# Holographic renormalized Entanglement and entropic $c-$function

Mitsutoshi Fujita<EMAIL_ADDRESS>School of Nuclear Science and
Technology, University of South China, Hengyang 421001, China  Song He
<EMAIL_ADDRESS>Center for Theoretical Physics and College of Physics, Jilin
University, Changchun 130012, People’s Republic of China  Max Planck Institute
for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476
Golm, Germany  Yuan Sun<EMAIL_ADDRESS>School of Physics and Electronics,
Central South University, Changsha 418003, China  Jun Zhang
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Alabama, 514 University Boulevard, Tuscaloosa, AL, 35487, USA

###### Abstract

We compute holographic entanglement entropy (EE) and the renormalized EE in
AdS solitons with gauge potential for various dimensions. The renormalized EE
is a cutoff-independent universal component of EE. Via Kaluza-Klein
compactification of $S^{1}$ and considering the low-energy regime, we deduce
the $(d-1)$-dimensional renormalized EE from the odd-dimensional counterpart.
This corresponds to the shrinking circle of AdS solitons, probed at large $l$.
The minimal surface transitions from disk to cylinder dominance as $l$
increases. The quantum phase transition occurs at a critical subregion size,
with renormalized EE showing non-monotonic behavior around this size. Across
dimensions, massive modes decouple at lower energy, while degrees of freedom
with Wilson lines contribute at smaller energy scales.

##  1 Introduction

Quantum entanglement entropy stands as a pivotal concept in quantum mechanics,
offering insight into the level of entanglement among distinct segments of a
quantum system. By quantifying the entanglement between different components,
this entropy provides a metric for the extent of shared information. Its
implications span various facets of quantum mechanics, encompassing quantum
information theory, black hole physics, and condensed matter physics. Notably,
the entanglement entropy of subsystem A quantifies the entangled degrees of
freedom within a given quantum field theory [1, 2]. Within the context of
condensed matter physics, this entropy displays divergence at critical
junctures of quantum critical phase transitions, assuming the role of an order
parameter [3]. This phenomenon encapsulates the geometric essence of field
theories, manifested in an area law that draws parallels between subregion
entanglement entropy and black hole entropy.

Introducing the Ryu-Takayanagi formula establishes a holographic counterpart
for entanglement entropy [5, 6, 7], emerging as a robust tool for dissecting
strongly coupled systems traditionally resistant to conventional analysis. In
specific contexts, this formula has served as an order parameter, signaling
the onset of confinement/deconfinement phase transitions within confining
gauge theories [10, 11, 12, 13, 14, 15]. 111On the other hand, holographic
quark anti-quark potential can distinguish confinement and topological phases
[16]. The transitions emanate from the interplay between two minimal surfaces,
resulting in the post-transition confinement phase entanglement entropy
becoming trivial at the infrared limit. Moreover, the holographic entanglement
entropy (HEE) emerges as a probing tool for phase transitions in holographic
superconductors [17, 18, 19, 20, 21, 22, 23] as well as for unveiling
topological phases of matter [24].

The entropic $c-$function concept provides deeper insights into entanglement
entropy in quantum systems [8, 9]. It represents the logarithmic derivative of
entanglement entropy with respect to subsystem size, revealing the intricate
interplay between entanglement and subsystem dimensions. The general entropic
$c-$function, proposed by [10], efficiently quantifies degrees of freedom in
confining theories and yields the central charge of the corresponding
conformal field theory (CFT) [25]. In the context of a quantum field theory
dual to an AdS soliton, the behavior of the entropic $c-$function, as it
decreases with increasing length, effectively serves as a probe for the
deconfinement phase transition. Recently, a study [26] computed the entropic
$c-$function for a striped entangling surface in the same background.
Intriguingly, this function displays non-monotonic behavior with increasing
background gauge field strength. Importantly, the entropic $c-$function for
the strip incorporates both A-type and B-type anomalies due to the coexistence
of these two anomaly types. [28] has derived constraints on anisotropic RG
flows from holographic entanglement entropy.

Our focus lies in assessing the degrees of freedom through entanglement
entropy using a spherical entangling surface, with a specific emphasis on
anomaly effects. The renormalized entanglement entropy, calculated from the
entanglement entropy of this spherical surface, offers a solution to the issue
of cutoff dependence [27]. This renormalized quantity, independent of the
cutoff, measures degrees of freedom in quantum entangled states at an energy
scale of $1/l$, yielding the central charge in conformal field theory (CFT).
In the context of four-dimensional (4D) CFT, it manifests as the A-type
anomaly, in alignment with the C-theorem: the renormalized entanglement
entropy decreases in the infrared (IR) limit as anticipated. [29] derived the
renormalized entanglement entropy for a kink region which reduced to a
universal positive finite term in the UV limit.

The computation of renormalized entanglement entropy for a spherical
entangling surface within the AdS soliton framework with a gauge potential
remains unexplored. The gauge potential’s interpretation in this background
involves a twisted boundary condition along a circle within the cigar
direction. This contributes to the negative Casimir energy of the dual field
theory, which can eventually become positive. An intriguing aspect emerges
from the interplay of Wilson lines, capable of inducing mass shifts in charged
particles [30]. It becomes desirable to capture such alterations through the
renormalized entanglement entropy. Conversely, contrasting the analysis
presented in [26] for a striped entangling surface in the same background, the
renormalized entanglement entropy in $R^{1,2}\times S^{1}$ quantum field
theory exhibits solely B-type anomaly characteristics. Generally, this
quantity doesn’t adhere to the C-theorem. Therefore, an engaging pursuit lies
in investigating this aspect, including scenarios in higher-dimensional cases.

This study focuses on introducing a holographic renormalized entanglement
entropy (HREE), which encompasses the finite portion of entanglement entropy.
We extensively investigate HREE’s behavior across diverse scenarios to uncover
the universal properties of quantum phase transitions. Our analysis reveals a
phase transition between disk and cylinder geometries. The dominance of the
disk type is evident for small $l$, while the cylinder shape prevails for
larger $l$. This critical size marks the occurrence of a quantum phase
transition. Notably, this transition is an outcome of the large $N$ limit and
is absent in free theory. The anticipated function of the renormalized
entanglement entropy is to quantify the degrees of freedom in the dual quantum
field theory (QFT). We probe HREE’s response to changes in operator mass and
gauge potential. Specifically, as the operator mass decreases, we expect HREE
to increase significantly for larger $l$, given the decoupling of massive
degrees of freedom.

The rest of this paper is organized as follows. In section II, we analyze the
UV structure of the entanglement entropy. We derive the renormalized
entanglement entropy of QFT dual to AdS solitons with gauge potential. We
discuss properties of the renormalized entanglement entropy in both odd and
even dimensions. Section III mainly focuses on holographic stress-energy
tensors in AdS solitons with the gauge potential. In sections IV and V, we
analyze the quantum phase transition of HREE in higher dimensional
backgrounds. We end with conclusions and prospects in section VI. Some
calculation details are presented in the appendices.

##  2 The UV structure of the entanglement entropy

The $d+1$-dimensional AdS solitons with gauge potential exhibit a geometry
akin to a cigar. In this setup, a compact circle gradually contracts to zero
size in the bulk, completing the geometry. This behavior is detailed in [26].
The dual theory on $R^{1,d-2}\times S^{1}$ transforms into a confining theory
with a discernible energy gap in this context. Notably, this theory
incorporates Wilson lines along the $S^{1}$ direction, which inherently alters
the boundary conditions.

We calculate the entanglement entropy in this theory. The spherical entangling
surface is chosen to have the topology $S^{d-3}\times S^{1}$, where the
entangling surface wraps another circle $S^{1}$ [31]. For $d=4$, $S^{1}\times
S^{1}$ is a cylinder with one identified direction, i.e., a torus. Because a
spherical entangling surface for QFT on $R^{1,d-1}$ has a different topology
$S^{d-2}$, we find that the UV scaling structure is different from those of
QFT on $R^{1,d-1}$. According to [31], the UV divergent structure of
entanglement entropy $S_{UV}$ is of the form

$\displaystyle S_{UV}=\dfrac{L_{\phi}}{R}S_{UV,0},$ (1)

where $S_{UV,0}$ is the UV structure of entanglement entropy of QFT
$S_{EE}^{(0)}$ on $R^{1,d-1}$ and $L_{\phi}$ is the periodicity along a circle
$S^{1}$ of the cigar.

