Quantum correlation in quark-gluon medium

Chanyong Parka***e-mail<EMAIL_ADDRESS>and Jung Hun Leeb†††e-mail :
<EMAIL_ADDRESS>

${}^{a}\,$Department of Physics and Photon Science, Gwangju Institute of
Science and Technology, Gwangju 61005, Korea

${}^{b}\,$Department of Physics and Astronomy, Sejong University, Seoul 05006,
Korea

ABSTRACT

We study thermodynamics and quantum correlations of the string cloud geometry
whose field theory dual is the quark-gluon medium. We found the novel
universality of the entanglement entropy first law in the high quark density
limit. We also showed that a correlation function generally decreases as the
entanglement entropy of the background medium increases due to the screening
effect of the background. We study the UV and IR effects of the medium on
phase transition behaviour observed in the holographic mutual information
using both perturbative and numerical computations. Moreover, by numerical
computation, we show that in the IR region the critical length obtained from
the mutual information behaves similar to the correlation length of the two-
point function.

##  1 Introduction

In quantum many-body systems the entanglement entropy nicely characterizes the
degrees of freedom for a pure state. This quantity provides one of the useful
tools to figure out quantum phenomena both in high energy physics and in
condensed matter physics. In the light of AdS/CFT Maldacena (1999); Witten
(1998), the entanglement entropy can be interpreted as the geometric object
called the minimal surface, which extends to the bulk AdS space Ryu and
Takayanagi (2006a, b); Hubeny  _et al._ (2007). This duality gives a specific
relationship between geometrical objects and informational quantities of the
quantum system, like entanglement entropy Takayanagi (2012); Li and Takayanagi
(2011); Fujita  _et al._ (2011); Ogawa and Takayanagi (2011); Nozaki  _et al._
(2013); Miyaji  _et al._ (2015), mutual information Molina-Vilaplana and
Sodano (2011); Allais and Tonni (2012); Molina-Vilaplana (2014); Mukherjee and
Narayan (2014); Casini  _et al._ (2015); Ziogas (2015), and complexity Brown
_et al._ (2016a, b); Bernamonti  _et al._ (2019, 2020).

In this paper, we consider a deformed conformal field theory (CFT) with a
nontrivial dual gravity theory called the generalized string cloud geometry,
which consists of the black hole geometry and uniformly distributed open
strings. Originally, the string cloud model has been proposed as a holographic
dust model which was represented by the Einstein gravity coupled with a one-
dimensional string instead of point particles Stachel (1980); Chakrabortty
(2011); Chakrabortty and Dey (2016); Park (2020). Recently, this model has
been paid attention to understanding various properties associated with quark-
gluon plasma (QGP) Lee  _et al._ (2009); Park (2010); Jo  _et al._ (2010);
Park (2012); Park and Hun Lee (2018), a strongly coupled thermal state of
matter, produced in the collision experiment of heavy nuclei Chakrabortty
(2011); Chakrabortty and Dey (2016). Our aim in this work is to investigate
macroscopic and microscopic quantum correlations in the heavy quark-gluon
medium.

For a system away from its critical point, the entanglement entropy can have
nontrivial corrections associated with the subsystem size Calabrese and Cardy
(2004); Park (2015, 2016); Kundu and Pedraza (2016); Kim and Park (2017);
Narayanan  _et al._ (2019); Park  _et al._ (2018); Park and Hun Lee (2020);
Park and Lee . In the entanglement entropy study, the subsystem size can be
reinterpreted as the inverse of the energy scale. From the viewpoint of the
renormalization group (RG) flow, unlike the momentum space RG usually utilized
in a quantum field theory (QFT), the size of the subsystem, which is
homologous to the minimal surface, is related to the real space RG flow used
in the condensed matter physics. Generally, the RG flow crucially depends on
the deformation we consider. In this work, we take into account two different
deformations, the density of open strings and black hole mass.

We expect that the excitation increases the degrees of freedom of the
underlying theory due to the screening effect between quark-gluon medium Kim
_et al._ (2007). Due to this reason, the entanglement entropy with excitation
increases along the RG flow in the UV region. We clarify this feature by
utilizing holographic entanglement entropy of the deformed theory.

One of the most important properties of entanglement entropy is that it has
information over correlation for the given system. A well-known example is the
mutual information. This quantity consists of the linear combination of
entanglement entropy. Another candidate which measures the correlation between
two disjoint subsystems is the entanglement wedge cross section Takayanagi and
Umemoto (2018). This is defined as the minimal length of the cross section
which divides the entanglement wedge into two parts. However, since for these
quantities, there is a lack of direct comparison target in the dual field
theory, we need to consider a more intuitive quantity that can be constructed
in both the boundary and bulk theories.

The entanglement entropy can be regarded as a quantum correlation because it
quantifies measures of the correlation between two macroscopic quantum
subsystems. In the semiclassical approximation, the two-point function can be
defined by a Feynman path integral over the geodesic length of a massive
particle propagating in the AdS space Susskind and Witten (1998);
Balasubramanian and Ross (2000); Louko  _et al._ (2000). Therefore, at least,
for two-dimensional CFT the two-point function can be written in terms of the
length of the minimal surface computed in the entanglement entropy
calculation. More precisely, these two different measures are identically
described by a one-dimensional geodesic curve anchored at the boundary two
points. These boundary points coincide with the boundary of the entangling
region for the entanglement entropy. Furthermore, it is also possible to
identify two boundary points with the positions of two local operators for the
two-point function. For this reason, by using the holographic renormalization
technique Balasubramanian and Kraus (1999); DeWolfe and Freedman (2000); de
Boer (2001); Bianchi  _et al._ (2002); Skenderis (2002) we can take into
account a two-point correlation function of two local operators which
corresponds to a microscopic quantum correlation.

