IR physics from the holographic RG flow

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

${}^{b}\,$College of General Education, Kookmin University, Seoul, 02707,
Korea

ABSTRACT

Applying the holographic method, we investigate an RG flow and IR physics
holographically when a two-dimensional conformal field theory is deformed by a
relevant scalar operator. To do so, we first assume an RG flow from a UV to
new IR CFT. On the dual gravity side, such an RG flow can be described by
rolling down of a bulk scalar field from an unstable to stable equilibrium
point. After considering a simple scalar potential allowing several local
extrema, we study the change of a ground state along the RG flow. We show that
the entanglement entropy at an IR fixed point leads to a logarithmic
divergence due to restoring of the conformal symmetry. We study how the change
of the ground state affects two-point functions. In the probe limit, we
numerically calculate the change of a conformal dimension caused by the
modification of the ground state. We further study the analytic form of the IR
conformal dimension which is perfectly matched to the numerical result.

##  1 Introduction

After the holography proposal, there were many attempts to understand various
strongly interacting systems on the dual gravity theory side. The holographic
relation or AdS/CFT correspondence is manifest for large supersymmetric
theories like $N=4$ Super Yang-Mills theory and ABJM theory Maldacena (1998a,
b); Witten (1998a, b); Gubser  _et al._ (2002); Aharony  _et al._ (2008);
Klebanov and Torri (2010). In these cases, IR physics is trivial because of
the conformal symmetry. To investigate nontrivial IR physics holographically,
one needs to break the conformal symmetry. In this work, we take into account
a relevant deformation of conformal field theory (CFT) and investigate new
macroscopic features at the IR fixed point by applying the holographic method.

The holography is one of the useful techniques to understand strongly
interacting systems. Although the dual gravity theories of maximally
supersymmetric gauge theories have been known, it still remains as an
unresolved issue to find the dual gravity of non-supersymmetric theories
appearing in nuclear and condensed matter physics. Nevertheless, if one can
find the dual gravity theories of real physical systems, one get more
information about new physics laws governing real systems. Since a relevant
deformation breaks the conformal symmetry, it usually generates a nontrivial
renormalization group (RG) flow and may allow a new IR CFT. In this case, the
appearance of the IR CFT is due to the non-perturbative RG flow involving all
quantum corrections. In the traditional quantum field theory (QFT),
unfortunately, it is very difficult to figure out such a non-perturbative RG
flow. However, one can investigate a non-perturbative RG flow by applying the
holographic method de Boer  _et al._ (2000); Petrini and Zaffaroni (1999);
Freedman  _et al._ (1999); de Haro  _et al._ (2001); Bianchi  _et al._ (2002);
Myers and Sinha (2011); Casini  _et al._ (2015); Gukov (2016); Kiritsis  _et
al._ (2017); Ghosh  _et al._ (2018); Park  _et al._ (2018); Gürsoy  _et al._
(2018); Ghosh  _et al._ (2019); Park and Hun Lee (2020).

Recently, quantum entanglement entropy has been widely studied in high energy
physics as well as quantum information and condensed matter theory Ryu and
Takayanagi (2006a, b); Klebanov  _et al._ (2008); Karch  _et al._ (2010);
Takayanagi (2011); Ogawa  _et al._ (2012); Erdmenger  _et al._ (2016a, b);
Rangamani and Takayanagi (2017); Takayanagi and Umemoto (2018). The
entanglement entropy is one of the important quantities to characterize
quantum correlation between subregions and plays a role of an order parameter
in quantum phase transitions Vidal  _et al._ (2003); Levin and Wen (2006). It
was well known that the entanglement entropy in the UV limit shows a universal
feature proportional to the area of the entangling surface Srednicki (1993);
Calabrese and Cardy (2004). For two-dimensional CFT, in particular, the
entanglement entropy diverges logarithmically and the coefficient of the
logarithmic divergence corresponds to the central charge of CFT. In the
holography studies, it was proposed that the entanglement entropy can be
described by the area of a minimal surface extending to the dual geometry Ryu
and Takayanagi (2006a, b); Casini  _et al._ (2011). It was also shown that the
holographic method correctly reproduces the area law of the entanglement
entropy in the UV limit. At a phase transition point in the IR region, another
logarithmic divergence can occur due to the restoration of a conformal
symmetry. This is also true at an IR fixed point. To see this, we consider a
three-dimensional Einstein-scalar gravity theory which corresponds to a two-
dimensional QFT flowing from a UV to IR CFT . On this background, we
investigate the RG flow of the entanglement entropy which is crucially
associated with the change of the ground state. We first numerically study the
RG flow of the entanglement entropy and shows that the entanglement entropy
diverges logarithmically at an IR fixed point. We also study the IR
entanglement entropy analytically by using perturbation near the IR fixed
point. We show how the entanglement entropy relies on the coupling constant
along the RG flow and find that the entanglement entropy diverges at a
critical coupling constant which is the value at the IR fixed point.

We further investigate the correlation function of a local operator when the
ground state changes along the RG flow. According to the AdS/CFT
correspondence, it was proposed that a geodesic length $L(t,x_{1};t,x_{2})$
connecting two local operators at the boundary is related to the two-point
function of the dual QFT Witten (1998a); Susskind and Witten (1998);
Solodukhin (1999); D’Hoker  _et al._ (1999); Liu and Tseytlin (1999);
Balasubramanian and Ross (2000); Louko  _et al._ (2000); Kraus  _et al._
(2003); Fidkowski  _et al._ (2004); Park and Lee (2021); Rodriguez-Gomez and
Russo (2021); Park (2024)

$\displaystyle\left\langle O(t,x_{1})\ O(t,x_{2})\right\rangle\sim
e^{-\Delta\,L(t,x_{1};t,x_{2})/R_{UV}},$ (1)

where $\Delta$ means a conformal dimension of a local operator $O(t,x_{1})$.
This holographic prescription describes two-point functions of the dual QFT in
the probe limit. The change of a ground state with the variation of coupling
constants can affect two-point functions and modifies the conformal dimension
at the IR fixed point which we call an anomalous dimension. We first
investigate the conformal dimension near the fixed points analytically and
then look into the variation of a conformal dimension numerically in the
entire energy range. We show that the analytic prescription is perfectly
matched to the numerical result.