Two UV scaling structures are related to each other. Using (1) and operating
differentiation on $S_{EE}$, the renormalized entanglement entropy (the UV-
independent part of the entanglement entropy) then becomes [31]

$\displaystyle
S_{ren}=\dfrac{1}{R}f_{d}(R\partial_{R})RS_{EE}=L_{d}(R\partial_{R})S_{EE}$
$\displaystyle=\begin{cases}&\dfrac{1}{(d-2)!!}R\partial_{R}(R\partial_{R}-2)\dots(R\partial_{R}-(d-3))S_{EE},\quad
d=\text{odd},\\\
&\dfrac{1}{(d-2)!!}(R\partial_{R}+1)(R\partial_{R}-1)\dots(R\partial_{R}-(d-3))S_{EE},\quad
d=\text{even},\end{cases}$ (2)

where $f_{d}$ defines the operation of the renormalized entanglement entropy
for $R^{1,d-1}$ [27]:

$\displaystyle
f_{d}(R\partial_{R})S_{EE}^{(0)}=\begin{cases}&\dfrac{1}{(d-2)!!}(R\partial_{R}-1)(R\partial_{R}-3)\dots(R\partial_{R}-(d-2))S_{EE}^{(0)},\quad
d=\text{odd},\\\
&\dfrac{1}{(d-2)!!}R\partial_{R}(R\partial_{R}-2)\dots(R\partial_{R}-(d-2))S_{EE}^{(0)},\quad
d=\text{even}.\end{cases}$ (3)

Recall that the first line in (2) is equivalent to the $d-1$ dimensional
renormalized entanglement entropy on $R^{1,d-2}$ (the second line in (3)) up
to coefficients.

Especially we obtain

$\displaystyle S_{ren}$ $\displaystyle=R\partial_{R}S_{EE}\quad\text{for
$d=3$},$ (4) $\displaystyle S_{ren}$
$\displaystyle=\dfrac{1}{3}R\partial_{R}(R\partial_{R}-2)S_{EE}\quad\text{for
$d=5$}.$ (5)

The formula (4) for QFT on $R^{1,1}\times S^{1}$ corresponds to the well-
established expression of the entropic $c$-function on $R^{1,1}$. Through
Kaluza-Klein reduction along $S^{1}$, the renormalized entanglement entropy
effectively embodies the 2-dimensional entropic $c-$function in the low-energy
limit. In systems respecting Lorentz symmetry, the 2-dimensional entropic
$c$-function is both non-negative and monotonically increasing. For $R\ll
L_{\phi}$ (in the UV limit), the renormalized entanglement entropy mirrors the
behavior of a 3-dimensional system. The subregion’s topology is not a disk $D$
but rather $L\times S^{1}$ with an interval $L$, while the entangling surface
forms $S^{1}$. Moving to formula (5) for QFT on $R^{1,3}\times S^{1}$, it
captures one variant of the 4-dimensional renormalized entanglement entropy on
$R^{1,3}$. Kaluza-Klein reduction along $S^{1}$ approximates the renormalized
entanglement entropy on $R^{1,3}$ in the low-energy regime. However, the
behavior of the 4-dimensional renormalized entanglement entropy can be either
negative or positive, displaying non-monotonic tendencies. In this scenario,
the subregion’s topology does not correspond to a ball $B^{4}$ but rather
$B^{3}\times S^{1}$, while the entangling surface takes the form of
$S^{2}\times S^{1}$.

###  2.1 Renormalized entanglement entropy of 4d QFT

This section examines the entanglement entropy and renormalized entanglement
entropy for various cases: a free scalar, Dirac fermion, and a 4-dimensional
conformal field theory (CFT) on $R^{1,2}\times S^{1}$. Additionally, we
provide an overview of the trace anomaly in general CFT, which is intricately
linked to the logarithmic term present in the entanglement entropy.

The entanglement entropy can be derived from the effective action $w=-\log Z$
in a $d(=4)$-dimensional manifold featuring conical singularities. By taking
the limit $n\to 1$, the entanglement entropy assumes an analytical expression
involving $w$ on a manifold with such singularities. Notably, the effective
action $w$ generally exhibits logarithmic divergence, which is connected to
the concept of conformal anomaly. We consider the infinitesimal rescaling
$g^{\mu\nu}\to(1-2\delta\lambda)g^{\mu\nu}$. We then have

$\displaystyle\dfrac{dw}{d\lambda}=-2g^{\mu\nu}\dfrac{\delta w}{\delta
g^{\mu\nu}}=-\int d^{4}x\sqrt{g}\langle T^{\mu}{}_{\mu}\rangle$ (6)

The equation stands as the trace anomaly, indicating the deviation from the
traceless condition $\langle T^{\mu}{}_{\mu}\rangle=0$ within Quantum Field
Theory (QFT) for Conformal Field Theory (CFT). The trace anomaly is
characterized by polynomials of the curvature tensor, a formulation contingent
on the dimension $d$. Notably, in odd dimensions, the trace anomaly must
satisfy the condition of vanishing.

When we define the length scale $R_{1}$ of the subregion, this scale is
related to rescaling the metric (6). Thus, one obtains the following formula:

$\displaystyle R_{1}\partial_{R_{1}}S_{A}=$ $\displaystyle\lim_{n\to
1}(-2\partial_{n}\int d^{d+1}xg_{\mu\nu}(x)\dfrac{\delta}{\delta
g_{\mu\nu}(x)}[w-nw|_{n=1}])$ (7) $\displaystyle\dfrac{1}{2\pi}\lim_{n\to
1}\partial_{n}\Big{(}\langle\int
d^{d+1}x\sqrt{g}T_{\mu}{}^{\mu}(x)\rangle_{M_{n}}-n\langle\int
d^{d+1}x\sqrt{g}T_{\mu}{}^{\mu}(x)\rangle_{M_{1}}\Big{)}.$

Here, the entanglement entropy has been replaced with a function of the
partition function in the manifold with conical singularities. The above-
mentioned formula relates the entanglement entropy to the trace anomaly.

To evaluate the entanglement entropy, we examine a subsystem $\sigma$ with a
cylindrical configuration where one direction is identified as
$\phi\sim\phi+L_{\phi}$. Interestingly, this subsystem aligns with the one in
QFT corresponding to $d+1$-dimensional AdS solitons. Due to this, a conical
singularity arises, characterized by a curvature tensor proportional to a
delta function. The resulting logarithmic contribution to the entanglement
entropy is expressed as $S_{EE}=s\log(\epsilon/R_{1})+\dots$, where $\epsilon$
represents the ultraviolet cut-off. Remarkably, this logarithmic term can also
be derived by integrating the entanglement entropy of a $3d$ free theory [32,
33]. $s$ is expressed in terms of extrinsic curvatures [34]. According to [32,
33], $s$ becomes

$\displaystyle
s=\dfrac{a}{180}\int_{\partial\sigma}d^{2}x\sqrt{h}E_{2}+\dfrac{c}{120}\int_{\partial\sigma}d^{2}x\sqrt{h}I_{2},$
(8)

where $E_{2}$ is the Euler density and $I_{2}$ is a Weyl invariant. Compared
with the normalization of [27], we have $a=360a_{4}$ and $c=120c_{4}$. $c=1$
for a real scalar and $c=6$ for Dirac fermion. Coefficients are consistent
with trace anomalies.

For CFT on a cylinder of length $L_{\phi}$ and radius $l$, $s$ becomes

$\displaystyle s=\dfrac{c}{240}\dfrac{L_{\phi}}{l}.$ (9)

By using the renormalized entanglement entropy for cylinder type topology in
(2), we obtain

$\displaystyle
S_{ren}=\dfrac{1}{2}(l\partial_{l}+1)(l\partial_{l}-1)S_{EE}=s=\dfrac{cL_{\phi}}{240l}.$
(10)

This formula shows that the renormalized entanglement entropy agrees with the
coefficient $s$. Furthermore, according to [27], $s$ agrees with the
renormalized entanglement entropy for a spherical entangling surface $S^{2}$
as follows:

$\displaystyle
S_{ren}=\dfrac{1}{2}R\partial_{R}(R\partial_{R}-2)S_{EE}=s=\dfrac{a}{90}.$
(11)

##  3 The $AdS$ soliton with the gauge potential

The AdS soliton is achieved through a double Wick rotation of the AdS black
hole, following the Einstein equation. It corresponds to the QFT system with
anti-periodic boundary conditions [35]. In our investigation, we apply this
approach to the Reissner Nordstrom $AdS$ black hole [36], performing an
analytical continuation of the metric in both temporal and spatial dimensions.
The metric of the $AdS$ soliton with the gauge potential becomes [26]

