# Temperature Dependence of Entanglement of Purification
in Presence of Chemical Potential

B<EMAIL_ADDRESS>Department of Physics, Shahid
Beheshti University, Tehran, Iran  M. Ali-
<EMAIL_ADDRESS>Department of Physics, Shahid
Beheshti University, Tehran, Iran  M<EMAIL_ADDRESS>IPM,
School of Particles and Accelerators, P.O. Box 19395-5531, Tehran, Iran

###### Abstract

Using holographic idea, we study the entanglement of purification in a field
theory with a critical point in intermediate and low temperature regime. This
theory includes temperature $T$ as well as chemical potential $\mu$. In the
intermediate regime, due to chemical potential, we observe that new terms
proportional to temperature square appear in the final result of entanglement
of purification or equivalently, apart from $T^{0}$ and $T^{4}$ terms in the
case of $\mu=0$, it contains the terms proportional to $T^{2}$. Our results
also indicate that the entanglement of purification, i.e. the correlation
between subsystems, can decrease or increase depending on the value of
$\frac{\mu}{T}$ when the other parameters are kept fixed. However, in the low
temperature limit, the correlation always decreases, comparing to the $\mu=0$
case, independent of the value of $\frac{\mu}{T}$ when the other parameters do
not alter. The existence of a critical point in the theory changes the
behavior of entanglement of purification in such a way that the entanglement
of purification experiences a maximum or minimum. Moreover, near the critical
point, we analytically show that the critical exponent is equal to 0.5 in both
regimes and also the term proportional to $T^{2}$ changes sign and becomes
negative in the intermediate regime.

###### Contents

  1. I Introduction
  2. II Review on the holographic background
  3. III Entanglement of purification
    1. III.1 Low temperature regime
    2. III.2 Intermediate temperature regime
  4. IV Critical exponent

##  I Introduction

How to calculate the dependence (correlation) between two subsystems has
always been an important question. In the case of pure state, described by
density matrix $\rho$, entanglement entropy, as a non-local observable in
quantum field theory, measures the quantum correlation between subsystem $A$
and its complementary, $\bar{A}$. The entanglement entropy of subsystem $A$ is
then defined as

$S_{A}=-{\rm{Tr}}(\rho_{A}\log\rho_{A}),$ (1)

where $S_{A}$ states how much entanglement exists between subsystems. The
reduced density matrix $\rho_{A}={\rm{Tr}}_{\bar{A}}(\rho)$ is obtained by
tracing over the degrees of freedom in the complementary subsystem. In the
case of mixed state, described by density matrix $\rho_{AB\equiv A\cup B}$,
the entanglement of purification (EoP) between two subsystems $A$ and $B$
measures the total (classical as well as quantum) correlation
Bhattacharyya:2019tsi ; arXiv:quant-ph/0202044v3 . In fact, the EoP is a
measure of correlation in terms of the entanglement of pure state. The EoP is
then defined as follows: consider a purification state
$|\psi\rangle_{ABA^{\prime}B^{\prime}}$ of a mixed state $\rho_{AB}$, that is
a pure state in an enlarged Hilbert space
${\cal{H}}_{A}\otimes{\cal{H}}_{B}\otimes{\cal{H}}_{A^{\prime}}\otimes{\cal{H}}_{B^{\prime}}$
with the following constraint

$\rho_{AB}={\rm{Tr}}_{A^{\prime}B^{\prime}}\big{[}\left(|\psi\rangle_{ABA^{\prime}B^{\prime}}\right)\left({}_{ABA^{\prime}B^{\prime}}\langle\psi|\right)\big{]}.$
(2)

Hence, the EoP is obtained by the minimal entanglement entropy over all
purifications $|\psi\rangle_{ABA^{\prime}B^{\prime}}$, i.e.

$E_{p}(\rho_{AB})=\underset{|\psi\rangle_{ABA^{\prime}B^{\prime}}}{\rm{min}}S_{AA^{\prime}},$
(3)

where $S_{AA^{\prime}}$ is the entanglement entropy corresponding to
$\rho_{AA^{\prime}}$ and
$\rho_{AA^{\prime}}={\rm{Tr}}_{BB^{\prime}}\big{[}\left(|\psi\rangle_{ABA^{\prime}B^{\prime}}\right)\left({}_{ABA^{\prime}B^{\prime}}\langle\psi|\right)\big{]}$.
This definition for the EoP reduces to the entanglement entropy of subsystem
$A$ if one considers the subsystem $B$ as complementary of subsystem $A$.

Gauge/gravity duality provides a remarkable connection between a classical
gravity in $d+1$ dimensions and a certain strongly coupled gauge theory in $d$
dimensions Maldacena:1997re . This duality has been applied to study different
areas of physics such as quantum chromodynamics phase transition, various
properties of quark-gluon plasma and condensed matter CasalderreySolana:2011us
; Hartnoll:2009sz . The gauge/gravity duality generally proposes an applicable
and simple geometrical interpretation on the gravity side for a complicated
phenomenon in the strongly coupled gauge theory. As an example, Ryu and
Takayanagi firstly proposed in Ryu:2006bv  that the entanglement entropy can
be computed from

$S_{A}=\frac{{\rm{Area}}(\Gamma_{A})}{4G_{N}},$ (4)

where $G_{N}$ is the $d$-dimensional Newton constant. $\Gamma_{A}$ is a
codimension-2 minimal surface, called RT-surface, whose boundary
$\partial_{A}\Gamma$ coincides with the boundary of the subregion $A$ on the
boundary of the gravity theory where, according to the gauge/gravity
dictionary the quantum field theory lives, i.e. $\partial_{A}\Gamma=\partial
A$. In fact, based on this duality in order to calculate the entanglement
entropy in a field theory we only need to compute an extremal surface in
gravity dual.