The rest of this paper is organized as follows. In Sec. 2, we take into
account a dual gravity theory which realizes the RG flow from a UV to IR fixed
point. We investigate the change of the entanglement entropy along the RG flow
which describes the modification of a ground state with varying a coupling
constant. In Sec. 3, we study how the change of the ground state affects two-
point functions and conformal dimensions in the probe limit. We conclude this
work with some discussions in Sec. 4.

##  2 Holographic dual of the RG flow

Let us begin with briefly discussing universal features of two-dimensional QFT
at critical points where the scale symmetry is restored Calabrese and Cardy
(2004); Peschel  _et al._ (1998); Kim  _et al._ (2017); Calabrese and Cardy
(2006). We take into account a transverse field Ising model whose Hamiltonian
is given by

$\displaystyle
H_{I}=-\sum_{n=1}^{L-1}\sigma^{x}_{n}-g\sum_{n=1}^{L-1}\sigma^{z}_{n}\sigma^{z}_{n+1},$
(2)

where $\sigma_{n}$ indicate the Pauli matrices at the site $n$. This model
describes paramagnet with $\left\langle\sigma_{n}^{z}\right\rangle=0$ for
$g=0$, while ferromagnet with $\left\langle\sigma_{n}^{z}\right\rangle=\pm 1$
appears as the ground state for $g=\infty$. Intriguingly, the transverse field
Ising model allows a second-order quantum phase transition at $g=1$. This
feature becomes manifest when we study entanglement entropy. For $g=\infty$,
since $\left\langle\sigma_{n}^{z}\right\rangle$ allows two accessible
configurations with opposite sign of magnetization, the resulting entanglement
entropy is proportional to $\log 2$.

At the quantum phase transition point, the entanglement entropy shows a
logarithmic divergence due to the scale symmetry. Applying the corner transfer
matrix method, the entanglement entropy is given by Calabrese and Cardy (2004)

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\epsilon\sum_{j=0}^{\infty}\frac{2j+1}{1+e^{(2j+1)\epsilon}}+\sum_{j=0}^{\infty}\log\left(1+e^{-(2j+1)\epsilon}\right)\quad{\rm
for}\ g<1,$ (3) $\displaystyle=$
$\displaystyle\epsilon\sum_{j=0}^{\infty}\frac{2j}{1+e^{2j\epsilon}}+\sum_{j=0}^{\infty}\log\left(1+e^{-2j\epsilon}\right)\quad{\rm
for}\ g>1,$

where the energy gap between the energy levels is given by

$\displaystyle\epsilon=\pi\frac{K(\sqrt{1-1/g^{2}})}{K(1/g)},$ (4)

and $K(1/g)$ is a complete elliptic integral of the first kind. Near the
critical point with $g=1$, the entanglement entropy is reduced to

$\displaystyle S_{E}\approx-\frac{1}{12}\log\left(1-\frac{1}{g}\right),$ (5)

which shows a logarithmic divergence as $g\to 1$. A similar feature also
occurs when a two-dimensional QFT flows into an IR fixed point where the
restoration of the conformal symmetry makes the entanglement entropy diverge.
Hereafter, we clarify this feature in the holographic setup.

We take into account a two-dimensional UV CFT deformed by a relevant operator.
Even when a UV physics is weakly interacting, the IR physics after the RG flow
can be strongly interacting. Therefore, knowing the correct IR physics
requires understanding a nonperturbative RG flow. To do so, we consider a
three-dimensional gravity theory with one bulk scalar field

$\displaystyle S=\frac{1}{2\kappa^{2}}\int d^{3}X\sqrt{-g}\left({\cal
R}-\frac{1}{2}\partial_{M}\phi\partial^{M}\phi-\frac{V(\phi)}{R_{UV}^{2}}\right).$
(6)

To describe the RG flow from a UV CFT to a new IR CFT, we introduce the
following simple scalar potential as a toy model

$\displaystyle
V(\phi)=2R_{UV}^{2}\Lambda_{UV}+\frac{M_{\phi}^{2}}{2}\phi^{2}+\frac{\lambda}{4}\phi^{4}=2R_{UV}^{2}\Lambda_{UV}+\frac{\lambda}{4}\phi^{2}\left(\phi^{2}-2\frac{m_{\phi}^{2}}{\lambda}\right),$
(7)

where $M_{\phi}^{2}=-m_{\phi}^{2}<0$, $\lambda>0$ and
$\Lambda_{UV}=-1/R_{UV}^{2}$ with a UV AdS radius $R_{UV}$. Note that here we
exploit dimensionless field $\phi$ and coupling constants, $m_{\phi}^{2}$ and
$\lambda$. The scalar potential considered here allows one local maximum,
$V=-2\Lambda_{UV}$ at $\phi=0$, and two degenerated local minima,
$V=2R_{UV}^{2}\Lambda_{UV}-m_{\phi}^{4}/4\lambda$ at $\phi_{\pm}=\pm
m_{\phi}/\sqrt{\lambda}$.