Entanglement entropy depends on the regularization scheme due to the UV
divergence structure. Mutual information can not only avoid this problem,
which is UV finite, but it also provides a useful tool to capture a first
order phase transition in the entanglement entropy Headrick (2010). In
particular, we survey the effect of the quark-gluon medium on phase transition
observed in holographic mutual information using both perturbative and
numerical computations.

The rest of this paper is organized as follows. In section 2, we review a
holographic model associated with the string cloud geometry and its
thermodynamic properties. In section 3, applying the holographic formulation
to deformed backgrounds, we study quantum corrections of the entanglement
entropy due to the effects of excitation and compute two-point correlations in
both UV and IR limits. In section 4, we extend the discussion of section 3 to
general d-dimensions. Further, we elaborate upon holographic mutual
information in section 5. Finally, we close this work with some concluding
remarks in section 6.

##  2 Holographic dual of quark-gluon plasma

In the holographic study, D-branes usually play an important role in making a
nontrivial gravity theory and in describing its dual field theory. When
$N_{c}$ $D3$-branes lie in a flat background spacetime on top of each other,
their gravitational backreaction allows a five-dimensional AdS space together
with a five-dimensional sphere in the near horizon limit. The radius of the
resulting AdS space is directly related to the number of $D3$-branes which on
the dual field theory side corresponds to the rank of the gauge group
$SU(N_{c})$. The five-dimensional AdS in the Poincare patch further
generalizes to an AdS black hole or black brane with an appropriate blackening
factor

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\left(-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+\delta_{ij}dx^{i}dx^{j}\right),$
(1)

where the blackening factor is expressed in terms of a black hole mass $m$

$\displaystyle f(z)=1-mz^{4}.$ (2)

According to the AdS/CFT correspondence, the pure AdS geometry becomes the
dual of a strongly interacting CFT at zero temperature, while the AdS black
hole is the dual of a CFT at finite temperature. Using the holographic
renormalization technique, the energy-momentum tensor of the dual CFT
automatically vanishes at zero temperature. At finite temperature, on the
other hand, the energy-momentum tensor is proportional to the black hole mass.
In this case, the energy-momentum tensor is proportional to $N_{c}^{2}$ and
traceless. This fact implies that the dual field theory of the black hole is
associated with the excitation of massless adjoint matter like gluons.
Therefore, we identify the black hole geometry with the gluon’s excitation of
the dual QFT.

Assuming that a probe $D3$-brane is located at the boundary, we can also take
into account open strings connecting $D3$-branes at the center an the probe
$D3$-brane. Since the end of an open string follows a fundamental
representation, it on the probe brane plays a role of a fundamental matter
like a quark with $N_{c}$ degrees of freedom. In this case, the fundamental
matter has an infinitely large mass. Combining the above fundamental and
adjoint matters allows us to investigate a quark-gluon plasma holographically.
For example, various time-dependent quantities in expanding universes was
studied in Park (2020, 2021). In this section, we holographically investigate
the entanglement entropy and two-point function of a local operator in the
quark-gluon plasma.

To describe fundamental matter living on the dual field theory side, we first
think of uniformly distributed open strings in a $(d+1)$-dimensional AdS
space. The corresponding gravity theory can be described by the following
action

$\displaystyle S=\frac{1}{16\pi G}\int d^{d+1}X\sqrt{-g}\left({\cal
R}-2\Lambda\right)+{\cal T}\sum_{i=1}^{N}\int
d^{2}\xi\sqrt{-h}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu\nu},$
(3)

where $g_{\mu\nu}$ and $h_{\alpha\beta}$ indicate a $(d+1)$-dimensional bulk
metric and an induced metric on an open string, respectively. The first part
including a negative cosmological constant $\Lambda$ allows a
$(d+1)$-dimensional AdS space, whereas the second part represents the action
of open strings with a string tension, ${\cal T}$. This model has been known
as the string cloud model Stachel (1980); Letelier (1979); Gibbons  _et al._
(2001); Herscovich and Richarte (2010); Chakrabortty (2011); Chakrabortty and
Dey (2016). Assuming that $N$ open strings are uniformly distributed, the
above action leads to the following Einstein equation

$\displaystyle{\cal R}_{\mu\nu}-\frac{1}{2}{\cal R}g_{\mu\nu}+\Lambda
g_{\mu\nu}=T_{\mu\nu},$ (4)

with

$\displaystyle T_{\mu\nu}=-\frac{16\pi G{\cal T}N}{V_{d-1}}\
\frac{\sqrt{-h}}{\sqrt{-g}}\ h_{\alpha\beta}\
\partial^{\alpha}X_{\mu}\partial^{\beta}X_{\nu},$ (5)

where $V_{d-1}$ indicates a $(d-1)$-dimensional volume perpendicular to the
open string’s worldvolume. Taking a static gauge with $\xi^{0}=t$ and
$\xi^{1}=z$, the stress tensor of open strings reduces to

$\displaystyle T_{\mu\nu}=-\frac{{16\pi G\cal T}N}{V_{d-1}}\
\frac{\sqrt{-h}}{\sqrt{-g}}\ h_{\alpha\beta}\
\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}.$ (6)

Noting that $h_{\alpha\beta}=g_{\alpha\beta}$ for $\alpha,\beta=0,1$ in the
static gauge, $T_{\mu\nu}=0$ for $\mu,\nu\neq 0,1$.

The open string configuration we considered allows us to find the following
analytic solution

$ds^{2}=\frac{R^{2}}{z^{2}}\left(-f(z)dt^{2}+\frac{1}{f(z)}dz^{2}+\delta_{ij}dx^{i}dx^{j}\right),$
(7)

with a nontrivial metric factor

$f(z)=1-\tau z^{d-1}-mz^{d}.$ (8)

where $\tau$ is the uniformly distributed string cloud density and can be
written as

$\displaystyle\tau=\frac{32\pi G{\cal T}N}{(d-1)RV_{d-1}}>0,$ (9)

where we only consider the positive density $\tau>0$ for which the metric
solution satisfies the weak and dominant energy conditions Chakrabortty
(2011); Chakrabortty and Dey (2016). On the dual field theory side, one can
identify $m$ and $\tau$ with the excitation energy of adjoint matter (gluons)
and the density of fundamental matter (quarks), respectively.

