\right],\hskip
14.22636ptx^{(j)}\in\left[-\frac{L_{2}}{2},\frac{L_{2}}{2}\right]~{}~{}\text{where}~{}j=2,3\\\
&B:~{}~{}x^{1}\equiv x\in\left[l+\frac{D}{2},\frac{D}{2}\right],\hskip
34.1433ptx^{(j)}\in\left[-\frac{L_{2}}{2},\frac{L_{2}}{2}\right]~{}~{}\text{where}~{}j=2,3\end{split}$
(75)

In the given setup, the surface with the minimum area, denoted as
$\Sigma_{\text{min}}$, which separates the subsystems $A$ and $B$, is
precisely the vertical surface positioned at $x=0$. The metric on the Cauchy
surface is described as follows:

$ds_{\Sigma_{min}}^{2}=e^{2A(z)}d{\vec{x}}_{2}^{2}+\frac{e^{2B(z)}}{h(z)}\frac{R^{4}}{z^{4}}dz^{2}$
(76)

Figure 7: Schematic diagram of the extremal surfaces corresponding to two
disjoint subsystems of equal length $l$ and separated by a distance $D$. The
surface $\Gamma_{W}$, marked in red, is the entanglement wedge

The EWCS is then computed by Chakrabortty:2020ptb

$E_{W}=\frac{L^{2}}{4G_{N}^{5}}\int_{z_{t}(D)}^{z_{t}(2l+D)}dz\sqrt{g_{mn}}$
(77)

The equation (76) defines the induced metric $g_{\text{mn}}$. The turning
points of the extremal surfaces we have examined are denoted as $z_{t}(2l+D)$
and $z_{t}(D)$. Consequently, when considering equations (76) and (77), we
obtain:

$E_{W}=\frac{L^{2}R^{2}}{4G_{N}^{5}}\int_{z_{t}(D)}^{z_{t}(2l+D)}{dz\frac{e^{2A(z)+B(z)}}{z^{2}\sqrt{h(z)}}}$
(78)

By employing (9) and the definition of the dimensionless parameter $\xi$, we
can express the integral above as follows, 333Note that in deriving equation
(79), we have applied the multinomial expansion, similar to our earlier
approach in section 4.

$\begin{split}E_{W}=\frac{L^{2}R^{3}}{4G_{N}^{5}}\int_{z_{t}(D)}^{z_{t}(2l+D)}dz\sum_{k=0}^{\infty}\sum_{j=0}^{k}\sum_{i=0}^{\infty}\frac{{(-1)}^{k+j}}{2}\frac{\Gamma(k+\frac{1}{2}){\xi}^{i+j+k}{(1+\xi)}^{j}}{\Gamma(i+1)\Gamma(\frac{3}{2}-i)\Gamma(j+1)\Gamma(k-j+1)}\frac{z^{2i+2j+2k-3}}{{z_{h}}^{2i+2j+2k}}\\\
=\frac{L^{2}R^{3}}{4G_{N}^{5}}\sum_{k=0}^{\infty}\sum_{j=0}^{k}\sum_{i=0}^{\infty}\frac{{(-1)}^{k+j}}{2}\frac{\Gamma(k+\frac{1}{2}){\xi}^{i+j+k}{(1+\xi)}^{j}}{\Gamma(i+1)\Gamma(\frac{3}{2}-i)\Gamma(j+1)\Gamma(k-j+1)}\frac{1}{(2i+2j+2k-2)}\\\
\times\Bigg{\\{}\frac{z_{t}(2l+D)^{2i+2j+2k-2}}{z_{h}^{2i+2j+2k}}-\frac{z_{t}(D)^{2i+2j+2k-2}}{z_{h}^{2i+2j+2k}}\Bigg{\\}}\end{split}$
(79)

In the following two subsections, we will examine the behavior of the EWCS
under varying temperature conditions, employing suitable approximations as
previously demonstrated. As indicated by the expression above, it becomes
evident that when $D$ significantly exceeds $l$, the EWCS completely
disappears.

###  7.1 Entanglement Wedge Cross Section at low temperature

As it is generally challenging to reverse the relationship between $z_{t}$ and
width $l$, and formulate a universal expression for EWCS based on boundary
parameters, we resort to specific thermal limits for this purpose. Therefore,
we examine EWCS in both low and high-temperature limits. In the low-
temperature limit, where $z_{t}(D)\ll z_{H}$ and $z_{t}(2l+D)\ll z_{H}$, we
can derive the following expressions for the turning points using equation
(32).

$z_{t}(D)=\frac{D}{a_{1}}\Bigg{\\{}1+\frac{\xi}{6a_{1}^{2}}{\left(\frac{D}{z_{h}}\right)}^{2}+\frac{1}{24a_{1}^{4}}\left[\frac{{\xi}^{2}}{6}\left(1-\frac{a_{3}}{2a_{2}}\right)-\frac{a_{2}}{a_{1}}(1+\xi)\right]{\left(\frac{D}{z_{h}}\right)}^{4}+\mathcal{O}{\left(\frac{D}{z_{h}}\right)}^{6}\Bigg{\\}}$
(80)
$\begin{split}z_{t}(2l+D)=\frac{2l+D}{a_{1}}\Bigg{\\{}1+\frac{\xi}{6a_{1}^{2}}{\left(\frac{2l+D}{z_{h}}\right)}^{2}+\frac{1}{24a_{1}^{4}}\left[\frac{{\xi}^{2}}{6}\left(1-\frac{a_{3}}{2a_{2}}\right)-\frac{a_{2}}{a_{1}}(1+\xi)\right]{\left(\frac{2l+D}{z_{h}}\right)}^{4}\\\
+\mathcal{O}{\left(\frac{2L+D}{z_{h}}\right)}^{6}\Bigg{\\}}\end{split}$ (81)

By substituting equations (80) and (81) into equation (79), one can derive the
expression for EWCS in the low-temperature regime. It’s important to note that
simplifying the EWCS at low temperatures can be achieved by applying a
binomial expansion to both turning points, considering terms up to the first
order. This expansion is feasible because when examining the coefficients of
$\mathcal{O}(1/z_{h}^{2})$, $\mathcal{O}(1/z_{h}^{4})$, etc., within the
parentheses in equations (80) and (81), it becomes evident that these terms
are smaller than $1$ in the low-temperature limit. Further simplifications in
the resulting equation (see Appendix B) can be achieved by truncating the
series at the lowest order, which corresponds to setting $i=j=k=0$. It is also
reasonable to assert that at low temperatures, both $D$ and $l$ are small,
allowing us to neglect higher exponents associated with these length scales.
Consequently, we can express the simplified version of the EWCS at low
temperatures as follows.