$\displaystyle ds_{d+1}^{2}=$
$\displaystyle\dfrac{L^{2}}{z^{2}}\Big{(}\dfrac{dz^{2}}{f_{d}(z)}+f_{d}(z)d\phi^{2}-dt^{2}+dR^{2}+R^{2}d\Omega_{d-3}\Big{)},$
(12) $\displaystyle A_{\phi}=$ $\displaystyle
a_{\phi}^{(0)}\Big{(}1-\Big{(}\dfrac{z}{z_{+}}\Big{)}^{d-2}\Big{)},$

where

$\displaystyle
f_{d}(z)=1-\Big{(}1+\tilde{\epsilon}z_{+}^{2}a_{\phi}^{2}\Big{)}\Big{(}\dfrac{z}{z_{+}}\Big{)}^{d}+\tilde{\epsilon}z_{+}^{2}a_{\phi}^{2}\Big{(}\dfrac{z}{z_{+}}\Big{)}^{2(d-1)}.$
(13)

Here, we set $\tilde{\epsilon}=-1$ 222$\tilde{\epsilon}=1$ for the Reissner
Nordstrom $AdS$ black hole., and define
$a_{\phi}^{2}=a_{\phi}^{(0)2}/\gamma^{2}$, where
$\gamma^{2}=\frac{(d-1)g_{e}^{2}L^{2}}{(d-2)\kappa^{2}}$ is a dimensionless
parameter. The gauge field $a_{\phi}^{(0)}$ acts as the source for the
conserved current and induces a non-zero VEV for the current, $\langle
J_{\phi}\rangle\neq 0$. Alternatively, this gauge field can be interpreted as
a Wilson line, altering the boundary condition (twisted boundary condition)
due to a gauge transformation. As the Wilson line vanishes at the tip of the
soliton ($z=z_{+}$), the gauge connection remains regular there. The radial
coordinate $z$ in (12) is confined to $z\leq z_{+}$, while the $\phi$
direction follows the periodicity $\phi\to\phi+1/M_{0}$ to prevent conical
singularities. The Kaluza-Klein mass $M_{0}$ of the $\phi$ direction is given
by

$\displaystyle M_{0}=\dfrac{1}{4\pi
z_{+}}\Big{(}d+(d-2)\dfrac{a_{\phi}^{2}}{z_{+}^{2}}\Big{)}.$ (14)

The formula (14) can also be rewritten in terms of $M_{0}$ and $a_{\phi}$ as
follows:

$\displaystyle z_{+}=\dfrac{d}{2\pi
M_{0}\pm\sqrt{4\pi^{2}M_{0}^{2}-d(d-2)a_{\phi}^{2}}}.$ (15)

There is also a minus branch. However, $z_{+}$ is divergent at small
$a_{\phi}$ in that case, and the background does not approach the $AdS_{d+1}$
soliton. It can be shown that the solution with the plus sign in (15) is
always more stable than the one in the minus branch.

The boundary stress tensor $T^{(0)}_{{\mu}{\nu}}$ for field theory dual to the
above background was computed in our previous work [26]. Here, we quote the
results of the boundary stress tensor for later use. For more details, please
refer to [26]

$T^{(0)}_{tt}=-T^{(0)}_{x^{i}x^{i}}=-\frac{L^{d-1}}{2\kappa^{2}}\frac{1}{z^{d}_{+}}\Big{(}1-{z^{2}_{+}a^{2}_{\phi}}\Big{)}=-\frac{L^{d-1}}{2\kappa^{2}}\frac{1}{z^{d}_{+}}\bar{{\alpha}}_{\phi},$
(16)
$T^{(0)}_{\phi\phi}=\frac{dL^{d-1}}{2\kappa^{2}}\frac{1}{z^{d}_{+}}\Big{(}1-{z^{2}_{+}a^{2}_{\phi}}\Big{)}\Big{(}-1+\frac{1}{d}\Big{)}.$
(17)

with

$\bar{a}_{\phi}=1-\Big{(}{z_{+}a_{\phi}}\Big{)}^{2}.$ (18)

The boundary energy then is [37]

$\displaystyle
M=-\dfrac{V_{d-2}}{M_{0}}\frac{L^{d-1}\bar{a}_{\phi}}{2\kappa^{2}z_{+}^{d}}.$
(19)

The boundary energy can change the sign when we change Wilson lines (gauge
potential).

$\displaystyle\begin{cases}M<0\quad\text{$z_{+}a_{\phi}<1$}\\\
M>0\quad\text{$z_{+}a_{\phi}>1$}.\end{cases}$ (20)

In other words, $M$ is negative for $a_{\phi}<2\pi M_{0}/(d-1)$, while it can
become positive when $a_{\phi}>2\pi M_{0}/(d-1)$. For $a_{\phi}=0$, it
realizes Casimir energy of $4d$ SYM theory on $R^{3}\times S^{1}$ [35]. This
behavior is analogous to Casimir energy of fields with twisted boundary
conditions in $2d$ CFT [30]. Casimir energy is different between the periodic
and anti-periodic boundary conditions.

##  4 The holographic entanglement entropy

In this section, we compute the entanglement entropy [5, 6]. The entangling
surface is specified by $z=0$ at $R=l$, and $0\leq\phi\leq L_{\phi}$ at a
constant time slice in the background (12). Its topology becomes $S^{1}\times
S^{d-3}$, where $S^{1}$ and $S^{d-3}$ be of radius $L_{\phi}$ and $l$
respectively. Note that the topology of the entangling surface differs from
the theory without the compactification of the $\phi$ direction. The surface
action becomes

$\displaystyle A=\int d^{d-1}x\mathcal{L}=\Omega_{d-3}L_{\phi}{L^{d-1}}\int
dz\dfrac{R^{d-3}}{z^{d-1}}\sqrt{1+f\dot{R}^{2}},$ (21)

where $\mathcal{L}=\sqrt{\det g_{ind}}$ and $g_{ind}$ is the induced metric.
The holographic entanglement entropy is given by

$\displaystyle S_{EE}=\dfrac{2\pi}{\kappa^{2}}A$ (22)

with $A$ minimized. Recall that $L^{d-1}/\kappa^{2}$ is dimensionless. We omit
the $AdS$ radius ($L=1$) for the convenience. We solve EOM derived from (21)
to obtain the minimal surface. The EOM of $R$ become 333EOM in terms of $z$ is
$\displaystyle-f(z)\left((d-1)Rz^{\prime}(R)^{2}+z(R)\left((d-3)z^{\prime}(R)+rz^{\prime\prime}(R)\right)\right)+(1-d)Rf(z)^{2}$
$\displaystyle-(d-3)z(R)z^{\prime}(R)^{3}=0.$ (23)

$\displaystyle
2z\left(d-R(z)f^{\prime}(z)R^{\prime}(z)-3\right)+f(z)(2(d-3)zR^{\prime}(z)^{2}-R(z)(-2(d-1)R^{\prime}(z)$
$\displaystyle+zf^{\prime}(z)R^{\prime}(z)^{3}+2zR^{\prime\prime}(z)))+2(d-1)f(z)^{2}R(z)R^{\prime}(z)^{3}=0.$
(24)

One should specify the IR boundary condition. In fact, there are two kinds of
RT surfaces, as drawn in Fig.(1) schematically. The turning point of the disk
type RT surface is $R(z_{t})=0$. Moreover, the surface of the disk type is
smooth at the bulk. The embedding scalar must satisfy
$R^{\prime}(z_{t})=\infty$. The surface ends at the tip of the soliton
$z_{t}=z_{+}$ for a cylinder case.

Figure 1: The RT surfaces corresponding to the small (left) and large (right)
subsystem in the AdS-Soliton background. If the subsystem is a spherical
region, then the RT surface has a disk (cylinder) topology for the small
(large) subsystem.

Varying $A$ in terms of $l$ and fixing $z=\epsilon$, the Hamiltonian-Jacobi
method (please refer to appendix A for a brief review of this method) becomes
[27]

$\displaystyle\dfrac{dA}{dl}=-H(z_{t})\dfrac{dz_{t}}{dl}-\Pi(\epsilon)\dfrac{dR(\epsilon)}{dl}=-\Pi(\epsilon)\dfrac{dR(\epsilon)}{dl},$
(25)

where

$\displaystyle\Pi=\dfrac{\partial\mathcal{L}}{\partial\dot{R}}=\Omega_{d-3}{L^{d-1}}L_{\phi}\dfrac{R^{d-3}f\dot{R}}{z^{d-1}\sqrt{1+f\dot{R}^{2}}},\quad
H=\Pi\dot{R}-\mathcal{L}=-\dfrac{\Omega_{d-3}L_{\phi}R^{d-3}{L^{d-1}}}{z^{d-1}\sqrt{1+f\dot{R}^{2}}}.$
(26)

The first term of (25) drops out due to the following IR boundary conditions

$\displaystyle R(z_{t})=0,\quad\dot{R}(z_{t})=\infty,\quad
H(z_{t})=0,\quad\text{for a disk},$
$\displaystyle\dfrac{dz_{t}}{dl}=\dfrac{d\epsilon}{dl}=0,\quad\text{for a
cylinder}.$

Equation (25) only depends on the solution near the $AdS$ boundary, and an
asymptotic expansion is useful.