Now, we investigate thermodynamics of the generalized string cloud geometry.
To do so, it is more convenient to rewrite the black mass in terms of $\tau$
and the horizon, $z_{h}$,

$\displaystyle m=\frac{1}{z_{h}^{d}}-\frac{\tau}{z_{h}}.$ (10)

Then, the Hawking temperature and Bekenstein-Hawking entropy read

$\displaystyle T_{H}$ $\displaystyle=$
$\displaystyle\frac{d-z_{h}^{d-1}\tau}{4\pi z_{h}},$ $\displaystyle S_{BH}$
$\displaystyle=$ $\displaystyle\frac{V_{d-1}}{4Gz_{h}^{d-1}}.$ (11)

Here, the positivity of temperature yields the upper bound of the black hole
horizon

$\displaystyle 0\leq z_{h}^{d-1}\leq\frac{d}{\tau}.$ (12)

Intriguingly, this black hole solution allows two horizons. Similar to a
charged black hole case, these two horizons become degenerate at zero
temperature and lead to an extremal limit. In the extremal limit, the
blackening factor has a double root like

$\displaystyle f(z)=\left(z_{h}-z\right)^{2}F(z),$ (13)

where $F(z)$ is regular even at the horizon. The existence of the extremal
limit becomes manifest due to the fact that the horizon at zero temperature is
located at a finite distance and, at the same time, $f(z_{h})$ and
$f^{\prime}(z_{h})$ automatically vanish. In the extremal limit, the near
horizon geometry reduces to $AdS_{2}\times R^{d-1}$ similar to the charged
black hole case.

To see more details, we consider a three-dimensional generalized string cloud
geometry

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\left(-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+dx^{2}\right),$
(14)

with

$f(z)=1-\tau z-mz^{2}.$ (15)

Denoting an event horizon as $z_{h}$, the black hole mass, as mentioned
before, is rewritten as

$\displaystyle m=\frac{1}{z_{h}^{2}}-\frac{\tau}{z_{h}}.$ (16)

Then, the blackening factor

$\displaystyle f(z)=1-\tau z+\left(\tau
z_{h}-1\right)\frac{z^{2}}{z_{h}^{2}},$ (17)

allows two roots, the event horizon $z_{h}$ and inner horizon $z_{in}$,

$\displaystyle z_{in}=\frac{z_{h}}{\tau z_{h}-1}.$ (18)

Recalling that the boundary and the center of the string cloud geometry are
located at $z=0$ and $z=\infty$ respectively, the assumption that $z_{h}$ is
the outer horizon implies $z_{h}\leq z_{in}$. Since $\tau$ and $z_{h}$ must be
positive, this constraint restricts the range of $\tau z_{h}$ to be $0\leq\tau
z_{h}\leq 2$. Now, we summarize the possible horizons relying on the parameter
regions

  * •

For $\tau z_{h}<1$ the inner horizon does not exist, so there is only one
horizon at $z=z_{h}$.

  * •

For $1\leq\tau z_{h}<2$, there are two non-degenerate horizons and $z_{h}$
corresponds to the outer horizon.

  * •

For $\tau z_{h}=2$, the outer and inner horizons become degenerate, which
corresponds to an extremal limit.

Although the range of $\tau z_{h}>2$ is not consistent with our previous
assumption, there still exists a black hole solution. In this case, $z_{h}$
and $z_{in}$ exchange their role. In other words, $z_{in}$ plays a role of the
outer horizon instead of $z_{h}$. Since the string cloud geometry is invariant
under the exchange of the inner and outer horizons, a similar parameter
dependence discussed before occurs. From now on, we focus on only the case of
$0\leq\tau z_{h}\leq 2$.

A black hole solution generally maps to a thermal system satisfying the
thermodynamics law. The string cloud geometry we considered can be also
reinterpreted as a thermal system. The regularity of the above metric at the
outer horizon leads to the Hawking temperature

$\displaystyle T_{H}=\frac{2-\tau z_{h}}{4\pi z_{h}}.$ (19)

Using this relation, we can also represent all thermodynamic quantities in
terms of $\tau$ and $T_{H}$ via

$\displaystyle z_{h}=\frac{2}{4\pi T_{H}+\tau}.$ (20)

As mentioned before, the extremal limit with $\tau z_{h}=2$ corresponds to the
zero temperature limit. In the black hole geometry, the thermal entropy is
described by the Bekenstein-Hawking entropy which is proportional to the area
of the horizon

$\displaystyle S_{BH}=\frac{LR}{4Gz_{h}},$ (21)

where $L$ is an appropriately regularized volume in the $x$-direction. Since
these two thermodynamic quantities must satisfy the first law of
thermodynamics, the energy contained in the system is given by

$\displaystyle E=\int T_{H}\ dS_{BH}=\frac{LR}{16\pi G}\frac{1-\tau
z_{h}}{z_{h}^{2}}+E_{0},$ (22)

where $E_{0}$ is introduced as an integral constant which does not affect on
the thermodynamics law. Since we are interested in a thermal system
corresponding to a black hole, we need to identify the above energy with the
thermal energy which must vanish at zero temperature. To interpret $E$ as a
thermal energy, $E$ must vanish at zero temperature satisfying $z_{h}=2/\tau$,
which fixes the integral constant to be $E_{0}=LR\tau^{2}/(64\pi G)$.
Therefore, the thermal energy of this system finally becomes

$\displaystyle E=\frac{LR}{64\pi G}\frac{(2-\tau
z_{h})^{2}}{z_{h}^{2}}=\frac{\pi LR}{4G}T_{H}^{2},$ (23)

which automatically satisfies the Stefan-Boltzmann law. For later convenience,
we define the energy density $\rho$ contained in the system

$\displaystyle\rho=\frac{\pi cT_{H}^{2}}{6},$ (24)

where $c=3R/2G$ is the central charge of two-dimensional CFT.