$E_{W}^{\text{low}}=\frac{L^{2}R^{3}}{4G_{N}^{5}}\Bigg{[}\frac{a_{1}^{2}}{2}\Bigg{\\{}\frac{1}{D^{2}}-\frac{1}{{(2l+D)}^{2}}\Bigg{\\}}+\frac{2}{a_{1}^{2}}\Bigg{\\{}\frac{{\xi}^{2}}{6}\left(1-\frac{a_{3}}{a_{2}}\right)-\frac{a_{2}}{a_{1}}(1+\xi)\Bigg{\\}}l(l+D)\frac{1}{z_{h}^{4}}+\mathcal{O}{\left(\frac{1}{z_{h}}\right)}^{6}\Bigg{]}$
(82)

In terms of temperature the above expression becomes

$E_{W}^{\text{low}}=\frac{L^{2}R^{3}}{4G_{N}^{5}}\Bigg{[}\frac{a_{1}^{2}}{2}\Bigg{\\{}\frac{1}{D^{2}}-\frac{1}{{(2l+D)}^{2}}\Bigg{\\}}+\frac{2}{a_{1}^{2}}\Bigg{\\{}\frac{{\xi}^{2}}{6}\left(1-\frac{a_{3}}{a_{2}}\right)-\frac{a_{2}}{a_{1}}(1+\xi)\Bigg{\\}}l(l+D){(\pi\hat{T})}^{4}+\mathcal{O}\left({\hat{T}}^{6}\right)\Bigg{]}$
(83)

Now, we will examine the outcomes we’ve acquired for the EWCS when the low
temperature in the background of a 1RCBH. As anticipated, the initial term
enclosed in curly braces, which is the temperature independent component,
implies that the EWCS will rise as the separation between the subsystems
diminishes, and in the limit where D approaches zero, it becomes unbounded. We
can further verify the validity of the previously obtained result by cross-
referencing it with the mutual information calculation in Ebrahim:2020qif . By
leveraging the connection between the EWCS and mutual information as discussed
in Umemoto:2019jlz , we can observe that, at low temperatures, the EWCS
exhibits an identical behavior to that of mutual information. This alignment
serves as strong confirmation for the accuracy of our findings. In the
critical limit, denoted as $\xi\to 2$, it is noteworthy that the EWCS remains
finite, mirroring the behavior reported for mutual information in
Ebrahim:2020qif .

###  7.2 Entanglement Wedge Cross Section at High Temperature

We can now examine the EWCS under high-temperature conditions. To achieve
this, there are two viable options regarding the boundary parameters $l$ and
$D$. If we opt for a scenario where $D$ is chosen to be very large but finite,
both the turning points corresponding to the extremal surfaces $\gamma_{D}$
and $\gamma_{2l+D}$ will move deeper into the bulk. Consequently, one can, in
principle, employ the near-horizon expansion for the turning points $z_{t}(D)$
and $z_{t}(2l+D)$. However, this approach yields a trivial outcome as, in the
limit where $D$ tends to infinity, the EWCS at high temperatures naturally
diminishes.

Alternatively, one can argue that to impose the high-temperature limit, we can
take the limit of $l$ approaching infinity while keeping $D$ fixed at a small
value. In this scenario, a non-zero, significantly large value for the EWCS is
expected, and this can be obtained by focusing on the near-horizon expansion
for the extremal surface $\gamma_{2l+D}$ exclusively. For the upcoming
calculations, we will be working within the former limit. Utilizing the
techniques employed in the previous sections, we commence with the following
expression for the turning point.

$z_{t}(D)=z_{h}(1+\epsilon)=z_{h}\left(1+k_{2}e^{-\sqrt{\frac{d(d-1)}{2}}z_{h}D}\right)$
(84)

where,

$k_{2}=\frac{1}{d}e^{\sqrt{\frac{d(d-1)}{2}}c_{1}}$ (85)
$c_{1}=\frac{2\sqrt{\pi}\Gamma\left(\frac{d}{2(d-1)}\right)}{\Gamma\left(\frac{1}{d-1}\right)}+\sum_{n=1}^{\infty}\Bigg{\\{}\frac{2}{(1+nd)}\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma(n+1)}\frac{\Gamma\left(\frac{d(n+1)}{2(d-1)}\right)}{\Gamma\left(\frac{nd+1}{2(d-1)}\right)}-\frac{\sqrt{2}}{\sqrt{d(d-1)}n}\Bigg{\\}}$
(86)

Note that we are working with $d=4$. Now we use the (84) in the last line of
(79) to obtain

$\begin{split}E_{W}^{\text{high}}=\frac{L^{2}R^{3}}{4G_{N}^{5}}\sum_{k=0}^{\infty}\sum_{j=0}^{k}\sum_{i=0}^{\infty}\frac{{(-1)}^{k+j}}{2}\frac{\Gamma(k+\frac{1}{2}){\xi}^{i+j+k}{(1+\xi)}^{j}}{\Gamma(i+1)\Gamma(\frac{3}{2}-i)\Gamma(j+1)\Gamma(k-j+1)}\frac{1}{(2i+2j+2k-2)}\\\
\times\Bigg{\\{}\frac{{\left(1+k_{2}e^{-\sqrt{6}z_{h}(2l+D)}\right)}^{2i+2j+2k-2}}{z_{h}^{2}}-\frac{{\left(1+k_{2}e^{-\sqrt{6}z_{h}D}\right)}^{2i+2j+2k-2}}{z_{h}^{2}}\Bigg{\\}}\end{split}$
(87)

Taking the binomial expansion in the above equation up to order $\epsilon$ and
suppressing the higher order terms one obtains the following expression for
EWCS at high temperature