We concentrate on the RG flow from a UV CFT to another IR CFT. Such an RG flow
in the holographic setup can be realized by a geometric solution interpolating
a local maximum to one of the local minima. A local maximum and minimum
correspond to an unstable or stable equilibrium point respectively, so the
scalar field rolls down from a maximum to minimum. In other words, a rolling
motion in gravity maps to an RG flow of dual QFT. From now on, we focus on the
RG flow from a UV CFT at $\phi=0$ to an IR CFT at $\phi_{+}=\phi_{IR}$. To
describe a relevant operator, we assume that the bulk field has a negative
mass square in the range of $0<-M_{\phi}^{2}=m_{\phi}^{2}<1$. Assuming that
the bulk field depends only on the radial coordinate, it does not break the
boundary Lorentz symmetry. The most general metric ansatz preserving the
boundary Lorentz symmetry is given by

$\displaystyle ds^{2}=dy^{2}+e^{2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}.$ (8)

In this case, UV and IR fixed points appear at $y=\infty$ and $-\infty$,
respectively.

A geometric solution interpolating two fixed points of the RG flow is obtained
by solving the equations of motion

$\displaystyle 0$ $\displaystyle=$ $\displaystyle A^{\prime
2}-\frac{1}{4}\phi^{\prime
2}-\frac{m_{\phi}^{2}}{4R_{UV}^{2}}\phi^{2}+\frac{\lambda}{8R_{UV}}\phi^{4}-\frac{1}{R_{UV}^{2}},$
(9) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
A^{\prime\prime}+A^{\prime 2}+\frac{1}{4}\phi^{\prime
2}-\frac{m_{\phi}^{2}}{4R_{UV}^{2}}\phi^{2}+\frac{\lambda}{8R_{UV}}\phi^{4}-\frac{1}{R_{UV}^{2}},$
(10) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\phi^{\prime\prime}+2A^{\prime}\phi^{\prime}+\frac{m_{\phi}^{2}}{R_{UV}^{2}}\phi^{2}-\frac{\lambda}{R_{UV}}\phi^{4},$
(11)

where the prime means a derivative with respect to $y$. Here, the first
equation is a constraint and the others determine dynamics of $\phi$ and $A$.
When we choose $R_{UV}=1$, $m_{\phi}=\sqrt{3}/2$ and $\lambda=0.1$, we depict
a numerical solution in Fig. 1. The numerical results show that $\phi$ and
$A^{\prime}$ approach $\phi_{IR}=2.7386$ and $A^{\prime}=1.3050$ at the IR
fixed point ($y\to-\infty$). Since a new AdS space appears at the IR fixed
point, $A^{\prime}=1.3050$ gives rise to an AdS radius $R_{IR}=0.7663$.

(a) Scalar field $\phi(y)$ (b) Metric factor $A(y)$

Figure 1: Profiles of (a) the scalar field $\phi(y)$ and (b) metric factor
$A(y)$ where we take $R_{UV}=1$, $m_{\phi}=\sqrt{3}/2$ and $\lambda=0.1$. In
this case, the deformation operator has a conformal dimension $\Delta=3/2$ and
the IR fixed point occurs with $\phi_{IR}=2.7386$ and $dA/dy=1.3050$ at
$y=-\infty$.

Recalling that a $(d+1)$-dimensional AdS space corresponds to the ground state
of a $d$-dimensional CFT, the above interpolating solution maps to the ground
state’s RG flow where the coupling constant and degrees of freedom vary.
Assuming that the boundary of bulk geometry is located at $\bar{y}$, the
boundary position in the holographic study is identified with the RG scale of
the dual QFT, $\mu=e^{A(\bar{y})}$. Moreover, the value of the scalar field at
the boundary, $\phi(\bar{y})$, is reinterpreted as the coupling constant at
the energy scale $\mu$. This prescription connects the geometrical
interpolating solution to the RG flow in the momentum space. There is another
holographic description for the RG flow. The entanglement entropy describing
macroscopic quantum correlations is determined as a function of a subsystem
size. In the RG flow of entanglement entropy, the RG scale is determined by a
subsystem size rather than the boundary position. This resembles the real
space RG flow widely used in condensed matter physics. Therefore, the
entanglement entropy describes the real space RG flow. As shown in the
previous transverse field Ising model, the entanglement entropy can be used as
an indicator for the quantum phase transition and fixed points.

Now, we investigate how the entanglement entropy modifies under the change of
a ground state. To do so, we discuss the entanglement entropy near the fixed
points analytically which may be useful to understand underlying theoretical
structures of the RG flow. At the fixed points, the interpolation geometry is
reduced to AdS spaces where the metric factor $A(y)$ has to be linearly
proportional to $y$. At the UV fixed point ($y\to\infty$), $A(y)$ is given by
$A(y)=y/R_{UV}$ with a UV AdS radius $R_{UV}$. Introducing a new UV coordinate

$\displaystyle z=R_{UV}\ e^{-y/R_{UV}},$ (12)

the UV metric at $z\to 0$ is rewritten as

$\displaystyle
ds_{UV}^{2}=\frac{R_{UV}^{2}}{z^{2}}\left(-dt^{2}+dx^{2}+dz^{2}\right).$ (13)

On the other hand, $A(y)$ at the IR fixed point ($y\to-\infty$) should be
$A(y)=y/R_{IR}$ where $R_{IR}$ is an IR AdS radius. Introducing a new IR
coordinate $\bar{z}$ as

$\displaystyle\bar{z}=R_{IR}\ e^{-y/R_{IR}},$ (14)

the IR metric for $\bar{z}\to\infty$ becomes

$\displaystyle
ds_{IR}^{2}=\frac{R_{IR}^{2}}{\bar{z}^{2}}\left(-d\bar{t}^{2}+d\bar{x}^{2}+d\bar{z}^{2}\right).$
(15)

Comparing (12) and (14), the UV coordinates can be related to the IR ones

$\displaystyle z=\frac{R_{UV}}{R_{IR}^{R_{IR}/R_{UV}}}\
\bar{z}^{R_{IR}/R_{UV}}\ ,\quad
dt=\frac{R_{IR}^{R_{UV}-R_{IR}}}{\bar{z}^{R_{UV}-R_{IR}}}d\bar{t}\quad{\rm
and}\quad dx=\frac{R_{IR}^{R_{UV}-R_{IR}}}{\bar{z}^{R_{UV}-R_{IR}}}d\bar{x}.$
(16)

On the dual QFT point of view, the first relation describes the change of the
RG scale. On the other hand, the last two relations describe the rescaling of
the boundary coordinates along the RG flow.