As an another example, in the context of information theory, a new quantity
which recently received a lot of interest in the gauge/gravity duality point
of view is EoP. As a matter of fact, it has been conjectured that the EoP is
holographically dual to entanglement wedge cross-section $E_{w}$ of
$\rho_{AB}$, as a measure of total correlation between $A$ and $B$, which is
defined Takayanagi:2017knl ; Nguyen:2017yqw

$E_{w}(\rho_{AB})=\frac{{\rm{Area}}(\Sigma_{AB}^{min})}{4G_{N}},$ (5)

where $\Sigma_{AB}^{min}$ is the minimal area surface in the entanglement
wedge $E_{w}(\rho_{AB})$ that ends on the RT-surface of $A\cup B$, the blue-
dashed line in figure 1. As a result, we have

$E_{p}(\rho_{AB})\equiv E_{w}(\rho_{AB}).$ (6)

It is also discussed in the literature that the entanglement wedge cross-
section $E_{w}$ of $\rho_{AB}$ may be identified with logarithmic negativity
Kudler-Flam:2018qjo , odd entropy Tamaoka:2018ned , entanglement distillation
Agon:2018lwq  and reflected entropy Dutta:2019gen . For more details, we refer
the interested reader to the original literature.

Figure 1: The gray region shows the entanglement wedge dual to $\rho_{AB}$.
The minimal surfaces, RT-surfaces, are denoted by $\Gamma$, the dashed curves.

In this paper, we are going to study the effect of chemical potential (and
temperature) on the EoP analytically. In other words, the gravity solution,
which is dual to the our state in the field theory, has charge as well as
mass. Moreover, since this theory enjoys a critical point we are able to
investigate the behavior of EoP near the critical point and obtain critical
exponent.

##  II Review on the holographic background

Since we would like to study a (3+1)-dimensional filed theory with a critical
point holographically, we start with the following 5-dimensional action
Gubser:1998jb

$S=\frac{1}{16\pi G_{N}^{(5)}}\int
d^{5}x\sqrt{-g}\left({\cal{R}}-\frac{f(\phi)}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\right),$
(7)

where $G_{N}^{(5)}$ is the 5-dimensional Newton constant. $g$ and ${\cal{R}}$
are the determinant of the metric and its corresponding Ricci scalar,
respectively. $F_{\mu\nu}$ is field strength of the gauge field and $\phi$
denotes dilaton field. The dilaton potential, $V(\phi)$, and the Maxwell-
Dilaton coupling, $f(\phi)$, are given by

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

Solving the equations of motion, one can find a solution, known as the 1RCBH
background, and it describes by the following metric

$\displaystyle
ds^{2}=e^{2A(z)}\bigg{(}-h(z)dt^{2}+d\vec{x}^{2}\bigg{)}+\frac{e^{2B(z)}}{h(z)}\frac{R^{4}}{z^{4}}dz^{2},$
(9)

where

$\displaystyle\begin{split}A(z)&=\ln\left(\frac{R}{z}\right)+\frac{1}{6}\ln\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right),\cr
B(z)&=-\ln\left(\frac{R}{z}\right)-\frac{1}{3}\ln\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right),\cr
h(z)&=1-\frac{M^{2}z^{4}}{R^{6}(1+\frac{Q^{2}z^{2}}{R^{4}})}.\cr\phi(z)&=-\sqrt{\frac{2}{3}}\ln\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right),\cr
A_{t}(z)&=\frac{MQz_{H}^{2}}{R^{4}\left(1+\frac{Q^{2}z_{H}^{2}}{R^{4}}\right)}-\frac{MQz^{2}}{R^{4}\left(1+\frac{Q^{2}z^{2}}{R^{4}}\right)},\end{split}$
(10)

where $A_{t}(z)$ is the time component of the gauge field which has been
chosen to be zero on the horizon and regular on the boundary. The above metric
describes a charged black hole background. $M$ and $Q$ are the black hole mass
and charge, respectively and $R$ is the $AdS$ radius supposed to be one
hereafter. The 3+1-dimensional spacetime, which the dual field theory lives
on, denotes by $(t,\vec{x}=(x_{1},x_{2},x_{3}))$ and $z$ is the radial bulk
coordinate and the strongly coupled field theory lives on the boundary at
$z\to 0$. The location of the black hole horizon is obtained from $h(z_{H})=0$
as follows

$\displaystyle z_{H}=\sqrt{\frac{Q^{2}+\sqrt{Q^{4}+4M^{2}}}{2M^{2}}}.$ (11)

The temperature $T$ and chemical potential $\mu$ of dual field theory are
given by

$\displaystyle\begin{split}T&=\frac{1}{2\pi
z_{H}}\left(\frac{2+Q^{2}z_{H}^{2}}{\sqrt{1+Q^{2}z_{H}^{2}}}\right),\cr\mu&=\frac{Q}{\sqrt{1+Q^{2}z_{H}^{2}}}.\end{split}$
(12)

The field theory, which is dual to metric (9) in the limit of zero charge, is
clearly conformal. One therefore expects that all physical quantities can be
written as a function of dimensionless parameter $\mu/T$. It is then shown
that the phase diagram of the dual field theory is simple and can be described
by a one-dimensional line which ends at a critical point, i.e.
$\mu/T=\pi/\sqrt{2}$ DeWolfe:2011ts ; Finazzo:2016psx . In other words, using
(12), one easily finds

$\displaystyle\xi^{\frac{1}{2}}\equiv
Qz_{H}=\sqrt{2}\left(\frac{1\pm\sqrt{1-\left(\frac{\mu/T}{\pi/\sqrt{2}}\right)^{2}}}{\frac{\mu/T}{\pi/\sqrt{2}}}\right).$
(13)

Since $Qz_{H}$ is real and nonnegative, the above equation indicates that
$\mu/T\in\left[0,\pi/\sqrt{2}\right]$. As a result, for a given value of
$\mu/T$, there are obviously two distinct values for $Qz_{H}$, which
parametrize two different branches of solutions. The thermodynamics of metric
(9) has been extensively discussed, for instance see DeWolfe:2011ts ;
Finazzo:2016psx ; Ebrahim:2020qif , and it is shown that the branch with
$Qz_{H}<\sqrt{2}\ (\mu/T<\pi/\sqrt{2})$ is stable one or in other words we
choose the minus sign in (13).