$E_{W}^{\text{high}}=\frac{L^{2}R^{3}}{4G_{N}^{5}}\sum_{k=0}^{\infty}\sum_{j=0}^{k}\sum_{i=0}^{\infty}\frac{{(-1)}^{k+j}}{2}\frac{\Gamma(k+\frac{1}{2}){\xi}^{i+j+k}{(1+\xi)}^{j}}{\Gamma(i+1)\Gamma(\frac{3}{2}-i)\Gamma(j+1)\Gamma(k-j+1)}\frac{k_{2}}{z_{h}^{2}}\Bigg{(}e^{-\sqrt{6}z_{h}(2l+D)}-e^{-\sqrt{6}z_{h}D}\Bigg{)}$
(88)

In terms of temperature, we can rewrite the above expression in the following
form

$E_{W}^{\text{high}}=\frac{L^{2}R^{3}}{4G_{N}^{5}}\sum_{k=0}^{\infty}\sum_{j=0}^{k}\sum_{i=0}^{\infty}\frac{{(-1)}^{k+j}}{2}\frac{\Gamma(k+\frac{1}{2}){\xi}^{i+j+k}{(1+\xi)}^{j}}{\Gamma(i+1)\Gamma(\frac{3}{2}-i)\Gamma(j+1)\Gamma(k-j+1)}{(\pi\hat{T})}^{2}\Bigg{(}e^{-\frac{\sqrt{6}(2l+D)}{\pi\hat{T}}}-e^{-\frac{\sqrt{6}D}{\pi\hat{T}}}\Bigg{)}$
(89)

Similar to the previous section, we will now refer to the calculation of
mutual information at high temperatures as presented in Ebrahim:2020qif . In
equation (89), we can apply the high-temperature limit by considering $D$ to
be large but finite. Consequently, as we take the limit $D\to\infty$, the EWCS
diminishes, as does the mutual information, as indicated in Ebrahim:2020qif .
This leads us to conclude that the expression we have derived for EWCS at
elevated temperatures is consistent. It’s worth noting, as mentioned earlier,
that working with the non-trivial limit in the boundary parameter could be an
intriguing exercise, demonstrating that this limit corresponds to a
substantial EWCS value. However, we defer this exploration to future research.

##  8 Holographic Mutual Information

The holographic dual of a thermofield double (TFD) state is essential for
studying information scrambling in a strongly coupled field theory in the
context of AdS/CFT correspondence. Entangled states, defined in a bipartite
Hilbert space comprised of the individual Hilbert spaces of two identical and
non-interacting copies of strongly coupled field theories, can serve as
examples of TFD states. Two entangled AdS black holes are holographically dual
to such a TFD state.

Figure 8:  Penrose diagram of the eternal blackhole. At t = 0, the spatial
extremal surface connecting two asymptotic boundaries of an eternal black hole
is denoted by the dashed line passing through the bifurcation point.

The outer region on both sides of the two-sided black hole in the Penrose
diagram fig.8 is made up of the right (R) and left (L) wedges, two causally
disconnected regions of spacetime. It is possible to have a non-local
correlation between two boundary theories that are separately residing in the
asymptotic boundaries of the R or L regions. An appropriate generalization of
mutual information (MI), known as thermo mutual information (TMI), was first
introduced in Morrison:2012iz , along with a generalization of holographic
mutual information (HMI), known as holographic thermo mutual information
(HTMI), can be used to describe a practical measure of such correlation. Later
holographically studied in Leichenauer:2014nxa ; Chakrabortty:2022kvq ;
Jahnke:2017iwi . Therefore the above definition is true for TMI, but the
entangling regions lie on the causally disconnected boundaries. An early time-
dependent perturbation that grows exponentially can destroy these correlations
holographically this perturbation is known as the shockwave created by a small
amount of energy in the asymptotic past. In the following sections, we will
study the HTMI without and with the shockwave.

###  8.1 Holographic Thermo Mutual Information (HTMI)

To determine the HTMI, we adopt the methodology presented in Jahnke:2017iwi ;
Chakrabortty:2022kvq ; Leichenauer:2014nxa , considering two strip-like
identical subsystems denoted as $A$ and $B$ with a width of $l$. Subsystem $A$
is positioned along the left ($L$) asymptotic boundary, while subsystem $B$ is
situated along the right ($R$) asymptotic boundary of the eternal black hole
at time $t=0$. In accordance with the RT proposal, $\mathcal{S}(A)$ and
$\mathcal{S}(B)$ are directly linked to the minimal surface areas of
$\gamma_{A}$ and $\gamma_{B}$ within the bulk, which correspond to the
entangling regions $A$ and $B$, respectively. We define the embedding for
$\gamma_{i}(i=A,B)$ as ($t=0,z,-l/2\leq x(z)\leq l/2,-L/2\leq x^{j}\leq L/2$
$j=2....d-1$). For the extremal surface corresponds to $\gamma_{A\cup B}$ can
be either $\gamma_{A}\cup\gamma_{B}$ or
$\gamma_{1}\cup\gamma_{2}=\gamma_{\text{wormhole}}$ where for $\gamma_{1}$ and
$\gamma_{2}$ the appropriate embedding is ($t=0,z,x=-l/2,-L/2\leq x^{j}\leq
L/2$) and ($t=0,z,x=l/2,-L/2\leq x^{j}\leq L/2$) respectively. $\gamma_{1}$
and $\gamma_{2}$ surfaces are connecting two asymptotic boundaries through the
bifurcation point of the eternal black hole denoted by the doted line in the
Fig.8. TMI becomes zero as
$\mathcal{A}(\gamma_{A}\cup\gamma_{B})\leq\mathcal{A}(\gamma_{\text{wormhole}})$,
and $I(A,B)$ is positive for the opposite situation. To find the area of
wormhole surface we need to follow the Hubeny-Rangamani-Takayanagi (HRT)
prescription Hubeny:2007xt . The induced metrics components for RT and HRT
surfaces are given by,