Near the UV fixed point, since the bulk field rapidly suppresses in the
asymptotic region ($z\to 0$) for $0<m_{\phi}^{2}<1$, it maps to a relevant
operator. In this case, the effect of $\lambda\phi^{4}$ is negligible. In the
asymptotic region, more manifestly, the profile of $\phi$ is approximately
determined by

$\displaystyle 0=\partial_{z}^{2}\phi-\frac{1}{z}\
\partial_{z}\phi+\frac{m_{\phi}^{2}}{z^{2}}\phi.$ (17)

The solution of this equation reads

$\displaystyle\phi(z)=c_{1}z^{2-\Delta_{UV}}+c_{2}z^{\Delta_{UV}},$ (18)

where the conformal dimension of the dual operator $O_{\phi}$ is given by

$\displaystyle\Delta_{UV}=1+\sqrt{1-m_{\phi}^{2}}.$ (19)

For $0<m_{\phi}^{2}<1$, the conformal dimension of the dual operator is in the
range of $1<\Delta_{UV}<2$. This implies that the dual operator is relevant.
Although the effect of a relevant deformation is negligible in the UV region,
it significantly modifies IR physics due to nonperturbative quantum
corrections. Near the UV fixed point, the RG scale is related to the radial
coordinate, $\mu\sim 1/z$. Therefore, $\phi(z)$ is reinterpreted as a coupling
constant at the RG scale $\mu$ on the dual QFT side. The $\beta$-function of
the coupling constant becomes

$\displaystyle\beta_{\phi}=\mu\frac{d\phi}{\delta\mu}=-(2-\Delta_{UV})\phi+\cdots,$
(20)

which is consistent with the fact that the dual operator has a conformal
dimension $\Delta_{UV}$.

A relevant operator usually causes a nontrivial RG flow and may lead to a new
IR fixed point where the conformal symmetry is restored. Since an IR fixed
point involving all quantum corrections is determined nonperturbatively, it is
difficult to account for IR physics on the traditional QFT side. However, the
holographic study can provide more information about IR physics. To see this,
we take into account an IR limit at $\bar{z}=\infty$ where the coupling
constant is given by

$\displaystyle\phi_{IR}=m_{\phi}/\sqrt{\lambda}.$ (21)

Expanding the scalar field with a small fluctuation $\delta\phi$

$\displaystyle\phi=\phi_{IR}+\delta\phi,$ (22)

the scalar potential is approximately near the IR fixed point

$\displaystyle
V\approx-2\left(\frac{8\lambda+m_{\phi}^{4}}{8\lambda}\right)+m_{\phi}^{2}\
\delta\phi^{2},$ (23)

where an IR cosmological constant is given by

$\displaystyle
2\Lambda_{IR}=-2\left(\frac{8\lambda+m_{\phi}^{4}}{8\lambda}\right)\frac{1}{R_{UV}^{2}}.$
(24)

Therefore, the IR AdS radius $R_{IR}$ is related to the UV one

$\displaystyle R_{IR}=\sqrt{\frac{8\lambda}{8\lambda+m_{\phi}^{4}}}\ R_{UV}.$
(25)

Plugging the numerical values, $R_{UV}=1$, $m_{\phi}=\sqrt{3}/2$ and
$\lambda=0.1$, we reproduce the numerical results obtained in Fig. 1,
$\phi_{IR}=2.7386$ and $R_{IR}=0.7663$. In this case, the deformation operator
has a conformal dimension $\Delta_{UV}=3/2$ at the UV fixed point. Note that
the IR AdS radius is always smaller than the UV one, $R_{IR}<R_{UV}$.
Recalling that the central charges of two-dimensional CFTs at the fixed points
are given by

$\displaystyle c_{UV}=\frac{3R_{UV}}{2G}\quad{\rm and}\quad
c_{IR}=\frac{3R_{IR}}{2G},$ (26)

$R_{IR}<R_{UV}$ implies the $c$-theorem of two-dimensional QFTs,
$c_{UV}>c_{IR}$.

Due to the restoration of the conformal symmetry at the IR fixed point, the
dual geometry is again given by an AdS space in (15). To determine the
$\phi$’s profile near the IR fixed point, we take into account the equation of
motion for $\delta\phi$

$\displaystyle 0=\partial_{\bar{z}}^{2}\delta\phi-\frac{1}{\bar{z}}\
\partial_{\bar{z}}\delta\phi-\frac{2\
m_{\phi}^{2}}{\bar{z}^{2}}\frac{R_{IR}^{2}}{R_{UV}^{2}}\ \delta\phi.$ (27)

The solution of this equation reads

$\displaystyle\delta\phi=d_{1}\,\bar{z}^{2+\delta_{IR}}-d_{2}\,\bar{z}^{-\delta_{IR}},$
(28)

where $d_{1}$ and $d_{2}$ are two integral constants and $\delta_{IR}$ is
given by