##  III Entanglement of purification

In the previous section, we reviewed the background which is dual to the
strongly coupled field theory with a critical point. Investigating properties
of various physical observables near the critical point is an attractive
question. Observables such as entanglement entropy, mutual information, and
quasinormal mode have been already discussed Ebrahim:2020qif ; Finazzo:2016psx
and in the following, we are interested in studying the EoP near a critical
point. Luckily, similar to the case of entanglement entropy, EoP has also a
simple holographic counterpart Takayanagi:2017knl ; Nguyen:2017yqw  and we in
fact need to calculate a minimal area (divided by 4$G_{N}^{(5)}$) which is
defined as the entanglement wedge cross section of a mixed state.

In order to compute the EoP holographically, we use background (9) and
consider a symmetric configuration of two strips at a given time slice as
subsystems $A$ and $B$, i.e. two parallel strips with equal widths $l$
extended along $x_{2}$ and $x_{3}$ directions with length
$L(\rightarrow\infty)$ and separated by a distance $l^{\prime}$, let’s say in
the $x_{1}$ direction, see figure 1. In this figure, one clearly sees that
$\Sigma_{AB}^{min}$ runs along the radial direction and connect the turning
points of minimal surfaces $\Gamma_{l^{\prime}}$ and $\Gamma_{l^{\prime}+2l}$.
Applying the holographic prescription introduced in Takayanagi:2017knl ;
Nguyen:2017yqw , the area of this hypersurface turns out to be

$\displaystyle
E_{w}=\frac{L^{2}}{4G_{N}^{(5)}}\int_{z^{*}_{l^{\prime}}}^{z^{*}_{l^{\prime}+2l}}dz\frac{e^{2A(z)+B(z)}}{z^{2}\sqrt{h(z)}},$
(14)

where $z^{*}_{l^{\prime}}$ and $z^{*}_{l^{\prime}+2l}$ denote the turning
point of $\Gamma_{l^{\prime}}$ and $\Gamma_{l^{\prime}+2l}$, respectively. To
find $z^{*}_{l^{\prime}}$, we consider the following configuration

$\displaystyle x_{1}(z)\equiv
x(z)\in[-\frac{l^{\prime}}{2},\frac{l^{\prime}}{2}]\ ,\ \ \ \ \ \ \
x_{i}\in[-\frac{L}{2},\frac{L}{2}]\ ,\ \ \ \ \ i=2,3,$ (15)

and by calculating the area of this configuration, we find

$\displaystyle
Area=2L^{2}\int_{0}^{z^{*}_{l^{\prime}}}dze^{3A(z)}\sqrt{x^{\prime}(z)^{2}+\frac{e^{2B(z)-2A(z)}}{z^{4}h(z)}}.$
(16)

Since there is no explicit $x(z)$ dependence in (16), the corresponding
Hamiltonian is constant and one can easily obtain

$\displaystyle\frac{l^{\prime}}{2}=\int_{0}^{z^{*}_{l^{\prime}}}dz\frac{e^{B(z)-A(z)}}{z^{2}h(z)}\frac{e^{3A(z^{*}_{l^{\prime}})}}{\sqrt{e^{6A(z)}-e^{A(z^{*}_{l^{\prime}})}}},$
(17)

where the constant is chosen to be
$\frac{e^{B(z)-A(z)}}{z^{2}\sqrt{h(z)}}\sqrt{e^{6A(z)}-e^{6A(z^{*}_{l^{\prime}})}}$.
Similar equation for $z^{*}_{l^{\prime}+2l}$ is obtained by replacing $l$ and
$z^{*}_{l^{\prime}}$ with $l^{\prime}+2l$ and $z^{*}_{l^{\prime}+2l}$ in above
equation, respectively. Substituting the explicit form of the metric
components (10) in (14) and (17), we have

$\displaystyle E_{p}\equiv
4G_{N}^{(5)}E_{w}=L^{2}\int_{z^{*}_{l^{\prime}}}^{z^{*}_{l^{\prime}+2l}}dz\frac{1}{z^{3}}\sqrt{1+\xi(\frac{z}{z_{H}})^{2}}\left(1+\xi(\frac{z}{z_{H}})^{2}-(1+\xi)(\frac{z}{z_{H}})^{4}\right)^{-\frac{1}{2}},$
(18)
$\displaystyle\frac{l^{\prime}}{2}=\int_{0}^{z^{*}_{l^{\prime}}}dz\left(1+\xi(\frac{z}{z_{H}})^{2}-(1+\xi)(\frac{z}{z_{H}})^{4}\right)^{-\frac{1}{2}}(\frac{z}{z^{*}_{l^{\prime}}})^{3}\left(\frac{1+\xi(\frac{z^{*}_{l^{\prime}}}{z_{H}})^{2}}{1+\xi(\frac{z}{z_{H}})^{2}}\right)^{\frac{1}{2}}\left[1-(\frac{z}{z^{*}_{l^{\prime}}})^{6}\left(\frac{1+\xi(\frac{z^{*}_{l^{\prime}}}{z_{H}})^{2}}{1+\xi(\frac{z}{z_{H}})^{2}}\right)\right]^{-\frac{1}{2}}.$
(19)

In order to calculate the above integrals, we need to use the following
(binomial and trinomial) expansions

$\displaystyle(1+x)^{r}$
$\displaystyle=\sum\limits_{n=0}^{\infty}\binom{r}{n}x^{n},\ \ \ \ \ \ \
|x|<1,$ (20a) $\displaystyle(1+x)^{-r}$
$\displaystyle=\sum\limits_{n=0}^{\infty}(-1)^{n}\binom{r+n-1}{n}x^{n},\ \ \ \
\ \ \ |x|<1,$ (20b) $\displaystyle(1+x+y)^{-r}$
$\displaystyle=\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^{k}(-1)^{k}\binom{r+k-1}{k}\binom{k}{n}x^{k-n}y^{n},\
\ \ \ \ \ \ |x+y|<1,$ (20c)

where $x$, $y$ and $r$ are real numbers and $r>0$. Using (20a) by identifying
$x=\xi(\frac{z}{z_{H}})^{2}$ and (20c) by identifying
$x=\xi(\frac{z}{z_{H}})^{2}$ and $y=-(1+\xi)(\frac{z}{z_{H}})^{4}$ and finally
by taking the integrals, it is straightforward to obtain