We compute the asymptotic expansion of the embedding scalar near $z=0$. The UV
behavior of $R(z)$ has the following ansatz:

$\displaystyle
R(z)=l+b_{0}\log\dfrac{z}{l}+\sum_{n=1}\Big{(}a_{n}+b_{n}\log\Big{(}\dfrac{z}{l}\Big{)}\Big{)}z^{n},$
(27)

where the log term arises in (27) similar to the Fefferman-Graham expansion of
fields in the $AdS$ spacetime [38, 39]. We can determine coefficients $a_{n}$
and $b_{n}$ after substituting the ansatz mentioned above into (4). Below we
will analyze the cases with $d=4,5,3$ in detail.

###  4.1 $d=4$

Let us begin with the $d=4$ case, where the boundary QFT lives on
$R^{1,2}\times S^{1}$, and the topology of the entangling surface becomes
$S^{1}\times S^{1}$. By substituting expansion of $R(z)$ near the boundary
$z=0$ (27) into equation of motion, one can obtain

$\displaystyle
R(z)=l-\dfrac{z^{2}}{4l}+a_{4}(l)z^{4}+\dfrac{z^{4}}{32l^{3}}\log\dfrac{z}{l}+\dots$
(28)

Here the higher order terms can be determined by parameters $l$ and
$a_{4}(l)$. The coefficient $a_{4}(l)$ can not be determined from the UV
expansion of the EOM. Instead, $a_{4}(l)$ has information determined by the IR
boundary condition.

Substituting (28) into (25), the $l$ derivative of the surface becomes

$\displaystyle\dfrac{dA}{2\pi
L_{\phi}dl}=-4la_{4}(l)-\dfrac{3}{32l^{2}}+\dfrac{1}{2\epsilon^{2}}-\dfrac{1}{8l^{2}}\log\Big{(}\dfrac{\epsilon}{l}\Big{)}+\dots$
(29)

The divergent structure of (29) is

$\displaystyle\dfrac{1}{2\epsilon^{2}}-\dfrac{1}{8l^{2}}\log\Big{(}\dfrac{\epsilon}{l}\Big{)}.$
(30)

Divergent pieces are defined up to a logarithmic term. Because the inside of
the log term should be dimensionless, there are no unique ways to remove it.
However, the renormalized entanglement entropy is finite and does not depend
on the cut-off. Thus, it has unique descriptions.

We compute the renormalized entanglement entropy, corresponding to DOF at an
energy scale $E\sim 1/l$. According to (2), the $4d$ renormalized entanglement
entropy becomes

$\displaystyle
S_{ren}=L_{4}(l\partial_{l})S=\dfrac{1}{2}(l\partial_{l}+1)(l\partial_{l}-1)S=\dfrac{1}{2}(l^{2}S^{\prime\prime}+lS^{\prime}-S),$
(31)

where we have used the commutation relation
$[\partial_{l},l\partial_{l}]=\partial_{l}$. Recall that the central charge of
$d=4$ $\mathcal{N}=4$ SYM is $a={\pi^{5}L^{8}}/{\kappa_{10}^{2}}={N^{2}}/{4}$,
where $\kappa_{10}^{2}=\pi^{3}L^{5}\kappa^{2}$. The renormalized entanglement
entropy depends on the entangling surface and the trace anomaly [27] as
follows:

$\displaystyle S_{ren}^{d=4}=2a_{4}\int_{\partial
A}d^{2}x\sqrt{h}E_{2}+c_{4}\int_{\partial A}d^{2}x\sqrt{h}I_{2},$ (32)

where $\partial A$ is the entangling surface (see also [6, 34]). In 4
dimensions, we have an A-type anomaly $a_{4}$ and a B-type anomaly $c_{4}$ on
the entangling surface. $E_{2}$ is the Euler density and $I_{2}$ is a Weyl
invariant. For the spherical entangling surface $\partial A=S^{2}$, $\int
d^{2}x\sqrt{h}E_{2}=2$ and the Weyl invariant is zero. Thus,
$S_{ren}^{d=4}=4a_{4}$. The renormalized entanglement entropy will satisfy the
C-theorem in that case. Because the entangling surface is $S^{1}\times S^{1}$
for QFT on $R^{1,2}\times S^{1}$, however, the Euler number is zero. Only the
B-type anomaly remains. The renormalized entanglement entropy can be non-
monotonic since there is no universal C-theorem for B-type anomalies.

To compute the renormalized entanglement entropy, we need $a_{4}(l)$, $l$
appearing in (28) and $S$ (not confused with anomaly $a_{4}$ in (32)). A
numerical result of $a_{4}(l)$ is obtained in Fig. 2. For pure imaginary
$a_{\phi}$, it leads to results of geometric entropy [40]. The geometric
entropy is related to entanglement entropy via the double Wick rotation [41,
42]. The disk shape is dominant for small $l$, and the cylinder shape is
dominant for large $l$. $a_{4}(l)$ becomes multi-valued near the phase
transition at a critical length between disk and cylinder surfaces. Multi-
valued behavior is also observed for other $a_{\phi}$.

Figure 2:  $a_{4}$ as a function of $l$. The disk shape dominates the behavior
for small $l$, while the cylinder shape dominates the behavior for large $l$.
Left: $a_{\phi}=\frac{i}{2}$. The critical length of the phase transition is
$l_{c}=0.66$. Right: $a_{\phi}=\frac{1}{\sqrt{2}}$. The critical length is
$l_{c}=1.26$.

Figure 3: Left: the finite part of on-shell action $A_{fin}$ for
$a_{\phi}=\frac{i}{2},\ 0,\ \frac{2}{3},\ \frac{1}{\sqrt{2}}$. The figure
shows that the entanglement entropy increases with the Wilson line $a_{\phi}$
increase. The quantum phase transition happens when $M_{0}l_{c}=0.21,\ 0.23,\
0.32,\ 0.4$. Right: closed-up figure of $A_{fin}$ for
$a_{\phi}=\frac{1}{\sqrt{2}}$.  Figure 4: $S_{ren}$ for $d=4$. The
renormalized entanglement entropy is plotted for several $a_{\phi}$.
$a_{\phi}=\frac{i}{2},\ 0,\ \frac{2}{3},\ \frac{1}{\sqrt{2}}$ from the left to
the right. The renormalized entanglement entropy non-monotonically behaves
near critical lengths. The quantum phase transition happens when
$M_{0}l_{c}=0.21,\ 0.23,\ 0.32,\ 0.4$.  Figure 5: $S_{ren}$ for $d=4$. The
renormalized entanglement entropy for several $M_{0}$: $M_{0}=1/\pi,\ 2/5,\
3/5$ from the right to the left. The renormalized entanglement entropy non-
monotonically behaves near critical lengths. The quantum phase transition
occurs when $M_{0}l_{c}=0.24,\ 0.27,\ 0.4$. The figure shows that massive
modes $Ml>1$ decouple others soon. The final states will be product states.

Substituting the boundary expansion (28) into the action (21) and expanding at
a small $z$, we obtain the following divergent part of $A=A_{fin}+A_{div}$:

$\displaystyle\dfrac{A_{div}}{2\pi
L_{\phi}}=\dfrac{l}{2\epsilon^{2}}+\dfrac{1}{8l}\log\Big{(}\dfrac{\epsilon}{l}\Big{)}.$
(33)

Thus, the divergent structure of entanglement entropy $A$ is

$\displaystyle A=2\pi
L_{\phi}\Big{(}\dfrac{l}{2\epsilon^{2}}+\dfrac{\log\epsilon}{8l}\Big{)}+S_{fin}(l),$
(34)

where the log dependence is included in the finite part $S_{fin}$.
$S_{fin}=A_{fin}-2\pi L_{\phi}\dfrac{1}{8l}\log l$.

To find minimal surfaces, one must compute the on-shell action of (21).
Minimal surfaces between the disk and the cylinder dominate the phase in Fig.
3. The quantum phase transition occurs at a critical length $l_{c}$. Yellow
and Blue curves show that the confinement occurs and decreases DOF [43, 10,
11]. Recall that $a_{\phi}$ increases Casimir energy of dual QFT, and then the
entanglement entropy increases with the increase of $a_{\phi}$.