$\displaystyle\delta_{IR}=\frac{\sqrt{8\lambda+m_{\phi}^{4}+16m_{\phi}^{2}\lambda}}{\sqrt{8\lambda+m_{\phi}^{4}}}-1.$
(29)

In this case, since $\delta_{IR}>0$, $\delta\phi$ diverges as
$\bar{z}\to\infty$. However, we assumed at the early stage that $\phi$ has a
fiinite value $\phi_{IR}$ at the IR fixed point, so the first term in (28)
must vanish. This implies that we should take $d_{1}=0$. Then, the resulting
profile of $\phi$ near the IR fixed point yields

$\displaystyle\phi(\bar{z})=\phi_{IR}-d_{2}\,\bar{z}^{-\delta_{IR}}+\cdots,$
(30)

where $d_{2}$ is given by a function of $c_{1}$ and $c_{2}$ in (18).
Exploiting this scalar profile, we can determine the IR RG scale as a function
of the coupling constant

$\displaystyle\mu\sim\frac{1}{\bar{z}}\approx\left(\frac{\phi_{IR}-\phi(\bar{z})}{d_{2}}\right)^{1/\delta_{IR}}.$
(31)

In addition, the $\beta$-function near the IR fixed point, where $\mu\sim
1/\bar{z}$, is given by

$\displaystyle\beta_{\phi}=\mu\frac{d\phi}{d\mu}=-\delta_{IR}\left(\phi_{IR}-\phi(\bar{z})\frac{}{}\right)+\cdots.$
(32)

At the IR fixed point with $\phi(\bar{z})=\phi_{IR}$, the $\beta$-function
automatically vanishes. This indicates the restoration of the conformal
symmetry and that the dual operator becomes a marginal one with an IR
conformal dimension, $\Delta_{IR}=2$.

To further study the RG flow of the ground state, let us investigate the RG
flow of the entanglement entropy. To desribe the entanglement entropy valid in
the entire energy scale, we calculate the entanglement entropy numerically. In
the previous interplation geometry (8), the entanglement entropy is governed
by the area of a minimal surface Ryu and Takayanagi (2006a, b)

$\displaystyle S_{E}=\frac{1}{4G}\int_{-\ell/2}^{\ell/2}dx\sqrt{y^{\prime
2}+e^{2A(y)}},$ (33)

where the prime means a derivative with respect to $x$. Using the conserved
quantity, the subsystem size and entanglement entropy are determined as
functions of a turning point $y_{t}$, at which $y^{\prime}=0$,

$\displaystyle\ell$ $\displaystyle=$
$\displaystyle\int_{y_{t}}^{\Lambda_{y}}dy\
\frac{2}{e^{2A(y)}\sqrt{e^{-2A_{t}}-e^{-2A(y)}}},$ (34) $\displaystyle S_{E}$
$\displaystyle=$ $\displaystyle\frac{1}{2G}\int_{y_{t}}^{\Lambda_{y}}dy\
\frac{e^{-A_{t}}}{e^{2A(y)}\sqrt{e^{-2A_{t}}-e^{-2A(y)}}},$ (35)

where $A_{t}=A(y_{t})$ and $\Lambda_{y}\to\infty$ denotes a UV cutoff. To
perform these integrals numerically, we take $G=1$, $R_{UV}=1$,
$m_{\phi}=\sqrt{3}/2$, $\lambda=0.1$ and $\Lambda_{y}=20$. Then, the
entanglement entropy as a function of $\phi(y_{t})=\phi_{t}$ is depicted in
Fig. 2(a). The numerical result shows that the entanglement entropy increases
logarithmically at the UV fixed point ($\phi_{t}=0$), where we get rid of a UV
divergence. At the IR fixed point ($\phi_{t}=\phi_{IR}=2.7386$), the
entanglement entropy again diverges logarithmically. This implies that a
logarithmic divergence of the entanglement entropy for two-dimensional QFT
naturally appears when the scaling symmetry is restored. The numerical result
in Fig. 2(b) shows that the coefficient of the logarithmic term approaches at
the IR fixed point

$\displaystyle-\frac{dS_{E}}{d\log(\phi_{IR}-\phi_{t})}=1.0316.$ (36)

Figure 2: (a) Entanglement entropy depending on the coupling constant and (b)
the coefficient of the IR entanglement entropy where we fix $G=1$. (a) The
entanglement entropy shows a logarithmic divergence at $\phi_{t}\to\phi_{IR}$.
(b) When $-\log(\phi_{IR}-\phi_{t})\to\infty$ near the IR fixed point,
$-dS_{E}/d\log(\phi_{IR}-\phi_{t})$ approaches $1.0316$ which is consistent
with the value expected by (46).

Although we calculated the entanglement entropy numerically, it is also
important to know the analytic form for characterizing the parameter
dependence. To do so, we reinvestigate the entanglement entropy analytically.
Assuming that a subsystem has a small size $\ell$ which is related to the
inverse of an RG scale, a minimal surface in the UV limit extends to only the
asymptotic region. Therefore, the dominant entanglement entropy in the UV
region is governed by

$\displaystyle S_{E}\approx\frac{R_{UV}}{4G}\int_{-\ell/2}^{\ell/2}dx\
\frac{\sqrt{z^{\prime 2}+1}}{z},$ (37)

where $z$ is the radial coordinate of the UV AdS geometry in (13). To evaluate
the entanglement entropy, we introduce a turning point $z_{t}$ at $x=0$, where
$z^{\prime}=0$. Then, the subsystem size and entanglement entropy in the UV
limit ($\ell\to 0$ and $z_{t}\to 0$) are reexpressed in terms of the turning
point