Applying various thermodynamic relation, the fundamental thermodynamic
quantities derived above allows us to get more information characterizing this
thermal system. First, the heat capacity of this system is given by

$\displaystyle C_{V}=\frac{LR(2-\tau z_{h})}{8Gz_{h}}\geq 0.$ (25)

Since the heat capacity is always positive in the considered parameter
regions, we can see that the thermal system derived from the string cloud
geometry is thermodynamically stable. Second, this system has the following
free energy and pressure

$\displaystyle F$ $\displaystyle=$ $\displaystyle-\frac{LR}{64\pi
G}\frac{4-\tau^{2}z_{h}^{2}}{z_{h}^{2}},$ (26) $\displaystyle P$
$\displaystyle=$ $\displaystyle\frac{R}{64\pi
G}\frac{4-\tau^{2}z_{h}^{2}}{z_{h}^{2}}.$ (27)

Now, let us discuss an equation of state parameter which is one of the
quantities characterizing the quark-gluon medium. Recalling that the energy
depends only on the temperature, the equation of state parameter reads

$\displaystyle\omega\equiv\frac{LP}{E}=1+\frac{\tau}{2\pi T_{H}}.$ (28)

For a two-dimensional QFT, $\omega=1$ indicates that the theory is conformal.
If we consider the case of $\tau=0$, the medium consists of only massless
gluons and leads to a CFT, as expected. In the very high temperature limit
with $T_{H}\to\infty$, the equation of state parameter again approaches $1$.
This is because in the limit of $T_{H}\to\infty$ the effect of the quark mass
is negligible. As a result, the limit of $T_{H}\to\infty$ describes a UV fixed
point. At finite temperature, intriguingly, the above result shows that the
equation of state parameter increases linearly with the quark density. In
other words, the pressure becomes high in the dense medium and in the low
temperature region.

##  3 Quantum correlation in the quark-gluon medium

In the previous section, we investigated thermodynamic properties of the
boundary thermal system dual to the string cloud geometry. Based on these
thermal properties, from now on, we study various quantum correlations in the
quark-gluon medium living in a two-dimensional spacetime. First, we focus on
the macroscopic correlation in the quark-gluon medium. To specify the
macroscopic quantum correlation, we exploit many different quantities like the
entanglement entropy, relative entropy, and mutual information, etc. In this
work, we focus on the entanglement entropy which describe the quantum
correlation between the inside and outside of the entangling surface. Since it
is not easy to calculate the entanglement entropy of an interacting system on
the QFT side, we will use the celebrated holographic technique proposed by Ryu
and Takayanagi (RT) Ryu and Takayanagi (2006a, b). The RT formula claims that
the entanglement entropy of a QFT can be determined by the area of minimal
surface extending to the dual bulk geometry.

To calculate the entanglement entropy holographically, we first divide the
boundary space into two parts, a subsystem and its complement. Assuming that
the subsystem lies in

$\displaystyle-\frac{l}{2}\leq x\leq\frac{l}{2},$ (29)

the corresponding entanglement entropy is described by a minimal surface
extending to the dual geometry

$\displaystyle S_{E}=\frac{R}{4G}\int_{-l/2}^{l/2}dx\ \frac{\sqrt{z^{\prime
2}+f}}{z\sqrt{f}}.$ (30)

Now, we introduce a turning point. Denoting the turning point as $z_{0}$, it
represents a maximum value which the minimal surface can reach. In other
words, the minimal surface extends only to $0\leq z\leq z_{0}$ in the dual
geometry. Due to the invariance of the entanglement entropy under $x\to-x$,
the turning point automatically satisfies $z(0)=z_{0}$ and $dz/dx=0$ at $x=0$.

In the above entanglement entropy, there exists a conserved quantity. Using
this conserved quantity, we can represent the subsystem size $l$ and the
entanglement entropy in terms of the turning point

$\displaystyle l$ $\displaystyle=$
$\displaystyle\int_{0}^{z_{0}}dz\frac{2z}{\sqrt{f(z_{0}^{2}-z^{2})}},$ (31)
$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\frac{R}{2G}\int_{\epsilon}^{z_{0}}dz\frac{z_{0}}{z\sqrt{f(z_{0}^{2}-z^{2})}},$
(32)

where $\epsilon$ is introduced as a UV cutoff which regularizes the UV
divergence. Since the string cloud geometry has the nontrivial blackening
factor, it is not easy to perform the above integrals exactly. Thus, from now
on we take into account two limiting cases characterizing the UV and IR
behaviors.

###  3.1 In a UV region

First, we consider the case with a very small subsystem size which describes
quantum correlation in the UV regime. On the dual string cloud geometry, this
small subsystem size limit is realized by taking the limit satisfying
$z_{0}/z_{h}\ll 1$ and $\tau z_{0}\ll 1$. After calculating the integral (31)
perturbatively in the small subsystem size limit, rewriting the turning point
in terms of $l$ gives rise to

$\displaystyle z_{0}=\frac{l}{2}\left(1-\frac{\pi}{16}\tau
l-\frac{2\pi}{c}\rho l^{2}+\cdots\right),$ (33)

where the ellipsis indicates higher order corrections and the energy density
$\rho$ was defined in (24). Using this turning point, the resulting
entanglement entropy becomes perturbatively

$\displaystyle
S_{E}=\frac{c}{3}\left(\log\frac{l}{\epsilon}+\frac{\pi}{16}\tau
l+\frac{\pi}{c}\rho l^{2}+\cdots\right).$ (34)

This result shows that the UV entanglement entropy grows up as the subsystem
size increases. At a given subsystem size, the entanglement entropy becomes
large when the string density $\tau$ and the excitation energy $\rho$ are
high, as expected. Defining a critical subsystem size as

$\displaystyle l_{c}=\frac{c\tau}{16\rho},$ (35)

we see that for $l\ll l_{c}$ the effect of the quark density becomes dominant
and the entanglement entropy increases linearly with the subsystem size $l$.
For $l\gg l_{c}$, on the other hand, the excitation energy becomes dominant
and the entanglement entropy increases by $l^{2}$.