$S_{fin}$ is different from $S_{ren}$ because the cut-off dependence is
removed at $S_{ren}$. Actually, $S_{fin}$ is related to $S_{ren}$ via

$\displaystyle
S_{ren}=\dfrac{1}{2}(l^{2}S^{\prime\prime}_{fin}+lS^{\prime}_{fin}-S_{fin}).$
(35)

Considering (29), $S_{fin}(l)$ satisfies the following relation:

$\displaystyle S_{fin}^{\prime}(l)=2\pi
L_{\phi}\Big{(}-4la_{4}(l)-\dfrac{3}{32l^{2}}+\dfrac{1}{8l^{2}}\log(l)\Big{)}.$
(36)

Due to (36), $S^{\prime\prime}_{fin}$ or $A_{fin}^{\prime\prime}$ becomes

$\displaystyle\dfrac{S_{fin}^{\prime\prime}}{2\pi L_{\phi}}=$
$\displaystyle\dfrac{5}{16l^{3}}-\dfrac{\log(l)}{4l^{3}}-4la_{4}^{\prime}(l)-4a_{4}(l),$
(37) $\displaystyle\dfrac{A_{fin}^{\prime\prime}}{2\pi L_{\phi}}=$
$\displaystyle-\dfrac{1}{16l^{3}}-4la_{4}^{\prime}(l)-4a_{4}(l).$

Recall that (37) is the finite part of the minimal surface $A$. The
renormalized entanglement entropy $S_{ren}$ is finite. Substituting (36) and
(37) into (35), $S_{ren}$ is rewritten as follows:

$\displaystyle\dfrac{\kappa^{2}S_{ren}}{4\pi^{2}L_{\phi}}=-4l^{2}a_{4}(l)-2l^{3}a_{4}^{\prime}(l)+\dfrac{7}{64l}-\dfrac{A_{fin}}{2}$
$\displaystyle=-4l^{2}a_{4}(l)-2l^{3}a_{4}^{\prime}(l)+2\int^{l}l^{\prime}a_{4}(l^{\prime})dl^{\prime}+\dfrac{1}{8l}+c_{r},$
(38)

The coefficient of $1/l$ comes from only the logarithmic term of $S$, which is
brought from the Weyl anomaly. The formula (4.1) also depends on a function of
$a_{4}(l)$ unlike the holographic entanglement entropy of the spherical
entangling surface in 4 dimensions [6]. For the spherical entangling surface,
here, HREE describes the A type anomaly in CFT: $S_{ren}^{d=4}=4a_{4}$. Recall
that QFT dual to the AdS soliton with gauge potential breaks conformal
invariance. The first three terms in (4.1) represent terms breaking conformal
invariance and the last term is to realize the small $l$ limit of HREE (CFT
behavior) [34].

Fig. 4 shows the renormalized entanglement entropy $S_{ren}$ for several
$a_{\phi}$. It becomes non-monotonic behavior, which is similar to a behavior
of GPPZ flow [27]. Intuitively, the renormalized entanglement entropy is also
a detector of the effective DOF of entangling states at the energy scale
$El\sim 1$. For large energy $E/M_{0}\sim 1/(M_{0}l)>1$, $S_{ren}$ decreases
as a function of $lM_{0}$. Because the degrees of freedom with Wilson lines
contribute to large $a_{\phi}$ and energy, however, $S_{ren}$ slowly decreases
until the critical length (see green and red curves). For small $E/M_{0}<1$,
the renormalized entanglement entropy can not detect effective DOF and can
almost become a constant. Even if the renormalized entanglement entropy
increases after the quantum phase transition, it satisfies a kind of
C-theorem: $S_{ren}(l\to 0)>S_{ren}(l\to\infty)$.

###  4.2 $d=5$

We proceed with $d=5$ case, where the AdS boundary expansion for $d=5$ has the
following form

$\displaystyle
R(z)=l-\dfrac{z^{2}}{3l}-\dfrac{5z^{4}}{54l^{3}}+z^{5}a_{5}(l)\dots,$ (39)

Similar to the case $a_{4}(l)$ discussed in the previous section, the
parameter $a_{5}(l)$ is not determined from the AdS boundary expansion but
determined from the IR boundary condition. Numerically, $a_{5}(l)$ is plotted
in Fig. 6 for fixed values of $M_{0}$ and $a_{\phi}$. Once we have the
expansion (39), by making use of (25), we can obtain the following derivative

$\displaystyle\frac{1}{\Omega_{2}L_{\phi}}\dfrac{dA}{dl}=\dfrac{2l}{3\epsilon^{3}}-5l^{2}a_{5}(l),$
(40)

where $\Omega_{2}=4\pi$. The first term is the cut-off dependent term. There
are no log divergences. The finite part of $dA/dl$ is determined by the second
term, which is similar to $d=3$ cases: a term corresponding to trace anomaly
is not present in odd dimensions.

Figure 6: $a_{5}$ as function of $l$ with
$T=\frac{1}{\pi},a_{\phi}=\frac{1}{2}$ in the case of $d$=5. The red line
corresponds to disk shaped RT surface and the blue line corresponds to a
cylinder shape. The critical length of the phase transition is $l_{c}=1.57$.

Figure 7:  Left: the finite part of $A$ for $a_{\phi}=i/2,\ 0,\ 1/2,\
2/\sqrt{15}$ from left to right. Right: close-up version of $a_{\phi}=1/2$
curve. There is a phase transition at the critical length.

Figure 8: Left: The renormalized entanglement entropy for $a_{\phi}=i/2,\ 0,\
1/2,\ 2/\sqrt{15}$. Renormalized entanglement entropy non-monotonically
behaves. For large $a_{\phi}$, DOF slowly decreases. Right: The renormalized
EE for $a_{\phi}=2/\sqrt{15}$. Mass is changed from $M_{0}=1/2,\ 2/5,\ 1/\pi$.
Massive modes quickly decrease.

Substituting the expansion (39) into the action (21) and expanding in terms of
small $z$, we can obtain the divergent part of entanglement entropy which has
the following divergent structure

$\displaystyle A(l)=4\pi
L_{\phi}\Big{(}\dfrac{l^{2}}{3\epsilon^{3}}-\dfrac{4}{9\epsilon}\Big{)}+A_{fin}(l).$
(41)

Since the second term in the parentheses does not depend on $l$,
$O(\epsilon^{-1})$ part is absent in (40). The $l$ dependence of $A(l)$
differs from the CFT one. We plotted the finite part of the entanglement
entropy in Fig. 7, where it is shown that the disk-shaped RT surface dominates
the behavior for small $l$ and cylinder shape dominates for large $l$. The
finite part $A_{fin}$ has the quantum phase transition at the critical length.
Making use of (40) and (41), the finite part satisfies the following relation

$\displaystyle\dfrac{A_{fin}^{\prime}(l)}{4\pi L_{\phi}}=-5l^{2}a_{5}(l).$
(42)

Next consider the 5-dimensional renormalized entanglement entropy defined in
(2), which is

$\displaystyle S_{ren}=\dfrac{1}{3}l\partial_{l}(l\partial_{l}-2)S.$ (43)

Substituting (41), the renormalized entanglement entropy can be rewritten as

$\displaystyle
S_{ren}(l)=\dfrac{l}{3}(-A_{fin}^{\prime}(l)+lA_{fin}^{\prime\prime}(l))=-\dfrac{5}{3}\cdot
4\pi L_{\phi}\cdot l^{3}(a_{5}(l)+la_{5}^{\prime}(l)).$ (44)

The renormalized entanglement entropy is plotted in Fig. 8 with different
$a_{\phi}$. The renormalized entanglement entropy counts the DOF of the
entangling states at an energy scale $E\sim 1/l$. It has non-monotonic
behaviors. The quantum phase transition happens at critical limits. And
$S_{ren}$ approaches to zero at large distances. Fig. 8 (right) shows that
massive DOF decouples other modes at low energy soon. This behavior is easily
seen when $a_{\phi}=0$, where scaling symmetry $M_{0}\to\lambda M_{0},\
l\to\lambda^{-1}l$ arises. This is the symmetry of the action and EOM. Scaling
symmetry simultaneously rotates $a_{\phi}$ when it is nonzero. The scaling
symmetry implies that the critical length is inversely proportional to
$M_{0}$: $l_{c}=c_{5}/M_{0}$. The quantum phase transition quickly occurs for
large masses, while it slowly occurs for small masses. After the phase
transition, the renormalized entanglement entropy gradually becomes constant.

##  5 $d=3$ (a striped shape)

In this section, we analyze the holographic entanglement entropy for $d=3$.
The configuration for $d=3$ is equal to a striped boundary shape. We start
with $d$ dimensional striped shapes, and $d=3$ is a special case. To consider
a striped shape, we replace $R^{2}d\Omega_{d-3}$ with $\sum dx_{\perp}^{2}$ in
$d+1$ dimensional $AdS$ soliton with a gauge field (12) as follows:

$\displaystyle
ds_{d+1}^{2}=\dfrac{L^{2}}{z^{2}}\Big{(}\dfrac{dz^{2}}{f_{d}(z)}+f_{d}(z)d\phi^{2}-dt^{2}+dR^{2}+\sum
dx_{\perp}^{2}\Big{)},$

where $R$ is along ($-\infty,\ \infty$) unlike the radial direction of polar
coordinate systems.