$G_{\text{in}}^{A}=G_{\text{in}}^{B}=\left(\frac{R_{\text{AdS}}^{2}}{z^{2}}\right)^{d-1}\left(\frac{1}{f(z)}+x^{\prime
2}\right),\qquad
G_{\text{in}}^{\text{wormhole}}=\left(\frac{R_{\text{AdS}}^{2}}{z^{2}}\right)^{d-1}\frac{1}{f(z)},$
(90)

where $x^{\prime}=0$ for $\gamma_{1}$ and $\gamma_{2}$. Now, HTMI is

$\begin{split}I(A,B)&=\frac{1}{4G_{N}^{5}}\left(\int_{-L/2}^{L/2}dx^{i}\right)\left[2\int_{0}^{z_{t}}dz\left(\sqrt{G_{in}^{A}}+\sqrt{G_{in}^{B}}\right)-4\int_{0}^{z_{h}}dz\sqrt{G_{in}^{A\cup
B}}\right]\\\
&=\frac{L^{2}R^{2}}{G_{N}^{5}}\left[\int_{0}^{z_{c}}\frac{e^{B(z)+2A(z)}dz}{z^{2}\sqrt{f(z)}}\sqrt{(\frac{e^{6A(z_{c})}}{e^{6A(z)}-e^{6A(z_{c})}}+1)}-\int_{0}^{z_{h}}\frac{e^{3A(z)}dz}{z^{2}\sqrt{f(z)}e^{A(z)-B(z)}}\right]\end{split}$
(91)

Due to the symmetric layout of the extremal surfaces, equation (91)
incorporates coefficients of both $2$ and $4$. The parameter $z_{t}$ denotes
the turning point of the RT surfaces associated with regions $A$ and $B$.

Figure 9: HTMI with respect to width $l$ for $T=1,R=1$ with different values
of $\xi$

The relationship between the HTMI and the width ($l$) of entangling region can
be determined through the use of the $l$ and $z_{t}$ relation.

$\frac{l}{2}=\int_{0}^{z_{t}}\frac{dz}{\sqrt{\left(\frac{Q^{2}z^{2}}{R^{4}}+1\right)\left(\frac{z_{t}^{6}\left(\frac{Q^{2}z^{2}}{R^{4}}+1\right)}{z^{6}\left(\frac{Q^{2}z_{t}^{2}}{R^{4}}+1\right)}-1\right)\left(1-\frac{z^{4}\left(\frac{Q^{2}z_{h}^{2}}{R^{4}}+1\right)}{z_{h}^{4}\left(\frac{Q^{2}z^{2}}{R^{4}}+1\right)}\right)}}$
(92)

The fig.9 illustrates that, as the width $l$ is increased, the HTMI also
increases. However, when we reach a specific value of $l$ (say critical width
$l_{c}$), the TMI is zero for any $l\leq l_{c}$. This critical value of width
dicreases as we raise the value of $\xi$. The TMI exhibited analogous
characteristics as reported in Chakrabortty:2022kvq ; Jahnke:2017iwi . In
Chakrabortty:2022kvq , it was noted that as the backreaction parameter $b$
increases, the critical width decreases, a trend also observed in
Jahnke:2017iwi  with the anisotropic parameter $a$. This occurs because, when
$l\leq l_{c}$, the HRT surface connecting points $A$ and $B$ accumulates a
greater surface area than the combined areas of the individual RT surfaces
associated with $A$ and $B$. Consequently, the selection for the $A\cup B$
surface will be equivalent to the sum of the areas of the RT surfaces
corresponding to $A$ and $B$.

It is evident that as the parameter $\xi$ increases, the critical width
$l_{c}$ decreases. As $\xi$ approaches the critical point of the theory, which
is $\xi\to 2$, all the values of the critical width $l_{c}$ for non-zero $\xi$
converge towards the value associated with $\xi=2$, while the critical width
corresponding to $\xi=0$ is significantly distant. The closer $\xi$ gets to
the critical value of $2$, the smaller the separation between the critical
widths becomes.

###  8.2 Holographic Thermo Mutual Information with shockwave

In this section, we examine the time-dependent behavior of the HTMI following
the application of a shockwave. The profile of the shockwave is $\alpha\approx
e^{\frac{2\pi}{\beta}t}$. The impact of the shock wave on the geometry can be
accounted by adjusting the Kruskal coordinate from $V$ to
$\hat{V}=V+\Theta(U)\alpha$, while leaving all other coordinates unchanged and
denoting them with a hat, as previously demonstrated in Leichenauer:2014nxa ;
Chakrabortty:2022kvq ; Jahnke:2017iwi . The function $\Theta(U)$ ensures that
the shockwave’s influence is confined to the left region of the Penrose
diagram shown in fig. 8, which is modified as depicted in fig.10.

Entanglement entropy exhibits UV divergences, while mutual information remains
unaffected by these divergences, as discussed in the preceding section. The
introduction of a shock wave can introduce new divergences into the system. We
investigate an early asymptotic pulse of energy generated at the boundary,
which acts as a minor inward disturbance entering the left (L) boundary of the
eternal black hole. This pulse experiences blue-shifting and evolves into a
shockwave as it progresses through time, eventually reaching the horizon at a
late time corresponding to the boundary time $t=0$. In light of this, it
proves advantageous to define the HTMI in the presence of the shockwave as,

$I(A:B;\alpha)=I(A,B;\alpha=0)-\mathcal{S}^{\text{reg}}_{A\cup B}(\alpha),$
(93)

$I(A:B;\alpha=0)$ has been previously calculated in equation (91), and in
order to mitigate the $\alpha$-independent UV divergences, we introduce a
regularized variant of HEE. $\mathcal{S}^{\text{reg}}_{A\cup
B}(\alpha)=\mathcal{S}_{A\cup B}(\alpha)-\mathcal{S}_{A\cup B}(\alpha=0)$. In
order to compute $\mathcal{S}^{\text{reg}}_{A\cup B}(\alpha)$ we choose a set
of time-dependent embeddings defined as $\\{t,z(t),x=-l/2,-L/2\leq x^{j}\leq
L/2\\}$ and $\\{t,z(t),x=l/2,-L/2\leq x^{j}\leq L/2\\}$. The area functional
corresponding to either of these time-dependent embeddings is given as,

$\mathcal{A}=L^{2}\int{dt\biggl{[}-e^{6A(z)}h(z)+\dot{z}^{2}\frac{R^{4}e^{2B(z)+4A(z)}}{z^{4}h(z)})\biggr{]}^{\frac{1}{2}}},~{}~{}\mathcal{L}=L^{2}\biggl{[}{-e^{6A(z)}h(z)+\dot{z}^{2}\frac{R^{4}e^{2B(z)+4A(z)}}{z^{4}h(z)}\biggr{]}^{\frac{1}{2}}}$
(94)

Figure 10: Penrose diagram after the shock wave. Red line shows the wormhole
surface with turning point $z_{0}$ connecting $A$ and $B$.