In the above entanglement entropy (34), the first term, which is independent
of $\tau$ and $\rho$ corresponds to the entanglement entropy $S_{0}$ of the
ground state. Therefore, the entanglement entropy caused by the quark density
and the excitation energy is given by

$\displaystyle\Delta S_{E}=S_{E}-S_{0}=\frac{c}{3}\left(\frac{\pi}{16}\tau
l+\frac{\pi}{c}\rho l^{2}+\cdots\right).$ (36)

Recalling that, after subtracting the ground state energy, the internal energy
of the subsystem $\Delta E$ is given by (23), we can define the entanglement
temperature $T_{E}$ as

$\displaystyle\frac{1}{T_{E}}\equiv\frac{\Delta S_{E}}{\Delta
E}=\frac{4Gc}{3\pi RT_{H}^{2}}\left(\frac{\pi\tau}{16}+\frac{\pi\rho
l}{c}\right)=\frac{c\,\pi}{48}\left(\frac{\tau}{\rho}+\frac{16}{c}l\right).$
(37)

It was well known that the entanglement temperature in the UV region is
universally proportional to the inverse of the subsystem size Bhattacharya
_et al._ (2013); Casini  _et al._ (2015). However, the entanglement entropy
studied in the quark-gluon medium shows a different possibility from the known
universality. For instance, if we consider the case of a small quark density
($\rho l\gg\tau$), the entanglement temperature is inversely proportional to
the subsystem size, as mentioned before. In the dense quark-gluon medium
satisfying $\tau\gg\rho l$, however, the entanglement temperature becomes
independent of the subsystem size. Intriguingly, this feature looks universal.
In other words, the same feature also appears in higher-dimensional cases, as
will be show later.

Now, we consider a microscopic two-point correlator of a local operator in the
quark-gluon medium. A two-point function is generally affected by the
interaction with the background medium, so that we can exploit a two-point
function as a measure probing the property of the background medium. Assuming
a local scalar operator whose conformal dimension is denoted by $\Delta_{O}$,
then the two-point function of this operator on the dual gravity side is
described by a geodesic curve whose two ends are attached to two local scalar
operators. Denoting the position of two local operators as $x_{1}$ and
$x_{2}$, the two-point function is given by

$\displaystyle\left\langle
O(x_{1})O(x_{2})\right\rangle=e^{-\Delta_{O}\,L(|x_{1}-x_{2}|)/R},$ (38)

where $L(|x_{1}-x_{2}|$ is a geodesic length connecting two local operators.
For a holographic three-dimensional geometry, the geodesic length and
entanglement entropy are associated with each other

$\displaystyle L(|x_{1}-x_{2}|)=4GS_{E}(|x_{1}-x_{2}|),$ (39)

where the subsystem size $l$ is replaced by the distance of two local
operators, $|x_{1}-x_{2}|$. In the UV region, using the above relation, the
two-point function of a scalar operator becomes in the quark-gluon medium

$\displaystyle\left\langle
O(x_{1})O(x_{2})\right\rangle\sim\frac{1}{|x_{1}-x_{2}|^{2\Delta_{O}}}\left(1-\frac{\pi\Delta_{O}}{8}\tau|x_{1}-x_{2}|-\frac{2\pi\Delta_{O}}{c}\rho|x_{1}-x_{2}|^{2}\right).$
(40)

Comparing this result with the CFT one implies that the two-point function in
the quark-gluon medium rapidly decreases with increasing the distance of two
operators. This is due to the screening effect caused by the background
matters. In a high density regime, the two-point function reduces by
$\tau|x_{1}-x_{2}|$, whereas it reduces by $\rho|x_{1}-x_{2}|^{2}$ in the low
quark density region.

###  3.2 In an IR region

In the previous section, we discussed the macroscopic and microscopic UV
quantum correlation in the quark-gluon medium. We showed that these two
quantum correlations lead to two different behaviors depending on the quark
density and the excitation energy. Now, we investigate IR quantum correlations
in the same quark-gluon medium. The turning point $z_{0}$ approaches the
horizon $z_{h}$ in the IR limit where the subsystem size diverges. The IR
entanglement entropy, therefore, can be rewritten as

$\displaystyle S_{E}=\lim_{z_{0}\to
z_{h}}\left(\frac{R}{4Gz_{0}}l+\frac{R}{2Gz_{0}}\int_{\epsilon}^{z_{0}}dz\frac{\sqrt{z_{0}^{2}-z^{2}}}{z\sqrt{f}}\right).$
(41)

The second term contains a UV divergence. After an appropriate renormalization
procedure, this second term becomes finite even in the IR limit. On the other
hand, the first does not have a UV divergence but diverges in the IR limit.
This implies that the first term gives rise to a leading contribution to the
IR entanglement entropy. Moreover, the first term is exactly the same as the
thermal entropy included in the subsystem. In the IR limit, the quantum
entanglement entropy flows to a thermal entropy with small quantum
corrections. The thermal entropy in the quark-gluon medium is represented as a
function of the quark density and temperature

$\displaystyle S_{th}=\frac{(4\pi T_{H}+\tau)R}{8G}l.$ (42)

From the entanglement entropy point of view, the temperature is well defined
only in the IR limit. As a result, the microscopic two-point function in the
IR regime reduces to

$\displaystyle\left\langle O(x_{1})O(x_{2})\right\rangle\sim
e^{-\Delta_{O}(4\pi T_{H}+\tau)|x_{1}-x_{2}|/2},$ (43)

where the correlation length should be identified as

$\displaystyle\xi_{c}\sim\frac{1}{4\pi T_{H}+\tau}.$ (44)

Due to the screening effect of the quark-gluon medium, the two-point function
is exponentially suppressed in the IR regime. This suppression becomes fast as
the temperature and quark’s density increase, as expected.