$\displaystyle\begin{split}E_{p}=\frac{L^{2}}{2}\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^{k}\sum\limits_{m=0}^{\infty}&\frac{\Gamma(k+\frac{1}{2})}{\Gamma(n+1)\Gamma(k-n+1)\Gamma(m+1)\Gamma(\frac{3}{2}-m)}\cr&\times\frac{(-1)^{k+n}\xi^{k-n+m}(1+\xi)^{n}}{2(n+k+m-1)}\frac{{z^{*}_{l^{\prime}+2l}}^{2(n+k+m-1)}-{z^{*}_{l^{\prime}}}^{2(n+k+m-1)}}{z_{H}^{2(n+k+m)}},\end{split}$
(21)

where the condition $n+k+m\neq 1$ must be satisfied. Moreover, one can check
that $|x|<1$ and $|x+y|<1$ for any allowable values of background parameters,
that is $\xi\in[0,2]$ and $z\ll z_{H}$. Similarly, identifying
$x=\xi(\frac{z}{z_{H}})^{2}$, $y=-(1+\xi)(\frac{z}{z_{H}})^{4}$ and using
(20c) and also identifying
$x=(\frac{z}{z^{*}_{l^{\prime}}})^{6}\left(\frac{1+\xi(\frac{z^{*}_{l^{\prime}}}{z_{H}})^{2}}{1+\xi(\frac{z}{z_{H}})^{2}}\right)$
and using (20b) and finally by taking the integrals, equation (19) leads to

$\displaystyle\begin{split}\frac{l^{\prime}}{2}=z^{*}_{l^{\prime}}\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^{k}\sum\limits_{m=0}^{\infty}\sum\limits_{j=0}^{\infty}&\frac{\Gamma(k+\frac{1}{2})\Gamma(j+m+\frac{1}{2})\Gamma(2+3j+k+n)}{\Gamma(n+1)\Gamma(k-n+1)\Gamma(3+3j+k+n+m)\Gamma(j+1)}\cr&\times(-1)^{k+n}\xi^{k-n+m}(1+\xi)^{n}\left[1+\xi(\frac{z^{*}_{l^{\prime}}}{z})^{2}\right]^{-m}(\frac{z^{*}_{l^{\prime}}}{z_{H}})^{2(m+n+k)},\end{split}$
(22)

and then it is easy to check that the conditions required to use the binomial
and trinomial expansions for negative powers are satisfied as follows

$\displaystyle\left(\frac{z}{z^{*}}\right)^{6}\left(\frac{1+\xi\left(\frac{z^{*}}{z_{H}}\right)^{2}}{1+\xi\left(\frac{z}{z_{H}}\right)^{2}}\right)<1\
\ \ \ \ \&\ \ \ \
\frac{\xi\left(\frac{z^{*}}{z_{H}}\right)^{2}}{1+\xi\left(\frac{z^{*}}{z_{H}}\right)^{2}}\left(1-\frac{z^{2}}{{z^{*}}^{2}}\right)<1.$
(23)

Now, we have to solve (22) for $z^{*}_{l^{\prime}}$ and use it in (21) to get
the EoP in terms of $l^{\prime}$. Unfortunately, it is not possible to solve
(22) analytically, and hence, from now on, we will mainly consider only some
orders of expansion in different regimes such as low ($Tl\ \&\ Tl^{\prime}\ll
1$) and intermediate ($Tl^{\prime}\ll 1\ll Tl$) temperature regimes.

###  III.1 Low temperature regime

In the limit of low temperature, i.e. $Tl\ \&\ Tl^{\prime}\ll 1$ or
equivalently ${z^{*}}_{l^{\prime}}\ \&\ {z^{*}}_{l^{\prime}+2l}\ll z_{H}$, the
extremal surfaces $\Gamma_{l^{\prime}}$ and $\Gamma_{l^{\prime}+2l}$ are
restricted to be near the boundary and therefore the leading contribution to
the EoP comes from the $AdS$ boundary. Finite temperature corrections, in the
presence of chemical potential, appear as sub-leading terms corresponding to
the deviation of the bulk geometry from pure $AdS$ and we will compute them
perturbatively. Thus, we expand (22) up to 4th order in
$\frac{z^{*}_{l^{\prime}}}{z_{H}}$ and finally have

$\displaystyle
l^{\prime}=z^{*}_{l^{\prime}}\left\\{a_{1}-\frac{a_{1}\xi}{6}(\frac{z^{*}_{l^{\prime}}}{z_{H}})^{2}+\left[\frac{a_{2}(1+\xi)}{2}+\frac{a_{3}\xi^{2}}{24}\right](\frac{z^{*}_{l^{\prime}}}{z_{H}})^{4}\right\\}+\mathcal{O}(\frac{z^{*}_{l^{\prime}}}{z_{H}})^{6},$
(24)

where $a_{1}$, $a_{2}$ and $a_{3}$ are given by

$\displaystyle\begin{split}a_{1}&=\frac{3\sqrt{\pi}\Gamma(\frac{5}{3})}{\Gamma(\frac{1}{6})},\
\ \ \ \ \ \
a_{2}=\frac{\sqrt{\pi}\Gamma(\frac{7}{3})}{4\Gamma(\frac{11}{6})},\cr
a_{3}&=\frac{3}{\sqrt{\pi}}\left[\Gamma(\frac{5}{6})\Gamma(\frac{5}{3})-\frac{3}{5}\Gamma(\frac{7}{6})\Gamma(\frac{7}{3})\right]-\frac{1}{70}{}_{3}F_{2}\left(\frac{3}{2},\frac{5}{3},\frac{7}{3};\frac{8}{3},\frac{10}{3};1\right).\end{split}$
(25)