Figure 9: Left: $s_{3}(l)=dA_{s}/(3L_{\phi}dl)$ as a function of $lM_{0}$. The
curve is for $a_{\phi}=i/2,\ 0,\ 1,\ 2/\sqrt{3}$ from the left to the right.
The phase transition happens at critical lengths $l_{c}=0.15,\ 0.19,\ 0.24,\
0.34$ in units of $1/M_{0}$ from the left to the right, respectively.
$s_{3}(l)$ has two values. The physical curve (an upper one) is consistent
with the strong subadditivity. Right: $S_{ren}$ for $d=3$. The curve is for
$a_{\phi}=0,\ 1,\ 2/\sqrt{3}$ from the left to the right. $S_{ren}$
monotonically decreases as a function of $lM_{0}$. $S_{ren}$ becomes 0 at a
large distance. It implies that theory is a product state there [44]. Figure
10: $S_{ren}$ for $d=3$. The curve is for $a_{\phi}=2/\sqrt{3}$. $M_{0}=3/5,\
2/5,\ 1/\pi$ from the left to the right. $S_{ren}$ monotonically decreases as
a function ozimf $lM_{0}$. The phase transition happens at critical lengths
$l_{c}M_{0}=0.19,\ 0.23,\ 0.34$ from the left to the right, respectively. It
implies that massive modes $M>1/l$ quickly decouple others and remain product
states.

We consider a strip with a length of $l$ along the $R$ direction ($-l/2\leq
R\leq l/2$) and choose $R=R(z)$ as an embedding scalar. The surface action
becomes

$\displaystyle A_{s}=\int d^{d-1}x\mathcal{L}=2V_{d-3}L_{\phi}{L^{d-1}}\int
dz\dfrac{1}{z^{d-1}}\sqrt{1+f\dot{R}^{2}},$ (45)

where $V_{d-3}$ was used to replace the volume of $d-3$ dimensional space
spanned by $x_{\perp}$. Factor 2 comes from two contributions of the minimal
surface. Note that the lagrangian density of (45) does not depend explicitly
on $R$. This simplifies the analysis for a rectangular region more than for a
circular region. The momentum doesn’t explicitly depend on $R$ for a striped
boundary shape. Solving the condition $\Pi=$const and imposing the IR boundary
condition $dz/dR|_{z=z_{t}}=0$, we have

$\displaystyle\dot{R}=\dfrac{1}{\sqrt{f_{d}(z)\Big{(}\dfrac{f_{d}(z)z_{t}^{2(d-1)}}{f_{d}(z_{t})z^{2(d-1)}}-1\Big{)}}}.$
(46)

This formula demonstrates that $z=z_{t}$ is the turning point
$dz/dR|_{z=z_{t}}=0$. The equation (46) gives the profile of the minimal
surface satisfying $R(\epsilon)=-l/2$ and $R(z_{t})=0$.

Using the $AdS$ boundary expansion $R^{\prime}(z)=ds_{d}(l)z^{d-1}+\dots$,
$R(z)$ is expanded as follows

$\displaystyle R(z)=-\dfrac{l}{2}+s_{d}(l)z^{d}\dots,$ (47)

where

$\displaystyle s_{d}(l)=\dfrac{\sqrt{f_{d}(z_{t})}}{dz_{t}^{d-1}}.$ (48)

The Hamilton-Jacobi equation for a striped shape has the same form as (25)

$\displaystyle\dfrac{dA_{s}}{dl}=-\Pi(\epsilon)\dfrac{dR(\epsilon)}{dl},$ (49)

where

$\displaystyle\Pi=\dfrac{\partial\mathcal{L}}{\partial\dot{R}}=2V_{d-3}L^{d-1}L_{\phi}\dfrac{f\dot{R}}{z^{d-1}\sqrt{1+f\dot{R}^{2}}}$
(50)

and $H(z_{t})=0$ due to the IR boundary condition. When (46) is substituted
into (25), the $l$ derivative of the surface becomes

$\displaystyle\dfrac{dA_{s}}{V_{d-3}L_{\phi}L^{d-1}dl}=ds_{d}(l)=\dfrac{\sqrt{f_{d}(z_{t})}}{z_{t}^{d-1}}.$
(51)

Note that the minimal surface must satisfy strong subadditivity. It requires
that the minimal surface must be concave [45] as follows:

$\displaystyle
A_{s}^{\prime\prime}=dV_{d-3}L_{\phi}L^{d-1}s_{d}^{\prime}(l)=V_{d-3}L_{\phi}L^{d-1}\dfrac{\partial
z_{t}}{\partial l}\partial_{z_{t}}(ds_{d}(l))\leq 0.$ (52)

Actually, one can show the following inequality:

$\displaystyle\partial_{z_{t}}(ds_{d}(l))\equiv\dfrac{h(z_{t})}{2\sqrt{f_{d}(z_{t})}}=\dfrac{(-2(d-1)z_{t}^{-d}+(d-2)(1-a_{\phi}^{2}z_{0}^{2})z_{0}^{-d})}{2\sqrt{f_{d}(z_{t})}}<0.$
(53)

because $h(z)$ satisfies the following condition: 444 When $a_{\phi}$ is pure
imaginary $a_{\phi}=ia_{\phi}^{\prime}$, the second condition is replaced with
$\displaystyle
h(z_{0})=-\dfrac{4\pi(\sqrt{4\pi^{2}+d(d-2)(a_{\phi}^{\prime}/M_{0})^{2}}-2\pi)}{(a_{\phi}^{\prime}/M_{0})^{2}(d-2)}<0.$
(54)

$\displaystyle h(0)=-\infty,\quad
h(z_{0})=(-d-(d-2)(a_{\phi}z_{0})^{2})z_{0}^{-d}<0,\quad
h^{\prime}(z)=2d(d-1)z^{-d-1}>0.$ (55)

According to [26], in addition, $\partial z_{t}/\partial l$ has two values. It
is positive for a minimal surface and negative for an unphysical curve.

Next, we consider the case in which the minimal surface is disconnected. We
focus on $d=3$. From

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\frac{\partial\mathcal{L}}{\partial\dot{R}}=\frac{\partial\mathcal{L}}{\partial
R}=0$ (56)

where

$\displaystyle\mathcal{L}\equiv\frac{1}{z^{2}}\sqrt{1+f\dot{R}^{2}},$ (57)

it is easy to see that one of the solutions is $R=\text{const}$, which is for
the disconnected minimal surface. Then the area of the surface is

$\displaystyle
A_{s}=2L_{\phi}\int_{\epsilon}^{z_{t}}\mathrm{d}z\frac{1}{z^{2}}=2L_{\phi}\left(\frac{1}{\epsilon}-\frac{1}{z_{t}}\right),$
(58)

and the corresponding entanglement entropy

$\displaystyle S_{EE}=\frac{4\pi
L_{\phi}}{\kappa}\left(\frac{1}{\epsilon}-\frac{1}{z_{t}}\right),$ (59)

where $z_{t}=z_{+}$. The finite part of $S_{EE}$ is $-4\pi L_{\phi}/(\kappa
z_{+})$. For disconnected surfaces, $s_{3}(l)=0$.

One can compute the renormalized entanglement entropy from the entanglement
entropy $S_{EE}$. We apply the renormalized entanglement entropy formula
defined in (2): $S_{ren}=l\partial_{l}S$ substituting $d=3$. Because dual
3-dimensional QFT theory is defined on $R^{1,1}\times S^{1}$, the renormalized
entanglement entropy will give the result of $2d$ QFT on $R^{1,1}$ in the low
energy limit.