Note that the Lagrangian density $\mathcal{L}$ lacks explicit time dependence,
consequently leading to the derivation of a conserved quantity based on
boundary conditions $\dot{z}=0|_{z=z_{0}}$, is
$\mathcal{P}=-L^{2}e^{3A(z_{0})}\sqrt{-h(z_{0})}$. Now, we can write,

$\dot{z}^{2}=\frac{z^{2}h(z)}{R^{4}e^{4A(z)+2B(z)}}\biggl{[}\frac{L^{4}h^{2}(z)e^{12A(z)}}{\mathcal{P}}+e^{6A(z)}h(z)\biggr{]}^{\frac{1}{2}}$
(95)

Substituting equation (95) in (94) the area functional,

$\mathcal{A}=L^{2}R^{2}\int_{0}^{z_{0}}dz~{}{\frac{e^{5A(z)+B(z)}}{z^{2}\sqrt{\mathcal{P}^{2}+L^{4}h(z)e^{6A(z)}}}}$
(96)

Also by integrating equation (95) we get,

$t(z)=\pm\int
dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)\sqrt{(1+\frac{L^{4}h(z)e^{6A(z)}}{\mathcal{P}^{2}})}}}$
(97)

Using the conserved momenta $\mathcal{P}$ in equation (96) and (97) we get,

$\mathcal{A}=L^{2}R^{2}\int_{0}^{z_{0}}dz~{}{\frac{e^{5A(z)+B(z)}}{z^{2}\sqrt{h(z)e^{6A(z)}-h(z_{0})e^{6A(z_{0})}}}},~{}~{}t(z)=\pm\int
dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}}.$
(98)

Figure 11: $S^{reg}$ vs $\frac{z_{0}}{z_{H}}$ for different $\xi$ with
$T=1,R=1$.

To determine the wormhole’s area, denoted as $\mathcal{A}(\gamma_{\text{w}})$,
we partition the integration variable $z$ from equation (98) into three
distinct regions. The initial region (I) initiates at the left boundary and
spans from there into the bulk, ultimately reaching the horizon. Subsequently,
the second region (II) commences at the horizon and concludes at $z=z_{0}$.
Lastly, the third region (III) commences at $z=z_{0}$, proceeds towards the
right (as depicted in fig.10), and extends until it encounters the horizon.
The direction of $t$ is contingent on the rate of change in $z$ along the
extremal surface.

Taking into account the three regions labeled as I, II, and III, the specific
expression for $\mathcal{A}(\gamma_{\text{w}})$ takes the following form:

$\begin{split}\mathcal{A}(\gamma_{\text{w}})=4L^{2}R^{2}&\biggl{[}\int^{z_{H}}_{0}{dz\biggl{(}\frac{e^{5A(z)+B(z)}}{z^{2}\sqrt{h(z)e^{6A(z)}-h(z_{0})e^{6A(z_{0})}}}}-\frac{e^{2A(z)+B(z)}}{z^{2}\sqrt{h(z)}}\biggr{)}\\\
&+2\int^{z_{0}}_{z_{H}}{dz\frac{e^{5A(z)+B(z)}}{z^{2}\sqrt{h(z)e^{6A(z)}-h(z_{0})e^{6A(z_{0})}}}}\biggr{]}\end{split}$
(99)

The regularized HEE is expressed as $S^{\text{reg}}{A\cup
B}(z_{0})=\frac{\mathcal{A}(\gamma_{w})}{4G_{N}^{5}}$. This regularized
entropy is depicted in Fig.11 as a function of the dimensionless parameter
$\frac{z_{0}}{z_{h}}$. It is evident from the plot that as the ratio
$z_{0}/z_{H}$ increases from unity, the value of $S^{\text{reg}}$ initiates
from zero at $z_{0}=z_{H}$ and progressively rises. For various nonzero values
of $\xi$, $S^{reg}$ exhibits varying changes; for smaller $\xi$ values, the
regularized entropy increases more significantly than it does for the maximum
permissible value of $\xi=2$.

By employing equations (93) and (99), we can express the HTMI as a function
dependent on $z_{0}$. Nevertheless, to examine how the HTMI changes concerning
the shock wave parameter $\alpha$, we must establish the connection between
$z_{0}$ and $\alpha$. In Fig.10, Region (I) is defined as the area between the
boundary point $(\hat{U},\hat{V})=(1,-1)$ and the point on the horizon
$(\hat{U},\hat{V})=(\hat{U}_{1},0)$. Region (II) spans from
$(\hat{U},\hat{V})=(\hat{U}_{1},0)$ to a point denoted as
$(\hat{U},\hat{V})=(\hat{U}_{2},\hat{V}_{2})$, while Region (III) extends from
$(\hat{U},\hat{V})=(\hat{U}_{2},\hat{V}_{2})$ to
$(\hat{U},\hat{V})=(0,\alpha/2)$. Region (II) from
$(\hat{U},\hat{V})=(\hat{U}_{1},0)$ to $z_{0}$
$(\hat{U},\hat{V})=(\hat{U}_{2},\hat{V}_{2})$ and region (III) from
$(\hat{U},\hat{V})=(\hat{U}_{2},\hat{V}_{2})$ to
$(\hat{U},\hat{V})=(0,\alpha/2)$.

Using the definition of Kruskal coordinates, it is possible to express the
variation of $\hat{U}=\pm e^{\frac{2\pi}{\beta}(z_{*}-t)}$ and $\hat{V}=\pm
e^{\frac{2\pi}{\beta}(z_{*}+t)}$ as,

$\begin{split}&\Delta\log\hat{U}^{2}=\log\hat{U}_{1}^{2}-\log{\hat{U}_{0}^{2}}=\frac{4\pi}{\beta}(\Delta
z_{*}-\Delta t)\\\
&\Delta\log\hat{V}^{2}=\log\hat{V}_{2}^{2}-\log{\hat{V}_{1}^{2}}=\frac{4\pi}{\beta}(\Delta
z_{*}+\Delta t)\end{split}$ (100)

$\begin{split}&\log{\hat{U}}=\frac{2\pi}{\beta}\int
dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}\Bigg{(}\frac{1}{\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}-1\Bigg{)}}\\\
&\log{\hat{V}}=\frac{2\pi}{\beta}\int
dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}\Bigg{(}\frac{1}{\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}+1\Bigg{)}}\end{split}$
(101)

where $z_{*}$ is defined as follows,

$z_{*}=-\int dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}}$ (102)

Note that in region (I), when $\dot{z}<0$, it leads to an overall negative
sign in the expression for $t$. Conversely, in region (II), the negative
numerical value of $h(z)$ corresponds to $\dot{z}>0$, and hence we introduce a
negative sign. Now, let’s consider the variation of $\hat{U}$ from the
boundary to the horizon.