##  4 Higher dimensional string cloud geometries

The previous study on quantum correlations can extend to higher-dimensional
cases. To check the universality of the UV entanglement entropy discussed
before, we investigate the UV entanglement entropy in a $d$-dimensional quark-
gluon medium holographically. In the higher-dimensional case, unlike the
previous two-dimensional case, we can take several different shapes of the
entangling surface. In this work, we focus on a strip-shaped entangling region
because the universal feature we are interested in is independent of the shape
of the entangling surface. To do so, we parameterize a strip-shaped region as

$\displaystyle-\frac{l}{2}\leq x=x_{1}\leq\frac{l}{2}\quad{\rm
and}\quad-\frac{L}{2}\leq x_{i}\leq\frac{L}{2},$ (45)

where $i$ runs from $2$ to $d-2$ and $L$ indicates an appropriated regularized
size of a total system. In a $(d+1)$-dimensional string cloud geometry, the
holographic entanglement entropy is determined by

$\displaystyle
S_{E}=\frac{R^{d-1}L^{d-2}}{4G}\int_{-l/2}^{l/2}dx\frac{\sqrt{f+z^{\prime
2}}}{z^{d-1}\sqrt{f}},$ (46)

where the blackening factor is given by

$\displaystyle f=1-\tau
z^{d-1}-\left(\frac{1}{z_{h}^{d}}-\frac{\tau}{z_{h}}\right)z^{d}.$ (47)

A conserved quantity similar to the previous case determines the subsystem
size and the entanglement entropy in terms of the turning point $z_{0}$

$\displaystyle
l=\frac{2}{z_{0}^{d-1}}\int_{0}^{z_{0}}dz\frac{z^{d-1}}{\sqrt{f}\sqrt{1-(z/z_{0})^{2(d-1)}}},$
(48) $\displaystyle
S_{E}=\frac{L^{d-2}R^{d-1}}{2G}\int_{\epsilon}^{z_{0}}dz\frac{1}{z^{d-1}\sqrt{f}\sqrt{1-(z/z_{0})^{2(d-1)}}}.$
(49)

###  4.1 In the UV region

In the UV region satisfying $z_{0}\ll z_{h}$ and $z_{0}\ll\tau^{1/(d-1)}$, the
entanglement entropy can be reexpressed as a perturbative expansion. For
$d=3$, the resulting entanglement entropy becomes

$\displaystyle
S_{E}=\frac{R^{2}}{4G}\frac{A_{1}}{\epsilon}-\frac{R^{2}\Gamma\left(\frac{3}{4}\right)^{4}}{4\pi
G}\frac{A_{1}}{l}+\frac{\pi^{2}R^{2}}{24G\Gamma\left(\frac{3}{4}\right)^{4}}V_{2}\tau+\frac{\pi^{2}R^{2}}{64G\Gamma\left(\frac{3}{4}\right)^{4}}\frac{V_{2}l}{z_{h}^{3}},$
(50)

where $A_{1}=2L$ and $V_{2}=lL$ are the area and volume of the two-dimensional
strip-shaped entangling region, respectively. Thus, the third and fourth terms
in the above entanglement entropy are proportional to the quark number and
excitation energy contained in the subsystem, respectively. For $d=4$, the
entanglement entropy in the UV region becomes

$\displaystyle
S_{E}=\frac{R^{3}}{8G}\frac{A_{2}}{\epsilon^{2}}-\frac{\sqrt{3}R^{3}\Gamma\left(\frac{2}{3}\right)^{5}\Gamma\left(\frac{5}{6}\right)^{2}}{2^{14/3}\pi^{2}G}\frac{A_{2}}{l^{2}}+\frac{\sqrt{3}\pi^{2}R^{3}\Gamma\left(\frac{4}{3}\right)}{2^{11/3}G\Gamma\left(\frac{2}{3}\right)^{5}}V_{3}\tau+\frac{81R^{3}\Gamma\left(\frac{4}{3}\right)^{4}}{80G\Gamma\left(\frac{2}{3}\right)^{5}}\frac{V_{3}l}{z_{h}^{4}},$
(51)

where $A_{2}=2L^{2}$ and $V_{3}=lL^{2}$ correspond to the area and volume of
the three-dimensional strip-shaped entangling region. Here, the parts, which
do not relies on $\tau$ and $\rho$, correspond to the ground state
entanglement entropy. The effect of excitation and quark density can be
extracted by subtracting this ground state entanglement entropy. Intriguingly,
the gluon’s excitation and quark density, regardless of the dimension of the
underlying theory, give rise to the similar contribution

$\displaystyle\Delta S_{E}\equiv
S_{E}-S_{0}=\frac{L^{d-2}R^{d-1}}{G}\left(g_{1}(\tau)\,l+g_{2}(\tau,\rho)\,l^{2}+\cdots\frac{}{}\right),$
(52)

where $S_{0}$ indicates the ground state entanglement entropy and $g_{1}$ and
$g_{2}$ are functions of $\tau$ and $\rho=E/V_{(d-1)}\sim 1/z_{h}^{d}$. This
fact leads to two distinct universal features in the quark-gluon medium. Since
the energy is an extensive quantity, it is proportional to the subsystem’s
volume ($\sim lL^{d-2}$). For example, the energy contained in the subsystem
for $d=3$ is in the high density limit ($\tau\to\infty$)

$\displaystyle E=\frac{\pi R^{2}T_{H}^{2}\sqrt{\tau}}{6\sqrt{3}G}V_{2},$ (53)

while it in the low density limit becomes

$\displaystyle E=\frac{8\pi^{2}R^{2}T_{H}^{3}}{27G}V_{2}.$ (54)