The renormalized entanglement entropy can be written in terms of $l$ and
$s_{3}(l)$ as follows:

$\displaystyle\dfrac{\kappa^{2}}{2\pi
L_{\phi}L^{2}}S_{ren}=\dfrac{ldA_{s}}{L_{\phi}L^{2}dl}=3ls_{3}(l)=\dfrac{l\sqrt{f(z_{t})}}{z_{t}^{2}},$
(60)

where $S=\frac{2\pi}{\kappa^{2}}A_{s}$. Note that the central charge of $3d$
dual CFT is $32L^{2}/(\pi G_{4})=2^{6}\sqrt{2}N^{3/2}\sqrt{k}/(3\pi)$, where
$k$ is the Chern-Simons level. For the disconnected surface, we see that the
entanglement entropy does not depend on the size $l$ of the entangling surface
at the boundary, which means

$\displaystyle S_{ren}=l\partial_{l}S_{EE}=0.$ (61)

We plotted $dA_{s}/(3L_{\phi}dl)=s_{3}(l)$ in Fig. 9 (left). Because the
concavity of the entanglement entropy [46][47] means $A_{s}^{\prime\prime}\leq
0$, $A_{s}^{\prime}$ monotonically decreases as a function of $l$. We plotted
the renormalized entanglement entropy in Fig. 9 (right). The renormalized
entanglement entropy detects the DOF of the entangling states at the energy
scale $El\sim 1$ again. Because massive degrees of freedom decouple in the low
energy limit, $S_{ren}$ decreases as a function of $l$ (with energy $E\sim
1/l$). 555Another entropic c function non-monotonically behaves in [26] and it
does not satisfy a c-theorem. The non-monotonic behavior occurs due to the
competition between a power of $l$ and $dA_{s}/dl$. This monotonic behavior
($S_{ren}^{\prime}(l)\leq 0$) will also be the consequence of Lorentz symmetry
and strong subadditivity [45]. The quantum phase transition happens at
critical lengths $l_{c}M_{0}=0.19,\ 0.24,\ 0.34$ for $a_{\phi}=0,\ 1,\
2/\sqrt{3}$, respectively. $S_{ren}$ becomes 0 at large distances. Fig. 10
changes $M_{0}$ after fixing $a_{\phi}$. It shows that massive modes
$M_{0}>1/l$ quickly decouple others at low energy, remaining product states.
When $a_{\phi}=0$, one can recover scaling symmetry $M_{0}\to\lambda M_{0}$,
$l\to\lambda^{-1}l$, in addition to bulk parameters. This is the symmetry of
the action and EOM. Scaling symmetry simultaneously changes $a_{\phi}$ if it
is nonzero. Thus, the critical length is the same as those for different
$M_{0}$ and is given by $M_{0}=c/l_{c}$, where $c$ is a constant. The phase
transition happens soon for large masses and slowly for small masses.

Our results agree with the fact that the renormalized entanglement entropy is
equal to the formula of the entropic $c$-function $l\partial_{l}S_{EE}$ on
$R^{1,1}$, which becomes observable in renormalizable theory. The entropic
$c$-function on $R^{1,1}$ counts DOF and depends on the radius $l(\sim 1/E)$
including the information in $S_{EE}(l_{1})-S_{EE}(l_{2})$ ($l_{1}>l_{2}$). It
is known that the $c$ function for a massive scalar field exponentially
decreases for large $r$ [8]. Finally, there remains no degree of freedom (zero
entropic $c$ function) at a large distance. Recall that the trace anomaly
exists in 2 dimensions. The monotonic quantity is the Euler term in the trace
anomaly there.

###  5.1 Small subregions

By employing the our previous results, where $z_{t}$ can be expressed in terms
of $l$ for small $l$ [26]

$\displaystyle z_{t}=$
$\displaystyle\dfrac{l}{2e_{-1}-2k_{-1}}+\dfrac{l^{4}\bar{a}_{\phi}\left(-4e_{-1}+4k_{-1}+\pi\right)}{256z_{+}^{3}\left(e_{-1}-k_{-1}\right){}^{5}}+\dfrac{l^{5}\left(1-\bar{a}_{\phi}\right)}{160z_{+}^{4}\left(e_{-1}-k_{-1}\right){}^{5}}$
(62)
$\displaystyle+\dfrac{l^{7}\bar{a}_{\phi}^{2}\left(-302e_{-1}k_{-1}+126e_{-1}^{2}-84\pi
e_{-1}+176k_{-1}^{2}+84\pi
k_{-1}+21\pi^{2}\right)}{172032z_{+}^{6}\left(e_{-1}-k_{-1}\right){}^{9}}+\dots,$

we obtain

$\displaystyle\dfrac{dA_{s}}{L_{\phi}{L^{2}}dl}=$
$\displaystyle\dfrac{4\left(e_{-1}-k_{-1}\right){}^{2}}{l^{2}}-\dfrac{l\pi\bar{a}_{\phi}}{16\left(z_{+}^{3}\left(e_{-1}-k_{-1}\right){}^{2}\right)}+\dfrac{9l^{2}\left(\bar{a}_{\phi}-1\right)}{40z_{+}^{4}\left(e_{-1}-k_{-1}\right){}^{2}}$
(63)
$\displaystyle+\dfrac{5l^{4}\bar{a}_{\phi}^{2}\left(80e_{-1}k_{-1}-80k_{-1}^{2}-21\pi^{2}\right)}{86016z_{+}^{6}\left(e_{-1}-k_{-1}\right){}^{6}}+\dots$
$\displaystyle=$
$\displaystyle\dfrac{8\pi^{3}}{l^{2}\Gamma\left(\frac{1}{4}\right)^{4}}-\dfrac{8l\bar{a}_{\phi}\Gamma\left(\frac{5}{4}\right)^{4}}{\pi^{2}z_{+}^{3}}+\dfrac{9(\bar{a}_{\phi}-1)l^{2}\Gamma\left(\frac{1}{4}\right)^{4}}{80\pi^{3}z_{+}^{4}}+\dfrac{5(20-21\pi)\bar{a}_{\phi}^{2}l^{4}\Gamma\left(\frac{1}{4}\right)^{12}}{688128\pi^{8}z_{+}^{6}}+\dots,$

where we have defined $\bar{a}_{\phi}=1-(z_{+}a_{\phi})^{2}$. For the notation
$e_{-1},k_{-1}$ please refer to [26] for more details. The expansion (63)
agrees with the holographic entanglement entropy in [26], and the formula (51)
is also valid for other dimensions realizing the holographic entanglement
entropy. The result is compared with our previous paper [26] in the appendix
B.

##  6 Summary and discussion

We computed holographic EE and the renormalized EE in the AdS soliton with the
gauge potential for several dimensions. The disk shape of the minimal surface
was dominant for small $l$, and the cylinder shape was dominant for large $l$
as similar to [31]. The quantum phase transition occurs at a critical size of
the subregion.

The renormalized EE, a universal part of EE independent of the cutoff, was
computed by operating differentiation on EE [27]. By containing modes with KK
mass and considering the low energy limit, we continuously derived from odd
dimensional renormalized EE to the formula of $d-1$ dimensional renormalized
EE. 666Note that the topology of the subregion is not one ball $B^{d-1}$ but
the ball and a circle $B^{d-2}\times S^{1}$ with the periodicity $L_{\phi}$.
Interpreting this $S^{1}$ as one perpendicular direction to $B^{d-2}$ with the
endpoint identified will be convenient. Actually, the $\phi$ circle shrinks to
zero at the tip of the AdS soliton ($z=z_{+}$), which is probed for large $l$.
The logarithmic term is absent since we don’t have a Weyl anomaly in odd
dimensions. This is a sort of topology change in the entanglement entropy. In
any dimension, massive modes $M_{0}l>1$ decouple others as a decrease of
energy as shown in Fig. 10 and then product states are retained. For high
energy limit ($l\ll L_{\phi}$), the renormalized EE recovers behaviors of the
original dimensions because the renormalized EE measures the degrees of
freedom in a state with energy $E\sim 1/l$. Because the degrees of freedom
with Wilson lines contribute to large $a_{\phi}$ and high energy, the
renormalized EE slowly changes until the critical length (see Fig. 9). The
paper [49] also tracked the entanglement entropy across dimensions and found
transitions.

In section 5, we analyzed striped surfaces for $d=3$. Our results demonstrated
that the renormalized EE is positive (non-negative) and satisfies the C
theorem. After dimensional reduction, the renormalized entanglement entropy of
2d QFT with Kaluza-Klein modes will also be consistent with the C theorem of
the 2-dimensional entropic $c-$function. We showed that when $d=3$, the
renormalized EE for the entangling boundary with a striped shape decreases
monotonically and jumps to zero when the size of the entangling surface
becomes large, which means that it probes a first-order phase transition and
the corresponding boundary field theory runs into a product state at the low
energy scale. Unlike the generalized entropic $c-$function studied in [26],
the renormalized EE here behaves monotonically even for the large value of
gauge potential $a_{\phi}$, and since $a_{\phi}$ can increase the degrees of
freedom and the increase of $a_{\phi}$ leads to the increase of the
renormalized EE, which implies that the renormalized EE is counting the
degrees of freedom (DOF) of the boundary field theory.

We analyzed the striped entangling surface in the previous analysis and
computed an entropic $c-$function. This entropic $c-$function is always
positive and non-monotonically behaves. The phase transition happens at a
critical length. The non-monotonic behavior is caused by the effective DOF of
Wilson lines along the $S^{1}$ direction. That is, effective DOF increases the
renormalized entanglement entropy. It implies that this Wilson line will
decrease the mass of particles such as glueballs [48] because particles of
small mass contribute to the entropic $c-$function at large $l$. On the other
hand, in higher dimensions, HREE can become negative near the phase transition
point, different from an entropic $c-$function. This will be considered as an
artifact in the large $N$ and strongly coupled limit. The two are similar in
the quantum phase transition’s presence and the Wilson lines’ effect. The
gauge potential will decrease the mass of particles such as glueballs. Thus,
more DOF will contribute to HREE and let it increase at a large distance.