$\hat{U}^{2}_{1}=\exp{\Bigg{[}\frac{4\pi}{\beta}\int_{0}^{z_{H}}dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}\Bigg{(}\frac{1}{\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}-1\Bigg{)}}\Biggr{]}}$
(103)
$\frac{\hat{U}^{2}_{2}}{\hat{U}_{1}^{2}}=\exp{\Bigg{[}\frac{4\pi}{\beta}\int_{0}^{z_{H}}dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}\Bigg{(}\frac{1}{\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}-1\Bigg{)}}\Bigg{]}}$
(104)

Figure 12: Shockwave parameter $\alpha(z_{0})$ v/s $\frac{z_{0}}{z_{H}}$ for
different values of $\xi$.

To find $\hat{U}_{2}$ consider a reference point at $\bar{z}$ where $z_{*}$ is
zero.

$\hat{V}_{2}\hat{U}_{2}=\exp{\left(\frac{4\pi}{\beta}z_{*}\right)}=\exp{\Bigg{[}-\frac{4\pi}{\beta}\int_{\bar{z}}^{z_{0}}dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}}\Bigg{]}}$
(105)

In region (III), where $\dot{z}>0$, yet $h(z)$ remains in the negative
numerical range, we introduce an overall negative sign to the variable $t$.
Consequently, the expression for the coordinate $\Delta\hat{V}$ in region
(III) adopts the following form:

$\frac{\alpha^{2}}{4\hat{V}_{2}^{2}}=\frac{4\pi}{\beta}\int_{0}^{z_{h}}dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}\Bigg{(}\frac{1}{\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}-1\Bigg{)}}$
(106)

From equation (105) and (106) we can write the relation between $\alpha$ and
$z_{0}$ as,

$\alpha(z_{0})=2\exp{(\mathcal{\eta}_{\text{I}}+\mathcal{\eta}_{\text{II}}+\mathcal{\eta}_{\text{III}})}$
(107)

where

$\begin{split}\mathcal{\eta}_{\text{I}}&=\frac{4\pi}{\beta}\int_{\bar{z}}^{z_{0}}dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}},~{}~{}~{}~{}\mathcal{\eta}_{\text{II}}=\frac{2\pi}{\beta}\int_{0}^{z_{h}}dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}\Bigg{(}\frac{1}{\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}-1\Bigg{)}}\\\
\mathcal{\eta}_{\text{III}}&=\frac{4\pi}{\beta}\int_{z_{h}}^{z_{0}}dz~{}{\frac{R^{2}e^{B(z)-A(z)}}{z^{2}h(z)}\Bigg{(}\frac{1}{\sqrt{1-\frac{h(z)}{h(z_{0})}e^{6A(z)-6A(z_{0})}}}-1\Bigg{)}}\end{split}$
(108)

Figure 13: TMI vs shock parameter $\alpha$ for different $\xi$ and $T=1,R=1$.

By utilizing the equation (107), it becomes possible to generate a graphical
representation of the shock wave parameter concerning the dimensionless
quantity $z_{0}/z_{H}$, as depicted in fig.12. It is noteworthy that, in
accordance with expectations, the shockwave parameter escalates as $z_{0}$
increases, and the pace of this elevation is contingent on the $\xi$
parameter. In instances where $\xi$ assumes larger values, the rate at which
$\alpha$ increases is comparatively slower in contrast to situations with
smaller $\xi$ values. In the end, in Fig. 13, the graph illustrates how the
HTMI changes concerning the shockwave parameter $\alpha$ for various nonzero
$\xi$ values. For distinct $\xi$ values, HTMI starts declining from a specific
initial value, each with its own rate, ultimately reaching zero at a critical
point, denoted as $\alpha=\alpha_{c}$. HTMI only exists for $\alpha$ values
less than or equal to $\alpha_{c}$. This critical value of $\alpha$ increases
as the $\xi$ parameter grows. It’s worth noting that as previously mentioned,
$\xi=2$ represents the theory’s critical point, and it’s evident that HTMI
remains finite at this critical point, echoing a similar observation made for
one-sided MI in Ebrahim:2020qif .

##  9 Summary and Discussions

In this work, we study the various measures for the entanglement structure of
mixed states and the properties of chaos in the four-dimensional
$\mathcal{N}=4$ super Yang-Mills theory at finite temperature T, charged under
a $U(1)$ subgroup of its $SU(4)$ R-symmetry with critical point. We use the
HLN, EWCS, HTMI to probe the entanglement structure near the critical point.
We also study the disruption of HTMI due the shockwave perturbation and
finally interpret our results in terms of boundary theory parameters.

We study the effect of parameter $\xi$ (related to the charge of black hole)
on the HLN in low and high temperature limits. In this analysis, we observe
that the corresponding RT surface dual to the boundary region A receives a
modification due to the presence of $\xi$. Moreover, for a fixed width $l$ of
the boundary region $A$, the RT surface goes deeper into the bulk for larger
value of $\xi$. For computing the HLN at low and high temperature, we consider
adjacent, disjoint and bipartite configuration of subsystems in the boundary.
It is straighforward to see that $\xi\to 0$ ($Q\to 0$) correctly reproduce the
results obtained for the AdSd+1 Schwarzschild black hole background. The HLN
exhibits a decreasing trend for adjacent configurations as the parameter $\xi$
increases at low temperatures and an increasing trend as temperature
approaches the high-temperature limit. For disjoint subsystems the HLN shows
an increasing behavior with $\xi$ at low temperature and vanishes for high
temperature. In the bipartite case the HLN shows an increasing behavior with
$\xi$ at low temperature and an decreasing behavior for high temperature. In
the field theory, the growth of the HLN can be understood as indicative of the
increasing quantum entanglement between two subsystems. As the critical limit
is approached ($\xi\to 2$) in all cases, HLN remains finite. A similar finding
was previously documented for HEE and HMI in the study by Ebrahim et al.
(Ebrahim:2020qif ).