$\displaystyle\ell(z_{t})$ $\displaystyle=$
$\displaystyle\int_{0}^{z_{t}}dz\frac{2z}{\sqrt{z_{t}^{2}-z^{2}}}=2z_{t}+\cdots,$
(38) $\displaystyle S_{E}(z_{t})$ $\displaystyle=$
$\displaystyle\frac{R_{UV}}{2G}\int_{\epsilon}^{z_{t}}dz\frac{z_{t}}{z\sqrt{z_{t}^{2}-z^{2}}}=\frac{R_{UV}}{2G}\log\frac{z_{t}}{\epsilon}+\cdots,$
(39)

where $\epsilon$ means a UV cutoff. Here, the ellipses mean higher order
corrections caused by the deformation in the asymptotic region. The
entanglement entropy relying on the subsystem size results in

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\frac{c_{UV}}{3}\log\frac{\ell}{\epsilon}+{\cal O}(\ell)\ .$
(40)

In this case, the leading contribution comes from a two-dimensional UV CFT
with small higher order corrections caused by the relevant deformation.

Recalling that the RG scale of the entanglement entropy is described by the
subsystem size and that the subsystem size is determined by the turning point
in (38), we can describe the RG scale of the entanglement entropy in terms of
the turning point, $\mu\sim 1/z_{t}$. As a result, the value of $\phi(z_{t})$
at the turning point can be reinterpreted as the coupling constant at the RG
scale $\mu$. In the UV region, the turning point $z_{t}$ is reexpressed in
terms of the coupling constant $\phi(z_{t})=\phi_{t}$

$\displaystyle
z_{t}\approx\left(\frac{\phi_{t}}{c_{1}}\right)^{1/(2-\Delta_{UV})}.$ (41)

Then, the entanglement entropy becomes in terms of the coupling constant

$\displaystyle S_{E}=\frac{c_{UV}}{3\
(2-\Delta_{UV})}\log\frac{\phi_{t}}{\phi_{\epsilon}}+{\cal
O}\left(\phi_{t}^{1/(2-\Delta_{UV})}\frac{}{}\right),$ (42)

where $\phi_{\epsilon}=\phi(\epsilon)$ means the coupling constant at the UV
fixed point.

Let us study the entanglement entropy in a large subsystem size limit
($\ell\to\infty$) which corresponds to the IR limit. The IR entanglement
entropy in the holographic setup is defined as the integration over $x$ in the
range of $-\ell/2<x<\ell/2$, so that it also contains the UV effect.
Nevertheless, the leading contribution to the IR entanglement entropy comes
from the minimal surface’s area extending to the IR region. Therefore, the
entanglement entropy near the IR fixed point is approximately given by

$\displaystyle
S_{E}\approx\lim_{\ell\to\infty}\frac{R_{IR}}{4G}\int_{-\ell/2}^{\ell/2}dx\
\frac{\sqrt{\bar{z}^{\prime 2}+1}}{\bar{z}},$ (43)

where $\bar{z}$ is the radial coordinate of (15) in the IR region. Similar to
the previous UV entanglement entropy, the leading subsystem size and IR
entanglement entropy up to the UV divergence are determined by the turning
point

$\displaystyle\ell(\bar{z}_{t})$ $\displaystyle\approx$
$\displaystyle\lim_{z_{t}\to\infty}2\bar{z}_{t},$ (44) $\displaystyle
S_{E}(\bar{z}_{t})$ $\displaystyle\approx$
$\displaystyle\lim_{z_{t}\to\infty}\frac{R_{IR}}{2G}\log\left(\bar{z}_{t}\right)=\frac{c_{IR}}{3}\log\left(\bar{z}_{t}\right).$
(45)

Therefore, the dominant IR entanglement entropy in terms of the coupling
constant reads

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle-\frac{c_{UV}\,\sqrt{8\,\lambda}}{3\,\left(\sqrt{8\lambda+m_{\phi}^{4}+16m_{\phi}^{2}\lambda}-\sqrt{8\lambda+m_{\phi}^{4}}\right)}\log\left(\phi_{IR}-\bar{\phi}_{t}\frac{}{}\right).$
(46)

where $\bar{\phi}_{t}=\phi(\bar{z}_{t})$. This shows that, similar to the
previous transverse field Ising model, a logarithmic divergence appears at the
IR fixed point due to restoration of the conformal symmetry. Using the
numerical values used in Fig. 1, the coefficient of the logarithmic term
results in

$\displaystyle-\frac{dS_{E}}{d\log(\phi_{IR}-\phi_{t})}=\frac{c_{UV}\,\sqrt{8\,\lambda}}{3\,\left(\sqrt{8\lambda+m_{\phi}^{4}+16m_{\phi}^{2}\lambda}-\sqrt{8\lambda+m_{\phi}^{4}}\right)}=1.0316.$
(47)

This is perfectly matched to the previous numerical result in (36).

##  3 Anomalous dimension of a local operator

Now, we investigate the RG scale dependence of correlation functions in order
to understand another nonperturbative feature of the RG flow. To do so, we
insert two local operators, $O_{\chi}(t_{1},x_{1})$ and
$O_{\chi}(t_{2},x_{2})$, into the ground state deformed by $O_{\phi}$. In the
probe limit where the backreaction of $O_{\chi}(t_{i},x_{i})$ with $i=1,2$ is
ignored, the background ground state, which is the dual of the previous
gravity theory, is not modified . The holographic setup considered here
describes how the change of the ground state affects two-point functions.
Supposed that two scalar operators $O_{\chi}$ have a conformal dimension
$\Delta_{UV}^{\chi}$ at the UV fixed point, then their two-point function is
given by

$\displaystyle\left\langle O_{\chi}(t_{1},x_{1})\
O_{\chi}(t_{2},x_{2})\right\rangle=\frac{1}{(-|t_{1}-t_{2}|^{2}+|x_{1}-x_{2}|^{2})^{\Delta_{UV}^{\chi}}}.$
(48)

At the IR fixed point, the restoration of the conformal symmetry results in
the same two-point function with a different IR conformal dimension. This is
because, during the nontrivial RG flow, nonperturbative quantum effects
crucially modify the conformal dimension. From now on, we discuss how to
explain the change of the conformal dimension along the RG flow with the aid
of holography.