Solving (24) for $z^{*}_{l^{\prime}}$ perturbatively up to 4th order in
$\frac{l^{\prime}}{z_{H}}$, we obtain

$\displaystyle
z^{*}_{l^{\prime}}=\frac{l}{a_{1}}\left\\{1+\frac{\xi}{6a_{1}^{2}}(\frac{l^{\prime}}{z_{H}})^{2}+\frac{1}{2a_{1}^{4}}\left[\frac{\xi^{2}}{6}(1-\frac{a_{3}}{2a_{1}})-\frac{a_{2}}{a_{1}}(1+\xi)\right](\frac{l^{\prime}}{z_{H}})^{4}\right\\}+\mathcal{O}(\frac{l^{\prime}}{z_{H}})^{6}.$
(26)

In the same way, one can find an expression for ${z^{*}}_{l^{\prime}+2l}$. It
is easy to see that its functionality is the same as (26) and one only needs
to substitute $z^{*}_{l^{\prime}}$ and $l^{\prime}$ with
${z^{*}}_{l^{\prime}+2l}$ and $l^{\prime}+2l$, respectively. Replacing
$z^{*}_{l^{\prime}}$ and ${z^{*}}_{l^{\prime}+2l}$ in (21), we obtain

$\displaystyle E_{p}=\frac{L^{2}a_{1}^{2}}{2}\left\\{\frac{1}{l^{\prime
2}}-\frac{1}{(l^{\prime}+2l)^{2}}\right\\}-\frac{2L^{2}\pi^{4}}{a_{1}^{2}}\left[\frac{\xi^{2}}{12}(\frac{a_{3}}{a_{1}}-1)+(\frac{a_{2}}{a_{1}}-\frac{1}{2})(1+\xi)\right]\left[\frac{(1+\xi)^{2}}{(1+\frac{\xi}{2})^{4}}\right]l(l^{\prime}+l)T^{4}+{...},$
(27)

where $\xi$ in terms of $\mu/T$ has been introduced in (13) and as a result
$E_{p}$ is a complicated function of $\mu/T$. We would like to emphasize a few
points about our result:

Figure 2: $\gamma(\xi)$ in terms of $\frac{\mu}{T}$.

  * •

The first term in the above equation is the leading term corresponding to the
EoP of our configuration in zero temperature field theory. Obviously, this
term is positive.

  * •

The terms which are proportional to $\xi^{k},\ k>0$ are coming from the effect
of non-zero chemical potential in the field theory. Note also that the
constant term $a_{3}$ only appears in this case.

  * •

The term $\frac{a_{2}}{a_{1}}-\frac{1}{2}>0$ is always positive in the
reliable regime of parameters and therefore temperature, for $\xi=0$,
decreases the correlation between our considered subsystems. Obviously, the
higher temperature, the bigger correction.

  * •

The term
$\gamma(\xi)=\left(\frac{\xi^{2}}{12}(\frac{a_{3}}{a_{1}}-1)+(\frac{a_{2}}{a_{1}}-\frac{1}{2})\xi\right)\frac{(1+\xi)^{2}}{(1+\frac{\xi}{2})^{4}}>0$
is always positive and similar to the case of finite temperature, non-zero
chemical potential decreases the correlation. We plot function $\gamma(\xi)$
in terms of $\frac{\mu}{T}$ in figure 2. It is clearly seen that this
correction depends on the value of $\frac{\mu}{T}$. Moreover, it shows that
for arbitrary value of temperature one may find two distinct states, labeled
with two different values of $\frac{\mu}{T}$, with the same value of
$\gamma(\xi)T^{4}$. In the case of zero chemical potential, there is one-to-
one correspondence between temperature and its related correction since it is
proportional to $T^{4}$. One may speculate the mentioned behavior happens
because of existence of critical point.

  * •

In the limit of $\xi=0$, we reproduce the previous results, see
BabaeiVelni:2019pkw . It seems that the minimum value of correction term
happens at $\xi=0$.

###  III.2 Intermediate temperature regime

In this subsection, we study another interesting limit called intermediate
regime. The limit is defined by $Tl^{\prime}\ll 1\ll Tl$ or equivalently
${z^{*}}_{l^{\prime}}\ll z_{H}$ and ${z^{*}}_{l^{\prime}+2l}\to z_{H}$. In
fact, the extremal surface $\Gamma_{l^{\prime}}$ is restricted to be near the
boundary and, on the contrary, the turning point of the extremal surface
$\Gamma_{2l+l^{\prime}}$ approaches the horizon, i.e.
${z^{*}}_{l^{\prime}+2l}\to z_{H}$. We define $u=\frac{z}{z_{H}}$ and rewrite
(18) as follows

$\displaystyle
E_{p}=\frac{L^{2}}{z_{H}^{2}}\int_{u_{l^{\prime}}}^{u_{l^{\prime}+2l}}du\frac{\sqrt{1+\xi
u^{2}}}{u^{3}\sqrt{1-u^{2}}\sqrt{1+(1+\xi)u^{2}}},$ (28)

where $u_{l^{\prime}}=\frac{z^{*}_{l^{\prime}}}{z_{H}}$ and
$u_{l^{\prime}+2l}=\frac{z^{*}_{l^{\prime}+2l}}{z_{H}}$. Therefore, the
intermediate temperature limit is equivalent to $u_{l^{\prime}}\ll 1$ and
$u_{l^{\prime}+2l}\to 1$. Using (28) and (20b) by identifying $x=-u^{2}$, we
get