We give analytic expressions for EWCS for 1RC black hole in the low and high
temperature limits that are consistent with the numerical result obtained in
Amrahi:2020jqg . We observe that, at low temperatures, the EWCS experiences a
correction attributed to the parameter $\xi$ and consequently it exhibits a
growth with respect to $\xi$. It’s worth noting that mutual information is
intricately connected to EWCS, as described in Umemoto:2019jlz . Our result of
EWCS also agrees with the numerical analysis of HMI reported in
Ebrahim:2020qif . For disjoint case, in the low-temperature regime, both the
HLN and EWCS exhibit a similar dependence on the boundary
parameter(characteristic length of the different regions) as well as the
temperature. In the high-temperature limit, these quantities vanish as stated
in Ebrahim:2020qif .

Moreover we notice that the entanglement between two subsystems of a TFD state
measured by TMI increases with the size of the subsystem. This is expected as
the larger the Hilbert space of individual subsystems more the correlation. If
we fix the size of subsystems, TMI increases as $\xi$ parameter approaches to
the higher values. Based on our analysis, we have noted that the two separate
subsystems do not manifest correlations regardless of the subsystem’s size. It
is only once a specific size, denoted as the critical width $l_{c}$, is
reached that total correlations start to emerge. As we already mentioned the
entanglement in the TFD state can be destroyed by the insertion of an operator
evolves in time. We have demonstrated the explicit disruption of holographic
TMI in the presence of a shockwave. Our findings suggest that the parameter
$\xi$ attempts to mitigate this disruption, indicating that the presence of
the $\xi$ parameter tends to stabilize the system and reduce its chaotic
behavior. For substantial values of $\xi$, holographic TMI exhibits a slower
rate of decay. In simpler terms, when $\xi$ is large, it takes more time for
TMI to completely vanish. This is in contrast to the findings in a recent
study Chakrabortty:2022kvq , where it was noted that TMI diminishes more
rapidly with increased backreaction.

Acknowledgement

We express our gratitude to Shankhadeep Chakrabortty for valuable comments on
the draft. SP acknowledges the support of Senior Research Fellowship from the
Ministry of Human Resource and Development, Government of India. SP expresses
gratitude to Dongmin Gang and Seoul National University for their generous
support and warm hospitality during a part of this work. DK wishes to extend
appreciation to Shankhadeep Chakrabortty and IIT Ropar for their support and
warm hospitality during the final stages of this project.

##  Appendix A Area of the Extremal Surface for Bipartite Systems

We provide a concise overview of employing the near horizon expansion
technique to estimate extremal surfaces within the bipartite subsystem, aiming
to determine the surface area. $\mathcal{A}_{B_{1}}$ and $\mathcal{A}_{A\cup
B_{1}}$ in the limit $L\to\infty$ it is convenient to start with the equation
(26) by rewriting it in the following form

$\mathcal{A}={\mathcal{A}}^{(1)}+{\mathcal{A}}^{(2)}+{\mathcal{A}}^{(3)}$
(109)

where we define the quantities
${\mathcal{A}}^{(1)},{\mathcal{A}}^{(2)},{\mathcal{A}}^{(3)}$ in the following
way

${\mathcal{A}}^{(1)}=\frac{L^{2}R^{3}}{{z_{t}}^{2}}\Bigg{\\{}\frac{3\xi}{2}{\left(\frac{z_{t}}{z_{h}}\right)}^{2}-{\left[1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right]}^{\frac{3}{2}}+\frac{1+\xi}{3\xi}{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\left[{\left(1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right)}^{\frac{3}{2}}-1\right]\Bigg{\\}}$
(110)
${\mathcal{A}}^{(2)}=\frac{L^{2}R^{3}}{{z_{t}}^{2}}\Bigg{\\{}\sum_{n=0}^{2}{\Lambda}_{2n0}\frac{\sqrt{\pi}\Gamma(n+1)}{\Gamma(n+3)}{\left(\frac{z_{t}}{z_{h}}\right)}^{4+2n}\\\
\times\left[1+(n+1)\left(1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right)\right]\Bigg{\\}}$
(111)
${\mathcal{A}}^{(3)}=\frac{L^{2}R^{3}}{{z_{t}}^{2}}\Bigg{\\{}\sum_{j=1}^{\infty}{\Lambda}_{000}\frac{\Gamma(j+\frac{1}{2})\Gamma(3j-1)}{\Gamma(j+1)\Gamma(3j+1)}\times\left[1+(3j-1)\left(1+\xi{\left(\frac{z_{t}}{z_{h}}\right)}^{2}\right)\right]\Bigg{\\}}$
(112)

note that we have truncated the series when writing ${\mathcal{A}}_{2}$ and
${\mathcal{A}}_{3}$ in order to obtain the lowest-order contribution, which is
our focus for the near-horizon expansion. Consequently, the higher-order
contributions will become superfluous for our analysis. We then employ
equation (63) within the context of equations (110), (111), and (112) to
derive the expression for extremal surfaces in the bipartite limit, as
presented in equation (66). In this context, we introduce the following
functions of $\xi$.