In the holographic setup, we consider a bulk field $\chi$ as the dual of
$O_{\chi}$. To realize the conformal dimension $\Delta_{UV}^{\chi}$, we assume
that $\chi$ has mass $-m_{\chi}^{2}/R_{UV}^{2}$ at the UV fixed point. In the
asymptotic region, the bulk field $\chi$ satisfies the following linear
differential equation in the probe limit

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-G}}\partial_{M}\left(\sqrt{-G}G^{MN}\partial_{M}\chi\right)+\frac{m_{\chi}^{2}}{R_{UV}^{2}}\chi$
(49) $\displaystyle=$
$\displaystyle\frac{z^{2}}{R_{UV}^{2}}\left(\partial_{z}^{2}\chi-\frac{1}{z}\
\partial_{z}\chi+\frac{m_{\chi}^{2}}{z^{2}}\chi\right).$

Assuming that the scalar field relies only on the radial coordinate, the mass
of $\chi$ determines the profile of $\chi$ as

$\displaystyle\chi=c_{1}\,z^{2-\Delta_{UV}^{\chi}}+c_{2}\,z^{\Delta_{UV}^{\chi}},$
(50)

where $c_{1}$ and $c_{2}$ are two integral constants and the dual operator
$O_{\chi}$ has a conformal dimension

$\displaystyle\Delta_{UV}^{\chi}=1+\sqrt{1-m_{\chi}^{2}}.$ (51)

In this case, it is worth noting that, since $\chi$ is governed by a linear
second order differential equation, $c_{1}$ and $c_{2}$ are not mixed during
the RG flow. Using the obtained UV conformal dimension, the two-point function
of $O_{\chi}$ at the UV fixed point results in

$\displaystyle\left\langle O_{\chi}(t_{1},x_{1})\
O_{\chi}(t_{2},x_{2})\right\rangle=\frac{1}{(-|t_{1}-t_{2}|^{2}+|x_{1}-x_{2}|^{2})^{\Delta_{UV}^{\chi}}}.$
(52)

Now, we take into account an IR correlation function. Since we consider the
probe limit, the equation of $\chi$ in (49) is still valid in the IR region
although the background geometry deforms. Rewriting the equation (49) in terms
of the IR coordinate $\bar{z}$ by using (16), we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{\bar{z}}^{2}\chi+\left(1-\frac{2R_{IR}}{R_{UV}}\right)\frac{1}{\bar{z}}\
\partial_{\bar{z}}\chi+\frac{R_{IR}^{2}}{R_{UV}^{2}}\frac{m_{\chi}^{2}}{\bar{z}^{2}}\chi$
(53)

Solving this equation, the profile of $\chi$ in the IR region reads

$\displaystyle\chi=\bar{c}_{1}\
\bar{z}^{2R_{IR}/R_{UV}-\Delta_{IR}^{\chi}}+\bar{c}_{2}\
\bar{z}^{\Delta_{IR}^{\chi}},$ (54)

where the IR conformal dimension is given by

$\displaystyle\Delta_{IR}^{\chi}=\frac{R_{IR}}{R_{UV}}\
\Delta_{UV}^{\chi}=\frac{\sqrt{8\lambda}}{\sqrt{8\lambda+m_{\phi}^{4}}}\
\left(1+\sqrt{1-m_{\chi}^{2}}\right).$ (55)

Comparing the obtained IR solution with the UV solution in (50), the integral
constants are determined as

$\displaystyle\bar{c}_{1}=\left(\frac{R_{UV}}{R_{IR}^{R_{IR}/R_{UV}}}\right)^{2-\Delta_{UV}^{\chi}}c_{1}\quad{\rm
and}\quad\bar{c}_{2}=\left(\frac{R_{UV}}{R_{IR}^{R_{IR}/R_{UV}}}\right)^{\Delta_{UV}^{\chi}}c_{2}.$
(56)

In the probe limit, the UV conformal dimension $\Delta_{UV}^{\chi}$ flows into
the IR one $\Delta_{IR}^{\chi}$ along the RG flow because of the absence of
mixing between $c_{1}$ and $c_{2}$. Therefore, we expect that the following
two-point function appears at the IR fixed point

$\displaystyle\left\langle O_{\chi}(t_{1},x_{1})\
O_{\chi}(t_{2},x_{2})\right\rangle=\frac{1}{(-|t_{1}-t_{2}|^{2}+|x_{1}-x_{2}|^{2})^{\Delta_{IR}^{\chi}}}.$
(57)

In this case, the anomalous dimension of $O_{\chi}$ is given by

$\displaystyle\gamma_{\chi}\equiv\Delta_{IR}^{\chi}-\Delta_{UV}^{\chi}=-\left(1-\sqrt{\frac{8\lambda}{8\lambda+m_{\phi}^{4}}}\right)\,\Delta_{UV}^{\chi}<0.$
(58)

and the conformal dimension decreases along the RG flow,
$\Delta_{UV}^{\chi}>\Delta_{IR}^{\chi}$.