$\displaystyle\begin{split}E_{p}&=\frac{L^{2}}{z_{H}^{2}}\sum\limits_{n=0}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{\pi}\Gamma(n+1)}\int_{u_{l^{\prime}}}^{u_{l^{\prime}+2l}}du\frac{u^{2n-3}\sqrt{1+\xi
u^{2}}}{\sqrt{1+(1+\xi)u^{2}}}\cr&=\frac{L^{2}}{2z_{H}^{2}}\int_{u_{l^{\prime}}}^{u_{l^{\prime}+2l}}du\frac{\left(2u^{-3}-u^{-1}\right)\sqrt{1+\xi
u^{2}}}{\sqrt{1+(1+\xi)u^{2}}}+\frac{L^{2}}{\sqrt{\pi}z_{H}^{2}}\sum\limits_{n=2}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}\int_{u_{l^{\prime}}}^{u_{l^{\prime}+2l}}du\frac{u^{2n-3}\sqrt{1+\xi
u^{2}}}{\sqrt{1+(1+\xi)u^{2}}}\end{split}$ (29)

Fortunately, we are able to take the above integrals analytically and it is
thus easy to find

$\displaystyle\begin{split}E_{p}&=\frac{L^{2}}{2z_{H}^{2}}\Bigg{\\{}-\frac{\sqrt{1+\xi
u_{l^{\prime}+2l}^{2}}\sqrt{1+(1+\xi)u_{l^{\prime}+2l}^{2}}}{u_{l^{\prime}+2l}^{2}}+\frac{\xi
u_{l^{\prime}+2l}^{2}}{2}F_{1}\left(1;\frac{1}{2},\frac{1}{2};2;-\xi
u_{l^{\prime}+2l}^{2},-(1+\xi)u_{l^{\prime}+2l}^{2}\right)\cr&+\sum\limits_{n=2}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{\pi}(n-1)\Gamma(n+1)}u_{l^{\prime}+2l}^{2n-2}F_{1}\left(n-1;-\frac{1}{2},\frac{1}{2};n;-\xi
u_{l^{\prime}+2l}^{2},-(1+\xi)u_{l^{\prime}+2l}^{2}\right)\cr&+\frac{\sqrt{1+\xi
u_{l^{\prime}}^{2}}\sqrt{1+(1+\xi)u_{l^{\prime}}^{2}}}{u_{l^{\prime}}^{2}}-\frac{\xi
u_{l^{\prime}}^{2}}{2}F_{1}\left(1;\frac{1}{2},\frac{1}{2};2;-\xi
u_{l^{\prime}}^{2},-(1+\xi)u_{l^{\prime}}^{2}\right)\cr&-\sum\limits_{n=2}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{\pi}(n-1)\Gamma(n+1)}u_{l^{\prime}}^{2n-2}F_{1}\left(n-1;-\frac{1}{2},\frac{1}{2};n;-\xi
u_{l^{\prime}}^{2},-(1+\xi)u_{l^{\prime}}^{2}\right)\Bigg{\\}},\end{split}$
(30)

where $F_{1}(a;b_{1},b_{2};c;x,y)$ is the Appell hypergeometric function of
two variables. For $|x|<1$ and $|y|<1$, the function $F_{1}$ is defined by the
double series

$\displaystyle
F_{1}(a;b_{1},b_{2};c;x,y)=\sum\limits_{n=0}^{\infty}\sum\limits_{m=0}^{\infty}\frac{(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}m!n!}x^{m}y^{n},$
(31)

where $(x)_{n}=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}$ is the Pochhammer symbol.
For other values of $x$ and $y$, the function $F_{1}$ is given by the
following series

$\displaystyle
F_{1}(a;b_{1},b_{2};c;x,y)=\sum\limits_{r=0}^{\infty}\frac{(a)_{r}(b_{1})_{r}(b_{2})_{r}(c-a)_{r}}{(c+r-1)_{r}(c)_{2r}r!}x^{r}y^{r}{}_{2}F_{1}\left(a+r,b_{1}+r;c+2r;x\right){}_{2}F_{1}\left(a+r,b_{2}+r;c+2r;y\right),$
(32)

where ${}_{2}F_{1}$ is the hypergeometric function. We should now check the
convergence of the above series. For large $n$, the infinite series in the
second line in (30) goes as

$\displaystyle\frac{u_{l^{\prime}+2l}^{2n-2}F_{1}\left(n;-\frac{1}{2},\frac{1}{2};n;-\xi
u_{l^{\prime}+2l}^{2},-(1+\xi)u_{l^{\prime}+2l}^{2}\right)}{n^{\frac{3}{2}}}=\frac{u_{l^{\prime}+2l}^{2n-2}}{n^{\frac{3}{2}}}\frac{\sqrt{1+\xi
u_{l^{\prime}+2l}^{2}}}{\sqrt{1+(1+\xi)u_{l^{\prime}+2l}^{2}}}.$ (33)

In the limit of $u_{l^{\prime}+2l}\to 1$, this term behaves as
$n^{-\frac{3}{2}}$. Therefore, the series is convergent and we can safely take
the mentioned limit. For another part of the intermediate regime, i.e.
$u_{l^{\prime}}\ll 1$, we expand (30) up to second order in $u_{l^{\prime}}$
and using (26) for $z^{*}_{l^{\prime}}$, one gets

$\displaystyle\begin{split}E_{p}=\frac{L^{2}}{2}\bBigg@{3}\\{&\frac{a_{1}^{2}}{l^{\prime
2}}+\Bigg{[}\sum\limits_{n=2}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{\pi}(n-1)\Gamma(n+1)}F_{1}\left(n-1;-\frac{1}{2},\frac{1}{2};n;-\xi,-(1+\xi)\right)+\frac{1}{2}-\sqrt{(1+\xi)(2+\xi)}\cr&+\frac{\xi}{2}\left(\frac{4}{3}+F_{1}\left(1;\frac{1}{2},\frac{1}{2};2;-\xi,-(1+\xi)\right)\right)\Bigg{]}\frac{(1+\xi)\pi^{2}}{(1+\frac{\xi}{2})^{2}}T^{2}\cr&+\frac{\pi^{4}}{a_{1}^{2}}\left[\frac{\xi^{2}}{12}\left(\frac{a_{3}}{a_{1}}-1\right)+\left(\frac{a_{2}}{a_{1}}-\frac{1}{2}\right)(1+\xi)\right]\frac{(1+\xi)^{2}}{(1+\frac{\xi}{2})^{4}}l^{\prime
<EMAIL_ADDRESS>(34)