$\alpha(\xi)=\left(\frac{3\xi}{2}-\frac{2}{3}(1+\xi)+\frac{k_{2}}{3\xi}(1-2\xi)(\xi-2)\right),~{}~{}\beta(\xi)=k_{2}\sqrt{6}\frac{(2\xi-1)(\xi-2)}{3\xi}$
(113)
$\begin{split}&\gamma(\xi)=\Bigg{(}u_{1}(\xi)-v_{1}(\xi)+x_{1}(\xi)+k_{2}\bigg{(}u_{2}(\xi)-v_{2}(\xi)+x_{2}(\xi)\bigg{)}\Bigg{)},~{}~{}\delta(\xi)=\Bigg{(}-u_{2}(\xi)+v_{2}(\xi)-x_{2}(\xi)\Bigg{)}\\\
&u_{1}(\xi)=\frac{3\xi^{2}}{16}(\xi^{2}+3\xi+2)\hskip
56.9055ptu_{2}(\xi)=\frac{3\xi^{2}}{16}(3\xi^{2}+6\xi+4)\\\
&v_{1}(\xi)=\frac{3\xi(1+\xi)}{24}(2\xi^{2}+5\xi+3)\hskip
19.91692ptv_{2}(\xi)=\frac{3\xi(1+\xi)}{24}(10\xi^{2}+21\xi+12)\\\
&x_{1}(\xi)=\frac{3{(1+\xi)}^{2}}{96}(\xi^{2}+7\xi+4)\hskip
22.76219ptx_{2}(\xi)=\frac{3{(1+\xi)}^{2}}{96}(21\xi^{2}+44\xi+24)\end{split}$
(114)
$\begin{split}\mu(\xi)=\sum_{j=1}^{\infty}\frac{3}{\sqrt{\pi}}\frac{\Gamma\left(j+\frac{1}{2}\right)\Gamma(3j-1)}{\Gamma(j+1)\Gamma(3j+1)}(1+\xi-(\xi+2)k_{2})\\\
\nu(\xi)=\sum_{j=1}^{\infty}\frac{3}{\sqrt{\pi}}\frac{\Gamma\left(j+\frac{1}{2}\right)\Gamma(3j-1)j}{\Gamma(j+1)\Gamma(3j+1)}(\xi+2)k_{2}\sqrt{6}\end{split}$
(115)

Using the above functions we obtain the following extremal areas

$\begin{split}{\mathcal{A}}_{B_{1}}=\frac{L^{2}R^{3}}{z_{h}^{2}}\Big{\\{}\alpha(\xi)+\gamma(\xi)+\mu(\xi)\Big{\\}}+\frac{L^{2}R^{3}}{z_{h}}\left(L-\frac{l}{2}\right)\Big{\\{}\beta(\xi)+\delta(\xi)+\nu(\xi)\Big{\\}}\\\
\mathcal{A}_{A\cup
B_{1}}=\frac{L^{2}R^{3}}{z_{h}^{2}}\Big{\\{}\alpha(\xi)+\gamma(\xi)+\mu(\xi)\Big{\\}}+\frac{L^{2}R^{3}}{z_{h}}\left(L+\frac{l}{2}\right)\Big{\\{}\beta(\xi)+\delta(\xi)+\nu(\xi)\Big{\\}}\end{split}$
(116)

By applying the equations mentioned earlier to the entanglement negativity
formula associated with the bipartite state in a low-temperature regime, we
derive equation (67). It’s worth noting that equation (67) is obtained through
the correlation between $\hat{T}$ and $z_{h}$. Additionally, we introduce the
following function for a more concise expression of the HLN.

$g(\xi)=\pi\big{(}\beta(\xi)+\delta(\xi)+\nu(\xi)\big{)}$ (117)

We employ the defined function $g(\xi)$ to examine the scenario as we approach
the limit where $\xi$ tends towards zero. In this limit, it becomes evident
from the equations mentioned earlier that (117) simplifies to the following
expression.

$g(\xi){\Bigg{|}}_{\xi\to
0}=\pi\Bigg{(}\frac{2}{3}k_{2}\sqrt{6}\frac{1}{\xi}-\frac{3}{4}+2k_{2}\sqrt{6}\sum_{j=1}^{\infty}\frac{3}{\sqrt{\pi}}\frac{\Gamma\left(j+\frac{1}{2}\right)\Gamma(3j-1)j}{\Gamma(j+1)\Gamma(3j+1)}\Bigg{)}$
(118)

Hence, if one’s primary concern is the temperature dependency of the function
$g(\xi)$ as $\xi$ approaches zero, an approximate approach involves focusing
on the initial term within the parentheses in the previous equation.
Consequently, it becomes evident that the function $g(\xi)$ exhibits a
proportionality to $T^{2}$.

Before concluding this appendix, it’s worth noting that equation (68)
introduces a constant $\mathcal{C}$. The precise value of this constant can be
determined by considering the coefficient of the initial term in equation
(118).

##  Appendix B Approximate EWCS at low temperature limit in terms of boundary
parameters

In this context, we derive the expression for the EWCS under the conditions of
low temperature. By inserting the equations for the turning points into the
overarching expression for the EWCS, as provided in equation (79), we derive
the ensuing series.

$\begin{split}E_{W}^{low}=\frac{L^{2}R^{3}}{4G_{N}^{5}}\sum_{k=0}^{\infty}\sum_{j=0}^{k}\sum_{i=0}^{\infty}\frac{{(-1)}^{k+j}}{2}\frac{\Gamma(k+\frac{1}{2}){\xi}^{i+j+k}{(1+\xi)}^{j}}{\Gamma(i+1)\Gamma(\frac{3}{2}-i)\Gamma(j+1)\Gamma(k-j+1)}\frac{z_{h}^{2-2i-2j-2k}}{(2i+2j+2k-2)}\\\
\times\Bigg{[}\Bigg{\\{}{\left(\frac{2l+D}{a_{1}}\right)}^{2i+2j+2k-2}-{\left(\frac{D}{a_{1}}\right)}^{2i+2j+2k-2}\Bigg{\\}}+(2i+2j+2k-2)\frac{\xi}{6z_{h}^{2}}\Bigg{\\{}{\left(\frac{2l+D}{a_{1}}\right)}^{2i+2j+2k}\\\
-{\left(\frac{D}{a_{1}}\right)}^{2i+2j+2k}\Bigg{\\}}+\frac{2i+2j+2k-2}{2z_{h}^{4}}\left(\frac{{\xi}^{2}}{6}\left(1-\frac{a_{3}}{2a_{2}}\right)\right)\Bigg{\\{}{\left(\frac{2l+D}{a_{1}}\right)}^{2i+2j+2k+2}-{\left(\frac{D}{a_{1}}\right)}^{2i+2j+2k+2}\Bigg{\\}}\\\
+\mathcal{O}{\left(\frac{1}{z_{h}}\right)}^{6}\Bigg{]}\end{split}$ (119)

As stated in Section 8, we have the option to conclude the series for
increased simplification by setting $i=j=k=0$. Consequently, this procedure
yields Equation (82) with ease.

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