There is another holographic method calculating two-point functions in the
probe limit. In the holographic setup, it was proposed that a two-point
function can be described by a geodesic length $L(t_{1},x_{1};t_{2},x_{2})$
connecting two local operators at the boundary Susskind and Witten (1998);
Solodukhin (1999); D’Hoker  _et al._ (1999); Liu and Tseytlin (1999);
Balasubramanian and Ross (2000); Park and Lee (2021)

$\displaystyle\left\langle O_{\chi}(t,x_{1})\
O_{\chi}(t,x_{2})\right\rangle\sim
e^{-\Delta_{UV}^{\chi}\,L(t,x_{1};t,x_{2})/R_{UV}},$ (59)

where $L(t,x_{1};t,x_{2})$ means a geodesic length. It is worth noting that
the procedure calculating two-point correlations for three-dimensional gravity
is similar to that used in the entanglement entropy calculation. This is
because the minimal surface in the entanglement entropy calculation is reduced
to a geodesic curve for three-dimensional gravity theory. However, this is not
the case for higher dimensional gravity theories. Using this proposal, the
known CFT’s correlators at zero and finite temperature are reproduced Park
(2021, 2023); Park  _et al._ (2022); Kim  _et al._ (2023).

Applying the holographic formula (59) to the previous interpolating geometry,
the geodesic length is determined by

$\displaystyle L(t,x_{1};t,x_{2})$ $\displaystyle=$
$\displaystyle\int_{y_{t}}^{\Lambda_{y}}dy\
\frac{2e^{-A_{t}}}{e^{2A(y)}\sqrt{e^{-2A_{t}}-e^{-2A(y)}}}.$ (60)

When we take the UV conformal dimension of $O_{\chi}$ to be
$\Delta^{\chi}_{UV}=1$, the RG flow of the conformal dimension of $O_{\chi}$
for $m_{\phi}=\sqrt{3}/2$ and $\lambda=0.1$ is plotted numerically in Fig. 3.
The numerical simulation shows that the conformal dimension monotonically
decreases from $\Delta^{\chi}_{UV}=1$ at the UV fixed point to
$\Delta^{\chi}_{IR}=0.7663$ at the IR fixed point (see Fig. 3). This numerical
IR conformal dimension can be understood by the previous analytic result

$\displaystyle\frac{\Delta^{\chi}_{IR}}{\Delta^{\chi}_{UV}}=\frac{\sqrt{8\lambda}}{\sqrt{8\lambda+m_{\phi}^{4}}}=0.7663.$
(61)

In the probe limit, the analytic prediction in (55) is perfectly matched to
the numerical result in Fig. 3.

Figure 3:  The RG flow of a conformal dimension of $O_{\chi}$. When the ground
state is deformed by $O_{\phi}$, the change of the ground state affects on
another operator $O_{\chi}$. Assuming that the UV conformal dimension is
$\Delta^{\chi}_{UV}=1$, the conformal dimension along the RG flow changes into
$\Delta^{\chi}_{IR}=0.7663$ at the IR fixed point. This is consistent with the
previous theoretical expectation in (55).

##  4 Discussion

In the present work, we have studied the RG flow of a two-dimension QFT from a
UV to IR fixed point by applying the holographic method. To describe such a RG
flow, we took into account a relevant operator deforming UV CFT. On the dual
gravity theory side, a UV fixed point corresponds to an unstable equilibrium
point (or local maximum) with a negative potential energy at which an AdS
space appears as a geometric solution. On the other hand, a new IR fixed point
maps to a stable equilibrium point (or local minimum) where another AdS space
becomes a geometric solution. To realize such a phenomenon, we consider a
simple scalar potential, as a toy model, which allows local maximum and
minima. Then, the RG flow from a UV to IR fixed point is reinterpreted as
rolling down of a scalar field from an unstable to stable equilibrium point on
the gravity theory side.

In general, a nonperturbative RG flow modifies the ground stat of QFT, which
also affects quantum correlation. Therefore, we expect that the entanglement
entropy nontrivially changes along the RG flow. After introducing the
holographic dual gravity realizing the RG flow from a UV to IR fixed point, we
investigated the RG flow of entanglement entropy by applying the holographic
formula. First, we numerically studied the change of entanglement entropy
depending on the energy scale. Noting that the bulk scalar field can be mapped
to a coupling constant on the dual QFT side, we also described the change of
entanglement entropy in terms of the coupling constant. To understand the RG
flow of entanglement entropy further, we also investigated the analytic
behavior of the entanglement entropy near the fixed points. Using the
perturbative expansion around the fixed points, we found the coupling constant
dependence of the entanglement entropy near the UV and IR fixed points. We
showed that the entanglement entropy of a two-dimensional QFT shows
logarithmic behaviors at fixed points due to the restoration of the conformal
symmetry. We explicitly showed that the analytic calculation near the fixed
points are perfectly matched to the numerical result.

We further studied the RG flow of a conformal dimension in the holographic
setup. To do so, we introduced two local operators and took into account the
probe limit where two local operators do not modify the ground state. Even in
this case, the change of the ground state affects the correlation function
between two operators. In the holographic dual gravity, a two-point function
in the probe limit can be described by a geodesic length connecting two local
operators. Using this holographic method, we investigated the numerical RG
flow of a two-point function when the ground state varies. To get more
analytic information about the change of the conformal dimension, we also
analytically studied the change of the two-point function near the fixed
points. After introducing another bulk scalar field as the dual of local
operators, we investigated its bulk equation of motion. From the solution of
this equation, we looked into the analytic form of the IR conformal dimension,
which is perfectly matched to the previous numerical result. These results
show that the holographic dual gravity can explain the RG flow of the dual QFT
correctly.

Acknowledgement

C. P. was supported by Mid-career Researcher Program through the National
Research Foundation of Korea (NRF) grant funded by the Korea government (No.
NRF-2019R1A2C1006639). J. H. L. was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government (No.
NRF-2021R1C1C2008737)

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