First of all, $l$ does not play any role in our final result. It seems
reasonable since in numerical analysis Amrahi:2020jqg  it was shown that for
large enough $l$, compared to $T^{-1}$, the EoP is independent of the length
of subsystems. Furthermore, our result shows that in the limit of zero
chemical potential, i.e. $\xi=0$, the terms which are proportional to $T^{2}$
disappear. Thus, due to non-zero $\mu$, a new contribution proportional to
temperature square becomes manifest and we call them non-perturbative
corrections since the temperature does not need to be small. However, the last
term is a perturbative effect since by assumption $l^{\prime}T$ is small
enough. Another important point is that in the presence of chemical potential
the correlation between two subsystems is sensitive to the value of
$l^{\prime}$ and temperature and not $l^{\prime}T$. Evidently, in the limit of
$\xi=0$, we reproduce the results obtained for the thermal case, see
BabaeiVelni:2019pkw .

The last term in (34), perturbative term, is positive. Then the terms which
are proportional to $T^{2}$ is written as $\alpha(\xi)T^{2}$, where

$\displaystyle\begin{split}\alpha(\xi)&=\Bigg{[}\sum\limits_{n=2}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{\pi}(n-1)\Gamma(n+1)}F_{1}\left(n-1;-\frac{1}{2},\frac{1}{2};n;-\xi,-(1+\xi)\right)+\frac{1}{2}-\sqrt{(1+\xi)(2+\xi)}\cr&+\frac{\xi}{2}\left(\frac{4}{3}+F_{1}\left(1;\frac{1}{2},\frac{1}{2};2;-\xi,-(1+\xi)\right)\right)\Bigg{]}\frac{(1+\xi)\pi^{2}}{(1+\frac{\xi}{2})^{2}}.\end{split}$
(35)

In figure 3, we plot $\alpha(\xi)$ and this figure indicates that
$\alpha(\xi)$ is finite and can be positive or negative. As a result, in the
limits we are considering here the term proportional to $l^{\prime-1}$ in (34)
is dominant and temperature, depending on the value of chemical potential,
increases or decreases the correlation between two subsystems.

Figure 3: $\alpha(\xi)$ in terms of $\frac{\mu}{T}$. The infinity in (34) is
considered to be $n=40000$.

##  IV Critical exponent

As it was mentioned, the field theory we are investigating in this paper
enjoys a critical point at $\frac{\mu}{T}^{*}=\frac{\pi}{\sqrt{2}}$. The
behavior of different observables near the critical point has been studied in
the literature DeWolfe:2011ts ; Finazzo:2016psx ; Ebrahim:2017gvk ;
Ebrahim:2018uky ; Ebrahim:2020qif ; Amrahi:2020jqg  and it reveals that this
behavior can be considered as $(\frac{\mu}{T}^{*}-\frac{\mu}{T})^{-\theta}$
where $\theta$ is the dynamical exponent and it is obtained to be 0.5. As
matter of fact, it was shown that various quantities, such as entanglement
entropy, mutual information, equilibration time and relaxation time, remain
finite as one moves towards the critical point but their slopes diverge at the
critical point and behave like $(\frac{\mu}{T}^{*}-\frac{\mu}{T})^{-\theta}$
Ebrahim:2017gvk ; Ebrahim:2018uky ; Ebrahim:2020qif . Now, we like to check
the behavior of EoP near this critical point for two different regimes
discussed earlier.

In order to find the slope of EoP near critical point, we start with

$\displaystyle\frac{dE_{p}}{d(\frac{\mu}{T})}=\frac{dE_{p}}{d\xi}\frac{d\xi}{d(\frac{\mu}{T})},$
(36)

where $\frac{d\xi}{d(\frac{\mu}{T})}$ can be easily found from (13).
$\frac{dE_{p}}{d\xi}$ is obtained by taking the derivative of (27) for low
temperature and of (34) for intermediate temperature. Therefore, after
expanding in power of $(\frac{\mu}{T}^{*}-\frac{\mu}{T})$, we have

Low temperature:

$\displaystyle\begin{split}\frac{dE_{p}}{d(\frac{\mu}{T})}=\frac{L^{2}\pi^{\frac{7}{2}}l(l^{\prime}+l)T^{4}}{2^{\frac{1}{4}}a_{1}^{3}}(a_{1}-a_{3})\left(\frac{\mu}{T}^{*}-\frac{\mu}{T}\right)^{-\frac{1}{2}}+{\cal{O}}\left((\frac{\mu}{T}^{*}-\frac{\mu}{T})^{0}\right)\end{split}$
(37)

Intermediate temperature:

$\displaystyle\frac{dE_{p}}{d(\frac{\mu}{T})}=C\left(\frac{\mu}{T}^{*}-\frac{\mu}{T}\right)^{-\frac{1}{2}}+\mathcal{O}\left((\frac{\mu}{T}^{*}-\frac{\mu}{T})^{0}\right),$
(38)

where

$\displaystyle\begin{split}C&=\frac{L^{2}}{2}\bBigg@{3}\\{\sum\limits_{n=2}^{\infty}\frac{-\pi\Gamma\left(n+\frac{1}{2}\right)T^{2}}{2^{\frac{5}{4}}(n-1)n\Gamma(n+1)}\cr&\times\left[nF_{1}\left(n-1;\frac{-1}{2},\frac{1}{2};n;-2,-3\right)+3(n-1)F_{1}\left(n;\frac{-1}{2},\frac{3}{2};n+1;-2,-3\right)-F_{1}\left(n;\frac{1}{2},\frac{1}{2};n+1;-2,-3\right)\right]\cr&+\frac{\pi^{\frac{3}{2}}T^{2}}{12\times
2^{\frac{1}{4}}}\left[13+9\sqrt{3}+12F_{1}\left(1;\frac{1}{2},\frac{1}{2};2;-2,-3\right)-9F_{1}\left(2;\frac{1}{2},\frac{3}{2};3;-2,-3\right)-9F_{1}\left(2;\frac{3}{2},\frac{1}{2};3;-2,-3\right)\right]\cr&-\frac{\pi^{\frac{7}{2}}l^{\prime
2}T^{4}}{12\times
<EMAIL_ADDRESS>(39)

As a result, our simple calculation shows that the slope of EoP diverges near
the critical point like other observables and dynamical critical exponent is
equal to 0.5.