The entanglement temperature in the low density limit ($\tau\to 0$) reduces to

$\displaystyle T_{E}=\frac{dE}{dS_{E}}\sim l^{-1}.$ (55)

This is the well-known universality of the entanglement temperature which is
originated from the relativistic excitation energy. We can also take into
account another case which allows a different universality. Taking a high
density limit of $\tau\to\infty$, we can ignore the effect of $\rho$ and the
quark density $\tau$ becomes dominant. Similar to the gluon’s excitation case,
we define the entanglement temperature which represents the change of the
entanglement entropy when the quark density varies. Then, the entanglement
temperature gives rise to another universality in the high quark density limit

$\displaystyle T_{E}=\frac{dE}{dS_{E}}\sim l^{0}.$ (56)

This implies that the entanglement temperature is independent of the subsystem
size which is independent of the dimension $d$. As a consequence, the
entanglement temperature in the UV regime can show two distinct universality
relying on the quark density.

###  4.2 Correlation in the IR regime

In the IR region, the subsystem has a very large size. Its size for $d=3$ is
determined as

$\displaystyle
l=2\int_{0}^{z_{0}}dz\frac{z^{2}}{\sqrt{f}\sqrt{z_{0}^{4}-z^{4}}},$ (57)

and the corresponding entanglement entropy reads

$\displaystyle
S_{E}=\frac{LR^{2}z_{0}^{2}}{2G}\int_{\epsilon}^{z_{0}}dz\frac{1}{z^{2}\sqrt{f}\sqrt{z_{0}^{4}-z^{4}}}.$
(58)

The entanglement entropy can be rewritten as the following form

$\displaystyle
S_{E}=\frac{R^{2}lL}{4Gz_{0}^{2}}+\frac{LR^{2}}{2Gz_{0}^{2}}\int_{\epsilon}^{z_{0}}\frac{\sqrt{z_{0}^{4}-z^{4}}}{z^{2}\sqrt{f}}.$
(59)

In the IR region with $l\to\infty$, the turning point approaches the horizon,
$z_{0}\to z_{h}$. In this IR region, the first term proportional to the size
of the subsystem diverges. The second term also has a divergence proportional
to $1/\epsilon$. This divergence corresponds to the UV divergence appearing in
the previous UV entanglement entropy. Ignoring this UV divergence, the second
term has no more divergence even in the case of $z_{0}=z_{h}$. Therefore, the
first term gives rise to the leading contribution to the IR entanglement
entropy. Intriguingly, this leading contribution is exactly the same as the
Bekenstein-Hawking entropy. This fact implies that the entanglement entropy
evolves to the thermal entropy along the RG flow.

We can also investigate a microscopic two-point function in the IR regime.
Regardless of the dimension $d$, the turning point determines the distance
between two local operators

$\displaystyle|x_{1}-x_{2}|=2\int_{0}^{z_{0}}\frac{z}{\sqrt{f}\sqrt{z_{0}^{2}-z^{2}}},$
(60)

which is the distance measured at the boundary. The length of the geodesic
curve extending to the dual geometry results in

$\displaystyle
D=2Rz_{0}\int_{\epsilon}^{z_{0}}\frac{1}{z\sqrt{f}\sqrt{z_{0}^{2}-z^{2}}}.$
(61)

Rewriting the geodesic length as the following form

$\displaystyle
D=\frac{R}{z_{0}}|x_{1}-x_{2}|+\frac{2R}{z_{0}}\int_{\epsilon}^{z_{0}}\frac{\sqrt{z_{0}^{2}-z^{2}}}{z\sqrt{f}},$
(62)

we see that, when we ignore the UV divergence, the first term gives rise to a
leading contribution in the IR regime ($z_{0}\to z_{h}$ and
$|x_{1}-x_{2}|\to\infty$). As a consequence, we finally obtain the following
IR two-point function

$\displaystyle\left\langle
O(x_{1})O(x_{2})\right\rangle=e^{-\Delta_{O}D/R}\sim
e^{-2\Delta_{O}|x_{1}-x_{2}|/z_{h}}.$ (63)

This result shows that, when the distance of two local operators increases,
the IR two-point function suppresses exponentially due to the screening effect
caused by the background quark-gluon medium.

##  5 Mutual information in the string cloud geometries

In this section, based on the holographic point of view, we survey another
quantum information quantity related to entanglement, which is called mutual
information. Mutual information measures how much the subsystems are
correlated with each other and is defined as

$\displaystyle I(A;B)=S(A)+S(B)-S(A\cup B),$ (64)

Where $S(A\cup B)$ is the entanglement entropy for a union of two subsystems.
Due to the subadditivity of the entanglement entropy, the mutual information
is always positive. When the mutual information becomes zero, $I(A;B)=0$,
holographically the minimal surface suddenly changes its bulk configuration
from a disconnected shape to a connected one. This feature allows to capture a
phase transition in which the critical length is determined by the ratio of
the size of the boundary subregions to their separation.