## Acknowledgments

We would like to thank X. Chen and P. Zhang for their helpful discussion. S.H.
would appreciate the financial support from the Fundamental Research Funds for
the Central Universities and Max Planck Partner Group and the Natural Science
Foundation of China (NSFC) Grants No. 12075101 and No. 12235016. This work is
also supported by the National Natural Science Foundation of China
(No.12105113)

##  Appendix A Hamilton-Jacobi equations

In the appendix, we give a brief review of the Hamilton-Jacobi method used in
analyzing minimal surfaces in (25). We introduce the following action

$\displaystyle S=\int^{t_{2}}_{t_{1}}\mathcal{L}(q,\dot{q},t)dt.$ (64)

We assume that the field $q$ can change at the boundary times $t_{1}$ and
$t_{2}$. Moreover, we allow changes of times $t_{1}(t_{2})$ into
$t_{1}^{\prime}(t_{2}^{\prime})$, respectively. The variation of the action
becomes

$\displaystyle\delta S=$
$\displaystyle\int^{t_{2}^{\prime}}_{t_{1}^{\prime}}dt\mathcal{L}(q^{\prime},\dot{q}^{\prime},t)dt-\int^{t_{2}}_{t_{1}}dt\mathcal{L}(q,\dot{q},t)dt$
$\displaystyle=$
$\displaystyle\int^{t_{2}^{\prime}}_{t_{2}}\mathcal{L}(q^{\prime},\dot{q}^{\prime},t)dt+\int^{t_{2}}_{t_{1}}(\mathcal{L}(q^{\prime},\dot{q}^{\prime},t)-\mathcal{L}(q,\dot{q},t))dt+\int^{t_{1}}_{t_{1}^{\prime}}\mathcal{L}(q^{\prime},\dot{q}^{\prime},t)dt$
$\displaystyle=$
$\displaystyle\mathcal{L}(q^{\prime},\dot{q}^{\prime},t_{2})\delta
t_{2}-\mathcal{L}(q^{\prime},\dot{q}^{\prime},t_{1})\delta
t_{1}+\dfrac{\partial\mathcal{L}}{\partial\dot{q}}\delta
q\Big{|}_{t_{2}}-\dfrac{\partial\mathcal{L}}{\partial\dot{q}}\delta
q\Big{|}_{t_{1}},$

where the EOM is used in the second line. We did not assume $\delta q=0$ at
the time boundary $t_{1}$ and $t_{2}$. We use the following transformation

$\displaystyle\dfrac{\partial\mathcal{L}}{\partial\dot{q}}\delta
q\Big{|}_{t_{i}}=\dfrac{\partial\mathcal{L}}{\partial\dot{q}_{i}}\delta
q_{i}-\dfrac{\partial\mathcal{L}}{\partial\dot{q}_{i}}\dot{q}_{i}\delta
t_{i},$ (65)

where $\delta q_{i}=q^{\prime}(t_{i}^{\prime})-q(t_{i})$. The variation of the
action becomes the total derivative as follows:

$\displaystyle\delta S=-\mathcal{H}\delta t_{2}+\mathcal{H}\delta
t_{1}+\dfrac{\partial\mathcal{L}}{\partial\dot{q}}\delta
q_{2}-\dfrac{\partial\mathcal{L}}{\partial\dot{q}}\delta q_{1}.$ (66)

It shows that

$\displaystyle\dfrac{\partial S}{\partial
t_{2}}=-\mathcal{H},\quad\dfrac{\partial S}{\partial
t_{1}}=\mathcal{H},\quad\dfrac{\partial S}{\partial
q_{2}}=\dfrac{\partial\mathcal{L}}{\partial\dot{q_{2}}}=p_{2},\quad\dfrac{\partial
S}{\partial q_{1}}=-\dfrac{\partial\mathcal{L}}{\partial\dot{q_{1}}}=-p_{1}.$
(67)

Thus, the motion in which (66) becomes the total derivative is possible.

##  Appendix B Hamilton-Jacobi equations for small $l$

We derive the Hamilton-Jacobi equation for a striped shape for $d=5,\ 6$ in
this appendix. When we restrict to the small size $l$ of the subregion, we can
use the analytic expression. The Hamilton Jacobi equation is given by eq. (49)
and eq. (51).

We consider $d=5$ first. The turning point, $z_{t}$ is expanded in the small
$l$ limit (see [26]),

$\displaystyle z_{t}=$
$\displaystyle\dfrac{5l\Gamma\left(\frac{9}{8}\right)}{2\sqrt{\pi}\Gamma\left(\frac{13}{8}\right)}-\dfrac{15625l^{6}\bar{a}_{\phi}\left(3\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{9}{8}\right)^{6}\Gamma\left(\frac{13}{8}\right)-5\Gamma\left(\frac{9}{8}\right)^{7}\Gamma\left(\frac{5}{4}\right)\right)}{1536\pi^{3}z_{h}^{5}\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{13}{8}\right)^{7}}$
(68)
$\displaystyle-\dfrac{1953125l^{9}(\bar{a}_{\phi}-1)\Gamma\left(\frac{9}{8}\right)^{9}}{2304\pi^{9/2}z_{h}^{8}\Gamma\left(\frac{13}{8}\right)^{9}}+O(l^{11}).$

The Hamilton-Jacobi equation becomes

$\displaystyle\dfrac{dA_{s}}{V_{2}L_{\phi}L^{4}dl}=\dfrac{\sqrt{f(z_{t})}}{z_{t}^{4}}$
(69) $\displaystyle=$
$\displaystyle\dfrac{16\pi^{2}\Gamma\left(\frac{13}{8}\right)^{4}}{625l^{4}\Gamma\left(\frac{9}{8}\right)^{4}}-\dfrac{25l\bar{a}_{\phi}\Gamma\left(\frac{9}{8}\right)^{2}\Gamma\left(\frac{5}{4}\right)}{12\sqrt{\pi}z_{h}^{5}\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{13}{8}\right)^{2}}+\dfrac{15625(\bar{a}_{\phi}-1)l^{4}\Gamma\left(\frac{9}{8}\right)^{4}}{288\pi^{2}z_{h}^{8}\Gamma\left(\frac{13}{8}\right)^{4}}+O(l^{6}).$

For $d=6$, $z_{t}$ is expanded in terms of small $l$ as follows:

$\displaystyle z_{t}=$
$\displaystyle\dfrac{3l\Gamma\left(\frac{11}{10}\right)}{\sqrt{\pi}\Gamma\left(\frac{8}{5}\right)}-\dfrac{2187\bar{a}_{\phi}l^{7}\left(7\Gamma\left(\frac{7}{10}\right)\Gamma\left(\frac{8}{5}\right)-12\Gamma\left(\frac{11}{10}\right)\Gamma\left(\frac{6}{5}\right)\right)\Gamma\left(\frac{11}{10}\right)^{7}}{70\pi^{7/2}z_{h}^{6}\Gamma\left(\frac{7}{10}\right)\Gamma\left(\frac{8}{5}\right)^{8}}$
(70)
$\displaystyle-\dfrac{885735(\bar{a}_{\phi}-1)l^{11}\Gamma\left(\frac{11}{10}\right)^{11}}{22\pi^{11/2}z_{h}^{10}\Gamma\left(\frac{8}{5}\right)^{11}}+O(l^{13}).$

The Hamilton-Jacobi equation becomes

$\displaystyle\dfrac{dA_{s}}{V_{3}L_{\phi}L^{5}dl}=\dfrac{\sqrt{f(z_{t})}}{z_{t}^{5}}=\dfrac{\pi^{5/2}\Gamma\left(\frac{8}{5}\right)^{5}}{243l^{5}\Gamma\left(\frac{11}{10}\right)^{5}}-\dfrac{18l\bar{a}_{\phi}\Gamma\left(\frac{11}{10}\right)^{2}\Gamma\left(\frac{6}{5}\right)}{7\sqrt{\pi}z_{h}^{6}\Gamma\left(\frac{7}{10}\right)\Gamma\left(\frac{8}{5}\right)^{2}}$
$\displaystyle+l^{5}(\bar{a}_{\phi}-1)\dfrac{4374\Gamma\left(\frac{11}{10}\right)^{5}}{11\pi^{5/2}z_{h}^{10}\Gamma\left(\frac{8}{5}\right)^{5}}+O(l^{7}).$
(71)

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