## References

  * (1) A. Bhattacharyya, A. Jahn, T. Takayanagi and K. Umemoto, “Entanglement of Purification in Many Body Systems and Symmetry Breaking,” Phys. Rev. Lett. 122 (2019) no.20, 201601 [arXiv:1902.02369 [hep-th]].
  * (2) B. M. Terhal, M. Horodecki, D. W. Leung and D. P. DiVincenzo, “The entanglement of purification,” J. Math.Phys. 43, 4286 (2002) [arXiv:quant-ph/0202044v3].
  * (3) J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38 (1999), 1113-1133 [arXiv:hep-th/9711200 [hep-th]].
  * (4) J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U. A. Wiedemann, “Gauge/String Duality, Hot QCD and Heavy Ion Collisions,” book:Gauge/String Duality, Hot QCD and Heavy Ion Collisions. Cambridge, UK: Cambridge University Press, 2014 5 doi:10.1017/CBO9781139136747 [arXiv:1101.0618 [hep-th]].
  * (5) S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics,” Class. Quant. Grav. 26 (2009), 224002 [arXiv:0903.3246 [hep-th]].
  * (6) S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006), 181602 [arXiv:hep-th/0603001 [hep-th]].
  * (7) T. Takayanagi and K. Umemoto, “Entanglement of purification through holographic duality,” Nature Phys. 14, no. 6, 573 (2018) [arXiv:1708.09393 [hep-th]].
  * (8) P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel and B. Swingle, “Entanglement of purification: from spin chains to holography,” JHEP 1801, 098 (2018) [arXiv:1709.07424 [hep-th]].
  * (9) J. Kudler-Flam and S. Ryu, “Entanglement negativity and minimal entanglement wedge cross sections in holographic theories,” Phys. Rev. D 99 (2019) no.10, 106014 [arXiv:1808.00446 [hep-th]].
  * (10) K. Tamaoka, “Entanglement Wedge Cross Section from the Dual Density Matrix,” Phys. Rev. Lett. 122 (2019) no.14, 141601 [arXiv:1809.09109 [hep-th]].
  * (11) C. A. Agón, J. De Boer and J. F. Pedraza, “Geometric Aspects of Holographic Bit Threads,” JHEP 05 (2019), 075 [arXiv:1811.08879 [hep-th]].
  * (12) S. Dutta and T. Faulkner, “A canonical purification for the entanglement wedge cross-section,” [arXiv:1905.00577 [hep-th]].
  * (13) S. S. Gubser, “Thermodynamics of spinning D3-branes,” Nucl. Phys. B 551 (1999), 667-684 [arXiv:hep-th/9810225 [hep-th]]; K. Behrndt, M. Cvetic and W. A. Sabra, “Nonextreme black holes of five-dimensional N=2 AdS supergravity,” Nucl. Phys. B 553 (1999), 317-332 [arXiv:hep-th/9810227 [hep-th]]; P. Kraus, F. Larsen and S. P. Trivedi, “The Coulomb branch of gauge theory from rotating branes,” JHEP 03 (1999), 003 [arXiv:hep-th/9811120 [hep-th]]; R. G. Cai and K. S. Soh, Mod. Phys. Lett. A 14 (1999), 1895-1908 doi:10.1142/S0217732399001966 [arXiv:hep-th/9812121 [hep-th]]; M. Cvetic and S. S. Gubser, “Phases of R charged black holes, spinning branes and strongly coupled gauge theories,” JHEP 04 (1999), 024 [arXiv:hep-th/9902195 [hep-th]]; M. Cvetic and S. S. Gubser, “Thermodynamic stability and phases of general spinning branes,” JHEP 07 (1999), 010 [arXiv:hep-th/9903132 [hep-th]].
  * (14) O. DeWolfe, S. S. Gubser and C. Rosen, “Dynamic critical phenomena at a holographic critical point,” Phys. Rev. D 84 (2011), 126014 [arXiv:1108.2029 [hep-th]].
  * (15) S. I. Finazzo, R. Rougemont, M. Zaniboni, R. Critelli and J. Noronha, “Critical behavior of non-hydrodynamic quasinormal modes in a strongly coupled plasma,” JHEP 01 (2017), 137 [arXiv:1610.01519 [hep-th]].
  * (16) H. Ebrahim and G. M. Nafisi, “Holographic Mutual Information and Critical Exponents of the Strongly Coupled Plasma,” Phys. Rev. D 102 (2020) no.10, 106007 [arXiv:2002.09993 [hep-th]].
  * (17) K. Babaei Velni, M. R. Mohammadi Mozaffar and M. H. Vahidinia, “Some Aspects of Entanglement Wedge Cross-Section,” JHEP 05 (2019), 200 [arXiv:1903.08490 [hep-th]].
  * (18) H. Ebrahim and M. Ali-Akbari, “Dynamically probing strongly-coupled field theories with critical point,” Phys. Lett. B 783 (2018), 43-50 [arXiv:1712.08777 [hep-th]].
  * (19) H. Ebrahim, M. Asadi and M. Ali-Akbari, “Evolution of Holographic Complexity Near Critical Point,” JHEP 09 (2019), 023 [arXiv:1811.12002 [hep-th]].
  * (20) B. Amrahi, M. Ali-Akbari and M. Asadi, “Holographic Entanglement of Purification near a Critical Point,” Eur. Phys. J. C 80 (2020) no.12, 1152 [arXiv:2004.02856 [hep-th]].