In order to study the effects of our holographic model on the mutual
information, we calculate this quantity for two disjoint subregions where
their sizes and separation length are given by $\ell_{1},\ell_{2}$ and $h$
respectively. Depending the relative size between subsystems and separation,
the mutual information becomes

$\displaystyle
I(A;B)=\left\\{\begin{array}[]{crr}S(\ell_{1})+S(\ell_{2})-S(\ell_{1}+\ell_{2}+h)-S(h)&\,\,\,\mbox{for}&h\ll\ell_{1},\ell_{2},\\\
0&\mbox{for}&h\gg\ell_{1},\ell_{2}.\end{array}\right.$ (67)

It is obvious that the critical length $h_{c}$, where $I(A;B)=0$, depends on
the two parameters $\tau$ and $\rho$ characterizing the string cloud geometry.
For small values of these parameters one can perturbatively find the critical
length as

$\displaystyle
h_{c}=(\sqrt{2}-1)\ell-\frac{\pi(\sqrt{2}-1)\tau}{16\sqrt{2}}\ell^{2}+\frac{\pi(c\pi(13-8\sqrt{2})\tau^{2}-2048\rho)}{1024\sqrt{2}c}\ell^{3}+\cdots,$
(68)

where we assume the two subsystems have the same size,
$\ell_{A}=\ell_{B}=\ell$ for simplicity. Depending on the values of $\tau$ and
$\rho$, the required critical length has a non trivial size compared to the
one in AdS. To show that, we consider the separation variable as a function of
$\tau$ and temperature. We observe that for the extremal case $T_{H}=0$, the
critical length has the maximum value. It implies that the upper bound of
$h_{c}$ where a phase transition occurs is determined by turning off the
effect of $\tau$ in the quark-gluon medium. Therefore, by turning up the
temperature (or increasing the density of energy) the critical length
decreases. Notice that the critical length is always shorter than the one in
pure AdS because the screening effect stemmed from the quark-gluon medium.

The above result (68) is only valid for the UV region where the minimal
surface is localized in the AdS boundary. To see the IR effect of the mutual
information, since there is no analytic form of the solution, it is needed to
compute (64) numerically for a given background geometry. To achieve these
numerics, we first rewrite the horizon in terms of the function of the
temperature $T_{H}$ and the density of string $\tau$ using (20).

In section 3.2 we studied the microscopic quantum correlation in an IR region
by exponentiating the minimal length whose boundary points are assumed to be
the positions of two local operators. For large values of the parameters,
$T_{H}$ and $\tau$, a two-point function decays exponentially. The mutual
information can also play a similar role in proving this kind of correlation.
In this case, the mutual information measures a collective correlation between
two subregions. From now, we call this the macroscopic quantum correlation. By
numerical computation, we found that in the IR region the critical length
$h_{c}$ behaves like the correlation length $\xi$ in (44), $h_{c}\sim\xi_{c}$,
for both the extremal and large $\tau$ limits, see the Fig.1.

##  6 Discussion

Using the entanglement entropy, we holographically studied various quantum
correlations in a quark-gluon medium. We considered a black hole geometry with
a string cloud in order to represent heavy quarks and gluon excitations
simultaneously. In the quark-gluon medium, quarks, gluons, and their
interaction generate a nontrivial RG flow. Along this RG flow, quantum
correlations usually vary and a UV theory flows to another IR theory. In this
work, we investigated how a UV theory in a medium changes into a new IR theory
by studying quantum correlations depending on the RG scale. At fixed points, a
theory becomes scale invariant and has an infinite correlation length.
However, a nontrivial RG flow generally breaks the scale symmetry given at
fixed points. The scale symmetry breaking makes the correlation length finite.
We looked into how the quark density and gluon’s excitation energy modify the
correlation length following the holographic propositions.

The entanglement entropy describes a quantum correlation between two
macroscopic subsystems. We found that the entanglement entropy in the UV
region always increases with increasing the quark density and gluon
excitation. This fact indicates that the increase of the quark density and
gluon excitation leads to increasing the degrees of freedom of the quark-gluon
medium. As a result, the entanglement entropy grows up in a dense medium. We
further took into account two local operators on this background medium and
investigated their microscopic two-point correlation function. The
entanglement entropy is governed by a minimal surface extending to the dual
geometry, while a microscopic two-point function is described by a geodesic
curve in the holographic setup. For a three-dimensional asymptotic AdS space,
in particular, a minimal surface reduces to a geodesic curve. Therefore, the
entanglement entropy of the background medium is closely related to a two-
point function of local operators. We studied a two-point function in the
quark-gluon medium and found that a microscopic two-point function decrease
when the quark density and gluon excitation grow up in the background medium.
This result means that the higher the density and the higher the temperature,
the stronger the screening effect of the background medium.

(a) $T_{H}=0$ (b) $T_{H}=0.001$

Figure 1: The plots of the critical length from $I(A;B)=0$ for the extremal
black hole (a) and for the black hole with $T_{H}\ll\tau$ (b). The dashed
green curve represents the microscopic correlation length $\xi_{c}$ in (44)
and the orange dots are numerical results.

Another interesting thing in this work is the universality of the entanglement
temperature in the medium. It was well known that the entanglement temperature
reveals the universality in the UV region. For a relativistic theory, the
entanglement temperature is inversely proportional to the subsystem size. In
the quark-gluon medium, however, the entanglement temperature reveals two
distinct universalities. In the low quark density limit where the gluon
excitation becomes dominant, the entanglement temperature shows the expected
universality, $T_{E}\sim 1/l$. In the high quark density limit, on the other
hand, we found that the entanglement temperature is independent of the
subsystem size. Recalling that quarks we considered are heavy, we expect that
the theory governing the dense medium deviates from a relativistic theory. Due
to this reason, intriguingly, the entanglement temperature of a dense medium
leads to a different universality, $T_{E}\sim l^{0}$.

We computed the holographic mutual information in the quantum system with the
quark-gluon medium. A discontinuous change of the minimal surface can be
interpreted as the first-order phase transition in the system. Utilizing both
perturbative and numerical computations, it is shown that the critical length
is diminished as the density and temperature increase which is the same as the
results in Sec.3.1. We also compare the microscopic IR correlation length with
the critical length obtained from the mutual information in the IR region. By
a numerical computation, we observed that in the IR region they are matched
almost exactly.

Acknowledgement

C. Park was supported by the National Research Foundation of Korea(NRF) grant
funded by the Korea government(MSIT) (No. NRF-2019R1A2C1006639). J. H. Lee was
supported by the National Research Foundation of Korea(NRF) grant funded by
the Korea government(MSIT) (No. NRF-2021R1C1C2008737).

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